1. Javad Lavaei Department of Electrical Engineering Columbia University Graph-Theoretic Algorithm for Nonlinear Power Optimization Problems

2. Outline Javad Lavaei, Columbia University 2  Convex relaxation for highly sparse optimization (Joint work with: Somayeh Sojoudi, Ramtin Madani, and Ghazal Fazelnia)  Optimization over power networks (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, Ramtin Madani, Baosen Zhang, Matt Kraning, Eric Chu, and Morteza Ashraphijuo)  Optimal decentralized control (Joint work with: Ghazal Fazelnia,Ramtin Madani, and Abdulrahman Kalbat)  General theory for polynomial optimization (Joint work with: Ramtin Madani and Somayeh Sojoudi)

3. Penalized Semidefinite Programming (SDP) Relaxation Javad Lavaei, Columbia University 3  Exactness of SDP relaxation:  Existence of a rank-1 solution  Implies finding a global solution  How to study the exactness of relaxation?

4. Example Javad Lavaei, Columbia University 4  Given a polynomial optimization, we first make it quadratic and then map its structure into a generalized weighted graph:

5. Complex-Valued Optimization Javad Lavaei, Columbia University 5  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 :

6. Treewidth Javad Lavaei, Columbia University 6  Tree decomposition:  We map a given graph G into a tree T such that:  Each node of T is a collection of vertices of G  Each edge of G appears in one node of T  If a vertex shows up in multiple nodes of T, those nodes should form a subtree  Width of a tree decomposition: The cardinality of largest node minus one  Treewidth of graph: The smallest width of all tree decompositions

7. Low-Rank SDP Solution Javad Lavaei, Columbia University 7 Real/complex optimization  Define G as the sparsity graph  Theorem: There exists a solution with rank at most treewidth of G +1  We propose infinitely many optimizations to find that solution.  This provides a deterministic upper bound for low-rank matrix completion problem.

8. Outline Javad Lavaei, Columbia University 8  Convex relaxation for highly sparse optimization (Joint work with: Somayeh Sojoudi, Ramtin Madani, and Ghazal Fazelnia)  Optimization over power networks (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, Ramtin Madani, Baosen Zhang, Matt Kraning, Eric Chu, and Morteza Ashraphijuo)  Optimal decentralized control (Joint work with: Ghazal Fazelnia,Ramtin Madani, and Abdulrahman Kalbat)  General theory for polynomial optimization (Joint work with: Ramtin Madani, Somayeh Sojoudi and Ghazal Fazelnia)

9. 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 9

10. Optimal Power Flow Javad Lavaei, Columbia University 10 Cost Operation Flow Balance

11. Project 1 Javad Lavaei, Columbia University 11  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)

12. Project 2 Javad Lavaei, Columbia University 12  Transmission networks may need phase shifters: Project 2: Find network topologies over which optimization is easy? (joint work with Somayeh Sojoudi, David Tse and Baosen Zhang)  Distribution networks are fine due to a sign definite property: PS

13. Project 3 Javad Lavaei, Columbia University 13 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 using ADMM.  It solves 10,000-bus OPF in 0.85 seconds on a single core machine.

14. Project 4 Javad Lavaei, Columbia University 14 Project 4: How to do optimization for mesh networks? (joint work with Ramtin Madani and Somayeh Sojoudi)  Observed that equivalent formulations might be different after relaxation.  Upper bounded the rank based on the network topology.  Developed a penalization technique.  Verified its performance on IEEE systems with 7000 cost functions.

15. Response of SDP to Equivalent Formulations Javad Lavaei, Columbia University 15 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.

16. Penalized SDP Relaxation Javad Lavaei, Columbia University 16  Use Penalized SDP relaxation to turn a low-rank solution into a rank-1 matrix:  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.  IEEE systems with 7000 cost functions  Modified 118-bus system with 3 local solutions (Bukhsh et al.)

17. Power Networks Javad Lavaei, Columbia University 17  Treewidth of a tree: 1  How about the treewidth of IEEE 14-bus system with multiple cycles? 2  How to compute the treewidth of a large graph?  NP-hard problem  We used graph reduction techniques for sparse power networks

18. Power Networks Javad Lavaei, Columbia University 18  Upper bound on the treewidth of sample power networks: Real/complex optimization  Theorem: There exists a solution with rank at most treewidth of G +1

19. Examples Javad Lavaei, Columbia University 19  Example: Consider the security-constrained unit-commitment OPF problem.  Use SDP relaxation for this mixed-integer nonlinear program.  What is the rank of X opt ? 1.IEEE 300-bus system: rank ≤ 7 2.Polish 3120-bus system: Rank ≤ 27 IEEE 14-bus system IEEE 30-bus systemIEEE 57-bus system  How to go from low-rank to rank-1? Penalization (tested on 7000 examples)

20. Outline Javad Lavaei, Columbia University 20  Convex relaxation for highly sparse optimization (Joint work with: Somayeh Sojoudi, Ramtin Madani, and Ghazal Fazelnia)  Optimization over power networks (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, Ramtin Madani, Baosen Zhang, Matt Kraning, Eric Chu, and Morteza Ashraphijuo)  Optimal decentralized control (Joint work with: Ghazal Fazelnia,Ramtin Madani, and Abdulrahman Kalbat)  General theory for polynomial optimization (Joint work with: Ramtin Madani, Somayeh Sojoudi and Ghazal Fazelnia)

21. Distributed Control Javad Lavaei, Columbia University 21  Computational challenges arising in the control of real-world systems:  Communication networks  Electrical power systems  Aerospace systems  Large-space flexible structures  Traffic systems  Wireless sensor networks  Various multi-agent systems Decentralized control Distributed control

22. Optimal Decentralized Control Problem Javad Lavaei, Columbia University 22  Optimal centralized control: Easy (LQR, LQG, etc.)  Optimal distributed control (ODC): NP-hard (Witsenhausen’s example)  Consider the time-varying system:  The goal is to design a structured controller to minimize

23. Graph of ODC for Time-Domain Formulation Javad Lavaei, Columbia University 23

24. Numerical Example Javad Lavaei, Columbia University 24 Mass-Spring Example

25. Distributed Control in Power Javad Lavaei, Columbia University 25  Example: Distributed voltage and frequency control  Generators in the same group can talk.

26. Outline Javad Lavaei, Columbia University 26  Convex relaxation for highly sparse optimization (Joint work with: Somayeh Sojoudi, Ramtin Madani, and Ghazal Fazelnia)  Optimization over power networks (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, Ramtin Madani, Baosen Zhang, Matt Kraning, Eric Chu, and Morteza Ashraphijuo)  Optimal decentralized control (Joint work with: Ghazal Fazelnia,Ramtin Madani, and Abdulrahman Kalbat)  General theory for polynomial optimization (Joint work with: Ramtin Madani, Somayeh Sojoudi, and Ghazal Fazelnia)

27. Polynomial Optimization Javad Lavaei, Columbia University 27  Sparsification Technique: distributed computation  This gives rise to a sparse QCQP with a sparse graph.  The treewidth can be reduced to 2. Theorem: Every polynomial optimization has a QCQP formulation whose SDP relaxation has a solution with rank 1, 2 or 3.

28. Conclusions Javad Lavaei, Columbia University 28  Convex relaxation for highly sparse optimization:  Complexity may be related to certain properties of a graph.  Optimization over power networks:  Optimization over power networks becomes mostly easy due to their structures.  Optimal decentralized control:  ODC is a highly sparse nonlinear optimization so its relaxation has a rank 1-3 solution.  General theory for polynomial optimization:  Every polynomial optimization has an SDP relaxation with a rank 1-3 solution.