## 4 Proposed LabVIEW Multisim Co-Simulation Platform
### Open-Loop Results of MMC Co-Simulation  
The open-loop NLC-based algorithm developed in LabVIEW for co-simulation is shown in Figure 4.  
The inserted number of SMs can be calculated by Equations (13) and (14). The round function \(round_{0.5}\) (x) will round the real number x to the nearest whole number according to the decimal fraction of x. If the decimal fraction of x is greater than 0.5, x is rounded up to the next whole number, or else it is rounded down to the next whole number. To understand the switching states, two cases [t\({}_{1}\) to t\({}_{2}\), t\({}_{2}\) to t\({}_{3}\)] are analyzed and shown in Figure 3b. In the first case [t\({}_{1}\) to t\({}_{2}\)], assuming \(V_{L}^{step}=MV_{d}\), then the reference values of the arm voltages and equivalent inner voltage of the phase at t = t\({}_{1}\) can be shown as Equations (15) and (16), respectively.  
\[\left\{\begin{array}{c}V_{L}^{ref}=(M+0.5)V_{d}\\ V_{U}^{ref}=[(N-M-1)+0.5]V_{d}\end{array}\right. \tag{15}\]  
\[V_{E}^{ref}=(M-0.5N+0.5)V_{d} \tag{16}\]  
In the first scenario, the step waves of the arm voltages and equivalent inner voltage are expressed as  
\[\left\{\begin{array}{c}V_{L}^{step}=MV_{d}\\ V_{U}^{step}=(N-M)V_{d}\end{array}\right. \tag{17}\]  
\[V_{E}^{step}=(M-0.5N)V_{d} \tag{18}\]  
Figure 4: Open-loop control algorithm development in LabVIEW for the MMC.  
In the second scenario, from \(t_{2}\) to \(t_{3}\), the reference values of the arm voltages and equivalent inner voltage are expressed as  
\[\left\{\begin{array}{l}V_{L}^{ref}=[(M-1)+0.5]V_{d}\\ V_{U}^{ref}=[(N-M)+0.5]V_{d}\\ \end{array}\right. \tag{19}\]  
\[V_{E}^{ref}=(M-0.5N-0.5)V_{d} \tag{20}\]  
The step waves of the arm voltages and equivalent inner voltage are shown as  
\[\left\{\begin{array}{l}V_{L}^{step}=(M-1)V_{d}\\ V_{U}^{step}=(N-M+1)V_{d}\\ \end{array}\right. \tag{21}\]  
\[V_{E}^{step}=(M-0.5N-1)V_{d} \tag{22}\]  
Comparing Equations (18) and (22), it can be observed that the step height in \(V_{E}^{step}\) is \(V_{d}\). Since the positive and negative DC voltage limits are \(\pm 0.5V_{dc}\), the maximum level in the equivalent inner voltage is equal to \(V_{dc}/V_{d}\) +1.  
The solver methods recommended for co-simulation are R-K 23 and 45 ODE solvers since they can auto-adjust the rate of the loop according to the complexity of the circuit. The MMC model was designed in the NI Multisim software, and a half-bridge SM topology was considered for co-simulation. The sub-circuit for each arm was made as shown in Figure 5. An open simple RL load is considered for co-simulation.  
Each arm is an MMC that is connected by inductors to reduce the current change when all the SMs in the arm are bypassed and to limit the fault current. In Multisim, on-page connectors are used to distribute the signals coming from LabVIEW since only 24 inputs/outputs are allowed in LabVIEW Multisim co-simulation; however, they can be increased using decoders. The MMC circuit designed in Multisim is shown in Figure 6.  
The open-loop MMC co-simulation was completed using an NLC control algorithm to obtain the results, and a simple RL load was considered. Capacitor voltage balancing was not considered. The output voltage waveform and current are shown in Figure 7.  
Figure 5: Sub-circuit of each arm in National Instruments (NI) Multisim.  
Figure 6: MMC circuit designed in NI Multisim.  
Figure 7: Three-phase 6-level output (**a**) voltage and (**b**) current waveforms of the MMC.