Parameterization of Linear Supply Functions in Nonlinear AC ...

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 2, MAY 2013

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Parameterization of Linear Supply Functions in Nonlinear AC Electricity Market Equilibrium Models—Part II: Case Studies Andreas G. Petoussis, Xiao-Ping Zhang, Senior Member, IEEE, Savvas G. Petoussis, and Keith R. Godfrey

Abstract—This study utilizes the primal-dual nonlinear interior point algorithm formulated in Part I of this two-series paper, to obtain electricity market supply function equilibrium solutions for the bi-level market problem of nonlinear power systems with AC meshed networks. The strategic offers of the market players are modeled by parameterizing the generation marginal cost functions using any of the four available parameterization methods to obtain profit-maximizing linear supply function bids. The analysis takes into account a variety of test systems and examines the relation between the equilibrium solutions obtained from the different parameterization methods and the role of network complexity. An analysis of multiple equilibria in AC network constrained systems is also provided. Index Terms—AC network modeling, electricity market equilibrium, network congestion, nodal prices, power system economics, SFE parameterization, supply function equilibrium.

I. INTRODUCTION

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HE linear supply function equilibrium (SFE) model [1] is a tool that can be used for the analysis of bid-based electricity pool markets with strategic players. Each player constructs optimal supply function bids by parameterizing the linear marginal cost functions of their generating units. Depending on whether the slope and intercept of the strategic bid functions are set equal to the true values or not, four different parameterization methods can be employed [2]. These have been discussed in detail in Part I of this paper [3]. The study in Part I developed a market equilibrium algorithm for solving the bi-level SFE problem using any of the four parameterization methods. To the best of our knowledge, no research has taken place on examining the impact of the SFE parameterization in applications with AC meshed networks or large systems. In this second part of the paper, the proposed model is utilized to show how the representation of the strategic Manuscript received July 06, 2011; revised November 05, 2011, February 21, 2012, May 03, 2012, and July 30, 2012; accepted August 06, 2012. Date of publication October 02, 2012; date of current version April 18, 2013. Paper no. TPWRS-00630-2011. A. G. Petoussis was with the School of Engineering, University of Warwick, Coventry, CV4 7AL, U.K., and now is with the Cyprus Transmission System Operator (TSO-Cyprus), Nicosia, CY-2057, Cyprus (e-mail: [email protected]; [email protected]). X.-P. Zhang is with the School of Electronic, Electrical and Computer Engineering, University of Birmingham, Birmingham, B15 2TT, U.K. (e-mail: [email protected]). S. G. Petoussis is with the Electricity Authority of Cyprus (EAC), Nicosia, CY-1399, Cyprus (e-mail: [email protected]). K. R. Godfrey is with the School of Engineering, University of Warwick, Coventry, CV4 7AL, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2012.2214244

bids using different parameterization methods affects the market equilibrium solutions. Several test cases, each involving a variety of systems under different network operational conditions are examined and the effects of the parameterization are correlated to the AC constraints and meshed network complexity. The nature of the different equilibrium solutions is investigated. This study provides results for, and discussions of, the case studies, as well as comparisons with the observations from the existing literature. The analysis emphasizes the following: 1) comparison of the market outcomes from the different parameterization methods and investigation of their interrelation; 2) examination of the change of the impact of the parameterizations when subject to different levels of transmission congestion, different bus voltage modes and different levels of load demand elasticity; 3) discrimination between the effects on small test systems and on larger more complex networks; and 4) investigation of multiple equilibria in AC network constrained systems. The next section provides a summary of the parameterization methods and the proposed market equilibrium algorithm. A detailed description is provided in Part I of this paper [3]. II. PARAMETERIZATION METHODS AND MARKET ASSUMPTIONS A. Summary of the Four Parameterization Methods The linear marginal cost function of the strategic firms is parameterized to construct the optimal , where and are the supply function bid parameterized bid terms that correspond to the true generation , and is the active cost coefficients and , with power generation. The four parameterization methods are: 1) Intercept-parameterization: the strategic players adjust the intercept , while keeping the slope constant. In this paper, this method is referred to as -parameterization. 2) Slope-parameterization: the strategic players adjust the slope , while keeping constant the intercept . This method is referred to as -parameterization. 3) (Slope intercept)-parameterization: the strategic players adjust both the slope and intercept, but in a fixed linear re, i.e., adjust the lationship, such that . This method is referred as -parameteriparameter zation.

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TABLE I RESULTS FOR THE 3-BUS TEST SYSTEM

4) (Slope-and-intercept)-parameterization: the strategic players adjust both the slope and intercept independently. This method is referred to as ( )-parameterization.

