Propagation of wind power induced fluctuations in power grids

Propagation of wind power induced fluctuations in power grids Hauke Haehne,1 Katrin Schmietendorf,1 Samyak Tamrakar,2 Joachim Peinke,1 and Stefan Kettemann2, 3 1

arXiv:1809.09098v1 [physics.soc-ph] 24 Sep 2018

Institute of Physics and Forwind, Carl von Ossietzky Universit¨ at Oldenburg, Ammerl¨ ander Heerstraße 114-118, 26129 Oldenburg 2 Jacobs University, Department of Physics and Earth Sciences, Campus Ring 1, 28759 Bremen, Germany 3 Division of Advanced Materials Science, Pohang University of Science and Technology (POSTECH), San 31, Hyoja-dong, Nam-gu, Pohang 790-784, South Korea (Dated: September 26, 2018) Renewable generators perturb the electric power grid with heavily non-Gaussian and correlated fluctuations. Recent experimental findings show short-term frequency fluctuations measured in the distribution grid which depend on wind power injection. In this Letter, we analyze the aspect of locality of such fluctuations. We show differences in the short-term response to wind power injection in parallel frequency measurements at two different spots and deduce an analytical expression for their propagation in power grids from linear response theory. We find that while the amplitudes of such fluctuations stay local, the non-Gaussian shape of increment distributions persists much wider in the grid– therewith emphasizing the importance of correct non-Gaussian modeling. The amount of grid inertia appears to play a crucial role for the propagation of such fluctuations in power grids.

Climate change has triggered a transition of electric energy supply towards renewable sources among which wind power plays a crucial role. Alone in 2017, 15.6 GW of new wind power capacity was installed in the European Union; now summing up to 168.7 GW, that is 18% of the total installed power generation capacity in the EU [1]. In contrast to the steady production of conventional power sources, wind power generation is highly volatile. Due to its continuing and increasing integration to the European power grid, maintaining a high power quality despite such fluctuations has become an important task [2]. For the analysis of fluctuations on a time scale θ of a stochastic process x(t), we make use of the statistics of increments ∆θ x(t) = x(t + θ) − x(t). Time series of wind power generation P (t) show non-Gaussian increment probability density functions (PDFs) p(∆θ P ) on short scales θ ∼ 1 sec [3, 4]. The tails of such PDFs severely deviate from the Gaussian distribution and depict an increased probability of large fluctuations on short time scales, Fig. 1 a. On time scales of one second and above, load frequency control compensates feed-in fluctuations of renewable generators and thus maintains a stable grid operation [6]. However, what is the impact of the characteristic non-Gaussian short-term fluctuations of wind power on the grid? Recent results from local distribution grid frequency measurements show that on time scales below one second, grid frequency fluctuations actually increase with increasing wind power production [7]. These findings are short-term effects occurring below one second with very low amplitudes. One second is approximately the time scale separating local from interarea modes in Small Signal Stability Analysis [8]. So, are such wind power induced fluctuations a global feature of the European power grid or, instead, rather a local one resulting from high wind power injection close to the measurement?

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FIG. 1. Short-term response to intermittent wind power feed-in depends on measurement position in the public power grid. (a) Distribution of power increments ∆θ P = P (t + θ) − P (t) of the average output P (t) of 12 wind turbines in a wind park (time scale θ = 1 sec) [5]. The tails deviate significantly from the Gaussian distribution (black reference curve). (b) Frequency increment distribution p(∆θ f |Pw ) conditioned on wind energy generation Pw in Germany, measured in Oldenburg (θ = 0.2 sec). All increments are measured in units of the standard deviation of the first Pw -bin. Increasing wind power generation Pw broadens the PDF. (c) Simultaneously measured data in G¨ ottingen show no such broadening with Pw . Again, θ = 0.2 sec. (d) Variance evolution of the distributions in (b) and (c) with Pw : While the Oldenburg measurements show a clear trend, the shortterm statistics seems to be independent of Pw in G¨ ottingen.

