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Javad Lavaei
Department of Electrical EngineeringColumbia University
Graph-Theoretic Algorithm for Nonlinear Power Optimization Problems
Outline
Javad Lavaei, Columbia University 2
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)
Penalized Semidefinite Programming (SDP) Relaxation
Javad Lavaei, Columbia University 3
Exactness of SDP relaxation:
Existence of a rank-1 solution
Implies finding a global solution
How to study the exactness of relaxation?
Example
Javad Lavaei, Columbia University 4
Given a polynomial optimization, we first make it quadratic and then map its structure into a generalized weighted graph:
Complex-Valued Optimization
Javad Lavaei, Columbia University 5
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:
Treewidth
Javad Lavaei, Columbia University 6
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
Low-Rank SDP Solution
Javad Lavaei, Columbia University 7
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.
Outline
Javad Lavaei, Columbia University 8
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)
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
Project 1
Javad Lavaei, Columbia University 11
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
Project 1: How to solve a given OPF in polynomial time? (joint work with Steven Low)
Project 2
Javad Lavaei, Columbia University 12
Transmission networks may need phase shifters:
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:
PS
Project 3
Javad Lavaei, Columbia University 13
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.
Project 4
Javad Lavaei, Columbia University 14
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.
Response of SDP to Equivalent Formulations
Javad Lavaei, Columbia University 15
P1 P2
Capacity constraint: active power, apparent power, angle difference, voltage difference, current?
Correct solution1. 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.
Penalized SDP Relaxation
Javad Lavaei, Columbia University 16
Use Penalized SDP relaxation to turn a low-rank solution into a rank-1 matrix:
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.
IEEE systems with 7000 cost functions Modified 118-bus system with 3 local solutions (Bukhsh et al.)
Power Networks
Javad Lavaei, Columbia University 17
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
Power Networks
Javad Lavaei, Columbia University 18
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
Examples
Javad Lavaei, Columbia University 19
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
IEEE 14-bus system IEEE 30-bus system IEEE 57-bus system
How to go from low-rank to rank-1? Penalization (tested on 7000 examples)
Outline
Javad Lavaei, Columbia University 20
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)
Distributed Control
Javad Lavaei, Columbia University 21
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
Optimal Decentralized Control Problem
Javad Lavaei, Columbia University 22
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
Distributed Control in Power
Javad Lavaei, Columbia University 25
Example: Distributed voltage and frequency control
Generators in the same group can talk.
Outline
Javad Lavaei, Columbia University 26
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)
Polynomial Optimization
Javad Lavaei, Columbia University 27
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.
Conclusions
Javad Lavaei, Columbia University 28
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.