Effect of inertia weight functions of pso in optimization of water distribution network

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online), Volume 6, Issue 5, May (2015), pp. 43-51 © IAEME
43
EFFECT OF INERTIA WEIGHT FUNCTIONS OF PSO IN
OPTIMIZATION OF WATER DISTRIBUTION NETWORK
Minakshi Shrivastava1
, Dr. Vishnu Prasad2
, Dr. Ruchi Khare3
1,2,3
Maulana Azad National Institute of Technology, Bhopal, 462051, India
ABSTRACT
Water distribution in adequate quantity at required pressure is the prime requirement of design
of network. It is found from the literature, that stochastic methods gives better results in optimizing
the water distribution network (WDN) than deterministic methods. The optimize solution obtained
from any stochastic method depends on the parameter values associated with that method. In the
present work a program is developed in MATLAB environment to optimize the water distribution
system using Particle Swarm Optimization (PSO) technique. The analysis is carried out using finite
element method. The validity of developed program is demonstrated by comparing results with a
benchmark water distribution network. Different values of inertia weight function in Partial swarm
optimization has been used to study the effect of this variant on the cost of water distribution network.
Key words: Optimization; Water Distribution Network; Particle Swarm Optimization; Inertia Weight
Function
1. INTRODUCTION
A safe supply of potable water is the basic necessity of mankind in the industrialized society,
therefore water supply systems are the most important public utility. A huge amount of money is
spent every year around the world for providing or upgrading drinking water facilities. The major
share of capital investment in a water supply system goes to the water conveyance and water
distribution network. [Wu et al,2001]. Therefore it is very important to optimized the network. The
methods used for optimization of water distribution network may be deterministic or stochastic
[Keedwell et al,2004l].Optimization of the network aims to find out the best diameter size from the
available commercial diameter sizes to get the minimum cost of the network within head and velocity
constraints. Conventional methods gives global optimal solution but they are tedious and time
consuming while the stochastic method are more efficient methods[[Afasher et al, 2003]].
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ISSN 0976 - 6480 (Print)
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Volume 6, Issue 5, May (2015), pp. 43-51
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44
It can be seen from the previous work that stochastic methods work successfully for water
distribution network. Various optimization techniques e.g. Genetic Algorithm, Ant Colony
optimization, artificial neural network etc have been successfully used to optimize the water
distribution network but they have their own limitations of rate of convergence, difference between
the shortest and the longest optimization response time etc [Izquierdoa, 2014]. Optimization of pipe
network by using GA also has limitations like poor convergence, unrealistic global optimum solution
etc. To overcome theses limitation, James Kennedy and Russell Eberhart [1995] developed the
concept of Particle Swarm Optimization (PSO) to overcome the limitations of GA.
Particle swarm optimization is a robust stochastic technique for optimization [Mahor et al,
2009]. In this method, the co-ordinates of each particle represent the possible solution and after each
iteration, the particle moves towards optimal solution. Particle swarm optimization is comparatively
topical, very simple and has very fast rate of convergence for various optimization problems.
Literature reveals that PSO is successfully worked for optimization of water distribution network. It is
seen that optimization of pipe network using PSO has been done taking a particular inertia weight
function and effect of other weight function on the optimized results are not studied.
In present work coding is done for optimizing water distribution network using PSO. The
network analysis is carried out by finite element method using Darcy Weishback head loss equation
the results are validated for two loop water distribution network. The results obtained are having good
agreement with the results obtained for the same network by using other stochastic methods of
optimization.
The inertia weighted function ‘w’ is one of the important parameter in particle swarm
optimization [Bansal et al, 2009, 2011]. In the present paper the cost optimization using eight
different variants of inertia weight function are compared and presented in graphical and tabulated
form.
2. PROBLEM FORMULATION
Objective function: It is to minimize the cost of water distribution network by selecting
commercially available pipe diameter and the objective function is given as;
1
( , )
m
i
MinZ C L D
=
= ∑
Where Z = total cost ,m = no of pipe in the network C= cost of the pipe having length L and
diameter D. The constraints to be satisfied for minimizing the cost of network are;
Constraint 1: Diameter constraint
The diameter of the pipe should be selected from the available commercial pipe diameters.
Commercially available pipe size and cost of unit length is given in table 1.
Constraint 2: Head constraint
The head available at each junction must be greater than the minimum required head to supply
sufficient amount of water at the junction.i.e.
minkH H≥
where k=1,2,3………………..n
Apart from these two constraints, the network is analysed to meet the following requirement.
1. The continuity equation i.e the amount of water entering to the junction must be equal to the
amount of water leaving from the junction.

