## Abstract

A water distribution network (WDN) is an essential component of an urban water supply scheme to deliver safe and adequate water to consumers under various operational conditions. This study focuses on the performance evaluation of an urban WDN using fuzzy logic-based aggregation of reliability, resilience, and vulnerability indices. To assess the individual performance indicators, this study advocates the pressure-dependent analysis (PDA) for hydraulic simulation. Furthermore, it advocates a fuzzy rule-based aggregated performance index (API) that will deliver the outcome in linguistic form and help the decision-maker to prioritize the maintenance of the WDN. The proposed method is illustrated with the help of a real-time WDN for part of Dire Dawa city in Ethiopia. It has been found that the API values for this network are 0.721 and 0.624, respectively, under normal and abnormal conditions, which are just satisfactory. It has been noticed that nodes J4, J44, J47, and J49 are critical from the overall low API. Efforts should be made to improve the hydraulic and residual chlorine conditions at these nodes to increase the API. It is felt that such a methodology will help the decision-makers to improve the performance of an existing urban WDN.

## HIGHLIGHTS

This study models a real urban WDN and assesses the nodal reliability, resilience, and vulnerability using a hydraulic simulator WaterGEMS.

This study proposes an aggregated performance index (API) as a single performance index for an urban WDN.

This study highlights the importance of the API as an aid to decision-making for maintenance purposes.

### Graphical Abstract

## INTRODUCTION

Performance assessment of an urban water distribution network (WDN) is a complex process but is an important aspect of delivering safe and adequate water supply to the consumers under normal and abnormal operating conditions. Hydraulic simulation in terms of flow, velocity, pressure, and residual disinfectant (chlorine) at various nodes helps the water utility manager and operators to understand, evaluate, and compare various alternatives, plans, and operation and maintenance options. Urban WDNs are complex infrastructure that interconnect various pipelines at crucial nodes. However, such pipelines are frequently vulnerable to pipe anomalies such as leakage, partial blockage, ill-junction, biofilm, corrosion, cavitation, air-pocket, and detachment (Duan 2018, 2020; Guo *et al.* 2021). Furthermore, the presence of pipe anomalies may cause reduction in water delivery, increase of head loss, and deterioration of water quality, thus leading to the reduction of the system performance (Che *et al.* 2018). In addition, water-hammer surges can potentially have adverse impacts on the performance of water supply infrastructures and the safety of users and operators (Annus *et al.* 2020; Trabelsi & Triki 2020). In the USA, the estimated number of pipe failures in WDNs per year is in the order of 240,000, wasting over 2 million gallons of drinking water/day (ASCE 2017). Besides, more than 30% of drinking water is lost from pipelines worldwide (Duan *et al.* 2020).

WDNs are asset-intensive systems that require effective management of components to ensure and sustain the delivery of the required service, which needs huge and reliable data on the performance of the network. Conditional assessment (CA) is the most used method to assess WDN components' performance (mainly water pipelines). Pipeline CA approaches can be classified into two broad categories: (i) direct, e.g., automation and non-destructive testing (NDT), and (ii) indirect, e.g., soil resistivity tests, flow testing, and water audits (Mazumder *et al.* 2018). Automation is crucial for integrating complex systems, and it is an advanced method of pipe CA, but it is expensive (Olsson 2021). On the other hand, NDT is the most widely used CA approach; however, it does not incorporate the hydraulic characteristics of the overall system. Furthermore, it is often expensive, time-consuming, laborious, and unable to address all the distress indicators (Mazumder *et al.* 2018).

Reliability, resilience, and vulnerability are the three independent characteristics of the system performance. These are the most widely used performance indicators (PIs) to quantify the system performance (Mazumder *et al.* 2018). The concept of these indices was introduced by Hashimoto *et al.* (1982) to evaluate the performance of water resource infrastructures (mainly for water supply reservoirs), and it was subsequently improved by several studies (Tabesh *et al*. 2010; Sandoval-Solis *et al.* 2011; Dziedzic & Karney 2016; Giustolisi 2020). In the context of the WDN, reliability is defined as the probability of the WDN delivering a sufficient amount of water at adequate pressure with acceptable water quality under a given condition (Shuang *et al.* 2014). The WDN is frequently vulnerable to mechanical, hydraulic, and water quality failure (Gheisi *et al.* 2016; Młyński *et al.* 2021); so, it is essential to incorporate all failures to evaluate the performance through the reliability index. Research works have been conducted to develop methods for reliability assessment for different failure types individually, i.e., mechanical-related (Mazumder *et al.* 2019), hydraulic-related (Giustolisi 2020), and water quality-related (Gupta *et al.* 2012) failures. Resilience is described as the ability of a WDN to recover its functionality after a disruptive or failure event (Mazumder *et al.* 2018; Albarakati *et al.* 2021). Recent works evaluate networks based on resilience by considering isolation valves and segments under pipe failure conditions (Liu & Kang 2022). Vulnerability is another metric that measures the susceptibility of a network from failure (Tornyeviadzi *et al.* 2021; Swain *et al.* 2022). Agathokleous *et al.* (2017) assessed vulnerability and topological robustness to evaluate WDNs.

