## ABSTRACT

Multi-reservoir systems that have diverse and conflicting objectives are challenging to design due to their uncertainties, non-linearities, dimensions and conflicts. The operation of multi-reservoir systems is crucial to increasing hydropower production. In this study, we have investigated the application and effectiveness of the new optimization algorithm MOAHA in multi-objective cascade reservoirs with conflicting objectives, and it has been investigated on a case-by-case basis on Karun cascade reservoirs (Karun 3, Karun 1, Masjed Soleyman and Gotvand). The suggested method (MOAHA) output with other optimization algorithms, MOALO, MOGWO and NSGA-II, were compared and evaluation criteria were used to select the best performance. Additionally, we employed the powerful TOPSIS method to determine the most suitable algorithm. The considered restrictions have also been observed. The results indicate that MOAHA's proposed method is better than the compared algorithms in solving optimal reservoir utilization problems in multi-reservoir water resource systems. The reduction of evaporation (losses) from the tank surface by 9% is accompanied by a 15% increase in hydropower energy production. MOAHA, scoring 0.90, is deemed the best algorithm in this study, whereas MOGWO, with a score of 0.10, is regarded as the least effective algorithm.

## HIGHLIGHTS

MOAHA's application in the optimal operation of multi-target cascade reservoirs with conflicting targets is introduced.

Comparing the performance of MOAHA with other optimization algorithms.

The MOAHA was more successful than the other three comparative algorithms.

Analysis of results and use of TOPSIS in selecting the best method.

MOAHA is a useful method for optimal exploitation of multi-purpose cascading reservoirs.

## INTRODUCTION

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.

## MATERIALS AND METHODS

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.

### Multi-objective MOAHA

Recently, Zhao *et al.* (2022) proposed a multi-objective algorithm called MOAHA^{1}, which has been shown to be a powerful optimizer, especially in cases of multi-objective reservoir exploitation. Below is a summary of MOAHA:

*n*iterations, a migration search is executed where the worst solutions ahead are randomly initialized in the search space, and the visit table is updated. After each iteration, non-dominated solutions for the new population are stored in the archive. If the archive size exceeds its predefined limit, the external archive method is called from (DECD)

^{2}. To achieve the maximum number of repetitions, DECD operators are executed sequentially. Finally, the archive containing non-dominated optimal solutions is returned as (PF)

^{3}. Figure 1 illustrates the basic structure of the multi-objective hummingbird algorithm. More details are provided in the references above.

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.

### Multi-objective gray wolf optimizer algorithm (MOGWO)

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.

### Multi-objective ant lion optimizer (MOALO)

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).

### Multi-objective genetic algorithm (NSGA-II)

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).

## OPTIMIZATION MODEL

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.

*t*is the time period,

*T*is the length of operation periods per month,

*Z*is the objective function that must be maximized or minimized, Re

*refers to the release volume of the reservoir per MCM and*

_{t}*S*refers to the storage volume of the reservoir in the period

_{t}*t*per MCM. The mass conservation equation in this problem is defined as follows:

*Q*indicates the tank input volume in the period

_{t}*t*per MCM, Sp

*is spillway of reservoir in period*

_{t}*t*per MCM and Loss

*is the evaporation from the tank surface in the period*

_{t}*t*per MCM. In Equations (3) and (4), there are additional restrictions related to tank storage and release that are mentioned:

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.

### Maximizing the total hydroelectric generation by hydropower plants

*et al.*2022):

*N*(

*i*,

*t*) is produced energy by reservoir

*i*in period

*t*per MW and PPC

*known as maximum capacity of the hydropower. Other terms and conditions that are considered in the optimization of hydropower production are defined as the following equations (Bozorg-haddad*

_{i}*et al.*2017):

*g*is the acceleration of gravity (9.81 m/s

^{2}),

*ei*is the efficiency of the tank power plant

*i*(fixed assumption for all periods), PF

*is the performance coefficient of the tank power plant*

_{i}*i*, MUI

*refers to the conversion factor of MCM to m*

_{t}^{3}/s in the period

*t*and TW(

*i*,

*t*) refers to the tail water level of tank

*i*in period

*t*in meters.

*H*(

*i*,

*t*) is the reservoir water level of

*i*at the beginning of period

*t*in meters,

*H*(

*i*,

*t*+ 1) is the reservoir water level of

*i*at the end of period

*t*in meters, TW(

*i*,

*t*) is the reservoir water level of

*i*in period

*t*in meters and

*a*0

*i*,

*a*1

*i*,

*a*2

*i*and

*a*3

*i*are constant conversion coefficients store in the tank to the respective height in the tank

*i*.Here, TW(

*i*,

*t*) is the tail water level of tank

*i*in period

*t*in meters and

*b*0

*i*,

*b*1

*i*,

*b*2

*i*and

*b*3

*i*are the fixed coefficients for transformation of the water output from the power plant to the bottom water level in tank

*i*.

