Synchronization dynamics on power grids in Europe and the United States

Dynamical simulation of the cascade failures on the Europe and United States (U.S.) high-voltage power grids has been done via solving the second-order Kuramoto equation. We show that synchronization transition happens by increasing the global coupling parameter $K$ with metasatble states depending on the initial conditions so that hysteresis loops occur. We provide analytic results for the time dependence of frequency spread in the large-$K$ approximation and by comparing it with numerics of $d=2,3$ lattices, we find agreement in the case of ordered initial conditions. However, different power-law (PL) tails occur, when the fluctuations are strong. After thermalizing the systems we allow a single line cut failure and follow the subsequent overloads with respect to threshold values $T$. The PDFs $p\left(N_f\right)$ of the cascade failures exhibit PL tails near the synchronization transition point $K_c$. Near $K_c$ the exponents of the PLs for the U.S. power grid vary with $T$ as $1.4\le \tau \le 2.1$, in agreement with the empirical blackout statistics, while on the Europe power grid we find somewhat steeper PLs characterized by $1.4\le \tau \le 2.4$. Below $K_c$, we find signatures of $T$-dependent PLs, caused by frustrated synchronization, reminiscent of Griffiths effects. Here we also observe stability growth following the blackout cascades, similar to intentional islanding, but for $K>K_c$ this does not happen. For $T<T_c$, bumps appear in the PDFs with large mean values, known as “dragon king” blackout events. We also analyze the delaying or stabilizing effects of instantaneous feedback or increased dissipation and show how local synchronization behaves on geographic maps.

Figure 1

Degree distribution of the Europe-HV network. The blue dashed line shows an exponential fit for $k\ge 15$, while the red long-dashed line is a a PL fit for $k>5$.

Figure 2

The adjacency matrix of the Europe-HV grid, with $A_{ij}=1$ displayed in black and white if otherwise (cf. Ref. [37] for the U.S.-HV grid data).

Figure 3

Number of nodes with distance $r$ or less from random seeds in the breadth-first search algorithm applied for the Europe-HV graph. The inset shows the local slopes, corresponding to the same data. Linear extrapolation to $1/r\to 0$ provides $d=2.6\left(1\right)$.

Figure 4

The frequency spread for $d=2$, obtained through numerical integration of Eq. (20) at different $\alpha$ and $K$ values for (a) $L=3000$ and (b) $L={10}^{12}$. For comparison, the dot-dashed lines show the full results of Eq. (1) with synchronized initial conditions and the dashed lines mark the algebraic decay law $t^{-d/2}$ of the first-order Kuramoto model.

Figure 5

Evolution of frequency spread order parameter on the 2D lattices as the function of $K$ and $\alpha$ values shown by the legends in case of disordered initial conditions. These curves exhibit PL tails within the exponent range $1.2\le \tau \le 1.5$. For comparison the dashed line shows the result for the U.S.-HV network at $K=20$ and $\alpha =3$, for which $\tau \simeq 1.5$ can be read off.

Figure 6

Evolution of frequency spread order parameter on the 3D lattices as the function of $K$ and $\alpha$ as shown by the legends in case of disordered initial conditions. These curves exhibit PL tails, with similar exponents $\tau \simeq 1.85$. For comparison the dashed line shows the result for the U.S.-HV at $K=20$ and $\alpha =3$, where $\tau \simeq 1.5$ can be read off.

Figure 7

Evolution of phase order parameter (top) and frequency spread order parameter (bottom) of the U.S. network at $K=20$ during thermalization (left) and after one subsequent line cut (right) with respect to different thresholds (right: $T=0.1,0.2,\cdots ,0.8$). Comparing to the normal case $\alpha =0.4$ (solid lines), a negative feedback $\alpha =3$ (dashed lines) essentially slows down the dynamics of the phase order parameter, while at the same time leads to a smaller frequency spread; similar effects are observed following one line cut.

