# A Dynamically Dimensioned Search Allowing a Flexible Search Range and Its Application to Optimize Discrete Hedging Rule Curves

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## Abstract

**:**

## 1. Introduction

- The DDS has constant specified-search boundaries of decision variables until the optimization is terminated.
- If the two decision variables are under an inequality constraint, the DDS-FSR can exclude infeasible areas for the decision variable by converting the upper or lower boundary for the decision variable to the current best solution for the other decision variable.

## 2. Methods

#### 2.1. Discrete Hedging Rules

- Normal: Release the monthly planned municipal water supply;
- Caution: Release the monthly contracted water supply with local governments/industrial complexes, etc.;
- Alert: Release the monthly actual usage surveyed on last year basis among the contractual water supply;
- Severe: Release 80% of the monthly actual usage.

#### 2.2. Dynamically Dimensioned Search Allowing a Flexible Search Range

#### 2.3. Reservoir Simulation and Optimization Model

- 1.
- The sum of water supply shortage for the total period T;
- 2.
- The penalty term to restrain the reversal of trigger volumes in drought phase severity;
- 3.
- The penalty term to restrain water supply failures within the optimization period.

## 3. Study Area and Data

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Historical and Reservoir Operation Records for Andong-Imha, Hapcheon, and Namgang Reservoirs

**Figure A1.**The historical records for Andong-Imha reservoir. The top plot shows the historical record of monthly inflow, the middle plot shows the reservoir operation records for storage, the bottom plot shows the monthly planned water supply and water supply records, and the areas filled with translucent red present the water supply shortage.

**Figure A2.**The historical records for Hapcheon reservoir. The top plot shows the historical record of monthly inflow, the middle plot shows reservoir operation records for storage, the bottom plot the monthly planned water supply and water supply records, and the areas filled with translucent red present the water supply shortage.

**Figure A3.**The historical records for Namgang reservoir. The top plot shows the historical record of monthly inflow, the middle plot shows reservoir operation records for storage, the bottom plot the monthly planned water supply and water supply records, and the areas filled with translucent red present the water supply shortage.

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**Figure 1.**Schematic diagram of the three heuristic algorithms. (

**a**) Genetic algorithm; (

**b**) Shuffled Complex Evolution-University Arizona; (

**c**) Dynamically dimensioned search algorithm.

**Figure 2.**The discrete hedging rule with the water rationing strategy of the Korean drought contingency plan. The left figure is the schematic diagram of the discrete hedging rule, and the right figure is the water rationing strategy in the Korean drought contingency plan.

**Figure 7.**Convergence processes of the objective function value under the condition of case 1 (r = 0.2, m = 100,000) through 10 random optimization trials with the DDS-FRS: (

**a**) Andong-Imha; (

**b**) Hapcheon; (

**c**) Namgang.

**Figure 8.**Convergence processes of the objective function value under the condition of case 1 (r = 0.2, m = 100,000) through 10 random optimization trials with the DDS: (

**a**) Andong-Imha; (

**b**) Hapcheon; (

**c**) Namgang.

**Figure 9.**Convergence processes of the objective function value under the condition of case 7 (r = 0.2, m = 10,000) through 10 random optimization trials with the DDS-FRS: (

**a**) Andong-Imha; (

**b**) Hapcheon; (

**c**) Namgang.

**Figure 10.**Convergence processes of the objective function value under the condition of case 7 (r = 0.2, m = 10,000) through 10 random optimization trials with the DDS: (

**a**) Andong-Imha; (

**b**) Hapcheon; (

**c**) Namgang.

**Figure 11.**The attempted candidate solutions during the discrete hedging rule optimization for HC using the DDS-FSR, (

**a**) the tried candidate solution at the initiation, (

**b**) the tried candidate solution at the 30,000th, (

**c**) the tried candidate solution at the 70,000th, (

**d**) the tried candidate solution at the 100,000th.

**Figure 12.**The attempted candidate solutions during the discrete hedging rule optimization for HC using the DDS, (

**a**) the tried candidate solution at the initiation, (

**b**) the tried candidate solution at the 30,000th, (

**c**) the tried candidate solution at the 70,000th, (

**d**) the tried candidate solution at the 100,000th.

**Figure 13.**The reservoir simulation results for HC. The top plot shows the simulated monthly storage; the bottom plot shows the simulated monthly available water and the derived hedging rule curves for HC.

