**2. Statistical characterization of water demand**

that illustrates the degree of uncertainty about the true value of the parameter. The out‐ comes, like nodal heads, are consequently also random variables, allowing the expression of

Uncertainty in demand and pressure heads was first explicitly considered by Lansey [13]. The authors developed a single-objective chance constrained minimization problem, which was solved using the generalized reduced gradient method GRG2. The obtained results showed that higher reliability requirements were associated to higher design costs when one

Xu and Goulter [14] proposed an alternative method for assessing reliability in WDS. The mean values of pressure heads were obtained from the deterministic solution of the network model. The variance values were obtained using the first-order second moment method (FOSM). The probability density function (PDF) of nodal heads defined by these mean and variance values was used to estimate the reliability at each node. The approach proved to be suitable for demands with small variability. Kapelan [15] developed two new methods for the robust design of WDS: the integration method and the sampling method. The integra‐ tion method consists in replacing the stochastic target robustness constraint (minimum pres‐ sure head) with a set of deterministic constraints. For that matter it is necessary to know the mean and standard deviation of the pressure heads. However, since pressure heads are de‐ pendent of the demands, it is not possible to obtain analytically the values for the standard deviations. Approximations of the values of the standard deviations are obtained by assum‐ ing the superposition principle, which makes it possible to estimate the contribution of the uncertainty in demand on the uncertainty of pressure heads. The sampling method is based on a general stochastic optimization framework, this is, a double looped process consisting on a sampling loop within an optimization loop. The optimization loop finds the optimal solution, and the sampling loop propagates the uncertainty in the input variables to the out‐

The aforementioned optimization problems are formulated as constrained single-objective problems, resulting in only one optimal solution (minimum cost), that provides a certain level of reliability. More recently, these optimization problems have been replaced with multi-objective problems. Babayan [16] formulated a multi-objective optimization problem considering two objectives at the same time: the minimization of the design cost and the maximization of the systems' robustness. Nodal demands and pipe roughness coefficients

At this point, all the aforementioned models assume nodal demands as independent ran‐ dom variables. However, in real-life demands are most likely correlated: demands may rise and fall due to the same causes. Kapelan [17] introduced nodal demands as correlated ran‐ dom variables into a multi-objective optimization problem. The authors verified that the op‐ timal design solution is more expensive when demands are correlated than the equivalent solution when demands are uncorrelated. A similar conclusion was achieved by Filion [18]. These results sustain that assuming uncorrelated demands can lead to less reliable network designs. Thus, even if increasing the complexity of optimization problems, demand correla‐

were assumed to be independent random variables following some PDF.

tion should always be taken into account in the design of WDS.

the networks' reliability.

108 Water Supply System Analysis - Selected Topics

of the variables of the problem was uncertain.

put variables, thus evaluating the potential solutions.

Recent studies on uncertainty in water distribution systems (WDS) refer that nodal demands are the most significant inputs in hydraulic and water quality models [19]. The variability of water demand affects the overall reliability of the model, the assessment of the spatial and temporal distributions of the pressure heads, and the evaluation of water quality along the different pipes. These uncertainties assume a different importance depending on the spatial and temporal scales that are considered when describing the network. The degree of uncer‐ tainty becomes more relevant when finer scales are reached, i.e., when small groups of users and instantaneous demands are considered. Thus, for a correct and realistic design and management, as well as simulation and performance assessment of WDS it is essential to have accurate values of water demand that take into account the variability of consumption at different scales. For that matter, the thorough description of the statistical properties of demand of the different groups of customers in the network, at specific temporal resolu‐ tions, is essential.

For a better understanding of this aspect, let us consider the distribution of the customers in a network. Figure 1 shows the network of a small town where the customers can be classi‐ fied mainly as residential.

If the consumers are assumed to be of the same type, the properties of demand can be con‐ sidered to be homogeneous in space, this is, they are independent of the particular consum‐ er that is taken into consideration. Regarding the temporal variability, the stochastic process can only be assumed to be stationary in time intervals during which the mean stays con‐ stant. Once the length of this time interval, *T* , is established, it is possible to determine the

*<sup>T</sup> <sup>q</sup>*(*i*, *<sup>t</sup>*) - *<sup>µ</sup>*<sup>1</sup>

For homogeneous and stationary demands, the expected values for the mean and variance,

The definition of the mean and variance for each type of consumer is not enough for a com‐ plete statistical characterization of demand. In order to obtain a realistic representation of the demand loads at the different nodes in a network; essential for the assessment of the net‐ work performance under conditions as close as possible to the actual working conditions, the correlation between nodal demands cannot be ignored. This correlation can be expressed

The cross-covariance,*covAB* , and cross-correlation coefficient,*ρAB* , between user *i*of group *A* and user *j*of group*B*, during the observation period*T* , are expressed, respectively, as fol‐

