World Journal of Engineering and Technology, 2013, 1, 49-58
Published Online November 2013 (http://www.scirp.org/journal/wjet)
http://dx.doi.org/10.4236/wjet.2013.13008
Open Access WJET
49
Pressure-Driven Demand and Leakage Simulation for Pipe
Networks Using Differential Evolution
Naser Moosavian, Mohammad Reza Jaefarzadeh
Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran.
Email: naser.moosavian@yahoo.com
Received September 1st, 2013; revised October 2nd, 2013; accepted October 8th, 2013
Copyright © 2013 Naser Moosavian, Mohammad Reza Jaefarzadeh. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original
work is properly cited.
ABSTRACT
Traditional techniques for hydraulic analysis of water distribution networks, which are referred to as demand-driven
simulation method (DDSM), are normally analyzed under the assumption that nodal demands are known and satisfied.
In many cases, such as pump outage or pipe burst, the demands at nodes affected by low pressures will decrease. There-
fore, hydraulic analysis of pipe networks under deficient pressure conditions using conventional DDSM may cause
large deviation from actual situations. In this paper, an optimization model is introduced for hydraulic analysis of water
distribution networks using a meta-heuristic method called Differential Evolution (DE) algorithm. In this methodology,
there is no need to solve linear systems of equations, there is a simple way to handle pressure-driven demand and leak-
age simulation, and it does not require an initial solution vector which is sometimes critical to the convergence. Also,
the proposed model does not require any complicated mathematical expression and operation.
Keywords: Hydraulic Analysis; Differential Evolution; Optimization Model
1. Introduction
In the recent past, several packages originally developed
for steady state analysis of looped water distribution sys-
tems. For instance, EPANET2 has been extended to in-
clude the possibility of “extended period simulations”
(EPS), namely the possibility of simulating long periods
of time by means of a succession of steady states, only
accounting for the change in storage of reservoirs occur-
ring from one time step to the next [1].
This model, which is used in current engineering prac-
tice, is based on the conventional Demand Driven Simu-
lation Method (DDSM). It assumes that nodal outflows
are fixed and are satisfied regardless of network pres-
sures. The assumption simplifies the mathematical solu-
tion of the problem but is not always appropriate because
it is clear that the amount of outflow at nodal outlets de-
pends on network pressures. If the pressure falls below a
minimum required level (due to some critical events such
as mechanical and hydraulic failures or excess demand),
the flow will be significantly reduced. Although some
nodes may be able to satisfy their demands, others may
meet the demand partially while the rest may fail and
may not provide any water at all. The assumption of
fixed nodal consumptions is therefore valid only under
normal conditions when the pressures can be expected to
be adequate to satisfy the stipulated demands. If the op-
eration of the system is simulated under pressure-critical
conditions, the relationship between pressure and outflow
should, therefore, be taken into account if the simulation
results are to be realistic [2-8]. Furthermore, water loss
via leakage constitutes a major challenge to the effective
operation of municipal distribution networks since it
represents not only diminished revenue for utilities, but
also undermined service quality [9] and wasted energy
resources [10]. A typical leakage control program usually
starts with a water audit based on available flow meas-
urements. Although this is an important first step, most
practical studies do not go beyond it. In order to assist in
leakage reduction and conduct more accurate analysis, a
hydraulic model capable of accounting for pressure-
driven (also known as head-driven) demand and leakage
flow at pipe level is introduced by Giustolisi et al. [11].
Meanwhile, there is still a chance to develop a new
method for pressure-driven demand and leakage simula-
tion in water distribution networks. In this paper, an op-
timization model is introduced for hydraulic analysis of
Pressure-Driven Demand and Leakage Simulation for Pipe Networks Using Differential Evolution
50
water distribution networks using a meta-heuristic algo-
rithm called Differential Evolution (DE). Analysis of
hydraulic networks can be achieved by treating it as an
optimization problem as shown by Arora [12], Hall [13],
and Collins et al. [14]. Arora considered a simple two-
piped loop while Collins et al. have based their approach
on rigorous theoretical background and developed
nonlinear optimization models, solutions of which yield
the hydraulic network analysis [15]. Collins’s model can
be minimized by application of differential evolution
algorithm. In this methodology, there is no need to solve
linear systems of equations, there is a simple way to han-
dle pressure-driven demand and leakage simulation, and
it does not require an initial solution vector which is
sometimes critical to the convergence. Also, the pro-
posed model does not require any complicated mathe-
matical expression and operation. In the next part,
Collins’s model is described.
2. Co-Content Model Approach
Arora [12] is the first researcher who suggested an ap-
proach based on the principle of conservation of energy.
This principle states: “Flow in the pipes of a hydraulic
network adjust so that the expenditure of the system en-
ergy is minimum.” Next, Collins et al. [14] proposed a
model termed the co-content model, that is based on
equations having the unknown nodal heads as the basic
unknowns, i.e., based on H equations. The unknown pipe
flows are expressed in terms of the nodal heads and the
known pipe resistances, so that the energy loss in pipe

