**2.2. The scaling laws approach in modelling water demand uncertainty**

Water demand uncertainty is made of both aleatory or inherent uncertainty, due to the natu‐ ral and unpredictable variability of demand in space and time, and epistemic or internal un‐ certainty, due to a lack of knowledge about it. Hutton [23] distinguishes epistemic uncertainty in two types. The first type concerns the nature of the demand patterns, and the lack of knowledge about this variability when modelling WDS both in time and space. This uncertainty is defined as 'two-dimensional' uncertainty since it is composed by both aleato‐ ry and epistemic uncertainty. It can be reduced with extended and expensive spatial and temporal data collection or through the employment of descriptive and predictive water de‐ mand models. The second type of epistemic uncertainty takes the spatial allocation of water demand into account when modelling WDS [24].

Dealing with the 'two-dimensional' uncertainty when modelling WDS, requires not only a complete statistical characterization of demand variability, but also the determination of the correlation among the different users and groups of users. The natural variability of demand can be expressed using probability density functions (PDF). A PDF is characterized by its shape (e.g. normal, exponential, gamma, among others) and by specific parameters like the population mean and variance. Thus, in order to represent uncertain water demand using a PDF, it is necessary to identify and estimate the values of these parameters. The considera‐ tion of different spatial and temporal aggregation levels induces changes in the PDF param‐ eters, often leading to a reduction of the uncertainty. The auto-correlation and crosscorrelation that characterize the water demand signals affect the extent to which the PDF parameters vary, and can introduce an additional sensitivity to the specific period of obser‐ vation in question.

In order to understand the effects of spatial aggregation and sampling intervals on the statis‐ tical properties of demand, it is possible to develop analytical expressions for the moments (mean, variance, cross-covariance and cross-correlation coefficient) of demand time series, at a fixed time sampling frequency∆*t*, of *n*aggregated users as a function of the moments of the single-user series sampled in the observation period *T*. These expressions are referred to as *"Scaling Laws"*, and can be expressed as:

$$E\mathbb{E}[m\_{\Delta t,T}(n)] = E\mathbb{E}[m\_{\Delta t,T}] \cdot n^a \cdot f\left(\Delta t, \, T\right) \tag{5}$$

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 both sampling rate and observation period.

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

Water demand uncertainty is made of both aleatory or inherent uncertainty, due to the natu‐ ral and unpredictable variability of demand in space and time, and epistemic or internal un‐ certainty, due to a lack of knowledge about it. Hutton [23] distinguishes epistemic uncertainty in two types. The first type concerns the nature of the demand patterns, and the lack of knowledge about this variability when modelling WDS both in time and space. This uncertainty is defined as 'two-dimensional' uncertainty since it is composed by both aleato‐ ry and epistemic uncertainty. It can be reduced with extended and expensive spatial and temporal data collection or through the employment of descriptive and predictive water de‐ mand models. The second type of epistemic uncertainty takes the spatial allocation of water

Dealing with the 'two-dimensional' uncertainty when modelling WDS, requires not only a complete statistical characterization of demand variability, but also the determination of the correlation among the different users and groups of users. The natural variability of demand can be expressed using probability density functions (PDF). A PDF is characterized by its shape (e.g. normal, exponential, gamma, among others) and by specific parameters like the population mean and variance. Thus, in order to represent uncertain water demand using a PDF, it is necessary to identify and estimate the values of these parameters. The considera‐ tion of different spatial and temporal aggregation levels induces changes in the PDF param‐ eters, often leading to a reduction of the uncertainty. The auto-correlation and crosscorrelation that characterize the water demand signals affect the extent to which the PDF parameters vary, and can introduce an additional sensitivity to the specific period of obser‐

In order to understand the effects of spatial aggregation and sampling intervals on the statis‐ tical properties of demand, it is possible to develop analytical expressions for the moments (mean, variance, cross-covariance and cross-correlation coefficient) of demand time series, at a fixed time sampling frequency∆*t*, of *n*aggregated users as a function of the moments of the single-user series sampled in the observation period *T*. These expressions are referred to

*<sup>E</sup> <sup>m</sup>*∆*t*,*<sup>T</sup>* (*n*) <sup>=</sup> *<sup>E</sup> <sup>m</sup>*∆*t*,*<sup>T</sup>* <sup>⋅</sup>*<sup>n</sup> <sup>α</sup>* <sup>⋅</sup> *<sup>f</sup>* (∆*t*, *<sup>T</sup>* ) (5)

**2.2. The scaling laws approach in modelling water demand uncertainty**

the following construction costs.

