## Abstract

In this paper, information entropy was proposed to measure water quality reliability in a water distribution system (WDS), which had been applied to evaluate hydraulic reliability in the WDS. In the water quality reliability evaluation, residual chlorine is a representative of water quality, and a first-order decay model was usually adopted. The water quality reliability (*R*) based on water quality entropy (WQE) and improved water quality reliability (*R _{d}*) based on improved water quality entropy (IWQE) were proposed and compared for three networks. The method was developed based on the EPANET toolkit and MATLAB environment. The results indicated that flow entropy (FE) is strongly related to WQE, and improved flow entropy (IFE) is also strongly related to IWQE. In addition,

*R*can reflect the effect of pipe velocity, whereas

_{d}*R*can only reflect the effects of pipe flow and the WDS layout. The novelty of this paper is to develop the entropy with consideration of the pipe velocity to measure water quality liability as a surrogate index, which can reduce the calculation load and can be applied to a nonlinear system. The proposed water quality reliability evaluation method based on information entropy can help design, analyze, and improve the water quality in the WDS.

## HIGHLIGHTS

Flow entropy (FE), diameter-sensitive flow entropy (DSFE), water quality entropy (WQE), and diameter-sensitive water quality entropy (DSWQE) were discovered.

Entropy was applied to water distribution systems (WDSs).

The difference between WQE and multiplier for WDSs was analyzed.

Water quality reliability analyses based on WQE and DSWQE were compared.

The relationship between entropy and characteristics of WDSs was discovered.

## ABBREVIATIONS

## INTRODUCTION

Reliability analysis of the water distribution system (WDS) is one of the most important aspects to guarantee undisrupted, enough, and safe water supply to consumers even during abnormal conditions (Tanyimboh *et al.* 2011). The reliability of the WDS is concluded into the following three groups: mechanical reliability, hydraulic reliability, and water quality reliability (WQR) (Kansal *et al.* 1995; Zhao *et al.* 2010). Substantial researches focused on two aspects of mechanical reliability and hydraulic reliability (Setiadi *et al.* 2005; Tanyimboh *et al.* 2011; Islam *et al.* 2014). Few research works put emphasis on the reliability of water quality (Zhao *et al.* 2010; Islam *et al.* 2014; Beker & Kansal 2022). However, the water quality in the WDS deteriorated with physical, chemical, and biological processes by traveling through pipe systems, which makes the research on WQR more essential and necessary. Since residual chlorine is one of the best indicators to represent water quality in the WDS, it is usually taken as the indicator to analyze the WQR in the WDS (Shafiqul Islam *et al.* 2014).

The main method to evaluate the WQR in the WDS is the simulation-based approach, which measures the probability that a failed WDS can supply the consumers with acceptable water quality (Gheisi *et al.* 2016). By taking account of the variation in water demand and chlorine, Monte Carlo Simulation (MCS) was applied to evaluate residual chlorine availability to represent WQR (Zhao *et al.* 2010; Li *et al.* 2013). Although MCS can obtain accurate reliability estimation, it is computationally inefficient due to a very large number of realizations. To improve the computational efficiency of the MCS, the first-order reliability method was proposed (Tolson *et al.* 2001). However, the first-order reliability method based on Taylor series expansion is more suitable for a linear system. As such, information entropy technology was introduced to analyze the reliability of the WDS, which is termed as flow entropy (FE) to measure the redundancy and flow uniformity of a WDS (Awumah *et al.* 1990, 1991). Thereafter, maximum FE for the WDS with single and multiple sources was proposed (Tanyimboh & Templeman 1993; Yassin-Kassab *et al.* 1999). The method can reduce the computational load since complex stochastic sampling and iteration procedures are not required (Liu *et al.* 2014). FE is proved to be strongly related to hydraulic reliability (Tanyimboh & Templeman 2000; Setiadi *et al.* 2005; Santonastaso *et al.* 2018). Sensitivity analysis between entropy and hydraulic reliability proved that the design of the WDS based on maximum entropy flow was more reliable than other designs (Tanyimboh & Setiadi 2008). Comparison among surrogate indicators of statistical entropy, network resilience, resilience index, and modified resilience index indicated that statistical entropy outperformed other indices (Tanyimboh *et al.* 2011). As such, FE was used for surrogate indicators of hydraulic reliability, and integrated into an optimization design based on reliability (Creaco *et al.* 2014; Singh & Oh 2015). Although entropy technology had been proposed and applied for more than two decades (Awumah *et al.* 1991; Yassin-Kassab *et al.* 1999; Singh & Oh 2015), it had not been widely used for measuring the WQR of the WDS. Tsalli entropy was proposed to measure the WQR of 26 layouts based on a WDS with eight nodes and a source, and a comparison between the Tsalli entropy and Shannon entropy was performed. The results indicated that the Shannon entropy can reflect the effect of loop number better than the Tsalli entropy (Wang & Zhu 2021). As such, the Shannon entropy was selected to measure the hydraulic reliabiltity and WQR. FE, improved flow entropy (IFE), water quality entropy (WQE), and improved water quality entropy (IWQE) were applied and compared for measuring the hydraulic reliability and WQR, respectively. The reason for introducing IFE and IWQE is that the flow velocity has significant effects on pipe flow and nodal water quality (chlorine concentration). For three WDSs, the relationships among FE, IFE, WQE, and IWQE were performed, respectively. In addition, the WQR *R* based on WQE and *R _{d}* based on IWQE were obtained, and the variation of

