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doi: 10.3934/naco.2021042
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A novel differential evolution algorithm for economic power dispatch problem

Pooja

Department of Electronics and Communication, University of Allahabad, Prayagraj, India

Received  March 2021 Revised  September 2021 Early access October 2021

Fund Project: The paper is handled by Gerhard-Wilhelm Weber as guest editor

In power systems, Economic Power dispatch Problem (EPP) is an influential optimization problem which is a highly non-convex and non-linear optimization problem. In the current study, a novel version of Differential Evolution (NDE) is used to solve this particular problem. NDE algorithm enhances local and global search capability along with efficient utilization of time and space by making use of two elite features: selfadaptive control parameter and single population structure. The combined effect of these concepts improves the performance of Differential Evolution (DE) without compromising on quality of the solution and balances the exploitation and exploration capabilities of DE. The efficiency of NDE is validated by evaluating on three benchmark cases of the power system problem having constraints such as power balance and power generation along with nonsmooth cost function and is compared with other optimization algorithms. The Numerical outcomes uncovered that NDE performed well for all the benchmark cases and maintained a trade-off between convergence rate and efficiency.

Keywords: Economic Power dispatch Problem, Control Parameters, Constraint Handling, Population Structure, Differential Evolution.
Mathematics Subject Classification: Primary: 68T20; Secondary: 90-08.
Citation: Pooja. A novel differential evolution algorithm for economic power dispatch problem. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2021042
References:
[1]

O. Abedinia, N. Amjady, A. Ghasemi and Z. Hejrati, Solution of economic load dispatch problem via hybrid particle swarm optimization with time–varying acceleration coefficients and bacteria foraging algorithm techniques, International Transactions on Electrical Energy Systems, 23(8) (2012), 1504-1522. 

[2]

B. R. Adarsh, T. Raghunathan, T. Jayabarathi and X. S. Yang, Economic dispatch using chaotic bat algorithm, Energy, 96 (2016), 666-675. 

[3]

W. M. Ali and H. Z. Sabry, Constrained optimization based on modified differential evolution algorithm, Information Sciences, 194 (2012), 171-208. 

[4]

B. V. Babu and R. Angira, Modified differential evolution (MDE) for optimization of nonlinear chemical processes, Comput. Chem. Engin, 30 (2006), 989-1002. 

[5]

A. Bhattacharya and P. K. Chattopadhyay, Solving complex economic load dispatch problems using biogeography–based optimization, Expert Systems with Applications, 37 (2010a), 3605-3615. 

[6]

A. Biswas, S. Dasgupta, B. K. Panigrahi, V. R. Pandi, S. Das, A. Abraham and Y. Badr, Economic load dispatch using a chemotactic differential evolution algorithm, 4th International Conference on Hybrid Artificial Intelligent Systems, LNAI, 5572 (2009), 252-260. 

[7]

J. Brest, V. Zumer and M. S. Maucecc, Control parameters in self-adaptive differential evolution, Bioinspired Optimization Methods and Their Applications, (2006), 35–44.

[8]

K. T. Chaturvedi, M. Pandit and L. Srivastava, Self–organizing hierarchical particle swarm optimization for nonconvex economic dispatch, IEEE Transactions on Power Systems, 23 (2008), 1079-1087. 

[9]

B. H. Choudhary and S. Rahman, A review of recent advances in economic dispatch, IEEE Trans. on Power System, 5 (1990), 1248-1259. 

[10]

S. Elsayed, M. F. Zaman and R. Sarker, Automated differential evolution for solving dynamic economic dispatch problems, Intelligent and Evolutionary Systems, Proceedings in Adaptation, Learning and Optimization, 5 (2016), 357-369. 

[11]

Z. L. Gaing, Particle swarm optimization to solving the economic dispatch considering generator constraints, IEEE Trans. on Power Systems, 18 (2003), 1187-1195. 

[12]

A. Goli, H. K. Zareh, R. Tavakkoli–Moghaddam and A. Sadeghieh, Application of robust optimization for a product portfolio problem using an invasive weed optimization algorithm, Numerical Algebra, Control & Optimization, 9 (2019), 187-209.  doi: 10.3934/naco.2019014.

[13]

A. Goli, H. Khademi–Zare, R. Tavakkoli–Moghaddam, A. Sadeghieh, M. Sasanian and R. M. Kordestanizadeh, An integrated approach based on artificial intelligence and novel meta–heuristic algorithms to predict demand for dairy products: a case study, Network: Computation in Neural Systems, 32 (2021), 1-35. 

[14]

A. Goli, H. K. Zareh, R. Tavakkoli–Moghaddam and A. Sadeghieh, A comprehensive model of demand prediction based on hybrid artificial intelligence and metaheuristic algorithms: A case study in dairy industry, Journal of Industrial and Systems Engineering, 11 (2018), 190-203. 

[15]

D. He, F. Wang and Z. Mao, A hybrid genetic algorithm approach based on differential evolution for economic dispatch with valve-point effect, International Journal of Electrical Power and Energy Systems, 30 (2008), 31-38. 

[16]

N. T. Hung, N. Hung, P. D. V. Nguyen and D. T. Viet, Application of improved differential evolution algorithm for economic and emission dispatch of thermal power generation plants, Proceedings of the 3rd International Conference on Machine Learning and Soft Computing, (2019), 93–98.

[17]

J. O. Kim, D. J. Shin, J. N. Park and C. Singh, Atavistic genetic algorithm for economic dispatch with valve point effect, Electric Power Systems Research, 62 (2002), 201-207. 

