Department of Electrical Engineering Columbia University GraphTheoretic Algorithm for Nonlinear Power Optimization Problems Outline Javad Lavaei Columbia University 2 Convex relaxation for highly sparse optimization ID: 277219
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Slide1
Javad LavaeiDepartment of Electrical EngineeringColumbia University
Graph-Theoretic Algorithm for Nonlinear Power Optimization ProblemsSlide2
OutlineJavad Lavaei, Columbia University2
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)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
ExampleJavad Lavaei, Columbia University
4
Given a polynomial optimization, we first make it quadratic and then map its structure into a generalized weighted graph:Slide5
Complex-Valued Optimization
Javad Lavaei, Columbia University5
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:Slide6
TreewidthJavad 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 decompositionsSlide7
Low-Rank SDP SolutionJavad 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.Slide8
OutlineJavad Lavaei, Columbia University8
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)Slide9
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
9Slide10
Optimal Power FlowJavad Lavaei, Columbia University10
Cost
Operation
Flow
Balance Slide11
Project 1Javad Lavaei, Columbia University11
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) Slide12
Project 2Javad Lavaei, Columbia University12
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:
PSSlide13
Project 3Javad Lavaei, Columbia University13
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.
Slide14
Project 4Javad Lavaei, Columbia University14
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.Slide15
Response of SDP to Equivalent FormulationsJavad Lavaei, Columbia University15
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.Slide16
Penalized SDP Relaxation Javad Lavaei, Columbia University16
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.)Slide17
Power NetworksJavad 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 networksSlide18
Power Networks
Javad Lavaei, Columbia University18
Upper bound on the treewidth of sample power networks:
Real/complex
optimization
Theorem:
There exists a solution with rank at most treewidth of
G
+1Slide19
ExamplesJavad 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
X
opt
?IEEE 300-bus system: rank ≤ 7Polish 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)Slide20
OutlineJavad Lavaei, Columbia University20
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)Slide21
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
T
raffic systems
W
ireless
sensor
networks
Various multi-agent
systems
Decentralized control
Distributed controlSlide22
Optimal Decentralized Control ProblemJavad Lavaei, Columbia University22
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 minimizeSlide23
Graph of ODC for Time-Domain FormulationJavad Lavaei, Columbia University23Slide24
Numerical ExampleJavad Lavaei, Columbia University24
Mass-Spring ExampleSlide25
Distributed Control in PowerJavad Lavaei, Columbia University25
Example: Distributed voltage and frequency control Generators in the same group can talk.Slide26
OutlineJavad Lavaei, Columbia University26
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)Slide27
Polynomial OptimizationJavad 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.Slide28
ConclusionsJavad Lavaei, Columbia University28
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.