Multivariate analysis of the pressure variation in intermittent water supply systems and the impact on demand satisfaction

In intermittent drinking water distribution systems, large volumes of the water are wasted due to leaks in the distribution networks. Similarly, user service is not always satis ﬁ ed in the time required to ﬁ ll the storage, nor with suf ﬁ cient pressure. Hence the importance of this study. Measuring the variability of pressure in the distribution network and determining the factors that in ﬂ uence the de ﬁ nition of a suf ﬁ cient minimum hours of service, is a ﬁ rst step to change to a continuous service 24/7, in order to minimize the volumes of lost water and meet demand. In total, 347 pressure sensors were placed in a network to detect changes in pressure and obtain data for 3 years. This study presents a new approach to determine the operating policy of the operating agency that provides the service intermittently. Two objectives are pursued: pressure variability – to minimize leaks – and de ﬁ ne the minimum hours of service. The analysis was performed using multivariate statistical techniques, including principal component analysis, correlation matrix and ANOVAs, to explore the association between objectives. The results obtained show that the pressure distribution has a Gaussian behavior and that the hours of service have a Poisson distribution.


INTRODUCTION
with the standard requires additional recharges that could affect the quality of the water, spoiling its potability (Christodoulou & Agathokleous ).
In continuous supply systems, water is provided 24 hours a day and the needs of the user are met in a timely manner, at the appropriate speed and pressure, in accordance with the demand scheme, allowing the system to operate under pressure and supply in peaks of demand per hour and per day.
When a system does not deliver water 24 hours a day to the end user, it is defined as an intermittent water supply (IWS) system, known in Mexico as 'tandeo' (Figure 1). In an intermittent system, the service is provided by cycles (duty cycle), (Taylor et al. ), which vary in areas and schedules, so the user must store water in tanks and cisterns (Cabrera-Béjar & Tzatchkov ), to cover their needs during water shortages (Totsuka et al. ; Milanés ).
The World Bank's Benchmarking International report on water and sanitation networks shows that only 16% of drinking water operating agencies in developing countries comply with a continuous 24-hour supply. The average supply estimate in these countries is only 16 hours a day (Van den Berg & Danilenko ).
The most frequent increase in physical water scarcity may be due to the following factors: climate change, increased demand, population growth and lack of investment in hydraulic infrastructure (maintenance and new works).
These factors cause the frequent appearance of intermittent supply systems (Tsegaye & Eckart ). The financial resources of the operating agencies are limited because the funds are often used in new infrastructure and not to modernize the existing one.
So, it is necessary to improve drinking water systems to make them profitable and sustainable (Piratla & Goverdhanam ).
During filling and emptying processes, depression or overpressure occur in the network, causing recurrent failures and leaks in the system, due to air pockets inside the pipes (PIGOO ).
The annual study of Mexico 2018 indicators shows that 22% of users of drinking water service systems have an intermittent supply, on average 13 hours per round. Therefore, IWS agencies present seven times more leaks in household intakes than those with a 24/7 service (PIGOO ).
Leaks damage the infrastructure and cause economic losses, excessive energy consumption and costs that are transferred to the end users (Casillas ). In addition, they do not meet end user demand (Taylor et al. ).
Although the immediate effect of a new leak or vacuum in the network causes the propagation of a transient wave in the pipe network, the effect disappears during the 'hours of vacuum.' When the assigned supply process begins, the wasted volume is considerable and requires a greater load for the fluid to displace the air contained in the network (Taylor et al. ). The research carried out develops a framework to identify the duration of the supply required in the sector, with the following objectives: a minimum pressure between 7 meters of water column (mwc) and 30 mwc, enough load to supply a two-storey house with storage on top; and avoid pressures less than 7 mwc, which generate vacuum or suction or greater than 30 mwc to reduce failures and losses of water volumes through leaks.
The definition of applicability of the technique of multivariate analysis (MA), is essentially the approach and study of variables in simultaneous resolution.
For example, we take a variable (pressure or leakage) and not only measure one aspect (quantitative or time span), but we consider several aspects and try to determine the relationship between the other variables in the statistical model.
Before defining the multivariate statistical techniques, the applicability of the techniques was first checked by analyzing the statistical power (Devore ).
The MA technique was chosen based on two possible criteria: the first one is known as independent methods, which are applied when it is of interest to investigate the associations between variables that are difficult to distinguish and the second one is identified as dependent methods which are used when the association between variables with component similarity is appreciated and they depend or are measured depending on the others.
To measure the association of variables and determine which factor has the greatest weight and make a 'quantitative' proposal to improve drinking water service in the city, the following should be considered: sufficient pressure variation in the service program to allow response to the user demand, as well as minimizing leaks to avoid losses in water volume (Fienberg ). Tools used: EXCEL, ORIGIN, MINITAB and SPSS.
The main analysis variable was found in the pressure variation in different sections of the network, whose distribution is influenced by two factors, mainly time and space.

Problem statement
This methodology was applied in the city of Chihuahua, with a population of approximately one million inhabitants.
Due to the growth of the urban area in the 1970s and 1990s, the intermittent supply operation policy was chosen.
There are large areas of influence or distribution of supply, identified by source and hours of service. These are classified according to the size, topographic characteristics and socioeconomic level of these areas. The following factors were considered: 'number of hours scheduled for water supply'; topographic features such as 'terrain elevation'; 'diameter of pipes' (classified in secondary network of 3' or less and primary network with diameters greater than 4') and area or 'surface' in which they distribute according to the type of source that feeds them (wells, tanks and pumping).
The operating policy has not had a significant variation in the last 5 years (no change in the physical delimitation of sectors). Therefore, it is considered that, in addition to measuring the association and determining the influence between the variables previously determined, it was relevant to see the effect of leaks in the system. For this reason, the leaks of the last 3 years were registered in the database of the Information and Service Center (ISC), classified as 'network leaks' and 'meter leaks'.

