# Reservoir Operation Management with New Multi-Objective (MOEPO) and Metaheuristic (EPO) Algorithms

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## Abstract

**:**

_{ba}) and logical operators (EPO

_{ad}) modified HR and SOP policies. Multi-Objective EPO (MPOEPO) and GEP with trigonometric functions were used to create a multi-objective policies formula. The results showed that the generation of the operation rules with EPO

_{ad}increased the dam reservoir Performance Indexes (Vulnerability and Reliability Indexes) compared to EPO

_{ba}. Moreover, HR application compared to SOP improves the mean dam reservoir’s Performance Indexes by about 12 and 33% in the baseline and 12 and 21% in the future period (climate change conditions), respectively. The MOO method (MOEPO) improved the Vulnerability and Reliability Indexes by about 36 and 25% in the baseline and by 31 and 26% in the future, respectively, compared to SOP.

## 1. Introduction

- Investigating the reservoir Performance Indexes through the change of operation policies (HR, SOP, and multi-objective optimization)
- Comparing operation policies generated by GEP logical and arithmetic operators
- Developing appropriate policies for the future period.

## 2. Materials and Methods

#### 2.1. Study Area and Input Data

#### 2.1.1. Baseline and Future Temperature and Precipitation

#### 2.1.2. Simulate Inflow to Dam Reservoir

#### 2.1.3. Future Agriculture and Domestic Demands

#### 2.2. Research Models

#### 2.2.1. Reservoir Operation Using Standard Operation Policy (SOP)

- MAE: mean absolute errors as objective function
- RSPt: total demand per month
- rspt: total output based on SOP (observational) in t period.

#### 2.2.2. Reservoir Operation Using Hedging Rule (HR)

_{p}) are obtained using the EPO optimization algorithm based on LSR minimization. These coefficients are the slope of the operation line in a given month (see Appendix A.2 for more details).

#### 2.2.3. Multi-Objective Optimization

#### 2.2.4. Gene Expression Programming

#### 2.2.5. Emperor Penguin Optimization (EPO) and Multi-Objective Emperor Penguin Optimization (MOEPO)

#### 2.2.6. Reservoir Operation Rule Generation Using EPO, MOEPO Algorithms and GEP

_{ad}) and elementary arithmetic operators (EPO

_{ba}) and the coupling of MOEPO with GEP by trigonometric functions. It is worth mentioning that the GEP model performs better in producing output formulas related to the HR and SOP, except for the four elementary arithmetic operators (×, ÷, +, −), there are several other operators such as the multi-criteria function (≤, ≥, <, >) operators of logical functions (if, and) and Boolean function were considered as logical operators. This condition is known as modified GEP in this study.

- The first scenario, development of baseline rules based on the volume of available water in the reservoir using EPO
_{ba}in the baseline condition. - The second scenario, development of baseline rules based on the volume of available water in the reservoir using the EPO
_{ad}in the baseline condition. - The third scenario, development of future rules based on the volume of available water in the reservoir using the EPO
_{ba}under future condition. - The fourth scenario, development of future rules based on the volume of available water in the reservoir using the EPO
_{ad}under future condition.

#### 2.3. Vulnerability and Reliability Indexes

- D
_{t}: Demand volume in the t period - D
_{max}: Maximum demand in the under-review period. - Re
_{t}: the released volume from the reservoir in the t period.

## 3. Results

#### 3.1. Integrate SOP and HR Using EPO_{ad} and EPO_{ba}

#### 3.1.1. Validation of the SOP Simulation with EPO

_{ad}and EPO

_{ba}algorithms. According to the Figure, the convergence rate of EPO

_{ad}and EPO

_{ba}are almost the same; both algorithms reach the final result of the objective function after about 400 iterations; EPO

_{ad}reaches the objective function of 0.32 while the EPO

_{ba}reaches the objective function of 0.75.

_{ad}and EPO

_{ba}algorithms are presented in Table 1. It shows the higher accuracy of EPO

_{ad}in simulating the SOP.

_{ba}and EPO

_{ad}approaches with the minimum objective function value for the baseline period are presented.

- RSP
_{t}: Total released water based on Reservoir System Policy in the t period - AW
_{t}: Available water in the dam reservoir (in the t period).

_{ad}algorithm will improve the performance in the baseline and future conditions.

_{ad}algorithm has a better performance in simulating the SOP, only the EPOad results (after this, referred to as the “SOP”) were used in the continuation.

