Feedforward Feedback Pitch Control for Wind Turbine Based...
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Research Article Feedforward Feedback Pitch Control for Wind Turbine Based on Feedback Linearization with Sliding Mode and Fuzzy PID Algorithm Haijun Ren , Hao Zhang, Guang Deng, and Bin Hou School of Advanced Manufacture and Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China Correspondence should be addressed to Haijun Ren; [email protected] Received 7 January 2018; Accepted 23 May 2018; Published 20 June 2018 Academic Editor: Kauko Leivisk¨ a Copyright © 2018 Haijun Ren et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Aſter exceeding rated power, variable speed variable pitch wind turbines need to keep output powers at rated value by adopting pitch angles. With the typical nonlinear characteristics of the wind turbines, it is difficult to control accurately by conventional linear controller. ough feedback control can realize the stability of the system, it is effective only when deviations are produced. As such, feedforward control can be applied to compensate the time-delay produced by feedback control. Accordingly, we propose a compound control strategy that combines feedback controller with feedforward controller in this paper. In feedback loop, we adopt fuzzy algorithm to adjust the parameters of PID controller. Furthermore, to overcome large variation of input wind speed, variable universe theory is proposed to optimize fuzzy algorithm. In feedforward loop, we propose feedback linearization to address nonlinear problem. Furthermore, sliding mode algorithm is supplied to improve the robustness of feedback linearization. erefore, feedforward loop can efficiently compensate time-delay deficiency of wind turbine systems. Simulation results show that the proposed controller can enhance the control accuracy and robustness of the system. 1. Introduction Wind energy is an important renewable energy so that wind turbines are rapidly developing in the world for decades [1– 4]. In the beginning, small wind turbine is the mainstream as its construction is simple and control algorithms are rarely used and therefore its power control greatly depends on aero- dynamics itself. With the development of modern control theories and power electronic technologies, variable speed variable pitch technology is widely used to adjust the output powers of wind turbines [5–9]. us, large wind turbine is rapidly developed and occupies the main commercial market. Usually, the working scope of wind turbine can be divided into three regions in Figure 1 [10, 11]. Region I is from cut in wind speed to rated speed of electric motor. In this stage, the optimal output power is extracted from wind by controlling electromagnetic torque. e process of working is shown in region II when wind speed is over the rated value. With the increasing of wind speed, output power is increasing. Output power reaches its nominal value when wind speed approaches the rated value. If wind speed continues to increase, the output power would be kept at the rated value for the requests of mechanism and security operation, which is called region III. In this region, it is an effective method that adjusts pitch angle to keep output power at constant. Various control algorithms have been greatly studied these years. Classic PID controller is widely employed in control system because its structure is simple and can be easily realized. However, wind turbine is a strongly nonlinear system so that linear PID controller cannot meet the requests of the system. Usually, PID algorithm is combined with intelligent algorithm to build a new controller whose control performance can be greatly improved. In [11], a nonlinear PI controller is proposed for realizing the optimum perfor- mance of wind turbine system. e introduced nonlinear controller consists of an extended-order state and perturba- tion observer (ESPO) with a classic PI controller. e ESPO can tackle nonlinear and disturbance problems. However, the presented algorithm needs the same accuracy system model as the feedback linearization method. In [12], an adaptive Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 4606780, 13 pages https://doi.org/10.1155/2018/4606780
Transcript of Feedforward Feedback Pitch Control for Wind Turbine Based...
Research ArticleFeedforward Feedback Pitch Control for Wind TurbineBased on Feedback Linearization with Sliding Mode and FuzzyPID Algorithm
Haijun Ren Hao Zhang Guang Deng and Bin Hou
School of AdvancedManufacture and Engineering Chongqing University of Posts and Telecommunications Chongqing 400065 China
Correspondence should be addressed to Haijun Ren renhjcqupteducn
Received 7 January 2018 Accepted 23 May 2018 Published 20 June 2018
Academic Editor Kauko Leiviska
Copyright copy 2018 Haijun Ren et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
After exceeding rated power variable speed variable pitch wind turbines need to keep output powers at rated value by adoptingpitch angles With the typical nonlinear characteristics of the wind turbines it is difficult to control accurately by conventionallinear controller Though feedback control can realize the stability of the system it is effective only when deviations are producedAs such feedforward control can be applied to compensate the time-delay produced by feedback control Accordingly we proposea compound control strategy that combines feedback controller with feedforward controller in this paper In feedback loop weadopt fuzzy algorithm to adjust the parameters of PID controller Furthermore to overcome large variation of input wind speedvariable universe theory is proposed to optimize fuzzy algorithm In feedforward loop we propose feedback linearization toaddress nonlinear problem Furthermore sliding mode algorithm is supplied to improve the robustness of feedback linearizationTherefore feedforward loop can efficiently compensate time-delay deficiency of wind turbine systems Simulation results show thatthe proposed controller can enhance the control accuracy and robustness of the system
1 Introduction
Wind energy is an important renewable energy so that windturbines are rapidly developing in the world for decades [1ndash4] In the beginning small wind turbine is the mainstreamas its construction is simple and control algorithms are rarelyused and therefore its power control greatly depends on aero-dynamics itself With the development of modern controltheories and power electronic technologies variable speedvariable pitch technology is widely used to adjust the outputpowers of wind turbines [5ndash9] Thus large wind turbine israpidly developed and occupies themain commercialmarket
Usually the working scope of wind turbine can be dividedinto three regions in Figure 1 [10 11] Region I is from cut inwind speed to rated speed of electric motor In this stage theoptimal output power is extracted from wind by controllingelectromagnetic torque The process of working is shown inregion II when wind speed is over the rated value With theincreasing of wind speed output power is increasing Outputpower reaches its nominal valuewhenwind speed approaches
the rated value If wind speed continues to increase theoutput power would be kept at the rated value for the requestsof mechanism and security operation which is called regionIII In this region it is an effective method that adjusts pitchangle to keep output power at constant
Various control algorithms have been greatly studiedthese years Classic PID controller is widely employed incontrol system because its structure is simple and can beeasily realized However wind turbine is a strongly nonlinearsystem so that linear PID controller cannot meet the requestsof the system Usually PID algorithm is combined withintelligent algorithm to build a new controller whose controlperformance can be greatly improved In [11] a nonlinearPI controller is proposed for realizing the optimum perfor-mance of wind turbine system The introduced nonlinearcontroller consists of an extended-order state and perturba-tion observer (ESPO) with a classic PI controller The ESPOcan tackle nonlinear and disturbance problems However thepresented algorithm needs the same accuracy system modelas the feedback linearization method In [12] an adaptive
HindawiMathematical Problems in EngineeringVolume 2018 Article ID 4606780 13 pageshttpsdoiorg10115520184606780
2 Mathematical Problems in EngineeringPo
wer
I II IIIV
Prated
Vcut-in VΩrated Vrated Vcut-out
Figure 1 The operation regions of wind turbine
pitch controller is adopted to solve system uncertainty byupdating parameters online The controller includes twoparts reference model and adaptive algorithm Adopting thismethod the generator speed can be kept at constant whileload is reduced In [13] a two-degree-of-freedom (2-DOF)controller is designed to adjust output power The controlstructure comprises a feedforward controller and a feedbackcontroller In feedforward loop a feedback linearization algo-rithm is proposed that can compensate the nonlinear featureOn the other hand a PID controller is adopted as feedbackloop which can tackle unknown disturbance and parametervariation In [14] a new intelligent genetic algorithm is usedto optimize proportional-integral-derivative parameters inwhich two operations are suggested in order to search thelocal minimum or the maximum value including increasingmutation rate and crossover point In [15] artificial neuralnetworks are adopted to solve system nonlinearization anddifficultly inverse function In [16] a controller for adjustingboth blade pitch and generator torque is suggested In thiscontrol method measurement values of wind speeds will notbe needed by means of designing a sliding surface to guar-antee the estimated aerodynamic power that is equal to theactual oneTherefore the structure of the system is simplifiedIn [17] a control strategy mixing1198672119867infin algorithm based onglobal exact linearization is suggested which can implementbetter performance in large range near operation points andalleviate loads of wind turbine In [18] in order to improvepitch systemperformance of wind turbine optimization esti-mation and compensation are synthetically proposed Firstlyan optimal PI controller is developed by utilizing directsearch optimization which has better control performance indelay-free pitch system Moreover a delay estimator is usedto estimate the interference in order to address time-delayproblem As delay-perturbation affects output value a signalcompensationmethod is further utilized to offset the error In[19] an adaptive sliding mode controller and a back-steppingcontroller based on pump displacement and stroke pistonof pitch system for wind turbine are proposed which canaccurately track the desired values of pitch angle In [20] afuzzy sliding mode controller is used to adjust pitch anglewithout accurate mathematical models In pitching fuzzycontroller is adopted to approximate the main controllerinput while fuzzy sliding mode controller is used to com-pensate the difference between the former two controllers
In [21] H Camblong et al firstly linearized wind turbinesystem at the operating point Subsequently linear quadraticGaussian (LQG) controllers are designed to reduce fatigueload and the suggestedmethod is verified by test experimentsAs wind speed is constantly changing in order to improvecontrol performance LPV gain scheduling is often adopted[22ndash26] However design of classic LPV algorithm requiresmuch time and cannot guarantee global stability ThereforeLPV usually combines variable gain scheduling with robustcontrolThe above all studied control algorithms for adjustingoutput power of wind turbine have already obtained excellentcontrol performance in different situations However somecontrol strategies can be further improved
As time-delay of pitch system the output power of windturbine shows overshoot phenomenon Although feedbackcontrol can fulfill tracking control it cannot overcomepower overshoot problem However overshoot could bedecreased by feedforward control in presetting control vari-ables Accordingly we adopt the same control structure asin [13] which consists of feedback control combining withfeedforward control In [13] a feedback control strategycombining with feedforward control was proposed How-ever in feedback loop only a conventional PID controllerwas applied as feedback controller which cannot meet thecontrol requirements of the nonlinear system To improvethe performance of the feedback controller we adopt avariable universe fuzzy algorithm to optimize the parametersof the PID controller In [13] feedback linearization theorywas proposed to overcome the nonlinear feature of windturbines in feedforward loop However the robustness ofthe feedback linearization controller should be improvedas it needs precise objective model Nevertheless slidingmode variable structure algorithm can overcome modeluncertainty problem for its adaptive capability Accordinglyin feedforward loop we adopt sliding mode algorithm tooptimize feedback linearization controller so that robustnesscan be enhanced
2 Model of Wind Turbine Generator System
Wind rotates turbine Then wind energy can be convertedinto electrical energy through electric generator motivatedby drive train that is connected to the turbine rotor Energyconversion system is depicted in Figure 2 Shaft powerextracted by wind turbine can be given from [11 12 27]
119875 = 119879119903120596119903 = 121205881205871198772V3119862119901 (120582 120573) (1)
119869119903 120596119903 = 119879119903 minus 119879119897 (2)
119862119901 (120582 120573) = (044 minus 00167120573) sin( 120587 (120582 minus 3)15 minus 03120573)minus 000184 (120582 minus 3) 120573
(3)
120582 = 120596119903119877V
(4)
where 119875 is the shaft power of wind turbine 119879119903 is theaerodynamic torque of turbine rotor 120596119903 is the rotor speed
Mathematical Problems in Engineering 3
M
gear boxwind
blade
generator
Figure 2 The block diagram of energy conversion of wind turbine
of turbine rotor 120588 is the air density 119877 is the radius ofturbine rotor V is the wind speed 119862119901(120582 120573) is the conversioncoefficient of wind energy 119869119903 is the rotational inertia ofturbine rotor 119879119897 is the torque of low speed shaft 120582 is the tip-speed ratio and 120573 is the pitch angle
The shaft power of wind turbine is proportional to thecube of wind speed known from (1) so that the outputpower will be strongly affected when wind speed increasesTherefore in order to keep output power at constant powerconversion coefficient will be decreased when wind speed isover the rated value In addition known from (3) and (4)power conversion coefficient is controlled by rotor speed ofturbine rotor and pitch angle of wind turbine when windspeed is changing After the output power of wind turbineis over the rated value rotor speed of electric generatoris constant Consequently adjusting pitch angle is a mainmethod for controlling the output power On the other handbased on (1) it is known that output power can also bemaintained at the rated value by adjusting torque
Due to the fact this paper focuses on algorithms ofadjusting power In order to simplify the model of thedrive train the system structure is assumed to be rigidthat will not affect researching wind turbine system Therelationship between low speed shaft and high speed shaft canbe expressed in [12]
119896 = 120596119892120596119903 =119879119897119879ℎ (5)
where 119896 is the gear ratio of the drive train 120596119892 is the angularspeed of high speed shaft and 119879ℎ is the torque of high speedshaft
In high speed section of the drive train the relationshipcan be expressed in [17]
119869119892 120596119892 = 119879ℎ minus 119879119890 (6)
where 119869119892 is the rotational inertia of electric generator and 119879119890is the torque of electric generator
The relationship can be gained from (5) as follows
119879119897 = 119896119879ℎ (7)
120596119892 = 119896120596119903 (8)
The relationship can be acquired based on (6) as
119879ℎ = 119869119892 120596119892 + 119879119890 (9)
Substituting (7) (8) and (9) into (2) the relationship canbe further written as
120596119903 (119869119903 + 1198962119869119892) = 119879119903 minus 119896119879119890 (10)
Controlling the pitch angle of variable pitch wind turbineis to maintain output power at nominal value when windspeed is over the rated value Pitch system can be expressedapproximately as a first-order inertia system in [17]
120573 = 1120591 (1205731 minus 120573) (11)
where 120573 is the pitch angle 120591 is the time constant and 1205731 isthe desired pitch angle When wind speed is changing thedesired pitch angle is the output value of controllerThereforethe real-time pitch angle can be calculated by using integralalgorithm based on (11)
According to double-fed generator equivalent circuitstator equation and rotor equation the expression of elec-tromagnetic torque can be expressed in
119879119890119898 = minus31199041199011199092119898119903212059611198882 11988021 + 3119901119909
2119898119903112059611198882 11988022
+ 311990111990911989812059611198882 [(11990311199032 + 11990411990921205901199091120590 + 1199041199092119898) sin12057212
minus (11990411990921205901199031 minus 11990911205901199032) cos12057212]11988011198802
(12)
where 119888 = radic(11990311199032 minus 11990411990921205901199091120590 + 1199041199092119898)2 + (11990411990921205901199031 + 11990911205901199032)2 119901 isthe pole pairs 1205961 is the synchronous angular velocity 119904 isthe slip rate 1198801 is the stator voltage 1198802 is the rotor voltage12057212 is the phase difference between stator and rotor 1199031 is thestatorwinding resistance1199091120590 is the leakage reactance 1199032 is therotorwinding resistance calculated to stator1199092120590 is the leakagecalculated to stator side and 119909119898 is the excitation reactance
The electromagnetic power of doubly fed motor can beexpressed in
119875119890 = 119879119890119898Ω1 = 1198791198901198981205961119901 = minus3119904119909211989811990321198882 11988021 + 3119909211989811990311198882 11988022
+ 31199091198981198882 [(11990311199032 + 11990411990921205901199091120590 + 1199041199092119898) sin12057212minus (11990411990921205901199031 minus 11990911205901199032) cos12057212]11988011198802
(13)
where Ω1 is the generator synchronous mechanical angularvelocity
3 Feedforward Feedback Controller Based onFeedback Linearization with Sliding Modeand Fuzzy Self-Tuning PID
Known from (3) the energy conversion coefficient 119862119901(120582 120573)is nonlinear function including 120582 and 120573 Therefore the shaft
4 Mathematical Problems in Engineering
feedback linearization
PID
wind turbine model
sliding modealgorithm
fuzzy rules
dedt
input scaling factor
output scaling factor
ＳＣＨ
ＳＩＯＮ
+
+
+
minus
u1
u2
Figure 3 The system control structure of wind turbine
power expression of wind turbine is nonlinear As windturbine system model is a typical nonlinear system the con-troller designed based on linear system is not the best choiceHowever nonlinear controller is a better idea In this paperwe propose a novel feedforward feedback algorithm whosefeedback section is fuzzy self-tuning PID algorithm andits feedforward section is feedback linearization algorithmHowever feedback linearization control needs an accuratesystem model and it is subject to parameter variationConsequently its application is limited As sliding modecontrol is not affected by parameter variation of systemand disturbances robustness of system can be improved bycombining sliding mode method with feedback linearizationalgorithm The chart that describes this algorithm is shownin Figure 3 Differential equations of wind turbine can beacquired from Section 2 as follows
120596119903 (119869119903 + 1198962119869119892) = 12120596119903 1205881205871198772V3119862119901 (120582 120573) minus 119896119879119892
120573 = 1120591 (1205731 minus 120573)(14)
Assuming state variables 1199091 = 120596119903 1199092 = 120573 control input119906 = 1205731 state equations can be given as
1199091 = 1119869119903 + 1198962119869119892 (121199091 120588120587119877
2V3119862119901 (120582 1199092) minus 119896119879119892)1199092 = 1120591 (119906 minus 1199092)
(15)
Then output equation can be expressed as follows
119910 = 1199091 (16)
31 Fuzzy Self-Tuning PIDAlgorithm Due to the nonlinearityand time-varying characteristics of the wind turbine theconventional PID cannot meet the control requirementsThe
fuzzy algorithm transforms input values by fuzzifier andmakes a judgment based on fuzzy rules which can control themore complex systems [28ndash32] In this paper we use the fuzzyalgorithm to adjust the parameters of the PID controller in thefeedback control loop to improve the control performance ofthe controller
Generator velocity will be a constant when wind speedis overrated value As drivetrain is considered stiffness windturbine velocity is also a constant However the velocitywill fluctuate in a very small range for there are randomdisturbances Accordingly we select wind turbine velocity asreference variable When pitching the input variables of thefuzzy controller are the wind turbine velocity errors and theerror changing ratesThe output variables are the proportionintegral and differential gains of the PID controller Errorand error changing rate corresponding to the fuzzy domainare taken as minus3 minus2 minus1 0 +1 +2 +3 Fuzzy rules arethe key of fuzzy control Not only the correctness of fuzzycontrol rules directly affects the performance of the controllerbut their number is also an important factor to measurecontroller performance Therefore we define seven fuzzysubsets and the language variables are negative big negativemiddle negative small zero positive small positive middlepositive big expressed as NB NM NS Z PS PM andPB respectivelyThese language variables cannot only clearlyexpress universe but the control rule number is appropriateFuzzy rules are shown in Tables 1ndash3 They can be furtherdescribed as follows
If e is NB and ec is NB then output is PBIf e is NB and ec is NM then output is PBIf e is NB and ec is NS then output is PM
The membership function determines the sensitivity ofthe fuzzy control performance If the membership functionchangesmore smoothly the stability of the controller is betterOn the other hand if the membership function is steeperthen the controller has high sensitivity Therefore when the
Mathematical Problems in Engineering 5
Table 1 The fuzzy rules of KP
eNB NM NS Z PS PM PB
NB PB PB PM PM PS Z ZNM PB PB PM PS PS Z NSNS PM PM PM PS Z NS NS
ec Z PM PM PS Z NS NM NMPS PS PS Z NS NS NM NMPM PS Z NS NM NM NM NBPB Z Z NM NM NM NB NB
Table 2 The fuzzy rules of KI
eNB NM NS Z PS PM PB
NB NB NB NM NM NS Z ZNM NB NB NM NS NS Z ZNS NB NM NS NS Z PS PS
ec Z NM NM NS Z PS PM PMPS NM NS Z PS PS PM PBPM Z Z PS PS PM PB PBPB Z Z PS PM PM PB PB
Table 3 The fuzzy rules of KD
eNB NM NS Z PS PM PB
NB PS NS NB NB NB NM PMNM PS NS NB NM NM NS ZNS Z NS NM NM NS NS Z
ec Z Z NS NS NS NS NS ZPS Z Z Z Z Z Z ZPM PB PS PS PS PS PS PBPB PB PM PM PM PS PS PB
error and error rate of change are largeGaussianmembershipfunction with smooth change is selected which makes thesystem change smoothly andhave good robustnessWhen theerror and error changing rate are both small the triangularmembership function with high sensitivity is proposed toimprove the system response The diagram of fuzzy systemwith two-input three-output is shown in Figure 4 Parametersof fuzzy controller are shown in Table 5
e is the error and ec is the error changing rateThe outputof PID controller is as follows
1199061 = 119870119875 (119890 (119905) + 1119879119868 int 119890 (119905) 119889119905 + 119879119863119889119890 (119905)119889119905 ) (17)
where 1199061 is the output of PID controller 119870119875 is the propor-tional coefficient 119890(119905) is the error119879119868 is the integral coefficientand 119879119863 is the differential coefficient
32 Variable Universe Theory Fuzzy PID controllers exhibitgood control performance Nevertheless when input vari-ables change a lot the control accuracy needs to be further
System fuzzpid 2 inputs 3 outputs 49 rules
e (7)
ec (7)
kp (7)
ki (7)
kd (7)
fuzzpid
(mamdani)
49 rules
Figure 4 The fuzzy system structure
improved The idea of variable universe is to make any inputcan correspond to the entire input universe by adding ascaling factor without increasing the fuzzy rules based onthe changes of the input [33] If the original basic universeis adopted control accuracy is not ideal Furthermore whenthe control error becomes smaller its corresponding rulescan be reduced However the rules corresponding to theinput can be increased by adjusting scaling factor if thevariable universe method is implemented In other wordsthe corresponding rule density is increased Therefore thecontrol accuracy is improvedThe key of the variable universeapproach is to determine the scaling factor which shouldmeet the following requirements
(1) Duality for (forall119909 isin 119883) 120572(119909) = minus120572(minus119909) is a necessarycondition
(2) Avoid zero 120572(0) = 120576(3) Monotonicity 120572(119909) is strictly monotonically increas-
ing from 0 to 119864(4) Coordination forall119909 isin 119883 119909 le 120572(119909)119864(5) Regularity 120572(plusmn119864) = 1Thereby the universe scaling factors are expressed in
120572 (119909) = (|119909|119864 )120583 + 120576 (18)
120578 (119909) = (|119909|119864 )1205831 ( 10038161003816100381610038161199101003816100381610038161003816119864119862)
1205832 + 120576 (19)
where 120572(119909) is the scaling factor of the input universe 120578(119909) isthe scaling factor of the output universe 119909 is the error 119910 is theerror rate of change 119864 is the value of the error universe 119864119862 isthe value of the error changing rate universe 120583 1205831 and 1205832 areconstants from 0 to 1 and 120576 is a full small positive number
The process of changing universe is shown in Figure 5
33 Design of Feedback Linearization of Nonlinear Model forWind Turbine Feedback linearization algorithm is to adda nonlinear compensation part in control input variablesThen a new equivalent controlled system can be gained Theoriginal system is transformed into linear form As feed-back linearization algorithm is based on strict mathematicalderivation and control input is expressed in output equation
6 Mathematical Problems in Engineering
NB NM NS ZE PS PM PB
minusE EO
O
O
(A)
(B)
(C)X
X
minus(x)E (x)E
minus(x)E (x)E
Universe expansion
Universe com
pression
Figure 5 The diagram of changing universe process
the nonlinear system model can be linearized by designingappropriate control rules Therefore control performance ofsystem can be improved distinctly
The affine nonlinear models of single input single outputare expressed as [13]
= 119891 (119909) + 119892 (119909) 119906119910 = ℎ (119909) (20)
where119909 = [1199091 1199092 sdot sdot sdot 119909119899]119879 are the 119899-dimensional state vectors119891(119909) and 119892(119909) are the fully smooth vector fields in 119877119899 119906 isin 119877119910 isin 119877 and ℎ(119909) is the fully smooth nonlinear functionThe expression can be gained as follows based on (15)
(16) and (20)
119891 (119909) = [1198911 (119909)1198912 (119909)]
= [[[[
1119869119903 + 1198962119869119892 (121199091 120588120587119877
2V3119862119901 (120582 1199092) minus 119896119879119892)minus11205911199092
]]]]
(21)
119892 (119909) = [[01120591]]
(22)
ℎ (119909) = 1199091 (23)
Known from (23) there is no direct relationshipbetween the output equation and the control inputTherefore
derivative of time for output equation must be implementedas
119910 = 119889ℎ (119909)119889119905 = 120597ℎ (119909)120597119909 120597119909120597119905= 120597ℎ (119909)120597119909 [119891 (119909) + 119892 (119909) 119906]= 120597ℎ (119909)120597119909 119891 (119909) + 120597ℎ (119909)120597119909 119892 (119909) 119906
(24)
120597ℎ (119909)120597119909 = [1 0] (25)
Equations (26) and (27) can be gained based on (25)
120597ℎ (119909)120597119909 119891 (119909) = 119871119891ℎ (119909) = 1198911 (119909) (26)
120597ℎ (119909)120597119909 119892 (119909) = 119871119892ℎ (119909) = 0 (27)
Substituting (26) and (27) into (24) it is seen that 119910does not include 119906 Then derivative of time for (24) will becontinued
119910 = 120597119871119891ℎ (119909)120597119909 120597119909120597119905 =120597119871119891ℎ (119909)120597119909 [119891 (119909) + 119892 (119909) 119906]
= 1198712119891ℎ (119909) + 119871119892119871119891ℎ (119909) 119906(28)
In (28)
1198712119891ℎ (119909) = 120597119871119891ℎ (119909)120597119909 119891 (119909)= [1205971198911 (119909)1205971199091
1205971198911 (119909)1205971199092 ] [1198911 (119909)1198912 (119909)]
= 1205971198911 (119909)1205971199091 1198911 (119909) +1205971198911 (119909)1205971199092 1198912 (119909)
(29)
119871119892119871119891ℎ (119909) = 120597119871119891ℎ (119909)120597119909 119892 (119909)
= [1205971198911 (119909)12059711990911205971198911 (119909)1205971199092 ][[
01120591]]= 1205971198911 (119909)1205971199092
1120591(30)
Therefore
119871119892119871119891ℎ (119909) = 0 (31)
From now on input variable 119906 is expressed in 119910 and thecondition for meeting feedback linearization is gained
34 Input-Output Feedback Linearization Based Sliding ModeControl Both linear controller and nonlinear controller areusually adopted to control nonlinear system However linearcontroller can only control linear system well In nonlinearsystem if linear controller is used nonlinear model willbe usually linearized Sliding mode control is a nonlinear
Mathematical Problems in Engineering 7
s=0slt0
sgt0
A BC
Figure 6 The schematic diagram of sliding mode hypersurface
algorithm and it can continuously adjust itself based on real-time state variables during dynamic process for system totrack the state trajectory of sliding mode [34 35] In thispaper we propose sliding mode algorithm to enhance therobustness of feedback linearization
Assume that there exists a system as follows
= 119891 (119909) 119909 isin 119877119899 (32)
In its state space there is a hypersurface 119904(119909) =119904(1199091 1199092 sdot sdot sdot 119909119899) = 0 depicted in Figure 6The state space is divided into two parts 119904 gt 0 and 119904 lt 0 in
hypersurface In switching surface 119904 = 0 the point in systemwill be close to termination point when it is near switchingsurface
If all points are termination points in a special field inthe switching surface once the points move to this area towhich they will be attracted it is called sliding mode areaThemovement of system in slidingmode area is called slidingmode action
As the moving point is termination point in sliding modearea the expression can be gained as follows when the pointmoves to a place near the sliding mode surface
In other words
lim119904rarr0+
119904 le 0 le lim119904rarr0minus
119904 (33)
The other expression is formulated as
lim119904rarr0119904 119904 le 0 (34)
Assume that a function is given by
V (1199091 1199092 sdot sdot sdot 119909119899) = [119904 (1199091 1199092 sdot sdot sdot 119909119899)]2 (35)
The derivative of time for (35) can be expressed as follows
1198891199042119889119905 = 2119904 119904 (36)
According to (34) (36) is negative semidefinite formTherefore (35) is a Lyapunov function of system Then thesystem is stable when 119904 = 0
According to these analyses above system performancecan be improved by combining sliding mode control withfeedback linear algorithm because of stable advantage ofslidingmode control To gain good tracking performance theerror between reference and output is defined as
119890 = 119910119894119899 minus 119910 (37)
where 119910119894119899 is the ideal tracked object and 119910 is the controloutput
Then sliding function can be obtained as follows
119904 (119909 119905) = ce (38)
where
c = [119888 1] (39)
119888 gt 0 (40)
e = [119890 119890]119879 (41)
According to the discussed study for wind turbine in 33the control rule can be designed as
1199062 = 1(1205971198911 (119909) 1205971199092) (1120591) (V
minus (1205971198911 (119909)1205971199091 1198911 (119909) +1205971198911 (119909)1205971199092 1198912 (119909)) + 120578 sgn (119904))
(42)
where V is the aided term 120578 ge 0 and sgn(119904) is the signfunction which is formulated by
sgn (119904) = 1 119904 gt 0minus1 119904 lt 0 (43)
Simulation results show that there is large fluctuationin output power and rotor speed when using sign functionIn order to reduce the fluctuation saturation function isimplemented to replace sign function Saturation function isdefined as follows Results are shown in Figures 7(a)ndash7(d) andFigures 8(a)ndash8(d)
119904119886119905 (119904) =
1 119904 gt 120575119904120575 |119904| le 120575minus1 119904 lt minus120575
(44)
Up to now the final control law can be expressed basedon (17) and (42) as follows
119906 = 1199061 + 1199062 (45)
4 Simulation Results and Analyses
Simulation experiments based on a MW scale wind turbineare carried out to verify the effectiveness of the proposedalgorithm in which parameters are depicted in Table 4 Thecontroller parameters are shown in Table 5
The simulation experiments for wind turbine pitchingcontrol are implemented using step wind and random windStep speed is shown in Figure 7(a) and random wind isdepicted in Figure 8(a)
Firstly Figures 7(b)ndash7(d) and Figures 8(b)ndash8(d) depictthe control results when using sign function and saturationfunction The results show that there are little differences in
8 Mathematical Problems in Engineering
Table 4 The parameters of the wind turbine
Symbols Parameters Values119875119903119886119905119890119889 Rated output power 3000KW119881119903119886119905119890119889 Rated wind speed 12ms119877 The Radius of the turbine rotor 475m119869119903 The inertia moment of the turbine rotor 6250000kgsdotm2119869119892 The inertia moment of the generator 15kgsdotm2119896 Drive ratio 80120591 Time constant 021199031 Stator winding resistance 00148Ω1199032 Rotor winding resistance calculated to stator 00370Ω1199091120590 Leakage reactance 00727Ω1199092120590 Leakage calculated to stator side 00863Ω119909119898 Excitation reactance 20000Ω
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Step wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19992
200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(d) Output powers using different functions in sliding mode
Figure 7 (a) Step wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions in slidingmode (d) output powers using different functions in sliding mode
Mathematical Problems in Engineering 9
Table 5 The parameters of the proposed controller
Symbols Parameters Values119888 Constant defined in sliding mode algorithm 10119890 119903119886119899119892119890 The universe of rotor deviation [-0003 0003]119890119888 119903119886119899119892119890 The universe of rotor deviation changing rate [-0008 0008]119870119875 119903119886119899119892119890 The universe of proportionality coefficent [-03 03]119870119868 119903119886119899119892119890 The universe of integral coefficient [-006 006]119870119863 119903119886119899119892119890 The universe of differential coefficient [-3 3]119864 The value of the error universe 0003119864119862 The value of the error changing rate universe 0008120583 The coefficient of universe scaling factor 091205831 The coefficient of universe scaling factor 091205832 The coefficient of universe scaling factor 04120576 A full small positive number 000001
