Ann-pso Aem Dore

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Application of Artificial Neural Networks and Particle

Swarm Optimization for the Management of GroundwaterResources

Shishir Gaur &Sudheer Ch &Didier Graillot &

B. R. Chahar &D. Nagesh Kumar

Received: 14 March 2011 /Accepted: 26 November 2012 /

Published online: 12 December 2012

# Springer Science+Business Media Dordrecht 2012

Abstract Ground management problems are typically solved by the simulation-optimization

approach where complex numerical models are used to simulate the groundwater flow and/or

contamination transport. These numerical models take a lot of time to solve the management

problems and hence become computationally expensive. In this study, Artificial Neural Net-

work (ANN) and Particle Swarm Optimization (PSO) models were developed and coupled for

the management of groundwater of Dore river basin in France. The Analytic Element Method

(AEM) based flow model was developed and used to generate the dataset for the training and

testing of the ANN model. This developed ANN-PSO model was applied to minimize thepumping cost of the wells, including cost of the pipe line. The discharge and location of the

pumping wells were taken as the decision variable and the ANN-PSO model was applied to find

out the optimal location of the wells. The results of the ANN-PSO model are found similar to

the results obtained by AEM-PSO model. The results show that the ANN model can reduce the

computational burden significantly as it is able to analyze different scenarios, and the ANN-

PSO model is capable of identifying the optimal location of wells efficiently.

Keywords Groundwater modeling. Groundwater management. Artificial neural network.

Analytic element method. Particle swarm optimization

Notations

The following symbols have been used in this paper:

Cwi Well installation cost (euros)

Cpn Capitalized cost of pipelines (euros)

Water Resour Manage (2013) 27:927941

DOI 10.1007/s11269-012-0226-7

S. Gaur (*) :D. GraillotUMR CNRS5600 EVS, Dpartement of Gosciences and Environnent, SPIN, Ecole Nationale Suprieure

des Mines, 158-Cours Fauriel, Saint-Etienne 42023, Francee-mail: [email protected]

S. Ch :B. R. ChaharDepartment of Civil Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India

D. N. Kumar

Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, India


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Cp Total cost of pumping (euros)

CpE Capitalized electricity cost (pumping cost)

Cpu Cost of pump units (euros)

Gt Global best value among all particles, gbest

hi Minimum water head on the periphery of theith

well (m)b Aquifer thickness (m)

K Hydraulic conductivity (m/s)

Li Pipe length for each well (m)

Nw Total number of wells

Pt Previous best value for each particle, pbest

Q Discharge from well (m3/s)

Qi Discharge from ith well (m3/s)

r The rate of interest (euros/euros/year)

RE The cost of the electricity per kilowatt-hour (euros/kwh)

Vmax Maximum velocity (m/s)

W Complex discharge function

Stream function (m3/s)

Rate of groundwater flow (m3/s)

h Groundwater head (m)

Discharge potential (m3/s)

Inertia weight

Density of the fluid (N/m3)

Combined efficiency of the pump and the prime mover

1 Introduction

The initial idea about ANN was introduced by McCulloch and Pitts (1943) by proposing the

model of a neuron. Later, the ANN got high recognition due to Rumelhart and McClelland

(1986). They discovered the mathematically rigorous theoretical framework for the ANN by

presenting the generalized delta rule, or back-propagation algorithm (BPA) and demonstrat-

ed its capability, in training a multilayer ANN. ANN have been applied in different areas of

hydrology such as rainfall-runoff modeling, groundwater management, stream flow fore-

casting, precipitation forecasting, hydrologic time series and reservoir operations. It wasfound that ANN based optimization model needs much less computational time and is more

flexible in comparison to mathematical programming methods.

The Groundwater management problems are typically solved by researchers through

simulation-optimization approach. In the simulation-optimization process, optimization

model employs simulation for getting the values of the groundwater head, velocity, concen-

tration etc. This repeated use of the flow model increases the computational burden

extensively and takes several hours to get the final solution. Different researchers (Rogers

and Dowla 1992; Johnson and Rogers1995; Coppola et al. 2003; Singh et al. 2004) used

ANN to substitute the computational by expensive numerical models. They solved thedifferent hydrological management problems using ANN and optimization algorithm and

found this combination to be faster and more robust.