#### 3.1.2. Validation of the HR Simulation with EPO

_{ad}and EPO

_{ba}algorithms are shown in Figure 5 and Table 3. As can be seen in the Figure, the EPO

_{ad}by objective function of 0.87 compare with EPO

_{ba}by objective function of 0.98 has a better performance in minimizing the MAE index. Examination of the results of the implementation of the HR by the algorithms shows that when using the EPO

_{ad}algorithm in simulating, the total output from the reservoir is more balanced with the demand, so the less objective function is obtained. Comparing the results in Table 2 also shows that the EPO

_{ba}outputs are more similar to the SOP.

_{ba}and EP

_{ad}approach with the minimum objective function value (see Equation (A6) for more details) for the baseline are presented.

#### 3.1.3. Comparison of the Results of HR and SOP in Extracting Allocation Rules in Four Scenarios

_{ad}diagram, the storage volume for 75% of the study period is more than 49 MCM and less than 120 MCM. The released volume from the reservoir for 25% of the study period is less than 20 MCM and more than 42 MCM. For 75% of the study period, the deficiencies amount is less than 1.2 MCM. Corresponding to the EPO

_{ba}diagram, the storage magnitude for 75% of the study period is more than 96 MCM and less than 171 MCM. For 50% of the study period, the released volume is between 22 and 40 MCM. For 75% of the study period, the amount of deficiencies is less than 1.8 MCM. According to the SOP diagram, for 50% of the months, the reservoir storage is about 92–41 MCM. The released volume for 75% of the study period is more than 18 MCM and less than 44 MCM. For 75% of the period, the deficit is less than 2.2 MCM.

_{ad}in Figure 7, the storage is over 33 MCM and less than 65 MCM in 75% of the study period. For 50% of the study period, the released volume is between 38 and 77 MCM. There are fewer than 1.9 MCM of deficiencies in 75% of the months. Based on EPO

_{ba}, in 75% of the study period, the storage is less than 68 MCM. For 75% of the months, the released water from the dam reservoir is less than 30 MCM. Deficiencies are less than 4.4 MCM in 75% of months. According to the SOP diagram, 50% of the time, the storage volume is between 12 and 30 MCM. In 75% of the months, the dam reservoir’s release is greater than 11 MCM and less than 29 MCM. Deficiencies are less than 8.2 MCM in 75% of months. As shown in part (a) of Figure 6 and Figure 7, the EPO

_{ad}performed better EPO

_{ba}in extracting the total output rule from the reservoir (total discharge, deficit and storage volume from the reservoir) in the baseline and future.

_{ad}, has increased the reliability index in the baseline by 11% and decreased the vulnerability index (improvement) by 40% compared to the EPO

_{ba}. These changes (using EPO

_{ad}instead of EPO

_{ba}algorithm) in future conditions (third and fourth scenarios) were 12% increasing the Reliability and 5% decreasing the vulnerability indexes. The results also show in the second and fourth scenarios (EPO

_{ad}) the reliability and vulnerability indexes will decline by 43% and increase by 44%, respectively. In other words, the status of reservoir performance indexes in the future will be much worse than the baseline.

#### 3.2. Comparison of the Results of Multi-Objective Optimization (MOEPO) and SOP in Extracting Allocation Rules in Four Scenarios

#### Operation Rules and Three Considered Scenarios

## 4. Discussion

## 5. Conclusions

_{ad}and EPO

_{ba}, was used for the simulation of this part. The validation results of algorithms in the extraction of the SOP indicated an appropriate performance of the EPO

_{ad}. In other words, the EPO

_{ad}improved the objective function by 57% over the EPO

_{ba}in reconstructing the SOP and decreased RMSE by 1.27%, and NS increased by 4% compared to the EPO

_{ba}.

_{ad}and EPO

_{ba}approach, it was used to derive the HR. The result of this part is also the EPO

_{ad}will improve the objective function by 15% compared to the EPO

_{ba}approach. Next, the optimal allocation rules (storge and deficit volume changes) based on EPO

_{ad}and EPO

_{ba}approaches for baseline and futures were compared in four scenarios. The results indicated the higher performance of the EPO

_{ad}.

