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An Unfixed Piecewise-Optimal Reactive Power-Flow Model and its Algorithm for AC-DC Systems Juan Yu, Wei Yan, Wenyuan Li, Fellow, IEEE, C. Y. Chung, Member, IEEE, and K. P. Wong, Fellow, IEEE
Abstract—A novel ac-dc optimal reactive power flow (ORPF) model with the generator capability limits is proposed. The objective is to minimize the total cost of generator reactive and active power outputs paid by an independent transmission company. In the model, the generator operation area is divided into four parts with each one having a different reactive output cost function. The boundaries between the four parts change with operation states of generators and are unfixed. The model is an unfixed piecewise mixed-integer nonlinear programming problem. A heuristic approach is introduced into the interior point method to solve the unfixed piecewise problem. The numerical results of six test systems that range in size from 14 to 2572 buses demonstrate the effectiveness of the proposed method. Index Terms—AC-DC power system, capability limits of synchronous generator, generator reactive power costs, interior point method (IPM), optimal reactive power flow (ORPF).
I. INTRODUCTION URING the past two decades, electric power systems around the world have been continuously experiencing important changes caused by the deregulation process. In the new environment of the electric market, reactive costs as an ancillary service are introduced to pay for reactive support devices [1]–[5]. The generator operation capacity is confined by the under-excitation, field current, armature current, and mechanical power limits [5]–[7]. In [5], the inequality constraints of generator-reactive outputs corresponding to these capability limits are considered. Reactive production may constrain the maximum active power output. This will reduce the opportunity of profits that generators may obtain from the active power market, resulting in the opportunity cost of generator reactive outputs [4]. It is obvious that the generator capability limits play an important role in calculating its opportunity cost. When reactive power outputs change, the limits and operating costs of generators are varied. In [2] and [5], the generator operation area is divided into three regions. However, [5] does not distinguish the opportunity costs of different regions in the model and algorithm. In [2], the price model is proposed
D
with binary variables for discrete selection of generators from any of the three regions. Unfortunately, reactive ancillary costs for generator leading phase operation are ignored. In addition, the mechanical power limits and the discrete characteristics of control variables are not considered in the models presented in References [2] and [5]. Most optimal power flow (OPF) and optimal reactive power flow (ORPF) studies have been carried out on ac systems. Very little work has been reported on OPF or ORPF of ac-dc systems. In [8], issues of OPF in an ac-dc system were discussed, but generation-reactive costs and the capability limits are totally overlooked. A novel ac-dc ORPF problem with the generator capability limits is proposed. The objective is to minimize the total cost of generator reactive and active power outputs paid by an independent transmission company. In the model, the generator operation area is divided into four parts with each one having a different capacity and opportunity costs of reactive outputs. The boundaries between the four parts change with operation states of generators and are unfixed. This is an essential characteristic of the proposed model. It is an unfixed piecewise mixed-integer nonlinear programming problem. Considerable optimization methods have been presented for solving a mixed-integer nonlinear programming problem [9]–[15], in which the interior point method (IPM) has been extensively developed and has a good convergence feature. Unfortunately, the existing IPM cannot solve the unfixed piecewise model. In this paper, a heuristic approach is introduced into IPM to handle the unfixed piecewise problem. The test results demonstrate that the improved IPM can deal with the unfixed piecewise problem effectively. The paper is organized as follows. Section II describes the generator capability limits and formulates a new ac-dc ORPF model. In Section III, the improved IPM is discussed. Test results are presented and compared in Section IV, followed by conclusions in Section V. II. MATHEMATICAL MODEL OF THE AC-DC ORPF
Manuscript received July 12, 2006; revised November 25, 2006. This work was supported by the National Natural Science Foundation of China under Contract 50577073 and by the Research Grants Council of Hong Kong (PolyU 5225/04E). Paper no. TPWRS-00423-2006. J. Yu and W. Yan are with the State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Electrical Engineering College of Chongqing University, Chongqing 400044, China (e-mail:
[email protected];
[email protected]). W. Li is with British Columbia Transmission Corporation, Vancouver, BC V7X 1V5, Canada (e-mail:
[email protected]). C. Y. Chung and K. P. Wong are with the Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/TPWRS.2007.907387
A. Capability Limits and Reactive Costs of Synchronous Generator The capability curve of synchronous generator is shown in Fig. 1. The internal area ABCD represents the normal operation region of the generator. The field heating limit is the circular arc CD with radius , centered on . The under-excitation limit is the beeline AB whose slope is ). The horizontal lines BC and AD are the maximum tg( and minimum active power outputs corresponding to mechanand . At the generator ical power limits, i.e.,
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Fig. 2. General bus representation with system components connected.
