The construction of low-impact development (LID) facilities has become an effective means to control urban waterlogging, but there is still a lack of scientific methods to achieve its accurate and reasonable planning and design. In this work, a high-precision hydrodynamic model was used to evaluate the construction effect of each LID facility scheme, and the fitted functional relationship was used to describe the law between the LID facility construction area and the construction effect. Finally, a genetic algorithm was used to automatically optimize the best LID facility construction scheme. Applying this method to the actual urban LID facility planning and construction, the optimal solution law is that the construction effect of single and combined LID facilities increases with the increase of construction cost. In the same low-cost construction scheme, the construction effect of combined LID facilities will be lower than that of single LID facilities, but with the continuous increase of construction cost, the construction effect of combined LID facilities will eventually be better than that of single LID facilities. According to this method, the decision-maker can get the optimal LID facility construction scheme to meet the actual engineering needs

  • Using the high-precision hydrodynamic model evaluated the construction effect of low-impact development (LID) facilities.

  • Based on the high-precision hydrodynamic model and the genetic algorithm, the new optimal design method for LID facilities is obtained.

  • This study demonstrated the optimal scheme for the construction of low-cost and high-cost LID facilities.

Graphical Abstract

Graphical Abstract
Graphical Abstract
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Graphical Abstract
Graphical Abstract
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Assisted decision-making, Genetic algorithm, Hydrodynamic model, Low-impact development (LID) facilities, Optimal design

With the quick development of human society, the process of urbanization is also advancing rapidly (Gu, 2019). The rapid development of urbanization brings convenience to human life, but it also poses a serious security threat (Octavianti, 2020; Teixeira et al., 2021). Among threats, the greater one to the life and property of the urban population is the risk of urban waterlogging caused by the increase of urban impervious underlying surface, the frequent occurrence of extreme weather, unreasonable wastewater discharge, and so on (Octavianti, 2020; Teixeira et al., 2021; Panagopoulos, 2022; Panagopoulos & Giannika, 2022). To control urban waterlogging and reduce the losses caused by urban waterlogging disasters, countries have begun to integrate the concept of low-impact development (LID) into urban planning and construction (Yawen et al., 2020), which means discrete and small-scale source control facilities are developed in urban planning to reduce the runoff generated by rainstorms and change the influence of humanistic structure at a minimum (Wu et al., 2014). The construction of LID facilities can effectively reduce urban runoff (Pei et al., 2020). However, to achieve the precise control of waterlogging, it is necessary to scientifically study the construction methods of LID facilities. At this stage, some advanced methods have had enlightening effects on the treatment of urban waterlogging (Dasineh et al., 2021; Ilderomi et al., 2022; Kumar et al., 2022). Most of the existing studies use hydrological models to evaluate the construction of LID when planning and designing LID facilities. Zhou et al. (2021) simulated the regional runoff control effect by setting different LID facility combinations and obtained the quantitative effect of different LID facility combinations on the runoff control in the study area. Li et al. (2021) simulated and calculated the rain runoff ptraditional tradition development (TD) and LID based on the storm flood management model (SWMM), established the relationship function between LID layout proportion and total runoff and construction cost by using the multiple regression method and obtained the optimization scheme of LID facility construction proportion and construction effect to reduce the maximum flood volume at the lowest cost. Sun et al. (2020) based on the second generation non-dominated sorted genetic (NSGA-II) algorithm, using the storm flood management model, took a certain area as an example, constructed a set of design schemes to minimize construction costs and maximize hydrological and water quality benefits, and quantitatively obtained the construction effect of LID facilities.

Based on the concept of the water cycle, the hydrological model calculates the precipitation, evaporation, runoff infiltration, and other processes in the region, which has the advantages of a comprehensive calculation process and fast calculation speed. However, when the hydrological model simulates the runoff generation and concentration process on the surface, the simulation results can only get the flow process at the outlet of the basin and cannot give the hydraulic characteristic elements of a specific location. Moreover, due to the weak ability of the hydrological model to describe complex terrain and strong empirical dependence on parameters, the accuracy of the simulation results is affected.

Hydrodynamic models usually calculate the waterlogging process on the surface by solving the two-dimensional Saint Venant equation. Compared with hydrological models, the calculation results are more accurate, the parameters are fewer, and more hydraulic information can be obtained. Meanwhile, the genetic algorithm has great robustness for solving nonlinear problems. Therefore, to get a better optimal design scheme for LID facilities, this study originally proposed an automatic optimization method for the LID facility construction scheme based on the high-precision hydrodynamic model and the genetic algorithm. Compared with the common optimization design method of LID facilities, this method can obtain a better optimization design scheme of LID facilities because of the introduction of the high-precision hydrodynamic model and the solution of the genetic algorithm.

