Evaluating evolutionary algorithms for simulating catchment response to river discharge

Rainfall-runoff models have gained significant importance among hydrologists for comprehending natural and anthropogenic factors that influence watershed characteristics during rainfall events (Dietze & Ozturk 2021). Numerous models are available for forecasting runoff from the watershed like TR-55 (Soil Conservation Service), HEC-1 (U.S. Army Corps of Engineers), Mike Hydro River (Danish Hydraulic Institute) and SWMM (US Environmental Protection Agency). Among these, the SWMM is the most widely used open-source model. The primary advantage of the SWMM is that it can simulate both hydrological and hydraulics models simultaneously. Jang et al. (2007) used the SWMM model to assess the hydrologic impact of urbanisation. Wang & Altunkaynak (2012) carried out comparisons and performance analyses between SWMM and fuzzy logic models. The predicted total runoffs from both models were found to agree rationally with the observed data. SWMM produced time-varying hydrographs, but fuzzy logic failed to generate such output due to methodological limitations. Krebs et al. (2013) used the SWMM model in a highly urbanised catchment and discussed a multiobjective multi-event calibration approach. Rosa et al. (2015) used SWMM to simulate runoff and nutrient export from a low impact development (LID) watershed and a watershed using traditional runoff controls. Relationships between long-term stream discharge and changes in groundwater use were analysed by Jun et al. (2010) with the help of SWMM − GE (storm water management model − groundwater edition). A wide variety of options are available in SWMM for editing input data, performing simulations of hydrologic, hydraulic and water quality nature problems and visualising the results. Several studies emphasised time-varying calibrating SWMM models, which consist of various factors that cannot be estimated legitimately from the catchment due to limitations in measurement. Estimating the value of hydrologic factors with the target that the model output is the closest possible to the catchment's response is the primary aim of the calibration. In modern times, auto-calibration of the model has gained recognition over manual calibration as it is proficient and efficient, whereas the latter is quite tedious (Gupta et al. 1998; Boyle et al. 2000; Vrugt et al. 2003; Kumar et al. 2018a, 2018b, 2019; Wolf et al. 2021). Calibration of the model should be a multiobjective problem so that it satisfies all the conditions that make it suitable for all the time periods. The selection of an objective function that precisely simulates all parts of a hydrograph, including low flows and peak flows, is a challenging task because each criterion may emphasise different types of data points (Gupta et al. 2009).

Artificial intelligence tools such as genetic algorithms (GA), fuzzy logic and artificial neural networks (ANN) are data-driven (Yamazaki et al. 2011; Singh et al. 2021) and offer an alternative solution to the problem. These methods are principally leaning towards optimisation (Savic 2005; Kim et al. 2006). Optimisation is the art of categorising the best among several feasible solutions through evaluation. The motivating force in optimisation is the objective function. Studies of optimisation methods encouraged the use of evolutionary algorithms (EA) based on populace-evaluations, like GA, shuffled complex evolution algorithms (SCEA) and simulated annealing (SA) (Wang 1991; Duan et al. 1994; Sumner et al. 1997; Orouji et al. 2016; Kumar et al. 2018a, 2018b, 2019, 2021; Attea et al. 2021). EA describes a broad variety of heuristic approaches to compete with evolution (Back et al. 2000). EA procedures consist of algorithms that employ a population of alternative outcomes or approaches, individually indicated by a probable decision vector. The fundamental reason behind the acceptance of EA is their competency in solving the nonlinear, discrete and multimodal, constrained, non-convex problems that have been challenged by a deterministic approach. The proficiencies of EA have been applied in solving various environmental and water resources problems like water distribution systems (Dandy et al. 1996), water resources management (Baltar & Fontane 2008) and hydrologic models calibration (Tolson & Shoemaker 2007).

