A Continuous Strategy Game for Power Transactions Analysis in Competitive Electricity Markets

Transcription

1 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER A Continuous Strategy Game for Power Transactions Analysis in Competitive Electricity Markets Jong-Bae Park, Balho H. Kim, Jin-Ho Kim, Man-Ho Jung, and Jong-Keun Park Abstract This paper presents a game theory application for analyzing power transactions in a deregulated energy marketplace such as PoolCo, where participants, especially, generating entities, maximize their net profits through optimal bidding strategies (i.e., bidding prices and bidding generations). In this paper, the electricity market for power transactions is modeled as a noncooperative game with complete information, where the solution is determined in a continuous strategy domain having recourse to the Nash equilibrium idea. In order to provide more apprehensible analysis, we suggest a new hybrid solution approach employing a 2-dimensional graphical approach as well as an analytical method. Finally, the proposed approach is demonstrated on a sample power system. Index Terms Continuous strategy, deregulation, game theory, PoolCo model, spot market. I. INTRODUCTION TRADITIONALLY, generation resources have been scheduled so as to minimize system-wide production costs while meeting various technical and operational constraints including demand-supply balance over the system [1] [3]. Recently, the electric power industries around the world are moving from the conventional monopolistic or vertically integrated environments to deregulated and competitive environments [2] [4], where each participant is concerned with profit maximization rather than system-wide costs minimization [3]. As a consequence, the conventional least-cost approaches for generation resource schedule can not exactly handle real-world situations any longer. Among the various trading mechanisms in a competitive electricity market, the PoolCo model is considered as one of the straightforward mechanisms to implement [2]. In the PoolCo model, each generating entity determines its bidding strategy for price and generation amount to sell, and the market sets the clearing price of the power transacted [2], [9]. In this environment, the generating entities have their concerns on the development of optimal decision-making procedures. Recently, there has been a considerable amount of works on game theory application to power system problems, especially Manuscript received January 17, 2000; revised March 23, J.-B. Park is with the Electrical Engineering Department, Konkuk University, Hwayang-dong 1, Hwangjin-gu, Seoul , Korea ( jbaepark@konkuk.ac.kr). B. H. Kim is with the School of Electrical Engineering, Hong-Ik University, Seoul, , Korea ( bhkim@wow.hongik.ac.kr). J.-H. Kim and J.-K. Park are with the School of Electrical Engineering, Seoul National University, Seoul, , Korea. M.-H. Jung is with the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, 1406 West Green St., Urbana IL USA. Publisher Item Identifier S (01) in competitive electricity environments. Maeda and Kaya applied game theories to analyze power transactions between customers and an electric utility using a cooperative, a Nash game, and a hypergame [5]. Haurie et al. described a two-player game theory to solve the cogeneration problem where demand elasticity was not considered [6]. Luh et al. formulated and resolved the load adaptive pricing problem using a closed-loop dynamic Stackelberg game theory [7]. Recently, Ferrero et al. modeled the power transactions as a static and complete information game where the cost information of each participant is shared among players and bidding prices are linked with generation output [8]. Also, Ferrero et al. applied a game theory to the problem of pricing electricity in a PoolCo system where they modeled the problem as a noncooperative game with incomplete information [9]. Bai et al.applied the Nash bargaining game to a transmission transaction problem for determining optimal transaction price and power of individual parties [10]. Cardell et al.addressed the market power and strategic interaction issues arising from the transmission network constraints, and solved by an analytical approach [11]. In this paper, we adopt the Nash game theory to model and analyze the transactions in a PoolCo market in a continuous strategy space and propose a new hybrid approach covering a 2-dimensional graphical analysis using the best response curves as well as an analytical method to determine the equilibriums. The significance of the proposed approach lies in that it can provide generating entities with very useful information on their bidding strategies. It can also provide a regulatory body such as ISO with information on the effects of the maximum bidding price (i.e., bidding price cap) on market equilibriums. II. FORMULATION OF POWER TRANSACTIONS FOR GAME THEORY APPLICATION A. Basic Assumptions for Game Theory Application The proposed approach assumes that; 1) the total amount of bidding generations is enough to provide the market demand (i.e.,, where denotes the bidding generation of entity, and denotes total system demand at time ), 2) the bidding generation of each player is less than the total system demand in a specific spot market (i.e.,, ), 3) the demand elasticity is not considered (i.e., ), 4) the transmission losses and network constraints are ignored, 5) all information such as cost function of each generating entity in the market is revealed in public implying a game with complete information. The last assumption holds for some countries such as Chile, Peru, and Korea [4], [12] /01$ IEEE

