Impact of Daily Operations of Cascade Hydropower Stations on Reservoir Flow Fluctuation Characteristics

Abstract

The daily operation of cascade hydropower stations induces periodic water level fluctuations (WLFs) that propagate as gravity waves, significantly affecting the hydrodynamics of reservoirs. Previous studies have mainly focused on the effects of individual stations, with little attention paid to the combined impacts of upstream and downstream operations. Taking the Wudongde Reservoir on the Jinsha River as a case study, we used a one-dimensional hydrodynamic model and cross-correlation analysis to simulate flow fluctuation patterns under joint daily operations. The results show that fluctuations from upstream stations attenuate rapidly in the reservoir, with greater attenuation during the dry season. Under joint operations, wave energy decayed exponentially near the reservoir tail and linearly in the main reservoir area, leading to a further reduction in the WLF amplitudes. The interactions between upstream- and downstream-propagating waves enhance energy dissipation. The wave type transitioned from kinematic to dynamic as the water depth increased. During the wet and dry seasons, the average wave velocities were approximately six and nine times higher, respectively, than those under natural conditions. Joint operations expand the range of potential slope instability but reduce the WLF rate compared to natural flows. These findings provide a scientific reference for optimising the daily operations of cascade hydropower stations and mitigating their ecological impacts.

1. Introduction

Global climate change is one of the most significant environmental issues and complex challenges facing humanity in the 21st century [1,2]. Ripple and colleagues clearly and unequivocally declared, with more than 15,000 scientist signatories from around the world, that planet Earth is facing a climate emergency [3]. Advocating for energy conservation and emission reduction has become a global trend for mitigating human impacts on the climate [4,5,6,7], with countries worldwide setting carbon reduction targets [8,9,10]. Among the measures taken, clean energy, especially multi-energy complementarity, has been proposed as a key strategy for driving the development of clean energy [11,12]. According to statistics from the National Energy Administration of China at the end of 2022, the proportions of wind, solar, and hydropower, the three clean energy sources, in the country’s total installed capacity were 14.2%, 15.3%, and 16.1%, respectively. Photovoltaic power generation faces issues of electrical instability and intermittency [13,14]. Hydro-photovoltaic (hydro-PV) complementarity can compensate for photovoltaic power generation by quickly starting and stopping the turbines of hydropower stations, thereby ensuring a more stable overall output [15,16,17]. The operation of hydro-PV complementarity is bound to alter the inflow and outflow processes of upstream and downstream cascade hydropower stations [18,19], significantly affecting the daily operations of these stations and reservoirs.

The discharge of inflow and outflow is largely controlled by human activities and often undergoes periodic changes due to variations in electricity demand between day and night [20] or navigation requirements for shipping [21]. The unsteady flow generated by daily operations can cause excessive fluctuations in water levels and flow velocities in downstream channels [22,23], affecting navigation for shipping and adversely impacting fish reproduction [24,25]. An increase in the fluctuation rate of the water level changes the original water-rock interaction environment and conditions of the bank slope, which may exacerbate the instability of the bank slope [26,27]. Additionally, daily operations lead to increased water level fluctuations (WLFs) and flow oscillations, which can accelerate the degradation rate of organic pollutants [28,29] and promote water exchange between river bays and main reservoirs, thereby improving the water quality of local water bodies.

Regarding the fluctuation characteristics of the unsteady flow generated by daily operations, Xie et al. found that daily operations can induce internal waves, causing high-frequency fluctuations in the thermocline layer water temperature [30]. Based on field measurements, Long et al. found that uneven power generation loads in cascade reservoirs can lead to instability in the outflow process, which can drive periodic fluctuations in the water level in front of the dam and propagate along the river in the form of gravity waves. They discovered that the WLFs generated by the daily operation of the Three Gorges Dam transmitted to the tributary (Xiangxi River) can induce standing waves within a period of 2 h [31]. Yang et al., in order to analyse the hydrodynamics in an impounded river in response to discharge regulation at upstream and downstream dams, developed a one-dimensional analytical model applied to study a 38 km long section of the Yangtze River located between the Three Gorges Dam and Gezhou Dam [32]. The model results indicated that a standing wave was generated by rapid discharge changes at both dams, and its amplitude was strongly dependent on the lag time between these changes. A comparison between this study and representative previous studies is shown in

Table 1. Comparison between this study and representative previous studies.

