## Abstract

Multi-objective reservoir operation presents a number of critical challenges that must be overcome for efficient management of water resources. The inherent contradiction between several goals, such as satisfying irrigation demand and maximizing hydropower generation, is one of the major issues. Trade-offs and compromises must be carefully considered to balance these objectives. To solve this problem, a study was carried out to optimize the operation of multi-objective reservoirs with two primary goals: minimizing irrigation deficits and maximizing hydropower generation. This study employs the self-adaptive multipopulation multi-objective Jaya algorithm (SAMP-MOJA), an improved version of the Jaya algorithm, to construct an optimal Pareto Front utilizing an a priori approach. The performance of SAMP-MOJA is compared to that of other algorithms such as multi-objective particle swarm optimization, multi-objective invasive weed optimization, and multi-objective Jaya algorithm. The results of this study demonstrate that the hydropower generated by the developed model surpasses 80% of the actual generation. The study's findings will aid in designing the most effective Pareto front possible.

## HIGHLIGHTS

A multi-objective model is developed to balance irrigation deficiency and maximum hydropower generation.

An improved version of the Jaya algorithm has been implemented over multi-objective reservoir operation.

Self-adaptive optimization method is used in the reservoir operation.

Optimization of multi-objective reservoir operation is carried out using the metaheuristic method.

Developed model outperforms better in comparison to several other algorithms.

## INTRODUCTION

Reservoirs are frequently made to serve multiple purposes, including irrigation, power generation, flood control, industrial needs, and river transportation (Kumar & Yadav 2018; Sharma *et al.* 2021). Multi-objective reservoir operation poses several significant difficulties and challenges that need to be addressed for effective water resources management. The major difficulties and challenges in multi-objective reservoir operation include balancing conflicting objectives, managing variability and uncertainty in water availability, addressing ecological considerations and downstream water requirements, managing storage constraints and operational limitations, and efficiently handling the complexity of the system (Hu *et al.* 2021; Lai *et al.* 2021; Mirmehdi *et al.* 2023). In the past, traditional optimization methods such as linear programming (LP), nonlinear programming (NLP), and dynamic programing (DP) were used to address these problems. These techniques, however, have some limitations. For example, LP considers a linear problem, whereas NLP needs a convex problem and may struggle with high-dimensional problems. However, DP has high computational requirements, making it unsuitable for large-scale optimization problems (Bozorgi *et al.* 2017). Traditional optimization methods face challenges when dealing with multi-objective problems, as well as nondifferentiable, nonconvex, and discontinuous functions (Kumar *et al.* 2023).

In the late 1970s, heuristic and metaheuristic optimization techniques were developed as a means to overcome the limitations of conventional optimization techniques. These heuristic techniques have increased in popularity in solving complex optimization problems with increasing computing power. Various researchers have used these techniques in reservoir operation, such as ant colony optimization (ACO) (Soghrati & Moeini 2020), firefly algorithm (Bozorg-Haddad *et al.* 2017), shuffled frog leaping algorithm (Li *et al.* 2018), differential evolution (DE) (Civicioglu & Besdok 2013), harmony search (Ranjbar *et al.* 2021), bat algorithm (BA) (Ethteram *et al.* 2018), hybrid model of support vector regression and fruitfly optimization algorithm (Sun *et al.* 2021), and evolutionary algorithms (EAs) (Du *et al.* 2022). The challenge faced in applying these methods is the government control over the dams. For example, in India, the ministry has set priorities for the operation of reservoir. Water for irrigation, minimum ecological needs, and drinking and domestic needs is given equal and ‘high priority’ according to the National Water Policy of 2012. The water allocation for power generation comes after these. The solution to this issue varies from nation to nation (Singh *et al.* 2023). Based on the precipitations received by the region and the need for water, the specific made solution is not possible. The dams are constructed for the storage of water and its maximum utilization during drought years (Kumar & Yadav 2022; Sharma *et al.* 2023).

