# Application of Optimization Techniques for Searching Optimal Reservoir Rule Curves: A Review

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Reservoir Simulation Models

_{ν}

_{,τ}is the discharge of the reservoir at time τ; SWA

_{τ}and EWA

_{τ}represent the available water at the start and end points at time τ, respectively; and D

_{τ}is the downstream water demand at time τ.

_{ν}

_{,τ}is the discharge of water during year ν and month τ (τ = 1 to 12, standing for January to December); D

_{τ}is the net downstream water demand during month τ; D

_{t}is the lower rule curve of month τ; D

_{t}+ C is the upper rule curve of month τ; and W

_{ν}

_{,τ}is the available water during year ν and month τ, as described in Equation (3):

_{v,τ}= S

_{v,τ−1}+ Q

_{v,τ}− R

_{v,τ}− E

_{τ}

_{ν}

_{,τ−1}is the stored water at the end of month τ − 1; Q

_{ν}

_{,τ}is the monthly inflow to the reservoir; and E

_{τ}is the evaporation loss.

## 3. Optimization Techniques for Reservoir Rule Curve Extraction

#### 3.1. Integrating Optimization Techniques and the Reservoir Simulation Model

#### 3.2. Objective Function of the Search Process

_{(avr)}is the average water shortage per year; Fre

_{(i)}is the frequency of water shortages; P

_{(avr)}is the average excess water per year; n represents an entire year; Sh

_{v}is the water shortage in year v; Y

_{shv}is the year of water shortage; Sp

_{v}is the release of excess water during year v; and Y

_{spv}is the year of the release of excess water.

#### 3.3. Optimizing the Points of the Rule Curves

_{1}, y

_{1}) should be higher than that in February (x

_{2}, y

_{2}) and then gradually decrease until June (x

_{6}, y

_{6}). From July (x

_{7}, y

_{7}), which marks the beginning of the flood season, the water level should be higher than that in June (x

_{6}, y

_{6}), and in August (x

_{8}, y

_{8}), the water level should rise higher than in July (x

_{7}, y

_{7}) and increase until the end of the rainy season in October (x

_{10}, y

_{10}) until the end of December (x

_{12}, y

_{12}). This pattern reflects the seasonal streamflow of Thailand. The smoothing function constraints are integrated into the fitness function of the reservoir simulation model to fit the rule curves, as presented below:

_{1}and y

_{1}represent the initial water level in January, which marks the beginning of the drought season. Similarly, x

_{5}and y

_{5}represent the water level in May; x

_{6}and y

_{6}represent the initial water level in June, which marks the end of the drought season and the beginning of the flood season, and marks the starting point of the lower rule curve for the flood season. Lastly, x

_{12}and y

_{12}represent the water level in December, which marks the end of the upper rule curve for the flood season.

## 4. Typically Applied Optimization Techniques

#### 4.1. Trial and Error Technique with the Reservoir Simulation Model

#### 4.2. Dynamic Programming

#### 4.3. Heuristic and Metaheuristic Algorithms

#### 4.3.1. Simulated Annealing Algorithm

#### 4.3.2. The Shuffled Frog Leaping Algorithm

_{1}, X

_{2}, …, X

_{n}}, created randomly within the feasible space. With the 24 decision variables for a single reservoir (upper and lower rule curves variables), the position of the ith frog is represented as X

_{i}= [x

_{i}

_{1}, x

_{i}

_{2}, …, x

_{i}

_{24}]

^{T}. Then, a set of rule curves is used in reservoir simulation and the release water is calculated by the simulation model considering these rule curves. Next, the release water is used to calculate the fitness function to evaluate the frog’s position. The fitness function is the minimum of the average water shortage subject to constraints on the simulation model, as illustrated in Equation (4).

#### 4.4. Evolutionary Algorithms

#### 4.4.1. Genetic Algorithm

#### 4.4.2. Differential Evolution

#### 4.4.3. Genetic Programing

#### 4.4.4. Cultural Algorithms

#### 4.5. Swarm Algorithms

#### 4.5.1. Particle Swam Optimization

#### 4.5.2. Cuckoo Search

_{1}, xi

_{2}, …, xi

_{24}]. Then, a set of rule curves is used in the reservoir simulation and the release water is calculated by the simulation model using these rule curves. The release water is then used to calculate the fitness function to evaluate a nest. A nest is then chosen randomly. A new set of rule curves is used in the reservoir simulation and the release water is used to calculate the fitness function again for Z(Xi+1). Next, the fitness function Z(Xi+1) and the fitness function Z(Xi) are compared: if Z(Xi+1) is larger than Z(Xi), return and choose a new nest, but if Z(Xi+1) is smaller than Z(Xi), replace Xi by Xi+1 and keep the new nest (accepted rule curves) for the next iteration. The next iteration is performed by choosing the new nest if the termination criterion is not satisfied. The process is then continued until the criterion is satisfied, as illustrated in Figure 11. In the case of climate change impact studies, CS has been used to optimize the multifunctional performance of reservoir systems. Its purpose is to meet downstream water needs and control potential flooding [106].

