1. College of Information Science and Engineering, Northeastern University, Shenyang 110819, China; 2. School of Science, University of Science and Technology Liaoning, Anshan 114051, China; 3. Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA; 4. College of Computer and Communication Engineering, Liaoning Shihua University, Fushun 113001, China; 5. Department of Computer Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China; 6. Department of Computer Science and Technology, Tongji University, Shanghai 201804, China

Abstract A modified cuckoo search (CS) algorithm is proposed to solve economic dispatch (ED) problems that have nonconvex, non-continuous or non-linear solution spaces considering valve-point effects, prohibited operating zones, transmission losses and ramp rate limits. Comparing with the traditional cuckoo search algorithm, we propose a self-adaptive step size and some neighbor-study strategies to enhance search performance. Moreover, an improved lambda iteration strategy is used to generate new solutions. To show the superiority of the proposed algorithm over several classic algorithms, four systems with different benchmarks are tested. The results show its efficiency to solve economic dispatch problems, especially for large-scale systems.

Fund:This work was supported in part by the National Key Research and Development Program of China (2017YFB0306400), in part by the National Natural Science Foundation of China (61573089, 71472080, 71301066), and Liaoning Province Dr. Research Foundation of China (20175032).

[1] J. C. Dodu, P. Martin, A. Merlin, and J. Pouget, "An optimal formulation and solution of short-range operating problems for a power system with flow constraints, " Proc. IEEE, vol. 60, no. 1, pp. 54-63, Jan. 1972. [2] A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control. Beijing: Tsinghua University Press, 2003, pp. 195. [3] J. Y. Fan and L. Zhang, "Real-time economic dispatch with line flow and emission constraints using quadratic programming, " IEEE Trans. Power Syst., vol. 13, no. 2, pp. 320-325, May 1998. [4] R. A. Jabr, A. H. Coonick, and B. J. Cory, "A homogeneous linear programming algorithm for the security constrained economic dispatch problem, " IEEE Trans. Power Syst., vol. 15, no. 3, pp. 930-936, Aug. 2000. [5] Z. X. Liang and J. D. Glover, "A zoom feature for a dynamic programming solution to economic dispatch including transmission losses, " IEEE Trans. Power Syst., vol. 7, no. 2, pp. 544-550, May 1992. [6] J. F. Bard, "Short-term scheduling of thermal-electric generators using lagrangian relaxation, " Oper. Res., vol. 36, no. 5, pp. 756-766, Sep. 1988. [7] X. L. Liang, W. F. Li, Y. Zhang, and M. C. Zhou, "An adaptive particle swarm optimization method based on clustering, " Soft Comput., vol. 19, no. 2, pp. 431-448, Feb. 2014. [8] X. W. Guo, S. X. Liu, M. C. Zhou, and G. D. Tian, "Disassembly sequence optimization for large-scale products with multiresource constraints using scatter search and petri nets, " IEEE Trans. Cybern., vol. 46, no. 11, pp. 2435-2446, Nov. 2016. [9] G. P. Singh and A. Singh, "Comparative study of krill herd, firefly and cuckoo search algorithms for unimodal and multimodal optimization, " Int. J. Intell. Syst. Appl., vol. 6, no. 3, pp. 35-49, Feb. 2014. [10] J. Li, J. Q. Zhang, C. J. Jiang, and M. C. Zhou, "Composite particle swarm optimizer with historical memory for function optimization, " IEEE Trans. Cybern., vol. 45, no. 10, pp. 2350-2363, Oct. 2015. [11] I. Chatterjee and M. C. Zhou, "Differential evolution algorithms under multi-population strategy, " in Proc. 