Automatic calibration of SWMM parameters based on multi-objective optimisation model

The storm water management model (SWMM) developed by the United States Environmental Protection Agency has a wide range of applications in simulating drainage pipe networks, non-point source pollution loads, and other areas (Rossman 2009; Hu et al. 2018). The SWMM has a complex built-in structure, numerous physical parameters, and uncertainty, often resulting in generalised outcomes with many parameters that cannot be obtained through direct measurements (Behrouz et al. 2019). Manually calibrated simulation results often exhibit significant discrepancies when compared with the measured results. Therefore, it is necessary to develop an automatic calibration model that is compatible with the SWMM interface to calibrate the parameters accurately. Previous approaches to parameter calibration in the SWMM primarily relied on single-dimensional calibration, often using manual trial and error methods, single-parameter sensitivity analysis, and comparisons with observable data (Wani et al. 2017). These methods frequently result in poor alignment with real-world conditions and inefficient calibration processes.

Parameter calibration primarily involves adjusting the physical parameters contained within the model to maximise the alignment of the simulation results with real-world conditions (Sun et al. 2013; Hu et al. 2018). In recent years, researchers have extensively applied various optimisation algorithms, such as genetic algorithms and multi-objective particle swarm optimisation, for the calibration of SWMM model parameters. For example, Liong et al. (1995) coupled SWMM with a genetic algorithm using peak flow as the objective function to optimise the eight parameters of a watershed model. This study demonstrated the suitability of genetic algorithms for optimising watershed model parameters. Wang (2010) utilised the particle swarm optimisation algorithm to optimise eight parameters in the hydrological module of the SWMM. They demonstrated that the particle swarm optimisation algorithm can approximate the optimal solution from a multidimensional space. Kang & Lee (2014) combined an adaptive penalty function with a hybrid complex evolution algorithm for SWMM parameter optimisation. The results indicated that constrained intelligent algorithms were more effective in improving simulation accuracy. However, the problem of underfitting (i.e. mismatch between the water level and flow process curves) remains significant. When the model encounters optimisation problems in complex high-dimensional systems, it often exhibits poor convergence and is prone to becoming stuck in local solutions (Kppen & Yoshida 2007). This can result in significant discrepancies between the optimised results and the actual conditions. Furthermore, most calibration methods are based on assumptions that deviate from the reality. For example, the notion of representing the entire rainfall–runoff process of an entire system using the flow process at the discharge outlet of a network is one-sided. The flow process at the discharge outlet should be considered to be independent of the water level at the network nodes and the flow process within the network pipes (Zhou 2018). Therefore, multi-objective continuous optimisation is expected to become the primary trend in the future calibration of drainage network parameters in SWMM. Among multi-objective continuous optimisation algorithms, the NSGA-3 algorithm is an extension of the NSGA-2 algorithm. It introduces a reference point-based sorting method, retains an elite strategy, and emphasises non-dominated relationships among population members. This addresses the convergence and diversity issues faced by traditional multi-objective optimisation algorithms. Consequently, this study employed a fast, non-dominated sorting genetic algorithm with a reference point-based selection operator (NSGA-3) for the multi-objective calibration of urban drainage network models, following a comparative selection of algorithms (Deb & Jain 2014; Jain & Deb 2014).

In this study, sensitive parameters and their value ranges were first determined using the Sobol and generalised likelihood uncertainty estimation (GLUE) methods (Zhao et al. 2009). This was done to narrow the solution space, prevent model distortion owing to extensive parameter ranges, and accelerate the model-solving process. Subsequently, multi-objective calibration using the NSGA-3 algorithm based on Pareto theory was employed to optimise the calibration with three events and three objectives. This approach yields a Pareto-frontier solution set that serves as a set of alternative parameter scenarios. A real rainfall event was selected, and the model was applied to an independent sub-catchment rainwater drainage network system in a city in Northwestern China. Multiple sets of Pareto-frontier solution sets were randomly chosen for model calibration effectiveness testing. A comparative analysis was conducted to assess the differences in the simulation accuracy between single-objective genetic algorithms and multi-objective continuous optimisation algorithms. This study aimed to validate the reliability of the automatic calibration effectiveness of the model, with the goal of providing technical support for the automatic calibration of SWMM model parameters and accurate simulation and application of the model.

