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## Optimal Power Flow in Direct Current Networks

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**Optimal Power Flow in Direct Current Networks**Lingwen Gan, Steven Low California Institute of Technology**Optimal power flow (OPF)**What is OPF? Schedule power plants, minimize electricity cost. Is OPF important? Solved every 15 minutes, underlies power system operation.**Challenges in solving OPF**Power flow governed by nonlinear (nonconvex) laws Three ways to deal with the challenge 1. Linearization / Approximation. [DC, …] 2. Local optima [Newton-Raphson, SQP, …] May guarantee global optimality 3. Convex relaxation [SDP, SOCP]**Outline**Formulate OPF and convex relaxations. Sufficient condition for exact convex relaxations. Modify OPF accordingly.**Terminologies in power network**Power network represented by a graph. bus line j i Direct current: V, s, y are real.**The OPF problem**min generation cost s V over power flow constraints s.t. j i power injection constraints Nonconvex! voltage constraints**Semi-definite programming(SDP) relaxation**Feasible set of OPF OPF SDP [Lavaei10] min Feasible set of SDP min s.t. s.t. …… …… …… remove rank 1**Second-order cone programming(SOCP) relaxation**Feasible set of SDP optimal solution Feasible set of SOCP Def: SOCP exact (every) optimal solution feasible for OPF.**Literature review**Counter examples exist. Sufficient conditions derived. Theorem [Lavaei10]: SDP exact. ×**A sufficient condition for exact SOCP**Affine in s. Theorem: at SOCP solutions SOCP exact. Proof: SOCP not exact exists SOCP solution (s,W) infeasible for OPF. construct (s’,W’) with Obj(s’,W’) < Obj(s,W). cannot be checked a priori.**A modified OPF**Impose as constraints! OPF-m Theorem: SOCP is alwaysexact for OPF-m.**Summary**Propose an SOCP relaxation for OPF in direct currentnetworks. Provide a sufficient condition for exact SOCP. Propose a modified OPF whose SOCP is always exact. Thank you!