# Simulating Marginal and Dependence Behaviour of Water Demand Processes at Any Fine Time Scale

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## Abstract

**:**

## 1. Introduction

## 2. Methodology and Key Components

#### 2.1. Modelling Rationale and Literature Review

#### 2.2. The bivariate Nataf Distribution Model

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Lemma**

**4.**

#### 2.3. Establishing the Target-Equivalent Correlation Relationship

#### 2.4. Modelling the Marginal Behaviour of Water Demand

#### 2.5. Demonstrating NDM Approach

#### 2.6. Moving from Random Variables to Stochastic Processes

#### 2.7. Modelling the Dependence Behaviour of Water Demand

#### 2.8. Summary of the Modelling Approach Step-by-Step

- Select and fit the suitable marginal distributions (i.e., continuous, mixed; see Section 2.4) and autocorrelation function (see Section 2.7) that better capture the distributional properties and autocorrelation structures, respectively, of the observed time series.
- Given the target autocorrelation structure ${\rho}_{\tau}$ (derive either from a theoretical autocorrelation model or as it is estimated directly from the data), establish the correlation transformation function and estimate the equivalent correlation coefficient ${\tilde{\rho}}_{\tau}$ up to the maximum specified lag $\tau $ (see Section 2.3).
- Identify the suitable auxiliary Gaussian linear stochastic model (e.g., AR($p$) and PAR($p$); see Section 2.6) and fit it on the basis of ${\tilde{\rho}}_{\tau}$.
- Generate a realization ${z}_{t}$ of the auxiliary process ${Z}_{t}$ at the Gaussian domain.
- Use the ICDF of the fitted distribution to map the synthetic time series ${z}_{t}$ to the actual domain (i.e., Equation (1)) and obtain the final realization ${x}_{t}$ of the target process ${X}_{t}$.

## 3. Case Studies

_{max}(L/min) or Q

_{max}(L/15-min); and (3) the maximum hourly demand, Q

_{max}(L/h), which is the highest hourly consumption of the day.

#### 3.1. Simulation of 1-min Water Demand

_{max}(L/min), as well as maximum hourly demand per day, Q

_{max}(L/h), produced by the model are in high agreement with the observed one, overestimating slightly the amounts with very low probability of exceedance.

#### 3.2. Simulation of 15-min Water Demand

_{max}(L/15-min), and the cdf of maximum hourly demand per day, Q

_{max}(L/h), respectively.

#### 3.3. Simulation of 1-h Water Demand

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Distribution | Probability Density Function |
---|---|

Gamma (G) | ${f}_{G}\left(x\right)=\frac{1}{\beta \Gamma \left(\gamma \right)}{\left(\frac{x}{\beta}\right)}^{\gamma -1}exp\left(-\frac{x}{\beta}\right),\text{}x0$ where $\beta >0$ is the scale parameter and $\gamma >0$ the shape parameter. |

Weibull (W) | ${f}_{W}\left(x\right)=\frac{\gamma}{\beta}{\left(\frac{x}{\beta}\right)}^{\gamma -1}exp\left(-{\left(\frac{x}{\beta}\right)}^{\gamma}\right),\text{}x0$ where $\beta >0$ is the scale parameter and $\gamma >0$ the shape parameter. |

Lognormal (LN) | ${f}_{LN}\left(x\right)=\frac{1}{\sqrt{\pi}\gamma x}exp\left(-l{n}^{2}{\left(\frac{x}{\beta}\right)}^{1/\gamma}\right),\text{}x0$ where $\beta >0$ is the scale parameter and $\gamma >0$ the shape parameter. |

Generalised Gamma (GG) | ${f}_{GG}\left(x\right)=\frac{{\gamma}_{2}}{\beta \Gamma \left({\gamma}_{1}/{\gamma}_{2}\right)}{\left(\frac{x}{\beta}\right)}^{{\gamma}_{1}-1}exp\left(-{\left(\frac{x}{\beta}\right)}^{{\gamma}_{2}}\right),\text{}x0$ where $\beta >0$ is the scale parameter, while ${\gamma}_{1}>0$ and ${\gamma}_{2}>0$ are the shapes parameters that control the behaviour of the left and right tail, respectively. |

Generalised Pareto (GPD) | ${f}_{GPD}\left(x\right)=\frac{1}{\beta}{\left(1+\frac{\gamma \left(x-c\right)}{\beta}\right)}^{\left(-1/\gamma -1\right)},\text{}xc$ where $\beta >0$ is the scale parameter, $\gamma $ the shape parameter and $c$ the location parameter (threshold). |

## Appendix B

**Figure A1.**Observed, theoretical and simulated empirical probability plots (Weibull plotting positions and in logarithmic scale) of the nonzero 1-min water demand for each hour of the day. Each plot displays the parameters of the best fitted distribution as well as the observed ${p}_{ND}$ and simulated ${\widehat{p}}_{ND}$ probability of no demand.

**Figure A2.**Observed, theoretical and simulated autocorrelation functions of 1-min water demand for each hour of the day. Each graph also displays the parameters of the fitted CAS.

**Figure A3.**Observed, theoretical and simulated empirical probability plots (Weibull plotting positions and in logarithmic scale) of the nonzero 15-min water demand for each hour of the day. Each plot displays the parameters of the best fitted distribution as well as the observed ${p}_{ND}$ and simulated ${\widehat{p}}_{ND}$ probability of no demand.