Study (Author, Year)	Study Area/Focus	Modelling Approach	Cascade System Considered	Hydro-PV Complementarity	Wave Interference Analysed	Key Contribution
Xie et al. [30]	Xiluodu reservoir	Field data + 2D	✗	✗	✗	Internal waves in deep run-of-river reservoirs
Long et al. [31]	Three Gorges Reservoir-Xiangxi River (tributary)	Field data	✗	✗	✗	Standing wave observation
Yang et al. [32]	TGD-Gezhouba river segment	1D Analytical model	✓	✗	✓	The high-frequency eigenwave
This study (Zhu et al. [27])	Wudongde Reservoir (deep V-shaped canyon)	1D	✓	✓	✓	Energy dissipation and wave interference under joint daily operations

.

There are relatively few studies on the impact of the daily operation of cascade power stations considering hydro-PV complementarity; in particular, the cumulative effects of joint daily scheduling by upstream and downstream cascade power stations on hydrodynamic characteristics in reservoir areas have not been reported. In this paper, the Wudongde Reservoir (WDDR) in the lower reaches of the Jinsha River was selected as the research object, and numerical simulation and cross-correlation analysis were used to comprehensively analyse the impact of joint hydro-PV complementarity on the hydrodynamic characteristics of the WDDR area for the daily operations of cascade power stations. The results provide a scientific reference for the formulation of scheduling schemes under hydro-PV complementarity.

2. Materials and Methods

2.1. Study Area

The Wudongde (WDD) power station is the first cascade power station in the lower reaches of the Jinsha River, and the valley in the reservoir area is narrow and characterised by V-shaped canyon morphology. The normal reservoir water level (RWL) is 975 m, and the corresponding reservoir backwater length is approximately 200 km. The total storage capacity is 7.408 billion m³, which has seasonal regulation capability. The total installed capacity of the power station is 10.2 million kilowatts. The reservoir area is rich in solar thermal resources. The annual average temperature is approximately 20 °C, the annual average solar radiation is approximately 2.22 × 10⁹ J/m², and the annual average sunshine hours are 2700 h.

As shown in Figure 1, the WDDR tail is located at the confluence of the Jinsha River and Yalong River. The Guanyinyan power station was built on the mainstream of the Jinsha River in the upper reaches of the WDD, and the Ertan power station was built on the Yalong River. With the adjustment of China’s energy structure, cascade power stations will participate in the peak regulation process of hydro-PV complementarity. Currently, the planned photovoltaic resources of the WDDR tail amount to 7.4 million kilowatts. With the development of photovoltaic resources, the ratio of hydropower to photovoltaic installed capacity will be continuously adjusted, which will significantly impact the daily schedules of cascade power stations.

2.2. Principles of the Hydrodynamic Model

A longitudinal one-dimensional unsteady flow model is used to simulate the hydrodynamics in the reservoir under the hydro-PV complementarity adopted for the daily operations of upstream and downstream cascade power stations, and the corresponding equations [33] are as follows:

Continuity equation (Navier–Stokes (N–S) equation):

$${\displaystyle \frac{\partial Z}{\partial t}}+{\displaystyle \frac{1}{B}}{\displaystyle \frac{\partial Q}{\partial x}}={\displaystyle \frac{1}{B}}{L}_q$$

(1)

The momentum equation:

$${\displaystyle \frac{\partial Q}{\partial t}}+{\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \frac{Q^2}{A}}\right)+\mathit{gA}{\displaystyle \frac{\partial Z}{\partial x}}+{\displaystyle \frac{gQ\left|Q\right|}{C^{2}AR}}=0$$

(2)

where Z is the water level, Q is the discharge, A is the area of the cross-sectional active flow, B is the top width of the wetted cross-section, L_q is the lateral inflow per unit length, g is the acceleration of gravity, C is the Chezy coefficient, and R is the hydraulic radius.

2.2.1. Model Setup

The study area covered the river section from the Sanduizi hydrological station (S98) to the WDD dam site (S01), simulating the impact of daily operations on the hydrodynamic conditions of this river section. The total length is 198 km, with a total of 98 measured cross-sections. During the calculation, these cross-sections were interpolated and divided into 446 sections with intervals ranging from 350 to 500 m. The upstream boundary conditions were determined by considering the hourly discharge of the Sanduizi hydrological station, and the downstream boundary conditions were determined by considering the hourly outflow from the WDD station.

2.2.2. Model Calibration and Validation

We selected measured data from 11 November 2020 to 13 November 2020 for model validation. The fluctuation range of inflow is between 2330 and 4710 m³/s, exhibiting a pattern of ‘two rises and two falls’ each day. The outflow fluctuates between 2540 and 4200 m³/s, with one rise and fall occurring each day, resulting in a daily variation of 1660 m³/s. This process is shown in Figure 2. The initial condition is the water level (965.5 m) of the WDD dam site (S01) at 00:00 on 11 November 2020.

