# A Novel Hybrid Evolutionary Data-Intelligence Algorithm for Irrigation and Power Production Management: Application to Multi-Purpose Reservoir Systems

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## Abstract

**:**

^{6}m

^{3}), and the amount of released water based on the new hybrid algorithm (NHA) is 141.25 (10

^{6}m

^{3}). Compared with the shark algorithm (SA), BA, weed algorithm (WA), PSO algorithm, and genetic algorithm (GA), the NHA decreased the computation time by 28%, 36%, 39%, 82%, and 88%, respectively, which represents an excellent enhancement result. The amount of released water based on the proposed hybrid method attains a more reliable index for the volumetric percentage and provides a more effective operation rule for supplying the irrigation demand. Additionally, the average demand for power production is 18.90 (10

^{6}kwh), whereas the NHA produces 18.09 (10

^{6}kwh) of power. Power production utilizing the NHA’s operation rule achieved a sufficient magnitude relative to that of stand-alone models, such as the BA, PSO, WA, SA, and GA. The excellent proficiency of the developed intelligence expert system is the result of the hybrid structure of the BA and PSO algorithm and the substitution of weaker solutions in each algorithm with better solutions from other algorithms. The main advantage of the proposed NHA is its ability to increase the diversity of solutions and hence avoid the worst possible solutions obtained using BA, that is, preventing a decrease in local optima. In addition, the NHA enhances the convergence rate obtained using the PSO algorithm. Hence, the proposed NHA as an intelligence model could contribute to providing reliable solutions for complex multi-purpose reservoir systems to optimize the operation rule for similar reservoir systems worldwide.

## 1. Introduction

#### 1.1. Background

^{6}m

^{3}and an annual average inflow of 415.23 × 10

^{6}m

^{3}. The released water based on the GA could meet downstream demand patterns effectively, and the annual average irrigation deficits based on the GA were 12% and 22% less than those achieved using the PSO algorithm and GA, respectively.

^{6}m

^{3}, respectively. The aim of these studies was to minimize irrigation deficiencies. Downstream demands were supplied based on a volumetric reliability index of approximately 90%, while the supply for the downstream irrigation demand based on the GA was accompanied by high deficiencies during the operation period of the reservoir. Another study focused on the Karoon4 reservoir and utilized the water cycle algorithm (WCA) to increase the benefit of hydropower generation based on the released water, and the results showed that compared with the PSO algorithm and the GA, the WCA increased the annual benefit of hydropower generation by approximately 30% and 40%, respectively [30]. For the same reservoir, Haddad et al. [31] tested the biography-based optimization (BBO) algorithm for increasing hydropower generation. The results showed the high ability of the BBO algorithm based on a fast convergence speed and highly accurate computations.

^{3}. The results indicated that compared with the PSO algorithm and the GA, the KA could increase the annual benefits of power generation by 12% and 15%, respectively. Additionally, the convergence velocity for the KA was considerable.

#### 1.2. Problem Statement and Novelty

#### 1.3. Research Objectives

## 2. Methodological Overview

#### 2.1. Bat Algorithm (BA)

- Echolocation is used by all bats, and this ability is helpful for identifying prey from obstacles.
- Bats fly at a random velocity, vl, and at a random location, xl. The frequency of a bat is fl. A0 and $\lambda $ represent the loudness and wavelength of bats, respectively.
- The loudness of bats varies from A0 (i.e., a large positive number) to Amin.

_{t}) and pulsation rate (r

_{l}) are updated in each iteration of the algorithms. The value for loudness decreases and the pulsation rate increases when the bats find their prey. The pulsation rate for the generated sounds is updated based on the following equation [47]:

#### 2.2. Particle Swarm Optimization (PSO) Algorithm

#### 2.3. New Hybrid Algorithm (NHA)

- The random parameters are initialized for two algorithms, and then the velocity and position vectors are considered for the BA and PSO algorithms;
- The objective function is individually calculated for the two algorithms, and then the best member is determined for the two algorithms;
- The velocity and position are updated for the BA based on Equations (1)–(3), and the velocity and position are updated based on Equations (6) and (7), respectively;
- The K agent, as the best member of each algorithm, is copied to the other algorithms, which are substituted with the worst solutions of the other algorithm;
- The convergence criteria are checked, and if the algorithm is satisfied, the algorithm finishes; otherwise, the algorithm returns to the second step.