TABLE II RESULTS FOR THE 5-BUS TEST SYSTEM:

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TABLE III RESULTS FOR THE 5-BUS TEST SYSTEM:

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B. Summary of Market Assumptions The proposed AC-network electricity market algorithm, presented in Part I of this paper, employs the primal-dual nonlinear interior point method to calculate the Nash equilibrium solution for bid-based pool markets. The market model considers profit-maximizing generating firms, which construct strategic linear supply function bids. The market clears based on a social welfare maximization scheme to determine nodal prices, while taking into account the network constraints and the strategic actions of the players. III. CASE STUDIES In order to examine the behavior of the market solution under the different types of SFE parameterization, numerical results were performed in test systems ranging from 3 to 57 buses. The test systems involve cases with and without transmission congestion, as well as different bus voltage modes, in order to show the interrelations of the network operating conditions to the effects of the different parameterization methods on the market outcome. The maximum and minimum limits for the parameterized terms in the strategic bids were set to , according to [4]. The limits for the bidding parameter were set to and as in [5]. It has been checked that varying the limits within a reasonable domain does not affect the equilibrium solution, while extremely large limits would not be credible. The nodal prices , cost coefficients and demand coefficients are measured in £/MWh; the cost coefficients and the demand coefficients in £/(MW) h; the profits and social welfare in £/hour; the parameter is dimensionless; and all the power quantities are calculated in p.u. The transformer tap-ratio can be optimized by the ISO within the range of 0.9 and 1.1. The

TABLE IV RESULTS FOR THE IEEE 30-BUS SYSTEM

results for the firms’ profits are indicated in the Tables by the firm’s title, e.g., . The results for the social welfare (S.W.) correspond to the true social welfare and not the quasi function from (6) in Part I. The left-hand column in each of Tables I–VI lists the

PETOUSSIS et al.: PARAMETERIZATION OF LINEAR SUPPLY FUNCTIONS IN NONLINEAR AC ELECTRICITY MARKET EQUILIBRIUM MODELS

TABLE V RESULTS FOR THE IEEE 57-BUS SYSTEM

TABLE VI RESULTS FOR LOAD DEMAND ELASTICITY ANALYSIS

Fig. 1. The 3-bus test system.

Parameterization Method, denoted by . The market results of the parameterization method that differ from the others for each case are presented in bold lettering. A. Market Analysis of the 3-Bus System The investigation begins with tests on the small 3-bus system shown in Fig. 1, which consists of 3 buses with load demand, 3 transmission lines and 2 generators owned by firms and . The inverse load demand functions are £/MWh and £/MWh, for . Three different case scenarios have been simulated using the four parameterization methods and the results are shown in Table I. Case 1 corresponds to normal operating conditions and no transmission congestion existing in the network, while the bus voltages can be optimized by the ISO within a wide domain of % from the rated value of 1 p.u.1 The results for the -, -, and -parameterizations are identical, while those for the 1In

Case 1, the upper limit of the bus voltage constraint is binding for the -parameterization solution. It should be stressed that at least one of the -solutions presented in this study is binding, thus voltage limits for all the the algorithm is able to converge to a local equilibrium point.

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-method show slight differences but still are very close to the others. For the -, -, and -parameterizations the bids submitted from both firms are much higher than their marginal costs. On the other hand, the coefficients for the -parameterization are much smaller than the true values, hence the bids are much smaller than the marginal cost. The bus voltage limit of % from 1 p.u. used for Case 1 is a rather wide limit. Thus, an equilibrium solution with a bus voltage outside of such limits would not be credible and a realistic market model should take into account voltage limits of at least such a range. By enforcing voltage limits in the market model, attention is brought to the issue of multiple equilibria. In the case of unconstrained markets the -parameterization is expected to result in a continuum of equilibrium points, but in Case 1 a local equilibrium solution has been obtained, since the multiplicity is eliminated by the voltage constraints, as the voltage values for the multiple equilibria fall far beyond the allowable voltage range. In order to examine this issue, further tests were carried out based on Case 1, by gradually increasing the voltage limits for the -solution. When the upper voltage limit is set to a value higher than +19% from 1 p.u., the algorithm does not converge and no solution can be obtained because there is no local equilibrium point, i.e., the solution oscillates between the manifold of equilibrium points from the multiple equilibria set. However, an equilibrium solution from this manifold would not be credible and cannot appear in practice, since no transmission system operator will allow the system to operate with such high voltages. Since the purpose of the proposed algorithm in this paper is to provide insight for the case of more realistic markets, the investigation of the non-practical multiple equilibria that lie beyond the voltage limits of % from 1 p.u. is outside of the scope of this research. Case 2 has the same input data as Case 1, but the MVA transmission limit for line 2-3 is set to p.u., which is binding for all parameterizations. Therefore, Case 2 has higher nodal prices, reduced active and reactive generation, and lower social welfare compared with Case 1. However, the purpose of this investigation is not the comparison of the different network operations but of the different parameterization types, for which the deviation of their corresponding solutions seems to be dependent on the network conditions. As can be seen in Table I, the results for Case 2 are similar for the -, -, and -parameterizations, while those for -parameterization differ. Again, the results of the -parameterization have small deviations from the others and the -bids are much smaller than the corresponding marginal costs. For Case 3, the bus voltage limits were lowered to % from 1 p.u. while the line congestion was still present to show that the level of the restrictions on the bus voltages further affects the deviation between the solutions of the different parameterizations. (Note that a case for which the voltage limits were set to % from 1 p.u. and no transmission congestion was present was also performed, but apart from the effects of the voltage mode there were no appreciable differences between the solutions of the four parameterizations, as happened in Case 1.) From the results for Case 3 in Table I it can be seen that, unlike Case 2, the solution of the -parameterization has become very similar with those of the - and -methods, while the results for the

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Fig. 2. The 5-bus test system.