[18]

P. Kumar, M. Pant and V. P. Singh, Two self adaptive variants of differential evolution algorithm for global optimization, Int. J. of Appl. Math. and Mech., 8 (2012), 22-34. 

[19]

J. Liu and J. Lampinen, A fuzzy adaptive differential evolution algorithm, Soft Computing, 9 (2005), 448-462. 

[20]

R. Lotfi, N. Mardani and G. W. Weber, Robust bi–level programming for renewable energy location, International Journal of Energy Research, 45 (2021), 7521-7534. 

[21]

R. Lotfi, Z. Yadegari, S. H. Hosseini, A. H. Khameneh, E. B. Tirkolaee and G. W. Weber, A robust time–cost–quality–energy–environment trade–off with resource–constrained in project management: A case study for a bridge construction project, Journal of Industrial & Management Optimization, 13 (2020).

[22]

R. Lotfi, Y. Z. Mehrjerdi, M. S. Pishvaee, A. Sadeghieh and G. W. Weber, A robust optimization model for sustainable and resilient closed–loop supply chain network design considering conditional value at risk, Numerical Algebra, Control & Optimization, 11 (2021), 221-253.  doi: 10.3934/naco.2020023.

[23]

E. Mezura–Montes, M. E. Miranda-Varela and R. C. Gómez-Ramón, Differential evolution in constrained numerical optimization: an empirical study, Information Sciences, 180 (2010), 4223-4262.  doi: 10.1016/j.ins.2010.07.023.

[24]

V. R. Pandi, B. K. Panigrahi, A. Mohapatra and M. K. Mallick, Economic load dispatch solution by improved harmony search with wavelet mutation, International Journal of Computational Science and Engineering, 6 (2011), 122-131. 

[25]

L. Ping, J. Sun and Q. Chen, Solving Power economic dispatch problem with a novel quantum–behaved particle swarm optimization algorithm, Mathematical Problems in Engineering, 2020 (2020), 1-11. 

[26]

Po oja, P. Chaturvedi, P. Kumar and A. Tomar, A novel differential evolution approach for constraint optimisation, Int. J. Bio–Inspired Computation, 12 (2018), 254-265. 

[27]

B. Y. Qu, Y. S. Zhu, Y. C. Jiao, M. Y. Wu, P. N. Suganthan and J. J. Liang, A survey on multi–objective evolutionary algorithms for the solution of the environmental/economic dispatch problems, Swarm and Evolutionary Computation, 38 (2018), 1-11. 

[28]

R. Rahmani, M. F. Othman, R. Yusof and M. Khalid, Solving economic dispatch problem using particle swarm optimization by an evolutionary technique for initializing particles, Journal of Theoretical and Applied Information Technology, 46 (2012), 526-536. 

[29]

A. Safari and H. Shayeghi, Iteration particle swarm optimization procedure for economic load dispatch with generator constraints, Expert Systems with Applications, 38 (2011), 6043-6048. 

[30]

N. Sinha, R. Chakrabarti and P. K. Chattopadhyay, Evolutionary programming techniques for economic load dispatch, IEEE Trans. Evol. Comput., 7 (2003), 83-94. 

[31]

R. Storn and K. Price, Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359.  doi: 10.1023/A:1008202821328.

[32]

M. F. Zaman, S. M. Elsayed, T. Ray and R. A. Sarker, Evolutionary algorithms for dynamic economic dispatch problems, IEEE Transactions on Power Systems, 31 (2016), 1486-1495. 

show all references

References:
[1]

O. Abedinia, N. Amjady, A. Ghasemi and Z. Hejrati, Solution of economic load dispatch problem via hybrid particle swarm optimization with time–varying acceleration coefficients and bacteria foraging algorithm techniques, International Transactions on Electrical Energy Systems, 23(8) (2012), 1504-1522. 

[2]

B. R. Adarsh, T. Raghunathan, T. Jayabarathi and X. S. Yang, Economic dispatch using chaotic bat algorithm, Energy, 96 (2016), 666-675. 

[3]

W. M. Ali and H. Z. Sabry, Constrained optimization based on modified differential evolution algorithm, Information Sciences, 194 (2012), 171-208. 

[4]

B. V. Babu and R. Angira, Modified differential evolution (MDE) for optimization of nonlinear chemical processes, Comput. Chem. Engin, 30 (2006), 989-1002. 

[5]

A. Bhattacharya and P. K. Chattopadhyay, Solving complex economic load dispatch problems using biogeography–based optimization, Expert Systems with Applications, 37 (2010a), 3605-3615. 

[6]

A. Biswas, S. Dasgupta, B. K. Panigrahi, V. R. Pandi, S. Das, A. Abraham and Y. Badr, Economic load dispatch using a chemotactic differential evolution algorithm, 4th International Conference on Hybrid Artificial Intelligent Systems, LNAI, 5572 (2009), 252-260. 

[7]

J. Brest, V. Zumer and M. S. Maucecc, Control parameters in self-adaptive differential evolution, Bioinspired Optimization Methods and Their Applications, (2006), 35–44.

[8]

K. T. Chaturvedi, M. Pandit and L. Srivastava, Self–organizing hierarchical particle swarm optimization for nonconvex economic dispatch, IEEE Transactions on Power Systems, 23 (2008), 1079-1087. 

