Optimization of Pressure Management in Water Distribution Systems Based on Pressure-Reducing Valve Control: Evaluation and Case Study

Abstract

Leakage in water distribution systems (WDS) is a major problem that results in substantial resource wastage. Pressure management, which is based on optimized control strategies using pressure-reducing valve installation, is considered one of the most effective methods for controlling network leakage and can be broadly classified into varied types, depending on the different control strategies. When facing diverse operational conditions in actual engineering, selecting the most suitable optimization strategy for the specific water supply network can greatly control leakage and efficiently utilize water resources. This article employs a greedy algorithm to iteratively determine the control sequence for pressure-reducing valves (PRVs), with the objective of minimizing surplus pressure. Subsequently, it proposed five evaluation indicators, namely, daily flow, daily leakage, pressure imbalance indicator, median daily pressure at nodes, and water age, which are used to evaluate and compare pressure management strategies in two sample networks. Finally, a suitable control strategy was successfully developed and implemented for an actual WDS of Y city in southern China, resulting in significant achievements. In conclusion, this paper embodies our attempt and contemplation in formulating pressure management strategies under diverse operating conditions in WDS, with the objective of providing guidance for practical engineering applications.

1. Introduction

A water distribution system (WDS), which serves as a crucial infrastructure for both livelihood and industrial production, possess a complex and extensive topological structure and constantly exhibit varying pressure state [1]. This presents a persistent challenge for water utilities, as they strive to ensure the crucial infrastructure is reliable [2].This is against a background of conflicting objectives relating to water network management: minimizing failure or minimizing the possibility of failure [3,4]. Furthermore, water utilities are deeply concerned about the economic advantages derived from operating WDS, in particular those relating to the reduction of water losses regarding the Economic Leakage Level [5,6,7,8]. Effective leakage control not only enhance the efficiency of water utilities, but also safeguards the water quality and infrastructure within the leakage-prone areas [9].

As cities grow, the amount of water leakage in their pipe networks increases. This is because water utility systems design their networks to maintain a minimum water pressure at all times, even during peak hours of consumption. However, this can lead to excessive pressure buildup during off-peak hours, which can cause pipes to crack or burst. Leakage is directly related to pressure, so high-pressure states in the network can significantly increase the frequency of leaks [10,11,12,13]. Although the leakage rate in the public water supply network has decreased from 2018 to 2021 in China, the leakage volume remained virtually constant at 80 billion m³, resulting in a significant waste of resources [14].

The International Water Association (IWA) has recommended a range of measures for leakage control methods that are widely recognized in the industry [15,16]. These measures focus on minimizing the economic level of leakage [6,17]. According to the methods proposed by the IWA, water utilities are actively engaged in developing new technologies to enhance the implementation of leakage reduction methods in WDSs [18]. Although expensive, pipe replacement is the most effective long-term solution among these methods for eliminating leakage at their source and reducing the amount of leakage while improving indicators such as water quality [19]. However, due to the high cost of this method, water suppliers prefer pipe rehabilitation when the damage to the pipe is minor, which has encouraged research into pipe rehabilitation materials and methods. The implementation of leak detection relies on the continuous monitoring of supervisory control and data acquisition (SCADA) systems, which determine the occurrence of pipe bursts based on flow and pressure information and provide timely location judgments in an effort to reduce the huge amount of leakage caused by bursts [20,21]. In addition to monitoring the occurrence of burst pipe incidents, leak detection also includes the monitoring of concealed leaks. Several researchers have worked on algorithms for the prediction of leaks and their locations, and have proposed many feasible optimisation algorithms [22,23]. Pressure management (PM) reduces leakage by lowering regional pressure levels within acceptable bounds to smooth out pressure variations by changing the pressure profile network [24]. Compared to pipe replacement, leak detection, and improved speed and quality of repairs, pressure management is more cost-effective and flexible [25,26,27]. Furthermore, pressure-managed networks can be adapted and updated to minimise leakage over a longer period [26,28,29].

PM is the process of adjusting the water pressure in a water distribution network to ensure that it is neither too high nor too low. This is important for a number of reasons: The prevention of leaks and bursts in pipes and ensuring that water is available at a sufficient pressure for all users [30]. The installation of pressure-reducing valves (PRVs) in over-pressured networks has proven to be an effective way of controlling leakage, and the application of pressure management techniques has reduced water supply costs by 20–55% [31]. There are four recognized PM methods for PRVs, which are based on the critical point states [32], namely fixed-outlet (FO), pressure-modulated, time-modulated (TM) and flow-modulated (FM) methods [24,33,34,35,36]. These techniques have been extensively studied by academics, and there are several case studies of their practical implementation, making them effective means of reducing pressure and leakage [37,38,39,40,41]. Previous research has revealed fixed-outlet pressure control does not achieve the goals for water loss [27,42]. Creaco et al., 2023 used a pressure-drop method in controlling PRVs for both remote and local modulation. The study revealed the ease of implementation and potential for modelling of flow-control valves in WDSs [43]. However, the majority of the studies focused on the isolated working of the instruments, and there are very limited studies involving the simultaneous simulation of multiple scenarios for the same pipe network [44,45]. And, there are few existing studies that combine the application and provide a detailed implementation process for all four methods simultaneously. This article provides a comprehensive description of the simulation process for the sample networks. The work also involves proposing a novel evaluation metric that differs completely from existing literature [4,13,24,46,47], along with deriving an expression for pressure imbalance.

