# Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect

Article

Yang, X., Hosseini, S. and Gandomi, A. 2012. Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect. Applied Soft Computing. 12 (3), pp. 1180-1186. https://doi.org/10.1016/j.asoc.2011.09.017
Type Article Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect Yang, X., Hosseini, S. and Gandomi, A. The growing costs of fuel and operation of power generating units warrant improvement of optimization methodologies for economic dispatch (ED) problems. The practical ED problems have non-convex objective functions with equality and inequality constraints that make it much harder to find the global optimum using any mathematical algorithms. Modern optimization algorithms are often meta-heuristic, and they are very promising in solving nonlinear programming problems. This paper presents a novel approach to determining the feasible optimal solution of the ED problems using the recently developed Firefly Algorithm (FA). Many nonlinear characteristics of power generators, and their operational constraints, such as generation limitations, prohibited operating zones, ramp rate limits, transmission loss, and nonlinear cost functions, were all contemplated for practical operation. To demonstrate the efficiency and applicability of the proposed method, we study four ED test systems having non-convex solution spaces and compared with some of the most recently published ED solution methods. The results of this study show that the proposed FA is able to find more economical loads than those determined by other methods. This algorithm is considered to be a promising alternative algorithm for solving the ED problems in practical power systems. Economic load dispatch; Valve loading effect; Firefly Algorithm; Metaheuristic Elsevier Applied Soft Computing 1568-4946 1872-9681 Mar 2012 17 Nov 2011 15 Nov 2012 22 Sep 2011 Published https://doi.org/10.1016/j.asoc.2011.09.017 WOS:000299324200018 English

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Chang, Y., Yi, J., Yan, W., Yang, X., Zhang, S., Gao, Y. and Wang, X. 2014. Oil supply between OPEC and non-OPEC based on game theory. International Journal of Systems Science. 45 (10), pp. 2127-2132. https://doi.org/10.1080/00207721.2012.762562
##### Bat algorithm is better than intermittent search strategy
Yang, X., Deb, S. and Fong, S. 2014. Bat algorithm is better than intermittent search strategy. Journal of Multiple-Valued Logic and Soft Computing. 22 (3), pp. 223-237.
##### Non-dominated sorting cuckoo search for multiobjective optimization
He, X., Li, N. and Yang, X. 2014. Non-dominated sorting cuckoo search for multiobjective optimization. 2014 IEEE Symposium on Swarm Intelligence (SIS). Orlando, Florida 09 - 12 Dec 2014 IEEE. pp. 1-7 https://doi.org/10.1109/SIS.2014.7011772
##### Computational optimization, modelling and simulation: past, present and future
Yang, X., Koziel, S. and Leifsson, L. 2014. Computational optimization, modelling and simulation: past, present and future. 5th workshop on Computational Optimization, Modelling and Simulations (COMS 2014) at the ICCS 2014. Cairns, Australia 10 - 12 Jun 2014 Elsevier. pp. 754-758 https://doi.org/10.1016/j.procs.2014.05.067
##### Discrete cuckoo search algorithm for job shop scheduling problem
Ouaarab, A., Ahiod, B., Yang, X. and Abbad, M. 2014. Discrete cuckoo search algorithm for job shop scheduling problem. 2014 IEEE International Symposium on Intelligent Control (ISIC). France 08 - 10 Oct 2014 IEEE. pp. 1872-1876 https://doi.org/10.1109/ISIC.2014.6967636
##### Nature-inspired framework for hyperspectral band selection
Nakamura, R., Garcia Fonseca, L., Dos Santos, J., Da S. Torres, R., Yang, X. and Papa, J. 2014. Nature-inspired framework for hyperspectral band selection. IEEE Transactions on Geoscience and Remote Sensing. 52 (4), pp. 2126-2137. https://doi.org/10.1109/TGRS.2013.2258351
##### A discrete firefly algorithm for the multi-objective hybrid flowshop scheduling problems
Marichelvam, M., Prabaharan, T. and Yang, X. 2014. A discrete firefly algorithm for the multi-objective hybrid flowshop scheduling problems. IEEE Transactions on Evolutionary Computation. 18 (2), pp. 301-305. https://doi.org/10.1109/TEVC.2013.2240304
##### Binary bat algorithm
Mirjalili, S., Mirjalili, S. and Yang, X. 2014. Binary bat algorithm. Neural Computing and Applications. 25 (3-4), pp. 663-681. https://doi.org/10.1007/s00521-013-1525-5
