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Water 2021, 13, 3098. https://doi.org/10.3390/w13213098 www.mdpi.com/journal/water
Article
Pumping Station Design in Water Distribution Networks
Considering the Optimal Flow Distribution between Sources
and Capital and Operating Costs
Jimmy H. Gutiérrez-Bahamondes 1, Daniel Mora-Meliá 2,*, Pedro L. Iglesias-Rey 3, F. Javier Martínez-Solano 3
and Yamisleydi Salgueiro 4
1 Doctorado en Sistemas de Ingeniería, Facultad de Ingeniería, Universidad de Talca, Camino Los Niches
Km 1, Curicó 3340000, Chile; jgutierrezb@utalca.cl
2 Departamento de Ingeniería y Gestión de la Construcción, Facultad de Ingeniería, Universidad de Talca,
Camino Los Niches Km 1, Curicó 3340000, Chile
3 Departamento de Ingeniería Hidráulica y Medio Ambiente, Universitat Politècnica de València, Camino de
Vera S/N, 46022 Valencia, Spain; piglesia@upv.es (P.L.I.-R.); jmsolano@upv.es (F.J.M.-S.)
4 Departamento de Ciencias de la Computación, Facultad de Ingeniería, Universidad de Talca, Camino Los
Niches Km 1, Curicó 3340000, Chile; ysalgueiro@utalca.cl
* Correspondence: damora@utalca.cl
Abstract: The investment and operating costs of pumping stations in drinking water distribution
networks are some of the highest public costs in urban sectors. Generally, these systems are
designed based on extreme scenarios. However, in periods of normal operation, extra energy is
produced, thereby generating excess costs. To avoid this problem, this work presents a new
methodology for the design of pumping stations. The proposed technique is based on the use of a
setpoint curve to optimize the operating and investment costs of a station simultaneously.
According to this purpose, a novel mathematical optimization model is developed. The solution
output by the model includes the selection of the pumps, the dimensions of pipelines, and the
optimal flow distribution among all water sources for a given network. To demonstrate the
advantages of using this technique, a case study network is presented. A pseudo-genetic algorithm
(PGA) is implemented to resolve the optimization model. Finally, the obtained results show that it
is possible to determine the full design and operating conditions required to achieve the lowest cost
in a multiple pump station network.
Keywords: optimization; water networks; pump station; setpoint curve; pseudo-genetic algorithm
1. Introduction
Optimization problems regarding the design and operation of water distribution
networks (WDNs) are very complex and important problems that affect the quality of life
of all people worldwide [1]. The demand for water increases rapidly with the growth of
the world’s population. Furthermore, climate change has increased water scarcity [2].
Electric power is one of the dominant costs incurred by water utilities, so reducing energy
consumption and conserving the available natural resources (such as water) are some of
society’s challenges [3]. The design of a cost-effective WDN is not a simple task, and the
operational performance of the designed network affects any city budget [4,5]. In this
regard, pumping stations (PSs) are expensive infrastructures.
Pump operating costs are a key aspect when a network is fed directly from
groundwater or does not have a high enough elevation for tanks to be installed [5]. In
these cases, the selection of pumps that best adapt to the system head curve is an
Citation: Gutiérrez-Bahamondes,
J.H.; Mora-Meliá, D.; Iglesias-Rey,
P.L.; Martínez-Solano, F.J.; Salgueiro,
Y. Pumping Station Design in Water
Distribution Networks Considering
the Optimal Flow Distribution
between Sources and Capital and
Operating Costs. Water 2021, 13,
3098. https://doi.org/10.3390/
w13213098
Academic Editor: Wencheng Guo
Received: 4 October 2021
Accepted: 1 November 2021
Published: 3 November 2021
Publisher’s Note: MDPI stays
neutral with regard to jurisdictional
claims in published maps and
institutional affiliations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license
(https://creativecommons.org/license
s/by/4.0/).
Water 2021, 13, 3098 2 of 15
important step, as doing so reduces excess head pressure [6]. Consequently, the efficient
design and operation of PSs could significantly reduce network total costs.
PS design involves the selection of pumps, accessories, and control systems.
Traditionally, the selection process is conducted by a catalogue that contains the
specifications of available pumps. The pumps that provide the required maximum
piezometric head are identified, and the selection is based, among other criteria, on energy
efficiency. Finally, the number of pumps that make up the station is calculated [7]. Then,
PS operation is determined by the on/off status of every pump at each time step and, in
the case of variable-speed pumps (VSPs), by modifying the rotational speed based on the
readings obtained from the metering devices.