2 848 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001 B. Costs and Benefits Analysis in a PoolCo Market In a centralized power system amenable to the least-cost procedures, the only information required is the operating cost function of each generator, usually approximated as the following form; (1) (2) where operating costs of entity with generation; minimum and maximum generation output of entity. In a deregulated power system, however, additional information such as market clearing price, bidding generations, allocated generations, and profits of the individual market participants, etc. would be required to model the market. In a competitive PoolCo marketplace, the profit of generation entity at time will be; where spot price at time ; allocated generation of entity. The allocated generation of entity is assumed to be determined by the following rules: 1) If the bidding price of entity is lower than the bidding price of the last-dispatched generating entity (i.e., spot price), the allocated generation is equal to bidding generation (i.e., ), 2) if generating entity becomes the last-dispatched one, then the allocated generation is the remaining system demand (i.e., where is a set of already-dispatched generating entities). If two or more entities have submitted the same bidding price for the last dispatch, the remaining demand is evenly allocated among them as long as possible, 3) if the bidding price of entity is higher than the market clearing price, then no generation is allocated. C. Details of the Proposed Game Model The power transaction activities in a PoolCo model are inherently a noncooperative game with incomplete information where each player does his best to maximize the profits. With the aforementioned assumptions and rules, the power transactions can be modeled as a game with complete information. In this paper, each generating entity is considered as a player, profits of each player constitute payoffs, and strategies of each player are composed of the bidding price ( ) and the bidding generation ( ) in a continuous space. In addition, the bidding price and the bidding generation of a player are treated independently to reflect the real-world situations. III. POWER TRANSACTIONS ANALYSIS IN A TWO-PLAYER GAME This section describes how the two players make decision on the bidding price and bidding generation in a PoolCo model. Through the analysis, each player can eventually obtain information on the bidding price and the bidding generation in a (3) continuous domain using Nash equilibrium idea. In addition, the regulatory agency can also understand the role of maximum bidding price (i.e., bidding price cap). The two prevailing approaches finding Nash equilibriums covering all continuous strategies of the two players are of; 1) analytical approaches, and 2) graphical approaches [13], [14]. It is not easy to apply an analytical approach to the problem with continuous decision variables (in our case, bidding prices and bidding generations of both players). Also, the conventional graphical approaches using the best response curves of both players require 4-dimensional (i.e., bidding prices and bidding generations of both players) representation of the curves, which makes the direct applications of the approaches practically impossible to obtain the Nash equilibriums. To resolve the problem more easily and efficiently, we suggest a new hybrid approach combining a 2-dimensional graphical approach and an analytical method. First, to exploit the graphical approach, we have devised and applied a bidding-generation-based decomposition approach to systematically find the best responses (behaviors) of each player. With this approach, one can obtain a set of candidate Nash equilibriums. Next, the obtained candidate Nash equilibriums are refined by the conventional analytical methods, finally, yielding a set of genuine Nash equilibriums. To analyze the 2-player power transactions game with the proposed approach, we decompose the game into 3 cases according to each player s bidding generation levels; 1) Case 1 deals with the situation where one player s bidding generation is greater than a half of the total demand, and the other s is greater than or equal to the total demand, 2) Case 2 where one s bidding generation is greater than, and the other s is less than a half of the total demand, and 3) Case 3 where both players bidding generations are exactly equal to a half of the system demand. In addition, for a systematic analysis of the game, we define the following 3 sets of generating entities; 1), containing generating entity whose bidding price is lower than that of the opponent, 2), containing generating entities whose bidding prices are all equal, 3), containing generating entity whose bidding price is higher than the opponent s. With these definitions, we can generally describe the spot price regardless of cases as follows; A. Case 1.,, for, or, In this case, the profits of each entity can be evaluated by (3), the spot price by (4), and the allocated generations by the allocation rules in secondary Section II-B. The allocated generations of player (hereafter, we denote the opponent as ) will be; if if (4) or or (5) To analyze power transactions game by the best response curves in a graphical manner, first it is needed to develop each

