Optimization of Low Impact Development Layouts for Urban Stormwater Management: A Simulation-Based Approach Using Multi-Objective Scatter Search Algorithm

Abstract

In recent years, the urgent need to mitigate stormwater runoff and address urban waterlogging has garnered significant attention. Low Impact Development (LID) has emerged as a promising strategy for managing urban runoff sustainably. However, the vast array of potential LID layout combinations presents challenges in quantifying their effectiveness and often results in high construction costs. To address these issues, this study proposes a simulation-optimization framework that integrates the Storm Water Management Model (SWMM) with advanced optimization techniques to minimize both runoff volume and costs. The framework incorporates random variations in rainfall intensity within the basin, ensuring robustness under diverse climatic conditions. By leveraging a multi-objective scatter search algorithm, this research optimizes LID layouts to achieve effective stormwater management. The algorithm is further enhanced by two local search techniques—namely, the ‘cost–benefit’ local search and path-relinking local search—which significantly improve computational efficiency. Comparative analysis reveals that the proposed algorithm outperforms the widely used NSGA-II (Non-dominated Sorting Genetic Algorithm II), reducing computation time by an average of 8.89%, 16.98%, 1.72%, 3.85%, and 1.23% across various scenarios. The results demonstrate the method’s effectiveness in achieving optimal LID configurations under variable rainfall intensities, highlighting its practical applicability for urban flood management. This research contributes to advancing urban sponge city initiatives by providing a scalable, efficient, and scientifically grounded solution for sustainable urban water management. The proposed framework is expected to support decision-makers in designing cost-effective and resilient stormwater management systems, paving the way for more sustainable urban development.

1. Introduction

In recent years, the escalating severity of environmental challenges has driven China to intensify its efforts in ecological governance and management. Among the critical issues within the urban ecological framework are the frequent occurrences of urban waterlogging and the prevalence of black and odorous water bodies in rivers and lakes, both of which pose significant threats to sustainable urban development. These problems primarily arise from the rapid expansion of impervious surfaces, such as buildings, roads, and pavements, which disrupt natural hydrological processes. Consequently, urban areas have experienced a decline in hydrological conditions, marked by recurrent waterlogging, deteriorating water quality, and inefficient utilization of rainwater resources [1,2,3].

Low-Impact Development (LID) has emerged as a promising complement to conventional gray infrastructure for managing urban runoff. LID strategies have gained widespread acceptance for their proven ability to mitigate the adverse effects of urbanization [4,5]. Furthermore, the sustainability, resilience, and adaptability of LID approaches are increasingly recognized for delivering significant benefits to society, the environment, and urban ecosystems [6,7]. For instance, studies have demonstrated that LID practices, such as rain gardens and permeable pavements, can reduce runoff volume by up to 30% while improving water quality [8]. Although LID is highly effective in reducing runoff, different layout configurations can produce varying outcomes [9,10]. Therefore, accurately calculating runoff is crucial to identifying the optimal layout combination, as highlighted by recent advancements in simulation-optimization frameworks [11,12].

However, the complexity of the runoff calculation process—driven by numerous influencing factors—makes it challenging to address solely through mathematical models. Consequently, simulation software is often required to accurately model and evaluate the effects of different LID configurations. Currently, a variety of drainage pipe network simulation models are available worldwide. Among these, the SWMM (Version: 5.2.2) stands out as a dynamic precipitation-runoff simulation model developed by the U.S. Environmental Protection Agency in 1970 [13]. Known for its free accessibility, robust data processing capabilities, and ability to simulate both hydraulic and water quality processes, SWMM has become widely adopted both domestically and internationally. Its open-source nature enables researchers to modify and customize the code to meet specific research requirements, further enhancing its utility.

For instance, Bah utilized SWMM5 to assess the feasibility of flood control strategies based on LID measures in an area near Conakry, the capital of Guinea [14]. Their simulations demonstrated that all LID measures—whether implemented individually or in combination—effectively reduced runoff peaks and nodal flooding, with combined measures yielding the most significant results. Similarly, Bruno José de Oliveira Sousa analyzed construction costs and runoff reduction across seven LID scenarios in a 32 km² watershed in central and western Brazil [15]. Their study considered permeable pavement (PP), rainwater harvesting (RWH), infiltration trenches (IT), and their combinations. They found that IT alone achieved a 15% reduction in runoff at a lower cost, equating to a 2.6% reduction in runoff per $1 million invested.

The vast number of possible LID layout combinations and the numerous parameters involved make exhaustive searches computationally impractical. This underscores the need for advanced intelligent search algorithms to efficiently identify optimal solutions. Recent studies have increasingly focused on addressing LID layout optimization challenges. Among these, the Multi-Objective Scatter Search (SS) algorithm has emerged as a promising solution due to its high search efficiency and suitability for combinatorial optimization problems [16].

Sheldon [17] developed a multi-objective stochastic optimization algorithm based on a random permutation strategy, offering valuable insights for improving local search efficiency in random allocation problems. Similarly, Laguna [16] demonstrated the effectiveness of multi-objective decentralized search algorithms in large search spaces, while Hakli [11] applied these algorithms to optimize facility layouts under cost constraints with uncertain parameters.

In LID research, multi-objective scatter search algorithms have gained traction. For instance, Duarte Lopes used the NSGA-II algorithm to optimize LID layouts in Brasilia, Brazil, considering runoff reduction and cost minimization [9]. Xie explored the applications, advantages, and limitations of multi-objective optimization (MOO) algorithms, such as GAs, NSGA-III, and PSO, in LID facility layout optimization, proposing future research directions to enhance urban stormwater management [7]. Lu introduced a framework for optimizing LID configurations based on spatial flood damage and life cycle costs (LCC), testing the effectiveness of DYCORS, TuRBO, and PSO [10].

Collectively, these studies highlight the potential of MOO algorithms in optimizing LID facility combinations and advancing stormwater management strategies.

Recent advancements in integrating algorithms with the SWMM have significantly enhanced simulation outcomes. For instance, Dell [18] developed SWMM-LITE, a low-impact technology assessment tool, and compared its hydrological outputs with the original SWMM and the National Rainwater Calculator (SWC). Yang incorporated a distributed time-varying gain model (DTVGM) into SWMM (DTVGM-SWMM) to simplify complexity while preserving hydrological process representation [19]. Jeung introduced SWMM-RL, a deep reinforcement learning model, to optimize SWMM parameters interactively in real environments using a reward–punishment system [20].

