Introduction

Currently, many environmental effects of climate change have been observed (“Effects” 2022). Greenhouse gas (GHG) emissions, primarily due to energy generation, are major contributors to climate change (“Causes of Climate Change” 2022). Various sectors use these energies to support the well-being of human society, particularly the water industry, which accounts for 3% of the global total energy consumption. The global total electricity consumption in the water and wastewater sectors is expected to increase by 33% over the next 20 years, primarily because of population growth and infrastructure development (Boutin and Bergerand 2013). Water distribution systems (WDSs) consume a significant amount of electricity in the USA. Public water supply utilities use an estimated 39.2 TWh of electrical energy per year, with pumping operations accounting for approximately 80% of the total consumption (Pabi et al. 2013). Water and wastewater utilities can consume up to 6% of regional electricity (Wakeel and Chen 2016). Hence, WDSs need efficient energy management to cope with increasing demand while mitigating impacts on climate change.

A WDS consists of various elements working together to transport water from its source to users. Some of these elements, such as tanks and pumps, are categorized as active elements that actively control and interact with the water being distributed. Considering the presence of these active physical components, WDS can adapt its energy consumption by optimizing its operational procedures, as demonstrated by demand response programs (Menke et al. 2016; Liu et al. 2020; Jalilian et al. 2022). The optimization of WDS components has been the subject of extensive research, with early works discussing the sizing of pipes, valves, tanks, and pumps in terms of the design cost (Lansey and Mays 1989). In the case of existing networks, it is feasible to apply this optimization to pumps, which are energy-intensive components, and tanks to reduce pump workload.

Pumps are an essential component in WDS, ensuring end users receive the required quantity of water at an acceptable pressure. As pump operation incurs significant expenses for water utilities, numerous efforts have been made to optimize their operation using a variety of objective and optimization techniques (Mala-Jetmarova et al. 2017). Conventionally, objective functions such as pump operation and maintenance cost, GHG emissions, water treatment cost, water quality, and pressure deficit are typically employed. Optimization procedures can also be divided into single objective (SO) and multi-objective optimization (MOO), with both focusing on minimizing costs. From the SO perspective, López-Ibáñez et al. (2008) applied ant colony optimization and compared it with a hybrid and simple genetic algorithm (GA) to reduce pump operating costs by optimizing the pump working periods. Bagirov et al. (2013) utilized a grid search technique to optimize pump scheduling and reduce operating expenses. Giustolisi et al. (2013) used a MOO strategy to minimize the pump operating cost and the cost of non-revenue water by optimizing the pump status with a GA. Stokes et al. (2015) addressed the MOO problem of minimizing the electricity cost for pump operation and GHG emissions from fuel use by optimizing pump schedules using NSGA-II.

Tanks are typically included in the WDS alongside pumps to help regulate pressure and provide various benefits. A WDS usually provides tanks with excess water storage capacity, enabling flexible pump scheduling (Sparn and Hunsberger 2015). Elevated water tanks can help smooth the network's daily pumping pattern and store water for emergencies. The design of a tank is based on its location and size and seeks to maximize the water supply’s reliability and quality while minimizing its cost. Before incorporating tanks into the WDS, optimization methods are required to ensure optimal benefits (Batchabani and Fuamba 2014). Murphy et al. (1994) used GA to optimize the tank location, volume, and minimum operational level to minimize the capital cost and satisfy hydraulic requirements. In addition to the previously mentioned tank decision variables, Prasad (2010) also considered the diameter–height ratio and emergency volume–total volume ratio in the GA to minimize capital and energy costs. On the MOO side, Farmani et al. (2005) examined the trade-off between total cost and network resilience, for which the tank location, maximum/minimum operational level, diameter, and bottom elevation were optimized using NSGA-II. Vamvakeridou-Lyroudia et al. (2007) considered the minimization of cost and maximization of benefit as objective functions for GA optimization. The work also considered several practical assumptions to ensure that the decision variables could be limited to the tank volume and minimum operational level.