**Figure 14.**The reservoir simulation results for HC. The top plot shows the simulated monthly release and total planned water supply; the bottom shows the simulated monthly drought phase.

**Table 1.**Conditions and equations of the water rationing for reservoir simulation with the discrete hedging rule.

Classification | Simulation Model | ||
---|---|---|---|

Drought Phases | Condition | Release | |

Release determination | Normal | ${S}_{t-1}+{I}_{t}>{V}_{1,p}$ | ${R}_{t}={D}_{p}$ |

Concern | ${V}_{2,p}<{S}_{t-1}+{I}_{t}\le {V}_{1,p}$ | ${R}_{t}={\alpha}_{1,p}{D}_{p}$ | |

Caution | ${V}_{3,p}<{S}_{t-1}+{I}_{t}\le {V}_{2,p}$ | ${R}_{t}={\alpha}_{2,p}{D}_{p}$ | |

Alert | ${V}_{4,p}<{S}_{t-1}+{I}_{t}\le {V}_{3,p}$ | ${R}_{t}={\alpha}_{3,p}{D}_{p}$ | |

Severe | ${V}_{5}<{S}_{t-1}+{I}_{t}\le {V}_{4,p}$ | ${R}_{t}={\alpha}_{,p4}{D}_{p}$ | |

Fail | ${S}_{t-1}+{I}_{t}\le {V}_{4,p}$ | ${R}_{t}=0$ |

Reservoir | Watershed Area (km ^{2}) | Annual Average Inflow (${10}^{6}$ m ^{3}) | Storage of Normal High Water Level (${10}^{6}$ m ^{3}) | Storage of Low-Water Level (${10}^{6}$ m ^{3}) | Active Capacity (${10}^{6}$ m ^{3}) |
---|---|---|---|---|---|

AD-IH | 2945 | 1613 | 1772 | 351.0 | 1421 |

HC | 925.0 | 637.4 | 724.1 | 151.0 | 599.0 |

NG | 2285 | 2105 | 182.4 | 16.15 | 166.3 |

Month | Andong-Imha | Hapcheon | Namgang | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\alpha}}_{\mathbf{1},\mathit{p}}$ | ${\mathit{\alpha}}_{\mathbf{2},\mathit{p}}$ | ${\mathit{\alpha}}_{\mathbf{3},\mathit{p}}$ | ${\mathit{\alpha}}_{\mathbf{4},\mathit{p}}$ | ${\mathit{\alpha}}_{\mathbf{1},\mathit{p}}$ | ${\mathit{\alpha}}_{\mathbf{2},\mathit{p}}$ | ${\mathit{\alpha}}_{\mathbf{3},\mathit{p}}$ | ${\mathit{\alpha}}_{\mathbf{4},\mathit{p}}$ | ${\mathit{\alpha}}_{\mathbf{1},\mathit{p}}$ | ${\mathit{\alpha}}_{\mathbf{2},\mathit{p}}$ | ${\mathit{\alpha}}_{\mathbf{3},\mathit{p}}$ | ${\mathit{\alpha}}_{\mathbf{4},\mathit{p}}$ | |||

January | 0.74 | 0.41 | 0.41 | 0.33 | 0.86 | 0.77 | 0.77 | 0.54 | 0.74 | 0.29 | 0.29 | 0.23 | ||

February | 0.75 | 0.42 | 0.42 | 0.34 | 0.91 | 0.82 | 0.82 | 0.58 | 0.75 | 0.28 | 0.28 | 0.22 | ||