Rogers and Dowla (1992) provided one of the first published approaches for optimal

groundwater remediation using artificial neural networks and a genetic algorithm. Ranjithan

et al. (1993) used ANN to simulate pumping index for a hydraulic conductivity realization,

to remediate groundwater at a contaminated site. Two subsets of 100 and 200 patterns

928 S. Gaur et al.


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(inputoutput vectors) were used respectively for training and testing a feed-forward ANN.

This problem was later expanded to a 3D multi-phase model by Rogers et al. (1995). Rogers

and Dowla (1994) used the approach of combining the Genetic Algorithm (GA) with ANN

which predicted the fitness measures of the generated pumping pattern of 20 wells. GA-

ANN optimization results were compared with other approaches of combining the ground-water flow model with a nonlinear programming with a quasi-Newton search method.

Johnson and Rogers (1995) used combined ANN and GA to select the optimal location of

pumping well for groundwater remediation. The networks were trained to predict mass-

extraction and contaminant information 2-D model SUTRA. Singh et al. (2004) performed

study for identification of unknown pollution sources, using an artificial neural network.

This study produced promising results even with large measurement errors.

Coppola et al. (2003) applied the ANN model for three types of groundwater prediction and

management problems. The study affirms that ANN is efficient to solve the variety of complex

groundwater management problems and overcome many of the problems and limitation

associated with other conventional physically based flow models. Rao et al. (2003) developed

a subsurface water management model by replacing the existing sharp interface flow model by

ANN model. The study concluded that the computational burden was significantly reduced by

replacing the simulator model with an ANN. Arndt et al. (2005) compared the results of a

computationally expensive finite-element simulation model with ANN-computed predictions.

The study concluded that 60 % of time reduction can be obtained using the ANN, compared to

the simulation-based solution. The study also concluded that the value of the objective function

obtained by the simulation based optimal solution was only 1 % better than that by the ANN.

Nikolos et al. (2008) combined the ANN with a differential evolution algorithm to replace the

finite-element numerical model to determine the optimal operational strategy for the pumpingwells to meet the water demand and maintain the water table at certain levels. They concluded

that ANN could significantly reduce the computational burden compared to the numerical

models. Artigue et al. (2012) use ANN for flash flood forecasting on ungauged basins in

southern France. Study demonstrates that efficient forecasting can be derived from a feed-

forward model using the available measured discharges. ANN applications have also been

observed in groundwater hydrology problems (Sreekanth and Datta2011; Trichakis et al. 2011;

Karthikeyan et al.2012; Chang et al.2012).

The Analytic Element Method (AEM) is a computational method based upon the

superposition of analytical expression to represent two dimensional vector fields. The AEMs

can superimpose hundreds of exact analytic solutions to solve groundwater flow problemsand are capable of simulating streams, lakes and complex boundary conditions (Strack

1989). Particle Swarm Optimization (PSO) is also an evolutionary computation technique

developed by Eberhart and Kennedy (1995). It has been applied to various fields of

engineering research and proved itself as an effective and efficient method. The application

of AEM-PSO has been found limited in the field of water resources and particularly

groundwater management. Matott et al. (2006) solved groundwater management problems

using the AEM based flow model. To solve pump and treat optimization (PATO) problems,

the AEM based flow model was used and the results generated with different optimization

techniques i.e. GA, SA, CG (Conjugate Gradient) and PSO were compared.This paper examines the capability of the ANN-PSO based simulation-optimization

model where the training dataset for the ANN model were generated by the AEM based

flow model. The paper also examines the applicability of the ANN-PSO model to identify

the minimum pumping cost for the wells. The discharges as well as location of the wells are

taken as the decision variables in the optimization function. The piping cost is also

considered in the cost function and its influence on the optimal solution is assessed. Both

Application of ANN-PSO in Ground Water 929


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ANN-PSO and AEM-PSO models are applied for the Dore river basin in France and the

results of the ANN-PSO model are compared with the AEM-PSO model.

