University of Cape Town

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1 Investgaton of the Impact of Demand Elastcty and System Constrants on Electrcty Market Usng Extended Cournot Approach by Jun Yan Thess Presented for the Degree of DOCTOR OF PHILOSOPHY n the Department of Electrcal Engneerng Unversty of Cape Town Unversty of Cape Town Supervsed by Prof. K.A. Folly May 2015

2 The copyrght of ths thess vests n the author. No quotaton from t or nformaton derved from t s to be publshed wthout full acknowledgement of the source. The thess s to be used for prvate study or noncommercal research purposes only. Publshed by the Unversty of Cape Town (UCT) n terms of the non-exclusve lcense granted to UCT by the author. Unversty of Cape Town

3 Acknowledgement It s a great pleasure for me to have the opportunty to work wth my supervsor, Prof. Komla Folly. I am deeply ndebted to hm for provdng me wth hs precous gudance, encouragement and constant support durng my study and research. I wsh to express my deep grattude to the Department of Electrcal Engneerng, especally to Prof. Alexander Petroanu and Prof. Trevor Gaunt, for ther valuable comments and contrbutons and constant assstance. I would lke to express my thanks fnancal support by the Natonal Research Foundaton of South Afrca and the TESP grants. I gratefully send my very specal thanks to my parents who have always gven me generous support and encouragement throughout my studes and lfe. My thanks also go to my wfe, Xaoyan Tan, for her contnuous support and patence whle I have studed and worked towards my Ph.D degree.

4 Abstract Snce late 1990s, prvatzaton and deregulaton n the South Afrcan Electrcty supply ndustry (ESI) have gradually been underway. Due to the ssues whch arose n the nternatonal markets, such as the case of UK lberalzaton, the level of market power was underestmated although the capacty dvestments have been substantal snce the early 1990s. Therefore, the deregulaton process s now at the stage of ntensve research and plannng. There are a number of models that are used to nvestgate the competton n the electrcty market. They nclude Bertrand model, Cournot model, Supply functon equlbrum model, conjectural varatons model, etc. These models dffer n ther modelng methodology and assumptons. As the quantty that s produced by ndependent power producers s a key market ndcator, the classcal Cournot competton under game theory has been often used n modelng the case wth the exstence of transmsson constrants to dentfy strategc behavors of the market partcpants. However, the classcal Cournot model focuses on fndng the Nash equlbrum as a soluton. In ths case, there s a hgh possblty of msmatch between the supply and demand of power. No power producer ntends to move away from the Nash equlbrum. The partcpaton of demand sde and the demand elastcty of power demand are underestmated. In today s electrcty supply ndustry, demand sde partcpaton s consdered an mportant factor that can nfluence the market performance and output effectvely. Demand elastcty shows the senstvty of demand sde to the market prce, and thus can provde potental adjustment of demand n the market. The purpose of ths research s to study the mpact of demand elastcty on power producers market competton output. An analytcal model, called Extended Cournot model s developed n ths thess based on the classcal Cournot model. Through the ntegraton wth conjectural varaton model, n whch power producers consder both the generaton and prce level, the extended Cournot model can analyze electrcty market results under the condtons of dfferent constrants.

5 In the classcal Nash Cournot model, capacty wthdrawal exsts n most cases especally when transmsson constrant occurs. In contrast, the newly developed analytcal model ensures that demand s always satsfed at all tme. Demand elastcty s ncorporated drectly nto the market results calculaton nstead of usng the market clearng prce. Ths approach enables the load demand to drectly obtan the market results by tunng ts demand elastcty. The ntenton s to show that demand sde should be more encouraged to partcpate n the market competton. In the classcal economc dspatch, the load demand s hghly nelastc. From the load curve, there s only a change of physcal volume of demand. The demand responsveness, whch s represented by demand elastcty, has been understated. In ths thess, the hypothess s that demand elastcty and system constrant have crtcal nfluence on the power producers competton results n terms of market clearng prce, ndvdual output and proft. Load demand can make use of demand elastcty to affect ts fnal payment to the market. Such ablty s expected to be lmted n the case where system constrants,.e. generaton lmts and transmsson lmts, exst. For smplcty, a small network and number of power producers are used n ths thess to nvestgate the effectveness of the Extended Cournot model. However, ths model can be appled to more complex networks wth dfferent market envronments. In order to compare the dfferent mpact of load demand s nfluence on market competton, the market results are calculated based on two cases,.e. change of physcal load demand wth constant demand elastcty and change of demand elastcty wth constant physcal load demand. In each case, three scenaros as descrbed below were modeled: 1) wthout generaton or transmsson constrants, 2) wth generaton constrants, and 3) wth generaton and transmsson constrants. Therefore, n total sx case studes were conducted to analyze the combned mpact of demand elastcty and system constrants on power producers market results together wth load payment. The optmzaton problem s solved based on the proft maxmzaton of all power producers. v

6 The market ndcators used for the nvestgaton nclude: Market clearng prce Power producers output Power producers proft Load payment To nvestgate the mpact of the change n demand elastcty on the margnal change of prce and power producers quantty, addtonal two ndcators are consdered: Dervatve of market clearng prce over demand elastcty Dervatve of ndvdual producton over demand elastcty In the case of change n total demand, the results show that change n total quantty does not affect market prce, n the scenaro wthout constrants. In contrast, market clearng prce s determned by the total quantty n the tradtonal Cournot model. However, the prce wll change when generaton and transmsson constrants reach a certan extent. Ths s manly due to the fact that the supply of the most economcal power producer s capablty to supply the load demand s dmnshed due to the above constrants. Any addtonal demand wll be allocated to the less economcal power producer wth hgh margnal cost. For the same reason, power producers profts and load payment wll ncrease when constrants are ntroduced. To valdate the above hypothess, smulatons were performed n ths thess. The analytcal model s valdated by comparng ts results to the results obtaned usng an ndustral electrcty market smulaton tool, known as Plexos. Plexos has the optons to nclude demand elastcty to model Cournot competton. Plexos also allows to ntroducton generaton and transmsson lmts n ts nput date sesson. Both the analytcal model and Plexos smulaton use the Nash Cournot competton approach. The smulaton results show that the analytcal model and Plexos software package are n agreement.. v

7 For some of the smulated scenaros, such as the one wth both generaton and transmsson constrants, there are varatons n the results between the analytcal model and Plexos output. The dscrepances n the results are manly caused by the varatons of nput data assumptons n the modelng of load demand. It s shown that the assumptons on margnal costs have sgnfcant nfluence on the power producers producton pattern n dfferent scenaros. v

8 Contents Acknowledgement.. Abstract. Lst of Tables...x Lst of Fgures..xv 1 Introducton Competton n the electrcty market Game theoretcal approach n modelng electrcty competton Hypothess Contrbutons of ths thess The contents of ths thess..7 2 Lterature revew Market competton related ssues Cournot modelng n electrcty market competton Demand elastcty 15 3 Model dervatons.18 v

9 3.1 Classc Cournot model and Nash equlbrum n game theory Dervaton of Extended Cournot model Extended Cournot model wth no constrants Derve extended Cournot model wth constrants under Kuhn-Tucker condtons Generaton capacty constrant Generaton and transmsson capacty constrants Smulaton results of analytcal model Case formulaton Margnal cost Generaton capacty Transmsson capacty Demand elastcty Case demonstraton Change of total demand wth constant demand elastcty Change of demand elastcty wth constant total demand Case 1 - Change of total demand wth constant demand elastcty Scenaro 1:- wth no system constrants Scenaro 2: wth generaton capacty constrant Scenaro 3:- wth generaton and transmsson capacty constrants Summary of case Case 2 - Change of demand elastcty wth constant total demand Scenaro 1: wth no capacty constrants Scenaro 2: wth generaton capacty constrant v

10 4.4.3 Scenaro 3: wth generaton and transmsson capacty constrant Summary of case Plexos smulaton results and the comparson wth analytcal model Bref descrpton of Plexos Comparson of Case 1 Scenaro 1 (change of total demand wth no constrant) Comparson of Case 1 Scenaro 2 (change of total demand wth generaton capacty constrant) Comparson of Case 1 Scenaro 3 (change of total demand wth generaton and transmsson capacty constrant) Comparson of Case 2 Scenaro 1 (change of demand elastcty wth no constrant) Comparson of Case 2 Scenaro 2 (change of demand elastcty wth generaton capacty constrant) Comparson of Case 2 scenaro 3 wth generaton and transmsson capacty constrants Conclusons and recommendatons Conclusons Recommendatons 94 References.96 Author s publcatons 106 x

11 Appendces.107 Appendx 1. Example of Plexos Excuton Log fle Case 1 Scenaro 1: Change of total demand (Q=100) wth constant demand elastcty (e= ) and wthout constrant 107 Appendx 2. Example of Plexos Excuton Log fle Case 1 scenaro 2: Change of total demand (Q=750) wth constant demand elastcty (e= ) and wth generaton capacty constrant..113 Appendx 3. Example of Plexos Excuton Log fle Case 1 scenaor 3: change of total demand (Q=700) wth constant demand elastcty (e= ) and wth generaton and transmsson capacty constrants 119 Appendx 4. Example of Plexos Excuton Log fle Case 2 scenaro 1: change of demand elastcty (e= ) wth constant total demand (Q=600) and wthout constrant..125 Appendx 5. Example of Plexos Excuton Log fle Case 2 scenaro 2: change of demand elastcty (e= ) wth constant total demand (Q=600) and wth generaton capacty constrant Appendx 6. Example of Plexos Excuton Log fle Case 2 scenaro 3: change of demand elastcty (e= ) wth constant total demand x

12 (Q=600MW) and wth generaton and transmsson capacty constrants Appendx 7. Values of and n case Appendx 8. Values of and n case x

13 Lst of Tables 3.1 Comparson between two dfferent Cournot models Comparson of ndvdual producton n case Comparson of market clearng prce n case Comparson of ndvdual profts n case Comparson of load payment n case Comparson of market clearng prce n case Comparson of ndvdual producton n case Comparson of ndvdual profts n case Comparson of load payment n case Comparson of dervatve of market clearng prce over demand elastcty n case Comparson of dervatve of ndvdual producton over demand elastcty n case Comparson of case 1 scenaro 1 wth change of Q Comparson of case 1 scenaro 2 wth change of Q Comparson of case 1 scenaro 3 wth change of Q x

14 5.4 Comparson of case 2 scenaro 1 wth change of e Comparson of case 2 scenaro 2 wth change of e Comparson of case 2 scenaro 3 wth change of e x

