# State Residualisation and Kron Reduction for Model Order Reduction of Energy Systems
## 4 Application and Results  
### 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, \(S_{b}\) and \(V_{b}\). For the grid, \(S_{b}=100\) MVA, while \(V_{b}\) 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.  
Figure 4: (**a**) IEEE 39-bus system with the 10 synchronous generators replaced by grid-forming converters. (**b**) schematic diagram of reduced model after implementing state residualisation and Kron reduction technique for Cases 1 and 2, with the fault applied at bus 19.  
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 \(E\) 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 \(E\) 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.