**1. Introduction**

Water is essential to all forms of life. The development of humanity is associated to the use of water, and nowadays, the constant availability and satisfaction of water demand is a fun‐ damental requirement in modern societies. Although water seems to be abundant on our planet, fresh water is not an inexhaustible resource and has to be managed in a rational and sustainable way. The demand for water is dynamic and influenced by various factors, from geographic, climatic and socioeconomic conditions, to cultural habits. Even within the same neighbourhood the user-specific water demand is elastic to price, condition of the water dis‐ tribution system (WDS), air temperature, precipitation, and housing composition (regarding only residential demand in this case). On top of all these factors, demand varies during the day and the week.

Traditionally, for WDS modelling purposes, water demand is considered as being determin‐ istic. This simplification worked relatively well in the past, since the major part of the stud‐ ies on water demands were conducted only with the objective of quantifying global demands, both on the present and on the long-term. With the development of optimal operating sched‐ ules of supply systems, hourly water demand forecasting started to become increasingly more important. Moreover, taking in consideration all the aforementioned factors that influ‐ ence water use, it is clear that demand is not deterministic, but stochastic. Thus, more recent‐ ly, in order to guarantee the requested water quantities with adequate pressure and quality, the studies began to focus on instantaneous demands and their stochastic structure.

© 2013 Vertommen et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **1.1. Descriptive and Predictive Models for Water Demand**

The first stochastic model for (indoor) residential water demands was proposed by Buch‐ berger and Wu [1]. According to the authors, residential water demand can be characterized by three parameters: frequency, duration and intensity, which in turn can be described by a Poisson rectangular pulse process (PRP). The adopted conceptual approach is relatively sim‐ ilar to basic notions of queuing theory: a busy server draws water from the system at a ran‐ dom, but constant, intensity and, during a random period of time. Residential demands were subdivided into deterministic and stochastic servers. Deterministic servers, including washing machines and toilets, produce pulses which are always similar. Stochastic servers, like water taps, instead produce pulses with great variability, and their duration and intensi‐ ty are independent. The PRP process found to best describe water demand is non-homoge‐ neous, i.e., when the pulse frequency is not constant in time. Different authors used real demand data to assess the adequacy of the non-homogeneous PRP model, achieving good results [2]. Moreover, the PRP model was confirmed to allow the characterization of the spa‐ tial and temporal instantaneous variability of flows in a network, unlike the traditional models that use spatial and temporal averages and neglect the instantaneous variations of demand. One drawback to the rectangular pulse based models is the fact that the total inten‐ sity is not exactly equal to the sum of the individual intensities of overlapping pulses, due to increased head loss caused by the increased flow [3]. This problem can however be solved by introducing a correction factor. The daily variability of demand represents another draw‐ back to the PRP model, since it can invalidate the hypothesis that pulses arrive following a time dependent Poisson process [2]. One possible solution to this question is to treat the time dependent non-homogeneous process as a piecewise homogeneous process, by divid‐ ing the day into homogeneous intervals [4]. Another solution consists in using an alternative demand model: the cluster Neyman-Scott rectangular pulse model (NSRP), proposed by Al‐ visi [5]. The model is similar to the PRP model, but the total demand and the frequency of pulses are obtained in different ways. In the PRP model the total water demand follows a Poisson process resulting from the sum of the single-user Poisson processes, with a single arrival rate. In the NSRP model, a random number of individual demands (or elementary demands) are aggregated in demand blocks. The origin of the demand blocks is given by a Poisson process, with a certain rate between the subsequent arrivals. The temporal distance between the origins of each of the elementary demands to the origin of the demand block, follows an exponential distribution with a different rate. The variation of these parameters during the day reflects the cyclic nature of demands. A good approximation of the statistical moments for different levels of spatial and temporal aggregation was achieved; however, the variance of demand becomes underestimated for higher levels of spatial aggregation.

