energies
Article
An Improved Hybrid Particle Swarm Optimization
and Tabu Search Algorithm for Expansion Planning
of Large Dimension Electric Distribution Network
Ali Ahmadian 1,2, * , Ali Elkamel 1,3 and Abdelkader Mazouz 4
1
2
3
4
*
College of Engineering, University of Waterloo, Waterloo, ON N2J 0A1, Canada
Department of Electrical Engineering, University of Bonab, Bonab 5551761167, Iran
Department of Chemical Engineering, The Petroleum Institute, Khalifa University of Science and Technology,
Abu Dhabi 127788, UAE
College of Business Administration, Al Ain University of Science and Technology, Al Ain 64141, UAE
Correspondence: ali.ahmadian@uwaterloo.ca or ali.ahmadian1367@gmail.com
Received: 22 June 2019; Accepted: 31 July 2019; Published: 8 August 2019
Abstract: Optimal expansion of medium-voltage power networks is a common issue in electrical
distribution planning. Minimizing the total cost of the objective function with technical constraints
make it a combinatorial problem which should be solved by powerful optimization algorithms. In this
paper, a new improved hybrid Tabu search/particle swarm optimization algorithm is proposed to
optimize the electric expansion planning. The proposed method is analyzed both mathematically
and experimentally and it is applied to three different electric distribution networks as case studies.
Numerical results and comparisons are presented and show the efficiency of the proposed algorithm.
As a result, the proposed algorithm is more powerful than the other algorithms, especially in larger
dimension networks.
Keywords: electric distribution network planning; optimization; particle swarm optimization;
tabu search
1. Introduction
Due to load demand growth, the different component of distribution networks should be planned
optimally. Many papers have been published in which the optimal distribution network components
have been planned using different techniques and algorithms. For example, the optimal planning of
energy storage in [1–3], wind distributed generation in [4,5], hybrid energy storage and distributed
generations in [6,7] have been proposed. One of the planning components is the expansion of
existent network so that the high voltage/medium voltage (HV/MV) substations are allocated and
medium voltage feeders are expanded optimally. The optimal planning is carried out considering
a complex objective function, where technical and economic constraints are taken into account.
The objective function, generally, includes the network investment cost and power loss cost. The main
constraints are also satisfaction of supply-demand requirements and grid nodes’ voltage and grid
lines’ capacity. In addition, the distribution network should have a radial configuration because of
protection systems constraints.
Different mathematical methods and optimization algorithms have been developed in the
literature to optimal expansion planning of the distribution network. The authors in [8] have proposed
a comprehensive review of classical models and issues for distribution planning. However, the classic
methods need high computational cost. The metaheuristic-based optimization algorithms can find a
global solution of the problem with an acceptable computation cost. Genetic algorithm (GA), particle
swarm optimization (PSO), Tabu search (TS), simulated annealing (SA), ant colony optimizations
Energies 2019, 12, 3052; doi:10.3390/en12163052
www.mdpi.com/journal/energies
Energies 2019, 12, 3052
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(ACO), and branch exchange (BE) are the most popular types of metaheuristic algorithms that are used
for distribution network planning in [9–15]. In addition, different hybrid methods have improved
in [16–22] for optimal distribution networks. Although, the metaheuristic algorithms have a good
ability to find the optimum solution, they may find the global optimum point when the problem
is more complex. The optimal planning of the distribution network becomes a complex problem if
the dimension of the network is high. Therefore, the conventional modern metaheuristic algorithms
cannot find the global optimum solution for high dimension distribution networks. In this paper, a new
improved hybrid algorithm is proposed that includes PSO and TS algorithms and some adaptations
are also taken into account. The proposed modified optimization algorithm is applied to the planning
of a single stage distribution network in different scenarios. The mathematical analysis and simulation
results show that the proposed modified optimization algorithm, in comparison with other algorithms,
has better performance especially in the high dimension distribution network.
2. Problem Statement
This section explains the mathematical formulation of the distribution network expansion planning.
The location and type of feeders as well as the allocation of new substations and/or the capacity
increment of existent substations are the results of the defined problem. The cost of objective function
is minimized under technical and economic constraints. The objective function includes the investment
cost and power loss cost. The optimal distribution network must supply the required load demand and
keep the grid nodes’ voltage within the allowable boundary. In addition, grid feeders’ capacity should
not be violated and the configuration of the network must be radial. Therefore, the mathematical
formulation can be written as Equation (1). The technical constraints are presented as Equations (2)–(5).
