## 3 Modelling and Simulation
### Mode #1 Harmonic Supply and Linear R-L Load
#### 3.1.4 Comparison of Power Components in 2-,3- and 5-Phase Supply Systems under RL Load  
Apparent, active, blind, and distortion power components during the transient state for 3-, 5-, and 2-phase systems are presented in Figure 11a-d. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Time** & \(\mathbf{S_{av}}\) **[VA]** & \(\mathbf{P_{av}}\) **[W]** & \(\mathbf{Q_{av}}\) **[Var]** & \(\mathbf{D_{av}}\) **[VAd]** & \(\mathbf{PF}\) **[–]** & \(\mathbf{A_{1}}\) **[A]** & \(\mathbf{\cos\varphi_{1}}\) **[W]** \\ \hline t = 0.1 s & 4600 & 3680 & 2760 & 0 & 0.8000 & 14.14 & 0.8000 \\ t = 0.2 s & 6371 & 3537 & 5307 & 0 & 0.5546 & 19.60 & 0.5546 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Classical calculation before and after the change (at steady states). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Time** & \(\mathbf{U_{a,rms}}\) **[V]** & \(\mathbf{U_{b,rms}}\) **[V]** & \(\mathbf{I_{a,rms}}\) **[A]** & \(\mathbf{I_{b,rms}}\) **[A]** & \(\mathbf{A_{1}}\) **[A]** & \(\mathbf{\varphi_{1}}\) **[\({}^{\circ}\)el.]** & \(\mathbf{\cos\varphi_{1}}\) **[–]** \\ \hline t = 0.1 s & 229.8 & 229.8 & 9.991 & 9.991 & 14.13 & 36.88 & 0.7947 \\ t = 0.2 s & 229.8 & 229.8 & 13.85 & 13.85 & 19.60 & 56.31 & 0.5547 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Simulation results before and after the change (at steady states). Figure 10: Phase voltages (**a**) and load currents (**b**) in \(\alpha,\beta\) coordinates. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Time** & \(\mathbf{S_{av}}\) **[VA]** & \(\mathbf{P_{av}}\) **[W]** & \(\mathbf{Q_{av}}\) **[Var]** & \(\mathbf{D_{av}}\) **[VAd]** & \(\mathbf{PF}\) **[–]** & \(\mathbf{THD_{i}}\) **[\%]** & \(\mathbf{P_{1 av}}\) **[W]** & \(\mathbf{P_{0 av}}\) **[W]** \\ \hline t = 0.1 s & 4591.9 & 3673.1 & 2755.7 & 0 & 0.7999 & 0.0 & 3649.3 & 0 \\ t = 0.2 s & 6366.7 & 3531.5 & 5297.5 & 0 & 0.5547 & 0.0 & 3533.3 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Simulation results before and after the change (at steady states). The simulation was performed under the same load phase parameters; therefore, the magnitudes are proportionately different. The waveforms of all power components per unit are very similar, but the interesting is that during the transient response, also a power distortion component is generated, Figure 11d. It is due to different arises of active and reactive (blind) power components (Figure 11b,c). Thus, during the transient state (approx. 3-4-time constants), the spectral density comprises different higher harmonics, which could cause an additional negative influence on the system. The distortion power calculated by three different methods (MAF, DFIR, and DIIR) for a 3-phase system is presented in Figure 12a. The time dependency of the corresponding current \(I_{d,rms}\) for the 3-phase system is shown in Figure 12b. In this simply case-harmonic supply, linear load--we can verify a time course \(I_{\text{DRms}}\) during transient using another method than \(p\)-\(q\) one. By decomposition of fundamental harmonic current into active and reactive (blind) components, we can calculate the RMS values of these currents. Knowing the RMS value of total harmonic fundamental, the RMS deformation component of the current can be determined as well as in transient states. Figure 11: Apparent (**a**), active (**b**), blind (**c**), and distortion (**d**) power components during transient states for 3-, 5-, and 2-phase systems under mains supplied linear RL load. So, knowing \(P_{av}(k)\) and \(Q_{av}(k)\) we get  
\[ \begin{bmatrix}i_{a}(k)\\ i_{\beta}(k)\end{bmatrix} =\frac{2}{3}\frac{1}{D} \begin{bmatrix}u_{a}(k)&-u_{ \beta}(k)\\ u_{\beta}(k)&u_{a}(k)\end{bmatrix}\begin{bmatrix}P_{av}(k)&p_{ac}(k)\\ Q_{av}(k)&q_{ac}(k)\end{bmatrix}  \tag{32}\]  
where \(p_{ac}(k)\) and \(q_{ac}(k)\) are calculated as (13 ab) using the moving average method.