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, PS
i distributes Q
i 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 L
1, L
2, and L
3 in
Figure 1b are parameterized as linear combinations of the diameters of the pipes:
where λ
p is a parameter defined in each case study, and ND
p 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.
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 V
d, 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.
2.3. 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):
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 F
a applying an interest rate r during N
p 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.
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 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 x
ij, considering the mathematical model of the network and the demand pattern, it is possible to calculate the vectors of the head HS
ij and flow QS
ij that must be supplied by PS
i 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 HS
ij and QS
ij 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 N
B,i for each PS
i. First, the maximum head H
i,max and flow Q
i,max are determined from the setpoint curve. Then, Equation (6) is used to calculate the flow rate Q
i,b supplied by a single pump for the head H
i,max.
In the previous equation, note that the parameters H
0,bi and A
b,i are determined from the characteristic curve of b
i. The number of pumps in PS
i is obtained by Equation (7).
where the result N
B,i is rounded up to the next integer. Finally, the number of VSPs (n
i) can be determined using Equation (8). There must always be at least one VSP.
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.
According to [
19], these values can be calculated as follows. The first term (C
pump) represents the cost of a pump according to Equation (10), where CP
0 and CP
1 depend on the case study.
The second term of Equation (9) constitutes the cost of the frequency inverter (C
inv) for each VSP. The third term (
) is explained in Equation (11), which considers pipes and accessories.
where for each PS, NB is the number of pumps, n
T is the number of union tees, n
e is the number of elbows, n
p is the number of pipes, and l
i 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 (C
control) according to Equation (12). Among these components, a pressure transducer (C
pressure), flowmeter (C
flowmeter), and programmable logic controller (C
PLC) are included.
where C
pressure and C
PLC correspond to the acquisition prices of the pressure switch and programmable logic controller, respectively.
The Cinv and Cfacility values 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 N
t is determined using Equation (13).
where for each PS
i, the parameters H
0,i, Ai, E
i, and F
i are the characteristic coefficients of the pump head and the performance curve and are extracted from an existing database depending on the pump model; Q
i,j,k represents the discharge of pump k during time step j in PS i; p
i,j is the energy cost, ϒ is the specific gravity of water, Δt
j is the discretization interval of the optimization period, and the numbers of FSPs and VSPs running at time step j are represented by m
i,j and n
i,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 HS
i,j. To achieve this, the parameter α is calculated according to Equation (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.
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
Table 1 and
Table 2, respectively. To calculate OPEX,
Table 3 presents the hourly electricity tariff for each PS in the MTF network.
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.
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 F
a = 7.92%. To calculate the length of the pipes (L
p), values of 5, 30, and 10 were used for λ
1, λ
2, and λ
3, respectively.
To calculate CAPEX, the CP
0 and CP
1 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) = ax
2 + 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 (C
pressure) and programmable logic controller (C
PLC), they assume a constant price of 570 and 372.44 EUR, respectively. Furthermore, the flowmeter is always installed in outlet pipe L
3 (
Figure 1b) and therefore has the same diameter as that pipe.