3 PARK et al.: A CONTINUOUS STRATEGY GAME FOR POWER TRANSACTIONS ANALYSIS IN COMPETITIVE ELECTRICITY MARKETS 849 Fig. 1. Profit functions of generating entity A with respect to. Fig. 3. Best response functions of generating entity A and B with consideration of bidding price caps in Case 1. Fig. 2. Profit function of generating entity A with respect to. player s profit function the bidding generations of both players are specified as two fixed-values. Figs. 1 and 2 show the profit functions of generating entity A (for notational convenience, we use A for player ) with respect to the bidding prices. The profit function of each player is linear to the bidding price with the slope being determined by the allocated generation. Fig. 1 illustrates the profit of entity A with respect to A is an element of set or. The intercept on -axis is associated with entity A s costs the allocated generations are determined by (5). Fig. 2 is another profit graph of entity A with respect to A is an element of set. Similarly, the profit functions of entity B can be developed and analyzed with respect to and. Since our model assumes no demand elasticity, if there exist no regulations on the bidding prices, each player will try to maximize its profits by setting its bidding price as infinity regardless of opponent s bidding strategy. Therefore, there exist(s) no rational equilibrium point(s) in this kind of energy marketplaces. To avoid these undesirable circumstances, it is required for a pool coordinator to set the maximum bidding price cap over each participant. This regulation on the maximum bidding prices results in the existence of rational Nash equilibriums in the energy marketplace as shown in Fig. 3. A set of the optimal bidding prices of entity A,, with respect to bidding price of entity B can be obtained by identifying the best response of entity A, while the bidding price of entity B being gradually increased. First, if the bidding price of entity B is sufficiently small, entity A will try to maximize its profits by bidding maximum price,. Second, the bidding price of entity B reaches to the point of where A s profits with its maximum price exactly equal to the profits by bidding the same price with B s, will be either or. As B s bidding price increases gradually from, the best response curve of entity A will be until the bidding price of entity B reaches the value of. Finally, the bidding price of entity B is greater than, the entity A can maximize its profit by bidding any price smaller than B s bidding price. The best response bidding price of entity B,, can be found in a similar manner, and depicted in Fig. 3. Now, the Nash equilibriums can be obtained from the intersection of the best response sets of entity A and B. However, the resultant equilibriums may be different from the true ones since the equilibriums are obtained under the restriction of specific pair of bidding generations. The candidate Nash solutions,, from Fig. 3 exist in the continuous space which is constituted of tripart components, and described in the following equation: where (6)

4 850 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001 To get the true Nash equilibriums over all the possible bidding generation strategies, it is necessary to probe the tripart components of (6). As a result, the second component of (6) vanishes away, and the proof is given in the Appendix. From now on, we will derive the Nash solutions including Nash bidding price and bidding generation in a continuous space especially for the first and the third components of (6). For the first component of (6), the bidding generation of player B can be determined from the optimality condition of entity B (i.e., ) that guarantees B s maximum profits the spot price is. Therefore, player B s bidding generation becomes. The calculated bidding generation of entity B should satisfy the following inequality to belong to Case 1 Here, the optimal can be adjusted to or according to location of the optimum point. Therefore, we will not consider the inequality constraint of (2) in the following developments. Until now, we have determined only the optimal bidding generation of player B. If any strategy of each player is to be a Nash equilibrium, it should satisfy the following inequality conditions (8) and (9). Inequality (8) deals with the condition that player A s Nash bidding price is to become player B s strategy is (7) where, Since A s bidding generation should be greater than a half of total demand in Case 1, the modified bidding generation of entity A should be; (12) We can simply expand this logic, and pick out the Nash equilibriums for the third component of (6), yielding the bidding price of entity B being that is the spot price. Next, the bidding generation of entity A can be obtained from the optimality condition as follows: (13) Also, the bidding price of player A can be obtained that is an