[9]

B. H. Choudhary and S. Rahman, A review of recent advances in economic dispatch, IEEE Trans. on Power System, 5 (1990), 1248-1259. 

[10]

S. Elsayed, M. F. Zaman and R. Sarker, Automated differential evolution for solving dynamic economic dispatch problems, Intelligent and Evolutionary Systems, Proceedings in Adaptation, Learning and Optimization, 5 (2016), 357-369. 

[11]

Z. L. Gaing, Particle swarm optimization to solving the economic dispatch considering generator constraints, IEEE Trans. on Power Systems, 18 (2003), 1187-1195. 

[12]

A. Goli, H. K. Zareh, R. Tavakkoli–Moghaddam and A. Sadeghieh, Application of robust optimization for a product portfolio problem using an invasive weed optimization algorithm, Numerical Algebra, Control & Optimization, 9 (2019), 187-209.  doi: 10.3934/naco.2019014.

[13]

A. Goli, H. Khademi–Zare, R. Tavakkoli–Moghaddam, A. Sadeghieh, M. Sasanian and R. M. Kordestanizadeh, An integrated approach based on artificial intelligence and novel meta–heuristic algorithms to predict demand for dairy products: a case study, Network: Computation in Neural Systems, 32 (2021), 1-35. 

[14]

A. Goli, H. K. Zareh, R. Tavakkoli–Moghaddam and A. Sadeghieh, A comprehensive model of demand prediction based on hybrid artificial intelligence and metaheuristic algorithms: A case study in dairy industry, Journal of Industrial and Systems Engineering, 11 (2018), 190-203. 

[15]

D. He, F. Wang and Z. Mao, A hybrid genetic algorithm approach based on differential evolution for economic dispatch with valve-point effect, International Journal of Electrical Power and Energy Systems, 30 (2008), 31-38. 

[16]

N. T. Hung, N. Hung, P. D. V. Nguyen and D. T. Viet, Application of improved differential evolution algorithm for economic and emission dispatch of thermal power generation plants, Proceedings of the 3rd International Conference on Machine Learning and Soft Computing, (2019), 93–98.

[17]

J. O. Kim, D. J. Shin, J. N. Park and C. Singh, Atavistic genetic algorithm for economic dispatch with valve point effect, Electric Power Systems Research, 62 (2002), 201-207. 

[18]

P. Kumar, M. Pant and V. P. Singh, Two self adaptive variants of differential evolution algorithm for global optimization, Int. J. of Appl. Math. and Mech., 8 (2012), 22-34. 

[19]

J. Liu and J. Lampinen, A fuzzy adaptive differential evolution algorithm, Soft Computing, 9 (2005), 448-462. 

[20]

R. Lotfi, N. Mardani and G. W. Weber, Robust bi–level programming for renewable energy location, International Journal of Energy Research, 45 (2021), 7521-7534. 

[21]

R. Lotfi, Z. Yadegari, S. H. Hosseini, A. H. Khameneh, E. B. Tirkolaee and G. W. Weber, A robust time–cost–quality–energy–environment trade–off with resource–constrained in project management: A case study for a bridge construction project, Journal of Industrial & Management Optimization, 13 (2020).

[22]

R. Lotfi, Y. Z. Mehrjerdi, M. S. Pishvaee, A. Sadeghieh and G. W. Weber, A robust optimization model for sustainable and resilient closed–loop supply chain network design considering conditional value at risk, Numerical Algebra, Control & Optimization, 11 (2021), 221-253.  doi: 10.3934/naco.2020023.

[23]

E. Mezura–Montes, M. E. Miranda-Varela and R. C. Gómez-Ramón, Differential evolution in constrained numerical optimization: an empirical study, Information Sciences, 180 (2010), 4223-4262.  doi: 10.1016/j.ins.2010.07.023.

[24]

V. R. Pandi, B. K. Panigrahi, A. Mohapatra and M. K. Mallick, Economic load dispatch solution by improved harmony search with wavelet mutation, International Journal of Computational Science and Engineering, 6 (2011), 122-131. 

[25]

L. Ping, J. Sun and Q. Chen, Solving Power economic dispatch problem with a novel quantum–behaved particle swarm optimization algorithm, Mathematical Problems in Engineering, 2020 (2020), 1-11. 

[26]

Po oja, P. Chaturvedi, P. Kumar and A. Tomar, A novel differential evolution approach for constraint optimisation, Int. J. Bio–Inspired Computation, 12 (2018), 254-265. 

[27]

B. Y. Qu, Y. S. Zhu, Y. C. Jiao, M. Y. Wu, P. N. Suganthan and J. J. Liang, A survey on multi–objective evolutionary algorithms for the solution of the environmental/economic dispatch problems, Swarm and Evolutionary Computation, 38 (2018), 1-11. 

[28]

R. Rahmani, M. F. Othman, R. Yusof and M. Khalid, Solving economic dispatch problem using particle swarm optimization by an evolutionary technique for initializing particles, Journal of Theoretical and Applied Information Technology, 46 (2012), 526-536. 

[29]

A. Safari and H. Shayeghi, Iteration particle swarm optimization procedure for economic load dispatch with generator constraints, Expert Systems with Applications, 38 (2011), 6043-6048. 

[30]

N. Sinha, R. Chakrabarti and P. K. Chattopadhyay, Evolutionary programming techniques for economic load dispatch, IEEE Trans. Evol. Comput., 7 (2003), 83-94. 