-parameterization differ from those by a noticeable deviation, for which the social welfare is higher than that of the other three methods. The above results lead to the conclusion that the four parameterizations behave in a different manner in the presence of transmission line congestion and that this impact depends on the bus voltage limits. Under normal operating conditions all methods give similar or identical solutions. When transmission congestion is introduced the -parameterization gives different market solution from the other three methods, while when the voltage limits are narrowed in the presence of congestion the -, -, and -solutions become similar, and differ from the -solution. For all tests, the -parameterization has at least a slight deviation from the other methods. In order to examine if these observations represent a general pattern, further cases are carried out on a 5-bus system. B. Market Analysis of the 5-Bus System The 5-bus system consists of 5 buses with load demand, 3 transmission lines, 2 on-load tap-changing transformer branches and 2 generators owned by firms and , as shown in Fig. 2. The inverse load demand functions are £/MWh for and £/MWh for . The numerical results for the 5-bus system are provided in Tables II and III. The tests on the 5-bus system begin with the benchmark Case 4, for which there is no transmission congestion in the network and the bus voltage limits are set to % from 1 p.u. Similarly to Case 1 of the smaller system, the market solutions from the -, -, and -parameterizations are very similar and there are small differences for the -solution, as shown in Table II. For the next two Cases, the transmission capacity limits of the entire system are gradually reduced to 2 p.u. and 1 p.u., respectively, to examine low and more intense transmission congestion conditions. When the system transmission limits are reduced to 2 p.u. for Case 5, transmission congestion exists in the transformer branch TR1 between buses 2–3. Again, as in Case 2, the results from the -parameterization are considerably different from those of the other methods, and the results for the - and -methods are identical, and differ by a small deviation from those of the -parameterization. Note that the social welfare for the -solution is higher than that of the other three methods (as in Case 3) despite the fact that the firms are entitled more degrees of freedom in building their strategies. This is discussed in Section III-C.

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For Case 6 where the transmission limits are further reduced to 1 p.u., congestion exists in the transformer branch TR1 and in line branch 1-5. The results of the -parameterization are now very close to those of the - and -methods, while the market outcome of the -parameterization shows a significant deviation. This behavior resembles Case 3 where the voltage limits are reduced in the presence of transmission congestion. In order to examine the interrelation between the similar observations for Case 3 that has reduced voltage limits and those for Case 6 that has intense transmission congestion, two more cases are examined on the 5-bus system for tighter bus voltage limits of % from 1 p.u. These are Cases 7 and 8, which are performed with no congestion and under intense transmission congestion respectively, as shown in Table III. By comparing Case 7 with Case 4, for which the voltage range has been reduced from % to % from 1 p.u., it can be seen that, unlike Case 4, the solution of the -parameterization in Case 7 differs from the others, as in the cases where low transmission congestion was present (Cases 2 and 5). However, here there is no transmission congestion. When the voltage range is reduced from % in Case 6 to % in Case 8 where the transmission limits in the network are set to 1.0 p.u., the overall power generation is reduced due to the voltage restrictions and, since less power is required to be transmitted in the network, one of the two congested lines is relieved. Therefore, there is only one transmission branch congested (TR1) in Case 8, but the deviation that existed in Case 6 between the solution of the -parameterization and the other methods was sustained. The observations for Cases 6 and 8 are compatible with those for Case 3, where only the -solutions show significant difference from those of the other parameterization methods. C. Discussion on the Results of the 3-Bus and 5-Bus Systems The numerical results from the test cases on the 3-bus and the 5-bus systems are in good agreement and have shown that the market equilibrium solutions from the different parameterization methods can be similar or have large deviations, depending on the presence and intensity of transmission congestion in conjunction with the level of the bus voltage limits. The - and -parameterization methods have been found to give very similar or identical solutions for all test cases independent of the operating network conditions. The solution of the -parameterization is different from the other three when there is transmission congestion up to a certain level, or, in the absence of transmission congestion, when there are tight limits for the bus voltages. The -parameterization method gives solutions with significant differences from the other three methods if the transmission congestion is intense, or if there is congestion in the presence of tight bus voltage limits. Nonetheless, the market solution from the -parameterization always has at least a small deviation from those of the other three methods. This may be the result of inherent characteristics of the less restricted format of the -bid that has more degrees of freedom compared to the other parameterization methods. According to the above observations it can be concluded that the presence and level of limitations associated with the power