Reddy & Kumar (2007) used the elitist-mutated particle swarm optimization (PSO) to address the multi-objective Bhadra reservoir scheme to develop reservoir strategies. The suggested model can be effectively used to utilize the water resources present in a multi-crop irrigation reservoir in the most efficient manner. The cuckoo optimization algorithm (COA) and the imperialist competitive algorithm (ICA) were used by Hosseini-Moghari *et al.* (2015) to solve the multireservoir system. These results suggest that COA and ICA perform more effectively than genetic algorithm (GA). COA, ICA, and GA objective functions had average values of 5.454, 6.461, and 6.869, respectively. For the purpose of controlling reservoir flooding at three Gorges reservoirs, Chen *et al.* (2017) used the nondominated sorting genetic algorithm III. Simulation's findings show that this approach can generate optimal, evenly distributed Pareto solutions. Afshar (2012) used adapted versions of PSO for the reservoir operation. The proposed algorithm can be extended to multireservoir systems and is not affected by the initial swarm or the swarming dimension. According to Ahmad *et al.* (2016), the application of ABC in optimizing reservoir operation resulted in more stable, effective, and quicker convergence compared to other methods, demonstrating its superiority. Trivedi & Shrivastava (2020) used elitism and different parameters for the PSO to evaluate the optimum operating policies of the reservoir. The model outperforms the other two heuristic approaches in terms of reliability, resilience, vulnerability indices, and root-mean-square error (RMSE) values.

Existing heuristic and metaheuristic optimization methods have some drawbacks despite their benefits. As per the existing literature, many evolutionary and swarm-based algorithms require parameter tuning (Niknam *et al.* 2012). These algorithms typically use two categories of parameters: algorithm-specific parameters and common control parameters. All population-based algorithms share common control parameters such as the number of iterations and population sizes. In this case, the population size refers to the total number of particles or the population that will be used to solve the problem, whereas the number of iterations denotes how many times the algorithm will execute to discover the optimal solution. Internal parameters are the parameters that are specific to the algorithm. Each algorithm has its own unique set of parameters, which must be fine-tuned before the algorithm is used. The performance of the algorithm may suffer from improper tuning of these parameters, and this raises the chances of getting caught up in the local optimum solution and increases processing time and cost (Zou *et al.* 2013). For instance, GA required fine-tuning of operation selection, crossover probability, and mutation probability. Inertia weight, cognitive, and social parameters are required by PSO. Crossover parameters and scaling factors needed to be tuned for DE. The number of accepted and tested solutions, the size of the hive, the queen's energy, and the initial speed are all requirements for honey-bee mating optimization (HBMO). Memory size tuning and pitch setting were necessary for harmony search (HS). The parameters scout, onlooker, and employed bees need to be tuned for ACB. Tuning is required for BA's essential parameters, including emission coefficient and wavelength. Several other algorithms have been developed and used for reservoir operations optimization, including the whale optimization algorithm, COA, improved BA, krill herd algorithm (Ehteram *et al.* 2017), ICA, and ACO, which also required the tuning of algorithm-specific parameters.

To overcome the issue of algorithm-specific parameters, Rao (2016) proposed the Jaya algorithm (JA), which eliminates the need for parameter tuning and reduces the user's workload. It is predicated on the idea of selecting the best option while avoiding the worst one (Rao *et al.* 2017). Rao *et al.* (2016) employed JA to enhance the micro-channel heat sink and discovered that it outperformed teaching learning-based optimization (TLBO) and MO EAs in terms of optimal solution and convergence. Rao & Keesari (2018) improved the planning of a wind farm using JA. According to the computational findings, JA outperformed other algorithms in terms of farm efficiency, total power generated, and cost per unit of electricity generated. Early JA users used it to address issues with water resources. Kumar & Yadav (2018) applied JA to a multireservoir operating system to maximize net benefits and found that JA outperformed in terms of global optimum solution and faster convergence. Kumar & Yadav (2020) found that JA outperformed TLBO, PSO, and DE for water release optimization in the Ukai reservoir. It was discovered that the JA had a faster rate of convergence than other algorithms. In the study by Chong *et al.* (2021), hedging-based JA was developed to achieve optimal reservoir operation. The outcomes demonstrated that, in terms of the best solution, the JA performed better than other algorithms. Kumar & Yadav (2019) used improved JA, i.e., elitist JA (EJA), to find an optimal crop pattern and found that the elitist JA was outperforming. When compared to elitist TLBO, it was found that EJA has a better and faster convergence property.