#### 2.4. Weed Algorithm (WA)

- Weeds are grown based on seeds, which are spread throughout the environment.
- Weeds that grow close to each other are known as a colony, and they can produce seeds based on their equality.
- Each produced seed distributes randomly throughout the environment.
- The algorithm finishes when the number of weeds reaches the maximum number.
- The different levels for the WA are based on the following levels:
- First, the initial population of the algorithm (Pinitial) is considered, and the position of each weed in the environment (i.e., search space) is considered a decision variable.
- The next level is known as the reproduction level. Reproduction causes new seeds to be produced from colonies, and the maximum and minimum numbers of seeds are $\left({N}_{0}{S}_{\mathrm{max}}\right)$ and $\left({N}_{0}{S}_{\mathrm{min}}\right)$, respectively (see Figure 3). Reproduction is an important level for the WA because there are two group solutions in the evolutionary algorithms. Appropriate solutions have a high chance of reproduction to continue the production of the best member for the next generation, and inappropriate solutions may have a weak chance of reproduction; however, they may have important information for the next levels of the algorithm. Thus, reproduction may be extended to inappropriate solutions that are not removed from the population, and they can continue their life based on suitable reproduction and the improvement in their quality. Some inappropriate solutions have important information, and this information can be used for the next levels of the algorithm.
- The produced seeds are distributed in the search space based on a normal distribution and zero mean.

_{max}, and there is competition among weeds because weeds of poor quality should be removed for population balance. Figure 4 shows the WA procedure.

#### 2.5. Shark Algorithm (SA)

- Injured fish are considered to be prey for sharks, as fish bodies distribute blood throughout the sea. Additionally, injured fish have negligible speeds compared with sharks.
- The blood is distributed into the sea regularly, and the effect of water flow is not considered for blood distribution.
- Each injured fish is considered as one blood production resource for sharks; therefore, the olfactory receptors help sharks find their prey.
- The initial population for sharks is shown by $\left[{X}_{1}^{1},{X}_{2}^{1},\dots ,{X}_{NP}^{1}\right],NP=population\left(size\right)$. Each solution candidate or shark position can have the following components based on the following equation:$${X}_{i}^{1}=\left[{x}_{i,1}^{1},{x}_{i,2}^{1},\dots {x}_{i,ND}^{1}\right],$$
^{1}_{ij}is the jth dimension of the shark position; and ND is the number of decision variables. The initial velocity for sharks is shown by ${V}_{i}^{1}=\left[{v}_{i,1}^{1},{v}_{i,2}^{1},\dots ,{v}_{i,ND}^{1}\right]$. The velocity components are considered based on the following equation:$${V}_{i}^{1}=\left[{v}_{i,1}^{1},{v}_{i,2}^{1},\dots ,{v}_{i,ND}^{1}\right],i=1,\dots NP,$$^{1}_{ij}is the jth dimension of the shark velocity. When the shark receives greater odour intensity, it moves faster towards its prey. Thus, if the odour intensity is considered an objective function, the velocity changes with the variation in the objective function based on the following equation:$${V}_{i}^{k}={\eta}_{k}.{R}_{1}.\nabla \left(OF\right){|}_{{x}_{i}^{k}},$$

#### 2.6. Genetic Algorithm (GA)

## 3. Case Study and Modelling Procedure

#### 3.1. Benchmark Function

#### 3.2. Multi-Purpose Reservoir Operation

^{3}. The irrigation area is 6367 ha, and the total area of the left and right bank canals is 87,512 ha. Figure 6a shows the schematization of the dam and reservoir’s basin, and Figure 6b shows the geographical location of the catchment area of the basin. The features of the reservoir can be seen in Table 2. Figure 6a shows the details for the system and Figure 6b shows the location of system on the river section. The command area for the river basin is 162,818 ha.