*R*and

*R*with the demand multiplier for three WDSs was analyzed. The novelty of this paper is to develop the entropy to measure WQR as a surrogate index, which can reduce the calculation load and can be applied to a nonlinear system.

_{d}In this paper, first, based on FE, the concepts of IFE, WQE, and IWQE were proposed. Second, WQR *R* and *R _{d}* were defined based on WQE and IWQE, respectively, and applied to three networks with various flows and layouts, respectively. Finally, the discussions and conclusions were drawn.

## METHOD

### Flow entropy/Improved flow entropy

*S*is the entropy,

*K*is an arbitrary positive constant, which is usually set to be 1.0,

*P*is the probabilistic associated with event

_{i}*i*,

*i*= 1, 2, 3, …,

*M*, which satisfies the condition of ,

*M*is the number of events.

*Tanyimboh*and

*Templeman*proposed the concept of conditional entropy, which is expressed by Equation (2) as follows.where

*S*is the FE of the system,

*S*

_{0}is the FE of sources,

*S*is the FE of node

_{i}*i*,

*N*is the number of nodes in the WDS, and is the ratio of flows into node

*i*() to the total flows supplied by all sources, which is expressed by ,

*T*

_{0}is the total flow for all the sources.

*S*

_{0}is expressed by Equation (3) as follows.where

*Q*

_{0i}is the flow at the source node

*i*, ,

*N*is the number of source nodes, and .

_{0}*i*(

*S*) is expressed by Equation (4a) as follows.where is the flow from node

_{i}*j*to node

*i*, ,

*NJ*is the number of nodes that flow into node

_{i}*i*, and . When only one pipe flows into node

*i*, the FE of node

*i*() is equal to 0.

Based on nodal IFE, the pipe with lower velocity, i.e., larger diameter under definite flow and less flow under definite diameter, results in larger entropy values. Larger pipe diameters and less flow can enhance the system's capability of coping with abnormal conditions, and decrease the probability of pipe bursts.

*S*characterizes the flow uniformity of the WDS, it remains to be identical when the layouts and the number of loops and pipes in the WDS are identified. However, the pipe velocity has a significant impact on reliability. As such, IFE is proposed, which is expressed by Equation (6) as follows.where is the IFE, represents the velocity of pipe from node

*j*to the node

*i*, and

*C*is a velocity constant of 1 m/s, as FE is a dimensionless quantity.

The definition of the IFE improved the FE by strengthening the relationship between reliability and entropy. As such, the IFE not only considers the (non) uniformity of flows in a network but also accounts for the impact of pipe velocity on entropy and reliability.