With the continuous development of the SWMM and the expanding scope of its applications, the model structure has become increasingly complex, requiring a richer set of parameters, which has led to higher parameter dimensions (Xia et al. 2008). Calibrating all parameters directly can easily lead to a ‘dimensionality curse’ due to the exponential increase in the computational complexity associated with higher-dimensional model solving. To address this, the Sobol method based on variance decomposition is initially employed in this section to examine the sensitivity of the hydrological module parameters in SWMM concerning peak timing, total runoff, and peak flow across varying rainfall intensities. This helps determine the subset of parameters that need to be calibrated. Additionally, parameter value ranges were constrained using the GLUE method. Subsequently, a multi-objective optimisation model was developed using the NSGA-3 algorithm, incorporating the reference point selection operator and Pareto theory (Guo et al. 2012). The NSGA-3 algorithm was applied to address the multi-objective optimisation problem, resulting in the derivation of the Pareto-frontier solution set, offering an alternative parameter set for the model.

The parameter sensitivity analysis aims to identify parameters significantly impacting the simulation results of the model, to optimise their values to prevent overfitting of specific hydrological model parameters, and to avoid transforming the optimisation process into a high-dimensional, nonlinear, and difficult-to-solve problem (Nagel et al. 2020).

In the Sobol global sensitivity analysis method, the use of Monte Carlo principles to compute the sensitivity arising from interactions between parameters is widely employed in large nonlinear models in environmental fields, such as water quality analysis simulation program (WASP), topography based hydrological model (TOPMODEL), soil and water assessment tool (SWAT), because of their robustness in producing results (Zhang et al. 2013). The core idea of this method is to decompose the function f(x) into a sum of increasing terms and samples to calculate the total variance of the response of the model to each parameter and the partial variance of each term. The SWMM is a high-dimensional nonlinear model, and it is not possible to directly calculate the variance of the model output results through analytical integration. Therefore, in this study, based on the Monte Carlo concept, numerical integration was employed to calculate the variance in the model output results. When the number of simulation samples reaches a specific value, the computed solutions closely approximate the analytical solutions. In this section, the runoff volume, peak flow, and peak timing are considered as the output variables for the parameter sensitivity analysis.

The specific calculation steps for the Sobol sensitivity indices in the SWMM for this study area are as follows.

• (1)
  The Latin hypercube sampling (LHS) method was used to perform two independent samplings within the parameter space, resulting in samples and ⁠, as shown in the following equation:

  

  (1)

  where the base sample size N was set to 5,000, and the number of SWMM model parameters p was 19. Building upon Equation (1), additional 19 new sample matrices, denoted as through ⁠, are obtained through a resampling process, where the first column of comes from the first column of ⁠, and the p-th column of comes from the column of ⁠.

  

  (2)
• (2)

  The 21 sample matrices obtained in Step 1, including and ⁠, are used to generate 105,000 parameter sets. These parameter sets were then input into the SWMM for simulation calculations, and the objective function values for the output variables were computed.

• (3)
  The objective function values obtained in step 2 are substituted into the following equations to calculate the sensitivity indices for each parameter X.

  

  (3)

  

  (4)

  

  (5)

  

  (6)

  

  (7)

  

  (8)

  where is the first-order global sensitivity index for the i-th parameter, is the variance caused by the joint effect of parameters other than on the model output results, and is the estimated value. The Sobol method assumes that the SWMM model can be represented in the form of function, where can be viewed as a discrete point in the rp dimension, and u represents the objective function value output by the model. Table 2 lists the main parameters of SWMM.
• (4)

  The sensitivities of the model parameters were ranked based on the numerical values of ⁠. Because Monte Carlo numerical integration is used for the calculations, whether is greater than zero can serve as a criterion for determining the sensitivity of the parameters (Saltelli 2002; Nossent et al. 2011).

