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State-Space Modeling and Performance Analysis of 
Variable-Speed Wind Turbine Based on a Model 

Predictive Control Approach 
 

Hussain Bassi 
Electrical Engineering Department  

Faculty of Engineering 
King Abdulaziz University 

Rabigh, Saudi Arabia 
hmbassi@kau.edu.sa  

Youssef Mobarak  
Electrical Engineering Department, Faculty of Engineering 

King Abdulaziz University, Rabigh, Saudi Arabia & 
Electrical Engineering Department, Faculty of Energy 

Engineering, Aswan University, Egypt 
ysoliman@kau.edu.sa 

 

  

Abstract—Advancements in wind energy technologies have led 
wind turbines from fixed speed to variable speed operation. This 
paper introduces an innovative version of a variable-speed wind 
turbine based on a model predictive control (MPC) approach. 
The proposed approach provides maximum power point tracking 
(MPPT), whose main objective is to capture the maximum wind 
energy in spite of the variable nature of the wind’s speed. The 
proposed MPC approach also reduces the constraints of the two 
main functional parts of the wind turbine: the full load and 
partial load segments. The pitch angle for full load and the 
rotating force for the partial load have been fixed concurrently in 
order to balance power generation as well as to reduce the 
operations of the pitch angle. A mathematical analysis of the 
proposed system using state-space approach is introduced. The 
simulation results using MATLAB/SIMULINK show that the 
performance of the wind turbine with the MPC approach is 
improved compared to the traditional PID controller in both low 
and high wind speeds.  

Keywords-blades; maximum power point trackers; modeling; 
predictive control; turbines   

I. INTRODUCTION  
Wind energy has known wide research interest over the past 

years [1-4]. Research focuses on various aspects such as 
reliability [5], system stability, security, low-voltage ride-
through faults, energy profile, current movement, the short 
circuit flows and responsive power ability [6]. Variable-speed 
wind turbines play a key role in energy harvesting. They 
enhance energy production and reduce drive train rotations as 
well as fluctuations in power production [7-9]. Design and 
implementation of the regulation method for wind energy 
conversion systems (WECS) is a complex process. Complexity 
results from the multi-input multi-output (MIMO) systems 
which consist of robustly attached variables. The nonlinear 
nature of the input power and the natural hindrances of the 
system differences also cause difficulties to the regulation 
design [10-12]. There are several control techniques that have 
been used to maximize wind turbine efficiency. In the industry, 
most of the commercial wind turbines are operated by 

controlling the blade pitch angle. These controllers are divided 
into main categories; linear and nonlinear. For instance, linear 
controllers are able to regulate wind turbines using pitch angle 
control after the system has been linearized. They are also 
extensively used together with quadratic Gaussian or quadratic 
regulator to control the output power [13-14]. The design 
process undertaken for the proportional integral derivative 
(PID) and the proportional integral (PI) has also been examined 
together with the regulators [15-18]. Since wind systems have 
highly nonlinear characteristics, the performance of the system 
may be diluted with the operating point diverge from the 
linearized range. 

Gain scheduling control is another linear technique that has 
been suggested as a way of introducing the linear features in 
the power systems. Although this technique outperforms PID 
and PD methods by widening the operation point range, it is 
limited in terms of the level of performance they offer to the 
wind turbines with non-linear characteristics [19-23]. Model 
predictive control (MPC) is one of the nonlinear control 
methods to improve the performance of the WECS that has 
attracted attention. There are two sub-categories derived within 
the electrical area of the MPC drives. The first category is 
based on the advancement of the conventional field-oriented 
control in which changes are made to the internal current loops 
[24-26]. It is done with a MPC regulator without removing the 
modulator. In the second sub-category, the modulator is 
removed and the MPC controls the inverter [27-30]. A different 
form of MPC is the Model Predictive Direct Torque Control 
(MPDTC). It enhances direct regulation of the torque as well as 
rotor flux in the machines [31-34]. The computational process 
of this technique may become tedious if the prediction horizon 
is extended massively. The MPC technique is a newer version 
which is responsible for controlling the machines stator 
currents [35-38]. 

In this paper, MPC approach is used to create a reasonable 
pitch angle regulator integrated in a variable speed wind 
turbine system. Mathematics concepts, MIMO and WECS are 
projected. The main objective of this study is to determine and 



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standardize the apprentice MPC system set in contrast to the 
PID system. 