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, *S*max_{(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 km^{2}, *A*_{(i, t)} and *A*_{(i, t+1)} is the reservoir area *i* at the beginning and the end of period *t* per km^{2}, respectively, *c*_{0i}, *c*_{1i}, *c*_{2i} and *c*_{3i} are constant coefficients that are obtained by the least square method.

### Minimizing the amount of evaporation from the surface of the reservoir

^{4}model (Mekonnen & Hoekstra 2012) has been selected to estimate the water consumption of cascade dams. This consumption is attributed to the increase in reservoir surface evaporation. Therefore, the chosen target function is the reduction of the evaporation rate, which is described as follows:where 10 is applied to convert millimeters to cubic meters per hectare,

*he*(

*t*) is the average annual water evaporation from the open water surface of the reservoir in month

*t*(mm/month) and

*Ae*

_{(t)}is the area of the free water surface of the reservoir in month

*t*(ha) (Wang

*et al.*2022).

### Performance analysis of algorithms

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.

#### Degree of constrained violation (CV)

*et al.*2021):

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

#### Maximum spread (MS)

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.

#### Mean ideal distance (MID)

*et al.*2021):

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.

#### Select best data mining algorithm

TOPSIS^{5} 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).

## CASE STUDY

^{2}, making it one of Iran's most fertile basins. Its coordinates range from longitude 47-30′ to 52-30′ east to latitude 30-20′ to 34-05′ north. The Karun River is crucial for downstream water supply and drainage. Four dams within the Karun basin – Gotvand, Masjed Soleyman, Karun 1 and Karun 3 Reservoirs – constitute an essential cascade in Iran's water resources, significantly influencing water supply and hydropower energy production. The output from Karun Dam 4 flows into the reservoir of Karun Dam 3. Subsequently, all dams within the Karun Basin are positioned sequentially downstream, with each reservoir followed by another dam. Karun 1 Dam (Shaheed Abbaspour) operates downstream of Karun 3 Dam, and the Godarlandar flow dam is situated in the Masjed Soleyman area downstream of Karun 1 Dam. The final dam within the catchment area of the Karun River is the Gotvand Dam. Figure 2 illustrates the entry and exit routes, as well as the interconnection of dams within the basin.

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).

Parameter . | Unit . | Karun 3 . | Karun 1 . | Masjed Soleyman . | Gotvand . |
---|---|---|---|---|---|

Number of power plant units | – | 8 | 8 | 8 | 4 |

Power of each unit | Megawatt (MW) | 250 | 250 | 250 | 375 |

Total power capacity | Megawatt (MW) | 2,000 | 2,000 | 2,000 | 1,500 |

Average annual energy production | Gigawatt hours (GWh) | 3,890.8 | 3,951.9 | 3,959.8 | 4,255.6 |

Efficiency of hydropower generation | % | 92.4 | 90 | 92 | 93 |

Performance coefficient | % | 32 | 23 | 23 | 22 |

Parameter . | Unit . | Karun 3 . | Karun 1 . | Masjed Soleyman . | Gotvand . |
---|---|---|---|---|---|

Number of power plant units | – | 8 | 8 | 8 | 4 |

Power of each unit | Megawatt (MW) | 250 | 250 | 250 | 375 |

Total power capacity | Megawatt (MW) | 2,000 | 2,000 | 2,000 | 1,500 |

Average annual energy production | Gigawatt hours (GWh) | 3,890.8 | 3,951.9 | 3,959.8 | 4,255.6 |

Efficiency of hydropower generation | % | 92.4 | 90 | 92 | 93 |

Performance coefficient | % | 32 | 23 | 23 | 22 |

. | Karun 3 . | Karun 1 . | Masjed Soleyman . | Gotvand . |
---|---|---|---|---|

Smax | 2,970 | 3,139 | 2,615 | 5,082 |

Smin | 1,250 | 1,275.3 | 210 | 500 |

. | Karun 3 . | Karun 1 . | Masjed Soleyman . | Gotvand . |
---|---|---|---|---|

Smax | 2,970 | 3,139 | 2,615 | 5,082 |

Smin | 1,250 | 1,275.3 | 210 | 500 |

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.

. | Karun 3 . | Karun 1 . | Masjed Soleyman . | Gotvand . |
---|---|---|---|---|

Average | 496.973 | 646.568 | 717.032 | 780.922 |

STDEV | 362.856 | 444.596 | 462.974 | 463.564 |

Coefficient of variation | 1.369 | 1.454 | 1.548 | 1.684 |

Skewness coefficient | 3.251 | 2.469 | 2.179 | 2.717 |

. | Karun 3 . | Karun 1 . | Masjed Soleyman . | Gotvand . |
---|---|---|---|---|

Average | 496.973 | 646.568 | 717.032 | 780.922 |

STDEV | 362.856 | 444.596 | 462.974 | 463.564 |

Coefficient of variation | 1.369 | 1.454 | 1.548 | 1.684 |

Skewness coefficient | 3.251 | 2.469 | 2.179 | 2.717 |

## RESULTS AND DISCUSSION

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.