This method evaluates the waterlogging control effect after the LID facility construction through the peak ponding reduction in the simulation results of the high-precision hydrodynamic model, fitted the construction area of LID facilities with the construction effect to obtain an empirical formula, combined the empirical formula with the unit cost of LID facilities to carry out the iterative calculation of genetic algorithm, and finally obtained the solution set of the lowest construction cost and best construction effect optimization scheme of LID facility optimal design. The construction laws of single and combined LID facilities under each construction cost were analyzed, and the relevant laws between the construction cost of LID facilities and the construction effect were obtained, which provided a set of new methods with strong applicability for the construction of urban LID facilities and the control of urban waterlogging.

Overview of the study area

The study area is located in the central core area of Xixian New Area, Xi'an City, Shaanxi Province, China. The area has a typical semi-humid continental monsoon climate, with concentrated rainfall in summer and mostly in the form of rainstorms, which is easy to cause natural disasters such as floods and waterlogging. The total area of the study area is 1.08 km2, with dense buildings, diverse land use types, crisscrossed road networks, complex underlying surfaces, and typical urbanization attributes. It is representative to use this area to study the optimal design of LID facilities. The overview of the study area is shown in Figure 1.
Fig. 1
Overview of the study area.
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Overview of the study area.

Fig. 1
Overview of the study area.
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Overview of the study area.

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High-precision coupled hydrodynamic model

2D surface hydrodynamic module

The 2D surface hydrodynamic module takes the two-dimensional shallow water equation as the governing equation, which ignores the kinematic viscosity term, turbulent viscosity term, wind stress, and Coriolis force. The vector form of the two-dimensional nonlinear shallow water equation is as follows (Hou et al., 2020):
formula
(1)
formula
(2)
where q is the variable vector, m2/s; h is the water depth, m; qx and qy are the unit width discharges in x and y directions, respectively, m3/(s·m); g is the gravity acceleration, m/s2; u and v are the velocity in x and y directions, m/s; F and G are the flux vectors in x and y directions, respectively; S is the source term vector; i is the infiltration and rainfall source term; zb is the bottom elevation, m; Cf = gn2/h1/3 is the Xie Cai coefficient, m1/2/s; n is the Manning coefficient, m1/3/s.

For the 2D surface hydrodynamic module, the finite volume method of the Godunov format is used to deal with complex spatial discretization, and the HLLC approximate Riemann solver is used to deal with the rapid flow and discontinuity of mass flux and momentum flux on the calculation unit interface, and the hydrostatic reconstruction method is used to deal with the common negative water depth at the dry and wet boundary of the hydrodynamic model, so as to ensure the stability of the model calculation. Meanwhile, the graphics processor (GPU)-accelerated calculation technology is introduced to accelerate the model's calculation process for ensuring the calculation accuracy and greatly improving the calculation efficiency (Hou et al., 2013).

1D pipe network hydrodynamic module

The one-dimensional pipe module simulates the pressurized flow in the pipe by calculating the one-dimensional Saint Venant equation. The solution process ignores the inertia force, and the finite difference method is used for numerical discretization. The solution equation of pipeline pressure flow is as follows:
formula
(3)
formula
(4)
where A is the cross-sectional area of the pipeline, m2; Q is the pipeline flow, m3/s; t is time, s; s is the distance of the fixed cross-section along the flow, m; Sf is the friction gradient, ⁠.

Coupling method of 1D and 2D hydrodynamic models

The coupling calculation between the 1D pipe network and the 2D surface is mainly carried out through catchment nodes such as rainwater wells. The weir flow formula or hole flow formula is used to calculate the amount of water flowing into rainwater wells on the surface.
formula
(5)
where Qin is the flow of surface water into the pipe network, m3/s; cw is the weir flow coefficient; co is the pore flow coefficient; Ci is the perimeter of the inlet of the rainwater well, m; h2D is the surface water depth, m, where h2D = Z2D– Z1D; Zb2D is the surface elevation; Z2D is the surface water level, m; Z1D is the water level elevation in the rainwater well, m; Ai is the sectional area of the inlet of the rainwater well, m2.
When the water depth in the rainwater well exceeds the elevation of the surface water, and the surface overflow occurs, the overflow is calculated by the orifice flow formula:
formula
(6)
where Qout is the overflow flow from the rainwater well to the surface, m3/s.

