# Comparison of Bottom-Up and Top-Down Procedures for Water Demand Reconstruction

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Top-Down Procedure

_{i}supplied at the i-th time step. The procedure is made up of two phases [26]. The first phase consists of using a stochastic [27] or non-parametric algorithm [28] to generate the total water demand time series of the area (i.e., one demand time series for each $\Delta t$ of the day). In the present work, the total demand at the generic time step of the day is sampled from a beta probability distribution with tunable bounds, which enables preserving mean, variance and skewness of the total demand time series [25]. A copula resorting is applied on the generated time series to preserve temporal correlations on the total demand at all temporal lags. These latter are derived from the measured time series through the Spearman index [29]. Specifically, a multivariate normal probability distribution, with means and standard deviations equal to 0 and 1 respectively, is used as copula [30]. The multivariate normal distribution is then used to generate time series expressing the rank cross correlations to be imposed on demand time series between users at all temporal lags.

_{h}|Q

_{h}), where Q

_{h}and q

_{h}are the random variables representing the aggregate demand in the h-th hour and the disaggregated demand in the h-th hour respectively. In the model presented in Nowak et al. [16], the conditional density function is carried out using a K-nearest neighbours (K-NN) approach applied on the basis of the observed aggregate series. Specifically, let us assume the length of the generated and measured aggregated time series respectively equal to ${n}_{d,g}\Delta {N}_{\Delta t}$ and ${n}_{d,m}\Delta {N}_{\Delta t}$, where ${n}_{d,g}$ and ${n}_{d,m}$ are the numbers of days of generated and measured time series respectively. K-nearest neighbours to each generated value of the aggregated series (${Q}_{n,h}^{gen}$, with $n=1:{n}_{d,g}$) are identified from the measured aggregate demands related to the same hour h (${Q}_{m,h}^{mea}$, with $m=1:{n}_{d,m}$). According to a heuristic approach, the optimal number K is equal to $\sqrt{{n}_{d,m}}$ [28]. However, the neighbours are computed based on the absolute value of the difference between the observed and generated aggregate values ($\Delta $). Therefore, the K values with the smallest $\Delta $ are selected. Then, after being reordered from the nearest to the farthest, the K-nearest neighbours are assigned a weight ${W}_{j}$ according to their position j in the reordered vector [28]:

#### 2.2. Bottom-Up Procedure

_{i}is finally obtained as the sum of the N values ${q}_{i}^{j}$ obtained after the copula based re-sorting.

#### 2.3. Case Studies

## 3. Results

#### 3.1. Results—Case Study 1

#### 3.2. Results—Case Study 2

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Patterns of aggregated measured hourly demand for 31 days for both Case study 1 (

**a**) and Case study 2 (

**b**).

**Figure 2.**Case study 1 (Application 1)—Comparison between measured and generated hourly demands at single user level: (

**a**) mean values $\mu $, (

**b**) standard deviation values $\sigma $, (

**c**) skewness $\gamma $, (

**d**) rank cross-correlations (black) $\rho $ and $\rho -lag0$ (gray).

**Figure 3.**Case study 1 (Application 1)—Comparison between measured and generated aggregated hourly demands: (

**a**) mean values $\mu $, (

**b**) standard deviation values $\sigma $, (

**c**) skewness $\gamma $, (

**d**) rank cross-correlations $\rho $.

**Figure 4.**Case study 1 (Application 2)—Comparison between measured and generated hourly demands at single node level: (

**a**) mean values $\mu $, (

**b**) standard deviation values $\sigma $, (

**c**) skewness $\gamma $, (

**d**) rank cross-correlations (black) $\rho $ and $\rho -lag0$ (gray).

**Figure 5.**Case study 1 (Application 2)—Comparison between measured and generated aggregated hourly demands: (

**a**) mean values $\mu $, (

**b**) standard deviation values $\sigma $, (

**c**) skewness $\gamma $, (

**d**) rank cross-correlations $\rho $.

**Figure 6.**Case study 2 (Application 2) - Daily temporal patterns of both mean $\mu $ (continuous lines) and intervals $\mu $ ± 0.5$\sigma $ (dotted lines) for measured (black lines) and generated (grey lines) aggregated demands: the generated demands were obtained applying the top-down approach (

**a**) and the bottom-up approach (

**b**) respectively.

**Table 1.**Comparison of mean values $\mu $, standard deviation values $\sigma $, skewness $\gamma $, rank cross-correlations $\rho $ and $\rho -lag0$ of measured and generated hourly demands, evaluating the fit in terms of R

^{2}at both single and aggregated scales, for both applications to the first case study and for both the top-down and the bottom-up procedures.

