TRANSMISSION EXPANSION PLANNING WITH RE-DESIGN
A Greedy Randomized Adaptive Search Procedure
Rosa Figueiredo
1
, Pedro Henrique Gonz´alez Silva
2
and Michael Poss
3
1
CIDMA, Departamento de Matem´atica, Universidade de Aveiro, Aveiro, Portugal
2
Instituto de Matem´atica e Estat´ıstica, Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brasil
3
GOM, D´epartement d’Informatique, Universit´e Libre de Bruxelles, Bruxelles, Belgique
Keywords:
Transmission expansion planning, Network design, Grasp.
Abstract:
Transmission expansion planning with re-design has been recently proposed in the literature to improve on
the classical transmission expansion planning by allowing to cut-off circuits while expanding the network.
Although the reductions in the solution costs are significant, the resulting mixed-integer linear programming
formulations are very difficult to solve exactly for large networks. In this work, we propose the first meta-
heuristic for the transmission expansion planing problem with re-design: a simple yet efficient GRASP meta-
heuristic. We show on realistic networks for which the optimal solutions are known that our method is able
to provide in short amounts of time feasible solutions as cheap as the optimal ones. Moreover, we are able to
compute a new feasible solution for benchmark instance Brazil Southeast that is cheaper than the best solution
from the literature.
1 INTRODUCTION
With the growth of energy demand over the years, it
becomes necessary for the managing entity to change
the electrical power system, adding new transmission
lines and power generators. Since transmission lines
are expensive to build, one would like to build new
generating units to tailor the supply of nearby con-
sumers. However, it is usually not possible or not
economical to build the new generating units close to
consumption centers so that they must be constructed
in distant places. Consider for instance the situation
of Brazil. The country possesses large resources in
hydropower. However, the later are usually located
far from main cities and industries. Therefore, it is
necessary to build new transmission circuits in order
to integrate all power plants into the electrical net-
work.
The decisions of the planning process yield an
optimization problem that must decide of both the
construction of the generating units and the transmis-
sion lines. This optimization problem is very hard to
solve, even approximately, so that most practitioners
split the problem into two smaller optimization prob-
lems. The first problem chooses the best generating
units and their emplacements. The second problem
designs the cheapest transmission network to connect
these units and the consumption centers. In this paper,
we focus on the problem of designing of the network,
given that the new power generating units are already
built.
Most works on transmission expansion planning
consider only the possibility of adding new circuits
to an existing transmission network, see the review
of (Latorre et al., 2003). Namely, all of the existing
circuits must be used in the expanded network. This
problem is denoted by (
TEP ) in what follows. Re-
cently, the authors of (Moulin et al., 2010) and (Kho-
daei et al., 2010) have independently shown that al-
lowing for cutting-off existing circuits while expand-
ing the network can lead to cheaper expansion plans.
Nevertheless, the resulting optimization problems are
extremely difficult to solve exactly for real-size net-
works. In (Moulin et al., 2010), the authors show that
the classical transmission expansion planning prob-
lem can be solved exactly for large-scale networks in
a couple of minutes using CPLEX 12 (IBM-ILOG,
2009). However, they show that the problem with re-
design (denoted by (
TEP
R
) in what follows) is signif-
icantly harder to solve exactly. In particular, they are
unable to compute the optimal solution for their larger
instance Brazil Southeast.
The extreme difficulty to solve ( TEP
R
) exactly is
the motivation for this work. We develop and test a
380
Figueiredo R., González Silva P. and Poss M..
TRANSMISSION EXPANSION PLANNING WITH RE-DESIGN - A Greedy Randomized Adaptive Search Procedure.
DOI: 10.5220/0003723403800385
In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 380-385
ISBN: 978-989-8425-97-3
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

meta-heuristic procedure that intends to provide good
solutions for ( TEP
R
) in a limited amount of time. Our
algorithm extends to ( TEP
R
) the GRASP developed
in (Binato et al., 2001a) for ( TEP ). We test the al-
gorithm on the real instances used in (Moulin et al.,
2010) and our algorithm finds the optimal solution
for a set of instances for which the optimal solution
is known. Moreover, it is able to provide in eleven
minutes a new solution for Brazil Southeast, cheaper
than the best solution found in the literature (Moulin
et al., 2010) after several hours of branch-and-bound
search.