This control system is responsible for adjusting the amount of water pumped
according to public demand at every moment [8]. Different approaches for improving PS
performance in WDNs have been explored [1,9,10]. One of the important problems
covered in the literature is pump scheduling optimization. This problem has been widely
studied [3,9]. Starting from a completed PS design, pump scheduling can be specified by
on/off pump switches during predefined equal time intervals. Traditionally, the objective
of such a problem is the minimization of energy consumption costs based on the operation
and/or maintenance costs of pumps [3]. These approaches do not allow for any structural
changes. Consequently, the obtainable energy savings are restricted by the previous PS
design, and insufficient or idle capacity may be generated. Therefore, the investment and
operating costs of PSs should be minimized together to achieve a complete solution to the
optimization problem [11].
In recent years, mathematical models have been developed to optimize both of the
aforementioned general objectives from diverse perspectives. The differences between
these models lie in the decision variables used to assemble their object functions and
constraints. In this regard, some previous researchers have focused their efforts on
optimizing the location of each PS, maximizing energy production [12,13], minimizing
losses by installing turbines to recover energy [14], or using the position and setting of a
PS, in addition to the working statuses of VSPs, as the decision variables [15]. Other
researchers have focused on analyzing the PS capacity to optimize the design and
operational costs of stations; the authors of [16] introduced a model with binary variables
for each design and operation option available in the face of a small number of demand
scenarios. The authors of [11] proposed a new design procedure that considers the
variable demand, equivalent flow, and equivalent volume to approximate the annual
costs of a given system. The authors of [17] calculated the design cost of an entire network
using a robust model based on the variability of the costs of multiple scenarios.
The main problem is that the methods proposed in the literature do not directly link
the internal design optimization of PSs with the optimization of the overall operation of
the corresponding WDN. Furthermore, the design is formulated while considering
extreme operating conditions. To bring these models closer to reality, the design of a PS
should evaluate its impact on the entire network, including other PSs, and consider
operational implications in the future. To avoid these problems, there are two important
stages that must be optimized together. The first stage is to determine the optimal flow
and head of a PS. Therefore, the PS should satisfy all demand and pressure requirements.
The second stage is to determine the best combination of the internal elements in PSs to
achieve these values.
In this context, the authors of [18] minimized the energy consumption of a network
after determining that water should be provided by each PS according to the demand
curve of the network using a novel optimization model based on a setpoint curve. The
heads and flows that must be supplied from each PS are determined, but the design of the
PS is not considered. From this viewpoint, considering the results of previous work, the
authors of [19] focused their research only on the internal configuration of a PS. The
authors proposed an alternative method in which the pumping group selection process
includes an estimate of the system operating cost based on the study of different control
Water 2021, 13, 3098 3 of 15
systems and operation schemes for the pumps before selecting the equipment. To achieve
this, investment (pumping equipment, hydraulic installations, and electrical and control
equipment) and operational costs are considered.
This work presents a novel methodology for the optimization of the design of PSs in
WDNs. Unlike the models developed in previous works, the proposed model combines
the overall optimization process for the distribution of flows and the analysis of the
internal components of each PS. It allows for energy and investment costs to be reduced
by adjusting the PS capacity according to demand and pump selection at once. For this
purpose, the calculation of a setpoint curve and the simulated operation of multiple pump
alternatives are combined to create a highly robust optimization model that adjusts the
flow rates provided by the pumps according to the given system’s demand curves.
Additionally, our work considers the resolution of the optimization model through
the implementation of a pseudo-genetic algorithm (PGA) that was presented by [20]. In
this regard, evolutionary algorithms have proved to be efficient in handling optimization
problems with respect to WDNs, especially when the size of the feasible solution space is
extremely large [1,21]. The evaluation of the hydraulic behavior of the resulting network
is analyzed using EPANET according to the specifications proposed in [22].
The remainder of the paper is organized as follows: Section 2 describes the proposed
methodology. First, the outline, notation, and formulation of the new mathematical model
are discussed. Then, the developed methodology is applied to a case study, and an
optimization method is implemented. Next, Section 3 provides the results, and a
discussion is detailed in Section 4. Finally, the conclusions of the research can be found in
Section 5.