The root mean squared error (RMSE) and mean absolute error (MAE) are used to describe the accuracy of the model.

$$RMSE=\sqrt{{\displaystyle \frac{\sum _{i=1}^n{(Y_O^i-Y_S^i)}^2}{n}}}$$

(3)

$$MAE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n|Y_O^i-Y_S^i|$$

(4)

The water level in 2020 was calibrated by adjusting the roughness coefficient. A comparison between the simulated and observed values is presented in Figure 3. MAE and RMSE are used to describe the accuracy of the model. These errors are within an acceptable range. The roughness coefficients are segmented and calibrated between 0.032 and 0.07 m. The model can simulate the peak and trough values of the WLF quite well, and the phases of the simulated peaks and troughs are generally consistent with the observed values. In section S82, located in the reservoir tail, the water level is impacted by inflow fluctuations, exhibiting a pattern similar to that of the inflow, with fluctuations ranging from 966.41 to 968.5 m and a maximum daily fluctuation of 2.09 m, lagging behind the inflow process. The water level of S01 in the pre-dam scenario is affected by the superposition of upstream and downstream operations, resulting in smaller fluctuations with a maximum daily fluctuation of 1 m. The constructed one-dimensional hydrodynamic model of the WDDR area can meet the needs for wave propagation research.

2.3. Wave Analysis Methods

2.3.1. Attenuation Rate Calculation Method

To analyse the attenuation process of fluctuations along a river channel, determining the power spectral density (PSD) is essential. The PSD of the WLF (calculated using the Welch method) can provide information on the distribution of the PSD at different frequencies. To better explore the impact of daily operation, the PSD along the river channel with a frequency of 1.16 × 10⁻⁵ Hz (a period of 24 h) is extracted, and then a graph of PSD–distance is obtained. The data are fitted with trend lines to obtain the attenuation coefficient, which is the attenuation rate (AR).

The fitting equation for exponential decay is

$$E_x=E_0·\mathrm{e}\mathrm{x}\mathrm{p}(-ax)$$

(5)

where a is the attenuation coefficient, and

The fitting equation for linear attenuation is as follows:

$$E_x/E_0=a-bx$$

(6)

where b denotes the attenuation coefficient.

2.3.2. Cross-Correlation Analysis

Cross-correlation analysis is a statistical method used to evaluate the correlation and lag relationships between two-time series. The lag period corresponding to the maximum cross-correlation coefficient is the lag time between the two-time series. Cross-correlation analysis is used to determine the propagation time of the wave between the two sections, and the wave velocity is then calculated.

$$CCF\left(k\right)={\displaystyle \frac{\sum _{t=1}^{n-k}(X_t-\overline{X})(Y_{t+k}-\overline{Y})}{\sqrt{\sum _{t=1}^n{(X_t-\overline{X})}^2\sum _{t=1}^n{(Y_t-\overline{Y})}^2}}}$$

(7)

where $X_t$ and $Y_t$ are the observed values of the two-time series at time t, $\overline{X}$ and $\overline{Y}$ are their mean values, respectively, n is the number of samples, k is the number of lag periods, and CCF(k) is the cross-correlation coefficient at lag k.

3. Results

3.1. Scenarios Setting

Considering the impact of the daily scheduling scheme, pre-dam operating water level, and wet and dry seasons on wave propagation in the WDDR area, the scenario settings are shown in

Table 2. Scenario settings.

Scenarios	Upstream Boundary	Downstream Boundary	Hydrological Period	Initial Condition
1	daily scheduling	natural outflow	wet season	natural water level
2			dry season	natural water level
3		constant outflow	wet season	RWL: 952 m
4			wet season	RWL: 975 m
5			dry season	RWL: 975 m
6	constant inflow	daily scheduling	wet season	RWL: 952 m
7			wet season	RWL: 975 m
8			dry season	RWL: 975 m
9	daily scheduling	daily scheduling	wet season	RWL: 952 m
10			wet season	RWL: 975 m
11			dry season	RWL: 975 m

. Three representative sections, S01, S30, and S58, are selected in the main reservoir area, which are 500 m, 50 km, and 100 km from the dam, respectively. Four representative sections, S77, S84, S93, and S98, are selected in the reservoir tail, which are 148 km, 168 km, 188 km, and 198 km away from the dam, respectively (the locations of each section are shown in Figure 1).

