# Risk Analysis of Reservoir Flood Routing Calculation Based on Inflow Forecast Uncertainty

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Basin and Forcing Data

^{2}, with a river length of about 86 km. Mountain and hill are two main landforms, and the topography decreases from south to north in the basin. The upstream mountainous area is characterized by steep slope, rapid-flow, and high forest coverage, while most area in the downstream basin is also covered by vegetation. The study area is dominated by the northern subtropics monsoon climate, with mean annual precipitation of about 1400 mm, but up to 60% of the precipitation occurs during the period of June–September. In the basin, the dominating runoff mechanism is saturation excess (Dunne) runoff.

^{3}/s if RWL < 130.07 m; (2) the discharge rate should be increased gradually when RWL > 130.07 m; and (3) all tunnels and spillways should be opened to maximize the discharge rate when RWL > 133 m. The relationship between RWL, reservoir water volume (RWV), and discharge rate is shown in Figure 2. In addition, the characteristic water levels for the Meishan Reservoir are also available [20]: dead water level = 107.07 m, normal water level = 126 m, top level of flood control = 133 m, design flood level = 139.17, and check flood level = 140.77 m.

#### 2.2. The Xinanjiang Model

_{P}= KC·E

_{0}) based on a three-layer soil moisture model, where E

_{0}is pan evaporation and KC is a ratio.

- when WU + P ≥ E
_{P},EU = E_{P}, EL = 0, ED = 0; - when WU + P < E
_{P}and WL ≥ C·WLM,EU = WU + P, EL = (E_{P}− EU)·WL/WLM, ED = 0; - when WU + P < E
_{P}and C·(EP − EU) ≤ WL < C·WLM,EU = WU + P, EL = C·(E_{P}− EU), ED = 0; - when WU + P < E
_{P}and WL < C·(EP − EU),EU = WU + P, EL = WL, ED = C·(E_{P}− EU) − EL;

#### 2.3. The HUP Model

_{0}is the given observed flow when it is forecasted; the variables H

_{n}and S

_{n}(n = 1, 2,…, N) are the actual flow process and the forecasted flow process of deterministic hydrologic model, respectively; and N is the lead time. The specific values h

_{n}and s

_{n}are used instead of the observed value H

_{n}and the estimated value S

_{n}, respectively. Based on the Bayesian theorem, when S

_{n}= s

_{n}, for any time n and observations H

_{0}= h

_{0}, the posterior density function of H

_{n}is as [26]:

_{n}takes the natural uncertainty of hydrologic elements into consideration; the likelihood function f

_{n}describes the uncertainty of hydrologic model and parameters constructed from the information of the samples; the posterior density function ${\Phi}_{n}$ is estimated using the total probability formula. Obviously, the posterior density function contains both the priori information and sample information. Equation (8) shows that the posterior density function of the actual flow H

_{n}is subjected to two values. One is measured flow H

_{0}when it begins to be forecasted, and the other is the deterministic forecasting value S

_{n}for the corresponding time. Therefore, the HUP provides probabilistic forecasts based on the deterministic forecasting model (Xinanjiang model).

_{0}) and the simulated flow with the lead time of one hour (S

_{1}) in the Meishan basin. The parameters of the log-Weibull distribution were estimated by the method of Moments.

#### 2.4. Risk Rate in Reservoir Flood Routing Calculation

_{1}, q

_{1}, and V

_{1}are the inflow, discharge rate, and RWV at time t

_{1}, respectively, while Q

_{2}, q

_{2}, and V

_{2}are the corresponding values at time t

_{2}; $\overline{Q}$ and $\overline{q}$ are the average inflow and outflow of the reservoir in the time step $\Delta t$; and $\Delta V$ is the change of RWV in the time step $\Delta t$. This equation ignores the water traveling time from the points that the basin runoff flows into to the flow-releasing structures. Generally, the relationship between discharge (q) and water level (Z) or water volume (V) can be described with a single function:

