Heterogeneity effects in power grid network models

We have compared the phase synchronization transition of the second-order Kuramoto model on two-dimensional (2D) lattices and on large, synthetic power grid networks, generated from real data. The latter are weighted, hierarchical modular networks. Due to the inertia the synchronization transitions are of first-order type, characterized by fast relaxation and hysteresis by varying the global coupling parameter $K$. Finite-size scaling analysis shows that there is no real phase transition in the thermodynamic limit, unlike in the mean-field model. The order parameter and its fluctuations depend on the network size without any real singular behavior. In case of power grids the phase synchronization breaks down at lower global couplings, than in case of 2D lattices of the same sizes, but the hysteresis is much narrower or negligible due to the low connectivity of the graphs. The temporal behavior of desynchronization avalanches after a sudden quench to low $K$ values has been followed and duration distributions with power-law tails have been detected. This suggests rare region effects, caused by frozen disorder, resulting in heavy-tailed distributions, even without a self-organization mechanism as a consequence of a catastrophic drop event in the couplings.

Figure 1

Structural representation of the synthetic networks. Left side: HV; right side: a radial cabinetwork. The highlighted red node connects the two “layers.” The network on the picture has $68850$ nodes and $68849$ edges.

Figure 2

Node degree distributions of the synthetic power grids generated for 2.5M, 1.5M, and 1M networks (right to left curves).

Figure 3

Admittance distribution of the power grids generated for 2.5M, 1.5M, and 1M networks (right to left curves).

Figure 4

Average number of nodes within topological distance $r$ in the $23\mathrm{M}$ graph. Dashed line shows a PL fit for $4<r<20$. Inset: Local slopes defined in Eq. (7).

Figure 5

Hysteresis in the steady-state order parameter in fully coupled networks of sizes $N=1000$ (black boxes) and $N=500$ (red diamonds). Error bars show standard error of the mean. Inset: ${\sigma }_R\left(K\right)$ peaks for the two different network sizes investigated.

Figure 6

Phase synchronization transition in the steady state in 2D networks of sizes $N=500\times 500$ (red bullets), $L=1000\times 1000$ (blue boxes) using $a=3$, and $N=500\times 500$ (green stars) using $a=1$. The red lines show the results using the adiabatic protocol, started from asynchronous (bottom) or synchronous (top) states in case of $N=500\times 500$ and $a=3$. Error bars show standard deviation of the mean. Inset: Time dependence or $R\left(t\right)$, in case of the $N=500\times 500$ lattice, for control parameters: $K=700$, 350, 200, 150, 100, 80, 60, 50 in case of synchronized initial condition (top to bottom curves) and for $K=700$, 100 desynchronized initial condition (top to bottom curves).

Figure 7

Steady-state order parameter for different power grid networks, using $a=3$ for 1M, 1.5M, and 2.5M power grids (top to bottom curves), obtained by the adiabatic protocol. Error bars show standard errors of the mean. Pink stars correspond to the U.S. HV power grid of size $N=4941$ for comparison. We can see a vanishing synchronization and hysteresis by increasing the size.

Figure 8

Fluctuation of the steady-state order parameter for different networks with $a=3$. Symbols: Red bullets and blue boxes are for 2D lattices of linear sizes: $L=500,1000$, respectively; up triangles are for 1M grids; down triangles are for 2.5M grids; stars correspond to the U.S. HV grid.

Figure 9

Avalanche duration distribution for $1000\times 1000$ lattices for $a=3$ at different coupling values: $K=10$, 5, 4, 3, 2, 1 (top to bottom solid curves). Dashed lines: PL fits for the distribution tails.

Figure 10

Avalanche duration distribution in the 1M power grid for $a=3$ and different coupling values $K=0.7$, 0.6, 0.5, 0.4, 0.2 (top to bottom solid curves). Dashed lines: PL fits for the tails.