Optimal Market Settlements Incorporating Voltage Stability Considerations and FACTS Devices

Transcription

1 Master of Scence Thess n the Internatonal Master Degree Programme, Electrc Power Engneerng v TCSC p v δ xc Y v δ Optmal Market Settlements Incorporatng Voltage Stablty Consderatons and FACTS Devces Thess for the Degree of Master of Scence Raúl Bachller Preto Department of Energy and Envronment Dvson of Electrc Power Engneerng CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden, 008

2 Optmal Market Settlements Incorporatng Voltage Stablty Consderatons and FACTS Devces Raúl Bachller Preto Examner: Dr. Tuan A. Le, Chalmers Unversty of Technology Supervsor: Dr. Tuan A. Le, Chalmers Unversty of Technology! Göteborg, Sweden 008

3 " # "$ % " &" $$ %$ ' ( )* +,,- Orgnal pcture n the cover extracted from: Danels S. Krschen and Goran Strbac. Fundamentals of Power System Economcs, John Wley & Sons, Ltd August 006 ISBN 10: (H/B) $.$ /$$ +,,-0 $ "%% )1 /"$ $ %/" 34+56!7& 1 "8 936:,;<4 ==+ 4,,,

4 Be a dreamer s the begnnng of our realzatons To all people I love v

5 Acknowledgments /$ 1 # /$ & % > $ "%% )1 /"$ $ %/"8 Frst of all, I wsh to thank my supervsor and examner Dr. Tuan A. Le at the department for the opportunty to explore n depth the nterestng area of electrcty market through ths work and hs lectures. A very specal grattude to my study coordnator Maln Blomqvst, Chalmers Studentcentrum Maskngränd, to be so knd wth me every tme I went to her offce. I am sure that wthout your help I could not have deal alone wth all admnstratve procedures related to the subects durng my two-year stay at Chalmers. I would lke to thank to my two frends Adela and Alba to be so patent wth me every tme I phoned them. You both are specal people. I am so happy to know you. Moreover, thanks to Andressa and Raquel for those nce coffees n the mddle of the last weeks before fnshng ths work that made me feel so good. Thanks a lot to those people I met who helped me n dfferent ways wthout beng aware of that. Naturally, the most heartfelt thanks to my parents to encourage me every day that I was far away from them. v

6 ( ) $ "%% )1 Chalmers Unversty of Technology Abstract In the past decade, many power utltes world-wde have been forced to change ther ways of dong busness, from vertcally ntegrated functonng to open-market systems. The reasons have been many, and dffered, across regons and countres. Reforms were undertaken by ntroducng commercal ncentves n generaton, transmsson and dstrbuton of electrcty, wth n many cases, large effcency gans. Though ths may seem farly straghtforward at frst glance, there are several complextes nvolved n restructurng and several new ssues have surfaced. Recent large-scale power system blackouts n the USA and Europe have gven us a wake-up call on the vulnerablty of our power systems that they are beng operated much closer to the lmts than ever before. Ths study ams at nvestgatng the changes n power market transacton levels when takng nto account the voltage stablty consderaton and FACTS devces n the market settlement scheme. The study s based on a securty constraned optmal power flow (SC- OPF) framework for a combned blateral-and-pool electrcty market. An IEEE 30-bus test system s used n the study n whch four separate cases are analyzed,.e., a base case, a base case wth securty margn and these two cases wth FACTS devces ncluded. It s found that the voltage securty constrants could help ndependent market operators (IMO) to nclude a suffcent margn for allowable power transactons to ensure the system securty whle maxmzng the socal welfare. It s also shown n the study that the use of FACTS devces (e.g., TCSC) can lead to an ncrease of up to 107% n avalable load capacty (ALC) for the same total transacton level (TTL) and an ncrease n TTL by up to about 3.5% when keepng the securty loadng factor of the system constant. Furthermore, the payment to the IMO s decreased by 4% due to congeston relef effects of FACTS. It can be concluded that power systems wll be operatng wth a larger securty margn wth the proposed market settlement. It s, however, mportant to note that the socal welfare s compromsed wth the ncreased securty margn. Keywords: Deregulated electrcty markets, voltage stablty, optmal power flow, FACTS devces, congeston management, socal welfare. v

7 Contents Chapter 1 Introducton 1.1 From regulaton to deregulaton: A background Types of electrcty markers System Relablty: An mportant concept Work outlne Chapter Voltage Stablty and Transfer Capablty.1 Voltage Stablty Voltage collapse revew Contnuaton Power Flow (CPF) P-V curves Avalable Transfer Capablty (ATC) A case study Voltage Stablty analyss: A bref revew Voltage collapse Optmal Power Flow ncludng voltage stablty consderatons Chapter 3 Modellng of Electrcty Markets: An Optmal Power Flow Formulaton 3.1 Market model based on OPF Formulaton of an Optmal Power Flow framework Modellng of transmsson lnes Modellng of Flexble Alternatng Current Transmsson Systems (FACTS) Statc model of TCSC Market models consdered Base OPF OPF wth FACTS v

8 3.4.3 SM-OPF SM-OPF wth FACTS Concluson Chapter 4 Market Settlement wth Voltage Stablty Consderatons and FACTS devces 4.1 Test system Market settlement structures Market settlement analyses Base OPF wthout FACTS devces Base OPF wth FACTS devces SM-OPF wthout FACTS devces SM-OPF wth FACTS devces Chapter 5 Conclusons 5.1 Conclusons Future research drectons Appendx A A Market and transmsson system data A.1 14 bus test system A. 30 bus test system Appendx B B Smulaton results References References v

9 Lst of fgures Fg 1.1 Monopoly model of electrcty market Fg 1. A wholesale electrcty market Fg 1.3 General structure of decentralsed or smple aucton market Fg 1.4 Market prce matchng n smple aucton markets Fg 1.5 General structure of hybrd market Fg 1.6 Classcal classfcaton of power system stablty Fg.1 Contnuaton Power Flow scheme Fg. Smple two bus grd Fg.3 P-V curves at dfferent power factors Fg.4 Graphcal representaton of ATC, ETC, TRM and TTC Fg.5 P-V curves for some representatve buses of the 14 bus test system Fg.6 P-V curve and representaton of MLC, ALC and current operaton ponts Fg. 7 P-V curve for bus 14 when only generator reactve power lmts Fg.8 P-V curve for bus 14 when only apparent power flow lmts through Fg.9 P-V curve for bus 14 for dfferent contngences takng nto account Fg 3.1 Lumped -equvalent model of a transmsson lne Fg 3. Smply FACTS classfcaton based on man effects Fg 3.3 Crcut dagram of dfferent FACTS devces Fg 3.4 Two mprovements due to TCSC Fg 3.5 Lumped -equvalent model of a transmsson lne wth TCSC Fg 3.6 Inecton model of TCSC Fg 4.1 Modfed IEEE 30 bus test power system Fg 4. Input for ISO n hybrd markets Fg 4.3 Studed market structures x

10 Fg 4.4 Locaton of most congestng lnes and TCSC unts Fg 4.5 Effects of TCSC Fg 4.6 Comparson of total transacton level (TTL) n dfferent scenaros Fg 4.7 Comparson of PAY_IMO n dfferent scenaros Fg 4.8 Congeston ratos n lnes (-4) and (10-1) Fg 4.9 Socal Welfare and TTL for dfferent compensaton levels Fg 4.10 Socal Welfare and TTL for dfferent compensaton levels Fg 4.11 Total transacton level (TTL) for dfferent loadng condtons Fg 4.1 LMP for dfferent loadng condtons Fg 4.13 Power demanded and Power suppled Fg 4.14 TTL comparson before and after usng FACTS devces Fg 4.15 Effects of usng FACTS devces Fg 4.16 Effect of FACTS devces on LMP and power supply Fg 4.17 TTL comparson between SM-OPF wth and wthout FACTS Fg 4.18 PAY_IMO comparson between SM-OPF wth and wthout FACTS Fg A bust test system Fg A. 30 bust test system x

11 Lst of tables Table.1 General results wthout consderng any lmt Table. General results consderng voltage lmts Table.3 General results consderng generator reactve power lmts Table.4 General results consderng power flow lmts Table.5 General results consderng dfferent lne outages Table.6 Classfcaton of voltage collapse approach Table.7 Classfcaton of statc methods Table.8 Classfcaton I of OPF wth voltage stablty crtera Table.9 Classfcaton II of OPF wth voltage stablty crtera Table.10 Dfferent ways of formulatng mult obectve functons Table.11 Formulaton of N-1 contngency Table 3.1 Benefts of FACTS devces for dfferent applcatons Table 4.1 Global results of base OPF Table 4. Results of base OPF wthout S lmts Table 4.3 More congested lnes and the effect of FACTS on them Table 4.4 Comparson between OPF and OPF-FACTS Table 4.5 Comparson between OPF and OPF-FACTS based on congeston Table 4.6 Man results for null and maxmum securty margn Table 4.7 Effects shown n Fg 4.15 of usng FACTS devces Table 4.8 Comparson between SM-OPF wthout FACTS and SM-OPF wth FACTS Table 4.9 Relatve varaton owng to FACTS Table 4.10 Securty maxmzaton and socal welfare maxmzaton Table A.1 Lne data 14 bus test system Table A. Bus and Load data 14 bus test system x

12 Table A.3 Lne data 30 bus test system Table A.4 Generators and Synchronous condensers data 30 bus test system Table A.5 Bus and Demand data 30 bus test system Table B.1 Results OPF-FACTS wthout apparent power flow lmts Table B. Results OPF-FACTS wth apparent power flow lmts Table B.3 Results SM-OPF wthout apparent power flow lmts and = Table B.4 Results SM-OPF wthout apparent power flow lmts and max Table B.5 Results SM-OPF wth apparent power flow lmts and = Table B.6 Results SM-OPF wth apparent power flow lmts and max Table B.7 Results SM-OPF-FACTS wthout apparent power flow lmts and = Table B.8 Results SM-OPF wthout apparent power flow lmts and max Table B.9 Results SM-OPF wth apparent power flow lmts and = Table B.10 Results SM-OPF wth apparent power flow lmts and max x

13 Lst of terms Acronyms: ALC ATC CBM CPF CPF-OPF DISCOS ESCOS ETM FACTS GAMS GENCOS IEEE ISO LIB LMP MCP MLC MO OPF PAY_IMO PCPDIPM PDIPM PSAT SM-OPF SNB STATCOM SVC TCPS TCSC TRANCOS TRM TTC TTL UPFC Avalable Load Capacty Avalable Transfer Capablty Capacty Beneft Margn Contnuaton Power Flow Contnuaton Power Flow Optmal Power Flow Dstrbuton Companes Energy Servce Companes Exstng Transmsson Commtments Flexble Alternatng Current Transmsson Systems General Algebrac Modelng System Generaton Companes Insttute of Electrcal and Electroncs Engneers Independent System Operator Lmt-Induced Bfurcaton Local Margnal Prce Market Clearng Prces Maxmum Loadng Condton Market Operator Optmal Power Flow Payment to the Independent Market Operator Predctor Corrector Prmal Dual Interor Pont Algorthm Prmal Dual Interor Pont Algorthm Power System Analyss Toolbox Securty Margn Optmal Power Flow Saddle-Node Bfurcaton Statc Compensators Statc Var Compensators Thyrstor Controlled Phase Shfter Thyrstor Controlled Seres Compensaton Transmsson Companes Transmsson Relablty Margn Total Transfer Capablty Total Transacton Level Unversal Power Flow Controllers x

14 Acronyms: f x p g h v δ r x B sh B G Y Power flow equatons Dependent varables Control varables Equalty constrants Inequalty constrants Bus voltage modules Bus voltage arguments Lne resstance Lne reactance Shunt chargng Lne suceptance Lne conductance Lne admttance modules ψ Lne admttance arguments k Multplers to desgnate the rate of load or generaton change P L Load actve powers Q L Load reactve powers P Generator actve powers G Q Generator reactve powers P s P P G D Q S P n Q n φ D Supply actve power bds Demand actve power bds Actve power flow Reactve power flow Apparent power flow Actve necton powers Reactve necton powers Power factor angle V lm Voltage lmts I lm Current lmts S lm Apparent power flow lmts C Supply bd prces S C D λ Demand bd prces Loadng parameter xv

15 Chapter 1 Introducton T he am of ths chapter s to provde a general descrpton of the process that electrcty sector has undergone snce the past decades whch has resulted n deregulated envronments. In addton, the three most common market structures, namely, centralzed, pool and hybrd, are descrbed and compared. Moreover, t s explaned the reasons why securty and economc are causes of great concern, more than ever before, due to the mplementaton of competence wthn ths ndustry. Therefore, concepts such as relablty, voltage stablty, voltage collapse and so forth are defned. Fnally, t s ponted out the organzaton of ths work. 1.1 From regulaton to deregulaton: A background In the past decades, the electrcty ndustry has undergone a worldwde restructurng process toward deregulaton, whch has nfluenced on several aspects of the market, such as busness, servce provded and securty [1]. Some years ago, most electrcty markets all over the world were structured vertcally based on monopoly rules. That means, man actvtes, such as generaton, transmsson and dstrbuton, were controlled by ether a sole or a reduce number of them, see Fg 1.1. The man obectve of these downward ntegrated utltes was to mnmze operatng cost satsfyng all constrants n the system. Governments or central authortes turned out to be pvotal partcpants n the regulated scenaro whch decded rates passed on to the consumers n return for the servce offered to all of them. The fact s that ths settlement worked properly for many years n the electrcty framework. However, dscrepances wth those pllars over whch phlosophy centralsed system were reled on started to be artculated by customers trggered manly due to two events. The frst one would be the ncrease n electrcty rates owng to a rse n fuel prces durng the seventes. The second fact that acted as catalyst for those who were not satsfed wth the monopoly orentaton of the sector was the postve results, n terms of prces, qualty and effcency, reported by other ndustral felds whch had been deregulated prevously, such us flght companes. Therefore, the dea of ntroducng competton n the electrcty sector took shape n the late seventes. On the other hand, t - 1 -