Objective function:
F=
P
xi,a × Li,a × CCon,i,a
P
+ y j × CSub (S j , SIns,j )
j
P P
P
+ {[ PLoss, f ,i + PLoss,s,j ]
i
t
(1)
i
j
1+Intr
×CE × ( 1+In f r ) × (1 + Incr)2(t−1)
Constraints:
X
Ps,i ≥ PLoss +
i
X
Lj
× 8760}
(2)
j
Vmin ≤ Vi ≤ Vmax
(3)
0 ≤ xi,a × Si,a ≤ Smax,i,a
(4)
Nb = Nl
(5)
where CCon,i,a = 0 for existent feeders and CSub is a discrete function.
In a radial configuration, each load node should be fed by only one substation. So feeding a load
from two different substations makes a loop in the network which is not allowed.
3. Proposed Optimization Algorithms
Although, the metaheuristic algorithms have a good ability to find the optimum solution, they may
find the global optimum point when the problem is more complex. The optimal planning of the
distribution network becomes a complex problem if the dimension of the network is high. Therefore,
the conventional modern metaheuristic algorithms cannot find the global optimum solution for high
dimension distribution networks. In this paper, a new improved hybrid algorithm is proposed that
includes PSO and TS algorithms and some adaptations are also taken into account. The details of the
proposed optimization algorithm are as follows.
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3.1. Tabu Search
TS is an intelligent search algorithm that is based on the neighborhood evaluations. Starting from
an initial solution, in each iteration the best neighbor solution is selected to move, although it increases
the objective function of the current solution. So, TS includes the hill-climbing movements, too.
Moreover, a Tabu list, as a memory, is used to prevent cycling movements. In practice, the probability
local search is used in TS and therefore there is no guarantee to achieve a global optimum especially in
large dimension problems [9]. However, TS is not used merely for the distribution network planning
problem and it needs improvements or commonly is used as an auxiliary algorithm [17,21].
3.2. Particle Swarm Optimization
PSO is a swarm intelligent algorithm that is based on the movement of some groups of particles
which share their explorations among themselves. This technique is motivated by simulation of
social behavior. Instead of using evolutionary operators to manipulate the individuals like in GA,
each individual in PSO called particle flies in the searching space with a velocity, which is dynamically
adjusted according to its own flying experience and its components’ flying experience. Assuming
that the searching space is n-dimensional, the ith particle of the swarm is presented by the vector
xi = (xi1 , xi2 , . . . , xin ) with a flying velocity vi = (vi1 , vi2 , . . . , vin ). Each particle maintains a memory of
its previous best position and swarm remembers its global best position. The particles are manipulated
according the Equations (6) and (7).
vi j = wvi j + c1 r(xpbest,i j − xi j ) + c2 r(x gbest,j − xi j )
(6)
xi j = xi j + vi j
(7)
where i = 1, 2, . . . , Np and j = 1, 2, . . . , n.
However, in discrete space Equation (7) is modified as Equation (8).
xi j = 1 if r < sigmoid(vi j )
where:
sigmoid(vi j ) =
1
1 + e−vij
(8)
(9)
From Equation (6) it is found that the particles’ velocity will be slowed down and the swarm is
congregating for x gbest if the latter two terms are zero. As a result, the swarm global best individual
x gbest may not be updated for a long time and the algorithm converges to a local optimum, especially
in smaller problems. So sometimes the restricted velocity is introduced [23] as Equation (10).
vmin ≤ vi j ≤ vmax
(10)
Moreover, using a dynamic mutation operator of GA in PSO, as a hybrid GA/PSO, instead
of the restricted velocity is introduced [22]. In such hybrid algorithm, the mutation rate must be
decreased during iterations because the constant mutation rate may disturb the converging. However,
the mutation operator is commonly needed in small size problems.
This section includes three parts: The first part describes the proposed algorithm for modification
of the particles’ movement; the proposed hybrid algorithm is explained in part two and the proposed
algorithm for expansion of substations with feeders is presented in the third part.
3.3. Proposed Movement Algorithm
The proposed algorithm is based on two new controller parameters (z and q0 ) and a Tabu list
(T) of branches as candidate feeders of the network. In this algorithm, the removing-from-network
movements are controlled and the left movements are restricted by the constraints of the problem.
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In the first step, T is empty and the velocity vector (vi ) of the moving individual (xi ) is calculated by
Equation (11).
