dezleg|nd func}ionarea {i serviciile sistemului energiei ... · 4.2.5 echilibrarea costurilor...

DEZLEG|ND FUNC}IONAREA {I SERVICIILE

SISTEMULUI ENERGIEI ELECTRICE

UNBUNDLING POWER SYSTEM

OPERATIONS AND SERVICES

JEAN CONSTANTINESCU

B U C U R E { T I2 0 1 1

E D I T O R J e a n C o n s t a n t i n e s c u

ISBN 978-606-92862-0-3

Coperta 1: Pablo Picasso, 1932

Tipar: PFA Monica Gheorghiu

[email protected]. j-constantinescu.org

Edi]ie \ngrijit` de prof. dr. ing. Sorin Hurdube]iu

Tehnoredactare: Cecilia B`l`nescu

ISBN 978-606-92862-0-3

DEZLEG|ND FUNC}IONAREA {I SERVICIILE

SISTEMULUI ENERGIEI ELECTRICE

UNBUNDLING POWER SYSTEM

OPERATIONS AND SERVICES

JEAN CONSTANTINESCU

B U C U R E { T I2 0 1 1

content

Introduction in inter-action, space and time unbundling of power system models and services

References

Part I. Inter-action or functional unbundling

1 Concept definition

2 Variable functional unbundling Case No. 1. “Decoupled”

Newton – Raphson method. NERA - 5 and NERA - 6 models

3 Process functional unbundling Case No. 2. Generation

reschedule with the strongest impact on the PS steady – state stability

Case No. 3. Power system load flows with minimum network losses.

NERA model Case No. 4. Extrapolation –

based load flows in long - term PS dynamics simulation.

LOTDYS.R model

4 Process functional unbundling in “technology – customer” system of electricity

4.1 Unbundled electric services and prices

4.2 Transmission service pricing. Benefit-Balanced Transmission Rates

4.2.1 Transmission revenue requirement

4.2.2 Transmission marginal costs

4.2.3 Transmission marginal costs based

4.2.4 Transmission marginal costs and revenue requirement

cuPrIns

Introducere în dezlegarea inter-ac]ional`, spa]ial` [i temporal` a modelelor [i serviciilor sistemului de energie electric`

Bibliografie

Partea I. Dezlegarea inter-ac]ional` sau func]ional`

1 Definirea conceptului

2 Dezlegarea func]ional` de variabile Exemplul nr. 1. Metoda

Newton – Raphson “decuplat`”. Modelele NERA - 5 [i NERA - 6

3 Dezlegarea func]ional` de procese Exemplul nr. 2. Reprogramarea

gener`rii avînd influen]` determinant` asupra stabilit`]ii statice a SEE

Exemplul nr. 3. Regimul permanent al SEE cu pierderi minime de putere în re]ea. Modelul NERA

Exemplul nr. 4. Regimuri determi-nate prin extrapolare în simularea dinamicii pe termen lung a SEE.

Modelul LOTDYS.R

4 Dezlegare func]ional` de pro-cese în sistemul “tehnologie – client” de energie electric`

4.1 Servicii [i pre]uri dezle-gate de energie electric`

4.2 Tarifarea serviciului de transport. Tarife de transport echilibrate cu beneficiile

4.2.1 Venitul necesar pentru transport

4.2.2 Costurile marginale de transport

4.2.3 Costurile marginale de transport

4.2.4 Costurile marginale de transport [i venitul necesar

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4.2.5 Echilibrarea costurilor servi ciului de transport cu o parte a beneficiului utilizatorilor re]elei

4.3 Impactul tarifelor nodale de transport asupra mecanismului pie]ei de energie electric`

4.4 Tarife de transport armo-nizate regional

Bibliografie

Partea a II-a. Dezlegarea spa]ial` sau structural`


2 Dezlegarea structural` (spa]ial`) de variabile

2.1 Metode iterative 2.2 Metode directe 2.2.1 Caz particular: metoda

“compensa]iei” Exemplul nr. 1. Solu]ia ecua]iilor lineare ale re]elei electrice Exemplul nr. 2. Regimul permanent simetric al re]elei electrice. Modelul DIAD Exemplul nr. 3. Regimurile tranzitorii electromecanice pe termen scurt ale SEE. Modelul RETRA

3 Dezlegarea structural` (spa]ial`) de procese

3.1 Analiza Nodal` – Dimo

4 Dezlegarea spa]ial` de proces a pie]ei [i sistemului de energie electric`

4.1 Dezlegarea în spa]iul nodurilor re]elei

4.2 Capacitatea de transport [i managementul congestiilor în proiectul nodal de pia]` de energie

4.2.5 Matching costs on transmission service with a part of grid users’ benefits

4.3 Impact of nodal transmission rates on the electric market mecha-nism

4.4 Transmission usage charge regionally harmonized

References

Part II. spatial or structural unbundling


2 Structural (spatial) unbundling of variables

2.1 Iterative methods 2.2 Direct methods 2.2.1 A particular case:

the “compensation” method Case No. 1. Solution of the linear ecuations of electric network Case No. 2. Symmetrical load flows within a power network. DIAD model Case No. 3. Short – term PS electromechanical transients. RETRA model

3 Process structural (spatial) unbundling

3.1 Dimo’s Nodal Analysis

4 Process spatial unbundling in electric market and system

4.1 Unbundling in the space of network nodes

4.2 Transmission capacity and congestion management in nodal electricity market design

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4.3 Capabilitatea de transfer a inter conexiunilor inter – zonale [i managementul restric]iilor pentru operarea în siguran]` a SEE

4.4 Pia]a de energie cu structur` nodal` [i re]elele inteligente

Bibliografie

Partea a III-a. Dezlegarea temporal` sau varia]ional`


2 Dezlegarea varia]ional` de variabile Exemplul nr. 1. Regimul tranzi-toriu pe termen mediu al SEE. Modelul TEMI Echivalen]ii dinamici ajustabili deriva]i din modelele REI-Dimo

3 Dezlegarea varia]ional` de procese Exemplul nr. 2. Capabilit`]i de transfer de energie electric` în noduri [i sec]iuni de re]ea relevante. Modelul SAMI

4 Dezlegarea temporal` de procese (func]ii) în pia]a [i în sistemul de energie electric`

Bibliografie

epilog – un model nodal com-plet dezlegat pentru opera]iile [i serviciile energiei electrice

1 Remarci de încheiere asupra paradigmei dezleg`rii

2 Proiectul nodal de pia]` de energie electric`

3 Foloasele pie]ei de energie cu structur` nodal`

4.3 Transfer capability of inter-area interconnections and constraint management in power system operational security

4.4 Nodal – structured electricity market and the Smart Grids

References

Part III. temporal or variational unbundling


2 Variational unbundling of variables Case No. 1. Mid-term dynamics simulation of the power system. TEMI model Adjustable dynamic equivalents based on REI-Dimo models

3 Process variational unbundling Case No. 2. Power transfer

capabilities of the network’s nodes and relevant interties.

SAMI model.

4 Temporal unbundling of processes (functions) in electric market and system

References

epilogue - A full unbundled nodal model of electricity operations and services

1 Concluding remarks on the unbundling paradigme

2 Nodal electricity market design

3 Gains of the nodal – structured electricity market

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AnneXes

1 numerical solution of the systems of linear equations with sparse coefficient matrix

1. Storage of sparse (coefficient) matrices in the computer memory

2. Solution of the linear system Ax > b by optimally ordered triangular factorization

3. Gaussian reduction of a system of linear equations Ax > b

2 Linear (topologic) graphs. Finding the loop-edge incidence matrix

1. Analytical definition of linear graph

2. Short glossary of graph components

3. Incidence matrices in a graph

4. Basic relationship between incidence matrices

5. A method for finding the loop - edge incidence matrix

3 systems of linear equations associated to a graph

4 the six – step algorithm of the direct structural unbundling

5 representation of synchronous generator in the Ps dynamics simulation

I. In RETRA model II. In TEMI model

6 symmetrical load-flow model of the power network in different hypothesis on the nodal power injections

gLossAry

AneXe

1 rezolvarea numeric` a siste-melor de ecua]ii lineare cu matrice spars` de coeficien]i

1. Memorarea matricelor lacunare în calculator

2. Rezolvarea prin factorizare triunghiular` ordonat` optim a sistemului linear Ax > b

3. Reducerea de tipul Gauss a unui sistem de ecua]ii lineare Ax > b

2 grafuri lineare (topologice). Determinarea matricei de inciden]` bucle-laturi

1. Defini]ie analitic` a grafului linear

2. Mic glosar de componente ale grafului

3. Matrice de inciden]` într-un graf

4. Rela]iile de baz` între matricele de inciden]`

5. Metod` de determinare a matricei de inciden]` bucle – laturi

3 sisteme de ecua]ii lineare asociate unui graf

4 Algoritmul în [ase pa[i al dezleg`rii structurale directe

5 reprezentarea generatorului sincron în regimurile tranzitorii ale see

I. |n modelul RETRA II. |n modelul TEMI

6 Modelul regimului permanent simetric al re]elei electrice în diferite ipoteze privind injec]iile nodale de putere

AbrevIerI

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Introducereîndezlegarea inter-ac]ional̀ , spa]ial̀[i temporal̀ amodelelor[i serviciilorsistemuluideenergieelectric`

“Dezlegarea” este cuvîntulsemnificativ al limbii române [i alaltor limbi de sorginte latin` pentruaflarea solu]iei unei probleme. Cu cîtmai încîlcit` este o chestiune, cu atîtmai potrivit` apare rezolvarea prin“dezlegare” (lat. disligare).

Dezleg`m ca s` simplific`m sau s`clarific`m, dar [i ca s` folosim ceea cecunoa[tem deja.

Dezleg`m în termenii abstrac]i aimatematicii, dar frecvent în cadrelesenzoriale ale spa]iului, timpului [iac]iunii, care repereaz` iremediabilgîndirea [i activitatea antropic` aomului.

Prin jocul întîmpl`rii, autorul s-a ocu-pat de energia electric`, mai cu seam`de aspectele de sistem ale energiei,inclusiv ale func]ionalit`]ii acestuisistem. Sistemul energiei, ca [i rela]iaacestuia cu alte sisteme antropice, auexercitat o atrac]ie puternic` pentruingineri, fizicieni, matematicieni, eco-nomi[ti, filozofi, sociologi [.a. Nu depu]ine ori, modelele [i solu]iile pentruSistemul Energiei Electrice (SEE) aufost, prin similitudine, surse deinspira]ie benefice.

Debutul în cercetarea SEE mi-afost puternic marcat de întîlnireacu un cercet`tor în domeniu, dar [ifilozof, pasionat de metoda

Introductionin inter-action, space and timeunbundling of power systemmodels and services

“Unbundling” is the meaningful wordto Romanian language, and to otherLatin origin languages as well, forreaching solution to an issue. Themore intricate a question, the righterdoes appear an “unbundling” approch(lat. disligare).

We are unbundling with a view tosimplifying or clarifying, yet to usingwhat we already know.

We are unbundling in abstractmathematical terms, yet frequently inthe sensorial frameworks of space,time and action, which irremediablybenchmark the man’s thinking andanthropic activity.

By game of chance, the author hasdealt with electricity, with the powersystem aspects particularly, includingthe system’s functionality. Electricpower system and its relatinshipswith other anthropic systems as wellhave exercised a strong attraction forengineers, mathematicians, econo-mists, philosophers, sociologists etc.Many a time, the Power System (PS)models and solutions have bysimilitude been a beneficial source ofinspiration.

As a beginner in PS research, I wasstrongly marked by a researcher inthe field, and a philosopher as well,passionately fond of diakoptics

9Unbundling Power System Operations and Services

Dezleg\nd func]ionarea [i serviciile sistemului energiei electrice10

diakoptic` propus` de Gabriel Kronîn anii 50’.

Am st`ruit de atunci în dezleg`ri,captivat de orice nou` confirmare. Amadunat în acest volum rezultate aleunor lucr`ri de cercetare efectuatede-a lungul a peste 40 de ani, avîndteza de doctorat ca reper distinct[JCO1]. Le-am ales pe acelea carep`streaz` semnifica]ie contemporan`limitînd prezent`rile la aspecteesen]iale. Presupunînd, deci, c`cititorul este familiarizat cumodelarea sistemelor de energie elec-tric`, am indicat numai bibliografia cem-a influen]at în aceste cercet`ri.Modelele [i metodele propuse nu sîntsimple exerci]ii teoretice. De regul`,ele au fost aplicate în programe decalcul, cele mai multe de produc]ie,pentru operarea [i planificarea SEEdin România.

Prin simpl` coinciden]`, interesulmeu constant pentru dezlegareamodelelor opera]ionale ale SEE s-a în-tîlnit cu recenta tendin]`, irezistibil`,c`tre de-reglementarea pie]elor de en-ergie electric`. Ori, competi]ia poate fiintrodus` numai dac` serviciile debaz` ale sistemului electric sînt dezle-gate în raport cu clien]ii. Sînt dezle-gate “pe vertical`” func]iunile deproducere, transport [i dispecerizare,distribu]ie [i furnizare, [i pre]urileaferente, iar “pe orizontal`”, societ`-]ile concurente, produc`toare [i furni-zoare. De dou` decenii, dezlegarea(unbundling-ul) sistemului “tehnologi-e – client” a devenit pretutindeni temazilei pentru energie electric`, iar au-

method that Gabriel Kron proposed inearly 50’s.

Since then, I have insisted uponunbundlings, being captivated by anynew confirmation. I have gathered inthis book findings of research workthat I carried out along over 40 years,with the doctoral thesis as adistinctive mark [JCO1]. I havechosen those that may have contem-porary significance without insistingon non-essential details. Assuming,hence, that the reader is familiarizedwith power system modeling, I haveindicated only the references that hadan impact on these researches. Theproposed models and methods are notsimply theoretical conjectures. Ingeneral, they were implemented incomputer codes, most of them beingproduction codes used in the opera-tion and planning of Romanian PS.

By mere coincidence, my steadyinterest for unbundling operationalmodels of the PSs met the recent,overwhelming, trend towardsderegulation of electric market. Or,competition can be only achieved if thebasic services of power system are un-bundled as against the electricity cus-tomers. Generation, transmission anddispatching, distribution and supplyfunctions, and associated prices, are“vertically” unbundled, whereas com-petitive generation and supply compa-nies are “horizonally” unbundled. Inthe last two decades, the unbundlingof “technology – customer” system haseverywhere become a news of the dayfor electricity, and the author has the

Unbundling Power System Operations and Services 11

chance to bring a modest contributionto this area of energy reform inRomania.

(For) What and howwe unbundle?

Facing a new mathematical, physicalor technical issue we are seeking forsimplifying or re-structuring proce-dures, so as to making use of solutionsat hand. We are trying to identify andretain the variables and issue compo-nents that are relevant or can be set-tled. A “competitive” selection wouldrender us evident the available appro-priate method for each identified com-ponent. Afterwards, we seek for the“system conditions” offering the neces-sary base for the articulation of indi-vidual solution pieces. Building asuccessful model for “interconnecting”the “partial” solutions, is actually thecritical step in the solution process.

The unbundling as a method of think-ing corresponds with the unbundlingof human actions that have been goingto specialization and “interconnec-tion”. The power system enrolls itselfin the general model of “anthropiza-tion”. The electric market participantsare competing each other with theirparticular solutions, and there is amarket integrating infrastructure.

Therefore, to our understanding, theunbundling is any procedure forsolving an operational issue of techno-logic or “technology – customer”system whereby the system model ispartitionned into components that areindependently solved.

torul a avut [ansa s`-[i aduc` omodest` contribu]ie în acest domeniual reformei energiei în Rom=nia.

(De) Ce [i cum dezleg`m?

|n fa]a unei probleme noi, matematice,fizice sau tehnice, cùt`m proceduri desimplificare sau de re-structurare, ast-fel încît s` putem folosi solu]ii la în-demîn`. |ncerc`m cu predilec]ie sìdentific`m [i s` re]inem variabilelesau p`r]ile problemei care sînt rele-vante sau rezolvabile. Pentru fiecarecomponent` identificat` cùt`m ometod` disponibil` potrivit`, aplicîndo selec]ie “competitiv`”. Cùt`m apoi“condi]iile de sistem” în baza c`roraputem articula solu]iile p`r]ilor indi-viduale. De fapt, pasul critic al proce-sului de rezolvare este reprezentat deconstruc]ia unui model reu[it de “in-terconectare” a solu]iilor “par]iale”.

Dezlegarea ca metod` de gîndirecorespunde cu dezlegarea ac]iuniloromului, care au tins spre specializare[i “interconectare”. Sistemul energieielectrice se înscrie în modelul generalal antropiz`rii. Participan]ii la pia]aenergiei, cu solu]iile lor particulare,se afl` în competi]ie [i exist` oinfrastructur` integratoare a pie]eienergiei.

A[adar, în]elegem prin dezlegare,orice procedeu de rezolvare a uneiprobleme opera]ionale de sistemtehnologic sau “tehnologie – client”prin parti]ionarea modelului sis-temului în componente, care sîntsolu]ionate independent.


Tipologia dezleg`rii

Problemele specifice ale func]ion`rii sis-temelor fizice se investigheaz` pe modelede calcul, sau modele de simulare.

Modelul dezlegat al sistemului puneîn eviden]` subsisteme mai mici alec`ror “sub-probleme” se rezolv` însuccesiune.

M`rimea sistemului este dat` dedimensiunea spa]iului st`rilor, unspa]iu vectorial finit – dimensional.|ntr-o interpretare mai pu]in rigu-roas`, în]elegem prin “dimensiune”num`rul de variabile de stare.

|n general, dezlegarea include etapeîn care sînt determinate solu]iilepentru subsisteme [i etape în caresolu]iile “par]iale” sînt “interconec-tate”.

Dezlegarea în care interconectareasemnific` “corectare” o vom denumide variabile.

Dezlegarea în care interconectareaapare ca simpl` reuniune de solu]iipar]iale, fiindc` nu este prejudiciat` oeroare de calcul prestabilit`, o vomdenumi de procese. |n general,(sub)procesele sînt independente dac`subsistemele con]in în exclusivitatetoate variabilele de stare relevante.

Dezlegarea în numai dou` subsistemeo vom denumi decuplare.

Strategia dezleg`rii este indicat` înprimul rînd pentru simplificareaproblemelor prin reducerea dimen-siunii.

The unbundling typology

Specific operational issues of the physi-cal systems are investigated on compu-tation models, or simulation models.

The unbundled system model rendersevident smaller - size subsystems,whose associated “sub-issues” aresequently solved.

The system’s size is given by thedimension of states’ space, a vectorialdimensional – finite space. In a lessrigorous interpretation, “dimension”would be the number of statevariables.

Generally, the unbundling includessteps in which solutions of thesubsystems are determined and stepsin which “partial” solutions are“interconnected”.

The unbundling would be of variablesif interconnection signifies“correction”.

The unbundling would be of processesif interconnection appears as simplyassembly of partial solutions, since apre-established computational accu-racy was ensured. In general, the(sub)processes are independent whenthe subsystems contain exclusivellyall relevant state variables.

Decoupling is a particular unbundlingwith only two subsystems.

Therefore, the unbundling strategy isrecommended firstly for its ability tosimplify the issues through reductionof their size.


Simpler issues, simpler solutions and,frequently overlooked, easier reason-ing on them. The models of electric PSbecame particularly complex due tophysical interconnection and increas-ing detailation of their constitutiveelements. In the same time, un-bundling joins the outstanding trendtowards microprocessor specializationand “parallelization” of computations.

The simulation of PS operationalregimes implies, generally, solutionof large – scale linear equations byoptimal triangular factorization.(Annex 1).

Through unbundling, the model equa-tions can be reorganized in a way thatfactorization generates matrix struc-tures that are more convenient to“parallel” processing.

Considering the physical nature of in-terdepencies that are candidating tounbundling, and the particular parti-tioning criterion, the following threeclasses of unbundling can be identi-fied [JCO1, JCO2]:

a) Inter-action (functional)unbundling

It is unbundling whereby subsystemsare the result of neglecting relativelyweak functional inter-actions. A sub-system would include relevant func-tional interdepences only: as againstcertain state variables or as againstfunctions of state variables, whichgenerally represent distinct processes.

Problemele mai simple au solu]ii maiu[or de calculat [i, deseori se scap` dinvedere, mai u[or de interpretat. Mode-lele SEE au devenit foarte complexedatorit` interconect`rii fizice [idetalierii crescînde a elementelor com-ponente. Dar, totodat`, dezlegarea con-cord` cu tendin]a marcant` despecializare a microprocesoarelor [i de“paralelizare” a calculelor.

Determinarea regimurilor de func-]ionare ale SEE implic`, de regul`,rezolvarea de ecua]ii lineare de maridimensiuni, prin factorizaretriunghiular` optim` (Anexa 1).

Prin dezlegare, ecua]iile modelelorpot fi reorganizate astfel încît factori-zarea s` genereze structuri de ma-trice, convenabile prelucr`riinumerice “paralele”.

}inînd cont de natura fizic` a inter-dependen]elor candidate la dezle-gare, [i de criteriul specific departi]ionare, se pot identifica ur-m`toarele trei tipuri de dezlegare[JCO1, JCO2]:

a) Dezlegarea inter-ac]ional`(func]ional`)

Este dezlegarea prin care subsis-temele sînt rezultatul neglij`rii inter-ac]iunilor func]ionale relativ slabe.Subsistemul va include numai inter-dependen]ele func]ionale relevante:fa]` de anumite variabile de stare, saufa]` de func]ii de variabile, care îngeneral reprezint` procese distincte.


Dezlegarea func]ional` poate fi limi-tat` la aspectele opera]ionale ale sis-temului, sau poate include [i serviciilesistemului pentru consumatorii de en-ergie electric`. Sistemul dezlegat“tehnologie – pia]`” sau “tehnologie –client” este de dat` recent` fiind oconsecin]` a dezleg`rii func]iunilorSEE, pe vertical` [i pe orizontal`.

b)Dezlegarea spa]ial` (structural`)

Este dezlegarea prin care subsis-temele sînt rezultatul descompuneriistructurii unui model de tip re]ea.Subsistemul include numai ele-mentele selec]ionate ale re]elei, deregul` într-un spa]iu bine delimitat.A[adar, dezlegarea spa]ial` se poateaplica sistemelor a c`ror structur`func]ional` este descris` de o re]ea(graf), cum este de pild` un SEE.

Abordarea rezolv`rii re]elelor mari,prin descompunere în subre]ele, a fostpropus` de c`tre Gabriel Kron în anul1950 sub numele de diakoptic` [GK].Termenul a fost introdus de c`trePhilip Staneley de la facultatea defilozofie a lui Union College, Schenec-tady, New York. El a rezultat din com-binarea cuvintelor grece[ti “kopto” – asec]iona [i “dia”, interpretat prinno]iunea de “sistem”. Kron [i-a bazatmetoda pe teoria circuitelor electrice“ortogonale”, care a suscitat în epoc` oîntreag` literatur` [AS, HH1].

Noi am ar`tat c` sîntem în prezen]aunei dezleg`ri structurale a modelu-lui sistemului, fizic sau topologic, cepoate fi dezvoltat` în cadrul teorieigenerale a sistemelor lineare [JCO1].

Functional unbundling can be con-tained to operations of the systemonly, or it can include the system’sservices to power consumers as well.The unbundled “technology - market”or “technology - customer” system isrelatively recent being a consequenceof vertical and horizontal unbundlingof the PS functions.

b) Spatial (structural) unbundling

It is unbundling whereby the subsys-tems are the result of decompositionof a network – structured model. Asubsystem would include the selectednetwork elements only, generallywithin a clear-born physical space.Hence, structural unbundling can beapplied to systems having an opera-tional structure described by a net-work (graph), as for instance a PS.

The approach of solving large net-works through subnetwork decompo-sition has been proposed by GabrielKron in the year 1950 as diakoptics[GK]. The term has been coined byPhilip Staneley from the Union Col-lege’s faculty of philosophy, Schenec-tady, New York. It resulted from acombination of Greek words “kopto” –to tear, and “dia” which was inter-preted as system. Kron based this ap-proach on the theory of “ortogonal”electric circuits that led to a wholeliterature [AS, HH1].

We have shown that is about a struc-tural (spatial) unbundling of a modelof physical system which can be de-veloped within the framework of gen-eral theory of linear systems [JCO1].


A special kind of structural un-bundling is ensured by the REI –Dimo models [PD]. Paul Dimo demon-strated that PS security requirementscan be analysed at one network nodeonly. The PS area of interest is thusreduced to an essential size and un-bundled from the rest of the system by“applied” voltages to border nodes.

The nodal paradigm that was intro-duced by Paul Dimo for the PS opera-tional analysis can be extended to theoverall “technology - market” systemof electricity, as well. We will showthat electricity and transmissionservice can be traded with referenceto one network node. Every nodalelementary electricity market isspatially unbundled, since one canfully define at the level of node boththe traded product and operationalsecurity standard, namely the powertransfer capability.

c) Temporal (variational)unbundling

It is unbundling whereby the subsys-tems are the result of separation onthe time domain of processes andvariables, during certain time periodsor over the whole simulation time.The state variables of subsystemmodel have a similar variation speed.

In the “technology - market” electric-ity system, the process temporal un-bundling signifies that preventive andcorrective operational function of thesystem operator is carried out in a se-quence of time-frames.

O dezlegare structural` special` esterealizat` de modelele REI – Dimo[PD]. Paul Dimo a demonstrat c` ce-rin]ele de siguran]` ale SEE se potanaliza dintr-un singur nod al re]elei.Zona de interes a SEE este astfel redu-s` la o dimensiune esen]ial` [i dezle-gat` de restul sistemului prin tensiuni“aplicate” la noduri de frontier`.

Paradigma nodal` introdus` de PaulDimo în analizarea func]ion`rii SEEfizic poate fi extins` la întregul sistem“tehnologie – pia]`” de energie elec-tric`. Vom ar`ta c` energia [i servi-ciul de transport se pot tranzac]ionacu referire la un nod al re]elei.Fiecare pia]` elementar` nodal` deenergie electric` este dezlegat`spa]ial, deoarece la nivelul nodului sepot defini pe deplin, atît produsultranzac]ionat, cît [i standardul de sig-uran]` opera]ional`, respectiv capa-bilitatea de transfer al energiei.

c) Dezlegarea temporal`(varia]ional`)

Este dezlegarea prin care subsistemelesînt rezultatul separ`rii în domeniultimpului a proceselor [i variabilelor, peanumite perioade de timp sau pe în-tregul interval de simulare. Variabilelede stare din modelul subsistemului auvitez` de varia]ie asem`n`toare.

|n sistemul “tehnologie – pia]`” de en-ergie electric`, dezlegarea temporal`a proceselor semnific` faptul c`func]ia opera]ional` a operatorului desistem, preventiv` [i corectiv`, esterealizat` în etape succesive.

[AS] Amari, Shun-ichi - Topological Foundation of Kron’s Tearing of Electric Network (R.A.A.G.Memoires, Vol. 3, 1962).

[GK] G.Kron-Diakoptics:ThePiecewiseSolutionofLarge–ScaleSystems(MacDonald,London,1963).

[HH1] H.H. Happ (editor) - Gabriel Kron and Systems Theory (Union College Press, Schenectady,New York, 1973).

[PD] P. Dimo - Nodal Analysis of Power Systems (Editura Academiei, Bucure[ti, România; Aba-cus Press, Tunbridge Wells, Kent, 1975).

[JCO1] J. Constantinescu - Contribu]ii privind aplicarea decupl`rii func]ionale [i structurale în mod-elele de calcul al regimurilor de func]ionare ale sistemelor electroenergetice complexe (Con-tributions to Functional and Structural Unbundling in the Operational Models of ComplexPower Systems– PhD thesis) (Tez` de doctorat, Institutul Politehnic Bucure[ti, facultatea deEnergetic ,̀ 1980).

[JCO2] J.Constantinescu -MidtermTransientProcesseswithDimo-REIModels.Part II (Rev.Roum.des Sci. Techn., Vol. 30, No. 3, pages 337 - 344, 1985)

[JCO3] J. Constantinescu- The Essential Role of Transmission Network in the Design of EnergyMarkets (Proceedings of the International Workshop "On the way to Transparency of energysystems", pp. 146 - 154, Ljubljana, 1997).

[JCO4] J. Constantinescu [ Paradigme noi pentru sistemul [i pia]a de energie electric` inspirate deAnaliza Nodal` Dimo (New Paradigmes of Electric System and Market Inspired from theDimo's Nodal Analysis) (Energetica 9, 2005).

The present work is an invitation to-wards unbundling power systemmodels in respect of action, space andtime depending on particular featuresof the given issue, as a first solutionoption. There is already available areach stock of specific solutions basedon smart models, which undoubtedlydemonstrated the practical usefulnessof this approach. We strongly believethat unbundling, when applied sys-tematically and with judgment, i.e.,based on a general concept, can bringsignificant efficiency and clearnessgains in the solution of powersystems’ operational and functionalissues.

Through unbundling, the bunch’sflowers lowers exhibit their beautyand significance.

Lucrarea este o invita]ie spre dezle-garea modelelor sistemului de energieelectric` în raport cu ac]iunile, spa]iul[i timpul, în func]ie de particula-rit`]ile problemei date, ca prim`op]iune de rezolvare. Exist` deja unfond bogat de rezolv`ri specificebazate pe modele inteligente, care audemonstrat concludent utilitatea prac-tic` a acestei abord`ri. Sîntem în-credin]a]i c` aplicarea sistematic` [icu discern`mînt a dezleg`rii, adic` înbaza unui concept general, poateaduce însemnate sporuri de eficien]`[i claritate în rezolvarea problemeloroper`rii [i dezvolt`rii sistemelor deenergie electric`.

Dezlegîndu-le, florile buchetuluibuchetului î[i arat` frumuse]ea [isemnifica]ia.

16 Dezleg\nd func]ionarea [i serviciile sistemului energiei electrice

Bibliografie / References

Partea IDezlegarea inter-ac]ional` saufunc]ional`

1. Definirea conceptului

Dezlegarea inter-ac]ional`, sau func-]ional`, în modelul sistemului fizic(tehnologic), sau sistemului “tehno-logie – client”, semnific` separarea desubsisteme prin omiterea interdepen-den]elor dintre (a) variabile sau (b)func]ii de variabile (procese).

Cazul (a) este dezlegare func]ional`de variabile, aplicabil` de regul` peetape, pentru economisirea [i cre[te-rea convergen]ei calculelor. De pild`,componentele “longitudinale” [i“transversale” ale (corec]iilor) tensiu-nilor nodale se pot determinaprintr-un procedeu pas cu pas, negli-jînd interdependen]ele de intensitateredus` dintre componente, pîn` laatingerea unei erori de calcul impuse.

Cazul (b) este dezlegarea func]ional`de procese. Este abordarea tradi]io-nal` în dezvoltarea de modele de sis-tem, care p`streaz` irevocabil numaivariabilele de stare [i interdependen-]ele relevante pentru o cerin]` par-ticular` a practicii. Astfel, în anumitesitua]ii, regimurile de func]ionare aleSEE se pot determina neglijîndu-serela]ia dintre injec]iile de putereactiv` [i reactiv` din nodurile re]elei.

|n sistemul “tehnologie-client” dezlegatfunc]ional al energiei electrice, impar-]ialitatea serviciului de transport [i desistem este premiza - cheie pentru

Part IInter-action or functionalunbundling

1. Concept definition

Inter-action, or functional unbund-ling, in the model of physical (tech-nology) system, or of “technology –customer” system, signifies separa-tion of subsystems by omitting inter-dependences between (a) variables or(b) functions of variables (processes).

Case (a) is the variable functionalunbundling, which is typically appliedin steps with a view to reducing thecomputation and increasing its con-vergence. For instance, the “longitudi-nal” and „transversal” (correction)voltage components can be deter-mined in a step-by-step procedure byneglecting the low intensity compo-nent interdependences, until rea-ching an imposed computation error.

Case (b) is the process functionalunbundling. It is the traditional ap-proach in the development of systemmodels, which irrevocably preservesonly the variables and interdepenciesthat are relevant for a particularpractical requirement. Thus, in cer-tain cases, the PS regimes can be de-termined by neglecting the relationshipbetween active and reactive powerinjections at network nodes.

In the functionally unbundled“technology-customer” system ofelectricity, impartiality of transmis-sion and system operation service is a

17Unbundling Power System Operations and Services


accesul nerestric]ionat la pia]a deenergie electric` [JCO9-11].

Serviciul de transport implic` respon-sabilit`]i conform standardelor desiguran]` [i calitate, [i poate fi livratindependent. Un contract între unproduc`tor [i un furnizor nu-l poateinfluen]a în niciun fel. Pe de alt`parte, particularit`]ile SEE impun caacest serviciu s` nu fie desp`r]it defunc]iunile privind planificarea opera-]ional` [i conducerea opera]ional`prin dispecer.

Fie modelul general pentru simu-larea unui proces în sistemul fizicdepinzînd de dou` seturi devariabile, descris de urm`toareaecua]ie matriceal`:

S` presupunem c` modelul poate fistructurat astfel încît s` fieeviden]iate dou` procese P1 [i P2 cafunc]ii distincte de variabile,

Pentru a evalua intensitatea rela]iei din-tre mul]imile de variabile sau func]ii devariabile (procese), vom presupune ovaria]ie suficient de mic` a procesului,care este indus` de modificarea unuiparametru oarecare [i descris` de ecua]iamatriceal̀ “linearizat`” a ecua]iei (2):

key prerequisite for introducing wideraccess to electricity market [JCO9-11].

The transmission service implies theresponsibilities for energy trans-mission according to well defined se-curity and quality standards, and canbe supplied independently. A contractin between a generator and a supplierdoes not influence it any way. On theother hand, the peculiarities of thepower system claim that this servicebe not split from the operationalplanning and dispatching functions.

Let be the general model for si-mulation of a process within the phy-sical system depending on twovariable sets, as described by thefollowing matrix equation:

(1)

Let us suppose that the model can berestructured so as to render evidenttwo processes P1 and P2 as distinctfunctions of variables,

(2)

In order to evaluate intensity of relationbetween the sets of state variables orfunctions of variables (processes), weconsider a small enough processvariation that is induced by a change ofsome parameter and described by the“linearized” matrix equation (2):

(3)


Or, in the inversed form,

(4)

We will distinguish two kinds of weakinter-depencies that can be neglectedand, correspondingly, two types ofFunctional Unbundling (FU):

- Variable FU- Process FU.

2. Variable functionalunbundling

When equations (3) feature,

(5)

the sub-matrices of (3) can beneglected resulting the general modelof variable functional unbundling, as:

(6)

Let us suppose that ∆P1 and ∆P2 aredetermined as mismatches for the givenconditions P10 [i P20 and approximationsx0 and y0. The two less – sized sets ofequations (6), as against the original set(3), can independently be solved, andthe abatements can be used to updatethe approximations x0 and y0.

Sau, în form` inversat`,

Vom distinge dou` categorii de inter-dependen]e reduse, care se pot neglija[i, corespunz`tor, dou` tipuri deDezleg`ri Func]ionale (DF):

- DF de variabile- DF de procese.

2. Dezlegarea func]ional`de variabile

Dac` în sistemul de ecua]ii (3) putemconsidera,

sub-matricele respective pot fi negli-jate, rezultînd modelul general al dez-leg`rii func]ionale de variabile:

Presupunem c` ∆P1 [i ∆P2 se deter-min` ca diferen]e - erori fa]` de condi-]ii date P10 [i P20 [i pentru aproxima]iix0 [i y0. Cele dou` sisteme de ecua]ii(6), de dimensiune redus` în raport cusistemul originar (3), pot fi rezolvateindependent, iar cu abaterile se potactualiza aproxima]iile x0 [i y0.

[i


De cele mai multe ori îns`, noile dife-ren]e ∆P’1 [i ∆P’2 nu convin, procesulde calcul necesitînd rezolvarea ite-rativ` a ecua]iilor (6), ajustîndu-sepas cu pas aproxima]iile x0 [i y0 pîn`cînd ∆P1 and ∆P2 sînt mai mici decîtun prag de eroare prestabilit.

Exemplul nr. 1. Metoda Newton –Raphson “decuplat`”.Modelele NERA - 5 [i NERA - 6

Cea mai r`spîndit` metod` pentrucalculul regimurilor permanente aleSEE a fost propus` de Stott [i Alsac[ST-AL] fiind cunoscut` ca metodaNewton – Raphson “decuplat`”. |nesen]`, modelul N-R decuplat este oconsecin]` a neglij`rii dependen]eireciproce dintre variabilele de starelongitudinale [i transversale, în numai pu]in cunoscutul model Newton –Raphson [JW-HH],

∆P [i ∆Q se determin` ca diferen]e -erori fa]` de injec]ii de puteri date P0

[i Q0, pe baza estim`rilor θ est > θ < ∆θ[i Uest > U < ∆U (Anexa 6). |n ecua-]iile (7), scrise în form` simbolic`, seneglijeaz` elementele sub-matricelorN [i J, iar la calculul elementelorsubmatricelor H [i L se aproximeaz`:UiGij sin(θi-θj)>0 [i cos(θi-θj)>1, în careGij sînt conductan]e în matriceaadmitan]elor nodale.

However, in almost all cases, the new mis-matches ∆P’1 and ∆P’2 are not acceptable,the computation process requires aniterative solution of the equations (6) anda step by step adjustment of x0 and y0

until the ∆P1 and ∆P2 are less than a pre-defined error threshold.

Case No. 1. “Decoupled” Newton –Raphson method.NERA - 5 and NERA - 6 models

The most widespread method fordetermination of the PS load flowswas originally proposed by Stott andAlsac, and is known as “decoupled”Newton – Raphson method [ST-AL].Essentially, the decoupled N – Rmodel is consequence of omission thereciprocal dependence betweenlongitudinal and transversal statevariables in the no less knownNewton – Raphson model [JW-HH]:

(7)

∆P and ∆Q are determined as mis-matches for the given power injectionsP0 and Q0, and estimations θest > θ < ∆θand Uest > U < ∆U (Annex 6). In thesymbolically written equations (7), theentries of sub-matrices N and J areneglected while those of H and L arecalculated under the approximationsUiGij sin(θi-θj)>0 and cos(θi-θj)>1, whereGij are conductances of the nodaladmittance matrix.


We proposed a model variant on aclearer basis noticing that,

(a) when computing the diagonalentries of H matrix, one canapproximate

however,(b) Qi can be neglected in the calcu-

lation of diagonal elements ofthe matrix L [JCO5].

Expressing the variables in polar coor-dinates, the following model is yielded:

(8)

where Bij are susceptances in thenodal admitanttance matrix, and the“0” index designates the quantities of“starting” state.

Equations (8) are alternatively anditeratively solved by optimaltriangular factorization until aconvergence criterion is met.

This model exhibits higher efficiency (con-vergence) as against conventional Stott –Alsac method, since no useless approxi-mation is introduced in functional un-bundling (decoupling) of variables. Forinstance, there is no justification to appro-ximate voltages with their nominal valueswhen angle corrections as well as diago-nal entries of Jii

θ in (8) are determined,

Noi am propus o variant` a modeluluipe o baz` mai clar` observînd c`,

(a) la calculul termenilor diagonaliai matricei H, se poate apro-xima

dar,(b) Qi se pot neglija în expresia ter-

menilor diagonali ai matricei L[JCO5].

Cu variabilele scrise în coordonatepolare, rezult` modelul urm`tor:

în care, Bij sînt susceptan]ele dinmatricea admitan]elor nodale iarindicele “0” desemneaz` m`rimile dinregimul “de plecare”.

Ecua]iile (8) se rezolv` alternativ [irepetat, prin factorizare triunghiular`optim`, pîn` la îndeplinrea criteriuluide convergen]`.

Fa]` de metoda clasic` Stott – Alsac, seob]ine un cî[tig de eficien]` (conver-gen]`), deoarece dezlegarea (decuplarea)func]ional` de variabile este aplicat`f`r` aproxima]ii inutile. Astfel, nu sejustific` aproximarea tensiunilor cuvalorile lor nominale la determinareacorec]iilor de unghiuri [i, de asemenea,a elementelor diagonale Jii

θ din (8),


De asemenea, renun]area la rapoar-tele exacte Uj0 / Uio, izvorît` dintr-oîn]elegere mai degrab` intuitiv` adezleg`rii (decupl`rii), introduceineficien]` de calcul, mai ales dac`regimurile se determin` în succesiune,unele din altele.

Programele de calcul NERA – 5 [iNERA – 6, pentru modele cu tensiuniexprimate în coordonate rectangulare[i respectiv polare, au fost aplicatemul]i ani în planificarea opera]ional`a SEE din România.

3Dezlegarea func]ional` deprocese

Dac` în sistemul de ecua]ii (4) avem,

sub-matricele respective pot fi negli-jate, rezultînd urm`torul model, aldezleg`rii func]ionale de procese:

Se verific` simplu c`, dac` însistemul (3) se pune alternativP1 (x,y) > 0 [i respectiv P2 (x,y) > 0,sub-matricele ∂x / ∂P1 [i ∂y / ∂P2

din (4) sînt nule, [i avem:

(9)

Additionally, renunciation to exactUj0 / Uio ratios that is based on arather intuitive understanding of theunbundling (decoupling), introducescomputation inefficiency when theload flows are determined insuccession, particularly.

The computer codes NERA – 5 andNERA – 6, for the models withvoltages expressed in rectangular andpolar co-ordinates respectively, havebeen used for years in operationalplanning of the Romanian PS.