The typical daily scheduled processes of the WDD power station during the wet and dry seasons under hydro-PV complementarity are illustrated in Figure 4. The discharge values used in the different scenarios are based on design assumptions derived from the typical regulation guidelines of the cascade system. These assumed values represent typical turbine operation flows under the corresponding RWL conditions and are used to ensure a consistent comparison across scenarios. The output of the hydropower station unit and the photovoltaic output show complementary anti-regulation characteristics, and the hydropower output is reduced during the period of sufficient light (11:00–16:00). In the wet season, the flow ranges from 5400 to 9830 m³/s during typical daily scheduled processes, with an average flow of 6787 m³/s and a variation of 4430 m³/s. In the dry season, the flow ranges from 960 to 4927 m³/s during typical daily scheduled processes, with an average flow of 2152 m³/s and a variation of 3967 m³/s.

The daily schedules of upstream cascade hydropower stations in the wet season and dry season are assumed to be consistent with those of the WDD hydropower station. The scenario settings are listed in Table 1. The simulation time for each scenario is 6 days. The initial error is eliminated by simulating a constant flow in the first three days, and the corresponding daily scheduling calculation is performed in the next three days. The calculation time step is 5 min.

As shown in Figure 5, the width of the river in the WDDR at the normal storage level varies from 200 m to 1400 m. The average river width of the 50 km pre-dam is 570 m, the average river width of the 50–100 km pre-dam is 825 m, and the average river width of the 100 km river channel in the reservoir tail is 350 m. The average water depth shows an overall increasing trend along the river channel; from upstream to downstream, the water depth gradually increases from 14 m to 180 m.

3.2. Characteristics of Water Level and Discharge Fluctuations During Daily Operation of the Upstream Power Stations

The simulation results for typical sections under daily operation of upstream power stations (DOUPS) are shown in Figure 6. The water level and discharge in the reservoir area exhibited daily fluctuations. During the DOUPS (Scenarios 1 and 4), as the inflow fluctuates with three rises and falls daily, the water level in the reservoir tail also shows a corresponding pattern of three rises and falls per day. The WLFs generated upstream propagate downstream in the form of gravity waves, causing successive WLFs in various downstream sections.

During the DOUPS and downstream natural outflow (Scenarios 1 and 2), the energy dissipation due to bottom friction leads to an overall decreasing trend in the daily WLFs along the river channel. The narrowing (or widening) of the terrain causes the wave energy to converge (or diverge), thereby increasing (or decreasing) the fluctuation amplitude. The overall trend of daily flow fluctuations along the river channel is decreasing.

Compared with the natural conditions, after the impoundment of the reservoir, the fluctuation amplitude is significantly weakened after the fluctuation is transmitted to the main reservoir area. The minimum daily variations in the water level near the dam in Scenarios 1 and 2 were 1.82 m and 2.25 m, respectively. The minimum daily variations in the water level in the reservoir area in Scenarios 3, 4, and 5 are 0.81 m, 0.79 m and 0.58 m, respectively.

In the wet season, when the reservoir is impounded at 952 m (Scenario 3), the daily variation in the water level in the approximately 50 km long tail upstream, unaffected by backwater, remains consistent with the natural outflow conditions (Scenario 1). The overall trend of the daily variation in the water level from the end of the backwater to 120 km from the dam monotonically decreased. When the reservoir is impounded at 975 m in the wet season and dry seasons (backwater to S98), the daily variation in the water level from the reservoir tail to the 140 km pre-dam decreases monotonically, and the AR of daily WLFs during the dry season is greater.

The water depth in the dry season is shallower than that in the wet season, and the WLFs caused by the flow change are more obvious. When natural outflow occurs, the daily variation in the water level in the dry season (scenario 2) is 2.25–7.43 m, and that in the wet season (Scenario 1) is 1.82–3.72 m; when the water storage is 975 m, the maximum daily variation in the water level along the river channel in the dry season (Scenario 5) is 5.34 m, and that in the wet season (Scenario 4) is 3.48 m.

3.3. Characteristics of Water Level and Discharge Fluctuations During Daily Operation of Downstream Power Stations

The simulation results (Scenario 7) of typical sections under the daily operation of downstream power stations (DODPS) are shown in Figure 7, and the WLF in the pre-dam area is the largest. From S01 to S98, the daily fluctuations in the water level and discharge gradually smoothed. From the pre-dam to the upstream, the daily variation in the water level in the reservoir area first decreases and then increases slowly. This is because the width of the reservoir increases upstream, and the water depth decreases, resulting in an increase in daily variation. Near the reservoir tail, the daily variation in the water level decreases rapidly because the reservoir tail has a greater slope, increasing the resistance to waves and weakening the wave energy. Additionally, the shallow depth near the reservoir tail enhances wave energy attenuation due to increased bottom friction.

