Department of Electrical Engineering Columbia University Convex Relaxation for Polynomial Optimization Application to Power Systems and Decentralized Control Outline Javad Lavaei Columbia University ID: 277220
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Javad LavaeiDepartment of Electrical EngineeringColumbia University
Convex Relaxation for Polynomial Optimization: Application to Power Systems and Decentralized ControlSlide2
OutlineJavad Lavaei, Columbia University2
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, Ramtin
Madani
, Baosen Zhang, Matt Kraning, Eric Chu)
Optimal decentralized control
(Joint work with: Ghazal Fazelnia and Ramtin Madani) General theory for polynomial optimization (Joint work with: Ramtin Madani, Somayeh Sojoudi and Ghazal Fazelnia)Slide3
Penalized Semidefinite Programming (SDP) RelaxationJavad 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? Slide4
Highly Structured OptimizationJavad Lavaei, Columbia University4
Abstract optimizations are NP-hard in the worst case. Real-world optimizations are highly structured:
Question:
How do structures affect tractability of an optimization?
Sparsity
:
Non-trivial structure:Slide5
ExampleJavad Lavaei, Columbia University
5
Given a polynomial optimization, we first make it quadratic and then map its structure into a generalized weighted graph:Slide6
ExampleJavad Lavaei, Columbia University6
Optimization:Slide7
Real-Valued OptimizationJavad Lavaei, Columbia University7
Edge
CycleSlide8
Complex-Valued Optimization
Javad Lavaei, Columbia University8 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:Slide9
Complex-Valued OptimizationJavad Lavaei, Columbia University9
Consider a real matrix
M
:
Polynomial-time solvable for weakly-cyclic bipartite graphs.Slide10
OutlineJavad Lavaei, Columbia University10
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, Ramtin
Madani
, Baosen Zhang, Matt Kraning, Eric Chu)
Optimal decentralized control
(Joint work with: Ghazal Fazelnia and Ramtin Madani) General theory for polynomial optimization (Joint work with: Ramtin Madani, Somayeh Sojoudi and Ghazal Fazelnia)Slide11
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
11Slide12
Resource Allocation: Optimal Power Flow (OPF)Javad Lavaei, Columbia University
12
OPF:
Given constant-power loads, find optimal
P
’s subject to:
Demand constraints Constraints on V’s, P’s, and Q’s.
Voltage
V
Complex power =
VI
*
=
P + Q i
Current
ISlide13
Broad Interest in Optimal Power FlowJavad Lavaei, Columbia University13
OPF-based problems solved on different time scales:Electricity marketReal-time operationSecurity assessmentTransmission 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 Slide14
Optimal Power FlowJavad Lavaei, Columbia University14
Cost Operation
Flow
Balance Slide15
Project 1Javad Lavaei, Columbia University15
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) Slide16
Project 2Javad Lavaei, Columbia University16
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:
PSSlide17
Project 3Javad Lavaei, Columbia University17
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.
Slide18
Project 4Javad Lavaei, Columbia University18
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.Slide19
Response of SDP to Equivalent FormulationsJavad Lavaei, Columbia University19
P
1
P
2
Capacity constraint:
active power, apparent power, angle difference, voltage difference, current?
Correct solution
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.Slide20
Low-Rank SolutionJavad Lavaei, Columbia University20Slide21
Penalized SDP Relaxation Javad Lavaei, Columbia University21
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.)Slide22
OutlineJavad Lavaei, Columbia University22
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, Ramtin
Madani
, Baosen Zhang, Matt Kraning, Eric Chu)
Optimal decentralized control
(Joint work with: Ghazal Fazelnia and Ramtin Madani) General theory for polynomial optimization (Joint work with: Ramtin Madani, Somayeh Sojoudi and Ghazal Fazelnia)Slide23
Distributed Control
Javad Lavaei, Columbia University23
Computational
challenges arising in the control of
real-world systems:
Communication networks
Electrical
power
systems
Aerospace systems
Large-space flexible structures
T
raffic systems
W
ireless
sensor
networks
Various multi-agent
systems
Decentralized control
Distributed controlSlide24
Optimal Decentralized Control ProblemJavad Lavaei, Columbia University24
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 minimizeSlide25
Two Quadratic Formulations in Static CaseJavad Lavaei, Columbia University
25 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:Slide26
Graph of ODC for Time-Domain FormulationJavad Lavaei, Columbia University26Slide27
Recovery of Rank-3 SolutionJavad Lavaei, Columbia University27
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. Slide28
Numerical Example (Decentralized Case)Javad Lavaei, Columbia University28
Rank: 1 or 2
Exactness in 59 trials
Rank: 1
99.8% optimality for
84 trials
SDP Relaxation
Penalized SDP
RelaxationSlide29
Distributed CaseJavad Lavaei, Columbia University29
100 random systems with standard deviation 3 for A and B, and 2 for X
0
.
Each communication link exists with probability 0.9.Slide30
OutlineJavad Lavaei, Columbia University29
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, Ramtin
Madani
, Baosen Zhang, Matt Kraning, Eric Chu)
Optimal decentralized control
(Joint work with: Ghazal Fazelnia and Ramtin Madani) General theory for polynomial optimization (Joint work with: Ramtin Madani, Somayeh Sojoudi, and Ghazal Fazelnia)Slide31
Polynomial OptimizationJavad Lavaei, Columbia University31
Consider an arbitrary polynomial optimization:
Fact 1:
Fact 2: Slide32
Polynomial OptimizationJavad Lavaei, Columbia University
32
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. Slide33
ConclusionsJavad Lavaei, Columbia University33
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.