lis , C nd 005 7 S ar o n pp al era outage cost. mine the optimal on line generation units such that the total fuel cost of the thermal units can be minimized [1]. tion of the system load.However, these policies do notmatch the stochastic nature of power systems. A more consistent the spinning reserve level. A good generation scheduling solved by UCPR would reduce the operating costs, er’s decomposition [5]. Although the features of these approaches are quite different, they were proposed in order either to decrease the computation time or to reduce the fuel costs of the power systems. Recently, with the appearance of artificial and computa- tional intelligence technologies, such methodologies as neu- * Corresponding author. Tel.: +886 3 5593142x3071; fax: +886 3 5573895. E-mail address:
[email protected] (T.-Y. Lee). Energy Conversion and Managem A standard unit commitment problem is often formulated subject to several constraints, including the real power operating limits of the generation units, electric power bal- ance and spinning reserve of the power system. Most utilities use deterministic criteria to determine the spinning reserve level and on line generation units. Their operating rules are to keep the spinning reserve level greater than the generation of the largest on line generator or a frac- increase the system reliability and maximize the energy effi- ciency of the generation units. Considerable efforts have been devoted to study the spinning reserve scheduling problem. Most of the results show that a considerable fuel cost saving or a reasonable spinning reserve schedule can be reached. These approaches include dynamic programming, Monte-Carlo simulation method [3], Lagrangian relaxation [4] and Bend- A 48 unit power system was used as a numerical example to test the new algorithm. The optimal scheduling of on line generation units could be reached in the testing results while satisfying the requirement of the objective function. � 2006 Elsevier Ltd. All rights reserved. Keywords: Iteration particle swarm optimization; Probabilistic reserve; Unit commitment; Outage cost 1. Introduction Because of the fast growing load of power systems and the large load gap between heavy load and light load peri- ods, the unit commitment problem has become a crucial task in the economic operation of a power system. The main objective of the unit commitment problem is to deter- and realistic method would be one based on probabilistic methods. A risk index based on such methods would enable a consistent comparison to bemade between various operat- ing strategies and the economics of such strategies [2]. Unit commitment with probabilistic reserve (UCPR) considers the effects of outage cost on the solution of the unit commitment problem and uses a risk index to evaluate Unit commitment with probabi Tsung-Ying Lee * Department of Electrical Engineering, Ming Hsing University of Science a Received 13 December 2 Available online Abstract This paper presents a new algorithm for solution of the nonline particle swarm optimization (IPSO). A new index, called iteratio improve the solution quality and computation efficiency. IPSO is a lem of a power system. The outage cost as well as fuel cost of therm the level of spinning reserve. The optimal scheduling of on line gen 0196-8904/$ - see front matter � 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2006.06.015 tic reserve: An IPSO approach hun-Lung Chen Technology, 1 Hsin-Hsing Road, Hsin Feng, Hsinchu 304, Taiwan, ROC ; accepted 26 June 2006 eptember 2006 ptimal scheduling problem. This algorithm is named the iteration best, is incorporated into particle swarm optimization (PSO) to lied to solve the unit commitment with probabilistic reserve prob- units was considered in the unit commitment program to evaluate tion units was reached while minimizing the sum of fuel cost and www.elsevier.com/locate/enconman ent 48 (2007) 486–493 Minim OCd where the ho sion and M ral networks, genetic algorithms and simulated annealing have also been applied to deal with the spinning reserve scheduling problem. In the neural network application, once the networks are well trained, the on line operation will be merely a simple arithmetic operation. A new gener- ation