The proposed framework was implemented in two WDN cases, namely Hanoi network (Fujiwara et al. 1990) and the PG network (Haghighi et al. 2014), to investigate the influence of correlations on the optimal design of WDN. The network layout, including the pipe and node IDs, as well as the expectation of nodal demands are shown in Fig. 2. The Hanoi network consists of 31 nodes, 34 pipes and one reservoir, which has a large amount of water demand, 5539 L/s. While the PG network consists of 45 nodes, 65 pipes and one reservoir, providing water supply by 342 L/s to an industrial park near the Persian Gulf in southwest of Iran with the service area of 120ha.

Figure 3 shows the information of nodal demand, nodal pressure and pipe diameter of the two WDNs. According to Figs. 3 (a) and 3(b), it is found that most of the nodal demand in Hanoi network ranges from 100L/s to 300L /s (the number of nodes accounts for 77.4%), while in PG network, the nodal demand is mostly concentrated in the range of 0-10L /s (the number of nodes accounts for 93%). The ratio of nodal demand to total network demand in the Hanoi network is concentrated between 2% and 5.5%, with three nodes {6, 17, 19} have the largest proportions of {6.8%, 6.7%, 6.4%}, respectively. In the PG network, the ratio of nodal demand to total network demand is concentrated between 1% and 3%, with nodes {9, 42} having the largest proportions of {20.75%, 13.56%}, respectively.

Regarding the pipe diameter distributions shown in Figs. 3(c) and 3(d), it is found that the pipe length of the Hanoi network is mostly concentrated around 500m-1500 m, while the length of the PG network is mostly concentrated around 100m-200 m. Therefore, the Hanoi network has a larger scale than the PG network in terms of nodal demand and pipe length, while the PG network has a greater number of nodes and pipes.

## 3.1 Influence of correlations between nodal water demand

In this section, the impact of correlation coefficient of nodal demand on the design results of WDN is investigated. It is assumed that the water demand at each node is a random variable (RV) following a normal distribution, with the mean value taken from the nodal demand in Fig. 3 and the coefficient of variation (CV) equal to 0.1. The correlation coefficient among all nodes is assumed to be the same and set to {0.2, 0.4, 0.6, 0.8, 1.0} for five correlation scenarios.

For the Hanoi network, the diameters of Pipe 1 and Pipe 2 are set to 2000mm, and the total water head at the reservoir is set to 60m, other parameters keep the same with that adopted by Fujiwara et al. (1990). The optimal design model as shown in Section 2 is used to determine the diameters of the remaining pipes (excluding Pipe 1 and Pipe 2). Each pipe has eight available diameter options ranges from 300mm to 1000mm with the interval of 100mm. In the PG network, each pipe has six available diameters ranges from 100mm to 600mm with the interval of 100mm. The NSGA-II algorithm is employed to solve the optimization model, with the following parameters: evolutionary generations (*Ne*) is 200, population size (*Np*) is 400, individual crossover probability is 0.9, and individual mutation probability is 0.1. The Monte Carlo sample size (*N*) is 100. Figure 4 shows the Pareto fronts obtained by the optimization model.

It can be seen from Fig. 4, the influence of correlation coefficient on design results varies according the level of reliability requirement. For Hanoi network, *P*R = 0.5 is the boundary of the influence. When *P*R <0.5, the design cost decreases with the increase of correlation coefficient ρ. When *P*R >0.5, the design cost increases as the correlation coefficient ρ increases. A similar pattern exists for the PG network. According to the data in Fig. 4, the Pareto fronts corresponding to *P*R = {0.1, 0.5, 0.95, 0.99} were selected to analyze the designs under different degree of correlation coefficient. The relative difference (RD) of design cost (|correlated- independent|/ independent) under varied degree of correlation coefficient are shown in Fig. 5, where the correlated case corresponding to the values with ρ > 0 and the independent case corresponding to the values with ρ = 0. It can be seen that when the WDN reliability *P*R = 0.99, the RD of design cost is large, and the maximum cost difference of Hanoi network and PG network is 12.69% and 4.1% respectively. When *P*R is 0.5, the RD of design cost is very small, and the max difference is only 0.82%. When *P*R is less than 0.5, the increase of correlation coefficient reduces the design cost. For example, when *P*R = 0.1, compared with the independent case, the design costs of Hanoi network and PG network considering correlation with ρ = 1.0 are reduced by 5% and 3%, respectively.