16 Chapter 1 Introducton seems that not all opnons are n favour of ths new trend snce t s ponted out that vulnerablty to volatlty of prces and prce spkes because of gamng actvtes could be much more frequent than n tradtonal centralsed structures. GENERATION TRANSMISITION DISTRIBUTION CONSUMER ENRGY FLOW MONEY FLOW Fg 1.1 Monopoly model of electrcty market. Deregulaton n electrcty markets [] can be analysed based on two ponts of vew. Frst, from the ndustry sde, the ntroducton of competton has, above all, altered the economcs perspectve of the sector. Second, from a controller standpont, besdes the latter consderaton, securty has rsen as an ssue of great concern, more than ever before, due to the reasons whch wll be ponted out n secton 1.3. The restructurng process has brought several new enttes to the market and has redefned the scope of others nherted from the vertcal structure. Thus, t s worth mentonng brefly ther man features n order to understand whch are the nteracton and purpose of them [] [7]. Therefore, the followng partcpants take part on most of deregulated markets: - GENCOS (Generaton Companes) - TRANCOS (Transmsson Companes) - DISCOS (Dstrbuton Companes) - ESCOS (Energy Servce Companes) - Customers - ISO (Independent System Operator) - MO (Market Operator) Generaton Companes (GENCOS) are the producers and sellers of electrcty, beng responsble of the nstallatons and equpments requred. Transmsson Companes (TRANCOS) own and operate the transmsson system ensurng the transportaton of electrcty from generators to customers. Ths actvty s stll regulated snce t s not proftable bult redundant facltes of ths nature. Dstrbuton Companes (DISCOS) own and operate local dstrbuton companes. They are allowed to buy electrcty through ether a spot/hybrd market, or drectly usng contracts wth generaton companes wth the am of supplyng that energy to end-use customers. Energy Servce - -

17 Chapter 1 Introducton Companes (ESCOS) work as dstrbuton companes n the market but they do not own local dstrbuton companes. They mght be large ndustral users, pool customers or prvate companes whch obectve conssts of purchasng power at the least cost. Customers are the last users of electrcty n the chan. As a result of deregulaton, they have at ther dsposal dfferent ways to buy electrcty. Independent System Operator (ISO) s responsble for ensurng the relablty and securty of the whole system. The role of the ISO depends on the market structure. Therefore, there are two man tasks whch can be managed by ths entty or be delegated to another. Normally, t provdes dfferent servces, such as emergency reserves or reactve power. On the other hand, ISO can be nvolved n the market transacton process (e.g. Ontaro). Fnally, Market Operator (MO) s n charge of market transacton when ths actvty s not handled by the ISO (e.g. old Calforna). Economcally, ths new framework could be analysed through the monetary flow establshed among the latter players [8]. In a vertcal structure, generator companes were taker prce snce they were allowed to fx the electrcty rates on account of ther domnant poston on the market. Thus the cash flow was practcally undrectonal, from the fnal consumer to the producer, beng dffcult to segregate the cost ncurred n ntermedate actvtes, such as generaton, transmsson and dstrbuton. However, n a horzontal structure, such as the one proposed by deregulaton, prces are determned ether by bd aucton agreements between GENCOS and DISCOS/ESCOS accordng to specfc market rules or drectly through contracts between the partcpants, see SUPPLIERS GENCO 1 GENCO... Wholesale seller BIDS TRANCOS ISO / MO PURCHASERS BIDS DISCO 1 DISCO... Wholesale customer Retaler Consumer 1 Consumer 3 Consumer 3... ELECTRICITY TRANSACTION Fg 1. A wholesale electrcty market

18 Chapter 1 Introducton That means, the revenues for producers are not ensured anymore, dependng manly on the prce strategc chosen, nstead. For ths reason, prce ndcators are crucal n order to be compettve and proftable [3][4][5][6]. Hence, t s expected that the latter nfluents postvely on customers n terms of reducton of payments and mprovement of servce qualty. Accordngly to the prevous statement, t seems clear that the amount of energy that the system has to deal wth have ncreased to date owng to the obectve of producers of maxmzng profts and the rapd ncrease performed by electrcty demand durng the last years. However, that would be nfeasble, f those standards of securty used n the regulated framework are not releved snce they were overszed n order to acheve the least level of rsk. Therefore, at ths pont, t has ust been ntroduced one of the man threats accentuated as a result of deregulaton, system relablty. 1. Types of electrcty markers As mentoned before the deregulaton process n electrcty sector, all over the world, s not unform, though goals are almost dentcal. Therefore, grade of mplementaton and model features can be vared. Thus, market models can be categorzed nto three man groups, namely, central markets, decentralsed markets and hybrd markets [9]. Followng, some of the most remarkable characterstcs of each one are ntroduced [10]. Centralsed markets (e.g. the old U.K. market, Chle 198 [11][1], PJM 1997 [14], New York 1998 [17], New England 1999 [18]) can be consdered as unt commtment and Optmal Power Flow (OPF) problems, where central market operaton and transmsson system operaton are montored by a sole authorty. Ths model s typcally mplemented on vertcal market structures. In decentralzed markets (e.g. U.K [13], Alberta 1996, Span 1998 [15], Calforna [16]), responsblty for operatng the system s shared between two enttes the two enttes ntroduced prevously, Market Operator (MO) and Independent System Operator (ISO), see Fg 1.3. The former determnes market schedules and the Market Clearng Prces (MCP) based on those bd submtted by partcpants usng a smple aucton mechansm. In ths case, a unque prce, determned by matchng the hghest bd demand wth the lower supply demand, s used for all transactons, as shown n Fg 1.4. MO ISO SUPPLY BIDS DEMAND BIDS Smple Matchng Process ECONOMIC Analyss Tools VIABILITY OF TRANSACTIONS FINAL MARKET SOLUTION Fg 1.3 General structure of decentralsed or smple aucton market. Securty and economc settlements are decoupled to each other

19 Chapter 1 Introducton $/kwh Sellers Cleared Prce Margnal Prce Buyers Market Energy kwh Fg 1.4 Market prce matchng n smple aucton markets. Fnally, hybrd (e.g. Ontaro 00 [19]) markets are based on spot prcng theory and OPF methods. In ths case market operaton and system operaton are not decoupled, Fg 1.5. Furthermore, the prce s not unform, beng affected by bd values and several other factors such as congeston and locaton. Commonly, t s sad that decentralzed markets seem more transparent to all partcpants though the need of two dfferent operators. However, the rapd development of computer scence provdes effcent tools that make hybrd models more attractve than tme ago. For ths reason, durng the last year several studes based on ths technque have been publshed wth the am of demonstratng a number of advantages and proposng nnovated technques manly related to mathematcal optmzaton algorthms and assessment of securty ssues, such as those lsted n secton.3 (e.g. [9][6][9][53]-[61]). ISO SUPPLY BIDS DEMAND BIDS OPF ECONOMIC SECURITY FINAL MARKET SOLUTION Fg 1.5 General structure of hybrd market. Both securty and economc ssues are management by a unque entty, the ISO usng OPF technques. 1.3 System Relablty: An mportant concept Relablty s a wdely used term related to many aspects of system operaton. Normally, t s comprsed of two concepts: securty and adequacy. On the one hand, adequacy, can be defned as the ablty of electrc systems to supply the aggregate electrcal demand and energy requrements of ther customers at all tmes, takng nto account scheduled and reasonably expected unscheduled outages of system elements

20 Chapter 1 Introducton On the other hand, securty s the ablty of the electrc systems to wthstand sudden dsturbances such as short crcuts or unantcpated loss of system elements [10]. Moreover, other mportant concept related to securty s the so called power system stablty [0][1][] whch can be defned as the capacty of a system to mantan an operatng equlbrum pont after beng subected to a dsturbance for gven ntal operatng condtons. In addton, some notons have to be taken nto account: montorng certan varables t s possble to determne the nature of the nstablty, the sze of the dsturbance has an nfluence on the tool used to address t and the tme framework avalable to allevate the problem s other essental. Furthermore, a classcal classfcaton of power stabltes extracted from [] s shown n Fg 1.6 Power System Stablty Rotor Angle Stablty Frequency Stablty Voltage Stablty Small- Dsturbance Angle Stablty Transent Stablty Large- Dsturbance Voltage Stablty Small- Dsturbance Voltage Stablty Short Term Short Term Long Term Short Term Long Term Fg 1.6 Classcal classfcaton of power system stablty. Brefly, small dsturbances can be assocated, for example to load changes, whle large dsturbances to fault condtons. Based on tme frame, short-term events are related manly wth dynamcs behavours but long-term events do not demand a so fast reacton. Angle stablty, s the capacty of synchronous generators to mantan synchronsm after beng subected to a dsturbance. Thus, t s related to the equlbrum among mechanc and electrcal varables. Frequency stablty, on the other hand, s the ablty of the system to mantan a steady frequency, after a sgnfcant mbalance between generaton and demand power occurs. Fnally, voltage stablty s the capacty of a power system to mantan steady voltages at all buses after a dsturbance from an ntal operatng condton. In ths work, the latter s the one analysed and appled

21 Chapter 1 Introducton Moreover, t s mportant to dstngush three concepts, such as voltage stablty, voltage nstablty and voltage collapse. The frst one, voltage stablty, s defned n the prevous paragraph. However, t s sad n [1] that a system enters a sate of voltage nstablty when a dsturbance, ncrease n load demand or change n system condton cause a progressve and uncontrollable drop n voltage. Normally, t s related to the nablty to meet the demand for reactve power. The heart of the problem s usually the voltage drop that occurs when actve power and reactve power flow through the nductve reactance assocated to the transmsson network. Fnally, voltage collapse s the process by whch the sequence of events accompanyng voltage nstablty leads to a low unacceptable voltage profle n a sgnfcant porton of the power system lead to a total or partal blackout Ths concept can be analysed from a statc or dynamc vewpont. The statc percepton s related to load flow problems and maxmum power that can be transferred before ths problem appear, whle the dynamc one requres more detaled, thus complex, models of those elements that consttute power networks, such as loads, generators, lnes and so forth. Furthermore, the dfference between transmsson congeston and system stablty should be ponted out. On the one hand, transmsson congeston could occur when the dspatchng of transacton or an unexpected power varaton due to, for example, a contngency lead to a volaton of some transmsson securty constrants, whch are typcally thermal lmts, voltage lmts and stablty lmts. On the other hand, system stablty, as sad before, s related to the capacty of the system to wthstand sudden dsturbances remanng as close as possble to the ntal condton that mght commt congeston. The man consequence of congeston s reducton of generaton n comparson to that that could be acheved n ts absence. Therefore, some amount of power s nether generated not delvered through the network to cover the expected demand. Hence, t leads to cheaper generaton but hgher cost of energy for the users. In facts, n deregulated markets models, the relef of ths problem can produce sgnfcant varaton n prces dependng on each partcpant and bds. What s more, t has been demonstrated the relaton between securty and cost, whch relevance has rsen due to the development of compettve electrcty markets. In ths work, senstvty or margnal parameters calculated wthn an optmzaton algorthm are used n order to determne the nfluence of the prevous notons on prces provdng, at the same tme, relevant strategc sgnals to all market partcpants. Certanly, the man am related to ths ssue s to be able to acheve a farly dstrbute securty costs among enttes nvolved valuatng postvely those whch do not contrbute to network congeston. It s possble to ndcate same real blackouts that have taken place n dfferent networks all around the world whch causes can be categorzed wthn the prevous dscusson [3][4][5]. For example, n 14 August 003, a loss of voltage stablty due to a seres of transmsson lne contngences and reactve power shortages produced a blackout n eght U.S. states and two Canadan regons. As a result, approxmately 50 mllon people were affected and 11% of the total load served n the Eastern Interconnecton of the North Amerca system was nterrupted (around 63 GW) whch - 7 -

22 Chapter 1 Introducton ended up costng $6.4 bllon to the U.S. economy. Unfortunately, ths has not been the sole case that authortes and market partcpants have had to cope wth recently. Thus, n 3 September 003, 4 mllon people were affected n Sweden and Demark. Moreover, fve days later, n 8 September 003, the Italan system lost 6400 MW. Therefore, these fgures hghlght the mportance of takng nto account serously securty lmts thoroughly and be aware of whch they are at dfferent loadng condtons. In short, n compettve electrcty markets, the so called zero-rsk mandate [6], characterstc n the regulated scenaro, s not sustanable any more snce customers are not wllng to pay hgh prces for power on account of reducng rsk beyond some reasonable lmt at any cost. For ths reason, t s crucal to montor and control securty ssues n nowadays deregulated scenaros wthout underestmatng the mportance of the new economc aspects nvolved. Ths work proposes a method to determne the most sutable soluton after a bd process takng nto account dfferent levels of securty margn decded by the ISO n hybrd markets. Thus, t wll be possble to maxmze the socal welfare, ntroduced n Chapter III, and ensure a certan power capacty to wthstand other knds of transacton of unscheduled events n the system. 1.4 Work outlne The present work s organsed n the followng manner: Chapter ntroduces one of the most mportant concepts related to securty ssues n nowadays energy management scenaro, the so call voltage stablty. A loadng parameter whch provdes the possblty to nclude operatonal margns wthn market models s the core of ths secton. Moreover, Contnuaton Power Flow (CPF) turns out to be a useful tool to depct PV curves whch can be used to demonstrate graphcally securty crtera. An extensve lterature revew of most mportant studes publshed to date about ths topc s presented. Fnally, a 14 bus test system s chosen to represent some of ts PV curves and determne voltage stablty lmts usng a computer tool called Power System Analyss Toolbox (PSAT). In Chapter 3, dfferent concepts related to optmal power flow market models are defned. Furthermore, FACTS devces are ntroduced from dfferent vewponts, such as typology, usage, characterstcs and so forth. Fnally, t s developed the mathematcal formulaton of the four models based on OPF analysed n Chapter 4. In Chapter 4 the results obtaned from those models proposed and developed n the prevous chapter are presented and commented. General Algebrac Modelng System (GAMS) s used to carry out the smulatons and MATLAB to represent graphcally the results usng an nterface between both programmes. In Chapter 5, man conclusons from ths study are presented. Moreover, dfferent future research drectons are provded

