Hydraulic classification of hydropeaking stages in a river reach

Hydropower is an important tool in the struggle for low‐emission power production. In the Nordic countries, hydropower operating conditions are expected to change and work more in conjunction with intermittent power production. This in turn might increase the amount of hydropeaking events in the reaches downstream of hydropower plants. The current work investigates the influence of highly flexible, high‐frequency hydropeaking on the hydrodynamics in the downstream reach. By quantifying four different dynamic stages in the study reach, the influence of the hydropeaking frequencies was investigated in the bypass reach of the Stornorrfors hydropower plant in the river Umeälven in northern Sweden. The hydrodynamics in the study reach were numerically modelled using the open source solver Delft3D. Eight different highly flexible future hydropeaking scenarios, varying from 12 to 60 flow changes per day, were considered. A method for identifying four hydropeaking stages—dewatering, dynamic, alternating and uniform —was introduced. The hydropeaking frequency directly decided the stage in most of the study reach. Furthermore, a Fourier analysis showed a significant difference between the stages and their corresponding power spectra. The classification of stages put forward in this work provides a novel, simple method to investigate the hydrodynamics due to hydropeaking in a river reach.

. As a result, rapid changes in power plant discharge, also called hydropeaking events, are expected to increase. With this background,the research consortium HydroFlex was started. The aim of HydroFlex was to develop technologies, for instance, a Francis turbine runner (Joy, Raisee, & Cervantes, 2022;Trivedi, Iliev, & Dahlhaug, 2020), permitting very flexible hydropower production while simultaneously investigating how an increase in hydropeaking frequency might affect the hydraulics in the downstream reaches. The scenarios of interest included as many as 60 flow changes (30 starts and stops) per day (HydroFlex, 2022).
Given the future scenarios discussed in this work, that is, a significant increase in daily hydropeaking events, the time between hydropeaking events, also called hydropeaking frequency, could prove to be important for the downstream hydrodynamics. The hydrodynamics of hydropeaking has been researched broadly in recent years. For instance, hydropeakings' effect on the dewatering of beaches was investigated using two-dimensional (2D) modelling (Burman, Hedger, Hellström, Andersson, & Sundt-Hansen, 2021;Juárez, Adeva-Bustos, Alfredsen, & Dønnum, 2019). Additionally, the magnitude of downstream vertical ramping velocities due to hydropeaking was investigated through 1D modelling in several river reaches in Austria (Hauer, Holzapfel, Leitner, & Graf, 2017). A Lagrangian approach to identify hydropeaking waves from discharge data was presented (Greimel et al., 2022). A recent work shows the need to further investigate the influence on the downstream reach for high-frequency hydropeaking.
The risk of dewatering for potential salmonid spawning habitat was reduced for high-frequency hydropeaking (Burman et al., 2021). Furthermore, the magnitude of the water level variation reduced as the hydropeaking frequency increased (Burman, Andersson, Hellström, & Angele, 2020), indicating that very high-frequency hydropeaking might reduce dewatering in the downstream parts of the reach.
Hence, it is of interest to further investigate the role of hydropeaking frequency and how it affects the hydrodynamics in the downstream reach. In this study, a calibrated numerical 2D model was used to predict the water depth dynamics in a bypassed reach when subject to the proposed scenarios corresponding to hydropeaking frequencies of up to 60 flow changes per day. The purpose of the study is hence 2-fold; first, to identify and quantify hydraulic stages that occur in the study reach due to the high-frequency hydropeaking. Second, to investigate the dynamics of the proposed hydraulic stages as a function of the hydropeaking frequency, using Fourier analysis.

| Study site
The chosen study site was the bypass reach in Stonorrfors, a reach measuring approximately 7 km in distance, in the river Umeälven, upstream of Umeå in northern Sweden. The hydropower plant in Stornorrfors is on average the most productive power plant in Sweden annually (Vattenfall, 2020). The catchment area is 26,568 km 2 (SMHI, 2022) and the mean annual flow through the power plant is 445 m 3 /s (Vattenfall, 2019). However, the operating conditions of the power plant do not affect the flow conditions in the bypass reach; except for cases when spilling is required (i.e., during the spring flood or turbine maintenance). The spillways of the dam lead into the bypass reach. In the winter months, the reach is mostly laid dry and only used in case of emergency spilling. In the summer months, a minimum flow of 21 m 3 /s is maintained through the fish ladder, which is used for the upstream migration of fish. Since the area around the bypass reach is a popular recreational area, the flow is often increased to 50 m 3 /s in the weekends by partially opening one of the spillways.
In the river Umeälven, the spring flood occurs in late May to early June. During the spring flood, the discharge annually reaches flow rates exceeding 1000 m 3 /s (Länsstyrelsen i Norrbotten, 2017). The fish ladder is mainly used by upstream migrating salmonids (Atlantic salmon and brown trout) as they migrate to the unregulated river Vindelälven, a protected national river with significant ecological value, which confluences with the river Umeälven approximately 10-km upstream of the dam (Vattenfall, 2020). To sustain the salmonid populations in the river Vindelälven, locally hatched juveniles are released into the reach yearly. An overview of the study reach in the proximity of Stornorrfors can be seen in Figure 1. The water level dynamics of the study reach was calibrated and validated for a hydropeaking cycle of 21 m 3 /s ! 50 m 3 /s ! 21 m 3 /s with a 5-min flow change time in a previous study (Burman et al., 2020). The well-defined flow conditions as well as validation data from previous work (Andersson et al., 2012;Angele & Andersson, 2018;Burman et al., 2020) performed in the reach made it a suitable study site.

