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An Efficient Computational Method for Nonlinear Power Optimization Problems

Javad Lavaei Department of Electrical Engineering Columbia University Joint work with Somayeh Sojoudi and Ramtin Madani. An Efficient Computational Method for Nonlinear Power Optimization Problems. Power Networks. Optimizations: Optimal power flow (OPF) Security-constrained OPF

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An Efficient Computational Method for Nonlinear Power Optimization Problems

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  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

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