Results Using Quasi-Steady Approach

The analysis using the quasi-steady approximate Lyapunov approach is carried out according to Section 4.4.3, andE is no longer considered constant. The pre-fault operating condition is however the same, yielding the pre-faultK’s and the initial conditions given by Tables 4.1 and 4.3 respectively.

Forward integration is then performed on the set of differential equations in (4.37a), inserted for (4.37b) and (4.37c). The stable e.p. from Table 4.3 is used as initial conditions, and the integration is carried out using theode45solver for differential equations in MATLAB. For each solution ofδ,E^postis to be calculated by (4.36).

At this point, some additional computational assumptions are made. At the time of fault-clearing, the magnitude of the PCC voltage is assumed to be less than 0.94p.u., and withD_q tuned to increase the reactive power set-point by 100 percent for a 10 percent reduction in voltage, the magnitude V_m,pcc < 0.94p.u. impliesQ_set > 0.6per unit. Also, it is assumed that when the fault is cleared, the active power set-point returns toP_set = P_ref = 0.8p.u.. The stipulated P_set = 0.8p.u. andQ_set >0.6p.u. yields a violation of the criterion in (3.13) for a total power ratingS_n = 1p.u., thus activating the reactive power set-point limitation in (3.14) which dictates Q_set = 0.6p.u.. Therefore, as equation (4.36) simplifies to (4.38).

0.6− Equation (4.38) is a second-order equation that can be solved forEat a givenδ. Therefore, for each solution ofδ,E^postis calculated using (4.38), always choosing the positive solution. With E^post obtained, the new post-faultK’s are calculated and used to find the new post-fault e.p., before the newV^cr is obtained by using (4.35). The value of the TEF is then calculated by (4.34) and compared toVcr.

A graphic illustration of the results is given in Figure 4.12, where the critical clearing time is found to be t_cc = 0.3450 s. At the time step of the integration corresponding to the critical clearing time, the solution ofδis obtained asδ=δcc= 108.3287°. The results of the stability analysis using the quasi-steady approximate Lyapunov approach are summarised in Table 4.8.

The dynamic system when using the AVR model is forwardly integrated for a clearing time both equal to- and 10 ms above the CCT found by the stability assessment above, for a fault initiated att= 0.3s. The response ofδis depicted in Figure 4.13.

Table 4.8: Results of stability analysis using quasi-steady approximate Lyapunov approach.

Parameter Value CCA,δ_cc 108.3287°

CCT,t_cc 0.3450s= 345.0ms

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Time [s]

V(x)

Energy Function

V(x) V_cr t_cc

Figure 4.12: Energy function vs. clearing time for AVR model using the quasi-steady approxim-ate Lyapunov method.

The need for the quasi-steady approximate Lyapunov approach when analysing the transient stability of the VSM was dedicated in Section 4.4.3 to the deteriorating effect of the AVR/RPL loop of the VSM. This was emphasised by the possibility of the back-EMF actually decreasing after the fault is cleared instead of increasing back to the nominal value, and thus contribute to further deteriorate the conditions. The argumentation above is confirmed to be valid for both the analytical system and the simulated system, as will be seen in Section 4.6.6, Figure 4.14c, whereEactually stabilises as low as0.61p.u. after the fault has been cleared, dropping from around0.9p.u. at the time of the fault-clearing. As such, an original Lyapunov method using the initialE of the post-fault system to assess the stability would yield a too optimistic result as the voltage drop of the post-fault system would not be considered. Thus, the theoretical need for the quasi-steady method is well confirmed also by simulation results.

However, before any analytical results were obtained using the quasi-steady method, some assumptions were made for the calculations, and it is therefore of interest to discuss these assumptions in light of the simulation results that will be provided in Section 4.6.6 before discussing the results. The first assumption is related to the PCC voltage being less than0.94p.u.

at the time of fault-clearing. Looking at Figure 4.14c, this assumption is clearly justified with the PCC voltage of both the analytical system and the simulated system dropping below0.9p.u.

0 0.5 1 1.5 2 2.5 3 3.5 4 Time [s]

0 20 40 60 80 100 120 140

Rotor angle [deg]

Stable vs. Unstable System

Stable system Unstable system

s

Figure 4.13:δfor stable and unstable system when using AVR model.

immediately after fault-initiation and continues to decrease during fault.

The second assumption revolves around the post-fault active power returning to0.8p.u. immedi-ately after clearing the fault. This assumption is justified by looking at Figure 4.14d, whereP returns to0.8p.u. after a very short transient reaching higher than0.8p.u. The final assumption of Q_set being limited to0.6p.u., which is truly at the core of the calculations carried out by (4.38), is confirmed looking at Figure 4.14e. Here, it is observed thatQimmediately follows the set-point and settles at0.6p.u. directly after clearing the fault.

