## 2 System Modeling
### Modeling of Turbine System  
For a flow stream of a given fluid flowing through an area \(A_{c}\) with density \(\rho\) and flow velocity \(u_{0}\), the amount of power available within that area \(P_{flow}\) can be calculated using (1). \[P_{flow}=\frac{1}{2}\rho A_{c}u_{0}^{3} \tag{1}\]  
The amount of power absorbed by the turbine is linearly related to the total power in the incoming flow stream through the functional area of the HKT through a term named coefficient of power \(C_{p}\), which is a function of the tip speed ratio \(\lambda\) (defined as \(\lambda=\omega L/u_{0}\), where \(\omega\) is the rotational speed of the turbine and \(L\) is the length of the turbine blades). Therefore, in terms of the HKT system, if it is assumed that the flow passes through the rotor swept area of the turbine \(A_{c}\) and the turbine is operating at a given tip speed ratio \(\lambda\), the power harvested by the turbine \(P_{turbine}\) can be estimated using (2). \[P_{turbine}=\frac{1}{2}\rho A_{c}C_{p}(\lambda)u_{0}^{3} \tag{2}\]  
The equation of motion that describes the rate of the rotational speed is shown in (3). This formula relates the net torque \(T_{net}\) (which is defined as the difference between the  
Figure 1: Hydrokinetic turbine system overview highlighting the turbine, PMSG, and generator-side converter subsystems. turbine torque \(T_{turbine}\), generator torque \(T_{gen}\), and friction torque \(T_{fric}\)) to the rate of change in \(\omega\) through the total moment inertia of the HKT system \(J\) (the sum of the moment of inertia of the turbine \(J_{turbine}\) and the generator \(J_{gen}\): \(J=J_{turbine}+J_{gen}\)). \[T_{net}=T_{turbine}-T_{gen}-T_{fric}=J\frac{d\omega}{dt} \tag{3}\]  
The torques \(T_{turbine}\) and \(T_{fric}\) are related to \(\omega\), as shown in (4), where \(C_{fric}\) is the coefficient of friction. \(T_{gen}\), on the other hand, is defined as a function of the current in the stator windings, as shown in Section 2.2, and is controlled through the proposed k-omega-squared control method. \[T_{turbine}=\frac{P_{turbine}}{\omega},\qquad T_{fric}=C_{fric}\cdot\omega \tag{4}\]  
To improve reliability and reduce system costs and complexity, it has been assumed that there is no gearing between the turbine and generator shafts, and there is no control over the pitch of the turbine blades. Region-Based Control  
The turbine modeled in this research has four regions of operation, which are defined by the flow velocity, as shown in Figure 2.  
The turbine operates in \"offline mode\" for flow velocities below the cut-in speed \(u_{ci}\) (Region 1) and above the cut-out speed \(u_{co}\) (Region 4). The turbine is blocked from rotating in these two operating regions using an external brake, which is modeled by increasing \(C_{fric}\) in this work. The turbine operates in the maximum power extraction region for flow velocities between \(u_{ci}\) and the rated turbine flow velocity \(u_{rated}\) (Region 2). In Region 2, the goal is generally to operate at the optimal tip speed ratio \(\lambda^{*}\) such that the \(C_{p}\) of the turbine is at its maximum value, often termed maximum power point tracking control. Operation in Region 3, on the other hand, is designed to ensure the turbine operates at rated power for flow velocities between \(u_{rated}\) and \(u_{co}\). Depending on the design of the turbine and its application environment, the harvested power can be regulated using pitch or speed control techniques [28]. As stated above, it is assumed that the turbine system in this work does not have pitch actuation. Therefore, speed control techniques must be used in Region 3 operation. Figure 2: Plots of normalized turbine power (\(P_{turbine}\)), torque (\(T_{turbine}\)), and rotational speed (\(\omega\)) compared to flow velocity (\(u_{0}\)), highlighting the operation of the turbine in each of the four regions of operation.