# Generating Scenarios of Cross-Correlated Demands for Modelling Water Distribution Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of Methodology

#### 2.1. Scaling Laws

#### 2.2. Generation of Scenarios

**Step****1.**- Create a random (S, N) dimensional matrix
**Z***, containing S Latin Hypercube Samples of size N from a standardized normal distribution, where S is the number of scenarios and N the number of the demand nodes in the WDN. For this purpose, the Matlab function lhsnorm was used. Owing to the finite size of the samples their correlation matrix**I***(Here, the asterisk is used to distinguish data and corresponding correlation matrices to be corrected) does not coincide with the identity matrix**I**, that is they are not independent. Then, the lower triangular Cholesky decomposition is applied to induce the desired correlation [27]. Specifically:$$\begin{array}{c}I=C\xb7{C}^{T}\\ {I}^{*}=E\xb7{E}^{T}\\ Z={Z}^{*}\xb7C\xb7{E}^{-1}\end{array}$$**Z**of perfectly independent S samples of size N from a standardized normal distribution is obtained. In order to obtain the Cholesky root, the Matlab function chol was used. **Step****2.**- Create a random (S, N) dimensional matrix
**G**containing S standardized normal samples with the correlation matrix**Corr**from the scaling laws for nodal demand. To this aim the desired correlation is induced in**Z**also applying the lower triangular Cholesky decomposition [27], that is:$$Corr=P\xb7{P}^{T}G=Z\xb7P\xb7{C}^{-1}$$ **Step****3.**- Transform the matrix
**G**in the (S, N) dimensional matrix**D***which complies with the desired marginal distributions at each demand ith node. Transformation is based on the inverse cumulative distribution function, CDF, of the desired marginals, F_{i}. Specifically, for each element D*_{i}of matrix D*, i.e., a non-normal random sample with the desired CDF, the following equation holds:$${D}_{i}^{*}={F}_{i}^{-1}\left(\mathsf{\Phi}\left({G}_{i}\right)\right)$$**G**and it is uniformly distributed. This procedure is known as the inverse transformation method [28]. Function $\mathsf{\Phi}\left({G}_{i}\right)$ can also be interpreted as a realization from the Gaussian copula. Applying the inverse CDF ${F}_{i}^{-1}$ to the uniform random variable $\mathsf{\Phi}\left({G}_{i}\right)$ ensures that ${D}_{i}^{*}$ is distributed according to ${\mathsf{\Phi}}_{i}$. Unfortunately, the transformation in Equation (1) is non-linear, and therefore the correlation matrix**Corr***of**D***is not equal to the desired correlation matrix**Corr**. **Step****4.**- Apply the Iman-Conover algorithm proposed by Ekström [29] in order to get a better approximation of the desired correlation matrix
**Corr**for the (S, N) matrix of nodal demand scenarios**D***. The algorithm is described in the following steps:- 4.1
- Calculate lower triangular Cholesky decomposition
**V**of**Corr**, i.e.,**Corr**=**V****∙V**^{T}. - 4.2
- Calculate lower triangular Cholesky decomposition
**Q**of**Corr***, i.e.,**Corr***=**Q****∙Q**^{T}. - 4.3
- Obtain
**T**such that**Corr**=**T****∙Corr****∙T**^{T}, can be calculated as**T**=**V****∙Q**^{−1}. - 4.4
- Obtain the matrix
**ScoreD***by rank-transforming**D**and convert to van der Waerden scores, defined as ${F}_{i}^{-1}\left(\mathsf{\Phi}\left(i/(N+1\right)\right)$ where ϕ is the CDF of the standard normal distribution, i is the assigned rank and N is the total number of samples. - 4.5
- Calculate the target scores matrix
**ScoreD**=**ScoreD***·**T**.^{T} - 4.6
- Match up the rank pairing in
**D***according to**ScoreD**, obtaining the new (S,N) dimensional matrix**D**containing the S scenarios of the N nodal demand in the WDS. The N samples are distributed according to the desired marginals and their correlation matrix is close to the correlation matrix derived from the scaling laws.

#### 2.3. Scenario Reduction

## 3. Application Example

#### 3.1. Theoretical WDN: The Apulian Network

_{1}] = 0.0043, was considered, in agreement with most of the experimental data from the case study of Latina. Table 1 also shows the number of consumption units for each water demand node. In order to highlight how the number of users per node and their relationships affect the generated demand scenarios two different frameworks were examined to which correspond respectively DemandA and DemandB column. The DemandA values are the same used by Giustolisi et al. [33] for Apulian network and the users’ number is consequent. Differently, DemandB values were defined assuming a smaller total number of users and a greater variability of the number of users per node.