### Water quality entropy/Improved water quality entropy

*E*is the overall WQE,

*E*

_{0}is the WQE of sources,

*E*is the WQE of node

_{i}*i*,

*p*is the weight of node

_{i}*i*, which can be expressed by , where is chlorine concentration at node

*i*.

*E*is expressed by Equation (8) as follows.where

_{0}*C*is the chlorine concentration at the source node

_{0i}*i*,

*W*

_{0}is the total chlorine content for all the sources, .

*j*,

*k*is the chlorine decay coefficient, and , , , are the travel time, length, velocity, and diameter of pipe from node

_{0}*j*to node

*i*, respectively.

For a specific pipe, lower velocity leads to longer residence time (i.e., water age), which is not beneficial for water quality due to chlorine decay. From Equations (10) to (14), it can be found that nodal WQE *E _{i}* decreased under the condition of lower velocity connected with node

*i*. As such, nodal WQE

*E*reflects the effect of pipe velocity on chlorine decay.

_{i}*E*was proposed, and expressed by Equation (16) as follows

_{d}Based on IWQE, the impact of pipe velocity has two sides. One side is that within Equation (16), lower velocity results in larger entropy values. Under definite pipe flow lower velocity means larger pipe diameter, which can reduce the occurrence of abnormal conditions, probability of pipe bursts, and water quality failure, IWQE improved WQE by strengthening the relationship between WQR and WQE. The other side is that in Equation (16), under definite pipe diameter lower velocity, means less flow, which results in longer residence time (i.e., water age) from node *j* to node *i,* and WQE at node *i* becomes lower since chlorine concentration at the end of pipe connecting from node *j* to node *i* () decreases due to the process of chlorine residual decay. The definition of WQE considers the effect of the chlorine decay and WDS layout, while the definition of IWQE considers also considers the effects of pipe velocity.

### Water quality reliability

*E*and

*E*to

_{d}*E*can be surrogate indices for assessing the WQR of the WDS, which are expressed by Equations (19a)–(19b) as follows.where

_{max}*R*is WQR based on the WQE, and is WQR based on the IWQE. The computation of

*R*and in the WDS was performed in MATLAB environment by calling EPANET toolkit. The water quality modeling in the WDS requires extended period simulation (EPS). In the EPS water quality simulation or EPANET, the quality time step is 5 minutes, which is much shorter than the hydraulic simulation time step of 1 h. The assumed initial chlorine concentrations were assigned to nodes, which does not happen in real WDS. The program was operated for 72 h to avoid the effect of initial chlorine concentrations on

*R*and , and the obtained values of WQR from the 49th hour to the 72nd hour were analyzed. The bulk decay and wall decay are considered to follow the first-order decay regulation in simulations.

## CASE STUDY

### Case 1

*et al.*2016). The bulk chlorine decay coefficient and the wall chlorine decay coefficients are −1.00 and −0.55/day, respectively. The initial chlorine concentration at the source node and demand nodes are 0.8 and 0.5 mg/L, respectively.

Hour . | Multiplier . | Hour . | Multiplier . | Hour . | Multiplier . |
---|---|---|---|---|---|

1 | 0.41 | 9 | 1.30 | 17 | 1.06 |

2 | 0.38 | 10 | 1.24 | 18 | 1.27 |

3 | 0.38 | 11 | 1.28 | 19 | 1.21 |

4 | 0.45 | 12 | 1.16 | 20 | 1.15 |

5 | 0.83 | 13 | 1.14 | 21 | 0.87 |

6 | 0.99 | 14 | 0.87 | 22 | 0.71 |

7 | 1.53 | 15 | 0.89 | 23 | 0.60 |

8 | 1.46 | 16 | 0.87 | 24 | 0.41 |

Hour . | Multiplier . | Hour . | Multiplier . | Hour . | Multiplier . |
---|---|---|---|---|---|