26th Wireless and Optical Communication Conf., Newark, NJ, USA, 2017, pp. 1-7. [12] A. Che, P. Wu, F. Chu, and M. C. Zhou, "Improved quantum-inspired evolutionary algorithm for large-size lane reservation, " IEEE Trans. Syst. Man Cybern. Syst., vol. 45, no. 12, pp. 1535-1548, May 2015. [13] Q. Kang, M. C. Zhou, and C. Xu, "Solving optimal power flow problems subject to distributed generator failures via particle swarm intelligence, " in Proc. Int. Conf. Advanced Mechatronic Systems, Tokyo, Japan, 2012, pp. 418-423. [14] W. Y. Dong and M. C. Zhou, "A supervised learning and control method to improve particle swarm optimization algorithms, " IEEE Trans. Syst. Man Cybern. Syst., vol. 47, no. 7, pp. 1135-1148, Jul. 2017. [15] X. H. Yuan, L. Wang, Y. B. Yuan, Y. C. Zhang, B. Cao, and B. Yang, "A modified differential evolution approach for dynamic economic dispatch with valve-point effects, " Energy Convers. Manage., vol. 49, no. 12, pp. 3447-3453, Dec. 2008. [16] S. K. Wang, J. P. Chiou, and C. W. Liu, "Non-smooth/non-convex economic dispatch by a novel hybrid differential evolution algorithm, " IET Gener., Transm. Distrib., vol. 1, no. 5, pp. 793-803, Sep. 2007. [17] M. A. Al-Betar, M. A. Awadallah, A. T. Khader, and A. L. A. Bolaji, "Tournament-based harmony search algorithm for non-convex economic load dispatch problem, " Appl. Soft Comput., vol. 47, pp. 449-459, Oct. 2016. [18] D. R. Dashti, A. Ghabeli, and S. M. Hosseini, "Solving static economic load dispatch using improved exponential harmony search optimisation, " Aust. J. Electr. Electron. Eng., vol. 13, no. 2, pp. 142-150, Nov. 2016. [19] M. A. Al-Betar, M. A. Awadallah, A. T. Khader, A. L. Bolaji, and A. Almomani, "Economic load dispatch problems with valve-point loading using natural updated harmony search, " Neural Comput. Appl., vol. 29, no. 10, pp. 767-781, Sep. 2016. [20] A. Bhattacharya and P. K. Chattopadhyay, "Biogeography-based optimization for different economic load dispatch problems, " IEEE Trans. Power Syst., vol. 25, no. 2, pp. 1064-1077, May 2010. [21] V. Raviprabhakaran and C. S. Ravichandran, "Enriched biogeography-based optimization algorithm to solve economic power dispatch problem, " in Proc. 5th International Conference on Soft Computing for Problem Solving, Singapore, 2016, pp. 875-888. [22] S. L. Yan, B. Gu, D. X. Tian, and S. J. Ai, "Hybrid biogeography constrained optimization for economic dispatch, " Electr. Power Sci. Eng., vol. 32, no. 3, pp. 31-36, Mar. 2016. [23] T. Jayabarathi, A. Yazdani, V. Ramesi, and T. Raghunathan, "Combined heat and power economic dispatch problem using the invasive weed optimization algorithm, " Front. Energy, vol. 8, no. 1, pp. 25-30, Mar. 2014. [24] A. K. Barisal and R. C. Prusty, "Large scale economic dispatch of power systems using oppositional invasive weed optimization, " Appl. Soft Comput., vol. 29, pp. 122-137, Apr. 2015. [25] J. B. Park, Y. W. Jeong, J. R. Shin, and K. Y. Lee, "An improved particle swarm optimization for nonconvex economic dispatch problems, " IEEE Trans. Power Syst., vol. 25, no. 1, pp. 156-166, Feb. 2010. [26] S. Hemamalini and S. P. Simon, "Dynamic economic dispatch using artificial bee colony algorithm for units with valve-point effect, " Eur. Trans. Electr. Power, vol. 21, no. 1, pp. 70-81, Jan. 2011. [27] J. Sun, V. Palade, X. J. Wu, W. Fang, and Z. Y. Wang, "Solving the power economic dispatch problem with generator constraints by random drift particle swarm optimization, " IEEE Trans. Ind. Inf., vol. 10, no. 1, pp. 222-232, Jun. 2013. [28] M. R. Narimani, "A new modified shuffle frog leaping algorithm