In 2021, a target was set to lower the leakage rate of China’s urban public water supply network to less than 9% by 2025, which is undoubtedly a great challenge considering that the leakage rate in 2021 was 12.75%. This study intends to guide the development of PM in China’s water supply network and effectively reduce the excess pressure in the network, thereby reducing leakage and burst rates. This paper aims to guide and inspire pressure management in various complex pipeline network conditions, aiming for the efficient utilization of urban water resources, through the comparison of PRV control strategies using numerical schemes and practical engineering cases.

2. Methods

2.1. Construction of WDS Model

The Water Network Tool for Resilience (WNTR) toolbox, an EPANET Multi-Species Extension Software-compatible Python package, was used in this study to model and analyze the resilience of water supply networks, and, thus, for constructing and simulating pipe network models. The WNTR-PDD kernel was used for the simulations.

These simulations based on the ZJ Network and KL Network were provided by the University of Kentucky’s Water Distribution Systems Toolkit [48,49]. Both the ZJ Network and KL Network are a single-water-source WDS. The water pressure in these networks was regulated by installing a pressure-reducing valve (PRV) downstream of the water source. The ZJ Network consists of 115 nodes and 165 pipes, with a total pipe length of 126.4 km. The KL Network comprises 480 nodes and 733 pipes, with a total pipe length of 206.5 km. Figure 1 presents the topology diagram of the two sample networks.

In the ZJ Network, the maximum base demand among all nodes is 0.246 m³/s, with 2.6% of the total number of nodes having a demand of 0 m³/s. In the KL Network, the respective values are 0.008 m³/s and 29.4%. The elevation ranges of all nodes in the ZJ Network and KL Network are [3.91 m, 10.42 m] and [349.91 m, 366.37 m], respectively. The respective reservoir heads are 60.0 m and 413.3 m. Three customer water usage patterns were added to the sample network to simulate the changes in the network water usage over the course of a day: residential, small commercial, and large commercial. The main initial state of the two networks is as follows:

ZJ Network:

• Initial daily flow rate: 128.448 × 10³ m³/d;
• Daily leakage rate: 33.048 × 10³ m³/d;
• Median daily pressure at nodes: 36.70 m.

KL Network:

• Initial daily flow rate: 63.324 × 10³ m³/d;
• Daily leakage rate: 19.872 × 10³ m³/d;
• Median daily pressure at nodes: 51.48 m.

Table 1. The composition of sample networks: (a) ZJ Network; (b) KL Network.

Diameter (mm)	Number of Pipes		Total Length (km)
	(a)	(b)	(a)	(b)
50	0	316	0	128.9
65	0	49	0	5.8
80	0	51	0	9.4
100	34	20	27.6	2.7
150	8	70	5.6	9.1
200	25	58	19.4	15.9
250	23	27	16.7	6.9
300	22	23	15.8	4.7
350	9	23	7.9	4.6
400	11	22	5.6	3.9
450	20	50	16.5	9.9
500	3	0	1.9	0
550	4	13	4.0	1.6
600	2	5	1.5	1.0
650	0	4	0	1.4
750	1	0	0.4	0
800	1	0	1.4	0
900	0	2	0	0.6
1100	2	0	2.0	0

presents the composition of their pipes.

In the ZJ Network, the maximum base demand among all nodes is 0.246 m³/s, with 2.6% of the total number of nodes having a demand of 0 m³/s. In the KL Network, the respective values are 0.008 m³/s and 29.4%. The elevation ranges of all nodes in the ZJ Network and KL Network are [3.91 m, 10.42 m] and [349.91 m, 366.37 m], respectively. The respective reservoir heads are 60.0 m and 413.3 m. Three customer water usage patterns were added to the sample network to simulate the changes in the network water usage over the course of a day: residential, small commercial, and large commercial. The main initial state of the two networks are as follows:

ZJ Network:

• Initial daily flow rate: 128.448 × 10³ m³/d;
• Daily leakage rate: 33.048 × 10³ m³/d;
• Median daily pressure at nodes: 36.70 m.

KL Network:

• Initial daily flow rate: 63.324 × 10³ m³/d;
• Daily leakage rate: 19.872 × 10³ m³/d;
• Median daily pressure at nodes: 51.48 m.

For each of the two sample networks, the leakage coefficient, a pseudo-random number obtained using the Mason rotation technique, was multiplied by the sum of the pipe diameters connected to the nodes to produce approximately 50 leakage points. The WNTR-PDD kernel model parameters were used, as shown in

Table 2. WNTR-PDD kernel model parameters.