This paper is structured as follows. The next sec-
tion describes the notation used throughout the pa-
per and a mathematical programming formulation for
( TEP
R
). Section 3 then presents the key aspects of the
GRASP developed herein. Finally, Section 4 presents
a numerical comparison of our meta-heuristic and the
integer programming formulation from (Moulin et al.,
2010).
2 MODEL AND NOTATIONS
2.1 Nomenclature
2.1.1 Sets and Parameters
B Set of buses.
Ω Set of all circuits.
Ω
0
Set of existing circuits.
Ω
1
Set of candidate circuits.
γ
k
Susceptance of circuit k.
c
k
Investment cost of circuit k.
f
k
Capacity of circuit k.
d
i
Load at bus i.
g
i
Maximal generation at bus i.
2.1.2 Variables
x
k
Indicates if circuit k is present.
f
k
Flow on circuit k.
g
i
Generation at bus i.
θ
i
Potential angle at bus i.
2.2 Mathematical Formulation
From the Combinatorial Optimization point of view,
the electrical network is an undirected graph (B, Ω)
where vertices i ∈ B are called buses and edges k ∈ Ω
are called circuits. The set of circuits is partitioned
into a subset Ω
0
, of existing circuits, and a subset of
candidate circuits, denoted by Ω
1
. For each circuit
k ∈ Ω, indices i(k) and j(k) denote, respectively, the
head and the tail of the circuit (which are chosen ar-
bitrarily), while γ
k
, f
k
and c
k
are the circuit suscep-
tance, capacity and cost, respectively. For each node
i ∈ B, δ
−
(i) := {k ∈ Ω : i(k) = i} and δ
+
(i) := {k ∈
Ω : j(k) = i}. The network can have parallel circuits,
k
1
, k
2
∈Ω, denoted by k
1
kk
2
, linking the same termi-
nal buses.
Different models can be found in the literature to
describe power flow in a transmission network (Bi-
enstock and Mattia, 2007). The most accurate one is
called the AC flow model. In that model, the voltage
at node i of the network is represented by a complex
number, U
i
e
√
−1θ
i
, where U
i
is the voltage amplitude
and θ
i
is the voltage angle at i. The power flowing
from i to j along the (undirected) edge (ij) is a com-
plicated function that involves quadratic terms in U
i
and U
j
multiplied by trigonometric functions of the
difference θ
i
−θ
j
. Expressing the flow in this accu-
rate way yields highly non-convex MINLPs, which
are extremely difficult to solve exactly even for very
small networks. For this reason, most approaches on
transmission expansion planning problem use the lin-
earized DC approach which defines the flow between
two buses as
f
k
= γ
k
(θ
i(k)
−θ
j(k)
). (1)
Using the DC model described in (1), ( TEP
R
) can
be written in the following form:
min
∑
k∈Ω
1
c
k
x
k
(2)
s.t.
∑
k∈δ
−
(i)
f
k
−
∑
k∈δ
+
(i)
f
k
= d
i
−g
i
i ∈B (3)
f
k
−x
k
γ
k
(θ
i(k)
−θ
j(k)
) = 0 k ∈ Ω (4)
− f
k
≤ f
k
≤ f
k
k ∈ Ω (5)
0 ≤ g
i
≤ g
i
i ∈B (6)
x
k
∈ {0, 1} k ∈ Ω. (7)
Objective function (2) contains the cost of circuits
pertaining to Ω
1
. Hence, the cost of cutting off exist-
ing circuits is assumed to be negligible. Constraints
(3) ensure that the incoming flow at each bus is equal
to the load requirement minus the generation at the
bus. Each constraint in (4) is defined by a disjunction
on the presence x
k
of circuit k. If x
k
= 1, the circuit is
present so that the associated equation in (4) defines
the flow for circuit k according to linearized Kirchoff
law. If x
k
= 0, circuit k is not present so that the as-
sociated equation in (4) yields a null flow for circuit
k. Finally, constraints (5) and (6) are bounds on the
TRANSMISSION EXPANSION PLANNING WITH RE-DESIGN - A Greedy Randomized Adaptive Search Procedure
381

real variables and constraints (7) force x to be a bi-
nary vector. Notice that a negative flow means that
electricity flows in the reverse direction on the circuit.
