# Optimal Spatial Design of Capacity and Quantity of Rainwater Harvesting Systems for Urban Flood Mitigation

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^{2}

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## Abstract

**:**

## 1. Introduction

**(**2014) [2] developed a regional-level and dimensionless analysis for designing a domestic RWHS. Moreover, regarding design using economic and dimensionless analysis-based optimization approach, Chiu et al. (2009) [3] optimized the most cost-effective rainwater tank volumes for different dwelling types using marginal analysis. Campisano and Modica (2012) [4] developed a dimensionless methodology for the optimal design of domestic RWHS. From these studies, we can find out that previous studies have scarcely designed the capacity of RWHS considering flood reduction benefits using an interdisciplinary integrated systematic analysis approach. In addition, the capacity design of RWHS is primarily limited to small communities and lacks full consideration of all metropolitan catchment with variations in spatial capacity and quantity design of RWHS.

## 2. Development of Methodology

#### 2.1. Procedures

#### 2.2. Development of Simulation Model for Spatial Arrangement of Quantity and Capacity

#### 2.2.1. Classified Methodology of Zonal Subregions for Design of Rainwater Harvesting System

#### 2.2.2. Design Methodology of Capacity and Quantity of Regular Rainwater Harvesting Systems

#### 2.2.3. Assessment Index of Designed Goodness

#### 2.2.4. Computation of Inundation Loss

#### 2.3. Introduction of SWMM

#### 2.3.1. Model Parameters and Routing

#### 2.3.2. The Rainwater Harvesting Function within Low-impact Development Components

^{2}(or 90 m

^{2}) of rooftop area captured.

#### 2.4. Development of Optimization Model

#### 2.4.1. Objective Function

_{r}is the number of rain barrels in subregion r, ${L}_{\text{non}}$ represents the flooding loss with no rain barrels established, and L

_{p}is the flooding loss at control point p. In addition, S

_{r}denotes the capacity of the rain barrel in subregion r, R is the quantity of the total subregion, and P represents the quantity at the flooding control point, wherein decision-making variables are the quantity of rain barrels in each N

_{r}and S

_{r}.

#### 2.4.2. Constraints

^{3}) and the roof area ${A}_{r}$ (m

^{2}).

#### 2.4.3. Solution of Optimization Model

**Figure 2.**Flowchart of optimizing the spatial design of capacity and quantity of rainwater harvesting systems using tabu search

#### 2.5. Development of BPNN-Based SWMM

#### 2.5.1. Model Structure of BPNN-based SWMM

_{0}), the meteorological-hydrological condition, and the spatial design pattern of the RWHS. We enter the trained BPNN single-moment calculation units and then obtain the simulation value at t + 1 from the output item. Accordingly, the cycle of continuous iterative calculations is repeated until the end of the moments, when the complete-event flooding and water levels of the drainage system can be simulated.

#### 2.5.2. Alternative Applicability Assessing Index of BPNN-based SWMM

## 3. Application

#### 3.1. Study Area

^{2}located in the southwest corner of the Taipei Basin. Its southern end has a high altitude and gradually lowers northward. In some areas, the Zhong-He District has extreme slope changes, which can lead to floods because of the locations of these changes at the intersections of mountainous terrain and the ground. Other areas are also vulnerable to flooding on account of their more gentle terrains or insufficient drainage capacities. Examples include the area near Jyu-Guang Road and Min-Siang Street, shown in Figure 3a; control point 1 (CP1), Guo-Guang Street; control point 2 (CP2), Min-Siang Street; and control point 3 (CP3), Jyu-Guang Road. These latter three locations are low lying such that the terrain height diagram can be shown in Figure 3b. It is, therefore, relatively difficult for the water to drain from these areas, causing flooding and life and property loss from rainstorms. Thus, these locations are set as control points for the flood damage assessment.

