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We consider a power system with $N$ transmission lines whose initial loads(i.e., power flows) $L_1, \ldots, L_N$ are independent and identicallydistributed with $P_L(x)$. The capacity $C_i$ defines the maximum flow allowedon 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 $p$-{\em fraction} of the lines, under ademocratic fiber bundle-like model. Namely, when a line fails, the load it wascarrying is redistributed equally among the remaining lines. Our contributionsare as follows: i) we show analytically that the final breakdown of the systemalways takes place through a first-order transition at the critical attack size$p^{\star}=1-\frac{E[L]}{\max\{P(L>x)(\alpha x + E[L ~|~ L>x])\}}~~~$; ii) wederive conditions on the distribution $P_L(x)$ for which the first order breakdown of the system occurs abruptly without any preceding diverging rate offailure; iii) we provide a detailed analysis of the robustness of the systemunder three specific load distributions: Uniform, Pareto, and Weibull, showingthat with the minimum load $L_{\textrm{min}}$ and mean load $E[L]$ fixed,Pareto distribution is the worst (in terms of robustness) among the three,whereas Weibull distribution is the best with shape parameter selectedrelatively large; iv) we provide numerical results that confirm our mean-fieldanalysis; and v) we show that $p^{\star}$ is maximized when the loaddistribution is a Dirac delta function centered at $E[L]$, i.e., when all linescarry the same load; we also show that optimal $p^{\star}$ equals$\frac{\alpha}{\alpha+1}$. This last finding is particularly surprising giventhat heterogeneity is known to lead to high robustness against random failuresin many other systems.

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