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Random wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
25
3
35
4
P(M
W)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
(d) Output powers using different functions in sliding mode
Figure 8 (a) Random wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions insliding mode (d) output powers using different functions in sliding mode
10 Mathematical Problems in Engineering
5
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
Local enlargeddrawing
4 5 6 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under step wind speed
9 10 11 12
2
2001Localenlargeddrawing
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under step wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
9 10 11 12
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) output power using different controllers under step wind speed
Figure 9 (a) Pitch angles using different controllers under step wind speed (b) rotor speed using different controllers under step wind speedand (c) output power using different controllers under step wind speed
the control when implementing the two different functionsunder the step wind speed However under random windspeed the control performance of the saturation functionadjusting output power is significantly better than that of thesign function in Figure 8(d)
Comparisons are further studied among which thereare conventional PID algorithm feedforward feedback con-trol combining PID with feedback linearization algorithmfeedforward feedback control based on fuzzy self-tuningPID with feedback linearization with sliding mode control(called fuzzy PID in feedback loop in Figures 9(a)ndash9(c)
and Figures 10(a)ndash10(c)) and the proposed controller inthis paper Results are depicted in Figures 9(a)ndash9(c) andFigures 10(a)ndash10(c) The wind speed is changed continuouslyto further validate pitching control algorithms which isdepicted in Figure 8(a)
At the initial moment control outputs fluctuate stronglywhich are caused by initial values of states However iffluctuations are in acceptable ranges of the physical structureof the wind turbine the initial responses will not affect theevaluation for the controller In Figure 7(a) at 119905 = 4 and119905 = 10 step wind speed occurs As thewind speeds exceed the
Mathematical Problems in Engineering 11
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
12 13 14 15
5 Local enlargeddrawing
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
8 9 10 11
1999
2
2001
Localenlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
13 14 15 16
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) Output power using different controllers under random wind speed
Figure 10 (a) Pitch angles using different controllers under random wind speed (b) rotor speed using different controllers under randomwind speed and (c) output power using different controllers under random wind speed
rated value pitch angles need to be adjusted to keep outputpower at constant
In Figure 9(a) the overshoot of the pitch angle is greaterwhile oscillation time is longer when adopting the conven-tional PID controller Although rotor speed also convergesat the rated value steady-state errors cannot be eliminatedwhich is approved in rotor speed chart depicted in Fig-ure 9(b) The rotor speeds of wind turbine are maintainedat constant when wind speed exceeds the rated value Dueto the steady-state errors of the pitch angle rotor speedshave deviations These results show that single linear PID
controller cannot track target well for wind turbine exhibitsstrongly nonlinear features
To address this problem we design a PID controllercombining with feedback linearization theory The PIDcontroller is feedback control while feedback linearizationis proposed as feedforward control Feedback control cankeep the system stable However the feedback control ishysteric Only when the deviation occurs the controller isimplemented to eliminate it Nevertheless feedforward canproduce a control variable in advanceThus the final outputsof the two controllers are proposed as the input of the wind
12 Mathematical Problems in Engineering
turbine system so that tracking accuracy of controller can beenhanced
Figures 9(a)ndash9(c) demonstrate that the control perfor-mance of feedforward feedback controller based on com-bining PID algorithm with feedback linearization algorithmis clearly superior to conventional PID controller in whichthe overshoot of pitch angle is smaller and adjusting timeof oscillating is shorter and furthermore the steady-statedeviation is eliminated which is also shown both in therotor speed diagram and in the output power diagram Tofurther improve control performance sliding mode controlis combined with feedback linear algorithm in feedforwardloop while the fuzzy self-tuning algorithm is used to adjustthe parameters of the PID controller in feedback loopFurthermore in order to decrease the effect when inputvariable and output variable change a lot variable universeapproach is proposed to optimize fuzzy rules Known fromthe results the deficiencies of controller produced by systemconstruct and parameters variation can be overcome FromFigures 9(a)ndash9(c) it is shown that the performance of thecontroller we propose in this paper is the best among thefour controllers whose overshoot is the least while it rapidlyconverges to constant without steady-state errors Althoughcontinuous step speed can evaluate controller performanceit still exhibits limitation
In order to further verify the effectiveness and universal-ity of the proposed controller in this paper we further adoptrandom wind speed Figure 8(a) depicts random wind speedFigures 10(a)ndash10(c) show the control results of the pitch anglerotor speed of turbine and output power produced by fourdifferent control algorithms under random wind speed Theresults are reasonable which are the same as those producedby step wind speed Therefore the control algorithm wepropose in this paper is the best among the described fouralgorithms and it can keep the output power at the rated valuewhen wind speed exceeds the nominal velocity
5 Conclusions
In this research a novel pitch control strategy for largevariable speed variable pitch wind turbines is studied aimingat maintaining the output power at constant so that theenergy quality is improved Known from the results as thesliding mode theory is combined with feedback linearizationalgorithm in feedforward loop the uncertainty of modeland time-varying of system parameters are overcome Fur-thermore with setting a control variable in advance theovershoot can be decreased when wind speed is varyingOn the other hand in feedback loop as variable universefuzzy algorithm is employed to optimize the parameters ofthe conventional PID controller the tracking performanceof the feedback controller is enhanced Meanwhile insteadof sign function saturation function is implemented todecrease oscillation of sliding mode algorithm and thus thefluctuations of output power and rotor speed of wind turbineare mitigated Both control accuracy and robustness usingthe proposed algorithm are the best among four differentcontrollers Another in future we plan to supply fuzzy
predictive adaptive algorithm in power control to furtherimprove system performance
Data Availability
The data used to support the findings of this study arecurrently under embargo while the research findings arecommercialized Requests for data 6 months after publica-tion of this article will be considered by the correspondingauthor
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National Natural ScienceFund of China under Grant no 51405052
References
[1] C-SWang andM-H Chiang ldquoA novel dynamic co-simulationanalysis for overall closed loop operation control of a large windturbinerdquo Energies vol 9 no 8 2016
[2] E Van Solingen J Beerens S P Mulders R De Breuker andJ W Van Wingerden ldquoControl design for a two-bladed down-wind teeterless damped free-yaw wind turbinerdquo Mechatronicsvol 36 pp 77ndash96 2016
[3] Y-J Chen and Y C Shiah ldquoExperiments on the performance ofsmall horizontal axis wind turbine with passive pitch control bydisk pulleyrdquo Energies vol 9 no 5 2016
[4] A Choudhry M Arjomandi and R Kelso ldquoMethods tocontrol dynamic stall for wind turbine applicationsrdquo Journal ofRenewable Energy vol 86 pp 26ndash37 2016
[5] H Li C Yang Y Hu X Liao Z Zeng and C Zhe ldquoAnimproved reduced-ordermodel of an electric pitch drive systemfor wind turbine control system design and simulationrdquo Journalof Renewable Energy vol 93 pp 188ndash200 2016
[6] E B Muhando T Senjyu A Uehara and T Funabashi ldquoGain-scheduledHinfin control forWECS via LMI techniques and para-metrically dependent feedback part II Controller design andimplementationrdquo IEEE Transactions on Industrial Electronicsvol 58 no 1 pp 57ndash65 2011
[7] F D Bianchi H De Battista and R JMantz ldquoRobustmultivari-able gain scheduled control of wind turbines for variable powerproductionrdquo International Journal of Systems Control vol 1 no3 pp 103ndash112 2010
[8] I Soslashnderby andMH Hansen ldquoOpen-loop frequency responseanalysis of a wind turbine using a high-order linear aeroelasticmodelrdquoWind Energy vol 17 no 8 pp 1147ndash1167 2014
[9] S Bououden M Chadli S Filali and A El Hajjaji ldquoFuzzymodel basedmultivariable predictive control of a variable speedwind turbine LMI approachrdquo Journal of Renewable Energy vol37 no 1 pp 434ndash439 2012
[10] J Chen T Lin CWen and Y Song ldquoDesign of a Unified PowerController for Variable-Speed Fixed-Pitch Wind Energy Con-version Systemrdquo IEEETransactions on Industrial Electronics vol63 no 8 pp 4899ndash4908 2016
Mathematical Problems in Engineering 13
[11] Y Ren L Li J Brindley and L Jiang ldquoNonlinear PI control forvariable pitch wind turbinerdquo Control Engineering Practice 2015
[12] Y Yuan and J Tang ldquoAdaptive pitch control of wind turbinefor load mitigation under structural uncertaintiesrdquo Journal ofRenewable Energy vol 105 pp 483ndash494 2017
[13] C-S Wang and M-H Chiang ldquoA novel pitch control systemof a large wind turbine using two-degree-of-freedom motioncontrol with feedback linearization controlrdquo Energies vol 9 no10 article no 791 2016
[14] Z Civelek E Cam M Luy and H Mamur ldquoProportional-integral-derivative parameter optimisation of blade pitch con-troller in wind turbines by a new intelligent genetic algorithmrdquoIET Renewable Power Generation vol 10 no 8 pp 1220ndash12282016
[15] A Dahbi N Nait-Said andM-S Nait-Said ldquoA novel combinedMPPT-pitch angle control for wide range variable speed windturbine based on neural networkrdquo International Journal ofHydrogen Energy vol 41 no 22 pp 9427ndash9442 2016
[16] M L Corradini G Ippoliti and G Orlando ldquoAn observer-based blade-pitch controller of wind turbines in high windspeedsrdquo Control Engineering Practice vol 58 pp 186ndash192 2017
[17] M Zhu J Liu Z Lin and H Meng ldquoMixed H2 Hinfin pitchcontrol of wind turbine generator system based on globalexact linearization and regional pole placementrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 5 pp921ndash930 2016
[18] R Gao and Z Gao ldquoPitch control for wind turbine systemsusing optimization estimation and compensationrdquo Journal ofRenewable Energy vol 91 pp 501ndash515 2016
[19] X-X Yin Y-G Lin W Li H-W Liu and Y-J Gu ldquoAdaptivesliding mode back-stepping pitch angle control of a variable-displacement pump controlled pitch system for wind turbinesrdquoISA Transactions vol 58 pp 629ndash634 2015
[20] M-H Chiang ldquoA novel pitch control system for a wind turbinedriven by a variable-speed pump-controlled hydraulic servosystemrdquoMechatronics vol 21 no 4 pp 753ndash761 2011
[21] H Camblong S Nourdine I Vechiu and G Tapia ldquoCom-parison of an island wind turbine collective and individualpitch LQG controllers designed to alleviate fatigue loadsrdquo IETRenewable Power Generation vol 6 no 4 p 267 2012
[22] F A InthamoussouHDe Battista andR JMantz ldquoLPV-basedactive power control of wind turbines covering the completewind speed rangerdquo Journal of Renewable Energy vol 99 pp996ndash1007 2016
[23] F D Bianchi R J Mantz and C F Christiansen ldquoGainscheduling control of variable-speed wind energy conversionsystems using quasi-LPVmodelsrdquoControl Engineering Practicevol 13 no 2 pp 247ndash255 2005
[24] F A Shirazi K M Grigoriadis and D Viassolo ldquoWind turbineintegrated structural and LPV control design for improvedclosed-loop performancerdquo International Journal of Control vol85 no 8 pp 1178ndash1196 2012
[25] F A Inthamoussou F D Bianchi H De Battista and R JMantz ldquoLPV wind turbine control with anti-windup featurescovering the complete wind speed rangerdquo IEEE Transactions onEnergy Conversion vol 29 no 1 pp 259ndash266 2014
[26] K Z Oslashstegaard J Stoustrup and P Brath ldquoLinear parametervarying control of wind turbines covering both partial loadand full load conditionsrdquo International Journal of Robust andNonlinear Control vol 19 no 1 pp 92ndash116 2009
[27] Y Lin L Tu H Liu and W Li ldquoHybrid power transmissiontechnology in a wind turbine generation systemrdquo IEEEASMETransactions onMechatronics vol 20 no 3 pp 1218ndash1225 2015
[28] S Bououden M Chadli F Allouani and S Filali ldquoA newapproach for fuzzy predictive adaptive controller design usingparticle swarmoptimization algorithmrdquo International Journal ofInnovative Computing Information and Control vol 9 no 9 pp3741ndash3758 2013
[29] S H Li and H Gu ldquoFuzzy adaptive internal model controlschemes for PMSM speed-regulation systemrdquo IEEE Transac-tions on Industrial Informatics vol 8 no 4 pp 767ndash779 2012
[30] A B Asghar and X Liu ldquoAdaptive neuro-fuzzy algorithmto estimate effective wind speed and optimal rotor speed forvariable-speed wind turbinerdquo Neurocomputing vol 272 pp495ndash504 2018
[31] Y Wang H Shen H R Karimi and D Duan ldquoDissipativity-based Fuzzy Integral SlidingModeControl ofContinuous-TimeT-S Fuzzy Systemsrdquo IEEE Transactions on Fuzzy Systems pp 1-1
[32] S Bououden M Chadli and H R Karimi ldquoAn ant colonyoptimization-based fuzzy predictive control approach for non-linear processesrdquo Information Sciences vol 299 pp 143ndash1582015
[33] NWang S-F Su J Yin Z Zheng andM J Er ldquoGlobal Asymp-totic Model-Free Trajectory-Independent Tracking Control ofan Uncertain Marine Vehicle An Adaptive Universe-BasedFuzzy Control Approachrdquo IEEE Transactions on Fuzzy Systems2017
[34] R Saravanakumar andD Jena ldquoValidation of an integral slidingmode control for optimal control of a three blade variable speedvariable pitch wind turbinerdquo International Journal of ElectricalPower amp Energy Systems vol 69 pp 421ndash429 2015
[35] C Evangelista P Puleston F Valenciaga and L M FridmanldquoLyapunov-designed super-twisting sliding mode control forwind energy conversion optimizationrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 538ndash545 2013
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Submit your manuscripts atwwwhindawicom
2 Mathematical Problems in EngineeringPo
wer
I II IIIV
Prated
Vcut-in VΩrated Vrated Vcut-out
Figure 1 The operation regions of wind turbine
pitch controller is adopted to solve system uncertainty byupdating parameters online The controller includes twoparts reference model and adaptive algorithm Adopting thismethod the generator speed can be kept at constant whileload is reduced In [13] a two-degree-of-freedom (2-DOF)controller is designed to adjust output power The controlstructure comprises a feedforward controller and a feedbackcontroller In feedforward loop a feedback linearization algo-rithm is proposed that can compensate the nonlinear featureOn the other hand a PID controller is adopted as feedbackloop which can tackle unknown disturbance and parametervariation In [14] a new intelligent genetic algorithm is usedto optimize proportional-integral-derivative parameters inwhich two operations are suggested in order to search thelocal minimum or the maximum value including increasingmutation rate and crossover point In [15] artificial neuralnetworks are adopted to solve system nonlinearization anddifficultly inverse function In [16] a controller for adjustingboth blade pitch and generator torque is suggested In thiscontrol method measurement values of wind speeds will notbe needed by means of designing a sliding surface to guar-antee the estimated aerodynamic power that is equal to theactual oneTherefore the structure of the system is simplifiedIn [17] a control strategy mixing1198672119867infin algorithm based onglobal exact linearization is suggested which can implementbetter performance in large range near operation points andalleviate loads of wind turbine In [18] in order to improvepitch systemperformance of wind turbine optimization esti-mation and compensation are synthetically proposed Firstlyan optimal PI controller is developed by utilizing directsearch optimization which has better control performance indelay-free pitch system Moreover a delay estimator is usedto estimate the interference in order to address time-delayproblem As delay-perturbation affects output value a signalcompensationmethod is further utilized to offset the error In[19] an adaptive sliding mode controller and a back-steppingcontroller based on pump displacement and stroke pistonof pitch system for wind turbine are proposed which canaccurately track the desired values of pitch angle In [20] afuzzy sliding mode controller is used to adjust pitch anglewithout accurate mathematical models In pitching fuzzycontroller is adopted to approximate the main controllerinput while fuzzy sliding mode controller is used to com-pensate the difference between the former two controllers
In [21] H Camblong et al firstly linearized wind turbinesystem at the operating point Subsequently linear quadraticGaussian (LQG) controllers are designed to reduce fatigueload and the suggestedmethod is verified by test experimentsAs wind speed is constantly changing in order to improvecontrol performance LPV gain scheduling is often adopted[22ndash26] However design of classic LPV algorithm requiresmuch time and cannot guarantee global stability ThereforeLPV usually combines variable gain scheduling with robustcontrolThe above all studied control algorithms for adjustingoutput power of wind turbine have already obtained excellentcontrol performance in different situations However somecontrol strategies can be further improved
As time-delay of pitch system the output power of windturbine shows overshoot phenomenon Although feedbackcontrol can fulfill tracking control it cannot overcomepower overshoot problem However overshoot could bedecreased by feedforward control in presetting control vari-ables Accordingly we adopt the same control structure asin [13] which consists of feedback control combining withfeedforward control In [13] a feedback control strategycombining with feedforward control was proposed How-ever in feedback loop only a conventional PID controllerwas applied as feedback controller which cannot meet thecontrol requirements of the nonlinear system To improvethe performance of the feedback controller we adopt avariable universe fuzzy algorithm to optimize the parametersof the PID controller In [13] feedback linearization theorywas proposed to overcome the nonlinear feature of windturbines in feedforward loop However the robustness ofthe feedback linearization controller should be improvedas it needs precise objective model Nevertheless slidingmode variable structure algorithm can overcome modeluncertainty problem for its adaptive capability Accordinglyin feedforward loop we adopt sliding mode algorithm tooptimize feedback linearization controller so that robustnesscan be enhanced
2 Model of Wind Turbine Generator System
Wind rotates turbine Then wind energy can be convertedinto electrical energy through electric generator motivatedby drive train that is connected to the turbine rotor Energyconversion system is depicted in Figure 2 Shaft powerextracted by wind turbine can be given from [11 12 27]
119875 = 119879119903120596119903 = 121205881205871198772V3119862119901 (120582 120573) (1)
119869119903 120596119903 = 119879119903 minus 119879119897 (2)
119862119901 (120582 120573) = (044 minus 00167120573) sin( 120587 (120582 minus 3)15 minus 03120573)minus 000184 (120582 minus 3) 120573
(3)
120582 = 120596119903119877V
(4)
where 119875 is the shaft power of wind turbine 119879119903 is theaerodynamic torque of turbine rotor 120596119903 is the rotor speed
Mathematical Problems in Engineering 3
M
gear boxwind
blade
generator
Figure 2 The block diagram of energy conversion of wind turbine
of turbine rotor 120588 is the air density 119877 is the radius ofturbine rotor V is the wind speed 119862119901(120582 120573) is the conversioncoefficient of wind energy 119869119903 is the rotational inertia ofturbine rotor 119879119897 is the torque of low speed shaft 120582 is the tip-speed ratio and 120573 is the pitch angle
The shaft power of wind turbine is proportional to thecube of wind speed known from (1) so that the outputpower will be strongly affected when wind speed increasesTherefore in order to keep output power at constant powerconversion coefficient will be decreased when wind speed isover the rated value In addition known from (3) and (4)power conversion coefficient is controlled by rotor speed ofturbine rotor and pitch angle of wind turbine when windspeed is changing After the output power of wind turbineis over the rated value rotor speed of electric generatoris constant Consequently adjusting pitch angle is a mainmethod for controlling the output power On the other handbased on (1) it is known that output power can also bemaintained at the rated value by adjusting torque
Due to the fact this paper focuses on algorithms ofadjusting power In order to simplify the model of thedrive train the system structure is assumed to be rigidthat will not affect researching wind turbine system Therelationship between low speed shaft and high speed shaft canbe expressed in [12]
119896 = 120596119892120596119903 =119879119897119879ℎ (5)
where 119896 is the gear ratio of the drive train 120596119892 is the angularspeed of high speed shaft and 119879ℎ is the torque of high speedshaft
In high speed section of the drive train the relationshipcan be expressed in [17]
119869119892 120596119892 = 119879ℎ minus 119879119890 (6)
where 119869119892 is the rotational inertia of electric generator and 119879119890is the torque of electric generator
The relationship can be gained from (5) as follows
119879119897 = 119896119879ℎ (7)
120596119892 = 119896120596119903 (8)
The relationship can be acquired based on (6) as
119879ℎ = 119869119892 120596119892 + 119879119890 (9)
Substituting (7) (8) and (9) into (2) the relationship canbe further written as
120596119903 (119869119903 + 1198962119869119892) = 119879119903 minus 119896119879119890 (10)
Controlling the pitch angle of variable pitch wind turbineis to maintain output power at nominal value when windspeed is over the rated value Pitch system can be expressedapproximately as a first-order inertia system in [17]
120573 = 1120591 (1205731 minus 120573) (11)
where 120573 is the pitch angle 120591 is the time constant and 1205731 isthe desired pitch angle When wind speed is changing thedesired pitch angle is the output value of controllerThereforethe real-time pitch angle can be calculated by using integralalgorithm based on (11)
According to double-fed generator equivalent circuitstator equation and rotor equation the expression of elec-tromagnetic torque can be expressed in
119879119890119898 = minus31199041199011199092119898119903212059611198882 11988021 + 3119901119909
2119898119903112059611198882 11988022
+ 311990111990911989812059611198882 [(11990311199032 + 11990411990921205901199091120590 + 1199041199092119898) sin12057212
minus (11990411990921205901199031 minus 11990911205901199032) cos12057212]11988011198802
(12)
where 119888 = radic(11990311199032 minus 11990411990921205901199091120590 + 1199041199092119898)2 + (11990411990921205901199031 + 11990911205901199032)2 119901 isthe pole pairs 1205961 is the synchronous angular velocity 119904 isthe slip rate 1198801 is the stator voltage 1198802 is the rotor voltage12057212 is the phase difference between stator and rotor 1199031 is thestatorwinding resistance1199091120590 is the leakage reactance 1199032 is therotorwinding resistance calculated to stator1199092120590 is the leakagecalculated to stator side and 119909119898 is the excitation reactance
The electromagnetic power of doubly fed motor can beexpressed in
119875119890 = 119879119890119898Ω1 = 1198791198901198981205961119901 = minus3119904119909211989811990321198882 11988021 + 3119909211989811990311198882 11988022
+ 31199091198981198882 [(11990311199032 + 11990411990921205901199091120590 + 1199041199092119898) sin12057212minus (11990411990921205901199031 minus 11990911205901199032) cos12057212]11988011198802
(13)
where Ω1 is the generator synchronous mechanical angularvelocity
3 Feedforward Feedback Controller Based onFeedback Linearization with Sliding Modeand Fuzzy Self-Tuning PID
Known from (3) the energy conversion coefficient 119862119901(120582 120573)is nonlinear function including 120582 and 120573 Therefore the shaft
4 Mathematical Problems in Engineering
feedback linearization
PID
wind turbine model
sliding modealgorithm
fuzzy rules
dedt
input scaling factor
output scaling factor
ＳＣＨ
ＳＩＯＮ
+
+
+
minus
u1
u2
Figure 3 The system control structure of wind turbine
power expression of wind turbine is nonlinear As windturbine system model is a typical nonlinear system the con-troller designed based on linear system is not the best choiceHowever nonlinear controller is a better idea In this paperwe propose a novel feedforward feedback algorithm whosefeedback section is fuzzy self-tuning PID algorithm andits feedforward section is feedback linearization algorithmHowever feedback linearization control needs an accuratesystem model and it is subject to parameter variationConsequently its application is limited As sliding modecontrol is not affected by parameter variation of systemand disturbances robustness of system can be improved bycombining sliding mode method with feedback linearizationalgorithm The chart that describes this algorithm is shownin Figure 3 Differential equations of wind turbine can beacquired from Section 2 as follows
120596119903 (119869119903 + 1198962119869119892) = 12120596119903 1205881205871198772V3119862119901 (120582 120573) minus 119896119879119892
120573 = 1120591 (1205731 minus 120573)(14)
Assuming state variables 1199091 = 120596119903 1199092 = 120573 control input119906 = 1205731 state equations can be given as
1199091 = 1119869119903 + 1198962119869119892 (121199091 120588120587119877
2V3119862119901 (120582 1199092) minus 119896119879119892)1199092 = 1120591 (119906 minus 1199092)
(15)
Then output equation can be expressed as follows
119910 = 1199091 (16)
31 Fuzzy Self-Tuning PIDAlgorithm Due to the nonlinearityand time-varying characteristics of the wind turbine theconventional PID cannot meet the control requirementsThe
fuzzy algorithm transforms input values by fuzzifier andmakes a judgment based on fuzzy rules which can control themore complex systems [28ndash32] In this paper we use the fuzzyalgorithm to adjust the parameters of the PID controller in thefeedback control loop to improve the control performance ofthe controller
Generator velocity will be a constant when wind speedis overrated value As drivetrain is considered stiffness windturbine velocity is also a constant However the velocitywill fluctuate in a very small range for there are randomdisturbances Accordingly we select wind turbine velocity asreference variable When pitching the input variables of thefuzzy controller are the wind turbine velocity errors and theerror changing ratesThe output variables are the proportionintegral and differential gains of the PID controller Errorand error changing rate corresponding to the fuzzy domainare taken as minus3 minus2 minus1 0 +1 +2 +3 Fuzzy rules arethe key of fuzzy control Not only the correctness of fuzzycontrol rules directly affects the performance of the controllerbut their number is also an important factor to measurecontroller performance Therefore we define seven fuzzysubsets and the language variables are negative big negativemiddle negative small zero positive small positive middlepositive big expressed as NB NM NS Z PS PM andPB respectivelyThese language variables cannot only clearlyexpress universe but the control rule number is appropriateFuzzy rules are shown in Tables 1ndash3 They can be furtherdescribed as follows
If e is NB and ec is NB then output is PBIf e is NB and ec is NM then output is PBIf e is NB and ec is NS then output is PM
The membership function determines the sensitivity ofthe fuzzy control performance If the membership functionchangesmore smoothly the stability of the controller is betterOn the other hand if the membership function is steeperthen the controller has high sensitivity Therefore when the
Mathematical Problems in Engineering 5
Table 1 The fuzzy rules of KP
eNB NM NS Z PS PM PB
NB PB PB PM PM PS Z ZNM PB PB PM PS PS Z NSNS PM PM PM PS Z NS NS
ec Z PM PM PS Z NS NM NMPS PS PS Z NS NS NM NMPM PS Z NS NM NM NM NBPB Z Z NM NM NM NB NB
Table 2 The fuzzy rules of KI
eNB NM NS Z PS PM PB
NB NB NB NM NM NS Z ZNM NB NB NM NS NS Z ZNS NB NM NS NS Z PS PS
ec Z NM NM NS Z PS PM PMPS NM NS Z PS PS PM PBPM Z Z PS PS PM PB PBPB Z Z PS PM PM PB PB
Table 3 The fuzzy rules of KD
eNB NM NS Z PS PM PB
NB PS NS NB NB NB NM PMNM PS NS NB NM NM NS ZNS Z NS NM NM NS NS Z
ec Z Z NS NS NS NS NS ZPS Z Z Z Z Z Z ZPM PB PS PS PS PS PS PBPB PB PM PM PM PS PS PB
error and error rate of change are largeGaussianmembershipfunction with smooth change is selected which makes thesystem change smoothly andhave good robustnessWhen theerror and error changing rate are both small the triangularmembership function with high sensitivity is proposed toimprove the system response The diagram of fuzzy systemwith two-input three-output is shown in Figure 4 Parametersof fuzzy controller are shown in Table 5
e is the error and ec is the error changing rateThe outputof PID controller is as follows
1199061 = 119870119875 (119890 (119905) + 1119879119868 int 119890 (119905) 119889119905 + 119879119863119889119890 (119905)119889119905 ) (17)
where 1199061 is the output of PID controller 119870119875 is the propor-tional coefficient 119890(119905) is the error119879119868 is the integral coefficientand 119879119863 is the differential coefficient
32 Variable Universe Theory Fuzzy PID controllers exhibitgood control performance Nevertheless when input vari-ables change a lot the control accuracy needs to be further
System fuzzpid 2 inputs 3 outputs 49 rules
e (7)
ec (7)
kp (7)
ki (7)
kd (7)
fuzzpid
(mamdani)
49 rules
Figure 4 The fuzzy system structure
improved The idea of variable universe is to make any inputcan correspond to the entire input universe by adding ascaling factor without increasing the fuzzy rules based onthe changes of the input [33] If the original basic universeis adopted control accuracy is not ideal Furthermore whenthe control error becomes smaller its corresponding rulescan be reduced However the rules corresponding to theinput can be increased by adjusting scaling factor if thevariable universe method is implemented In other wordsthe corresponding rule density is increased Therefore thecontrol accuracy is improvedThe key of the variable universeapproach is to determine the scaling factor which shouldmeet the following requirements
(1) Duality for (forall119909 isin 119883) 120572(119909) = minus120572(minus119909) is a necessarycondition
(2) Avoid zero 120572(0) = 120576(3) Monotonicity 120572(119909) is strictly monotonically increas-
ing from 0 to 119864(4) Coordination forall119909 isin 119883 119909 le 120572(119909)119864(5) Regularity 120572(plusmn119864) = 1Thereby the universe scaling factors are expressed in
120572 (119909) = (|119909|119864 )120583 + 120576 (18)
120578 (119909) = (|119909|119864 )1205831 ( 10038161003816100381610038161199101003816100381610038161003816119864119862)
1205832 + 120576 (19)
where 120572(119909) is the scaling factor of the input universe 120578(119909) isthe scaling factor of the output universe 119909 is the error 119910 is theerror rate of change 119864 is the value of the error universe 119864119862 isthe value of the error changing rate universe 120583 1205831 and 1205832 areconstants from 0 to 1 and 120576 is a full small positive number
The process of changing universe is shown in Figure 5
33 Design of Feedback Linearization of Nonlinear Model forWind Turbine Feedback linearization algorithm is to adda nonlinear compensation part in control input variablesThen a new equivalent controlled system can be gained Theoriginal system is transformed into linear form As feed-back linearization algorithm is based on strict mathematicalderivation and control input is expressed in output equation
6 Mathematical Problems in Engineering
NB NM NS ZE PS PM PB
minusE EO
O
O
(A)
(B)
(C)X
X
minus(x)E (x)E
minus(x)E (x)E
Universe expansion
Universe com
pression
Figure 5 The diagram of changing universe process
the nonlinear system model can be linearized by designingappropriate control rules Therefore control performance ofsystem can be improved distinctly
The affine nonlinear models of single input single outputare expressed as [13]
= 119891 (119909) + 119892 (119909) 119906119910 = ℎ (119909) (20)
where119909 = [1199091 1199092 sdot sdot sdot 119909119899]119879 are the 119899-dimensional state vectors119891(119909) and 119892(119909) are the fully smooth vector fields in 119877119899 119906 isin 119877119910 isin 119877 and ℎ(119909) is the fully smooth nonlinear functionThe expression can be gained as follows based on (15)
(16) and (20)
119891 (119909) = [1198911 (119909)1198912 (119909)]
= [[[[
1119869119903 + 1198962119869119892 (121199091 120588120587119877
2V3119862119901 (120582 1199092) minus 119896119879119892)minus11205911199092
]]]]
(21)
119892 (119909) = [[01120591]]
(22)
ℎ (119909) = 1199091 (23)
Known from (23) there is no direct relationshipbetween the output equation and the control inputTherefore
derivative of time for output equation must be implementedas
119910 = 119889ℎ (119909)119889119905 = 120597ℎ (119909)120597119909 120597119909120597119905= 120597ℎ (119909)120597119909 [119891 (119909) + 119892 (119909) 119906]= 120597ℎ (119909)120597119909 119891 (119909) + 120597ℎ (119909)120597119909 119892 (119909) 119906
(24)
120597ℎ (119909)120597119909 = [1 0] (25)
Equations (26) and (27) can be gained based on (25)
120597ℎ (119909)120597119909 119891 (119909) = 119871119891ℎ (119909) = 1198911 (119909) (26)
