Comparative analysis of three soft computing techniques for optimizing reservoir rule curves

The rising demand for water poses a significant challenge in Iran, a country characterized by its arid and semi-arid climate. Surface water reservoirs are a crucial component of water supply systems in this country. In arid regions like Iran, over 90% of renewable water is used by the agricultural sector, primarily sourced from surface water reservoirs. Therefore, effective reservoir operation is vital under both normal and extreme conditions, such as droughts (Anvari et al. 2017).

Beyond simply managing river flows, reservoirs provide vital services to society, including hydroelectric power generation, reliable water supply, and flood control. To maximize their effectiveness, reservoirs must be strategically designed to capture excess runoff during wet seasons for use in drier periods. This proactive approach ensures that sufficient water is released to meet downstream demands, thereby enhancing overall water availability and management amid growing challenges (Cai et al. 2011; Anvari et al. 2014; Sasireka & Neelakantan 2017).

As illustrated in Figure 1, full demand fulfillment is pursued when the water available (WA) falls within the range [LRC_m, URC_m], with LRC_m and URC_m representing the lower and upper RCs for month m, respectively. If WA exceeds URC_m, supply surpasses demand; however, if WA is below LRC_m, no water is released. So, identifying the optimal reservoir storage capacity (RSC) is a crucial design parameter that significantly impacts both water supply and environmental flow requirements. Additionally, it enhances confidence in water release decisions (Adeloye et al. 2016).

Various simulation-based methodologies determine the necessary RSC for planned water supply, including the Ripple mass curve method, sequent peak algorithm (SPA), modified SPA (MSPA), extended deficit analysis (EDA), behavior analysis (BA), and the empirical method by Vogel and Stedinger (McMahon & Adeloye 2005). Each method is chosen based on reservoir needs, data availability, and desired outcomes. For example, SPA uses historical inflow data to determine maximum storage required to meet demand without shortages, identifying the highest cumulative deficit. MSPA extends SPA by incorporating varying demand patterns and environmental flow requirements. The Ripple mass curve method plots cumulative inflow against demand, with the difference between the highest and lowest points representing required storage. BA simulates reservoir operations under various scenarios to assess RCs' performance in meeting demand and maintaining levels, involving iterative testing and adjustment (Montaseri 1999; McMahon & Adeloye 2005; Adeloye et al. 2016; Kosasaeng et al. 2022; Moghaddasi et al. 2022; Anvari et al. 2023).

Traditional optimization techniques like linear programming (LP), non-linear programming (NLP), and dynamic programming (DP) are widely used in water reservoir operations. However, they have several drawbacks compared to soft computing (SC) approaches such as genetic algorithms (GAs), flower pollination algorithm (FPA), tabu search (TS), particle swarm optimization (PSO), harmony search (HS), Harris Hawks optimization (HHO), honey-bee mating optimization (HBMO), simulated annealing (SA), ant colony (AC), bat algorithm, and gravitational search algorithm (GSA) (Goldberg 1998; Chang et al. 2005a, b; Kumar & Reddy 2006; Chen et al. 2007; Jalali et al. 2007; Haddad et al. 2008; Madadgar & Afshar 2009; Wang et al. 2009; Fallah-Mehdipour et al. 2011; Jothiprakash et al. 2011; Ostadrahimi et al. 2012; Bozorg-Haddad et al. 2014; Ahmadianfar et al. 2016; Spiliotis et al. 2016; Kangrang et al. 2018; Marchand et al. 2019; Ranjbar et al. 2021; Moghaddasi et al. 2022; Techarungruengsakul & Kangrang 2022; Anvari et al. 2019). Traditional optimization techniques have several disadvantages, including the assumption of linearity for both the objective function and decision variables (as seen in LP), sensitivity to input variability and the curse of dimensionality (common in DP), increased computational complexity with a growing number of variables and constraints, the tendency to converge on local optima (characteristic of NLP), and challenges in managing uncertainty (Anvari et al. 2014; Sharifi et al. 2021; Lai et al. 2022; Beiranvand & Ashofteh 2023). In contrast, SC techniques are adaptive, flexible, and capable of handling complex optimization problems. These methods are robust against uncertainties and can optimize multiple conflicting objectives (Soundharajan et al. 2016; Darakantong & Kangrang 2019; Papazoglou & Biskas 2023).

