Abstract-The optimal power flow (OPF) problem with discrete control variables is an NP-hard problem in its exact formulation. To cope with the immense computational-difficulty of this problem, we propose an ordinal optimization theory-based algorithm to solve for a good enough solution with high probability. Aiming for hard optimization problems, the ordinal optimization theory, in contrast to heuristic methods, guarantee to provide a top % solution among all with probability more than 0.95. The approach of our ordinal optimization theory-based algorithm consists of three stages. First, select heuristically a large set of candidate solutions. Then, use a simplified model to select a subset of most promising solutions. Finally, evaluate the candidate promising-solutions of the reduced subset using the exact model. We have demonstrated the computational efficiency of our algorithm and the quality of the obtained solution by comparing with the competing methods and the conventional approach through simulations.Index Terms-Discrete control variables, nonlinear programming, optimal power flow, ordinal optimization. NOMENCLATURE-dimensional vector of discrete control variables such as switching shunt capacitor banks and transformer taps.-dimensional vector of continuous variables consisting of real and reactive power generation, real and imaginary parts of bus complex voltage. Sample space of . Real and reactive power flow balance equations. Inequality constraints such as thermallimit security constraints, security limits on voltage magnitude, and real and reactive power generation limits.

In this paper, we present a new method for solving nonlinear multicommodity network flow problems with convex objective functions. This method combines a well-known projected Jacobi method and a new dual projected pseudo-quasi-Newton (DPPQN) method which solves multicommodity flow quadratic subproblems induced in the projected Jacobi method. The DPPQN method is a dual Newtontype method that differs very much from the conventional Lagrangian Newton method; our method fully exploits the structural advantages of network-type linear equality constraints to obtain a constant sparse approximate Hessian matrix with a decoupling structure and includes a novel finite-iteration successive projection and (truncated) seal algorithm to resolve the difficulty caused by coupling capacity constraints. The DPPQN method also consists of two decomposition effects, the commodity decomposition effect and the arc decomposition effect, which resolve the potential numerical difficulties caused by large dimensions. We show the convergence of our method including the convergence of the finite-iteration successive projection and (truncated) seal algorithm. Compared with the Frank-Wolfe with PARTAN algorithm in which a price-directive decomposition method is used to solve linearized multicommodity flow problems, our method is dramatically faster in terms of the CPU time on a Sparc-10 workstation at solving numerous nonlinear multicommodity network flow examples.

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