schedule can, thus, be obtained immediately. This method was proved feasible. However, the network needs to be retrained whenever the scenario varies [6]. Simulated annealing mimics the physical operation of an annealing process. It was also presented to solve the near optimal solution of the spinning reserve scheduling problem. It is easy to implement, yet the complicated annealing schedule is closely related to the optimization performance. A poor tuning of the annealing schedule may inadvertently affect the performance of simulated annealing [7]. The genetic algorithm is inspired by the principles of natural evolution. It is very popular in solving the optimization problem in power systems. The drawbacks of this approach are attrib- uted to the long computing time and the complicated pro- cess in coding and decoding the problem [8]. Particle swarm optimization (PSO) was originally pre- sented byKennedy andEberhart in 1995 [9]. It was originally inspired by observation of the behaviors of bird blocks and fish schools. The main advantages of PSO are its simple con- cept, computational efficiency and easy implementation. PSO has been successfully applied to various fields of power system optimization, such as economic dispatch, reactive and voltage control and power system stabilizer design. In this paper, an efficient algorithm, which is modified from particle swarm optimization [9] is developed to solve the UCPR problem of power systems. A new index, called iteration best, is incorporated into PSO to improve the solution quality and computation efficiency. The effects of outage cost as well as fuel cost are considered in the unit commitment program to evaluate the level of spinning reserve. Besides, the lowest spinning reserve level is restricted at least to be the generation of the largest on line generation unit. This new algorithm can minimize the sum of fuel cost and outage cost when the optimal on line gen- eration scheduling is reached. Finally, a 48 unit power system was used as a numerical example to test the new algorithm. Results show that the solution of the UCPR problem can be achieved when the minimum of the summation of fuel cost and outage cost during the study periods is reached. Finally, these results are also compared with the results of dynamic program- ming and another artificial algorithm to show the effective- ness of the presented approached. 2. Problem formulation and the objective function The outage cost as well as fuel cost of generation units should be considered in power system operation [2]. The new approach presented in this paper solves the optimal hourly on line unit number while minimizing the sum of fuel T.-Y. Lee, C.-L. Chen / Energy Conver cost and outage cost of a power system during the study per- iod. The objective function of this problem is expressed as lue of lost load [2], N is the number of on line units at hour d and w1 is a weighting factor representing the percentage of outage cost in total social cost. TCd is the total operating cost of the power system at hour d, OCd is the outage cost of the power system at hour d, FCn is the fuel cost function of thermal unit n, PGn,d is the generation of thermal unit n at hour d, SCn,d is the start up cost of unit n at hour d and An, Bn, Cn are constants. EENSd is the expected energy not supplied at hour d. The concept of EENS was originally formulated for long term planning studies [2]. It is applied here to determine how many MWs would have to be disconnected in the event of some outage of generation capacity and for how long the disconnection would take place. The optimization is subject to: (a) Real power balanceX n PGn;d ¼ Loadd ; n 2 on line units ð5Þ (b) Real power operating limits of generation units Pminn 6 PGn;d 6 Pmaxn ð6Þ (c) Spinning reserve (10 min) constraint LSRd PMaxðPGn;dÞ; n 2 on line units ð7Þ (d) Minimum up time and down time of generation units Tonn;d�1 P tupn; when Toffn;d ¼ 1; d 2 D ð8Þ Toffn;d�1 P tdownn; when Tonn;j ¼ 1; d 2 D ð9Þ (e) Start up cost of generation units SCn;d ¼ d0 1� e Toffn;d d1 � � þ d2; d 2 D ð10Þ Start up costs are only incurred when a transition from state ‘‘off’’ to state ‘‘on’’ of the generation unit occurs. (f) Ramping speed of generation units