Multiscale Stochastic Characterisation of Residential Water Demand for Sustainable Network Design

Abstract

Residential water demand exhibits marked variability over time and across users, with direct implications for the sustainable and robust design of water distribution networks. In this study, a high-resolution experimental dataset is analysed to characterise the statistical structure of hourly consumption, deriving mean and variance scaling laws, cross-correlations between user groups, and probability density functions (PDFs) of aggregated demand. The results show that demand does not behave as an independent process. During the morning peak (07:00–08:00), the distribution does not converge to a unimodal shape as aggregation increases, but exhibits a clear bimodality for aggregation levels larger than approximately N ≈ 400 users. This behaviour indicates the presence of two synchronised consumption regimes and a non-negligible average correlation. In contrast, during the evening and night slots, unimodal distributions (Gamma or Lognormal) emerge, consistent with largely independent contributions and limited synchronisation. For comparison, a simplified Poisson Rectangular Pulse (PRP) model is evaluated. While this model reproduces the mean flow rate, it does not capture the observed variance, underscoring the need for models that account for heterogeneity and user correlations. The scaling laws, correlations, and empirical PDFs derived in this study provide a quantitative basis for generating probabilistic demand scenarios, supporting the sustainable, resilient, and robust design of water distribution networks.

1. Introduction

Water demand is one of the most critical variables in the design, management, and planning of Water Distribution Networks (WDNs). It directly determines pressure values, flow rates, and energy efficiency, affecting both the infrastructure and operational strategies. Inappropriate water demand modelling can lead to significant errors in hydraulic simulation, loss calculation, and service reliability assessment [1,2]. Therefore, correct demand modelling is essential for sustainable water management capable of ensuring security of supply, energy efficiency, and resilience to climate and socio-economic changes.

In urban areas, the interaction of climatic, demographic, technological, and behavioural factors determines the complexity of water demand [3,4]. The residential component represents the largest share of total water demand (60–70%) [5] and is also the most variable and difficult to model [6]. It is shaped by individual habits, household composition, technological infrastructure, and the socio-cultural context, manifesting as an intermittent and impulsive behaviour characterised by brief consumption events alternating with periods of inactivity. The aggregation of such events at larger time scales generates the variability observed at the district or WDN level [7].

The intermittent nature of the phenomenon has driven research towards stochastic modelling approaches capable of representing demand as a random rather than a deterministic process. Models based on average values or peak coefficients remain widespread in design practice; however, they cannot capture the real temporal structure of consumption and tend to underestimate variability and local peaks [8]. Conversely, probabilistic approaches reproduce the statistical distribution of consumption durations, intensities, and frequencies, improving the physical coherence of simulations and the quality of risk analyses [7,8].

Among the fundamental contributions to stochastic demand modelling are the works of Buchberger and Wu [9], who introduced the Poisson Rectangular Pulse (PRP) model, and Buchberger and Wells [10], who later refined it. This approach represents demand as a sequence of random elementary pulses, each characterised by duration and intensity distributed according to probabilistic laws. This formalism enables high-resolution demand simulation and the generation of synthetic demand time series consistent with the physics of the phenomenon.

A later development came with the SIMDEUM model (SIMulation of Demand, an End-Use Model) by Blokker et al. [11], which introduced a bottom-up approach based on water end-uses. This approach integrates user behaviour with the technical characteristics of equipment, allowing the distinction between consumption types and the evaluation of efficiency policies. SIMDEUM represents one of the first integrated applications combining statistical and behavioural modelling. In parallel, several studies have highlighted the presence of long-range correlations and nonlinear aggregation effects [12,13,14,15,16]. These findings indicate that consumption variability does not scale proportionally with the number of aggregated users but follows power law relationships typical of multifractal processes. Further developments, such as those by Di Palma et al. [17], introduced the Overall Pulse stochastic model to analyse user aggregation and parameter calibration across groups of consumers.

A growing body of research has also investigated the empirical distributions of water demand at different spatial aggregation levels. Studies such as those by Gargano et al. [18] and Kossieris and Makropoulos [19] revealed that marginal demand distributions typically consist of a discrete component (zero demand) and a continuous one (often Gamma, Weibull, or Lognormal). These works showed that asymmetry, tail behaviour, and intermittency strongly depend on the aggregation level, although the link between observed statistical features (PDFs, CDFs) and the physical parameters of stochastic models remains only partially understood.

Recent contributions, such as Zhang et al. [20], revisited the PRP from an uncertainty quantification perspective, showing that parameter estimation reliability depends on both the number of users and data density. While confirming the PRP’s capability to reproduce realistic urban demand profiles, these studies also highlighted its limitations in representing heterogeneous or correlated users, suggesting the need for broader investigations into the statistical origin of demand variability.