120597ℎ (119909)120597119909 119892 (119909) = 119871119892ℎ (119909) = 0 (27)
Substituting (26) and (27) into (24) it is seen that 119910does not include 119906 Then derivative of time for (24) will becontinued
119910 = 120597119871119891ℎ (119909)120597119909 120597119909120597119905 =120597119871119891ℎ (119909)120597119909 [119891 (119909) + 119892 (119909) 119906]
= 1198712119891ℎ (119909) + 119871119892119871119891ℎ (119909) 119906(28)
In (28)
1198712119891ℎ (119909) = 120597119871119891ℎ (119909)120597119909 119891 (119909)= [1205971198911 (119909)1205971199091
1205971198911 (119909)1205971199092 ] [1198911 (119909)1198912 (119909)]
= 1205971198911 (119909)1205971199091 1198911 (119909) +1205971198911 (119909)1205971199092 1198912 (119909)
(29)
119871119892119871119891ℎ (119909) = 120597119871119891ℎ (119909)120597119909 119892 (119909)
= [1205971198911 (119909)12059711990911205971198911 (119909)1205971199092 ][[
01120591]]= 1205971198911 (119909)1205971199092
1120591(30)
Therefore
119871119892119871119891ℎ (119909) = 0 (31)
From now on input variable 119906 is expressed in 119910 and thecondition for meeting feedback linearization is gained
34 Input-Output Feedback Linearization Based Sliding ModeControl Both linear controller and nonlinear controller areusually adopted to control nonlinear system However linearcontroller can only control linear system well In nonlinearsystem if linear controller is used nonlinear model willbe usually linearized Sliding mode control is a nonlinear
Mathematical Problems in Engineering 7
s=0slt0
sgt0
A BC
Figure 6 The schematic diagram of sliding mode hypersurface
algorithm and it can continuously adjust itself based on real-time state variables during dynamic process for system totrack the state trajectory of sliding mode [34 35] In thispaper we propose sliding mode algorithm to enhance therobustness of feedback linearization
Assume that there exists a system as follows
= 119891 (119909) 119909 isin 119877119899 (32)
In its state space there is a hypersurface 119904(119909) =119904(1199091 1199092 sdot sdot sdot 119909119899) = 0 depicted in Figure 6The state space is divided into two parts 119904 gt 0 and 119904 lt 0 in
hypersurface In switching surface 119904 = 0 the point in systemwill be close to termination point when it is near switchingsurface
If all points are termination points in a special field inthe switching surface once the points move to this area towhich they will be attracted it is called sliding mode areaThemovement of system in slidingmode area is called slidingmode action
As the moving point is termination point in sliding modearea the expression can be gained as follows when the pointmoves to a place near the sliding mode surface
In other words
lim119904rarr0+
119904 le 0 le lim119904rarr0minus
119904 (33)
The other expression is formulated as
lim119904rarr0119904 119904 le 0 (34)
Assume that a function is given by
V (1199091 1199092 sdot sdot sdot 119909119899) = [119904 (1199091 1199092 sdot sdot sdot 119909119899)]2 (35)
The derivative of time for (35) can be expressed as follows
1198891199042119889119905 = 2119904 119904 (36)
According to (34) (36) is negative semidefinite formTherefore (35) is a Lyapunov function of system Then thesystem is stable when 119904 = 0
According to these analyses above system performancecan be improved by combining sliding mode control withfeedback linear algorithm because of stable advantage ofslidingmode control To gain good tracking performance theerror between reference and output is defined as
119890 = 119910119894119899 minus 119910 (37)
where 119910119894119899 is the ideal tracked object and 119910 is the controloutput
Then sliding function can be obtained as follows
119904 (119909 119905) = ce (38)
where
c = [119888 1] (39)
119888 gt 0 (40)
e = [119890 119890]119879 (41)
According to the discussed study for wind turbine in 33the control rule can be designed as
1199062 = 1(1205971198911 (119909) 1205971199092) (1120591) (V
minus (1205971198911 (119909)1205971199091 1198911 (119909) +1205971198911 (119909)1205971199092 1198912 (119909)) + 120578 sgn (119904))
(42)
where V is the aided term 120578 ge 0 and sgn(119904) is the signfunction which is formulated by
sgn (119904) = 1 119904 gt 0minus1 119904 lt 0 (43)
Simulation results show that there is large fluctuationin output power and rotor speed when using sign functionIn order to reduce the fluctuation saturation function isimplemented to replace sign function Saturation function isdefined as follows Results are shown in Figures 7(a)ndash7(d) andFigures 8(a)ndash8(d)
119904119886119905 (119904) =
1 119904 gt 120575119904120575 |119904| le 120575minus1 119904 lt minus120575
(44)
Up to now the final control law can be expressed basedon (17) and (42) as follows
119906 = 1199061 + 1199062 (45)
4 Simulation Results and Analyses
Simulation experiments based on a MW scale wind turbineare carried out to verify the effectiveness of the proposedalgorithm in which parameters are depicted in Table 4 Thecontroller parameters are shown in Table 5
The simulation experiments for wind turbine pitchingcontrol are implemented using step wind and random windStep speed is shown in Figure 7(a) and random wind isdepicted in Figure 8(a)
Firstly Figures 7(b)ndash7(d) and Figures 8(b)ndash8(d) depictthe control results when using sign function and saturationfunction The results show that there are little differences in
8 Mathematical Problems in Engineering
Table 4 The parameters of the wind turbine
Symbols Parameters Values119875119903119886119905119890119889 Rated output power 3000KW119881119903119886119905119890119889 Rated wind speed 12ms119877 The Radius of the turbine rotor 475m119869119903 The inertia moment of the turbine rotor 6250000kgsdotm2119869119892 The inertia moment of the generator 15kgsdotm2119896 Drive ratio 80120591 Time constant 021199031 Stator winding resistance 00148Ω1199032 Rotor winding resistance calculated to stator 00370Ω1199091120590 Leakage reactance 00727Ω1199092120590 Leakage calculated to stator side 00863Ω119909119898 Excitation reactance 20000Ω
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Step wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19992
200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(d) Output powers using different functions in sliding mode
Figure 7 (a) Step wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions in slidingmode (d) output powers using different functions in sliding mode
Mathematical Problems in Engineering 9
Table 5 The parameters of the proposed controller
Symbols Parameters Values119888 Constant defined in sliding mode algorithm 10119890 119903119886119899119892119890 The universe of rotor deviation [-0003 0003]119890119888 119903119886119899119892119890 The universe of rotor deviation changing rate [-0008 0008]119870119875 119903119886119899119892119890 The universe of proportionality coefficent [-03 03]119870119868 119903119886119899119892119890 The universe of integral coefficient [-006 006]119870119863 119903119886119899119892119890 The universe of differential coefficient [-3 3]119864 The value of the error universe 0003119864119862 The value of the error changing rate universe 0008120583 The coefficient of universe scaling factor 091205831 The coefficient of universe scaling factor 091205832 The coefficient of universe scaling factor 04120576 A full small positive number 000001
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Random wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
25
3
35
4
P(M
W)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
(d) Output powers using different functions in sliding mode
Figure 8 (a) Random wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions insliding mode (d) output powers using different functions in sliding mode
10 Mathematical Problems in Engineering
5
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
Local enlargeddrawing
4 5 6 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under step wind speed
9 10 11 12
2
2001Localenlargeddrawing
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under step wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
9 10 11 12
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) output power using different controllers under step wind speed
Figure 9 (a) Pitch angles using different controllers under step wind speed (b) rotor speed using different controllers under step wind speedand (c) output power using different controllers under step wind speed
the control when implementing the two different functionsunder the step wind speed However under random windspeed the control performance of the saturation functionadjusting output power is significantly better than that of thesign function in Figure 8(d)
Comparisons are further studied among which thereare conventional PID algorithm feedforward feedback con-trol combining PID with feedback linearization algorithmfeedforward feedback control based on fuzzy self-tuningPID with feedback linearization with sliding mode control(called fuzzy PID in feedback loop in Figures 9(a)ndash9(c)
and Figures 10(a)ndash10(c)) and the proposed controller inthis paper Results are depicted in Figures 9(a)ndash9(c) andFigures 10(a)ndash10(c) The wind speed is changed continuouslyto further validate pitching control algorithms which isdepicted in Figure 8(a)
At the initial moment control outputs fluctuate stronglywhich are caused by initial values of states However iffluctuations are in acceptable ranges of the physical structureof the wind turbine the initial responses will not affect theevaluation for the controller In Figure 7(a) at 119905 = 4 and119905 = 10 step wind speed occurs As thewind speeds exceed the
Mathematical Problems in Engineering 11
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
12 13 14 15
5 Local enlargeddrawing
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
8 9 10 11
1999
2
2001
Localenlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
13 14 15 16
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) Output power using different controllers under random wind speed
Figure 10 (a) Pitch angles using different controllers under random wind speed (b) rotor speed using different controllers under randomwind speed and (c) output power using different controllers under random wind speed
rated value pitch angles need to be adjusted to keep outputpower at constant
In Figure 9(a) the overshoot of the pitch angle is greaterwhile oscillation time is longer when adopting the conven-tional PID controller Although rotor speed also convergesat the rated value steady-state errors cannot be eliminatedwhich is approved in rotor speed chart depicted in Fig-ure 9(b) The rotor speeds of wind turbine are maintainedat constant when wind speed exceeds the rated value Dueto the steady-state errors of the pitch angle rotor speedshave deviations These results show that single linear PID
controller cannot track target well for wind turbine exhibitsstrongly nonlinear features
To address this problem we design a PID controllercombining with feedback linearization theory The PIDcontroller is feedback control while feedback linearizationis proposed as feedforward control Feedback control cankeep the system stable However the feedback control ishysteric Only when the deviation occurs the controller isimplemented to eliminate it Nevertheless feedforward canproduce a control variable in advanceThus the final outputsof the two controllers are proposed as the input of the wind
12 Mathematical Problems in Engineering
turbine system so that tracking accuracy of controller can beenhanced
Figures 9(a)ndash9(c) demonstrate that the control perfor-mance of feedforward feedback controller based on com-bining PID algorithm with feedback linearization algorithmis clearly superior to conventional PID controller in whichthe overshoot of pitch angle is smaller and adjusting timeof oscillating is shorter and furthermore the steady-statedeviation is eliminated which is also shown both in therotor speed diagram and in the output power diagram Tofurther improve control performance sliding mode controlis combined with feedback linear algorithm in feedforwardloop while the fuzzy self-tuning algorithm is used to adjustthe parameters of the PID controller in feedback loopFurthermore in order to decrease the effect when inputvariable and output variable change a lot variable universeapproach is proposed to optimize fuzzy rules Known fromthe results the deficiencies of controller produced by systemconstruct and parameters variation can be overcome FromFigures 9(a)ndash9(c) it is shown that the performance of thecontroller we propose in this paper is the best among thefour controllers whose overshoot is the least while it rapidlyconverges to constant without steady-state errors Althoughcontinuous step speed can evaluate controller performanceit still exhibits limitation
In order to further verify the effectiveness and universal-ity of the proposed controller in this paper we further adoptrandom wind speed Figure 8(a) depicts random wind speedFigures 10(a)ndash10(c) show the control results of the pitch anglerotor speed of turbine and output power produced by fourdifferent control algorithms under random wind speed Theresults are reasonable which are the same as those producedby step wind speed Therefore the control algorithm wepropose in this paper is the best among the described fouralgorithms and it can keep the output power at the rated valuewhen wind speed exceeds the nominal velocity
5 Conclusions
In this research a novel pitch control strategy for largevariable speed variable pitch wind turbines is studied aimingat maintaining the output power at constant so that theenergy quality is improved Known from the results as thesliding mode theory is combined with feedback linearizationalgorithm in feedforward loop the uncertainty of modeland time-varying of system parameters are overcome Fur-thermore with setting a control variable in advance theovershoot can be decreased when wind speed is varyingOn the other hand in feedback loop as variable universefuzzy algorithm is employed to optimize the parameters ofthe conventional PID controller the tracking performanceof the feedback controller is enhanced Meanwhile insteadof sign function saturation function is implemented todecrease oscillation of sliding mode algorithm and thus thefluctuations of output power and rotor speed of wind turbineare mitigated Both control accuracy and robustness usingthe proposed algorithm are the best among four differentcontrollers Another in future we plan to supply fuzzy
predictive adaptive algorithm in power control to furtherimprove system performance
Data Availability
The data used to support the findings of this study arecurrently under embargo while the research findings arecommercialized Requests for data 6 months after publica-tion of this article will be considered by the correspondingauthor
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National Natural ScienceFund of China under Grant no 51405052
References
[1] C-SWang andM-H Chiang ldquoA novel dynamic co-simulationanalysis for overall closed loop operation control of a large windturbinerdquo Energies vol 9 no 8 2016
[2] E Van Solingen J Beerens S P Mulders R De Breuker andJ W Van Wingerden ldquoControl design for a two-bladed down-wind teeterless damped free-yaw wind turbinerdquo Mechatronicsvol 36 pp 77ndash96 2016
[3] Y-J Chen and Y C Shiah ldquoExperiments on the performance ofsmall horizontal axis wind turbine with passive pitch control bydisk pulleyrdquo Energies vol 9 no 5 2016
[4] A Choudhry M Arjomandi and R Kelso ldquoMethods tocontrol dynamic stall for wind turbine applicationsrdquo Journal ofRenewable Energy vol 86 pp 26ndash37 2016
[5] H Li C Yang Y Hu X Liao Z Zeng and C Zhe ldquoAnimproved reduced-ordermodel of an electric pitch drive systemfor wind turbine control system design and simulationrdquo Journalof Renewable Energy vol 93 pp 188ndash200 2016
[6] E B Muhando T Senjyu A Uehara and T Funabashi ldquoGain-scheduledHinfin control forWECS via LMI techniques and para-metrically dependent feedback part II Controller design andimplementationrdquo IEEE Transactions on Industrial Electronicsvol 58 no 1 pp 57ndash65 2011
[7] F D Bianchi H De Battista and R JMantz ldquoRobustmultivari-able gain scheduled control of wind turbines for variable powerproductionrdquo International Journal of Systems Control vol 1 no3 pp 103ndash112 2010
[8] I Soslashnderby andMH Hansen ldquoOpen-loop frequency responseanalysis of a wind turbine using a high-order linear aeroelasticmodelrdquoWind Energy vol 17 no 8 pp 1147ndash1167 2014
[9] S Bououden M Chadli S Filali and A El Hajjaji ldquoFuzzymodel basedmultivariable predictive control of a variable speedwind turbine LMI approachrdquo Journal of Renewable Energy vol37 no 1 pp 434ndash439 2012
[10] J Chen T Lin CWen and Y Song ldquoDesign of a Unified PowerController for Variable-Speed Fixed-Pitch Wind Energy Con-version Systemrdquo IEEETransactions on Industrial Electronics vol63 no 8 pp 4899ndash4908 2016
Mathematical Problems in Engineering 13
[11] Y Ren L Li J Brindley and L Jiang ldquoNonlinear PI control forvariable pitch wind turbinerdquo Control Engineering Practice 2015
[12] Y Yuan and J Tang ldquoAdaptive pitch control of wind turbinefor load mitigation under structural uncertaintiesrdquo Journal ofRenewable Energy vol 105 pp 483ndash494 2017
[13] C-S Wang and M-H Chiang ldquoA novel pitch control systemof a large wind turbine using two-degree-of-freedom motioncontrol with feedback linearization controlrdquo Energies vol 9 no10 article no 791 2016
[14] Z Civelek E Cam M Luy and H Mamur ldquoProportional-integral-derivative parameter optimisation of blade pitch con-troller in wind turbines by a new intelligent genetic algorithmrdquoIET Renewable Power Generation vol 10 no 8 pp 1220ndash12282016
[15] A Dahbi N Nait-Said andM-S Nait-Said ldquoA novel combinedMPPT-pitch angle control for wide range variable speed windturbine based on neural networkrdquo International Journal ofHydrogen Energy vol 41 no 22 pp 9427ndash9442 2016
[16] M L Corradini G Ippoliti and G Orlando ldquoAn observer-based blade-pitch controller of wind turbines in high windspeedsrdquo Control Engineering Practice vol 58 pp 186ndash192 2017
[17] M Zhu J Liu Z Lin and H Meng ldquoMixed H2 Hinfin pitchcontrol of wind turbine generator system based on globalexact linearization and regional pole placementrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 5 pp921ndash930 2016
[18] R Gao and Z Gao ldquoPitch control for wind turbine systemsusing optimization estimation and compensationrdquo Journal ofRenewable Energy vol 91 pp 501ndash515 2016
[19] X-X Yin Y-G Lin W Li H-W Liu and Y-J Gu ldquoAdaptivesliding mode back-stepping pitch angle control of a variable-displacement pump controlled pitch system for wind turbinesrdquoISA Transactions vol 58 pp 629ndash634 2015
[20] M-H Chiang ldquoA novel pitch control system for a wind turbinedriven by a variable-speed pump-controlled hydraulic servosystemrdquoMechatronics vol 21 no 4 pp 753ndash761 2011
[21] H Camblong S Nourdine I Vechiu and G Tapia ldquoCom-parison of an island wind turbine collective and individualpitch LQG controllers designed to alleviate fatigue loadsrdquo IETRenewable Power Generation vol 6 no 4 p 267 2012
[22] F A InthamoussouHDe Battista andR JMantz ldquoLPV-basedactive power control of wind turbines covering the completewind speed rangerdquo Journal of Renewable Energy vol 99 pp996ndash1007 2016
[23] F D Bianchi R J Mantz and C F Christiansen ldquoGainscheduling control of variable-speed wind energy conversionsystems using quasi-LPVmodelsrdquoControl Engineering Practicevol 13 no 2 pp 247ndash255 2005
[24] F A Shirazi K M Grigoriadis and D Viassolo ldquoWind turbineintegrated structural and LPV control design for improvedclosed-loop performancerdquo International Journal of Control vol85 no 8 pp 1178ndash1196 2012
[25] F A Inthamoussou F D Bianchi H De Battista and R JMantz ldquoLPV wind turbine control with anti-windup featurescovering the complete wind speed rangerdquo IEEE Transactions onEnergy Conversion vol 29 no 1 pp 259ndash266 2014
[26] K Z Oslashstegaard J Stoustrup and P Brath ldquoLinear parametervarying control of wind turbines covering both partial loadand full load conditionsrdquo International Journal of Robust andNonlinear Control vol 19 no 1 pp 92ndash116 2009
[27] Y Lin L Tu H Liu and W Li ldquoHybrid power transmissiontechnology in a wind turbine generation systemrdquo IEEEASMETransactions onMechatronics vol 20 no 3 pp 1218ndash1225 2015
[28] S Bououden M Chadli F Allouani and S Filali ldquoA newapproach for fuzzy predictive adaptive controller design usingparticle swarmoptimization algorithmrdquo International Journal ofInnovative Computing Information and Control vol 9 no 9 pp3741ndash3758 2013
[29] S H Li and H Gu ldquoFuzzy adaptive internal model controlschemes for PMSM speed-regulation systemrdquo IEEE Transac-tions on Industrial Informatics vol 8 no 4 pp 767ndash779 2012
[30] A B Asghar and X Liu ldquoAdaptive neuro-fuzzy algorithmto estimate effective wind speed and optimal rotor speed forvariable-speed wind turbinerdquo Neurocomputing vol 272 pp495ndash504 2018
[31] Y Wang H Shen H R Karimi and D Duan ldquoDissipativity-based Fuzzy Integral SlidingModeControl ofContinuous-TimeT-S Fuzzy Systemsrdquo IEEE Transactions on Fuzzy Systems pp 1-1
[32] S Bououden M Chadli and H R Karimi ldquoAn ant colonyoptimization-based fuzzy predictive control approach for non-linear processesrdquo Information Sciences vol 299 pp 143ndash1582015
[33] NWang S-F Su J Yin Z Zheng andM J Er ldquoGlobal Asymp-totic Model-Free Trajectory-Independent Tracking Control ofan Uncertain Marine Vehicle An Adaptive Universe-BasedFuzzy Control Approachrdquo IEEE Transactions on Fuzzy Systems2017
[34] R Saravanakumar andD Jena ldquoValidation of an integral slidingmode control for optimal control of a three blade variable speedvariable pitch wind turbinerdquo International Journal of ElectricalPower amp Energy Systems vol 69 pp 421ndash429 2015
[35] C Evangelista P Puleston F Valenciaga and L M FridmanldquoLyapunov-designed super-twisting sliding mode control forwind energy conversion optimizationrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 538ndash545 2013
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Mathematical Problems in Engineering 3
M
gear boxwind
blade
generator
Figure 2 The block diagram of energy conversion of wind turbine
of turbine rotor 120588 is the air density 119877 is the radius ofturbine rotor V is the wind speed 119862119901(120582 120573) is the conversioncoefficient of wind energy 119869119903 is the rotational inertia ofturbine rotor 119879119897 is the torque of low speed shaft 120582 is the tip-speed ratio and 120573 is the pitch angle
The shaft power of wind turbine is proportional to thecube of wind speed known from (1) so that the outputpower will be strongly affected when wind speed increasesTherefore in order to keep output power at constant powerconversion coefficient will be decreased when wind speed isover the rated value In addition known from (3) and (4)power conversion coefficient is controlled by rotor speed ofturbine rotor and pitch angle of wind turbine when windspeed is changing After the output power of wind turbineis over the rated value rotor speed of electric generatoris constant Consequently adjusting pitch angle is a mainmethod for controlling the output power On the other handbased on (1) it is known that output power can also bemaintained at the rated value by adjusting torque
Due to the fact this paper focuses on algorithms ofadjusting power In order to simplify the model of thedrive train the system structure is assumed to be rigidthat will not affect researching wind turbine system Therelationship between low speed shaft and high speed shaft canbe expressed in [12]
119896 = 120596119892120596119903 =119879119897119879ℎ (5)
where 119896 is the gear ratio of the drive train 120596119892 is the angularspeed of high speed shaft and 119879ℎ is the torque of high speedshaft
In high speed section of the drive train the relationshipcan be expressed in [17]
119869119892 120596119892 = 119879ℎ minus 119879119890 (6)
where 119869119892 is the rotational inertia of electric generator and 119879119890is the torque of electric generator
The relationship can be gained from (5) as follows
119879119897 = 119896119879ℎ (7)
120596119892 = 119896120596119903 (8)
The relationship can be acquired based on (6) as
119879ℎ = 119869119892 120596119892 + 119879119890 (9)
Substituting (7) (8) and (9) into (2) the relationship canbe further written as
120596119903 (119869119903 + 1198962119869119892) = 119879119903 minus 119896119879119890 (10)
Controlling the pitch angle of variable pitch wind turbineis to maintain output power at nominal value when windspeed is over the rated value Pitch system can be expressedapproximately as a first-order inertia system in [17]
120573 = 1120591 (1205731 minus 120573) (11)
where 120573 is the pitch angle 120591 is the time constant and 1205731 isthe desired pitch angle When wind speed is changing thedesired pitch angle is the output value of controllerThereforethe real-time pitch angle can be calculated by using integralalgorithm based on (11)
According to double-fed generator equivalent circuitstator equation and rotor equation the expression of elec-tromagnetic torque can be expressed in
119879119890119898 = minus31199041199011199092119898119903212059611198882 11988021 + 3119901119909
2119898119903112059611198882 11988022
+ 311990111990911989812059611198882 [(11990311199032 + 11990411990921205901199091120590 + 1199041199092119898) sin12057212
minus (11990411990921205901199031 minus 11990911205901199032) cos12057212]11988011198802
(12)
where 119888 = radic(11990311199032 minus 11990411990921205901199091120590 + 1199041199092119898)2 + (11990411990921205901199031 + 11990911205901199032)2 119901 isthe pole pairs 1205961 is the synchronous angular velocity 119904 isthe slip rate 1198801 is the stator voltage 1198802 is the rotor voltage12057212 is the phase difference between stator and rotor 1199031 is thestatorwinding resistance1199091120590 is the leakage reactance 1199032 is therotorwinding resistance calculated to stator1199092120590 is the leakagecalculated to stator side and 119909119898 is the excitation reactance
The electromagnetic power of doubly fed motor can beexpressed in
119875119890 = 119879119890119898Ω1 = 1198791198901198981205961119901 = minus3119904119909211989811990321198882 11988021 + 3119909211989811990311198882 11988022
+ 31199091198981198882 [(11990311199032 + 11990411990921205901199091120590 + 1199041199092119898) sin12057212minus (11990411990921205901199031 minus 11990911205901199032) cos12057212]11988011198802
(13)
where Ω1 is the generator synchronous mechanical angularvelocity
3 Feedforward Feedback Controller Based onFeedback Linearization with Sliding Modeand Fuzzy Self-Tuning PID
Known from (3) the energy conversion coefficient 119862119901(120582 120573)is nonlinear function including 120582 and 120573 Therefore the shaft
4 Mathematical Problems in Engineering
feedback linearization
PID
wind turbine model
sliding modealgorithm
fuzzy rules
dedt
input scaling factor
output scaling factor
ＳＣＨ
ＳＩＯＮ
+
+
+
minus
u1
u2
Figure 3 The system control structure of wind turbine
power expression of wind turbine is nonlinear As windturbine system model is a typical nonlinear system the con-troller designed based on linear system is not the best choiceHowever nonlinear controller is a better idea In this paperwe propose a novel feedforward feedback algorithm whosefeedback section is fuzzy self-tuning PID algorithm andits feedforward section is feedback linearization algorithmHowever feedback linearization control needs an accuratesystem model and it is subject to parameter variationConsequently its application is limited As sliding modecontrol is not affected by parameter variation of systemand disturbances robustness of system can be improved bycombining sliding mode method with feedback linearizationalgorithm The chart that describes this algorithm is shownin Figure 3 Differential equations of wind turbine can beacquired from Section 2 as follows
120596119903 (119869119903 + 1198962119869119892) = 12120596119903 1205881205871198772V3119862119901 (120582 120573) minus 119896119879119892
120573 = 1120591 (1205731 minus 120573)(14)
Assuming state variables 1199091 = 120596119903 1199092 = 120573 control input119906 = 1205731 state equations can be given as
1199091 = 1119869119903 + 1198962119869119892 (121199091 120588120587119877
2V3119862119901 (120582 1199092) minus 119896119879119892)1199092 = 1120591 (119906 minus 1199092)
(15)
Then output equation can be expressed as follows
119910 = 1199091 (16)
31 Fuzzy Self-Tuning PIDAlgorithm Due to the nonlinearityand time-varying characteristics of the wind turbine theconventional PID cannot meet the control requirementsThe
fuzzy algorithm transforms input values by fuzzifier andmakes a judgment based on fuzzy rules which can control themore complex systems [28ndash32] In this paper we use the fuzzyalgorithm to adjust the parameters of the PID controller in thefeedback control loop to improve the control performance ofthe controller
Generator velocity will be a constant when wind speedis overrated value As drivetrain is considered stiffness windturbine velocity is also a constant However the velocitywill fluctuate in a very small range for there are randomdisturbances Accordingly we select wind turbine velocity asreference variable When pitching the input variables of thefuzzy controller are the wind turbine velocity errors and theerror changing ratesThe output variables are the proportionintegral and differential gains of the PID controller Errorand error changing rate corresponding to the fuzzy domainare taken as minus3 minus2 minus1 0 +1 +2 +3 Fuzzy rules arethe key of fuzzy control Not only the correctness of fuzzycontrol rules directly affects the performance of the controllerbut their number is also an important factor to measurecontroller performance Therefore we define seven fuzzysubsets and the language variables are negative big negativemiddle negative small zero positive small positive middlepositive big expressed as NB NM NS Z PS PM andPB respectivelyThese language variables cannot only clearlyexpress universe but the control rule number is appropriateFuzzy rules are shown in Tables 1ndash3 They can be furtherdescribed as follows
If e is NB and ec is NB then output is PBIf e is NB and ec is NM then output is PBIf e is NB and ec is NS then output is PM
The membership function determines the sensitivity ofthe fuzzy control performance If the membership functionchangesmore smoothly the stability of the controller is betterOn the other hand if the membership function is steeperthen the controller has high sensitivity Therefore when the
Mathematical Problems in Engineering 5
Table 1 The fuzzy rules of KP
eNB NM NS Z PS PM PB
NB PB PB PM PM PS Z ZNM PB PB PM PS PS Z NSNS PM PM PM PS Z NS NS
ec Z PM PM PS Z NS NM NMPS PS PS Z NS NS NM NMPM PS Z NS NM NM NM NBPB Z Z NM NM NM NB NB
Table 2 The fuzzy rules of KI
eNB NM NS Z PS PM PB
NB NB NB NM NM NS Z ZNM NB NB NM NS NS Z ZNS NB NM NS NS Z PS PS
ec Z NM NM NS Z PS PM PMPS NM NS Z PS PS PM PBPM Z Z PS PS PM PB PBPB Z Z PS PM PM PB PB
Table 3 The fuzzy rules of KD
eNB NM NS Z PS PM PB
NB PS NS NB NB NB NM PMNM PS NS NB NM NM NS ZNS Z NS NM NM NS NS Z
ec Z Z NS NS NS NS NS ZPS Z Z Z Z Z Z ZPM PB PS PS PS PS PS PBPB PB PM PM PM PS PS PB
error and error rate of change are largeGaussianmembershipfunction with smooth change is selected which makes thesystem change smoothly andhave good robustnessWhen theerror and error changing rate are both small the triangularmembership function with high sensitivity is proposed toimprove the system response The diagram of fuzzy systemwith two-input three-output is shown in Figure 4 Parametersof fuzzy controller are shown in Table 5
e is the error and ec is the error changing rateThe outputof PID controller is as follows
1199061 = 119870119875 (119890 (119905) + 1119879119868 int 119890 (119905) 119889119905 + 119879119863119889119890 (119905)119889119905 ) (17)
where 1199061 is the output of PID controller 119870119875 is the propor-tional coefficient 119890(119905) is the error119879119868 is the integral coefficientand 119879119863 is the differential coefficient
32 Variable Universe Theory Fuzzy PID controllers exhibitgood control performance Nevertheless when input vari-ables change a lot the control accuracy needs to be further
System fuzzpid 2 inputs 3 outputs 49 rules
e (7)
ec (7)
kp (7)
ki (7)
kd (7)
fuzzpid
(mamdani)
49 rules
Figure 4 The fuzzy system structure
improved The idea of variable universe is to make any inputcan correspond to the entire input universe by adding ascaling factor without increasing the fuzzy rules based onthe changes of the input [33] If the original basic universeis adopted control accuracy is not ideal Furthermore whenthe control error becomes smaller its corresponding rulescan be reduced However the rules corresponding to theinput can be increased by adjusting scaling factor if thevariable universe method is implemented In other wordsthe corresponding rule density is increased Therefore thecontrol accuracy is improvedThe key of the variable universeapproach is to determine the scaling factor which shouldmeet the following requirements
(1) Duality for (forall119909 isin 119883) 120572(119909) = minus120572(minus119909) is a necessarycondition
(2) Avoid zero 120572(0) = 120576(3) Monotonicity 120572(119909) is strictly monotonically increas-
ing from 0 to 119864(4) Coordination forall119909 isin 119883 119909 le 120572(119909)119864(5) Regularity 120572(plusmn119864) = 1Thereby the universe scaling factors are expressed in
120572 (119909) = (|119909|119864 )120583 + 120576 (18)
120578 (119909) = (|119909|119864 )1205831 ( 10038161003816100381610038161199101003816100381610038161003816119864119862)
1205832 + 120576 (19)
where 120572(119909) is the scaling factor of the input universe 120578(119909) isthe scaling factor of the output universe 119909 is the error 119910 is theerror rate of change 119864 is the value of the error universe 119864119862 isthe value of the error changing rate universe 120583 1205831 and 1205832 areconstants from 0 to 1 and 120576 is a full small positive number
The process of changing universe is shown in Figure 5
33 Design of Feedback Linearization of Nonlinear Model forWind Turbine Feedback linearization algorithm is to adda nonlinear compensation part in control input variablesThen a new equivalent controlled system can be gained Theoriginal system is transformed into linear form As feed-back linearization algorithm is based on strict mathematicalderivation and control input is expressed in output equation
6 Mathematical Problems in Engineering
NB NM NS ZE PS PM PB
minusE EO
O
O
(A)
(B)
(C)X
X
minus(x)E (x)E
minus(x)E (x)E
Universe expansion
Universe com
pression
Figure 5 The diagram of changing universe process
