# Comparing Simple Flood Reservoir Operation Rules

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## Abstract

**:**

## 1. Introduction

## 2. Methods

_{p,in}), peak outflow (Q

_{p,out}), peak flow reduction (Δq

_{p}), total flow duration (D), maximum inflow duration (d), flood storage volume (V), and ramping rates (r

_{1},r

_{2}). The non-shaded region is the reservoir’s outflow over time, which reaches a maximum of Q

_{p,out}. Flood volume is the integral of each hydrograph shape. The quantity Δq

_{p}is the maximum reduction in peak flow, defined as the maximum inflow minus the maximum outflow. The percent peak reduction is calculated by dividing the maximum reduction in peak flow by the maximum inflow (Δq

_{p}/Q

_{p,in}). The rising slope r

_{1}and recession slope r

_{2}are the rise and fall of the flow into the reservoir per unit time. The hydrograph’s total flow duration includes all time periods in which the flow exceeds a base (zero for this study).

**Figure 1.**Four simple hydrographs and optimal peak flow reductions for a fixed flood storage volume [2]. (

**a**) Triangular; (

**b**) Abrupt wave; (

**c**) Flood pulse; (

**d**) Broad peak.

_{1}= r

_{2}). The total flow duration of each hydrograph D (length of the base on the time axis) is set to be 100 time periods for all runs. The broad peak hydrograph maintains a maximum inflow for 60 time periods in all cases. All four hydrograph shapes have the same flood volume for each case. The model’s downstream channel capacity can vary. The reservoir model presented does not consider effects of hydrology or attenuation of the flood wave as it passes through a simulated reservoir. The hydrographs used in analysis are deterministic and idealized rather than based on a probability distribution of precipitation and runoff.

## 3. Results and Discussion

#### 3.1. Minimize Flood Peak (MFP) with Perfect Forecasting

_{p}) can be calculated using the peak inflow and volume for a given flood storage capacity, as in Table 1 for these simple hydrograph shapes.

_{p,out}) are calculated for each hydrograph. As inflow begins to increase, releases increase to match inflow. If storage is not empty, the reservoir would initially release more than the inflow to create additional flood storage. Once inflow reaches the calculated ideal peak outflow, the outflow is capped at this optimal rate and held constant until after the flood peak. Once the peak outflow is reached, the reservoir begins to store water as inflow exceeds reservoir outflow. Inflow increases to a maximum and then begins to decrease, with the reservoir continuing to store water. Eventually, the inflow decreases to less than the ideal peak outflow, and the reservoir begins to drain. When enough water has been released to evacuate the flood storage volume, reservoir outflow is once again synchronized with inflow.

**Table 1.**Maximum peak reduction and ideal peak outflow for simple hydrographs given peak inflow (Q

_{p,in}), flood duration (D), maximum inflow duration (d) and flood storage volume V Δq

_{p}= Q

_{p,in}− Q

_{p,out.}

Shape | Triangular | Abrupt wave | Flood pulse | Broad peak |
---|---|---|---|---|

Stored volume V = | DΔq_{p} | |||

Peak flow reduction Δq_{p} = | ||||

Peak flow Q_{p,out} = |

**Figure 2.**Minimize flood peak with perfect forecasting for (

**a**) Abrupt wave hydrograph and (

**b**) Broad peak hydrograph.

**Table 2.**Modeled results of the minimize flood peak with a perfect forecast operating rule. Storage = 3000 units, Percent of the hydrograph contained = 30%.

Hydrograph shape | Triangle | Abrupt wave | Pulse | Broad peak |
---|---|---|---|---|

% Peak reduction | 55 | 55 | 30 | 36 |

Optimal outflow | 90.5 | 90.5 | 70 | 80.3 |

#### 3.2. Minimize Flooding Frequency (MFF)

**Figure 5.**Minimize flood frequency (flooding prevented) for (

**a**) Triangular hydrograph and (

**b**) Broad peak hydrograph.

**Figure 6.**Minimize flood frequency rule (flooding occurs) for (

**a**) Triangular hydrograph and (

**b**) Broad peak hydrograph.

**Figure 7.**Efficiency of the MFF rule in reducing peak flow normalized by the peak flow reduction of the MFP rule, for varying channel capacities normalized by the ideal peak outflow of the MFP rule.

#### 3.3. Short Forecast Peak Minimization (SFPM)

_{t}), the SFPM rule increases the current release rate (R

_{t-1}) in the next time-step (to R

_{t}) so peak release over the forecast period (f) is minimized. The forecast flows from t until the end of the forecast (t + f)(I

_{t}(τ)) are optimally stored over the forecast period f for a peak-minimizing release:

**Figure 8.**Short forecast peak minimization operation rule (with flooding prevented) for (

**a**) Triangular hydrograph and (

**b**) Broad peak hydrograph.

**Figure 9.**Short forecast peak minimization (when flooding occurs) for (

**a**) Triangular hydrograph and (

**b**) Broad peak hydrograph.

**Figure 10.**Comparison of minimize flood frequency (MFF) and short forecast peak minimization (SFPM) rules for triangle and abrupt wave hydrographs.

**Figure 11.**Comparison of Minimize Flood Frequency (MFF) and Short Forecast Peak Minimization (SFPM) rules for Flood Pulse and Broad Peak Hydrographs.

**Figure 12.**Effect Forecast Period on Operation using the SFPM Rule for (

**a**) Triangular Hydrograph and (

**b**) Broad Peak Hydrograph.

#### 3.4. Peak Inflow: Outflow Curves

**Figure 13.**Peak inflow vs. peak outflow for (

**a**) Triangular hydrograph and (

**b**) Abrupt wave hydrograph (

**c**) Flood pulse hydrograph and (

**d**) Broad peak hydrograph.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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## Share and Cite

**MDPI and ACS Style**

Connaughton, J.; King, N.; Dong, L.; Ji, P.; Lund, J.
Comparing Simple Flood Reservoir Operation Rules. *Water* **2014**, *6*, 2717-2731.
https://doi.org/10.3390/w6092717

**AMA Style**

Connaughton J, King N, Dong L, Ji P, Lund J.
Comparing Simple Flood Reservoir Operation Rules. *Water*. 2014; 6(9):2717-2731.
https://doi.org/10.3390/w6092717

**Chicago/Turabian Style**

Connaughton, James, Natalie King, Licheng Dong, Patrick Ji, and Jay Lund.
2014. "Comparing Simple Flood Reservoir Operation Rules" *Water* 6, no. 9: 2717-2731.
https://doi.org/10.3390/w6092717