Optimal power flow over tree networks

Abstract

Abstract-The optimal power flow (OPF) problem is critical to power system operation but it is generally non-convex and therefore hard to solve. Recently, a sufficient condition has been found under which OPF has zero duality gap, which means that its solution can be computed efficiently by solving the convex dual problem. In this paper we simplify this sufficient condition through a reformulation of the problem and prove that the condition is always satisfied for a tree network provided we allow over-satisfaction of load. The proof, cast as a complex semi-definite program, makes use of the fact that if the underlying graph of an n × n Hermitian positive semi-definite matrix is a tree, then the matrix has rank at least n − 1. I. MOTIVATIONOptimal operation of a power grid has been extensively studied since the pioneering work of Carpentier [1] in 1962. The general optimal power flow (OPF) problem seeks to minimize some cost function, such as power loss, generation cost and/or user utilities, subject to capacity and network constraints on the voltages, powers (real and reactive) and the loads [2]- [4]. The general OPF problem is non-convex and NP hard. Given the practical importance of the problem there has been a lot of research into efficient solution algorithms, and historically the most common solution techniques have relied on linear programming techniques [5], [6]. Researchers have also proposed a number of relaxations to make the OPF problem more tractable. The simplest of which is the DC Power Flow problem, which is widely used because it is a linear program and thus easy to solve. However this approximation makes a number of assumptions that are not always valid in a real power circuit. A number of studies have sought to characterize the instances where the DC approximation is acceptable, e.g. Recently there has been some effort toward convexifying the full AC problem. Jabr provided a conic quadratic model of radial distribution systems [13] and meshed networks [14] and demonstrated an efficient solution method to these problems using an interior point method for convex conic quadratic programming. The implementation of this method on a distribution system containing various reactive power components such as tap changers and shunt capacitors was also studied [15] [17] and studies a second order cone relaxation to determine the optimal control strategy for the multi-timescale problem of simultaneous optimal inverter and shunt capacitor control and conjectures that the relaxation is exact. In all of these works the relaxations seem to perform well but no guarantees are provided as to the ability to recover the solution of the original problem nor is there a characterization of the worst-case distance from their solution to that of the original problem. Bai et al. show in [19], [20] that under certain conditions the OPF problem can be cast as a semi-definite program. This idea was further refined and extensively analyzed by Lavaei et al. in [21], [22] who proved that these conditions always hold for resistiv...

Branch flow model: Relaxations and convexification

“…We verified the exactness of the SDP relaxation by checking if the solution matrix was of rank one [13], [16]. In all test cases, the SDP relaxation was exact and hence the optimal objective values reported were indeed the optimal value of OPF (10)- (11).…”

“…This result is extended in [14] to include other variables and constraints and in [15] to exploit the sparsity of power networks and phase shifters for convexification of OPF. In [16], [17], it is proved that the sufficient condition of [13] always holds for a radial (tree) network, provided the bounds on the power flows satisfy a simple pattern. See also [18] for a generalization.…”

“…We also report the optimal objective values of OPF with and without phase shifters in Tables II and III. The optimal values of OPF without phase shifters were obtained by implementing the SDP formulation and relaxation proposed in [16] for solving OPF (10)- (11). We verified the exactness of the SDP relaxation by checking if the solution matrix was of rank one [13], [16].…”

Exact convex relaxation of OPF for radial networks using branch flow model

“…Moreover, if SDR fails to provide exact relaxations, the solutions produced by the SDR are physically meaningless in those cases. Remarkably, it turns out that if the network is radial, then the sufficient condition of [12] always holds, provided that the bounds on the power flows satisfy a simple pattern [14], [15], [16]. This is important as almost all distribution systems are radial networks.…”

Reduced-Complexity Semidefinite Relaxations of Optimal Power Flow Problems

“…An SDR of (10) is given by (7) where X ∈ H n E,c is a convex relaxation of the constraint X = xx H . Recall that applying the conversion method to (7) yields the equivalent problem (9). We refer to (9) as "full conversion" which is closely related to the sparse SDR technique of Waki et al [31] for polynomial optimization with structured sparsity.…”