**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 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 the networks' reliability.

The robust design of WDS has gained popularity over the last years. Researchers have been focusing on methods and algorithms to solve the stochastic optimization problems, and great improvements have been made in this aspect. However, the quantification of the un‐ certainty itself has not been addressed. Values for the variance and correlation of nodal de‐ mands are always assumed and no attention is being paid in properly quantifying these parameters. The optimization problems could be significantly improved if more realistic val‐

Water Demand Uncertainty: The Scaling Laws Approach

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

109

This work addresses the need to understand in which measure the statistical parameters de‐ pend on the number of aggregated users and on the temporal resolution in which they are estimated. It intends to describe these dependencies through scaling laws, in order to derive the statistical properties of the total demand of a group of users from the features (mean, variance and correlation) of the demand process of a single-user. Being part of the first au‐ thor's PhD research, which aims the development of descriptive and predictive models for water demand that provide insight into peak demands, extreme events and correlations at different spatial and temporal scales, these models will, in future stages, be incorporated in decision models for design purpose or scenario evaluation. Through this approach, we hope

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‐

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‐

to develop more realistic and reliable WDS design and management solutions.

ues for the uncertainty would be taken into account.

**2. Statistical characterization of water demand**

tions, is essential.

fied mainly as residential.

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 of the variables of the problem was uncertain.

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‐ put variables, thus evaluating the potential solutions.

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 were assumed to be independent random variables following some PDF.

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

The robust design of WDS has gained popularity over the last years. Researchers have been focusing on methods and algorithms to solve the stochastic optimization problems, and great improvements have been made in this aspect. However, the quantification of the un‐ certainty itself has not been addressed. Values for the variance and correlation of nodal de‐ mands are always assumed and no attention is being paid in properly quantifying these parameters. The optimization problems could be significantly improved if more realistic val‐ ues for the uncertainty would be taken into account.

This work addresses the need to understand in which measure the statistical parameters de‐ pend on the number of aggregated users and on the temporal resolution in which they are estimated. It intends to describe these dependencies through scaling laws, in order to derive the statistical properties of the total demand of a group of users from the features (mean, variance and correlation) of the demand process of a single-user. Being part of the first au‐ thor's PhD research, which aims the development of descriptive and predictive models for water demand that provide insight into peak demands, extreme events and correlations at different spatial and temporal scales, these models will, in future stages, be incorporated in decision models for design purpose or scenario evaluation. Through this approach, we hope to develop more realistic and reliable WDS design and management solutions.
