# Multi-Objective Operation of Cascade Hydropower Reservoirs Using TOPSIS and Gravitational Search Algorithm with Opposition Learning and Mutation

^{1}

^{2}

^{3}

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^{*}

## Abstract

**:**

## 1. Introduction

## 2. Enhanced Gravitational Search Algorithm (EGSA)

#### 2.1. Gravitational Search Algorithm (GSA)

_{j}is the random number uniformly distributed in the range of [0,1]. $Kbest$ is the number of agents with better fitness values.

_{i}is the random number uniformly distributed in the range of [0,1].

#### 2.2. Opposition Learning Strategy to Improve the Convergence Speed of the Swarm

^{1}for short) of x could be easily obtained by ${x}^{1}=a+b-x$. Based on previous references, when the optimal objective function value was unknown, the opposite directions of the current solution could increase the probability of finding better solutions [56,57,58]. The opposition learning strategy could enlarge the search region in the three-dimensional space. Thus, the modified opposition learning strategy based on the social learning mechanism in PSO is proposed to improve the convergence speed of the swarm, which could be expressed as below:

#### 2.3. Partial Mutation Strategy to Enhance the Individual Diversity

_{1}is the random number uniformly distributed in the range of [−0.5, 0.5].

#### 2.4. Elastic-Ball Modification Strategy to Promote Solution Feasibility

#### 2.5. Execution Procedure of the Proposed EGSA Method

- Step 1: Set the values of the computational parameters and then randomly generate the initial swarm in the problem space.
- Step 2: Calculate the fitness values of all the agents in the current population, and then update the personal best-known of each agent and the global best-known agent of the swarm.
- Step 3: Calculate the correlated variables (like the gravitational coefficient, mass, and acceleration) to update the velocity and position values of all the agents.
- Step 4: Execute the opposition learning strategy to increase the convergence speed of the swarm.
- Step 5: Execute the partial mutation search strategy to enhance the individual diversity.
- Step 6: Execute the elastic-ball modification strategy to promote the solution feasibility.
- Step 7: Repeat Step 2–6 until the stopping criterion is met, and then the global optimal position is regarded as the final solution of the optimization problem.

## 3. Numerical Experiments to Verify the Performance of the EGSA Method

#### 3.1. Benchmark Functions

_{min}is the optimal objective value in theory. For all test functions, the optimal value of F

_{8,}−418.9 × D, varies with the dimension; while the optimal values for other functions are 0. Meanwhile, the benchmark functions are divided into two categories—unimodal functions (F

_{1}–F

_{8}) with one global optimum, and multimodal functions (F

_{9}–F

_{12}) with multiple local optimums. The unimodal functions are used to test the convergence speed of the algorithm, while the multimodal function can test the ability of jumping out of the local optimum, which can fully verify the performance of the evolutionary methods.

#### 3.2. Parameters Settings

_{1}and c

_{2}) were set as 2.0, respectively.

#### 3.3. Comparison with Other Evolutionary Algorithms in Small-Scale Problems

#### 3.3.1. Result Comparison in Multiple Runs

_{5}, the WOA performance was tied with EGSA in F

_{9}and defeated by EGSA in the other functions. To sum up, the proposed method could obtain better results than the other evolutionary algorithms in the 12 test functions.

#### 3.3.2. Box and Whisker Test

_{4}–F

_{5}, which indicated that this method easily fell into the state of premature convergence in the search process; the SCA algorithm exhibited outliers in most functions but had a distribution of relatively discrete solutions in F

_{3}, which indicated that SCA was not ideal in terms of robustness. Additionally, the EGSA method had a more concentrated solution distribution and fewer outliers in all functions, demonstrating its satisfying robustness and search ability. Thus, the performances of the EGSA method were superior to the comparative methods in the 12 functions.

#### 3.3.3. Wilcoxon Nonparametric Test

^{+}than R

^{–}in all comparisons, while all values of p were smaller than 0.05. This case proved that the EGSA method outperformed the other methods in a statistical manner.

#### 3.3.4. Convergence Analysis

_{1}–F

_{4}as examples, three methods (GSA, PSO, and DE) quickly fail into local optima because their best-so-far solutions change slightly in the evolutionary process; both EGSA and SCA converge to a smaller objective at the initial stage, but EGSA can quickly find better solutions at iteration 800, while SCA fails to make it. Additionally, for multimodal functions, EGSAs achieve the global optimal solution in both F

_{9}and F

_{11}, and better results than other methods in F

_{8}, F

_{10}, and F

_{12}. Thus, the above analysis fully proves that the presented method has a superior convergence speed and global search ability.

