Abstract
This paper shows how to apply the perturbation theory for nonlinear programming problems to the study of the optimal power flow problem. The latter is the problem of minimizing losses of active power over a very high voltage power networks. In this paper, the inverse of the square root of the reference voltage of the network is viewed as a small parameter. We call this scheme the very high voltage approximation.
After some proper scaling, it is possible to formulate a limiting problem, that does not satisfy the Mangasarian-Fromovitz qualification hypothesis. Nevertheless, it is possible to obtain under natural hypotheses the second order expansion of losses and first order expansion of solutions. The latter is such that the computation of the active and reactive parts are decoupled. We also obtain the high order expansion of the value function, solution and Lagrange multiplier, assuming that interactions with the ground are small enough. Finally we show that the classical direct current approximation may be justified and improved using the framework of very high voltage approximation.
Similar content being viewed by others
References
A. Auslender and R. Cominetti, “First and second order sensitivity analysis of nonlinear programs under directional constraint qualification conditions,” Optimization, vol. 21, pp. 351–363, 1990.
A.R. Bergen, Power Systems Analysis, Prentice-Hall: Englewood Cliffs, New Jersey, 1986.
J.F. Bonnans, “Mathematical study of very high voltage power networks I: The optimal DC power flow problem,” SIAM J. Optimization, vol. 7, pp. 979–990, 1997.
J.F. Bonnans, “Mathematical study of very high voltage power networks II: The AC power flow problem,” SIAM J. Applied Mathematics, vol. 58, pp. 1547–1567, 1998.
J.F. Bonnans, A.D. Ioffe, and A. Shapiro, “Développement de solutions exactes et approchées en programmation non linéaire,” Comptes Rendus Acad. Sci. Paris, Série I, vol. 315, pp. 119–123, 1992.
J.F. Bonnans and A. Shapiro, “Optimization problems with perturbations, A guided tour,” SIAM Review, vol. 40, pp. 202–227, 1998.
J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag: New York, 2000.
J.F. Bonnans and A. Sulem, “Pseudopower expansion of solutions of generalized equations and constrained optimization problems,” Mathematical Programming, vol. 70, pp. 123–148, 1995.
J. Gauvin and R. Janin, “Directional behaviour of optimal solutions in nonlinear mathematical programming,” Mathematics of Operations Research, vol. 13, pp. 629–649, 1988.
B. Gollan, “On the marginal function in nonlinear programming,” Mathematics of Operations Research, vol. 9, pp. 208–221, 1984.
S.M. Robinson, “Strongly regular generalized equations,” Mathematics of Operations Research, vol. 5, pp. 43–62, 1980.
A. Shapiro, “Sensitivity analysis of nonlinear programs and differentiability properties of metric projections,” SIAM J. Control and Optimization, vol. 26, pp. 628–645, 1988.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bonnans, J.F. Mathematical Study of Very High Voltage Power Networks III: The Optimal AC Power Flow Problem. Computational Optimization and Applications 16, 83–101 (2000). https://doi.org/10.1023/A:1008781604329
Issue Date:
DOI: https://doi.org/10.1023/A:1008781604329