Convex Relaxation for Polynomial Optimization:
Application to Power Systems and
Decentralized Control
Javad Lavaei
Department of Electrical Engineering
Columbia University
Outline
 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)
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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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Highly Structured Optimization
 Abstract optimizations are NP-hard in the worst case.
 Real-world optimizations are highly structured:
 Sparsity:
 Non-trivial structure:
 Question: How do structures affect tractability of an optimization?
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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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Example
Optimization:
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Real-Valued Optimization
Edge
Cycle
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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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Complex-Valued Optimization
 Consider a real matrix M:
 Polynomial-time solvable for
weakly-cyclic bipartite graphs.
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Outline
 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)
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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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Resource Allocation: Optimal Power Flow (OPF)
Voltage V
Current I
Complex power = VI*=P + Q i
OPF: Given constant-power loads, find optimal P’s subject to:
 Demand constraints
 Constraints on V’s, P’s, and Q’s.
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Broad Interest in Optimal Power Flow
 OPF-based problems solved on different time scales:
 Electricity market
 Real-time operation
 Security assessment
 Transmission 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
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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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Low-Rank 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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Outline
 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)
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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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Two Quadratic Formulations in Static Case
 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:
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Graph of ODC for Time-Domain Formulation
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Recovery of Rank-3 Solution
 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.
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Numerical Example (Decentralized Case)
SDP Relaxation
Penalized SDP Relaxation
 Rank: 1 or 2
 Rank: 1
 Exactness in 59 trials
 99.8% optimality for 84 trials
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Distributed Case
 100 random systems with
standard deviation 3 for A and B, and 2 for X0.
 Each communication link exists with probability 0.9.
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Outline
 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)
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Polynomial Optimization
 Consider an arbitrary polynomial optimization:
 Fact 1:
 Fact 2:
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Polynomial Optimization
 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.
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Conclusions
 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.
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