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Multivariable robust blade pitch control design to reject periodic loads on wind turbines

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Abstract

The demand on sustainable operations of large-scale wind turbines necessitates the concurrent advancement of power regulation and load mitigation through blade pitch control. Traditional collective pitch control (CPC) mechanisms can only deal with symmetric disturbances. The advent of individual pitch control (IPC) provides new opportunities to mitigate asymmetric or periodic loads on blades. Nevertheless, difficulties in control synthesis remain. In order for IPC to be truly effective, the complicated dynamic coupling between turbine components has to be accounted for. Moreover, wind turbine dynamics is highly nonlinear, and significant modeling uncertainties exist. In this research, a multivariable robust IPC framework is developed, aiming at rejecting periodic loads. The inter-blade coupling is explicitly modeled to provide response characteristics in the frequency domain. Subsequently, the structured singular values (μ)-synthesis strategy is adopted, as it shows distinct capability of dealing with periodic loads. In particular, weighting functions can be tailored to suppress response peaks at periodic frequencies with guaranteed robustness. Systematic case investigations indicate that, with the proposed IPC strategy, one can achieve significant periodic load mitigation as well as fatigue alleviation in speed-varying wind fields.
Multivariable Robust Blade Pitch Control Design to Reject Periodic Loads on Wind Turbines
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Abstract
      2      .  3  2      
32
*&, 32  
    *4&,    3                 2
5             4      4&    2      
   23 2    2     3
2                  6  3    2
34224&37
0.26
    8  28     *9,.
      3 2   3 2  
8432
 7   8 3  2   
34&3
.3
Keywords: :2  2 0   2  9.

1. Introduction
;3223
 8 2 22
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 *&,  3    2  0   3  
   3  <-.+>        
    2     A  6  ;  B4%C          3
.68*A.CB ,
223<$> %2327
2323
3<)!>%32
*%,23</))>
43323;20
   2  2 3       <)#>   
.324*4&,
3        2            7       
<)#>2
BDE  3     4 <)=>      
;    3    .  .     3      
3=P*,6 4
<)->4&233.223
3  33 6
32223236222
3 <->  %    3  3   2    3  
36
;  2  2   2
A3243
3 323  3  4&
2
2   : 2  
3%
8;
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 2 
322 8
73232<#@>
 37
3 :2
 3 3    C = 2   
22.32:2
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32323:.
2 *I,   8
233:;
  *
,.  2 2
7          3       4        3

.78%
2 22 23
2
;332I
 #2237 =
          -  
; /
2. Wind Turbine Modeling
2.1 General equations of motion
%3273 6
/$=J I;K3 <#+>32;32'
;632632G3;63
237.%;32
22326
  3 22 3 
8227
%  3  
  3         
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4 3   A % 2 *A% , 
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; 282 3LA *
,6;
3.8 <=#> 2023
3  3    3 LA    LA      
3LA.LA%30
322
2322
       A    2     2  ;  2.  3      
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      3 2 <@ #@>  4  2   3 
3 <#@==>3 
4;32
 3        ;    A%  
63
d d
d d
 
 
x Ax Bu B u
y Cx Du D u
*/,
3x2yu
d
u
2
2 )  
2 #  
2 =  
 
2 )  
2 #  
2 =  
 
2 )  2 
2 # 
 
 
 
 
 
 
 
 
 
 
 
x
y 2 
2 =  2 
 
 
 
 
 
 
*",
2 )  
2 #  
2 =  
; 2. 3 
; 3 
 3 
d
 
 
 
 
 
 
 
 
 
 
u
u
*@,
4 3 2    23.   8     
  3  2         6 
 4 2.          8 
6   G 3    2.     3   
8<#$>4333
22
;2.3
2.2 Multi-blade coordinate (MBC) transformation
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B43 L
33
3348*/,2A% 322
6 <=->4
23233
                3      3     .2    *I,
                       .  
*363,
3  3    I          .
 2 <=/>  C   3   I  3 2   
  23  ;  2      ;          2
2<=">;326
I<=->
2626
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   
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   
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 
 
*+,
 
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  
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  
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T
   
   
 
 
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   
 
   
 
   
 
 
   
 
   
 
   
 
*$,
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%    * 3 ,  A%   ; .
 .2 ;   .        2
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2 2 2 2 2 2 2
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2.6%32B4
326332
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2.3 Open-loop transfer functions
% ##I B4 6
;332 72
564
33233 ;
2 G3 3
2 ; :
)+H)-H)"H#!H
##HA)A#
    .  8    23              3