*<sup>A</sup>*, *t*) −*µA qB*( *j*

As known, the WDS need to guarantee minimum working conditions, this is, the minimum pressure requirements have to be satisfied at each node even under maximum demand loading conditions. If all the consumers in the network are of the same type, it seems reason‐ able to assume a perfect correlation between demands, and to simplify the analysis of the network by assigning the same demand pattern to all the consumers. The synchronism of demands is the worst scenario that can occur on a network, causing the widest pressure fluctuations at the nodes. The assumption of a perfect correlation for design purposes results in reliable networks, but it also requires the increase of the pipe diameters, which conse‐ quently increases the networks cost. In fact, as mentioned earlier, each consumer has his

*<sup>ρ</sup>AB* <sup>=</sup> *covAB σ<sup>A</sup>* ⋅ *σ<sup>B</sup>*

<sup>2</sup> , obtained from *N* observations, provide the mean and variance of the process.

, of the demand signal of the single-user*i*, as followed:

*<sup>T</sup> q*(*i*, *t*)*dt* (1)

<sup>2</sup>*dt* (2)

Water Demand Uncertainty: The Scaling Laws Approach

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

111

*<sup>B</sup>*, *t*) −*µB dt* (3)

(4)

2

*σ*1 <sup>2</sup> <sup>=</sup> <sup>1</sup> *T ∫* 0

*<sup>µ</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> *T ∫* 0

through the cross-covariance and cross-correlation coefficient functions.

*covAB* <sup>=</sup> <sup>1</sup> *T ∫* 0 *T qA*(*i*

temporal mean, *µ*1, and variance,*σ*<sup>1</sup>

**2.1. Correlation between consumers**

*E µ*<sup>1</sup> and*E σ*<sup>1</sup>

lowed:

**Figure 1.** Spatial variability of customers in a real distribution network, from Magini *et al*. [25]. The number of custom‐ ers is outlined at various locations.

The most peripheral pipe serves the inhabitants of one single building. When moving up‐ wards in the network, the number of customers increases reaching a maximum of 1258 cus‐ tomers near the tank. Obviously, as a consequence, the mean flow increases from the peripheral building to the tank. The increase of the variance of the flow is, however, less ob‐ vious. For larger networks and more densely populated towns, the difference between the number of customers that are close and far from the tank, and consequently, the variations of the mean and variance of the flow is even more pronounced.

Another important aspect when modelling a network is the choice of the adequate temporal resolution. This choice depends on the characteristics of the available measurement instru‐ ments and on the type of analysis to perform. When modelling the peripheral part of a net‐ work, characterized by a significant temporal variation of demand, it is important to adopt fine temporal resolutions, i.e., in the order of seconds. For the estimation of peak flows in design problems Tessendorff [20] suggests the use of different temporal resolutions on dif‐ ferent sections of the network: the author suggests the use of a 15 second time interval for customer installation lines, two minutes for service lines, 15 minutes for distribution lines, and 30 minutes for mains and secondary feeders. The statistical properties of water demand are affected by the considered temporal resolution. The use of longer sampling intervals causes an inevitable loss of information about the signals, resulting in lower estimates for the variance [21, 22]. This aspect is particularly relevant at the peripheral pipes of the net‐ work that, as aforementioned, are characterized by large demand fluctuations. Therefore, understanding the spatial and temporal scaling properties of water demand is essential to build a stochastic model for water consumption.

Water demand can be described by a stochastic process in which *q*(*i*, *t*)represents the de‐ mand of water of the single-user *i*at time instant*t*. In order to estimate the statistical proper‐ ties of water demand, it is necessary to have a historical series of observations, extended to sufficiently wide number of users of each type. From this data it is then possible to estimate the mean and variance of the process.

If the consumers are assumed to be of the same type, the properties of demand can be con‐ sidered to be homogeneous in space, this is, they are independent of the particular consum‐ er that is taken into consideration. Regarding the temporal variability, the stochastic process can only be assumed to be stationary in time intervals during which the mean stays con‐ stant. Once the length of this time interval, *T* , is established, it is possible to determine the temporal mean, *µ*1, and variance,*σ*<sup>1</sup> 2 , of the demand signal of the single-user*i*, as followed:

$$\mu\_1 = \frac{1}{T} \mathbb{J}\_0^T q(\mathbf{i}, \mathbf{t}) dt \tag{1}$$

$$
\sigma\_1^2 = \frac{1}{T} \zeta\_0^T \mathbb{I} \{ q(\mathbf{i}\_\prime \ t) \mathbf{ -} \boldsymbol{\mu}\_1 \}^2 dt \tag{2}
$$

For homogeneous and stationary demands, the expected values for the mean and variance, *E µ*<sup>1</sup> and*E σ*<sup>1</sup> <sup>2</sup> , obtained from *N* observations, provide the mean and variance of the process.
