Energy and Power Engineering, 2013, 5, 975-979
doi:10.4236/epe.2013.54B187 Published Online July 2013 (http://www.scirp.org/journal/epe)
Distribution Network Expansion Planning Based on
Multi-objective PSO Algorithm
Chunyu Zhang, Yi Ding, Qiuwei Wu, Qi Wang, Jacob Østergaard
Center for Electric Power and Energy, Technical University of Denmark, Copenhagen, Denmark
Received January, 2013
This paper presents a novel approach for electrical distribution network expansion planning using multi-objective parti-
cle swarm optimization (PSO). The optimization objectives are: investment and operation cost, energy losses cost, and
power congestion cost. A two-phase multi-objective PSO algorithm was proposed to solve this optimization problem,
which can accelerate the convergence and guarantee the diversity of Pareto-optimal front set as well. The feasibility and
effectiveness of both the proposed multi-objective planning approach and the improved multi-objective PSO have been
verified by the 18-node typical system.
Keywords: Distribution Network Expansion Planning; Two-phase; Multi-objective PSO
The restructure and deregulation of the global power
industry have introduced fundamental changes to the
practices of power system planning. Conventional single
optimization objective approach such as the minimiza-
tion of the investment cost is no longer suitable for the
new market operation environment. The multi-objective
planning approach has become necessary in order to take
into account new problems caused by the competitive
power market environment [1,2].
Over the past decade, a large amount of researches and
literatures have been accomplished in multi-objective
distribution network expansion planning approach. A
detailed review of different methods can be found in
The distribution system planning problem dimension
increases with number of nodes. Normally, numerical
optimization tools such as nonlinear programming (NLP)
 and dynamic programming  have been used to
solve this problem with lower node systems. In multi-
objective problems, there are some specific disadvan-
tages in using these analytical solution strategies, i.e.,
curse of dimensionality, non-differentiability, discon-
tinuous objective space etc . Moreover, to get a set of
solutions (as with Pareto-optimality principle), any nu-
merical method requires several trial runs.
Considering the complex solution space, non-convex
and nonlinear mixed integer objective functions, the so-
lution of multi-objective distribution network planning
problem is difficult to gain by many traditional optimiza-
tion methods. Therefore, several intelligent algorithms
have been used to enhance the performance of distribu-
tion expansion planning process, including greedy algo-
rithms, genetic algorithms, particle swarm optimization,
and evolutionary optimization. Particle swarm optimiza-
tion (PSO) is one of the most widely used multi-point
search algorithms using stochastic behavior. PSO is de-
veloped by inspiring with social behavior that is ob-
served in nature such as flocks of birds and schools of
fish . Recently, PSO has been gained more and more
attention, due to its powerful search ability in function
optimization. PSO is adopted to the multi-objective op-
timization widely. The main differences from single-
objective PSO are the proper introduction of archive to
reserve Pareto-optimal candidates and the appropriate
selection of guide particles for multi-objective optimiza-
tion . For example, Mostaghim and Teich  pro-
posed a Sigma method. This method selected the best
local guides for each particle, which mainly improved the
convergence to the Pareto front. But, the method could
not gain good convergence rate and uniform diversity
simultaneously. Coello and Lechgua  introduced the
global guide selection method based on Pareto optimality
and hypercube in the objective function space to main-
tain the diversity of the particles. However, its conver-
gence rate was low . The performance of multi-ob-
jective PSO optimization algorithm needs to be im-
In this paper, a distribution network expansion plan-
ning approach is proposed to optimize three objectives:
the investment and operation cost, energy losses cost,
Copyright © 2013 SciRes. EPE
C. Y. ZHANG ET AL.
and power congestion cost. Accordingly, a two- phase
multi-objective PSO is also proposed, which accelerates
the convergence and guarantees the diversity of Pareto-
optimal front set as well. Case study based on the 18-
node system is conducted to demonstrate the feasibility
and effectiveness of both the proposed planning approach
and the improved multi-objective PSO algorithm.
2. Problem Formulation
2.1. Multi-Objective Planning Model
The power system restructuring forces a change in duties
and objectives of traditional planning and it compels to
consider several objectives that are in mutual conflict.
The proposed planning model minimizes different cost
functions related to the cost of investment and operation
s1(x), cost of energy losses s2(x) and cost of power con-
gestion s3(x) in the form of multi-objective optimization
where g(x) and h(x) are equality and inequality con-
straints of the problem.