45
1
j n
j
Q
=
=
∑ = 0 ……… (a)
Where n=no of pipe meeting at any junction
2. Net head loss within the loop must be zero.
1
p nop
LP
p
H
=
=
∑ = 0 ……….. (b)
Where nop= no of pipes in any loop
Particle Swarm Optimization
Partical swarm optimization is bio inspired met heuristic technique for optimization. James
Kennedy and Russell Eberhart developed the concept of PSO in 1995 [Shrivastava et al, 2014]. In this
method the co-ordinates of each particle represents the possible solution. After each iteration the
objective function is evaluated to help particle to move towards optimal solution
In this method the initial population is generated randomly and based on the objective function
it is updated for the best optimal solution. If the initial position of the particle is xi(t),then after the
next iteration it will move to the next position of xi(t+1). The particle moves toward the best optimal
solution using velocity update from vi(t) to vi(t+1) as in equation 1 and equation 2.[Izquierdoa et al,
2014].
vi(t+1) = w*vi(t) +C1*R1*(pbest- xi(t))+C2*R2*(gbest- xi(t)) -------I
xi(t+1) = xi(t)+ vi(t+1) -------II
Where C1 and C2 are the positive constants named as cognitive learning rate and social
learning rate respectively, they are used to accelerate the particle towards the optimal solution. R1 and
R2 are the uniform random number ranging from 0 to 1. 'pbest' is the best solution obtained by the
individual particle and gbest is the best value of objective function from the entire swarm. ‘w’ is the
inertia weight function.
3. METHODOLOGY
The MAT LAB environment is used for developing program for optimizing water distribution.
The program is developed in two parts. The first part calculates the head and flow in the network for
the given set of diameters by using Finite Element methods whereas the optimization is done in the
second part. The flow chart for the complete calculation is shown in fig. 1.
Table 1: Commercially available HDPE pipe dia and their cost [Babu Jinesh et al, 2009]
S.No Commercially available dia size(mm) Cost per unit length (units)
1 25.40 2
2 50.80 5
3 76.20 8
4 101.60 11
5 152.40 16
6 203.20 23
7 254.00 32
8 304.80 50
9 355.60 60
10 406.40 90
11 457.20 130
12 508.00 170
13 558.80 300
14 609.60 550

46
Iteration1:k
Fig.1 Flowchart of optimization using PSO
The two loop network used for validation is shown fig.2. It has 8 pipe and 7 nodes (including
reservoir). Head at node 1 is to be maintained as 210 m [Babu Jinesh et al, 2009]. Commercially
available diameters, used for the network are given in table 1. Elevation and demand at each junction
and pipe length for each pipe is shown in fig 2. Minimum head at each junction is required to be
maintained as 30 m.
As seen in fig.2 that all pipes of network are of constant length of 1000m. The finite element
method is used for calculating head and velocity in pipe network, using Darcy Weishbach equation:
for head loss.
hf= f*L*V2
/ (2*D*g) -------(III)
yes
yes
Initialize:Partical and their flight velocity
No
No
Analysis : analyze the network using FEM
Evaluate: fitness value
Initialize:Pbest and gbest
Iteration=1:k
Partical 1:s
Compute flight velocity
Update partical position
Analyze the network
Evaluate fitness value
Update pbest position
Update gbest position
Terminate and print the result
Is the last partical?
Termination criteria meet?

47
Fig.2 Two loop water distribution network [Babu Jinesh et al, 2009]
Optimization of the network for minimum cost is done for eight different variants of inertia
weight function of PSO [Bansal et al, 2011].Following variants of inertia weight are used in present
work:
Type Name of Inertia Weight Function Formulae of Inertia Weight
1 Constant Inertia Weight ω1=.7
2 Random inertia weight 1
(1)
.5
2
rand
w = +
3 Linear decreasing inertia weight
( max min
max1
w - w )
w = - [ ] iterationw
maxiteration
×
4 Logrithmic inertia weight 1
1
.5 (1 )
1 log( )
w
iteration
= × +
+
5.
Natural exponent inertia weight strategy
e-1 PSO
max
10
1 m in max min
( ).e
iter
iteration
w w w w
  
  
  −
  
  
  
= + −
6
Natural exponent inertia weight strategy
e2-pso
2
max
4
1 min max min
( ).e
iter
iteration
w w w w
    
  −  
  
   
= + −
7.
Simulated annealing inertia weight 1 min max min
( ) ^ ( 1)w w w w iterλ= + − × −
Here ƛ = 0.95
8. Time varying inertia weight 1 max min min
(max )
[( ) ]
max
iteration iter
w w w w
iteration
−
= − × +

48
4. RESULTS AND DISCUSSION
The optimization of network shown in fig 1 is done with the minimum pressure head
constraint of 30 m at all juctions of network. PSO is used for optimization and coding is done in
matlab environment. The result obtained after 100 iterations, for 15 trials of different swarm sizes are
shown in Fig. 3:
Fig.3 Variation of cost with swarm size
Table 2: Comparison of diameter and Head with the previous results of the same net work
Pipe no
Dia(present
work)
(m)
Dia(Jinesh et al
2014)
(m)
Node
no
Head(present
work)(m)
Head(Jinesh et
al 2014)(m)
1 0.457 0.457 1 210.00 -
2 0.254 0.254 2 53.1931 53.25
3 0.4064 0.4064 3 30.2319 30.46
4 0.1016 0.1016 4 43.3534 43.45
5 0.4064 0.4064 5 33.4958 33.81
6 0.2540 0.2540 6 30.2767 30.44
7 0.254 0.254 7 30.4233 30.55
8 0.254 0.254
410000
420000
430000
440000
450000
460000
470000
480000
490000
500000
0 10 20 30 40 50 60 70 80 90 100
Cost(units)
Swarm size
Variation of cost with swarm size

