Introduced flow variability and its propagation downstream of hydropower stations in Sweden

Hydropower production contributes to 97% of the renewable electricity production worldwide (Göthe et al. 2019). As such, it has an immense role in the mitigation of climate change. In addition to this direct value as a source of renewable electricity, it also contributes to balancing power as needed depending on how well the production of wind and other fluctuating sources meet the energy demand. However, this might result in substantial fluctuations in streamflow downstream of power plants. These fluctuations can create zones that are wetted and dried at short time intervals, ultimately becoming lost as a habitat to most species. Macrophytes may then die, affecting also invertebrates and fish. Young fish may be stranded, and important areas for fish spawning might be lost (Tallaksen & Lanen 2004). To reduce ecological impacts, restrictions for hydropower production are enforced by regulatory frameworks, such as the Water Framework Directive in Europe. In Sweden, there will be a particular focus on the ecological impacts of hydropower regulations over the current and coming decade when all hydropower licences are being revised (Swedish Agency for Marine and Water Management et al. 2019).

Restrictions for hydropower production for improved ecology should consider flow fluctuations downstream of hydropower regulations and how quickly these are attenuated in the downstream direction. Because ecological impacts are expected to depend on the frequency and amplitude of flow variability, a useful measure of flow variability is obtained by using the Fourier transformation. This transformation decomposes the variability into a frequency domain characterized by periodic waves, or modes, of different frequencies with their respective amplitudes (deviations from the local average flow). The flow variability introduced by a hydropower plant will be attenuated as the water moves downstream, and by analyzing the variability in the frequency domain, the attenuation rates can be assessed separately for these different modes. Especially the modes with large amplitudes are of interest and we call the one with the highest amplitude the dominant mode. The attenuation depends on the type of disturbance. Andersson (2020) investigated the damping of perturbations caused by an upstream step-wise change in the discharge rate. Applying the Fourier transformation to observed or simulated time series, as done previously in Sweden by, e.g. Wörman et al. (2010), provides more information on the damping of flow perturbations of different periods caused by hydropower regulations. The same authors provided expressions for an exponential decay in the downstream direction, depending on the period (frequency) as well as on the celerity and dispersion coefficient. Even though exponential decay is reported in the literature and is the classical solution to the diffusion problem (Henderson 1966), and despite its usefulness in performing quick estimates of attenuation, we have not found verified decay rates for instream attenuation that can be applied in Swedish regulated rivers.

As opposed to the attenuation of rainfall over a watershed, which is usually dominated by processes outside individual river reaches (Rinaldo et al. 1991; Robinson et al. 1995; White et al. 2004), we are interested in the attenuation of hydropower-induced flow perturbations that are acting directly on river flow. It is reasonable to assume that mainly instream processes will attenuate these disturbances. In Sweden, rivers regulated for hydropower production often resemble cascades with low-gradient river stretches of slope less than 10⁻³ m/m between the dams, because most of the head difference is used for production. The question is then how efficient attenuation is in these rivers and how it is best evaluated. The hydrological rainfall–runoff model HYPE (Lindström et al. 2010) and the diffusion wave equation (Henderson 1966) have both been applied frequently to describe attenuation in Swedish-regulated rivers. These models consider different physical mechanisms of attenuation, and no model intercomparison study has been performed to assess which model best describes the amplitude reduction downstream of hydropower regulations. Therefore, there is a lack of understanding of the dominant processes for attenuation in Swedish-regulated rivers, which hampers the assessment of ecological impacts.

HYPE is used operationally for Sweden's national flood forecasting and drought warnings, and is also used to assist in the planning of hydropower operations, evaluate hydropower and climate impacts, and calculate the leakage of nitrogen and phosphorus (e.g. Grimvall et al. 2014). The instream translation and attenuation of flood waves is represented by a simple combination of a linear channel and a linear reservoir within each sub-catchment, i.e. the attenuation depends on storage effects similar to lakes, which become increasingly important at decreasing Peclet number (Pe) in rivers (Rinaldo et al. 1991). The simulated flow has been carefully calibrated to observations as a continuous effort by the Swedish Meteorological and Hydrological Institute (SMHI) over the past decade (e.g. Lindström 2016), but the attenuation has not been previously evaluated in terms of the decay rates of Fourier modes along river reaches.