the nonlinear system model can be linearized by designingappropriate control rules Therefore control performance ofsystem can be improved distinctly
The affine nonlinear models of single input single outputare expressed as [13]
= 119891 (119909) + 119892 (119909) 119906119910 = ℎ (119909) (20)
where119909 = [1199091 1199092 sdot sdot sdot 119909119899]119879 are the 119899-dimensional state vectors119891(119909) and 119892(119909) are the fully smooth vector fields in 119877119899 119906 isin 119877119910 isin 119877 and ℎ(119909) is the fully smooth nonlinear functionThe expression can be gained as follows based on (15)
(16) and (20)
119891 (119909) = [1198911 (119909)1198912 (119909)]
= [[[[
1119869119903 + 1198962119869119892 (121199091 120588120587119877
2V3119862119901 (120582 1199092) minus 119896119879119892)minus11205911199092
]]]]
(21)
119892 (119909) = [[01120591]]
(22)
ℎ (119909) = 1199091 (23)
Known from (23) there is no direct relationshipbetween the output equation and the control inputTherefore
derivative of time for output equation must be implementedas
119910 = 119889ℎ (119909)119889119905 = 120597ℎ (119909)120597119909 120597119909120597119905= 120597ℎ (119909)120597119909 [119891 (119909) + 119892 (119909) 119906]= 120597ℎ (119909)120597119909 119891 (119909) + 120597ℎ (119909)120597119909 119892 (119909) 119906
(24)
120597ℎ (119909)120597119909 = [1 0] (25)
Equations (26) and (27) can be gained based on (25)
120597ℎ (119909)120597119909 119891 (119909) = 119871119891ℎ (119909) = 1198911 (119909) (26)
120597ℎ (119909)120597119909 119892 (119909) = 119871119892ℎ (119909) = 0 (27)
Substituting (26) and (27) into (24) it is seen that 119910does not include 119906 Then derivative of time for (24) will becontinued
119910 = 120597119871119891ℎ (119909)120597119909 120597119909120597119905 =120597119871119891ℎ (119909)120597119909 [119891 (119909) + 119892 (119909) 119906]
= 1198712119891ℎ (119909) + 119871119892119871119891ℎ (119909) 119906(28)
In (28)
1198712119891ℎ (119909) = 120597119871119891ℎ (119909)120597119909 119891 (119909)= [1205971198911 (119909)1205971199091
1205971198911 (119909)1205971199092 ] [1198911 (119909)1198912 (119909)]
= 1205971198911 (119909)1205971199091 1198911 (119909) +1205971198911 (119909)1205971199092 1198912 (119909)
(29)
119871119892119871119891ℎ (119909) = 120597119871119891ℎ (119909)120597119909 119892 (119909)
= [1205971198911 (119909)12059711990911205971198911 (119909)1205971199092 ][[
01120591]]= 1205971198911 (119909)1205971199092
1120591(30)
Therefore
119871119892119871119891ℎ (119909) = 0 (31)
From now on input variable 119906 is expressed in 119910 and thecondition for meeting feedback linearization is gained
34 Input-Output Feedback Linearization Based Sliding ModeControl Both linear controller and nonlinear controller areusually adopted to control nonlinear system However linearcontroller can only control linear system well In nonlinearsystem if linear controller is used nonlinear model willbe usually linearized Sliding mode control is a nonlinear
Mathematical Problems in Engineering 7
s=0slt0
sgt0
A BC
Figure 6 The schematic diagram of sliding mode hypersurface
algorithm and it can continuously adjust itself based on real-time state variables during dynamic process for system totrack the state trajectory of sliding mode [34 35] In thispaper we propose sliding mode algorithm to enhance therobustness of feedback linearization
Assume that there exists a system as follows
= 119891 (119909) 119909 isin 119877119899 (32)
In its state space there is a hypersurface 119904(119909) =119904(1199091 1199092 sdot sdot sdot 119909119899) = 0 depicted in Figure 6The state space is divided into two parts 119904 gt 0 and 119904 lt 0 in
hypersurface In switching surface 119904 = 0 the point in systemwill be close to termination point when it is near switchingsurface
If all points are termination points in a special field inthe switching surface once the points move to this area towhich they will be attracted it is called sliding mode areaThemovement of system in slidingmode area is called slidingmode action
As the moving point is termination point in sliding modearea the expression can be gained as follows when the pointmoves to a place near the sliding mode surface
In other words
lim119904rarr0+
119904 le 0 le lim119904rarr0minus
119904 (33)
The other expression is formulated as
lim119904rarr0119904 119904 le 0 (34)
Assume that a function is given by
V (1199091 1199092 sdot sdot sdot 119909119899) = [119904 (1199091 1199092 sdot sdot sdot 119909119899)]2 (35)
The derivative of time for (35) can be expressed as follows
1198891199042119889119905 = 2119904 119904 (36)
According to (34) (36) is negative semidefinite formTherefore (35) is a Lyapunov function of system Then thesystem is stable when 119904 = 0
According to these analyses above system performancecan be improved by combining sliding mode control withfeedback linear algorithm because of stable advantage ofslidingmode control To gain good tracking performance theerror between reference and output is defined as
119890 = 119910119894119899 minus 119910 (37)
where 119910119894119899 is the ideal tracked object and 119910 is the controloutput
Then sliding function can be obtained as follows
119904 (119909 119905) = ce (38)
where
c = [119888 1] (39)
119888 gt 0 (40)
e = [119890 119890]119879 (41)
According to the discussed study for wind turbine in 33the control rule can be designed as
1199062 = 1(1205971198911 (119909) 1205971199092) (1120591) (V
minus (1205971198911 (119909)1205971199091 1198911 (119909) +1205971198911 (119909)1205971199092 1198912 (119909)) + 120578 sgn (119904))
(42)
where V is the aided term 120578 ge 0 and sgn(119904) is the signfunction which is formulated by
sgn (119904) = 1 119904 gt 0minus1 119904 lt 0 (43)
Simulation results show that there is large fluctuationin output power and rotor speed when using sign functionIn order to reduce the fluctuation saturation function isimplemented to replace sign function Saturation function isdefined as follows Results are shown in Figures 7(a)ndash7(d) andFigures 8(a)ndash8(d)
119904119886119905 (119904) =
1 119904 gt 120575119904120575 |119904| le 120575minus1 119904 lt minus120575
(44)
Up to now the final control law can be expressed basedon (17) and (42) as follows
119906 = 1199061 + 1199062 (45)
4 Simulation Results and Analyses
Simulation experiments based on a MW scale wind turbineare carried out to verify the effectiveness of the proposedalgorithm in which parameters are depicted in Table 4 Thecontroller parameters are shown in Table 5
The simulation experiments for wind turbine pitchingcontrol are implemented using step wind and random windStep speed is shown in Figure 7(a) and random wind isdepicted in Figure 8(a)
Firstly Figures 7(b)ndash7(d) and Figures 8(b)ndash8(d) depictthe control results when using sign function and saturationfunction The results show that there are little differences in
8 Mathematical Problems in Engineering
Table 4 The parameters of the wind turbine
Symbols Parameters Values119875119903119886119905119890119889 Rated output power 3000KW119881119903119886119905119890119889 Rated wind speed 12ms119877 The Radius of the turbine rotor 475m119869119903 The inertia moment of the turbine rotor 6250000kgsdotm2119869119892 The inertia moment of the generator 15kgsdotm2119896 Drive ratio 80120591 Time constant 021199031 Stator winding resistance 00148Ω1199032 Rotor winding resistance calculated to stator 00370Ω1199091120590 Leakage reactance 00727Ω1199092120590 Leakage calculated to stator side 00863Ω119909119898 Excitation reactance 20000Ω
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Step wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19992
200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(d) Output powers using different functions in sliding mode
Figure 7 (a) Step wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions in slidingmode (d) output powers using different functions in sliding mode
Mathematical Problems in Engineering 9
Table 5 The parameters of the proposed controller
Symbols Parameters Values119888 Constant defined in sliding mode algorithm 10119890 119903119886119899119892119890 The universe of rotor deviation [-0003 0003]119890119888 119903119886119899119892119890 The universe of rotor deviation changing rate [-0008 0008]119870119875 119903119886119899119892119890 The universe of proportionality coefficent [-03 03]119870119868 119903119886119899119892119890 The universe of integral coefficient [-006 006]119870119863 119903119886119899119892119890 The universe of differential coefficient [-3 3]119864 The value of the error universe 0003119864119862 The value of the error changing rate universe 0008120583 The coefficient of universe scaling factor 091205831 The coefficient of universe scaling factor 091205832 The coefficient of universe scaling factor 04120576 A full small positive number 000001
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Random wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
25
3
35
4
P(M
W)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
(d) Output powers using different functions in sliding mode
Figure 8 (a) Random wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions insliding mode (d) output powers using different functions in sliding mode
10 Mathematical Problems in Engineering
5
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
Local enlargeddrawing
4 5 6 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under step wind speed
9 10 11 12
2
2001Localenlargeddrawing
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under step wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
9 10 11 12
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) output power using different controllers under step wind speed
Figure 9 (a) Pitch angles using different controllers under step wind speed (b) rotor speed using different controllers under step wind speedand (c) output power using different controllers under step wind speed
the control when implementing the two different functionsunder the step wind speed However under random windspeed the control performance of the saturation functionadjusting output power is significantly better than that of thesign function in Figure 8(d)
Comparisons are further studied among which thereare conventional PID algorithm feedforward feedback con-trol combining PID with feedback linearization algorithmfeedforward feedback control based on fuzzy self-tuningPID with feedback linearization with sliding mode control(called fuzzy PID in feedback loop in Figures 9(a)ndash9(c)
and Figures 10(a)ndash10(c)) and the proposed controller inthis paper Results are depicted in Figures 9(a)ndash9(c) andFigures 10(a)ndash10(c) The wind speed is changed continuouslyto further validate pitching control algorithms which isdepicted in Figure 8(a)
At the initial moment control outputs fluctuate stronglywhich are caused by initial values of states However iffluctuations are in acceptable ranges of the physical structureof the wind turbine the initial responses will not affect theevaluation for the controller In Figure 7(a) at 119905 = 4 and119905 = 10 step wind speed occurs As thewind speeds exceed the
Mathematical Problems in Engineering 11
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
12 13 14 15
5 Local enlargeddrawing
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
8 9 10 11
1999
2
2001
Localenlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
13 14 15 16
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) Output power using different controllers under random wind speed
Figure 10 (a) Pitch angles using different controllers under random wind speed (b) rotor speed using different controllers under randomwind speed and (c) output power using different controllers under random wind speed
rated value pitch angles need to be adjusted to keep outputpower at constant
In Figure 9(a) the overshoot of the pitch angle is greaterwhile oscillation time is longer when adopting the conven-tional PID controller Although rotor speed also convergesat the rated value steady-state errors cannot be eliminatedwhich is approved in rotor speed chart depicted in Fig-ure 9(b) The rotor speeds of wind turbine are maintainedat constant when wind speed exceeds the rated value Dueto the steady-state errors of the pitch angle rotor speedshave deviations These results show that single linear PID
controller cannot track target well for wind turbine exhibitsstrongly nonlinear features
To address this problem we design a PID controllercombining with feedback linearization theory The PIDcontroller is feedback control while feedback linearizationis proposed as feedforward control Feedback control cankeep the system stable However the feedback control ishysteric Only when the deviation occurs the controller isimplemented to eliminate it Nevertheless feedforward canproduce a control variable in advanceThus the final outputsof the two controllers are proposed as the input of the wind
12 Mathematical Problems in Engineering
turbine system so that tracking accuracy of controller can beenhanced
Figures 9(a)ndash9(c) demonstrate that the control perfor-mance of feedforward feedback controller based on com-bining PID algorithm with feedback linearization algorithmis clearly superior to conventional PID controller in whichthe overshoot of pitch angle is smaller and adjusting timeof oscillating is shorter and furthermore the steady-statedeviation is eliminated which is also shown both in therotor speed diagram and in the output power diagram Tofurther improve control performance sliding mode controlis combined with feedback linear algorithm in feedforwardloop while the fuzzy self-tuning algorithm is used to adjustthe parameters of the PID controller in feedback loopFurthermore in order to decrease the effect when inputvariable and output variable change a lot variable universeapproach is proposed to optimize fuzzy rules Known fromthe results the deficiencies of controller produced by systemconstruct and parameters variation can be overcome FromFigures 9(a)ndash9(c) it is shown that the performance of thecontroller we propose in this paper is the best among thefour controllers whose overshoot is the least while it rapidlyconverges to constant without steady-state errors Althoughcontinuous step speed can evaluate controller performanceit still exhibits limitation
In order to further verify the effectiveness and universal-ity of the proposed controller in this paper we further adoptrandom wind speed Figure 8(a) depicts random wind speedFigures 10(a)ndash10(c) show the control results of the pitch anglerotor speed of turbine and output power produced by fourdifferent control algorithms under random wind speed Theresults are reasonable which are the same as those producedby step wind speed Therefore the control algorithm wepropose in this paper is the best among the described fouralgorithms and it can keep the output power at the rated valuewhen wind speed exceeds the nominal velocity
5 Conclusions
In this research a novel pitch control strategy for largevariable speed variable pitch wind turbines is studied aimingat maintaining the output power at constant so that theenergy quality is improved Known from the results as thesliding mode theory is combined with feedback linearizationalgorithm in feedforward loop the uncertainty of modeland time-varying of system parameters are overcome Fur-thermore with setting a control variable in advance theovershoot can be decreased when wind speed is varyingOn the other hand in feedback loop as variable universefuzzy algorithm is employed to optimize the parameters ofthe conventional PID controller the tracking performanceof the feedback controller is enhanced Meanwhile insteadof sign function saturation function is implemented todecrease oscillation of sliding mode algorithm and thus thefluctuations of output power and rotor speed of wind turbineare mitigated Both control accuracy and robustness usingthe proposed algorithm are the best among four differentcontrollers Another in future we plan to supply fuzzy
predictive adaptive algorithm in power control to furtherimprove system performance
Data Availability
The data used to support the findings of this study arecurrently under embargo while the research findings arecommercialized Requests for data 6 months after publica-tion of this article will be considered by the correspondingauthor
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National Natural ScienceFund of China under Grant no 51405052
References
[1] C-SWang andM-H Chiang ldquoA novel dynamic co-simulationanalysis for overall closed loop operation control of a large windturbinerdquo Energies vol 9 no 8 2016
[2] E Van Solingen J Beerens S P Mulders R De Breuker andJ W Van Wingerden ldquoControl design for a two-bladed down-wind teeterless damped free-yaw wind turbinerdquo Mechatronicsvol 36 pp 77ndash96 2016
[3] Y-J Chen and Y C Shiah ldquoExperiments on the performance ofsmall horizontal axis wind turbine with passive pitch control bydisk pulleyrdquo Energies vol 9 no 5 2016
[4] A Choudhry M Arjomandi and R Kelso ldquoMethods tocontrol dynamic stall for wind turbine applicationsrdquo Journal ofRenewable Energy vol 86 pp 26ndash37 2016
[5] H Li C Yang Y Hu X Liao Z Zeng and C Zhe ldquoAnimproved reduced-ordermodel of an electric pitch drive systemfor wind turbine control system design and simulationrdquo Journalof Renewable Energy vol 93 pp 188ndash200 2016
[6] E B Muhando T Senjyu A Uehara and T Funabashi ldquoGain-scheduledHinfin control forWECS via LMI techniques and para-metrically dependent feedback part II Controller design andimplementationrdquo IEEE Transactions on Industrial Electronicsvol 58 no 1 pp 57ndash65 2011
[7] F D Bianchi H De Battista and R JMantz ldquoRobustmultivari-able gain scheduled control of wind turbines for variable powerproductionrdquo International Journal of Systems Control vol 1 no3 pp 103ndash112 2010
[8] I Soslashnderby andMH Hansen ldquoOpen-loop frequency responseanalysis of a wind turbine using a high-order linear aeroelasticmodelrdquoWind Energy vol 17 no 8 pp 1147ndash1167 2014
[9] S Bououden M Chadli S Filali and A El Hajjaji ldquoFuzzymodel basedmultivariable predictive control of a variable speedwind turbine LMI approachrdquo Journal of Renewable Energy vol37 no 1 pp 434ndash439 2012
[10] J Chen T Lin CWen and Y Song ldquoDesign of a Unified PowerController for Variable-Speed Fixed-Pitch Wind Energy Con-version Systemrdquo IEEETransactions on Industrial Electronics vol63 no 8 pp 4899ndash4908 2016
Mathematical Problems in Engineering 13
[11] Y Ren L Li J Brindley and L Jiang ldquoNonlinear PI control forvariable pitch wind turbinerdquo Control Engineering Practice 2015
[12] Y Yuan and J Tang ldquoAdaptive pitch control of wind turbinefor load mitigation under structural uncertaintiesrdquo Journal ofRenewable Energy vol 105 pp 483ndash494 2017
[13] C-S Wang and M-H Chiang ldquoA novel pitch control systemof a large wind turbine using two-degree-of-freedom motioncontrol with feedback linearization controlrdquo Energies vol 9 no10 article no 791 2016
[14] Z Civelek E Cam M Luy and H Mamur ldquoProportional-integral-derivative parameter optimisation of blade pitch con-troller in wind turbines by a new intelligent genetic algorithmrdquoIET Renewable Power Generation vol 10 no 8 pp 1220ndash12282016
[15] A Dahbi N Nait-Said andM-S Nait-Said ldquoA novel combinedMPPT-pitch angle control for wide range variable speed windturbine based on neural networkrdquo International Journal ofHydrogen Energy vol 41 no 22 pp 9427ndash9442 2016
[16] M L Corradini G Ippoliti and G Orlando ldquoAn observer-based blade-pitch controller of wind turbines in high windspeedsrdquo Control Engineering Practice vol 58 pp 186ndash192 2017
[17] M Zhu J Liu Z Lin and H Meng ldquoMixed H2 Hinfin pitchcontrol of wind turbine generator system based on globalexact linearization and regional pole placementrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 5 pp921ndash930 2016
[18] R Gao and Z Gao ldquoPitch control for wind turbine systemsusing optimization estimation and compensationrdquo Journal ofRenewable Energy vol 91 pp 501ndash515 2016
[19] X-X Yin Y-G Lin W Li H-W Liu and Y-J Gu ldquoAdaptivesliding mode back-stepping pitch angle control of a variable-displacement pump controlled pitch system for wind turbinesrdquoISA Transactions vol 58 pp 629ndash634 2015
[20] M-H Chiang ldquoA novel pitch control system for a wind turbinedriven by a variable-speed pump-controlled hydraulic servosystemrdquoMechatronics vol 21 no 4 pp 753ndash761 2011
[21] H Camblong S Nourdine I Vechiu and G Tapia ldquoCom-parison of an island wind turbine collective and individualpitch LQG controllers designed to alleviate fatigue loadsrdquo IETRenewable Power Generation vol 6 no 4 p 267 2012
[22] F A InthamoussouHDe Battista andR JMantz ldquoLPV-basedactive power control of wind turbines covering the completewind speed rangerdquo Journal of Renewable Energy vol 99 pp996ndash1007 2016
[23] F D Bianchi R J Mantz and C F Christiansen ldquoGainscheduling control of variable-speed wind energy conversionsystems using quasi-LPVmodelsrdquoControl Engineering Practicevol 13 no 2 pp 247ndash255 2005
[24] F A Shirazi K M Grigoriadis and D Viassolo ldquoWind turbineintegrated structural and LPV control design for improvedclosed-loop performancerdquo International Journal of Control vol85 no 8 pp 1178ndash1196 2012
[25] F A Inthamoussou F D Bianchi H De Battista and R JMantz ldquoLPV wind turbine control with anti-windup featurescovering the complete wind speed rangerdquo IEEE Transactions onEnergy Conversion vol 29 no 1 pp 259ndash266 2014
[26] K Z Oslashstegaard J Stoustrup and P Brath ldquoLinear parametervarying control of wind turbines covering both partial loadand full load conditionsrdquo International Journal of Robust andNonlinear Control vol 19 no 1 pp 92ndash116 2009
[27] Y Lin L Tu H Liu and W Li ldquoHybrid power transmissiontechnology in a wind turbine generation systemrdquo IEEEASMETransactions onMechatronics vol 20 no 3 pp 1218ndash1225 2015
[28] S Bououden M Chadli F Allouani and S Filali ldquoA newapproach for fuzzy predictive adaptive controller design usingparticle swarmoptimization algorithmrdquo International Journal ofInnovative Computing Information and Control vol 9 no 9 pp3741ndash3758 2013
[29] S H Li and H Gu ldquoFuzzy adaptive internal model controlschemes for PMSM speed-regulation systemrdquo IEEE Transac-tions on Industrial Informatics vol 8 no 4 pp 767ndash779 2012
[30] A B Asghar and X Liu ldquoAdaptive neuro-fuzzy algorithmto estimate effective wind speed and optimal rotor speed forvariable-speed wind turbinerdquo Neurocomputing vol 272 pp495ndash504 2018
[31] Y Wang H Shen H R Karimi and D Duan ldquoDissipativity-based Fuzzy Integral SlidingModeControl ofContinuous-TimeT-S Fuzzy Systemsrdquo IEEE Transactions on Fuzzy Systems pp 1-1
[32] S Bououden M Chadli and H R Karimi ldquoAn ant colonyoptimization-based fuzzy predictive control approach for non-linear processesrdquo Information Sciences vol 299 pp 143ndash1582015
[33] NWang S-F Su J Yin Z Zheng andM J Er ldquoGlobal Asymp-totic Model-Free Trajectory-Independent Tracking Control ofan Uncertain Marine Vehicle An Adaptive Universe-BasedFuzzy Control Approachrdquo IEEE Transactions on Fuzzy Systems2017
[34] R Saravanakumar andD Jena ldquoValidation of an integral slidingmode control for optimal control of a three blade variable speedvariable pitch wind turbinerdquo International Journal of ElectricalPower amp Energy Systems vol 69 pp 421ndash429 2015
[35] C Evangelista P Puleston F Valenciaga and L M FridmanldquoLyapunov-designed super-twisting sliding mode control forwind energy conversion optimizationrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 538ndash545 2013
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4 Mathematical Problems in Engineering
feedback linearization
PID
wind turbine model
sliding modealgorithm
fuzzy rules
dedt
input scaling factor
output scaling factor
ＳＣＨ
ＳＩＯＮ
+
+
+
minus
u1
u2
Figure 3 The system control structure of wind turbine
power expression of wind turbine is nonlinear As windturbine system model is a typical nonlinear system the con-troller designed based on linear system is not the best choiceHowever nonlinear controller is a better idea In this paperwe propose a novel feedforward feedback algorithm whosefeedback section is fuzzy self-tuning PID algorithm andits feedforward section is feedback linearization algorithmHowever feedback linearization control needs an accuratesystem model and it is subject to parameter variationConsequently its application is limited As sliding modecontrol is not affected by parameter variation of systemand disturbances robustness of system can be improved bycombining sliding mode method with feedback linearizationalgorithm The chart that describes this algorithm is shownin Figure 3 Differential equations of wind turbine can beacquired from Section 2 as follows
120596119903 (119869119903 + 1198962119869119892) = 12120596119903 1205881205871198772V3119862119901 (120582 120573) minus 119896119879119892
120573 = 1120591 (1205731 minus 120573)(14)
Assuming state variables 1199091 = 120596119903 1199092 = 120573 control input119906 = 1205731 state equations can be given as
1199091 = 1119869119903 + 1198962119869119892 (121199091 120588120587119877
2V3119862119901 (120582 1199092) minus 119896119879119892)1199092 = 1120591 (119906 minus 1199092)
(15)
Then output equation can be expressed as follows
119910 = 1199091 (16)
31 Fuzzy Self-Tuning PIDAlgorithm Due to the nonlinearityand time-varying characteristics of the wind turbine theconventional PID cannot meet the control requirementsThe
fuzzy algorithm transforms input values by fuzzifier andmakes a judgment based on fuzzy rules which can control themore complex systems [28ndash32] In this paper we use the fuzzyalgorithm to adjust the parameters of the PID controller in thefeedback control loop to improve the control performance ofthe controller
Generator velocity will be a constant when wind speedis overrated value As drivetrain is considered stiffness windturbine velocity is also a constant However the velocitywill fluctuate in a very small range for there are randomdisturbances Accordingly we select wind turbine velocity asreference variable When pitching the input variables of thefuzzy controller are the wind turbine velocity errors and theerror changing ratesThe output variables are the proportionintegral and differential gains of the PID controller Errorand error changing rate corresponding to the fuzzy domainare taken as minus3 minus2 minus1 0 +1 +2 +3 Fuzzy rules arethe key of fuzzy control Not only the correctness of fuzzycontrol rules directly affects the performance of the controllerbut their number is also an important factor to measurecontroller performance Therefore we define seven fuzzysubsets and the language variables are negative big negativemiddle negative small zero positive small positive middlepositive big expressed as NB NM NS Z PS PM andPB respectivelyThese language variables cannot only clearlyexpress universe but the control rule number is appropriateFuzzy rules are shown in Tables 1ndash3 They can be furtherdescribed as follows
If e is NB and ec is NB then output is PBIf e is NB and ec is NM then output is PBIf e is NB and ec is NS then output is PM
The membership function determines the sensitivity ofthe fuzzy control performance If the membership functionchangesmore smoothly the stability of the controller is betterOn the other hand if the membership function is steeperthen the controller has high sensitivity Therefore when the
Mathematical Problems in Engineering 5
Table 1 The fuzzy rules of KP
eNB NM NS Z PS PM PB
NB PB PB PM PM PS Z ZNM PB PB PM PS PS Z NSNS PM PM PM PS Z NS NS
ec Z PM PM PS Z NS NM NMPS PS PS Z NS NS NM NMPM PS Z NS NM NM NM NBPB Z Z NM NM NM NB NB
Table 2 The fuzzy rules of KI
eNB NM NS Z PS PM PB
NB NB NB NM NM NS Z ZNM NB NB NM NS NS Z ZNS NB NM NS NS Z PS PS
ec Z NM NM NS Z PS PM PMPS NM NS Z PS PS PM PBPM Z Z PS PS PM PB PBPB Z Z PS PM PM PB PB
Table 3 The fuzzy rules of KD
eNB NM NS Z PS PM PB
NB PS NS NB NB NB NM PMNM PS NS NB NM NM NS ZNS Z NS NM NM NS NS Z
ec Z Z NS NS NS NS NS ZPS Z Z Z Z Z Z ZPM PB PS PS PS PS PS PBPB PB PM PM PM PS PS PB
error and error rate of change are largeGaussianmembershipfunction with smooth change is selected which makes thesystem change smoothly andhave good robustnessWhen theerror and error changing rate are both small the triangularmembership function with high sensitivity is proposed toimprove the system response The diagram of fuzzy systemwith two-input three-output is shown in Figure 4 Parametersof fuzzy controller are shown in Table 5
e is the error and ec is the error changing rateThe outputof PID controller is as follows
1199061 = 119870119875 (119890 (119905) + 1119879119868 int 119890 (119905) 119889119905 + 119879119863119889119890 (119905)119889119905 ) (17)
where 1199061 is the output of PID controller 119870119875 is the propor-tional coefficient 119890(119905) is the error119879119868 is the integral coefficientand 119879119863 is the differential coefficient
32 Variable Universe Theory Fuzzy PID controllers exhibitgood control performance Nevertheless when input vari-ables change a lot the control accuracy needs to be further
System fuzzpid 2 inputs 3 outputs 49 rules
e (7)
ec (7)
kp (7)
ki (7)
kd (7)
fuzzpid
(mamdani)
49 rules
Figure 4 The fuzzy system structure
improved The idea of variable universe is to make any inputcan correspond to the entire input universe by adding ascaling factor without increasing the fuzzy rules based onthe changes of the input [33] If the original basic universeis adopted control accuracy is not ideal Furthermore whenthe control error becomes smaller its corresponding rulescan be reduced However the rules corresponding to theinput can be increased by adjusting scaling factor if thevariable universe method is implemented In other wordsthe corresponding rule density is increased Therefore thecontrol accuracy is improvedThe key of the variable universeapproach is to determine the scaling factor which shouldmeet the following requirements
(1) Duality for (forall119909 isin 119883) 120572(119909) = minus120572(minus119909) is a necessarycondition
(2) Avoid zero 120572(0) = 120576(3) Monotonicity 120572(119909) is strictly monotonically increas-
ing from 0 to 119864(4) Coordination forall119909 isin 119883 119909 le 120572(119909)119864(5) Regularity 120572(plusmn119864) = 1Thereby the universe scaling factors are expressed in
120572 (119909) = (|119909|119864 )120583 + 120576 (18)
120578 (119909) = (|119909|119864 )1205831 ( 10038161003816100381610038161199101003816100381610038161003816119864119862)
1205832 + 120576 (19)
where 120572(119909) is the scaling factor of the input universe 120578(119909) isthe scaling factor of the output universe 119909 is the error 119910 is theerror rate of change 119864 is the value of the error universe 119864119862 isthe value of the error changing rate universe 120583 1205831 and 1205832 areconstants from 0 to 1 and 120576 is a full small positive number
The process of changing universe is shown in Figure 5
33 Design of Feedback Linearization of Nonlinear Model forWind Turbine Feedback linearization algorithm is to adda nonlinear compensation part in control input variablesThen a new equivalent controlled system can be gained Theoriginal system is transformed into linear form As feed-back linearization algorithm is based on strict mathematicalderivation and control input is expressed in output equation
6 Mathematical Problems in Engineering
NB NM NS ZE PS PM PB
minusE EO
O
O
(A)
(B)
(C)X
X
minus(x)E (x)E
minus(x)E (x)E
Universe expansion
Universe com
pression
Figure 5 The diagram of changing universe process
the nonlinear system model can be linearized by designingappropriate control rules Therefore control performance ofsystem can be improved distinctly
The affine nonlinear models of single input single outputare expressed as [13]
= 119891 (119909) + 119892 (119909) 119906119910 = ℎ (119909) (20)
where119909 = [1199091 1199092 sdot sdot sdot 119909119899]119879 are the 119899-dimensional state vectors119891(119909) and 119892(119909) are the fully smooth vector fields in 119877119899 119906 isin 119877119910 isin 119877 and ℎ(119909) is the fully smooth nonlinear functionThe expression can be gained as follows based on (15)
(16) and (20)
119891 (119909) = [1198911 (119909)1198912 (119909)]
= [[[[
1119869119903 + 1198962119869119892 (121199091 120588120587119877
2V3119862119901 (120582 1199092) minus 119896119879119892)minus11205911199092
]]]]
(21)
119892 (119909) = [[01120591]]
(22)
ℎ (119909) = 1199091 (23)
Known from (23) there is no direct relationshipbetween the output equation and the control inputTherefore
derivative of time for output equation must be implementedas
119910 = 119889ℎ (119909)119889119905 = 120597ℎ (119909)120597119909 120597119909120597119905= 120597ℎ (119909)120597119909 [119891 (119909) + 119892 (119909) 119906]= 120597ℎ (119909)120597119909 119891 (119909) + 120597ℎ (119909)120597119909 119892 (119909) 119906
(24)
120597ℎ (119909)120597119909 = [1 0] (25)
Equations (26) and (27) can be gained based on (25)
120597ℎ (119909)120597119909 119891 (119909) = 119871119891ℎ (119909) = 1198911 (119909) (26)
120597ℎ (119909)120597119909 119892 (119909) = 119871119892ℎ (119909) = 0 (27)
Substituting (26) and (27) into (24) it is seen that 119910does not include 119906 Then derivative of time for (24) will becontinued
119910 = 120597119871119891ℎ (119909)120597119909 120597119909120597119905 =120597119871119891ℎ (119909)120597119909 [119891 (119909) + 119892 (119909) 119906]
= 1198712119891ℎ (119909) + 119871119892119871119891ℎ (119909) 119906(28)
In (28)
1198712119891ℎ (119909) = 120597119871119891ℎ (119909)120597119909 119891 (119909)= [1205971198911 (119909)1205971199091
1205971198911 (119909)1205971199092 ] [1198911 (119909)1198912 (119909)]
= 1205971198911 (119909)1205971199091 1198911 (119909) +1205971198911 (119909)1205971199092 1198912 (119909)
(29)
119871119892119871119891ℎ (119909) = 120597119871119891ℎ (119909)120597119909 119892 (119909)
= [1205971198911 (119909)12059711990911205971198911 (119909)1205971199092 ][[
01120591]]= 1205971198911 (119909)1205971199092
1120591(30)
Therefore
119871119892119871119891ℎ (119909) = 0 (31)
From now on input variable 119906 is expressed in 119910 and thecondition for meeting feedback linearization is gained
34 Input-Output Feedback Linearization Based Sliding ModeControl Both linear controller and nonlinear controller areusually adopted to control nonlinear system However linearcontroller can only control linear system well In nonlinearsystem if linear controller is used nonlinear model willbe usually linearized Sliding mode control is a nonlinear
Mathematical Problems in Engineering 7
s=0slt0
sgt0
A BC
Figure 6 The schematic diagram of sliding mode hypersurface
