An Efficient Computational Method for Nonlinear Power Optimization Problems

1. JavadLavaeiDepartment of Electrical EngineeringColumbia UniversityJoint work with SomayehSojoudiand RamtinMadani An Efficient Computational Method for Nonlinear Power Optimization Problems

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. Local Solutions P1 P2 OPF Local solution: $1502 Global solution: $338 Javad Lavaei, Columbia University 4

5. Local Solutions Source of Difficulty: Power is quadratic in terms of complex voltages. • Anya Castillo et al. • Ian Hiskens from Umich: • Study of local solutions by Edinburgh’s group Javad Lavaei, Columbia University 5

6. 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 6

7. 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 7

8. Geometric Intuition: Two-Generator Network Javad Lavaei, Columbia University 8

9. 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 9

10. 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 10

11. AC Transmission Networks • How about AC transmission networks? • May not be true for every network • Various sufficient conditions • AC transmission network manipulation: • High performance (lower generation cost) • Easy optimization • Easy market (positive LMPs and existence of eq. pt.) PS Javad Lavaei, Columbia University 11

12. Phase Shifters • Blue: Feasible set (PG1,PG2) • Green: Effect of phase shifter • Red: Effect of convexification • Minimization over green = Minimization over green and red (even with box constraints) Javad Lavaei, Columbia University 12

13. Phase Shifters • Simulations: • Zero duality gap for IEEE 30-bus system • Guarantee zero duality gap for all possible load profiles? • Theoretical side: Add 12 phase shifters • Practical side: 2 phase shifters are enough • IEEE 118-bus system needs no phase shifters (power loss case) Phase shifters speed up the computation: Javad Lavaei, Columbia University 13

14. Response of SDP to Equivalent Formulations • Capacity constraint: active power, apparent power, angle difference, voltage difference, current? P2 P1 Equivalent formulations behave differently after relaxation. SDP works for weakly-cyclic networks with cycles of size 3 if voltage difference is used to restrict flows. Correct solution Javad Lavaei, Columbia University 14

15. Low-Rank Solution Javad Lavaei, Columbia University 15

16. Penalized SDP Relaxation • How to turn a low-rank solution into a rank-1 solution? • Extensive simulations show that reactive power needs to be corrected. • 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 16

17. 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 17

18. Penalized SDP Relaxation Javad Lavaei, Columbia University 18

19. Problem of Interest • Abstract optimizations are NP-hard in the worst case. • Real-world optimizations are highly structured: • Sparsity: • Non-trivial structure: • Question: How does the physical structure affect tractability of an optimization? Javad Lavaei, Columbia University 19

20. Example 1 Trick: SDP relaxation: • Guaranteed rank-1 solution! Javad Lavaei, Columbia University 20

21. Example 1 Opt: • Sufficient condition for exactness: Sign definite sets. • What if the condition is not satisfied? • Rank-2 W(but hidden) • NP-hard Javad Lavaei, Columbia University 21

22. Sign Definite Set • Real-valued case: “T “ is sign definite if its elements are all negative or all positive. • Complex-valued case: “T “ is sign definite if T and –T are separable in R2: Javad Lavaei, Columbia University 22

23. Exact Convex Relaxation • Each weight set has about 10 elements. • Due to passivity, they are all in the left-half plane. • Coefficients: Modes of a stable system. • Weight sets are sign definite. Javad Lavaei, Stanford University Javad Lavaei, Columbia University 17 23

24. Formal Definition: Optimization over Graph Optimization of interest: (real or complex) Define: • SDP relaxation for y and z (replace xx* with W). • f (y , z) is increasing in z (no convexity assumption). • Generalized weighted graph: weight set for edge (i,j). Javad Lavaei, Columbia University 24

25. Real-Valued Optimization Edge Cycle Javad Lavaei, Columbia University 25

26. Complex-Valued Optimization • Main requirement in complex case: Sign definite weight sets • SDP relaxation for acyclic graphs: • real coefficients • 1-2 element sets (power grid: ~10 elements) Javad Lavaei, Columbia University 26

27. Conclusions • Focus: OPF with a 50-year history • Goal: Find a global solution efficiently • Obtained provably global solutions for many practical OPFs • Developed various theories for distribution and transmission networks • Still some open problems to be addressed Javad Lavaei, Columbia University 27