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

Abstract

Abstract-This two-part paper details a theory of solvability for the power flow equations in lossless power networks. In Part I, we derive a new formulation of the lossless power flow equations, which we term the fixed-point power flow. The model is stated for both meshed and radial networks, and is parameterized by several graph-theoretic matrices -the power network stiffness matrices -which quantify the internal coupling strength of the network. The model leads immediately to an explicit approximation of the high-voltage power flow solution. For standard test cases, we find that iterates of the fixedpoint power flow converge rapidly to the high-voltage power flow solution, with the approximate solution yielding accurate predictions near base case loading. In Part II, we leverage the fixed-point power flow to study power flow solvability, and for radial networks we derive conditions guaranteeing the existence and uniqueness of a high-voltage power flow solution. These conditions (i) imply exponential convergence of the fixed-point power flow iteration, and (ii) properly generalize the textbook two-bus system results.Index Terms-Power flow equations, complex networks, power systems, circuit theory, optimal power flow, fixed point theorems. I . I N T R O D U C T I O NThe power flow equations describe the balance and flow of power in a synchronous AC power system. The solutions of these equations (also called operating points of the network) describe the configurations of voltages and currents which (i) are physically consistent with Kichhoff's and Ohm's laws, and (ii) meet the prescribed boundary conditions, specified in terms of fixed power injections or fixed voltage levels at particular nodes in the network. Knowledge of the current system operating point is crucial, as is understanding how the current operating point will change as control actions are taken or as unexpected contingencies occur. As such, the power flow equations are embedded in nearly every power system analysis or control problem, including optimal power flow and its security-constrained variants, transient and voltage stability assessment, contingency screening, short-circuit analysis, and wide-area monitoring/control [1].As the equations are nonlinear, the existence of real-valued solutions is not guaranteed: lightly loaded networks typically possess many solutions [2], while a network which is loaded sufficiently will possess none. Despite this potential for both multiple reasonable solutions and infeasibility, typically there is a single desirable solution, characterized by high voltage magnitudes at buses and small inter-bus current flows. This solution is often termed stable, as it behaves in a manner Aside from intellectual merit, there are at least two important engineering motivations for understanding solvability. The first is to better understand the convergence of iterative numerical algorithms for solving power flow equations. When a power flow solver diverges, it may be because of a numerical instability in the algorithm, an ini...

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