Figure 8

Evolution of phase order parameter (top) and frequency spread order parameter (bottom) of the Europe network at $K=80$ during thermalization (left) and after one subsequent line cut (right) with respect to different thresholds (right: $T=0.1,0.2,0.4,0.6,0.8$). Comparing to the normal case $\alpha =0.4$ (solid lines), a negative feedback $\alpha =3$ (dashed lines) essentially slows down the dynamics of the phase order parameter, while at the same time leads to a smaller frequency spread; similar effects are observed following one line cut.

Figure 9

Fluctuations of $R(t\to \infty )$ in case of the U.S.-HV power grid at the end of the thermalization process. Both for the normal $\alpha =0.4$ and the $\alpha =3$ dissipation cases. Partial synchronization transition occurs at $K_c\simeq 22\left(2\right)$. The inset shows the decay of $\mathrm{\Omega }\left(t\right)$ starting from disordered states at $\alpha =3$ and for different global couplings, as shown by the legends. The curves are multiplied by a factor $t^{1.5}$ in order to see the scaling at $K_c=20$.

Figure 10

Relative change of the steady-state Kuramoto order parameter at $\alpha =0.4$ as the consequence of the cascade in the U.S.-HV power grid. The $R(T=1)$ values are set to be the reference points and $R\left(T\right)/R(T=1)$ is plotted. The gray dashed line marks the baseline for the emergence of islanding effects. The inset shows $\sigma \left[R\right(T,K\left)\right]$.

Figure 11

Probability distribution of line failures for $K=30$, $\alpha =0.4$ and different thresholds marked by the legends in case of the U.S.-HV power grid. Dashed lines show the power-law fit for the scaling region, determined by visual inspection.

Figure 12

Probability distribution of line failures for $K=20$, $\alpha =0.4$ and different thresholds marked by the legends in case of the U.S.-HV power grid. DK cascade failures at $T=0.3$ can be observed.

Figure 13

The steady-state values of the Kuramoto order parameter for the Europe-HV network. Start from ordered or disordered initial states causes a hysteresis loop, closing at $K\sim {10}^6$ for $\alpha =0.4$. For $\alpha =3$ we could not reach the closure point as for this large value the adaptive solver becomes very slow. There exists infinite intermediate branches if we initialize the phases as ${\theta }_i\left(0\right)\in (0,{\theta }_{\max })$ with different ${\theta }_{\max }$ values. The inset shows $\sigma \left(R\right)$, in which the peaks of the solid curves indicate $K_c\simeq 100$.

Figure 14

Probability distribution of line failures for different thresholds for $K=60$ shown in the legends in case of the Europe-HV power grid. Dashed lines show power-law fits for the scaling region, determined by visual inspection. One can observe DK bumps for low thresholds.

Figure 15

Probability distribution of line failures for different thresholds for $K=80$ shown in the legends in case of the Europe-HV power grid. Dashed lines show power-law fits for the scaling region determined by visual inspection. One can observe DK bumps for low thresholds.

Figure 16

Relative change of the steady-state Kuramoto order parameter at $\alpha =0.4$ the consequence of the cascade in the Europe-HV grid. The $R(T=1)$ values are set to be the reference points and $R\left(T\right)/R(T=1)$ is plotted. The gray dashed line marks the baseline for the emergence of islanding effects. The inset shows $\sigma \left[R\right(T,K\left)\right]$.

Figure 17

Local Kuramoto results encoded by the color map as $1-r_i$. Red corresponds low local synchronization, green to high synchronization. The width of gray edges is proportional to the amplitude of the power flow.

Figure 18

Probability distribution of line failures for different thresholds for $K=20$ shown in the legends in case of the Europe-HV power grid.

Figure 19

The stationary frequency spread and the corresponding standard deviation show that the frequency entrainment has a transition point at $K_c^{\prime }\simeq 20$ for the U.S.-HV grid (left) and at $K_c^{\prime }\simeq 20$ for the Europe-HV grid (right). This transition point coincides with $K_c$ for the U.S.-HV case but differs from that of the Europe-HV case, as estimated from the peaks of $\sigma \left(R\right)$; cf. Figs. 9 and 13.