WT is composed of different parts to achieve kinetic-to-electric energy conversion. The wind kinetic energy is converted to mechanical energy by the blades. The mechanical energy is transmitted through the drive train to the generator, which converts mechanical energy into electric energy. This conversion is supported by power converters that deliver the power from the generator to the grid, as shown in Figure 1 [12] .

The physical definition for kinetic energy is

K E = 1 2 m V 2 (1)

But the power definition P is the kinetic energy per unit time:

P = 1 2 m ˙ V 2 = 1 2 d m d t V 2 (2)

where, V is the speed and m ˙ is the air mass that passes the disc of wind turbine rotor in a unit length of time, e.g., m ˙ = ρ A V ∞

Thus, the available power in the wind stream is

P a v a i l = 1 2 ρ A V ∞ 3 = 0.5 π ρ R t 2 V ∞ 3 (3)

where P a v a i l is the wind power, ρ is the air mass density, π R t 2 = A is the swept area of turbine rotor, and V ∞ is the wind speed. The mechanical power that can be extracted from the wind power is

P m e c h = P a v a i l C p ( λ , β ) (4)

where, C p ( λ , β ) is, the power coefficient, the ratio of the mechanical power to available power, β is the pitch angle of the blade, and λ is the Tip-Speed- Ratio (TSR) can be obtained by

λ = Ω w t r R t V ∞ (5)

where, Ω w t r is the low-speed shaft and R t is turbine rotor radius.

Wind turbine operation regions are typically divided into four primary regions, as seen in Figure 2. Region 1 spans operation from startup to the cut-in wind speed. In Region 2 the turbine is said to be in sub-rated region when wind speeds are between cut-in and rated speed. The primary goal is to capture as much power as possible. In Region 3 wind speeds are high enough to drive the generator at its rated power output; in this case, the goal is to regulate speed and power safely at rated levels. Region 4 occurs when the turbine shuts down to prevent damage due to high wind speeds [13] .

The quantity C p ( λ , β ) is a nonlinear function, difficult to measure and predict, has different values between each turbine and another depending on turbine parameters and design. The theoretical upper limit for C p is Betz Limit equals to 0.59. The C p is obtained based on the exponential function in [14] .

C p ( λ , β ) = 0.5176 ( 116 λ i − 0.4 β − 5 ) ∗ e ( − 21 λ i ) + 0.0068 λ (6)

where, 1 λ i = 1 λ + 0.08 β − 0.035 β 3 + 1 .

The power coefficient C p ( λ , β ) is a function of blade bitch angle and tip speed ratio. The maximum value of C p equals to 0.479 at β = 0 and λ = 8 , so for region II operation, the pitch angle is settled to zero and the C p is now a function of TSR only C p ( λ ) . Then, the obtained power coefficient is shown in Figure 3(a) and Figure 3(b).

The model has two masses corresponding to WT rotor and generator, respectively. The dynamic equations of motion for the drive-train are written with respective to the generator mass as given below. Table 1 illustrates all parameters definitions [15] .

T ′ w t r = J ′ w t r d Ω w t r d t + D ′ s h a f t ( Ω ′ w t r − Ω g e n ) + K ′ s h a f t ( θ ′ w t r − θ g e n ) (7)

− T g e n = J g e n d Ω g e n d t + D ′ s h a f t ( Ω g e n − Ω ′ w t r ) + K ′ s h a f t ( θ g e n − θ ′ w t r ) (8)

d θ ′ w t r d t = Ω ′ w t r (9)

d θ g e n d t = Ω g e n (10)

The dynamic equations of induction machine in reference frame can be written using either fluxes, currents or both as state-variables. The four loop equations described from the equivalent circuits where the variables i d s , i q s , i d r and i q r represent the state variables and V d s , V q s , V d r and V q r represent the input vector of the state-space model in the dq reference frame. So, the state and input vectors are respectively:

x = [ x 1 ( t ) x 2 ( t ) x 3 ( t ) x 4 ( t ) ] T = [ i d s i q s i d r i q r ] T (11)

u = [ V d s V q s V d r V q r ] T (12)

For SCIG, the d and q components of the rotor voltage set to zero, i.e. ( V R d = V R q = 0 ) since the rotor of squirrel cage machine has slot-embedded bars that are shorted by end rings. The nonlinear fourth-order state model of SCIG can be presented as in the equations below [3] [16] .

