An Improved Quantum-Behaved Particle Swarm Optimization Method for Economic Dispatch Problems with Multiple Fuel Options and Valve-Points Effects
Abstract
:1. Introduction
2. Formulation of the ED Problem
2.1. The ED Problem with Valve-Point Effects
2.2. ED Problem with Multiple Fuels and Valve-Point Effects
3. The Proposed SQPSO Algorithm
3.1. Conventional Particle Swarm Optimization
- and represent the position and velocity of individual i at generation t;
- w is the inertia weight parameter that controls the momentum of particles;
- c1 and c2 are positive constants, which balance the need for local and global search;
- rand() is a random number between 0 and 1.
3.2. Quantum-Behaved Particle Swarm Optimization
- (1)
- xij (t + 1) is denoted as the position of the jth dimension of the ith particle for the next generation t + 1.
- (2)
- Pij (t) is the local attractor to make sure SQPSO can converge, which is defined as follows:
- (3)
- In this paper, β is called the constriction-expansion coefficient, and it is linearly decreasing when the iteration increases:
- (4)
- u and k are two random numbers uniformly distributed in (0,1).
3.3. The Proposed Quantum-Behaved Particle Swarm Optimization
- (1)
- Initialize the population, which are generated randomly within the minimum and maximum output of each generator, using the following equations:
- (2)
- Constraint handling for real power balance. Since the individuals of the population are created randomly and with the evolution of particles, newly generated individual may violate the constraints. Therefore, it is important to keep all the individual variables within their feasible ranges. Hence, the following procedure is adopted by the SQPSO to modify the value of new generated variables to satisfy the power balance constraint.
- (3)
- Parameter setting. There are two parameters in SQPSO, one is the constriction-expansion coefficient which decreases from 1.0 to 0.5 linearly. Another parameter is the introduced selective probability (SP). In this paper, the SP for SQPSO increases from 0.5 to 0.8 linearly using the following equation:
- (4)
- Evaluate the objective function value of each particle.
- (5)
- Update pbest. Compare each particle’s objective function value with its pbest. If the current value is better than the pbest value, set the pbest value to the current value.
- (6)
- Update gbest. Determine best gbest of the swarm as the minimum pbest of all particles.
- (7)
- Calculate the Mbest, constriction-expansion coefficient β according to Equation (11) and Equation (12), respectively.
- (8)
- Calculate the local attractor according to the Selective probability operator proposed in this paper.
- (9)