II.  WECS MODEL 
WECS changes electric energy from kinetic energy through 

a process where air streams are converted into electric energy. 
The conversion process takes place when wind causes the 
blades to rotate which in turn spins the turbines. The spinning 
rotor drives the machine which then produces electricity. 
Currently, many wind turbines are built on the lateral design 
and consists of either two or three blades [2]. Energy 
transformation in the wind turbines takes place in four sub-
systems as illustrated below [4]: 
• Electric generator model: this works together with the basics 

required for local grid connection. MPC or PID controls the 
inputs to generator model and pitch actuator system to 
improve wind turbine efficiency. 

• Aerodynamic model: these are the blades for the turbine. 
Input factor consist of the speed of wind, how fast the rotor 
spins and the pitch angle. The output on the other hand 
consists of the rotation of the rotor which in turn produces 
electricity.  

• Pitch actuator model: this part consists of a mechanical servo 
that rotates the turbine blades against the wind to regulate the 
generator rotor speed. 

• Drive train model: involves the speed shafts, the turbine hub, 
and the speed multiplier. The input in this section constitutes 
the rotor rotation and speed multiplier which drives the train 
model. The rotor speed and shaft rotation make up the 
output. 

Figure 1 shows the WECS model as a three-fundamental 
system that portrays a standard practice in WECS control. Tr 
and Tg are the rotor torque and generator torque input, and ωr 
and ωg are the revolving speed of the rotor and the generator 
speed. 

 

 
Fig. 1.  Aerodynamic, mechanical and electrical models of the WECS. 

Wind turbines have evolved form the traditional constant 
speed to the more efficient variable speed design. There are 
assumptions that wind turbines which generate levels of energy 
greater than the recorded speed are not cost effective. The 
energy curve is divided into three specific areas [10]. Area I 
constitutes mild wind speed lower that the rated power of the 
turbines, therefore the turbine is operated at the highest 

possible level. Area II is the transformation area and 
responsible for reducing the level of noise. Lastly, area III 
experiences high wind speed and it is where the turbine derives 
their highest level of power. At this level, the wind turbines are 
regulated within the limits of the optimal energy. The variable-
speed variable-pitch WECS is made up of two operating areas 
[11]. First, the partial load regime consists of all the different 
wind speeds and rotors of the wind turbine responsible for 
achieving the highest level of performance. Secondly, the full 
load regime consists of wind turbines which are run when the 
speeds level are high or below the vco. The control systems in 
this area controls the output energy and the speeds of generator 
based on their expected speed levels. It is known as the ratio of 
recorded output of energy to the free stream-like power 
flowing in the same area. The tip speed ratio λ is defined as 
the speed of the wind turbine at the highest part of the blade to 
the free stream speed of wind. The speed on the tip of the 
blade has to reach a certain speed so that the wind turbine can 
generate maximum energy [10]. Utilizing an efficient control 
technique such as MPC will produce this desirable outcome. 
The highest output energy released does not flow in the same 
direction as the highest torque level. Both follow different 
laws with the output energy following the cube law while the 
torque takes the square law. 

A. Wind Turbine Aerodynamics System 
The aerodynamic system converts wind energy into useful 

mechanical energy. A thorough discussion about wind turbine 
aerodynamics can be found by reading [16]-[20]. It is 
important to understand the equations about the torque and 
how power is captured by the blades. Figure 2 illustrates three 
aspects (wind, energy curve, and responsiveness to wind 
fluctuations) of capturing wind energy. The energy and torque 
derived from the wind is shown in the equations below: 

  2 3w P w
1

P = C λ,β ρπR v
2

(1) 

 Ta
3 2

w
1

T = C λ,β ρπR v
2

 (2) 

Where:         
Pw is the extracted power from the wind, Ta is the 

aerodynamic torque of the rotor, ρ is the air density, R is the 
radius of the blades, vw is the wind velocity, CP is the power 
coefficient, β is the pitch angle of the rotor blades and CT is 
the aerodynamic coefficient.  

 
Fig. 2.  Block diagram of the aerodynamic wind turbine model 

CP and CT are nonlinear functions with respect to the tip 
speed ratio and the pitch angle and have the following relation, 
CP(λ,β)= λ CT(λ,β). Where λ is tip speed ratio which is the ratio 
between the blade tip speed and the wind speed upstream the 
rotor and is given as: 



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r

w

ω R
λ=

v
(3) 

where ωr is the rotational speed of the rotor in (rad/s) and 
vw is effective wind speed (m/s). 

