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Graph-Theoretic Algorithm for Nonlinear Power
Optimization Problems
Javad Lavaei
Department of Electrical Engineering
Columbia University
Outline
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)
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Penalized Semidefinite Programming (SDP) Relaxation
Exactness of SDP relaxation:
Existence of a rank-1 solution
Implies finding a global solution
How to study the exactness of relaxation?
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Example
Given a polynomial optimization, we first make it quadratic and then map its
structure into a generalized weighted graph:
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Complex-Valued Optimization
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:
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Treewidth
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
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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.
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Outline
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)
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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)
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Optimal Power Flow
Cost
Operation
Flow
Balance
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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
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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
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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.
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Project 4
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.
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Response of SDP to Equivalent Formulations
Capacity constraint: active power, apparent
power, angle difference, voltage difference, current?
P1
P2
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
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Penalized SDP Relaxation
Use Penalized SDP relaxation to turn a low-rank solution into a rank-1 matrix:
IEEE systems with 7000 cost functions
Modified 118-bus system with 3 local
solutions (Bukhsh et al.)
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.
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Power Networks
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
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Power Networks
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
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Examples
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
How to go from low-rank to rank-1? Penalization (tested on 7000 examples)
IEEE 14-bus system
IEEE 30-bus system
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IEEE 57-bus system
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Outline
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)
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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
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Optimal Decentralized Control Problem
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
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to minimize
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Graph of ODC for Time-Domain Formulation
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Numerical Example
Mass-Spring Example
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Distributed Control in Power
Example: Distributed voltage and frequency control
Generators in the same group can talk.
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Outline
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)
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Polynomial Optimization
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.
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Conclusions
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.
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