maxðPminn;PGn;d�1 � T60�RDnÞ 6 PGn;d 6 minðPmaxn;PGn;d�1 þ T60�RUnÞ ð11Þ where Loadd is the load at hour d, Pminn is the mini- mum generation limit of thermal unit n, Pmaxn is the generation rating of thermal unit n, LSRd is the min- imum spinning reserve level at hour d, Tonn,d is the TSC is the total social cost of the power system, D is ur number of the scheduling period, VOLL is the va- TCd ¼ XN n¼1 ½FCnðPGn;dÞ þ SCn;d � ð3Þ FCnðPGn;dÞ ¼ An þ Bn � PGn;d þ Cn � PG2n;d ð4Þ ize TSC ¼ XD d¼1 ðTCd þ w1�OCdÞ ð1Þ ¼ EENSd � VOLL ð2Þ anagement 48 (2007) 486–493 487 on time of unit d at hour d, Toffn,d is the off time of unit n at hour d, tupn is the minimum up time of unit calculation, RUn is the ramping up rate of unit n, (6) 4. Ite A Eq. ( compu (12). This type of PSO is named IPSO in this paper. V kþ1j ¼ where has bee resents that pull each particle toward Ibest. (15) w x = 9.0 nated t conditi where A f ðx; yÞ ¼ x sin4xþ 1:1y sin2y; x 2 ½0;10�; y 2 ½0;10� ð15Þ sion and Ma according to Eqs. (12) and (13). Repeat steps (2)–(5) until a criterion is met. ration particle swarm optimization RDn is the ramping down rate of unit n, T60 = 60 min is the UC time step. Constraints (5) and (6) can be considered by solving the economic dispatch problem at each time stage. Constraints (7)–(11) will be handled during the process of the iteration particle swarm optimization (IPSO). 3. Particle swarm optimization Particle swarm optimization is a parallel search tech- nique with characteristics of high performance and ease of implementation. Originally, it mimics the sociality of bird blocks and fish schools. Through a tracking of two best values, i.e. Pbest and Gbest, the global optimum may be achieved by this optimization technique [9]. Pbest is the best value of the fitness function of every particle of the population considered. Gbest is the best value of the fitness function that has been achieved so far by any particle. Eqs. (12) and (13) show the mechanism to modify the speed and particle location in the PSO. V kþ1j ¼ V kj þ c1� rand� ðPbestkj � X kj Þ þ c2� rand� ðGbestk � X kj Þ ð12Þ X kþ1j ¼ X kj þ V kþ1j ð13Þ where V kj is the velocity of particle j in iteration k, X k j is the position of particle j in iteration k, Pbestkj is the best value of fitness function that has been achieved by particle j be- fore iteration k, Gbestk is the best value of fitness function that has been achieved so far by any particle, c1 and c2 rep- resent the weighting of the stochastic acceleration terms that pull each particle toward Pbest and Gbest positions, rand means a random variable between 0.0 to 1.0. The process of implementing the PSO is as follows [9]: (1) Create an initial population of particles with random positions and velocities within the solution space. (2) For each particle, calculate the value of the fitness function. (3) Compare the fitness of each particle with its Pbest. If the current solution is better than its Pbest, then replace its Pbest by the current solution. (4) Compare the fitness of all the particles with Gbest. If the fitness of any particles is better than Gbest, then replace Gbest. (5) Update the velocity and position of all particles n, tdownn is the minimum down time of unit n, coeffi- cients d0, d1 and d2 are parameters for start up costs 488 T.-Y. Lee, C.-L. Chen / Energy Conver new index named ‘‘Iteration Best’’ is incorporated in 12) in this paper to improve the solution quality and 5. Solution method and implementation of IPSO The main computational processes of the algorithm pre- sented in this paper to solve the UCPR problem of power systems are discussed in the following subsections. This algorithm is an implementation of IPSO. Step 1: Initialize the IPSO parameters Set up the set of parameters of IPSO, such as, number of particles Q = 120, weighting factors c1 = 0.01, c2 = 0.01, c3 = c1 Æ (1 � e�c1Æk), maximum number of iterations ITmax = 4000. Step 2: Create an initial population of particles randomly Each particle contains the real power generation of the generators. Eq. (16) shows a particle q, X kq ¼ Pk1;1 P k 1;2 � � � Pk1;D Pk2;1 P k 2;2 � � � Pk2;D .. . .. . . . . .. . PkN ;1 P k N ;1 � � � PkN ;D 2 666664 3 777775; q ¼ 1; 2; 3; . . . ;Q ð16Þ where Pkn;d is the generation