# Optimal Determination and Dynamic Control Analysis of the Graded and Staged Drought Limit Water Level of Typical Plateau Lakes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Determination Method of Lake Graded and Staged Drought Limit Water Level

#### 2.1. Grading and Staging of Lake Drought Limited Water Level

#### 2.1.1. Grading of Drought Limited Water Level

#### 2.1.2. Staging of Drought Limited Water Level

#### 2.2. Analysis of Lake Types

#### 2.3. Calculation of Lake Hydrological Elements

#### 2.3.1. Determination of Lake Ecological Water Level

#### 2.3.2. Calculation of Socio-Economic Water Demand Outside the Lake

#### 2.3.3. Calculation of Lake Inflow Runoff

#### 2.3.4. Lake Surface Evapotranspiration Loss

_{Φ20}is the evaporation measured by a 20 mm diameter pan; k is the correction coefficient of lake evaporation; A is the lake area.

#### 2.4. Calculation of Lake Drought Limit Water Level

- (1)
- Lakes with water supply function

- (2)
- Lakes without water supply function

#### 2.5. Compared with the Water Level Determined by Other Optimization Algorithms

- Objective function:

- 2.
- Constraints:

- 3.
- Decision variables

- 4.
- Model solution

#### 2.6. Correction and Rationality Analysis of Lake Drought Limit Water Level

#### 2.6.1. Lake Drought Limit Water Level Correction

- (1)
- Comprehensive management constraint correction

- (2)
- Ecological protection constraint correction

- (3)
- Correction of staging value

#### 2.6.2. Rationality Analysis of Drought Limited Water Level

^{4}m

^{3}; $\overline{W}$ is the average monthly lake water storage for many years,10

^{4}m

^{3}.

## 3. Dynamic Control Method of Lake Drought Limit Water Level

#### 3.1. Hydrological Prediction of Lake Inflow Runoff

#### 3.2. Dynamic Control of Lake Drought Limit Water Level

## 4. Case Study

#### 4.1. Research Overview

^{2}, and the lake volume is 1.62 billion m

^{3}. There is a natural levee in the north of the lake, which divides Dianchi Lake into two parts, the north and the south, commonly known as Caohai and Waihai. Waihai accounts for 96.1% of the total area of Dianchi Lake and is the main body of Dianchi Lake. The normal height water level of Waihai is 1887.5 m, the lowest working water level is 1885.5 m, the corresponding water level in the dry year is 1885.2 m, the limited water level in the flood season is 1887.2 m, and the highest water level in 20 years is 1887.5 m.

#### 4.2. Calculation of Hydrological Elements of Drought Limited Water Level

#### 4.2.1. Determination of Ecological Water Level in Dianchi Lake

#### 4.2.2. Calculation of Socio-Economic Water Demand Outside the Lake

^{3}/ha; when at the frequency of P = 95%, the multi-year average agricultural comprehensive irrigation water quota is 4753.83 m

^{3}/ha, and the monthly agricultural comprehensive irrigation water quota is shown in Figure 8. In the planning year of 2030, the quota of agricultural comprehensive irrigation water in the Dianchi Lake basin will change greatly. The quota of agricultural comprehensive irrigation water at the frequency of P = 75% is 5535.00 m

^{3}/ha, and the agricultural comprehensive irrigation water quota at the frequency of P = 95% is 4725.66 m

^{3}/ha. All meet the most stringent water resources management water efficiency red line control regulations implemented by the state. The monthly water demand according to the proportion of monthly agricultural irrigation water is shown in Table 4.

#### 4.2.3. Lake Inflow Runoff Calculation

#### 4.2.4. Lake Surface Evapotranspiration Loss

#### 4.3. Stage Division and Calculation of Drought Limited Water Level

#### 4.4. Correction and Rationality Analysis of Drought Limited Water Level

#### 4.4.1. Correction of Drought Limited Water Level

- (1)
- Comprehensive management constraint correction

- (2)
- Correction of staging value

#### 4.4.2. Rationality Analysis of Drought Limited Water Level

#### 4.5. Hydrological Prediction Analysis of Dianchi Lake Inflow Runoff

#### 4.6. Analysis of Dynamic Control Results of Drought Limited Water Level in Dianchi Lake

^{3}of water shortage. And there are 67 months that the monthly measured lake inflow runoff was less than 95% frequency lake inflow runoff. If the drought water level is not adjusted, it will result in a water shortage of approximately 790 million m

^{3}, as shown in Figure 13a,b.

