**2.1. Correlation between consumers**

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

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

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

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

of the mean and variance of the flow is even more pronounced.

build a stochastic model for water consumption.

the mean and variance of the process.

ers is outlined at various locations.

110 Water Supply System Analysis - Selected Topics

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

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‐ lowed:

$$cov\_{AB} = \frac{1}{T} \int\_0^T \mathbb{I}\_A(i\_{A'} \ t) - \mu\_A \mathbb{II}\_B(j\_{B'} \ t) - \mu\_B \mathbf{J} dt \tag{3}$$

$$\rho\_{AB} = \frac{\alpha v\_{AB}}{\sigma\_A \cdot \sigma\_B} \tag{4}$$

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 own demand pattern based on specific needs and habits, without knowing what other con‐ sumers are doing at the same time. This means that demand signals in real networks are cor‐ related, but are not synchronous. Thus, in order to obtain the optimum design of a network, it is essential to estimate the accurate level of correlation between the consumers. On the other hand, to estimate accurately the spatial correlation between demands, it is necessary to collect and analyse historical series, resulting in additional costs in the design phase. How‐ ever, these additional costs will most certainly be compensated by the achieved reduction of the following construction costs.

Where*E m*∆*t*,*<sup>T</sup>* (*n*) is the expected value of the moment *m*for *n*users for the time interval*T* ; *E m*∆*t*,*<sup>T</sup>* is the expected value of the moment *m*for the single-user for the same time interval; *α*is the exponent of the scaling law; and *f* (∆*t*, *T* )is a function that expresses the influence of

The development of the scaling laws is based on the assumption that the demand can be de‐ scribed by a homogeneous and stationary process, which implies that the *n*aggregated users are of the same type (residential, commercial, industrial, etc.), and that the statistical proper‐ ties of demand, mean and variance, can be assumed constant in time. The scaling laws for the mean, variance, and lag1 covariance were derived by Magini [25]. The expected value of

> *T ∫* 0 *<sup>T</sup>* ∑ *j*=1 *n*

Where *E µ*1 is the expected demand value for the single user or 'unit mean'. This expres‐ sion shows that the mean demand increases linearly with the number of users according to a factor of proportionality equal to the expected value of the single user and is independent of

In order to estimate the expected value of the demand variance it is necessary to consider the covariance function *cov*(*s*, *τ*)of the single-user demand at the spatial and temporal lags,

cov*Δ<sup>t</sup>*

riance function. This expression shows that the expected value for the sample variance of the *n*-users process depends on the correlation structure of the single-user demands. The

negligible when*T* > >*θ*, being *θ* a parameter, connected to the cross-correlation of the de‐ mands and similar to the scale of fluctuation for the auto-correlation of a single signal.

*covΔ<sup>t</sup>*

*σ*1,*Δ<sup>t</sup>* <sup>2</sup> *j*

<sup>2</sup>and*τ* =*τ*<sup>1</sup> - *τ*2, respectively. The following expression is obtained (see [26] for the

(*s*,0)−cov*Δt*(*s*, *τ*) *dτ*1*dτ*<sup>2</sup>

cov*Δt*(*s*, *τ*)*dτ*1*dτ*<sup>2</sup>

(*s*, 0) is the covariance function at lag*s* =0, and *cov*∆*t*(*s*, *τ*) is the space-time cova‐

*<sup>T</sup> cov*∆*t*(*s*, *<sup>τ</sup>*)*dτ*1*dτ*2 decreases as the period of observation *<sup>T</sup>* increases, becoming

(*s*, 0) is independent from *τ*1 and*τ*2, and assumes the following values:

(*s*) *j* <sup>1</sup> ≠ *j* 2

> <sup>1</sup> = *j* 2

*E q*( *j*, *τ*) *dτ* =*n* ⋅*E µ*<sup>1</sup> (6)

Water Demand Uncertainty: The Scaling Laws Approach

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

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(7)

(8)

*<sup>q</sup>*( *<sup>j</sup>*, *<sup>τ</sup>*)*d<sup>τ</sup>* <sup>=</sup> <sup>1</sup>

both sampling rate and observation period.

*<sup>E</sup> <sup>µ</sup>*∆*t*,*<sup>T</sup>* (*n*) <sup>=</sup>*<sup>E</sup>* <sup>1</sup>

the sampling rate and observation period.

*s* = *j* <sup>1</sup> - *j*

Where *cov*∆*<sup>t</sup>*

The term *cov*∆*<sup>t</sup>*

term 1 *<sup>T</sup>* <sup>2</sup> *∫* 0 *T ∫* 0

mathematical passages):

*E σΔ<sup>t</sup>*,*<sup>T</sup>*

<sup>2</sup> (*n*) <sup>=</sup> <sup>1</sup>

=∑ *i* <sup>1</sup>=1 *n* ∑ *i* <sup>2</sup>=1 *n*

*<sup>T</sup>* <sup>2</sup> *<sup>∫</sup>* 0 *T ∫* 0 *T* ∑ *i* 1=1 *n* ∑ *i* 2=1 *n*

cov*Δ<sup>t</sup>*

*covΔ<sup>t</sup>*

(*s*,0)={

(*s*,0)<sup>−</sup> <sup>1</sup> *<sup>T</sup>* <sup>2</sup> *<sup>∫</sup>* 0 *T ∫* 0 *T*

the total mean demand *q*(*n*, *t*)can be expressed as followed:

*T ∫* 0 *<sup>T</sup>* ∑ *j*=1 *n*