3 Process functional unbundling

When equations (4) feature,

(10)

the respective sub-matrices can beneglected and the model of processfunctional unbundling results, as:

(11)

It can be simply verified that, ifalternatively P1 (x,y) > 0 andP2 (x,y) > 0 in the system (3), thesub-matrices ∂x / ∂P1 and ∂y / ∂P2

in (4) are nought, and it results:

[i


(12)

The two sets of equations (11), less –sized as against the original system (3),can be solved independently. If the sub– processes P1 and P2 interrelate eachother unsignificantly, that is, if thealternance P1 (x,y) > 0 and P2 (x,y) > 0is actually valid when the processes P2

and P1 respectively are determined, thecomputation algorithm of processfunctional unbundling does not requireiterative solution of equations (11).

In the PS operation, it is notorious theweak inter-action between the activepowers that are controled by thevoltage angles, and the reactivepowers that are controled by thevoltage magnitudes. Similarly, afunctional process unbundling thatled to confirmed practical resultsresorts to the invariance of generatoractive load as against the change ofnetwork regime (∆Pg > 0), a solutionthat was used for development of theSAMI model (Case No. 2, Part III).

Cele dou` sisteme de ecua]ii (11), dedimensiune redus` în raport cu sistemuloriginal (3), pot fi rezolvate independent.Dac` sub-procesele P1 [i P2 se influen-]eaz` între ele nesemnificativ, adic`dac` alternan]a P1 (x,y) > 0 [i P2 (x,y) >0 este real valabil` la determinareaproceselor P2 [i respectiv P1, algoritmulde calcul al dezleg`rii func]ionale deprocese nu necesit` rezolvarea repetat`(iterativ`) a ecua]iilor (11).

|n func]ionarea SEE, este de noto-rietate interac]iunea redus` dintreputerile active, controlate de unghiu-rile tensiunilor, [i puterile reactive,controlate de amplitudinea tensiu-nilor. De asemenea, o dezlegarefunc]ional` de proces ce a condus larezultate practice confirmate recurgela invarian]a sarcinii active a gene-ratorului la modificarea regimuluire]elei (∆Pg > 0), solu]ie folosit` ladezvoltarea modelului SAMI(Exemplul nr. 2, Partea a III-a).


Exemplul nr. 2. Reprogramareagener`rii avînd influen]`determinant` asupra stabilit`]iistatice a SEE

Selectarea automat` a puterii genera-te în cre[tere / sc`dere, cu efect deter-minant asupra stabilit`]ii statice, estede mare interes în evalu`rile privindsiguran]a func]ion`rii SEE, în parti-cular în analiza contingen]elor. Pro-cedeul pe care îl propunem este bazatpe un indice (unghiular) de stabilitatederivat din “criteriul practic” ∂P / ∂δ,conjugat cu echivalen]i ai sistemuluiredu[i fundamental. Este un model demarj` de stabilitate care concretizeaz`de fapt o dezlegare de proces în contro-lul gener`rii de putere, în baza c`reiamodificarea puterii active este deter-minat` esen]ial de varia]ia unghiuluiintern δ al generatorului. Indicele seob]ine ca raport între criteriul practicpentru starea analizat` a SEE [i unulsimilar al st`rii de referin] ,̀ în care toateunit`]ile generatoare s\nt completdesc`rcate de sarcin` activ .̀

Se folose[te un model echivalent desistem foarte simplu, în care fiecaregenerator este conectat la o bar` deputere infinit`. Echivalen]ii sînt calcu-la]i pe baza puterilor active generate [ia tensiunilor nodale ale re]elei cores-punz`toare st`rii de start. De fapt,reactan]a echivalent` a sistemului re-flect` curentul de scurtcircuit (sta]io-nar) furnizat de c`tre sistem în nodulde conectare a generatorului la re]ea.

Case No. 2. Generationreschedule with the strongestimpact on the PS steady – statestability

The automated selection of up-ward /down – ward power generation, with adeterminant effect on the PS steady –state stability, is of much interest inthe PS security assessment, in thecontingencies analysis particularly. Wepropose a procedure that is based onthe (angular) stability index derivedfrom the “practical criterion” ∂P / ∂δ, inconjunction with system equivalentsthat are reduced fundamentally. It isactually a model of stability marginthat puts the process unbundling in thepower generation control in a concreteform, on the base that the change ofactive power is essentially determinedby the variation of generator’s internalangle δ. The index is established as aratio between the practical criteria ofthe given PS state and of a referenceone, in which all generating units arecompletely unloaded.

A very simple equivalent system modelis used, in which every generating unitis connected to an infinite powercapacity busbar. The active generatingpowers and the network’s bus voltagesof the starting state are used for thedetermination of equivalents. Actually,the equivalent system reactancereflects the (steady-state) short-circuitpower that is delivered by the system inthe corresponding connection grid node.


The algorithm

Step 1. For each node connectinggenerators, an equivalent internalreactance Xg of aggregated generatingunit is determined.

Step 2. For each node connectinggenerators, an equivalent systemreactance X is found by Gaussianreduction of the nodal susceptancematrix (Annex 1, §2).

Step 3. For each aggregatedgenerating unit, a relative stabilityindex (∂P / ∂δ)* is computed as follows:

(13)

Step 4. Generating units are rankedin the ascending order of stabilityindex

Step 5. The load of selected gene-rating units is to be rescheduled asfollows:

- The first units in the orderedlist are downloaded (with agiven power increment) up toreaching a (given) amount ofpower

- Likewise, there would be up-loa-ded the last units from the list.

Algoritmul

Pasul 1. Pentru fiecare nod conectîndgeneratoare, se determin` reactan]aechivalent` intern` Xg a grupuluigenerator agregat.

Pasul 2. Pentru fiecare nod conectîndgeneratoare, se calculeaz` reactan]aechivalent` X a sistemului prinreducerea de tipul Gauss a matriceisusceptan]elor nodale (Anexa 1, §2).

Pasul 3. Pentru fiecare grup generatoragregat, se calculeaz` indicele relativ destabilitate (∂P / ∂δ)*, dup` cum urmeaz :̀

Pasul 4. Grupurile generatoare seordoneaz` în ordinea cresc`toare aindicelui de stabilitate

Pasul 5. Sarcina grupurilor genera-toare selectate urmeaz` s` fie repro-gramat` astfel:

- Primele grupuri din listaordonat` sînt desc`rcate (cu unincrement de sarcin` specificat)pîn` la atingerea unei (anumite)puteri cumulate

- Asem`n`tor, se încarc` ultimelegrupuri din list`.


Reprogramarea este transmis`codului de calcul SAMI (Partea a III-a,Exemplul nr. 2) ce determin` indexulintegral de stabilitate ISSM al SEE,traiectoriile variabilelor de stare pîn`la starea limit` [i Capabilit`]ile deTransfer pentru Interconexiunile inter– zonale Relevante pre-definite.

Tabelul 1.1 arat` indicativ indicii destabilitate de ordonare pentru o re]eareal` a sistemului na]ional, avînd20 noduri, 25 de elemente [i 8 grupurigeneratoare [JCO-MH3].

Tabelul 1.1. Indici de stabilitate static`(unghiular`) în ordine ascendent`

Nodul generatorului (∂P / ∂δ)*

13 0,1842 0,3891 0,63516 0,65415 0,7637 0,82312 0,91111 0,941

The reschedule is sent to the SAMIcode (Part III, Case No. 2) thatdetermines the PS integral stabilityindex ISSM, the trajectories of statevariables up to the limit state and theTransfer capabilities of Relevant inter– area Interconnections that are pre-defined.

The Table 1.1 shows indicatively thestability ordering indices for a realnetwork of the national PS with20 nodes, 25 elements and 8 genera-ting units [JCO-MH3].

Table 1.1. Steady – state angle stabilityindices in the ascending order

Generator’s node (∂P / ∂δ)*

13 0.1842 0.3891 0.635

16 0.65415 0,7637 0.823

12 0.91111 0.941


Case No. 3. Power system loadflows with minimal networklosses.

NERAmodel

The general issue of PS optimal ope-ration, with a minimal cost ofend-use electricity, could be modeledand solved with particular difficulty.The practice has consecrated theapproach of process functionalunbundling, which is based on therelative independence of the (proces-ses of) generation active power dis-patch and reactive power distributionwithin the network, respectively.

The model NERA [JCO-MH2] reducesthe losses in an electric network byoptimizing the reactive power distri-bution. The direction towards optimalstate is given by the slope or gradientof objective function (f) representingthe amended power losses of thenetwork. The penalty functionexpresses the “soft” constraints, as aµ-weighted sum of squared voltagedeviations as against their limits incertain load nodes (NS

’ ),

(14)

The initial state (x,u)0 corresponds toa possible operational regime of thesystem, the following ones arestep-by-step estimated, as follows:

(15)

Exemplul nr. 3. Regimulpermanent al SEE cu pierderiminime de putere în re]ea.

Modelul NERA

Problema general` a func]ion`riioptime a SEE, corespunz`toare costu-lui minim al energiei electrice la con-sumator, comport` dificult`]ideosebite de modelare [i solu]ionare.Practica a consacrat abordarea prindezlegare func]ional` de procese,bazat` pe independen]a relativ` a(proceselor) reparti]iei sarcinii activepe generatoare [i, respectiv,distribu]iei puterii reactive în re]ea.

Modelul NERA [JCO-MH2] reducepierderile de putere în re]ea prinoptimizarea distribu]iei puterilorreactive. Direc]ia spre starea optim`este dat` de panta sau gradientulfunc]iei-obiectiv (f) reprezentîndpierderea de putere în re]ea,amendat`. Func]ia de penalizareexprim` restric]iile “slabe”, ca sum`ponderat` a p`tratelor abaterilortensiunilor fa]` de limitele impuse înunele noduri de sarcin` (NS

’ ),

Starea ini]ial` (x,u)0 corespunde uneisitua]ii posibile de func]ionare asistemului, urm`toarele se estimeaz`pas cu pas, astfel:


unde dk este vectorul direc]iei de c`uta-re (a st`rii optime), Bk este lungimeapasului de avans, x este vectorul varia-bilelor de stare, i.e. tensiunile depen-dente, iar u este vectorul variabilelor dereglaj, respectiv tensiuni independente[i rapoarte de transformare aletransformatoarelor cu reglaj în sarcin`.

A[a-zicîndul model al gradientului“redus” [6] include în vectorul –direc]ie numai derivatele func]iei f înraport cu tensiunile,

în care, matricea hesian,

a fost exprimat` în func]ie degradientul “redus”,

Gradientul se calculeaz` f`r` dificul-t`]i deosebite prin rezolvarea siste-mului de ecua]ii al regimuluipermanent, pe baza modeluluiNewton – Raphson (Anexa 6), [i aunor ecua]ii lineare avînd matricelejacobian Jx [i Ju din acela[i modelN-R [DO-TI], astfel:

where dk is the vector of searchdirection (towards optimal state),Bk is the advance step length, x isthe vector of state variables, i.e. thedependant voltages, and u is thevector of control variables asindependant voltages and under loadtransformer ratios.

The so – called model of “reduced”gradient [6] includes in the directionvector the derivatives of function f inrespect to voltages only,

(16)

in which, the Hessean matrix,

(17)

was expressed in terms of the“reduced” gradient,

(18)

The gradient is calculated with noparticular difficulty by solving theload – flow set of equations, on thebase of Newton – Raphson model(Annex 6), and linear equationswith the Jx and Ju Jacobeans fromthe same N-R model [DO-TI], asfollows:


(19)

In (19), H(x,u)>0 represents theequality constraints in the form ofnodal power equations, and λ, thevector of Lagrange multipliers asso-ciated to H, or of the sensitivities of fin respect to the nodal powerinjections.

The flowchart in Figure 1.1 illustratesthe main steps of computationalgorithm.

Fig 1.1 Main steps of the reduced gradientmethod

|n (19), H(x,u)>0 reprezint` restric]iilede egalitate sub forma ecua]iilorbilan]urilor nodale de puteri, iar λ,vectorul multiplicatorilor Lagrangeasocia]i cu H, sau al sensibilit`]ilorfunc]iei f în raport cu injec]iile nodalede puteri.

Principalele etape ale algoritmului decalcul sînt ilustrate în schema logic`din Figura 1.1.

Fig 1.1 Principalele etape ale metodeigradientului redus


Aplica]iile pe scheme reale ale SENau ar`tat c`,

- Starea optim` se ob]ine de regul`dup` 4 – 5 itera]ii (regimuri)succesive

- Pierderile de putere în re]ea scad cu3 – 5% în regimul sta]ionar optimizat

- Coeficien]ii de penalizare mariaccelereaz` conformitatea ten-siunilor cu restric]iile, dar potdeprecia uneori convergen]a cal-culelor.

Figura 1.2 exemplific` varia]ia pier-derilor de putere în itera]iile deoptimizare, cu [i f`r` limit`ri impusetensiunilor nodale. Au fost alese dou`re]ele de transport func]ionînd latensiunile de 220 [i 400 kV: (a) de67 noduri, în regim de vîrf de sarcin`,[i (b&c) de 45 noduri, în regim de gol[i respectiv de vîrf de sarcin`.

Fig 1.2 Varia]ia pierderilor de putere înprocesul de optimizare, cu [i f`r` restric]ii latensiuni

Applicatins on the real schemes ofnational PS have shown that,

- Optimal state is normallyreached in 4 – 5 successiveiterations (regimes)

- The network’s power losses decreaseby 3 – 5% in the optimal state

- Large penalizing coefficients canaccelerate the conformity of voltagesas against the constraints, however,in certain cases, the computationconvergence might be depreciated.

Figure 1.2 exemplifies variation of thenetwork power losses along opti-mization iterations, with and withoutlimitations on the nodal voltages.There were chosen two 220 & 400 kVtransmission networks: (a) of67 nodes in a peak load regime, and(b&c) of 45 nodes, in shallow andpeak load regimes, respectivelly.

Fig 1.2 Power losses variation along theoptimization process, with and withoutvoltage constraints


Extending the process functionalunbundling to variables too, a modelof particular interest for practice isobtained [JCO1].

Thus, the model (18) is reshaped usingthe second set of quations (8), as follows:

(20)

where JU is the coefficient matrix inthe second set of quations (8), kT arethe adjustable transformer ratios,while G and L signify indices ofgenerators and loads.

The gradient is determined aftercalculation of λQ multipliers, from thefirst set of equations (20), and theirreplacement in the second set. Equations(20) can be much easier solved than (19),and the only approximation that canhave an impact on the final rezult is theneglect of ∂f / ∂θ, a relatively insig-nificant component, generally. The vec-tor ∂f / ∂UL that is really important, iscalculated with no approximation.

Extinzînd dezlegarea func]ional` deprocese [i la variabile, se ob]ine ovariant` a modelului de interespractic evident [JCO1].

Astfel, folosind setul al doilea deecua]ii (8), modelul (18) devine,

\n care, JU este matricea de coeficien]idin al doilea set de ecua]ii (8), kT sîntrapoartele de transformare reglabile,iar indicii G [i L semnific`generatoare [i sarcini.

Gradientul se determin` dup` calcu-larea multiplicatorilor λQ, din primulsistem (20), [i înlocuirea acestora înal doilea sistem. Ecua]iile (20) sîntmai u[or de rezolvat decît (19), iarsingura aproxima]ie care poateinfluen]a rezultatul final este negli-jarea ∂f / ∂θ, o component` îngeneral nesemnificativ`. Vectorul∂f / ∂UL, cel cu adev`rat important,se calculeaz` f`r` aproxima]ie.


Exemplul nr. 4. Regimurideterminate prin extrapolare însimularea dinamicii pe termenlung a SEE.

Modelul LOTDYS.R

Dinamicile pe termen lung ale SEEsînt provocate de perturba]ii succesive,de regul` altele decît scurtcircuitele, [isînt simulate pe intervale de timp deordinul minutelor.

Modelul LOTDYS originar propus dec`tre EPRI [GenE] neglijeaz` osci-la]iile rapide dintre ma[inile sincrone,în schimb, reprezint` în detaliudinamica instala]iilor termo- [ihidro-mecanice din centraleleelectrice.

|n esen]`, modelul presupune o dubl`dezlegare: una func]ional` de procese,prezentat` mai jos, [i altavaria]ional` (temporal`), care estedescris` în Partea a III-a, Capitolul 2.

Frecven]a sistemului este variabil`,dar aceea[i în toate punctele SEE:

în care, Ji este constanta de iner]ie agrupului generator i (MW.sec / MVA),PMi [i PEi sînt puterile mecanic` [ielectric`, iar Pacc este puterea “deaccelerare” a sistemului.

Case No. 4. Extrapolation –based load flows inlong-term PS dynamicssimulation.

LOTDYS.R model

Long-term PS dynamics are pro-voked by succesive disturbances,seldom short-circuits, and are simu-lated on time intervals of the orderof minutes.

The original LOTDYS model that wasproposed by EPRI [GenE] neglects thefast oscillations among synchronousgenerators while representing indetail the dynamics of thermal- andhydro- mechanical equipment withinthe power plants.

Essentially, the model assumes adouble unbundling: a process functionalone that is treated hereafter, and avariational (temporal) unbundling,which is the scope of Part III, Chapter 2.

The system frequency is variable,however the same in every PS point:

(21)

in which, Ji is the inertial constant ofi generating unit (MW.sec / MVA), PMi

and PEi are the mechanical andelectric powers, and Pacc is the PS’s“acceleration” power.


The assumpion of a single fequencyimplies an unique speed change forall rotors of the synchronousmachines, i.e. the total powerunbalance is distributed on thegenerators in proportion to the inertiaconstants of their rotors,

(22)

Determination of PMi requires theintegration in the time domain of aset of ordinary differential equationsof the form,

(23)

In the equation system (23), x de-notes the vector of “inertial slow” va-riables characterizing the turbinedynamics at load variation, whilethe “fast” oscillations of z electricvariables in the models ofgenerators and network areneglected.

Dynamic response of a steam turbinedepends on the boiler, turbine andadjoining controls dynamic charac-teristics. Hydro turbines and theirload control systems are alsorepresented with relative simpleEPRI models.

Due to quasi-steady state conditionimposed to electric variables, theknowledge of PEi implies determi-nation of the system load flows, on the

Ipoteza frecven]ei unice implic` ace-ea[i varia]ie a vitezei rotoarelor ma[i-nilor sincrone, respectiv faptul c`dezechilibrul total de puteri este re-partizat pe generatoare propor]ionalcu constantele de iner]ie alerotoarelor,

Determinarea puterilor mecanice PMi

necesit` integrarea în domeniul tim-pului a unui sistem de ecua]ii diferen-]iale ordinare de forma,

|n sistemul de ecua]ii (23), x desem-neaz` vectorul variabilelor “iner]ialelente” caracterizînd dinamica turbi-nelor la varia]ia de sarcin`, în timp ceoscila]iile “rapide” ale m`rimilorelectrice z din modelele genera-toarelor electrice [i ale re]elei sîntneglijate.

Dinamica turbinei de abur depinde decaracteristicile dinamice ale caza-nului, turbinei [i sistemelor aferentede reglaj. Turbinele hidro [i sistemelelor de control se reprezint`, de aseme-nea, prin modele EPRI relativ simple.

Datorit` condi]iei de regim cvasi-sta]ionar impus` variabilelor electri-ce, cunoa[terea PEi necesit`determinarea regimului permanent alsistemului, pe baza metodei


Newton – Raphson, dar reformulat`(Anexa 6) astfel încît s` fie respectat`ecua]ia (22):

|ns`, în intervalele în care regimul semodific` nesemnificativ, PEi [iimplicit puterea de accelerare se potdetermina prin extrapolare, pe bazaunui regim anterior “zp”,

unde D este un coeficient deamortizare global, iar ∆ω estevaria]ia frecven]ei sistemului.

Pentru determinarea lui D [i avectorului de coeficien]i KG, ecua]ia(25) se compar` cu urm`toareaecua]ie, exact`:

i ∈ G (mul]imea generatoarelor)j ∈ L (mul]imea sarcinilor)

Ecua]ia (26) se dezvolt` exprimînd∆PL [i ∆PLS (varia]ia pierderilor totaleale re]elei) în func]ie de varia]iiletensiunilor nodale [i ale frecven]ei,dup` modelul,

base of Newton – Raphson method,yet re-written (Annex 6) so as theequation (22) is observed:

(24)

However, in those time intervals withinsignificant regime change, the PEi

and acceleration power implicitely areestimated by extrapolation, on thebase of a previous regime “zp”,

(25)

where D is an global dampingcoefficient and ∆ω is the system’sfrequency change.

In view of determining D and thecoefficient vector KG, equation (25) iscompared with the following exactequation:

(26)

i ∈ G (generator set)j ∈ L (load set)

Equation (26) is developped expres-sing ∆PL and ∆PLS (variation of thetotal network losses) in terms of fre-quency and nodal voltages variations,following the model,

(27)


where ∆U and ∆ω are determined onthe base of Newton – Raphson loadflow model.

The original computation procedure,which resorts to large – size inversematrices, can be significantly simpli-fied [JCO1] by applying a processfunctional unbundling technique tothe Newton – Raphson model:

(28)

(29)

Finally, after identification, thecoefficients in demand acquire thefollowing expressions:

(30)

In which, the coefficient vectors K3, K4

and K6 are determined by solving thefollowing linear equations (withsparse coefficient matrices) throughoptimal factorization:

(31)

unde ∆U [i ∆ω se determin` pe bazaecua]iilor din modelul Newton –Raphson al regimului permanent.

Procedeul de calcul original, carerecurge la matrice inverse de mari di-mensiuni, poate fi simplificat apre-ciabil [JCO1] aplicînd modeluluiNewton – Raphson tehnica dezleg`riifunc]ionale de procese:

|n final, dup` identificare, coefi-cien]ii c`uta]i cap`t` urm`toareleexpresii:

|n care, vectorii - coeficienti K3, K4 [iK6 se determina rezolvînd prinfactorizare optimal` urm`toareleecua]ii lineare (cu matrice lacunar`de coeficien]i):


4 Dezlegare func]ional` deprocese în sistemul “tehnologie– client” de energie electric`

4.1 Servicii [i pre]uri dezlegate deenergie electric`

Dezlegarea func]ional` a sistemului“tehnologie – pia]`” de energie, saudemonopolizarea, pe criteriul servi-ciului furnizat, este de dat` relativrecent`.

|ntr-un mediu de monopol în alimen-tarea cu energie electric`, consu-matorii cu tarife reglementatecump`r` un serviciu legat, la un pre]cu ridicata, ne-defalcat de c`tre socie-tatea furnizoare pe activit`]ile de ge-nerare, transport, distribu]ie [ifurnizare, cu ratele aferente derecuperare a capitalului.

Marfa “kilowatt-or`” este livrat`direct consumatorilor finali sau fur-nizorilor cu am`nuntul, la un pre]reglementat “adecvat”. Formula depre] de monopol,

presupune c` exist` un cost c “re-

(32)

(33)

4 Process functional unbundlingin “technology – customer”system of electricity

4.1 Unbundled electricity servicesand prices

Functional unbundling of theelectricity “technology - cutomer”system, or de-monopolization, on thecriterion of supplied service, is ofrelatively recent date.

In a monopolistic electrictricity en-vironment, the tariff customers pur-chase a bundled electric service atregulated bulk sales tariff price thatincludes generation, transmission,distribution and supply costs, andreturn on the capital employed byelectric company for the performanceof these functions.

The “kilowatt-hour” commodity isdirectly supplied to end users or toretail sellers at an “adequate” regu-lated price. The monopolistic priceformulae,

(34)

is goverened by the assumptions thatthe cost c is “reasonable” and the rate


of return on the capital r is “reasona-ble” too. Establishment of regulatedbulk price is quite difficult due to thebundled structure of both the servicesand service provider as well as thecompanies’ superior information andexpertise as against the regulator’s.

The issue is simplified and solutionbecome more efficient in a compe-titive environment of unbundledservices [SH]. Eligible customersthat choose to leave the regulatedtariff regime buy unbundled electricservices at market prices includingdistinct components for:

- wholesale power including de-mand and energy charges forbalancing services

- ancillary system services- use and operation of the

transmission system- use and operation of the distribu-

tion system- supply activities- operation of the market.

The price formulae of competitiveelectricity market is:

(35)

Expression (35) ignores “justified”costs for energy generation and sup-ply. In a market with open, transpa-rent and unbiased competition, thereis high incentive for electricitygenerators and traders to increasetheir efficiency and reduce costs.

Politically driven price caps onelectricity prices that are below costs

zonabil” [i c` rata r de recuperare acapitalului investit este, de asemenea,“rezonabil`”. Reglementarea pre]uluicu ridicata este dificil` datorit` struc-turii integrate, atît a serviciilor, cît [ia furnizorului, precum [i informa]iei[i expertizei superioare a companiilorfa]` de acelea ale reglementatorului.

|ntr-un mediu concuren]ial de serviciidezlegate [SH], problema sesimplific`, iar solu]ia devine maieficient`. Consumatorii eligibili, carealeg renun]area la tarifele reglemen-tate, cump`r` servicii dezlegate alealiment`rii cu energie electric`, lapre]uri de pia]` distincte, pentru:

- energia achizi]ionat` pe pia]aangro, inclusiv serviciile de echi-librare pentru capacitate [i energie

- serviciile tehnologice de sistem- folosirea [i operarea sistemului

electric de transport- folosirea [i operarea sistemului

electric de distribu]ie- activit`]ile de furnizare- operarea pie]ei de energie electric`.

Formula pre]ului pie]elor concuren-]iale este:

Expresia (35) ignor` costurile “justifi-cate” pentru produc`tori [i furnizori.Dac` în pia]` exist` concuren]`deschis`, transparent` [i impar]ial`,produc`torii [i furnizorii devin pu-ternic motiva]i s`-[i creasc` eficien]a[i s`-[i reduc` costurile.

Concuren]a nu trebuie perturbat` de


plafoane de pre] sub costuri, stabilitepe criterii politice, care suprim` sem-nalele de pre] necesare pentru rea-lizarea de noi capacit`]i sau folosireaeficient` a energiei de c`tre con-sumatori.

|ntr-un sistem dezlegat al energieielectrice, transportul [i distribu]ia deenergie ramîn reglementate. Acesteasînt servicii de monopol naturaldeoarece costurile fixe sînt prepon-derente [i economiile de scar` continu`s` fie prezente. Pentru asigurareaviabilit`]ii, costurile operatorilor, inclu-siv ratele de recuperare a capitaluluiinvesti]iilor noi [i de modernizare,trebuie s` fie acoperite din venituri.

|ns`, este avantajos ca dezlegareafunc]ional` a sistemului “tehnologie –pia]`” de energie s` fie aprofundat`chiar [i la nivelul func]iei detransport. Vom ar`ta în continuare c`transportul de energie este un serviciucu valoare ad`ugat` multipl`, careinclude compensarea pierderilor înre]ea, compensarea congestiilor detransport [i, de asemenea, serviciul deacces la pia]a de energie.

should not interfere with competitionand suppress price signals thatencourage construction of newcapacity and energy efficiencyupgrades by the consumers.

In an unbundled electricity systemthe power transmission and distri-bution still remain fully regulated.These are natural monopoly servicesbecause the fixed costs are largelypreponderant and economies of scaleare still present. The costs includingreturn on new investments or up-grades should match the revenueswith a view to ensuring theoperators’ sustainability.

However, it would be rewarding to godeeply with functional unbundling ofenergy “technology & market”system even at the level of transmis-sion function. It will be shown here-after that power transmission is amultiple added-value service that in-cludes the compensation of networklosses and of transmission congestionas well as the access to marketservice.


4.2 Nodal price and the costs of anunbundled transmission service

4.2.1 Transmission revenuerequirement

The TSO requires revenues forcovering the input costs regardinggrid losses, O&M, depreciation, andreturn on assets, as input resources.On the other hand, the input costs arecurrently partitioned into operatingcosts and capital (asset) costs. TheTSO faces additional input costswhen the transmission congestionsimpose limitations to market tradearrangements. These are the effect ofreplacement in the market of cheapgeneration by out-of-merit one, due toPS security reasons. Based onCongestion Management (CM)strategies, the TSO can choose eitherto make the market participants up aloss or to remove the grid congestionsby grid reinforcements. Consequently,the CM can generate either operatingcosts or asset costs. For PS operatio-nal purposes, the TSO purchasesancillary system services from thequalified generators, which includesecondary and tertiary frequencyregulation reserves, voltage regula-tion within the network and blackstart PS restoration [JCO11]. TheTSO also utilizes demand and energyresources available in the balancingmarket with a view to ensuring (a) PSintegrity, (b) security and quality ofenergy supply, (c) appropriate PSgenerating reserve, and (d)management of network constraints.

4.2 Pre]ul nodal [i costurile unuiserviciu de transport dezlegat

4.2.1 Venitul necesar pentrutransport

OTS are nevoie de venituri pentruacoperirea costurilor de intrareprivind pierderile de energie în re]ea,mentenan]a [i operarea, amortiz`rile[i ratele asupra activelor. Pe de alt`parte, costurile de intrare pot fi deoperare [i de active. OTS suport`costuri de intrare suplimentare atuncicînd impune limit`ri în tranzac]iile cuenergie datorit` congestiilor de trans-port. Acestea sînt efectul înlocuirii înpia]` a produc]iei mai ieftine cu unamai scump`, din motive de securitatea SEE. Prin strategii de Managemental Congestiilor (MCO), OTS poatealege, fie s` desp`gubeasc` partici-pan]ii la pia]a angro care sufer`pierderi, fie s` elimine congestiileprin înt`rirea re]elei. MCO genereaz`a[adar fie costuri de operare fiecosturi de active. Pentru operareasistemului, OTS cump`r` servicii desistem tehnologice de la produc`toricalifica]i, care includ rezervele dereglaj secundar [i ter]iar de frecven]`,reglajul tensiunii în re]ea [i serviciulde pornire pentru restaurarea SEN[JCO11]. De asemenea, OTS folose[teresursele de capacitate [i de energiedin pia]a de echilibrare în scopulasigur`rii (a) integrit`]ii SEE, (b)securit`]ii [i calit`]ii aliment`rii cuenergie, (c) rezervei normate de pro-duc]ie în centralele SEE [i (d) ma-nagementului restric]iilor de re]ea.


|n general, costul total de intrarepentru OTS, [i necesarul de venitcorespunz`tor (Revenue Requirement- RR), pot fi estimate [i reglementate“ex-ante”, pe perioade de reglemen-tare, în baza unor scenarii adecvatede func]ionare [i dezvoltare a re]elei.

OTS vinde clien]ilor s`i un pachet deservicii, respectiv:

- Servicii de operare a sistemului- Echilibrarea balan]elor de

energie- Compensarea Pierderilor [i a

Congestiilor în re]ea (CP&CC)- Serviciul de Acces la re]ea / Pia]`

(AP).

Serviciile CP, CC [i AP reprezint`împreun` serviciul pentru operarea [iexploatarea sistemului electric detransport.

Componentele CP [i CC sînt în gene-ral recunoscute de c`tre sistemele dereglementare. Dar serviciul AP nueste nici recunoscut [i nici definit, înciuda faptului c` tocmai acesta adeterminat apari]ia re]elei. Noi ampropus o component` tarifar` pentruoportunitate de pia]` [JCO5], comple-mentar` pl`]ilor pentru CP [i CC.

Tariful de transport este instru-mentul prin care se strînge venitulnecesar RR. RR trebuie s` asigurefunc]ionarea sigur` a re]elei [i dez-voltarea sustenabil` a acesteia, curespectarea urm`toarelor cerin]e:

1. RR s` fie alocat echitabil peclien]ii re]elei prin tariful detransport.

Generally, the total grid cost of inputsfor the TSO, and the correspondingRevenue Requirement (RR), can beestimated and regulated “ex-ante”, fora regulatory period, consideringappropriate scenarios of gridoperation and development.

The TSO is selling to its users amultiple service, namely:

- Ancillary system services- Power balancing service- Loss Compensation and

Congestion Compensation(LC&CC) services

- Access to grid / Market (AM)service.

The LC, CC and AM service com-ponents may altogether be called asservice for transmission system useand operation.

The LC&CC components are ack-nowledged by almost regulatory sys-tems. Neither recognition nordefinition was given to the AMservice, though it may be said that itwas the very reason of grid birth. Wehave proposed a market opportunityfee within transmission rate [JCO5],as a complement to the LC&CC fees.

The transmission rate is an instru-ment for gathering the required re-venue RR. The RR must ensure safeoperation of the grid and its sus-tainable development, provided thatfollowing requirements are met:

1. The RR is fairly allocated on thegrid customers through the gridrate.


2. The RR equals the regulatedgrid cost of inputs.

3. The transmission rate sends along - run locational signal togrid customers, as relevant aspossible.

4.2.2 Transmission marginal costs

The principle of fair cost allocationdoes require the TSO to charge thetransmission customers depending onthe impact of service on the total gridcost.

As for the LC&CC services, only theinfluences of power feedingin/taking off the grid on the totalcost are relevant. These node-discriminated influences areessentially the grid marginal costs.Economics of the market shows thatthe social optimum is reached whenthe prices of goods and services areset at marginal costs and thesystems are economically “adapted”so as to producing a certain amountof goods or services at a minimalcost. The marginal cost, or themarket price, is the requiredoptimal tariff.

Marginal cost represents the deriva-tive of the total cost z in respect toresource vector b,

(36)

ensuring optimal use of the availableresources be they power generatingand network capacities or consumerdemands.

2. RR s` egaleze costurile regle-mentate de intrare.

3. Tariful de transport s` trimit`clien]ilor re]elei semnale loca-]ionale pe termen lung, cît mairelevante posibil.

4.2.2Costurilemarginaledetransport

Principiul aloc`rii echitabile a cos-turilor impune ca OTS s`-[i taxezeclien]ii de transport pe m`surainfluen]ei serviciului asupra costuluitotal al re]elei.

|n ceea ce prive[te serviciile CP [iCC, relevante sînt numai influen]eleinjec]iilor de putere în nodurile deintrare / ie[ire asupra costului total.Aceste influen]e discriminate nodalsînt în esen]` reprezentate de c`trecosturile marginale ale re]elei. Teoriaeconomic` a pie]ei arat` c` optimulsocial se ob]ine atunci cînd bunurile[i serviciile sînt vîndute la costurilelor marginale, situa]ie în caresistemele se “adapteaz`” economicastfel încît s` produc` la cost minim oanumit` cantitate de bunuri [iservicii. Tariful optim dorit estecostul marginal sau pre]ul de pia]`.

Costul marginal reprezint` derivatacostului total z în raport cu vectorul –resurse b,

asigurînd folosirea optim` a resur-selor disponibile, fie acestea capacit`]ide produc]ie sau de transport, fiecerere de energie.


Costul marginal al serviciului de trans-port în nodul k exprim` cre[tereaincremenal` a costului re]elei la schim-barea cu o unitate a puterii active injec-tate în nod (generat` sau consumat`).Acesta este, de regul`, disponibil cainforma]ie derivat` în procedura deoptimizare a reparti]iei sarcinii în SEE:

unde,MCG – costul marginal al gener`riiZ – vectorul restric]iilor de re]eaµ – vectorul multiplicatorilor Kuhn [i

Tucker asocia]i cu restric]iile deinegalitate (congestiile detransport)

L – pierderile de putere în re]eaPk – injec]ia de putere în nodul k.

Vom ar`ta mai jos c` exist` [i oop]iune alternativ` de determinare acosturilor marginale de transport, încare, în locul simul`rii reparti]ieioptime a sarcinii, se folosesc pre]uriledin pie]ele centralizate.

A[adar, costul marginal de transportîn nodul k are dou` componente, carese raporteaz` la pierderile în re]ea [ila congestia de transport, dup` cumurmeaz`:

Un sistem electric de transport este“congestionat” dac` func]ioneaz` lalimita respect`rii uneia sau mai multorrestric]ii de SEE. Managementul con-

Marginal cost for the transmissionservice in a node k expresses theincrement of grid cost when changingthe node’s active power injection(generation or demand) with a unit. Itis typically available as a by-information in the optimal PSdispatch procedure:

(37)

where,MCG – generation marginal costZ – vector of network constraintsµ – vector of Kuhn and Tucker

multipliers associated to gridinequality constraints(transmission congestions)

L – power losses within the networkPk – power injection at the node k.

We will show that there is yet analternative option to determine themarginal costs of transmission, inwhich, instead of simulating the opti-mal PS dispatch, prices on the cen-tralized energy markets are used.

Hence, the transmission marginalcost in the node k has twocomponents addresing the networklosses and transmission congestion,as follows:

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A power transmission system issaid to be "congested” when itoperates at the limit of one or morePS constraint. Congestion


management consists in controllingthe system so that no powertransfer limits are exceeded (detailsin Part II, §4.4.2).

According to operational and marketrules, the TSO prevents congestionsby redispatching or commitment tostart-up of the least-cost out-of-meritgenerators, or by curtailing certainenergy transactions. Application ofCM procedures generally leads todegraded transactions and to ad-ditional costs for the TSO, which mayhave to pay compensation for theconsequences of congestions. Accor-ding to the principle of cost recovery,the TSO is entitled to applycongestion fees on transactions at thegrid nodes with non-zero MCco (wherecongestions are generated).

The transactions’ effect on the gridcost is measured by G > MCG, at thegeneration (source, entry) node, andby L > MCL at the load (sink, exit)node.

4.2.3 Transmission marginal costsbased on marketprices of electricity

Let us suppose that time unit is onehour, the trading interval of both thecentralized electricity market andmarket of balance energy, so the MWand MWh numbers are equal to eachother. One can easily notice that:

MCG is actually the Electricity MarketPrice (EMP) that is cleared on acentralized market

gestiilor const` în reglarea sistemuluiastfel încît s` nu existe limite detransfer de putere dep`[ite (detalii înPartea a II-a, §4.4.2).

Conform regulilor de opera]ionale [i depia]`, OTS previne congestiile prin re-dispecerizare sau pornirea din rezerv`a generatoarelor cu cost minim, dardin afara ordinii de merit, sau prinlimitarea unor tranzac]ii. |n general,aplicarea procedurilor de MCO degra-deaz` tranzac]iile cu energie [i inducecosturi suplimentare pentru OTS, carepl`te[te compensa]ii pentru consecin-]ele congestiilor. Potrivit principiuluirecuper`rii costurilor, OTS este în-drept`]it s` aplice taxe de congestieasupra tranzac]iilor din nodurile dere]ea cu termeni MCco diferi]i de zero(unde se genereaz` congestii).

Efectul tranzac]iilor asupra costurilorre]elei se m`soar` prin G > MCG, înnodurile de generare (surs`, de intra-re) [i prin L > MCL în nodurile desarcin` (recep]ie, ie[ire).

4.2.3 Costurile marginale detransport bazate pe pre]uride pia]` ale energiei

S` presupunem c` unitatea de timpeste ora, intervalul de tranzac]ionareatît pe pia]a centralizat` de energieelectric`, cît [i pe cea de echilibrare,astfel încît num`rul de MW este egalcu num`rul de MWh. Vom observa c`:

MCG este chiar pre]ul de închidere alpie]ei de energie (EMP - Electri-city Market Price)


∂L/∂Pkeste cota din injec]ia (unitar`) deputere Pk ce se pierde în elemen-tele re]elei, respectiv, coeficientulde pierderi (LCF – LossesCoefficient) pentru nodul k.

Deci, primul termen al ecua]iei (36)poate fi estimat ca,

∂Z/∂Pkeste cota din injec]ia (unitar`)de putere Pk ce trebuie înlo-cuit` pentru respectareatuturor restric]iilor detransfer de putere, respectiv,coeficientul de congestie alnodului de re]ea k.

|nlocuirea optim` de putere se facela pre]ul energiei de echilibrare(PBE - Price of Balance Energy) pe opia]` de echilibrare în care toateunit`]ile dispecerizabile trimit oferteobligatorii de reglare. Restric]iile sepot defini în raport (i) cu sumacircula]iilor de putere prin anumitecoridoare , sau (ii) chiar cu injec]iade putere în nodul k (aceast`capacitate de transfer nodal` sau de“punct de racord” se va detalia înPartea a II-a, §4.2.1).

|n ipoteza (i), al doilea termenal ecua]iei (37) se poate estimaastfel:

undeMBEdown este pre]ul energiei pe pia]a

∂L/∂Pk is the share of (unitary) Pk

node injection which is lostacross the network elements,that is, the Losses Coefficient(LCF) of the network node k.

Hence, the first term of equation (36)can be estimated as,

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∂Z/∂Pkis the share of (unitary)Pk node injection that needsreplacement in order to removeall the network’s powertransfer constraints, that is, acongestion coefficient of thenetwork node k.

The optimal power inter - change istypically carried out at the Price ofBalance Energy (PBE) on a marketof balance energy in which alldispatchable generating units sendbinding regulating offers. Constraintdefinition can address either (i)aggregate power flows across certainnetwork paths, or (ii) the very powerinjection at node k (this “driving –point” or “point of connection”transfer capability approach will bedetailed at Part II, §4.2.1).

Under the hypothesis (i), the secondterm of equation (37) can beestimated as,

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whereMBEdown is energy price on the


market of balance energy fordown-ward regulation

MBEup is energy price on the marketof balance energy for up-wardregulation.

PTDF as congestion coefficient is thePower Transfer Distribution Factor,i.e. fraction of the Pk node powerinjection as transaction from one areato another. The PTDF matrix relatingall node power injections to individualline flows is generally available fromthe system operator’s EMS / SCADAplatform.

Under the hypothesis (ii), the unitarycongestion coefficient allows for astraightforward estimation of the se-cond term in equation (37) as,

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To determine the MCk of every trans-mission network node, the TSO’splanning department should assessthe losses and congestion coefficients,and prices on centralized markets forboth the electricity and balance ener-gy, for every trading interval of everyday or of typical days within a year.These interval - based marginal costsshould be then weighted by corres-ponding amounts of energy to becomenodal rates for power transmission (Gfor feed in power and L for take outpower). For the sake of simplicity, theGs and Ls are typically averaged forgroups of similar network nodes orzones with no internal congestions.

de echilibrare pentrureglare în sc`dere

MBEup este pre]ul energiei pe pia]a deechilibrare pentru reglare încre[tere.

