**2. Aerodynamics, electrical, and drive train modeling**

#### **2.1 Aerodynamics modeling**

The wind is the net result of the pressure gradient force, gravity, Coriolis, centrifugal and friction forces acting on the atmosphere. Since wind speed usually varies from one location to another and also fluctuates over time in a stochastic way, J.G. Slootweg [17] proposed a mathematical model that takes some landscape parameters to generate a wind speed *Vm*ð Þ*t* in (meter/sec) sequence for any location, as per equation one:

$$V\_w(t) = V\_{w\\_d}(t) + V\_{w\\_r}(t) + V\_{w\\_g}(t) + V\_{w\\_t}(t) \tag{1}$$

Where, *Vwa*ð Þ*t* is a constant component, *Vwr*ð Þ*t* is a common ramp component, *Vwg*ð Þ*t* is a gust component, and *Vwt*ð Þ*t* is a turbulence component in (meter/sec).

The operation of a wind turbine can be characterized by its mechanical power output *Pm* through a cross-sectional area *A* normal to the wind as a function of wind speed *Vw* [18].

$$P\_m = 0.5\rho A V\_w^3 C\_P(a, \beta) \tag{2}$$

Where *<sup>ρ</sup>* is the air mass density, *<sup>A</sup>* <sup>¼</sup> *<sup>π</sup>r*<sup>2</sup> is the turbine swept area, *<sup>r</sup>* is the turbine radius, and *Vw* is the wind speed. *Cp* is a nonlinear function of *λ* and *β* is referred to as the performance coefficient and is smaller than 0.59, given by [18, 19].

$$C\_p = 0.5 \left[ \frac{rC\_f}{\lambda} - 0.022\beta - 2 \right] e^{-0.25 \frac{rC\_f}{\lambda}} \tag{3}$$

Where *β* is the turbine pitch angle, *λ* is the tip-speed ratio, and *Cf* is the blade design constant *λ* is defined by:

$$
\lambda = \frac{r o\_{tur}}{V\_w} \tag{4}
$$

Where *ωtur* is the rotational angular speed of turbine blades in mechanical rad/sec.

$$P\_{m\max} = \left(\frac{0.5\rho A r^3 C\_{P\max}}{\lambda\_{opt}^3}\right) w\_{trr}^3 = K\_{opt} w\_{trr}^3 \tag{5}$$

Eq. (5) indicates that *Pmax* is proportional to the cube of turbine speed, hence the mechanical torque *Tmmax* is:

$$T\_{m\max} = K\_{opt} w\_{tur}^2 \tag{6}$$

#### **2.2 Electrical modeling**

A wounded-rotor induction motor can operate as a double-fed induction motor (DFIM) with the stator side windings openly attached to the three-phase power grid/load and the rotor side windings attached to a side-by-side moderately measured (20–30) % rating power converter as shown in **Figure 1**. As shown in **Figure 2**, an

**Figure 1.** *Grid-connected DFIG [20].*

induction motor works on the interface principle between the stator and rotor magnetomotive forces (MMF). The stator side windings current produce an MMF revolving at power grid side frequency especially including an MMF in the rotor side windings. The rotor speed does not compliment the stator side MMF. This induced rotor MMF will rotate at the so-called slip frequency which possesses the subsequent value [21]:

$$
\alpha\_{slip} = \alpha\_{mmf}^{rotor} = \alpha\_{mmf}^{star} - \alpha\_{rotor} \tag{7}
$$

Where, *ωslip* is the slip frequency, corresponding to the frequency of rotor current and voltage, *ωstator mmf* is the stator or the grid frequency in (rad/sec), *ωrotor* is the rotor rotating frequency (rad/sec). In both sub-synchronous and super-synchronous operations, the DFIM machine can be operated either as a motor (0 < slip<1) with positive rotor torque or a generator (slip<0) with negative rotor torque. The Park and Clark transform allowing the transformation of time-dependent variables into constant values. The per-unit electromagnetic torque equation expressed in d-q park reference is given by [22]:

$$T\_e = \wp\_{ds} I\_{qs} - \wp\_{qs} I\_{ds} = \wp\_{qr} I\_{dr} - \wp\_{dr} I\_{qr} = L\_m \left( I\_{qs} I\_{dr} - I\_{ds} I\_{qr} \right) \tag{8}$$

Neglecting the power losses associated with the stator and rotor resistances, the active and reactive stator powers for the DFIG are [22]:

$$P\_s = \left(\frac{3}{2}\right) \left(V\_{ds}I\_{ds} + V\_{q3}I\_{q3}\right) \tag{9}$$

$$Q\_s = \left(\frac{3}{2}\right) \left(V\_{qs}I\_{ds} - V\_{ds}I\_{qs}\right) \tag{10}$$

And the active and reactive rotor powers are given by:

$$P\_r = \left(\frac{3}{2}\right) \left(V\_{dr}I\_{dr} + V\_{qr}I\_{qr}\right) \tag{11}$$

$$Q\_r = \left(\frac{3}{2}\right) \left(V\_{qr}I\_{dr} - V\_{dr}I\_{qr}\right) \tag{12}$$

The overall system equations can also be re-written with relation to the rotating frames [21]:

$$P\_T = P\_s + P\_r = \frac{3}{2} \left( V\_{qr}' I\_{qr}' + V\_{dr}' I\_{dr}' + V\_{ds} I\_{ds} + V\_{qs} I\_{qs} \right) \tag{13}$$

$$Q\_T = Q\_s + Q\_r = \frac{3}{2} \left( V\_{qr}' I\_{qr}' - V\_{dr}' I\_{dr}' + V\_{ds} I\_{ds} - V\_{qs} I\_{qs} \right) \tag{14}$$