##### Bat algorithm based on simulated annealing and Gaussian perturbations
He, X., Ding, W. and Yang, X. 2014. Bat algorithm based on simulated annealing and Gaussian perturbations. Neural Computing and Applications. 25 (2), pp. 459-468. https://doi.org/10.1007/s00521-013-1518-4
##### Discrete cuckoo search algorithm for the travelling salesman problem
Ouaarab, A., Ahiod, B. and Yang, X. 2014. Discrete cuckoo search algorithm for the travelling salesman problem. Neural Computing and Applications. 24 (7-8), pp. 1659-1669. https://doi.org/10.1007/s00521-013-1402-2
##### Cuckoo search: recent advances and applications
Yang, X. and Deb, S. 2014. Cuckoo search: recent advances and applications. Neural Computing and Applications. 24 (1), pp. 169-174. https://doi.org/10.1007/s00521-013-1367-1
##### Chaotic bat algorithm
Gandomi, A. and Yang, X. 2014. Chaotic bat algorithm. Journal of Computational Science. 5 (2), pp. 224-232. https://doi.org/10.1016/j.jocs.2013.10.002
##### An empirical study of test effort estimation based on bat algorithm
Srivastava, P., Bidwai, A., Khan, A., Rathore, K., Sharma, R. and Yang, X. 2014. An empirical study of test effort estimation based on bat algorithm. International Journal of Bio-Inspired Computation. 6 (1), pp. 57-70. https://doi.org/10.1504/IJBIC.2014.059966
##### Bio-inspired computation: success and challenges of IJBIC
Yang, X. and Cui, Z. 2014. Bio-inspired computation: success and challenges of IJBIC. International Journal of Bio-Inspired Computation. 6 (1), pp. 1-6. https://doi.org/10.1504/IJBIC.2014.059969
##### A wrapper approach for feature selection based on Bat Algorithm and Optimum-Path Forest
Rodrigues, D., Pereira, L., Nakamura, R., Costa, K., Yang, X., Souza, A. and Papa, J. 2014. A wrapper approach for feature selection based on Bat Algorithm and Optimum-Path Forest. Expert Systems with Applications. 41 (5), pp. 2250-2258. https://doi.org/10.1016/j.eswa.2013.09.023
##### Improved cuckoo search algorithm for hybrid flow shop scheduling problems to minimize makespan
Marichelvam, M., Prabaharan, T. and Yang, X. 2014. Improved cuckoo search algorithm for hybrid flow shop scheduling problems to minimize makespan. Applied Soft Computing. 19, pp. 93-101. https://doi.org/10.1016/j.asoc.2014.02.005
##### A bio-inspired algorithm for identification of critical components in the transportation networks
Zhang, X., Adamatzky, A., Yang, H., Mahadaven, S., Yang, X., Wang, Q. and Deng, Y. 2014. A bio-inspired algorithm for identification of critical components in the transportation networks. Applied Mathematics and Computation. 248, pp. 18-27. https://doi.org/10.1016/j.amc.2014.09.055
##### A firefly-inspired method for protein structure prediction in lattice models
Maher, B., Albrecht, A., Loomes, M., Yang, X. and Steinhofel, K. 2014. A firefly-inspired method for protein structure prediction in lattice models. Biomolecules. 4 (1), pp. 56-75. https://doi.org/10.3390/biom4010056
##### True global optimality of the pressure vessel design problem: a benchmark for bio-inspired optimisation algorithms
Yang, X., Huyck, C., Karamanoglu, M. and Khan, N. 2013. True global optimality of the pressure vessel design problem: a benchmark for bio-inspired optimisation algorithms. International Journal of Bio-Inspired Computation. 5 (6), pp. 329-335. https://doi.org/10.1504/IJBIC.2013.058910
##### Applications and analysis of bio-inspired eagle strategy for engineering optimization
Yang, X., Karamanoglu, M., Ting, T. and Zhao, Y. 2014. Applications and analysis of bio-inspired eagle strategy for engineering optimization. Neural Computing and Applications. 25 (2), pp. 411-420. https://doi.org/10.1007/s00521-013-1508-6
##### Flower pollination algorithm: a novel approach for multiobjective optimization
Yang, X., Karamanoglu, M. and He, X. 2014. Flower pollination algorithm: a novel approach for multiobjective optimization. Engineering Optimization. 46 (9), pp. 1222-1237. https://doi.org/10.1080/0305215X.2013.832237
##### Random walks, Lévy flights, Markov chains and metaheuristic optimization
Yang, X., Ting, T. and Karamanoglu, M. 2013. Random walks, Lévy flights, Markov chains and metaheuristic optimization. in: Future information communication technology and applications: ICFICE 2013 Springer Netherlands.