#### 3.2. Generation of Demand Scenarios

#### 3.2.1. DemandA

#### 3.2.2. DemandB

#### 3.3. Reduction of Demand Scenarios

#### 3.3.1. DemandA

#### 3.3.2. DemandB

#### 3.4. Hydraulic Simulation with Scenarios from DemandA

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 7.**Reduced demand scenarios, DemandA. Reduced nodal scenarios -> green line; average nodal values -> black dotted line; scenario with max probability -> red dotted line.

**Figure 8.**Reduced demand scenarios’ probabilities, DemandA. Inside the bar is the number of the rank order of the generated scenario.

**Figure 9.**Reduced demand scenarios, DemandB. Reduced nodal scenarios -> green line; average nodal values -> black dotted line; scenario with max probability -> red dotted line.

**Figure 10.**Reduced demand scenarios’ probabilities, DemandB. Inside the bar is the number of the rank order of the generated scenario.

**Figure 12.**Reduced pressure headlines scenarios, DemandA. Total reduced scenarios -> green line; average nodal values -> black dotted line; scenario with max probability -> red dotted line.

**Figure 13.**Reduced pressure scenarios’ probabilities DemandA. Inside the bar is the number of the rank order of the generated demand scenario.

PIPES | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Pipe Number | Start Node | End Node | Length (m) | C Hazen Williams | D (m) | Pipe Number | Start Node | End Node | Length (m) | C Hazen Williams | D (m) |

1 | 1 | 2 | 348.5 | 100 | 0.327 | 18 | 1 | 19 | 583.9 | 100 | 0.164 |

2 | 2 | 3 | 955.7 | 100 | 0.29 | 19 | 5 | 18 | 452 | 100 | 0.229 |

3 | 3 | 4 | 483 | 100 | 0.1 | 20 | 6 | 16 | 794.7 | 100 | 0.1 |

4 | 3 | 9 | 400.7 | 100 | 0.29 | 21 | 7 | 15 | 717.7 | 100 | 0.1 |

5 | 2 | 4 | 791.9 | 100 | 0.1 | 22 | 8 | 14 | 655.6 | 100 | 0.258 |

6 | 1 | 5 | 404.4 | 100 | 0.368 | 23 | 15 | 14 | 165.5 | 100 | 0.1 |

7 | 5 | 6 | 390.6 | 100 | 0.327 | 24 | 16 | 15 | 252.1 | 100 | 0.1 |

8 | 6 | 4 | 482.3 | 100 | 0.1 | 25 | 17 | 16 | 331.5 | 100 | 0.1 |

9 | 9 | 10 | 934.4 | 100 | 0.1 | 26 | 18 | 17 | 500 | 100 | 0.204 |

10 | 11 | 10 | 431.3 | 100 | 0.184 | 27 | 17 | 21 | 579.9 | 100 | 0.164 |

11 | 11 | 12 | 513.1 | 100 | 0.1 | 28 | 19 | 23 | 842.8 | 100 | 0.1 |

12 | 10 | 13 | 428.4 | 100 | 0.184 | 29 | 21 | 20 | 792.6 | 100 | 0.1 |

13 | 12 | 13 | 419 | 100 | 0.1 | 30 | 20 | 14 | 846.3 | 100 | 0.184 |

14 | 22 | 13 | 1023.1 | 100 | 0.1 | 31 | 9 | 11 | 164 | 100 | 0.258 |

15 | 8 | 22 | 455.1 | 100 | 0.164 | 32 | 23 | 21 | 427.9 | 100 | 0.1 |

16 | 7 | 8 | 182.6 | 100 | 0.29 | 33 | 19 | 18 | 379.2 | 100 | 0.1 |

17 | 6 | 7 | 221.3 | 100 | 0.29 | 34 | 24 | 1 | 158.2 | 100 | 0.368 |

NODES | |||||||||||

node ID | elevation (m) | users A | DemandA (l/s) | users B | DemandB (l/s) | ||||||