Given this context, the present study focuses on the statistical characterisation of residential water demand as a function of spatial aggregation, analysing how the shape of the probability distribution (PDF) and related statistical descriptors evolve with the number of aggregated users. The term “multiscale” refers to the analysis of residential water demand at different levels of spatial aggregation based on the number of users. Aggregation affects the scaling of variance, correlations, and empirical distributions.

The Poisson Rectangular Pulse (PRP) model is employed as an exploratory tool to assess its ability to reproduce multiscale behaviours and to interpret the relationship between stochastic parameters and the empirically observed variability.

This work thus contributes to bridging the gap between data-driven statistical analysis and stochastic process representation, providing a foundation for integrating demand uncertainty and scale effects into the sustainable design of water distribution systems.

Based on the above, this article is organised as follows:

• Section 2 describes the data sources and the processing and analysis procedures used.
• Section 3 illustrates the results obtained, with particular reference to the scaling analysis and probability distributions corresponding to the different levels of aggregation, and discusses the emergence of bimodality. A comparison between the observed empirical distributions (PDF) and the simulations based on the PRP model is also shown.
• Section 4 discusses the implications of the observed bimodal behaviour for the generation of demand scenarios and for the robust and sustainable design of water distribution networks, through an application example.

Finally, Section 5 presents the concluding remarks, focusing on the key implications of the study for the sustainable design of WDNs.

2. Materials and Methods

2.1. Hourly Water Demand and Spatial Aggregation Analysis

The experimental investigation was carried out using water consumption data recorded on a sample of 82 residential users in an IACP (Istituto Autonomo Case Popolari) building located in the city of Latina (Italy), monitored by high-time-resolution electronic meters. The data acquisition and validation methods are described in detail in the works of Guercio et al. [12] and Magini et al. [14]. The measured data were appropriately processed to ensure a constant time step of 60 s, thus providing high-frequency values of the instantaneous demand of each user.

The dataset considers only weekdays during the winter period, from 15 January to 11 March, encompassing 39 monitored weekdays. Weekends were excluded to reduce variability associated with occasional household behaviours. For each user and each weekday, the time series corresponding to each hourly interval is treated as an independent realisation of the same stochastic process, represented by 60 consecutive 1-min samples. In total, 3127 hourly realisations were retained after data quality control, out of a theoretical maximum of 3198 (82 × 39), with 71 hourly series discarded due to missing or corrupted data.

Processing the 24-h data enabled the reconstruction of the average hourly profile for all users, including zero consumption during the night. Total hourly consumption was calculated as the integrated demand over each hourly window, obtained by summing the average minute volumes across all users and all minutes of the hour, Figure 1.

The analysis of the dataset indicates that the daily peak demand occurs between 7:00 and 8:00 am, with an overall average value of approximately 88.9 L/h, while the nighttime minimum is recorded between 2:00 and 4:00 am. These time slots are therefore selected as representative of peak and minimum demand conditions for the subsequent analyses. During the peak period, the main household activities are typically concentrated on personal hygiene, meal preparation, and cleaning.

To analyse the effects of spatial aggregation, individual demands were combined into groups of increasing size (N = 5, 10, 20, 50, 100, 150, 200, 250, …, 1200), simulating the behaviour of larger sets of users. This allowed us to investigate aggregation-dependent changes in the main statistical properties. This approach, already adopted by Magini et al. [14] and Vertommen et al. [21], also enables the investigation of the transition of these quantities from discrete and intermittent regimes to continuous and aggregated regimes.

Although the primary focus was on the daily peak hour, 7:00–8:00 am, the study was extended to the following additional representative time slots:

• A nighttime slot (3:00–4:00 am), where average consumption is minimal and isolated uses or network losses prevail;
• Two daytime slots (2:00–3:00 pm and 5:00–6:00 pm), the first corresponding to the midday peak and the second to the afternoon minimum, which is characterised by a withdrawal level close to the daily average;
• An evening slot (7:00–8:00 pm), approaching the highest evening consumption.

These time slots represent the typical consumption patterns associated with different user behaviours.

Starting from these time slots, the analysis focused on identifying the scaling laws that describe the change in variance as the size of the user group increases and on studying the cross-correlations between different levels of aggregation. These indicators allow us to quantify the degree of dependence between consumer behaviours, highlighting the presence of phenomena of synchronisation or statistical independence.

In parallel, the empirical probability distributions (PDFs) of demand were analysed to assess how the distribution shape evolves with increasing spatial aggregation.