[31]

R. Storn and K. Price, Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359.  doi: 10.1023/A:1008202821328.

[32]

M. F. Zaman, S. M. Elsayed, T. Ray and R. A. Sarker, Evolutionary algorithms for dynamic economic dispatch problems, IEEE Transactions on Power Systems, 31 (2016), 1486-1495. 

Figure 1.  Convergence graph between total cost and #FE for EPP: (a) 6-unit generating system, (b) 15-unit generating system, (c) 40-unit generating system
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Table 1.  Survey on related work
Reference Problem Algorithms Case Study
[17] ELD with valve point effect GA 13 generating units
[30] ELD EP 3, 13, and 40 generating units
[11] ELD with generator constraints PSO 6, 15, and 40 generating units
[15] ELD with valve-point effect HGA 13 and 40 generating units
[8] ELD SPSO 6, 15 and 40 generating units
[6] ELD CDE 6 and 13 generating units
[5] ELD BBO 6, 10, 20 and 40 generating units
[29] ELD with generator constraints IPSO 6 and 15 generating units
[24] ELD IHSWM 40 generating units
[28] ELD MPSO 3, 6, 15, and 40 generation units
[1] ELD HPSOTVAC 6, 15 and 38 generating units
[10] Dynamic ELD ADE (1) 5-unit thermal system with Ploss for a 24-hours planning horizon;
(2) 10-unit thermal system without Ploss for a 12-hours planning horizon;
(3) 10-unit thermal system without Ploss for a 24-hours planning horizon
[2] ELD CBA 6, 13, 20, 40 and 160 generating units
[32] Dynamic ELD EA (1) 5-unit thermal problems with and without Ploss
(2) 10-unit thermal problems with and without Ploss;
(3) 7-unit hydro-thermal problem without Ploss;
(4) 19-unit solar-Cthermal system without Ploss;
(5) 6-unit wind-Cthermal system with Ploss
[27] Environmental/ Economic Dispatch MOEA (1) 6-generator 30-bus standard test system;
(2) 13-generator 57-bus system;
(3) 3-generator system;
(4) 6-generator system;
(5) 14-generator 118-bus system;
(6) 40-generator system;
(7) 10-generator system
[16] Economic and Emission Dispatch IDE 6 generating units
[25] ELD QPSO 6, 15 and 40 generating units
[20] Renewable Energy Location Robust Bi-Level Programming Locating renewable energy sites
This research EPP NDE 6, 15 and 40 generating units
ELD: Economic Load Dispatch
EPP:EconomicPower dispatch
GA: Genetic Algorithm
EP: Evolutionary Programming
PSO: Particle Swarm Optimization
HGA: Hybrid Genetic Algorithm approach based on Differential Evolution
SPSO: Self-organizing Hierarchical Particle Swarm Optimization
CDE: Chemotactic Differential Evolution Algorithm
BBO: Biogeography-Based Optimization
IPSO: Iteration Particle Swarm Optimization
IHSWM: Improved Harmony Search with Wavelet Mutation
MPSO: Particle Swarm Optimization by Evolutionary Technique
HPSOTVAC: Hybrid Particle Swarm Optimization with Time-Varying Acceleration Coefficients
ADE: Automated Differential Evolution
CBA: Chaotic Bat Algorithm
EA: Evolutionary Algorithms
MOEA: Multi-Objective Evolutionary Algorithms
IDE: Improved Differential Evolution Algorithm
SG-QPSO: Novel Quantum-Behaved Particle Swarm Optimization Algorithm
NDE: Novel Differential Evolution
Table Options

