Fuzzy logic-based integrated performance evaluation of a water distribution network

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.

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.

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 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).

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.

Finally, the overall reliability index of the jth 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.

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).

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

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.

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.

Dire Dawa city's WDN is selected to demonstrate the proposed methodology. It consists of nine districts that cover a total area of 86 km². 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).

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.

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.

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.

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.

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.

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.

The authors declare that there are no conflicts of interest.

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