The insights derived from these analyses, i.e., scaling laws, correlations, and distributions, highlight the complexity and multiscale nature of residential water consumption, providing an empirical basis for the development of stochastic models that consistently represent the uncertainty and aggregation dynamics of water demand.

2.2. Variance Scaling Law

Water demand data have been analysed at different spatial aggregation scales to evaluate how the variance of the aggregate flow rate increases with the number of co-occurring users, N. In accordance with the formulation proposed by Magini et al. [14], the empirical relationship is expressed in the form:

$${\sigma }_{\mathit{agg}}^2(N)={\sigma }_1^2 N^{\alpha }$$

(1)

where ${\sigma }_{\mathit{agg}}^2(N)$ represents the variance of the aggregate signal of N users, ${\sigma }_1^2$ represents the variance of the individual user, and the exponent α quantifies the degree of statistical dependence between the elementary demand signals (α = 1 for complete independence, α = 2 for perfect correlation).

This law describes how overall variability increases as more users are aggregated, reflecting the correlation structure linking individual behaviours.

In this study, for each representative time slot defined in Section 2.1, spatial aggregation was simulated by randomly combining independent realisations drawn from the set of 3127 hourly time series. Each realisation corresponds to a 60-min demand time series with a 1-min resolution, associated with a specific user on a specific weekday. Realisations are not obtained by concatenating data across days, and no artificial users are generated. Aggregation is, therefore, performed on a set of independent user/day time profiles.

2.3. Analytical Relationship Between Variance Scaling and Correlations

If we consider the aggregated demands of N users, having the same variance ${\sigma }_0^2$ and sharing the same mean pairwise correlation ${\rho }_0$, the following relation holds [22,23]:

$${\sigma }_{\mathit{agg}}^2(N)={\sigma }_0^2\left[N+N\left(N-1\right){\rho }_0\right]$$

(2)

or equivalently

$${\displaystyle \frac{{\sigma }_{\mathit{agg}}^2(N)}{N}}={\sigma }_0^2[1+(N-1){\rho }_0]$$

(3)

Equation (3) is linear in N.

Then, performing a linear regression of $Y\left(N\right)={\displaystyle \frac{{\sigma }_{\mathit{agg}}^2(N)}{N}}$ versus N:

$$Y(N)=b_0+b_{1}N$$

(4)

Furnishes an empirical estimation of the basic parameters:

$$\begin{array}{c}{\sigma }_0^2=b_0+b_1, {\rho }_0={\displaystyle \frac{b_1}{b_0+b_1}}.\end{array}$$

(5)

Thus, the variance scaling law immediately yields an estimate of the average pairwise correlation among users. Here, ${\rho }_0$ represents the correlation parameter inferred from the variance scaling law under the assumption of homogeneous pairwise dependence.

Inverting Equation (3) gives an effective mean pairwise correlation among the N(N − 1)/2 user pairs within the group:

$${\rho }_{eff}(N)=\left({\displaystyle \frac{{\sigma }_{agg}^2}{{N\sigma }_0^2}}-1\right)/\left(N-1\right)$$

(6)

This quantity coincides with the average pairwise correlation among all user pairs within the same group and, therefore, is directly comparable with the empirical mean correlation coefficient, ${\rho }_{mean}$, estimated from the data.

Now, assuming two disjoint groups of users of size $N_1$ and$N_2,$respectively, the expected correlation between their aggregate demands$S_{N_1}$ and $S_{N_2}$is [21]:

$$\rho (N_1,N_2)={\rho }_0 {\displaystyle \frac{N_1{N}_2}{\sqrt{N_1[1+(N_1-1){\rho }_0] N_2[1+(N_2-1){\rho }_0]}}}$$

(7)

This equation analytically links the one-dimensional scaling law of Section 2.2 to the two-dimensional correlation structure represented by the corresponding correlation abaci (see Section 3).

Equations (6) and (7) describe two complementary aspects: i.e., ρ_eff(N) is the average pairwise correlation within a single group, ρ(N₁, N₂) is the correlation between two aggregated groups. When N₁ = N₂ = N:

$$\rho (N,N)={\rho }_0 {\displaystyle \frac{N}{1+(N-1){\rho }_0}}$$

(8)

2.4. The Poisson Rectangular Pulse (PRP) Model

After defining the empirical and theoretical framework used to characterise aggregation effects, variance scaling, and correlations, a stochastic demand generator is introduced to assess whether these characteristics can be reproduced.

