Low-Rank Solution of Convex Relaxation for Optimal Power Flow Problem

1. JavadLavaeiDepartment of Electrical EngineeringColumbia UniversityJoint work with SomayehSojoudiand RamtinMadani Low-Rank Solution of Convex Relaxation for Optimal Power Flow Problem

2. Power Networks • Optimizations: • Optimal power flow (OPF) • Security-constrained OPF • State estimation • Network reconfiguration • Unit commitment • Dynamic energy management • Issue of non-convexity: • Discrete parameters • Nonlinearity in continuous variables • Transition from traditional grid to smart grid: • More variables (10X) • Time constraints (100X) Javad Lavaei, Columbia University 2

3. Broad Interest in Optimal Power Flow • OPF-based problems solved on different time scales: • Electricity market • Real-time operation • Security assessment • Transmission planning • Existing methods based on linearization or local search • Question: How to find the best solution using a scalable robust algorithm? • Huge literature since 1962 by power, OR and Econ people Javad Lavaei, Columbia University 3

4. Summary of Results Project 1:How to solve a given OPF in polynomial time? (joint work with Steven Low) • A sufficient condition to globally solve OPF: • Numerous randomly generated systems • IEEE systems with 14, 30, 57, 118, 300 buses • European grid • Various theories: Itholds widely in practice Project 2:Find network topologies over which optimization is easy? (joint work with Somayeh Sojoudi, David Tse and Baosen Zhang) • Distribution networks are fine. • Every transmission network can be turned into a good one. Javad Lavaei, Columbia University 4

5. Summary of Results Project 3:How to design a distributed algorithm for solving OPF? (joint work with Stephen Boyd, Eric Chu and Matt Kranning) • A practical (infinitely) parallelizable algorithm • It solves 10,000-bus OPF in 0.85 seconds on a single core machine. Project 4:How to do optimization for mesh networks? (joint work with RamtinMadani and Somayeh Sojoudi) • Developed a penalization technique • Verified its performance on IEEE systems with 7000 cost functions Javad Lavaei, Columbia University 5

6. Geometric Intuition: Two-Generator Network Javad Lavaei, Columbia University 6

7. Optimal Power Flow Cost Operation Flow Balance • Extensions: • Other objective (voltage support, reactive power, deviation) • More variables, e.g. capacitor banks, transformers • Preventive or corrective contingency constraints Javad Lavaei, Columbia University 7

8. Various Relaxations OPF • SDP relaxation: • IEEE systems • SC Grid • European grid • Random systems Dual OPF SDP • Exactness of SDP relaxation and zero duality gap are equivalent for OPF. Javad Lavaei, Columbia University 8

9. Response of SDP to Equivalent Formulations • Capacity constraint: active power, apparent power, angle difference, voltage difference, current? P2 P1 Correct solution Equivalent formulations behave differently after relaxation. Problem D has an exact relaxation. Javad Lavaei, Columbia University 9

10. Weakly-Cyclic Networks • Theorem: SDP works for weakly-cyclic networks with cycles of size 3 if voltage difference is used to restrict flows. • Observation: A lossless 3-bus system has a non-convex flow region but a convex injection region. Javad Lavaei, Columbia University 10

11. Highly Meshed Networks • Theorem: The injection region is non-convex for a single cycle of size 5 or more. • How to deal with highly meshed networks or systems with large cycles? • If we can’t find a rank-1 solution, it’s still plausible to obtain a low-rank solution: • Approximate a low-rank solution by a rank-1 matrix thru eig decomposition. • Fine-tune a low-rank solution using a local search algorithm. • Is there a low-rank solution for real-world systems? Javad Lavaei, Columbia University 11

12. Low-Rank Solution Javad Lavaei, Columbia University 12

13. Penalized SDP Relaxation • How to turn a low-rank solution into a rank-1 solution? • Consider a PSD matrix with some free entries. • Maximization of the sum of the off-diagonal entries results in a rank-1 solution. • Lossless networks: • Active power is in terms of Im{W}. • Reactive power is in terms of Re{W}. • Hence, penalization of reactive power is helpful. Javad Lavaei, Columbia University 13

14. Penalized SDP Relaxation • Extensive simulations show that reactive power needs to be corrected. • Intuition: Among many pairs (PG,QG)’s with the same first component, we want to find one with the best second component. • Penalized SDP relaxation: • Penalized SDP relaxation aims to find a near-optimal solution. • It worked for IEEE systems with over 7000 different cost functions. • Near-optimal solution coincided with the IPM’s solution in 100%, 96.6% and 95.8% of cases for IEEE 14, 30 and 57-bus systems. Javad Lavaei, Columbia University 14

15. Penalized SDP Relaxation • Let λ1 and λ2 denote the two largest eigenvalues of W. • Correction of active powers is negligible but reactive powers change noticeably. • There is a wide range of values for ε giving rise to a nearly-global local solution. Javad Lavaei, Columbia University 15

16. Penalized SDP Relaxation Javad Lavaei, Columbia University 16

17. Conclusions • Focus: OPF with a 50-year history • Goal: Find a near-global solution efficiently • Equivalent formulations may lead to different relaxations (best formulation = use voltage difference for line capacity). • Existence of low-rank solutions for power networks. • Recovery of a rank-1 solution thru a penalization (by correcting reactive power). • Simulations performed on 7000 problems. Javad Lavaei, Columbia University 17