23 Chapter Voltage Stablty and Transfer Capablty V oltage stablty, defned n Chapter 1 as the ablty of power systems to reman bus voltages wthn certan acceptable ntervals ether f t s operated under normal condtons or undergo some contngences, has to be consdered serously owng to the consequences of voltage stablty fals that can lead to blackouts. Therefore, t s essental to understand those basc concepts and tools related to ths phenomenon. Thus, t would be possble to use these analyses to enhance the new energy management envronment..1 Voltage Stablty.1.1. Voltage collapse revew A loadng parameter, λ, s used n power system analyses n order to apply a general mathematcal theory to classfy nstabltes, namely, bfurcaton theory [0]. Moreover, ths methodology reports quanttatve nformaton n the neghbourhood of partcular ponts, such as collapse ponts and unstable ponts. Therefore, system equatons need to nclude, besdes state varables, a new set of parameters, λ, as follows: f ( x, λ) = 0 (.1) Several studes have been publshed based on ths theory appled on power system where dfferent formulatons usng the latter parameter can be outlned [7][8]. Therefore, a classcal model could be the followng one: P L Q P L G = P + λ( k P ) (.) L L L 0 = Q + λ( k Q ) (.3) L L L 0 = P (1 + λk ) (.4) G G 0-9 -

24 Chapter Voltage Stablty and Transfer Capablty where k L k G are multplers to desgnate the rate of load or generaton change at bus as λ changes; P L 0 Q L 0 PG 0 are the ntal load and generaton assocated to the current operaton pont; P Q, terms multpled by the loadng factor are called power drectons. L L However, smplfcatons of the prevous expressons lead to the followng two alternatves whch can be related to electrcty market [9]: P P G L = P + λp (.5) 0 G L S D = P 0 + λp (.6) P = 1+ λ )( P + P ) (.7) G ( G 0 S ( 1+ )( PL P 0 D P = λ + ) (.8) L where λ s the loadng parameter; PL P 0 G are load and generaton for the current 0 operatng pont; P P are power supply and demand at each bus. S D In [9] s demonstrated that load drectons, powers multpled by λ, n (.5)- (.6) depend only on the market partcpants, beng ths formulaton approprated to determne the mpact of aucton on securty and to mnmze that effect. Nevertheless, (.7)-(.8) optmzes the aucton results and the transacton outsde the bd process to mprove the system securty. Accordng to the mathematcal theory ntroduced prevously, t s possble to dstngush two types of sngular ponts assocated to the condton of Jacoban matrx, namely, SNB (Saddle-Node Bfurcaton) and LIB (Lmt-Induced Bfurcaton) [0][8][9]. The latter s related to the dsappearance of steady-state solutons when system control lmts are reached, for example maxmum generator reactve power lmts. The former s characterzed by two equlbrums, one stable and one unstable, beng the maxmum power transfer capacty when not other boundares get actve before. Therefore, both of them mght lead to voltage collapse and consequently to those problems ntroduce n Chapter 1. However, t s worth mentonng that voltage stablty lmt not always s assocated to ether SNBs or LIBs snce bus voltage lmts (related to V lm ), thermal lmts (related to I lm ) or maxmum transmsson lmts (related to S lm ) can be reached before to the other two bfurcaton ponts. Furthermore, LIBs not have to concde wth SNBs [9]

25 Chapter Voltage Stablty and Transfer Capablty.1.. Contnuaton Power Flow (CPF) It s consdered that CPF (Contnuaton Power Flow) s an effcent and useful tool to determne the so called P-V curves and maxmum loadng ponts [0][8][9]. Moreover, nformaton provded by ths technque can be used to calculate a seres of senstvty factors of dfferent operatng ponts respect to the loadng parameter. Therefore, t s possble to understand how system varables are affected by the latter factor and thus establsh whch should be vared n order to acheve stablty mprovement. Brefly, a general formulaton of CPF, conssts of an teratve process comprsed by two steps, predctor and corrector. Frst, the predctor step estmates a new soluton from an ntal pont usng a tangent vector. Second, the corrector step that can be a local parameterzaton or a perpendcular ntersecton s used to locate the exact soluton C usng a modfed power flow to calculate the proper value of λ. Fg.1 represents graphcally ths method. predctor corrector Bus Voltage crtcal pont Load Fg.1 Contnuaton Power Flow scheme P-V curves The so called P-V curves are useful graphcal representatons that depct the evoluton of voltage at dfferent bus n power electrcal systems [0][30]. It s possble to demonstrate the concept of these curves usng a smple two bus grd, as shows n Fg.. v1 δ 1 =0 z 1 ψ R v δ S = P + cos( θ ) Fg. Smple two bus grd. Q

26 Chapter Voltage Stablty and Transfer Capablty From the two-port equatons: ψ 1 1 V1 V V P = cos( δ ψ ) cos( ) (.9) Z Z Q V1 V V = sn( δ ψ ) sn( ) (.10) Z Z ψ 1 1 Usng the next trgonometrc relaton cos (x) + sn (x) = 1 wth (.9) and (.10), ( P 4 + Q )Z ( P cos( ψ ) + Q sn( ψ )) V V V + V 0 (.11) 1 1 = From (.11) assumng constant power factor Q = P tg(cos( )), θ V b ± b 4ac = x = (.1) a P-V curves are generated for dfferent values of P keepng constant the power factor n (.11)-(.1). It s possble to observe n Fg.3 that for partcular values of load actve power and power factor, two dfferent voltage levels are determned. The one at the top s sad stable snce t s assocated to hgher voltage and lower current than the other. Thus the power system can be operated at ths pont. When both ponts concde, a SNB s reached. Moreover, as the compensaton s accentuated both voltage and maxmum power transfer lmt ncrease n comparson to a lower or lag compensaton. However, for hgh lead power factors, t s more complcated to determne the dfferent between the feasble and nfeasble soluton. Indeed, voltage can even rse as the actve load actve power does as a result of domnant capactance effect. Furthermore, other type of curves, namely, P-Q curves have been analysed n [30] to determne steady-state voltage stablty lmt, though further research s needed n order to demonstrate ts features n large systems. Load voltage magntude [pu] la 0. 9 la unty 0.95 lead pf=0.9 1 Load actve 3 power 4 5 [pu] 6 7 Fg.3 P-V curves at dfferent power factors

27 Chapter Voltage Stablty and Transfer Capablty.. Avalable Transfer Capablty (ATC) Accordng to [31] ATC s a measure of the transfer capablty remanng n the physcal transmsson network for further commercal actvty over and above already commtted uses Ths term s defned through the Total Transfer Capablty (TTC), Transmsson Relablty Margn (TRM), exstng transmsson commtments and the Capacty Beneft Margn (CBM): Where [31][6], ATC = TTC TRM ETC (.13) Total Transfer Capablty (TTC) s defned as the amount of electrc power that can be transferred over the nterconnected transmsson network n a relable manner whle meetng all of a specfc set of defned pre- and postcontngency system condtons. { P, P P } TTC = mn. (.14) max max, I V lm lm max S Transmsson Relablty Margn (TRM) s defned as that amount of transmsson transfer capablty necessary to ensure that the nterconnected transmsson network s secure under a reasonable range of uncertantes n system condtons. Capacty Beneft Margn (CBM) and Exstng Transmsson Commtments (ETC) s defned as that amount of transmsson transfer capablty reserved by load servng enttes to ensure access to generaton from nterconnected systems to meet generaton relablty requrements. In Fg.4 the prevous concepts are explaned graphcally usng P-V curves of a system power system for dfferent scenaros accordng to whch lmt domnates. It s demonstrated that n order to make a thoroughly analyss, t s mportant to take nto account contngences snce the power transfer s remarkable lower than n absence of them. Hence, those measures mplemented toward to enhance power system securty can cover a wder range of probable stuatons. It s possble to smplfy (.13) when no contngences are taken nto account or not ntensve detaled analyss s needed, as follows: lm ALC = MLC TTL (.15) where ALC (Avalable Load Capacty) would be related to ATC, MLC (Maxmum loadng Condton) would be smlar to TTC and TTL (Total Transacton Level) would be assocated wth TRM

28 Chapter Voltage Stablty and Transfer Capablty Where the relaton wth the loadng parameter λ s establshed through the next expressons usng the concepts ntroduced n ALC = MLC TTL (.15(.15). MLC = ( 1+ λ C ) PL (.16) ALC = MLC PL = MLC TTL (.17) TTL P (.18) = L ALC = λ P = λ TTL (.19) C L C (a) (b) (c) Fg.4 Graphcal representaton of ATC, ETC, TRM and TTC. (a) Voltage lmts domnate. (b) Thermal lmts domnate. (c) Voltage stablty lmts domnate. (source [7] )

29 Chapter Voltage Stablty and Transfer Capablty..1. A case study The prevous concepts have been demonstrated usng PSAT (Power System Analyss Toolbox) [3] and a 14 bus test system 0 whch detals are presented n Appendx A.1. Moreover, a Contnuaton Power Flow (CPF) technque s performed n order to generate dfferent PV curves n several scenaros. In Fg.1Fg.5 t s possble to observe P-V curves assocated to three relevant buses of the test system. In ths case, no lmts have been taken nto account. Therefore, the stablty lmt s defned only by the maxmum power transfer that s related to the SNB ntroduced before. Numerc values are gven n Table.1. Operatng pont SNB Fg.5 P-V curves for some representatve buses of the 14 bus test system wthout consderng any of the followng lmts Vlm, Qglm, Slm Table.1 General results wthout consderng any lmt. base load 59 MW max pu MLC MW ALC MW However, when voltage lmts are ntroduced n the algorthm, results dffer from the latter smulaton. In ths case, the maxmum loadng level before any lmt get actve s lower than the one calculated wthout lmts (-6.33 %) Thus, the remanng transfer capablty s reduced by the same proporton. The frst bus whch voltage level s equal to one of the lmts s bus 14. Therefore, Fg.6 depcts the voltage profle of node ndcatng the defnton of ALC and MLC. Table. contans the numerc values assocated to ths smulaton

30 Chapter Voltage Stablty and Transfer Capablty Operatng pont Vlm Base load ALC MLC Fg.6 P-V curve and representaton of MLC, ALC and current operaton ponts for bus 14 when only voltage lmts are taken nto account Table. General results consderng voltage lmts. base load [MW] 59 max [pu] 3.80 MLC [MW] ALC [MW] On the other hand, when only generator reactve power lmts are mposed, effects on the system are even more sgnfcant, as t s demonstrated n Fg. 7 and Table.3. The maxmum loadng parameter reduced by 65.1% n comparson to the smulaton wthout takng nto account any lmt. Generator s the frst unt to reaches ts upper lmt. Operatng pont New maxmum loadng condton Fg. 7 P-V curve for bus 14 when only generator reactve power lmts are taken nto account

31 Chapter Voltage Stablty and Transfer Capablty Table.3 General results consderng generator reactve power lmts. base load [MW] 59 max [pu] 1.41 MLC [MW] 64.7 ALC [MW] Furthermore, power flow lmtaton through lnes has been also smulated reportng the results represented n Fg.8 and lsted n Table.4. The lne from bus 5 to bus 6 s the frst one whch reaches ts lmt. In ths case the maxmum loadng parameter suffers a declne by whch s smlar to the effect owng to ntroduce generator reactve power lmts. Operatng pont New maxmum loadng condton Fg.8 P-V curve for bus 14 when only apparent power flow lmts through lnes are taken nto account. Table.4 General results consderng power flow lmts. base load [MW] 59 max [pu] MLC [MW] 643. ALC [MW] 384. Fnally, n Fg.9 changes n PV curves owng to some contngences n terms of lne outages are depcted takng nto account voltage and power flow boundares. Therefore, for the outage of lne 1- the lmt s determned by the power flow through lne 1-5, whle for the outage of branch -4 the power flow through lne 5-6 lmted the maxmum loadng level. Table.5 contans a resume of the man results

32 Chapter Voltage Stablty and Transfer Capablty Operatng pont Fg.9 P-V curve for bus 14 for dfferent contngences takng nto account voltage and power flow lmts. Table.5 General results consderng dfferent lne outages. BASE CASE LINE 1- OUT LINE -4 OUT base load [MW] max [pu] MLC [MW] ALC [MW] Voltage Stablty analyss: A bref revew.3.1. Voltage collapse Snce voltage collapse usually leads to blackouts or extremely low voltage n sgnfcant areas of power systems, several methods have been proposed to date whch can be classfed n two dfferent groups, namely, statc and dynamc [7]. Table.6 contans the most remarkable features of both categores. Therefore, t seems nterestng to revew those methodologes publshed related to ths matter [34]

