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July  2014, 10(3): 761-776. doi: 10.3934/jimo.2014.10.761

## A hydrothermal problem with non-smooth Lagrangian

 1 University of Oviedo, Department of Mathematics, E.P.I, Campus of Viesques, Gijón, 33203, Spain, Spain, Spain, Spain

Received  September 2012 Revised  June 2013 Published  November 2013

This paper deals with the optimization of a hydrothermal problem that considers a non-smooth Lagrangian $L(t ,z,z^{\prime})$. We consider a general case where the functions $L_{z^{\prime}}(t ,\cdot,\cdot)$ and $L_{z}(t ,\cdot ,\cdot)$ are discontinuous in $\{(t,z,z^{\prime})/z^{\prime}=\phi(t,z)\}$, which is the borderline point between two power generation zones. This situation arises in problems of optimization of hydrothermal systems where the thermal plant input-output curve considers the shape of the cost curve in the neighborhood of the valve points. The problem shall be formulated in the framework of nonsmooth analysis, using the generalized (or Clarke's) gradient. We shall obtain a necessary minimum condition and we shall generalize the known result (smooth transition) that the derivative of the minimum presents a constancy interval. Finally, we shall present an example.
Citation: Luis Bayón, Jose Maria Grau, Maria del Mar Ruiz, Pedro Maria Suárez. A hydrothermal problem with non-smooth Lagrangian. Journal of Industrial and Management Optimization, 2014, 10 (3) : 761-776. doi: 10.3934/jimo.2014.10.761
##### References:
 [1] N. Amjady and H. Nasiri-Rad, Solution of nonconvex and nonsmooth economic dispatch by a new Adaptive Real Coded Genetic Algorithm, Expert Syst. Appl., 37 (2010), 5239-5245. doi: 10.1016/j.eswa.2009.12.084. [2] L. Bayón, J. M. Grau, M. M. Ruiz and P. M. Suárez, Nonsmooth Optimization of Hydrothermal Problems, J. Comput. Appl. Math., 192 (2006), 11-19. doi: 10.1016/j.cam.2005.04.048. [3] L. Bayón, J. M. Grau, M. M. Ruiz and P. M. Suárez, An optimization problem in deregulated electricity markets solved with the nonsmooth maximum principle, Int. J. Comput. Math., 86 (2009), 237-249. doi: 10.1080/00207160701864483. [4] L. Bayón, J. Grau, M. M. Ruiz and P.M. Suárez, A Constrained and Nonsmooth Hydrothermal Problem, Appl. Math. Comput., 209 (2009), 10-18. doi: 10.1016/j.amc.2008.06.013. [5] L. Bayón, J. M. Grau, M. M. Ruiz and P. M. Suárez, Algorithm for calculating the analytic solution for economic dispatch with multiple fuel units, Comput. Math. Appl., 62 (2011), 2225-2234. doi: 10.1016/j.camwa.2011.07.008. [6] C. L. Chiang, Genetic algorithm for static power economic dispatch, Computer Science and Information Engineering, 2009 WRI World Congress (2009), 646-650. doi: 10.1109/CSIE.2009.440. [7] F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983. [8] L. S. Coelho and V. C. Mariani, Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with valve-point effect, IEEE Trans. Power Syst., 21 (2006), 989-996. [9] A. D. Ioffe and R. T. Rockafellar, The Euler and Weierstrass conditions for nonsmooth variational problems, Calc. Var. Partial Dif., 4 (1996), 59-87. doi: 10.1007/BF01322309. [10] D. Liu and Y. Cai, Taguchi method for solving the economic dispatch problem with nonSmooth cost functions, IEEE Trans. Power Syst., 20 (2005), 2006-2014. doi: 10.1109/TPWRS.2005.857939. [11] P. D. Loewen and R. T. Rockafellar, New necessary conditions for the generalized problem of Bolza, SIAM J. Control Optim., 34 (1996), 1496-1511. doi: 10.1137/S0363012994275932. [12] C. Marcelli, Variational problems with nonconvex, noncoercive, highly discontinuous integrands: characterization and existence of minimizers, SIAM J. Control Optim., 40 (2002), 1473-1490. doi: 10.1137/S036301299936141X. [13] C. Marcelli, E. Outkine and M. Sytchev, Remarks on necessary conditions for minimizers of one-dimensional variational problems, Nonlinear Anal., 48 (2002), 979-993. doi: 10.1016/S0362-546X(00)00228-5. [14] J. B. Park, K. S. Lee, J. R. Shin and K. Y. Lee, A particle swarm optimization for economic dispatch with nonsmooth cost functions, IEEE Trans. Power Syst., 20 (2005), 34-42. doi: 10.1109/TPWRS.2004.831275. [15] J. L. Troutman, Variational Calculus with Elementary Convexity, Springer, New York, 1983. [16] M. T. Tsai, H. J. Gow and W. M. Lin, A novel stochastic search method for the solution of economic dispatch problems with non-convex fuel cost functions, Int. J. Elec. Power, 33 (2011), 1070-1076. doi: 10.1016/j.ijepes.2011.01.026. [17] R. Vinter and H. Zheng, The extended Euler-Lagrange condition for nonconvex variational problems, SIAM J. Control Optim., 35 (1997), 56-77. doi: 10.1137/S0363012995283133. [18] A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control, Wiley-Interscience, 1996. doi: 10.1016/0140-6701(96)88715-7. [19] X. Yuan, L. Wang, Y. Zhang and Y. Yuan, A hybrid differential evolution method for dynamic economic dispatch with valve-point effects, Expert Syst. Appl., 36 (2009), 4042-4048. doi: 10.1016/j.eswa.2008.03.006. [20] K. Zare, M. T. Haque and E. Davoodi, Solving non-convex economic dispatch problem with valve point effects using modified group search optimizer method, Electr. Pow. Syst. Res., 84 (2012), 83-89. doi: 10.1016/j.epsr.2011.10.004.