Model Parameters	Value
Accuracy	0.01
Maximum iterations	40
Nominal pressure	16 m
Minimum pressure	0 m
Minimum service pressure	10 m

.

The relationship between pressure and nodal water demand was described using a segmentation function in an equation [50]. This paper utilized an exponential leakage model ($\mathrm{Q}=\mathrm{C}{\mathrm{p}}^{\mathsf{\gamma }}$, in which Q, C, p, and $\mathsf{\gamma }$ represent leakage rate (m³/s), discharge coefficient, pressure (m), and the pressure exponent, respectively) for the pressure–leakage relationship, and determined the approximate range of its pressure exponent based on relevant FAVAD theories [51,52].

2.2. PRV Control Strategy Based on Greedy Algorithm

To effectively utilize the function of PRV control in urban water supply networks, the control mode of PRVs should be optimized according to the operation of the network to achieve a better pressure and leakage reduction effect. In urban water supply networks, if the set value of the PRV is taken as the decision variable (independent variable) and the pressure at the most unfavorable point as the dependent variable (target quantity), it is difficult to find a clear functional relationship between the two variables when the network is in operation, which is similar to many optimization problems.

The laws of operation of the pipe network conform to the law of conservation of energy and the principle of entropy increase. If the pressure at the inlet of the pipe network decreases, the pressure at the nodes in the network cannot rise without the action of external forces. In other words, the direction of pressure change at the nodes should be the same as the direction of change in the set value of the PRV. In a stable model, the PRV setting, that is, the pressure at the most unfavorable point, should be monotonically increased as a function of time. To simulate this characteristic relationship, the concept of a greedy algorithm can be used to determine the optimal control sequence for the PRV in the model. Thus, a stepwise iterative method based on the greedy algorithm was developed to find the optimal solution to the PRV control sequence optimization problem. Figure 2 shows the algorithm flowchart.

A starting PRV control sequence A was entered as the input to initiate the model and iteration. Simulation was carried out to obtain the surplus water pressure sequence B for the most unfavorable point. Following a continuous step iteration that concluded when sequence B was within the tolerance range, the optimal sequence A was obtained, and the iteration was terminated.

According to the abovementioned approach, simulations were performed for the ZJ urban pipe network model under the following simulation conditions:

• The length of the PRV control sequence should be the same as the length of the daily water usage variation factor sequence, which is 24.
• The sequence of post-valve pressure values at the initial operating conditions of the model shall be used as the initial sequence A, as shown in

  Table 3. Post-valve pressure value for the initial working condition of the ZJ urban pipe network.

  Time	Pressure (m)	Time	Pressure (m)	Time	Pressure (m)
  0	50.511	8	48.506	16	48.289
  1	50.933	9	47.274	17	48.013
  2	50.974	10	47.144	18	47.618
  3	51.042	11	47.181	19	48.101
  4	51.024	12	46.66	20	47.846
  5	51.02	13	46.97	21	48.069
  6	50.344	14	47.653	22	48.57
  7	49.022	15	47.954	23	49.67

  (pressure in the table indicates pressures after PRV).
• The endpoint tolerance should be set to 1, i.e., the iteration ends when all the rich pressure values fall within the set [0, 1).

The control sequence A in the ZJ urban pipe network model went through five iterations under the simulation conditions before reaching the tolerance condition at the sixth solution. Figure 3 shows the variation of the control sequence and the sequence of residual pressure values between the two simulations during the iterative process. The lower red line represents the sequence of the most unfavorable point of surplus pressure values and the upper red line represents the sequence of PRV settings. The figure shows that during the iterative process, the surplus pressure values initially dropped rapidly, almost touching the tolerance line (green line), then gradually dropped close to the tolerance line, and, finally, all values fell below the calibrated 1 m tolerance line. At this point, the surplus pressure values were all positive. The sequence of PRV settings ended up in the upper blue part of the graph, representing the optimal PRV control sequence.

2.3. Methods for Evaluating the Effectiveness of Pressure Management Controls

In order to compare the differences, strengths, and weaknesses of different strategies, a series of evaluation metrics was developed. The indicator values were used as a basis to evaluate the results of the operation of the strategy in the model. There are two factions in a water supply network: supply and demand. The consumer expects the network to provide sufficient water pressure and the pressure to be as stable as possible. On the other hand, the water supplier wants to optimize the operation of the network and reduce water supply costs as much as possible while still meeting the requirements of the consumer. From these two perspectives, the following indicators were developed to evaluate the network:

• Daily flow ${\mathrm{Q}}_{\mathrm{day}}$

$${\mathrm{Q}}_{\mathrm{day}}=\sum _{\mathrm{i}=1}^{24}{\mathrm{q}}_{\mathrm{res}-\mathrm{i}}$$

(1)

where:

• ${\mathrm{Q}}_{\mathrm{day}}$: Total daily flow rate of the pipe network, indicating the total amount of water flowing into the pipe network in a day (m³);
• ${\mathrm{q}}_{\mathrm{res}-\mathrm{i}}$: Pipe network inlet flow for time i (m³/h).

