# Dynamic Stability Analysis of Isolated Power System
## 3 Response of the Isolated EPS to a Load Change
### Response of Isolated EPS to a Load Increase
The value of factor \(k\) is determined by knowing the sum of \(P_{N}\) values of all generators operating in the isolated EPS with primary frequency control enabled:  
\[k=\frac{\sum(P_{N\;FCR})}{2} \tag{2}\]  
The \(k\) factor absolute value of a single power plant is:  
\[k_{PP}=\frac{P_{N}}{f_{N}}.\frac{1}{\delta} \tag{3}\]  
where \(P_{N}\) is the nominal value of the active power of the generator, which is different for each generator, \(f_{N}\) is the nominal frequency of 50 Hz, and \(\delta\) is droop relative value (%). Then, the \(k\) factor absolute value of multiple power plant is:  
\[k_{sum\;PP}=\sum_{1}^{i}\biggl{(}\frac{P_{N\;i}}{f_{N\;i}}\biggr{)}.\biggl{(} \frac{1}{\delta_{i}}\biggr{)} \tag{4}\]  
The total \(k\) factor value is equal to:  
\[k_{total}=k_{sum\;PP}+k_{self\;regulation} \tag{5}\]  
where \(k_{self\;regulation}\) is the self-regulation \(k\) factor of a load, the frequency-dependent load value (MW/Hz). Calculations of the factor \(k\) are useful:  
1. With small frequency deviations (up to 0.2-0.25 Hz);
2. With the low dead-band of operating generators (\(\geq\)10 mHz);
3. When the power plants in operation have a sufficient reserve of active power;
4. When the EPS is in a quasi-static state (\(\Delta f\) varies up to 0.2-0.3 Hz). Calculations of the factor \(k\) are performed only when the operation of the isolated EPS is stable. In order to ensure the capability of an isolated EPS to operate independently, sufficient power reserves are needed to balance the isolated system and to perform frequency control [16; 17; 18; 19; 24]. The FCR (Frequency Containment Reserves) during normal EPS isolated operation should be = 1.1 \(P_{max\_infeed}\) (where \(P_{max\_infeed}\) is the biggest infeed in the EPS). In a case where the dimensioning fault in the EPS during isolated operation is 200 MW, then in order to ensure stable frequency control during isolated operation, the amount of FCR in the EPS should be around \(\pm\)220 MW. It is expected that at least ca. \(\pm\)50 MW of FCR is initially received from the neighboring transmission system operator (TSO) through the HVDC link. The rest of the needed FCR must be deployed locally by the isolated EPS. The FCR of the isolated EPS is 200 MW. The full activation time of the FCR is 30 s, and full activation causes frequency deviation (or \(\pm\)200 mHz). Response of Isolated EPS to a Load Decrease When HVDC BtB Converter's Frequency Control Function Is Off  
A frequency increase in the isolated EPS is simulated by a soaring change of the active power flow through an HVDC BtB converter or with the operating generators of the system from 10 to 50 MW, in 10 MW increments. In the digital simulation, it was assumed that one unit (case A) was operating in SC (synchronous condenser mode). The simulation results are shown in Figure 8a,b. When changing the active power transmitted by the HVDC BtB converter, there are no oscillations in the active power. The short-term frequency tolerance of 200 mHz of the isolated EPS is reached only in the event of an increase in active power by 30 MW or more. Frequency increments up to 50.20 Hz correspond to the load decreasing by 25 MW. The minus sign here represents active power export from the EPS. The isolated power system frequency change values are given in Table 4[16; 17; 18; 19; 24]. After estimating the maximum possible disturbance (30 MW) with a calculated inertia of 7777 MW, the reaction of the isolated EPS when two (case B) SC (synchronous condenser) units are operating in the system was analyzed. The inertia of the isolated EPS is 5793 + (SC1) 992 + (SC2) 992 MWs [15; 21; 22; 25; 26]. The simulation results are shown in Figure 9. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Power Change, MW** & **Frequency, Hz** & **Power of HVDC Converter, MW** \\ \hline \(-\)10 & 50.08 & \(-\)215 \\ \(-\)20 & 50.17 & \(-\)205 \\ \(-\)30 & 50.23 & \(-\)195 \\ \(-\)40 & 50.31 & \(-\)185 \\ \(-\)50 & 50.39 & \(-\)175 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Frequency change in the isolated power system.