Report

Graph-Theoretic Algorithm for Nonlinear Power Optimization Problems Javad Lavaei Department of Electrical Engineering Columbia University Outline 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) Javad Lavaei, Columbia University 2 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 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 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 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 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 Javad Lavaei, Columbia University 6 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 Outline 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) Javad Lavaei, Columbia University 8 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 Optimal Power Flow Cost Operation Flow Balance Javad Lavaei, Columbia University 10 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: It holds widely in practice Javad Lavaei, Columbia University 11 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 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 Project 4 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. Javad Lavaei, Columbia University 14 Response of SDP to Equivalent Formulations Capacity constraint: active power, apparent power, angle difference, voltage difference, current? P1 P2 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. Correct solution Javad Lavaei, Columbia University 15 Penalized SDP Relaxation Use Penalized SDP relaxation to turn a low-rank solution into a rank-1 matrix: IEEE systems with 7000 cost functions Modified 118-bus system with 3 local solutions (Bukhsh et al.) 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 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 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 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? 1. IEEE 300-bus system: rank ≤ 7 2. Polish 3120-bus system: Rank ≤ 27 How to go from low-rank to rank-1? Penalization (tested on 7000 examples) IEEE 14-bus system IEEE 30-bus system Javad Lavaei, Columbia University IEEE 57-bus system 19 Outline 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) Javad Lavaei, Columbia University 20 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 Decentralized control Distributed control Javad Lavaei, Columbia University 21 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 Javad Lavaei, Columbia University to minimize 22 Graph of ODC for Time-Domain Formulation Javad Lavaei, Columbia University 23 Numerical Example Mass-Spring Example Javad Lavaei, Columbia University 24 Distributed Control in Power Example: Distributed voltage and frequency control Generators in the same group can talk. Javad Lavaei, Columbia University 25 Outline 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) Javad Lavaei, Columbia University 26 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 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