Heuristic algorithms have proven particularly effective for LID combinatorial optimization with SWMM. Wang and Zhang combined SWMM with NSGA-II, optimizing LID facility costs and linking landscape configuration for rainwater management [5,21]. Li [22] proposed SWMM-FTC, replacing SWMM’s routing module with a flow transport chain (FTC) and using GDE3 for LID layout optimization. Zhao [8] developed LSTM-SWMM, a hybrid model combining SWMM with LSTM neural networks for efficient urban flood simulation. Du [12] demonstrated the superiority of DSRC over PSO and MOPSO in SWMM parameter optimization, evaluating performance using metrics like NSE, REv, REp, and EPt. Liu [3] introduced BK-SWMM, a multi-scale stormwater model framework that leverages Bayesian Information Criterion (BIC) and K-means clustering to analyze SWMM uncertainty parameters.

This study builds on these advancements by integrating SWMM with a multi-objective optimization algorithm to address the computational complexity of LID layout optimization, offering a robust and efficient solution for urban stormwater management.

This study integrates the SWMM with a heuristic algorithm to optimize LID layouts. A conceptual model is developed to minimize LID costs while accounting for plot-specific characteristics. The Multi-Objective Decentralized Search Algorithm is employed, incorporating a path-relinking local search method to enhance convergence, avoid local optima, and reduce computation time. Beyond average rainfall intensity, the study addresses stochastic rainfall conditions by transforming the problem into sub-catchment-specific rainfall intensity scenarios using the expected value method. Extensive simulations and optimizations are conducted for both average and stochastic rainfall models. The novelty of this research lies in its integration of SWMM with MODSA for multi-objective optimization under stochastic rainfall conditions, offering a robust and efficient solution for LID layout optimization. Results demonstrate the algorithm’s effectiveness in minimizing runoff and costs under varying rainfall conditions, advancing urban stormwater management strategies.

2. Experiments and Methods

2.1. Site Description

The study area selected for this research is the Hot Spring District of Fuzhou City, China. This area was chosen due to its representative urban drainage challenges, which are common in rapidly urbanizing regions. Fuzhou’s Hot Spring District is characterized by a high population density, extensive impervious surfaces, and frequent urban waterlogging events, making it an ideal case study for evaluating the effectiveness of LID strategies. The district’s unique combination of urban development and natural hydrological features provides a microcosm of the broader challenges faced by cities worldwide in managing stormwater runoff and mitigating flood risks. The total area of this district is 12.0 hm², with a land area of 7.8 hm². The types of land surface in the demonstration area include roofs, green spaces, and roads, which account for 28%, 42%, and 30% of the land area, respectively. The hydraulic parameters involved are listed in

Table 1. Relevant parameter setting of hydraulic model.

Underlying Surface	Flow Generation Model	Runoff Coefficient	Initial Penetration Rate (mm/h)	Stable Permeability (mm/h)	Attenuation Rate (mm/h)	Confluence Model	Confluence Parameter
roofing	Fixed	0.98	-	-	-	SWMM	0.012
pavement	Fixed	-	-	-	-	SWMM	0.01
greenbelt	Horton	-	79.38	12.7	4.34	SWMM	0.2
massif	Horton	-	79.38	13.42	4.34	SWMM	0.3

.

Using the collected data on topography, land use, and the pipe network, the study area is modeled and subdivided into 423 sub-catchment areas. The area also includes 560 inspection well nodes, 560 pipeline sections, and 3 rainwater discharge outlets. The initial model parameter values are based on typical values from the user manual. For calibration and verification, six sets of rainfall data from a rain gauge station in Fuzhou during 2017, along with the corresponding water quality and quantity data, are selected.

2.2. Simulation Model Description

The SWMM simulation model under the LID layout scheme divides the urban area into multiple catchment areas based on elements such as terrain data, road networks, and building layouts. Each catchment area is further subdivided into smaller-to-capture specific characteristics of land use, runoff generation, confluence, and pipeline layout. SWMM is capable of simulating the stormwater transport process in urban drainage systems [6,23], considering various factors ranging from runoff generation and confluence to drainage flow transportation.

The drainage area consists of both permeable and impermeable surfaces, with impermeable surfaces categorized into two types: those with low-lying storage capacity and those without. After rainfall, each drainage area receives a specific temporal and spatial distribution of precipitation, which then undergoes two stages—runoff and confluence—before entering pipelines or river channels and ultimately flowing to the outlet of the catchment area. The flood process at the outlet of each catchment is determined through discharge calculations, and these are superimposed to obtain the overall flood situation for the catchment. This model is used for LID layout simulations in SWMM to predict the stormwater formation process. The software can also monitor and record various parameters such as flow rate, water level, and water quality in each pipeline and channel. The connectivity between LID facilities contributes to a higher percentage of runoff reduction [24]. A conceptual model of the SWMM simulation under the LID layout combination scheme is shown in Figure 1.

In SWMM, the hydrological process simulation of LID is achieved through the combination of vertical functional layers. These layers are primarily divided into surface, pavement, soil, aquifer, drainage, and drainage cushion layers. The LID parameters are set based on the area of a single plot, allowing the software to flexibly set the scale and area of selected facilities across different catchment areas. This provides a calculation basis for the optimal selection and deployment of LID measures. In this study, seven types of LID facilities were mainly selected [25], and the necessity and selectivity of the corresponding vertical functional layers served as a reference for subsequent LID configurations.

The simulation of water balance, water quantity, and water quality in LID systems involves numerous parameters. Accurate configuration of these parameters is critical for developing effective pollution control schemes. Based on model manuals and engineering cases [25,26,27,28,29], this study summarizes the parameters for each functional layer of LID, providing a valuable reference for optimizing LID configurations.

In China, the concept of sponge cities is still emerging, and many regions lack standardized guidelines for LID construction. Unlike traditional drainage systems, which have well-established cost databases, LID cost estimation faces challenges due to varying site conditions, climate, topography, regulations, and construction technologies. Construction costs are a critical factor in decision-making for urban stormwater management [30]. To address this, this study compiles LID cost data from regions such as Xixian New District, Pingxiang, Hangzhou, Chongqing, Beijing, and Shanghai, along with the relevant literature [29,31]. Hangzhou’s cost standard is adopted in SWMM, with the area correction factor set to the default value of 1.