Over the recent studies carried out in the past, stochastic exploratory optimisation philosophies, particularly EAs, have discovered their broad uses in water resources. Numerous issues have been settled in the water resource spectrum by utilising various EAs on the ground. Among them are GA (Azamathulla & Zahiri 2012; Anand et al. 2018; El Fels et al. 2019; Khadr & Schlenkhoff 2021), SCEA (Duan et al. 1993), particle swarm optimisation (PSO) (Moghaddam et al. 2016; Chen et al. 2020; Bozorg-Haddad et al. 2021; Torkomany et al. 2021), shuffled frog leaping algorithm (SFLA) (Eusuff et al. 2006) and differential evolution (Storn & Price 1997). Some researchers used GAs (Khadr & Schlenkhoff 2021), PSO (Chen et al. 2020; Jahandideh-Tehrani et al. 2020; Bozorg-Haddad et al. 2021) and weed optimisation algorithm (Asgari et al. 2016; Azizipour et al. 2016) in optimising reservoir operation and some others used SA, multiswarm particle swarm optimisation (MSPSO) (Ostadrahimi et al. 2012) for multi-reservoir system operation. For modelling the stage-discharge relationship in Pahang River, Azamathulla et al. (2011) used gene-expression programming (GEP), an extension of genetic programming (GP). Moeini & Afshar (2019) coupled the ant colony optimisation algorithm with nonlinear programming in sewer networks for the optimal design. El-Ghandour et al. (2020) applied PSO for determining the optimal design of irrigation canals and compared it with probabilistic global search lausanne (PGSL) and classical optimisation methods for verification. Yazdi (2018) integrated the MOEA (multipurpose evolutionary algorithm) with SWMM to develop a model and estimated cost-effective repair actions in the event of a structural failure in a key element of the network. The model developed is a resilience-based remediation approach proposed for urban drainage systems using improved bottleneck structures. The algorithm's performance depends on the problem's objective because some algorithms can better resolve a certain problem relative to others. No specific optimisation algorithm functions admirably for all water resource issues (Labadie 2004). Therefore, the hydrologist must understand the edge of ambiguity related to the model output that helps to evaluate the model's efficiency.

SWMM is widely used in urban areas. Its application in a natural watershed is not very popular, and there is a paucity of information in this area. Rai et al. (2017) applied the SWMM model in the Brahmani River basin to simulate river floods. In the present study, the upper Damodar River basin's rainfall-runoff model was developed with the SWMM. This model was used to access its applicability in a natural watershed and the physiography impact on runoff. In the present study, the GIS (Geographic Information Systems) based SWMM model was set up to verify the viability of the developed model as a flow emulator in the upper Damodar River basin's catchments. Three EAs, i.e. GA, PSO and SFLA, were employed. This study aims at assessing the performance of the evolutionary algorithm during parameter optimisation under limited model iterations options for the river system. The study involves the attuning of SWMM by coupling the SWMM model with the MATLAB setting internally. The key target of this study is to determine the best algorithm for the optimisation of a given rainfall-runoff model. Statistical parameters like Nash–Sutcliffe effectiveness (NSE), percent bias (PBIAS) and root-mean-square error–observations standard deviation ratio (RSR) were used to analyse the results obtained from the algorithms. NSE is sensitive to the peak value, PBIAS checks the average trend of model results with the equivalent observed data, and RSR checks the noise pattern. On a global scale, several techniques successfully applied coupling open-source software with EA for river hydraulics. However, we have employed the same at a local scale for Damodar River Basin in India. The outcomes provide significant information to hydrologic experts for evaluating the catchment response to river discharge.

Data availability is the most critical issue of the hydrological and hydraulic study, and its collection from different sources is the most challenging task. Poor support and the absence of good observed information associated with hydrology and hydraulics are the significant limitations of the study related to these areas (Singh et al. 2020, 2022). For the rainfall-runoff model data, hydrological, meteorological and topographical data are essential. These data have substantial effects on the hydrological process. Daily discharge and rainfall data were collected from Damodar Valley Corporation (DVC), Maithon Jharkhand, India. Topographical data derived from digital elevation model (DEM) were used to understand the topographical influence on the spatial extent of runoff. Freely available land use (LU/LC) data for 2015–2016 were collected from the BHUVAN website of the Government of India. The generic data collected are summarised in Table 1.

SWMM is an open-source tool commonly used for development, analysis and design associated with stormwater runoff, sewers and urban planning throughout the world. It helps create cost-effective stormwater control solutions. In its latest version, it is adept at computing hydrologic and hydraulic methods for both urban as well as non-urban regions. The hydrology package incorporates all the modules such as surface and sub-surface flow, climatology, evaporation, infiltration and snow. Spatial unpredictability in these procedures is accomplished by distributing the basin into the zone of a smaller sub-watershed. Every zone comprises its segment of pervious as well as impervious sub-areas. Surface flow can be routed between sub-watersheds or among entrance points of a drainage system. SWMM executes specific events and continuous events anticipating the spatio-temporal variability in the topographical feature of the catchment. Surface runoff is evaluated using the nonlinear reservoir method, which uses the Manning's equation to evaluate surface runoff. The Green–Ampt model, Horton Model or curve number, calculates infiltration losses in the SWMM model. Back pressure and pressure flow effects are also taken into account in the model. SWMM gives a coordinated setting for observing the outputs in a standard format.