15 Lst of Fgures 4.1 Smplfed three-bus benchmark DC network An example of 3-bus network wth transmsson network constrants wth total demand of 600MW Change of total demand wth constant demand elastcty Change of demand elastcty wth fxed total quantty demand wth constant margnal cost Market clearng prce wth 100MW change nterval of total demand, where demand elastcty stays constant at Indvdual outputs wth change of total demand, where demand elastcty stays constant at Indvdual profts wth change of total demand, where demand elastcty stays constant at Load payments wth change of total demand, where demand elastcty stays constant at Market clearng prce wth constant demand elastcty under generaton constrant Indvdual producton wth constant demand elastcty under generaton constrant Indvdual profts wth constant demand elastcty under generaton constrant Load payment wth constant demand elastcty under generaton constrant xv

16 4.13 Market clearng prce wth constant demand elastcty under generaton and transmsson constrants Indvdual producton wth constant demand elastcty under generaton and transmsson constrants Indvdual profts wth constant demand elastcty under generaton and transmsson constrants Load payment wth constant demand elastcty under generaton and transmsson constrants Market clearng prce wth the change of demand elastcty, and total demand stays constant at 600MW Market prce wthout extreme lower bound of elastcty, and total demand stays constant at 600MW Indvdual outputs wth change of demand elastcty, and total demand stays constant at 600MW Indvdual profts wth change of demand elastcty wthout lower bound of elastcty, and total demand stays constant at 600MW Change of load payment wth change of demand elastcty wthout lower bound of elastcty, and total demand stays constant at 600MW Dervatve of market clearng prce over demand elastcty Dervatve of ndvdual quantty output over demand elastcty..62 xv

17 4.24 Indvdual producton wth change of demand elastcty under generaton constrant Market clearng prce wth change of demand elastcty under generaton constrant Indvdual proft wth change of demand elastcty under generaton constrant Load payment wth change of demand elastcty under generaton constrant Dervatve of market clearng prce over demand elastcty Dervatve of ndvdual producton over demand elastcty Indvdual producton wth change of demand elastcty under generaton and transmsson constrants Marke clearng prce wth change of demand elastcty under generaton and transmsson constrants Indvdual proft wth change of demand elastcty under generaton and transmsson constrants Load payment wth change of demand elastcty under generaton and transmsson constrants Dervatve of market clearng prce over demand elastcty under generaton and transmsson constrants...73 xv

18 4.35 Dervatve of ndvdual producton over demand elastcty under generaton and transmsson constrants Illustraton of Cournot Models n Plexos wth PoolCo and Blateral models Comparson of network confguraton between analytcal model and Plexos (a) Results from Analytcal model calculaton n case1 scenaro1 wth total demand at 700MW (b) Results from Plexos smulaton n case1 scenaro1 wth total demand at 700MW Results from analytcal model and Plexos wth generaton capacty constrant (a) Results from Analytcal model calculaton n case 1 scenaro 2 wth total demand at 700MW (b) Results from Plexos smulaton n case 1 scenaro 2 wth total demand at 700MW Results from analytcal model and Plexos wth transmsson capacty constrant of 240MW on lne L (a) Results from Analytcal model calculaton n case 1 scenaro 3 wth total demand at 700MW (b) Results from Plexos smulaton n case 1 scenaro 3 wth total demand at 700MW 85 xv

19 Notaton Roman symbols a Intercept of the demand functon d Percentage of Q goes through lne L jk b C C n P Q Q Slope of the demand functon Margnal cost of GenCo Average margnal cost Number of GenCos n the market Market clearng prce Indvdual output of GenCo Total producton output max Q GenCo s maxmum generaton capacty T jk Transmsson capacty of lne L jk Greek symbols The multpler determned by the relatonshp between d Q n 1 and T jk. Indvdual proft of GenCo for 1,2, 3 n Demand elastcty The multpler determned by the relatonshp between Q and max Q, Abbrevatons DG ECM GenCo Dstrbuton generaton Extended Cournot model Generaton company xv

20 HHI ISO KKT LMP MPE NEM PIPA RSM RTOs Herfndahl-Hrschman Index Independent system operator Karush-Kuhn-Tucker Locatonal margnal prce Mathematcal program wth equlbrum Natonal electrcty market Penalty Interor Pont Algorthm Response surface method Regonal transmsson organzatons xx

21 Chapter 1 Introducton 1.1 Competton n the electrcty market The electrcty supply ndustry has been undergong contnuous deregulaton over the last twenty years. Durng ths restructurng process, the exercse of market power has been found n a number of electrcty markets [1]. Market power s usually defned as a market suppler that owns the ablty to drve the prce above the compettve level, or control output for a sgnfcant perod of tme. The exercse of market power creates neffcences n both resource producton and allocaton. The most common effects of market power nclude less producton n the market whch results n lower consumpton, and hgher prce levels compared to margnal cost. One of the man threats of market power s dmnshng consumer surplus whch transfers socal benefts from the consumers to the supplers [1]. Many methods have been developed to assess market power, such as Lerners ndex and the Herfndahl-Hrschman Index (HHI ndex) [1, 2]. More recent research studes have rased the concern of lack of market dynamcs n the above ndces. It has been found n many studes that generators are more lkely to exercse market power wthn the presence of potental network constrants [3, 4]. Accordng to the market mechansm, the electrcty markets have been defned by two man models: the PoolCo model and the Blateral model [1, 3]. The PoolCo model s more centralzed, n whch the ndependent system operator determnes the economcs of dspatch and mantans system securty. The Blateral model s a decentralzed mechansm n whch the market determnes the economc dspatch and electrcty prce. A hybrd model, whch s a combnaton of the PoolCo and Blateral models, 1

22 has also been suggested as a soluton to complement the central dspatch wth blateral contracts [1]. It s found that generators have more potental to behave strategcally n the PoolCo model compared to the Blateral model. The strategc behavor can be modeled through mperfect competton usng game theory. Prevous research has shown that generators can nfluence the transmsson constrants and market prce wth ther generaton output decsons [2, 3, 4, 5]. 1.2 Game theoretcal approach n modelng electrcty competton The basc strateges for generators to play n a market nclude prce, quantty, and supply functon. Under the regulatons, any explct prce ncrease by generators wll be very senstve and mght be penalzed by the system operators or regulators. On the other hand, changng the generaton supply curve s more mplct and less lkely to be detected. The most usual theory used to analyze prce and quantty gamng and competton n electrcty markets s game theory [6]. In game theory, a prcng game s usually defned as a Bertrand game, and a quantty game s defned as a Cournot game. Players n the Bertrand game wll usually rase ther bds above ther margnal costs. For the classc Bertrand model, there are a number of assumptons as follows [7]: 1) Product homogenety and dentcal unt costs. 2) No capacty constrants. 3) Generaton Companes (GenCos) choose prce smultaneously. 4) Compettors have the ncentve to undercut each other s prces vgorously, whch results n a compettve outcome. 2

23 However, when network constrants exst, Generaton Companes may not choose to prce at a compettve level. Therefore, t s beleved that the nature of the above assumptons may affect market equlbrum sgnfcantly. In the case of hgh demand together wth the exstence of sgnfcant constrants, Cournot competton s often used to predct power producers behavor [7]. The classc Cournot model of competton enables the power producers to compete aganst each other wth quantty strateges. The assumptons under classc Cournot competton are as follows [7]: 1) Homogeneous products and dentcal unt producton costs. 2) GenCos bd once n the market and choose quanttes non-cooperatvely and smultaneously. 3) A prce wll be determned that equates demand and supply. The frst assumpton s the same as that of the Bertrand game. The mplcaton of the second assumpton s that power producers choose ther producton level based on ther own nterest, and such choce s ndependent of the choces of other power producers. Ths s why t s known as a non-cooperatve game. The choce each power producer makes s also based on the antcpaton of other power producers quanttes. The Cournot model utlzes two functons, the proft functon and demand functon. The proft functon ncludes the market prce, total and ndvdual producton, and ndvdual margnal cost. When demand functon s ntroduced, the frst order of proft maxmzaton wll yeld the results of ndvdual quanttes. Once quanttes are chosen, market prce wll be determned. When there s only one unque equlbrum, a Nash-Cournot equlbrum exsts. Although the outcome of the classc Cournot competton model s often questonable as the supply sde ultmately chooses the prce level, Cournot equlbrum s stll favored n nvestgatng mperfect competton n varous ndustres, ncludng the 3

24 electrcty market. The wthholdng of output by the power producers n order to rase the market prce s a typcal feature n Cournot competton. Nash-Cournot equlbrum s a well-known strategy adopted by market players durng market competton [8]. Ths thess focuses on the quantty game whch s the foundaton of Nash-Cournot equlbrum. The reason for choosng the Nash-Cournot competton model s that t focuses on quantty strateges whch wll nfluence market prce. One of the man features of Cournot competton s ts ablty to ncorporate quantty as part of prce n a demand functon. In the demand functon, prce s a functon of quantty demanded n the market. The functon descrbes an nverse relatonshp between prce and quantty. Ths creates opportuntes for the generators to wthhold capacty n order to rase the market prce. Dependng on the level of elastcty of demand, the reducton n generaton wll result n a relatvely greater ncrease n prce and the proft of the ndvdual generator wll ncrease. Both demand functon and elastcty of demand n Cournot competton have mportant mplcatons n electrcty market modelng. The Cournot model s found to be applcable for nterpretng the electrcty market features and has been wdely used to represent generator behavor such as the matrx approach that s used n [9] to solve a three-player game. However, t s argued that the presence of capacty constrants should not be the necessary motvaton for Cournot modelng. Ths means that, n most studes that are usng Cournot competton to analyze generator behavors, capacty constrant has not been consdered as a constrant n the models. More realstcally, there wll be arbtragers, or traders, nvolved n the electrcty market tradng process. More attenton should be pad to ths as the Cournot competton ncludes conjectural varaton, n whch generators decde the generaton and prce level on the assumpton that arbtragers wll not change ther purchases. The arbtragers are treated exogenously n the Cournot Conjectural varaton model [10]. 4

25 In the classcal Nash Cournot model, capacty wthdrawal exsts n most cases, especally when transmsson constrant occurs. On the contrary, the analytcal model ensures that demand s always satsfed at all tmes. Demand elastcty s ncorporated drectly nto the market results calculaton nstead of usng the market clearng prce. Ths approach enables the load demand to drectly obtan the market results by tunng ts demand elastcty. The ntenton s to show that there should be more encouragement of demand sde partcpaton n market competton. In the classcal economc dspatch, the load demand s hghly nelastc. From the load curve, there s only change n physcal volume of demand. The demand responsveness, whch s represented by demand elastcty, has been understated. 1.3 Hypothess The research hypothess s that demand elastcty and system constrants wll have a crtcal nfluence on the power producers competton results n terms of market clearng prce, ndvdual output and proft. Load demand can utlze demand elastcty to affect ts fnal payment to the market. Such ablty s expected to be lmted n the case where system constrants,.e. generaton lmt and transmsson lmt, exst. Extended Cournot model can be used for analyzng the mpact of demand elastcty and system constrants on electrcty market outputs. 1.4 Contrbutons of ths thess From the lterature revew, t can be concluded that the mpact of dfferent types of constrants on GenCo market strateges has not been suffcently addressed. In addton, the mpact of demand elastcty has not been fully nvestgated. The purpose of ths thess s to develop an Extended Cournot model (ECM) based on conjectural varatons models. One of the objectves of ths thess s to use elastcty n the ECM wth dfferent constrants, such as generaton and transmsson constrants. The man advantage of ECM s the ablty to nterpret the relatonshp between 5