Henderson (1966) proposed that the hydromechanical response in rivers can be evaluated using the diffusion wave equation, and this has also been applied to describe attenuation in Swedish-regulated catchments (e.g. Wörman et al. 2010). This equation is sometimes called the kinematic wave equation, but we avoid this terminology because it has also been used to describe waves without dispersive effects (e.g. Maidment, 1992). The kinematic celerity is, however, used in both models. The diffusion wave equation has two parameters, the flow celerity and the dispersion coefficient. These are assumed to be approximately constant in a river if the water depth does not change much (Wörman et al. 2010), and in this case, the equation is linear. However, applying constant parameters also assumes that the friction slope consists mainly of the river bed slope, which according to Henderson (1966) would require bed slopes larger than 10⁻³ m/m, a condition often not met in regulated rivers in Sweden. Rinaldo et al. (1991) found relationships between the celerity and dispersion coefficient that depend on friction properties, water depth, and the bottom slope (for perturbations from uniform flow). However, this approach underestimates the dispersion coefficient and overestimates the celerity if the bed slope is too small. Another reason the dispersion coefficient may be underestimated is that additional dispersion can occur due to channel irregularities (Henderson 1966). It is therefore uncertain how well the model from Rinaldo et al. (1991) will perform for the rivers considered in this study. White et al. (2004) and Robinson et al. (1995) studied the dependence of attenuation on the Horton–Strahler order and return period, both of which can be interpreted to some extent as a dependence on the overall discharge rate. They found that the hydraulic dispersion coefficient increased with increasing order and return period, i.e. with increased discharge, which is consistent with a dispersion coefficient that increases with celerity and water depth, as found by Rinaldo et al. (1991).

In addition to instream attenuation, Sweden has about 100,000 lakes with a surface area of more than 0.01 km². Lakes may substantially contribute to the damping of flood waves (e.g. Quin & Destouni, 2018). As far as the authors know, there is no overview of how lakes contribute to the damping of flow variability of different frequencies, and this adds uncertainty about the total attenuation in catchments with many lakes.

The specific aims of this study are to (i) assess the flow variability caused by hydropower in all of Sweden, (ii) assess the rate at which this variability is attenuated in rivers, (iii) improve the understanding of the processes that are responsible for instream attenuation, (iv) assess what models can be used to quantify the attenuation rate, and (v) to assess the role of lakes for attenuation.

To meet the aims of this study, analyses were performed on discharge data obtained from various sources as shown in Table 1 and described below in further detail. The dominant mode was assessed based on observations and simulations with HYPE. Simulations without hydropower were then analyzed to separate the impact of hydropower. The attenuation was evaluated from these data complemented by other models to facilitate the understanding of important processes for attenuation, and to assess what models can provide reliable results. Specifically, HYPE simulations with only instream processes (here called HYPE_riv) were performed to separate the impacts of instream processes from those of variable runoff, and HYPE results were compared to results from the diffusion wave equation to further increase the understanding of dominant attenuation processes and assess what model best describes observation data.

Observed discharge rates at a daily time step were collected from SMHI's database. In six river stretches, these were used to compare the simulated and observed attenuation in rivers, see Figure 1(b). In addition, observations were used to update simulation results at 579 locations (Figure 1(b)) as explained below.

Daily discharge was calculated using the hydrological model HYPE with the Swedish parameter set S-HYPE (version 2016f), whereby Sweden and surrounding inflow areas are discretized into approximately 40,000 sub-catchments. A 10-year initialization period was used. HYPE is a rainfall–runoff model that is calibrated to observations (e.g. Lindström 2016), mainly using parameters that vary by land use and soil type, sometimes (also here) with regional adjustments. Therefore, it can also be used in catchments that lack observations. The version used here has mean NSE (Nash–Sutcliffe Efficiency) of 0.80, mean KGE (Kling–Gupta Efficiency) of 0.84, and mean volume error of −0.9%, measured over a subset of 283 stations that belong to the official SMHI network, over the calibration period 2009–2018. These statistics were obtained without using station updates to the observed flow, see below.