algorithm and it can continuously adjust itself based on real-time state variables during dynamic process for system totrack the state trajectory of sliding mode [34 35] In thispaper we propose sliding mode algorithm to enhance therobustness of feedback linearization
Assume that there exists a system as follows
= 119891 (119909) 119909 isin 119877119899 (32)
In its state space there is a hypersurface 119904(119909) =119904(1199091 1199092 sdot sdot sdot 119909119899) = 0 depicted in Figure 6The state space is divided into two parts 119904 gt 0 and 119904 lt 0 in
hypersurface In switching surface 119904 = 0 the point in systemwill be close to termination point when it is near switchingsurface
If all points are termination points in a special field inthe switching surface once the points move to this area towhich they will be attracted it is called sliding mode areaThemovement of system in slidingmode area is called slidingmode action
As the moving point is termination point in sliding modearea the expression can be gained as follows when the pointmoves to a place near the sliding mode surface
In other words
lim119904rarr0+
119904 le 0 le lim119904rarr0minus
119904 (33)
The other expression is formulated as
lim119904rarr0119904 119904 le 0 (34)
Assume that a function is given by
V (1199091 1199092 sdot sdot sdot 119909119899) = [119904 (1199091 1199092 sdot sdot sdot 119909119899)]2 (35)
The derivative of time for (35) can be expressed as follows
1198891199042119889119905 = 2119904 119904 (36)
According to (34) (36) is negative semidefinite formTherefore (35) is a Lyapunov function of system Then thesystem is stable when 119904 = 0
According to these analyses above system performancecan be improved by combining sliding mode control withfeedback linear algorithm because of stable advantage ofslidingmode control To gain good tracking performance theerror between reference and output is defined as
119890 = 119910119894119899 minus 119910 (37)
where 119910119894119899 is the ideal tracked object and 119910 is the controloutput
Then sliding function can be obtained as follows
119904 (119909 119905) = ce (38)
where
c = [119888 1] (39)
119888 gt 0 (40)
e = [119890 119890]119879 (41)
According to the discussed study for wind turbine in 33the control rule can be designed as
1199062 = 1(1205971198911 (119909) 1205971199092) (1120591) (V
minus (1205971198911 (119909)1205971199091 1198911 (119909) +1205971198911 (119909)1205971199092 1198912 (119909)) + 120578 sgn (119904))
(42)
where V is the aided term 120578 ge 0 and sgn(119904) is the signfunction which is formulated by
sgn (119904) = 1 119904 gt 0minus1 119904 lt 0 (43)
Simulation results show that there is large fluctuationin output power and rotor speed when using sign functionIn order to reduce the fluctuation saturation function isimplemented to replace sign function Saturation function isdefined as follows Results are shown in Figures 7(a)ndash7(d) andFigures 8(a)ndash8(d)
119904119886119905 (119904) =
1 119904 gt 120575119904120575 |119904| le 120575minus1 119904 lt minus120575
(44)
Up to now the final control law can be expressed basedon (17) and (42) as follows
119906 = 1199061 + 1199062 (45)
4 Simulation Results and Analyses
Simulation experiments based on a MW scale wind turbineare carried out to verify the effectiveness of the proposedalgorithm in which parameters are depicted in Table 4 Thecontroller parameters are shown in Table 5
The simulation experiments for wind turbine pitchingcontrol are implemented using step wind and random windStep speed is shown in Figure 7(a) and random wind isdepicted in Figure 8(a)
Firstly Figures 7(b)ndash7(d) and Figures 8(b)ndash8(d) depictthe control results when using sign function and saturationfunction The results show that there are little differences in
8 Mathematical Problems in Engineering
Table 4 The parameters of the wind turbine
Symbols Parameters Values119875119903119886119905119890119889 Rated output power 3000KW119881119903119886119905119890119889 Rated wind speed 12ms119877 The Radius of the turbine rotor 475m119869119903 The inertia moment of the turbine rotor 6250000kgsdotm2119869119892 The inertia moment of the generator 15kgsdotm2119896 Drive ratio 80120591 Time constant 021199031 Stator winding resistance 00148Ω1199032 Rotor winding resistance calculated to stator 00370Ω1199091120590 Leakage reactance 00727Ω1199092120590 Leakage calculated to stator side 00863Ω119909119898 Excitation reactance 20000Ω
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Step wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19992
200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(d) Output powers using different functions in sliding mode
Figure 7 (a) Step wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions in slidingmode (d) output powers using different functions in sliding mode
Mathematical Problems in Engineering 9
Table 5 The parameters of the proposed controller
Symbols Parameters Values119888 Constant defined in sliding mode algorithm 10119890 119903119886119899119892119890 The universe of rotor deviation [-0003 0003]119890119888 119903119886119899119892119890 The universe of rotor deviation changing rate [-0008 0008]119870119875 119903119886119899119892119890 The universe of proportionality coefficent [-03 03]119870119868 119903119886119899119892119890 The universe of integral coefficient [-006 006]119870119863 119903119886119899119892119890 The universe of differential coefficient [-3 3]119864 The value of the error universe 0003119864119862 The value of the error changing rate universe 0008120583 The coefficient of universe scaling factor 091205831 The coefficient of universe scaling factor 091205832 The coefficient of universe scaling factor 04120576 A full small positive number 000001
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Random wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
25
3
35
4
P(M
W)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
(d) Output powers using different functions in sliding mode
Figure 8 (a) Random wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions insliding mode (d) output powers using different functions in sliding mode
10 Mathematical Problems in Engineering
5
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
Local enlargeddrawing
4 5 6 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under step wind speed
9 10 11 12
2
2001Localenlargeddrawing
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under step wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
9 10 11 12
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) output power using different controllers under step wind speed
Figure 9 (a) Pitch angles using different controllers under step wind speed (b) rotor speed using different controllers under step wind speedand (c) output power using different controllers under step wind speed
the control when implementing the two different functionsunder the step wind speed However under random windspeed the control performance of the saturation functionadjusting output power is significantly better than that of thesign function in Figure 8(d)
Comparisons are further studied among which thereare conventional PID algorithm feedforward feedback con-trol combining PID with feedback linearization algorithmfeedforward feedback control based on fuzzy self-tuningPID with feedback linearization with sliding mode control(called fuzzy PID in feedback loop in Figures 9(a)ndash9(c)
and Figures 10(a)ndash10(c)) and the proposed controller inthis paper Results are depicted in Figures 9(a)ndash9(c) andFigures 10(a)ndash10(c) The wind speed is changed continuouslyto further validate pitching control algorithms which isdepicted in Figure 8(a)
At the initial moment control outputs fluctuate stronglywhich are caused by initial values of states However iffluctuations are in acceptable ranges of the physical structureof the wind turbine the initial responses will not affect theevaluation for the controller In Figure 7(a) at 119905 = 4 and119905 = 10 step wind speed occurs As thewind speeds exceed the
Mathematical Problems in Engineering 11
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
12 13 14 15
5 Local enlargeddrawing
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
8 9 10 11
1999
2
2001
Localenlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
13 14 15 16
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) Output power using different controllers under random wind speed
Figure 10 (a) Pitch angles using different controllers under random wind speed (b) rotor speed using different controllers under randomwind speed and (c) output power using different controllers under random wind speed
rated value pitch angles need to be adjusted to keep outputpower at constant
In Figure 9(a) the overshoot of the pitch angle is greaterwhile oscillation time is longer when adopting the conven-tional PID controller Although rotor speed also convergesat the rated value steady-state errors cannot be eliminatedwhich is approved in rotor speed chart depicted in Fig-ure 9(b) The rotor speeds of wind turbine are maintainedat constant when wind speed exceeds the rated value Dueto the steady-state errors of the pitch angle rotor speedshave deviations These results show that single linear PID
controller cannot track target well for wind turbine exhibitsstrongly nonlinear features
To address this problem we design a PID controllercombining with feedback linearization theory The PIDcontroller is feedback control while feedback linearizationis proposed as feedforward control Feedback control cankeep the system stable However the feedback control ishysteric Only when the deviation occurs the controller isimplemented to eliminate it Nevertheless feedforward canproduce a control variable in advanceThus the final outputsof the two controllers are proposed as the input of the wind
12 Mathematical Problems in Engineering
turbine system so that tracking accuracy of controller can beenhanced
Figures 9(a)ndash9(c) demonstrate that the control perfor-mance of feedforward feedback controller based on com-bining PID algorithm with feedback linearization algorithmis clearly superior to conventional PID controller in whichthe overshoot of pitch angle is smaller and adjusting timeof oscillating is shorter and furthermore the steady-statedeviation is eliminated which is also shown both in therotor speed diagram and in the output power diagram Tofurther improve control performance sliding mode controlis combined with feedback linear algorithm in feedforwardloop while the fuzzy self-tuning algorithm is used to adjustthe parameters of the PID controller in feedback loopFurthermore in order to decrease the effect when inputvariable and output variable change a lot variable universeapproach is proposed to optimize fuzzy rules Known fromthe results the deficiencies of controller produced by systemconstruct and parameters variation can be overcome FromFigures 9(a)ndash9(c) it is shown that the performance of thecontroller we propose in this paper is the best among thefour controllers whose overshoot is the least while it rapidlyconverges to constant without steady-state errors Althoughcontinuous step speed can evaluate controller performanceit still exhibits limitation
In order to further verify the effectiveness and universal-ity of the proposed controller in this paper we further adoptrandom wind speed Figure 8(a) depicts random wind speedFigures 10(a)ndash10(c) show the control results of the pitch anglerotor speed of turbine and output power produced by fourdifferent control algorithms under random wind speed Theresults are reasonable which are the same as those producedby step wind speed Therefore the control algorithm wepropose in this paper is the best among the described fouralgorithms and it can keep the output power at the rated valuewhen wind speed exceeds the nominal velocity
5 Conclusions
In this research a novel pitch control strategy for largevariable speed variable pitch wind turbines is studied aimingat maintaining the output power at constant so that theenergy quality is improved Known from the results as thesliding mode theory is combined with feedback linearizationalgorithm in feedforward loop the uncertainty of modeland time-varying of system parameters are overcome Fur-thermore with setting a control variable in advance theovershoot can be decreased when wind speed is varyingOn the other hand in feedback loop as variable universefuzzy algorithm is employed to optimize the parameters ofthe conventional PID controller the tracking performanceof the feedback controller is enhanced Meanwhile insteadof sign function saturation function is implemented todecrease oscillation of sliding mode algorithm and thus thefluctuations of output power and rotor speed of wind turbineare mitigated Both control accuracy and robustness usingthe proposed algorithm are the best among four differentcontrollers Another in future we plan to supply fuzzy
predictive adaptive algorithm in power control to furtherimprove system performance
Data Availability
The data used to support the findings of this study arecurrently under embargo while the research findings arecommercialized Requests for data 6 months after publica-tion of this article will be considered by the correspondingauthor
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National Natural ScienceFund of China under Grant no 51405052
References
[1] C-SWang andM-H Chiang ldquoA novel dynamic co-simulationanalysis for overall closed loop operation control of a large windturbinerdquo Energies vol 9 no 8 2016
[2] E Van Solingen J Beerens S P Mulders R De Breuker andJ W Van Wingerden ldquoControl design for a two-bladed down-wind teeterless damped free-yaw wind turbinerdquo Mechatronicsvol 36 pp 77ndash96 2016
[3] Y-J Chen and Y C Shiah ldquoExperiments on the performance ofsmall horizontal axis wind turbine with passive pitch control bydisk pulleyrdquo Energies vol 9 no 5 2016
[4] A Choudhry M Arjomandi and R Kelso ldquoMethods tocontrol dynamic stall for wind turbine applicationsrdquo Journal ofRenewable Energy vol 86 pp 26ndash37 2016
[5] H Li C Yang Y Hu X Liao Z Zeng and C Zhe ldquoAnimproved reduced-ordermodel of an electric pitch drive systemfor wind turbine control system design and simulationrdquo Journalof Renewable Energy vol 93 pp 188ndash200 2016
[6] E B Muhando T Senjyu A Uehara and T Funabashi ldquoGain-scheduledHinfin control forWECS via LMI techniques and para-metrically dependent feedback part II Controller design andimplementationrdquo IEEE Transactions on Industrial Electronicsvol 58 no 1 pp 57ndash65 2011
[7] F D Bianchi H De Battista and R JMantz ldquoRobustmultivari-able gain scheduled control of wind turbines for variable powerproductionrdquo International Journal of Systems Control vol 1 no3 pp 103ndash112 2010
[8] I Soslashnderby andMH Hansen ldquoOpen-loop frequency responseanalysis of a wind turbine using a high-order linear aeroelasticmodelrdquoWind Energy vol 17 no 8 pp 1147ndash1167 2014
[9] S Bououden M Chadli S Filali and A El Hajjaji ldquoFuzzymodel basedmultivariable predictive control of a variable speedwind turbine LMI approachrdquo Journal of Renewable Energy vol37 no 1 pp 434ndash439 2012
[10] J Chen T Lin CWen and Y Song ldquoDesign of a Unified PowerController for Variable-Speed Fixed-Pitch Wind Energy Con-version Systemrdquo IEEETransactions on Industrial Electronics vol63 no 8 pp 4899ndash4908 2016
Mathematical Problems in Engineering 13
[11] Y Ren L Li J Brindley and L Jiang ldquoNonlinear PI control forvariable pitch wind turbinerdquo Control Engineering Practice 2015
[12] Y Yuan and J Tang ldquoAdaptive pitch control of wind turbinefor load mitigation under structural uncertaintiesrdquo Journal ofRenewable Energy vol 105 pp 483ndash494 2017
[13] C-S Wang and M-H Chiang ldquoA novel pitch control systemof a large wind turbine using two-degree-of-freedom motioncontrol with feedback linearization controlrdquo Energies vol 9 no10 article no 791 2016
[14] Z Civelek E Cam M Luy and H Mamur ldquoProportional-integral-derivative parameter optimisation of blade pitch con-troller in wind turbines by a new intelligent genetic algorithmrdquoIET Renewable Power Generation vol 10 no 8 pp 1220ndash12282016
[15] A Dahbi N Nait-Said andM-S Nait-Said ldquoA novel combinedMPPT-pitch angle control for wide range variable speed windturbine based on neural networkrdquo International Journal ofHydrogen Energy vol 41 no 22 pp 9427ndash9442 2016
[16] M L Corradini G Ippoliti and G Orlando ldquoAn observer-based blade-pitch controller of wind turbines in high windspeedsrdquo Control Engineering Practice vol 58 pp 186ndash192 2017
[17] M Zhu J Liu Z Lin and H Meng ldquoMixed H2 Hinfin pitchcontrol of wind turbine generator system based on globalexact linearization and regional pole placementrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 5 pp921ndash930 2016
[18] R Gao and Z Gao ldquoPitch control for wind turbine systemsusing optimization estimation and compensationrdquo Journal ofRenewable Energy vol 91 pp 501ndash515 2016
[19] X-X Yin Y-G Lin W Li H-W Liu and Y-J Gu ldquoAdaptivesliding mode back-stepping pitch angle control of a variable-displacement pump controlled pitch system for wind turbinesrdquoISA Transactions vol 58 pp 629ndash634 2015
[20] M-H Chiang ldquoA novel pitch control system for a wind turbinedriven by a variable-speed pump-controlled hydraulic servosystemrdquoMechatronics vol 21 no 4 pp 753ndash761 2011
[21] H Camblong S Nourdine I Vechiu and G Tapia ldquoCom-parison of an island wind turbine collective and individualpitch LQG controllers designed to alleviate fatigue loadsrdquo IETRenewable Power Generation vol 6 no 4 p 267 2012
[22] F A InthamoussouHDe Battista andR JMantz ldquoLPV-basedactive power control of wind turbines covering the completewind speed rangerdquo Journal of Renewable Energy vol 99 pp996ndash1007 2016
[23] F D Bianchi R J Mantz and C F Christiansen ldquoGainscheduling control of variable-speed wind energy conversionsystems using quasi-LPVmodelsrdquoControl Engineering Practicevol 13 no 2 pp 247ndash255 2005
[24] F A Shirazi K M Grigoriadis and D Viassolo ldquoWind turbineintegrated structural and LPV control design for improvedclosed-loop performancerdquo International Journal of Control vol85 no 8 pp 1178ndash1196 2012
[25] F A Inthamoussou F D Bianchi H De Battista and R JMantz ldquoLPV wind turbine control with anti-windup featurescovering the complete wind speed rangerdquo IEEE Transactions onEnergy Conversion vol 29 no 1 pp 259ndash266 2014
[26] K Z Oslashstegaard J Stoustrup and P Brath ldquoLinear parametervarying control of wind turbines covering both partial loadand full load conditionsrdquo International Journal of Robust andNonlinear Control vol 19 no 1 pp 92ndash116 2009
[27] Y Lin L Tu H Liu and W Li ldquoHybrid power transmissiontechnology in a wind turbine generation systemrdquo IEEEASMETransactions onMechatronics vol 20 no 3 pp 1218ndash1225 2015
[28] S Bououden M Chadli F Allouani and S Filali ldquoA newapproach for fuzzy predictive adaptive controller design usingparticle swarmoptimization algorithmrdquo International Journal ofInnovative Computing Information and Control vol 9 no 9 pp3741ndash3758 2013
[29] S H Li and H Gu ldquoFuzzy adaptive internal model controlschemes for PMSM speed-regulation systemrdquo IEEE Transac-tions on Industrial Informatics vol 8 no 4 pp 767ndash779 2012
[30] A B Asghar and X Liu ldquoAdaptive neuro-fuzzy algorithmto estimate effective wind speed and optimal rotor speed forvariable-speed wind turbinerdquo Neurocomputing vol 272 pp495ndash504 2018
[31] Y Wang H Shen H R Karimi and D Duan ldquoDissipativity-based Fuzzy Integral SlidingModeControl ofContinuous-TimeT-S Fuzzy Systemsrdquo IEEE Transactions on Fuzzy Systems pp 1-1
[32] S Bououden M Chadli and H R Karimi ldquoAn ant colonyoptimization-based fuzzy predictive control approach for non-linear processesrdquo Information Sciences vol 299 pp 143ndash1582015
[33] NWang S-F Su J Yin Z Zheng andM J Er ldquoGlobal Asymp-totic Model-Free Trajectory-Independent Tracking Control ofan Uncertain Marine Vehicle An Adaptive Universe-BasedFuzzy Control Approachrdquo IEEE Transactions on Fuzzy Systems2017
[34] R Saravanakumar andD Jena ldquoValidation of an integral slidingmode control for optimal control of a three blade variable speedvariable pitch wind turbinerdquo International Journal of ElectricalPower amp Energy Systems vol 69 pp 421ndash429 2015
[35] C Evangelista P Puleston F Valenciaga and L M FridmanldquoLyapunov-designed super-twisting sliding mode control forwind energy conversion optimizationrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 538ndash545 2013
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Mathematical Problems in Engineering 5
Table 1 The fuzzy rules of KP
eNB NM NS Z PS PM PB
NB PB PB PM PM PS Z ZNM PB PB PM PS PS Z NSNS PM PM PM PS Z NS NS
ec Z PM PM PS Z NS NM NMPS PS PS Z NS NS NM NMPM PS Z NS NM NM NM NBPB Z Z NM NM NM NB NB
Table 2 The fuzzy rules of KI
eNB NM NS Z PS PM PB
NB NB NB NM NM NS Z ZNM NB NB NM NS NS Z ZNS NB NM NS NS Z PS PS
ec Z NM NM NS Z PS PM PMPS NM NS Z PS PS PM PBPM Z Z PS PS PM PB PBPB Z Z PS PM PM PB PB
Table 3 The fuzzy rules of KD
eNB NM NS Z PS PM PB
NB PS NS NB NB NB NM PMNM PS NS NB NM NM NS ZNS Z NS NM NM NS NS Z
ec Z Z NS NS NS NS NS ZPS Z Z Z Z Z Z ZPM PB PS PS PS PS PS PBPB PB PM PM PM PS PS PB
error and error rate of change are largeGaussianmembershipfunction with smooth change is selected which makes thesystem change smoothly andhave good robustnessWhen theerror and error changing rate are both small the triangularmembership function with high sensitivity is proposed toimprove the system response The diagram of fuzzy systemwith two-input three-output is shown in Figure 4 Parametersof fuzzy controller are shown in Table 5
e is the error and ec is the error changing rateThe outputof PID controller is as follows
1199061 = 119870119875 (119890 (119905) + 1119879119868 int 119890 (119905) 119889119905 + 119879119863119889119890 (119905)119889119905 ) (17)
where 1199061 is the output of PID controller 119870119875 is the propor-tional coefficient 119890(119905) is the error119879119868 is the integral coefficientand 119879119863 is the differential coefficient
32 Variable Universe Theory Fuzzy PID controllers exhibitgood control performance Nevertheless when input vari-ables change a lot the control accuracy needs to be further
System fuzzpid 2 inputs 3 outputs 49 rules
e (7)
ec (7)
kp (7)
ki (7)
kd (7)
fuzzpid
(mamdani)
49 rules
Figure 4 The fuzzy system structure
improved The idea of variable universe is to make any inputcan correspond to the entire input universe by adding ascaling factor without increasing the fuzzy rules based onthe changes of the input [33] If the original basic universeis adopted control accuracy is not ideal Furthermore whenthe control error becomes smaller its corresponding rulescan be reduced However the rules corresponding to theinput can be increased by adjusting scaling factor if thevariable universe method is implemented In other wordsthe corresponding rule density is increased Therefore thecontrol accuracy is improvedThe key of the variable universeapproach is to determine the scaling factor which shouldmeet the following requirements
(1) Duality for (forall119909 isin 119883) 120572(119909) = minus120572(minus119909) is a necessarycondition
(2) Avoid zero 120572(0) = 120576(3) Monotonicity 120572(119909) is strictly monotonically increas-
ing from 0 to 119864(4) Coordination forall119909 isin 119883 119909 le 120572(119909)119864(5) Regularity 120572(plusmn119864) = 1Thereby the universe scaling factors are expressed in
120572 (119909) = (|119909|119864 )120583 + 120576 (18)
120578 (119909) = (|119909|119864 )1205831 ( 10038161003816100381610038161199101003816100381610038161003816119864119862)
1205832 + 120576 (19)
where 120572(119909) is the scaling factor of the input universe 120578(119909) isthe scaling factor of the output universe 119909 is the error 119910 is theerror rate of change 119864 is the value of the error universe 119864119862 isthe value of the error changing rate universe 120583 1205831 and 1205832 areconstants from 0 to 1 and 120576 is a full small positive number
The process of changing universe is shown in Figure 5
33 Design of Feedback Linearization of Nonlinear Model forWind Turbine Feedback linearization algorithm is to adda nonlinear compensation part in control input variablesThen a new equivalent controlled system can be gained Theoriginal system is transformed into linear form As feed-back linearization algorithm is based on strict mathematicalderivation and control input is expressed in output equation
6 Mathematical Problems in Engineering
NB NM NS ZE PS PM PB
minusE EO
O
O
(A)
(B)
(C)X
X
minus(x)E (x)E
minus(x)E (x)E
Universe expansion
Universe com
pression
Figure 5 The diagram of changing universe process
the nonlinear system model can be linearized by designingappropriate control rules Therefore control performance ofsystem can be improved distinctly
The affine nonlinear models of single input single outputare expressed as [13]
= 119891 (119909) + 119892 (119909) 119906119910 = ℎ (119909) (20)
where119909 = [1199091 1199092 sdot sdot sdot 119909119899]119879 are the 119899-dimensional state vectors119891(119909) and 119892(119909) are the fully smooth vector fields in 119877119899 119906 isin 119877119910 isin 119877 and ℎ(119909) is the fully smooth nonlinear functionThe expression can be gained as follows based on (15)
(16) and (20)
119891 (119909) = [1198911 (119909)1198912 (119909)]
= [[[[
1119869119903 + 1198962119869119892 (121199091 120588120587119877
2V3119862119901 (120582 1199092) minus 119896119879119892)minus11205911199092
]]]]
(21)
119892 (119909) = [[01120591]]
(22)
ℎ (119909) = 1199091 (23)
Known from (23) there is no direct relationshipbetween the output equation and the control inputTherefore
derivative of time for output equation must be implementedas
119910 = 119889ℎ (119909)119889119905 = 120597ℎ (119909)120597119909 120597119909120597119905= 120597ℎ (119909)120597119909 [119891 (119909) + 119892 (119909) 119906]= 120597ℎ (119909)120597119909 119891 (119909) + 120597ℎ (119909)120597119909 119892 (119909) 119906
(24)
120597ℎ (119909)120597119909 = [1 0] (25)
Equations (26) and (27) can be gained based on (25)
120597ℎ (119909)120597119909 119891 (119909) = 119871119891ℎ (119909) = 1198911 (119909) (26)
120597ℎ (119909)120597119909 119892 (119909) = 119871119892ℎ (119909) = 0 (27)
Substituting (26) and (27) into (24) it is seen that 119910does not include 119906 Then derivative of time for (24) will becontinued
119910 = 120597119871119891ℎ (119909)120597119909 120597119909120597119905 =120597119871119891ℎ (119909)120597119909 [119891 (119909) + 119892 (119909) 119906]
= 1198712119891ℎ (119909) + 119871119892119871119891ℎ (119909) 119906(28)
In (28)
1198712119891ℎ (119909) = 120597119871119891ℎ (119909)120597119909 119891 (119909)= [1205971198911 (119909)1205971199091
1205971198911 (119909)1205971199092 ] [1198911 (119909)1198912 (119909)]
= 1205971198911 (119909)1205971199091 1198911 (119909) +1205971198911 (119909)1205971199092 1198912 (119909)
(29)
119871119892119871119891ℎ (119909) = 120597119871119891ℎ (119909)120597119909 119892 (119909)
= [1205971198911 (119909)12059711990911205971198911 (119909)1205971199092 ][[
01120591]]= 1205971198911 (119909)1205971199092
1120591(30)
Therefore
119871119892119871119891ℎ (119909) = 0 (31)
From now on input variable 119906 is expressed in 119910 and thecondition for meeting feedback linearization is gained
34 Input-Output Feedback Linearization Based Sliding ModeControl Both linear controller and nonlinear controller areusually adopted to control nonlinear system However linearcontroller can only control linear system well In nonlinearsystem if linear controller is used nonlinear model willbe usually linearized Sliding mode control is a nonlinear
Mathematical Problems in Engineering 7
s=0slt0
sgt0
A BC
Figure 6 The schematic diagram of sliding mode hypersurface
algorithm and it can continuously adjust itself based on real-time state variables during dynamic process for system totrack the state trajectory of sliding mode [34 35] In thispaper we propose sliding mode algorithm to enhance therobustness of feedback linearization
Assume that there exists a system as follows
= 119891 (119909) 119909 isin 119877119899 (32)
In its state space there is a hypersurface 119904(119909) =119904(1199091 1199092 sdot sdot sdot 119909119899) = 0 depicted in Figure 6The state space is divided into two parts 119904 gt 0 and 119904 lt 0 in
hypersurface In switching surface 119904 = 0 the point in systemwill be close to termination point when it is near switchingsurface
If all points are termination points in a special field inthe switching surface once the points move to this area towhich they will be attracted it is called sliding mode areaThemovement of system in slidingmode area is called slidingmode action
As the moving point is termination point in sliding modearea the expression can be gained as follows when the pointmoves to a place near the sliding mode surface
In other words
lim119904rarr0+
119904 le 0 le lim119904rarr0minus
119904 (33)
The other expression is formulated as
lim119904rarr0119904 119904 le 0 (34)
Assume that a function is given by
V (1199091 1199092 sdot sdot sdot 119909119899) = [119904 (1199091 1199092 sdot sdot sdot 119909119899)]2 (35)
The derivative of time for (35) can be expressed as follows
1198891199042119889119905 = 2119904 119904 (36)
According to (34) (36) is negative semidefinite formTherefore (35) is a Lyapunov function of system Then thesystem is stable when 119904 = 0
According to these analyses above system performancecan be improved by combining sliding mode control withfeedback linear algorithm because of stable advantage ofslidingmode control To gain good tracking performance theerror between reference and output is defined as
119890 = 119910119894119899 minus 119910 (37)
where 119910119894119899 is the ideal tracked object and 119910 is the controloutput
Then sliding function can be obtained as follows
119904 (119909 119905) = ce (38)
where
c = [119888 1] (39)
119888 gt 0 (40)
e = [119890 119890]119879 (41)
According to the discussed study for wind turbine in 33the control rule can be designed as
1199062 = 1(1205971198911 (119909) 1205971199092) (1120591) (V
minus (1205971198911 (119909)1205971199091 1198911 (119909) +1205971198911 (119909)1205971199092 1198912 (119909)) + 120578 sgn (119904))
(42)
where V is the aided term 120578 ge 0 and sgn(119904) is the signfunction which is formulated by
sgn (119904) = 1 119904 gt 0minus1 119904 lt 0 (43)
Simulation results show that there is large fluctuationin output power and rotor speed when using sign functionIn order to reduce the fluctuation saturation function isimplemented to replace sign function Saturation function isdefined as follows Results are shown in Figures 7(a)ndash7(d) andFigures 8(a)ndash8(d)
119904119886119905 (119904) =
1 119904 gt 120575119904120575 |119904| le 120575minus1 119904 lt minus120575
(44)
Up to now the final control law can be expressed basedon (17) and (42) as follows
119906 = 1199061 + 1199062 (45)
4 Simulation Results and Analyses
Simulation experiments based on a MW scale wind turbineare carried out to verify the effectiveness of the proposedalgorithm in which parameters are depicted in Table 4 Thecontroller parameters are shown in Table 5
The simulation experiments for wind turbine pitchingcontrol are implemented using step wind and random windStep speed is shown in Figure 7(a) and random wind isdepicted in Figure 8(a)
Firstly Figures 7(b)ndash7(d) and Figures 8(b)ndash8(d) depictthe control results when using sign function and saturationfunction The results show that there are little differences in
8 Mathematical Problems in Engineering
Table 4 The parameters of the wind turbine
Symbols Parameters Values119875119903119886119905119890119889 Rated output power 3000KW119881119903119886119905119890119889 Rated wind speed 12ms119877 The Radius of the turbine rotor 475m119869119903 The inertia moment of the turbine rotor 6250000kgsdotm2119869119892 The inertia moment of the generator 15kgsdotm2119896 Drive ratio 80120591 Time constant 021199031 Stator winding resistance 00148Ω1199032 Rotor winding resistance calculated to stator 00370Ω1199091120590 Leakage reactance 00727Ω1199092120590 Leakage calculated to stator side 00863Ω119909119898 Excitation reactance 20000Ω
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Step wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19992
200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(d) Output powers using different functions in sliding mode
Figure 7 (a) Step wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions in slidingmode (d) output powers using different functions in sliding mode
Mathematical Problems in Engineering 9
Table 5 The parameters of the proposed controller
Symbols Parameters Values119888 Constant defined in sliding mode algorithm 10119890 119903119886119899119892119890 The universe of rotor deviation [-0003 0003]119890119888 119903119886119899119892119890 The universe of rotor deviation changing rate [-0008 0008]119870119875 119903119886119899119892119890 The universe of proportionality coefficent [-03 03]119870119868 119903119886119899119892119890 The universe of integral coefficient [-006 006]119870119863 119903119886119899119892119890 The universe of differential coefficient [-3 3]119864 The value of the error universe 0003119864119862 The value of the error changing rate universe 0008120583 The coefficient of universe scaling factor 091205831 The coefficient of universe scaling factor 091205832 The coefficient of universe scaling factor 04120576 A full small positive number 000001
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Random wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
25
3
35
4
P(M
W)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
(d) Output powers using different functions in sliding mode
Figure 8 (a) Random wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions insliding mode (d) output powers using different functions in sliding mode
10 Mathematical Problems in Engineering
5
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
Local enlargeddrawing
4 5 6 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under step wind speed
9 10 11 12
2
2001Localenlargeddrawing
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under step wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
9 10 11 12
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) output power using different controllers under step wind speed
Figure 9 (a) Pitch angles using different controllers under step wind speed (b) rotor speed using different controllers under step wind speedand (c) output power using different controllers under step wind speed
the control when implementing the two different functionsunder the step wind speed However under random windspeed the control performance of the saturation functionadjusting output power is significantly better than that of thesign function in Figure 8(d)
