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A Solution to the Unit Commitment Problem Using Imperialistic Competition Algorithm Moosa Moghimi Hadji, Student Member, IEEE, and Behrooz Vahidi, Senior Member, IEEE
Abstract—This paper presents a new approach via a new evolutionary algorithm known as imperialistic competition algorithm (ICA) to solve the unit commitment (UC) problem. In ICA the initial population individuals (countries) are in two types: imperialists and colonies that all together form some empires. Imperialistic competitions among these empires converge to a state in which there exists only one empire. In the proposed ICA for the UC problem, the scheduling variables are coded as integers; therefore, the minimum up/down-time constraints can be handled directly. A new method for initializing the countries is proposed. To verify the performance of the proposed algorithm, it is applied to systems with number of generating units in range of 10 up to 100 in one-day scheduling period. Index Terms—Generation scheduling, imperialistic competition algorithm, optimization methods, unit commitment problem.
NOMENCLATURE Number of units. Scheduling horizon (in hours). System load demand at hour . System reserve at hour . Number of operating cycles for each unit. Duration of operating cycle for unit . Operation status of unit at hour . Output power of unit at hour . Maximum output power of unit . Minimum output power of unit . Maximum output power of unit at hour . Minimum output power of unit at hour . Maximum up-time limit of unit . Minimum down-time limit of unit . Ramp-up rate of unit . Ramp-down rate of unit . Start-up cost of unit . Hot start cost of unit . Cold start cost of unit . Manuscript received July 31, 2010; revised December 11, 2010 and March 12, 2011; accepted May 18, 2011. Date of publication June 27, 2011; date of current version January 20, 2012. Paper no. TPWRS-00615-2010. The authors are with the Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2011.2158010
Cold start hour of unit . Fuel cost of unit . Total fuel cost. Initial state of unit . I. INTRODUCTION
O
NE of the most important problems in power system operation is unit commitment (UC). The UC is a nonlinear, non-convex, large scale, mixed integer problem to determine a start-up and shut-down schedule of generating units at minimum operating cost. In this problem, the demand and reserve requirements and the constraints of the generating units must be satisfied [1], [2]. It should be mentioned that solving this problem can be computationally expensive in large power systems. Several solutions have been proposed for this problem that can be divided into two classes [3]: the first one is numerical optimization techniques such as priority list (PL) [4], [5], dynamic programming (DP) [6], [7], Lagrangian relaxation (LR) [8]–[10], branch-and-bound [11], and mixed integer programming [12]. These methods are fast and simple, but most of them suffer from numerical convergence and solution quality problems [1]. The second class is stochastic search methods such as genetic algorithms (GA) [13]–[15], evolutionary programming (EP) [16], particle swarm optimization (PSO) [2], [17], ant colony optimization [18], and bacterial foraging (BF) [19]. These methods can successfully handle complex nonlinear constraints and provide high-quality solutions, but all of them suffer from the curse of dimensionality [1]. In [20], another method called Straightforward (SF) is presented which decomposes the UC problem into three subproblems. This method is based on linearization of quadratic cost functions of generating units. Atashpaz-Gargari and Lucas first introduced Imperialistic competition algorithm (ICA) in 2007 [21]. This method is inspired by the imperialistic competition. All the countries are divided into imperialists and colonies. The colonies are divided among imperialists. The imperialists and colonies all together form some empires. Then the imperialistic competition among the empires begins. This hopefully causes the colonies to converge to the global optimum of the objective function. At the end of imperialistic competition, only one empire remains. The colonies of this empire are in the same position as the imperialist of this empire and have the same cost [21]. Recently ICA has been successfully applied for solving some optimization problems [22], [23]. This paper presents an integer-coded ICA for solving the UC problem. A new initialization method based on priority list is proposed. The coding scheme presented in [24] is used. The
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minimum up/down time constraints are always satisfied and there is no need of using a penalty function for these constraints. The effectiveness of the proposed methodology is evaluated on several case studies and the results have been presented.
