A combination approach for optimization operation of multi-objective cascade reservoir systems (Case study: Karun reservoirs)

Due to drought and decrease in rainfall, as well as the lack of water resources and the exhaustion of non-renewable energies, reservoirs are efficient structures that influence water resource management. Dams represent one of the available water sources, requiring management and planning for their optimal exploitation. By exploiting the dams, it is possible to meet the water needs of downstream reservoirs and to use the water in the dams to produce electric energy, which is one of the most important renewable energies. It is also important to prevent and reduce evaporation from the tank surface as dam losses, in order to prevent water wastage. Determining the optimal solutions for reservoir exploitation is a challenging engineering problem. Due to the challenges and problems of optimal exploitation of the reservoir, meta-heuristic methods were used so that they could be employed instead of definitive solutions. The meta-heuristic method involves a variety of operators that are continuously used to minimize or maximize objective functions (Beheshti & Shamsuddin 2013). Among the complexities of optimal exploitation of multi-objective reservoirs, we can refer to the decision variables for each exploitation objective, defined limits, conflicting objective functions and the possible nature of inputs (Sharifi et al. 2022). Several optimization strategies have been developed over the past few decades to address optimization problems and challenges. Optimization algorithms have been inspired by a range of aspects of nature, social behavior and the crowding of living organisms (Zhao et al. 2022).

One of the important and fundamental issues that has been the focus of researchers in recent decades is that of energy shortage due to its non-renewable nature. The presence of this energy fosters progress in various aspects (Ehteram et al. 2017). These days, electricity is not only a vital factor for human beings but also a factor for society's progress. Optimizing the use of reservoirs and providing suitable solutions is one of the most effective ways to produce more energy. By managing water resources and having proper planning, multi-purpose reservoirs and water behind dams can be used to produce hydropower energy. It is estimated that using reservoirs can provide 20% of the world's electricity (Tayebiyan et al. 2019). The task of obtaining relationships and operational rules for multi-purpose reservoir hydropower plants is very complex. With the advancement of recent technologies, some of these problems have been solved to some extent in these multiple reservoirs (Ahmadianfar et al. 2023).

The Water Cycle Optimization Algorithm (WCA) was utilized to optimize the efficiency of multiple reservoirs in the Gorgan River Basin. The results have demonstrated that the employed solution (WCA) has a better performance than the two solutions of genetic algorithm (GA) and particle swarm algorithm (PCO) (Qaderi et al. 2018). Recently for proper management of water stored in reservoirs and hydroelectric power generation electricity, various evolutionary algorithms and mathematical models have been used, and the results show that the shark algorithm has provided better results (Ehteram et al. 2018b). A hybrid PSO algorithm with differential evolution (DE) has been developed to improve power generation at a power station (Ahmadianfar et al. 2019). Optimization of multi-reservoir systems has been achieved by using the combined procedure of whale-genetic algorithms, while the powerful multi-criteria decision-analysis solution, i.e. TOPSIS, has been utilized to consider the proficiency and conclusion of the solutions. From the results obtained, it can be concluded that the combined whale-genetic algorithm has better results than the genetic algorithm and whale only (Mohammadi et al. 2019b). Optimal and convenient solutions for the real performance of the tank have been discovered through the use of the Nash bargaining method and evolutionary algorithms (genetic algorithms and ant colony). Observing the outputs shows that the method employed has the ability to meet downstream needs (Sharifazari et al. 2021). For problems of optimal utilization and solving mathematical functions in real single-tank systems (case study of Dez Dam) and hypothetical systems with 4 and 10 reservoirs, the mixed gray wolf optimization algorithm (SGWO) developed based on the evolutionary hybrid optimization algorithm has been used. The superior performance of the proposed method compared to the gray wolf algorithm can be observed by examining the obtained output (Masoumi et al. 2022). In the moot point of optimal exploitation of the reservoir, to approach the final solution and the absolute optimum of the solutions and to reduce the error and increase the accuracy in calculating the final solution, the IASO algorithm was used. The results obtained show that the use of this solution (IASO) brings the results 95–99% closer to the response of the improved global system (Moslemzadeh et al. 2023). Humboldt's method for optimizing squids was created by them. The optimization algorithm's high performance and accuracy in engineering problems are demonstrated by its results (Anaraki & Farzin 2023).