Application—Procedure | Single Demand μ | Single Demand σ | Single Demand γ | Single Demand ρ | Single Demand $\mathit{\rho}-\mathit{l}\mathit{a}\mathit{g}0$ | Aggregated Demand μ | Aggregated Demand σ | Aggregated Demand γ | Aggregated Demand ρ |
---|---|---|---|---|---|---|---|---|---|

Application 1—Top-down | 1 | 0.95 | 0.96 | 0 | 0.89 | 1 | 1 | 0.94 | 1 |

Application 2—Top-down | 1 | 0.94 | 0.95 | 0 | 0.85 | 1 | 1 | 0.94 | 1 |

Application 1—Bottom-up | 1 | 1 | 0.96 | 1 | - | 1 | 1 | 0.59 | 0.37 |

Application 2—Bottom-up | 1 | 1 | 0.79 | 1 | - | 1 | 0.92 | 0.72 | 0.69 |

**Table 2.**Comparison of mean values $\mu $, standard deviation values $\sigma $, skewness $\gamma $, rank cross-correlations $\rho $ and $\rho -lag0$ of measured and generated hourly demands, evaluating the fit in terms of R

^{2}at both single and aggregated scales, for each application of top-down procedure to Case study 2.

Application | Single Demand μ | Single Demand σ | Single Demand γ | Single Demand ρ | Single Demand $\mathit{\rho}-\mathit{l}\mathit{a}\mathit{g}0$ | Aggregated Demand μ | Aggregated Demand σ | Aggregated Demand γ | Aggregated Demand ρ |
---|---|---|---|---|---|---|---|---|---|

Application 1 | 1 | 0.99 | 0.94 | 0.54 | 0.92 | 1 | 1 | 0.98 | 1 |

Application 2 | 1 | 0.98 | 0.92 | 0.00 | 0.90 | 1 | 1 | 0.98 | 1 |

Application 3 | 1 | 0.97 | 0.92 | 0.00 | 0.88 | 1 | 1 | 0.98 | 1 |

**Table 3.**Comparison of mean values $\mu $, standard deviation values $\sigma $, skewness $\gamma $, rank cross-correlations $\rho $ and $\rho -lag0$ of measured and generated hourly demands, evaluating the fit in terms of R

^{2}at both single and aggregated scales, for each application of bottom-up procedure to Case study 2.

Application | Single Demand μ | Single Demand σ | Single Demand γ | Single Demand ρ | Aggregated Demand μ | Aggregated Demand σ | Aggregated Demand γ | Aggregated Demand ρ |
---|---|---|---|---|---|---|---|---|

Application 1 | 1 | 1 | 0.90 | 1 | 1 | 0.97 | 0.64 | 0.96 |

Application 2 | 1 | 1 | 0.88 | 1 | 1 | 0.98 | 0.40 | 0.94 |

Application 3 | 1 | 1 | 0.90 | 1 | 1 | 0.98 | 0.34 | 0.93 |

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**MDPI and ACS Style**

Fiorillo, D.; Creaco, E.; De Paola, F.; Giugni, M. Comparison of Bottom-Up and Top-Down Procedures for Water Demand Reconstruction. *Water* **2020**, *12*, 922.
https://doi.org/10.3390/w12030922

**AMA Style**

Fiorillo D, Creaco E, De Paola F, Giugni M. Comparison of Bottom-Up and Top-Down Procedures for Water Demand Reconstruction. *Water*. 2020; 12(3):922.
https://doi.org/10.3390/w12030922

**Chicago/Turabian Style**

Fiorillo, Diana, Enrico Creaco, Francesco De Paola, and Maurizio Giugni. 2020. "Comparison of Bottom-Up and Top-Down Procedures for Water Demand Reconstruction" *Water* 12, no. 3: 922.
https://doi.org/10.3390/w12030922