- Update the particle’s position using Equation (9)
- (10)
- Check if the stop criterion satis fied?
- (11)
- If not, then go to step 2.
- (12)
- Else, the searching process is stopped.
4. Experimental Results
4.1. Benchmark Functions
Name | Function | Dim | Range | Opt |
---|---|---|---|---|
Sphere | 40 | [−100,100] | 0 | |
Jason | 40 | [−100,100] | 0 | |
Griewank | 40 | [−600,600] | 0 | |
Rosenbrock | 40 | [−2.048,2.048] | 0 | |
Rastrigrin | 40 | [−5.12,5.12] | 0 |
Function | f1 (Sphere) | f2 (Jason) | f3 (Griewank) | f4 (Rosenbrock) | f5 (Rastrigrin) | |
---|---|---|---|---|---|---|
Algorithm | Mean (Best) | Mean (Best) | Mean (Best) | Mean (Best) | Mean (Best) | |
EGA [11] | 2.743 × 10−10 (0) | 8.865 × 10−8 (3.748 × 10−22) | 1.042 × 10−4 (7.952 × 10−13) | 0.84 (6.537 × 10−4) | 2.257 (6.537 × 10−4) | |
DPSO [11] | 5.403 × 10−7(4.532 × 10−14) | 2.595 × 10−6 (1.173 × 10−12) | 1.322 × 10−3 (2.167 × 10−10) | 28.094 (1.150 × 10−2) | 28.826 (19.899) | |
HPSO [11] | 1.319 × 10−6 (2.824 × 10−10) | 6.735 × 10−3 (1.503 × 10−10) | 2.546 × 10−3 (5.136 × 10−9) | 28.995 (2.346 × 10−2) | 29.956 (15.393) | |
IPSO [11] | 1.524 × 10−7 (3.406 × 10−11) | 1.350 × 10−5 (2.107 × 10−10) | 2.224 × 10−3 (1.454 × 10−10) | 27.13 (2.339 × 10−2) | 31.906 (15.064) | |
IQPSO [11] | 1.085 × 10−23 (0) | 2.078 × 10−23 (0) | 3.221 × 10−7 (0) | 2.19 × 10−2 (2.717 × 10−9) | 0.521 (1.075 × 10−4) | |
PSO | 2.885 × 10−21 (1.774 × 10−23) | 1.4526 × 10−21 (4.413 × 10−24) | 8.0215 × 10−3 (0) | 56.1057 (12.4904) | 34.6046 (20.8941) | |
DE | 1.3727 × 10−47 (3.9244 × 10−49) | 1.0097 × 10−30 (0) | 3.4506 × 10−4 (0) | 12.4830 (6.6779) | 56.7802 (14.9244) | |
QPSO | 5.054 × 10−26 (7.333 × 10−31) | 5.3011 × 10−30(0) | 8.1 (0) | 48.4957 (25.2717) | 25.7895 (13.9294) | |
SQPSO | 6.5759 × 10−74 (1.8122 × 10−89) | (0) (0) | 2.217 × 10−7 (0) | 32.68016 (14.7115) | 13.7105 (3.9798) |
4.2. ED Problem with Valve-Point Effects
Methods | Generation cost($/H) | Standard Deviation | ||
---|---|---|---|---|
Minimum | Mean | Maximum | ||
IFEP [1] | 122,624.35 | 123,382 | 125,740.63 | NR |
GA-PS-SQP [22] | 121,458.14 | 122,039 | NR | NR |
PC-PSO [23] | 121,767.90 | 122,461.30 | 122,867.55 | NR |
SOH-PSO [23] | 121,501.14 | 121,853.57 | 122,446.3 | NR |
NPSO [24] | 121,704.74 | 122,221.37 | 122,995.10 | NR |
NPSO-LRS [24] | 121,664.43 | 122,209.32 | 122,981.59 | NR |
PSO-GM [25] | 121,845.98 | 122,398.38 | 123,219.22 | 258.44 |
CBPSO-RVM [25] | 121,555.32 | 122,281.14 | 123,094.98 | 259.99 |
ICA-PSO [26] | 121,422.17 | 121,428.14 | 121,453.56 | NR |
ACO [5] | 121,532.41 | 121,606.45 | 121,679.64 | 45.58 |
APSO(2) [27] | 121,663.52 | 122,153.67 | 122,912.40 | NR |
HDE [28] | 121,813.26 | 122,705.66 | NR | NR |
ST-HDE [28] | 121,698.51 | 122,304.30 | NR | NR |
