## Abstract

This study evaluates three contemporary evolutionary algorithms, namely, shark, genetic, and particle swarm algorithms, for optimization in reservoir operation and water supply. The Klang gate dam in Malaysia is selected as the case study to optimize reservoir operation. The key objective of this study is the minimization of water deficits based on demands and released water. The global solution of the problem is computed based on software Lingo and the average solution of the shark algorithm is able to attain 99% of global solution. As well, the shark algorithm can furnish demand values at a faster convergence rate than both genetic and particle swarm algorithms. The reliability index and resiliency index, as useful indices in water resource management, are used and the values of these indices have the highest percent for the shark algorithm, indicating its superiority over other evolutionary algorithms.

## NOTATION

*C*penalty coefficient

*D*demand

*f*_{si}residency index

*F*_{si}the number of periods of happened failure

*I*inflow

*k*the maximum number of shark movement toward the position

*loss*loss value for reservoir operation

*m*number of points of local search

*MG*million gallons

best position of particle

*R*released water

*S*storage

*S*_{max}maximum storage

*S*_{min}minimum storage

*V*velocity

*w*_{f}criteria's weight

*X*decision variable

*Z*the new shark position after local search

constant value

a random number between 0 and 1

objective function

random number

inertia coefficient

volumetric reliability

limiter velocity

constriction coefficient

vulnerability index

residency index

time interval

## INTRODUCTION

Increasing drought periods and limiting resources cause many difficulties for decision-makers in managing resources. Water scarcity in different regions of the world has emerged to become a serious problem (Bozorg Haddad *et al.* 2008a, 2011). Construction of large dams can be one choice for water supply, but it may not be the best approach in term of water demands. This is because the construction of large dams entails high costs and, as such, cannot be the first choice. In fact, the best method for providing water and energy supply should be an economical one (Oliveira & Loucks 1997; Fallah-Mehdipour *et al.* 2013a; Kumar *et al.* 2016). Water resource management is a science which studies the resource operation based on economical approaches and optimization. It is important to have the highest efficiency for resources operation. The problems of water resource management can be defined in the framework of an optimization problem (Ngoc *et al.* 2013). The aim of different projects can be defined based on the objective function and, in fact, the effective variable for the problems can be defined based on decision variables. As well, the limitations of projects or problems can be defined based on different constraints. Water resource management attempts to provide comprehensive programming for regulating different resources such as water and energy (Fallah-Mehdipour *et al.* 2011, 2013b). This science uses different mathematical models for reservoir and power plant operations to determine the amount and time of water release for downstream consumers (Ostadrahimi *et al.* 2012; Shirzad & Tabesh 2016). Linear, nonlinear, and dynamic programming methods are used as mathematical models for the problems of reservoir and power plant operation (Kumar & Reddy 2006). Yet these methods have their inherent limitations, which result in difficulties in defining new methods for their replacement. For example, if a certain problem has a nonlinear objective function, linear programming cannot be used as the solving tool and, also, for problems with many decision variables, dynamic programming method requires high computational time. Thus, researchers attempt to find more powerful methods than the conventional methods such as nonlinear programming method or dynamic programming method. Evolutionary algorithms or meta-heuristic algorithms are new approaches for solving complex problems in water resource management and in reservoir and power plant operation. Tospornsampan *et al.* (2005) used a simulated annealing method for reservoir operation to supply irrigation deficits and the results were compared with those by genetic algorithm (GA). Results indicated that a simulated annealing method could furnish demands better than a GA. Jothiprakash *et al.* (2011) generated rule curves for reservoir operation based on GA and results showed that the GA method had the higher reliability index than nonlinear programming method for water supply. Jalali *et al.* (2007) used ant colony optimization method for multi-reservoir operation. The aim of the study was the maximization of the generated power from power plants. Results indicated that ant colony optimization method could generate more power for downstream consumers. Bozorg Haddad *et al.* (2008b) studied reservoir operation based on honey bee mating optimization method. Results showed that the method had faster convergence speed and more satisfactory downstream demands than the GA. Its higher power production and faster convergence indicated that the diploid method demonstrated effective performance in hydropower operation. Ghimire & Reddy (2013) used particle swarm algorithm for the generation of rule curves for hydropower operation. Results indicated that the percentage of power deficit was decreased with particle swarm algorithm. Bozorg Haddad *et al.* (2015) used bat algorithm for the optimization of hydropower and results showed the annual power production was more than its counterparts by genetic and particle swarm algorithms. Also, Bozorg Haddad *et al.* (2015) used the water cycle algorithm for reservoir operation. Results indicated that the water cycle algorithm could supply water demands with less vulnerability index, yet these new meta-heuristic methods demonstrated high performance for water supply in water resource management problems.