- The storage constraint is as follows:$${S}_{\mathrm{min}}\le {S}_{t}\le {S}_{\mathrm{max}},$$
- The power production constraints are as follows:$${k}_{1}{R}_{l,t}{H}_{l,t}\le {E}_{1,\mathrm{max}},$$$${k}_{2}{R}_{r,t}{H}_{r,t}\le {E}_{2,\mathrm{max}},$$$${k}_{3}{R}_{bt}{H}_{bt}\le {E}_{3,\mathrm{max}},$$
- The canal capacity constraints are as follows:$${R}_{l,t}\le {C}_{l,\mathrm{max}},$$$${R}_{r,t}\le {C}_{r,\mathrm{max}},$$
- The irrigation demands are as follows:$${D}_{l,t}^{\mathrm{min}}\le {R}_{l,t}\le {D}_{l,r}^{\mathrm{max}},$$$${D}_{r,t}^{\mathrm{min}}\le {R}_{r,t}\le {D}_{r,t}^{\mathrm{max}},$$
- The steady storage constraint is as follows:$${S}_{13}={S}_{1}.$$

- The decision variables for the left canal, right canal, and riverbed are initialized based on the initial matrix for the NHA. In fact, the released water for the downstream demands is considered as the initial population.
- The storage reservoir can be calculated based on the continuity equation, and the different constraints should be checked.
- If the constraints are not satisfied, the penalty functions are considered as violations; then, the objective function is calculated based on Equation (31).
- Then, the NHA process is considered for the optimization process based on the independent performances of the BA and PSO algorithm in the NHA.
- The convergence criteria are checked, and if the algorithm is satisfied, it finishes; otherwise, the algorithm returns to the second step.

## 4. Modelling Evaluation Indexes

- Volumetric reliability index. This index is based on the ratio of released water to irrigation demands. Thus, a high percentage of this index represents the high performance of each algorithm.$${\alpha}_{V}=1-\frac{{N}_{t=1}^{T}\left({D}_{t}>{R}_{t}\right)}{T},$$
- Vulnerability index. This index represents the maximum intensity of the failure that occurred during the operation period of a system. The periods for which irritation demands are not met are known as failure periods or critical periods, and maximum deficiency occurrences during these periods are represented by the vulnerability index; thus, a low percentage for this index is preferable [35].$$\lambda =Ma{x}_{t=1}^{T}\left(\frac{{D}_{t}-{R}_{t}}{{D}_{t}}\right)\times 100.$$
- Resiliency index. This index represents the existing speed of a system from failure. Thus, a high percentage for this index is preferable. This index shows the flexibility of different algorithms versus the critical periods when they should manage the system well [35].$${\gamma}_{i}=\frac{{f}_{si}}{{F}_{i}},$$
_{si}is the number of failure series that occurred; and ${F}_{i}$ is the number of failure periods that occurred. These indexes were used to evaluate the percentage of success of the examined optimization algorithms based on their achieved operation rules to minimize the gap between the water release and water demand. Furthermore, to evaluate the performance of each algorithm with respect to the computational time needed for convergence, the time consumption for each algorithm to achieve the operation rule was determined. The best algorithm is the one that could achieve the global optima in less time for convergence.

## 5. Results, Discussion, and Application Analysis

#### 5.1. Benchmark Functions

#### 5.2. Sensitivity Analysis for the NHA

_{1}= c

_{2}) are equal to 2, and the inertia weight is 0.7. Other accurate values for the other algorithms can be seen in Table 5, Table 6, Table 7 and Table 8. The population size for the SA is 30, and the velocity limit for this method is 4. The mutation and crossover probabilities are 0.70 and 0.60, respectively. The size populations for P

_{initial}and P

_{max}based on the WA are 10 and 30, respectively. Additionally, other parameters can be seen in Table 5, Table 6 and Table 7.

