Emergent Failures and Cascades in Power Grids: A Statistical Physics Perspective

We model power grids transporting electricity generated by intermittent renewable sources as complex networks, where line failures can emerge indirectly by noisy power input at the nodes. By combining concepts from statistical physics and the physics of power flows and taking weather correlations into account, we rank line failures according to their likelihood and establish the most likely way such failures occur and propagate. Our insights are mathematically rigorous in a small-noise limit and are validated with data from the German transmission grid.

Figure 1

(a) Nominal line flows $|{\nu }_{\ell }|$ at 11 am. (b) True overload probabilities ${\log }_{10}\mathbb{P}(|f_{\ell }|\ge 1)$ at 11 am. (c) Top 5% of most likely lines to fail (red) at 11 am, according to (3), and nominal injections from renewable sources.

Figure 2

(a) After the emergent failure of line 27 (red), six additional lines (orange) fail, 4 pm. (b) Most likely power injection ${\mathbf{p}}^{(\ell )}$ causing the isolated failure of line 720 (red) and subsequent failures (orange). The bus sizes reflect how much ${\mathbf{p}}^{(\ell )}$ deviates from $\mathit{\mu }$ at 11 am (red, positive deviations; blue, negative). Left, with correlation in noise; right, without correlation in noise (setting to 0 all the off-diagonals of ${\mathbf{\Sigma }}_p$).

Figure 3

Left: Most likely power injections ${\mathbf{p}}^{(\ell )}$ leading to the failure of line $\ell$ (orange), visualized using the color and size of the nodes (red, positive deviations; blue, negative), together with power flows $f_k^{(\ell )}$. Right: Situation after the power flow redistribution with three subsequent failures and the values ${\tilde{f}}_k^{(\ell )}=f_k^{(\ell )}-1$, $k\ne \ell$.