The higher the RWL, the smaller the daily variation in the water level. The daily variation in the pre-dam water level at 952 m RWL (Scenario 6) is 1.34 m, and that at 975 m RWL (Scenario 7) is 1.09 m. This is because the increased water depth enhances the reservoir’s storage-buffering capacity, thereby reducing the sensitivity of the water level to unit discharge variations. When the RWL is 975 m, the daily variation in the water level during the wet season is larger than that during the dry season. The daily variation in the pre-dam water level in Scenario 8 is 0.72 m, which is smaller than the 1.09 m in Scenario 7. First, flow variation in the wet season is inherently larger than that in the dry season. Second, under the same flow variation, scenarios with larger inflows cause greater variation in the water level.

3.4. Characteristics of Water Level and Discharge Fluctuations During Joint Daily Operation of Cascade Power Stations

The simulation results for the typical sections are shown in Figure 8, and the daily fluctuations along the river channel are shown in Figure 9. Due to the significantly smaller depth and width of the river near the reservoir tail than those in the main reservoir area, the discharge fluctuations caused by the daily operation of the upstream and downstream power stations result in more pronounced WLFs near the reservoir tail than near the dam site.

During the joint daily operation of cascade power stations (JDOCPS) (Figure 9d), when the reservoir is impounded at 975 m in the wet season (Scenario 10) and dry season (Scenario 11), the 50 km section of the reservoir tail and 150 km section of the main reservoir area are affected by the inflow and outflow of the reservoir, respectively, and the daily WLF decreases monotonically, reaching a minimum near S77 (150 km pre-dam), with amplitudes of 0.08 m and 0.20 m, respectively. The minimum amplitude position is related to the RWL, and the reduction amplitude is associated with the scheduled flow. In the wet season, when the reservoir is impounded at 952 m (Scenario 9), the daily WLF in the 50 km upstream section is unaffected by backwater and remains consistent with the natural outflow. The WLF in the remaining reaches is the same as that in Scenarios 10 and 11, reaching a minimum near S69 (125 km pre-dam), and the amplitude is 0.53 m.

4. Discussion

4.1. Comparison of the Effects of Three Daily Operation Modes on Reservoir Hydrodynamics

To better explore the cumulative effect of the JDOCPS on the hydrodynamics of the reservoir area, the results shown in Figure 10 are further investigated. The three daily operation modes during the wet season at 952 m RWL and 975 m RWL and during the dry season at 975 m RWL have varying degrees of impact on the study river section. The DOUPS has the greatest influence on the WLF in the reservoir tail, and the DODPS has the greatest influence on the WLF in the main reservoir area. Compared to the daily operation of individual upstream and downstream power stations, the JDOCPS has a superimposed effect because the fluctuation amplitude of the reservoir tail is smaller than that of the DOUPS, and the fluctuation amplitude of the main reservoir area is smaller than that of the DODPS. The reasons for this are discussed from the perspective of wave energy in Section 4.2. Compared with the separate operation of upstream and downstream hydropower stations, the daily fluctuations in the discharge and water level of the JDOCPS tend to be consistent, and the daily variations from the reservoir tail and pre-dam gradually decrease along the river channel. However, the impact on the daily discharge fluctuation in the reservoir is a positive superposition effect, and the impact on the daily WLF is a negative superposition effect. This is because the discharge in the reservoir is affected by the simultaneous daily operation of the inflow and outflow of the reservoir, and the maximum daily variation in the reservoir is not smaller than that of the DOUPS or DODPS (the daily operation flow at the dam site or the reservoir tail is a constant value).

4.2. Wave Decay Characteristics

This section analyses the differences caused by the three daily operation modes from the perspective of wave energy. The PSD for Scenario 10 is shown in Figure 11. PSDs of the six sections along the reach under the six operating conditions is shown in Figure 12. Figure S1 shows the PSD of the WLF corresponding to a 24 h period (frequency of 1.16 × 10⁻⁵ Hz) along the river channel under three daily operation modes. Using the method described in Section 2.3.1, the attenuation form and attenuation coefficient of each river section under each scenario were obtained.

During the JDOCPS, the wave energy decays exponentially from the reservoir tail to the downstream region. The exponential attenuation coefficients of Scenarios 9, 10, and 11 are 0.1247, 0.0727, and 0.138, respectively, which are higher than the attenuation coefficients of Scenarios 3, 4, and 5 at the reservoir tail during the DOUPS of 0.0732, 0.0573, and 0.0911, respectively. The wave energy shows obvious linear attenuation from the pre-dam to the upstream region. The linear attenuation coefficients of Scenarios 9, 10, and 11 are 1 × 10⁻⁶, 3 × 10⁻⁷, and 6 × 10⁻⁸, respectively, while the wave energy obtained for the three conditions of the DODPS (Scenarios 6, 7, and 8) shows no obvious attenuation in the reservoir area.

