Component Analysis and Accuracy Improvement of Linear Power Flow Equation Based on Legendre Polynomial Expansion

The linearization of the power flow equation is widely used in power industry. Existing linearization approaches are generally regarded as nonlinear state variable space transformation, where the original power flow equation can be approximately reformulated as the linear forms in transformed state variable spaces. However, the optimality of linear power flow equations cannot be theoretically compared in different spaces and is only verified based on case studies. In this paper, based on Legendre polynomial expansion, power flow equations are uniformly transformed into a single state variable space, where dimensions correspond to certain orders of Legendre expansion. Through component analysis of power flow equations on each dimension, we find that the accuracy of the linear power flow equation can be improved by formulating an appropriate state variable space transformation to reduce Legendre component loss. Based on this, a linearization method minimizing Legendre component loss is formulated. Compared with the regular case-based linearization, the proposed method does not depend on empirical data, and can theoretically guide the selection of the linear power flow equation within the specified operating bound. The accuracy performance is verified in OPF calculation based on numerous IEEE and Polish test systems.