2.

Daily leakage ${\mathrm{Q}}_{\mathrm{day}-\mathrm{leak}}$

$${\mathrm{Q}}_{\mathrm{day}-\mathrm{leak}}=\sum _{\mathrm{j}\in \mathrm{J}}{\mathrm{q}}_{\mathrm{j}-\mathrm{leak}}=\sum _{\mathrm{j}\in \mathrm{J}}\sum _{\mathrm{i}=1}^{24}{\mathrm{q}}_{\mathrm{ij}-\mathrm{leak}}$$

(2)

where:

• ${\mathrm{Q}}_{\mathrm{day}-\mathrm{leak}}$: Daily leakage of the pipe network, indicating the total leakage of the network in a day (m³);
• J: Collection of pipe network nodes;
• ${\mathrm{q}}_{\mathrm{j}-\mathrm{leak}}$: Daily leakage at node j (m³);
• ${\mathrm{q}}_{\mathrm{ij}-\mathrm{leak}}$: Leakage at time i for node j (m³/h).

3.

Pressure imbalance indicator ${\mathrm{P}}_{\mathrm{equ}}$

${\mathrm{P}}_{\mathrm{equ}}$ is a novel evaluation index introduced in this study to assess the overall pressure balance in a WDN. It mainly consists of the following three components.

• i.

  Nodal daily overpressure factor f_j

In the theoretical framework of the pressure-driven model, each node is assigned a nominal pressure [50]. In accordance with this theoretical framework, when the pressure at a node surpasses or equals the nominal pressure, the flow rate at that node becomes a constant value. The first component, ${\mathrm{f}}_{\mathrm{j}}$, represents the average proportion of node pressures exceeding the nominal pressure within a day. A smaller positive value indicates preferable performance. The excess pressure beyond the nominal value does not affect the flow rate at the node and increases the overall pressure load in the WDNs.

$${\mathrm{f}}_{\mathrm{j}}=\frac{1}{24}\sum _{\mathrm{i}=1}^{24}\frac{{\mathrm{H}}_{\mathrm{ij}}-{\mathrm{H}}_{\mathrm{j}}^{\mathrm{nom}}}{{\mathrm{H}}_{\mathrm{j}}^{\mathrm{nom}}}$$

(3a)

where:

• ${\mathrm{f}}_{\mathrm{j}}$: The average value of the ratio of the surplus nodal pressure to the nominal pressure during a day. When the value is positive, the smaller the value, the better;
• ${\mathrm{H}}_{\mathrm{ij}}$: Pressure of node j at time i (m);
• ${\mathrm{H}}_{\mathrm{j}}^{\mathrm{nom}}$: Nominal pressure at node j (m).

• ii.

  Standard deviation of nodal day pressure ${\mathsf{\sigma }}_{\mathrm{j}}$

The standard deviation of nodal day pressure is used to characterize the fluctuation of water pressure that users experience within a day. A smaller value indicates greater stability at the node, which is desirable.

$${\mathsf{\sigma }}_{\mathrm{j}}=\sqrt{\sum _{\mathrm{i}=1}^{24}\frac{{\left({\mathrm{H}}_{\mathrm{ij}}-\overline{{\mathrm{H}}_{\mathrm{j}}}\right)}^2}{24}}$$

(3b)

where:

• ${\mathsf{\sigma }}_{\mathrm{j}}$: The volatility of the water pressure received by the user during the day. The smaller the value the more stable the node;
• ${\mathrm{H}}_{\mathrm{ij}}$: Pressure value of node j at time i (m);
• $\overline{{\mathrm{H}}_{\mathrm{j}}}$: Average daily pressure at node j (m).

• iii.

  Nodal day water demand ${\mathrm{q}}_{\mathrm{j}}$:

The Nodal day water demand varies among different nodes in the piping network. Nodes with higher Nodal day water demand serve a larger proportion of the overall user population in the network, signifying their greater significance. Therefore, this study employs Nodal day water demand as the weight of each node in the formula. The higher the water demand at a node, the higher its weight. Specifically, if the Nodal day water demand is zero, indicating a transfer node, its weight is also zero, contributing nothing to the ${\mathrm{P}}_{\mathrm{equ}}$ indicator.

$${\mathrm{q}}_{\mathrm{j}}=\sum _{\mathrm{i}=1}^{24}{\mathrm{q}}_{\mathrm{ij}}$$

(3c)

where:

${\mathrm{q}}_{\mathrm{j}}$: Nodal day water demand for node j (m³);

${\mathrm{q}}_{\mathrm{i}\mathrm{j}}$: Actual water demand at time i for node j (m³/h).

In summary, by defining pressure imbalance as the sum of the products of the three indicators, a smaller value is desirable. This indicates that the WDN maintains a more balanced pressure throughout the day, enabling it to operate under a lower load.