March | 0.74 | 0.42 | 0.42 | 0.34 | 0.79 | 0.71 | 0.71 | 0.50 | 0.74 | 0.30 | 0.30 | 0.24 | ||

April | 0.76 | 0.43 | 0.43 | 0.34 | 0.65 | 0.56 | 0.56 | 0.39 | 0.73 | 0.39 | 0.37 | 0.25 | ||

May | 0.83 | 0.40 | 0.38 | 0.32 | 0.74 | 0.61 | 0.61 | 0.42 | 0.73 | 0.40 | 0.38 | 0.24 | ||

June | 0.89 | 0.35 | 0.33 | 0.28 | 0.85 | 0.63 | 0.63 | 0.44 | 0.88 | 0.88 | 0.73 | 0.11 | ||

July | 0.86 | 0.34 | 0.33 | 0.27 | 0.74 | 0.55 | 0.55 | 0.39 | 0.89 | 0.89 | 0.66 | 0.09 | ||

August | 0.88 | 0.28 | 0.27 | 0.22 | 0.75 | 0.53 | 0.53 | 0.37 | 0.90 | 0.90 | 0.66 | 0.09 | ||

September | 0.85 | 0.34 | 0.33 | 0.27 | 0.95 | 0.78 | 0.78 | 0.55 | 0.84 | 0.73 | 0.56 | 0.14 | ||

October | 0.78 | 0.42 | 0.42 | 0.34 | 0.83 | 0.73 | 0.73 | 0.51 | 0.75 | 0.28 | 0.28 | 0.22 | ||

November | 0.77 | 0.45 | 0.45 | 0.36 | 0.84 | 0.75 | 0.75 | 0.53 | 0.75 | 0.28 | 0.28 | 0.22 | ||

December | 0.74 | 0.43 | 0.43 | 0.34 | 0.87 | 0.78 | 0.78 | 0.55 | 0.74 | 0.29 | 0.29 | 0.23 |

**Table 4.**The historical records for the total water shortage and the optimization period to derive the hedging rule curves for each reservoir.

Reservoir | Optimization Period | Total Water Supply Shortage (${10}^{6}$ ${\mathbf{m}}^{3}$) |
---|---|---|

AD-IH | January 1992∼ December 2020 | 9481 |

HC | January 1989∼ December 2020 | 4562 |

NG | January 2002∼ December 2020 | 765 |

Case | Maximum Number of Function Evaluations (m) | Neighborhood Perturbation Size (r) |
---|---|---|

1 | 100,000 | 0.2 |

2 | 100,000 | 0.1 |

3 | 50,000 | 0.2 |

4 | 50,000 | 0.1 |

5 | 20,000 | 0.2 |

6 | 20,000 | 0.1 |

7 | 10,000 | 0.2 |

8 | 10,000 | 0.1 |

**Table 6.**The search boundary of decision variables to optimize the hedging rule curves from the DDS-FSR and the DDS: S-LWL is the storage of low-water level; S-NHWL is the storage of normal high water level; ${V}_{dp,p}^{best}$ is the best solution at the current iteration.

Trigger Volume | DDS-FSR | DDS | |||
---|---|---|---|---|---|

Lower Bound | Upper Bound | Lower Bound | Upper Bound | ||

${V}_{1,p}$ | ${V}_{2,p}^{best}$ | S-NHWL | S-LWL | S-NHWL | |

${V}_{2,p}$ | ${V}_{3,p}^{best}$ | ${V}_{1,p}^{best}$ | |||

${V}_{3,p}$ | ${V}_{4,p}^{best}$ | ${V}_{2,p}^{best}$ | |||

${V}_{4,p}$ | S-LWL | ${V}_{3,p}^{best}$ |

**Table 7.**Comparison of statistics of the converged objective function values between the DDS and DDS-FRS for the cases: the underlined value is superior to the other (unit: ${10}^{6}$${\mathrm{m}}^{3}$).