Infiltration module

To more accurately describe the infiltration capacity of various land use types, it is obtained by coupling the Green–Ampt infiltration model into the source term of two-dimensional shallow water equation. The Green–Ampt conception of the infiltration process is one in which infiltrated water moves vertically downward in a saturated layer, beginning at the surface (Hou et al., 2018). To describe the characteristics of soil water infiltration, the Green–Ampt infiltration model was applied according to the basic assumptions (Hsu & Hilpert, 2011):
formula
(7)
where fp represents the infiltration rate, cm/min; Ks is the saturated hydraulic conductivity, cm/min; and are the initial soil moisture and saturated water content, respectively, cm3/cm3; Sf represents the humid front suction, cm; tp represents the start time of inundation after rainfall begins, min; R is the rainfall intensity, cm/min; and Ip is cumulative infiltration, cm, where Ip = tpR.

In order to reflect the construction effect of common LID facilities, permeable pavement, and rainwater gardens are selected for the optimization design. The construction effect is obtained by adjusting its area and infiltration parameters. The Green–Ampt infiltration model is still used to describe the infiltration process of LID facilities. The infiltration formula is shown in Equation (7). Relevant parameters can be measured or referred to local standards and specifications.

The methods of LID facility optimization

Figure 2 is the key flow chart of the LID facility optimization design method based on the genetic algorithm and the hydrodynamic model-assisted decision-making in this work.
Fig. 2
Flow chart of LID facility optimization design.
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Flow chart of LID facility optimization design.

Fig. 2
Flow chart of LID facility optimization design.
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Flow chart of LID facility optimization design.

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Fig. 3
Schematic diagram of LID facility layout method.
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Schematic diagram of LID facility layout method.

Fig. 3
Schematic diagram of LID facility layout method.
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Schematic diagram of LID facility layout method.

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Genetic algorithm

A genetic algorithm is a method to search for the optimal solution by simulating the natural evolution process. The algorithm can input all kinds of linear, nonlinear, univariate, or multivariate functions, and it has good adaptability to various constraints, which makes the algorithm widely used in engineering optimization (Li et al., 2015).

Based on the genetic algorithm, taking the maximum reduction of peak water accumulation, the minimum construction cost of LID facilities, and the maximum construction area of LID facilities as the objective functions, and taking the maximum construction area of LID facilities as the constraint conditions, the optimal Pareto solution set is obtained through the iterative calculation to realize the optimal design of LID facilities.

The method of LID facility layout

(1) Using the coupled hydrodynamic model to obtain the waterlogging risk map under the original conditions in the study area. (2) Identifying the local place with serious waterlogging in the waterlogging risk map and then selecting several initial construction points of LID facilities in combination with land use types and actual engineering needs. (3) Taking these initial construction points as the center, the corresponding LID facilities are added outward in turn until the maximum boundary of LID facilities can be built, is reached, and the different construction area conditions of various LID facilities are obtained. The layout method of Lid facilities for each additional area condition is shown in Figure 3.

Objective function and constraint condition

  • (1)
    LID facility cost function:
    formula
    (8)
    where Ni is the construction area of type i LID facilities, m2; Mi is the construction unit price of type i LID facilities, yuan/m2.
  • (2)
    Surface peak ponding reduction function:
    formula
    (9)
    where F0 is the total accumulated water volume at the peak ponding time before the construction of LID facilities, m3, f(Ni) is the total peak surface water volume after the construction of LID facilities, m3.
  • (3)
    Overall objective function of LID facility optimization design:
    formula
    (10)
  • (4)

    Constraint conditions

In the actual project, the construction boundary of LID is definite, so it is necessary to restrict the area conditions of the optimization model. The reconstruction area of the rainwater garden mainly depends on the urban green space and the available open space around the building. The reconstruction area of a pervious pavement depends on the area of urban pedestrian traffic roads, squares, and other facilities. The following LID facility constraints can be determined according to the corresponding facilities in the study area:
formula
(11)
where N1 and N2 are the actual construction area of pervious pavement and rainwater garden, m2; N1max and N2max are the maximum area of permeable pavement and rainwater garden in the study area, respectively, m2.

Hydrodynamic model building

In this study, a high-precision numerical model coupled with 1D and 2D hydrodynamic processes is used to simulate the process of waterlogging in the study area. The input data of the model mainly include rainfall, terrain, land use type, infiltration, pipe network, and other model parameters.

