Javad Lavaei Department of Electrical Engineering Columbia University

1. Javad LavaeiDepartment of Electrical EngineeringColumbia University Graph-Theoretic Algorithm for Arbitrary Polynomial Optimization Problems with Applications to Distributed Control, Power Systems, and Matrix Completion Joint work with:RamtinMadani, Ghazal Fazelnia, AbdulrahmanKalbat and MortezaAshraphijuo(Columbia University)Somayeh Sojoudi (NYU Langone Medical Center)

2. Outline • Theory: Convex relaxation • Application 1: Optimization for power networks • Application 2: Optimal decentralized control • Implementation: High-performance solver handling 1B variables • Theory: General polynomial optimization • Application: Matrix completion Javad Lavaei, Columbia University

3. Penalized Semidefinite Programming (SDP) Relaxation • Exactness of SDP relaxation: • Existence of a rank-1 solution • Implies finding a global solution Javad Lavaei, Columbia University 2

4. Sparsity Graph • Example: . . . 1 2 n Javad Lavaei, Columbia University 3

5. Treewidth Tree decomposition Vertices Bags of vertices • Treewidth of graph: The smallest width of all tree decompositions • Rank of W at optimality ≤ Treewidth +1 • How to find it? (answered later) Javad Lavaei, Columbia University 4

6. Outline • Theory: Convex relaxation • Application 1: Optimization for power networks • Application 2: Optimal decentralized control • Implementation: High-performance solver handling 1B variables • Theory: General polynomial optimization • Application: Matrix completion Javad Lavaei, Columbia University

7. Optimization for Power Systems • Optimization: • 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 • Challenge: ~90% of decisions are made in day ahead and ~10% are updated iteratively during the day so a local solution remains throughout the day. Cost local global Production Javad Lavaei, Columbia University 6

8. Contingency Analysis Contingency • Contingency Analysis: • Assume a line is disconnected. • Many generators cannot change productions quickly. • The flows over other lines would increase. • This triggers a cascading failure. • Secure operation: Design an operating point such that the network survives under certain line or generator outages. Limited correction by a generator • Challenge 1: Number of constraints is prohibitive (our project with Ross Baldick proposes a new technique to address this). • Challenge 2: How to find the best operating point given the nonlinearity of the problem? Javad Lavaei, Columbia University 8

9. Security-Constrained Optimal Power Flow (SCOPF) Cost for pre-contingency case Power flow equations for pre- and post- contingencies Physical and network limits for pre- and post- contingencies Preventive and corrective actions Javad Lavaei, Columbia University 9

10. Power Networks • What graph is important for SCOPF? • Sparsity graph: Disjoint union of pre- and post contingency graphs: Contingency 1 Pre-contingency Contingency 2 • Observation: treewidth of sparsity graph of SCOPF = treewidth of power network • Treewidth: IEEE 300 bus: 6, Polish 3120 ≤ 24, New York State ≤ 40. Javad Lavaei, Columbia University 10

11. Decomposed SDP 0 0 0 Decomposed SDP Full-scale SDP Javad Lavaei, Columbia University 5

12. Power Networks • Decomposed relaxed SCOPF: SDP problem with small-sized constraints 0, 0, 0 • Reduction of the number of variables for a Polish system from ~9,000,000 to ~90K. • Result: Rank of W at optimality ≤ Treewidth +1 • Stronger Result: Rank of W at optimality ≤ maximum rank of bags • By-product: Lines of network in not-rank-1 bags make SDP fail. • Penalized SDP: Penalize the loss over problematic lines Javad Lavaei, Columbia University 11

13. Example 1: • Performance of penalized relaxed OPF on IEEE and Polish systems: Javad Lavaei, Columbia University 12

14. Example 2: • Performance of penalized relaxed OPF on Bukhsh’s examples: http://www.maths.ed.ac.uk/OptEnergy/LocalOpt/ Javad Lavaei, Columbia University 13

15. Example 3: • New England 39 bus system under 10 contingencies: • IEEE 300 bus system under 1 contingency corresponding to 3 outages: Javad Lavaei, Columbia University 14

16. Outline • Theory: Convex relaxation • Application 1: Optimization for power networks • Application 2: Optimal decentralized control • Implementation: High-performance solver handling 1B variables • Theory: General polynomial optimization • Application: Matrix completion Javad Lavaei, Columbia University

17. Motivation • 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 controller: • A group of isolated local controllers • Distributed controller: • Partially interacting local controllers Javad Lavaei, Columbia University 15

18. Optimal Distributed control (ODC) • Stochastic ODC: Find a structured control for the system: • to minimize the cost functional: disturbance noise • Finite-horizon ODC: Deterministic system with the objective: • Infinite-horizon ODC: Terminal time equal to infinity. Javad Lavaei, Columbia University 16