Each sub-catchment in HYPE may have a main river that is fed by rivers upstream, by runoff from the sub-catchment itself (via a local river and lake), and to a smaller degree by precipitation on the water surface itself. At the outlet of a sub-catchment, a lake that receives water from the main river can be modeled. Such lakes will impact the overall attenuation of the upstream flow perturbations. Additionally, a hydropower station that regulates the outflow from the lake can be introduced. The discharge may also be impacted by abstractions and releases due to, e.g. municipal use and by water transfer such as the transfer of hydropower production water to a sub-catchment further downstream. The HYPE code and parameters are described elsewhere (e.g. Lindström et al. 2010; Strömqvist et al. 2012) and only the most relevant aspects for this work are presented here, i.e. the description of hydropower regulations and river and lake routing.

Hydropower regulations: HYPE can model the outflow from regulated lakes as a combination of hydropower production and spill (Arheimer & Lindström 2014). The production water is sometimes released further downstream. Dates with different production rates can be set, and the spill is calculated by a rating curve, in general as a nonlinear function of water depth above a threshold. Careful and incremental adjustments to the model description of Sweden's hydropower regulations have been performed over the past decade, and seasonal regulations are now very well described. Hydropower regulations of shorter duration, such as on the weekly time scale, have not been introduced directly into the model parameterization or equations. However, most of these regulations can also be introduced directly into the model by replacing the simulated flow with the observed value for historical periods, as was performed here. In a total of 579 stations (Figure 1(a)), the calculated discharge was substituted by the observed discharge at every time step where available data exists, impacting the simulation of the downstream flow. These station-updates give us a unique opportunity to study the attenuation of short-term regulations over the entire country.

Natural regime: Stream discharges simulated for the current regulated conditions were compared to stream discharges simulated assuming natural lake outflow, the so-called HYPE-QN model (Arheimer & Lindström 2014 and Table 1). The purpose of the regulation is to even out the flow, and store water for electricity production, primarily in the winter. In the QN version, this storage is removed, and natural flow conditions are simulated. The outflow is modeled using rating curves that describe observations prior to the building of dams, if observations from this period were available, or otherwise by rating curves that describe spill at regulated conditions. In addition, man-made diversions to hydropower plants are removed. Natural flow simulations have been verified against reconstruction calculations of natural flow performed by the hydropower industry (see e.g. Arheimer & Lindström 2014).

Here, Δt (days) is the time-step size and S(0) (m³) is the linear reservoir storage volume before inflow this day.

Here, q_L(t) (m³/day) is the lake outflow, h(t) (m) is the height above a threshold for outflow, K is a constant (m^3−p/day) that will equal the outflow when the water is 1 m above the threshold, and p (–) is an exponent. The exponent has been fitted individually where observations are available (Lindström 2016). The most frequent value, p = 2, consistent with a parabolic shape of the outlet, was then used in ungauged basins.

Streamflow time series were created from an artificial inflow at the upstream end of the river that was then routed through the river using HYPE's equations for river transport and attenuation (linear channel and linear reservoir) alone, without other processes such as precipitation, evaporation, or local runoff.

At the upstream end of the river, a sinusoidal inflow with a mean of 10 m³/s, amplitude of 1 m³/s, and with the chosen period was inserted during a simulation time corresponding to 10 such periods. The outflow at a distance downstream of this periodic inflow was calculated. Different river lengths between 1 and 20 km were tested, but as very minor differences in the attenuation rate were observed, only results from the 10-km river length are shown in the Results section.

Here, y (m) is the local and temporal water height of the river, t (s) is time, x (m) is location along the river, v (m/s) is celerity, and D_L (m²/s) is the dispersion coefficient. Note that the differential equation is valid for the perturbation in water height as well as for the actual height because the perturbation can be calculated by subtraction of the average height from the actual height. We assume that the perturbation of discharge is proportional to the perturbation of water height, such that y can represent a discharge perturbation (unit m³/s). This means that we assume that the change in river width when the height fluctuates is negligible compared to the mean river width, which is often a good approximation, and that the change in celerity with the perturbations is also negligible (cf. Henderson 1966). Hence, we let y be the flow perturbation of a given frequency ω (–).