#### 5.3. Ten Random Results for Evolutionary Algorithms

#### 5.4. Computed Irrigation Deficiencies

^{6}m

^{3}), and the average amounts of released water for the NHA, SA, BA, WA, PSO algorithm and GA are 141.25, 140.33, 138.75, 135.43, 134.12 and 133.21 (10

^{6}m

^{3}), respectively. Thus, the NHA can supply the irrigation demand as a primary priority in this problem. The volumetric reliability, vulnerability and resiliency indexes were used for more detailed information and a deep comparative analysis of all implemented algorithms. The high percentage for the volumetric reliability index found for the NHA showed that irrigation demands can be supplied for more operation periods; therefore, the volume of released water can respond to downstream irrigation demands. In fact, the volumetric reliability index based on the NHA is 5%, 8%, 17%, 18% and 31% greater than that based on the SA, BA, WA, PSO algorithm and GA, respectively.

#### 5.5. Computational Power Production

^{6}kwh), and the average amount of produced power based on the NHA is 18.08 (10

^{6}kwh), while it is 17.99, 17.32, 16.96, 16.32, and 15.34 (10

^{6}kwh) for the SA, BA, WA, PSO algorithm, and GA, respectively (see Figure 6b). Thus, the NHA can produce more power to supply the demand (Table 10). Additionally, the correlation coefficient for the NHA is greater than that for other algorithms, and the root mean square error (RMSE) and mean absolute error (MAE) have the smallest values in the NHA among the evaluated algorithms based on the difference between demand and power production. Additionally, the NHA has a better performance than the MOGA and the MOPSO algorithms based on the lower values for the error indexes and higher correlation values.

## 6. Conclusions

^{6}m

^{3}), and the NHA can release 141.25 (10

^{6}m

^{3}), which represents a much higher level of accuracy over comparable models. The average demand for power production is 18.08 (10

^{6}kwh), and the produced power using the NHA is 17.99 (10

^{6}kwh), which represents the capability of the NHA for applied applications.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**New hybrid algorithm (NHA) diagram of the hybridized particle swarm optimization (PSO)–bat algorithm (BA) with a communication strategy.

**Figure 8.**Comparison of the fitness value and number of function evaluation (NFE) for different algorithms. GA—genetic algorithm.

**Figure 9.**(

**a**) Released water for downstream irrigation and (

**b**) power production for downstream demand.

Test Problem | Objective Function | Search Range | Optimum Value | Dimension | Characteristic | Acceptable Error (AE) |
---|---|---|---|---|---|---|

Schwefel function [52] | ${f}_{1}\left(x\right)={{\displaystyle \sum _{i=1}^{D}\left({\displaystyle \sum _{j=1}^{i}{x}_{j}}\right)}}^{2}$ | [−100, 100] | 0 | 30 | Unimodal | 1.0 × 10^{−3} |

Rastrigin [52] | ${f}_{2}\left(x\right)=10D+{\displaystyle \sum _{i=1}^{D}\left|{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)\right|}$ | [−5.12, 5.12] | 0 | 30 | Multimodal | 5.0 × 10^{−1} |

Dekkers and Aarts [52] | ${f}_{3}\left(x\right)={10}^{5}{x}_{1}^{2}+{x}_{2}^{2}-\left({x}_{1}^{2}+{x}_{2}^{2}\right)+{10}^{-5}{\left({x}_{1}^{2}+{x}_{2}^{2}\right)}^{4}$ | [−20,20] | −24,777 | 2 | Unimodal | 1.0 × 10^{−5} |

Step function [52] | ${f}_{4}\left(x\right)={{\displaystyle \sum _{i=1}^{D}\left(\left|{x}_{i}+0.5\right|\right)}}^{2}$ | [−100, 100] | 0 | 30 | Unimodal | 1.0 × 10^{−3} |

Axis parallel function [52] | ${f}_{5}\left(x\right)={\displaystyle \sum _{i=1}^{D}i{x}_{I}^{2}}$ | [−5.12, 5.12] | 0 | 30 | Unimodal | 1.0 × 10^{−5} |