Main model data

Rainfall data come from the Chicago rainfall formula of the Xixian new area with local characteristic rainfall attributes, as shown in Equation (12). As LID facilities mainly alleviate urban waterlogging in a low recurrence period (Zuo et al., 2017), Chicago design rainfall data with a total rainfall duration of 2 h in the return period of five years are used as the main input rainfall data of this study (Hou et al., 2020):
formula
(12)
where q is the rainstorm intensity, L/(s·hm2); P is the return period, a; t is the rainfall duration, min.
The terrain data in the study area are 1 m high-precision digital elevation model (DEM) data. The data come from the digital interpretation of the terrain elevation cruise measurement of the study area by unmanned aerial vehicle. The area is finally divided into 1,351 × 1,300, a total of 1,756,300 computing grids. The terrain data of the study area are shown in Figure 4.
Fig. 4
Terrain data.
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Terrain data.

The land use type data required by the model is obtained by the maximum likelihood classification of the orthophoto map of the study area, which is divided into five different land use types, including building, road, bare land, forest land, and grassland. The land use type data of the study area are shown in Figure 5.
Fig. 5
Land use type data.
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Land use type data.

Fig. 5
Land use type data.
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Land use type data.

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In order to reflect the layout optimization effect of typical LID facilities, based on the concept of LID of urban sponge construction, this work selects two LID facilities that have a great impact on the process of runoff generation and concentration, pervious pavement, and rainwater garden for the optimization simulation design study. The infiltration process of each land use type is described by the Green–Ampt model and the specific parameters are determined according to the relevant literature and the calibration value of measured rainfall parameters.

The pipe network layout in the study area is generalized into 58 conduits, 52 junctions, and three outlets. The Manning coefficient of conduits is taken as 0.017.

Model validation and parameter calibration

In order to improve the accuracy and reliability of the model, the model was calibrated and verified by using the measured rainfall data of the Xixian new area meteorological station on August 25, 2016, as well as the survey of submerged depth and submerged area. The measured rainfall data in the area are shown in Figure 6.
Fig. 6
Measured rainfall data.
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Measured rainfall data.

Fig. 6
Measured rainfall data.
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Measured rainfall data.

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The simulation errors of the three survey monitoring points under the measured rainfall are shown in Table 1. From this table, it can be seen that the simulated results of waterlogging monitoring locations are highly consistent with the measured results.

Table 1

Comparison between simulated ponding and measured water.

Monitoring pointWaterlogging area (m2) (simulated/measured)Waterlogging depth (m) (simulated/measured)
P1 916/>1,000 36/>25 
P2 1,566/>1,600 40/>40 
P3 1,734/>1,800 52>48 
Monitoring pointWaterlogging area (m2) (simulated/measured)Waterlogging depth (m) (simulated/measured)
P1 916/>1,000 36/>25 
P2 1,566/>1,600 40/>40 
P3 1,734/>1,800 52>48 
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The relative error of waterlogging's area and depth is 4.7 and 16.2%, respectively. Meanwhile the standard deviation of waterlogging's area and depth is 0.027 and 0.177, respectively. The comparison results show that the simulated results of urban waterlogging are consistent with the actual monitoring results, which shows that the simulation results of the model for urban waterlogging are relatively reliable. Finally, the relevant parameters determined according to the measured rainfall rate are shown in Table 2.

Table 2

Land use types and parameters.

ParametersForest landBuildingRoadBare landGrasslandPervious pavementRainwater garden
Ks(mm/s) 0.0105 0 0 0.00405 0.01 0.04 0.027 
S(mm) 19.17 0 0 4.21 6.39 80 60 
D-value of Ks and S 0 0 0.3 0.3 0.3 0.3 0.3 
Manning coefficient 0.4 0.015 0.014 0.08 0.24 0.03 0.24 
ParametersForest landBuildingRoadBare landGrasslandPervious pavementRainwater garden
Ks(mm/s) 0.0105 0 0 0.00405 0.01 0.04 0.027 
S(mm) 19.17 0 0 4.21 6.39 80 60 
D-value of Ks and S 0 0 0.3 0.3 0.3 0.3 0.3 
Manning coefficient 0.4 0.015 0.014 0.08 0.24 0.03 0.24 
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Simulation results of the hydrodynamic model

Due to the large area and complex land use types in this study area, the area constraints should be determined according to the actual situation of the study area when LID facilities are designed, planned, and constructed. In this study, the reconstruction method of LID facilities is as follows: the grassland in the original land use types and the bare land near the buildings are newly built into the rainwater garden, and the bare land outside the roads is newly built into the permeable pavement. As the urban trunk road carries heavy traffic activities, if it is replaced with permeable pavement, the daily maintenance cost is huge, and the original traffic activities are greatly affected. Therefore, the LID facilities of the regional traffic trunk road will not be changed. Thus, the maximum construction area of permeable pavement is 225,622 m2, and the maximum construction area of a rainwater garden is 135,175 m2. According to the constraints in Section 2.3.2, this is taken as the upper limit of the maximum construction area of the optimization model.