2. Materials and Methods
2.1. Model Outline
In this section, a novel optimization model for designing PSs while considering
optimal operating conditions is proposed and mathematically described in detail.
Specifically, this model determines the configuration of each PS, including the number of
fixed-speed pumps (FSPs) and VSPs, and the pump model according to an available
database. The optimization model adjusts the PS design to the optimal distribution of
flows, which is calculated during each period within the optimization process.
It is important to highlight that the proposed methodology requires some available
data: (a) a WDN model calibrated for different demand conditions, (b) a modular design
for the PS, (c) knowledge of the demand patterns, and (d) an existing database to select
the correct pump model. The database must include the model, price, and head and
efficiency curves for each pump.
Next, a general scheme for the problem to be solved is presented in Figure 1.
Specifically, Figure 1a shows a general case with three PSs. During each period, PSi
distributes Qi of water from the total flow. These flows vary over time according to the
demand pattern of the WDN. Figure 1b shows the details of the basic modular design of
a PS used in this optimization model. A PS consists of several lines in parallel with one
pump installed in each. Every pump has two isolation valves and a check valve. There are
also two isolation valves at the ends (inlet and outlet) of the PS. The lengths L1, L2, and L3
in Figure 1b are parameterized as linear combinations of the diameters of the pipes:
(1)
where ʎp is a parameter defined in each case study, and NDp is the nominal diameter (ND)
of the corresponding pipe p, which is used for defining the diameters of elements such as
isolation valves or check valves.
Water 2021, 13, 3098 4 of 15
(a)
(b)
Figure 1. (a) General scheme and (b) PS modular design.
This methodology calculates the optimal flow rates provided by each PS following
the methodology presented by [18]. Next, the number of pumps required is determined
according to the model. Then, considering the design velocity Vd, the NDs are selected,
and finally, the lengths of the pipes are determined using Equation (1). That is, using the
basic modular design from Figure 1b after the model and number of pumps are
determined automatically leads to a specific PS design.
The optimization problem seeks to minimize the capital expenditures (CAPEX) and
operating expenses (OPEX) of the general scheme presented in Figure 1 while considering
an optimal distribution of flows. First, the decision variables and the mathematical
notation of the proposed model are presented. Next, the calculation of OPEX and CAPEX
from the decision variables is explained in detail. A case study exemplifies the model
implementation in a network with three PSs and a database of pump models. Finally, the
optimization method used in this work is briefly presented.
2.2. Mathematical Notation
The above problem can be posed as a mathematical optimization model, where the
decision variables are related to the distribution of flows among the different sources and
the configuration of each PS. On the one hand, xij defines the percentage of the flow
supplied from PS i (PSi) at each time step j. The parameters Nt and Nps represent the total
number of time steps and total number of PSs, respectively; mi indicates the number of
FSPs; and bi corresponds to the identifier of the pump model to be installed in PSi.
Once these values are known, it is possible to calculate the maximum flow for each
PS, the number of total pumps (NB,i), the number of VSPs (ni), and the dimensions of each
pipeline Lp. In short, it is possible to fully define the design of the PS.
2.3. Optimization Model
The optimization problem seeks to minimize both the CAPEX and the OPEX of the
system. The objective function is detailed in Equations (2) and (3):
(2)
(3)
where F represents the total annualized cost of the project. To calculate the loss of value
of the assets over the useful life of the project, CAPEX is amortized by the factor Fa
Water 2021, 13, 3098 5 of 15
applying an interest rate r during Np periods. OPEX represents the total operational
expenses throughout the life of the project.
Obviously, the optimization model is restricted by continuity and momentum
equations and by minimum head requirements in the demand nodes. Additionally, the
model is constrained by Equations (4) and (5). These equations guarantee that the total
flow supplied by the PS is equal to the flow demanded during each period.
(4)
(5)
The optimization model calculates the CAPEX and OPEX from the values of the
decision variables in each iteration of the algorithm. Figure 2 shows a flowchart of the
complete model.
.
Figure 2. Calculation of OPEX and CAPEX.
Figure 2 shows how the decision variables and the model input requirements are
related. The dashed lines represent four intermediate steps required to calculate the
CAPEX and OPEX. After solving the mathematical model, it is possible to determine the
setpoint curve for each PS, the complete PS design, the dimensions of all pipes, the number
of pumps, the respective models, and the PS control systems.