The expression for pressure imbalance becomes

$${\mathrm{P}}_{\mathrm{equ}}=\sum _{\mathrm{j}\in \mathrm{J}}{\mathrm{f}}_{\mathrm{j}}{\mathsf{\sigma }}_{\mathrm{j}}{\mathrm{q}}_{\mathrm{j}}=\frac{1}{48\sqrt{6}}\sum _{\mathrm{j}\in \mathrm{J}}\left[\sum _{\mathrm{i}=1}^{24}\left(\frac{{\mathrm{H}}_{\mathrm{ij}}-{\mathrm{H}}_{\mathrm{j}}^{\mathrm{nom}}}{{\mathrm{H}}_{\mathrm{j}}^{\mathrm{nom}}}{·\mathrm{q}}_{\mathrm{ij}}\right)·\sqrt{\sum _{\mathrm{i}=1}^{24}{\left({\mathrm{H}}_{\mathrm{ij}}-\overline{{\mathrm{H}}_{\mathrm{j}}}\right)}^2}\right]$$

(3)

where:

• ${\mathrm{P}}_{\mathrm{equ}}$: pressure imbalance indicator.

4.

Median daily pressure at nodes ${\mathrm{P}}_{\mathrm{mid}}$

$${\mathrm{P}}_{\mathrm{mid}}=\mathrm{median}(\overline{{\mathrm{p}}_{\mathrm{j}}})$$

(4)

where:

• ${\mathrm{P}}_{\mathrm{mid}}$: Median daily value of nodal pressure (m). This indicator can characterize the median level of water pressure received by all consumers during the day;
• $\overline{{\mathrm{p}}_{\mathrm{j}}}$: Average daily pressure at node j (m).

5.

Water age

The residence time of water in the pipe network directly affects the quality of the effluent at the consumer’s end. Thus, the change in water age under different operating strategies is also one of the indicators for evaluating the effectiveness of the control strategy.

2.4. Model Implementation of the Four Control Methods

The control strategy described in the previous sections generated an optimal PRV control sequence, wherein the set value of the PRV was changed hourly while maintaining synchronization with the change in the water usage pattern. However, in actual network operation, this time interval is not commonly used for control throughout the day, and, therefore, it needs to be discussed in relation to different network conditions. Currently, there are four widely recognized control modes for PRV, which are as follows: FOPM, pressure-modulated PM (PMPM), TMPM, and FMPM, which have been already introduced.

2.4.1. FOPM

The key to this control method was to maintain the pressure at the critical point above the minimum service pressure throughout the day. When the valve operation was stable, the post-valve pressure fluctuated around the set value. To obtain a precise solution, the optimal control sequence A can be determined via the methodology outlined in Section 2.2. The resulting maximum pressure value can be utilized as a constant post-pressure value for the valve throughout the day.

2.4.2. TMPM

In the present study, a control period was defined as a day (24 h) split into several successive periods, each with a corresponding PRV setpoint. The time periods were combined with the corresponding control values to form control segments, each with a fixed valve back pressure control, and all control segments were combined in sequence to form a control mode. The number of segments in an actual pipe network does not normally exceed 8.

TMPM consists of a time segment and a segment pressure. In this study, the control model was based on the optimal control sequence discussed in the previous section and the segment pressure was solved using the time segments to form the final control model. Figure 4 shows the solution flowchart for the segmental pressure sequence.

The segment point control sequence C was a sequence of integers with each value representing the start time of the next interval (included) and the end time of the previous interval (not included). For example, if the sequence was 3, then the corresponding segments were {[3, 10), [10, 20), [20, 3)}.

2.4.3. FMPM

A series of operational data were recorded during the optimization process by applying the PRV control strategy based on the greedy algorithm to the sample pipe network. This helped in generating a series of ΔP-Q values (post-valve flow and PRV-critical-point pressure drop), from which a complete fitted curve y containing the most data points was obtained, with the data points fluctuating in the vicinity of y. This y curve was used to solve for the flow–pressure curve in the control. The equation is obtained as follows:

$${\mathrm{f}}_{\mathrm{q}}\left(\mathrm{q}\right)=\mathrm{y}\left(\mathrm{q}\right)+{\mathrm{H}}_{\min }$$

(5)

where:

• fq: Flow–pressure control curves;
• ${\mathrm{H}}_{\min }$: Minimum service pressure at the critical point (10 m);
• $\mathrm{y}$: Total fitted curve of the post-valve flow PRV to the least favorable pressure drop value in the sample pipe network.

To avoid pressure shortfalls, a control safety margin was set in the flow-based control P_safe, considering the pressure fluctuations, and added to fq. Therefore, the post-valve pressure setting at a given moment of flow q becomes:

$${\mathrm{P}}_{\mathrm{q}}={\mathrm{f}}_{\mathrm{q}}\left(\mathrm{q}\right)+{\mathrm{P}}_{\mathrm{safe}}$$

(6)

where:

• P_q: Target PRV control value (m);
• ${\mathrm{f}}_{\mathrm{q}}$: Flow–pressure control curves;
• P_safe: Control of safety margins (m).

Figure 5 shows the flow of FMPM employed in the model.