The effectiveness of LID measures is primarily evaluated based on runoff reduction and pollution control. This study summarizes data from related studies and engineering cases, including runoff reduction rates and pollutant removal rates, to assess LID performance [32,33,34]. LID performance varies significantly depending on climate, site conditions, design, and operational management. Constructed wetlands, infiltration ponds, grassed swales, and porous pavements generally perform better than traditional pipe networks, which typically remove 10–30% of TSS, 5–10% of COD and BOD5, and 10–20% of metals [29]. LID measures demonstrate superior control over urban rainfall runoff pollution compared to traditional systems.

The diverse structures and functions of LID measures require careful consideration of site-specific conditions during layout design. Optimal LID layouts aim to maximize benefits in stormwater quantity and quality control, ecological impact, and social benefits while minimizing costs. Key parameters include the location, connectivity, size, and type of LID facilities [2]. Studies show that implementing LID measures across 50% of an area can reduce runoff volume by 2–12%. For example, bioretention systems reduce overflow volume by 25% and lower stormwater management costs [35]. Grassed swales achieve average removal rates of 69% for TSS, 46% for TP, and 56% for TN [36].

This study integrates the structural, economic, and technical characteristics of various LID measures, drawing on practical application experiences and guidelines from domestic and international sources. Regional benchmarks for each LID measure are established based on these references [37], providing a foundation for comparison and selection.

In the hydraulic model of the pipe network and river channel within SWMM, there are certain universal characteristic parameters for the area that cannot be directly collected from data but must be determined through model calibration. This step is essential to ensure that the simulation model closely reflects the actual on-site conditions. Since one of the optimization objectives in this study is total runoff, runoff data are chosen as the key input for determining model parameters. In particular, the simulation of runoff indices for each sub-catchment affects the total runoff indices, making the parameters influencing total runoff the primary focus of this part of the study.

Two evaluation indices are commonly selected in the parameter determination of the hydraulic model: NSE (Nash-Sutcliffe Efficiency) and PBIAS (Percent Bias) [38,39]. The NSE is used to assess the overall performance of simulation results and is a widely accepted objective function for hydraulic models. Meanwhile, the PBIAS index reflects the deviation between the simulated and observed values, with lower values indicating better model accuracy [40]. This index is often applied in various simulation contexts. Specific grading indicators for these indices are shown in

Table 2. Analysis of performance evaluation indicators and measures.

Performance Evaluation Index	Objective Function Classification
	NSE	PBIAS
Excellent	(0.75, 1.00]	(0, 0.1]
Good	(0.65, 0.75]	(0.1, 0.15]
average	(0.50, 0.65]	(0.15, 0.25]
range	(0, 0.50]	>0.25

, and their formulas are as follows:

Nash efficiency coefficient:

$$NSE=1-{\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^n{(C_{obs,i}-C_{sim,i})}^2}{{{\displaystyle \sum }}_{i=1}^n{(C_{obs,i}-C_{m,obs})}^2}}$$

(1)

Root mean square error coefficient:

$$PBIAS=\sqrt{{\left[{\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^n\left(C_{obs,i}-C_{sim,i}\right)}{{{\displaystyle \sum }}_{i=1}^n{C}_{obs,i}}}\right]}^2}$$

(2)

where C_obs,i is the observed value at instant i, C_sim,i is the analog value at instant i, C_m,obs is the average value of the observed values, and n is the number of measured values, Cobs,i is the observed value at instant i, C_sim,i is the analog value at instant i, C_m,obs is the average value of the observed values at instant i, and n is the number of measured values.

When the performance evaluation indexes of NSE and PBIAS are both in the excellent and good interval, it is proven that the simulation model is similar to the actual situation, belongs to a relatively good simulation model, meets the simulation requirements, and can enter the next simulation calculation process. However, if it is in the average or poor interval, the parameters need to be further adjusted.

2.3. Mathematical Models Under Different Rainfall Conditions

2.3.1. Mathematical Model Under Average Rainfall Intensity

In the SWMM simulation model with LID layout, the average rainfall intensity represents a theoretical, hypothesized rainfall mode where precipitation is evenly distributed across the basin. Given that reducing runoff is a key objective for LID layouts, and considering that the cost of LID construction significantly exceeds that of standard construction and modifications, cost is also a primary consideration. Therefore, the optimization objective for the LID measure layout in this paper is to minimize both total runoff and LID construction costs.

The study area comprises 423 sub-catchment areas, with 60 of these capable of supporting LID installations. When evaluating an LID layout scheme, five attributes must be considered: land use type, groundwater level, service area, slope rate of confluence, and river buffer distance. The mathematical model is expressed as follows:

$$minQ=SWMM\left(g,T_{ij},q_i,F_{ij}\right)$$

(3)

$$minC={{\displaystyle \sum }}_j{{\displaystyle \sum }}_i{S}_{ij}{F}_{ij}{T}_{ij}$$

(4)

$$\sum F_{ij}\ge F_j,\forall i,\forall j$$

(5)

$$F_{ij}\ge 0,\forall i,\forall j$$

(6)

$$T_{ij}=\left\{0,1\right\},\forall i,\forall j$$

(7)

where g represents rainfall, i represents the number of sub-catchment area, j represents optional LID type, Q represents total runoff, $q_i$ represents rainfall intensity of sub-catchment area i, C represents the total cost of LID layout combination scheme, $S_{ij}$ represents the price per square meter of j LID type set in sub-catchment area i. $F_j$ represents the minimum area of LID layout of type j, $F_{ij}$ represents the area of sub-catchment area i of LID layout of type j, and decision variable $T_{ij}$ represents the setting of type j LID in sub-catchment area.

2.3.2. Mathematical Model Under Random Rainfall Intensity

In reality, rainfall distribution is often uneven. Thus, the concept of random rainfall intensity is introduced, which accounts for the random variation pattern of rainfall within the basin under actual conditions. This approach more accurately reflects real rainfall scenarios and enhances the simulation of rainwater runoff.

Specifically, frequency analysis of multi-year rainfall data can yield design rainfall for various durations at different design frequencies. These data are then combined randomly to determine the design rainfall intensity for each period. In this study, the stochastic rainfall intensity model is converted into a deterministic model using the Chicago rainfall process formula.

The Chicago rainfall model uses a statistical formula to design a typical rainfall process. The commonly used rainfall intensity formula for drainage pipe design in our country is:

$$\widehat{q_i}={\displaystyle \frac{167A\left(1+G_i\log p\right)}{{\left(t_d+b\right)}^c}},\forall i$$

(8)

where A is the rainfall of one year in the recovery period; $G_i$ is the rainfall variation parameter of sub-catchment area i, which is one of the parameters reflecting the intensity variation degree of rainfall in different return periods. p is the designed rainfall; $t_d$ is rainfall duration; b and c are constants, which together reflect the design rainfall of the return period with the decreasing intensity of duration extension.

