Javad Lavaei Department of Electrical Engineering Columbia University Optimization over Graph with Application to Power Systems

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)

Javad Lavaei, Columbia University 3

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)


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


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


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


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


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


Trick:

SDP relaxation:

Guaranteed rank-1 solution!

Example 1


Opt:

Sufficient condition for exactness: Sign definite sets.

What if the condition is not satisfied?

Rank-2 W (but hidden)

NP-hard

Example 2


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


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


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


Edge

Cycle

Complex-Valued Optimization


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


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


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:


Javad Lavaei, Stanford University 17Javad Lavaei, Columbia University 21

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.


Javad Lavaei, Stanford University 17Javad Lavaei, Columbia University

PS

22

AC transmission network manipulation: High performance (lower generation cost) Easy optimization Easy market

Injection Regions


Injection region for acyclic networks:

When can we allow equality constraints? Need to study Pareto front

Generalized Network Flow (GNF)


injections

flows

Goal:

limits

Assumption: • fi(pi): convex and increasing• fij(pij): convex and decreasing

Convexification of GNF


Convexification:

Feasible set without box constraint:

It finds correct injection vector but not necessarily correct flow vector.

Convexification of GNF


Feasible set without box constraint:

Correct injections in the feasible case.

Why decreasing flow functions?

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

Documents

Javad Lavaei Department of Electrical Engineering Columbia University Optimization over Graph with Application to Power Systems