1 | 6.4 | 932 | 10.86 | 155 | 1.8 | ||||||

2 | 7 | 1461 | 17.03 | 427 | 4.98 | ||||||

3 | 6 | 1282 | 14.95 | 48 | 0.56 | ||||||

4 | 8.4 | 1224 | 14.28 | 129 | 1.5 | ||||||

5 | 7.4 | 869 | 10.13 | 1270 | 14.81 | ||||||

6 | 9 | 1316 | 15.35 | 486 | 5.67 | ||||||

7 | 9.1 | 782 | 9.11 | 63 | 0.73 | ||||||

8 | 9.5 | 901 | 10.51 | 766 | 8.93 | ||||||

9 | 8.4 | 1045 | 12.18 | 63 | 0.73 | ||||||

10 | 10.5 | 1249 | 14.57 | 45 | 0.52 | ||||||

11 | 9.6 | 848 | 9.88 | 964 | 11.24 | ||||||

12 | 11.7 | 650 | 7.58 | 354 | 4.12 | ||||||

13 | 12.3 | 1303 | 15.2 | 122 | 1.42 | ||||||

14 | 10.6 | 1162 | 13.55 | 185 | 2.15 | ||||||

15 | 10.1 | 791 | 9.23 | 81 | 0.94 | ||||||

16 | 9.5 | 960 | 11.2 | 55 | 0.64 | ||||||

17 | 10.2 | 984 | 11.47 | 968 | 11.29 | ||||||

18 | 9.6 | 928 | 10.82 | 55 | 0.64 | ||||||

19 | 9.1 | 1258 | 14.68 | 45 | 0.52 | ||||||

20 | 13.9 | 1142 | 13.32 | 416 | 4.85 | ||||||

21 | 11.1 | 1255 | 14.63 | 567 | 6.61 | ||||||

22 | 11.4 | 1030 | 12.01 | 137 | 1.59 | ||||||

23 | 10 | 886 | 10.33 | 920 | 10.73 | ||||||

24 (Reservoir) | 36.4 | 24258 | 282.86 | 8321 | 96.97 |

Statistical Parameter | Value |
---|---|

${\mu}_{average}\text{}\left(\mathrm{L}/\mathrm{min}\right)$ | 0.365 |

${\mu}_{peak\text{}hour}\text{}\left(\mathrm{L}/\mathrm{min}\right)$ | 0.700 |

${\sigma}_{peak\text{}hour}\left(\mathrm{L}/\mathrm{min}\right)$ | 0.870 |

scaling law exponent α | 1.230 |

INPUT | OUTPUT | INPUT | OUTPUT | ||||||
---|---|---|---|---|---|---|---|---|---|

Node ID | a | b | a | b | Node ID | a | b | a | b |

1 | 125.20 | 0.0868 | 125.21 | 0.0868 | 13 | 162.06 | 0.0938 | 162.07 | 0.0938 |

2 | 176.99 | 0.0963 | 177.01 | 0.0963 | 14 | 148.38 | 0.0914 | 148.39 | 0.0914 |

3 | 160.04 | 0.0935 | 160.06 | 0.0934 | 15 | 110.35 | 0.0836 | 110.36 | 0.0836 |

4 | 154.44 | 0.0925 | 154.45 | 0.0925 | 16 | 128.09 | 0.0874 | 128.10 | 0.0874 |

5 | 118.63 | 0.0855 | 118.65 | 0.0855 | 17 | 130.55 | 0.0879 | 130.56 | 0.0879 |

6 | 163.30 | 0.0940 | 163.32 | 0.0940 | 18 | 124.79 | 0.0868 | 124.80 | 0.0868 |

7 | 109.38 | 0.0834 | 109.39 | 0.0834 | 19 | 157.73 | 0.0930 | 157.75 | 0.0930 |

8 | 121.98 | 0.0862 | 122.00 | 0.0862 | 20 | 146.41 | 0.0910 | 146.42 | 0.0910 |

9 | 136.73 | 0.0892 | 136.75 | 0.0892 | 21 | 157.44 | 0.0930 | 157.46 | 0.0930 |

10 | 156.86 | 0.0929 | 156.88 | 0.0929 | 22 | 135.22 | 0.0889 | 135.24 | 0.0889 |

11 | 116.42 | 0.0850 | 116.43 | 0.0850 | 23 | 120.42 | 0.0858 | 120.43 | 0.0858 |

