Dynamic Control of Flood Limited Water Levels for Parallel Reservoirs by Considering Forecast Period Uncertainty

Abstract

The objective of this study is to achieve the dynamic optimization of the flood limited water level (FLWL) in parallel reservoirs, using Luhun Reservoir and Guxian Reservoir as case studies. The innovation lies in establishing a dynamic control optimization model for the FLWL of parallel reservoirs, considering the uncertainty in the forecasting period of the flood forecast due to the varying locations of the rainstorm center from upstream to downstream. To commence, the Fisher optimal segmentation method is employed for flood season staging to determine the staged FLWL of each reservoir. Subsequently, considering the uncertainty in the foresight period, the upper range of the dynamic FLWL is determined through the improved pre-discharge capacity constraint method and Monte Carlo simulation. Finally, a multi-objective optimization model is established to determine the optimal dynamic FLWL control operation scheme for parallel reservoirs, utilizing the Non-Dominated Sorting Genetic Algorithm II (NSGA-II). This model takes into account both downstream flood control requirements and the water supply benefits of the parallel reservoirs. Through the optimization of the scheme, the water supply of the parallel reservoirs can be augmented by 15,347.6 m³ during the flood season. This optimization effectively achieves a harmonious balance between flood control and water supply, holding significant implications for mitigating drought risks amid changing conditions.

1. Introduction

As the economy and society continue to grow, the demand and supply imbalance of water resources has intensified [1]. Addressing water scarcity due to drought has become an urgent challenge [2]. Reservoirs, as important man-made structures, play a crucial role in fully utilizing flood resources and significantly increasing the capacity for integrated water resource development and utilization [3]. However, the escalating impacts of global climate change have led to a significant increase in the frequency, extent, and intensity of regional droughts [4,5]. As a result, the problem of water scarcity has become more serious and drought-induced water shortages have hindered economic development and the improvement of living standards [6,7]. Both the water-scarce population and its proportion within the total population have exhibited a rising trend [8]. Therefore, how to further develop and utilize flood resources has become an important issue to be solved in reservoir research.

FLWL operation management has been transformed from static to dynamic. In recent years, various studies have explored the dynamic control of FLWL in multiple reservoirs. Chen et al. [9] proposed a dynamic FLWL optimization model that successfully achieved flood control and power generation in Qingjiang River terrace reservoirs. Zhou et al. [10] extended the water resources dynamic control model from single and terrace reservoirs to a mixed reservoir system. They analyzed joint scheduling scenarios for the Three Gorges Reservoir and Qingjiang River Terrace Reservoir, illustrating that joint and dynamic scheduling significantly increased the flood utilization rate of the mixed-gradient reservoir system. However, most of the previous calculations for the foreseeable period in the dynamic FLWL control are based on experience in order to determine a fixed value, which can be a better simulation of the actual situation, but since the impact of its uncertainty is not enough to consider, this study takes into account the location of the center of the rainstorm for the foreseeable flood period and the impact of the uncertainty of the dynamic control factors on the results, so that the results of the dynamic control can have a wider range of applicability.

Reservoirs serve various crucial functions, such as flood control, power generation, and navigation. While flood control remains paramount, pursuing other functionalities holds immense practical significance. Addressing the multi-objective optimization problem is a key challenge in reservoir scheduling research, and substantial experience has accumulated in this domain [11,12,13]. Chen et al. [14] utilized an improved multi-objective particle swarm algorithm to construct a multi-objective reservoir cluster in the Ganjiang River Basin, considering power generation, water supply, and ecological aspects. Lu et al. [15] created a water–sediment optimization model for the Yellow River’s Sanmenxia and Xiaolangdi cascade reservoirs, considering objective differences at different times. They provided a joint optimal water–sediment dispatching scheme for varying incoming water and sand scenarios.

As the number of objectives and reservoirs increases, the dimensionality of multi-objective optimization problems escalates, leading to the curse of dimensionality. In such scenarios, traditional nonlinear optimization methods are susceptible to local optima and struggle to identify global optimal solutions [16]. In recent years, intelligent algorithms [17] and evolutionary algorithms [18] have offered potential solutions to this problem. Among the available computational methods, the particle swarm optimization algorithm [19], ant colony algorithm [20], artificial neural networks [21], and especially genetic algorithms [22,23,24], along with their improved versions [25,26,27], have been widely adopted in optimal reservoir scheduling and management due to their notable advantages.

This study aims to provide a scientific approach for dynamically controlling the FLWL in parallel reservoirs through multi-objective optimization. Parallel reservoirs are reservoirs located on different rivers or on different tributaries of the same river. Focusing on Luhun Reservoir and Guxian Reservoir, situated on the Yellow River tributaries of Yi and Luo Rivers in Henan Province, China, this research commences by applying the Fisher optimal segmentation method to delineate the flood season stages. By incorporating the typical flood process line method and considering reservoir safety and downstream flood control, this study derives the staged FLWL for each reservoir, constituting the lower limit for dynamic FLWL. Subsequently, leveraging the Monte Carlo simulation, this research determines the upper range of the staged dynamic FLWL for each reservoir. This determination factors in the shifting rainstorm center in the upstream and downstream during various foreseeable periods, coupled with forecasting and dispatching uncertainties. Finally, an effective multi-objective optimization model, aiming to minimize downstream flood peaks and maximize water supply, is established. The NSGA-II algorithm, integrating an elite strategy, is employed to find the optimal upper limit value for the dynamic FLWL. This informs the formulation of practical reservoir operation schemes and control thresholds for dynamic FLWL, tailored to real-world operational contexts.

2. Materials and Methods

2.1. Study Area

The Yiluo River is situated in the northwest of Henan Province, flowing into the Yellow River from southwest to northeast. It is one of the tributaries of the Yellow River, possessing abundant water resources. Floods in the Yiluo River are significant contributors to the overall floods occurring between the Sanmenxia and Huayuankou regions along the Yellow River. The Yiluo River basin experiences uneven distribution of annual precipitation in both time and space. The mountainous areas receive higher rainfall, making them the rainy regions, while the river valley and nearby hilly areas receive comparatively less rainfall, making them less rainy regions. The precipitation is concentrated in the months of July and August, accounting for approximately 85% of the total precipitation during the flood season. The average annual temperature in the Yiluo River Basin is 7.8–13.9 °C, and the average annual rainfall is 690 mm. The control area of the basin’s outlet Heishiguan station accounts for about 98% of the basin’s area, with an average annual runoff of 146 m³/s. The predominant form of precipitation is heavy rainfall, characterized by large volume, extensive coverage, concentrated occurrence, and prolonged duration. Downstream areas are particularly vulnerable to flood threats, especially during the summer, which may lead to severe disasters and losses.