the ABF region has to satisfy (11) because it is on the beeline in the CDE region has to satisfy AB. Similarly, (12). The detailed derivations of these two equations are shown in the Appendix. B. Unfixed Piecewise AC-DC ORPF
Fig. 1. Capability curve of synchronous generator.
operation point , the capability of the active power , is confined by the above limits. output, represented by Apparently, the inequality of holds. , , , and represent the terminal voltage, the synchronous reactance, the magnitude of maximum internal voltage, and the maximum armature current, respectively. According to the reactive power coordinate in Fig. 1, the generator operation area ABCD can be divided into four regions: , BGHF region GCEH region , ABF region and CDE region . In this paper, the generator reactive power ancillary service costs include the capacity cost and opportunity cost of the generator. The capacity cost corresponding to the four regions is given by [4]:
for for
0( 0(
and and
region) region) (1)
Fig. 2 is a representation of a power system bus with the connected generator, load, reactive compensator, converter and transmission branches (lines and off- or on- load tap changing , , , and transformers). are the powers of the generator, load, reactive compensator and converter. and are the voltage magnitude and angle. , , , and are the dc voltage, dc current, commutation resistance, and converter transformer turns ratio. For convenience buses are connected of description, it is assumed that the first to converters and the other ones are not. Based on (1) and (2), a new unfixed piecewise ac-dc ORPF and are the model is proposed as shown in (3)–(25). firing angle cosine and power factor angle of the converter, is the gating delay angle for the rectifier or the exwhere is the turns ratio of tinction advance angle for the inverter. on-load tap changing transformers (LTCs) or converter trans, , and (except for the formers. In the model, swing bus) are known quantity, , , , , , , and are state variables, is continuous control variables, and are discrete control variables and
is the unit capacity cost of the generator. where To provide the reactive power, a generator may have to decrease the active power output because of the capability limits, which will in turn reduce the opportunity of profits that the generator may obtain from the active power market. This is known as the opportunity cost, which can be expressed approximately as references [3] and [5] (2) is the profit rate of active power production, usually where is the cost between 0.05 and 0.10. function of the active power production, in which , , and are the economic parameters. is the function of which is releand the corresponding capability limits. In the vant to GCEH and BGHF regions, , and thus . However, the generator is limited by the under-excitation limit and the field heating limit in the ABF , and CDE regions, respectively, so that and . Note that the point in Authorized licensed use limited to: CHONGQING UNIVERSITY. Downloaded on December 8, 2008 at 04:38 from IEEE Xplore. Restrictions apply.
