Results Using Classical Model Without Damping

4.6.3.1 Equal Area Criterion

When using the Equal Area Criterion (EAC) derived in Section 4.3, the classical model is used, i.e.E is considered constant. Thus, the post-faultK’s are equal to their pre-fault counterparts in Table 4.1, meaning also the post-fault equilibrium points are identical to the pre-fault equilibrium points in Table 4.2. During fault, the grid voltage is0.1p.u. yielding the calculatedK_1p^f ,K₂^f and K₃^f in Table 4.4.

The Critical Clearing Angle (CCA),δcc, is then found by solvingg(δcc) = Aa−Ad,max = 0 using thefsolvefunction in MATLAB. HereA_aand A_d,maxare calculated using equations (4.23a) and (4.23b) respectively withδ_max =δ_u0^post =δ_u0^pre.

Table 4.4: FaultK’s assumingE =constant.

K’s during fault Value

K_1p^f 0.1585 p.u.

K₂^f 0.1240 p.u.

K₃^f =K₃^post =K₃ 0.1282 rad

Using the EAC, the CCA is found to beδ_cc = 89.5836°. TheP −δcurve for both the pre/post-fault state and the pre/post-fault-state are depicted in Figure 4.7. Here, also the stable and unstable equilibrium points are plotted, in addition to the critical clearing angle. The two areas, i.e.

the acceleration area and the deceleration area, that are equal at the critical clearing angle, are also indicated as A_a and A_d respectively. As seen in Figure 4.7, P_e^f is dependant on δ, i.e.

P_e^f 6=constant. The critical clearing time can therefore not be found analytically using (4.24), and the result of the stability analysis using the EAC is thus summarised in Table 4.5.

Table 4.5: Results of stability analysis using EAC.

Parameter Value CCA,δ_cc 89.5836°

Figure 4.7:P −δcurve for the system, including critical clearing angle and indicated areas.

4.6.3.2 Transient Energy Function Without Damping

The TEF method without damping uses the same system dynamics as the EAC, and the pre-, fault-and post-faultK’s are therefore the same as in Tables 4.1 and 4.4, with subsequent equilibrium points as in Table 4.2. Using (4.33), the critical value of the TEF is calculated to beV^cr = 0.8041.

Forward integration is then performed on the set of differential equations in (4.10) 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 of δand∆ω of the integration, the value of the TEF is calculated by (4.26) and compared to V^cr. A graphic illustration of the results is given in Figure 4.8, where the critical clearing time is found to bet_cc = 0.0445s. At the time step of the integration corresponding to the critical clearing time, the solution ofδis obtained asδ = δ_cc = 89.3926°. The results of the stability analysis using the TEF without damping are summarised in Table 4.6.

Table 4.6: Results of stability analysis using TEF without damping.

Parameter Value

V^cr 0.8041

CCA,δ_cc 89.3926°

CCT,t_cc 0.0445s= 44.5ms

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Time [s]

0 0.5 1 1.5 2 2.5

V(x)

Energy Function

V(x) V_cr t_cc

Figure 4.8: Energy function vs. clearing time for classical model without damping.

The dynamical system when using the classical model without damping is 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 4.9.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time [s]

-50 0 50 100 150 200 250

Rotor angle [deg]

Stable vs. Unstable System

Stable system Unstable system

s

Figure 4.9:δfor stable and unstable system when neglecting damping.

4.6.3.3 Performance of the Classical Model Without Damping

The Equal Area Criterion and the Transient Energy Function without damping both use the same mathematical model as a foundation for analysing the system stability. As such, it is expected that the two methods should yield very similar results for system stability limits. Comparing the CCA found in Tables 4.5 and 4.6, the results confirms this similarity with a difference in CCA of only0.191°, which for this application is considered negligible. The results do however emphasise the large drawback of the EAC method, as no CCT can be found for the given fault scenario using the EAC. With this said, the EAC as depicted in Figure 4.7 serves as an intuitive method of finding the critical clearing angle when that is the parameter of interest. Furthermore, the curves in Figure 4.7 reaffirms the equilibrium points as the intersections between the power referencePref and the electrical powerPe.

The limitation related to finding the CCT is clearly overcome by the transient energy function method, which by means of the critical value ofV is able to determine the CCT as visualised in Figure 4.8. Here, as expected from the theory, the critical clearing time is found as the time when the value of the TEF exceeds the critical value. As such, it can be argued that the EAC serves as a good introduction to understanding the core problem of transient angular instability, using visual representation as an aid, but that the TEF method is a better approach to determining the actual stability limit of the system when the critical clearing time is of interest.

With that being said, the CCTt_cc = 44.5ms identified in Figure 4.8 is considered a very short clearing time compared to typical clearing times for power systems, equalling approximately 2 cycles. Furthermore, looking at Table 4.11 in Section 4.6.7 and comparing the limits as found by the EAC and TEF without dampening to the simulation results, the deviation from the actual CCT is found to be 88.47% which is a substantial divergence.

Even though this seems like a large difference, the deviation is expected considering two major

aspects impacting the results. Firstly, neglecting the damping is expected to negatively impact the stability results. This is natural as the acceleration and deceleration of the rotor angle are increased and decreased respectively. While this is true also for the conventional SG, the effect of neglecting the damping when analysing the VSM can be particularly large as the VSM can have a tuned inertiaJ that is way smaller than what typically found in the conventional SG. This makes the dampening/frequency-droop govern a significant part of the dynamic response. As such, neglecting the damping when analysing the VSM could potentially yield a much larger deviation from the actual system than when neglecting the damping when analysing the conventional SG.

Secondly, the transient energy function is expected to yield a conservative result. This attribute is inherent to the method and comes from the fact that for any dynamic system there exists a set of potential Lyapunov functions that can be utilised. All of these Lyapunov functions have their own attraction regions for which a set of initial conditions will render the system stable, i.e.

satisfy the stability requirements. Thus, each Lyapunov function that can be found for a given system usually only covers a sub-set of the actual stability area, and thereby tends to yield a pessimistic assessment of the system stability. This attribute can however be argued to be positive in the sense that the analysis is less likely to yield a stable result for an unstable situation, which in worst case could jeopardise system integrity. A too conservative result on the other hand can be argued to be negative as it might lead to protections systems being set to trip prematurely, which may affect customers and increase the risk of major disruptions in the power supply.

According to the theory in Section 2.8.2, the rotor angle would enter periodic oscillating swings around the equilibrium point if damping is neglected. This is due to the value ofV(δ,∆ω)being constant after fault-clearing so that the state trajectory depicted in Figure 4.5a becomes a closed curve around the s.e.p.. Recalling Section 2.8.2, the system is then considered stable but not asymptotically stable. The dynamic response depicted in Figure 4.9 is completely in accordance with this theory, where it is clearly visible that the rotor angle of the stable system starts to oscillate with constant swings around the stable e.p., while the rotor angle of the unstable system diverges.

Furthermore, the rotor angle of the stable system in Figure 4.9 continues to increase after the fault is cleared as the deceleration process takes some time. However,∆ω = 0is achieved at

≈ 150°, just before the u.e.p. at156.1805° as per the requirement for stability, and the rotor angle then accelerates back towards the s.e.p.. The rotor angle of the unstable system on the other hand is seen decelerating until reaching the point where∆ω is the smallest, which coincides with the u.e.p. at156.1805°. When the rotor angle passes156.1805°, the mechanical power is again larger than the electrical, as per theP −δcurve, and the rotor angle can be seen to start accelerating again, thus diverging from the stable state.