International Journal of Nonlinear Analysis and Applications
Optimization of the nonlinear model of neural network training in predicting thermal efficiency of solar concentrator with simulated annealing algorithm
Document Type : Research Paper
Authors
1 Faculty of Mechanical Engineering, Semnan University, Semnan, Iran
2 School of Engineering, Damghan University, Damghan, Iran
Abstract
Nowadays, artificial neural networks are widely used to solve large-scale and complex problems. The purpose of this study is to use artificial intelligence techniques such as artificial neural networks and simulated annealing algorithm, to optimize the solar energy system in order to maximize its economic benefits. Here, a new nonlinear computational model has been presented to predict the thermal performance of compound parabolic concentrator (CPC). In this regard, three models of artificial neural network (ANN) including radial-basis function (RBFANN), multi-layer perception (MLPANN) as well as adaptive neuro fuzzy inference system (ANFIS) are used to identify the nonlinear relationship between input and output parameters of the system. The optimal nonlinear structure of the model is determined through the simulated annealing (SA) method. Validation of the proposed models is performed on a CPC. The results show that all the three models are efficient. In particular, statistical analyses show that the ANFIS model is more accurate in the process of predicting thermal efficiency. So, the given models can be effectively applied in practical fields.
Keywords
[1] E.H.L. Aarts and P.J.M. van Laarhoven, Simulated annealing, Simulated Annealing: Theory and applications,
Springer, 1987.
[2] M.H. Ahmadi, M.A. Ahmadi, R. Bayat, M. Ashouri and M. Feidt, Thermo-economic optimization of Stirling heat
pump by using non-dominated sorting genetic algorithm, Energy Conver. Manag. 91 (2015), 315–322.
[3] M.H. Ahmadi, M.A. Ahmadi, F. Pourfayaz and M. Bidi, Thermodynamic analysis and optimization for an irreversible heat pump working on reversed Brayton cycle, Energy Conver. Manag. 110 (2016), 260–267.
[4] S. Babaie-Kafaki, R. Ghanbari and N. Mahdavi-Amiri, Two effective hybrid metaheuristic algorithms for minimization of multimodal functions, Int. J. Comput. Math. 88 (2011), no. 11, 2415–2428.
[5] S. Babaie-Kafaki, R. Ghanbari and N. Mahdavi-Amiri, An efficient and practically robust hybrid metaheuristic
algorithm for solving fuzzy bus terminal location problems, Asia-Pacific J. Oper. Res. 29 (2012), no. 2, 1250009.
[6] S. Babaie-Kafaki, R. Ghanbari and N. Mahdavi-Amiri, Hybridizations of genetic algorithms and neighborhood
search metaheuristics for fuzzy bus terminal location problems, Appl. Soft Comput. 46 (2016), 220–229.
[7] S. Babaie-Kafaki and S. Rezaee, A randomized nonmonotone adaptive trust region method based on the simulated
annealing strategy for unconstrained optimization, Int. J. Intell. Comput. Cyber. 12 (2019), no. 3, 389–399.
[8] S. Babaie-Kafaki and S. Rezaee, A randomized adaptive trust region line search method, Int. J. Optim. Control:
Theor. Appl. (IJOCTA) 10 (2020), no. 2, 259–263.
[9] D. Bertsimas and J.N. Tsitsiklis, Introduction to linear optimization, vol. 6, Athena Scientific Belmont, MA, 1997.[10] P.A. Castillo, J. Carpio, J.J. Merelo, A. Prieto, V. Rivas and G. Romero, G-Prop: Global optimization of multilayer
perceptrons using GAs, Neurocomput. 35 (2000), no. 1-4, 149–163.
[11] Y. Da and X.R. Ge, An improved PSO-based ANN with simulated annealing technique, Neurocomput. 63 (2005),
527–533.
[12] J.A. Duffe and W.A. Beckham, Solar engineering of thermal processes, 3, Wiley New York, 2006.
[13] W.L. Dong, X. Li and Z. Peng, A simulated annealing-based Barzilai–Borwein gradient method for unconstrained
optimization problems, Asia-Pacific J. Oper. Res. 36 (2019), no. 4, 195–207.
[14] B. Hajek, Cooling schedules for optimal annealing, Math. Oper. Res. 13 (1988), no. 2, 311–329.
[15] D. Henderson, Sh. H. Jacobson and A.W. Johnson, The theory and practice of simulated annealing, Handbook of
metaheuristics, Springer, 2003.