Download as excel

Reference Problem Algorithms Case Study
[17] ELD with valve point effect GA 13 generating units
[30] ELD EP 3, 13, and 40 generating units
[11] ELD with generator constraints PSO 6, 15, and 40 generating units
[15] ELD with valve-point effect HGA 13 and 40 generating units
[8] ELD SPSO 6, 15 and 40 generating units
[6] ELD CDE 6 and 13 generating units
[5] ELD BBO 6, 10, 20 and 40 generating units
[29] ELD with generator constraints IPSO 6 and 15 generating units
[24] ELD IHSWM 40 generating units
[28] ELD MPSO 3, 6, 15, and 40 generation units
[1] ELD HPSOTVAC 6, 15 and 38 generating units
[10] Dynamic ELD ADE (1) 5-unit thermal system with Ploss for a 24-hours planning horizon;
(2) 10-unit thermal system without Ploss for a 12-hours planning horizon;
(3) 10-unit thermal system without Ploss for a 24-hours planning horizon
[2] ELD CBA 6, 13, 20, 40 and 160 generating units
[32] Dynamic ELD EA (1) 5-unit thermal problems with and without Ploss
(2) 10-unit thermal problems with and without Ploss;
(3) 7-unit hydro-thermal problem without Ploss;
(4) 19-unit solar-Cthermal system without Ploss;
(5) 6-unit wind-Cthermal system with Ploss
[27] Environmental/ Economic Dispatch MOEA (1) 6-generator 30-bus standard test system;
(2) 13-generator 57-bus system;
(3) 3-generator system;
(4) 6-generator system;
(5) 14-generator 118-bus system;
(6) 40-generator system;
(7) 10-generator system
[16] Economic and Emission Dispatch IDE 6 generating units
[25] ELD QPSO 6, 15 and 40 generating units
[20] Renewable Energy Location Robust Bi-Level Programming Locating renewable energy sites
This research EPP NDE 6, 15 and 40 generating units
ELD: Economic Load Dispatch
EPP:EconomicPower dispatch
GA: Genetic Algorithm
EP: Evolutionary Programming
PSO: Particle Swarm Optimization
HGA: Hybrid Genetic Algorithm approach based on Differential Evolution
SPSO: Self-organizing Hierarchical Particle Swarm Optimization
CDE: Chemotactic Differential Evolution Algorithm
BBO: Biogeography-Based Optimization
IPSO: Iteration Particle Swarm Optimization
IHSWM: Improved Harmony Search with Wavelet Mutation
MPSO: Particle Swarm Optimization by Evolutionary Technique
HPSOTVAC: Hybrid Particle Swarm Optimization with Time-Varying Acceleration Coefficients
ADE: Automated Differential Evolution
CBA: Chaotic Bat Algorithm
EA: Evolutionary Algorithms
MOEA: Multi-Objective Evolutionary Algorithms
IDE: Improved Differential Evolution Algorithm
SG-QPSO: Novel Quantum-Behaved Particle Swarm Optimization Algorithm
NDE: Novel Differential Evolution
Table 2.  Cost coefficient and bound values for 6-unit system
Unit no. $ a_i $ (MW) $ b_i $ (MW) $ c_i $ (MW) $ pwr_i^{min} $ (MW) $ pwr_i^{max} $ (MW)
1 0.007 7 240 100 500
2 0.0095 10 200 50 200
3 0.009 8.5 220 80 300
4 0.009 11 200 50 150
5 0.008 10.5 220 50 200
6 0.0075 12 190 50 120
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Unit no. $ a_i $ (MW) $ b_i $ (MW) $ c_i $ (MW) $ pwr_i^{min} $ (MW) $ pwr_i^{max} $ (MW)
1 0.007 7 240 100 500
2 0.0095 10 200 50 200
3 0.009 8.5 220 80 300
4 0.009 11 200 50 150
5 0.008 10.5 220 50 200
6 0.0075 12 190 50 120
Table 3.  Ramp rate and prohibited zones limits for 6-unit system
Unit no. $ UR_i $ $ DR_i $ $ pwr_i(0) $ Zone-1 Zone-2
1 80 120 440 210-240 350-380
2 50 90 170 90-110 140-160
3 65 100 200 150-170 210-240
4 50 90 150 80-90 110-120
5 50 90 190 90-110 140-150
6 50 90 110 75-85 100-105
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Unit no. $ UR_i $ $ DR_i $ $ pwr_i(0) $ Zone-1 Zone-2
1 80 120 440 210-240 350-380
2 50 90 170 90-110 140-160
3 65 100 200 150-170 210-240
4 50 90 150 80-90 110-120
5 50 90 190 90-110 140-150
6 50 90 110 75-85 100-105
Table 4.  Experimental results of NDE, DE and other comparative algorithms for 6-unit system
Unit no. NDE DE GA PSO
1 441.8657 446.7157 474.8066 447.497
2 169.6242 172.7829 178.6363 173.3221
3 259.2367 259.1119 262.2089 263.4745
4 139.5649 142.234 134.2826 139.0594
5 160.22 165.8878 151.9039 165.4761
6 105.0001 88.735 74.1812 87.128
$ pwr_L $ 13.0289 12.4673 13.0217 12.9584
Total output power 1, 276.03 1, 275.47 1, 276.03 1, 276.01