The Poisson Rectangular Pulse (PRP) model is therefore used here as a reference stochastic framework to interpret the observed behaviours and assess the consistency between empirical and simulated data. It represents instantaneous water demand as the superposition of random rectangular pulses generated by a homogeneous Poisson process, characterised by an average arrival rate of pulses (events) λ (s⁻¹). Each pulse is described by two independent random variables: duration D (s) and intensity H (L s⁻¹), generally modelled using Gamma or Lognormal distributions [9,12].

The flow process can be expressed as:

$$q(t)=\sum _{i=1}^{N(t)}{H}_i {\delta }_i(t)$$

where N(t) is the number of pulses at time t, H_i represents the intensity of the i-th pulse and δ_i(t) is an indicator function defined as:

$${\delta }_i(t)=\{\begin{array}{cc}1,& \mathrm{if} T_i\le t<T_i+D_i\\ 0,& \mathrm{otherwise}\end{array}$$

The average flow rate of the process is therefore equal to:

$$\overline{q}=\lambda DH$$

The statistical behaviour of the signal depends on its degree of intermittency and the overlap of the pulses, both summarised by the dimensionless parameter:

$$\rho =\lambda D$$

which represents the average fraction of time in which multiple consumption events overlap. Values of ρ < 1 correspond to an intermittent regime, dominated by periods of inactivity, while for ρ > 1, demand tends toward a quasi-continuous regime, characterised by an approximately Gaussian distribution. This framework, introduced by Buchberger and Wu [9] and subsequently extended by Guercio et al. [12], forms the basis for the stochastic and multiscale representation of water demand.

The model parameters, λ, D, and H, were estimated from the experimental data using a two-step iterative procedure. In the first step, the method of moments equated the first- and second-order statistics (mean, variance, and autocorrelation) between observed and simulated data [24,25]. In the second step, a distribution-based method minimised the first-order Wasserstein distance between the empirical and simulated distributions, considering both the discrete (zero consumption) and continuous components [26,27].

The optimisation was implemented in MATLAB (R2024b) using a combination of the Pattern Search (global search) and Fmincon (local refinement) functions. To ensure the physical consistency of the model, positivity constraints were imposed on the parameters as well as on the respect of the mean flow rate.

3. Results

3.1. Preliminary Analysis of Hourly Demand Profiles (1-Min Resolution) Data

For each hourly time slot examined, a preliminary statistical analysis was conducted on the 3127 independent instantaneous water consumption realisations (series of 60 samples at 1-min intervals). Independence is assumed across hourly realisations, while temporal autocorrelation at the 1-min scale is an intrinsic property of the demand process and characterises each realisation.

Table 1. Hourly water demand statistics: mean μ, variance σ², coefficient of variation CV, and intermittency p₀.

Hour (hh)	Mean (L·min−1)	σ2 ((L·min−1)2)	CV (-)	p0 (-)
03–04 am	0.05	0.11	6.64	0.94
07–08 am	0.19	0.73	4.438	0.85
02–03 pm	0.29	1.16	3.74	0.8
05–06 pm	0.25	0.98	3.877	0.81
07–08 pm	0.31	1.35	3.784	0.79

summarises the hourly statistics for the five time slots considered (3:00 am–4:00 am, 7:00 am–8:00 am, 2:00 pm–3:00 pm, 5:00 pm–6:00 pm, 7:00 pm–8:00 pm). Intermittency is quantified through the parameter p₀, defined as the empirical probability of zero demand at 1-min resolution. As expected, the nighttime slot has the lowest average consumption (Mean = 0.05 L min⁻¹) and the highest intermittency (p₀ = 0.94, CV = 6.640), consistent with a regime dominated by quiet states and sporadic demands. As household activity increases, daytime and evening time slots show higher mean values and a reduction in intermittency; among these, the evening time slot is the most intense, while the afternoon ones assume intermediate values. The 7:00 am–8:00 am time window shows an average consumption of 0.19 L min⁻¹ with p₀ = 0.85 and CV = 4.438, reflecting a still markedly intermittent profile, but with significant signs of synchronisation as shown below.

The Pearson cross-correlation coefficients between pairs of realisations, reported in

Table 2. Statistics of the cross-correlation coefficients ρ in each hourly time slot: mean $\overline{\rho }$, standard deviation ρ_std, minimum value ρ_min, maximum ρ_max, ρ_−95% and ρ_+95% extreme of 95% confidence interval.

