Fuzzy Modeling of Urban Water Supply Crisis

1. Introduction

According to data from the World Meteorological Organization, global water consumption increased more than six-fold in less than a century, more than double the rate of the population growth, and continues to grow considering rising consumption in the agricultural, industrial and domestic sectors [1]. These data lead to the conclusion that in the coming years the global situation of water reserves will move towards a crisis, both in quantitative and qualitative aspects, if adequate water management actions are not taken. More recently, urban water supply crises (UWC) have been observed, a context characterized by water scarcity, as well as damage to the environment and population health, especially among poor populations. As human populations continue to grow, these problems are likely to become more frequent and serious. One example is the case of New York City’s water supply, which is facing a crisis. The social and economic development of New York City from the 1970s led to a sudden crisis in the city’s water supply system in the 1990s [2]. Other examples were also reported, such as in Palestine, where a UWC case was observed caused mainly by the inadequate access to freshwater resources and inappropriate management [3]. The city of Tijuana, in Mexico, has shown the highest rates of economic growth in the country, resulting in a rapid increase of water demand and consequently the emergence of a UWC [4]. From 1998 to 2000, the city of Campina Grande in Brazil faced a WSC caused by severe periods of drought and the complete absence of freshwater resource management [5]. This UWC caused serious water rationing in Campina Grande that lasted one year. This is not a unique case in Brazil as frequent water rationing has been observed in the cities of Recife and São Paulo [6]. In the Brazilian Federal District, rapid non-planned urbanization and land changes have had a considerable impact on water resources. From 2016 to 2018, the city of Brasilia also went through a UWC situation, and governance, regulation, as well as management support strategies in the urban and rural environment were implemented [7].

In another study, an investigation into Qatar’s sustainability crisis, originating from high levels of water, electricity and food consumption was carried out [8]. The high levels of consumption were made possible by the significant wealth of hydrocarbons, redistributive water governance of a generous rentier state and structural dependence on imported food and subsidies on food production. In this state, the water crisis is silent because it does not cause interruptions in supply or public discontent. The possible solution comes from programs that integrate the water, energy and food sectors [8].

The imprecise and ambiguity are inherent to the water supply system, e.g., pressures, flow rate supply and consumption, age and characterization of pipe resistance. Some researches that addressed this imprecise using Fuzzy Logic are briefly described below. A study to accommodate aleatory uncertainty was performed using stochastic analysis to represent the input uncertainties and to estimate resulting uncertainty in nodal pressures and pipe flows) [9]. Results of Fuzzy analyses for two realistically sized water distribution networks show that the proposed method performs with an acceptable level of accuracy and greatly reduces computational time [9].

The need to improve predictive models of hydraulic transients in water systems (water supply networks) was the subject of another study [10]. For this purpose, triangular Fuzzy numbers are used to represent the input uncertainties. Then, to obtain the extreme pressure heads in each location of the network and at each level of uncertainty, four independent optimization problems are solved. The results is found computationally fast and promising for real applications [10].

The Fuzzy Logic is a perfect tool among white-box methods (models based on laws of physics, chemistry, others, that govern the dynamic behavior of the system) for risk analysis due to the capabilities in dealing with limited data, subjective and temporal variables, and modeling expert opinions, among others [11]. For this reason, Fuzzy Logic was used as comprehensible framework for assessing water supply risk based on existing and under-construction projects in Mashhad city, Iran. The results showed that the framework based on the Fuzzy and possibility theory is befitting to information gathered from experts. Such a framework can provide a simple method to apply the proposed methodology to other water management projects, where, despite the high level of investment, there is no clear idea of the risks and their consequences [11].

Mathematical modeling is a well-known tool for water management. However, when considering UWC, there are some limitations of the conventional mathematical modeling that are related to vague and ambiguous data (e.g. real water-loss, real water availability, setting water tariffs, among others). In theory, the UWC problem can be formulated as a mathematical problem. In practice, rules are considered to be a more practical method. However, the combination of mathematical programming and Fuzzy rule has rarely been discussed in the literature [12]. The aim of this chapter is to describe the develop a mathematical model for UWC, dealing with this ambiguity and real data uncertainty.

2. Theoretical background

UWC modeling was essentially based on the definitions of water crisis and Fuzzy Logic. The topics are presented briefly below.