The waterlogging risk map adopts the water depth map under the return period of five-year rainfall, as shown in Figure 7. Integrating the ponding depth, land use type, and building damage characteristics determines the initial construction point of LID facilities as shown in Figure 8. This study took the three construction initial points as the construction center and successively added various LID facilities to the outer boundary until getting the maximum constructable boundary. For capturing the relationship between the construction area of LID facilities and the construction effect accurately, according to the layout method of LID facilities in section 2.3.2, increased the area of two LID facilities five times successively, and finally got six groups of simulation results for each single facility of pervious pavement and rainwater garden, as well as 36 groups of simulation results for the combined LID facilities, as shown in Table 3.
Table 3

Simulation results of different LID facility construction cases.

Case numberPervious pavement area (m2)Rainwater garden area (m2)Peak ponding reduction (m3)Case numberPervious pavement area (m2)Rainwater garden area (m2)Peak ponding reduction (m3)
Case 1 19,012 0 433.6 Case 19 107,591 109,955 2,265.5 
Case 2 19,012 30,588 678.1 Case 20 164,091 0 2,124.8 
Case 3 0 30,588 316.2 Case 21 164,091 30,588 2,368.5 
Case 4 39,787 0 858 Case 22 164,091 90,411 2,697.7 
Case 5 39,787 30,588 1,101 Case 23 164,091 109,893 2,725 
Case 6 39,787 90,411 1,431.6 Case 24 164,091 117,184 2,742.3 
Case 7 107,591 0 1,665.3 Case 25 225,622 0 2,796.8 
Case 8 107,591 30,588 1,908.8 Case 26 225,622 30,588 3,041.3 
Case 9 107,591 90,411 2,238 Case 27 225,622 90,411 3,371.1 
Case 10 225,622 109,893 3,398.3 Case 28 19,012 109,955 1,238.9 
Case 11 225,622 117,184 3,415.6 Case 29 19,012 117,184 1,285.4 
Case 12 225,622 135,175 3,443.4 Case 30 39,787 109,893 1,532.1 
Case 13 164,091 135,175 2,809.5 Case 31 39,787 117,184 1,578.4 
Case 14 107,591 135,175 2,380.8 Case 32 107,591 117,184 2,310.5 
Case 15 39,787 135,175 1,648.7 Case 33 0 90,411 783.1 
Case 16 19,012 135,175 1,355.7 Case 34 0 109,955 886.5 
Case 17 0 135,175 1,003.1 Case 35 0 117,184 932.8 
Case 18 19,012 90,411 1,138.5 Case 36 0 0 0 
Case numberPervious pavement area (m2)Rainwater garden area (m2)Peak ponding reduction (m3)Case numberPervious pavement area (m2)Rainwater garden area (m2)Peak ponding reduction (m3)
Case 1 19,012 0 433.6 Case 19 107,591 109,955 2,265.5 
Case 2 19,012 30,588 678.1 Case 20 164,091 0 2,124.8 
Case 3 0 30,588 316.2 Case 21 164,091 30,588 2,368.5 
Case 4 39,787 0 858 Case 22 164,091 90,411 2,697.7 
Case 5 39,787 30,588 1,101 Case 23 164,091 109,893 2,725 
Case 6 39,787 90,411 1,431.6 Case 24 164,091 117,184 2,742.3 
Case 7 107,591 0 1,665.3 Case 25 225,622 0 2,796.8 
Case 8 107,591 30,588 1,908.8 Case 26 225,622 30,588 3,041.3 
Case 9 107,591 90,411 2,238 Case 27 225,622 90,411 3,371.1 
Case 10 225,622 109,893 3,398.3 Case 28 19,012 109,955 1,238.9 
Case 11 225,622 117,184 3,415.6 Case 29 19,012 117,184 1,285.4 
Case 12 225,622 135,175 3,443.4 Case 30 39,787 109,893 1,532.1 
Case 13 164,091 135,175 2,809.5 Case 31 39,787 117,184 1,578.4 
Case 14 107,591 135,175 2,380.8 Case 32 107,591 117,184 2,310.5 
Case 15 39,787 135,175 1,648.7 Case 33 0 90,411 783.1 
Case 16 19,012 135,175 1,355.7 Case 34 0 109,955 886.5 
Case 17 0 135,175 1,003.1 Case 35 0 117,184 932.8 
Case 18 19,012 90,411 1,138.5 Case 36 0 0 0 
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Fig. 7
Waterlogging risk map.
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Waterlogging risk map.

