Abimbola M. Jubril, Philip O. Ogunbona

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The emission function associated with the economic-emission dispatch problem contains exponential functions that model the emission pollutants. This paper presents a strategy of solving the economic-emission dispatch problem whereby the exponential function is approximated by a rational function that permits reduction to a standard polynomial optimization problem. This is reformulated as a hierarchy of semidefinite relaxation problems using the moment theory and the resulting SDP problem is solved. Different degrees of rational functional approximation were considered. The approach was tested on the IEEE 30-bus test systems to investigate its effectiveness. Solutions obtained were compared with those from some of the well known evolutionary methods. Results showed that SDP has inherently good convergence property and a lower but comparable diversity property.


rational polynomial approximation, semidefinite program, multiobjective optimization, economic dispatch, emission dispatch


multiobjective optimization; polynomial optimization; rational function approximations; economic dispatch; emission dispatch; semidefinite programming; weighted sum method

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M. A. Abido: A niched pareto genetic algorithm for multiobjective environmental/economic dispatch Int. Jour. Elect. Power Ener. Sys.,25(2),2003, 97–105.

——, A novel multiobjective evolutionary algorithm for environmental/economic power dispatch, Elec. Power Sys. Res., 65(1), 2003, 71–81.

——, Environmental/economic power dispatch using multiobjective evolutionary algorithms, IEEE Trans. Power Syst.,18(4), 2003, 1529–1537.

J. B. Park, K. S. Lee, J. R. Shin, and K. Y. Lee: A particle swarm optimization for economic dispatch with nonsmooth cost function, IEEE Trans. Power Syst. 20(1),2005, 34–42.

J. G. Vlachogiannis and K. Y. Lee: Economic load dispatch: a comparative study on heuristic optimization techniques with an improved coordinated aggregationbased pso, IEEE Trans. Power Syst.24(2), 2009, 99–1001.

I. A. Farhat and M. E. El-Hawary: Multi-objective economic-emission optimal load dispatch using bacterial foraging algorithm. In ser. 25th IEEE Canadian Conf. Elect. and Comp. Eng. (CCECE), 2012, pp. 1–5.

R. allah Hooshmand, M. Parastegari, and M. J. Morshed: Emission, reserve and economic load dispatch problem with non-smooth and non-convex cost functions using the hybrid bacterial foraging-nelder- mead algorithm, Appl. Ener. 89(1),2012, 443–453.

K. K. Mandal and N. Chakraborty: Effect of control parameters on differential evolution based combined economic emission dispatch with valve-point loading and transmission loss, Int J. Emerg. Elect Power Sys. 9(4), 2008, 1–18.

M. Madrigal and H. Quintana: Semidefinite programming relaxations for 0,1 power dispatch problems. In Proc. IEEE Power Eng. Soc. Summer Meeting Conf., Edmonton, Canada, July 1999, pp. 697–701.

R. Fuentes-Loyola and V. H. Quintana: Medium-term hydothermal coordination by semidefinite programming, IEEE Trans. Power Syst.,18(4),2003, 1515–1522.

X. Bai, H. Wei, K. Fujisawa, and Y. Wang:Semidefinite programming for optimal power flow problem, Int. Jour. Elect. Power Ener. Sys. 30,2008, 383–392.

D. Jibetean: Global optimization of rational multivariate functions Centrum voor Wiskunde en Informatica, PNA, Tech. Rep. PNA-R0120, October 2001.

O. Devolder, F. Glineur, and Y. Nesterov: Solving infinite-dimensional optimization problems by polynomial approximation, Universite catholique de Louvain, Center for Operations Research and Econometrics, Belgium, discussion paper 2010/29, June 2010.

J. B. Lasserre: Global optimization with polynomials and the problem of moments, SIAM Jour. Opt.,11(3),2001, 796–817.

B. P. Flannery, S. Teukolsky, W. H. Press, and W. T. Vetterling: Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press, 1992.

F. Guo, L. Wang, and G. Zhou: Minimizing rational functions by exact jacobian sdp relaxation applicable to finite singularities, Tech. Rep. arXiv:1205.6442v1 [math.OC], May 2012.

P. A. Parrilo and B. Sturmfel: Minimizing polynomial functions. In DIMACS Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science, Basu and L. Gonzalez-Vega, Eds., 60. American Mathemtical Society, March 2003, pp. 83–100.

S. Boyd and L. Vandenberghe: Semidefinite programming relaxations of nonconvex problems in control and combinatorial optimization In: Communications, Computation, Control and Signal Processing: A Tribute to Thomas Kailath, A. Paulraj, V. Roychowdhuri, and C. Schaper, Eds., Kluwer, 1997, ch. 15, pp. 279–288.

R. T. Marler and S. J. Arora: The weighted sum method for multiobjective optimization: New insights, Structural and Multidisciplinary Optimization, 41(6),2009, 853–862.

A. M. Jubril: A nonlinear weight selection in weighted sum for convex multiobjective optimization, FACTA UNIV.(NIS) Series Maths. and Inform. 27(3),2013,357–372.

L. Vandenberghe and S. Boyd: Semidefinite programming, SIAM Review, 38,1996, 49–95.

I. A. Farhat and M. E. El-Hawary: Interior point methods application in optimum operational scheduling of electric power systems, IET Gen., Trans. and Distr., 3()11, 2009,1020–1029.

F. Alizadeh: Interior point methods in semidefinite programming with application to combinatorial optimization,SIAM Jour. Opt. 5, 1995, 13–51.

A. S. Nemirovski and M. J. Todd: Interior-point methods for optimization Acta Numerica, 2008, 191–234.

D. Jibetean and E. de Klerk: Global optimization of rational functions: a semidefinite programming approach, Math. Program. Ser. A(106), 2006, 93–109.

F. Bugarin, D. Henrion, and J.-B. Lasserre: Minimizing the sum of many rational functions, arXiv.1102.4954v1 [math.OC], Tech. Rep., February 2011.

D. Henrion and J.-B. Lasserre: Convergent relaxations of polynomial matrix inequalities and static output feedback, 51(2), 2006, 192–202.

M. Laurent: Sums of Squares, Moment Matrices and Polynomial Optimization ser. Emerg. Appl. Alge. Geom. IMA, 2009. 149, ch. 3, pp. 157–270.

——, “Semidefinite programming in combina- torial and polynomial optimization,” NAW, vol. 5, no. 4, pp. 256–262, December 2008.

D. Henrion, J.-B. Lasserre, and J. Lofberg: GloptiPoly 3: Moments, Optimizationand Semidefinite Programming, 3rd ed., May 2007.

J. F. Sturm and the Advanced Optimization Laboratory at McMaster University Canada, SeDuMi version 1.1R3, Advanced Optimization Laboratory at McMaster University, Canada, October 2006.

M. A. Abido: Multiobjective evolutionary algorithms for electric power dispatch problem IEEE Trans. Evol. Comput. 10(3),2006, 315–329.

A. Farag, S. Al-Baiyat, and T. C. Cheng. Economic load dispatch multiobjective optimization procedures using linear progamming techniques. IEEE Trans. Power Syst., 10(2),May 1995,731–738.

R. T. F. A. King, H. C. S. Rughooputh, and K. Deb: Evolutionary multi-objective environmental/ economic dispatch: Stochastic vs deterministic approaches. KanGAL, Tech. Rep. 2004019, 2004.



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