x
x
E is given by [15]


11
1
n
ij
xxx n
x
HH
EQh R


 (1)
In which Rx is the hydraulic resistance function, hx is
head loss in pipe x, i
H
and are pressure heads in
node i and node j.
j
H
Now consider the network of Figure 1, with the
known and unknown parameters as shown therein. Let
the unknown nodal heads at nodes 3, 4, and 5 be H3, H4,
[1]
(1) (2)
[3]
[4]
(3)
(4)
[5]
(5) [2]
Figure 1. Schematic representation of the looped pipe net-
work with 5 pipes.
and H5, respectively. Herein also consider a ground node
G with fixed known level H0G, as shown in Figure 1.
The nodes 3, 4, and 5 are connected to the ground node
G with pseudo pipes, carrying the known nodal outflows
q3, q4, and q5 as shown in Figure 1.
The co-content optimization model is expressed as

 




11 11
01 334
11
12
11 11
54 35
11
34
11
02 5
33 0
1
5
44 0
550
.
11
11
11
nn
nn
nn
nn
n
G
n
G
G
HH HH
Min CHRR
HH HH
RR
HH qH H
n
R
qH H
n
qHH
n











 



 


(2)
01
H
and 02
H
are known pressure heads for source
nodes. The first five terms of the objective function rep-
resent the energy loss in real pipes 1, of the net-
work, respectively, and the last three terms show
, 5
11n
times the energy loss in the pseudo pipes [15].
It should be noted that there are no constraints and there-
fore an unconstrained model in three decision variables is
made. For minimization of optimization model, which
are partially differentiating in unknown heads, the node-
flow continuity equations are created. Therefore, the so-
lution of the co-content model gives the values of the
unknown heads such that the node-flow continuity rela-
tionships are satisfied [15]. For simplicity, oG
H
can be
taken as zero, so that the General co-content model can
be expressed as

11
1
1
Min. 1
n
ij
j
j
n
xj
x
HH
CHq H
n
R





(3)
Collins et al. [14] suggested the solution of the NLP
optimization of the model. Their method were 1) the
Frank-Wolfe method; 2) a piece-wise linear approxima-
tion; and 3) the convex simplex method. These methods
are highly depends on initial guesses and in some cases
they converged to an incorrect solution [14].
3. Head Dependent Analysis
In the common approaches, it is presumed that the nodal
demands are always satisfied at all demand nodes, irre-
spective of the available HGL values at demand nodes
[15]. But in practice, if the head at a node is insufficient,
a reduction in the water flowing from the tap is expected
and, in the worst case, the discharge that can be drafted
will be zero, regardless to the actual demand [1]. There
are several solutions in the literature for these conditions.
Open Access WJET
Pressure-Driven Demand and Leakage Simulation for Pipe Networks Using Differential Evolution 51
Wagner et al. [16] and Chandapillai [17] suggested a
parabolic relationship between required nodal head and
minimum head. Their relationships are
min
min
min
min
0j
1
p
j*
jj j
*
*
jj
HH
HH
qqH HH
HH
qHH