Most approaches to the optimization of WDS are centered on hydraulics and cost. However, urban water supplies are typically inefficient under energy and carbon emission constraints (Babel et al. 2021; Ma et al. 2022; Molinos-Senante et al. 2022; Qiu et al. 2022). Therefore, an energy-based approach should be used to adequately address energy consumption in the water distribution sector. In WDS, energy efficiency can be enhanced by reducing water losses (Surendra et al. 2021; Kowalska et al. 2022; Spedaletti et al. 2022), implementing water saving devices (Berger et al. 2017; Zaeid 2018), and optimizing treatment and supply (Arai et al. 2013), among many other measures. As the WDS provides services in the form of water volume, the water–energy relationship can be measured in terms of energy intensity (EI) (kWh m−3). In the context of a WDS, EI is the amount of energy required to transport a unit volume of water. Several prior studies have quantified the system-level energy intensity of the WDS (Twomey and Webber 2012; Stokes-Draut et al. 2017). A more recent study by Liu and Mauter (2021) proposed the concept of marginal energy intensity (MEI), which quantifies the energy intensity at a specific location and time, allowing for a more detailed spatial and temporal analysis of energy usage in a WDS. The concept of local energy intensity is especially useful when dealing with active WDS components (e.g., pumps and tanks) that can exert varying influences on the system during its operation.

Most previous works were focused solely on reducing energy consumption. However, given that a WDS is a service-providing infrastructure, it is appropriate to consider both the energy investment and the user benefits. This study proposes efficient returned pressure (ERP) as a new metric that considers the service pressure received by users in terms of energy intensity. The new metric advances the concept of MEI (Liu and Mauter 2021) by adding a hydraulic factor of available service pressure. Each node's energy intensity is calculated based on the connected network elements (pipes, pumps, and tanks) at a specific time step. Then, the relationship between the service pressure and energy consumption is analyzed, with each node assigned a weight based on its contribution to the network using a multi-criteria decision analysis (MCDA), considering energy, reliability, and pressure uniformity. The network ERP is then computed as the product of the energy intensity, pressure, and weighting. The proposed ERP can determine whether the energy investment is satisfied with adequate pressure for network users, thus making it a viable optimization objective. Typically, the MO optimization efforts in WDS concern the cost and energy and result in a cost reduction equivalent to a decrease in energy and pressure. Using ERP to optimize against cost will produce system designs that balance the required energy and service level. Furthermore, as ERP captures the relationship of energy and pressure in a single metric, it reduces the computational complexity during the optimization process, compared to considering all three objectives simultaneously. In addition to calculating the proposed metric, the application of ERP is demonstrated by employing it as an optimization objective in a MOO scheme for three different network configuration scenarios. A comparison with the conventional modified resilience index (MRI) (Jayaram and Srinivasan 2008) was conducted to evaluate the performance of the newly proposed ERP.

The remainder of this paper is organized as follows. The methodology employed in this study is described in  "Methodology." section. The application and performance of the proposed metric are described in "Application." section. Finally, the conclusions of this study are presented in "Conclusions." section.

Methodology

Several variables should be considered to determine the network ERP. An initial hydraulic simulation was performed to obtain all network conditions at each time step of the daily operation period. From the simulation data, the energy intensity and available pressure at all demand nodes were then derived. Energy intensity was calculated using a backtracking process to identify the contribution of all elements associated with the delivery of water to the nodes. The daily averages of both variables (i.e., energy intensity and pressure) were then analyzed using a pressure-energy (P-E) chart, with MCDA used to calculate the individual weights. The system ERP is the product of pressure, energy intensity, and weight. The calculation process is illustrated in Fig. 1, with the key variables highlighted in bold boxes.

Fig. 1
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Process of ERP calculation

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Energy intensity

The energy intensity of a WDS is defined as the amount of energy required to deliver one unit of water at a specific time step (Liu and Mauter 2021). Calculation occurs primarily in the network links, such as the connection between the two pressure points. It is computed based on the difference in head between the points; hence, energy intensity exists for each linking element, such as pumps, pipes, and valves. In this section, \(l\), \(t\), \(n\), and \(j\) represent the sets of links, times, nodes, and junctions, respectively. According to the EPANET water network model (Rossman 2000), nodes represent any vertices in the network (reservoir, junction, and tank), whereas junctions refer to a specific node where water enters or exits the network. The energy intensity of the pipes and valves is calculated using Eq. 1.