2.4.4. PMPM

PMPM requires the establishment of one or more pressure-monitoring points within the network, usually at the potential critical point. The PRV controller receives real-time pressure feedback and modulates the network pressure changes by increasing the outlet pressure of the PRV when the pressure is low and decreasing it when the pressure is high. This method of control is very flexible as it does not rely on pre-set curves or values. However, the method is heavily dependent on real-time communication, and if reliable information on the critical point pressure is not available, pressure adjustment will be difficult to continue.

In this study, the PRV setpoint in the pipe network model was adjusted every 5 min depending on the pressure at the critical point, as modelled in actual operation. The setpoint was determined using the following formula:

$${\mathrm{P}}_{\mathrm{v}2}={\mathrm{P}}_{\mathrm{v}1}-({\mathrm{H}}_{\mathrm{cri}}-{\mathrm{H}}_{\min })$$

(7)

where:

• ${\mathrm{P}}_{\mathrm{v}2}$: Target PRV control value (m);
• ${\mathrm{P}}_{\mathrm{v}1}$: Previous PRV control value (m);
• ${\mathrm{H}}_{\mathrm{cri}}$: Pressure at the critical point (m);
• ${\mathrm{H}}_{\min }$: Minimum service pressure at the critical point (m).

Figure 6 shows the process for PMPM used in the model.

3. Results and Case Study

3.1. ZJ Network

A total of six schemes were proposed for the four control methods to model the ZJ Network. Among them, the control of different segmentation methods in the TMPM was selected to participate in the evaluation and comparison of their optimal solutions, with a total of three solutions. Herein, the segmentation method with the smallest sum of surplus pressures was selected for each period, similar to that for the KL Network. The segmentation points are as follows:

• Two time segments: 7, 23;
• Three time segments: 7, 14, 23;
• Four time segments: 7, 9, 14, 23.

The simulation results for the remaining three control methods were unique. Figure 7 and

Table 4. ZJ Network control working conditions: (a) FOPM; (b) two-time-segment control; (c) three-time-segment control; (d) four-time-segment control; (e) FMPM; (f) PMPM. Q_day (m³).

Evaluation Metrics	Qday (m3)	Qday-leak (m3)	${\mathbf{P}}_{\mathbf{equ}}$	${\mathbf{P}}_{\mathbf{mid}}\left(\mathbf{m}\right)$
a	121,176	25,884	380,844	29.70
b	113,868	19,116	274,536	22.69
c	111,816	17,172	171,720	20.59
d	110,988	16,380	129,744	19.73
e	111,456	16,596	146,592	19.96
f	106,560	13,716	59,724	16.63

show the evaluation indicators for all scenarios.

Using the four approaches for the ZJ Network, the following six options were found:

• FOPM was the least effective, with ${\mathrm{Q}}_{\mathrm{day}}$, ${\mathrm{Q}}_{\mathrm{day}-\mathrm{leak}}$, ${\mathrm{P}}_{\mathrm{equ}}$, and ${\mathrm{P}}_{\mathrm{mid}}$ being the highest of the six options.
• According to the TMPM, ${\mathrm{Q}}_{\mathrm{day}}$, ${\mathrm{Q}}_{\mathrm{day}-\mathrm{leak}}$, ${\mathrm{P}}_{\mathrm{equ}}$, and ${\mathrm{P}}_{\mathrm{mid}}$ decreased sequentially with the increase in the time segments.
• The flow performance according to the FMPM was superior than the average of time controls of the 2 and 3 segments, and its performance in terms of ${\mathrm{P}}_{\mathrm{equ}}$, which was close to the 4 segments time control.
• The performance of PMPM was the best among all four methods, with the lowest values for all indicators.

3.2. KL Network

The control method used for this sample pipe network was the same as that used for the KL Network. The segmentation points in the time-based segmentation control are as follows:

• Two time segments: 7, 0;
• Three time segments: 7, 20, 0;
• Four time segments: 7, 14, 18, 0.

The simulation results for the remaining three control methods were unique. Figure 8 and

Table 5. KL Network control working conditions: (a) FOPM; (b) two-time-segment control; (c) three-time-segment control; (d) four-time-segment control; (e) FMPM; (f) PMPM.

Evaluation Metrics	Qday (m3)	Qday-leak (m3)	${\mathbf{P}}_{\mathbf{equ}}$	${\mathbf{P}}_{\mathbf{mid}}\left(\mathbf{m}\right)$
a	52,632	9324	40,536	27.15
b	51,768	8532	47,232	25.11
c	52,308	8424	43,920	24.91
d	51,480	8280	40,716	24.56
e	51,984	8640	41,868	25.47
f	51,012	7920	35,784	23.66

show the evaluation indicators for all scenarios Table 5.