Considering the actual WDN design, the required service reliability *P*R of node is generally larger than 0.95. Therefore, to reach the service reliability *P*R>0.95, the correlation of nodal water demand will lead to the increase in the construction cost, and the greater the correlation coefficient ρ, the higher the construction cost. It can be seen that with the increase in the level of reliability requirement and the increase in the degree of correlation coefficient, the slope of Pareto front curve decreases, indicating that the reliability benefit obtained by unit cost investment decreases. To further analyze the reasons behind this phenomenon, the Monte Carlo samples for the reliability levels *P*R = {0.1, 0.5, 0.9} of the Hanoi network were further investigated. Node 12 in the Hanoi network was selected, which represents the node with the worst service pressure. The probability density function (PDF) of the water pressure at Node 12 were plotted in Fig. 6. It can be observed that nodal demand correlation increases the variability of node pressures in the Monte Carlo samples, which corresponding to a wider shape of the PDF curve. At a reliability level of *P*R = 0.1 and the correlation degree of ρ = 1.0, the mean pressure of Node 12 is 18.45m, which is lower than the corresponding value of 26.02m when *P*R = 0.1 and ρ = 0. At a reliability level of *P*R = 0.9 and the correlation degree of ρ = 1.0, the mean pressure is 34.15m, which is higher than the corresponding value of 30.25m when *P*R = 0.9 and ρ = 0. For a reliability level of *P*R = 0.5 and the correlation degree of ρ = 1.0, the mean pressure Node 12 is 29.02m, which is close to the corresponding value of 27.87m when *P*R = 0.5 and ρ = 0. The number of Monte Carlo samples that meet the pressure requirements is basically the same, therefore the construction cost of the two cases is similar as shown in Figs. 4 and 5.

## 3.2 Influence of WDN characteristics on optimal design

To reach the optimal design of WDN with larger reliability levels (such as *P*R>0.90), the relative difference of construction cost between correlated and independent demands for Hanoi is greater than that of PG network. This is due to the differences in demand and pipe length between the two WDN cases. For the design solutions corresponding to the reliability level *P*R = 0.95, the statistics of nodal pressures in the Monte Carlo samples are shown in Fig. 7.

According to Figs. 3 and 7, In the Hanoi network, the nodes with greater water demand have greater coefficient of variation (CV) of node water pressure, which leads to lower reliability of node water pressure, such as Node 12 has a larger water demand 261.1 L/s and corresponding to a greater CV of water pressure 0.167. The Hanoi network has 31 nodes with a total water demand of 5538.9 L/s. Approximately 77% of the nodes have a demand above 100 L/s, and the total length of pipes is approximately 39.42 km. On the other hand, in the PG network, 93% of the nodes have a demand below 10 L/s, and the total pipe length is approximately 9.9 km. In Fig. 7(c), the CVs of water pressures at nodes in the Hanoi network ranges from 0.06 to 0.167 for correlation coefficients ρ = 1.0. In Fig. 7(d), the CVs of water pressures at nodes in the PG network is smaller than that in Hanoi network, where the CVs always below 0.04.

According to Figs. 7(a) and 7(b), the correlation coefficient does not affect the mean value of nodal pressure. Additionally, as shown in Fig. 7(d), the nodes {8 ~ 10, 28, 29, 36, 40 ~ 45} in the PG network have slightly larger CVs compared to other nodes. These nodes are either located further away from the reservoir or situated at the end of the network. As a result, the uncertainty in water demand brings greater impact on the pressure fluctuations at these nodes.

## 3.3 Influence of varied degree of uncertainty and correlation

In this section, the differences in design results of WDN under various uncertainty levels are investigated. The uncertainty levels of the random variables (RVs), represented by the coefficient of variation (CV), are set to {0.1, 0.2, 0.3}, and the correlation coefficient (ρ) between nodal demands is set to {0, 0.5}. The Pareto fronts of the optimal design of Hanoi pipe network are shown in Fig. 8. As shown in the Figs. 8(a) and 8(b), the design cost decreases as the CV increases when *P*R < 0.35, while the design cost increases with a larger CV when *P*R > 0.35. This pattern is similar to that shown in Fig. 4, but the difference is that the threshold reliability *P*R has changed, with the *P*R threshold 0.5 for the PG network and 0.35 for the Hanoi network.