1 | 0.41 | 9 | 1.30 | 17 | 1.06 |

2 | 0.38 | 10 | 1.24 | 18 | 1.27 |

3 | 0.38 | 11 | 1.28 | 19 | 1.21 |

4 | 0.45 | 12 | 1.16 | 20 | 1.15 |

5 | 0.83 | 13 | 1.14 | 21 | 0.87 |

6 | 0.99 | 14 | 0.87 | 22 | 0.71 |

7 | 1.53 | 15 | 0.89 | 23 | 0.60 |

8 | 1.46 | 16 | 0.87 | 24 | 0.41 |

### Case 2

*et al.*2011; Islam

*et al.*2014). Water is supplied by gravity from two elevated reservoirs with a total head of 90 and 85 m, respectively. The detailed nodal demands, elevations, and pipe information can be found in literature works (Islam

*et al.*2011, 2014). The water demand multiplier, the bulk chlorine decay coefficient, the wall chlorine decay coefficient, the initial chlorine concentrations at source nodes, and the initial chlorine concentrations at demand nodes are the same as Case 1.

### Case 3

*et al.*1987). The elevations of nodes 10, 65, and 165 are 10 ft (3.05 m), 215 ft (65.575 m), and 215 ft (65.575 m), respectively. The WDS has 19 connection nodes and 40 pipes. The pump has a shutoff head value of 300 ft (91.5 m), a maximum flow rate of 8,000 GPM (0.506 m

^{3}/s). Detailed information of pipe length, diameter, and nodal elevation can be found in the literature (Walski

*et al.*1987). The water demand multiplier, the bulk chlorine decay coefficient, the wall chlorine decay coefficient, the initial chlorine concentrations at source nodes, and the initial chlorine concentrations at demand nodes are the same as Case 1.

## RESULTS AND DISCUSSION

### Application to Case 1

To analyze the variation of nodal inflow, chlorine concentration, FE, IFE, WQE, IWQE, *R*, and *R _{d}*, the average node degree (AND) is introduced. The AND is a basic measure of the connectivity reflecting the overall topological similarity of the network to perfect grids or lattice-like structures. The average value of the node degree distribution can be expressed as , where <

*k*> is AND,

*m*is the number of pipes, and

*n*is the number of nodes in the WDS. The values of AND are 2.2, 3.2, and 4.2 for Case 1, Case 2, and Case 3, respectively, which are related to the nodal chlorine concentrations due to the mixing process that occurred at nodes.

In Figure 5(b), the values of average nodal FE remain to be the same value of 0.053, and the average nodal IFE values range from 0.076 to 0.304 larger than the values of average nodal FE. The value of average nodal WQE almost unchanged ranging from 0.05 to 0.053 with little variation, and the average nodal IWQE values range from 0.075 to 0.290, which are much larger than average nodal WQE. The average nodal FE and WQE have a relationship with the layout of the WDS, which reflect the uniformity of nodal flow and chlorine concentration in the WDS. In addition, the values of average nodal IFE have a negative relationship with the demand multiplier, which is different from the nodal inflow. Since the average nodal IFE is affected by pipe velocities, which increase with the increase of nodal inflow. The values of average nodal IWQE also have a negative relationship with the demand multiplier, which is the same with chlorine concentration. The reason is that the average nodal IWQE is affected by the decay process, the WDS layout, and pipe flows. With the increase of demand multiplier, the decaying process leads to the increase of average nodal IWQE, and the WDS layout and pipe flow leads to the increase or decrease of average nodal IWQE. The effect of the WDS layout and pipe flow is larger than the decay process, which leads to the values of average nodal IWQE having negative relationship with the demand multiplier.

In Figure 5(c), the values of FE remain to be the same value of 0.005, and the values of IFE range from 0.008 to 0.03. The values of WQE range from 0.006 to 0.009, while the IWQE values range from 0.009 to 0.05. The values of FE, IFE, WQE, and IWQE are less than average for nodal FE, IFE, WQE, and IWQE. In addition, the variation trends of FE, IFE, and IWQE are the same with the variation trends of average nodal FE, IFE, and IWQE, while the variation trends of WQE are almost contrary to the variation trends of average nodal WQE. The reason is that in Equations (5), (6), (15), and (16), the values of FE, IFE, WQE, and IWQE are not only affected by average nodal FE, IFE, WQE, and IWQE, but also affected by flow and residual chlorine distribution represented by *pf _{i}* and

*p*. The values of four entropy measures decreased in the order of IWQE > IFE > WQE > FE. In Case 1, the demand multiplier only affects IFE WQE, and IWQE, while having no effects on FE. The values of IFE and IWQE also have a negative relationship with the demand multiplier. The reason is the same with the reason for average nodal IFE and IWQE.