dish washer, kitchen tap, shower, outside tap, washing machine, WC). The statistical distri‐ bution for the frequency of each end-use was retrieved from survey information from the Netherlands. The duration and intensity were determined, partly from the survey and part‐ ly from technical information on water-using appliances. From the retrieved information, a diurnal pattern could be built for each user. Users represent a key point in the model and are divided into groups based on household size, age, gender and occupation. Simulation results were found to be in good agreement with measured demand data. The End-Use model has also been combined with a network solver, obtaining good results for the travel

Water Demand Uncertainty: The Scaling Laws Approach

http://dx.doi.org/10.5772/51542

107

The PRP and the End-Use model were compared against data from Milford, Ohio. The ach‐ ieved results showed that both models compare well with the measurements. The End-Use model performs better when simulating the demand patterns of a single family residence, while the PRP models describes more accurately the demand pattern of several aggregated residences [8]. The main difference between the models is the number of parameters they use: the PRP model is a relatively simple model that has only a few parameters, while the End-Use model has a large number of parameters. However, the End-Use model is very flexible towards the input parameters, which also have a clearer physical meaning and hence more intuitive to calibrate. The PRP model describes the measured flows very well. From the analytic description provided by the PRP model, a lot of mathematical deductions can be made. Thus, one can classify the PRP model as a descriptive model with a lot of po‐ tential to provide insight into some basic elements of water use, such as peak demands [9]and cross-correlations [10]. The End-Use model is a Monte Carlo type simulation that can be used as a predictive model, since it produces very realistic demand patterns. The End-Use model can be applied in scenario studies to show the result of changes in water using appliances and human behaviour. Possible improvements to the model include the incorpo‐ ration of leakage, the consideration of demands as a function of the network pressure and the application of the model outside the Netherlands [11]. Li [10]studied the spatial correla‐ tion of demand series that follow PRP processes. It was verified that while time averaged demands that follow a homogeneous PRP process are uncorrelated, demands that follow a non-homogenous PRP process are correlated, and that this correlation increases with spatial and temporal aggregation. A similar conclusion about the correlation was achieved by

times, maximum flows, velocities and pressures [7].

Moughton [12]from field measurements.

**1.2. Uncertainty and reliability-based design of water distribution systems**

The problem of WDS design consists in the definition of improvement decisions that can op‐ timize the system given certain objectives. As aforementioned, in the earliest works regard‐ ing the optimal design of water distribution systems (WDS), input parameters, like water demand, were considered as being deterministic, often leading to under-designed networks. A robust design, allowing a system to remain feasible under a variety of values that the un‐ certain input parameters can assume, can only be achieved through a probabilistic ap‐ proach. In a probabilistic analysis the input parameters are considered to be random variables, i.e., the single values of the parameters are replaced with statistical information

The aforementioned models are mainly descriptive. More recently, Blokker and Vreeburg [6] developed a predictive end-use model, based on statistical information about users and end uses, which is able to forecast water demand patterns with small temporal and spatial scales. In this model, each end-use is simulated as a rectangular pulse with specific probabil‐ ity distribution functions for the intensity, duration and frequency, and a given probability of use over the day. End-uses are discriminated into different types (bath, bathroom tap, dish washer, kitchen tap, shower, outside tap, washing machine, WC). The statistical distri‐ bution for the frequency of each end-use was retrieved from survey information from the Netherlands. The duration and intensity were determined, partly from the survey and part‐ ly from technical information on water-using appliances. From the retrieved information, a diurnal pattern could be built for each user. Users represent a key point in the model and are divided into groups based on household size, age, gender and occupation. Simulation results were found to be in good agreement with measured demand data. The End-Use model has also been combined with a network solver, obtaining good results for the travel times, maximum flows, velocities and pressures [7].

