1. Low-Rank Solution of Convex Relaxation for Optimal Power Flow Problem
Javad Lavaei Department of Electrical Engineering Columbia University Joint work with Somayeh Sojoudi and Ramtin Madani

2. Javad Lavaei, Columbia University
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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3. 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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4. Javad Lavaei, Columbia University
Summary of Results
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
Project 2: Find network topologies over which optimization is easy? (joint work with Somayeh Sojoudi, David Tse and Baosen Zhang)
Distribution networks are fine.
Every transmission network can be turned into a good one.
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5. Javad Lavaei, Columbia University
Summary of Results
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
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6. Geometric Intuition: Two-Generator Network
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7. Javad Lavaei, Columbia University
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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8. Javad Lavaei, Columbia University
Various Relaxations
OPF
SDP relaxation:
IEEE systems
SC Grid
European grid
Random systems
Dual OPF
SDP
Exactness of SDP relaxation and zero duality gap are equivalent for OPF.
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9. Response of SDP to Equivalent Formulations
Capacity constraint: active power, apparent power, angle difference, voltage difference, current? P1 P2
Correct solution
Equivalent formulations behave differently after relaxation.
Problem D has an exact relaxation.
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10. Weakly-Cyclic Networks
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.
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11. Highly Meshed Networks
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?
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12. Javad Lavaei, Columbia University
Low-Rank Solution
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13. Penalized SDP Relaxation
How to turn a low-rank solution into a rank-1 solution?
Consider a PSD matrix with some free entries.
Maximization of the sum of the off-diagonal entries results in 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.
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14. 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:
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.
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15. Penalized SDP Relaxation
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.
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16. Penalized SDP Relaxation
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17. Javad Lavaei, Columbia University
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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