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Maximizing the power capture is an important issue to the turbines that are installed in low wind speed area. In this paper, we focused on the modeling and control of variable speed wind turbine that is composed of two-mass drive train, a Squirrel Cage Induction Generator (SCIG), and voltage source converter control by Space Vector Pulse Width Modulation (SPVWM). To achieve Maximum Power Point Tracking (MPPT), the reference speed to the generator is searched via Extremum Seeking Control (ESC). ESC was designed for wind turbine region II operation based on dither-modulation scheme. ESC is a model-free method that has the ability to increase the captured power in real time under turbulent wind without any requirement for wind measurements. The controller is designed in two loops. In the outer loop, ESC is used to set a desired reference speed to PI controller to regulate the speed of the generator and extract the maximum electrical power. The inner control loop is based on Indirect Field Orientation Control (IFOC) to decouple the currents. Finally, Particle Swarm Optimization (PSO) is used to obtain the optimal PI parameters. Simulation and control of the system have been accomplished using MATLAB/Simulink 2014.

Wind energy is a fast-growing interdisciplinary field that encompasses multiple branches of engineering and science. Wind energy has gained tremendous attention over the past two decades as one of the most promising renewable energies for the future, environmentally friendly solutions, pollution-free and inexhaustible sources [

According to speed control, WTs are classified into two types: fixed and variable speed WT. Fixed speed turbines are using SCIG directly connected to grid without using power electronic converters. The speed of the generator is constant regardless of the wind speed. Variable-speed turbines are currently the most used turbines that use different types of generators connected to grid through the power converters. Variable-speed turbines allow the rotational speed to be continuously adapted and controlled in such a manner that the turbine operates constantly at its highest level of aerodynamic efficiency then the maximum energy can be extracted relying upon the wind speed variations.

Maximizing power capture for variable speed Wind Turbine (WT) by means of control theory has been a popular topic for the researchers. According to WT operation regions that will be explained in the next section, the researchers applied different control methodologies by using generator speed control to harvest maximum energy in region II, using blade pitch control to reduce mechanical loads and maintenance cost in region III, or a combination of both. Model-based strategies have been studies for region II control [

To overcome some difficulties with model-based control methods and reduce the dependency on the mathematical model, other researchers proposed different model-free control methods for Region II operation. Johnson et al. [

In this paper, Extremum Seeking Control (ESC) scheme [

The approach in this paper focuses on the control of variable speed WT composed of two-mass drive train, SCIG, and voltage source converter control by SPVWM. The controller is designed in two loops, in the outer loop ESC has been used to set a desired reference for PI control of the generator speed and extract the maximum electrical power within high wind speed variation. The inner control loop is based on IFOC for the currents decoupling. The following Sections are organized as a follow. In Section 2, the design of wind energy conversion system has presented with concentration on the dynamic equations of SCIG in dq reference frame and two-mass drive train. In Section 3, the controller design was discussed. In Sections 4 and 5, Simulation results and conclusions are presented respectively.

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

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

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 [

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

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.

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)

Parameters | Description | Unit | Referred to the mass of the generator |
---|---|---|---|

k g e a r | Gearbox ratio | ||

J g e n | Generator inertia | kg・m^{2} | |

J w t r | Wind turbine rotor inertia | kg・m^{2} | J ′ w t r = J w t r k g e a r |

K s h a f t | Shaft stiffness | N・m/rad | K ′ s h a f t = K s h a f t k g e a r 2 |

D s h a f t | Shaft damping coefficient | N・m・s/rad | D ′ s h a f t = D s h a f t k g e a r 2 |

T g e n | Generator torque | N・m | |

T w t r | Wind turbine rotor torque | N・m | T ′ w t r = T w t r k g e a r |

Ω g e n | Speed of the high speed shaft | rad/s | |

Ω w t r | Speed of the low speed shaft | rad/s | Ω ′ w t r = k g e a r ⋅ Ω w t r |

θ g e n | High speed shaft angular position | rad | |

θ w t r | Low speed shaft angular position | rad | θ ′ w t r = k g e a r ⋅ θ w t r |

T s h a f t | Internal shaft torque | N.m |

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 [

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 [

{ 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 [

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

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 [^{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 [

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

The modeling of variable speed wind turbine for mechanical and electrical components has been built based on Equations (1)-(17), the currents of the

The applied method | K p | K i |
---|---|---|

PI gains tuned using trial and error | 2100 | 1000 |

PI gains tuned using PSO algorithm | 2199.0295 | 1093.6790 |

squirrel cage induction generator is decoupled using IFOC based on Equations (18)-(22) and ESC was presented as explained before. MATLAB/Simulink block diagram is shown in

The input signal to the whole model is the wind speed which has been modeled and developed based on the Kaimal spectra in [

The Simulation has been solved using fixed-step ode4 solver with sampling time ( τ s = 1 e − 5 ) . The initial values for the speed of generator (high-speed shaft) and the speed of turbine rotor (low-speed shaft) are 156.5, 1.874 rad/s respectively.

The other parameters values are initially set to zero.

In this paper, the development of MPPT with ESC to harvest maximum energy captured in the sub-rated speed region has been presented. The simulation results show the strength of ESC to get the position of the operating point neglecting the model dynamics equations as compared with the conventional MPPT methods that highly depend on the dynamic equations and speed measurements. For its being a self-optimizing model-free method, ESC has provided good convergence rate despite the time-varying in wind speed. The output electrical power was used as a performance index to the ESC scheme which shows more interest in energy capture than using the mechanical power. We examined different kinds of dither signals for ESC to achieve slow perturbation to the system and the non-harmonic wind signal with first order transfer function was the better one. The rotor flux orientation control was implemented in inner control loop to achieve the required decoupling between currents and electromagnetic torque. The PI gains values in speed control loop is tuned firstly by trial and error then improved with PSO, where the obtained parameters gains attain the optimal energy captured.

Rafaat, S.M. and Hussein, R. (2018) Power Maximization and Control of Variable-Speed Wind Turbine System Using Extremum Seeking. Journal of Power and Energy Engineering, 6, 51-69. https://doi.org/10.4236/jpee.2018.61005