PTDF (Power Transfer DistributionFactor), coeficientul de congestie, estefactorul de reparti]ie a puterii de trans-fer, respectiv cota din injec]ia Pk catranzac]ie de la o zon` la alta. MatriceaPTDF, rela]ionînd toate injec]iile nodalecu circula]iile pe linii, este în generaldisponibil` de la platforma EMS /SCADA a operatorului de sistem.

|n ipoteza (ii), coeficientul unitar decongestie permite estimarea direct` acelui de-al doilea termen al ecua]iei(37) ca,

Pentru a determina MCk al fiec`ruinod al re]elei de transport, direc]ia deplanificare a OTS trebuie s` evaluezecoeficien]ii de pierderi [i de congestie,[i pre]urile pe pie]ele centralizate deenergie [i de echilibrare, pentru fiecareinrterval de tranzac]ionare al fiec`reizile sau al zilelor tipice dintr-un an.Aceste costuri marginale de intervaltrebuie apoi ponderate cu cantit`]ile deenergie respective, pentru a devenitarife nodale de transport (G pentruintroducere [i L pentru extragere deputere din re]ea). Pentru simplitate, G[i L sînt, de regul`, mediate pe grupede noduri similare sau zone de re]eaneafectate de congestii.


4.2.4 Costurile marginale detransport [i venitul necesar

Valorile G [i L reprezint` aloc`ri aleRR asupra clien]ilor re]elei.

Cu resursele oferite de G [i L, [i pebaza unei analize cost – beneficiu,OTS alege dac` pl`te[te diferen]elede pre] datorate re-dispeceriz`rii saulimit`rii tranzac]iilor, sau dac` inves-te[te în înt`rirea re]elei. Aceast` me-tod` de integrare a pre]uluicongestiilor în pre]ul energiei este, celmai probabil, calea optim` destabilire a semnalelor de pre]loca]ionale. Investitorii în noi proiectede generare sau de întreprindericonsumatoare de energie devinrepede con[tien]i de tarifele dife-ren]iate pe care le vor pl`ti în diferitelocuri de acces la re]ea.

|n general, tarifele de transport G [iL pot fi stabilite apriori pentruperioade de reglementare, pe bazaunor scenarii plauzibile defunc]ionare [i dezvoltare a re]elei.Din motive practice, ele se mediaz`pe intervale de timp [i grupe denoduri (zone).

|ns`, aplicînd costurile marginaleMClo [i MCco tuturor tranzac]iilor degenerare [i de consum, în toatenodurile re]elei [i pe fiecare intervalde tranzac]ionare, se ob]ine un venitILC mai mic decît cel reglementat RR

4.2.4 Transmission marginal costsand revenue requirement

Gs and Ls represent the allocations ofRR on grid customers.

With money from the Gs and Ls,and based on a cost - benefitanalysis, the TSO chooses whetherto pay price differences due to re-dispatch or curtailed transactions,or to invest in grid reinforcement.This method of integrationcongestion pricing with energypricing is most likely the best way toestablish effective locational prices.The investors in new generationprojects or large energy consumingplants would clearly be aware ofdifferentiated transmission ratesthey have to pay for access tonetwork in different locations.

In general, the G and L transmissionrates can be set up ex-ante forregulatory periods, based onplausible scenarios of networkoperation and development. Forpractical reasons, they are averagedfor time intervals and groups ofsimilar nodes (zones).

However, the MClo and MCco

marginal costs, when applied togeneration or load transactions atevery node for every trading/timeinterval, give rise to less ILC incomethan the regulated RR:


where P is active power [MW] in atrading interval.

Matching costs and revenues fortransmission service by sustainableallocations, on the basis of L and Gfees, is a key issue of electricitymarket design.

4.2.5 Matching costs ontransmission service witha part of grid users’benefits

(a) Benefits estimated from theexistence of each networkfacility

Paper [PA-FR-PG] presents a versionadopted in Argentina. A comple-mentary charge is allocated tonetwork users in proportion to thebenefit that each one receives fromthe existence of each network facility.The benefit for a consumer is thereduction in its total electricitycharges based on spot prices, ignoringelasticity of demand to prices. For agenerator, the benefit is the income atspot prices less generation costs.

The method requires simulating oneyear of operation for each corridor (orline) for which a charge is to be alloca-ted. The model is run with and with-

unde P este puterea activ` [MW] înintervalul de tranzac]ionare.

Echilibrarea costurilor [i veniturilorserviciului de transport, prin aloc`risustenabile pe baza tarifelor G [i L,este o problem` - cheie a proiect`riipie]ei de energie electric`.

4.2.5 Echilibrarea costurilorserviciului de transport cu oparte a beneficiuluiutilizatorilor re]elei

(a) Estimarea beneficiilordatorate fiec`rui echipamentde transport

Lucrarea [PA-FR-PG] prezint` solu]iaadoptat` în Argentina. Utilizatorilorre]elei li se impune un tarif adi]ional,propor]ional cu beneficiul pe care îlob]in din existen]a fiec`rui element alre]elei. Beneficiul unui consumatoreste dat de reducerea pre]ului integralal energiei fa]` de pre]ul spot, negli-jîndu-se elasticitatea cererii în raportcu pre]urile. La produc`tor, beneficiuleste egal cu venitul dat de pre]ul spotdin care se scade costul gener`rii.

Metoda necesit` simularea func]io-n`rii pe un an pentru fiecare coridor(sau linie) care contribuie la tarifuladi]ional. Simularea este efectuat`

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pe modelul caracteristic alsistemului, cu [i f`r` coridor, pentrufiecare zi a anului, calculîndu-seastfel beneficiile [i tarifele adi]ionalepentru fiecare consumator rezultatedin existen]a facilit`]ilor respectivede re]ea.“Factorii de Beneficiu” de alocare (BF)sînt actualiza]i periodic, de exemplula trei ani.

Lucrarea conchide c` metoda dealocare este eficient` sub aspecteconomic, atît pe termen scurt ,cît [ipe termen lung, deoarece tarifulcomplementar este în general maimic decît beneficiul total net pe care-l ob]in utilizatorii re]elei cîndre]eaua este planificat` optim.

Lucrarea [RU-AR] concluzioneaz`îns` c`, “în ciuda avantajelor, imple-mentarea metodei BF, prezentat` în[PA-FR-PG], înt=mpin` serioasedificult`]i:a) Definirea [i calcularea beneficiilor

produc`torilor sînt ambigue [inecesit` date care, de regul`, sîntindisponibile.

b) |ndep`rtarea [i reintroducereaunei linii pentru a calcula benefi-ciile asociate are sens numai dac`se studiaz` proiectul unei linii noi.”

Autorii sugereaz` c` “este prefe-rabil` o procedur` alternativ`metodei BF, care s` foloseasc` om`rime “proxim`” beneficiiloreconomice, pentru a se evitadificult`]ile de implementaremen]ionate”.

out the corridor for typical power sys-tem operation of every day for thewhole year in order to calculate thebenefit for each consumer gainedfrom each facility, and the correspon-ding complementary transmissioncharge.Allocation “Benefit Factors” (BF) areupdated periodically, for exampleevery three years.

It is concluded that the allocationmethod is economically efficientboth in the short and in the longrun, since the complementarycharge is generally smaller than thetotal net benefit for the networkusers of an optimally plannedfacility.

Paper [RU-AR] concludes howeverthat “despite its advantages, the BFmethod, presented in [PA-FR-PG],has serious implementationproblems:a) Definition and computation of

generation benefits may beambiguous and may require datathat are not easily available.

b) Removing and restoring a line tocompute the associate benefits onlymake sense when the project of anew line is being studied”.

The authors suggested “the conve-nience of developing alternate proce-dures to the BF method which wouldmake use of a “proxy” magnitude ofthe economic benefits, to avoid theabove mentioned implementationdifficulties”.


Paper [AS] recognizes that the “supe-riority of [the] BF method is charge al-location based on economic concepts”.The method would assign the fixedcosts of transmission based on the eco-nomic benefit that each network faci-lity provides to each agent. “If thebenefit of consumer k decreases orbecomes negative due to removing theline l, the presence of that line makes apositive benefit for the consumer in-deed, and he is fond of participating inthe charge of that line. On the contraryif the consumer’s benefit increases orget positive due to the absence of theline l, he minds having that line in thetransmission system, but the presenceof the line is necessary for intercon-nectivity of the system. This does notmean that this consumer is remune-rated for the presence of the line l”. Thepaper points out that simulating theoperation of the system with andwithout a facility for a whole year is a“serious implementation problem”.

An alternate Critical Capacity (CrC)method is proposed in which the fixedcost of transmission is paid by firmservice customers based on the bene-fit they derive from certain transmis-sion facilities. The CrC of a line isused in the planning pattern, whichwould cause “a considerable reduc-tion in computational burden” to cal-culate the benefit each agent derivesfrom the line instead of removing andinserting that in the system as in theBF method. The CrC method doesalso imply the simulation of powersystem operation, but only once, forthe planning pattern.

Lucrarea [AS] recunoa[te faptul c`“superioritatea metodei BF const` înalocarea tarifar` bazat` pe concepteeconomice”. Metoda atribuie costurilefixe ale transportului în func]ie decontribu]ia fiec`rui element de re]eala beneficiul utilizatorilor re]elei.“Dac` beneficiul consumatorului kdescre[te sau devine negativ odat`cu îndep`rtarea liniei l, prezen]aliniei creaz` într-adev`r un beneficiupozitiv, iar consumatorul dore[te s`participe la finan]area ei. Dimpo-triv`, dac` beneficiul consumatoruluicre[te sau devine pozitiv ca urmare aabsen]ei liniei l, el se îndoie[te denecesitatea liniei în sistemul detransport, de[i linia poate fi necesar`pentru interconectarea sistemului.Prezen]a liniei nu aduce îns` valoaread`ugat` consumatorului”. Se maiobserv` c` simularea func]ion`riisistemului pe un an, cu [i f`r`elementul de re]ea, este “o problem`serioas` de implementare”.

Se propune metoda Capacit`]iiCritice (CCr), în care costul fix altransportului este pl`tit numai dec`tre clien]ii fermi în func]ie de bene-ficiul ob]inut din anumite facilit`]i detransport. CCr a unei linii este folo-sit` în modelul de planificare, fapt ceaduce “ o reducere considerabil` aefortului” pentru calculul beneficiuluipe care îl are fiecare utilizator dinexisten]a liniei, f`r` a fi nevoie de în-dep`rtarea [i inserarea ei în sistem,ca în metoda BF. {i metoda CCr im-plic` simularea func]ion`rii sistemu-lui de transport, dar numai o singur`dat`, în modelul de planificare.


(b) Estimarea beneficiilor dinaccesul integral la re]ea

Lucr`rile noastre [JCO5, JCO-CP-CA, JCO6, JCO13] au propus recupe-rarea costurilor pentru servireaclien]ilor de transport prinperceperea Tarifelor de Oportunit`]ide Pia]` (TOP), propor]ionale cubeneficiile pe care utilizatorii le ob]indin accesul integral la re]ea. Dinpunctul de vedere al utilizatorului,care angajeaz` întreaga re]ea, bene-ficiile sale din serviciul re]elei nu sîntdate de elemente individuale. Vîn-z`torii [i cump`r`torii de energieprofit` de diferen]ele dintre pre]urileofertate [i pre]ul pie]ei (Figura 1.3).

Fig 1.3 Beneficiile pe care le ob]in clien]ii detransport din participarea la pia]a de energieelectric`

|n consecin]`, se pot stabili compo-nente justificate de “echilibrare” atarifului sub forma cotelor - p`r]i dinbeneficiile reale pe care le ob]in par-ticipan]ii la pia]`. |n fiecare interval

(b) Market benefits arising fromfull network access

Our papers [JCO5, JCO-CP-CA,JCO6, JCO13] proposed to recoverthe costs of serving transmissionusers by charging Market Opportu-nities Fees (MOF), as a proportion ofthe benefits grid users get from fullnetwork access. From a grid user'sviewpoint, being able to see the entirenetwork, not merely individual ele-ments of it ensures the benefit to theuser as a particular network service.Electricity sellers and buyers takean opportunity of the differencesbetween their offered prices and themarket price (Figure 1.3).

Fig 1.3 The benefits transmission custo-mers get from participation to electricitymarket

Consequently, appropriate “balan-cing” components of the rate may beestablished as a pro-rata with thesereal benefits provided to marketparticipants. In each trading interval,


the TSO would apportion the marketbenefits with a view to getting ratesconsistent with the revenue require-ment. Normally, the market benefitis significantly greater than theassociated fee paid for transmissionservice. The market benefits areevident in the centralizedmarketplaces, for both the spot andbilateral deals. They could also beestablished at standard values in themarkets of negotiated bilateralcontracts. Due to a smaller risk, tran-sactions would migrate to centralizedmarkets thus increasing market in-tegration and liquidity. No powersystem simulations are needed inthis approach, although demandelasticity could be taken into account.

The solution of the profit maximiza-tion problem in a perfect competitivemarket can explain the reasonbehind this procedure. ILC wouldequal RR only if all grid capacity wascontinuously used. This is a referenceoption for the most economic gridsolution, in which the congestioncomponents of marginal costs MCco

are equal to the incremental costs ofgrid reinforcements to remove thecongestions. It may be said that forthe ideal grid case the short-runmarginal costs equal the long-runmarginal costs; in other words, theoperationally-based cost of congestionbecomes asset-based instead.

However, the grid is generallydesigned and operated with surpluscapacity. This is not necessarily a de-sign inconsistency, generating stran-

de tranzac]ionare, OTS va încasa unprocentaj din beneficiile de pia]` pen-tru a asigura congruen]a tarifelor cunecesarul s`u de venit. Beneficiul depia]` este, în mod normal, semnifica-tiv mai mare decît plata coresponden-t` pentru serviciul de transport.Beneficiile de pia]` sînt evidente pepie]ele centralizate atît spot, cît [i detranzac]ii bilaterale. Ele pot fi, deasemenea, stabilite la valori stan-dard, [i pentru pie]ele de contractebilaterale negociate. Datorit` risculuimai mic, tranzac]iile vor migra sprepie]ele centralizate, cu efect încre[terea integr`rii [i lichidit`]iipie]ei. Aceast` abordare nu necesit`simul`ri ale func]ion`rii SEE [i poate]ine seama de elasticitatea cererii.

Procedeul se bazeaz` pe solu]ia pro-blemei maximiz`rii profitului pe opia]` concuren]ial` ideal`. ILC vaegala RR numai dac` este folosit`continuu capacitatea integral` are]elei. Este op]iunea de referin]`,pentru cea mai economic` solu]ie dere]ea, în care componentele decongestie ale costurilor marginaleMCco sînt egale cu costurileincrementale ale înt`ririlor de re]eapentru evitarea congestiilor. Se poatespune c` în cazul re]elei idealecosturile marginale de termen scurtsînt egale cu cele de termen lung; cualte cuvinte, costurile opera]ionalede congestie devin costuri de active.

Dar, re]eaua este în general proiectat`[i exploatat` cu excedent de capacita-te. Aceast` practic` nu este o inconsis-ten]` real` de proiectare, generatoare


de costuri “e[uate”, care trebuie socia-lizate clien]ilor re]elei. Dimpotriv`, eaeste o politic` normal` de dezvoltare are]elei, menit` s` complementeze ser-viciile CP [i CC cu acela de acces lapia]`, în vederea cre`rii de oportu-nit`]i de pia]` [JCO6]. Ca furnizor alserviciului de acces la pia]`, OTS vaavea dreptul s` pretind` un tarif deoportunitate bazat pe un cost de opor-tunitate (α), pentru fiecare tranzac]iecu energie electric`. Tarifele deoportunitate se adaug` tarifelor G [i Lîn nodurile respective, astfel încît ILC

s` fie ajustat la valoarea reglementat`RR. |n acest scop, se calculeaz` α ca ocot` – parte din beneficiul (b) altranzac]iei unui produc`tor (G trans)sau a unui consumator / furnizor (Ltrans), adic` diferen]a dintre pre]ulofertat [i pre]ul pie]ei de energie:

α × k > bG sau α × k > bL

unde,

Costul de oportunitate este a[adar unraport între efectul economic pierdutfa]` de solu]ia de re]ea ideal` [icî[tigul ob]inut cînd re]eaua real` areexcedent de capacitate.

ded costs that may be “socialized” togrid customers. On the contrary, it isa normal network development policy,which aims to complement the LCand CC services with that of access tothe market, with a view to creatingmarket opportunities [JCO6]. As asupplier of the service of access to themarket, the TSO ought to be able torequest an opportunity fee based onopportunity cost (α), for each electri-city transaction. The opportunity feesare added to Gs and Ls rates in thecorresponding nodes, so as to have theILC adjusted to the value of regulatedRR. To this goal, α is calculated as apro-rata with the benefit (b) from thegeneration transaction (G trans) orthe load transaction (L trans), that is,the difference between the offeredprice and the market price of energy:

α × k > bG or α × k > bL

where,

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The opportunity cost is basically aratio between the lost economic effectin the best (ideal) grid solution andthe gain obtained from the grid beingeffectively oversized.

efectul economic pierdut \n solu]ia de re]ea cea mai bun` (ideal`)c=[tigul \n alternativa adoptat` (beneficiul utilizatorilor de re]ea

α >

lost economic effect in the best (ideal) grid solutiongain in the adopted alternative (benefit of grid users)

α >


Although transaction-dependent, theex-post opportunity fees do not giverise to a regulatory issue since theyare rigorously adjusted to theregulated RR.

4.3 Impact of nodal transmissionrates on the electric marketmechanism

The balancing component of thetransmission rate in the MOF modelmay be derived from the net benefit ofmarket participant [JCO7, JCO13],considering the posted G and L com-ponents included in the electricityprice-offer (Figure 1.4).

Fig. 1.4 Nodal energy pricing and benefit ofmarket participant (seller)

De[i dependente de tranzac]ie, tari-fele de oportunitate ex-post nu dauna[tere unei probleme dereglementare deoarece ele asigur`riguros venitul RR reglementat.

4.3 Impactul tarifelor nodale detransport asupra mecanismuluipie]ei de energie electric`

Componenta de echilibrare a tarifuluide transport din modelul TOPtrebuie determinat` pe baza benefi-ciului net al participantului la pia]`[JCO7, JCO13], considerînd compo-nentele postate G [i L incluse înoferta de pre] (Figura 1.4).

Fig. 1.4 Pre]ul nodal al energiei [i beneficiulunui participant la pia]` (vînz`tor)


Clien]ii re]elei vor pl`ti componenteleG [i L direct OTS, în timp ce alocarea[i plata componentei de oportunitate(α) poate fi asigurat` de c`treplatforma de tranzac]ionare a pie]ei.Ca op]iune alternativ`, dar mai pu]inpredictibil`, mecanismul detranzac]ionare al pie]ei centralizatede energie va stabili toatecomponentele tarifului de transportpe baza pre]urilor de pia]` pentruenergie, dup` cum s-a ar`tat la §4.2.3.

Acest mecanism al pie]ei de energieelectric`, discriminat` nodal, desco-per` adev`ratele ordine de merit înofertele de vînzare [i cump`rare, pre-]uri de închidere a pie]ei [i volumetotale de tranzac]ii. Se înt`re[te astfelsemnalul loca]ional al tarifului deacces la re]ea [i, totodat`, se optimi-zeaz` costul agregat al achizi]iei deenergie, care include [i serviciul detransport. Pre]ul fictiv de închidere dinpia]a tradi]ional` este înlocuit cu unpre] real al pie]ei angro. Aceast`agregare de pre] contribuie la cre[tereaeficien]ei pie]ei [i la dezvoltarea uneiinfrastructuri ra]ionale de pia]`.

4.4 Tarife de transportarmonizate regional(mecanismul ITC)

|ntr-un cadru de reglementare multi-OTS, armonizat, componentele tarifareG [i L trebuie stabilite pe baza de regulicommune privind serviciul, configura]ia[i costurile re]elei. G, la punctul de“intrare” în re]ea, [i L la cel de “ie[ire”,se colecteaz` de c`tre OTS din ]araimportatoare [i respectiv, exportatoare.

Grid customers would pay Gs and Lsdirectly to the TSO, while allocationand payment of opportunity fees (α)would be ensured by the tradingmechanism of the electricity market.As an alternative option, howeverless predictable, the tradingmechanism of centralized energymarket would establish allcomponents of transmission ratebased on electricity market prices, asshown at §4.2.3.

This mechanism of the node-discri-minated electricity market discoversthe real merit order of sell and buyoffers, market clearing prices and totalvolumes of energy transactions. Itwould strengthen the locational signalof the rate for access to grid whileoptimizing the aggregated cost ofenergy acquisition that includes thetransmission service as well. Thefictitious clearing price of the tradi-tional marketplace is thus replaced bya real wholesale price of energy. Thisprice aggregation would help increasemarket efficiency and develop arational market infrastructure.

4.4 Transmission usage chargeregionally harmonized(ITC mechanism)

In a multi-TSO regulatory environ-ment that is harmonized, the Gs andLs are supposed to be establishedfollowing the same rules of grid service,grid configuration and cost definition.G in the network’s “entry” point, and Lin the “exit” one, can be collected by theTSOs in the exporting and importing


Suma acestora trebuie s` acopere între-gul cost al serviciului în interconexiu-nea multi – OTS. Un mecanism decompensare inter – OTS (Inter-TSOCompensation, ITC), redistribuiepl`]ile în func]ie de efectele tranzitelor.

Mecanismul ITC actual (ETSO) folo-se[te chei de împ`r]ire [i factorii dedistribu]ie a puterii transferate (Po-wer Transfer Distribution Factors,PTDF) ca factori de ponderare. Efec-tul a[a-zicîndelor congestiitransfrontier` (TF) se compenseaz`separat de mecanismul ITC.

|n pia]a regional` Nordpool, exist` unvenit din decontarea contractelor spotîn situa]iile cu congestie “cu o ziînainte”. Venitul net este distribuitc`tre OTS pe baza unei formuleconvenite.

Raportul preg`tit pentru ComisiaEuropean` de c`tre Frontier Econo-mics [i Consentec [FE&C],recomand` integrarea veniturilor dincongestii într-o schem` nou` ITC,pentru a putea fi rezolvate problemelede investi]ii ale interconexiunilor.

|n opinia noastr`, un singurmecanism ITC poate s` distribuietoate veniturile colectate, inclusivvenitul din congestii TF [JCO12].Schema ITC propus` ar acoperi toatecosturile determinate de tranzitele deputere fiind pe deplin congruent` cuproiectul de pia]` bazat pe accesulnodal la re]ea, deoarece numai OTSparticip` la pl`]ile care privescefectele tranzitelor transfrontier`.

country respectively. Their sum ismeant to recover all service cost of themulti-TSO interconnection. An Inter-TSO Compensation (ITC) mechanism,redistributes payments by countries inthe region based on effects of transits.

The current (ETSO) mechanismof ITC is based on sharing keysand Power Transfer DistributionFactors (PTDFs) as weightingfactors. The effect of so-calledCross – Border (CB) congestionsis compensated separately from theITC.

On the NordPool regional market,there is a net income in settlement ofspot contracts in situations with day –ahead congestion. The net income isdistributed to TSOs based on aformula agreed on.

The Report prepared for the EuropeanCommission by Frontier Economicsand Consentec [FE&C], recommendsthat congestion revenues andintegration with a newly designed ITCscheme could overcome investmentproblems of interconnectors.

In our view, a sole ITC mechanismcould distribute all collected revenues,including CB congestion income[JCO12]. The proposed ITC schemewould cover all incurred costs ofpower transits being fully consistentwith the market design that is basedon nodal acces to the network, sincethe TSOs only would participate tothe payments related to the effects ofcross – border power exchanges.


Actualele mecanisme “concuren]iale”explicite1 pentru alocarea costurilor[i rentei congestiei TF de tip “punct-la-punct”, administrate de c`treoperatori de pia]`, stimuleaz` OTSs` deplaseze congestia la grani]e,nefiind capabile optimizeze MCO înraport cu adev`ratele restric]iiopera]ionale ale SEE. Ele pot fiînlocuite de mecanismul dedesp`gubire bazat pe tarifele detransport, ca în context na]ional, re-nun]îndu-se la istorica tratarediferen]iat` a interconexiunilor TF.

Venitul din transportul TF se poatedetermina ca diferen]` întreîncas`rile totale cu [i f`r` schimburi.Acesta poate fi împ`r]it între OTSpropor]ional cu circula]iile de puteri

The current explicit1 “competitive”mechanisms for allocation of the“point -to-point” CB congestion costsand rents, that are administrated bythe market operators, stimulateTSOs to shift congestion to theborders, while failing to optimize theCM in respect to real PS operationalconstraints. They can be replaced bythe transmission rate-basedreimbursement mechanism, as inthe national context, giving up thehistorical differentiated treatment ofCB tie lines.

The revenue from CB transmissioncan be determined as a differente inbetween the total receipts, with andwithout exchanges. This would beshared by the TSOs as a pro-rata

1 In South East Europe, capacity limits ofcross-border interconnections are stillestablished in different and rather non-transparent ways, while capacity usagevaries from stated capacity. Pursuant tothe commitments under the “EnergyCommunity” Treaty, the regional TSOsdecided to implement explicit co-ordinated auctioning. A CoordinatingAuctioning Office (CAO) wouldadminister yearly/monthly/dailyauctions and compensation payments.After 5 years, the dry-runs are notfinalized, while the CAO is notoperational. Allocating assets owned bydifferent companies and revenue-sharing is a sensitive issue. The 15thAthens Forum on November 25, 2009„took note of the remaining challengesand encouraged the responsible bodiesto actively promote ... proper definitionof transmission capacities, andconvergence and consistency acrossborders”.

1 |n Europa de Sud-Est, limitele de capacitatepentru interconexiunile transfrontier` sîntstabilite înc` diferit [i mai degrab` ne-transparent, iar capacit`]ile folosite variaz`fa]` de cele declarate. Ca urmare ahot`rîrilor tratatului „ComunitateaEnergiei”, OTS din regiune au decis s`aplice metoda licita]iilor explicite,coordonate. Un Oficiu de coordonare alicita]iilor (Coordinating Auctioning Office,CAO) ar trebui s` administreze licita]iianuale / lunare / zilnice [i pl`]ile încompensa]ie aferente. Dup` 5 ani, exerci]iilede simulare nu sînt înc` finalizate, iar CAOnu este opera]ional. Alocarea de activeapar]inînd unor companii diferite [iîmp`r]irea venitului acestora se dovede[te oproblem` sensibil`. Al 15 - lea Forum de laAtena din noiembrie 2009 „ a luat not` deprovoc`rile r`mase [i a încurajat p`r]ileresponsabile s` promoveze activ...definireaadecvat` a capacit`]ii de transport ,convergen]a [i congruen]a în comer]ultransfrontier cu energie”


TF m`surate la grani]e, în valoareabsolut`, ponderate cu G [i L alenodurilor de grani]` ale re]elei, înesen]` pre]uri marginale (shadowprices). Este o metod` de plat` clar` aOTS ce înregistreaz` costuri caurmare a efectului cumulat altranzitelor, congruent` cu principiilede baz` ale tarif`rii transportului deenergie electric`. OTS ar fi stimulates` declare [i s` construiasc` capa-cit`]i de interconexiune mai mari.

with CB metered (absolute) flows,weighted by the Gs & Ls at CB gridnodes that are essentially marginal(shadow) prices. This is a clear wayto reimburse the TSOs thatregister costs due to cumulatedeffect of transits, which isconsistent with the basic principlesof electricity transmission. TheTSOs would be stimulated todeclare and build highinterconnection capacity.


[AS] Alireza Sedaghati - Cost of Transmission System Usage Based on an EconomicMeasure, (IEEE Transactions, Vol. PAS – 21, No. 2, May 2006).

[DO-TI] H.W. Dommel, W. Tinney - Optimal Power Flow Solutions (IEEE Transactions,Vol. PAS – 87, 1968).

[FE&C] Improving Incentives for Investment in Electricity Transmission Infrastructure (aReport prepared for the EC by Frontier Economics and Consentec, November2008).

[GenE] General Electric, Long-term Power System Dynamics (EPRI Research Project 90-7/1974, Electric utility engineering operation, Final Report).

[JW-HH] J. Ward, H. Hale - Digital Computer Solution of Power Flow Problems (Trans.AIEE 75, 1956).

[MH-FG] M. Huneault, F. D. Galiana - A Survey of the Optimal Power Flow Literature(IEEE Transactions, Vol. PAS – 6, No. 2, May 1991).

[PA-FR-PG] I.J. Pérez-Arriaga, F.J. Rubio and J.F. Puerta Gutiérrez et al. - Marginal Pricing ofTransmission Services: An Analysis of Cost Recovery (IEEE Transactions, Vol.PAS – 10, No. 1, Feb. 1995).

[RU-AR] F.J. Rubio, I.J. Pérez-Arriaga - Marginal Pricing of Transmission Services: AComparative Analysis of Network Cost Allocation Methods (IEEE Transactions,Vol. PAS – 15, No. 1, Feb. 2000).

[SH] S. Hunt - Making Competition Work in Electricity (John Wiley & Sons, Inc., 2002).

[ST-AL] B. Stott, O. Alsac - Fast Decoupled Load Flow (IEEE Transactions, Vol. PAS – 93,No. 3, 1974).

[JCO-CP-CA] J. Constantinescu, C. Popescu, C. Aldea - An Efficiency – Oriented TransmissionCharge Allocation Based on Transmission Service Consumers Benefits (CIGREPaper No 38 – 204, Session 1998).


[JCO-MH1] J. Constantinescu, M. Homos - Algoritm [i program pentru calculul regimurilorpermanente ale sistemelor electroenergetice prin metoda decupl`rii (A DecoupledMethod for Determination of Power Systems Load - flows) (Energetica 6, 1976).

[JCO-MH2] J. Constantinescu, M. Homos - Algoritm [i program pentru calculul regimurilorpermanente ale sistemelor electroenergetice cu pierderi minime de putere activ`(Power System Load - flows with Minimal Network Losses. Algorithm andComputer Program) (Energetica 2, 1975).

[JCO-MH3] J. Constantinescu, M. Homos - Metod` practic` de calcul pentru studiul stabilit`]iistatice a sistemelor electroenergetice complexe (A Practical Method for the Studyof Steady - state Stability of the Complex Power Systems) (Energetica 9, 1977).

[JCO1] J. Constantinescu - Contribu]ii privind aplicarea decupl`rii func]ionale [istructurale în modelele de calcul ale regimurilor de func]ionare ale sistemelorelectroenergetice complexe (Contributions to Functional and StructuralUnbundling in the Operational Models of Complex Power Systems– PhD thesis)(Tez` de doctorat, Institutul Politehnic Bucuresti, facultatea de Energetic`, 1980).

[JCO5] J. Constantinescu - Tarif pentru serviciul de transport cu dubl` determinare:costul transportului [i beneficiile ob]inute din participarea la pia]a energiei(Transmission Rate with a Double Determination: Service Cost and Customers'Benefits) (Energetica 1, 1998).

[JCO6] J. Constantinescu - Transmission Fees in a Multi – Service / Multi – TSOInterconnection (Paper 211, 2ND CIGRE/IEEE PES Symposium, New Orleans,2005).

[JCO7] J. Constantinescu - A Unitary Approach of Operational Security and MarketDesign in a Multi –TSO Interconnection (CIGRE Session 2008, Report C5 – 303).

[JCO8] J. Constantinescu - Romanian Electricity Sector Reform / Market Opening andChallenges (IEEE PES Winter Meeting, Toronto, Ca, 2003).

[JCO9] J. Constantinescu - Reforma sectorului energiei electrice în Romania (PowerIndustry Reform in Romania) (Energetica 6, 1998).

[JCO10] J. Constantinescu - Semnifica]ia serviciilor de sistem pe pia]a energiei electrice(Significance of Ancillary System Services on the Electric Market) (Energetica 4,2001).

[JCO11] J. Constantinescu - The Essential Role of Transmission Network in the Design ofEnergy Markets / Zonele de competitie [i de serviciu public in industria energieielectrice. Rolul esen]ial al re]elei electrice în organizarea pie]ei energiei(Proceedings of the International Workshop "On the way to Transparency ofenergy systems", pp. 146 - 154, Ljubljana, 1997; Energetica 7, 1997).

[JCO12] J. Constantinescu - What Kind of Regional Market Integration to AccommodateLarge Scale RES? (EURELECTRIC Annual Conference, Bucharest, 15 - 16 June2009).

[JCO13] Jean Constantinescu - An Electricity Market Mechanism Based On Benefit-Balanced Transmission Rates (Un mecanism pentru pia]a de energie electric`bazat pe tarife de transport echilibrate cu beneficiile) (VGB PowerTech 03/2010,pp. 93 – 95).


Partea II. Dezlegarea spa]ial` saustructural`


Dezlegarea Spa]ial` sau Structural`(DS) este aplicabil` modelelorsistemelor fizice ale c`ror structuri potfi descrise de un graf sau re]ea.Modelul sistemului se dezleag` încomponente asociate cu sub-grafuri(subre]ele) pentru a i se determinasolu]ia de ansamblu pe baza solu]iilorindividuale ale componentelor.

DS este de variabile sau de procesedup` cum, în procesul de solu]ionare,componentele sistemului r`mîndependente unele fa]` de altele, saudevin definitiv independente.

La limit`, pot fi individualizatecomponente asociate unui singur nodde re]ea.

2 Dezlegarea structural`(spa]ial`) de variabile

}inînd seama de modul în care serezolv` modelele lineare alefunc]ion`rii sistemului, deosebim dou`clase de metode de dezlegare spa]ial`de variabile [JCO1], [i anume:

a) Metode iterative, ce ob]in solu]iaîntr-un proces de corectare avariabilelor pas-cu-pas, pîn` laîndeplinirea unui criteriu deconvergen]`

b) Metode directe, în care solu]iaexact` se determin` dintr-odat`, f`r` itera]ii.

Part II. Spatial or structuralunbundling


Spatial or Structural Unbundling(SU) addresses the models ofphysical systems whose structurescan be described by a graph ornetwork. The system model isunbundled in components associatedto sub-graphs (subnetworks) in viewof finding its solution on the base ofcomponents’ individual solutions.

There would be variable SU or processSU as, in the solution process, thesystem components remain dependentagainst each other or they becomeirrevocably independent.

At the limit, components associatedto one network node only can beindividualized.

2 Structural (spatial) unbundlingof variables

We can distinguish two classesof methods for variable spatialunbundling considering the way inwhich the linear models of systemoperation are solved [JCO1], namely:

a) Iterative methods, that reachthe solution in a step-by-stepprocess of variable correction,up to meeting a convergencecriterion

b) Direct methods, in which theexact solution is reached all atonce, with no iterations.


2.1 Iterative methods

The techniques of the first cathegoryrepresent an extension of Seidel-Gaussmethod for solving linear or near linearset of equations. Coefficient sub-matricesthat are associated to subnetworkequations are treated like the simplematrix elements in the original method.Subnetworks can be the result of cuttingeither the network elements or thenetwork nodes. In the nodal1 linearequationsYn

kUnk = In

k associatedto subnetworks, the injections at theborder nodes include the currents acrossi-j tie lines that are removed as well(Figure 2.1):

Fig 2.1 Sub-networks alternatively solved (aSeidel – Gauss approach)

In [BC], the network is partitionnedinto tree – shaped subnetworks, whichallows efficient calculation.

2.1 Metode iterative

Procedeele din prima categorie re-prezint` o extensie a metodei Seidel –Gauss de solu]ionare a sistemelor deecua]ii lineare sau aproximativ linea-re. Sub-matricele de coeficien]i aso-ciate ecua]iilor subre]elelor sînttratate ca [i elementele matricealesimple din metoda original`.Subre]elele rezult` prin sec]ionarea,fie de elemente de re]ea, fie de noduride re]ea. |n ecua]iile nodale1 lineareasociate cu subre]elele, Yn

kUnk = In

k,injec]iile în nodurile de grani]` includ[i curen]ii prin interconexiunile i-j cese îndep`rteaz` (Figura 2.1):

Fig 2.1 Rezolvarea alternativ` a sub-re]elelor(abordare de tip Seidel – Gauss)

|n [BC], re]eaua este parti]ionat` însubre]ele de tip arbore, care permit uncalcul avantajos.

1 The nodal equations constitute the currentmathematical model for determination of thenetwork power flows because equations can eas-ily be written, there is a one - to - one correspon-dence in between the structure of coefficientmatrix and network configuration, and thisstructure is advantageous for calculations. Wewill show hereafter that still the “direct” struc-tural unbundling re-brings to actual interest theequations of independent loops, as a mathemati-cal model for interconnection of the subnetworks.

1 Ecua]iile nodale alc`tuiesc modelul matematicuzual pentru determinarea regimurilorre]elelor electrice deoarece acestea se scriu cuu[urin] ,̀ exist` o coresponden]` direct` întrestructura matricei de coeficien]i [i configura]iare]elei, iar aceast` structur` este avantajoas`pentru efectuarea calculelor. Vom ar`ta aici, c`totu[i, dezlegarea structural̀ “direct`” re-aduceîn aten]ie ecua]iile buclelor independente, camodel matematic al interconect`riisubre]elelor.


Dac` subre]elele se definesc prinsec]ionarea nodurilor [DL], în nodurilesec]ionate se introduc surse fictive decurent pentru p`strarea injec]iilornodale neschimbate (Figura 2.2).

Fig 2.2 Echilibrarea bilan]urilor de putere lanodurile de grani]` prin curen]i imprima]i fictivi

Subre]elele sînt solu]ionate pe rînd,iterativ, ajustîndu-se injec]iile lagrani]` pîn` la anularea valorii tuturorcuren]ilor fictivi. Este de la sine în]elesc` eficien]a acestor metode euristicedepinde în mare masur` de „violen]a”sec]ion`rilor.

2.2 Metode directe

Gabriel Kron a propus în anul 1950 oteorie a re]elelor lineare vaste, cunos-cut` sub numele de diakoptics [GK].

Termenul a fost propus de c`tre PhilipStaneley de la facultatea de filozofie a luiUnion College, Schenectady, New York,[i provine din combinarea cuvintelorgrece[ti „kopto” – a t`ia, [i „dia”, care afost interpretat drept „sistem”. Kron[i-a bazat modelul pe teoria circuitelorelectrice „ortogonale”, ce a alimentat oîntreag` literatur` de discu]ii [AB,HH1, HH2, MAL, RO, AS-HB etc.].

When subnetworks are the result ofnode cuts [DL], fictitious currentsources are connected to the bordercutted nodes in order to keep the nodalinjections unchanged (Figure 2.2).

Fig 2.2 Balancing power injections at bordernodes with fictitious driving currents

The subnetworks are solved one by one,iteratively, by adjusting the borderinjections up to getting zero value for allfictitious currents. It goes without sayingthat the efficiency of these euristicmethods is largely dependent on the„violence” of network cuts.

2.2 Direct methods

A theory of linear vast networks has beenproposed by Gabriel Kron in the year1950 and it is known asdiakoptics [GK].

The term was proposed by PhilipStaneley with the philosophy faculty ofUnion College, Schenectady, New York,and originates from a combination of thegreek words „kopto” – to tear, and „dia”,which was interpreted as „system”. Kronbased his model on „orthogonal”electrical circuit theory, which suppliedan extensive discussion literature [AB,HH1, HH2, MAL, RO, AS-HB, etc.].


|n procesul de solu]ionare, în generaldirect (ne-iterativ), re]eaua sedescompune în p`r]i convenabile,matricele impedan]elor nodale asociatesubre]elelor se inverseaz` individual [ise recombin` pentru a se ob]inematricea invers` pentru re]eauaîntreag` [GK]. Kron asociaz` laturiechivalente fiec`rui element diagonalal matricei de impedan]e [i mutualit`]iîntre aceste laturi, elementelor nedia-gonale. Laturile echivalente convergîntr-un nod comun, unul singur înprimul model, cîte unul pentru fiecaresub-re]ea în al doilea model. Odat`determinat` inversa matricei deimpedan]elor nodale, solu]ia se ob]inesimplu pentru orice condi]ie ini]ial`.Kron a extins aria de aplicare ametodei stabilind „circuite echivalente”pentru o clas` larg` de fenomene fizice:cîmpul electromagnetic, cîmpuri decurgere fluid` compresibile [iincompresibile, procese în reactoarelenucleare, spectrele de vibra]ie alemoleculelor, probleme de radia]ie, deelasticitate, de programare linear` [. a.Prin analogie cu re]elele folosite de ella reprezentarea ecua]iilor diferen]iale,Kron a delimitat dou` modele de baz`:Poisson [i „ecua]ie de difuzie”. Acesteaau fost redefinite de c`tre colaboratoruls`u apropiat H. Happ [HH2] ca: (a)model în care subre]elele sîntinterconectate radial (torn subdivisionsradially attached), [i (b) model în caresubre]elele rezult` complet separat(torn subdivisions not attached).

Noi am mai interpretat abordarea luiKron ca dezlegare structural`(spa]ial`) a unui model descris de

In the solution process, in generaldirect (non-iterrative), the network isdecomposed in convenient subnet-works, the nodal impedance matricesasociated to sub-networks are inversedindividually and recombined in view ofgetting the inverse matrix of the wholenetwork [GK]. Kron associatesequivalent network branches to eachdiagonal entry of the impedancematrix, and inter-branches mutualitiesto the non-diagonal elements. Theequivalent branches are linked to acommon node, a single one in the firstmodel, and one for every sub-networkin the second model. Once the inverseof nodal impedance matrix is deter-mined, the solution is reached easilyfor any initial condition. Kron extendedthe method’s area of applicationestablishing „equivalent circuits” for alarge class of physical phenomena:electromagnetic fields, compressible andun-compressible fields of fluid flow,processes in nuclear reactors, vibrationspectra of molecules, issues of radiation,elasticity, linear programming, etc.Based on his networks forrepresentation of differential equations,Kron identified two basic models:Poisson and „diffusion equation”. Thesewere rephrased by his close collaboratorH. Happ [HH2] as: (a) a model inwhich the subnetworks are radiallyinterconnected (torn subdivisionsradially attached), and (b) a model inwhich the subnetworks are fully sepa-rated (torn subdivisions not attached).