The dynamics of high-voltage AC-grids are – in the simplest model – captured by networks of inert phasecoupled oscillators [9]. The impact of perturbations on

2 such models is a current field of research in physics: With the help of linear response theory, the spreading of singular [10, 11], harmonic [12], and stochastic [13] perturbations were analyzed and evaluated. Modified Fokker-Planck Equations have shown to be useful to predict steady state frequency distributions obtained from measurements in highly meshed grids [14]. Further, the probability of outages caused by fluctuating perturbations was evaluated in [15, 16]. All of these results show that locally induced stochastic perturbations break the perfect global oscillator synchronization and cause small node-dependent deviations in the frequency fluctuation statistics. How do fluctuations of renewable generators propagate in the grid? We contribute to this fundamental question with the analysis of short term increments ∆θ f = f (t + θ) − f (t) of the grid frequency f (t). We present grid frequency measurements from two different positions in Germany: Oldenburg, in the north-western region of Germany with a high share of wind energy injection close to the measurement, and, in 213 km straight line distance towards the center of Germany, G¨ ottingen with less concentrated wind energy injection [17]. We show that conditioning the variance of the short-term frequency increment statistics on the amount of injected wind power in Germany reveals a qualitatively different behavior at both measurement positions. Further, we study linear chains of synchronous machines and provide an analytical expressions for the variance of the frequency increment PDF at each node. Our numerical investigations show that variance and shape of frequency increment PDFs propagate qualitatively differently in linear oscillator chains; therewith supporting our experimental findings. Data Analysis.– We use 10 kHz voltage samplings simultaneously measured in Oldenburg and G¨ottingen from July 25th, 2017, till March 13th, 2018, to derive grid frequency time series f (t) with a time resolution of 5 Hz. We provide details on our measurement techniques in the Supplement. We characterize the short term frequency fluctuations in terms of width (variance) and shape (kurtosis) of their increment PDFs. In an earlier measurement period [7], we have observed that p(∆θ f ) broadens with increased amount of wind energy Pw fed to the grid in Germany. We use publicly available power generation data [18] to obtain conditioned PDFs p(∆θ f |Pw ). In our new data set, we were able to perfectly reproduce our earlier results in Oldenburg, Fig. 1 b, while in the parallel G¨ottingen measurements, we see no such effect, Fig. 1 c. For comparison, we show the evolution of the conditioned variance σ 2 (θ, Pw ) := var(p(∆θ f |Pw )) with Pw (Fig. 1 d) clearly confirming the different short-term response to wind power feed-in at the two measurement positions. Both short-term increment PDFs p(∆θ f ) show non-

Gaussian tails, characterized by a kurtosis k > 3. The kurtosis is defined as k=

h(∆θ f − h∆θ f i)4 i . h(∆θ f − h∆θ f i)2 i2

(1)

Conditioning the kurtosis k on Pw in the same manner as with the variance reveals no clear trend with Pw . However, we observe, averaged over Pw , a higher kurtosis in Oldenburg (k = 4) than in G¨ottingen (k = 3.4). In summary, we find a different response of the width of the short-term increment PDFs to Pw at the two measurement positions. Further, the PDFs have different shapes. To explore how width and shape of p(∆θ f ) evolve with distance to the volatile feed-in, we consider a simple power grid model driven by a stochastic signal and derive an analytic expression for the variance of the increment statistics from linear response theory. A network of synchronous machines.– We consider the Synchronous Motor Model [9] for high-voltage AC grids. The phase angle ϑi (t) is, in this model, governed by the Swing Equation, τ 2α ¨ i + 2τ α˙ i = −

N X j=1

J Pi γ 2 ω0

J Kij sin(ϑ0i − ϑ0j + αi − αj ). γ 2 ω0

(2)

where αi (t) = ϑi (t) − ϑ0i is the deviation of ϑi (t) from the fixed point ϑ0i . Pi denotes the generated or consumed power at node i. In the following we fix γ = JΓ = 105 kgm2 /s and K = 0.5 GW. Further, ω0 = 2π · 50 Hz. The internal time scale τ follows from τ = 1/Γ = J/γ. Let us now consider the case in which a subset of nodes {j} is driven by a noisy signal Pj (t). The production or consumption at one of the perturbed nodes j decomposes as Pj (t) = Pj0 + δPj (t) into a constant value Pj0 corresponding to the fixed point and a stochastic perturbation δPj (t) with hδPj (t)i = 0. If the system stays close to its fixed point of operation, |αi |  1, we may calculate the response of phase αi (t) during the driving δΠj (t) := J/(γ 2 ω0 )δPj (t) as Zt αi (t) = −∞

dt0 X δΠj (t0 )Gij (t0 − t) τ j

(3)

where the propagator is defined by Gij (t0 − t) =

N −1 X

√ φni φ∗ t0 −t √ nj (−σ)e(1+σ 1−Λn ) τ , 2 1 − Λn n=0 σ=±1

X

(4) see Supplement for a step-by-step derivation. Here, Λn ∈ R are eigenvalues and φ ∈ RN the corresponding

