GraphTheoretic Algorithm for Nonlinear Power Optimization Problems Javad

Graph-Theoretic Algorithm for Nonlinear Power Optimization Problems Javad Lavaei Department of Electrical Engineering Columbia University

Outline v Convex relaxation for highly sparse optimization (Joint work with: Somayeh Sojoudi, Ramtin Madani, and Ghazal Fazelnia) v 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) v Optimal decentralized control (Joint work with: Ghazal Fazelnia , Ramtin Madani, and Abdulrahman Kalbat) v General theory for polynomial optimization (Joint work with: Ramtin Madani and Somayeh Sojoudi) Javad Lavaei, Columbia University 2

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

Example v 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 v Real-valued case: “T “ is sign definite if its elements are all negative or all positive. v Complex-valued case: “T “ is sign definite if T and –T are separable in R 2: Javad Lavaei, Columbia University 5

Treewidth q Tree decomposition: q We map a given graph G into a tree T such that: v Each node of T is a collection of vertices of G v Each edge of G appears in one node of T v If a vertex shows up in multiple nodes of T, those nodes should form a subtree q Width of a tree decomposition: The cardinality of largest node minus one q Treewidth of graph: The smallest width of all tree decompositions Javad Lavaei, Columbia University 6

Low-Rank SDP Solution Real/complex optimization q Define G as the sparsity graph q Theorem: There exists a solution with rank at most treewidth of G +1 q We propose infinitely many optimizations to find that solution. q This provides a deterministic upper bound for low-rank matrix completion problem. Javad Lavaei, Columbia University 7

Outline v Convex relaxation for highly sparse optimization (Joint work with: Somayeh Sojoudi, Ramtin Madani, and Ghazal Fazelnia) v 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) v Optimal decentralized control (Joint work with: Ghazal Fazelnia , Ramtin Madani, and Abdulrahman Kalbat) v General theory for polynomial optimization (Joint work with: Ramtin Madani, Somayeh Sojoudi and Ghazal Fazelnia) Javad Lavaei, Columbia University 8

Power Networks q Optimizations: § Optimal power flow (OPF) § Security-constrained OPF § State estimation § Network reconfiguration § Unit commitment § Dynamic energy management q Issue of non-convexity: § Discrete parameters § Nonlinearity in continuous variables q Transition from traditional grid to smart grid: § More variables (10 X) § Time constraints (100 X) Javad Lavaei, Columbia University 9

Optimal Power Flow Cost Operation Flow Balance Javad Lavaei, Columbia University 10

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

Project 2: Find network topologies over which optimization is easy? (joint work with Somayeh Sojoudi, David Tse and Baosen Zhang) q Distribution networks are fine due to a sign definite property: q Transmission networks may need phase shifters: PS Javad Lavaei, Columbia University 12

Project 3: How to design a distributed algorithm for solving OPF? (joint work with Stephen Boyd, Eric Chu and Matt Kranning) q A practical (infinitely) parallelizable algorithm using ADMM. q It solves 10, 000 -bus OPF in 0. 85 seconds on a single core machine. Javad Lavaei, Columbia University 13

Project 4: How to do optimization for mesh networks? (joint work with Ramtin Madani and Somayeh Sojoudi) q Observed that equivalent formulations might be different after relaxation. q Upper bounded the rank based on the network topology. q Developed a penalization technique. q Verified its performance on IEEE systems with 7000 cost functions. Javad Lavaei, Columbia University 14

Response of SDP to Equivalent Formulations q Capacity constraint: active power, apparent power, angle difference, voltage difference, current? P 1 P 2 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 q Use Penalized SDP relaxation to turn a low-rank solution into a rank-1 matrix: q IEEE systems with 7000 cost functions q Modified 118 -bus system with 3 local solutions (Bukhsh et al. ) q 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 q Treewidth of a tree: 1 q How about the treewidth of IEEE 14 -bus system with multiple cycles? 2 q How to compute the treewidth of a large graph? v NP-hard problem v We used graph reduction techniques for sparse power networks Javad Lavaei, Columbia University 17

Power Networks q Upper bound on the treewidth of sample power networks: Real/complex optimization q Theorem: There exists a solution with rank at most treewidth of G +1 Javad Lavaei, Columbia University 18

Examples q Example: Consider the security-constrained unit-commitment OPF problem. q Use SDP relaxation for this mixed-integer nonlinear program. q What is the rank of Xopt? 1. IEEE 300 -bus system: rank ≤ 7 2. Polish 3120 -bus system: Rank ≤ 27 q 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 v Convex relaxation for highly sparse optimization (Joint work with: Somayeh Sojoudi, Ramtin Madani, and Ghazal Fazelnia) v 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) v Optimal decentralized control (Joint work with: Ghazal Fazelnia , Ramtin Madani, and Abdulrahman Kalbat) v General theory for polynomial optimization (Joint work with: Ramtin Madani, Somayeh Sojoudi and Ghazal Fazelnia) Javad Lavaei, Columbia University 20

Distributed Control q Computational challenges arising in the control of real-world systems: v Communication networks v Electrical power systems v Aerospace systems v Large-space flexible structures v Traffic systems v Wireless sensor networks v Various multi-agent systems Decentralized control Distributed control Javad Lavaei, Columbia University 21

Optimal Decentralized Control Problem q Optimal centralized control: Easy (LQR, LQG, etc. ) q Optimal distributed control (ODC): NP-hard (Witsenhausen’s example) q Consider the time-varying system: q 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 q Example: Distributed voltage and frequency control q Generators in the same group can talk. Javad Lavaei, Columbia University 25

Outline v Convex relaxation for highly sparse optimization (Joint work with: Somayeh Sojoudi, Ramtin Madani, and Ghazal Fazelnia) v 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) v Optimal decentralized control (Joint work with: Ghazal Fazelnia , Ramtin Madani, and Abdulrahman Kalbat) v General theory for polynomial optimization (Joint work with: Ramtin Madani, Somayeh Sojoudi, and Ghazal Fazelnia) Javad Lavaei, Columbia University 26

Polynomial Optimization q Sparsification Technique: distributed computation q This gives rise to a sparse QCQP with a sparse graph. q 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 v Convex relaxation for highly sparse optimization: q Complexity may be related to certain properties of a graph. v Optimization over power networks: q Optimization over power networks becomes mostly easy due to their structures. v Optimal decentralized control: q ODC is a highly sparse nonlinear optimization so its relaxation has a rank 1 -3 solution. v General theory for polynomial optimization: q Every polynomial optimization has an SDP relaxation with a rank 1 -3 solution. Javad Lavaei, Columbia University 28