Decentralized calibration process for distributed water resources systems using the self-adaptive multi-memory melody search algorithm

Throughout the world, water resources simulation models are employed as a vital tool for investigating and analyzing quantitative and qualitative variations in water resources and the impacts of climate change on water resources among other applications (Razavi et al. 2012; Tsakiris & Alexakis 2012; Benedini & Tsakiris 2013; Tena et al. 2019; Rani et al. 2020). These models in fact simulate the current and/or possible future statuses of water resources and supply of various demands in strategic regions. Accordingly, the success of these models depends on the accurate determination of the model parameters (calibration), as well as on the analysis of the parameters' uncertainties (Kouchi et al. 2017). It is common practice to investigate the effects of the different parameters on hydrologic models of watersheds, and several studies have been conducted in this area in recent years (Eckhardt & Arnold 2001; Huard & Mailhot 2006; Bárdossy 2007; Tolson & Shoemaker 2007; Milzow et al. 2011; Razavi & Tolson 2013; Arsenault et al. 2014; Haberlandt & Radtke 2014; Kan et al. 2019). However, the calibration of simulation models for water resources management in watersheds can be considered as a research gap. The reason for this gap can be that most studies have used lumped models to simulate watershed resources management for which calibration has often been deemed insignificant. However, distributed models and the modeling of demand centers distributed across the watershed are needed for an accurate investigation of some complex water resources systems and for the analysis of their details (Ashrafi & Dariane 2017). It is expected that the calibration of quantitative management models for water resources systems is also highly important in these cases. Distributed models perform the calculations based on many values of the different parameters, some of which cannot often be measured or estimated accurately. As a result, the values of such parameters must be determined based on model calibration in order to simulate a specific system (Razavi & Tolson 2013). Robust optimization algorithms must be used to find the optimized values of the simulation models in order to improve the accuracy of the results obtained by the model and to reduce model output uncertainty (Krauße et al. 2012).

Calibration refers to calculations for minimizing the difference between the model outputs and corresponding observational values based on a specific objective function (Qin et al. 2016). Therefore, the calibration of simulation models can be defined as an optimization problem. Observational statistics and information of control points across the basin can be used to address the non-uniqueness of calibration problem solutions (Abbaspour et al. 2018). However, the calibration of models even with a small number of parameters may be associated with high computational costs (Arsenault et al. 2014). Manual calibration is the simplest but complex time-consuming process. Manual calibration is often impossible in large-scale models and results in locally optimal solutions dependent on the experience of users (Eckhardt & Arnold 2001). Automated calibration is also used for calibrating large-scale models allowing global optimal solutions (Tolson & Shoemaker 2007; Piotrowski et al. 2017). In this method, the intended computational model is coupled with a directed optimization algorithm, and the optimal values of the model parameters are determined through repetitive simulations. Optimization algorithms seek to find those values of parameters for which the difference between the computational results and observational values recorded within a certain interval is minimized (Moradkhani & Sorooshian 2009). Local search methods have been used in initial versions of automated calibration leading to solutions close to the initial solution (Nash & Sutcliffe 1970; Sorooshian & Gupta 1983). The initial local search algorithms include derivative-based methods such as the quasi-Newton method and free derivative algorithms such as the Nelder-mean simplex method. The main drawback of these techniques is their failure to achieve the global optimal solution leading to unreal solutions in calibrating large-scale complex models. With an increase in the software and hardware computational power and complexity of simulation models, more advanced and controllable optimization algorithms were gradually employed (Razavi et al. 2021). For instance, the following metaheuristic algorithms were used for calibrating the basin-level hydrological models: adaptive random sampling (Masri et al. 1980), controlled random search (Price 1978), the multi-start simplex, genetic algorithm (Wang 1991), simulated annealing (Thyer et al. 1999), and the shuffled complex evolution (SCE) algorithm (Duan et al. 1993), SCE algorithm (Van Griensven & Meixner 2007), real-value coding genetic algorithm (Wu et al. 2012), a chaos genetic algorithm and simulated annealing (Wang et al. 2012), Harmony Search algorithm (Arsenault et al. 2014), hybrid particle swarm optimization (Okkan & Kirdemir 2020), differential evolution algorithm and particle swarm optimization (Piotrowski et al. 2017), and melody search (MeS) algorithm (Ashrafi & Mahmoudi 2019).

