Javad Lavaei

Department of Electrical EngineeringColumbia University

Graph-Theoretic Algorithm for Nonlinear Power Optimization Problems

Outline

Javad Lavaei, Columbia University 2

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)

Penalized Semidefinite Programming (SDP) Relaxation

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Exactness of SDP relaxation:

Existence of a rank-1 solution

Implies finding a global solution

How to study the exactness of relaxation?

Example

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Given a polynomial optimization, we first make it quadratic and then map its structure into a generalized weighted graph:

Complex-Valued Optimization

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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:

Treewidth

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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

Low-Rank SDP Solution

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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.

Outline

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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)

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

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Cost

Operation

Flow

Balance

Project 1

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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)

Project 2

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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:

PS

Project 3

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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.

Project 4

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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.

Response of SDP to Equivalent Formulations

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P1 P2

Capacity constraint: active power, apparent power, angle difference, voltage difference, current?

Correct solution1. 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.

Penalized SDP Relaxation

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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.)

Power Networks

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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

Power Networks

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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

Examples

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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

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)

Outline

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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)

Distributed Control

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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

Optimal Decentralized Control Problem

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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 minimize

Graph of ODC for Time-Domain Formulation

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Numerical Example

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Mass-Spring Example

Distributed Control in Power

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Example: Distributed voltage and frequency control

Generators in the same group can talk.

Outline

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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)

Polynomial Optimization

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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.

Conclusions

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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.