$${\text{EL}}_{l,t} = \rho \cdot g \cdot H_{l,t} \cdot \frac{1}{{3.6 \cdot 10^{6} }}$$
(1)

where \({EL}_{l,t}\) is the energy intensity at time \(t\) for link \(l\) (kWh m−3), \(\rho\) is the mass density of water (1000 kg/m3), \(g\) is the gravitational acceleration (9.81 m/s2), and \({H}_{l,t}\) is the head loss occurring in link \(l\) at time \(t\) (m). In the case of pumps, where mechanical energy is converted into hydraulic energy, an additional variable is required to represent efficiency. When the linking element is a pump, Eq. 2 is used:

$${\text{EL}}_{l,t} = \rho \cdot g \cdot H_{l,t} \cdot \eta^{ - 1} \cdot \frac{1}{{3.6 \cdot 10^{6} }}$$
(2)

where \(\eta\) is the pump efficiency and \({H}_{l,t}\) is the head gained by the pump.

In a single time step, the energy intensity for all links is calculated. To calculate the nodal energy intensity, a backtracking procedure is then used to identify all the connected upstream elements for each specific node. The backtracking procedure begins at the concerned node. The flow path is then traced based on the flow direction until the source (a reservoir and/or tank) is reached. It is also necessary to identify the ratio of the node inflow in each linked element. After determining the node flow path, the nodal energy intensity can be calculated as the summation of \({EL}_{l,t}\) multiplied by the flow ratio, as follows:

$${\text{EN}}_{n,t} = \mathop \sum \limits_{l} {\text{EL}}_{l,t} \cdot \frac{{Qin_{l - n,t} }}{{Qin_{n,t} }},$$
(3)

where \({\mathrm{EN}}_{n,t}\) is the energy intensity at time \(t\) for node \(n\) (kWh m−3), \({Qin}_{n,t}\) is the inflow to node \(n\) at time \(t\) (m3/s), and \({Qin}_{l-n,t}\) is the amount of inflow to node \(n\) contained in linked element \(l\) at time \(t\) (m3/s).

The energy intensity of the tanks is calculated only during the filling period. During a single filling time step, the energy intensity in the tank is calculated similarly to the nodal energy intensity. However, as the tank stores water with different energy intensities at different times, the energy intensity of the tank during the filling period is a mixture of its previous state of energy intensity and the newly received water, which is expressed as follows:

$${\text{ET}}_{s,t} = \frac{{V_{s,t - 1} \cdot {\text{ET}}_{s,t - 1} + Vin_{s,t} \cdot {\text{ET}}in_{s,t} }}{{V_{s,t - 1} + Vin_{s,t} }};\left( {s \subset n} \right),$$
(4)

where \({\mathrm{ET}}_{s,t}\) and \({\mathrm{ET}}_{s,t-1}\) are the energy intensities of tank \(s\) at time \(t\) and \(t-1\) (kWh m−3), respectively; \({V}_{s,t-1}\) is the water storage of tank \(s\) at time \(t-1\) (m3); \({Vin}_{s,t}\) is the volume of water received by tank \(s\) at time \(t\) (m3); and \({ETin}_{s,t}\) is the energy intensity of the supplied water at time \(t\) for tank \(s\) (kWh m−3).

Finally, knowing all the energy intensities of the connected elements during a specific time step, the junction energy intensity for a given time can be calculated as the summation of the energy intensity of the linking elements (including pumps) and tanks along the flow path, as follows:

$${\text{EJ}}_{j,t} = {\text{EN}}_{j,t} + \mathop \sum \limits_{s} {\text{ET}}_{s,t} \cdot \frac{{Qin_{s - j,t} }}{{Qin_{j,t} }};\left( {s \subset n;j \subset n;s \notin j} \right),$$
(5)

where \({\mathrm{EJ}}_{j,t}\) is the energy intensity of junction \(j\) at time \(t\) (kWh m−3), \({Qin}_{s-j,t}\) is the amount of inflow provided by tanks \(s\) to junction \(j\) at time \(t\) (m3/s), and \({Qin}_{j,t}\) is the inflow to junction \(j\) at time \(t\) (m3/s).

For an example WDS, calculations were performed for two different conditions—when the tank is filling and when it is discharging—with a simulation time step of one hour. The initial tank water storage is assumed to be 1000 m3 and the energy intensity is 0.04 kWh m−3. In the first condition, the connection of network elements and their status during the tank filling period are shown in Fig. 2. The calculation of link and node energy intensities is provided in Tables 1 and 2, respectively. As seen in Table 2, Node C shows the highest energy intensity and Tank T is the second highest energy intensity point.