METHODS
Due to the permanence of the policy of operation and review of the daily behavior of the pressure in the 347  The independent variables were classified between those that are constant and those that are variable, in the 3-year period.
1. Those that remain unchanged over time are: Hours of service (HS), Supply hours in ranges, to observe the effect on spatial distribution and pressure magnitude.
As for the flow rates, the HS were classified according to the openings of the supply source for the analyzed sector. The agency considers as its most common operating policy to assign the following supply schedules (Duty cycle): • 4H, up to 4 hours (generally in a single block) • 8H, up to 8 hours (divided into two blocks, morning and evening) • 12H, up to 12 hours (divided into two blocks, morning and evening) • 16H, up to 16 hours (divided into two blocks, morning and evening) • 24H, continuous service 24 hours. FA is a data dimensionality reduction technique whose purpose is to find the minimum of factors or variables that explain the information contained in the sampled data. In the FA all the variables have the same role, since they are independent and, a priori, there is no conceptual dependence between them.
The grouping or classification of pressure was made based on the research of previous studies (Al-Ghamdi ), where the physical relationship between the leak rate and pressure was determined. This relationship can be expressed in general form of the equation: where Q is the leakage rate (litres/second), and P is the pressure (bar). The coefficient 'a' and exponent 'n' are constants determined from the field investigations. The ANOVA technique is one of the most widely used methods for analyzing data from experimental designs. It is used when looking to contrast more than two means of the sampled groups (Pressure results groups against Hours of Service groups) (Table 1).   ANOVA between groups is performed, called one-way ANOVA, considering pressures and independent flow rates/hours of service.

RESULTS AND DISCUSSION
As the behavior of the demand for drinking water service by the user is documented, the supply has a pattern of a Gaussian curve, daily and hourly; its pattern depends on the time, regardless of the location characteristics of the user. For this reason, the maximum hourly and daily demand curves in the 24/7 service are considered to define the operation of the drinking water distribution systems to the communities.
To determine the validity of the statistical analysis, in the intermittent supply that followed the Gaussian behavior, a normality tests were first performed by observations, (measurements at each of the 347 points) the results of this analysis indicated that only 10 sampling points reveal a behavior pattern that deviates from the 'normal' line during hours of service of less than 4 hours, and whose supply sources are Tanks B and Cerro Grande, as shown in the following figure. These 10 points show a lognormal or poissonic model ( Figure 5).
For each group of data or observations of the variables described, it is corroborated that they comply with the assumptions of the classical statistical tests (normality, homogeneity, independence).
However, more than 80% meet the test of normality in observations, so the tests were carried out by factors or variables, since it is considered to have sufficient statistical capacity to do so.   is 'logical' (Table 3).
Through this analysis, it is corroborated that the number of hours supplied is the most important variable or factor to maintain the pressures that arise in the assumption of   Between the two components the variability variance in more than 68% is explained. In this way, most of the data structure can be captured in two underlying dimensions ( Figure 7).
One-way ANOVA, this procedure is done by analysis by Factor (Pressure) and by Treatment (Hours of Service/Duty Cycle) (Table 4).
Considering the pressures as an answer ( Figure 8) the behavior of the pressures is completely Gaussian; only a few points move away from the Gaussian line.
There are also significant differences between the pressures worked by Tukey and Fisher (Figure 9) that is to say, it is concluded that the groups associated by pressure, if they present a significant difference that is influenced by hours of service.
It is clear that the main and the most influential factor is the pressure of 30 mwc and the one with the least influence is the pressure equal to and less than 0.
The significant differences between the group means were obtained with a 95% reliability.
This exercise confirms that the importance of the hours of service (treatment) is still significant and that the main effects are due to the factor ( Figure 10).
The summary of the results in this exercise shows that the behavior of the hours of service are not Gaussian, rather a Poisson distribution. They move away from the Gaussian line (semilog) by a large number of points to the ends.
Adjustment of the exponential treatment equation (hours of service): where: The results of the exercise produced by MINITAB are as follows: ANOVA considering both factors: (Pressure) and    It was found that both models (observed and adjusted) have a normal distribution behavior in terms of pressures, which gives us the certainty that the adjusted and experimental regression value is reliable on the significance of the factors.
However, regarding the hours of service, it was found that their behavior is a Poisson distribution or exponential order.
It is important to highlight that the treatment (4H) and the factors (P0, P7, Pm30) have a negative impact, in other words, this pressure range should be avoided.
In conclusion, despite the fact that the hours of service are granted in blocks of 4 hours and in this way the Operating Agency increases or decreases the supply to the sectors in a linear manner, in reality the quality of the service does not behave in a linear manner, as the operating policy orders, but the quality of the service decreases exponentially.
It is also observed that 4 hours of service are insufficient since, in many cases, it takes 2 hours to reach the end user, so a minimum of 6 hours of service is recommended if supplied in a single block, since in blocks of 8 hours, 2 hours of the evening shift are used to fill the domestic storages, which is displayed with the obtained coefficient similar to that of 4 hours (Equation (5)).

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

REFERENCES
Al-Ghamdi, A. S.  Leakage-pressure relationship and leakage detection in intermittent water distribution systems. Journal