According to the parameter data queried in the “Water Supply and Drainage Design Manual” of Beijing Municipal Engineering Design and Research Institute, and according to the rainfall duration of 240 min, the formula can be substituted:

$$\widehat{q_i}={\displaystyle \frac{2001\left(1+G_i\log p\right)}{{\left(240+8\right)}^{0.71}}},\forall i$$

(9)

$G_i$ is the rainfall variation parameter of sub-catchment area i. According to the slope rate and other conditions set in SWMM software (Version: 5.2.2) of each sub-catchment area in Fuzhou hot spring area, the corresponding range is distributed in (0.75–0.85). Through the above formula, the random rainfall intensity can be transformed into the determined rainfall intensity of each sub-catchment area.

2.4. Multi-Objective Scatter Search Algorithm

Given that the problem is NP-hard and SWMM simulations are time-consuming, this paper proposes a multi-objective scatter search algorithm combined with a local search algorithm. This approach effectively enhances search efficiency, identifies high-quality solutions within a short timeframe, and avoids becoming trapped in local optima.

The flowchart of the scatter search algorithm is presented in Figure 2. In the first step, the LID layout is binary-encoded and decoded. The second step involves initialization, where feasible solutions are generated by calculating the non-dominated crowding degree. Subsequently, two local search algorithms based on dominant attributes are introduced to accelerate convergence and refine the local search process.

2.4.1. Encoding and Decoding

There are seven possible types of LID facility placements, along with one case where no placement is implemented within each sub-catchment area. Consequently, each solution is binary-coded, with the length of the gene fragment for each sub-catchment area set to 3. Each gene represents one of eight scenarios, corresponding to the eight possible placement configurations. The correspondence between gene codes and LID types is shown in

Table 3. LID types correspond to gene codes.

LID Facility Placement Type	Genetic Code
No LID facility set (0)	000
Bioretention tank (1)	001
Green Roof (2)	010
Porous pavement (3)	011
Rain barrel (4)	100
Rain Garden (5)	101
Seepage ditch (6)	110
Grass planting ditch (7)	111

.

Each LID layout scheme can be considered a chromosome, represented as follows:

$$T^r=[T_1,T_2,T_3\dots T_n]$$

(10)

where r ∈ R, R denotes all the combination schemes of LID layout. $T^r$ represents the RTH gene fragment within these combinations. Where $T_1$ represents the first gene fragment in this combination scheme, which indicates the type of LID layout in the first sub-catchment area. Since there are n sub-catchment areas where LIDs can be arranged, each chromosome consists of n gene fragments.

2.4.2. Initialization

A diverse generation method for initial solutions is employed to create a wide range of LID layout schemes, forming the foundation for the initial search. In this study, a piecewise approach is used to generate initial solutions while controlling the total cost, thereby providing more varied combinations within specified cost intervals.

Initial solution generation setting: the total cost is divided into 50 equal intervals, with one initial scheme generated for each interval. For each LID placement scenario, which involves 60 sub-catchment regions, every possible LID placement scenario is treated as a chromosome. These 60 sub-catchment regions correspond to the 60 gene segments comprising a chromosome. Each gene segment has a length of 3, resulting in a chromosome length of 180. Thus, the total number of distinct chromosomes in the population is $2^{180}$.

2.4.3. Improving Initial Solution

To refine the initial solutions, the feasibility of each plan is evaluated, and the fitness function values are calculated. The number of initial solutions is set at 50; if this threshold is not met, the process loops back to the generation step. Using regional basic data, site selection is analyzed by comparing the regional applicability index and its benchmarks.

Initially, the solution space is immense, with a size of 423! * 5!. However, by excluding non-applicable areas such as roads and water systems, the remaining 60 sub-catchment regions available for LID placement reduce the solution space to 60! * 5!.

Next, the seven LID measures identified earlier are applied for preliminary site selection. Constraints on LID placement are incorporated into a standard tool as threshold conditions for analysis and selection. This reduces the search range for the initial solution from 60! * 5! to 60! * 2!.

2.4.4. Reference Set Update

Each iteration of the solution set for the reference set corresponds to a combination of LID layout schemes, with each scheme linked to two objective function values: total runoff and total cost. This makes it impossible to apply single-objective comparison methods. The solution set of LID layout combinations within the entire reference set undergoes non-dominated sorting, and each solution is assigned a corresponding level. This level serves as the basis for subsequent updates to the reference set.

Each LID layout combination scheme is represented by a chromosome $T^r$, and its objective function values are calculated using simulation software.

The number of chromosomes $n^r$ that dominate $T^r$ in the solution set is compared with the number of chromosomes $R_r$ dominated by $T^r$. In this study, both cost and total runoff are considered as objectives to be minimized. The domination rule is as follows:

$$Q_1<Q_{1}and{C}_1<C_1$$

(11)

After this comparison, it can be concluded that $T^1$ dominates $T^2$. As a result, $n^r$ for $T^2$ is incremented by 1, and $T^2$ is added to the set dominated by $T^1$. Through this iterative process, the dominating solution set R_r and the number of dominated solution sets $n^r$ for each LID layout combination scheme (chromosome $T^r$) are determined, allowing the computation of the domination level (rank 1).

For non-dominated solutions at rank 2, the crowding degree of the reference solution set at this level is calculated. The concept of crowding degree refers to the density of individuals in the solution space that share the same non-dominated rank (rank 2). In the multi-objective scatter search algorithm, this measure serves as a criterion to select better solutions within the same rank, which benefits the reference set during subsequent processes such as path relinking and other operations.

2.4.5. Subset Generation

In the multi-objective scatter search algorithm, generating guided solution subsets enables more uniform exploration of the solution space and facilitates the discovery of high-quality solutions within a shorter time frame. By continuously generating and refining candidate solutions, the algorithm gradually approaches the optimal solution.

In this study, given that the length of a single chromosome is 180, a binary flip is performed on a randomly selected fragment of length 6 from 50 chromosomes in the reference solution set. This operation generates 50 new chromosomes, which serve as the new guided solutions.

2.4.6. Path Relinking

Path relinking is a method for identifying better solutions during the local search process. The core idea is to establish a heuristic path between two distinct local optimal solutions. By constructing a search path across different LID layout schemes, the method performs combinatorial searches along the path to uncover improved solutions. This approach has proven effective in addressing complex optimization problems.