It is easy to see that ( TEP ) can be obtained from
( TEP
R
) by adding the set of constraints
x
k
= 1 k ∈ Ω
0
. (8)
The practical interest of ( TEP
R
) is that its optimal so-
lution cost may be strictly smaller than the one of
( TEP ). We compare in Table 1 the optimal costs for
several realistic networks; the “≤” in the last line in-
dicates that the optimal cost of ( TEP
R
) is not known
for that instance. More details on these instances are
provided in the numerical experiments section.
Table 1: Optimal costs computed by (Moulin et al., 2010).
Name ( TEP ) ( TEP
R
)
Garver 110 110
IEEE24 152 152
South 154.4 146.2
South Red 72.87 63.2
Southeast 424.8 ≤ 405.9
Table 1 shows that for the three last instances,
there is a clear benefit in allowing for re-design. How-
ever, problem ( TEP
R
) is much harder to solve than
( TEP ) mainly because it contains more binary vari-
ables. In fact, while it is easy to solve ( TEP ) with
modern solvers for realistic networks, ( TEP
R
) is still
very challenging, see (Moulin et al., 2010). The
objective of this work is therefore to provide good
feasible solutions for ( TEP
R
) computable in a short
amounts of time.
3 ALGORITHM DESCRIPTION
We present in this section our meta-heuristic GRASP.
Our algorithm follows closely the line of the GRASP
from (Binato et al., 2001a) which we adapted to han-
dle the possibility of cutting-off some of the circuits.
GRASP algorithms typically consists in two phases,
see (Feo and Resende, 1995) for more details. In the
construction phase, one intelligently constructs an ini-
tial solution via an adaptive randomized greedy func-
tion, see Algorithm 1. In the local search phase, one
tries to improve the constructed solution by looking at
its neighborhood, see Algorithm 2. These two phases
are described for ( TEP
R
) in Sections 3.1 and 3.2 be-
low.
3.1 Construction Phase
The objective of this phase is to build a network that
can attend the demand of each consumption site. For-
mally, we want to construct a set of circuits
ˆ
Ω ⊆ Ω
with ˆx
k
= 1 for k ∈
ˆ
Ω and 0 otherwise, such that there
exists (θ, f, g) ∈ R
|B|
2
×|Ω|
that satisfies constraints
(3)–(6) for ˆx. To check the feasibility of a given cir-
cuits set
ˆ
Ω, it will be useful to consider the following
feasibility test:
FEAS (
ˆ
Ω) := min
∑
i∈B
r
i
s.t.
∑
k∈
ˆ
δ
−
(i)
f
k
−
∑
k∈
ˆ
δ
+
(i)
f
k
= d
i
−g
i
−r
i
i ∈ B (9)
f
k
−γ
k
(θ
i(k)
−θ
j(k)
) = 0 k ∈
ˆ
Ω (10)
− f
k
≤ f
k
≤ f
k
k ∈
ˆ
Ω
0 ≤ g
i
≤ g
i
i ∈ B,
where r
i
describes the shortage at bus i,
ˆ
δ
−
(i) := {k ∈
ˆ
Ω : i(k) = i} and
ˆ
δ
+
(i) := {k ∈
ˆ
Ω : j(k) = i} for each
node i ∈ B. Constraints (9) are different from (3) in
two aspects. First, for each i ∈ B sets δ
+
(i) and δ
−
(i)
are replaced by
ˆ
δ
+
(i) and
ˆ
δ
−
(i) to consider only the
flows for circuits in
ˆ
Ω. Second, each constraint in (9)
features a shortage variable r
i
in its right-hand side
to match the demand if the whole load of bus i can-
not be supplied by the network defined by
ˆ
Ω. Con-
straints (10) are derived from constraints (4) by set-
ting x
k
to one for each k ∈
ˆ
Ω and relaxing constraints
for k ∈ Ω\
ˆ
Ω. The last two sets of constraints are
bounds on the flows and on the generations. We see
that FEAS (
ˆ
Ω) = 0 implies that
ˆ
Ω is a feasible circuits
set in the sense above.
To start the construction phase, we set
ˆ
Ω :=
/
0.