This methodology can be described through three steps. The first step determines the
setpoint curve using the methodology proposed by [18]. The setpoint curve represents the
minimum head required at each PS for a certain flow distribution. From the distribution
of flows xij, considering the mathematical model of the network and the demand pattern,
it is possible to calculate the vectors of the head HSij and flow QSij that must be supplied
by PSi at time step j. Then, the control system is adjusted in such a way that the output of
the PSs is always equal to the HSij and QSij values of the setpoint curve. Therefore, it is
possible to guarantee compliance with the minimum pressure restrictions for all nodes of
the network.
The second step calculates the total number of pumps NB,i for each PSi. First, the
maximum head Hi,max and flow Qi,max are determined from the setpoint curve. Then,
Equation (6) is used to calculate the flow rate Qi,b supplied by a single pump for the head
Hi,max.
Water 2021, 13, 3098 6 of 15
(6)
In the previous equation, note that the parameters H0,bi and Ab,i are determined from
the characteristic curve of bi. The number of pumps in PSi is obtained by Equation (7).
,
(7)
where the result NB,i is rounded up to the next integer. Finally, the number of VSPs (ni)
can be determined using Equation (8). There must always be at least one VSP.
(8)
The third step defines the PS design. A PS is completely defined when the maximum
demand of the PS, the number of pumps, and the pump model selected in the database
are known. Then, it is possible to calculate both the CAPEX and OPEX.
The CAPEX is calculated according to Equation (9), representing the total investment
costs for each PS.
(9)
According to [19], these values can be calculated as follows. The first term (Cpump)
represents the cost of a pump according to Equation (10), where CP0 and CP1 depend on
the case study.
(10)
The second term of Equation (9) constitutes the cost of the frequency inverter (Cinv)
for each VSP. The third term () is explained in Equation (11), which considers pipes
and accessories.
(11)
where for each PS, NB is the number of pumps, nT is the number of union tees, ne is the
number of elbows, np is the number of pipes, and li is the length of pipe i.
Specifically, the previous equation considers the costs of isolation valves (CSV), check
valves (CCV), pipes (Cpipe), elbows (Celbow), and union tees (CT).
Finally, the fourth term represents all control components (Ccontrol) according to
Equation (12). Among these components, a pressure transducer (Cpressure), flowmeter
(Cflowmeter), and programmable logic controller (CPLC) are included.
(12)
where Cpressure and CPLC correspond to the acquisition prices of the pressure switch and
programmable logic controller, respectively.
The Cinv and Cfacilityvalues can be expressed as second-degree polynomial curves as
functions of the pump power P (kW). Similarly, CSV, CCV, Cpipe, Celbow, CT, and Cflowmeter are
functions of the ND and fit to second-degree polynomial curves. All coefficients of the
polynomials incorporated in the previous equations depend on the case study.
Finally, to calculate the OPEX, the cost of the total electrical power consumed by all
pumps running in the WDN during time step Nt is determined using Equation (13).
(13)
Water 2021, 13, 3098 7 of 15
where for each PSi, the parameters H0,i, Ai, Ei, and Fi are the characteristic coefficients of
the pump head and the performance curve and are extracted from an existing database
depending on the pump model; Qi,j,k represents thedischarge of pump k during time step
j in PS i; pi,j is the energy cost, ϒ is the specific gravity of water, Δtj is the discretization
interval of the optimization period, and the numbers of FSPs and VSPs running at time
step j are represented by mi,j and ni,j, respectively. These values depend on the selected
pump model and the system selected to control the operation point. In this work, pumps
are controlled by adjusting their heads to the setpoint curve HSi,j. To achieve this, the
parameter α is calculated according to Equation (14).
(14)
2.4. Case Study
To apply the methodology described above, one case study was conducted. Figure 3
shows the topology of a new WDN called the MTF network.
Figure 3. MTF network.
The MTF network has 3 PSs (PS1, PS2, and PS3), 15 consumption nodes, and 25 pipes.
A hydraulic analysis was carried out for one day, and the time was discretized in periods
of one hour. The minimum pressure at the nodes is 20 m, the demanded average flow rate
is 100 L/s, and the roughness coefficient is 0.1. Information about the nodes and pipelines
is detailed in Tables 1 and 2, respectively. To calculate OPEX, Table 3 presents the hourly
electricity tariff for each PS in the MTF network.
Water 2021, 13, 3098 8 of 15
Table 1. Case study network: node information.
