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On the Non-Convexity Degree of Lossy Optimal Power Flow Models: Numerical Studies

The non-convexity of an AC optimal power flow (AC OPF) problem has been presented by giving some examples, but the non-convexity of general AC OPF has not been presented. This paper proposes that transmission losses will cause the non-convexity of the feasible injection region (FIR) of general AC OPF problems. It shows that network lossless is the necessary condition for a convex FIR, but it is not a sufficient condition. It is also shown that the FIR of lossless AC OPF problems can be non-convex. A numerical method is proposed to calculate the non-convexity degree of a set and is applied to show that a seemingly convex FIR is, in fact, nonconvex. It also seeks to show that network losses can lead to the non-convexity of the FIR. Large network losses lead to a high degree of non-convexity. It is also shown that the higher the network losses, the higher the degree of non-convexity for the FIR. The conclusions are illustrated in 9-, 33-, and 57-bus systems.