Inexactness of Second Order Cone Relaxations for Calculating Operating Envelopes

As the number of distributed energy resources participating in power networks increases, it becomes increasingly more important to actively manage network constraints to ensure safe operation. One proposed method that has gained significant attention and implementation, particularly in Australia, is the use of dynamic operating envelopes. Operating envelopes represent net export limits set by the system operator on every node in the distribution network that change as system conditions change. They are calculated using an optimal power flow problem, frequently using a linearization or relaxation of the nonlinear power flow equations. This paper presents two case studies and some numerical analysis to explain why a second order cone relaxation of the power flow equations will lead to ineffective operating envelopes. A modification to the objective function which allows the second order cone relaxation to nearly recover the solution to the nonlinear formulation is also presented.