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A Majorization-Minimization Approach to
Design of Power Transmission Networks
Jason K. Johnson and Michael Chertkov
Abstract—We propose an optimization approach to design
cost-effective electrical power transmission networks. That is,
we aim to select both the network structure and the line
conductances (line sizes) so as to optimize the trade-off between
network efficiency (low power dissipation within the transmis-
sion network) and the cost to build the network. We begin with
a convex optimization method based on the paper “Minimizing
Effective Resistance of a Graph” [Ghosh, Boyd & Saberi]. We
show that this (DC) resistive network method can be adapted to
the context of AC power flow. However, that does not address
the combinatorial aspect of selecting network structure. We
approach this problem as selecting a subgraph within an over-
complete network, posed as minimizing the (convex) network
power dissipation plus a non-convex cost on line conductances
that encourages sparse networks where many line conductances
are set to zero. We develop a heuristic approach to solve
this non-convex optimization problem using: (1) a continuation
method to interpolate from the smooth, convex problem to
the (non-smooth, non-convex) combinatorial problem, (2) the
majorization-minimization algorithm to perform the necessary
intermediate smooth but non-convex optimization steps. Ulti-
mately, this involves solving a sequence of convex optimization
problems in which we iteratively reweight a linear cost on line
conductances to fit the actual non-convex cost. Several examples
are presented which suggest that the overall method is a good
heuristic for network design. We also consider how to obtain
sparse networks that are still robust against failures of lines
and/or generators.
I. INTRODUCTION
The power grid of today was not systematically planned
but grew in a piecemeal fashion. In spite of this it is
largely reliable, arguably among the greatest engineering
achievements of the 20th century. However, this status quo
is now challenged with increased demand and stress on the
aging network leading to extremely costly and growing-in-
scale blackouts and operational problems. A shift towards
renewable sources of energy will further stress the grid as
these resources are intermittent and thus not reliable in the
traditional sense. These changes emphasize the importance
of incorporating new and extending existing infrastructure
in a systematic way. In this paper we present a proof of
principles study suggesting an efficient algorithmic approach
for optimal or close to optimal power grid design.
A. Motivation
A key challenge in updating and extending the power grid
is determining where to place new transmission, generation
J. Johnson and M. Chertkov are both with the Center for Nonlinear
Studies and Theoretical Division T-4 of Los Alamos National Laboratory,
Los Alamos, NM 87544. M. Chertkov is also affiliated with the New
Mexico Consortium, Los Alamos, NM 87544. jasonj@lanl.gov,
chertkov@lanl.gov
and storage facilities or in some cases how to design a
new grid from scratch. Specifically, the present theoretical
study was motivated by the national challenge of integrating
renewables into operation of the existing US grid. Renew-
able generation, such as wind and solar, are intermittent.
Moreover, regions where wind is plentiful often lack ade-
quate transmission lines. Effective and reliable exploitation
of renewables requires planning. The National Renewable
Energy Laboratory’s (NREL) WinDS project [13], [14] is
an excellent first step, however, it does not account for
power flow stability or grid resiliency. A study of the WinDS
solution performed at LANL [15] has discovered that it
results in an often infeasible electric grid suggesting there
is a problem in generating globally optimal solutions that
accommodate intermittent renewable generation.
Our paper develops an approach towards the challenging
problem of planning cost-effective and robust extensions of
the power grid to accommodate growing demand and long-
term addition of renewables. Our approach may also provide
a starting point for practical planning approaches such as the
one proposed in [15].
B. Related Work
The initial inspiration for our approach was the convex
network optimization methods of Ghosh, Boyd and Saberi
[10]. Building on earlier work [5], they consider the problem
of minimizing the total resistance of an electrical network
subject to a linear budget on line conductances, where they
interpret the total resistance metric as the expected power dis-
sipation within the network under a random current model.
We extend their work by also selecting the network structure.