Case | Reservoir | DDS | DDS-FSR | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Best | Mean | Worst | St. Dev | Best | Mean | Worst | St. Dev | |||

1 | AD-IH | 3831 | 3912 | 3996 | 43 | 3806 | 3874 | 4008 | 56 | |

HC | 1983 | 2001 | 2058 | 21 | 1941 | 1972 | 1988 | 18 | ||

NG | 0 | 3 | 8 | 4 | 0 | 0 | 0 | 0 | ||

2 | AD-IH | 3814 | 3887 | 3964 | 48 | 3803 | 3859 | 3907 | 32 | |

HC | 1948 | 1985 | 2047 | 24 | 1959 | 1995 | 2079 | 33 | ||

NG | 0 | 2 | 9 | 4 | 0 | 1 | 9 | 3 | ||

3 | AD-IH | 3935 | 4058 | 4345 | 111 | 3843 | 3885 | 3990 | 43 | |

HC | 1984 | 2025 | 2192 | 59 | 1953 | 1987 | 2030 | 23 | ||

NG | 8 | 15 | 26 | 7 | 0 | 5 | 9 | 4 | ||

4 | AD-IH | 3876 | 3993 | 4202 | 91 | 3836 | 3891 | 3935 | 29 | |

HC | 1955 | 2006 | 2111 | 42 | 1979 | 2000 | 2028 | 18 | ||

NG | 0 | 8 | 17 | 6 | 0 | 1 | 8 | 2 | ||

5 | AD-IH | 3948 | 4361 | 4867 | 330 | 3964 | 4132 | 4673 | 238 | |

HC | 2014 | 2072 | 2155 | 44 | 1975 | 2027 | 2112 | 47 | ||

NG | 17 | 28 | 43 | 8 | 9 | 19 | 26 | 5 | ||

6 | AD-IH | 3978 | 4366 | 6072 | 583 | 3902 | 4088 | 4537 | 186 | |

HC | 1983 | 2056 | 2196 | 60 | 1998 | 2045 | 2175 | 51 | ||

NG | 8 | 18 | 34 | 9 | 0 | 11 | 26 | 8 | ||

7 | AD-IH | 4337 | 4814 | 6510 | 590 | 4066 | 4301 | 4650 | 202 | |

HC | 2016 | 2157 | 2451 | 126 | 1985 | 2067 | 2116 | 36 | ||

NG | 25 | 40 | 59 | 12 | 17 | 27 | 43 | 6 | ||

8 | AD-IH | 4193 | 4568 | 4977 | 254 | 4040 | 4450 | 4893 | 207 | |

HC | 2022 | 2078 | 2190 | 59 | 2017 | 2073 | 2179 | 45 | ||

NG | 18 | 36 | 51 | 10 | 17 | 27 | 42 | 8 |

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**MDPI and ACS Style**

Jin, Y.; Lee, S.; Kang, T.; Kim, Y.
A Dynamically Dimensioned Search Allowing a Flexible Search Range and Its Application to Optimize Discrete Hedging Rule Curves. *Water* **2022**, *14*, 3633.
https://doi.org/10.3390/w14223633

**AMA Style**

Jin Y, Lee S, Kang T, Kim Y.
A Dynamically Dimensioned Search Allowing a Flexible Search Range and Its Application to Optimize Discrete Hedging Rule Curves. *Water*. 2022; 14(22):3633.
https://doi.org/10.3390/w14223633

**Chicago/Turabian Style**

Jin, Youngkyu, Sangho Lee, Taeuk Kang, and Yeulwoo Kim.
2022. "A Dynamically Dimensioned Search Allowing a Flexible Search Range and Its Application to Optimize Discrete Hedging Rule Curves" *Water* 14, no. 22: 3633.
https://doi.org/10.3390/w14223633