33 Chapter Voltage Stablty and Transfer Capablty Table.6 Classfcaton of voltage collapse approach. STATIC APPROACH DYNAMIC APPROACH - Quantfcaton of how close a partcular operatng pont s to the steady state voltage collapse pont and estmaton of the steady state voltage stablty lmt for the current operatng pont. - Requres not hgh rates of computaton tme and provdes senstve nformaton. - Provdes a closer nsght nto the system. - Analysng how dfferent utltes affect the voltage stablty form the vewpont of dynamc. - Requres a set of dfferental equaton to model exctaton elements. - Demands hgh rates of computaton and analyss tme. The statc approach of the problem can be dvded n two man categores such as, ndex and optmzaton methods [34][35][36]. Table.7 summarzes the man characterstcs. Table.7 Classfcaton of statc methods. SATATIC INDEX OPTIMIZATION - Provdes nformaton about the proxmty of voltage collapse usng dfferent knd of ndcators. - Smple and easy programmng - Determnes optmal control parameters to maxmze load margns to voltage collapse. - Wde range of purpose - Accurate results Index methods can be classfed n the followng way accordng to the lterature: Based on a normal load flow soluton (.e. L-ndex) [37][38][39][40]. Based on located power flow soluton pars [41]. Based on senstvty analyss (.e. VQ senstvtes) [39][4] Based on Newton-Raphon power flow Jacoban matrx (.e. sngular values, egenvalues) [43][44] Local qualtatve ndces (.e. load flow feasblty) [45] Normally, ndexes based on load flow soluton are smpler than those whch used senstvty analyss or sngular values owng to the straghtforward of methods related to load flow equatons. Arguably, a dsadvantage of usng mnmum sngular values as ndexes s the large amount computaton tme requred n performng a sngular value factorzaton for large matrxes. However these problems could be

34 Chapter Voltage Stablty and Transfer Capablty balanced because of the fact that the hgh senstvty of mnmum sngular value near nstablty lmts. In addton, t has been proved that one drawback of smulaton methods s the slowly convergence when the collapse regon s approached. For ths reason, t s complcated to determne the dstance between the operatng pont and the voltage collapse pont. Moreover, prevous studes are requred to determne the step for each teraton. However, the nvestgaton of voltage collapse usng load flow feasblty does not rely on load flow or optmal load flow smulatons. Therefore, these problems are over come. On the other hand, t s not possble to make so clear classfcaton of optmzaton methods owng to the wde range of purpose of ths technque [46]. Therefore some of the most mportant examples gathered n the lterature are lsted below. Optmal shunt and seres compensaton parameter settngs to maxmze the dstance to a saddle-node bfurcaton [47]. Reactve power margn from the pont of vew of voltage collapse s determned usng nteror pont methods. A barrer functon s used to ncorporate lmts [48] Determne the closest bfurcaton to the current operaton pont on the hyperspace of bfurcaton ponts [49]. Usng the maxmum load capablty of power system s examned usng nteror pont methods [50]. Interor pont optmzaton technque s used to determne the optmal PV generator settngs to maxmze the dstance to voltage collapse [51][5]. The dynamc approach [7] to voltage s emergng wth new studes. As a matter of fact, the role of reactve power n mantanng proper voltage profle n the system began recevng attenton. Lkewse, the mportance of dynamcs of the machnes, excters, tap changers as well as loads was found to effect voltage stablty sgnfcantly. The maor challenge for these methods s to demonstrate suffcent practcal applcablty n real system management. Thus, t s ustfed that although the classfcaton shown n Table.6, nteractons between dfferent knd of stabltes s possble..3.. Optmal power flow ncludng voltage stablty consderatons In ths secton a revew of the most mportant OPF formulatons publshed to date whch nclude securty constrants related to voltage stablty are presented. Therefore, Table.8 and Table.9 show a classfcaton of these methods accordng to several crtera

35 Chapter Voltage Stablty and Transfer Capablty Table.8 Classfcaton of OPF wth voltage stablty crtera accordng to market type, obectve functon and type of soluton. SIMPLE AUCTION MARKET HYBRID MARKET SIMPLE MULTI PURE NO PURE IT IS POSSIBLE TYPE OF MARKET OBJECTIVE FUNCTION MARKET SOLUTION N-1 SIMPLE ACTION SYSTEM WITH REDISPATH BASIC OPTIMAL POWER FLOW MAXIMUM LOADING DISTANCE SECURITY CONSTRAINED OPF-BASED ELECTRICITY MARKET VOLTAGE STABILITYCONS TRAINED OPF MARKET MODEL MIXED CPF-OPF TECHNIQUE X X X X X X X X X X X X X X X X X X X X X Accordng to Table.3, t s possble to observe that most studes have focused on hybrd markets [9]. Moreover, obectve functons can be classfed nto dfferent typology, namely, smple or mult. Smple obectve functons usually ntroduce economc terms such as, generaton cost, transmsson losses or socal beneft, whle multple obectve functon usually combne economc and securty ssues. The latter can be formulated n dfferent ways [53], such s shown n Table.10. It s nterestng to observe that voltage stablty constraned OPF-market model can use both smple and multple obectve functons. The soluton provded by these methods can be ether pure or no pure [9][54]. In other words, when the algorthm generates a set of solutons t s sad no pure market soluton; otherwse the market soluton s pure. Generally, mult obectve formulatons (.e. lnear combnaton) produce no pure market solutons because the explctly parameter dependence. For ths reason some technques, such as mxed CPF-OPF, tres to overcome ths problem

36 Chapter Voltage Stablty and Transfer Capablty Table.9 Classfcaton of OPF wth voltage stablty crtera accordng to how the voltage stablty s managed ITERATIVE COMPUTATION OF ATC AND REDISPATCH BUS VOLTAGE LEVEL INTRODUCTION OF A SECONDSET OF EQUATIONS ACTIVE POWER TRANSFER LIMITS ATC (avalable trans. capabl.) MULTIOPJECTIVE FUNCTION LOADING PARAMETER SIMPLE ACTION SYSTEM WITH REDISPATH BASIC OPTIMAL POWER FLOW MAXIMUM LOADING DISTANCE SECURITY CONSTRAINED OPF-BASED ELECTRICITY MARKET VOLTAGE STABILITYCONS TRAINED OPF MARKET MODEL MIXED CPF-OPF TECHNIQUE X X X X X X X X X X X X X X X X X Table.9 demonstrates that each technology of OPF can use more than one mechansm to ntroduce voltage stablty wthn the model. In all OPF model the bus voltage level s lmted usng a constrant that fxes upper and lower values. However, that s not enough to ensure voltage securty. For ths reasons another constrants are used [55][56]. Therefore, some methods ntroduce a second set of equatons whch are related to a loadng parameter dfferent to the current operatng pont [9] [53][54][57]. Ths dea conssts on measure and mantans the dstance between both operaton ponts. Arguably those algorthms whch used actve power transfer lmts calculated by off-lne procedures are easer than others; however they mght not reflect accurately the current stuaton studed. Undoubtedly, the ncorporaton of ATC (avalable transfer capablty) [58][31][6][58] s the most extended method snce mprove the performance of the actual power system state. Normally, ths magntude s tght lnked to the loadng parameter. Fnally, the loadng parameter can be ether fxed before solvng the OPF or calculated as a varable. Furthermore, mostly all knd of OPF formulaton can be extended to a mult-perod framework [9]. In that case, tme s ncluded to take nto account the varaton of the parameters n each perod. - -

37 Chapter Voltage Stablty and Transfer Capablty Table.10 Dfferent ways of formulatng mult obectve functons FORMULATION OF MULTI OBJECTIVE FUNCTION Lnear combnaton Fxed loadng margn Lnear combnaton wth a fxed loadng margn Modfed goal programmng On the other hand, the ntroducton of N-1 contngency analyss s one of the man goals of the proposed technques n ths feld snce t can provde more realstc solutons [54][57]. Therefore, two of the methods lsted before take nto account ths crteron. Table.11 summarzes these approaches. Normally, t s necessary teratve methods to programmng N-1 analyss [59]. Table.11 Formulaton of N-1 contngency. SECURITY CONSTRAIN OPF- BASED ELECTRICITY MARKET MIXED CPF-OPF TECHNIQUE ITERATIVE METHOD WITH N-1 CONTINGENCY CRITERIO MULTIPLE VSC-OPF WITH CONTINGENCY RANKING - Determnng the lowest SACT ( system wde avalable transfer capablty) - Usng senstvty factors. - The securty margn s determned usng an N-1 contngency crteron. Fnally, n [1] s demonstrated a tght relaton between reactve power, actve power and voltage nstablty phenomena. Therefore, smlar models to those ndcated prevously consder reactve power and voltage stablty together [60][61]

38 Chapter 3 Modellng of Electrcty Markets: An Optmal Power Flow Formulaton T hs chapter descrbes n detals the formulatons of the mathematcal models of deregulated electrcty markets. The models are based on an optmal power flow (OPF) framework. These nclude base OPF (Optmal Power Flow), base OPF wth FACTS (Optmal Power Flow usng Flexble Alternatng Current Transmsson Systems), SM-OPF (Securty Margn Optmal Power Flow) and SM-OPF wth FACTS (Securty Margn Optmal Power Flow usng Flexble Alternatng Current Transmsson Systems). The advantages of usng FACTS n power transmsson systems are ponted out. Thyrstor Controlled Seres Compensaton (TCSC), has been chosen among others n order to demonstrate the effectveness of usng FACTS devces n power operaton and market results. 3.1 Market model based on OPF As mentoned before three market structures are the most common around the world, namely centralzed or pool markets, decentralzed or smple aucton markets and hybrd or OPF-based markets [][8]. The mportance of the second and thrd model has arsen durng the last two decades owng to those deregulaton processes that electrcty sector has undergone. Therefore, t has been decded that ths work s gong to be focused on one of them, the hybrd or OPF-based market Formulaton of an Optmal Power Flow framework The OPF formulaton was ntroduced n the early 1960s by Carpenter and t has turned out to be a useful tool to analyse power systems [6]. It s characterzed to be a nonlnear programmng problem consstng of one obectve functon that must be optmzed n accordance wth a set of assocated equalty and nequalty constrants, as follows: - 4 -

39 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton Maxmze f ( p) (3.1) subect to g( x, p) = 0 (3.) hmn h( x, p) (3.3) h( x, p) h (3.4) max pmn p (3.5) p p max (3.6) n where f s the obectve functon, x R are the dependent varables, such as bus m voltage phasors and p R are the control varables,.e., power demand and supply bds h R n P D and P respectvely, S l R the nequalty constrants. g n l R R are the equalty constrants and Naturally, computer tools are of utmost mportance snce obtanng a soluton would be unfeasble n another way on account of the huge number of varables and nolnear equatons. Nowadays, the most used mathematcal method to solved non lneal programmng problems n power system s the nteror pont method [63][64]. The man reason of ths s ther computatonal advantages when large systems have to be analyzed snce nclude several operatonal and control lmts. Two are the most popular nteror pont methods, the prmal-dual nteror pont algorthm (PDIPM) and the predctor-corrector prmal-dual nteror pont algorthm (PCPDIPM). The maor dfferent between both s the ntroducton of nonlnear terms nto complementary equatons by the former. Bascally, all of them are based manly on Lagrandgan formulaton and Kuhn Tucker s Condtons [9]. These algorthms reports a seres of multplers that can be defned as margnal ndcators assocated to those varables that appear on the obectve functon. Therefore, t s possble to quantfy the senstvty of the obectve functon to a change n supply-demand market result and to the changes n unt operatng lmts [54]. The latter nformaton s of great mportance for the market partcpants snce the strateges could be reled on that. The obectve functon, f, used n ths work s the socal welfare defned as: =CDPD maxmze socal welfare f ( p) C P (3.7) where CS represents supply prce bbs, CD modelled demand prce bds, PS ntroduces amount of power supply and PD corresponds to bulk of power demand. It represents the maxmzaton of socal welfare related to producton and consumpton wthn the framework of electrcty markets [8]. It s comprsed of two terms. The frst one, namely consumer surplus, s the sum of accepted demand bds. In other words, power demand cleared tmes the bdng prce assocated to. The second term, producer surplus, corresponds to the accepted supply bds, defned as t the latter but usng cost and power producton values nstead. S S - 5 -

40 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton Moreover, the equalty constrants, g, represent the standard power flow equatons to both real and reactve power whch are reled on dependent and control varables. On the other hand, h, restrct lower and upper lmts of dfferent varables such as, voltage, real and reactve power outputs and transmsson lnes loadngs. 3. Modellng of transmsson lnes The model used to represent transmsson lnes s the lumped -equvalent, as s descrbed n the Fg 3.1 [70]: v δ Bus- P Q Y = G + B P Q Bus- v δ B sh B sh Fg 3.1 Lumped -equvalent model of a transmsson lne and power flowng through t The real and reactve power flow from bus- to bus- can be wrtten as follows: [ G cos( δ δ ) + B sn( δ δ )] ) v v [ G sn( δ δ ) B cos( δ δ )] P = v G v v (3.8) Q = v ( B + B (3.9) sh Smlarly, the real and reactve power flow from bus- to bus- s formulated, P Q [ G cos( δ δ ) B sn( δ δ )] ) + v v [ G sn( δ δ ) + B cos( δ δ )] = v G v v (3.10) = v ( B + B (3.11) r x where G = and B = r + x r + x sh - 6 -

41 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton 3.3 Modellng of Flexble Alternatng Current Transmsson Systems (FACTS) As mentoned before, one of the most mportant ssues that has to be addressed after deregulaton s the fact that power systems are beng operated closer to ther lmts than ever, snce all trade partcpants try to maxmze ther benefts wthout havng to care about system condtons. Therefore, power transfers have undergone a faster growth than transmsson capacty. Thus, operaton of power systems has become remarkably more complcated, wth a hgher probablty of sufferng unscheduled contngences and ncrease n losses. All thngs consdered, systems are more nsecure. Consequently, the newcomer Independent System Operator (ISO) mght have to face these knds of problems n real-tme operaton to facltate those transactons set ahead followng specfc market rules. Partcularly, one of these problems s congeston n some transmsson lnes when bulk of power flowng through them exceeds upper lmts. Thereby, ISO, as responsble for power system securty has to releve that congeston to ensure a secure state. Manly there are two technques at ISO s dsposal to manage ths stuaton whch are the followng ones [65] : 1. Cost-free measures: a. Out-agng of congested lnes b. Operaton of transformer tops phase shfters c. Operaton of FACTS devces partcularly seres devces. Non-cost free measures: a. Re-dspatch of generaton n a manner dfferent from the natural settng pont of the market. Some generators back down whle others ncrease ther output. The effect of ths s that generators no longer operate at equal ncremental costs. b. Curtalment of loads and the exercse of (not-cost-free) load nterrupton optons. In ths work FACTS devces are used to releve congeston and analyse effects on market results owng to the number of advantages assgned to them compared to the other technques, such as better utlzaton of exstng transmsson system assets, ncreased transmsson system relablty and avalablty, ncreased dynamc and transent grd stablty and reducton of loop flows, ncreased qualty of supply for senstve customers and envronmental benefts [65][66][67]. For example, FACTS as a cost-free opton does not nclude economcal ssues related to GENCOS and DISCOS. Moreover, they can be placed n exstng transmsson systems savng the expenses assocated to rebuldng tasks

