1 / 33

Javad Lavaei Department of Electrical Engineering Columbia University

Javad Lavaei Department of Electrical Engineering Columbia University. Convex Relaxation for Polynomial Optimization: Application to Power Systems and Decentralized Control. Outline. Convex relaxation for highly structured optimization (Joint work with: Somayeh Sojoudi)

roger
Télécharger la présentation

Javad Lavaei Department of Electrical Engineering Columbia University

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Javad LavaeiDepartment of Electrical EngineeringColumbia University Convex Relaxation for Polynomial Optimization: Application to Power Systems and Decentralized Control

  2. Outline • 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, RamtinMadani, Baosen Zhang, Matt Kraning, Eric Chu) • Optimal decentralized control • (Joint work with: Ghazal Fazelnia and RamtinMadani) • General theory for polynomial optimization • (Joint work with: RamtinMadani, Somayeh Sojoudi and Ghazal Fazelnia) 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. Highly Structured Optimization • Abstract optimizations are NP-hard in the worst case. • Real-world optimizations are highly structured: • Sparsity: • Non-trivial structure: • Question: How do structures affect tractability of an optimization? Javad Lavaei, Columbia University 4

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

  6. Example Optimization: Javad Lavaei, Columbia University 6

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

  8. 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 8

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

  10. Outline • 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, RamtinMadani, Baosen Zhang, Matt Kraning, Eric Chu) • Optimal decentralized control • (Joint work with: Ghazal Fazelnia and RamtinMadani) • General theory for polynomial optimization • (Joint work with: RamtinMadani, Somayeh Sojoudi and Ghazal Fazelnia) Javad Lavaei, Columbia University 10

  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) Voltage V Current I Complex power = VI*=P + Q i • OPF: Given constant-power loads, find optimal P’s subject to: • Demand constraints • Constraints on V’s, P’s, and Q’s. Javad Lavaei, Columbia University 12

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

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

  15. 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 15

  16. 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 16

  17. 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 17

  18. 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 18

  19. 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 19

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

  21. 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 21

  22. Outline • 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, RamtinMadani, Baosen Zhang, Matt Kraning, Eric Chu) • Optimal decentralized control • (Joint work with: Ghazal Fazelnia and RamtinMadani) • General theory for polynomial optimization • (Joint work with: RamtinMadani, Somayeh Sojoudi and Ghazal Fazelnia) Javad Lavaei, Columbia University 22

  23. 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 23

  24. 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 24

  25. Two Quadratic Formulations in Static Case • 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: Javad Lavaei, Columbia University 25

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

  27. Recovery of Rank-3 Solution • 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. Javad Lavaei, Columbia University 27

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

  29. Distributed Case • 100 random systems with standard deviation 3 for A and B, and 2 for X0. • Each communication link exists with probability 0.9. Javad Lavaei, Columbia University 29

  30. Outline • 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, RamtinMadani, Baosen Zhang, Matt Kraning, Eric Chu) • Optimal decentralized control • (Joint work with: Ghazal Fazelnia and RamtinMadani) • General theory for polynomial optimization • (Joint work with: RamtinMadani, Somayeh Sojoudi, and Ghazal Fazelnia) Javad Lavaei, Columbia University 29

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

  32. Polynomial Optimization • 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. Javad Lavaei, Columbia University 32

  33. Conclusions • 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. Javad Lavaei, Columbia University 33

More Related