## 4. EGSA for the Multi-Objective Operation of Cascade Hydropower Reservoirs

#### 4.1. Mathematical Model

#### 4.1.1. Objective Functions

#### 4.1.2. Physical Constraints

#### 4.2. Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS)

#### 4.3. Details of EGSA for Multi-Objective Operation of Cascade Hydropower Reservoirs

#### 4.3.1. Individual Structure and Swarm Initialization

#### 4.3.2. Heuristic Constraint Handling Method

#### 4.3.3. Calculation of the Modified Objective Values

^{’}and F

^{’}).

_{1}th inequality constraint; ${e}_{{l}_{2}}(\xb7)\le 0$ is the l

_{2}th equality constraint; ${c}_{{l}_{1}}$ and ${c}_{{l}_{2}}$ are the penalty coefficients for the relevant inequality or equality constraint. ${L}_{g}$ and ${L}_{e}$ denote the number of inequality and equality constraints, respectively.

#### 4.3.4. Execution Procedures of the EGSA Method for the Target Problem

## 5. Case Studies

#### 5.1. Engineering Background

#### 5.2. Case Study 1: Power Generation of Cascade Hydropower Reservoirs

#### 5.2.1. Robustness Testing of Different Evolutionary Algorithms

#### 5.2.2. Comparison of the Optimal Results Obtained by Different Evolutionary Algorithms

^{4}kW·h, 3.2 × 10

^{4}kW·h, 8.61 × 10

^{4}kW·h, and 8.5 × 10

^{4}kW·h improvements in power generation in Case 1, respectively. Additionally, among all hydroplants, the GPT hydroplant with a huge installed capacity produced the largest proportion of power generation in all solutions, which was in line with the actual operation situation of the hydropower system [80,81,82,83]. Thus, the above analysis demonstrated the effectiveness and rationality of the scheduling process obtained by the proposed method.

#### 5.2.3. Convergence Analysis of Different Evolutionary Algorithms

#### 5.3. Case Study 2: Peak Operation of Cascade Hydropower Reservoirs

#### 5.3.1. Robustness Testing of Different Evolutionary Algorithms

#### 5.3.2. Comparison of the Optimal Results Obtained by Different Evolutionary Algorithms

#### 5.3.3. Rationality Analysis of the Best Results Obtained by the Different Evolutionary Algorithms

#### 5.4. Case Study 3: Mutli-Objective Operation of Cascade Hydropower Reservoirs

#### 5.4.1. Comparative Analysis of the Optimal Results Obtained by the Different Methods with 100 Weights

_{1}for power generation was increased from 0 to 1.0 at the same interval of 0.01, while the weight w

_{2}for peak operation was set as 1.0 – w

_{1}. From Figure 11, it was observed that the total generation was gradually decreasing with the increasing objective value of peak operation, which implied that there was an obvious conflict between power generation and peak operation. Additionally, the solutions of PSO and GSA were dominated by that of the EGSA method, which meant that the EGSA method had a higher probability to obtain the Pareto optimal solutions than the PSO and GSA method. Thus, the proposed method can generate the near-optimal Pareto solutions to approximate the relationship between power generation and peak shaving in practice.

#### 5.4.2. Rationality Analysis of the Best Results Obtained by the Different Evolutionary Algorithms

## 6. Conclusions

- (1)
- Due to the loss of the population diversity, the conventional GSA method suffered from severe premature convergence shortcomings. The proposed method based on the three modified strategies (opposite learning strategy, partial mutation strategy and elastic-ball modification strategy) could effectively improve the convergence speed, swarm diversity, and solution feasibility of the standard GSA method, respectively.
- (2)
- For the original complex multi-objective optimization problem, balancing power generation and peak operation requirements, the famous TOPSIS method was used to transform it into the relatively simple single-objective problem, which could help make an obvious reduction in the modeling difficulty of the multi-objective decision.
- (3)
- There was a competitive relationship between the generation benefit of the hydropower enterprise and the peak operation of the power system. In other words, an increasing value of one objective would obviously reduce another objective value. Thus, it was necessary for operators to carefully determine the scheduling schemes so as to effectively balance the practical requirements of power generation enterprises and power grid companies.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Convergence trajectories of the different methods for 12 benchmark functions with 30 variables.

**Figure 4.**The flowchart of enhanced gravitational search algorithm (EGSA) for multi-objective operation of cascade hydropower reservoirs.

