Residential water demand is a random variable which influences greatly the performance of municipal water distribution systems (WDSs). The water request at network nodes reflects the behavior of the residential users, and a proper characterization of their water use habits is vital for the hydraulic system modeling. This study presents a stochastic approach for the characterization of the daily residential water use. The proposed methodology considers a unique probabilistic distribution – mixed distribution – for any time during the day, and thus for any entity of the water demanded by the users. This distribution is obtained by the merging of two cumulative distribution functions taking into account the spike of the cumulative frequencies for the null requests. The methodology has been tested on three real water distribution networks, where the water use habits are different. Experimental relations are given to estimate the parameters of the proposed stochastic model in relation to the users number and to the average daily trend. Numerical examples for a practical application have shown the effectiveness of the proposed approach in order to generate the time series for the residential water demand.

## LIST OF SYMBOLS AND ACRONYMS

*C*(_{D}*t*)[–] demand coefficient at time

*t*- CDF
cumulative distribution function

- CE
Castelfranco Emilia (one of three case studies)

- CV(
*t*)[–] variation coefficient of the demand coefficient at time

*t* - DMA
demand monitoring area

*F*_{o}[–] probability of occurrence of null request

*F**[–] CDF of the flow demand when it is different to zero

maximum value of the probability of null water demand

- Fr
Franeker (one of three case studies)

- L
logistic distribution

- MD
mixed distribution

- N
normal distribution

*N*_{us}[–] number of users

- PSG
Piedimonte San Germano (one of three case studies)

*Q(t)*[L

^{3}/T] demanded water flow at time*t*[–] mean demand coefficient at time

*t*[–] minimum threshold value of the demand coefficient

[L

^{3}/T] mean daily water demand[L

^{3}/T] mean water demand at time*t*[–] deviation standard of demand coefficient at the time

*t*

## INTRODUCTION

Water distribution system (WDS) simulation models have been improved drastically with advances in computational algorithms, machine speed and data storage.

For instance, hydraulic simulators have recently been developed by taking into account the functional relationship between delivered flows and nodal pressures. This is of particular importance when pressures are below the minimum level required to meet the desired water demand (e.g. Todini 2003; Giustolisi *et al.* 2008). In addition, some WDS models are able to calculate unsteady flow processes in WDSs (e.g. Vítkovský *et al.* 2006; Ferrante *et al.* 2014) or to detect pollution sources in the network (e.g. Preis & Ostfeld 2006; Liserra *et al.* 2014). Calibration techniques have been proposed for determining pipe roughness coefficient more realistically (e.g. Walski 1983; Greco & del Giudice 1999; Jung *et al.* 2014).

But, the performance of a WDS model is highly dependent on water demands. Hence, the value and utility of these WDS modeling advances is determined by the degree to which the WDS simulation is able to reproduce real water demands. For this aim the water demand models have to be capable of representing the random nature of the water requirements.

Bao & Mays (1990) used a Monte Carlo sample generator to model water demand uncertainty in a WDS reliability study. Subsequently, several approaches were proposed to incorporate water demand uncertainty in WDS hydraulic reliability (e.g. Gargano & Pianese 2000; Babayan *et al.* 2006; Chung *et al.* 2009), in sizing urban reservoirs (e.g. Nel & Haarhoff 1996; Van Zyl *et al.* 2008) and in the optimal rehabilitation/design of hydraulic networks (e.g. Kapelan *et al.* 2005; Tricarico *et al.* 2006; Tricarico *et al.* 2014).

A key challenge for practical application, however, is obtaining reliable estimates of the parameters of the corresponding cumulative distribution functions (CDFs) that need to be implemented for water demand generation.

Unfortunately, in the technical literature, there are only a few studies which reproduce water demand variability with CDF parameters that were estimated from experimental data (Alvisi *et al.* 2003; Blokker *et al.* 2010). This is probably due to the difficulty of getting experimental data, which can effectively represent the water requirements of a specific number of residential users.

The target of this paper is to define a practical stochastic approach which allows us to model realistically the residential water demand in relation to the users number and thus to generate daily time series. This is referred to an appreciable number of aggregated users (200–1,250), therefore the water demand can be handled as a unique random variable that is continuous positive.

