Javad Lavaei

Department of Electrical EngineeringColumbia University

Joint work with Somayeh Sojoudi and Ramtin Madani

Low-Rank Solution of Convex Relaxation for Optimal Power Flow Problem

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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Broad Interest in Optimal Power Flow

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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

Summary of Results

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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)

Distribution networks are fine. Every transmission network can be turned into a good one.

Project 2: Find network topologies over which optimization is easy? (joint work with Somayeh Sojoudi, David Tse and Baosen Zhang)

Summary of Results

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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 It solves 10,000-bus OPF in 0.85 seconds on a single core machine.

Project 4: How to do optimization for mesh networks? (joint work with Ramtin Madani and Somayeh Sojoudi)

Developed a penalization technique Verified its performance on IEEE systems with 7000 cost functions

Geometric Intuition: Two-Generator Network

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Optimal Power Flow

Cost

Operation

Flow

Balance

Extensions: Other objective (voltage support, reactive power, deviation) More variables, e.g. capacitor banks, transformers Preventive or corrective contingency constraints

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Various Relaxations

Dual OPF SDP

OPF

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SDP relaxation: IEEE systems SC Grid European grid Random systems

Exactness of SDP relaxation and zero duality gap are equivalent for OPF.

Response of SDP to Equivalent Formulations

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P1 P2

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

Correct solution 1. Equivalent formulations behave differently after relaxation.

2. Problem D has an exact relaxation.

Weakly-Cyclic Networks

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Theorem: SDP works for weakly-cyclic networks with cycles of size 3 if voltage difference is used to restrict flows.

Observation: A lossless 3-bus system has a non-convex flow region but a convex injection region.

Highly Meshed Networks

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Theorem: The injection region is non-convex for a single cycle of size 5 or more.

How to deal with highly meshed networks or systems with large cycles?

If we can’t find a rank-1 solution, it’s still plausible to obtain a low-rank solution:

Approximate a low-rank solution by a rank-1 matrix thru eig decomposition.

Fine-tune a low-rank solution using a local search algorithm.

Is there a low-rank solution for real-world systems?

Low-Rank Solution

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Penalized SDP Relaxation

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Consider a PSD matrix with some free entries.

Maximization of the sum of the off-diagonal entries results in a rank-1 solution.

How to turn a low-rank solution into a rank-1 solution?

Lossless networks: Active power is in terms of Im{W}. Reactive power is in terms of Re{W}. Hence, penalization of reactive power is helpful.

Penalized SDP Relaxation

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Penalized SDP relaxation:

Extensive simulations show that reactive power needs to be corrected.

Intuition: Among many pairs (PG,QG)’s with the same first component, we want to find one with the best second component.

Penalized SDP relaxation aims to find a near-optimal solution.

It worked for IEEE systems with over 7000 different cost functions.

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.

Penalized SDP Relaxation

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Let λ1 and λ2 denote the two largest eigenvalues of W.

Correction of active powers is negligible but reactive powers change noticeably.

There is a wide range of values for ε giving rise to a nearly-global local solution.

Penalized SDP Relaxation

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Conclusions

Focus: OPF with a 50-year history

Goal: Find a near-global solution efficiently

Equivalent formulations may lead to different relaxations (best formulation = use voltage difference for line capacity).

Existence of low-rank solutions for power networks.

Recovery of a rank-1 solution thru a penalization (by correcting reactive power).

Simulations performed on 7000 problems.

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