A Majorization-Minimization Approach to Design of Power Transmition Networks
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7/30/2019 A Majorization-Minimization Approach to Design of Power Transmition Networks
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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
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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
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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).
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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 .
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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).
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Four Examples Optimal Network Design
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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.
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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.
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Four Examples Sparse Network Design
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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...
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Four Examples Robust Network Design
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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!
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