Henan Province has taken measures to address various risk challenges by establishing several significant reservoirs, among them, Luhun Reservoir and Guxian Reservoir. Luhun Reservoir controls a watershed area of 3492 km², accounting for 58% of the total watershed area of 6029 km² of the Yi River. The Guxian Reservoir controls a basin area of 5370 km², accounting for 44.6% of the basin area of the Luo River. Information on geomorphology and reservoir construction in the Yiluo River Basin is provided in Supplementary Materials. The rising riverbed caused by the significant issue of sediment siltation in the lower reaches of the Yellow River exacerbates the severity of flooding disasters and exerts a substantial impact on the river’s ecology [28]. The coordinated management and utilization of these two large reservoirs, Luhun and Guxian, effectively alleviate flood control pressure in the middle and lower reaches of the Yiluo River and reduce the flood risk in the downstream region of the Yellow River. In this paper, the parallel reservoirs Luhun Reservoir and Guxian Reservoir in Yiluo River Basin are studied as the scope of this study. Figure 1 illustrates the distribution of water systems and hydrological stations in the study area.

2.2. Data Sources

The hydrological data utilized in this research consisted of daily precipitation spanning a period of 61 years, from 1961 to 2021. The daily precipitation dataset was acquired from the National Meteorological Information Center of the China Meteorological Administration: ”http://data.cma.cn” (accessed on 15 March 2023). The main parameters of the distribution function in Monte Carlo simulations taken from the article [29]. The reservoir characteristic level parameters were acquired from local watershed management agencies. The characteristic parameters of Luhun Reservoir and Guxian Reservoir are shown in

Table 1. Characteristic water levels of Luhun and Guxian Reservoir.

Reservoir	Dead Pool Level (m)	Original FLWL (m)	Design Flood Level (m)	Check Flood Level (m)	Crest Elevation (m)	Requisitioned Land Water Level (m)
Luhun	298	317.5	327.50	331.80	333	321.5
Guxian	518	530	548.55	551.02	553	543.2

.

The scheduling and utilization principles for Luhun Reservoir and Guxian Reservoir are as described in detail in Supplementary Materials.

2.3. Methods

2.3.1. Staged FLWL

Designing FLWL based on the maximum annual design flood overlooks the prevalence of medium and small floods during actual operations. Adopting a fixed FLWL throughout the flood season [30] inevitably leads to wastage of flood resources, contrary to the current imperative of comprehensive water resource utilization. The staged FLWL accounts for the distinct characteristics of floods, enabling full utilization of flood resources throughout the flood season while maintaining the established flood control standard. Various methods exist to implement flood staging, including the qualitative analysis method [31], set pair analysis [32,33], change-point analysis [34], fuzzy set analysis [35], fractal clustering method [36], and Fisher’s optimal segmentation method [37,38], among others. By analyzing flood season staging results, the derived staged FLWL enables efficient utilization of flood resources throughout the season, especially crucial for reservoir operations in the later stages to prevent water scarcity towards the end of the flood season [10].

The Fisher optimal segmentation is a cluster analysis method applied to ordered time sample sequences. It is designed to minimize the total deviations of the samples, aiming for the least intraclass variation and the greatest interclass variation [39]. As a clustering approach for ordered samples, it can accommodate multiple factor indicators while preserving the original order, thus determining the optimal number of segments and flood season segmentation outcomes based on the defined objective function B(n,k). Specific formulas and their definitions can be found in Supplementary Materials. In this study, the Fisher segmentation method is adopted to stage the flood season, following the specific steps outlined in [40].

2.3.2. Dynamic Control of FLWL

(1)

Improved pre-discharge capacity constraint method

The improved pre-discharge capacity constraint method refers to how much the FLWL will be floated upward when the discharge capacity is available during the effective foresight period [41]. The improved pre-discharge capacity constraint method extends the original approach by incorporating rainfall forecast information. It relies on the water balance principle and is governed by the following basic formula:

$$w_{yx}=\sum _{t_{cu}}^{t_{cu}+T_y}\left[q_{out}\left(t\right)-Q_{in}\left(t\right)\right]\times \mathsf{\Delta }t$$

(1)

where $w_{yx}$ is the amount of water that can be pre-discharged during the effective foresight period; $Q_{in}\left(t\right)$ is the forecasted inlet flow process during the effective foresight period; $q_{out}\left(t\right)$ is the process of discharging flow during the effective foresight period; $\mathsf{\Delta }t$ is the length of the calculation time period; $t_{cu}$ is the starting moment of the implementation of the reservoir’s real-time dynamic control; $t_{cu}=t_0+T_{cu}$, $t_0$ is the time of the operational forecast, $T_{cu}$ is the sum of staff time for information transfer, decision-making, gate opening, $T_y$ is the effective pre-discharged time after considering flood forecast and rainfall forecast information.

According to the obtained $w_{yx}$, the upper limit value of the dynamic control bound of the FLWL ${Z_d}^{+}$ is deduced from Equation (2).

$${Z_d}^{+}=f\left(V\left(Z_d\right)+w_{yx}\right)$$

(2)

where $f\left(\mathrm{*}\right)$ is the relationship between reservoir capacity and reservoir; $Z_d$ denotes the lower limit of the dynamic control bound.

(2)

Monte Carlo simulation

The Monte Carlo method, also known as random sampling or statistical test method, belongs to a branch of mathematics. Monte Carlo simulation is an analytical method that combines mathematical statistics, expert judgment, reservoir flood control algorithms, and the Monte Carlo method, which offers robust simulation capabilities and can be integrated with various methodologies [29]. The key factors influencing the upward fluctuation of the FLWL include the forecast period of incoming floods and rainfall, the uncertainty related to forecast accuracy, and the time lag in reservoir dispatching. In this study, the Monte Carlo simulation technique is employed to simulate and analyze the range of the upward fluctuation period of the rainstorm center, considering the uncertainties associated with forecasting accuracy and the time lag in reservoir dispatching. The aim is to identify the range of the upper limit for the dynamic control bound of the FLWL.

