4.1. General Structure of Modified IEEE 39-Bus System
The IEEE 10-generator, a 39-bus system [
35] with 10 synchronous generators replaced by grid-forming converters, is considered, as shown in
Figure 4a. Each GFM is modelled based on its defined base values,
and
. For the grid,
, while
is the same across the system. All equations for the electrical components are described in
Section 2. The physical and control parameters of the GFMs are the same as in [
4,
29], as shown in
Table 1, while the remaining parameters are the same as in [
35]. Note that the maximum current for the GFMs is tuned to be 1.2 pu, by incorporating virtual impedance current-limiting control.
Two short-circuits and one line-tripping event are simulated, and the state residualisation method is applied to the system in each case. The short-circuit is modelled as a shunt resistance, which switches from a very large to a very small value when the event is triggered. The resistance that creates the short-circuit is considered as the input for calculating the residualisation matrix in (42). Similarly, the line-tripping event is modelled by a series connected resistance that becomes very large when the event happens, with the resistance again considered as the residualisation matrix.
In order to demonstrate that the proposed technique achieves more accurate transient simulations than the traditional phasor approximate model, along with reducing the simulation time, the following cases are created:
Case 1: A bolted three-phase short-circuit is applied at bus 19 at 1 s, and cleared at 1.15 s.
Case 2: The same as Case 1, but the short-circuit is 10 times less severe, and the length (and series and shunt impedance) of line B16–19 is increased by a factor of 18 (a sufficiently large scaling that ensures that the system remains stable for different events and disturbances), so as to focus the impact on generators G4 and G5.
Case 3: Line B26–29 is disconnected at 1 s. In order to create a noticeable impact on the nearby generators, B26–29 is shortened by a factor of 10, while line B28–29 is lengthened by a factor of 10.
Different from [
25], where only one converter is observed, the currents of all converters in the system are observed as outputs, as part of determining the residualisation matrix
in (42), noting that the location of an event is not known a priori. Since the system topologies are different under the three events, the state residualisation needs to be determined separately in each case.
Here, the converter currents are chosen as the observed output variables, since (1) converters have a much lower overcurrent capability relative to synchronous machines, and (2) converter fast transient currents need to be captured with good accuracy to ensure converter integrity.
4.2. Application of State Residualisation Technique
The state categorisation described in
Section 3.1.2 is applied to the three test cases, with
selected as 0.75 for the short-circuit events in Cases 1 and 2, and 0.55 for the line-tripping event in Case 3, which results in three different grouping formulas, as shown in
Table 2. The principle for choosing
is not overly critical, but it should be based on the severity of an event, such that the less severe the event in question, the smaller
should be, since for a less-severe event the states will be more likely to have equal participation in a mode. Hence, choosing a smaller
for a less-severe event can disjoint the system states and separate them into different groups. In this way, the state variables that are most relevant to an event are grouped into individual groups, while the remainder are grouped into a few larger groups. In general, simply choosing
as 0.75 for short-circuits, and as 0.55 for line-tripping, leads to satisfactory grouping.
Subsequently, modal state residualisation of (42) in
Section 3.1.3 is applied, by choosing the desired size
as equal to the phasor approximate model. Three reduced models are obtained, which are summarised in
Table 3. For example, for Case 1, the variables to be retained are almost the same as for Case 2, except for the remaining GFMs, which are less impacted by the short-circuit. The result is not surprising given that the short-circuit is applied at the same bus, and the topology of each system is very similar. However, for Case 3, as the system topology is more noticeably different from the previous two cases, the variables to be retained are different.
It is seen that state residualisation helps to identify zones to reduce, while high-level details are retained elsewhere, depending on the simulated event. It is confirmed that those parts of the system that are located far from the simulated event are generally modelled with fewer details than those parts that are located nearby.
4.3. 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
Figure 5,
Figure 6 and
Figure 7, respectively, noting that a different reduced model should be used for each simulated event.
Figure 5,
Figure 6 and
Figure 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 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.
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 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 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.
Figure 8,
Figure 9 and
Figure 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, and hence the impact of the simulation event as an input can be captured in the linearised model. Note that the aim of presenting
Figure 8,
Figure 9 and
Figure 10 is not to conduct detailed small-signal analysis but to show the match degree of the reduced and phasor models to the full EMT model.
Figure 8 and
Figure 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
and
of G3 and G4 (
95%)) and critical oscillation modes.
Figure 8 and
Figure 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
is not matched and extra poles −0.3 and −1.15 are added, which is also reflected in the time simulation results in
Figure 5 and
Figure 6.
Figure 10 shows that both (state residualisation and phasor) reduced models miss some critical poles, e.g.,
and
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.
Figure 8a,
Figure 9a, and
Figure 10a all show that the reduced model keeps the most-left poles, which, however, failed in the phasor model. It, with the above results of preserving critical poles and adding some extra pole, thus illustrates that the reduced model is able to keep the more energetic poles, through achieving the target, (51), of minimising the energy error for each the simulated event input to the observed variables output.
Three main conclusions can be drawn from the above dynamic simulation results. Firstly, the most appropriate reduced model depends on the location and nature of the simulated event. Secondly, by making the model order the same, state residualisation can capture event transients with better accuracy than phasor approximation models, particularly for short-circuit events where converter overcurrents are of most concern. Finally, for short-circuits, the phasor approximation model does not achieve good accuracy, and it is particularly concerning that the fast initial transients (very important for power converters) may be missed.