Description | Quantity |
---|---|

Gross storage capacity | 2025 Mm^{3} |

Live storage capacity | 1784 Mm^{3} |

Dead storage capacity | 241 Mm^{3} |

Average annual inflow | 2845 Mm^{3} |

Left bank canal capacity | 10 m^{3}/s |

Right bank canal capacity | 71 m^{3}/s |

Left bank turbine capacity | 2000 kW |

Right bank turbine capacity (Phase2) | 13,200 kW |

Riverbed turbine capacity (Phase3) | 24,000 kW |

**Table 3.**Experimental results using benchmark functions. SD—standard deviation; ME—mean error; ANFE—average number of function evaluations; SR—success rate; NHA—new hybrid algorithm.

Function | Algorithms | SD | ME | ANFE | SR |
---|---|---|---|---|---|

f_{1} | Differential Evolution Algorithm | 1.42 × 10^{−4} [52] | 8.68 × 10^{−4} [52] | 27,378 [52] | 100 |

Artificial Bee Colony Algorithm | 2.02 × 10^{−4} [52] | 7.54 × 10^{−4} [52] | 35,091 [52] | 100 | |

Particle Swarm Optimization | 6.72 × 10^{−5} | 9.34 × 10^{−4} | 45,914.5 | 100 | |

Bat Algorithm | 5.12 × 10^{−5} | 6.12 × 10^{−4} | 231,245 | 100 | |

Shark Algorithm | 5.01 × 10^{−5} | 5.25 × 10^{−4} | 209,878 | 100 | |

Genetic Algorithm | 1.34 × 10^{−5} | 9.56 × 10^{−4} | 37,094 | 100 | |

Spider Monkey Algorithm | 2.12 × 10^{−6} [52] | 5.65 × 10^{−5} | 19,878 [52] | 100 | |

Krill Algorithm | 2.22 × 10^{−6} [52] | 7.12 × 10^{−5} | 18,235 [52] | 100 | |

NHA | 5.25 × 10^{−7} | 8.12 × 10^{−6} | 14,224 | 100 | |

f_{2} | Differential Evolution Algorithm | 4.93 [52] | 2.09 × 10^{−3} [53] | 200,000 [52] | 98 |

Artificial Bee Colony Algorithm | 3.14 × 10^{−4} [52] | 7.48 × 10^{−4} [53] | 87,039 [52] | 98 | |

Particle Swarm Optimization | 1.35 × 10^{+1} | 2.98 × 10^{−3} | 200,000 | 98 | |

Bat Algorithm | 3.24 × 10^{−5} | 3.12 × 10^{−5} | 54,223 | 98 | |

Shark Algorithm | 4.56 × 10^{−7} | 4.12 × 10^{−6} | 45,221 | 98 | |

Genetic Algorithm | 8.78 | 2.12 × 10^{−3} | 205,000 | 98 | |

Spider Monkey Algorithm | 6.12 × 10^{−8} [53] | 5.12 × 10^{−7} [53] | 32,124 [53] | 98 | |

Krill Algorithm | 7.91 × 10^{−7} [53] | 6.12 × 10^{−7} [53] | 35,125 [53] | 100 | |

NHA | 9.12 × 10^{−9} | 7.12 × 10^{−8} | 310,191 | 100 | |

f_{3} | Differential Evolution Algorithm | 1.12 × 10^{−3} | 4.09 × 10^{−1} | 2725.5 | 100 |

Artificial Bee Colony Algorithm | 5.25 × 10^{−3} | 4.09 × 10^{−1} | 2567 | 85 | |

Particle Swarm Optimization | 5.64 × 10^{−3} | 4.02 × 10^{−1} | 4979 | 85 | |

Bat Algorithm | 4.12 × 10^{−4} | 3.12 × 10^{−2} | 1285 | 85 | |

Shark Algorithm | 5.12 × 10^{−5} | 3.22 × 10^{−2} | 1100 | 98 | |

Genetic Algorithm | 1.12 × 10^{−2} | 4.12 × 10^{+1} | 1400 | 98 | |

Spider Monkey Algorithm | 5.78 × 10^{−5} | 2.12 × 10^{−4} | 987 | 98 | |

Krill Algorithm | 5.45 × 10^{−3} | 3.12 × 10^{−5} | 765 | 98 | |

NHA | 1.14 × 10^{−6} | 1.12 × 10^{−6} | 654 | 100 | |

f_{4} | Differential Evolution Algorithm | 1.12 × 10^{+2} | 2.19 × 10^{+1} | 180,000 | 84 |