_{ad}resulted in an 11% increase in reliability, a 40% decrease in vulnerability indexes in the baseline. Additionally, in the third and fourth scenarios, the use of EPO

_{ad}increased the reliability index by 12% and reduced the vulnerability index by 5%. The results showed that in the second and fourth scenarios using EPO

_{ad}(future period), the reliability and vulnerability indexes in the future compared to the baseline will increase by 43% and decrease by 44%, respectively. Meanwhile, in the first and third scenarios, the above indexes will decrease by 44% and increase by 26%, respectively. In other words, the status of Reservoir Performance Indexes in both algorithms declined in future compared to the baseline period.

- The boolean function increased the accuracy and performance of the generated allocation rules.
- The multi-objective optimization policy, SOP, and HR were classified from the most to the least based on improving the Performance Indexes.
- To increase the performance of the dam reservoir, it is necessary to generate particular management policies for each interval.
- The suggestions for future study are:
- Comparing this Metaheaustric algorithm with other well-known in terms of solving time consumption, convergence, etc.
- Investigating the Agriculture adaptation strategies (Deficit Irrigation, Changing cultivation date, etc.) in improving the system performance.
- Investigating other decision variables in Performance Indexes.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Standard Operation Policy

- AW
_{t}: Available water volume during period t - S
_{t}: Reservoir storage volume in the t period - Q
_{t}: Inflow volume during the period t

- E
_{t}: Evaporation depth from the surface of the reservoir during the t period - A
_{t}and A_{t+1}: the reservoir surface areas at the beginning and end of the t th period, which use Equations (A3) and (A4), respectively.$${A}_{t}={a}_{0}+{a}_{1}{S}_{t}\forall t=1,2,\dots ,T$$$${A}_{t+1}={a}_{0}+{a}_{1}{S}_{t+1}\forall t=1,2,\dots ,T$$_{0}and a_{1}are constant coefficients of the surface-volume curve of the reservoir.

- rsp
_{t}: total output based on SOP (observational) in t period - D: the average volume of demand over the entire period of operation.
- S
_{max}: the maximum volume or reservoir capacity (constant number).

#### Appendix A.2. Hedging Rule

#### Appendix A.2.1. Objective Function

- LSR: Long-term Shortage Ratio (as objective function)
- RSPH
_{t}: total output (sum of release and overflow) based on the HR in t period. - D
_{t}: the demand in t. - D
_{max}: the highest demand during t.

#### Appendix A.2.2. Constraint

- S
_{t}: the reservoir storage in t. - S
_{min}: reservoir dead volume.

#### Appendix A.3. Multi Objective Optimization of Dam Reservoir Operation

- F(u
_{1}): Objective function related to the vulnerability index - F(u
_{2}): Objective function related to the reliability index - D
_{t}: Demand volume in the t period - D
_{max}: Maximum demand in the under-review period. - $R{e}_{t}$: the released volume from the reservoir in t period.

_{min}is the minimum volume or dead volume of the reservoir, which can be obtained from the continuity equation (Equation (A19)). In this equation, time steps are considered monthly.

_{t+}

_{1}and S

_{t}are reservoir storage volumes at the beginning and end of t and t + 1 periods, respectively, Q

_{t}is amount of inflow volume to the reservoir during the t period, R

_{et}is the volume of release from the reservoir during the t period, SP

_{t}is the amount of overflow volume from the reservoir at the beginning of the t period (Equation (A20)) and LE

_{t}is the volume of losses due to evaporation from reservoir surface during t period.

_{max}is the maximum volume of reservoir capacity and S

_{t+}

_{1}is reservoir storage volume at the beginning and end of the t + 1 the period.

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**Figure 2.**Average monthly inflow volume to the reservoir, average monthly evaporation depth, and the average monthly volume of water demand in the baseline and under climate change conditions.

**Figure 6.**Change in output, storage, deficit volume using EPO

_{ad}and EPO

_{ba}algorithms in the first and second scenarios: (

**a**) Released water and demand Volume (MCM); (

**b**) Reservoir storage (MCM); (

**c**) Deficit, demand and reservoir inflow volumes (MCM).

**Figure 7.**Change in output, storage, deficit volume using EPO

_{ad}and EPO

_{ba}algorithms in the third and fourth scenarios: (

**a**) Released water and demand Volume (MCM); (

**b**) Reservoir storage (MCM); (

**c**) Deficit, demand and reservoir inflow volumes (MCM).

**Figure 8.**Comparison of the Pareto curve of two objective functions (vulnerability and reliability Indexes) in the baseline and future periods.

