# Probability-Based Rule Curves for Multi-Purpose Reservoir System in the Seine River Basin, France

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Introducing the Case Study: Seine River Basin, France

^{2}, with four large reservoirs situated on the Yonne, Seine, Aube, and Marne Rivers. The Pannecière reservoir located on the Yonne River is known as an inline reservoir, while the other three reservoirs (Seine, Aube, and Marne) are bypass reservoirs. The other main features of these reservoirs are summarized in Table 1. These reservoirs, which are under the management of the Seine Grands Lacs (SGL), responsible for managing the infrastructure system in the Seine River Basin, France, are operated in a parallel mode for flood protection and water shortage relief in the Parisian region.

#### 2.2. Data Sources

#### 2.3. Objective Function

^{+}operator represents the maximum value between its argument and zero; Q

_{up}refers to inflow from upstream; Q

_{down}stands for the sum of reservoir release (R

_{t}) and lateral flow from downstream sub-catchment; Q

_{t}is inflow during time step t; q

_{min}and q

_{max}are minimum and maximum inflow to reservoirs; u

_{min}and u

_{max}are minimum and maximum limits of flow at the upstream; d

_{min}and d

_{max}are minimum and maximum limits of flow at the downstream; and r

_{min}and r

_{max}are minimum and maximum release from reservoirs.

_{min}) which is required at point A for environmental protection between sections A and B (see Figure 3). When considering the low-flow season (Q

_{up}is low), the priority of supply will be given to section A; however, if high flow occurs (Q

_{up}is high), the maximum flow at point A will be restricted by a threshold value (u

_{max}). Thereafter, the excess water will be diverted to a nearby reservoir—the so-called bypass reservoir—as much as possible to avoid any damage to the river between sections A and B.

_{min}to d

_{max}. In cases of low-flow events, the supply (R

_{t}) will be maximized at the outlet to satisfy downstream needs, whereas the supply will be limited if the flow at point B reach the d

_{max}cutoff point.

_{min}and s

_{max}are the minimum and maximum reservoir storage, respectively, and S

_{t}is the estimated storage at time step t.

_{x}(for flood protection) or greater or equal to low-flow thresholds, l

_{x}(for low-flow protection), as described in Equations (8) and (9), respectively.

#### 2.4. Application of Dynamic Programming for Optimization of Reservoir Operation

_{1}, R

_{1}}, …, {Q

_{T}, R

_{T}} such that the total cost of the system J in the objective function (Equation (1)) is optimized. It should be noted that decision variables for the bypass reservoir system are not only limited to the release (R) but also for inflow (Q) (see Equations (2) and (4)). In view of continuous-time variational control problems, it can be described when the control is assumed to be piecewise constant in time and appropriate transformations is transformed into the discrete-time case [50]. To deal with the above problem, the iterative functional equation that determines the optimal control for any admissible state at any stage (t) is a way to dynamic programming solution. Referring to Equation (1), the minimum-cost function (ξ) is defined for all states S and all times t = 1, …, T, as can be shown in Equation (10) [51].

_{t+1}= Φ, where Φ is a function of water balance of reservoirs. The iterative functional equation can then be written as:

_{t}, R

_{t}}, can be defined as the values of Q

_{t}and R

_{t}in which the minimum in Equation (11) is achieved. The determination of ξ(S, t) and $\widehat{U}$(S, t) are expressed in terms of ξ(S, t + 1), and the problem can then be solved backward in t. Considering the terminal boundary condition, the minimum cost function can be rewritten as:

_{T}for each time step of the simulation period. In this study, the optimization was seeded with an empty reservoir volume (S

_{T}= 0) for the optimization of the lower rule curve and with a fully filled reservoir (S

_{T}= S

_{max}) for the generation of the upper rule curve. As for the worst-case scenarios, this assumption will help the optimization to maximize as much as possible the space available in each reservoir for ensuring that enough water is available to meet downstream needs during low-flow situations while preventing a surplus of water under high-inflow conditions. In this case, the optimization algorithm was applied to find out the optimal storage level of each day of the year. It is important to note that since the evaporation loss depends largely on climate and the surface area of stored water, the evaporation rate was then neglected as the reservoirs considered in this study are rather small, and also to simplify and avoid complexities in the analysis. Likewise, the seepage loss was also ignored due to insufficient data acquired from the field. However, both losses can be a very significant factor in water balance calculation, which should be considered in future applications.

#### 2.5. Joint Operation of Multiple Reservoir System

_{all}is the total downstream water demand from all reservoirs.

_{t}, R

_{t}}.

