# Development of a Multiobjective Automatic Parameter-Calibration Framework for Urban Drainage Systems

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Datasets

#### 2.1. Study Area

^{2}. The land-use characteristics of this network include 80% residential area and 16 and 4% public and road areas, respectively.

#### 2.2. Datasets

^{3}/10 min). One year of historical data for 13–14 July 2013 (i.e., historical urban flood events in Korea) were used to develop the MAPC framework, including rainfall and outflow (e.g., total system discharge). The measurement data (i.e., rainfall (hyetograph) and outflow (hydrograph)) used in this study are shown in Figure 2. Notably, two rainfall events were considered to calibrate and validate the results obtained using the MAPC framework.

## 3. Modeling Methodology

#### 3.1. Stormwater Management Model (SWMM)

#### 3.2. Step I: Sensitivity Analysis (SA)

#### 3.3. Step II: Objective Selection Process (OSP)

^{2}) were used to confirm inherent correlation between MAIs. A set of two MAIs should not be simultaneously considered in the multiobjective calibration model if they are highly correlated and aligned with the regression line (i.e., high R-squared coefficient). Considering one indicator in the objective function can automatically minimize/maximize the other indicator by their inherent correlation without explicitly and additionally considering the other in the formulation.

#### 3.4. Step III: SWMM Parameter-Calibration Model

#### 3.5. Step IV: Performance Evaluation

## 4. Application Results

#### 4.1. Sensitivity Analysis (SA)

_{width}, Ф

_{imp}, and CN were identified as the parameters that increase the outflow. k

_{width}is multiplied when calculating the surface runoff in the SWMM governing equation [29]. Therefore, an increase in k

_{width}may lead to an increase in outflow. An increase in Ф

_{imp}and CN, which determine the infiltration amount, influences the increase in outflow [49]. As n

_{imp}, n

_{perv}, and n

_{conduit}increase, the total outflow decreases. As n

_{imp}, n

_{perv}, and n

_{conduit}increase, losses caused by friction occur [50], which reduce the outflow. Figure 4 shows that all the parameters considered in the sensitivity analysis were consistently altered. Therefore, even if the parameter search range is entered as a continuous range in the SWMM parameter-calibration model, this range would not have a large impact on the result [39].

_{imp}and CN were found to be highly sensitive to the model output. These two parameters directly affect the amount of rainfall that is converted to surface runoff through infiltration. In contrast, k

_{width}, n

_{imp}, n

_{perv}, and n

_{conduit}did not show large changes in outflow compared to parameter changes. However, because these parameters affect the peak outflow, further investigation is required. When the characteristics of the study network (urban catchment) were considered, a lack of influence of k

_{width}, n

_{imp}, and n

_{conduit}could not be determined. Although outliers were found for n

_{perv}, their impact was not large because the target network had the characteristic of a high infiltration rate. Therefore, n

_{perv}was excluded from the parameters calibrated using the SWMM parameter-calibration model.

#### 4.2. Selection of Two Objectives

^{2}).

^{2}for the relationships between 500 MAI calculation results extracted randomly from 1000 SWMM runs with randomly adjusted parameters. In the results, for a pair of MAIs considered good, objective functions of the SWMM parameter-calibration model should be depicted in a space where the trade-off relationship shows the optimum value of each MAI. Among the pairs of MAIs considered in this study, 18 sets exist, including RMSE-PB, RMSE-MaxAE, TVE-PFE, and TVE-NSE, which can be selected as two objective functions. Most pairs of MAIs show the trade-off relationship on the lower left side. However, if NSE is included, the trade-off relationship is shown on the upper right side; this is because among the MAIs considered in this study, NSE is the only MAI with good model performance when large. The coefficient of determination (R

^{2}) of these sets is between 0.1 and 0.8.

^{3}/s), enabling an easy analysis of the derived solutions. Furthermore, as the MAIs show the characteristics of the outflow curve intuitively, the hydrographs of the validation and calibration SWMMs are expected to be easily examined.

#### 4.3. Comparison between Pareto-Optimal Solutions

^{3}/s, PFE = 0.59 m

^{3}/s). In this study, three solutions (S-1, S-6, and S-30) were selected based on the reference solutions. S-1 and S-30 are the closest solutions to the reference solutions of TVE and PFE, respectively. S-6 is the ideal solution to the intersection point where the reference solutions meet the Pareto-optimal solutions.

_{width}and n

_{imp}are highly calibrated compared with the other solutions. Thus, it was confirmed that the peak flow of the study network can be adjusted using k

_{width}and n

_{imp}. As the characteristics of the urban network were well reflected, Ф

_{imp}of all solutions, including S-1, S-6, and S-30, was calibrated to be high. CN and n

_{conduit}were calibrated to be high in S-6, unlike S-1 and S-30, implying that CN and n

_{conduit}are parameters that play a decisive role in the search for the trade-off section between TVE and PFE. The overall results revealed that the calibrated value of each parameter obtained from the Pareto-optimal solutions displayed a consistent tendency for the system characteristics (e.g., hillslope width factor, curve number, etc.).