(3)
(4)
(5)
(6) (7)
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(8) (9) (10) (11)
(12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) where , , , , and denote the number of system buses, generator nodes, reactive compensator nodes, LTCs, and , , , and represent converters, respectively; the number of generators operating in the GCEH, BGHF, ABF, and CDE regions, respectively; and Ns are the active power and are the th price and the swing bus number; row and th column element of the system admittance matrix and are the number and dc network conductance matrix; of poles and series-connected six-pulse bridges of each pole in and represent the maximum and the converter; . minimum limits of The objective function in (3) consists of four parts. The first one is the generator active power cost, the second and third terms are the generator reactive capacity cost in the GCEH and CDE regions and in the BGHF and ABF regions, and the last one is the generator opportunity cost in the ABF and CDE regions. The active and reactive powers at buses are given in (4) and , the converter at bus is a rectifier; if (5), where if , the converter at bus is an inverter; and if , no converter is connected at bus . Thus, for and for . Equations (6)–(10) are the dc power-flow equations [17]. The dc voltage relationships are expressed in (6) and (7), . Equations (8) and (9) are the active and reactive powers flowing into or out of the converter at bus . Equation (10) is the dc current injection at bus in terms of dc network conductance and dc voltage. The limits of the active output capability in the ABF and CDE regions are given in (11) and (12), respectively. The generator-reactive power bound constraints corresponding to the four regions are
, , , , , , , , given in (13)–(16). are respectively restricted by (17)–(25). In this model, each generator can operate in one of the four regions depending on its operating point. In other words, a generator can operate across the boundary between the neighboring , , and are not fixed numregions and thus bers during the resolution of the model. In addition, the boundand between the four regions change aries defined by during the resolution and thus are with the terminal voltage unfixed, which is explained in the Appendix. Therefore, the objective function (3) and the constraints (11)–(17) are undetermined. As a consequence, the proposed model is an unfixed piecewise mixed-integer nonlinear programming problem. III. IMPROVED IPM To solve the unfixed piecewise nonlinear programming problem, an improved IPM combined with a heuristic approach (HEUIPM) is introduced in this section. The heuristic approach is used to search the generator regions and determine the values , , , , , and so that the unfixed of piecewise model can be converted into a general mixed-integer nonlinear model at each step, which IPM can solve. A. IPM For General Nonlinear Problem When the generator operating regions and their boundaries , , , , , and are are known, i.e., once obtained, the proposed unfixed piecewise model can be transformed into the conventional nonlinear formulation expressed as in the following general form: (26) (27) (28) (29) represents the equality (4)–(12); and where represent the functions in the inequality (13)–(25); and are the upper and lower bounds in the inequality (13)–(25); is the vector of optimal variables. The general nonlinear problem given in (26)-(29) can be solved using the IPM described in References [12] and [16]. In this method, slack variables and Lagrange multipliers are introduced for dealing with the inequality and equality constraints, and the logarithmic barrier functions are used to treat the non-negativity conditions of the slack variables. Then, the ORPF problem can be transformed to minimize the Lagrangian function without constraints.
(30)
where and are the vectors of the Lagrangian multipliers for the equality and inequality and constraints in the optimal problem (26)-(29); are the vectors of the slack variables; , and are the number of , , and , respectively; and is the barrier parameter.
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Based on the Karush-Kuhn-Tucker (KKT) first-order conditions of the problem (30), a set of nonlinear algebraic equations is formed and then solved using the Newton-Raphson algorithm. An outline of the IPM procedure is as follows:
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TABLE I BASIC INFORMATION FOR TEST SYSTEMS
STEP 1: Initialize the primal and dual variables, and set the ; the Lagrange multipliers , slack variables , , ; and the barrier parameter . Also set the , initial iteration count max number of iteration , and the convergence mismatch . STEP 2: Predicting process. Drop and the second-order error terms, and then form and solve the pure Newton system to obtain the affine direction:
Correct and estimate the second-order error terms, and then form the correction equation. STEP 3: Correcting process. Solve the correcting Newton system to get the actual search direction:
Compute the step length in the search direction and then update the primary and dual variables:
Increment iteration count, i.e.,
.
STEP 4: If the following two convergence criteria are simultaneously satisfied, then output results and stop. Otherwise, go to STEP 2. a) The complementary gap must be less than
b) The maximum norm of the KKT conditions must be less than
STEP 2: Determine (3) and (11)–(17), transform the unfixed piecewise model into the general nonlinear one, and update and using IPM. STEP 3: If of each generator does not reach the bound or , end the calculation. Otherwise, switch to value reaches the neighboring region for the generators whose or . The rules for region change are as follows: if a generator is in the ABF region (or CDE region), the new region is the BGHF region (or GCEH region). If a generator is in the GCEH region (or BGHF region), the new region is the CDE reaches the upper limit, region (or GCEH region) when reaches the and the BGHF region (or ABF region) when lower limit. STEP 4: If either of the following two convergence criteria is , , satisfied, end the calculation. Otherwise, renew , , and , and then go to STEP 2. a) The new region is the one that has been searched in the previous iterations. b) The maximum number of searches for generator operation regions is reached. IV. SIMULATION RESULTS
B. Improved IPM (HEUIPM) HEUIPM is proposed to handle the unfixed piecewise optimal problem presented in Section II, which the IPM cannot directly solve. Its procedure is summarized as follows. STEP 1: Search the initial generator regions: ignore the generator reactive power costs in the objective function (3); use the maximum and minimum reactive power outputs as the bound constraints of to replace corresponding to (13)–(16); solve this general nonlinear model using IPM to and ; calculate and with (33) and (34) obtain in the Appendix; and then determine the initial region of each , , and using generator and the numbers of , and .
