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.

/pdf/branch-flow-model-relaxations-and-convexification-part-ii-3yqj9g9dqx.pdf

Branch Flow Model: Relaxations and Convexification—Part II

This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relationships among them. Part II presents sufficient conditions under which the convex relaxations are exact.

/pdf/convex-relaxation-of-optimal-power-flow-part-i-formulations-52chepgnj0.pdf

Convex Relaxation of Optimal Power Flow—Part I: Formulations and Equivalence

This paper shows that the load flow problem of a radial distribution system can be modeled as a convex optimization problem, particularly a conic program. The implications of the conic programming formulation are threefold. First, the solution of the distribution load flow problem can be obtained in polynomial time using interior-point methods. Second, numerical ill-conditioning can be automatically alleviated by the use of scaling in the interior-point algorithm. Third, the conic formulation facilitates the inclusion of the distribution power flow equations in radial system optimization problems. A state-of-the-art implementation of an interior-point method for conic programming is used to obtain the solution of nine different distribution systems. Comparisons are carried out with a previously published radial load flow program by R. Cespedes

Radial distribution load flow using conic programming

This paper proposes a mixed-integer conic programming formulation for the minimum loss distribution network reconfiguration problem. This formulation has two features: first, it employs a convex representation of the network model which is based on the conic quadratic format of the power flow equations and second, it optimizes the exact value of the network losses. The use of a convex model in terms of the continuous variables is particularly important because it ensures that an optimal solution obtained by a branch-and-cut algorithm for mixed-integer conic programming is global. In addition, good quality solutions with a relaxed optimality gap can be very efficiently obtained. A polyhedral approximation which is amenable to solution via more widely available mixed-integer linear programming software is also presented. Numerical results on practical test networks including distributed generation show that mixed-integer convex optimization is an effective tool for network reconfiguration.

Minimum Loss Network Reconfiguration Using Mixed-Integer Convex Programming

This paper presents a new sparse formulation for the solution of unbalanced three-phase power systems using the Newton-Raphson method. The three-phase current injection equations are written in rectangular coordinates resulting in an order 6n system of equations. The Jacobian matrix is composed of 6/spl times/6 block matrices and retains the same structure as the nodal admittance matrix. Practical distribution systems were used to test the method and to compare its robustness with that of the backward/forward sweep method.

https://www.researchgate.net/profile/N_Martins/publication/3266183_Three-phase_power_flow_calculations_using_the_current_injection_method/links/0046352c1ba6120432000000.pdf

Three-phase power flow calculations using the current injection method

This paper presents a method to solve the power flow problem in radial distribution networks. In the proposed formulation, the load flow equations are written in terms of new variables resulting in a set of 3N equations (2N linear plus N quadratic) for a network with N+1 buses. A computationally efficient solution scheme based on the Newton-Raphson method is proposed, and possible simplifications are discussed. Finally, a comparative evaluation of the proposed method's performance against existing algorithms is provided.

E. Romero Ramos

Papers

Reliable load flow technique for radial distribution networks