1. Javad Lavaei Department of Electrical Engineering Columbia University Convex Relaxation for Polynomial Optimization: Application to Power Systems and Decentralized Control

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

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. Highly Structured Optimization Javad Lavaei, Columbia University 4  Abstract optimizations are NP-hard in the worst case.  Real-world optimizations are highly structured :  Question: How do structures affect tractability of an optimization?  Sparsity:  Non-trivial structure:

5. Example Javad Lavaei, Columbia University 5  Given a polynomial optimization, we first make it quadratic and then map its structure into a generalized weighted graph:

6. Example Javad Lavaei, Columbia University 6 Optimization:

7. Real-Valued Optimization Javad Lavaei, Columbia University 7 Edge Cycle

8. Complex-Valued Optimization Javad Lavaei, Columbia University 8  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 :

9. Complex-Valued Optimization Javad Lavaei, Columbia University 9  Consider a real matrix M:  Polynomial-time solvable for weakly-cyclic bipartite graphs.

10. Outline Javad Lavaei, Columbia University 10  Convex relaxation for highly structured optimization (Joint work with: Somayeh Sojoudi)  Optimization over power networks (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, Ramtin Madani, Baosen Zhang, Matt Kraning, Eric Chu)  Optimal decentralized control (Joint work with: Ghazal Fazelnia and Ramtin Madani)  General theory for polynomial optimization (Joint work with: Ramtin Madani, Somayeh Sojoudi and Ghazal Fazelnia)

11. 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 11

12. Resource Allocation: Optimal Power Flow (OPF) Javad Lavaei, Columbia University 12 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

13. Broad Interest in Optimal Power Flow Javad Lavaei, Columbia University 13  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

14. Optimal Power Flow Javad Lavaei, Columbia University 14 Cost Operation Flow Balance

15. Project 1 Javad Lavaei, Columbia University 15  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)

16. Project 2 Javad Lavaei, Columbia University 16  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

17. Project 3 Javad Lavaei, Columbia University 17 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.

18. Project 4 Javad Lavaei, Columbia University 18 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.

19. Response of SDP to Equivalent Formulations Javad Lavaei, Columbia University 19 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.

20. Low-Rank Solution Javad Lavaei, Columbia University 20

21. Penalized SDP Relaxation Javad Lavaei, Columbia University 21  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.)

22. Outline Javad Lavaei, Columbia University 22  Convex relaxation for highly structured optimization (Joint work with: Somayeh Sojoudi)  Optimization over power networks (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, Ramtin Madani, Baosen Zhang, Matt Kraning, Eric Chu)  Optimal decentralized control (Joint work with: Ghazal Fazelnia and Ramtin Madani)  General theory for polynomial optimization (Joint work with: Ramtin Madani, Somayeh Sojoudi and Ghazal Fazelnia)

23. Distributed Control Javad Lavaei, Columbia University 23  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

24. Optimal Decentralized Control Problem Javad Lavaei, Columbia University 24  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

25. Two Quadratic Formulations in Static Case Javad Lavaei, Columbia University 25  Formulation in time domain:  Stack the free parameters of K in a vector h.  Define v as:  Formulation in Lypunov domain:  Consider the BMI constraint:  Define v as:

26. Graph of ODC for Time-Domain Formulation Javad Lavaei, Columbia University 26

27. Recovery of Rank-3 Solution Javad Lavaei, Columbia University 27  How to find such a low-rank solution?  Nuclear norm technique fails.  Add edges in the controller cloud to make it a tree:  Add a new vertex and connect it to all existing nodes.  Perform an optimization over the new graph.

28. Numerical Example (Decentralized Case) Javad Lavaei, Columbia University 28  Rank: 1 or 2  Exactness in 59 trials  Rank: 1  99.8% optimality for 84 trials SDP Relaxation Penalized SDP Relaxation

29. Distributed Case Javad Lavaei, Columbia University 29  100 random systems with standard deviation 3 for A and B, and 2 for X 0.  Each communication link exists with probability 0.9.

30. Outline Javad Lavaei, Columbia University 29  Convex relaxation for highly structured optimization (Joint work with: Somayeh Sojoudi)  Optimization over power networks (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, Ramtin Madani, Baosen Zhang, Matt Kraning, Eric Chu)  Optimal decentralized control (Joint work with: Ghazal Fazelnia and Ramtin Madani)  General theory for polynomial optimization (Joint work with: Ramtin Madani, Somayeh Sojoudi, and Ghazal Fazelnia)

31. Polynomial Optimization Javad Lavaei, Columbia University 31  Consider an arbitrary polynomial optimization:  Fact 1:  Fact 2:

32. Polynomial Optimization Javad Lavaei, Columbia University 32  Technique: distributed computation  This gives rise to a sparse QCQP with a sparse graph.  Question: Given a sparse LMI, find a low-rank solution in polynomial time?  We have a theory for relating the rank of W to the topology of its graph.

33. Conclusions Javad Lavaei, Columbia University 33  Convex relaxation for highly structured optimization:  Complexity of SDP relaxation can be related to properties of a generalized 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.