| Hydropeaking scenarios
The study reach is technically not subject to hydropeaking, since the flow conditions in the bypass reach are not necessarily related to the power production of the power plant. However, rapid flow fluctuations still occur up to 10 times per week with current-day operating conditions (Länsstyrelsen i Norrbotten, 2022). Seven model scenarios of varying hydropeaking frequencies, ranging from 12 to 60 flow changes per day, were investigated. The discharge varied between 50 and 21 m 3 /s, which were the two most common discharges that occur in the reach in the summer. The corresponding hydrographs that were used can be seen in Figure 2. Each scenario spanned a period of 6 h. A sufficient spin-up time was used so that the steady state corresponding to 50 m 3 /s was reached before the first flow change occurred. Each flow change took 5 min (increase or decrease) since this was the most observed flow change time in the public discharge data, which had a temporal resolution of 1 min (Länsstyrelsen i Norrbotten, 2017). These hydropeaking events per day is not planned in the reach currently, but is rather a benchmark put forward by HydroFlex for a future flexible hydropower plant (HydroFlex, 2022). Hence predicting the effects of such high frequent hydropeaking is of interest.

| Numerical modelling
The hydrodynamics was modeled using the open-source solver Delft3D (Deltares, 2014). Delft3D solves the equations governing all fluid dynamics, the Navier-Stokes equations, by making some assumptions.
First, the Navier-Stokes equations are depth averaged so that only two, depth averaged, velocity components are solved for. Second, the Boussinesq approximation, that is, the assumption of approximately constant density is applied (density effects are in general not important in river flows). Lastly, it is assumed that the horizontal length scales are much larger than the water depth length scales, which is often the case in large scale river simulations. By applying these assumptions, the Navier-Stokes equations reduce to shallow water equations (SWE) The seven different hydrographs used in the study with hydropeaking frequency varying from 12 to 60 flow changes per day. bathymetry presented in Burman et al. (2020); in the study, the depth was measured in eight points along the reach; afterward the Manning number, in proximity to each measurement point, was chosen in such a way that the water depth error was minimized. The model required three boundary conditions; one upstream, one downstream and one wall condition. In each simulation, the upstream condition was set to one of the corresponding hydrographs in Figure 2. The wall condition was set to free slip, the recommended boundary condition for largescale simulations. Finally, the downstream condition was a Neumann condition with a slope of the water surface set to 0.001. The advantage of using the Neumann condition was that no information on the downstream water level was required. Furthermore, a threshold depth of 0.1 m was used, that is, when the water depth was less than 0.1 m, the node was considered to be above the water line. In a previous work (Burman et al., 2020;Burman, Andersson, & Hellström, 2019), Richardson extrapolation indicated that a mesh of approximately 450,000 nodes proved sufficient to provide a mesh independent solution and was hence used in this work.