Fig. 7
Waterlogging risk map.
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Waterlogging risk map.

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Fig. 8
Schematic diagram of the initial construction point.
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Schematic diagram of the initial construction point.

Fig. 8
Schematic diagram of the initial construction point.
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Schematic diagram of the initial construction point.

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Optimization function

In order to quantitatively characterize the construction law of LID facilities, it is necessary to fit the construction area and effect of LID facilities with appropriate functions according to the simulation results, which can be applied to the genetic algorithm. The construction area and effect fitting image of single and combined LID facilities are shown in Figure 9, and the determination coefficient R2 of the fitting formula is shown in Table 4.
Table 4

Fitting formula and R2.

LID facilitiesFitting formulaR2
Pervious pavement  0.994 
Rainwater garden  0.999 
Combined LID facilities  0.932 
LID facilitiesFitting formulaR2
Pervious pavement  0.994 
Rainwater garden  0.999 
Combined LID facilities  0.932 
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Fig. 9
Fitting function images of various LID facilities. (a) Fitting function image of pervious pavement. (b) Fitting function image of rainwater garden. (c) Fitting function image of combined LID facilities.
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Fitting function images of various LID facilities. (a) Fitting function image of pervious pavement. (b) Fitting function image of rainwater garden. (c) Fitting function image of combined LID facilities.
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Fitting function images of various LID facilities. (a) Fitting function image of pervious pavement. (b) Fitting function image of rainwater garden. (c) Fitting function image of combined LID facilities.

Fig. 9
Fitting function images of various LID facilities. (a) Fitting function image of pervious pavement. (b) Fitting function image of rainwater garden. (c) Fitting function image of combined LID facilities.
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Fitting function images of various LID facilities. (a) Fitting function image of pervious pavement. (b) Fitting function image of rainwater garden. (c) Fitting function image of combined LID facilities.
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Fitting function images of various LID facilities. (a) Fitting function image of pervious pavement. (b) Fitting function image of rainwater garden. (c) Fitting function image of combined LID facilities.

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In the table, x1 and x2 are the construction area of pervious pavement and rainwater garden, respectively, and y1, y2, and y3 are the peak ponding reduction of pervious pavement, rainwater garden, and combined LID facilities, respectively.

Those fitting functions are nonlinear functions, which means that the optimization process will have amounts of calculation time, in order to ensure the accurate, uniform distribution of the optimization objective function value and save the operation time as much as possible. Finally, we determined the genetic algorithm optimization parameters as shown in Table 5.

Table 5

Genetic algorithm parameters.

ParametersOptimal individual coefficientPopulation sizeMaximum evolutionary algebraStop algebraFitness function deviation
Values 0.3 100 200 200 10−8 
ParametersOptimal individual coefficientPopulation sizeMaximum evolutionary algebraStop algebraFitness function deviation
Values 0.3 100 200 200 10−8 
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The construction price of LID facilities in the optimization model consults the technical guide for sponge city construction and the actual construction cases of cities with the same economic level, and the unit price of pervious pavement is 200 yuan/m2, and the unit price of rainwater garden is 500 yuan/m2.

Optimization results and analysis

Through the genetic algorithm, the cost of LID facilities and waterlogging control effect in the study area are optimized, and the Pareto optimal solution sets of several single and combined LID construction schemes are obtained. The calculation results of the optimization model are shown in Figure 10. Each optimal solution includes the construction area, total construction cost, and peak ponding reduction of the corresponding LID facilities. To accurately evaluate the urban waterlogging control effect of the centralized LID facilities obtained from the optimization algorithm, three schemes for each single and combined LID facility are selected for comparative analysis according to the construction cost from small to large.
  • (1)

    Optimization results of single LID facilities

Fig. 10
Pareto solution set of combined LID facilities. (a) Pareto solution set of pervious pavement. (b) Pareto solution set of rainwater garden. (c) Pareto solution set of combined LID facilities.
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Pareto solution set of combined LID facilities. (a) Pareto solution set of pervious pavement. (b) Pareto solution set of rainwater garden. (c) Pareto solution set of combined LID facilities.
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Pareto solution set of combined LID facilities. (a) Pareto solution set of pervious pavement. (b) Pareto solution set of rainwater garden. (c) Pareto solution set of combined LID facilities.