Min cost (＄/hr) 15, 444.09 15, 449.48 15, 459 15, 450
Mean cost (＄/hr) 15, 448.48 15, 452.28 15, 469 15, 454
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Unit no. NDE DE GA PSO
1 441.8657 446.7157 474.8066 447.497
2 169.6242 172.7829 178.6363 173.3221
3 259.2367 259.1119 262.2089 263.4745
4 139.5649 142.234 134.2826 139.0594
5 160.22 165.8878 151.9039 165.4761
6 105.0001 88.735 74.1812 87.128
$ pwr_L $ 13.0289 12.4673 13.0217 12.9584
Total output power 1, 276.03 1, 275.47 1, 276.03 1, 276.01
Min cost (＄/hr) 15, 444.09 15, 449.48 15, 459 15, 450
Mean cost (＄/hr) 15, 448.48 15, 452.28 15, 469 15, 454
Table 5.  Cost coefficient and bound values for 15-unit system
Unit no. $ a_i $ (MW) $ b_i $ (MW) $ c_i $ (MW) $ pwr_i^{min} $ (MW) $ pwr_i^{max} $ (MW)
1 0.000299 10.1 671 455 150
2 0.000183 10.2 574 455 150
3 0. 001126 8.8 374 130 20
4 0. 001126 8.8 374 130 20
5 0.000205 10.4 461 470 150
6 0.000301 10.1 630 460 135
7 0.000364 9.8 548 465 135
8 0.000338 11.2 227 300 60
9 0.000807 11.2 173 162 25
10 0. 001203 10.7 175 160 25
11 0. 003586 10.2 186 80 20
12 0. 005513 9.9 230 80 20
13 0.000371 13.1 225 85 25
14 0.001929 12.1 309 55 15
15 0.004447 12.4 323 55 15
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Unit no. $ a_i $ (MW) $ b_i $ (MW) $ c_i $ (MW) $ pwr_i^{min} $ (MW) $ pwr_i^{max} $ (MW)
1 0.000299 10.1 671 455 150
2 0.000183 10.2 574 455 150
3 0. 001126 8.8 374 130 20
4 0. 001126 8.8 374 130 20
5 0.000205 10.4 461 470 150
6 0.000301 10.1 630 460 135
7 0.000364 9.8 548 465 135
8 0.000338 11.2 227 300 60
9 0.000807 11.2 173 162 25
10 0. 001203 10.7 175 160 25
11 0. 003586 10.2 186 80 20
12 0. 005513 9.9 230 80 20
13 0.000371 13.1 225 85 25
14 0.001929 12.1 309 55 15
15 0.004447 12.4 323 55 15
Table 6.  Ramp rate and prohibited zones limits for 15-unit system
Unit no. $ UR_i $ $ DR_i $ $ pwr_i(0) $ Zone-1 Zone-2 Zone-3
1 180 120 400 150-150 150-150 150-150
2 180 120 300 185-255 305-335 430-450
3 130 130 105 20-20 20-20 20-20
4 130 130 100 20-20 20-20 20-20
5 80 120 90 180-200 305-335 390-430
6 80 120 400 230-255 335-395 430-455
7 80 120 350 135-135 135-135 135-135
8 65 100 95 60-60 60-60 60-60
9 60 100 105 60-60 25-25 25-25
10 60 100 110 25-25 25-25 25-25
11 80 80 60 20-20 20-20 20-20
12 80 80 40 30-40 55-65 20-20
13 80 80 30 25-25 25-25 25-25
14 55 55 20 15-15 15-15 15-15
15 55 55 20 15-15 15-15 15-15
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Unit no. $ UR_i $ $ DR_i $ $ pwr_i(0) $ Zone-1 Zone-2 Zone-3
1 180 120 400 150-150 150-150 150-150
2 180 120 300 185-255 305-335 430-450
3 130 130 105 20-20 20-20 20-20
4 130 130 100 20-20 20-20 20-20
5 80 120 90 180-200 305-335 390-430
6 80 120 400 230-255 335-395 430-455
7 80 120 350 135-135 135-135 135-135
8 65 100 95 60-60 60-60 60-60
9 60 100 105 60-60 25-25 25-25
10 60 100 110 25-25 25-25 25-25
11 80 80 60 20-20 20-20 20-20
12 80 80 40 30-40 55-65 20-20
13 80 80 30 25-25 25-25 25-25
14 55 55 20 15-15 15-15 15-15
15 55 55 20 15-15 15-15 15-15
Table 7.  For 15-unit system (Experimental outcomes and comparison)
Unit no. NDE DE MPSO GA PSO IPSO
1 446.3316 454.998 455 415.31 455 455
2 366.7521 379.996 455 359.72 380 380
3 127.9874 129.9991 130 104.43 130 129.97
4 129.1781 129.9899 130 74.99 130 130
5 165.6126 169.9968 286.4128 380.28 170 169.93
6 423.1885 429.9944 460 426.79 460 459.88
7 415.0202 429.9944 465 341.32 430 429.25
8 132.9585 120.1228 60 124.79 60 60.43
9 124.8283 47.6016 25 133.14 71.05 74.78
10 82.68406 146.0069 37.5603 89.26 159.85 158.02
11 68.83981 79.99735 20 60.06 80 80
12 71.96576 79.9997 80 50 80 78.57
13 37.01169 25.0118 25 38.77 25 25
14 25.78839 16.8516 15 41.94 15 15
15 24.5418 20.6135 15 22.64 15 15
$ Pow_L $ 29.8923 31.1792 28.9734 38.278 30.908 30.858
Total Output Power 2, 659.89 2, 661.20 2, 658.97 2, 668.40 2, 660.90 2, 660.80
Mean Cost (＄/Hr) 32, 561.89 32, 747.16 32, 569.95 33, 113 32, 708 32, 709
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Unit no. NDE DE MPSO GA PSO IPSO
1 446.3316 454.998 455 415.31 455 455
2 366.7521 379.996 455 359.72 380 380
3 127.9874 129.9991 130 104.43 130 129.97
4 129.1781 129.9899 130 74.99 130 130