1
2
(11)
vi j = wvi j + c1 r xi j − xpbest,i j +c2 r xi j − x gbest,j
In the next step, all selected branches of xi with high velocity and not the selected branches of xi
with low velocity are moved to the Tabu list with a probability of q0 if they are not selected in the best
global position. Controller q0 is a constant parameter that determines the importance of intensification
1 it is
and diversification of the algorithm. The criterion of velocity is: sigmond
− z, where 0 ≤ z0≤ 1 and
sigmond
a constant parameter that controls the intensification of the Tabu list.
Finally, a new individual is generated of not-Tabu branches and it is substituted as a moved
individual if it satisfies the constraints. The flowchart of this algorithm is presented in Figure 1.
Start
Input xi position
Empty T
Calculate vi using Eq. (11)
j=1
Yes
Sigmoid-z < r
Xij=0
No
Yes
Sigmoid-z > r
No
No
Yes
Xgbest,j = 0
or
q0 < r
Yes
Move xij to T
Yes
Xgbest,j = 0
or
q0 < r
No
No
j=j+1
No
j<n?
Yes
H=A-T
Generate a new xi using components of H
Run load flow
No
Are Eqs. (2)-(5)
satisfied?
Yes
Stop
Figure 1. Flowchart of proposed algorithm for movement of particles.
3.4. Hybrid TS/PSO Algorithm
In this paper, a PSO-based hybrid TS/PSO is proposed for the distribution network expansion
planning. The hybrid algorithm includes the neighborhood search for the best position of each group
xpbest,i (that: i = 1, 2, . . . , N g ) and set the best neighbor as the new best position of group, if it satisfies
the constraints and also it improves the previous best position. This process repeats in each iteration
after moving the particles. Flowchart of the hybrid algorithm is presented in Figure 2.
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Start
Generate initial particles
xi (i = 1,2, …, Np) randomly that satisfy Eqs. (2)-(5)
Select xpbest and xgbest
Calculate velocity of all positions by Eq. (11)
Move particles using the proposed algorithm
Update xpbest and xgbest
Generate d neighbors of
xpbest,i (i = 1,2, …, Ng) that satisfy Eq. (2)-(5)
Select the best neighbor xnbest,i of xpbest,i (i = 1,2, …, Ng)
No
F(x nbest,i ) < F(x pbest,i )?
Yes
&
x pbest,i = xnbest,i
F(x pbest,i ) = F(x nbest,i )
Update xgbest
Is stop criterion
satisfied?
No
Yes
Stop
Figure 2. Flowchart of the hybrid TS/PSO algorithm.
Local searches in TS/PSO can be supposed as intelligent mutations that control the diversification
and they do not disturb the converging. This may be important in small dimension problems or in the
converging conditions for large problems.
3.5. Proposed Algorithm for Substations Expansion
In the distribution network expansion planning because of the load growth, it may be needed
to add some new HV/MV substations to the network. Added substations are selected among some
candidate substations. In this paper, a two-phase and one-switching algorithm is proposed to expand
both substations and feeders of the distribution network. In phase one, each group of particles loads a
random feasible configuration of substations and evaluates the candidate configuration of substations
{1 (1
) } {1 [1 (1
) ] }
using the proposed hybrid algorithm for feeders expansion. In this phase, q0 ≈ 0.5 and also mutation
in the substations’ states is needed to avoid the soon converging to a local optimum in the substations
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search space. In these conditions, it can be proofed that the optimum mutation rate is: mbest = N1 [24].
Sub
After enough iteration, the algorithm is switched to phase two that uses the selected information of
the first phase. In the switching step, the selected substations of the best solution resulting of phase
one are selected for all groups in phase two and all other information is reset. In the second phase,
0.5 << q0 < 1 and m = 0 and the search space is of the feeders’ configuration. Here, a step-decreasing
mutation rate is used because of the difference between two search spaces. The algorithm can be stated
as follows.
First Phase:
Search in substations’ configuration space based on feeders’ configurations search;
1.
2.
3.
Using mutation in substations
Using the proposed hybrid algorithm in feeders
Set a medium q0
Switching Step:
After enough iteration;
1.
2.
3.
Set m = 0
Set SSub,i = SSub,best (that: i = 1, 2, . . . , Np )
Reset all feeders’ positions and velocity vectors
Second Phase:
Search in feeders’ configurations space based on the best selected substations;
1.
2.