 


0
(4)
H* is the required nodal head. This formulation is eas-
ily handled to co-content model without any mathemati-
cal complexity.
4. Leakage Simulation
Water losses via leakages constitute a major challenge to
the effective operation of municipal WDN since they
represent not only diminished revenue for utilities, but
also undermined service quality [9] and wasted energy
resources [10]. In order to conduct more accurate analy-
sis of a WDN, such as a better estimate of flow through
the network (with respect to both satisfied demand and
losses through leakage), a hydraulic analysis based on
capable of accounting for pressure-driven (also known as
head-driven) demand and leakage flow at the pipe level
should prove invaluable. To reach this goal, a leakage
model is expressed as follows [3]

if 0
0if
k
kk kk
k-leak
k
βlPP
q
P
(5)
Where Pk = average pressure in the pipe computed as
the mean of the pressure values at the end nodes I and j
of the kth pipe; and lk = length of that pipe. Variables αk
and βk = two leakage model parameters [11]. The alloca-
tion of leakage to the two end nodes can be performed in
a number of ways [18]. Here the nodal leakage flow
qjleak is computed as the sum of qkleak flows of all pipes
connected to node j as follows:

--
1if 0
12
20if
k
kk kk
jleakkleak
kk
k
lP P
qq
P

0
 (6)
where
2
kij
PPP . This formulation is also easily
handled to co-content model without any mathematical
complexity.
5. Application of Differential Evolution
Algorithm for Minimizing Co-Content
Model
For the hydraulic analysis, this study introduces Differ-
ential Evolution (DE) algorithm. Because the algorithm
was originally developed for solving optimization prob-
lems, the hydraulic network analysis wasintroduced into
an optimization problem (co-content model). One ad-
vantage of the DE algorithm is the fact that it does not
require an initial solution vector which is sometimes
critical to the convergence. Also, application of DE algo-
rithm in co-content model does not require any compli-
cated mathematical expression and operation. In this mo-
del, pressure-driven demand and leakage can be simu-
lated.
5.1. Differential Evolution (DE)
Differential evolution (DE) is a simple powerful and
population-based stochastic optimization algorithm that
outperforms many meta-heuristic algorithms on numeri-
cal single objective optimization problems. In DE each
decision variable is represented in the chromosome by a
real number. The DE algorithm requires only three con-
trol parameters: weight factor (F), crossover rates (CR),
and population size (NP). The initial population is ran-
domly generated by uniformly distributed random num-
bers using the maximum and minimum limitation of each
decision variable. Then the fitness values of all the indi-
viduals of population are calculated to find out the best
individual xbest,G of current generation, where G is the
index of generation. Three main steps of DE, mutation,
crossover, and selection were performed sequentially and
were repeated during the optimization cycle [19].
The steps in the procedure of DE are shown in Figure
2. They are as follows:
Terminate
Competitions
Repeat steps
3, 4, and 5
NI
Step 6: Check stopping criterion
Step 1: Initialize Parameters Step 2: Generate samples
C(H): Objective function
Hi: Decision variable
(pipe diameter)
N
: Number of decision variables
P: Number of population
CR = Crossover probability
F = Mutation rate
N
I = Number of Iterations
i = 1:P
Randomly generate the solutions
Evaluate objective function
Calculate cost function C(H)
Step 4: DE algorithm
For i = 1:P (number of population)
Select a solution vector from the population
Compute the objective function
Crossover Mutation
Step 5: Selection
If new solution vector
Is better than Old one.
Update solution vector
Figure 2. DE procedure for minimization of co-content model.
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Pressure-Driven Demand and Leakage Simulation for Pipe Networks Using Differential Evolution
52
Step 1. Initialize problem and algorithm parameters
Step 2. Samples Generation
Step 3. Start Iterative Process
Step 3.1. Mutation Operator
Step 3.2. Cross-Over Operator
Step 4. Selection
Step 5. Check The Stopping Criterion.
5.2. Step 1. Initialize the Problem and Algorithm
Parameters
In Step 1, the optimization problem is specified as fol-
lows:

11
1
1
Min. 1
n
ij
j
j
n
x
x
HH
CHq H
n
R





j
(7)
Where
CH is an objective function; H is the set of
each decision variable. In this paper, the objective func-
tion is the co-content model; the unknown heads are the
decision variables.
5.3. Step 2. Samples Generation
The initial population, initial values of the mutation fac-
tor, F and initial values of the crossover rate, CR for the
DE is created arbitrarily by following formula:
 



min1maxmin
min2 maxmin
min 3maxmin
,, ,
,0
,
H
ijHij HijHij
FjFF F
CR CRCRCR
 
 
 
(8)
where 123
= independently generated random num-
bers in the range of
τ,τ,τ
0,1 .
min ,
H
ij and Hmax(i, j) are
maximum and minimum limits of variable j and node i.
min
F
and max
F
are maximum and minimum limits of
mutation factor. min and max are maximum and
minimum limits of crossover rate. Then, the fitness val-
ues of all the individuals of population are cal-
culated. The position matrix of the population of genera-
tion G can be represented as:
CR CR

CH

11 1
112
22 2
212
1
N
GN
nPop nPop
nPop N
CHH H
CHHH
P
CHH











(9)
N is the number of unknown nodes.
5.4. Step 3. Start Iterative Process
In this step, two main steps of DE, mutation, and cross-
over, are performed sequentially and new solution vec-
tors are created.
5.4.1. Mutation Operator
In this step, mutation operator is used, for each solution
vector in the population, to create new solutions in DE
according to the following formula:


new ,, ,,
H
ijHiCFHiA HiB (10)
A, B, and C are random solution vectors.
5.4.2. Cross-Over Operator
In the crossover operator, the new vector is generated by
choosing some parts of mutation vector, and other parts
come from the target vector. The crossover operator of
DE is shown as follows:


new
new ,ifrand
,,otherwise
H
ij CR
Hij Hij
(11)
where CR represents the crossover probability. If ran-
dom number rand is larger than CR value, the component
of mutation vector will be chose to the trial vector. Oth-
erwise, the component of target vector is selected to the
trial vectors. The mutation and crossover operators are
used to diversify the search area of optimization prob-
lems [19].
5.5. Step 4. Selection
The trial vector is carried to the next generation only if it
yields a reduction in the value of the objective function
in the case of the minimization problem. Otherwise, the
target vector will be selected for the next generation.
The population of the next generation is selected as
follows:
  




new new
new if
otherwise
H
jCHjCHj
Hj
Hj
(12)
where
CH j represents the cost of the jth individ-
ual in the current generation. The F selections for the
next generation is given by
min2maxmin
,1FjGFF F
  (13)
where G is the generation number. It should be noted that
0G
in the initial generation.
5.5. Step 5. Check the Stopping Criterion
In this section, Steps 3, 4 and 5 are repeated until the
termination criterion is satisfied.
6. Numerical Examples
In this section, the hydraulic analyses for several condi-
tions in some water distribution networks are done. All
of computations were executed in MATLAB program-
ming language environment with an Intel(R) Core(TM) 2
Duo CPU P8700 @ 2.53 GHz and 4.00 GB RAM. In this
study proposes the use of mass balance and energy bal-
Open Access WJET
Pressure-Driven Demand and Leakage Simulation for Pipe Networks Using Differential Evolution 53
ance in the network for demonstrating the effectiveness
of DE in comparison with other methods.
The average of mass and energy balance is shown by δ
and is calculated by following formula:

1
1
connectedto
through
mean ,
2, ,
n
ij
j
n
ik
jk
HH
abs q
R
jN









(14)
In all numerical examples ,
min max
min and max . To check the performance
of the DE for the minimization of co-content model, ten
optimization runs were performed using different random
initial solutions in all examples.
0.2CR CR
0.2F0.8F
6.1. Numerical Example 1
In order to demonstrate the advantages of the proposed
model in pressure-driven demand condition, the simpli-
fied water distribution network shown in Figure 3, was
used. For the sake of simplicity, the same Hazen-Wil-
liams roughness coefficient C = 130 was assumed for all
the 14 pipes of identical length of 1000 m, while no mi-
nor losses have been added. The following diameters
have been used in the example: 500 mm (P-2); 400 mm
(P-1); 300 mm (P-4, P-7); 250 mm (P-10); 200 mm (P-3,
P-5, P-6, P-13); 150 mm (P-8, P-9, P-11, P-12, P-14).
The nodal demands are q2 = 1, q3 = 1, q4 = 2, q5 = 15, q6
= 15, q7 = 10, q8 = 5 (m3/min). Without loss of general-
Reservoir
1 2
3
3
2
4
4 8
5 6 7
6
9
5
10 11 12 13
14
7 8
Figure 3. Schematic representation of the looped pipe net-
work used in the numerical example 1.
ity, in this example, the minimum head requirement *
i
H
has been assumed equal to the ground elevation Zi [1]. So
the relationship between required nodal head and mini-
mum head is:
0
j
j
j
j
jj
H
Z
qqZ H
(15)
Todini [1] proposed a three steps approach for solving
this network and its solution is reported in the 4th col-
umn of Table 1. In proposed methodology, pressure-
driven model can be applied in hydraulic analysis with-
out any mathematical formulation. In this situation, an
if-then rule is added to co-content model and optimiza-
tion process is conducted. The DE technique is applied to
solve this problem in three cases. DE model parameters
selected are as follows: number of decision variables = 7;
number of population for case 1 = 10, case 2 = 20, case 3
= 20; number of iteration for case 1 = 1000, case 2 =
1000 and case 3 = 5000. The bound variables were set
between 50 and 140. The best, worst and average solu-
tions of DE algorithm in three cases are shown in Table
2. This table compares the average of mass and energy
balance of the three cases with those obtained using To-
dini algorithm. As it can be seen in Table 2, DE found
the optimal solution more accurately than Todini method
in all cases. Results of the best performance of DE and
convergence history are reported in Table 1 and Figure
4, respectively.
As you can see in the Figure 4, after about 400 itera-
tions the parameter δ becomes convergent and then it
doesn’t change. The minimum value of δ calculated by
DE algorithm is 2.07E-02, while the value obtained for
this parameter, by the method introduced by Todini
equals to 2.76E-02. Values of δ at each node are com-
pared in the seventh and eighth columns of Table 1, us-
ing the two proposed methods and the method of Todini.
0100 200300 400500 600 700800 9001000
10
-2
10
-1
10
0
Number of It erat ion
Average o
f
Mass and E nergy Bal anc e
Figure 4. Convergence history of numerical example 1.
Open Access WJET
Pressure-Driven Demand and Leakage Simulation for Pipe Networks Using Differential Evolution
Open Access WJET
54
Table 1. Head and parameter δ in numerical example 1.
DE 3 steps DE 3 steps DE 3 steps
Node Z(m)
H(m) H(m) [2] H-Z H-Z [1] δ δ [1]
1 140 140 140 0 0 0 0
2 80 129.304 130.07 49.304 50.07 7.93e-10 0.0003
3 90 132.288 132.76 42.288 42.76 7.24e-09 0.004