Yet, we interpreted the Kron’sapproach as structural (spatial)unbundling of a model described by a


ecua]ii lineare c`ruia i se poate asociaun graf [JCO1, JCO14].

Fie deci sistemul urm`tor de ecua]iilineare care trebuie rezolvat:

S este o matrice patrat`, simetric` [inesingular` de ordinul n, iar h0 unvector dinRn. Ecua]iile (1) se pot asociaunui graf [i sînt echivalente sistemuluide ecua]ii urm`tor (Anexa 3):

Pe baza ecua]iilor (2) se arat` (Anexa4) c` solu]ia x0 a ecua]iilor (1) se poatedetermina [i din urm`torul sistem deecua]ii:

avînd matricea de coeficien]ibloc-diagonal`. Sistemul (3) este unmodel dezlegat structural [i poate firezolvat mult mai u[or decît (1), cu unalgoritm de calcul în [ase pa[i:

set of linear equations that can beassociated to a graph [JCO1, JCO14].

Let therefore be the following set of linearequations that has to be solved:

(1)

S is a quadratic, symmetrical andnonsingular matrix of the order n, andh0 is a vector fromRn.Equations (1) canbe assoaciated to a graph and areequivalent to the following set (Annex 3):

(2)

Onthebasisof equations (2) it canbeshown(Annex4) that thesolutionx0ofequations (1) canbedetermined fromthefollowingsetof equationsaswell:

(3)

with a bloc – diagonal matrix ofcoefficients. The set (3) is a structuralunbundledmodel and can be solvedconsiderably easier than (1), following asix – step computation algorithm:

(4)


\n care,

iar matricea c este diagonal`.

Avantajele dezleg`rii structurale, i.e.,reducerea dimensiunii modelului [iposibilitatea prelucr`rii paralele, sîntconsecin]a structurii bloc-diagonale amatriceiM1. GrafulG0 asociat cumatriceaM1 este compus din subgrafuriconexe, interconectate radial printr-unnod de referin]`P0 (Figura 2.3).

Fig 2.3 GrafulG0 asociat cu matricea bloc-dia-gonal`M1

La pa[ii A1 [i A5, se rezolv` sisteme deecua]ii de dimensiune redus`, deoareceordinul submatricelor dinM1 este maimic decît al lui S din (1). |n plus,elementele vectorului Ttx01, de la pasulA2, se determin` simplu, ca diferen]eîntre elemente din x01, în timp ceelementele lui h0S, de la pasul A4, sîntponderile laturilor „de interconexiune”ale unui graf complementarGc.

in which:

(5)

and the matrix c is diagonal.

The convenience of structuralunbundling, i.e., the reduction of modelsize and possibility of parallel processing,derive from the bloc-diagonal shape ofM1.The associate graphG0 of the matrixM1

is made up of connex subgraphs, radiallyinterconnected through a reference nodeP0 only (Figure 2.3).

Fig 2.3 The graphG0 associated to bloc-diago-nal matrixM1

At the steps A1 and A5, reduced-sizesets of equations are solved, becausethe order ofM1 submatrices is lowerthan that of S from (1). Yet, the entriesof vector Ttx01 in the stepA2 can besimply determined as differences inbetween entries of x01, whereas theentries of h0S, in the stepA4, are theweights of „interconnection” edges of acomplementary graphGc.


Dificultatea determin`rii matriceideinterconectareNS a fost cauzane-atractivit`]ii dezleg`rii structuraledirecte, ca metod` de rezolvare asistemelor mari. Aceast` dificultate poatefi surmontat` [JCO14-15] observînd c`NS este rezultatul reducerii de tipulGauss (Anexa 1) a matricei ponderilorbuclelor independente ale grafuluiG0UGc, pîn` la ciclurile definite delaturile de interconexiune (desenatepunctat în figura 2.3).

Pe aceast` baz`, au fost dezvoltatedou` procedee de calcul eficiente,prezentate în Anexa 4.

2.2.1 Caz particular:metoda „compensa]iei”

Dac` mul]imeaΣ1

care un singur ele-

ment, cu di= -Sqm , iar b=0, atunci,- M1 se ob]ine direct din S prin înde-p`rtarea elementului nediagonalSqm

- h0 = α; α = 0-NS sereducelaunsingurelementnS.

|n acest caz, solu]ia sistemului (1) sepoate scrie direct, astfel:

Ecua]iile (6) stau la baza a[a-zicîndeimetode a compensa]iei, larg folosit` înanaliza contingen]elor.

The difficulty in calculating theinterconnectionmatrix NS was res-ponsible for non-attractivity of directstructural unbundling as a method forsolving large systems. This difficultycan be overpassed [JCO14-15] noticingthatNS is a result of Gaussion reduction(Annex 1) of theweightmatrix ofindependent loops for the graphG0UGc,up to the loops that are defined byinterconnection edges (the dotted linesin the Figure 2.3).

On this basis, two efficientcomputation procedures weredeveloped as presented in the Annex 4.

2.2.1 A particular case:the „compensation”method

When the setΣ1

chas an one element

only, with di= -Sqm , and b=0, then,- M1 can be obtained directly from Sby removing the non-diagonalentry Sqm

- h0= α; α = 0- NS has an only entry nS.

In this case, the solution of equationset (1) can be written directly as,

(6)

The equations (6) are the basis ofso-called compensationmethod, which islargely used in the contingency analysis.


Astfel:- Se pleac` de la o solu]ie x01 asistemuluiM1x01 = h0

- Se caut` solu]ia ecua]iilorS(x01 + x02) = h0

- în care S difer` deM1 numai prinelementul Sqm.

Variabilelew din ultimele ecua]ii (6) sedetermin` cu u[urin]` deoareceM1

este deja factorizat` iar T este unvector lacunar.

Observa]ie: scalarulnS se determin`simplu dac` sînt disponibile elementelenqq, nmm [inqm ale inversei matriceiM1,

Exemplul nr. 1. Solu]ia ecua]iilorlineare ale re]elei electrice

Ecua]ii de forma (2) pot reprezentamodelul cunoscut al re]elei electrice,

cu elementele de re]ea descrise înFigura 2.4.

Thus:- A starting solution x01 of the setM1x01 = h0 is known

- The solution of equation setsS(x01 + x02) = h0 is sought

- in which S differs fromM1 by theentry Sqm only.

The variabilelesw from the lastequations (6) can be easily determined,sinceM1 is already factorized and T isa sparse vector.

Note: the scalar nS can simply be foundif the entries nqq, nmm and nqm of theinversedM1 are available,

(7)

Case No. 1. Solution of the linearecuations of electric network

Equationsintheform(2)canrepresentthefamiliarmodeloftheelectricnetwork:

(8)

with the network elements asdescribed in the Figure 2.4.


Fig 2.4 Cele dou` reprezent`ri ale elementuluide re]ea electric`

E [i j sînt vectorii tensiunilor /curen]ilor imprima]i de latur` iar z [iy sînt matricele impedan]elor /admitan]elor de latur`. A [i B sîntmatricele de inciden]` „noduriindependente – laturi” [i respectiv,„bucle (cicluri) independente – laturi”descrise in Anexa 2. Re]eaua sepresupune f`r` mutualit`]i, prinurmare z [i y sînt matrice diagonale.

Solu]ia ecua]iilor (8) poate fi g`sit`rezolvînd urm`torul sistem de ecua]iiechivalent:

Este sistemul de ecua]ii nodale avîndtensiunile nodale ca variabile de baz`,omologul ecua]iilor (1).

Adoptînd urm`toarele ipoteze:

sistemul dezlegat structural în forma(3) este,

Fig 2.4 The two representations of an electricnetwork element

E and j are the vectors of branch drivingvoltages /currents while z and y are thematrices of branch impedances / admit-tances.A andB are the „independentnode – branch”, and „independent loop(circuit) – branch” incidence matricesrespectively, that are described in theAnnex 2. The network are supposedwith no mutualities, consequently z andy are diagonal matrices.

The solution of equations (8) can befound by solving an equivalent set ofequations, as follows:

(9)

This is the set of nodal equations withthe nodal voltages as basic variables, thehomologous of equations (1).

Under the following hypotheses:

(10)

the structural unbundled system inthe form (3) is,


Ecua]iile (11) se rezolv` prin folosireaalgoritmului (4), în [ase pa[i,profitîndu-se de structurabloc-diagonal` a matricei Y1:

|n pasul A1 avem tot ecua]ii nodale,dar scrise pentru subre]ele, iar în A3,ecua]ii de bucle independente. Atîtcuren]ii nodali In de la pasul A1, cît [icuren]ii de bucl` iS prin elementelecare interconecteaz` subre]elele, de lapasul A3, au valorile din re]eaua real`(ne-descompus`). Trebuie subliniatfaptul c` aceast` proprietate a mode-lului poate fi folosit` în controlul circu-la]iilor de putere prin interconexiuni.

Matricea ZS se calculeaz` prin redu-cerea de tipul Gauss a matriceiimpedan]elor de bucl`, descris` înAnexa 4 [i mai în detaliu în [JCO14].

Exemplul nr. 2. Regimulpermanent simetric al re]eleielectrice.

Modelul DIAD

Kron [i-a elaborat teoria pentrumodelul ZI =U, al impedantelornodale, modelul de baz` pentru deter-minarea regimului permanent la aceavreme. De[i benefic`, diakoptica nu aputut compensa neajunsurile matricei

(11)

Equations (11) can be solved with thesix – step algorithm (4), takingadvantage of the bloc-diagonal shapeof matrix Y1:

(12)

At the stepA1 there are still nodalequations, however written for thesubnetworks, while atA3, they areequations of independent loops. Thenodal currents In at the stepA1, as wellas the loop currents iS across thesubnetwork tie lines, at the stepA3,have their values in the real (integral)network. Stress is laid on this modelfeature that can be used in the control ofpower flows across the interconnections.

The matrixZS is obtained by theGaussean reduction of loop impedancematrix as described in the Annex 4, andin more detail in [JCO14].

Case No. 2. Symmetricalload flows within a powernetwork.

DIADmodel

Kron established his theory for the modelZI=Uof nodal impedances, the basicmodel for determination of load flows bythose times. In spite of its clear com-putation advantages, diakoptics failed tocompensate the drawbacks of non-sparce


Z, ne-lacunară [i dificil de calculat.Dup` ce metoda Newton – Raphson,inclusiv procedeul factoriz`rii optimale[WT-JW], a demonstrat o eficien]` netsuperioar`, în întreaga lume s-a impusmodelul admitan]elor nodale YU = I.

|n cercet̀ rile noastre privind modeleledecuplate structural am determinatregimul permanent la nivelul subre]elelorprin mai multe metode, inclusiv N-R.Ne-a interesat, în mod special, unprocedeu eficient de „interconectare” asolu]iilor par]iale (etapa A3), problem`critic` a dezleg`rii structurale [JCO14,JCO16]. |ntre timp, odezlegare func]io-nal̀ de variabilepropus` de Stott [i Alsacsub numele de metoda N-R „decuplat̀ ” s-a dovedit imbatabil̀ pentru aplica]ii lare]elele obi[nuite, cu rezisten]ele multmai mici decît reactan]ele.

Modeluluinostrudezlegat func]ional(Partea I,Exemplulnr.1) i sepoateaplica[i odezlegarestructural̀ .Rezult̀ ometod` [JCO1]eficient̀ subaspectulvolumuluidecalcule, [i interesant̀ înceeaceprive[te reglareacircula]iilordeputereprinelementelede leg t̀ur̀ a subre]elelor.

Fie deci cele dou` sisteme de ecua]iilineare (8) (Partea I-a), ale modeluluiN-R „decuplat”:

Fiecare set de ecua]ii lineare (13) sedezleag` structural în forma (3) [i serezolv` cu algoritmul în 6 pa[i (4). Depild`, aplicat primului set de ecua]ii

and difficult to calculate Z matrix. Whenthe Newton-Raphson method includingoptimal factorization technique [WT-JW1]demonstrated a much higher efficiency,the modelYU=Iof nodal admitancesimposed itself worldwide.

Inourresearchonstructuralunbundledmodels, the loadflowsof individualsubnetworksweredeterminedwithdifferentmethods, includingtheN-Rone.Wewere interestedparticularly inanefficientprocedure for „interconnection” thepartialsolutions (stepA3), thecritical issueofstructuralunbundling [JCO14,JCO16].Meanwhile,a functionalunbundlingofvariables, thatwasproposedbyStottandAlsacasthe„decoupled”N-Rmethod,provedtobeunbeatablewhenappliedtotheordinarynetworks,withresitancesmuchlessthanreactances.

Yet, a structural unbundling can beapplied to our functional unbundledmodel (Part I, Case No. 1). Theresulted method [JCO1] iscomputation – efficient and interestingas far as the control of power flowsover subnetwork tie lines is concerned.

Let therefore be the two sets of linearequations (8) (Part I), of the„decoupled” N-R model:

(13)

Each set of the linear equations (13) isstructurally unbundled in the form (3),and solved with the six – step algorithm(4). For instance, when applied to the first


set of equations (13), the algorithm (4)can be written as,

(14)

The matricesNS are determinedfollowing the previously describedprocedure. The computations areodered so as to have done a significantshare of optimal triangularization ofthe matrices J1

θ and J1U all at once with

determination of theNSθ andNS

U.Equations in the steps A1 and A5 aresolved only partially (Figure 2.5), sincefor the calculation of ∆θ’’ (∆u’’respectively), in the step A2, only thelast elements of ∆Θ1 (∆U1) are required.Noticeably, δp”(δq”) could beconveniently adjusted, if control of theactive (reactive) power flows over thesubnetwork tie lines was required.

(13), algoritmul (4) se poate scrieastfel:

MatriceleNS se determin` cu procedeuldescris anterior. Calculele se ordoneaz`astfel încît odat` cu determinarea luiNS

θ [iNSU s` se efectueze [i o parte

însemnat` din triangularizareamatricelor J1

θ [i J1U, cu metoda

factoriz`rii optimale. Ecua]iile de lapa[ii A1 [i A5 se solu]ioneaz` doarpar]ial (Figura 2.5), întrucît la calculul∆θ’’ (respectiv ∆u’’), de la pasul A2,intervin numai ultimele elemente din∆Θ1 (∆U1). De remarcat faptul c`δp”(δq”) se pot ajusta convenabil, dac`se impune reglajul circula]iilor deputere activ` (reactiv`) prin elementelece interconecteaz` subre]elele.


Fig 2.5 Procesele de factorizare în modelulN-R dezlegat func]ional [i structural

Exemplul nr. 3. Regimuritranzitorii electromecanicepe termen scurt ale SEE

Modelul RETRA

|n modelul regimurilor tranzitorii petermen scurt, perturba]iile sînt finite[i de intensitate relativ mare, deregul` scurtcircuite. Regimul SEE sepresupune sinusoidal simetric,frecven]a sistemului este constant`, ca[i parametrii circuitelor, iar procesuleste asociat cu varia]ia vitezeirotoarelor generatoarelor. Re]eaua [iconsumatorii se descriu prin ecua]iialgebrice, lineare sau nelineare,ma[inile sincrone [i sistemele lor dereglare se reprezint` printr-un sistem

Fig 2.5 Factorization processes in the func-tional & structural unbundled N-R model

Case No. 3. Short – termPS electromechanicaltransients

RETRAmodel

In the model of short-term PSelectromechanical transients,disturbances are of finite size and ofrelativelly high intensity, short-circuitsin general. The PS regime is supposedsymmetrical sinusoidal, the system fre-quency as well as the circuit parametersare constant, and the process isassociated to variation of the rotor speedof generators. The network and loadsare modeled by algebraic equations,linear and nonlinear, synchronuousgenerators and their adjacent control


systems are represented by a mixed setof equations, differential and algebraic,linear and nonlinear (Eqs. (23), Part I).

At every moment t on the time scale,the network and generator models are„interconnected” each other by means of„interface” equations that ensure thebalance of nodal power injections aswell as d-q (Park) - R-I (rectangular)conversion of variables.

Taking as a criterion the moment whenthe interface supplies to the networkmodel the values of machine variables,one can distinguish two methods (i)with a time lag , equal to the simulationstep, and (ii) with no time lag, as ourmodel is. In the RETRA model[JCO17], the equations can be solvedsimultaneously owing to the implicitintegration procedure of „trapezes”,which is simple and robust. The „alge-brized” differential equations ofsynchronous machines, includinggovernors and voltage regulators ones,acquire the following form (details inthe Annex 5):

(15)

The equation (15), actually a„steady-state” generator model at thecomputation step t, should beexpressed in theR – I reference framein order to aggregate it with thenetwork model. The following currentcomponents at the generators’terminals as expressed in theR – Icoordinates can be distinguished:

de ecua]ii mixt: diferen]iale [ialgebrice, lineare [i nelineare (ecua-]iile (23), Partea I-a).

La fiecare moment t pe scara timpului,modelele re]elei [i ale ma[inilor se„interconecteaz`” prin ecua]ii „deinterfa]`”, care asigur` atît bilan]ulinjec]iilor nodale, cît [i conversiavariabilelor din coordonatele d-q (Park)în sistemul R-I (rectangular).

}inînd seama de momentul la careinterfa]a furnizeaz` modelului re]eleivalorile variabilelor ma[inilor,distingem dou` metode (i) cu decalajtemporal, egal cu pasul de simulare [i(ii) f`r` decalaj temporal, a[a cumeste modelul nostru. |n modelulRETRA [JCO17], ecua]iile se potrezolva simultan datorit` folosiriiprocedeului implicit de integrare, al„trapezelor”, simplu [i robust.Ecua]iile diferen]iale „algebrizate”ale ma[inilor sincrone, inclusiv aleregulatoarelor de tura]ie [i alesistemelor de excita]ie, cap`t` formaurm`toare (detalii in Anexa 5),

Ecua]ia (15), în definitiv un model„sta]ionar” al generatorului la pasul t,trebuie exprimat` în sistemul dereferin]`R – I, pentru a se puteaagrega cu modelul re]elei.|n coordonateR – I, se pot distingeurm`toarele componente alecuren]ilor la bornelegeneratoarelor:


Curen]ii I10 exprim` efectul a[a-zicîndei „anizotropii” dup` axele d-q.Ace[tia, împreun` cu I0 se adaug`componentelor constante ale curen]ilorde sarcin` formînd împreun` termenulliber IMOD al ecua]iilor re]elei. Curen]iiI11 [i componentele lineare ale curen-]ilor de sarcin` se adaug` elementelordiagonale ale matricei admitan]elornodale ale re]elei YMOD.

Rezult` un sistem de ecua]ii lineareavînd structura sistemului (7) alecua]iilor nodale,

Ecua]iile (17), rescrise în forma (11)corespunz`toare modelului dezlegatstructural, se rezolv` cu algoritmul în[ase pa[i (12). Matricele YMOD1 [i ZS sefactorizeaz` numai în etapa deini]ializare (t=0), [i la intervalele detimp în care intervin modific`ri înstructura sistemului.

De asemenea, la etapele A1 [i A5,ecua]iile se rezolv` numai par]ial: unadin cele dou` substitu]ii de variabile,cea „înainte” pentru A1, [i cea „înapoi”pentru A5, se parcurge numai pentruvariabilele asociate cu nodurile deinterconexiune. Etapele A2, A4 [i A6 serealizeaz` prin logica de programare,iar ZS se determin` pe baza matricelortriangularizate ale subsistemelor, cuprocedeul prezentat în Anexa 4.

(16)

The currents I10 express the effect of theso-called „anisotropy’” along the axesd-q. These, together with I0, are added tothe constant components of the loadcurrents forming together the free termIMOD in network equations. The currentsI11 and the linear components of the loadcurrents are added to the diagonalelements of the network nodal admit-tance matrix of the network,YMOD.

A set of linear equations with thesame pattern as the nodal equations(7), results:

(17)

Equations (17), re-written as (11) thatcorrespond to the structural unbundledmodel, are solved with the six – stepalgorithm (12). The matrices YMOD1 andZS are factorized in the step ofinitialization (t=0) only, and at the timeintervals with changes in the system’sstructure.

At the steps A1 and A5, the equationsare solved only partially: one of thetwo variable substitutions, „forward”in A1 and „backward” in A5, is doneonly for the variables associated tointerconnection nodes. The steps A2,A4 and A6 are carried out throughprogamming logic, while ZS is deter-mined on the base of triangularizedmatrices of the subsystems, with theprocedure presented in the Annex 4.


Prelucr`rile se efectueaz` în suc-cesiunea descris` de diagrama logic`din Figura 2.6.

Fig 2.6 Schema logic` simplificat`a programului de calcul RETRA

Processing is carried out in a sequencethat is described by flowchart in Figure2.6.

Fig 2.6 Simplified flowchartof RETRA code


Din cîte [tim, RETRA este singurul codde produc]ie pentru simularearegimurilor tranzitorii electro-mecaniceale SEE bazat pe un model de sistemdezlegat structural.

Pentru reprezentarea obi[nuit` a SEEna]ional, cu 910 noduri de re]ea [i 95 degrupuri generatoare, rezultatelesimul`rii au fost riguros acelea[i, atîtpentru modelul dezlegat structural, cît[i pentru modelul nedezlegat. Dar,duratele de calcul (CPU) [i spa]iulnecesar de memorare în calculator aufost semnificativ mai mici în abordareadezlegat`, chiar [i în absen]a prelucr`riiparalele.

3 Dezlegarea structural` (spa]ial`)de procese

3.1 Analiza Nodal` - Dimo

Modelele REI-Dimo reduc la o dimen-siune esen]ial` zonele de interes aleSEE dezlegîndu-le de restul sistemuluiprin frontiere cu tensiuni controlate[PD, JCO18]. Tensiunile aplicate lanodurile de frontier` re]inute auacelea[i valori ca în re]eaua real`.Restructurarea sistemului se efec-tueaz` prin folosirea re]elelor de„Bilan] Energetic Nul” (BEN), cu carese formeaz` noduri REI fictive degenerare, de sarcin` sau mixte (a sevedea Partea a III-a, Exemplul nr.1).

Cel mai simplu model REI-Dimo, sauechivalent REI de sistem, estereprezentat de schema REI pentru unnod. Subre]eaua radial` din Figura 2.7devine independent` fa]` de starea

To our knowledge, RETRA is the onlyproduction code for the simulation of PSelectro-mechanical dynamics based ona system model that is structurallyunbundled.

For the ordinary representation of thenational PS, with 910 network nodesand 95 generating units, thesimulation results were rigorously thesame, for both the structurallyunbundled and bundled system.However, the CPU times and thenecessary computer storage weresignificantly less in the unbundledapproach, even in the absence ofparallel processing.

3 Process structural (spatial)unbundling

3.1 Dimo’s Nodal Analysis

The REI – Dimo models reduce the PSareas of interest to an essential sizeunbundling them from the rest of thesystem along with voltage - controlledborders [PD, JCO18]. The voltagesapplied to the retained border nodeshave the same values in the realnetwork. System restructuring iscarried by the use of networks with„Zero Power Balance” (ZPB) that helpforming fictitious REI nodes ofgeneration, load or mixed generation &load (see Part III, Case No.1).

The most simple REI-Dimo model, orREI equivalent of the system, isrepresented by the REI net for onenode only. The radial sub-network inFigure 2.7 is independent as against


restului re]elei, datorit` tensiunilorreglate, „aplicate” la nodurile de eifrontier`.

Fig 2.7 Modelul REI – Dimo

Schema REI [i „imaginea nodal`”aferent` se construiesc „liniarizînd”curen]i de scurtcircuit, în sarcin` [i îngol, în orice nod de sarcin`, real saufictiv. Cu ajutorul celor dou` „conceptefigurative standardizate”, în nod se„vizualizeaz`” starea sistemului [ievolu]iile posibile ale acesteia, unelecaracteristici de structur` ale SEE [i,în general, rela]iile dintre cauze [iefecte în func]ionarea normal` sauperturbat` a unui SEE complex.Comparînd unghiurile [i m`rimeavectorilor curen]ilor de scurtcircuitcare se compun în imaginea nodal`, ungraf standardizat, se pot cunoa[te prin„ra]ionament direct” starea critica aSEE [i perturba]iile care înr`uta]escpericulos starea sistemului, efectelevaria]iilor de frecven]` [i de tensiuneasupra stabilit`]ii sistemului, [i calita-

the state of the rest of network due tothe controlled voltages that are„applied” to its border nodes.

Fig 2.7 REI-Dimo model

The REI net and its “nodal image” arebuilt by “linearizing” the short-circuitcurrents, at load and at no-load, in everyreal or fictitious load node.Based on the two “standardized andfigurative concepts”, at the node can be“vizualized” the system’s state and itspotential evolution, some structuralfeatures of the system and, in general,the relationships in between causes andeffects in normal and disturbed operationof a complex PS. By comparing anglesand magnitudes of the vectors ofshort-circuit currents that composethemself as the nodal image, astandardized graph, one can findthrough “direct reasoning” the PS criticalstate as well as disturbances that canworsen dangerously the system state,the effects of frequency and voltagevariations on the system stability, and


tea reglajului sistemului în ansamblu.Paul Dimo a demonstrat c` st`rile [icerin]ele de siguran]` [i reglaj alesistemului se pot determina [i analizadintr-un singur nod al re]elei.

4 Dezlegarea spa]ial` de procesa pie]ei [i sistemuluide energie electric`

4.1 Dezlegarea în spa]iul nodurilorre]elei

Paradigma nodal` introdus` de PaulDimo poate fi extins` [i sistemului„tehnologie – client” al energieielectrice. Energia [i serviciul detransport se pot reglementa [itranzac]ionata cu referire la un nod alre]elei. Tranzac]iile cu energie electricàu ca efect injec]ii de putere în nodurilere]elei, adic` transferuri nodale deputere între re]ea [i clien]ii acesteia.

Tot rezultatul unei tranzac]ii nodaleeste [i serviciul de transport, în esen]ùn serviciu de acces la re]ea. |n ParteaI, §4.4.2, s-a ar`tat c` func]ia detransport, pre]ul [i costultransportului, pot fi definite completla nivelul nodului de re]ea. Opertorulde Transport [i de Sistem (OTS) oferàccesul puterii la sistemul de trans-port, spre nodul re]ea sau dinsprenodul de re]ea, asigurîndu-se c` para-metrii sistemului respect` normele desecuritate. Pentru aceasta, OTSîntreprinde m`suri consumatoare deresurse, [i anume:

- Compensarea pierderilor inerentede energie în re]ea [i a efectelorcongestiilor de transport

the quality of the entire PS control. PaulDimo demonstrated that the PS statesas well as the system security andcontrol requirements can be determinedand analysed at one network node only.

4 Process spatialunbundling in electricmarket and system

4.1 Unbundling in the space ofnetwork nodes

The nodal paradigm that was introducedby Paul Dimo can be extended to the„technology - customer” system ofelectricity, as well. Electricity andtransmission service can be regulatedand traded with reference to onenetwork node. Electricity transactionslead to power injections in the networknodes, that is, nodal power transfers inbetween the network and its customers.

Result of a nodal transaction is thetransmission service as well, essentiallyan access to network service. In Part I,§4.4.2, there was shown thattransmission function, transmissionrate and cost, can be fully defined at thelevel of one network node.Transmission System Operator (TSO)offers the access of power to transmis-sion system, towards the network nodeor from the network node, ensuringhimself that the system parametersmeet the security standards. To thisaim, the TSO undergoes resource –consumming actions, as follows:

- Compensating the inherent powerlosses of the network and theeffects of transmission congestions


- Realizarea în avans de excedentede capabilitate a re]elei în vedereacre`rii de oportunit`]i de pia]` înbeneficiul clien]ilor de transport.

OTS acoper` aceste cheltuieli printarifele nodaleG [iL. |ntr-un sistem demari dimensiuni, admi]înd c` esteoperat de mai multe OTS, este dificil deadministrat tarife pentru fiecare nodreal. Din motive practice, se grupeaz` pezone tarifare nodurile avînd costurimarginale care nu difer` semnificativîntre ele [i la care, de regul̀ , nu seînregistreaz` congestii. Tarifele “zonale”de transport se pot calcula în prim`aproxima]ie ca medii ponderate ale tari-felor nodale. Tarife zonale de transportmai precise se pot determina îns` pemodelele echivalente REI Dimo, princi-pial riguroase în jurul st`rii de referin]`[JCO22]. Generatoarele pot fi reprezen-tate fie distinct fie agregate dup` diversecriterii, inclusiv dup` afilierea la o anu-mit` companie de energie. Consumatoriide energie pot fi grupa]i similar, în timpce liniile de interconexiune de interes potfi p`strate distinct în model.

Mai mult, putem spune c` îns`[i pia]ade energie electric` se poate dezlegaspa]ial deoarece pentru fiecare dinpie]ele elementare nodale se poate defini[i un standard de siguran]`opera]ional .̀ Vom ar`ta în cele ceurmeaz` c` acesta este capabilitatea detransfer a energiei în nod, ca poart` deacces în sistemul de transport. Cerin]eleunei pie]e eficiente [i cele de siguran]`în func]ionare ale SEE se pot astfelcorela direct unele cu altele.Subestimarea capacit`]ii de transport

- Oversizing in advance thenetwork capability in view of cre-ating market opportunities in thebenefit of transmission customers.

The TSO covers these costs with therevenue from the G and L nodal rates. In alarge – scale PS, admitting that it is multi– TSO operated, it is difficult toadministrate nodal rates for every realnode. For practical reasons, the nodes withmarginal costs that are insignificantlydiffer from each other, and generally donot encounter congestions, are grouped ontariff zones. At a rough guess, the „zonal”transmission rates are the weightedaverages of corresponding nodal rates.More accurate zonal rates can be howeverdetermined on the equivalent REI Dimomodels that are principially rigorousaround the system’s reference state[JCO22]. The power generators can berepresented either distinctly or aggregatedon different criteria, including theaffiliation to a certain company. Powerconsumers can be similarly grouped whilethe interconnectors of particular interestcan be distinctly modeled.

Yet, we can say that electricity marketitself can be spatially unbundledbecause an operational securitystandard can be defined for every nodalelementary market as well. We willshow hereafter that this is the powertransfer capability of the node as a gateof access to transmission system. Therequirements of an efficient powermarket and those of PS securityoperation can be therefore each otherdirectly correlated. Underestimation oftransmission capability reduces


reduce atît eficien]a pie]ei, cît [i gradulde folosire a re]elei. Supraestimareacapacit`]ii de transport duce fie larestric]ii opera]ionale, adic` la ineficien]̀de pia]̀ , fie, în cazuri limit ,̀ la colapsulSEE. Influen]a tranzac]iilor în pia]a deenergie electric` asupra siguran]eisistemului este problema central` a ali-ment`rii cu energie generînd numeroasecontroverse, în special dup` colapsurilede sistem din anul 2003 [JCO21].

Comisia European` caut` rezolvareaacesteia în cre[terea transparen]eipie]elor, tratarea egal` a tranzac]iilor[i înt`rirea re]elelor [EurC].

4.2 Capacitatea de transport [imanagementul congestiilor înproiectul nodal de pia]` deenergie electric`

4.2.1Capacitateade transport încomer]ul cuenergie electric`

Pentru comer], capacitatea detransport (Transmission Capacity, TC)este volumul - limit` de tranzac]ii cuenergie care asigur` protejarea atât aintegrit`]ii func]ion`rii SEE, cît [i afiec`rui element al re]elei [NERC].

|n interpretarea noastr` [JCO6, JCO20,JCO23], TC este în esen]̀ capacitatenodal̀ de transfer de energie electric` sau„de racord” („Driving - Point” powerTransfer Capability, DP-TC).

|n Partea a III-a, Exemplul nr. 2, estedescris` o metod` de calcul robust`pentru determinarea capacit`]ii DP,aplicat` îndelung în practic`.

efficiency of both the power market andnetwork use. Overestimation of trans-mission capability leads to eitheroperational constraints, that is, marketinefficiency, or, in certain cases, to PScollapse. The impact of electricitymarket transactions on the PS securityrepresents the central issue of the elec-tricity supply while generating nume-rous controversies, particularly afterthe blackouts of 2003 [JCO21].

The European Commission is seeking thesolution in the increase of market trans-parency, equal treatment of transactionsand enhancement of networks [EurC].

4.2 Transmission capacity andcongestionmanagement innodal electricitymarket design

4.2.1 Transmission Capacity (TC)for electricity trade

Transmission Capacity (TC) forelectricity trade is a capping thresholdfor the volume of energy transactions,meant to protect the integrity of PSoperation and of individual networkelements [NERC].

In our interpretation [JCO6, JCO20,JCO23], TC is essentially a„Driving-Point” (DP) or„Point-of-Connection” power transfercapability of the network.

Part III, Case No. 2, describes a robustcomputation method for determinationof the DP capacity with extensivepractical usage.


Clien]ii / tranzac]iile de transportrezerv` p`r]i din TC, pentru intro-ducere sau extragere de energie în / dinre]ea, în ordinea de merit atranzac]iilor de energie. Altfel spus,energia se va vinde odat` cu eliberareade capacitate de transport.Corespunz`tor, congestia (comercial`)este o limitare a tranzac]iei detransport la punctul de acces însistemul de transport ce reflect`diferite restric]ii ale SEE în st`rineafectate de contingen]e, impuse destandardele opera]ionale (Figura 2.8).Congestia apare dup` încheiereatranzac]iei, iar compensarea efectuluicongestiei în pia]` este în respon-sabilitatea OTS, care devine astfelinteresat s` aplice proceduri de costminim pentru ManagementulCongestiilor (MCO).

Fig 2.8 Modelul nodal (DP, de punct deconectare) de acces la re]eaua de transport

Transmission users / transactionswould rent shares of TC either to feedin or to take out network power, in themerit order of energy deals. In otherwords, electricity will be sold all at oncewith release of transmission capacity.Accordingly, DP (commercial)congestion is a limitation ontransmission transactions at the accesspoint to transmission system thatreflects a variety of PS constraints inthe pre-contingency states forcompliance to operational standards(Figure 2.8). Congestion occurs afterthe conclusion of a transaction, whilecompensation for the effect ofcongestion in the market is theresponsibility of the TSO, who istherefore interested to apply least-costCongestion Management (CM)procedures.

Fig 2.8 Nodal (DP, Point of Connection) modelof access to transmission network


Aceste proceduri mobilizeaz` resursedin Pie]ele de Servicii de Sistem (PSS)[i de Echilibrare (PE), inclusiv din PEregional`. OTS cump`r` pe acestepie]e energia de echilibrare necesar` [ioptimizeaz` modific`rile de injec]iinodale pentru îndep`rtarea restric]ieiopera]ionale, eviden]iat` de analiza decontingen]e.

Dar, actualele reglement`ri UE, înparticular cele pentru comer]ultransfrontier [ReurCB, ETSO], definescTC ca o capabilitate de transfer deputere „Punct-la-Punct” (PP), de la ozon` de re]ea surs` la o zon` receptor.In modelul PP de serviciu de transport,congestia nu este eviden]iat` la locultranzac]iilor de energie ([i de transport),ci la anumite interconexiuni inter-zonale(critice, transfrontiere). |n aceast`abordare, interconexiuneatransfrontier` sau critic` este elementulde re]ea restric]ionat. Efectul congestieiasupra pre]urilor este determinat prinlicitarea capacit`]ii interconexiuniiînainte de încheierea tranzac]iei detransport, [i este transferat automatconsumatorilor de transport. Conceptulde capacitate bazat pe interconexiuneacritic` s-a dovedit problematic curînddup` implementare. |n mecanismulpie]ei a crescut rapid num`rul deinterconexiuni congestionate, în timp ceOTS s-a ar`tat neinteresat în eliminareacauzei congestiilor.

Modelele DP [i PP de capacitate detransport se pozi]ioneaz` diferit fa]` decerin]ele – cheie ale accesului la re]ea [iproiect`rii pie]ei (figura 2.9), [i anume:

These procedures are mobilizingresources from ancillary SystemServices Market (SSM) and BalancingMarket (BM) including the regionalone (RBM). The TSO buys on thesemarkets the necessary balance energyand optimizes injection changes thatobviate operational constraint asreflected by contingencies analysis.

However, for cross-border trade inparticular, the current Europeanregulation [REurCB, ETSO] definesthe TC as “Point-to-Point” (PP) powertransfer capability from a sourcenetwork area to a sink network area.In the PP model of transmissionservice, congestion is not renderedevident at the point where energy (andtransmission) transactions take place,but at certain inter-areainterconnections. In this approach, thecross-border or critical powerinterconnection is the constrainednetwork element. The congestion priceeffect is determined by auctioning anestimated capacity prior to conclusionof transmission deals, and is passedthrough to the transmission customer.The path – based capacity conceptencountered problems soon afterimplementation. In the marketmechanism, the number of paths onwhich congestion appeared grewrapidly while the TSO appeared notinterested to remove the cause ofcongestions.

DP and PP capacity models treat thekey requirements of grid access andmarket design differently (Figure 2.9),namely:


a) Inteligibilitatea informa]ieide capacitate

TC publicat` trebuie s` fie în]eleas` dec`tre participan]ii la pia]`. Vizibili-tatea ridicat` a interac]iunii pia]` –sistem fizic poate remodela [i sprijinioperatorii în rezolvarea problemelortehnice [i comerciale. În timp ceinforma]ia DP –TC este u[or de în]eles,un participant la pia]` fiind capabils`-[i optimizeze portofoliul de pia]`,limita PP – TC nu este inteligibil` unuiutilizator obi[nuit al re]elei. Limitelecircula]iei de putere prin elementele„critice” de re]ea sunt de în]eles doarpentru OTS. O capacitate inteligibil`,stabilit` transparent, ajut` deasemenea participan]ii la pia]` s`-[icontroleze mai eficient riscurile.

b) Predictibilitatea capacit`]iire]elei

Pentru un orizont de timp de un an,capacitatea de transport trebuie s` fiepredictibil̀ [i, aproape f̀ r` excep]ie,acest deziderat este realizat numai delimitele asupra injec]iilor de putere înnodurile de re]ea. |n timp ce capacitatea„de racord” este relativ stabil̀ [i se poatecalcula cu metode robuste (metodaprezentat` în Partea a III-a, Exemplulnr. 2, a fost folosit` intensiv în aplica]iireale), limitele fizice ale interconexiunilor[i influen]a tranzac]iilor asupracircula]iilor de putere într-o re]ea buclat`sînt în general volatile chiar [i pe durataunei luni. Predictibilitatea deplin` acapacit`]ii la punctul de racord va fi pebun` dreptate tr`s`tura definitorie aviitoarelor re]ele inteligente (SmartGrids).

a)Meaningfulness of transmissioncapacity

TC should be assessed and posted so asto be understandable for market players.High visibility of the market – PSinteraction can remodel and support theoperators when solving their operationaland commercial issues. While the DP-TCinformation is fairly easy to understand,allowing a market player to optimize itsmarket portfolio, this cannot be said ofPP-TC capacity that does not appeal tothe average grid user. Power flow limitsfor “critical” network elements aremeaningful only for TSOs. Transparencyand meaningfulness in the capacitycalculation process is also helpful formarket players, in managing their risksmore effectively.

b) Predictability of transmissioncapacity

For a certain time horizon, transmis-sion capacities should be as pre-dictable as possible, and almostwithout exception this is accomplishedonly by limits on power injections atnodes. While a Point-of-Connectioncapacity is relatively stable and can becalculated with robust methods (themethod presented in Part III, CaseNo.2 was intensively used in real-lifeapplications), the physical limits ofinterconnectors and impacts oftransactions on the line flows over themeshed grid with random line outagesare generally volatile even on amonthly basis. Full predictability ofcapacity at the point of network con-nection would actually be the definingfeature of the future SmartGrids.


c)Cre[terea eficien]ei re]elei [i aintegr`rii pie]ei de energie

Capacitatea DP se potrive[te naturalaloc`rii implicite, care este realizat`chiar în cadrul licita]iilor de energie.Alocarea TC participan]ilor la pia]`prin licitatii explicite nu motiveaz`OTS-urile s` foloseasc` la nivel ridicatcapacitatea sistemului de transport.De altfel, mecanismul aloc`rii explicitea „drepturilor” de transport [RM-HR]este incongruent cu natura de monopola tranzac]iilor de transport. Cîndenergia [i capacitatea de transport seliciteaz` separat, se ajunge la frag-mentare a pie]ei [i ineficien]` a re]elei.Regenerabilele cu acces intermitent nutrebuie s` se confrunte cu o procedur`de alocare a capacit`]ii de interco-nexiune „bazat` pe mecanismelepie]ei”, imposibil de respectat.

d) Planificare [i utilizare congruenteale re]elei

Sistemul de transport este înt`rit înprimul rând pentru a permite conectareade noi facilit`]i de generare a energiei,iar OTS trebuie s` garanteze capacitateaacordat` în condi]ii opera]ionale [i depia]` în general neprecizate. Deci,capacitatea DP [i nu cea de tip PP esteinforma]ia relevant` în planificareare]elei. Dac` standardele de proiectare [ifunc]ionare sînt diferite, tranzac]iile dinpia] ,̀ din ce în ce mai ample [i maiimpredictibile, fie pot pune integritateaSEE în pericol, fie pot fi limitate excesiv.

