Optimization over Graph with Application to
Power Systems
Javad Lavaei
Department of Electrical Engineering
Columbia University
Acknowledgements
Caltech:
Steven Low
Somayeh Sojoudi
Stanford University:
Stephen Boyd
Eric Chu
Matt Kranning
UC Berkeley:
David Tse
Baosen Zhang
 J. Lavaei and S. Low, "Zero Duality Gap in Optimal Power Flow Problem," IEEE Transactions on Power
Systems, 2012.
 J. Lavaei, D. Tse and B. Zhang, "Geometry of Power Flows in Tree Networks,“ in IEEE Power & Energy
Society General Meeting, 2012.
 S. Sojoudi and J. Lavaei, "Physics of Power Networks Makes Hard Optimization Problems Easy To Solve,“
in IEEE Power & Energy Society General Meeting, 2012.
 M. Kraning, E. Chu, J. Lavaei and S. Boyd, "Message Passing for Dynamic Network Energy
Management," Submitted for publication, 2012.
 S. Sojoudi and J. Lavaei, "Semidefinite Relaxation for Nonlinear Optimization over Graphs with
Application to Optimal Power Flow Problem," Working draft, 2012.
 S. Sojoudi and J. Lavaei, "Convexification of Generalized Network Flow Problem with Application to
Optimal Power Flow," Working draft, 2012.
Power Networks (CDC 10, Allerton 10, ACC 11, TPS 12, ACC 12, PGM 12)
 Optimizations:
 Resource allocation
 State estimation
 Scheduling
 Issue: Nonlinearities
 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
 Interested companies: ISOs, TSOs, RTOs, Utilities, FERC
 OPF solved on different time scales:
 Electricity market
 Real-time operation
 Security assessment
 Transmission planning
 Existing methods based on local search algorithms
 Can save $$$ if solved efficiently
 Huge literature since 1962 by power, OR and Econ people
Recent conference by Federal Energy Regulatory Commission about OPF
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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 parallel 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 relate the polynomial-time solvability of an optimization to its
structural properties? (joint work with Somayeh Sojoudi)
Project 5: How to solve generalized network flow (CS problem)? (joint work with Somayeh
Sojoudi)
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Problem of Interest
 Abstract optimizations are NP-hard in the worst case.
 Real-world optimizations are highly structured:
 Sparsity:
 Non-trivial structure:
 Question: How does the physical structure affect tractability of an optimization?
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Example 1
Trick:
SDP relaxation:
 Guaranteed rank-1 solution!
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Example 1
Opt:
 Sufficient condition for exactness: Sign definite sets.
 What if the condition is not satisfied?
 Rank-2 W (but hidden)
 NP-hard
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Example 2
Opt:
Acyclic Graph
Real-valued case: Rank-2 W (need regularization)
 Complex-valued case:
 Real coefficients: Exact SDP
 Imaginary coefficients: Exact SDP
 General case: Need sign definite sets
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Sign Definite Set
 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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Formal Definition: Optimization over Graph
Optimization of interest:
(real or complex)
Define:
 SDP relaxation for y and z (replace xx* with W) .
 f (y , z) is increasing in z (no convexity assumption).
 Generalized weighted graph: weight set
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for edge (i,j).
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Real-Valued Optimization
Edge
Cycle
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Complex-Valued Optimization
 Main requirement in complex case: Sign definite weight sets
Generalization of Bose et al.
work from Caltech
 SDP relaxation for acyclic graphs:
 real coefficients
 1-2 element sets (power grid: ~10 elements)
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Complex-Valued Optimization
 Purely imaginary weights (lossless power grid):
Generalization of Zhang &
Tse’s work from UC Berkeley
 Consider a real matrix M:
 Polynomial-time solvable for weakly-cyclic bipartite graphs.
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Optimal Power Flow
Cost
Operation
Flow
Balance
 Express the last constraint as an inequality.
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Exact Convex Relaxation
 OPF: DC or AC
 Networks: Distribution or transmission
 Result 1: Exact relaxation for DC/AC distribution and DC transmission.
 Energy-related optimization:
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Exact Convex Relaxation
 Each weight set has about 10 elements.
 Due to passivity, they are all in the left-half plane.
 Coefficients: Modes of a stable system.
 Weight sets are sign definite.
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Exact Convex Relaxation
 AC transmission network manipulation:
 High performance (lower generation cost)
 Easy optimization
 Easy market
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Injection Regions
 Injection region for acyclic networks:
 When can we allow equality constraints? Need to study Pareto front
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Generalized Network Flow (GNF)
injections
flows
limits
 Goal:
Assumption:
• fi(pi): convex and increasing
• fij(pij): convex and decreasing
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Convexification of GNF
Feasible set without box constraint:
 Convexification:
 It finds correct injection vector but not necessarily correct flow vector.
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Convexification of GNF
Feasible set without box constraint:
 Correct injections in the feasible case.
 Why decreasing flow functions?
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Conclusions

Motivation: OPF with a 50-year history

Goal: Develop theory of optimization over graph

Mapped the structure of an optimization into a generalized weighted graph

Obtained various classes of polynomial-time solvable optimizations

Talked about Generalized Network Flow

Passivity in power systems made optimizations easier
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