##### Are motorways rational from slime mould's point of view?
Adamatzky, A., Akl, S., Alonso-Sanz, R., Van Dessel, W., Ibrahim, Z., Ilachinski, A., Jones, J., Kayem, A., Martínez, G., De Oliveira, P., Prokopenko, M., Schubert, T., Sloot, P., Strano, E. and Yang, X. 2013. Are motorways rational from slime mould's point of view? International Journal of Parallel, Emergent and Distributed Systems. 28 (3), pp. 230-248. https://doi.org/10.1080/17445760.2012.685884
##### Advances in simulation-driven optimization and modelling
Koziel, S., Leifsson, L. and Yang, X. 2012. Advances in simulation-driven optimization and modelling. Journal of Computational Methods in Science and Engineering. 12 (1-2), pp. 1-4. https://doi.org/10.3233/JCM-2012-0400
##### Metaheuristic algorithms for self-organizing systems: a tutorial
Yang, X. 2012. Metaheuristic algorithms for self-organizing systems: a tutorial. in: Self-Adaptive and Self-Organizing Systems (SASO), 2012 IEEE Sixth International Conference on IEEE Conference Publications. pp. 249 -250
##### Integrating nature-inspired optimization algorithms to K-means clustering
Tang, R., Fong, S., Yang, X. and Deb, S. 2012. Integrating nature-inspired optimization algorithms to K-means clustering. in: Seventh International Conference on Digital Information Management (ICDIM), IEEE Conference Publications. pp. 116-123
##### Wolf search algorithm with ephemeral memory
Rui, T., Fong, S., Yang, X. and Deb, S. 2012. Wolf search algorithm with ephemeral memory. in: Seventh International Conference on Digital Information Management (ICDIM), 2012 IEEE Conference Publications. pp. 165 -172
##### Metaheuristic applications in structures and infrastructures
Gandom, A., Yang, X., Talatahari, S. and Alavi, A. 2013. Metaheuristic applications in structures and infrastructures. Elsevier.
##### Multi-objective flower algorithm for optimization
Yang, X., Karamanoglu, M. and He, X. 2013. Multi-objective flower algorithm for optimization. 2013 International Conference on Computational Science . Barcelona, Spain. 05 - 07 Jun 2013 Elsevier. pp. 861- 868 https://doi.org/10.1016/j.procs.2013.05.251
##### Optimal test sequence generation using firefly algorithm
Srivatsava, P., Mallikarjun, B. and Yang, X. 2013. Optimal test sequence generation using firefly algorithm. Swarm and Evolutionary Computation. 8, pp. 44-53. https://doi.org/10.1016/j.swevo.2012.08.003
##### Multiobjective firefly algorithm for continuous optimization
Yang, X. 2013. Multiobjective firefly algorithm for continuous optimization. Engineering with Computers. 29 (2), pp. 175-184. https://doi.org/10.1007/s00366-012-0254-1
##### Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems
Gandomi, A., Yang, X. and Alavi, A. 2013. Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Engineering with Computers. 29 (1), pp. 17-35. https://doi.org/10.1007/s00366-011-0241-y
##### Multiobjective cuckoo search for design optimization
Yang, X. and Deb, S. 2013. Multiobjective cuckoo search for design optimization. Computers and Operations Research. 40 (6), pp. 1616-1624. https://doi.org/10.1016/j.cor.2011.09.026
##### Flower pollination algorithm for global optimization
Yang, X. 2012. Flower pollination algorithm for global optimization. in: Unconventional Computation and Natural Computation: 11th International Conference, UCNC 2012, Orléan, France, September 3-7, 2012. Proceedings Berlin Springer.