A prototype implementation of HEUIPM has been developed in MATLAB and tested on the IEEE Test Systems with 14, 30, 57, and 118 buses, and two actual utility systems in China (Chongqing power grid with 171 buses and China Southern power grid with 2572 buses). The computer set is Intel SaiYang processor 2.0 GHz and EMS 256 MB. Table I displays the basic information for each test system and their dc systems are described as follows: IEEE 14 bus system: ac lines between buses 2 and 4, 2 and 5, 4 and 5, and 9 and 10 are replaced by dc links. This system has 5 converter stations. IEEE 30 bus system: ac lines between buses 2 and 4, 2 and 6, and 4 and 6 are replaced by dc links. This system is a 3terminal dc mesh system.
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TABLE II RESULTS OBTAINED USING HEUIPM
TABLE III RESULTS CORRESPONDING TO THE MAXIMUM AND MINIMUM OBJECTIVE VALUE
IEEE 57 bus system: The ac line between buses 8 & 9 is replaced by a dc link. This system is a 2-terminal dc system. IEEE 118 bus system: ac lines between buses 26 and 30 and 80 and 77 are replaced by dc links. This system has 4 converter stations. Chongqing power grid with 171 buses (Ch-171): a 500kV ac line is replaced by a dc link. This system is a 2-terminal dc system. China Southern power grid with 2572 buses (S-2572): this system has three HVDC links whose resistance values are 22.16 , 10.00 and 10.00 , respectively. The commutating reactance values of six converters are 44.55 , 35.30 , 22.50 , 22.47 , 20.82 and 20.82 , respectively. In all the cases, the lower and upper limits on the load bus (p.u.) at voltages were set at 0.95 and 1.05 (p.u.), those on at 0.9 and 1.1 (p.u.). The limits on 1.0 and 1.1 (p.u.), those on of the LTC and converter transformers were set at 0.9 and 1.1 (p.u.). The bounds on the firing angles were assumed to be 5 and 60 for the rectifiers and 15 and 50 for the inverters. The active power price, the generator reactive capacity price and the profit rate of active power production were 200RMB/MW per hour, 40RMB/MVAR per hour and 0.07, respectively. and were assumed to be 10% and 5% above their rated is set at 75 . values respectively. The optimization results using HEUIPM are listed in Table II. It is observed that HEUIPM offers good convergence in solving
the proposed unfixed piecewise model. The number of searches for the generator operation regions and the iteration counts in each search directly affect the CPU time, which is not always proportional to the system scale. It can be seen from Table II that the proposed heuristic approach can properly handle unfixed boundaries between generator operation regions. The IEEE14 and IEEE30 systems require two searches, the IEEE57 and Ch-171 systems require three searches, and IEEE118 and S-2572 systems require four and six searches, respectively. The number of searches in all the cases is less than seven. The minimum objective values are lower than the maximum ones obtained in the first search in which the generator reactive costs have been ignored. The HEUIPM technique can search the valid generator operation regions fast and obtain an effective optimization solution for the proposed unfixed piecewise ac-dc ORPF model. The results obtained using HEUIPM are shown in Table III. In comparison between the two states corresponding to the maximum and minimum objective values, it can be observed in the from Table III that the generator reactive outputs IEEE14 and IEEE30 systems are down to 0 MVAR from in the other systems 58.24 MVAR and 39.56 MVAR. The are also decreased by 27.6%, 44.2%, 58.3%, and 94.4%, in the respectively. The reactive compensator outputs IEEE14, IEEE30, Ch-171 and S-2572 systems are respectively increased by 40.11MVAR, 27.67 MVAR, 333.57 MVAR, and 5866.41MVAR to balance the reactive decrement of the genera-
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YU et al.: UNFIXED PIECEWISE-ORPF MODEL AND ITS ALGORITHM FOR AC-DC SYSTEMS
tors. On the contrary, in the IEEE57 and IEEE118 systems are respectively decreased by 167.36MVAR and 106.03MVAR . Bebecause of the large decline of system reactive losses sides, compared with the results in the first search, the reactive flowing into the converters are cut down by 10.52%, power 29.20%, 34.17%, 46.76%, 34.74%, and 14.88% for the six studied systems, respectively. The results also demonstrate that the reactive flows obtained using the ac-dc ORPF model with and without the generator reactive costs are much different and the variance will have a great effect on the operation and control of the converters. The ORPF model with the generator reactive costs is a useful computing tool for ac-dc power systems.