Hour (hh)	ρmean (-)	ρstd (-)	ρmin (-)	ρmax (-)	ρ−95% (-)	ρ+95% (-)
03–04 am	0	0.195	−0.76	0.98	−0.049	0.057
07–08 am	0.032	0.24	−0.447	0.999	0.028	0.071
02–03 pm	0.005	0.202	−0.515	0.998	−0.002	0.028
05–06 pm	−0.001	0.204	−0.557	0.997	−0.012	0.024
07–08 pm	0.006	0.21	−0.659	1	−0.001	0.029

, calculated over 60-min windows, are mostly centred around zero. The (7:00–8:00 am) time slot is the only one with a significantly positive $\overline{\rho }$ value, indicating a modest synchrony between time profiles, probably linked to common morning routine activities. The other time slots show $\overline{\rho }$ value~0, sometimes even slightly negative, confirming that, over the 60-min time slot, the synchrony of the peaks is weak. As expected for intermittent series, the minimum and maximum extreme values reflect random misalignments and occasional similarities between profiles, without indicating true physical anticorrelation/correlation. The 95% confidence intervals on $\overline{\rho }$ confirm the low average coherence of the different realisations, except for the morning time slot, where it remains modest, but not negligible.

3.2. Scaling Laws

Demand data were analysed across different spatial aggregation scales. Variance scaling was evaluated according to Equation (1) following the aggregation procedure described in Section 2.2. For each N, results were averaged over repeated random extractions. The following aggregation levels were considered:

$$N=\{5, 10, 20, 50, 75, 100, 125, 150, 200, 250, 300, 400, 500, 750, 1000, 1200\}$$

For each value of N, synthetic aggregations were generated by summing, minute by minute, N demands, without repetition within the group. The aggregate variance was computed as the population variance of the resulting signal and then averaged over 800 independent extractions to reduce stochastic variability. The scaling exponent α was estimated through ordinary least squares (OLS) regression applied to the log–log scaled data, including the 95% confidence interval and the coefficient of determination R².

The results are reported in

Table 3. Scaling exponent α estimated for different representative hourly time slots. For each period, the table reports the fitted exponent α, the lower and upper bounds of the 95% confidence interval (α_−95% and α_+95%), and the coefficient of determination R².

Hour (hh)	σ12 ((L·min−1)2)	α (-)	α−95% (-)	α+95% (-)	R2 (-)
03–04 am	0.11	0.986	0.97	1.002	0.999
07–08 am	0.731	1.355	1.265	1.445	0.987
02–03 pm	1.186	1.079	1.051	1.108	0.998
05–06 pm	0.977	0.997	0.99	1.004	1
07–08 pm	1.346	1	0.993	1.006	1

, where ${\sigma }_1^2$ indicates the variance of the single user and α is the estimated scaling exponent. During the night (3:00–4:00), α ≈ 1.05 and β ≈ 0.95, indicating nearly independent behaviour among households, consistent with the low probability of event overlap. In the morning peak time slot (7:00–8:00), α increases to 1.36 (β ≈ 0.64), highlighting greater statistical synchronisation due to the temporal concentration of household activities. During the daytime and evening hours (14:00–20:00), α approaches 1 (β ≈ 1), indicating the prevalence of independent behaviour and a progressive attenuation of variability.

Figure 2 shows the scaling law in detail for the 7:00–8:00 am time slot, while Figure 3 reports the local slope τ(N) of the variance scaling law estimated using rolling OLS regression:

$$\tau \left(N\right)={\displaystyle \frac{d\left(log{\sigma }^2\right)}{d\left(logN\right)}}$$

(9)

Local slope analysis is introduced to further explore the scale-dependent behaviour of aggregation. Unlike a single global exponent, the local slope allows us to identify transitions between independent and synchronised regimes as the number of aggregated users increases. In this time range, τ(N) increases from values close to 1 up to about 1.8 for aggregations of 200–400 users, indicating that the correlation between the elementary series increases with the aggregation scale rather than decreases.

This suggests the presence of an emergent synchronisation effect, in which the probability of temporal coincidence of consumption events increases as the number of users considered increases.

On the contrary, in the other daytime and evening time slots, the estimated scaling exponent remains close to 1 without systematic variation, indicating the absence of significant correlations and the prevalence of independent behaviour (Figure 4 and Figure 5).

Overall, the scale analysis confirms the daily transition from an independent stochastic regime (typical of night and evening hours) to a partially correlated one during peak hours, reflecting the cyclical evolution of use behaviours and the multiscale structure of residential water demand.

3.3. Correlation Analysis

Correlation between users is a key indicator of the statistical consistency of water demand and allows us to quantify the extent to which individual consumption tends to synchronise over time.

To analyse this structure, cross-correlations between hourly demand profile series were calculated, considering disjoint groups of users of equal size, N.