Flood forecast accuracy: Based on the analysis of available data and information, it is observed that the relative flood errors of Luhun and Guxian reservoirs do not follow a normal distribution. However, there is no significant difference for the lognormal distribution, with distribution parameters of $\mu$ = 0.0384, $\sigma$ = 0.1330 for Luhun reservoir and $\mu$ = 0.0203, $\sigma$ = 0.0897 for Guxian reservoir, specific parameter values are given in [29].

The sample forecast error ${\epsilon }_y$ obeys the density function:

$$f\left(x\right)=\frac{1}{\sqrt{2\pi }\sigma }\mathit{exp}[-\frac{(\mathit{ln}{\epsilon }_y-\mu {)}^2}{2{\sigma }^2}]$$

(3)

Scheduling lag: Scheduling lag is caused by various factors, including the transmission of forecast information and operational instructions, decision-making meetings, equipment reliability and operation, operator skills and expertise, and others. In this study, a triangular distribution is proposed to be used to estimate the risk of schedule delay.

The scheduling lag ${\epsilon }_d$ obeys the density function:

$$f\left(x\right)=\left\{\begin{array}{c}\begin{array}{c}\frac{2\left({\epsilon }_d-a\right)}{\left(b-a\right)\left(c-a\right)}\begin{array}{cc}& \left(a\le {\epsilon }_d\le b\right)\end{array}\\ \frac{2\left(c-{\epsilon }_d\right)}{\left(c-b\right)\left(c-a\right)}\begin{array}{cc}& \left(b\le {\epsilon }_d\le c\right)\end{array}\\ 0\begin{array}{ccccc}& & & & \end{array}({\epsilon }_d<a,{\epsilon }_d>c)\end{array}\end{array}\right.$$

(4)

where $a$, $b$, and $c$ express the minimum, most probable, and maximum values of the risk variable ${\epsilon }_d$ respectively. Here, the minimum value $a$ = 0 h, the most probable value $b$ = 0 h, and the maximum value $c$ = 3 h are taken.

Forecasting period: The determination of the forecasting center’s location was based on the rate determination of the Lushi Xinanjiang model and the flood simulation results of the Luhun Reservoir. By analyzing the forecasting data, it was established that the forecasting period of the heavy rainfall centers for both Guxian Reservoir and Luhun Reservoir, from upstream to downstream, ranged from 12 h to 4 h. Given the stochastic nature of the forecasting center’s location, a uniform distribution was employed for the simulation process.

2.3.3. Optimization Model of Dynamic FLWL in Parallel Reservoirs

The optimization model aims to enhance the dynamic FLWL in parallel reservoirs by optimizing the upper range of these levels across different foreseeable periods. The primary objectives are to meet downstream flood control requirements and increase water supply availability. The foremost concern is flood safety, encompassing dam and downstream area safety. However, the main focus of this study is to effectively address potential drought disasters and maximize flood resource utilization while maintaining flood safety. As such, the starting water level for dynamic regulation during the flood season serves as the upper limit value of the dynamic FLWL, with the objective function focusing on optimizing the benefits by increasing water supply during regulation.

(1)

Objective functions

Objective 1: flood control

The objective function based on the maximum peak reduction criterion is constructed by selecting the Heishiguan section after the confluence of Yiluo River as the flood control point for the parallel reservoir group.

$$min{F}_1=min\left[max\sum _{j=1}^k\sum _{i=1}^n{q}_{i,t}\right]tϵ\left(t_0,t_D\right)$$

(5)

where $t_0, t_D$ are the time of the beginning and end; $q_{i,t}$ denotes the ith reservoir discharge process; $n$ denotes the number of reservoirs, and $k$ denotes the total number of time periods for the discharge process.

Objective 2: water supply

Dynamic control of the FLWL of parallel reservoirs can further improve the use of flood resources on the basis of the staged FLWL by constructing the total water supply increase in the two reservoirs over the staged FLWL as an objective function:

$$max{F}_2=\sum _{i=1}^{n}V\left(Z_d^{+}\right)-V\left(Z_d\right)$$

(6)

where $V\left(Z_d^{+}\right)$ and $V\left(Z_d\right)$ denote the ith reservoir capacity corresponding to the upper and lower dynamic control limits, respectively.

(2)

Constraints

• Water balance constraints:

$$V_i\left(t\right)=V_i\left(t-1\right)+\left[\frac{Q_i\left(t\right)+Q_i\left(t-1\right)}{2}-\frac{q_i\left(t\right)+q_i\left(t-1\right)}{2}\right]∆t$$

(7)

where $V_i\left(t-1\right)$ and $V_i\left(t\right)$ denote the amount of water stored at the beginning and end of time period t in the ith reservoirs; $Q_i\left(t-1\right)$ and $Q_i\left(t\right)$ denote the incoming flow at the beginning and end of time period t in the ith reservoirs, respectively; and $q_i\left(t-1\right)$ and $q_i\left(t\right)$ denote the discharged flow at the beginning and end of time period t in the ith reservoirs, respectively, and $∆t$ denotes the length of the time period.

2.

Reservoir characteristic level constraints:

$$Z_{min}\left(t\right)\le Z\left(t\right)\le Z_{max}\left(t\right)$$

(8)

$$Z_{end}=Z_d^{+}$$

(9)

where $Z\left(t\right)$ denotes the water level of the ith reservoir at time t; $Z_{min}\left(t\right)$ and $Z_{max}\left(t\right)$ denote the permissible minimum and maximum water levels of the ith reservoir at time $t$, respectively; and $Z_{end}$ denotes the water level of the ith reservoir at the end of the reservoir scheduling period.

3.