112 Water Supply System Analysis - Selected Topics

demand into account when modelling WDS [24].

as *"Scaling Laws"*, and can be expressed as:

vation in question.

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

$$E\left[\boldsymbol{\mu}\_{\Delta t,T}\left(\boldsymbol{n}\right)\right] = E\left[\frac{1}{T}\mathbb{J}\_0^T \sum\_{j=1}^n q\{j\_j,\tau\}d\tau\right] = \frac{1}{T}\mathbb{J}\_0^T \sum\_{j=1}^n E\left[q\{j\_j,\tau\}\right]d\tau = n \cdot E\left[\boldsymbol{\mu}\_1\right] \tag{6}$$

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 the sampling rate and observation period.

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

$$\begin{aligned} \mathbb{E}\left[\sigma\_{\text{Al},T}^{2}\{\mathbf{r}\}\right] &= \frac{1}{T^{2}}\Big|\_{0}^{T} \int\_{0}^{T} \sum\_{l\_{1}=1}^{n} \sum\_{l\_{2}=1}^{n} \left[\text{cov}\_{\text{Al}}(\text{s},0) - \text{cov}\_{\text{Al}}(\text{s},\tau)\right] d\tau\_{1} d\tau\_{2} \\ &= \sum\_{l\_{1}=1}^{n} \sum\_{l\_{2}=1}^{n} \left[\text{cov}\_{\text{Al}}(\text{s},0) - \frac{1}{T^{2}}\int\_{0}^{T} \int\_{0}^{T} \text{cov}\_{\text{Al}}(\text{s},\tau) d\tau\_{1} d\tau\_{2}\right] \end{aligned} \tag{7}$$

Where *cov*∆*<sup>t</sup>* (*s*, 0) is the covariance function at lag*s* =0, and *cov*∆*t*(*s*, *τ*) is the space-time cova‐ 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 term 1 *<sup>T</sup>* <sup>2</sup> *∫* 0 *T ∫* 0 *<sup>T</sup> cov*∆*t*(*s*, *<sup>τ</sup>*)*dτ*1*dτ*2 decreases as the period of observation *<sup>T</sup>* increases, becoming 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.

The term *cov*∆*<sup>t</sup>* (*s*, 0) is independent from *τ*1 and*τ*2, and assumes the following values:

$$cov\_{\Lambda t} \text{(s,0)} = \begin{vmatrix} cov\_{\Lambda t} \text{(s)} & j\_1 \neq j\_2 \\ \sigma\_{1,\Lambda t}^2 & j\_1 = j\_2 \end{vmatrix} \tag{8}$$

### 114 Water Supply System Analysis - Selected Topics

Where *cov*∆*<sup>t</sup>* (*s*) is the spatial cross-covariance between different single-user demands, and *σ*1,∆*<sup>t</sup>* 2 the variance of the single user. For large values of*T* , equation (7) can be simplified into:

$$\begin{aligned} \mathrm{E}\left[\sigma\_{\mathrm{Al},T}^{2}\left(n\right)\right] &= \sum\_{j\_{1}=1}^{n} \sum\_{j\_{2}=1}^{n} \mathrm{cov}\_{\mathrm{Al}}\{\mathrm{s},0\} = \sum\_{j\_{1}=1}^{n} \sum\_{j\_{2}=j\_{1}}^{n} \mathrm{cov}\_{\mathrm{Al}}\{\mathrm{s},0\} + \sum\_{j\_{1}=1}^{n} \sum\_{j\_{2}\neq j\_{1}}^{n} \mathrm{cov}\_{\mathrm{Al}}\{\mathrm{s},0\} \\ &= n \cdot \sigma\_{1,\mathrm{Al}}^{2} + n \cdot \left(n-1\right) \cdot \mathrm{cov}\_{\mathrm{Al}}\{\mathrm{s}\} \end{aligned} \tag{9}$$

This equation represents the scaling law for the variance, neglecting the bias that can be caused when using small the demand series (short observation periods).