4 70 109.587 110.96 39.587 40.96 5.56e-10 0.0021
5 80 80.000 88.54 0.000 8.54 0.0576 0.034
6 90 90.000 91.45 0.000 1.45 0.0069 0.0173
7 90 90.000 90.00 0.000 0.00 0.0803 0.106
8 100 88.922 90.43 11.078 9.57 2.54e-09 0.0439
Table 2. Average of mass and energy balance for numerical example 1.
Mass and Energy Balance (δ)
DE
best worst mean std
Number of population 10
Number of iteration 1000
2.07E-02 9.01E-02 4.14E-02 2.37E-02
Number of population 20
Number of iteration 1000
2.07E-02 2.08E-02 2.07E-02 3.38E-05
Number of population 20
Number of iteration 5000
2.07E-02 2.07E-02 2.07E-02 6.20E-06
Three Steps Approach [1] Maximum Accuracy 2.76E-02
As you can see at all the nodes, in calculating the mini-
mum value of δ, the proposed method works better than
the Todini method.
6.2. Numerical Example 2
The second considered network is a real planned network
designed for an industrial area in Apulian town (Southern
Italy). The network layout is shown in Figure 5 and the
corresponding data are provided in Table 3. With respect
to the leakages, they have been assumed as pressure-
driven (see Equation (5)) since they are implemented in
the pressure-driven network simulation model as above
described [11]. The parameter β = 1.0632 × 10 7 and α
= 1.2, as reported in Giustolisi et al. [11] for this network.
Giustolisi et al. [11] proposed a hydraulic simulation
model, which fully integrates a classic hydraulic simula-
tion algorithm, such as that of Todini and Pilati [20]
found in EPANET 2, with a pressure-driven model that
entails a more realistic representation of leakage. They
applied their model in this network and results are dem-
Figure 5. Schematic representation of the looped pipe net-
work used in the numerical example 4.
onstrated in Table 4. The DE technique is applied to
solve this problem and DE model parameters selected are
as follows: number of decision variables = 23; number of
population for all case 1 = 50, case 2 = 50, case 3 = 20,
Pressure-Driven Demand and Leakage Simulation for Pipe Networks Using Differential Evolution 55
Table 3. Hydraulic data relevant to the numerical example
2.
Pipe L (m) D (mm) Pipe L (m)D (mm) Pipe L (m)D (mm)
1 348.5 327 12 428.4184 23 165.5 100
2 955.7 290 13 419 100 24 252.1100
3 483 100 14 1023.1100 25 331.5100
4 400.7 290 15 455.1164 26 500204
5 791.9 100 16 182.6290 27 579.9 164
6 404.4 368 17 221.3290 28 842.8 100
7 390.6 327 18 583.9164 29 792.6 100
8 482.3 100 19 452 229 30 846.3184
9 934.4 100 20 794.7100 31 164258
10 431.3 184 21 717.7100 32 427.9100
11 513.1 100 22 655.6258 33 379.2100
34 158.2368
case 4 = 20 and case 5 = 50; number of iteration for case
1 = 500, case 2 = 1000, case 3 = 10,000, case 4 = 5000
and case 4 = 2500. In proposed method, there is no need
to change mathematical formulation for hydraulic analy-
sis. An if-then rule is added to co-content model and op-
timization process is done easily. In this example, the
bound variables were set between 0 and 36.4.
Table 5 compares different cases of algorithm DE for
minimization of model. The best result is related to the
case in which the number of population is equal to 50
and the number of iterations equal to 2500. After per-
forming 10 different runs, the best value of δ is obtained
equal to 0.0026, while the best result is obtained equal to
0.0181 in Giustolis method. The results of two men-
tioned methods are compared in the fifth and sixth col-
umns of Table 4. In this table, the best result is shown in