To avoid excessive model distortion and maintain the parameters within their original physical meanings during calibration, it is essential to constrain the parameters to a specific range. This not only limits the parameter search space but also enhances the optimisation efficiency of multi-objective algorithms (Ragab et al. 2020). To achieve this, the GLUE method was introduced to determine the range of values for sensitive parameters in the SWMM model (Dai & Li 2020). The idea of this method is to use the parameter ranges provided in <the SWMM User Manual> as prior distributions for the sensitive parameters to be calibrated. This was performed using random sampling to create parameter sets, which were then used for the model simulations. Furthermore, a likelihood function was defined, and the likelihood values were calculated based on a comparison between the simulated results from each parameter set and the observed values. Likelihood values are assigned to each parameter set. When the likelihood values were used to represent the calibration probability, a threshold was empirically set, and only parameter sets with likelihood values exceeding this threshold were retained for further use in the model. Finally, a scatter plot was generated to illustrate the correspondence between the parameter values and likelihood values under a specific rainfall intensity, helping to determine the range of values for the sensitive parameters.

Owing to the inherent limitations of manual parameter tuning and single-objective drainage network optimisation algorithms, such as simple genetic and particle swarm optimisation algorithms, the parameter-calibrated models generated using these methods often exhibit poor agreement with real-world data (Wang & Zhou 2009). In this study, a multi-objective combinatorial optimisation was performed. The convergence of the multi-objective optimisation algorithm NSGA-2 tends to be less effective when dealing with three or more objective functions, making it prone to becoming trapped in local solutions (Kppen & Yoshida 2007). To address this issue, this study considers the use of a fast, non-dominated sorting genetic algorithm with a reference point-based selection operator, known as NSGA-3, to solve the multi-objective calibration problem of urban drainage network models.

The distributions of the Pareto solutions obtained using each algorithm were compared with the corresponding standard Pareto front. The NSGA-3 algorithm exhibits some advantages in terms of solution-set convergence, uniformity, and coverage. The changes in the hypervolume metric indicate that as the number of iterations increases, the hypervolume value corresponding to the NSGA-3 algorithm continues to rise and eventually stabilises. In contrast, the particle swarm optimisation, BrogMOEA, and NSGA-2 algorithms all have slower hypervolume values and longer times to reach stability, with less desirable convergence characteristics.

To avoid the randomness associated with analysing a single rainfall event and ensure the robustness of the results, three short-duration rainfall events observed in the study area in July and August 2019 (with different rainfall intensities, as shown in Table 1) were used as boundary conditions for the model inputs. The sensitivities of the hydrological and hydraulic parameters (listed in Table 2) in the SWMM model for three different levels of rainfall events were assessed with respect to three main categories of output variables: sensitivity indices for runoff volume, peak flow, and peak timing. The sensitivity analysis method described in section 1.1 was used to calculate the first-order sensitivity and total-order sensitivity of various key parameters under Rain1, Rain2, and Rain3 rainfall conditions. The calculation results are listed in Table 3 (owing to space limitations, this paper only presents the results for Rain1).

Table 2

The main parameters and physical meaning of the SWMM model

Parameter code .	Parameter name .	Physical significance .	Parameter code .	Parameter name .	Physical significance .
1	N-Imperv	Manning's coefficient for impervious area	11	N-pev	Manning's coefficient for permeable
2		Sub-watershed slope ratio factor	12	Dstore-Perv	Storage depth in the permeable portion of the sub-watershed depressions (mm)
3		Sub-watershed width ratio factor	13	Percent routed	Flow exchange ratio between permeable and impermeable areas
4	S-Imperv	Impervious area storage capacity (mm)	14	S-per	Permeable area storage capacity (mm)
5	Max rate	Initial infiltration rate (mm/h)	15	Manning-N	Manning's coefficient for pipes
6	Min rate	Steady-state infiltration rate (mm/h)	16	%Slope	Average slope of the sub-watershed
7	Con-Mann1	Manning's roughness coefficient for concrete pipe section	17	Exponent	Index characterising the functional relationship between water depth or head and flow velocity
8	Decay	Percolation attenuation coefficient	18	Initial depth	Initial water depth at the node at the beginning of the simulation (mm)
9	Pct-Zero	Percentage of impervious area without depressions	19	Drytime	Time required for soil to become dry after wetting
10	%Imperv-A	Percentage of root impermeability	…	…	…