for non-smooth economic dispatch, " World Appl. Sci. J., vol. 12, no. 6, pp. 803-814, Jan. 2011. [29] P. Roy and A. Chakrabarti, "Modified shuffled frog leaping algorithm with genetic algorithm crossover for solving economic load dispatch problem with valve-point effect, " Appl. Soft Comput., vol. 13, no. 11, pp. 4244-4252, Nov. 2013. [30] J. J. Liang, A. K. Qin, P. N. Suganthan, and S. Baskar, "Comprehensive learning particle swarm optimizer for global optimization of multimodal functions, " IEEE Trans. Evol. Comput., vol. 10, no. 3, pp. 281-295, Jun. 2006. [31] N. Lynn and P. N. Suganthan, "Heterogeneous comprehensive learning particle swarm optimization with enhanced exploration and exploitation, " Swarm Evol. Comput., vol. 24, pp. 11-24, Oct. 2015. [32] X. S. Yang and S. Deb, "Cuckoo search via lévy flights, " in Proc. World Congress on Nature and Biologically Inspired Computing, Kitakyushu, Japan, 2009. [33] X. S. Yang and S. Deb, "Engineering optimisation by cuckoo search, " Int. J. Math. Modell. Numer. Optim., vol. 1, no. 4, pp. 330-343, Dec. 2010. [34] H. Rakhshani and A. Rahati, "Snap-drift cuckoo search: a novel cuckoo search optimization algorithm, " Appl. Soft Comput., vol. 52, pp. 771-794, Mar. 2017. [35] A. F. Ali and M. A. Tawhid, "A hybrid cuckoo search algorithm with Nelder Mead method for solving global optimization problems, " SpringerPlus, vol. 5, pp. 473, Apr. 2016. [36] G. G. Wang, A. H. Gandomi, X. J. Zhao, and H. C. E. Chu, "Hybridizing harmony search algorithm with cuckoo search for global numerical optimization, " Soft Comput., vol. 20, no. 1, pp. 273-285, Jan. 2016. [37] X. T. Li and M. Yin, "A particle swarm inspired cuckoo search algorithm for real parameter optimization, " Soft Comput., vol. 20, no. 4, pp. 1389-1413, Apr. 2016. [38] L. Huang, S. Ding, S. H. Yu, J. Wang, and K. Lu, "Chaos-enhanced Cuckoo search optimization algorithms for global optimization, " Appl. Math. Modell., vol. 40, no. 5-6, pp. 3860-3875, Mar. 2016. [39] X. T. Li and M. H. Yin, "Modified cuckoo search algorithm with self adaptive parameter method, " Inf. Sci., vol. 298, pp. 80-97, Mar. 2015. [40] W. H. Han, J. Xu, M. Z. Zhou, G. Y. Tian, P. Wang, X. H. Shen, and E. Hou, "Cuckoo search and particle filter-based inversing approach to estimating defects via magnetic flux leakage signals, " IEEE Trans. Magn., vol. 52, no. 4, pp. 6200511, Apr. 2016. [41] W. H. Han, X. S. Lu, M. C. Zhou, X. H. Shen, J. X. Wang, and J. Xu, "An evaluation and optimization methodology for efficient power plant programs, " IEEE Trans. Syst., Man, Cybern: Syst., vol. PP, no. 99, pp. 1-10, Jul. 2017. [42] M. Basu and A. Chowdhury, "Cuckoo search algorithm for economic dispatch, " Energy, vol. 60, pp. 99-108, Oct. 2013. [43] D. N. Vo, P. Schegner, and W. Ongsakul, "Cuckoo search algorithm for non-convex economic dispatch, " IET Gener., Transm. Distrib., vol. 7, no. 6, pp. 645-654, Jun. 2013. [44] A. B. S. Serapião, "Cuckoo search for solving economic dispatch load problem, " Intell. Control Autom., vol. 4, no. 4, pp. 385-390, Jul. 2016. [45] T. T. Nguyen, D. N. Vo, and B. H. Dinh, "Cuckoo search algorithm for combined heat and power economic dispatch, " Int. J. Electr. Power Energy Syst., vol. 81, pp. 204-214, Oct. 2016. [46] M. A. Mellal and E. J. Williams, "Cuckoo optimization algorithm with penalty function for combined heat and power economic dispatch problem, " Energy, vol. 93, pp. 1711-1718, Dec. 2015. [47] E. Valian, S. Tavakoli, S. Mohanna, and A. Haghi, "Improved cuckoo search for reliability optimization problems, " Comput. Ind. Eng., vol. 64, no. 1, pp. 459-468, Jan. 