Introducing the Pearson cross-correlation coefficient given by*<sup>ρ</sup>* <sup>=</sup> *cov xy <sup>σ</sup>xσ<sup>y</sup>* , and considering *ρΔ<sup>t</sup>* as the cross-correlation coefficient between each couple of single-user demands, the spatial covariance can be expressed as*covΔ<sup>t</sup>* =*ρΔ<sup>T</sup> σ*1,*Δ<sup>t</sup>* <sup>2</sup> , and Equation (9) becomes:

$$E[\sigma\_{\Delta t}^2(n)] = n^2 \cdot \rho\_{\Delta t} \cdot \sigma\_{1\Delta t}^2 + n \cdot \left[1 \cdot \rho\_{\Delta t}\right] \cdot \sigma\_{1\Delta t}^2 \tag{10}$$

If demands are perfectly correlated in space then *ρ*∆*t* is equal to one, and equation (10) is simplified into:

$$E\left[\sigma\_{\Delta t}^2(n)\right] = n^2 \cdot \sigma\_{1,\Delta t}^2\tag{11}$$

The variance function, *γ*(*Δt*), measures the reduction of the variance of the instantaneous

<sup>2</sup> is the variance of the instantaneous signal for the single user. Introducing the var‐

<sup>2</sup> ⋅*γ*(*Δt*) (14)

Water Demand Uncertainty: The Scaling Laws Approach

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

<sup>2</sup> ⋅*φ*(∆*t*) (15)

(*s*, *τ*) *dτ*1*dτ*<sup>2</sup>

(16)

115

signal when the sampling interval *Δt*increases [27], as followed:

iance function in equation (13), the following is obtained:

*<sup>E</sup> covAB*,*Δt*(*na*, *nb*) <sup>=</sup> <sup>1</sup>

=∑ *i*=1 *n* ∑ *j*=1 *n*

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

Neglecting the term <sup>1</sup>

riance becomes quadratic.

*<sup>E</sup> <sup>ρ</sup>AB*,∆*t*(*na*, *nb*) <sup>=</sup> *<sup>E</sup> Cov AB*,∆*t*(*na*, *nb*)

by:

is given by:

*E σ*∆*<sup>t</sup>*

Similarly, the expected value of the cross-covariance is given by:

*<sup>T</sup>* <sup>2</sup> *∫* 0 *T ∫* 0 *T* ∑ *i*=1 *na* ∑ *j*=1 *nb*

cov*AB*,*Δ<sup>t</sup>*

Where *σ*<sup>1</sup>

*σ*1,∆*<sup>t</sup>* <sup>2</sup> <sup>=</sup>*σ*<sup>1</sup>

<sup>2</sup> (*n*) <sup>=</sup>*<sup>n</sup> <sup>α</sup>* <sup>⋅</sup>*σ*<sup>1</sup>

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

*covAB*,*Δ<sup>t</sup>*

between the demands of *na* aggregated users of group A and *nb* aggregated users of group B

Where, *E ρab*,∆*<sup>t</sup>* is the expected Pearson cross-correlation coefficient between the single-user demands of the two groups; and *σa*,∆*t* and *σb*,∆*t*are the standard deviations of the single-user demands of groups *A*and*B*, respectively, at the sampling rate∆*t*. The expected value of the cross-covariance increases according to the product between the number of users of each group. In the particular case in which both groups have the same statistical properties, i.e., they belong to the same process, and assuming that*na* =*nb*, the scaling law of the cross-cova‐

