# Fluctuations and Uncertainty of Wind and Solar Power Generation
```{admonition} Part of a series: Uncertainty in Future Energy Systems.
Follow reading [here](energy_systems_intro.md)
```
## Introduction
Future energy systems will rely on renewable power sources, especially wind and solar power.
Their operation depends on the weather and is thus highly variable and uncertain.
In this article, we will discuss fluctuations in renewable power generation occurring on different time scales.
## Fluctuations on Different Time Scales
First, there are deterministic periodic variations.
Obviously, there is a strong diurnal variation in solar power generation, which is simply zero during the night.
Notably, wind power generation may also show a weak deterministic diurnal variation {cite}`emeis2014current`.
The seasonal variability is also apparent for solar power.
Solar power generation is higher in summer, as days are longer and the solar zenith angle is smaller.
Wind power typically shows a pronounced seasonal variation, too, depending on the location.
In Europe, wind power generation is higher in the winter such that the seasonal variations of wind and solar power generation balance {cite}`heide2010seasonal`.
Stochastic fluctuations of wind and solar power on time scales up to one hour were reviewed in {cite}`anvari2016short`.
One finds that generation is highly intermittent.
That is, the power output can change substantially from low to maximal values and back in a short period of time.
This behavior can be quantified in terms of the increments
$$
\begin{align}
X_\tau(t) &= P(t+\tau) - P(t),
\end{align}
$$
where $P(t)$ is the power generation at time $t$.
{numref}`fig_anvari_increments` shows the increment statistics for different values of $\tau$ for wind and solar power reproduced from {cite}`anvari2016short`.
It is obvious that the statistics is far from Gaussian and exhibits heavy-tails, such that large values of $X_\tau$ occur far more often than expected from ordinary normal statistics.
```{figure} figures/energy_systems_main/anvari2016cropped.jpg
:align: center
:name: fig_anvari_increments
Empirical probability density functions of the increments of power power generation $X_\tau(t) = P(t+\tau) - P(t)$ for solar photovoltaics (left panel) and wind power (right panel).
Increments are given in the unit of the respective standard deviation ${\sigma }_{\tau }$ and PDFs are shifted in the vertical direction to improve visibility.
Figure is a cropped version taken from {cite}`anvari2016short`, which is licensed under the CC BY 3.0 DEED license.
```
Fluctuations of wind power generation on short time scales are driven by turbulence {cite}`milan2013turbulent` and thus share many statistical properties with other turbulent processes.
In particular, the power spectrum $S(f)$ follows the characteristic Kolmogorov law of turbulence {cite}`kolmogorov1968local`,
$$
\begin{align}
S(f) \propto f^{-5/3},
\end{align}
$$
up to a frequency of $f \approx 0.1 {\rm Hz}$.
At higher frequencies, the internal dynamics and inertia of the wind turbine must be taken into account such that the power generation $P(t)$ does not simply follow the momentary wind speed $v(t)$ {cite}`milan2013turbulent`.
The intermittency of solar power generation on short time scales is driven by cloud dynamics {cite}`tabar2014kolmogorov`.
On a partly cloudy day, the generation of a single photovoltaic panel will rapidly jump from a high to a low value if a cloud moves over the panel.
To model this flickering behavior, {cite}`anvari2016short` suggest a stochastic process with a bistable potential.
```{figure} figures/energy_systems_main/weber_composite.png
:align: center
:name: fig_weber_wind
(a) Time series of the wind speed (i.e., wind speed $v(t)$) at the location of the Alpha Ventus wind farm in the North Sea from the ERA-Interim reanalysis {cite}`dee2011era`.
(b) The distribution of the wind speed $v(t)$ is well described
by a Weibull distribution. The figure shows a histogram of the data (blue) and the most likely Weibull distribution (red) with shape parameter $\alpha \approx 2.36$ and scale parameter $ \beta \approx 9.66$.
(c) Histogram (blue) of the duration $d$ of periods with low wind power generation (velocity $v < 4 m/s$) at the location Harthaeuser Wald in Germany.
