A quadratic approximation of AC power flows in transformed polar coordinates

A mathematical model is the backbone of any computer-aided simulation. In electrical power systems, the well-known AC power flow equations provide the foundation for deriving various mathematical models. Depending on the representation space for the state variables and the quantity used for computing the nodal balances, the model exhibits distinct characteristics and showcases different nonlinearities. A voluminous amount of literature exists regarding alternative models to the AC power flow equations. Nevertheless, a proposal with an explicit closed quadratic form that operates in a polar voltage-based setting has not been proposed yet. To address this gap, this paper presents a novel quadratic approximation in transformed polar coordinates that applies to both single-phase and unbalanced three-phase networks with radial and meshed topologies. Importantly, the power flow problem is expressed as a set of quadratic forms of the same size as the set of nonlinear AC power flow equations, such that there is no need for additional equations to include operational set points. Furthermore, the approximate optimal power flow problem is formulated as a quadratically-constrained quadratic programming problem. The formulation is packaged and publicly released as part of the widely used tool for power systems research and education, Matpower , along with several contributions related to unbalanced three-phase modeling and features for handling quadratic equations from the simulation and optimization standpoints. Results obtained from fifty test cases with different topologies and numbers of buses demonstrate the proposed formulation is a low-error approximation to AC power flows, which is robust to varying the reactance-to-resistance ratios. • Quadratic approximation of power flow equations based on polar coordinates. • Quadratically constrained quadratic programming for optimal power flow. • Evaluation of quadratic forms using compact matrix computations. • Exact single-phase to three-phase data conversion for Matpower test cases. • Unbalanced three-phase power flow analysis using Matpower .