Calibration of the model is performed to evaluate the value of catchment factors with the model output target which is closest possible to the catchments response. In modern times, auto-calibration of models has picked up fame over manual calibration as it is proficient and efficient, whereas later one is a tedious method (Boyle et al. 2000; Kumar et al. 2019; Johny et al. 2020). EAs, i.e. GA, PSO and SFLA, were employed in the present study due to their consistency in testing universal optimums and simplicity in execution. These algorithms were engaged for calibration and validation purposes in the SWMM model on a daily scale. For every parameter, 2,000 samples were produced by utilising the Latin hypercube testing strategy. The Latin hypercube sampling (LHS) is a brilliant effective method that comprises stratified selection. A 3-D FEM groundwater model was developed by Hardyanto & Merkel (2007) for demonstrating the suitability of LHS for susceptibility and affectability analysis. The internal coupling of these algorithms was done in the MATLAB platform for auto-tuning the SWMM. From 1 June 2002 to 31 October 2006, calibration was done for discharge/gauge data. Validation was done from 1 June 2007 to 31 October 2011. Precipitation and discharge flow time-series data used in the present study are at a daily scale. The Horton method was used for infiltration modelling because the equation involved in the infiltration method required information on LU and soil present in the watershed. Computational time is significant because simulation is performed multiple times by the optimisation algorithms for converging the output to the optimal point. For routing, the dynamic wave method was operated.

Moreover, long-term simulations were considered for the calibration of the model because of better continuous simulation results. Therefore, the time period taken into account for the model's calibration and validation in the present study will bring substantial enhancements. Further, the performance of the model was estimated by matching the simulated model output with the observed data based on statistical parameters.

Two types of parameters involved in hydrodynamic models are categorised as measured and inferred factors. Those factors which have been assessed directly from the model come under inferred parameters. In contrast, river geometry, river depth, river catchment area, and elevations come under the measured parameters category (Choi & Ball 2002). Measured parameters are free from error, but inferred parameters need some modification during the calibration, as expressed by Choi & Ball (2002). Hence, inferred parameters like channel and watersheds roughness coefficient, pervious and impervious percentage area, infiltration parameters and storage (depression) were modified during the calibration procedure. The calculated watershed's area was given as SWMM input using ArcGIS tools. Further, the percentage imperviousness area, river slope and watersheds width were calculated together with LU and DEM support in ArcGIS.

For the calibration of the SWMM model, ten parameters of SWMM were considered. For various subcatchments, some parameters have different values, such as slope, percentage imperviousness and width. Therefore, a test value was assigned as an initial trial to these parameters, followed by varying the parameters as per the interval decided for calibration. It is assumed that SWMM parameters follow a uniform distribution (Muleta & Nicklow 2005). As per the engineering assessment and previous studies, the upper and lower range of the SWMM parameters were decided (Huber & Dickinson 1992; Temprano et al. 2007; Rossman 2010). Table 2 shows a list of possible parameters and possible settings. Statistical parameters of discharge data (Burgan et al. 2017) at five gauging stations are given in Table 3.

In Holland (1975), GA originally presented, motivated by the Darwinian theory of evolution and survival of the fittest means genetic mechanisms of natural species evolution. The simple correspondence of GA is that the population symbolises a set of solutions; every individual in the population is equivalent to each solution in a group of solutions. The objective function of each solution characterises an individual's fitness. Fitness is the only criteria for the emergence of a new population from the past one. With the goal that the fittest individuals have the most noteworthy likelihood of producing the next generation, transition rules are applied. Selection, crossover, and mutation are genetic operators that are used in the process of transition. The ongoing execution advancements of computational frameworks and parallel computing strategies have enhanced the attention and made these sorts of optimisation quite extraordinary. The GA utilises the original results to generate second-generation individuals, which are assessed again individually. The most significant property is strength, and this denotes an imitation of nature's adaptation algorithm of choice. Scientifically, it implies that it is conceivable to discover an answer regardless of whether the information does not encourage finding such an answer. GA is generally applied to non-differentiable functions and capacities with numerous nearby optima.

