Fragility of the semidefinite relaxation for the optimal power flow problem

We investigate semidefinite relaxation of the optimal power flow problem. There is a gap between the theoretical validation and practical behavior of the relaxed solution. Several sufficient conditions determine the class of the problems such that semidefinite relaxation gives the exact solutions but it fails for the equivalent problems. On the other hand for the large number of problems zero duality gap is observed without any reasonable explanation. We demonstrate that certain changes in the objective function and/or constraints may lead to the failure of semidefinite relaxations. Moreover, the number of constraints determines the dimension of the feasibility set and adding/cancelling some constraints influences the tightness of relaxation as well. We present simulation for the small 3bus system and IEEE14 test system.