Fig. 2
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Example network element interaction during tank filling

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Table 1 Example link energy intensity calculation during tank filling
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Table 2 Example node energy intensity calculation during tank filling
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The following example demonstrates the calculation during tank discharge, which immediately follows the tank filling period in the preceding example. During this time period, there is an increase in demand, resulting in a lower total head. The tank is discharged to support the demand in Nodes B and C, as shown in Fig. 3. The link and node energy intensity calculations for this period are presented in Tables 3 and 4, respectively. The flow ratio calculation for Node C is shown in Fig. 4.

Fig. 3
figure 3

Example network element interaction during tank discharging

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Table 3 Example link energy intensity calculation during tank discharging
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Table 4 Example node energy intensity calculation during tank discharging
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Fig. 4
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Example flow ratio calculation for Node C

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Efficient returned pressure (ERP)

After calculating the individual energy intensity for all demand nodes, the system ERP is computed. Following the results of the previous section, there exists time-series data of the available pressure and energy intensity for each demand junction. The time-series data are flattened into a temporal average value for each junction to ensure that they can be plotted in the P-E chart, which illustrates the relationship between the available pressure and the energy intensity. The points should be weighted to quantify the system-average ERP value because they are classified differently depending on the quadrants in which they are located and thus have different influence values on the system, as shown in Fig. 5.

Fig. 5
figure 5

Pressure–energy quadrant classification (P-E chart)

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The points in Q-IV should be favored less, while the points in Q-II should be weighted higher. The MCDA was applied to the data to provide this weighting. MCDA evaluates the points using the criteria of energy (E), reliability (R), and pressure dispersal (D). In terms of energy (E), points with lower energy intensity were favored. Reliability (R) refers to the ratio of available pressure to minimum required pressure at the junctions, which indicates the reliability of the service under abnormal conditions, with a larger ratio being preferred. Finally, pressure dispersal (D) assesses the contribution of a specific junction pressure to system pressure uniformity, with a lower value preferred. The pressure dispersal for each node was calculated by measuring the difference between system pressure standard deviation and the system pressure standard deviation excluding the concerned junction. By removing the junction pressure, the standard deviation of the system pressure is changed. If the change is minimal, then the pressure at junction \(j\) has a small impact on the pressure uniformity of the system. The pressure dispersal is expressed as follows:

$$D_{j} = \left| {P\sigma - P\sigma_{ex,j} } \right|,$$
(6)

where \({D}_{j}\) is the pressure dispersal for node \(j\), \(P\sigma\) is the standard deviation of the system pressure, and \({P\sigma }_{ex,j}\) is the standard deviation of the system pressure excluding node \(j\). The importance of each criterion is set to be 0.4 (E), 0.3 (R), and 0.3 (D), putting more priority on the energy intensity. The calculation was performed using the inverse to emphasize the difference in value. High available pressure is usually accompanied by a poor contribution to network pressure uniformity, which pulls the weight down. An example of weight calculation is shown in Fig. 6. The graphical results of this analysis are presented and discussed in the application section.

Fig. 6
figure 6

Example of MCDA calculation

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Finally, by coalescing the pressure, energy intensity, and weight, the ERP can be calculated using Eq. 7. ERP describes the network performance in terms of the average pressure gained from a unit of energy intensity.

$${\text{ERP}} = {\raise0.7ex\hbox{${\mathop \sum \nolimits_{j} \frac{{\overline{{P_{j} }} }}{{\overline{{EJ_{j} }} }} \cdot w_{j} }$} \!\mathord{\left/ {\vphantom {{\mathop \sum \nolimits_{j} \frac{{\overline{{P_{j} }} }}{{\overline{{EJ_{j} }} }} \cdot w_{j} } {\mathop \sum \nolimits_{j} w_{j} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\mathop \sum \nolimits_{j} w_{j} }$}},$$
(7)

where \(\mathrm{ERP}\) is the efficient returned pressure (m / (kWh m−3)), \(\overline{{P }_{j}}\) is the time-averaged pressure for junction \(j\) (m), \(\overline{{EJ }_{j}}\) is the time-averaged energy intensity for junction \(j\) (kWh m−3), and \({w}_{j}\) is the weighting value for junction \(j\).