In this study, the shark algorithm is used as one of the meta-heuristic and solving tools for reservoir optimization. The shark algorithm was used for benchmark functions and complex problems, and results showed the shark algorithm had high potential for applications in engineering and optimization problems (Ehteram *et al.* 2017a). The shark algorithm has been employed for the maximization of power production in a power plant. Results showed that the power production based on the shark algorithm was higher than conventional meta-heuristic algorithms (Ehteram *et al.* 2017b).

The Klang gate dam is used as a case study for irrigation demands supply. The objective function for this problem is the minimization of irrigation deficit for downstream consumers. As well, three kinds of inflow are considered in this problem to understand the supply routes of water demands based on reservoir operation and meta-heuristic algorithm. The GA and particle swarm algorithm, being successful algorithms in water resource management, are employed as benchmarks for comparison with results by the shark algorithm. In addition, different indices such as reliability, vulnerability, and resiliency index as widespread indices in water resource management, are used to evaluate the performances of different methods.

## SHARK ALGORITHM

Sharks have powerful olfaction so that they can find their prey based on this ability. Sharks move to the odor resource and when the concentration of odor is greater, the shark moves faster because it has found the food resource. The shark algorithm acts based on the following assumptions:

Fish in the water are considered as prey for sharks. Each fish is considered to have an injured body so that it produces blood from itself. Thus, the movement of the fish is slow and it is approximately fixed.

The blood produced from the fish is distributed based on a regular form. The water movement and tides do not have any effect on blood distribution. The shark follows the odor particle and particles which are close to injured fish have more odor concentration. Thus, the shark can find prey based on odor particle.

A blood resource is generated by a fish and the shark attempts to find this blood resource as its prey.

*Of*is the objective function,

*R*

_{1}is a random number between 0 and 1, is a constant between 0 and 1, and k

_{max}is the maximum number of shark movements toward the position.

The momentum coefficient causes increased search diversity and the chance of attaining global solution increases with the consideration of this term.

*m*is the number of points of local search, and

*R*

_{3}is a coefficient between −1 and 1.

## GENETIC ALGORITHM

The GA is a well-known algorithm in engineering and optimization files. Real coded GA is used for continuous variables and has three important steps: elitist strategy, crossover, and mutation. The elitist strategy keeps the best members of the population for increasing convergence chance. Crossover operator combines two chromosomes with one another. Mutation operator leads to the increase of the population diversity.

### Elitist operator

Elitist operator keeps the best members to be considered directly in the next generation and thus, through this operator, the convergence chance is increased. Crossover and mutation operators are then used for generating new populations.

### Crossover operator

*u*) is first considered and parameter is generated based on the following equation: where and is computed based on the following equation: where

*x*denote the parents of children chromosomes; parameter is the distribution index and a larger value of this parameter denotes that the children chromosomes are closer to the parent chromosomes (global solutions);

*x*and

_{i}^{l}*x*are the upper and lower limit of decision variables. The children candidates are considered based on the following equation: where

_{i}^{u}*y*are the children chromosomes.

### Mutation operator

*X*=

*x*

_{1},

*x*

_{2}, …,

*x*n) is selected and then mutation is considered based on the following equation: and is computed based on the following equation: where

*t*is the maximum iteration.

## PARTICLE SWARM ALGORITHM

*i*is shown as . As well, index g indicates the best particles in a swarm in all iterations. The velocities and positions are updated based on the following equations: where

*q*is the inertia coefficient,

*c*is the acceleration coefficient,

*r*is a random value between 0 and 1, and is the constriction coefficient.

For the particle swarm algorithm equals 1.

The initial population is considered and initial velocities and positions are generated. Then, the objective function is computed for each particle and the local solution and global solution are selected in any iteration and all iterations, respectively. Velocities and positions are then updated and the algorithm continues to satisfy the specified criteria.