**Figure 3.**Study area: (

**a**) spatial distribution of drainage system, zonal subregions for design of rainwater harvesting system using the fuzzy C-means cluster algorithm and the low-lying control points; (

**b**) terrain height above sea level.

#### 3.2. Analyzed Results of the Simulation Model for Spatial Design of Quantity and Capacity

#### 3.2.1. Classified Results of Zonal Subregions for Design of RWHS

^{2}to 722 m

^{2}; distance for clustering number 4, ranges from 547 m

^{2}to 1075 m

^{2}; distance for clustering number 5, ranges from 391 m

^{2}to 1117 m

^{2}; and distance for clustering number 6, ranges from 375 m

^{2}to 1134 m

^{2}. The average distance between each combination of two central locations for clustering number 3 to 6 are 656 m

^{2}, 714 m

^{2}, 673 m

^{2}and 685 m

^{2}, respectively. Hence, clustering number 4 can cover wider designed area than the other clustering numbers with most efficient zonal mode.

#### 3.2.2. Spatial Designed Results of Specific Representative Regular RWHS

^{2}(A

_{1}), 82.4 m

^{2}(A

_{2}), 108.5 m

^{2}(A

_{3}) and 152.0 m

^{2}(A

_{4}), respectively, to ensure all designs of volume and arranged density can actually be applied to the building of Zhong-He drainage area. The adopted return period (T) of $\widehat{{P}_{T}^{RP}}$ are 2, 5, 25, 50 and 100 years, and the designed rainfall duration is 6 hours. Finally, the designed regular volume of rain barrel are ${A}_{1}\cdot \widehat{{P}_{2\text{year}}^{RP}}$, ${A}_{2}\cdot \widehat{{P}_{5\text{year}}^{RP}}$, ${A}_{2}\cdot \widehat{{P}_{\text{50year}}^{RP}}$, ${A}_{3}\cdot \widehat{{P}_{\text{50year}}^{RP}}$, ${A}_{4}\cdot \widehat{{P}_{\text{25year}}^{RP}}$ and ${A}_{4}\cdot \widehat{{P}_{100\text{year}}^{RP}}$ that the values are 3.03 m

^{3}(S

_{1}), 6.14 m

^{3}(S

_{2}), 9.12 m

^{3}(S

_{3}), 12.01 m

^{3}(S

_{4}), 15.05 m

^{3}(S

_{5}) and 18.05 m

^{3}(S

_{6}), respectively, to ensure that the designed volume can handle all kinds magnitude of storm rainwater of return periods.

^{2}under the rain barrel per household (Case X-1); one for every 82.4 m

^{2}(Case X-2); one for every 108.5 m

^{2}(Case X-3); and one for every 152.0 m

^{2}(Case X-4). The rain barrel quantity of each case with each spatial arrangement is shown in Table 1. In the capacity design, we divided the capacity of rain barrels into 3.03 m

^{3}(Case X-Y-1), 6.14 m

^{3}(Case X-Y-2), 9.12 m

^{3}(Case X-Y-3), 12.01 m

^{3}(Case X-Y-4), 15.05 m

^{3}(Case X-Y-5), and 18.05 m

^{3}(Case X-Y-6) to compare the simulated effect of flood detention.

Case No. | Case X-1 | Case X-2 | Case X-3 | Case X-4 |
---|---|---|---|---|

Case 1 | 3194 | X | ||

Case 2 | 472 | 317 | 236 | 167 |

Case 3 | 306 | 204 | 153 | 108 |

Case 4 | 395 | 262 | 198 | 136 |

#### 3.2.3. Calibration and Validation of US EPA SWMM

Title | Surface Roughness Coefficient | Horton-Based Max/Min Infiltration Rates/Decay Constant | Manning’s Roughness Coefficient for Conduits |
---|---|---|---|

Region 1 | 0.015~0.033 | 3.5~4.6 (mm/h)/0.9~1.6 (mm/h)/1.8~2 (1/h) | 0.013~0.016 |

Region 2 | 0.013~0.037 | 3.6~5.8 (mm/h)/1.1~1.9 (mm/h)/1.6~1.9 (1/h) | 0.015~0.017 |

Region 3 | 0.013~0.025 | 3~3.5 (mm/h)/0.5~0.9 (mm/h)/2~2.2 (1/h) | 0.011~0.015 |

Region 4 | 0.012~0.021 | 3.3~3.9 (mm/h)/0.7~1.2 (mm/h)/1.9~2 (1/h) | 0.012~0.014 |

#### 3.2.4. Simulated Analytical Results of the Designed Spatial Arranged Regular Cases for RWHS

Date | Time Series Number (HOUR) | Representative Return Period of Total Precipitation | |||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | ||

1 July 2009 | 4.0 | 96.5 | 11.0 | 3.0 | 2.5 | 0 | 50 year |

12 August 2009 | 2.5 | 101.5 | 20.0 | 3.5 | 1.5 | 0 | 100 year |

21 June 2010 | 4.0 | 0 | 4.5 | 44.5 | 48.0 | 28.5 | 125 year |

12 August 2012 | 70.5 | 49.5 | 1.5 | 0 | 0 | 0 | 75 year |

^{3}. The unit inundation loss of CP

_{2}is obviously larger than the other two control points.