IQPSO [21] | 121,448.21 | 122,225.07 | NR | NR |
FCASO [30] | 121,516.47 | 122,082.59 | NR | NR |
CASO [30] | 121,865.63 | 122,100.74 | NR | NR |
CPSO-SQP [31] | 121,458.54 | 122,028.16 | NR | NR |
CPSO [31] | 121,865.23 | 122,100.87 | NR | NR |
DE | 121,805.56 | 122,142.97 | 122,466.75 | 151.88 |
PSO | 121,956.18 | 122,459.36 | 122,785.73 | 209.12 |
QPSO | 121,487.27 | 121,750.48 | 121,991.99 | 111.68 |
CQPSO | 121,463.39 | 121,732.98 | 121,778.74 | 79.38 |
SQPSO | 121,434.41 | 121,723.22 | 121,881.51 | 104.29 |
Unit | Methods | ||||
---|---|---|---|---|---|
DE | PSO | QPSO | CQPSO | SQPSO | |
P1 | 111.8012 | 113.9945 | 113.6426 | 113.9999 | 110.9173 |
P2 | 111.5734 | 110.9343 | 111.9581 | 113.9999 | 111.7807 |
P3 | 95.79661 | 100.748 | 97.56082 | 120.0000 | 97.56128 |
P4 | 182.4958 | 179.1588 | 179.7457 | 179.7333 | 179.7005 |
P5 | 87.27856 | 97.0000 | 88.53738 | 96.9999 | 93.37496 |
P6 | 140.0000 | 140.0000 | 139.9981 | 140.0000 | 139.9862 |
P7 | 300.0000 | 300.0000 | 299.989 | 300.0000 | 259.8548 |
P8 | 285.2077 | 300.0000 | 284.9879 | 299.9999 | 284.9466 |
P9 | 286.9856 | 299.9040 | 284.7968 | 293.3932 | 284.5976 |
P10 | 130.0000 | 130.0000 | 130.0093 | 130.0000 | 130.0493 |
P11 | 94.25143 | 94.0000 | 94.02522 | 94.0000 | 168.807 |
P12 | 94.61699 | 94.0000 | 94.0286 | 94.0000 | 94.00315 |
P13 | 125.7718 | 125.0000 | 125.0323 | 125.0000 | 214.7713 |
P14 | 393.1819 | 393.9392 | 394.2728 | 394.2794 | 394.2986 |
P15 | 395.1001 | 394.1116 | 394.2987 | 394.2794 | 304.61 |
P16 | 393.7253 | 304.3765 | 394.3071 | 304.5196 | 394.2632 |
P17 | 487.6391 | 500.0000 | 489.3179 | 489.2794 | 489.363 |
P18 | 491.819 | 490.6004 | 489.2953 | 489.2795 | 489.5688 |
P19 | 512.8806 | 513.8928 | 511.3082 | 511.2794 | 511.2797 |
P20 | 511.7995 | 514.1406 | 511.3473 | 511.2794 | 511.3193 |
P21 | 524.2502 | 524.3505 | 523.3044 | 523.2796 | 523.2616 |
P22 | 523.9075 | 523.4735 | 523.3182 | 523.2796 | 523.3642 |
P23 | 519.8336 | 529.2841 | 523.3638 | 523.2796 | 523.2587 |
P24 | 527.6248 | 547.3133 | 523.3677 | 550.0000 | 523.3996 |
P25 | 523.9776 | 522.9096 | 523.2928 | 523.2795 | 523.2836 |
P26 | 523.2693 | 524.9206 | 523.3083 | 523.2798 | 523.2817 |
P27 | 10.3912 | 10.0000 | 10.01133 | 10.0000 | 10.00975 |
P28 | 10.0000 | 10.0000 | 10.08587 | 10.0000 | 10.0344 |
P29 | 10.0335 | 10.0000 | 10.00228 | 10.0000 | 10.00645 |
P30 | 92.73803 | 91.53567 | 90.21066 | 96.9999 | 88.52085 |
P31 | 187.1519 | 190.0000 | 189.9984 | 190.0000 | 189.9972 |
P32 | 189.9415 | 190.0000 | 189.9968 | 190.0000 | 189.9834 |
P33 | 189.4094 | 190.0000 | 189.9988 | 190.0000 | 189.9822 |
P34 | 197.3705 | 199.9374 | 199.9794 | 199.9999 | 165.321 |
P35 | 199.2062 | 198.4492 | 199.9942 | 200.0000 | 199.9666 |