PSO, first presented by Kennedy & Eberhart (1995), is motivated by animal actions or conduct to answer optimisation problems. It is centred on the consistent process of crowd correspondence to share particular information when a gathering of birds searches for foodstuff or moves in a way through space, albeit all birds don't have an idea where the best position is. But from the idea of social conduct, if any fellow can discover a desirable way to go, the remaining individuals will follow rapidly. In PSO, every single associate of the population is known as a particle, and the population is known as a swarm. It does not require any gradient information to be optimised and uses only basic mathematical operators. The PSO procedure utilises a swarm of particles that navigate a multidimensional pursuit space to search out optima. Every particle is a potential result and is impacted by neighbours' capabilities and themselves. All the particles reserve their specific best performance. They also know the best performance of their association. If the superior solution is found, the swarm will start to push towards this new solution. But this methodology is known to be simply trapped in local minima (Fang et al. 2007; Reddy & Adarsh 2010). The PSO strategy is becoming famous because of its straightforwardness of execution, just as the capacity to join a reasonable solution quickly. It only utilised basic mathematical operators, not the gradient information of the function for optimisation.

SFLA is a metaheuristic optimisation technique that comprises features of regional investigation and global info interchange. It predominantly associates the benefits of both the genetic-based memetic algorithms (MAs) and PSO algorithms. SFLA involves a technique of probable solutions which is distributed among subgroups described as memeplexes. Inside every memeplex, each frog holds opinions that the opinions of different frogs can impact, and the thoughts can advance through a procedure of memetic development. The SFLA executes at the same time self-determining local investigation in individual memeplexes using a PSO like method. To certify a universal search, the virtual frogs are shuffled and restructured into new memeplexes in a procedure likely to employ the shuffled complex evolution algorithm following a definite number of memeplex progression steps. However, if there is no substantial improvement in the method, the procedure starts again from the generation phase. The processes of local searches and shuffling carry on until defined performance norms of convergence are fulfilled.

During calibration and validation, all the parameters (Table 2) were simulated in SWMM. The gauge/discharge data of the river measured at the gauge station were compared with the simulated output data. The objective function used in the study is NSE given in Equation (1). NSE chosen as an objective function, as suggested by Nash & Sutcliffe (1970), recommended that it is a normalised statistic that evaluates the relative magnitude of the residual alteration compared to the measured data alteration in hydrologic modelling for determining the best model output. Identifying an objective function that meets all requirements means precisely predicting low and peak flows along all parts of the hydrograph is difficult since each criterion may emphasise various types of simulated and observed behaviours. So other efficiency criteria like PBIAS (Equation (2)) and RSR (Equation (3)) are employed during the calibration and validation to check the correctness of the model results.

The first step was to digitise the river extent using a field survey, Google Maps and GIS software. The river shapefile is imported into PCSWMM software that uses the SWMM model engine. The initial step in developing an SWMM is a delineation of the catchment area. Delineation of the catchment area was performed utilising spatial analyst tools in PCSWMM software that used the SWMM engine (developed by Computational Hydraulics International, Canada). Flow direction and flow accumulation layers created in ArcMap show the direction and accumulation points in the streams. Streams were classified based on flow accumulation. At the point of substantial change in flow accumulation, pour points were identified on the streams indicating approximately the number of subcatchments to be generated. The watershed was delineated following this entire process. In the model, water flowed from the common node to the next node by using conduit links from each part of a subcatchment. In the model, the watershed delineation was done, which divided the basin (21,815.079183 km² in the area) into 641 subcatchments. The DEM (10 × 10 m) and catchment delineation was performed, and the Damodar River basin catchment was developed.

GA, SFLA and PSO algorithms coupled with the SWMM model to calibrate (1 June 2002 to 31 October 2006) and validate (01 June 2007 to 31 October 2011) flow at five different discharge-recording stations, namely Barkisuriya, Palgang, Nandadih, Phusro and Damodar Bridge. The behaviour of the algorithm is arbitrated on three conditions: (1) based on the statistical parameters value attained, (2) optimal value achieved in minimum iterations and (3) to achieve the best optimal value, least simulation time required. The simulation time also considered checking the behaviour of algorithms as generation cycles numbers are different for each algorithm. We fixed the number of simulation iterations at 2000.