Using the four approaches for the KL Network, a total of six options were found:

• FOPM delivered the worst leakage reduction effect, with the highest ${\mathrm{Q}}_{\mathrm{day}}$ and ${\mathrm{Q}}_{\mathrm{day}-\mathrm{leak}}$. In particular, its ${\mathrm{P}}_{\mathrm{mid}}$ was the highest among the six options, while its ${\mathrm{P}}_{\mathrm{equ}}$ performance was close to that of all five options except for the performance of PMPM, which gave a lower value.
• According to the time-based control method, ${\mathrm{Q}}_{\mathrm{day}}$, ${\mathrm{Q}}_{\mathrm{day}-\mathrm{leak}}$, ${\mathrm{P}}_{\mathrm{equ}}$, and ${\mathrm{P}}_{\mathrm{mid}}$ decreased sequentially as the time segments increased; however, the difference was not significant and the performance of the three solutions was relatively similar. In comparison to TMPM, FOPM did not exhibit a significant difference, as observed in the ZJ network state.
• FMPM was slightly less effective than the TMPM, but its performance in terms of ${\mathrm{P}}_{\mathrm{equ}}$ was close to the average of the TMPM.
• PMPM was the best among all options, with the lowest values for all indicators.

Comparing the results of the two sample networks, it can be concluded that the performance of the different control methods for the pipe network was closely related to the characteristics of the network itself.

3.3. Sensitivity Analysis

In the previous simulation study, the effectiveness of different control strategies was evaluated in two sample networks, and the metrics mentioned earlier were used to assess their performance. To validate the sensitivity of these PM methods to different equivalent roughness values, this article investigated the variation in roughness (H-W coefficient) of the ZJ Network from 130 to 140 [53] (within the reference range of cast-iron pipes according to the Epanet official documentation), with a step size of 2. All other conditions remained consistent with the previous study mentioned. Figure 9 displays the sensitivity analysis results for four metrics.

The results indicate that as the H-W coefficient increases, indicating a decrease in pipeline roughness, there is a consistent downward trend for the same metric across all scenarios. Despite the variation in pipeline roughness, the PMPM consistently maintains the best performance among all control strategies. The performance ranking of the other control strategies remains largely consistent with the original setting (roughness = 130) in terms of performance.

3.4. Comparison of Water Age

It has been observed that after the pressure management of the network, the drop in pressure and flow rate can result in an increase in water age at some nodes [54]. In studies conducted by Patelis et al. (2020), it was discovered that the age of the water was growing as the pressure decreases and the water age was declining as the water pressure was increased downstream, a finding corroborated by earlier findings of Kravvari et al. (2018) in investigating the variations in residual chlorine and water age as the network is sectorized and the water pressure is managed using PRVs [30,54,55]. Thus, the ability to keep the water age within a reasonable and safe range is also a factor to be considered when choosing a control strategy [30,54,56]. It should be noted that increased water age is often caused by changes in demand, system operation, and system design—an example being unreasonable network topology. Conclusively, if pressure control can be carried out in conjunction with network rehabilitation, it can maximize the management of pressure which can lower leakage rates and improve water quality for the consumer.

Table 6. Results of water age simulations.

Model	Maximum Water Age Node	Time of Occurrence	Water Age/h
ZJ urban pipe network	16	7:00	8.56
	16	7:00	8.88
	92	8:00	9.77
	92	8:00	9.79
	92	8:00	9.85
	92	8:00	9.85
	12	9:00	10.3
KL urban pipe network	1191	11:00	13.88
	1191	11:00	14.69
	1191	11:00	14.78
	1191	11:00	14.81
	1191	11:00	14.81
	1191	11:00	14.71
	1191	11:00	14.99

lists the water age for the different control methods. The data for the two models in the table correspond to the initial state, FOPM, two-time-segment control, three-time-segment control, four-time-segment control, FMPM and PMPM, from top to bottom.

It has been observed that after the pressure management of the network, the drop in the pressure and flow rate can result in an increase in water age at some nodes. Thus, the ability to keep the water age within a reasonable and safe range is also a factor to be considered when choosing a control strategy. It should be noted that excessive water age is often caused by unreasonable network topology. Conclusively, if pressure control can be carried out in conjunction with network rehabilitation, it can maximize the management of pressure which can lower leakage rates and improve water quality for the consumer.

3.5. Case Study

This study conducted numerical experiments for field validation in an area with low terrain and high pressure at the network inlet in Y city. Figure 10 is a topological diagram of the selected area.

According to historical data, the inlet pressure at the network’s total metering point should not fall below 0.4644 MPa. By maintaining a pressure of at least 0.35 MPa at the inlet, normal water usage can be ensured for all users in the area. This results in a minimum surplus pressure value of 11.44 m, indicating a potential pressure drop of at least 11.44 m.