Parameter code .	Parameter name .	Physical significance .	Parameter code .	Parameter name .	Physical significance .
1	N-Imperv	Manning's coefficient for impervious area	11	N-pev	Manning's coefficient for permeable
2		Sub-watershed slope ratio factor	12	Dstore-Perv	Storage depth in the permeable portion of the sub-watershed depressions (mm)
3		Sub-watershed width ratio factor	13	Percent routed	Flow exchange ratio between permeable and impermeable areas
4	S-Imperv	Impervious area storage capacity (mm)	14	S-per	Permeable area storage capacity (mm)
5	Max rate	Initial infiltration rate (mm/h)	15	Manning-N	Manning's coefficient for pipes
6	Min rate	Steady-state infiltration rate (mm/h)	16	%Slope	Average slope of the sub-watershed
7	Con-Mann1	Manning's roughness coefficient for concrete pipe section	17	Exponent	Index characterising the functional relationship between water depth or head and flow velocity
8	Decay	Percolation attenuation coefficient	18	Initial depth	Initial water depth at the node at the beginning of the simulation (mm)
9	Pct-Zero	Percentage of impervious area without depressions	19	Drytime	Time required for soil to become dry after wetting
10	%Imperv-A	Percentage of root impermeability	…	…	…

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Table 3

Sensitivity calculation results of parameters of Rain1 rainfall condition

Sensitivity order .	Total runoff .			Peak flow .			Peak time .
	Parameter name .	.	.	Parameter name .	.	.	Parameter name .	.	.
1	N-Imperv	0.642	0.713	N-Imperv	0.511	0.681	N-Imperv	0.573	0.611
2		0.254	0.362		0.432	0.523		0.511	0.593
3		0.020	0.105		0.411	0.519		0.492	0.532
4	S-Imperv	0.011	0.081	Min rate	0.343	0.429	Con-Mann1	0.436	0.497
5	Max rate	0.002	0.073	Con-Mann1	0.201	0.389	S-Imperv	0.329	0.411
6	Con-Mann1	−0.002	0.033	S-Imperv	0.104	0.198	Min rate	0.204	0.367
7	Min rate	−0.004	0.044	%Imperv-B	0.050	0.102	Pct-Zero	0.122	0.254
…	…	…	…	…	…	…	…	…	…
15	Pct-Zero	−0.011	0.065	Decay	−0.004	0.091	Max Rate	−0.006	0.052
…	…	…	…	…	…	…	…	…	…
19	%Imperv-B	−0.020	0.066	Min rate	−0.015	0.052	Decay	−0.018	0.043

Sensitivity order .	Total runoff .			Peak flow .			Peak time .
	Parameter name .	.	.	Parameter name .	.	.	Parameter name .	.	.
1	N-Imperv	0.642	0.713	N-Imperv	0.511	0.681	N-Imperv	0.573	0.611
2		0.254	0.362		0.432	0.523		0.511	0.593
3		0.020	0.105		0.411	0.519		0.492	0.532
4	S-Imperv	0.011	0.081	Min rate	0.343	0.429	Con-Mann1	0.436	0.497
5	Max rate	0.002	0.073	Con-Mann1	0.201	0.389	S-Imperv	0.329	0.411
6	Con-Mann1	−0.002	0.033	S-Imperv	0.104	0.198	Min rate	0.204	0.367
7	Min rate	−0.004	0.044	%Imperv-B	0.050	0.102	Pct-Zero	0.122	0.254
…	…	…	…	…	…	…	…	…	…
15	Pct-Zero	−0.011	0.065	Decay	−0.004	0.091	Max Rate	−0.006	0.052
…	…	…	…	…	…	…	…	…	…
19	%Imperv-B	−0.020	0.066	Min rate	−0.015	0.052	Decay	−0.018	0.043