Suppose two chromosomes are chosen as follows:

$$T^1=\left[0,1,0,0,1,0,0,0,0,1,1,0,1, \dots 1,0,1\right]$$

(12)

$$T^2=\left[0,1,1,1,0,0,0,0,0,1,1,0,1, \dots 1,0,1\right]$$

(13)

By decoding genes can be expressed as follows.

$$T^{1*}=\left[2,2,0,6, \dots , 5\right]$$

(14)

$$T^{2*}=\left[3,4,0,6, \dots , 2\right]$$

(15)

If the selected gene positions in the combination scheme are [3,4,5], path relinking is performed at these positions, generating a subset for further analysis.

2.4.7. “Cost–Benefit” Local Search

The effectiveness of LID measures is primarily reflected in runoff reduction and pollution control. This paper focuses on the impact of LID on runoff reduction rates. The actual performance of LID measures depends not only on their structural characteristics but also on factors such as climate, site conditions, design specifics, and operation and management practices. Consequently, LID performance varies significantly across different locations, as indicated by the wide range of reported data. To enhance search efficiency, a “cost–benefit” local search algorithm is proposed, which integrates the cost and runoff reduction efficiency of various LID measures. The specific mathematical formulation is as follows:

$${\sigma }_{km}={\displaystyle \frac{Q_k-Q_{km}}{C_{km}}},\forall k,m$$

(16)

Here, ${\sigma }_{km}$ is the “cost–benefit” coefficient, $Q_k$ is the runoff after placing type M LID facilities on the sub-catchment where LID facilities can be placed, $Q_{km}$ is the runoff after placing type m LID facilities on the sub-catchment where LID facilities can be placed, and $C_{km}$ is the cost of placing type M LID facilities on the sub-catchment where LID facilities can be placed.

By calculating the “cost–benefit” coefficient for each sub-catchment, the types of LID facilities can be prioritized. This transforms the combinatorial optimization problem into an allocation problem, thereby improving search efficiency.

After path relinking, the algorithm avoids being trapped in local optima. Additionally, new solutions can be generated through crossover combinations, effectively expanding the search space. However, not all subset solutions generated by path relinking are feasible. Therefore, feasibility screening is necessary before calculating the fitness function during simulation.

3. Results and Discussion

3.1. Sample Data

This study compiles LID cost data from Xixian New District, Pingxiang, Hangzhou, Chongqing, Beijing, and Shanghai, integrating the relevant literature and project data [29,31]. Additionally, unit cost data for seven types of LID facilities were collected. For the simulation and optimization of LID layouts in this study, the cost standard from Hangzhou was adopted. Due to the limited availability of cost–benefit data for LID projects in China, the labor and material costs derived from related foreign case studies and calculations may differ significantly from domestic values. This highlights the need for further comparative studies and validation [15].

Based on 2021 rainfall data obtained from the Fuzhou Meteorological Bureau, probabilistic analysis indicated that rainfall distribution closely follows a normal distribution. To enhance model parameters calibration, rainfall events were selected to ensure a relatively uniform distribution within the rainfall return period. Five representative rainfall events were chosen for analysis: 1 February, 26 March, 5 September, 1 November, and 6 December 2021. The corresponding rainfall probabilities are illustrated in Figure 3.

Among these events, the rainfall on 5 September 2021, with a recorded precipitation of 50.6 mm, had the highest occurrence probability. Based on the rainfall return period distribution, four additional rainfall events with recorded precipitation levels of 13 mm, 33.2 mm, 66.3 mm, and 153.7 mm were selected for analysis in this study.

3.2. Model Verification

3.2.1. Average Rainfall Intensity Model Verification

Following the parameter ranges recommended in the SWMM user manual, four parameters were selected for calibration: characteristic width, characteristic slope, Manning’s coefficient of permeable surfaces, and the percentage of impermeable surfaces. The manual trial-and-error method was employed for calibrating these parameters, while default values were used for the remaining parameters. The minimum and maximum values for the calibrated parameters are detailed in

Table 4. Parameters.

Parameters to Be Determined	Parameter Variation Range
	Minimum Value	Maximum Value
Feature width/m	10	50
Characteristic slope/%	0.05	0.1
Manning coefficient of water penetration	0.11	0.15
Percentage of impervious water without storage/%	25	50

.

The parameter calibration was based on the rainfall event with 50.6 mm of precipitation recorded on 5 September 2021. Given that the two objectives, NSE (Nash-Sutcliffe Efficiency) and PBIAS (Percent Bias), show no clear correlation and are relatively equivalent, their combined evaluation requires a comprehensive approach due to the complexity of involving two objective functions. To address this, a composite index was introduced to evaluate the overall performance of the objective functions. The calculation formula for the composite index is as follows, where a smaller value indicates better simulation results:

Composite index = −α₁ NSE + α₂ PBIAS × 10

(17)

where weights were assigned to the two objectives, with NSE and PBIAS each weighted equally at 0.5.

The results of the hydraulic model parameter calibration are shown in Figure 4a. After calibration, the NSE efficiency coefficient was calculated to be 0.90, and the PBIAS coefficient was 0.05. The calibrated hydraulic model was then applied to four other measured rainfall events for validation, with the results illustrated in Figure 4. The NSE efficiency coefficients and PBIAS coefficients for the calibration model applied to the four rainfall events are summarized in

Table 5. NSE and PBIAS efficiency coefficients of random rainfall intensity.

Evaluation Index	13 mm	33.2 mm	50.6 mm	66.3 mm	153 mm
NSE	0.8855	0.8876	0.8706	0.8504	0.8744
PBIAS	0.06428	0.06928	0.0908	0.0591	0.0456

. The NSE coefficients were 0.89, 0.88, 0.87, and 0.90, while the PBIAS coefficients were 0.065, 0.061, 0.070, 0.054, and 0.044, respectively. According to the relevant research [29], a hydrological and hydraulic simulation model is considered to perform very well if the NSE efficiency coefficient exceeds 0.85 and the PBIAS coefficient is less than 0.1. These results demonstrate that the hydraulic model developed in this study performs effectively and is suitable for subsequent simulation and research.

3.2.2. Random Rainfall Intensity Model Verification

Using the deterministic parameter solutions, the hydraulic model’s performance was validated against random rainfall intensities across five events, as shown in Figure 4a–e. The NSE efficiency coefficients for the calibration model applied to these random rainfall intensity events were 0.8855, 0.8876, 0.8706, 0.8504, and 0.8744, respectively. The corresponding PBIAS coefficients were 0.06428, 0.06928, 0.0908, 0.0591, and 0.0456, respectively, as summarized in Table 5.