bold, and it is considered the method DE has calculated
the best value of δ at 17 nodes and Gistulishi method has
calculated it at 5 nodes. The convergence process of al-
gorithm DE has been shown in two forms in Figures 6
and 7. The absolute value of δ is calculated for each it-
eration in Figure 6 and the amount of objective function
C(H) is calculated for each iteration in Figure 7. The
algorithm becomes convergent after 2500 iterations.
6.3. Numerical Example 3
Figure 8 shows the network from Mallick et al. [21].
The network consists of 2 reservoirs, 13 nodes and 21
pipes. The detailed properties are shown in Tables 6 and
7. It is supposed that the desired pressure for each node
(H*) is 30 m, and the minimum pressure (Hmin) is 10 m
[22]. The pipe leakage coefficients are β = 5 × 10 7 and
Table 4. Head and parameter δ in numerical example 2.
Node H
number
q (l/s)
(m) [1]
H (m) δ [11] δ
1 10.86326.9 33.29 0.1547426
0.0046
2 17.03424.81 31.83 0.02131049
0.00648
3 14.94721.3 27.39 0.0477137 0.00498
4 14.2817.22 25.34 0.0220368 0.00504
5 10.13323.54 30.89 0.0261836 0.00378
6 15.3520.1 29.02 0.04038949
0.00517
7 9.11418.91 27.94 0.0171474 0.00275
8 10.5117.9 27.34
0.0022701 0.00351
9 12.18217.85 26.35
0.0029365 0.00378
10 14.57912.66 23.24 0.008277 0.0043
11 9.00716.23 25.95 0.03155407
0.00258
12 7.57510.12 22.05 0.0027315 0.00213
Node H
number
q (l/s)
(m) [11]
H (m) δ [11] δ
13 15.210.03 22.45 0.01259978
0.00418
14 13.5515.41 25.95 0.06347619
0.00418
15 9.22614 24.17 0.0091379 0.00287
16 11.214.36 24.05 0.0070886 0.00357
17 11.46915.3 25.42
0.0001028 0.00354
18 10.81818.83 28.38 0.01188886
0.00384
19 14.67519.35 28.39
5.43E-05 0.00505
20 13.31810.01 23.79 0.0377624 0.00398
21 14.63111.48 22.35
0.00274996 0.00411
22 12.01214 25.46 0.00390141
0.0036
23 10.32610.45 20.11 0.0085767 0.00296
24 36.45 36.45
α = 1.18, for this network. The DE technique is applied
to solve this problem and DE model parameters selected
are as follows: number of decision variables = 13; num-
ber of population for all cases = 20; number of iteration
for case 1 = 1000, case 2 = 2500, case 3 = 5000 and case
4 = 10,000. The bound variables were set between 0 and
60.96.
In this example, the network has been analyzed ac-
cording to two cases. In the first case, the dependence of
pressure on the demand is not included, but in the latter
case, the analysis is performed by taking into account the
dependence of pressure on the demand. Table 8 shows
Open Access WJET
Pressure-Driven Demand and Leakage Simulation for Pipe Networks Using Differential Evolution
56
05001000 1500 2000 2500
10-3
10-2
10-1
100
Number of Iterat i on
Average of M ass and E nergy Bal ance
Figure 6. Convergence history of numerical example 2 (case
1).
0500 1000 1500 2000 2500
10
1.2
10
1.3
10
1.4
Number of Iterat i on
Obje ctive Functi on C(H)
Figure 7. Convergence history of numerical example 2 (case
2).
Figure 8. Schematic representation of the looped pipe net-
work used in the numerical example 3.
Table 5. Average of mass and energy balance for numerical
example 2.
Mass and Energy Balance (δ)
DE
best worst mean std
Number of
population 50
Number of
iteration 500
6.00E-03 1.76E-02 1.14E-023.60E-03
Number of
population 50
Number of
iteration 1000
2.60E-03 1.01E-02 5.90E-032.20E-03
Number of
population 20
Number of
iteration 10,000
3.90E-03 4.00E-03 4.00E-034.08E-05
Number of
population 20
Number of
iteration 5000
3.20E-03 3.80E-03 3.60E-031.88E-04
Number of
population 50
Number of