According to Singh & Woolhiser (2002), selecting a suitable efficient optimization algorithm plays a key role in the calibration process. There has been recently a great interest in the use of metaheuristic algorithms to develop automated calibration models (Arsenault et al. 2014). Although these algorithms do not guarantee a global optimum (Li et al. 2016), they have multiple advantages such as ease of programming and application, ease of linking to simulation models, controllable searching process, and the use of efficient search operators to reduce the computational volume. Moreover, the efficiency and performance of metaheuristic algorithms in solving calibration problems are dependent on the determination of algorithm parameters in addition to the structure of algorithm operators (Piotrowski et al. 2019). This can be considered as strength but considerably increases the computational volume and thereby costs because before applying metaheuristic optimizer algorithms, the optimal values of their parameters can be determined via an extensive sensitivity analysis with high computational costs. Hence, optimization algorithms with the minimum number of parameters are preferred for calibrating applied models with long execution times (Okkan & Kirdemir 2020). In the meantime, the use of self-adaptive algorithms can be very useful to reduce the computational cost of the calibration process.

In some hydrologic modeling attempts, the calibrated model does not necessarily yield the best results for the verification period (Beven & Freer 2001; Madsen 2003; Brocca et al. 2011), and it is important to address this issue also for quantitative models of water resources system management. In most previous studies, distributed models for catchments were calibrated based on the hydrographic statistics related to a catchment outlet point (Awol et al. 2018). Some hydrologic models of catchment basins have been calibrated with data from several hydrometric stations scattered across the basin using an optimization algorithm (Zhang et al. 2010; Leta et al. 2016). Here, a decentralized methodology is introduced for calibrating a distributed simulation model for the Great Karun River basin using metaheuristic algorithms and the data from several hydrometric stations. This methodology can be used to fine-tune large-scale models for water resources systems under similar conditions.

The calibration of hydraulic simulation models is essential for fine-tuning model parameters to be consistent with the studied hydraulic phenomenon (Muleta & Nicklow 2005). However, in water resources simulation models, model calibration in fact aims to prepare the model for simulating the system. Nevertheless, it is important not to expect water resources simulation models to replicate the past behavior of the system exactly and completely. The reason is that the system management history is not necessarily based on management principles considered in the modeling, and many underlying assumptions made in model development could have been violated (Ashrafi & Mahmoudi 2019). Therefore, in the calibration of water resources simulation models, achieving a certain level of accuracy that suggests the correct implementation of the schema of the watershed and determines the approximate values of the variable parameters of the model will suffice.

Most water resources management models in Iranian basins have been developed based on integrated modeling of demand centers regardless of their distribution across the basin (Kim & Heo 2006; Dariane & Sarani 2013; Abadi et al. 2015; Ahmadianfar et al. 2017, 2019, 2021; Ehteram et al. 2018; Karami et al. 2019; Ashrafi et al. 2020; Azizipour et al. 2021; Rahimi et al. 2020). However, for the exact determination of consequences of management policies, the distribution and scattering of demand sites at the basin level should be considered (Ashrafi & Dariane 2017). Numerous studies have been conducted on modeling water resources systems by focusing on quality flow parameters (Emamgholizadeh et al. 2014; Keshavarzi et al. 2015; Jafari et al. 2019; Khorasani et al. 2020; Shirnezhad et al. 2021; Behboudian et al. 2021). Multiple studies have been conducted on Iranian basins to investigate the effect of climate changes on water resources and to analyze their results (Marofi et al. 2012; Jamali et al. 2013; Dehghani et al. 2014; Moazami et al. 2016; Adib et al. 2020). In these studies, the simulation model for operating water resources at the basin level has not been calibrated, and exact calculations with acceptable details have not been presented. There are also studies on hydrological and hydraulic models in which the aim is not to manage water resources systems and calibrate the simulation models. In most studies, there are no details on water resources systems and the distribution of water demands at the basin level or have been modeled briefly with multiple assumptions. To address discrepancies in inter-basin water transfer, numerous studies have been carried out using simulation and optimization models for water resources systems (Sadegh et al. 2010; Nikoo et al. 2012; Toosi & Samani 2012, 2014; Hashemi et al. 2013; Abed-Elmdoust & Kerachian 2014; Safavi et al. 2015). Most studies aimed to distribute water between adjacent basins based on the overall profitability of each basin or supply criteria for the basin. Accordingly, the details on the flow distribution at the basin level and the distribution of water demand centers in each region were not considered.