#### 4.5.3. Tabu Search Algorithm

#### 4.5.4. Firefly Algorithm

#### 4.5.5. Flower Pollination Algorithm

#### 4.5.6. Gray Wolf Optimizer

#### 4.5.7. Wind-Driven Optimization

#### 4.5.8. Ant Colony Optimization

#### 4.5.9. Honey-Bee Mating Optimization

## 5. Suitable Future Rule Curves

#### 5.1. Climate Change

#### 5.2. Land Use Changes

#### 5.3. SWAT Model

^{2}(coefficient of determination), RE (relative error), and E

_{ns}(Nash–Suttclife simulation efficiency). In this study, the calibration of SWAT was conducted by adjusting the eight hydrologic parameters: Alpha_BF, Gwqmn, Gw_Revap, Sol_Awc, Epco, Esco, Ch_N2, and Gw_delay.

^{2}, RE, and E

_{ns}) could be accepted. In order to predict the future runoff at the Ubolratana Dam station between 2015 and 2064, the calibrated SWAT was simulated using inputs from the climate data of PRECIS, which also had decreased tolerances, and the land use maps of the CA Markov model. The simulated outcomes showed that the future average annual runoffs from the A2 and B2 scenarios were 4028.91 and 4580.50 MCM, respectively. Figure 17 and Figure 18 indicate an increase in the runoff into the Ubolratana Reservoir in the future periods relative to the baseline period. Subsequently, SWAT-derived runoff was used as input for the development of reservoir rule curves linked to the reservoir simulation models and various optimization techniques were used to solve problems by searching optimal solutions based on many constraints and objective functions.

#### 5.4. Participation of Stakeholders

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Heuristic integration of the optimization algorithm and reservoir simulation model to search for the rule curves.

**Figure 8.**The computational dimensionality of DP/PPO to search for the optimal rule curves: (

**a**) iteration l = 1 and (

**b**) iteration l = 2, 3, 4, etc.

**Figure 9.**Heuristic and metaheuristic algorithms with reservoir simulation to search for the optimal rule curves.

**Figure 10.**Evolutionary algorithm integrated with the reservoir simulation model to search for the optimal rule curves.

**Figure 11.**Swarm algorithms integrated with the reservoir simulation model to search for the optimal rule curves.

**Figure 12.**Future climate trends compared to the baseline A2 scenario [9].

**Figure 14.**Simulated land use maps for 2015–2064: (

**a**) 2024, (

**b**) 2034, (

**c**) 2044, (

**d**) 2054, and (

**e**) 2064 [9].

**Figure 17.**Baseline and future yearly Streamflow into the Ubolratana Reservoir [9].

**Figure 18.**Future runoff into the Ubolratana Reservoir [9].

**Figure 19.**Optimal rule curves of the Ubolratana Reservoir (A2 Scenario) adjusted after stakeholder participation [9].

**Figure 20.**Optimal rule curves of the Ubolratana Reservoir (B2 Scenario) adjusted after stakeholder participation [9].

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**MDPI and ACS Style**

Kangrang, A.; Prasanchum, H.; Sriworamas, K.; Ashrafi, S.M.; Hormwichian, R.; Techarungruengsakul, R.; Ngamsert, R.
Application of Optimization Techniques for Searching Optimal Reservoir Rule Curves: A Review. *Water* **2023**, *15*, 1669.
https://doi.org/10.3390/w15091669

**AMA Style**

Kangrang A, Prasanchum H, Sriworamas K, Ashrafi SM, Hormwichian R, Techarungruengsakul R, Ngamsert R.
Application of Optimization Techniques for Searching Optimal Reservoir Rule Curves: A Review. *Water*. 2023; 15(9):1669.
https://doi.org/10.3390/w15091669

**Chicago/Turabian Style**

Kangrang, Anongrit, Haris Prasanchum, Krit Sriworamas, Seyed Mohammad Ashrafi, Rattana Hormwichian, Rapeepat Techarungruengsakul, and Ratsuda Ngamsert.
2023. "Application of Optimization Techniques for Searching Optimal Reservoir Rule Curves: A Review" *Water* 15, no. 9: 1669.
https://doi.org/10.3390/w15091669