According to review studies, a comprehensive study has not been conducted on the optimal exploitation of the cascade reservoirs of the Karun basin regarding the maximum production of hydroelectric energy and the reduction of losses from the reservoir. The meta-heuristic algorithm of the artificial hummingbird (AHA), inspired by living organisms, is considered one of the most widely used methods in the field of optimal exploitation of single-purpose reservoirs. However, since the application and ability of this algorithm are limited to single-purpose reservoirs, AHA is expanded to solve the difficulty of multi-objective problems, and the developed method is called the multi-objective artificial hummingbird algorithm (MOAHA).

Among the benefits of MOAHA, the following can be mentioned:

• (1) Having an external archive to store and maintain Pareto optimal solutions. (2) A crowding method is based on dynamic elimination. (3) Integration with a non-dominated sorting method to update the solutions and improve the Pareto solutions. Therefore, in the current study, we have investigated the application and effectiveness of the new optimization algorithm (MOAHA) in multi-objective cascade reservoirs, with conflicting objectives. It has been examined on a case-by-case basis on the Karun cascade reservoirs (Karun 3, Karun 1, Masjed Soleyman and Gotvand). The evaluation criteria used in this study are CV, MS and MID, which measure the effectiveness and performance of algorithms. Choosing the foremost solutions based on these evaluation criteria is a difficult task, so multi-criteria decision-making is employed. The powerful TOPSIS multi-criterion method is used for selection. The obtained results demonstrate the high potential of this algorithm for optimizing the cascade reservoir system and producing hydropower energy.

Section 2 introduces the proposed method (MOAHA) and the inspiration behind this algorithm. Section 3 includes the investigated objective functions, solution evaluation criteria and TOPSIS. Section 4 presents the case study and data usage. Section 5 presents the results and discussion. Section 6 presents the conclusion.

Energy management for multi-reservoir system operation optimization (MSOO) is a critical concern for decision-makers. Therefore, it is essential to develop a comprehensive plan and management strategy for the optimal operation of MSOO. Utilizing heading policies can help generate more power to meet downstream demand. Additionally, ensuring sufficient storage capacity during critical periods such as droughts is crucial. To determine the optimal release of water and reservoir storage based on heading policies, heading parameters need to be obtained using optimization algorithms. Consequently, optimizing the operation of MSOO based on heading policies to increase power generation is a significant topic.

By implementing a program to manage multi-reservoir systems, we can enhance energy production. In this study, we employed the multi-objective optimization method (MOAHA) and integrated it with TOPSIS to increase hydropower generation.

Recently, Zhao et al. (2022) proposed a multi-objective algorithm called MOAHA¹, which has been shown to be a powerful optimizer, especially in cases of multi-objective reservoir exploitation. Below is a summary of MOAHA:

Pareto front: In many optimization problems, trying to optimize multiple objectives (usually in opposites) is simultaneous; therefore, the optimal function of the definition must consider all criteria. The nature of the issues: It is multi-purpose so that the comparison of answers cannot be easily made. PF is a technique that divides the responses into two categories: failed and non-failed categories obtain the optimal response. Front Pareto can be defined as a set of non-dominated solutions.

Crowding distance measures the density of non-dominated solutions within a solution's neighborhood. It is frequently employed in MOAs to maintain the distribution of obtained solutions on the PF.

The single-objective gray wolf algorithm was introduced in 2014 (Mirjalili et al. 2014), followed by the multi-objective gray wolf algorithm in 2016 (Mirjalili et al. 2016), inspired by the hunting behavior of gray wolves. This algorithm illustrates the strong social structure observed in gray wolf packs. The gray wolf optimizer (GWO) belongs to the meta-heuristic swarm intelligence methods and has recently been tailored to tackle a wide array of optimization challenges. It stands out among other artificial intelligence methods due to its attributes such as a low number of parameters, simplicity, ease of use and flexibility. One notable feature of this algorithm is its optimal convergence, achieved by striking a balance between exploration and exploitation stages during the search process. The multi-objective gray wolf optimizer (MOGWO) incorporates two novel components for multi-objective optimization: an archive of non-dominated best solutions and an alpha, beta and gamma leader selection strategy essential for storing non-dominated Pareto solutions. For more detailed information, please refer to the provided references.