P36 | 198.9157 | 200.0000 | 199.9942 | 200.0000 | 200.0000 |
P37 | 109.5043 | 110.0000 | 110.0000 | 110.0000 | 110.0000 |
P38 | 110.0000 | 110.0000 | 109.9926 | 110.0000 | 109.9984 |
P39 | 108.1849 | 110.0000 | 109.9915 | 110.0000 | 109.992 |
P40 | 512.3655 | 512.0254 | 511.3299 | 511.2794 | 511.2849 |
Total Demand | 10,500 | 10,500 | 10,500 | 10,500 | 10,500 |
Total Cost | 121,805.5647 | 121,956.1827 | 121,487.2762 | 121,463.3942 | 121,434.4071 |
4.3. The ED Problem with Multi-Fuel Option and Valve-Point Effects
Methods | Generation cost ($/H) | Standard Deviation | Average CPU times | ||
---|---|---|---|---|---|
Minimum | Mean | Maximum | |||
CGA_MU [2] | 624.7193 | 627.6087 | 633.8652 | NR | 26.64 |
IGA_MU [2] | 624.5178 | 625.8692 | 630.8705 | NR | 7.32 |
ACO [5] | 623.9000 | 624.3500 | 624.7800 | NR | 8.35 |
ED-DE [32] | 623.8290 | 623.8807 | 623.8894 | NR | NR |
ARCGA [33] | 623.8281 | 623.8495 | 623.8814 | NR | NR |
NPSO [24] | 624.1624 | 625.2180 | 627.4237 | NR | NR |
NPSO-LRS [24] | 624.1273 | 624.9985 | 626.9981 | NR | NR |
PSO-GM [25] | 624.3050 | 624.6749 | 625.0854 | 0.1580 | NR |
CBPSO-RVM [25] | 623.9588 | 624.0816 | 624.2930 | 0.0576 | NR |
APSO [27] | 624.0145 | 624.8185 | 627.3049 | NR | 0.52 |
GA [34] | 624.5050 | 624.7419 | 624.8169 | 0.1005 | 18.3 |
TSA [34] | 624.3078 | 635.0623 | 624.8285 | 1.1593 | 9.71 |
DSPSO–TSA [34] | 623.8375 | 623.8625 | 623.9001 | 0.0106 | 3.44 |
DE | 623.9280 | 624.0068 | 624.0653 | 0.0271 | 0.625 |
PSO | 624.0120 | 624.2055 | 624.4376 | 0.0889 | 0.308 |
QPSO | 623.8766 | 623.9639 | 624.4163 | 0.0688 | 0.315 |
CQPSO | 623.8476 | 623.8652 | 623.8885 | 0.0151 | 0.318 |
SQPSO | 623.8319 | 623.8440 | 623.8605 | 0.0107 | 0.324 |
Unit | CQPSO | DE | PSO | QPSO | SQPSO | |||||
---|---|---|---|---|---|---|---|---|---|---|
Output | Fuel | Output | Fuel | Output | Fuel | Output | Fuel | Output | Fuel | |
(MW) | type | (MW) | type | (MW) | type | (MW) | type | (MW) | type | |
P1 | 217.567 | 2 | 220.8058 | 2 | 220.8058 | 2 | 218.587 | 2 | 218.5939 | 2 |
P2 | 211.7117 | 1 | 211.7154 | 1 | 211.7154 | 1 | 210.4723 | 1 | 211.2166 | 1 |
P3 | 279.6489 | 1 | 280.7032 | 1 | 280.7032 | 1 | 280.7087 | 1 | 281.6653 | 1 |
P4 | 240.5800 | 3 | 239.7713 | 3 | 239.7713 | 3 | 239.3708 | 3 | 238.9676 | 3 |
P5 | 276.3749 | 1 | 277.2203 | 1 | 277.2203 | 1 | 279.6347 | 1 | 279.9345 | 1 |
P6 | 239.6394 | 3 | 238.9671 | 3 | 238.9671 | 3 | 240.7144 | 3 | 239.2363 | 3 |
P7 | 290.0985 | 1 | 289.0121 | 1 | 289.0121 | 1 | 290.1244 | 1 | 287.7275 | 1 |
P8 | 240.8488 | 3 | 240.175 | 3 | 240.175 | 3 | 239.6396 | 3 | 239.6394 | 3 |
P9 | 427.6622 | 3 | 425.4145 | 3 | 425.4145 | 3 | 423.8487 | 3 | 427.1502 | 3 |