[16] J.S. Jang, ANFIS: adaptive-network-based fuzzy inference system, IEEE Trans. Syst. Man Cyber. 23 (1993), no.
3, 665–685.
[17] J.S.R. Jang, C.T. Sun and E. Mizutani, Neuro-fuzzy and soft computing: A computational approach to learning
and machine intelligence, IEEE Trans. Automatic Control 42 (1997), no. 10, 1482–1484.
[18] M. Kahani, M.H. Ahmadi, A. Tatar and M. Sadeghzadeh, Development of multilayer perceptron artificial neural
network (MLP-ANN) and least square support vector machine (LSSVM) models to predict Nusselt number and
pressure drop of TiO2/water nanofluid flows through non-straight pathways, Numer. Heat Transfer Part A: Appl.
74 (2018), no. 4, 1190–1206.
[19] D. Karaboga and B. Basturk, A powerful and efficient algorithm for numericalfunction optimization: artificial
bee colony (ABC) algorithm, J. Glob. Optim. 39 (2007), no. 3, 459–471.
[20] A. Kasaeian, M. Ghalamchi, M.H. Ahmadi and M. Ghalamchi, GMDH algorithm for modeling the outlet temperatures of a solar chimney based on the ambient temperature, Mech. Ind. 18 (2017), no. 2, 216.
[21] A.Y.S. Lam and V.O.K. Li, Chemical-reaction-inspired metaheuristic for optimization, IEEE Trans. Evo. Comput.
14 (2009), no. 3, 381–399.
[22] T. Liao, K. Socha, M.A. Montes de Oca, T. Sttzle and M. Dorigo, Ant colony optimization for mixed-variable
optimization problems, IEEE Trans. Evo. Comput. 18 (2013), no. 4, 503–518.
[23] R. Loni, A. Kasaeian, K. Shahverdi, E. Askari Asli-Ardeh, B. Ghobadian and M.H. Ahmadi, ANN model to
predict the performance of parabolic dish collector with tubular cavity receiver, Mech. Ind. 18 (2017), no. 4, 408.
[24] I.G. Manuel, R.P. Luis and H.L. Jesus ,Evaluation of thermal parameters and simulation of a solar-powered,
solid-soition chiller whith a CPC collector, Renewable Energy 34 (2009), no. 3, 570–577.
[25] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, Equation of state calculations by fast computing
machines, J. Chem. Phys. 21 (1953), no. 6, 1087–1092.
[26] N.A. Mohammad, Sh.R. Aliakbari and A.H. Eshraghniaye Jahromi, A particle swarm-BFGS algorithm for nonlinear programming problems, Comput. Oper. Res. 40 (2013), no. 4, 963–972.
[27] A. Mohebbi, M. Taheri and A. Soltani, A neural network for predicting saturated liquid density using genetic
algorithm for pure and mixed refrigerants, Int. J. Refrig. 31 (2008), no. 8, 1317–1327.
[28] M. Mustapha, M.W. Mustafa, S.N. Khalid, I. Abubakar and A.M. Abdilahi, Correlation and wavelet-based shortterm load forecasting using Anfis, Indian J. Sci. Technol. 9 (2016), no. 46, 1–8.
[29] J.M. Ortiz-Rodriguez, M. del Rosario Martinez-Blanco and H. Vega-Carrillo, Evolutionary artificial neural networks in neutron spectrometry, IntechOpen, 2011.
[30] M.M. Papari, F. Yousefi, J. Moghadasi, H. Karimi and A. Campo, Modeling thermal conductivity augmentation
of nanofluids using diffusion neural networks, Int. J. Thermal Sci. 50 (2011), no. 1, 44–52.
[31] I. Pence, M.C. Cesmeli, F.A. Senel and B. Cetisli, A new unconstrained global optimization method based on
clustering and parabolic approximation, Expert Syst. Appl. 55 (2016), 493–507.
[32] R. Prasad Parouha and K. Nath Das, A memory based differential evolution algorithm for unconstrained opti-mization, Appl. Soft Comput. 38 (2016), 501–517.
[33] E. Rashedi, H. Nezamabadi–pour and S. Saryazdi, GSA: a gravitational search algorithm, Information sciences,
179 (2009), no. 13, 2232–2248.
[34] C.R. Reeves, Modern heuristic techniques, Modern Heuristic Search Meth. 1 (1996), 1–25.