The red line shows the most likely exponential distribution given by the exponential constant $\lambda_e$, which strongly underestimates the tails of the distribution for long durations. Figure reproduced from {cite}`weber2019wind`, which is licensed under the CC BY 4.0 DEED license.
```
The variability of wind and solar power on time scales of hours to weeks is intensively studied in research and engineering (see, e.g., {cite}`staffell2018increasing`).
This variability is related to the atmospheric dynamics on the synoptic scale, which is of the order of magnitude of $1000$ km.
Typically, high- and low-pressure systems seen on weather maps are on the synoptic scale.
[Numerical weather forecasts](./uq_weather_prediction.md) are essential to predict the variability of renewable power generation on the synoptic time scale {cite}`emeis2018wind,ahmed2020review`. A particular challenge in this area is the prediction of ramp events (i.e., events where the generation changes quickly) and their timing {cite}`gallego2015review`.
Energy system simulation and optimization models typically use historical weather data, more precisely reanalysis datasets {cite}`dee2011era,staffell2016using` or synthetic time series from generative models {cite}`Cramer2022`.
The statistics of the wind speed $v(t)$ on this time scale is well described by a Weibull distribution as shown in {numref}`fig_weber_wind`.
Wind speeds can then be converted to wind power generation via the power curve of the respective wind turbine which is typically given by
$$
\begin{align}
P(v) = P_{\rm rated} \times \; \left\{ \begin{array}{l l l}
0 & & v < v_i \; \mbox{or} \; v_i > v_o \\
\frac{v^3 - v_{i}^3}{v_r^3-v_i^3}
& \; \mbox{if} \; & v_i \le v \le v_r \\
1 & & v_r < v \le v_o \; ,
\end{array} \right.
\end{align}
$$
where $v_i$, $v_r$ and $v_o$ are the cut-in, rated and cut-out velocities of the turbine, respectively. $P_{\rm rated}$ refers to the rated power of the wind turbine.
In addition to the overall statistics, temporal and spatial correlations are important for power system operation.
For instance, long periods of low renewable generation constitute a major challenge and typically require fossil backup power plants.
The persistence of periods with constantly low or high wind power generation in Europe has been studied in {cite}`weber2019wind`.
Assuming that the events that cross a threshold follow a Poisson process, the resulting persistence statistics should follow an exponential distribution.
It was shown that the duration of such periods is not well described by an exponential distribution but rather features heavy tails as shown in {numref}`fig_weber_wind`.
Remarkably, long periods can be observed for very different weather regimes.
For example, extremely long low wind periods were observed for a strong Azores High and extended ridge towards Central Europe, a blocking condition, and a cold high pressure centre over Eastern Europe.
```{figure} figures/energy_systems_main/pone.0190707.g003.png
:align: center
:name: fig_wohland_interannual
Natural variability of wind generation in Germany relative to 2016.
(a) Time series of wind generation and (b) its empirical distribution for a fictitious fixed wind turbine fleet that is kept constant to focus on the impact of wind variability on wind power generation.
Blue denotes calculations {cite}`wohland2018natural` and green indicates results from the renewables.ninja dataset {cite}`staffell2016using`.
The dashed brown line indicates the ratio of 2015 to 2016 of the redispatch volume (i.e., the amount of a change in power output at different power plants that is requested by grid operators to prevent congestion) in the German power transmission grid.
Figure reproduced from {cite}`wohland2018natural`, which is licensed under the CC BY 4.0 DEED license.
```
Renewable power generation displays considerable inter-annual variability.
{numref}`fig_wohland_interannual` shows how annual wind power generation in Germany would vary if the wind turbine fleet would be fixed.
One observes that variations of $\pm 20 \%$ from one year to another occur regularly.
The inter-annual variability has considerable impacts on energy system operation and must be taken into account in any energy system simulation or optimization model {cite}`collins2018impacts`.
Renewable power generation is further affected by climatic variations on even longer time scales {cite}`wohland2019significant,wohland2020european`. For instance, wind power resources in Western Europe are affected by the North Atlantic Oscillation (NAO).
## References
```{bibliography}
:filter: docname in docnames
```
## Authors
Dirk Witthaut, and Philipp Böttcher
## Contributors
Jonas Bauer, Annette Möller, and Lars Nerger