The proposed model leads to an important simplification in respect to the approaches which are capable of reproducing the residential water demand of the end users such as the *Poisson Rectangular Pulse* (PRP – Buchberger & Wu 1995; Buchberger & Wells 1996), or the *Neyman-Scott Rectangular Pulse* (NSRP – Alvisi *et al.* 2003; Alcocer-Yamanaka *et al.* 2012) but without losing accuracy. In fact, the end user approaches need to model the random phenomena (usually at least three phenomena: frequency, duration and intensity of the residential request) which contribute to defining the water demand for each user. This issue arises also when the aim of the end user approach is to model the total request of the dwellings, neglecting the demands at the single taps, as the *Overall Pulse model* (Gargano *et al.* 2016).

In addition, in the end user models it is necessary to aggregate the water demand of single flats in order to represent the water demand for WDS nodes, i.e. for several users (*bottom-up approach*). This operation can be influenced by important scale effects, especially when the user number increases. Indeed, the estimation of the parameters for an aggregated number of users *N _{us}* on the basis of the parameters of a single consumer – at most a few – represents an arduous issue (e.g. Moughton

*et al.*2006; Filion

*et al.*2007; Magini

*et al.*2008), which can limit the practical applications of the end user approaches in modeling the residential water demand, especially when the inhabitant number is relevant.

This issue is avoided when the residential water demand is directly modeled as a clustered request of a significant number of users (e.g. Tricarico *et al.* 2007; Gato-Trinidad & Gan 2012). Indeed, the proposed stochastic model represents the aggregated water demand of users whose number ranges roughly between 200 and 1,250.

It is worth noting that the range of the considered number of users, even if restricted, is nevertheless of significance for characterizing the water demand at WDS nodes.

Instead of considering several probability distributions in relation to the time of the day, it is herein suggested a novel distribution which can be applied generally for any time of the day. This specific probabilistic distribution, called mixed distribution (MD), and the relationships for estimating its parameters are proposed in order to achieve an approach concretely applicable to generate synthetic time series of water demand during the day.

The proposed approach allows us to model the residential water demand in relation to the daily time, assuming the simplifying hypothesis to disregard the time correlation effects. However this assumption does not prejudice the effectiveness of the proposed approach, as is shown in the following sections by the comparison between the observed data and the generated time series.

The proposed relationships for the estimation of the MD parameters are in relation to the user number and the average water request during the day. Therefore, the implementation of the proposed model requires only the knowledge of the daily trend of the water demand, which is related to the considered users, but it is usually known by the water companies (e.g. by measuring the outflows of the municipal reservoirs).

In order to obtain results of general application, the actual time series of the water demand of users with different life styles were analyzed. In fact, a monitoring system for the WDS of a small town in Southern Italy (Piedimonte San Germano (PSG)) was carried out to obtain reliable data on residential water request. These data have been then compared with the time series of two other towns: Castelfranco Emilia (I) (Alvisi *et al.* 2003) and Franeker (NL) (Blokker *et al.* 2006).

The residential water request has been considered dimensionless by means of the daily mean water demand.

The flow request was measured [L/s] and it was averaged on a 1 min time interval. This interval allows us to obtain a detailed description of the water demand, and a 1 min interval is not so approximate as to induce appreciable scale effects caused by the time aggregation.

## MONITORING SYSTEMS AND SAMPLE DATA

Nowadays the possibility of installing economic and reliable water demand monitoring systems allows us to have in-situ laboratories on real WDSs. But in order that the collected data be effectively representative of the water demand, the WDSs need to be redundant, i.e. with pressures at all nodes of the network higher than the minimum required level.

Moreover the number of users is a relevant parameter to describe the water demand (e.g. Mays 1999; Martinez-Solano *et al.* 2008; Gargano *et al.* 2012), therefore the measured flows were analyzed in relation to the relative served inhabitants. Hence, attention has to be focused on the measurement point locations, where the number of users is often difficult to estimate due to the looped structure of the WDSs.

Measurement points should be thus allocated along the pipes which supply a circumscribed demand monitoring area (DMA). DMAs are present when the topological WDS scheme – or part of it – is branched, or when in the network it is possible to detect areas which are connected with the rest of the system by means of a limited number of pipes (e.g. Buchberger & Nadimpalli 2004). The DMA characterization requires an accurate census in order to define the number of users requiring water.

Only when the pressure is adequate, can the flows collected be considered an expression of the residential water demand; therefore, the monitoring systems must record jointly the pressure and the flow rate.