When the reservoir is impounded at 952 m and 975 m in the wet season and 975 m in the dry season, the wave energy obtained for the JDOCPS dissipates faster than that for the DOUPS or DODPS, and the energy reaches a minimum near the reservoir areas S69 (952 m) and S77 (975 m). This is because the two waves propagating upstream and downstream are affected by the topography and bottom friction. At contact, complex interactions occur, resulting in a continuous decrease in the wave energy. This trend may be attributed to wave interference, where the wave cancellation effect gradually increases as the phase difference between the two waves approaches 180°, thereby decreasing the wave energy. If the phase difference mutates at a certain point, it may enhance the wave cancellation effect, thereby causing the wave energy to decrease more rapidly. To further illustrate the interference mechanism between the upstream and downstream-propagating waves, a schematic diagram is provided (see Figure S2).

4.3. Wave Velocity Variation

The wave generated by reservoir operation is analogous to a flooding wave. Flood waves in natural river channels are classified as kinematic waves; however, when floods enter reservoirs, they theoretically exhibit both kinematic and dynamic wave characteristics. The movement of kinematic waves resembles congestion phenomena in high-density traffic flow areas, with their upstream and downstream boundaries moving in the flow direction over time. The propagation of kinematic waves is primarily influenced by factors such as river channel topography, flow velocity, and discharge. In contrast, dynamic waves represent a mode of material or energy transfer controlled by external driving forces, whose propagation is influenced mainly by factors such as external driving forces, medium properties (such as water density and viscosity), and the river channel topography. In kinematic wave calculations, both the inertial term and the additional slope term resulting from changes in water depth along the flow path, i.e., pressure gradient terms, are ignored. In the reservoir tail, flood waves are primarily kinematic waves, and their propagation velocity can be expressed as W = ku (1 ≤ k ≤ 2), where u represents the flow velocity. Dynamic waves retain all terms in the momentum equation (inertial terms, additional slope terms, resistance terms, and bed slope terms), and flood waves in the main reservoir area are primarily dynamic waves, with the wave speed expressed as u ± √gy (where y is the water depth). Generally, the dynamic wave velocities are much greater than the kinematic wave velocities.

To illustrate the differences in wave transmission under various hydraulic conditions more clearly, the WDDR is divided into four sections. The wave velocity and propagation time for each section under the 11 different scenarios were calculated using the method described in Section 2.3.2, as shown in

Table 3. Segmental statistics of the wave velocity and propagation time for each scenario.

Scenarios	Propagation Time (h)				Wave Velocity (m/s)
	Reach 1	Reach 2	Reach 3	Reach 4	Reach 1	Reach 2	Reach 3	Reach 4
1	2.75	2.92	3.75	2.67	5.05	4.53	3.73	5.20
2	4.25	4.25	5.25	3.33	3.26	3.11	2.66	4.16
3	2.75	0.83	-	-	5.05	15.86	-	-
4	1.75	-	-	-	7.93	-	-	-
5	1.67	-	-	-	8.33	-	-	-
6	-	1.00	0.50	0.25	-	13.22	27.95	55.48
7	1.25	0.67	0.42	0.25	11.10	19.83	33.54	55.48
8	1.17	0.58	0.42	0.25	11.89	22.66	33.54	55.48
9	2.67	1.17	0.58	0.25	5.20	11.33	23.96	55.48
10	1.58	0.50	0.83	0.25	8.76	26.43	16.77	55.48
11	1.42	0.42	0.58	0.33	9.79	31.72	23.96	41.61

Notes: “-” indicates that this value cannot be determined. Reach 1: S98 to S77 (49.95 km), Reach 2: S77 to S58 (47.58 km), Reach 3: D58 to S30 (50.31 km), Reach 4: S30 to S1 (49.93 km).

and Figure 13. Simultaneously, the average flow velocities, wave velocities of kinematic waves, and dynamic waves for each section were calculated to assess the wave characteristics of the four sections under different scenarios (Table S1). For Scenarios 3–5, the propagation times and wave speeds for some river sections could not be determined. This is because as kinematic waves propagate upstream and downstream, they also undergo reflection and superposition, resulting in wave overlap and causing the water level in the reservoir area to rise and fall almost simultaneously.

