Advertisement

Optimization and Engineering

, Volume 4, Issue 4, pp 309–336 | Cite as

A Primal-Dual Interior-Point Method to Solve the Optimal Power Flow Dispatching Problem

  • Rabih A. Jabr
Article

Abstract

This paper presents a primal-dual path-following interior-point method for the solution of the optimal power flow dispatching (OPFD) problem. The underlying idea of most path-following algorithms is relatively similar: starting from the Fiacco-McCormick barrier function, define the central path and loosely follow it to the optimum solution. Several primal-dual methods for OPF have been suggested, all of which are essentially direct extensions of primal-dual methods for linear programming. Nevertheless, there are substantial variations in some crucial details which include the formulation of the non-linear problem, the associated linear system, the linear algebraic procedure to solve this system, the line search, strategies for adjusting the centring parameter, estimating higher order correction terms for the homotopy path, and the treatment of indefiniteness. This paper discusses some of the approaches that were undertaken in implementing a specific primal-dual method for OPFD. A comparison is carried out with previous research on interior-point methods for OPF. Numerical tests on standard IEEE systems and on a realistic network are very encouraging and show that the new algorithm converges where other algorithms fail.

power scheduling optimisation non-linear non-convex path-following 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Alan and J.W. H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall: Englewood Cliffs, London, 1981.Google Scholar
  2. K. C. Almeida, F. D. Galliana, and S. Soares, “A general parametric optimal power flow,” IEEE Trans. Power Syst. vol. 9, no. 1, pp. 540–547, 1994.Google Scholar
  3. E. D. Andersen, J. Gondzio, C. Meszaros, and X. Xu, “Implementation of interior point methods for large scale linear programming,” in Interior Point Methods of Mathematical Programming T. Terlaky (Ed.), Kluwer Academic Publishers: The Netherlands, 1996.Google Scholar
  4. J. R. Bunch and B. N. Parlett, “Direct methods for solving symmetric indefinite systems of linear equations,” SIAM J. Num. Anal. vol. 8, no. 4, pp. 639–651, 1971.Google Scholar
  5. R. H. Byrd, J. C. Gilbert, and J. Nocedal, “A trust region method based on interior point techniques for nonlinear programming,” Math. Program. vol. 89, no. 1, Ser. A, pp. 149–185, 2000.Google Scholar
  6. E. D. Castronuovo, J. M. Campagnolo, and R. Salgado, “Optimal power flow solutions via interior point method with high performance computation techniques,” in Proceedings of the 13th Power Systems Computation Conference 99, Trondheim, Norway, June 28–July 2nd 1999 vol. 2, pp. 1207–1213.Google Scholar
  7. R. M. Chamberlain, M. J. D. Powell, C. Lemarechal, and H. C. Pedersen, “The watchdog technique for forcing convergence in algorithms for constrained optimization,” Math. Program. Study vol. 16, pp. 1–17, 1982.Google Scholar
  8. J. Czyzyk, S. Mehrotra, M. Wagner, and S. J. Wright, “PCx user guide (version 1.1),” Optimisation Technology Centre, Technical Report OTC 96/01, November 3, 1997.Google Scholar
  9. A. S. El-Bakry, R. A. Tapia, T. Tsuchiya, and Y. Zhang, “On the formulation and theory of Newton interior point method for nonlinear programming,” J. Optim. Theo. Appl. vol. 89, no. 3, pp. 507–541, 1996.Google Scholar
  10. A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley: New York, 1968.Google Scholar
  11. R. Fletcher, Practical Methods of Optimisation 2nd edition., Wiley: Chichester, 1987.Google Scholar
  12. R. Fletcher and S. Leyffer, “Nonlinear programming without a penalty function,” University of Dundee Numerical Analysis Report, NA/171, September 22, 1997.Google Scholar
  13. D. M. Gay, M. L. Overton, and M. H. Wright, “A primal-dual interior method for nonconvex nonlinear programming,” Computing Sciences Research Center, Bell Laboratories, Murray Hill, New Jersey 07974, Technical Report 97–4–08, July 29, 1997.Google Scholar