A further preprocessing procedure has to be applied when the observed data are affected by leakage phenomena. In fact, in this circumstance, the time series have to be filtered from the water losses in order to represent the effectively demanded flows.

On the basis of these circumstances, a specific monitoring system which involves the real WDS of a small town – PSG – was realized by the Laboratorio di Ingegneria delle Acque of the Università di Cassino e del Lazio Meridionale.

The monitoring system consists of four measurement points, each with a pressure cell, and an electromagnetic flow meter, and all probes were connected to a data logger. The measurements are recorded continuously with an acquisition frequency of up to 1 Hz. The strategic location of the meters allowed measurement of the water demand for four DMAs with different numbers of users, *N _{us}*: 239, 777, 981 and 1,220.

The size of the water users has been measured by means of the number of the supplied inhabitants *N _{us}*, to which the relationships refer in the following sections.

*Q*(

*t*) the flow demand during the day. Hence, all the following statistics are referred to the dimensionless random variable of Equation (1).

In order to check the robustness of the results of this research, the proposed approach and the relative relationships have been tested with the water demand of users which present lifestyles totally different in respect to PSG inhabitants.

Indeed, the recorded time series of two further monitoring systems have been analyzed:

Castelfranco Emilia (Italy) (CE) with a number of users equal to 596 (Alvisi

*et al.*2003);Franeker (The Netherlands) (Fr) with a number of users equal to 1,150 (Blokker

*et al.*2006).

These two monitoring systems were realized specifically for studying the water demand, hence they gave reliable time series for the water request. In addition, CE and Fr monitoring systems, as the field laboratory of PSG, provide a fine description of the water demand during the day because the time step Δ*t* equals 1 min.

The assumption of a fine time step is precautionary in respect to the extreme water demands. Indeed, Tricarico *et al.* (2007) observed that when the time step is equal to 1 h, the peak demands are underestimated around the 20% for Δ*T* = 1 min. Moreover Buchberger & Nadimpalli (2004) demonstrated that the probability of null request decreases exponentially with the increasing Δ*T*.

On this basis, all the following relations and considerations assume Δ*t* = 1 min.

It is worth noting that in-situ laboratories with the above mentioned characteristics are rare, therefore it is difficult to obtain reliable time series of the flow demand.

Table 1 summarizes the principal characteristic of the field laboratories and of the observed time series, which were used for the statistical elaborations and inferences.

. | PSG (I) . | Fr (NL) . | CE (I) . |
---|---|---|---|

N _{us} | 239-777-981-1220 | 1,150 | 596 |

Monitored period | 5 Jan.- 30 Mar. 2004 7 Jan.- 31 Mar. 2006 | 23 Jan. - 31 Mar. 2006 | 7 Jan. - 20 Mar. 2000 |

Type of counter | Flow meter | Flow meter | Volumetric meter |

Frequency of sampling | 1 Hz | 1/60 Hz | 1/60 Hz |

Number of used days for the statistics | 95 | 50 | 50 |

. | PSG (I) . | Fr (NL) . | CE (I) . |
---|---|---|---|

N _{us} | 239-777-981-1220 | 1,150 | 596 |

Monitored period | 5 Jan.- 30 Mar. 2004 7 Jan.- 31 Mar. 2006 | 23 Jan. - 31 Mar. 2006 | 7 Jan. - 20 Mar. 2000 |

Type of counter | Flow meter | Flow meter | Volumetric meter |

Frequency of sampling | 1 Hz | 1/60 Hz | 1/60 Hz |

Number of used days for the statistics | 95 | 50 | 50 |

Because different consumptions were observed between the weekends and the weekdays, only the latter were analyzed.

For all the monitored urban areas it can be assumed that the users are exclusively residential and mainly characterized by an indoor water use. Users are mainly represented by middle class families where the major economical occupation is constituted by industrial workers.

The significant differences among the daily trends of the considered users allowed the development of severe tests for the proposed stochastic model.

## THE MIXED DISTRIBUTION

The mixed distribution allows us to model the random component of the residential water demand by means of a unique probabilistic CDF, whatever the daily time.

*C*(

_{D}*t*)

*=*0). Indeed, observed cumulative frequencies of minimum water demand (Figure 2) have pointed out the presence of a vertical mass spike for

*C*

_{D}*=*0.

Obviously, the need to consider the spike for *C _{D}* = 0 is more evident when the number of users considered is negative. The null water requirement is quite probable for a reduced number of users, and this is not just during the night hours.

