Propagation of wind-power-induced fluctuations in power grids

Renewable generators perturb the electric power grid with heavily non-Gaussian and time correlated fluctuations. While changes in generated power on timescales of minutes and hours are compensated by frequency control measures, we report subsecond distribution grid frequency measurements with local non-Gaussian fluctuations which depend on the magnitude of wind power generation in the grid. Motivated by such experimental findings, we simulate the subsecond grid frequency dynamics by perturbing the power grid, as modeled by a network of phase coupled nonlinear oscillators, with synthetically generated wind power feed-in time series. We derive a linear response theory and obtain analytical results for the variance of frequency increment distributions. We find that the variance of short-term fluctuations decays, for large inertia, exponentially with distance to the feed-in node, in agreement with numerical results both for a linear chain of nodes and the German transmission grid topology. In sharp contrast, the kurtosis of frequency increments is numerically found to decay only slowly, not exponentially, in both systems, indicating that the non-Gaussian shape of frequency fluctuations persists over long ranges.

Figure 1

(a) Distribution of power increments ${\mathrm{\Delta }}_{\theta }P=P(t+\theta )-P\left(t\right)$ of the aggregated output $P\left(t\right)$ of 12 wind turbines in a wind park (timescale $\theta =1$ s) [5] and Gaussian distribution as a reference in black. Each turbine has a rated power $P_r\approx 2$ MW. Power increments are given in units of the standard deviation ${\sigma }_{\mathrm{\Delta }P}=0.023$ MW of $p\left({\mathrm{\Delta }}_{\theta }P\right)$. (b) Frequency increment distribution $p\left({\mathrm{\Delta }}_{\theta }f\right|P_w)$ conditioned on wind energy generation $P_w$ in Germany, measured in Oldenburg ($\theta =0.2$ s). $p\left({\mathrm{\Delta }}_{\theta }f\right|P_w)$ broadens with increasing $P_w$. (c) Simultaneously measured data in Göttingen. Standard deviations of $p\left({\mathrm{\Delta }}_{\theta }f\right)$ are ${\sigma }_{\mathrm{\Delta }f,o}=0.58$ mHz in Oldenburg and ${\sigma }_{\mathrm{\Delta }f,g}=0.73$ mHz in Göttingen. (d) Variance evolution of the distributions in (b) and (c) with $P_w$, normalized with the first $P_w$ bin.

Figure 2

Evolution of the variance $〈{\mathrm{\Delta }}_{\theta }{\omega }_i^2〉$ of increment PDFs $p\left({\mathrm{\Delta }}_{\theta }{\omega }_i\right)$ ($\theta =0.01$ s) in a chain system of $N=20$ oscillators from linear response theory Eq. (5) (solid lines) and direct Runge-Kutta simulations (symbols). Node $j=1$ is driven by the perturbation signal $\delta P\left(t\right)$. We choose $\theta =0.01$ s because $\delta P\left(t\right)$ shows the largest deviations from the Gaussian distribution on small timescales.

Figure 3

(a) Variance $〈{\mathrm{\Delta }}_{\theta }{\omega }_i^2〉$ from Runge-Kutta simulations of a chain of $N=50$ oscillators (symbols) compared to an exponential decay with slope $m=-1/\xi =-2\sqrt{{\omega }_0}\gamma /\sqrt{JK}$ (straight lines). (b) Eigenvalues ${\mathrm{\Lambda }}_n$ for increasing inertia $J$ (color coded, ${\mathrm{\Lambda }}_n$ increase with increasing $J$). The dashed line marks ${\mathrm{\Lambda }}_n=1$. (c) Exponential fits (orange data with error bars) of $m$ converge to our analytical prediction $m=-1/\xi$ [Eq. (7), solid blue line] for increasing inertia $J$. The vertical dashed line marks $J_c$. Error bars correspond to the $2\sigma$ confidence bound of the exponential fit.

Figure 4

Kurtosis $k$ of $p\left({\mathrm{\Delta }}_{\theta }{\omega }_i\right)$ in a longer chain of $N=100$ oscillators. The frequency increment distribution $p\left({\mathrm{\Delta }}_{\theta }{\omega }_i\right)$ deforms only slowly towards an (almost) Gaussian distribution (insets, from left to right, $i=2$, 20, 50, $\theta =0.01$ s). Results were obtained from direct Runge-Kutta simulations.

Figure 5

(a) SciGrid topology of the 380-kV grid in Germany [28] with homogeneous parameters as in the preceding simulations. The fluctuating perturbation $\delta P_i\left(t\right)$ is injected at node $j=109$ (highlighted in red). (b) Variance of the short-term increments ${\mathrm{\Delta }}_{\theta }{\omega }_i$ plotted against distance $d(i,j)$ to the perturbation. Here, we choose the shortest path distance and average the variances for all nodes with the same $d(i,j)$. The timescale is $\theta =0.01$ s. (c) Evolution of the kurtosis $k\left(i\right)$ of the increments ${\mathrm{\Delta }}_{\theta }{\omega }_i$. The error bars correspond to the standard deviation of the kurtosis of nodes with the same distance $d(i,j)$. In (b), we omit error bars due to the logarithmic $y$ axis.