Recent studies underscore the effectiveness of SC techniques in water resource management. For example, Adeloye et al. (2016) analyzed Hedging-Integrated RCs for the Indian Pong reservoir, using SPA to design the RC. Their application of hedging rules and GA reduced the vulnerability index from 61 to 20%. Another study evaluated the reservoir's performance under climate change, demonstrating that static hedging rules decreased vulnerability from over 60% to under 25% while maintaining volume-based reliability. Although dynamic rules provided minor improvements, their complexity renders static rules the more efficient option. Recent advancements in hydrological modeling have greatly improved water resource management in Iran. Anvari et al. (2022) developed the ANN-IBGSA model to forecast 1-month ahead streamflow for the Karun River. This model combines an artificial neural network (ANN) with an improved binary gravitational search algorithm (IBGSA) to minimize the sum of root mean square error (RMSE) and coefficient of determination (R²) while identifying optimal predictors. The ANN-IBGSA model outperformed forecasts from 2013, achieving average improvements of 9.91% in RMSE, 11.85% in R², and 9.13% in mean absolute error (MAE). Building on this work, Anvari et al. (2023) enhanced drought-related reservoir operations in the Aharchay basin by integrating a hedging rule-based reservoir operation model (HRROM) with a climate-based irrigation scheduling model (CBISM), collectively known as HRROM-CBISM. The HRROM optimizes long-term decisions for the Sattarkhan reservoir by evaluating various streamflow scenarios, while the CBISM allocates irrigation based on fluctuating agricultural demands and evapotranspiration. This integrated approach employs three optimization techniques of LP, NLP, and PSO to maximize the total income of the Aharchay agricultural network, considering climatic factors and water supply. Their findings indicated that the HRROM-CBISM significantly improved time-based (α_t) and volume-based (α_v) reliability indices by 20 and 44%, respectively, compared to the traditional standard operating policy (SOP). The average values of α_t, α_v and vulnerability (V) for SOP were 0.33, 0.51, and 0.48, respectively. With the HRROM-CBISM, the average values of α_t, α_v and vulnerability (V) were about 0.5, 0.55, and 0.45, respectively. The literature on soft computing techniques in water resource management identifies the GSA as an effective heuristic method based on physical principles. GSA excels in solving complex optimization problems through gravitational attraction, achieving a superior balance between exploration and exploitation. This results in improved convergence rates with fewer parameters compared to traditional algorithms like PSO and GA, enhancing its robustness. However, it is important to recognize that no single algorithm is universally superior for all optimization tasks (Rashedi et al. 2018; Hashemi et al. 2021).

This study presents a novel methodology for deriving reservoir RCs by integrating the SPA with advanced optimization techniques, specifically PSO, GSA, and GAs. By utilizing monthly demand and discharge data from 1987 to 2018, we first establish optimal RCs through the SPA. These foundational curves are then enhanced using hybrid approaches -PSO–SPA, GSA–SPA, and GA–SPA to effectively minimize total water shortages. Additionally, this research incorporates environmental considerations by calculating and comparing reservoir performance indices, thereby providing a holistic evaluation of the proposed methodologies.

This paper is structured as follows: the subsequent section details the case study, data, methodology, and formulation of the optimization models employed in this research. Following that, we present and discuss the results obtained. Finally, the paper concludes with a summary of the key findings and implications of the study.

As depicted in Figure 3, the reservoir reaches its peak inflow in April, while the lowest inflow is observed in October. Notably, during the summer months (June, July, August) and autumn months (October, November, December), inflow levels are comparatively lower than in other seasons. Conversely, spring and winter experience the highest inflows, contributing approximately 87% of the total annual inflow during these periods.