OTS trebuie s` fie preg`tit s` preia pesocoteala sa efectul restric]iiloropera]ionale asupra tranzac]iilor cuenergie, cump`rînd serviciile tehnice

c) Increase in network efficiency andmarket integration

DP capacity makes a natural model forimplicit allocation of grid capacity.Implicit allocation is ensured within theenergy auctions themselves. Allocationof TC to market players in explicitauctions is no incentive for TSOs toallow greater transmission systemutilization. As a matter of fact, explicitallocation mechanism of transmission“rights” [RM-HR] is inconsistent withthe monopolistic nature of transmissiontransactions. Market fragmentationand network inefficiency can beintroduced when electricity andtransmission capacity are auctionnedseparately. Short notice renewablesshould not face an impossible to meet“market – based” procedure ofallocation interconnection capacity.

d)Consistent planning and usageof the network

Transmission system is enhancedprimarily with a view to connecting newpower facilities, and the TSO shallguarantee the approved capacity underrather indeterminate operational andcommercial conditions. Hence, DP and notPP transmission capacity is the relevantinformation in transmission grid planning.If standards of planning and operationwere not the same, larger and lesspredictable power market transactionscould either jeopardize system’s integrityor be eccessively limited.

The TSO should be prepared to take onhis own account the effect of operationalconstraints on the electricitytransactions, buying the necessary


de sistem necesare [i energia deechilibrare, aplicînd proceduri optimede compensare a congestiei [i defolosire a re]elei. OTS este interesat s`înt`reasc` re]eaua [i s` minimizezeabaterile de la func]ionarea normal`.

Fig 2.9 Cerin]e – cheie privind capacitatea detransport pentru accesul la re]ea [i proiectulde pia]`

4.2.2Managementul congestiilor înproiectul nodal de pia]` deenergie electric`

|n context na]ional, de regul̀ , OTSdesp`gube[te participan]ii la pia]̀ pentruorice tranzac]ie degradat̀ ca urmare aMCO. Venitul din tarifele de transportacoper` costul suplimentar alre-dispeceriz`rii gener`rii, în condi]iiledescuraj̀ rii efective a specula]iilor produ-c t̀orilor. Tarifele nodale de transportG [iLau la baz` costurile marginale alere]elei [i, de asemenea, beneficiile con-sumatorilor de transport din participareala pia]a de energie, conform propuneriinoastre (Partea I, §4.2.2).

ancillary system services and balancingenergy, and applying optimal proceduresfor congestion compensation and use ofthe network. The OTS is interested toenhance the network and minimizeabattment from normal operation.

Fig 2.9 Key requirements of grid access andpower market design as regards transmissioncapacity)

4.2.2 Congestionmanagementin nodal electricity marketdesign

In the national context, normally, theTSO reimburses the market players forany degraded transaction as a result ofCM. Transmission-rate-based incomeshould cover the extra cost of gene-ration redispatch, on the assumptionthat generators’ gaming is properlyprevented.G and L nodal transmissionrates are based on the grid marginalcosts and, conformably to our proposal(Part I, §4.2.2), on the benefits of gridusers from their participation to theelectric market as well.


Responsabilitatea OTS trebuie s`r`mîn` neschimbat` [i în contextregional armonizat, multi – OTS.G [iL trebuie stabilite pe baza unui modelcomun de re]ea [i a acelora[i reguli deefectuare a serviciului [i de definire acosturilor. OTS din ]ara exportatoare,[i respectiv importatoare, colecteaz`Gîn punctul de intrare [i L în cel deie[ire. Suma acestora trebuie s`recupereze întregul cost al serviciuluide transport în interconexiunea multi -OTS. Fiecare OTS va desp`gubi propriiclien]i afecta]i pentru orice schimbarea venitului din pia]` ca urmare aMCO. Volumul acestor desp`gubiripoate fi minimizat prin folosireamecanismelor pie]elor de echilibrare,na]ional` [i regional`, în timp ceefectele tranzitelor asupra costurilor seredistribuie între OTS prin meca-nismul ITC.

4.3Capabilitatea de transfera interconexiunilorinter-zonale [imanagementulrestric]iilor pentru operarea însiguran]` a SEE

OTS trebuie s` previn` inadec-van]a opera]ional` a SEE verificînddac` nivelul de siguran]` al siste-mului respect` criteriul ‘N-1’ [i, înanumite situa]ii de planificare ope-ra]ional`, criteriul suplimentar‘N-2’. Conform reguliloropera]ionale, OTS controleaz` re-stric]iile SEE printr-un proces de-cizional în doi pa[i, [i anume:

- Programarea circula]iilor deenergie pentru func]ionarea SEEîn siguran]` dup` terminarea zilei

The TSO responsibility should remainunchanged in a multi-TSO harmonizedenvironment as well. TheGs and Lsare supposed to be established on acommon grid model, following thesame rules of grid service and costdefinition.G at the entry point and Lat the exit point can be collected by theTSOs in the exporting and importingcountries, respectively. The sum ofGand L is meant to recover all the cost ofservice of the multi-TSO intercon-nection. Each TSO may reimburse itsown customers for any change inmarket revenue due to CM. With themechanisms of area and regionalbalancing markets in place, the TSOscan minimize the volume of thesereimbursements, while the ITCmechanism redistributes the costeffects of transits among the TSOsinvolved.

4.3. Transfer capability ofinter-area interconnectionsand constraint managementin power system operationalsecurity

The TSO has to prevent PS operationalinadequacy by verifying that the observedlevel of security complies with the ‘N-1’security criterion as well as with theadditional criterion of taking into accountcertain ‘N-2’ situations in operationalplanning studies. According to operationalrules, the TSO manages PS constraints ina two-step decision-making process,namely:

- Scheduling power flows forreliable PS operation aftercompletion of the transaction day


de tranzac]ionare [i primireanotific`rii tranzac]iilor (inclusiv areprogram`rilor de mentenan]`).Evaluarea predictiv` de siguran]`trebuie realizat` pentru fiecareschem` de dispecerizare, încondi]ii normale sau decontingen]e. Rezolvarearestric]iilor poate conduce lareprogramare sau la limit`ri întranzac]ii.

- Dispecerizarea SEE, care includemonitorizarea re]elei [i aplicareade m`suri corective în situa]ii derestric]ii, ca de pild`redispecerizare, pornire degrupuri generatoare sau limitarede tranzac]ii.

St`rile de func]ionare ale SEE sînt îngeneral verificate pe baza criteriilor(marjelor) de stabilitate, static` [itranzitorie, sau de limite termicepentru echipament.

Din ra]iuni de control, aceste criteriisînt aplicate de regul` sub formaCapabilit`]ilor de Transfer pentruInterconexiunile inter-zonaleRelevante(CTIR). Acestea pot fi artere electrice detransport majore, interconexiuni cusistemele vecine sau coridoare critice dere]ea. Operatorul de sistem din Româniacontroleaz` începînd cu mijlocul anilor ’70adecvan]a circula]iilor totale de puterefa]̀ de CTIR în sec]iunile critice alere]elei, inclusiv transfrontiere, iar autorula dezvoltat metodologia specific ,̀ careeste în esen]̀ bazat̀ pe criteriul marjeide stabilitate static`(Partea a III-a, Exemplul nr. 2).

and receiving notification oftransactions (and maintenancerescheduling, if that is the case).Predictive security assessment mustbe performed for each candidatedispatch scheme, under normalconditions and contingencies.In certain cases,relieving constraints can lead torescheduling of generation or tocurtailment of transactions.

- Dispatching the PS, whichincludes transmission monitoringand, when PS constraints aredetected, applying corrective/remedial actions, such as redis-patching, committing generationor curtailing transactions.

PS operating states are generallychecked as against specific steady-stateand transient stability criteria(margins) and equipment thermalratings.

For control reasons, these criteria aretypically implemented in terms ofpower Transfer Capabilities forRelevant inter-area Interconnections(TCRI). These are major interties, tielines with neighbouring systems orgrid critical paths. The Romaniansystem operator has been controllingthe adequacy of line-flows as againstTCRIs for the critical network pathsincluding cross-border interties sincethe mid-70s, and the author developedthe specific methodology, which isessentially based on the PSsteady-state stability margin(Part III, Case No. 2).


Limitele de transfer prin interco-nexiunile relevante se predetermin` [ise tabeleaz` pentru contingen]ele criticedin sistem [i pentru diferite rezerve destabilitate. De regul`, limitele detransfer se stabilesc luînd în calcul toatecontingen]ele din clasaN-1 [i unele dinclasaN-2, [i pentru rezerve destabilitate static` a SEE de 8% [i 20%.

Capabilitatea de Transfer a Intercone-xiunii inter-zonale este de fapt atins`prin „înr`ut`]irea” în cel mai sever [itotu[i realist mod a st`rii de func]ionarea sistemului, într-o abordare de tip PP.

Deoarece CTIR sînt dependente de st̀ riparticulare ale SEE, iar unele inter-conexiuni inter-zonale se pot suprapune,CTIR pot fi în]elese numai de c t̀re OTS.Controlul transferului de putere prininterconexiuni apar]ine opera]iilor OTSprivind siguran]a SEE, fiind mult maicomplex decît solu]iile de MCO „bazate pemecanisme de pia]̀ ” ale operatorilor depia]̀ . De remarcat faptul c` [i OTScontroleaz` restric]iile din SEE pecriteriul costului minim al congestiilor. |nacest scop, acesta achizi]ioneaz` resursede pe pie]ele concuren]iale de servicii desistem [i de echilibrare, [i aplic` proceduride cost minim pentru conformarea larestric]ii. Aceste proceduri sînt bazate fiepe simularea func]ion`rii sistemului, fiepe sisteme expert, care memoreaz` [ifolosesc cuno[tin]e ob]inute în simul̀ rianterioare.

Incertitudinile de operare pot fi reduseprin coordonare inter -OTS a manage-mentului restric]iilor, inclusiv a calcululuicapabilit̀ ]ilor de transfer. Coordonarea

Transfer limits across relevantinterconnections are estimated andtabulated for critical contingencies inthe system and different stabilitymargins. Typically, the transferlimits are determined by studying allN-1 contingencies and some N-2contingencies, and for 8% and 20%PS steady - state stability margins.

The Transfer Capability of inter-areaInterconnection is actually reached by„worsening” in the most severe but alsorealistic way the system operationalstate, under a PP approach.

Since the TCRIs depend on theparticular state of the PS, and certaininter-area interconnections overlap,only the TSOs can understand theTCRIs. The control of power transferacross interconnections belongs to theTSO’s security operations, being largelymore complex than the “market-based”CM solutions of market operators.Noticeably, the TSO also manages thePS constraints so as to optimize thetotal cost of congestion. To this aim, theTSO purchases necessary resources onthe ancillary system services andbalancing markets, and appliesleast-cost procedures to clearconstraints. These procedures arebased either on simulation of systemoperation, or on expert systems thatstore and use the knowledge gatheredfrom ex-ante simulations.

Operation uncertainties can bedecreased by inter - TSO co-ordinationof constraint management, includingcalculation of transfer capabilities.


SEE pan – european poate fi îmbun t̀̀ ]it`prin punerea la dispozi]ia OTS a unorregimuri calculate de func]ionare pentruîntreaga re]ea de 400 kV, la cele douòrizonturi temporale de interes: cu o ziînainte [i pentru timpul (cvasi)real.

|n plan regional, un Oficiu demanagement [i monitorizare acongestiilor (Congestion Managementand Monitoring Office, CMMO) arputea sprijini OTS s`-[i coordonezeprogramarea SEE [i m`surile deremediere în cazuri de restric]iicomune. CMMO ar putea chiar sàdministreze un sistem de programareregional` vizînd maximizarea folosiriire]elei.

4.4 Pia]a de energie cu structur`nodal` [i re]elele inteligente

Exist` mari a[tept`ri [CIGRE1] caa[a-zicîndele Re]ele Inteligente (SmartGrids) s` r`spund` tuturor nevoilorclien]ilor integrînd eficient [i resurseleregenerabile (Tabelul 1).

Adev`rata provocare pentru re]eleleinteligente (SG) este asigurarea uneicapacit`]i predictibile pentru oriceclient al re]elei, inclusiv pentruregenerabilele cu func]ionareintermitent`. Adic`, s` garantezeclientului capacitatea necesar` lapunctul de racord la re]ea, f`r` vreocondi]ionare de locul partenerului sùde afaceri. Acest scop de primìmportan]` al SG este \n deplin înacord cu proiectul de pia]` bazat pemodelul nodal de acces la re]ea.

Coordination of the pan-European PScan be improved by making availableto all TSOs a frequently updatedsnapshot of the whole 400 kV intercon-nection network in both time frames ofinterest: day-ahead and near real-time.

In the regional context, aCongestion Management andMonitoring Office (CMMO)may assist TSOs to coordinatePS scheduling, and takeremedial actions when constraintsinvolving several TSOs aredetected. The CMMO could evenadminister a regional schedulingsystem aiming at maximizing theuse of network.

4.4 Nodal – structured electricitymarket and the Smart Grids

It is widely expected [CIGRE1] thatthe so-called SmartGrids wouldrespond to all grid customers needsand integrate efficiently the RES aswell (Table 1).

The real challenge for the SmartGrids(SGs) would be to provide apredictable capacity at the connectionpoint for any grid customer, includinga short notice RES. That is, toguarantee the user capacity need atthe very point of network connection,without any reference to location ofthe user’s business partners. Thisprimary purpose of SGs is fullyconsistent with electricity marketdesign based on a nodal model ofaccess to the network.


Platforma Tehnologic` European`privind Smart Grids (SG)

EURELECTRIC

Electric Power Research Institute dinSUA

SG integreaz` inteligent ac]iunile tuturorclien]ilor re]elei

SG integreaz` inteligent comportamentul[i ac]iunile tuturor clien]ilor re]elei

SG trebuie s` vin` în întîmpinareanevoilor întreprinderilor pe pia]a energiei[i ale companiilor de utilit`]i de energie

European Technology Platform onSmartGrids

EURELECTRIC

Electric Power Research Institute ofUS

SG can intelligently integrate the actionsof all network users

SG can intelligently integrate thebehaviour and actions of all network users

SG should be responsive to energy-marketand utility business-enterprise needs.

Tabelul 1 Cerin]e definitorii ale re]elelor inteligente

Table 1 Defining requirements of the Smart Grids



[AB] A. Brameller, Practical Diakoptics for Electrical Networks (Chapman and Hall, London,1969)

[AS-HB] A.M. Sasson, H.E. Brown,Deductions and Clarification on the Diakoptics Approach(PICA Conference Proceedings, 1971)

[BC] B. Carré, Solution of Load – Flow Problems by Partitioning Systems into Trees (IEEETransactions, Vol. PAS – 87, 1968)

[CIGRE1] CIGRE WG “Network of the Future”, Electricity Supply Systems of the Future,DraftReport, V.07, November 2010

[DL] Dy Liacco, Control of Power Systems via the Multi-level Concept (SRC – 68 – 19, SystemsResearch Center, Case Western Reserve University, June 1968)

[ETSO] ETSO, Procedures for Cross-border Transmission Capacity Assessments(October 2001)

[EurC] European Commission Energy Package, January 10, 2007. No. 6 - Communication fromthe Commission to the Council and the European Parliament – Internal Market for Gasand Electricity; No. 7 COM (2006) Gas and Electricity Infrastructure – PrioritaryInterconnection Plan

[GK] G. Kron,Diakoptics: The Piecewise Solution of Large-scale Systems (MacDonald, London,1963)

[HH1] H.H. Happ (editor)Gabriel Kron and Systems Theory (Union College Press, Schenectady,New York, 1973)

[HH2] Z – Diakoptics – Torn Subdivisions Radially Attached (IEEE Transactions, Vol. PAS – 86,June 1967)

[MAL] M.A. Laughton,Decomposition Techniques for Load – flow Analysis Using NodalImpedance Matrix (IEE Proceedings, 115, No. 4, 1968)

[NERC] Available Transfer Capability Definitions and Determination (NERC Technical Report,June 1996)

[PD] P. Dimo,Nodal Analysis of Power Systems (Editura Academiei, Bucure[ti, România; Aba-cus Press, Tunbridge Wells, Kent, 1975)

[REurCB]Regulation of the European Parliament and Council of the European Union onConditions for Access to the Network for Cross-border Exchanges in Electricity andRepealing Regulation (EC) No. 1228/2003

[RM-HR] R. Mendez, H. Rudnick, Congestion Management and Transmission Rights inCentralized Electric Markets (IEEE Transactions, Vol. PAS – 19, No. 2, 2004)


[RO] R. Onodera,Diakoptics and Codiakoptics of Electrical Networks (RAAG Memoirs, Vol. 2,1958)

[WT-JW] W.F. Tinney, J.W. Walker,Direct Solutions of Sparse Network Equations by OptimallyOrdered Triangular Factorization (IEEE Transactions, Vol. PAS – 55, November 1967)

[JCO1] J. Constantinescu, Contribu]ii privind aplicarea decupl`rii func]ionale [i structurale înmodelele de calcul ale regimurilor de func]ionare ale sistemelor electroenergetice complexe(Contributions to Functional and Structural Unbundling in the Operational Models ofComplex Power Systems– PhD Thesis) (Tez` de doctorat, Institutul Politehnic Bucure[ti,facultatea de Energetic`, 1980)

[JCO14] J. Constantinescu,Diakoptica -metodăde analizăa re�elelor electrice (Diakoptics –AMethodforPowerNetworkAnalyses) (Energetica, nr. 10, 1971)

[JCO15] J. Constantinescu,Metod` de analiz` a sistemelor electrice prin descompunere însubre]ele (AMethod for Analyzing Large Power Systems by Decomposition) (St. Cerc.Energ. Electr., nr. 3, pg. 675 - 691, 1973)

[JCO16] J. Constantinescu, Aplicarea metodei buclelor la calculul re]elelor electrice. Re]ele reduseechivalente (A Loop Method for Electric Networks Computations. Equivalent ReducedNetworks) (Studii [i cercet`ri în energetic`, nr. 3, 1971)

[JCO17] J. Constantinescu, Study of the Transient Processes in Large-scale Power Systems (Rev.Roum. des Sci. Techn., Vol. 27, No. 2, pages 211 - 227, 1982)

[JCO18] J. Constantinescu, Adjustable REI - Dimo Equivalents for Longer-term Power SystemDynamics Symulation (Energetica, nr. 6B, 1994)

[JCO19] J. Constantinescu, Pia]a centralizat` cu discriminare spa]ial` ca proiect-]int` de pia]`a energiei electrice la nivel na]ional [i regional (Centralized Market with SpatialDiscrimination, a Target Project for National and Regional Electricity Markets)(Energetica, nr. 2, 2007)

[JCO20] J. Constantinescu,Determination of the Real Net Transfer Capacities and CapacityReserves (CIGRE Paper C2 – 305, Session 2006)

[JCO6] J. Constantinescu, A Unitary Approach of Operational Security andMarket Design in aMulti -TSO interconnection (CIGRE - Paper C5 - 303, Session 2008)

[JCO21] J. Constantinescu, Colapsul de sistem electroenergetic: experien]a SUA din 14 - 15august 2003, a Italiei din 28 septembrie 2003 [i riscurile de c`dere de sistem în România(Power System Blackout: Experience from the USA and Italy in the Year 2003, andPotential Risks for Romania) (Energetica, nr. 11, 2003)

[JCO22] J. Constantinescu, Transmission Service on a Deregulated Electricity Market (CIGREBlack Sea El-Net Regional Meeting, Paper II.5.5, June 2001)

[JCO23] J. Constantinescu, Electricity Market Design Based on a Nodal Model of Access to theNetwork (Editor Jean Constantinescu, ISBN 978-973-0-09601-9, Bucure[ti, 2010)


Partea III. Dezlegarea temporal`sau varia]ional`


Prin Dezlegare temporal` sau Varia-]ional` (DV), modelul unui sistem fizicsau tehnologic se separ` în subsistemeîn domeniul (pe scara) timpului. |naceast` lucrare, ne vom m`rgini lamodelele pentru simularea regimurilortranzitorii ale SEE, în care m`rimile destare variaz`, fie ca oscila]ii electrome-canice, fie ca modific`ri aperiodice.

DV poate fi de variabile sau de procesedup` cum subsistemele r`mîndependente unele fa]` de altele înprocesul solu]ion`rii, sau sfîr[esc prina fi rezolvate independent.

Vom aborda, de asemenea, [i dezle-garea temporal` a proceselor(func]iilor) în modelul „tehnologie-client”al sistemului de energie electric`.

2 Dezlegarea varia]ional`de variabile

La începuturi, modelarea proceselortranzitorii electromecanice ale SEE avizat dinamica rotoarelor ma[inilorsincrone. Mai tîrziu, proceseletranzitorii de tip electromecanic aufost individualizate ca dinamici pe timpscurt, mediu [i lung (Short-, Medium-and Long- Term Dynamics), în func]iede durata total` a simul`rii. |n esen]`,modelele STD, MTD [i LTD sedeosebesc prin natura [i intensitateaac]iunilor perturbatoare, ipotezeleprivind desf̀ [urarea procesului [i

Part III. Temporal or variationalunbundling


Temporal or Variational Unbundling(VU), in a model of physical ortechnological system, separatessubsystems on the time – domain. Inthis work, we will confine ourself to themodels for the simulation of PSdynamics, in which the state quantitiesvary either as electromechanicaloscillations, or as aperiodic changes.

There would be a variable VU, as in thesolution process the subsystems stillremain dependent against each other, ora Process VU, when the subsystems endby being solved independently.

We will also approach the temporalunbundling of processes (functions)within the „technology - customer”model of electricity system.

2 Variational unbundlingof variables

At the beginning, modeling of PSelectromechanical transients wasfocused on dynamics of the rotors ofsynchronous machines. Later on, thetransient processes of electromecha-nical type were individualized as theShort-, Medium- and Long- TermDynamics, depending on the totalduration of the simulation. Essentially,the STD, MTD and LTD models aredifferent from one another as far as thenature and intensity of disturbingactions, the hypotheses on process


componentele sistemului ce sînt repre-zentate.

Modelul STD sau al „primei lovituri”este potrivit pentru investigareaefectelor scurtcircuitelor asuprastabilit`]ii cinetice a sistemului. El estefolosit înc` înainte de apari]iacalculatoarelor electronice [EWK, FD-CC]. Perioada de simulare este decîteva secunde, iar criteriul de baz` alstabilit`]ii este oscila]ia relativ` dintrema[inile sincrone.

Evenimentele ce au avut loc pe duratacolapsurilor SEE au pus în eviden]`limitele modelului STD. |n particular,acesta nu poate reprezenta impactulvaria]iei în limite largi a tensiunilor [ifrecven]ei asupra: dezechilibrelor inter-zonale de puteri, comport`rii echipa-mentului [i deconect`rilor (c`derilor) încascad`.

Cercet`rile efectuate în anii 70’ în SUAde c`tre EPRI au consacrat modelulLTD [GenE1, PD-JCO-MP-IR-IN] (a sevedea [i Exemplul nr. 3, Partea I-a), ceintroduce simplific`ri dar [i detalieride modelare fa]` de reprezent`rileSTD conven]ionale. Scopul model`riiLTD nu necesit` simularea mi[c`riirelative dintre ma[inile electrice, nicila nivel de zon`, nici de sistem.Dinamicile din sistem sînt rezultatulevenimentelor [i perturba]iilor „încascad`”, în general nu scurtcircuite,iar simularea este prelungit` peintervale de ordinul minutelor sauzecilor de minute.

development, and the representedsystem components are concerned.

The STD or „the first hit” model ismeant for investigation the short –circuit effects on kinetic stability of thesystem. It has been used well inadvance the electronic computers cameinto being [EWK, FD-CC]. Thesimulation period is of few seconds,and the relative swing amongsynchronous machines is the basicstability criterion.

The events that happened during thePS black-outs have rendered evidentthe limits of STD model.Particularly, this cannot estimatethe impact of large voltage andfrequence excursions upon: inter –area power unbalances, behavior ofpower equipment and cascadingfailures.

Researches carried out in the 70’s by theEPRI in the USA consecrated the LTDmodel [GenE1, PD-JCO-MP-IR-IN] (seealso the Case No. 3, Part I) thatintroduced model simplification as wellas detailedness as against conventionalSTD system representation. The scope ofLTD modelling does not require simula-tion of the relative swinging of powerunits against each other, neither at area,nor at overall system level. The PS tran-sients are the result of „cascading”disturbances and events, noshort-circuits generally, while simulationprocess is prolonged for time intervals ofminutes or even tens of minutes.


Modelul dinamicilor pe termen mediu(MTD) combin` reprezent`ri STD [iLTD relevînd informa]ia esen]ial` atîtpentru procesele tranzitorii rapide, cît[i pentru cele lente.

Modelele matematice ale regimurilortranzitorii ale SEE [PK], care sînt deordin înalt [i rigide sub aspectulanalizei numerice, pot fi dezlegate înfunc]ie de viteza de varia]ie temporalà variabilelor de stare. |n[PKO-JA-JW-JC] este prezentatprocedeul „perturba]iei singulare” dereducere a ordinului modelelor [i deseparare pe scara timpului. Ipoteza debaz` este constan]a variabilelor „lente”,din modelele echipamentului primar [iale sistemelor de reglare aferente, pedurata determin`rii dinamicilor rapide[i, alternativ, starea cvasi – sta]ionarà variabilelor „rapide”, din modelele\nf`[ur`rilor generatoarelor [isistemelor de excita]ie, pe intervalelede timp pe care se integreaz`variabilele lente.

Notînd cu x [i z vectorii variabilelorlente, respectiv rapide, modelul pentrusimularea regimului tranzitoriu de tipelectromecanic al SEE se exprimàstfel:

Modelul (1) se rescrie astfel încît s` fieeviden]iate dou` procese cu vitez` devaria]ie temporal` distinct`:

The model of Mid – Term – Dynamics(MTD) is a combination of STD and LTDrepresentations, providing the essentialinformation of both the short- and long-term transient processes.

Mathematical models of PS dynamics[PK], that are of high order and rigidfrom the numerical viewpoint, can beunbundled in accordance with thespeed of temporal variation of the statevariables. In [PKO-JA-JW-JC] ispresented the „singular perturbation”procedure for reducing the model orderand separation of time scale. The basichypothesis is constancy of the „slow”variables, associated to models ofprimary equipment and its control,during the simulation of „fast”transients, and, alternatively, thequasi-steady state of the „fast”variables, associated to generators’windings and excitation systemsmodels, during the time intervals atwhich the slow variables aredetermined.

Denoting by x and z the vectors of slowand fast variables respectively, themodel for the simulation ofelectromechanical PS transients canbe expressed as follows:

(1)

The model (1) can be rewritten so as torender evident two processes withdistinct speed of temporal variation:


Efectuînd schimbarea de variabile,

se ob]ine o separare a sc`rilor de timpcorespunz`toare celor dou` categorii devariabile (τ > 0 corespunde unuimoment particular t’ de pe axa t).

La limit` (ε > 0), rezult` modelul dina-micilor de termen lung în spa]iul „t”,

Indicele S arat` c` vectorul-solu]ie xS

se determin` pentru variabile zS

cvasi-sta]ionare.

|n spa]iul „τ”, ecua]iile (2) se pot rescrieastfel:

iar trecînd la limit` (ε→ 0, dx / dτ > 0)

în care t0 > t’, iar x0 reprezint` valorileconstante ale lui x în momentul t0.

(2)

Substituting the change of variables,

(3)

a separation of time scales corres-ponding to the two types of variablesis obtained (τ > 0 corresponds to aparticular moment t’ on the t-axis).

On the boundary (ε> 0), the long – termdynamics model in the field „t” results as,

(4)

The index S shows that the solution –vector xS is determined for variables zS

under steady – state condition.

In the field „τ”, the equations (2) can bere-written as:

(5)

andpassingtotheboundary,(ε→0,dx/dτ>0)

(6)

where t0 > t’ and x0 stand for theconstant values of x at the moment t0.


In relation (3), it can be noticed that if(ε→ 0), the increase of t is negligible(x can be considered as invariable) andis large enough so that z may get closerto the stationary values zS.

Case No. 1.Mid-term dynamicssimulation of the power system.

TEMImodel

The mid-term dynamics simulationmodel does not exhibit sub-transientvariations of the state variablesassociated to generator’s windings andexcitation system neither the slowdynamics of time-lagging primaryequipment. The MTD model reflectsthe conditions of keeping generators insynchronism and the effects of severepower imbalances, as well.

Tothisaim,thereismodeledaprocessof„medium”speed,whichis furtheron(variational)unbundledduetoitsparticulardegreeofcomplexity.Twodynamicsresult,whoseassociatedvariablesareinterconnected intheendofeachlargerintegrationstep [JCO2,JCO24].

Hence, integration in the timedomain is organized in two nestedloops, corresponding to variable setsthat are variationally unbundled, asfollows:

a) Equations in form (6) of the „fast”variables are integrated by meansof trapezoidal method (that has aquite good numerical stability), fork times within the interval ∆T,with x > xT- and ∆t > ∆T/k, in therange of 0.05 seconds.

Este de observat c` în rela]ia (3), dac`(ε→ 0), cre[terea lui t este neglijabil`(x poate fi considerat invariabil) darsuficient` pentru ca z s` se aproprie devalorile sta]ionare zS.

Exemplul nr. 1. Regimul tranzitoriupe termenmediualSEE.

Modelul TEMI

Modelul MTD al regimului tranzitoriude „termen mediu” nu re]ine varia]iilesub-tranzitorii ale m`rimilor electricedin înf`[ur`rile generatorului sau dinsistemul de excita]ie, nici dinamicilelente asociate cu echipamentul avîndtimp de r`spuns lung. |ns`, modelulMTD reflect` atît condi]iile de p`strarea generatoarelor în sincronsim, cît [iefectele dezechilibrelor severe de puteri.

|n acest scop, se modeleaz` un procesde vitez` „medie”, care este în conti-nuare dezlegat (varia]ional) din cauzagradului înalt de complexitate. Rezult`dou` dinamici ale c`ror variabile sîntinterconectate la finele fiec`rui pasmare de integrare [JCO2, JCO24].

A[adar, algoritmul de integrare îndomeniul - timp este organizat în dou`cicluri, exterior [i interior, corespunz`-toare celor dou` mul]imi de variabiledezlegatevaria]ional,dup`cumurmeaz̀ :

a) Ecua]iile de forma (6) ale varia-bilelor „rapide” se integreaz` cuajutorul metodei trapezelor (careare o stabilitate numeric`suficient de bun`), de k ori înintervalul ∆T, cu x > xT- [i∆t > ∆T/k, de circa 0,05 secunde.


b) Ecua]iile de forma (4) ale varia-bilelor „lente” se integreaz` prinmedoda Euler modificat` (sau ometod` Runge – Kutta) cu un pasde integrare ∆T cu un ordin dem`rime mai mare (0,2 – 0,5 s).Pentru variabilele z, se poateadopta drept „condi]ie sta]ionar`”media aritmetic` a valorilor pedurata intervalului precedent ∆T.

R`spunsul dinamic al blocului cazan-turbin` cu supraînc l̀zire, la varia]iile desarcin ,̀ este determinat decaracteristicile dinamice ale cazanului,turbinei [i sistemelor aferente pentrucontrolul sarcinii, ca [i de modificareacapabilit`]ii serviciilor auxiliare înprocesul tranzitoriu. S-a folosit modelulEPRI pentru unit`]ile termoelectrice.Sînt reprezentate numai buclele decontrol care determin` admisia aburuluiîn turbin ,̀ iar pentru modelarearegulatoarelor de tura]ie sînt mai multeop]iuni. Tot modele EPRI, relativ simple,sînt folosite la reprezentarea turbinelorhidroelectrice, de înalt` sau joas` c`dere.Ecua]iile diferen]iale care descriufunc]ionarea echipamentul „lent” seintegreaz` cu o metod` Eulermodificat ,̀ în timp ce ecua]iile ma[inilorsincrone, inclusiv ale regulatoarelor deexcita]ie, respectiv componentele„rapide” ale modelului general, sînt„algebrizate” cu metoda trapezelor.Rezult` modelul „sta]ionar” al genera-torului sincron la pasul de calcul t, deforma urm`toare (detalii în Anexa 5):

b) Equations in form (4) of the „slow”variables are solved by means ofmodified Euler method (or aRunge – Kutta method) with anintegration step ∆T of superiororder of magnitude (0.2 – 0.5 s).For the variables z, one can adoptas „stationary condition” thearithmetic mean of their valuesduring the previous ∆T.

The dynamic response of the boiler-reheat steam turbine block at loadvariation is determined by the dynamiccharacteristics of the boiler, turbine andthe adjoining load control systems as wellas by the change in the capability ofancillary services during the transientprocess. We have represented thesecharacteristics as in the EPRI LTD modelof thermal units. Control loops aremodeled that affect the admission ofsteam into the turbine only, while for thespeed governors more model options areavailable. High or low head hydroturbines and their load control systemsare also represented with the relativesimple EPRI models. The differentialequations that describe operation of the„slow” equipment are integrated with themodified Euler method. Distinctively, theequations of synchronous machinesincluding the voltage regulators, i.e. the„fast” components of general model, are„algebrized” by means of trapezoidal rule.The result is a „steady – state” generatormodel at the computation step t, of thefollowing form (details in the Annex 5):

(7)


Generatorul dintr-un nod al re]elei sereprezint` a[adar printr-un curentnodal, exprimat în coordonaterectangulare (R-I) astfel:

Ecua]iile re]elei se rezolv` prin metodaNewton – Raphson (Anexa 6). Efecteleanizotropiei polilor generatoarelor [ineliniarit`]ile nu constituie o problem`pentru aceast` metod .̀ Estereprezentat` satura]ia circuitelormagnetice [i, de asemenea, dependen]ade tensiune [i de frecven]` a sarcinilorconsumatorilor. Submatricele Jacobia-nului conven]ional sînt reformulate dincauza varia]iei cu tensiunea [i frecven]aa injec]iilor nodale de puteri (generare –g, [i consum – l). Aceast` ipotez` conducela sc`derea algebric` a termenilor

din elementele diagonale ale sub-matricelor H, N, J [i L. La derivarea înraport cu amplitudinea tensiunii, se]ine seama de ecua]iile regulatoarelorde tensiune ale generatoarelor.

Ecua]iile sînt solu]ionate prinfactorizare triunghiular` optim`.

Ac]iunea dispozitivelor de protec]ie [ide automatizare este modelat` fieimplicit, prin scenariul definit deutilizator, fie prin reprezentareacaracteristicilor lor de declan[are.

(8)

A generator in a given network node isthus represented by a node currentinjection that is expressed in the (R-I)rectangular co-ordinates as:

(9)

The network equations are solved withthe Newton – Raphson load flow method(Annex 6). It can tackle with noparticular difficulty the generator’ssaliency and nonlinearity issues. Thesaturation of generator’s magneticcircuits are represented, as well as theload dependence on voltage andfrequency. The sub-matrices of con-ventional Jacobean are reformulated dueto voltage and frequency dependence ofthe node power injections (generation –g, and load – l). This assumption leads toalgebraic subtraction of the terms

from the diagonal entries of H, N, Jand L sub-matrices. The derivation asagainst voltage magnitude takes intoaccount the equations of generators’voltage control.

The solution of equations is carried outby optimal triangular factorization.

The action of PS protective and controldevices can be modeled eitherimplicitly, as a user - defined scenario,or by representation of their trippingcharacteristics.


Echivalen]ii dinamici ajustabilideriva]i dinmodelele REI-Dimo

|n reprezent`rile de mari dimensiuniale SEE, complexitatea modeluluiTEMI conduce la un volum apreciabilde date, afectate de incertitudini.Scenariile de evenimente devin, deasemenea, complexe [i greupredictibile. |n plus, într-o re]ea cunumeroase componente exist` un riscsemnificativ de neidentificare asitua]iilor critice din cauza gamei largide traiectorii posibile ale regimului.

Pentru a se dep`[i aceste dificult`]i,modelul TEMI include [i o func]ie dereducere a schemei SEE, prin dezle-gare structural` (de procese), pe bazaechivalen]ilor REI Dimo (Partea a II-a,§3.1). Echivalen]ii REI pentru zonelede re]ea de interes secundar potreduce spectaculos dimensiunea zoneiSEE în studiu, f`r` compromitereapreciziei solu]iei (Figura 3.1a).

Fig 3.1a. |nlocuirea zonelor de re]ea de interessecundar prin echivalen]i REI

Adjustable dynamic equivalentsbased on REI – Dimomodels

For large – scale PS representations,the complexity of TEMI model leads toa great amount of data, largellyaffected by uncertainties. The eventsscenarios become also complex andhardly predictable. Yet, in a powernetwork with numerous componentsthere is a significant risk of non-identification critical situations due tothe wide range of possible regimetrajectories.

In order to overpass these difficulties,the TEMI model includes a function ofreduction the PS scheme as well, by(process) structural unbundling, that isbased on the Dimo’s REI equivalents(Part II, §3.1). The REI equivalents forthe network areas of secondary interestcan drastically reduce the size of PS areaof study, without essential prejudice onsolution accuracy (Figure 3.1a).

Fig 3.1a. Replacement of the network areas ofsecondary interest by REI equivalents


Integrarea echivalen]ilor REI înmodelul dinamicilor pe termen mediunecesit` îns`,

- o grupare adecvat` a unit`]ilorgeneratoare [i a sarcinilor care setransfer` în noduri REI (problemacoeren]ei), [i

- o reprezentare dinamic` adecvat`a generatoarelor [i sarcinilorconsolidate (agregate) înechivalen]ii din nodurile REIrespective (problema echivalen]eiparametrilor).

O cerin]` deosebit` prive[te ajustareasistematic` a echivalentului pe duratasimul`rii, deoarece variabilelemodelului au varia]ii într-un domeniumai larg decît în alte simul`ri. Estebine-cunoscut faptul c` orice modelechivalent este riguros numai înregimul care a servit ca baz` pentrudeterminarea sa. Pentru acest regimini]ial, caracteristicile unit`]ilorgeneratoare, sarcinilor [ielementelor de re]ea din modelul reduspot fi acordate cu cele din modeluloriginar.

Dat̀ fiind neliniaritatea accentuat̀ acomponentelor SEE, solu]iile nu pot firiguroase pentru st̀ rile neini]iale alesistemului. Se pot ob]ine îns` rezultateacceptabile pentru practic` folosindipoteze simplificatoare, sau chiarproceduri empirice, cu condi]ia ca acesteas` concorde cu caracteristicile de baz` alesistemului [i ale studiului, [i cu rezultateledisponibile din \ncerc`ri pe viu.

The integration of REI equivalentswithin mid-term dynamics modelrequires, however,

- a proper grouping of generatingunits and loads that aretransfered to REI nodes(coherence issue), and

- a proper dynamic representationof consolidated (aggregated)generators and loads by therespective equivalents in the REInodes (parameter equivalenceissue).

A particular requirement addressessystematic adjustment of theequivalent over the simulation time,since the model variables haveexcursions within a domain larger thanin other system simulations. It iswell-known that any equivalent modelis rigurous in the very regime thatserved as a base for its derivation, only.For this initial regime, thecharacteristics of generating units,loads and network elements in thereduced model can be matched withthose in the original model.

Due to strong nonlinearity of the PScomponents, rigurous solutionscannot be found for the noninitialsystem states. Simplifyinghypotheses and even empiricalprocedures can yield fairly goodresults for practice providing thatthey are harmonized with the systemand study particularities, and withthe available field tests results.


Parametrii echivalen]i ai sarcinilor

Sarcinile consolidate pot fi reprezentateprintr-o caracteristic` simpl` putere /tensiune (U) [i frecven]` (ω),

Nodurile de sarcin` consolidate (i) segrupeaz` în baza criteriului varia]ieiconforme tensiune, rezultînd urm`toriiparametri echivalen]i:

Parametrii echivalen]iai generatoarelor

Constantade iner]ieesteparametrul-cheieînoricesimularederegimuri tranzitoriielectromecanice.Dac̀ seadopt̀ modeleunitare, experien]aarat̀ c̀ unparametrukealmodeluluigeneratorului echivalentpoate fi aproximatastfel:

Constanta de iner]ie ageneratorului REI se determin`astfel încît viteza ma[iniiechivalente s` reprezinte o medie avitezelor generatoarelor consolidate.

Load equivalent parameters

Consolidated loads can be representedby a simple voltage (U) and frequency(ω) / power characteristics, as:

(10)

The nodes with consolidated loads (i)are grouped on the basis of conformevoltage variation criterion, resultingthe following equivalent parameters:

(11)

Generating unit equivalentparameters

Inertia constant is a key parameter inany electromechanical dynamicssimulation. If unitarian models areadopted, experience shows that amodel parameter ke of the equivalentgenerator can be approximated as,

(12)

Inertia constant of the REIgenerator is determined so that thespeed of equivalent machinerepresents an average ofconsolidated generators’ speeds.


Considering as coherence criterionthe same acceleration at everyconsolidated unit, we have:

(13)

However, the equivalent inertiaconstant should reflect the location ofthe replaced generators within thenetwork. To this goal, a procedure hasbeen developed [JCO25] on theobservation that, in relation to rotoracceleration (a), the generated activepower can be expressed as a sum oftwo components, namely:

(14)

in which Pg0 is generated at a > 0(initial state) and Pa > –T . a.

Thus, initially, the components Pg0 onlyare used (linerized) at computation ofthe elements of zero power balance(ZPB) network, while the components(–Tm

. am) are formally localized in therespective nodes m (Figure 3.1).

Fig 3.1 „Acceleration“ components of gener-ated powers

Considerînd drept criteriu decoeren]` aceea[i accelera]ie lafiecare grup consolidat, avem:

Dar, constanta echivalent` de iner]ietrebuie s` reflecte localizarea în re]eaa generatoarelor înlocuite. |n acestscop, a fost dezvoltat un procedeu[JCO25] bazat pe observa]ia c`, înraport cu accelera]ia rotorului (a),puterea activ` generat` se poateexprima ca sum` de dou` compo-nente, respectiv:

în care Pg0 este generat la a > 0 (stareaini]ial`), iar Pa > –T . a.