The water demand model can be thus considered as a combination of two distributions: the first one referred to as a random discrete variable, the second one as a continuous random variable (Gargano *et al.* 2014). In particular, the first distribution describes the event of null demanded flow, while the second distribution represents the circumstance for which the flow demand is different from zero.

The mixed distributions are effective when the analyzed random events are the summation of two phenomena. For instance, in the hydrological realm the two-component extreme value distribution is effective to model the flood peaks, as a product of the mixture of two types of storms (Rossi *et al.* 1984).

*C*, the resulting CDF is:If

_{D}*F*

_{o}is the occurrence probability that the water demand is null (), and

*F** represents the CDF of the flow demand when it is not null (), Equation (2) can be rewritten as:Equation (3) has been obtained by observing that the is the probability of a certain event, and that the two conditions of flow demand and are incompatible and exhaustive events.

*F*

_{o}and

*F**, the value of which depends on the daily time:Equation (3′) represents the application of the MD for the context of a daily stochastic model.

### Probability of null water demand, *F*_{o}(*t*)

The event for which the water demand in a WDS node is equal to zero or different by zero might be modeled by means of the simple Bernoulli distribution, where the probabilities of the two incompatible and exhaustive events are respectively and the complement .

On the basis of the field data recorded the *F*_{o}(*t*) probability was estimated. It is appropriate, nevertheless, to highlight that as the peculiarity of the users' habits affects significantly the daily trend , so it affects also the probability *C _{D}*(

*t*) = 0 estimation, as demonstrated in the following sections.

Moreover, it was shown (Buchberger & Nadimpalli 2004) that the *F*_{o}(*t*) value depends significantly on the interval time Δ*t* with which the day has been discretizated and on the number of users supplied.

The following analysis of *F*_{o} value has been developed for Δ*t* = 1 minute and for a number of users ranging between 200 and 1,250 *N _{us}*.

*F*

_{o}(

*t*) values estimated by means of the observed time series (Table 1), for which it is evident – as it was expected – that the probability of null water request assumes significant values during the night time, and the

*F*

_{o}(

*t*) trend depends significantly on the users' habits.

In addition, Figure 3 shows that the *F*_{o} increases when the number of users decreases.

The *F*_{o} values during the rest of the day are substantially null and this depends on considered *N _{us}*. Hence, for a minus number of users minus respect to that investigated, the

*F*

_{o}(

*t*) should present non null values also in other daily time.

Therefore, the maximum value of the probability of null water demand , decreases with the increasing the number of supplied users.

_{o}could be estimated in relation to the users number by means of:Equation (4), valid for

*N*= 200 ÷ 1,250, shows that tends rapidly to 1 for small user numbers. In this condition, the null water request defined during the night hours becomes practically a certain event.

_{us}*F*

_{o}can be estimated in relation to the mean values of the demand coefficients, . Indeed, the experimental data can be represented by the exponential relationship:The

*F*

_{o}probability value promptly decreases towards the null value already for the flows close to the mean daily value of the demand coefficient (Figure 5). This trend is also more evident for the increase of the user number.

It is worth noting that the condition *F*_{o} = 1 (which implies ) represents a theoretical limit, because the event of null water request cannot be a certain event, even during the night time. Therefore, the Equation (5) becomes unfounded when the value of leads to a probability of null water request close to 1. Therefore, from the observed data it is clear that a threshold value () of should be defined in such a way as to limit inferiorly the Equation (5), hence Equation (5) is valid for .

### The distributions for not null water demand, *F**(*t*)

The identification of the distributions suitable to model the not null water demand was carried out by statistical inferences that have considered the data of the actual users (CE, Fr and PSG), eliding the null water requests from the experimental samples.

The L distribution, in respect to the N distribution, has the advantage that the probability density function can be integrated, although it has the same trend of the Normal model (Swamee 2002; Ashkar & Aucoin 2012).

Both the distributions (N and L) need to be truncated being *C _{D}* > 0. In fact for low values of the probability of a negative value of

*C*by Equation (7) is not negligible.

_{D}The choice of taking into account bi-parametric models has been guided by the need to find models that permit reliable estimation of few parameters and thus lend themselves to a practical application.