Study network

The proposed method was applied to the OBCL-1 WDS, a medium-sized benchmark network for Harrisburg, USA (Vasconcelos et al. 1997; Hernandez et al. 2016). The network originally featured a single reservoir connected to a pumping station located in a high-elevation area. In this study, the original network was modified by changing the location of the reservoir and pumping station to a lower-elevation area. The layout of this network is shown in Fig. 7, and the details of the elements are listed in Table 5.

Fig. 7
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Layout of OBCL-1 network

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Table 5 Characteristics of OBCL-1 network
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Application

All analyses in this study were simulated using Python 3 programming language (Van Rossum 1995). The hydraulic calculation was supported by the Water Network Tool for Resilience 0.4.1 Python package (Klise et al. 2017) to utilize the EPANET engine. The optimization was conducted using the PYMOO framework (Blank and Deb 2020).

Optimization of network elements

Three cases of network configurations were created, with their various active components serving as decision variables for optimization. The three configurations applied are as follows: Case 1, a source-connected single pump; Case 2, a combination of source-connected pump and booster pump; and Case 3, a combination of source-connected pump and tank. All optimizations were conducted using the NSGA-II algorithm provided in PYMOO. The objectives are to maximize the ERP and minimize the total annual cost, which is the sum of the annual capital cost (pump and tank) and the pump operation cost, accounting for the lifetime of each specific network element (Marchi et al. 2014). Given the design criteria, all resulting Pareto fronts of the cases are shown in Fig. 8.

Fig. 8
figure 8

Pareto fronts of all cases by considering a annual total cost, b annual pumping cost, and c limited ERP range

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Each optimization case generated Pareto fronts from 100 design points selected by the algorithm. Figure 8a depicts the Pareto accounting for all costs, i.e., the annual capital costs of the pump and the annual pumping costs for Cases 1 and 2, and the additional capital cost of the tank for Case 3. As the cases feature different elements, the pumping cost is used as a benchmark because it occurs in all cases. The Pareto with the extracted pumping cost is shown in Fig. 8b. For a closer look at the comparable range, Fig. 8c shows the representative design points for each case at an annual pumping cost of approximately $65,000.

The Pareto fronts have different ranges owing to the different network elements. Compared with Case 1, Case 2 provides a broader range of possible ERP because of the two configurable pumps. Case 3 has a narrower range owing to more restrictions on the pumping behavior to work concurrently with the tank and prevent operation failure. As the ERP is the ratio of available pressure to energy intensity, the growth of both parameters affects its value in the optimization problem. It can be seen in the Pareto front of all cases that ERP increases with an increase in pumping cost. As there is no parameter change on the demand side, the increase in pumping cost is driven by selecting a larger pump for supply, which effectively induces a higher energy intensity. With an increase in energy intensity, the ERP still increases because the available pressure increases at a higher rate. This behavior means that when using ERP as an optimization objective, the algorithm attempts to find the highest pressure while maintaining the minimum energy intensity possible at a specific annual cost. Alternatively, the ERP describes the lowest possible energy investment while maintaining the highest possible service pressure. In this sense, the Pareto front provides options that can be chosen depending on the available economic investment and desired level of service, which are effective in terms of energy investment. Detailed discussions for individual cases at each representative point (shown in Fig. 8c) and the overall discussion are presented in the following subsections.

Case 1: source-connected single pump

In the first case, no modifications are made to the network elements. Optimization is performed to determine the best efficiency point (BEP) of the source-connected pump, with the flow and head of the single-point pump curve serving as decision variables. The constraints on the optimization are a minimum pressure of 15 m at all demand junctions, and the pumping flow is not allowed to exceed 120% of its BEP. Plotting the calculated average energy intensity and pressure of every demand junction creates the P-E chart shown in Fig. 9a, with the centroid plotted as the brown dotted line intersection. Based on the quadrants from the P-E chart, the positions of the corresponding junctions in the network are shown in Fig. 9b.