  14. J. R. Gilbert, C. Moler, and R. Schreiber, “Sparse matrices in Matlab: Design and implementation,” SIAM J. Mat. Anal. Appl. vol. 13, no. 1, pp. 333–356, 1992.Google Scholar
  15. N. M. Gould and P. L. Toint, “A note on the convergence of barrier algorithms to second-order necessary points,” Math. Program. vol. 85, pp. 433–438, 1999.Google Scholar
  16. J. J. Grainger and W. D. Stevenson, Power System Analysis, McGraw-Hill, Inc: Singapore, 1994.Google Scholar
  17. N. J. Higham, “Computing a nearest symmetric positive semidefinite Matrix,” Lin. Algeb. Appl. vol. 103, pp. 103–118, 1988.Google Scholar
  18. R. A. Jabr and A. H. Coonick, “Homogeneous interior point method for constrained power scheduling,” IEE Proc. C, Gener. Transm. Distrib. vol. 147, no. 4, pp. 239–244, 2000.Google Scholar
  19. F. G. M. Lima, S. Soares, A. Santos Jr., K. C. Almeida, and F. D. Galiana, “Optimal power flow based on a general nonlinear parametric approach,” in Proceedings of the 13th Power Systems Computation Conference 99, Trondheim, Norway, June 28 – July 2nd 1999 vol. 1, pp. 495–502.Google Scholar
  20. D. G. Luenberger, Nonlinear Programming 2nd edition, Addison-Wesley: Reading; Massachusetts; London, 1984.Google Scholar
  21. I. J. Lustig, R. E. Marsten, and D. F. Shanno, “On implementing Mehrotra's predictor-corrector interior-point method for linear programming,” SIAM J. Optim. vol. 2, no. 3, pp. 435–449, 1992.Google Scholar
  22. K. R. C. Mamandur and R. O. Chenoweth, “Optimal control of reactive power flow for improvements in voltage profiles and for real power loss minimization,” IEEE Trans. Power Appar. Syst. vol. PAS-100, pp. 3185–3194, 1981.Google Scholar
  23. S. Mehrotra, “On the implementation of a primal-dual interior point method,” SIAM J. Optim. vol. 2, no. 4, pp. 575–601, 1992.Google Scholar
  24. A. Monticelli and W.-H. E. Liu, “Adaptive movement penalty method for the newton optimal power flow,” IEEE Trans. Power Syst. vol. 7, no. 1, pp. 334–342, 1992.Google Scholar
  25. D. F. Shanno and R. J. Vanderbei, “Interior-point methods for nonconvex nonlinear programming: Orderings and higher-order methods,” Math. Program. vol. 87, no. 2, pp. 265–280, 2000.Google Scholar
  26. D. I. Sun, B. Ashley, B. Brewer, A. Hughes, and W. F. Tinney, “Optimal power flow by Newton approach,” IEEE Trans. Power Appar. Syst. vol. PAS-103, no. 10, pp. 2864–2880, 1984.Google Scholar
  27. G. L. Torres and V. H. Quintana, “An interior-point method for nonlinear optimal power flow using voltage rectangular coordinates,” IEEE Trans. Power Syst. vol. 13, no. 4, pp. 1211–1218, 1998.Google Scholar
  28. R. J. Vanderbei, Linear Programming: Foundations and Extensions, Kluwer Academic Publishers: Boston, 1998.Google Scholar
  29. R. J. Vanderbei, “LOQO: An interior point code for quadratic programming,” Statistics and Operations Research, Princeton University, New Jersey, USA, Technical Report SOR-94–15, Revised: Nov. 30, 1998.Google Scholar
  30. R. J. Vanderbei and D. F. Shanno, “An interior-point algorithm for nonconvex nonlinear programming,” Comp. Optim. Appl. vol. 13, pp. 231–252, 1999.Google Scholar
  31. H. Wei, H. Sasaki, J. Kubakawa, and R. Yokoyama, “An interior point non-linear programming for optimal power flow problems with a novel data structure,” IEEE Trans. Power Syst. vol. 13, no. 3, pp. 870–877, 1998.Google Scholar
  32. S. J. Wright, Primal Dual Interior Point Methods, SIAM: Philadelphia, 1997.Google Scholar
  33. Y. C. Wu, A. S. Debs, and R. E. Marsten, “A direct nonlinear predictor-corrector primal-dual interior-point algorithm for optimal power flows,” IEEE Trans. Power Syst. vol. 9, no. 2, pp. 876–883, 1994.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Rabih A. Jabr
    • 1
  1. 1.Department of Electrical, Computer and Communication EngineeringNotre Dame UniversityZouk MosbehLebanon

Personalised recommendations