2.2. Investment and Operation Cost
The total cost function (s1) for investment and operation
of DG units is given in (2). The costs of installing DGs in
candidate buses of a distribution network are considered
as follow. It is assumed that DG units can be installed in
all load buses; however, the best sites are determined
according to the optimization process. Yearly investment
cost of each technology is determined based on discount
rate and payment period according to (3). As an incentive
program for renewable sources, this parameter is calcu-
lated using a low discount value. Besides, feed-in tariffs
increase the capacity factor o of DG technologies.
Cj DGijCj DGijjj
where ICj and OCj are the yearly investment and hourly
operating costs of jth technology of DG-unit, respec-
tively, TICj is the total investment cost of jth DG tech-
nology, d and t are, respectively, discount rate and pay-
ment period, N is the number of load buses in the distri-
bution system; PDG,i j is the capacity of jth technology of
DG-unit at bus i, T is yearly operating hours, Aj is the
availability factor related to jth technology; and aj is the
average capacity factor of jth DG technology considering
incentive effects of feed-in tariff policy.
2.3. Energy Losses Cost
This objective function (s2) attempts to minimize the
total cost of the energy losses in the distribution network
due to installation of DG units. This function is strongly
related to the locations of DG units in the distribution
system. The power flow in the feeder connecting buses i
and j is used to formulate the energy loss function as fol-
where N is the total number of buses in the distribution
system; V is the bus voltage, Zij and Pij are the impedance
and power flow of branch i–j, respectively; Pf is the sys-
tem power factor and k is the expected price of electric-
2.4. Power Congestion Cost
The congestion cost (s3) is given by
where li , lj is the locational marginal price at bus i, j,
which are the Lagrange multipliers or shadow prices of
the power flow constraints. Pf is the large penalty factor.
W is the total artificial generator (shed load) in normal
3. Two-phase Multi-objective PSO
3.1. Classical PSO
Particle swarm optimization is a relatively new technique
for optimization of continuous non-linear functions. The
individuals in a PSO have their own positions and ve-
locities. Each particle moves in the search space with
velocity, which is dynamically adjusted and balanced
based on its own best movement (pb) and the best
movement of the group (gb). If the best previous position
of the i-th particle is recorded and represented as pi=( pb1,
pb2, pb3, …, pbd) and the global best position is recorded
and represented as gi= (gb1, gb2, gb3, …, gbd), the modifi-
cation of velocity (vid) and position (xid) of the i-th parti-
cle can be calculated by the current velocity and the dis-
tance from pi to gi as shown in the following formulas,
112 2idii idi id
where w is known as the inertia weight, c1 and c2 are two
Copyright © 2013 SciRes. EPE
C. Y. ZHANG ET AL. 977
positive constants, r1 and r2 are random numbers in the
range of [0, 1].
The important part in multi-objective PSO is to deter-
mine the global best particle gi for each particle i of the
population. In single objective PSO, the global best par-
ticle is determined easily by selecting the particle in the
best position. In multi-objective optimization problems,
each particle of the population should select its global
best particle from the set of Pareto-optimal solution.
3.2. Two-phase Multi-objective PSO Algorithm
1) The steps of two-phase multi-objective PSO.
Step 1: Initialize the parameters of two-phase multi-
objective PSO, including size of the swarm P, capacity of
the archive A, PSO coefficients w, c1 and c2, and maxi-
mum iterations Z, t=0;
Step 2: Randomly initialize the position and velocity
of each particle in set P and set A. Set the initial position
as the individual best position pi of each particle;
Step 3: For t=1 to Z,
a) Update archive,
b) Select the global bests (gi) from A for each particle
in the set P based on the strategy of two-phase guided,
c) Update position and velocity of every particle ac-
cording to the (7)−(8),
d) If t < 0.85Z, the particle is mutated according to the
e) Each particle in set P has a new location, if the cur-
rent location is dominated by its personal best location
(pi), then the previous location is kept, otherwise, the
current location is set as the personal best location. If the
particles are mutually non-dominated, one particle is
2) Strategy of two-phase guided.
The selection of gi is the key of PSO algorithm. For
multi-objective problems, gi is related to the convergence
speed of the algorithm and the diversity of solution. A
two-phase guide gi selection strategy is proposed in this
paper. Sigma method is adopted to the first half of the
evolution because of its fast convergence rate, and an
approximated front of Pareto is found. And then, a pro-
posed ideal optimal particle method is adopted to the
remaining half of the evolution, which will promote the
diversity of solution.
a) Sigma method
The Sigma method involves choosing the guide parti-
cle based on the similarity of the angular position in ob-
jective space. This method was proposed by Mostaghim
and Teich , which was proved to have good conver-
gence speed. For two-dimensional objective space, δ is
The guide particle of particles in set P is the particle
that has the closest δ in set A.
b) Optimal particle method
’ is regarded as the optimal value of the jth
objective, and the point (f1, f2, …, fn) is called ideal point.