^{3}, and the net benefit for each year was 4.61 × 10

^{5}US dollars; (3) In each case, the rain barrel’s best capacity was between 12 and 15 m

^{3}; greater benefits were produced when the rain barrel was set in the easily flooded area.

#### 3.3. Construction Results of BPNN-based SWMM

#### 3.3.1. Training and Validating Results of Single-moment Simulation

**Figure 9.**Training results of single-moment full pipe percentage simulation of water flow: (

**a**) at CP1; (

**b**) at CP2; and (

**c**) at CP3.

**Figure 10.**Validation results of single-moment full pipe percentage simulation of water flow: (

**a**) at CP1; (

**b**) at CP2; and (

**c**) at CP3.

#### 3.3.2. Sensitivity Analysis Result

^{3}) is within the range from 0.44% to 2.15%. The change in upstream quantity and capacity of RWHS (Regions 3 and 4) make more sensitivity to the other low-lying subregions (Regions 1 and 2) that coincide with the analytical results of simulation method (Section 3.2.4), so the developed model is available for the embedded optimizing process.

**Table 4.**Sensitivity analysis of the backpropagation neural network (BPNN)-based water flow simulation model.

Input | Average Variance of Output (Full Pipe Percentage) | ||
---|---|---|---|

of CP3 at t + 1 | of CP2 at t + 1 | of CP1 at t + 1 | |

Precipitation at time t (change in 0.2 mm/min) | 3.46% | 5.15% | 5.26% |

Full pipe percentage of CP3 at t (change in 1%/min) | 1.12% | 0.51% | 0.72% |

Full pipe percentage of CP2 at t (change in 1%/min) | 0.59% | 1.00% | 0.53% |

Full pipe percentage of CP1 at t (change in 1%/min) | 0.96% | 0.98% | 1.00% |

Arranged quantity of Region 1 (change in 100 number) | 3.22% | 2.68% | 2.74% |

Arranged quantity of Region 2 (change in 100 number) | 2.38% | 3.54% | 1.81% |

Arranged quantity of Region 3 (change in 100 number) | 2.40% | 5.02% | 3.42% |

Arranged quantity of Region 4 (change in 100 number) | 4.40% | 3.30% | 6.75% |

Arranged capacity of Region 1 (change in 3 m^{3}) | 1.43% | 1.19% | 0.88% |

Arranged capacity of Region 2 (change in 3 m^{3}) | 0.49% | 0.86% | 0.44% |

Arranged capacity of Region 3 (change in 3 m^{3}) | 1.43% | 2.15% | 1.75% |

Arranged capacity of Region 4 (change in 3 m^{3}) | 1.16% | 1.29% | 1.62% |

#### 3.3.3. Validation Results of the Entire-event Iterative Simulation

^{2}), and the second design capacity (6 m

^{3}).

Unsteady Simulated Events | Guo-Guang Street (CP1) | Min-Xiang Street (CP2) | Ju-Guang Road (CP3) | |||
---|---|---|---|---|---|---|

MAE (%) | CC | MAE (%) | CC | MAE (%) | CC | |

Designed Case 3-1-3 on 1 July 2009 | 5.7 | 0.974 | 7.0 | 0.975 | 9.2 | 0.974 |

Designed Case 4-4-1 on 1 July 2009 | 5.2 | 0.984 | 7.5 | 0.981 | 9.6 | 0.975 |

Designed Case 2-1-2 on 12 August 2009 | 3.8 | 0.995 | 3.0 | 0.994 | 10.1 | 0.963 |

Designed Case 3-2-4 on 12 August 2009 | 4.1 | 0.995 | 3.6 | 0.994 | 9.4 | 0.973 |

