# Operational Resilience Metrics for Complex Inter-Dependent Electrical Networks
## 2 Operational Resilience Metric for Electrical Distribution Networks  
This section describes how to link the QoS time behavior with the measure of the resilience property. The estimation of the proposed QoS requires a model of the EDN allowing the assessment of the QoS as a function of the many different properties (that were introduced in the previous section) that influence the response of the system to a perturbation [3; 4]. Formally, the EDN is represented by a node-weighted directed graph \(\mathcal{G}=\{\mathcal{V},\mathcal{E},\mathcal{W}\}\) without loops and a telecommunication network \(\mathcal{B}\) providing tele-control functionality containing a number of BTS \(b_{i}\) s.t. \(b_{i}\in\mathcal{B}\) with \(|\mathcal{B}|=X\), where:  
* vertex set \(\mathcal{V}=\mathcal{Q}\cup\mathcal{R}\cup\mathcal{A}\cup\mathcal{D}\) with \(|\mathcal{V}|=N\);
* \(v_{i}\in\mathcal{Q}\) represents an electric primary station (PS) containing high-to-medium tension transformers equipped with remote control functionality provided by a proprietary telecommunication network;
* \(v_{i}\in\mathcal{R}\) represents an electric secondary station (\(SS\)) containing medium-to-low tension transformers equipped with remote controlled switches (that depend on some BTS \(b_{i}\);
* \(v_{i}\in\mathcal{A}\) represents an electric secondary station (\(SS\)) containing medium-to-low tension transformers equipped with automatic switches;
* \(v_{i}\in\mathcal{D}\) represents an electric secondary station (\(SS\)) containing medium-to-low tension transformers without remote control functionality;
* edge set \(\mathcal{E}=\{e_{ij}\}\) where the generic \(e_{ij}\) represents the portion of an electrical line connecting the two electric stations \(v_{i}\) and \(v_{j}\);
* weight set \(\mathcal{W}=\{on,disc,dam\}\) associated to each vertex \(v_{i}\) where: (i) _on_ represents a physically intact and fully available station, (ii) _disc_ represents a physically intact and functionally unavailable station;
* each \(b_{i}\in\mathcal{B}\) is electrically supplied by a specific \(v_{i}\) s.t. \(v_{i}\in\{\mathcal{R}\cup\mathcal{D}\}\). In the following, the QoS function described in Section 1 is first formalized in order to describe and analyze the behavior of a medium-voltage (MV) line \(l\) over the time horizon \([0,t_{f}]\). Then, the formulas to compute the operational resilience of the overall electrical distribution network are described. The QoS function, describing a perturbed MV line, is represented by a piecewise linear function (Figure 3). Let us assume that a perturbation affecting some electrical stations \(v_{i}\) occurs at time \(t_{0}\) (with \(0<t_{0}<t_{f}\)) and that, following the recovery operations, the MV line returns fully operational at time \(t_{0}+T\leq t_{f}\). Let us consider the set \(\mathcal{S}(t)=\{s_{1}(t),...,s_{L}(t)\}\) containing the local states \(s_{i}(t)\in\mathcal{W}\) of each electric station \(v_{i}\) in the interval \(t\in[0,t_{f}]\). Let us define the function \(QoS_{l}(t):\mathcal{S}(t)\rightarrow[0,1]\) measuring the overall state of the MV line \(L\) for \(t\in[0,t_{f}]\) s.t. :  
* \(QoS_{l}(\mathcal{S}(t))=M=1\Rightarrow\forall\;\;v_{i}\;\;s_{i}(t)=on\;\;,\;0 \leq t<t_{0},\;\;t\geq t_{0}+T\). The QoS of the line reaches the maximum value if the line is not perturbed or when, after a perturbation, the line is completely restored within a time interval \(T\) (pre and post disturbance green lines of Figure 3). * \(QoS_{l}(\mathcal{S}(t))=m=0\;\;for\;\;t_{0}<t<t_{1}\). The QoS for the line \(l\) reaches the minimum value at time \(t_{0}\) when there exists at least one \(v_{i}\) in the line \(l\) is in the _dam_ state. For EDN medium-voltage lines the QoS almost instantly reaches the minimum value (i.e., the zero value) because of the opening of protection devices that disconnect all electrical SS in the interested line. * \(QoS_{l}(\mathcal{S}(t))=r\;\;with\;\;m<r<M\;\;\;\text{for}\;\;\;t_{0}\leq t<t_{1}\).