Function | D | Range | f_{min} |
---|---|---|---|

${F}_{1}(x)={\displaystyle \sum _{i=1}^{n}{x}_{i}{}^{2}}$ | 30 | [−100,100] | 0 |

${F}_{2}\left(x\right)={\displaystyle \sum _{i=1}^{n}\left|{x}_{i}\right|}+{\displaystyle \prod _{i=1}^{n}\left|{x}_{i}\right|}$ | 30 | [−10,10] | 0 |

${F}_{3}(x)={\displaystyle \sum _{i=1}^{n}({\displaystyle \sum _{j=1}^{i}{x}_{j}}}{)}^{2}$ | 30 | [−100,100] | 0 |

${F}_{4}\left(x\right)=\mathrm{max}\left\{\left|{x}_{i}\right|,1\le i\le n\right\}$ | 30 | [−100,100] | 0 |

${F}_{5}(x)={\displaystyle \sum _{i=1}^{n-1}[100{({x}_{i+1}^{}-{x}_{i}^{2})}^{2}+{({x}_{i}^{}-1)}^{2}]}$ | 30 | [−30,30] | 0 |

${F}_{6}(x)={\displaystyle \sum _{i=1}^{n}{({x}_{i}+0.5)}^{2}}$ | 30 | [−100,100] | 0 |

${F}_{7}(x)={\displaystyle \sum _{i=1}^{n}i{x}_{i}^{4}}+random[0,1)$ | 30 | [−1.28,1.28] | 0 |

${F}_{8}(x)={\displaystyle \sum _{i=1}^{n}-{x}_{i}}\mathrm{sin}(\sqrt{\left|{x}_{i}\right|})$ | 30 | [−500,500] | −418.9 × D |

${F}_{9}(x)={\displaystyle \sum _{i=1}^{n}[{x}_{i}{}^{2}-10\mathrm{cos}(2\pi {x}_{i})+10]}$ | 30 | [−5.12,5.12] | 0 |

$\begin{array}{l}{F}_{10}(x)=-20\mathrm{exp}(-0.2\sqrt[]{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}})-\mathrm{exp}(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\mathrm{cos}(2\pi {x}_{i}))}\\ \hspace{1em}\hspace{1em}\hspace{1em}+20+e\end{array}$ | 30 | [−32,32] | 0 |

${F}_{11}(x)=\frac{1}{4000}{\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}-{\displaystyle \prod _{i=1}^{n}\mathrm{cos}(\frac{{x}_{i}}{\sqrt{i}}}})+1$ | 30 | [−600,600] | 0 |

$\begin{array}{l}{F}_{12}\left(x\right)=\frac{\pi}{n}\{10{\mathrm{sin}}^{2}(\pi {y}_{1})+{\displaystyle \sum _{i=1}^{n-1}}{({y}_{i}-1)}^{2}[1+10{\mathrm{sin}}^{2}(\pi {y}_{i+1})]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{({y}_{n}-1)}^{2}\}+{\displaystyle \sum _{i=1}^{n}u({x}_{i}},10,100,4)\\ \hspace{1em}\hspace{1em}\hspace{1em}{y}_{i}=1+\frac{{x}_{i}+1}{4},u({x}_{i},a,k,m)=\{\begin{array}{l}k{({x}_{i}-a)}^{m}\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{x}_{i}>a\\ 0\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}a\le {x}_{i}<a\\ \hspace{1em}k{(-{x}_{i}-a)}^{m}\text{\hspace{1em}\hspace{1em}}{x}_{i}<-a\end{array}\end{array}$ | 30 | [−50,50] | 0 |

Function | Item | CS | MCS | LSA | GWO | FA | WOA | ALO | DE | PSO | SCA | GSA | EGSA |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

F_{1} | Ave. | 9.06 × 10^{1} | 1.01 × 10^{0} | 4.81 × 10^{−8} | 6.59 × 10^{−28} | 1.20 × 10^{−2} | 1.41 × 10^{−30} | 2.59 × 10^{−10} | 7.80 × 10^{−6} | 4.58 × 10^{−7} | 2.87 × 10^{−35} | 4.00 × 10^{−9} | 6.96 × 10^{−134} |

Std. | 2.62 × 10^{1} | 2.72 × 10^{−1} | 3.40 × 10^{−7} | 6.34 × 10^{−5} | 4.30 × 10^{−3} | 4.91 × 10^{−30} | 1.65 × 10^{−10} | 3.14 × 10^{−6} | 3.51 × 10^{−7} | 1.26 × 10^{−34} | 2.92 × 10^{−9} | 3.70 × 10^{−133} | |

F_{2} | Ave. | 9.70 × 10^{0} | 1.81 × 10^{−1} | 3.68 × 10^{−2} | 7.18 × 10^{−17} | 3.73 × 10^{−1} | 1.06 × 10^{−21} | 1.84 × 10^{−6} | 3.73 × 10^{−4} | 1.19 × 10^{−4} | 2.21 × 10^{−27} | 5.48 × 10^{−5} | 5.21 × 10^{−69} |

Std. | 1.98 × 10^{0} | 3.31 × 10^{−2} | 1.56 × 10^{−1} | 2.90 × 10^{−2} | 1.01 × 10^{−1} | 2.39 × 10^{−21} | 6.58 × 10^{−7} | 9.05 × 10^{−5} | 8.03 × 10^{−5} | 6.99 × 10^{−27} | 2.24 × 10^{−5} | 2.09 × 10^{−68} | |