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Under Rain1, for runoff volume, the top four sensitivity-ranked parameters were N-Imperv, ⁠, ⁠, and S-Imperv, with N-Imperv having a significantly higher sensitivity index than the others, indicating that variations in N-Imperv have a more significant impact on the variance of the model results. In contrast, the other parameters have a more negligible influence. For the peak flow, the most highly ranked sensitivity parameters were N-Imperv, ⁠, ⁠, and max rate. Regarding the peak timing, the sensitivity analysis results were similar to those of the peak flow, with the top four parameters being N-Imperv, ⁠, ⁠, and Con-Mann1.

The simulation results indicated that rainfall intensity has a significant impact on the sensitivity of the SWMM model parameters. Under heavy rainfall conditions, exemplified by Rain1 and Rain2, parameters related to infiltration-excess runoff, notably N-Imperv, exhibited sensitivity. In contrast, under lower rainfall conditions, such as Rain3, the S-Imperv became a sensitive parameter. In cases of lower rainfall intensity, the depressed storage of the impervious area could hold a certain amount of rainfall, resulting in less surface runoff. However, as the rainfall intensity increases, these depression areas may quickly fill up, and changes in their values have a relatively minor impact on the overall surface runoff. Therefore, parameters such as N-Imperv exhibit sensitivity. The sensitivity analysis results for Rain2 and Rain3 are listed in Table 4.

Table 4

Sensitivity calculation results of parameters of Rain2 and Rain3 rainfall condition

.	Sensitivity order .	Total runoff .			Peak flow .			Peak time .
		Parameter name .	.	.	Parameter name .	.	.	Parameter name .	.	.
Rain2	1	N-Imperv	0.623	0.701	N-Imperv	0.505	0.678	N-Imperv	0.579	0.623
	2		0.212	0.287		0.432	0.523		0.523	0.567
	3		0.023	0.121		0.410	0.512		0.467	0.556
	4	S-Imperv	0.013	0.072	Min rate	0.323	0.443	Con-Mann1	0.423	0.445
	…	…	…	…	…	…	…	…	…	…
Rain3	1	S-Imperv	0.645	0.781	S-Imperv	0.513	0.669	S-Imperv	0.536	0.697
	2	N-Imperv	0.452	0.523	N-Imperv	0.321	0.481	N-Imperv	0.473	0.511
	3		0.276	0.378		0.232	0.323		0.489	0.502
	4		0.017	0.123		0.389	0.398		0.192	0.232
	…	…	…	…	…	…	…	…	…	…

.	Sensitivity order .	Total runoff .			Peak flow .			Peak time .
		Parameter name .	.	.	Parameter name .	.	.	Parameter name .	.	.
Rain2	1	N-Imperv	0.623	0.701	N-Imperv	0.505	0.678	N-Imperv	0.579	0.623
	2		0.212	0.287		0.432	0.523		0.523	0.567
	3		0.023	0.121		0.410	0.512		0.467	0.556
	4	S-Imperv	0.013	0.072	Min rate	0.323	0.443	Con-Mann1	0.423	0.445
	…	…	…	…	…	…	…	…	…	…
Rain3	1	S-Imperv	0.645	0.781	S-Imperv	0.513	0.669	S-Imperv	0.536	0.697
	2	N-Imperv	0.452	0.523	N-Imperv	0.321	0.481	N-Imperv	0.473	0.511
	3		0.276	0.378		0.232	0.323		0.489	0.502
	4		0.017	0.123		0.389	0.398		0.192	0.232
	…	…	…	…	…	…	…	…	…	…

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The results indicate that land use in the study area has a particular influence on parameter sensitivity. In this area, rooftops and hardened surfaces account for 82% of the total construction area, resulting in a high proportion of impermeable surfaces, making N-Imperv highly sensitive. For permeable areas, the max rate, min rate, and decay parameters in the Horton infiltration model were also considered sensitive. This is because, during heavy rainfall, the runoff rate significantly exceeds the infiltration rate in permeable areas, leading to surface runoff. Sensitive parameters for total runoff and peak flow showed a certain degree of similarity, suggesting that these parameters are correlated with their impacts on peak flow and total runoff in the SWMM model. Con-Mann1 consistently exhibited high sensitivity in the simulation process, particularly in relation to the peak time. This was because the roughness of the conduits affected the flow velocity within the pipes, thereby influencing the peak time.