Under natural conditions (Scenarios 1 and 2), the wave velocity increased and decreased along the river channel due to the influence of topography. Compared with natural conditions, the wave velocity increases gradually with increasing water depth owing to the influence of backwater after impoundment; therefore, the wave velocity in the main reservoir area is significantly greater than that in the reservoir tail. In the wet and dry seasons, the average wave velocities of the entire river reach at the normal RWL are approximately six times and nine times those at the natural RWL, respectively. The wave velocity in the reservoir tail (Section 1) is 1.6–3.6 times greater than that under natural conditions, while in the pre-dam section (Section 4), it was 10.0–13.3 times greater. The wave propagation time significantly decreases after reservoir impoundment relative to natural conditions: in the wet season, the wave propagation time in the entire river reach is shortened from 12.1 h to 2.6–2.7 h, while in the dry season, it is shortened from 17.1 h to 2.3–2.8 h.

At the 975 m RWL, the propagation time during the wet season (Scenarios 4, 7, and 10) generally increases compared to that during the dry season (Scenarios 5, 8, and 11). This is because the wet season corresponds to a greater discharge, causing longer river segments with kinematic wave characteristics in the tail section and a slower wave velocity. Overall, under the three operation modes corresponding to 975 m of storage (Scenarios 4, 7, and 10), the wave velocity is faster than that under 952 m of storage (Scenarios 3, 6, and 9), as higher water levels result in deeper water depths and faster wave velocities.

4.4. Impact of Daily Operation on Slope Stability in Reservoir Areas

A rapid WLF within a day has an impact on reservoir bank slope stability and endangers reservoir safety. Under the influence of daily operations, when the RWL decreases, the pore pressure on the slope surface and slope bottom decreases, leading to directional seepage flow. As a result, fine soil particles are carried away by seepage forces, leaving behind a large-pore framework. A decrease in soil compaction implies a decrease in stiffness and strength, which may result in slope instability [34]. Among these factors, the rate of decrease in the RWL had a significant impact on slope stability, with the stability coefficient of the reservoir bank slopes decreasing as the rate of decrease in the RWL increased. According to previous studies, the critical rate of decrease in RWL is approximately 0.5 m/h [34]. In Figure 14, the maximum rate of decrease in the water level along the river channel under the 11 scenarios is calculated, which is similar to the daily fluctuation trend of the water level. The possible instability range is determined by identifying river segments where the rate of decrease in the RWL exceeds the critical value of 0.5 m/h. The possible instability ranges for the 11 scenarios are listed in

Table 4. Potential instability ranges for each scenario.

Scenario	1	2	3	4	5	6	7	8	9	10	11
Reservoir tail (km)	71.5	100.23	49.95	28.81	23.81	0	0	0	48.98	32.49	27.91
Main reservoir area (km)	0	0	0	0	0	13.07	4.72	0	20.17	14.03	0
Total (km)	71.5	100.23	49.95	28.81	23.81	13.07	4.72	0	69.15	46.52	27.91

.

The discharge fluctuation of the DOUPS causes a large fluctuation in the water level near the reservoir tail. The possible instability range is basically concentrated in the reservoir tail. Closer to the dam site, the wave gradually attenuates and flattens, and the maximum rate of decrease in the water level gradually decreases. The possible instability range caused by natural conditions (Scenarios 1 and 2) is larger than that after water storage (Scenarios 3, 4, and 5). During the DODPS (Scenarios 6, 7, and 8), the maximum rate of decrease in the water level gradually decreases from the pre-dam to the reservoir tail. The possible instability range is basically concentrated in the pre-dam section. The possible instability range caused by the JDOCPS (Scenarios 9, 10, and 11) may occur in the reservoir tail and pre-dam sections.

The possible instability range caused by 975 m of water storage in the wet season is smaller than that caused by 952 m of water storage. When the water storage is 975 m, the possible instability range caused by the wet season is larger than that caused by the dry season. When the hydrological season and the water level are consistent, the possible instability range caused by the JDOCPS is the largest, the possible instability range caused by the DODPS is the smallest, and the possible instability range caused by the DOUPS is between the above two operation modes.

In general, reservoir impoundment reduces the WLF rate to a certain extent, increases the stability of bank slopes, and reduces the risk of instability; however, the location and range of the instability area change with different operation modes. The possible instability range caused by the JDOCPS is the largest.

4.5. Limitations and Future Work

These results suggest that ecological flow regulations should consider not only the minimum discharge requirements but also the rate and magnitude of short-term flow fluctuations, which are critical for fish spawning and bank stability. The modelling framework developed in this study provides a useful tool for supporting short-term flow prediction and hydrodynamic risk assessment. However, its practical application faces several challenges, including the need for real-time multi-source data acquisition, high computational efficiency, and effective coordination across different management agencies within the basin.

