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Javad Lavaei Department of Electrical Engineering Columbia University

Javad Lavaei Department of Electrical Engineering Columbia University. Graph-Theoretic Algorithm for Nonlinear Power Optimization Problems. Outline. Convex relaxation for highly sparse optimization (Joint work with: Somayeh Sojoudi , Ramtin Madani , and Ghazal Fazelnia )

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Javad Lavaei Department of Electrical Engineering Columbia University

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  1. Javad LavaeiDepartment of Electrical EngineeringColumbia University Graph-Theoretic Algorithm for Nonlinear Power Optimization Problems

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

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

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

  5. Complex-Valued Optimization • 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 R2: Javad Lavaei, Columbia University 5

  6. Treewidth • 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 Gappears 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 Javad Lavaei, Columbia University 6

  7. Low-Rank SDP Solution 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. Javad Lavaei, Columbia University 7

  8. Outline • Convex relaxation for highly sparse optimization • (Joint work with: SomayehSojoudi, RamtinMadani, and Ghazal Fazelnia) • Optimization over power networks • (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, RamtinMadani, Baosen Zhang, Matt Kraning, Eric Chu, and MortezaAshraphijuo) • Optimal decentralized control • (Joint work with: Ghazal Fazelnia ,RamtinMadani, and AbdulrahmanKalbat) • General theory for polynomial optimization • (Joint work with: RamtinMadani, Somayeh Sojoudi and Ghazal Fazelnia) Javad Lavaei, Columbia University 8

  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 Cost Operation Flow Balance Javad Lavaei, Columbia University 10

  11. Project 1 Project 1:How to solve a given OPF in polynomial time? (joint work with Steven Low) • A sufficient condition to globally solve OPF: • Numerous randomly generated systems • IEEE systems with 14, 30, 57, 118, 300 buses • European grid • Various theories: Itholds widely in practice Javad Lavaei, Columbia University 11

  12. Project 2 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: • Transmission networks may need phase shifters: PS Javad Lavaei, Columbia University 12

  13. Project 3 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. Javad Lavaei, Columbia University 13

  14. Project 4 Project 4:How to do optimization for mesh networks? (joint work with RamtinMadani 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. Javad Lavaei, Columbia University 14

  15. Response of SDP to Equivalent Formulations • Capacity constraint: active power, apparent power, angle difference, voltage difference, current? P2 P1 Equivalent formulations behave differently after relaxation. SDP works for weakly-cyclic networks with cycles of size 3 if voltage difference is used to restrict flows. Correct solution Javad Lavaei, Columbia University 15

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

  17. Power Networks • 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 Javad Lavaei, Columbia University 17

  18. Power Networks • 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 Javad Lavaei, Columbia University 18

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

  20. Outline • Convex relaxation for highly sparse optimization • (Joint work with: SomayehSojoudi, RamtinMadani, and Ghazal Fazelnia) • Optimization over power networks • (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, RamtinMadani, Baosen Zhang, Matt Kraning, Eric Chu, and MortezaAshraphijuo) • Optimal decentralized control • (Joint work with: Ghazal Fazelnia ,RamtinMadani, and AbdulrahmanKalbat) • General theory for polynomial optimization • (Joint work with: RamtinMadani, Somayeh Sojoudi and Ghazal Fazelnia) Javad Lavaei, Columbia University 20

  21. Distributed Control • 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 Distributed control Decentralized control Javad Lavaei, Columbia University 21

  22. Optimal Decentralized Control Problem • 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 Javad Lavaei, Columbia University 22

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

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

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

  26. Outline • Convex relaxation for highly sparse optimization • (Joint work with: SomayehSojoudi, RamtinMadani, and Ghazal Fazelnia) • Optimization over power networks • (Joint work with: Steven Low, David Tse, Stephen Boyd, Somayeh Sojoudi, RamtinMadani, Baosen Zhang, Matt Kraning, Eric Chu, and MortezaAshraphijuo) • Optimal decentralized control • (Joint work with: Ghazal Fazelnia ,RamtinMadani, and AbdulrahmanKalbat) • General theory for polynomial optimization • (Joint work with: RamtinMadani, Somayeh Sojoudi, and Ghazal Fazelnia) Javad Lavaei, Columbia University 26

  27. Polynomial Optimization • 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. Javad Lavaei, Columbia University 27

  28. Conclusions • 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. Javad Lavaei, Columbia University 28

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