5 165.6126 169.9968 286.4128 380.28 170 169.93
6 423.1885 429.9944 460 426.79 460 459.88
7 415.0202 429.9944 465 341.32 430 429.25
8 132.9585 120.1228 60 124.79 60 60.43
9 124.8283 47.6016 25 133.14 71.05 74.78
10 82.68406 146.0069 37.5603 89.26 159.85 158.02
11 68.83981 79.99735 20 60.06 80 80
12 71.96576 79.9997 80 50 80 78.57
13 37.01169 25.0118 25 38.77 25 25
14 25.78839 16.8516 15 41.94 15 15
15 24.5418 20.6135 15 22.64 15 15
$ Pow_L $ 29.8923 31.1792 28.9734 38.278 30.908 30.858
Total Output Power 2, 659.89 2, 661.20 2, 658.97 2, 668.40 2, 660.90 2, 660.80
Mean Cost (＄/Hr) 32, 561.89 32, 747.16 32, 569.95 33, 113 32, 708 32, 709
Table 8.  Cost coefficient and bound values for 40-unit system
Unit no. $ a_i $ (MW) $ b_i $ (MW) $ c_i $ (MW) $ pwr_i^{min} $ (MW) $ pwr_i^{max} $ (MW)
1 0.00708 9.15 1728.3 114 36
2 0.00313 7.97 647.85 114 36
3 0.00313 7.95 649.69 120 60
4 0.00313 7.97 647.83 190 80
5 0.00313 7.97 647.81 97 47
6 0.00298 6.63 785.96 140 68
7 0.00298 6.63 785.96 300 110
8 0.00284 6.66 794.53 300 135
9 0.00284 6.66 794.53 300 135
10 0.00277 7.1 801.32 300 130
11 0.00277 7.1 801.32 375 94
12 0.52124 3.33 1055.1 375 94
13 0.52124 3.33 1055.1 500 125
14 0.52124 3.33 1055.1 500 125
15 0.0114 5.35 148.89 500 125
16 0.0016 6.43 222.92 500 125
17 0.0016 6.43 222.92 500 220
18 0.0016 6.43 222.92 500 220
19 0.0001 8.95 107.87 550 242
20 0.0001 8.62 116.58 550 242
21 0.0001 8.62 116.58 550 254
22 0.0161 5.88 307.45 550 254
23 0.0161 5.88 307.45 550 254
24 0.0161 5.88 307.45 550 254
25 0.00313 7.97 647.83 550 254
26 0.00708 9.15 1728.3 550 254
27 0.00313 7.97 647.85 150 10
28 0.00313 7.95 649.69 150 10
29 0.00313 7.97 647.83 150 10
30 0.00313 7.97 647.81 97 47
31 0.00298 6.63 785.96 190 60
32 0.00298 6.63 785.96 190 60
33 0.00284 6.66 794.53 190 60
34 0.00284 6.66 794.53 200 90
35 0.00277 7.1 801.32 200 90
36 0.00277 7.1 801.32 200 90
37 0.52124 3.33 1055.1 110 25
38 0.52124 3.33 1055.1 110 25
39 0.52124 3.33 1055.1 110 25
40 0.0114 5.35 148.89 550 242
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Unit no. $ a_i $ (MW) $ b_i $ (MW) $ c_i $ (MW) $ pwr_i^{min} $ (MW) $ pwr_i^{max} $ (MW)
1 0.00708 9.15 1728.3 114 36
2 0.00313 7.97 647.85 114 36
3 0.00313 7.95 649.69 120 60
4 0.00313 7.97 647.83 190 80
5 0.00313 7.97 647.81 97 47
6 0.00298 6.63 785.96 140 68
7 0.00298 6.63 785.96 300 110
8 0.00284 6.66 794.53 300 135
9 0.00284 6.66 794.53 300 135
10 0.00277 7.1 801.32 300 130
11 0.00277 7.1 801.32 375 94
12 0.52124 3.33 1055.1 375 94
13 0.52124 3.33 1055.1 500 125
14 0.52124 3.33 1055.1 500 125
15 0.0114 5.35 148.89 500 125
16 0.0016 6.43 222.92 500 125
17 0.0016 6.43 222.92 500 220
18 0.0016 6.43 222.92 500 220
19 0.0001 8.95 107.87 550 242
20 0.0001 8.62 116.58 550 242
21 0.0001 8.62 116.58 550 254
22 0.0161 5.88 307.45 550 254
23 0.0161 5.88 307.45 550 254
24 0.0161 5.88 307.45 550 254
25 0.00313 7.97 647.83 550 254
26 0.00708 9.15 1728.3 550 254
27 0.00313 7.97 647.85 150 10
28 0.00313 7.95 649.69 150 10
29 0.00313 7.97 647.83 150 10
30 0.00313 7.97 647.81 97 47
31 0.00298 6.63 785.96 190 60
32 0.00298 6.63 785.96 190 60
33 0.00284 6.66 794.53 190 60
34 0.00284 6.66 794.53 200 90
35 0.00277 7.1 801.32 200 90
36 0.00277 7.1 801.32 200 90
37 0.52124 3.33 1055.1 110 25
38 0.52124 3.33 1055.1 110 25
39 0.52124 3.33 1055.1 110 25
40 0.0114 5.35 148.89 550 242
Table 9.  For 40-unit system (Experimental outcomes and comparison)
Unit no. NDE DE IHSWM HPSOTVAC SPSO BBO
1 90.59008 113.035 113.9088 113.9907 113.97 110.8158
2 102.8105 110.0641 110.9064 113.2932 114 111.0896
3 95.41785 96.508 97.402 120 109.19 97.40261
4 160.0214 180.7317 179.7332 175.0364 179.77 179.7549
5 81.69155 87.3028 88.7117 91 97 88.20832
6 107.7849 110.2516 139.9991 140 91.01 139.9886
7 278.066 259.0112 259.6372 260.3635 259.87 259.5935
8 268.3649 284.6521 284.6106 288.1256 286.99 284.6174
9 266.8508 286.1955 284.6024 286.9435 284.09 284.6479
10 228.3208 131.3615 130 130 204.05 130.0298
11 286.0884 243.8422 168.7992 170 168.4 94.01459
12 159.1006 168.7969 168.806 170 94 94.26367
13 309.307 302.4883 214.7593 210.0287 212.3 304.5153