Fig. 10
Pareto solution set of combined LID facilities. (a) Pareto solution set of pervious pavement. (b) Pareto solution set of rainwater garden. (c) Pareto solution set of combined LID facilities.
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Pareto solution set of combined LID facilities. (a) Pareto solution set of pervious pavement. (b) Pareto solution set of rainwater garden. (c) Pareto solution set of combined LID facilities.
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Pareto solution set of combined LID facilities. (a) Pareto solution set of pervious pavement. (b) Pareto solution set of rainwater garden. (c) Pareto solution set of combined LID facilities.

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Figure 11 shows the comparison effect of optimization solutions among the cases of single LID facilities. It can be seen from the optimization results that under the low-cost construction case, for the two kinds of LID facilities constructed separately, the control effect of waterlogging cannot meet the optimal requirements due to the small construction area of LID facilities. But with the increase of LID facility construction area and cost, the peak ponding reduction of urban rainfall also decreases. If the decision-makers need to construct a single LID facility in the city, this optimal solution method can be selected according to the specific engineering needs and actual conditions to carry out the LID facility construction of the city.
Fig. 11
Comparative analysis of single LID facilities. (a) Construction cost of pervious pavement. (b) Peak ponding reduction of pervious pavement. (c) Construction cost of rainwater garden. (d) Peak ponding reduction of rainwater garden.
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Comparative analysis of single LID facilities. (a) Construction cost of pervious pavement. (b) Peak ponding reduction of pervious pavement. (c) Construction cost of rainwater garden. (d) Peak ponding reduction of rainwater garden.
View largeDownload slide

Comparative analysis of single LID facilities. (a) Construction cost of pervious pavement. (b) Peak ponding reduction of pervious pavement. (c) Construction cost of rainwater garden. (d) Peak ponding reduction of rainwater garden.

Fig. 11
Comparative analysis of single LID facilities. (a) Construction cost of pervious pavement. (b) Peak ponding reduction of pervious pavement. (c) Construction cost of rainwater garden. (d) Peak ponding reduction of rainwater garden.
View largeDownload slide
Comparative analysis of single LID facilities. (a) Construction cost of pervious pavement. (b) Peak ponding reduction of pervious pavement. (c) Construction cost of rainwater garden. (d) Peak ponding reduction of rainwater garden.
View largeDownload slide

Comparative analysis of single LID facilities. (a) Construction cost of pervious pavement. (b) Peak ponding reduction of pervious pavement. (c) Construction cost of rainwater garden. (d) Peak ponding reduction of rainwater garden.

Close modal

However, if we only deploy a single LID facility, it has great limitations for the effects of waterlogging control capacity in cities because of the different properties of various LID facilities. In this work, the peak ponding reduction under the maximum permeable pavement construction area is 2,796.8 m3, the rainwater garden is 1,003.1 m3, and the combined LID facility is 3,443.4 m3. Therefore, if only constructing the single LID facilities in the region cannot achieve the best effect of waterlogging, the decision-maker should carry out the combined LID facility construction.

  • (2)

    Combined LID facility optimization results

Figure 12 shows the comparison effect of optimization solutions among the cases of combined LID facilities. From the relevant data information of the three cases, it can be seen that the trend of urban waterlogging control effect is the same as the single LID facilities, showing the law that the peak ponding reduction increases with the increase of construction cost. However, according to the combined optimization solution set shown in Figure 10(c) that, with the increase of the construction cost of the combined LID facilities, the peak ponding reduction in the study area first increases slowly and then increases rapidly after the cost reaches 62,160,000 yuan. The main reason is that, compared to the sole LID facility scheme which has a powerful waterlogging control ability with the combined LID facility scheme, the LID facilities with weak waterlogging control ability in the combined LID facilities weaken the overall construction effect of the combined LID facility scheme. Finally, the combined LID facilities have a worse effect on waterlogging control than the single LID facilities in the same low-cost conditions.
Fig. 12
Comparative analysis of combined LID facilities. (a) Construction cost of combined LID facilities. (b) Peak ponding reduction of combined LID facilities.
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Comparative analysis of combined LID facilities. (a) Construction cost of combined LID facilities. (b) Peak ponding reduction of combined LID facilities.

Fig. 12
Comparative analysis of combined LID facilities. (a) Construction cost of combined LID facilities. (b) Peak ponding reduction of combined LID facilities.
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Comparative analysis of combined LID facilities. (a) Construction cost of combined LID facilities. (b) Peak ponding reduction of combined LID facilities.