Reliability, resilience, and vulnerability are three independent system performance parameters. Most of the researchers have assessed and reported these separately or simply by taking the average to evaluate the overall performance. For example, Loucks (1997) presented a sustainability index (SI) for aggregating the three PIs which has been applied by Aydin *et al.* (2014). Dziedzic & Karney (2016) introduced an energy efficiency-based overall performance measure to evaluate the WDN using a connectivity index along with the other three PIs. These indices are aggregated using a simple geometric averaging method. However, there can be an ambiguity or eclipsing problem associated with each parameter (Samee & Tyagi 2000) and may lead to the problem of visualization of overall performance. Keeping this in mind, this study highlights the need for adopting a more scientific way to aggregate these parameters rather than just averaging/simply reporting. It is believed that fuzzy-based aggregation of three performance parameters can lead to a better single performance parameter. The fuzzy-based method is recommended as it can easily consider the qualitative and quantitative indicator values through the membership functions (MFs). The other methods can only take either quantitative or qualitative but not both. Therefore, it is useful to propose a better and acceptable aggregating methodology to help the decision-makers improve the WDN performance.

Fuzzy logic is a modern mathematical, efficient, and valuable tool that can handle linguistic variable, unclear, uncertain, or ambiguous information, in addition to integrating the PIs (Gupta & Bhave 2007). Ameyaw & Chan (2016) applied fuzzy logic to allocate risks in water infrastructure projects of the public–private partnership. Tornyeviadzi *et al.* (2021) used the fuzzy logic approach for nodal vulnerability assessment of WDNs. Fuzzy logic is efficient and performs better than other available methods for integrating PIs (Gupta & Bhave 2007; Ameyaw & Chan 2016).

Based on the above limitation, this paper aims to propose an aggregated performance index (API) for assessing the overall WDN performance using fuzzy logic under all ranges of operation mode (normal and a pipe failure state). The parts of Dire Dawa city (Ethiopia) are assigned for this study to test the validity of the present methodology.

## PROPOSED METHODOLOGY

Figure 1 depicts the flow chart methodology for assessing the API of the WDN based on fuzzy logic. The method has two parts: (1) to define and calculate PIs (reliability, resilience, and vulnerability) and (2) to aggregate PIs to a single API. The approaches begin by estimating hydraulic (water flow and nodal pressure) and water quality (residual chlorine concentration) parameters in normal and abnormal conditions needed to determine each PI. For this purpose, the WaterGEMS coupled with ArcGIS is applied for modeling and simulating the WDN. Furthermore, an efficient approach to the gradient method for the pressure demand dependent (PDD) analysis model developed by Wu and Walski (2006), integrated into the WaterGEMS model, is used to predict the pressure-dependent flow.

### Estimation of PIs

In this study, two scenarios (states) have been investigated. In the first state, performance has been evaluated under normal working conditions, where all the pipes are under regular operation, whereas in the second state, performance has been assessed under failure conditions (abnormal) when one pipe fails at a time. This study considers that one pipe failed at a time in sequence. Hence, there are 113 failure scenarios in total pertinent to each pipe (i.e., simulation was carried out for 113 pipes individually). However, an unlimited number of failure scenarios are possible, and the probability of multiple pipe failures simultaneously is very low (Tabesh *et al.* 2010). Multiple pipe failures rarely occur in the WDN, when natural disasters such as earthquake, landslide, and floods happen. Therefore, single pipe failure scenarios have been assumed for this study.

### Reliability index

Reliability is defined as the system's capacity to satisfy the consumer's needs (i.e., in terms of water flow, pressure, and water quality) under all operative conditions at a given period that can arise during an operation's working life (Gheisi *et al.* 2016). This study estimates the reliability index using the availability of pipes, hydraulic, and water quality aspects under normal and abnormal working conditions. Reliability can be a combination of mechanical, hydraulic, and water quality reliability (Tabesh *et al.* 2010).