26 elastcty of demand and other key market ndcators, for example, market prce, ndvdual generaton, and ndvdual proft. Wth the ntroducton of constrants, the dfferent mpact of elastcty on the market can be shown [61], [62], [63], [64], and [65]. The man contrbutons of ths thess are summarzed as follows. The Extended Cournot model developed n ths thess consders more factors affectng the outcomes of market competton, such as demand elastcty, generaton capacty constrant, transmsson constrant as compared to the classcal Cournot model. As a result, the mportance of demand responsveness has been hghlghted and nvestgated. Sgnfcant mpact of demand elastcty on the market outputs has been dentfed. Ths thess has provded a tool for quantfyng the mpact of the demand sde to the market outputs, whch ncludes market clearng prce, ndvdual producton and proft, and load payment. In the case of change n demand elastcty, the proportonal change of market prce and ndvdual producton aganst demand elastcty has been quantfed. The results show how demand sde partcpaton can effectvely nfluence the market outputs. It s also found that, when the generaton and transmsson constrants are mplemented, the mpact of demand elastcty on the market outputs wll be affected. It has been found that demand elastcty has much hgher mpact on the market clearng prce at ts lower bound sesson. Ths ndcates that the demand sde can effectvely reduce the market clearng prce by becomng more responsve. The Extended Cournot model proposed n ths thess can be used to analyze mperfect competton on a scenaro-to-scenaro bass, gven specfc assumptons. In a scenaro wthout any system constrants, ths model can be used to predct realstc market results. Therefore, unlke the classcal cournot 6

27 model, the Extended Cournot model can be appled to stuatons wth and wthout system constrants. 1.5 The contents of ths thess Ths thess s organzed as follows: Chapter 2 gves a comprehensve lterature revew on the ssues of electrcty market competton, game theory, Nash Cournot competton, network congeston, and demand elastcty. Chapter 3 presents the dervaton of the Extended Cournot model whch s ntegrated wth elastcty of demand. It ncludes two cases, namely, change of total demand, and change of demand elastcty. System constrants have been nvestgated separately as generaton and transmsson capacty constrants. Chapter 4 analyses the calculaton results wth two cases: 1) change n total demand and 2) change n demand elastcty. For each case, three scenaros were nvestgated,.e., ) wthout constrant, ) wth generaton capacty constran, ) wth generaton and transmsson constrants,. The purpose s to show the dfferent mpact demand elastcty has on market results wth and wthout system constrants durng market competton. Chapter 5 compares the smulaton results from a market smulator, Plexos, wth the results from Chapter 4. Plexos has the opton to nclude demand elastcty when modelng market competton. Therefore, t s a practcal to compare the results from the Extended Cournot model wth the ones from Plxos. Chapter 6 gves the conclusons and dscusses the recommendatons for future potental research. 7

28 Chapter 2 Lterature revew Ths lterature revew has covered a number of market-related ssues n the electrcty supply ndustry. Frst, market competton, whch ncludes prcng and transmsson congeston analyss, s revewed as these marketng elements are the foundaton of the research hypothess. Second, Game Theory and Nash Cournot competton and ther applcaton to the electrcty market are dscussed n order to compare them wth the Extended Cournot model proposed n ths thess. Thrd, demand responsveness that has been consdered as an mportant market varable, s dscussed. 2.1 Market competton related ssues The analyss of decson-makng process s becomng more mportant to the olgopolstc electrcty market competton. Smple learnng rules have been mplemented n dynamc models for strategc bddng [11]. In [12], the authors defne near-equlbrum as a case where generators maxmze ther profts and consumers maxmze ther economc utltes. They try to analyze the equlbrum n a compettve market wth sngle tme perod usng locatonal margnal prce (LMP). It s suggested that the market near-equlbrum s an effcent approach for market montorng on the generators profts. In [13], the author emphaszes the benefts of LMP compared to the actual market. It uses the Ontaro market as a case study to prove that LMP can sgnfcantly mprove both system relablty and market performance. Reference [14] analyzes changes n LMPs wth respect to operatonal parameters,.e., demands, generator cost parameters, and voltage bounds. It s beleved that the changes n LMPs as parameters vary, provdng nsght nto the functonng and behavor of the electrc energy system. Ths nformaton on senstvty mght help producers and consumers to establsh ther respectve bddng strateges, and the 8

29 regulator to assess the degree of compettveness of the electrcty market. Smple analytcal expressons are developed n [14] to compute LMP senstvtes wth respect to changes n demands throughout an electrc power network. In [15], a novel LMP polcy for dstrbuton system wth sgnfcant penetraton of dstrbuton generaton (DG) n a compettve electrcty market s presented. A new LMP method s based on remuneratng DG unts for ther partcpaton n reduced levels of energy losses n dstrbuton systems brought about by partcpaton of all DG unts n meetng demand. Snce the LMP predcton methods used n [15] contan error, uncertanty modelng s mplemented for modelng the effect of error n predcton on the LMP calculaton. In [16], the authors nvestgate the margnal costs and prces n the electrcty ndustry. The ntenton s to dentfy whether the consumers wll be charged at the prce levels that are ndcatve of the costs of supply. It s argued that approprate market prces can send sgnals about both short run and long run costs. There s agreement wth the opnon that a prce that s hgher than margnal cost ndcates the exercse of market power. Ths can be understood as the recovery of long term captal costs. Reference [17] assesses the benefts and costs of the formaton of regonal transmsson organzatons (RTOs) by evaluatng the margnal loss prcng scheme. It s argued that, for generators and consumers to receve approprate short and long term sgnals n the transmsson network, the loss component must be prced usng margnal cost methods. A case study on the transmsson losses n New York s used to llustrate the mportance of margnal cost of transmsson losses that wll affect the generator s bddng behavors. In [18], the author assesses the market based prce dfferentals n zonal and LMP market desgn. He tres to use LMP to reflect the actual costs of delverng energy by usng an accurate network model to prce n both congeston and losses wth the market desgn smulatons at the Calforna Independent System Operator. The major 9

30 fndng s that dependng on the locaton of congeston, the smulatons show that LMP wll not create sustaned adverse mpact on market prces. Dfferences n LMPs are a result of ncreased transparency of prces that show the value of energy throughout the grd. A one snap-shot Nash equlbrum s presented n [19] to analyze the market competton wth transmsson constrants. In ths model the authors demonstrate how to analyze the mpact of transmsson capacty on competton. The model s appled to the western U.S. electrcty market. The results show that transmsson constrants enhance the exercse of market power. Reference [20] focuses on congeston management and rsk management ssues n a practcal case n Europe. A coordnated congeston mechansm has been developed by the authors n order to acheve hgher level of market competton, maxmum nterconnecton usage, and decreased rsk for market partcpants. An analyss for the mpact of network constrants on the market equlbrum s presented n [21]. The authors solved a two-level optmzaton problem n the model they desgned. Frstly, the ndependent system operator (ISO) dspatches generaton and sets transmsson prce through solvng optmal power flow. Secondly, the ndvdual generators use Nash-Supply functon equlbrum strategy to submt ther generaton bds after takng nto account the ISO s decson. It s found that the generators have the ntenton to bd for generaton close to the constrant boundary. Ths resulted n mult-nash equlbrums, whch means that generators may change ther bddng behavors. In [22], suppler equlbrum strategy s analyzed by consderng transmsson constrants. The Cournot model s used to smulate the market competton wth a 2- bus and a 3-bus network. The results show that, when there s potental transmsson constrant, there mght be multple equlbrums n the Cournot competton. Another fndng s that the transmsson capacty must be far hgher than the actual power flow 10

31 on the specfc lne n order to obtan the same equlbrum wthout transmsson constrant. In [23] and [24], bddng strateges from the supply sde under possble network constrant have been nvestgated to assess the level of competton and the market performance. The authors found that strategc behavor may mtgate the serousness of congeston, whle congeston offers addtonal opportuntes for gamng to the producers. To characterze LMPs that appear n the payment cost objectve functon, the authors of [25] establshed and embedded Karush-Kuhn-Tucker (KKT) condtons of economc dspatch as constrants. They nvestgated the payment cost mnmzaton problem wth transmsson capacty constrants as formulated. A regularzaton method was used to satsfy constrant qualfcatons. Ther numercal results show that payment cost mnmzaton leads to consumer payment savngs as compared to bd cost mnmzaton for the same set of supply bds. References [11] to [18] have focused on olgopolstc competton, and market prcng mechansms, but they dd not nvestgate further on the transmsson constrants or the nvolvement of demand sde. References [19] to [25] have nvestgated the mpact of transmsson constrants on a compettve electrcty market. The constrants presented are focused on transmsson sde. However generaton capacty constrant should also be nvestgated. 2.2 Cournot modelng n electrcty market competton A study on the market power of generators n the electrcty market wth transmsson constrants s done n [26]. It s argued that many authors do not obtan the same results, nor do they ndcate ther assumptons n the Cournot models. A smplfed duopoly model wth one transmsson lne s presented n the paper. A transmsson lmt s appled to the model and a lnear demand functon s defned. Wth the 11

32 nvolvement of the System Operator, t s concluded that there s no drect relaton between the use of transmsson capacty and the transmsson prce outcome. A two-settlement equlbrum n compettve electrcty markets s formulated n [27]. In ths model, each generaton GenCo solves a Mathematcal Program wth Equlbrum Constrants (MPE). The model ncludes forward and spot markets. It mplements a two-soluton approach and apples these solutons to a 6 node and 8- lne model. Two approaches, teratve Response Surface Method (RSM) and teratve Penalty Interor Pont algorthm (PIPA), are compared. The authors expect that the generators market power s encouraged by rasng the forward prce to ncrease profts. Other suppler sde modelng mechansm, such as the supply functon equlbrum model, are also used to compare the market equlbrum wth the one from Cournot competton model n the case of transmsson constrant [28, 29]. It was found that strategc generators were able to capture all the congeston rental of the constraned lne. They recommend that market rules be desgned to restrct the bddng flexblty and eventually reduce the equlbrum prces. In [30], Cournot equlbrum s nvestgated n a smple network. The network conssts of three buses, and each bus has one generator and one load. The study analyzes the Cournot competton wth the exstence of transmsson constrants. The results from the model show that generators can sgnfcantly affect the lne flows and market prce through strategc behavors. When transmsson constrants exsted, the market outcome became much more uncertan. A demonstraton on the mportance of actve partcpaton of transmsson rghts owners n a compettve market to acheve an effcent outcome s addressed n [31]. It s argued that, even wthout market concentraton, congeston, and passve transmsson rghts wll cause mplct colluson among generators. As a result, the market prce wll be hgher than the margnal cost. The prce devaton can also result n short and long term neffcency. 12