## CASE STUDY

*D*is the demand and

*x*is the released water.

There are certain constraints for reservoir operation as follows.

Different indies are used for evaluating and comparing the performance of different methods:

Months . | High inflow . | Medium inflow . | Low inflow . | Demand . |
---|---|---|---|---|

January | 1,506.89 | 760.85 | 123.12 | 1,298.64 |

February | 1,901.08 | 1,024.49 | 259.34 | 1,083.09 |

March | 2,831.70 | 1,646.31 | 923.34 | 1,152.45 |

April | 2,919.74 | 1,959.92 | 764.88 | 1,173.11 |

May | 2,974.20 | 1,768.32 | 938.31 | 1,198.73 |

June | 2,825.20 | 1,355.22 | 447.95 | 1,271.73 |

July | 2,717.32 | 1,618.95 | 645.61 | 1,258.14 |

August | 2,948.26 | 1,644.53 | 816.78 | 1,260.41 |

September | 3,368.12 | 1,859.86 | 631.15 | 1,160.45 |

October | 3,545.83 | 2,316.25 | 654.35 | 1,204.14 |

November | 3,838.47 | 2,342.89 | 1,021.79 | 1,213.09 |

December | 2,699.30 | 1,455.70 | 340.69 | 1,290.59 |

Months . | High inflow . | Medium inflow . | Low inflow . | Demand . |
---|---|---|---|---|

January | 1,506.89 | 760.85 | 123.12 | 1,298.64 |

February | 1,901.08 | 1,024.49 | 259.34 | 1,083.09 |

March | 2,831.70 | 1,646.31 | 923.34 | 1,152.45 |

April | 2,919.74 | 1,959.92 | 764.88 | 1,173.11 |

May | 2,974.20 | 1,768.32 | 938.31 | 1,198.73 |

June | 2,825.20 | 1,355.22 | 447.95 | 1,271.73 |

July | 2,717.32 | 1,618.95 | 645.61 | 1,258.14 |

August | 2,948.26 | 1,644.53 | 816.78 | 1,260.41 |

September | 3,368.12 | 1,859.86 | 631.15 | 1,160.45 |

October | 3,545.83 | 2,316.25 | 654.35 | 1,204.14 |

November | 3,838.47 | 2,342.89 | 1,021.79 | 1,213.09 |

December | 2,699.30 | 1,455.70 | 340.69 | 1,290.59 |

*Resiliency index*: The resiliency index shows the model's efficiency to recover a failure. A higher percent for this index is more suitable and the following equation shows the resiliency index: where is the residency index, is the number of series of happened failure, is the number of periods of happened failure.

### Evaluation of the meta-heuristic methods

*nc*is the number of indices,

*w*and

_{e}*w*are the criteria's weight.

_{f}## RESULTS AND DISCUSSION

Table 2 shows the sensibility analysis for reservoir operation. The population size for shark algorithm, based on the best value of the objective function, is 30 and the values of are 20 and 0.76, respectively. For particle swarm algorithm, the population size, inertial weight, and acceleration coefficient are 50, 0.78, and 2, respectively. For GA, the population size, crossover probability, mutation probability, and mutation rate are 30, 0.76, 0.87, and 0.001, respectively. The objective function is computed based on the best parameter values in each method. The Lingo software is an optimization and engineering software tailored especially for water resource management. The linear programming method of this software is used for the computation of a global solution and the evaluation of average solutions of different meta-heuristic algorithms based on a global solution. The global solution for this problem based on Lingo software is 1.245. Table 3 shows ten random results for different meta-heuristic algorithms. The average solution for shark algorithm is 1.247, which denotes 99% of the global solution. The average solutions for genetic and particle swarm algorithms are 1.544 and 1.457, denoting 0.80 and 0.85 of the global solution, respectively. The variation coefficient for shark algorithm is 10 and 15 times smaller than the corresponding values for the particle swarm algorithm and GA, respectively. Figure 3 shows the pattern of water release for January in the entire operation period. The volume of released water is based on the average value of January in the operation period. Three kinds of inflow are considered for each month and ten groups are considered for reservoir storage. The first objective was to produce a curve that can show the optimum releases for an inflow defined group. The objective of these techniques was to generate a release policy that decreases the water deficit as far as possible, keeping the release domains for every month and safe storage volume. Thus, here, the storage condition was also provided with release options to obtain a clear picture of the reservoir system. Figure 3 shows calibrated curves for the month of January and this month was selected because the demand for this month has the most value based on Table 1. Shark algorithm meets demands for high inflow when the storage is 2,153.71 MG while the particle swarm and GAs meet demands when storage values are 2,658.94 and 2,678.87 MG for January, respectively. Shark algorithm method meets the demand for medium inflow when the storage is 3,163.78 MG (Figure 3). Yet particle swarm and GAs meet demands when reservoir storages are 3,668.2 MG and 3,724.12 MG for January, respectively. Shark algorithm meets demands for low inflow when the storage is 3,678.94 MG while genetic and particle swarm algorithms meet demand when the storage values are 3,787.33 MG and 3,711.21 MG for January, respectively. Thus, it can be seen that the demands are supplied with the shark algorithm earlier than those with the GA and particle swarm algorithm. Figure 3 shows that the shark algorithm release options are reached and follow the demand line more quickly and closely than other algorithms and capacity above demand will be reserved for next year.