_{t}are the risk rate and RWL at time t, respectively; and Z

_{control}is the defined safety control water level that could be set as the characteristic reservoir water level. The generalized concept of “risk” involves not only the flooding probability (i.e., the hazard), but also the exposure loss. The risk rate used here describes the hazard, but no exposure loss was involved. The calculation of risk rates during a reservoir flood routing process is shown in Figure 3. Obviously, when using deterministic inflow process, Z(t), there would not be any risk during the reservoir flood routing calculation. However, there would be the probability of Z

_{t}> Z

_{control}described by PDF of water level at each time step. This probability is different during the reservoir flood routing calculation process, and the maximum value is defined as the risk rate for the entire process:

## 3. Results and Discussion

#### 3.1. Calibration and Validation of Xinanjiang Model

_{RE}, %), flood volume error (FV

_{RE}, %), peak flow occurrence time error (OT

_{RE}, hours) and Nash–Sutcliffe coefficient (NS). In the calibration, the NS varies from 0.68 to 0.92, with an average of 0.78, showing a good agreement with the observed hydrographs (Figure 4a). In the detection of peaks, only one event does not meet the accuracy requirement of |PK

_{RE}| < 20%, and most events have smaller peak errors. Furthermore, the simulated peak occurrence times were also close to the observation, with a maximum bias of 3 h. This proved the applicability of the model in peak detection. In addition, the FV

_{RE}statistic also had acceptable accuracy when only one event moderately exceeded the limit line of |FV

_{RE}| < 20%.

_{RE}and OT

_{RE}measures. It is worth mentioning that PK

_{RE}had a narrower range in validation than in calibration. This may be caused by the relatively smaller number of flood events in the validation than in the calibration period. In addition, the parameter uncertainty in the calibration (and random biases) may provide a better accuracy of a certain index (PK

_{RE}in this case) in the validation. For the Meishan basin, the accuracy of peaks slightly decreased in the measure of PK

_{RE}, but was still within the 20% limit. The difference of model performance between these two hydrologic stations could also be found in the measure of FV

_{RE}. The maximum bias of flood volume at the Huangnizhuang station was FV

_{RE}= −4.3% for the 16th flood event (Figure 4b), while the large biases of the Meishan basin could be found at the 3rd and 9th events (−21.0% and 24.4% for FV

_{RE}, respectively, Figure 4c). As a whole, although there were several large biases in the validations, the calibrated parameters could be assumed reasonable for the Meishan basin with the consideration of all test results.

#### 3.2. Inflow Forecast Uncertainty Analysis

_{RE}= −2.55% and NS = 0.95 for all ten flood events. The simulated peaks also showed high accuracies in both the value and corresponding occurrence time, while PK

_{RE}ranged from −12.3% to −0.5% with an absolute value of 6.84%, and the average bias of occurrence time was one hour. Obviously, the mean forecast of HUP showed a better agreement than the Xinanjiang model, whose main biases could be found in low flows and the peak occurrence time.

#### 3.3. Risk Rate Calculation

^{3}/s for all time steps. Thus, only four events with relatively large flows (i.e., the 5th, 7th, 8th, and 9th floods) are shown in Figure 6. At each time step, both deterministic and probabilistic forecasts (such as the 50th percentiles and 90% confidence intervals, respectively) could be obtained based on the PDFs of RWL. Results suggested that the forecast uncertainty increased from low to high RWL values, indicating by the width of uncertainty band.

_{control}), the risk rates can be calculated. In general, the Z

_{control}value is defined as a characteristic reservoir water level; e.g., top level of flood control, design flood level, or check flood level. For the cases in this study, however, the possible RWL values of all selected flood events (derived from its PDF) were far less than these characteristic levels; i.e., P

_{r}= 0 (no risk). In order to demonstrate the risk rate calculation procedure, we defined the Z

_{control}value as a relatively low water level (Z

_{control}= 125.5 m) and calculated the risk rate at each time step (Figure 6). Results suggested that the risk rate largely depends on the forecast uncertainty of RWL. High flows and wide uncertainty bands were more likely to produce high risk rates, such as the 7th flood shown in Figure 6b. As a negative example, in the 9th flood, the PDF of RWL at peak time indicated that RWL had no possibility of reaching the safety control water level, and thus the risk rate for this flood was zero. The risk rates of the reservoir flood routing calculation could provide reliable references for flood control and reservoir regulation decision-making.