The difficulty comes with the population size, which necessitates a sensitivity analysis to determine the correct population size. The improved version of the self-adapting multipopulation Jaya algorithm (SAMP-JA) speeds up the process of determining the appropriate population size. Since the population is divided into various subpopulation groups, the search space is more diverse, reducing user burden. The objective is to segment the search space into various zones by allocating different subgroups. The effectiveness of the algorithm in this case depends on selecting numbers of subpopulation. This problem has been solved using self-adaptive. The algorithm's self-adaptation feature aids in successfully tracking the situation and adjusting the amount of subpopulations as the problem's environment changes, such as increasing or decreasing the number of subpopulations.

To solve the multi-objective reservoir operation problem, self-adaptive multipopulation multi-objective Jaya algorithm (SAMP-MOJA) was used for India's Ukai reservoir system. Ukai reservoir serves as a primary source for irrigation, domestic use, power generation, and flood control purposes. The goal of this study is to increase hydropower generation while minimizing irrigation deficits. Both objectives conflict with each other. To reduce the irrigation deficit, more water must be provided for irrigation purposes. However, plenty of storage capacity in the reservoir is necessary for maximizing hydropower output. The comparative analysis of SAMP-MOJA's efficiency is done by comparing it with other algorithms like MOJA, multi-objective particle swarm optimization (MOPSO), and multi-objective invasive weed optimization (MOIWO). The study's findings will help in the development of an optimal Pareto front. The Pareto front obtained from the optimization process can guide decision-making and enable the selection of the optimal operating strategy that meets both objectives.

The vital contributions of this work are as follows:

Implement SAMP-MOJA that adapts to deciding the appropriate population size more quickly.

The performance of SAMP-MOJA is inspected using a multi-objective reservoir operation problem.

## RESEARCH METHODOLOGY

The research methodology consisted of the following steps:

- (1)
Problem formulation: The study identified the research problem, which is the effective management of multi-objective reservoir operation with competing goals. The problem was formulated into a mathematical model that can optimize the operation of the Ukai reservoir.

- (2)
Data collection: The study collected data related to the Ukai reservoir such as inflow, demand, storage capacity, and other relevant data required for the mathematical model.

- (3)
Model development: The mathematical model was developed using the collected data, incorporating the two primary objectives of minimizing irrigation deficits and maximizing hydropower production.

- (4)
Optimization method: In the study, the SAMP-MOJA algorithm is used to optimize the Ukai reservoir operation problem.

- (5)
Algorithm comparison: SAMP-MOJA's performance was compared to those of other algorithms such as MOIWO, MOPSO, and MOJA.

- (6)
Pareto front development: By utilizing an a priori approach, the study created an ideal Pareto Front utilizing the SAMP-MOJA algorithm.

- (7)
Performance evaluation: The performance of the developed model was evaluated using various performance indicators such as hydropower generated, reliability, vulnerability index, and error metrics.

- (8)
Result analysis: The results were analyzed to determine the effectiveness of the developed model and based on that conclusions are drawn.

### Self-adaptive multipopulation multi-objective Jaya algorithm

- (i)
Specify the optimal population size and number of iterations for the algorithm.

- (ii)
Determine the initial solutions that lie within the variables bounds.

- (iii)
Split the sample into

*m*groups, where*m*is initially set to 2, and then update it based on the function's value. - (iv)
Choose the finest and the worst solutions from the group of population for the next iteration.

- (v)
Here, the finest and worst answers are represented by and, respectively, and the most recent solution is . The terms and assist the algorithm in moving toward the best solution and preventing the worst solution, respectively. The values of and are randomly selected from [0,1].

- (vi)
Compare the objective function values of the original solution and the updated solution. If the updated solution is improved than the original one, then

*m*is increased by 1 (i.e., ), which allows for greater search area exploration. Or*m*is reduced by 1, i.e., , which aids in the exploration of the search space. In this case, . - (vii)
It involves checking if the maximum number of generations has been reached. If it has, the cycle terminates, otherwise, it continues to run. Any duplicate solutions are replaced by freshly generated solutions to maintain diversity. Figure 2 shows the SAMP-MOJA flowchart.

