1. Javad Lavaei Department of Electrical Engineering Columbia University Optimization over Graph with Application to Power Systems

2. Acknowledgements Caltech: Steven Low Somayeh Sojoudi Stanford University: Stephen Boyd Eric Chu Matt Kranning UC Berkeley: David Tse Baosen Zhang  J. Lavaei and S. Low, "Zero Duality Gap in Optimal Power Flow Problem," IEEE Transactions on Power Systems, 2012.  J. Lavaei, D. Tse and B. Zhang, "Geometry of Power Flows in Tree Networks,“ in IEEE Power & Energy Society General Meeting, 2012.  S. Sojoudi and J. Lavaei, "Physics of Power Networks Makes Hard Optimization Problems Easy To Solve,“ in IEEE Power & Energy Society General Meeting, 2012.  M. Kraning, E. Chu, J. Lavaei and S. Boyd, "Message Passing for Dynamic Network Energy Management," Submitted for publication, 2012.  S. Sojoudi and J. Lavaei, "Semidefinite Relaxation for Nonlinear Optimization over Graphs with Application to Optimal Power Flow Problem," Working draft, 2012.  S. Sojoudi and J. Lavaei, "Convexification of Generalized Network Flow Problem with Application to Optimal Power Flow," Working draft, 2012.

3. Power Networks (CDC 10, Allerton 10, ACC 11, TPS 12, ACC 12, PGM 12) Javad Lavaei, Columbia University 3  Optimizations:  Resource allocation  State estimation  Scheduling  Issue: Nonlinearities  Transition from traditional grid to smart grid:  More variables (10X)  Time constraints (100X)

4. Resource Allocation: Optimal Power Flow (OPF) Javad Lavaei, Columbia University 4 OPF: Given constant-power loads, find optimal P’s subject to:  Demand constraints  Constraints on V’s, P’s, and Q’s. OPF: Given constant-power loads, find optimal P’s subject to:  Demand constraints  Constraints on V’s, P’s, and Q’s. Voltage V Complex power = VI * =P + Q i Current I

5. Broad Interest in Optimal Power Flow Javad Lavaei, Columbia University 5  Interested companies: ISOs, TSOs, RTOs, Utilities, FERC  OPF solved on different time scales:  Electricity market  Real-time operation  Security assessment  Transmission planning  Existing methods based on local search algorithms  Can save $$$ if solved efficiently  Huge literature since 1962 by power, OR and Econ people Recent conference by Federal Energy Regulatory Commission about OPF

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 parallel 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 relate the polynomial-time solvability of an optimization to its structural properties? (joint work with Somayeh Sojoudi) Project 5 : How to solve generalized network flow (CS problem)? (joint work with Somayeh Sojoudi)

8. Problem of Interest Javad Lavaei, Columbia University 8  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:

9. Example 1 Javad Lavaei, Columbia University 9 Trick: SDP relaxation:  Guaranteed rank-1 solution!

10. Example 1 Javad Lavaei, Columbia University 10 Opt:  Sufficient condition for exactness: Sign definite sets.  What if the condition is not satisfied?  Rank-2 W (but hidden)  NP-hard

11. Example 2 Javad Lavaei, Columbia University 11 Opt:  Real-valued case: Rank-2 W (need regularization)  Complex-valued case:  Real coefficients: Exact SDP  Imaginary coefficients: Exact SDP  General case: Need sign definite sets Acyclic Graph

12. Sign Definite Set Javad Lavaei, Columbia University 12  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 :

13. Formal Definition: Optimization over Graph Javad Lavaei, Columbia University 13 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:

14. Real-Valued Optimization Javad Lavaei, Columbia University 14 Edge Cycle

15. Complex-Valued Optimization Javad Lavaei, Columbia University 17  SDP relaxation for acyclic graphs:  real coefficients  1-2 element sets (power grid: ~10 elements)  Main requirement in complex case: Sign definite weight sets Generalization of Bose et al. work from Caltech

16. Complex-Valued Optimization Javad Lavaei, Columbia University 18  Purely imaginary weights (lossless power grid):  Consider a real matrix M:  Polynomial-time solvable for weakly-cyclic bipartite graphs. Generalization of Zhang & Tse’s work from UC Berkeley

17. Optimal Power Flow Javad Lavaei, Columbia University 19 Cost Operation Flow Balance  Express the last constraint as an inequality.

18. Exact Convex Relaxation  Result 1: Exact relaxation for DC/AC distribution and DC transmission. Javad Lavaei, Stanford University 17 Javad Lavaei, Columbia University 20  OPF: DC or AC  Networks: Distribution or transmission  Energy-related optimization:

19. Exact Convex Relaxation Javad Lavaei, Stanford University 17 Javad Lavaei, Columbia University 21  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.

20. Exact Convex Relaxation Javad Lavaei, Stanford University 17 Javad Lavaei, Columbia University PS 22  AC transmission network manipulation:  High performance (lower generation cost)  Easy optimization  Easy market

21. Injection Regions Javad Lavaei, Columbia University 23  Injection region for acyclic networks:  When can we allow equality constraints? Need to study Pareto front

22. Generalized Network Flow (GNF) Javad Lavaei, Columbia University 24 injections flows  Goal: limits Assumption: f i (p i ): convex and increasing f ij (p ij ): convex and decreasing

23. Convexification of GNF Javad Lavaei, Columbia University 25  Convexification: Feasible set without box constraint:  It finds correct injection vector but not necessarily correct flow vector.

24. Convexification of GNF Javad Lavaei, Columbia University 26 Feasible set without box constraint:  Correct injections in the feasible case.  Why decreasing flow functions?

25. Conclusions Javad Lavaei, Columbia University 27  Motivation: OPF with a 50-year history  Goal: Develop theory of optimization over graph  Mapped the structure of an optimization into a generalized weighted graph  Obtained various classes of polynomial-time solvable optimizations  Talked about Generalized Network Flow  Passivity in power systems made optimizations easier