# An Event-Based Stochastic Parametric Rainfall Simulator (ESPRS) for Urban Stormwater Simulation and Performance in a Sponge City

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data

^{2}, Figure 1) is an urban drainage area in Fengxi New City, a designated UNESCO Ecohydrology demo site [34] and a pilot sponge city in China. Many LIDs, such as green roofs, permeable pavements, and bioretention cells, have been built since 2013 to improve the stormwater management infrastructure. WR8 has an elevation of 380.5–384.3 m and covers a layer of non-self-weight collapsible loess. The stormwaters in WR8 are drained to the Fenghe River through the outfall in Figure 1d. The main types of land cover use are residential, educational, and transportation land. It is situated in a warm temperate zone with a continental monsoon semi-arid and semi-humid climate. The annual average precipitation is 552.0 mm, and the rainfall is uneven throughout the year. The precipitations are primarily concentrated in the wet seasons, i.e., from July to September, (accounting for 50–60% of the annual total). We obtained most of the data for applying the storm water management model (SWMM) [35] from the Fengxi Management Committee [36]. In addition, we collected the rainfall time series of 2000 and 2002–2017 at the Xianyang hydrological station (Figure 1c) from the Xi’an University of Technology [37].

## 3. Methodology

#### 3.1. Event-Based Stochastic Parametric Rainfall Simulator (ESPRS)

#### 3.1.1. Rainfall Fractions (RFs)

_{i,j}is the rainfall depth in the jth period for the ith rainfall event, mm; and b

_{i,j}is the ratio of precipitation in the jth period to the total precipitation for the ith rainfall event. The RF matrix for α measured precipitation events is expressed as

#### 3.1.2. Probability Distribution Functions (PDFs)

_{j}; μ, σ, and ξ are the parameters of a PDF, which need to be estimated. Thus, we obtained 60 fitting functions (12 periods, each with 5 functions).

#### 3.1.3. Goodness-of-Fit Statistics

_{i}are the theoretical and empirical frequency values of the ith sample, respectively; and k is the number of parameters of a PDF that needs to be estimated, which are 2, 2, 2, 2, and 1, respectively (see Equations (4)–(8)).

_{i}is the value of the ith sample point (note that the samples were sorted in ascending order); and F is the cumulative distribution function of the target distribution.

_{1}, c

_{2}, c

_{3}, c

_{4}, and c

_{5}are coefficients determined by ${A}_{\alpha}^{2}$.

#### 3.1.4. Stochastic Rainfall Fraction Series

_{p}is the set of measured rainfall events that peak at the pth period.

#### 3.1.5. Stochastic and Chicago Rainfall Time Series

#### 3.2. Storm Water Management Model (SWMM)

#### 3.3. Performance Evaluation

_{m}), peak time (t

_{m}), flood duration (t

_{D}), variation coefficient (Q

_{CV}), flood rise rate (Q

_{r}), and flood drop rate (Q

_{d}).

^{3}/s; and Q

_{0}and Q

_{end}are the flow rate at the onset time (t

_{0}) and end time (t

_{end}) of the flood, respectively.

_{r}and t

_{p}are the lag time and rainfall peak time, respectively. The t

_{p}of stochastic precipitation events approximately equals the peak period’s middle moment.

## 4. Results and Discussion

#### 4.1. Stochastic Rainfall Fractions

#### 4.2. Catchment Outflow

#### 4.3. Low-Impact Development Facilities

## 5. Conclusions

- The ESPRS outperformed the Chicago method in predicting extreme precipitation events for urban stormwater simulation and control. Fixed rain patterns can hardly represent the actual temporal structures of precipitation events, especially the extreme ones, leading to uncertainties. Using an ESPRS has strong potential for revealing the influence of temporal rainfall characteristics on hydrological responses.
- The rainfall peak period, rainfall peak fraction, and cumulative rainfall depth at the peak period are control factors for an ESPRS.
- Rear-type precipitation events with high peak fractions present the most negative pattern for outflow control by LIDs.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Map of the study area for simulation: (

**a**) Shannxi Province, China; (

**b**) Weihe River No. 8 system zone (WR8), Fengxi New City; (

**c**) Study area map of WR8 in storm water management model; and (

**d**) Aerial photographs of WR8.

**Figure 2.**Event-based stochastic parametric rainfall simulator (ESPRS) framework and storm water management model simulation.

**Figure 3.**Rainfall fractions (RF) of 12 periods and their upper and lower limits. For a period, the upper (lower) limit of RF equals the maximum (minimum) value of the measured RFs. For a period, the upper (lower) limit of peak RF means is the maximum (minimum) value of the measured RFs of precipitation events that peaked in that period. The points with different colors represent the fractions of observed rainfalls.

**Figure 4.**Rainfall fraction and catchment outflow time series in no- and full-LID scenarios under Chicago and stochastic precipitation events. The no-LID denotes low-impact development facilities (LIDs) are not constructed, and full-LID denotes LIDs are constructed as much as possible.

**Figure 5.**Variation coefficients of Chicago and stochastic rainfall time series and variation coefficients of corresponding outflow time series in no- and full-LID scenarios. * represents the median.

