Dynamic modeling of the electric transportation network

– We introduce a model for the dynamic self-organization of the electric grid. The model is characterized by a conserved magnitude, energy, that can travel following the links of the network to satisfy nodes’ load. The load fluctuates in time causing local overloads that drive the dynamic evolution of the network topology. Our model displays a transition from a fully connected network to a configuration with a non-trivial topology and where global failures are suppressed. The most efficient topology is characterized by an exponential degree distribution, in agreement with the topology of the real electric grid. The model intrinsically presents selfinduced break-down events, which can be thought as representative of real black-outs. Introduction. – The electric grid is a critical infrastructure for our economy and society. Recent events, ranging from the large-scale black-outs a few years ago to the California crisis today [1], highlight the need to enhance the insight into the electric grid, complementing the traditional technological analysis [2] with new transversal points of view. Our approach is to study the electric grid macroscopic behavior rather than to dissect individual events. At that macroscopic scale, the electric grid exhibits behaviors typical of complex systems. For instance, on the basis of 15 years time series of transmission system black-outs of the U.S. electric network [3, 4], it has been proposed that the electric grid may be a self-organized critical system, operating at or near a critical point. A signature of the electric grid is that it can be represented as a complex network, where nodes are the generators and the links the transmission lines. Recent research in complex networks has shown that a detailed knowledge of the topology of a communication or transportation network is essential for the understanding of cascading failures [5–8]. While some of these studies have focused on the topological robustness of the underlying network to random failures and targeted attacks [9], other research have considered dynamic processes on static networks [10, 11]. However, these studies do not consider the network as a dynamic entity whose evolution is driven by the action of the nodes [12–17]. In this paper, we present a dynamic model aiming to describe the growth and evolution of a transportation network. The network growth relies on the need of resource distribution in a heterogenous environment. The model. – We consider N dynamic elements located at the nodes of a two-dimensional square lattice. Each element i is characterized by its size si, drawn from a probability distribution p(s). To each element i are associated two dynamic variables: the load (energy consumption) l i , and the supply (available energy) f t i . We assume that i) the load of element (∗) E-mail: [email protected] A. Scirè et al.: Dynamic modeling of the electric transportation network 319 i is described by l i = mi+ √ miξ t i , where mi is a constant value and ξ t i represents a fluctuation term; and ii) initially at t = 0, the available energy and the constant load at each element are proportional to its size si, f i = frsi , (1) mi = mrsi , (2) where fr andmr are constant values and in general fr > mr. Starting from an initial condition where each element is isolated, the network grows as follows. At each time step: 1) If the load overcomes the supply at element i (f t i < l t i), a failure occurs. After a failure, the failing element i chooses a target neighbor through a wiring strategy. Following the empirical results observed in communication networks with spatial constraints [18], the target node j is chosen such that it maximizes the function π(i, j) = sβj dγij , (3) where dij represents the Euclidean distance between the two elements, and the exponents β and γ indicate the preference for size and proximity, respectively, in the wiring. 2) The supply at element i is updated according to fi. f t+1 i =   f t i if ki = 0, l i + ∑ j∈V(i) f j−l j kj if ki = 0, (4) where kj is the number of links possessed by the element j, j running over the neighbors of the element i, V(i). The prescription given by eq. (4) allows the energy to be distributed through the links of the network as it is needed, depending on the instantaneous load of each linked element, making next overloads more unlikely. Each Nsteps time steps, the links are actually set and the network is consequently updated. The choice of the time step for the network construction is a further degree of freedom of our model. In real electric grids, the time scales of the fluctuating demand are much faster than the time scales at which the network is modified. Therefore, we choose to update the network each Nsteps 1 time steps of the local dynamics. The initial total supply E and the total load L in the system are given by