F_{3} | Ave. | 3.84 × 10^{3} | 4.62 × 10^{2} | 4.32 × 10^{1} | 3.29 × 10^{−6} | 1.81 × 10^{3} | 5.39 × 10^{−7} | 6.06 × 10^{−10} | 2.94 × 10^{4} | 2.33 × 10^{0} | 9.74 × 10^{1} | 3.61 × 10^{2} | 4.30 × 10^{−119} |

Std. | 7.24 × 10^{2} | 1.23 × 10^{2} | 2.99 × 10^{1} | 7.91 × 10^{1} | 6.60 × 10^{2} | 2.93 × 10^{−6} | 6.34 × 10^{−10} | 4.13 × 10^{3} | 1.14 × 10^{0} | 2.24 × 10^{2} | 1.45 × 10^{2} | 2.29 × 10^{−118} | |

F_{4} | Ave. | 7.23 × 10^{0} | 1.73 × 10^{0} | 1.49 × 10^{0} | 5.61 × 10^{−7} | 7.67 × 10^{−2} | 7.26 × 10^{−2} | 1.36 × 10^{−8} | 1.43 × 10^{0} | 5.35 × 10^{−1} | 2.66 × 10^{0} | 9.48 × 10^{−2} | 5.58 × 10^{−70} |

Std. | 6.76 × 10^{−1} | 5.12 × 10^{−1} | 1.30 × 10^{0} | 1.32 × 10^{0} | 1.46 × 10^{−2} | 3.97 × 10^{−1} | 1.81 × 10^{−9} | 3.21 × 10^{−1} | 1.33 × 10^{−1} | 4.16 × 10^{0} | 3.92 × 10^{−1} | 2.12 × 10^{−69} | |

F_{5} | Ave. | 4.98 × 10^{0} | 5.81 × 10^{1} | 6.43 × 10^{1} | 2.68 × 10^{1} | 1.28 × 10^{2} | 2.79 × 10^{1} | 3.47 × 10^{−1} | 4.45 × 10^{2} | 1.64 × 10^{2} | 2.74 × 10^{1} | 3.45 × 10^{1} | 2.69 × 10^{1} |

Std. | 1.75 × 10^{0} | 3.31 × 10^{1} | 4.38 × 10^{1} | 6.99 × 10^{1} | 2.79 × 10^{2} | 7.64 × 10^{−1} | 1.10 × 10^{−1} | 1.55 × 10^{2} | 1.55 × 10^{2} | 3.85 × 10^{−1} | 3.65 × 10^{1} | 9.72 × 10^{−2} | |

F_{6} | Ave. | 4.31 × 10^{4} | 4.44 × 10^{3} | 3.34 × 10^{0} | 8.17 × 10^{−1} | 0 | 3.12 × 10^{0} | 2.56 × 10^{−10} | 7.94 × 10^{−6} | 9.29 × 10^{−7} | 5.30 × 10^{−2} | 4.99 × 10^{−9} | 8.23 × 10^{−15} |

Std. | 7.21 × 10^{3} | 7.52 × 10^{2} | 2.09 × 10^{0} | 1.26 × 10^{−4} | 0 | 5.32 × 10^{−1} | 1.09 × 10^{−10} | 3.39 × 10^{−6} | 1.93 × 10^{−6} | 4.98 × 10^{−2} | 3.74 × 10^{−9} | 4.61 × 10^{−15} | |

F_{7} | Ave. | 2.46 × 10^{−2} | 9.10 × 10^{−3} | 2.41 × 10^{−2} | 2.21 × 10^{−3} | 3.52 × 10^{−2} | 1.43 × 10^{−3} | 4.29 × 10^{−3} | 1.35 × 10^{−1} | 3.55 × 10^{0} | 7.48 × 10^{−3} | 3.05 × 10^{−2} | 4.76 × 10^{−4} |

Std. | 7.90 × 10^{−3} | 2.20 × 10^{−3} | 5.72 × 10^{−3} | 1.00 × 10^{−1} | 2.40 × 10^{−2} | 1.15 × 10^{−3} | 5.09 × 10^{−3} | 2.93 × 10^{−2} | 4.79 × 10^{0} | 6.96 × 10^{−3} | 1.59 × 10^{−2} | 2.32 × 10^{−4} | |

F_{8} | Ave. | −8.98 × 10^{3} | −9.80 × 10^{3} | −8.00 × 10^{3} | −6.12 × 10^{3} | −5.90 × 10^{3} | −5.08 × 10^{3} | −1.61 × 10^{3} | −7.95 × 10^{3} | −6.39 × 10^{3} | −5.66 × 10^{3} | −2.75 × 10^{3} | −1.19 × 10^{4} |

