Spatial uniformity in the power-grid system

Robust synchronization is indispensable for stable operation of a power grid. Recently, it has been reported that a large number of decentralized generators, rather than a small number of large power plants, provide enhanced synchronization together with greater robustness against structural failures. In this paper, we systematically control the spatial uniformity of the geographical distribution of generators and conclude that the more uniformly generators are distributed, the more enhanced synchronization occurs. In the presence of temporal failures of power sources, we observe that spatial uniformity helps the power grid to recover stationarity in a shorter time. We also discuss practical implications of our results in designing the structure of a power-grid network.

Figure 1

Ensemble average of the order parameter $\langle m\rangle$ versus coupling strength $K$ when power generators are randomly placed on a two-dimensional square lattice of size $N=L\times L$. (a) We fix the generator density $N_{\mathrm{gen}}/N\approx 0.2$ and vary the system size: $N_{\mathrm{gen}}=9$, 25, and 49 for $L=7$, 11, and 15, respectively). (b) We fix the system size $L=15$ and vary the number of generators: $N_{\mathrm{gen}}=25$, 49, and 81. From (a) and (b), we conclude that the level of synchrony is a decreasing function of the system size $L$ and an increasing function of the number of generators $N_{\mathrm{gen}}$.

Figure 2

Method 1: Control of spatial uniformity by changing the available region for generators. The two-dimensional square lattice of linear size $L$ is divided into two regions (gray and yellow). The yellow region between the outer square of size $L_2$ and the inner square of size $L_1$ is allowed for generator locations. Among all $S=L_2^2-L_1^2$ points in the available yellow region, $N_{\mathrm{gen}}$ points are randomly selected as positions of generators. (b) An example realization of generator positions (brown circles). Small black squares are the locations of consumers.

Figure 3

We change the spatial uniformity as depicted in Fig. 2 for a given number of generators $N_{\mathrm{gen}}=49$ for a power grid of size $L=15$. The synchronization order parameter $\langle m\rangle$ versus the coupling constant $K$ (a) for various values of $L_2$ at fixed $L_1=0$ and (b) for various values of $L_1$ at fixed $L_2=L=15$. (c, d) Normalized histograms $P\left(f\right)$ for the power transmitted through each link at $K=7$, corresponding to (a) and (b), respectively, are displayed as a color map. The lines for $\overline{f}\equiv {\sum }_{f}fP\left(f\right)$ and for $f^{*}\equiv argmaxP\left(f\right)$ are also shown. It is clearly shown that as generators are more uniformly spread in space [as $L_2$ is increased in (c) and as $L_1$ is decreased in (d)], $f$ tends to have small values. (e, f) Scatterplots of $m$ versus $n_c$ [see Eq. (5)] at $K=7$, corresponding to (c) and (d), respectively.

Figure 4

Scatterplot for $m$ versus $n_c$ at $K=7$ for $L=15$ and $N_{\mathrm{gen}}=49$. To elaborate the effect of $n_c$ for a fixed value of $S=L_2^2-L_1^2$, we use the parameter values $(L_1,L_2)=(5,9),(13,15)$ for $S=56$, $(3,9),(7,11)$ for $S=72$, and $(1,11),(7,13)$ for $S=120$. Positive correlations between $m$ and $n_c$ are seen. The number of samples is 200, and the ensemble averages of $m$ and $n_c$ are represented as black squares, together with the error bars.

Figure 5

(a) Temporal failure of a power generator is implemented as ${\overline{P}}_i=0$ for $200\le t\le 210$. (b) Recovery time $\tau$ versus fraction $f_{\mathrm{off}}$ of failed generators (see the text) for $L_2=7$ and 15. We have used $L_1=0$ and $K=8$. More uniformly distributed generators ($L_2=15$) exhibit shorter recovery times than less uniformly distributed generators ($L_2=7$).

Figure 6

Method 2: Control of spatial uniformity by imposing superlattices. For a two-dimensional power-grid network of size $16\times 16$, we placed superlattices of sizes $l\times l$ with $l=4$, 8, and 16 and generators are packed near the center of each superlattice unit cell. The total number of generators is $N_{\mathrm{gen}}=64$. Positions of generators (consumers) are shown by brown circles (black squares) as in Fig. 2.

Figure 7

The spatial uniformity is changed as in Fig. 6, for $N_{\mathrm{gen}}=64$ and $L=16$. (a) $m$ versus $K$ for superlattices of size $l=4$, 8, and 16. As $l$ is increased, generators are more clustered, and the level of synchrony becomes worse. (b) $m$ versus $n_c$ at $K=7$.

Figure 8

A simple transmission line model. The circuit contains a resistor $R$ and an inductor $L$. The limit $R\to 0$ is taken when we compute the active power of the circuit.

Figure 9

(a) For the system discussed in Fig. 2, we place additional generators of large powers (denoted as larger circles) at fixed positions along the boundary to mimic a realistic distribution of larger plants along the coastline. (b) Scatterplots of $m$ versus $n_c$ for various vales of $L_2$ at fixed values of $L_1=0$ and $K=7$. Black squares represent averages over 200 sample runs.