PETOUSSIS et al.: PARAMETERIZATION OF LINEAR SUPPLY FUNCTIONS IN NONLINEAR AC ELECTRICITY MARKET EQUILIBRIUM MODELS

transmission and the bus voltages affect the market solution obtained from the four different supply function parameterization methods. Low level restrictions result in deviations on the solution of the -parameterization and tighter restrictions affect the solution of the -parameterization. It can also be seen that the effect of transmission congestion is more intense than that of voltage restrictions. In order to check that such observations occur regularly under the corresponding conditions, equilibrium solutions were obtained for several 3-bus and 5-bus sample systems using the four parameterization methods. The comparison of the solutions from the different parameterizations for 20 sample systems, which verifies the above findings, is presented in the Appendix. In contrast to the observations on the unconstrained and linearized systems in [2], [6], and [7], the producers’ surplus in all the -test cases for the AC systems examined here is found to be lower than those of the other three parameterization methods. This may be explained as follows. Even though the strategic bids of any of the -, -, and -methods can be obtained using the -parameterization, they may not correspond to an equilibrium point since a player may be able to implement another -strategy that will result in higher profit (if the strategies of its rivals remain constant). The rivals will then anticipate this action and sequentially alter their strategies for their benefit, always subject to the network and market constraints present. As various strategies are restricted by these constraints, the equilibrium point resulting from the strategic interactions between the players at which no firm will be able to further increase its profit may correspond to a lower surplus for all firms even though the -method allows for more degrees of freedom in constructing the strategic bid. The investigations in [2] and [6] show that the profits of one of the multiple -SFE points equal those of the Cournot solution and quote the concept of focal equilibrium, which is mutually beneficial and preferred by all players. However, the introduction of the AC network modeling in the current study seems to prevent such results. This finding is supported by the study in [8], where it is shown how the possible equilibria in practical looped-network systems are reduced to a very small number, in most cases to a unique equilibrium, while if a second equilibrium exists it may be eliminated by introducing elastic load demand (as is the case in the study of this paper). For an illustration on how a group of equilibrium strategies are eliminated with the introduction of network constraints see the four graphs in [2], where a simple restriction on the output of one generator in a 2-firm structure with no network loops removes a large number of possible -equilibrium strategies. It may be postulated that a complex combination of AC network constraints may lead to the elimination of many (if not all) equilibria, including a possible focal point. However, in order to corroborate such conclusions an AC Cournot model should be implemented in a similar algorithmic framework and compared with the linear SFE model for unconstrained and constrained test cases. This, in conjunction with an investigation of different formats of supply curves, may be an interesting topic for further research. Nevertheless, it appears that the -strategies do not always result in lower social welfare compared with the other parameterization methods in the presence of AC constraints.

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Another observation is that an equilibrium -strategy in the context of AC network modeling may require a bid that is much lower than the corresponding marginal cost function, as the particular player will gain profit due to the level of market price, which depends on several factors other than just the firms’ strategies (e.g., network constraints and topology). A similar observation is made in [9] where equilibrium bids for block offers that are lower than the corresponding marginal costs are obtained. In such cases, a player may expect an increase in profit that results from the increase in the market clearing price caused by the strategic actions of some other player. However, the strategy of the latter player may be constructed only if the former player bids lower than its marginal cost, for the particular market outcome to be an equilibrium point. Furthermore, by observing the cases with moderate limitations on transmission and bus voltages for which the -parameterization gives different market solutions than the other three methods (Cases 2, 5, and 7), it can be seen that the producers’ surplus for -parameterization is always higher than that of the other methods. This is in agreement with the observations in [10], where market solutions for parameterizations of slopes and intercepts are compared for a 4-bus linearized DC system with transmission limits for active power. The authors of [10] postulate that by using the -parameterization method the resulting equilibria resemble Cournot market outcomes, which might be the case under certain restrictions. This also holds for the results in [6], where the slope manipulation results in higher profits than the intercept-parameterization in an unconstrained system. Last, but not least, it was shown that the continuum of multiple equilibria that appears in small unconstrained systems for the -parameterization falls far beyond the practical bus voltage limits, thus any of these multiple equilibria cannot be regarded as a realistic market solution. Therefore, the introduction of voltage constraints assists in rejecting the non-realistic equilibria and in identifying the more practical equilibrium points, which happen to be local SFE solutions. These solutions, which do not involve comparison complications due to multiplicity issues, provide a much better comprehension in terms of mathematical analysis. In addition to the cases provided above, several other tests not shown here were performed. These have examined the effect on the deviation between the solutions from the four parameterization methods for power factor adjustments and for variation of other network constraints, such as the active and reactive generation limits. However, no significant difference was observed between the resulting market solutions from the different parameterization methods (as in Cases 1 and 4). In order to examine if the above conclusions are valid in the case of larger and more realistic systems, simulations on the IEEE 30-bus and 57-bus systems were also performed. D. Market Analysis of the IEEE 30-Bus System The IEEE 30-bus system [15] consists of 30 buses of which 22 have load demand, 37 transmission lines, 4 on-load tap-changing transformers, and 6 generators, each one forming an individual strategic firm (entitled to ). For this and for other larger systems the algorithm was unable to converge to an equilibrium solution when the -parameterization