2.2 Fuzzy logic

The advent of Fuzzy Logic originates from the need of a method that can systematically express inaccurate, vague, ill-defined quantities [15]. For example, instead of using a complex mathematical model, industrial controllers based on Fuzzy Logic can be implemented using knowledge from human operators, or heuristic knowledge. This makes the control action using Fuzzy Logic as good as using the raw knowledge (generally better) and always consistent. In decision analysis, Fuzzy Logic can be used in tasks in which individual variables are not defined in exact terms. Some examples of this have been provided in the literature [16, 17]. In the environmental area, the population’s preference for water conservation action, the assessment of the performance of environmental education programs and policies in preventing forest fires are examples of individual variables not defined in exact terms. The properties of Fuzzy Systems are demonstrated in several publications, as, for example, in [18]. An introduction to the theory of Fuzzy sets is presented below.

In general, a Fuzzy Set can be summarized as follows [15, 18, 19]:

• Definition 1: a subset A of a set X is said to be Fuzzy Set if μA:X→01, where μAdenote the degree of belongingness of A in X.

• Definition 2: a fuzzy set A of set X is said to be normal if μAx=1, ∀x∈X.

• Definition 3: the height of A is defined and denoted as hA=supμA, ∀x∈X.

• Definition 4: the α-cut and strong α-cut is defined and denoted respectively as αA=x/μAx≥α, μA+=x/μAx>α.

• Definition 5: let a∼, b∼be two fuzzy numbers, their sum is defined and denoted as μa∼+b∼z=supminz=u+vμa∼uμb∼v, where 0≤λ∈R.

• Definition 6: if a Fuzzy number a∼is fuzzy set A on R, it must possess at last following three properties: (i) μAx=1; (ii) x∈R/μa∼x>αis a closed interval for every α∈ [0,1]; (iii) x∈R/μa∼x>0is a bounded and it is denoted by aλLaλR.

• Theorem 1: a fuzzy set A on R is convex if and only if μAλx1+1−λx2≥minμAx1μAx2, for all x1,x2∈Xand for all λ∈01where min denotes the minimum operator.

• Theorem 2: let a∼be a fuzzy set on R, the a∼∈fRif and only μa∼satisfies (Eq. 2):

μa∼x=1,for x∈mnLx,for x<mRx,for>nE2

Where: Lxis the right continuous monotone increasing function, 0≤Lx≤1and limx→∞Lx=0; Rxis a left continuous monotone decreasing function, 0≤Rx≤1, limx→∞Rx=0.

The Fuzzy Linear Programming Problem (FLPP) with decision variables and coefficient matrix of constraints are in Fuzzy nature (Eqs. 3–6).

z∼=max∑j=1nc∼jxjE3

subject to

∑j=1nA∼ijxj≤B∼iE4

1≤i≤mE5

xi>0E6

The triangular Fuzzy numbers A which can be represented by three crisp numbers s, l, r (Eqs. 7–10).

fixj=max∑j=1nc∼jxjE7

subject to

∑x≥0sijlijrijxij≤tiuiviE8

1≤i≤mE9

1≤j≤nE10

Where: c∼=csclcr, A∼=sijlijrijand B∼=tiuiviare Fuzzy numbers.

• Theorem 3: For any two triangular Fuzzy numbers A∼=s1l1r1and B∼=s1l2r2, A∼≤B∼if and only if s1≤s2,ss−l1and s1+r1≤s2+r2. Above problem can be rewritten (Eqs. 11–15).

fixj=max∑j=1nc∼jxjE11

subject to

∑j=1nsijxij≤tiE12

∑j=1nsij−lijxj≤ti−uiE13

∑j=1nsij−lijxj≤ti−viE14

xi≥0E15

As examples of membership functions for a Fuzzy number m∼, such as approximately m=10, a triangular membership function (Eq. 16) and a bell-shaped membership function (Eq. 17) is widely used.

μm∼=01−x−ma,a>0E16

μm∼=e−bx−m2,b≥1E17

Such membership functions are illustrated in Figures 1 and 2. More information about Fuzzy Mathematical Programming is available in the literature [15, 16, 17, 18, 19, 20, 21, 22, 23].

3. Methodology

The methodology of this work comprised the following steps: (1) identifying the influencing factors in UWC; (2) proposing a conceptual model for UWC; (3) data collection and data simulation; (4) optimizing the proposed conceptual model parameters (calibration); and (5) assessing conceptual model performance (verification).

For step (1), a literature review was conducted related to WSC management.

For step (2), the definition of UWC found in the literature was taken and the Fuzzy Non-Linear Programming (FNLP) was used [12, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].