For each group, the Pearson correlation coefficient ${\rho }_{ij}$ was estimated between all pairs of users, and the mean, minimum, and maximum values were subsequently obtained (${\rho }_{\mathit{mean}}$, ${\rho }_{\mathit{min}}$, ${\rho }_{\mathit{max}}$).

The results are summarised in the correlation abaci, which show the evolution of these indicators as the number of aggregate users increases. The abaci describe the consistency between groups of different sizes $(N_1,N_2)$, highlighting how the correlations between individual series vary significantly over the course of the day (Figure 6).

During the night hours (03:00–04:00 am), most users show mean ${\rho }_{\mathit{mean}}$ values close to zero, indicating statistically independent behaviour. In this time slot for the largest aggregations, ${\rho }_{\mathit{mean}}$does not reach 0.010. Even in the afternoon, these values remain low, also when the number of users in the pairs increases. Only during the morning peak hour, the mean correlation ${\rho }_{\mathit{mean}}$significantly increases, approaching values larger than 0.8 for large aggregates.

The correlation abacus can be interpreted as a two-dimensional graphical representation of the variance-scaling law, described in Section 3.2.

In Figure 7 and Figure 8, model-based ${\rho }_{eff}\left(N\right)$and data-based ${\rho }_{mean}(N)$ are compared in the morning and afternoon time slots. During the morning peak hour, these values increase with N and approach 1, indicating that aggregate quantities become more correlated than those of individual users. A further observation is that the parameter obtained from the regression (Equation (5)) is not independent of the empirical correlations measured in Section 3.1. Indeed, these quantities are directly comparable.

In detail, in the (7:00–8:00) am time range, the empirical mean cross-correlation increases to ≈0.3, while ${\rho }_{\mathit{eff}}\left(N\right)$ reaches values between 0.25 and 0.32 for N between 200 and 400, i.e., in the same scale in which the exponent α shows its maximum increase. In the evening hours, 7:00–8:00 pm, both values remain close to zero, highlighting almost independent consumption.

Figure 7 and Figure 8 compare the empirical and theoretical cross-correlation curves for the 7:00–8:00 am and 7:00–8:00 pm time slots. At the same time,

Table 4. Comparison between the empirical ρ_mean and theoretical ρ_eff cross-correlation coefficients for the different time slots. The values of ρ_eff are derived from the scaling law as given by Equation (6).

Hour (hh)	ρmean Empirical (-)	ρstd Empirical (-)	ρeff (-)
03–04 am	0.02	0.05	0.01–0.03
07–08 am	0.3	0.09	0.25–0.32
02–03 pm	0.12	0.07	0.10–0.15
05–06 pm	0.08	0.06	0.05–0.10
07–08 pm	0.04	0.05	0.00–0.05

summarises the average values estimated by both approaches for all time slots considered.

Overall, the results confirm that both scaling laws and experimental abaci can adequately describe the demand correlation structure.

3.4. Analysis of Empirical PDFs

The analysis of the empirical distributions of aggregate consumption further confirms the results of the scaling laws and cross-correlations.

For each representative time slot (03:00–04:00 am, 07:00–08:00 am, 02:00–03:00 pm, 05:00–06:00 pm, and 07:00–08:00 pm), the probability density functions (PDFs) of aggregated water demand were estimated and compared against several candidate distributions: Exponential, Gamma, Weibull, Lognormal, and, when required to capture bimodality, a two-component Lognormal Mixture model.

The optimal model was selected using the Bayesian Information Criterion, BIC [28,29].

The results,

Table 5. Prevalent PDF type for each time slot.

Hour (hh)	Small Aggregates (N ≤ 50)	Medium/Large Aggregates (N > 50)
3–4 am	Lognormal/Exponential	Gamma
7–8 am	Lognormal/Weibull/Gamma	Gamma/Lognormal Mixture (2) from N ≈ 400
2–3 pm	Lognormal/Weibull/Gamma	Gamma
5–6 pm	Lognormal/Weibull/Gamma	Gamma
7–8 pm	Lognormal/Weibull/Gamma	Gamma

and Figure 9 and Figure 10, show a clear evolution in the distribution’s shapes with the aggregation scale.

During nighttime hours (3:00–4:00 am), consumption is rare and small: the PDF exhibits a strong positive skew and is well represented by an Exponential or Lognormal distribution for small aggregates.

This reflects the presence of a discrete non-consumption component combined with a few supply events, often associated with users with irregular habits or automatic devices.

The prevalence of the Lognormal distribution over the Gamma distribution suggests that demand does not arise from a simple additive process of independent events, but from a multiplicative combination of factors: number, duration, and intensity of flows, that together determine the overall distribution of consumption.