The blades are designed to ensure the highest energy 
coefficient outcome at the best speed on the tip speed ratio λopt. 
The energy coefficient is based on the speed at the tip and will 
fall under 0≤CP≤0.4. However, it will not be able to attain the 
maximum value Cp-Betz, referred to as Betz Limit, Cp<Cp-
Betz=0.593 [18]. The equation below illustrates utmost 
collected energy and aerodynamic torque of the wind turbine: 

  2 3max P,max opt w
1

P = C λ ,β ρπR v
2

(4) 

  5a,max P,max opt3
opt

1
T = C λ ,β ρπR

2λ
(5) 

B. Pitch Actuator System  
This is a non-linear system that rotates the wind turbine 

blades. It is created using the saturation of the amplitude which 
can be modeled as mathematical dynamic equations [23]. 
Figure 3 demonstrates the first-order system of the actuator. 
The equation below illustrates the changing performance of the 
pitch actuator. 

* 1 1
d   

   
    
   

(6) 

where: βmin≤ β ≤βmax, βmin
*≤β*≤βmax

*, β the actual pitch angle 
(0-45o) with ±10% max and β* the desired pitch angle. 

After calculating the desired pitch angle, the error in the 
closed loop has to go through a pitch angle limiter to bound the 
rate of change on the blades angle which results in increasing 
the blades lifetime [24]. 

 

 
Fig. 3.  Pitch angle actuator model  

C. Drive Train System 
Figure 4 shows the model made up of two mass mechanical 

models which are connected using a shaft with damping and 
stiffness coefficients. This shaft can be modeled as a torsion 
spring connecting two masses [26]. The mechanical model 
using the motion laws can be described as: 

  T T s
T

1
ω = T - K δθ+Bδω

J
 (7) 

θ=δω (8) 

 1 1g s g
G gear

K B T
J n

  
 

   
 
 

(9) 

T gδθ=θ -θ (10) 

g
T

gear

ω
δω=ω -

n
(11)

D. Generator System  
It is important to model only the generator side converter 

considering the control system design of the wind turbine. The 
following equation represents the overall component of the 
generator torque control action by the generator side for 
second-order dynamic system: 

 
 

2
g ng

* 2 2 2
g ng ng ng

T s ω
=

T s s +2ξ ω s+ω
   (12) 

Where Tg
* is the command value of the generator torque 

(wng ≈ 40 r/s) is the natural frequency, and (ξng ≈ 0.7) is the 
damping ratio. 

 

 
Fig. 4.  Two mass model for the driven train  

III. LINEARIZED MODEL OF WECS 
According to the rotor, the aerodynamic rotation, non-linear 

function of wind speed, pitch angle and rotor speed can be 
formulated into linear equations as below: 

( ) ( ) ( )a w r w r v r w w β r wδT =-β ω ,v ,β δw +-β ω ,v ,β δv +-β ω ,v ,β δβ (13)
Where: 

   

   

   

r w

r w

r w

a
w r w ω ,v ,β

r

a
v r w ω ,v ,β

w

a
β r w ω ,v ,β

δT
β ω ,v ,β = |

δw

δT
β ω ,v ,β = |

δv

δT
β ω ,v ,β = |

δβ



For state space equation: 

X=AX+BU+GW  (14) 
Y=CX  (15) 

Where: 



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r w sh g

* * T
g

w

Τ

T
g

]X=[δw δv δT       δT δβ

U=[δT δβ

Y=[ δv δP

]

]



Then: 

2

0 0

0 0 0 0 0

0

0 0 0 0

0 0 0

A

1

1
0

w

T T T

shsh w sh sh
D sh

T T g T

g

r

N

J J J

NDND N D D
K N K

J J J J













 
 
 
 
 

  
         

 
 

 
 

 
 
 

 

0 1 0 0 0
1 ,  ,     

0 0

0 0

0 0 0

0

0
0

0

   
0 0 1 0

1

v

T

sh v
r

g T

r

J

B G and CND

J










   
   
   
   

    
       

    
   
   
   

  
 

Equation (13) shows the wind speed model based on (17) 
[28]. The model consists of tower shadow, rotational sampling 
and wind shear effects. The working positions are ascertained 
by the regulatory strategy. However, it has limitations such as 
energy, rotor, wind speed and pitch angle [32]. Equations (14) 
and (15), increase and assess the various wind speeds. 
Equations (14) and (15) are the linear model of the entire 
system. The state vector and the regulatory input are the 
assessed results. The system is described as MIMO which has a 
changing dynamics whenever there is a change of wind speeds. 