Occupying 4.2% of the country's land area, the Great Karun Watershed is one of Iran's largest water resources. Furthermore, the basin stretches across Khuzestan, Kohgiluyeh and Boyer-Ahmad, Chaharmahal and Bakhtiari, Lorestan, Hamadan, Esfahan, Fars, and Markazi Provinces. Given the large number of storage reservoirs and hydroelectric power plants and the particular inter-basin water transfer considerations, this watershed is undoubtedly the most important and complex water resources system in Iran, making the use of modern management systems indispensable for its management and utilization. Most previous studies on this watershed have been theoretical and have only modeled the reservoirs in the system without considering the details related to the demand sites (e.g. Taghian et al. 2014; Ahmadi Najl et al. 2016; Taghian & Ahmadianfar 2018).

It should be noted that the literature on calibrating hydrological models and rainfall–runoff models of basins was reviewed. To the best of our knowledge, there are a few studies on calibrating simulation models of water resources systems. The present study introduces a multi-memory optimization algorithm based on the principles of the MeS algorithm for the purpose of performing the distributed calibration and comparing its results with those of other optimization algorithms. The main goal of this research is not about introducing a general powerful optimization scheme; rather, we are attempting to find the best algorithm for performing distributed water resources model calibration. Hence, the proposed metaheuristic algorithm has been designed for this aim and its capabilities for solving the current problem have been evaluated compared to other well-known metaheuristics. After comparison, the best algorithm will be used to develop a calibrated water resources model for the Great Karun River basin in southwest Iran. Such a model will be a realistic and practical instrument for consumption management and achievement of optimal water resources allocation management. At the end, the effects of the calibration dimensions on computational cost and its impact on the performance of the results are investigated by forming various calibration models with different dimensions. It is so important to choose the correct calibration dimension, where the computational cost enhances by increasing the problem dimensions.

There are several parameters associated with the development of simulation models of water resources systems because of the multiplicity and interrelatedness of system components. It is crucial to perform calibration calculations using field data and engineering experience to quantify such variables. Performing these calculations manually is an extremely difficult task and yet the results obtained are not accurate. Therefore, in this study, a decentralized intelligent algorithm is proposed to determine the best values of model parameters. Figure 1 shows the general flowchart of the proposed approach.

Most of the studies to calibrate water resources models utilize statistics of the latest downstream hydrometric station. The lack of a unique response to such problems is one of the challenges that such methods encounter. This method also works in models in which the demand sites are integrated. However, in models where the demand sites are distributed throughout the basin (named distributed simulation models), there is a great necessity for decentralized calibration. In this study, the purpose of decentralized calibration is to work with statistics of multiple and distributed hydrometric stations within the system to perform the calibration process. The water evaluation and planning system (WEAP) model is adopted to simulate the distribution of water resources system of the Great Karun Watershed.

Generally, the input of each simulation model of water resources systems includes input variables (such as surface flow values, precipitation, evaporation, and different values of consumption) and known and unknown model parameters. Known parameters are those whose accurate values can be extracted and specified, while for unknown parameters, the accurate value is not known or cannot be extracted, and we can only specify approximate values and rational ranges. Several optimization algorithms can be utilized to calibrate simulation models. In this study, a modified metaheuristic algorithm is utilized relying on significant interactions between various memories of the MeS algorithm, which is described below.

In this research, the RMSE function is considered as the main objective function, and the other functions will be examined in the following sections. Also, the unknown parameters that we are attempting to determine their precise values include return flows, flow losses at different intervals throughout the basin, fraction of diverted flow at the Band-e Mizan intersection node, and demand–supply priority at the basin level. The precise values of these parameters need to be determined in the calibration process to gain more consistency between model results and what has been happened in real condition.