#### 2.6. Identification of Optimal Rule Curves

## 3. Results

#### 3.1. Efficient Water Allocation under Reservoir Joint Operation

#### 3.2. Annual Optimal Rule Curves

#### 3.2.1. The Proposed URC for Flood Control

^{3}/s at the Paris gauging station, as depicted in Figure 6. When comparing the current management practices, the URC of Aube reservoir shows a lower reservoir storage during the summer from August to September than in July, as indicated by the red dot. The results of the 50- and 100-year return periods suggest impounding up to the gross storage capacity of the reservoir during May to August, which is higher than the current operation management. It can be said that the proposed upper rule curve would help to ensure that the effects of water shortage can be attenuated during the summer months, while lowering the reservoir storage prior to the flood season could ensure extra capacity for flood storage. To compare the proposed URC of Seine reservoir with its current operation management, a relatively low storage capacity at 50- and 100-year return periods was observed during the dry period, which can lead to increasing the reservoir storage capacity and offering space for accommodating floodwater. Interestingly, although both reservoirs are close to each other, the mean annual inflow of the Seine is almost double that of the Aube and it is the main water supply contributor for Aube and Pannecière reservoirs in the Seine tributary (see Figure 1). Focusing on the Pannecière, the smallest reservoir in the Seine River Basin that regulates the ecosystems and habitats along the Yonne and Seine tributaries, the adjustment of URC can cause a reduction in water supply during the summer compared to the current management. The withdrawal at the red dot in November, for example, can be considered an oversupply, as recommended by the SGL (personal communication, 15 May 2020), in which it would probably cause excessive water in the Seine tributary and subsequently flooding in Paris City. It was also found that the water demand from the Pannecière is marginal at the rate of 8 MCM per year, and this is due to a significant contribution from the Seine reservoir. Regarding the Marne, which has the largest capacity in the Seine River Basin, it is known as the main source of water supply and flood control in the Marne sub-basin and Paris region. Similarly to the other reservoirs, under the new derived URC of the Marne and at high return periods, there was a substantial withdrawal rate during the summer months. The reducing pattern is slightly similar with the current management policy, as denoted in the 50- and 100-year return periods. Subsequently, the flood peak entering the Marne can be absorbed especially for large flood frequency events due to the increase in available reservoir storage capacity. Obviously, it can be seen in the proposed URC that the reservoir storage capacities under higher return periods are lower than the lower ones. The reservoirs are recommended to operate at lower storage levels, especially for flows with higher return periods, in order to enable operation in high-flow conditions. Then, it is important to mention that the provision of empty storage space available in the reservoirs should be considered to give extra room for flood attenuation in downstream areas.

#### 3.2.2. The Proposed LRC for Low-Flow Support Objective

^{3}/s at Paris gauging station estimated by the Bernard and Bos-Levenbach approximation. There is no doubt that the characteristic components of the LRC of all major reservoirs is coincide with the URC, i.e., the rising limb can be seen during the rainy months, while the falling limb is more evident in summer. To avoid water shortage and ensure sufficient water quantity during abnormally dry periods, the storage in the buffer zone can only be used to satisfy essential and high priority needs. However, improper and rapid release of storage volume from this zone may result in a dam failure due to a rapid drawdown and a geological failure. Hence, it can be said that a single optimal solution may not satisfy the tradeoff between different objectives, but rather a set of potential solutions is required. Overall, the revised LRC was proposed to provide a guarantee of adequate reservoir capacity through the dry season without affecting the reliability of meeting various demands. It can be seen that for all LRCs under different return periods, there was a seemingly significant difference between the current operating reservoir storage and the proposed one, especially in the month of July (the Marne is the largest and the Pannecière is the smallest) and almost no difference in the month of November.

#### 3.3. Web-Based Application for Optimization of Multiple Reservoir Operation

## 4. Conclusions

^{3}/s at the Paris gauging station, the URC of Aube reservoir was proposed to postpone the water supply from July to August until September. For the Seine reservoir, since a relatively high withdrawal rate at 50- and 100-year return periods was noticed during the dry season, its storage capacity would then be increased, offering additional space for floodwater. The adjustment of URC of Pannecière reservoir would also cause a reduction in water supply during the summer compared to the current management. The withdrawal in November could be considered oversupply, since it would cause excessive water in the Seine tributary and Paris City. Under the new derived URC of the Marne reservoir, at high return periods, there was a substantial withdrawal rate during the summer months, which would increase the available reservoir storage capacity for flood absorption. Above all, considering all of the proposed URCs, it was found that reservoir storage under higher return periods is allowed to go below the lower return periods, in order to enable operation under high-flow conditions. Under the low-flow threshold of 81 m