42 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton FACTS technology can be classfed wthn the power electronc feld whch performs a rapd mprovement day by day. Therefore, nowadays t s a relevant area of study wth prosperous applcatons to power transmsson systems among others. Thus, t s possble to lst some of the most mportant advantages of usng these devces [66][67]: - Greater control of power transmtted. - Secure loadng of transmsson lnes to level nearer to ther thermal lmts. - Greater ablty of transfer between controlled areas. - Preventon of cascadng outages. - Dampng of power system oscllatons. The latter group of features s reled on the followng concept. As s well known, power flowng through an ac lne s determned as a functon of manly three varables or parameters, such as phase angle, lne end voltage and lne mpedance. Tradtonal methods to allevate congeston and other related events do not perform any drect nfluent on these varables. Therefore, ther effcency and effcacy n ths framework could be lmted n comparson to FACTS devces that brng the opportunty to controllng any of the prevous varables as demonstrated n Fg 3.. v δ P Y v δ Parallel compensators P = v x v sn( δ δ Seres-parallel compensators ) Seres compensators Fg 3. Smply FACTS classfcaton based on man effects It s possble to pont out dfferent typologes of FACTS devces [68], such as those ncluded n the next lst and n Fg 3.3. Moreover Table 3.1 shows benefts of these components for dfferent applcatons [67]. - Statc Var Compensators (SVC) - Statc Compensators (STATCOM) - Thyrstor Controlled Seres Capactors (TCSC) - Unversal Power Flow Controllers (UPFC) - Thyrstor Controlled Phase Shfter (TCPS) - 8 -

43 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton Table 3.1 Benefts of FACTS devces for dfferent applcatons. * *** better Load flow control Voltage control Transent stablty Dynamc stablty SVC * *** * ** STATCOM * *** ** ** TCSC ** * *** ** UPFC *** *** ** ** (source: Klaus Habur and Donal O Leary For Cost Effectve and Relable Transmsson of Electrcal Energy ) v δ TCPS B v δ TCSC x c Y v δ (a) (b) v δ TCPS * 1: t Y v δ v δ v s1 UPFC Y v δ (c) q (d) T Fg 3.3 Crcut dagram of dfferent FACTS devces. (a) TCPS can be seen as a varable shunt suceptance. (b) TCSC can be seen as a controllable reactance. (c) TCPS can be seen as an equvalent deal transformer wth complex taps. (d) UPFC controls three parameters: magntude and angle of the nserted voltage, vs1 s1, and the magntude of the current q Numerous studes have been publshed to combne the use of both FACTS devces and OPF. Thus, n [69] a new genetc algorthm/partcle swarm optmzaton searchng method (GA/PSO) s developed to search optmal locaton for FACTS devces and assocated optmal system settngs usng OPF wth dfferent obectves functon, such as fuel cost mnmzaton, voltage profle mprovement and voltage stablty enhancement. Moreover, n [65][70][75][77] congeston management n - 9 -

44 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton deregulated power system ncludng FACTS devces s treated. In [7] an Interor-Pont based OPF s presented when FACTS are ntroduced n deregulaton envronment, whle n [73] s descrbed an optmzaton method reled on sequental quadratc programmng (SQP). Authors n [74] propose a two step method whch conssts on a power flow control problem and a conventonal OPF problem. Furthermore, n [76] s descrbed how to modfy a based OPF formulaton n order to nclude dfferent types of FACTS devces usng power model necton. On the other hand, probably one drawback of FACTS devces s the hgh nvestment related to the nstallaton. For ths reason, t s necessary to perform dfferent studes n order to fnd out whch s the most effcent locaton taken nto account dfferent factors. Therefore, some sensblty ndexes have been performed n order to determne the optmal soluton accordng to the latter pont. Thus, n [70][78] are used factors based on reducton of total system reactve power loss and real power flow varaton. Fnally, [70][71], cost-beneft analyses are proposed to evaluate the economcal ustfcaton of usng FACTS devces for congeston management. Fnally, n [68] t s presented an exhaustve revew of these devces role n deregulaton electrc power systems from dfferent vewponts, such as benefts and techncal problems, whle n [67] s carred out a comprehensve comparson of dfferent types of FACTS based on several crtera provdng real examples of power systems where these components have been mplemented, as well as the reported results Statc model of TCSC Thyrstor Controlled Seres Compensaton (TCSC) has been chosen, among other types of FACTS, to be mplemented n both models OPF and SM-OPF. Bascally, TCST n steady state can be defned as a varable capactance whch contrbutes wth reactance xc n seres wth the lne mpedance. Therefore, the man contrbuton of these devces conssts on reduce the nductance character of those transmsson lnes where they are located, and thus nfluent on [68]: current control, dampng oscllaton, transent and dynamc stablty, voltage stablty, fault current lmtng. As a result of the nfluent on voltage stablty, ths typology has been chosen to be ntroduced n the models presented n ths work. Fg 3.4 demonstrates two effects of ntroducng TCST devces n power systems. As a matter of nterest, t s possble to trace power systems that have been equpped wth TCST devces all over the world yeldng postve results [67]. For example, the lne that nterconnects the North grd and the South grd n Brazl snce 1999, whch cover more than 95% of the electrc power transmsson n the country, ncludes TCST. It has been demonstrated, among others, reducton of losses, stablzaton of the lne and mtgaton of resonance phenomena. Moreover, n the Kayenta Substaton, Arzona, Western Area Power Admnstraton (WAPA) system, USA, was nstalled a TCSC unt n order to overcame a bottleneck lmtaton n the power system transmsson. As a result, the power transfer ncrease by 33% mantanng relable system operaton and wthstand successfully certan contngences

45 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton 1 pu V mn Voltage Wthout TCSC Wth TCSC Power Wth TCSC P m A DEC A ACC Wthout TCSC P 1 P Power δ 0 δc δ (a) (b) P power 0 pre-fault generator angle V voltage c angle at fault clearng generator angle A ACC acceleratng energy A DEC retardng energy. Fg 3.4 Two mprovements due to TCSC. (a) Improvement of voltage profle. (b) Improvement of stablty margn There are two methods to nclude FACTS wthn the electrcal equatons. The frst one conssts of modfyng the admttance matrx through the contrbuton of xc. The second alternatve, whch s used n ths work, s based on a power necton model [70][76] whch would avod changng the orgnal power system admttance matrx as s demonstrated below: v δ Bus- Z = r + x Bus- v δ - x c B sh B sh Fg 3.5 Lumped -equvalent model of a transmsson lne wth TCSC

46 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton The real and reactve power flow from bus- to bus-, and from bus- to bus- of a transmsson lne modelled accordng to the equvalent crcut havng a TCSC connected shown n Fg 3.5, are: P Q P Q ' ' [ G cos( δ δ ) + B sn( δ δ )] ' ' v v [ G sn( δ δ ) B cos( δ δ )] ' ' [ G cos( δ δ ) B sn( δ δ )] ' ' ) + v v [ G sn( δ δ ) + B cos( δ δ )] c ' = v G v v (3.1) c ' = v ( B + B ) (3.13) sh c ' = v G v v (3.14) c ' = v ( B + B (3.15) sh Furthermore, t s possble to calculate the actve and reactve power loss n that lne usng the followng expressons, P Q loss loss ' ' = P + P = G ( v + v ) v v G cos( δ δ ) (3.16) ' ' = Q + Q = ( v + v )( B + B ) + v v B cos( δ δ ) (3.17) r ' where G = r + ( x x ) and ( x ' xc ) B = c r + ( x x ) sh c v δ Bus- Z = r + x Bus- v δ S n S n Fg 3.6 Inecton model of TCSC The equvalent model for the lne depcted n Fg 3.5 can be represented as a lne wthout seres capactances assocated to TCSC and wth power necton at the recevng and sendng ends of the branch, Fg 3.6. Therefore, the apparent power nected at bus- and bus- can be wrtten as, S = P + Q (3.18) n n n n n S = P + Q (3.19) n The actve and reactve power flow nectons used n (3.18) and (3.19) are calculated accordng to the followng expressons, - 3 -

47 Chapter 3 P n Q P n n Q n c Modellng of Electrcty Markets: An optmal Power Flow Formulaton [ G cos( δ δ ) + B sn( δ δ )] v [ G sn( δ δ ) B cos( δ δ )] [ G cos( δ δ ) B sn( δ δ )] v [ G sn( δ δ ) + B cos( δ δ )] = P P = v G v v (3.0) c = Q Q = v B v (3.1) c = P P = v G v v (3.) c = Q Q = v B + v (3.3) where G = ( r x r ( x c c + x )( r x + ( x ) x c ) ) and B xc ( r x + xcx ) = ( r + x )( r + ( x x ) c ) 3.4 Market models consdered Base OPF The base OPF model used n ths work s defned accordng to the general formulaton presented n secton as follows: a Obectve functon. Mn Z = ( Cd Pd Cs Ps) (3.4) The obectve functon used s the socal welfare descrbed n secton It should be observed that n ths case the expresson s mnmsed nstead of beng maxmsed. The result s dentcal snce all rght hand terms are multpled by -1. However the sgn of margnal values change, though ther meanngs are the same. b Power flow equatons Ps Pd = n = 1 v Qg Pd tgφ = n v = 1 v Y v cos( δ δ ψ ) Y sn( δ δ ψ ) It s assume that the power factor s constant for each load, tg( φ ) = cte. (3.5) (3.6) c Apparent power flow lmtaton S = P + Q S (3.7) max [ G cos( δ δ ) + B sn( δ δ )] ) v v [ G sn( δ δ ) B cos( δ δ )] P = v G v v (3.8) Q = v ( B + B (3.9) sh

48 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton Smlarly, S P Q are calculated substtutng by and vce versa n equatons (3.7) (3.8) (3.9) d Supply and demand bds blocks Ps Pd Ps (3.30) mn Ps max Pd (3.31) mn Pd max e Generaton reactve power lmts Qgmn Qg Qg (3.3) max f Voltage lmts V v (3.33) mn V max π δ π (3.34) The need of keepng bus voltages wthn a certan range s ustfed to facltate voltage regulaton and enhance securty n the operaton of transmsson systems. Therefore the upper lmt s related to avod nsulaton falures. However, the lower lmt s consderer more arbtrary [8]. In ths model the securty constrants can be assocated to voltages constrants and transmsson lnes loadngs whch can be used as a measure of congeston rate at each lne. However, t s not possble to quantfy drectly how much dstanced the actual operatng pont s from the one whch defnes the commencement of nfeasblty regon. Therefore, ths method s useful to determne, n a proper manner, a current system state but not nformaton about securty ssues s reported explctly whch s consdered a drawback for the market partcpants snce t could be a lmtaton to determne ther strateges. Fnally, a fragment of the code wrtten en GAMS to programme ths model s shown n the next frame

49 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton EQUATIONS SOCIALW z obectve functon for market: socal welfare PBAL(N) QBAL(N) P_flow(N,NC) Q_flow(N,NC) S_flow(N,NC) actve power balance n each bus - current state [pu] reactve power balance n each bus - current state [pu] Pflow actve power lne flow - current state [pu] Qflow actve power lne flow - current state [pu] apparent power lne flow - current state [pu] ; * OBJECTIVE FUNCTION SOCIALW.. z=e=(-1)*(sum(n$(bus(n,'pdmax') ne 0),0*BUS(N,'A')*sqr(pbd(N))+BUS(N,'B')*pbd(N)+0*BUS(N,'C'))- sum(g$(gdata(g,'gen_type') eq 1),0*GDATA(G,'A')*sqr(pbs(G))+GDATA(G,'B')*pbs(G)+0*GDATA(G,'C'))); * POWER FLOW EQUATIONS PBAL(N).. sum(g$(gn(g,n) and (GDATA(G,'GEN_TYPE') eq 1)),((pbs(G)/SB)))- ((pbd(n)/sb)$(bus(n,'pdmax') ne 0))=e=sum(NC$(CONEX(N,NC) or CONEX(NC,N) or (ord(n) eq ord(nc))),momatrix(n,nc)*v(nc)*v(n)*cos(d(n)-d(nc)- ARMATRIX(N,NC))); QBAL(N).. sum(g$(gn(g,n)),(qg(g)/sb))- (((pbd(n)/sb))*(bus(n,'ql')/bus(n,'pl')))$(bus(n,'pl') ne 0)=e=sum(NC$(CONEX(N,NC) or CONEX(NC,N) or (ord(n) eq ord(nc))),momatrix(n,nc)*v(nc)*v(n)*sn(d(n)-d(nc)-armatrix(n,nc))); * ACTIVE, REACTIVE AND APPARENT POWER THROUGH LINES P_flow(N,NC)$((CONEX(N,NC) or CONEX(NC,N))).. Pflow(N,NC)=e=((- RMATRIX(N,NC))*(sqr(v(N))-v(N)*v(NC)*cos(d(N)-d(NC)))-(- IMATRIX(N,NC))*v(N)*v(NC)*sn(d(N)-d(NC))); Q_flow(N,NC)$((CONEX(N,NC) or CONEX(NC,N))).. Qflow(N,NC)=e=(-1*sqr(v(N))*(- IMATRIX(N,NC)+LINE(N,NC,'Y')/+LINE(NC,N,'Y')/)-v(N)*v(NC)*((- RMATRIX(N,NC))*sn(d(N)-d(NC))-(-IMATRIX(N,NC))*cos(d(N)-d(NC)))); S_flow(N,NC)$(CONEX(N,NC) or CONEX(NC,N)).. sqrt(sqr(pflow(n,nc))+sqr(qflow(n,nc)))=l=fs*((line(n,nc,'smax')/(sb))+ (LINE(NC,N,'Smax')/(SB))); MODEL opf / SOCIALW PBAL QBAL P_flow Q_flow S_flow /; SOLVE opf USING nlp MINIMIZE z;