IV. PROBLEM DESCRIPTION AND PROPOSED MPC APPROACH 
The main task is to regulate the rotor speed of the turbine 

which will in turn enhance the smoothness in the turbines 
making them efficient; especially at the limited load regime. 
The process however has a limitation which occurs during the 
creation of the regulator. The regulator is able to completely 
enhance the changed efficiency while at the same time 
reducing the temporary loads. The outcome indicated that 
variances existed between two objectives which required a 
balancing position [34]. The objective of regulation when 
operating at full load regime is to control the generator 
resulting energy and equally control the generator at set 
standards. The greatest limitation at this point is the big 
percentages of variations in the turbines energy. The 
percentages are caused by fluctuations in wind speed. The 
limitation also leads to a fluctuation in the drive train torsion 
torque as well as output electric power delivered to the grid. 
This power must be regulated thereby reducing the WECS 
apparatus and the flicker troubles [36]. 

Ordinary tests on the full load and partial regimes occur 
when the systems dynamics are not linear and viable variations 
occur at the operational center. Another test occurs when the 
system has cyclic aerodynamic torque difference and the 
turbine blades are rotating at three times their speed. This 
occurrence is known as the rotation sampling effect which is 
caused by uneven wind. Similarly, regulatory systems must 
identify regulatory goals kept at a certain level when facing 
limitations. For example, the highest level 1 is based on 
amplitude, speed pitch servos, generator energy and the speed 
of the turbines. Normally, variable speed and pitch consist of 
two working areas.  The first is undertaken in partial load 
region with the aim of deriving the highest level of energy by 
regulating the turbine rotor speed. The second working area is 
the full load region where regulatory processes are able to alter 
the resulting energy as well as the speed of the generator at 
standardized figures. This creates difficulties especially in the 
WECS design since the MIMO system is not linear. In 
addition, factors such as the stochastic and sustainable 
differences of the speed of the wind and the restrictions by the 
system limit must be considered. The model illustrated above 
provides further complications to the regulatory design process. 
When assessed in the same level as the fixed speeds wind 
turbines, the variable speeds wind turbine has greater benefits 
such as increased production of energy on lower costs [37]. 

MPC control method can alter, regulate and predict 
perceptions of predictive models. It can also alter limitations 
and different weights. When varying scenarios are run on the 
system, which could either be linear or non-linear, the MPC 
can assess the regulatory performance. This can be seen in the 
altering of chemical processes and electromechanical systems. 
It is most beneficial when used in predictive model and is 
relevant in deriving regulatory actions by reducing certain 
functions which are objective. MPC optimization takes into 
consideration reduction of the variances existing in the 
predicted and reference parameters. The regulatory attempt is 
also based on the determined limitations.  The MPC takes the 
following approach: 

• The expected outputs for the anticipated prospect n, are 
predicted at each instant, t using the process model. 

• For k=1, …, n, the predicted outputs y=(t+k(t)) depends on 
the known values up to instant t (past inputs and outputs) and 
on the future control signals u=(t+k(t)), for k=0, ...,n-1, are 
calculated out of the process. 

• The future control signals set is determined by an enhancing 
categorization to keep the process as close to the trajectory of 
the reference w=(t+k). 

• The reference trajectory can be the set point itself or a close 
approximation of it.  

• Typically, this standard takes the appearance of a quadratic 
role of the errors between the predicted output signal and 
reference trajectory.  

• The regulatory indication u(t) is taken to the system while 
the next regulatory indicator calculated the rejected, since the 
next sampling instant y(t+1) is known and the step 1 is 
repeated with this new value. 



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MPC model is vital in the prediction of future outputs 
arising from current values and expected future regulation 
action based on general composition as illustrated in Figure 5. 
When using the linear model, over prediction are based on 
open and compulsory answers. The optimizer allows for the 
selection of the most beneficial regulation systems hence 
reducing cost functions. This is carried out by getting the 
weighted sum of the square predicted inaccuracy as well as 
square predictable regulatory figures. They are calculated at the 
high and low predicted possibilities and are known as the 
regulation perspective. The regulatory perspective allow the 
reduction allow the reduction in the predicted regulation based 
on the Δu(k+j) for J≥Nu. The w(k+j). This signifies the 
location route over the regulatory perspective N. 

 

 
Fig. 5.  MPC Controller 

Limitations placed on the regulatory indicators, the outputs, 
and regulatory indicator varying can be included in the 
following equation: 

( ) ( ) ( ) ( )( ) ( ) ( )( )
2

1

2 2

1 2 u
1

J N ,N ,N = β j y k+j -w k+j + λ j u k+ (16)j -1
uNN

j N j= =
å å

 
If N1, N2 fall at the bottom and highest prediction level on 

top of the results, then Nu is the regulatory possibility, β(j), λ(j) 
are weighting variables. Limitation on the regulatory 
indicators, results as well as regulatory indicators varying can 
be included in equation below: 

 
 

 

min max

min max

min max

u u k u

u u k u

y y k y

 

    

 



The outcome of (16) derives the best order of regulation 
indicators based on the perspective N and at the same time 
recognizing limitation on (17). The MPC has many benefits 
such as the ability to control different systems, easy to use on 
complex systems, and offer prediction even if the outcome is 
delayed. Similarly, it can bring about the predicted results in 
systems which are closed loop. It can also be used to achieve 
the best outcomes with recognition of limitations. Although the 
benefits are diverse, it requires skills for anyone to model the 
system and handle its limitations. The MPC model is more 

complex as compared to PID since it requires more 
calculations. 