ID
Elevation (m)
Base Demand (L/s)
ID
Elevation (m)
Base Demand (L/s)
N2
8
5
N11
7
5
N3
8
4
N12
7
10
N4
5
3
N13
5
5
N5
8
4
N14
4
2
N6
4
3
N15
3
10
N7
2
8
N16
3
15
N8
5
7
PS2
4
-
N9
6
10
PS3
0
-
N10
2
9
PS1
23
-
Table 2. Case study network: pipeline information.
Pipe ID
Node 1
Node 2
Length (m)
Diam. (mm)
Pipe ID
Node 1
Node 2
Length (m)
Diam. (mm)
1
N2
N3
200
200
13
N12
N8
300
100
2
N3
N4
150
150
14
N12
N5
250
150
3
N4
N5
150
150
15
N9
N13
250
100
4
N5
N2
200
250
16
N6
N14
100
150
5
N6
N7
200
150
17
N4
N13
98
200
6
N8
N9
400
100
18
N4
N15
300
100
7
N7
N8
300
100
19
N15
N16
500
100
8
N9
N6
300
100
20
N3
N16
400
150
9
N9
N5
250
100
21
PS1
N2
1500
300
10
N8
N10
300
100
22
PS2
N11
125
100
11
N11
N12
300
100
23
N13
N14
52
150
12
N10
PS2
125
100
24
N13
PS3
100
100
Table 3. Electricity for the case study (€/kWh).
Time (h)
PS1
PS2
PS3
1–8
0.094
0.092
0.09
9–18
0.133
0.131
0.129
19–22
0.166
0.164
0.162
23–24
0.133
0.131
0.129
In Table 1, the base demand represents the average or nominal demand for water at
the junction. A time pattern is used to characterize time variation in demand, providing
multipliers that are applied to the base demand to determine actual demand in a given
time period. The demand patterns for the 24 h of a day are presented in Figure 4.
Water 2021, 13, 3098 9 of 15
Figure 4. Demand pattern for the MTF network.
To perform the optimization process, a database with 67 pump models was used. The
maximum flow rate of the pumps in the database varied between 9 L/s and 50.7 L/s. The
maximum head fluctuated between 15.8 m and 105 m, and the maximum efficiency was
in the range of 39% to 84%. The annualized costs of these models were calculated using
an interest rate of 5% per year and a projection time of 20 years, as indicated by [7]. This
led to an amortization factor Fa = 7.92%. To calculate the length of the pipes (Lp), values of
5, 30, and 10 were used for ʎ1, ʎ2, and ʎ3, respectively.
To calculate CAPEX, the CP0 and CP1 values from Equation (10) are 142.88 and 0.5437,
respectively, for efficiency values less than or equal to 65% or 203.14 and 0.6115 otherwise.
The coefficients of the polynomial equations used to calculate Equation (9) are expressed
in the format f(x) = ax2 + bx + c. The independent variable (x) can be represented by P or
ND. The values of coefficients a, b, and c are shown in Table 4. These values were
determined by [19]. In the case of the pressure transducer (Cpressure) and programmable
logic controller (CPLC), they assume a constant price of 570 and 372.44 EUR, respectively.
Furthermore, the flowmeter is always installed in outlet pipe L3 (Figure 1b) and therefore
has the same diameter as that pipe.
Table 4. Case study coefficients for calculating CAPEX equations.
f(x)
x
a
b
c
Cinv
P(W)
−0.603
116.08
168.19
CSV
ND (mm)
0.01
1.53
11.82
CCV
0.006
0.25
14.55
Cpipe
0.0004
−0.15
8.01
Celbow
0.02
−4.23
269.49
CT
0.001
−1.74
144.24
Cflowmeter
0.051
−7.91
716.64
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
12345678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Multiplier
Time (time period = 1.0 hour)
Water 2021, 13, 3098 10 of 15
2.5. Optimization Method
The proposed optimization model requires the use of a computational method to
solve a given problem. The solution space of the problem to be solved in this case study
is equal to 10104 possible combinations, and the objective function is not linear. Therefore,
the use of traditional deterministic optimization methodologies is not possible. Because
of the huge space of solutions and the complexity of the objective function, it is advisable
to use a heuristic method [5]. In the literature, there are many available algorithms that
can be adapted to solve the proposed mathematical model. In this case, the meta-heuristic
algorithm chosen is a pseudo-genetic algorithm (PGA) developed by the authors, whose
details and particularities can be found in [20].
