# Storage size estimation for volatile renewable power generation: an application of the Fokker–Planck equation

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## Abstract

The time series of the volatile wind and solar power production in Germany are analysed with regard to a storage system with a finite capacity. The dimensions of this system are estimated in such a way that the storage is available at any time with a certain probability of failure. It is shown that, under simplifying assumptions, the storage filling equation belongs to the equations describing the Brownian motion of particles in the viscous limit. Using the associated Fokker–Planck equation, we show that the storage content is exponentially distributed in a first approximation and the failure probability is calculated from the distribution function. It is shown that the wind and solar power production together with a storage facility is in principle capable of providing a safe base load for longer periods of time if the mean inflow of the storage facility is more than $$20\%$$ larger than the mean outflow.

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## Data Availability Statement

The power generation data time series analysed in the current study are available from the authors on request. This manuscript has associated data in a data repository [Authors’ comment: All data included in this manuscript are available on request by contacting with the corresponding author.].

## Notes

1. https://www.entsoe.eu/

2. Regarding the difference in the definition of autocorrelation function and autocovariance function see for example [3]: In engineering it is common to define the autocorrelation function $$R_{xx}(\tau )$$ of a time function x(t) as

\begin{aligned} R_{xx}(\tau )=E \langle {x}(t)\,x(t+\tau )\rangle, \end{aligned}
(5)

while the autocovariance function $$C_{xx}(\tau )$$ is assigned to the function x(t) being centred by its expected value:

\begin{aligned} C_{xx}(\tau )=E \langle (x(t)-E_x)\,(x(t+\tau )-E_x)\rangle \end{aligned}
(6)
3. If the time function x(t) exists for times $$T+\tau$$ instead of only for a time interval of length T then

\begin{aligned} \tilde{R}_{xx}(\tau )=\frac{1}{T}\int \limits _0^T x(t) x(t+\tau )\,\text {d}t \end{aligned}
(8)

is an alternative definition of the autocorrelation function estimate [3]. Here the ACF is estimated from a random sequence over 50 years (i.e. $$N=6100$$), with an averaging interval with $$=5588$$ elements. The ACF time series thus has 512 elements which corresponds to a duration of more than 4 years.

4. This constant corresponds to the diffusion constant $$D_d$$ in diffusion processes, whereby $$D_d$$ is usually defined as half the value, i.e. $$C\cong 2\,D_d$$.

5. Schwabl has shown that the differential equation Eq. (22) is associated with the Smoluchowski-equation for the distribution of the Brownian particle displacements [8].

6. Due to the coordinate transformation in Eq. (26) lower average filling levels lead to higher mean values.

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## Acknowledgements

Our special thanks go to Prof. Michael Thorwart for the critical discussion and to Rolf Schuster for supplying the time series data.

## Author information

Authors

### Contributions

All the authors were involved in the preparation of the manuscript. All the authors have read and approved the final manuscript.

### Corresponding author

Correspondence to Detlef Ahlborn.

## Ethics declarations

### Conflict of interest

The authors confirm that we have no financial or other interest in the subject of the work in which we were involved, which may be considered as constituting a real, potential or apparent conflict of interest. The work presented was not subject of any public or other funding and it is the private work of the authors.

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Ahlborn, D., Ahlborn, F. Storage size estimation for volatile renewable power generation: an application of the Fokker–Planck equation. Eur. Phys. J. Plus 138, 401 (2023). https://doi.org/10.1140/epjp/s13360-023-04008-y