(a) . | |||||
---|---|---|---|---|---|

Population size . | Objective function . | M . | Objective function . | . | Objective function . |

10 | 1.345 | 10 | 1.298 | 0.56 | 1.289 |

30 | 1.247 | 20 | 1.247 | 0.66 | 1.254 |

50 | 1.312 | 30 | 1.310 | 0.76 | 1.247 |

70 | 1.298 | 40 | 1.312 | 0.86 | 1.314 |

(b)
. | |||||

Population size
. | Objective function
. |
. c_{1=}c_{2} | Objective function
. |
. w | Objective function
. |

10 | 1.871 | 1.8 | 1.612 | 0.58 | 1.512 |

30 | 1.657 | 1.9 | 1.545 | 0.68 | 1.505 |

50 | 1.451 | 2.0 | 1.457 | 0.78 | 1.457 |

70 | 1.520 | 2.1 | 1.567 | 0.88 | 1.545 |

(c)
. | |||||

Population size
. | Objective function
. | Crossover probability
. | Objective function
. | Mutation probability
. | Objective function
. |

10 | 1.698 | 0.66 | 1.598 | 0.66 | 1.612 |

30 | 1.544 | 0.76 | 1.544 | 0.76 | 1.544 |

50 | 1.672 | 0.86 | 1.587 | 0.86 | 1.589 |

70 | 1.712 | 0.96 | 1.589 | 0.96 | 1.611 |

(a) . | |||||
---|---|---|---|---|---|

Population size . | Objective function . | M . | Objective function . | . | Objective function . |

10 | 1.345 | 10 | 1.298 | 0.56 | 1.289 |

30 | 1.247 | 20 | 1.247 | 0.66 | 1.254 |

50 | 1.312 | 30 | 1.310 | 0.76 | 1.247 |

70 | 1.298 | 40 | 1.312 | 0.86 | 1.314 |

(b)
. | |||||

Population size
. | Objective function
. |
. c_{1=}c_{2} | Objective function
. |
. w | Objective function
. |

10 | 1.871 | 1.8 | 1.612 | 0.58 | 1.512 |

30 | 1.657 | 1.9 | 1.545 | 0.68 | 1.505 |

50 | 1.451 | 2.0 | 1.457 | 0.78 | 1.457 |

70 | 1.520 | 2.1 | 1.567 | 0.88 | 1.545 |

(c)
. | |||||

Population size
. | Objective function
. | Crossover probability
. | Objective function
. | Mutation probability
. | Objective function
. |

10 | 1.698 | 0.66 | 1.598 | 0.66 | 1.612 |

30 | 1.544 | 0.76 | 1.544 | 0.76 | 1.544 |

50 | 1.672 | 0.86 | 1.587 | 0.86 | 1.589 |

70 | 1.712 | 0.96 | 1.589 | 0.96 | 1.611 |

Run . | Shark algorithm . | Particle swarm algorithm . | Genetic algorithm . |
---|---|---|---|