For more information on JA, refer to the study by Venkata Rao (2016), invasive weed optimization algorithm (IWO) can be learned from Karimkashi & Kishk (2010), and PSO details are available in the study by Eberhart & Kennedy (1995).

### Multi-objective optimization

MO is a technique used for solving decision-making problems that involve optimizing multiple objective functions simultaneously. Unlike single-objective optimization, which aims to find the best solution for a single objective, MO seeks to identify a set of trade-off solutions that offer the best balance between conflicting objectives. When there is no single optimal option and decisions must be made by considering the trade-offs between many goals, this approach is especially beneficial. A set of points in the objective space known as the ‘Pareto frontier’ represents the best trade-off solutions that cannot be improved without surrendering another objective in a MO problem.

*et al.*2019):

### Statistical efficiency

The statistical efficiency indicators listed in the following subsections were used to assess the efficacy of EAs for optimal reservoir operation.

#### Root-mean-square error

#### Mean absolute error

### Performance evaluation indicators

The following evaluation indicators are used to determine EAs' capacity for optimal reservoir operation.

#### Reliability index

*et al.*2018):where the reliability index is denoted by . A higher percentage for the indicator is preferable.

#### Vulnerability index

*et al.*2018):where is the vulnerability index and

*i*is the total time period, i.e., .

## STUDY AREA DESCRIPTION

There are two power plants at the Ukai reservoir. The first power plant, which has been in operation since 1974 and has a capacity of 300 MW, is located on the main dam. The second power plant on ULBMC, known as the ULBMC mini-station, was built in 1988 and has a 5 MW capacity. Together, these two power plants contribute significantly to the region's power supply. The Ukai reservoir inflow box plot is shown in Figure S1 (Supplementary Material). It can be noted that during the monsoon periods, the maximum inflow is observed, and in September, the peak inflow is observed. The mean, minimum, and maximum are nearly the same during the nonmonsoon months, which demonstrates that the data are normally distributed. The reservoir depends entirely on the inflow of the monsoon. Thus, to fulfil demand during nonmonsoon periods, water storage is essential. Table 1 presents the Ukai basin silent features. The data required for this study were collected from the Ukai division and the Surat irrigation department. For the analysis, the following data were used:

Monthly inflows from the reservoir from 1972 to 2016

Monthly storage capacity from 1972 to 2016

Monthly discharge from 1976 to 2016

Irrigation demand for all canal releases, cultivable command areas, and generated power from 1976 to 2016

Monthly evaporation rates from 1972 to 2016

Reservoir areas from 1972 to 2016

Industrial demands and domestic demands from 1990 to 2016

Features . | Readings . |
---|---|

Geographical location | 72°33′ to 78°17′ E longitude |

20°9′ to 22° N latitude | |

Average rainfall | 820.07 mm |

Highest elevation | 1,556 |

Highest dam | Ukai |

Number of irrigation projects | Major, 13, medium, 68 |

No of watersheds | 100 |

No of flood forecasting site | 3 |

No of villages | 9,443 |

No of subbasin | 3 |

Road width on the spillway | 6.706 m |

Catchment area at Ukai | 62,225 km^{2} |

Type of spillway | Radial |

Top of dam | 111.252 m |

Features . | Readings . |
---|---|

Geographical location | 72°33′ to 78°17′ E longitude |

20°9′ to 22° N latitude | |

Average rainfall | 820.07 mm |

Highest elevation | 1,556 |

Highest dam | Ukai |

Number of irrigation projects | Major, 13, medium, 68 |

No of watersheds | 100 |

No of flood forecasting site | 3 |

No of villages | 9,443 |

No of subbasin | 3 |

Road width on the spillway | 6.706 m |

Catchment area at Ukai | 62,225 km^{2} |

Type of spillway | Radial |

Top of dam | 111.252 m |

## MATHEMATICAL MODEL FORMULATIONS

This section focuses on reservoir operations with multiple objectives. The research aims to reduce irrigation shortfalls while increasing hydropower generation. These objectives are conflicting. The following are the objectives.