### Hydraulic reliability

*AV*) is defined as the probability of the operating state of the pipe

_{i}*i*(Tabesh

*et al.*2009) and can be calculated using Equation (1). In the WDN, it is challenging to determine the availability of pipes, as there is insufficient data on failure and repair of elements. However, studies developed a regression equation to estimate availability based on its relationship with the length and diameter of the pipe (Cullinane

*et al.*1992). For instance, Kansal (1996) proposed Equations (2a–c) for stationary values of the mean time between failure (

*MTTF*), mean time to repair (

_{i}*MTTR*), and break rate for developing countries, especially in Indian conditions. The following equation is adopted to calculate the availability of pipes for this study.where

_{i}*N*is the break rate (breaks/km/length),

_{i}*D*is the diameter of the pipe (mm), and

*t*is the age of the pipeline (year), and assuming that the network is 25 years old for this study, and

*L*is the length of the pipe (km).

_{i}*et al*. (2010) proposed a method to determine the hydraulic reliability for a pipe failure condition that incorporates mechanical and hydraulic failures. According to the authors, hydraulic reliability at a node

*j*(

*HRel*(

*FS*)

*) is calculated by multiplying the probability of a failure with the hydraulic reliability factor under the event of multiple failure scenarios. This study adopted the approach by Tabesh*

_{j}*et al*. (2010). The author has mathematically described the approach in Equation (3a).

*re*(0) is the nodal hydraulic reliability factor when all pipes are working as calculated in Equation (3c), and

_{j}*re*(

_{j}*l*) represents the hydraulic reliability factor with

*i*specified pipes unavailable and is determined as follows:where

*P*(0) is the probability of no pipe failure and determined using Equation (3b),

*P*(

*l*) is the probability of only one pipe failure and is calculated by Equation (3b),

*m*is the probability of simultaneous breakage of pipes, and

*UAV*is the unavailability of the pipe

_{i}*i*(

*UAV*

_{i}*=*

*1–AV*).where is the available flow at node

_{i}*j*when pipe

*i*is unavailable and is the available flow at node

*j*when all pipes are working.

*j*(

*HRel*(

*NS*)

*j*). It is described as the ratio of available flow and required flow and is expressed as follows:where is the water flow available to the

*j*th node at time

*t*; is the required demand of

*j*th node at time

*t*.

### Water quality reliability

Most of the reliability analysis considers only mechanical and hydraulic aspects. However, with growing water quality concerns, water quality reliability (WQR) has become equally important. WQR is defined as the proportion of time the system supplies safe water quality in the WDN at a given time and place (Kansal *et al*. 2004). Water quality in the WDN is a function of various parameters. Hence, the residual chlorine concentration is considered to calculate the reliability related to the water quality aspect.

*et al.*2012). This paper has adopted the method proposed by Tabesh

*et al*. (2010) under a single pipe failure state. The author determined WQR by multiplying the probability of occurrence for each event with the WQR factor and mathematically described it as follows:where

*WQRel*(

*FS*)

*is the water quality reliability at node*

_{j}*j*for failure state,

*rel*(

_{j}*f*)

^{(0)}is the WQR factor at the normal operating condition at node

*j*, and

*rel*(

_{j}*f*)

^{(l)}is the WQR factor for a single pipe failure.

*et al*. 2004; Gupta

*et al.*2012). The authors consider the WQR factors as one, if the available residual chlorine concentration is within the acceptable limit range, if not considered zero. However, it is essential to consider the variation of values between the boundaries to determine the factors. For this study, new WQR factors are proposed based on four values (levels) of chlorine concentration, such as the minimum acceptable level of chlorine concentration (

*C*), the lower amount of desired concentration (

^{mLoc}*C*), the upper level of desired concentration (

^{lDes}*C*), and the upper maximum acceptable concentration (

^{uDes}*C*) as shown in Figure S1 (see Supplementary Material). Equation (7) developed from Supplementary Figure S1 is designed to calculate the WQR factor, in which the values of

^{uLoc}*C*,

^{mLoc}*C*,

^{lDes}*C*, and

^{uDes}*C*are 0.2, 0.35, 0.5, and 0.7 mg/l, respectively.where

^{uLoc}*rel*(

_{j}*f*) is the WQR factor at node

*j*and the other parameters are described above.

*et al*. 2004; Gupta

*et al.*2012) and is described as follows:where

*WQRel*(

*NS*)

*is the WQR for normal working conditions,*

_{j}*K*is the coefficient (0 or 1),

*C*is the available residual chlorine concentration at node

_{j,t}*j*at time

*t*; is the required demand of

*j*th node at time

*t*; and

*T*is the total time the model simulated (

*T*= 24 h).

Finally, the overall reliability index of the *j*th node should be a combination of hydraulic and WQR at a given time. It is calculated separately using a simple arithmetic average of hydraulic and WQR for normal and abnormal states.