Artificial Bee Colony Algorithm | 1.18 × 10^{+1} | 1.19 × 10^{+1} | 170,000 | 84 | |

Particle Swarm Optimization | 6.70 × 10^{+2} | 2.80 × 10^{−3} | 200,000 | 84 | |

Bat Algorithm | 5.70 × 10^{−3} | 1.12 × 10^{−4} | 180,000 | 84 | |

Shark Algorithm | 4.71 × 10^{−3} | 5.45 × 10^{−5} | 160,000 | 84 | |

Genetic Algorithm | 6.14 × 10^{+3} | 1.21 × 10^{−2} | 210,000 | 84 | |

Spider Monkey Algorithm | 1.45 × 10^{−4} | 3.12 × 10^{−5} | 180,000 | 84 | |

Krill Algorithm | 1.23 × 10^{−5} | 4.21 × 10^{−5} | 165,000 | 84 | |

NHA | 2.12 × 10^{−6} | 2.12 × 10^{−7} | 140,000 | 98 | |

f_{5} | Differential Evolution Algorithm | 1.31 × 10^{−6} | 4.90 × 10^{−1} | 2741 | 100 |

Artificial Bee Colony Algorithm | 2.00 × 10^{−6} | 4.87 × 10^{−1} | 4811 | 100 | |

Particle Swarm Optimization | 6.12 × 10^{−7} | 4.75 × 10^{−1} | 4912 | 100 | |

Bat Algorithm | 2.12 × 10^{−8} | 2.22 × 10^{−3} | 1811 | 100 | |

Shark Algorithm | 1.11 × 10^{−8} | 2.12 × 10^{−4} | 1712 | 100 | |

Genetic Algorithm | 1.21 × 10^{−5} | 3.21 × 10^{−4} | 5121 | 100 | |

Spider Monkey Algorithm | 2.12 × 10^{−8} | 5.12 × 10^{−3} | 1001 | 100 | |

Krill Algorithm | 1.14 × 10^{−8} | 5.45 × 10^{−4} | 987 | 100 | |

NHA | 1.41 × 10^{−9} | 6.78 × 10^{−5} | 567 | 100 |

Size Population | Objective Function | W (Inertia Coefficient) | Objective Function | c_{1} = c_{2} | Objective Function | Maximum Frequency | Objective Function | Minimum Loudness | Objective Function |
---|---|---|---|---|---|---|---|---|---|

10 | 2.45 | 0.30 | 2.21 | 1.6 | 2.34 | 1 | 2.11 | 0.3 | 2.23 |

30 | 2.24 | 0.50 | 2.00 | 1.8 | 2.12 | 2 | 2.00 | 0.5 | 2.05 |

50 | 1.98 | 0.70 | 1.98 | 2.0 | 1.98 | 3 | 2.14 | 0.7 | 2.0 |

70 | 2.01 | 0.90 | 2.12 | 2.2 | 2.12 | 4 | 2.16 | 0.90 | 2.1 |

Size Population | Objective Function | β_{k} (Velocity Limiter) | Objective Function | α_{k} | Objective Function |
---|---|---|---|---|---|

10 | 2.45 | 2 | 2.44 | 0.20 | 2.55 |

30 | 2.12 | 4 | 2.12 | 0.40 | 2.12 |

50 | 2.24 | 6 | 2.34 | 0.60 | 2.67 |

70 | 2.36 | 8 | 2.44 | 0.80 | 2.78 |

Pinitial | Objective Function | Pmax | Objective Function | N0Smax | Objective Function |
---|---|---|---|---|---|

