Convexity Analysis and Linear Approximation of Voltage Magnitude - Power Injection Mapping for Distribution Systems with Unknown Parameters

In many optimization problems for distribution systems, the operational constraints such as power flow equations and bus voltage constraints should be considered. A popular approach is to formulate the problem as a convex optimization problem assuming that the exact network parameters and topology are known. However, in most cases, the exact network parameters and topology are not known and difficult to infer. With the developments of automation and digitalization of the distribution power grid, smart meters provide abundant useful information and data, such as the injection power and bus voltage, etc., which can help us gain knowledge of the network. In this paper, an idea to linearly approximate the voltage magnitude - power injection mapping with known measurement data is proposed as an attempt to solve distribution system optimization problems with unknown parameters and topology. First, the convexity of the voltage magnitude - power injection mapping is analyzed, based on which the linear approximation of the mapping is proposed. Abundant tests validate the feasibility of the mapping linear approximation. Factors that may influence linearization effects are also investigated and the associated recommendation is provided.