Then, an iterative algorithm tries to construct a cir-
cuits set
ˆ
Ω such that FEAS (
ˆ
Ω) = 0 by adding one cir-
cuit at the time to
ˆ
Ω. In order to choose the circuit
to add at each step, we compute a greedy function
that approximates the benefit from adding each cir-
cuit individually. Then, using the greedy function we
rank the circuits in order to construct the candidate
listCL ⊆Ω, that is, the set of most promising circuits.
Finally, we randomly choose an element fromCL and
add it to
ˆ
Ω. This procedure is repeated until either
a feasible solution is found or all circuits have been
added. In the latter situation, no feasible solution has
been found. In a second step, we try to withdraw cir-
cuits one by one according to their costs. The whole
procedure is resumed in Algorithm 1.
We are left to describe our greedy function and
how to construct CL. Let π ∈ R
|B|
be the set of dual
variables associated to constraints (9). The authors of
(DeChamps et al., 1979) have shown that the benefits
of susceptance changes can be estimated by
ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems
382

Algorithm 1: Construction phase.
ˆ
Ω is empty;
repeat
Evaluate the greedy function for each k ∈ Ω\
ˆ
Ω;
Build CL;
k is randomly selected from CL;
ˆ
Ω :=
ˆ
Ω∪{k};
until FEAS(
ˆ
Ω) = 0 or CL is empty;
if FEAS(
ˆ
Ω) > 0 then
No feasible solution found, abort;
else
repeat
Rank circuits in
ˆ
Ω by decreasing cost c;
k is the first element of the list;
ˆ
Ω :=
ˆ
Ω\{k};
until FEAS(
ˆ
Ω) > 0;
ˆ
Ω :=
ˆ
Ω∪{k};
return
ˆ
Ω
Π
k
= (π
i(k)
−π
j(k)
)(θ
i(k)
−θ
j(k)
) (11)
for each k ∈ Ω\
ˆ
Ω. We construct CL by ordering the
circuits in Ω\
ˆ
Ω by decreasing value of Π and taking
the 70% best circuits in the list. Algorithm 1 may not
always find a feasible solution. This is due to the fact
that (11) provides only an intuitive indication on the
promising new circuits, it is not a rigorous measure.
For this reason, the elements are selected randomly
from CL.
The main difference between our algorithm and
the one from (Binato et al., 2001a) is the following:
their algorithm starts with
ˆ
Ω = Ω
0
and always con-
sider that potential new circuits belong to Ω
1
. In op-
position, our algorithm starts with an empty set
ˆ
Ω and
considers potential new circuits in the whole set Ω.
3.2 Local Search
Since the greedy function used in the construction
phase does not involve the cost of the circuits added
at each iteration, it is likely to end up with a costly
circuits set
ˆ
Ω. In particular, there may exist a cheaper
circuit set Ω
∗
that satisfies FEAS (Ω
∗
) = 0 as well and
that has structure very similar to
ˆ
Ω. The concept of
similar structure can be formalized by the introduc-
tion of neighborhoods. Generally speaking, a neigh-
borhood is a mapping that associates a set of circuit
set to each Ω
∗
⊆Ω. The local search phase tries there-
fore to improve the constructed solution
ˆ
Ω by looking
at its neighborhood.
Consider a circuit set Ω
∗
⊆ Ω. In this work,
we use the following neighborhoods: neighborhood
N
n
(Ω
∗
) of Ω
∗
is the set that contains all circuit sets
with the same number of circuits as Ω
∗
, among which
Algorithm 2: Local search phase using N
n
.
Let Ω
∗
⊆Ω be the circuit set constructed with
Algorithm 1;
repeat
ˆ
Ω := Ω
∗
;
foreach Ω
′
∈ N
n
(Ω
∗
) do
Compute FEAS (Ω
′
);
if FEAS(Ω
′
) = 0 and c(Ω
′
) < c(
ˆ
Ω) then
ˆ
Ω := Ω
′
;
if c(
ˆ
Ω) < c(Ω
∗
) then Ω
∗
:=
ˆ
Ω;
else Ω
∗
is a local solution;
until Ω
∗
is a local solution;
exactly n are different from the circuits in Ω
∗
. It is
defined formally as
N
n
(Ω
∗
) := {Ω
′
s.t. Ω
′
⊆Ω, |Ω
′
|= |Ω
∗
|, |Ω
∗
\Ω
′
|= n}.