where is obtained by (5) with . This is 10-min maximum response-rate-constrained active power of the th generating unit. • Minimum up/down times: if if
II. UC PROBLEM FORMULATION The objective of the UC problem is to minimize total operating cost of the generating units during the scheduling horizon, while both unit and system constraints must be satisfied. Total operating cost consists of fuel, start-up, and shut-down costs of the generating units [25]. The output powers of the units in each hour are determined by economic dispatch. In general, the fuel cost can be calculated using the fuel cost functions of the units as (1): (1) where , , and are fuel cost coefficients of generator . Then the total fuel costs for an H-hour load horizon will be (2) Start-up costs are expressed as an exponential or linear function of the time [25]. A simplified time-dependent start-up cost is also adopted as follows [17]: if if
(3) where is the unit’s down time. Finally shut-down cost can be considered as a fixed cost for each unit per shut down. However, the shut-down costs are not considered in this paper. The UC problem has several constrains that must be satisfied: • Generating units initial operating status • Maximum and minimum power limits: (4) • Ramp up/down rates The response-rate constraints of the unit, limits the power generation by the following time-dependent operating limits:
(8)
where is a sign integer which represents the continuous ON/OFF status duration of the th cycle of the th unit. The sum of the absolute values of for all units must be equal to the scheduling horizon: (9)
III. IMPERIALISTIC COMPETITION ALGORITHM Atashpaz Gargari and Lucas introduced the imperialistic competition algorithm which is inspired by imperialistic competitions. They applied this method to some of benchmark cost functions [21]. Similar to other evolutionary algorithms, the ICA starts with an initial population that is called countries. Some of the best countries that have the best objective functions are selected to be imperialists. The rest of the countries form the colonies of these imperialists. The colonies are divided among the imperialists based on the imperialists’ power. The power of an imperialist in a minimization problem is inversely proportional to its cost function. Then the colonies move toward their relevant imperialist and the position of the imperialists will be updated if necessary. In the next stage, the imperialistic competition among the empires begins, and through this competition, the weak empires are eliminated. The imperialistic competition will gradually lead to an increase in the power of powerful empires and a decrease in the power of weaker ones. Finally the weak empires that are not able to improve their position will be collapsed. These competitions among the empires will cause all the countries to converge to a state in which there exists only one empire in the word and all the other countries are colonies of this empire. A. Generating Initial Empires
(5)
At the beginning of the algorithm, an initial population called countries should be created. In an -dimensional optimization problem, the position of the th country is defined as , where is total number of the countries. The cost function of the countries can be found by evaluating the function at the variables . Then the cost of the th country is as follows:
(6)
(10)
where is the UC time step. • Power balance of the system:
• Spinning reserve of the system: (7)
is the cost of the th country. where Then of the most powerful countries are selected to form empires. The remaining countries will be the colonies of these empires. In the next step, the colonies must be divided among the imperialists based on their power. The imperialists’
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D. Calculating Total Power of an Empire The total power of an empire depends on both the power of the imperialist and the power of its colonies. But it is mainly affected by the power of the imperialist. The total power of an empire is defined as
(15)
Fig. 1. Movement of the colonies toward their relevant imperialist in a randomly deviated direction [21].
powers are inversely proportional to their cost function in a minimization problem. The initial number of colonies of an empire is directly proportional to its power. To do this, the normalized costs of the empires are defined as (11) where is the cost of the th imperialist and is its normalized cost. The normalized power of each imperialist is defined by
(12)
is the total power of the th empire and is a posiwhere tive number which is considered to be less than 1. In fact, represents the role of the colonies in determining the total power of an empire. E. Imperialistic Competition In this stage, the imperialistic competition begins and all the empires try to take possession of the colonies of other empires. This competition is modeled by picking some of the weakest colonies (usually one) of the weakest empires and making a competition among all empires to possess these (this) colonies. Each of the empires will have a likelihood of taking possession of these colonies based on their total powers; therefore, the more powerful empires have greater chance to possess the mentioned colonies. To do this, the possession probability of each empire must be found. The normalized total power of each empire is calculated as follows:
Then the initial number of colonies of an empire will be
(16) (13)
is the initial number of colonies of the th empire. where For each imperialist, of the colonies are randomly selected and are given to it. These colonies along with their imperialist form the empire.
is normalized total power of the th empire. where The possession probability of each empire is given by
(17)
B. Moving the Colonies Toward Their Imperialists In this stage, the colonies start to move toward their relevant imperialists. The positions of the colonies of the th empire are updated as follows: (14) where is the position of the th colony of the th imperialist, rand is a random number in (0,1), is a weight factor, and is the position of the th imperialist. To search different points around the imperialist, a random amount of deviation is added to the directionof movement. Fig. 1 shows the movement of a colony toward its relevant imperialist in the new direction. In this figure, is a random angle between where is the parameter that adjusts the deviation from the original direction. C. Updating Positions of the Imperialists During the previous stage, a colony may reach to a position with lower cost than that of the imperialist. In such a case, the positions of the imperialist and that colony must be exchanged. Then the rest of the colonies of this empire move toward the new position of the imperialist.