5 | 3.69 | 10 | 3.55 | 3 | 3.78 |

10 | 3.12 | 30 | 3.12 | 5 | 3.34 |

15 | 3.24 | 50 | 3.28 | 7 | 3.12 |

20 | 3.36 | 70 | 3.32 | 9 | 3.44 |

Size Population | Objective Function | Mutation Probability | Objective Function | Crossover Probability | Objective Function |
---|---|---|---|---|---|

10 | 5.12 | 0.30 | 4.88 | 0.20 | 4.69 |

30 | 4.98 | 0.50 | 4.55 | 0.40 | 4.34 |

50 | 4.15 | 0.70 | 4.15 | 0.60 | 4.12 |

70 | 4.55 | 0.90 | 4.24 | 0.80 | 4.24 |

**Table 8.**Ten random results for the proposed hybrid evolutionary algorithm and the stand-alone algorithms.

Run | NHA | SA | BA | WA | PSO | GA |
---|---|---|---|---|---|---|

1 | 1.99 | 2.12 | 2.45 | 3.16 | 3.45 | 4.15 |

2 | 1.98 | 2.12 | 2.47 | 3.12 | 3.51 | 4.24 |

3 | 1.98 | 2.12 | 2.49 | 3.12 | 3.45 | 4.26 |

4 | 1.98 | 2.12 | 2.45 | 3.12 | 3.45 | 4.15 |

5 | 1.98 | 2.14 | 2.45 | 3.12 | 3.45 | 4.15 |

6 | 1.98 | 2.12 | 2.45 | 3.12 | 3.45 | 4.15 |

7 | 1.98 | 2.12 | 2.45 | 3.12 | 3.45 | 4.15 |

8 | 1.98 | 2.12 | 2.45 | 3.12 | 3.45 | 4.15 |

9 | 1.98 | 2.12 | 2.45 | 3.12 | 3.45 | 4.15 |

10 | 1.98 | 2.12 | 2.45 | 3.12 | 3.45 | 4.15 |

Average solution | 1.98 | 2.12 | 2.45 | 3.12 | 3.45 | 4.17 |

Coefficient variation | 0.001 | 0.002 | 0.005 | 0.004 | 0.005 | 0.006 |

Time | 50 | 70 | 79 | 83 | 91 | 94 |

**Table 9.**Evaluation of different algorithms for irrigation demands based on different indexes. NHA—new hybrid algorithm; SA—shark algorithm; BA—bat algorithm; WA—weed algorithm; PSO—particle swarm optimization; GA—genetic algorithm; MOGA—multi-objective GA; MOPSO—multi-objective PSO.

Index | Equation | NHA | SA | BA | WA | PSO | GA | MOGA | MOPSO |
---|---|---|---|---|---|---|---|---|---|

Correlation Coefficient | $r=\frac{{\displaystyle \sum _{t=1}^{T}\left({D}_{t}-{\overline{D}}_{t}\right).\left({R}_{t}-{\overline{R}}_{t}\right)}}{\sqrt{{\displaystyle \sum _{t=1}^{T}{\left({D}_{t}-{\overline{D}}_{t}\right)}^{2}.{\displaystyle \sum _{t=1}^{T}\left({R}_{t}-{\overline{R}}_{t}\right)}}}}$ | 0.93 | 0.91 | 0.86 | 0.87 | 0.75 | 0.67 | 0.74 | 0.83 |

Root Mean Square Error (RMSE) (10 ^{6} m^{3}) | $RMSE=\sqrt{\frac{{\displaystyle \sum _{t=1}^{T}{\left({D}_{t}-{R}_{t}\right)}^{2}}}{T}}$ | 5.1 | 7.2 | 8.8 | 9.3 | 10.5 | 11.8 | 9.6 | 8.7 |

Mean absolute Error (10 ^{6} m^{3}) | $MAE=\frac{{\displaystyle \sum _{t=1}^{T}\left|{D}_{t}-{R}_{t}\right|}}{T}$ | 4.3 | 5.59 | 6.1 | 7.1 | 6.9 | 6.4 | 6.3 | 6.1 |