X ˙ = A ( Ω g e n ) ⋅ X + B ⋅ u (13)

Y ≡ T g e n = 3 P p L m 2 ( x 2 x 3 − x 1 x 4 ) (14)

T g e n = 3 P p L m 2 ( i q s i d r − i d s i q r ) (15)

where,

A ( Ω g e n ) = [ − R s σ L s ω s + P p Ω g e n L m 2 σ L s L r L m R r σ L s L r P p Ω g e n L m σ L s − ( ω s + P p Ω g e n L m 2 σ L s L r ) − R s σ L s − P p Ω g e n L m σ L s L m R r σ L s R r L m R s σ L s L r − P p Ω h L m σ L r − R r σ L r ω s − P p Ω g e n σ P p Ω g e n L m σ L r R s L m σ L s L r − ω s + P p Ω g e n σ − R r σ L r ] (16)

B = [ 1 σ L s 0 − L m σ L s L r 0 0 1 σ L s 0 − L m σ L s R r ] T (17)

while, the Equation (15) can be rewritten with rotor flux variables instead of rotor current as in Equation (18) to be prepared for IFOC.

T g e n = 3 P p L m 2 L r ( i q s ψ d r − i d s ψ q r ) (18)

Electrical generators are usually controlled by power electronics converters. Power converters are controlled using a switching strategy. Space vector modulation (SVM) is one of the preferred real-time modulation techniques and it is widely used for digital control of voltage source converters. SVM allows reducing commutation losses and the harmonic distortion of output voltage waveform, giving higher amplitude modulation indexes. The SVM control is designed space vector combinations to get a reference voltage V r e f . The summation of the voltage multiplied by the time interval of chosen space vectors equal the product of the reference voltage V r e f and sampling period T z by knowing in which sector the reference voltage vector is located, we select the two adjacent basic voltage space vectors and one zero-vector. For example, when V r e f falls into Sector I, it can be synthesized by V 1 , V 2 and V 0 that gives the equation below. The detail explanation is found in [17] and its references, where our implementation is done based on them.

{ V 1 T 1 + V 2 T 2 + V 0 T 0 = V r e f T z T z = T 1 + T 2 + T 0 (19)

FOC can be classified in two design approaches 1) Direct FOC and 2) Indirect FOC, according to the implementation method that is used to estimate the rotor flux ψ r and the synchronous position θ s . The rotor flux orientation is one of the most popular schemes used in AC drives control and WECS control due to its simple and easy implementation. The necessity of using FOC is the decoupled control of the rotor flux and electromagnetic torque of the machine to achieve high dynamic performance. The synchronous frequency equals to the summation of machine rotor frequency ω r and slip frequency ω s l then by the integration of them, the position θ s can be obtained.

θ s = ∫ ​     ω s = ∫ ​ ( ω r + ω s l ) = θ r + θ s l (20)

For rotor flux orientation, the mathematical model of the SCIG in d-q frame is given in Equations (11)-(18). The decoupling control can be achieved by aligning the stator flux component of the current i d s on the d-axis, and the torque component of the current i q s should align the q-axis, that leads to ψ d r = 0 , ψ d r = ψ r [3] [12] [16] [18] . Then,

T g e n = 3 P p L m 2 L r ( i q s ψ d r ) (21)

L r R r d ψ r d t + ψ r = L m i d s (22)

A new method for driving the operating point to the energy maximum, based on Extremum Seeking (ES) was introduced to overcome challenges with the conventional MPPT algorithm and remove the dependence of the plant modeling. ESC is a class of self-optimizing control strategy which can search for unknown time varying input for optimizing a performance index of nonlinear plant. In this Work, the electrical power was used as the performance index and our objective is to maximize this function f ( u ) by a proper selection of dither signal frequency ω , Low Pass Filter (LPF) frequency ω l and High Pass Filter (HPF) frequency ω h , and integrator gain k to set the control u ^ in the real time. ESC is interpreted as single-input setting with dither signal (usually sinusoidal signal S ( t ) ) added to the estimated u ^ . The schematic diagram of whole variable speed WT including control system, mechanical and electrical components, and MPPT method using ESC is shown in Figure 4 [8] [11] .