As a consequence, the expected value of the Pearson cross-correlation coefficient between the demands of *na* aggregated users of group A and *nb* aggregated users of group B, is given

> *<sup>E</sup> <sup>σ</sup>A*,∆*t*(*na*) <sup>⋅</sup> *<sup>E</sup> <sup>σ</sup>B*,∆*t*(*nb*) <sup>=</sup> *na* <sup>⋅</sup> *nb* <sup>⋅</sup> *<sup>E</sup> <sup>ρ</sup>ab*,∆*<sup>t</sup> na*

(*s*,0)−cov*AB*,*Δ<sup>t</sup>*

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

*E cov AB*,∆*t*(*na*, *nb*) =*na* ⋅*nb* ⋅*E ρab*,∆*<sup>t</sup>* ⋅*E σa*,∆*<sup>t</sup>* ⋅*E σb*,∆*<sup>t</sup>* (17)

(1 + *E ρa*,∆*<sup>t</sup>* ⋅ *na* - 1 ) ⋅ *nb*

(1 <sup>+</sup> *<sup>E</sup> <sup>ρ</sup>b*,∆*<sup>t</sup>* <sup>⋅</sup> *nb* - <sup>1</sup> ) (18)

*<sup>T</sup> cov AB*,∆*t*(*s*, *<sup>τ</sup>*)*dτ*1*dτ*2, *<sup>t</sup>*he expected value of the cross-covariance

If demands are uncorrelated in space then *ρΔt*is equal to zero, and equation (9) is simplified into:

$$E[\sigma\_{\Delta t}^2(n)] = n \cdot \sigma\_{1,\Delta t}^2\tag{12}$$

Since the cross-correlation coefficient can assume values between 0 and1, equations (10) and (11) represent the maximum and minimum expected values for the variance. Equation (9) can be simplified into a more generic form given by:

$$E\left[\sigma\_{\Delta t}^2(n)\right] = n^a \cdot \sigma\_{1,\Delta t}^2\tag{13}$$

Where1≤*α* ≤2.

In conclusion, it can be stated that the variance in the consumption signal of a group of users *n*, homogeneous in type, is proportional to the mean variance of the single-user according to an exponent, which varies between 1 and 2. The value of the scaling exponent depends on the structure of the spatial correlation, i.e., the correlation that exists between the different consumptions during the observation period: if demands are uncorrelated in space, the scal‐ ing law is linear, if demands are perfectly correlated in space, the scaling law is quadratic.

The variance function, *γ*(*Δt*), measures the reduction of the variance of the instantaneous signal when the sampling interval *Δt*increases [27], as followed:

$$
\sigma\_{1,\Delta t}^2 = \sigma\_1^2 \cdot \gamma \{\Delta t\} \tag{14}
$$

Where *σ*<sup>1</sup> <sup>2</sup> is the variance of the instantaneous signal for the single user. Introducing the var‐ iance function in equation (13), the following is obtained:

$$E\left[\sigma\_{\Delta t}^{2}(n)\right] = n^{\alpha} \cdot \sigma\_{1}^{2} \cdot \varphi\left(\Delta t\right) \tag{15}$$

Similarly, the expected value of the cross-covariance is given by:

Where *cov*∆*<sup>t</sup>*

simplified into:

Where1≤*α* ≤2.

into:

*E σΔ<sup>t</sup>*,*<sup>T</sup>*

114 Water Supply System Analysis - Selected Topics

=*n* ⋅*σ*1,*Δ<sup>t</sup>*

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

covariance can be expressed as*covΔ<sup>t</sup>* =*ρΔ<sup>T</sup> σ*1,*Δ<sup>t</sup>*

*E σ*∆*<sup>t</sup>*

can be simplified into a more generic form given by:

*covΔ<sup>t</sup>*

<sup>2</sup> <sup>+</sup> *<sup>n</sup>* <sup>⋅</sup> (*<sup>n</sup>* <sup>−</sup>1) <sup>⋅</sup> *covΔt*(*s*)

(*s*,0)= ∑ *j* 1=1 *n* ∑ *j* 2= *j* 1 *covΔ<sup>t</sup>*

caused when using small the demand series (short observation periods).