2 The Study Area

The Dore River catchment, situated in the eastern part of the Massif-Central in France

(Fig.1), was considered to set up the pumping wells. The average annual rainfall recorded at

the rain gauge station of the basin in 780 mm. The Dore River is an important tributary of the

Allier River. The average annual flow of the Dore River is 20.2 m3/s which can vary from

50 m3/s, at high flow conditions, to 5 m3/s in dry periods whereas the river stage fluctuates

less than 1 m. A major part of the area is covered by fluvial quaternary sediments underlain

by marl and clay. Depending on the rate of clay deposits, the hydraulic conductivity varies

from 1103 to 3103m/s. The location of different hydrological features and other

required data were extracted from the geological maps to the scale 1/50,000 provided by

the BRGM (National Service for Geological Survey). A total of 12 piezometric measure-

ments are available in the study area which shows the hydraulic gradient in the North

direction. The water levels in the two rivers were observed at 7 and 4 different locations and

those were used to develop the AEM based groundwater flow model.

3 Objective Function and Constraints

In this study, the objective function is defined to identify the minimum pumping and pipingcost for installation of new pumping wells. The total cost for new system of pumping wells

consists of the cost of well installation, piping cost and cost of pumping. The different parts

of the total cost are as follows,

Well Installation Cost: In this study all the wells were considered as of the same depth

of 12 m and diameter of 3 m. Therefore the cost of each well was taken as constant and

the total cost was calculated by multiplying the installation cost of one well by the

number of wells.

Piping Cost: The piping cost depends on the location of new wells along with many other

factors. In this study, the piping length was considered from the wells to a single storage tank andall the pipes were considered as of the same diameter and material. The water from all the wells

will be stored in the storage tank and transported to the city. The piping cost can be given as

Cpp A2XNwi1

Li 1

whereLi (m) 0pipe length from ith well to the storage tank andA2 0total charges for per meter

piping including cost laying and fixing at site. In Eq. (1) assumed that each well is individually

connected to the storage tank.

Pumping Cost: The major factors that influence the pumping cost depend on the volume

of water to be pumped, density of the water, hydraulic head, efficiency of the pump and

energy cost (Sharma and Swamee 2006; Moradi et al. 2003). The total cost of pumping (Cpin euros) consists of the cost of pump units (Cpu in euros) and the capitalized electricity cost

(pumping cost) (CpEin euros) including the annual repair and maintenance cost and hence

can be expressed (Swamee and Sharma1990) for a single well as

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Cpi Cpu CpE kPgQH

8:76REgQHrT

2

#

#

#

##

#

#

#

#

#

#

!.

!.

!.

!.

!.!.

!.

!.

!.

!.

!.

!. OW-1

OW-8

OW-6

OW-5

OW-4

OW-3

OW-2

OW-12

OW-11

OW-10

OW-9

OW-7

685000.000000

685000.000000

687000.000000

687000.000000

2108000.

000000

2108000.

000000

2110000.

000000

2110000.

000000

2112000

.000000

2112000

.000000

0 0.5 1 1.50.25

Kilometers

Legend!. Observation well

# River Head

River

Fig. 1 Location map of the study area



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where

rT 1 r T 1

r 1 r T 3

H0pumping head (m), which is equal to the head from water table in aquifer to the height of

storage tank including head losses in pipes; 0combined efficiency of the pump and the

prime mover;RE 0the cost of the electricity per kilowatt-hour (euros/kwh);r 0the rate of

interest expressed as euros for euros per year (euros/euros/year),T 0total life of the project

(years) andkP0cost of a pump unit per kwh (euros), which can be obtained by interpolating

values from a curve between cost and pump capacity. For long life of project (T)Eq.(3)

givesrT01/r. The pump parameters have been ascertained from market surveys of branded

pumps of various heads and capacities. Thus, the total pumping cost can be expressed as