##### Parameter estimation from laser flash experiment data
Wright, L., Yang, X., Matthews, C., Chapman, L. and Roberts, S. 2011. Parameter estimation from laser flash experiment data. in: Computational optimization and applications in engineering and industry studies in computational intelligence Berlin Springer.
##### Benchmark problems in structural optimization
Gandomi, A. and Yang, X. 2011. Benchmark problems in structural optimization. in: Koziel, S. and Yang, X. (ed.) Computational optimization, methods and algorithms Springer.
##### Computational optimization: an overview
Yang, X. and Koziel, S. 2011. Computational optimization: an overview. in: Computational Optimization, Methods and Algorithms Springer.
##### Engineering optimisation by cuckoo search
Yang, X. and Deb, S. 2010. Engineering optimisation by cuckoo search. International Journal of Mathematical Modelling and Numerical Optimisation. 1 (4), pp. 330 -343. https://doi.org/10.1504/IJMMNO.2010.03543
##### Engineering optimization: an introduction with metaheuristic applications
Yang, X. 2010. Engineering optimization: an introduction with metaheuristic applications. New Jersey John Wiley & Sons.
##### Metaheuristics in water, geotechnical and transport engineering
Yang, X., Gandomi, A., Talatahari, S. and Alavi, A. 2012. Metaheuristics in water, geotechnical and transport engineering. Elsevier.
##### Bat algorithm for topology optimization in microelectronic applications
Yang, X., Karamanoglu, M. and Fong, S. 2012. Bat algorithm for topology optimization in microelectronic applications. 1st International Conference on Future Generation Communication Technology. London, UK 12 - 14 Dec 2012 IEEE. pp. 150-155 https://doi.org/10.1109/FGCT.2012.6476566
##### Rare events forecasting using a residual-feedback GMDH neural network
Fong, S., Nannan, Z., Wong, R. and Yang, X. 2012. Rare events forecasting using a residual-feedback GMDH neural network. in: Seventh International Conference onDigital Information Management (ICDIM), 2012 IEEE Conference Publications. pp. 464-473
##### Cuckoo search for business optimization applications
Yang, X., Deb, S., Karamanoglu, M. and He, X. 2012. Cuckoo search for business optimization applications. National Conference on Computing and Communication Systems (NCCCS). Durgapur, West Bengal, India 21 - 22 Nov 2012 IEEE. https://doi.org/10.1109/NCCCS.2012.6412973
##### Bat algorithm for constrained optimization tasks
Gandomi, A., Yang, X., Alavi, A. and Talatahari, S. 2012. Bat algorithm for constrained optimization tasks. Neural Computing and Applications. 22 (6), pp. 1239-1255. https://doi.org/10.1007/s00521-012-1028-9
##### Efficiency analysis of swarm intelligence and randomization techniques
Yang, X. 2012. Efficiency analysis of swarm intelligence and randomization techniques. Journal of Computational and Theoretical Nanoscience. 9 (2), pp. 189-198. https://doi.org/10.1166/jctn.2012.2012
##### Modelling of a pulsating heat pipe and start up asymptotics
Yang, X. and Luan, T. 2012. Modelling of a pulsating heat pipe and start up asymptotics. Procedia Computer Science. 9, pp. 784-791. https://doi.org/10.1016/j.procs.2012.04.084
##### Computational optimization, modelling and simulation: smart algorithms and better models
Yang, X., Koziel, S. and Leifsson, L. 2012. Computational optimization, modelling and simulation: smart algorithms and better models. Procedia Computer Science. 9, pp. 852-856. https://doi.org/10.1016/j.procs.2012.04.091