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was employed. Hence, only the solutions of the other three parameterization methods are provided for this system as shown in Table IV. Discussion on possible causes for the non-convergence of the -solution are provided in Section III-F. Convergence problems for the -parameterization method were also reported in [4]. Based on the conclusions from the results on the 3-bus and 5-bus systems, it was investigated whether the -parameterization method would have produced different results on the 30-bus system for a particular level of transmission congestion. The tests started from transmission limits of 2.0 p.u. for the entire network (Case 9), where there was no transmission congestion. The limits were gradually reduced and it was found that within the domain of 1.16 to 1.10 p.u., for which 3 transmission lines were congested, there was a deviation between the solution from the -parameterization and the results from the - and -methods. Note that the range of transmission limits that resulted in different -solutions for the 30-bus system is much smaller than those for the 3-bus and 5-bus systems. Case 10 presented in Table IV corresponds to an intermediate case for a limit of 1.13 p.u. When the transmission limits were further -parameterizations reduced, the solutions of the -, -, and became very similar to each other. Case 11 shows the solutions for a limit of 0.4 p.u. For the latter case it can be presumed that, -solution exists and could be obtained, it would have if an been different from those of the other three methods, following the pattern observed for the smaller systems. During the procedure of reducing the transmission limits, it was observed that for certain values the number of congested lines in the solutions from the different parameterizations was not the same, and hence their results were dissimilar since they corresponded to different network operations. Similar observations were also made when the bus voltage limits were varied. Further elaboration on this takes place in the analysis of the IEEE 57-bus system below. However, in all cases with the same number of congested lines, the parameterization solutions were following the pattern discussed above. E. Market Analysis of the IEEE 57-Bus System The larger IEEE 57-bus system [15] is tested to show the complications that arise in relation to the solutions provided from the different parameterization methods for more realistic situations. This system consists of 57 buses of which 42 have load demand, 63 transmission lines, 17 on-load tap-changing transformers, and 7 generators, each one forming an individual to ). Again, results are presented strategic firm (entitled for the -, -, and -parameterizations, as shown in Table V. The last column of the table provides the number of congested for each equilibrium solution. transmission lines Case 12 corresponds to normal operating conditions with no transmission congestion in the network and as for all the other systems all the parameterization methods result in very similar equilibrium solutions. When the MVA transmission limits of the system are reduced to 1.7 p.u. in order to impose transmission congestion (Case 13), the number of congested transmission lines for each parameterization method is different as shown in the last column of Table V. Hence, the three solutions

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for Case 13 correspond to different network operating conditions. Different cases with a variety of reduced limits for particular transmission lines and for the entire network were examined, but the three parameterization methods resulted in dissimilar network operations in most cases. This also happens for other larger systems. Even though for the smaller systems the four parameterization methods were giving solutions that correspond to the same operating conditions under network congestion and at least two solutions were in numerical agreement, for the larger systems this is not the case. Similar observations are made for variations of the bus voltage limits, since the voltages have a direct impact on the reactive transmission flows. The occurrence that the parameterization methods give solutions that correspond to different network operations only for the larger systems may be due to the fact that there is a wider combination of alternative routes for the power flows to follow. In larger and more complex networks, the active and reactive power flows can be redistributed to suit a more appropriate scenario for the ISO objective (while shifting generation from one unit to another in order to comply with the voltage and power flow limits) for only a small change in the supply function bids submitted from the strategic firms (in this case resulting from the different form of the parameterized bids). An observation for dissimilar congestion configurations for the different parameterization methods has also been made in [4], for a linearized DC 39-bus system. F. Discussion on the Non-Convergence of the for Larger Systems

-Solution

As far as the -parameterization in the case of the 57-bus system is concerned, as a transmission congestion configuration is changing for an adjustment in a bid during the iterative -strategy may become more profitable process, a different for some player, and the successive resulting situation can correspond to a different transmission congestion configuration that -strategy for a different may give rise to a more profitable player, and so on. This may be a possible reason for the conver-parameterization for large systems gence problems of the as pure Nash equilibria may not exist. In such cases the market players may choose probabilistic mixed strategies. For the definition of mixed strategies equilibria in electricity markets refer to [8]. -parameterization may Convergence problems for the also be attributed to the existence of multiple equilibria as discussed in Section II-B of Part I of this paper.2 However, even though multiple -equilibria are expected to appear in the form of a continuum in unconstrained electricity market problems, models with voltage constraints and AC network representation tend to yield local equilibrium points, as discussed in Sections III-A and III-C. According to [8], these local points are not to be expected in large numbers for realistic situations under the restrictive conditions of the AC model and this is further supported by the analysis performed 2This discussion in Part I refers to the case where a numerical algorithm is not able to identify an equilibrium solution, if multiple solutions exists and the sought solution is not at least locally unique, due to oscillations in the convergence of the iterative process caused by a continuum of equilibria [4].

PETOUSSIS et al.: PARAMETERIZATION OF LINEAR SUPPLY FUNCTIONS IN NONLINEAR AC ELECTRICITY MARKET EQUILIBRIUM MODELS

Fig. 3. Illustration of different load demand elasticities.

in Section IV below. In order to exclude the possibility for multiple equilibria, further tests were carried out on the 57-bus -parameterization, by gradually reducing system for the the allowable range of the bus voltages, as done in Section III-A for the 3-bus system, but no solution could be obtained; for very tight voltage limits the convergence procedure could not be initialized properly. Hence, it is more likely that the non-convergence of the -parameterization is attributed to the elimination of pure strategy equilibria. Discussions on this topic take place in [11], where it is conjectured that the introduction of tight network constraints in complicated systems may induce inconsistent strategic behavior from the market participants resulting in no pure Nash equilibria. In our study this inconsistent behavior is characterized by optimal strategies that correspond to different number of congested transmission lines and hence different market operating conditions. The elimination of pure strategy equilibria due to the presence of transmission constraints has also been demonstrated in [12] using the Cournot model. G. Effect of Load Demand Elasticity Further analysis was done to gain some insight on the role of load demand elasticity in relation to the differences between the equilibrium solutions from the four parameterization methods. (The effects of varying the load demand elasticity on the market outcome are not analyzed, as this is out of the scope of this study.) Since the consumers’ inverse load demand function is , the behavior of load demand given by is characterized by the demand coefficients and (see Part I [3] for elaboration). These coefficients represent a measure of the demand elasticity, therefore, the elasticity can be mathematically adjusted by varying one of them. By definition, demand is considered to be elastic if a given percentage change in price produces a larger percentage change in demand, while the demand is considered inelastic if the relative change in demand is smaller than the relative change in price [13]. In this analysis, the demand elasticity is adjusted by varying the quadratic load demand coefficient . A smaller value of indicates a more elastic load demand, while a larger corresponds to less elastic demand. This effect is illustrated in Fig. 3, where, for different values of coefficient , a given change in price corresponds to different changes in load demand. , the changes in demand are , hence For