-100
0
100 tilt
0
180
360
-100
0
100
10 010 2
-720
0
720
yaw
10 010 2
Uncertain model
Nominal model
Bode Diagram
Frequency (Hz)
A)833
23*)+H3,3
*)-H)"H#!H##H3,
A )3 833
:32
               8        ;  
2)+H 3A% :3)-H)"
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H#! H##H 382A% 
3332B4
*)+H,3323
8 22.
3
-50
0
50
100
ud avg
-720
0
720
1440
-100
0
100
10 0
-720
0
720
ud tilt
10 0
ud yaw
10 0
Uncertain model
Nominal model
Bode Diagram
Frequency (Hz)
A#823
23*)+H3,
3*)-H)"H#!H##H3,
A#382*;
2, 3  : 2 2 ; 
23*

d
u
3
d
u
,@!I+!I;3*

d
u
,
       #! I  =! I 3     
 2 2    2 2  
37
3. Robust Individual Control Formulation
6  3 2       
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    nP 8 *3 P    8     
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 % 3 2     20   2 6 
 3  3 3      <-> 3 3  2 
0  
 2 02; :
 3   27 5CB .2 3 2    
L3    23 3   
38
3.1 Baseline controllers
4       3
22.
*E &4,<#$>E &43;2
    2  2    :     3  3
2283023
      2    &4.2   
3..2 <)=>
: 72
3%3&4
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   ! 
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)P    4 3  3  8   # P =P  -P 8
3    3.8   4  3        3
2   6  I   3    4 
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2
3.2 Proposed control structure for individual pitch control
% ;67   
3243
      3             2  ;    
         A = 3 3   
273
       L                    3
            2  2; 
     4       2   
     3    6   I    
     3  27  
I3

G3
H

H

3..32323
34223
  3      0    8        8      
A226
;
H
2
r
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M
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A=27
3.3 Structured singular value (µ)-synthesis
4      3        ;        *9,.  
22;3A=

μ
.3222
  2           3
7   3       M. 
8
3A-:
2222G
3              2           2    
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332.<=@>
&2*C&,3A63
2w2238=P*
6,3206 S
      8     4            3  6
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W
    
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W W I
3
p
W
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l
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   .
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 =P *!"G;,80#P-PR8
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  
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  G 8 2 -P   
223
3 3 
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23 2   
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       2   
23:
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W W I
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u
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3
32323223 
82
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A/383
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3
2  2         6
  2         <=@>     
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         2      
..2
3320 3
3   2   ; 
  3  2         A "  4  2
238.33
 .        =P 8    3  
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A@.8333
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!)@+-C C&
  3     A @   2     3
        3      4                     
   L    2                8  
.383
      3  8   :    = P 8      .    
2                 .  .      2

4. Results and Performance Comparison
                      
362H 7
 5CBA%  <=#>   
27322 <#$>2
322 <-!>3)!.
I323323
26
  2 . *E &4, <#$> &4.2
 <)=>2243
2&4.22
E &4 
E &43</>
4.1 Nominal performance around the operating point
:      6               :    
323)+H3)-J2
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*,*2,
*,*,
A+.E &4&4.2
2*,:*)+H23,N*2,E
N*,&N*,I)3
A+*,323A+*2,S+*,3.
2)2)3:
2)  22# 2= 
AA +*2,322&4.2
2        62              
  D  C      
&4.22!!+#=
!!@-=    !!@/-                    3
 2    2   
        % 3  A +*,    2  
   8      3   5  
38 C 
        &4.2            2    
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8*B,3
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4.2 Robust performance
62333
 :3.
2)2)3)-H23
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 !!/$+!!/-"!!/-@   )+H 23
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C &4.2
          2        3    A  $*,  
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4.3 Load mitigation and fatigue alleviation
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    &  2 3  4
2& 2+@"$J+@+$J
2
4  3   2       &    =P 8
2 =P 8  223   =-

A 3 6  2    3 3     
)+H:.3)-H)"H)+H#!H
##H8*B,A))3
C= 3
2 3B 2 3 B3 2. 
B32.BA)) %
          2   I    &4            2
23B 
3 #!J    3     2  3  B  2
 2 2    3        2 /J  
32.B66223
      3 2 .  B   2 =J.)!J 2  
   L    2     ;
233

