# State Residualisation and Kron Reduction for Model Order Reduction of Energy Systems
## 4 Application and Results
### Dynamic Simulation Results and Analysis
(**a–j**) the current of the GFMs at bus 39 and 30–38, respectively, as shown in Figure 4.  
Figure 6: \(Cont.\)  
Figure 6: Converter current for GFMs located at buses 30–39 for the full EMT, phasor, and proposed reduced model for a 150 ms short-circuit applied at bus 19 at 1 s under Case 2. (**a–j**) the current of the GFMs at bus 39 and 30–38, respectively, as shown in Figure 4.  
Figure 7: _Cont._  
It is also seen that for other GFMs in Cases 1-3, the reduced models tend to give a faster initial current rise (which is also seen in the phasor models) and tend to slightly underestimate or overestimate the maximum current, which is, however, not critical for respecting converter integrity. The reason for this is because the dynamics of a large part of the network and the \(L_{f}C_{f}\) filter of the less-impacted converters are not modelled. Note that in Cases 1 and 2, the current of G4, and also G5, is greater than the 1.2 pu limit, which is due to the \"soft\" virtual impedance-based current-limiting control, which cannot strictly limit the converter current. Table 4 shows the average absolute error of the peak/valley for the first oscillation cycle of the current for the 10 GFMs relative to the equivalent oscillation for the full EMT model from 1.00-1.01 s for the phasor and state residualisation approaches for Cases 1-3. It is seen for the severe fault in Case 1 that the average error in the phasor model is 3.8 times greater than that for the reduced model; for the less-severe fault in Case 2, the average  
Figure 7: Converter current for GFM located at buses 30–39 for the full EMT, phasor, and proposed reduced model when tripping line B26–29 at 1 s under Case 3. (**a–j**) the current of the GFMs at bus 39 and 30–38, respectively, as shown in Figure 4.  
error for the phasor model is 14.9 times worse than the reduced model; while for line disconnection event in Case 3, the phasor model actually performs better than the reduced model, although both models can be used for Case 3, since Figure 7 shows that the current of the most relevant converter, G9, is accurately represented in both models (although the reduced model performs a little better with more accurate capture of the timing of the current drop and waveforms of the current oscillations), and other generators with small impact are also reasonably represented. Figures 8-10 compare the poles of the linearised EMT, phasor, and reduced model of the faulted systems for Cases 1-3, respectively. The linearised A matrix is obtained by using the _Linearize_ tool in Dymola software, which performs a linearisation based on the real-time operating point at a specific time point. The time point is set immediately after an event is applied,  
Figures 8 and 9 show that the critical poles (i.e., rightmost poles that dominate the system transients) of the reduced models are well overlapped with those of the full EMT model, while the phasor models fail to represent the unstable poles 0.3 (participation factor calculations show that the two unstable poles are mainly contributed by \(\delta_{\mathrm{rgsm}}\) and \(p_{m}\) of G3 and G4 (\(\geq\)95%)) and critical oscillation modes. Figures 8 and 9 also show that the state residualisation reduced model in Case 2 appears to better preserve the critical poles over Case 1, as in Case 1 the pole pair \(-2.4\pm j\)0.42 is not matched and extra poles \(-0.3\) and \(-1.15\) are added, which is also reflected in the time simulation results in Figures 5 and 6. Figure 10 shows that both (state residualisation and phasor) reduced models miss some critical poles, e.g., \(-1.64\pm j\)0.0075 and \(-1.645\), as seen in Figure 10c. However, these dominant poles are seen to be very close with each other (within a 0.01 distance difference) and hence both models can be used, which is also confirmed by the time domain results in Figure 7 as the current of G9 with the severest impact is accurately represented in both models. Figures 8a, 9a, and 10a all show that the reduced model keeps the most-left poles, which, however, failed in the phasor model.