and As shown in Fig. 1, are the end points of the beeline AB and the circular CD, respectively. We can obtain the following equations from (31) and (32): (33) (34) Because , , and are constant, and can be calculated using the terminal voltage with (30) and (31). REFERENCES
V. CONCLUSIONS According to the generator capability limits and reactive power coordinate, the generator operation area is divided into four regions. The capacity and opportunity cost functions for reactive outputs in each region are developed. An unfixed piecewise nonlinear mixed-integer ORPF model is established to minimize the total cost of the generator active power output and reactive ancillary service. A heuristic approach combined with IPM is presented for solving the unfixed piecewise optimization model. Numerical simulations on the IEEE Test Systems with 14, 30, 57, and 118 buses, Chongqing power grid with 171 buses and China Southern power grid with 2572 buses demonstrate effectiveness and efficiency of the proposed method. The results indicate that the reactive flows obtained using the ac-dc ORPF model with the generator reactive costs is much different from those without considering the costs. The generator reactive costs have great impacts on the operation state of the converters and all reactive sources. APPENDIX The beeline AB passes through the point and its slope is so that any point
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on AB satisfies (31)
When the generator operates in the ABF region, that is, the generator is limited by the under-excitation limit AB, must be on the beeline AB, and thus the following equation holds:
Therefore, for all the generators in the ABF region, we have (11). The radius of the circular arc CD centered on is so that any point on CD satisfies (32) When the generator operates in the CDE region, is on CD. Similarly, for all the generators in the CDE region, we have (12).
[1] S. Ahmed and G. Strbac, “A method for simulation and analysis of reactive power market,” IEEE Trans. Power Syst., vol. 15, no. 3, pp. 1047–1052, Aug. 2000. [2] K. Bhattacharya and J. Zhong, “Reactive power as an ancillary service,” IEEE Trans. Power Syst., vol. 16, no. 2, pp. 294–300, May 2001. [3] X. Lin, A. K. David, and C. W. Yu, “Reactive power optimization with voltage stability consideration in power market systems,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 150, no. 3, pp. 305–310, May 2003. [4] J. W. Lamont and J. Fu, “Cost analysis of reactive power support,” IEEE Trans. Power Syst., vol. 14, no. 3, pp. 890–898, Aug. 1999. [5] M. J. Rider and V. L. Paucar, “Application of a nonlinear reactive power pricing model for competitive electric markets,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 151, no. 3, pp. 407–414, May 2004. [6] C. W. Taylor, Power System Voltage Stability. New York: McGrawHill, 1994. [7] W. D. Rosehart, C. A. Cañizares, and V. H. Quintana, “Effect of detailed power system models in traditional and voltage stability constrained optimal power flow problems,” IEEE Trans. Power Syst., vol. 18, no. 1, pp. 27–35, Feb. 2003. [8] C. N. Lu, S. S. Chen, and C. M. Ong, “The incorporation of HVDC equations in optimal power flow methods using sequential quadratic programming techniques,” IEEE Trans. Power Syst., vol. 3, no. 3, pp. 1005–1011, Aug. 1988. [9] A. G. Bakirtzis, P. N. Biskas, and C. E. Zoumas, “Optimal power flow by enhanced genetic algorithm,” IEEE Trans. Power Syst., vol. 17, no. 2, pp. 229–236, May 2002. [10] J. R. O. Soto, C. R. R. Domellas, and D. M. Falcao, “Optimal reactive power dispatch using a hybrid formulation: genetic algorithm and interior point,” in Proc. IEEE Power Tech’2001 Conf., Porto, Portugal, 2001, pp. 5–8. [11] W. Yan, F. Liu, C. Y. Chung, and K. P. Wong, “A hybrid genetic algorithm interior point