12 | 94.86 | 0.0799 | 94.87 | 0.0799 | - | - | - | - | - |

E[ρ_{1}] = 0.0043 | ||
---|---|---|

ρ Input Correlation Matrix (Scaling Laws) | ρ Output Correlation Matrix (Scenarios) | |

min | 0.7542 | 0.7542 |

average | 0.8147 | 0.8147 |

max | 0.8568 | 0.8567 |

INPUT | OUTPUT | INPUT | OUTPUT | ||||||
---|---|---|---|---|---|---|---|---|---|

Node ID | a | b | a | b | Node ID | a | b | a | b |

1 | 31.46 | 0.0575 | 31.46 | 0.0575 | 13 | 26.16 | 0.0544 | 26.16 | 0.0544 |

2 | 68.64 | 0.0726 | 68.65 | 0.0726 | 14 | 36.05 | 0.0599 | 36.05 | 0.0599 |

3 | 12.76 | 0.0439 | 12.76 | 0.0439 | 15 | 19.08 | 0.0495 | 19.09 | 0.0495 |

4 | 27.31 | 0.0551 | 27.31 | 0.0551 | 16 | 14.17 | 0.0453 | 14.17 | 0.0453 |

5 | 158.89 | 0.0933 | 158.90 | 0.0932 | 17 | 128.91 | 0.0876 | 128.92 | 0.0876 |

6 | 75.83 | 0.0748 | 75.84 | 0.0748 | 18 | 14.17 | 0.0453 | 14.17 | 0.0453 |

7 | 15.73 | 0.0467 | 15.73 | 0.0467 | 19 | 12.14 | 0.0433 | 12.14 | 0.0432 |

8 | 107.65 | 0.0830 | 107.66 | 0.0830 | 20 | 67.28 | 0.0721 | 67.28 | 0.0721 |

9 | 15.73 | 0.0467 | 15.73 | 0.0467 | 21 | 85.39 | 0.0775 | 85.40 | 0.0775 |

10 | 12.14 | 0.0433 | 12.14 | 0.0432 | 22 | 28.60 | 0.0559 | 28.61 | 0.0559 |

11 | 128.50 | 0.0875 | 128.51 | 0.0875 | 23 | 123.96 | 0.0866 | 123.97 | 0.0866 |

12 | 59.41 | 0.0695 | 59.42 | 0.0695 | - | - | - | - | - |

E[ρ_{1}] = 0.0043 | ||
---|---|---|

ρ Input Correlation Matrix (Scaling Laws) | ρ Output Correlation Matrix (Scenarios) | |

min | 0.1627 | 0.1627 |

average | 0.4329 | 0.4330 |

max | 0.8261 | 0.8261 |

Node ID | Scenario 3720 | Scenario 5945 | Scenario 4492 | Mean Scenario | Node ID | Scenario 3720 | Scenario 5945 | Scenario 4492 | Mean Scenario |
---|---|---|---|---|---|---|---|---|---|

1 | 10.87 | 11.31 | 10.29 | 10.73 | 13 | 15.20 | 15.66 | 14.98 | 14.94 |

2 | 17.04 | 16.88 | 15.71 | 17.27 | 14 | 13.56 | 13.43 | 13.09 | 13.53 |

3 | 14.96 | 14.67 | 14.15 | 14.66 | 15 | 9.23 | 8.96 | 8.44 | 8.90 |

4 | 14.28 | 13.82 | 13.32 | 14.35 | 16 | 11.20 | 10.85 | 10.40 | 11.01 |

5 | 10.14 | 9.97 | 10.73 | 10.40 | 17 | 11.48 | 12.04 | 10.77 | 11.59 |

6 | 15.35 | 15.19 | 14.71 | 16.08 | 18 | 10.83 | 10.61 | 10.31 | 10.64 |

7 | 9.12 | 8.94 | 8.31 | 8.71 | 19 | 14.68 | 14.65 | 14.11 | 14.71 |

8 | 10.51 | 10.64 | 9.92 | 10.24 | 20 | 13.32 | 14.10 | 13.45 | 13.40 |

9 | 12.19 | 12.38 | 11.26 | 12.85 | 21 | 14.64 | 14.35 | 14.39 | 13.89 |

10 | 14.57 | 14.33 | 14.19 | 14.41 | 22 | 12.02 | 12.34 | 11.12 | 12.19 |

11 | 9.89 | 9.53 | 9.53 | 9.96 | 23 | 10.34 | 10.53 | 10.12 | 10.37 |

12 | 7.58 | 7.58 | 7.33 | 7.25 | - | - | - | - | - |

Node ID | Scenario 6232 | Scenario 9392 | Scenario 9550 | Mean Scenario | Node ID | Scenario 6232 | Scenario 9392 | Scenario 9550 | Mean Scenario |
---|---|---|---|---|---|---|---|---|---|