33 In [32], the authors modeled Nash equlbrum through relaxaton procedure and appled t to an objectve functon, known as the Nkado-Isoda functon. The Nkado-Isoda functon s defned as a transformaton of an equlbrum problem nto an optmzaton problem. The relaxaton algorthm s used to calculate Nash equlbrum under constrant. The Nash equlbrum s appled to the IEEE 30-bus system. The emphass s on the mportance of prce-demand elastcty. As elastcty changes, generators wll also change ther ways n order to collude. The strategc behavors of the generators that may result n transmsson congeston s dscussed n [33]. The author calculates Nash-Cournot equlbrum n a crtcally congested 6 bus network. The fndng s that Nash-Cournot equlbrum does exst n the case of transmsson congeston. Both [34] and [35] used a smplfed three generaton power system model to nvestgate olgopolstc competton n the electrcty market. The authors n [34] conducted a set of experments on the olgopolstc markets wth three generatng companes, and ther results show that market competton wll converge nto the results between perfect competton equlbrum and Nash Cournot equlbrum. In [35], a smple numercal example wth three generaton companes s demonstrated to show the basc dea of the authors proposed method. They focused on the analytcal study of the equlbrum of N-company generaton market. They obtaned Nash equlbrum n a unform prce aucton n the spot market. In [36], an analyss s conducted on the potental market power n a restructured New Jersey electrcty market. The calculaton of Nash-Cournot equlbrum s done wth dfferent demand levels durng peak and off-peak perods. A prce responsve demand curve s also ncluded. It s found that there s a threshold level of demand at whch the Cournot competton wll result n hgher market prce f the actual demand s hgher than the threshold level. When the demand level s lower than the threshold level, the effect of a Nash Cournot game on the prce s relatvely much smaller. A 13

34 rapd prce ncrease s dentfed when there s a potental transmsson constrant between New Jersey and the rest of the PJM market. Two Cournot models of mperfect competton among generators, one wth arbtrage (Blateral model) and the other wth arbtrage (PoolCo model), are formulated n [37] and [38]. A congeston prcng scheme for transmsson s concluded. The mportance of ncludng the transmsson network, whle modelng the Cournot competton and the choce of transmsson prcng scheme, s addressed. A new algorthm s used n [39] to calculate the Cournot equlbrum wthout the presence of transmsson congeston. The Cournot game s transformed nto a three level decson-makng process wth economc sgnal exchange. The advantages and dsadvantages between two popular methods (.e., Cournot Equlbrum and Supply Functon Equlbrum (SFE)) are compared to model competton. The model s tested by usng a market comprsng the generatng unts of the IEEE 3-area RTS9. References [40] and [41] have compared Cournot competton wth Bertrand competton. In [40], the Australa Natonal Electrcty Market (NEM) was utlzed as an example test system to valdate the authors method. The comparson between Cournot model and other models, such as the Bertrand model, shows that the Cournot model has a more reasonable perfomance n the NEM. The comparson between Bertrand and Cournot competton n [41] shows that Bertrand competton can also yeld Cournot outcomes dependng on both the strategc varables and the context n whch those varables are employed. Capacty wthhold by generaton companes has been nvestgated under Nash Cournot competton n [42] and [43]. The smultaneous move and sequental-move games n appled mathematcs are used to model the nteractons of the generaton companes, transmsson network and market operator. The results from the smulatons n the sx-bus Garver s example system and the IEEE 14-bus system show that market power can be successfully modeled n the transmsson augmentaton algorthm [42]. The authors n [43] consdered physcal wthholdng of capacty n the 14

35 energy market and the HHI ndex s utlzed to measure market concentraton. They found that market desgn s crucal to determne the possble exstence of capacty wthhold and market power. Other papers, such as [44], [45] and [46] focused on electrcty markets that are cleared by mert order usng pure strategy Nash equlbrum. They analyzed how market power s nfluenced by the number and sze of the competng generaton companes. There are other studes done on Cournot competton, such as [47], [48], [49], [50], [51] and [52]. Reference [47] modeled Cournot prces wth generator avalablty and demand uncertanty, but wth no consderaton of transmsson congeston. Reference [48] used an agent-based test bed, AMES (a market software package), to study the power market operatons subject to generaton constrants, transmsson constrants, and strategc behavors. Reference [49] nvestgated Nash Cournot equlbrum n power markets n both PoolCo and Blateral models. The authors found that Cournot competton among producers yelds the same outcomes for both market desgns. A compettve co-evolutonary algorthm s used to model agents nteractons n the market by fndng the Nash Cournot Equlbrum [50]. A Game theoretcal approach s also compared wth other decson makng methods, such as cost beneft analyss [51]. A drect computaton s developed to calculate Nash Equlbrum of electrcty markets usng the supply functon model [52]. A large number of smulatons have been done n [26] to [52] on the Cournot competton wth the presence of transmsson constrants. However, they dd not specfy the mportant mpact that demand elastcty could have on market compettons. 2.3 Demand elastcty The responsveness of customers to prce changes s characterzed by ther prce elastcty of demand [53]. Demand responsveness plays a vtal role n ncreasng 15

36 effcency and reducng prce volatlty n the electrcty markets. The authors of [53] nvestgated demand-bd prce senstvty and supply-offer prce caps on LMPs. They modeled a restructured wholesale power market operatng through tme subject to transmsson lne constrants, generaton capacty constrants and strategc trader behavor The authors of [54] modeled demand response programs n the power markets. They developed an extended responsve load economc model based on prce elastcty and customer beneft functon. In [55], the authors nvestgated the nfluence of prce responsve demand shftng bddng on congeston and LMPs n pool-based day-ahead electrcty markets. They compared the prce responsve demand and conventonal prce responsve and prce takng bds. They found that the conventonal prce takng bds encourages the GenCos to exercse market power and may lead to uncontrolled LMPs and system congeston. Authors of [56] found that ncrement n the demand elastcty provdes the expected postve results n market performance n terms of the Lerner ndex and the reduced congeston n the network. Smlarly, the results from the smulatons n [57] whch attempt to quantfy the effect of demand response show that market clearng prce tends to reduce wth ncreasng level of demand shftng. However, the study conducted n [58] shows that, although demand response provdes an mportant contrbuton to the electrcty market, ts contrbuton has lmted mpact on relablty of supply sde resource and the capacty market. There s also a study that focuses on the analyss of a compettve power market wth constant elastcty functon, usng Cournot game to determne the market equlbrum and prce level [59]. The behavor of customers under dfferent demand response programs s modeled consderng ncentve and penalty mechansms [60]. In [53] to [60], demand elastcty, sometmes also called demand response, s ntroduced n the dscusson of power market. But the mpact of change n the physcal quantty demanded and demand elastcty has not yet been clarfed. 16

37 The followng ssues need further nvestgatons: The system constrants can be consdered n two dfferent areas,.e. generaton constrants and transmsson constrants. Most of the studes that menton constrant are only focused on transmsson constrant. The generaton capacty tself also mples a constrant to the electrcty market and network. Especally when the margnal costs are dfferent among generators, the generaton capacty constrant does sgnfcantly affect the outcome of Cournot equlbrum. The ssue of generaton capacty constrant has not yet been dentfed fully n the lterature. The prce elastcty of demand has been mentoned n several papers, but none of them has ncluded t as an nput varable nto ther mathematcal models. The elastcty of demand s a key factor to help nclude dynamcs nto the calculatons of Cournot competton. The effect of change of demand elastcty on the ndvdual generaton of each generator and ther profts needs to be nvestgated The prce dervatve aganst elastcty has also not been addressed. One of the key ndcators n measurng generators strategc behavors s the market prce level. As the elastcty of demand changes, market prce wll also change as consumers may change ther demand wth the response to the change n prce. The ndvdual quantty dervate aganst elastcty has not been nvestgated. The ndvdual quantty refers to the generaton of each generator, whch s determned by the Cournot equlbrum. As the elastcty of demand s ncluded, the total quantty demanded by the customer may change, and ths wll result n a change n the correspondng total generaton. Therefore, the share of the total generaton among the generators wll be dfferent. 17

38 Chapter 3 Model dervatons 3.1 Classc Cournot model and Nash equlbrum n game theory In the classc model of Cournot competton, players compete wth each other through ther quantty bds. Ths competton model apples to the game comprsng two or more players. Wthn Cournot competton, the most popular soluton concept s called Nash equlbrum. Nash equlbrum exsts when each player has chosen a strategy and no player can beneft by changng hs strategy whle the other players keep thers unchanged. There are two mportant assumptons for Cournot competton. Frstly, one player decdes hs strategy or decson on quantty bd, takng nto account the others quantty decsons. Ths means that the player s payoff or proft depends not on hs own quantty but also the quanttes of other players. Another assumpton s that players decde ther quantty bds smultaneously. The smultaneous aspect mples that each player chooses hs quantty wthout knowng the choces of others [6], [31], [34], [35], [36], [40]. A two-stage game between expected proft maxmzng GenCos has shown an unque equlbrum n the Cournot competton. In the frst stage, GenCos decde ther producton smultaneously and ndependently. In the second stage, GenCos choose prce smultaneously and ndependently and the demand s allocated among GenCos at the market clearng prce level. If the demand functon s concave, the Cournot outcome s the unque equlbrum outcome. The result shows an mportant fndng that quantty competton s a choce of scale wth GenCos competng by means of ther cost functons. The weght of each GenCo s own cost over average GenCos sum costs determnes the quantty produced by each GenCo [41]. 18