is possession probability of the th empire. Then the where vector is formed to divide the mentioned colonies among the empires: (18) After that, a random vector with the same size as ated as follows:
is cre(19)
where (0,1). At last, the
are randomly generated numbers in vector is formed by subtracting
from
:
(20) The mentioned colonies will be handed to an empire which has the maximum relevant index in vector.
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TABLE I CONFIGURATION OF A COUNTRY FOR TEN-UNIT SYSTEM AND THE UNIT COMMITMENT SCHEDULE FOR 24 h
F. Eliminating the Powerless Empires The powerless empires will collapse in the imperialistic competition. Different criteria can be defined for collapse mechanism. In this paper, an empire is assumed collapsed when it loses all of its colonies. G. Convergence After some imperialistic competitions, all the empires except the most powerful one will collapse and all of the countries under their possession become colonies of this empire. All the colonies have the same positions and the same costs and there is no difference between the colonies and their imperialist. In such a case, the algorithm stops. IV. IMPLEMENTATION OF ICA FOR UC PROBLEM A. Country Position Definition The position of a country in integer-coded ICA for UC problem consists of a sequence of ON/OFF cycle durations of the units during the time horizon of the problem. A positive integer in the Country vector represents duration of continuous ON status of that unit, while a negative integer represents duration of continuous OFF status of that unit. The unit’s minimum up/down time and the load peaks affect the number of unit’s ON/OFF cycles during the UC horizon [24]. A daily load profile with two peaks is shown in Fig. 2 for determining the number of ON/OFF cycles of the units [19], [24]. As it can be seen in this figure, the number of ON/OFF cycles for peak load units is 5. The numbers of cycles of the base and intermediate load units are lower, but for simplicity, they are considered to be the same and equal to the number of the cycles of peak load units (i.e., 5 for one-day scheduling).
Fig. 2. Base-load, medium-load, and peak-load operating cycles.
Therefore, a country in ICA implementation for one-day scheduling is a vector with elements. Definition of Country from ON/OFF cycle duration of units and the unit commitment schedule are illustrated in Table I. B. Creating Initial Population (Countries) A new initialization method is proposed in this paper. In this method, the units are divided into three classes. The units are divided in base-load, medium-load, and peak-load on the basis of similar characteristics, such as up/down time requirements, output power capacity, and start up costs. Since the base-load units are always in ON state, and peak load units are turned on in load peaks based on their fuel costs, the optimal state of medium-load units must be found. To do this, some priority lists
MOGHIMI HADJI AND VAHIDI: A SOLUTION TO THE UNIT COMMITMENT PROBLEM USING IMPERIALISTIC COMPETITION ALGORITHM
are created. The base-load units are in the highest priority based on their fuel costs. The peak-load units are in the lowest priority based on their fuel costs. A random permutation of the mediumload units is created and placed between the base-load and peakload units. The initial population is created using these priority lists. It should be noted that if only one priority list is used for the whole scheduling period, the search space will be restricted and the results are not appropriate, especially in large scale systems. To overcome this drawback, different priority lists are created and used for sequence of turning on or off the units. As it has been mentioned in [24], the duration of the first cycle of each unit must be initialized so that the units continue their operating mode (ON/OFF) of the last cycle of the previous scheduling day for at least as many hours as required to satisfy the minimum up/down time constraints. Also the time constraints for the rest of the operating cycles must be satisfied. Therefore, in the next step, the time constraints must be checked for all of the operating cycles. The method of satisfying these constraints will be presented in constraint handling subsection. C. Cost Function The cost function used in ICA is sum of total production costs (fuel costs and start-up/shut down costs) over the scheduling horizon and a penalty function that penalizes the violation of system constraints. The fuel costs are calculated using (1). The shut down costs are not considered in this paper. The start-up costs are calculated as follows: (21) where tion.
is total start-up cost and is the unit step funccan be calculated using the following equation: if if
.