#### 4.4. Multiobjective Calibration and Validation

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

UDS | Urban drainage system |

SWMM | Stormwater management model |

NSHS | Non-dominated sorting harmony search |

MAPC | Multiobjective automatic parameter-calibration |

MAIs | Model accuracy indicators |

PI | Performance indicator |

SA | Sensitivity analysis |

OSP | Objective selection process |

MCS | Monte Carlo sampling |

RMSE | Root-mean-square error |

TVE | Total volume error |

PFE | RMSE of peak flow error |

NSE | Nash-Sutcliffe efficiency coefficient |

APD | Absolute peak difference |

PB | Percent bias |

MAE | Mean absolute error |

MaxAE | Maximum absolute error |

TMSE | Total mean squared error |

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**Figure 2.**Two types of measurement data in the Yongdap drainage area: (

**a**) calibration (13 July 2013); (

**b**) validation (14 July 2013). The input measurement data are two independent rainfall events, comprising hyetograph (rainfall event) and hydrograph (outflow), respectively. The hyetograph is depicted as a gray invert bar plot and the hydrograph is depicted as the black scatter plot.

**Figure 4.**Sensitivity results between system outflow and (

**a**) hillslope width factor; (

**b**) impervious fraction in urban land; (

**c**) curve number; (

**d**) Manning’s n on impervious; (

**e**) Manning’s n on pervious; (

**f**) Manning’s n on pipe roughness.

**Figure 5.**Box and whisker plot to quantify the variation between outflow and each parameter obtained from Step I.

**Figure 6.**Set of scatterplots for the impact relationship between a MAI and other MAIs obtained from Step II.

**Figure 7.**Pareto-optimal solution proposed by the SWMM parameter-calibration model (Step III) and comparison of the model and reference solutions of two objectives.

**Figure 9.**Comparison between observed outflow and simulated outflow for the SWMM parameter-calibration model: (

**a**) calibration (13 July 2013) and (

**b**) validation (14 July 2013).

**Figure 10.**Hydrograph (including observed and simulated outflow data) obtained from several solutions (i.e., S-1, S-6, and S-30) among the Pareto-optimal solutions: (

**a**) calibration (13 July 2013); (

**b**) validation (14 July 2013).

Parameter | Description (Unit) | Prior Distribution |
---|---|---|

k_{width} | Hillslope width factor (m) | *U (30, 170) |

Ф_{imp} | Impervious fraction in urban land (%) | *U (60, 100) |

CN | Curve number (-) | *U (53, 75) |

n_{imp} | Manning’s n on impervious (-) | *U (0.03, 0.05) |

n_{perv} | Manning’s n on pervious (-) | *U (0.03, 0.05) |

n_{conduit} | Manning’s n on pipe roughness (-) | *U (0.011, 0.017) |

Indicator Names (Abbreviations) | Formulations |
---|---|

Root-mean-square error (RMSE) | $\sqrt{\frac{1}{n}{\displaystyle \sum _{t=1}^{1}}{\left({Q}_{obs\left(t\right)}-{Q}_{sim\left(t\right)}\right)}^{2}}$ |

Total volume error (TVE) | $\frac{1}{{Q}_{sim\left(t\right)}}{\displaystyle \sum _{t=1}^{1}}\left|{Q}_{obs\left(t\right)}-{Q}_{sim\left(t\right)}\right|$ |

RMSE of peak flow error (PFE) | $\sqrt{\frac{1}{{n}_{P}}{\displaystyle \sum _{t=1}^{1}}{\left({P}_{obs}-{P}_{sim}\right)}^{2}}$ |

Nash-Sutcliffe efficiency coefficient (NSE) | $1-\frac{\frac{1}{n}{\sum}_{t=1}^{1}{\left({Q}_{obs\left(t\right)}-{Q}_{sim\left(t\right)}\right)}^{2}}{\frac{1}{n}{\sum}_{t=1}^{1}{\left({Q}_{obs\left(t\right)}-\overline{{Q}_{obs}}\right)}^{2}}$ |

Absolute peak difference (APD) | $\left|\underset{1\le t\le n}{\mathrm{max}}({O}_{obs\left(t\right)})-\underset{1\le t\le n}{\mathrm{max}}({O}_{sim\left(t\right)})\right|$ |

Percent bias (PB) | $100\times \left|\frac{\frac{1}{n}({\sum}_{t=1}^{n}{Q}_{obs\left(t\right)}-{\sum}_{t=1}^{n}{Q}_{sim\left(t\right)})}{\frac{1}{n}{\sum}_{t=1}^{n}{Q}_{obs\left(t\right)}}\right|$ |

Mean absolute error (MAE) | $\frac{1}{n}{\displaystyle \sum _{t=1}^{n}}\left|{Q}_{obs\left(t\right)}-{Q}_{sim\left(t\right)}\right|$ |

Maximum absolute error (MaxAE) | $\underset{1\le t\le n}{\mathrm{max}}\left|{O}_{obs\left(t\right)}-{Q}_{sim\left(t\right)}\right|$ |

Total mean squared error (TMSE) | $\frac{1}{n}{\displaystyle \sum _{t=1}^{1}}\left({Q}_{obs}\left(t\right)-{Q}_{sim}\left(t\right)\right)$ |

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**MDPI and ACS Style**

Kim, S.W.; Kwon, S.H.; Jung, D.
Development of a Multiobjective Automatic Parameter-Calibration Framework for Urban Drainage Systems. *Sustainability* **2022**, *14*, 8350.
https://doi.org/10.3390/su14148350

**AMA Style**

Kim SW, Kwon SH, Jung D.
Development of a Multiobjective Automatic Parameter-Calibration Framework for Urban Drainage Systems. *Sustainability*. 2022; 14(14):8350.
https://doi.org/10.3390/su14148350

**Chicago/Turabian Style**

Kim, Seon Woo, Soon Ho Kwon, and Donghwi Jung.
2022. "Development of a Multiobjective Automatic Parameter-Calibration Framework for Urban Drainage Systems" *Sustainability* 14, no. 14: 8350.
https://doi.org/10.3390/su14148350