Unlike the traditional version of the genetic algorithm, the PGA is based on the
integer coding of its solutions. The software was implemented in the Java language
following the specification of [22]. This system can conduct massive simulations and is
integrated with the hydraulic network solver EPANET using the programmer’s toolkit
[23].
Different parameters guide the search process of the algorithm, so the obtained
results are sensitive to the values of these parameters. The main parameters of the
proposed PGA are the population size (P), crossover frequency (Pc), and mutation
frequency (Pm). We have adopted the optimal parameter calibration range obtained in
previous works for the PGA [22,24–26]. To ensure a minimum level of statistical
confidence of the results, 500 experiments were performed and analyzed.
3. Results
Through the proposed methodology, the designs and operations of the PSs in the
case study were optimized. Table 5 details the dimensions of ND1, ND2, and ND3 and the
lengths of pipes L1, L2, and L3 according to the modular design presented in Figure 1b.
Furthermore, Table 5 shows the number of FSPs and VSPs (mi, ni) and the characteristic
and efficiency curve coefficients H0, A, E, and F. In addition, the last row displays the
selected model pump from the database.
Table 5. Pump station designs for the case study.
PS1
PS2
PS3
ND1
300
200
150
(mm)
ND2
125
125
150
ND3
300
200
150
L1
1.5
1
0.75
(m)
L2
3.75
3.75
4.5
L3
3
2
1.5
mi
0
0
0
ni
5
2
1
H0
38.0800
53.7000
104.9800
A
−0.02094
−0.06002
−0.04438
E
0.07130
0.10899
0.05182
F
0.00167
0.00364
0.00107
Model Id
GNI 50–16/10
GNI 100–20/50
GNI 50–26/40
Figure 5 details the flow that must be supplied by at time step according
to the calculated setpoint curve. The pattern of daily consumption is represented by a
dotted line. Additionally, the number of active pumps is included in the middle of each
bar of the graph. If the color associated with a PS does not appear in a bar, it means that it
is not necessary to turn on any pumps during that period. The results show that PS1
supplies a large portion of the total flow. However, depending on the required flow and
Water 2021, 13, 3098 11 of 15
head calculated, the mathematical model determines the optimal combination of active
pumps and its speed of rotation to be able to supply exactly the demand of each period.
Figure 5. Optimized case study operations.
To clarify, Table 6 shows the details of the rotation speed fraction of the pumps (α),
used to calculate the efficiency (η) according to Equation (15).
(15)
where Q is the flow driven by the PS, n is the number of pumps, and E and F are the
efficiency curve coefficients. It is important to highlight that, despite the loss of efficiency
of the pumps, the level of power consumption in the PSs is the minimum to ensure that
all nodes of the network reach the minimum pressure of the network (20 m). The flows Qij
and heads HSij are also presented for each PSi during a 24 h period.
Table 6. PS designs for the case study.
PS1
PS2
PS3
T(h)
α
η
QS(l/s)
HS(m)
α
η
QS(l/s)
HS(m)
α
η
QS(L/s)
HS(m)
1
0.714
0.413
25.500
5.773
0.629
0.593
4.500
20.020
-
-
-
-
2
0.436
0.756
10.000
5.137
-
-
-
-
-
-
-
-
3
0.436
0.756
10.000
5.137
-
-
-
-
-
-
-
-
4
0.436
0.756
10.000
5.137
-
-
-
-
-
-
-
-
5
0.436
0.756
10.000
5.137
-
-
-
-
-
-
-
-
6
0.603
0.526
20.000
5.483
-
-
-
-
-
-
-
-
7
0.714
0.413
25.500
5.773
0.629
0.593
4.500
20.020
-
-
-
-
8
0.863
0.671
98.780
15.564
0.706
0.812
11.220
19.231
-
-
-
-
9
0.916
0.736
114.900
20.888
0.782
0.814
24.300
24.011
0.591
0.591
10.800
31.436
10
0.972
0.741
119.765
23.924
0.921
0.787
32.640
29.544
0.694
0.629
17.595
36.784
11
0.916
0.736
114.900
20.888
0.782
0.814
24.300
24.011
0.591
0.591
10.800
31.436
12
0.920
0.741
113.440
21.460
0.885
0.794
30.720
27.944
0.662
0.630
15.840
34.838
11 1 1 1 11
455554 4 5443 3 44543
1 1
1
2222
2 2
2
21
2 2
12
2
2
1
1
1
11
1 1
1
1 1
1
1