Designed Case 3-3-1 on 21 June 2010 | 12.4 | 0.974 | 10.1 | 0.991 | 17.1 | 0.936 |

Designed Case 4-3-6 on 21 June 2010 | 11.1 | 0.970 | 7.0 | 0.991 | 15.8 | 0.956 |

Designed Case 2-2-5 on 12 August 2012 | 5.1 | 0.951 | 6.9 | 0.951 | 7.2 | 0.967 |

Designed Case 4-1-6 on 12 August 2012 | 5.9 | 0.924 | 9.5 | 0.914 | 9.8 | 0.954 |

Designed Case 4-3-2 on 12 August 2012 | 5.5 | 0.941 | 7.9 | 0.940 | 7.0 | 0.964 |

Average | 6.5 | 0.968 | 7.0 | 0.970 | 10.6 | 0.963 |

**Figure 11.**Unsteady continuous simulated results of Case 4 during the “12 August 2012 Flood Event” using BPNN-based SWMM.

#### 3.4. Optimization Results

_{r}and N

_{r}was 1 and 10, respectively. The initial solution was the best design approach of the simulation method of Case 4-1-4 (one rain barrel for every 50 m

^{2}; capacity of 12 m

^{3}).

^{6}US dollars; and of optimized design, was 0.27 × 10

^{6}US dollars. The optimized spatial design of RWHS could reduce 72% of inundation losses according to the four simulated flood events. Besides, the annual net benefit of the best solution in the simulation method was 4.61 × 10

^{5}US dollars (Figure 13), and the annual net benefit of hybrid simulation-optimization method was 5.20 × 10

^{5}US dollars (12.75% better than using the single simulation method), which is quite good. It indicates that the optimization model developed by our institute can search for the optimal solutions for spatial quantity and capacity arrangement of RWHS with consideration of flood retention benefits.

**Figure 12.**Optimized design results of capacity and quantity of rainwater harvesting systems for Zhong-He drainage system.

**Figure 13.**Comparison on full pipe percentage of water flow and flooding volume between optimal rainwater harvesting systems (RWHS) design, best design of simulation method and original no design: (

**a**) CP1; (

**b**) CP2; and (

**c**) CP3.

Flood Event | 1 July 2009 | 12 August 2009 | 21 June 2010 | 12 August 2012 | Average |
---|---|---|---|---|---|

Inundated loss while no installing rain barrels (US dollars) | 1.03 × 10^{6} | 1.63 × 10^{6} | 0.58 × 10^{6} | 0.91 × 10^{6} | 1.04 × 10^{6} |

Inundated loss of optimized design (US dollars) | 0.25 × 10^{6} | 0.33 × 10^{6} | 0.22 × 10^{6} | 0.27 × 10^{6} | 0.27 × 10^{6} |

Decreased inundated loss (US dollars) | 0.79 × 10^{6} | 1.30 × 10^{6} | 0.36 × 10^{6} | 0.64 × 10^{6} | 0.77 × 10^{6} |

Benefit percentage of flood mitigation (%) | 76.2% | 79.8% | 62.1% | 70.0% | 72.0% |

Cost (US dollars) | 0.25 × 10^{6} | ||||

Annual net benefit (US dollars) | 0.52 × 10^{6} |

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Huang, C.-L.; Hsu, N.-S.; Wei, C.-C.; Luo, W.-J.
Optimal Spatial Design of Capacity and Quantity of Rainwater Harvesting Systems for Urban Flood Mitigation. *Water* **2015**, *7*, 5173-5202.
https://doi.org/10.3390/w7095173

**AMA Style**

Huang C-L, Hsu N-S, Wei C-C, Luo W-J.
Optimal Spatial Design of Capacity and Quantity of Rainwater Harvesting Systems for Urban Flood Mitigation. *Water*. 2015; 7(9):5173-5202.
https://doi.org/10.3390/w7095173

**Chicago/Turabian Style**

Huang, Chien-Lin, Nien-Sheng Hsu, Chih-Chiang Wei, and Wei-Jiun Luo.
2015. "Optimal Spatial Design of Capacity and Quantity of Rainwater Harvesting Systems for Urban Flood Mitigation" *Water* 7, no. 9: 5173-5202.
https://doi.org/10.3390/w7095173