# Synchronous Generator Stability Characterization for Gas Power Plants Using Load Rejection Tests
## 2 Theoretical Models
### SG Model: GENSAL  
The SG is shaped using the GENSAL model, which is included in Figure 2. Table 1 shows all the parameters related to Figure 2. Most of these parameters are given by the manufacturer.  
Figure 1: Electrical scheme and simulation model correlation for an SG connected to an infinite power bus.  
Figure 2: GENSAL model for the SG.  
However, three parameters need to be calculated, which include two terms related to the saturation factors and the inertia of the power plant. These factors are encompassed by the turbine and the alternator.  
On the one hand, the saturation parameters are calculated according to expressions (1) and (2) as:  
\[S_{(1.0)}=\frac{A_{10}-B_{10}}{B_{10}} \tag{1}\]  
\[S_{(1.2)}=\frac{A_{12}-B_{12}}{B_{12}} \tag{2}\]  
The values \(A_{10}\), \(B_{10}\), \(A_{12}\) and \(B_{12}\) are parameters that are required to be calculated for the modeling of a synchronous generator. These parameters are obtained from the no-load characteristic, which can be found in the generator datasheet. Figure 3 shows an example of a typical saturation test plot that is used to obtain the no-load characteristic.  
On the other hand, during the load rejection test, the inertia of the system can be determined. The kinetic model of the system can be represented by expression (3):  
\[\omega_{r}(t)=\frac{1}{2H}\int T_{a}(t)dt=\frac{1}{2H}\int\left[T_{mce}(t)-T_{ elce}(t)\right]dt \tag{3}\]  
where \(\omega_{r}(t)\) is the rotor speed, \(T_{mce}(t)\) is the shaft torque due to the turbine, \(T_{elce}(t)\) is the electrical torque due to the network load and \(H\) is the machine inertia.  
At the moment of load rejection, the electrical torque becomes null (\(T_{elce}(t)=0\)) but the mechanical torque remains. The inertia of the machine determines the frequency variation. Therefore, the values of \(\Delta t\) and \(\Delta f\) used to determine the inertia must be calculated as the difference between the steady state and the first sample obtained after opening the main  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Parameter** & **Units** \\ \hline d-axis open circuit transient time constant & \(T_{d0}^{\prime}\) & [s] \\ d-axis open circuit sub-transient time constant & \(T_{d0}^{\prime\prime}\) & [s] \\ q-axis open circuit sub-transient time constant & \(T_{d0}^{\prime\prime}\) & [s] \\ Machine inertia & \(H\) & [s] \\ Speed damping & \(D\) & [s] \\ d-axis synchronous reactance & \(X_{d}\) & [pu] \\ q-axis synchronous reactance & \(X_{q}\) & [pu] \\ d-axis transient reactance & \(X_{d}^{\prime\prime}\) & [pu] \\ Sub-transient reactance & \(X_{d}^{\prime\prime}\) & [pu] \\ Leakage reactance & \(X_{l}\) & [pu] \\ Saturation factor at 1.0 pu voltage & \(S_{(1.0)}\) & \\ \hline \hline \end{tabular}
\end{table}
Table 1: GENSAL model parameters.  
Figure 3: SG no-load characteristic.  
circuit breaker. By performing this, the influence of the speed governor on the frequency change is not taken into consideration.  
Using per-unit values of the machine, the machine inertia, \(H\), can be calculated through (4). Also, a graphical example is plotted in Figure 4.  
\[H=\frac{P_{mcc}[pu]\cdot\Delta t[s]}{2\cdot\Delta f[pu]} \tag{4}\]  
The \(P_{mcc}\) parameter given in (4) is calculated as:  
\[P_{mcc}[pu]=\frac{P_{test}}{S_{b}} \tag{5}\]  
where \(P_{test}\) is the active power delivered to the grid before the load rejection, \(S_{b}\) is the machine rated power, \(\Delta t\) is the time difference between the steady-state measurement and the load rejection measurement and \(\Delta f\) is the frequency difference between the steady-state measurement and the load rejection measurement.