Astfel, ini]ial, la calculul elementelorre]elei de bilan] energetic nul (BEN)sînt folosite (linearizate) numai compo-nentele Pg0, în timp ce componentele(–Tm

. am) sînt localizate numai formalîn nodurile m respective (Figura 3.1).

Fig 3.1 Componentele de „de accelera]ie“ aleputerilor generate


Modelul re]elei este apoi redus prineliminare Gauss, componentele(Tm

. am) fiind „aruncate” succesiv lanodurile vecine. |n final, la fiecare nodREI se acumuleaz` o component` deputere (–Te

. ae). Dar, în baza criteriuluide coeren]` adoptat, procedura sepoate aplica direct constantelor deiner]ie, rezultînd astfel Te.

Actualizarea modelelor REI-Dimope durata procesului tranzitoriu

Tr`s`tura caracteristic` a re]elei BEN,definitorie pentru modelele REI – Dimo,este absen]a pierderilor de putere. Dac`tensiunea nodurilor O’ se alege devaloare zero, aceast` proprietate sepoate exprima prin ecua]ia urm`toare:

Pe durata st`rilor succesive ale pro-cesului tranzitoriu, modelul poate s`-[imen]in` aceast` proprietate, ca ocerin]` de adecvan]`, dac`:

sau,

La o schimbare conform` a tensiunilor,

The network model is then reduced byGaussian elimination, with compo-nents (Tm

. am) being successively„thrown” to neighbouring nodes. Finally,a power component (–Te

. ae) would beaccumulated at every REI node. Yet, onthe adopted coherence criterion basis,the procedure can directly be applied forinertia constants, thus yielding Te.

Updating REI – Dimo models duringtransient process

The characteristic feature of ZPBnetwork, lying at the foundation of REI– Dimo models, is absence of the powerlosses. When voltage value of the O’nodes is chosen zero, this property isexpressed by the following equation:

(15)

During subsequent states of thetransient process, the model canpreserve this basic property, as anadequacy requirement, if:

(16)

or,

(17)

Under a conform change of voltages,

(18)

[i dac` tensiunea nodurilor O’ r`mîneegal` cu zero,

- termenii ecua]iilor (16) [i (17)reprezint` înc` injec]iile de putere[i

- puterea echivalent` la nodul REIreprezint` cu precizie injec]iile deputere din nodurile (m).

Abaterea de la conformitate a ten-siunilor [i, prin urmare, de la modeleechivalente corecte, este eviden]iat̀ deapari]ia pierderilor de putere în re]eleleBEN [i de schimbarea tensiunilor no-durilor O’. Ultima informa]ie st̀ la bazametodei de actualizare a modelului. |nacest scop, în echivalen]i se insereaz` re-]ele de „Restabilire a Tensiunii Zero”(RTZ). Corec]ia este realizat̀ astfel încîts` fie p`strate (i) tensiunile Ui la nodurilede grani]̀ [i (ii) caracteristicile putere –tensiune Se > f(Ue) la nodurile REI.

Algoritmul poate fi schi]at astfel:

Pasul 1. Calculeaz` noua Ue’ care

men]ine U0’, respectiv:

Pasul 2. Calculeaz` modific`rile curen]ilorla nodurile de grani]̀ (i) [i admitan]ele decorec]ie corespunz t̀oare,

Pasul 3. Calculeaz` admitan]a decorec]ie la nodurile (e),


and providing that the voltage of O’nodes remains equal to zero,

- the terms of equations (16) and (17)still represent power injections,and

- the equivalent power at a REInode represents accurately thepower injections in the (m) nodes.

Abatement from voltage conformityand therefore, from accurate equiva-lent models, is rendered evident byemerging losses in the ZPB networks,and change of node O’ voltages as well.The last information stays at the basisof model update. To this aim, the „ZeroVoltage Restoration” (ZVR) networksare inserted within the equivalents.Correction is carried out so that (i) Uivoltages of border nodes and (ii) power- voltage characteristics at Se > f(Ue)REI nodes are not altered.

The algorithm can be schetched as follows:

Step 1. Compute a new Ue’ to maintain

U0’, namely:

(19)

Step 2. Compute the changes of currentsat (i) border nodes, and correspondingcorrection admittances,

(20)

Step 3. Compute the correctionadmittance at (e) nodes,


Pasul 4. Insereaz` re]elele RTZ înmodelele echivalente (Figura 3.2) [ielimin` nodurile O” prin transform`ristea - triunghi.

Fig 3.2 Re]eaua de Restabilire a TensiuniiZero (RTZ)

Rezultate

Programul de calcul TEMI a intrat înpractica curent` de planificareopera]ional` la Dispecerul EnergeticNa]ional, oferind rezultate consonantecu fenomenele din SEN [i alte tipuri deanalize.

Figura 3.3 ilustreaz` prin diagrame pescara timpului evolu]ia unei tensiuninodale, a tensiunilor la dou` noduri O’[i a unei puteri reactive generate[detalii în JCO25].

(21)

Step 4. Insert the ZVR networks withinthe equivalent models (Figure 3.2) andremove the nodes O” by wye-deltatransformations.

Fig 3.2 Zero Voltage Restoration (ZVR)network

Results

The computer code TEMI wascurrently used by the National PowerControl center in operation planningproviding results consonant withactual PS phenomena and other typesof analyses.

Figure 3.3 illustrates by time-scalediagrams the evolution of a busvoltage, of the voltages of two O’ nodesand of a generator reactive power[details in JCO25].


Fig 3.3 (a) Evolu]ia tensiunii unui nod în mod-elul originar [i în modelele reduse

Fig 3.3 (a) Evolution of a bus voltage in theoriginal and reduced models)


Fig 3.3 (b) Evolu]ia unei puteri reactive gener-ate în modelul originar [i în modelele reduse

Fig 3.3 (b) Evolution of a reactive power gener-ation in the original and reduced models)


Re]eaua reprezentat` cuprinde268 noduri de 400 kV, 220 kV [i 110 kV,[i 49 unit`]i generatoare, reale sauechivalente. Sarcina total̀ a sistemului afost de circa 6.000 MW. Simularea a fostrepetat` pe dou` modele reduse: unulavînd o re]ea în studiu de 92 noduri,17 generatoare [i 2 echivalen]i REI-Dimo,cel̀ lalt avînd o re]ea în studiu de39 noduri, 10 generatoare [i 4 echivalen]iREI.

Scenariul de incidente a presupusie[irea din func]iune a unuitransformator de 400 kV / 220 kVînc`rcat la 215 MW. Avaria a condus ladeclan[area unei linii electrice de 220 kVce evacua puterea a 3 unit`]i hidro-energetice. Dup` 2,5 secunde,2 generatoare sînt automat deconectatede la re]ea pentru oprirea func]ion`rii înregim asincron.

3. Dezlegarea varia]ional`de procese

Procesele tranzitorii electromecaniceale SEE descrise de modelele STD [iLTD sînt pe deplin dezlegate varia]io-nal (subsisteme rezolvate independent)prin ipoteze simplificatoare potrivite cuscopul simul`rii.

Pe de o parte, în modelul STD, putereamecanic` la axul ma[inilor estemen]inut` constant`, neglijîndu-sedinamica întregului echipament termo-[i hidromecanic din centralele electrice.Sarcina consumatorului de energie estereprezentat` simplu printr-oimpedan]` fix`, iar parametrii re]eleisînt constan]i pe durata simul`rii.

The represented network comprises268 nodes of 400 kV, 220 kV and110 kV, and 49 real or equivalentgenerating units. The total system loadwas roughly 6,000 MW. The simulationwas repeated for two reduced models:one with a study area of 92 nodes,17 generators and 2 REI-Dimoequivalents, and the other with a studyarea of 39 nodes, 10 generators and4 REI equivalents.

The incident scenario supposes outof step of a 400 kV / 220 kVtransformer unit loaded at 215 MW.This failure leads to tripping of a220 kV power line that evacuatesthe output of 3 hydropower units.After 2.5 seconds, 2 units areautomatically disconnected from thegrid to arrest their asynchronousregime.

3. Process variationalunbundling

The PS electromechanical dynamics thatare described by STD and LTD modelsare fully variationally unbundled (sub-systems solved independently) by meansof simplfying prerequisites suited to thesimulation goal.

On one hand, in the STD model, themechanical power driving the machines’rotors is kept constant by neglectingdynamics of all power plant thermal-and hydro-mechanical equipment.The load of electric consumer is simplyrepresented by a fixed impedance, andthe network parameters are keptconstant during the simulation.


Pe de alt` parte, în modelul LTDregimul tranzitoriu din înf`[ur`rilegeneratoarelor este neglijat. Semodeleaz` îns` în detaliu echipamentuldin centralele electrice avînd influen]`semnificativ` asupra sarcinii grupurilor,ca [i dependen]a sarcinii consumato-rilor, parametrilor re]elei [i ai genera-toarelor de tensiune [i de frecven] .̀

O dezlegare varia]ional` clasic` deproces se aplic` la evaluarea limiteide stabilitate static` a SEE. Regimulsistemului este deplasat spre stareacritic` pe traiectorii care excludoscila]ia variabilelor de stare.

Exemplul nr. 2.Capabilit`]i detransfer de energie electric` înnoduri [i sec]iuni de re]earelevante.

Modelul SAMI

Metodaserve[te, înprimulrînd, laevaluarea înc̀ rc̀ rii limit̀ înnodurilere]elei,definit̀ dreptcapacitatedetrans-port, de regul̀ pentru fiecare nod (Parteaa II-a, §4.3). Este marja cuprins` întrepunctul curent de func]ionare a sistemului[i cel de colaps, în care sistemul devineinstabil. Aceasta se mai nume[te, instabi-litate „de tensiune” la o sarcin` superioar`capabilit̀ ]ii de transfer a re]elei.Asigurarea c` regimul SEE respect̀limitele, este tradi]ionala siguran]̀ înfunc]ionare. Metoda poate servi în egal̀masur` [i la determinarea Capabilit̀ ]ii deTransfer pentru Interconexiunile inter-zonale Relevante (CTIR), ob]inute prinînsumarea circula]iilor pe linii în stareafinal̀ (stabil̀ ) a SEE.

On the other hand, in the LTD modelthe transients within generatorwindings are neglected. Instead, thosepower plant facilities with significantinfluence on the units’ load aremodeled, as well as the impact ofvoltage and frequency variations onconsumers’ load, network andgenerator parameters.

A classical variational unbundling ofprocesses is applied in assessment ofthe PS steady-state stability limit. Thesystem’s regime is displaced towardscritical state along trajectories with nooscillation of the state variables.

Case No. 2.Power transfercapabilities of thenetwork’s nodes and relevantinterties.

SAMImodel

The method aims first to assess thegrid node loadability that was definedas transmission capacity, typically foreach bus (Part II, §4.3). It is a marginbetween the current point of systemoperation and the point of collapse,where the system becomes unstable.That is, the point of so-called “voltage”instability for a load higher than thepower transfer capability of thenetwork. Ensuring that PS operateswithin its limits has traditionally beenreferred to as power system security.The method can equally determine thepower Transfer Capabilities ofRelevant inter-area Interconnections(TCRI) by summing up the line flowsin the final (stable) PS state.


Limita de înc`rcare a re]elei se poateatinge prin modific`ri incrementale,dar fine, ale puterilor active generate [iconsumate, puterilor reactive consu-mate, sau tensiunilor de excita]ie alegeneratoarelor - m`rimi de stare caredetermin` puterile reactive generate -\n diferite scenarii de contingen]eN-1 sau N-2 [JCO-MH3, JCO20,JCO26, JCO-CA].

„|nr`ut`]irea” ipotetic` a regimuluiSEE, ca urmare a cre[terii / descre[-terii pas – cu – pas a injec]iilor deputeri în anumite noduri, urmeaz` defapt un proces de deplasare dezlegatvaria]ional, datorit` condi]iilorcvasi-sta]ionare impuse variabilelorde stare. Acestea urmeaz` traiectoriiraportate la un timp fictiv t. Se omiteîns` dinamica m`rimilor iner]iale [ioscila]iile induse de aceasta.

O stare a SEE este descris` de siste-mul de p ecua]ii diferen]iale de ordinulîntîi (5), scris în forma normal`,

în care, func]iile fi nu depind explicit devariabila temporal` t. Un punct deechilibru (sta]ionar) este definit prininvarian]a în timp a variablelor de stare

[i are coordonatele egale cu solu]iaurm`torului sistem de ecua]ii:

The limit of network loading can bereached by changing incrementally,however smoothly, either active powersof generators and loads, reactivepowers of loads, or generators’ fieldvoltages - state parameters determinigreactive power generations - undervarious N-1 and N-2 contingencyscenarios [JCO-MH3, JCO20, JCO26,JCO-CA].

The hypothetic „worsening” of PS regime,as a result of step – by – step increases /decreases of the power injections atcertain nodes, follows a desplacementprocess which is virtually variationallyunbundled due to the quasi – staticconditions imposed for the state variables.These are following trajectories as againsta fictitious time t. Yet, the dynamics ofinertial variables are omitted, and theinduced PS oscillations, as well.

A PS state is described by the set of pfirst – order differential equations (5)in the normal form,

(22)

in which, fi does not explicitely dependon the temporal variable t. An equilbrium(steady-state) point is defined by the timeinvariance of the state variables,

(23)

and has as co-ordinates the solution ofthe following equation set:

(24)


|n (24), x semnific` componentelepolare U [i θ ale tensiunilor nodale, iarf, injec]iile nodale de puteri active [ireactive; au disp`rut variabilele careintervin numai \n regimul tranzitoriu.Fiind neliniar, sistemul (24) are maimulte solu]ii, corespunz`toare unuinum`r egal de st`ri de echilibru.Problema stabilit`]ii unei st`ri deechilibru const` în verificarea reveniriiei în pozi]ia ini]ial` dup` o varia]ietemporar` [i ipotetic` a unuiparametru al sistemului, care esteconstant în mod normal. La evaluareastabilit`]ii statice, perturba]ia trec`-toare este de m`rime infinitezimal`. |nipoteza neglij`rii oscila]iilor proprii alesistemului, pierderea stabilit`]ii se faceprin varia]ii aperiodice ale variabilelorde stare. |n acest caz, este demonstrat[VV] c`:

-Starea de echilibru este stabil̀ dac`Termenul Liber al Ecua]ieiCaracteristice (TLEC) a sistemului(23) linearizat are acela[i semn caîntr-o stare cunoscut̀ ca fiind stabil̀ .

- TLEC este egal cu determinantulmatricei de coeficien]i a sistemuluiob]inut prin linearizarea ecua]iilor(24).

Starea de raportare cea mai stabil` asistemului, notat` cu indicele „0”, esteaceea în care toate generatoarele sîntcomplet desc`rcate de putere activ`.

Notînd cu π(t) vectorul perturba]iilornodale, ecua]iile puterilor în regimurileînr`ut`]ite sînt de forma,

In (24), x signifies the polar componentsU and θ of the nodal voltages, whereas fare nodal active and reactive powerinjections; there are no state variablesthat are present during transient regimeonly. Being nonlinear, the set (24) hasmultiple solutions and adjacentequilibrium states. The stability issue ofan equilibrium state consists of thecheck in return in its initial conditionsafter a temporary and hypotheticalvariation of a system parameter which isnormaly constant. Assessment of steady– state stability is generally checked fora transitional and infinitesimaldisturbance. If the system oscillationsare neglected, the loss of stability wouldtake place as aperiodic displacements ofthe state variables. Under theseconditions, it is demonstrated [VV]that:

- The equilibrium state is stableif the Free Term of Cha-racteristic Equation (FTCE) ofthe set (23) has the same sign asin a certified stable state.

- The FTCE equals thedeterminant of coefficientmatrix in the linearizedequations of (24).

The certified (most) stable state,denoted by the index „0” corresponds tothe situation in which all power unitsgenerate no active power.

Noting the nodal disturbances by thesparse vector π(t), the node power balanceequations can be written as follows:

(25)f(x(t), π(t))=0


Sau în form` explicit`,

unde perturba]iile de stare se defi-nesc prin urm`toarele func]ii simple:

S-a notat cu: gi – generatoarele dinnodul i la care se modific` treptattensiunile de excita]ie sau puterile active(a se vedea [i procedura de selec]ie auto-mat` a înc`rc`rii / desc`rc`rii de sarcin`descris` în Exemplul nr. 2, Partea I),l – simbol de sarcin` [i cup, q, e – varia]iile incrementale ale P, Q, E.

Derivînd func]ia implicit` (25) înraport cu x [i π se ob]ine o ecua]ie deforma urm`toare:

în care, punctul semnific` derivatapar]ial` în raport cu „t”.

|n cazul considerat, ecua]ia (28) sescrie astfel:

Or, explicitelly,

(26)

where the state disturbances are definedby the following simple functions:

(27)

gi denotes generators in node i whosefield excitations or active power aregradually changed (to see the procedureof automated selection up-ward /down-ward power generation, describedat the Case No. 2, Part I), l – the load’ssymbol and p, q, e – incrementalvariations of P, Q, E.

The derivative of composite function(25) with respect to x and π yields anequation of the following form:

(28)

where, upper dots symbolize thepartial derivatives with respect to „t”.

In the considered case, the equation(28) is written as follows:

(29)


eg, pg, pl [i ql sînt vectorii lacunari aiperturba]iilor nodale incrementale,elementele matricei de coeficien]i -Jacobian sînt similare acelora dinmodelul Newton – Raphson (laelementele diagonale se mai adaug`îns` derivatele puterilor de sarcin`),iar elementele matricei diagonale Jg auurm`toarea expresie:

Reprezentarea generatoarelor prinnumai termenul Jg a simplificatapreciabil modelul (29). Este urmareapremizei invarian]ei sarcinii activegenerate fa]` de schimbarea tensiuniire]elei (Anexa 6, §3), adic` în modelulSAMI s-a aplicat [i o dezlegarefunc]ional` de proces.

Din ecua]ia matriceal` (29) se excludrîndul [i coloana asociate derivateiunghiului de tensiune ales careferin]`.

Prin integrarea sistemului de ecua]iidiferen]iale ordinare (29) în spa]iul t,se ob]in traiectoriile urmate devariabilele x în procesul de înr`ut`]irea regimului SEE.

Metoda previne instabilitatea calcululuinumeric care precede instabilitatea SEE[i modelarea inadecvat̀ a sarcinilor [igener`rilor din modelele tradi]ionale pen-tru determinarea circula]iilor de putere.

Este de subliniat faptul c` matriceaJacobian nu este Jacobianul clasic din

eg, pg, pl and ql are the sparse vectors ofthe incremental nodal disturbances,the entries of coefficient matrix –Jacobean are similar to those in theNewton-Raphson model (the deriva-tives of load powers are yet added tothe diagonal matrix elements),whereas the Jg diagonal elements havethe following expression:

(30)

The representation of generators by theonly Jg term considerably simplified themodel (29). It follows from theprerequisite of active power generationinvariance as against the networkvoltage change (Annex 6, §3), that is, inthe SAMI model a process functionalunbundling was applied, as well.

The row and the column associated tothe derivative of voltage angle that istaken as a reference are excluded fromthe matrix equation (29).

By integrating the system of ordinarydifferential equations (29) in the tspace, the trajectories that are followedby the x’s during PS regime decay areobtained.

The method prevents the numericalinstability preceding the PSinstability, and inadequate load andgenerator modelling in traditionalload flow models.

It is worth stressing that the Jacobeanmatrix is not the Newton - Raphson


classical Jacobean (Annex 6). Thediagonal matrix entries include specificcomponents referring to load andgenerators, that are represented withtheir steady – state voltage/powercharacteristics.

After each integration step, the(aperiodic) stability is verified.As a checking criterion, the authorhas proposed [CIGRE2] theIntegral Steady-State StabilityMargin index (ISSM), with valuesranging from 0 to 1, irrespective ofthe PS size:

(31)

For similar characteristics ofgenerations and loads,

- In the process of regime decay(eqs. (28)), and

- At checking PS stability on thebase of ISSM

the Jacobeans ∂f / ∂x in (28) and Jx in(31) are identical.

Jx is the mere free term ofcharacteristic equation in the PSdynamic model. It is a measure of PSsteady - state stability under thecondition of aperiodic change ofvoltages when exceeding the stabilitylimit (voltage stability).

This hypothesis leads to a conservativeestimation of the stability limit.

The Jacobean J0 corresponds to ahypothetical state of the PS, which is

metoda Newton-Raphson (Anexa 6). |ntermenii diagonali î[i fac prezen]acomponente specifice datoratesarcinilor [i generatoarelor, ce sîntreprezentate cu caracteristicile lorstatice tensiune/putere.

Dup` fiecare pas de integrare, se verific`stabilitatea sistemului (de tip aperiodic).Drept criteriu de verificare, autorul apropus [CIGRE2] indicele integral almarjei de stabilitate static` (IntegralSteady-State Stability Margin index -ISSM), cu valori cuprinse între 0 [i 1,indiferent de dimensiunea sistemului:

Pentru caracteristici similare alegener`rilor [i ale sarcinilor,

- |n procesul de înr`ut`]ire aregimului (ecua]iile (28)) [i

- La verificarea stabilit`]iiregimului cu criteriul ISSM

matricele Jacobian ∂f / ∂xdin (28) [i Jx

din (31) sînt identice.

Jx este chiar termenul liber al ecua]ieicaracteristice din modelul dinamic alSEE. Acesta este o m`sur` astabilit`]ii statice în condi]iiledeplas`rii aperiodice a tensiunilordup` trecerea de regimul limit`(stabilitate de tensiune).

Ipoteza duce de regul` la o estimareconservativ` a limitei de stabilitate.

Jacobianul J0 corespunde unei st`riipotetice a SEE, care este complet


desc`rcat de putere activ`. Fiind ceamai stabil`, aceasta este aleas` castare de referin]`.

Capabilit`]ile de Transfer pentru In-terconexiunile inter – zonale Relevantese calculeaz` în starea stabil` final` aSEE, prin însumarea circula]iilor peliniile corespunzatoare [i se tabeleaz`pentru contingen]ele critice. De regul`,CTIR se determin` cu marje asigur`-toare, studiind toate contingen]ele detipul N-1 [i unele de tipul N-2.

Figura 3.4 ofer` o indica]ie asupraevolu]iei ISSM în studiul transferuluimaxim de putere printr-un coridor dere]ea [JCO26].

Fig 3.4 Evolu]ia indicelui ISSM pe modelultest CIGRE 38 – 02 – 11

Pentru a se înr`ut`]i condi]iile destabilitate static`, modelul test CIGRE

completely unloaded. Being the moststable one, this state is chosen as areference.

Transfer Capabilities of Relevant inter-area Interconnections are calculated inthe final stable PS state, by summingup the corresponding line flows, and aretabulated for critical contingencies onthe system. Typically, the TCRIs areestablished with a secure margin, bystudying all N-1 contingencies andsome N-2 contingencies.

Figure 3.4 offers an indication withrespect to evolution of the ISSM whenstudying the maximum power transferacross a network corridor [JCO26].

Fig 3.4 Evolution of ISSM index on the CIGRE38 – 02 – 11 test model

To worsen the steady – state stabilityconditions, the CIGRE 38 – 02 – 11 test


38 – 02 – 11 acompaniaz` cre[tereacircula]iei prin liniile selectate cucontingen]e de linie (L), [i respectiv degenerator (G).

Au fost reprezentate urm`toarelecaracteristici „sarcin` – tensiune”:

Rezumînd, capabilitatea de transfer are]elei poate fi evaluat` pe bazastabilit`]ii statice (de tensiune [i deunghi) a SEE. Indicele ISSM, de staredat` (nu de abatere mare, conformdefinirii din [CIGRE2]) înlocuie[te„criteriile practice”, prea simple pentrua fi sigure. Valoarea indicelui ISSM,cuprins` între 0 [i 1 pentru domeniulde stabilitate, constituie o informa]iecomplet` asupra marjei de stabilitatestatic` a sistemului. Mai adaug`m c`metoda propus` pentru determinareacircula]iilor de puteri, esen]ialneconven]ional`, determin`întotdeauna în patru pa[i de calcul osolu]ie de încredere, fiecare pasechivalînd cu o itera]ieNewton-Raphson sub aspectultimpului de calcul CPU.

model, accompanies the increase ofpower flows along the selected lines byadditional line (L) and generator (G)contingencies.

The following load – voltage character-istics have been represented:

(32)

In summary, the network powertransfer capability can be assessed onthe basis of PS steady - state(voltage&angle) stability. The ISSMindex as a given state based one (not alarge deviation based index as definedin [CIGRE2]) is meant to replace the“practical criteria”, too simple to besafe. The value of ISSM, ranging from0 to 1 within the stability domain, con-stitutes a comprehensive informationon the system steady – state stabilitymargin. Additionally, the proposedload flow method, fullyunconventional, can always find atrustworthy solution in fourcomputation steps, each step beingequivalent to a Newton – Raphsoniteration as far as CPU time isconcerned.


4 Dezlegare temporal` de procese(func]ii) în pia]a [i în sistemulde energie electric`

In practica curent`, func]iaopera]ional` a OTS se realizeaz`într-un proces dezlegat temporal îndou` etape (Partea II, §4.3), [i anume:

a) Planificarea opera]ional` cu o zi– înainte, adic` programarea(optim` a) sistemului dup`închiderea pie]elor de energie cuo zi înainte [i autoprogramareaparticipan]ilor, [i

b) Dispecerizarea SEE, sau opera-rea în timp – real a sistemului,inclusiv monitorizarea re]elei detransport.

Datorit` incertitudinilor, OTS ar puteas` adînceasc` dezlegarea oper`riisistemului în intervale mai scurte detimp, la momente intermediare întreprogramarea cu o zi înainte [i timpulreal. Cu cîteva ore înainte de timpulreal, incertitudinea descre[te, dar, estenevoie ca aceste func]ii s` fie executatemai rapid. Devine necesar` efectuareacalculelor în paralel, inclusiv prinrecurgerea la tehnici de dezlegarefunc]ional` [i structural`. Analizele desiguran]` a func]ion`rii se efectueaz`pentru fiecare schem` candidat` ladispecerizare identificîndu-se liniileelectrice care ating limitele termice [iestimîndu-se marjele de stabilitate alesistemului. Pe baza acestor analize,OTS determin` ac]iunile optimepreventive [i corective care satisfaccriteriile de siguran]` opera]ional` peintervalul (de dezlegare temporal`) încare el presupune c` starea sistemului

4 Temporal unbundling ofprocesses (functions) in electricmarket and system

In the current practice, the operationalfunction of TSO is carried out in atwo-step temporal decoupled process(Part II, §4.3), namely:

a) Day-ahead operational planningi.e. power system (optimal)scheduling, after completion ofday – ahead electric markets andparticipants’ auto scheduling,and

b) Dispatching the PS, or near real– time system operation,including monitoring of trans-mission network.

Due to the uncertainty, the TSO coulddeepen unbundling of system operationin shorter time-frames, at intermediatemoments in time between day-aheadoperational scheduling and real-time. Afew hours ahead of real time, theuncertainty is decreasing, however,there is a need to run faster thesefunctions. Parallelization of compu-tations including the use of functionaland structural unbundling techniqueswould be required. Security analysesare performed for each candidatedispatch scheme in order identify powerlines that reach their thermal ratingsand estimate the power system steady– state and transient stability margins.Based on these analyses, the TSOdetermines the optimal preventive andcorrective actions needed to satisfy theoperational security criteria during aninterval (of temporal unbundling) forwhich the TSO assumes that changes


nu se schimb` semnificativ. OTSdesp`gube[te participan]ii la pia]`pentru orice tranzac]ie degradat` caurmare a respect`rii restric]iilor.

Puterea transferat` prin sec]iunilecritice ale re]elei se poate sc`dea prinrecurgerea la variate scenarii defunc]ionare, implicînd:

a) M`suri la produc`tor:modificarea puterii generate deanumite centrale sau în zone deimport [i de export.

b) M`suri în re]ea: solu]iile practicerecurg la folosirea dispozitivelor decompensare: condensatoare - serie;condensatoare - [unt; compen-satoare statice (SVC); controlulautomat al Autotransformatoa-relor cu Reglaj sub Sarcin` (ARS).

c)M s̀uri laconsumator:blocareacontroluluiARSpentruprevenireainstabilit̀ ]iidetensiune;desc̀ rcareadesarcin ;̀automatizareadistribuit̀ .

Limitele de transfer pot fi crescutec`tre limitele de capacitate termic`dac` se dispune de sisteme automatede control, care cresc marja destabilitate a sistemului.

Abordarea specific` a aspectelortehnice ale transportului,manageriate de OTS [i a aspectelorcomerciale, ce intereseaz` operatorulde pia]` (PX) [i participan]ii la pia]`,este eviden]iat` în Figura 3.5.

|n aceast` privin]`, actualaReglementare 1228 a UE privindmanagementul congestiilor nu poate fi

in the system state are insignificant.The TSO may reimburse the marketplayers for any degraded transaction asa result of constraint clearing.

The power transfer on the criticalinterties may be decreased by means ofvarious possible operating scenarios,implying:

a)Generation–basedcountermeasures:changingthegeneratedpoweratappropriatepowerstationsorintheimportingandtheexportingareas.

b) Transmission network-basedcountermeasures: the mostpractical solutions involve com-pensation devices: series capa-citors; shunt capacitor banks;SVCs; automatic control ofnetwork LTC autotransformers.

c) Load – based countermeasures:blocking of LTC to combatvoltage instability; load shedding;distribution automation.

The transfer limits could be increasedtowards the thermal capacity limit ifautomatic control systems wereavailable to increase the PS stabilitymargin.

Specificapproachof transmissiontechnicalaspects thataremanagedbytheTSO,andof transmissioncommercialaspects thatareof interest for thePowerExchange(PX)andelectricmarketplayersaswell isrenderedevident in theFigure3.5.

In this respect, the current EURegulation 1228 regarding congestionmanagement cannot be accepted


acceptat` f`r` rezerve, deoarece induceo interferare a operatorilor de pia]` înopera]iile OTS.

O separare clar` a responsabilit`]iloropera]ionale [i comerciale este esen]ial`pentru folosirea [i dezvoltarea eficient`[i în condi]ii de siguran]` a sistemuluide transport de energie electric .̀Capabilitatea opera]ional` de transfer însec]iuni de re]ea nu este acela[i lucru cucapacitatea de transport pentru pia]` dinalocarea implicit ,̀ [i la fel, restric]iile înfunc]ionarea SEE nu pot fi confundatecu congestiile de transport. DeoareceCTIR depind de starea particular` defunc]ionare a SEE [i, de regul ,̀interconexiunile interzonale sesuprapun unele cu altele, numai OTSpot în]elege [i controla fluxurile deputere prin re]ea.

Tabelul 3.1 prezint` sintetic aceast`distinc]ie, ca [i activit`]ile operatorului depia]` [i ale operatorului de sistem într-uncadru clasic de func]ionare.

unreservedly since it induces interfer-ence of market operators within TSOsoperations.

A clear separation of operational andcommercial responsibilities may beessential for efficient and reliable useand development of the powertransmission system. Operationaltransfer capability across networkpaths is not the same as tradabletransmission capacity in implicitallocation approach, and alike, thetransmission power system constraintsshould not be mingled withtransmission congestions. Since theTCRIs depend on the particular stateof the PS and, typically, inter-area in-terconnections overlap each other, onlythe TSOs can understand and controlthe power flows accross the network.

Table 3.1 shows synthetically thisdistinction, as well as activities ofmarket operator and power systemoperator in the classical framework.

Fig 3.5 Accesul la re]eaua de transport înmecanismul pie]ei [i în opera]iile OTS

Fig 3.5 Transmission access in marketmechanism and TSO’s operations


Tabelul 3.1 Reflectarea capacit`]ii de transport [i a restric]iilor SEEîn proiectarea pie]ei [i în siguran]a func]ion`rii sistemului

I. Capacitatea de transport ca problem` de proiectare a pie]ei

Sînt implica]i atît OTS cît [i consumatorii de transport.Modelul nodal de transfer al energiei, o paradigm` nou` a proiectului de pia]`:

- Tranzac]ii [i pre]uri nodale, atît pentru energie cît [i pentru serviciul de transport- Capacitate nodal` de transport: capabilitate net` de transfer la punctul de conectare(Driving-Point) (DP - NTC)

- DP - NTC este evaluat` [i postat` anual, lunar, cu o zi înainte- DP – NTC reflectat` în tranzac]iile de transport [i în cele de energie cu o zi înainte:

Tranzac]ii de transport DP-TC ca [i capacitate de transport este o prevedere atranzac]iei de transport:- Congestiile nu sînt vizibile la încheierea tranzac]iilor- OTS desp`gube[te clien]ii de transport pentru efectelenerespect`rii TC pe baza resurselor oferite de tarifele (nodale)de transport G [i L.

Pia]a de energie Operatorul de pia]` verific` ofertele de vînzare [i cump`rare,„cu o zi înainte” (PX) ajustîndu-le fa]` de limitele NTC ale re]elei (nodale),

pentru fiecare interval de tranzac]ionareSistemul de tranzac]ionare stabile[te pre]ul de închidere al pie]eiParticipan]ii la pia]` trimit confirm`rile de tranzac]ii, obiec]iile[i notific`rile fizice la P`r]ile Responsabile cu Echilibrarea, PX [iOTS

II. Capabilitatea de transfer inter - zonal` ca problem` de siguran]`opera]ional` a SEE

Sînt implicate numai OTS.Modelul de transfer surs` – receptor pentru controlul schimburilor de energie inter – zonale:

- Restric]iile opera]ionale ale SEE sînt reflectate de capabilitatea de transport „punct –la – punct” (PP –TC)

- Congestia, abatere de la DP-TC cauzat` de nerespectarea standardului PP-TC- Managementul (optim al) Congestiilor în activit`]ile opera]ionale ale sistemului:

Programarea OTS preg`te[te programul de generare – consum [i determin`opera]ional` disponibilitatea Serviciilor tehnice de Sistem (SS)a SEE Rezolvarea Congestiilor (RC) ca solu]ie de cost minim pe baza

resurselor din Pia]a de Echilibrare (PE) [i din pia]a de SSProgramarea coordonat` regional` cu sprijinul CMMO

Dispecerizarea în RC pe baza resurselor fizice din pie]ele PE [i SStimp real a SEE


Table 3.1 Transmission capacity and PS constraints as reflected in themarketdesign and system operational security

I. Transmission capacity as amarket design issue

Both the TSOs and transmission customers are involved.Nodal power transfer model, a new paradigm of market design:

- Node - related transactions and prices for both the energy and transmission service- Node – related transmission capacity: Driving – Point (or Point of Connection) Net powerTransfer Capability (DP - NTC)

- DP - NTC assessed and posted yearly, monthly, day – ahead- DP – NTC as used in transmission transactions and day – ahead electricitytransactions:

Transmission DP – TC as Transmission Capacity is a transmissiontransactions transaction provision:

- Congestions are not visible when transactions are concluded- TSO should reimburse transmission customers for the effects ofTC non-compliance based on resources from G and L (nodal)transmission rates

„Day – ahead” Market Operator verifies buy- and sell – offers, and adjustselectric market (PX) offers for the (nodal) NTC grid limits, for each trading interval

Transaction system establishes market clearing priceMarketparticipantssendtradeconfirmations,objectionsandphysicalnotifications to Balance Responsible Parties, PX and TSO

II. Inter – area transfer capability as a PS operational security issue

TSOs involved, only.Source to sink power transfer model for control of inter – area power exchanges:

- Operational PS security constraints as reflected by the „Point – to – Point” TransmissionCapability (PP –TC)

- Congestion as abatement from DP –TC due to PS security constraints- (Optimal) Congestion Management in PS Operational activities:

PS Operational TSO prepares Generation and Demand Schedule, determinesScheduling availability of ancillary System Services (SS)

Congestions Relief (CR) as least cost solution with physicalresources from the Balancing Market (BM) and SS marketRegional coordinated scheduling with assistance from theCMMO

Near real-time CR with physical resources from the BM and SS marketsPS Dispatch



[CIGRE2] CIGRE TF 38.02.11 – Indices Predicting Voltage Collapse Including DynamicPhenomena (1994)

[EWK] E.W. Kimbark, Power System Stability, John Wiley & Sons, New York, 1948; 1956

[FD-CC] F. P. DeMello, C. Concordia, Concepts of Synchronous Machine Stability as Affected byExcitation Control (IEEE Transactions, Vol. PAS – 88, April 1969)

[GenE] General Electric, Long-term Power System Dynamics (EPRI Research Project 90-7/1974,Electric utility engineering operation, Final Report)

[PD-JCO-MP-IR-IN] P. Dimo, J. Constantinescu, M. Pomârleanu, I. Radu, I. Nicolae,Determinarea comport`rii dinamice a sistemului energetic pe intervale maride timp, urmare a unor perturba]ii succesive în sistem ( Long-term PowerSystem Dynamics Simulation Following Cascade Disturbances in theSystem) (Energetica 10 - 11, 1980)

[PK] P. Kundur, Power System Stability and Control (McGraw – Hill, New York, 1994)

[PKO-JA-JW-JC] P.V. Kokotovic. J.J. Allemong. J.R. Winkelman, J.H. Chow, SingularPerturbation and Iterative Separation of Time Scales (Automatica 16,pp 23 – 33, 1980)

[VV] V. Venikov et al., Estimation of Electrical Power System Steady - State Stability, (IEEETransactions, Vol. PAS – 94, May / June 1975)

[JCO-CA] J. Constantinescu, C. Aldea, Sistem opera]ional pentru supravegherea în timp real asiguran]ei regimului SEN (An Operative System for Real-time Monitoring ofOperational Security of the National Power System) (Energetica 4B, 1995)

[JCO-MH3] J. Constantinescu, M. Homos, Metod` practic` de calcul pentru studiul stabilit`]iistatice a sistemelor electroenergetice complexe (A Practical Method for the Study ofSteady - state Stability of the Complex Power Systems) (Energetica 9, 1977)

[JCO2] J. Constantinescu, Midterm Transient Processes with Dimo - REI Models. Part II (Rev.Roum. des Sci. Techn., Vol. 30, No. 3, pages 337 - 344, 1985)

[JCO20] J. Constantinescu, Determination of the Real Net Transfer Capacities and CapacityReserves (CIGRE Paper C2 – 305, Session 2006)

[JCO24] J. Constantinescu, Midterm Transient Processes with Dimo - REI Models (Rev. Roum.des Sci. Techn., Vol. 28, No. 1, pages 45 - 60, 1983)

[JCO25] J. Constantinescu, Adjustable REI – Dimo Equivalents for Longer – term Power SystemDynamics Simulation (Energetica 6B, 1994)

[JCO26] J. Constantinescu, Practical Assessement of the Power System Stability Margins(Rev. Roum. des Sci. Techn., Vol. 39, No. 2, pages 217 - 222, 1994)


Epilog – Unmodel nodal completdezlegat pentru opera]iile[i serviciile energiei electrice

1 Remarci de încheiere asupraparadigmei dezleg`rii

Metodele de rezolvare prin dezlegaresimplific` modelele de calcul prinignorarea temporar` sau irevocabil` derela]ii între variabile de stare saufunc]ii de variabile (procese).Dezlegarea de variabile prezint`interes la etape de calcul, în timp cedezlegarea de procese ramîne efectiv`pîn` la ob]inerea solu]iei finale.

Avem dezlegare func]ional`,structural` sau varia]ional` dup` cuminterdependen]ele care se neglijeaz`sînt de intensitate redus`, întrestructuri diferite ale modelului, sauîntre variabile (procese) avînd vitezediferite de varia]ie în timp.Modeleleinteligente reprezint` de fapt o com-bina]ie optim` de dezleg`ri de procese[i de variabile, în func]ie de problemaconcret` ce trebuie rezolvat`.

De fapt, chiar apari]ia modelului decalcul poate fi considerat` ca prim`atestare a conceptului de dezlegare.Orice model este stabilit în baza unorpremize privitoare la relevan]a m`-rimilor [i interdependen]elorprocesului avut în vedere.

|n ceea ce prive[te func]ionareaSEE, avem urm`toarele exempletipice:a) Modelarea reparti]iei sarciniiactive [i reactive în re]ea [i între

Epilogue - A full unbundled nodalmodel of electricityoperations and services

1 Concluding remarks on theunbundling paradigm

The unbundling – based solution me-thods simplify computation models bytemporary or irrevocably ignoring rela-tionships among the state variables orfunctions of variables (processes). Thevariable unbundling is of interest atcomputation steps while processunbundling does remain effective up toreaching the final solution.

One can distinguish functional,structural or variational unbundling asthe neglected interdepencies are of lowintensity, between different modelstructures, or between variables(processes) of different speeds oftemporal variation. The smart modelsrepresent an optimal combination ofprocess and variable unbundling,depending on the particular issue thathas to be solved.

Actually, the very birth of the com-putation model can be taken as thefirst attestation of unbundling concept.Any model is derived on the base ofcertain prerequisites concerning therelevance of quantities and interde-pencies of the considered process.

As far as the PS operation isconcerned, the following typicalexamples can be given:a) Modeling of the active andreactive power distribution


unit`]ile generatoare: neglijareainterac]iunii (reduse), fie dintreamplitudinea tensiunilor [iputerile active, fie dintreunghiurile tensiunilor [i puterilereactive, [i

b)Modelarea regimurilor tranzitoriide tip electro-mecanic ale SEE, determen scurt,mediu [i lung:neglijarea, fie a oscila]iilor lente aleechipamentului termo- [i hidro-mecanic al grupurilor energetice, înfavoareamodel̀ rii proceselorrapide din înf̀ [ur`rile generatoa-relor [i sistemelor de excita]ie, fie aoscila]iilor rapide dintrema[inilesincrone, în favoarea reprezent`riiîn detaliu a dinamicii instala]iilortermo- [i hidro-mecanice dincentralele electrice, ca [i aimpactului varia]iei în limite largia tensiunilor [i frecven]ei.