(22)
Then the total production cost will be (23) The penalty function used in this paper is similar to that used in [19] and [24]. This function has two terms. Equation (24) is the first term that is used to penalize spinning reserve constraint violations, and (25) is the second term that is used to penalize excessive capacity for situation in which the load is less than the sum of the minimum outputs of all committed units at a specific hour:
π (24)
π (25)
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Fig. 3. Flowchart of modifying the first cycle of a unit.
where is the unit ramp function and is a multiple of the maximum operating cost of system over the scheduling period [24]: (26) where is a constant coefficient that has been considered equal to 1. Finally the ICA cost function is formed as follows:
π
π
(27)
D. Constraint Handling After creating initial population (i.e., countries), the initial empires are created. Now the movement of colonies toward their relevant imperialists begins. As said before, the position of the colonies is updated using (14). Since in UC problem the parameters must be integer, the Round function is applied to new positions of the colonies: (28) As said before, the sum of the absolute values of the cycles must be equal to the scheduling period (i.e., 24 in one-day scheduling) for each unit. Also the minimum up/down time constraints must be satisfied. A method for satisfying these constraints simultaneously is used in this paper. The method starts with the first cycle of each unit . The absolute value of must be in the range [1,24] and if this value violates this range, it must be brought to the nearest limit. Then must be checked with respect to the unit’s initial state . The process of modifying for is presented in Fig. 3. The concept of modifying process for is similar. Now the next cycles must be checked. A new variable called res is used to ensure that the sum of absolute values of cycles of the units is equal to scheduling period. For each cycle of a unit, res is the difference of scheduling period and sum of the absolute values of the previous cycles of this unit. For example
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TABLE II DATA FOR THE TEN-UNIT SYSTEM
Fig. 4. Flowchart of modifying the cycles of a unit.
in a one-day scheduling problem for , res is 24and for , res is 24. The process of modifying for is presented in Fig. 4. The concept of modifying process for is similar. The absolute value of the last cycle of each unit is always equal to the res. After satisfying the time constraints, the cost function for this new position is calculated. For calculating the cost function, economic dispatch (ED) must be done for each hour. As said before, the penalty functions of (24) and (25) are used in ED. If the colony has reached to a position with lower cost than that of the imperialist, the positions of the imperialist and that colony must be exchanged. After moving all the colonies and updating the imperialists’ positions, the imperialistic competition begins, and if there is a powerless empire, it should be eliminated. Then the colonies move toward their imperialists again. This procedure will continue in this way until the stopping criterion is satisfied. In this paper, the algorithm will stop if there is only one empire. V. TEST RESULTS The proposed ICA was tested on systems with 10, 20, 40, 60, 80, and 100 generating units considering a time horizon of 24 h. The required data for the ten generating unit system is given in Table II. The load data for this system is given in Table III. For the 20-generating unit system, the data of the ten generating unit system was duplicated and the load data was multiplied by 2. For problems with more generating units, the problem data were scaled appropriately. The spinning reserve of the system was assumed 10% of hourly load in all cases. Optimal parameters of ICA for the ten-unit system which are obtained through several runs are , , , and . For systems with more units, the same parameters are used, except and that may be increased. When some colonies are in a powerful empire, they move toward their relevant imperialist in several iterations. If the posi-
TABLE III LOAD DEMAND FOR 24 h
tion of this imperialist does not change in some consecutive iterations, these colonies become too close to the imperialist. Therefore, they lose their search capability and just increase the execution time until the position of the imperialist changes. To overcome this drawback, parameter which adjusts the deviation from the original direction is set to higher values in . The convergence of the algorithm for ten-generating unit system with 100, 200, and 300 initial countries is shown in Fig. 5. Also the mean operation costs and mean execution times for these cases are presented in Table IV. As it can be seen, increasing the number of initial countries from 100 to 200 noticeably improves the convergence and the mean operation cost, but increases the solution time. Further increasing the number of initial countries only increases the solution time. The convergence does not improve so much; also the improvement of mean operation cost is not noticeable. The best result for the ten generating unit system obtained by ICA is compared with that of GA [14], ICGA [24], hybrid particle swarm optimization (HPSO) [2], and bacterial foraging (BF) [19] in Table V.
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TABLE VII COMPARISON OF EXECUTION TIMES OF ICGA, ICA, AND BF
VI. CONCLUSION
Fig. 5. Convergence characteristics of the ICA for different numbers of initial countries (Nc).