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the demand that corresponds to has a higher elasticity than that of . Based on these assumptions, the different parameterization solutions of test Case 3 are compared in Table VI with those of Case 14, which differs only on the load demand elasticity. The demand elasticity in Case 14 is higher than in Case 3, because coefficient is smaller at all buses. As observed from Table VI, the difference that existed in Case -solution and the other parameterization 3 between the results was eliminated and all solutions became very similar, despite the fact that there is transmission congestion and tight voltage limits present. Therefore, the relation between the equilibrium solutions from the different parameterizations is similar to that with no transmission congestion in the network. When the same network conditions were tested using lower demand elasticity than that of Case 3 (i.e., with higher coefficients), -solution was sustained. Thus, it is the difference of the observed that the effect of increasing the elasticity of demand on the difference between the solutions of the four parameterizations is similar to that of alleviating the stress in the network. Several other test cases were performed, resulting in similar observations: no differences between the parameterization solutions for high load demand elasticities even with high network -solution for low elasticities. In the stress, and different absence of transmission congestion, the same equilibrium solution was obtained from all parameterization methods for a wide range of elasticity degrees (for example Case 1 with coefficients ranging from 0.05 to 0.08). In relation to the examined parameterization methods, the load demand elasticity has demonstrated a behavior that resembles the pattern observed for the network stress. This may be justified by the fact that a less elastic load demand allows a smaller range of active power load demand values to appear in the market equilibrium solution, thus acting as a network constraint, while a more elastic demand behaves in a manner similar to widening the limits of a constraint. H. General Discussion of Results The impact of the parameterization method chosen on the equilibrium solution for more realistic systems is evident, in terms of market results and nature of the equilibrium strategies that in turn may affect the convergence of the algorithm. It should be noted that if multiple equilibria exist for each parameterization method, these are comparable to the other methods only if they correspond to the same network operating conditions (i.e., to the same number of congested transmission lines), as is the case for all the market results examined in Cases 1 to 12 and 14. Even though it cannot be guaranteed that the presented solutions are unique Nash equilibria, the fact that a pattern is observed for the relationship between the results of the four parameterization methods for systems of different topologies, in addition to the arguments in the beginning of Section III of Part I based on the study in [8], strengthen the notion that these equilibrium points, or the equilibrium market outcomes in terms of prices, scheduled power generation, profits and social welfare, may be unique. This is further examined in Section IV, where the issue of multiple equilibria in different feasible regions is also addressed.

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TABLE VII COMPARISON OF SOLUTIONS FOR DIFFERENT STARTING POINTS (CASE 1)

IV. INVESTIGATION OF THE NASH SFE SOLUTIONS FROM THE INTERIOR POINT ALGORITHM The initialization of the strategic bids for the iterative process of the primal-dual nonlinear interior point algorithm takes place by setting the parameterized bid terms and equal to the true generation cost coefficients and , respectively, and the bidding parameter equal to unity. In order to check if the SFE solutions obtained from the implemented algorithm depend on the starting point of the bidding strategies and if multiple equilibria exist, cases with different starting points for the bidding variables are performed. The results for firm and social welfare in Case 1, where no transmission congestion exists, are provided in Table VII for the - and -parameterizations, where is the set of bid terms and the subscript “0” denotes initial conditions. From the results in Table VII it can be seen that the different starting points for all the bidding variables yield almost exactly the same power generation, nodal price and social welfare, while the values of the firm’s profit differ by less than 0.7%. These minor differences are caused by truncation errors during the iterative process of the algorithmic procedure. The value of the bidding variable for the -parameterization is the same in two decimal places for all starting points. However, the supply function bids for the -parameterization exhibit two different behaviors: one with a large intercept and very small slope and a second with a large slope and a very small intercept. Despite the difference in the -strategies, the market outcome is the same for both occasions, revealing the existence of two equivalent SFE points. The number of the existing equivalent pure strategy equilibria (two) is compatible with the conclusions of the study performed in [8], which states that very few equilibrium points should be expected in looped systems with realistic network representation. It is irrelevant which -equilibrium point to choose for a comparison of the -results with those of the other parameterization solutions, since the market outcome obtained is unique. If a case with transmission line binding constraints is examined, non-comparable multiple equilibrium points may exist in