_{i}In Figure 5(d), the values of *R* and *R _{d}* for Case 1 are obtained by performing Equations (18) and (19), which range from 0.095 to 0.129 and from 0.138 to 0.739, respectively. The values of

*R*and

*R*have a negative relationship with demand multipliers, which are the same with the variation trends of WQE and IWQE. When the demand multiplier is the greatest at the 7th hour (1.53) with higher user demands, the

_{d}*R*and

*R*reach almost the lowest value of 0.097 and 0.138, respectively. When the water demand multiplier is the lowest during periods from the 2nd hour to the 3rd hour (0.38) with less user demands, the

_{d}*R*and

*R*reach almost the highest value of 0.126 and 0.739, respectively. The values of

_{d}*R*and

*R*are also affected by two aspects. One aspect is that when the water demand multiplier increases, the water age becomes shorter which causes the rise of

_{d}*R*and

*R*in the WDS. The other aspect is that when the water demand multiplier is higher, the pipe flow increases, which may increase or decrease the values of

_{d}*R*and

*R*for a specific WDS layout. For Case 1, the effect of pipe flow on

_{d}*R*and

*R*is greater than the decay process, which leads to the negative relationship between

_{d}*R*/

*R*and the demand multiplier. In addition, the value of

_{d}*R*is greater than

_{d}*R*which proved that

*R*can reflect the effect of the water demand multiplier more significantly than

_{d}*R*.

### Application to Case 2

In Figure 6(b), the values of average nodal FE and IFE range from 0.226 to 0.283 and from 0.268 to 0.856, respectively, which are larger than Case 1. The value of average nodal WQE and IWQE range from 0.225 to 0.281 and from 0.266 to 0.849, respectively, which are also larger than Case 1. The results indicated that the uniformity of nodal flow and chlorine concentration in Case 2 is better than Case 1. In addition, due to the effect of pipe velocity, the values of average nodal IFE and IWQE are greater than average nodal FE and WQE, respectively. The average nodal FE and WQE have a negative relationship with demand multipliers except for the times from the 6th hour (0.99) to the 20th hour (1.15), and the average nodal IFE and IWQE have a negative relationship with demand multiplier. The results indicated that at times with relatively greater demand multipliers for Case 2 with AND value larger than Case 1 the WDS layout has an effect on the average nodal FE and WQE values. In addition, both WDS layout and pipe velocity have effects on the average nodal IFE and IWQE.

In Figure 6(c), the values of FE and IFE range from 0.382 to 0.858 and from 0.633 to 1.189, respectively. The values of WQE and IWQE range from 0.395 to 0.898 and from 0.672 to 1.110, respectively, which are larger than Case 1. The variation trends of FE, IFE, WQE, and IWQE are in accordance with the variation trends of average nodal FE, IFE, WQE, and IWQE except for the times with the lower multipliers from the 21st hour (0.87) to the 5th hour (0.83). The results indicated that the effects of *pf _{i}* and

*p*are more significant at times with lower multipliers. Different from Case 1, the values of four entropy measures decreased in the order of IFE > IWQE > WQE > FE. For Case 2 the demand multipliers affect not only IFE, WQE, and IWQE, but also FE. The values of FE and WQE have a positive relationship with the demand multiplier, while the values of IFE and IWQE have a negative relationship with the demand multiplier except for the times from the 21st hour to the 5th hour. The results indicated for Case 2 with the relatively larger flow and AND, the decay process, the WDS layout, and distribution of flow and residual chlorine have effects on the values of FE and WQE, and the pipe velocity as well as effects on the values of IFE and IWQE.