*N*= 1,150 (Figure 6(a)) and PSG

_{us}*N*= 777 (Figure 6(b)) users – between the N and L distributions and the observed cumulative frequencies at 5.00 a.m., the time at which the

_{us}*F*

_{o}probability is significant for both PSG and Fr users.

### Tests of the MD

The comparison between the field data and the mixed distribution (Equation (3′)) have demonstrated the effectiveness of the proposed distribution in representing the flow demand during the whole day.

*N*= 777) and CE (

_{us}*N*= 596) have been reported in Figure 7 as an example of the satisfying results that have been obtained also in reference to the other monitored users number. The less number of the experimental points in the CE plots depends on the degree of accuracy of the water meters used for Castelfranco Emilia monitoring system (volumetric metres).

_{us}In Figure 7 the observed cumulative frequencies are compared with the mixed distribution (continuous line), for which the not null water request was described by means of the L distribution (Equation (7)). The parameters of the distribution were estimated on the basis of experimental data and the relative values are reported in Table 3 (columns , and ). The plots with dot lines in Figure 7 furthermore represent the MD CDFs, where Equation (5) gave the probability of null request (in Table 3, column ) and the standard deviations (in Table 3, column ) has been estimated by means of the relation (Equation (8)) described in the following section.

Users . | t [hh.mm]
. | F_{o}(t) [%]
. | [%] . | [–] . | σ_{CD}(t) [–]
. | [–] . |
---|---|---|---|---|---|---|

PSG N = 777 _{us} | 1.00 a.m. | 76.2 | 72.8 | 0.080 | 0.072 | 0.070 |

PSG N = 777 _{us} | 9.00 a.m. | 0.0 | 0.2 | 1.630 | 0.336 | 0.293 |

PSG N = 777 _{us} | 5.00 p.m. | 0.0 | 1.9 | 1.016 | 0.372 | 0.214 |

CE N = 596 _{us} | 5.00 a.m. | 69.9 | 53.3 | 0.211 | 0.123 | 0.117 |

CE N = 596 _{us} | 1.00 p.m. | 0.0 | 1.1 | 1.510 | 0.467 | 0.307 |

CE N = 596 _{us} | 9.00 p.m. | 1.0 | 0.8 | 1.637 | 0.436 | 0.323 |

Users . | t [hh.mm]
. | F_{o}(t) [%]
. | [%] . | [–] . | σ_{CD}(t) [–]
. | [–] . |
---|---|---|---|---|---|---|

PSG N = 777 _{us} | 1.00 a.m. | 76.2 | 72.8 | 0.080 | 0.072 | 0.070 |

PSG N = 777 _{us} | 9.00 a.m. | 0.0 | 0.2 | 1.630 | 0.336 | 0.293 |

PSG N = 777 _{us} | 5.00 p.m. | 0.0 | 1.9 | 1.016 | 0.372 | 0.214 |

CE N = 596 _{us} | 5.00 a.m. | 69.9 | 53.3 | 0.211 | 0.123 | 0.117 |

CE N = 596 _{us} | 1.00 p.m. | 0.0 | 1.1 | 1.510 | 0.467 | 0.307 |

CE N = 596 _{us} | 9.00 p.m. | 1.0 | 0.8 | 1.637 | 0.436 | 0.323 |

The MD is able to manage the spike of cumulative frequencies of the water demand that assumes remarkable values during the night (see in particular plots of Figure 7 for h1.00–5.00 am).

In particular, the robustness of the proposed approach was checked by means the Kolmogorov–Smirnov test reported in Figure 8 as an example for Fr and CE, considering the N distribution to describe the component of Equation (3′). The tests were done every half hour of the day, and Figure 8 shows the Kolmogorov–Smirnov parameter *D*, which exceeds the threshold-significant level = 5% – for only a few points.

## ESTIMATION OF THE PARAMETERS OF THE MD

The use of bi-parametric models for representing the *F** component implies the need to define practical relations for estimating the CDF parameter, i.e. mean and standard deviation.

As described above, the mean value represents the daily water demand trend and it is a mirror of the users habits. Therefore, its pattern traces the lifestyle of the inhabitants (i.e. the start and the end time of the prevailing productive activities of the town; the average duration of the lunch break, etc.). Moreover, the following relations help to obtain a reliable daily trend:

the ratio between the daily integral of the (summation for the time discretization) and the daily time is equal to 1;

several equations in the technical literature allow us to estimate the peak phenomenon in relation to the user number (e.g. Molino

*et al.*1991; Tricarico*et al.*2007; Martínez-Solano*et al.*2008);the minimum peak of the water demand can be estimated by means of Equation (6).