Pîn` acum, dezlegarea a fost practicat`sub formametodelor euristice „dedecuplare“ (i.e. dezlegare func]ional` adou` seturi de variabile sau func]ii devariabile), în scopul reduceriivolumului de calcule [i/sau cre[terii[ansei de a ob]ine o solu]ie. GabrielKron, prin abordarea sa „diakoptic`“(i.e. dezlegare structural` de variabile),a deschis o direc]ie revolu]ionar`, careconst` în descompunerea unei re]ele înp`r]i convenabile, urmat` derecombinarea solu]iilor individuale alep`r]ilor pentru a se ob]ine solu]iageneral`. Modelele REI-Dimoreprezint` un caz particular dedezlegare structural`, în care nodurileREI sînt delimitate prin frontiere cutensiuni controlate („imprimate“).

within the network and amongthe generating units: the neglectof (low) interaction, either bet-ween voltagemagnitude and ac-tive powers, or between voltageangles and reactive powers, and

a) Modeling of the short-, medium-and long-term PS electro-mechanical dynamics: theneglect of either slow oscillationsof the thermal- and hydro-mechanical equipment of powerunits in the benefit of modelingthe fast processes within thegenerators’ windings and controlsystems, or of the fast oscillationsamong synchronous machines, infavour of detailed representationof the thermal and hydroinstalations as well as theimpacts of large voltage andsystem frequency excursions.

So far, the unbundling was employedas decoupled euristic methods (i.e.func]ional unbundling of two variablesets or processes), with the aim to re-ducing the computation volume and/orincreasing the chance of reaching asolution. Gabriel Kron, with his„diakoptics“ approach (i.e. a structuralvariable decoupling) opened arevolutionary direction, which consistsof decomposition of a network inconvenient pieces that is followed by aremix of the individual partialsolutions so as to get the overallsolution. The REI – Dimomodelsrepresent a particular case ofstructural unbundling, in which theREI nodes are delimited by the borderswith controlled („impressed“) voltages.


In the electric supply system, thecurrent trends privilege the marketmechanism – based solutions, in thedetriment of classical optimizationmethods. For instance, the real - timeoptimization of generation powerdispatch, a standard procedure of theold EMS / SCADA systems, is nowreplaced by the merit order from theelectricity markets, day – ahead orintraday. Under this context, it isnatural that the unbundling methodsmigrate from the models of physicalsystem (technology) to the aggregated„technology – customer“ system.All at once with simpler issues thereare increased responsibility andincentives of the operators ofunbundled services for improvedefficiency.

In the functionally unbundled„technology – customer“ system ofelectricity, the TSO’s service has to beimpartial, responsible and regulated.Impartiality of this service guaranteeswider access to electricity market.Responsibilities are determined by thewell defined security and qualityoperational standards while thenecessity of regulation arises from theservice’s natural monopoly nature.

The power transmission is a multipleadded-value service that includes thecompensation of network losses and oftransmission congestion, as well asthe access to market service. In ourview, a market opportunity fee withinthe transmission rate should supportthe acces to market service[Part I, §4.2.5].

|n sistemul energiei electrice,tendin]ele actuale privilegiaz`rezolv`rile prin mecanisme de pia]`, îndauna unor metode clasice deoptimizare. De pild`, optimizarea întimp real a reparti]iei sarcinii pegrupurile generatoare, o procedur`standard a vechilor sisteme deconducere prin dispecer(EMS/SCADA), este acum înlocuit` cuordinea de merit din pie]ele de energie,cu o zi-înainte sau intrazilnice. |n acestcontext, este firesc ca metodeledezleg`rii s` fie extinse de la modelelesistemului fizic (ale tehnologiei) spreansamblul „tehnologie-client“. Odat` cusimplificarea problemelor, cre[te [iresponsabilitatea [imotiva]ia opera-torilor serviciilor dezlegate pentruîmbun`t`]irea eficien]ei.

|n sistemul dezlegat func]ional„tehnologie – client“ al energiei electrice,serviciul OTS-ului trebuie s` fieimpar]ial, responsabil [i reglementat.Impar]ialitatea acestui serviciu garan-teaz` accesul nerestric]ionat la pia]a deenergie electric .̀ Responsabilit`]ile sîntdeterminate de standardelor opera-]ionale de siguran]` [i calitate, în timpce cerin]a de reglementare deriv` dinnatura demonopol natural a serviciului.

Transportul de energie este un serviciucu valoare adaugat`multipl`, careinclude compensarea pierderilor în re-]ea, compensarea congestiilor de trans-port [i, de asemenea, serviciul de accesla pia]a de energie. Serviciul de acces lapia]` trebuie sus]inut de o component`tarifar` specific`, pe care am numit-otax` de oportunitate [Partea I, §4.2.5].


2 Proiectul nodal de pia]` deenergie electric`

Paradigma nodal` introdus` de PaulDimo poate fi extins` [i la sistemul„tehnologie – client“ al energieielectrice. Energia, serviciul detransport [i capacitatea de transport sepot reglementa [i tranzac]ionata cureferire la un nod al re]elei.Tranzac]iile cu energie electric` au caefect transferuri nodale de putere întrere]ea [i clien]ii acesteia. Serviciul detransport este în esen]` un serviciu deacces la re]ea cu pre] [i cost definitecomplet la nivelul nodului de re]ea.

Tariful nodal de transport, cu compo-nentele de cost marginal G, la punctualde „intrare“ în re]ea, [i L la cel de„ie[ire“, la care trebuie ad`ugat` com-ponenta de oportunitate, este alocatechitabil pe clien]ii re]elei si transmiteacestora semnale loca]ionale relevante.

Costurile marginale de transport se potstabili prin simularea regimului optimsau pe baza pre]urilor de pia]` aleenergiei. |n a doua op]iune, OTS tre-buie s` evalueze coeficien]ii de pierderi[i de congestie, [i pre]urile pe pie]elecentralizate de energie, inclusiv deechilibrare, pentru fiecare interval detranzac]ionare al fiec`rei zile sau alzilelor tipice dintr-un an.

Clien]ii re]elei vor pl`ti componenteleG [i L direct OTS, în timp ce alocarea[i plata componentei de oportunitatepoate fi asigurat` chiar de c`tre plat-forma de tranzac]ionare a energiei.|ntr-un cadru de reglementare

2 Nodal electricity marketdesign

The nodal paradigm that was intro-duced by Paul Dimo can be extended tothe „technology - customer“ system ofelectricity, too. Electricity, transmissionservice and transmission capacity aswell can be regulated and traded withreference to a network node. Electricitytransactions lead to nodal powertransfers between the network and itscustomers. The transmission service isessentially an access to network servicewith the rate and cost fully defined atthe level of network node.

The nodal transmission rate with itsmarginal cost components, G in thenetwork’s „entry“ point, and L in the„exit“ one, to which may be added theopportunity component, is fairly allo-cated on the grid customers and sendsthem relevant locational signals.

Transmission marginal costs can beestablished through simulation of theoptimal regime or on the base ofmarket prices of electricity. In thesecond option, the TSO should assessthe losses and congestion coefficients,and prices on the centralized marketsincluding that of balance energy, ofevery trading interval, for every day orfor typical days within a year.

Grid customerswould pay theGs andLsdirectly to the TSO,while the allocationand payment of opportunity feeswouldbe ensured by the very tradingmechanism of electricity. In amulti-TSOregulatory environment that is harmo-


nized, theGs andLs can be collected bytheTSOs in the exporting and importingcountry, respectively. An Inter-TSOCompensation (ITC)mechanism redis-tributes payments by countries in theregion based on the cumulative effect oftransits, including that of CB congestion.

An operational security standard canbe defined for every nodal elementarymarket as well: this is the DP powertransfer capability at the node.Consequently, the electricity can besold all at once with release oftransmission capacity. In this view, thecongestion is any limitation ontransmission transaction at the accesspoint to transmission system, andreflects all PS operational constraints.

Compensation for the effect ofcongestion in the market is theresponsibility of the TSO.

In the national context, tipically, theTSO reimburses the market playersfor any degraded transaction as aresult of CM.

TheTSO ismobilizing resources from theancillarySystemServices andBalancingMarkets. TheTSObuys on thesemarkets thenecessary balance energyand optimizes injection changes thatrelief operational constraint. Trans-mission-rate-based income should coverthe extra cost of generation redispatch.TheTSOresponsibility should remainunchanged in amulti-TSOharmonizedenvironment aswell. EachTSOmayreimburse its own customers for anychange inmarket revenue due to theCM.

multi-OTS, armonizat, G [i L secolecteaz` de c`tre OTS din ]ara im-portatoare [i respectiv, exportatoare.Un mecanism de compensare inter-OTS redistribuie pl`]ile în func]ie deefectul cumulat al tranzitelor, inclusival congestiilor TF.

Pentru fiecare din pie]ele elementarenodale se poate defini [i un standardde siguran]` opera]ional`; acesta estecapabilitatea DP de transfer a energieiîn nod. Prin urmare, energia se poatevinde odat` cu eliberarea de capacitatede transport. |n aceast` viziune, con-gestia este orice limitare de tranzac]iede transport la punctul de acces însistemul de transport [i reflect` toaterestric]iile opera]ionale ale SEE.

Compensarea efectului congestieiîn pia]` este în responsabilitateaOTS.

|n context na]ional, de regul`, OTSdesp`gube[te participan]ii la pia]`pentru orice tranzac]ie degradat` caurmare aMCO.

Acesta mobilizeaz` resurse din Pie]elede Servicii de Sistem [i de Echilibrare[i optimizeaz` modific`rile de injec]iinodale pentru îndep`rtarea restric]ieiopera]ionale. Venitul din tarifele detransport acoper` costul suplimentaral re-dispeceriz`rii gener`rii. Respon-sabilitatea OTS trebuie s` r`mîn`neschimbat` [i în context regionalarmonizat. Fiecare OTS va desp`gubiproprii clien]i afecta]i pentru oriceschimbare a venitului din pia]` caurmare aMCO.


Predictibilitatea deplin` a capacit`]ii lapunctul de racord va fi pe bun` drep-tate tr`s`tura definitorie a viitoarelorre]ele inteligente (Smart Grids).

Controlul fluxurilor de putere prinre]ea [i managementul restric]iilorSEE, inclusiv a capabilit`]ii de transfera interconexiunilor inter – zonale,pentru operarea în siguran]` asistemului, sînt, de asemenea, înresponsabilitatea TSO.

OTS trebuie s` se asigure c` nivelulde siguran]` al sistemului respect`criteriul „N-1“ [i, în anumite situa]iide planificare opera]ional`, criteriulsuplimentar „N-2“. St`rile defunc]ionare ale SEE sînt în generalverificate pe baza criteriilor (marjelor)de stabilitate, static` [i tranzitorie,sau de limite termice pentruechipament.

Capabilitatea de Transfer pentruInterconexiunea inter-zonal`Relevant`, în]eleas` numai de c`treOTS, nu poate fi tranzac]ionat` înpia]` drept capacitate de transport.CTIR se pot cre[te prin recurgerea lavariate scenarii de func]ionare,implicînd m`suri la produc`tori, înre]ea [i la consumatori.

Ac]iunile preventive [i corective ale OTSpentru satisfacera criteriilor desiguran]` opera]ional ,̀ se realizeaz` deregul` printr-un proces în dou` etape:a) Programarea sistemului, dup`închiderea pie]ei de energie cu ozi-înainte, [i

Full predictability of capacity at thepoint of network connection wouldactually be the defining feature of thefuture SmartGrids.

The control of power flows across thenetwork and the PS constraintmanagement including that of transfercapability of inter-area interconnec-tions, with the purpose of ensuring thesystem’s operational security, are TSOresponsibilities as well.

The TSO has to insure himself that theobserved level of security complies withthe „N-1“ security criterion as well aswith the additional criterion of takinginto account certain „N-2“ situations inoperational planning studies. PSoperating states are generally checkedas against the specific steady-state andtransient stability criteria (margins)and the equipment thermal ratings.

The Transfer Capability of Relevantinter-area Interconnection that ismeaningful for the TSO only, cannot betraded in the market as transmissioncapacity. The TCRI may be increasedby means of various possible operatingscenarios, implying generation-,transmission network- and load- basedcountermeasures.

The TSO’s preventive and correctiveactions with the aim to satisfying theoperational security criteria, are typi-cally carried out in a two-step process:a) Power system scheduling, aftercompletion of day – aheadelectric market, and


b) Dispatching the PS, or nearreal-time system operation.

Co-ordination of regionalscheduling and dispatching couldbe carried out by a CongestionManagement and Monitoring Office(CMMO).

From the foregoing we conclude thatthe future power system should beseen in a larger than pure technicalperspective, in the sense of mitigatingphysical system security requirementswith those of the electric market. Themarket design may include properassign of the resources and incentivesfor the TSOs to invest in the network,and of the roles market and transmis-sion operators in the management ofelectric supply system. Certainly, themarket integration and flexibility thatare currently rather deficient should beincreased, however not by substitutingTSOs for the market operators in theoperation of power interconnectors,because this can jeopartize the PSoperational security.

The market design that is based on thenodal access to the network carries outa clear separation of operational andcommercial servicies and offers theright responsibility to the TSO.

3 Gains of the nodal – structuredelectricity market

On the basis ofmarketmechanism thatwas described in Part I, §4.3, and oftransmission capacitymodel aswell(Part II, §4.2 here above), it may be

b) Dispecerizarea SEE, sau opera-rea în timp real a sistemului.

Coordonareaprogram`rii [i dispeceriz`riiregionale ar putea fi realizat̀ de c t̀reunOficiudemanagement [imonitorizare acongestiilor (CongestionManagementandMonitoringOffice -CMMO)

Din cele ar`tate mai sus conchidem c`viitorul sistem de energie electric`trebuie v`zut într-o perspectiv` mailarg` decît cea pur tehnic`, în sensulconcilierii cerin]elor securit`]iisistemului fizic cu acelea ale pie]ei deenergie. Proiectul de pia]` trebuie s`includ` desemnarea adecvat` aresurselor [i motiva]iilor pentru OTSpentru a investi în re]ea, [i a roluriloroperatorilor de pia]` [i de transport înmanagementul sistemului energieielectrice. Trebuie într-adev`r crescuteintegrarea [i flexibilitatea pie]ei,deficitare în prezent, dar nu prinsubstitirea OTS de c`tre operatorii depia]` în operarea interconexiunilorelectrice, deoarece se poate periclitafunc]ionarea în siguran]` a SEE.

Proiectul de pia]` bazat pe accesulnodal la re]ea realizeaz` o separareclar` a serviciilor opera]ionale [icomerciale, [i ofer` OTSresponsabilitatea necesar`.

3 Foloasele pie]ei de energiecu structur` nodal`

|n baza mecanismului de pia]` descrisîn Partea I, §4.3 [i a modelului nodalde capacitate de transport (Partea II,§4.2) putem conchide c` sistemul de


alimentare cu energie electric` se poatepe deplin dezlega în pie]e elementarenodale. Tranzac]iile, pre]urile [icapacitatea infrastructurii pie]ei deenergie electric` au natur` „nodal`“.|n aceast` interpretare, congestia estelimitarea capacit`]ii contractuale detransport (nodale), detectate înprogramarea opera]ional` cu o ziînainte sau pe durata dispeceriz`rii.

Modelul nodal de acces la re]ea vagenera urm`toarele avantajeimportante:- Tranzac]iile migreaz` c`tre pie]elecentralizate, în beneficiulintegr`rii [i lichidit`]ii pie]ei,pentru c` energia se vinde odat`cu capacitatea de transport nece-sar`.- Se reduc costurile: pe termenscurt, datorit` îmbun`t`]iriiprogram`rii sistemului; pe termenlung, datorit` mai buneiamplas`ri a centralelor.- Participan]ii la pia]` pot s`-[ioptimizeze mai u[or portofoliul depia]` deoarece capacitatea nodal`de transport este inteligibil` [imai predictibil`, oglindindcapacit`]ile comerciale proprii.- OTS devine mai interesat s`aplice proceduri de cost minim îneliminarea congestiilor deoareceva purta întreaga responsabilitatepentru compensarea efectulcongestiilor în pia]`.- Se u[ureaz` managementulriscurilor pentru OTS datorit`coordon`rii calculului capacit`]iire]elei cu tranzac]iile de transport[i cu procedurile opera]ionale.

gather that electricity supply system canbe fully unbundled in elementary nodalmarkets. Transactions, prices, andinfrastructure capacity of an electricitymarket are essentially „nodal“ in nature.In this interpretation, congestion sig-nifies any limitation of (nodal) contrac-tual capacity for transmission that areidentified in day-ahead operationalscheduling or during power dispatch.

Particularly, the nodal model of accessto the network would yield thefollowing significant advantages:- Transactions would migrate tocentralized markets thusincreasing market integration andliquidity since electricity is sold allat once with release of networkcapacity.- The costs will be reduced: in theshort run due to improvedscheduling of generation and in-terconnectors’ flows; in the longerrun, due to better plant locations.- Market players can more easilyoptimize their market portfoliobecause the nodal transmissioncapacity is meaningful and morepredictable, mirroring their owntradable capacity.- The TSOwould be more interes-ted to apply least-cost congestionmanagement procedures since hewill bear full responsibility forcompensation of congestion effectin the market.- The TSO’s management of riskswould be more effective due tocoordination of capacitycalculation with transmissiontransactions and operations.


- Transmission capacity‘pancaking’, with congestionsmoved to the borders, woulddisappear because TSOs onlywould be responsible for powerflows across interconnections.- The investor confidence andconsumer support for themarketswould increase against a back-ground of reduced temptation for po-litical and regulatory interventions.-Nodal transmission rateswould giveadequate resources for theTSOs toincreasenetwork capacitywhiletheir integrationwithanewlydesigned ITCschemewouldovercome investmentproblemsofinterconnectors.- There will be enhancedmarket in-tegration of intermittent RESs thatare critically dependent on shortnotice access to regionalmarkets.

Therefore, the nodal – structuredelectricity market can enhanceflexibility, reliable use anddevelopment of the power network,and market-efficient integration aswell, owing to consistency of marketand power system security procedures.The mechanism of nodal electricitymarket discovers the real marketprices and transactions volumes.

We should recognize, however, thatimplementation of the nodal marketdesign that impact on revenue streamswould be a significant reform in the EUelectricity regulatory system. Conse-quently, the reform should be agreedby all Member States, the followingprinciples particularly:

- Se evit` pl`]ile redundante pentrucapacitatea de transport, prindeplasarea restric]iilor la grani]e(‚pancaking’ effect), deoarecenumai OTS vor fi responsabile decircula]iile prin interconexiuni.- Spore[te încrederea investitorilor[i sprijinul consumatorilor fa]` depia]` pe fondul reducerii tenta]ieipentru interven]ii politice sau prinreglement`ri.- Tarifele nodale de transport voroferi OTS resurse adecvate pentrudezvoltarea re]elei, iar integrareaacestora cu un noumecanism ITCva rezolva problemele privindinvesti]iile în interconexiuni.- Se intensific` integrarea în pia]` aresurselor regenerabileintermitente, dependente deaccesul rapid la pie]ele regionale.

A[adar, pia]a de energie electric`structurat` nodal intensific`flexibilitatea, siguran]a func]ion`rii [idezvoltarea re]elei, precum [iintegrarea pie]ei, datorit` congruen]eiprocedurilor comerciale [i de siguran]òpera]ional`. Mecanismul pie]ei nodalede energie electric` descoperàdev`ratele pre]uri de pia]` [i volumede tranzac]ii.

Trebuie îns` recunoscut faptul cìmplementarea modelului nodal depia]` de energie, influen]înd fluxul devenituri, ar fi o reform` semnificativ` asistemului de reglementari al UE. Deaceea, toate Statele Membre trebuie sàccepte aceast` reform`, în particularurm`toarele principii:


a) Compensarea efectuluicongestiei în pia]` înresponsabilitatea unic` a OTS;aceasta nu este în prezent oregul` general` în UE.

b) Tratament nediferen]iat alcongestiilor în comer]ul intern [itransfrontier.

a) Compensation for the effect ofcongestion in the market as aresponsibility of TSO only;currently, this is not a generalrule within EU.

a) No different treatment ofcongestions in domestic andcross-border trade.


Anexa 1

Rezolvarea numeric` a sistemelorde ecua]ii lineare cumatricelacunar` de coeficien]i

1. Memorareamatricelorlacunare în calculator

Matricele de coeficien]i ale ecua]ilordin modelele SEE sînt de ordin mare,dar au un num`r redus de elementenenule pe un rînd, în general dou` pîn`la 10, adic` sînt lacunare. A[adar, omatrice lacunar` con]ine în principalelemente nule. Termenul a fot introdusde c`tre HarryM. Markowitz(Wikipedia).

Omatrice lacunar`A se memoreaz` [ise prelucreaz` în form` comprimat`:[irul de coordonate vectorialeVAdiferite de zero [i listele de indici destructur` IS, necesari pentruidentificarea elementelor luiVA. Prinevitarea opera]iilor aritmetice cuelemente nule ale matricei se ob]ineconomii însemnate de calcul [i despa]iu dememorare în calculator. Dac`A este simetric`, trebuie memorateelementele sub- sau supradiagonale.Mai este de dorit ca indicii IS s`localizeze informa]ia direct, s` fie înnum`rminim [i u[or adaptabili la oricemodificare de structur` amatricei, peparcursul procesului de calcul.

Pot fi folosite diferite structuriparticulare de date. Vom prezenta încontinuare (a) lista IS înl`n]uit`, [i (b)lista dezvoltat` [i aplicat` cu pre-dilec]ie de c`tre autor.

Annex 1

Numerical solution of the setsof linear equations with sparsecoefficient matrix

1. Storage of sparce (coefficient)matrices in the computer

The coefficient matrices in theequations of PSmodels are of highorder, however with a reduce numberof elements, two to ten non – zeroentries on a row generlly, that is, theyare sparse. Hence, a sparse matrix is amatrix populated primarily with zeros.The term itself was coined by HarryM.Markowitz (Wikipedia).

A sparsematrix A is stored andprocessed in a compressed form: the(coordinate) vector array of non – zeroentriesVA and the lists of structureindices IS that are used for identificationof theseVA entries. Avoiding arithmeticoperationswith zero - value entries ofthematrix, significant savings incomputation and data storage can beobtained. For a symmetricalmatrixA,either lower-left or upper-rightnon-diagonal entries are generallystored. It is also recommended that ISindices identify information directly, andbe in aminimumnumber and easilyadjusted to any change of the structureofAmatrix during computation process.

Different particular data structurescan be used. We will present bellow (a)a chained IS list, and (b) the list ofindices that was developed andfrequently used by the author.


a. Lista de indici înl`n]uite

Fiecare element nenul din VA esteasociat cu doi indici: (i) - al rînduluidin matricea A [i (ij) - al urm`toruluitriplet (i, aij, ij) de pe aceea[i coloan`(ij>0, dac` aij este ultimul element ne-nul de pe coloan`). Procedeul permiteadaptarea cu u[urin]` a listei lamodificarea structurii matricei decoeficien]i: pentru inserarea unuielement nenul, se introduce un noutriplet (Figura A.1) [i se modific` unsingur indice (ij).

Fig A.1 Liste de indici înl`n]uite

Asem`n`tor, se pot refolosi loca]iile cedevin disponibile la anumite etape aleprelucr`rii. O variant` alternativ` aacestui tip de IS se ob]ine plasînd

a. Chained list of indices

Every non-zero entry of VA isassociated to two indices: (i) - of therow in A and (ij) – of the next triplet(i, aij, ij) on the same column (ij>0, ifaij is the bound non-zero element ofthe column). The procedure allowseasy adjust of the list to any changeof the structure of coefficient matrix:when a non – zero entry is inserted,a new triplet is introduced (FigureA.1) and only one (ij)-th index ischanged.

Fig A.1 Chained lists of indices

Likewise, locations that becomeavailable at certain computation stepscan be re-used. An alternative ISoption is reached by arranging in


succession multiple entries on thesame column of A; the resultedsequence can be addressed by an only(ij) – th index. In this way, storagespace is saved, however the scheme isless flexible to further changes.

b. List of column indices and a listof indices addressing positions ofrows’ starting elements

TheentriesofVAare inone–to–onecorrespondencewith theentriesof IC, a listcolumnindices.Asecond list ILdesignatespositionsof rows’ startingentries inVAand ICaswell.Whentherowsofmatrixareprocessed inan„optimal“order,anadditional listNL indicates thisorder inthe list IL (FigureA.2).

Fig A.2 Author’s preferred list of indices

consecutiv mai multe elemente de peaceea[i coloan` a luiA; secven]a astfelob]inut` poate fi referit` printr-unsingur indice (ij). Se realizeaz` o econo-mie de spa]iu dememorare, dar schemaestemai pu]in flexibil` lamodific`ri.

b. List` de indici de coloan` [i list`de indici adresînd pozi]iileelementelor de început de rînduri

Elementele VA sînt în coresponden]`direct` cu elementele listei IC, de indicide coloan`. O a doua list` IL desem-neaz` începuturile de rînduri în VA [ide asemenea IC. Dac` rîndurile matri-cei se prelucreaz` în ordine „optimal`“,listei IL i se asociaz` o nou` list`NL,care indic` aceast` ordine (Figura A.2).

Fig A.2 Lista de indici preferat` a autorului


Volumul de indici este minim [i per-forman]ele de calcul sînt maxime dac`:- Elementele matricei sînt ordonatepe fiecare rînd; dac` matricea estenesimetric`, se mai introduce olist` de indici, care indic` pozi]iaîn rînd a elementelor diagonale- Accesul la elementele matricei seface pe rînduri- Structura matricei nu se modific`pe durata calculelor (pentruevitarea transla]iilor în VA [i IC).

Primele dou` condi]ii se realizeazù[or prin logica de programare. Pentruîndeplinirea celei de a treia, sedetermin` structura final` a matricei,într-o etap` preg`titoare. De pild`,atunci cînd se rezolv` un sistem deecua]ii prin factorizare optimal` (a sevedea mai jos), se simuleazèlimin`rile Gauss în ordine optim`(factorizare simbolic`), numai pe bazalistelor de indici reflectînd structuramatricei, în vederea identific`riiapari]iei elementelor noi. Elementelenoi sînt elementele matricei care ini]ialerau nule devenind diferite de zero înprocesul de factorizare. Regula este:dac` exist` perechile de indici ik [i jk,atunci apare ij nou, dac` nu exist`deja. Pentru evitarea cùt`rilor înlistele de indici, se poate folosi,temporar, un sistem de indexare maiamplu.

Thevolumeof indices isminimumandthecomputationperformance ismaximumif:- Thematrix entries are placed inorder on the every row; if thematixis unsymmetrical, an additional listof indices indicates the rowpositions of diagonal entries- Access to matrix elements is doneon the rows- Thematrix structure remainsunchanged during computations (toavoid translations inVA and IC).

The first two requirements can besimplymet through the programminglogic. To fulfil the third one, the finalmatrix structure is determined, in apreparatory step. For instance, whenthe set of equations is solved throughoptimal factorization (as shown bellow),the Gaussian eliminations in optimalorder are simulated (symbolicfactorization), on the basis of indiceslists reflecting the matrix structureonly, with a view to identifying thefill - in. The fill-in are those matrixentries which change from an initialzero to a non-zero value during theexecution of factorization. The rule is: ifthere existes the pair of indices ik andjk if the pair of indices ik and jk exists.A more developed system of indices canbe used temporarily for faster lookupwithin the list of indices.


2. Rezolvarea prin factorizaretriunghiular` ordonatòptim a sistemului linearAx > b

Procedeul de eliminare (factorizaretriunghiular`) Gauss constituie meto-da cu cea mai larg` aplicabilitate înrezolvarea sistemelor de ecua]ii linearecu matrice lacunar` de coeficien]i, saude reducere a dimensiunii acestora.Eficien]a programelor pentru simu-larea regimurilor SEE depinde, înprimul rînd, de performan]a rutinelorlor de „factorizare optim`“.

Cu algoritmul clasic de factorizare,solu]ia sistemului Ax > b se ob]ine întrei etape:- Factorizarea matricei A într-unprodus de dou` matrice: inferior,respectiv superior triunghiulare(A > LU)- Rezolvarea sistemului Lw > b, nu-mit` [i substitu]ia „înainte“- Rezolvarea sistemuluiUx > w, sausubstitu]ia „înapoi“.

Pentru omatriceA simetric ,̀L [iUdifer` numai prin elementele diagonale.|n calculator sememoreaz` elementeleluiU [i diagonala luiL. DacÀ estenesimetric ,̀ iar sistemul nu se rezolv`repetat, pentru b diferi]i, primele douètape pot fie combinate în vedereaeconomisirii spa]iului dememoraredestinat luiL. Este cazulmetodeiNewton-Raphson de determinare aregimurilor permanente. Substitu]ia„înainte“, descris` schematic în figuraA.3, se poate efectua fie în varianta a), fieîn varianta b), [i are ca rezultatU [iw.

2. Solution of the linearset Ax > b by optimallyordered triangularfactorization

The Gaussian elimination (triangularfactorization) procedure constitutesthe most applicable method forsolving the sets of linear equationswith sparce matrix of coefficients, orfor reduction of their size. Theefficiency of computer codes for thesimulation of PS regimes depends,firstly, on the efficiency of their„optimal factorization“ routines.

With the classical factorizationalgorithm, the solution of the set Ax > bis reached in the following three steps:- Factorization of the matrix A as aproduct of two matrices: inferiorand superior triangular,respectively (A > LU)-Solution of the system Lw > b, orthe „forward“ substitution-Solution of the systemUx > w, orthe „backward“ substitution.

For a symmetricalmatrixA, theL andUdiffer fromone another throughdiagonalentries only. In the computer the entriesofU and the diagonal ofLare stored. IfAis unsymmetrical, and the set is notsolved repeatedly, for different b, the firsttwo steps could bemerged in viewofsaving the storage space for theL. It isthe case ofNewton-Raphsonmethod fordetermination of the network load flows.The „forward“ substitution, schematicallydescribed in the figureA.3, can be carriedout in one of the options a) or b), andleads toU andw.


Fig A.3 Dou` procedee de factorizaretriunghiular`

Varianta a) prezint` avantajul c`numai un rînd k din A coexist` înmemoria calculatorului cuU în forma-re. Acest mod de conducere a calcululuia contribuit în mare m`sur` lasuccesul metodei Newton-Raphson.Dac` A este nesimetric`, [i sistemulse rezolva repetat, pentru diferi]i b,se memoreaz` atît L, c\t [iU.

Nu vommai insista aici asupraopera]iilor de factorizare propriu-zise,bine-cunoscute. Subliniem c` numaisubstitu]ia „înainte“ genereaz`elemente noi în matrice. Pentruminimizarea num`rului acestora, se

Fig A.3 Two procedures of triangularfactorization)

The advantage of option a) is the factthat an only row k of A co-existes in thecomputer memory with the emergingU. This way of processing didessentially contribute to the success ofNewton – Raphsonmethod. If A isunsymmetrical, and the set isrepeatedly solved, for different b, boththe L andU should be stored.

We will not insist here on the detailedfactorizing operations that are well –known. We should notice that only the„forward“ substitution generates newmatrix entries. In view of minimizingtheir number, a symbolic factorization


simuleaz` \n prealabil factorizareasimbolic`, stabilindu-se ordinea optim`de prelucrare a ecua]iilor. Odat` cuordonarea, se identific` elementelematriceale noi [i se indexeaz` loca]iileacestora. Factorizarea este optim`atunci c\nd în final se ajunge lanum`rul minim ra]ional de elementenoi, f`r` a fi afectat` semnificativprecizia numeric` (la fiecare pas,elementul diagonal „pivot“ trebuie s`fie mai mare decît un prag). Lamatricele mari, solu]ia ideal` ar cereun num`r inacceptabil de mare deinspect`ri. S-a dovedit c` un compromisavantajos îl realizeaz` criteriul propusde c`tre Tinney [iWalker [WT-JW], aldeciziei „optimului limitat“: la fiecarepas de factorizare, se alege ecua]ia(rîndul dinmatrice) avînd cel mai micnum`r de elemente nenule noi. Pentruun spor de eficien]`, noi am introdus ocerin]` adi]ional`, [i anume, num`rulminim de elemente noi carese genereaz` la fiecare pas.

Optimalitatea factoriz`rii LU cap`t` onou` semnifica]ie în modelele SEEdezlegate structural (Partea II). Astfel,un obiectiv al dezleg`rii structurale (cua sa matrice bloc-diagonal` decoeficien]i asociat` unor subre]ele)poate fi [i num`rul minim de elementenenule generate de elimin`rile de tipGauss. Pe ansamblu, acesta ar trebuis` fie semnificativ mai mic decît încazul în care sistemul se rezolv` caîntreg. |n acest sens, prezint` interesidentificarea de subre]eleleinterconectate printr-un num`r cît mairedus de elemente.

139

is simulated in advance so as toestablish the optimal order in whichequations should be processed. Duringthe ordering, the fillin is identified andassociated storage locations areestablished, as well. Optimalfactorization should lead finally to arational minimum number of matrixentries without deteriorating thenumerical accuracy (at each step, the„pivotal“ diagonal entry should belarger than a threshold). For the largematrices, the ideal solution wouldrequire an unacceptable large numberof inspections. An advantageouscompromise proved to be met by thecriterion of „limited optimum” decisionproposed by Tinney andWalker[WT-JW]: at every factorization stepthe equation (matrix row) that has theleast number of non – zero entries ischosen. To increase efficiency, we haveintroduced as additionalrequirement the minimum number ofnew entries that are generated at eachstep.

For the structural unbundledmodels ofPS (Part II), optimal LU factorizationcan acquire a new significance. Thus,an objective of the structuralunbundling (with its bloc-diagonalcoefficient matrice and adjacent sub-networks) could also be the minimumnumber of non-zero entries generatedby the Gaussian eliminations. On thewhole, this number should besignificantly less than in case when theset is solved as an integer. In thissense, identification of subnetworksthat are interconnected by a minimumnumber of elements is of interest.


3. Reducerea de tipul Gauss aunui sistem de ecua]ii lineareAx > b

Metoda permite ob]inerea unui sistemde ecua]ii echivalent de dimensiunimai mici (cu mai pu]ine variabile) [ist` la baza marii majorit`]i amodelelor echivalente de SEE.

Eliminarea de variabile de tipul Gaussse efectueaz` cu un algoritm pas-cu-pas.La pasul k, variabilele xk-1 se elimin` dinecua]iilem (m>k, k+1,...,n) rezultînd unsistem echivalent al c`rui ordin esteredus cu o unitate.

|npractic` s-auconsacraturm t̀oareledou`procedeedeelimin`ri de tipulGauss:

P1. Procedeul substitu]iei par-]iale „înainte“ din factori-zarea LU. Cele r ecua]ii, [irespectiv rînduri ale matricei,asociate cu variabilele p`strate,se ordoneaz` ultimele (FiguraA.4). Se efectueaz` par]ialsubstitu]ia „înainte“ din facto-rizarea LU, pentru k>2,...,n-r.Procedeul este eficient doardac` r este suficient de mic.

Fig A.4 Reducerea uneimatrice prin elimin`ri detipul Gauss

3. Gaussian reduction of a set oflinear equationsAx > b

Themethod can acquire lower – sizeequivalent set of equations (with a lessnumber of variables) and stands at thebasis of a good many of the equivalentPSmodels.

Gaussian elimination of the variables iscarried out in a step-by-step algorithm.At the step k, the variabiles xk-1areeliminated from the equationsm (m>k, k+1,...,n) resulting an equivalentsystemwith aunity - dimished order.

The following two procedures ofGaussian eliminationswere consecrated:

P1.Theprocedureof partial„forward“ substitutionofLUfactorization.The r equations,andmatrix rows respectively,associated to conserved variables,are the last to be ordered (FigureA.4). The „forward“ substitutionofLU factorization is partiallycarried out, for k>2,...,n-r. Theprocedurewould be eficient if rwas sufficiently small.

Fig A.4 Matrixreduction by Gaussianeliminations


P2. The „build-and-eliminate“procedure. It is based on thenotice that at a step “ do generateeffects only the non - zero entries(i,k-1) and (k-1,j). Therefore, atthis step there are no storagerequirements for the row i -thand the column j - th with noincidences to axes k-1.Additionally, at the end of step k,the information of the row andthe column k-1-th is no morenecessary, and consequently, thecorresponding storage space canbe re-allocated. The procedure isquite simple, and isadvantageous in the applicationswhere a full solution of the setAx>b is not needed at all.

In numerous applications reducedmodels ofmatricesA1, A2, ... aredetermined that differ one against otherby only few entries. In these cases, it ispreferrable to calculate one reducedmatrix only and to re-use it fordetermination of all the others. Thesolution is described in [JCO27] andsupposes reconstitution of (eliminated)equations that were affected by changes.To do so, the only requirement is to havestored the diagonal entries of the fullprocessedmatrix, before carrying outthe last step of procedureP1 orP2.

P2.Procedeul „construie[te-[i-elimin`“.Acesta se bazeaz` peobserva]ia c` la un pas k producefecte numai elementele (i,k-1) [i(k-1,j) diferite de zero. A[adar, laacest pas nu este necesar s`existe loca]ii dememorie re-zervate pentru rîndul i [i coloanaj care nu au inciden]e la axelek-1. De asemenea, dup`parcurgerea pasului k,informa]ia de pe linia [i coloanak-1numai este necesar ,̀ spa]iulrespectiv dememorie putînd firealocat. Acest procedeu estedestul de simplu [i este avantajosîndeosebi în aplica]iile care nunecesit` [i rezolvarea integral̀ asistemuluiAx>b.

|n multe aplica]ii sînt necesare modelereduse ale unor matrice A1, A2, ... caredifer` între ele prin numai cîtevaelemente. |n aceste cazuri, este depreferat calculul unei singure matricereduse [i refolosirea acesteia ladeterminarea celorlate. Solu]ia estedescris` în [JCO27] [i presupunereconstituirea ecua]iilor (eliminate)afectate de modific`ri. Pentru aceasta,va trebui numai memorarareatermenilor diagonali ai întregii matriceprelucrate, înainte de efectuareaultimului pas al reducerii P1 sau P2.

References

[WT-JW]W.F. Tinney, J.W.Walker,Direct Solutions of Sparse Network Equations by OptimallyOrdered Triangular Factorization (IEEETransactions, Vol. PAS – 55, November 1967)

[JCO27] J. Constantinescu, Re]ea redus` echivalent` pentru analiza „în timp real“ a sistemelorelectrice extinse (AnEquivalent ReducedNetwork for Real Time Analysis of Large PowerSystems) (Energetica 11, 1973)


Anexa 2

Grafuri lineare (topologice).Determinarea matriceide inciden]` bucle - laturi

1. Defini]ie analitic`a grafului linear

Fie omul]ime de puncteX={x1,x2,...,xn}care se unesc între ele dup` o anumit`regul` de coresponden] .̀ Figura carerezult` reprezint` un graf, iar punctelese numesc vîrfuri. Perechile de vîrfuriconectate printr-un arc xij = (xi,xj) sîntnumite adiacente, în timp cemul]imeaperechilor adiacente, rela]ie deadiacen]`. Perechile de v\rfuri senumesc arce, dac` sînt direc]ionate,sau laturi dac` nu sînt direc]ionate.

Coresponden]a dintre xi este o apli-ca]ie Γ a mul]imii X în ea îns`[i. Dac`xj ∈ Γxi, coresponden]a se face de lavîrful ini]ial xi la vîrful final xj. Notîndcu Umul]imea arcelor {xij}, graful sepoate defini [i prin simbolul G =(X,U). Un graf este complet dac`pentru orice pereche de vîrfuri xi [i xj

exist` xij∈U [i xji∈U, altfel spus, oricenod este legat la orice alt nod.Reprezentarea „sagital`” a grafului seob]ine reprezentînd geometric grafuldefinit analitic.

Fiec`rei laturi a unui graf i se poateatribui o pondere (cost, distan]`,transmitan]`, rezisten]` etc.). |n gra-furile cu ponderi de laturi, conexiunile- pereche au valori numerice. Un grafcu ponderi de laturi se nume[te re]ea.|n teoria re]elelor, care ne intereseaz`

Annex 2

Linear (topologic) graphs.Finding the loop - edge incidencematrix

1. Analytical definitionof linear graph

Let be a array of pointsX={x1,x2,...,xn}which are connected to each other inaccordance to a certain rule ofcorrespondence. The resulting figure is agraph,while the points are called vertices.Pairs of vertices connected by an edgexij= (xi,xj) are said to beadjacentwhile theset of adjacent pairs,adjacency relation.Pairs of connected vertices (endvertex)are calledarcs, if directed, or edges, ifundirected.

Correspondence among xi is anapplication Γ of the set X in itself. Ifxj ∈ Γxi, the correspondence is fromthe initial vertex xi to the final verticexj. Noting with U the array of edges{xij}, the graph can be defined by thesymbol G = (X,U). A graph is completeif for any pair of vertices xiand xj

there exist xij∈U and xji∈U, i.e. everynode is linked to every other node.„Sagital” representation is ageometrical representation of theanalitically defined graph.

To each edge of the graph a weight(cost, distance, transmitance, resistan-ce, etc.) can be assigned. For thegraphs with weights the pairwiseconnections have numerical values. Agraph with weighted edges is called anetwork. In the network theory, which


is of interest for us in this work,vertices are known as nodes and edgesas branches, or network elements.

2. Short glossary of graphcomponents

A subgraph of a graph G is a graphwhose vertex set is a subset of that ofG, and whose adjacency relation is asubset of that of G restricted to thevertex subset.

A route or chaine is a sequence ofedges and vertices from one vertex toanother. Any given edge or vertexmight be used more than once.