### Minimize annual irrigation deficiency

### Maximize the production of hydropower

*p*power is calculated as follows:where is the discharge in cubic meter per second, is the head in meters,

*p*is power in watts, is water density (1,000 kg/m

^{3}), and

*g*is the acceleration due to gravity (9.81 m/s

^{2}).

*t*, and is the plant efficiency. and express the river bed turbines and ULBMC net head at time

*t*. The following constraints are applied to the objectives.

### Continuity constraints

*t*. represents the amount of water stored in the reservoir at the end of period

*t*. refers to the volume of water inflow into the reservoir during period

*t*in million cubic meters. represents the amount of water that evaporates from the reservoir during period

*t*in million cubic meters. represents the spill overflow during period

*t*in million cubic meters.

### Evaporation constraints

*t*; and represent reservoir areas in at

*t*and , respectively.

### Storage constraints

*t*in million cubic meters.

### Maximum constraints on power production

*t*must be less than or the same as the peak power output:where is the maximum output of riverbed turbines and is the maximum power produced at the ULBMC, during the time

*t*, expressed in kilowatt-hours. The correlation between the water head and the reservoir's storage capacity is shown in Equation (20). There are actually a number of elements that affect the water head in a reservoir despite the appearance that is exclusively influenced by storage capacity. One of the criteria that influences the water head () in Equation (20) is storage capacity (). To determine the water head, there are other factors and bases besides this one. The equation also takes into account the complex relationship between storage capacity and water head by including additional coefficients (, , , , and ). These coefficients reflect the reservoir's numerous hydraulic and geometric features, including its size, shape, and terrain, which affect the head–discharge relationship. These coefficients allow the formula to account for the reservoir system's nonlinear behavior and give a more accurate depiction of the water head as a function of storage capacity:where , , , , and are the storage height coefficients.

### Constrictions on canal capacity

### Overflow restrictions

### Constrictions on irrigation releases

### Water downstream releases

*t*.

## MODEL APPLICATION

A multi-objective model was developed to reduce irrigation deficits while maximizing hydropower generation. The weight for minimizing irrigation deficits is taken as , while the weight for maximizing hydropower generation is . To generate the Pareto front, the values of and were changed in a predetermined order. In each attempt, a single optimal solution will be found, and the value of each objective will be calculated (Afshar *et al.* 2009). A SAMP-MOJA was implemented for the monthly time step to solve the Ukai multi-objective reservoir operation. The proposed method's effectiveness is compared with the MOJA, MOPSO, and MOIWO. The algorithms were written in MATLAB R2014b software.

### Weights

In this study, five models were developed based on the combination of and . Since irrigation is the main purpose of the dam, irrigation is weighted more. So, for the irrigation, 50, 60, 70, 80, and 90% were chosen randomly. The combinations of weights are as follows: ; ; ; ; and . To apply the aforementioned models, average monthly input into a reservoir measured over 45 years for every calendar month (from 1972 to 2016) is used.

## PARAMETERS

The sensitivities analysis has been carried out using a variety of combinations of common control parameters, including population size (e.g., 25, 50, 75, and 100). The performance of algorithms was analyzed over the course of 10 independent runs. The evaluation criteria are the termination criteria which are: functional evaluation = population size × iteration number. It was discovered that all algorithms significantly improved for population's size 50, and SAMP-MOJA has done much better for population's size 100. For IWO, the initial and final standard deviations were set as 0.6 and 0.001, respectively. The lowest and highest numbers of seeds were preserved at 0 and 3, respectively, and the modulation index was kept at 2. The PSO's internal parameters were set to 1, 1.3, and 2.0 for the inertia weight, cognitive parameter, and social parameter, respectively. The values of internal parameters were acquired using sensitivity analysis. For SAMP-MOJA and MOJA, there were no inner parameters available.

## RESULTS AND DISCUSSION

### Pareto-optimal solution

^{3}, respectively. For model 4, maximum power generation were 1,032.225, 999.702, 924.437, and 802.157 MkWh

_{,}respectively, and minimum irrigation deficiency , , 371.448, and , respectively. It can be shown that the results obtained using SAMP-MOJA is better for each model in comparison with the results suggested by MOJA, multi-objective particle swarm optimization (MOPOS), and MOIWO in terms of both the objectives.