TABLE VIII COMPARISON OF SOLUTIONS FOR DIFFERENT STARTING POINTS (CASE 2)

different feasible regions. In order to illustrate the existence of such points, the results for firm and social welfare in Case 2, where a transmission limit exists in line 2-3, are provided in Table VIII for the -parameterization. From Table VIII it can be seen that when the bidding variable is initially set to the true intercept or to 30 the algorithm converges to the same equilibrium point with the transmission line 2-3 being congested (numerical error is less than 0.8%). In contrast, when is initialized to the larger value of 80, a different equilibrium point results, which corresponds to a different feasible region since line 2-3 is not congested. This equilibrium is not comparable with the original Case 2, as it corresponds to different network operating conditions. However, it can be seen that the solution for is not preferred by either the strategic firms or the ISO, since both the profits and the social welfare are lower than for the original Case 2. Therefore, the solution that corresponds to the original Case 2 is the preferred equilibrium solution. Note that all the examined cases in this study correspond to the same network operation, i.e., to the same feasible region, and therefore are comparable. Furthermore, no multiple non-equivalent equilibria have been observed in the same feasible region. Thus, if the algorithm provides a unique equilibrium or equivalent -equilibria in the same feasible region for a wide range of starting points, the resulting unique market outcome (power generation, prices, profits and social welfare) can be regarded as the global solution of the market problem. If multiple equilibria are observed in different feasible regions of the decision space, then an algorithmic scheme as in [8] or [14] should be employed for determining the global solution. V. CONCLUSION This paper has examined the impact of the choice of parameterization method for the linear SFE model, on the market equilibrium solution. The solutions for all the parameterization methods, for the same case, were very similar for no transmission congestion and no strict voltage limits. As the network stress worsens, the solutions for the -parameterization become different from the other methods when the stress in the network is low, while in the presence of tighter network limits the -parameterization yields solutions dissimilar to those from the other parameterizations. However, no pattern was observed for larger systems, because each parameterization method was resulting in different number of congested lines and hence to dissimilar network operational conditions. The analysis has considered unique or equivalent equilibrium points within the same feasible region of the decision space to

PETOUSSIS et al.: PARAMETERIZATION OF LINEAR SUPPLY FUNCTIONS IN NONLINEAR AC ELECTRICITY MARKET EQUILIBRIUM MODELS

show that the relation between the equilibrium solutions from the four parameterization methods is directly dependent on the presence and severity of the transmission congestion in conjunction with the level of the bus voltage limits and load demand elasticity. The issue of multiple equilibria in different regions has also been addressed to show that, depending on the initial conditions of the strategic variables, equilibria for the same parameterization method may be obtained in different regions. Global solutions and local Nash equilibrium traps may be identified by observation for small systems, while the implementation of a search algorithm for the identification of multiple equilibria in different regions and the determination of the global equilibrium solution in large complex AC-network constrained systems would be of particular interest. The proposed primal-dual nonlinear interior point algorithm has been proven to be superior in terms of computational performance for the -, -, and -parameterization methods, while it has difficulties in solving large systems using the -parameterization. These difficulties may be the result of the nonexistence of pure strategy equilibria due to inconsistent bidding caused by the complicate looped network structure. However, the -algorithm is able to successfully solve small test systems and provide equivalent SFE points with unique market outcome. In addition, it was shown that the continuum of multiple equilibria that appears in the solutions of unconstrained market problems corresponds to bus voltages that lie far beyond practical voltage limits. Thus, the introduction of voltage constraints in the market model assists in rejecting the non-realistic multiple equilibrium solutions. APPENDIX Figures of the percentage deviation of the equilibrium solutions obtained from the four different parameterization methods by the application of the primal-dual interior point algorithm on sample 3-bus and 5-bus test systems are presented. The parameters of the sample systems that differ from those of the test systems used for the numerical results in Section III include the generation cost coefficients, the load power factor, the load demand coefficients, the transmission limits, etc. The solutions from the different parameterization methods are assumed to correspond to the same equilibrium point if all of the individual nodal prices, the system active and reactive power generation, the system active and reactive power load demand, the system producers’ surplus and the social welfare do not deviate more than a reasonable tolerance from each other. This deviation tolerance, which includes the truncation error during the iterative process of the algorithmic procedure, is assumed to be 3% for the 3-bus system and 5% for the 5-bus system.3 Twenty sample systems were tested; ten for each of the 3-bus and 5-bus system configurations. The sample tests include cases where all parameterization methods provide very similar solutions, i.e., the network limits do not affect normal operation, and cases where the - or the -solution deviates from the other 3The number of variables updated after each iteration and the dimensions of the Newton matrix for the 3-bus system are 378 and 108 108, respectively, while for the 5-bus system these are 808 and 204 204, respectively. For details on the formulation of the algorithm refer to Part I of this paper [3].

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TABLE IX PERCENTAGE DEVIATIONS FOR SAMPLE SYSTEMS: SIMILAR SOLUTIONS

TABLE X PERCENTAGE DEVIATIONS FOR SAMPLE SYSTEMS: DISSIMILAR SOLUTIONS

three (i.e., cases of low or heavy network stress, respectively). The largest and average deviations of the set of the abovementioned quantities for the sample cases of the similar solutions are presented in Table IX, while the largest and average deviations for the solutions that deviate from each other are presented in Table X. The information on the latter cases refers to the deviations between the different solution and the other three (termed as “Deviation of dissimilar solutions” in Table X), as well as to the deviations between the three similar solutions (termed as “Deviation of similar solutions”). ACKNOWLEDGMENT The authors would like to thank the six anonymous referees for their valuable comments and discussions. REFERENCES [1] R. Green, “Increasing competition in the British electricity spot market,” J. Ind. Econ., vol. 44, no. 2, pp. 205–216, Jun. 1996. [2] R. Baldick, “Electricity market equilibrium models: The effect of parametrization,” IEEE Trans. Power Syst., vol. 17, no. 4, pp. 1170–1176, Nov. 2002. [3] A. G. Petoussis, X.-P. Zhang, S. G. Petoussis, and K. R. Godfrey, “Parameterization of linear supply functions in nonlinear AC electricity market equilibrium models—Part I: Literature review and equilibrium algorithm,” IEEE Trans. Power Syst., vol. 28, no. 2, pp. 650–658, May 2013. [4] X. Hu and D. Ralph, “Using EPECs to model bilevel games in restructured electricity markets with locational prices,” Oper. Res., vol. 55, no. 5, pp. 809–827, Sep.–Oct. 2007. [5] X. Wang, Y. Li, and S. Zhang, “Oligopolistic equilibrium analysis for electricity markets: A nonlinear complementarity approach,” IEEE Trans. Power Syst., vol. 19, no. 3, pp. 1348–1355, Aug. 2004.