^{3}and 4.08 billion m

^{3}, respectively, as shown in Figure 13c,d. When taking drought warning measures, the actual measured lake inflow runoff is greater than the predicted water inflow in 56 months and 21 months of raising the drought warning water level and the drought water level, respectively, indicating that there may be approximately 312 million m

^{3}and 101 million m

^{3}of water that has not been fully utilized.

## 5. Conclusions

- (1)
- According to the characteristics and different types of lakes, the staged method of lake drought limit water level is optimized, and finally, a practical calculation method of the lake’s graded and staged drought limit water level with the ecological water level, lake inflow runoff, and evapotranspiration loss as the main calculation factors are proposed.
- (2)
- The determination method and dynamic control optimization model of drought limit water level are applied to Dianchi Lake in Yunnan Province, and the monthly and staged drought limit water levels of Dianchi Lake are obtained. Considering the uncertainty of the drought process in Dianchi Lake, the percentage of precipitation anomaly and water storage anomaly are used as the meteorological and hydrological drought indexes for the rationality analysis of the drought limit water level of Dianchi Lake.
- (3)
- In order to solve the control problem of lake drought limit water level, a hydrological forecasting model of lake inflow runoff is constructed. Based on the optimal control theory, the SCSSA-Elman neural network model is used for optimization prediction.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The dynamic control process of drought limits water level for medium and long-term hydrological prediction of lake inflow runoff.

**Figure 8.**Monthly agricultural comprehensive irrigation water quota at the frequency of P = 75% and P = 95%.

**Figure 9.**Weight vector of SOFM-ANN model with different iterations of Dianchi Lake drought limit water level.

**Figure 10.**Monthly drought limit water level and staged drought limit water level of Dianchi Lake after correction.

**Figure 11.**Comparison of consistency between the monthly drought limit water level and meteorological and hydrological drought.

**Figure 12.**Comparison of consistency between the staged drought limit water level and meteorological and hydrological drought results.

**Figure 13.**Estimated monthly water shortage of Dianchi Lake from 1954 to 2016. (

**a**) Dynamic control of drought warning water level (P = 75%) was not adopted; (

**b**) Dynamic control of drought water level (P = 95%) was not adopted; (

**c**) Adopting dynamic control of drought warning water level (P = 75%); (

**d**) Adopting dynamic control of drought water level (P = 95%).

Number | Method | Method Description | Formula | Regulation |
---|---|---|---|---|

1 | Hydrologic frequency analysis method | According to the measured sequence data of the lake’s monthly water level, P-III theoretical curve fitting is carried out. | The water levels at P = 75% and P = 95% inflow frequencies were selected. | The determination of the ecological water level is based on the calculation results of drought warning water level (P = 75%) and drought water level (P = 95%). The maximum outsourcing value is the ecological water level. |

2 | Lake morphology method | Establish a fitting curve between the lake water level and area, and compare the water level corresponding to the maximum value with the lowest water level. If close, it is the lowest ecological water level. | $\left\{\begin{array}{c}{\mathsf{\partial}}^{2}f\left(Z\right)/{\mathsf{\partial}}^{2}Z=0\\ {Z}_{\mathrm{min}}^{-}\le Z\le {Z}_{\mathrm{min}}^{+}\end{array}\right.$ | |

3 | Biological minimum Space method | The minimum water level required for the survival and reproduction of organisms in lakes | ${Z}_{\mathrm{min}}=\left\{\begin{array}{c}{\overline{Z}}_{\mathrm{elevation}}\\ +\\ \u25b3{Z}_{\mathrm{water}\mathrm{level}\mathrm{increase}}^{\mathrm{biological}\mathrm{minimum}}\end{array}\right.$ | |

4 | Ecological water level method | Long series (≥50 a) water level data are statistically analyzed, and the average water level is obtained by multiplying the ecological water level coefficient. | $\theta =\overline{{Z}_{Worst}^{HF}}/\left(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{Z}_{i}}\right)$ | |