Mirjalili introduced the ant lion optimizer (ALO) algorithm in 2017 (Mirjalili et al. 2017), inspired by the hunting behavior of ants in nature. The prey hunting process in ALO consists of five main steps: random walking of ants, trap construction, ants trapping, prey hunting and trap reconstruction. The algorithm calculates the balance of ants and those in the trap, as well as the events that occur during the hunting process. Ants are driven to move around randomly in search of food, which may be hunted by other ants. For further details on the MOALO algorithm, please refer to the main article by Mirjalili et al. (2017).

The NSGA-II is a multi-objective genetic algorithm introduced by Deb et al. (2002). While the NSGA-II algorithm has addressed many of the shortcomings of classical multi-objective evolutionary algorithms, it still has some limitations, which are outlined below:

• 1- Computational complexity.

• 2- Lack of utilization of the elitist method in obtaining the solution.

The elitism method involves retaining the best individuals from the previous generation in the reproduction population after applying genetic algorithm operators. This helps in convergence to accelerate optimal responses and improve search efficiency. Additionally, the algorithm incorporates a congestion distance operator, which partially addresses the aforementioned issues. This operator eliminates the need for users to define parameters to maintain diversity among population members and has lower computational complexity. For more detailed information about multi-objective genetic algorithms, please refer to the provided reference (Bennett et al. 1999).

In Karun's multi-reservoir system optimization simulation model, two objective functions were considered:

• (1) Maximizing the total production potential of hydropower plants.

• (2) Minimizing the amount of evaporation from the surface of the reservoir.

This study was conducted over a period of 96 months (2012–2020). In this problem, the decision variables include evaporation height, rainfall height, monthly demand volume and river flow.

In the above limits, S_Min is minimum allowed reservoir's storage and S_Max indicates maximum allowed reservoir's storage. Re_(Min) indicates the minimum release volume of the authorized tank and Re_(Max) indicates the maximum release volume of the authorized tank. In optimal operation, the used decision variable is the release volume of the reservoir.

The decision variable is the volume of release from the tank.

In the above equations, Q_{(i, t)} is reservoir inflow volume of the reservoir i in the period t per MCM, Sp_{(i, t)} is reservoir spillway i in the period t per MCM, Loss_{(i, t)} is total loss of reservoir i in the period t per MCM, Smax_(i) is maximum volume of reservoir i, Ev_{(i, t)} is net evaporation from reservoir i in the period t (evaporation minus precipitation per mm), Ā_{(i, t)} is the reservoir average area i in the period t per km², A_{(i, t)} and A_{(i, t+1)} is the reservoir area i at the beginning and the end of period t per km², respectively, c_0i, c_1i, c_2i and c_3i are constant coefficients that are obtained by the least square method.

Three performance measures of degree of constrained violation (CV), maximum spread (MS) and mean ideal distance (MID) were used to evaluate the efficiency of the multi-target MOAHA, as given by the following equations.

Here, ng + nh is the sum of the first and second constraints. The range of this criterion is between zero and infinity.

In this relationship, m is the number of target functions, Max f(i) is the maximum value of the objective function and Min f(i) is the minimum value of the objective function. This measure quantifies the dispersion of non-dominated solutions or the PF. A wider coverage and greater dispersion of the PF indicate better performance. The range of this criterion is from zero to infinity, with larger values indicating better dispersion and coverage.

In the above equation, np is the number of non-dominated solutions and F_1i and F_2i are the values of the first and second target functions of the i-th non-dominant solutions. Clearly, a lower MID value gives a better achievement. In this criterion, lower values are preferred as they indicate a shorter average distance from the ideal point. The ideal point is located at (0, 0), representing when both objective functions are zero. Therefore, achieving the lowest possible value for this criterion is desirable. The lowest achievable value is zero, while the highest positive value approaches infinity.