Std. | 1.98 × 10^{2} | 5.31 × 10^{2} | 6.69 × 10^{2} | 4.09 × 10^{3} | 6.56 × 10^{2} | 6.96 × 10^{2} | 3.14 × 10^{2} | 3.33 × 10^{2} | 1.29 × 10^{3} | 3.52 × 10^{2} | 4.00 × 10^{2} | 3.13 × 10^{2} | |

F_{9} | Ave. | 2.94 × 10^{2} | 1.35 × 10^{2} | 6.28 × 10^{1} | 3.11 × 10^{−1} | 2.63 × 10^{1} | 0 | 7.71 × 10^{−6} | 1.32 × 10^{2} | 7.87 × 10^{1} | 1.87 × 10^{−9} | 1.67 × 10^{1} | 0 |

Std. | 1.43 × 10^{1} | 2.16 × 10^{1} | 1.49 × 10^{1} | 4.74 × 10^{1} | 9.15 × 10^{0} | 0 | 8.45 × 10^{−6} | 8.78 × 10^{0} | 2.91 × 10^{1} | 1.02 × 10^{−8} | 3.82 × 10^{0} | 0 | |

F_{10} | Ave. | 1.93 × 10^{1} | 1.21 × 10^{1} | 2.69 × 10^{0} | 1.06 × 10^{−13} | 5.12 × 10^{−2} | 7.40 × 10^{0} | 3.73 × 10^{−15} | 1.37 × 10^{−3} | 5.99 × 10^{−4} | 1.34 × 10^{1} | 3.25 × 10^{−5} | 3.64 × 10^{−15} |

Std. | 3.50 × 10^{−1} | 7.52 × 10^{−1} | 9.11 × 10^{−1} | 7.78 × 10^{−2} | 1.37 × 10^{−2} | 9.90 × 10^{0} | 1.50 × 10^{−15} | 8.85 × 10^{−4} | 5.43 × 10^{−4} | 9.64 × 10^{0} | 1.10 × 10^{−5} | 1.08 × 10^{−15} | |

F_{11} | Ave. | 2.12 × 10^{2} | 8.32 × 10^{0} | 7.24 × 10^{−3} | 4.49 × 10^{−3} | 5.84 × 10^{−3} | 2.89 × 10^{−4} | 1.86 × 10^{−2} | 3.36 × 10^{−3} | 6.98 × 10^{−3} | 1.86 × 10^{−5} | 4.34 × 10^{0} | 0 |

Std. | 3.97 × 10^{1} | 1.54 × 10^{0} | 6.70 × 10^{−3} | 6.66 × 10^{−3} | 1.43 × 10^{−3} | 1.59 × 10^{−3} | 9.55 × 10^{−3} | 1.06 × 10^{−2} | 8.16 × 10^{−3} | 1.02 × 10^{−4} | 1.66 × 10^{0} | 0 | |

F_{12} | Ave. | 1.47 × 10^{0} | 1.38 × 10^{−1} | 3.58 × 10^{−1} | 5.34 × 10^{−2} | 2.40 × 10^{−4} | 3.40 × 10^{−1} | 9.74 × 10^{−12} | 6.87 × 10^{−1} | 2.01 × 10^{−2} | 8.52 × 10^{−3} | 9.59 × 10^{−2} | 5.30 × 10^{−17} |

Std. | 3.61 × 10^{−1} | 2.86 × 10^{−1} | 7.44 × 10^{−1} | 2.07 × 10^{−2} | 1.00 × 10^{−4} | 2.15 × 10^{−1} | 9.33 × 10^{−12} | 3.05 × 10^{−1} | 2.26 × 10^{−2} | 4.69 × 10^{−3} | 1.75 × 10^{−1} | 1.92 × 10^{−17} |

**Table 3.**The results of Wilcoxon test for a single problem for statistical significance level at alpha = 0.05.

Function | EGSA–CS | EGSA–MCS | EGSA–LSA | EGSA–GWO | EGSA–FA | EGSA–WOA | EGSA–ALO | EGSA–DE | EGSA–PSO | EGSA–SCA | EGSA–GSA |
---|---|---|---|---|---|---|---|---|---|---|---|

F_{1} | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA |

F_{2} | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA |

F_{3} | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA |

F_{4} | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA |

F_{5} | CS | EGSA | EGSA | GWO | EGSA | EGSA | ALO | EGSA | EGSA | EGSA | EGSA |

F_{6} | EGSA | EGSA | EGSA | EGSA | FA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA |

F_{7} | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA |

F_{8} | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA |

F_{9} | EGSA | EGSA | EGSA | EGSA | EGSA | Tie | EGSA | EGSA | EGSA | EGSA | EGSA |

F_{10} | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA |

F_{11} | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA |

F_{12} | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA | EGSA |

W/T/L | 11/0/1 | 12/0/0 | 12/0/0 | 11/0/1 | 11/0/1 | 11/1/0 | 11/0/1 | 12/0/0 | 12/0/0 | 12/0/0 | 12/0/0 |

**Table 4.**The results of Wilcoxon signed-rank test for statistically significance level at alpha = 0.05.

Item | EGSA–CS | EGSA–MCS | EGSA–LSA | EGSA–GWO | EGSA–FA | EGSA–WOA | EGSA–ALO | EGSA–DE | EGSA–PSO | EGSA–SCA | EGSA–GSA |
---|---|---|---|---|---|---|---|---|---|---|---|

R^{+} | 72 | 78 | 78 | 70 | 77 | 77.5 | 67 | 78 | 78 | 78 | 78 |

R^{−} | 6 | 0 | 0 | 8 | 1 | 0.5 | 11 | 0 | 0 | 0 | 0 |

p-value | 6.84 × 10^{−3} | 4.88 × 10^{−4} | 4.88 × 10^{−4} | 1.22 × 10^{−2} | 9.77 × 10^{−4} | 9.77 × 10^{−4} | 2.69 × 10^{−2} | 4.88 × 10^{−4} | 4.88 × 10^{−4} | 4.88 × 10^{−4} | 4.88 × 10^{−4} |

Significant | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ |

**Table 5.**Statistical results of the 5 methods in 20 independent runs for 4 cases of power generation (10

^{4}kW·h).

Case | Method | Best | Worst | Average | Standard Deviation | Range |
---|---|---|---|---|---|---|

Case 1 | DE | 5388.10 | 5386.25 | 5387.19 | 0.56 | 1.85 |

PSO | 5393.67 | 5391.27 | 5392.49 | 0.57 | 2.40 | |

SCA | 5388.26 | 5384.34 | 5385.65 | 0.89 | 3.92 | |

GSA | 5388.37 | 5385.34 | 5386.76 | 0.88 | 3.03 | |

EGSA | 5396.87 | 5396.81 | 5396.86 | 0.02 | 0.06 | |

Case 2 | DE | 6439.12 | 6434.22 | 6436.14 | 1.43 | 4.90 |

PSO | 6445.48 | 6442.35 | 6444.12 | 0.79 | 3.13 | |

SCA | 6439.06 | 6435.25 | 6437.07 | 1.03 | 3.81 | |

GSA | 6440.17 | 6437.00 | 6438.55 | 0.79 | 3.17 | |

EGSA | 6450.34 | 6450.33 | 6450.33 | 0.01 | 0.01 | |

Case3 | DE | 4977.94 | 4976.38 | 4977.17 | 0.40 | 1.56 |

PSO | 4981.62 | 4979.71 | 4980.58 | 0.55 | 1.91 | |

SCA | 4975.51 | 4973.58 | 4974.54 | 0.58 | 1.93 | |

GSA | 4976.74 | 4973.83 | 4975.25 | 0.71 | 2.91 | |

EGSA | 4984.56 | 4984.54 | 4984.55 | 0.01 | 0.02 | |

Case4 | DE | 4433.08 | 4431.68 | 4432.32 | 0.44 | 1.40 |

PSO | 4435.89 | 4434.23 | 4435.11 | 0.49 | 1.66 | |

SCA | 4431.51 | 4428.62 | 4429.58 | 0.70 | 2.89 | |

GSA | 4431.61 | 4429.07 | 4430.37 | 0.64 | 2.54 | |

EGSA | 4438.33 | 4438.32 | 4438.33 | 0.01 | 0.01 |

Runoff | Method | HJD | DF | SFY | WJD | GPT | Sum |
---|---|---|---|---|---|---|---|