Comparisons are further studied among which thereare conventional PID algorithm feedforward feedback con-trol combining PID with feedback linearization algorithmfeedforward feedback control based on fuzzy self-tuningPID with feedback linearization with sliding mode control(called fuzzy PID in feedback loop in Figures 9(a)ndash9(c)
and Figures 10(a)ndash10(c)) and the proposed controller inthis paper Results are depicted in Figures 9(a)ndash9(c) andFigures 10(a)ndash10(c) The wind speed is changed continuouslyto further validate pitching control algorithms which isdepicted in Figure 8(a)
At the initial moment control outputs fluctuate stronglywhich are caused by initial values of states However iffluctuations are in acceptable ranges of the physical structureof the wind turbine the initial responses will not affect theevaluation for the controller In Figure 7(a) at 119905 = 4 and119905 = 10 step wind speed occurs As thewind speeds exceed the
Mathematical Problems in Engineering 11
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
12 13 14 15
5 Local enlargeddrawing
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
8 9 10 11
1999
2
2001
Localenlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
13 14 15 16
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) Output power using different controllers under random wind speed
Figure 10 (a) Pitch angles using different controllers under random wind speed (b) rotor speed using different controllers under randomwind speed and (c) output power using different controllers under random wind speed
rated value pitch angles need to be adjusted to keep outputpower at constant
In Figure 9(a) the overshoot of the pitch angle is greaterwhile oscillation time is longer when adopting the conven-tional PID controller Although rotor speed also convergesat the rated value steady-state errors cannot be eliminatedwhich is approved in rotor speed chart depicted in Fig-ure 9(b) The rotor speeds of wind turbine are maintainedat constant when wind speed exceeds the rated value Dueto the steady-state errors of the pitch angle rotor speedshave deviations These results show that single linear PID
controller cannot track target well for wind turbine exhibitsstrongly nonlinear features
To address this problem we design a PID controllercombining with feedback linearization theory The PIDcontroller is feedback control while feedback linearizationis proposed as feedforward control Feedback control cankeep the system stable However the feedback control ishysteric Only when the deviation occurs the controller isimplemented to eliminate it Nevertheless feedforward canproduce a control variable in advanceThus the final outputsof the two controllers are proposed as the input of the wind
12 Mathematical Problems in Engineering
turbine system so that tracking accuracy of controller can beenhanced
Figures 9(a)ndash9(c) demonstrate that the control perfor-mance of feedforward feedback controller based on com-bining PID algorithm with feedback linearization algorithmis clearly superior to conventional PID controller in whichthe overshoot of pitch angle is smaller and adjusting timeof oscillating is shorter and furthermore the steady-statedeviation is eliminated which is also shown both in therotor speed diagram and in the output power diagram Tofurther improve control performance sliding mode controlis combined with feedback linear algorithm in feedforwardloop while the fuzzy self-tuning algorithm is used to adjustthe parameters of the PID controller in feedback loopFurthermore in order to decrease the effect when inputvariable and output variable change a lot variable universeapproach is proposed to optimize fuzzy rules Known fromthe results the deficiencies of controller produced by systemconstruct and parameters variation can be overcome FromFigures 9(a)ndash9(c) it is shown that the performance of thecontroller we propose in this paper is the best among thefour controllers whose overshoot is the least while it rapidlyconverges to constant without steady-state errors Althoughcontinuous step speed can evaluate controller performanceit still exhibits limitation
In order to further verify the effectiveness and universal-ity of the proposed controller in this paper we further adoptrandom wind speed Figure 8(a) depicts random wind speedFigures 10(a)ndash10(c) show the control results of the pitch anglerotor speed of turbine and output power produced by fourdifferent control algorithms under random wind speed Theresults are reasonable which are the same as those producedby step wind speed Therefore the control algorithm wepropose in this paper is the best among the described fouralgorithms and it can keep the output power at the rated valuewhen wind speed exceeds the nominal velocity
5 Conclusions
In this research a novel pitch control strategy for largevariable speed variable pitch wind turbines is studied aimingat maintaining the output power at constant so that theenergy quality is improved Known from the results as thesliding mode theory is combined with feedback linearizationalgorithm in feedforward loop the uncertainty of modeland time-varying of system parameters are overcome Fur-thermore with setting a control variable in advance theovershoot can be decreased when wind speed is varyingOn the other hand in feedback loop as variable universefuzzy algorithm is employed to optimize the parameters ofthe conventional PID controller the tracking performanceof the feedback controller is enhanced Meanwhile insteadof sign function saturation function is implemented todecrease oscillation of sliding mode algorithm and thus thefluctuations of output power and rotor speed of wind turbineare mitigated Both control accuracy and robustness usingthe proposed algorithm are the best among four differentcontrollers Another in future we plan to supply fuzzy
predictive adaptive algorithm in power control to furtherimprove system performance
Data Availability
The data used to support the findings of this study arecurrently under embargo while the research findings arecommercialized Requests for data 6 months after publica-tion of this article will be considered by the correspondingauthor
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National Natural ScienceFund of China under Grant no 51405052
References
[1] C-SWang andM-H Chiang ldquoA novel dynamic co-simulationanalysis for overall closed loop operation control of a large windturbinerdquo Energies vol 9 no 8 2016
[2] E Van Solingen J Beerens S P Mulders R De Breuker andJ W Van Wingerden ldquoControl design for a two-bladed down-wind teeterless damped free-yaw wind turbinerdquo Mechatronicsvol 36 pp 77ndash96 2016
[3] Y-J Chen and Y C Shiah ldquoExperiments on the performance ofsmall horizontal axis wind turbine with passive pitch control bydisk pulleyrdquo Energies vol 9 no 5 2016
[4] A Choudhry M Arjomandi and R Kelso ldquoMethods tocontrol dynamic stall for wind turbine applicationsrdquo Journal ofRenewable Energy vol 86 pp 26ndash37 2016
[5] H Li C Yang Y Hu X Liao Z Zeng and C Zhe ldquoAnimproved reduced-ordermodel of an electric pitch drive systemfor wind turbine control system design and simulationrdquo Journalof Renewable Energy vol 93 pp 188ndash200 2016
[6] E B Muhando T Senjyu A Uehara and T Funabashi ldquoGain-scheduledHinfin control forWECS via LMI techniques and para-metrically dependent feedback part II Controller design andimplementationrdquo IEEE Transactions on Industrial Electronicsvol 58 no 1 pp 57ndash65 2011
[7] F D Bianchi H De Battista and R JMantz ldquoRobustmultivari-able gain scheduled control of wind turbines for variable powerproductionrdquo International Journal of Systems Control vol 1 no3 pp 103ndash112 2010
[8] I Soslashnderby andMH Hansen ldquoOpen-loop frequency responseanalysis of a wind turbine using a high-order linear aeroelasticmodelrdquoWind Energy vol 17 no 8 pp 1147ndash1167 2014
[9] S Bououden M Chadli S Filali and A El Hajjaji ldquoFuzzymodel basedmultivariable predictive control of a variable speedwind turbine LMI approachrdquo Journal of Renewable Energy vol37 no 1 pp 434ndash439 2012
[10] J Chen T Lin CWen and Y Song ldquoDesign of a Unified PowerController for Variable-Speed Fixed-Pitch Wind Energy Con-version Systemrdquo IEEETransactions on Industrial Electronics vol63 no 8 pp 4899ndash4908 2016
Mathematical Problems in Engineering 13
[11] Y Ren L Li J Brindley and L Jiang ldquoNonlinear PI control forvariable pitch wind turbinerdquo Control Engineering Practice 2015
[12] Y Yuan and J Tang ldquoAdaptive pitch control of wind turbinefor load mitigation under structural uncertaintiesrdquo Journal ofRenewable Energy vol 105 pp 483ndash494 2017
[13] C-S Wang and M-H Chiang ldquoA novel pitch control systemof a large wind turbine using two-degree-of-freedom motioncontrol with feedback linearization controlrdquo Energies vol 9 no10 article no 791 2016
[14] Z Civelek E Cam M Luy and H Mamur ldquoProportional-integral-derivative parameter optimisation of blade pitch con-troller in wind turbines by a new intelligent genetic algorithmrdquoIET Renewable Power Generation vol 10 no 8 pp 1220ndash12282016
[15] A Dahbi N Nait-Said andM-S Nait-Said ldquoA novel combinedMPPT-pitch angle control for wide range variable speed windturbine based on neural networkrdquo International Journal ofHydrogen Energy vol 41 no 22 pp 9427ndash9442 2016
[16] M L Corradini G Ippoliti and G Orlando ldquoAn observer-based blade-pitch controller of wind turbines in high windspeedsrdquo Control Engineering Practice vol 58 pp 186ndash192 2017
[17] M Zhu J Liu Z Lin and H Meng ldquoMixed H2 Hinfin pitchcontrol of wind turbine generator system based on globalexact linearization and regional pole placementrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 5 pp921ndash930 2016
[18] R Gao and Z Gao ldquoPitch control for wind turbine systemsusing optimization estimation and compensationrdquo Journal ofRenewable Energy vol 91 pp 501ndash515 2016
[19] X-X Yin Y-G Lin W Li H-W Liu and Y-J Gu ldquoAdaptivesliding mode back-stepping pitch angle control of a variable-displacement pump controlled pitch system for wind turbinesrdquoISA Transactions vol 58 pp 629ndash634 2015
[20] M-H Chiang ldquoA novel pitch control system for a wind turbinedriven by a variable-speed pump-controlled hydraulic servosystemrdquoMechatronics vol 21 no 4 pp 753ndash761 2011
[21] H Camblong S Nourdine I Vechiu and G Tapia ldquoCom-parison of an island wind turbine collective and individualpitch LQG controllers designed to alleviate fatigue loadsrdquo IETRenewable Power Generation vol 6 no 4 p 267 2012
[22] F A InthamoussouHDe Battista andR JMantz ldquoLPV-basedactive power control of wind turbines covering the completewind speed rangerdquo Journal of Renewable Energy vol 99 pp996ndash1007 2016
[23] F D Bianchi R J Mantz and C F Christiansen ldquoGainscheduling control of variable-speed wind energy conversionsystems using quasi-LPVmodelsrdquoControl Engineering Practicevol 13 no 2 pp 247ndash255 2005
[24] F A Shirazi K M Grigoriadis and D Viassolo ldquoWind turbineintegrated structural and LPV control design for improvedclosed-loop performancerdquo International Journal of Control vol85 no 8 pp 1178ndash1196 2012
[25] F A Inthamoussou F D Bianchi H De Battista and R JMantz ldquoLPV wind turbine control with anti-windup featurescovering the complete wind speed rangerdquo IEEE Transactions onEnergy Conversion vol 29 no 1 pp 259ndash266 2014
[26] K Z Oslashstegaard J Stoustrup and P Brath ldquoLinear parametervarying control of wind turbines covering both partial loadand full load conditionsrdquo International Journal of Robust andNonlinear Control vol 19 no 1 pp 92ndash116 2009
[27] Y Lin L Tu H Liu and W Li ldquoHybrid power transmissiontechnology in a wind turbine generation systemrdquo IEEEASMETransactions onMechatronics vol 20 no 3 pp 1218ndash1225 2015
[28] S Bououden M Chadli F Allouani and S Filali ldquoA newapproach for fuzzy predictive adaptive controller design usingparticle swarmoptimization algorithmrdquo International Journal ofInnovative Computing Information and Control vol 9 no 9 pp3741ndash3758 2013
[29] S H Li and H Gu ldquoFuzzy adaptive internal model controlschemes for PMSM speed-regulation systemrdquo IEEE Transac-tions on Industrial Informatics vol 8 no 4 pp 767ndash779 2012
[30] A B Asghar and X Liu ldquoAdaptive neuro-fuzzy algorithmto estimate effective wind speed and optimal rotor speed forvariable-speed wind turbinerdquo Neurocomputing vol 272 pp495ndash504 2018
[31] Y Wang H Shen H R Karimi and D Duan ldquoDissipativity-based Fuzzy Integral SlidingModeControl ofContinuous-TimeT-S Fuzzy Systemsrdquo IEEE Transactions on Fuzzy Systems pp 1-1
[32] S Bououden M Chadli and H R Karimi ldquoAn ant colonyoptimization-based fuzzy predictive control approach for non-linear processesrdquo Information Sciences vol 299 pp 143ndash1582015
[33] NWang S-F Su J Yin Z Zheng andM J Er ldquoGlobal Asymp-totic Model-Free Trajectory-Independent Tracking Control ofan Uncertain Marine Vehicle An Adaptive Universe-BasedFuzzy Control Approachrdquo IEEE Transactions on Fuzzy Systems2017
[34] R Saravanakumar andD Jena ldquoValidation of an integral slidingmode control for optimal control of a three blade variable speedvariable pitch wind turbinerdquo International Journal of ElectricalPower amp Energy Systems vol 69 pp 421ndash429 2015
[35] C Evangelista P Puleston F Valenciaga and L M FridmanldquoLyapunov-designed super-twisting sliding mode control forwind energy conversion optimizationrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 538ndash545 2013
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Submit your manuscripts atwwwhindawicom
6 Mathematical Problems in Engineering
NB NM NS ZE PS PM PB
minusE EO
O
O
(A)
(B)
(C)X
X
minus(x)E (x)E
minus(x)E (x)E
Universe expansion
Universe com
pression
Figure 5 The diagram of changing universe process
the nonlinear system model can be linearized by designingappropriate control rules Therefore control performance ofsystem can be improved distinctly
The affine nonlinear models of single input single outputare expressed as [13]
= 119891 (119909) + 119892 (119909) 119906119910 = ℎ (119909) (20)
where119909 = [1199091 1199092 sdot sdot sdot 119909119899]119879 are the 119899-dimensional state vectors119891(119909) and 119892(119909) are the fully smooth vector fields in 119877119899 119906 isin 119877119910 isin 119877 and ℎ(119909) is the fully smooth nonlinear functionThe expression can be gained as follows based on (15)
(16) and (20)
119891 (119909) = [1198911 (119909)1198912 (119909)]
= [[[[
1119869119903 + 1198962119869119892 (121199091 120588120587119877
2V3119862119901 (120582 1199092) minus 119896119879119892)minus11205911199092
]]]]
(21)
119892 (119909) = [[01120591]]
(22)
ℎ (119909) = 1199091 (23)
Known from (23) there is no direct relationshipbetween the output equation and the control inputTherefore
derivative of time for output equation must be implementedas
119910 = 119889ℎ (119909)119889119905 = 120597ℎ (119909)120597119909 120597119909120597119905= 120597ℎ (119909)120597119909 [119891 (119909) + 119892 (119909) 119906]= 120597ℎ (119909)120597119909 119891 (119909) + 120597ℎ (119909)120597119909 119892 (119909) 119906
(24)
120597ℎ (119909)120597119909 = [1 0] (25)
Equations (26) and (27) can be gained based on (25)
120597ℎ (119909)120597119909 119891 (119909) = 119871119891ℎ (119909) = 1198911 (119909) (26)
120597ℎ (119909)120597119909 119892 (119909) = 119871119892ℎ (119909) = 0 (27)
Substituting (26) and (27) into (24) it is seen that 119910does not include 119906 Then derivative of time for (24) will becontinued
119910 = 120597119871119891ℎ (119909)120597119909 120597119909120597119905 =120597119871119891ℎ (119909)120597119909 [119891 (119909) + 119892 (119909) 119906]
= 1198712119891ℎ (119909) + 119871119892119871119891ℎ (119909) 119906(28)
In (28)
1198712119891ℎ (119909) = 120597119871119891ℎ (119909)120597119909 119891 (119909)= [1205971198911 (119909)1205971199091
1205971198911 (119909)1205971199092 ] [1198911 (119909)1198912 (119909)]
= 1205971198911 (119909)1205971199091 1198911 (119909) +1205971198911 (119909)1205971199092 1198912 (119909)
(29)
119871119892119871119891ℎ (119909) = 120597119871119891ℎ (119909)120597119909 119892 (119909)
= [1205971198911 (119909)12059711990911205971198911 (119909)1205971199092 ][[
01120591]]= 1205971198911 (119909)1205971199092
1120591(30)
Therefore
119871119892119871119891ℎ (119909) = 0 (31)
From now on input variable 119906 is expressed in 119910 and thecondition for meeting feedback linearization is gained
34 Input-Output Feedback Linearization Based Sliding ModeControl Both linear controller and nonlinear controller areusually adopted to control nonlinear system However linearcontroller can only control linear system well In nonlinearsystem if linear controller is used nonlinear model willbe usually linearized Sliding mode control is a nonlinear
Mathematical Problems in Engineering 7
s=0slt0
sgt0
A BC
Figure 6 The schematic diagram of sliding mode hypersurface
algorithm and it can continuously adjust itself based on real-time state variables during dynamic process for system totrack the state trajectory of sliding mode [34 35] In thispaper we propose sliding mode algorithm to enhance therobustness of feedback linearization
Assume that there exists a system as follows
= 119891 (119909) 119909 isin 119877119899 (32)
In its state space there is a hypersurface 119904(119909) =119904(1199091 1199092 sdot sdot sdot 119909119899) = 0 depicted in Figure 6The state space is divided into two parts 119904 gt 0 and 119904 lt 0 in
hypersurface In switching surface 119904 = 0 the point in systemwill be close to termination point when it is near switchingsurface
If all points are termination points in a special field inthe switching surface once the points move to this area towhich they will be attracted it is called sliding mode areaThemovement of system in slidingmode area is called slidingmode action
As the moving point is termination point in sliding modearea the expression can be gained as follows when the pointmoves to a place near the sliding mode surface
In other words
lim119904rarr0+
119904 le 0 le lim119904rarr0minus
119904 (33)
The other expression is formulated as
lim119904rarr0119904 119904 le 0 (34)
Assume that a function is given by
V (1199091 1199092 sdot sdot sdot 119909119899) = [119904 (1199091 1199092 sdot sdot sdot 119909119899)]2 (35)
The derivative of time for (35) can be expressed as follows
1198891199042119889119905 = 2119904 119904 (36)
According to (34) (36) is negative semidefinite formTherefore (35) is a Lyapunov function of system Then thesystem is stable when 119904 = 0
According to these analyses above system performancecan be improved by combining sliding mode control withfeedback linear algorithm because of stable advantage ofslidingmode control To gain good tracking performance theerror between reference and output is defined as
119890 = 119910119894119899 minus 119910 (37)
where 119910119894119899 is the ideal tracked object and 119910 is the controloutput
Then sliding function can be obtained as follows
119904 (119909 119905) = ce (38)
where
c = [119888 1] (39)
119888 gt 0 (40)
e = [119890 119890]119879 (41)
According to the discussed study for wind turbine in 33the control rule can be designed as
1199062 = 1(1205971198911 (119909) 1205971199092) (1120591) (V
minus (1205971198911 (119909)1205971199091 1198911 (119909) +1205971198911 (119909)1205971199092 1198912 (119909)) + 120578 sgn (119904))
(42)
where V is the aided term 120578 ge 0 and sgn(119904) is the signfunction which is formulated by
sgn (119904) = 1 119904 gt 0minus1 119904 lt 0 (43)
Simulation results show that there is large fluctuationin output power and rotor speed when using sign functionIn order to reduce the fluctuation saturation function isimplemented to replace sign function Saturation function isdefined as follows Results are shown in Figures 7(a)ndash7(d) andFigures 8(a)ndash8(d)
119904119886119905 (119904) =
1 119904 gt 120575119904120575 |119904| le 120575minus1 119904 lt minus120575
(44)
Up to now the final control law can be expressed basedon (17) and (42) as follows
119906 = 1199061 + 1199062 (45)
4 Simulation Results and Analyses
Simulation experiments based on a MW scale wind turbineare carried out to verify the effectiveness of the proposedalgorithm in which parameters are depicted in Table 4 Thecontroller parameters are shown in Table 5
The simulation experiments for wind turbine pitchingcontrol are implemented using step wind and random windStep speed is shown in Figure 7(a) and random wind isdepicted in Figure 8(a)
Firstly Figures 7(b)ndash7(d) and Figures 8(b)ndash8(d) depictthe control results when using sign function and saturationfunction The results show that there are little differences in
8 Mathematical Problems in Engineering
Table 4 The parameters of the wind turbine
Symbols Parameters Values119875119903119886119905119890119889 Rated output power 3000KW119881119903119886119905119890119889 Rated wind speed 12ms119877 The Radius of the turbine rotor 475m119869119903 The inertia moment of the turbine rotor 6250000kgsdotm2119869119892 The inertia moment of the generator 15kgsdotm2119896 Drive ratio 80120591 Time constant 021199031 Stator winding resistance 00148Ω1199032 Rotor winding resistance calculated to stator 00370Ω1199091120590 Leakage reactance 00727Ω1199092120590 Leakage calculated to stator side 00863Ω119909119898 Excitation reactance 20000Ω
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Step wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19992
200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(d) Output powers using different functions in sliding mode
Figure 7 (a) Step wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions in slidingmode (d) output powers using different functions in sliding mode
Mathematical Problems in Engineering 9
Table 5 The parameters of the proposed controller
Symbols Parameters Values119888 Constant defined in sliding mode algorithm 10119890 119903119886119899119892119890 The universe of rotor deviation [-0003 0003]119890119888 119903119886119899119892119890 The universe of rotor deviation changing rate [-0008 0008]119870119875 119903119886119899119892119890 The universe of proportionality coefficent [-03 03]119870119868 119903119886119899119892119890 The universe of integral coefficient [-006 006]119870119863 119903119886119899119892119890 The universe of differential coefficient [-3 3]119864 The value of the error universe 0003119864119862 The value of the error changing rate universe 0008120583 The coefficient of universe scaling factor 091205831 The coefficient of universe scaling factor 091205832 The coefficient of universe scaling factor 04120576 A full small positive number 000001
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Random wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
25
3
35
4
P(M
W)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
(d) Output powers using different functions in sliding mode
Figure 8 (a) Random wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions insliding mode (d) output powers using different functions in sliding mode
10 Mathematical Problems in Engineering
5
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
Local enlargeddrawing
4 5 6 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under step wind speed
9 10 11 12
2
2001Localenlargeddrawing
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under step wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
9 10 11 12
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) output power using different controllers under step wind speed
Figure 9 (a) Pitch angles using different controllers under step wind speed (b) rotor speed using different controllers under step wind speedand (c) output power using different controllers under step wind speed
the control when implementing the two different functionsunder the step wind speed However under random windspeed the control performance of the saturation functionadjusting output power is significantly better than that of thesign function in Figure 8(d)
Comparisons are further studied among which thereare conventional PID algorithm feedforward feedback con-trol combining PID with feedback linearization algorithmfeedforward feedback control based on fuzzy self-tuningPID with feedback linearization with sliding mode control(called fuzzy PID in feedback loop in Figures 9(a)ndash9(c)
and Figures 10(a)ndash10(c)) and the proposed controller inthis paper Results are depicted in Figures 9(a)ndash9(c) andFigures 10(a)ndash10(c) The wind speed is changed continuouslyto further validate pitching control algorithms which isdepicted in Figure 8(a)
At the initial moment control outputs fluctuate stronglywhich are caused by initial values of states However iffluctuations are in acceptable ranges of the physical structureof the wind turbine the initial responses will not affect theevaluation for the controller In Figure 7(a) at 119905 = 4 and119905 = 10 step wind speed occurs As thewind speeds exceed the
Mathematical Problems in Engineering 11
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
12 13 14 15
5 Local enlargeddrawing
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
8 9 10 11
1999
2
2001
Localenlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
13 14 15 16
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) Output power using different controllers under random wind speed
Figure 10 (a) Pitch angles using different controllers under random wind speed (b) rotor speed using different controllers under randomwind speed and (c) output power using different controllers under random wind speed
rated value pitch angles need to be adjusted to keep outputpower at constant
In Figure 9(a) the overshoot of the pitch angle is greaterwhile oscillation time is longer when adopting the conven-tional PID controller Although rotor speed also convergesat the rated value steady-state errors cannot be eliminatedwhich is approved in rotor speed chart depicted in Fig-ure 9(b) The rotor speeds of wind turbine are maintainedat constant when wind speed exceeds the rated value Dueto the steady-state errors of the pitch angle rotor speedshave deviations These results show that single linear PID
controller cannot track target well for wind turbine exhibitsstrongly nonlinear features
To address this problem we design a PID controllercombining with feedback linearization theory The PIDcontroller is feedback control while feedback linearizationis proposed as feedforward control Feedback control cankeep the system stable However the feedback control ishysteric Only when the deviation occurs the controller isimplemented to eliminate it Nevertheless feedforward canproduce a control variable in advanceThus the final outputsof the two controllers are proposed as the input of the wind
12 Mathematical Problems in Engineering
turbine system so that tracking accuracy of controller can beenhanced
Figures 9(a)ndash9(c) demonstrate that the control perfor-mance of feedforward feedback controller based on com-bining PID algorithm with feedback linearization algorithmis clearly superior to conventional PID controller in whichthe overshoot of pitch angle is smaller and adjusting timeof oscillating is shorter and furthermore the steady-statedeviation is eliminated which is also shown both in therotor speed diagram and in the output power diagram Tofurther improve control performance sliding mode controlis combined with feedback linear algorithm in feedforwardloop while the fuzzy self-tuning algorithm is used to adjustthe parameters of the PID controller in feedback loopFurthermore in order to decrease the effect when inputvariable and output variable change a lot variable universeapproach is proposed to optimize fuzzy rules Known fromthe results the deficiencies of controller produced by systemconstruct and parameters variation can be overcome FromFigures 9(a)ndash9(c) it is shown that the performance of thecontroller we propose in this paper is the best among thefour controllers whose overshoot is the least while it rapidlyconverges to constant without steady-state errors Althoughcontinuous step speed can evaluate controller performanceit still exhibits limitation
In order to further verify the effectiveness and universal-ity of the proposed controller in this paper we further adoptrandom wind speed Figure 8(a) depicts random wind speedFigures 10(a)ndash10(c) show the control results of the pitch anglerotor speed of turbine and output power produced by fourdifferent control algorithms under random wind speed Theresults are reasonable which are the same as those producedby step wind speed Therefore the control algorithm wepropose in this paper is the best among the described fouralgorithms and it can keep the output power at the rated valuewhen wind speed exceeds the nominal velocity
5 Conclusions
In this research a novel pitch control strategy for largevariable speed variable pitch wind turbines is studied aimingat maintaining the output power at constant so that theenergy quality is improved Known from the results as thesliding mode theory is combined with feedback linearizationalgorithm in feedforward loop the uncertainty of modeland time-varying of system parameters are overcome Fur-thermore with setting a control variable in advance theovershoot can be decreased when wind speed is varyingOn the other hand in feedback loop as variable universefuzzy algorithm is employed to optimize the parameters ofthe conventional PID controller the tracking performanceof the feedback controller is enhanced Meanwhile insteadof sign function saturation function is implemented todecrease oscillation of sliding mode algorithm and thus thefluctuations of output power and rotor speed of wind turbineare mitigated Both control accuracy and robustness usingthe proposed algorithm are the best among four differentcontrollers Another in future we plan to supply fuzzy
predictive adaptive algorithm in power control to furtherimprove system performance
Data Availability
The data used to support the findings of this study arecurrently under embargo while the research findings arecommercialized Requests for data 6 months after publica-tion of this article will be considered by the correspondingauthor
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National Natural ScienceFund of China under Grant no 51405052
References
[1] C-SWang andM-H Chiang ldquoA novel dynamic co-simulationanalysis for overall closed loop operation control of a large windturbinerdquo Energies vol 9 no 8 2016
[2] E Van Solingen J Beerens S P Mulders R De Breuker andJ W Van Wingerden ldquoControl design for a two-bladed down-wind teeterless damped free-yaw wind turbinerdquo Mechatronicsvol 36 pp 77ndash96 2016
[3] Y-J Chen and Y C Shiah ldquoExperiments on the performance ofsmall horizontal axis wind turbine with passive pitch control bydisk pulleyrdquo Energies vol 9 no 5 2016
[4] A Choudhry M Arjomandi and R Kelso ldquoMethods tocontrol dynamic stall for wind turbine applicationsrdquo Journal ofRenewable Energy vol 86 pp 26ndash37 2016
[5] H Li C Yang Y Hu X Liao Z Zeng and C Zhe ldquoAnimproved reduced-ordermodel of an electric pitch drive systemfor wind turbine control system design and simulationrdquo Journalof Renewable Energy vol 93 pp 188ndash200 2016
[6] E B Muhando T Senjyu A Uehara and T Funabashi ldquoGain-scheduledHinfin control forWECS via LMI techniques and para-metrically dependent feedback part II Controller design andimplementationrdquo IEEE Transactions on Industrial Electronicsvol 58 no 1 pp 57ndash65 2011
[7] F D Bianchi H De Battista and R JMantz ldquoRobustmultivari-able gain scheduled control of wind turbines for variable powerproductionrdquo International Journal of Systems Control vol 1 no3 pp 103ndash112 2010
[8] I Soslashnderby andMH Hansen ldquoOpen-loop frequency responseanalysis of a wind turbine using a high-order linear aeroelasticmodelrdquoWind Energy vol 17 no 8 pp 1147ndash1167 2014
[9] S Bououden M Chadli S Filali and A El Hajjaji ldquoFuzzymodel basedmultivariable predictive control of a variable speedwind turbine LMI approachrdquo Journal of Renewable Energy vol37 no 1 pp 434ndash439 2012
[10] J Chen T Lin CWen and Y Song ldquoDesign of a Unified PowerController for Variable-Speed Fixed-Pitch Wind Energy Con-version Systemrdquo IEEETransactions on Industrial Electronics vol63 no 8 pp 4899ndash4908 2016
Mathematical Problems in Engineering 13
[11] Y Ren L Li J Brindley and L Jiang ldquoNonlinear PI control forvariable pitch wind turbinerdquo Control Engineering Practice 2015
[12] Y Yuan and J Tang ldquoAdaptive pitch control of wind turbinefor load mitigation under structural uncertaintiesrdquo Journal ofRenewable Energy vol 105 pp 483ndash494 2017
[13] C-S Wang and M-H Chiang ldquoA novel pitch control systemof a large wind turbine using two-degree-of-freedom motioncontrol with feedback linearization controlrdquo Energies vol 9 no10 article no 791 2016
[14] Z Civelek E Cam M Luy and H Mamur ldquoProportional-integral-derivative parameter optimisation of blade pitch con-troller in wind turbines by a new intelligent genetic algorithmrdquoIET Renewable Power Generation vol 10 no 8 pp 1220ndash12282016
[15] A Dahbi N Nait-Said andM-S Nait-Said ldquoA novel combinedMPPT-pitch angle control for wide range variable speed windturbine based on neural networkrdquo International Journal ofHydrogen Energy vol 41 no 22 pp 9427ndash9442 2016
[16] M L Corradini G Ippoliti and G Orlando ldquoAn observer-based blade-pitch controller of wind turbines in high windspeedsrdquo Control Engineering Practice vol 58 pp 186ndash192 2017
[17] M Zhu J Liu Z Lin and H Meng ldquoMixed H2 Hinfin pitchcontrol of wind turbine generator system based on globalexact linearization and regional pole placementrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 5 pp921ndash930 2016
[18] R Gao and Z Gao ldquoPitch control for wind turbine systemsusing optimization estimation and compensationrdquo Journal ofRenewable Energy vol 91 pp 501ndash515 2016
[19] X-X Yin Y-G Lin W Li H-W Liu and Y-J Gu ldquoAdaptivesliding mode back-stepping pitch angle control of a variable-displacement pump controlled pitch system for wind turbinesrdquoISA Transactions vol 58 pp 629ndash634 2015
[20] M-H Chiang ldquoA novel pitch control system for a wind turbinedriven by a variable-speed pump-controlled hydraulic servosystemrdquoMechatronics vol 21 no 4 pp 753ndash761 2011
[21] H Camblong S Nourdine I Vechiu and G Tapia ldquoCom-parison of an island wind turbine collective and individualpitch LQG controllers designed to alleviate fatigue loadsrdquo IETRenewable Power Generation vol 6 no 4 p 267 2012
[22] F A InthamoussouHDe Battista andR JMantz ldquoLPV-basedactive power control of wind turbines covering the completewind speed rangerdquo Journal of Renewable Energy vol 99 pp996ndash1007 2016
[23] F D Bianchi R J Mantz and C F Christiansen ldquoGainscheduling control of variable-speed wind energy conversionsystems using quasi-LPVmodelsrdquoControl Engineering Practicevol 13 no 2 pp 247ndash255 2005