P10 | 275.8686 | 1 | 276.2151 | 1 | 276.2151 | 1 | 276.8994 | 1 | 275.8686 | 1 |
Pd | 2,700 | 2,700 | 2,700 | 2,700 | 2,700 | |||||
Total Cost | 623.8476 | 623.928 | 624.012 | 623.8766 | 623.8319 |
Demand | Method | Generation cost ($/H) | Standard Deviation | Average CPU times | ||
---|---|---|---|---|---|---|
Minimum | Mean | Maximum | ||||
2400 | DE | 481.9030 | 481.9527 | 482.0231 | 0.0285 | 0.4781 |
PSO | 482.0807 | 484.1717 | 491.5540 | 2.5598 | 0.3625 | |
QPSO | 481.9235 | 483.4540 | 492.6059 | 2.2612 | 0.3562 | |
CQPSO | 481.7469 | 481.7711 | 481.7974 | 0.0180 | 0.3683 | |
SQPSO | 481.7320 | 481.7440 | 481.7591 | 0.0068 | 0.3390 | |
2500 | DE | 526.4154 | 526.4771 | 526.5379 | 0.0244 | 0.5156 |
PSO | 526.4849 | 527.5594 | 535.1762 | 1.3761 | 0.3578 | |
QPSO | 526.3758 | 527.5720 | 534.9611 | 1.4797 | 0.3328 | |
CQPSO | 526.2537 | 526.2839 | 526.3229 | 0.0187 | 0.3453 | |
SQPSO | 526.2447 | 526.2556 | 526.2897 | 0.0079 | 0.3500 | |
2600 | DE | 574.5489 | 574.6371 | 574.9653 | 0.0916 | 0.5984 |
PSO | 574.6194 | 576.0185 | 589.1900 | 2.5451 | 0.3515 | |
QPSO | 574.5857 | 575.7198 | 589.1281 | 2.0467 | 0.3315 | |
CQPSO | 574.4492 | 574.6538 | 574.7928 | 0.1439 | 0.3576 | |
SQPSO | 574.3866 | 574.5076 | 574.7659 | 0.1640 | 0.3484 |
5. Conclusions
Acknowledgments
References
- Sinha, N.; Chakrabarti, R.; Chattopadhyay, P.K. Evolutionary programming techniques for economic load dispatch. IEEE Trans. Evolut. Comput. 2003, 7, 83–94. [Google Scholar] [CrossRef]
- Chiang, C.L. Improved genetic algorithm for power economic dispatch of units with valve-point effects and multiple fuels. IEEE Trans. Power Syst. 2005, 20, 1690–1699. [Google Scholar] [CrossRef]
- Park, J.B.; Lee, K.S.; Shin, J.R. A particle swarm optimization for economic dispatch with non-smooth cost functions. IEEE Trans. Power Syst. 2005, 20, 34–42. [Google Scholar] [CrossRef]
- Balamurugan, R.; Subramanian, S. Hybrid integer coded differential evolution-dynamic programming approach for economic load dispatch with multiple fuel options. Energy Convers. Manag. 2008, 49, 608–614. [Google Scholar] [CrossRef]
- Pothiya, S.; Ngamroo, I.; Kongprawechnon, W. Ant colony optimisation for economic dispatch problem with non-smooth cost functions. Int. J. Electr. Power Energy Syst. 2010, 32, 478–487. [Google Scholar] [CrossRef]
- Basu, M. A simulated annealing-based goal-attainment method for economic emission load dispatch of fixed head hydrothermal power systems. Electr. Power Energy Syst. 2005, 27, 147–153. [Google Scholar] [CrossRef]
- Kennedy, J.; Eberhart, R. Particle Swarm Optimization. In Proceedings of the IEEE International Conference on Neural Networks Proceedings, Perth, WA, Australia, 27 November–1 December 1995; pp. 1942–1948.