We also define c(Ω
∗
) :=
∑
k∈Ω
∗
c
k
. We say that Ω
∗
is a
local solution for neighborhood N
n
if c(Ω
∗
) ≤ c(Ω
′
)
for each Ω
′
∈ N
n
(Ω
∗
) that satisfies FEAS (Ω
′
) = 0.
The main ingredient of the local search phase is to
look iteratively at the neighborhood of the current so-
lution for a solution with lower cost until we are stuck
at a local solution. The procedure is schematized in
Algorithm 2.
The bottleneck of Algorithm 2 is located at the
solution of the linear programs necessary to compute
FEAS (Ω
′
) for each Ω
′
∈ N
n
(Ω
∗
). If Ω does not con-
tain parallel circuits, then the neighborhood size of a
circuit set Ω
∗
is equal to

n
|Ω
∗
|

n
|Ω\Ω
∗
|

=
|Ω
∗
|!
n!(|Ω
∗
|−n)!
|Ω\Ω
∗
|!
n!(|Ω\Ω
∗
|−n)!
,
(12)
that is, the neighborhood size grows extremely fast
with the value of n. In all our cases studies, the net-
works contain parallel circuits so that (12) must be re-
placed by a smaller number that depends on the num-
ber of parallel circuits allowed for each pair of buses.
Although the resulting number is smaller than (12), it
is still prohibitive when n > 1. For this reason, we
limit our tests to n = 1 and n = 2 in our cases stud-
ies. Notice that additional tricks have been used in
this work to decrease the number of elements from N
n
that need to be considered, see (Binato et al., 2001a).
4 NUMERICAL EXPERIMENTS
In this section, we compare the solution times and the
best feasible solutions found using the GRASP de-
scribed in this paper and the integer programming for-
mulation proposed in (Moulin et al., 2010).
TRANSMISSION EXPANSION PLANNING WITH RE-DESIGN - A Greedy Randomized Adaptive Search Procedure
383

4.1 Data
We compare the two solution methods on the realis-
tic networks whose main characteristics are provided
in Table 2. The first column indicates the name of
the instance, the next two columns provide the num-
ber of buses and circuits, respectively, while the last
column provides a reference where the complete in-
stance details can be found. The difference between
instances South and South Red concerns the maxi-
mum generations available: South Red allows for re-
dispatch while South does not allow for re-dispatch,
that is, the maximal generations for South have been
scaled down to meet the total demand
∑
i∈B
d
i
.
Table 2: Networks Characteristics.
Name |B| |Ω| Reference
Garver 6 96 (Garver, 1970)
IEEE24 24 140 (Alguacil et al., 2003)
South 46 299 (Binato, 2000)
South Red 46 299 (Binato, 2000)
Southeast 79 405 (Binato, 2000)
4.2 Integer Programming
The non-linearity of ( TEP
R
) arises from constraints
(4) that turn the problem into a Non-Convex Mixed-
Integer Linear Program. Although this class of
programs has witnessed a tremendous attention in
the past years (Belotti et al., 2008), they are still
very challenging to solve to optimality, especially
for problems as large as ( TEP
R
). For this rea-
son, most approaches for the transmission expansion
planning problem use instead linearizations that in-
troduce “big-M” coefficients, see (Bahiense et al.,
2001; Binato et al., 2001b; Moulin et al., 2010; Vil-
lanasa, 1984), among others. One exception is (Rider
et al., 2008) who tackle the non-convex MINLP by
a branch-and-bound algorithm. Although their ap-
proach provides interesting results for their cases
studies, they cannot guarantee the optimality of their
solution because of the non-convexityof the problem.
In this work, we follow the traditional approach
that linearizes (4) with the help of “big-M” coeffi-
cients. According to (Moulin et al., 2010), an efficient
approach replaces constraints (4) with
−M
k
(1−x
k
) ≤ f
k
−γ
k
(θ
i(k)
−θ
j(k)
) ≤ M
k
(1−x
k
),
for each k ∈Ω, and constraints (5) with
−x
k
f
k
≤ f
k
≤ x
k
f
k
for each k ∈Ω.