50 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton 3.4. OPF wth FACTS The OPF model presented before s modfed n order to nclude FACTS devces. As mentoned before Thyrstor Controlled Seres Compensaton (TCSC) has been chosen n order to study the effects on the market results. Accordng to the prevous secton the OPF ncludng TCSC s defned as follows: a Obectve functon: Socal welfare Mn Z = ( Cd * Pd Cs* Ps) (3.35) b Power necton for TCSC P n Q n [ G cos( δ δ ) + B sn( δ δ )] [ G sn( δ δ ) B cos( δ δ )] = v G v v (3.36) = v B v v (3.37) where xtcsc r ( xtcsc x ) G = (3.38) ( r + x )( r + ( x x ) ) TCSC xtcsc ( r x + xtcsc x ) B = (3.39) ( r + x )( r + ( x x ) ) TCSC c Power flow equatons Ps Pd = n = 1 v n v = 1 Y cos( δ δ ψ ) P (3.40) Qg Pd tgφ = v v Y sn( δ δ ψ ) Q (3.41) n n In ths case the power flow equaton (3.6) and (3.7) are modfed n order to nclude the charge n the lne due to TCSC accordng to the power necton model presented n secton d Apparent power flow lmtaton S = P + Q S (3.4) max [ G cos( δ ) + B sn( δ )] Pn ) v v [ G sn( δ ) B cos( δ )] Q P = v G v v δ δ (3.43) Q ( B + Bsh n = v δ δ (3.44) The latter expressons nclude the effects of TCSC n terms of power necton

51 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton e Supply and demand bds blocks Ps Pd Ps (3.45) mn Ps max Pd (3.46) mn Pd max f Generaton reactve power lmts Qgmn Qg Qg (3.47) max g Voltage lmts V v (3.48) mn V max π δ π (3.49) h Compensaton lmts 0 x 0.8 (3.50) TCSC x The compensaton lmt assocated to TCSC unts s lmted to avod resonance problems wth other elements n the system that could produce dramatc operatng problems. Fnally, a fragment of the code wrtten en GAMS to programme ths model s shown n the next frame. EQUATIONS SOCIALW z obectve functon for market: socal welfare eq1 delta_g conductance change from to owng to FACTS - current state [pu] eq delta_b suceptance change from to owng to FACTS - current state [pu] eq3 delta_g conductance change from to owng to FACTS - crtcal state [pu] eq4 delta_b suceptance change from to owng to FACTS - crtcal state [pu] eq5 Pn eq6 Qn PBAL(N) QBAL(N) P_flow(N,NC) Q_flow(N,NC) S_flow(N,NC) ; actve power nected at bus N for FACT model - current state [pu] actve power nected at bus N for FACT model - current state [pu] actve power balance n each bus - current state [pu] reactve power balance n each bus - current state [pu] Pflow actve power lne flow - current state [pu] Qflow actve power lne flow - current state [pu] apparent power lne flow - current state [pu] * OBJECTIVE FUNCTION SOCIALW.. z=e=(-1)*(sum(n$(bus(n,'pdmax') ne 0),0*BUS(N,'A')*sqr(pbd(N))+BUS(N,'B')*pbd(N)+0*BUS(N,'C'))- sum(g$(gdata(g,'gen_type') eq 1),0*GDATA(G,'A')*sqr(pbs(G))+GDATA(G,'B')*pbs(G)+0*GDATA(G,'C')));

52 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton * POWER INJECTION MODEL TCSC eq1(n,nc)$(tcsc(n,nc)).. delta_g(n,nc)=e=(- 1)*MOMATRIX(N,NC)*cos(ARMATRIX(N,NC))*((-1)*(*LINE(N,NC,'X')- xtcsc(n,nc))*xtcsc(n,nc))/(sqr(line(n,nc,'r'))+sqr(line(n,nc,'x')- xtcsc(n,nc))); eq(n,nc)$(tcsc(n,nc)).. delta_b(n,nc)=e=-xtcsc(n,nc)*(sqr(line(n,nc,'r'))- sqr(line(n,nc,'x'))+xtcsc(n,nc)*line(n,nc,'x'))/((sqr(line(n,nc,'r'))+sqr(line (N,NC,'X')))*(sqr(LINE(N,NC,'R'))+sqr(LINE(N,NC,'X')-xtcsc(N,NC)))); eq3(n,nc)$(tcsc(n,nc)).. delta_g(nc,n)=e=delta_g(n,nc); eq4(n,nc)$(tcsc(n,nc)).. delta_b(nc,n)=e=delta_b(n,nc); eq5(n,nc)$(tcsc(n,nc) or TCSC(NC,N)).. Pn(N,NC)=e=sqr(v(N))*delta_G(N,NC)- v(n)*v(nc)*(delta_g(n,nc)*cos(d(n)-d(nc))+delta_b(n,nc)*sn(d(n)-d(nc))); eq6(n,nc)$(tcsc(n,nc) or TCSC(NC,N)).. Qn(N,NC)=e=-sqr(v(N))*delta_B(N,NC)- v(n)*v(nc)*(delta_g(n,nc)*sn(d(n)-d(nc))-delta_b(n,nc)*cos(d(n)-d(nc))); * POWER FLOW EQUATIONS PBAL(N).. sum(g$(gn(g,n) and (GDATA(G,'GEN_TYPE') eq 1)),((pbs(G)/SB)))- ((pbd(n)/sb)$(bus(n,'pdmax') ne 0))=e=sum(NC$(CONEX(N,NC) or CONEX(NC,N) or (ord(n) eq ord(nc))),momatrix(n,nc)*v(nc)*v(n)*cos(d(n)-d(nc)- ARMATRIX(N,NC)))-sum(NC$(TCSC(N,NC) or TCSC(NC,N)),Pn(N,NC)); QBAL(N).. sum(g$(gn(g,n)),(qg(g)/sb))- (((pbd(n)/sb))*(bus(n,'ql')/bus(n,'pl')))$(bus(n,'pl') ne 0)=e=sum(NC$(CONEX(N,NC) or CONEX(NC,N) or (ord(n) eq ord(nc))),momatrix(n,nc)*v(nc)*v(n)*sn(d(n)-d(nc)-armatrix(n,nc)))- sum(nc$(tcsc(n,nc) or TCSC(NC,N)),Qn(N,NC)); * ACTIVE, REACTIVE AND APPARENT POWER THROUGH LINES P_flow(N,NC)$((CONEX(N,NC) or CONEX(NC,N))).. Pflow(N,NC)=e=((- RMATRIX(N,NC))*(sqr(v(N))-v(N)*v(NC)*cos(d(N)-d(NC)))-(- IMATRIX(N,NC))*v(N)*v(NC)*sn(d(N)-d(NC))-Pn(N,NC)$(TCSC(N,NC) or TCSC(NC,N))); Q_flow(N,NC)$((CONEX(N,NC) or CONEX(NC,N))).. Qflow(N,NC)=e=(-1*sqr(v(N))*(- IMATRIX(N,NC)+LINE(N,NC,'Y')/+LINE(NC,N,'Y')/)-v(N)*v(NC)*((- RMATRIX(N,NC))*sn(d(N)-d(NC))-(-IMATRIX(N,NC))*cos(d(N)-d(NC)))- (Qn(N,NC)$(TCSC(N,NC) or TCSC(NC,N)))); S_flow(N,NC)$(CONEX(N,NC) or CONEX(NC,N)).. sqrt(sqr(pflow(n,nc))+sqr(qflow(n,nc)))=l=fs*((line(n,nc,'smax')/(sb))+(line(n C,N,'Smax')/(SB))); MODEL opf / SOCIALW PBAL QBAL eq1 eq eq3 eq4 eq5 eq6 P_flow Q_flow S_flow /; SOLVE opf USING nlp MINIMIZE z;

53 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton SM-OPF Ths s a recent way to ncorporate securty ssues n OPF formulaton. It s possble to fnd through the lterature provded n Chapter several ways to address ths topc. Therefore, the model used n ths work s based on dfferent researches carred out to date. The man dstncton between ths model and the base OPF presented before s a new set of equatons depended on a loadng parameter λ, descrbed n Chapter, that drves the system to ts maxmum loadng condton [9]. As a result, t s possble to dstngush two operatonal states. The frst one, namely current operatng pont, correspond to the actual workng pont defned by specfc values for each varable, whle the second one s used to perform securty ssues. That s, demonstrate how far the system can be settled usng predetermne λ ensurng feasblty. Moreover, K s used to dstrbute the actve losses assocated to the latter state. It s sad that the system stops beng feasble because t s not possble to fulfl all constrants mposed to the model snce some varables reach ether a lower or upper lmt. It s mportant to emphasze that there s no reason for the system to become nfeasble when the frst lmt s acheved by some varable snce the capacty of rearrangng the others to keep the feasblty though the market results could be not as benefcal as before that stuaton. Furthermore, dfferent parameter are defned as a result of the addton of λ, such as Total Transacton Load (TTL), Avalable Transacton Capacty (ATC) and Maxmum Transacton Level (MTL) whch are developed below based on the comments ponted out n Chapter. P G = Ps (3.51) P = (3.5) L Pd G' (1 + + P = λ K ) P (3.53) Lc P L Gc G P = ( 1+ λ) (3.54) MLC = ( 1+ λ C ) PL (3.55) ALC = MLC PL = MLC TTL (3.56) TTL P (3.57) = L ALC = λ P = λ TTL (3.58) C L C The ALC ndex s often used as an ndcator of addtonal power that can be securty transferred by the transmsson network. It s worth mentonng that snce the obectve functon used s the maxmzaton of socal welfare whch reles only on g

54 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton varables assocated to the current operatng pont, the loaded state does not ensure that premse. However, the most mportant thng provded to both operators and market partcpants through ths model s the capacty of analysng consequences of dfferent acton regardng to securty ssues and thus plannng decsons. Normally t s consdered that λ has to be wthn certan boundares. Therefore, a lower lmt s ntroduced n order to ensure a mnmum level of securty and the upper one to mpose the counterpart. As wll be demonstrated the upper boundary s determned ether by low local margnal prces (LMP) [9][79] or nfeasblty problems, though both of them are farly related to each other. Fnally, other mportant concept s the total prce pad to the Independent Market Operator (PAY_IMO) whch s related to the socal welfare and congeston payment. LMP are bascally the Lagrangan multplers of the actve power flow equatons ncluded n the models. Therefore each bus s charactersed by dfferent prces. That s, market partcpants pay for ther consumpton or get pad for ther productons accordng to bds as well as congestons cases n the network. The relaton between socal welfare and market operator payment s explaned below: where: PAY _ IMO = LMP Pn balance = LMP ( PD - PS ) (3.59) LMP LMP LMP P n S D S = ρ = C + µ (3.60) P S S µ S max PS mn P = ρ = C + µ µ ρ tan( φ ) (3.61) P D D D PD max PD mn Q D = LMP LMP (3.6) = P P (3.63) G D D Moreover, PAY _ IMO = = = LMP P n balance LMP ( P P ) = Bd part + Congeston part (3.64) D S where Bd part = f (C,C ), Congeston part = f ( µ P, µ ) and µ are the S multplers of P s and Ps n Lagrange formulaton. D mn P max However the latter formulaton for local margnal prces s vald only for a sngle OPF snce the loadng parameter λ used n SM-OPF affects them as can see below. Therefore, t could be complcated to compare the payment calculated by OPF and SM-OPF

55 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton LMP LMP S D = ρ = C + µ µ ρ (1 + λ K ) (3.65) P S P max P mn cp + S S S S = ρ = C + µ µ ρ (1 + λ) ρ (1 + λ) tan( φ ) (3.66) P D D PD max PD max cp D g cq D D Therefore, the SM-OPF model s defned as follows: a Obectve functon: Socal welfare Mn Z = ( Cd * Pd Cs* Ps) (3.67) b Power flow equatons Ps Pd = n = 1 v Qg Pd tgφ = n v = 1 v Y v cos( δ δ ψ ) (1 + λ + Kg) Pg (1 + λ) Pd = v Qg c (1 + λ) Pd tgφ = n Y = 1 v sn( δ δ ψ ) c v n = 1 c Y c v c Y c cos( δ δ ψ ) c c sn( δ δ ψ ) c (3.68) (3.69) (3.70) (3.71) where (3.70) and (3.71) represent the new set of power flow equaton descrbed prevously n ths secton whch new varables are dstngushed by the sub ndex c c Apparent power flow lmtaton S = P + Q S (3.7) max [ G cos( δ δ ) + B sn( δ δ )] ) v v [ G sn( δ δ ) B cos( δ δ )] P = v G v v (3.73) Q = v ( B + B (3.74) sh S P Q c c c = P + Q S (3.75) c c c c c max [ G cos( δ c δ c ) + B sn( δ c δ c )] v v [ G sn( δ δ ) B cos( δ δ )] = v G v v (3.76) c = v ( B + B ) (3.77) sh c c c c c c Accordngly, there are lmts n apparent power flow assocated to the operatng pont defned by the loadng parameter λ

