A study of the water supply system failure in terms of the seasonality: analysis by statistical approaches

Among the many factors affecting water supply system failures are the weather conditions that change over the year. Since this is an important research issue, as part of this study, investigation of water supply system failure seasonality by the selected statistical approaches was presented. The basis of the research was monthly number of the pipelines’ failures from the multi-year period of 2007–2017 of the municipal water network located in southern Poland. Mann–Kendall test results proved decreasing seasonal trend of the failure rate indexes λ. In turn, the results of the Colwell indexes’ calculations allowed it to be stated that seasonal course of the water pipelines’ failure events can be relatively easy to predict. As it turned out, it is difficult to determine unambiguously the impact of a given period of the year on the water pipeline failure events’ occurrence. However, greater failure-free operation of the water pipelines may be expected in spring and summer months than in autumn and winter months. Because using Colwell indexes for seasonality analysis has no limitations compared to other methods, Colwell indexes may be considered as reliable tools for the assessment of the seasonal course of the water pipelines’ failure events.

along with reducing leakages through pressure regulators' installation or active leakage control is considered as a priority for water loss reduction (Tsanov et al. ). It is also important to pay attention to the drinking water quality safety; this is a key element of safe water systems' use and this issue is also related to the technical conditions of water pipelines. Also, it should be mentionedas the paper by Zhang () showsmany environmental studies from the last two decades have paid more and more attention to the problem of drinking water pollution, as one of the main water pollution problems. Returning to the point, it must be noted that, especially corrosion of internal pipelines' surfaces creates the risk of secondary potable water pollution. As is known, this is not the only problem concerning corrosion; In order to avoid all the above problems, first, it is necessary to design and build water networks carefully.
Apart from that, proper operation and technical support of the water system facilities, especially their systematic maintenance and current rehabilitation, is very important as well as planning water networks' renovation. In addition, an effective water supply system operation should be supported by using highly reliable and complex automation technology (Olsson   summer months was stated. In general, for seasonality investigations, autocorrelation analysis or Fourier analysis is usually used, but because these methods have some limitations, sometimes, using them is difficult. Therefore, used in this paper, among others, the Colwell indexes were proposed as alternative tools for seasonality analysis. This method has not been used in water supply failures' assessment before, which makes this paper novel.

CASE STUDY
Statistical analysis of the water supply system failure seasonality was made based on the operational data of the water network located in the city of Nowy Są cz (Poland) (Figure 1(a)). The water network in Nowy Są cz was built in 1912. In the first year of its existence, the source of the transported water was underground water supplied via a 28 km long water pipelines to 180 farms. Nowadays, water from the infiltration intake and the surface intake is transported to about 71,000 consumers by main and distribution pipelines with a total length of about 313 km.
Grey cast iron and ductile cast iron are the prevailing materials in the material structure of the considered network; these materials represent 50% of all pipelines. The rest of the water system is made of PE and PVC (35%) and steel (15%).
In 2008, the analysed water supply network was divided into eight smaller measurement zones, where temporary water consumption and water pressure may be controlled.
This action was aimed at significant acceleration of the detection of the water pipes' failures and immediate failures' removal; thus, it tended to result in a significant reduction of  water losses. Currently, the water supply system in Nowy Są cz is divided into the eight measurement zones (from zone A to zone H) (Figure 1(b)). Since in zone G, water pipe failures were noted only in July and in September 2015, zone G was not considered in this paper.

METHODOLOGY
The input data for this study included monthly number of ( where S is Mann-Kendall test statistic, n is the length of the data set in time series (number of observations), i is the number of the previous element in time series, j is the number of the next element in time series, x i , x j are the time series elements in chronological order.
After Mann-Kendall test statistic S determination, standardized test statistic Z was calculated according to Equation (4): where Z is standardized test statistic, Var(S) is variance, determined using Equation (5): The main assumption of the Mann-Kendall test is lack of the autocorrelation in data series. In the case of the months' analysis, such dependence may be observed more often than in the case of the seasons' analysis. While the autocorrelation is stated, this may lead to the Then, correction of the standardized test statistic Z* was determined similarly like Z parameter by using Equation (4).
where Var*(S) is correction of the variance, n n Ã s represents a correction due to the autocorrelation and was calculated as follows: where n Ã s is an 'effective' number of the observations (due to the autocorrelation), ρ S (i) is the autocorrelation function of the ranks of the observations.
In the next stage of the study, the assessment of the failure rate indexes λ seasonality was made using Colwell indexes. These include predictability P and its two components: constancy C and contingency M; they can take values from 0 to 1. Predictability P is a measure of the regularity of a given phenomenon. Constancy C describes a susceptibility of the variable to remain unchanged throughout the whole analysed period of time. Constancy C assumes maximum value, when the tested variable has the same value in each analysed period of time. Contingency where P is predictability, C is constancy, M is contingency, s is the number of the analysed class intervals describing a Additionally, for each month, the average values of the probability R(t) of failure-free operation of the water supply system were determined using Equation (11) (Kapur & Pecht ): where R(t) is the probability of failure-free operation of the water supply system, t is the considered period of time.  Table 1.

As
Based on the results summarized in Table 1, it was found that for the considered ten-year period, the changes of the failure rate indexes λ of the analysed water supply system were at high and very high level. This is evidenced   The assessment of seasonal variability of the water network failure rate indexes λ was performed using Colwell indexes: predictability P and its componentsconstancy C and seasonality (contingency) M. In addition, C/P and M/P ratios were determined (Figures 3-6).
Comparing with each other the tested measurement zones by months' analysis (Figure 3   Finally, as part of the statistical data analysis, the average monthly values of the probability R(t) of failure-free operation of the tested water pipelines in each measurement zone were determined ( Figure 7).
As can be seen in Figure 7,  it is difficult to determine clearly the impact of a given period of the year on the water pipelines' failure events' occurrence. Nevertheless, calculations of the probability R(t) proved that greater failure-free operation of the water  In addition, an attempt may be made to use the Colwell indexes for statistical description of other water supply systems' operational characteristics, for which, seasonality is an important factor.

CONFLICTS OF INTEREST
The authors declare that there are no conflicts of interest regarding the publication of this paper.

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