39 However, the classcal Cournot model ntally does not consder the constrants, such as generaton capacty and transmsson network constrants. It seeks the equlbrum based on the supply sde, for example, the margnal costs of the GenCos have sgnfcant mpact on the ndvdual output. There s a hgh possblty that demand may not be 100% satsfed. Although demand functon, wth demand ntercept and slope, s used for calculaton, the demand elastcty has not been specfed. In ths Chapter, an Extended Cournot model s developed. It ncludes the consderaton of elastcty of demand as an mportant nput to the model tself. In ths thess, system constrants are modeled as generaton and transmsson constrants. Demand elastcty s ncluded whch allows load to express ts wllngness to reduce consumpton when prce changes. Therefore, the nvestgaton s focused on the mpact of generaton constrants, transmsson constrants and demand elastcty on GenCo market strateges. Table 3.1 represents a comparson between the Cournot classcal model and the Cournot extended model. In the Cournot extended model, demand elastcty becomes a very crtcal nput varable. Wth the analyss of prce dervate and generaton quantty dervatve, t s possble to demonstrate the mpact of margnal change n elastcty of demand on the market prce and quantty produced by each generator. Table 3.1 Comparson between two dfferent Cournot models Classcal Cournot model Extended Cournot model Input Margnal cost Margnal cost varables Total demand Total demand Demand elastcty Generaton constrants Transmsson constrants Output Generaton Generaton varables Market clearng Market clearng prce prce Prce dervatve Generaton quantty dervatve 19

40 3.2 Dervaton of Extended Cournot model Cournot competton has been used extensvely n duopoly competton, wth two players, and olgopoly competton, wth multple players, to nvestgate how GenCos are maxmzng ther profts by usng quantty as a strategy. The maxmzaton problem of ndvdual proft n the case of three GenCos s derved as follows: Max ( ) (3.1) where, 1,2, 3 n - ndvdual proft of GenCo As each GenCo normally acts n ts own self-nterest, the proft maxmzaton functon of a company n an olgopoly competton wth three GenCos can be descrbed as follows: P Q C * Q (3.2) where, - ndvdual proft of GenCo P - market clearng prce Q - ndvdual output of GenCo Q - total producton output C - margnal cost of GenCo In the competton of a Cournot model, the Nash Cournot equlbrum s the soluton of the game. However, the assumpton s that each GenCo chooses ts own quantty level assumng that the quanttes of other GenCo are fxed [8]. As each GenCo 20

41 maxmzes ts own proft aganst ts quantty, Nash Cournot equlbrum only requres the frst order (F.O.) partal dervatve of Equaton (3.2) to be equal to 0 (.e., 0 Q ) to ensure the proft maxmzaton. Settng the partal dervatve of Equaton (3.2) to equal to zero gves the followng: Q P Q P P * Q C * Q C * Q * Q P C 0 Q (3.3) P In Equaton (3.3), s the prce dervatve. It shows the margnal change n market Q prce over the margnal change n quantty produced by all GenCo. From Equaton (3.3), the followng s obtaned C P Q (3.4) P Q P The relatonshp between market clearng prce and total quantty,.e. Q normally consdered as negatve, whch means that when prce ncreases, the total quantty demanded at the load s gong to decrease. Therefore, Equaton (3.4) can be smplfed to the followng,, s Q P C 0 (3.5) Equaton (3.5) mples that a GenCo s quantty change has an nverse relatonshp wth ts margnal cost. Ths means that a GenCo wth hgher cost wll produce less output. Ths descrbes the rule of quantty dstrbuton among the GenCos n the quantty game. When a GenCo s costs rse, ts producton wll decrease. Ths also ndcates a potental relatonshp between GenCo s costs and the output of the other 21

42 GenCos. For example, the ncrease n GenCo s costs may result n an ncrease n the producton of the other GenCos. In a gven demand functon, the market clearng prce can be gven as: P a bq a b n Q 1 where, P - market clearng prce a - ntercept of the demand functon b - slope of the demand functon (3.6) Q - ndvdual quantty producton of GenCo 3.3 Cournot Extended model wth no constrants The objectve functon n the case wth no constrants s Equaton (3.1). After takng the frst dervatve, Equaton (3.7) s the condton for proft maxmzaton. Elastcty of demand s defned as follows: Q %Q Q Q P P (3.7) % P P P Q P Q P Q where, - Demand elastcty, % Q - Percentage change n quantty demanded by consumer % P - Percentage change n market prce P Q - Prce dervatve, usually, t can also be noted as P. 22

43 From Equaton (3.7), the followng s obtaned P Q P Q (3.8) P To ntroduce elastcty of demand nto Equaton (3.3), Q n Equaton (3.3) s replaced by P of equaton (3.8). The followng Equaton (3.9) s then obtaned, Q P Q ( ) P C 0 Q (3.9) From Equaton (3.9), the followng Equaton (3.10) s obtaned Q Q ( C P) (3.10) P As the total demand equals the total supply, whch s the sum of all ndvdual producton, we have Q n Q 1 where n - the number of GenCos n the market. Q n Equaton (3.11) s replaced by followng s obtaned, (3.11) Q ( C P) of Equaton (3.10), and the P Q n n Q Q [( C P) ] C nq (3.12) P P 1 1 Usng Equaton (3.12), the followng Equaton (3.13) s obtaned: 23

44 P ( n 1) n C 1 (3.13) Usng Equaton (3.10), and by replacng P wth (3.13), the followng Equaton (3.14) s obtaned ( n C n 1) 1 as shown n Equaton C ( n 1) Q ( n C 1 n C 1 ) Q (3.14) The above equatons are based on the assumptons that demand elastcty, total demand Q, and margnal cost of all generaton companes C are known, and, the market prce P and ndvdual output Q are unknown. Accordng to Equaton (3.13), the market prce s determned by the margnal cost, elastcty of demand, and the number of GenCos. It can be seen that market prce s ndependent of the total quantty demanded by the load. Ths s another key dfference between the Extended Cournot model and the classcal Cournot model. In the classcal Cournot model, the market prce s hghly dependent on the total quantty that s expressed n a demand functon. Whereas n the Extended Cournot model, market prce s not dependng on the total quantty. In Equaton (3.14), GenCo s ndvdual quantty s determned by margnal cost, elastcty of demand, number of GenCos and total demand. Note that the sgn of demand elastcty s negatve, whch represents the nverse relatonshp between market prce and quantty. For example, comparng the levels of demand elastcty between and -0.5, t s stated that the elastcty of demand s 24

45 hgher at than the one at -0.5 n absolute value. The prce elastcty of -1.0 means that a 1% ncrease n prce, wll result n a 1% decrease n quantty. The partal dervatve of market prce wth respect to demand elastcty s gven by Equaton (3.15): n C P 1 ( n 1) 2 (3.15) The partal dervatve of GenCo quantty wth respect to demand elastcty s gven by Equaton (3.16): Q CQ C Q (3.16) where C n C n s the average margnal cost Rearrangng Equaton (3.16) gves: Q CQ C Q C C 1Q (3.17) Once elastcty of demand s gven, the quantty of power generated has an nverse relatonshp wth the margnal cost. It means that the lower the margnal cost of the th GenCo, the hgher the quantty produced. It looks smlar to the classcal economc dspatch, however Equaton (3.17) does not take nto account the weght of ndvdual capacty Q n the measure of average margnal cost C. In prncple, the maxmum 25

46 capacty of the GenCos can be dfferent. In ths thess, for smplcty purpose, the maxmum capactes of the GenCos are set to be all equal. Equaton (3.17) also shows that output ncreases when the cost of the th GenCo s lower than the average cost, and decreases when the cost of the th GenCo s hgher than the average cost. It remans constant (dervatve equals to zero) when the cost of the th GenCo s equal to the average cost,.e. C C. 3.4 Derve Extended Cournot model wth constrants under Kuhn-Tucker condtons The Kuhn-Tucker theorem s used to solve the proft maxmzaton problem of the players n the Extended Cournot model settng wth constrants. In the Extended Cournot model, partal dervatve s taken for ndvdual proft aganst ndvdual quantty producton. Ths s not an optmzaton of the sum of total profts. Accordng to the Kuhn-Tucker theorem, the non-lnear programmng problem can be descrbed as follows [65]: max f ( x) subject to g ( x) 0 = 1,, n (3.18) where f(x) s the objectve functon and g (x) s the constrant functon. The assumpton s that all functons are dfferentable. There are three condtons to hold: 1) x * s feasble then, there exst multplers 0, 1,..., n, such that 2) ( x*) 0 1,..., n, and g 3) f ( x*) g ( x*) 0 m 1 26

47 where f (x*) s maxmzaton of the objectve functon The Lagrange functon can be shown to be [65]: n L( x, ) f ( x) g ( x) f ( x) g ( x) (3.19) 1 Two multplers, and, are ntroduced for the generaton capacty and transmsson capacty constrants, n sectons and 3.4.2, respectvely Generaton capacty constrant s assumed to be the multpler determned by the relatonshp between max Q, then, the followng s obtaned Q and max ( Q Q ) 0, 0 (3.20) where, max Q - GenCo s maxmum generaton capacty Referrng to the Kuhn Tucker theorem, the term ( Q Q ) should be zero whch means that both factors can be zero. For Equaton (3.20), ths means that f a generator max s fully dspatched,.e. Q max Q, then the multpler can be zero or non-zero n ths case. But f a generator s not fully dspatched,.e., for Equaton (3.20) to hold. Q max Q, equals to zero Ths means that can be consdered as a penalty mposed on the GenCos when ther real outputs are equal to the maxmum. In ths way, the penalty wll be accounted for when calculatng the margnal cost of a GenCo. For example, f GenCo 1 s producng at maxmum capacty, ts margnal cost becomes C 1 1. Ths makes GenCo 1 more 27

48 expensve and not as economcal as before. The same prncple apples to other GenCos. The maxmzaton problem s the proft maxmzaton of each generator, whch s descrbed as follows: Max (3.21) max subject to Q Q 0 max ( Q Q ) =0, 0 Therefore, the Lagrange functon can be rewrtten as follows: max L(, ) ( Q ) g( Q ) ( P Q C Q ) ( Q Q ) (3.22) max s.t. Q Q 0 ( Q max Q ) 0 The followng s defned, ( 1,..., t ) max max 1 max t ( Q Q ) [( Q Q ),...,( Q Q )] 1 (3.23) Based on equaton (3.22), the ndvdual proft functon can be rewrtten to the followng, max ( P Q C Q ) ( Q Q ) (3.24) By takng the frst order dervatve of equaton (3.24.) wth respect to Q, the followng Equaton (3.25) s obtaned: 28

49 Q P Q Q P C 0 (3.25) wth the same dervaton process n secton (3.3), the followng s obtaned, n ( Q Q)( C ) 1 max0, C (3.26) ( n 1) Q n P ( C ) (3.27) n 1) ( 1 Q ( C )( n 1) Q Q n C 1 (3.28) The above equatons are based on the assumptons that demand elastcty, total demand Q, and margnal cost of all generaton companes C and the sum of ndvdual generaton capacty multpler are known, and, the market prce P, ndvdual output Q and ndvdual generaton capacty multpler are unknown. Accordng to Equaton (3.27), the market prce s dependng on. In Equaton (3.28), Q s determned by and. To nvestgate the mpact of the margnal change of prce and quantty wth respect to the change of elastcty of demand, dervatves on Equatons (3.27) and (3.28) were taken and the followng was obtaned: 29

