Fuzzy Guided Constructive Heuristic Applied to Transmission System Expansion Planning
ABSTRACT This work presents a constructive heuristic algorithm that uses fuzzy decision making to solve the transmission system expansion planning problem. The fuzzy system is used as a guide to circumvent some critical problems found in constructive heuristics that employ sensitivity index. The sensitivity index is derived from the resolution of relaxed models, and works as a guide to circuit addition. The heuristic presented in this paper is based on the well known branchandbound algorithm. Fuzzy decision making is used to decide the instant to divide the problem into two new subproblems. Tests have been conducted with part of real Brazilian systems in order to verify the efficiency of the proposed method.

Conference Paper: A Heuristic method based on the branch and cut algorithm to the transmission system expansion planning problem
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ABSTRACT: In this paper we introduce a new heuristic algorithm to the centralized longterm transmission system expansion planning problem. The proposed method introduces cuts (new constraints) to the mathematical model to force generating feasible solutions to the DC model. The application of the strategy resulted in good solutions for the DC model. The computational results demonstrate the efficiency of the proposed method when applied to IEEE24 bus system, Brazilian Southern System and Colombian system.Power and Energy Society General Meeting, 2011 IEEE; 01/2011
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Fuzzy Guided Constructive Heuristic Applied to
Transmission System Expansion Planning
Aldir S. Sousa and Eduardo N. Asada Member, IEEE
Abstract—This work presents a constructive heuristic algo
rithm that uses fuzzy decision making to solve the transmission
system expansion planning problem. The fuzzy system is used as a
guide to circumvent some critical problems found in constructive
heuristics that employ sensitivity index. The sensitivity index
is derived from the resolution of relaxed models, and works
as a guide to circuit addition. The heuristic presented in this
paper is based on the well known branchandbound algorithm.
Fuzzy decision making is used to decide the instant to divide the
problem into two new subproblems. Tests have been conducted
with part of real Brazilian systems in order to verify the efficiency
of the proposed method.
Index Terms—Transmission system expansion planning, Fuzzy
Decision Making, Constructive Heuristic Algorithm.
I. INTRODUCTION
T
deregulation and the disaggregation of the power companies
aimed the creation of a competitive market environment, in
which the electric power is considered as a commodity. The
planning of transmission systems can be divided in short,
medium and long term plan. As the planning gets closer to
a short term, detailed analysis such as those involving voltage
limits, reactive power management, stability, construction de
tails must be considered. For a long term planning, the main
objective is to obtain the backbone of the power transmission.
The idea is to determine the location of transmission lines
and/or transformers to be installed in a specified time horizon
in order to meet the desired operating conditions at minimum
cost. The traditional planning, also known as centralized
planning, takes into account the existence of a centralized
system in a regulated monopoly structure. A simplification of
the problem is obtained considering a “static” approach, which
represents a planning for a single estimate of power demand.
The base year topology, candidate circuits, generation and
load data for the planning horizon and investment constraints,
represent the basic data. A more realistic model is represented
by the existence of several planning stages in which it is also
necessary to determine the period when the new circuits would
be installed (multistage planning). The latter formulation also
represents a much more complex problem to solve.
Let us consider the centralized planning with the static
approach. This case represents the less complex planning
HE power system operation and management have un
dergone through deep changes in the recent decades. The
This work was supported by The State of S˜ ao Paulo Research Foundation
(FAPESP), CNPq and CAPES
Eduardo N. Asada is with S˜ ao Carlos School of Engineering, University of
S˜ ao Paulo, CEP 13566590, S˜ ao Carlos, SP, Brasil, (email: easada@usp.br)
Aldir S. Sousa is with S˜ ao Carlos School of Engineering, Univer
sity of S˜ ao Paulo, CEP 13566590, S˜ ao Carlos, SP, Brasil, (email:
aldirss@sel.eesc.usp.br)
model. However, the mathematical modeling is still complex.
It is usually formulated as a mixed integer nonlinear problem.
