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## Optimal Power Flow in Radial Networks

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**Optimal Power Flow in Radial Networks**Lingwen Gan1, Na Li2, Ufuk Topcu3, Steven Low1 1California Institute of Technology 2Harvard University 3University of Pennsylvania**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]**Convex relaxations of OPF**Feasible set of OPF OPF is Quadratic Constrained Quadratic Programming Feasible set of SDP optimal solution standard relaxation Semi-Definite Programming (SDP) [Bai08, Javad10] Feasible set of SOCP radial network Second-Order Cone Programming (SOCP) [Farivar11, Javad12] Sufficient conditions derived for exact convex relaxation.**Outline**Formulate OPF. Sufficient condition for exact SOCP. Modify OPF accordingly. Numerical case studies.**Terminologies in power network**Power network represented by a graph. bus line Radial j i**The OPF problem**min generation cost j i s S l v over power flow constraints s.t. Nonconvex! injection constraints voltage constraints**The SOCP relaxation**OPF SOCP [Farivar11] feasible set of SOCP feasible set of OPF optimal solution Def: SOCP is exact (every) optimal solution feasible for OPF.**When is relaxation exact?**Counter examples exist. j i Sufficient conditions derived for radial networks. Type 1: Type 2: Type 3: s satisfy some (mild) constraints.**A sufficient condition**Theorem 1: C1 + C2 SOCP exact. C1: C2: What is C1? Introduce for every line. root Impose .**A sufficient condition**Theorem 1: C1 + C2 SOCP exact. C1: C2: is affine in s. SOCP if line loss is small.**Interpretation of C1 and C2**C1 + C2 SOCP exact. Hold for various networks with sufficient margin. Impose C2 in the first place.**A modified OPF**Impose C2 in the first place. OPF-m Theorem: C1 SOCP exact for OPF-m.**Numerical studies**C1 SOCP exact for OPF-m. How widely does C1 hold? How big is the modification? IEEE 13-, 34-, 37-, 123-bus networks; SCE147-, 56-bus networks. 1Southern California Edison, a utility company in southern California, USA.**C1 holds with sufficient margin**Yes C1 holds? No margin > 1 C1 holds. distributed generation & shunt capacitor larger margin C1 holds more “safely.” C1 margin margin > 1 Without distributed generation, margin is very big (>10). With big distributed generation, margin is still sufficient.**OPF-m is similar to OPF**The value ε quantifies the gap between OPF-m and OPF.**Summary**C1 Checked a priori. Holds for all test networks. + power power current C2 Motivate modified OPF problem. SOCP is exact