0
20
40
60
80
100
120
140
160
180
200
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Q (l/s)
Time (h)
PS1 PS2 PS3 Water demand network
Water 2021, 13, 3098 12 of 15
13
0.870
0.683
97.790
16.301
0.830
0.804
27.720
25.506
0.631
0.628
14.490
32.491
14
0.870
0.683
97.790
16.301
0.830
0.804
27.720
25.506
0.631
0.628
14.490
32.491
15
0.920
0.741
113.440
21.460
0.885
0.794
30.720
27.944
0.662
0.630
15.840
34.838
16
0.863
0.673
98.410
15.662
0.761
0.814
23.400
22.903
0.552
0.534
8.190
29.053
17
0.863
0.671
98.780
15.564
0.706
0.812
11.220
19.231
-
-
-
-
18
0.809
0.584
76.600
11.240
0.773
0.815
23.400
23.909
-
-
-
-
19
0.809
0.584
76.600
11.240
0.773
0.815
23.400
23.909
-
-
-
-
20
0.863
0.671
98.780
15.564
0.706
0.812
11.220
19.231
-
-
-
-
21
0.863
0.673
98.410
15.662
0.761
0.814
23.400
22.903
0.552
0.534
8.190
29.053
22
1.000
0.748
120.175
25.942
0.992
0.772
36.480
32.888
0.986
0.534
33.345
52.681
23
0.863
0.673
98.410
15.662
0.761
0.814
23.400
22.903
0.552
0.534
8.190
29.053
24
0.793
0.588
74.880
10.901
0.800
0.758
15.120
20.663
-
-
-
-
Table 7 details the CAPEX and OPEX values for the best solution found. First, on the
dotted line, all CAPEX terms of Equation (9) are exposed. The OPEX costs are shown
according to Equation (13). The last row details the annualized costs according to Equation
(2) for each PS. In this case study, the calculated coefficient Fa is 0.0791. The total annualized
cost is highlighted in the lower right corner of the table.
Table 7. Cost details of the obtained solution.
PS1
PS2
PS3
Total
CAPEX
Cpump
€ 51,225
€ 20,357
€ 8698
€ 80,280
Cinv
€ 5024
€ 7274
€ 3108
€ 15,406
CFacility
€ 16,928
€ 4809
€ 2547
€ 24,284
CControl
€ 3297
€ 1543
€ 1048
€ 5887
OPEX
€ 19,964
€ 5750
€ 4578
€ 30,292
Fa x CAPEX + OPEX
€ 26,020
€ 8441
€ 5798
€ 40,259
Finally, the curve on Figure 6 shows the evolution of objective function value during
the process optimization. The best solution is marked by points, while the shaded area
represents the represents the solution intervals for the 500 experiments.
Figure 6. Optimization process evolution.
Water 2021, 13, 3098 13 of 15
4. Discussion
The application of the new methodology aims to find the optimal economic design
of PS1, PS2, and PS3, minimizing the sum of CAPEX and OPEX. In addition to the
complete design of the PSs, the solution determines the operational conditions of each
pump for each time step of the day.
The calculation of the setpoint curve is the key to the methodology presented. By
delivering exactly the energy established by the setpoint curve, it is possible to ensure
compliance with the predefined head requirements using the minimum amount of
energy. This information is then used to design all PSs at once.
The traditional design method takes only the highest point of demand [7].
Consequently, the distribution of flows is considered an isolated optimization stage and
does not consider the effects of the operations during the rest of the daily periods.
Therefore, the process of finding the lowest cost solution is complex and ineffective. In
contrast, the new mathematical model considers the possible energy consumption of
WDN operations over the analysis time for each possible design.
In the case study, if only the period with the highest demand is considered (time = 22
h), there are many design combinations for each PS that would allow for the flow rates
and heads to be achieved. However, optimizing total costs is not successful. For example,
if the pump model selected for PS2 in Table 5 is used for PS1 and PS2, it would be possible
to meet the head and flow requirements; however, the total cost would be EUR 50,334.
That is, the cost would be 25% more expensive than that of the suggested implementation.
In this way, if only the model selected for PS3 is used, the total cost would be EUR 70,980,
which represents an increase of 76% compared to the cost of the best solution found.