Drainage flow capacity constraints:

$$q\left(t\right)\le q_{max}\left(Z\left(t\right)\right)$$

(10)

$$q\left(t\right)\le Q_{dl}$$

(11)

where $q\left(t\right)$ denotes the downstream discharge flow at moment $t$ of the ith reservoir; $q_{max}\left(Z\left(t\right)\right)$ denotes the maximum discharge capacity corresponding to the ith reservoir level $Z\left(t\right)$ at moment $t$ of the ith reservoir; and $Q_{dl}$ denotes the downstream safe flow.

4.

River constraints: for the river flood evolution constraints, the Muskingum method is used.

$$Q\left(t\right)=C0·q\left(t\right)+C1·q\left(t-1\right)+C2·Q\left(t-1\right)$$

(12)

where C0, C1, C2 are parameters of the Muskingum method, which satisfy the relation C0 + C1 + C2 = 1; $Q\left(t-1\right)$ and $Q\left(t\right)$ represent the flow rate of the downstream section at the beginning and the end of the calculation period; and $q\left(t-1\right)$ and $q\left(t\right)$ represent the inflow rate of the upstream section at the beginning and the end of the calculation period.

5.

Non-negative constraints:

The study calculations are all non-negative.

2.3.4. Optimization Solution Algorithm

This parallel reservoir optimization model addresses both flood control and water supply objectives, presenting a multi-objective constrained optimization problem. To tackle this, the commonly employed NSGA-II is utilized. The NSGA-II is a widely used evolutionary algorithm for efficiently solving multi-objective optimization problems. It is characterized by low computational effort, inclusion of elitism, and the absence of parameter sharing [42]. The NSGA-II algorithm incorporates a fast non-dominated sorting method and an elite strategy, ensuring both the diversity of solutions and the iterative retention of elite operators. This combination contributes to effective convergence and the identification of Pareto optimal frontiers in solving multi-objective optimization problems [43]. The utilization of NSGA-II enables a holistic evaluation of two objective functions—flooding and power generation—resulting in the identification of the Pareto optimal frontier. This approach circumvents the challenge of determining suitable weights when converting multi-objectives into a single objective function. Additionally, it provides a multitude of executable solutions for practical operation in multi-objective parallel reservoir systems.

The calculation steps of NSGA-II are as follows:

1:

Generate an initial population based on decision variables and population size.

2:

Evaluate the fitness of each chromosome according to the objective function.

3:

Set the number of genetic generations.

4:

Prepare the mating pool.

5:

Perform crossover and mutation operations.

6:

Evaluate the new offspring according to the fitness.

7:

Merge parent and new offspring for non-dominated sorting.

8:

Calculate the crowding degree.

9:

Select from low to high according to the non-dominated sorting rank, if the rank is the same then select the individual with higher crowding degree so that the selected individual is the same as the population number.

10:

Iterate the genetic count and repeat steps 4 to 9 until the count reaches the maximum number of generations.

11:

Generate Pareto frontier set.

2.3.5. Methodology Flow Chart

To elucidate the methodological logic, we presented a technical roadmap in Figure 2, aiming to provide a visual representation of the research methodology’s sequential flow.

3. Results

3.1. Staged FLWL for Parallel Reservoirs

In this study, the daily precipitation from the Dongwan and Lushi hydrological stations spanning the years 1961 to 2021 are utilized for the flood season staging of the Luhun and Guxian reservoirs. The rainy season in both reservoirs mainly occurs from July to September, and to encompass the entire flood season, the study period is set from 1 July to 30 October. The calculation time unit is chosen as 5 days to ensure independence and seasonality in the rainfall indicators. The selected indicators for constructing the characteristic matrix include the multi-year average maximum 1-day rainfall, multi-year average maximum rainfall days, and multi-year average precipitation coefficient of variation.

Table 2. Result of flood season division of the Luhun and Guxian reservoirs.

k	Luhun Classification	Guxian Classification
2	1–10, 11–24	1–12, 13–24
3	1–10, 11–20, 21–24	1–12, 13–20, 21–24
4	1–10, 11–16, 17–20, 21–24	1–12, 13–15, 16–20, 21–24
…	…	…

Note: k indicates number of phases.

presents the flood season staging results for Luhun Reservoir and Guxian Reservoir, respectively. This article exclusively presents the outcomes for segment number k = 4 and preceding values. For the complete table, please refer to Supplementary Materials. Figure 3 displays the curves of B(n,k)-k and γ(k)-k for Luhun Reservoir and Guxian Reservoir, respectively, where γ(k) is the slope of the objective function B(n,k) in segment k. For the Fisher optimal segmentation method, the number of segments corresponding to the position of the inflection point of B(n,k) is the optimal number of segments, and in order to facilitate the finding of the optimal segmentation result, the largest value of γ(k) corresponding to the number of segments is used as the optimal number of segments. The curves reveal that both reservoirs exhibit a peak value for γ(k) at k = 3, and B(n,k)-k displays a distinctive inflection point at the same value of k = 3. Hence, the optimal number of segments for the flood season of both reservoirs is determined to be 3. For Luhun Reservoir, the corresponding flood season segments are 1–10, 11–20, and 21–24. These segments represent the periods from 1 July to 20 August, 21 August to 10 October, and 11 October to 31 October, respectively. As for Guxian Reservoir, the corresponding segments are 1–12, 13–20, and 21–24, which correspond to the pre-flood period from 1 July to 31 August, the first post-flood period from 1 September to 10 October, and the second post-flood period from 11 October to 31 October.

Based on the staging results, the interstage sampling method was used, and the curve configuration was performed using the P-III curve for the staging results, and the results of the designed floods at different frequencies for each staging were obtained as shown in

Table 3. The results of the frequency analysis.

Reservoirs	Periods	10,000 PD	1000 PD	100 PD	20 PD	5 PD
Luhun	Pre-flood	16,540	11,950	7470	4350	2030
	First post-flood	7580	5460	3210	1973	773
	Second post-flood	6659	4797	2820	1706	679
Guxian	Pre-flood	11,800	8520	5310	3170	1480
	First post-flood	8040	5790	3670	2160	830
	Second post-flood	6349	4572	2898	1706	655

Note: 10,000 PD is the design peak flow rate of m³/s for 1 in 10,000 years, others are the same.

.