## 4. Conclusions

_{RE}, OT

_{RE}, FV

_{RE}, and NS) meet the accuracy requirements. In addition, the validation in the entire Meishan basin also proved the model applicability in the reservoir inflow forecasts; (2) The total uncertainty (except for the input uncertainty) in the inflow forecasts was quantified by the HUP model, providing the PDF of inflow and corresponding PFF results (e.g., 90% confidence intervals) at each time step; (3) Based on the posterior PDFs of inflow forecasts derived by the HUP model, a large number of inflow samples that were randomly extracted from inflow’s PDF could generate the RWL (and its PDF) during flood events through the reservoir scheduling simulation. Thus, the risk rate of RWL exceeding the safety control level (Z

_{control}) could be calculated for the references in the reservoir flood control and regulation decision-making.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**Relationship between water level, water volume and discharge rate of the Meishan Reservoir. RWL: reservoir water level.

**Figure 3.**Sketch of the calculation of risk rates during a reservoir flood routing process. PDF: probability density function.

**Figure 4.**Comparisons between simulated and observed flows of flood events: (

**a**) calibration at the Huangnizhuang station; (

**b**) validation at the Huangnizhuang station; and (

**c**) validation in the entire Meishan basin. The box-plots present the distributions of four statistics (peak flow error, PK

_{RE}; flood volume error, FV

_{RE}; peak flow occurrence time error, OT

_{RE}; and Nash–Sutcliffe coefficient, NS) for all considered flood events. The numbers on the top axes—with the two-digit format for flood ID and three-digit for ending time step—separate the artificially attached series of flood events.

**Figure 5.**Hydrologic Uncertainty Processor (HUP) forecasts compared to the observations and the simulations of Xinanjiang model. Green boxes magnify the zone of peak time steps. The same flood events are used as shown in Figure 4c. The numbers on the bottom axes—with the two-digit format of for flood ID and three-digit for ending time step—separate the artificially attached series of flood events.

**Figure 6.**Risk rate calculation of selected flood events for the Meishan Reservoir. The grey small graph is the probability density function (PDF) of reservoir water level (RWL) at the peak occurrence time step.

**Table 1.**Calibrated parameter values of the Xinanjiang model for flood events shown in Figure 4a.

Evapotranspiration | Runoff Production | Runoff Separation | Runoff Concentration |
---|---|---|---|

KC = 0.998 | WM = 120 mm | SM = 10 mm | CI = 0.98 |

WUM = 20 mm | B = 0.4 | EX = 1.2 | CG = 0.85 |

WLM = 60 mm | IM = 0.1 | KG = 0.45 | CS = 0.01 |

C = 0.2 | KI = 0.25 | L ^{1} |

^{1}The “lag” parameter, L, is set to 3 h for two upstream sub-basins above the Huangnizhuang station (in the lower left zone of Figure 1), and to one hour for all other sub-basins.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Li, B.; Liang, Z.; Zhang, J.; Chen, X.; Jiang, X.; Wang, J.; Hu, Y.
Risk Analysis of Reservoir Flood Routing Calculation Based on Inflow Forecast Uncertainty. *Water* **2016**, *8*, 486.
https://doi.org/10.3390/w8110486

**AMA Style**

Li B, Liang Z, Zhang J, Chen X, Jiang X, Wang J, Hu Y.
Risk Analysis of Reservoir Flood Routing Calculation Based on Inflow Forecast Uncertainty. *Water*. 2016; 8(11):486.
https://doi.org/10.3390/w8110486

**Chicago/Turabian Style**

Li, Binquan, Zhongmin Liang, Jianyun Zhang, Xueqing Chen, Xiaolei Jiang, Jun Wang, and Yiming Hu.
2016. "Risk Analysis of Reservoir Flood Routing Calculation Based on Inflow Forecast Uncertainty" *Water* 8, no. 11: 486.
https://doi.org/10.3390/w8110486