Values of flow loss in the ith interval during the period (⁠⁠) include infiltration ⁠, stream evaporation ⁠, and illegal withdrawals that occur at some intervals. Values of flow loss along the Karun, Gargar, Shoteyt, Dez, and Great Karun rivers have been modeled as a percentage of the flow flowing across the rivers (⁠⁠) at a total of 13 different intervals (⁠⁠). Based on the available statistics, the maximum value of flow loss in this area was considered (⁠⁠).

As compared with previous studies, the modeling approach adopted in this study is considering more details of the system in modeling and aims to reach a distributed model. The WEAP software is, therefore, used to develop the simulation model of a Great Karun system. All the basic elements of the system, including the collection and return points, are seen in the configuration as shown in Figure 2. Flow collection points include the supplying water required by demand elements such as agricultural lands, municipal water-distribution networks, aquaculture farms, industrial demands, and water conveyance channels.

According to the configuration shown in Figure 2, the system consists of 6 storage reservoirs, 2 regulatory dams, 6 hydroelectric power plants, 5 inter-basin water transfer systems, 5 head flows, 9 municipal water-distribution networks, 17 irrigation networks, 8 aquaculture demands, and 6 industrial demands. Also, 11 hydrometric stations are considered at the basin level to control the accuracy of the simulation model performance.

Table 1 presents the characteristics of the Great Karun reservoirs.

Figure 3 shows the total water demands in the system. Municipal and aquaculture water demands are almost the same in all seasons. The inter-basin water transfer varies monthly, and the highest values are observed from February to July. As shown, the highest water uses are observed in agricultural and industrial sectors. The highest monthly water demands in the system are observed from May to October. However, the maximum natural flow of rivers in the basin is observed from December to May. This indicates the role of reservoir management in supplying the regulatory flow required for supplying water uses.

In this study, the basic capabilities of MeS algorithm are implemented to derive an efficient algorithm for calibrating distributed water resources simulation models. MeS algorithm has been recently introduced by Ashrafi & Dariane (2017) as a powerful optimization method (Ashrafi & Kourabbaslou 2015). MeS algorithm simulates the performed processes of a group of musicians with emphasis on interactive relations occurring between music players, while they are looking for a better succession of pitches within a melody. In such a group, players can lead each other to obtain a better-adjusted note (variable in optimization problem) within their melodies (solutions in optimization problem). Each music player sounds his/her melody separately, while he/she can choose better notes by hearing good melodies in the group. Hence, getting influenced by the other players, each music player improves the melody step by step in the group and consequently, he/she can influence the others (Ashrafi & Kourabbaslou 2015). The conceptual interactive relations among different memories are demonstrated in Figure 4.

In MeS algorithm, each melody is analogous to a solution vector, each pitch within the melodic line to a decision variable, possible range of pitches to the value ranges of decision variables for randomization (which can be varied in different iterations), and finally, qualities of melodies are analogous to the values of objective functions. The MeS algorithm applies three parameters including player memory considering rate (PMCR), pitch adjusting rate (PAR), and distance bandwidth (bw). The PMCR parameter is the selection probability for picking up variables from the algorithm memory randomly to improvise new solutions. The selected variables can be adjusted using PAR and bw as the probability and the maximum range of the adjusting process, respectively. Pitch adjusting operator is designed to avoid getting trapped in local optimal solutions. In adaptive versions, various adaptive techniques can be adopted for evaluating proper values of the algorithm parameters to match different phases of evolution and enhance the algorithm efficiency. More details about MeS algorithm can be found in Ashrafi & Kourabbaslou (2015) and Ashrafi et al. (2017). In this study, the ability of MeS algorithm in modeling interactive relationships among different memories is adopted to design an efficient optimization structure for water resources model calibration.

The unknown parameters of the water resources system model are considered as decision variables of the optimization model. In the designed algorithm, decision variables of the model are classified into several categories based on their nature, and the optimal values of each category of variables are determined in one of the algorithm's memories. In each memory, one set of improvised decision variables and the values of the decision variables of the other categories of the best solution found are replaced in their specific memory. Figure 5 shows a schematic diagram of the proposed optimization algorithm for a problem with three memories.