In terms of network model, usually for a long term planning
the DC load flow model is used even though the ac model
could be employed as well [1]. Even when the network is
simplified, for instance, using the DC load flow model, it still
represents a very hard problem due to the number of variables,
constraints and also possibilities that increases according to
the system size. As a consequence, it is regularly treated as a
combinatorial problem.
Two major approaches have been found in the technical
literature: 1) The problem is solved with exact methods such
as the Benders decomposition and 2) by approximate methods.
Methods belonging to (1) have proof of convergence and
also a good performance with small and medium systems,
as well as the upper and lower bounds for the solutions.
However, for large systems, the convergence is obtained with
more difficulty and depends on the fine tuning of the method
(usually, a classical nonlinear optimization problem). Among
the exact methods, the branch and bound [2] and Benders
decomposition are most utilized [3], [4]. The approximate
methods are represented by heuristic and metaheuristic algo
rithms. The principal feature is the gradual construction of
the solution, which results in a less complex programming,
however, the convergence to the optimal solution is uncertain.
In this class of method, the constructive heuristic algorithms
[5] and metaheuristics such as genetic algorithm, simulated
annealing among others have been reported in the technical
literature [6]–[9].
II. STATIC DC MODEL FOR LONG TERM TRANSMISSION
PLANNING
When a single planning horizon is considered, the mathe
matical model is as follows:
Minv =
?
(i,j)∈Ω
cijnij
(1)
s.t.
Sf + g = d
fij− γij(no
fij ≤ (no
0 ≤ g ≤ g
(2)
(3)
(4)
ij+ nij)(θi− θj) = 0
ij+ nij)fij
0 ≤ r ≤ d
(5)
0 ≤ nij≤ nij
nij integer
fij and θj unbounded
(i,j) ∈ Ω
(6)
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where
v
cij
θj
γij
no
investment cost in m.u. (monetary unit)
cost of a circuit in path i − j (m.u.)
voltage angle in bus j
susceptance of the circuit in path i − j
number of circuits of the original topology in path
i − j
number of circuits added into path i − j
maximum number of circuits in path i − j
power flow in path i − j
maximum power flow limit of a circuit in path i−j
power generation in bus k
maximum power generation in bus k
Transpose of the branchnode incidence matrix
vector of artificial generators (load shedding)
vector of maximum generation in buses
vector of loads
set of all paths
Expression (1) specifies the minimization of the objective
function, constraint (2) represents Kirchhoff’s Current Law
and constraint (3) is the Kirchhoff’s Voltage Law. The rest
of the constraints represent the operational limits of the
components. This formulation is the same of the one presented
in [3]. Another approach to deal with the difficulty in solving
the earlier problem is to apply simplifications. If constraint
(3) is ignored, the problem turns into the transportation model
and only the active power is considered [10]. It must be
noticed that the suppression of constraint (3) makes linear the
mathematical model while the complete model (DC model) is
nonlinear due to the product of integer and real variables in
(3). We call as hybrid any intermediary model between the DC
model and the transportation model. Therefore, any model that
considers only part of the constraints (3), represents a hybrid
model. In this context, it is possible to formulate a hybrid
linear model or a hybrid nonlinear model as in [11]. In this
work we have employed the hybrid linear model as will be
discussed further in the next sections.
ij
nij
nij
fij
fij
gk
gk
S
r
g
d
Ω
III. CONSTRUCTIVE HEURISTIC ALGORITHM (CHA)
Since Garver [10] presented the constructive heuristic algo
rithm with the transportation model, several new CHA have
been proposed to the transmission expansion planning. Some
of them are: the algorithm based on minimum effort criterion
[12] and the algorithm based on minimum load shedding
criterion [13]. Taking a closer look on these algorithms, one
can notice that the main difference among them relies on the
sensitivity index that is used to guide the search. A construc
tive heuristic algorithm adds one transmission line in each
iteration based on specified criterion, which is usually based
on a performance index. The performance index provides the
information of how much the system would improve if a
specific transmission line is inserted into the system. Normally,
the calculation of such index requires the resolution of linear
program or nonlinear program. When the stopping criterion is
met, the algorithm stops. As can be observed, this is a simple
strategy to determine the expansion of the system and usually
it has a good performance for small systems. The critical point
of the classical CHA is that it may get stuck fastly in poor
local optimal solutions.
