On reactive power flow and voltage stability in microgrids

This paper focuses on reactive power flow and voltage stability in electrical grids. We provide novel analytical understanding of the solutions to the classic nonlinear polynomial equations describing the decoupled reactive power flow. As of today, solutions to these equations can be found only via numerical methods. Yet an analytical understanding would be beneficial to the rigorous design of future electrical grids. This paper has two main contributions. First, for sufficiently high reference voltages, we guarantee the existence of a high-voltage solution for the reactive power flow equations and provide its approximate analytical expression. We bound the approximation error in terms of network topology and parameters. Second, we consider a recently proposed droop control strategy for voltage stabilization in a microgrid equipped with inverters. For sufficiently high reference voltages, we prove the existence and the exponential stability of a high-voltage fixed point of the closed-loop dynamics. We provide an approximate expression for this fixed point and bound the approximation error in terms of the network topology and parameters. Finally, we validate the accuracy of our approximations through numerical simulation of the IEEE 37 standard test case.