# Forecasting Quarterly Inflow to Reservoirs Combining a Copula-Based Bayesian Network Method with Drought Forecasting

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Procedures of Quarterly Inflow Forecasting

#### 2.2. Copula-Based Bayesian Network

**x**) that evolve over time (e.g., streamflow or drought states) can be shown in a DAG and their probabilistic queries can be represented within a Bayesian network. The joint probability density function of the set of random variables in vector

**x**$\left({x}_{{t}_{1}},{x}_{{t}_{2}}\cdots ,{x}_{{t}_{n}}\right)$ forming a Bayesian network can be written as the product of individual density functions conditional on their parent variables [25,26]. If the dependency ordering of random variables exactly follows the temporal sequence, and the parent variables of ${x}_{{t}_{i}}$ is the set of all prior variables $\left({x}_{{t}_{i-1}},{x}_{{t}_{i-2}}\cdots ,{x}_{{t}_{1}}\right)$, the joint probability density function of the

**x**can be written as Equation (1).

^{6}m

^{3}, calculated using Equation (3). According to the probability distribution, when the second quarter inflow is 100 × 10

^{6}m

^{3}, the third quarter inflow with the greatest probability density function value is around 480 × 10

^{6}m

^{3}. The next-quarter inflow with the greatest probability density function value is normally used as the inflow forecast value when drought forecasting is not performed. A quarterly inflow prediction that incorporates drought forecasting will be explained later.

#### 2.3. Drought Forecasting Using Drought Index

#### 2.4. Quarterly Inflow Forecasting Combined with Drought Forecasting

- (1)
- Use the probability distribution of the next quarter inflow (like that in Figure 3) to calculate the cumulative probability that corresponds to a specific inflow of the next quarter.
- (2)
- Use the cumulative probability to calculate the SII.
- (3)
- When the calculated SII is the same as the lower bound value of each drought stage, set the corresponding quarterly inflow as the quarterly inflow that represents that drought stage (for D4, the inflow where the SII is −2.5).
- (4)
- Repeat Steps (1) to (3) for all quarters and quarterly inflows.

## 3. Results

#### 3.1. Two Selected Dams in This Study

^{2}. Its storage capacity is 2900 × 10

^{6}m

^{3}, making it the largest multipurpose dam in the Han River basin. The average yearly flow to the dam is 1750 × 10

^{6}m

^{3}and provides 1213 × 10

^{6}m

^{3}of water for use every year (Table 2). The Soyanggang dam supplies industrial water every month and agricultural water from March to October, and does not provide instream flow (Figure 5a). The Andong dam is located on the Nakdong River, and its basin area is 1584 km

^{2}. The storage capacity of the dam, the largest multipurpose dam in the Nakdong River basin, is 1248 × 10

^{6}m

^{3}. The average yearly flow to the dam is 940 × 10

^{6}m

^{3}and provides 926 × 10

^{6}m

^{3}of water for use every year (Table 2). The Andong dam supplies industrial water and instream flow every month and a relatively large amount of agricultural water from April to October (Figure 5b).

^{6}m

^{3}, and the standard deviation is 608 × 10

^{6}m

^{3}. The smallest third quarter inflow was recorded in 2014. At that time, the third quarter inflow was 513 × 10

^{6}m

^{3}, which is around 35% of the average inflow. Additionally, in 2015, the third quarter inflow of the Soyanggang dam was 578 × 10

^{6}m

^{3}, which is 40% of the average inflow. The average third quarter inflow of the Andong dam is 607 × 10

^{6}m

^{3}, and the standard deviation is 278 × 10

^{6}m

^{3}. In 2015, when the third quarter inflow of the Andong dam was the smallest, it was 98 × 10

^{6}m

^{3}, which is 16% of the average inflow. Unlike the Soyanggang dam, the third quarter inflow of the Andong dam in 2013 was smaller than that in 2014. The third quarter inflow in 2013 was 243 × 10

^{6}m

^{3}, which is 40% of the average inflow. As can be realized from the two dam's inflow states, the drought that occurred in the Republic of Korea from 2013 to 2015 was fairly severe. We used the quarterly inflow data from the first year of observation to 2010 to train the copula-based Bayesian network model. The quarterly inflow data from 2011 and 2016 were used to verify the forecast results from the trained copula-based Bayesian network module.

#### 3.2. Quarterly Inflow Forecasting Curves Conforming to Drought Stages

#### 3.3. Quarterly Drought Forecast Results

#### 3.4. Quarterly Inflow Forecast Results

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**An example of probability density function of the third quarter inflow conditioned on the second quarter inflow of 100 × 10

^{6}m

^{3}.

**Figure 2.**Probability density function of the third quarter SII conditioned on the second quarter SII of −1.

**Figure 3.**Example of estimating representative quarterly inflows corresponding to lower bounds of SII ranges of drought stages.