As the calculations are shown to be based on well-founded assumptions, the results can be more thoroughly discussed. From Figure 4.12 it is observable that, in contrast to what was the case for the classical model, the critical value of the Lyapunov function,V^cr, is no longer constant but decreases with the clearing time. This is expected as a new critical value is calculated for each new post-faultEresulting from (4.38). The decrease can be explained by the fact that a longer clearing time, i.e. a higherδ, yields a lower post-faultEby (4.38) and thus worsened stability conditions. Thus, it is seen that the stability is limited not so much by the increased value ofV, but rather the rapid decrease in V^cr. This is in accordance with the theory which states that a lowerE^postyields a lowerP_ecurve with a subsequent smaller deceleration area, meaning that the system can go unstable for an angle that the system would have remained stable at if the back-EMF was constant at the pre-fault value.

The found CCT of345.0ms is to a larger extent inside the expected range of what a critical clearing time might be for a typical power system, and is thus deemed more credible than the previously found critical clearing times. Furthermore, when comparing the CCA and CCT in Table 4.8 with the results of the Simulink simulation in Table 4.11 in Section 4.6.7, two observations can be made. Firstly, the deviation from the CCT of the simulated system is only 41.1ms = 10.64% or approximately two cycles. This is quite small considering the

simplifications of the model/procedure, such as neglecting the filter capacitance of the system.

This simplification will however be addressed later. With an error of approximately 2 cycles, the functioning of the method can be said to be very good, clearly taking into account the rapid response of the Reactive Power Loop. It can therefore be argued that a method of analysing the transient stability of a VSM analytically has been established in a satisfactory manner, partly based on classical stability analysis added with modifications to accommodate the distinctive characteristic of the Synchronverter VSM.

Secondly, the stability limits found by the quasi-steady method can be compared to the stability limits of the analytical system as found by the full-forward numerical integration. Such a comparison is of interest as it will provide insight into the quasi-steady method’s performance related to the system it is actually analysing, i.e. the full Synchronverter dynamics without the filter capacitance as modelled in (4.15). Using the results in Table 4.11 in Section 4.6.7 it can be calculated that the discrepancy between the quasi-steady approximate Lyapunov method and the full-forward numerical integration is only7.2ms = 2.04%for the CCT and≈1.5° for the CCA. This is a highly accurate prediction for such a dynamical system, and testifies further to the functioning of the modified, quasi-steady approximate Lyapunov method for analysing the stability of a VSM which is particularly influenced by the reactive power loop, i.e. the Automatic Voltage Regulator (AVR) of the VSM. It can thus be argued that by including the filter capacitance into the analytical model, the deviation towards the simulated system can be further decreased from41.1ms.

Looking at Figure 4.13, it is clear that the responses of the stable and unstable system share many of the characteristics with the responses depicted in Figure 4.11. Both of the unstable power angles quickly accelerates and diverges from the stable state, while both of the stable power angles settles back to the s.e.p.. A notable difference is however that while the stable system in Figure 4.11 settles back from a clearing angle around150° in approximately 2 seconds from the time of clearing, the stable system in Figure 4.13 uses almost 3 seconds to settle back down from a clearing angle of approximately109°. The system in Figure 4.13 is thereby using a longer time to settle back with fewer degrees to go. This means that a side effect of the RPL is not only to worsen the stability conditions, but also to slow down the process of returning the power angle to the s.e.p. after fault-clearing.

A final observation/clarification that should be discussed regarding the quasi-steady method relates to the post-fault equilibrium points. Recalling the procedure explained in Section 4.4.3, each newly calculatedEis used to re-calculate theK’s with subsequent new post-fault equilib-rium points. However, as seen in Figure 4.13, also the dynamics including the AVR will return to the same post-fault e.p. as the other models. This is as expected, due to the AVR eventually bringingEback to its nominal value, yielding the same post-fault e.p. as ifEhad been constant all along. With this in mind, the new post-fault e.p. found, and used, by the quasi-steady method only relates to theE that the VSM settles at immediately after fault-clearing as calculated by (4.38).

As such, this newly calculated e.p. is only used to assess the operating conditions immediately after fault-clearing, and thus has nothing to to with the post-fault e.p. in the long term. This is important to bear in mind if e.g. calculations related to the N-1 criterion are of interest, where the post-fault e.p. could have a large influence on the system’s ability to remain stable for the next contingency. This is however more relevant to other types of system/fault configurations than the one discussed in this thesis.