Similarly, the same comparison can be made for all combination options for the 67 models
in the database.
The solution obtained indicates that the algorithm did not select any FSP. Although
the implementation of VSPs is more expensive and incorporates the loss of efficiency
introduced by frequency inverters, these pumps allow for the reduction of excessive
energy consumption. Therefore, this turns into lower economic costs. Nevertheless, these
results are not generalizable to all networks because the number of pumps in each of them
depends on the conditions of each WDN and the current energy prices.
Despite the benefits of the work presented, it does present some limitations. In this
work, the control system of pumps fits their heads to a calculated setpoint curve. VSPs are
used first, and then FSPs are applied. Therefore, to extend the proposed model, it would
be interesting to incorporate other control systems. Additionally, the calculation of OPEX
is presented through a second-degree equation; however, the incorporation of a different
exponent does not pose a major problem. That is, it does not affect the methodology
presented.
5. Conclusions
The design of PSs in a WDN is a very complex and critical problem for any city. Water
pumping operations represent a large fraction of the electrical power consumed in urban
zones. The investment cost of each PS is very high and directly influences the future
energy consumption of the overall network.
Traditionally, this type of design problem is solved by considering only the extreme
consumption points or by optimizing the design and operation objectives in two
independent stages. This approach does not allow for the capacity of the PS to be adapted
to actual operating conditions. Therefore, excess energy is generated, so the costs of WDN
grow significantly. To avoid this, a new methodology was proposed in this paper for the
design of PSs in networks with multiple pumped water sources and no storage.
A mathematical optimization model was developed to simultaneously minimize the
sum of the OPEX and CAPEX incurred by the network through the use of a setpoint curve.
Based on the results obtained from the developed model, it is possible to determine the
Water 2021, 13, 3098 14 of 15
optimal selection of pump models from a database, the numbers of VSPs and FSPs, the
lengths and diameters of the pipes, the distribution of flows, and the statuses of the pumps
during each period.
A case study was presented, and the results obtained from the application of the PGA
were satisfactory. The main benefits of using this new methodology are clearly
demonstrable. The use of the setpoint curve ensured compliance with the requests made
throughout the WDN and reduced energy excesses by reducing the differences between
the consumption curves and the heads used by the pumps. Furthermore, the design of
each PS considered the impact of that station on the entire network, including the effects
on the rest of the PSs in the long term.
As discussed in the paper, the presented work has some limitations. Only one control
system for the pumps is contemplated. That is, VSPs follow the setpoint head, while FSPs
only have on/off states. However, the inclusion of new pump control methods does not
require any change in the general methodology.
Author Contributions: All authors contributed extensively to the work presented in this paper.
Conceptualization, J.H.G.-B., P.L.I.-R., and F.J.M.-S.; data curation, J.H.G.-B., Y.S., and F.J.M.-S.;
formal analysis, J.H.G.-B., P.L.I.-R., and F.J.M.-S.; funding acquisition D.M.-M.; investigation, J.H.G.-
B., P.L.I.-R., F.J.M.-S., and D.M.-M.; methodology J.H.G.-B., P.L.I.-R., F.J.M.-S., and D.M.-M.; project
administration P.L.I.-R. and F.J.M.-S.; resources, J.H.G.-B., P.L.I.-R., F.J.M.-S., and D.M.-M.;
software, J.H.G.-B., P.L.I.-R., Y.S., and F.J.M.-S.; supervision, P.L.I.-R. and D.M.-M.; validation,
J.H.G.-B., P.L.I.-R., F.J.M.-S., Y.S., and D.M.-M.; visualization, J.H.G.-B., P.L.I.-R., F.J.M.-S., and D.M.-
M.; writing—original draft, J.H.G.-B., P.L.I.-R., F.J.M.-S., and D.M.-M.; writing—review and editing,
J.H.G.-B., P.L.I.-R., F.J.M.-S., and D.M.-M. All authors have read and agreed to the published version
of the manuscript.
Funding: This research was funded by the Program Fondecyt Regular, grant number 1210410.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Acknowledgments: This work was supported by the Program Fondecyt Regular (Project
N°1210410) of the Agencia Nacional de Investigación y Desarrollo (ANID), Chile. It is also
supported by CONICYT PFCHA/DOCTORADO BECAS CHILE/2018–21182013.
Conflicts of Interest: The authors declare no conflict of interest.
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