The torque expression and the stator reactive power, which are the control objectives of the rotor-side converter control, are shown in Eqs. 17 and 18. Where, *p* is the number of pole pairs of the generator, *Iqs* and *Iqr* are the *q* component of the stator and rotor current, *Ids* and *Idr* are the *d* component of the stator and rotor current, *Vqs* and *Vds* are the *q* and *d* components of the stator voltage. The stator and rotor flux linkages in the synchronous reference frame are expressed as [23]:

$$
\Delta \varphi\_s = L\_s I\_s + L\_m I\_r \tag{15}
$$

$$
\mu \mu\_r = L\_m I\_s + L\_r I\_r \tag{16}
$$

The electromagnetic torque can be expressed using the *d* � *q* components as following [23].

$$T\_m = \frac{3}{2} p \frac{L\_m}{L\_s} \left(\wp\_{qs} I\_{dr} - \wp\_{ds} I\_{qr}\right) \tag{17}$$

$$\mathcal{Q}\_s = \frac{3}{2} \left( V\_{qs} I\_{ds} - V\_{ds} I\_{qs} \right) \tag{18}$$

#### **2.3 Drive train model**

Considering the mechanical aspect of the wind turbine, the mechanical representation of the drive train of the entire wind turbine is complex. Following four types of the drive train in wind turbine models are generally used [24].

1.Six mass drive train model.


Of the above four types of drive train models, the one that was modeled and implemented is the simplified form of the two mass-shaft model power train systems as shown in **Figure 3** consisting of a shaft and gearbox. As per the two-mass model of the drive train system described in [24], all masses are grouped into low and highspeed shafts. The inertia of the low-speed shaft comes mainly from the rotating blades and the inertia of the high-speed shaft. The input to the model for a two-mass system is established as torque *TA*, which is gained by the aerodynamics methodology and the generator response torque *Te*. The target is the deviations in the rotor speed *ω<sup>r</sup>* and the generator speed *ωg*. The deviations in the mechanically compelled torque *Tm*, the generator torque response *Te*, and torque loss owing to friction *Tfric*, causes the variation of angular velocity *ω* <sup>∙</sup> *<sup>g</sup>* [24].

$$T\_m - T\_e - T\_{f\text{ric}} = j\_\text{g} \times \dot{o}\_\text{g} \tag{19}$$

The change in the angular speed *ω*\_*<sup>r</sup>* is caused by the difference between the aerodynamic torque *TA* and shaft torque *Ts* at a low speed shaft [24].

$$T\_A - T\_s = j\_r \times \dot{o}\_r \tag{20}$$

$$
\dot{\alpha}\_{\text{g}} = \rho\_{\text{g}}'' \,\text{and } \dot{\alpha}\_{r} = \rho\_{r}'' \tag{21}
$$

*Tm* and *Ts* are connected by the gear ratio*n*, as *Ts* ¼ *nTm*

$$T\_s = K\_s \cdot \Delta \rho + D\_s \cdot \rho = K\_s \cdot \Delta \rho + D\_s \left(\alpha\_r - \alpha\_\text{g}/n\right) \tag{22}$$

Where *Ks* is the stiffness constant and *Ds* is the damping constant of the shaft. Considering a two-mass free-swinging system the Eigen frequency is as follows:

$$\dot{o}\_r = \frac{1}{J\_r} \left( T\_A - D\_S \cdot o\_r + \frac{D\_S}{n} \dot{o}\_\mathbf{g} - K\_S \int (o\_r - \frac{o\_\mathbf{g}}{n}) dt \right) \tag{23}$$

$$\dot{o}\_{\rm g} = \frac{1}{J\_{\rm g}} \left( -T\_{\epsilon} - \left( D\_{\rm g} + \frac{D\_{\rm S}}{n^2} \right) o\_{\rm g} + \frac{D\_{\rm S}}{n} \dot{o}\_{\rm r} - \frac{K\_{\rm s}}{n} \left[ \left( o\_{\rm r} - \frac{o\_{\rm g}}{n} \right) dt \right] \tag{24}$$

**Figure 3.** *Schematic drawing of the two mass shaft drive train model [24].*

Where, *Ks* is the stiffness constant and *Ds* is the damping constant of the shaft. Considering a two-mass free-swinging system, the Eigen frequency is given as:

$$
\alpha\_{as} = 2\pi f\_{as} = \sqrt{\frac{K\_s}{J\_{gs}}} \tag{25}
$$

The total inertia of the free-swinging system on the low-speed is calculated by:

$$J\_{\rm gee} = \frac{J\_r \cdot J\_\mathbf{g} \cdot n^2}{J\_r + J\_\mathbf{g} \cdot n^2} \tag{26}$$

So, the stiffness constant *Ks* and the damping constant *Ds* of the low-speed shaft, with *ξ<sup>s</sup>* as logarithmic decrement is:

$$K\_s = J\_{ge} \cdot \left(2\pi f\_{as}\right)^2\tag{27}$$

$$D\_s = 2\xi\_s \cdot \sqrt{\frac{K\_s l\_{\rm ges}}{\xi\_s^2 + 4\pi^2}}\tag{28}$$