Algorithm . | Weights . | Maximum power generation () . | Minimum irrigation deficiency . |
---|---|---|---|

SAMP-MOJA | 1,002.446 | ||

1,014.489 | |||

1,029.389 | |||

1,032.225 | |||

1,036.842 | |||

MOJA | 978.724 | ||

981.671 | |||

998.509 | |||

999.702 | |||

1,002.268 | |||

MOPSO | 905.214 | 145.677 | |

919.324 | 173.469 | ||

924.437 | 367.408 | ||

957.103 | 371.448 | ||

976.010 | 1,296.01 | ||

MOIWO | 793.079 | ||

794.543 | |||

796.345 | |||

802.157 | |||

822.571 |

Algorithm . | Weights . | Maximum power generation () . | Minimum irrigation deficiency . |
---|---|---|---|

SAMP-MOJA | 1,002.446 | ||

1,014.489 | |||

1,029.389 | |||

1,032.225 | |||

1,036.842 | |||

MOJA | 978.724 | ||

981.671 | |||

998.509 | |||

999.702 | |||

1,002.268 | |||

MOPSO | 905.214 | 145.677 | |

919.324 | 173.469 | ||

924.437 | 367.408 | ||

957.103 | 371.448 | ||

976.010 | 1,296.01 | ||

MOIWO | 793.079 | ||

794.543 | |||

796.345 | |||

802.157 | |||

822.571 |

*Note*: Bold values indicate the optimal results.

### Convergence plot by different algorithms

*y*-axis shows the combined objective function solutions, and the

*x*-axis shows the number of iterations, or how many times the algorithm has run. It has been observed that SAMP-MOJA and MOJA converge to an optimal solution faster and more rapidly than MOPOS and MOIWO. MOPSO had a better performance than MOIWO. The MOIWO convergence rate was very slow compared to SAMP-MOJA, MOJA, and MOPSO. The average times to run the models were 15–16 min for all SAMP-MOJA. For all MOJA, the models took an average of 16–17 min to run. For all MOPSO and MOIWO, the models took 2021 min to run on average.

*y*-axis is the hydropower generation in megakilowatt-hours, and the

*x*-axis is the actual yearly hydropower generation and results of the models. The models were named in the short form in the graph as SAMP-JA1 to SAMP-JA5 for the SAMP-MOJA (model 1) to SAMP-MOJA (model 5). For other algorithms, similar abbreviations are used. The SAMP-MOJA, MOJA, and MOPSO have been observed to perform better than 80% of the actual generation. But for the years 1981, 1983, 1988, 1989, and 1990, the actual power generation was higher as compared to the models, this is due to the large inflow into the dam.

### Statistical efficient parameters

The results of water demand and release for irrigation based on various error indices for various algorithms are shown in Table 3. The RMSEs for SAMP-MOJA, MOJA, MOIWO and MOPSO for model 5 were , , , and 10.39 Mm^{3}, respectively. RMSE values acquired by SAMP-MOJA and MOJA were found to be smaller for all weights compared to MOIWO and MOPSO. The MAEs for SAMP-MOJA, MOJA, MOIWO, and MOPSO calculated for model 5 were , , _{,} and 4.12 Mm^{3}, respectively. SAMP-MOJA had a comparatively small MAE than MOJA, MOIWO, and MOPSO.

Error index . | Models . | Weights . | SAMP-MOJA . | MOJA . | MOPSO . | MOIWO . |
---|---|---|---|---|---|---|

RMSE (Mm^{3}) | Model 1 | 3.48 | ||||

Model 2 | 3.80 | |||||

Model 3 | 5.53 | |||||

Model 4 | 5.56 | |||||

Model 5 | 10.39 | |||||

MAE (Mm^{3}) | Model 1 | 3.79 | ||||

Model 2 | 4.34 | |||||

Model 3 | 5.46 | |||||

Model 4 | 5.32 | |||||

Model 5 | 4.12 |

Error index . | Models . | Weights . | SAMP-MOJA . | MOJA . | MOPSO . | MOIWO . |
---|---|---|---|---|---|---|