### Resilience index

*et al.*2021). In this study, the widely used resilience concept proposed by Hashimoto

*et al.*(1982) and used by Khatavkar & Mays (2019) is adopted for both normal and abnormal conditions of the WDN. Hence, it is simple in the calculation and handles different aspects by changing parameters. The resilience index can be developed based on the three types of failures (i.e., water flow, pressure, and water quality failure). Resilience related to flow is determined based on water flow failure at nodes (i.e., when the available flow is less than the required demand). Similarly, resilience related to pressure is calculated based on pressure failure (i.e., when the nodal pressure is less than the minimum and greater than the maximum acceptable limit). Whereas resilience related to water quality is determined based on failure at a node (i.e., when the residual chlorine concentration at a node is out of the accepted limit range). Two states are defined for the WDN at a node at a given time

*t*, such as the state of satisfactory (

*S*) and failure (

*F*). When the system moves from time

*t*to

*t*+ 1, the network may migrate to other states or remain in some states. In the process of resiliency determination, the minimum pressure (20 m), maximum pressure (50 m) values, and desirable acceptable limit values (0.2–0.5 mg/l) for the concentration of residual chlorine are recommended and assigned. Therefore, the resilience index related to parameters (nodal pressure, water flow, and residual chlorine at nodes) is calculated as follows:where

*Res*(

_{j}*P*),

*Res*(

_{j}*Q*), and

*Res*(

_{j}*C*) represent resiliency related to nodal pressure, flow, and water quality aspect at node

*j*, respectively. The

*x*ɛ

_{t}*S*is the estimated value of the pressure, available flow, or residual chlorine at a given time

*t*member to (

*S*).

*x*

_{t−}_{1}ɛ

*F*is the estimated value of the pressure, available flow, or residual chlorine at a given time

*t–*1 belonging to failure state (

*F*), and

*T*is the number of time steps in the analysis (24 h is used for this study). The resilience index is determined for normal and abnormal working conditions of the WDN. Finally, the overall resilience index at a node

*j*is determined by aggregating

*Res*(

_{j}*P*),

*Res*(

_{j}*Q*), and

*Res*(

_{j}*C*) using the simple arithmetic average.

### Vulnerability index

This paper applied the Hashimoto *et al.* (1982) concept of vulnerability with little improvement, and it measures the failure severity of the WDN. Vulnerability is calculated based on the threshold deviation experienced during a failure event at a particular node. Three vulnerability indices are estimated separately for normal and pipe failure scenarios based on the values of nodal pressure, available flow, and residual chlorine at nodes. Vulnerability related to the pressure at node *j* is described in Equation (11).

*et al.*2016). The vulnerability index related to pressure is determined based on the nodal head. Nodes with maximum and minimum pressure above or below acceptable limits are vulnerable to pressure failure. For this study, minimum allowable pressure (

*H*) and maximum acceptable pressure (

_{min}*H*) are assumed as 20 and 50 m, respectively. Thus, the vulnerability related to the pressure at node

_{max}*j*is calculated as follows:where

*Vul*(

_{j}*P*) is the vulnerability related to the pressure at demand of the

*j*th node,

*F*is the vulnerability coefficient (0 or 1) for

_{j}*H*<

_{j}*H*, and is the vulnerability coefficient (0 or 1) for

_{min}*H*≥

_{j}*H*

_{max.}*Vul*(

_{j}*Q*) is the vulnerability related to flow at node

*j*and

*K*represents a coefficient (0 or 1).

_{j}*C*) and (

^{mLoc}*C*) values are 0.2 and 0.5 ml/g (Gupta

^{uDes}*et al.*2012), respectively. Thus, the vulnerability of the water quality aspect is calculated as follows:where

*Vul*(

_{j}*C*) is the vulnerability related to water quality at the

*j*th node,

*Z*is a coefficient of residual (0 or 1) applied when

_{j}*C*<

_{j}*C*and

^{juDes}*Z**is the coefficient applied when

*C*≥

_{j}*C*

^{uDe}^{s}desirable residual chlorine concentration. Finally, the overall vulnerability index at a node

*j*is calculated by aggregating all pressure, flow, and water quality vulnerabilities using a simple arithmetic average.

### Integration of PIs using the fuzzy logic approach

Once all PIs are determined, the next step would be aggregating those indices to the single API. In this study, the fuzzy set approach is applied for aggregating PIs. Fuzzy logic is a mathematical theory introduced by Zadeh (1965) and designed to address the human cognitive system's vagueness. Performance indices integrate to the API using the following three steps in the fuzzy logic approach:

Step 1: definition of MFs and fuzzification of PIs

For a TFN, *a*, *b*, and *c* represent the minimum, the most likely, and the maximum value parameters, respectively. For a ZFN, *a* is the minimum value, *d* is the maximum value, and *b* and *c* are the two values that represent the interval of the possible values.