In consideration of the patterns of water usage, a comprehensive control methodology is suggested to formulate corresponding pressure regulation strategies, delineated on a time-of-day basis and encompassing multiple control modes, as follows:

• During the period of 0:00–5:00, since the accumulated water usage by users is relatively low and the head loss in the pipeline network is also limited, FOPM was proposed. Based on the recorded data, the maximum head loss was determined to be 2.58 m, and maintaining a pressure of 0.27 MPa at the network inlet would suffice to meet the water demand.
• During the period of 5:00–14:00, water usage by customers varies greatly, resulting in significant fluctuations in maximum head loss at the critical point during different time periods. Our data analysis found no clear correlation between flow rate and pressure. As a result, we recommend using PMPM.
• During the period of 14:00–18:00, the instantaneous flow rate at the inlet exhibits a relatively smooth variation, and the fluctuation of the head loss in the pipeline network is minor. Therefore, it is advisable to adopt FOPM. Furthermore, the maximum head loss recorded during the observation period is 8.86 m, and maintaining the pressure at the inlet of the pipeline network at 0.33 MPa would suffice to meet the water demand.
• The period of 18:00–24:00, was considered as the peak water usage period, as the results calculated for each period were relatively close. Considering the complexity and stability of control implementation, employing FOPM is a potentially superior option. Thus, the ultimate decision was to regulate the system by fixing the inlet pressure value at 0.315 MPa.

To summarize the pressure regulation strategies,

Table 7. Pressure regulation strategies of case study.

Time	WDS Inlet Pressure (MPa)	Time	WDS Inlet Pressure (MPa)
0:00–5:00	0.27	10:00–11:00	0.317
5:00–6:00	0.265	11:00–12:00	0.297
6:00–7:00	0.279	12:00–13:00	0.307
7:00–8:00	0.33	13:00–14:00	0.307
8:00–9:00	0.33	14:00–18:00	0.33
9:00–10:00	0.33	18:00–24:00	0.315

shows the final pressure control scheme. Based on the exponential leakage model mentioned in Section 2.1, this article utilized historical engineering measurement data to fit the corresponding parameters. The values obtained were C = 0.007 and $\mathsf{\gamma }$ = 1.5. In accordance with the established PM strategy and in conjunction with the exponential leakage model, the total leakage volume of the pipeline network within a 30-day period was determined to be 851.21 m³. When normalized by historical water consumption, the resulting reduction in non-revenue water amounted to 1217.97 m³, thereby yielding a decrease in the apparent loss rate from 22.75% to 10.28%. Assuming a water tariff of 4 CNY per unit, this region stands to reduce its monthly water supply losses by 4871.87 CNY.

Comparing the simulated scenarios in the sample work, in the case study, we need to consider more complex real-life situations, such as the following:

• PMPM relies on stable, real-time pressure feedback and requires reliable real-time communication with the target pressure measurement point or data center, which increases the cost of the equipment. It may require the establishment of a municipal power supply to maintain the long-term operation of the pressure-reducing station.
• In addition to requiring support from the relevant electrical power infrastructure, the implementation of FMPM poses significant challenges due to the reliance on reliable flow–pressure curves. Consequently, the adoption of FMPM was not feasible in the case study due to these hurdles.
• It was observed that the underperforming FOPM, which initially showed unfavorable results in simulations, ended up being the preferred option in the majority of instances. This can be attributed to various objective factors, such as the water consumption habits of users and the level of infrastructure development in the water distribution network.

It is worth mentioning that during the peak demand period (18:00–24:00), the pressure setpoint for the PRV at the critical point was lower compared to the non-peak period (14:00–18:00). Based on the investigation conducted in actual engineering scenarios, it was observed that during the peak demand period, the demand of high-rise users at the critical point was minimal. Additionally, after implementing the proposed pressure control schemes, no user complaints regarding water supply experience were received. This suggests that the phenomenon of having lower pressure setpoints during the peak demand period compared to the non-peak period is reasonable.

4. Conclusions

After suggesting the use of a greedy algorithm to obtain an optimized control sequence for PRV, this paper presented the implementation steps of the four existing PRV control strategies in detail. This paper proposed the use of five evaluation metrics, including a novel evaluation metric called Pequ, to assess the effectiveness of PM in WDS. Through the evaluation of two sample networks, significant differences were observed between them, which effectively reflect the state of WDS under different PRV control strategies. This contribution enhances the scientific and comprehensive evaluation of PM, having positive impacts on systems’ sustainability. Furthermore, it is observed, in the case study in Section 3.5, that practical engineering has more flexibility in selecting strategies when faced with complex situations compared to numerical simulations.

By examining indicators related to flow and pressure, the paper finds that among all the control strategies, PMPM exhibits the most prominent performance, despite variations in the characteristics of the WDS. Meanwhile, it is observed that the effectiveness of other control strategies varies between the two sample networks. For example, the performance of FOPM in the KL Network differs less from the other control strategies compared to its significant difference in the ZJ Network. Considering that the KL Network consistently exhibited higher surplus pressure throughout the day and had a smaller range compared to the ZJ Network, it is suggested that the pressure distribution of the network itself should be taken into account when selecting control strategies. This study also observes that in PM, a trade-off consideration is required when evaluating water quality indicators such as water age, as well as indicators related to flow and pressure.

Future research should aim to evaluate the benefits of PM beyond what we studied. This includes asset life extension and the reduction of carbon dioxide (CO₂) emissions. Researchers can use a multi-objective optimization approach to evaluate these benefits.