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##### References:
 [1] N. Amjady and H. Nasiri-Rad, Solution of nonconvex and nonsmooth economic dispatch by a new Adaptive Real Coded Genetic Algorithm, Expert Syst. Appl., 37 (2010), 5239-5245. doi: 10.1016/j.eswa.2009.12.084. [2] L. Bayón, J. M. Grau, M. M. Ruiz and P. M. Suárez, Nonsmooth Optimization of Hydrothermal Problems, J. Comput. Appl. Math., 192 (2006), 11-19. doi: 10.1016/j.cam.2005.04.048. [3] L. Bayón, J. M. Grau, M. M. Ruiz and P. M. Suárez, An optimization problem in deregulated electricity markets solved with the nonsmooth maximum principle, Int. J. Comput. Math., 86 (2009), 237-249. doi: 10.1080/00207160701864483. [4] L. Bayón, J. Grau, M. M. Ruiz and P.M. Suárez, A Constrained and Nonsmooth Hydrothermal Problem, Appl. Math. Comput., 209 (2009), 10-18. doi: 10.1016/j.amc.2008.06.013. [5] L. Bayón, J. M. Grau, M. M. Ruiz and P. M. Suárez, Algorithm for calculating the analytic solution for economic dispatch with multiple fuel units, Comput. Math. Appl., 62 (2011), 2225-2234. doi: 10.1016/j.camwa.2011.07.008. [6] C. L. Chiang, Genetic algorithm for static power economic dispatch, Computer Science and Information Engineering, 2009 WRI World Congress (2009), 646-650. doi: 10.1109/CSIE.2009.440. [7] F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983. [8] L. S. Coelho and V. C. Mariani, Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with valve-point effect, IEEE Trans. Power Syst., 21 (2006), 989-996. [9] A. D. Ioffe and R. T. Rockafellar, The Euler and Weierstrass conditions for nonsmooth variational problems, Calc. Var. Partial Dif., 4 (1996), 59-87. doi: 10.1007/BF01322309. [10] D. Liu and Y. Cai, Taguchi method for solving the economic dispatch problem with nonSmooth cost functions, IEEE Trans. Power Syst., 20 (2005), 2006-2014. doi: 10.1109/TPWRS.2005.857939. [11] P. D. Loewen and R. T. Rockafellar, New necessary conditions for the generalized problem of Bolza, SIAM J. Control Optim., 34 (1996), 1496-1511. doi: 10.1137/S0363012994275932. [12] C. Marcelli, Variational problems with nonconvex, noncoercive, highly discontinuous integrands: characterization and existence of minimizers, SIAM J. Control Optim., 40 (2002), 1473-1490. doi: 10.1137/S036301299936141X. [13] C. Marcelli, E. Outkine and M. Sytchev, Remarks on necessary conditions for minimizers of one-dimensional variational problems, Nonlinear Anal., 48 (2002), 979-993. doi: 10.1016/S0362-546X(00)00228-5. [14] J. B. Park, K. S. Lee, J. R. Shin and K. Y. Lee, A particle swarm optimization for economic dispatch with nonsmooth cost functions, IEEE Trans. Power Syst., 20 (2005), 34-42. doi: 10.1109/TPWRS.2004.831275. [15] J. L. Troutman, Variational Calculus with Elementary Convexity, Springer, New York, 1983. [16] M. T. Tsai, H. J. Gow and W. M. Lin, A novel stochastic search method for the solution of economic dispatch problems with non-convex fuel cost functions, Int. J. Elec. Power, 33 (2011), 1070-1076. doi: 10.1016/j.ijepes.2011.01.026. [17] R. Vinter and H. Zheng, The extended Euler-Lagrange condition for nonconvex variational problems, SIAM J. Control Optim., 35 (1997), 56-77. doi: 10.1137/S0363012995283133. [18] A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control, Wiley-Interscience, 1996. doi: 10.1016/0140-6701(96)88715-7. [19] X. Yuan, L. Wang, Y. Zhang and Y. Yuan, A hybrid differential evolution method for dynamic economic dispatch with valve-point effects, Expert Syst. Appl., 36 (2009), 4042-4048. doi: 10.1016/j.eswa.2008.03.006. [20] K. Zare, M. T. Haque and E. Davoodi, Solving non-convex economic dispatch problem with valve point effects using modified group search optimizer method, Electr. Pow. Syst. Res., 84 (2012), 83-89. doi: 10.1016/j.epsr.2011.10.004.

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