56 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton d Supply and demand bds blocks Ps Pd Ps (3.78) mn Ps max Pd (3.79) mn Pd max e Generaton reactve power lmts Qgmn Qg mn Qg Qg (3.80) max Qg c Qg (3.81) max f Voltage lmts V V v (3.8) mn V max mn v c V max (3.83) π δ π (3.84) π δ π c (3.85) Fnally, a fragment of the code wrtten en GAMS to programme ths model s shown n the next frame. EQUATIONS SOCIALW z obectve functon for market: socal welfare PBAL(N) QBAL(N) PBALC(N) QBALC(N) P_flow(N,NC) Q_flow(N,NC) S_flow(N,NC) actve power balance n each bus - current state [pu] reactve power balance n each bus - current state [pu] actve power balance n each bus - crtcal state [pu] reactve power balance n each bus - crtcal state [pu] Pflow actve power lne flow - current state [pu] Qflow actve power lne flow - current state [pu] apparent power lne flow - current state [pu] P_flow_c(N,NC) Pflow_c actve power lne flow - crtcal state [pu] Q_flow_c(N,NC) Qflow_c actve power lne flow - crtcal state [pu] S_flow_c(N,NC) apparent power lne flow - crtcal state [pu] ; * OBJECTIVE FUNCTION SOCIALW.. z=e=(-1)*(sum(n$(bus(n,'pdmax') ne 0),0*BUS(N,'A')*sqr(pbd(N))+BUS(N,'B')*pbd(N)+0*BUS(N,'C'))- sum(g$(gdata(g,'gen_type') eq 1),0*GDATA(G,'A')*sqr(pbs(G))+GDATA(G,'B')*pbs(G)+0*GDATA(G,'C'))); * POWER FLOW EQUATIONS ACTUAL OPERATING POINT PBAL(N).. sum(g$(gn(g,n) and (GDATA(G,'GEN_TYPE') eq 1)),((pbs(G)/SB)))

57 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton ((pbd(n)/sb)$(bus(n,'pdmax') ne 0))=e=sum(NC$(CONEX(N,NC) or CONEX(NC,N) or (ord(n) eq ord(nc))),momatrix(n,nc)*v(nc)*v(n)*cos(d(n)-d(nc)-armatrix(n,nc))); QBAL(N).. sum(g$(gn(g,n)),(qg(g)/sb))- (((pbd(n)/sb))*(bus(n,'ql')/bus(n,'pl')))$(bus(n,'pl') ne 0)=e=sum(NC$(CONEX(N,NC) or CONEX(NC,N) or (ord(n) eq ord(nc))),momatrix(n,nc)*v(nc)*v(n)*sn(d(n)-d(nc)-armatrix(n,nc))); * POWER FLOW EQUATIONS f() PBALC(N).. (1+lamda+kgo)*sum(G$(GN(G,N) and (GDATA(G,'GEN_TYPE') eq 1)),((pbs(G)/SB)))- (1+lamda)*( (pbd(n)/sb)$(bus(n,'pdmax') ne 0))=e=sum(NC$(CONEX(N,NC) or CONEX(NC,N) or (ord(n) eq ord(nc))),momatrix(n,nc)*vc(nc)*vc(n)*cos(dc(n)- dc(nc)-armatrix(n,nc))); QBALC(N).. sum(g$(gn(g,n)),(qgc(g)/sb))- (1+lamda)*(((pbd(N)/SB)*(BUS(N,'QL')/BUS(N,'PL'))))$(BUS(N,'PL') ne 0)=e=sum(NC$(CONEX(N,NC) or CONEX(NC,N) or (ord(n) eq ord(nc))),momatrix(n,nc)*vc(nc)*vc(n)*sn(dc(n)-dc(nc)-armatrix(n,nc))); * ACTIVE, REACTIVE AND APPARENT POWER THROUGH LINES - ACTUAL OPERATING POINT P_flow(N,NC)$((CONEX(N,NC) or CONEX(NC,N))).. Pflow(N,NC)=e=((- RMATRIX(N,NC))*(sqr(v(N))-v(N)*v(NC)*cos(d(N)-d(NC)))-(- IMATRIX(N,NC))*v(N)*v(NC)*sn(d(N)-d(NC))); Q_flow(N,NC)$((CONEX(N,NC) or CONEX(NC,N))).. Qflow(N,NC)=e=(-1*sqr(v(N))*(- IMATRIX(N,NC)+LINE(N,NC,'Y')/+LINE(NC,N,'Y')/)-v(N)*v(NC)*((- RMATRIX(N,NC))*sn(d(N)-d(NC))-(-IMATRIX(N,NC))*cos(d(N)-d(NC)))); S_flow(N,NC)$(CONEX(N,NC) or CONEX(NC,N)).. sqrt(sqr(pflow(n,nc))+sqr(qflow(n,nc)))=l=fs*((line(n,nc,'smax')/(sb))+(line(nc, N,'Smax')/(SB))); * ACTIVE, REACTIVE AND APPARENT POWER THROUGH LINES - f() P_flow_c(N,NC)$(CONEX(N,NC) or CONEX(NC,N)).. Pflow_c(N,NC)=e=((- RMATRIX(N,NC))*(sqr(vc(N))-vc(N)*vc(NC)*cos(dc(N)-dc(NC)))-(- IMATRIX(N,NC))*vc(N)*vc(NC)*sn(dc(N)-dc(NC))); Q_flow_c(N,NC)$(CONEX(N,NC) or CONEX(NC,N)).. Qflow_c(N,NC)=e=(-1*sqr(vc(N))*(- IMATRIX(N,NC)+LINE(N,NC,'Y')/+LINE(NC,N,'Y')/)-vc(N)*vc(NC)*((- RMATRIX(N,NC))*sn(dc(N)-dc(NC))-(-IMATRIX(N,NC))*cos(dc(N)-dc(NC)))); S_flow_c(N,NC)$(CONEX(N,NC) or CONEX(NC,N)).. sqrt(sqr(pflow_c(n,nc))+sqr(qflow_c(n,nc)))=l=fs*((line(n,nc,'smax')/(sb))+(line (NC,N,'Smax')/(SB))); MODEL opf / SOCIALW PBAL QBAL PBALC QBALC P_flow Q_flow S_flow P_flow_c Q_flow_c S_flow_c /; SOLVE opf USING nlp MINIMIZE z;

58 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton SM-OPF wth FACTS At ths pont, FACTS devces are ncluded n the prevous SM-OPF model n order to analyse ther nfluent as t was done n the base OPF case. One more tme, the methodology used s the necton power model explan n Secton 3.3 Therefore, followng the same reasonng proposed when the SM varaton was developed, a new set of varables are needed assocated to the state defned by the loadng parameter λ for FACTS expressons. The followng equatons defned ths model. a. Obectve functon: Socal welfare Mn Z = ( Cd * Pd Cs* Ps) (3.86) b. Power necton for TCSC P n Q P n cn Q cn [ G cos( δ δ ) + B sn( δ δ )] [ G sn( δ δ ) B cos( δ δ )] vc [ Gc cos( δ c δ c ) + Bc sn( δ c δ c )] v [ G sn( δ δ ) B cos( δ δ )] = v G v v (3.87) = v B v v (3.88) c c = v G v (3.89) c c c = v B v (3.90) c c c c c c c where xtcsc r ( xtcsc x ) G = (3.91) ( r + x )( r + ( x x ) ) TCSC xctcscr ( xctcsc x ) Gc = (3.9) ( r + x )( r + ( x x ) ) ctcsc xtcsc ( r x + xtcsc x ) B = (3.93) ( r + x )( r + ( x x ) ) TCSC xctcsc ( r x + xctcsc x ) Bc = (3.94) ( r + x )( r + ( x x ) ) ctcsc c. Power flow equatons Ps Pd = n = 1 v n v = 1 Y cos( δ δ ψ ) P (3.95) Qg Pd tgφ = v v Y sn( δ δ ψ ) Q (3.96) n ( 1+ c + Kg) Pg (1 + λc ) Pd = vc vc Y cos( δ c δ c ψ ) Pc n = 1 Qg c λ (3.97) n n n ( 1+ c ) Pdtgφ = vc vc Y sn( δ c δ c ψ ) Qcn = 1 λ (3.98)

59 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton d. Apparent power flow lmtaton S = P + Q S (3.99) max [ G cos( δ ) + B sn( δ )] Pn ) v v [ G sn( δ ) B cos( δ )] P = v G v v δ δ (3.100) Q ( B + Bsh Qn = v δ δ (3.101) S P Q c c c = P + Q S (3.10) c c c c c max [ G cos( c δ c ) + B sn( δ c c )] Pc n vc vc [ G sn( c δc) B cos( δc c) ] Qc = v G v v δ δ (3.103) c ( B + Bsh ) n = v δ δ (3.104) e. Supply and demand bds blocks Ps Pd Ps (3.105) mn Ps max Pd (3.106) mn Pd max f. Generaton reactve power lmts Qgmn Qg mn Qg Qg (3.107) max Qg c Qg (3.108) max g. Voltage lmts V V v (3.109) mn V max mn v c V max (3.110) π δ π (3.111) π δ π c (3.11) Compensaton lmts 0 xtcsc 0.8 x (3.113) 0 x 0.8 (3.114) ctcsc x Fnally, a fragment of the code wrtten en GAMS to programme ths model s shown n the next frame

60 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton EQUATIONS SOCIALW z obectve functon for market: socal welfare eq1 delta_g conductance change from to owng to FACTS - current state [pu] eq delta_b suceptance change from to owng to FACTS - current state [pu] eq3 delta_g conductance change from to owng to FACTS - crtcal state [pu] eq4 delta_b suceptance change from to owng to FACTS - crtcal state [pu] eq1_c delta_gc conductance change from to owng to FACTS - crtcal state [pu] eq_c delta_bc suceptance change from to owng to FACTS - crtcal state [pu] eq3_c delta_gc conductance change from to owng to FACTS - crtcal state [pu] eq4_c delta_bc suceptance change from to owng to FACTS - crtcal state [pu] eq5 Pn eq6 Qn actve power nected at bus N for FACT model - current state [pu] actve power nected at bus N for FACT model - current state [pu] eq5_c Pn_c actve power nected at bus N for FACT model - current state [pu] eq6_c Qn_c actve power nected at bus N for FACT model - current state [pu] PBAL(N) QBAL(N) PBALC(N) QBALC(N) P_flow(N,NC) Q_flow(N,NC) S_flow(N,NC) actve power balance n each bus - current state [pu] reactve power balance n each bus - current state [pu] actve power balance n each bus - crtcal state [pu] reactve power balance n each bus - crtcal state [pu] Pflow actve power lne flow - current state [pu] Qflow actve power lne flow - current state [pu] apparent power lne flow - current state [pu] P_flow_c(N,NC) Pflow_c actve power lne flow - crtcal state [pu] Q_flow_c(N,NC) Qflow_c actve power lne flow - crtcal state [pu] S_flow_c(N,NC) apparent power lne flow - crtcal state [pu] ; * OBJECTIVE FUNCTION SOCIALW.. z=e=(-1)*(sum(n$(bus(n,'pdmax') ne 0),0*BUS(N,'A')*sqr(pbd(N))+BUS(N,'B')*pbd(N)+0*BUS(N,'C'))- sum(g$(gdata(g,'gen_type') eq 1),0*GDATA(G,'A')*sqr(pbs(G))+GDATA(G,'B')*pbs(G)+0*GDATA(G,'C'))); * POWER INJECTION MODEL TCSC - ACTUAL OPERATING POINT eq1(n,nc)$(tcsc(n,nc)).. delta_g(n,nc)=e=(-1)*momatrix(n,nc)*cos(armatrix(n,nc))*((- 1)*(*LINE(N,NC,'X')- xtcsc(n,nc))*xtcsc(n,nc))/(sqr(line(n,nc,'r'))+sqr(line(n,nc,'x')-xtcsc(n,nc))); eq(n,nc)$(tcsc(n,nc)).. delta_b(n,nc)=e=-xtcsc(n,nc)*(sqr(line(n,nc,'r'))- sqr(line(n,nc,'x'))+xtcsc(n,nc)*line(n,nc,'x'))/((sqr(line(n,nc,'r'))+sqr(line(n,nc,'x')))*(sqr(line(n,nc,'r'))+sqr(line(n,nc,'x')-xtcsc(n,nc)))); eq3(n,nc)$(tcsc(n,nc)).. delta_g(nc,n)=e=delta_g(n,nc); eq4(n,nc)$(tcsc(n,nc)).. delta_b(nc,n)=e=delta_b(n,nc); * POWER INJECTION MODEL TCSC - f() eq1_c(n,nc)$(tcsc(n,nc)).. delta_gc(n,nc)=e=(-1)*momatrix(n,nc)*cos(armatrix(n,nc))*((- 1)*(*LINE(N,NC,'X')- xtcsc_c(n,nc))*xtcsc_c(n,nc))/(sqr(line(n,nc,'r'))+sqr(line(n,nc,'x')- xtcsc_c(n,nc))); eq_c(n,nc)$(tcsc(n,nc)).. delta_bc(n,nc)=e=-xtcsc_c(n,nc)*(sqr(line(n,nc,'r'))- sqr(line(n,nc,'x'))+xtcsc_c(n,nc)*line(n,nc,'x'))/((sqr(line(n,nc,'r'))+sqr(line (N,NC,'X')))*(sqr(LINE(N,NC,'R'))+sqr(LINE(N,NC,'X')-xtcsc_c(N,NC)))); eq3_c(n,nc)$(tcsc(n,nc)).. delta_gc(nc,n)=e=delta_gc(n,nc); eq4_c(n,nc)$(tcsc(n,nc)).. delta_bc(nc,n)=e=delta_bc(n,nc);