of unit n at time stage d average computation time in a run. Fig. 1 and Table 1 show the trajectory of all particles in a typical run in IPSO. It is observed in Fig. 1 and Table 1 that six particles would reach the global best value after 500 iterations in this run. The results of PSO are AF = �16.5 andAT = 85 ms. The computation time of IPSO is less than that of PSO and the average particle value of IPSO is less than that of PSO. The improvement in performance of IPSO is confirmed. hose global best value is f(x,y) = �18.5544 when 4 and y = 8.673. A 10 particle population is coordi- o find the solution. After 10 runs, the final solution ons achieved by IPSO are AF = �18.2, AT = 77 ms, F is the average of the particle values and AT is the A simple test was performed to test the performance of PSO and IPSO on a IBM/PC Pentium IV 2.8 GHz. In this test, PSO and IPSO are applied to find the minimum of Eq. V kj þ c1� rand� ðPbestkj � X kj Þ þ c2� rand � ðGbestk � X kj Þ þ c3� rand� ðIbestk � X kj Þ ð14Þ IbestK is the best value of the fitness function that n achieved by any particle in iteration k and c3 rep- the weighting of the stochastic acceleration terms tation time. Eq. (14) shows the new form of Eq. nagement 48 (2007) 486–493 in iteration k, D is the study period, N is the number of generators, Q is the number of particles. The gen- Step 3: 0 2 4 6 8 10 0 2 4 6 8 10 X Axis Y A xi s A xi s 0 2 4 6 8 10 0 2 4 6 8 10 X Axis Y Ax is 6 8 10 0 2 4 6 8 10 0 Y A xi s b) 150th iter SGkd ¼ CSkq;4 ¼ TTkn;d ¼ CSkq;5 ¼ RCkn;d ¼ T.-Y. Lee, C.-L. Chen / Energy Conversion and Management 48 (2007) 486–493 489 ð17Þ where V kn;d is the velocity of a particle q. It represents a movement of the generation of unit n at time stage erations of the generators have to satisfy the requirements of Eqs. (5)–(11).Eq. (17) shows the velocity of a particle q, V kq ¼ V k1;1 V k 1;2 � � � V k1;D V k2;1 V k 2;2 � � � V k2;D .. . .. . . . . .. . V kN ;1 V k N ;1 � � � V kN ;D 2 666664 3 777775; q ¼ 1; 2; 3; . . . ;Q Y 0 2 4 0 2 4 6 8 10 X Axis Fig. 1. Trajectory of particles in a simple test: (a) initial, ( d in iteration k. Evaluate the fitness of the particles For each particle, calculate the value of the fitness function. The fitness function is an index to evalu- ate the fitness of the particles. Eq. (18) shows the fitness function of the UCPR problem. P n ðPkn;d þRUn � 10Þ; when Loadd þ LSRd > P n ðPkn;d þRU Loadd þ LSRd ; otherwise ( c4� TTkn;d tupn � Tonn;d�1; when Toffn;d ¼ 1 and Tonn;d�1 < t tdownn � Toffn;d�1; when Tonn;d ¼ 1 and Toffn;d�1 < t 0; otherwise 8>< >: c5�RCkn;d P kn;d � ðPkn;d�1 þRUn � T60Þ; when Pkn;d > Pkn;d�1 þRUn � Pkn;d�1 � ðPkn;d �RDn � T60Þ; when Pkn;d < Pkn;d�1 �RDn � 0; otherwise 8>< >: FTkq ¼ TSCþ X5 b¼1 CSkq;b ð18Þ and CSkq;1 ¼ c1� XD d¼1 XN n¼1 Pkn;d � Loadd ����� ����� ð19Þ CSkq;2 ¼ c2� jPkn;d � PNnj ð20Þ PNn ¼ Pmaxn; when Pkn;d > Pmaxn Pminn; when Pkn;d < Pminn P kn;d ; otherwise 8>< >: ð21Þ 2 4 6 8 10 X Axis ation, (c) 300th iteration and (d) 500th iteration. CSkq;3 ¼ c3� ðLoadd þ LSRd � SGkdÞ ð22Þ where CSkq;1 is the penalty function for Eq. (5), CS k q;2 is the penalty function for Eq. (6), CSkq;3 is the pen- alty function for Eq. (7), CSkq;4 is the penalty func- tion for Eqs. (8) and (9), CSkq;5 is the penalty n � 10Þ; n 2 on line units ð23Þ ð24Þ upn downn ð25Þ ð26Þ T60 T60 ð27Þ Step 6: .30 .30 .93 .55 sion applied to update the position of the particles. The position of a particle is the generation of the generators. End conditions function for Eq. (11), c1 to c5 are the penalty factors of the constraints and FTkq is the fitness of particle q in iteration k. Step 4: Record and update the best values The three best values are recorded in the searching process. Each particle keeps track of its coordinate in the solution space that is associated with the best solution it has reached so far. This value is recorded as Pbest. Another best value to be recorded is Gbest, which is the overall best value obtained so far by any particle. Ibest is the best value of the fit- ness function that has been achieved by any particle in the present iteration. Pbest, Gbest and Ibest are the generations of generators in this study. This step also updates Pbest, Gbest and Ibest. At first, we compare the