[24] F A Shirazi K M Grigoriadis and D Viassolo ldquoWind turbineintegrated structural and LPV control design for improvedclosed-loop performancerdquo International Journal of Control vol85 no 8 pp 1178ndash1196 2012
[25] F A Inthamoussou F D Bianchi H De Battista and R JMantz ldquoLPV wind turbine control with anti-windup featurescovering the complete wind speed rangerdquo IEEE Transactions onEnergy Conversion vol 29 no 1 pp 259ndash266 2014
[26] K Z Oslashstegaard J Stoustrup and P Brath ldquoLinear parametervarying control of wind turbines covering both partial loadand full load conditionsrdquo International Journal of Robust andNonlinear Control vol 19 no 1 pp 92ndash116 2009
[27] Y Lin L Tu H Liu and W Li ldquoHybrid power transmissiontechnology in a wind turbine generation systemrdquo IEEEASMETransactions onMechatronics vol 20 no 3 pp 1218ndash1225 2015
[28] S Bououden M Chadli F Allouani and S Filali ldquoA newapproach for fuzzy predictive adaptive controller design usingparticle swarmoptimization algorithmrdquo International Journal ofInnovative Computing Information and Control vol 9 no 9 pp3741ndash3758 2013
[29] S H Li and H Gu ldquoFuzzy adaptive internal model controlschemes for PMSM speed-regulation systemrdquo IEEE Transac-tions on Industrial Informatics vol 8 no 4 pp 767ndash779 2012
[30] A B Asghar and X Liu ldquoAdaptive neuro-fuzzy algorithmto estimate effective wind speed and optimal rotor speed forvariable-speed wind turbinerdquo Neurocomputing vol 272 pp495ndash504 2018
[31] Y Wang H Shen H R Karimi and D Duan ldquoDissipativity-based Fuzzy Integral SlidingModeControl ofContinuous-TimeT-S Fuzzy Systemsrdquo IEEE Transactions on Fuzzy Systems pp 1-1
[32] S Bououden M Chadli and H R Karimi ldquoAn ant colonyoptimization-based fuzzy predictive control approach for non-linear processesrdquo Information Sciences vol 299 pp 143ndash1582015
[33] NWang S-F Su J Yin Z Zheng andM J Er ldquoGlobal Asymp-totic Model-Free Trajectory-Independent Tracking Control ofan Uncertain Marine Vehicle An Adaptive Universe-BasedFuzzy Control Approachrdquo IEEE Transactions on Fuzzy Systems2017
[34] R Saravanakumar andD Jena ldquoValidation of an integral slidingmode control for optimal control of a three blade variable speedvariable pitch wind turbinerdquo International Journal of ElectricalPower amp Energy Systems vol 69 pp 421ndash429 2015
[35] C Evangelista P Puleston F Valenciaga and L M FridmanldquoLyapunov-designed super-twisting sliding mode control forwind energy conversion optimizationrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 538ndash545 2013
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Mathematical Problems in Engineering 7
s=0slt0
sgt0
A BC
Figure 6 The schematic diagram of sliding mode hypersurface
algorithm and it can continuously adjust itself based on real-time state variables during dynamic process for system totrack the state trajectory of sliding mode [34 35] In thispaper we propose sliding mode algorithm to enhance therobustness of feedback linearization
Assume that there exists a system as follows
= 119891 (119909) 119909 isin 119877119899 (32)
In its state space there is a hypersurface 119904(119909) =119904(1199091 1199092 sdot sdot sdot 119909119899) = 0 depicted in Figure 6The state space is divided into two parts 119904 gt 0 and 119904 lt 0 in
hypersurface In switching surface 119904 = 0 the point in systemwill be close to termination point when it is near switchingsurface
If all points are termination points in a special field inthe switching surface once the points move to this area towhich they will be attracted it is called sliding mode areaThemovement of system in slidingmode area is called slidingmode action
As the moving point is termination point in sliding modearea the expression can be gained as follows when the pointmoves to a place near the sliding mode surface
In other words
lim119904rarr0+
119904 le 0 le lim119904rarr0minus
119904 (33)
The other expression is formulated as
lim119904rarr0119904 119904 le 0 (34)
Assume that a function is given by
V (1199091 1199092 sdot sdot sdot 119909119899) = [119904 (1199091 1199092 sdot sdot sdot 119909119899)]2 (35)
The derivative of time for (35) can be expressed as follows
1198891199042119889119905 = 2119904 119904 (36)
According to (34) (36) is negative semidefinite formTherefore (35) is a Lyapunov function of system Then thesystem is stable when 119904 = 0
According to these analyses above system performancecan be improved by combining sliding mode control withfeedback linear algorithm because of stable advantage ofslidingmode control To gain good tracking performance theerror between reference and output is defined as
119890 = 119910119894119899 minus 119910 (37)
where 119910119894119899 is the ideal tracked object and 119910 is the controloutput
Then sliding function can be obtained as follows
119904 (119909 119905) = ce (38)
where
c = [119888 1] (39)
119888 gt 0 (40)
e = [119890 119890]119879 (41)
According to the discussed study for wind turbine in 33the control rule can be designed as
1199062 = 1(1205971198911 (119909) 1205971199092) (1120591) (V
minus (1205971198911 (119909)1205971199091 1198911 (119909) +1205971198911 (119909)1205971199092 1198912 (119909)) + 120578 sgn (119904))
(42)
where V is the aided term 120578 ge 0 and sgn(119904) is the signfunction which is formulated by
sgn (119904) = 1 119904 gt 0minus1 119904 lt 0 (43)
Simulation results show that there is large fluctuationin output power and rotor speed when using sign functionIn order to reduce the fluctuation saturation function isimplemented to replace sign function Saturation function isdefined as follows Results are shown in Figures 7(a)ndash7(d) andFigures 8(a)ndash8(d)
119904119886119905 (119904) =
1 119904 gt 120575119904120575 |119904| le 120575minus1 119904 lt minus120575
(44)
Up to now the final control law can be expressed basedon (17) and (42) as follows
119906 = 1199061 + 1199062 (45)
4 Simulation Results and Analyses
Simulation experiments based on a MW scale wind turbineare carried out to verify the effectiveness of the proposedalgorithm in which parameters are depicted in Table 4 Thecontroller parameters are shown in Table 5
The simulation experiments for wind turbine pitchingcontrol are implemented using step wind and random windStep speed is shown in Figure 7(a) and random wind isdepicted in Figure 8(a)
Firstly Figures 7(b)ndash7(d) and Figures 8(b)ndash8(d) depictthe control results when using sign function and saturationfunction The results show that there are little differences in
8 Mathematical Problems in Engineering
Table 4 The parameters of the wind turbine
Symbols Parameters Values119875119903119886119905119890119889 Rated output power 3000KW119881119903119886119905119890119889 Rated wind speed 12ms119877 The Radius of the turbine rotor 475m119869119903 The inertia moment of the turbine rotor 6250000kgsdotm2119869119892 The inertia moment of the generator 15kgsdotm2119896 Drive ratio 80120591 Time constant 021199031 Stator winding resistance 00148Ω1199032 Rotor winding resistance calculated to stator 00370Ω1199091120590 Leakage reactance 00727Ω1199092120590 Leakage calculated to stator side 00863Ω119909119898 Excitation reactance 20000Ω
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Step wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19992
200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(d) Output powers using different functions in sliding mode
Figure 7 (a) Step wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions in slidingmode (d) output powers using different functions in sliding mode
Mathematical Problems in Engineering 9
Table 5 The parameters of the proposed controller
Symbols Parameters Values119888 Constant defined in sliding mode algorithm 10119890 119903119886119899119892119890 The universe of rotor deviation [-0003 0003]119890119888 119903119886119899119892119890 The universe of rotor deviation changing rate [-0008 0008]119870119875 119903119886119899119892119890 The universe of proportionality coefficent [-03 03]119870119868 119903119886119899119892119890 The universe of integral coefficient [-006 006]119870119863 119903119886119899119892119890 The universe of differential coefficient [-3 3]119864 The value of the error universe 0003119864119862 The value of the error changing rate universe 0008120583 The coefficient of universe scaling factor 091205831 The coefficient of universe scaling factor 091205832 The coefficient of universe scaling factor 04120576 A full small positive number 000001
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Random wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
25
3
35
4
P(M
W)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
(d) Output powers using different functions in sliding mode
Figure 8 (a) Random wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions insliding mode (d) output powers using different functions in sliding mode
10 Mathematical Problems in Engineering
5
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
Local enlargeddrawing
4 5 6 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under step wind speed
9 10 11 12
2
2001Localenlargeddrawing
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under step wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
9 10 11 12
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) output power using different controllers under step wind speed
Figure 9 (a) Pitch angles using different controllers under step wind speed (b) rotor speed using different controllers under step wind speedand (c) output power using different controllers under step wind speed
the control when implementing the two different functionsunder the step wind speed However under random windspeed the control performance of the saturation functionadjusting output power is significantly better than that of thesign function in Figure 8(d)
Comparisons are further studied among which thereare conventional PID algorithm feedforward feedback con-trol combining PID with feedback linearization algorithmfeedforward feedback control based on fuzzy self-tuningPID with feedback linearization with sliding mode control(called fuzzy PID in feedback loop in Figures 9(a)ndash9(c)
and Figures 10(a)ndash10(c)) and the proposed controller inthis paper Results are depicted in Figures 9(a)ndash9(c) andFigures 10(a)ndash10(c) The wind speed is changed continuouslyto further validate pitching control algorithms which isdepicted in Figure 8(a)
At the initial moment control outputs fluctuate stronglywhich are caused by initial values of states However iffluctuations are in acceptable ranges of the physical structureof the wind turbine the initial responses will not affect theevaluation for the controller In Figure 7(a) at 119905 = 4 and119905 = 10 step wind speed occurs As thewind speeds exceed the
Mathematical Problems in Engineering 11
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
12 13 14 15
5 Local enlargeddrawing
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
8 9 10 11
1999
2
2001
Localenlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
13 14 15 16
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) Output power using different controllers under random wind speed
Figure 10 (a) Pitch angles using different controllers under random wind speed (b) rotor speed using different controllers under randomwind speed and (c) output power using different controllers under random wind speed
rated value pitch angles need to be adjusted to keep outputpower at constant
In Figure 9(a) the overshoot of the pitch angle is greaterwhile oscillation time is longer when adopting the conven-tional PID controller Although rotor speed also convergesat the rated value steady-state errors cannot be eliminatedwhich is approved in rotor speed chart depicted in Fig-ure 9(b) The rotor speeds of wind turbine are maintainedat constant when wind speed exceeds the rated value Dueto the steady-state errors of the pitch angle rotor speedshave deviations These results show that single linear PID
controller cannot track target well for wind turbine exhibitsstrongly nonlinear features
To address this problem we design a PID controllercombining with feedback linearization theory The PIDcontroller is feedback control while feedback linearizationis proposed as feedforward control Feedback control cankeep the system stable However the feedback control ishysteric Only when the deviation occurs the controller isimplemented to eliminate it Nevertheless feedforward canproduce a control variable in advanceThus the final outputsof the two controllers are proposed as the input of the wind
12 Mathematical Problems in Engineering
turbine system so that tracking accuracy of controller can beenhanced
Figures 9(a)ndash9(c) demonstrate that the control perfor-mance of feedforward feedback controller based on com-bining PID algorithm with feedback linearization algorithmis clearly superior to conventional PID controller in whichthe overshoot of pitch angle is smaller and adjusting timeof oscillating is shorter and furthermore the steady-statedeviation is eliminated which is also shown both in therotor speed diagram and in the output power diagram Tofurther improve control performance sliding mode controlis combined with feedback linear algorithm in feedforwardloop while the fuzzy self-tuning algorithm is used to adjustthe parameters of the PID controller in feedback loopFurthermore in order to decrease the effect when inputvariable and output variable change a lot variable universeapproach is proposed to optimize fuzzy rules Known fromthe results the deficiencies of controller produced by systemconstruct and parameters variation can be overcome FromFigures 9(a)ndash9(c) it is shown that the performance of thecontroller we propose in this paper is the best among thefour controllers whose overshoot is the least while it rapidlyconverges to constant without steady-state errors Althoughcontinuous step speed can evaluate controller performanceit still exhibits limitation
In order to further verify the effectiveness and universal-ity of the proposed controller in this paper we further adoptrandom wind speed Figure 8(a) depicts random wind speedFigures 10(a)ndash10(c) show the control results of the pitch anglerotor speed of turbine and output power produced by fourdifferent control algorithms under random wind speed Theresults are reasonable which are the same as those producedby step wind speed Therefore the control algorithm wepropose in this paper is the best among the described fouralgorithms and it can keep the output power at the rated valuewhen wind speed exceeds the nominal velocity
5 Conclusions
In this research a novel pitch control strategy for largevariable speed variable pitch wind turbines is studied aimingat maintaining the output power at constant so that theenergy quality is improved Known from the results as thesliding mode theory is combined with feedback linearizationalgorithm in feedforward loop the uncertainty of modeland time-varying of system parameters are overcome Fur-thermore with setting a control variable in advance theovershoot can be decreased when wind speed is varyingOn the other hand in feedback loop as variable universefuzzy algorithm is employed to optimize the parameters ofthe conventional PID controller the tracking performanceof the feedback controller is enhanced Meanwhile insteadof sign function saturation function is implemented todecrease oscillation of sliding mode algorithm and thus thefluctuations of output power and rotor speed of wind turbineare mitigated Both control accuracy and robustness usingthe proposed algorithm are the best among four differentcontrollers Another in future we plan to supply fuzzy
predictive adaptive algorithm in power control to furtherimprove system performance
Data Availability
The data used to support the findings of this study arecurrently under embargo while the research findings arecommercialized Requests for data 6 months after publica-tion of this article will be considered by the correspondingauthor
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National Natural ScienceFund of China under Grant no 51405052
References
[1] C-SWang andM-H Chiang ldquoA novel dynamic co-simulationanalysis for overall closed loop operation control of a large windturbinerdquo Energies vol 9 no 8 2016
[2] E Van Solingen J Beerens S P Mulders R De Breuker andJ W Van Wingerden ldquoControl design for a two-bladed down-wind teeterless damped free-yaw wind turbinerdquo Mechatronicsvol 36 pp 77ndash96 2016
[3] Y-J Chen and Y C Shiah ldquoExperiments on the performance ofsmall horizontal axis wind turbine with passive pitch control bydisk pulleyrdquo Energies vol 9 no 5 2016
[4] A Choudhry M Arjomandi and R Kelso ldquoMethods tocontrol dynamic stall for wind turbine applicationsrdquo Journal ofRenewable Energy vol 86 pp 26ndash37 2016
[5] H Li C Yang Y Hu X Liao Z Zeng and C Zhe ldquoAnimproved reduced-ordermodel of an electric pitch drive systemfor wind turbine control system design and simulationrdquo Journalof Renewable Energy vol 93 pp 188ndash200 2016
[6] E B Muhando T Senjyu A Uehara and T Funabashi ldquoGain-scheduledHinfin control forWECS via LMI techniques and para-metrically dependent feedback part II Controller design andimplementationrdquo IEEE Transactions on Industrial Electronicsvol 58 no 1 pp 57ndash65 2011
[7] F D Bianchi H De Battista and R JMantz ldquoRobustmultivari-able gain scheduled control of wind turbines for variable powerproductionrdquo International Journal of Systems Control vol 1 no3 pp 103ndash112 2010
[8] I Soslashnderby andMH Hansen ldquoOpen-loop frequency responseanalysis of a wind turbine using a high-order linear aeroelasticmodelrdquoWind Energy vol 17 no 8 pp 1147ndash1167 2014
[9] S Bououden M Chadli S Filali and A El Hajjaji ldquoFuzzymodel basedmultivariable predictive control of a variable speedwind turbine LMI approachrdquo Journal of Renewable Energy vol37 no 1 pp 434ndash439 2012
[10] J Chen T Lin CWen and Y Song ldquoDesign of a Unified PowerController for Variable-Speed Fixed-Pitch Wind Energy Con-version Systemrdquo IEEETransactions on Industrial Electronics vol63 no 8 pp 4899ndash4908 2016
Mathematical Problems in Engineering 13
[11] Y Ren L Li J Brindley and L Jiang ldquoNonlinear PI control forvariable pitch wind turbinerdquo Control Engineering Practice 2015
[12] Y Yuan and J Tang ldquoAdaptive pitch control of wind turbinefor load mitigation under structural uncertaintiesrdquo Journal ofRenewable Energy vol 105 pp 483ndash494 2017
[13] C-S Wang and M-H Chiang ldquoA novel pitch control systemof a large wind turbine using two-degree-of-freedom motioncontrol with feedback linearization controlrdquo Energies vol 9 no10 article no 791 2016
[14] Z Civelek E Cam M Luy and H Mamur ldquoProportional-integral-derivative parameter optimisation of blade pitch con-troller in wind turbines by a new intelligent genetic algorithmrdquoIET Renewable Power Generation vol 10 no 8 pp 1220ndash12282016
[15] A Dahbi N Nait-Said andM-S Nait-Said ldquoA novel combinedMPPT-pitch angle control for wide range variable speed windturbine based on neural networkrdquo International Journal ofHydrogen Energy vol 41 no 22 pp 9427ndash9442 2016
[16] M L Corradini G Ippoliti and G Orlando ldquoAn observer-based blade-pitch controller of wind turbines in high windspeedsrdquo Control Engineering Practice vol 58 pp 186ndash192 2017
[17] M Zhu J Liu Z Lin and H Meng ldquoMixed H2 Hinfin pitchcontrol of wind turbine generator system based on globalexact linearization and regional pole placementrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 5 pp921ndash930 2016
[18] R Gao and Z Gao ldquoPitch control for wind turbine systemsusing optimization estimation and compensationrdquo Journal ofRenewable Energy vol 91 pp 501ndash515 2016
[19] X-X Yin Y-G Lin W Li H-W Liu and Y-J Gu ldquoAdaptivesliding mode back-stepping pitch angle control of a variable-displacement pump controlled pitch system for wind turbinesrdquoISA Transactions vol 58 pp 629ndash634 2015
[20] M-H Chiang ldquoA novel pitch control system for a wind turbinedriven by a variable-speed pump-controlled hydraulic servosystemrdquoMechatronics vol 21 no 4 pp 753ndash761 2011
[21] H Camblong S Nourdine I Vechiu and G Tapia ldquoCom-parison of an island wind turbine collective and individualpitch LQG controllers designed to alleviate fatigue loadsrdquo IETRenewable Power Generation vol 6 no 4 p 267 2012
[22] F A InthamoussouHDe Battista andR JMantz ldquoLPV-basedactive power control of wind turbines covering the completewind speed rangerdquo Journal of Renewable Energy vol 99 pp996ndash1007 2016
[23] F D Bianchi R J Mantz and C F Christiansen ldquoGainscheduling control of variable-speed wind energy conversionsystems using quasi-LPVmodelsrdquoControl Engineering Practicevol 13 no 2 pp 247ndash255 2005
[24] F A Shirazi K M Grigoriadis and D Viassolo ldquoWind turbineintegrated structural and LPV control design for improvedclosed-loop performancerdquo International Journal of Control vol85 no 8 pp 1178ndash1196 2012
[25] F A Inthamoussou F D Bianchi H De Battista and R JMantz ldquoLPV wind turbine control with anti-windup featurescovering the complete wind speed rangerdquo IEEE Transactions onEnergy Conversion vol 29 no 1 pp 259ndash266 2014
[26] K Z Oslashstegaard J Stoustrup and P Brath ldquoLinear parametervarying control of wind turbines covering both partial loadand full load conditionsrdquo International Journal of Robust andNonlinear Control vol 19 no 1 pp 92ndash116 2009
[27] Y Lin L Tu H Liu and W Li ldquoHybrid power transmissiontechnology in a wind turbine generation systemrdquo IEEEASMETransactions onMechatronics vol 20 no 3 pp 1218ndash1225 2015
[28] S Bououden M Chadli F Allouani and S Filali ldquoA newapproach for fuzzy predictive adaptive controller design usingparticle swarmoptimization algorithmrdquo International Journal ofInnovative Computing Information and Control vol 9 no 9 pp3741ndash3758 2013
[29] S H Li and H Gu ldquoFuzzy adaptive internal model controlschemes for PMSM speed-regulation systemrdquo IEEE Transac-tions on Industrial Informatics vol 8 no 4 pp 767ndash779 2012
[30] A B Asghar and X Liu ldquoAdaptive neuro-fuzzy algorithmto estimate effective wind speed and optimal rotor speed forvariable-speed wind turbinerdquo Neurocomputing vol 272 pp495ndash504 2018
[31] Y Wang H Shen H R Karimi and D Duan ldquoDissipativity-based Fuzzy Integral SlidingModeControl ofContinuous-TimeT-S Fuzzy Systemsrdquo IEEE Transactions on Fuzzy Systems pp 1-1
[32] S Bououden M Chadli and H R Karimi ldquoAn ant colonyoptimization-based fuzzy predictive control approach for non-linear processesrdquo Information Sciences vol 299 pp 143ndash1582015
[33] NWang S-F Su J Yin Z Zheng andM J Er ldquoGlobal Asymp-totic Model-Free Trajectory-Independent Tracking Control ofan Uncertain Marine Vehicle An Adaptive Universe-BasedFuzzy Control Approachrdquo IEEE Transactions on Fuzzy Systems2017
[34] R Saravanakumar andD Jena ldquoValidation of an integral slidingmode control for optimal control of a three blade variable speedvariable pitch wind turbinerdquo International Journal of ElectricalPower amp Energy Systems vol 69 pp 421ndash429 2015
[35] C Evangelista P Puleston F Valenciaga and L M FridmanldquoLyapunov-designed super-twisting sliding mode control forwind energy conversion optimizationrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 538ndash545 2013
Hindawiwwwhindawicom Volume 2018
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Mathematical Problems in Engineering
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Submit your manuscripts atwwwhindawicom
8 Mathematical Problems in Engineering
Table 4 The parameters of the wind turbine
Symbols Parameters Values119875119903119886119905119890119889 Rated output power 3000KW119881119903119886119905119890119889 Rated wind speed 12ms119877 The Radius of the turbine rotor 475m119869119903 The inertia moment of the turbine rotor 6250000kgsdotm2119869119892 The inertia moment of the generator 15kgsdotm2119896 Drive ratio 80120591 Time constant 021199031 Stator winding resistance 00148Ω1199032 Rotor winding resistance calculated to stator 00370Ω1199091120590 Leakage reactance 00727Ω1199092120590 Leakage calculated to stator side 00863Ω119909119898 Excitation reactance 20000Ω
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Step wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19992
200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(d) Output powers using different functions in sliding mode
Figure 7 (a) Step wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions in slidingmode (d) output powers using different functions in sliding mode
Mathematical Problems in Engineering 9
Table 5 The parameters of the proposed controller
Symbols Parameters Values119888 Constant defined in sliding mode algorithm 10119890 119903119886119899119892119890 The universe of rotor deviation [-0003 0003]119890119888 119903119886119899119892119890 The universe of rotor deviation changing rate [-0008 0008]119870119875 119903119886119899119892119890 The universe of proportionality coefficent [-03 03]119870119868 119903119886119899119892119890 The universe of integral coefficient [-006 006]119870119863 119903119886119899119892119890 The universe of differential coefficient [-3 3]119864 The value of the error universe 0003119864119862 The value of the error changing rate universe 0008120583 The coefficient of universe scaling factor 091205831 The coefficient of universe scaling factor 091205832 The coefficient of universe scaling factor 04120576 A full small positive number 000001
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Random wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
25
3
35
4
P(M
W)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
(d) Output powers using different functions in sliding mode
Figure 8 (a) Random wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions insliding mode (d) output powers using different functions in sliding mode
10 Mathematical Problems in Engineering
5
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
Local enlargeddrawing
4 5 6 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under step wind speed
9 10 11 12
2
2001Localenlargeddrawing
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under step wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
9 10 11 12
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) output power using different controllers under step wind speed
Figure 9 (a) Pitch angles using different controllers under step wind speed (b) rotor speed using different controllers under step wind speedand (c) output power using different controllers under step wind speed
the control when implementing the two different functionsunder the step wind speed However under random windspeed the control performance of the saturation functionadjusting output power is significantly better than that of thesign function in Figure 8(d)
Comparisons are further studied among which thereare conventional PID algorithm feedforward feedback con-trol combining PID with feedback linearization algorithmfeedforward feedback control based on fuzzy self-tuningPID with feedback linearization with sliding mode control(called fuzzy PID in feedback loop in Figures 9(a)ndash9(c)
and Figures 10(a)ndash10(c)) and the proposed controller inthis paper Results are depicted in Figures 9(a)ndash9(c) andFigures 10(a)ndash10(c) The wind speed is changed continuouslyto further validate pitching control algorithms which isdepicted in Figure 8(a)
At the initial moment control outputs fluctuate stronglywhich are caused by initial values of states However iffluctuations are in acceptable ranges of the physical structureof the wind turbine the initial responses will not affect theevaluation for the controller In Figure 7(a) at 119905 = 4 and119905 = 10 step wind speed occurs As thewind speeds exceed the
Mathematical Problems in Engineering 11
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
12 13 14 15
5 Local enlargeddrawing
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
8 9 10 11
1999
2
2001
Localenlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
13 14 15 16
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) Output power using different controllers under random wind speed
Figure 10 (a) Pitch angles using different controllers under random wind speed (b) rotor speed using different controllers under randomwind speed and (c) output power using different controllers under random wind speed
rated value pitch angles need to be adjusted to keep outputpower at constant
In Figure 9(a) the overshoot of the pitch angle is greaterwhile oscillation time is longer when adopting the conven-tional PID controller Although rotor speed also convergesat the rated value steady-state errors cannot be eliminatedwhich is approved in rotor speed chart depicted in Fig-ure 9(b) The rotor speeds of wind turbine are maintainedat constant when wind speed exceeds the rated value Dueto the steady-state errors of the pitch angle rotor speedshave deviations These results show that single linear PID
controller cannot track target well for wind turbine exhibitsstrongly nonlinear features
To address this problem we design a PID controllercombining with feedback linearization theory The PIDcontroller is feedback control while feedback linearizationis proposed as feedforward control Feedback control cankeep the system stable However the feedback control ishysteric Only when the deviation occurs the controller isimplemented to eliminate it Nevertheless feedforward canproduce a control variable in advanceThus the final outputsof the two controllers are proposed as the input of the wind
12 Mathematical Problems in Engineering
turbine system so that tracking accuracy of controller can beenhanced
Figures 9(a)ndash9(c) demonstrate that the control perfor-mance of feedforward feedback controller based on com-bining PID algorithm with feedback linearization algorithmis clearly superior to conventional PID controller in whichthe overshoot of pitch angle is smaller and adjusting timeof oscillating is shorter and furthermore the steady-statedeviation is eliminated which is also shown both in therotor speed diagram and in the output power diagram Tofurther improve control performance sliding mode controlis combined with feedback linear algorithm in feedforwardloop while the fuzzy self-tuning algorithm is used to adjustthe parameters of the PID controller in feedback loopFurthermore in order to decrease the effect when inputvariable and output variable change a lot variable universeapproach is proposed to optimize fuzzy rules Known fromthe results the deficiencies of controller produced by systemconstruct and parameters variation can be overcome FromFigures 9(a)ndash9(c) it is shown that the performance of thecontroller we propose in this paper is the best among thefour controllers whose overshoot is the least while it rapidlyconverges to constant without steady-state errors Althoughcontinuous step speed can evaluate controller performanceit still exhibits limitation
In order to further verify the effectiveness and universal-ity of the proposed controller in this paper we further adoptrandom wind speed Figure 8(a) depicts random wind speedFigures 10(a)ndash10(c) show the control results of the pitch anglerotor speed of turbine and output power produced by fourdifferent control algorithms under random wind speed Theresults are reasonable which are the same as those producedby step wind speed Therefore the control algorithm wepropose in this paper is the best among the described fouralgorithms and it can keep the output power at the rated valuewhen wind speed exceeds the nominal velocity
5 Conclusions
In this research a novel pitch control strategy for largevariable speed variable pitch wind turbines is studied aimingat maintaining the output power at constant so that theenergy quality is improved Known from the results as thesliding mode theory is combined with feedback linearizationalgorithm in feedforward loop the uncertainty of modeland time-varying of system parameters are overcome Fur-thermore with setting a control variable in advance theovershoot can be decreased when wind speed is varyingOn the other hand in feedback loop as variable universefuzzy algorithm is employed to optimize the parameters ofthe conventional PID controller the tracking performanceof the feedback controller is enhanced Meanwhile insteadof sign function saturation function is implemented todecrease oscillation of sliding mode algorithm and thus thefluctuations of output power and rotor speed of wind turbineare mitigated Both control accuracy and robustness usingthe proposed algorithm are the best among four differentcontrollers Another in future we plan to supply fuzzy
predictive adaptive algorithm in power control to furtherimprove system performance
Data Availability
The data used to support the findings of this study arecurrently under embargo while the research findings arecommercialized Requests for data 6 months after publica-tion of this article will be considered by the correspondingauthor
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National Natural ScienceFund of China under Grant no 51405052
References
[1] C-SWang andM-H Chiang ldquoA novel dynamic co-simulationanalysis for overall closed loop operation control of a large windturbinerdquo Energies vol 9 no 8 2016
[2] E Van Solingen J Beerens S P Mulders R De Breuker andJ W Van Wingerden ldquoControl design for a two-bladed down-wind teeterless damped free-yaw wind turbinerdquo Mechatronicsvol 36 pp 77ndash96 2016
[3] Y-J Chen and Y C Shiah ldquoExperiments on the performance ofsmall horizontal axis wind turbine with passive pitch control bydisk pulleyrdquo Energies vol 9 no 5 2016
[4] A Choudhry M Arjomandi and R Kelso ldquoMethods tocontrol dynamic stall for wind turbine applicationsrdquo Journal ofRenewable Energy vol 86 pp 26ndash37 2016
[5] H Li C Yang Y Hu X Liao Z Zeng and C Zhe ldquoAnimproved reduced-ordermodel of an electric pitch drive systemfor wind turbine control system design and simulationrdquo Journalof Renewable Energy vol 93 pp 188ndash200 2016
[6] E B Muhando T Senjyu A Uehara and T Funabashi ldquoGain-scheduledHinfin control forWECS via LMI techniques and para-metrically dependent feedback part II Controller design andimplementationrdquo IEEE Transactions on Industrial Electronicsvol 58 no 1 pp 57ndash65 2011
[7] F D Bianchi H De Battista and R JMantz ldquoRobustmultivari-able gain scheduled control of wind turbines for variable powerproductionrdquo International Journal of Systems Control vol 1 no3 pp 103ndash112 2010
[8] I Soslashnderby andMH Hansen ldquoOpen-loop frequency responseanalysis of a wind turbine using a high-order linear aeroelasticmodelrdquoWind Energy vol 17 no 8 pp 1147ndash1167 2014
[9] S Bououden M Chadli S Filali and A El Hajjaji ldquoFuzzymodel basedmultivariable predictive control of a variable speedwind turbine LMI approachrdquo Journal of Renewable Energy vol37 no 1 pp 434ndash439 2012
[10] J Chen T Lin CWen and Y Song ldquoDesign of a Unified PowerController for Variable-Speed Fixed-Pitch Wind Energy Con-version Systemrdquo IEEETransactions on Industrial Electronics vol63 no 8 pp 4899ndash4908 2016