Using the proposed hybrid algorithm in feeders
Set q0 ≈ 1(depends on network size)
4. Mathematical Modeling
In this section, the mathematical model of the proposed methodology is presented. To analyze the
proposed algorithm and its advantages a parameter called success probability (PSuc ) is used. Success
probability is defined as the probability of changing at least one incorrect component and not changing
any of the correct components of a solution. Here, PSuc is defined for the best individuals of the
groups and the success at least in one of them. The similar success probability can be defined for other
individuals too; however, it is neglected in this paper. So the success probability mathematically is as
Equation (12).
PSuc = {1 − (1 − PCh )i×N g } × {1 − [1 − (1 − PCh )n−i ]
Ng
}
(12)
where i is the number of incorrect components and PCh is the probability of changing a component by
the algorithm’s operators in an iteration. Changing the probability of the improved PSO algorithm is
as Equation (13).
PCh = σ × (
1
1
− z) + (1 − σ) × (
− z) × (1 − q0 )
1 + e−v1
1 + e−v2
(13)
where 0 ≤ σ ≤ 1 is the variance of the all best solutions of groups from the global best solution and v1
and v2 are components of the velocity vector when xi j , x gbest and xi j = x gbest , respectively and they
are obtained of Equation (11). It is found of Equation (11) that: vmin ≤ v2 ≤ v1 ≤ vmax . Whereas the
initial population of the PSO algorithm is generated randomly so we expect that i = n2 in Equation (12)
for the first iteration and i is decreased partly during the iterations of the algorithm.
min
2
1
max
2
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0
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1
Using 0 < q0 < 1 in Equation (13) describes that if the best solution of the group is similar to the
global best, it is changed with lower probability. As a result, if the parameter q0 increases the variance
(1 0 )
decreases as Equation (14).
σ ∝ ( 1 − q0 )
(14)
Equations (12)–(14) can describe the influence of problem dimension on the success probability of
PSO. Figure 3 presents the influence of n on the PSuc of
PSO
improved
.5
0.1and the 0proposed
10 conventional
1 5
2
n
10 as
PSO using Equations (12)–(14), N g = 10, v1 = 5, v2 = 0.1, z = 0.5 and i = 10
in Equation (12)
normal conditions.
Figure 3. Influence of problem dimension and q0 on success probability of the proposed PSO.
Figure 3 shows that PSuc of the conventional PSO (q0 = 00) is high enough for low dimension
0
networks (5 < n < 50), but is poor when the dimension is high. Increasing q0 in the improved PSO
5
50
algorithm increases the success probability and shifts the maximum point of it to the higher dimension
problems. Therefore it is more powerful for large-scale distribution networks planning. However,
decreasing the controller parameter z may disturb the success of the algorithm. The influence of
parameter z on the success probability in various problem dimensions is presented in Figure 4, using
n
Equations (12)–(14), N g = 10, v1 = 5, v2 = 0.1, z = 0.5, i = 10
and q0 = 0.98.
0.5
5
0
.
1
0.98
10
1
2
0
10
Figure 4. Influence of parameter z on the success probability of the proposed algorithm in various
problem dimensions (q0 = 0.98).
0
0.98
The figure shows that the success probability in high dimension problems (200 ≤ n ≤ 300) is poor
200
300
if parameter z decreases (0 ≤ z ≤ 0.4). Increasing z to 0.6, improves the success probability
in n = 300.
0
0.4
300
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However, increasing z is not needed in low dimension problems. Additionally, it is found of Equation
(13) that z should be increased if q0 is decreased.
Adding local searches to the improved PSO increases the probability of changing incorrect
components and it decreases the probability of changing correct components. The probability of
changing a component in one neighborhood search, using the branch exchange algorithm, is equal to n2 .
So, the probability of not changing a correct component in any d local searches, after it was changed by
d
the PSO, is equal to (1 − n2 ) and the probability of not changing an incorrect component in any d local
d
searches, after it was not changed by the PSO, is equal to (1 − n2 ) . As a result, the success probability
of hybrid algorithm is as Equation (15).
d i×N g
PSuc = {1 − [(1 − PCh ) × (1 − n2 ) ]
{1 − [1 − (1 − PCh × (1 −
}×
2 d
n) )
n−i N g
(15)
] }
where PCh is obtained of Equation (13).
To analyze the advantage of using local searches in PSO, in convergence conditions for high
dimension problems, suppose that in Equations (13) and (15), we have n >> 1, i << n, σ << 1, d ≥ 1,
q0 ≈ 1 and so PCh ≈ ( 1+e1−v2 − z) × (1 − q0 ) << 1, then the success probability is as Equation (16).