Each memory is assigned to a set of decision variables. In the first iteration of the algorithm, several solutions are randomly generated and distributed in different memories (initialization step). Hereinafter and in each iteration, a new solution is generated in each memory and if it is better than the worst solution in the memory, then the worst solution will be replaced by the new improvised solution. As shown in Figure 5, to generate a new solution per memory in each iteration of the algorithm, the number of variables assigned to the same memory is generated based on the triple operators of the MeS algorithm (memory consideration, pitch adjustment, and randomization). The value of the other variables is accurately picked from the best-achieved solution stored in other memories. In other words, the value of each decision variable of the best solution stored in the specific memory of that variable is accurately transferred to other memories to generate new solutions. Such a transfer is regarded as the influence of the performance of memories on each other, which is the basic characteristic of the MeS algorithm. The proposed structure significantly coordinates the MiMeS optimization algorithm and the calibration problem of water resources models (and many other engineering issues).

A framework consisting of 10 computational steps is designed for solving the calibration optimization problem through MiMeS algorithm as follows.

Step1. Initializing the optimization problem, and categorizing the solution vector

Step2. Initializing all memories based on their specified categories and possible ranges of variables

Step3. Evaluating initial random solutions stored in memories

Step4. Setting the initial values of adaptive parameters (i.e. PMCR_ini, PAR_ini, and bw_ini)

Step5. Improvising a new solution from each memory using proposed adaptive improvisation scheme:

 for i^th memory

  for j^th variable

    if, whereis the set of variables specified to the i^th memory

     if r₁<PMCR_t

      Memory Consideration: choose a j^th variable from the i^th memory randomly

      if r₂<PAR_t

        Pitch Adjustment: adjust the chosen variable using a specified jump rate

      End if

     else

      Randomization: generate a random value within the possible variable range

     end

    else

     for k^th memory

      if

       x_i=the i^th variable of the best solution in k^th memory

      end if

    next k

   end if

  next j

 next i

Step6. Evaluating new solutions

Step7. Updating all memories

Step8. Determining the best solution of each memory and their variable values

Step9. Determining possible variable ranges for next randomization employing the proposed procedure

Step10. Resetting the adaptive parameters values if a learning period is over

Step11. Repeat steps 5–10 until the stop criterion for the algorithm is satisfied

When the stop condition is satisfied at the end of the algorithm, the best-obtained solution is extracted from the solutions available in all memories as the final response. The ability of the MeS algorithm to search for solving various optimization problems has been demonstrated (Ashrafi & Kourabbaslou 2015). The use of one memory specifically to search for optimal values of the same decision variables and applying the optimal values of other variables from other memories diminishes the length of the solution string and increases the computational capability of the algorithm. The method proposed by Ashrafi & Kourabbaslou (2015) has been adopted to update the allowable range for the initialization of decision variables. The values of the algorithm parameters are updated self-adaptively. More information is provided in Ashrafi et al. (2017) on the self-adaptive MeS model. In the proposed algorithm, a certain memory is allocated to update the algorithm parameters.

The modeling performed in this study is based on the monthly time step. Based on the current water year in Iran's climate, the start and end of the water year in the simulation model are defined, respectively, October and September. According to the data available at the stations and the date of starting operations in the Gotvand-e-Olya Dam, we used the data of 10 consecutive years (from October 2006 to September 2016) for the calibration period. Since the policies over the operation of the main reservoirs influence main inflows of the system and also there is no precise pattern (curve or relation) of the operation policy applied to the Dez and Gotvand-e-Olya reservoirs in past years, we are not able to simulate the behavior of these storage elements in the water resources system of the basin following management principles. Water resources simulation models frequently follow a standard operation policy to supply demands that are similar to what happened in reality but are not always true. Contrarily, the time series of the output of reservoirs are accessible only in particular historical periods. Therefore, the best approach to measure the model's variable parameters (model calibration) is excluding reservoirs and considering the reservoir's outflows as inflows of the system's head flows. Accordingly, this can eliminate the adverse effect of differences in reservoir performance between model and reality from the model calibration process. Given the availability of values of historical flows in the downstream of reservoirs during the intended period, the water resources system was modeled in this modeling without considering the system reservoirs. Finally, the historical monthly data of the system in four consecutive years (from October 2016 to September 2020) are applied for the verification of the calibrated model.