analogy of (10) as follows: (14) where, Similarly, inequality (9) is associated with the condition of player B s Nash bidding price player s A strategy is given. Here, the following inequality (9) should be reflected only. Otherwise, the inequality (9) is not necessary since the spot price is For the bidding generation of entity B, we can derive the following results just like as entity A under the assumption that (15) (8) where, (9) The solution of (8) can be obtained as follows, and it embeds more restrictions to the bidding price of entity B where, (10) Also, the inequality condition (9) yields the following result that restricts the bidding generation of entity A (11) As a result, the Nash equilibriums in Case 1 covering bidding prices and bidding generations of both players can be summarized as shown in (16) at the bottom of the next page. B. Case 2.,, for or In this case, the profit function of player can be determined by applying the following equation if if. (17)

5 PARK et al.: A CONTINUOUS STRATEGY GAME FOR POWER TRANSACTIONS ANALYSIS IN COMPETITIVE ELECTRICITY MARKETS 851 maximum bidding price and the other bids a sufficiently lower bidding price, (18) where is the bidding price of entity B obtained from the equation of. To elicit Nash equilibriums covering bidding prices and bidding generations in a continuous space, we should follow the same procedures described in Case 1. For the simplicity of analysis, we assume that player A bids its generation more than half of the total demand. For the first row of (18), the spot price is determined as and the bidding generation of player B can be obtained from the optimality condition [i.e., ]. Since we have assumed, the bidding generation of entity B should satisfy the following inequality condition (19) Fig. 4. Best response functions of generating entity A and B with consideration of bidding prices cap in Case 2. As in Case 1, we can develop the best response curves of the players the bidding generations are predetermined to certain values as depicted in Fig. 4. From the intersection of both entities best response curves, one can obtain a set of candidate Nash equilibriums on bidding prices in Case 2. Contrary to Case 1, the supposed Nash equilibriums the bidding generations of A and B are fixed with certain values are composed of two feasible regions as shown in (18). Each equilibrium set occurs one player bids its By applying (8) and (9) for the Case 2, we can obtain the range of player B s bidding price as well as the range of player A s bidding generation, which are the same as (10) and (11), respectively. In Case 2, the total bidding generations should be equal to or greater than system demand yielding an additional inequality condition for player A (20) For the second row in (18), the bidding price of entity B is, the spot price. Next, the bidding generation of entity A (16)

6 852 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001 ( ) can be obtained from the optimality condition [i.e., ]. Also, the bidding price of player A have the same result as in Case 1. The bidding generation of entity B is ( and ); To elicit Nash equilibriums, we should follow the same procedures described in Cases 1 and 2. In (24), the spot price is and the bidding generation of player A can be obtained from the optimality condition satisfying the following equality constraint: (21) However, there is another constraint on that reflects the supply and demand condition. This will be shown in (22) and (23) at the bottom of the page. Consequently, the Nash equilibriums in Case 2 over all the bidding strategies are given as (23). (25) Applying the same procedures in Cases 1 and 2 yields the following Nash equilibriums of both players C. Case 3. for In this case, the allocated generation to each generation is equal to a half of the total demand. For the simplicity of analysis, we suppose that is smaller than. (This also holds for the reverse case.) Under this assumption, just like as Cases 1 and 2, the resultant set of candidate Nash solutions is given as follows: (24) (26) (22) or (23) or

7 PARK et al.: A CONTINUOUS STRATEGY GAME FOR POWER TRANSACTIONS ANALYSIS IN COMPETITIVE ELECTRICITY MARKETS 853 TABLE I TEST SYSTEM INPUT DATA TABLE II PAYOFF TABLE OF THE SAMPLE SYSTEM IN CASE 1 IV. NUMERICAL EXAMPLES This section verifies the results derived in Section III for a sample power system. The system demand,, is assumed to be 300 [MW], and other necessary data including each participant s cost coefficients, and generation limits are given in Table I. The economic dispatch to this problem yields 196 [MW] and 104 [MW] as the optimal generations of generator A and B, respectively. Also, the corresponding system marginal cost,, is turned out to be 91 [$/MWh]. Then, the profits of each generating entity are [$] and 3675[$], respectively. Also, the total profits of both players are [$]. There can be several alternatives setting the bidding price cap of each player, but we set the price cap to the marginal production cost at the maximum output of each generator: [$/MWh] [$/MWh]. A. Tests for Case 1 In this case, let the bidding generation of one player be equal to or greater than 150 [MW], and that of the other player bids greater than 150 [MW]. Then, there exist two regions for Nash