As illustrated in Figure 4, the GA process starts with initializing a population (P₀) of candidate solutions, represented as chromosomes. These individuals are evaluated using a fitness function. Selection mechanisms, such as roulette wheel or tournament selection, identify high-performing solutions for reproduction. Offsprings are generated through genetic operators like crossover probability rate (P_c), which recombines parent chromosomes, and mutation probability (P_m), which introduces variability by altering genes randomly. The new individuals replace less fit members of the population, and this process iterates until a termination condition is met. Common termination criteria include reaching a maximum number of generations, achieving population convergence, or attaining a predefined fitness threshold. Key parameters influencing GA performance include population size, crossover rate, mutation rate, and the number of generations, all of which require careful tuning to balance exploration and exploitation in the search space.

According to Figure 5, the PSO algorithm involves initializing particle positions and velocities, evaluating the objective function, updating personal and global bests, and modifying velocities and positions. Key parameters include the number of particles, inertia weight (for exploration and exploitation), and acceleration coefficients (balancing personal and social influence). Termination conditions typically involve reaching a maximum number of iterations, achieving a satisfactory fitness level, or minimal changes in the global best function. The number of parameters that need optimization, which are as follows, determines the dimensions of the search space:

• The position of each particle for calculating the fitness value (X_i);

• The velocity of each particle defined as the rate of position change (V_i);

• The previous best positions (X_i−1).

  Furthermore, the position of the particle, which has the best fitness value, is referred to as the ‘global best position’ and is indicated by G. During the search process, the present positions of all particles (X_i) are assessed by employing the fitness function based on the following equation:

  

  (9)

The second factor is the previous best position of the given particle, which is referred to as the ‘particle memory’ or the ‘cognitive component’. As shown in Equation (16), this term is used to adapt the velocity toward the best position that the particle can have (C₁r₁(P_i(t) − X_i(t))). The third factor is the position of the best fitness value which is named the ‘social component’ (C₂r₂(G − X_i(t))) and is employed to adjust the velocity toward the global best position in all particles (Hassan et al. 2005).

This study utilized several accuracy indices to assess the operational performance of the Zarrineh Rud reservoir, including time-based and volume-based reliability measures (α_t and α_v, respectively), vulnerability (V), mean annual shortage (in MCM), and total shortage (in MCM). These indices are directly related to water scarcity and are defined according to Hashimoto et al. (1982) and Jain (2010) (see Table 1).

• - Time-based reliability (α_t):α_t is defined as the percentage of time that the system operates without failure (Hashimoto et al. 1982) and is calculated according to Equation (22), where t is the index for year, N indicates the number of years, D_t represents the downstream demands, R_t shows the release from the reservoir, and Z_t is an indicator equal to one when the demands are satisfied and equal to zero otherwise.

• - Volume-based reliability (α_v):α_v is the proportion of the total time under consideration during which a reservoir is able to meet the full demand without any shortages (Equation (23)).

• - Vulnerability (V): The definition of vulnerability used in the current study is the average period shortfall as a ratio of the average period demand (Equation (24)). In this equation, V indicates vulnerability and R_t represents the actual release during t.

Table 1

The accuracy indices for reservoir operation

Equation .	No. .	Name of Equation .
Time-based reliability (αt)	(22)
Volume-based reliability (⁠⁠)	(23)
Vulnerability (V)	(24)

Equation .	No. .	Name of Equation .
Time-based reliability (αt)	(22)
Volume-based reliability (⁠⁠)	(23)
Vulnerability (V)	(24)

View Large

In this section, the Zarrineh Rud reservoir was initially designed using the SPA method to fulfill existing demands without failure, based on various historical runoff scenarios. For this design, both drinking and agricultural demands were taken into account to determine reservoir storage. The results indicated that the calculated capacity (K_a) was 467 MCM, which increased to 573 MCM when factoring in the dead storage.

The parameters of the algorithms were set as shown in Table 2. For all algorithms, T is set to 500. For the GA, we utilized a population size of 30. The crossover operator was implemented using the two-point crossover method, while the selection operator was based on the roulette wheel selection method. For the PSO, the initial population size (number of particles) is 30, c₁ and c₂ are set to 2, w_max to w_min are considered as 0.9 and 0.2, respectively. The final parameters of the GA, PSO, and GSA are detailed in Table 2. These values were determined through trial and error and by conducting experiments within a range of values.