### Setting the parameters for the used algorithms

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).

Parameters . | Algorithm . |
---|---|

Max It = 100 | MOAHA |

Archive Size = 100 | |

N Pop = 100 | |

Max It = 100 | MOGWO |

Archive Size = 100 | |

N Pop = 100 | |

alpha = 0.1 | |

N Grid = 10 | |

beta = 4 | |

gamma = 2 | |

Max It = 100 | MOALO |

N Pop = 100 | |

Archive Max Size = 100 | |

Max It = 100 | NSGA-II |

N Pop = 100 | |

Archive Max Size = 100 | |

Mutation Rate = 0.02 | |

Mutation Percentage = 0.4 | |

Crossover Percentage = 0.7 |

Parameters . | Algorithm . |
---|---|

Max It = 100 | MOAHA |

Archive Size = 100 | |

N Pop = 100 | |

Max It = 100 | MOGWO |

Archive Size = 100 | |

N Pop = 100 | |

alpha = 0.1 | |

N Grid = 10 | |

beta = 4 | |

gamma = 2 | |

Max It = 100 | MOALO |

N Pop = 100 | |

Archive Max Size = 100 | |

Max It = 100 | NSGA-II |

N Pop = 100 | |

Archive Max Size = 100 | |

Mutation Rate = 0.02 | |

Mutation Percentage = 0.4 | |

Crossover Percentage = 0.7 |

### Analysis of results and interpretation

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.

Solution . | . | Karun 3 . | Karun 1 . | Masjed Soleyman . | Gotvand . |
---|---|---|---|---|---|

MOAHA | Max | 43,238.299 | 44,405.295 | 24,782.351 | 30,787.524 |

Min | 3,955.389 | 5,549.212 | 2,752.180 | 3,262.079 | |

Avg | 8,487.695 | 10,496.963 | 4,855.936 | 7,242.739 | |

MOALO | Max | 28,726.711 | 30,807.450 | 12,848.063 | 16,832.296 |

Min | 4,003.327 | 5,613.173 | 2,359.682 | 3,189.621 | |

Avg | 8,380.008 | 10,038.019 | 4,219.643 | 5,569.475 | |

MOGWO | Max | 24,109.790 | 36,247.080 | 15,213.824 | 19,930.790 |

Min | 3,235.161 | 6,520.159 | 2,735.287 | 3,631.233 | |

Avg | 6,805.714 | 10,351.428 | 4,300.072 | 5,828.037 | |

NSGA-II | Max | 36,008.828 | 43,425.228 | 23,068.021 | 28,651.382 |

Min | 0 | 4,116.108 | 3,741.268 | 4,277.670 | |

Avg | 3,257.102 | 9,831.089 | 4,661.120 | 6,863.954 |

Solution . | . | Karun 3 . | Karun 1 . | Masjed Soleyman . | Gotvand . |
---|---|---|---|---|---|

MOAHA | Max | 43,238.299 | 44,405.295 | 24,782.351 | 30,787.524 |

Min | 3,955.389 | 5,549.212 | 2,752.180 | 3,262.079 | |

Avg | 8,487.695 | 10,496.963 | 4,855.936 | 7,242.739 | |

MOALO | Max | 28,726.711 | 30,807.450 | 12,848.063 | 16,832.296 |

Min | 4,003.327 | 5,613.173 | 2,359.682 | 3,189.621 | |

Avg | 8,380.008 | 10,038.019 | 4,219.643 | 5,569.475 | |

MOGWO | Max | 24,109.790 | 36,247.080 | 15,213.824 | 19,930.790 |

Min | 3,235.161 | 6,520.159 | 2,735.287 | 3,631.233 | |

Avg | 6,805.714 | 10,351.428 | 4,300.072 | 5,828.037 | |

NSGA-II | Max | 36,008.828 | 43,425.228 | 23,068.021 | 28,651.382 |

Min | 0 | 4,116.108 | 3,741.268 | 4,277.670 | |

Avg | 3,257.102 | 9,831.089 | 4,661.120 | 6,863.954 |

*S*max. The minimum storage volume in reservoirs is equal to: Karun 3 = 856.5283645, Karun 1 = 1053.945118 Masjed Soleyman = 985.0660638 and Gotvand = 1991.13663. Upon inspection, it can be observed that the minimum storage volume obtained from

*S*min has also decreased in Karun 3 Dam.

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.

## CONCLUSION

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.

## DATA AVAILABILITY STATEMENT

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

## CONFLICT OF INTEREST

The authors declare there is no conflict.

Multi-objective artificial hummingbird algorithm.

Dynamic elimination-based crowding distance.

Pareto front.

Reservoir water footprint.

Technique for Order Preference by Similarity to Ideal Solution.

Water Resources Management32, 2539–2560.