TOPSIS⁵ is a highly practical technique for addressing multi-objective decision-making problems (Ren et al. 2014). It utilizes multiple criteria to facilitate decision-making. By assigning different weights to each criterion, this method has been employed in selecting the most suitable data mining algorithm. The approach was initially introduced in 1995 by Yoon & Hwang (1995).

TOPSIS is a technique for prioritizing methods based on their similarity to select the ideal solution. One of the advantages of this method is its easy and straightforward process for evaluating performance. Additionally, regardless of the number of features, the number of steps remains constant when using this method. TOPSIS involves comparing a collection of solutions and selecting the most efficient one based on the weight of each criterion in the decision matrix (Shannon 2001). In this method, the alternative that has the least distance to the positive ideal solution (Q⁺) and the greatest distance to the negative ideal solution (Q⁻) is considered the best alternative (Kadkhodazadeh et al. 2022).

Finally, the solutions are classified by rating. To learn more about TOPSIS, refer to Zavadskas et al. (2006) and Ehteram et al. (2018a).

The multi-reservoir system described is one of the primary sources of hydropower energy production in the country, boasting a capacity exceeding 7,500 MW. The specifications of this system are detailed in Tables 1 and 2. It contributes to over 90% of the national hydropower supply, with a nominal annual electricity production reaching 16,058.1 gigawatt hours (GWh).

The average monthly inflow of cascade reservoirs is presented in Table 3, showing that Gotvand Dam has the highest inflow amount, while Karun 3 Dam has the lowest. Additionally, the table provides the standard deviation, coefficient of variation and skewness of the reservoirs' inflow.

Electricity production plays a crucial role in human life, necessitating a well-structured plan to enhance its generation. Optimal operation of multi-purpose reservoirs is imperative for increasing electricity production. This research focuses on optimizing the efficiency of multiple and multi-purpose reservoirs in Iran using optimization algorithms. Investigating the optimal utilization of cascade reservoirs holds significant importance for effective water resources management. The aim of this study is to demonstrate how the MOGWO can be effectively employed to optimize cascade reservoirs and hydropower plant operations. The primary objectives include maximizing hydropower generation and minimizing reservoir evaporation losses.

All parameters utilized for the various algorithms in this research are optimized according to the main articles. These algorithms are commonly employed for solving engineering problems, as outlined in Table 4 of the respective studies (Mirjalili et al. 2014, 2016).

The most, least and average amount of hydropower energy produced are shown in Table 5. According to the results obtained by MOAHA algorithm, Dam Karun 3 = 43,238.299, Karun 1 = 44,405.295, Masjed Soleyman = 24,782.351 and Gotvand = 3,024,987, it has performed better and has produced the highest amount of hydropower energy. Also, the worst results obtained are related to MOALO. The MOAHA algorithm has produced 15% more hydropower energy than the MOALO algorithm and 9% more than the MOGWO algorithm.

The effectiveness of the solutions (algorithms) was evaluated using three criteria: MS, CV and MID. Based on the results obtained from the evaluation criteria, the MOAHA algorithm has delivered excellent results. The best value in the MOAHA algorithm is MS = 34,146,776, while the worst value in the MOALO algorithm is MS = 177,056,949. CV = 2,203,185 in the MOAHA algorithm is the best value, while CV = 7,971,643 in the MOGWO algorithm is the worst value. MID = 850,960.117 in the MOAHA algorithm is the best value, while MID = 5,415,977.367 in the MOALO algorithm is the worst value. It is worth noting that due to the infinity of the second target function, the NSGA-II algorithm was unable to calculate MID.

To compare the effectiveness and efficiency of algorithms on objective functions, we utilized a well-known ranking method called TOPSIS. This method ranks the methods used for optimal reservoir exploitation by assigning different weights to each criterion, as outlined in the third chapter's formulas. In this study, MOAHA achieved the highest score of 0.90, indicating its superiority, while NSGA-II obtained the lowest score of 0.10, signifying its inferior performance among the algorithms evaluated.

Today, the use of optimization tools such as meta-heuristic algorithms is common in solving complex and extensive water resource management engineering problems. These methods are favored for their speed, simplicity and higher accuracy.