**Figure 6.**Catchment peak outflow rate in no- and full-LID scenarios under Chicago and stochastic precipitation events with different peak periods. * represents the median.

**Figure 7.**Time lag in no- and full-LID scenarios under Chicago and stochastic precipitation events. * represents the median.

**Figure 8.**Rise and drop rates of flood time series in (

**a**) no- and (

**b**) full-LID scenarios under Chicago and stochastic precipitation events. * represents the median.

**Figure 9.**Catchment total outflow in no- and full-LID scenarios under Chicago and stochastic precipitation events. + represents the outlier.

**Table 1.**Performances of five probability distribution functions for the rainfall fraction of the first period.

Distribution Function | AIC ^{1} | ${{\mathit{A}}_{\mathit{\alpha}}^{2}}^{1}$ | p-Value ^{1} |
---|---|---|---|

Normal | −151.09 | 3.08 | 0.005 |

Weibull | −191.93 | 1.16 | 0.010 |

Generalized extreme value | −247.46 | 0.93 | 0.017 |

Gamma | −198.76 | 0.82 | 0.040 |

Lognormal | −233.94 | 0.31 | 0.549 |

^{1}Smaller values of AIC and ${A}_{\alpha}^{2}$ represent better fits, and a p-value greater than 0.05 indicates that the data obey the target distribution.

Period | Best Fitting Form | Formula | Bounds of Rainfall Fraction (x) |
---|---|---|---|

1 | Lognormal | ${f}_{1}\left(x\right)=\frac{0.0156}{x}{e}^{\frac{-{\left(\mathrm{lnx}+3.401\right)}^{2}}{0.936}}$ | 0.01–0.19 |

2 | Lognormal | ${f}_{2}\left(x\right)=\frac{0.0112}{x}{e}^{\frac{{-\left(lnx+2.635\right)}^{2}}{1.832}}$ | 0.01–0.44 |

3 | Gamma | ${f}_{3}\left(x\right)=\frac{0.028}{{\mathsf{\Gamma}}_{\left(0.04\right)}}{x}^{-0.96}{e}^{-2.503x}$ | 0.02–0.29 |

4 | Lognormal | ${f}_{4}\left(x\right)=\frac{0.0164}{x}{e}^{\frac{{-\left(\mathrm{ln}x+2.304\right)}^{2}}{0.855}}$ | 0.03–0.43 |

5 | Gamma | ${f}_{5}\left(x\right)=\frac{0.028}{{\mathsf{\Gamma}}_{\left(0.028\right)}}{x}^{-0.972}{e}^{-3.665x}$ | 0.03–0.22 |

6 | Generalized extreme value | ${f}_{6}\left(x\right)=0.027{e}^{-{\left(1+0.01x\right)}^{-100}}$ | 0.02–0.28 |

7 | Gamma | ${f}_{7}\left(x\right)=\frac{0.028}{{\mathsf{\Gamma}}_{\left(0.031\right)}}{x}^{-0.969}{e}^{-3.128x}$ | 0.02–0.22 |

8 | Lognormal | ${f}_{8}\left(x\right)=\frac{0.018}{x}{e}^{\frac{{-\left(\mathrm{ln}x+2.571\right)}^{2}}{0.73}}$ | 0.02–0.33 |

9 | Generalized extreme value | ${f}_{9}\left(x\right)=0.027{e}^{-{\left(1+0.209x\right)}^{-4.7847}}$ | 0.02–0.23 |

10 | Lognormal | ${f}_{10}\left(x\right)=\frac{0.0164}{x}{e}^{\frac{{-\left(lnx+2.992\right)}^{2}}{0.874}}$ | 0.01–0.22 |

11 | Lognormal | ${f}_{11}\left(x\right)=\frac{0.0144}{x}{e}^{\frac{{-\left(\mathrm{ln}x+3.138\right)}^{2}}{1.149}}$ | 0.01–0.40 |

12 | Lognormal | ${f}_{12}\left(x\right)=\frac{0.012}{x}{e}^{\frac{{-\left(lnx+3.722\right)}^{2}}{1.574}}$ | 0.00–0.13 |

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## Share and Cite

**MDPI and ACS Style**

Yang, Y.; Xu, X.; Liu, D.
An Event-Based Stochastic Parametric Rainfall Simulator (ESPRS) for Urban Stormwater Simulation and Performance in a Sponge City. *Water* **2023**, *15*, 1561.
https://doi.org/10.3390/w15081561

**AMA Style**

Yang Y, Xu X, Liu D.
An Event-Based Stochastic Parametric Rainfall Simulator (ESPRS) for Urban Stormwater Simulation and Performance in a Sponge City. *Water*. 2023; 15(8):1561.
https://doi.org/10.3390/w15081561

**Chicago/Turabian Style**

Yang, Yuanyuan, Xiaoyan Xu, and Dengfeng Liu.
2023. "An Event-Based Stochastic Parametric Rainfall Simulator (ESPRS) for Urban Stormwater Simulation and Performance in a Sponge City" *Water* 15, no. 8: 1561.
https://doi.org/10.3390/w15081561