fitness of each particle with its Pbest. If the current solution is better than its Pbest, then replace Pbest by the current solution. Second, Ibest is compared with the value of any par- ticle in the current iteration. If the current solution is better than Ibest, then replace Ibest by the best value in the current iteration. Finally, the fitness of all particles is compared with Gbest. If the fitness of any particle is better than Gbest, then replace Gbest. Step 5: Update the velocity and position of the particles Eq. (12) is applied to update the velocity of the par- ticles. The velocity of a particle represents a move- ment of the generation of the generators.Eq. (13) is Table 1 Results of a simple test case Fitness Q Iteration 1 2 3 4 5 Initial 4.0897 3.5671 �14.5441 7.285 �11 150 �7.4188 �2.2255 �14.5441 �3.2319 �11 300 �15.0779 �15.0537 �14.7083 �15.0593 �14 500 �15.0779 �15.0537 �18.5544 �15.0897 �18 Global best value of Eq. (14) is �18.5544. Q: particle number. 490 T.-Y. Lee, C.-L. Chen / Energy Conver Check the end condition. If it is reached, the algo- rithm stops, otherwise, repeat steps 3–5 until the end conditions are satisfied. In this study, the ‘‘end conditions’’ of IPSO are: (1) The total operating cost between two consecu- tive iterations is unchanged or the variation of operating cost is within a permitted range. (2) The Gbest between two consecutive iterations is unchanged for 20 iterations. (3) The variation of Gbest is within a permitted range. (4) The maximum number of iterations is reached. 6. Numerical examples A 48 unit power system with 168 h load is used as an example to test the proposed algorithm. The largest unit in this system is a 900 MW unit, the VOLL was set as 135 NT$/kW h [10] and the weighting factor w1 in Eq. (1) was set as 1.0. The parameters of IPSO are selected as: the number of particle Q = 120, weighting factors c1 = 0.01 and c2 = 0.01, c3 = c1 Æ (1 � e�c1Æk), ITmax = 4000, penalty factors c1 to c5 are selected as 5000. The results show that the solution of UCPR problem was reached within 24 min on an IBM/PC-PentiumIV/ 2.8G personal computer while satisfying the requirements of the objective function and all constraints. The minimum of the total social cost is NT$3,323,567,000. Table 2 shows the solution of IPSO after 10 runs under different numbers of particles. It is observed that the aver- age value after 10 runs decreased when the particle number increased. The execution time is proportional to the num- ber of particles. In order to evaluate the performance of IPSO, three other approaches were also applied to solve this case. They are DP, GA and PSO. Table 3 shows the TSC and compu- tation time of these algorithms after 10 runs. It is found that the average solution of IPSO is less than that of the other algorithms and the computation time of IPSO is the least. Fig. 2 shows the optimization procedure of IPSO in comparison with the GA and PSO. It is seen that the TSC solved by IPSO decreases very soon, before the 200th iterations. The final result of the IPSO is also better 6 7 8 9 10 19 0.1122 7.0357 2.7971 �4.5577 �2.4174 19 �8.1865 �11.6751 �6.5899 �9.9091 �2.4174 02 �15.1045 �15.0064 �11.6478 �15.0682 �15.0948 44 �18.5544 �16.0557 �18.5544 �18.5544 �18.5544 and Management 48 (2007) 486–493 than that of GA or PSO. These results demonstrate the high performance of IPSO in this case. As previously described in this paper, the number of on line units and spinning reserve level of power systems are closely related with not only the fuel costs but also the outage cost. In order to observe the effect of outage cost on the UCPR problem, the results of the IPSO method were compared with that of the traditional fixed spinning reserve method, where the spinning reserve setting is kept constant during the study period. Table 4 shows the results, where the spinning reserve equal to 900 MW means the spinning reserve is kept at least 900 MW during the study period. Table 2 Solution of IPSO after 10 runs under different particle numbers Run 80 particles 120 particles 160 particles TSC Time (s) TSC Time (s) TSC Time (s) 1 3350.027 1066.5 3323.567 1446.7 3323.572 1894.6 2 3324.682 1072.0 3348.903 1438.6 3323.567 1925.2 3 3338.898 1093.2 3348.679 1471.9 3348.565 1951.2 4 3330.960 1119.9 3323.571 1405.3 3323.567 1873.2 5 3324.682 1068.7 3323.650 1419.4 3348.564 1958.7 6 3343.196 1087.4 3323.567 1451.1 3323.575 1899.2 7 3324.682 1066.5 3323.568 1406.5 3323.567 1976.2 8 3325.758 1090.4 3326.488 1447.0 3323.567 1916.0 9 3349.692 1095.6 3323.567 1415.4 3323.594 1921.4 10 3349.687 1092.5 3348.564 1441.8 3323.643 1959.4 1.4 T.