50 ( 1) n C P n (3.29) Q C Q C Q ) ( (3.30) where s the average value of and s defned as follows, n n n 1 (3.31) and C as defned prevously. Equatons (3.30) s rearranged as: Q C C Q C Q C Q 1 ) ( (3.32) Generaton and transmsson capacty constrants After the ntroducton of generaton capacty lmt as a constrant, another constrant s mposed on the transmsson lne between bus j and bus k. The dea s to create a potental transmsson constrant for transmttng power from the most economcal GenCo to the load. It s necessary to note that the transmsson lmt between buses j and k can also be modeled wth constrant. The maxmzaton problem s stll the proft maxmzaton of each generator as descrbed n Equaton (3.21). However t s subject to the followng: subject to 0 max Q Q

51 31 ) ( max Q Q =0, n jk Q d T where, d - percentage of Q goes through lne jk L jk T - transmsson capacty of lne jk L The possble transmsson constrant s assumed on lne jk L, and ths constrant apples to every GenCo s proft functon. Therefore, the general expresson of the lagrangan functons s as follows: ) ( ) ( ) ( ) ( ) ( ) ( ),, ( n jk n n jk n a Q T Q Q Q C Q P a Q T Q g Q L (3.33) For the transmsson capacty constrant, s assumed as the multpler determned by the relatonshp between n a Q 1 and jk T. After takng frst dervatve, the followng s obtaned: 0 1 n Q d C P Q Q P (3.34) wth the same dervaton process n secton (3.3), the followng s obtaned,

52 32 ) ) ( ( 1) ( 1 1 n n Q a C n P (3.35) Q a Q C n a C Q n n 1 1 ) ( 1) )( ( (3.36) n n a C Q n a C Q Q 1) ( ) ) ( )( ( 0, max 1 1 max 1 (3.37) Agan, when frst dervate s taken on Equatons (3.35) and (3.36) n respect to demand elastcty, the followng s obtaned: ) ( ) ) ( ( n a C P n n (3.38) Q a C a C n Q n n 1 ) ( ) ( 1 1 (3.39) Based on the above equatons, smulatons have been conducted usng the Extended Cournot model wll be dscussed n chapters 4 and 5.

53 Chapter 4 Smulaton results of analytcal model In order to llustrate the hypothess more clearly and easly, a smplfed three-bus benchmark network as shown n Fgure 4.1 s used. It should be mentoned that n a few papers [30], [31], [33] a smlar smplfed power system network was used to model Nash Cournot competton n the presence of transmsson constrants. In prncple, the extended Cournot competton can be appled to a more complcated power system network. Fg. 4.1 Smplfed three bus benchmark DC network As shown n Fgure 4.1, there s one GenCo on each node, and a load s placed on node 3 where the most expensve generator s located. It s mportant to note that the poston of load n the network s crtcal to the market competton output. The reason to place the load on bus 3 s to dstance t from the more economcal generators, whch are generators 1 and 2. The load poston s mportant n the case of mposng transmsson constrants, n whch the load may not be able to acqure as much power as t wshes from the more economcal generators. The smulatons performed n ths thess covered market operatons under the followng two cases: 33

54 Case 1: Change n total demand Case 2: Change n elastcty of demand In each case, three dfferent scenaros are presented based on the level of constrants that were ntroduced, whch nclude: Scenaro 1: No constrants Scenaro 2: Generaton capacty constrants Scenaro 3: Transmsson and generaton capacty constrants To present the market competton results, the focus s on the followng output varables for analyss GenCos ndvdual output GenCos ndvdual profts Market clearng prce Load payment 4.1 Case formulaton For these smulatons, the followng assumptons were made: The change of quantty demanded does not affect the market clearng prce n a Nash-Cournot game as dscussed earler, gven that demand elastcty s constant. Ths s dfferent from the tradtonal understandng that the change of demand wll cause an ncrease n prce due to the decrease n the supply sde, whch s generally an economc dspatch approach. Wth the ntroducton of system constrants, the mpact of demand elastcty on market competton wll be more lmted. The demand sde s usually expected to utlze ts demand elastcty to bd n order to reduce the market clearng prce and the load payment. The exstence of systems constrants, especally 34

55 the transmsson constrants can sgnfcantly offset the effect of demand elastcty Margnal cost C The margnal costs (C ) n Rand (R ) terms of GenCos are assumed as follows: C1 R100 C2 R150 C3 R200 It s possble to change the dfference nterval between the GenCos costs as they may sgnfcantly affect the allocaton of demand among GenCos. However ths s not the focus of ths thess and can therefore be nvestgated n future research Generaton capacty The maxmum outputs of the GenCo are denoted as O. In scenaro 1, wth no constrants, the maxmum generaton capactes of three GenCos are: max max max 1 2 3, O O O 1000MW In scenaros 2 and 3, the maxmum generaton capactes of three GenCos are: max max max O O O 250MW It s assumed that due to the exstence of generaton capacty constrants for all GenCos, the mpact of demand elastcty on market competton wll decrease. The multpler can be consdered as a penalty added to the margnal cost when a specfc GenCo s fully dspatched. Ths wll cause an ncrease n market clearng prce when one or more GenCos are supplyng at full capacty. 35

56 4.1.3 Transmsson capacty In scenaro 3, wth generaton and transmsson constrants, for example as shown n Fgure 4.2, a network constrant wth a maxmum transmsson capacty of 225 MW s appled to lne L 13, whch s the transmsson lne between bus 1 and bus 3. The maxmum transmsson capactes for lne L 12 and L 23 are set to be at 500MW. Fg. 4.2 An example of 3-bus network wth transmsson network constrants wth total demand of 600 MW In general, f the percentages of Q 1 and Q2goes through lne L13 ncrease, whch are d 1 and d 2, respectvely, the flow of power from bus 1 to bus 3 va lne L 13 wll ncrease. Ths wll affect the proporton of power to be delvered from GenCo 1 to the load. The value of d, whch s the percentage of for further research calculaton and nterpretaton. Q goes through lne L jk, can be changed 36

57 In the calculatons n ths thess, t s assumed that all three lnes have the same mpedance to ensure that the power flow satsfes Krchhoff Law. Ths s also due to the consderaton of a smplfed network Demand elastcty To defne wthn whch area elastcty can vary, based on Equatons (3.13), the condton for the prce to be postve s that, n 1 0, Snce n 1 s the denomnator n Equaton (3.13), t should be less than, but not equal to, zero. In ths study, the number of GenCos s three, therefore the value of n equals to 3. Then, based on the above equaton, 1 3 In Equaton (3.14), the condton for the ndvdual quantty Q to be postve s that, C ( n 1) 0 n C 1 For n=3, n 1 C R100 / MW R150 / MW R200 / MW R450 / MW C Therefore, wthout consderng Q, 3* C

58 To meet the above condton, C s replaced wth the margnal cost of GenCo 1, GenCo 2 and GenCo 3, respectvely. Then s obtaned as follows: 4 3 Therefore, the area of elastcty s found as follows: In the calculaton, t s found that the market results for value of demand elastcty wthn the area of to become extremely volatle. There s large change n the output varables, such as market clearng prce, ndvdual quanttes, aganst the demand elastcty. Ths s due to the fact that the mpact of demand elastcty, at the lower bound of ts value, s sgnfcantly hgh on the above mentoned varable. Therefore, the results wth the area of demand elastcty between to are consdered n ths thess. It s necessary to pay attenton to the negatve sgn: whch s lower (more negatve) than , but economcally speakng, elastcty s hgher (n absolute value) at Case Dscusson Change of total demand wth constant demand elastcty The assumptons n ths secton are as follows: Demand elastcty stays constant, wth (there s no partcular reason for choosng the demand elastcty at , t can be any value between to ) 38

59 Margnal cost stays constant at R100/MW, R150/MW and R200/MW, respectvely Total demand vares from 100 MW to 800MW wth 100 MW nterval The change of total demand can be explaned by Fgure 4.3. The total demand curves n Fgure 4.3 have the same ntercepton on the vertcal axs, whch represents the hghest market clearng prce. The horzontal axs represents the total demand. When the demand elastcty s constant, a horzontal dashed lne shows that prce stays the same at P* at dfferent total demand. The elastcty of demand s the same at pont A, B, C and D. Market clearng prce (Rand/MW) P* A B C D Demand 1 Demand 2 Demand 3 Demand 4 Q1 Q2 Q3 Q4 Quantty (MW) Fg. 4.3 Change of total demand wth constant demand elastcty Change of demand elastcty wth constant total demand The assumptons n ths secton are as follows: 39

60 Demand elastcty vares from to Margnal cost stays constant wth R100/MW, R150/MW and R200/MW, respectvely Total demand stays constant at 600 MW The change of demand elastcty can be descrbed n Fgure 4.4. The demand elastcty at pont E, F, G and H are dfferent from each other. The vertcal dashed lne represents the same quantty demanded at Q*. As demand elastcty changes wth constant total demand, dfferent market clearng prces are obtaned. Market clearng prce (Rand/MW) P4 H P3 P2 P1 G F E Q* Quantty (MW) Fg. 4.4 Change of demand elastcty wth fxed total quantty demand and constant margnal cost 40

61 4.3 Case 1 - Change of total demand wth constant demand elastcty Scenaro 1 - wth no system constrants Fgure 4.5 shows the value of the market clearng prce over the change of total demand. It s found that the market prce does not change as total demand ncreases. Accordng to the conventonal electrcty supply ndustry, as demand ncreases, prce wll change. For example, prce at peak demand wll be hgher than at off-peak demand. 350 Market clearng prce (Rand/MW) Total demand (MW) Fg. 4.5 Market clearng prce wth 100MW change nterval of total demand, where demand elastcty stays constant at However, n the case of Extended Cournot model competton, the market clearng prce s not determned by the total demand. As shown n Equaton (3.13), market clearng prce s determned by the demand elastcty and the sum of the margnal cost of all GenCos. In the assumpton, the margnal costs of GenCos are constant. The 41

62 results match the expectaton that the market clearng prce does not change f total demand changes. Fgure 4.6 shows the level of GenCos ndvdual outputs when total demand changes. All three GenCos ndvdual outputs ncrease as total demand ncreases. The addtonal demand s allocated among GenCos accordng to ther margnal cost. Each GenCo s output s determned by the weght of ts margnal cost over the sum of all the GenCos margnal costs. Indvdual producton (MW) Total demand (MW) Q1 Q2 Q3 Fg. 4.6 Indvdual outputs wth change of total demand, where demand elastcty stays constant at The results show that the lower the margnal cost of a GenCo, the more output wll be requred from ths GenCo. Based on the assumpton, GenCo 1 s the most economcal generator wth the lowest margnal cost. The output of GenCo 1 s the hghest. Conversely, GenCo 3 s the most expensve generator wth the hghest margnal cost, and ts output s the lowest. Although the quantty allocaton n Cournot competton s determned partly by the margnal costs of GenCos, t s fundamentally dfferent 42