_{i}In Figure 6(d) the values of *R* and *R _{d}* for Case 2 range from 0.366 to 0.818 and from 0.621 to 1.011, respectively. The values of

*R*and

*R*in Case 2 are much higher than Case 1. The variation trends of

_{d}*R*and

*R*are also the same with the variation trends of WQE and IWQE. The relationship between

_{d}*R*as well as

*R*and water demand multiplier is different from Case 1. The value of

_{d}*R*has a positive relationship with the demand multiplier while

*R*has a negative relationship with the demand multiplier except for the times from the 21st hour to the 5th hour.

_{d}### Application to case 3

In Figure 7(b), the values of average nodal FE and IFE range from 0.469 to 0.538 and from 1.257 to 4.027, respectively. The value of average nodal WQE and IWQE range from 0.425 to 0.503 and from 0.957 to 3.520, respectively. The values of average nodal IFE and average nodal IWQE are larger than average nodal FE and WQE. The average nodal FE, IFE, WQE, and IWQE are larger than Case 1 and Case 2. The results indicated that the uniformity of nodal flow and chlorine concentration in Case 3 is better than Case 1 and Case 2 due to the relatively larger AND value of 4.2. Similar with Case 2, the average nodal FE and WQE have a negative relationship with demand multipliers except for the times from the 6th hour (0.99) to the 20th hour (1.15), and the average nodal IFE and IWQE have a negative relationship with demand multiplier. The results indicated that at times with relatively greater demand multipliers for Case 3 with AND value larger than Case 1 and Case 2 the pipe flow has an effect on the average nodal FE and WQE values. In addition, both pipe flow, the WDS layout, and pipe velocity have effects on the average nodal IFE and IWQE.

In Figure 7(c), the values of FE and IFE range from 0.979 to 1.530 and from 2.079 to 2.769, respectively. The values of WQE and IWQE range from 0.894 to 1.472 and from 1.775 to 2.208, respectively. The variation trends of FE and WQE are in accordance with the variation trends of average nodal FE and WQE except for the times with the lower multipliers from the 21st hour (0.87) to the 5th hour (0.83), while the variation trends of IFE and IWQE are in accordance with the variation trends of average nodal IFE and IWQE. The results indicated that average nodal FE and WQE were affected by both the values of *pf _{i}* and

*p*and the flow in the WDS. For Case 2 with a relatively larger flow, the average nodal FE, WQE, IFE, and IWQE were all affected by the values of

_{i}*pf*and

_{i}*p*at times with lower multipliers. For Case 3 with the relatively smaller flow, the average nodal FE and WQE were affected at times with lower multipliers, while the average nodal IFE and IWQE were not affected by

_{i}*pf*and

_{i}*p*. The values of four entropy measures decreased in the order of IFE > IWQE > FE > WQE. The values of FE and WQE have a positive relationship with the demand multiplier, while the values of IFE and IWQE have a negative relationship with the demand multiplier.

_{i}In Figure 7(d), the values of *R* and *R _{d}* range from 0.512 to 0.842 and from 1.015 to 1.245, respectively. The values of WQR are higher than Case 1 and Case 2. The variation trends of

*R*and

*R*are also the same with the variation trends of WQE and IWQE. The relationship between

_{d}*R*as well as

*R*and water demand multiplier is different from Case 1 and Case 2. The value of

_{d}*R*has a positive relationship with the demand multiplier while

*R*has a negative relationship with the demand multiplier. As such,

_{d}*R*can only reflect the effects of the decay process, pipe flow, and the WDS layout, while

*R*can reflect the effects of pipe velocity as well.