Introducing the Pearson cross-correlation coefficient given by*<sup>ρ</sup>* <sup>=</sup> *cov xy*

<sup>2</sup> (*n*) <sup>=</sup>*<sup>n</sup>* <sup>2</sup> <sup>⋅</sup>*ρ*∆*<sup>t</sup>* <sup>⋅</sup>*σ*1,∆*<sup>t</sup>*

*E σ*∆*<sup>t</sup>*

*E σ*∆*<sup>t</sup>*

*E σ*∆*<sup>t</sup>*

*σ*1,∆*<sup>t</sup>*

(*s*) is the spatial cross-covariance between different single-user demands, and

(*s*,0) + ∑ *j* <sup>1</sup>=1 *n* ∑ *j* <sup>2</sup>≠ *j* 1 *covΔ<sup>t</sup>* (*s*,0)

<sup>2</sup> , and Equation (9) becomes:

<sup>2</sup> <sup>+</sup> *<sup>n</sup>* <sup>⋅</sup> <sup>1</sup> - *<sup>ρ</sup>*∆*<sup>t</sup>* <sup>⋅</sup>*σ*1,∆*<sup>t</sup>*

(9)

*<sup>σ</sup>xσ<sup>y</sup>* , and considering *ρΔ<sup>t</sup>*

<sup>2</sup> (10)

<sup>2</sup> (11)

<sup>2</sup> (12)

<sup>2</sup> (13)

2 the variance of the single user. For large values of*T* , equation (7) can be simplified into:

This equation represents the scaling law for the variance, neglecting the bias that can be

as the cross-correlation coefficient between each couple of single-user demands, the spatial

If demands are perfectly correlated in space then *ρ*∆*t* is equal to one, and equation (10) is

<sup>2</sup> (*n*) <sup>=</sup>*<sup>n</sup>* <sup>2</sup> <sup>⋅</sup>*σ*1,∆*<sup>t</sup>*

<sup>2</sup> (*n*) <sup>=</sup>*<sup>n</sup>* <sup>⋅</sup>*σ*1,∆*<sup>t</sup>*

<sup>2</sup> (*n*) <sup>=</sup>*<sup>n</sup> <sup>α</sup>* <sup>⋅</sup>*σ*1,∆*<sup>t</sup>*

In conclusion, it can be stated that the variance in the consumption signal of a group of users *n*, homogeneous in type, is proportional to the mean variance of the single-user according to an exponent, which varies between 1 and 2. The value of the scaling exponent depends on the structure of the spatial correlation, i.e., the correlation that exists between the different consumptions during the observation period: if demands are uncorrelated in space, the scal‐ ing law is linear, if demands are perfectly correlated in space, the scaling law is quadratic.

Since the cross-correlation coefficient can assume values between 0 and1, equations (10) and (11) represent the maximum and minimum expected values for the variance. Equation (9)

If demands are uncorrelated in space then *ρΔt*is equal to zero, and equation (9) is simplified

$$\begin{aligned} \operatorname{E}\Big[\operatorname{cov}\_{AB,\operatorname{A}t}\{\boldsymbol{\eta}\_{\omega'},\boldsymbol{\eta}\_{b}\}\Big] &= \frac{1}{T^{\frac{3}{2}}}\Big[\int\_{0}^{T}\sum\_{i=1}^{\frac{n}{2}}\sum\_{j=1}^{n\_{b}}\Big[\operatorname{cov}\_{AB,\operatorname{A}t}\{\boldsymbol{s},0\}-\operatorname{cov}\_{AB,\operatorname{A}t}\{\boldsymbol{s},\boldsymbol{\tau}\}\Big]d\,\boldsymbol{\tau}\_{1}d\,\boldsymbol{\tau}\_{2} \\ &= \sum\_{i=1}^{\frac{n}{2}}\sum\_{j=1}^{n}\Big[\operatorname{cov}\_{AB,\operatorname{A}t}\{\boldsymbol{s},0\}-\frac{1}{T^{\frac{2}{3}}}\Big[\int\_{0}^{T}\operatorname{cov}\_{AB,\operatorname{A}t}\{\boldsymbol{s},\boldsymbol{\tau}\}d\,\boldsymbol{\tau}\_{1}d\,\boldsymbol{\tau}\_{2}\Big] \end{aligned} \tag{16}$$

