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Javad Lavaei
Department of Electrical EngineeringColumbia University
Optimization over Graph with Application to
Power Systems
Acknowledgements
Caltech:Steven Low
Somayeh Sojoudi
Stanford University:Stephen Boyd
Eric ChuMatt 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)
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Optimizations: Resource allocation State estimation Scheduling
Issue: Nonlinearities
Transition from traditional grid to smart grid: More variables (10X) Time constraints (100X)
Resource Allocation: Optimal Power Flow (OPF)
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OPF: Given constant-power loads, find optimal P’s subject to: Demand constraints Constraints on V’s, P’s, and Q’s.
Voltage V
Complex power = VI*=P + Q i
Current I
Broad Interest in Optimal Power Flow
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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
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 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)
Problem of Interest
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Abstract optimizations are NP-hard in the worst case.
Real-world optimizations are highly structured:
Question: How does the physical structure affect tractability of an optimization?
Sparsity: Non-trivial structure:
Example 1
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Trick:
SDP relaxation:
Guaranteed rank-1 solution!
Example 1
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Opt:
Sufficient condition for exactness: Sign definite sets.
What if the condition is not satisfied?
Rank-2 W (but hidden)
NP-hard
Example 2
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Opt:
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
Acyclic Graph
Sign Definite Set
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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:
Formal Definition: Optimization over Graph
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Optimization of interest:
(real or complex)
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 for edge (i,j).
Define:
Real-Valued Optimization
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Edge
Cycle
Complex-Valued Optimization
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SDP relaxation for acyclic graphs:
real coefficients
1-2 element sets (power grid: ~10 elements)
Main requirement in complex case: Sign definite weight sets
Generalization of Bose et al.
work from Caltech
Complex-Valued Optimization
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Purely imaginary weights (lossless power grid):
Consider a real matrix M:
Polynomial-time solvable for weakly-cyclic bipartite graphs.
Generalization of Zhang &
Tse’s work from UC Berkeley
Optimal Power Flow
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Cost
Operation
Flow
Balance
Express the last constraint as an inequality.
Exact Convex Relaxation
Result 1: Exact relaxation for DC/AC distribution and DC transmission.
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OPF: DC or AC
Networks: Distribution or transmission
Energy-related optimization:
Exact Convex Relaxation
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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.
Exact Convex Relaxation
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PS
22
AC transmission network manipulation: High performance (lower generation cost) Easy optimization Easy market
Injection Regions
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Injection region for acyclic networks:
When can we allow equality constraints? Need to study Pareto front
Generalized Network Flow (GNF)
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injections
flows
Goal:
limits
Assumption: • fi(pi): convex and increasing• fij(pij): convex and decreasing
Convexification of GNF
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Convexification:
Feasible set without box constraint:
It finds correct injection vector but not necessarily correct flow vector.
Convexification of GNF
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Feasible set without box constraint:
Correct injections in the feasible case.
Why decreasing flow functions?
Conclusions
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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