# Improving the Energy Conversion Efficiency for Hydrokinetic Turbines Using MPPT Controller
## 2 Design of the MPPT Control Algorithm  
The proposed system consists of a permanent magnet three-phase generator connected to the hydrokinetic turbine, a three phase uncontrolled rectifier, a DC boost converter with MPPT control to extract maximum available power, and a buck converter with CC-CV control to impose the desired voltage and current on the load. A simulation was carried out in order to test the MPPT algorithm. The block diagram is presented in Figure 2.  
Figure 1: Power curves for a shrouded hydrokinetic turbine at different water velocities [21]. Figure 2: Hydrokinetic turbine with maximum power point tracking (MPPT) control algorithm. In order to test the MPPT algorithm, we built the individual blocks on the basis of the corresponding equations of each component. The hydrokinetic turbine is modeled on the basis of the following equation:  
\[P_{turbine}=\frac{1}{2}.\rho\cdot\pi R^{2}\cdot\omega^{3}\cdot C_{p} \tag{1}\]  
where  
\(\rho\)--water density (kg/m\({}^{3}\));  
\(R\)--turbine radius (m);  
\(v\)--water velocity (m/s);  
\(C_{p}\)--turbine power coefficient, also called Betz coefficient. In this case, a turbine with fixed blade angle is considered; thus, \(C_{p}\) is only a function of tip speed ratio:  
\[\lambda=\omega_{gen}\cdot\frac{R}{\varphi} \tag{2}\]  
where \(\omega_{gen}\) is the angular velocity of the generator (rad/s). The permanent magnet generator is described by the following equations written in the synchronous rotating coordinate system d - q:  
\[E_{d}=(\psi_{PM}-L_{d}\cdot i_{d})\cdot\omega_{gen} \tag{3}\]  
\[E_{q}=L_{q}\cdot i_{q}\cdot\omega_{gen} \tag{4}\]  
where \(E_{d}\) and \(E_{q}\) are the induced voltages, \(\psi_{PM}\) is the permanent magnet maximum flux, \(i_{d}\) and \(i_{q}\) are the direct and quadrature currents, \(L_{d}\) and \(L_{q}\) are the synchronous inductances, and \(\omega_{gen}\) is the generator synchronous speed, which depends on the mechanical speed of the generator as follows:  
\[\omega_{gen}=p\cdot\omega_{gen} \tag{5}\]  
where \(p\) is the number of pole pairs of the generator. To complete the model, we had to add the mechanical equations of the rotating moving parts (the friction terms were small compared to the total inertia so they were ignored):  
\[M_{gen}-M_{turbine}=\frac{J}{p}.\frac{d}{dt}\omega_{gen} \tag{6}\]  
\[M_{gen}=1.5p\cdot\left(\psi_{PM}+\left(L_{d}-L_{q}\right)i_{d}\right)i_{q} \tag{7}\]  
where  
\(M_{gen}\)--generator shaft torque (Nm);  
\(M_{turbine}\)--hydro turbine torque (Nm);  
J--moment of inertia of all the moving parts (kgm\({}^{2}\));  
\(p\)--generator number of pole pairs. The three-phase rectifier from Figure 3 was simulated using the concept of commutation functions [22]. Figure 1:0 Three-phase rectifier. In this case, the state of each diode of the rectifier is represented by the following general equation:  
\[G_{abc}= \begin{cases}1\ (on),&i_{abc}\geq 0\\ 0\ (off),&i_{abc}<0\end{cases}  \tag{8}\]  
Using this representation, the rectified current and voltage can be written according to the following formulas:  
\[i_{R}=g_{\alpha}\cdot i_{\alpha}+i_{\beta}\cdot g_{\beta} \tag{9}\]  
\[u_{R}=(g_{\alpha}\frac{di_{\alpha}}{dt}+g_{\beta}\frac{di_{\beta}}{dt})\cdot L _{dc}+R_{dc}\cdot i_{R}+E_{dc} \tag{10}\]  
where \(\alpha\) and \(\beta\) are components of both the phase currents and the switching functions, and are calculated using the CLARKE matrix transformation:  
\[\left[ \begin{array}{c}\alpha\\ \beta\\ 0\end{array} \right]=\frac{2}{3}\left[ \begin{array}{ccc}1&-0.5&-0.5\\ 0&\sqrt{3}/2&-\sqrt{3}/2\\ 0.5&0.5&0.5\end{array} \right]\left[ \begin{array}{c}a\\ b\\ c\end{array} \right] \tag{11}\]  
where \(L_{dc}\) is the DC current smoothing inductance, \(R_{dc}\) is the inductance equivalent resistance, and \(E_{dc}\) is the DC voltage. In the above equations, the zero component was not used because it was assumed that the generator produces a balanced three-phase voltage system. The two DC/DC converters were modeled according to the same principle, state space representation.