-Y. Lee, C.-L. Chen / Energy Conversion and Management 48 (2007) 486–493 491 Average 3336.226 1085.2 333 TSC(NT$1,000,000). Table 3 The total social cost found by the IPSO method is the least one shown in Table 4. The objective function is thus reached, and the applicability of the IPSO method is reconfirmed. Performance of different algorithms Algorithms DP GA {population size = 120, probability of crossover = 100%, mutation rate = PSO {Q = 120, c1 = 0.01, c2 = 0.01} IPSO {Q = 120, c1 = 0.01, c2 = 0.01, c3 = c1 Æ (1 � e�c1Æk)} 3320000 3370000 3420000 3470000 3520000 0 1000 2000 3000 4000 Iteration TS C, N T$ 1,0 00 IPSO PSO GA Fig. 2. Optimization procedure of the IPSO. 20 25 30 35 40 45 50 55 60 1 25 49 73 Tim N u m be r o f O n - Li ne U n its IPSO method S.R.=900MW Fig. 3. Number of on line units under 12 1434.5 3328.578 1920.0 Fig. 3 shows the number of on line thermal units under different spinning reserve level setting in Table 3. Compar- ing the number of on line units for fixed SR = 900 MW and the IPSO method, it is seen that the number of on line units for SR = 900 MW and the IPSO method is similar during Average of TSC (NT$1,000,000) Average of computation time (s) 3352.177 10615.3 5%} 3373.399 3299.3 3358.493 1752.3 3331.412 1434.5 Table 4 Operating cost under different spinning reserve settings Spinning reserve Fuel cost (NT$) Outage cost (NT$) Total social cost (NT$) 900 MWa 2,851,634,000 1,469,025,000 4,320,659,000 1200 MWb 2,924,728,000 731,195,000 3,655,923,000 IPSO methodc 3,024,405,000 299,162,000 3,323,567,000 a Spinning reserve is kept at least as much as 900 MW. b Spinning reserve is kept at least as much as 1200 MW. c Determine spinning reserve by the IPSO method. 97 121 145 e, hour S.R.=1200MW load*0.003 different spinning reserve settings. Tim W diff sion and Management 48 (2007) 486–493 500 1000 1500 2000 2500 3000 3500 4000 1 25 49 73 Sp in n in g R es er v e, M W IPSO method S.R.=900M Fig. 4. Spinning reserve under 0.7 492 T.-Y. Lee, C.-L. Chen / Energy Conver light load hours. The reason is that 900 MW is the lowest spinning reserve level allowable in this system. Therefore, in order to save the total fuel cost of the generation units and serve the lowest spinning reserve requirement, the IPSO method regulates the number of on line units during light load hours automatically. Besides, a larger number of on line units are committed during heavy load periods when the IPSO method is applied. This will reduce the out- age cost and will cause an improvement in system security. Comparing the number of on line units for fixed SR = 1200 MW and the IPSO method, it is seen that a large number of on line units, which will cause an increase in fuel cost, are committed during the light load hours when the spinning reserve is fixed and set as 1200 MW; however, less units are needed during light load periods; such as the result of the IPSO method. Fig. 4 shows a comparison of the spinning reserve for the numbers of on line units in Fig. 3. The spinning reserve of the IPSO method is observed to be greater than the fixed SR = 900 MW during heavy load periods and similar to the fixed SR = 900 MW during light load periods in this 0 0.1 0.2 0.3 0.4 0.5 0.6 1 25 49 73 Tim R isk In de x PSO methd S.R.=1200MW Fig. 5. Hourly risk index under diff 97 121 145 e, hour S.R=1200MW load*0.2 erent spinning reserve settings. figure. This means an enhancement of system security or a reduction of outage cost during heavy load periods, and a security guarantee, both during light load hours and heavy load hours, can be achieved when the IPSO method is applied to determine the spinning reserve of power systems. Since the decrease in outage cost is greater then the increase in fuel costs during heavy load hours, the total social cost of the power system during the scheduling period can be minimized. Fig. 5 shows the hourly risk index (unit commitment risk, UCR) [2] in Fig. 3. The hourly risk index of fixed SR = 1200 MW is seen to be smaller than the results of 97 121 145 e, hour S.R.