For the third step (step 3), a case study was simulated in Brazil, more precisely in the Federal District, taking into account predictions made by some researchers. The necessary data were collected from the Brazilian Federal District’s water supplier and sanitation company (CAESB), from the National Institute of Meteorology in Brazil (INMET) and from the Brazilian Federal District’s Government (GDF). The simulation was conducted assuming the hypothesis of a significant increase in the water consumption in coming years, as well as managerial and political stagnation of the water supplier and sanitation company. The time series analysis refers to three years (2007, 2008 and 2009). The period for model calibration was considered for two years and the verification period was considered for one year. Data were normalized prior to optimization of conceptual model parameters. This was done to restrict their range within the interval of −1.0 to +1.0 to eliminate the difference among scales measuring the influencing factors, according to Eq. (18).

Where: x_norm is the normalized value; x₀ is the original value; x_max is the maximum value; x_min is the minimum value.

For step (4), the minimization of the sum of squared errors and the Differential Evolution & Particle Swarm Optimization algorithm (DEPS) was adopted, using the spreadsheet from Open Office (Calc-Solver). As a justification for using the DEPS algorithm, it presents good performance to obtain global optimums, based on the synergy between Particle Swarm Optimization (PSO) and Differential Evolution (DE) [25].

In step (5), some tests used to model performance assessment were proposed, among them, the correlation coefficient (r), the determination coefficient (R²), the average relative error percentage (AREP) and graphic observed values versus estimated values.

4. Results and discussions

A bibliographic review was conducted by [26] and resulted in a list of influential factors presented below:

• Population growth rate.

• Human population density.

• Socio-economic level.

• Education level.

• Industry level.

• Ambient temperature.

• Relative humidity.

• Rainfall.

• Seasonality.

• Size and topographic characteristics of the city.

• Percentage of water metering.

• Water tariffs.

• Type of water tariff policies.

• Existence of wastewater collection systems.

• Human Development Index.

• Pressure in the water distribution network.

• Existence of conservation habits.

• Number/type of hydro-sanitary equipment per household.

• Constructed area per household.

• Number of rooms.

• Abundance or scarcity of water sources.

• Water-loss.

• Social representation and identification of each family.

• Existence and type of municipal water resources policy.

• Acceptance of the population to water conservation and rational use actions.

• Typology of land use.

• Type of consumers.

• Type of municipality.

• Predominant function of urban environment.

• Existence of policies to promote water conservation.

• Intermittence in the water supply system.

• Energy consumption.

• Existence of regulatory policy on water consumption.

• Existence of environmental education program.

• Dissemination of the belief: water is an inexhaustible resource and low-priced.

The UWC has appeared as an inadequate ratio between water consumption and water supply [12]. Water consumption and water availability are vague and ambiguous terms, because they are dependent on haziness measures, qualitative factors, scarcity data and low-quality data [27].

Design and analysis of water distribution networks (WDNs) are laden with uncertainty. There is natural randomness, such as variations in reservoir elevation heads, and there is epistemic randomness, i.e., incomplete knowledge, imprecise data, and linguistic ambiguity. Both are associated with the characterization of pipe resistance, nodal demands, and hydraulic responses [9]. The analysis of water distribution networks has to take into account the variability of users’ water demand and the variability of network boundary conditions, e.g. the presence of local private tanks and intermittent distribution [28].

Some of these factors were considered in this paper, the faulty water metering (C∼and A∼), the imprecise value of the average pressure (p∼) in the water distribution network and the imprecise value of water-loss (l∼) in the water supply system. A fuzzy mathematical model of the UWC is written as Eq. (19).

Where: C∼is a fuzzy mathematical model that represents the average value of the water consumption; A∼is a fuzzy mathematical model that represent the average value of the water availability.

For a mathematical representation of the C∼, a fuzzy mathematical model was proposed considering some influencing factors in water consumption: the ambient temperature, the relative humidity, rainfall, collected revenues, unemployment indicator, and average pressure in the water distribution network. The C∼model was based on some assumptions, as follows: the existence of a non-linear relationship among the influencing factors, the existence of a haziness of ±10%, that is C∼=CsClCr, in observed values of water consumption (error of household’s water meter), the increase in the water consumption with the increase in the ambient temperature, the existence of a relationship between relative humidity and the rainfall, the reduction of water consumption with the increase relative humidity, the increase in the water consumption with the increase in the revenues collected, the reduction in water consumption with the increase of the unemployment indicator, the increase in water consumption with the increase of the average pressure (p) in the water distribution network and the existence of a haziness of ±5% in observed values of the average pressure in the water distribution network, that is p∼=psplpr.