• (1) This research investigates the applicability and effectiveness of the new MOAHA optimization algorithm in multi-objective cascade reservoirs with conflicting objectives. It is examined case by case on the Karun cascade reservoirs (Karun 3, Karun 1, Masjed Soleyman and Gotvand).

• (2) The outputs obtained from the optimization algorithms MOGWO, MOALO and NSGA-II are compared with those of MOAHA. The effectiveness and efficiency of the algorithms are estimated and evaluated using three evaluation criteria: MS, MID and CV.

• (3) The results obtained from the MOAHA algorithm indicate the highest amount of electric energy produced as follows: 43,238/299 for Karun 3, 44,405/295 for Karun 1, 24,783/351 for Masjed Soleyman and 302,498 for Gotvand Dam. Conversely, the worst results were obtained from the MOALO algorithm, with the amount of electrical energy produced being 28,726/711 for Karun 3, 30,807/450 for Karun 1, 12,848/063 for Masjed Soleyman and 16,832/296 for Gotvand Dam.

• (4) According to the output of the algorithms, the rates of evaporation obtained are as follows:

• • – MOAHA algorithm: 4,264,738/976 mm

  • – MOGWO algorithm: 4,393,686/843 mm

  • – MOALO algorithm: 4,630,959/259 mm

  • – Multi-objective genetic algorithm: Infinity

Based on these results, it can be concluded that the MOAHA algorithm exhibits the best performance in reducing the evaporation rate, while the NSGA-II algorithm performs the worst.

• (1) The results indicate that the MOAHA algorithm can enhance electric energy production by 15% while simultaneously decreasing evaporation from the reservoir surface by 10%. This research employs multi-objective approaches, including the MOAHA, MOALO, MOGWO and NSGA-II algorithms, in optimizing cascaded reservoir systems.

• (2) The effectiveness of the MOAHA algorithm in optimally exploiting reservoirs is evident from the comparison of Pareto solutions using three criteria. In other words, the results obtained from MOAHA exhibit distribution and uniformity, indicating its capability to provide well-balanced solutions across multiple objectives.

• (3) Upon examining the obtained results, it is evident that the performance of NSGA-II is unfavorable in reducing the evaporation rate. In this objective function, NSGA-II is less efficient compared to other algorithms evaluated in the study.

• (4) The results obtained by the evaluation criteria to compare the performance of the algorithms according to the values of 34,146,776 MS in the MOAHA algorithm and 177,056,949 MS in the MOALO algorithm are the best MOAHA algorithm and the most inappropriate MOALO algorithm. Also, CV = 2,203,185 was the best value in the MOAHA algorithm, while CV = 7,971,643 was the worst value in the MOGWO algorithm. MID = 117.85096 was the best value in the MOAHA algorithm, but MID = 367.5415877 was the worst value in the MOALO algorithm. Due to the infinity of the second objective function, the NSGA-II algorithm was unable to calculate the MID.

• (5) In addition, to address the challenge of selecting the best solution among data mining algorithms based on accuracy and calculation time criteria, the powerful multi-criteria solution TOPSIS has been employed. In this research, the MOAHA algorithm achieved the highest score of 0.90, indicating its superiority, while the MOGWO algorithm obtained the lowest score of 0.10, signifying its inferior performance. Therefore, the proposed model can serve as a valuable solution for studies related to the optimal exploitation of multi-purpose cascade reservoirs.

• (6) Due to the No Free Lunch Theorem (NFL) (Wolpert & Macready 1997), one optimization algorithm can perform reasonably in a range of optimization problems, but not all (Anaraki & Farzin 2023; Valikhan Anaraki & Farzin 2024). The performance of algorithms in different fields and subjects and even different data can work differently and it is not always possible to conclude that, for example, the gray wolf algorithm shows the same performance and result in all subjects.

Based on the results obtained, particularly considering the accuracy and uniformity of the PF of the algorithms, it can be concluded that the AHA algorithm proves to be a suitable method for the exploitation of multi-purpose reservoirs.

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

The authors declare there is no conflict.