V. RESULTS AND DISCUSSIONS 
The considered 600 kW turbine system consists of three 

blades and pitch regulator. The feature of the MPC is an 
advantage over traditional PID controller including nonlinear 
control action. It can also reduce the noise, disturbance and 
parameters variations. The limitations for partial-load and full-
load operations with MPC and with PID controllers are similar. 
They include wind velocity, angular momentum, pitch angle, 
generator power as well as torque output for 300 seconds. The 
system parameters are given in the appendix. 

A. Partial-Load Operation 
Figure 6 shows the results from the reputation exercise 

conducted for 300 seconds of partial-load WECS process. The 
outcomes reveal that the MPC and the PID regulators had 
identical tracing performance as observed in the speed and 
energy of the generator. This is resulted from the consistency 
of the return speed from the generator in tracking the maximum 
point tracking. Nevertheless, amounts of these speed and 
energy variations reduce with the use of the MPC control 
strategy. Therefore, the system has a better performance with 
MPC approach compared with traditional PID control. 

B. Full-Load Operation 
Figure 7 shows a duplicate outcome key for an operation 

that consists of a full load. It is important in examining the 
strength of the suggested MPC against uncertainties in the 
parametric. The power production when using the MPC leads 
to a more even and decreased fluctuation levels in the power 
output.  The value of the decrease on the fluctuation of power 
production is enabled and enhanced based on the speed 
variations in the generator when compared to the PID control 
method as shown in Figure 7.Since the speed of wind, up to the 
first 50 seconds, is below the rated value of wind speed (10m/s) 
as shown in Figure 7a. In this period, the pitch angle is forced 
to level zero and the torque devolved is affected. From 50 to 
218 seconds time period, the angle pitch is adjusted in order to 
extract the maximum power from the wind turbine above the 
rated wind speed as shown in Figure 7d. 

Compared to the PID controller that forces the pitch angle 
to zero at early time of 218 seconds, the MPC approach 
enhances the wind power exploitation. Table I shows the 
evaluation of performance in the two systems and this arises 
because of the sustainable differences in the wg* which follows 
the MPPT. However, the scales of these fluctuations reduce 
with the utilization of the MPC regulation approach. When the 
MPC is compared with other methods based on performance, 
the MPC is vital for either strong or mild wind speeds. Table I 
illustrates average speed in the generator, the average energy, 
as well as the highest pitch motion. The outcomes from Table I 
indicate captured power shut by union at the mild or strong 
wind speeds. Nevertheless, the highest level of the pitch angle 
was reduced by the MPC regulator as compared to the PID. 
The requested method proved most valuable as compared to 
other methods based on the level of performance. 



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Fig. 6.  Partial-load operation 

TABLE I.  OUTPUT PARAMETERS FOR MPC AND PID CONTROLLERS 

Mild wind velocity Strong wind velocity 
Item 

MPC PID MPC PID 
Power output, pu. 0.97 0.73 0.94 0.87 
Pitch angle, deg. 26 13 34 26 

Generator speed, pu 0.89 0.81 0.89 0.78 

 

VI. CONCLUSION  
The main objective of this paper is to examine the variable 

speed and the pitch turbine regulatory designs of WECS, with 
the goal of extracting maximum wind energy and enhancing 
the progress against full load and partial load operation. The 
paper illustrates how the MPC controller regulates the wind 

turbine and introduces a system which is dynamic based on its 
speed, pitch wind turbine as well as limitations in the of the 
rotation and pitch angle regulator. The system is modeled and 
simulated using MATLAB/SIMULINK. The obtained results 
at partial-load and full-load operations show that the studied 
system with the proposed MPC approach has a better 
performance compared to the traditional PID control in terms 
of speed response and energy extraction. 

 

 

 

 

 

Fig. 7.  Full-load operation 

ACKNOWLEDGMENT 

This project was funded by the Deanship of Scientific Research 
(DSR), King Abdulaziz University, Jeddah, under Grant no. (829-236-
D1435). The authors, therefore, acknowledge with thanks DSR's 
technical and financial support. 



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