iteration 2500
2.60E-03 5.50E-03 3.00E-035.88E-04
Giustolisi
Algorithm [11]Maximum Accuracy 1.81E-02
Table 6. Pipe characteristics of Sample network from Mal-
lick et al. [21].
Pipe Number L (m) D (mm) C
1 609.6 762 130
2 243.8 762 128
3 1524 609 126
4 1127.76 609 124
5 1188.72 406 122
6 640.08 406 120
7 762 254 118
8 944.88 254 116
9 1676.4 381 114
10 883.92 305 112
11 883.92 305 110
12 1371.6 381 108
13 762 254 106
14 822.96 254 104
15 944.88 305 102
16 579 305 100
17 487.68 203 98
18 457.2 152 96
19 502.92 203 94
20 883.92 203 92
21 944.88 305 90
Open Access WJET
Pressure-Driven Demand and Leakage Simulation for Pipe Networks Using Differential Evolution 57
Table 7. Nodes properties of Sample network from Mallick
et al. [21].
Node Number q (L/S) Elevation (m)
1 0 27.43
2 59 33.53
3 59 28.96
4 178 32
5 59 30.48
6 190 31.39
7 178 29.56
8 91 31.39
9 0 32.61
10 0 34.14
11 30 35.05
12 30 36.58
13 0 33.53
Table 8. Average of mass and energy balance for numerical
example 3.
Mass and Energy Balance (δ)
DE
best worst mean std
Number of
population 20
Number of
iteration 1000
1.90E-031.11E-02 5.30E-032.60E-03
Number of
population 20
Number of
iteration 2500
1.46E-047.03E-04 4.55E-041.72E-04
Number of
population 20
Number of
iteration 5000
1.06E-061.37E-06 5.00E-064.32E-06
Number of
population 20
Number of
iteration 10,000
1.18E-082.90E-08 1.75E-086.29E-09
how the accuracy of parameter δ depends on the iteration
number of convergence in the first case. As it can be seen,
the value of δ reaches the accuracy 1e-2 after 1000 itera-
tions, the accuracy 1e-4 after 2500 iterations, 1e-6 after
5000 iterations and 1e-8 after 10,000 iterations. Figure 9
compares the nodal pressures in the mentioned two cases.
As it can be observed, if the dependence of pressure on
the demand is not included, a negative pressure is made
at the nodes 6, 8, 11 and 12, but all pressures in the sec-
ond case are greater than 5 meters and the minimum
pressure of 10 meters has been partly supplied in most of
the nodes. Figure 10 shows the process of changing the
Figure 9. Simulation results with and without pressure-
driven demand.
01000 20003000 4000 5000 60007000 80009000 10000
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Number of Iterati on
A v erage of Mass and E nergy Bal an c e
Figure 10. Convergence history of numerical example 3.
parameter δ towards the number of iterations.
7. Conclusions
The purpose of this paper has been to introduce a novel
methodology for hydraulic analysis of water distribution
systems under deficient pressure conditions considering
the pressure-driven demand and leakage. The methodol-
ogy is illustrated using three networks with different
layouts.
The overall results indicate that the proposed method
has the capability to handle various pipe networks prob-
lems without changing in model and mathematical for-
mulation. Application of DE to co-content model can
solve pressure-driven demand and leakage simulation
with applying if-then rules in co-content model. The ad-
vantage of the proposed methodology is its flexibility in
employing different formulation and specifying parame-
ters related to pressure-driven demand. Another advan-
tage of this method is that it can be easily developed for
Open Access WJET
Pressure-Driven Demand and Leakage Simulation for Pipe Networks Using Differential Evolution
Open Access WJET
58
common users to undertake deficient pressure conditions.
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