*,*2,
#-
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/@"
/@@
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/+!
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/+#
/+=
/+-
/+/
/+"
/+@
/++
/+$
/$!
/$)
/$#
/$=
/$-
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
*,*,
A))23B*,23B*2,
32.B*,32.B*,
&42 ' '&4
 'C2
5. Conclusion
          3  2      3        2
%  7     3        3
22C=
3%322
223  .2  *I, 
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Acknowledgment
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References
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... They modeled the inter-blade coupling to provide response characteristics in the frequency domain. In this research, Systematic case investigations demonstrate that, with the proposed IPC strategy, one can achieve significant periodic load mitigation as well as fatigue alleviation in speed-varying wind fields [14]. In the same year, Civelek published a paper on optimizing fuzzy logic (Takagi-Sugeno) blade pitch angle controller in wind turbines by genetic algorithm. ...
... Using definitions (15)(16), model (14) leads to the following model (17): ...
... In the third step of the proposed algorithm, the optimization problem P1 is mentioned, which is entirely described in the following: Therefore, the proposed algorithm will design a proper output feedback controller for model (14) under conditions (4-6). ...
Article
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A pitch-regulated wind turbine has an exclusive pitch activator for every single blade, and it is possible to send various pitch angle demands to each blade. They possess a controller to perform this task, and the problem of delay-dependent robust stability with polytopic-type uncertainties of these time-varying delay systems must be resolved. This paper deals with the dynamic output feedback robust stabilization of the large wind turbine generator in the presence of time-varying delay and polytopic uncertainty. Two critical assumptions are considered for the turbine model: uncertain parameters and the blade-pitch control input actuated by a time-varying unknown delay parameter. A set of intervals are considered for the uncertain and delay parameters, which are assumed to be given and known. Then, a novel algorithm is proposed to design a proper controller for this system based on the Lyapunov-Krasovskii functional approach. The proposed controller simultaneously compensates for the effects of both delay parameters and uncertain parameters. To validate the results in this study, two simulation examples are proposed considering different turbines to compare the performance of the designed controller with previously designed controllers. The results reveal the proposed controller's superiority over the existing controller.
... On the other hand, mechanical stresses are another important challenge, which require powerful optimal or adaptive approaches to protect WTs [7]. Therefore, some pitch angle controllers have been proposed for WT blades between cut-out and rated wind speeds [6][7][8][9][10][11][12][13][14]. In [6], a digital controller was designed; classical controllers such as PID (proportional-integral-derivative) are proposed in [7,8]; a PID controller with an adaptive self-tuning regulator (STR) was constructed in [9]; a gain-scheduled PID controller was designed in [10]; a PI controller scheme is shown in [11]; and a combination of adaptive and PI controllers is presented in [12]. ...
... Therefore, some pitch angle controllers have been proposed for WT blades between cut-out and rated wind speeds [6][7][8][9][10][11][12][13][14]. In [6], a digital controller was designed; classical controllers such as PID (proportional-integral-derivative) are proposed in [7,8]; a PID controller with an adaptive self-tuning regulator (STR) was constructed in [9]; a gain-scheduled PID controller was designed in [10]; a PI controller scheme is shown in [11]; and a combination of adaptive and PI controllers is presented in [12]. Some simple nonlinear feedback controllers are proposed in [13,14]. ...
... ω r = −K g n g ω r − T g + T ls n g (8) or: n g 2 J g ...
Article
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To achieve the maximum power from wind in variable-speed regions of wind turbines (WTs), a suitable control signal should be applied to the pitch angle of the blades. However, the available uncertainty in the modeling of WTs complicates calculations of these signals. To cope with this problem, an optimal controller is suitable, such as particle swarm optimization (PSO). To improve the performance of the controller, fractional order PSO (FPSO) is proposed and implemented. In order to construct this approach for a two-mass WT, we propose a new state feedback, which was first applied to the turbine. The idea behind this state feedback was based on the Taylor series. Then, a linear model with uncertainty was obtained with a new input control signal. Thereafter, the conventional PSO (CPSO) and FPSO were used as optimal controllers for the resulting linear model. Finally, a comparison was performed between CPSO and FPSO and the fuzzy Takagi–Sugeno–Kang (TSK) inference system. The provided comparison demonstrates the advantages of the Taylor series with combination to these controllers. Notably, without the state feedback, CPSO, FPSO, and TSK fuzzy systems cannot stabilize WTs in tracking the desired trajectory.
... Meanwhile, the complex motion of wind turbines has higher requirements for the structural load, and how to reduce structural cost and maintenance cost is also the main research content. At present, many control methods (Tang et al., 2022;Sarkar et al., 2020;Yuan et al., 2020;Zhang and Plestan, 2021) can reduce the blade load/fatigue and stabilize the platform, but based on the fixed design layout, these control methods have limited ability to reduce the cost. Therefore, novel layout design is the most fundamental solution to reduce the cost, and this section investigates its practical significance according to the corresponding wind energy conversion capacity and rough blade cost estimation of three design methods. ...
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With the continuously increasing installed capacity of offshore wind turbines, highly adaptable vertical axis wind turbines (VAWTs) are facing new opportunities. Large-scale offshore platforms need a megawatt installed capacity, at least 1∼2 MW, which requires the feasibility analysis of aerodynamic characteristics for large-scale VAWTs. Due to the unsteady aerodynamic phenomenon, i.e. the dynamic stall phenomenon, the power output of VAWTs is very sensitive to the variations of Reynolds number and reduced frequency, which are closely related to the scale of wind turbines. In order to explore the large-scale VAWTs for offshore platforms, three design methods are proposed: Increasing the Reynolds number; Decreasing reduced frequency; Forming an array. Their feasibility and economies are verified by a high-resolution numerical method, and the results show that increasing the Reynolds number could improve their power coefficients to 0.259 for a one-blade VAWT, while an excessive decrease of reduced frequency would lead to power losses. The form of installing one or more levels of VAWT arrays on a single platform is a better design scheme with averaged power coefficient of 0.326 and a blade weighing 2.7 tons per 10 MW, and for large offshore platforms, it could reach the lowest levelized cost of energy.
... The control is combined with the IPC to reduce 1p frequency of blades fatigue loads, and its performance has been investigated by simulation studies. A multivariable robust IPC framework control by accounting for inter-blade coupling is presented in (Yuan et al., 2020). To deal with periodic loads, a structured singular values (μ)-synthesis strategy is employed where weighting functions are tailored to suppress oscillations at periodic frequencies. ...
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