A QCQP and SDP Formulation of the Optimal Power Flow Including Renewable Energy Resources

The optimal power flow (OPF) problem is a relevant subject for the secure and economic power systems operation. For instance, OPF can be used to reduce the power system technical losses. A reduction in 0.1% in the losses accounts for near 50 billion USD in cost savings. Likewise, OPF can be used along with power production forecast tools, for renewable energy sources, to evaluate their impact in the grid security operation planning. In an OPF problem, an objective function related with demand supply generation cost, power line losses, or violation limits is optimized subject to several system security constraints. These constraints are related with branch power flow limits, voltages limits, power injection limits, among others. The exact model of the objective function and constraints of the OPF problem is neither linear nor convex. In this paper it is presented the OPF problem in a Quadratically Constrained Quadratic Program (QCQP) approach. Objective function is based on a quadratic function of the bus voltages and constraints are formulated as Quadratic forms. A two-bus system is used to demonstrates the non-convexity of the OPF problem. Also, it is presented the Rank- 1 convex relaxation of the OPF problem which relaxes the QCQP model into a positive Semidefinite Programming (SDP) model. Once the OPF problem is relaxed into a SDP convex form, a global optimal solution can be obtained. An application example for the OPF problem is presented for the IEEE 14 system. The QCQP and the SDP results are compared and discussed.