# State Residualisation and Kron Reduction for Model Order Reduction of Energy Systems
## 4 Application and Results
### Dynamic Simulation Results and Analysis  
For each case, three mode types are compared using time domain simulations: full order EMT model (order 308), phasor model (order 130, with the network modelled as algebraic equations), and reduced-order models based on the state residualisation results summarised in Table 3.  
The simulations are performed using the Modelica [36] language as implemented using Dymola 2022 software. Modelica is very convenient when simulating DAE systems and for state residualisation, as the model equations are directly written. A variable integration time step is applied, such that a larger step size can be used when state variables representing fast dynamics are residualised (i.e., turned into algebraic variables). The integration tolerance is set as 0.0001. For Cases 1-3, comparison simulation results of the 10 GFM currents are shown in Figures 5-7, respectively, noting that a different reduced model should be used for each simulated event. Figures 5-7 show that by choosing the model order to be the same as the phasor approximate model, that the reduced models better capture the GFM transients (as they retain the most relevant dynamics) over the RMS models. More precisely, the reduced models accurately simulate the GFMs that are the most impacted by the short-circuit and line-tripping events, i.e., G4 and G5 for Cases 1 and 2, and G9 for Case 3, while the phasor models miss the initial current peaks, and so would suggest that the converter current was within acceptable limits, when, in fact, it was not. For the other GFMs in Cases 1 and 2, the reduced models also capture the transients of the first peak current of the converters with better accuracy than the phasor approximated models, except G6 in Case 1 where the first peak current of G6 is slightly lower than with the phasor model. It is seen that the former correctly predict the trends of the first oscillation of the converter's current with much smaller first peak errors, while the latter sometimes gives opposite directions for the first oscillations (for example for G1, G8 and G9 in Case 1) and tends to have a larger error for the first peak (e.g., the error is 0.1-0.3 pu for G2, G3, G7, and G10 in Case 1). Similarly, in Case 3, for the other GFMs, i.e., not G9, the low-frequency current oscillations in the full \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **Retained State Variables (The Remainder Are Residualised)** & **MOR Order** \\ \hline \multirow{4}{*}{Case 1} & All state variables for G4 and G5, except \(q_{m}\) for G5 & \\  & \(\delta_{\text{bsm}}\) and \(q_{m}\) for the remaining GFMs & 53 \\  & Branch B16–19, transformers T20–34, T20–34 and T19–20, load L20 & \\ \hline \multirow{4}{*}{Case 2} & All state variables for G4 and G5, except \(q_{m}\) for G5 & \\  & \(\delta_{\text{ssm}}\), \(p_{m}\), \(q_{ADd}\), \(M_{ADp}\), \(\xi_{d}\), \(\xi_{d}\), \(\sigma_{d}\), \(\sigma_{q}\) for the remaining GFMs & 101 \\  & Branch B16–19, transformers T20–34, T20–34 and T19–20, load L20 & \\ \hline \multirow{4}{*}{Case 3} & All state variables for G9 & \\  & \(\delta_{\text{ssm}}\), \(p_{m}\), \(q_{m}M_{ADd}\), \(M_{ADp}\), \(\xi_{d}\), \(\xi_{d}\), \(\sigma_{d}\), \(\sigma_{q}\) for the remaining GFMs & 112 \\ \cline{1-1}  & Branches B26–29, B26–28, B28–29, transformer T29–38, loads L29, L28 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimal reduced-order model obtained by using state residualisation for the different events. EMT model are better matched by the reduced model over the phasor model, although they both miss the first peaks, as seen in Figure 7, where in the phasor model the phases of the low-frequency oscillations are almost opposite to those in the full EMT model. Figure 5: \(Cont.\)  
Figure 5: 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 1.