After selecting MF forms, the next step is determining the linguistic variable and its scale. Rahman (2007) proposed the linguistic variables and scales. The author presents three types of linguistic variables and scales such as (1) three-point scale (poor–adequate–good), (2) four-point scale (poor–fair–satisfactory–good), and (3) five-point scale (poor–fair–satisfactory–good–excellent). This study adopted the third linguistic variable, representing the linguistic variable scale in a better way. The linguistic variables are subjective (Zadeh 1965). For this study, the value for the linguistic variable is assigned based on the literature and consulting expertise in the fields. Based on these methods, Table 1 and Figure S3 in Supplementary Material indicate fuzzy numbers of PIs and API with their associated linguistic variables and values.

Step 2: fuzzy inference and aggregation

Linguistic variable . | Poor . | Fair . | Satisfactory . | Good . | Excellent . | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Thresholds . | a . | b . | c . | d . | a . | b . | c . | a . | b . | c . | a . | b . | c . | a . | b . | c . | d . | |

Input (%) | Rel | 0 | 0 | 0.2 | 0.4 | 0.2 | 0.4 | 0.6 | 0.4 | 0.6 | 0.85 | 0.60 | 0.85 | 0.95 | 0.85 | 0.95 | 1 | 1 |

Res | 0 | 0 | 0.4 | – | 0.2 | 0.4 | 0.6 | 0.4 | 0.6 | 0.80 | 0.60 | 0.80 | 0.99 | 0.80 | 1.00 | 1 | – | |

L. Var. | Very low | low | Medium | High | Very High | |||||||||||||

Vul. | 0 | 0 | 0.1 | – | 0 | 0.1 | 0.2 | 0.1 | 0.2 | 0.4 | 0.2 | 0.4 | 0.6 | 0.4 | 0.6 | 1 | 1 | |

Poor | Fair | Satisfactory | Good | Excellent | ||||||||||||||

Output (%) | API | 0 | 0.2 | 0.4 | – | 0.2 | 0.4 | 0.6 | 0.4 | 0.6 | 0.85 | 0.6 | 0.85 | 0.95 | 0.85 | 0.95 | 1 | 1 |

The proposed scale of the API | ||||||||||||||||||

The granularity | Poor | Fair | Satisfactory | Good | Excellent | |||||||||||||

Ranges (API) | 0.01–0.40 | 0.41–0.60 | 0.61–0.80 | 0.81–0.90 | 0.91–1.00 |

Linguistic variable . | Poor . | Fair . | Satisfactory . | Good . | Excellent . | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Thresholds . | a . | b . | c . | d . | a . | b . | c . | a . | b . | c . | a . | b . | c . | a . | b . | c . | d . | |

Input (%) | Rel | 0 | 0 | 0.2 | 0.4 | 0.2 | 0.4 | 0.6 | 0.4 | 0.6 | 0.85 | 0.60 | 0.85 | 0.95 | 0.85 | 0.95 | 1 | 1 |

Res | 0 | 0 | 0.4 | – | 0.2 | 0.4 | 0.6 | 0.4 | 0.6 | 0.80 | 0.60 | 0.80 | 0.99 | 0.80 | 1.00 | 1 | – | |

L. Var. | Very low | low | Medium | High | Very High | |||||||||||||

Vul. | 0 | 0 | 0.1 | – | 0 | 0.1 | 0.2 | 0.1 | 0.2 | 0.4 | 0.2 | 0.4 | 0.6 | 0.4 | 0.6 | 1 | 1 | |

Poor | Fair | Satisfactory | Good | Excellent | ||||||||||||||

Output (%) | API | 0 | 0.2 | 0.4 | – | 0.2 | 0.4 | 0.6 | 0.4 | 0.6 | 0.85 | 0.6 | 0.85 | 0.95 | 0.85 | 0.95 | 1 | 1 |

The proposed scale of the API | ||||||||||||||||||

The granularity | Poor | Fair | Satisfactory | Good | Excellent | |||||||||||||

Ranges (API) | 0.01–0.40 | 0.41–0.60 | 0.61–0.80 | 0.81–0.90 | 0.91–1.00 |

*if-then*’ structure using combinations of the three indices. The rule, in its turn, is made up of an antecedent that defines the conditions and a consequent that describes the action. For each input linguistic variable of the model, in the antecedent, there is a form type (

*x*is

*La*), where

*x*can be an indicator (PIs) and

*La*is a linguistic label revealing a fuzzy set (i.e.,

*La*= poor/very low, fair/low, satisfactory/medium, good/high, or excellent/very high as presented in Table 1), while the consequent (

*Lc*) determines the condition of outputs (API) (i.e., represents similarly to the antecedent ‘

*La*’ as described as Equation (15)). Moreover, Figure S4 in the Supplementary Material shows the method of determining the equivalent performance of different combinations of the PIs. The number of rules covering all the different performance combinations proposed to be 125 rules (see Figure S5 in Supplementary Material) (5

^{3}), (i.e., 5 represents linguistic variables and 3 is the three PIs).where

*Rule*is the

^{i}*i*th rule;

*Rel*,

*Res*, and

*Vul*are the fuzzy set inputs (PIs) and

*API*. Furthermore, two types of fuzzy rule systems are available, namely Mamdani and Sugeno. For this paper, the Mamdani fuzzy rules type is selected and applied in the fuzzy model in the MATLAB, which is easier to understand, efficient, and more accurate than the Sugeno type.

