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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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