=900MW load*3.0e-5 erent spinning reserve settings. Table 5 Operating costs under different VOLL VOLL (NT$/kW h) Fuel cost (NT$) Outage cost (NT$) Total social cost (NT$) 135 3,024,405,000 299,162,000 3,323,567,000 162 3,042,060,000 338,007,000 3,380,067,000 190 3,064,942,000 374,949,000 3,439,891,000 225 3,065,043,000 424,702,000 3,489,745,000 trol. John Wiley & Sons Inc.; 1984. 3000 3500 4 ,0 00 e r T.-Y. Lee, C.-L. Chen / Energy Conversion and Management 48 (2007) 486–493 493 the IPSO method during light load hours; however, a larger risk index can be acceptable due to the lower risk during light load hours for the purpose of reducing fuel costs. This can be achieved by the IPSO method. On the contrary, a smaller hourly risk index is preferred during heavy load hours for the purpose of reducing the outage cost. This also can be reached by the IPSO method as shown in the results. In order to study the effect of VOLL on the selection of spinning reserve, the VOLL was changed from 135 NT$/ kW h to 225 NT$/kW h. The results listed in Table 5 show that all costs increase when VOLL is increased. Finally, the weighting factor w1 in Eq. (1) was varied to study how it affects the results of UCPR solution. Fig. 6 shows the results of this study. In this figure, TC means the total fuel costs, OC means the outage cost and TSC means the total social cost in Eq. (1). It is observed from this figure that all costs increase when w1 increases. The slope of the outage cost is a little greater then that of the fuel cost. From the test results shown above, the IPSO method seems to be a potential alternative in solving the UCPR problem. The developed computer program can also be applied as a useful tool for the power company in solving the UCPR problem. 7. Conclusions 0 500 1000 1500 2000 2500 0 0.2 0. Co st s, N T$ 1,0 00 TC Fig. 6. Effects of w1 on th This paper shows how a probabilistic reserve can be applied in the unit commitment problem to evaluate the spinning reserve requirement that can help power systems to overcome unscheduled generators outages and major load forecasting errors without load shedding. A highly complex problem of unit commitment with probabilistic reserve can be solved by the proposed IPSO method. The results are compared with the results of apply- ing the DP, GA and PSO methods. The test results demon- strate the effectiveness of IPSO in solving the unit commitment with probabilistic reserve problem. [2] Billinton R, Allan RN. Reliability evaluation of power systems. 2nd ed. Plenum Press; 1996. [3] Thalassinakis EJ, Dialynas EN. A Monte-Carlo simulation method for setting the under frequency load shedding relays and selecting the spinning reserve policy in autonomous power systems. IEEE Trans Power Syst 2004;19(4):2044–52. [4] Gooi HB, Mendes DP, Bel KRW, Kirschen DS. Optimal scheduling of spinning reserve. IEEE Trans Power Syst 1999;14(4):1485–92. [5] Moya OE. A spinning reserve, load shedding, and economic dispatch solution by Bender’s decomposition. IEEE Trans Power Syst 2005;20(1):348–88. [6] Li F, Chen C. Sizing a flexible spinning reserve level with artificial neural networks. In: IEEE power engineering society winter meeting, The presented method determines the number of on line units and spinning reserve level automatically based on the outage costs of the power system. The test results shown above reveal that the spinning reserve level determined by the proposed method is both economic and safe. The com- puter program developed in this paper can, therefore, be a powerful tool in power system operating analysis. Cur- rently, this program is being experimentally added into an energy management system as auxiliary software to sup- ply service to power companies. References [1] Wood AJ, Wollenberg BF. 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Unit commitment with probabilistic reserve: An IPSO approach Introduction Problem formulation and the objective function Particle swarm optimization Iteration particle swarm optimization Solution method and implementation of IPSO Numerical examples Conclusions References