| Classifcation of hydraulic stages
The water depth dynamics in the reach was separated into four different hydraulic stages; a dynamic stage, a dewatering stage, an alternating stage and a uniform stage. In the dynamic stage, the depth oscillated approximately sinusoidally, but never reached the steady state for either 50 m 3 /s or 21 m 3 /s. The "dewatering" stage was observed in nodes where the depth time series was discontinuous, that is, during the flow decrease, the node dried out. In the alternating stage, the depth achieved the steady state for both the 50 and 21 m 3 /s before a new flow change occurred. Finally, the uniform stage was observed in downstream parts of the reach where the difference in maximum and minimum depth was low, that is, the depth is approaching a uniform value. In Figure 3, examples corresponding to each different stage can be seen for the case of 60 flow changes per day. The matter of classifying the stage in each node was rather straight forward: • Dynamic: Nodes that are within the depth range of 1:01d min < d < 0:99d max for the entire time series, were considered to be in the dynamic range.
• Dewatering: During simulation, nodes that had discontinuous depth time series were considered to be in the dewatering stage, see • Alternating: If a node reached the steady state depth for both Q = 50 m 3 /s and Q = 21 m 3 /s, it was considered to be in the steady stage.
• Uniform: When the nodes where in the range d max À d min < 0:25 d 50 À d 21 ð Þ , that is, the maximum difference in depth in the node is 25% of the difference in depth for the steady states, the nodes were considered to be in the uniform stage.
For a node to be considered for classification, the node had to have a non-constant depth during the cycle, this was done to remove puddles that formed during initialization of the simulation. These nodes would otherwise have been classified as uniform. Due to the hysteresis of the increase-decrease cycle, that is, the depth decrease time is longer than the depth increase time, there exists an edge case where a node will reach steady state for the Q = 50 m 3 /s flow and not reach the Q = 21 m 3 /s steady state (Burman et al., 2020). These nodes were considered to be in the "dynamic" stage. Furthermore, the cutoff threshold of 25% for the uniform state was deemed reasonable for this study. This threshold should be chosen based on the characteristics of each individual study reach.

| Fourier analysis
A Fourier series is the expansion of a periodic function using an orthogonal base of sine and cosine functions. Assuming that a function has a period T, then the geometric Fourier series is where the Fourier coefficients a n and b n are analyzing how many terms of the Fourier series (Equation 1) that was required to recreate the original depth time series, it was possible to quantify the relative impact of the hydropeaking frequency on the suggested hydraulic stages. In this work, the fast Fourier transform (FFT) pack provided by the SciPy package was used (Scipy.org, 2022).
By then identifying how many peaks occurred, as well as the corresponding frequency, it was possible to recreate the temporal signal using Equation (1).

| Distribution of hydraulic stages as a function of hydropeaking frequency
Each node that qualified for classification went through the process described in the method for each of the hydrographs in Figure 2. In order to aid visualization, each state was then given a color; green for the dynamic stage, red for the dewatering state, blue for the alternating stage and yellow for the uniform stage. The resulting spatial distribution of the stages for all hydropeaking frequencies can be seen in

| Fourier analysis
The simulated water depth time series were Fourier transformed in four points along the study reach, for each model scenario. The Fourier points were chosen in such a way that all hydraulic stages, except dewatering, were represented; the longitudinal distribution can be seen in Figure 1. The resulting power spectra in each point can be seen in Figure 6. The most upstream point (Figure 6a), was in a location were the alternating stage occurred for all hydropeaking scenarios. The remaining points (Figure 6b-d) were, except for low hydropeaking frequencies, in the dynamic or uniform stage. The power spectra are dependent on the location and the stage. All spectra show a dominant peak roughly corresponding to the hydropeaking frequency (since the flow change occurs over five minutes, this shifts the peak). In point 1, there are prominent peaks for all scenarios correlating to approximately Ω 2 ≈ 3 Â Ω 1 , where Ω 1 is the first dominant peak and Ω 2 is the first trailing peak, which is then followed by a tail of smaller peaks. Looking at the location of Ω 1 (denoted in Figure 6a) for 60 flow changes per day in point 1, it can be seen that

| Distribution of hydraulic stages as a function hydropeaking frequency
The distribution of the hydraulic stages in the study reach, expectedly, There is a trend that the proportion of the wetted area in the dewatering stage is decreasing with an increase in hydropeaking frequency. This is due to a decrease in the spanwise extent of the reach, which occurs as a result of the longitudinal dampening in the reach; an effect which is well documented in the literature (Greimel et al., 2022;Hauer, Schober, & Habersack, 2013;Juárez et al., 2019).
This in turn means that the dewatering is inherently connected with the dynamic or uniform stages. An increase in hydraulic stages that never reach steady state, reduces the proportion of the dewatering stage in the reach. A reduction of the proportion of dewatering stage could aid in avoiding stranding events or dewatering of redds. It is, however, likely that the increase in hydropeaking frequency would have additional negative impacts due hydropeaking as described in that is, Flodmark et al. (2004); Bakken, Forseth, et al. (2016);Vollset et al. (2016). Additionally, hydrofibrilation, a phenomena mainly caused by run-of-the-river plants, is similar to hydropeaking in frequency but smaller in magnitude (Greimel et al., 2015). to the hydraulics of the study reach but not included in 2D models (Pisaturo et al., 2017). Although field measurements of the water depth dynamics are difficult to obtain due to the nature of the model scenarios, it would be of great interest to perform such measurements to validate the water depth dynamics of this study.