TABLE IV COMPARISON OF MEAN COST AND MEAN EXECUTION TIME OF ICA WITH DIFFERENT NUMBER OF INITIAL COUNTRIES FOR TEN-UNIT SYSTEM
In this paper, a new evolutionary algorithm known as ICA for solving the UC problem is proposed. A new strategy for creating initial population is introduced. Also in the UC problem, minimum up/down time constraints is considered while producing the feasible solution and therefore no penalty function is used for these constraints. The efficiency of the proposed method is proved by considering one-day scheduling period with number of units from ten up to 100. The results obtained by the proposed ICA are compared with the results of some previous methods. The numerical results show that ICA can find high-quality solutions within a reasonable time. REFERENCES
TABLE V COMPARISON OF BEST RESULT OF ICA WITH METHODS FOR TEN-UNIT SYSTEM
THE
OTHER
TABLE VI COMPARISON OF OPERATING COST OF DIFFERENT METHODS
The best results obtained by ICA are compared with those of LR [14], GA [14], ICGA [24], BF [19], and MILP [26] for systems with ten up to 100 generating units in Table VI. The execution times of ICA and available execution times of the other methods are presented in Table VII. The proposed ICA is implemented on an Intel Pentium IV 2-GHz CPU with 512 Mbytes of RAM which previously was used for implementation of the BF. The execution times of ICGA are obtained from an Intel Pentium IV 1.6-GHz CPU. As it can be seen in Table VII, the ICA can find more qualified solutions with much lower execution times in comparison with BF. Although the proposed approach requires more execution time in comparison with ICGA, the quality of the final results are much better than the results obtained by ICGA.
[1] V. S. Pappala and I. Erlich, “A new approach for solving the unit commitment problem by adaptive particle swarm optimization,” in Proc. IEEE PES General Meeting, Pittsburgh, PA, Jul. 20–24, 2008, pp. 1–6. [2] T. O. Ting, M. V. C. Rao, and C. K. Loo, “A novel approach for unit commitment problem via an effective hybrid particle swarm optimization,” IEEE Trans. Power Syst., vol. 21, no. 1, pp. 411–418, Feb. 2006. [3] Y. W. Jeong, J. B. Park, S. H. Jang, and K. Y. Lee, “A new quantuminspired binary PSO for thermal unit commitment problems,” in Proc. IEEE 15th Int. Conf. Intelligent System Applications to Power Systems, Curitibia, Brazil, Nov. 8–12, 2009, pp. 1–6. [4] R. M. Burns and C. A. Gibson, “Optimization of priority lists for a unit commitment program,” in Proc. IEEE Power Eng. Soc. Summer Meeting, 1975, Paper A 75 453-1. [5] G. B. Sheble, “Solution of the unit commitment problem by the method of unit periods,” IEEE Trans. Power Syst., vol. 5, no. 1, pp. 257–260, Feb. 1990. [6] W. L. Snyder, Jr., H. D. Powell, Jr., and J. C. Rayburn, “Dynamic programming approach to unit commitment,” IEEE Trans. Power Syst., vol. 2, no. 2, pp. 339–350, May 1987. [7] Z. Ouyang and S. M. Shahidehpour, “An intelligent dynamic programming for unit commitment application,” IEEE Trans. Power Syst., vol. 6, no. 3, pp. 1203–1209, Aug. 1991. [8] F. Zhuangand and F. D. Galiana, “Toward a more rigorous and practical unit commitment by Lagrangian relaxation,” IEEE Trans. Power Syst., vol. 3, no. 2, pp. 763–770, May 1988. [9] F. N. Lee, “A fuel-constrained unit commitment method,” IEEE Trans. Power Syst., vol. 4, no. 3, pp. 691–698, Aug. 1989. [10] S. Virmani, C. Adrian, K. Imhof, and S. Mukherjee, “Implementation of a Lagrangian relaxation based unit commitment problem,” IEEE Trans. Power Syst., vol. 4, no. 4, pp. 1373–1380, Nov. 1989. [11] A. I. Cohen and M. Yoshimura, “A branch-and-bound algorithm for unit commitment,” IEEE Trans. Power App. Syst., vol. PAS-102, pp. 444–451, Feb. 1983. [12] J. A. Muckstadt and R. C. Wilson, “An application of mixed-integer programming duality to scheduling thermal generating systems,” IEEE Trans. Power App. Syst., vol. PAS-, pp. 1968–1978, 1968. [13] G. B. Sheble and T. T. Maifeld, “Unit commitment by genetic algorithm and expert-system,” Elect. Power Syst. Res., vol. 30, no. 2, pp. 115–121, Jul./Aug. 1994. [14] S. A. Kazarlis, A. G. Bakirtzis, and V. Petridis, “A genetic algorithm solution to the unit commitment problem,” IEEE Trans. Power Syst., vol. 11, no. 1, pp. 83–92, Feb. 1996.