1 | 1.81 | 1.81 | 1.75 | 1.80 | 13 | 1.42 | 1.52 | 1.27 | 1.39 |

2 | 4.98 | 5.01 | 4.80 | 4.71 | 14 | 2.16 | 2.41 | 1.58 | 2.13 |

3 | 0.56 | 0.46 | 0.64 | 0.48 | 15 | 0.94 | 1.19 | 0.99 | 1.06 |

4 | 1.50 | 1.52 | 1.39 | 1.55 | 16 | 0.64 | 0.66 | 0.65 | 0.58 |

5 | 14.82 | 15.03 | 14.55 | 14.92 | 17 | 11.29 | 11.34 | 11.24 | 11.16 |

6 | 5.67 | 5.38 | 5.38 | 5.62 | 18 | 0.64 | 0.60 | 0.83 | 0.64 |

7 | 0.73 | 0.56 | 0.59 | 0.68 | 19 | 0.52 | 0.49 | 0.45 | 0.55 |

8 | 8.94 | 8.82 | 8.45 | 8.82 | 20 | 4.85 | 5.04 | 4.86 | 4.91 |

9 | 0.73 | 0.65 | 0.76 | 0.69 | 21 | 6.61 | 7.05 | 6.65 | 6.59 |

10 | 0.52 | 0.50 | 0.34 | 0.46 | 22 | 1.60 | 1.72 | 1.53 | 1.10 |

11 | 11.25 | 11.45 | 11.04 | 11.24 | 23 | 10.73 | 11.30 | 11.12 | 10.64 |

12 | 4.13 | 4.14 | 4.04 | 4.05 | - | - | - | - | - |

Node ID | Scenario 7577 | Scenario 6780 | Scenario Mean Pressure | Scenario 3720 | Scenario Mean Demand | Node ID | Scenario 7577 | Scenario 6780 | Scenario Mean Pressure | Scenario 3720 | Scenario Mean Demand |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 25.965 | 25.962 | 25.963 | 25.966 | 25.965 | 13 | 15.754 | 15.904 | 15.684 | 15.839 | 15.723 |

2 | 28.077 | 28.052 | 28.061 | 28.085 | 28.070 | 14 | 17.727 | 17.850 | 17.661 | 17.697 | 17.692 |

3 | 24.548 | 24.543 | 24.512 | 24.592 | 24.534 | 15 | 18.135 | 18.278 | 18.001 | 18.107 | 18.041 |

4 | 21.829 | 21.852 | 21.714 | 21.963 | 21.755 | 16 | 21.023 | 21.222 | 21.042 | 21.138 | 21.068 |

5 | 25.280 | 25.314 | 25.254 | 25.289 | 25.267 | 17 | 22.946 | 23.040 | 22.908 | 22.930 | 22.927 |

6 | 24.875 | 24.929 | 24.829 | 24.888 | 24.849 | 18 | 22.933 | 22.966 | 22.895 | 22.906 | 22.911 |

7 | 22.453 | 22.522 | 22.383 | 22.449 | 22.407 | 19 | 25.149 | 25.185 | 25.147 | 25.139 | 25.155 |

8 | 21.545 | 21.618 | 21.454 | 21.522 | 21.483 | 20 | 21.019 | 21.193 | 20.997 | 21.021 | 21.026 |

9 | 20.647 | 20.691 | 20.592 | 20.694 | 20.620 | 21 | 18.794 | 18.859 | 18.768 | 18.731 | 18.785 |

10 | 19.421 | 19.553 | 19.366 | 19.501 | 19.404 | 22 | 15.926 | 16.363 | 15.908 | 16.111 | 15.959 |

11 | 18.808 | 18.864 | 18.731 | 18.855 | 18.764 | 23 | 21.919 | 21.976 | 21.894 | 21.862 | 21.909 |

12 | 17.846 | 17.902 | 17.700 | 17.901 | 17.747 | - | - | - | - | - | - |

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Magini, R.; Boniforti, M.A.; Guercio, R. Generating Scenarios of Cross-Correlated Demands for Modelling Water Distribution Networks. *Water* **2019**, *11*, 493.
https://doi.org/10.3390/w11030493

**AMA Style**

Magini R, Boniforti MA, Guercio R. Generating Scenarios of Cross-Correlated Demands for Modelling Water Distribution Networks. *Water*. 2019; 11(3):493.
https://doi.org/10.3390/w11030493

**Chicago/Turabian Style**

Magini, Roberto, Maria Antonietta Boniforti, and Roberto Guercio. 2019. "Generating Scenarios of Cross-Correlated Demands for Modelling Water Distribution Networks" *Water* 11, no. 3: 493.
https://doi.org/10.3390/w11030493