## 2 Unbalanced Load Effect on Operation of a Low-Power Synchronous Machine  
The flow chart of the investigated computer model, implemented using the Simulink and SimPowerSystems packages of the MATLAB R2022b environment, is shown in Figure 1. The \"synchronous machine\" block was used to simulate the generator of the DG unit, the parameters of which are shown in Table 1. A steam turbine (the \"steam turbine\" block in Figure 1) was used as the primary engine, the mathematical model of which with respect to the shaft power of the turbine and the regulator opening was characterized by the following transfer function:  
\[W(s)=\frac{1}{T_{\text{T}}s+1} \tag{1}\]  
where \(T_{\text{T}}\) is the time constant of the turbine, determined by the lag in the conversion of steam energy into mechanical energy (on average, \(T_{\text{T}}\) is 0.2 s), and \(s\) is a complex variable. The excitation system model was formed based on the equations describing the thyristor converter and the input amplifier [25]. The exciter of the synchronous generator was modeled by the following transfer function, which is typical for a thyristor exciter [26]: \(\frac{1}{0.0025s+1}\). In this case, the model of the excitation system (the \"excitation system\" block in Figure 1) additionally takes into account the amplifier and the limiting block. The block diagram of the excitation system model is shown in Figure 2.  
Figure 1: Flow chart of the MATLAB model of the generator serving an unbalanced load. To regulate the generator voltage and rotor speed, we used the microprocessor-driven ARE and ARS models (the \"MARE\" and \"automatic regulator of rotor speed\" blocks in Figure 1), detailed in [26]. The voltage regulators were set up using the technique described in [27]. The measurements of non-sine waveforms and unbalance were performed using standard MATLAB SymPowerSystems blocks: total harmonic distortion and three-phase sequence analyzer. The generator in the model served an active and inductive load of 1.8 + j0.6 MV-A. The unbalanced currents and voltages were introduced by increasing the load between phases \(a\) and \(b\). The load between phases \(b\), \(c\) and \(c\), \(a\) was reduced by an appropriate amount to maintain total power. The results of computer-aided simulations are summarized in Table 2 and illustrated in Figures 3 and 4. \begin{table}
\begin{tabular}{l l} \hline
**Parameter** & **Value** \\ \hline Power, kVA & 3125 \\ \hline Power factor \(\cos\phi\) & 0.8 \\ \hline Nominal voltage \(U_{\text{NOM}}\), V & 6300 \\ \hline Frequency \(f\), Hz & 50 \\ \hline Synchronous speed, rpm & 1000 \\ \hline Ohmic resistance of the stator \(R_{s}\), per unit & 0.00901 \\ \hline Leakage inductance of the stator \(L_{l}\), per unit & 0.045 \\ \hline Direct axis inductance of the stator \(L_{md}\), per unit & 2.43 \\ \hline Quadrature axis inductance of the stator \(L_{mq}\), per unit & 1.69 \\ \hline Resistance of the field winding \(R_{f}\), per unit & 0.0021 \\ \hline Inductance of the field winding of the rotor \(L_{lfd}\), per unit & 0.506 \\ \hline Direct axis resistance of the damper winding \(R_{kl}\)’, per unit & 0.2998 \\ \hline Direct axis inductance of the damper winding \(L_{lhd}\)’, per unit & 3.742 \\ \hline Quadrature axis resistance of the damper winding \(R_{kql}\), per unit & 0.0254 \\ \hline Quadrature axis inductance of the damper winding \(L_{lql}\)’, per unit & 0.2388 \\ \hline Inertia constant \(H\), s & 2.279 \\ \hline Friction coefficient \(F\), per unit & 0.009 \\ \hline Number of pole pairs \(p\) & 3 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the synchronous machine SGS-14-1-100-6U. Figure 2: Block diagram of the thyristor excitation system model in MATLAB: \(U_{ARE}\)—signal coming from ARE; \(V_{f}\)—signal coming to generator field winding; limit—voltage limitation block.