Case 1 | DE | 726.73 | 646.19 | 527.41 | 1106.65 | 2381.12 | 5388.10 |

PSO | 728.66 | 647.76 | 526.51 | 1107.99 | 2382.75 | 5393.67 | |

SCA | 727.92 | 644.60 | 527.68 | 1106.77 | 2381.29 | 5388.26 | |

GSA | 728.35 | 647.35 | 526.77 | 1107.15 | 2378.75 | 5388.37 | |

EGSA | 728.84 | 646.94 | 530.33 | 1108.10 | 2382.66 | 5396.87 | |

Case 2 | DE | 869.46 | 770.94 | 632.25 | 1322.47 | 2844.00 | 6439.12 |

PSO | 871.70 | 774.09 | 627.59 | 1325.38 | 2846.72 | 6445.48 | |

SCA | 871.15 | 771.70 | 628.34 | 1322.76 | 2845.11 | 6439.06 | |

GSA | 871.48 | 774.01 | 627.42 | 1324.03 | 2843.23 | 6440.17 | |

EGSA | 871.58 | 773.17 | 633.84 | 1324.91 | 2846.84 | 6450.34 | |

Case 3 | DE | 234.57 | 400.46 | 461.27 | 1052.33 | 2829.31 | 4977.94 |

PSO | 235.21 | 402.56 | 459.19 | 1054.13 | 2830.53 | 4981.62 | |

SCA | 233.99 | 401.01 | 459.94 | 1052.01 | 2828.56 | 4975.51 | |

GSA | 235.02 | 402.45 | 457.71 | 1053.20 | 2828.36 | 4976.74 | |

EGSA | 235.50 | 401.67 | 462.62 | 1054.23 | 2830.54 | 4984.56 | |

Case 4 | DE | 208.46 | 355.81 | 411.02 | 936.99 | 2520.8 | 4433.08 |

PSO | 208.58 | 358.21 | 409.24 | 938.46 | 2521.4 | 4435.89 | |

SCA | 208.26 | 356.74 | 409.21 | 937.08 | 2520.22 | 4431.51 | |

GSA | 208.82 | 357.66 | 408.43 | 937.47 | 2519.23 | 4431.61 | |

EGSA | 209.13 | 357.35 | 411.88 | 938.51 | 2521.46 | 4438.33 |

Season | Item | Best | Worst | Average | Standard Deviation | Range |
---|---|---|---|---|---|---|

Spring | DE | 32,130.69 | 32,183.80 | 32,153.90 | 16.22 | 53.11 |

PSO | 32,052.91 | 32,127.22 | 32,084.17 | 20.24 | 74.31 | |

SCA | 32,160.67 | 32,230.89 | 32,194.00 | 22.32 | 70.22 | |

GSA | 32,173.70 | 32,224.22 | 32,199.76 | 14.41 | 50.52 | |

EGSA | 31,986.47 | 31,986.62 | 31,986.53 | 0.04 | 0.15 | |

Summer | DE | 33,006.63 | 33,093.05 | 33,051.56 | 23.51 | 86.42 |

PSO | 32,962.41 | 33,057.03 | 32,993.63 | 22.72 | 94.62 | |

SCA | 33,032.40 | 33,129.77 | 33,100.08 | 24.05 | 97.37 | |

GSA | 33,069.24 | 33,134.56 | 33,097.29 | 19.83 | 65.32 | |

EGSA | 32,888.05 | 32,888.16 | 32,888.10 | 0.03 | 0.11 | |

Autumn | DE | 36,681.47 | 36,752.01 | 36,719.49 | 17.51 | 70.54 |

PSO | 36,595.43 | 36,714.71 | 36,667.14 | 30.32 | 119.28 | |

SCA | 36,745.91 | 36,840.43 | 36,781.14 | 24.73 | 94.52 | |

GSA | 36,738.72 | 36,835.34 | 36,802.57 | 27.25 | 96.62 | |

EGSA | 36,536.95 | 36,537.06 | 36,537.00 | 0.03 | 0.11 | |

Winter | DE | 35,825.89 | 35,890.14 | 35,857.52 | 14.79 | 64.25 |

PSO | 35,762.63 | 35,850.26 | 35,811.97 | 23.97 | 87.63 | |

SCA | 35,822.72 | 35,950.51 | 35,896.25 | 32.32 | 127.79 | |

GSA | 35,879.31 | 35,965.39 | 35,931.63 | 22.63 | 86.08 | |

EGSA | 35,671.70 | 35,671.80 | 35,671.74 | 0.03 | 0.10 |

Season | Method | Item | Peak | Valley | Peak-valley | Average | Standard Deviation |
---|---|---|---|---|---|---|---|

