Abstract
We investigate methods for solving high-dimensional nonlinear optimization problems which typically occur in the daily scheduling of electricity production: problems with a nonlinear, separable cost function, lower and upper bounds on the variables, and an equality constraint to satisfy the demand. If the cost function is quadratic, we use a modified Lagrange multiplier technique. For a nonquadratic cost function (a penalty function combining the original cost function and certain fuel constraints, so that it is generally not separable), we compare the performance of the gradient-projection method and the reduced-gradient method, both with conjugate search directions within facets of the feasible set. Numerical examples at the end of the paper demonstrate the effectiveness of the gradient-projection method to solve problems with hundreds of variables by exploitation of the special structure.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Talukdar, S. N., andWu, F. F.,Computer-Aided Dispatch for Electrical Power Systems, Proceedings of the IEEE, Vol. 69, pp. 1212–1231, 1981.
Van Den Bosch, P. P. J., andHonderd, G.,A Solution of the Unit Commitment Problem via Decomposition and Dynamic Programming, IEEE Transactions, Vol. PAS-104, pp. 1684–1690, 1985.
Murty, K. G.,Linear and Combinatorial Programming, Wiley, New York, New York, 1976.
Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming, Sequential Unconstrained Minimization Techniques, Wiley, New York, New York, 1968.
Lootsma, F. A.,A Survey of Methods for Solving Constrained Minimization Problems via Unconstrained Minimization, Numerical Methods for Nonlinear Optimization, Edited by F. A. Lootsma, Academic Press, London, England, pp. 313–348, 1972.
Bitran, G. R., andHax, A. C.,On the Solution of Convex Knapsack Problems with Bounded Variables, Survey of Mathematical Programming, Edited by A. Prékopa, North-Holland, Amsterdam, Holland, Vol. 1, pp. 357–367, 1979.
Hestenes, M. R., andStiefel, E.,Methods of Conjugate Gradients for Solving Linear Systems, National Bureau of Standards, Report No. 1659, 1952.
Rosen, J. B.,The Gradient Projection Method for Nonlinear Programming, Part 1, Linear Constraints, SIAM Journal on Applied Mathematics, Vol. 8, pp. 181–217, 1960.
Wolfe, P.,Methods of Nonlinear Programming, Edited by J. Abadie, North-Holland, Amsterdam, Holland, pp. 97–131, 1967.
Abadie, J.,The GRG Method for Nonlinear Programming, Design and Implementation of Optimization Software, Edited by H. J. Greenberg, Sijthoff and Noordhoff, Alphen aan den Rijn, Holland, pp. 335–362, 1978.
Lasdon, L. S., andWaren, A. D.,Generalized Reduced Gradient Software for Linearly and Nonlinearly Constrained Problems, Design and Implementation of Optimization Software, Edited by H. J. Greenberg, Sijthoff and Noordhoff, Alphen aan den Rijn, Holland, pp. 363–396, 1978.
Schittkowski, K.,Nonlinear Programming Codes, Springer, Berlin, Germany, 1980.
Fletcher, R., andReeves, C. M.,Function Minimization by Conjugate Gradients, Computer Journal, Vol. 7, pp. 149–154, 1964.
Van Den Bosch, P. P. J.,Short-Term Optimization of Thermal Power Systems, Delft University of Technology, Department of Electrical Engineering, PhD Thesis, Delft, Holland, 1983.
Lootsma, F. A.,The Algol 60 Procedure Minifun for Solving Nonlinear Optimization Problems, Design and Implementation of Optimization Software, Edited by H. J. Greenberg, Sijthoff and Noordhoff, Alphen aan den Rijn, Holland, pp. 397–445, 1978.
Author information
Authors and Affiliations
Additional information
Communicated by J. Abadie
Rights and permissions
About this article
Cite this article
Van Den Bosch, P.P.J., Lootsma, F.A. Scheduling of power generation via large-scale nonlinear optimization. J Optim Theory Appl 55, 313–326 (1987). https://doi.org/10.1007/BF00939088
Issue Date:
DOI: https://doi.org/10.1007/BF00939088