Fig. 9
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a P-E chart and b quadrant layout of OBCL network for Case 1

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The P-E chart shows that most junctions are in Q-II and Q-IV. Q-II contains junctions located near the source, where they require minimal energy intensity to obtain a sufficiently high pressure. Junctions further downstream with rising elevation are primarily located in Q-IV, describing their high energy intensity requirement with a low-pressure return. The Q-I junctions describe the area on the left where the elevation is comparable to that of the Q-II junctions, but they are located further downstream; thus, they have similar pressure but require greater energy intensity. Q-III contains junctions between Q-II and Q-IV, as seen in the transition in Fig. 9b. As the network is supplied with only a pumped source, it creates the opposing problem of junctions located near the source requiring less energy intensity but having too much pressure. In contrast, the energy intensity increases as the available pressure decreases downstream; this relationship is clearly illustrated by the downward trend in the P-E chart.

Accounting for the three criteria in the MCDA, weighting was properly distributed. Note the weight for each junction is seen in different colors in Fig. 9a. As expected, the majority of the heavier weighted (i.e., favorable) junctions were in Q-II. These junctions represent the desired high pressure and low energy intensity for a high service level with low energy consumption, whereas most junctions in Q-IV have a low weighting value (i.e., unfavorable). Adding the pressure dispersal criterion gives more weight to junctions that contribute less variation to the pressure distribution value, reflected in the chart, where the top-left point no longer has the highest weighting value and is instead shifted toward the center. Considering all the nodes in the chart, the average ERP of this network was 126.87 m/kWh m−3. Applying an average energy intensity of 0.43 kWh m−3, the efficient available pressure is 54.69 m. The annual capital cost for the pump is 5425 $/yr and the annual pumping cost is 64,757 $/yr, for a total annual cost of $70,183. Due to the lack of additional elements, the first case will also serve as a reference for the subsequent two cases.

Case 2: combination of source-connected pump and booster pump

In the second case, a booster pump is added to the network. The booster pump is expected to alleviate the source-connected pump load and increase the pressure distribution of the network. The booster pump is installed in the middle of the network (as shown in Fig. 10b), which is the transitional area or Q-III in Case 1. Owing to the addition of the booster pump, two parallel pipes beside it must be closed to ensure that the water flow is not disrupted, thus creating a bridge connection with the downstream network (as shown in Fig. 10b). Four decision variables are applied in this case—the pair of points in the BEP of the single-point curve of each of the two pumps. This optimization also considers the same constraints as in Case 1. The P-E chart and layout of the quadrants are presented in Fig. 10.

Fig. 10
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a P-E chart and b quadrant layout of OBCL network for Case 2

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Compared with Case 1, some differences in the P-E chart can be observed. The downward trend chart is divided into two parts. Relative to the flow direction, the pressure changes from high to low with a low energy intensity (Q-II to Q-III) before the flow meets the booster pump. Subsequently, there is a jump in the required energy intensity owing to the booster pump (Q-I and Q-IV). The junctions closer to the booster pump (Q-I) exhibit a higher pressure than the previous part of the network (Q-III), and the pressure decreases as it travels further from the booster pump (Q-IV). Several junctions located in Q-IV in Case 1 are moved to Q-I, as they now have a higher pressure provided by the booster pump. Similarly, several Q-II junctions in Case 1 are now moved to Q-III because of the lower pressure provided by the smaller source-connected pump. The P-E chart demonstrates the effect of adding a booster pump as the junctions are now distributed more evenly across all quadrants compared to the crowding of Q-II and Q-IV in Case 1. Overall, this configuration provides a more uniform pressure distribution over the network than that in Case 1. However, the energy intensity exhibits a wider spatial range compared with Case 1. The annual capital cost for pumps is 7591 $/yr and the pumping cost is 65,104 $/yr, amounting to 72,695 $/yr. With that level of investment, the ERP for this case is calculated to be 122.75 m/kWh m−3, slightly lower than that of Case 1. With an average energy intensity of 0.44 kWh m−3, the efficient available pressure for the network is 53.75 m.

Case 3: combination of source-connected pump and tank

In the third case, a tank is added to the network without a booster pump, as shown in Fig. 11b. The tank is assumed to be cylindrical with a diameter-to-height ratio of 1:1. Based on the P-E chart of Case 1, the tank location is determined to be in the center and connected to the nearest supply line. For this case, five decision variables are used: two for the source-connected pump BEP and three for the tank. The decision variables for the tank are the diameter (height), initial water level, and bottom elevation. Constraints similar to those in the previous cases are applied, with additional constraints for the tank. These constraints include a 0.3 m limit for tank water level difference between the start and end of the daily operation cycle and a 50 m maximum top elevation for the tank. The resulting P-E chart and quadrant layout for Case 3 is shown in Fig. 11.