Step 3: defuzzification process and determination of the API

Defuzzification is the reverse process of fuzzification. It is defined as the process of converting the fuzzified output into a crisp value that is considered the most representative of the fuzzy set output. The process is needed to translate the meaning of the fuzzy output and their membership values and resolve conflicts between differing results. Defuzzification can be performed with several methods: centroid, the bisector of area method, a maximum operation, mean maximum membership, first of maximum, mean of maximum, and last of maximum. Among those, the centroid method is the most widely applied and acceptable method for the process of defuzzification and is used and applied to manipulate the API in a fuzzy logic toolbox in the MATLAB for this study. After the fuzzy logic model is developed in the fuzzy toolbox in the MATLAB, PIs for all nodes are inserted into the MATLAB (fuzzy logic toolbox) through coding and the result of API at node *j* is saved in the Excel.

*API*is the aggregated performance index of the network,

_{Network}*API*is the aggregated performance index of node

_{j}*j*, and

*W*is the weightage of

_{j}*j*th node and assumes equal weightage for all nodes (

*W*= 1).

_{j}### Comparison of fuzzy-based (API) PIs integration methods with others

The comparison of available methods to integrate PIs with the proposed API is crucial to prove whether the new approach is better or not. For this purpose, the proposed metric (API) was compared to the other integration methods, namely the SI, developed by Loucks (1997), and simple average methods (geometric average), and their equations are presented in Table 2.

Methods . | Equations . | Literatures . |
---|---|---|

Simple averaged value (geometric mean) | Dziedzic & Karney (2016) | |

SI | Loucks (1997) |

Methods . | Equations . | Literatures . |
---|---|---|

Simple averaged value (geometric mean) | Dziedzic & Karney (2016) | |

SI | Loucks (1997) |

## RESULTS AND DISCUSSION

### Case study

Dire Dawa city's WDN is selected to demonstrate the proposed methodology. It consists of nine districts that cover a total area of 86 km^{2}. It is located in the Awash River Basin within the eastern rift margin lowlands of Ethiopia. The city's specific geographical location is 41.768°–41.891° north latitude and 9.574°–9.643° east longitude. From an operational perspective, the existing WDN of Dire Dawa is separated into three pressure zones (Zones I, II, and III). Zone II networks were selected for this study (Figure 3). The sources of the WDN for this area are deep well located at the ‘Boren’ and ‘Sabian’ well fields. The entire WDN of this zone consists of 74 nodes and 113 pipes with a 22.75 km length, with the pipe diameter varying between 90 and 500 mm and pipe material of Poly Vinyl Chloride (PVC) and Ductile Iron (DI).

### Performance evaluation in normal and abnormal conditions

A WDN is evaluated based on the proposed API under normal and abnormal conditions. Figure 2 indicates the nodal PIs (reliability, resilience, and vulnerability) results and the API value with and without pipe failure (normal and abnormal). The results show that the nodal performance values for PIs and API for the normal working state (API = 0.721) are higher than the pipe failure state (API = 0.624) and vice versa for the vulnerability index. The WDN can provide sufficient water at the required pressure and a desirable quality at a given time in the normal state compared to a failure condition. In addition, in a normal state, the network takes less time to recover from failure (resilience) and is less susceptible to failure than the failure state. Furthermore, as the available flow pressure and residual chlorine are reduced, reliability and resilience indices decrease and the vulnerability index increases, thereby reducing the API. Hence, the pipe failure may adversely affect the network performance, which decreases the water flow, reduces pressure, increases head loss, and causes deterioration of water quality in the network (Gheisi & Naser 2014; Duan *et al.* 2020; Essaidi & Triki 2021). These may minimize nodal PIs and API in a failure state, indicating a poor network performance.