The first term (the fraction) in Equation (8) is the Laplace transform of A₀ sin(ωt) and the second term relates to the damping. The flow perturbation in the time domain is the inverse of the full expression, corresponding to the convolution of the inverse of the two terms. We obtained this inverse numerically for different ω.

Here, c = 0.56 ((m³/s)^1−f) and f = 0.3405 (–).

We will refer to this as the observed celerity.

When studying attenuation of the flow perturbations obtained from observations and simulations, we first applied the Fourier transform to convert these time series to Fourier modes of different periods T=2π/ω with their respective perturbation amplitudes A. The choice of using amplitudes (unit of flow), rather than variance, was made to facilitate intuitive interpretation of results.

We chose to disregard temporal changes of the spectra, such as seasonal variations, which could have been analyzed by, e.g. wavelet analysis (Schaefli et al. 2007).

Here, rivlen (m) is the main river length in a sub-catchment. If water flows between the outlet of sub-catchment j–1 to the outlet of sub-catchment j then it will have passed through the entire main river of sub-catchment j. The corresponding σ was assigned to sub-catchment j, because it represents processes occurring there. If this sub-catchment receives flow from more than one upstream sub-catchment, the one with largest overall variance was used in the definition of growth rate. This is an intuitive choice and is also consistent with the expected average amplitude, because the smaller perturbation can either add to the larger one (when they are in phase) or subtract from it (when they have opposite phase). However, we also note that if perturbations of the sub-catchments typically occur with the same phase, the overall impact might be somewhat additive instead. In cases where hydropower production water is released further downstream in a river, the simulated σ may have very large negative or positive values as the river enters and exits the part of the river that is bypassed by production water, and the observed σ that compares observation data upstream and downstream of the bypassed part of the river may be close to zero due to the absence of river attenuation processes, because the total length of the river section between comparison locations was always used in the definition.

Note the similarity of this equation with Equation (9).

Here, k_L (1/day) is the recession coefficient of the lake at the mean flow. As Equation (21) shows, the attenuation factor over the lake is /⁠, i.e. it depends on the parameters describing the rating curve as well as the lake area, in addition to the average flow and frequency of the perturbation studied. For example, perturbations of period 7 days (⁠ 10⁻⁵ s⁻¹) and mean flow 100 m³/s in a lake with p = 2, K=q_m/4 and α_L = 10 km² (a large lake extending, e.g. 10 km × 1 km) would give k_L = 10⁻⁵ s⁻¹. Then the damping factor is 1/⁠, such that about 70% of the amplitude remains after the perturbation passes the lake.

Here, we first describe the dominant modes in regulated rivers of Sweden, based on the daily time series from HYPE. After this, HYPE's decay rates along a few chosen rivers are examined in more detail and compared with observations as well as with results from the diffusion wave equation. The attenuation in rivers is then presented at the national scale, followed by results on attenuation in lakes.

By using station updates in the hydrological simulations, flow variability due to hydropower is captured also over short time periods, only limited by the daily time step. In the river sections between these updates, the simulations typically show attenuation. In this section, the simulated attenuation, in terms of the exponential decay rate, is compared with attenuation shown by observations, focusing on regulated rivers. In regulated catchments of Sweden, it is rare to find two hydrological stations along a river with no flow regulation occurring between these stations. This makes the comparison of simulated and observed attenuation challenging. When these regulations between stations increase the perturbation amplitude (positive σ), we disregard them from the comparison, since we are interested in the river attenuation. However, not all regulations perturb all periods, either by choice of the managers, by limitations caused by the available regulation volume or other restrictions.

The HYPE_riv growth rates are very close to the full simulation results for periods up to 14 days, which means that the simulated river decay is indeed due to attenuation within rivers and not significantly impacted by other processes. The 15-day mode is not shown but its amplitude increases also between the last stations. On this time scale, other processes also impact the simulated attenuation.

In summary, the comparison with observations shows that attenuation of periods from 7 days and larger is well described by HYPE, that σ is determined by instream processes for periods up to 14 days, and that almost no attenuation could be observed in some rivers, likely due to regulations between observations and/or bypassed river channels.