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[15] T. Senjyu, H. Yamashiro, K. Shimabukuro, K. Uezato, and T. Funabashi, “Fast solution technique for large-scale unit commitment problem using genetic algorithm,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 150, no. 6, pp. 753–760, Nov. 2003. [16] K. A. Juste, H. Kita, E. Tanaka, and J. Hasegawa, “An evolutionary programming solution to the unit commitment problem,” IEEE Trans. Power Syst., vol. 14, no. 4, pp. 1452–1459, Nov. 1999. [17] Z. L. Gaing, “Discrete particle swarm optimization algorithm for unit commitment,” in Proc. IEEE Power Eng. Soc. General Meeting, Jul. 2003, vol. 1, pp. 13–17. [18] N. S. Sisworahardjo and A. A. El-Keib, “Unit commitment using the ant colony search algorithm,” in Proc. Large Engineering System Conf. Power Engineering (LESCOPE 02), 2002, pp. 2–6. [19] M. Eslamian, S. H. Hosseinian, and B. Vahidi, “Bacterial foragingbased solution to the unit-commitment problem,” IEEE Trans. Power Syst., vol. 24, no. 3, pp. 1478–1488, Aug. 2009. [20] S. H. Hosseini, A. Khodaei, and F. Aminifar, “A novel straightforward unit commitment method for large-scale power systems,” IEEE Trans. Power Syst., vol. 22, no. 4, pp. 2134–2143, Nov. 2007. [21] E. A. Gargari and C. Lucas, “Imperialist competitive algorithm: An algorithm for optimization inspired by imperialistic competition,” in Proc. IEEE Congr. Evolutionary Computation, 2007. [22] A. Biabangard-Oskouyi, E. Atashpaz-Gargari, N. Soltani, and C. Lucas, “Application of imperialist competitive algorithm for materials property characterization from sharp indentation test,” Int. J. Eng. Simulat., to be published. [23] A. Khabbazi, E. Atashpaz-Gargari, and C. Lucas, “Imperialist competitive algorithm for minimum bit error rate beamforming,” Int. J. Bio-Inspired Computation (IJBIC), vol. 1, no. 1–2, pp. 125–133, 2009. [24] I. G. Damousis, A. G. Bakirtzis, and P. S. Dokopoulos, “A solution to the unit commitment problem using integer-coded genetic algorithm,” IEEE Trans. Power Syst., vol. 19, no. 2, pp. 1165–1172, May 2004. [25] A. J. Wood and B. F. Wollenberg, Power Generation Operation and Control. New York: Wiley, 1984. [26] M. Carrion and J. M. Arroyo, “A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem,” IEEE Trans. Power Syst., vol. 21, no. 3, pp. 1371–1378, Aug. 2006.
Moosa Moghimi Hadji (S’10) was born in Tehran, Iran, in 1986. He received the B.S. degree in electrical engineering from Iran University of Science and Technology, Tehran, Iran, in 2008. He is now pursuing the M.S. degree in electrical engineering from Amirkabir University of Technology, Tehran, Iran. His research interests include power market, optimization techniques, and artificial intelligence and their application in power systems.
Behrooz Vahidi (M’00–SM’04) was born in Abadan, Iran, in 1953. He received the B.S. degree in electrical engineering from Sharif University of Technology, Tehran, Iran, in 1980 and M.S. degree in electrical engineering from Amirkabir University of Technology, Tehran, Iran, in 1989. He also received the Ph.D. degree in electrical engineering from the University of Manchester Institute of Science and Technology, Manchester, U.K., in 1997. From 1980 to 1986, he worked in the field of high voltage in industry as a chief engineer. From 1989 to the present, he has been with the Department of Electrical Engineering of Amirkabir University of Technology, where he is now a Professor. His main fields of research are high voltage, electrical insulation, power system transient, lightning protection, and pulse power technology. He has authored and co-authored more than 220 papers and five books on high voltage engineering and power system.