3 eigenvectors of the generalized Graph Laplacian matrix L (definition in Supplement). In contrast to prior results [11, 13, 19], our expression applies to any stochastic signal δΠj (t) and does not rely on a pure analysis of power spectra. An expression similar to Eq. (3) has been deduced in [20] independently from our work. For simplicity, we now focus on the case where the system (2) is driven at a single node j. With the help of Eq. (3), we derive an expression for the variance of the frequency increment PDF on time scale θ, which only depends on the auto-correlation function of the increment time series acf(|δ|) = h∆θ Πj (t)∆θ Πj (t + δ)i: h∆θ ωi2 i

−∞ Z

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−dδ acf(|δ|)∂t Gij (t˜)∂t Gij (t˜ − δ), τ

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(5) where ∂t Gij denotes the partial derivative of the propagator Gij (t0 − t) with respect to t. Again, we provide the calculation details in the Supplement. Localization of increment PDF variances.– To test our approach (5) and analyze the propagation and localization of frequency increment statistics, we consider linear chains of N coupled oscillators obeying Swing Eq. (2). We emphasize that our linearization approach is principally applicable to more complex networks, but here, we want to focus on the propagation from oscillator to oscillator and hence choose chains to exclude effects resulting from complex network topologies. In our setup, solely node j = 1 is driven by a stochastic signal δΠj (t), which we obtain by means of a Langevin-type stochastic differential equation and subsequent modification of the power spectrum [15] (details on this procedure are provided in the Supplement). As a result, the time series δΠj (t) reproduces the key features of wind power generation data: extreme events, temporal correlations, characteristic power spectrum with -5/3 decay and heavy-tailed increment statistics. We have var(δPj ) = 2.3 MW and choose a grid with no initial power transfer, i.e. Pi0 = 0 for all i. Increment variances obtained from direct numerical simulations of a chain of N = 20 oscillators show good agreement with our response theory, Fig. 2. Apart from small deviations at the feed-in node j = 1 itself, our theory accurately predicts the width of the short-term increment distribution. The results confirm our conjecture concerning the locality of fluctuations: The absolute amplitude of fluctuations decreases rapidly a few nodes away from the perturbation, Fig. 2 inset. In the semi-logarithmic plot, however, we observe – disregarding boundary effects – an exponential decay of the variance with distance to the perturbation. The impact of reduced grid inertia J is of considerable importance for future power grids fed by a high share of

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FIG. 2. Variance of short-term frequency increments decreases exponentially in a chain system of N = 20 oscillators. Variance h∆θ ωi2 i of increment pdf (θ = dt = 0.01 sec) from linear response theory Eq. (5) (full lines) and direct Runge-Kutta simulations (squares). Despite small deviations at the perturbed node j = 1 itself (inset, linear plot) the theoretical prediction fits the numerical simulation very well and captures also boundary effects (large axes, semilogarithmic plot). Decreasing the inertia J (color coded) leads to a faster decay of the fluctuations along the chain.

renewable sources providing no inertia per se [21]. Decreasing the inertia J in our model system leads, as expected, to higher fluctuation amplitudes, Fig. 2 inset. However, as the semi-logarithmic plot reveals, decreasing J while letting the damping γ constant causes faster decay of the increment variance. So, how exactly does the exponential decay depend on the system parameters? Increasing the inertia J while keeping γ constant increases the eigenvalues Λn . For chain-like grids with a large number of nodes N  1 and all nonzero Λn > 1, the expression for the variance of short-term frequency increments, Eq. (5), simplifies to   i−1 1 h∆θ ωi2 i = exp − h∆θ δP12 i, (6) JKω0 ξ where h∆θ δP12 i is the second moment of the increment PDF of the disturbance at site j = 1. We have used the Kolmogorov power spectrum S(f ) ∝ f −5/3 of δP1 (t) and provide the calculations in the Supplement. Thus, we confirm analytically that the second moment of the frequency increments is exponentially decaying with distance i from the position of the√disturbance √ and find the correlation length ξ = vτ /2 = JK/(2 ωγ). We confirm the correlation length ξ with numerical investigations of slightly longer chains (N = 50) to reduce finite-size effects, Fig. 3: Once J exceeds a critical value Jc = ωγ 2 N 2 /π 2 K ≈ 1.6 · 106 kgm2 , all nonzero eigenvalues Λn are larger than one (Fig. 3 b) and the slope m of the exponential decay approaches

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FIG. 4. Non-Gaussian shape of frequency increment distributions is a long-range effect. Kurtosis of p(∆θ ωi ) in a longer chain of N = 100 oscillators. The frequency increment distribution p(∆θ ωi ) deforms only slowly towards an (almost) Gaussian distribution (insets, from left to right i = 2, 20, 50, θ = 0.01 sec). Results were obtained from direct Runge-Kutta simulations. As in the analysis of the variance propagation in chain grids, decreasing grid inertia J appears to evoke a faster supression of non-Gaussian fluctuations.