**Figure A4.**Observed, theoretical and simulated autocorrelation functions of 15 min water demand for each hour of the day. Each graph also displays the parameters of the fitted CAS.

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**Figure 1.**Comparison between observed and theoretical cumulative distribution functions (Weibull plotting positions) of the five distributions under study for the indicative example.

**Figure 2.**Numerical example demonstrating the generation of correlated random variables according to NDM approach. (

**a**) Established relationship between target and equivalent correlations as well as scatter plots and marginal distributions of correlated variables at normal (

**b**), uniform (

**c**) and real (

**d**) domain.

**Figure 3.**Simulation of 1-min residential water demand: (

**a**) Observed 1-min data of a typical day. (

**b**) Synthetic 1-min time series of a randomly selected day. (

**c**)–(

**e**) Observed, theoretical and simulated distribution functions (Weibull plotting positions and in logarithmic scale) of the nonzero water demand values for three representative hours of the day; each graph also displays the parameters of the fitted distribution as well as the observed ${p}_{ND}$ and simulated ${\widehat{p}}_{ND}$ values of probability of no demand. (

**f**)–(

**h**) Observed, theoretical and simulated autocorrelation functions of three representative hours of the day; each graph also displays the parameters of CAS. (

**i**)–(

**k**) the established relationship between the equivalent ${\rho}_{Z}$ and target ${\rho}_{X}$ correlation for the mixed-type distribution as well as the distribution fitted on nonzero demand values.

**Figure 4.**Observed and simulated cumulative distribution functions (Weibull plotting positions and in logarithmic scale) of: (

**a**) the entire 1-min water demand series, (

**b**) total water volume per day, (

**c**) maximum nonzero 1-min demand per day, (

**d**) maximum nonzero hourly demand per day.

**Figure 5.**Simulation of 15-min residential water demand: (

**a**) Observed 15-min data of a typical day. (

**b**) Synthetic 15-min time series of a randomly selected day. (

**c**)–(

**e**) Observed, theoretical and simulated distribution functions (Weibull plotting positions and in logarithmic scale) of the nonzero water demand values for three representative hours of the day; each graph also displays the parameters of the fitted distribution as well as the observed ${p}_{ND}$ and simulated ${\widehat{p}}_{ND}$ values of probability of no demand. (

**f**)–(

**h**) Observed, theoretical and simulated autocorrelation functions of three representative hours of the day; each graph also displays the parameters of CAS. (

**i**)–(

**k**) the established relationship between the equivalent ${\rho}_{Z}$ and target ${\rho}_{X}$ correlation for the mixed-type distribution as well as the distribution fitted on nonzero demand values.

**Figure 6.**Observed and simulated cumulative distribution functions (Weibull plotting positions and in logarithmic scale) of: (

**a**) the entire 15-min water demand series, (

**b**) total water volume per day, (

**c**) maximum nonzero 15-min demand per day, (

**d**) maximum nonzero hourly demand per day.

**Figure 7.**Observed, theoretical and simulated empirical probability plots (Weibull plotting positions and in logarithmic scale) of the nonzero hourly water demand for each hour of the day. Each plot displays the parameters of the best fitted distribution as well as the observed ${p}_{ND}$ and simulated ${\widehat{p}}_{ND}$ probability of no demand.

**Figure 9.**(

**a**) Observed hourly water demand data of 10 days. (

**b**) Synthetic hourly water demand data of 10 days. (

**c**) Observed and simulated cumulative distribution functions (Weibull plotting positions and in logarithmic scale) the entire hourly water demand series. (

**d**) Observed and simulated cumulative distribution functions (Weibull plotting positions and in logarithmic scale) of total water volume per day.

**Table 1.**Comparison between observed and theoretical summary statistics of the five distributions under study for the indicative example.

Observed | G | W | LN | G-GPD | |
---|---|---|---|---|---|

Mean, μ | 1.751 | 1.751 | 1.751 | 1.751 | 1.742 |

L-variation, τ_{2} | 0.720 | 0.720 | 0.720 | 0.720 | 0.695 |

L-skewness, τ_{3} | 0.592 | 0.559 | 0.592 | 0.658 | 0.577 |

L-kurtosis, τ_{4} | 0.317 | 0.295 | 0.358 | 0.474 | 0.342 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kossieris, P.; Tsoukalas, I.; Makropoulos, C.; Savic, D.
Simulating Marginal and Dependence Behaviour of Water Demand Processes at Any Fine Time Scale. *Water* **2019**, *11*, 885.
https://doi.org/10.3390/w11050885

**AMA Style**

Kossieris P, Tsoukalas I, Makropoulos C, Savic D.
Simulating Marginal and Dependence Behaviour of Water Demand Processes at Any Fine Time Scale. *Water*. 2019; 11(5):885.
https://doi.org/10.3390/w11050885

**Chicago/Turabian Style**

Kossieris, Panagiotis, Ioannis Tsoukalas, Christos Makropoulos, and Dragan Savic.
2019. "Simulating Marginal and Dependence Behaviour of Water Demand Processes at Any Fine Time Scale" *Water* 11, no. 5: 885.
https://doi.org/10.3390/w11050885