Comparing the first-order sensitivity and total-order sensitivity of the sensitive parameters, it is observed that some parameters have significantly higher total-order sensitivity than first-order sensitivity, as seen with Con-Mann1 for peak flow. This suggests that this parameter interacts with other parameters during the simulation process, leading to increased variance in the results. Owing to the complexity of the natural environment in the study area, it is often challenging to calibrate the parameters precisely. Instead, parameter estimation relies on a comparison of field monitoring data with historical data, and some hydrological and hydraulic parameters in the SWMM exhibit complementarity or correlation. This can lead to the ‘different parameters, same effect’ issue during model calibration. Therefore, the GLUE method is required to determine the range of values for the parameters within the parameter set.

Through the analysis of the scatter plots showing the correspondence between model-sensitive parameter values and likelihood values in Figure 9, it can be observed that except for and ⁠, which exhibit significant differences in NS values across different value ranges, the NS values of the other eight sensitive parameters do not show a distinct trend of concentration. During the simulation, the prior distributions of the selected sensitive parameters were assumed to be uniform. However, from the posterior distributions, it is evident that except for and ⁠, the remaining parameters do not exhibit regions of high NS concentration. This suggests the presence of the ‘different parameters, same effect’ phenomenon, meaning that the impact on the actual simulation results is not solely due to individual parameters but rather a combination of multiple parameters.

Based on the calculations mentioned earlier, the parameter ranges with likelihood values greater than 0.8 were selected as the upper and lower bounds for the constraints of the multi-objective optimisation model. Narrowing the parameter space improved the efficiency of the algorithm. The selected sensitivity intervals for the optimisation parameter sets are listed in Table 6.

Table 6

Determine the range of parameter (rain1)

Parameter code .	Parameter name .	Physical significance .	Sensitivity range .
1	N-Imperv	Manning's roughness coefficient for impervious area	(0.02, 0.07)
2		Sub-watershed slope ratio factor	(0.7, 1.2)
3		Sub-watershed width ratio factor	(0.7, 0.9)
4	S-Imperv	Impervious area detention capacity(mm)	(0.5, 1.85)
5	Max rate	Initial infiltration rate (mm·h−1)	(50.8, 101.6)
6	Min rate	Steady-state infiltration rate(mm·h−1)	(7.6, 15.2)
7	Con-Mann1	Manning's roughness coefficient for the concrete pipe section	(0.013, 0.017)
8	Decay	Infiltration attenuation coefficient	(2, 7)
9	Pct-Zero	Percentage of non-depressed impervious area	(20, 35)
10	%Imperv-A	Roof impermeable percentage	(92, 100)

Parameter code .	Parameter name .	Physical significance .	Sensitivity range .
1	N-Imperv	Manning's roughness coefficient for impervious area	(0.02, 0.07)
2		Sub-watershed slope ratio factor	(0.7, 1.2)
3		Sub-watershed width ratio factor	(0.7, 0.9)
4	S-Imperv	Impervious area detention capacity(mm)	(0.5, 1.85)
5	Max rate	Initial infiltration rate (mm·h−1)	(50.8, 101.6)
6	Min rate	Steady-state infiltration rate(mm·h−1)	(7.6, 15.2)
7	Con-Mann1	Manning's roughness coefficient for the concrete pipe section	(0.013, 0.017)
8	Decay	Infiltration attenuation coefficient	(2, 7)
9	Pct-Zero	Percentage of non-depressed impervious area	(20, 35)
10	%Imperv-A	Roof impermeable percentage	(92, 100)

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To validate the parameter calibration results, the optimised parameter sets were used in the SWMM to obtain simulated results for all hydrological and hydraulic elements, including the water levels at all nodes and flow rates in the pipes. The reliability of the model was tested under different rainfall scenarios, specifically heavy rain (24-h rainfall between 50.0 and 99.9 mm), moderate rain (24-h rainfall between 10.0 and 24.9 mm), and light rain (24-h rainfall less than 10 mm). The parameters were calibrated using Rain1, Rain2, and Rain3, and the model performance was evaluated.