For all five events, the NSE coefficients exceeded 0.85, and the PBIAS coefficients were below 0.1, indicating that the model’s performance under random rainfall intensity conditions is comparable to that under average rainfall intensity conditions. This demonstrates the model’s reliability and suitability for subsequent analysis of random rainfall intensities.

3.3. Simulation Optimization Analysis of Average Rainfall Intensity Model

3.3.1. Comparative Analysis of “Cost–Benefit” Local Search

The proposed “cost–benefit” local search algorithm enhances the efficiency of the multi-objective decentralized search process. To evaluate its performance, five different rainfall scenarios were selected, and the efficiency and solution quality of the local search algorithm with and without the “cost–benefit” component were compared.

In Figure 5, the blue scatter plot represents the Pareto frontier obtained using the local search algorithm, while the orange scatter plot corresponds to the Pareto frontier derived from the multi-objective scatter search algorithm without the local search component. It is evident that for the dual-objective model considered in this study, the Pareto frontier generated by the multi-objective decentralized search algorithm with the “cost–benefit” local search significantly outperforms the frontier obtained without the local search. This confirms the effectiveness of the proposed algorithm in addressing the problem’s dual objectives.

3.3.2. Optimization Analysis

To further validate the advantages of the proposed algorithm, simulations were conducted under rainfall conditions of 13 mm, 33.2 mm, 50.6 mm, 66.3 mm, and 153.7 mm following model validation. Both the multi-objective decentralized search algorithm and NSGA-II were applied to optimize the simulation model. A comparison of the dual-objective Pareto frontiers for LID layout solutions, focusing on reducing total runoff while balancing cost increases, is presented in Figure 6.

The results show that under identical rainfall conditions and maximum iteration limits, the multi-objective decentralized search algorithm consistently outperforms NSGA-II in generating superior frontier solutions. Further analysis reveals that the Pareto frontier achieved by the decentralized search algorithm is more optimized than that obtained through NSGA-II.

To quantify the advantages of the proposed algorithm, a comparative analysis was conducted using Pareto frontier evaluation metrics.

1. Extensiveness index: Spread [41] (expressed by D2 in the formula [42]). This index mainly represents the wide distribution of Pareto frontiers. The higher the evaluation index value, the better the diversity of Pareto frontiers. The formula is as follows:

$${\mathrm{D}}_2\left(X\right)={\displaystyle \frac{d_f+d_l+{{\displaystyle \sum }}_{i=1}^{\left|X\right|-1}\left|\overline{d}-d_i\right|}{d_f+d_l+\left(\left|X\right|-1\right)\overline{d}}}$$

(18)

where di represents the Euclidean distance of the adjacent solutions of the Pareto front in the D2 index, $\overline{d}$ Yes $d_i$ The average value, and the calculation formula for the conversion between the two is as follows:

$$d_i=\left({{\displaystyle \sum }}_{i=1}^{\left|X\right|}\left(f_k\left(x_i\right)-f_x\left(x_{i+1}\right)\right)\right)$$

(19)

$$\overline{d}={\displaystyle \frac{1}{\left|X\right|-1}}{{\displaystyle \sum }}_{k=1}^{\left|X\right|}{d}_i$$

(20)

where the symbol corresponding to it $d_f$ and $d_l$ is the Euclidean distance of the extreme solution and other solutions in the Pareto front. The calculation formula looks like this:

$$d_f={\left({{\displaystyle \sum }}_{k=1}^{\left|X\right|}{\left(f_k^{max}-max\left\{f_k\left(x_i\right)\right\}\right)}^2\right)}^{0.5}$$

(21)

$$d_l={\left({\displaystyle \sum }_{k=1}^{\left|X\right|}{\left(f_k^{min}-min\left\{f_k\left(x_i\right)\right\}\right)}^2\right)}^{0.5}$$

(22)

2. Uniformity index: Minimal Spacing (expressed by MS in the formula) [43]. This index mainly represents the minimum distance standard deviation from each solution of the Pareto front to other solutions. The smaller the evaluation index value, the better the uniformity of the Pareto front. The formula is as follows:

$$\mathrm{MS}\left(\mathrm{X}\right)=\sqrt{{\displaystyle \frac{1}{\left|\mathrm{X}\right|-1}}{{\displaystyle \sum }}_{\mathrm{i}=1}^{\left|\mathrm{X}\right|}{\left( \overline{\mathrm{d}} -{\mathrm{d}}_{\mathrm{i}}\right)}^2}$$

(23)

wherein $d_i$ and $\overline{d}$ The meaning and algorithm of the representation are the same as those in the above index.

3. Distribution index: Hypervolume (super volume index, referred to by HV) [44].

$$\mathrm{HV}=\mathsf{\delta }\left({{\displaystyle \cup }}_{\mathrm{i}=1}^{\left|\mathrm{X}\right|}{\mathrm{v}}_{\mathrm{i}}\right)$$

(24)

where $\mathsf{\delta }$ represents a Lebesgue measure, used to measure volume, $\left|X\right|$ denotes the number of solutions in the Pareto front, while $v_i$ it represents the size of the super volume corresponding to the i-th solution in the Pareto front. By calculating the space area between the Pareto front and the reference point, the convergence and distribution of the solution are evaluated. The larger the HV value, the better the overall performance of the algorithm.

Table 6. Evaluation of three Pareto frontier indicators of average rainfall intensity under different rainfall levels.

Rainfall	D2		MS		HV
	Multi-Target Decentralized Search	NSGA II	Multi-Target Decentralized Search	NSGA II	Multi-Target Decentralized Search	NSGA II
13 mm	1.4458	1.3995	0.0415	0.0984	91.3421	93.3186
33.2 mm	1.4428	1.4820	0.0256	0.1184	56.5876	48.9282
50.6 mm	1.5006	1.4982	0.0099	0.0299	127.1488	72.4342
66.3 mm	1.4643	1.4621	0.0010	0.0592	169.3381	125.1995
153.7 mm	1.5080	1.4821	0.0177	0.0121	56.1358	61.9430
Mean value	1.4723	1.46544	0.01914	0.0636	100.11048	80.3647
Optimal number	4	1	4	1	3	2

provides a comparison of the three evaluation metrics—Spread (D₂), Minimal Spacing (MS), and Hypervolume (HV)—for the multi-objective decentralized search algorithm and the NSGA-II algorithm under identical rainfall conditions. Values representing better performance are highlighted in bold.