The morning peak period (7:00–8:00 am) marks a transitional regime with the appearance of a clear bimodality for aggregations greater than 400 users.

The Lognormal Mixture distribution is consistent with the presence of two distinct sub-periods of demand within the hour, e.g., consumption related to personal hygiene concentrated in the first part of the time slot, and that associated with domestic activities, such as breakfast and cooking, in the second.

This double peak is consistent with the high level of synchronisation observed in Section 3.3, and represents the maximum degree of collective coherence of the day.

In the central hours, 2:00–3:00 pm, synchronisation decreases and the distributions return to unimodality, with Gamma or Lognormal shapes depending on the level of aggregation, indicating a more regular demand regime with limited variance.

In the late afternoon and evening time slots, 5:00–6:00 pm and 7:00–8:00 pm, variability increases again, but the correlation remains weak or non-existent; the PDFs remain unimodal and are well represented by Gamma for most scales, consistent with independent or partially anticorrelated behaviours.

Overall, the empirical distributions show a well-defined regime transition from highly asymmetric exponential or lognormal forms (rare and independent events) to bimodal shapes during peak hours, to nearly symmetric Gamma distributions for large aggregations.

This behaviour reflects the combined effect of statistical aggregation and temporal synchronisation, already highlighted in the previous paragraphs: the former tends to stabilise variability, the latter to concentrate consumption in common time windows, generating the observed multimodal structures.

Based on this empirical evidence, the next section analyses the ability of the Poisson Rectangular Pulse (PRP) model to reproduce the observed statistical behaviours, with particular attention to the transition between intermittent and aggregate regimes.

3.5. Comparison of PDFs from Empirical and Simulated Data

To evaluate the ability of the Poisson Rectangular Pulse (PRP) model, described in Section 2, to reproduce the statistical characteristics of aggregate demand, a simulation was performed during the morning peak hour. In this period, experimental data show the most evident transition towards a bimodal distribution starting from approximately N ≈ 400 aggregate users (Section 3.4).

In this application, the PRP is used assuming independent and identically distributed users, considering two distinct time windows (lags) within the hour, with parameters λ (frequency), D (duration), and H (intensity) estimated separately over 0–30 min and 30–60 min. This choice is motivated by the fact that, in the measured data, the observed bimodality arises probably from the presence of two distinct usage patterns within the same hour. The adoption of two windows, therefore, constitutes a simple attempt to introduce temporal heterogeneity.

The calibration of the λ, D, and H parameters was conducted starting from the elementary series sampled per minute, imposing consistency of the average demand per user and with durations and intensities following exponential distributions.

Owing to the temporal resolution of the data, the elementary pulses of the PRP model cannot be directly identified from the measurements. For this reason, it is not possible to apply traditional methods based on the recognition of individual pulses. In this study, a two-step estimation procedure was therefore adopted: (1) imposing consistency between the observed mean and variance and those generated by the model (method of moments); (2) refining the shape of the distribution using the Wasserstein distance between the PDFs.

The simulations were run with a time step of 1 s, then aggregated to 60 s to obtain the same resolution as the measured data.

The comparison was performed at the N = 400 scale, corresponding to the first clearly bimodal regime in the real data. For this aggregation, the measured data provide an average demand equal to

$${\mu }_{meas}\approx 75.6 \mathrm{L} \mathrm{m}\mathrm{i}{\mathrm{n}}^{-1},{\sigma }_{meas}^2\approx 693 (\mathrm{L} \mathrm{m}\mathrm{i}{\mathrm{n}}^{-1}{)}^2.$$

The calibrated PRP produced

$${\mu }_{sim}\approx 94.7 \mathrm{L} \mathrm{m}\mathrm{i},$$

with a relative error $\approx 25\%$, and a variance

$${\sigma }_{sim}^2\approx 135 (\mathrm{L} \mathrm{m}\mathrm{i}{\mathrm{n}}^{-1}{)}^2,$$

that is, only 20% of the observed one.

Figure 11 shows the comparison between the aggregate distributions. The measured distribution presents a marked bimodality, with a first peak at low demand values and a second peak associated with the overlap of more intense domestic uses.

The PDF obtained from the PRP model is concentrated in a narrower range of values and exhibits a bimodal profile due to the two-window modelling.

This result highlights a structural limitation of the adopted PRP formulation. Since all users generate independent, uncorrelated impulses, aggregate variability rapidly decreases, and the process converges to a nearly linear behaviour with scale—that is, the exponent α of the scaling law tends to 1. This prevents the reproduction of the observed level of variability and the correct scaling of the bimodal regime, although the model can generate a bimodal shape.

