#### 5.2. Large Signal Stability Analysis of the VSG-IIDG

The metallic short-circuit fault happens in the middle of the cable at the time of 3 s and is cleared after variable time delays. Once the fault is cleared, $\Delta \omega $ and ${\theta}_{ig}$ deviate from the equilibrium point at various degrees (different values of ${x}_{1}$ and ${x}_{2}$). It means operating points with different values of the Lyapunov function $V({x}_{1},{x}_{2})$.

Figure 7 shows the contour plots of the Lyapunov function of operating points with different

$\Delta \omega $ and

${\theta}_{ig}$. Deviating from the equilibrium point at various degrees, the values of the Lyapunov function are diverse. For the points inside the stability domain, the value of the Lyapunov function is under the critical value

M, which indicates that the system is asymptotic stability when operated in these points.

Figure 8 shows that the stability domain is in a conical type. Whichever disturbance that has angular frequency and angle limited in the taper can recuperate the system.

Figure 8 shows the operational trajectories of five stable points (p1–p5) whose Lyapunov function value is less than the critical value

M. The abscissa axis of time starts after the fault is cleared. These curves display the deviation of angular frequency and angle from a disturbance condition. It can be observed that after the fault is cleared, these operational trajectories tend to the equilibrium points where

$\Delta \omega ={\theta}_{ig}=0$. Even though these operational trajectories deviate from the equilibrium point at various degrees, as time pass by, the system finally runs in a zero-deviation state.

When it comes to the unstable operating points, it is quite different. Take an unstable point

m and a stable operating point

n as example. The point

m comes when the fault is cleared at 3.15 s and the deviation

$\Delta \omega =11.5$,

${\theta}_{ig}=0.56$. While the point

n comes when the fault is cleared at 3.1 s and the deviation

$\Delta \omega =0.75$,

${\theta}_{ig}=-0.1$. As shown in

Figure 7, the Lyapunov function values of point

m and

n are on the different sides of the contour plane of the critical stability energy

M. The Lyapunov function value of point

n is smaller than the critical stability energy

M, which indicated it is inside the stability domain. However, the Lyapunov function value of point

m is greater than the critical stability energy

M and it is an unstable point.

Figure 9 shows the comparison of simulations of the unstable point

m and the stable operating point

n. Due to different fault clearing time of these two points, the extent of disturbance varies (different values of

$\Delta \omega $ and

${\theta}_{ig}$). As

Figure 9a shows, the operational trajectory of the stable point

n experiences damped oscillations before reaching the equilibrium stability where

$\Delta \omega ={\theta}_{ig}=0$. The angular frequency gets slowly back to the reference value and the angle difference between the VSG-IIDG and the network reduces to zero. However, the trajectory of the unstable point

m is divergent and of extreme volatility. Even when the fault is cleared, the angular frequency can’t keep in synchronism. The system can’t recuperate and tends to instability. The simulations shown in

Figure 9 verify the analysis results.

#### 5.3. The Impact of Parameters on the Stability Domain

The cable impedance, load power, and virtual inertia have a significant influence on the stability of VSG-IIDG [

27,

28]. The stability domain with different parameters will be explained in the next sections. The stability domain is determined by the boundary of the equation

V(

x) =

M. The boundary quantifies the extent of disturbances that the VSG-IIDG system can endure. In the simulation, only one parameter is changed and the others are kept the same in different scenarios.

Figure 10a depicts the impact of the impedance of cable

${Z}_{L}$ (

${R}_{L}+j{X}_{L}$) on the stability domain. As impedance decreases, the stability domain expands in the ratio of equality. The increase in impedance means the connection between the IIDG and the network is less. When the supporting function of the network is weakened, the IIDG system is apt to instability.

Figure 10b shows the change of the stability domain when load power

${P}_{D}$ decreases. When load power decreases, the stability domain tends to increase. This is due to the fact that a larger load power adds more burden to the system. This is similar to the power angle stability of conventional synchronous generators, where lower power causes relative increase of the acceleration area and out-of-step situations will be rare.

The effect of different virtual inertia H on the stability domain is shown in

Figure 10c. As the virtual inertia increases, the stability domain is prone to be smaller. What’s more, the shrink doesn’t happen in the ratio of equality. The shape of boundary changes and a peak appears.

Figure 10c shows that the rising virtual inertia has a negative effect on the large signal stability of the VSG-IIDG. In large signal stability of conventional synchronous generators, the change of inertia does not contribute to the change of the stability boundary [

29]. However, it’s not the case in the VSG algorithm and

Figure 10c witnesses a marked distortion of the stability domain as virtual inertia increases.

It should be noted that the rising virtual inertia will also slow down the response speed of the IIDG system since it can be seen as an integration constant. Hence the rising virtual inertia leads to a decrease not only in the distance between an operating point and the stability boundary but also the speed running from this point to the boundary. When the distance and speed are reduced simultaneously, it’s not sure whether the time duration will be shortened or not. That means, the critical cleaning time of the fault may not decrease even with a large virtual inertia (a smaller stability domain), since the response speed of the IIDG system is slow and there is still enough time to diagnose and handle the fault.

Table 3 presents the stability of the VSG-IIDG with different values of load power, cable impedance, and virtual inertia when the clearing time is 0.1 s. As shown in the table, when active power of local load increases, the IIDG system tends to instability. When load power and control parameter

H remain unchanged, the IIDG system is unstable with larger cable impedance. Also, when the inertia is too large, the stability of the system goes from stable to unstable. The results show in

Table 3 indicate that the IIDG system is apt to be unstable with larger load power and cable impedance or smaller virtual inertia.

#### 5.4. Sensitivity Analysis of Parameters

Investigation of the large signal stability by ad hoc variations of the parameters is challenging, especially when several parameters act at the same time. Sensitivity analysis is helpful in identify which parameter should be modified in an easier way. Such studies are of high importance, considering the assessment of large signal stability in different scenarios with large expected variations in grid configurations, operating conditions, and system parameters.

This paper draws on the experience of transient stability analysis in conventional power system [

30]. The sensitivity analysis of the load power, cable impedance, and virtual inertia on the stability domain area is performed. The definition of the sensitivity

${k}_{sen}$ is in partial differential equation as:

where,

${S}_{d}$ is the area of stability domain,

y is the parameter to be analyzed (load power

${P}_{D}$, cable impedance

${Z}_{L}$, and virtual inertia

H). The results are shown in

Table 4, respectively.

It can be seen that the stability domain is mainly sensitive to load power, cable impedance, and virtual inertia. The sensitivity does not change a lot when different values of load power and cable impedance are adopted. When virtual inertia is kept at a small value, the area of stability domain changes slowly as inertia varies. It should be pointed out that only the virtual inertia can be modified to improve the stability of a control strategy. The load power and cable impedance cannot be influenced by the control. However, the load power and cable impedance can be selected within the proper range during the design of the system.