**1.1. Descriptive and Predictive Models for Water Demand**

106 Water Supply System Analysis - Selected Topics

The first stochastic model for (indoor) residential water demands was proposed by Buch‐ berger and Wu [1]. According to the authors, residential water demand can be characterized by three parameters: frequency, duration and intensity, which in turn can be described by a Poisson rectangular pulse process (PRP). The adopted conceptual approach is relatively sim‐ ilar to basic notions of queuing theory: a busy server draws water from the system at a ran‐ dom, but constant, intensity and, during a random period of time. Residential demands were subdivided into deterministic and stochastic servers. Deterministic servers, including washing machines and toilets, produce pulses which are always similar. Stochastic servers, like water taps, instead produce pulses with great variability, and their duration and intensi‐ ty are independent. The PRP process found to best describe water demand is non-homoge‐ neous, i.e., when the pulse frequency is not constant in time. Different authors used real demand data to assess the adequacy of the non-homogeneous PRP model, achieving good results [2]. Moreover, the PRP model was confirmed to allow the characterization of the spa‐ tial and temporal instantaneous variability of flows in a network, unlike the traditional models that use spatial and temporal averages and neglect the instantaneous variations of demand. One drawback to the rectangular pulse based models is the fact that the total inten‐ sity is not exactly equal to the sum of the individual intensities of overlapping pulses, due to increased head loss caused by the increased flow [3]. This problem can however be solved by introducing a correction factor. The daily variability of demand represents another draw‐ back to the PRP model, since it can invalidate the hypothesis that pulses arrive following a time dependent Poisson process [2]. One possible solution to this question is to treat the time dependent non-homogeneous process as a piecewise homogeneous process, by divid‐ ing the day into homogeneous intervals [4]. Another solution consists in using an alternative demand model: the cluster Neyman-Scott rectangular pulse model (NSRP), proposed by Al‐ visi [5]. The model is similar to the PRP model, but the total demand and the frequency of pulses are obtained in different ways. In the PRP model the total water demand follows a Poisson process resulting from the sum of the single-user Poisson processes, with a single arrival rate. In the NSRP model, a random number of individual demands (or elementary demands) are aggregated in demand blocks. The origin of the demand blocks is given by a Poisson process, with a certain rate between the subsequent arrivals. The temporal distance between the origins of each of the elementary demands to the origin of the demand block, follows an exponential distribution with a different rate. The variation of these parameters during the day reflects the cyclic nature of demands. A good approximation of the statistical moments for different levels of spatial and temporal aggregation was achieved; however, the variance of demand becomes underestimated for higher levels of spatial aggregation.

The aforementioned models are mainly descriptive. More recently, Blokker and Vreeburg [6] developed a predictive end-use model, based on statistical information about users and end uses, which is able to forecast water demand patterns with small temporal and spatial scales. In this model, each end-use is simulated as a rectangular pulse with specific probabil‐ ity distribution functions for the intensity, duration and frequency, and a given probability of use over the day. End-uses are discriminated into different types (bath, bathroom tap,

The PRP and the End-Use model were compared against data from Milford, Ohio. The ach‐ ieved results showed that both models compare well with the measurements. The End-Use model performs better when simulating the demand patterns of a single family residence, while the PRP models describes more accurately the demand pattern of several aggregated residences [8]. The main difference between the models is the number of parameters they use: the PRP model is a relatively simple model that has only a few parameters, while the End-Use model has a large number of parameters. However, the End-Use model is very flexible towards the input parameters, which also have a clearer physical meaning and hence more intuitive to calibrate. The PRP model describes the measured flows very well. From the analytic description provided by the PRP model, a lot of mathematical deductions can be made. Thus, one can classify the PRP model as a descriptive model with a lot of po‐ tential to provide insight into some basic elements of water use, such as peak demands [9]and cross-correlations [10]. The End-Use model is a Monte Carlo type simulation that can be used as a predictive model, since it produces very realistic demand patterns. The End-Use model can be applied in scenario studies to show the result of changes in water using appliances and human behaviour. Possible improvements to the model include the incorpo‐ ration of leakage, the consideration of demands as a function of the network pressure and the application of the model outside the Netherlands [11]. Li [10]studied the spatial correla‐ tion of demand series that follow PRP processes. It was verified that while time averaged demands that follow a homogeneous PRP process are uncorrelated, demands that follow a non-homogenous PRP process are correlated, and that this correlation increases with spatial and temporal aggregation. A similar conclusion about the correlation was achieved by Moughton [12]from field measurements.