We impose sparsity on that structure in a manner similar
to a number of methods that modify a convex optimization
problem by adding some non-convex regularization to obtain
sparser solutions, such as in compressed sensing [6]–[8]
or edge-preserving image restoration [12]. The method of
Candes et al [6] is especially relevant to our approach.
They recommend the majorization-minimization algorithm
[11] as a heuristic approach to sparsity-favoring non-convex
optimization.
Another important element of our approach is that we
follow a similar strategy as in the graduated non-convexity
algorithm [1] in that we solve a sequence of optimization
problems that interpolates from a convex relaxation of the
actual non-convex problem. A somewhat similar approach
has been used to obtain sparse transport networks [2].
arXiv:1004.2285v1 [math.OC] 13 Apr 2010
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VI. SUMMARY AND FUTURE WORK
In summary, we have developed an optimization approach
to design electric power transmission networks with the aim
of balancing network efficiency versus the cost of building
the network. At the core of our methods lies a convex net-
work optimization problem generalizing methods of [5], [10].
We also have proposed non-convex extensions of this basic
line-sizing problem to further encourage network sparsity.
This allows the heuristic design of the network structure
G by seeking a sparse solution within an over-complete
graph. We developed a heuristic solution technique using
smoothed annealing and majorization-minimization methods
[1], [6], [11]. So far, the experimental results obtained by
these methods in toy problems have yielded very reasonable
networks that appear to be optimal or near-optimal solutions
of the proposed optimization problem.
There are many possible extensions of the basic optimiza-
tion model we have developed. We begin by listing some
straight-forward extensions that may give a better fit to real-
world applications:
• Modeling renewable power generation. This may be
handled as we have treated consumer nodes, but with
positive¯bi.
• Incorporating constraints on the maximum output of
generators. This is especially relevant in the multiple-
generator setting.
• Modeling power storage capabilities (e.g. hybrid and
electric vehicles). These are nodes of the network that
may absorb power from the network when there is a
surplus and then re-emit this power when demand is
high.
• Allowing for load shedding. For a number of reasons,
it may become necessary that not all of the demand
for power can be met so that load shedding becomes
necessary. It would be good to treat this somehow both
in our random current model and in how we model the
handling of line and/or generator failures.
• Putting (convex) constraints on the power dissipation
and/or current per line. These is important to avoid
over-loading lines in the first place (to avoid cascading
failures).
A less trivial direction to explore is that of directly treating
the AC power flow problem (rather then using the leading
order DC approximation). However, so far it is unclear if
this can be usefully treated within a convex optimization
framework.
Another direction to explore concerns developing more
efficient algorithms. The methods we are using so far all
involve convex optimization procedures with per-iteration
complexity that grows essentially as O(n3) (fixing the degree
of G). While this is sufficient for moderately sized problems,
it is probably not practical for large-scale networks with
millions of nodes. Part of the problem here is that our current
formulation does not make maximal use of the sparsity of
the graph G. We anticipate that more scalable algorithms
(e.g., O(n3/2) for planar graphs) may be possible using
formulations that introduce auxiliary variables so as to allow
Newton’s method to use sparse linear solvers.
Otherwise, it would be interesting to better motivate the
heuristics developed for non-convex optimization from a
theoretical perspective. Specifically, it would be useful to
identify some sufficient conditions for optimality of those
heuristics (in special cases) or to bound the sub-optimality
of the approximate solutions. So far, we have not found a
good approach to these questions.
ACKNOWLEDGMENTS
We are thankful to all the participants of the “Optimization
and Control for Smart Grids” LDRD DR project at Los
Alamos and Smart Grid Seminar Series at CNLS/LANL for
multiple fruitful discussions, and especially to S. Backhaus
for critical reading of the manuscript. Research at LANL
was carried out under the auspices of the National Nuclear
Security Administration of the U.S. Department of Energy
at Los Alamos National Laboratory under Contract No.
DE C52-06NA25396. M. Chertkov acknowledges partial
support of NMC via NSF collaborative grant CCF-0829945
on “Harnessing Statistical Physics for Computing and Com-
munications”.
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