Besides, the GA model also depicts high flows. We further examined the GA technique performance based on NSE, RSR and PBIAS values (Table 4). All the values were in the acceptable range (Santhi et al. 2001; Singh et al. 2005) even though the time taken by GA to converge was longer. We further noted that with the increase in the number of variables, the GA performance declines. Our findings are in good congruence with Raphael & Smith (2003). The time needed to complete the computation work was minimum for SFLA among the three algorithms. Still, the iteration number was higher than the other two algorithms to attain the objective range. The simulated discharge follows the monitored flow trends (Figure 4(a)–4(e)). The results obtained against the statistical criteria are within respectable limits with the recorded data. Thus, it is suggested that SFLA performance is akin to GA. The PSO performance is satisfactory compared with the other two algorithms. In addition, NSE, RSR and PBIAS values are in the satisfactory range (Table 4). The PSO simulated results match the observed flow but under-predict it. In addition, the time required for simulation is longer.

Model complication performs a significant task in algorithm performance. Primarily, model parameters are inter-reliant, which means that the parameters cannot be simulated individually and thus, the algorithm's performance faces challenges. The performance of the GA technique was excellent and effective during SWMM parameters calibration, and a similar observation was stated by Wan & James (2002). The PSO algorithm performed satisfactorily, but its convergence speed is faster than the other two algorithms. For SFLA, the algorithm requires larger iterations to converge the model, and the performance on statistical parameters was outstanding.

In the present work, the upper Damodar River watershed was divided into five sub-catchments (based on five gauging locations), and five subcatchments were considered. In all five subcatchments, the algorithm's performance was good. But none of the algorithms could exhibit stable execution superior among the others for all the five subcatchments. To some extent, this shows the complexity of the basin characteristics during calibration. Although SWMM has been used to calibrate all the five subcatchments' parameters, the subcatchment's properties are different in all the cases, confirming unambiguously unique results for the chosen algorithms. Among all the sub-catchments, the performance of GA was superior, with the best objective function value attained during model iterations.

Altogether, three optimisation techniques have been employed to test which algorithms converge faster and their capability to reach the best objective minima. During any optimisation situation, an algorithm's converging speed is very vital. For illustration, when the two techniques give a similar objective function value, a technique that converges faster will be the hydrologist's first choice. This is primarily applicable to those models, which take longer simulation time. When employed on any watersheds, complex distributed models might result in longer simulation time. In such circumstances, convergence speed is essential over objective function minimum value. Figure 5 shows that the two algorithms, i.e. GA and SFLA, attain nearly identical NSE values; however, they attain 95% of that NSE value in unique levels. A case of this is the Damodar Bridge station, where the value of NSE for GA and SFLA is almost similar, but still, GA (1,710 iterations) was the first choice because the algorithm took 155 iterations which are less than SFLA (1,865 iterations). GA should be preferred over SFLA for longer simulation models.

Computationally comprehensive, yet efficient and robust, three EAs were employed for SWMM parameter calibrations in a river system. Our results highlight that GA provided model iteration equivalent to 2,000 and performed better among the three algorithms trailed by SFLA. SFLA showed performance comparative to GA with a higher convergence speed value, whereas PSO performed satisfactorily. We recommend the PSO algorithms considering limited simulation time, while GA and SFLA should be preferred when satisfactory computational resources are offered. Interestingly, if GA is preferred for SWMM parameters optimisation with no restrictions on iteration, the results see no improvement after 1,800 iterations. However, employing GA and PSO to enhance SWMM parameters, a population with a small size is preferred. All the EAs have specific attributes and combining these into a single complete algorithm could be further investigated in the near future. Indeed, the findings of this study will be helpful while selecting optimisation techniques that can perform better with SWMM.

The authors would like to acknowledge the authorities of IIT(ISM), Dhanbad, for providing financial support to Prof. V.V. Govind Kumar under the FRS project (FRS (120)/2017–18/ME) for carrying out this research work. Authors duly acknowledge Computational Hydraulics International, Canada, for providing PCSWMM software to carry out this research study. The authors would like to thank DVC Maithon for providing generic data to carry out this research. AA acknowledges the joint funding support from the University Grant Commission (UGC) and DAAD under the Indo-German Partnership in Higher Education (IGP).

Data may be obtained from the authors upon request.

The code may be obtained from the authors upon request.

All authors contributed equally to the manuscript.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.