**4. Resilience metrics**

*Infrastructure Management and Construction*

*Electrical distribution grid SCADA system and tlc dependencies.*

station—BTS hereafter—of the telecommunication network). In turn, BTS are supplied by the energy provided by the SS of the EDN, thus configuring a dependency loop (no energy on a specific BTS, no telecontrol functionality of this BTS in favor of other SS of the network). In this work, we suppose that BTS do not have power backup, i.e., we will simulate the worst possible case. This implies that if a certain BTS depends on a certain substation SS that is in a damaged (or disconnected) state,

Each SS can be modeled as a finite state machine as shown in **Figure 3**. In normal conditions, the SS is in the initial "functioning" state. Starting from this state, the

• Failure state: when a failure in the SS occurs, transition 1 is activated. The SS

• Not functioning state: in case of a contingency, the protection devices of the grid will disconnect a number of secondary substations that will change their state from "functioning" to "not functioning." For example, when a SS moves into the failure state, all SS on the same line move into their not functioning state. A SS

that specific BTS will immediately stop functioning.

*Secondary substation (SS) finite state model.*

secondary substation (SS) can move into two different states:

remains in this state for the expected failure duration;

**38**

**Figure 3.**

**Figure 2.**

Let us assume to have an EDN characterized by its topology, with nodes *N* and links *L* corresponding to electrical stations and electrical lines, respectively. The function representing the *functioning state* for all the elements of the EDN is referred to as *F*:

$$F(\mathbf{N}, L, \mathbf{t}) = \mathbf{O} \forall \mathbf{t} \tag{1}$$

if all elements *N* and *L* are in a *functioning state* and all telecontrol functionalities are active. Let us now introduce a perturbation function *P* that can change the state of one EDN element from the *functioning state* to one of the other possible states. In such a case

$$P \colon F(N, L, t) \to F'(N, L, t) \tag{2}$$

where *F*′ (*N,L,t*) *>* **0** for *t* ∈ [*0,T*] and zero elsewhere. For the sake of simplicity, we will apply the perturbation *P* only to the electrical secondary station (referred to as SS). Time *T* represents the time when all elements have been repaired and the network comes back to its fully functional state *F(N, L, t) = 0*. A perturbation *P*, in principle, could affect one (or more) electrical station and bring it (or them) from the functioning state to the not functioning or the failure states.

The damage of a SS consequent to the introduction of *P* produces a sequence of perturbations on the network. These consist in the disconnection of other nodes along the line due to instantaneous opening of protection switches. The damaged nodes are replaced by power generators (PGs) to ensure electrical continuity to the node's customers. The damaged nodes will not be repaired in the time space of the simulation, but their function will be restored through the settlement of PGs. The disconnected nodes, in turn, are reconnected either through a telecontrol operation (if available) or by dispatching technical crews to provide manual reconnection. All such interventions require specific times, which are considered when defining a restoration sequence of interventions. The impact of the perturbation *P* on the EDN is measured using a key performance indicator (KPI) that is currently used by the Italian Energy Authority to estimate the level of service continuity of an EDN. Such KPI is expressed as the number of disconnected customers *ni* of the *i-th* EDN node times the duration *i* of its disconnection. Such a value is expressed in terms of *kilominutes* (i.e., 103 minutes). Thus, if the damage of the *i-th* SS of the network will result in the disconnection of *m* SS, each one for a time *<sup>j</sup>* ( *j =* **1**,*m*), the overall KPI outage metrics will be measured in terms of *i* that is defined as follows:

$$
\Gamma\_i = \sum\_{j=1}^{m} n\_j \,\,\tau\_j \tag{3}
$$

For a given perturbation *P*, the integral over the simulated time span of Eq. (3) represents the perturbed functional state of the grid defined in Eq. (2):

$$\int\_{\mathbf{0}}^{\mathbf{T}} \mathbf{F}^{'} \left(\mathbf{N}, \mathbf{L}, \mathbf{t}\right) d\mathbf{t} = \Gamma\_{i} \tag{4}$$

*i* , thus, represents the *impact* that the damage of an EDN element (the *i-th* node) can produce, by using an official KPI as a metric. The larger the value of *<sup>i</sup>* , the weaker the capability of the network to withstand the perturbation in terms of impacts produced on the EDN customers. In general the value of *<sup>i</sup>* depends on different factors (described in detail in Section 4) ranging from the topology of the network and the employed technologies to the efficiency of operator restoration procedures; therefore, it would not be inappropriate to correlate the value of *<sup>i</sup>* with the inverse of the resilience concept *R*. In other terms

$$\mathcal{R}^{-1} \propto \Gamma\_i \tag{5}$$

We can generalize the concept by checking the EDN behavior versus all possible perturbations. The overall operational network resilience will be thus associated with the inverse of the value of the integral of the distribution function of all the *<sup>i</sup>* values (*D*()) resulting from the failure of each one of the *N* nodes of the EDN (normalized with respect to the total number of nodes *N*):

$$\mathbf{R} \propto \frac{\mathbf{1}}{\epsilon \,\Gamma \,\mathrm{\,\,\mathrm{\,\,\mathrm{\,\mathrm{\,\mathrm{\,\mathrm{\,\mathrm{\,\mathrm{\mathrm{\,\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\cdot}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} \,\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\,\mathrm{\mathrm{\text{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\cdot}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} \,\mathbf} $$

The higher the impact, the lower is the resulting operational resilience of the EDN network.