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976
6480(Print), ISSN 0976 – 6499(Online), Volume 6, Issue 5, May (2015), pp.
Fig.4.variation of cost with no of runs for various swarm size
It is observed from the fig.3 that cost of network is varying with swarm size and the least cost
of network is obtained for swarm size of 50.The cost obtained with number of runs for each swarm
size is represented graphically in fig.4, which shows that the least cost is at swarm size 50 .The
pressure head at junction are closely matching with the results obtained by jinesh et
Further the network is optimized for 50 swarm size and 15 runs by using eight variant
weighted function. It is observed that the
(Fig.5). It is least i,e, Rs 4,19,000 for 4th type of
as it is highest in case of random inertia weight (2nd type) varient.
convergence is high in case of logarithmic
work is also observed as 419000 units.
Fig 5.Variation
441000
462000
390000
400000
410000
420000
430000
440000
450000
460000
470000
1 2
cost(units)
Variation of cost for different variants
0
200000
400000
600000
800000
1000000
1200000
1400000
1600000
0 5
Cost(units)
Variation of cost with no. of runs
International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976
6499(Online), Volume 6, Issue 5, May (2015), pp. 43-51 © IAEME
49
variation of cost with no of runs for various swarm size
swarm size of 50.The cost obtained with number of runs for each swarm
pressure head at junction are closely matching with the results obtained by jinesh et
Further the network is optimized for 50 swarm size and 15 runs by using eight variant
nction. It is observed that the network cost considerably varies with type of variant
(Fig.5). It is least i,e, Rs 4,19,000 for 4th type of variant which is logarithmic
in case of random inertia weight (2nd type) varient. It is because
logarithmic inertia weight (Bansal et al). Minimum cost in previous
ed as 419000 units.
Variation of cost for different types of variants
447000
419000
462000
441000
459000
3 4 5 6 7
Type of variant
10 15 20
No . of runs
Variation of cost with no. of runs
© IAEME
variation of cost with no of runs for various swarm size
swarm size of 50.The cost obtained with number of runs for each swarm
pressure head at junction are closely matching with the results obtained by jinesh et al.
Further the network is optimized for 50 swarm size and 15 runs by using eight variants of
network cost considerably varies with type of variant
logarithmic Inertia weight, where
It is because the rate of
Minimum cost in previous
459000
447000
8
swarm size 10
swarm size 30
swarm size 50
swarm size 70
swarm size 90

50
Table 3:Diameter and flow for different pipe.
No of pipe Diameter(m) Flow(L/sec.)
1 0.457 0.311
2 0.254 0.0934
3 0.4064 0.1899
4 0.1016 0.0092
5 0.4064 0.1474
6 0.2540 0.0557
7 0.254 0.0656
8 0.254 0.0002
The optimum set of diameters (for best result i.e. 4th
variant) for each pipe of two loop network
is given in table 3. The pressure head at each junctions (the pressure head at junction 1 is fixed i.e,
210m) is depicited in table 4. It is seen that the lowest value of pressure heads are at type 4 inertia
weight (Logrithmic Inertia weight). This is because no. of iterations and swarm size is fixed for all
types of inertia weights in this analysis, but the rate of convergence is high for logrithmic inertia
weight. Others may also achieve the low pressure head values at higher number of iterations.
Table 4: Values of pressure head at all junctions for different types of Inertia weights
Node
No.
Pressure
head for
Type 1
variant
Pressure
head for
Type 2
variant
Pressure
head for
Type 3
variant
Pressure
head for
Type 4
variant
Pressure
head for
Type 5
variant
Pressure
head for
Type 6
variant
Pressure
head for
Type 7
variant
Pressure
head for
Type 8
variant
2 53.1931 53.1931 53.1931 53.1931 55.9806 53.1931 53.1931 53.1931
3 39.6279 37.5225 33.9689 30.2319 38.1285 39.6279 38.9799 33.9689
4 43.3403 44.5384 44.8001 43.3534 45.028 43.3403 43.4327 44.8001
5 38.3232 42.0028 41.2598 33.4958 44.972 38.3232 44.2504 41.2598
6 30.2666 30.3286 30.4528 30.2767 30.764 30.2666 30.232 30.4528
7 30.4258 30.5032 30.002 30.4233 30.6955 30.4258 30.0902 30.002
5. CONCLUSION
The present pipe network optimization concludes that the type of weighted integral plays an
important role in network optimization. The logrithmic weighted integral is most suitable for fast
convergence and cost minimization in case of water distribution network. However the suitable
swarm size and number of iterations are also significant in PSO. It can be seen that forth variant gives
best result with minimum cost of 419000 unit.
Notations
Hk - head available at each junction
Hmin - Min head required to be available at the junction
K - total no. of junction in the network
hf - head loss
f - friction factor
L - length of the pipe
V - velocity in the pipe
D - diameter of the pipe

51
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