- Sun, J.; Xu, W.B.; Feng, B. A Global Search Strategy of Quantum-Behaved Particle Swarm Optimization. In Proceedings of the IEEE Conference on Cybernetics and Intelligent Systems, Singapore, 1–3 December 2004; pp. 111–116.
- Fang, W. A review of quantum-behaved particle swarm optimization. IETE Tech. Rev. 2010, 27, 336–348. [Google Scholar] [CrossRef]
- Sun, J.; Wu, X.J.; Palade, V.; Fang, W.; Lai, C.-H.; Xu, W.B. Convergence analysis and improvements of quantum-behaved particle swarm optimization. Inf. Sci. 2012, 193, 81–103. [Google Scholar] [CrossRef]
- Sun, J.; Fang, W.; Palade, V.; Wu, X.J.; Xu, W.B. Quantum-behaved particle swarm optimization with Gaussian distributed local attractor point. Appl. Math. Comput. 2011, 218, 3763–3775. [Google Scholar] [CrossRef]
- Sun, J.; Fang, W.; Wu, X.J.; Palade, V.; Xu, W.B. Quantum−behaved particle swarm optimization: Analysis of the individual particle’s behavior and parameter selection. Evolut. Comput. 2012, 20, 349–393. [Google Scholar] [CrossRef]
- Luitel, B.; Venayagamoorthy, G.K. Particle swarm optimization with quantum infusion for system identification. Eng. Appl. Artif. Intell. 2010, 23, 635–649. [Google Scholar] [CrossRef]
- Sun, J.; Liu, J.; Xu, W. Using quantum-behaved particle swarm optimization algorithm to solve non-linear programming problems. Int. J. Comp. Math. 2007, 84, 261–272. [Google Scholar] [CrossRef]
- Sun, C.F.; Lu, S.F. Short-term combined economic emission hydrothermal scheduling using improved quantum-behaved particle swarm optimization. Expert Syst. Appl. 2010, 37, 4232–4241. [Google Scholar] [CrossRef]
- Leandro dos, S.C.; Viviana, C.M. Particle swarm approach based on quantum mechanics and harmonic oscillator potential well for economic load dispatch with valve-point effects. Energy Convers. Manag. 2008, 49, 3080–3085. [Google Scholar] [CrossRef]
- Chakraborty, S.; Senjyu, T.; Yona, A.; Saber, A.Y.; Funabashi, T. Solving economic load dispatch problem with valve-point effects using a hybrid quantum mechanics inspired particle swarm optimization. IET Gener. Transm. Distrib. 2011, 5, 1042–1052. [Google Scholar] [CrossRef]
- Park, J.B.; Jeong, Y.W.; Shin, J.R.; Lee, K.Y. An improved particle swarm optimization for nonconvex economic dispatch problems. IEEE Trans. Power Syst. 2010, 25, 156–166. [Google Scholar] [CrossRef]
- Shi, Y.; Eberhart, R. Empirical Study of Particle Swarm Optimization. In Proceedings of the 1999 Congress on Evolutionary Computation (CEC 99), Washington, DC, USA, 6–9 July 1999; Volume 3, pp. 1945–1950.