61 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton eq5(n,nc)$(tcsc(n,nc) or TCSC(NC,N)).. Pn(N,NC)=e=sqr(v(N))*delta_G(N,NC)- v(n)*v(nc)*(delta_g(n,nc)*cos(d(n)-d(nc))+delta_b(n,nc)*sn(d(n)-d(nc))); eq6(n,nc)$(tcsc(n,nc) or TCSC(NC,N)).. Qn(N,NC)=e=-sqr(v(N))*delta_B(N,NC)- v(n)*v(nc)*(delta_g(n,nc)*sn(d(n)-d(nc))-delta_b(n,nc)*cos(d(n)-d(nc))); eq5_c(n,nc)$(tcsc(n,nc) or TCSC(NC,N)).. Pn_c(N,NC)=e=sqr(vc(N))*delta_Gc(N,NC)- vc(n)*vc(nc)*(delta_gc(n,nc)*cos(dc(n)-dc(nc))+delta_bc(n,nc)*sn(dc(n)- dc(nc))); eq6_c(n,nc)$(tcsc(n,nc) or TCSC(NC,N)).. Qn_c(N,NC)=e=-sqr(vc(N))*delta_Bc(N,NC)- vc(n)*vc(nc)*(delta_gc(n,nc)*sn(dc(n)-dc(nc))-delta_bc(n,nc)*cos(dc(n)- dc(nc))); * POWER FLOW EQUATIONS ACTUAL OPERATING POINT PBAL(N).. sum(g$(gn(g,n) and (GDATA(G,'GEN_TYPE') eq 1)),((pbs(G)/SB)))- ((pbd(n)/sb)$(bus(n,'pdmax') ne 0))=e=sum(NC$(CONEX(N,NC) or CONEX(NC,N) or (ord(n) eq ord(nc))),momatrix(n,nc)*v(nc)*v(n)*cos(d(n)-d(nc)-armatrix(n,nc)))- sum(nc$(tcsc(n,nc) or TCSC(NC,N)),Pn(N,NC)); QBAL(N).. sum(g$(gn(g,n)),(qg(g)/sb))- (((pbd(n)/sb))*(bus(n,'ql')/bus(n,'pl')))$(bus(n,'pl') ne 0)=e=sum(NC$(CONEX(N,NC) or CONEX(NC,N) or (ord(n) eq ord(nc))),momatrix(n,nc)*v(nc)*v(n)*sn(d(n)-d(nc)-armatrix(n,nc)))- sum(nc$(tcsc(n,nc) or TCSC(NC,N)),Qn(N,NC)); * POWER FLOW EQUATIONS f() PBALC(N).. (1+lamda+kgo)*sum(G$(GN(G,N) and (GDATA(G,'GEN_TYPE') eq 1)),((pbs(G)/SB)))- (1+lamda)*( (pbd(n)/sb)$(bus(n,'pdmax') ne 0))=e=sum(NC$(CONEX(N,NC) or CONEX(NC,N) or (ord(n) eq ord(nc))),momatrix(n,nc)*vc(nc)*vc(n)*cos(dc(n)- dc(nc)-armatrix(n,nc)))-sum(nc$(tcsc(n,nc) or TCSC(NC,N)),Pn_c(N,NC)); QBALC(N).. sum(g$(gn(g,n)),(qgc(g)/sb))- (1+lamda)*(((pbd(N)/SB)*(BUS(N,'QL')/BUS(N,'PL'))))$(BUS(N,'PL') ne 0)=e=sum(NC$(CONEX(N,NC) or CONEX(NC,N) or (ord(n) eq ord(nc))),momatrix(n,nc)*vc(nc)*vc(n)*sn(dc(n)-dc(nc)-armatrix(n,nc)))- sum(nc$(tcsc(n,nc) or TCSC(NC,N)),Qn_c(N,NC)); * ACTIVE, REACTIVE AND APPARENT POWER THROUGH LINES - ACTUAL OPERATING POINT P_flow(N,NC)$((CONEX(N,NC) or CONEX(NC,N))).. Pflow(N,NC)=e=((- RMATRIX(N,NC))*(sqr(v(N))-v(N)*v(NC)*cos(d(N)-d(NC)))-(- IMATRIX(N,NC))*v(N)*v(NC)*sn(d(N)-d(NC))-Pn(N,NC)$(TCSC(N,NC) or TCSC(NC,N))); Q_flow(N,NC)$((CONEX(N,NC) or CONEX(NC,N))).. Qflow(N,NC)=e=(-1*sqr(v(N))*(- IMATRIX(N,NC)+LINE(N,NC,'Y')/+LINE(NC,N,'Y')/)-v(N)*v(NC)*((- RMATRIX(N,NC))*sn(d(N)-d(NC))-(-IMATRIX(N,NC))*cos(d(N)-d(NC)))- (Qn(N,NC)$(TCSC(N,NC) or TCSC(NC,N)))); S_flow(N,NC)$(CONEX(N,NC) or CONEX(NC,N)).. sqrt(sqr(pflow(n,nc))+sqr(qflow(n,nc)))=l=fs*((line(n,nc,'smax')/(sb))+(line(nc, N,'Smax')/(SB))); * ACTIVE, REACTIVE AND APPARENT POWER THROUGH LINES - f() P_flow_c(N,NC)$(CONEX(N,NC) or CONEX(NC,N)).. Pflow_c(N,NC)=e=((- RMATRIX(N,NC))*(sqr(vc(N))-vc(N)*vc(NC)*cos(dc(N)-dc(NC)))-(- IMATRIX(N,NC))*vc(N)*vc(NC)*sn(dc(N)-dc(NC))-Pn_c(N,NC)$(TCSC(N,NC) or TCSC(NC,N))); Q_flow_c(N,NC)$(CONEX(N,NC) or CONEX(NC,N)).. Qflow_c(N,NC)=e=(-1*sqr(vc(N))*(- IMATRIX(N,NC)+LINE(N,NC,'Y')/+LINE(NC,N,'Y')/)-vc(N)*vc(NC)*((- RMATRIX(N,NC))*sn(dc(N)-dc(NC))-(-IMATRIX(N,NC))*cos(dc(N)-dc(NC)))- (Qn_c(N,NC)$(TCSC(N,NC) or TCSC(NC,N))));

62 Chapter 3 Modellng of Electrcty Markets: An optmal Power Flow Formulaton S_flow_c(N,NC)$(CONEX(N,NC) or CONEX(NC,N)).. sqrt(sqr(pflow_c(n,nc))+sqr(qflow_c(n,nc)))=l=fs*((line(n,nc,'smax')/(sb))+(line (NC,N,'Smax')/(SB))); MODEL opf / SOCIALW PBAL QBAL PBALC QBALC eq1 eq eq3 eq4 eq1_c eq_c eq3_c eq4_c eq5 eq6 eq5_c eq6_c P_flow Q_flow S_flow P_flow_c Q_flow_c S_flow_c /; SOLVE opf USING nlp MINIMIZE z; 3.5 Conclusons In ths secton t has been presented a methodology to addressng securty threats arose from deregulaton markets. Accordngly, dfferent models have been proposed. As a startng pont a base OPF has been developed whle dfferent mprovements have led t to the SM-OPF where t s possble to obtan a drect measurement of a securty margn whch could turn out to be useful ndcator for operators to ensure and montor the system relablty. Furthermore, t seems that the relevance of hybrd or OPF-base markets can be growth owng to the numerous modfcatons that are beng proposes nowadays. However, the mathematc formulaton of the problem can be a drawback snce t reduces the smplcty and transparency for market partcpants

63 Chapter 4 Market Settlement wth Voltage Stablty Consderatons and FACTS Devces I n ths chapter, the results obtaned from the models presented on the prevous secton are analysed. The analyses focus on behavour of man power system varables, management of securty ssues and ts effects on market settlements and benefts of FACTS devces nstalled on some crtcal lnes. A modfed IEEE 30-bus test system s used to demonstrate the results. The General Algebrac Modellng System (GAMS/MINOS) s used to model and solve the optmzaton problems nvolved and the MATLAB software s used to perform graphcal representatons through an nterface between both programs [8]. 4.1 Test system A modfed IEEE 30-bus test system [33] s used n ths work. It s comprsed of 30 buses, 41 lnes, 3 generators, 3 synchronous condensors, shunt capactors, 1 loads and 6 transformers. Furthermore, for smplcty, bd prces for power have been consdered constant, between 31 $/MWh and 37 $/MWh, beng farly close to possble margnal beneft and margnal producton cost for partcpants. In addton, t has been decded to establsh boundares for the mnmum amount of power, both generated and consumed, submtted to the market process n order to smulate reasonable loadng values accordng to the apparent power lne lmts based on those used n [83]. All data are presented n Appendx B. As mentoned before, n the power system presented there are 3 generators and 1 loads. In order to smulate a compettve market structure, t has been consdered the generators as GENCOS and the loads as DISCOS/ESCOS. In all cases, t s assumed that they are ndvdual enttes whch are operated separately. However, t would not have mplcated any loss of generalty f some unts had been putted nto groups belongng ether to the same GENCO or DISCO/ESCO accordng to, for example, geographcal or busness crtera. A modfed sngle lne dagram of the system s shown n Fg

64 Chapter 4 Market Settlement wth Voltage Stablty Consderatons and FACTS Devces GENCO 1 GENCO G G C GENCO 3 G C C G GENCO C SYNCHRONOUS CONDENSERS? DISCOS/ESCOS Fg 4.1 Modfed IEEE 30 bus test power system. It s mportant to menton that the test system represents a transmsson grd and not a dstrbuton network. It s necessary to pont out ths dstncton snce at ths level wthn the power transacton chan GENCOS, ESCOS and DISCOS are manly the players allowed to partcpate n the wholesale market. That means end-customers are not nvolved drectly. However, GENCOS, ESCOS and DISCOS could base ther strateges usng hstorc electrcty consumpton trends of these users. Owng to ths, technques to predct accurately these fgures turn out to be complcated [8]. Therefore, n ths framework both GENCOS and DISCOS/ESCOS submt to the market operator ther requrements whch are comprsed of dfferent terms. On the one hand, the amount of energy that each partcpant would be wllng to ether sells or purchase through ths mechansm whch s bounded wthn a mnmum and maxmum quantty. The lower lmt can be determned accordng to forecast studes of demand based on hstorc fgures, whle the upper one can be decded on account of techncal constrants or corporatve decsons. Moreover, the bd coeffcents represent the margnal cost that generaton companes ncurred to produce electrcty and the margnal benefts

65 Chapter 4 Market Settlement wth Voltage Stablty Consderatons and FACTS Devces reported to the customers. In ths work, cost functons and beneft functons, formulated through the last two bd parameters, are lnear to smplfy calculatons. Therefore, margnal values for both partcpants correspond drectly to the value B ndcated n Table A.4 Appendx A. The meanng of these margnal values can be nterpreted as follows. From a producer vewpont t would be the cost of producng the next unt of power, whle for a customer t would be the beneft reported for an addtonal unt purchased. On the other hand, as explaned n secton 1., n hybrd markets both economcal and securty ssues are management and montored by only one entty, an ndependent market operator. Therefore, ths authorty needs to know whch the operatng lmts of the dfferent unts nvolved n the systems are n order to verfy them at the same tme that the power transfer s calculated. For ths reason, reactve lmts are sent to the operator, for nstance. Fg 4. represents schematcally the nformaton requred by the operator from the partcpants where further nformaton represents, for example, the avalablty and locaton of ancllary servces. SUPPLY BIDS SUPPLY/ DEMAND LIMITS ISO Further nformaton DEMAND BIDS TRANSMISSION LIMITS Fg 4. Input for ISO n hybrd markets 4. Market settlement structures The four market settlement structures are represented n Fg 4.3 whch mathematcal formulatons were shown through Chapter 3. The frst market structure, whch was developed n secton 3.4.1, s a base OPF. Structure II, an extenson of the prevous one, ncludes FACTS devces n order to enhance the transmsson system accordng to the notons provded n sectons 3.3 and On the other hand, n structure III and structure IV, defned n sectons and respectvely, besdes FACTS contrbutons, ts s explored the consequences of ntroducng a second set of power flow equatons to smulate a new operatng pont assocated to securty loadng margns

66 Chapter 4 Market Settlement wth Voltage Stablty Consderatons and FACTS Devces STRUCTURE I OPF WITHOUT FACTS STRUCTURE III SM-OPF WITHOUT FACTS STRUCTURE II OPF WITH FACTS STRUCTURE IV SM-OPF WITH FACTS Fg 4.3 Studed market settlement structures 4.3 Market settlement analyses Base OPF wthout FACTS devces Two dfferent cases have been analysed. The frst smulaton does not consder loadng lnes lmts. That means not upper boundares are mposed to the apparent power through each lne. The second one, that flow s constraned. The former can be assocated to that stuaton when loadng lne lmts for all branches are large enough to not be acheved, whle for the latter that lmtaton s sgnfcant and t can nfluent n the global system operaton. Moreover, t s mportant to be aware that there are other requrements that must be fulflled n order to ensure a feasble operaton of the system, such as voltage range and power submtted levels. Lkewse, t s nterestng to demonstrate the nfluent of loadng lmts n the system snce ths s one remarkable dfference wth other knds of market structures. That s, market settlement and system requrements are defned at the same tme nstead of perform two steps lke n the smple aucton model. In Table 4.1 t s possble to observe the total transacton level (TTL), actve losses and payment to the ndependent market operator (PAY_IMO) for both cases, wth and wthout apparent power lmtaton. Therefore, wthout flow upper boundares, TTL decreases by.1%. That entals the system varables are forced to modfy ther values n order to satsfy those new condtons ntroduced n the model. Moreover, actve losses and PAY_IMO decrease snce a lower bulk of power s transmtted n the second case