The input and the output to plant become:

u = u ^ + cos ( ω t ) (23)

y = f ( u ^ + cos ( ω t ) ) (24)

where, ω is the dither frequency equals to 20.6 rad/s. Then Taylor series expansion for Equation (24) becomes:

y = f ( u ^ + cos ( ω t ) ) = f ( u ^ ) + cos ( ω t ) ∂ f ∂ u + H . O . T (25)

HPF filters out the DC term while passes the AC term then by multiplying with the demodulation signal M ( t ) = cos ( ω t ) , that yields:

cos ( ω t ) × ( cos ( ω t ) ∂ f ∂ u + H . O . T ) = ∂ f ∂ u + H . O . T (26)

LPF retains DC term to estimate the gradient of the objective function with respect to the control input u while filters out AC term from Equation (26). The loop is closed with an integrator to drive the gradient to zero in steady state. Several signals were examined as dither signals for ESC. The cosine signal was applied as a harmonic signal and wind speed signal was applied as non-har- monic signal. In addition, the HPF and LPF have been implemented in both as 1^st and 2^nd order transfer functions.

The 1^st and 2^nd order transfer function for HPF is given below with cutoff frequency ω h = 3 rad/s:

G 1st_HPF = s s + ω h ,     G 2nd_HPF = s 2 s 2 + 2 × 0.58 × ω h s + ω h 2 (27)

The 1^st and 2^nd order transfer function for LPF is given below with cutoff frequency ω l = 8 rad/s:

G 1st_LPF = ω l s + ω l ,     G 2nd_LPF = ω l 2 s 2 + 2 × 0.6 × ω l s + ω l 2 (28)

By feedbacking the electrical power to ES, the optimal speed value is obtain to achieve the maximum energy captured. Different types of dither signals have been examined to get the best reference speed to the system since the selection of the dither signal is an important issue in the application of ESC [19] [20] . Figure 5 shows the effect of choosing different type of dither signals to generate the reference speed. The response in blue color (using 1^st order filters and cosine signal as dither signal for ES) suffers from ripples and rises above the optimal speed 157.5 rad/s in the periods (16 - 20 s), while the responses in green (using 2^nd order filters and cosine signal as a dither signal for ES) and pink (using 2^nd order filters and wind signal as a dither signal for ES) have high overshoots at 0.5 s. Moreover the green response is decreasing to 156.5 rad/s at 16 s. The response in red color (using 1^st order filters and wind signal as a dither signal for ES) was found to be the best to choose as a reference speed.

The PI controller is the more commonly used control strategy in electrical machines for speed control. There are many different algorithms to find suitable parameters of the controller. To achieve the closed loop speed control, we firstly select one of the generated speed references from the above sub-section to be the speed command for PI controller. The PI parameters are found using trial and error method to control the speed of generator then Particle Swarm Optimization (PSO) has been presented to determine the optimal PI parameters and ensure the maximum power captured. PSO is an off-line optimization algorithm based on population. PSO algorithm gives high quality solutions and stable

convergence characteristics within shorter calculation time. In PSO, particles fly around in a multidimensional search space then during its flight each particle adjusts its position according to its own experience (P-best), and according to the experience of a neighboring particle (G-best), made the best position encountered by itself and its neighbor [21] .

The velocity of each agent can be modified by the following equation:

V k + 1 = i n e r t i a ⋅ V k + c 1 ⋅ r a n d ⋅ ( P b e s t − x k ) + c 2 ⋅ r a n d ⋅ ( G b e s t − x k ) (29)

The current position can be modified by the following equation:

x k + 1 = x k + V k + 1 (30)

The performance index (ITSE) is chosen as cost function for PSO and given by.

ITSE = ∫ 0 ∞ t ⋅ e ( t ) 2 d t (31)

As mentioned before, the closed loop control for regulating the speed of the generator had accomplished using PI controller. The gain values for K_p and K_i were tuned manually then improved using PSO. The presented data in Figure 6 and Table 2 had been obtained at the following values: iteration = 20, swarm size = 30, inertia = 0.9, and c 1 = c 2 = 2 .