Escape routes, weak links, and desynchronization in fluctuation-driven networks

Shifting our electricity generation from fossil fuel to renewable energy sources introduces large fluctuations to the power system. Here, we demonstrate how increased fluctuations, reduced damping, and reduced intertia may undermine the dynamical robustness of power grid networks. Focusing on fundamental noise models, we derive analytic insights into which factors limit the dynamic robustness and how fluctuations may induce a system escape from an operating state. Moreover, we identify weak links in the grid that make it particularly vulnerable to fluctuations. These results thereby not only contribute to a theoretical understanding of how fluctuations act on distributed network dynamics, they may also help designing future renewable energy systems to be more robust.

Figure 1

Fluctuating power input may desynchronize a synchronous generator. (a),(b) When the input power $P$ fluctuates, the generator can lose synchrony to the grid after an escape time $\tau$. (c) The generator dynamics corresponds to the motion of a particle in a tilted washboard. The fluctuations can drive the particle over the potential barrier of height $\mathrm{\Delta }U$. (d) Kramer's escape rate theory predicts the escape process for Gaussian white noise. Theoretical prediction, Eq. (5), (black lines) perfectly predicts the mean escape times $\overline{\tau }$ for intermediate damping ($D=4\times {10}^{-5}{\mathrm{s}}^2$) as checked by direct numerical simulations (symbols). The averaged escape time crucially decreases with increasing load $P$ of the system. (e) Weaker damping $D$ undermines system robustness, which can become a major problem in future power grids. (Parameters: $H=4\mathrm{s},K=1$, and $\overline{P}=0.95$, if not stated otherwise.)

Figure 2

Where does the system escape? (a) Exemplary escaping process of the basin of attraction in phase space. The state trajectory (thin black line) crosses the boundary of the basin of attraction (red line) in the vicinity of but not at the saddle point (red disk). The stable fixed point is indicated by a green disk. (b),(c) Probability distribution of exit points, i.e., the crossing points of the trajectory and the basin boundary. Numerical results (histogram) compared to the theoretical prediction, Eq. (6) (dashed black line). With increasing noise the distribution becomes broader, i.e., a crossing at some distance from the saddle point (red disk) becomes more probable. (Parameters: $K=1$ and $\overline{P}=0.95$ as in Fig. 1.)

Figure 3

The easiest escape route determines the escape time $\overline{\tau }$. (a) Two identical generators are coupled to a third node representing the bulk grid via transmission lines with capacity $K_1=1$ (constant) and $K_2$ (variable), respectively. The transmitted power on both lines is ${\overline{P}}_{1,2}=0.95$. Power fluctuates on all nodes independently. (b) The mean escape time $\overline{\tau }$ as a function of the noise amplitude $\sigma$. Disks represent numerical values; the solid lines are fits to extract the scaling exponent. (c) In this scenario, the exponent in Kramer's rate is determined by the lowest barrier $\mathrm{\Delta }U$, Eq. (7) of the two-dimensional potential landscape, which is determined by $min\{K_1,K_2\}$, i.e., the weaker of the two transmission line capacities. Thus it increases with $K_2$ as long as $K_2\le K_1$ but depends only on $K_1$ for $K_2>K_1$. A comparison of numerical results obtained from exponential fits to the data (disks) and the analytical value of the potential barrier $\mathrm{\Delta }U$ (with constant $c$) shows very good agreement. (d) Imbalance of the two escape routes: $p_2-p_1$ with $p_{1,2}$ being the probability that link 1 or 2 is overloaded first, as a function of $K_2/K_1$ and the noise amplitude. For weak noise there is a sharp transition at $K_2/K_1=1$, which smears out for stronger noise. Panels (b)–(d) use a rescaled noise $\tilde{\sigma }=40\sigma$.

Figure 4

Vulnerable links predicted by the topology of saddle points. Top: (a) The stable phase-locked fixed points in a model power grid with four generators (filled circles) and eight consumers (open circles). Shown is the phase difference ${\delta }_i-{\delta }_j$ along the transmission lines. (b) Probability that a transmission line is overloaded first ($|{\delta }_i\left(t\right)-{\delta }_j\left(t\right)|$ crosses $\pi /2$) when the grid becomes unstable due to a fluctuating power input. Four vulnerable transmission lines are identified. Bottom: The vulnerable transmission lines can be traced back to four different saddle points with comparably low potential barrier. All saddle points have exactly one transmission line (darkest line in each plot) with $|{\delta }_i-{\delta }_j|>\pi /2$, corresponding to the vulnerable lines identified in (b). The color scale shows the phase differences as in panel (a). The networks consist of four generators ($•,P_j=+2P_0$) and eight consumers ($\circ ,P_j=-P_0$); all lines have capacity $K=24/19\times P_0$.