- Coelho, L.S.; Mariani, V.C. Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with valve-point effect. IEEE Trans. Power Syst. 2006, 21, 989–996. [Google Scholar] [CrossRef]
- Meng, K. Quantum-inspired particle swarm optimization for valve-point economic load dispatch. IEEE Trans. Power Syst. 2010, 25, 215–222. [Google Scholar] [CrossRef]
- Alsumait, J.S.; Sykulski, J.K. A hybrid GA-PS-SQP method to solve power system valve-point economic dispatch problems. Appl. Energy 2010, 87, 1773–1781. [Google Scholar] [CrossRef]
- Chaturvedi, K.T.; Pandit, M.; Srivastava, L. Self-organizing hierarchical particle swarm optimization for nonconvex economic dispatch. IEEE Trans. Power Syst. 2008, 23, 1079–1087. [Google Scholar] [CrossRef]
- Selvakumar, A.I.; Thanushkodi, K. A new particle swarm optimization solution to nonconvex economic dispatch problems. IEEE Trans. Power Syst. 2007, 22, 42–51. [Google Scholar] [CrossRef]
- Lu, H.Y. Experimental study of a new hybrid PSO with mutation for economic dispatch with non-smooth cost function. Electr. Power Energy Syst. 2010, 32, 921–935. [Google Scholar] [CrossRef]
- Vlachogiannis, J.K.; Lee, K.Y. Economic load dispatch—A comparative study on heuristic optimization techniques with an improved coordinated aggregation-based PSO. IEEE Trans. Power Syst. 2009, 24, 991–1001. [Google Scholar] [CrossRef]
- Selvakumar, A.I.; Thanushkodi, K. Anti-predatory particle swarm optimization: Solution to nonconvex economic dispatch problems. Electr. Power Syst. Res. 2010, 78, 2–10. [Google Scholar] [CrossRef]
- Wang, S.K.; Chiou, J.P.; Liu, C.W. Non-smooth/non-convex economic dispatch by a novel hybrid differential evolution algorithm. IET Gener. Transm. Distrib. 2007, 1, 793–803. [Google Scholar] [CrossRef]
- Lee, K.Y.; Sode-Yome, A.; Park, J.H. Adaptive hopfield neural networks for economic load dispatch. IEEE Trans. Power Syst. 1998, 13, 519–525. [Google Scholar] [CrossRef]
- Cai, J.J.; Li, Q.; Li, L.X.; Peng, H.P.; Yang, Y.X. A fuzzy adaptive chaotic ant swarm optimization for economic dispatch. Electr. Power Energy Syst. 2012, 34, 154–160. [Google Scholar] [CrossRef]
- Cai, J.J.; Li, Q.; Li, L.X.; Peng, H.P.; Yang, Y.X. A hybrid CPSO-SQP method for economic dispatch considering the valve-point effects. Energy Convers. Manag. 2012, 53, 175–181. [Google Scholar] [CrossRef]
- Wang, Y.; Li, B.; Weise, T. Estimation of distribution and differential evolution cooperation for large scale economic load dispatch optimization of power systems. Inf. Sci. 2010, 180, 2405–2420. [Google Scholar] [CrossRef]
- Amjady, N.; Nasiri-Rad, H. Nonconvex economic dispatch with AC constraints by a new real coded genetic algorithm. IEEE Trans. Power Syst. 2009, 24, 1489–1502. [Google Scholar] [CrossRef]
- Khamsawang, S.; Jiriwibhakorn, S. DSPSO-TSA for economic dispatch problem with nonsmooth and noncontinuous cost functions. Energy Convers. Manag. 2010, 51, 365–375. [Google Scholar] [CrossRef]
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Share and Cite
Niu, Q.; Zhou, Z.; Zhang, H.-Y.; Deng, J. An Improved Quantum-Behaved Particle Swarm Optimization Method for Economic Dispatch Problems with Multiple Fuel Options and Valve-Points Effects. Energies 2012, 5, 3655-3673. https://doi.org/10.3390/en5093655
Niu Q, Zhou Z, Zhang H-Y, Deng J. An Improved Quantum-Behaved Particle Swarm Optimization Method for Economic Dispatch Problems with Multiple Fuel Options and Valve-Points Effects. Energies. 2012; 5(9):3655-3673. https://doi.org/10.3390/en5093655
Chicago/Turabian StyleNiu, Qun, Zhuo Zhou, Hong-Yun Zhang, and Jing Deng. 2012. "An Improved Quantum-Behaved Particle Swarm Optimization Method for Economic Dispatch Problems with Multiple Fuel Options and Valve-Points Effects" Energies 5, no. 9: 3655-3673. https://doi.org/10.3390/en5093655