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
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
 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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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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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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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
OPF
Dual OPF
SDP
 SDP relaxation:
 IEEE systems
 SC Grid
 European grid
 Random systems
 Exactness of SDP relaxation and zero duality gap are equivalent for OPF.
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Response of SDP to Equivalent Formulations
 Capacity constraint: active power, apparent power, angle difference, voltage difference, current?
P1
P2
Correct solution
1.
Equivalent formulations behave
differently after relaxation.
2.
Problem D has an exact relaxation.
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Weakly-Cyclic Networks
 Theorem: SDP works for weaklycyclic 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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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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Low-Rank Solution
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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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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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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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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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