What adaptive neuronal networks teach us about power grids

Power grid networks, as well as neuronal networks with synaptic plasticity, describe real-world systems of tremendous importance for our daily life. The investigation of these seemingly unrelated types of dynamical networks has attracted increasing attention over the past decade. In this paper, we provide insight into the fundamental relation between these two types of networks. For this, we consider well-established models based on phase oscillators and show their intimate relation. In particular, we prove that phase oscillator models with inertia can be viewed as a particular class of adaptive networks. This relation holds even for more general classes of power grid models that include voltage dynamics. As an immediate consequence of this relation, we discover a plethora of multicluster states for phase oscillators with inertia. Moreover, the phenomenon of cascading line failure in power grids is translated into an adaptive neuronal network.

Figure 1

Hierarchical multicluster states in networks of coupled phase oscillators with inertia. The panels [(a), (b)], [(c), (d)], and [(e), (f)] show the temporally averaged phase velocities $\langle {\dot{\varphi }}_j\rangle$, phase snapshots ${\varphi }_j\left(t\right)$, and the pseudocoupling matrices ${\chi }_{ij}\left(t\right)$, respectively, at $t=10000$. In panel (e), the oscillator indices are sorted in increasing order of their mean phase velocity. The states were found by numerical integration of (3) with identical oscillators $P_i=0, h\left(\varphi \right)=-\sigma \gamma sin(\varphi +\alpha )$, and uniform random initial conditions ${\varphi }_i\left(0\right)\in (0,2\pi ), {\psi }_i\left(0\right)\in (-0.5,0.5)$. The parameter $\alpha$ is a phase lag of the interaction [66]. Parameters are [(a), (c), (e)] globally coupled network, $M=1, \gamma =0.05, \sigma =0.016, \alpha =0.46\pi$; [(b), (d), (f)] nonlocally coupled ring network with coupling radius $P=40, M=1, \gamma =0.3, \sigma =0.033, \alpha =0.8\pi , N=100$.

Figure 2

Temporal evolution of the pseudocoupling matrix ${\chi }_{ij}$ for the phase oscillator model with inertia. The pseudocoupling matrix is shown at (a) $t=100$, (b) $t=1750$, (c) $t=5000$, and (d) $t=10000$ and the color code is chosen as in Fig. 1. Starting from an incoherent state in panel (a), the largest cluster is formed first (b), and the other clusters are then successively formed depending on their size. All parameters and the initial condition as in Figs. 1, 1 and 1.

Figure 3

(a) Map of the German ultra-high-voltage power grid consisting of 95 net generators (green squares) and 343 net consumers (red dots) connected by 662 bidirectional transmission lines (black lines). (b) The histogram shows the distribution of the net power $P_i$ for each generator (green) and consumer (red). The data displayed in panels (a) and (b) are taken from the ELMOD-DE data set reported in Ref. [70].

Figure 4

(a) Mean phase velocity $\langle {\dot{\varphi }}_i\rangle$ for each node in the German power grid network presented in Fig. 3. (b) Mean phase velocity $\langle {\dot{\varphi }}_i\rangle$ vs natural frequency ${\omega }_i=P_i/\gamma$ for each node in the German ultra-high-voltage power grid. The dashed line shows the relation $\langle {\dot{\varphi }}_i\rangle ={\omega }_i$. (c) Temporal evolution (time in seconds) of two typical elements from the pseudocoupling matrix $\chi$ introduced in (7). One element corresponds to a link between a solitary node and a node from the synchronous cluster (blue) and the other element corresponds to a link between two nodes of the synchronous cluster (orange). Setup: $\alpha =0, \sigma =800\mathrm{MW}, t=600\mathrm{s}$, uniformly distributed random initial conditions $\varphi \in (0,2\pi ), \dot{\varphi }\in (-1,1)$.

Figure 5

Panels (a) and (b) show a map of the German ultra-high-voltage power grid as given in Fig. 3. The solitary nodes presented in Fig. 4 are displayed in blue. For each transmission line, we display in panels (a) and (b) the normalized absolute value of the average pseudocoupling weight $\overline{|\langle {\chi }_{ij}\left(t\right)\rangle |}$ and the normalized amplitude of ${\chi }_{ij}\left(t\right)$, i.e., $\overline{max({\chi }_{ij}\left(t\right))-min({\chi }_{ij}\left(t\right))}$, respectively. The bar denotes the normalization for each value to the maximum for all $i,j=1,\cdots ,N$. The temporal evolution is evaluated over an averaging window of 100 time units. All values for each line are displayed in the color code given by the color bars above the figures. Panels I (a) and I (b) provide enlargements of panels (a) and (b) for the solitary node $i=235$ [red shading in panels (a) and (b)], respectively. Panels II (a) and II (b) provide enlargements of panels (a) and (b) for the solitary node $i=214$ [blue shading in panels (a) and (b)], respectively. The black nodes in the enlargements represent the consumer and generator of the power grid network. All parameters are as in Fig. 4.

Figure 6

Cascading line failures in a network of five adaptively coupled phase oscillators (1) and (2) with $f\left(\varphi \right)=-2sin\varphi , g\left(\varphi \right)=-cos\varphi , ({\omega }_1,\cdots ,{\omega }_5)=(-1.2,1.8,-1.2,-1.2,1.8)$, and $\epsilon =0.01$. Panel (a) shows the connectivity structure of the nodes before (left inset) and after the cascading line failures (right inset), and the number of line failures occurring during the numerical simulation is presented vs time. The line which is cut off due to an abrupt line failure at $t=0.5$ is indicated by a lightning symbol. Panel (b) shows the transient dynamics of the absolute value of the local synaptic input $F_{ij}$ during the cascading line failures. The colors of lines correspond to the colors of the links displayed in panel (a). As initial condition, we choose a stable equilibrium of (1) and (2) with connectivity structure as shown in the left inset of panel (a). The dotted horizontal line indicates the threshold $K=1.04$ above which the lines fail. The dashed horizontal lines correspond to the stable equilibria after the link between 2 and 4 is cut off.