14 346.2984 394.1255 394.2774 390.0677 393.76 394.264
15 382.4365 306.0514 304.5207 307.6247 303.62 304.5057
16 354.3069 394.3787 394.2762 300.0056 392.05 394.2472
17 442.9098 489.715 489.2787 487.0486 489.49 489.3273
18 440.5635 402.2923 489.2875 485.0793 489.35 489.3047
19 503.2849 516.0751 511.2827 510.541 512.39 511.3087
20 481.891 512.4736 511.2768 511.3472 511.21 511.2495
21 506.4032 524.042 523.2884 524.9522 522.61 523.3217
22 496.2304 524.0563 523.2794 526 523.65 523.3144
23 512.3193 524.3457 523.2772 523.9211 523.06 523.3629
24 511.7555 524.2132 523.2928 525.612 520.72 523.2883
25 507.4037 525.7952 523.3047 521.02 524.86 523.2989
26 493.267 522.6361 523.2872 520.1457 525.22 523.2802
27 10.30096 10.1786 10 10 10 10.02817
28 19.53625 12.3112 10.0022 10 10 10.00321
29 10.68301 10.8716 10.0018 10 10 10.0288
30 87.11869 90.3572 88.362 89.7002 87.64 88.14595
31 184.1883 187.5783 190 190 190 189.9913
32 174.4745 167.4291 190 190 190 189.9888
33 169.4674 177.4801 189.9935 190 190 189.9998
34 171.0925 166.2373 164.7992 167.0209 200 164.8452
35 176.4929 185.5927 164.8923 200 167.18 192.9876
36 172.1041 173.4381 164.864 200 172.12 199.9876
37 94.5799 89.3767 110 110 110 109.9941
38 85.26457 91.0112 110 110 110 109.9992
39 93.85836 91.6815 109.9965 110 95.58 109.9833
40 493.3801 512.0169 511.2828 511.1323 510.85 511.2794
$ Pow_{loss} $ 0 0.001 0 0 0 0.28
Total Output Power 10, 500.00 10, 500.00 10, 500.00 10, 500.00 10, 500.00 10, 500.28
Min Cost (＄/Hr) 1, 21, 721.62 1, 21, 974.50 1, 21, 416.26 1, 21, 070.64 1, 22, 049.66 1, 21, 479.50
Mean Cost (＄/Hr) 1, 21, 992.20 1, 22, 580.30 1, 21, 553.42 1, 21, 075.74 1, 22, 327.36 1, 21, 512.05
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Unit no. NDE DE IHSWM HPSOTVAC SPSO BBO
1 90.59008 113.035 113.9088 113.9907 113.97 110.8158
2 102.8105 110.0641 110.9064 113.2932 114 111.0896
3 95.41785 96.508 97.402 120 109.19 97.40261
4 160.0214 180.7317 179.7332 175.0364 179.77 179.7549
5 81.69155 87.3028 88.7117 91 97 88.20832
6 107.7849 110.2516 139.9991 140 91.01 139.9886
7 278.066 259.0112 259.6372 260.3635 259.87 259.5935
8 268.3649 284.6521 284.6106 288.1256 286.99 284.6174
9 266.8508 286.1955 284.6024 286.9435 284.09 284.6479
10 228.3208 131.3615 130 130 204.05 130.0298
11 286.0884 243.8422 168.7992 170 168.4 94.01459
12 159.1006 168.7969 168.806 170 94 94.26367
13 309.307 302.4883 214.7593 210.0287 212.3 304.5153
14 346.2984 394.1255 394.2774 390.0677 393.76 394.264
15 382.4365 306.0514 304.5207 307.6247 303.62 304.5057
16 354.3069 394.3787 394.2762 300.0056 392.05 394.2472
17 442.9098 489.715 489.2787 487.0486 489.49 489.3273
18 440.5635 402.2923 489.2875 485.0793 489.35 489.3047
19 503.2849 516.0751 511.2827 510.541 512.39 511.3087
20 481.891 512.4736 511.2768 511.3472 511.21 511.2495
21 506.4032 524.042 523.2884 524.9522 522.61 523.3217
22 496.2304 524.0563 523.2794 526 523.65 523.3144
23 512.3193 524.3457 523.2772 523.9211 523.06 523.3629
24 511.7555 524.2132 523.2928 525.612 520.72 523.2883
25 507.4037 525.7952 523.3047 521.02 524.86 523.2989
26 493.267 522.6361 523.2872 520.1457 525.22 523.2802
27 10.30096 10.1786 10 10 10 10.02817
28 19.53625 12.3112 10.0022 10 10 10.00321
29 10.68301 10.8716 10.0018 10 10 10.0288
30 87.11869 90.3572 88.362 89.7002 87.64 88.14595
31 184.1883 187.5783 190 190 190 189.9913
32 174.4745 167.4291 190 190 190 189.9888
33 169.4674 177.4801 189.9935 190 190 189.9998
34 171.0925 166.2373 164.7992 167.0209 200 164.8452
35 176.4929 185.5927 164.8923 200 167.18 192.9876
36 172.1041 173.4381 164.864 200 172.12 199.9876
37 94.5799 89.3767 110 110 110 109.9941
38 85.26457 91.0112 110 110 110 109.9992
39 93.85836 91.6815 109.9965 110 95.58 109.9833
40 493.3801 512.0169 511.2828 511.1323 510.85 511.2794
$ Pow_{loss} $ 0 0.001 0 0 0 0.28
Total Output Power 10, 500.00 10, 500.00 10, 500.00 10, 500.00 10, 500.00 10, 500.28
Min Cost (＄/Hr) 1, 21, 721.62 1, 21, 974.50 1, 21, 416.26 1, 21, 070.64 1, 22, 049.66 1, 21, 479.50
Mean Cost (＄/Hr) 1, 21, 992.20 1, 22, 580.30 1, 21, 553.42 1, 21, 075.74 1, 22, 327.36 1, 21, 512.05
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