1 | 1.247 | 1.457 | 1.545 |

2 | 1.246 | 1.449 | 1.545 |

3 | 1.247 | 1.449 | 1.544 |

4 | 1.247 | 1.457 | 1.544 |

5 | 1.247 | 1.457 | 1.544 |

6 | 1.247 | 1.457 | 1.544 |

7 | 1.247 | 1.457 | 1.544 |

8 | 1.247 | 1.457 | 1.544 |

9 | 1.247 | 1.457 | 1.544 |

10 | 1.247 | 1.457 | 1.544 |

Average | 1.247 | 1.457 | 1.544 |

Variation coefficient | 0.0002 | 0.002 | 0.003 |

Global solution | 1.245 |

Run . | Shark algorithm . | Particle swarm algorithm . | Genetic algorithm . |
---|---|---|---|

1 | 1.247 | 1.457 | 1.545 |

2 | 1.246 | 1.449 | 1.545 |

3 | 1.247 | 1.449 | 1.544 |

4 | 1.247 | 1.457 | 1.544 |

5 | 1.247 | 1.457 | 1.544 |

6 | 1.247 | 1.457 | 1.544 |

7 | 1.247 | 1.457 | 1.544 |

8 | 1.247 | 1.457 | 1.544 |

9 | 1.247 | 1.457 | 1.544 |

10 | 1.247 | 1.457 | 1.544 |

Average | 1.247 | 1.457 | 1.544 |

Variation coefficient | 0.0002 | 0.002 | 0.003 |

Global solution | 1.245 |

Another point is related to the convergence behaviors of different algorithms. Figure 4 shows that shark algorithm can converge earlier than the GA and particle swarm algorithm. As such, the quality and velocity of convergence for the shark algorithm are better than those for particle swarm and GAs. To find the closest curve to a demand for a specific inflow category, the root mean square error (RMSE) is used and the RMSE for every month is given in Table 4 for medium inflow (medium inflow is selected because no violation of storage consistency has occurred). Shark optimization algorithm has the lowest error in this case study. Table 4 shows that total RMSE for the shark algorithm is 42 MG and the corresponding indexes for particle swarm and GAs are 53 MG and 63 MG, respectively. Thus, the shark algorithm decreases RMSE compared to the particle swarm algorithm and GA by about 20 and 33%, respectively.

Month . | Shark algorithm (MG) . | Particle swarm algorithm (MG) . | Genetic algorithm (MG) . |
---|---|---|---|

January | 8.78 | 9.11 | 10.22 |

February | 6.54 | 7.49 | 8.14 |

March | 5.56 | 6.12 | 7.23 |

April | 5.45 | 6.78 | 7.14 |

May | 7.87 | 8.11 | 8.44 |

June | 6.54 | 7.12 | 8.11 |

July | 3.44 | 3.89 | 4.01 |

August | 1.28 | 1.78 | 1.86 |

September | 1.18 | 1.48 | 1.78 |

October | 0.98 | 1.01 | 1.11 |

November | 0.87 | 0.98 | 1.04 |

December | 0.86 | 0.89 | 0.99 |

Total | 49.35 | 54.68 | 60.07 |

Month . | Shark algorithm (MG) . | Particle swarm algorithm (MG) . | Genetic algorithm (MG) . |
---|---|---|---|