Evaluating performance at nodes is valuable for identifying the performance variation among different network parts (Figure 3) that needs to select the critical regions (nodes) for further operational and maintenance. Results of the case study identify the critical nodes (J4, J44, J47, and J49) that should be prioritized to improve their performances, whose indices are presented in Table 3. The main responsible factors that cause deterioration of the critical nodes' performance are (1) they are located at high elevation and, hence, obtain low pressure and flow and (2) they are far from the chlorine boosting station (Tank 1), and thus, they receive water with less residual chlorine concentration. So, the nodes with low performance cannot deliver sufficient quantity and quality of water for the user in the given time, which minimizes customer satisfaction and needs. Furthermore, it is necessary to classify the nodes into the level of performance. For instance, Figure 3 results reveal that several nodes (51 of 74 nodes) have API in the range of 0.61–0.80 (satisfactory performance); 17 nodes are under low performance (0.41–0.60); two nodes have a good performance (0.81–0.90), and four nodes have a poor performance (API < 0.40). The analysis proved that those nodes that received minimum pressure (less than 20 m) and maximum pressure (above 50 m) during all simulation periods have a poor performance and adversely affect the quantity and quality of water delivered to the consumer. A reinforcement plan is essential for improving the performance in that critical region. There are several options recommended for enhancing the performance in the crucial area (nodes), such as replacing new pipes near nodes, establishing a new water path (parallel pipe) from down steam section, constructing new tanks for emergency water sources, and the chlorine boosting station near the incident section.

Nodes . | Rel . | Res . | Vul . | API . |
---|---|---|---|---|

J4 | 0.258 | 0.297 | 0.580 | 0.277 |

J47 | 0.336 | 0.055 | 0.454 | 0.330 |

J49 | 0.293 | 0.051 | 0.542 | 0.341 |

J-44 | 0.360 | 0.057 | 0.360 | 0.401 |

Nodes . | Rel . | Res . | Vul . | API . |
---|---|---|---|---|

J4 | 0.258 | 0.297 | 0.580 | 0.277 |

J47 | 0.336 | 0.055 | 0.454 | 0.330 |

J49 | 0.293 | 0.051 | 0.542 | 0.341 |

J-44 | 0.360 | 0.057 | 0.360 | 0.401 |

### Performance evaluation using single PIs and API for failure state

Under failure state, the results of reliability computation show that the network could fulfill its function, i.e., ability to deliver sufficient water at desirable pressure and water quality by 59.3% (*Rel* = 0.593) that has a fair performance level. Similarly, the resilience analysis indicates that the network has a 47.4% (i.e., *Res* = 0.474) probability of quickly meeting demand after an instance of insufficient supply, low pressure, or quality that has a fair performance level (Table 4). In addition, the case study results indicate that the network has 18.3% (*Vul* = 0.183) susceptibility towards failures that is in the range of medium vulnerability. Performance evaluations based on individual PIs differ significantly from each other. Moreover, when the network is evaluated using the integration of three PIs (API), the results reveal that the network fulfills its designated functions by 62.4% (API = 0.624), which has a satisfactory performance.

Node . | Rel . | Res . | Vul . | PIs integration methods . | ||
---|---|---|---|---|---|---|

SI . | Geometric average . | API . | ||||

J47 | 0.336 (P) | 0.055 (P) | 0.454 (H) | 0.010 (P) | 0.213 (P) | 0.330 (P) |

J6 | 0.498 (F) | 0.093 (P) | 0.264 (L) | 0.146 (P) | 0.324 (P) | 0.536 (F) |

J57 | 0.635 (S) | 0.313 (P) | 0.111 (L) | 0.435 (F) | 0.561 (F) | 0.669 (S) |

J66 | 0.598 (F) | 0.452 (F) | 0.120 (L) | 0.613 (S) | 0.620 (S) | 0.733 (S) |

Network | 0.593 (F) | 0.474 (F) | 0.183 (M) | 0.219 (P) | 0.596 (F) | 0.624 (S) |

Node . | Rel . | Res . | Vul . | PIs integration methods . | ||
---|---|---|---|---|---|---|

SI . | Geometric average . | API . | ||||

J47 | 0.336 (P) | 0.055 (P) | 0.454 (H) | 0.010 (P) | 0.213 (P) | 0.330 (P) |

J6 | 0.498 (F) | 0.093 (P) | 0.264 (L) | 0.146 (P) | 0.324 (P) | 0.536 (F) |

J57 | 0.635 (S) | 0.313 (P) | 0.111 (L) | 0.435 (F) | 0.561 (F) | 0.669 (S) |

J66 | 0.598 (F) | 0.452 (F) | 0.120 (L) | 0.613 (S) | 0.620 (S) | 0.733 (S) |

Network | 0.593 (F) | 0.474 (F) | 0.183 (M) | 0.219 (P) | 0.596 (F) | 0.624 (S) |

P, poor performance; F, fair performance; S, satisfactory performance; L, low performance; M, medium performance; H, high performance.