PSuc
2id
i
[N g − (1 − q0 ) × (n − 2d)] 2dN g ( )
n
n
(16)
where the following approximation is used:
(1 + α)β ≈ 1 + αβ
(17)
where α << 1 << β.
It is found of Equation (16) that in converging conditions ( ni << 1), if d = 1 then PSuc << 1,
but using d >> 1 increases the success probability. As a result, increasing the local search size is more
efficient than increasing the group size, until d < N g . Figure 5 presents the influence of the local search
n
50
.
size on PSuc in various problem dimensions using Equations (13)–(15) where i = 50
Figure 5. Success probability of hybrid algorithm versus problem dimension in various local search sizes.
Figure 5 shows that using TS in PSO increases the PSuc in all problem dimensions; however the
increment is less in larger problems.
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5. Case Study
In order to show the capability of the proposed algorithm for solving the distribution network
expansion planning problem, three different distribution networks as case studies are presented.
The first case is a 23-bus network with one existent HV/MV substation, for expanding only the feeders.
The second case is a 48-bus network with one existent HV/MV substation, for expanding only the
feeders. The third case is a 71-bus network with one existent and five candidate HV/MV substations
and it includes both substations and feeders expansion. Economic data are shown in Table 1 and more
details of the cases are presented in Table 2. More details about the case studies including networks
configuration, candidate substation positions, etc. can be found in [25,26].
Table 1. Economic data for numerical studies.
Economic Parameters
Value
Electrical Energy Cost ($/MWh)
Inflation Rate
Interest Rate
50
0.12
0.07
Table 2. Information of three cases for numerical studies.
Parameters of Cases
Case 1
Case 2
Case 3
Voltage Level (kV)
Number of Load Nodes
Number of Existent Substations
Number of Candidate Substations
Number of Existent Feeders
Number of Candidate Feeders
Average Load per Node (kW)
Power Factor
Annual Increment Load Rate
Time Period (years)
33
22
1
0
8
41
320
0.9
0.01
20
20
47
1
0
16
119
315
0.9
0.03
15
20
65
1
5
24
178
800
0.9
0.03
15
The proposed algorithm is applied to the three cases in three statuses. In the first status,
conventional PSO is applied to the cases. The Improved PSO (IPSO) without hybridization is applied in
the second statuses. The hybrid Tabu Search/Improved PSO (TS/IPSO) is applied to the cases in the third
status. Moreover, the proposed two-phase algorithm in Section 3.5 for substations expansion is used in
the third case for all statuses. Controller parameters of algorithms are presented in Table 3, regarding
the mathematical analysis in Section 4. The results of each algorithm are presented in Figure 6, which
shows the objective function and convergence time of the algorithms in three case studies.
Table 3. Controller parameters of algorithms in three case studies.
Value
Controller Parameter
w
(c1 , c2 )
Ng
(vmin , vmax )
m
Z*
q0 *
d*
Case 1
Case 2
Case 3
0.9
(1, 0.1)
10
(0.1, 5)
0
0.6
0.8
5
0.9
(1, 0.1)
10
(0.01, 10)
0
0.5
0.95
10
0.9
(1, 0.1)
10
(0.01, 10)
0.1
0.5
0.5, 0.995
10
* All of them are defined in the Nomenclature.
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Figure 6. Comparison of conventional PSO, Improved PSO (IPSO) and hybrid Tabu Search/Improved
PSO (TS/IPSO) algorithms in three case studies.
The results show that objective functions of PSO, IPSO and TS/IPSO are the same in case 1
(where
41 n = 41), but the convergence time is decreased by improvement and hybridization. However,
the objective function is decreased by IPSO in cases 2 and 3 (where n =119
119 and n =183
183, respectively)
and it is decreased more by the hybrid TS/IPSO. As a result, the proposed algorithm is more efficient in
larger-scale distribution networks expansion planning.
To show the advantage of the proposed algorithm compared with the other algorithms, six modern
algorithms are applied to the same three case studies which are described in Table 2 and the results are
compared with the proposed hybrid TS/IPSO algorithm’s result. Figures 7–9 present the comparison of
the objective functions obtained of SA [12], TS [23], improved GA (IGA) [18,19], hybrid SA/TS [20],
hybrid TS/IGA [21], hybrid GA/ACS [13] and proposed algorithm in three case studies.