Despite these contributions, the current study has several limitations. First, model validation was constrained to a 3-day period during the dry season due to the availability of high-resolution monitoring data. While this period reflects typical daily operations under hydro-PV complementarity, the model’s performance under other hydrological regimes, such as during flood events or emergency operations, remains untested. This temporal limitation may restrict the generalizability of our results. Future work will focus on expanding the validation across multiple seasons and dispatch scenarios to enhance robustness.

Second, the selected RWLs of 952 and 975 m were chosen to represent historically frequent low and normal storage conditions under current management strategies. Within this operational range, the results consistently indicate that deeper water enhances attenuation and accelerates wave propagation. Although the study did not simulate more extreme RWLs, the observed trends imply that these mechanisms are strongly governed by underlying reservoir morphology and flow regimes. To improve the generalizability of the findings, future simulations should explore a broader spectrum of RWLs and cross-sectional geometries.

Third, while the current study demonstrates that the 1D unsteady flow model is sufficient to capture the primary dynamics of short-term fluctuations in the WDDR, it inherently lacks the ability to resolve lateral and vertical flow structures. Future research should, therefore, consider two- or three-dimensional hydrodynamic modelling for localised zones, such as river confluences, lateral bays, and dam-adjacent areas, where complex flow patterns may occur. These efforts will help quantify the limitations of the 1D model and further refine our understanding of hydrodynamic processes under hydro-PV complementary operations. Additionally, coupling such models with ecological response modules could improve the predictive capacity for habitat-scale flow impacts.

Lastly, it is important to recognise that the proposed modelling framework is particularly well-suited for steeply incised cascade reservoirs, such as those in the Jinsha River Basin. In contrast, low-gradient systems with wide floodplains or deltaic features are subject to more complex hydrodynamic interactions, including significant lateral exchanges and floodplain inundation, which may require higher-dimensional modelling approaches. Although the analytical strategy used here is conceptually transferable, the model structure and parameterisation must be adapted to suit different geomorphological settings.

5. Conclusions

In this study, the Wudongde Reservoir on the Jinsha River was used as a case study to investigate the hydrodynamic impacts of daily cascade operations under hydro-PV complementarity. A one-dimensional unsteady flow model was employed to simulate reservoir-scale flow fluctuations, and cross-correlation analysis was used to examine wave propagation, attenuation, and energy dissipation across different hydrological seasons, reservoir water levels (RWLs), and operational modes. The main findings and their implications for cascade hydropower management are as follows:

(1)

Daily inflow and outflow regulation induces bi-directional wave propagation—downstream from the reservoir tail and upstream from the pre-dam—resulting in dynamic water level fluctuations along the channel. Wave amplitude and attenuation are strongly influenced by the channel topography, water depth, and bed resistance.

(2)

Reservoir impoundment significantly accelerates wave propagation. Compared to natural river conditions, the wave velocity under the normal storage level increases by approximately six times during the wet season and nine times during the dry season, reflecting the effect of increased water depth and reduced bottom friction.

(3)

Constructive and destructive wave interference occurs under the joint daily operation of cascade power stations (JDOCPS). Minimum wave amplitudes emerged at characteristic locations (e.g., near S69 or S77), depending on the RWL and discharge schedule. These interference patterns can be used to identify the low-energy zones in the reservoir.

(4)

Wave energy dissipation under JDOCPS is more pronounced than that under separate operations, showing an exponential decay in the reservoir tail and a linear decay in the main reservoir. The strongest attenuation occurs at the wave interference zones, indicating that coordinated scheduling can help reduce the intensity of flow fluctuations.

(5)

Slope instability risk is concentrated near the dam site and reservoir tail, where the maximum rate of decrease in the water level exceeds the critical thresholds. Although impoundment mitigates short-term water level variations, JDOCPS can expand the spatial extent of instability-prone areas compared with independent upstream/downstream operations.

These findings provide quantitative evidence for improving the short-term cascade scheduling strategies. By identifying the key regions of wave interference and energy attenuation, operators can optimise reservoir regulation to minimise harmful water level fluctuations, reduce the risk of slope instability, and support the ecological stability of reservoir shorelines. Specifically, the results suggest the following: (i) adaptive RWL management strategies based on the hydrological season and wave interference characteristics, aiming to reduce WLF amplitude and enhance slope stability; and (ii) integration of the proposed 1D hydrodynamic model into real-time forecasting systems to predict short-term water level variations, which can support early warning, emergency response, and slope monitoring.

The modelling framework developed in this study may also serve as a valuable reference for other deeply incised cascade systems operating under high-frequency renewable-energy scenarios.