The analysis indicates that the performance result is biased when the network performance has been assessed through either of the three PIs individually. For example, the J66 has a fair performance when it is evaluated using reliability or resilience and has low vulnerability (Table 4). Is this fair or low? One cannot answer. However, the API can lead to the most acceptable result (satisfactory) that incorporates all PIs. Moreover, PIs (Rel, Res, and Vul) cannot address all the characteristics of the WDN effectively. In addition, these are independent measures, and hence, different networks cannot be compared using district indices. There can be an ambiguity or eclipsing problem associated with each parameter and may lead to visualization of the overall performance (Samee & Tyagi 2000). This indicates the superiority of fuzzy-based API compared to individual performance parameters. Therefore, it is crucial to capture all the network characteristics using a comprehensive measure, which is well addressed by the API.

### Comparison of the fuzzy-based method (API) with others

The proposed metric (API) was compared to the other integration methods developed by Loucks (1997) (i.e., SI) and the simple averaged value method (geometric mean). The performance of the integration methods for selected nodes and networks is presented in Table 4. The result shows a significant variation between the average value method (geometric mean) and the fuzzy-based (API) method. Moreover, the simple average value approach had underestimated the results (J6, J47, J57, and network performance in Table 4) because the aggregation carried out through simple average does not highly consider the relationship of the parameters. However, the analysis proves that all parameters have a relationship (Figure 4). When reliability increases, resilience increases, whereas vulnerability decreases and vice versa for most nodes. The proposed API follows the same trend with reliability and resilience and the opposite for vulnerability index as presented in Table 4 and shown in Figure 4. The proposed fuzzy-based integration method (API) considers the relationship between PIs, unlike the average value method. A highly reliable system is likely to be resilient and less vulnerable to failure, and hence, WDN performs better (Gheisi *et al.* 2016). Moreover, if any PI has zero value, the geometric aggregation becomes zero, which is not true in practice. Therefore, it is concluded that the fuzzy-based integration provides a sound output compared to the average value methods.

The comparison between the API and the SI indicates a significant variation between the proposed API and SI, as evident from Table 4 and Figure 5. The aggregated result provided by the SI is lower than the API. This may be due to the fact that the SI was aggregated by multiplying the indexes, which produced low and biased results, especially for lower values of reliability and resilience (J47 and J6). Additionally, if the system is 100% vulnerable, the SI turns out to be zero. However, the system's performance is zero if all the parameters have the worst possible values. The fuzzy logic method is superior to these integration techniques based on the above limitation. Hence, the proposed fuzzy-based integration method (i.e., API) has three advantages: (1) considers the relationship between parameters; (2) provides better outcomes for all ranges of the performance index value; and (3) describes the results in a qualitative and quantitative manner.

### Sensitivity analysis

The sensitivity of PIs (reliability, resilience, and vulnerability) was assessed by the local sensitivity analysis technique, i.e., one-factor-at-a-time (Table 5). Sensitivity of the PIs was carried out by changing each of the single PIs, i.e., by increasing 10 and 20%, while keeping other PIs constant and evaluating the changes in the output. Based on this analysis, the reliability index was recognized as the most sensitive PI.

. | . | Network API variation in (%) . | |
---|---|---|---|

PIs . | % increase . | 10% . | 20% . |

Rel | 1.83 | 3.83 | |

Res | 1.54 | 2.37 | |

Vul | 1.50 | 2.32 |

. | . | Network API variation in (%) . | |
---|---|---|---|

PIs . | % increase . | 10% . | 20% . |

Rel | 1.83 | 3.83 | |

Res | 1.54 | 2.37 | |

Vul | 1.50 | 2.32 |

## CONCLUSIONS

The performance evaluation of an urban WDN should reflect the system's capacity to deliver safe and adequate water supply to the consumers under various operating conditions. The performance can be assessed in terms of reliability, resilience, and vulnerability. However, from the decision-maker's point of view, it is desired to aggregate these individual performance parameters in a single PI. This study has demonstrated the use of a fuzzy-based API for evaluating the WDN under normal and abnormal operating conditions. The methodology and the API are easy to understand and will help the decision-makers prioritize the areas for up-gradation/maintenance based on the low performance and relative importance in terms of their contribution to the overall API. Fuzzy logic is easy to understand, and rule-bases can be easily programmed in MATLAB. This paper suggests multiple pipe failure analyses and the validity of the API in practice as the future scope of this research.

## CONFLICTS OF INTEREST

The authors declare that there are no conflicts of interest.

## DATA AVAILABILITY STATEMENT

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

## REFERENCES

*Reliability Analysis of Water Distribution System*

*Ph.D. Thesis*

*Condition Assessment Tool for Elements of Drinking Water Treatment Plant*

*MS Thesis*