Spring | Original | 13,477.93 | 10,101.60 | 3376.33 | 11,910.93 | 1281.95 | |

DE | Optimization | 10,952.90 | 7716.77 | 3236.13 | 9242.92 | 789.32 | |

Reduction | 2525.03 | 2384.83 | 140.20 | 2668.01 | 492.63 | ||

PSO | Optimization | 10,050.98 | 8169.90 | 1881.08 | 9231.64 | 640.11 | |

Reduction | 3426.95 | 1931.7 | 1495.25 | 2679.29 | 641.84 | ||

SCA | Optimization | 10,752.18 | 7809.30 | 2942.88 | 9239.61 | 926.13 | |

Reduction | 2725.75 | 2292.3 | 433.45 | 2671.32 | 355.82 | ||

GSA | Optimization | 10,737.30 | 7592.29 | 3145.01 | 9229.75 | 1058.60 | |

Reduction | 2740.63 | 2509.31 | 231.32 | 2681.18 | 223.35 | ||

EGSA | Optimization | 9428.01 | 8987.22 | 440.79 | 9232.28 | 165.13 | |

Reduction | 4049.92 | 1114.38 | 2935.54 | 2678.65 | 1116.82 | ||

Summer | Original | 14,119.78 | 10,342.98 | 3776.80 | 12,170.94 | 1302.21 | |

DE | Optimization | 11,151.13 | 8535.00 | 2616.13 | 9505.54 | 670.54 | |

Reduction | 2968.65 | 1807.98 | 1160.67 | 2665.40 | 631.67 | ||

PSO | Optimization | 10,610.47 | 8392.24 | 2218.23 | 9492.57 | 673.27 | |

Reduction | 3509.31 | 1950.74 | 1558.57 | 2678.37 | 628.94 | ||

SCA | Optimization | 10,749.92 | 7591.09 | 3158.83 | 9500.12 | 839.87 | |

Reduction | 3369.86 | 2751.89 | 617.97 | 2670.82 | 462.34 | ||

GSA | Optimization | 11,379.10 | 7774.71 | 3604.39 | 9493.04 | 1028.32 | |

Reduction | 2740.68 | 2568.27 | 172.41 | 2677.90 | 273.89 | ||

EGSA | Optimization | 9759.71 | 9234.99 | 524.72 | 9492.46 | 172.66 | |

Reduction | 4360.07 | 1107.99 | 3252.08 | 2678.48 | 1129.55 | ||

Autumn | Original | 14,773.84 | 10,786.00 | 3987.84 | 13,220.16 | 1528.83 | |

DE | Optimization | 12,615.24 | 8928.53 | 3686.71 | 10,553.86 | 880.28 | |

Reduction | 2158.6 | 1857.5 | 301.1 | 2666.3 | 648.6 | ||

PSO | Optimization | 11,234.24 | 9366.35 | 1867.89 | 10,547.10 | 613.62 | |

Reduction | 3539.6 | 1419.65 | 2119.95 | 2673.06 | 915.21 | ||

SCA | Optimization | 11,891.57 | 8142.23 | 3749.34 | 10,552.90 | 1099.32 | |

Reduction | 2882.27 | 2643.77 | 238.50 | 2667.26 | 429.51 | ||

GSA | Optimization | 12,649.33 | 8347.14 | 4302.19 | 10,541.04 | 1193.12 | |

Reduction | 2124.51 | 2438.86 | −314.35 | 2679.12 | 335.71 | ||

EGSA | Optimization | 10,760.16 | 10,182.25 | 577.91 | 10,545.07 | 221.99 | |

Reduction | 4013.68 | 603.75 | 3409.93 | 2675.09 | 1306.84 | ||

Winter | Original | 14,913.26 | 11,028.24 | 3885.02 | 12,971.28 | 1489.76 | |

DE | Optimization | 11,709.53 | 8772.72 | 2936.81 | 10,303.38 | 955.49 | |

Reduction | 3203.73 | 2255.52 | 948.21 | 2667.90 | 534.27 | ||

PSO | Optimization | 11,884.52 | 8637.32 | 3247.20 | 10,293.13 | 970.91 | |

Reduction | 3028.74 | 2390.92 | 637.82 | 2678.15 | 518.85 | ||

SCA | Optimization | 12,391.09 | 8667.77 | 3723.32 | 10,301.54 | 1052.86 | |

Reduction | 2522.17 | 2360.47 | 161.70 | 2669.74 | 436.90 | ||

GSA | Optimization | 11,946.42 | 8414.17 | 3532.25 | 10,293.50 | 1174.01 | |

Reduction | 2966.84 | 2614.07 | 352.77 | 2677.78 | 315.75 | ||

EGSA | Optimization | 10,568.93 | 10,012.57 | 556.36 | 10,295.50 | 208.98 | |

Reduction | 4344.33 | 1015.67 | 3328.66 | 2675.78 | 1280.78 |

© 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**

Feng, Z.-k.; Liu, S.; Niu, W.-j.; Jiang, Z.-q.; Luo, B.; Miao, S.-m. Multi-Objective Operation of Cascade Hydropower Reservoirs Using TOPSIS and Gravitational Search Algorithm with Opposition Learning and Mutation. *Water* **2019**, *11*, 2040.
https://doi.org/10.3390/w11102040

**AMA Style**

Feng Z-k, Liu S, Niu W-j, Jiang Z-q, Luo B, Miao S-m. Multi-Objective Operation of Cascade Hydropower Reservoirs Using TOPSIS and Gravitational Search Algorithm with Opposition Learning and Mutation. *Water*. 2019; 11(10):2040.
https://doi.org/10.3390/w11102040

**Chicago/Turabian Style**

Feng, Zhong-kai, Shuai Liu, Wen-jing Niu, Zhi-qiang Jiang, Bin Luo, and Shu-min Miao. 2019. "Multi-Objective Operation of Cascade Hydropower Reservoirs Using TOPSIS and Gravitational Search Algorithm with Opposition Learning and Mutation" *Water* 11, no. 10: 2040.
https://doi.org/10.3390/w11102040