Neglecting the term <sup>1</sup> *<sup>T</sup>* <sup>2</sup> *∫* 0 *T ∫* 0 *<sup>T</sup> cov AB*,∆*t*(*s*, *<sup>τ</sup>*)*dτ*1*dτ*2, *<sup>t</sup>*he expected value of the cross-covariance between the demands of *na* aggregated users of group A and *nb* aggregated users of group B is given by:

$$E\left[cov\_{AB,\Delta t}(n\_{a'}, n\_b)\right] = n\_a \cdot n\_b \cdot E\left[\rho\_{ab,\Delta t}\right] \cdot E\left[\sigma\_{a\Delta t}\right] \cdot E\left[\sigma\_{b,\Delta t}\right] \tag{17}$$

Where, *E ρab*,∆*<sup>t</sup>* is the expected Pearson cross-correlation coefficient between the single-user demands of the two groups; and *σa*,∆*t* and *σb*,∆*t*are the standard deviations of the single-user demands of groups *A*and*B*, respectively, at the sampling rate∆*t*. The expected value of the cross-covariance increases according to the product between the number of users of each group. In the particular case in which both groups have the same statistical properties, i.e., they belong to the same process, and assuming that*na* =*nb*, the scaling law of the cross-cova‐ riance becomes quadratic.

As a consequence, the expected value of the Pearson cross-correlation coefficient between the demands of *na* aggregated users of group A and *nb* aggregated users of group B, is given by:

$$\mathbb{E}\left[\rho\_{AB,\Delta t}(n\_{\mu^\*} \cdot n\_b)\right] = \frac{\mathbb{E}\left[\mathbb{C}\boldsymbol{\alpha}\_{AB,\Delta t}(n\_{\mu^\*} \cdot n\_b)\right]}{\mathbb{E}\left[\boldsymbol{\sigma}\_{A,\Delta t}(n\_b)\right] \cdot \mathbb{E}\left[\boldsymbol{\sigma}\_{B,\Delta t}(n\_b)\right]} = \frac{n\_x \cdot n\_b \cdot \mathbb{E}\left[\rho\_{ab,\Delta t}\right]}{\sqrt{n\_x^{\dagger} \left(1 + \mathbb{E}\left[\rho\_{a,\Delta t}\right] \cdot \left[\boldsymbol{\tau}\_x \cdot 1\right]\right)} \cdot \sqrt{n\_y \left(1 + \mathbb{E}\left[\rho\_{b,\Delta t}\right] \cdot \left[\boldsymbol{\tau}\_y \cdot 1\right]\right)}}\tag{18}$$

This equation shows that this coefficient depends separately on the spatial aggregation lev‐ els of each group, *na* and *nb*, and not only on their product as happens for the cross-cova‐ riance. If *na = nb = n* equation [18] becomes:

$$E\left[\rho\_{AB,\Delta t}(n)\right] = \frac{n \cdot E\left[\rho\_{ab,\Delta t}\right]}{\sqrt{1 + \left(n \cdot 1\right) \cdot E\left[\rho\_{x,\Delta t}\prod\left(1 + \left(n \cdot 1\right) \cdot E\left[\rho\_{b,\Delta t}\right]\right]\right]}}\tag{19}$$

