Promises of Conic Relaxation for Contingency-Constrained Optimal Power Flow Problem

This paper is concerned with the security-constrained optimal power flow (SCOPF) problem, where each contingency corresponds to the outage of an arbitrary number of lines and generators. The problem is studied by means of a convex relaxation, named semidefinite program (SDP). The existence of a rank-1 SDP solution guarantees the recovery of a global solution of SCOPF. We prove that the rank of the SDP solution is upper bounded by the treewidth of the power network plus one, which is perceived to be small in practice. We then propose a decomposition method to reduce the computational complexity of the relaxation. In the case where the relaxation is not exact, we develop a graph-theoretic convex program to identify the problematic lines of the network and incorporate the loss over those lines into the objective as a penalization (regularization) term, leading to a penalized SDP problem. We perform several simulations on large-scale benchmark systems and verify that the global minima are at most 1% away from the feasible solutions obtained from the proposed penalized relaxation.

[1]  Javad Lavaei,et al.  Relationship between power loss and network topology in power systems , 2010, 49th IEEE Conference on Decision and Control (CDC).

[2]  Charles R. Johnson,et al.  Positive definite completions of partial Hermitian matrices , 1984 .

[3]  R. Jabr.  Exploiting Sparsity in SDP Relaxations of the OPF Problem , 2012, IEEE Transactions on Power Systems.

[4]  Katsuki Fujisawa,et al.  Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results , 2003, Math. Program..

[5]  Steven H. Low,et al.  Convex Relaxation of Optimal Power Flow—Part II: Exactness , 2014, IEEE Transactions on Control of Network Systems.

[6]  Ufuk Topcu,et al.  On the exactness of convex relaxation for optimal power flow in tree networks , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[7]  Javad Lavaei,et al.  Geometry of power flows in tree networks , 2012, 2012 IEEE Power and Energy Society General Meeting.

[8]  Lorenz T. Biegler,et al.  Global optimization of Optimal Power Flow using a branch & bound algorithm , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[9]  Ian A. Hiskens,et al.  Sparsity-Exploiting Moment-Based Relaxations of the Optimal Power Flow Problem , 2014, IEEE Transactions on Power Systems.

[10]  J. Lavaei,et al.  Physics of power networks makes hard optimization problems easy to solve , 2012, 2012 IEEE Power and Energy Society General Meeting.

[11]  Martin S. Andersen,et al.  Reduced-Complexity Semidefinite Relaxations of Optimal Power Flow Problems , 2013, IEEE Transactions on Power Systems.

[12]  Paul A. Trodden,et al.  Local Solutions of the Optimal Power Flow Problem , 2013, IEEE Transactions on Power Systems.

[13]  Steven H. Low,et al.  Convex Relaxation of Optimal Power Flow—Part I: Formulations and Equivalence , 2014, IEEE Transactions on Control of Network Systems.

[14]  David Tse,et al.  Distributed algorithms for optimal power flow problem , 2011, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[15]  Javad Lavaei,et al.  Exactness of Semidefinite Relaxations for Nonlinear Optimization Problems with Underlying Graph Structure , 2014, SIAM J. Optim..

[16]  Arie M. C. A. Koster,et al.  Treewidth computations II. Lower bounds , 2011, Inf. Comput..

[17]  K. Mani Chandy,et al.  Optimal power flow over tree networks , 2011, 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[18]  K. Fujisawa,et al.  Semidefinite programming for optimal power flow problems , 2008 .

[19]  Monique Laurent,et al.  A new graph parameter related to bounded rank positive semidefinite matrix completions , 2012, Math. Program..

[20]  Javad Lavaei.  Zero duality gap for classical opf problem convexifies fundamental nonlinear power problems , 2011, Proceedings of the 2011 American Control Conference.

[21]  Dzung T. Phan,et al.  Lagrangian Duality and Branch-and-Bound Algorithms for Optimal Power Flow , 2012, Oper. Res..

[22]  J. Lavaei,et al.  Convex relaxation for optimal power flow problem: Mesh networks , 2013, 2013 Asilomar Conference on Signals, Systems and Computers.

[23]  David Tse,et al.  Geometry of injection regions of power networks , 2011, IEEE Transactions on Power Systems.

[24]  Eilyan Bitar,et al.  Nondegeneracy and Inexactness of Semidefinite Relaxations of Optimal Power Flow , 2014, 1411.4663.

[25]  Arie M. C. A. Koster,et al.  Treewidth computations I. Upper bounds , 2010, Inf. Comput..

[26]  Kazuo Murota,et al.  Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework , 2000, SIAM J. Optim..

[27]  Jesse T. Holzer,et al.  Implementation of a Large-Scale Optimal Power Flow Solver Based on Semidefinite Programming , 2013, IEEE Transactions on Power Systems.

[28]  Albert Y. S. Lam,et al.  An Optimal and Distributed Method for Voltage Regulation in Power Distribution Systems , 2012, IEEE Transactions on Power Systems.

[29]  T. Markham,et al.  A Generalization of the Schur Complement by Means of the Moore–Penrose Inverse , 1974 .

[30]  S. Low,et al.  Zero Duality Gap in Optimal Power Flow Problem , 2012, IEEE Transactions on Power Systems.

[31]  D. Ernst,et al.  Contingency Filtering Techniques for Preventive Security-Constrained Optimal Power Flow , 2007, IEEE Transactions on Power Systems.

[32]  F. Fomin,et al.  New upper bounds on the decomposability of planar graphs , 2006 .

[33]  Monique Laurent,et al.  Polynomial Instances of the Positive Semidefinite and Euclidean Distance Matrix Completion Problems , 2000, SIAM J. Matrix Anal. Appl..

[34]  Pablo A. Parrilo,et al.  Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization , 2007, SIAM Rev..

[35]  Daniel K. Molzahn,et al.  Examining the limits of the application of semidefinite programming to power flow problems , 2011, 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton).