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

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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.].



  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

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

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

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

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

    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.


  1. D. Ahlborn, Principal component analysis of West European wind power generation Eur. Phys. J. Plus 135, 568 (2020)

    Article  Google Scholar 

  2. T. Linnemann, G. Vallana, Wind Energy in Germany and Europes Status, potentials and challenges for baseload application Part 2: European Situation in 2017. VGB PowerTech 3, 1–17 (2019)

    Google Scholar 

  3. J.S. Bendat, A.G. Piersol, Random Data Analysis and Measurement Procedures, 4th edn. (Wiley, New York, 2010)

    Book  MATH  Google Scholar 

  4. A. Papoulis, S.U. Pallai, Probability Random Variables and Stochastic Processes (Mac Graw Hill, New York, 2002)

    Google Scholar 

  5. G. Grimmet, D. Stirzaker, Probability and Random Processes, 3rd edn. (Oxford University Press, Oxford, 2001)

    Google Scholar 

  6. N. Pottier, Nonequilibrium Statistical Physics: Linear Irreversible Processes (Oxford University Press, Oxford, 2009)

    MATH  Google Scholar 

  7. H. Risken, The Fokker Planck Equation, 2nd edn. (Springer, Berlin, 1989)

    MATH  Google Scholar 

  8. F. Schwabl, Statistische Mechanik (Springer, New York, 2006)

    Book  MATH  Google Scholar 

  9. M. Smoluchowsi, Drei Vorträge über Diffusion. Brownsche Molekularbewegung und Koagulation von Kolloidteilchen Physikalische Zeitschrift 77, 530–599 (1916)

    Google Scholar 

  10. R. Fürth, Wärmeleitung und Diffusion in Die Differentialgleichungen der Physik Vol. 2, editet by Frank, Ph. and Mises, R.v. (Braunschweig, Vieweg) pp. 526–626 (1935)

  11. P.A.P. Moran, A theory of dams with continuous input and a general release rule. J. Appl. Probab. 6(1), 88–98 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Zielinski, An application of the Fokker–Planck equation in stochastic reservoir theory. Appl. Math. Comput. 15, 123–136 (1984)

    MathSciNet  MATH  Google Scholar 

  13. O. Ruhnau, S. Qvist, Storage requirements in a 100% renewable electricity system: Extreme events and inter-annual variability (ZBW - Leibniz Information Centre for Economics, Kiel, 2021)

    Google Scholar 

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

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Correspondence to Detlef Ahlborn.

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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).

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