As the results clearly show, it is evident that the multi-objective decentralized search algorithm outperforms NSGA-II on most evaluation metrics. The multi-objective decentralized search algorithm demonstrates superior performance in terms of diversity (Spread), uniformity (Minimal Spacing), and overall distribution (Hypervolume) of the Pareto front.

This analysis confirms that the multi-objective decentralized search algorithm generates more optimized solutions than NSGA-II, validating its effectiveness for the dual-objective optimization problem in this study.

3.3.3. Comparison of Runoff Reduction Rates Under Different Rainfall Conditions

The comparative analysis in Section 3.3.2 reveals that the Pareto frontier solutions of both algorithms converge when the cost reaches 30 million yuan. To delve deeper into this observation, we analyze the total runoff across different rainfall conditions, as shown in Figure 7.

Due to the significant variation in total runoff under five distinct rainfall conditions, it is challenging to make direct comparisons. Therefore, we convert the total runoff into a runoff reduction rate for easier comparison. The formula for calculating the runoff reduction rate is as follows:

$$r=1-{\displaystyle \frac{{\mathrm{Q}}_{\mathrm{a}}}{{\mathrm{Q}}_0}}\times 100\%$$

(25)

where r is the total runoff reduction rate, Q_a is the total runoff at different costs levels, and Q₀ is the total runoff when the cost is 0, that is, the total runoff in the absence of LID facilities. The comparison is shown in Figure 8.

When converted to runoff reduction rates, the data reveal a clear trend: both total runoff and runoff reduction rates decrease as rainfall intensity increases. This indicates that under heavy rainfall conditions, the capacity of LID facilities to reduce runoff diminishes. Specifically, LID layouts demonstrate greater effectiveness in mitigating runoff during smaller to moderate rainfall events, whereas their performance weakens under heavy rainfall conditions. Additionally, the comparison reveals that the runoff reduction rate reaches a high level when the total cost is 30 million yuan, regardless of the rainfall condition. Beyond this cost, further increases in the cost result in only a marginal increase in the reduction rate, suggesting that investing more than 30 million yuan in LID facilities is not effective for further runoff reduction, particularly in the hot spring area.

This study also compares the calculation time between the two algorithms under different rainfall conditions. From the results, it is evident that the multi-objective decentralized search algorithm is, on average, 8.89%, 16.98%, 1.72%, 3.85%, and 1.23% faster than NSGA-II under the respective rainfall conditions. This suggests that the multi-objective decentralized search algorithm not only outperforms NSGA-II in terms of computation time but also yields better and more stable optimization results. However, as rainfall intensity increases and runoff escalates, the runoff reduction rate for both algorithms decreases, and the performance of the two algorithms gradually converges.

3.3.4. Optimization of Layout Scheme

The LID layout scheme with a total cost of 30 million yuan was selected and imported into SWMM for verification simulations under a 13 mm rainfall condition. The simulation results, shown in Figure 9, Figure 10 and Figure 11, indicate significant reductions in total runoff, peak flow, and water quality levels for the 30 million yuan LID layout scheme under 13 mm rainfall.

To further assess the performance of this scheme, it was tested under the remaining four rainfall scenarios using SWMM. The comparison of total runoff, peak discharge, and water quality concentration across different rainfall conditions is presented in Figure 12, Figure 13 and Figure 14.

The results from the simulation across five different rainfall conditions demonstrate that the comprehensive LID layout scheme is effective in reducing total runoff, peak discharge, and water quality concentrations. Notably, the runoff reduction rate achieved 52.8% under 13 mm rainfall. Moreover, the water quality indicators show clear improvements with minimal sensitivity to changes in rainfall conditions. These findings underscore the overall effectiveness and resilience of the LID layout scheme in mitigating both runoff and water quality challenges across varying rainfall scenarios.

3.4. Simulation Optimization Research of Stochastic Rainfall Intensity Model

3.4.1. Optimization Analysis

For rainfall conditions of 13 mm, 33.2 mm, 50.6 mm, 66.3 mm, and 153.7 mm after model verification, the rainfall intensity for each sub-catchment area was determined using a random rainfall function. The simulation-optimization research of the stochastic rainfall intensity model highlights the importance of incorporating variability and uncertainty into urban hydrological modeling. The proposed framework not only improves the robustness and cost-effectiveness of LID layouts but also provides a scalable solution for sustainable urban water management. By addressing the challenges of stochastic rainfall and computational efficiency, this study contributes to the advancement of urban sponge city initiatives. The multi-objective decentralized search algorithm and NSGA-II were then applied to optimize the simulation model. Figure 15 presents a comparison of the two-objective Pareto frontier solutions for the LID layout, illustrating the trade-off between total runoff reduction and cost increase.

Through the comparison, it can be seen that under the condition of random rainfall intensity, the multi-objective decentralized search algorithm is clearly outperforms NSGA-II in Pareto frontier solution, and the effect is better than that under the average rainfall intensity model. To quantify the advantages of the multi-objective decentralized search algorithm under random quantifiable rainfall intensity conditions, three evaluation indicators of Pareto frontier of multi-objective optimization under random rainfall intensity are compared and analyzed.

In

Table 7. Evaluation of three Pareto frontier indexes of random rainfall intensity under different rainfall.

Rainfall	D2		MS		HV
	Multi-Target Decentralized Search	NSGA II	Multi-Target Decentralized Search	NSGA II	Multi-Target Decentralized Search	NSGA II
13 mm	1.5066	1.4722	0.0082	0.0669	161.3868	55.4978
33.2 mm	1.5104	1.4707	0.0132	0.0148	200.9086	52.7769
50.6 mm	1.4952	1.4799	0.0479	0.0804	414.0026	280.0833
66.3 mm	1.5568	1.4552	0.1446	0.1523	381.4610	91.6152
153.7 mm	1.5169	1.4716	0.0568	0.0584	231.7102	52.9767
Mean value	1.51718	1.46992	0.05414	0.07456	277.89384	106.58998
Optimal number	5	0	5	0	5	0

, bolded values represent the superior results for each index when comparing the multi-objective decentralized search algorithm and the NSGA-II algorithm under identical rainfall conditions. The table reveals that across all rainfall conditions, the multi-objective decentralized search algorithm outperforms NSGA-II for each of the three evaluation metrics. Furthermore, as shown in Section 3.3.2, the multi-objective decentralized search algorithm demonstrates greater optimization stability under random rainfall intensity scenarios, which closely resemble real-world conditions, indicating its suitability for LID layout simulation optimization research. In the comparative analysis, it is found that the Pareto frontiers of the two algorithms tend to coincide under the condition of random rainfall intensity as under the condition of average rainfall intensity. The frontier solutions align when the cost is 30 million yuan.