In compliance with dam safety requirements, the maximum reservoir level of both Luhun Reservoir and Guxian Reservoir should not surpass the design flood level and calibration flood level during extreme floods events such as 1000-year floods, and 10,000-year floods. Additionally, considering the flood control objectives for downstream of the Yiluo River, the maximum reservoir level of the reservoirs during flood regulation should not exceed 321.5 m for Luhun Reservoir and 543.2 m for Guxian Reservoir during a 20-year flood event. Furthermore, to meet the requisitioned land water level flood control requirements, the highest reservoir level during flood regulation should not surpass the requisitioned land water level during a flood of one in five years. To comprehensively account for the risk implications of adjusting the FLWL in the reservoir, each frequency design flood regulation algorithm is evaluated across three distinct phases. Through comparing the results of all the frequency design flood regulation algorithms within each phase, the lowest FLWL that meets the flood control requirements for all frequencies is determined and set as the FLWL for that specific phase. By utilizing flood regulation algorithms designed for floods of different frequencies at various stages, we comprehensively consider the risks associated with adjusting the FLWL. This approach ensures that the reservoir can effectively meet the prescribed flood control requirements during floods of varying frequencies at different stages. The specific characteristic water level values are presented in Table 1. Based on these flood control requirements and scheduling rules for different frequency flood events, the FLWL for each regulation phase of the reservoirs were determined as shown in

Table 4. Results of staged FLWL.

Luhun	Periods	1 June–20 August	21 August–10 October	11 October–31 October
	FLWL (m)	317.5	318.6	319.0
Guxian	Periods	1 June–31 August	1 September–10 October	11 October–31 October
	FLWL (m)	530.0	532.0	534.0

and shown in Figure 4 using each reservoir staging node as a horizontal coordinate.

3.2. The Upper Limit Range of the Dynamic Control of FLWL

According to the improved pre-discharge capacity constraint method, this study employs a 24 h short-term rainfall forecast for pre-discharge. Based on the rainfall forecast information, a pre-discharge of 500 m³/s is implemented in the event of light or moderate rainfall, while a pre-discharge of 800 m³/s is adopted in the case of heavy rainfall or torrential rainfall. Additionally, a pre-discharge of 1000 m³/s is implemented based on the maximum safe downstream flow rate during the flood forecasting period. The flood forecasting period, ranging from downstream to upstream of the rainstorm center, spans from 4 h to 12 h. Considering the uncertainties associated with the flood forecasting accuracy and scheduling hysteresis, the Monte Carlo simulation technique is utilized to determine the upper limit range of the staged FLWL. The lower limit of the upper limit range corresponds to the dynamic control level in the 4 h flood forecasting period, while the upper limit of the upper limit range corresponds to the dynamic control level in the 12 h flood forecasting period. These determinations are made to meet the flood control requirements effectively. The final results of each stage of dynamic control are shown in

Table 5. Results of the dynamic control of the FLWL.

Luhun	Periods	1 June–20 August	21 August–10 October	11 October–31 October
	FLWL (m)	317.5~317.9–318.4	318.6~319–319.4	319.0~319.3–319.5
Guxian	Periods	1 June–31 August	1 September–10 October	11 October–31 October
	FLWL (m)	530.0~530.8–532.3	532.0~532.7–534.1	534.0–534.3–534.8

and Figure 5 using each reservoir staging node as a horizontal coordinate.

Upon scrutinizing the results presented in Table 5 and Figure 5, the upper limit range of the dynamic control of the FLWL is determined based on different foreseeable periods, leading to the ranges of 317.9–318.4 m for Luhun Reservoir and 530.8–532.3 m for Guxian Reservoir during the pre-flood period, 319–319.4 m for Luhun Reservoir and 532.7–534.1 m for Guxian Reservoir in the first post-flood period, and 319.3–319.5 m for Luhun Reservoir and 534.3–534.8 m for Guxian Reservoir in the second post-flood period.

In this study, considering that most reservoirs often face small and medium-sized floods during their operation, the typical flood process observed in 1982 is selected as an example to demonstrate the dynamic FLWL regulation process under a 12 h foresight period of a 20-year flood from the pre-flood periods. The typical flood event in 1982 in the Yiluo River Basin is characterized by a flood frequency of 1.29%, with a return period of 78 years. The purpose is to illustrate the application of the improved pre-discharge capacity constraint method, which utilizes Monte Carlo simulation technology for advanced flow release. Among them, the number of Monte Carlo simulations is 1000 times. This is performed to fully consider the impact of uncertainty on the operational risk of the reservoir. The criterion for adjusting the FLWL dynamically in each phase is to simulate the number of times exceeding the flood control standard as 0. Figure 6 and Figure 7 depict the dynamic flood regulation process for Luhun Reservoir and Guxian Reservoir, respectively, showcasing the controlled release of flow based on the improved pre-discharge capacity constraint method. Additionally, Figure 8 displays the flood process before and after the downstream Heishiguan section is dispatched in response to the actions taken by the parallel reservoirs.

From Figure 6 and Figure 7, it is evident that Luhun Reservoir and Guxian Reservoir implement proactive water release based on rainfall and flood forecasts. By conducting pre-discharge before the flood peak, their flood control capacity is enhanced effectively. This approach allows the reservoirs to operate optimally during the flood season while ensuring the safety of both the reservoirs and meeting downstream flood control requirements. After the flood, the water levels are brought back to the upper limit of the dynamic FLWL.

The existing operation rules of the Luhun Reservoir and Guxian Reservoir are shown in Supplementary Materials. Before the flood, when the incoming water is less than 1000 m³/s, the reservoirs will release as much water as comes under the existing operation rules until it exceeds 1000 m³/s for flood control and regulation. From Figure 6 and Figure 7, it can be seen that according to the results of the rainfall forecast and flood forecast, the reservoir can be operated at the upper limit of the dynamic FLWL, and before the incoming water is less than 1000 m³/s, by increasing the discharge flow in advance, it can realize that the FLWL can be lowered back to the lower limit of the FLWL before the flood, and after the flood, it can be gradually restored to the upper limit of the FLWL. As can be seen from Figure 8, downstream of the Heishiguan flood process, this method makes the flow increase before the flood, but does not make the flood peak flow increase, and does not increase the risk of flood control.