Likewise, a Fuzzy mathematical behavior of the A∼was proposed, considering the total water-loss in water supply system and the intermittence in the water supply system. The A∼model was based on the following assumptions as influencing factors: the existence of a non-linear relationship among the influencing factors, the existence of a haziness of ±10%, that is A∼=AsAlAr, in observed values of the water supply (error of waterworks’ water meter), the reduction of the water supply with the increase in the intermittence in the water supply system, the reduction in the water supply with the increase in the total water-loss (l) in the water supply system and the existence of a haziness of ±10% in observed values of the total water-loss, that is l∼=lslllr.

The membership functions for the fuzzy numbers C∼and A∼are approximately equal to the observed values of water consumption and water supply, following a triangular membership function as shown in Figures 3 and 4.

The membership functions for the fuzzy numbers (p∼) and (l∼) were considered approximately equal to the observed values of the average pressure in the water distribution network and total water-loss in the water supply system, following a pattern of a triangular membership function as shown in Figures 5 and 6.

All assumptions, from C∼and A∼, made are according to previous research [4, 27, 28, 29, 30, 31]. Eqs. (20) to (27) compose the proposed FNLP model.

subject to

subject to

Where: z∼Cis the objective function representing the calibration of C∼; C∼E,iis a fuzzy number representing the estimated water consumption; C∼O,iis a Fuzzy number representing the observed water consumption; CO,i,sand CO,i,rare the lower bound and upper bound of Fuzzy triangular numbers in the set C∼O,i, as is shown in Figure 1; z∼Ais the objective function representing the calibration of A∼; A∼E,iis a Fuzzy number representing the estimated water availability; A∼O,iis a Fuzzy number representing the observed water availability; A∼O,i,sand A∼O,i,rare the lower bound and upper bound of Fuzzy triangular numbers in the set A∼O,i, as is shown in Figure 1; β0, β1, …, β8 are parameters, x1 is the ambient temperature; x2 is the relative humidity; x3 is the rainfall; x4 is the amount of collected revenues; x5 is the unemployment indicator; p∼is the average pressure in the water distribution network; l∼is the total water-loss in the water supply system; x6 is the intermittence in the water supply system.

Eq. (28) and (29) show the results of the parameter optimization in the proposed conceptual model when applying the previously described data normalized according to Eq. 4.

The collected and simulated data and its respective descriptions are shown in Table 1. Table 2 and Figures 7–12 show the results of the model performance.

The calibration and verification results indicated that the proposed conceptual model has shown good agreement for the collected and simulated data, when considering some previous research. For instance, a study on water demand developed in the cities of Oklahoma and Tulsa Oklahoma State, USA, resulted in a statistical model to explain water demand with R² range within the interval of 0.140 a 0.920 [35].

In a household water demand study in the northwest of Spain, the price, billing, climatic, and sociodemographic variables were used as explanatory variables and the results showed a R² range within the interval of 0.198 to 0.891 [36].

Another study aiming to predict future water consumption from Istanbul, Turkey, was developed using the Takagi Sugeno Fuzzy method for modeling monthly water consumption, and the overall prediction presented an AREP of less than 10% [37]. In this study, AREP ranged from −40% to 5% indicating an opportunity to improve the model developed. In another regional water study, in the case of Tijuana, in Northwest Mexico, the purpose was to analyze monthly water consumption dynamics, and the empirical estimation results were considered fairly satisfactory with the R² range from 0.4582 to 0.5932 [4].

The analysis in Figure 12, agreeing with the AREP of −40%, reinforces the difficulty of the model developed in the C∼Padjustment. The reasons for this difficulty are varied, including problems with data quality (simulation, filling in gaps), the need to incorporate influential variables, the need to improve Fuzzy modeling. Therefore, we encourage continuous improvement of water consumption forecasting models focusing on optimizing the pertinence functions for the case (format, error rate rates, others).

In general, when considering the quality of the models developed (C∼Pe A∼P), according to Table 2 and Figures 7–12, the viability of the FNLP can be verify. The Fuzzy Logic allowed inaccuracies inherent to the water system to be captured and incorporated into conventional mathematical programming techniques. The satisfactory results agree with previous studies [9, 27, 28].

5. Conclusions and recommendations

Acknowledgments

The authors would like to express their gratitude for the financial support from the Brazilian agencies CNPq (project No. 556084/2009-8), CAPES and DPP-UnB.

Conflict of interest

The authors declare no conflict of interest.