# A Reversible Hydropump-Turbine System
## 3 An Energy-Based Model Relative to the Reversible Hydropump-Turbine System
### Motor Configuration of the Asynchronous Machine  
In the Figure 2, the asynchronous machine model configured as a motor is presented as in [15; 16; 23; 24] in the d-q axis so the parameters of time dependence have been rewritten in direct and quadrature axes. Consequently, the explicit time-dependent variable is not presented anymore. We recapitulate here the PH modelling approach relative to the asynchronous machine configured as a motor and generator previously published in [17]. The nonlinear dynamics of the asynchronous motor are described as follows:  
\[\dot{\lambda}_{s_{d}} =\omega_{e}\lambda_{s_{q}}+V_{s_{d}}-R_{s}i_{s_{d}}, \tag{6}\] \[\dot{\lambda}_{s_{q}} =-\omega_{e}\lambda_{s_{d}}+V_{s_{q}}-R_{s}i_{s_{q}},\] (7) \[\dot{\lambda}_{r_{d}} =(\omega_{e}-\omega_{r})\lambda_{r_{q}}+V_{r_{d}}-R_{r}i_{r_{d}},\] (8) \[\dot{\lambda}_{r_{q}} =-(\omega_{e}-\omega_{r})\lambda_{r_{d}}+V_{r_{q}}-R_{r}i_{r_{q}},\] (9) \[\dot{p}_{M} =\delta n_{p}\lambda_{r}^{\top}j_{2}i_{r}-R_{m_{M}}\omega_{m}- \tau_{L}, \tag{10}\]  
where state variable \(p_{M}\in\mathbb{R}\left[\text{kgm}^{2}/s\right]\) represents the generalized momentum. The stator and rotor vector fluxes are given, respectively, by \(\lambda_{s}=\text{col}\left(\lambda_{s_{d}},\lambda_{s_{q}}\right)\) and \(\lambda_{r}=\text{col}\left(\lambda_{r_{d}},\lambda_{r_{q}}\right)\), where \(\lambda_{s}\in\mathbb{R}^{2\times 1}\left[\text{Wb}\right]\), \(\lambda_{r}\in\mathbb{R}^{2\times 1}\left[\text{Wb}\right]\), and \(J_{2}\) is a skew-symmetric matrix such that the following is the case. \[J_{2}= \begin{bmatrix}0&-1\\ 1&0\end{bmatrix} . \tag{11}\]  
Figure 1: General diagram of reversible hydropump–turbine system. Furthermore, in (6)-(10), we have \(V_{\text{s}_{\text{t}}}\), \(V_{\text{s}_{\text{g}}}\), \(V_{\text{r}_{\text{d}}}\), \(V_{\text{r}_{\text{q}}}\) [\(V\)] and \(i_{\text{s}_{\text{d}}}\), \(i_{\text{s}_{\text{q}}}\), \(i_{\text{r}_{\text{d}}}\), \(i_{\text{r}_{\text{q}}}\) [\(A\)], which are the stator and rotor voltages and currents for each axis, respectively. Moreover, \(\omega_{r}\left[\text{rad}/\text{s}\right]\), \(\omega_{e}\left[\text{rad}/\text{s}\right]\), and \(\omega_{m}\left[\text{rad}/\text{s}\right]\) are the rotor, synchronous, and mechanical shaft velocities, respectively. In addition to this, the system's dissipation factors, i.e., the stator and rotor resistances and the rotor friction coefficient, are given by passive elements \(R_{s}\left[\Omega\right]\), \(R_{r}\left[\Omega\right]\), and \(R_{m_{M}}\left[\text{kgm}^{2}/\text{s}\right]\). \(\tau_{L}\left[Nm\right]\) is the torque input to the mechanical shaft. The electrical domain's active elements are represented by the stator and rotor self-inductance \(L_{s}\left[\text{H}\right]\) and \(L_{r}\left[\text{H}\right]\), respectively, and the rotor and stator mutual-inductances \(L_{m}\left[\text{H}\right]\). Moreover, the number of pole pairs of the system is represented by \(n_{p}\), and a constant rotational inertia \(J_{m_{M}}\left[\text{kgm}^{2}\right]\) from the mechanical domain. Finally, the stator voltage is given by \(V_{s}\left[\text{V}\right]\), while \(V_{r}=0\left[\text{V}\right]\) represents the rotor's voltage for a squirrel-cage induction motor. In Table 1, all key values and physical quantities are provided for simulation purposes.