The sensitivity analysis of the GSA is presented here, focusing on four parameters that influence the sensitivity of reservoir performance to variations in each GSA parameter. These parameters are the maximum number of iterations (T), the number of agents (N), the gravitational constant (G₀), and the gravitational constant decay factor (α). The GSA for the reservoir optimization problem is run 10 times for each parameter set, and the mean performance value is obtained across these runs. The value of T depends on the complexity of the problem. Low values of T would not provide enough time for the objects to explore the search space and high values increase computational complexities. In most GSA studies, the value of N is typically set to 50. In this study, the performance of GSA with different values of N in the range [10–100] was evaluated, and the optimal setting of N = 30 was selected. The other main parameters of GSA, G₀ andα, control the exploration and exploitation of the algorithm.

Figures 10 and 11 illustrate that both GSA–SPA and PSO–SPA exhibit similar behavior in the URC, although some differences are observed in the LRC. The results presented highlight critical insights into the performance of the GSA–SPA and PSO–SPA algorithms in reservoir management, emphasizing their strengths and limitations within specific contexts. Both algorithms exhibit similar behavior in managing the URC, suggesting that they are equally effective in addressing upper boundary constraints. However, differences in the LRC behavior indicate variability in their ability to handle lower boundary conditions, which may influence water allocation decisions during periods of scarcity.

The comparative analysis of reservoir performance indices for these three optimization-based techniques provided additional depth to the evaluation. The reduced variation in vulnerability for GSA–SPA and PSO–SPA suggests that both algorithms consistently avoid critical system failures, enhancing their reliability under diverse scenarios. However, the high variation in total shortage for GA–SPA highlights challenges in achieving uniform water allocation across time periods, an aspect that requires further attention.

The performance of GSA–SPA demonstrates distinct advantages over the historical SPA (SPA-Hist), with a 3 and 2% improvement in time-based (α_t) and volume-based (α_v) reliability indices, respectively. These enhancements reflect a more reliable and efficient water supply system, a critical factor in ensuring sustainable reservoir management. Additionally, the significant reduction in mean annual and total shortages by approximately 8% highlights the practical benefits of GSA–SPA in mitigating water deficits, potentially improving water availability for downstream users. In comparing GSA–SPA and PSO–SPA, it is evident that while both algorithms are effective, GSA–SPA offers superior performance in reducing shortages and improving reliability indices. However, PSO–SPA's simpler implementation and computational efficiency may make it a preferred choice in scenarios where computational resources are limited or rapid solutions are required.

From a practical perspective, the application of these advanced algorithms in reservoir management can provide critical support for decision-making, particularly in regions prone to hydrological variability and water stress. Their ability to balance supply and demand while minimizing shortages and improving reliability indices underscores their potential to enhance water resource sustainability. However, their limitations, such as variability in performance indices and computational demands, must be carefully considered when choosing an algorithm for specific reservoir management scenarios.

In November, the moderate positive correlation (R = 0.55) between R and AW suggests that reservoir releases are moderately influenced by the AW. This indicates that the model prioritizes maintaining a balance between water availability and operational needs, ensuring sufficient supply without over-reliance on stored water. The weak positive correlation between R and Q (R = 0.28) implies that inflow exerts some influence on release decisions, though its effect is less pronounced. This may reflect the algorithm's strategy of integrating inflow as a secondary consideration, prioritizing AW for immediate operational decisions. The low positive correlation between Q and AW (R = 0.35) suggests a limited, but not negligible, relationship between Q and AW in the reservoir. This could indicate that the reservoir's storage dynamics or retention policies effectively moderate the impact of seasonal inflows, preventing excessive dependence on inflow variability. Such a relationship is particularly critical for ensuring resilience during low inflow periods or unpredictable climatic conditions.