Close modal
Figure 13 shows the waterlogging control effect of the single and combined LID facilities under the construction cost of 31.5 million yuan. Comparing the three LID construction cases, we can clearly get that the waterlogging control effect of sole permeable pavement is 136% higher than combined LID facilities. The existence of this result means the construction effect of the combined LID facility scheme is weaker than that of a single LID facility. The main reason for the result is that among the low-cost combined LID facilities, the construction area of LID facilities with a strong pervious effect is relatively small. In this combined LID facility scheme, the construction area of pervious pavement with a strong pervious effect is 2,429.6 m2, and the construction area of the rainwater garden is 61,771.0 m2. Therefore, comparing the scheme of sole permeable pavement with that of combined LID facilities, the latter shows a relatively lower effect of waterlogging control. However, the result does not mean that we should give up the construction scheme of combined LID facilities. If we have sufficient construction funds, with the gradual investment of construction costs, the waterlogging control effect of combined LID facilities will substantially exceed that of individual LID facilities eventually. Like the data in Table 3, under the maximum construction area of two kinds of LID facilities, the construction effect of the combined LID facility scheme is 23.1, and 243.3% higher than that of sole permeable pavement and rainwater garden.
Fig. 13
Comparative analysis of the single and combined LID facilities in the same low-cost case.
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Comparative analysis of the single and combined LID facilities in the same low-cost case.

Fig. 13
Comparative analysis of the single and combined LID facilities in the same low-cost case.
View largeDownload slide

Comparative analysis of the single and combined LID facilities in the same low-cost case.

Close modal

Overall, in the actual project, if we want to save costs while ensuring the construction effect, we should adopt the construction scheme of a single LID facility with strong water permeability. If our cost is slightly sufficient, we can adopt this method in this paper to obtain the optimal scheme with the lowest cost and the best construction effect. If the cost is not a problem, then we can transform all appropriate areas within the scope of study into LID facilities. Then, the maximum effect of waterlogging control will be achieved.

To get the planning and construction scheme of LID facilities more accurately and effectively, this work proposed a new optimal design method for LID facilities based on the genetic algorithm and the high-precision hydrodynamic model, which applies this method in the actual area of LID facilities construction and obtains the following conclusions:

  • The high-precision hydrodynamic model is used to evaluate the construction effect of LID facilities. The simulation effect of the model is evaluated through the measured rainfall and survey data. The relative error of waterlogging's area and depth is 4.7 and 16.2%, and the standard deviation of waterlogging's area and depth is 0.027 and 0.177, which shows that the evaluation method is effective and reliable.

  • Case results display that the construction effect of the sole permeable pavement scheme is 136% higher than that of the combined LID facility scheme in the same low construction cost. But, under the maximum construction area of two kinds of LID facilities, the construction effect of the combined LID facility scheme is 23.1 and 243.3% higher than that of the sole permeable pavement scheme and rainwater garden scheme. Therefore, to ensure the construction effect of LID facilities in the low-cost scheme, the construction scheme of the sole LID facility with strong water permeability can be adopted. If the construction cost is sufficient, the method in this work can be considered to obtain the combined LID facility construction scheme

  • Through the principle demonstration of the method and the analysis of the case, the high-precision hydrodynamic model can evaluate the construction effect of LID facilities more truly, and the genetic algorithm can obtain the optimal LID facility construction plan. The optimal design method for the LID facilities is accurate and reasonable and can be applied to the construction of the LID facility in the cities.

In the analysis of the results of this work, the effect of waterlogging control increases with the increase of LID facilities, which is consistent with the study results by Zhou et al. (2021). The law of the optimal solution set obtained by the genetic algorithm is consistent with the relevant scholars' study (Li et al., 2015, 2021), which shows that the method is accurate and effective for the optimal design of LID facilities. However, this work obtains the accurate relationship between the construction area of combined LID facilities and the construction effect through the method of fitting function, which has some limitations. Because of the excessive independent variables, it is difficult to guide the appropriate functional relationship to describe the quantitative relationship between the construction area of combined LID facilities and the construction effect when carrying out the optimal design of three or more LID facilities. Therefore, the future study can take the high-dimensional function fitting method as the study object and ultimately can get a better LID facility optimization design method.

This work was partly supported by the National Natural Science Foundation of China (52009104), the National Natural Science Foundation of China (52079106), and the Sino-German Mobility Programme (Grant No. M-0427).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.

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