[35] R.M. Rizk-Allah, E.M. Zaki and A.A. El-Sawy, Hybridizing ant colony optimization with firefly algorithm for
unconstrained optimization problems, Appl. Math. Comput. 224 (2013), 473–483.
[36] M. Roozbeh, S. Babaie-Kafaki and A. Naeimi Sadigh, A heuristic approach to combat multicollinearity in least
trimmed squares regression analysis, Appl. Math. Model. 57 (2018), 105–120.
[37] S.A. Sadatsakkak, M.H. Ahmadi, R. Bayat, S.M. Pourkiaei and M. Feidt, Optimization density power and thermal
efficiency of an endoreversible Braysson cycle by using non-dominated sorting genetic algorithm, Energy Conver.
Manag. 93 (2015), 31–39.
[38] J.R.M. Smits, W.J. Melssen, L.M.C. Buydens and G. Kateman, Using artificial neural networks for solving
chemical problems: part I. Multi-layer feed-forward networks, , Chemom. Intell. Lab. Syst. 22 (1994), no. 2,
165–189.
[39] M. Tahani, M. Vakili and S. Khosrojerdi, Experimental Evaluation and ANN Modeling of Thermal Conductivity
of Graphene Oxide Nanoplatelets/Deionized Water Nanofluid, Int. Commun. Heat Mass Transfer 76 (2016), no.
76, 358–365.
[40] S. Toghyani, M.H. Ahmadi, A. Kasaeian, A.H. Mohammadi, Artificial neural network, ANN-PSO and ANN-ICA
for modelling the Stirling engine, Int. J. Ambient Energy 37 (2016), no. 5, 456–468.
[41] M.D. Toksari, Ant colony optimization for finding the global minimum, Appl. Math. Comput. 176 (2006), no. 1,
308–316.
[42] M.D. Toksari, A heuristic approach to find the global optimum of function, J. Comput. Appl. Math. 209 (2007),
no. 2, 160–166.
[43] M.D. Toksari, Minimizing the multimodal functions with ant colony optimization approach, Expert Syst. Appl.
36 (2009), no. 3, 6030–6035.
[44] M.D. Toksari and E. Guner, Solving the unconstrained optimization problem by a variable neighborhood search,
J. Math. Anal. Appl. 328 (2007), no. 2, 1178–1187.
[45] V.J.P. Vilar, L.X. Pinho, A.M.A. Pintor and R. Boaventura, Treatment of textile waste waters by solar-driven
advanced oxidation processes, Solar Energy 85 (2011), no. 9, 1927–1934.
[46] Z. Wang, C.D. Massimo, M.T. Tham and A.J. Morris, A procedure for determining the topology of multilayer feed
forward neural networks, Neural Networks 7 (1994), no. 2, 291–300.
[47] X.S. Yang, Nature-inspired optimization algorithms, J. Comput. Sci. 46 (2020), 101–104.
Springer, 1987.
[2] M.H. Ahmadi, M.A. Ahmadi, R. Bayat, M. Ashouri and M. Feidt, Thermo-economic optimization of Stirling heat
pump by using non-dominated sorting genetic algorithm, Energy Conver. Manag. 91 (2015), 315–322.
[3] M.H. Ahmadi, M.A. Ahmadi, F. Pourfayaz and M. Bidi, Thermodynamic analysis and optimization for an irreversible heat pump working on reversed Brayton cycle, Energy Conver. Manag. 110 (2016), 260–267.
[4] S. Babaie-Kafaki, R. Ghanbari and N. Mahdavi-Amiri, Two effective hybrid metaheuristic algorithms for minimization of multimodal functions, Int. J. Comput. Math. 88 (2011), no. 11, 2415–2428.
[5] S. Babaie-Kafaki, R. Ghanbari and N. Mahdavi-Amiri, An efficient and practically robust hybrid metaheuristic
algorithm for solving fuzzy bus terminal location problems, Asia-Pacific J. Oper. Res. 29 (2012), no. 2, 1250009.
[6] S. Babaie-Kafaki, R. Ghanbari and N. Mahdavi-Amiri, Hybridizations of genetic algorithms and neighborhood
search metaheuristics for fuzzy bus terminal location problems, Appl. Soft Comput. 46 (2016), 220–229.
[7] S. Babaie-Kafaki and S. Rezaee, A randomized nonmonotone adaptive trust region method based on the simulated
annealing strategy for unconstrained optimization, Int. J. Intell. Comput. Cyber. 12 (2019), no. 3, 389–399.