From equation (19) it is possible to observe that the expected value *E ρAB*,∆*t*(*na*, *nb*) increases with the number of users, *na*and*nb*, reaching the following limit value:

$$E\left[\boldsymbol{\rho}\_{AB,\boldsymbol{\omega}t}\left(\boldsymbol{n}\_{a^{\boldsymbol{\nu}}},\boldsymbol{n}\_{b}\right)\right] = \varprojlim\_{\boldsymbol{n}\_{a}\to\boldsymbol{\omega}} E\left[\boldsymbol{\rho}\_{AB,\boldsymbol{\omega}t}\left(\boldsymbol{n}\_{a^{\boldsymbol{\nu}}},\boldsymbol{n}\_{b}\right)\right] = \frac{E\left[\boldsymbol{\rho}\_{ab,\boldsymbol{\omega}t}\right]}{\sqrt{\mathbb{E}\left[\boldsymbol{\rho}\_{a,\boldsymbol{\omega}t}\right] \cdot E\left[\boldsymbol{\rho}\_{b,\boldsymbol{\omega}t}\right]}}\tag{20}$$

Since by definition*E ρab*,*Δ<sup>t</sup>* ≤1, the maximum value that the expected value of the cross-cor‐ relation coefficient between the single-user demands of group *A*and *B*can assume is:

$$E\left[\boldsymbol{\rho}\_{AB,\Delta t}\right]\_{\max} = \sqrt{\left(E\left[\boldsymbol{\rho}\_{a,\Delta t}\right] \cdot E\left[\boldsymbol{\rho}\_{b,\Delta t}\right]\right)}\tag{21}$$

**Figure 2.** Scaling laws of*E* ρ*AB*(*n*) , for different values ofρ*<sup>a</sup>* ⋅ρ*b*.

**Figure 3.** Scaling laws of*E* ρ*AB*(*n*) , for different values ofρ*ab*.

In the particular case in which both groups of users have the same statistical properties, i.e., they belong to the same process, and assuming*na* =*nb* =*n*, the scaling law for the cross-corre‐

1 + (*n* - 1) ⋅ *E ρ*∆*<sup>t</sup>*

From equation (22) it is clear that the cross-correlation coefficientincreases with the number of aggregated users, tending to one. This limit value is reached as sooner as the cross-corre‐

(22)

Water Demand Uncertainty: The Scaling Laws Approach

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

117

lation coefficient, considering no differences in the sampling time intervals, is:

*<sup>E</sup> <sup>ρ</sup>AB*,∆*t*(*n*) <sup>=</sup> *<sup>n</sup>* <sup>⋅</sup> *<sup>E</sup> <sup>ρ</sup>*∆*<sup>t</sup>*

lation coefficient, *E ρ*∆*<sup>t</sup>* , between the single-user demands is higher.

From equation (21) it is also possible to observe that the Pearson cross-correlation coefficient between the *na* aggregated users of group *A*and the *nb* aggregated users of group *B*depends on both the cross-correlations inside each group and the cross-correlation between the groups. Therefore, it seems interesting to investigate the way in which these two aspects, one at a time, affect the expected value of the cross-correlation when the number of aggre‐ gated users increases and for a fixed sampling rate∆*t*. In order to do so let us first consider a fixed value of *ρab* and varying values of *ρ<sup>a</sup>* and*ρb*. Figure 2 shows the graphical results for *ρab* =0.1 and different pairs of *ρa* and*ρb*.

As expected, all the curves have a common starting point, since *ρab* is fixed. According to equation (19) a gradual flattening of the curves and a reduction of the shape ratio *ρAB*,*lim* / *ρab* can be noticed when the product *ρ<sup>a</sup>* ⋅*ρb* increases. Let us now consider a different case in which *ρa* and *ρb* are fixed and *ρab*varies. The results are shown graphically in figure 3. The curves have now different starting points and equal shape ratios*ρAB*,*lim* / *ρab*. Increasing *ρab* produces only an upward shift of the curves, extending their transient.