##### Free lunch or no free lunch: that is not just a question?
Yang, X. 2012. Free lunch or no free lunch: that is not just a question? International Journal on Artificial Intelligence Tools. 21 (3). https://doi.org/10.1142/S0218213012400106
##### Coupled eagle strategy and differential evolution for unconstrained and constrained global optimization
Gandomi, A., Yang, X., Talatahari, S. and Deb, S. 2012. Coupled eagle strategy and differential evolution for unconstrained and constrained global optimization. Computers and Mathematics with Applications. 63 (1), pp. 191-200. https://doi.org/10.1016/j.camwa.2011.11.010
##### Bat algorithm: a novel approach for global engineering optimization
Yang, X. and Gandomi, A. 2012. Bat algorithm: a novel approach for global engineering optimization. Engineering Computations. 29 (5), pp. 464-483. https://doi.org/10.1108/02644401211235834
##### Evolutionary boundary constraint handling scheme
Gandomi, A. and Yang, X. 2012. Evolutionary boundary constraint handling scheme. Neural Computing and Applications. 21 (6), pp. 1449-1462. https://doi.org/10.1007/s00521-012-1069-0
##### Two-stage eagle strategy with differential evolution
Yang, X. and Deb, S. 2012. Two-stage eagle strategy with differential evolution. International Journal of Bio-Inspired Computation. 4 (1), pp. 1-5. https://doi.org/10.1504/IJBIC.2012.044932
##### Accelerated particle swarm optimization and support vector machine for business optimization and applications
Yang, X., Deb, S. and Fong, S. 2011. Accelerated particle swarm optimization and support vector machine for business optimization and applications. Fong, S. (ed.) NDT 2011: International Conference on Networked Digital Technologies. Macau, China 11 - 13 Jul 2011 Springer. pp. 53-66 https://doi.org/10.1007/978-3-642-22185-9_6
##### Bat algorithm for multi-objective optimisation
Yang, X. 2011. Bat algorithm for multi-objective optimisation. International Journal of Bio-Inspired Computation. 3 (5), pp. 267-274. https://doi.org/10.1504/IJBIC.2011.042259
##### Mixed variable structural optimization using Firefly Algorithm
Gandomi, A., Yang, X. and Alavi, A. 2011. Mixed variable structural optimization using Firefly Algorithm. Computers and Structures. 89 (23-24), pp. 2325-2336. https://doi.org/10.1016/j.compstruc.2011.08.002
##### A new metaheuristic bat-inspired algorithm
Yang, X. 2010. A new metaheuristic bat-inspired algorithm. in: González, J., Pelta, D., Cruz, C., Terrazas, G. and Krasnogor, N. (ed.) Nature Inspired Cooperative Strategies for Optimization (NISCO 2010) Berlin Springer.
##### Eagle strategy using Lévy walk and firefly algorithm for stochastic optimization
Yang, X. and Deb, S. 2010. Eagle strategy using Lévy walk and firefly algorithm for stochastic optimization. in: González, J., Pelta, D, Cruz, C., Terrazas, G. and Krasnogor, N. (ed.) Nature Inspired Cooperative Strategies for Optimization (NICSO 2010) Springer. pp. 101-111
##### Optimization and data mining for fracture prediction in geosciences
Shi, G. and Yang, X. 2010. Optimization and data mining for fracture prediction in geosciences. Procedia Computer Science. 1 (1), pp. 1359-1366. https://doi.org/10.1016/j.procs.2010.04.151
##### Firefly algorithm, stochastic test functions and design optimisation
Yang, X. 2010. Firefly algorithm, stochastic test functions and design optimisation. International Journal of Bio-Inspired Computation. 2 (2), pp. 78-84. https://doi.org/10.1504/IJBIC.2010.032124
##### Oil and gas assessment of the Kuqa depression of Tarim Basin in western China by simple fluid flow models of primary and secondary migrations of hydrocarbons
Shi, G., Zhang, Q., Yang, X. and Mi, S. 2010. Oil and gas assessment of the Kuqa depression of Tarim Basin in western China by simple fluid flow models of primary and secondary migrations of hydrocarbons. Journal of Petroleum Science and Engineering. 75 (1-2), pp. 77-90. https://doi.org/10.1016/j.petrol.2010.10.009
##### Firefly algorithms for multimodal optimization
Yang, X. 2009. Firefly algorithms for multimodal optimization. in: Stochastic Algorithms: Foundations and Applications; 5th International Symposium, SAGA 2009, Sapporo, Japan, October 26-28, 2009. Proceedings Springer.
##### Cuckoo search via Lévy flights
Yang, X. and Deb, S. 2009. Cuckoo search via Lévy flights. in: World Congress on Nature & Biologically Inspired Computing (NaBIC 2009) IEEE Publications. pp. 210-214