## 5 CTMC Estimation of improvement in Blackout Recovery Time with Autarkic Microgrids  
In this section, continuous time Markov chain (CTMC) modelling of the power grid and communications network is used to quantify the expected improvement in the blackout recovery time using the autarkic microgrid method [6] compared to the conventional top-down black start method. CTMC is a state space based stochastic modelling technique to calculate the reliability and availability of a system given the stochastic operational status of its constituent components [19]. It is specifically used for systems in which the operational state of a component depends statistically on the operational state of other components. Grid restoration after a large-scale blackout is conventionally done by first starting power stations with black start capability, then expanding the re-energised grid to include more large synchronous power stations, and expanding the re-energised grid. In this way, MV and LV levels are re-energised hierarchically and top-down, i.e., from the transmission level. The CTMC model of conventional grid restoration is based on the following assumptions, as shown in Figure 5a:  
* the DERs in the DG act independently without external control and without external communications;
* in case of blackout, they disconnect from the power line;
* the substations receive control signals from, and send measurements to, a DG grid control centre;
* the areas enclosed by dotted lines correspond to the grids served by the substations;  
Figure 4: LTE Latency Measurements in Test Setup. * the substations and the control centre have a private communications network available, and this network continues to operate during blackouts. In the autarkic microgrid blackout recovery method the MV/LV grid is divided into a set of autarkic microgrids, as shown in Figure 5b, which are islanded in case of blackout. The DERs remain connected to the grid and continue to use their MAS MTC devices to communicate with each other and with the DG control centre during the blackout [6]. To simplify the modelling, we assume each microgrid has its own eNodeB equipped with an emergency power supply. Two test cases are modelled: Test Case 1 compares the blackout recovery times of the conventional and autarkic microgrid methods; Test Case 2 compares the availability of communications in the autarkic microgrid with and without the IOPS function. * **Test Case 1, Scenario 1: Conventional hierarchical top-down blackout recovery:** no use is made of DERs present in the MV and LV levels to assist in the blackout recovery. The simplifying assumption is made that the communications network continues to function in the blackout. The blackout recovery steps are: to re-energise the grid control centres (SCADA) and bring up generation units; to re-energise the transmission grid; and to re-energise the DG and loads. The different CTMC states and mean state transition times are tabulated in Tables 1 and 4 and shown in Figure 6. The state transition sequence and the mean time taken for individual steps of the restoration process are derived based on the recovery from the Italian blackout of 2003 [20] and the US-Canada blackout of 2003 [21]. \begin{table}
\begin{tabular}{c c} \hline \hline
**State** & **State Meaning** \\ \hline S1 & Blackout \\ S2 & SCADA\_Up, Generator\_Dn,Network\_Dn, Load\_Dn \\ S3 & SCADA\_Dn, Generator\_Up,Network\_Dn, Load\_Dn \\ S4 & SCADA\_Up, Generator\_Up,Network\_Dn, Load\_Dn \\ S5 & SCADA\_Up, Generator\_Up,Network\_Up, Load\_Dn \\ S6 & Grid Restored \\ \hline \hline \end{tabular}
\end{table} 
Table 1: CTMC States in conventional hierarchical top-down blackout recovery (Scenario 1). Figure 5: (**a**) Conventional grid restoration (hierarchical top-down blackout recovery) and (**b**) autarkic microgrid restoration (each islanded microgrid autonomously recovers from blackout).