Branch Flow Model: Relaxations and Convexification—Part I

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.

Stochastic reactive power management in microgrids with renewables

“…Beyond mean value convergence from (25), the bound in (26) assures that h(q g T ) remains close to the optimum h(q g ) with high probability. According to the online convex optimization terminology, the algorithm in Table I enjoys sublinear regret [29].…”

Investigation of Maximum Possible OPF Problem Decomposition Degree for Decentralized Energy Markets

“…Of course, if the recent efforts in convexifying the OPF problem (e.g. [39,40,41]) find general application then any such convergence issues would be obsolete. Another question of interest is convergence performance in degenerate cases.…”

Community Microgrid Scheduling Considering Network Operational Constraints and Building Thermal Dynamics

“…Considering the power flow direction as depicted in Figure 2. Equation (27) - (30) represents the DistFlow equations, which have been verified in [26,29]. The nodal voltage drop along a feeder is represented by (27).…”

“…In the third category, the non-linear/non-convex items in the D-OPF problems are directly approximated or relaxed into linear/convex form. Thus, the D-OPF is formulated as linear programming [24,25]; or semidefinite programming (SDP) [26]; or second-order cone programming (SOCP) using either polar coordinates [27] or the DistFlow model [28,29] by several convex relaxations. The relaxed convex D-OPF preserves the accuracy of the power flow model while improving the solution efficiency, and thus, it has become more and more popular in recent years.…”