method for optimal reactive power flow,” IEEE Trans. Power Syst., vol. 21, no. 3, pp. 1163–1169, Aug. 2006. [12] M. Liu, S. K. Tso, and Y. Cheng, “An extended nonlinear primal-dual interior-point algorithm for reactive-power optimization of large-scale power systems with discrete control variables,” IEEE Trans. Power Syst., vol. 17, no. 4, pp. 982–991, Nov. 2002. [13] V. H. Quintana, “Mini-lecture book on interior-point applications to power systems,” in Proc. 1999 PICA Conf., Santa Clara, CA. [14] M. E. El-Hawary, “Optimal power flow solution techniques, requirement, and challenges,” in IEEE-OES, 96-TP-111–0, 1996. [15] W. Li, Secure and Economic Operation of Power Systems. Chongqing, China: Chongqing Univ. Publishing House, 1989. [16] W. Yan, J. Yu, D. C. Yu, and K. Bhattarai, “A new optimal reactive power flow model in rectangular form and its solution by predictor corrector primal dual interior point method,” IEEE Trans. Power Syst., vol. 21, no. 1, pp. 61–67, Feb. 2006. [17] J. Arrillaga and B. Smith, AC-DC Power System Analysis. London, U.K.: IEE, 1998. Juan Yu was born in Jingzhou, China, on December 17, 1980. She received the Ph.D. degree in electrical engineering from Chongqing University, Chongqing, China, in 2007. Her research interests include reactive optimal problem, risk assessment, and state estimation in power systems.
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Wei Yan received the Ph.D. degree in electrical engineering from Chongqing University, Chongqing, China, in 1999. Currently, he is a Professor and Associate Chairman of the Electrical Power Department in the Electrical Engineering College, Chongqing University. His research interests include optimal operation and control in power systems.
Wenyuan Li (F’02) is currently a Principal Engineer at British Columbia Transmission Corporation (BCTC), Vancouver, BC, Canada, and an Advisory Professor at Chongqing University, Chongqing, China. He was the author of four books and considerable technical papers in power system operation, planning, optimization, and reliability assessment. Dr. Li was the winner of the 1996 “Outstanding Engineer” awarded by the IEEE Canada.
K. P. Wong (F’02) received the M.Sc, Ph.D., and D.Eng. degrees from the Institute of Science and Technology, University of Manchester, Manchester, U.K., in 1972, 1974, and 2001, respectively. He has been with The University of Western Australia since 1974. He is currently Chair Professor and Head of Department of Electrical Engineering, The Hong Kong Polytechnic University. He has published numerous research papers in power systems and in the applications of artificial intelligence and evolutionary computation to power system planning and operations. His current research interests include evolutionary optimization in power, power market analysis, power system planning and operation in deregulated environments, and power quality. Dr. Wong received three Sir John Madsen Medals (1981, 1982, and 1988) from the Institution of Engineers Australia, the 1999 Outstanding Engineer Award from IEEE Power Chapter Western Australia, and the 2000 IEEE Third Millennium Award. He is a Fellow of the IEE, HKIE, and IEAust.
C. Y. Chung (M’01) received the B.Eng. degree (with first-class honors) and the Ph.D. degree in electrical engineering from The Hong Kong Polytechnic University, Hong Kong, China. He then worked in the Electrical Engineering Department, University of Alberta, Edmonton, AB, Canada, and Powertech Labs, Inc., Surrey, BC, Canada. Currently, he is an Assistant Professor in the Department of Electrical Engineering, The Hong Kong Polytechnic University. His research interests include power system stability/control, computational intelligence applications, and power markets.
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