Damping described by the diffusion wave is also shown in the figure, here using celerity v = 1 m/s as a first example. To make a fair comparison with HYPE_riv results employing a daily time step, the FFT solution was calculated for a daily time step, however tests showed that a finer time step produced very similar results, that results were insensitive to changed river length between 1 and 100 km (10 km is used in the figure), and that the analytical solution (Equation (19)) produced very similar results. Perturbations of period less than 10 days are less attenuated with the diffusion wave compared to the verified HYPE_riv solution, and this applies regardless of the value of D_L, with the chosen celerity. The closest agreement between the diffusion wave results and the confirmed HYPE_riv results for periods 7–11 days was obtained using D_L on the order 10¹⁰ m²/day (10⁵ m²/s).

For each contour of σ, the region above the maximum celerity is not permissible, because for a given celerity v, a higher dispersion coefficient D_L should always produce more negative σ. The existence of a non-permissible region can also be identified directly from Equation (20) because larger D_L in the denominator will give smaller absolute value of σ if the impact of the denominator is larger than that of the nominator, which depends on v and T. Comparing Figure 6(a) and 6(b), σ is not permissible in the approximate range log10(Pe) < 0, i.e. Pe < 1, meaning when the time for diffusion is less than the time for advection. It is intuitive that the diffusion wave equation cannot be used when the dispersive effects would dominate, because the dispersion of a wave is caused by different travel times of moving water in a stream. When attenuation is due to volume storage more similar to lakes, other processes dominate.

We therefore need to find combinations of (v, D_L) that are both permissible and produce the observed σ. Table 2 shows example combinations and their resulting σ and permissibility, based on characteristics of river Västerdalälven and nine hypothetical rivers that have realistic bed slopes and discharge rates (also shown in Figure 7). Here, the celerity and water depth were first estimated based on river discharge (Equations (13) and (14)), and then Rinaldo's model (Equation (11)) was used to calculate the dispersion coefficient from the celerity, water depth and bed slope.

The choice of using a calibrated water velocity (Equation (14)) to represent wave celerity v (m/s) in Table 1 was made because it can easily be calculated for all of Sweden, and was much closer to the observed celerity (Equation (15)) than Rinaldo's celerity (Equation (11)). For example, the lower stretch of river Västerdalälven had observed celerity 0.6 m/s in a flood situation in the summer of 2010. The peak flow was approximately 400 m³/s, which gives velocity 0.7 m/s according to Equation (14), very close to the observed peak celerity. On the other hand, Rinaldo's celerity from Equation (11) with friction factor from Equation (12) gives an overestimated celerity 3 m/s, here using n = 0.03. We also made comparisons in two more rivers with the same conclusion: In May 1995, the observed celerity in river Vindelälven, a tributary to Umeälven (slope 0.001 m/m), was 1.1 m/s, as compared to nearly the same water velocity 1.0 m/s from Equation (14) but much higher celerity 5.8 m/s from Rinaldo's Equation (11). In May 2018, the observed celerity in Öre älv, south of Umeälven (slope 0.002 m/m), was 0.7 m/s, as compared with the same value for water velocity from Equation (14) but celerity 5.7 m/s from Rinaldo's Equation (11).

River Västerdalälven has larger slope (10⁻³ m/m) and river Indalsälven has smaller slope (10⁻⁵ m/m) than the slope 10⁻⁴ m/m required by the diffusion wave model to obtain observed σ in the permissible range. Hence, damping in these rivers could not be described by the diffusion wave model. Instead, this model would predict an attenuation one order of magnitude too low for river Västerdalälven, only reaching closer to the observed value if the dispersion coefficient was increased by one order of magnitude. For the very low-gradient river Indalsälven, the diffusion wave solution is not even permissible. Hence, the slope does not predict σ in these rivers, contrary to what the diffusion wave equation with Rinaldo's D_L would predict.

