Results of Enhanced Control Loops: Power Correction Loop

Before adding the PCL to the system, the coefficient M must be set based on (5.4). For this thesisM is set so that the active power set-point is reduced by25%of base power for a100%

reduction in grid voltage,∆V =V_b. Using this,M can be calculated as in (5.17).

M = 0.25S_b

ω_nV_b = 1.4125 (5.17)

Furthermore, the voltage threshold activating the PCL is set toV_set = 0.5p.u.. WithM = 1.4125, the reduction in the active power set-point for the given reduction in grid voltage,∆V = 0.9p.u., is calculated to be∆P_m,corr = 225kW= 0.225p.u.. The PCL is added to the grid side controls in the Simulink model as depicted in Figure 5.1, and the Simulink model of the controller added with the PCL can be found in Appendix C.2, Figures C.9 and C.10.

Using the quasi-steady approximate Lyapunov approach on the system added with the PCL, the critical clearing time is found analytically to be t_cc = 0.5663s, with a corresponding critical clearing angle δ_cc = 108.3255°. Graphic illustrations of the analytic results are provided in Figure 5.5, where a clearing time equal to the CCT and a clearing time15ms above the CCT are used in Figure 5.5b.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Time [s]

V(x)

Energy Function

V(x) Vcr t_cc

(a) Energy function vs. clearing time for system added with PCL.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time [s]

0 20 40 60 80 100 120 140

Rotor angle [deg]

Stable vs. Unstable System

Stable system Unstable system

s

(b)δfor stable and unstable analytical system added with PCL.

Figure 5.5: Analytical stability assessment of system added with PCL.

Furthermore, full forward numerical integration of the system added with the PCL yielded a critical clearing time equal to t_cc = 0.5801 s, with a corresponding critical clearing angle δ_cc = 110.0725°. To see how the the PCL affects the dynamic response, and to find the actual CCT, the system added with the Power Correction Loop was simulated using the same procedure as in Section 4.6.6 until the actual CCT was obtained. Simulation results yielded at_cc = 0.6444 s, with a corresponding critical clearing angleδ_cc = 112.4382°, and the stability limits as found by the stability analysis of the system added with the PCL are summarised in Table 5.1.

Table 5.1: Stability limits for system added with power correction loop, using different types of assessment methods.

Method CCA CCT ∆CCT[%]

Quasi-steady approx. Lyapunov 108.3255° 0.5663s= 566.3ms 12.12%

Full Forward Integration 110.0725° 0.5801s= 580.1ms 9.98%

Simulink Simulation 112.4382° 0.6444s= 644.4ms 0%

Using the quantification methods in (5.1) and (5.2), the improvement in the stability limit,I, and the stability margin,M^stab, as a result of the added power correction loop can be calculated as in (5.18a) and (5.18b). The results are further summarised for easy identification in Table 5.2.

I =

0.6444−0.3861 0.3861

·100% = 66.9% (5.18a)

Mstab =

0.6444−0.3861 0.6444

·100% = 40.08% (5.18b)

Table 5.2: Quantified stability improvement for PCL system.

Parameter Value

Stability limit improvement,I 66.9%

Stability margin,Mstab 40.08%

As previously explained, the PCL has the major advantage of easy implementation in both the quasi-steady approximate Lyapunov method and the full-forward numerical integration, meaning analytical stability analysis can be carried out without heavy re-modelling of the analytical equations. This is possible because the PCL does not affect steady-state operation, nor the post-fault system, as the loop will only be active when the fault is on.

Implementing the PCL does however require additional hardware equipment, such as additional sensors and communication devices, to provide the control system with the grid voltage. This could potentially increase the CAPEX of a project by some minor amount, but will in most cases be far less expensive than installing e.g. braking resistors. Another factor is the potential delays that might occur in communications when the grid voltage is being communicated over longer distances. These delays are not taken into account in the analysis performed here but should be considered for a real application.

Looking at the obtained results in Figures 5.5a and 5.5b, many of the same observations that were pointed out for the original system also holds for the system added with the PCL. As such, they will not be treated in detail again, but it can be noted that the significant decrease in the critical value of the Lyapunov function can still be seen in Figure 5.5a, and that the stable system in Figure 5.5b is asymptotically stable.

When added with the power correction loop, the critical clearing time of the simulated system is improved by258.3ms compared to the original system, which yields an improvement in the stability limits, I, equal to66.9% as seen from Table 5.2. Furthermore, the stability margin M^stabis found to be40.8%, and the PCL thus achieves the first of the two objectives by indeed making the original system more stable. From the results of the stability analysis in Table 5.1 it is evident that even with the added power correction loop, the quasi-steady method provides adequate estimates for the system stability, having a deviation of 12.12% from the CCT of the Simulink model. The full-forward numerical integration has, as expected, a slightly better accuracy deviating by only9.98%. It can thus be concluded that the analytical analysis is still feasible for the system added with a PCL.

The critical clearing angles listed in Figures 5.1 provides some interesting insight into the system characteristics for both the Simulink model and the analytic model. When comparing the CCA obtained by each method in Table 5.1 to their respective counterparts in Table 4.11, they are found to be practically identical with the largest difference being0.2330° for the full forward integration method. This means that the CCA of the system added with a PCL is the same as for the original system but with a66.9%improvement in the CCT. This observation can be seen in relation to the fact that the PCL affects the faulted system by decreasing the power imbalance yielding a smaller speed deviation∆ω, while the available post-fault deceleration area remains unchanged. As such, it takes a longer time to inject the same amount of kinetic energy. With a smaller speed deviation,δuses a longer time to reach the CCA, and thus the CCT is increased.

The response of the system added with the PCL when pushed to its critical clearing time will be discussed in detail in Section 5.3.4. It can however be mentioned already here that the system response is to a large extent identical to what was seen for the original system, only with a longer clearing time. This is especially true looking at Figure 5.11i, where the problem of the excessively high post-fault current is still present when the system is pushed to its limits, thus making the PCL system fail the second objective. It is however reasonable to believe that if simulated for a clearing time equal to the CCT of the original system, the response would indeed be far better, drastically reducing the operating time with a current above 1 per unit.

As a final remark it can be noted that the power correction coefficientM was set to reduce the active power set-point by 25% of base power for a 100% reduction in the grid voltage, yielding a reduction of0.225p.u. for the fault investigated. From the response in Figure 4.14d, it can be deduced that the difference between the set-point of0.8p.u. and the injected power is approximately0.5p.u. during fault for the original system. Based on this deduction, it can be argued thatM could have been tuned more aggressively, e.g. reducing the set-point closer to 0.4p.u. for the same fault. This could potentially increase the stability limits beyond what was achieved here.

The response of the simulated system equipped with the PCL when fault is cleared att_cc = 0.6444 is depicted in Figures 5.11 and 5.12 in Section 5.3.4, where the response is compared to the responses of the system equipped with a virtual resistor and the system equipped with both a virtual resistor and damper windings.