Otherwise, the daily trend can be deduced by means of in-situ measurements in a few points of the WDS. For instance, the water companies often have available flow meters on the outlet pipe of the urban reservoirs, which produce useful time series.

The standard deviation parameter can be obtained by means of a practical relationship herein suggested which allows us to estimate the variation coefficient CV(*t*) in relation to the user number.

According to Equation (8) CV also decreases with *N _{us}*, in agreement with other experimental evidences (e.g. Moughton

*et al.*2006; Gato-Trinitade & Gan 2012).

It is worth noting that at the increase of the number of users and of the demand coefficient the asymptotic value of Equation (8) is CV = 0.1. This result is in line with other experimental studies relative to the peak phenomenon (Gargano & Pianese 2000; Tricarico *et al.* 2007).

The dot line and the continuous line of Figure 7 represent the CDFs of the MD that was obtained respectively by estimating the CV(*t*) by means of Equation (8) and on the basis of experimental data. Hence, the plots of Figure 7 show that Equation (8) implies admissible errors in the estimations of the coefficient of variation.

## NUMERICAL EXAMPLE

The proposed approach, together with the suggested relationships and criteria to estimate the parameters, allows effortlessly the generation of the synthetic time series of the water demand applying the Monte Carlo method (Metropolis & Ulam 1949).

The residential water request of the different case studies examined has been taken into consideration for reporting a numerical example. Hence synthetic data of these users were generated by means of the proposed approach and the relative relationships to estimate the parameters.

Solely as an example, the numerical results obtained for Fr (*N _{us}* = 1,150) and CE (

*N*= 596) have been herein reported.

_{us}*F*

_{o}(

*t*) and CV(

*t*) were estimated for a time interval of 24 h (Figure 10) with a time interval Δ

*t*= 1 min. Moreover, the probability of the occurrence of the maximum null water demand was estimated by Equation (4) (the peaks of the plots in Figure 10(a)–10(c)).

The plots of Figure 10 show that *F*_{o} and CV increase with reducing number of users, in particular during the night.

*t*) is the generated probability for time

*t*.

## CONCLUSION

A concrete contribution towards the generation of time series of the residential water demand during the day has been described in this paper. The fundamental importance of modelling the water demand accurately is widely recognized as an undeniable prerequisite for effective WDS analysis. The probabilistic approaches existing on the topic are not reliable, or they are too complex to be implemented in practice by the decision makers of the water company.

This is often due to the absence of a suitable experimental activity that allows to both test the effectiveness of the probabilistic models proposed, and provide indications for the estimation of the distribution parameters.

Hence, by means of statistical inferences on real residential demand data an independent stochastic approach and relative parameters able to represent the residential water requirements for 200–1,250 users have been herein suggested. This range represents the limits of the validity for the proposed stochastic model. However, the above mentioned validity extremes do not limit in a substantial way the effectiveness of the approach, which allows us to consider the whole water demand as a continuous and positive random variable.

With the aim of applying the obtained results to a wide range of cases, variegated sample data were examined allowing us also to study the incidence of the different life styles on water consumption. However, the obtained results are in line with other experimental analysis on water demand of real residential users.

The random component of the stochastic approach is modeled by means of a novel distribution (mixed distribution) that is of effectiveness in describing the water demand for the entire day. The MD, obtained by merging two distributions, describes the residential water demand, regardless of its entity. In this way the proposed probabilistic model is able to manage at the same time a discrete random variable – necessary to take into account the null request during the night hours – and a continuous positive random variable, for the not null demanded flows.

In addition, some relationships are proposed in order to obtain robust estimations of the parameters of the mixed distribution, once the number of supplied users and the daily mean trend are known.

These equations, and the proposed stochastic approach, permit us to generate, by means of the Monte Carlo simulator, reliable sample data of the demanded water flows in a simple way.

The tests showed that the assumptions do not limit the effectiveness of the proposed stochastic approach.

## ACKNOWLEDGEMENTS

The authors thank Prof. Marco Franchini and Dr Stefano Alvisi and Dr Mirjam Blokker for providing water demand data for Castelfranco Emilia and Franeker monitoring systems. The writers would also like to acknowledge Prof. Steven G. Buchberger for his fruitful suggestions.