One of the difficulties of the resulting MILP arises
from the symmetry existing among variables x
k
that
are associated with parallel circuits. Basically, paral-
lel circuits yield feasible points that are indistinguish-
able by the objective function. In order to break this
symmetry, we order parallel circuits by introducing
the partial order ≺ among elements of Ω and intro-
duce the precedence constraint x
k
≥ x
h
for each pair
k, h ∈ Ω such that h ≺ k, see (Moulin et al., 2010) for
the details.
4.3 GRASP
In practice, our GRASP starts by running Algorithm 1
until a feasible solution is found. Then, a local search
is performed as described in Algorithm 2. Many it-
erations of Algorithm 1 were needed, ranging from
8 for Garver to 4078 for Southeast for the results
presented in Table 3. Consequently, the whole algo-
rithm spends significantly more time in the construc-
tion phase, with between 15% and 25% of the total
solution time spent in the local search.
4.4 Results
Both approaches have been coded in Xpress
Mosel 3.2.1 with the solver Xpress Optimizer
21.01.06 (FICO, 2009) on a computer using a proces-
sor Pentium Core 2 Quad Q6600 de 2.4 GHz and 4GB
of RAM memory. In addition, we have also tested
the integer programming approach with the Interac-
tive Optimizer of CPLEX 12.2 (IBM-ILOG, 2009).
The best feasible solutions obtained with the
GRASP are given in Table 3. The second column of
the table shows the best feasible solution reported in
the literature, see (Moulin et al., 2010), which are op-
timal for all networks but Southeast. The third column
shows the best feasible solution obtained using the
construction phase only, while the next two columns
showthe best solutions after performinga local search
with neighborhood N
1
and N
2
, respectively. For all
instance but Southeast, the solutions found after any
of the local search are the optimal ones. More in-
terestingly, the local search with N
2
found a cheaper
solution for Southeast than the best feasible solution
reported in the literature.
Table 3: Results for GRASP after the construction phase,
the local search with N
1
, and the local search with N
2
.
Name Best Cons N
1
N
2
Garver 110 130 110 110
IEEE24 152 194 152 152
South 146.2 172.8 146.2 146.2
South Red 63.2 75.81 63.2 63.2
Southeast 405.9 406.8 405.9 392.8
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384

In Table 4, we provide CPU times in seconds for
the GRASP using N
2
and CPU times in seconds or
duality gaps (when it is available) for the MILP for-
mulations (a time limit of ten hours has been set) us-
ing Xpress and CPLEX, respectively. Of course, the
GRASP only computes a feasible solution while the
MILP formulations also provide a guarantee of opti-
mality. Hence, we do not intend to compare directly
these times but we rather want to check that GRASP
is fast enough for the difficult instances. We see from
Table 4 that GRASP finds good feasible solutions for
the difficult instances South and Southeast in a cou-
ple of seconds and minutes, respectively, while both
solvers can hardly solve these instances within 10
hours of computing time. The best solution found
by CPLEX for Southeast has a cost of 527.70, while
Xpress could find no feasible solution within the time
limit.
Table 4: Running times in seconds (or duality gaps) for the
MILP formulation and for the GRASP.
Name GRASP Xpress CPLEX
Garver 3.27 0.062 0.27
IEEE24 0.92 6.47 5.94
South 7.41 7.15% 32319
South Red 7.68 48.8 31
Southeast 632.17 – 71.41%
ACKNOWLEDGEMENTS
Michael Poss is supported by an “Actions de
Recherche Concert´ees” (ARC) projet of the “Com-
munaut´ee franc¸aise de Belgique” and is a research fel-
low of the “Fonds pour la Formation `a la Recherche
dans l’Industrie et dans l’Agriculture” (FRIA). Rosa
Figueiredo is supported by FEDER founds through
COMPETE–Operational Programme Factors of Com-
petitiveness and by Portuguese founds through the
CIDMA (University of Aveiro) and FCT, within
project PEst-C/MAT/UI4106/2011 with COMPETE
number FCOMP-01-0124-FEDER-022690. Pedro
Henrique Gonz´alez Silva thanks financial support of
a master scholarship from brazilian funding agency
CAPES.
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