January | 8.78 | 9.11 | 10.22 |

February | 6.54 | 7.49 | 8.14 |

March | 5.56 | 6.12 | 7.23 |

April | 5.45 | 6.78 | 7.14 |

May | 7.87 | 8.11 | 8.44 |

June | 6.54 | 7.12 | 8.11 |

July | 3.44 | 3.89 | 4.01 |

August | 1.28 | 1.78 | 1.86 |

September | 1.18 | 1.48 | 1.78 |

October | 0.98 | 1.01 | 1.11 |

November | 0.87 | 0.98 | 1.04 |

December | 0.86 | 0.89 | 0.99 |

Total | 49.35 | 54.68 | 60.07 |

Table 5 shows the value of different indices for different meta-heuristic algorithms. The shark algorithm has a higher value of reliability index (namely, 97%) compared with the GA. In other words, the shark algorithm can supply demands with higher reliability index after the particle swarm algorithm. The resiliency index for the shark algorithm is 44% and the corresponding indexes for GA and particle swarm algorithm are 29 and 35%, respectively. As such, the system based on the shark algorithm can exit from a failure faster than other methods. The vulnerability index for the shark algorithm is 14% while the corresponding indexes for the GA and particle swarm algorithm are 24 and 21%, respectively. These indexes have been computed for a period of 10 years (1994–2003). Figure 5 indicates that the shark algorithm corresponds to demand values better than the cases of GA and particle swarm algorithm. Thus, the shark algorithm has high potential in reservoir operation. Also, Figure 5(d) shows the average annual deficit of the reservoir by the applied algorithms. According to Figure 5(d), the severity of deficits computed by the shark algorithm is lower than other algorithms. The maximum deficits in the reservoir according to the shark algorithm, particle swarm algorithm, and GA were 17.12, 96.24, and 100 GB. The shark algorithm provided the lowest deficit during this study, indicating the greater performance of this algorithm for minimizing deficit.

Method . | Reliability index . | Vulnerability index . | Resiliency index . |
---|---|---|---|

Shark algorithm | 97% | 14% | 44% |

Particle swarm algorithm | 99% | 21% | 39% |

Genetic algorithm | 85% | 24% | 25% |

Method . | Reliability index . | Vulnerability index . | Resiliency index . |
---|---|---|---|

Shark algorithm | 97% | 14% | 44% |

Particle swarm algorithm | 99% | 21% | 39% |

Genetic algorithm | 85% | 24% | 25% |

In general, all of these optimization results indicate that the best algorithm should not be selected based only on one index or on one kind of objective function. Also, the system used in this study is a single reservoir, which could limit the performance and versatility of the method when a different study location or different flow conditions are considered. Therefore, it can be postulated that the selection of the best method is likely to be more complex for multi-reservoir systems. As a result, we conclude that an effective tool should be found by means of comparing different evolutionary methods such that a multi-criteria decision model is determined and applied for making better decisions in the optimization of reservoir operations.

Next, we checked the normalized decision-making parameters ( and for different methods) in Table 6 where the weight of all indices is considered based on the same value.

(a) . | ||||
---|---|---|---|---|

Algorithms . | Reliability index . | Vulnerability index . | Resiliency index . | RMSE . |

Shark algorithm | 0.97 | 1 | 1 | 1 |

Particle swarm algorithm | 1 | 0.66 | 0.88 | 0.90 |

Genetic algorithm | 0.87 | 0.58 | 0.56 | 0.81 |

(b) . | ||||

Part of decision-making index . | Shark algorithm . | Genetic algorithm . | Particle swarm algorithm . | |

0.9925 | 0.705 | 0.8612 | ||

0.9999 | 0.691 | 0.8541 |

(a) . | ||||
---|---|---|---|---|

Algorithms . | Reliability index . | Vulnerability index . | Resiliency index . | RMSE . |

Shark algorithm | 0.97 | 1 | 1 | 1 |

Particle swarm algorithm | 1 | 0.66 | 0.88 | 0.90 |

Genetic algorithm | 0.87 | 0.58 | 0.56 | 0.81 |

(b) . | ||||

Part of decision-making index . | Shark algorithm . | Genetic algorithm . | Particle swarm algorithm . | |

0.9925 | 0.705 | 0.8612 | ||

0.9999 | 0.691 | 0.8541 |

Table 7(a) shows the values of for different algorithms tested in the present study. Eleven initial values are considered for and, then, the values for different methods are computed and, finally, the rank of each method is determined based on the overall value. Evidently, the shark algorithm has the highest rank based on all of the values.

(a) . | ||||
---|---|---|---|---|

Fraction . | . | . | . | |

= 0 | 0.9925 | 0.8541 | 0.7076 | |

= 0.1 | 0.9932 | 0.8548 | 0.7118 | |

= 0.2 | 0.9939 | 0.8555 | 0.7160 | |

= 0.3 | 0.9947 | 0.8562 | 0.72032 | |

= 0.4 | 0.9954 | 0.8569 | 0.7245 | |

= 0.5 | 0.9962 | 0.8576 | 0.7288 | |

= 0.6 | 0.9969 | 0.8583 | 0.7330 | |

= 0.7 | 0.9976 | 0.8590 | 0.7372 | |

= 0.8 | 0.9984 | 0.8597 | 0.7415 | |

= 0.9 | 0.9991 | 0.8604 | 0.7457 | |

= 1.0 | 0.9999 | 0.8612 | 0.75 | |

Rank | 1 | 2 | 3 | |

(b) . | ||||

Contender A
. | Contender B
. | Number of victories for A
. | Number of victories for B
. | Winner
. |