2013. [48] S. Walton, O. Hassan, K. Morgan, and M. R. Brown, "Modified cuckoo search: a new gradient free optimisation algorithm, " Chaos Solitons Fractals, vol. 44, no. 9, pp. 710-718, Sep. 2011. [49] X. S. Yang, "Nature-inspired optimization algorithms, " in Nature-Inspired Optimization Algorithms, X. S. Yang, Ed. Oxford: Elsevier, 2014. [50] R. N. Mantegna, "Fast, accurate algorithm for numerical simulation of Lévy stable stochastic processes, " Phys. Rev. B, vol. 49, no. 5, pp. 4677-4683, May 1994. [51] S. Das, S. S. Mullick, and P. N. Suganthan, "Recent advances in differential evolution——an updated survey, " Swarm Evol. Comput., vol. 27, pp. 1-30, Apr. 2016. [52] M. Sydulu, "A very fast and effective noniterative ‘λ-logic based’ algorithm for economic dispatch of thermal units, " in Proc. IEEE Region 10 Conference TENCON 99, vol. 2, pp. 1434-1437, 2002. [53] P. K. Singhal, R. Naresh, V. Sharma, and K. N. Goutham, "Enhanced lambda iteration algorithm for the solution of large scale economic dispatch problem, " in Proc. Recent Advances and Innovations in Engineering, Jaipur, India, 2014, pp. 1-6. [54] P. Aravindhababu and K. R. Nayar, "Economic dispatch based on optimal lambda using radial basis function network, " Int. J. Electr. Power Energy Syst., vol. 24, no. 7, pp. 551-556, Oct. 2002. [55] Z. L. Gaing, "Particle swarm optimization to solving the economic dispatch considering the generator constraints, " IEEE Trans. Power Syst., vol. 18, no. 3, pp. 1187-1195, Aug. 2003. [56] Z. L. Gaing, "Closure to "discussion of ‘particle swarm optimization to solving the economic dispatch considering the generator constraints’", " IEEE Trans. Power Syst., vol. 19, no. 4, pp. 2122-2123, Nov. 2004. [57] D. C. Secui, "A new modified artificial bee colony algorithm for the economic dispatch problem, " Energy Convers. Manage., vol. 89, pp. 43-62, Jan. 2015. [58] M. Modiri-Delshad, S. H. Aghay Kaboli, E. Taslimi-Renani, and N. A. Rahim, "Backtracking search algorithm for solving economic dispatch problems with valve-point effects and multiple fuel options, " Energy, vol. 116, pp. 637-649, Dec. 2016. [59] L. Wang and L. P. Li, "An effective differential harmony search algorithm for the solving non-convex economic load dispatch problems, " Int. J. Electr. Power Energy Syst., vol. 44, no. 1, pp. 832-843, Jan. 2013. [60] A. I. Selvakumar and K. Thanushkodi, "A new particle swarm optimization solution to nonconvex economic dispatch problems, " IEEE Trans. Power Syst., vol. 22, no. 1, pp. 42-51, Feb. 2007. [61] S. Pothiya, I. Ngamroo, and W. Kongprawechnon, "Application of multiple tabu search algorithm to solve dynamic economic dispatch considering generator constraints, " Energy Convers. Manage., vol. 49, no. 4, pp. 506-516, Apr. 2008. [62] B. R. Adarsh, T. Raghunathan, T. Jayabarathi, and X. S. Yang, "Economic dispatch using chaotic bat algorithm, " Energy, vol. 96, pp. 666-675, Feb. 2016. [63] X. S. Yang, Nature-Inspired Optimization Algorithms. Amsterdam, Holland: Elsevier Science Publishers B. V., 2014. [64] N. Sinha, R. Chakrabarti, and P. K. Chattopadhyay, "Evolutionary programming techniques for economic load dispatch, " IEEE Trans. Evol. Comput., vol. 7, no. 1, pp. 83-94, Feb. 2003. [65] A. Srinivasa Reddy and K. Vaisakh, "Shuffled differential evolution for large scale economic dispatch, " Electr. Power Syst. Res., vol. 96, pp. 237-245, Mar. 2013. [66] M. S. P. Subathra, S. E. Selvan, T. A. A. Victoire, A. H. Christinal, and U. Amato, "A hybrid with cross-entropy method and sequential quadratic programming to solve economic load dispatch problem, " IEEE