RMSE (Mm^{3}) | Model 1 | 3.48 | ||||

Model 2 | 3.80 | |||||

Model 3 | 5.53 | |||||

Model 4 | 5.56 | |||||

Model 5 | 10.39 | |||||

MAE (Mm^{3}) | Model 1 | 3.79 | ||||

Model 2 | 4.34 | |||||

Model 3 | 5.46 | |||||

Model 4 | 5.32 | |||||

Model 5 | 4.12 |

### Performance evaluation measures

The results of water demand and releases for irrigation based on various performance metrics for various algorithms are shown in Table 4. In comparison to MOPOS, the reliability index scores for SAMP-MOJA and MOJA are greater, indicating improved irrigation water delivery and optimal reservoir operation. The lower vulnerability index value of SAMP-MOJA also shows that the severity of the failure is smaller than that of MOPSO, MOJA, and MOIWO. The reservoir operation using SAMP-MOJA can, therefore, better prevent water shortage crises compared to MOPSO and MOIWO.

Performance index (%) . | Models . | Weights . | SAMP-MOJA . | MOJA . | MO PSO . | MO IWO . |
---|---|---|---|---|---|---|

Reliability index | Model 1 | 100 | 100 | 99.26 | 99.99 | |

Model 2 | 100 | 100 | 99.15 | 99.99 | ||

Model 3 | 99.99 | 99.99 | 98.93 | 99.99 | ||

Model 4 | 99.99 | 99.99 | 98.96 | 99.99 | ||

Model 5 | 99.99 | 99.99 | 99.19 | 99.97 | ||

Vulnerability index | Model 1 | 3.81 | ||||

Model 2 | 6.99 | |||||

Model 3 | 8.61 | |||||

Model 4 | 8.34 | |||||

Model 5 | 33.24 |

Performance index (%) . | Models . | Weights . | SAMP-MOJA . | MOJA . | MO PSO . | MO IWO . |
---|---|---|---|---|---|---|

Reliability index | Model 1 | 100 | 100 | 99.26 | 99.99 | |

Model 2 | 100 | 100 | 99.15 | 99.99 | ||

Model 3 | 99.99 | 99.99 | 98.93 | 99.99 | ||

Model 4 | 99.99 | 99.99 | 98.96 | 99.99 | ||

Model 5 | 99.99 | 99.99 | 99.19 | 99.97 | ||

Vulnerability index | Model 1 | 3.81 | ||||

Model 2 | 6.99 | |||||

Model 3 | 8.61 | |||||

Model 4 | 8.34 | |||||

Model 5 | 33.24 |

### Discussion

The developed multi-objective reservoir operation helped in reducing irrigation deficiencies by supplying irrigation water and maximized hydropower generation by taking into account reservoir operational constraints. The optimal Pareto front has been developed based on the weights of priorities. The optimal Pareto front can be used to make decisions and operate the reservoir as a multi-objective in the desired alternative priorities. The reservoir has varying inflows every year. Some years are dry, while some may be wet years. The alternatives proposed with different weights are helpful in such conditions. For example, for the dry year and less storage in the reservoir alternative with a higher weight for irrigation may be preferred over an alternative with higher hydropower weights. It is also observed that the hydropower generated by the developed model outperforms 80% of the actual generation. The developed model has a better reservoir storage policy and can supply more water for irrigation to produce more crop yield. The optimal release of water results in less irrigation deficits, which also contributes to hydropower. More hydropower can be generated if more water is released by the bed, but the deficiencies in ULBMC irrigation increase and the surplus water enters the ocean. Higher levels of weight are assigned to irrigation in the objective function so that irrigation deficits are minimized first, and then hydropower is maximized. The best optimal solution should satisfy both objectives. SAMP-MOJA performs admirably in both areas, contributing to lower irrigation deficits and increased hydropower generation. More importantly, the proposed SAMP-MOJA method requires less CPU time. Moreover, SAMP-MOJA and MOJA involve only common controlling parameters compared with MOIWO and MOPSO where both algorithm-specific and common controlling parameters are present. In particular, the operational policies of the SAMP-MOJA are superior to those of the MOJA, MOIWO, and MOPSO. Finally, it is recommended that the SAMP-MOJA can be used to solve MO problems as an efficient alternative technique.