63 from the conventonal economc dspatch. The conventonal cost-based dspatchng mechansm determnes the output of generators based on ther margnal costs. The most economcal generator wll be utlzed as much as possble to satsfy the load before the more expensve generator s accepted. The Cournot competton s a quantty game, or a quantty competton. The equlbrum s based on the proft maxmzaton of each ndvdual GenCo. At the equlbrum pont, the output levels of all GenCos are optmal. Even the more expensve GenCo wll be dspatched before the less expensve GenCo s fully dspatched. The change of output levels n Fgure 4.6 shows that any addtonal demand wll be satsfed wth the same pattern, or proporton, among the GenCos. Fgure 4.7 shows the GenCos ndvdual profts aganst the change of total demand. It can be seen from Fgure 4.7 that ndvdual output, Q, ncreases as total demand, Q, ncreases. Therefore, ndvdual profts ncrease as total demand ncreases. The proportonal ncrease of each GenCo s proft has a smlar pattern to the example wth an ncrease n quantty. Indvdual profts (Rand) π π π Total demand (MW) π1 π2 π3 Fg. 4.7 Indvdual profts wth change of total demand, where demand elastcty stays constant at

64 Fgure 4.8 shows the change of load payments as total demand changes and demand elastcty stays constant ( ). As load payment s equal to the market clearng prce multpled by the total quantty demanded, the load payment ncreases as total demand ncreases Load payment (Rand) Load payment Total demand (MW) Fg. 4.8 Load payments wth change of total demand, where demand elastcty stays constant at Scenaro 2: wth generaton capacty constrant Wth generaton capacty constrant, the market clearng prce ncreases as total demand ncreases. When total demand was low, the allocaton of output among GenCos was the same as the one wthout capacty constrant, as equals to zero. 44

65 Fgure 4.9 shows the change of market clearng prcng when total generaton reaches ts lmt. As shown n Fgure 4.9, when total demand ncreases, the most economcal GenCo wll be dspatched fully,.e. Q1 Q1 and wll be non-zero. The quantty allocaton among the three GenCos for a total demand of 600 MW s 250 MW, 207 MW and 143 MW, respectvely. The market clearng prce reaches P=R311 at ths demand level. The ncrease of market clearng prce becomes non-lnear at an acceleratng rate. Ths means that, when the level of total demand becomes close to the total generaton capacty lmts of all three GenCos, the proportonal ncrease of the market clearng prce s hgher than the ncrease of total demand. Market clearng prce (Rand/MW) Total demand (MW) Fg. 4.9 Market clearng prce wth constant demand elastcty under generaton constrant 45

66 Fgure 4.10 presents the change of ndvdual producton gven the generaton capacty constrant. As shown n Fgure 4.10, before reachng the maxmum generaton capacty, all GenCos output levels are the same as the ones wthout generaton capacty constrants. When the capacty lmt of a specfc GenCo s reached, ts output wll reman constant. When GenCo 1 s fully dspatched, GenCo 2 and GenCO 3 both ncrease ther outputs to meet the addtonal demand. For example, after the total demand reaches 600 MW, GenCo 1 s fully dspatched; GenCo 2 and GenCo 3 are producng 207 MW and 143 MW, respectvely. The proportonal ncrease of all GenCos has changed due to the generaton capacty constrant. 300 Indvdual producton (MW) Total demand (MW) Q1 Q2 Q3 Fg Indvdual producton wth constant demand elastcty under generaton constrant 46

67 The ndvdual proft change aganst total demand s shown n Fgure The ndvdual proft s determned by both market clearng prce and ndvdual producton. When both varables ncrease, the GenCos proft wll ncrease at an accelaratng rate. Ths apples to GenCo 2 and GenCo 3 n the case when GenCo 1 s fully dspatched, as the market clearng prce wll only ncrease when one of the GenCo s generaton capacty lmt s reached. In ths case, GenCo 1 s proft wll only ncrease as the market clearng prce ncreases as ts producton reaches maxmum. But the profts of GenCo 2 and GenCo 3 wll ncrease dramatcally as both ther producton and market clearng prce are ncreasng Indvdual profts (Rand) Total demand (MW) π1 π2 π3 Fg Indvdual profts wth constant demand elastcty under generaton constrant 47

68 As shown n Fgure 4.12, the change pattern of load payment s also affected by the generaton capacty constrant. Load payment depends on total quantty demanded and the market clearng prce. Ths ndcates that load demand s bearng the cost of system constrant Load payment (Rand) Total demand (MW) Fg Load payment wth constant demand elastcty under generaton constrant 48

69 4.3.3 Scenaro 3: wth generaton and transmsson capacty constrants Fgure 4.13 presents the market clearng prce wth both generaton and transmsson constrants. By comparng wth the prce level n scenaros 1 and 2, the level of market clearng prce s the hghest among three scenaros. From the smulatons, the total demand can only ncrease up to 735 MW under both constrants. Ths means that, wth the exstence of transmsson constrants, GenCo 1 and/or GenCo 2 may not be able to be fully dspatched. Ths has also pushed the market clearng prce hgher as there s no margnal unt that can be suppled by the more economcal GenCos, that s, GenCo 1 and GenCo 2, to the load at bus 3. Market prce (Rand/MW) Total demand (MW) Fg Market clearng prce wth constant demand elastcty under generaton and transmsson constrants 49

70 It s nterestng to see that the ndvdual producton of GenCo 1 frst ncreases to ts maxmum capacty and then decreases as shown n Fgure GenCo 2 and GenCo 3 both ncrease consstently untl they reach ther generaton capacty lmts. Ths pattern has clearly shown the mpact of transmsson capacty constrant on the market results,.e. the producton of GenCo 1 s forced to decrease as total demand ncreases n order to avod reachng the maxmum transmson rate of lne L Indvdual producton (MW) Q Q Q Total demand (MW) Fg Indvdual producton wth constant demand elastcty under generaton and transmsson constrants 50

71 Fgure 4.15 shows the ndvdual profts under generaton and transmsson constrants. The ndvdual profts of three GenCos are the same as scenaro 2 untl transmsson constrant starts to take effect. In Fgure 4.15, although GenCo1 has less producton due to the transmsson constrant, ts proft s stll hgher than n the other two cases. Ths s due to the hgher ncrease n market clearng prce compared to a lower declne n producton. GenCo2 and GenCo3 have also benefted from the exstence of transmsson constrants by supplyng more producton wth a hgher market clearng prce. Indvdual profts (Rand) π π π Total demand (MW) Fg Indvdual profts wth constant demand elastcty under generaton and transmsson constrants 51

72 Load payment under generaton and transmsson constrant s shown n Fgure Load payment s hgher than when there are no constrans and when there s generaton constrant only. The man contrbuton towards the hgher load payment s the ncrease n the market clearng prce. Ths ndcates that load demand s dsadvantaged by overpayng for the same capacty t requres from the network Load payment (Rand) Load payment Total demand (MW) Fg Load payment wth constant demand elastcty under generaton and transmsson constrants Summary of case 1 The comparson of market results from all scenaros under case 1 s presented n ths secton. Snce the maxmum total demand vares from no constrant (800MW) to the one wth generaton capacty constrant (750MW) and transmsson capacty constrant (R735MW), the total demand range that s used for comparson n ths secton s from 100MW to 700MW. 52

73 Table 4.1 shows the ndvdual producton n case 1. For GenCo 1, producton reaches maxmum at 250MW when total demand ncreases to 600MW n scenaro 2 and scenaro 3. GenCo 1 s producton decreases when total demand reaches 700MW n scenaro 3 n order to avod transmsson congeston. A smlar stuaton apples to GenCo 2, when total demand ncreases to 700MW, ts producton reaches ts maxmum capacty at 250MW n scenaro 2, but also needs to reduce to 233MW n scenaro 3 n order to avod the transmsson congeston on lne L 13. Although GenCo 3 has the hghest margnal cost, t takes advantage of ts poston of beng at the same bus as the load demand. When total demand ncreases to 700MW, GenCo 3 s producton reaches 224MW n scenaro 3, compared to 156MW and 200MW n scenaros 1 and 2, respectvely. Q 1 Table 4.1 Comparson of ndvdual producton n case 1 Total Demand (MW) Scenaro 1 (MW) Scenaro 2 Scenaro 3 Scenaro 1 (MW) Scenaro 2 Scenaro 3 Scenaro 1 (MW) Scenaro 2 Scenaro Q 2 Q 3 53

74 Table 4.2 shows the comparson of market clearng prce from all three scenaros n case 1. In scenaro 1, wthout constrants, market clearng prce stays unchanged. Wth the ntroducton of generaton and transmsson capacty constrants n scenaros 2 and 3, market clearng prce ncreases once the generaton and transmsson capacty lmts are reached. When total demand ncreases to 700MW, market clearng prce n scenaro 3 s R384/MW, hgher than R300/MW n scenaro 1 and R350/MW n scenaro 2, respectvely. Total Demand (MW) Table 4.2 Comparson of market clearng prce n case 1 Scenaro 1 Scenaro 2 (Rand/MW) (Rand/MW) Scenaro 3 (Rand/MW) Table 4.3 shows the comparson of ndvdual profts n case 1. As ndvdual proft s determned by the change of market clearng prce and ndvdual producton, wth the ncrease of market clearng prce n scenaros 2 and 3, ndvdual profts tend to ncrease. Ths s especally so when demand ncreases to 700MW, as all three GenCos reach hghest profts n scenaro 3 due to the presence of generaton and transmsson constrants. π 1 Table 4.3 Comparson of ndvdual profts n case 1 Total Demand (MW) Scenaro 1 (Rand) Scenaro 2 Scenaro 3 Scenaro 1 (Rand) Scenaro 2 Scenaro 3 Scenaro 1 (Rand) Scenaro 2 Scenaro π 2 π 3 54

75 Table 4.4 presents the comparson of load payment n case 1. Wth the same level of total demand, after the ntroducton of generaton and transmsson constrants, load payment becomes hgher. When total demand ncreases to 700MW, load payment reaches the hghest level of R268,799 n scenaro 3. Total Demand (MW) Table 4.4 Comparson of load payment n case 1 Scenaro 1 (Rand) Scenaro 2 (Rand) Scenaro 3 (Rand) Case 2 - Change of demand elastcty wth constant total demand Scenaro 1: wth no capacty constrants The market clearng prce changes as elastcty changes as shown n Fgure