Cp XNwi1

CpiXNwi1

kP gQiHi

8:76REgQiHirT

4

Total Cost: Thus the total cost can be calculated by adding the above explained three

components. As the well cost is constant, it is not required in the optimization function. Also,

the problem constraints were finalized with the help of water authority officials and stake-

holders. The First constraint prescribes the maximum and minimum discharge limit of a

single well. The second constraint prescribes the minimum discharge limits by all wells. The

third constraint was taken to limit the drawdown of groundwater under permissible limit

defined by stakeholders. The fourth constraint accounts for the minimum distance of the

wells from the river to ensure the minimum retention time for the groundwater in the aquiferand to avoid the influence of pumping on the water level in the river. The fifth constraint

incorporates the minimum distance between the wells in order to provide a protective zone

around the wells. The final form of the objective function is,

MinimizeXNwi1

A2Li kPgQiHi

8:76REgQiHirT

b1Ph b2PQ

( ) 5

Subject to

Qi;min hi;min 5c

xi;yi 6 Ai 5d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixi xj 2

yi yj 2q

Sw;min 5e

932 S. Gaur et al.


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wells and zones of different hydraulic conductivity). The detailed description of the analytic

element method can be found in Strack (1989). In this study, ground water flow model based on

analytic element theory was developed by MATLAB 7.0 (Math Works 2001) and applied to the

case study. The details regarding the development of AEM model for the study area are

explained in Gaur et al. (2011).

4.2 Particle Swarm Optimization

The PSO (Eberhart and Kennedy 1995) is a stochastic, swarm-based evolutionary computer

algorithm for the solution of optimization problems. In PSO, the system is initialized with a

population of particles which represent the potential solution. In simple terms, each particle is

moves through a multidimensional search space, where the position of each particle is adjusted

according to its own experience and that of its neighbors. In this process, each particle keeps a

track of its coordinates in the problem space which are associated with the best solution (fitness)

that it has achieved so far (the fitness value is also stored). Anotherbestvalue that is tracked

by the global version of the particle swarm optimizer is the overall best value, and its location,

obtained so far by any particle in the population. To find the new position of each particle at

each iteration (time step), a velocity term is computed on the basis of experience of particles.

The velocity term and corresponding new location are defined as,

vtij c wvt1ij c1r1 P

t1ij x

t1ij

c2r2 G

t1j x

t1ij

h i 7

xtij x

t

1ij v

tij 8

wherec1andc2are the acceleration constants,r1and r2are random real numbers between 0

to 1. Pt andGtdenote the individual and global best values of particles. In this study, a PSO

model has been developed on MATLAB platform. Different parameters of PSO have been

taken on the basis of literature review and performing the sensitivity analysis.

4.3 ANN Parameters

A typical ANN structure is defined by three types of layers which consisting the different

types of neurons (Hsu et al. 1995) i.e. input neurons, output neurons and hidden neurons.The hidden neurons connect the input layer neurons to output layer. The hidden neurons can

be arranged in one or more hidden layers. The choice for the number of hidden neurons/

layers and the selection of ANN parameters play a key role in the ANN performance. Very

less number of neurons in the hidden layer does not allow the network to produce accurate

maps from the input to the desired output, whereas too many neurons can result in over

fitting (ASCE Task Committee on Application of Artificial Neural Networks in Hydrology

2000). As there are no specific rules for the design the architecture of an ANN (Anmala et al.

2000), a trial-and-error procedure has been adopted in this study by following the guidelines

suggested by various researchers (Atiya and Ji 1997; Morshed and Kaluarachchi1998). The

back propagation technique with Levenberg-Marquardt (L-M) method was used to train the

ANN. A multilayer network with one hidden layer can be used to approximate almost any

function if enough neurons are provided in the hidden layer, and hence, only one hidden

layer has been adopted in this study. The performance of the developed ANN model is

measured on the basis of coefficient of correlation (R2), RMSE and Nash Sutcliffe efficiency.

The stopping criteria for the training were defined on the basis of (1) the specific maximum

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number of epochs (2) if the performance gradient falls below the minimum gradient. The

termination of the training process of the network is justified when the minimum perfor-

mance gradient falls below 1010, because the BPNN performance does not improve even if

further training continues (Hagan et al.1996).

5 Application of AEM, ANN and PSO

In this study ANNs have been created for each scenario. Number of ANNs in each scenario

is equal to number of wells in that scenario. Three scenarios were considered in this study by

choosing the set of four, five and six wells. (a) AEM: Initially the AEM model was used to

compute the groundwater head with the given random discharges and coordinates for those

sets of wells. The random discharges for the pumping wells, for each scenario, were

generated using a random number generator, within the range of 80 m3/hr to 300 m3/hr.