^{3}/s at Paris gauging station, the trend of the LRC of all main reservoirs appeared to correspond with the trend of the URC, as an increasing trend was noticed during the rainy season and the decreasing trend was noted in summer. Clearly, when comparing all proposed LRCs under different return periods with the existing ones, a clear difference was seen in the month of July (with the largest in the Marne and the smallest in the Pannecière), while almost no difference was found in November. In brief, the optimal operation rule curves suggested a reduction in water supply from reservoirs in the dry season, as water shortage is not the primary constraint for Seine River Basin. In addition, the web-based support system for reservoir operation Interactive Reservoir MAnagement Risk Assessment (IRMaRA) was also developed and deployed for public access in an easily accessible and user-friendly format, which enables users to execute the application and view the results. The obtained results from this study revealed that DP provides considerably higher computational efficiency, which enables the optimal joint operation of the multi-reservoir system to be achievable. Eventually, a set of proposed optimal rule curves based on the current reservoir storage and possible inflow scenarios can be considered as guidelines for formulating reservoir sustainability plans of the Seine River Basin.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Historical mean monthly inflow and release from the four large reservoirs located in the Seine River Basin between 1900 and 2009.

**Figure 4.**Basic steps for the application of the optimization and simulation approach for searching optimal operation of reservoir systems.

**Figure 5.**The plots of proportions of (

**a**) expected inflow to each reservoir and (

**b**) allocated water from each reservoir.

**Figure 6.**Proposed optimal upper and lower rule curves for four major reservoirs located in the Seine River Basin based on the threshold of 950 m

^{3}/s and 81 m

^{3}/s at the Paris gauging station, respectively.

Elements | Pannecière | Seine | Aube | Marne |
---|---|---|---|---|

Type of reservoir | Inline | Bypass | Bypass | Bypass |

Water surface area (km^{2}) | 5.2 | 23.2 | 23.2 | 28 |

Catchment area (km^{2}) | 220 | 2380 | 1650 | 2900 |

Maximum dam height (m) | 49.0 | 25.0 | 22.5 | 20.0 |

Crest length (m) | 352 | 5700 | 3500 | 20,300 |

Gross storage capacity (MCM) | 82.5 | 219.5 | 183.5 | 364.5 |

Live storage capacity (MCM) | 74.0 | 212.9 | 181.2 | 354.5 |

Dead storage capacity (MCM) | 8.5 | 6.6 | 2.3 | 10.0 |

Monitoring Station | River | Impact | Low Flow Thresholds (m^{3}/s) | High Flow Thresholds (m^{3}/s) | |||||
---|---|---|---|---|---|---|---|---|---|

Vigilance | Alert | Reinforced Alert | Crisis | Vigilance | Regular | Exceptional | |||

Arcis-sur-Aube | Aube | A | 6.3 | 5 | 4 | 3.5 | 110 | 260 | 400 |

Mery-sur-Seine | Seine | S | 7.3 | 5 | 4 | 3.5 | 140 | 170 | 400 |

Nogent-sur-Seine | Seine | A + S | 25 | 20 | 17 | 16 | 180 | 280 | 420 |

Gurgy | Yonne | P | 14 | 12.5 | 11 | 9.2 | 220 | 340 | 400 |

Courlon-sur-Yonne | Yonne | P | 23 | 16 | 13 | 11 | 550 | 700 | 900 |

Alfortvile | Seine | A + S + P | 64 | 48 | 41 | 36 | 850 | 1200 | 1400 |

Chalons-sur-Marne | Marne | M | 12 | 11 | 9 | 8 | 330 | 520 | 700 |

Noisiel | Marne | M | 32 | 23 | 20 | 17 | 350 | 500 | 650 |

Paris | Seine | A + S + P + M | 81 | 60 | 51 | 45 | 950 | 1600 | 2000 |

**Note:**A = Aube, M = Marne, P = Pannecière, S = Seine.

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## Share and Cite

**MDPI and ACS Style**

Dau, Q.V.; Kangrang, A.; Kuntiyawichai, K.
Probability-Based Rule Curves for Multi-Purpose Reservoir System in the Seine River Basin, France. *Water* **2023**, *15*, 1732.
https://doi.org/10.3390/w15091732

**AMA Style**

Dau QV, Kangrang A, Kuntiyawichai K.
Probability-Based Rule Curves for Multi-Purpose Reservoir System in the Seine River Basin, France. *Water*. 2023; 15(9):1732.
https://doi.org/10.3390/w15091732

**Chicago/Turabian Style**

Dau, Quan Van, Anongrit Kangrang, and Kittiwet Kuntiyawichai.
2023. "Probability-Based Rule Curves for Multi-Purpose Reservoir System in the Seine River Basin, France" *Water* 15, no. 9: 1732.
https://doi.org/10.3390/w15091732