## 3 Sequential Restoration Strategy  
### Problem Formulation  
The object function is used to minimize the total time taken to restore generators [30]. It should select the generator that can be restored in the shortest time. The object function can be denoted with the following equation:  
\[min\sum_{i=2}^{N}\Delta t_{\text{GEN}_{i+1}}-t_{\text{GEN}_{i}} \tag{3}\]  
where \(N\) is the total number of generators; \(t_{\text{GEN}_{i}}\) is the time to re-start the \(i\)th generator at the \(i\)th step; and \(\Delta t_{\text{GEN}_{i+1}}-t_{\text{GEN}_{i}}\) means the time taken to start the \(i+1\)th generator from the \(i\)th generator. The object function can be formulated as the minimization of the restoration time. While restoring the system, we must satisfy the following constraints.  
* Solving power flow equations \[PowerFlow(P_{\text{GEN}},\;P_{\text{LO}},Q_{\text{GEN}},Q_{\text{LO}})=0\] (4)
* No violation of generation, transmission and voltage limits \[V_{\text{MIN\_BU}}\leq V_{\text{BU}}\leq V_{\text{MAX\_BU}}\] (5) \[P_{\text{MIN\_LO}}\leq P_{\text{LO}}\leq P_{\text{MAX\_LO}}\] (6) \[P_{\text{MIN\_GEN}}\leq P_{\text{GEN}}\leq P_{\text{MAX\_GEN}}\] (7)  
The second equation means the system balance \(P_{\text{GEN}},\;P_{\text{LO}}\) are the active power of the generator and load, and \(Q_{\text{GEN}},\;Q_{\text{LO}}\) are the reactive power of the generator and load, respectively. Equations (5)-(7) represent the need to maintain a range of voltage and power; \(V_{\text{BU}}\) is the voltage of the bus. \(V_{\text{MIN\_BU}},\;V_{\text{MAX\_BU}}\) denote the minimum and maximum, respectively. \(P_{\text{LO}}\) means the power of the load, and \(P_{\text{MIN\_LO}}\) and \(P_{\text{MAX\_LO}}\) are the minimum and maximum, respectively. In the same way, Equation (5) is the range of the generator.  
The criteria for selecting the next generator are required as in the following equations: \(\sum P_{\text{gen}}^{i}\) is the amount of output in MW from the \(i\)th generator; \(P_{\text{start}}^{i+1}\) is the power needed to start the \(i+1\)th generator. When the generation amount of the system is large enough to supply the cranking power of the generator to be committed, the sequential process of the restoration is performed. If the generation  
Figure 3: The example of applying the enhanced Dijkstra algorithm.  
amount of the system is insufficient, it will take some time to meet the cranking power. In summary, we can select the next generator when the total amount of power to the \(i\)th is greater than the cranking power of the \(i+1\)th generator.  
\[\sum P_{\text{gen}}^{i}-\ P_{\text{start}}^{i+1}\ \geq 0 \tag{8}\]  
To restore the generator in a short amount of time, the generator must calculate the time required for the re-energizing. The estimation of the time for cranking the generator is performed in each step. The time can be formulated as the following equation. The generator that takes the shortest time to start is selected as the next generator by comparing the time calculated as shown below:  
\[t_{\text{GEN}_{i}}=ST_{\text{GEN}}^{i}+ETL_{i}^{i-1}+ET\gamma_{i}^{i-1} \tag{9}\]  
where \(ST_{GEN}^{i}\) is the start-up time for the \(i\)th generator; \(ETL_{i}^{i-1}\) is the energizing time of the transmission lines to restore the path from \(i-1\)th to \(i\)th; \(ET\gamma_{i}^{i-1}\) is the restoration time of the transformers to the \(i\)th generator; \(ST_{\text{GEN}}^{j}\) may be described in the equation and Figure 4 as follows.  
\[ST_{\text{GEN}}=T_{\text{start}}+\frac{P_{\text{start}}}{Rr} \tag{10}\]  
When supplying power from the grid at time \(T_{\text{on}}\), the generator starts power generation after the time required for cranking the generator. In the case of a black start unit that can be started without any external power from the power system, the \(T_{\text{on}}\) value is zero. Once the generator starts power generation at time \(T_{\text{start}}\), it generates the power up to the maximum power according to its ramp rate, \(Rr\).