1) Hybrid Linear Model: Let us consider the hybrid linear
model, which has been used in this work and represents a
linearized version of [11]. Differently from the DC model,
in this formulation only the existing transmission lines must
follow the Kirchhoff’s Voltage Law. The candidate lines,
which are represented by variables nij are not subject to
the nonlinearity of KVL. If variable nij is considered a real
variable, instead of integer variable, this model becomes a
Linear Programming problem. Since lines added to the base
topology must follow the two Kirchhoff’s laws, this model
generates solutions that are also feasible for the DC model.
minv
=
?
(i,j)
cijnij
(7)
s.t.
S f + Sofo+ g = d
fo
ij(θi− θj) = 0 ∀(i,j) ∈ Ω1
fo
ij
∀(i,j) ∈ Ω1
fij ≤ fijnij
∀(i,j) ∈ Ω
0 ≤ g ≤ g
ij− γo
ij ≤ fijno
0 ≤ nij≤ nij
nij
integer
fij
unconstrained
θj
unconstrained ∀j ∈ Ω2
Basically, the variables are the same of DC model, except
Sowhich is the branchnode incidence matrix of the existing
lines of the base topology; fois the power flow in the circuits
of the base topology; S is the branchnode incidence matrix of
the candidate circuits, f is the vector of power flows through
the added branches; γo
the base topology; θj are the voltage angles in buses of the
base topology; Ω1is the set of existing circuits; Ω is the set
of candidate circuits;Ω2is the set of all buses.
2) Calculation of sensitivity index: An approach to calcu
late the sensitivity index is to relax the integrality constraint of
variables nij that represent the number of transmission lines
in path i − j. Instead of considering it as an integer variable,
it is made a real variable. According to the selected network
model, the problem becomes a linear program problem (LP) or
nonlinear problem (NLP). For the hybrid model shown earlier
the problem becomes a LP problem. We can select the path
to add a line based on the maximum value of the resulting
nij variable as defined in [10]. Considering this criterion the
following sensitivity index is calculated.
ijis the susceptance of the circuits of
SIij= nijfij
(8)
where nij is the value obtained from the LP. The circuit
corresponding to the largest SIij is chosen, or:
Selected path ij = max{nijfij;nij?= 0}
(9)
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The initial topology (or the base topology) with the addition
of the new line according to the sensitivity index will form the
current topology for the next iteration.
A. Deficiencies of the CHA
As pointed earlier, the CHA may find the optimal or
suboptimal solutions of small or medium systems. However,
for large systems the CHA based on the sensitivity index
may end up far from the optimal solution. According to the
reference [14], the principal reasons for the CHA to deviate
the search to low quality solutions are:
1) Selection of high cost lines;
2) Selection of lines based on small values of nij.
The deviation tends to occur in two moments during the
search. The first one occurs in the first iterations when high
cost lines (which usually also present the highest transmission
capacity) are selected. When this happen, other possibilities in
volving smaller lines, also cheaper ones, are rather decreased.
The second critical moment is in the last iterations, when
the values of variable nij resulting from the solution of the
corresponding LP or NLP problem approaches to zero. It
might exist a situation in which the current topology still
requires additional transmission lines, however, the decision of
where to install would have been taken based on tiny values of
nij. The accuracy of such choice is low because the sensitivity
index gives an indication based on low values of nijand after
the path i − j is selected an integer value of line is added
instead of the fraction of a line as indicated by the nij. In order
to deal with the two situations depicted above a constructive
heuristic in branchandbound structure with fuzzy decision
making is adopted in this paper.
IV. CHA WITH FUZZY DECISION MAKING
A CHA usually converges rapidly in few iterations. How
ever, for large systems the tendency is to converge in poor
solutions. The metaheuristics usually require a considerable
amount of computing time and provide good quality solutions.