**Figure 9.**Comparison of: (

**a**) Released volume; (

**b**) Storage volume; (

**c**) Deficit volume, corresponding to the first scenario.

**Figure 10.**Comparison of: (

**a**) Released volume; (

**b**) Storage volume; (

**c**) Deficit volume, corresponding to the second scenario.

**Figure 11.**Comparison of: (

**a**) Released volume; (

**b**) Storage volume; (

**c**) Deficit volume, corresponding to the third scenario.

Algorithm | MAD ^{1} | MSE ^{2} | RMSE ^{3} | MAPE ^{4} | R(XY) ^{5} | NS ^{6} | MAE ^{7} | R^{2} | SSE ^{8} |
---|---|---|---|---|---|---|---|---|---|

EPO_{ad} | 0.774 | 1.513 | 1.230 | 4.487 | 0.999 | 0.995 | −0.774 | 0.997 | 381.386 |

EPO_{ba} | 0.511 | 12.276 | 3.504 | 2.124 | 0.98 | 0.961 | −0.24 | 0.961 | 3093.48 |

^{1}Mean Absolute Deviation.

^{2}Mean Square Error.

^{3}Root Mean Square Error.

^{4}Mean Absolute Percentage Error.

^{5}Correlation Coefficient between X and Y.

^{6}Nash–Sutcliffe.

^{7}Mean Absolute Error.

^{8}Sum of Squared Errors.

Scenarios | Reliability (%) | Vulnerability (%) |
---|---|---|

First | 43.56 | 9.44 |

Second | 55.88 | 6.73 |

Third | 29.74 | 23.45 |

Fourth | 36.65 | 14.65 |

Algorithm | MAD | MSE | RMSE | MAPE | R(XY) | NS | MAE | R^{2} | SSE |
---|---|---|---|---|---|---|---|---|---|

EPO_{ad} | 1.734 | 22.838 | 4.779 | 10.245 | 0.97 | 0.927 | −0.98 | 0.93 | 5755.17 |

EPO_{ba} | 1.113 | 9.656 | 3.107 | 5.308 | 0.99 | 0.966 | −0.98 | 0.95 | 2433.26 |

Sc. | Reliability % | Vulnerability % | Parameter Changes in Scenarios | Reliability Change % | Vulnerability Change % |
---|---|---|---|---|---|

First | 48.41 | 8.15 | Comparison of the first and second | 11.60 | −40.03 |

Second | 54.76 | 5.82 | Comparison of the third and fourth | 12.44 | −5.51 |

Third | 33.49 | 11.11 | Comparison of the first and third | −44.55 | 26.64 |

Fourth | 38.25 | 10.53 | Comparison of the second and fourth | −43.16 | 44.73 |

Method | Sc. | Vulnerability % | Reliability % | Parameter Changes in Scenarios | Vulnerability Change % | Reliability Change % |
---|---|---|---|---|---|---|

MOEPO | First Scenario | 4.33 | 48 | Comparison of the first and third | 0 | −4.3478 |

Second Scenario | 5.98 | 46 | Comparison of the Second and third | −38.106 | 0 | |

Third Scenario | 4.33 | 46 | Comparison of the first and second | 27.592 | −4.3478 | |

SOP | Baseline Condition | 14 | 48 | Comparison of the first scenario and Baseline condition | −35.714 | 25 |

Future Condition | 16 | 46 | Comparison of the Second scenario and Baseline condition | −31.25 | 26.087 |

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## Share and Cite

**MDPI and ACS Style**

Yoosefdoost, I.; Basirifard, M.; Álvarez-García, J.
Reservoir Operation Management with New Multi-Objective (MOEPO) and Metaheuristic (EPO) Algorithms. *Water* **2022**, *14*, 2329.
https://doi.org/10.3390/w14152329

**AMA Style**

Yoosefdoost I, Basirifard M, Álvarez-García J.
Reservoir Operation Management with New Multi-Objective (MOEPO) and Metaheuristic (EPO) Algorithms. *Water*. 2022; 14(15):2329.
https://doi.org/10.3390/w14152329

**Chicago/Turabian Style**

Yoosefdoost, Icen, Milad Basirifard, and José Álvarez-García.
2022. "Reservoir Operation Management with New Multi-Objective (MOEPO) and Metaheuristic (EPO) Algorithms" *Water* 14, no. 15: 2329.
https://doi.org/10.3390/w14152329