To investigate the threshold value of reliability in the Pareto fronts, the Monte Carlo samples of water pressures at Node 44 in the PG network were selected for analysis. Actually, Node 44 located at the far terminal of the network with the worst availability of water pressure. The probability distribution function (PDF) and cumulative distribution function (CDF) plots of water pressure samples at Node 44 are shown in Fig. 9.

In Figs. 9(a) to 9(c), the mean value of pressures at Node 44 under different uncertainty levels (CVs) are labeled in the figure legend. As shown in Fig. 9(b), the mean pressures for three CVs are approximately equal when *P*R = 0.5, which is consistent with phenomena that the intersection of the three CVs’ Pareto frontiers at *P*R = 0.5 as shown in Fig. 8(c). As shown in Figs. 9(d) to 9(f), the intersection points of the CDF plots for the three CVs are located nearby the value (1- *P*R), which indicates the intersection points of the CDF plots depend on the reliability level. The impact of uncertainty level (CVs) on the PDF shapes follows a similar pattern to the impact of correlation level (ρ) on the PDF shapes, as shown in Fig. 6 and described in Section 3.1.

Table 1 shows the relative difference (RD) of construction cost corresponding optimal designs between varied uncertainty levels (CVs). For the Hanoi network, achieving the same reliability level of *P*R = 0.95 with independent nodal demands (ρ = 0), compared to the CV = 0.1, the RD of design cost for CV = 0.2 and CV = 0.3 are 3.20% and 4.80%, respectively. In the case of a correlation coefficient ρ = 0.5, compared to the CV = 0.1, the RD of design cost for CV = 0.2 and CV = 0.3 are 8.53% and 15.51%, respectively. Therefore, it is evident that an increase in the uncertainty of nodal demands leads to an increase in the design cost, while the correlations between nodal demands further results in an additional increment in the optimal design costs.

Table 1

Relative difference (RD) of design cost between different uncertainty levels of water demand (Compared with CV = 0.1)

WDN | ρ | *P*R = 0.85 | *P*R = 0.9 | *P*R = 0.95 |

CV = 0.1 | CV = 0.2 | CV = 0.3 | CV = 0.1 | CV = 0.2 | CV = 0.3 | CV = 0.1 | CV = 0.2 | CV = 0.3 | |

Hanoi | 0 | 0 | 1.61% | 3.23% | 0 | 3.05% | 4.03% | 0 | 3.20% | 4.80% | |

0.5 | 0 | 5.56% | 11.11% | 0 | 7.09% | 14.17% | 0 | 8.53% | 15.51% | |

PG | 0 | 0 | 2.08% | 3.99% | 0 | 3.12% | 4.60% | 0 | 3.52% | 4.36% | |

0.5 | 0 | 2.31% | 5.19% | 0 | 3.27% | 6.61% | 0 | 3.71% | 9.48% | |

To investigate the reasons for the cost increment in network design due to the increased uncertainty and correlation of nodal water demand, the design results of pipe diameters in PG network at *P*R = 0.95 are shown in Fig. 10. According to Figs. 10(a) and 10(b), compared to the pipe diameter corresponding to CV = 0.1, the diameters of 23 pipes change in the optimal design solutions corresponding to CV = 0.3, where 14 pipes corresponding an increment in diameter, generally by 100mm. As shown in Figs. 10(b) and 10(c), compared to the pipe diameter corresponding to the uncertain scenario with ρ = 0, the diameters of 37 pipes change in the optimal design solutions corresponding to ρ = 0.5, where 20 pipes have a diameter increment ranging from 100mm to 200mm. From the perspective of changes in the diameter of the pipes, the correlation of nodal water demand has a greater impact on the design results. In the independent scenario shown in Fig. 10(a), the backbone of network PG consists of two main pipelines, depicted by yellow and orange colors. Conversely, in the correlation scenario with ρ = 0.5 as shown in 10Figure (c), there is only one main pipeline depicted by red and orange colors. Comparison between Figs. 10(a) and 10(c) shows that, there are Significant changes in the diameters of pipe serving the nodes with higher water demand (Such as Nodes 27, 28, 29, 31) and those situated far from the source (Such as Nodes 42, 43, 45), where the larger pipe diameter is evident. These changes in pipe diameters underscore the influence of water demand uncertainty and correlation on the optimal design of WDN. Additionally, the volume of nodal water demand and the distance to the source are also crucial factors that shape the optimal design.

