Robustness of power systems under a democratic-fiber-bundle-like model

We consider a power system with $N$ transmission lines whose initial loads (i.e., power flows) $L_1,...,L_N$ are independent and identically distributed with $P_L\left(x\right)=\mathbb{P}\left[L\le x\right]$. The capacity $C_i$ defines the maximum flow allowed on line $i$ and is assumed to be given by $C_i=(1+\alpha )L_i$, with $\alpha >0$. We study the robustness of this power system against random attacks (or failures) that target a $p$ fraction of the lines, under a democratic fiber-bundle-like model. Namely, when a line fails, the load it was carrying is redistributed equally among the remaining lines. Our contributions are as follows. (i) We show analytically that the final breakdown of the system always takes place through a first-order transition at the critical attack size $p^{☆}=1-\frac{\mathbb{E}\left[L\right]}{{max}_x\left(\mathbb{P}\left[L>x\right](\alpha x+\mathbb{E}\left[L|L>x\right])\right)}$, where $\mathbb{E}\left[·\right]$ is the expectation operator; (ii) we derive conditions on the distribution $P_L\left(x\right)$ for which the first-order breakdown of the system occurs abruptly without any preceding diverging rate of failure; (iii) we provide a detailed analysis of the robustness of the system under three specific load distributions—uniform, Pareto, and Weibull—showing that with the minimum load $L_{\min }$ and mean load $\mathbb{E}\left[L\right]$ fixed, Pareto distribution is the worst (in terms of robustness) among the three, whereas Weibull distribution is the best with shape parameter selected relatively large; (iv) we provide numerical results that confirm our mean-field analysis; and (v) we show that $p^{☆}$ is maximized when the load distribution is a Dirac delta function centered at $\mathbb{E}\left[L\right]$, i.e., when all lines carry the same load. This last finding is particularly surprising given that heterogeneity is known to lead to high robustness against random failures in many other systems.

Figure 1

We demonstrate the distinction between an abrupt first-order rupture and a first-order rupture that is preceded by a diverging failure rate. $p_L\left(x\right)$ is assumed to be of uniform density over the range $[L_{\min },L_{\max }]=[10,50]$. In both plots, red (lower) curves stand for the case where $\alpha =0.2$, whereas blue (upper) curves represent $\alpha =1.2$. Panel (a) shows $\mathbb{P}\left[L>x\right]\left(\alpha x+\mathbb{E}\left[L|L>x\right]\right)$, whereas panel (b) plots the corresponding variation of $n_{\infty }\left(p\right)$ with attack size $p$. We observe that for $\alpha =0.2$ (red), $\mathbb{P}\left[L>x\right]\left(\alpha x+\mathbb{E}\left[L|L>x\right]\right)$ takes its maximum at the point $x=L_{\min }=10$. As a result, we see an abrupt first-order transition of $n_{\infty }\left(p\right)$ as it suddenly drops to zero at the point $p=p^{☆}=0.0625$, while decaying linearly as $1-p$ up until that point. The case where $\alpha =1.2$ is clearly different as $\mathbb{P}\left[L>x\right]\left(\alpha x+\mathbb{E}\left[L|L>x\right]\right)$ is now maximized at $x=17.6>L_{\min }$. As expected from our discussion, this ensures that the total failure of the system occurs after a diverging failure rate is observed. This divergence is clearly seen in panel (b), where the dashed line corresponds to the $1-p$ curve.

Figure 2

The two stage breakdown of the system is demonstrated, where $L_1,...,L_N$ are drawn from Weibull distribution [see Eq. (19)] with $k=0.8, \lambda =150, L_{\min }=10, \alpha =0.2$. We plot the relative final size $n_{\infty }\left(p\right)$ as a function of the attack size $p$. The inset magnifies the region where the system goes through a series of a first-order, a second-order, and then again a first-order transition.

Figure 3

We plot $n_{\infty }\left(p\right)$ vs $p$ under six different cases. Analytical results are represented by lines, whereas empirical results (obtained through averaging over 500 independent runs) are represented by symbols. We set $N=100000, L_{\min }=10$, and $\mathbb{E}\left[L\right]=30$. For the case when $L_1,...,L_N$ follow a Weibull distribution, we take the shape parameter to be $k=2$, leading to a scale parameter $\lambda =22.5676$. We see that numerical results match the analytical results very well.

Figure 4

We plot $n_{\infty }\left(p\right)$ vs $p$ when $L_1,...,L_N$ follow a Weibull distribution with $L_{\min }=10, \mathbb{E}\left[L\right]=30$. We set $N=100000$ and $\alpha =0.7$. Analytical results are represented by lines, whereas empirical results (obtained through averaging over 500 independent runs) are represented by symbols. Again, we see that numerical results match the analytical results pretty well. (Inset) We plot the number of iterations (i.e., the number of load redistribution steps) needed for reaching the steady state under the same setting for $k=2,4,6$. As would be expected, the number of iterations diverge near the corresponding critical attack size $p^{☆}$ in each case.

Figure 5

We plot the maximum attack size $p^{☆}$, when $L_1,...,L_N$ follow a Weibull distribution with $L_{\min }=10, \mathbb{E}\left[L\right]=30$, as a function of the shape parameter $k$ of the Weibull distribution. We set $N=100000$ and consider four tolerance parameters: $\alpha =0.1,0.3,0.5,0.7$. The curves correspond to analytical results computed directly from (13).

Figure 6

We plot the final system size $n_{\infty }\left(p\right)$ as a function of the attack size $p$, when $L_1=\cdots =L_N=\mathbb{E}\left[L\right]$. For the numerical results, we take $N=100000, \mathbb{E}\left[L\right]=30$, and consider four tolerance parameters: $\alpha =0.2,0.7,1.0,2.0$. Each data point (represented by a symbol) is the result of averaging over 500 independent runs. The lines correspond to analytical results computed directly from (13). We see a perfect agreement between analysis and experiments. In all cases, the system breaks down abruptly through a first-order transition at $p^{☆}=\frac{\alpha }{\alpha +1}.$

Figure 7

We plot the number of lines failed at every load redistribution step $t=1,2,...,$ after the initial attack. Simulation is performed for a single system with $N={10}^7$ lines, where loads are uniformly distributed over $[10,50]$ and tolerance parameter is set to $\alpha =1.2$. Our analysis shows that the critical attack size is $p^{☆}=0.3256$ under this setting. For the subcritical attack size of $p=0.32$ we observe an exponentially decaying rate of failure $R\left(t\right)$. For attack sizes larger than $p^{☆}$, we observe a decaying rate of failure up until a certain iteration step, after which the rate of failure increases monotonically at every step. These observations are in parallel with the corresponding results obtained for the standard fiber-bundle model [40], Chap. V-B].