3.3. Multi-Objective Optimization Model

The uncertainty associated with the foresight period poses challenges in determining the precise upper limit of the dynamic FLWL, making it unsuitable for direct reservoir operation guidance. To enable the dynamic scheduling of reservoirs, a fixed upper limit value within the upper limit range must be established for use as the upper limit level during dynamic reservoir scheduling. Hence, our study aims to identify the optimal upper limit value of the dynamic FLWL for parallel reservoir operation. This objective is accomplished by considering a comprehensive range of factors, including minimizing downstream flood peaks and maximizing the potential increase in water supply. This study seeks to strike a balance between the flood control risk and water supply benefits to ensure effective and efficient reservoir operation.

To evaluate the dynamic regulation model of parallel reservoirs, a case study is conducted using the example of a 10,000-year mega flood during the pre-flood period. To address the multi-objective optimization problem involving the flood control risk and water supply benefit, the NSGA-II is employed for computation. In the calculation process, specific parameters are set, such as a population size of 50, a maximum number of iterations of 100, a crossover rate of 0.8, and a mutation rate of 0.05. The results are presented in the form of a Pareto frontier, as shown in Figure 9.

As can be seen from Figure 8, the points in blue are the Pareto frontier solutions, all of which strictly dominate the other solutions in red, constituting the global Pareto optimal solution. The Pareto frontier analysis led to the selection of three distinct scenarios for further evaluation. In Scenario 1, the primary objective is to achieve the best downstream peak shaving, while in Scenario 2, the focus is on maximizing the increase in water supply. For the compromise Scenario 3, the 50th percentile frontier solution is chosen, and in cases where an even number of solutions exists, the solution with a larger water supply is prioritized. The results of these scenarios are presented in

Table 6. Results of the optimization program.

Scenario	Luhun Reservoir (m)	Guxian Reservoir (m)	Flood Peak (m3/s)	Increased Water Supply (m3)
1	531.0	318.4	15,601.2	5200.8
2	532.3	318.4	16,086.1	7670.1
3	531.7	318.4	15,842.9	6581.9

for a detailed assessment and comparison.

According to the optimization results presented in Table 6, it is observed that the optimal upper limit of the dynamic FLWL for Luhun Reservoir is determined to be 318.4 m. For Guxian Reservoir, the best downstream peak reduction effect is achieved when the upper limit of the dynamic FLWL is set to 531.0 m, while the largest increase in water supply occurs when the upper limit is 532.3 m. Additionally, the optimal compromise program, considering both the flood control risk and water supply benefit, is obtained with an upper limit of 531.7 m for the dynamic FLWL of Guixian Reservoir. These results provide a clear understanding of the trade-offs and implications associated with different upper limit values, aiding decision-makers in choosing the most suitable approach for reservoir operation, which optimizes flood management while maximizing water resource utilization.

According to the results obtained from the upper range of the dynamic FLWL for each phase of the parallel reservoirs as shown in Table 5, their respective Pareto frontiers were generated. Subsequently, the optimal compromise scheme was selected, taking into account both the flood control risks and water supply benefits. The summarized results are shown in

Table 7. Optimization results of dynamic control of FLWL.

Luhun	Periods	1 June–20 August	21 August–10 October	11 October–31 October
	FLWL (m)	317.5~318.4	318.6~319.4	319.0~319.4
Guxian	Periods	1 June–31 August	1 September–10 October	11 October–31 October
	FLWL (m)	530.0~532.3	532.0~533.3	534.0~534.7

for each reservoir staging node as a horizontal coordinate.

Through a comprehensive multi-objective optimization model, the flood control risks and water supply benefits associated with elevating the FLWL in parallel reservoirs were thoroughly considered to optimize the daily operation strategy of these reservoirs. The dynamic control bound of the FLWL, guiding the daily reservoir operation, were determined within the upper limit range. The optimization results are shown in Figure 10.

For Luhun Reservoir, the dynamic control bounds are set at 317.5–318.4 m during the pre-flood season, 318.6–319.4 m during the first post-flood period, and 319.0–319.4 m in the second post-flood period. On the other hand, for Guxian Reservoir, the dynamic control bounds are established as 530.0–532.3 m during the pre-flood season, 532.0–533.3 m during the first post-flood period, and 534.0–534.7 m in the second post-flood period. When comparing the optimized dynamic control bound with the static staged FLWL, the water supply can be increased by 6581.9 m³ in the pre-flood season, 5634.9 m³ in the first post-flood period, and 3130.8 m³ in the second post-flood period, without compromising the safety of the flood control.

4. Discussion

4.1. Reasonableness of Staged FLWL

The Luhun Reservoir and Guxian Reservoir currently operate with static, fixed, staged FLWL. Both reservoirs delineate the pre-flood and post-flood periods using 20 August as the dividing point, as indicated in

Table 8. Current staging program for reservoir operation.

Luhun	Periods	1 June–20 August	21 August–31 October
	FLWL (m)	317.5	319.0
Guxian	Periods	1 June–31 August	21 August–31 October
	FLWL (m)	530	534.0

.

The findings from Table 4 and Table 8 are compared and analyzed. By implementing the regulation of the staged FLWL for Luhun Reservoir and Guxian Reservoir, the varying characteristics of the flood season rainfall can be adequately addressed to effectively manage the flood risk and meet the reservoir storage demands at different times. During the pre-flood periods, the FLWLs are set at 317.5 m for Luhun Reservoir and 530.0 m for Guxian Reservoir, enabling an efficient response to potential flood peaks and enhancing the flood control capacity. During the pre-flood season, the results obtained through the Fisher optimal segmentation method align with the current operational FLWL. Consequently, the staging outcomes do not impact the primary flood control procedures in this season, maintaining consistent flood risk levels. In the first post-flood period, the FLWL is elevated to 318.6 m for Luhun Reservoir and 532.0 m for Guxian Reservoir. In the second post-flood period, these levels are further raised to 319.0 m for Luhun Reservoir and 534.0 m for Guxian Reservoir. Through a comparison with the existing operation and the application of the Fisher optimal segmentation method, the post-flood period can be effectively subdivided into two phases, progressively increasing the FLWL to match the current operational standards.