The same happens with classical optimization methods. As a
conclusion, an algorithm that stands between the effortless
programming of CHA that provides good solutions as the
metaheuristics is desired. Therefore, we propose a hybrid
algorithm that blends these two characteristics as in [5].
Moreover, we introduce the fuzzy decision making to adjust
adaptively and dynamically the search. The main idea is to
obtain high quality solutions with low computational cost.
A. Fuzzy Decision Making
The human reasoning accepts and naturally process ap
proximate or inaccurate information gathered from different
ways. However, in a computing system, the modeling and
treatment of approximate information is a complex task. For
example, the adjustment of the room temperature with the air
conditioning system would be a simple task if we set to turn
the air conditioning on when the temperature reaches 77.0◦F
and to turn off when it goes below 68.0◦F. However, if the
the adjustment is made according to the sensation of the user,
Fig. 1.Branching and creation of two subproblems
which depends on many factors, it would be totally different.
For example, we would decide to turn on when the room is too
“hot” and keep cooling until a“mild temperature” is reached.
Fuzzy logic deals with uncertain information by treating them
not as a binary information like black or white, or true or
false, but also allowing the degree of truth of a condition that
can range between 0 and 1. For linguistic variables which
translate the state, it would exist intermediary states such as
“not so cold” or “very hot”. Therefore, there is a variable
grade of membership that describes such situation. Lofti Zadeh
[15] proposed a theory to correctly express the grade of
membership and created a translation between the world of
exact representation (crisp information) and the imprecise
information from the real world. We will use the concepts
from Fuzzy Set Theory to improve the CHA.
The use of fuzzy sets in transmission expansion planning
is increasing gradually. Most of the papers published in this
topic focus on the use of fuzzy sets theory to model and to
deal with the uncertainties from different operation conditions
due to power market behavior, reliability. Or, to deal with
some conflicting objectives that may affect the cost of the plan.
Some of these papers can be found in [16], [17] In this paper,
the use of fuzzy logic is slightly different and is employed to
enhance the algorithm by introducing approximate logic to a
conventional constructive heuristic algorithm.
1) Motivation: In the method proposed in [5], the algorithm
has to be tuned to each system in order to provide the best
performance. This task may not be easy, because it depends on
the sensitivity of the user on the characteristics of the system
and may result in irregular performance of the algorithm.
Usually, parameter adjustment has been the trickiest part for
heuristic methods and also represents the subject of criticism
for those methods. The main motivation of this work was
to extract the main characteristics that allow an automatic
parameter adjustment of the constructive heuristic algorithm
by using fuzzy logic. As a result, the method becomes less
sensitive to different systems being solved, thus presenting a
stable performance.
2) Inclusion of Fuzzy Logic into the CHA: As mentioned
earlier, we will use a CHA structure similar to [5], with the aim
to avoid getting trapped in local optimal solutions. The fuzzy
system will make a decision to create new subproblems to test
according to the value of variable nij and its relative cost. If
the system considers that the selected nijfor addition is very
small in the problem being solved, then another subproblem
(or another alternative possibility) is created (as in Figure
1) and is also tested. This strategy is based on the divide
and conquer technique, which consists of dividing complex
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012345678
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
TY
SM
MD
L
VL
Fig. 2.
Fuzzy variable value of nij.
00.10.2 0.30.40.50.60.70.8 0.91
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
TY
SM
MD
L
VL
Fig. 3.
Fuzzy variable cost of line i − j.
problems into smaller ones to facilitate its resolution.
Therefore, instead of setting up or controlling manually the
parameters that will adjust the search, the fuzzy system will act
based on information about the cost of transmission lines and
also on the results given by the solution of LP that represent
the model.
