## 2 Mathematical Description of the Power Supply System  
The calculating algorithm for the dynamic modes of the power supply system is multi-stage [12; 13]. At first, the nodal voltages calculation [14] is performed using equations in a single generalized form of Equation (1). Then, the differential equations are solved, and the currents are found. The procedure is repeated at each step of the numerical integration of differential equations. \[p\mathbf{I}_{i}=-\mathbf{A}_{i}\mathbf{U}_{i}-\mathbf{B}_{i}\mathbf{I}_{i}- \mathbf{H}_{i}, \tag{1}\]  
where \(\mathbf{I}_{i}\) is a currents vector of element \(i\), \(p\mathbf{I}_{i}\) is a currents derivative vector of element \(i\), \(\mathbf{U}_{i}\) is the vector of voltages applied at the terminals of the element \(i\), \(\mathbf{A}_{i}\) and \(\mathbf{B}_{i}\) are matrices whose dimension depends on the coordinate system in which the structural element is modeled, as well as on whether these equations are complete or simplified, and \(\mathbf{H}_{i}\) is the vector of impact on the element \(i\)[15]. The main models of elements in the transformed coordinates \(\mathrm{d}\) and \(\mathrm{q}\) of the PSS are shown (the generally accepted notations are used when writing systems of mathematical description):  
* Main power system: \(\mathbf{A}=\left( \begin{array}{cc}1/x_{PS}&0\\ 0&1/x_{PS}\end{array} \right)\), \(\mathbf{B}=\left( \begin{array}{cc}0&-\omega\\ \omega&0\end{array} \right)\), \(\mathbf{H}=\left( \begin{array}{c}-E_{d}/x_{PS}\\ -E_{q}/x_{PS}\end{array} \right)\);
* Synchronous generator (SG): \[\mathbf{A}=\left[\left( \begin{array}{cc}x_{d}&0\\ 0&x_{d}\end{array} \right)-\left( \begin{array}{cc}x_{ad}&x_{ad}&0\\ 0&0&x_{aq}\end{array} \right)\left( \begin{array}{cc}x_{f}&x_{ad}&0\\ x_{ad}&x_{D}&0\\ 0&0&x_{Q}\end{array} \right)^{-1}\left( \begin{array}{cc}x_{ad}&0\\ x_{ad}&0\\ 0&x_{aq}\end{array} \right)\right]^{-1},\] \[\mathbf{B}=\mathbf{A}\times\left( \begin{array}{cc}r&\omega x_{q}&-r_{f} \frac{x_{D}x_{ad}-x_{ad}^{2}}{x_{f}x_{D}-x_{ad}^{2}}&\omega x_{aq}\\ -\omega\;x_{d}&r&-\omega x_{ad}&-r_{Q}\frac{x_{aq}}{x_{Q}}\end{array} \right), \mathbf{H}=\mathbf{A}\times\left( \begin{array}{c}\frac{x_{D}x_{ad}-x_{ad}^{2} }{x_{f}x_{D}-x_{ad}^{2}}\\ 0\end{array} \right)U_{f};\]
* Power transformer (T): \(\mathbf{A}=-\left( \begin{array}{cc}1/x_{T}&0\\ 0&1/x_{T}\end{array} \right)\), \(\mathbf{B}=\left( \begin{array}{cc}r_{T}/x_{T}&-\omega\\ \omega&r_{T}/x_{T}\end{array} \right)\), \(\mathbf{H}=0\);
* Cable (CL) and overhead (OL) power lines (PL): \(\mathbf{A}=-\left( \begin{array}{cc}1/x_{PL}&0\\ 0&1/x_{PL}\end{array} \right)\), \(\mathbf{B}=\left( \begin{array}{cc}r_{PL}/x_{PL}&-\omega\\ \omega&r_{PL}/x_{PL}\end{array} \right)\), \(\mathbf{H}=0\);
* Static load (SL): \(\mathbf{A}=-\left( \begin{array}{cc}1/x_{SL}&0\\ 0&1/x_{SL}\end{array} \right)\), \(\mathbf{B}=\left( \begin{array}{cc}r_{SL}/x_{SL}&\omega\\ -\omega&r_{SL}/x_{SL}\end{array} \right)\), \(\mathbf{H}=0\);\[\bullet \text{Induction motor}\cdot\mathbf{A}=-\left[\left( \begin{array}{cc}x_{s} &0\\ 0&x_{s}\end{array} \right)-\left( \begin{array}{cc}x_{m}&0\\ 0&x_{m}\end{array} \right)\left( \begin{array}{cc}x_{r}&0\\ 0&x_{r}\end{array} \right)^{-1}\left( \begin{array}{cc}x_{m}&0\\ 0&x_{m}\end{array} \right)\right]^{-1},\] \[\mathbf{B}=\mathbf{A}\times\left( \begin{array}{cccc}r&\omega\ x_ {s}-\frac{\omega_{2}x_{m}^{2}}{x_{r}}&-r_{2}\frac{x_{m}}{x_{r}}&(\omega\ -\omega_{2})x_{m}\\ \frac{\omega_{2}x_{m}^{2}}{x_{r}}-\omega\ x_{s}&r&(\omega_{2}\ -\omega\ )x_{m}&-\frac{r_{2}x_{m}}{x_{r}}\end{array}  \right),\omega_{2}=(\omega\ -\omega_{IM}),\mathbf{H}=0.\]  
It is advisable to use simplified identification models, which, however, should adequately reproduce the energy transients and the nonlinearity of the characteristics of GTU for the study of them as part of a complex model. The GTU interacts with the SG mechanically; thus, bringing the models of GTUs to a single generalized form (1) is not required [16,17].