**Figure 4.**Locations of the Soyanggang and Andong dams. (

**a**) Soyanggang dam region; (

**b**) The Republic of Korea; (

**c**) Andong dam region.

**Figure 7.**Histogram against the five distributions of the quarterly inflow volumes of the Soyanggang dam during the training period of 1974–2010: (

**a**) first quarter; (

**b**) second quarter; (

**c**) third quarter; (

**d**) fourth quarter.

**Figure 8.**Histogram against the five distributions of the quarterly inflow volumes of the Andong dam during the training period of 1977–2010: (

**a**) first quarter; (

**b**) second quarter; (

**c**) third quarter; (

**d**) fourth quarter.

**Figure 9.**Quarterly inflow forecasting curves conforming to drought stages for the Soyanggang dam: (

**a**) first quarter; (

**b**) second quarter; (

**c**) third quarter; (

**d**) fourth quarter.

**Figure 10.**Quarterly inflow forecasting curves conforming to drought stages for the Andong dam: (

**a**) first quarter; (

**b**) second quarter; (

**c**) third quarter; and (

**d**) fourth quarter.

**Figure 13.**The results of quarterly inflow forecasting without or with drought forecast under the threshold probability of drought occurrence to 50%: (

**a**) Soyanggang dam; (

**b**) Andong dam.

**Figure 14.**The results of quarterly inflow forecasting without or with drought forecast under the threshold probability of drought occurrence ranging 50–55% for the Andong dam.

SII Range | Drought Category |
---|---|

2.00 ≤ SII | Extreme wet |

1.50 ≤ SII < 2.00 | Very wet |

1.00 ≤ SII < 1.50 | Moderately wet |

0.00 ≤ SII < 1.00 | Near normal |

−1.00 ≤ SII < 0.00 | Mild drought (D1) |

−1.50 ≤ SII < −1.00 | Moderate drought (D2) |

−2.00 ≤ SII < −1.50 | Severe drought (D3) |

SII < −2.00 | Extreme drought (D4) |

Engineering Data | Soyanggang Dam | Andong Dam |
---|---|---|

Basin area (km^{2}) | 2703 | 1584 |

Year completed | 1973 | 1976 |

Total storage capacity (× 10^{6} m^{3}) | 2900 | 1248 |

Average yearly inflow (× 10^{6} m^{3}) | 1750 | 940 |

Yearly water supply (× 10^{6} m^{3}) | 1213 | 926 |

industrial water | 1200 | 450 |

agricultural water | 13 | 300 |

instream flow | - | 176 |

Dam | Distribution | Quarter | Threshold Value | |||
---|---|---|---|---|---|---|

First | Second | Third | Fourth | |||

Soyanggang | Lognormal | 0.0724 | 0.0636 | 0.0898 | 0.1242 | 0.219 |

Gamma | 0.1062 | 0.0898 | 0.1110 | 0.1583 | ||

Gumbel | 0.2189 | 0.2029 | 0.1534 | 0.2334 | ||

Weibull | 0.1169 | 0.1127 | 0.1261 | 0.1601 | ||

Gaussian | 0.1691 | 0.1570 | 0.1438 | 0.2082 | ||

Andong | Lognormal | 0.0915 | 0.1001 | 0.1057 | 0.0813 | 0.232 |

Gamma | 0.1267 | 0.1123 | 0.0771 | 0.1072 | ||

Gumbel | 0.2457 | 0.1930 | 0.1288 | 0.2396 | ||

Weibull | 0.1394 | 0.1038 | 0.0736 | 0.1380 | ||

Gaussian | 0.1871 | 0.1205 | 0.0932 | 0.1713 |

Dam | First and Second Quarters | Second and Third Quarters | Third and Fourth Quarters | Fourth and First Quarters |
---|---|---|---|---|

Soyanggang | 0.4781 | 0.0543 | –0.1770 | 0.0739 |

Andong | 0.3767 | 0.4828 | 0.1281 | 0.1831 |

Cases | Absolute Error for the Third Quarter 2014 (%) | Absolute Error for the Third Quarter 2015 (%) | Range of Absolute Error for the Third Quarter (%) | Range of Absolute Error for All Quarters (%) |
---|---|---|---|---|

BN forecast without drought forecast | 103.6 | 93.3 | 4.1–103.6 | 3.2–103.6 |

BN forecast with drought forecast by 50% criteria | 14.2 | 18.6 | 4.1–48.0 | 3.2–100.1 |

Cases | Absolute Error for the Third Quarter 2013 (%) | Absolute Error for the Third Quarter 2015 (%) | Range of Absolute Error for the Third Quarter (%) | Range of Absolute Error for All Quarters (%) |
---|---|---|---|---|

BN forecast without drought forecast | 169.7 | 355.6 | 2.2–355.6 | 1.2–355.6 |

BN forecast with drought forecast by 50% criteria | 169.7 | 32.1 | 32.1–169.7 | 1.2–169.7 |

BN forecast with drought forecast by 55% criteria | 169.7 | 9.4 | 9.4–169.7 | 1.2–169.7 |

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**MDPI and ACS Style**

Kim, K.; Lee, S.; Jin, Y.
Forecasting Quarterly Inflow to Reservoirs Combining a Copula-Based Bayesian Network Method with Drought Forecasting. *Water* **2018**, *10*, 233.
https://doi.org/10.3390/w10020233

**AMA Style**

Kim K, Lee S, Jin Y.
Forecasting Quarterly Inflow to Reservoirs Combining a Copula-Based Bayesian Network Method with Drought Forecasting. *Water*. 2018; 10(2):233.
https://doi.org/10.3390/w10020233

**Chicago/Turabian Style**

Kim, Kwanghoon, Sangho Lee, and Youngkyu Jin.
2018. "Forecasting Quarterly Inflow to Reservoirs Combining a Copula-Based Bayesian Network Method with Drought Forecasting" *Water* 10, no. 2: 233.
https://doi.org/10.3390/w10020233