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FIG. 3. Variance of increment distribution √ decays √ exponentially with correlation length ξ = JK/(2 ωγ) in high inertia systems. (a) Variance h∆θ ωi2 i from RungeKutta simulations of a chain of N = 50 oscillators (squares) compared√to an exponential decay with slope m = −1/ξ = √ −2 ωγ/ JK (straight lines). (b) With increasing inertia J and constant γ, the eigenvalues Λn increase (colored triangles). For J > Jc = ωγ 2 N 2 /π 2 K ≈ 1.6 · 106 kgm2 , all Λn 6= 0 are larger than one. (c) Exponential fits (orange data with errorbars) of m converge to our prediction m = 1/ξ (straight blue line) for increasing inertia J. The vertical dashed line marks Jc . Error bars correspond to the 2σ confidence bound of the exponential fit.

m = −1/ξ (Fig. 3 c). In the case of low inertia J < Jc , we have eigenvalues 0 < Λn < 1. Such Λn cause modes which decay more slowly with relaxation rates Γn < 1/τ . The propagator Eq. (4) then gets the form of a diffusion propagator [10] and we find that the variance of frequency increments decays with distance more slowly, with a power law. We observe the transition to diffusive behavior in Fig. 3 a for J = 105 kg m2 , where the exponential decay only lasts up to node i ≈ 15. Farther away, the slowly decaying soft modes dominate. Kurtosis evolution.– Finally, we analyze the evolution of the shape of the increment PDFs in terms of their kurtosis k. Interestingly, the non-Gaussian shape persists much longer than the absolute fluctuation amplitudes, Fig. 4. In numerical simulations of a chain of N = 100 oscillators, we observe that the short-term increment PDF deforms towards an almost Gaussian

distribution only after approximately 40 nodes, see also the insets of Fig. 4. We observe a linear decay of k with distance until k ≈ 3, which corresponds to a Gaussian distribution. Again, the grid inertia seems to play a crucial role: Decreasing the inertia leads to higher kurtosis values, which means heavier tails, but also to a faster deformation toward a Gaussian distribution. Conclusion.– In this Letter, we have analyzed the spreading of short-term frequency fluctuations in power grids induced by wind power feed-in. We have presented parallel distribution grid frequency measurements from one region with a large installed capacity of wind power generators, Oldenburg, and another region, G¨ ottingen, with a rather small installed capacity [17]. We have shown that while the short-term fluctuations in Oldenburg increase with increasing wind power generation in Germany, the G¨ottingen measurements do not show this trend. Motivated by the thus indicated locality of frequency fluctuations, we have presented an analytical expression for the propagation of fluctuations in terms of the width of frequency increment statistics. We have evaluated our analytics numerically with perturbation time series from a stochastic model which reproduces key features of wind power production fluctuations. Our investigations of chains of synchronous machines show that the amplitudes of such short-term fluctuations damp out very fast and – as expected from our measurement – form a local stochastic property of the frequency. We have derived an approximation of the exponential decay of the fluctuations as a function of the system parameters which possibly helps to improve the impact of such fluctuations in real power grids. The tails of frequency increment distributions p(∆θ f )

5 characterize the probability of large fluctuations on time scale θ. If the kurtosis exceeds 3, such fluctuations are heavily underestimated by a Gaussian distribution. Our numerical results show that, in contrast to the absolute amplitude of fluctuations, such heavy-tailed shapes of p(∆θ f ) persist on much wider spatial scales. The frequency increment distributions of our measurements show such heavy tails; expressed by a kurtosis k = 4 in Oldenburg and k = 3.4 in G¨ ottingen. Even though we cannot clearly pinpoint the reason for the difference in kurtosis to lie in wind power feed-in, this fact still emphasizes the necessity of correct non-Gaussian models of renewable energy fluctuations for a reliable operation of future power grids.

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