The observed correlations are indicative of the GSA–SPA model's robust capacity for adaptive reservoir management. The ability to effectively utilize AW while maintaining moderate sensitivity to Q demonstrates its practicality in regions prone to hydrological variability. By ensuring a balanced approach to release decisions, the model supports sustainable water resource management, particularly in scenarios involving competing demands or seasonal constraints. From a practical perspective, these findings underscore the importance of prioritizing AW as a key driver in reservoir operations, ensuring consistent supply to meet downstream demands. The moderate and weak correlations also suggest that Q can be leveraged as a supplementary resource, particularly during wet seasons, while minimizing vulnerability during dry periods. Additionally, the low correlation between Q and AW highlights the model's potential in enhancing water retention strategies, supporting long-term sustainability. This adaptive capability is particularly relevant for managing reservoirs in monsoon-dependent or arid regions, where inflow can be unpredictable.

Surface water reservoirs are indispensable in arid and semi-arid regions of Iran. They capture excess runoff during wet seasons, ensuring a reliable water supply during dry periods. These reservoirs are vital for supporting agriculture through irrigation, mitigating drought impacts, controlling floods, generating hydropower, and maintaining local ecosystems. Consequently, they play a crucial role in sustaining water needs and supporting various sectors in these regions. Therefore, enhancing the performance of reservoir RCs is essential for guiding operators on optimal water release based on the AW at the start of each month.

This study aimed to evaluate the performance of various optimal reservoir RCs for the Zarrineh Rud reservoir, an important sub-basin of Lake Urmia basin, using GA, PSO, and GSA. To achieve this, daily meteorological and hydrometric data were collected from selected stations upstream of the dam over a 26-year period (1990–2016). The failure-free approach of the SPA was utilized to establish historical RCs as a benchmark. Considering the drinking and agricultural water demands, the active storage and its RC were simulated. Subsequently, the optimal RC was determined through GA–SPA, PSO–SPA, and GSA–SPA, with the objective of minimizing downstream water shortages.

The results revealed that the GSA–SPA outperformed the historical SPA, achieving increases in α_t and α_v of 2 and 1.5%, respectively. Furthermore, the GSA–SPA demonstrated superior performance in reducing mean annual and total shortages, decreasing these shortages by 8% compared to the historical SPA. Overall, the average improvements in α_t, α_v, and total shortages were 2, 1.4, and 6.5%, respectively, across all optimization methods, highlighting their superiority over the simulation approach. Additionally, the correlation graphs illustrating the relationships between R, Q, and AW for the optimal GSA–SPA model effectively depicted the variations in these variables in response to changing seasonal demands, corroborating our findings.

In conclusion, while this research demonstrates positive outcomes in optimizing reservoir RCs, it also presents notable limitations. A key constraint is the assumption of constant water demands for agricultural crops, which may not accurately reflect the dynamic nature of agricultural water use influenced by climate variability and differing crop types. Furthermore, the parameter tuning process for the optimization models can be resource-intensive, potentially hindering their practical application in real-time decision-making scenarios.

Despite these challenges, the findings of this study hold significant applicability beyond the Zarrineh Rud reservoir, providing a valuable framework for optimizing reservoir operations in other regions facing similar climatic conditions and water management issues. The insights derived from this research can guide policymakers and water resource managers in developing strategies to enhance water supply reliability and address shortages effectively. The demonstrated improvements in reservoir performance metrics indicate that optimization techniques such as GA, PSO, and GSA can significantly enhance water resource management, particularly in areas where water scarcity is a critical concern.

Looking ahead, future research should focus on exploring the relationship between the proposed optimization models and streamflow variability. Developing inflow scenarios that reflect the statistical properties of observed streamflow data could yield a deeper understanding of water availability and demand fluctuations. This approach would strengthen reservoir management strategies, especially in light of changing climate conditions and increasing water demands. Additionally, expanding research to encompass diverse geographical contexts and varying hydrological conditions could further validate the generalizability of these findings and contribute to more effective water resource management practices.

The authors ensure that they have written entirely original works, and if the authors have used the work and words of others, they have been appropriately cited or quoted.

S.A. and M.M. contributed to the conceptualization of the methodology, data validation, and model execution. S.A. also drafted the original manuscript and created the figures and tables, while reviewing, editing, and validating the work. E.R. was instrumental in developing the GSA model. A.E.-S. provided valuable contributions in addressing the reviewers' comments.

The data were not obtained using questionnaires.

The program used here can be made available (by the corresponding author), upon reasonable request.

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

The authors declare there is no conflict.