Figure 7. Comparison of the proposed algorithm with other modern algorithms in case 1.
Figure 8. Comparison of the proposed algorithm with other modern algorithms in case 2.
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Figure 9. Comparison of the proposed algorithm with other modern algorithms in case 3.
The comparisons show that all algorithms obtain the same solution in case 1, however only hybrid
TS/IGA and the proposed TS/IPSO algorithms obtain the global optimum in case 2 which is larger than
case 1. According to Figure 9, the proposed algorithm obtains the best solution among all algorithms
in case 3 which is larger than case 2. This figure also shows the efficiency of the proposed two-phase
algorithm for substations and feeders expansion in large-scale networks compared with the other
modern algorithms.
6. Conclusions
Particle swarm optimization is a modern algorithm which is extremely applied to various
combinatorial optimization problems. However, it needs diversification in low dimension problems
such as HV/MV substations expansion and it needs intensification in high dimension problems such as
the distribution feeders expansion. In this paper, a modified PSO algorithm which is improved by
increasing diversification in substations expansion and increasing intensification in feeders expansion,
is proposed. Moreover, local searches, as intelligent mutations, were added to the modified PSO
to improve the algorithm in converging conditions for feeders expansion. As a result, particles’
movements of the improved PSO move the hybrid algorithm to proper areas and local searches find
the best point of the area. Finally, combining the algorithms for substations and feeders expansion in
the proposed serial two-phase algorithm makes the proposed PSO-based hybrid TS/IPSO algorithm a
powerful method for large-scale distribution networks expansion planning. Mathematical analysis
and numerical comparisons show that the proposed TS/IPSO algorithm has better performance in
comparison with other algorithms including SA, TS, IGA, SA/TS, TS/IGA, and GA/ACS. Moreover,
the ability of the proposed methodology is more highlighted in case studies 2 and 3, where the
dimension of the electric network is high.
Author Contributions: A.A. has modelled mathematically, simulated and analyzed; A.E. and A.M. have modelled
economic aspects of the problem, edited and proofread.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
Nomenclature
A
CCon,a
CSub
CE
c1 , c2
d
F
H
Incr
In f r
The list of all candidate branches
Fix cost of installation feeder of size a ($/km)
Cost function of installing a substation ($)
Electrical energy cost ($/MWh)
Positive constants as learning factors
Neighborhood search size for TS
Objective function ($)
The list of allowed branches to be added
Annual increment load rate
Inflation rate
Energies 2019, 12, 3052
Intr
Lj
Li,a
m
mbest
n
Nb
Ng
Nl
Np
NSub
PSuc
PS,i
PLoss
PLoss, f ,i
PLoss, f , j
q0
r
SIns, j
Si,a
Smaxi,a
Sj
SSub,i
SSub,best
T
t
Vi
vi
Vmax
vmax
Vmin
vmin
w
xi
xi,a
x gbest
x gbest, j
xpbest,i
xnbest,i
yj
z
σ
α, β
Interest rate
Load demand of node j (MW)
Length of feeder i of size a (km)
Mutation rate for substations’ states
Optimal mutation rate for substations’ states
Number of candidate components (Dimension of the
problem)
Number of branches in the candidate network
Number of all groups
Number of load nodes
Number of all positions
Number of candidate HV/MV substations
Success probability
Capacity of substation i (MW)
Total power loss in network (MW)
Power loss of feeder i (MW)
Power loss of substation j (MW)
Real variable that determines the relative importance
of the exploitation over the exploration
Random function generating random numbers
uniformly distributed within the range [0, 1]
Capacity of substation j that was installed before
(MVA)
Power flow of feeder i of size a (MVA)
Maximum allowed power flow of feeder i of size a
(MVA)
Total capacity of substation j (MVA)
Substations’ state of solution i
Substations’ state of the best solution
Tabu list of branches
Number of operation year
Voltage of node i (p.u.)
i-th velocity vector
Maximum allowed operation voltage (p.u.)
Maximum allowed velocity
Minimum allowed operation voltage (p.u.)
Minimum allowed velocity
Inertia weight
i-th particle position vector
Binary decision variable associated to the installation
of feeder i of size a
Global best position vector
j-th component of the global best position vector
Previous best position vector of particle i
The best neighbor position of particle i
Binary decision variable associated to the installation
of substation j
Real variable that controls the Tabu list of PSO
Variance of particles population
Fixed coefficients
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