The useful advice and insights could be provided to parties involved in the planning and decision-making processes for energy and water management. The findings help to build effective and sustainable water allocation methods that ensure optimal exploitation of water resources while taking into account the needs of various sectors and ecosystems.

- (a)
Water management strategies: The multi-objective reservoir operating model that has been built provides important information about water management tactics for the Ukai reservoir. The model offers decision-makers a variety of optimal options based on various weighting combinations while taking into account the dual goals of avoiding irrigation deficits and optimizing hydropower generation. By using this information, reservoir managers may allocate water more effectively, meet agricultural needs, and increase hydropower output all while guaranteeing resource efficiency.

- (b)
Reservoir operation optimization: The study's findings show how the SAMP-MOJA algorithm can be used to achieve optimal reservoir operating. The model aids reservoir managers in creating operation strategies that balance water supply, storage, and hydropower generation by including a variety of constraints, including continuity, evaporation, storage, power production, canal capacity, and downstream discharges. The model's use by reservoir managers can increase the effectiveness of water release schedules, which will increase the amount of water available for irrigation, hydropower production, and downstream ecosystems.

- (c)
Drought and water scarcity management: In areas that are prone to drought, the proposed model's capacity to reduce irrigation deficits is very beneficial. By effectively allocating water resources during dry spells, the model's optimal solutions can help manage circumstances where there is a shortage of water. While taking into account the available water storage, inflow, and environmental limits, decision-makers can prioritize the requirement for irrigation. This can assist environmentally friendly water management techniques and lessen the negative effects of droughts on agriculture.

- (d)
Renewable energy planning: The significance of renewable energy sources is shown by the study's emphasis on maximizing hydropower generation. The model assists in determining the best release techniques to maximize hydropower production while taking environmental constraints into account by optimizing reservoir operations. Planning and policy decisions relating to renewable energy can be influenced by this information, helping to create a more sustainable and dependable energy mix.

### Limitations and future directions of the study

#### Limitations of the study

- (a)
The suggested model is specific to the Ukai reservoir; the findings might not be directly applicable to other reservoir systems with different properties. To validate the concept in various geographic contexts, more study is required.

- (b)
The analysis uses a fixed set of input data from 1972 to 2016, which might not take future changes in the climate and patterns of water consumption into consideration. The model's robustness would be improved by incorporating recent data and taking climate change scenarios into account.

#### Future directions

- (a)
The generalizability and scalability of the developed model could be better understood by investigating how it might be applied to other reservoirs and water management systems.

- (b)
Researching the financial aspects of reservoir operation, such as cost–benefit analysis and the value of ecosystem services, will help us gain a deeper understanding of the trade-offs between irrigation and hydropower production.

- (c)
Addressing the socioeconomic effects of reservoir operation, especially those on nearby populations and downstream ecosystems, would aid in the creation of management plans that are more environmentally and socially responsible.

## CONCLUSIONS

In this study, an a priori approach was used to develop a Pareto-optimal solution for the multi-objective reservoir operation of the Ukai reservoir. To overcome algorithm-specific parameters, the improved version of JA was used, which does not require any algorithm-specific parameters to tune, and therefore, the user's burden gets reduced. The improved version of SAMP-JA helps in deciding the appropriate population size more quickly. The performances of SAMP-MOJA were compared with MOJA, MOPSO, and MOIWO. The findings clearly indicate SAMP-MOJA's superiority over the other algorithms. In chosen weights, the SAMP-MOJA convergence was better than other algorithms. SAMP-MOJA model has minimum RMSE and MAE. SAMP-MOJA's increased reliability and low vulnerability index value show better irrigation water supply and less failure severity. The SAMP-MOJA algorithm's main advantages are that it is relatively easy to implement, requiring only common controlling parameters to be adjusted, and that it can handle interdependence relationships among decision variables effectively, producing quick and effective Pareto-optimal solutions. The vulnerability index value shows better irrigation water supply and less failure severity. Finally, SAMP-MOJA was found to be outperforming MOJA, MOPOS, and MOIWO.

## FUNDING

No funds, grants, or other support were received.

## ETHICS APPROVAL

This research does not contain any studies with human participants or animals performed by any of the authors.

## 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.

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