Shark algorithm | Particle swarm algorithm | 11 | 0 | Shark algorithm |

Particle swarm | Genetic algorithm | 11 | 0 | Particle swarm algorithm |

(c) . | ||||

Method
. | Rank
. | |||

Shark algorithm | 1 | |||

Particle swarm algorithm | 2 | |||

Genetic algorithm | 3 | |||

(d) . | ||||

Method
. | CPU (time: s)
. | |||

Shark algorithm | 50.6 | |||

Particle swarm algorithm | 60.1 | |||

Genetic algorithm | 80.5 |

(a) . | ||||
---|---|---|---|---|

Fraction . | . | . | . | |

= 0 | 0.9925 | 0.8541 | 0.7076 | |

= 0.1 | 0.9932 | 0.8548 | 0.7118 | |

= 0.2 | 0.9939 | 0.8555 | 0.7160 | |

= 0.3 | 0.9947 | 0.8562 | 0.72032 | |

= 0.4 | 0.9954 | 0.8569 | 0.7245 | |

= 0.5 | 0.9962 | 0.8576 | 0.7288 | |

= 0.6 | 0.9969 | 0.8583 | 0.7330 | |

= 0.7 | 0.9976 | 0.8590 | 0.7372 | |

= 0.8 | 0.9984 | 0.8597 | 0.7415 | |

= 0.9 | 0.9991 | 0.8604 | 0.7457 | |

= 1.0 | 0.9999 | 0.8612 | 0.75 | |

Rank | 1 | 2 | 3 | |

(b) . | ||||

Contender A
. | Contender B
. | Number of victories for A
. | Number of victories for B
. | Winner
. |

Shark algorithm | Particle swarm algorithm | 11 | 0 | Shark algorithm |

Particle swarm | Genetic algorithm | 11 | 0 | Particle swarm algorithm |

(c) . | ||||

Method
. | Rank
. | |||

Shark algorithm | 1 | |||

Particle swarm algorithm | 2 | |||

Genetic algorithm | 3 | |||

(d) . | ||||

Method
. | CPU (time: s)
. | |||

Shark algorithm | 50.6 | |||

Particle swarm algorithm | 60.1 | |||

Genetic algorithm | 80.5 |

To further explore the accuracy of the shark algorithm, Table 7(b) presents a pair contest based on Copeland's procedure for the 11 values of . In accordance with Copeland's procedure, the present results demonstrate the superiority of the shark algorithm over the genetic and particle swarm algorithms, which can be confirmed by the final ranks assigned in Table 7(c).

The objective value of the cost function and the average CPU run times for each algorithm (based on a PC with i5 CPU GHz/4B, ram/500 GB HDD) is presented in Table 7(d). CPU run time of the shark algorithm (50.6 s) was the least of all the algorithms.

## CONCLUSIONS

In this paper, a newly developed meta-heuristic evolutionary algorithm (known as the shark algorithm) is proposed for the optimization of the Klang gate dam's reservoir operation. The analysis presented revealed that the operation policies and results which have been obtained using the shark algorithm approach (compared to the GA and the improved particle swarm optimization algorithm (IPSOA) approaches) could lead to better water release plans that are likely to help minimize the mismatch existent with the water users' demands. Evaluation of the shark algorithm performance has been carried out for a historical period and the results show that this approach carries a greater robustness compared to the other methods for all inflow scenarios. Furthermore, the results show that the shark algorithm approach provided a closer solution to the global solution and was ranked in first position based on the multi-criteria decision model, followed by lower ranks assigned to the IPSOA and GA approaches. In addition, the shark algorithm approach outperformed the IPSOA and GA approaches in attaining a greater resilience index and a lower vulnerability index for all inflow scenarios relevant to the reservoir. In a situation where the low inflow scenario could be unsatisfactory in terms of meeting the water demands pattern in an adequate manner, the shark algorithm approach for water management was found to perform better than the other two optimization algorithms. This shows that the shark algorithm approach is considerably robust, especially in terms of challenging water management situations (e.g., low inflow scenarios).