Realistic reproduction of multimodal regimes requires introducing multiple user populations with distinct parameters, or models capable of generating correlations between users. However, the approach adopted in this study provides an important framework for interpreting the role of the parameters λ, D, and H and for quantifying the limitations of impulsive models in simulating aggregate water demand in the presence of marked synchronisation.

4. Discussion

As an example that highlights the importance of the results presented in this study, the classic water distribution network proposed by Alperovitz & Shamir [30] is considered, here adopted in the 6-node configuration shown in Figure 12, assigning to each node a number of users equal to [340, 400, 280, 623, 434, 525]. On this network, instantaneous snapshots of the nodal demand were generated, preserving the correlation structure between nodes that derives from the scaling laws and is therefore dependent on the number of users per node [31]. The objective is to evaluate the effect of the choice of the marginal distributions (unimodal vs. bimodal) on the generation of demand scenarios for robust design of WDNs.

The distributions obtained show that, while keeping the nodal correlation imposed by the scaling model unchanged, the shape of the marginal distributions has a decisive influence on the representation of demand variability (Figure 13).

For nodes with high numbers of users (N ≳ 300), real aggregates exhibit a structural bimodality that unimodal models cannot capture. This leads, in the unimodal case, to distributions that are excessively concentrated and have shorter right tails; in contrast, the bimodal approach reproduces more faithfully the dispersion and the extreme-demand scenarios, as is evident in the more populated nodes (e.g., Node 6).

These differences become crucial in the context of robust design, since, as highlighted by Cunha et al. [32], the quality and representativeness of the demand scenarios directly determine the ability of multi-objective models to estimate resilience and performance in critical conditions. Ignoring bimodality, therefore, implies a systematic underestimation of the most severe events, compromising the reliability of the design solutions; on the contrary, integrating them through bimodal marginal models allows one to more realistically explore the region of high demands to support a truly robust design process.

5. Conclusions

This study proposes a multiscale analysis of instantaneous residential water demand, considering scaling laws for variance, spatial cross-correlation, and probability distributions for empirical consumption series. The possibility of using a stochastic Poisson Rectangular Pulse (PRP) simulation model is also examined. Results obtained from over 3000 independent simulations clearly show that domestic water demand exhibits strong temporal variability and scale-dependent statistical behaviour, determined by the interaction between intermittency, synchronisation, and aggregation.

On an hourly scale, demand alternates between two distinct regimes: an intermittent and weakly correlated regime, typical of nighttime and evening hours, and a partially synchronised regime that emerges during the morning peak hours. The scaling analysis shows that the variance of aggregated demands grows according to a power law with an exponent α ranging from approximately 1 (independent users) to values close to 1.4 during the morning peak, reflecting a significant increase in the concomitance of consumption events. Correlation abaci confirm this behaviour; while in most time slots the average correlation is negligible, in the time slot 7:00–8:00 am, average correlations close to 0.03 are observed for pairs with number of users between 200 and 400 users. These empirical correlations are consistent with the theoretical values obtained from scaling laws, showing that the correlation structures observed in the abaci derive analytically from the behaviour of variance as the population grows.

A consistent picture also emerges from the study of empirical PDFs. For low levels of aggregation, nighttime demand is dominated by asymmetric patterns (Exponential or Lognormal), reflecting rare and isolated consumption. During the morning peak, for large clusters (N ≥ 400), a clear bimodality appears, indicative of the presence of two distinct subperiods within the hour, dependent on the diversification of uses. During the day and evening, demand remains unimodal and well described by predominantly gamma-type distributions, consistent with the low synchronisation between users.

The comparison with the PRP model highlights the structural limitations of models that generate independent pulses. Although the PRP model is able to reproduce the mean and bimodal shape of marginal distributions when intra-hour inhomogeneity is explicitly considered, it tends to systematically underestimate the aggregate variance. This limitation stems directly from the PRP formulation: user independence pushes the scaling exponent α toward 1, preventing the model from capturing the level of variability observed under synchronised conditions during peak hours. This suggests that realistic modelling of peak-hour demand requires the use of models capable of generating inter-user dependencies, such as stochastic–behavioural approaches.

Overall, the combination of scaling laws, empirical schedules, and distribution analysis highlights the multiscale structure of residential water demand: the statistical properties of demand are not fixed, but vary both over time and with aggregation scale, and these variations can be described through variance–correlation relationships. This has direct implications for stochastic modeling, scenario generation, and the sustainable design of resilient water networks.