A Majorization-Minimization Approach to Design of Power Transmition Networks

7/30/2019 A Majorization-Minimization Approach to Design of Power Transmition Networks

1/12

A Majorization-Minimization Approach to

Design of Power Transmission Networks1

Mini-Workshop on Optimization and Control

Theory for Smart Grids

Jason Johnson, Michael Chertkov (LANL, CNLS & T-4)

August 10, 2010

1

To appear at the 49th IEEE Conference on Decision and Control,http://arxiv.org/abs/1004.2285
http://find/http://goback/


2/12

Introduction

Objectivedesign the network, both the graph structure andthe line sizes, to achieve best trade-off between efficiency (low

power loss) and construction costs.

Our approach was inspired by work of Ghosh, Boyd and

Saberi, Minimizing effective resistance of a graph, 09.

Overview

resistive network model (DC approximation)

convex network optimization (line sizing) non-convex optimization for sparse network design

robust network design
http://find/


3/12

Resistive Network ModelBasic current flow problem (power flow in AC grid):

graph G injected currents bi

sources bi > 0 sinks bi < 0

junction points bi = 0 line conductances ij 0

ui

ij

Network conductance matrix:

Ki,j() =

k=i ik, i = j

ij, {i,j} G

Solve K()u = b for node potentials ui. Line currents given by

bij = ij(uj ui).
http://find/


4/12

Resistive Power Loss

For connected G and > 0,

u = K1b (K + 11T)1b.

Power loss due to resistive heating of the lines:

L() =ij

ij(ui uj)2 = uTK()u = bTK()1b

For random b, we obtain the expected power loss:

L() = bTK()1b = tr(K()1 bbT) tr(K()1B)

Importantly, this power loss is a convex function of .
http://find/


5/12

Convex Network Optimization

We may size the lines (conductances) to minimize power loss

subject to linear budget on available conductance:

minimize L()

subject to 0

T

C

is cost of conductance on line .

Alternatively, one may find the most cost-effective network:

min0{L() + T}

where 1 represents the cost of power loss (accrued overlifetime of network).
http://find/


6/12

Four Examples Optimal Network Design


7/12

Imposing Sparsity (Non-convex continuation)We now add a zero-conductance cost on lines:

min0 {L() + T

+ T

()}

where (t) is (element-wise) unit step function.

We smooth this combinatorial problem to a continuous one,replacing the discrete step by a smooth one:

(t) =

t

t +

t0 1

0

1

The convex optimization is recovered as and thecombinatorial one as 0.
http://find/


8/12

Majorization-Minimization Algorithm

Approach inspired by Candes and Boyd, Enhancing sparsity by

reweighted L1 optimization, 2009.

Uses majorization-minimization heuristic for non-convexoptimization. Iteratively linearize about previous solution:

(t+1) = arg min0

{L() + [ + ((t))]T}

results in a sequence of convex network optimization problemswith reweighted .

Monotonically decreases the objective and (almost always)converges to a local minimum.


9/12

Four Examples Sparse Network Design


10/12

Adding Robustness

To obtain solutions that are robust of failure of k lines, wereplace L() by the worst-case power dissipation afterremoving k lines:

L\k() = maxz{0,1}m|1Tz=k

L((1 z) )

This is still a convex function. It is tractable to compute onlyfor small values of k.

Method can be extended to also treat failures of generators.

Solved using essentially the same methods as before...
http://find/


11/12

Four Examples Robust Network Design


12/12

Conclusion

A promising heuristic approach to design of power transmission

networks. However, cannot guarantee global optimality.

Future Work:

Bounding optimality gap?

Use non-convex continuation approach to place generators possibly useful for graph partitioning problems

adding further constraints (e.g. dont overload lines)

better approximations to AC power flow?

Thanks!
http://find/

Documents

A Majorization-Minimization Approach to Design of Power Transmition Networks