Syst. J., vol. 9, no. 3, pp. 1031-1044, Sep. 2015. [67] T. T. Nguyen and D. N. Vo, "The application of one rank cuckoo search algorithm for solving economic load dispatch problems, " Appl. Soft Comput., vol. 37, pp. 763-773, Dec. 2015. [68] S. Khamsawang and S. Jiriwibhakorn, "DSPSO-TSA for economic dispatch problem with nonsmooth and noncontinuous cost functions, " Energy Convers. Manage., vol. 51, no. 2, pp. 365-375, Feb. 2010. [69] S. Pothiya, I. Ngamroo, and W. Kongprawechnon, "Ant colony optimisation for economic dispatch problem with non-smooth cost functions, " Int. J. Electr. Power Energy Syst., vol. 32, no. 5, pp. 478-487, Jun. 2010. [70] K. Bhattacharjee, A. Bhattacharya, and S. H. N. Dey, "Oppositional real coded chemical reaction optimization for different economic dispatch problems, " Int. J. Electr. Power Energy Syst., vol. 55, pp. 378-391, Feb. 2014. [71] J. G. Vlachogiannis and K. Y. Lee, "Economic load dispatch——a comparative study on heuristic optimization techniques with an improved coordinated aggregation-based PSO, " IEEE Trans. Power Syst., vol. 24, no. 2, pp. 991-1001, May 2009. [72] C. T. Su and C. T. Lin, "New approach with a Hopfield modeling framework to economic dispatch, " IEEE Trans. Power Syst., vol. 15, no. 2, pp. 541-545, May 2000. [73] M. Moradi-Dalvand, B. Mohammadi-Ivatloo, A. Najafi, and A. Rabiee, "Continuous quick group search optimizer for solving non-convex economic dispatch problems, " Electr. Power Syst. Res., vol. 93, pp. 93-105, Dec. 2012. [74] M. Modiri-Delshad and N. A. Rahim, "Solving non-convex economic dispatch problem via backtracking search algorithm, " Energy, vol. 77, pp. 372-381, Dec. 2014. [75] A. Bhattacharya and P. K. Chattopadhyay, "Solving complex economic load dispatch problems using biogeography-based optimization, " Expert Syst. Appl., vol. 37, no. 5, pp. 3605-3615, May 2010. [76] N. Ghorbani and E. Babaei, "Exchange market algorithm for economic load dispatch, " Int. J. Electr. Power Energy Syst., vol. 75, pp. 19-27, Feb. 2016. [77] T. Niknam, H. D. Mojarrad, and H. Z. Meymand, "Non-smooth economic dispatch computation by fuzzy and self adaptive particle swarm optimization, " Appl. Soft Comput., vol. 11, no. 2, pp. 2805-2817, Mar. 2011. [78] V. Hosseinnezhad, M. Rafiee, M. Ahmadian, and M. T. Ameli, "Species-based quantum particle swarm optimization for economic load dispatch, " Int. J. Electr. Power Energy Syst., vol. 63, pp. 311-322, Dec. 2014. [79] N. Amjady and H. Sharifzadeh, "Solution of non-convex economic dispatch problem considering valve loading effect by a new modified differential evolution algorithm, " Int. J. Electr. Power Energy Syst., vol. 32, no. 8, pp. 893-903, Oct. 2010. [80] X. S. Yang, S. S. Sadat Hosseini, and A. H. Gandomi, "Firefly algorithm for solving non-convex economic dispatch problems with valve loading effect, " Appl. Soft Comput., vol. 12, no. 3, pp. 1180-1186, Mar. 2012. [81] X. W. Guo, S. X. Liu, M. C. Zhou, and G. D. Tian, "Dual-objective program and scatter search for the optimization of disassembly sequences subject to multiresource constraints, " IEEE Trans. Autom. Sci. Eng., vol. PP, no. 99, pp. 1-13, Aug. 2017. [82] X. Y. Lu, M. C. Zhou, A. C. Ammari, and J. C. Ji, "Hybrid Petri nets for modeling and analysis of microgrid systems, " IEEE/CAA J. of Autom. Sinica, vol. 3, no. 4, pp. 349-356, Oct. 2016. [83] J. J. Zhang, F. Y. Wang, Q. Wang, D. Z. Hao, X. J. Yang, D. W. Gao, X. Y. Zhao, and Y. C. Zhang, "Parallel dispatch: a new paradigm of electrical power system dispatch, " IEEE/CAA J. of Autom. Sinica, vol. 5, no. 1, pp. 311-319, Jan. 2018.