The coordinates for the pumping wells were also generated using randomly within the limits

of the study area by using the random number generator. The AEM model thus computes the

groundwater head at the periphery of the wells. (b) ANN: The dataset generated from AEM

model was used for the training of ANN. The coordinates and the discharge of the wells

were taken as input and groundwater head at the periphery of the well was taken as output

for the testing of the ANN model. (c) PSO: The developed AEM and ANN models were

integrated with PSO for evaluation of the cost function. Particle Swarm Optimization

Algorithm first generates the random coordinates and discharge values of the wells with in

the given range. Further, the objective function is evaluated. The cost obtained by the each

particle is compared with all the other particles and it keeps track of the best solution of allparticles this is known as gbest. Further the particle keeps track of solution obtained by each

particle which is known as pbest, thus the optimal solution is obtained by PSO. The

constraints of the problem were handled by penalty function approach, where the specific

penalty is incorporated for violation of constraints (Han and Mangasarian1979)

At each scenario a two stage optimization search is performed. In the first stage, ANN is

developed for the entire area with coarser grid. Then the ANN is integrated with PSO to find

the optimal well location and optimal discharge values. In the second stage, a fine grid

search is performed by training the ANN for a smaller region which is obtained on the basis

of results of the coarse grid search. Then the ANN is again integrated with PSO to find the

optimal well location and optimal discharge values. Results obtained by ANN-PSO modelare then compared with those from AEM-PSO. The set of optimal solutions generated by

ANN-PSO search is actually based on simulation that introduces a certain degree of error,

the final step of the methodology is therefor to present the optimal set of solutions to AEM

for verification. The simulation results along with the fitness value of the optimal solution

can then be used to decide the relative ranking amongst the final set of solutions.

6 Model Application

Initially the ANN model was trained with the training dataset generated for the whole study

area, ANN-PSO has been applied to minimize the cost for the set of four wells. The results of

the ANN-PSO model were found very inferior in comparison to the results obtained by the

AEM-PSO model. The number of iterations for the convergence of the PSO model was also

found very high. To overcome this problem and to increase the accuracy and efficiency of

the model, two modifications were done.



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First modification was made in the velocity term of the PSO model to deal with the

decision variable of the well coordinates. Normally, the velocity term produces the value in

decimal number which increases the iteration of PSO model exponentially, particularly in

identifying the optimal location of wells. To overcome this problem, velocity term was

restricted to the integer number and units positionwas rounded up to 5 or 10 accordingly. Itmeans that every new search location will be at least 5 m away from the previous location.

This modification helped to reduce the number of iterations significantly. A scenario for the

set of 4 wells has shown that normal AEM-PSO takes more than 2,500 iterations to

converge, whereas the modified model gets converged in less than 1,000 iterations.

In the second modification, the ANN-PSO model was applied in two stages to find the

optimal solution. In the first stage, the training data set was generated for the whole study

area and ANN model was trained for the same. Further, the result of ANN-PSO model was

compared with AEM-PSO model. The ANN-PSO was found unable to converge early to the

best solution, as compared to the AEM-PSO, as ANN model for the larger part was found

inferior to handle the decision variable based on the well coordinates. The failure occurred

due to the limited availability of dataset for training of ANN model near the storage tank. To

overcome this problem, the second stage was applied in which a new dataset was generated

for the smaller part and the ANN model was trained again. The area for the smaller part was

defined based on the results from the ANN-PSO in the first stage. A circle of 100 m radius

was considered around the each well location obtained in the first stage. A new dataset for

these locations was developed and the ANN model was trained into the same. The results

generated by both ANN-PSO and AEM-PSO were compared. The training dataset for the

ANN model was also generated by considering first modifications in PSO. The AEM

model has also been run to generate the dataset from the same values.The following values of different parameters, in the cost function, were considered in

simulation-optimization process. A1 was assumed as 4,000 , which adopts steel casing and

12 m as depth of well. Based on the experience of field experts A2was adopted as 140/meter.