The closer the value of M-NSE is to 1, the more reliable and valuable the model. Typically, when M-NSE > 0.75, the simulated values of the model are considered to have a high level of agreement with the observed values when using the optimised parameter set, which can be used for practical engineering purposes (Servat & Dezetter 1991).

Based on the results of the model, under the Rain4 rainfall event (date: 16 September 2019, rainfall duration: 55 min, total rainfall: 16.3 mm, rainfall intensity: heavy rain), using the parameters calibrated with Rain1 (date: 14 July 2019, rainfall duration: 32 min, total rainfall: 10.8 mm, rainfall intensity: heavy rain), the M-NSE values for the J_44 water level process are all above 0.80, and the M-NSE values for the C_27 pipe flow process are also above 0.78, indicating ideal simulation performance. When the model was inputted with rainfall of other intensities (moderate rain and light rain), the M-NSE values for the J_44 water level process were all above 0.82, and the M-NSE values for the C_27 pipe flow process were also above 0.79, indicating excellent model verification results (the verification process is not detailed here because of space limitations). In summary, under this optimal parameter set, the simulation results of the model were reasonably reliable and could be applied in practical engineering.

During the optimisation process using SGA, the basic parameters were set as follows: population size of 350, number of iterations as 100, crossover probability of 0.7, mutation probability of 0.15, and the constraint conditions were based on the parameter ranges determined in section 2.3. The optimal parameter set based on the SGA was obtained through simulations, as listed in Table 8.

Table 8

Optimised parameter set of SGA

Parameter code .	Parameter name .	SGA optimisation value .	Parameter code .	Parameter name .	SGA optimisation value .
1	N-Imperv	0.0382	6	Min rate	9.3947
2		0.9732	7	Con-Mann1	0.0140
3		0.8293	8	Decay	4.3281
4	S-Imperv	1.5732	9	Pct-Zero	30.2811
5	Max rate	90.8753	10	%Imperv-A	97.5266

Parameter code .	Parameter name .	SGA optimisation value .	Parameter code .	Parameter name .	SGA optimisation value .
1	N-Imperv	0.0382	6	Min rate	9.3947
2		0.9732	7	Con-Mann1	0.0140
3		0.8293	8	Decay	4.3281
4	S-Imperv	1.5732	9	Pct-Zero	30.2811
5	Max rate	90.8753	10	%Imperv-A	97.5266

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This study proposes a parameter optimisation and calibration method for the SWMM model and applies it to the hydrological parameter optimisation of an independent sub-catchment stormwater drainage pipe network system in a residential area of a northwest city. The simulation results demonstrate good performance. The design of the parameter calibration method proposed in this study has the following innovative points, providing new ideas for parameter calibration of related hydrological models in the future:

• (1)

  This study provides a quantitative interpretation of the results of sensitivity parameter screening and ranking of the SWMM model. The premise of the parameter calibration is to determine the sensitive parameters in the study area. The core task was to qualitatively or quantitatively evaluate the impact of changes in the model input parameters on the output results. Many scholars have conducted sensitivity analyses of SWMM parameters, but most of them use qualitative methods (Sun et al. 2014; Akdoğan & Güven 2016; Tsai et al. 2017; Swathi et al. 2019). The calculation results can only be used for screening and ranking sensitivity parameters and cannot provide quantitative explanations for the results. In this study, considering the range of values for various parameters in the SWMM, a quantitative analysis of the sensitivity of the SWMM model parameters in the study area was conducted using the Sobol method based on variance decomposition. This method calculates the effect of each parameter on the variance in the model output results and assesses the sensitivity of each parameter.