Fig. 11
figure 11

a P-E chart and b layout of OBCL network for Case 3

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Compared to the pattern of the P-E chart in Case 1, no major differences are observed, except in Q-III, indicating that the addition of a tank has a minimal effect on the specific area of the network. However, Case 3 still shows an improvement in the distribution of junctions to the other quadrants. The tank helps reduce the energy intensity for the junctions supported by it, which are on the upper-right side of the network. The addition of a tank provides a pressure similar to that of Case 1, with a lower energy intensity requirement. Owing to the reduced energy intensity, several junctions in Q-IV for Case 1 move to Q-III for Case 3. Although the pattern is similar to that of Case 1, observing the centroid position, the average pressure is found to increase, whereas the required energy intensity is reduced.

The annual capital costs for the source-connected pump and tank are 5087 $/yr and 29,526 $/yr, respectively, and the annual pumping cost is 64,994 $/yr, for a total annual cost of $ 99,608. With such investment, the ERP for this case is calculated to be 133.92 m/kWh m−3, slightly higher than that of Case 1. Considering an average energy intensity of 0.42 kWh m−3, the effective pressure is 56 m, which is an improvement compared with Case 1.

Overall comparison of cases

There was no overlap between the Pareto fronts of the three cases, indicating that there was no configuration in different cases that would produce the same amount of ERP and total annual cost. Case 3 was the most expensive option in terms of annual cost, followed by Case 2, while Case 1's operation with a single source-connected pump was the least expensive to achieve the same ERP value. If the comparison is isolated to the annual pumping cost, Case 3 takes over first place to be the least costly option. There is a possibility that the Pareto for Cases 1 and 2 intersect at a higher ERP value, as evidenced by the comparison of annual total cost and pumping cost. At a similar pumping cost level represented by the chosen points, the ERP ranged from 122 to 133 m/kWh m−3, a slight difference because the medium-sized network does not require much energy intensity. To evaluate the performances of the cases, additional comparisons can be made with other measurable parameters, such as GHG emissions, pressure uniformity, and water age. Emissions were calculated based on GHG equivalencies to the consumed electricity (“Greenhouse Gases” 2022). Pressure uniformity is the standard distribution of the pressure in the network. The age of the water received by the junctions can vary depending on the time of day, especially when the tank is involved. This value for water age represents the average water age at all junctions during daily operations. Using Case 1 as the reference (i.e., a value of 1.0), the performance comparison of each chosen point is detailed and visualized in Table 6 and Fig. 12.

Table 6 Details of performance comparison of three applied cases
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Fig. 12
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Performance change relative to Case 1

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Relative to Case 1, there was not much difference in ERP and emissions for Cases 2 and 3. Case 3 shows a 0.3% higher pumping cost and has a higher ERP than Case 1, but its emission is 5% higher. The cost is proportional to the daily electricity tariff, whereas the emission is proportional to the pump’s energy consumption. Owing to the operation of a tank, the pump must carry a greater flow to fill the tank when the demand in the network is low, typically occurring when the electricity price is lower. The water age degraded in Case 3 because of the storage of water in the tank, while there was a slight improvement in Case 2 because of the pipe closure. The pressure uniformity is similar for Cases 1 and 3 but significantly improved in Case 2 with the booster pump operation. Considering these parameters, Case 2 was the preferred configuration for the study network.

Comparison to conventional index

The preceding section discussed the application and behavior of ERP as an optimization objective. As an energy-based metric, ERP should perform better than the conventional index when energy is one of the essential aspects of performance. In this study, the performance of ERP is compared with the MRI (Jayaram and Srinivasan 2008), a conventional index used for WDS optimization in terms of maximizing surplus pressure for reliability. Three parameters were selected for comparison: pumping cost, GHG emissions, and pressure uniformity. Pumping costs and GHG emissions were chosen based on the economic benefits and environmental impact, respectively. Pressure uniformity is also included, as it is one of the factors considered in the ERP calculation. The optimization using MRI as an objective follows the same procedure and parameters as the ERP-objective optimization to provide a fair comparison. A comparison of the results is shown in Figs. 13–15. It should be noted that in Fig. 15, no uniformity comparison is made for Case 1 because the usage of one pump will not make any difference to the uniformity throughout the possible configuration.