The cost of pump unit was obtained from the table between cost and pump power. Due to

availability of pumps for specific pump power, the pump power for each well was computed

and the appropriate pump was selected from the available pumps in market which has the same

or higher pump power than the required pump power. The values of other parameters selected

wereRE00.08 euros/kwh;09,810 N/m3;080 %;r06 % euros/euros/year andT025 years.

The reference location of the storage tank was fixed after consulting with local authority having

coordinates asX0687,000andY0218,000for computing the pipe length.In the first constraint, on the basis of aquifer properties and availability of pumps, the

discharge limit was selected at 100 m3/hr


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computational time. The ANN-PSO model takes 45 min for 700 iterations of model

convergence, whereas AEM-PSO model takes approximately 56 h for the same number

of iterations.

The trainlmalgorithm of L-M technique toolbox is used for updating the weights and

biases (Hagan et al.1996). Similarly, the transfer functions of both the hidden layer and the

output layer were chosen from the sigmoid transfer functions tansig, log-sigmoid logsig

and the linear purelin. However, the sigmoid transfer function in the hidden layer and the

linear/sigmoid transfer function in the output layer were found better for these networks. The

log-sigmoid transfer function was used and results were compared with the results computed

by the linear transfer function. The sigmoid transfer function proved better than the linear

transfer function, as it reduced the mean square error by 0.34 %.

Fig. 2 R2-value for different number of datasets for the thr

Fig. 3 Optimal cost by AEM-PSO and ANN-PSO models at two sta

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8 Sensitivity Analysis

The sensitivity analysis for the linearly varying inertia weights in PSO has been performed

with different combinations of 1.01.0, 1.00.4, 0.90.4 and 0.80.4. It was found that

choosing the inertia weight 0.80.4 helps to converge the PSO model with less number ofiterations. In this study, the model has been run for different values of the acceleration

constants and 2.01.5 has been found as the good value of the acceleration constant whereas

the 25 particles has been found to converge the model fast. Influence of the piping cost on

the cost function was examined by performing the sensitivity analysis. The sensitivity of the

model with respect to piping cost was examined by changing the pipe cost as 100, 140 and

180 euros per meter. The results show that a decline in the cost of pipe pushes the wells

further from the reference location with reduction in the value of cost function. In this case

pumping cost dominates, which decreases with drawdown and hence the model tries to

minimize the interference between the wells by shifting the wells further from the reference

location. On the other hand, an increase on cost of pipes forces the wells to shift nearer the

reference location leading to increment in the overall cost function. Table2presents lists the

total pumping cost for varying cost of piping. The sensitivity analysis was also performed

without considering the piping cost in the objective function. The result shows that the

pumping wells get shifted in the south part of the study area and near to the river side where

the aquifer has more potential and hence less drawdown and pumping cost, although

increased length of pipes and associated piping cost are involved. This indicates that

consideration of the piping cost in the objective function influences the location of the

pumping wells and hence it is important to include piping cost in the objective function.

9 Summary and Conclusions

In this study, the ANN model was compared to the AEM based simulation model which

were both embedded into the PSO algorithm to minimize the pumping well cost. The

locations of the pumping wells (coordinates) were considered as decision variables. The

piping cost was also considered in the cost function. The ANN was trained using datasets

generated by the AEM model.

The results also show that once the ANN model is properly trained, it is capable to produce

the results similar to the AEM model. Further, the developed ANN model can be used toanalyze different scenarios without spending considerable time for simulation of flow model

using AEM. The results of the ANN-PSO model show that a single model for whole study area

is not capable of producing good results whereas the development of the ANN model for the

smaller area, in two stages, can produce better results. Development of the ANN models for the

smaller area needs fewer amounts of data duly accounting for the interference between the

wells. More important, the ANN-PSO model is capable of finding out the optimal location for

Table 2 Sensitivity analysis for

the cost of piping Set of wells Piping cost

100 (euros/m) 140 (euros/m) 180 (euros/m)

4 236758 276636 326534

5 229712 275951 325569

6 257090 314908 355641



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the pumping wells. The study also suggests that the proposed modifications in the PSO model

are effective to reduce the number of iterations. In future, coupling of pipe network model with

ANN-PSO model can help to find much better pipe network configuration.

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