equilibriums. First, let player A bid its maximum price of 116 [$/MWh]. In this case, player B s bidding generation becomes [MW]. Since this result violates the inequality constraints of (16), there exist no Nash equilibriums. Next, let player B bid its maximum price of 170 [$/MWh], then, the optimal bidding generation of player A is [MW]. This satisfies the inequality condition of (16), but exceeds its generation limit. Therefore, the bidding generation of player A should be adjusted to 250 [MW], and consequently the bidding price of player A is obtained from the Nash equilibrium sets in (16) as follows: bidding generation is greater than 150 [MW]. Therefore, the Nash equilibrium in this case is and the corresponding profits of the players are [$] and [$], respectively. From the definition of Nash equilibrium, if a player changes his strategy from the equilibrium point, the corresponding profits will decrease. For example, suppose that player A changes its strategy from to while player B stays in the Nash equilibrium. In this case, player A s allocated generations is 240 [MW] and B s 60 [MW], respectively, which results in [$] and [$]. Also suppose that player B changes its strategy from to while player A stands in the equilibriums. The changed profits are [$] and [$], respectively. These two changes of strategies have resulted in the decrease of the corresponding player s profits as summarized in Table II. B. Tests for Case 2 Let one player bid its generation greater than 150 [MW] and the other bid less than 150 [MW]. In this case, the Nash solutions can be obtained from (23). To verify this, let player A bid its maximum price of 116 [$/MWh]. Then, player B s optimal bidding generation is. This satisfies the inequality constraint in (19) yielding; Therefore, one set of the Nash equilibriums in this case is The bidding generation of player B should also be equal to or greater than 150 [MW] since A s and the corresponding profits of the players are [$] and [$], respectively.

8 854 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001 TABLE III PAYOFF TABLE IN CASE 2 WITH AS MARGINAL PRICE As discussed in Section IV-A, an example payoff table in this case can be obtained as the following Table III. Next, let player B bid its maximum price of 170 [$/MWh]. From the fact that the procedure for the Nash solutions is exactly the same as Case 1, except that the bidding generation of player B should be less than a half of the total system demand, one can obtain the following Nash equilibriums, The corresponding profits of both players are calculated as [$] and [$], respectively. C. Tests for Case 3 This case considers the Nash equilibriums the bidding generations of both players are exactly equal to 150 [MW]. Since the bidding price caps for each player are set to [$/MWh] and [$/MWh], there exists no Nash equilibrium as discussed in Section III. However, a Nash equilibrium can be obtained by adjusting the bidding price caps so as to satisfy the following conditions a hybrid framework combining a graphical approach and an analytical method amenable to the problems in a continuous strategy space. A numerical example is given to demonstrate the approach, and as a result, the Nash equilibriums can be obtained and are proved to be stable in a discrete sense. Although the proposed approach requires more investigations on the several assumptions and for -player incomplete game application, we expect the proposed approach can provide players with useful information on determining their optimal bidding strategies as well as can provide the regulatory agency with an opportunity to understand the role of bidding price cap. APPENDIX The candidate Nash solutions of (6) in Case 1 are based on a pair of fixed bidding generations,. The second component of (6) does not generally become Nash equilibriums in case the bidding generations are expanded from a set of fixed-points to a set of continuous space. If there exists any strategy (i.e., bidding price and bidding generation) of entity A or B that can increase his profits while the opponent still remains in his strategy, the assumed solutions are not the Nash equilibriums. For the mathematical proof, we assume that player A reduce his bidding price from the points. In this case, the system marginal price remains as the, and the player A can maximize the profits by bidding generation as. Thus, the net incremental of A s profits can be calculated as follows: or (27) or (28) The constraint (27) justifies the situation that the optimal bidding generation of player ( the system marginal price is determined by player ) is equal to a half of demand, and the constraint (28) can be easily derived from (9). A set of Nash solution is V. CONCLUSION In this paper, we propose an approach to analyzing the power transactions in a deregulated marketplace. The proposed approach models the market as a noncooperative, two-player game with complete information seeking the Nash equilibriums for the optimal strategies (bidding price and bidding generation). The novelty of the suggested approach lies in that it provides As shown in the above equation, there exist player A s strategies that can increase his profits since the coefficient is positive if. Therefore, the second component of (5) does not become equilibrium points.