19. Motivation: Maximum Penetration of Renewable Energy 39-Bus New England System Four Communication Topologies Javad Lavaei, Columbia University 17

20. Motivation: Maximum Penetration of Renewable Energy • Assumption: Every generator has access to the rotor angle and frequency of its neighbors (if any). • Problem: Design a near-global distributed controller for adjusting the mechanical power • We solve three ODC problems for various values of alpha (gain) and sigma (noise). Javad Lavaei, Columbia University 18

21. Two Formulations of SODC SODC: Convex Reformulated SODC: Non-convex (NP-hard) Convex Non-convex but quadratic Javad Lavaei, Columbia University 19

22. Quadratic Formulation in Static Case • SDP relaxation for SODC: W • Theorem: W has rank 1, 2, or 3 at optimality for finite-horizon ODC, infinite-horizon ODC, and SODC. Lyapunov domain Time domain Javad Lavaei, Columbia University 20

23. Computationally-Cheap SDP Relaxation • Dimension of SDP variable = O(n2) • Is lower bound tight enough? • Can penalize the trace of W in the objective to make it penalized SDP. • Goal: Design a new SDP relaxation such that: • Dimension of SDP variable = O(n) • The entries of W are automatically penalized in the problem. Javad Lavaei, Columbia University 21

24. First Stage of SDP Relaxation Automatic penalty of the trace of W Lyapunov Inversion of variables Exactness: Rank n Direct and two-hop pattern Javad Lavaei, Columbia University 22

25. Two-Stage SDP Relaxation • Stage 1: Solve SDP relaxation. • Stage 2: Recover a near-global controller. • Direct Method: Read off the controller from the SDP solution. • Indirect Method: Read off G from the SDP solution and solve a convex program. • ϒ helps to make the problem feasible, but it’s penalized to keep its value low. Javad Lavaei, Columbia University 23

26. Mass-Spring System • Mass-spring system: • Goal: Design two constrained controllers for 10 masses. • Solution: (under various level of measurement noise) Javad Lavaei, Columbia University 24

27. Outline • Theory: Convex relaxation • Application 1: Optimization for power networks • Application 2: Optimal decentralized control • Implementation: High-performance solver handling 1B variables • Theory: General polynomial optimization • Application: Matrix completion Javad Lavaei, Columbia University

28. Low-Complex Algorithm for Sparse SDP Slides for this section are removed. Javad Lavaei, Columbia University 25

29. Outline • Theory: Convex relaxation • Application 1: Optimization for power networks • Application 2: Optimal decentralized control • Implementation: High-performance solver handling 1B variables • Theory: General polynomial optimization • Application: Matrix completion Javad Lavaei, Columbia University

30. Polynomial Optimization • Vertex Duplication Procedure: • Edge Elimination Procedure: • This gives rise to a sparse QCQP with a sparse graph. • The treewidth can be reduced to 1. • Theorem: Every polynomial optimization has a QCQP formulation whose SDP relaxation has a solution with rank 1 or 2. Javad Lavaei, Columbia University 28

31. Outline • Theory: Convex relaxation • Application 1: Optimization for power networks • Application 2: Optimal decentralized control • Implementation: High-performance solver handling 1B variables • Theory: General polynomial optimization • Application: Matrix completion Javad Lavaei, Columbia University

32. Matrix Completion x1 Matrix completion: Fill a partially know matrix to a low-rank matrix (PSD or not PSD) x2 x3 x4 ? x5 x6 ? x7 ? x8 0 x9 x10 x11 ? • There are rank-1 solutions. • Minimize an arbitrary nonzero sum of X entries to get a rank-1 solution. Javad Lavaei, Columbia University 29

33. Matrix Completion ? ? ? 0 x1 ? x2 x3 x4 x5 x6 ? ? x7 x8 x9 x10 • Minimize a nonzero sum of X entries if all blocks are nonsingular and square. Javad Lavaei, Columbia University 30

34. Matrix Completion ? 0 0 x1 x2 ? x3 x4 u1 x5 x6 u2 u3 x7 x8 u4 • Minimize a nonzero sum of X entries by choosing U’s to be 1. • General theory based on msr, OS, and treewidth for a general non-block case. • Form a penalized SDP or nuclear-norm minimization to find a low-rank solution. Javad Lavaei, Columbia University 31

35. Conclusions • Theory: Low-rank optimization • Applications: Power, Control, nonlinear optimization,… • Implementation: High-performance solver handling 1B variables • Two of our solvers posted online and another one is forthcoming. • Collaboration with industry for demonstration on real data. Javad Lavaei, Columbia University 32