Javad Lavaei. Department of Electrical Engineering Columbia University
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1 Graph Theoretic Algorithm for Nonlinear Power Optimization Problems Javad Lavaei Department of Electrical Engineering Columbia University Joint work with: Ramtin Madani, Ghazal Fazelnia and Abdulrahman Kalbat (Columbia University) Somayeh Sojoudi (New York University)
2 Software System Decision making towards real-time operation Slow time scale: Optimize the operating cost Centralized or distributed optimization Fast time scale: Regulate the signals Distributed control Approach: SDP relaxation, matrix completion, tree decomposition, low-rank optimization Javad Lavaei, Columbia University 2
3 Optimization 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 Cost 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. local global Production Javad Lavaei, Columbia University 3
4 Resource Allocation: Optimal Power Flow (OPF) Current I Voltage V 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 4
5 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 5
6 Penalized Semidefinite Programming (SDP) Relaxation Exactness of SDP relaxation: Existence of a rank-1 solution Implies finding a global solution Javad Lavaei, Columbia University 6
7 Optimal Power Flow Cost Operation Flow Balance Javad Lavaei, Columbia University 7
8 Old 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 8
9 Old 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 9
10 Old 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 10
11 Graph-Theoretic SDP Relaxation Issues 1: What if we get a low-rank but not rank-1 solution? Issue 2: How to deal with a computationally-expensive SDP? Approach: 1. Use a graph-theoretic approach to break down complexity 2. This also tells what lines of the network cause non-convexity 3. We first sparsify SDP and then penalize problematic lines Example borrowed from Bukhsh et al.: 1. Modify IEEE 118-bus system with 3 local solutions: , and Our method finds the best one. Javad Lavaei, Columbia University 11
12 Treewidth Tree decomposition: Treewidth of graph: The smallest width of all tree decompositions Treewidth: 1 for a tree, 2 for IEEE 14-bus and 2-26 for IEEE and Polish systems Javad Lavaei, Columbia University 12
13 Power Networks Break down the complexity through sparsification: W = positive semidefinite submatrices of W induced by bags of tree = positive semidefinite Reduction of the number of parameters for a Polish system from ~9,000,00 to ~100K. Result 1: Rank of W at optimality Treewidth +1 (valid for SC-UC-OPF) Result 2: Rank of W at optimality maximum rank of bags (valid for SC-UC-OPF) Result 3: Lines of network in high-rank bags are the source of nonzero duality gap Javad Lavaei, Columbia University 13
14 Power Networks SDP: no penalty Tier 1: uniform penalty Total reactive loss Select loss Tier 2: non-uniform penalty Javad Lavaei, Columbia University 14
15 Power Networks Several bad examples have been contrived by Buksh et al. Javad Lavaei, Columbia University 15
16 Power Networks We have written a solver in MATLAB to find a near-global solution. Computation time for Polish System with ~3100 buses: ~2.4 min in MOSEK (low precision). Javad Lavaei, Columbia University 16
17 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 17
18 Optimal Distributed control (ODC) Optimal centralized control: Easy (LQR, LQG, etc.) Optimal distributed control (ODC): NP-hard (Witsenhausen s example) Stochastic ODC: Find a structured control for the system: disturbance noise to minimize the cost functional: Rank of an expanded SDP relaxation of SODC =1, 2 or 3. How to find a computationally-cheap relaxation? Javad Lavaei, Columbia University 18
19 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 19
20 Second Stage of SDP Relaxation Direct Recovery Method: Recover the controller from the SDP solution. Indirect Recovery Method: Recover the Lyapunov matrix from the SDP solution and pass it to a second SDP problem to design a controller. Note: Trace of the to-be low-rank W is penalized in the objective by the noise covariance. Implication: The higher the noise level, the better the rank enforcement. Theorem: The relaxation is always exact in the centralized case (= Riccati equations) Javad Lavaei, Columbia University 20
21 IEEE 39 Bus (New England Power System) Maximization of penetration of renewables: Adjust the mechanical power of each generator based on the angle and frequency of neighboring generators to minimize variations. G8# 37# 30# G10# 25# 26# 28# 29# 2# 27# 38# 18# 17# 24# G9# 1# G1# 3# 16# 15# 21# 35# 22# G6# New England System 39# 4# 14# 5# 6# 12# 19# 23# 9# 8# 7# G2# 31# 11# 13# 10# 32# 20# G5# 34# G4# 33# G7# 36# G3# Javad Lavaei, Columbia University 21
22 Four Communication Topologies Javad Lavaei, Columbia University 22
23 Near-Global Controllers Various levels of controller gain Various levels of noise Javad Lavaei, Columbia University 23
24 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 24
25 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 2. Theorem: Every polynomial optimization has a QCQP formulation whose SDP relaxation has a solution with rank 1 or 2. Javad Lavaei, Columbia University 25
26 Conclusions Focus: Optimization and control Goal: Design efficient algorithms Two thrusts: Global optimization Distributed control We have developed two solvers (available on my website). Javad Lavaei, Columbia University 26
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