• (2)

  During the construction of the objective function, the multi-objective continuous optimisation criteria of the SWMM model system were adopted. Typically, researchers and engineers base the determination of the objective function for parameter calibration optimisation on the belief that the effluent flow hydrograph at the end of the pipe network system can fully represent the entire rainfall–runoff process of the pipe network system (Behrouz et al. 2019; Gao et al. 2023). However, when the effluent flow hydrograph is accurately fitted, the water level at the nodes and flow in the pipe segments of the drainage pipe network system may not necessarily match the observed data. This implied that the effluent flow, node water level, and pipe network flow processes were mutually independent. The starting point of this study was based on the assumption of the mutual independence of these processes. Each process was separately summarised as an objective function, and the final three-objective mathematical model was optimised and solved.

• (3)

  By combining the concepts of uncertainty and optimisation, this study integrates the uncertainty of the model with the optimisation of a multi-objective mathematical model based on the obtained posterior distribution. Two completely different approaches have emerged for the optimisation and calibration of hydrological model parameters. On one hand, there is the uncertainty approach, which stems from the uncertainty of the model itself. However, there is an optimisation approach based on the optimal objective function of the model. The former considers that the parameter values of the model are uncertain but follow a particular distribution, which can be determined through uncertainty methods, such as Bayesian theory, to obtain the posterior distribution of parameter sets (Chillkoti et al. 2018). The latter assumes that the specific parameter set for the model in a particular case study area is deterministic with only one true parameter set. Even if it cannot be directly obtained through measurements, it can be calibrated using automatic optimisation algorithms. Based on the uncertainty approach, this study used the generalised likelihood uncertainty estimation (GLUE) method to obtain the posterior distribution of the parameter sets. On this basis, a multi-objective optimisation algorithm was employed to obtain the Pareto-front solution set. Although optimisation algorithms were used in this study, the obtained solutions lie on the Pareto front, combining the concepts of uncertainty and optimisation to obtain the optimisation solution of the multi-objective mathematical model based on the obtained posterior distribution.

This study proposes an automatic calibration method for the SWMM model parameters. Sensitivity analysis of the parameters in the study area was conducted using the Sobol method, which is based on quantitative analysis. This involves identifying sensitive parameters, determining their sensitivity ranges, and utilising the NSGA-3 multi-objective optimisation algorithm to perform optimal calibration. The method is based on a multi-objective continuous optimisation criterion that combines the concepts of uncertainty and optimisation.

To verify the effectiveness of the calibration method, numerical experiments were conducted using an independent rainwater drainage network system in a residential area of a city in northwest China. The simulation results were in line with the expectations. To further test the performance of the algorithm, the proposed SWMM parameter multi-objective calibration method was compared with a single-objective genetic algorithm calibration method in terms of simulation accuracy. Under specific optimisation parameter settings, it was found that the simulation accuracy of the multi-objective continuous optimisation algorithm was significantly better than that of the single-objective genetic algorithm. The research findings can provide technical support for the automatic calibration of the SWMM model parameters and the accurate simulation and application of the model.

Regarding the multi-objective continuous optimisation calibration of the SWMM model parameters, the selection of monitoring points in the objective function requires further consideration. This study only selected nodes at the intersections of the main streets, where the inflow was relatively large, as monitoring points for water levels. Subsequent research should propose a systematic and optimised scheme for the arrangement of monitoring points and choose the optimal number and locations more scientifically and economically.

This work was supported by the National Key Research and Development Program Projects for the 14th Five-Year Plan (2021YFC3200200), National Natural Science Foundation Projects (52025093, 51979284), and Free Exploration Project of the State Key Laboratory of Watershed Water Cycle Simulation and Regulation (SKL2022TS01).

The authors have no relevant financial or non-financial interests to disclose.

T. W. conceptualized the whole article, developed the methodology, investigated the work, arranged the software, rendered support in data curation, and wrote the original draft preparation; L. Z. arranged the software, visualized the project, wrote the review, and edited the article; J. Z. investigated the work and arranged the software; L. W. supervised the work, wrote the review and edited the article; Y. Z. wrote the review and edited the article; K. L. wrote the review and edited the article.

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

The authors declare there is no conflict.