5 | Lowest water level method | The lower limit of ecosystem water level, that is, the average water level of the driest year, is taken as the lowest ecological water level. | ${Z}_{\mathrm{min}}=\lambda \left({\displaystyle \sum _{i=1}^{n}{Z}_{i}}/n\right)$ |

Drought Type | Index | Drought Level | ||||
---|---|---|---|---|---|---|

No Drought | Mild Drought | Medium Drought | Heavy Drought | Serious Drought | ||

Meteorological drought | Percentage of precipitation anomaly ${\nu}_{1}$/% | >−4 | [−12, −4] | [−20, −13] | [−28, −21] | <−28 |

Hydrological drought | Percentage of water storage anomaly ${\nu}_{2}$/% | >−10 | [−30, −10] | [−50, −31] | [−80, −51] | <−80 |

**Table 3.**Calculation results of monthly water level of Dianchi Lake at P = 75% and P = 95% frequencies (unit: m).

Frequency | January | February | March | April | May | June |
---|---|---|---|---|---|---|

P = 75% | 1886.56 | 1886.49 | 1886.36 | 1886.20 | 1886.02 | 1886.03 |

P = 95% | 1886.19 | 1886.12 | 1885.99 | 1885.83 | 1885.65 | 1885.66 |

Frequency | July | August | September | October | November | December |

P = 75% | 1886.17 | 1886.43 | 1886.59 | 1886.66 | 1886.67 | 1886.63 |

P = 95% | 1885.80 | 1886.06 | 1886.22 | 1886.29 | 1886.30 | 1886.26 |

Month | January | February | March | April | May | June |
---|---|---|---|---|---|---|

Distribution proportion | 9.82 | 8.23 | 9.41 | 9.16 | 15.01 | 9.08 |

P = 75% | 3427.62 | 2872.14 | 3283.41 | 3196.31 | 5239.63 | 3169.96 |

P = 95% | 2926.32 | 2452.08 | 2803.20 | 2728.83 | 4473.31 | 2706.34 |

Month | July | August | September | October | November | December |

Distribution proportion | 6.92 | 4.11 | 3.13 | 8.40 | 7.73 | 9.00 |

P = 75% | 2414.81 | 1432.83 | 1093.51 | 2931.99 | 2697.28 | 3140.51 |

P = 95% | 2061.63 | 1223.28 | 933.58 | 2503.17 | 2302.79 | 2681.20 |

**Table 5.**Monthly inflow runoff calculation results of Dianchi Lake at a frequency of P = 75% and P = 95% (unit: 10

^{4}m

^{3}).

Frequency | January | February | March | April | May | June |
---|---|---|---|---|---|---|

P = 75% | 2661.49 | 2725.55 | 2263.68 | 1842.71 | 3791.75 | 7050.88 |

P = 95% | 1890.04 | 2044.16 | 1587.96 | 1266.87 | 2615.00 | 4776.40 |

Frequency | July | August | September | October | November | December |

P = 75% | 9752.77 | 12,196.34 | 7390.05 | 5719.88 | 3353.31 | 2735.52 |

P = 95% | 6751.92 | 8443.62 | 5010.21 | 3877.89 | 2294.37 | 1942.62 |

**Table 6.**The calculation results of monthly average evapotranspiration in Dianchi Lake at the frequency of P = 75% and P = 95% (unit: 10

^{4}m

^{3}).

Frequency | January | February | March | April | May | June |
---|---|---|---|---|---|---|

P = 75% | 78.16 | 102.16 | 136.45 | 165.81 | 160.85 | 145.22 |

P = 95% | 68.71 | 87.40 | 123.25 | 151.85 | 145.12 | 132.86 |

Frequency | July | August | September | October | November | December |

P = 75% | 139.44 | 138.61 | 122.29 | 101.08 | 82.76 | 70.16 |

P = 95% | 130.64 | 126.94 | 111.99 | 92.48 | 72.87 | 61.77 |

**Table 7.**Monthly staging results of SOFM-ANN model with different iterations of drought-limited water level in Dianchi Lake.