| Fourier analysis
Based on the number of peaks in the power spectrum, one would expect that the amount of Fourier coefficients required to resolve the water depth time series would be more for the alternating than for the dynamic stage. An example of this can be seen when looking at the power spectrum for Fourier point 4 ( Figure 6d). For 44 flow changes per day in Figure 6d, it can be assumed that a Fourier series with two coefficients will converge on the simulated response. This is also observed in Figure 7. In the case of 12 flow changes per day, when the point is in the alternating stage, the power spectrum indicates that more Fourier coefficients needs to be considered in the expansion, which is also seen in Figure 7a.
The power spectra that contained more peaks also required more coefficients to have a good fit. Due to the hysteresis in the increase-decrease event (Burman et al., 2020), the shape of the depth time series is not completely sinusoidal in the dynamic stage, this is reflected in the amount of Fourier coefficients required to recover the original signal in Figure 7b,c. The dynamics in the uniform stage can be completely recovered using a Fourier series expansion of the hydropeaking frequency, using as many coefficients as the spectra indicates. However, the alternating and dynamic stage requires more coefficients than the spectra would indicate to capture the depth dynamics. This indicates that the dynamics is dependent on additional effects and variables than the hydropeaking frequency in these stages. Which can be expected based on the trailing peaks in the Fourier transform. One such effect, the hysteresis in the increase-decrease cycle causes, likely adds additional spikes to the spectrum. The hysteresis in turn depends on 3D effects, such as turbulence, as well as longitudinal distance from the spillways (Burman et al., 2020).
Additionally, it is seen that the shift in frequency for the first trailing peak (Ω 2 ) differs between Fourier point 1 and the remaining points. This difference may be due to several reasons. It is possible that in point 1, hydropeaking waves are being reflected from the pool, which occurs shortly downstream of Fourier point 1. Another part of the explanation could be that the local geometry plays an important part.
Regarding the classification of the hydraulic stages; it could prove fruitful to look at the power spectra to obtain a less arbitrary way of classification. Consider Figure 6a, where the magnitude of Ω 1 is approximately independent of the hydropeaking frequency; this could be taken as an indication that the node is in the alternating stage.
When the magnitude of the leading peak becomes dependent on the hydropeaking frequency (Figure 6b-d), the node is either in the dynamic or uniform stage. One way to distinguish these two stages would be to count the number of peaks in the spectrum. For instance, in point 4 (Figure 6d), the spectra corresponding to the uniform stage only contain the leading peak Ω 1 .
Lastly, even though HydroFlex have suggested scenarios corresponding to up to 60 flow changes per day, it is very unlikely that such extreme hydropeaking frequencies will occur in the Nordics. A study investigating future European energy scenarios concluded that, in the most extreme scenario, the Stornorrfors power plant might only be subject to hydropeaking frequencies of 6 flow changes per day (Wirtz, Siemonsmeier, Schönefeld, & Moser, 2020).

| CONCLUSIONS
As the role of hydropower is expected to change to incorporate more flexible power production, it is of interest to investigate how such flexibility, in terms of increased hydropeaking frequency, affects the reach downstream of such operating conditions. By performing 2D simulations of seven model scenarios, four different hydraulic stages that occur in the study reach were quantified. The spatial distribution of the hydraulic stages varied substantially with the hydropeaking frequency. With an increase in hydropeaking frequency, the proportion of the study reach in the dynamic stage increased while the alternating stage decreased. For the scenarios with the most extreme hydropeaking frequencies, the uniform stage is apparent in the lower parts of the reach, this stage is also related to a decrease in proportion of dewatering. A Fourier analysis gave further insights into the dynamics of the different hydraulic stages. After a Fourier transform, the alternating stage showed several prominent peaks, where the leading peak (Ω 1 ) corresponded to the hydropeaking frequency and the trailing peak (Ω 2 ) approximately coincided with an integer multiple of the hydropeaking frequency. In the dynamic stage, only two prominent peaks were observed and the magnitude of the peaks reduced with increasing hydropeaking frequency. In the downstream parts of the reach, the Ω 2 peak disappeared, which indicated that the uniform state was occurring. These features observed in the Fourier transform could form an alternative, less arbitrary, method of defining the hydraulic stages in the reach. The Fourier analysis also showed that the dynamics of uniform stage can be completely characterized by the hydropeaking frequency. Future research regarding the influence of hydropeaking frequency should take into consideration model scenarios of varying discharge. Moreover, a study investigating the suggested hydraulic stages in a another study reach subject to different flow conditions would be crucial for additional understanding. Additionally, large scale 3D CFD simulations might provide valuable insights into the fluid dynamics that are lost using 2D methods. Furthermore, the results presented in this work is based on validation data for specifically one depth increase-decrease cycle (Burman et al., 2020). In the future, it would be important to apply the methodology suggested herein on a reach where the hydropeaking frequency allows for such analysis. This would highlight the broader applicability of the methods suggested in this work.