To estimate the performance of the proposed algorithm (MiMeS) for the calibration process, the results obtained are compared with those of SaMeS (Ashrafi et al. 2017), HS (Lee & Geem 2005), SGHS (Pan et al. 2010), GA (Hınçal et al. 2011), PSO (Haddad et al. 2013), and EMPSO (Afshar 2009) algorithms. For a fair comparison, the parameters of different algorithms are chosen to reach an equal number of objective function evaluations for all algorithms. Moreover, the values ​​of the algorithm parameters are selected according to sensitivity analysis and what was suggested in the main references. The parameter of the number of fitness evaluations (NoFE) is selected in this study based on the convergence diagram for all algorithms, ensuring that all algorithms reach their best performance. Table 2 shows the optimal values for the parameters of all algorithms used in this study. Considering the adaptive algorithm proposed in this study, there is no need for heavy sensitivity analysis calculations to find the optimal values for algorithm parameters. This is of great importance in calibrating the simulation models with long execution times.

Table 3 shows the results of this comparison where the best solutions for calibration and verification periods are related to the proposed algorithm, although it has had the most computational time. In Table 3, NDV indicates the number of decision variables, and NoFE stands for the number of performed fitness evaluations. The least running time belongs to the HS algorithm, but the answer is far from the optimal range. It is more obvious for the verification process. SGHS, EMPSO, and GA algorithms have reached the optimal results in the calibration period with better running times than MiMeS, although the best solution has been reported for the proposed algorithm. The calibrated models by SGHS and GA algorithms have not suitable performances in the verification period, while the EMPSO and MiMeS algorithms have acceptable performances based on the verification results. Figure 6 shows the linear correlation graph of the MiMeS model results with the observed data at different stations.

With an increase in dimensions of the optimization problem, the algorithms encounter more challenges to find the optimal solution. Contrarily, by accurately estimating the larger number of algorithm parameters, the model can be optimally calibrated, and thus better value can be achieved from the objective function. Therefore, the efficiency of the algorithm applied will be directly influenced by the dimensions of the problem. To investigate this phenomenon, the system simulation model of the Great Karun was calibrated in 15 different combinations of decision variables, and the results were compared. In each case, a large set of model parameters is removed from the calibration process and replaced with the expected mean for those parameters. Table 4 summarizes the dimensional characteristics and results of different calibrated models for simulating system in the verification period (from October 2016 to September 2020).

Considering the high computational costs of the calibration of distributed models, it is essential to determine the effect of calibrating different variables of a water resources system on the precision of simulation model results. The results of this study revealed which variables of the water resources simulation model have a higher priority to be calibrated.

In Model 01, all the parameters of the model are obtained from the calibration process and it is assumed that the results of this model have the highest consistent with the reality of the system. The obtained values of the NSE, R², MAE, and RMSE indices indicate this consistency. In Model 16, however, none of the parameters are obtained from the calibration model and its results are known as results of the base model. Based on traditional modeling in the region, the return flow is 30% (for traditional irrigation networks and water rights), 55% (for drainage sugarcane farms), 75% (for aquaculture demands), and 0% (for industries that are not allowed to discharge into the river). In Model 02, all the parameters modeled in the main problem are derived from the calibration model except for the return flow values that are considered equal to default values based on the data available. In Model 06, beside the return flow values, the flow loss values are also excluded from the calibration process and their mean values are replaced in the model. In Model 07, beside the return flow values, the demand–supply priorities are also excluded from the calibration process. In these models, municipal demands are prioritized over industrial demands (and industrial demands over agricultural demands), but spatial distribution is not considered for these priorities and, for example, all agricultural demands at the basin level have the same priority.

In this analysis, it is assumed that Model 01, as the most comprehensive model, has the greatest consistency with the reality of the system, and hence the consistency of the other models to the model's responses indicates their better performance. Here, is the relative difference of the worst reliability of the demand–supply in the jth model, is the least reliability of the demand–supply in the base model, is the worst reliability of the demand–supply for the jth model, is the relative difference of the maximum vulnerability in the jth model, is the maximum vulnerability of the jth model, is the maximum vulnerability of the base model, is the reliability of the ith demand–supply in the jth model, is the running time of the jth model, is the highest running time of the calibration model related to the complete model (Model 01), is the running time of the base model (Model 16), is the relative increase of the running time of the calibration model than the base model, and is the number of demand sites in each model. Table 5 shows the time characteristics needed to run different models and also the difference in relative performance indices of these models compared to each other.