**Figure 2.** Scaling laws of*E* ρ*AB*(*n*) , for different values ofρ*<sup>a</sup>* ⋅ρ*b*.

This equation shows that this coefficient depends separately on the spatial aggregation lev‐ els of each group, *na* and *nb*, and not only on their product as happens for the cross-cova‐

1 + (*n* - 1) ⋅ *E ρa*,∆*<sup>t</sup>* 1 + (*n* - 1) ⋅ *E ρb*,∆*<sup>t</sup>*

*<sup>E</sup> <sup>ρ</sup>AB*,*Δt*(*na*, *nb*) <sup>=</sup> *<sup>E</sup> <sup>ρ</sup>ab*,*Δ<sup>t</sup>*

*E ρAB*,∆*<sup>t</sup> max* = (*E ρa*,∆*<sup>t</sup>* ⋅*E ρb*,∆*<sup>t</sup>* ) (21)

From equation (19) it is possible to observe that the expected value *E ρAB*,∆*t*(*na*, *nb*) increases

Since by definition*E ρab*,*Δ<sup>t</sup>* ≤1, the maximum value that the expected value of the cross-cor‐

From equation (21) it is also possible to observe that the Pearson cross-correlation coefficient between the *na* aggregated users of group *A*and the *nb* aggregated users of group *B*depends on both the cross-correlations inside each group and the cross-correlation between the groups. Therefore, it seems interesting to investigate the way in which these two aspects, one at a time, affect the expected value of the cross-correlation when the number of aggre‐ gated users increases and for a fixed sampling rate∆*t*. In order to do so let us first consider a fixed value of *ρab* and varying values of *ρ<sup>a</sup>* and*ρb*. Figure 2 shows the graphical results for

As expected, all the curves have a common starting point, since *ρab* is fixed. According to equation (19) a gradual flattening of the curves and a reduction of the shape ratio *ρAB*,*lim* / *ρab* can be noticed when the product *ρ<sup>a</sup>* ⋅*ρb* increases. Let us now consider a different case in which *ρa* and *ρb* are fixed and *ρab*varies. The results are shown graphically in figure 3. The curves have now different starting points and equal shape ratios*ρAB*,*lim* / *ρab*. Increasing *ρab*

produces only an upward shift of the curves, extending their transient.

relation coefficient between the single-user demands of group *A*and *B*can assume is:

(19)

(*<sup>E</sup> <sup>ρ</sup>a*,*Δ<sup>t</sup>* <sup>⋅</sup> *<sup>E</sup> <sup>ρ</sup>b*,*Δ<sup>t</sup>* ) (20)

(*n*) <sup>=</sup> *<sup>n</sup>* <sup>⋅</sup> *<sup>E</sup> <sup>ρ</sup>ab*,∆*<sup>t</sup>*

with the number of users, *na*and*nb*, reaching the following limit value:

*na*→*∞ nb*→*∞*

riance. If *na = nb = n* equation [18] becomes:

116 Water Supply System Analysis - Selected Topics

*E ρAB*,∆*<sup>t</sup>*

*E ρAB*,*Δt*(*na*, *nb*) = lim

*ρab* =0.1 and different pairs of *ρa* and*ρb*.

**Figure 3.** Scaling laws of*E* ρ*AB*(*n*) , for different values ofρ*ab*.

In the particular case in which both groups of users have the same statistical properties, i.e., they belong to the same process, and assuming*na* =*nb* =*n*, the scaling law for the cross-corre‐ lation coefficient, considering no differences in the sampling time intervals, is:

$$E\left[\rho\_{AB,\Delta t}\left(n\right)\right] = \frac{n \cdot E\left[\rho\_{\Delta t}\right]}{1 + \left(n \cdot 1\right) \cdot E\left[\rho\_{\Delta t}\right]}\tag{22}$$

From equation (22) it is clear that the cross-correlation coefficientincreases with the number of aggregated users, tending to one. This limit value is reached as sooner as the cross-corre‐ lation coefficient, *E ρ*∆*<sup>t</sup>* , between the single-user demands is higher.