In summary, the diffusion wave could produce observed attenuation for certain combinations of v and D_L but these combinations were not consistent with Rinaldo's celerity, which was overestimated, or dispersion coefficient, which was underestimated. This can be explained by the use of bed slope to represent the friction slope in rivers where this approximation is known to be questionable, i.e. when the bed slope is around 10⁻³ m/m or less (Henderson 1966), in addition to neglecting channel irregularities in the estimate of D_L. We expect that similar results hold for other modes, albeit with other ranges of acceptable v and D_L. These ranges can be calculated for any period T by expanding terms in Equation (20).

As opposed to the above discussion, HYPE simulations were able to reproduce attenuation in rivers of both very small (Indalsälven) and larger (Västerdalälven) slope, the latter slope also being quite small due to the impact of regulations that use up much of the depth difference in regulated rivers.

The figure or corresponding equations can be used to obtain a quick estimate of required lake area or parameters for the outflow section, whereas actual constructions would require additional consideration to lake slopes etc.

Despite rivers acting almost like lakes, attenuation is not quick enough to make much of a difference in many rivers, in relation to the short river stretches that are undisturbed. Therefore, ecological values may not increase significantly away from hydropower stations in many regulated rivers in Sweden. However, not all regulations impose perturbations of all periods. For example, in Västerdalälven, the downstream stretch has more than 100 km river length with undisturbed decay of the weekly perturbation, reducing its amplitude from about 9 to 6 m³/s. In Indalsälven downstream of the large, regulated lake Storsjön, some regulations do further impact the 7-day mode and some do not, but here the overall impact is a more or less constant amplitude of this mode.

The analysis of lakes showed that the processes governing lake attenuation are the parameters describing the outflow restrictions (rating curve) as well as the lake area, in addition to the average flow and the period of the perturbation studied. The average flow varies throughout the year. Variability during low-flow periods can be especially harmful for ecology due to very low resulting minimum flows. The lake attenuation may be of larger relative importance during these critical periods since the requirements of lake size or outflow section are reduced in these instances, related to the nonlinear shape of the rating curve. There are some existing unregulated lakes in regulated rivers, e.g. one lake in river Skellefteälven. Although these lakes have the potential to reduce variability, this may not always be desirable from an economic standpoint of downstream production. For example, a reduction of the 7-day mode would reduce flow on weekdays, when the electricity price is higher, and increase flow on weekends, when the price is lower.

Our work demonstrates how a well-calibrated national hydrological model can be used to understand the dominant physical river processes and be of use for management decisions. By analyzing discharge variability and its attenuation along regulated streams from observations, HYPE simulations and the diffusion wave model, and by performing an analysis of the attenuation in lakes, we found that:

• Flow variability caused by hydropower in Swedish rivers is often dominated by regulations at a weekly time scale, when analyzing variability of time-scales from 2 days up to 1 month.

• To reduce the amplitude of these weekly perturbations in rivers by a factor of two, around 180 km of undisturbed river is required. The actual reduction factors are often smaller, due to limited distances between hydropower regulations.

• Attenuation of flow variability introduced by hydropower in the downstream direction along rivers is mainly due to instream lake-alike processes.

• The simple river routing method in HYPE using a linear channel and a linear reservoir described the flow attenuation very well, and gave much better results than the diffusion wave analogy for Swedish rivers. The reason for this is probably that the river slopes between hydropower dams is small. The difficulties in applying the diffusion wave model may also be due to neglecting the impact of channel irregularities, an effect that is difficult to parameterize.

• Lakes may contribute significantly to attenuation of the 7-day mode, but only when the flow is on the order 100 m³/s or less. Longer modes than 7 days require larger lakes for a given amplitude reduction.

Apart from the direct use of the results for management decisions, the results presented herein can also be used to introduce short-term regulations in hydrological simulators without the need for station updates. This is needed for forecasts, such as deterministic forecasts (the 7-day mode typically has a minimum value on Sundays) or stochastic seasonal forecasts where it can improve indicators such as the number of days below or above a threshold. Another important avenue for future research is to assess sub-daily flow variability that might become increasingly important in a future energy system.

All relevant data are available from an online repository or repositories. Data from the flow gauges used for this work can be downloaded at https://www.smhi.se/data/hydrologi/vattenwebb and model outputs can be downloaded from https://vattenwebb.smhi.se/archive/.

The authors declare there is no conflict.