Liu et al. [44] conducted the flood staging of the Three Gorges Reservoir to realize the exploration of dividing the flood season of the Three Gorges Reservoir from two to three periods, and to further understand the incoming water law and seasonal characteristics of the Three Gorges Reservoir. This paper, like the conclusion of the study on the Three Gorges Reservoir, adopts a more scientific staging method to further explore the staging characteristics of the flood season. This strategic adjustment enhances the utilization of the staged flooding nature, thereby improving the system’s resilience to flood risks. This facilitates optimal water storage in the reservoirs and reserves a surplus of water supply, preventing a situation where there is insufficient water storage in the later flood season. As a result, the efficiency of the water resource utilization is significantly improved.

However, the current staged FLWL approach represents a static control method that may not fully exploit the potential of modern forecasting technologies and methods. To enhance the utilization of flood resources, a shift towards dynamic regulation can be considered, which involves integrating forecast information to achieve real-time adjustments of the FLWL. By leveraging advanced forecasting technology, the FLWL can be dynamically and flexibly adjusted based on real-time rainfall forecasts and reservoir storage conditions, enabling more accurate adaptation to varying hydrological conditions. Such a dynamic control strategy would offer greater flexibility and scientific precision in reservoir operation, further optimizing water resource management, and better fulfilling the evolving needs of society and people’s livelihoods.

4.2. Dynamic Control of FLWL for Parallel Reservoirs

4.2.1. Impact of Uncertainty of the Foresight Period

In this study, the upper range of the dynamic control for the FLWL, influenced by the uncertainty of the rainstorm center location, is determined through the coupled dynamic regulation method of the improved pre-discharge capacity constraint and Monte Carlo simulation. Relative to the static staged FLWL, the augmentation range of the FLWL following dynamic control in the pre-flood season, first post-flood period, and second post-flood period is observed as 0.4–0.9 m, 0–0.4 m, and 0.3–0.5 m for Luhun Reservoir, and 0.8–2.3 m, 2.7–4.1 m, and 4.3–4.8 m for Guxian Reservoir. Compared with the static staged FLWL, the enhanced FLWL range following dynamic control is noted as 0.4–0.9 m, 0–0.4 m, and 0.3–0.5 m for Luhun Reservoir, and 2.7–4.1 m, 4.3–4.8 m, and 4.3–4.8 m for the Guxian Reservoir. Consequently, the FLWL of each stage is improved to varying extents. The conclusions are consistent with those of Chen et al. [9] on the dynamic FLWL of the Qingjiang River cascade reservoir, Li et al. [45] in the Three Gorges reservoir, and Zhou et al. [10] on the dynamic FLWL of the Qingjiang River cascade reservoir and the Three Gorges mixed reservoir. The dynamic FLWL method can increase the FLWL, which is the safe water level for flood season operation, in all phases of the flood season without increasing the risk of flood control. The results also show that the dynamic FLWL is not only applicable to cascade reservoirs but also to parallel reservoirs. The optimization of daily reservoir operation through the dynamic control of FLWL offers a promising strategy to effectively manage flood control risks while maximizing water supply benefits.

Considering the uncertainty of the foresight period for the dynamic control of the FLWL, it can be found that the foresight period has different degrees of influence on the upper limit value of the dynamic control of each phasing stage, which further indicates that considering the uncertainty of the foresight period for the dynamic control of the FLWL is of great significance and research value. This dynamic control strategy further enhances the FLWL during the flood season, effectively increasing water storage in the reservoirs and improving the available water supply during the flood season.

4.2.2. Multi-Objective Optimization

Figure 9 illustrates the Pareto fronts and initial solution sets of the two objective functions. From the graph, it is evident that the range of the Pareto front corresponds to the range of the maximum downstream flood peak, spanning from 15,601.2 to 16,086.1 m³/s, while the potential increase in the water supply ranges from 5200.8 to 7670.1 m³. The Pareto frontier curve highlights a positive correlation between the incremental water supply and the downstream maximum flood peak. Increasing the upper dynamic FLWL enables a higher water supply during the flood season but also raises the downstream maximum flood peak, thus augmenting the flood risk. Bai et al. [46] similarly concluded that an increased water supply would exacerbate the pressure of flood risk through the interrelationship of the four target dimensions. The Pareto frontier provides insight into the optimal upper limit value of the dynamic FLWL under varying conditions of flood risk and water supply benefit. These findings offer valuable guidance for reservoir operation, allowing for a more effective balance between flood control requirements and water supply advantages during practical reservoir management. In this paper, the compromise between flood control and water supply is used as the optimization scheme, although flood control and water supply requirements are considered comprehensively, the influence of decision-making opinions on the optimization results is ignored. Jia et al. [47] fully considered the influence of decision preference information on the optimization results to determine the optimal compromise scheme for reservoir cluster operation. In order to improve the reasonableness of the optimization results, further research is needed to improve the optimization method.

The observed increase in the water supply before and after floods during the flood season underscores the potential gains achieved through optimization. The substantial augmentation in the water supply without compromising the flood control safety is a significant accomplishment. This not only translates to immediate benefits in the water availability but also enhances the resilience of the region to potential droughts and uncertain climatic patterns in the future. Moreover, by integrating a comprehensive multi-objective optimization model, this study addresses the evolving challenges of water resource management. The consideration of flood control risks alongside water supply benefits provides a holistic approach, ensuring the balanced and sustainable utilization of water resources. The optimized operational strategies derived from this study not only contribute to efficient reservoir management but also lay the foundation for robust decision-making frameworks in water resource planning. This significant improvement in the utilization efficiency of flood resources during the flood season enhances the ability to cope with potential drought risks and adapt to uncertain future climate conditions. As such, this study provides a viable solution and scientific basis to address the increasing challenges of water resource situation.