## 3.4 Comparison between partial inter-group and overall correlations

In this section, the effects of partial inter-group correlation and overall correlation of nodal demands on the design results are discussed. In actual water consumption patterns, the water demand at different types of user nodes (e.g., residential, commercial and industrial) behaves partial inter-group correlation rather than overall correlation with a uniform coefficient. In the Hanoi network, the user nodes are randomly divided into three groups: A, B, and C. The intra-group correlation coefficient (ρintra) and the inter-group correlation coefficient (ρinter) are set to ρintra = 0 and ρinter = 0.5, respectively. Similarly, for the PG network, the user nodes are divided into two groups: A and B, where Nodes 1 to 22 are assigned to Group A, and nodes 23 to 45 are assigned to Group B. Nodes in Group A are closer to the source, while nodes in Group B are far away from the source. The correlation scenarios of nodal water demand for different groups are shown in Table 2, where C1 corresponds to independent scenario, C2 corresponds to partial inter-group correlation scenario and C3 corresponds to overall correlation scenario.

For the Hanoi network, as shown in Fig. 11(a), when the reliability level *P*R > 0.5, the cost ranking for the three cases is C3 > C2 > C1, corresponding to the design cost of {129 billion CNY, 126 billion CNY, 124 billion CNY}at *P*R ≥ 0.95, respectively. Compared to the independent scenario (C1), the cost increment rates for partial inter-group correlation scenario (C2) and overall correlation scenario (C3) are 1.61% and 4.03%, respectively.

Table 2

Correlation cases of nodal water demand

WDN | Case | Intra-group correlation coefficient ρintra | Inter-group correlation coefficient ρinter | Description |

Group A | Group B | Group C |

Hanoi | C1 | 0 | 0 | 0 | 0 | Independent |

C2 | 0.5 | 0.5 | 0.5 | 0 | Partial inter-group correlation |

C3 | 0.5 | 0.5 | 0.5 | 0.5 | Overall correlation |

PG | C1 | 0 | 0 | - | 0 | Independent |

C2 | 0.5 | 0.5 | - | 0 | Partial inter-group correlation |

C3 | 0.5 | 0.5 | - | 0.5 | Overall correlation |

For the PG network, as shown in Fig. 11(b), the cost ranking for the three scenarios is C3 > C1 > C2 and the design costs for the three cases are {5.19 billion CNY, 4.99 billion CNY, 4.82 billion CNY} at *P*R ≥ 0.95, respectively. Compared to the case C1, the cost increment rates for C2 and C3 are − 3.41% and 4.01%, respectively. Therefore, the design cost for the partial inter-group correlation (C2) is lower than that for the independent scenario (C1), primarily attributed to the grouping of nodal correlation. Specifically, in the PG network, nodes in Group A are closer to the source, whereas nodes in Group B are far away. In contrast, the nodes in the Hanoi network are grouped randomly.

The pipe design results corresponding to the three cases of independent, partial inter-group correlation, and overall correlation in the Hanoi network are shown in Fig. 12. Compared to C1, the diameters of pipes {12, 16, 18, 19, 26} increased in C2, and the diameters of pipes {12, 14, 16, 17, 18, 19, 26, 27, 28, 30} increased in C3. The common characteristic of the pipes with increased diameter in C2 and C3 is that they are either connected to the nodes with higher water demand (such as Node 17 and Node 25) or connected to the nodes far away from the source (such as Node 12).

The above analysis indicates that ignoring the influence of partial inter-group correlation and directly evaluating based on overall correlation may lead to higher costs and overly conservative design results. In practical design of WDN, different water consumption patterns of users actually reveal grouped nodal water demand and present the partial inter-group correlation. In order to obtain economically rational design solutions for actual WDN, it is necessary to differentiate the partial inter-group correlation and overall correlation of nodal water demand.