# A Reversible Hydropump-Turbine System
## 3 An Energy-Based Model Relative to the Reversible Hydropump-Turbine System
### Motor Configuration of the Asynchronous Machine
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline IM Stator resistance & \(R_{\text{s}}\) & 1.115 (\(\Omega\)) \\ \hline IN Rotor resistance & \(R_{r}\) & 1.083 (\(\Omega\)) \\ \hline IM Stator inductance & \(L_{s}\) & 0.20965 (H) \\ \hline IM Rotor inductance & \(L_{r}\) & 0.20965 (H) \\ \hline IM Magnetization inductance & \(L_{m}\) & 0.2037 (H) \\ \hline IM Number of pole pairs & \(n_{p}\) & 2 \\ \hline IM Rotational inertia & \(J_{m_{M}}\) & 0.02 (kg m\({}^{2}\)) \\ \hline IM Rotor friction coefficient & \(R_{m_{M}}\) & 0.005752 (kg m\({}^{2}\)/s) \\ \hline IM Nominal voltage & \(V_{s}\) & 460 (V\({}_{\text{rms}}\)) \\ \hline IM Operating frequency & f & 60 (Hz) \\ \hline IM Mechanical speed & \(\omega_{m}\) & 1750 (rpm) \\ \hline IM Number of phases factor & \(\delta\) & 3/2 \\ \hline Pump rotational inertia & \(J_{m_{p}}\) & 0.008 (kg m\({}^{2}\)) \\ \hline Pump friction coefficient & \(R_{m_{p}}\) & 0.005752 (kg m\({}^{2}\)/s) \\ \hline Water density & \(\rho\) & 998.2 (kg m\({}^{2}\)) \\ \hline Gravitational constant & \(g\) & 9.81 (m/s\({}^{2}\)) \\ \hline Fluid dynamic viscosity & \(\mu\) & 0.00089 (nm/s\({}^{2}\)) \\ \hline Pipe length & \(L_{p}\) & 30 (m) \\ \hline Pipe height & \(h_{p}\) & 30 (m) \\ \hline Reservoir height & \(h_{r}\) & 5 (m) \\ \hline Pipe crossection area & \(A_{p}\) & 0.005 (m\({}^{2}\)) \\ \hline Reservoir crossection area & \(A_{r}\) & 5 (m\({}^{2}\)) \\ \hline Pump Speed-flow constant & \(K_{h}\) & 191,000 (rad/m\({}^{3}\)) \\ \hline Turbine Speed-flow constant & \(K_{h}\) & 91,000 (rad/m\({}^{3}\)) \\ \hline Pump Speed-pressure constant & \(K_{o}\) & 50 (kg/m) \\ \hline Turbine Speed-pressure constant & \(K_{o}\) & 25 (kg/m) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Key parameters of the reversible hydropump–turbine system. According to [25], the model is extended here with a parameter (constant) that represents the asynchronous machine' phases defined as \(\delta>0\) and shown in (10). In [25], the phase constant is \(\delta=\{1,3/2\}\) for a two- and a three-phase asynchronous machine, respectively. We follow a PH framework of [15; 16]; then, a matrix form can represent a asynchronous machine configured as a motor. The underlying dynamics (6)-(10) (d-q reference frame) are shown below. The Hamiltonian (energy) function is described as follows:  
\[\tilde{H}(\lambda,p_{M})=\frac{1}{2}(\delta{}J_{m_{M}})^{-1}p_{M}^{2}+\frac{1} {2}\lambda^{\top}L^{-1}\lambda \tag{12}\]  
where an inductance matrix \(L\) is defined as follows:  
\[L= \begin{bmatrix}L_{s}&0&L_{m}&0\\ 0&L_{s}&0&L_{m}\\ L_{m}&0&L_{r}&0\\ 0&L_{m}&0&L_{r}\end{bmatrix} , \tag{13}\]  
and with momenta \(p_{M}\) as in (10), constant \(\delta\), and the system's fluxes \(\lambda\in\mathbb{R}^{2\times 2}\) is given by \(\lambda=\text{col}(\lambda_{r},\lambda_{s})\). Then, the system's PH framework of the asynchronous machine configured as a motor (see (6)-(10)) is as follows:  
\[\Sigma_{M} \begin{cases}\hat{x}=\left[\tilde{J}(\tilde{x})-\tilde{R}(\tilde{x} )\right]\
abla_{\tilde{x}}\tilde{H}(\tilde{x})+\tilde{g}(\tilde{x})\tilde{u} \\ \tilde{g}=\tilde{g}^{\top}(\tilde{x})\
abla_{\tilde{x}}\tilde{H}(\tilde{x}), \end{cases}  \tag{14}\]  
where \(\tilde{J}(\tilde{x})\) is the skew-symmetric matrix:  
\[\tilde{J}(\tilde{x})= \begin{bmatrix}0&0&0\\ 0&0&\delta{}n_{p}J_{2}\lambda_{r}\\ 0&\delta{}n_{p}\lambda_{r}^{\top}J_{2}&0\end{bmatrix} ; \tag{15}\]  
and the positive semi-definite matrix \(\tilde{R}(\tilde{x})\) is defined as follows. Figure 2: In the reference frame of a synchronously rotating asynchronous machine, we can represent the two phases d-q via an equivalent circuit. \[\tilde{R}(\tilde{x})= \begin{bmatrix}R_{s}I_{2}&0&0\\ 0&R_{r}I_{2}&0\\ 0&0&\delta R_{m_{M}}\end{bmatrix} . \tag{16}\]  
Finally, the input matrix is described as follows: \(\tilde{g}(\tilde{x})\) is  
\[\tilde{g}(\tilde{x})= \begin{bmatrix}I_{2}&-I_{2}\lambda_{s}&0\\ 0&-I_{2}\lambda_{r}&0\\ 0&0&-1\end{bmatrix} ; \tag{17}\]  
with \((\tilde{u},\tilde{y})\) being the so-called port pair.