4.3. Limitations and Implications

This study exhibits three noteworthy potential limitations. Firstly, the rationalization of the risk assessment methodology warrants enhancement. This investigation centers on the influence of diverse forecasting periods on dynamic control, incorporating considerations of the flood forecast accuracy and scheduling lag through the Monte Carlo simulation. While this study extensively investigates the influences of various risk factors, it overlooks the comparative analysis of emerging methodologies, such as credibility theory [30] and Bayes’ theorem [48]. The comprehensiveness of the credibility theory, facilitated by its ability to integrate information from multiple sources, enhances the overall scope of the risk assessment. The Bayesian theorem, with its capability to continuously update risk estimates based on new observational data, excels in capturing the dynamic changes in information, making it particularly noteworthy in risk evaluation. A comprehensive comparison of different risk analysis methods can deepen the credibility and rationality of the model. Further optimization is necessary to enhance the reliability and rationality of the conclusions drawn from the risk assessment. Secondly, refinement is required in optimizing the multi-objective model. The establishment of a multi-objective optimization model in this paper aims at determining the optimal FLWL dynamic control scheme. However, the model solely accounts for flood control and water supply objectives, neglecting a detailed analysis of other pertinent factors, including power generation and ecological considerations within the reservoir. Further research is needed to explore the comprehensive implications of reservoir operation. Lastly, the determination and optimization of the solution method necessitate further exploration. This study exclusively employs the NSGA-II for model optimization, lacking a comparative analysis of the impacts of different solution methods on the optimization results, such as particle swarm algorithms [49] and ant colony algorithms [50]. Furthermore, a more in-depth analysis of various algorithms, including genetic algorithms, particle swarm optimization, and ant colony optimization, is needed. This analysis should encompass the considerations of convergence, stability, optimality, robustness, and other metrics to determine the algorithm that is most suitable for the model.

In this study, an improved pre-discharge capacity constraint method is used to realize the dynamic adjustment of the reservoir FLWL through the advanced discharge of water from the reservoir by integrating the rainfall and flood forecast information. The methodology’s foundational principles exhibit generalizability to cascade reservoirs and reservoir groups. The analysis of the forecasting accuracy and scheduling lag is predicated on regional statistical data. As the forecasting accuracy improves and systematic, intelligent scheduling decision-making advances, the widespread adoption and credibility of regional statistical data form a robust foundation for the application and dissemination of the research method. Diverging from the considerations for parallel reservoirs, which predominantly focus on the downstream confluence risk, the investigation of this methodology for cascade reservoirs and reservoir groups necessitates heightened attention to reservoir interactions. It demands a nuanced consideration of the connectivity between upstream and downstream reservoirs and the incorporation of compensatory characteristics to further enhance the method’s rationality and applicability universally.

This study holds significant implications for sustainable development. Firstly, through an in-depth investigation into the dynamic control of FLWL in parallel reservoirs, we can enhance the effective management of water resources, facilitating their sustainable utilization. This contributes to the overall efficiency of reservoirs, ensuring an ample water supply during drought periods. Secondly, this research contributes to enhancing the multifunctionality of reservoirs. By considering different objectives, including flood control, water supply and, in the future, further consideration of power generation and ecological protection, we can achieve the comprehensive utilization of reservoirs and maximize their social, economic, and environmental benefits. Such multifunctionality aids in realizing the comprehensive objectives of sustainable development, catering to varied needs across different domains. Furthermore, this study can improve the transparency of reservoir operations and the scientific basis of decision-making. By establishing advanced dynamic control models, coupled with sophisticated forecasting and decision support systems, we can more accurately predict changes in the reservoir water levels and formulate corresponding operational strategies. This helps enhance the efficiency of reservoir management, mitigate the adverse impacts of unforeseen events on reservoir operations, and thereby promote the sustainable utilization of water resources. In summary, the in-depth exploration of the dynamic control of the FLWL in parallel reservoirs in this study provides robust support for sustainable development in various aspects, including water resource management, flood control, and disaster reduction. It serves as a scientific foundation for future hydraulic engineering and water resource management decision-making.

5. Conclusions

(1)

Utilizing the Fisher optimal segmentation method, this study optimizes the flood season staging results for Luhun Reservoir and Guxian Reservoir, leading to the deduction of the FLWL for each stage. The refinement introduced transforms the existing two-phase staging scheme of the two reservoirs into a more nuanced three-phase scheme, which further subdivides the current post-flood periods into two distinct periods. This refined staging methodology enhances the consideration of flood staging characteristics, allowing for a gradual increase in the FLWL and a concurrent reduction in the risk associated with flood control.

(2)

The research introduces the improved pre-discharge capacity constraint method, taking into account the uncertainty in the forecasting accuracy and scheduling lag time, and incorporates the Monte Carlo simulation technology to achieve the dynamic control of the staged FLWL for the parallel reservoirs. By considering different foreseeable periods of the rainstorm center location in the upstream and downstream, the upper limit range of the dynamic FLWL is determined. Finally, an optimization model is developed, aiming to minimize the downstream flood peak while maximizing the water supply. The NSGA-II is employed to solve the optimization model, leading to the determination of optimal FLWL dynamic control bound to guide the daily operation of the parallel reservoirs. For Luhun Reservoir, the dynamic control bounds are 317.5–318.4 m in the pre-flood period, 318.6–319.4 m in the first post-flood period, and 319.0–319.4 m in the second post-flood period. For Guxian Reservoir, the dynamic control bounds are 530.0–532.3 m in the former flood season, 532.0–533.3 m in the first post-flood period, and 534.0–534.7 m in the second post-flood period. Implementing the dynamic control of the FLWL allows for an increase in the water supply of 15,347.6 m³ throughout the entire flood season when compared with the staged FLWL.

This research provides valuable insights for flood control and water resource utilization at Luhun Reservoir and Guxian Reservoir through the implementation of a dynamic control strategy. By optimizing flood control and water supply objectives using multi-objective optimization models, this practical approach enhances water resource utilization and management efficiency. The methodology employed in this study is primarily founded on the framework of parallel reservoirs, wherein the optimization of dynamic control for the FLWL is achieved through an examination of risk scenarios at the downstream Heishiguan confluence location. While initially applied to parallel reservoirs, the proposed method model demonstrates broad and general applicability. It presents an avenue for extending optimization efforts to the dynamic control of the FLWL in cascade reservoirs, considering the interconnections and influences among them. As the reservoir group system is progressively established, this methodology not only offers insights into and methods for optimizing the FLWL within the reservoir group but also contributes to ideas and approaches for system optimization in the broader context of reservoir group dynamics.