1. Javad Lavaei Department of Electrical Engineering Columbia University Various Techniques for Nonlinear Energy-Related Optimizations

2. Acknowledgements Caltech: Steven Low, Somayeh Sojoudi Columbia University: Ramtin Madani UC Berkeley: David Tse, Baosen Zhang Stanford University: Stephen Boyd, Eric Chu, Matt Kranning  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 11, 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. Summary of Results Javad Lavaei, Columbia University 5  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)

6. Summary of Results Javad Lavaei, Columbia University 6 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 5 : How to relate the polynomial-time solvability of an optimization to its structural properties? (joint work with Somayeh Sojoudi) Project 6 : How to solve generalized network flow (CS problem)? (joint work with Somayeh Sojoudi) Project 4: How to do optimization for mesh networks? (joint work with Ramtin Madani)

7. Convexification Javad Lavaei, Columbia University 7  Flow:  Injection:  Polar:  Rectangular:

8. Convexification in Polar Coordinates Javad Lavaei, Columbia University 8  Imposed implicitly (thermal, stability, etc.)  Imposed explicitly in the algorithm Similar to the condition derived in Ross Baldick’s book

9. Convexification in Polar Coordinates Javad Lavaei, Columbia University 9  Idea:  Algorithm:  Fix magnitudes and optimize phases  Fix phases and optimize magnitudes

10. Convexification in Polar Coordinates Javad Lavaei, Columbia University 10  Can we jointly optimize phases and magnitudes?  Observation 1: Bounding |V i | is the same as bounding X i.  Observation 2: is convex if m ≥ 2.  Observation 3: is convex for a large enough m.  Observation 4: |V i | 2 is convex for m ≤ 2. Change of variables: Assumption (implicit or explicit):

11. Convexification in Polar Coordinates Javad Lavaei, Columbia University 11 Strategy 1: Choose m=2  Q ij and Q i become convex in both phases and magnitudes.  P ij and P i become convex after the following approximation:  Justification: Active power is sensitive to phases and not magnitudes. Strategy 2: Choose m large enough  P ij, Q ij, P i and Q i become convex after the following approximation: Replace |V i | 2 with its nominal value.

12. Convexification in Polar Coordinates Javad Lavaei, Columbia University 12  DC OPF: Linearization around phases with fixed voltage magnitudes.  Active power becomes linear in phases.  Reactive power becomes constant!  Can we approximate active power using DC OPF but keep nonlinearity in reactive power?

13. Example 1 Javad Lavaei, Columbia University 13 Trick: SDP relaxation:  Guaranteed rank-1 solution!

14. Example 1 Javad Lavaei, Columbia University 14 Opt:  Sufficient condition for exactness: Sign definite sets.  What if the condition is not satisfied? Rank-2 W (but hidden) Complex case:

15. Formal Definition: Optimization over Graph Javad Lavaei, Columbia University 15 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:

16. Highly Structured Optimization Javad Lavaei, Columbia University 16 Edge Cycle

17. Convexification in Rectangular Coordinates Javad Lavaei, Columbia University 17 Cost Operation Flow Balance  Express the last constraint as an inequality.

18. Convexification in Rectangular Coordinates  Partial results for AC lossless transmission networks. Javad Lavaei, Stanford University 17 Javad Lavaei, Columbia University 18

19. Phase Shifters Javad Lavaei, Stanford University 17 Javad Lavaei, Columbia University PS 19  Practical approach: Add phase shifters and then penalize their effects.  Stephen Boyd’s function for PF:

20. Integrated OPF + Dynamics Javad Lavaei, Stanford University 17 Javad Lavaei, Columbia University 20  Swing equation:  Define:  Linear system:  Synchronous machine with interval voltage and terminal voltage.

21. Sparse Solution to OPF Javad Lavaei, Stanford University 17 Javad Lavaei, Columbia University 21  Unit commitment: 1- 2-  Unit commitment: 1- 2-  Sparse solution to OPF: 1- 2- Sparse vector  Minimize:

22. Lossy Networks Javad Lavaei, Stanford University 17 Javad Lavaei, Columbia University 22  Assumption (implicit or explicit):  Conjecture: This assumptions leads to convexification in rectangular coordinates.  Partial Result: Proof for optimization of reactive powers.  Relationship between polar and rectangular?

23. Lossless Networks Javad Lavaei, Stanford University 17 Javad Lavaei, Columbia University 23 (P1,P2)(P1,P2) (P 12,P 23,P 31 ) Lossless 3 bus (P 1,P 2,P 3 ) for a 4-bus cyclic Network: Theorem: The injection region is never convex for n ≥ 5 if  Consider a lossless AC transmission network.  Current approach: Use polynomial Lagrange multiplier (SOS)

24. OPF With Equality Constraints Javad Lavaei, Columbia University 24  Injection region under fixed voltage magnitudes:  When can we allow equality constraints? Need to study Pareto front

25. Generalized Network Flow (GNF) Javad Lavaei, Columbia University 25 injections flows  Goal: limits Assumption: f i (p i ): convex and increasing f ij (p ij ): convex and decreasing

26. Convexification of GNF Javad Lavaei, Columbia University 26  Convexification: Feasible set without box constraint  It finds correct injection vector but not necessarily correct flow vector.

27. Conclusions Javad Lavaei, Columbia University 27  Motivation: OPF with a 50-year history  Goal: Find a good numerical algorithm  Convexification in polar coordinates  Convexification in rectangular coordinates  Exact relaxation in several cases  Some problems yet to be solved.