76 It can be seen from Fgure 4.17 that the market clearng prce s extremely hgh when demand elastcty s low. The prce was R750, 150/MWh at the extreme lower bound of elastcty. The prce level vares aganst demand elastcty, between the values of and , as shown on the horzontal axs. Ths pattern explans the reason why only the value of demand elastcty between to s chosen for smulaton. As menton above, n ths thess, the area of demand elastcty s calculated between to When the number of GenCos and the margnal costs of GenCos change, the area of demand elastcty wll change accordngly Market clearng prce (Rand/MW) Demand elastcty Fg Market clearng prce wth the change of demand elastcty, where total demand stays constant at 600MW 56

77 Fgure 4.18 shows a clearer change of market prce after zoomng area of demand elastcty between to , whch cannot be seen clearly n Fgure Ths apples to all subsequent analyss n ths thess Market clearng prce (Rand/MW) Demand elastcty Fg Market prce wthout extreme lower bound of elastcty, where total demand stays constant at 600MW In Fgure 4.18, the pattern of prce s smlar to the one n fgure Wth the enlarged vew, t can be seen that market clearng prce approaches R200/MW at demand elastcty of It means that, as the demand becomes more responsve, market prce drops. Ths ndcates that the demand sde can effectvely reduce the market clearng prce by becomng more responsve. 57

78 Fgure 4.19 shows the ndvdual producton of GenCos aganst the change of demand elastcty. As shown n Fgure 4.19, the ndvdual output ncreases for GenCo 1, t stays the same for GenCo 2 and decreases for GenCo 3. The reason for ths type of change s that, when demand s more responsve to prce, the most economcal generator wll be dspatched more fully frst. In contrast, the more expensve a generator s, the less t wll be dspatched. Margnal cost plays an mportant role n the determnaton of the output change pattern. The margnal cost of GEenCo 2 s the mean of the margnal costs of GenCo 1 and GenCo 3. Therefore, ts output level does not change. If the margnal cost of GenCo 2 s close to that of GenCo 1, ts output wll ncrease. On the other hand, f the margnal cost of GenCo 2 s close to the one of GenCo 3, ts output wll decrease. Indvdual producton (MW) Demand elastcty Q1 Q2 Q3 Fg Indvdual outputs wth change of demand elastcty, where total demand stays constant at 600MW 58

79 In Fgure 4.20, when demand becomes more responsve to prce, the more expensve GenCo wll not be dspatched as fully as before. When demand elastcty s close to for example, GenCo 3 s not dspatched, and ts proft becomes zero. The profts of GenCo 1 and GEenCo 2 decrease due to the decrease n market prce Indvdual proft (Rand) Demand elastcty π1 π2 π3 Fg Indvdual profts wth change of demand elastcty wthout lower bound of elastcty, where total demand stays constant at 600MW 59

80 Fgure 4.21 shows that load can effectvely change ts payment by changng ts elastcty. It means that load payment can be reduced excessvely f the demand s more responsve to market prce. When the load s more elastc, t pays lower market prce per unt for more quantty demanded. Load payment (Rand) Demand elastcty Fg 4.21 Change of load payment wth change of demand elastcty wthout lower bound of elastcty, where total demand stays constant at 600MW 60

81 Fgure 4.22 shows the margnal change of market clearng prce over demand elastcty. As mentoned above, the absolute value of demand elastcty determnes ts level of mpact. As demand becomes more elastc, for example changes from to , ts margnal mpact on market clearng prce decreases untl t s equal to the margnal cost of the most expensve GenCo. It can be seen that demand elastcty has much hgher mpact on the market clearng prce at ts lower bound sesson P/ ε (Rand/MW) Demand elastcty Fg Dervatve of market clearng prce over demand elastcty 61

82 In Fgure 4.23, the shape of the dervatves of ndvdual quantty output over demand elastcty for three GenCos are presented. As the three GenCos have the same dfference nterval of ther margnal costs, the margnal change of GenCo2 output, Q 2, s not affected by the change of demand elastcty as ts margnal cost equals to the average cost. Therefore, ts dervatve equals to zero. The outputs of GenCo 1 and GenCo 3 have constant and nverse pattern. It s mportant to note that as the value of demand elastcty s negatve, the margnal change of the ndvual producton over demand elastcty should be ntepreted nversely. It means that the magnal change of demand elastcty wth 1 unt wll cause 200 MW ncrease of Q 1, zero change of Q 2 and a 200 MW decrease of Q 3. The change of demand elastcty s 0.1 n each nterval, and has 10 ntervals n total. Therefore, the changes n ndvdual producton of Q1, Q2 and Q3, are +20, 0 and - 20, respectvely. Fgrue 4.23 also shows that the magnal mpact of demand elastcty on ndvdual producton s constant at dfferent level of demand elastcty Q/ ε (MW) Demand Elastcty Q1/ ε Q2/ ε Q3/ ε Fg Dervatve of ndvdual quantty output over demand elastcty 62

83 4.4.2 Scenaro 2: wth generaton capacty constrant Fg presents the ndvdual producton wth change of demand elastcty under generaton constrant. The pattern of ndvdual producton of all GenCos changes after the demand elastcty reaches as shown n Fgure As GenCo 1 s producton stops ncreasng, the addtonal load demand s allocated between GenCo 2 and GenCo 3. When demand elastcty reaches ts maxmum of , GenCo 1 and GenCo 2 both reach ther maxmum capacty. The rest of the load demand s suppled by GenCo 3. Ths s dfferent from the case n whch system constrant does not exst. The contnuous supply from the most expensve generaton company,.e. GenCo 3, s the man contrbuton to the hgh market clearng prce. The hgher prce sgnal should be captured as an alert of a supply shortage, especally of the lower cost generaton supply. 300 Invdual producton (MW) Demand elastcty Q1 Q2 Q3 Fg Indvdual producton wth change of demand elastcty under generaton constrant 63

84 The market clearng prce s decreasng as the demand becomes more responsve. However, the mpact of demand elastcty on market clearng prce has been lmted due to the generaton capacty constrant. As shown n Fgure 4.25, due to the effect of the multpler of the relatonshp between the GenCo s actual generaton quantty and maxmum generaton capacty, mu, market clearng prce s relatvely hgher after the demand elastcty reaches , compared to the one n scenaro 1. When demand elastcty s wthn ts upper bound, the effect of the multpler reaches ts maxmum. Even when the demand elastcty has reached ts hghest level, at , the market clearng prce does not go down to R200/MW as does GenCo 3 s margnal cost because of the exstence of the multpler. Therefore, the market clearng prce stays hgher than n scenaro Market clearng prce (Rand/MW) Demand elastcty Fg Market clearng prce wth change of demand elastcty under generaton constrant 64

85 Fgure 4.26 shows the ndvdual proft wth change of demand elastcty under generaton constrant. The ndvdual profts of all GenCos are decreasng durng the ncrease of demand elastcty n absolute value as shown n Fgure GenCo 3 s stll obtanng proft even when the demand elastcty reaches ts maxmum. GenCo 1 obtans less proft under generaton constrant, after demand elastcty reaches , compared to scenaro 1. The sum of ndvdual profts wth generaton capacty constrant s hgher than those wthout constrant. Ths s manly due to the hgher market clearng prce and the contnuous supply from the more expensve generaton,.e. GenCo3. In general, GenCo 1 s worse off wth the generaton capacty constrant due to the fact that ts dspatched generaton declnes. GenCo 2 and GenCo 3 are better off wth the generaton capacty constrant. Ths can be explaned by the fact that after reachng ts generaton capacty lmt, GenCo 1 keeps supplyng the same amount of power at ts low prce. Therefore, ts proft decreases more quckly than n the case where there s no constrant Indvdual profts (Rand) Demand elastcty π1 π2 π3 Fg Indvdual proft wth change of demand elastcty under generaton constrant 65

86 Fgure 27 shows the change of load payment aganst the change of demand elastcty. Wth the ncrease of demand elastcty, as shown n Fgure 4.27, load payment decreases smlarly to the scenaro wthout constrant. However, the decreasng rate of load payment s less than n the case wthout generaton capacty constrant. Snce the total quantty demand remans unchanged at 600MW, as demand elastcty ncreases, the load payment s heavly dependent on the market clearng prce. The market clearng prce wth generaton capacty constrant ncludes the consderaton of. Therefore, the decrease of market prce n the case wth generaton capacty constrant s not as much as the one wthout constrant. Ths has resulted n hgher load payment n the case wth generaton capacty constrant Load payment (Rand) Demand elastcty Fg Load payment wth change of demand elastcty under generaton constrant 66

87 The pattern of the dervatve of market clearng prce over demand elastcty s presented n Fgure In the case of generaton capacty constrant, the dervatve of market clearng prce over demand elastcty s not only determned by the sum of margnal cost and the value of demand elastcty, but also on the value of mu. It s mportant to note that, although the value of the dervatve s hgher than the one n the case wthout constrant, t does not necessary mean that demand elastcty has caused a hgher change n the market clearng prce n ths case. Ths s manly due to the penalty effect on the generaton constrant from the supply sde P/ ε (Rand/MW) Demand elastcty Fg Dervatve of market clearng prce over demand elastcty 67

88 In Fgure 4.29, the dervatve of ndvdual producton over demand elastcty under generaton capacty constrant s presented. The value of dervatves wthout generaton capacty constrant s constant for all GenCos, but the values become non lnear after the value of µ becomes non zero n the case wth generaton capacty constrant. As demand elastcty ncreases further, the mpact of demand elastcty becomes weaker on the ncrease n GenCo 1 s producton. Over a certan pont, GenCo 2 s margnal producton also starts to ncrease. The decreasng rate of GenCo 3 s margnal producton over demand elastcty s less than n the case wthout constrant. Dfferent from scenaro 1, the dervatve of ndvdual producton over demand elastcty under generaton capacty constrant s no longer constant after the level of demand elastcty ncreases over Q/ ε (MW) Demand elastcty Fg Dervatve of ndvdual producton over demand elastcty 68

89 4.4.3 Scenaro 3: wth generaton and transmsson capacty constrant Fgure 4.30 shows ndvdual producton wth change of demand elastcty under generaton and transmsson constrants. Indvdual producton of all GenCos changes n the begnnng and remans almost the same when demand elastcty s beyond Ths means that demand elastcty has no further mpact on ndvdual productons at and above a certan pont. It s beleved that there s no room for the more economcal GenCos, (.e. GenCo 1 and GenCo 2 ) to supply any addtonal MW to the load due to the transmsson constrant. For a quantty game, ths wll be the optmal soluton among the GenCos Indvdual producton (MW) Demand elastcty Q1 Q2 Q3 Fg Indvdual producton wth change of demand elastcty under generaton and transmsson constrants 69