The fuzzy system proposed in this paper is based on three
variables and 25 fuzzy rules. The variables used are value
of nij, cost of line in path i − j, variable branching which
defines the action of to branch or not to branch. The set of
linguistic values for variables value of nij and cost of line in
path i − j are Tiny (TY), Small (SM), Medium (MD), Large
(L) and Very Large (VL), which are represented by a vector
of real values between 0 and 1. Each element of the vector
is called discretizing point. In this paper, all fuzzy variables
presents 1000 discretizing points. Figures 2, 3 e 4 represent
the variables fuzzy value of nij, cost of line in path i−j and
branching, respectively.
In the following some of fuzzy rules are presented:
If (N is small) and (C is very large)
then perform branching
) and (C is tiny
then perform branching
If (N is medium)
00.10.20.30.40.50.6 0.7 0.80.91
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Branching
Not Branching
Fig. 4.
Fuzzy variable branching.
If (N is medium) and (C is large)
then Do not branch
If (N is large ) and (C is tiny)
then Do not branch
After identifying the active rules, an aggregation operation
is made with those ones to obtain a final set for values of nij
and for the cost of the lines. The aggregation operation used
was defined as follows:
argaggr = max(µi(xj))
(10)
for i = 1...number of fuzzy rules and j = 1...number of
discretizing points.
In the deffuzification (process to convert fuzzy variables to
quantifiable values according to fuzzy logic, the smallest of
maximum (SMax) criterion has been used:
SMax = min
x{max(µ(x))}
(11)
After defuzzification, if SMax is greater or equal 0.5 means
that the problem must be branched into two new subproblems.
An incumbent solution will be the one with the lowest invest
ment value found during the resolution.
B. Algorithm
The algorithm is divided in two phases:
1) Phase I : In this phase, lines are added into the system
until the corresponding investment is zero (v = 0);
2) Phase II: After adding new lines in phase I, we try to
refine the solution by testing the removal of all added
lines. As the inclusion has been made based on an
index, some of these lines may have become superfluous.
Therefore in the end of the process we check all again.
1) Phase I:
a) Initialization:
Adopt a mathematical model (DC, AC or Hybrid)
and the CHA . Solve the corresponding model (LP
or NLP). If the resulting nij are integer, STOP.
For an LP problem, the solution is global optimal.
Otherwise, store the value of the objective function
as inferior limit vinf for the next subproblems
and proceed to Step b; Initialize the list of open
problems.
b) Selection of the variable for separation: Identify the
separation variable nk
indicated by sensitivity index).
If SMax < 0.5 then generate the subproblem set
ting npq= 0 and add it to the list of subproblems.
Otherwise, If SMax ≥ 0.5 add one line to the path
pq and return to step b. Generate a subproblem
making nk
cost of the selected circuit.
c) Selecting and solving a candidate subproblem:
Select the last subproblem added to list (in the
LIFO order). If there are no more subproblems in
the list of open problems, the incumbent is the best
solution found to the chosen model; therefore stop.
pq(the circuit of path (p,q)
pq= nk
pq+ 1 and update v1 adding the
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Otherwise, solve the selected problem (LP or NLP)
and calculate the objective function v2. Store the
inferior limit of the branch subproblem as vinf=
v1+ v2, and go to Step d.
d) Tree pruning tests:
After solving the subproblem using an LP (or
NLP), pruning is executed if any of the following
tests are true:
Test 1: vinf≥ v∗, where v∗is the value of the incum
bent solution;
Test 2: LP (or NLP) solution is unfeasible.
Test 3: The optimal solution from the LP (or NLP)
yields v2 = 0. This means that a feasible
solution has been found for the chosen model. In
this case, verify whether the objective function
of the current subproblem (v1) is smaller than
the incumbent. If yes, then make v∗= vinf
solution and apply Test 1 again for all candidates
not pruned yet.
e) If a subproblem was pruned in Step 4 go to Step
3, otherwise, go to Step 2.
2) Phase II
a) Simulate the removal of each added line in the base
topology.