[8] S. Babaie-Kafaki and S. Rezaee, A randomized adaptive trust region line search method, Int. J. Optim. Control:
Theor. Appl. (IJOCTA) 10 (2020), no. 2, 259–263.
[9] D. Bertsimas and J.N. Tsitsiklis, Introduction to linear optimization, vol. 6, Athena Scientific Belmont, MA, 1997.[10] P.A. Castillo, J. Carpio, J.J. Merelo, A. Prieto, V. Rivas and G. Romero, G-Prop: Global optimization of multilayer
perceptrons using GAs, Neurocomput. 35 (2000), no. 1-4, 149–163.
[11] Y. Da and X.R. Ge, An improved PSO-based ANN with simulated annealing technique, Neurocomput. 63 (2005),
527–533.
[12] J.A. Duffe and W.A. Beckham, Solar engineering of thermal processes, 3, Wiley New York, 2006.
[13] W.L. Dong, X. Li and Z. Peng, A simulated annealing-based Barzilai–Borwein gradient method for unconstrained
optimization problems, Asia-Pacific J. Oper. Res. 36 (2019), no. 4, 195–207.
[14] B. Hajek, Cooling schedules for optimal annealing, Math. Oper. Res. 13 (1988), no. 2, 311–329.
[15] D. Henderson, Sh. H. Jacobson and A.W. Johnson, The theory and practice of simulated annealing, Handbook of
metaheuristics, Springer, 2003.
[16] J.S. Jang, ANFIS: adaptive-network-based fuzzy inference system, IEEE Trans. Syst. Man Cyber. 23 (1993), no.
3, 665–685.
[17] J.S.R. Jang, C.T. Sun and E. Mizutani, Neuro-fuzzy and soft computing: A computational approach to learning
and machine intelligence, IEEE Trans. Automatic Control 42 (1997), no. 10, 1482–1484.
[18] M. Kahani, M.H. Ahmadi, A. Tatar and M. Sadeghzadeh, Development of multilayer perceptron artificial neural
network (MLP-ANN) and least square support vector machine (LSSVM) models to predict Nusselt number and
pressure drop of TiO2/water nanofluid flows through non-straight pathways, Numer. Heat Transfer Part A: Appl.
74 (2018), no. 4, 1190–1206.
[19] D. Karaboga and B. Basturk, A powerful and efficient algorithm for numericalfunction optimization: artificial
bee colony (ABC) algorithm, J. Glob. Optim. 39 (2007), no. 3, 459–471.
[20] A. Kasaeian, M. Ghalamchi, M.H. Ahmadi and M. Ghalamchi, GMDH algorithm for modeling the outlet temperatures of a solar chimney based on the ambient temperature, Mech. Ind. 18 (2017), no. 2, 216.
[21] A.Y.S. Lam and V.O.K. Li, Chemical-reaction-inspired metaheuristic for optimization, IEEE Trans. Evo. Comput.
14 (2009), no. 3, 381–399.
[22] T. Liao, K. Socha, M.A. Montes de Oca, T. Sttzle and M. Dorigo, Ant colony optimization for mixed-variable
optimization problems, IEEE Trans. Evo. Comput. 18 (2013), no. 4, 503–518.
[23] R. Loni, A. Kasaeian, K. Shahverdi, E. Askari Asli-Ardeh, B. Ghobadian and M.H. Ahmadi, ANN model to
predict the performance of parabolic dish collector with tubular cavity receiver, Mech. Ind. 18 (2017), no. 4, 408.
[24] I.G. Manuel, R.P. Luis and H.L. Jesus ,Evaluation of thermal parameters and simulation of a solar-powered,
solid-soition chiller whith a CPC collector, Renewable Energy 34 (2009), no. 3, 570–577.
[25] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, Equation of state calculations by fast computing
machines, J. Chem. Phys. 21 (1953), no. 6, 1087–1092.
[26] N.A. Mohammad, Sh.R. Aliakbari and A.H. Eshraghniaye Jahromi, A particle swarm-BFGS algorithm for nonlinear programming problems, Comput. Oper. Res. 40 (2013), no. 4, 963–972.
[27] A. Mohebbi, M. Taheri and A. Soltani, A neural network for predicting saturated liquid density using genetic
algorithm for pure and mixed refrigerants, Int. J. Refrig. 31 (2008), no. 8, 1317–1327.
