1. 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

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 Javad Lavaei, Columbia University 3  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

4. Local Solutions Javad Lavaei, Columbia University 4 Local solution: $1502 Global solution: $338 OPF P1P1 P2P2

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

6. Summary of Results Javad Lavaei, Columbia University 6  A sufficient condition to globally solve OPF:  Numerous randomly generated systems  IEEE systems with 14, 30, 57, 118, 300 buses  European grid  Various theories: It holds widely in practice Project 1: How to solve a given OPF in polynomial time? (joint work with Steven Low)  Distribution networks are fine.  Every transmission network can be turned into a good one. Project 2: Find network topologies over which optimization is easy? (joint work with Somayeh Sojoudi, David Tse and Baosen Zhang)

7. Summary of Results Javad Lavaei, Columbia University 7 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 Ramtin Madani and Somayeh Sojoudi)  Developed a penalization technique  Verified its performance on IEEE systems with 7000 cost functions

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 Dual OPFSDP OPF Javad Lavaei, Columbia University 10  SDP relaxation:  IEEE systems  SC Grid  European grid  Random systems  Exactness of SDP relaxation and zero duality gap are equivalent for OPF.

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 Javad Lavaei, Columbia University 12  Blue: Feasible set (P G1,P G2 )  Green: Effect of phase shifter  Red: Effect of convexification  Minimization over green = Minimization over green and red (even with box constraints)

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) Javad Lavaei, Columbia University 13 Phase shifters speed up the computation:

14. Response of SDP to Equivalent Formulations Javad Lavaei, Columbia University 14 P1P1 P2P2  Capacity constraint: active power, apparent power, angle difference, voltage difference, current? Correct solution 1. Equivalent formulations behave differently after relaxation. 2.SDP works for weakly-cyclic networks with cycles of size 3 if voltage difference is used to restrict flows.

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

16. Penalized SDP Relaxation Javad Lavaei, Columbia University 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 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.

17. Penalized SDP Relaxation Javad Lavaei, Columbia University 17  Let λ 1 and λ 2 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.

18. Penalized SDP Relaxation Javad Lavaei, Columbia University 18

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

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

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

22. Sign Definite Set Javad Lavaei, Columbia University 22  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 R 2 :

23. Exact Convex Relaxation Javad Lavaei, Stanford University 17 Javad Lavaei, Columbia University 23  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.

24. Formal Definition: Optimization over Graph Javad Lavaei, Columbia University 24 Optimization of interest: (real or complex)  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). Define:

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

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

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