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[6] H. Chen, K. P. Wong, C. Y. Chung, and D. H. M. Nguyen, “A coevolutionary approach to analyzing supply function equilibrium model,” IEEE Trans. Power Syst., vol. 21, no. 3, pp. 1019–1028, Aug. 2006. [7] H. Haghighat, H. Seifi, and A. R. Kian, “Gaming analysis in joint energy and spinning reserve markets,” IEEE Trans. Power Syst., vol. 22, no. 4, pp. 2074–2085, Nov. 2007. [8] P. F. Correia, T. J. Overbye, and I. A. Hiskens, “Searching for noncooperative equilibria in centralized electricity markets,” IEEE Trans. Power Syst., vol. 18, no. 4, pp. 1417–1424, Nov. 2003. [9] T. Li and M. Shahidehpour, “Strategic bidding of transmission-constrained GENCOs with incomplete information,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 437–447, Feb. 2005. [10] C. A. Berry, B. F. Hobbs, W. A. Meroney, R. P. O’Neil, and W. R. Stewart, Jr, “Understanding how market power can arise in network competition: A game theoretic approach,” Util. Policy, vol. 8, pp. 139–158, 1999. [11] Y. Liu and F. F. Wu, “Impacts of network constraints on electricity market equilibrium,” IEEE Trans. Power Syst., vol. 22, no. 1, pp. 126–135, Feb. 2007. [12] L. B. Cunningham, R. Baldick, and M. L. Baughman, “An empirical study of applied game theory: Transmission constrained Cournot behavior,” IEEE Trans. Power Syst., vol. 17, no. 1, pp. 166–172, Feb. 2002. [13] D. S. Kirschen and G. Strbac, Fundamentals of Power System Economics. New York: Wiley, 2004. [14] Y. S. Son and R. Baldick, “Hybrid coevolutionary programming for Nash equilibrium search in games with local optima,” IEEE Trans. Evol. Computat., vol. 8, no. 4, pp. 305–315, Aug. 2004. [15] [Online]. Available: http://www.ee.washington.edu/research/pstca/.

Andreas G. Petoussis received the B.Eng. (1st Class Honors) and Ph.D. degrees in electrical engineering from the University of Warwick, Coventry, U.K., in 2005 and 2009, respectively. He is a Consultant Electrical Engineer with the Cyprus Transmission System Operator, Nicosia, Cyprus. His research interests include electricity market equilibrium analysis, dynamic modeling of isolated power systems, and grid-connected renewable energy systems.

Xiao-Ping Zhang (M’95–SM’06) received the B.Eng., M.Sc., and Ph.D. degrees in electrical engineering from Southeast University, China, in 1988, 1990, and 1993, respectively. He is currently a Professor in Electrical Power Systems at the University of Birmingham, Birmingham, U.K., and he is also Director of the University Institute for Energy Research and Policy. Before joining the University of Birmingham, he was an Associate Professor in the School of Engineering at the University ofWarwick, Coventry, U.K. From 1998 to 1999, he was visiting UMIST. From 1999 to 2000, he was an Alexander-von-Humboldt Research Fellow with the University of Dortmund, Germany. He worked at China State Grid EPRI on EMS/DMS advanced application software research and development between 1993 and 1998. He is co-author of the monograph Flexible AC Transmission Systems: Modeling and Control (New York: Springer, 2006 and 2012). Prof. Zhang is an Editor of the IEEE TRANSACTIONS ON POWER SYSTEMS.

Savvas G. Petoussis received the B.Eng. and Ph.D. degrees in electrical engineering at the University of Warwick, Coventry, U.K., in 2002 and 2006, respectively. He is an Electrical Network Engineer with the Electricity Authority of Cyprus (EAC), Networks Business Unit, Nicosia, Cyprus. He received the IEE Prize Award for his excellent performance achievement in his B.Eng. degree studies

Keith R. Godfrey received the B.Sc. and Ph.D. degrees from the University of London, London, U.K., and the D.Sc. degree from the University of Warwick, Coventry, U.K. He is an Emeritus Professor and member of the Systems, Measurement and Modelling Research Group in the School of Engineering at the University of Warwick, Coventry, U.K. He is author/co-author of more than 200 papers. Prof. Godfrey was awarded the IEE Snell Premium (2000) and the Honeywell International Medal (2000–2001) of the Institute of Measurement and Control, for distinguished contributions to control engineering. He is a member of the IFAC Technical Committees on Biomedical Engineering and Control, and on Modeling, Identification and Signal Processing.