[28] M. Mustapha, M.W. Mustafa, S.N. Khalid, I. Abubakar and A.M. Abdilahi, Correlation and wavelet-based shortterm load forecasting using Anfis, Indian J. Sci. Technol. 9 (2016), no. 46, 1–8.
[29] J.M. Ortiz-Rodriguez, M. del Rosario Martinez-Blanco and H. Vega-Carrillo, Evolutionary artificial neural networks in neutron spectrometry, IntechOpen, 2011.
[30] M.M. Papari, F. Yousefi, J. Moghadasi, H. Karimi and A. Campo, Modeling thermal conductivity augmentation
of nanofluids using diffusion neural networks, Int. J. Thermal Sci. 50 (2011), no. 1, 44–52.
[31] I. Pence, M.C. Cesmeli, F.A. Senel and B. Cetisli, A new unconstrained global optimization method based on
clustering and parabolic approximation, Expert Syst. Appl. 55 (2016), 493–507.
[32] R. Prasad Parouha and K. Nath Das, A memory based differential evolution algorithm for unconstrained opti-mization, Appl. Soft Comput. 38 (2016), 501–517.
[33] E. Rashedi, H. Nezamabadi–pour and S. Saryazdi, GSA: a gravitational search algorithm, Information sciences,
179 (2009), no. 13, 2232–2248.
[34] C.R. Reeves, Modern heuristic techniques, Modern Heuristic Search Meth. 1 (1996), 1–25.
[35] R.M. Rizk-Allah, E.M. Zaki and A.A. El-Sawy, Hybridizing ant colony optimization with firefly algorithm for
unconstrained optimization problems, Appl. Math. Comput. 224 (2013), 473–483.
[36] M. Roozbeh, S. Babaie-Kafaki and A. Naeimi Sadigh, A heuristic approach to combat multicollinearity in least
trimmed squares regression analysis, Appl. Math. Model. 57 (2018), 105–120.
[37] S.A. Sadatsakkak, M.H. Ahmadi, R. Bayat, S.M. Pourkiaei and M. Feidt, Optimization density power and thermal
efficiency of an endoreversible Braysson cycle by using non-dominated sorting genetic algorithm, Energy Conver.
Manag. 93 (2015), 31–39.
[38] J.R.M. Smits, W.J. Melssen, L.M.C. Buydens and G. Kateman, Using artificial neural networks for solving
chemical problems: part I. Multi-layer feed-forward networks, , Chemom. Intell. Lab. Syst. 22 (1994), no. 2,
165–189.
[39] M. Tahani, M. Vakili and S. Khosrojerdi, Experimental Evaluation and ANN Modeling of Thermal Conductivity
of Graphene Oxide Nanoplatelets/Deionized Water Nanofluid, Int. Commun. Heat Mass Transfer 76 (2016), no.
76, 358–365.
[40] S. Toghyani, M.H. Ahmadi, A. Kasaeian, A.H. Mohammadi, Artificial neural network, ANN-PSO and ANN-ICA
for modelling the Stirling engine, Int. J. Ambient Energy 37 (2016), no. 5, 456–468.
[41] M.D. Toksari, Ant colony optimization for finding the global minimum, Appl. Math. Comput. 176 (2006), no. 1,
308–316.
[42] M.D. Toksari, A heuristic approach to find the global optimum of function, J. Comput. Appl. Math. 209 (2007),
no. 2, 160–166.
[43] M.D. Toksari, Minimizing the multimodal functions with ant colony optimization approach, Expert Syst. Appl.
36 (2009), no. 3, 6030–6035.
[44] M.D. Toksari and E. Guner, Solving the unconstrained optimization problem by a variable neighborhood search,
J. Math. Anal. Appl. 328 (2007), no. 2, 1178–1187.
[45] V.J.P. Vilar, L.X. Pinho, A.M.A. Pintor and R. Boaventura, Treatment of textile waste waters by solar-driven
advanced oxidation processes, Solar Energy 85 (2011), no. 9, 1927–1934.
[46] Z. Wang, C.D. Massimo, M.T. Tham and A.J. Morris, A procedure for determining the topology of multilayer feed
forward neural networks, Neural Networks 7 (1994), no. 2, 291–300.
[47] X.S. Yang, Nature-inspired optimization algorithms, J. Comput. Sci. 46 (2020), 101–104.
)
Volume 13, Issue 2
July 2022Pages 2947-2960
- Receive Date: 29 December 2020
- Revise Date: 02 March 2021
- Accept Date: 04 March 2021