_{d}### Nodal mean WQE and nodal mean IWQE

*R*for node 16, node 27, and node 29 are not equal to 0 with values of 0.549, 0.641, and 0.413, respectively, which only occupied 0.098% of the total number of connection nodes. In Case 2, during the simulation process, the mean WQE for ten nodes including node 3, node 6, node 7, node 10, node 12, node 17, node 22, node 23, node 24, and node 26 are equal to zero shown in (Figure 8(a)). For node 8, node 11, node 14, node 15, node 16, node 19, node 20, and node 27, the WQE is greater than 0.5, which occupied 29.6% of the connection nodes. The regulations for WQE and IWQE for nodes are different, and the greatest WQE is at node 20, while the greatest IWQE is at node 27. The reason is that node 27 is at the end of the WDS with less velocity and pipe flow, which leads to the greatest IWQE. The result indicated that IWQE can reflect the effect of pipe velocity comprehensively.

The mean WQE of connection nodes of the Anytown network is shown in Figure 8(b). During the simulation process, the mean WQE for three nodes including node 20, node 40, and node 55 is equal to zero. For node 50, node 75, node 80, node 90, node 100, node 120, and node 130, node 140, and node 150, the values of WQE are greater than 0.5, which occupied 47.4% of the total connection nodes. Similar with Case 2, the regulations of WQE and IWQE for nodes are different, and the greatest WQE is at node 140, while the greatest IWQE is at node 80. The reason is that WQE considers the effect of the WDS layout and distribution of flow and residual chlorine, and IWQE also takes into account of pipe velocity.

### Sensitivity analysis

*R*and

*R*under various optimal designs of pipe diameters for Case 3 (Siew

_{d}*et al.*2016) is performed (shown in Figure 9). Compared with Figure 7(d) for Case 3 (Original designs), the values of

*R*and

*R*based on WQE and IWQE decreased after optimization with the objectives of minimizing total cost and maximizing system performances. In addition, the variation trend of

_{d}*R*based WQE is similar to that of

*R*based on IWQE, which is different from

_{d}*R*and

*R*for the original design of Case 3 (shown in Figure 7(d)). The results indicated that the variation of

_{d}*R*based on IWQE is associated with the layout of the WDS, while the variation of

_{d}*R*based on WQE is almost not affected by the layout of the WDS, which can be explained by the definition of

*R*and

*R*. The relationship between optimization layout and WQR assessment should be researched in depth in the future study.

_{d}## CONCLUSIONS

In this paper, based on FE, the concepts of IFE, WQE, and IWQE based on the Shannon entropy were proposed, and *R* and *R _{d}* based on WQE and IWQE were defined and applied to three networks. The results indicated that: (1) Generally, the

*R*values have a negative relationship with the demand multiplier for the three cases, while the

_{d}*R*values have a negative relationship, partly negative relationship, and positive relationship with the demand multiplier for Case 1, Case 2, and Case 3, respectively; (2) The FE, IFE, WQE, IWQE,

*R*, and

*R*increased in the order of Case 1 < Case 2 < Case 3, which is in accordance with the order of AND values for the three cases; (3) The FE has a strong relationship with WQE, and IFE has a strong relationship with IWQE, and the values of IFE and IWQE are much larger than values of FE and WQE; (4) The nodal inflow increase with the demand multiplier, while chlorine decay is affected by decay process, pipes' flow and number connecting to the node. The results indicated that

_{d}*R*based on the WQE can only reflect the effects of the decay process, pipe flow, and the WDS layout, while

*R*based on the IWQE can reflect the effects of velocity as well. As such,

_{d}*R*can be a more suitable surrogate indicator for WQR than

_{d}*R*.

In this paper, the WQR based on the WQE and IWQE was analyzed and compared based on the traditional demand-driven hydraulic and water quality model. However, it cannot deal with the effects of deficiency in pressure on the water quality through the WDS. In addition, the variation of *R _{d}* is different under the various layout of the WDS. The method proposed is difficult to be applied to real WDS with the various layout since in the real WDS, the water quality in the WDS is affected by many factors, such as pipe burst, chlorine decay, and pipe failure. As such, in future research, the WQR under pressure-deficient conditions and under various layouts should be researched and quantified. The proposed method can help analyze the reliability of the WDS from viewpoint of water quality and can help improve the design of WDS.

## ACKNOWLEDGEMENT

This work was funded by the Natural Science Foundation of Jiangsu Province (Grant No. BK20191147).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.