Recent advances in computational methods for the power flow equations

Abstract

The power flow equations are at the core of most of the computations for designing and operating electric power systems. The power flow equations are a system of multivariate nonlinear equations which relate the power injections and voltages in a power system. A plethora of methods have been devised to solve these equations, starting from Newton-based methods to homotopy continuation and other optimization-based methods. While many of these methods often efficiently find a high-voltage, stable solution due to its large basin of attraction, most of the methods struggle to find low-voltage solutions which play significant role in certain stability-related computations. While we do not claim to have exhausted the existing literature on all related methods, this tutorial paper introduces some of the recent advances in methods for solving power flow equations to the wider power systems community as well as bringing attention from the computational mathematics and optimization communities to the power systems problems. After briefly reviewing some of the traditional computational methods used to solve the power flow equations, we focus on three emerging methods: the numerical polynomial homotopy continuation method, Gröbner basis techniques, and moment/sum-of-squares relaxations using semidefinite programming. In passing, we also emphasize the importance of an upper bound on the number of solutions of the power flow equations and review the current status of research in this direction.

A Theory of Solvability for Lossless Power Flow Equations—Part I: Fixed-Point Power Flow

“…In all cases, branch and shunt conductances were set to zero, in line with our main theoretical assumption. 9 The cycle-space matrix C ∈ {−1, 0, 1} |E|×c was generated using C = null(A,"r") in MATLAB. Simulation results are shown in Table I.…”

“…To examine performance in more heavily loaded networks, each case was loaded along the base case direction 90% of the way to the power flow insolvability boundary, as determined by continuation power flow cpf in MATPOWER. The previous experiments were repeated, and the results are 9 For context on this assumption, the mean branch R/X ratios of the networks in Table I are Figure 4 repeats the comparison for heavy loading. In this case the NR requires only one additional iteration to reach machine precision, while the FPPF and FDLF iteration counts double; this is consistent across all cases.…”

“…Expanding k(v) by substituting the block incidence matrix (2), the block D matrix (9), and the partitioned h(v) from (16), we obtain…”

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“…(iii) polynomial reformulation and semi-definite relaxation of the power-flow equations (1) [7,65,66,68]; and (iv) affine approximation of the sets F(d) and C which is discussed next.…”

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“…This section presents an alternative to the iterations in (12). The PF equations can be arranged into a different fixed-point iteration after dividing (7) by v n to get…”