Fig. 13
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Distribution of potential pumping costs for a Case 1, b Case 2, and c Case 3

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All measurements indicate that ERP-based optimization results show improved performance with a lower value compared to MRI-based optimization results. Overall, the ERP-based optimization results exhibit lower or similar minimal, median, and maximal values. The possible pumping cost and emission (Figs. 13 and 14) in Case 1 do not show much variance, but Cases 2 and 3 show clear differences between the two optimization schemes. The ERP-based optimization scheme outperformed MRI-based optimization. Figure 15a demonstrates that the ERP performs significantly better in Case 2 when it optimally adjusts the pumps to balance the network pressure owing to its consideration of pressure uniformity. The high maximum value exhibited in the MRI-based optimization is due to the choice of a large booster pump because MRI tends to pursue surplus pressure in the network. The range for uniformity is small in Case 3 (Fig. 15b); although it is not significant, the ERP-based optimization in Case 3 still results in lower values, indicating that the inclusion of pressure uniformity when assigning weights to the ERP calculation has a considerable benefit. As a more complex measurement than MRI, ERP-based optimization produces better designs in terms of cost (economic), GHG emissions (environmental), and uniformity (hydraulic) aspects.

Fig. 14
figure 14

Distribution of potential GHG emissions for a Case 1, b Case 2, and c Case 3

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Fig. 15
figure 15

Distribution of potential pressure uniformity for a Case 2, b Case 3

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Conclusions

Considering the climate impact of energy generation, energy-intensive industries, such as the water industry, must manage their energy usage efficiently. As a service-providing infrastructure, a WDS must account for its service quality when aiming for energy usage reduction. This study proposed an ERP as a metric for the service pressure level that also considers the required energy intensity in a WDS. The calculation of ERP includes the temporal calculation of energy intensity, which requires backtracking to identify all the network elements contributing to a particular node. The relationship between the average pressure and energy intensity of the demand junctions is plotted in a P-E chart, where the junctions are given a specific weight. MCDA was applied to consider the energy, reliability, and pressure uniformity factors to weight each point in the chart. By gathering the service pressure, energy intensity, and weight of each junction, the ERP of the network can be calculated to describe the available service pressure achieved per unit of energy intensity.

The usage of the ERP metric was demonstrated in a MOO approach using NSGA-II to optimize the cost (capital and operation of pump and tank) and ERP in three different network configurations. Upon examining each case's potential design, it was discovered that the P-E chart reflects the effect of the various network configurations. The correlation between the available pressure and energy intensity of junctions is dependent on their position and proximity to other network elements and can be categorized as quadrants on the P-E chart. Moreover, the location of a group of junctions can be correlated with their supply network components. The chart can identify problem areas in the network where energy investment is not met with adequate pressure, thereby providing information for network adjustments or rehabilitation. Optimization using ERP as an objective provides low energy intensity solutions at each cost point while maintaining the maximum possible service pressure, which aligns with the WDS management objective. The performance of the proposed metric was further established by conducting an additional analysis where it was compared with a conventional MRI index in optimization. The ERP-based optimization produced more beneficial results in terms of economic cost, GHG emissions, and network pressure uniformity. Overall, this study has provided the calculation methodology and usage of ERP for WDS. The ERP proves to be a viable metric for comparing the network designs or optimization of WDS by considering the relationship between energy and pressure, which are both important factors in water networks. The inclusion of energy intensity and level of service in a water network design will be of significant attention to achieve a green and resilient system.

However, this study has some limitations regarding network supply and uncertainty. The ERP metric is only applied to pumped networks because gravity-fed networks are unaffected by active components and have inherently lower energy requirements. Assuming a normal network condition, uncertainties are not considered in any part of the calculation. This study introduces ERP and provides a simple usage example. As only a single study network was used, future studies could analyze ERP behavior in larger networks with various configurations. Instead of the daily averaged system ERP, the temporal value could be used in a more advanced optimization to optimize features such as the working time of multiple pumps. The MCDA section can also be modified to incorporate additional factors that are deemed necessary, thereby modifying the weighting process. As uncertainty is not factored into this study, the ERP behavior under uncertainty (e.g., water demand and pipe roughness) could also be investigated.