A path is a route that does not passany edge more than once. If the pathdoes not pass any vertex more thanonce, it is a simple path.

If some route exists from everyvertex to every other, the graph isconnected.

A vertex / edge cut is a set ofvertices / edges whose removaldisconnects the remainingsubgraph.

A loop or cycle is a path which endsat the vertex where it began. A loopcan or cannot have an orientation.

A (complete) tree is a connectedsubgraph that containes all thegraph’s vertices and no loops. Anarborescence is a tree, in generaloriented, that has all verticesreachable from a single vertex

în aceast` lucrare, vîrfurile sîntcunoscute ca noduri iar laturile, caelemente de re]ea.

2. Mic glosar de componente alegrafului

Subgrafulunui grafG este un graf avîndmul]imeade vîrfuri ca submul]imeaaceleia a luiG [i rela]ia de adiacen]` casubmul]imea aceleia a luiG, limitat` lasubmul]imea vîrfurilor.

Subgraful formatdeo succesiunede laturicarepermite trecereade launvîrf laaltulsenume[te lan].Oricevîrf sau latur`dat(̀ ) sepoate folosi demaimulte ori.

Calea este un lan] care nu trece demai multe ori printr-o latur`. Caleaeste simpl` dac` nu trece printr-unnod de mai multe ori.

Graful în care pentru oricare dou`noduri exist` un lan] se nume[te grafconex.

O sec]ionare de vîrfuri / laturi estemul]imea de vîrfuri / laturi a c`reiîndep`rtare realizeaz` o separare însubgraful remanent.

Calea care pleac` dintr-un vîrf [iajunge la acela[i vîrf se nume[te ciclu.Ciclul poate avea sau nu o orientare.

Arborele (complet) este un subgrafconex care con]ine toate vîrfurilegrafului dar nici un ciclu. Avem oarborescen]`, de regul` într-un arboreorientat, dac` toate nodurile sîntconectate la un singur nod (r`d`cin`).


Un co – arbore este complementularborelui într-un graf.

Sec]ionarea fundamental̀ este este osec]ionare de laturi care include o singur`latur` aunui arbore complet al grafului.O sec]ionare fundamental̀ k este linearindependent̀ deoarece vectorul asociatvk, cum elemente vi

k = 1 sau vik = 0dup`

cummuchia i apar]ine saunu sec]ion`rii,este linear independent.

Bucla fundamental` se formeaz` prinad`ugarea unei laturi (coard`) care nuapar]ine arborelui complet al grafului.Similar, se arat` c` buclelefundamentale sînt linear independente.

Subgrafuri independente

Dou` subgrafuri sînt disjuncte dac`nu au vîrfuri în comun ([i deci nicilaturi). Avem omul]ime de grafuridisjuncte, dac` acestea sînt disjuncteunul fa]` de altul. Un graf poate fidescompus în mai multe subgrafuriindependente dac` mul]imeavîrfurilor sale se poate parti]iona însubmul]imi disjuncte una fa]` decealalt`. Astfel de submul]imiindependente se numescmul]imi- parti]ie sau p`r]i. Un grafeste k-parti]ie dac` poate fidescompus în cel pu]in k p`r]i.

3. Matrice de inciden]`într-un graf

Se spune c` vîrfurile adiacente, ini]ial[i final, sînt incidente la latur`, ori,echivalent, latura este incident` lacele dou` vîrfuri.

(the root). A co – tree is thecomplement of tree in a graph.

A principal edge cut is an edge cutwhich includes an only edge of acomplete tree of the graph. Aprincipal edge cut k is linear inde-pendent since the associate vector vk,withm elements vi

k = 1 or vik = 0 as the

edge i belongs or does not belong tothe edge cut, is linear independent.

A principal loop is formed by addingan edge (chord) which does not belongto the complete tree of the graph.Likewise, it can be shown that theprincipal loops are linear independent.

Independent subgraphs

Two subgraphs are vertex disjoint ifthey have no vertices (and thus, alsono edges) in common. A set of disjointsubgraphs are assumed to be pairwisevertex disjoint. A graph can bedecomposed into independentsubgraphs in the sense that the entirevertex set of the graph can bepartitioned into pairwise disjointsubsets. Such independent subsetsare called partite sets, or simply parts.A graph that can be decomposed intonot less than k partite sets isk-partite.

3. Incidence matricesin a graph

The adjacent, initial and final,vertices are said to be incident to thatedge, or, equivalently, that edge isincident to those two vertices.


The graph is fully represented by thevertex – edge incidence matrix A ofsize |V| (number of vertices) x |E|(number of edges) where the entrycontains the endvertex data ofconventionally oriented edge (1 – ini-tial vertex; -1 – final vertex;0 - not incident). Matrix A is indepen-dent of the choice of tree, however thetree data can be introduced by amatrix partitioning as against thetree – edges and the chord - edges: A= [Aa Ac]. Quadratic submatrix Aa isalways nonsingular, and its deter-minant equals ±1. Generally, Ac isrectangular and singular.

(Principal) Loop – edge incidencematrix B has an entry equal to 0, if theedge is not incident to the loop, or ±1, ifthe edge and loop are incident to eachother and have the same / contrary(external) orientation. B does notcontains complete data on the networkstructure, and can be partitioned as[Ba 1c], in which Ba is generally rectan-gular and nonsingular. Likewise, the(principal) edge cut – edge incidencematrix C it may be defined.

4. Basic relationship betweenincidence matrices

Half of the adjacent edges of aprincipal edge cut - principal looppair have the same external orien-tation, as against the edge cut as wellas the loop, and consequently wehave,

(1)

Graful este deplin reprezentat dema-triceade inciden]̀ noduri - laturiA cudimensiunea |N| (num`rul denoduri) x |L|(num`rul de laturi), [i avînd ca elementeinforma]iaprivind extremit̀ ]ile laturei,orientat̀ conven]ional (1 – vîrf ini]ial;-1 – vîrf final; 0 –ne-inciden]̀ ). MatriceaAnudepindedealegereaarborelui, însìnforma]iaprivindarborele poate fiintrodus`printr-o parti]ionareamatriceiîn raport cu laturile - arbore (ramuri) [ilaturile - coardàle grafului:A= [Aa Ac].Submatriceapatrat̀ Aaeste întotdeaunanesingular ,̀ determinantul sù fiind egalcu±1.Aceste în general dreptunghiular`[i singular .̀

Matricea B de inciden]` bucle (funda-mentale) - laturi are un element egal cu0, dac` latura nu este incident` labucl`, sau cu ±1, dac` latura [i buclasînt incidente [i au orientarea (extern`)aceea[i / contrar`.B nu con]ineîntreaga informa]ie privind structurare]elei, [i poate fi parti]ionat` ca [Ba 1c],în careBaeste în general dreptun-ghiular` [i nesingular`. Similar, sedefine[te [imatricea C de inciden]`sec]ion`ri fundamentale - laturi.

4. Rela]iile de baz` întrematricele de inciden]`

Jum`tate din laturile comune ale uneiperechi sec]ionare fundamental` -bucl` fundamental` au aceea[iorientare extern`, atît fa]` de sec-]ionare, cît [i fa]` de bucl`, [i înconsecin]` avem,


Dup` parti]ionare, matricele deinciden]` în raport cu ramurile [irespectiv, coardele, rela]iile (1) se potre-scrie astfel,

5. Metod` de determinare amatricei de inciden]` bucle -laturi

Matriceade inciden]̀ bucle (fundamentale)- laturiserve[te lacalculareamatriceiimpedan]elorbuclelor, careesteesen]iala înabordareanoastr̀ dedezlegarestructural̀(ParteaaII-a,Capitolul2).

Se determin` mai înt\i matricea D deinciden]` bucle - laturi a unui grafauxiliar, format din arborele completal grafului la care se adaug` (n-1)corzi ce unesc un nod de referin]` cufiecare nod al grafului. Corzileauxiliare au nodul de referin]` caextremitate ini]ial`.

Pentru graful auxiliar, matricea deinciden]` vîrfuri - laturi areurm`toarea form` particular`,

Iar matricea de inciden]` bucle -laturi se scrie astfel,

By partitionning the incidencematricesas against the tree – edges and chord –edges respectivelly, the relations (1) canbe re-written as:

(2)

5. A method for findingthe loop - edge incidencematrix

The (principal) loop - edge incidencematrix serves to the calculation ofloop impedance matrix, which isessential in our approach of structuralunbundling (Part II, Chapter 2).

Firstly, there is determined theloop-edge incidence matrix D of anauxiliary graph, that is made up ofthe complete graph tree and (n-1)chords joining a reference vertex withevery vertex of the graph. Theauxiliary chords have the referencevertex as initial endvertex.

For the auxiliary graph, the vertex -edge incidence matrix has thefollowing particular form,

(3)

And, the loop - edge incidence matrixcan be written as,

(4)


By introducing (3) and (4) in the firstrelation (1), the key information in (2)for determination of B is found,namely:

(5)

Thus, on the base of matrix D, asimple and efficient algorithm forfinding the matrix B can bedeveloped, as follows [JCO16]:

The vertices are labeled so as to pairevery one with at least a vertex ofsmaller label number. The co – tree isfound by inspecting the list of edges,and selecting those with endvertexthat was met at least once. Thenon-selected edges form anarborescence. The tree-edges are fullydefined by their final endvertex.

Conventionally, the edge is smaller – to– large (initial – to – final) endvertexoriented, and the loop in accordancewith orientation of its definition chord.

The rows of matrixA, not including thatassociated to the arborescence root, aresupposed to be arranged in the order oftree – branches labels. Consequently,the entries of submatrixAahave thefollowing values:

(6)

Multiplication of matrix Act with

(Aa-1)t = Dc

t is equivalent to a sequenceof row additions.

Introducînd (3) [i (4) în prima rela]ie(1) se ob]ine informa]ia – cheie din (2)pentru determinarea lui B,respectiv:

Astfel, pe baza matricei D, se poatedezvolta un algoritm simplu [i efi-cient pentru determinarea matriceiB, dup` cum urmeaz` [JCO16]:

Vîrfurile se numeroteaz` astfel încîtorice vîrf s` fie conectat la cel pu]in unaltul cu num`rmai mic. Co-arborele sedetermin` prin parcurgerea listeilaturilor, selectînd pe acelea ale c`rornoduri au fost ambele întîlnite cel pu]ino dat`. Laturile ne-selectate formeaz` oarborescen]`. Ramurile sînt completdefinite de extremit`]ile lor finale.

Conven]ional, latura este orientat̀ de lanodul cunum`rmaimic (ini]ial) sprenodul cunum`rmaimare (final), iarbucladup`orientarea coardei dedefini]ie.

Rîndurile matricei A, din care s-aeliminat acela asociat cu r`d`cinaarborescen]ei, se presupun aranjateîn ordinea numerelor ramurilor.Elementele submatricei Aaau în acestcaz urm`toarele valori:

|nmul]irea matricei Act cu (Aa

-1)t = Dct

este echivalent` cu o succesiune deadun`ri de rînduri.


Algoritmul include urm`toareleetape:a) Se stabile[te o arborescen]` [ico-arborele asociat. Se anuleaz`fiecare element al matricei Ca.Fiec`rei coloane i se asociaz` ocoard` (bucl`) [i fiec`rui rînd, oramur`.

b) Se formeaz` Act înscriind \n

fiecare coloan`: +1 în rîndul aso-ciat cu vîrful ini]ial al coardei, [i-1 în rîndul asociat cu vîrfulfinal.

c) |ncepînd cu ultimul, fiecare rîndj se adun` algebric la rîndulramurii i care are ca vîrf final(de defini]ie) vîrful ini]ial alramurii j.Se determin` astfel inciden]elela bucle ale ramurilor care nuau noduri comune cu coardelede defini]ie a buclelor.

Actually, the algorithm includes thefollowing steps:a) An arborescence and its co – treeare established. Every entry ofthe matrix Ca is canceled. Achord (loop) is associated toevery column, and an edge – treeto every row.

b)Act is formed by filling in everycolumn: +1 in the row associatedto the chorde’s initial vertex, and-1 in the row associated to itsfinal vertex.

c) Starting with the last one, everyrow j is added algebraically to therow of tree - edge iwhich has asfinal endvertex (of definition) theinitial endvertex of tree - edge j.In this way, are determined theincidences to loops of the treeedges that have no adjacencies tothe loop definition chords.


[DR] D. Reinhard, Graph Theory (3rd edition ed. Graduate Texts in Mathematics, vol. 173,Springer-Verlag, 2005)

[SCS] S.C. S`vulescu, Grafuri [i re]ele electrice (Graphs and Power Networks) (EdituraTehnic`, Bucure[ti, 1994)

[JCO16] J. Constantinescu, Aplicarea metodei buclelor la calculul re]elelor electrice. Re]elereduse echivalente (A Loop Method for Electric Networks Computations. EquivalentReduced Networks) (Studii [i cercet`ri în energetic`, nr. 3, 1971)


Anexa 3

Sisteme de ecua]ii lineareasociate unui graf

Se arat` condi]iile în care rezolvareasistemului de ecua]ii,

în care S este o matrice patrat`,simetric` [i nesingular` de ordinul n,iar h0 este un vector din Rn,

se poate reduce la rezolvarea siste-mului (echivalent) urm`tor:

Sistemului (1) i se poate asociagraful,

\n care:Σ0c estemul]imeavîrfurilorPi (i=0,1,2...n)

Σ1 - mul]imea laturilor (Pi,Pj);Σ1a - mul]imea laturilor - arbore (P0,Pi);Σ1c - mul]imea laturilor - coard`, care

îndepline[te rela]ia:

|n raport cu Σ1c, se poate defini o mul-]ime de cicluri independente, astfel:

Annex 3

Systems of linear equationsassociated to a graph

There are shown the requirements thatsolution of the set of equations,

(1)

in which S is a quadratic, symmetri-cal and non-singular matrix of theorder n, and h0 is a vector in Rn,

can be reduced to solution of thefollowing (equivalent) set:

(2)

A graph G can be associated to theset (1),

(3)

where:Σ0cis the set of Pivertices (i=0,1,2...n)

(4)

Σ1 - the set of (Pi,Pj) edges;Σ1a - the set of (P0,Pi) tree - edges;Σ1c - the set of chord - edges meeting

the following relation:

(5)

With reference to Σ1c, a set of indepen-dent loops can be defined, as follows:


(6)

A diagonalmatrix di is associated to thematrixS from (1), defined as follows:

(7)

The vertex – edge and loop - edgeincidence matrices, A and B, expres-sing surjective applications from E1 toE0 and from E1 to E2, respectivelly,

(8)

meet the following relationships:

(9)

Under the conditions,

(10)

it can be checked up that the systems(1) and (2) are equivalent, if:

(11)

Se asociaz` matricei S din (1) omatrice diagonal` di, definit` astfel:

Matricele de inciden]` vîrfuri - laturi(A) [i bucle - laturi (B), exprimîndaplica]ii surjective de la E1 la E0 [irespectiv, de la E1 la E2,

îndeplinesc urm`toarele rela]ii:

|n condi]iile,

se poate verifica c` sistemele (1) [i (2)sînt echivalente, dac`:


Anexa 4

Algoritmul în [ase pa[i al dezle-g`rii structurale directe

(Partea a II-a, §2.2.2)

Sistemul de ecua]ii,

care este echivalent lui S.x0 = h0, seaduce la o form` avantajoas` pentrucalcul, cu o sub-matrice M1

bloc-diagonal`, astfel:

Fie: Ek’, Ek

”subspa]ii în Ek (k=0,1,2),mul]imile de laturi [i respectiv debucle independente Σ1 [i Σ2 ale unuigraf linear, matricele de inciden]` A[i B (definite ca în Anexele 2 [i 3), [imatricele π de proiec]ie, respectiv i deinjec]ie,

a) Dac` din Σ2’ se ia o bucl`, [i în

Β1’se includ toate laturile buclei,

[ib) Dac` din Σ1

’ se ia o latur`, [i înΒ0

’se includ ambele vîrfuriadiacente laturii,

Annex 4

The six – step algorithm of thedirect structural unbundling

(Part II, §2.2.2)

The following set of equations,

(1)

which is echivalent to S.x0 = h0, isbrought to a form that is advantage-ous for calculations, with abloc-diagonal sub-matrix M1, namely:

(2)

Let be:Ek’,Ek

”sub-spaces inEk (k=0,1,2),the sets of edges Σ1 and Σ2 ofindependent loops of a linear graph,the incidence matrices A and B (asdefined in the Annexes 2 and 3), andthe projection matrix π and theinjection one i,

(3)

a) If a loop is taken from Σ2’, and in

Β1’are included all the loop

edges, andb) If an edge is taken from Σ1

’, andin Β0

’ are included the two endvertices,


atunci sînt satisf`cute urm`toarelerela]ii:

[i de asemenea,

Se aplic` primelor dou` ecua]ii (1)proiec]iile π2

’ [i π0’, în ipoteza c`

vectorii a [i b satisfac condi]iileurm`toare:

Se ob]ine,

\n care s-a notat,

A’ (A”) [i B’ (B”) rezult` din matricelede inciden]` A [i B anulîndelementele de pe coloanele asociate culaturile din Σ1

” ([i respectiv din Σ1’).

|n (7), x’, x”, y’, y” se înlocuiesc cu,

then the following relations aresatisfied:

(4)

and also,

(5)

Projections π2’ and π0

’ are applied to thefirst two equations (1), with vectors aand b satisfying the followingrequirements,

(6)

The result is,

(7)

where it was noted,

(8)

A’ (A”) and B’ (B”) result from the inci-dence matrices A and B by cancellingentries on the columns associated toedges in Σ1

” (and in Σ1’ respectivelly).

In (7), x’, x”, y’, y” are replaced with,


Rezult`,

unde, s-au mai introdus urm`toarelenota]ii:

Vom avea ecua]iile în forma (2)dac`,

Rezolvarea ecua]iilor (2) permite caprin rela]iile (9) s` se determine x’, x”,y’, y” iar, în final, solu]ia sistemului (1):

(9)

It yields,

(10)

where, additional notations have beenintroduced:

(11)

We will reach the equations in theform (2) if,

(12)

Solution of equations (2) and relations(9) allows determination of x’, x”, y’, y”

and, finally, solution of set (1):

(13)


The sought solution of equation setS.x0 = h0 satisfies the following system:

(14)

with

(15)

Solution of the set (2) can beorganized in six computation steps(the six - step algorithm), as follows:

(16)

in which

(17)

The particular computation convenienceof the method is determined by the bloc– diagonal structure of matrixM1,which can be associated to a k-partitegraph (Annex 2) i.e. a set of kindependent disjoint subgraphs orsubnetworks, not taking into accountthe reference vertex (node)P0.

In the particular case when,

(18)

the equations (2) do coincide with theoriginal set S.x0 = h0 (Annex 3).

The size of equation sets in the stepsA1andA5 is smaller than that of original

Solu]ia c`utat`, a sistemului deecua]ii S.x0 = h0, satisface sistemul,

cu

Rezolvarea sistemului (2) se poateorganiza în [ase pa[i de calcul (algo-ritmul în [ase pa[i), dup` cum urmeaz :̀

în care

Avantajul de calcul deosebit almetodei rezult` din structurabloc-diagonal` a matricei M1, carepoate fi asociat` unui graf k – parti]ie(Anexa 2), respectiv unei mul]imi de ksub-grafuri (subre]ele) disjuncteindependente, f`cînd abstrac]ie devîrful (nodul) de referin]` P0.

|n cazul particular în care,

ecua]iile (2) coincid cu sistemuloriginar S.x0 = h0 (Anexa 3).

Dimensiunea sistemelor de ecua]ii de lapa[iiA1 [iA5 este mai mic` decît a


sistemului original. |n plus, deoarece Tse identific` cuA”, înmul]irile de lapa[iiA2 [iA4 pot fi substituite prinopera]ii simple. Elementele vectoruluiTtx01 asociate cu λi = (Pq, Pm) sedetermin` ca diferen]` între elementeledin x01 asociate cu Pq, respectiv Pm, întimp ce elementele lui h0S asociate cu q[im sînt egale cu - y”km [i respectiv y”km.

Rela]ia urm`toare define[te o matrice„de interconectare“ NS

Determinarea matricei NS se simpli-fic` mult dac`,

NS devine în acest caz o matrice deponderi de bucle Nr

c, rezultat alreducerii de tipul Gauss a matriceiponderilor buclelor independente alegrafului, la buclele definite de laturile- coard` de interconectare.

Un procedeu alternativ [i mai eficientconstruie[te NSdirect pe graful redus,care con]ine numai nodurile deinterconectare, astfel:

- Din M1 se elimin` (Gauss) toateliniile [i coloanele asociate nodu-rilor interne ale Σ0

- Matricea redus`, de asemeneabloc-diagonal`, se asociaz` cumul]imile Σ0

r [i Σ1a,r.

NS se ob]ine din matricea ponderilorde bucle pentru graful redus (Σ0

r, Σ1r)

cu Σ1r = Σ1

a,rUΣ1

c, dup` reducerea detipul Gauss la buclele definite delaturile din Σ1

c.

system. Moreover, becauseT is identicalwithA”, mere simple operations cansubstitute the multiplications inA2 andA4. The entries of the vectorTtx01

associated to are determined asdifferences in between entries of x01as-sociated toPq, andPm respectivelly, whilethe entries ofh0S associated to q andmare equal to - y”km and y”km respectivelly.

An „interconnection“ matrix NS isdefined by the relation

(19)

Determination of the matrix NS ismuch simplified when,

(20)

In this case, the NS becomes a loopweight matrix Nr

c, a result ofGaussian reduction of the weightmatrix of graph independent loops tothe loops defined by interconnection(chords) edges.

An alternative and even more efficientprocedure builds theNS directly on a thereduced graph that contains theinterconnectionvertices only, as follows:

- FromM1 there are eliminated(Gauss) all rows and columnsassociated to internal vertices of Σ0

- The reduced matrix, block-dia-gonal as well, is associated to thesets and Σ0

r and Σ1a,r.

NS is yielded from the loop weightmatrix of reduced graph (Σ0

r, Σ1r) with

Σ1r = Σ1

a,rUΣ1

c, after Gaussianreduction to the loops defined by theedges in Σ1

c.


Anexa 5

Reprezentarea generatoruluisincron în regimurile tranzitoriiale SEE

I. |n modelul RETRA

Ecua]iile care descriu comportareagrupurilor generatoare în regimultranzitoriu de tip electromecanic, petermen scurt, sînt exprimate în sis-temul de referin]` d-q [DWO]. Axa qprecede axa d în direc]ia de rota]ie arotorului. Un vector A avînd compo-nentele AR [i AI în sistemul dereferin]` R-I este reprezentat caAq + j Ad = (AR + j AI) e-jδ în sistemuld – q.

Annex 5

Representation of synchronousgenerator in the PS dynamicssimulation

I. In RETRA model

The equations describing theshort-term electromechanicaltransient performance of generatingunits are expressed in the d-q direct-quadrate reference frame [DWO].The q axis precedes the direct axis din the rotation direction of the rotor.A vector A with the components AR

and AI in the R-I reference frame isrepresented as Aq + j Ad = (AR + j AI) e-jδ

in the d – q frame.

(1)

(2)

(3)

(4)

(5)


Ecua]iile diferen]iale (2) [i (3) seintegreaz` (algebrizeaz`) prin metoda„trapezelor“. Ecua]iile rezultate sefolosesc la înlocuirea lui eq

”, eq’[i ed

”

din (1), ajungîndu-se la modelul„sta]ionar“ al generatorului la pasulde calcul t, dup` cum urmeaz`[JCO17]:

în care xD”, xQ

”, Fd [i Fqdepind deparametrii modelului [i de valorile luied

”, eq”, eq

’, id, iq la pasul de calculprecedent (t-∆t).

II. |n modelul TEMI

Ecua]iile generatorului sincron sîntscrise în ipoteza c` pe rotor sîntdispuse dou` circuite electrice:unul real reprezentînd înf`[urareade excita]ie [i un circuit echivalent,prin fierul rotorului, ce determin`comportarea tranzitorie a ma[inii peaxa q.

(6)

(7)

The differential equations (2) and (3)are integrated (algebrized) by the„trapezoidal“ technique. Theresulted equations are used tosubstitute the eq

”, eq’and ed

” from (1),yielding the „steady-state“ model ofgenerator at the computation step t,as follows [JCO17]:

(8)

where xD”, xQ

”, Fd and Fq depend onthe model parameters and on thevalues of ed

”, eq”, eq

’, id, iq at theprevious computation step (t-∆t).

II. In TEMI model

The equations of synchronous gene-rator are written on the assumptionthat on the rotor two electric circuitsare disposed: a real one representingthe field winding and an equivalentcircuit within the rotor iron thatdetermines the transient behavior ofmachine on the q axis.

(9)

(10)


Ecua]iile algebrice (9) [i (10) secombin` într-o singur` ecua]iematriceal`, astfel:

iar ecua]iile diferen]iale (11) se inte-greaz` (algebrizeaz`) prin metoda„trapezelor“, rezultînd:

ad, aq, Fd, Fq depind de parametriimodelului [i de valorile lui ed

’, eq’, id,

iq la pasul de calcul precedent (t-∆t).Modelul „sta]ionar“ al generatoru-lui la pasul de calcul t rezult` dinecua]iile (15) [i (16), astfel[JCO24]:

cu,

(11)

(12)

(13)

(14)

The algebraic equations (9) and (10)are combined into a sole matrixequation, as follows:

(15)

while the differential equations (11)are integrated by the „trapezoidal“technique thus yielding,

(16)

ad, aq, Fd, Fq depend on the modelparameters, and on the values of ed

’,eq

’, id, iq at the previous computationstep (t-∆t). The „steady-state“ modelof generator at the computation stept results from the equations (15) and(16), as follows [JCO24]:

(17)

with,

(18)


Simboluri

xd, xq - reactan]ele generatorului pe axeledirect` [i în cuadratur` (în u.r.)

xd”, xq” - reactan]ele subtranzitorii, direc-t` [i în cuadratur` (în u.r.)

xd’, xq’ - reactan]ele tranzitorii, direct` [iîn cuadratur` (în u.r.)

Td0’, Tq0

’ - constante de timp de gol peaxele direct` [i în cuadratur`(în secunde)

Td0”, Tq0

” - constante de timp subtran-zitorii de gol pe axele direct` [iîn cuadratur` (în secunde)

ef - tensiune de excita]ie (în u.r.)id, iq (ud, uq) - componentele direct` [i în

cuadratur` ale curentuluistatoric (tensiunii la borne)(în u.r.)

ed”, eq”, ed’, eq’, - tensiuni electromotoare decalcul pe axele direct` [i încuadratur` (subtranzitorii[i tranzitorii) (în u.r.)

s - alunecarea rotoruluiCm -cuplullaarbore(puteredeintrare)(înu.r.)Ce - cuplul electric (putere de ie[ire) (în u.r.)T = 2H - constantadeiner]ie (MW.s/MVA)δ - pozi]ia relativ` a rotorului în raport cu

sistemul de referin]` rotitor sincronω - viteza unghiular` (frecven]a) (în u.r.)

Nomenclature

xd, xq - direct and quadrate axis genera-tor reactances (in p.u.)

xd”, xq” - subtransient direct and qua-drate axis reactances (in p.u.)

xd’, xq’ - transient direct and quadrateaxis reactances (in p.u.)

Td0’, Tq0

’ - direct and quadrate axistransient open – circuit timeconstants (in seconds)

Td0”, Tq0

” - direct and quadrate axissubtransient open – circuittime constants (in seconds)

ef - field voltage (in p.u.)id, iq (ud, uq) - direct and quadrate axis

component of the statorcurrent (terminal voltage)(in p.u.)

ed”, eq”, ed’, eq’, - computation electromotivevoltage on the direct andquadrate axes (subtransientand transient) (in p.u.)

s - rotor slipCm - shaft torque (power input) (in p.u.)Ce - electrical torque (power output) (in p.u.)T = 2H - inertia constant (MW.s/MVA)δ - generator rotor position relative to

synchronous rotating reference frameω - angular velocity (frequency) (in p.u.)

Bibliografie References

[DWO] D.W. Olive, Digital simulation of synchronous machine transients (IEEE Transactions,Vol. PAS – 87, August 1968)

[JCO17] J. Constantinescu, Study of the Transient Processes in Large - Scale Power Systems(Rev. Roum. des Sci. Techn., Vol. 27, No. 2, pages 211 - 227, 1982)

[JCO24] J. Constantinescu, Midterm Transient Processes with Dimo - REI Models (Rev.Roum. des Sci. Techn., Vol. 28, No. 1, pages 45 - 60, 1983)


Anexa 6

Modelul regimului permanentsimetric al re]elei electrice îndiferite ipoteze privind injec]iilenodale de putere

1. Ecua]iile injec]iilor nodale deputere

Ecua]iile care descriu bilan]ulxde putere în nodurile re]elei seexprim` ca sum` algebric` de treicomponente:

n fiind num`rul de noduri al re]elei.

Pij, Qij sînt puterile active [i reactivetransmise din nodul i nodurilor veci-ne. Exprimînd componentele tensiu-nilor nodale în sistemul de coordonate“polare” , puterile Pij, Qij

se pot scrie astfel:

în care Gij, Bij sînt elemente alematricei admitan]elor nodale.

Pli + j . Qli este puterea consumuluiagregat în nodul i. Pentru regimurile

Annex 6

Symmetrical load - flow model ofthe power network in differenthypothesis on the nodal powerinjections

1. Equations of the nodal powerinjections

The equations describing the powerbalance at the network nodes can beexpressed as an algebraic sum ofthree components:

(1)

n being the number of network nodes.

Pij,Qij are the active and reactive powersthat are sent from the i node to theneighbouring ones. By expressing thenode voltage components in the “polar”system of coordinates as ,the powersPij,Qij , can be written as:

(2)

in which Gij, Bijare the entries in thenodal admittance matrix.

Pli + j .Qli is the power of aggregate loadat node i. For the „steady – state“ load


flows as system states in the long-termPS dynamics (Part I, Case No. 4), thedependence of load on the voltage andfrequency can be expressedby simple characteristics in theform,

(3)

For the permanent load flows that arefrequency independent, the load-voltage variation can be formulatedby relations with two terms only, as:

(4)

For limited voltage variations and aconvenient coefficient choice, thisrepresentation can also approximatethree - component expressions.

The „linear“ component depending onU2, can be included in thecorresponding diagonal entry ofadmittance matrix, as follows:

(5)

Pgi + j .Qgi is the i node injected powerfrom the generator gi. By neglecting theresitances of electric circuits and thepoles’ non-uniformity, the generatedactive and reactive powers aredescribed by the following equations:

„sta]ionare“ reprezentînd st`ri îndinamicile pe termen lung ale SEE(Partea I-a, Exemplul nr. 4),dependen]a sarcinii de tensiune [ifrecven]` poate fi exprimat` decaracteristici simple, de forma,

Pentru regimurile permanente inde-pendente de frecven]`, varia]ia sarciniicu tensiunea poate fi formulat` printr-orela]ii cu numai doi termeni, astfel:

Pentru varia]ii nu prea mari aletensiunii [i o alegere convenabil` a coefi-cien]ilor, cu aceast` reprezentare se potaproxima [i expresii cu trei componente.

Componenta „linear`“, dependent` deU2, se poate introduce în termenuldiagonal corespunz`tor al al matriceiadmitan]elor nodale, astfel:

Pgi + j . Qgi este puterea injectat` înnodul i de generatorul gi. Neglijîndrezisten]ele circuitelor electrice [ineuniformitatea rotorului, puterilegenerate - activ` [i reactiv` - sîntdescrise de urm`toarele ecua]ii:


\n care Xgi este reactan]a intern` ageneratorului, majorat` cu reactan]atransformatorului.

Egi este tensiunea din “spatele” lui Xgi,propor]ional` cu tensiunea deexcita]ie dup` o rela]ie de forma:

unde Kg este coeficientul total deamplificare al sistemului de reglare atensiunii, iar δgi este unghiul dintrefazorii tensiunilor Egi[i Ui.

2. Modelul conven]ionalNewton – Raphson pentruinjec]ii constante de puteriîn nodurile de re]ea

Sistemul linearizat de ecua]ii din meto-da “Newton – Raphson” pentru injectiinodale de puteri independente detensiune, poate fi scris simbolic astfel:

Elementele celor patru submatrice alematricei Jacobian reprezint̀ derivate par-]iale ale puterilor nodale active [i reactiveîn raport cu componentele tensiunii:

(6)

in which Xgi is the generator’sinternal rectance that is incrementedby that of the transformer.

Egi is the voltage from “behind” of Xgi,depending on the field voltage asfollows:

(7)

where Kg is the total gain coefficient ofthe voltage regulation set, and δgi is theangle in between the Egi and Ui voltagephasors.

2. The conventionalNewton – Raphson modelfor constant power injectionsat the network nodes

The linearized set of equations of the“Newton-Raphson” method forvoltage-independent power injections,can be symbolically written as:

(8)

The elements of the four Jacobeansubmatrices represent the partialderivatives of nodal powers in respectto the voltage components:

(9)


and can analytically be expressed as,

(10)

∆P and ∆Q are determined as mis-matches as against given powerinjectionsP0 andQ0, on the base ofestimations θest = θ + ∆θ andUest =U+ ∆U.

The equation associated to the voltageof reference node should beeliminated from the set (8).

3. The Newton-Raphsonmodel for voltage- andfrequency- dependant powerinjections

The voltage- and frequency-depen-dance of generation (g) and load (l)power injections leads to algebraicsubtracting the diagonal entries of H,N, J and L Jacobean sub-matrices sub– matrices, of the following terms.

At the derivation of generation powers inrespect to the voltage magnitude the

avînd urm`toarele expresii analitice:

∆P [i ∆Q se determin` ca diferen]e-erori fa]` de injec]ii de puteri date P0

[i Q0, pe baza estim`rilor θ est = θ + ∆θ[i Uest = U + ∆U .

Din sistemul (8) trebuie eliminat`ecua]ia asociat` cu unghiul tensiuniinodului de referin]`.

3. Modelul Newton-Raphsonpentru injec]ii de puterivariabile cu tensiunea [ifrecven]a

Varia]ia injec]iilor generatoarelor (g)[i a sarcinilor (l) cu tensiunea [ifrecven]a necesit` sc`dereaurm`torilor termeni din elementelediagonale ale sub-matricelorH, N, J [i L ale Jacobianului:

La derivarea puterilor generate înraport cu tensiunea trebuie ]inut

Kij = 1(0), j = i (j ≠ i)


seama [i de reglajul de tensiune, depild` dup` rela]ia simpl`

Modelul se simplific` apreciabil dac` seadopt` invarian]a sarcinii active a gene-ratoarelor fa]` de schimbarea tensiu-nilor re]elei ∆Pg = 0 ca premiz` de baz`.

Prin aceast` dezlegare func]ional` deproces, influen]a generatoarelor înmodelul N-R se reprezint` modificînddoar elementele Liiale Jacobianului,dup` cum urmeaz`:

Invarian]a puterii active generate lamodificarea regimului re]elei a fostconfirmat` în numeroasele aplica]iipractice ale modelului SAMI(Exemplul nr. 2, Partea a III-a).

4. Modelul Newton – Raphsonpentru modific`ri impuseputerilor generate

Dac` la schimbarea de regim perma-nent, puterile electrice generate PE

trebuie s` respecte anumi]i factori dealocare F, de pild`, inversele constan-telor de iner]ie ale rotoarelor(Partea I-a, Exemplul nr. 4),

modelul Newton – Raphson trebuiereformulat astfel:

voltage control should be considered, aswell, for instance as a simple equation

The model can be considerablysimplified if the invariance ofgenerator active power ∆Pg = 0 isadopted as a basic prerequisite.

This is a process functionalunbundling that allows for onlychanges of Lii in the N-R Jacobeanmatrix due to the impact ofgenerators, as follows:

(11)

The invariance of generator activepower as against the changes ofnetwork regime was well confirmed bynumerous practical applications of theSAMI model (Case No. 2, Part III).

4. The Newton – Raphson modelfor imposed changes ofgeneration powers

If, after a change of load flowsdistribution, the electric powers PE ofgenerators have to follow certainallocation factors F, for instance, theinverse of inertia constants(Part I, Case No. 4),

(12)

the Newton – Raphson model shouldbe reformulated, as:


(13)

where,

(14)

(15)

The index r designates the generationnode (equation) with a referencevoltage for the angles.

unde,

Indicele r desemneaz` nodul degenerare (ecua]ia) cu tensiunea dereferin]` pentru unghiuri.


[JCO-MH1] J. Constantinescu, M. Homos, Algoritm [i program pentru calculul regimurilorpermanente ale sistemelor electroenergetice prin metoda decupl`rii (A DecoupledMethod for Determination of Power Systems Load - flows) (Energetica 6, 1976)

[JCO-MH3] J. Constantinescu, M. Homos, Metod` practic` de calcul pentru studiul stabilit`]iistatice a sistemelor electroenergetice complexe (A Practical Method for the Studyof Steady - state Stability of the Complex Power Systems) (Energetica 9, 1977)

[JCO24] J. Constantinescu, Mid-term Transient Processes with Dimo - REI Models (Rev.Roum. des Sci. Techn., Vol. 28, No. 1, pages 45 - 60, 1983).

Available Transmission Capacity

“Benefit Factors” methodCoordinating Auctioning OfficeCross – Border Critical Capacity methodCongestion ManagementCongestion Management and Monitoring Office

“Driving-Point” or “point-of-connec-tion” power transfer capability

Electricity Market PriceEnergy Management System

Electric Power Research Institute - USAFree Term of Characteristic EquationFunctional Unbundling “Generation” and “Load” nodaltransmission ratesIndex of Column of a matrixLC&CC Income

Index of Row of a matrixIndex of Structure of a matrixIntegral Steady-State Margin stability index

Inter-TSO Compensation mechanism

Loss Compensation, Congestion Compensation and Access to Market power transmission services

Losses Coefficientunder Load Tap – Changing autotransformers

Long Term DynamicsLower and Upper Gaussian triangularization

Market of Balance Energy

Marginal Cost

ARSATCBENBF

CAOCBCrCCM

CMMO

CP&CC&AP

CTIR

DFDP

DSDV

EMPEMS

EPRI

FTCEFU

G&L

ICILC

IRIS

ISSM

ITC

LC&CC&AM

LCFLTC

LTDLU

MBE MC

MCO

Auto transformator cu Reglaj sub Sarcin`Capacitate disponibil` de transportRe]ea de Bilan] Energetic Nul Metoda “Factorilor de Beneficiu”

Oficiu de coordonare a licita]iilor

Metoda Capacit`]ii Critice

Oficiu de management [i monitorizare a congestiilorServicii de Compensare a

Pierderilor [i a Congestiilor în re]ea [i Acces la Pia]`Capabilitate de Transfer pentru Interconexiune inter-zonal` Relevant` Dezlegare Func]ional`Capabilitate nodal` de transfer, sau de punct de racordare Dezlegarea Spa]ial` sau Strucural`Dezlegare Varia]ional` sau Temporal`Pre] de pia]` al energieiSistem informatic de conducere prin dispecer

Tarife nodale de transport, de Generare [i consum (Load)Indice de Coloan` a unei matriceVenit pentru acoperirea pierderilor în re]ea [i a congestiilor de transportIndice de Rînd al unei matriceIndice de Structur` a unei matriceMarj` integral` de stabilitate static` (indice de stabilitate)Mecanismul de Compensare inter - OTS

Coeficient de pierderi

Regimuri tranzitorii pe termen lung Triangularizare gaussian` LU: factorizarea unei matrice ca produs între o matrice superior (Upper) [i alta inferior (Lower) triunghiular` Pia]a de echilibrare a energiei electriceManagementul CongestiilorCost marginal

ABREVIERI GLOSSARY

Market Opportunity FeeMedium Term DynamicsNewton – Raphson methodNet Transmission CapacityOperation & Maintenance

Price of Balance Energy“Point-to-Point” power transfer capabilityPower SystemPower Transfer Distribution FactorPower ExchangeThe accronim of REI - Dimo model R – Radial, E – Equivalent, I – IndependentRenewable Enery SourceRectangular coordinates

Revenue Requirement

Supervisory Control and Data Acquisition SystemSouth-East Europe

Smart GridAncillary System Service

Short Term DynamicsStructural UnbundlingStatic VAr Compensator

Transmission CapacityPower Transfer Capability of

Relevant inter-area Interconnection

Transmission System OperatorZero Power Balance network“Zero Voltage Restoration” networkcoordinate Vector Array

Variational or TemporalUnbundling

Regimuri tranzitorii pe termen mediuMetoda Newton – Raphson

Capacitatea net` de transportOperare [i Mentenan]`Operator de Transport [i de SistemPre]ul energiei de echilibrareCapabilitate de transfer

“Punct-la-Punct”

Factor de reparti]ie a puterii de transferBurs` de energieAcronim pentru modelul REI-Dimo R – Radial, E – Echivalent, I – Independent

Coordonate rectangulareReal` - Imaginar`Cerin]a de venitRe]ea de “Restabilire a Tensiunii Zero”

Sistem de comand` [i achizi]ie de date

Sistem de Energie Electric`Surs` de Energie Regenerabil`Re]ea inteligent`Serviciu tehnic de sistemRegimuri tranzitorii pe termen scurt

Compensator static reglabil de energie reactiv`

Termen Liber al Ecua]iei Caracteristice Transfrontier`Tarif de Oportunitate de Pia]`

MOFMTDNR

NTCO&MOTSPBEPP

PSPTDF

PXREI

RESR-I

RRRTZ

SCADA

SEESERSGSS

STDSU

SVC

TCTCRI

TLECTF

TOPTSOZPBZVRVAVU

dezleg|nd func}ionarea {i serviciile sistemului energiei ... · 4.2.5 echilibrarea costurilor...

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