According to Table 5, increasing the number of variables in the calibration process has a direct effect on increasing the computational cost (running time), while necessarily does not linearly improve the values of the model's performance indices. Therefore, the implementation of a very complete model with large dimensions is not always affordable, and smaller models may be preferred depending on the available computational capacity. Figure 7 shows the graph for the computational time against relative changes in the simulation efficiency indices of the water resources system.

The results presented in Table 5 indicate that the calibration of simulation models of the water resources system is necessary for distributed models. As illustrated in Figure 7, the smallest reliability of the system's supply–demand in the most comprehensive model compared to the base model, in which none of its parameters are obtained through calibration, has changed by about 9% and its maximum vulnerability by about 14%. These indicate significant values ​​that cannot be neglected. However, the calibration process is not often important in lumped water resources models. Also, according to Tables 3 and 4, the calibration of different parameters will differently affect the model results. By comparing models 1–5, the highest influence on the model results is with calibrating the flow loss values and demand priorities, and the least impact is with calibrating the percentage of return flows. Similar results are obtained by comparing the results of models 12–15 but when only one set of parameters is calibrated, then a fraction of diverted flow in the Band-e Mizan has the least effect on the model results. It can be concluded that an increase in the calibration model dimensions (and thereby computational costs) does not necessarily increase the final model results. For example, in estimating vulnerability, Model 04, which increased the computational time up to 70% of the overall model, gives the same result as Model 14, which increased the computational time up to 30% of the overall model. It was also found that Model 15, which increased the computational time up to 11% of the overall model, did not affect the increased accuracy in estimating the reliability and/or vulnerability of water uses. Moreover, this study shows which model parameters should be exactly determined and more concerned. According to the results, the maximum accuracy should be, respectively, obtained in calibrating flow losses, prioritizing water demands, the percentage of return flows, and the percentage of flow splitting at Band-e Mizan. It can be generally concluded that the optimization of the model calibration dimensions depends on the computational capabilities and the preferable accuracy associated with calculations.

The robustness of the algorithm is of special importance to solve large-scale time-consuming problems such as calibrating water resources simulation models. It is further highlighted by comparing the results of different algorithms. For example, for the algorithm proposed in this study that reached the best solution, the coefficient of determination (R²) is twice that of the HS algorithm, while its computational time is approximately 11% more than that of the HS algorithm. On the other hand, the proposed algorithm is relatively complex with the longest computational time. This is more apparent for more complex models. Accordingly, algorithms such as SGHS are preferred that the difference of their results with the proposed algorithm is only 4%, whereas its computational time is 10% lower than the algorithm proposed in this study.

Considering multiple unknown parameters affecting water resources system simulation and uncertainty regarding the determination of such parameters, it was shown in the study the necessity of calibrating distributed simulation models used for water resources systems. The lack of calibration may lead to results far from reality. It was also found that the calibration of different parameters differently affects the precision of model results, and the relationship between increased computational costs of calibration models and increased precision of the results is not completely linear. In other words, an increase in the calibration model dimensions and thereby the computational cost does not necessarily increase the accuracy of model results. In the meantime, selecting intended parameters for calibration requires sufficient knowledge of modelers to achieve desirable results.

There are two major drawbacks to executing the proposed method. First, it is not a common method to develop distributed models for simulating water resources systems because of the need for a great deal of data on water uses and high-resolution system configuration, which is not available in most cases, and estimating their values is associated with high uncertainty. Second, processes governing the performance of water resources systems are significantly affected by management systems in addition to hydrological changes. Therefore, the results of the calibrated water resources model are reliable until the operating strategy is constant, and the model should be recalibrated by changing the operating and management strategies.

The authors gratefully acknowledge support by the Shahid Chamran University of Ahvaz (SCU) under grant SCU.EC98.31254. Moreover, the authors thank the Stockholm Environment Institute (SEI) for providing a free license of WEAP software for noncommercial research.

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