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Finding a global solution to the optimal power flow (OPF) problem isdifficult due to its nonconvexity. A convex relaxation in the form ofsemidefinite programming (SDP) has attracted much attention lately as it yieldsa global solution in several practical cases. However, it does not in allcases, and such cases have been documented in recent publications. This paperpresents another SDP method known as the moment-sos (sum of squares) approach,which generates a sequence that converges towards a global solution to the OPFproblem at the cost of higher runtime. Our finding is that in the smallexamples where the previously studied SDP method fails, this approach finds theglobal solution. The higher cost in runtime is due to an increase in the matrixsize of the SDP problem, which can vary from one instance to another. Numericalexperiment shows that the size is very often a quadratic function of the numberof buses in the network, whereas it is a linear function of the number of busesin the case of the previously studied SDP method.

Application of the Moment-SOS Approach to Global Optimization of the OPF Problem