The optimal particle xI of the following equation is the
ideal optimal particle
For a complex multi-objective optimization problem,
the optimal value of the each objective and the corre-
sponding optimal point tends to be unknown advance. In
this paper, the previous optimal value of each objective
in iterative process is regarded as fj
The process in which the guide particles are selected
from set A is described as the following steps:
Step 1: Calculate the optimal value fj
’ and the corre-
sponding optimal particle x’ in set A;
Step 2: According to (10), obtain the ideal optimal
Step 3: Compute the average of the optimal particle x’
and the ideal optimal particle xI, and regard and average
value as gi.
The value gi represents the optimal information of the
each objective, achieves the global optimization of the
algorithm, and is helpful to achieve the local depth
c) Selecting guided particle strategy
Sigma method which has faster convergence rate is
adopted to the first n1-th generation of the evolution, and
the ideal optimal particle method is adopted to the re-
maining (Z−n1)-th generation.
3) Strategy of mutation.
The use of mutation operator in this Multi-objective
PSO is needed because the algorithm may converge to
local optimal fronts. Mutation probability (Pm) is reduced
with the iteration of the algorithm according to
where Cg is the number of current generation. For each
particle, the variable mr is a random number in the range
of [0, 1].
If mr<Pg, the particle is randomly selected for mutation
where μ points out the direction in mutation and θ con-
trols the distance covered by a jump. In this paper, μ=3
and θ is set as ±1 randomly. If the solution is beyond its
boundary by mutation, it is moved to the corresponding
Copyright © 2013 SciRes. EPE
C. Y. ZHANG ET AL.
Copyright © 2013 SciRes. EPE
5. Conclusions 3.3. Program Flow
The major multi-objective distribution network problem
modules and the general flow of the program are shown
in Figure 1. The Fuzzy satisfying decision making ap-
proach  is introduced in this program.
A multi-objective distribution network expansion ap-
proach is proposed, the objectives are the investment and
operation cost, energy losses cost, and power congestion
cost. And its solving algorithm based on the two-phase
multi-objective PSO is also proposed in this paper.
4. Case Study
Case study has been carried out on the 18-node system,
to prove the effectiveness of both the proposed multi-
objective planning approach and two-phase multi-objec-
tive PSO. The parameter details of the 18-node system
can be found in . The system has 28 right-of-ways,
the active power transmission limit of each line is 50MW,
and line cost is $130,500/km.
The best three planning schemes of the 18-node sys-
tem are presented in detail in Figure 2 and Table 1,
which all indicate the network of the 18-node system has
a relative tightly linked structure. Further comparison
between M1 and M2 shows that, under the almost same
total cost, M2 is definitely better than M1 due to its lower
energy losses cost and power congestion cost, with
strong future adaptability. Compared with M3, M2 has
extremely high power congestion cost and obviously
double energy losses cost, but M3 with a notable increase
on total cost. Figure 1. M multi-objective distribution network flow.
Figure 2. Topology of the expanded network of the 18-node system.
Table 1. Best planning schemes of the 18-node system.
Schemes M1 M
New added distribution network lines 7-9, 9-10 8-9, 9-10(2), 1-2, 6-14, 7-9,
7-13, 16-17, 14-15
1-11, 4-16, 5-12, 6-13, 6-14, 7-15,
10-18, 11-12, 12-13, 14-15, 16-17, 17-18
Investment and operation cost (×103$) 2043.01 10452.73 12250.56
Energy losses cost (×103$) 5583.37 572.25 282.45
Power congestion cost (×103$) 2832.43 252.40 34.66
Total cost (×103$) 10458.81 10397.38 12837.67
C. Y. ZHANG ET AL. 979
The planning results of the 18-node system show that,
for a practical system, the proposed multi-objective dis-
tribution network expansion method can effectively en-
hance the distribution capacity by adding specific new
lines under the variety conditions of future uncertainties.
Considering efficiency, reliability, and economic, the
best planning schemes can be put forward by the pro-
posed two-phase multi-objective PSO, which shows its
superiority as well.
Further research can focus on the multi-stage and mul-
ti-objective model, which should consider the uncertainty
of the bidding parameters and other uncertain factors in
distribution expansion problem.
The authors gratefully acknowledge the financial sup-
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