A convex relaxation approach for power flow problem

Abstract

A solution to the power flow problem is imperative for many power system applications and several iterative approaches are employed to achieve this objective. However, the chance of finding a solution is dependent on the choice of the initial point because of the nonconvex feasibility region of this problem. In this paper, a non-iterative approach that leverages a convexified relaxed power flow problem is employed to verify the existence of a feasible solution. To ensure the scalability of the proposed convex relaxation, the problem is formulated as a sparse semi-definite programming problem. The variables associated with each maximal clique within the network form several positive semidefinite matrices. Perturbation and network reconfiguration schemes are employed to improve the tightness of the proposed convex relaxation in order to validate the existence of a feasible solution for the original non-convex problem. Multiple case studies including an ill-conditioned power flow problem are examined to show the effectiveness of the proposed approach to find a feasible solution.

A Basic AC Power Flow Based on the Bus Admittance Matrix Incorporating Loads and Generators Including Slack Bus

“…Δi Δi u u (20) For the slack bus, (20) must not be applied (since both active and reactive power of the slack generator are unknow quantities) so that:…”

“…The corrected load vector uL1,c is obtainable from ( 17) and can determine, by (18), the correcting load current vector ΔiL1,c , which enters into (19) so giving the correcting generator current vector ΔiG1,c. Since the generator corrections must involve the reactive power, the quadrature component ΔiG1,c_q is derived by means of (20). This procedure is iterated until convergence, i.e.…”

“…It is also worth reminding the approaches based on the AC power flow representation in conic format [18,19] and convex relaxation technique [20][21][22][23]. Both are very different from the author's approach: [18] is an analytical approach based on convex relaxation which is tightened by using appropriate perturbation functions along with a network reconfiguration scheme; [20] is still an approximation of the AC power flow equations.…”

“…It is also worth reminding the approaches based on the AC power flow representation in conic format [18,19] and convex relaxation technique [20][21][22][23]. Both are very different from the author's approach: [18] is an analytical approach based on convex relaxation which is tightened by using appropriate perturbation functions along with a network reconfiguration scheme; [20] is still an approximation of the AC power flow equations. The present method is not a numerical approach solving a non-linear problem but a circuital one which makes large use of admittance matrixes and of correcting current vectors without any perturbation functions, any reconfiguration scheme, any convex relaxation and conic format.…”

Accurate and Efficient Derivative-Free Three-Phase Power Flow Method for Unbalanced Distribution Networks

“…Thus, the power flow problem constitutes a steady-state study where electrical variables (voltages and currents) are determined for a particular load and generation condition [3,4]. The main characteristic of the power flow problem is the conformation of a non-linear system of equations as a result of the presence of products of variables and trigonometric functions, which demands the use of numerical methods to find a solution [5,6]. Typically, the power flow problem in electrical distribution grids has been addressed only with single-phase equivalents in the literature [7].…”