9 PARK et al.: A CONTINUOUS STRATEGY GAME FOR POWER TRANSACTIONS ANALYSIS IN COMPETITIVE ELECTRICITY MARKETS 855 REFERENCES [1] A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control: John Wiley & Sons, Inc., [2] P. F. Penner, Electric Utility Restructuring: A Guide to the Competitive Era. Vienna, VA: Public Utilities Reports, Inc., [3] F. Nishimura, R. D. Tabors, M. D. Ilic, and J. R. Lacalle-Melero, Benefit optimization of centralized and decentalized power systems in a multi-utility environment, IEEE Trans. PWRS, vol. 8, no. 3, pp , Aug [4] H. Rudnick, Pioneering electricity reform in South America, IEEE Spectrum, pp , Aug [5] A. Maeda and Y. Kaya, Game theory approach to use of noncommercial power plants under time-of-use pricing, IEEE Trans. Power Systems, vol. 7, no. 3, pp , Aug [6] A. Haurie, R. Loulou, and G. Savard, A two-player game model of power cogeneration in New England, IEEE Trans. Automatic Control, vol. 37, no. 9, pp , Sept [7] P. B. Luh, Y.-C. Ho, and R. Muralidharan, Load adaptive pricing: An emerging tool for electric utilities, IEEE Trans. Automatic Control, vol. 27, no. 2, pp , Apr [8] R. W. Ferrero, S. M. Shahidehpour, and V. C. Ramesh, Transaction analysis in deregulated power systems using game theory, IEEE Trans. Power Systems, vol. 12, no. 3, pp , Aug [9] R. W. Ferrero, J. F. Rivera, and S. M. Shahidehpour, Application of games with incomplete information for pricing electricity in deregulated power pools, IEEE Trans. Power Systems, vol. 13, no. 1, pp , Feb [10] X. Bai, S. M. Shahidehpour, V. C. Ramesh, and E. Yu, Transmission analysis by Nash game method, IEEE Trans. Power Systems, vol. 12, no. 3, pp , Aug [11] J. B. Cardell, C. C. Hitt, and W. W. Hogan, Market power and strategic interaction in electricity networks, Resources and Energy Economics, vol. 19, pp , [12] KEMA Consulting, Technical advisory report on KEPCO restructuring programme: Selection of the cost-based pool principles,, July [13] H. S. Bierman and L. Fernandez, Game Theory with Economic Applications: Addison-Wesley, [14] D. Fudenberg and J. Tirole, Game Theory. Cambridge, MA: The MIT Press, Jong-Bae Park received the B.S., M.S., and Ph.D. degrees from Seoul National University in 1987, 1989, and 1998, respectively. For , he worked as a researcher of Korea Electric Power Corporation (KEPCO), and since 1998 he has been an Assistant Professor at Anyang University, Korea. His research interests are power system planning and economic studies. Balho H. Kim received the B.S. from Seoul National University, South Korea and the M.S. and Ph.D. degrees from the University of Texas at Austin. Currently, he is an Assistant Professor at Hong-Ik University, South Korea. His research interests include power system planning and operation, and real-time pricing. Jin-Ho Kim received the B.S. and M.S. degrees from Seoul National University in 1995, 1997, respectively. He is currently working for Ph.D. degree at Seoul National University. His research interests are analysis of power market and power system operation. Man-Ho Joung received the B.S. and M.S. degrees from Seoul National University in 1995, 1997, respectively. For , he worked as a power system operator of Korea Electric Power Corporation (KEPCO). He is currently working for Ph.D. degree at University of Illinois at Urbana-Champaign. His research interests are power system operation and planning. Jong-Keun Park received the B.S. degree from Seoul National University in 1973 and the M.S. and Ph.D. degrees from University of Tokyo in 1979, 1982, respectively. He is a professor of School of Electrical Engineering at Seoul National University. His present interests are control and protection in FACTS, power economics, and application of artificial intelligent to power systems.