Iteration Times | Ⅰ | Ⅱ | Ⅲ |
---|---|---|---|

5 | (1 2 3 4 12) | (7 8 9) | (5 6 10 11) |

10 | (1 2 3 4 5 11 12) | (7 8) | (6 9 10) |

50 | (1 2 3 4 11 12) | (6 7 8 9 10) | (5) |

100 | (1 2 3 4 11 12) | (6 7 8 9) | (5 10) |

200 | (1 2 3 4 11 12) | (6 7 8 9) | (5 10) |

500 | (1 2 3 4 11 12) | (6 7 8 9) | (5 10) |

1000 | (1 2 3 4 11 12) | (6 7 8 9) | (5 10) |

5000 | (1 2 3 4 11 12) | (6 7 8 9) | (5 10) |

Staging Method | Non-Flood Season | Main Flood Season | Transition Period |
---|---|---|---|

Fuzzy set analysis method | 1–4, 12 | 6–9 | 5, 10–11 |

Genetic analysis methods | 1–4, 11, 12 | 7–8, 10 | 5–6, 9 |

System clustering method | 1–4, 11, 12 | 6–9 | 5, 10 |

Fisher optimal segmentation method | 1–4, 11, 12 | 6–9 | 5, 10 |

SOFM-ANN model method | 1–4, 11, 12 | 6–9 | 5, 10 |

Frequency | January | February | March | April | May | June |
---|---|---|---|---|---|---|

Drought warning water level (P = 75%) | 1886.59 | 1886.50 | 1886.40 | 1886.25 | 1886.07 | 1886.03 |

Drought water level (P = 95%) | 1886.23 | 1886.14 | 1886.06 | 1886.07 | 1886.09 | 1886.02 |

Frequency | July | August | September | October | November | December |

Drought warning water level (P = 75%) | 1886.17 | 1886.43 | 1886.59 | 1886.66 | 1886.67 | 1886.65 |

Drought water level (P = 95%) | 1886.02 | 1886.06 | 1886.22 | 1886.29 | 1886.30 | 1886.29 |

Frequency | January | February | March | April | May | June |
---|---|---|---|---|---|---|

Drought warning water level | 1886.52 | 1886.45 | 1886.32 | 1886.16 | 1885.98 | 1885.99 |

Drought water level | 1886.09 | 1886.02 | 1885.89 | 1885.73 | 1885.55 | 1885.56 |

Frequency | July | August | September | October | November | December |

Drought warning water level | 1886.13 | 1886.39 | 1886.55 | 1886.62 | 1886.63 | 1886.59 |

Drought water level | 1885.70 | 1885.96 | 1886.12 | 1886.19 | 1886.20 | 1886.16 |

CSA- Elman | WOA- Elman | SOA- Elman | SSA- Elman | SCSSA- Elman | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Training Samples | Testing Samples | Training Samples | Testing Samples | Training Samples | Testing Samples | Training Samples | Testing Samples | Training Samples | Testing Samples | |

MAE | 195.481 | 10.088 | 70.312 | 9.355 | 39.325 | 27.757 | 83.929 | 43.354 | 37.444 | 9.951 |

RMSE | 343.032 | 12.566 | 102.271 | 12.821 | 64.531 | 37.961 | 108.608 | 50.345 | 57.988 | 10.534 |

MAPE | 0.259 | 0.084 | 0.095 | 0.084 | 0.052 | 0.138 | 0.106 | 0.047 | 0.034 | 0.063 |

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## Share and Cite

**MDPI and ACS Style**

Ge, Q.; Gu, S.; Wang, L.; Chen, G.; Chen, J.
Optimal Determination and Dynamic Control Analysis of the Graded and Staged Drought Limit Water Level of Typical Plateau Lakes. *Water* **2023**, *15*, 2580.
https://doi.org/10.3390/w15142580

**AMA Style**

Ge Q, Gu S, Wang L, Chen G, Chen J.
Optimal Determination and Dynamic Control Analysis of the Graded and Staged Drought Limit Water Level of Typical Plateau Lakes. *Water*. 2023; 15(14):2580.
https://doi.org/10.3390/w15142580

**Chicago/Turabian Style**

Ge, Qiang, Shixiang Gu, Liying Wang, Gang Chen, and Jinming Chen.
2023. "Optimal Determination and Dynamic Control Analysis of the Graded and Staged Drought Limit Water Level of Typical Plateau Lakes" *Water* 15, no. 14: 2580.
https://doi.org/10.3390/w15142580