Volumetric Reliability Index% | ${\alpha}_{V}=\frac{{\displaystyle \sum _{t=1}^{T}{R}_{t}}}{{\displaystyle \sum _{t=1}^{T}{D}_{t}}}\times 100$ | 95% | 90% | 87% | 78% | 75% | 64% | 77% | 79% |

Resiliency Index% | ${\gamma}_{i}=\frac{{f}_{si}}{{F}_{i}}$ | 45% | 40% | 38% | 35% | 33% | 29% | 35% | 34% |

Vulnerability Index | $\lambda =Ma{x}_{t=1}^{T}\left(\frac{{D}_{t}-{R}_{t}}{{D}_{t}}\right)\times 100$ | 14% | 20% | 21% | 23% | 24% | 25% | 22% | 21% |

_{t}: demand; ${\overline{D}}_{t}$: average demand; R

_{t}: released water; and ${\overline{R}}_{t}$: average released water.

Index | Equation | NHA | SA | BA | WA | PSO | GA | MOGA (Reddy, 2006) | MOPSO (Reddy, 2006) |
---|---|---|---|---|---|---|---|---|---|

Correlation Coefficient | $r=\frac{{\displaystyle \sum _{t=1}^{T}\left({P}_{dt}-{\overline{P}}_{dt}\right).\left({P}_{st}-{\overline{P}}_{st}\right)}}{\sqrt{{\displaystyle \sum _{t=1}^{T}{\left({P}_{dt}-{\overline{P}}_{obt}\right)}^{2}.{\displaystyle \sum _{t=1}^{T}\left({P}_{st}-{\overline{P}}_{st}\right)}}}}$ | 93% | 90% | 87% | 75% | 69% | 65% | 73% | 75% |

Root Mean Square Error (RMSE) (106 kwh) | $RMSE=\sqrt{\frac{{\displaystyle \sum _{t=1}^{T}{\left({P}_{obt}-{P}_{st}\right)}^{2}}}{T}}$ | 3.1 | 4.9 | 4.2 | 3.8 | 4.2 | 3.7 | 3.5 | 3.8 |

Mean Absolute Error (MAE) (106 kwh) | $MAE=\frac{{\displaystyle \sum _{t=1}^{T}\left|{P}_{obt}-{P}_{st}\right|}}{T}$ | 3.2 | 4.1 | 3.8 | 3.6 | 3.4 | 3.5 | 3.3 | 3.4 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yaseen, Z.M.; Ehteram, M.; Hossain, M.S.; Fai, C.M.; Binti Koting, S.; Mohd, N.S.; Binti Jaafar, W.Z.; Afan, H.A.; Hin, L.S.; Zaini, N.;
et al. A Novel Hybrid Evolutionary Data-Intelligence Algorithm for Irrigation and Power Production Management: Application to Multi-Purpose Reservoir Systems. *Sustainability* **2019**, *11*, 1953.
https://doi.org/10.3390/su11071953

**AMA Style**

Yaseen ZM, Ehteram M, Hossain MS, Fai CM, Binti Koting S, Mohd NS, Binti Jaafar WZ, Afan HA, Hin LS, Zaini N,
et al. A Novel Hybrid Evolutionary Data-Intelligence Algorithm for Irrigation and Power Production Management: Application to Multi-Purpose Reservoir Systems. *Sustainability*. 2019; 11(7):1953.
https://doi.org/10.3390/su11071953

**Chicago/Turabian Style**

Yaseen, Zaher Mundher, Mohammad Ehteram, Md. Shabbir Hossain, Chow Ming Fai, Suhana Binti Koting, Nuruol Syuhadaa Mohd, Wan Zurina Binti Jaafar, Haitham Abdulmohsin Afan, Lai Sai Hin, Nuratiah Zaini,
and et al. 2019. "A Novel Hybrid Evolutionary Data-Intelligence Algorithm for Irrigation and Power Production Management: Application to Multi-Purpose Reservoir Systems" *Sustainability* 11, no. 7: 1953.
https://doi.org/10.3390/su11071953