V. TESTS AND RESULTS
In order to test the efficiency of the algorithm, tests with two
realistic systems have been set up: Southern Brazilian system
of 46 buses, 79 circuits and demand of 6,880MW [14], and
(2) reduced NorthNortheastern Brazilian system of 87 buses,
183 circuits and demand of 20,600 MW (considering only
stage P1) [18]. These systems represent benchmark systems
and have been widely used in transmission expansion planning
problems.The hybrid linear model of [11] which solutions
are feasible for the DC model has been used. Despite its
dimension, the Southern system presents medium complexity
for this problem and the NorthNortheastern system a high
complexity system. The optimal solution for the Northeastern
system is still unknown, even for the simplest planning model.
This is due to the elevated number of isolated buses and points
of high load demand.
A. Southern Brazilian System
For this system, two situations can be analyzed:
1) Planning with generation redispach
2) Planning without generation redispach (generation is
fixed throughout the planning).
In the following the results for Southern Brazilian system
with generation redispach.
1) Southern Brazilian System with redispach (S1): An
optimal solution to this system considering the DC model is $
70,289,000 (US$) which was found with Benders decompo
sition after solving thousands of LP [3]. The CHA with fuzzy
decision making found a feasible solution in 8 iterations. The
CHA has found the same solution for DC model after 222
iterations. The proposed addition is the following:
n13−20= 1, n20−23= 1, n20−21= 2, n42−43= 1,
n46−06= 1, n05−06= 2.
It was not possible to remove any line in Phase 2.
2) Southern Brazilian System without redispach (S2): For
this case, the optimal solution is v = 154,420,000 (US$),
which was found by Benders Decomposition after solving
thousands of LP [3] .
The CHA with fuzzy decision making found a feasible
solution after 13 iterations and the optimal solution has been
found after 322 iterations with the investment value of v =
154,420,000 (US$) with the following topology.
n20−21= 1, n42−43= 2, n46−06= 1, n19−25= 1,
n31−32= 1, n28−30= 1, n26−29= 3, n24−25= 2,
n29−30= 2, n05−06= 2.
It was not possible to remove any line in Phase 2.
B. The NorthNortheastern Brazilian System (NN)
The data of this system allows multistage planning (two
stages  Plan P1 and P2), however, only studies considering
generation without redispach is possible. This system is known
from its great complexity and the global optimal solution is
still unknown [14]. One of the reasons for the complexity
is the existence of many isolated buses that requires the
multiple insertion of lines to make the operation feasible.
The best solution published in the technical literature for the
DC model is 1,360,000,000 (US$), for plan P1, which was
found after solving 300,000 LP with an enhanced version of
Genetic Algorithm [19]. The CHA with fuzzy decision making
found an expansion plan for the stage P1 with the value of
1,482,842,000 (US$) after solving 30545 LP.
In the phase II, lines n69−87and n27−53have been removed.
Finally the investment resulted in v = 1,455,856,000 (US$),
with the following topology.
n02−87= 2, n03−71= 1, n03−87= 2, n04−05= 1,
n04−69= 1, n05−58= 2, n05−68= 1, n13−15= 3,
n14−59= 1, n15−16= 2, n15−46= 1, n16−44= 3,
n16−61= 1, n18−50= 6, n18−74= 3, n20−21= 2,
n20−38= 1, n22−23= 1, n22−58= 2, n23−24= 1,
n25−55= 2, n26−54= 1, n30−31= 1, n30−63= 2,
n36−46= 2, n40−45= 1, n41−64= 2, n43−55= 1,
n43−58= 1, n48−49= 1, n49−50= 2, n52−59= 1,
n53−54= 1, n54−63= 1, n61−64= 1, n61−85= 2,
n67−71= 2, n71−72= 1, n72−73= 1, n73−74= 1.
Considering the complexity of this system and also the
number of LP solved by the algorithm, we can conclude that
the method presents quality for the generation of good quality
topologies that can be used to initialized advanced methods
such as the metaheuristics.
The measure of a performance improvement using heuristic
methods is a complicated issue because depending on the
method, the strategies and the way that a candidate solution
is coded (or represented) may differ completely. However,
for the planning problem, we have adopted the number of
LP solved as a measure of efficiency. The reason for this is
due to the fact that the resolution of the mathematical model