## 2 Unbalanced Load Effect on Operation of a Low-Power Synchronous Machine
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Parameter** & \multicolumn{6}{c}{**Value**} \\ \hline \(k_{2l}\), \% & 0 & 2.99 & 5.97 & 8.97 & 12.0 & 15.0 & 18.0 \\ \hline \(k_{2l}\), \% & 0 & 0.61 & 1.21 & 1.81 & 2.42 & 3.02 & 3.63 \\ \hline \(k_{lla}\), \% & 0.001 & 0.33 & 0.66 & 0.99 & 1.31 & 1.62 & 1.93 \\ \hline \(k_{llb}\), \% & 0.001 & 0.34 & 0.67 & 1.01 & 1.35 & 1.70 & 2.04 \\ \hline \(k_{llc}\), \% & 0.001 & 0.34 & 0.68 & 1.02 & 1.37 & 1.72 & 2.07 \\ \hline \(l_{f}^{\text{(mid)}}\), per unit & 0.79 & 0.79 & 0.79 & 0.79 & 0.79 & 0.79 & 0.79 \\ \hline \(l_{f}^{\text{(max)}}\), per unit & 0.79 & 0.80 & 0.81 & 0.82 & 0.84 & 0.85 & 0.86 \\ \hline \(l_{mf}\), per unit & 0 & 0.011 & 0.023 & 0.035 & 0.046 & 0.057 & 0.069 \\ \hline \(l_{A}\), per unit & 0.430 & 0.44 & 0.449 & 0.453 & 0.461 & 0.469 & 0.477 \\ \hline \(l_{B}\), per unit & 0.430 & 0.44 & 0.441 & 0.447 & 0.453 & 0.459 & 0.466 \\ \hline \(l_{C}\), per unit & 0.430 & 0.42 & 0.404 & 0.391 & 0.378 & 0.365 & 0.352 \\ \hline \(dl\), \% & 0 & 2 & 4.5 & 6.2 & 8.3 & 10.4 & 12.5 \\ \hline \hline \end{tabular} Note: \(k_{2l}\)—unbalance ratio of negative sequence currents in the stator, \(\%\); \(k_{2l}\)—unbalance ratio of negative sequence voltages, \(\%\); \(k_{lle}\), \(k_{llc}\)—total harmonic distortion of voltages, \(\%\); \(l_{f}^{\text{(mid)}}\), \(l_{f}^{\text{(max)}}\)—mean and maximum values of excitation current, per unit; \(l_{mf}\)—amplitude of forced oscillations with a frequency of 100 Hz; \(l_{A}\), \(l_{B}\), \(l_{C}\)—effective values of stator currents, per unit; \(dl\)—maximum phase current difference, \(\%\).  
\end{table} 
Table 2: Experimental data for the case of the generator model in MATLAB serving a unbalanced load. Figure 3: Waveforms of stator currents (**a**) and rotor currents (**b**) of the generator at \(k_{2l}=11.96\%\). The average excitation current of the generator serving unbalanced loads was practically unchanged. However, as a result of the unbalance, currents with a frequency of 100 Hz were induced in the field winding, and the amplitude of forced current oscillations in the field winding increased linearly (Figure 4c). This is due to the fact that the unbalanced currents of the three-phase stator winding create an elliptical field, which is  
Figure 4: Generator load flow as a function of the current unbalance ratio \(k_{2I}\). (**(a)**—\(k_{U}=f\) (\(k_{2I}\)), \(k_{2U}=f\) (\(k_{2I}\)); (**b)**—\(I=f\) (\(k_{2I}\)); (**c)**—\(I_{f}=f\) (\(k_{2I}\))). equivalent to the sum of two fields rotating at the same speed in opposite directions. The field component corresponding to the negative sequence currents rotates relative to the rotor at double synchronous speed and induces double frequency currents in the rotor circuits, which leads to additional losses. The growth of the current unbalance ratio \(k_{2I}\) was accompanied by an increase in currents in individual phases and the level of voltage unbalance at the terminals of the synchronous generator, as well as the appearance of harmonic distortions. These circumstances must be taken into account when integrating DG units into RPSSs. Failure to take these factors into account can lead to rapid wear and tear of electrical equipment and even to breaking down of generators. Studies were also carried out on a physical model of a low-power turbogenerator. A schematic and photo of the experimental setup are shown in Figure 5. The turbogenerator model was realized as coupled electric machines. The main nominal parameters of the electric machines of the physical model were as follows (Figure 2):  
1. Synchronous machine: - Active power \(P=100\) W; - Power factor \(\cos\!\varphi=1\); - Stator winding voltage \(U=220\) V; - Nominal rotor speed \(n=1500\) rpm. 2. Turbine modeled as a direct current machine: - Power \(P=90\) W; - Armature voltage \(U_{a}=220\) V; - Field winding voltage \(U_{f}=220\) V; - Rotation speed \(n=1500\) rpm.