3.4.2. Comparison of Runoff Reduction Rate Under Different Rainfall Conditions

Similarly to the scenario of average rainfall intensity, this study compares runoff reduction rates across various rainfall conditions, as illustrated in Figure 16. It is evident that both total runoff and runoff reduction rates decrease as rainfall increases. This indicates that under heavy rainfall conditions, the capacity of Low-Impact Development (LID) facilities to reduce runoff diminishes. Notably, at a rainfall intensity of 50.6 mm, the runoff reduction rate is significantly higher compared to other rainfall levels. This suggests that implementing LID measures in the hot spring area is particularly effective at mitigating runoff at this specific rainfall intensity. However, as rainfall increases beyond this point, the efficiency of runoff reduction declines.

Furthermore, the comparison chart reveals that under different rainfall scenarios, the runoff reduction rate reaches a high level when the total investment in LID facilities reaches 30 million yuan. Beyond this threshold, further increases in expenditure result in only marginal improvements in reduction rates. Thus, investing more than 30 million yuan in LID layouts does not significantly enhance the overall effectiveness.

Natural factors, such as soil type, topography, groundwater levels, rainfall patterns, runoff characteristics, geological features, and other hydrological conditions, play a critical role in determining the priority sites for LID implementation. Socio-economic constraints also heavily influence these decisions. Since the impacts of LID practices vary depending on local conditions, their implementation should be based on thorough site assessments to maximize both cost efficiency and functional benefits. This highlights the need for a spatial allocation framework that evaluates local conditions, identifies areas with the greatest need for resources, and locations where benefits can be maximized [45]. However, as urban demands evolve, a single LID strategy often proves inadequate for addressing diverse design requirements, and manually exploring combined LID schemes is time-consuming and labor-intensive.

To effectively address the challenges of multi-objective optimization in LID layout planning, robust screening methods are essential to quickly identify optimal allocations based on multiple criteria. To date, numerous GIS-based frameworks, models, and tools have been developed to integrate hydrological, hydraulic, ecological, and socio-economic factors [45].

The multi-objective decentralized search algorithm employed in this study tackles these challenges by incorporating a novel diversity maintenance mechanism and a problem decomposition strategy. These features enhance both convergence and global optimization by simultaneously identifying multiple solutions. While this algorithm is not widely applied in urban planning, future research could broaden its application in this field and related areas. Furthermore, integrating simulation models and algorithms with deep learning or other advanced computational techniques could improve both the efficiency and accuracy of model outcomes, offering a promising direction for further exploration.

4. Conclusions

This study focused on optimizing Low-Impact Development (LID) layouts using SWMM simulations to enhance urban hydrological responsiveness and strengthen flood defense capabilities. The primary goal was to address the dual challenges of minimizing stormwater runoff and reducing construction costs while ensuring effective urban flood management. The key findings and contributions of this research are as follows:

1. Effective Modeling of Rainfall Runoff and Pollutant Transport: The SWMM simulations successfully modeled rainfall runoff and pollutant transport within the catchment area under various LID layout scenarios. The model accurately quantified total runoff across pipes, channels, storage/treatment facilities, diversion structures, and external sources, providing a reliable basis for evaluating LID performance.

2. Dual-Objective Optimization Achieved: Simulation experiments conducted in the Hot Spring area of Fuzhou City demonstrated the effectiveness of the proposed optimization scheme in addressing dual-objective problems. The framework successfully balanced the trade-off between reducing runoff volume and minimizing costs, achieving optimal LID configurations that meet practical urban flood management needs.

3. Robustness Under Variable Rainfall Conditions: By incorporating random rainfall intensity variations, the optimization process produced results that closely align with real-world scenarios. This approach ensured high consistency between optimized outcomes and practical conditions, highlighting the method’s adaptability to diverse and unpredictable climatic conditions.

4. Superior Computational Efficiency: The proposed multi-objective scatter search algorithm, enhanced by cost–benefit local search and path-relinking techniques, outperformed the widely used NSGA-II algorithm. It reduced computation time by an average of 8.89%, 16.98%, 1.72%, 3.85%, and 1.23% across different scenarios, demonstrating significant improvements in computational efficiency.

5. Practical Applicability and Scalability: The SWMM-based LID layout optimization method showcased robust global search capabilities and holds substantial application value for urban flood management. The approach is scalable and can be integrated into urban intelligent management systems to enable real-time monitoring, prediction, and optimization of waterlogging management strategies.

By addressing the target problem of urban waterlogging and stormwater management, this study provides a scientifically grounded and cost-effective solution for sustainable urban development. The proposed framework not only enhances flood defense capabilities but also contributes to urban ecological protection and resilience. Future work will focus on integrating this optimization approach into smart city systems, enabling dynamic and adaptive urban water management in response to changing environmental conditions.

Limitations

This study has several limitations that should be acknowledged. First, the assumption of uniform soil properties across the study area may not reflect real-world conditions, where heterogeneous infiltration rates can significantly impact LID optimization. Variations in soil permeability and saturation levels could alter the effectiveness of infiltration-based LID measures, such as rain gardens and permeable pavements, leading to uncertainties in the simulation results. Future studies should incorporate spatially distributed soil data to improve model accuracy.

Second, variations in these parameters may affect the convergence speed, solution quality, and robustness of the Multi-Objective Scatter Search Algorithm. A detailed sensitivity analysis would help identify optimal parameter settings and reduce uncertainties in the optimization process.

Additionally, potential sources of error in the experimental data collection, such as measurement inaccuracies in rainfall intensity, runoff volume, and pollutant concentrations, could introduce uncertainties into the developed models and simulation results. These errors may arise from instrument calibration, sampling methods, or environmental variability during data collection. Future work should include error quantification and uncertainty analysis to enhance the reliability of the findings. Software: Name: SWMM, Version: 5.2.2, Supplier: U.S. Environmental Protection Agency (EPA), Manufacturer: U.S. Environmental Protection Agency (EPA), Country and City of Origin: United States, Washington, D.C. Brand Name: EPA SWMM, Year of Release: 2021.