## 5 Simulation Results  
An MMC with a few levels is sufficient for a proof-of-concept of the deadbeat predictive current control, then, the simulation model of the SRPC consists of twenty-four SMs per each leg (13-levels), and it was developed under _PSIM_ v9.1 simulation tool, where the main parameters are presented in Tables 1 and 2. The main objective is to perform a comparative study when using the deadbeat predictive control for circulating current reduction in MMC (presented in Figure 6a), and when using the conventional PI controller (presented in Figure 6b) to track the reference signals of \(i_{rx}\)', and \(i_{ry}\)'. In addition, and in order to show the advantages of the presented deadbeat predictive current control methodology, simulation results are presented under transient load conditions. The value of the MMC arm inductor directly influences the control system in terms of capability to track the compensation current references. As the current ripples depend on the voltage applied on the MMC arm inductor, the value of the inductor and the time that the voltage is applied, the MMC arm inductor can be designed by considering the factors of current ripples suppression, and the speed of signal tracking. A low value of capacitance in the SM implies high voltage ripples. However, a high value of capacitance implies costly and bulky MMC. In this context, the previous aspects were \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Power grid voltage & \(U_{AB}\) & 110 kV \\ Traction grid voltage & \(U_{\phi}\); \(\sigma\in\{x,y\}\) & 25 kV \\ Fundamental frequency & \(f\) & 50 Hz \\ Section \(x\) load power & \(P_{Lx}\) & 1.75 MW \\ Section \(y\) load power & \(P_{Ly}\) & 2.62 MW \\ V/V transformer turns ratio & \(K_{V}\) & 110:25 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation model parameters. Figure 8: Command generation for each two-level SM applied to a phase-shifted PWM. considered in the selection process of the SM capacitance and the MMC arm inductors, where the values presented in Table 2 were selected based on a trade-off between the size, cost, current ripples and voltage ripples. Since the power factor of the electric locomotives in high-speed trains is close to unitary and the current harmonics are quite small [26], the high-speed railway electric locomotive is modeled as a resistive-inductive load in parallel with an uncontrollable rectifier bridge connected to the secondary side of the locomotive transformer, as shown in Figure 9[26]. The output of the uncontrollable rectifier bridge is connected to resistor \(R_{2}\) and inductor \(L_{2}\), which are in series. The turns ratio of the locomotive transformer is 25:1.5. The locomotive model parameters are selected as \(R_{1}=2\)\(\Omega\), \(R_{2}=3.3\)\(\Omega\) and \(L_{2}=0.8\) mH. At these values, section \(x\) load power is near 1.75 MW. Section \(y\) load power is chosen to be 150% higher than section \(x\) load power, as presented in Table 1.  
This case study presents the results for two scenarios: (a) When using the deadbeat predictive current control methodology; (b) When replacing the Equations (23)-(26) in Figure 6a by conventional PI controllers, as shown in Figure 6b, to correct the error signals of \(i_{rx}^{*}\ -\ i_{rx}\) and \(i_{ry}^{*}\ -\ i_{ry}\), as explained in [6]. The results for both cases are presented after considering the same control parameters (similar PI parameters) of the MMC DC-link voltage presented in Figure 6, averaging voltage balancing control (for each MMC leg) presented in Figure 7a, and the individual voltage balancing control (for each SM) presented in Figure 7b. Figure 10a shows the three-phase currents before compensation. The currents are imbalanced and have NSC of currents. On the other hand, Figure 10b presents the same currents after the SRPC compensates reactive power and shifts half of the active power difference between catenary sections. At that case, the three-phase currents are balanced and sinusoidal.