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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 & 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.  doi: 10.1016/j.eswa.2009.12.084.  Google Scholar

[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.  doi: 10.1016/j.cam.2005.04.048.  Google Scholar

[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.  doi: 10.1080/00207160701864483.  Google Scholar

[4]

L. Bayón, J. Grau, M. M. Ruiz and P.M. Suárez, A Constrained and Nonsmooth Hydrothermal Problem,, Appl. Math. Comput., 209 (2009), 10.  doi: 10.1016/j.amc.2008.06.013.  Google Scholar

[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.  doi: 10.1016/j.camwa.2011.07.008.  Google Scholar

[6]

C. L. Chiang, Genetic algorithm for static power economic dispatch,, Computer Science and Information Engineering, (2009), 646.  doi: 10.1109/CSIE.2009.440.  Google Scholar

[7]

F. H. Clarke, Optimization and Nonsmooth Analysis,, John Wiley & Sons, (1983).   Google Scholar

[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.   Google Scholar

[9]

A. D. Ioffe and R. T. Rockafellar, The Euler and Weierstrass conditions for nonsmooth variational problems,, Calc. Var. Partial Dif., 4 (1996), 59.  doi: 10.1007/BF01322309.  Google Scholar

[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.  doi: 10.1109/TPWRS.2005.857939.  Google Scholar

[11]

P. D. Loewen and R. T. Rockafellar, New necessary conditions for the generalized problem of Bolza,, SIAM J. Control Optim., 34 (1996), 1496.  doi: 10.1137/S0363012994275932.  Google Scholar

[12]

C. Marcelli, Variational problems with nonconvex, noncoercive, highly discontinuous integrands: characterization and existence of minimizers,, SIAM J. Control Optim., 40 (2002), 1473.  doi: 10.1137/S036301299936141X.  Google Scholar

[13]

C. Marcelli, E. Outkine and M. Sytchev, Remarks on necessary conditions for minimizers of one-dimensional variational problems,, Nonlinear Anal., 48 (2002), 979.  doi: 10.1016/S0362-546X(00)00228-5.  Google Scholar

[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.  doi: 10.1109/TPWRS.2004.831275.  Google Scholar

[15]

J. L. Troutman, Variational Calculus with Elementary Convexity,, Springer, (1983).   Google Scholar

[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.  doi: 10.1016/j.ijepes.2011.01.026.  Google Scholar

[17]

R. Vinter and H. Zheng, The extended Euler-Lagrange condition for nonconvex variational problems,, SIAM J. Control Optim., 35 (1997), 56.  doi: 10.1137/S0363012995283133.  Google Scholar

[18]

A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control,, Wiley-Interscience, (1996).  doi: 10.1016/0140-6701(96)88715-7.  Google Scholar

[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.  doi: 10.1016/j.eswa.2008.03.006.  Google Scholar

[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.  doi: 10.1016/j.epsr.2011.10.004.  Google Scholar

show all references

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.  doi: 10.1016/j.eswa.2009.12.084.  Google Scholar

[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.  doi: 10.1016/j.cam.2005.04.048.  Google Scholar

[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.  doi: 10.1080/00207160701864483.  Google Scholar

[4]

L. Bayón, J. Grau, M. M. Ruiz and P.M. Suárez, A Constrained and Nonsmooth Hydrothermal Problem,, Appl. Math. Comput., 209 (2009), 10.  doi: 10.1016/j.amc.2008.06.013.  Google Scholar

[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.  doi: 10.1016/j.camwa.2011.07.008.  Google Scholar

[6]

C. L. Chiang, Genetic algorithm for static power economic dispatch,, Computer Science and Information Engineering, (2009), 646.  doi: 10.1109/CSIE.2009.440.  Google Scholar

[7]

F. H. Clarke, Optimization and Nonsmooth Analysis,, John Wiley & Sons, (1983).   Google Scholar

[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.   Google Scholar

[9]

A. D. Ioffe and R. T. Rockafellar, The Euler and Weierstrass conditions for nonsmooth variational problems,, Calc. Var. Partial Dif., 4 (1996), 59.  doi: 10.1007/BF01322309.  Google Scholar

[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.  doi: 10.1109/TPWRS.2005.857939.  Google Scholar

[11]

P. D. Loewen and R. T. Rockafellar, New necessary conditions for the generalized problem of Bolza,, SIAM J. Control Optim., 34 (1996), 1496.  doi: 10.1137/S0363012994275932.  Google Scholar

[12]

C. Marcelli, Variational problems with nonconvex, noncoercive, highly discontinuous integrands: characterization and existence of minimizers,, SIAM J. Control Optim., 40 (2002), 1473.  doi: 10.1137/S036301299936141X.  Google Scholar

[13]

C. Marcelli, E. Outkine and M. Sytchev, Remarks on necessary conditions for minimizers of one-dimensional variational problems,, Nonlinear Anal., 48 (2002), 979.  doi: 10.1016/S0362-546X(00)00228-5.  Google Scholar

[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.  doi: 10.1109/TPWRS.2004.831275.  Google Scholar

[15]

J. L. Troutman, Variational Calculus with Elementary Convexity,, Springer, (1983).   Google Scholar

[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.  doi: 10.1016/j.ijepes.2011.01.026.  Google Scholar

[17]

R. Vinter and H. Zheng, The extended Euler-Lagrange condition for nonconvex variational problems,, SIAM J. Control Optim., 35 (1997), 56.  doi: 10.1137/S0363012995283133.  Google Scholar

[18]

A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control,, Wiley-Interscience, (1996).  doi: 10.1016/0140-6701(96)88715-7.  Google Scholar

[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.  doi: 10.1016/j.eswa.2008.03.006.  Google Scholar

[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.  doi: 10.1016/j.epsr.2011.10.004.  Google Scholar

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