Results Using Transient Energy Function With Damping

The TEF added with the damping term is based on the classical model with damping. Here,E is still considered constant and the pre-, fault- and post-faultK’ are therefore the same as in Tables 4.1 and 4.4, with subsequent equilibrium points as in Table 4.2. Using (4.35), the critical value of the TEF is calculated to beV^cr = 85.8290.

Forward integration is then carried out on the set of differential equations in (4.12) inserted for theK’s in Table 4.4 and using the stable e.p. from Table 4.2 as initial conditions. The integration is carried out using theode45solver for differential equations in MATLAB. For each solution ofδand∆ωof the integration, the value of the TEF is calculated by (4.34) and compared toVcr.

A graphic illustration of the results is given in Figure 4.10, where the critical clearing time is found to be t_cc = 0.6079 s. At the time step of the integration corresponding to the critical clearing time, the solution ofδis obtained asδ=δ_cc= 155.3479°. The results of the stability analysis using the TEF with damping are summarised in Table 4.7.

Table 4.7: Results of stability analysis using TEF with damping.

Parameter Value V^cr 85.8290 CCA,δ_cc 155.3479°

CCT,t_cc 0.6079s= 607.9ms

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time [s]

0 50 100 150 200 250 300

V(x)

Energy Function

V(x) V_cr t_cc

Figure 4.10: Energy function vs. clearing time for classical model with damping.

The dynamic system when using the classical model with damping was forwardly integrated for a clearing time 1 ms below and 1 ms above the CCT found by the stability assessment above, for a fault initiated att= 0.1s. The response ofδis depicted in Figure 4.11.

The Transient Energy Function including damping yielded, as seen in Table 4.7, a much more optimistic result of the system stability. As the classical model with damping has the same stable and unstable equilibrium points as the classical model without damping, the first observation that can be discussed is that the critical clearing angleδ_cc = 155.3479° in Table 4.7 is dramatically close to the unstable post-fault e.p. δ_s0^post = 156.1805°. This implies that the full deceleration of the rotor angle takes place in the course of less than1°.

The possibility of such an event can be explained using two different factors, both contributing to the result. Firstly, the damping clearly restrains the acceleration of the rotor angle, as the slope in

0 0.5 1 1.5 2 2.5 3 Time [s]

-50 0 50 100 150 200 250

Rotor angle [deg]

Stable vs. Unstable System

Stable system Unstable system

s

Figure 4.11:δfor stable and unstable system when including damping.

Figure 4.11 is much more gradual than for the case without damping. As such, less deceleration is required. Secondly, a large damping coefficient along with relatively small inertia for the VSM makes the damping govern the response. The larger the damping/drooping, or smaller the inertia, the faster the speed of the VSM will be able to change. Thus, for the full deceleration process to take place in the course of1°, the influence of the damping on the dynamic response is very large, further emphasising the impact of damping when analysing the VSM with small inertia, as discussed for the model neglecting the damping.

Again, by comparing V to the critical value Vcr, the critical clearing time was obtained as visualised in Figure 4.10. The CCT of607.9ms is, in sharp contrast to the CCT of the system without damping, considered very high. This is also confirmed when looking at Table 4.11 in Section 4.6.7, where it is evident that the TEF with damping has yielded a CCT57.45%above the CCT of the simulated system, and thereby deemed an unstable system as stable. The analysis has thus failed its main purpose of obtaining the stability limits to ensure the safe operation of the power system. Using the classical model including damping for a Synchronverter with relatively small inertia and relatively large damping coefficient could therefore involve rather large deviations in the “wrong” direction from the actual stability limits, making the method less suitable for such a stability analysis. This emphasises the need for the quasi-steady method tested in the next section.

When looking at Figure 4.11, two large differences from Figure 4.9 can be noticed. The first one is the already discussed slope of the rotor angle during fault, which in addition to being much more gradual, seems to be linear. This linearity can be explained by the large damping limiting∆ωto reach its maximum value almost instantly before restraining∆ωat this constant maximum level for the duration of the fault, making the increase inδlinear.

The second large difference is seen in the rotor angle for the stable system, which, for the model

including damping, falls back to the s.e.p. after the fault-clearing. This is in line with the described theory in Section 2.8.2, stating that if damping is present, the rotor angle will settle back to the equilibrium point. This is because the value ofV(δ,∆ω)decreases after clearing the fault due to the damping so that the state trajectories depicted in Figure 4.5b spirals back to the s.e.p.. Recalling Section 2.8.2, the system is then considered asymptotically stable. For the unstable system, the response is identical to that in 4.9 with the rotor angle accelerating fast after passing the u.e.p..