Branch Flow Model: Relaxations and Convexification—Part II

Abstract

We propose a branch flow model for the analysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks, we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact but the angle relaxation may not be exact, and we provide a simple way to determine if a relaxed solution is globally optimal. We propose convexification of mesh networks using phase shifters so that OPF for the convexified network can always be solved efficiently for an optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Part I introduces our branch flow model, explains the two relaxation steps, and proves the conditions for exact relaxation. Part II describes convexification of mesh networks, and presents simulation results.

Optimal reactive power flow with exact linearized transformer model in distribution power networks

“…A simple way to modify the upper limit of SVA given by [12] is dropping the upper limit of SVA. Thus, we solve rxORPF according the following procedure:…”

“…Steven Low et al further studied the OPF problem based on the relaxation and provided various sufficient conditions to guarantee the exactness of the relaxation [12,13].…”

“…Recently, solving AC-constrained OPF (ACOPF) based on convex relaxation is widely concerned as its ability to find global optima and polynomial time complexity [10][11][12][13], especially second-order cone (SOC) relaxation-based method to solve ACOPF in radial power networks. The method built on Branch Flow Model and relaxed quadratic constraints to SOC constraints which transformed ORPF to an efficiently solvable second-order cone programming (SOCP) problem.…”

Fully distributed multi‐area economic dispatch method for active distribution networks

“…However, a piecewise linear expression will result in integer variables in the model, which means that it will remain non-convex. Although non-convex OPF provides more details of the operation of the distribution system, the high computational cost inhibits deployment in practical applications (which remains the case even if some recent techniques are used to transform OPF into a convex form [19,30,31,[33][34][35]). Furthermore, non-convexity is an obstacle for a distributed implementation because distributed optimisation algorithms typically require convexity to ensure convergence.…”

“…Model 5 can be solved in a distributed manner using ADMM [17]. Let l A ij denotes the Lagrangian multiplier corresponding to the equality constraint in (31). Because of the favourable decomposability of the original Lagrangian, the augmented Lagrangian of Model 5 can be written as…”