Mathematical Problems in Engineering 13
[11] Y Ren L Li J Brindley and L Jiang ldquoNonlinear PI control forvariable pitch wind turbinerdquo Control Engineering Practice 2015
[12] Y Yuan and J Tang ldquoAdaptive pitch control of wind turbinefor load mitigation under structural uncertaintiesrdquo Journal ofRenewable Energy vol 105 pp 483ndash494 2017
[13] C-S Wang and M-H Chiang ldquoA novel pitch control systemof a large wind turbine using two-degree-of-freedom motioncontrol with feedback linearization controlrdquo Energies vol 9 no10 article no 791 2016
[14] Z Civelek E Cam M Luy and H Mamur ldquoProportional-integral-derivative parameter optimisation of blade pitch con-troller in wind turbines by a new intelligent genetic algorithmrdquoIET Renewable Power Generation vol 10 no 8 pp 1220ndash12282016
[15] A Dahbi N Nait-Said andM-S Nait-Said ldquoA novel combinedMPPT-pitch angle control for wide range variable speed windturbine based on neural networkrdquo International Journal ofHydrogen Energy vol 41 no 22 pp 9427ndash9442 2016
[16] M L Corradini G Ippoliti and G Orlando ldquoAn observer-based blade-pitch controller of wind turbines in high windspeedsrdquo Control Engineering Practice vol 58 pp 186ndash192 2017
[17] M Zhu J Liu Z Lin and H Meng ldquoMixed H2 Hinfin pitchcontrol of wind turbine generator system based on globalexact linearization and regional pole placementrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 5 pp921ndash930 2016
[18] R Gao and Z Gao ldquoPitch control for wind turbine systemsusing optimization estimation and compensationrdquo Journal ofRenewable Energy vol 91 pp 501ndash515 2016
[19] X-X Yin Y-G Lin W Li H-W Liu and Y-J Gu ldquoAdaptivesliding mode back-stepping pitch angle control of a variable-displacement pump controlled pitch system for wind turbinesrdquoISA Transactions vol 58 pp 629ndash634 2015
[20] M-H Chiang ldquoA novel pitch control system for a wind turbinedriven by a variable-speed pump-controlled hydraulic servosystemrdquoMechatronics vol 21 no 4 pp 753ndash761 2011
[21] H Camblong S Nourdine I Vechiu and G Tapia ldquoCom-parison of an island wind turbine collective and individualpitch LQG controllers designed to alleviate fatigue loadsrdquo IETRenewable Power Generation vol 6 no 4 p 267 2012
[22] F A InthamoussouHDe Battista andR JMantz ldquoLPV-basedactive power control of wind turbines covering the completewind speed rangerdquo Journal of Renewable Energy vol 99 pp996ndash1007 2016
[23] F D Bianchi R J Mantz and C F Christiansen ldquoGainscheduling control of variable-speed wind energy conversionsystems using quasi-LPVmodelsrdquoControl Engineering Practicevol 13 no 2 pp 247ndash255 2005
[24] F A Shirazi K M Grigoriadis and D Viassolo ldquoWind turbineintegrated structural and LPV control design for improvedclosed-loop performancerdquo International Journal of Control vol85 no 8 pp 1178ndash1196 2012
[25] F A Inthamoussou F D Bianchi H De Battista and R JMantz ldquoLPV wind turbine control with anti-windup featurescovering the complete wind speed rangerdquo IEEE Transactions onEnergy Conversion vol 29 no 1 pp 259ndash266 2014
[26] K Z Oslashstegaard J Stoustrup and P Brath ldquoLinear parametervarying control of wind turbines covering both partial loadand full load conditionsrdquo International Journal of Robust andNonlinear Control vol 19 no 1 pp 92ndash116 2009
[27] Y Lin L Tu H Liu and W Li ldquoHybrid power transmissiontechnology in a wind turbine generation systemrdquo IEEEASMETransactions onMechatronics vol 20 no 3 pp 1218ndash1225 2015
[28] S Bououden M Chadli F Allouani and S Filali ldquoA newapproach for fuzzy predictive adaptive controller design usingparticle swarmoptimization algorithmrdquo International Journal ofInnovative Computing Information and Control vol 9 no 9 pp3741ndash3758 2013
[29] S H Li and H Gu ldquoFuzzy adaptive internal model controlschemes for PMSM speed-regulation systemrdquo IEEE Transac-tions on Industrial Informatics vol 8 no 4 pp 767ndash779 2012
[30] A B Asghar and X Liu ldquoAdaptive neuro-fuzzy algorithmto estimate effective wind speed and optimal rotor speed forvariable-speed wind turbinerdquo Neurocomputing vol 272 pp495ndash504 2018
[31] Y Wang H Shen H R Karimi and D Duan ldquoDissipativity-based Fuzzy Integral SlidingModeControl ofContinuous-TimeT-S Fuzzy Systemsrdquo IEEE Transactions on Fuzzy Systems pp 1-1
[32] S Bououden M Chadli and H R Karimi ldquoAn ant colonyoptimization-based fuzzy predictive control approach for non-linear processesrdquo Information Sciences vol 299 pp 143ndash1582015
[33] NWang S-F Su J Yin Z Zheng andM J Er ldquoGlobal Asymp-totic Model-Free Trajectory-Independent Tracking Control ofan Uncertain Marine Vehicle An Adaptive Universe-BasedFuzzy Control Approachrdquo IEEE Transactions on Fuzzy Systems2017
[34] R Saravanakumar andD Jena ldquoValidation of an integral slidingmode control for optimal control of a three blade variable speedvariable pitch wind turbinerdquo International Journal of ElectricalPower amp Energy Systems vol 69 pp 421ndash429 2015
[35] C Evangelista P Puleston F Valenciaga and L M FridmanldquoLyapunov-designed super-twisting sliding mode control forwind energy conversion optimizationrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 538ndash545 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 9
Table 5 The parameters of the proposed controller
Symbols Parameters Values119888 Constant defined in sliding mode algorithm 10119890 119903119886119899119892119890 The universe of rotor deviation [-0003 0003]119890119888 119903119886119899119892119890 The universe of rotor deviation changing rate [-0008 0008]119870119875 119903119886119899119892119890 The universe of proportionality coefficent [-03 03]119870119868 119903119886119899119892119890 The universe of integral coefficient [-006 006]119870119863 119903119886119899119892119890 The universe of differential coefficient [-3 3]119864 The value of the error universe 0003119864119862 The value of the error changing rate universe 0008120583 The coefficient of universe scaling factor 091205831 The coefficient of universe scaling factor 091205832 The coefficient of universe scaling factor 04120576 A full small positive number 000001
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
13
135
14
145
15
155
16
v(m
s)
(a) Random wind speed
sign functionsaturation function
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(b) Pitch angles using different functions in sliding mode
sign functionsaturation function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(c) Rotor speeds using different functions in sliding mode
sign functionsaturation function
25
3
35
4
P(M
W)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
(d) Output powers using different functions in sliding mode
Figure 8 (a) Random wind speed (b) pitch angles using different functions in sliding mode (c) rotor speeds using different functions insliding mode (d) output powers using different functions in sliding mode
10 Mathematical Problems in Engineering
5
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
Local enlargeddrawing
4 5 6 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under step wind speed
9 10 11 12
2
2001Localenlargeddrawing
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under step wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
9 10 11 12
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) output power using different controllers under step wind speed
Figure 9 (a) Pitch angles using different controllers under step wind speed (b) rotor speed using different controllers under step wind speedand (c) output power using different controllers under step wind speed
the control when implementing the two different functionsunder the step wind speed However under random windspeed the control performance of the saturation functionadjusting output power is significantly better than that of thesign function in Figure 8(d)
Comparisons are further studied among which thereare conventional PID algorithm feedforward feedback con-trol combining PID with feedback linearization algorithmfeedforward feedback control based on fuzzy self-tuningPID with feedback linearization with sliding mode control(called fuzzy PID in feedback loop in Figures 9(a)ndash9(c)
and Figures 10(a)ndash10(c)) and the proposed controller inthis paper Results are depicted in Figures 9(a)ndash9(c) andFigures 10(a)ndash10(c) The wind speed is changed continuouslyto further validate pitching control algorithms which isdepicted in Figure 8(a)
At the initial moment control outputs fluctuate stronglywhich are caused by initial values of states However iffluctuations are in acceptable ranges of the physical structureof the wind turbine the initial responses will not affect theevaluation for the controller In Figure 7(a) at 119905 = 4 and119905 = 10 step wind speed occurs As thewind speeds exceed the
Mathematical Problems in Engineering 11
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
12 13 14 15
5 Local enlargeddrawing
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
8 9 10 11
1999
2
2001
Localenlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
13 14 15 16
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) Output power using different controllers under random wind speed
Figure 10 (a) Pitch angles using different controllers under random wind speed (b) rotor speed using different controllers under randomwind speed and (c) output power using different controllers under random wind speed
rated value pitch angles need to be adjusted to keep outputpower at constant
In Figure 9(a) the overshoot of the pitch angle is greaterwhile oscillation time is longer when adopting the conven-tional PID controller Although rotor speed also convergesat the rated value steady-state errors cannot be eliminatedwhich is approved in rotor speed chart depicted in Fig-ure 9(b) The rotor speeds of wind turbine are maintainedat constant when wind speed exceeds the rated value Dueto the steady-state errors of the pitch angle rotor speedshave deviations These results show that single linear PID
controller cannot track target well for wind turbine exhibitsstrongly nonlinear features
To address this problem we design a PID controllercombining with feedback linearization theory The PIDcontroller is feedback control while feedback linearizationis proposed as feedforward control Feedback control cankeep the system stable However the feedback control ishysteric Only when the deviation occurs the controller isimplemented to eliminate it Nevertheless feedforward canproduce a control variable in advanceThus the final outputsof the two controllers are proposed as the input of the wind
12 Mathematical Problems in Engineering
turbine system so that tracking accuracy of controller can beenhanced
Figures 9(a)ndash9(c) demonstrate that the control perfor-mance of feedforward feedback controller based on com-bining PID algorithm with feedback linearization algorithmis clearly superior to conventional PID controller in whichthe overshoot of pitch angle is smaller and adjusting timeof oscillating is shorter and furthermore the steady-statedeviation is eliminated which is also shown both in therotor speed diagram and in the output power diagram Tofurther improve control performance sliding mode controlis combined with feedback linear algorithm in feedforwardloop while the fuzzy self-tuning algorithm is used to adjustthe parameters of the PID controller in feedback loopFurthermore in order to decrease the effect when inputvariable and output variable change a lot variable universeapproach is proposed to optimize fuzzy rules Known fromthe results the deficiencies of controller produced by systemconstruct and parameters variation can be overcome FromFigures 9(a)ndash9(c) it is shown that the performance of thecontroller we propose in this paper is the best among thefour controllers whose overshoot is the least while it rapidlyconverges to constant without steady-state errors Althoughcontinuous step speed can evaluate controller performanceit still exhibits limitation
In order to further verify the effectiveness and universal-ity of the proposed controller in this paper we further adoptrandom wind speed Figure 8(a) depicts random wind speedFigures 10(a)ndash10(c) show the control results of the pitch anglerotor speed of turbine and output power produced by fourdifferent control algorithms under random wind speed Theresults are reasonable which are the same as those producedby step wind speed Therefore the control algorithm wepropose in this paper is the best among the described fouralgorithms and it can keep the output power at the rated valuewhen wind speed exceeds the nominal velocity
5 Conclusions
In this research a novel pitch control strategy for largevariable speed variable pitch wind turbines is studied aimingat maintaining the output power at constant so that theenergy quality is improved Known from the results as thesliding mode theory is combined with feedback linearizationalgorithm in feedforward loop the uncertainty of modeland time-varying of system parameters are overcome Fur-thermore with setting a control variable in advance theovershoot can be decreased when wind speed is varyingOn the other hand in feedback loop as variable universefuzzy algorithm is employed to optimize the parameters ofthe conventional PID controller the tracking performanceof the feedback controller is enhanced Meanwhile insteadof sign function saturation function is implemented todecrease oscillation of sliding mode algorithm and thus thefluctuations of output power and rotor speed of wind turbineare mitigated Both control accuracy and robustness usingthe proposed algorithm are the best among four differentcontrollers Another in future we plan to supply fuzzy
predictive adaptive algorithm in power control to furtherimprove system performance
Data Availability
The data used to support the findings of this study arecurrently under embargo while the research findings arecommercialized Requests for data 6 months after publica-tion of this article will be considered by the correspondingauthor
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National Natural ScienceFund of China under Grant no 51405052
References
[1] C-SWang andM-H Chiang ldquoA novel dynamic co-simulationanalysis for overall closed loop operation control of a large windturbinerdquo Energies vol 9 no 8 2016
[2] E Van Solingen J Beerens S P Mulders R De Breuker andJ W Van Wingerden ldquoControl design for a two-bladed down-wind teeterless damped free-yaw wind turbinerdquo Mechatronicsvol 36 pp 77ndash96 2016
[3] Y-J Chen and Y C Shiah ldquoExperiments on the performance ofsmall horizontal axis wind turbine with passive pitch control bydisk pulleyrdquo Energies vol 9 no 5 2016
[4] A Choudhry M Arjomandi and R Kelso ldquoMethods tocontrol dynamic stall for wind turbine applicationsrdquo Journal ofRenewable Energy vol 86 pp 26ndash37 2016
[5] H Li C Yang Y Hu X Liao Z Zeng and C Zhe ldquoAnimproved reduced-ordermodel of an electric pitch drive systemfor wind turbine control system design and simulationrdquo Journalof Renewable Energy vol 93 pp 188ndash200 2016
[6] E B Muhando T Senjyu A Uehara and T Funabashi ldquoGain-scheduledHinfin control forWECS via LMI techniques and para-metrically dependent feedback part II Controller design andimplementationrdquo IEEE Transactions on Industrial Electronicsvol 58 no 1 pp 57ndash65 2011
[7] F D Bianchi H De Battista and R JMantz ldquoRobustmultivari-able gain scheduled control of wind turbines for variable powerproductionrdquo International Journal of Systems Control vol 1 no3 pp 103ndash112 2010
[8] I Soslashnderby andMH Hansen ldquoOpen-loop frequency responseanalysis of a wind turbine using a high-order linear aeroelasticmodelrdquoWind Energy vol 17 no 8 pp 1147ndash1167 2014
[9] S Bououden M Chadli S Filali and A El Hajjaji ldquoFuzzymodel basedmultivariable predictive control of a variable speedwind turbine LMI approachrdquo Journal of Renewable Energy vol37 no 1 pp 434ndash439 2012
[10] J Chen T Lin CWen and Y Song ldquoDesign of a Unified PowerController for Variable-Speed Fixed-Pitch Wind Energy Con-version Systemrdquo IEEETransactions on Industrial Electronics vol63 no 8 pp 4899ndash4908 2016
Mathematical Problems in Engineering 13
[11] Y Ren L Li J Brindley and L Jiang ldquoNonlinear PI control forvariable pitch wind turbinerdquo Control Engineering Practice 2015
[12] Y Yuan and J Tang ldquoAdaptive pitch control of wind turbinefor load mitigation under structural uncertaintiesrdquo Journal ofRenewable Energy vol 105 pp 483ndash494 2017
[13] C-S Wang and M-H Chiang ldquoA novel pitch control systemof a large wind turbine using two-degree-of-freedom motioncontrol with feedback linearization controlrdquo Energies vol 9 no10 article no 791 2016
[14] Z Civelek E Cam M Luy and H Mamur ldquoProportional-integral-derivative parameter optimisation of blade pitch con-troller in wind turbines by a new intelligent genetic algorithmrdquoIET Renewable Power Generation vol 10 no 8 pp 1220ndash12282016
[15] A Dahbi N Nait-Said andM-S Nait-Said ldquoA novel combinedMPPT-pitch angle control for wide range variable speed windturbine based on neural networkrdquo International Journal ofHydrogen Energy vol 41 no 22 pp 9427ndash9442 2016
[16] M L Corradini G Ippoliti and G Orlando ldquoAn observer-based blade-pitch controller of wind turbines in high windspeedsrdquo Control Engineering Practice vol 58 pp 186ndash192 2017
[17] M Zhu J Liu Z Lin and H Meng ldquoMixed H2 Hinfin pitchcontrol of wind turbine generator system based on globalexact linearization and regional pole placementrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 5 pp921ndash930 2016
[18] R Gao and Z Gao ldquoPitch control for wind turbine systemsusing optimization estimation and compensationrdquo Journal ofRenewable Energy vol 91 pp 501ndash515 2016
[19] X-X Yin Y-G Lin W Li H-W Liu and Y-J Gu ldquoAdaptivesliding mode back-stepping pitch angle control of a variable-displacement pump controlled pitch system for wind turbinesrdquoISA Transactions vol 58 pp 629ndash634 2015
[20] M-H Chiang ldquoA novel pitch control system for a wind turbinedriven by a variable-speed pump-controlled hydraulic servosystemrdquoMechatronics vol 21 no 4 pp 753ndash761 2011
[21] H Camblong S Nourdine I Vechiu and G Tapia ldquoCom-parison of an island wind turbine collective and individualpitch LQG controllers designed to alleviate fatigue loadsrdquo IETRenewable Power Generation vol 6 no 4 p 267 2012
[22] F A InthamoussouHDe Battista andR JMantz ldquoLPV-basedactive power control of wind turbines covering the completewind speed rangerdquo Journal of Renewable Energy vol 99 pp996ndash1007 2016
[23] F D Bianchi R J Mantz and C F Christiansen ldquoGainscheduling control of variable-speed wind energy conversionsystems using quasi-LPVmodelsrdquoControl Engineering Practicevol 13 no 2 pp 247ndash255 2005
[24] F A Shirazi K M Grigoriadis and D Viassolo ldquoWind turbineintegrated structural and LPV control design for improvedclosed-loop performancerdquo International Journal of Control vol85 no 8 pp 1178ndash1196 2012
[25] F A Inthamoussou F D Bianchi H De Battista and R JMantz ldquoLPV wind turbine control with anti-windup featurescovering the complete wind speed rangerdquo IEEE Transactions onEnergy Conversion vol 29 no 1 pp 259ndash266 2014
[26] K Z Oslashstegaard J Stoustrup and P Brath ldquoLinear parametervarying control of wind turbines covering both partial loadand full load conditionsrdquo International Journal of Robust andNonlinear Control vol 19 no 1 pp 92ndash116 2009
[27] Y Lin L Tu H Liu and W Li ldquoHybrid power transmissiontechnology in a wind turbine generation systemrdquo IEEEASMETransactions onMechatronics vol 20 no 3 pp 1218ndash1225 2015
[28] S Bououden M Chadli F Allouani and S Filali ldquoA newapproach for fuzzy predictive adaptive controller design usingparticle swarmoptimization algorithmrdquo International Journal ofInnovative Computing Information and Control vol 9 no 9 pp3741ndash3758 2013
[29] S H Li and H Gu ldquoFuzzy adaptive internal model controlschemes for PMSM speed-regulation systemrdquo IEEE Transac-tions on Industrial Informatics vol 8 no 4 pp 767ndash779 2012
[30] A B Asghar and X Liu ldquoAdaptive neuro-fuzzy algorithmto estimate effective wind speed and optimal rotor speed forvariable-speed wind turbinerdquo Neurocomputing vol 272 pp495ndash504 2018
[31] Y Wang H Shen H R Karimi and D Duan ldquoDissipativity-based Fuzzy Integral SlidingModeControl ofContinuous-TimeT-S Fuzzy Systemsrdquo IEEE Transactions on Fuzzy Systems pp 1-1
[32] S Bououden M Chadli and H R Karimi ldquoAn ant colonyoptimization-based fuzzy predictive control approach for non-linear processesrdquo Information Sciences vol 299 pp 143ndash1582015
[33] NWang S-F Su J Yin Z Zheng andM J Er ldquoGlobal Asymp-totic Model-Free Trajectory-Independent Tracking Control ofan Uncertain Marine Vehicle An Adaptive Universe-BasedFuzzy Control Approachrdquo IEEE Transactions on Fuzzy Systems2017
[34] R Saravanakumar andD Jena ldquoValidation of an integral slidingmode control for optimal control of a three blade variable speedvariable pitch wind turbinerdquo International Journal of ElectricalPower amp Energy Systems vol 69 pp 421ndash429 2015
[35] C Evangelista P Puleston F Valenciaga and L M FridmanldquoLyapunov-designed super-twisting sliding mode control forwind energy conversion optimizationrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 538ndash545 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
10 Mathematical Problems in Engineering
5
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
Local enlargeddrawing
4 5 6 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under step wind speed
9 10 11 12
2
2001Localenlargeddrawing
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under step wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
9 10 11 12
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) output power using different controllers under step wind speed
Figure 9 (a) Pitch angles using different controllers under step wind speed (b) rotor speed using different controllers under step wind speedand (c) output power using different controllers under step wind speed
the control when implementing the two different functionsunder the step wind speed However under random windspeed the control performance of the saturation functionadjusting output power is significantly better than that of thesign function in Figure 8(d)
Comparisons are further studied among which thereare conventional PID algorithm feedforward feedback con-trol combining PID with feedback linearization algorithmfeedforward feedback control based on fuzzy self-tuningPID with feedback linearization with sliding mode control(called fuzzy PID in feedback loop in Figures 9(a)ndash9(c)
and Figures 10(a)ndash10(c)) and the proposed controller inthis paper Results are depicted in Figures 9(a)ndash9(c) andFigures 10(a)ndash10(c) The wind speed is changed continuouslyto further validate pitching control algorithms which isdepicted in Figure 8(a)
At the initial moment control outputs fluctuate stronglywhich are caused by initial values of states However iffluctuations are in acceptable ranges of the physical structureof the wind turbine the initial responses will not affect theevaluation for the controller In Figure 7(a) at 119905 = 4 and119905 = 10 step wind speed occurs As thewind speeds exceed the
Mathematical Problems in Engineering 11
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
12 13 14 15
5 Local enlargeddrawing
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
8 9 10 11
1999
2
2001
Localenlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
13 14 15 16
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) Output power using different controllers under random wind speed
Figure 10 (a) Pitch angles using different controllers under random wind speed (b) rotor speed using different controllers under randomwind speed and (c) output power using different controllers under random wind speed
rated value pitch angles need to be adjusted to keep outputpower at constant
In Figure 9(a) the overshoot of the pitch angle is greaterwhile oscillation time is longer when adopting the conven-tional PID controller Although rotor speed also convergesat the rated value steady-state errors cannot be eliminatedwhich is approved in rotor speed chart depicted in Fig-ure 9(b) The rotor speeds of wind turbine are maintainedat constant when wind speed exceeds the rated value Dueto the steady-state errors of the pitch angle rotor speedshave deviations These results show that single linear PID
controller cannot track target well for wind turbine exhibitsstrongly nonlinear features
To address this problem we design a PID controllercombining with feedback linearization theory The PIDcontroller is feedback control while feedback linearizationis proposed as feedforward control Feedback control cankeep the system stable However the feedback control ishysteric Only when the deviation occurs the controller isimplemented to eliminate it Nevertheless feedforward canproduce a control variable in advanceThus the final outputsof the two controllers are proposed as the input of the wind
12 Mathematical Problems in Engineering
turbine system so that tracking accuracy of controller can beenhanced
Figures 9(a)ndash9(c) demonstrate that the control perfor-mance of feedforward feedback controller based on com-bining PID algorithm with feedback linearization algorithmis clearly superior to conventional PID controller in whichthe overshoot of pitch angle is smaller and adjusting timeof oscillating is shorter and furthermore the steady-statedeviation is eliminated which is also shown both in therotor speed diagram and in the output power diagram Tofurther improve control performance sliding mode controlis combined with feedback linear algorithm in feedforwardloop while the fuzzy self-tuning algorithm is used to adjustthe parameters of the PID controller in feedback loopFurthermore in order to decrease the effect when inputvariable and output variable change a lot variable universeapproach is proposed to optimize fuzzy rules Known fromthe results the deficiencies of controller produced by systemconstruct and parameters variation can be overcome FromFigures 9(a)ndash9(c) it is shown that the performance of thecontroller we propose in this paper is the best among thefour controllers whose overshoot is the least while it rapidlyconverges to constant without steady-state errors Althoughcontinuous step speed can evaluate controller performanceit still exhibits limitation
In order to further verify the effectiveness and universal-ity of the proposed controller in this paper we further adoptrandom wind speed Figure 8(a) depicts random wind speedFigures 10(a)ndash10(c) show the control results of the pitch anglerotor speed of turbine and output power produced by fourdifferent control algorithms under random wind speed Theresults are reasonable which are the same as those producedby step wind speed Therefore the control algorithm wepropose in this paper is the best among the described fouralgorithms and it can keep the output power at the rated valuewhen wind speed exceeds the nominal velocity
5 Conclusions
In this research a novel pitch control strategy for largevariable speed variable pitch wind turbines is studied aimingat maintaining the output power at constant so that theenergy quality is improved Known from the results as thesliding mode theory is combined with feedback linearizationalgorithm in feedforward loop the uncertainty of modeland time-varying of system parameters are overcome Fur-thermore with setting a control variable in advance theovershoot can be decreased when wind speed is varyingOn the other hand in feedback loop as variable universefuzzy algorithm is employed to optimize the parameters ofthe conventional PID controller the tracking performanceof the feedback controller is enhanced Meanwhile insteadof sign function saturation function is implemented todecrease oscillation of sliding mode algorithm and thus thefluctuations of output power and rotor speed of wind turbineare mitigated Both control accuracy and robustness usingthe proposed algorithm are the best among four differentcontrollers Another in future we plan to supply fuzzy
predictive adaptive algorithm in power control to furtherimprove system performance
Data Availability
The data used to support the findings of this study arecurrently under embargo while the research findings arecommercialized Requests for data 6 months after publica-tion of this article will be considered by the correspondingauthor
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National Natural ScienceFund of China under Grant no 51405052
References
[1] C-SWang andM-H Chiang ldquoA novel dynamic co-simulationanalysis for overall closed loop operation control of a large windturbinerdquo Energies vol 9 no 8 2016
[2] E Van Solingen J Beerens S P Mulders R De Breuker andJ W Van Wingerden ldquoControl design for a two-bladed down-wind teeterless damped free-yaw wind turbinerdquo Mechatronicsvol 36 pp 77ndash96 2016
[3] Y-J Chen and Y C Shiah ldquoExperiments on the performance ofsmall horizontal axis wind turbine with passive pitch control bydisk pulleyrdquo Energies vol 9 no 5 2016
[4] A Choudhry M Arjomandi and R Kelso ldquoMethods tocontrol dynamic stall for wind turbine applicationsrdquo Journal ofRenewable Energy vol 86 pp 26ndash37 2016
[5] H Li C Yang Y Hu X Liao Z Zeng and C Zhe ldquoAnimproved reduced-ordermodel of an electric pitch drive systemfor wind turbine control system design and simulationrdquo Journalof Renewable Energy vol 93 pp 188ndash200 2016
[6] E B Muhando T Senjyu A Uehara and T Funabashi ldquoGain-scheduledHinfin control forWECS via LMI techniques and para-metrically dependent feedback part II Controller design andimplementationrdquo IEEE Transactions on Industrial Electronicsvol 58 no 1 pp 57ndash65 2011
[7] F D Bianchi H De Battista and R JMantz ldquoRobustmultivari-able gain scheduled control of wind turbines for variable powerproductionrdquo International Journal of Systems Control vol 1 no3 pp 103ndash112 2010
[8] I Soslashnderby andMH Hansen ldquoOpen-loop frequency responseanalysis of a wind turbine using a high-order linear aeroelasticmodelrdquoWind Energy vol 17 no 8 pp 1147ndash1167 2014
[9] S Bououden M Chadli S Filali and A El Hajjaji ldquoFuzzymodel basedmultivariable predictive control of a variable speedwind turbine LMI approachrdquo Journal of Renewable Energy vol37 no 1 pp 434ndash439 2012
[10] J Chen T Lin CWen and Y Song ldquoDesign of a Unified PowerController for Variable-Speed Fixed-Pitch Wind Energy Con-version Systemrdquo IEEETransactions on Industrial Electronics vol63 no 8 pp 4899ndash4908 2016
Mathematical Problems in Engineering 13
[11] Y Ren L Li J Brindley and L Jiang ldquoNonlinear PI control forvariable pitch wind turbinerdquo Control Engineering Practice 2015
[12] Y Yuan and J Tang ldquoAdaptive pitch control of wind turbinefor load mitigation under structural uncertaintiesrdquo Journal ofRenewable Energy vol 105 pp 483ndash494 2017
[13] C-S Wang and M-H Chiang ldquoA novel pitch control systemof a large wind turbine using two-degree-of-freedom motioncontrol with feedback linearization controlrdquo Energies vol 9 no10 article no 791 2016
[14] Z Civelek E Cam M Luy and H Mamur ldquoProportional-integral-derivative parameter optimisation of blade pitch con-troller in wind turbines by a new intelligent genetic algorithmrdquoIET Renewable Power Generation vol 10 no 8 pp 1220ndash12282016
[15] A Dahbi N Nait-Said andM-S Nait-Said ldquoA novel combinedMPPT-pitch angle control for wide range variable speed windturbine based on neural networkrdquo International Journal ofHydrogen Energy vol 41 no 22 pp 9427ndash9442 2016
[16] M L Corradini G Ippoliti and G Orlando ldquoAn observer-based blade-pitch controller of wind turbines in high windspeedsrdquo Control Engineering Practice vol 58 pp 186ndash192 2017
[17] M Zhu J Liu Z Lin and H Meng ldquoMixed H2 Hinfin pitchcontrol of wind turbine generator system based on globalexact linearization and regional pole placementrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 5 pp921ndash930 2016
[18] R Gao and Z Gao ldquoPitch control for wind turbine systemsusing optimization estimation and compensationrdquo Journal ofRenewable Energy vol 91 pp 501ndash515 2016
[19] X-X Yin Y-G Lin W Li H-W Liu and Y-J Gu ldquoAdaptivesliding mode back-stepping pitch angle control of a variable-displacement pump controlled pitch system for wind turbinesrdquoISA Transactions vol 58 pp 629ndash634 2015
[20] M-H Chiang ldquoA novel pitch control system for a wind turbinedriven by a variable-speed pump-controlled hydraulic servosystemrdquoMechatronics vol 21 no 4 pp 753ndash761 2011
[21] H Camblong S Nourdine I Vechiu and G Tapia ldquoCom-parison of an island wind turbine collective and individualpitch LQG controllers designed to alleviate fatigue loadsrdquo IETRenewable Power Generation vol 6 no 4 p 267 2012
[22] F A InthamoussouHDe Battista andR JMantz ldquoLPV-basedactive power control of wind turbines covering the completewind speed rangerdquo Journal of Renewable Energy vol 99 pp996ndash1007 2016
[23] F D Bianchi R J Mantz and C F Christiansen ldquoGainscheduling control of variable-speed wind energy conversionsystems using quasi-LPVmodelsrdquoControl Engineering Practicevol 13 no 2 pp 247ndash255 2005
[24] F A Shirazi K M Grigoriadis and D Viassolo ldquoWind turbineintegrated structural and LPV control design for improvedclosed-loop performancerdquo International Journal of Control vol85 no 8 pp 1178ndash1196 2012
[25] F A Inthamoussou F D Bianchi H De Battista and R JMantz ldquoLPV wind turbine control with anti-windup featurescovering the complete wind speed rangerdquo IEEE Transactions onEnergy Conversion vol 29 no 1 pp 259ndash266 2014
[26] K Z Oslashstegaard J Stoustrup and P Brath ldquoLinear parametervarying control of wind turbines covering both partial loadand full load conditionsrdquo International Journal of Robust andNonlinear Control vol 19 no 1 pp 92ndash116 2009
[27] Y Lin L Tu H Liu and W Li ldquoHybrid power transmissiontechnology in a wind turbine generation systemrdquo IEEEASMETransactions onMechatronics vol 20 no 3 pp 1218ndash1225 2015
[28] S Bououden M Chadli F Allouani and S Filali ldquoA newapproach for fuzzy predictive adaptive controller design usingparticle swarmoptimization algorithmrdquo International Journal ofInnovative Computing Information and Control vol 9 no 9 pp3741ndash3758 2013
[29] S H Li and H Gu ldquoFuzzy adaptive internal model controlschemes for PMSM speed-regulation systemrdquo IEEE Transac-tions on Industrial Informatics vol 8 no 4 pp 767ndash779 2012
[30] A B Asghar and X Liu ldquoAdaptive neuro-fuzzy algorithmto estimate effective wind speed and optimal rotor speed forvariable-speed wind turbinerdquo Neurocomputing vol 272 pp495ndash504 2018
[31] Y Wang H Shen H R Karimi and D Duan ldquoDissipativity-based Fuzzy Integral SlidingModeControl ofContinuous-TimeT-S Fuzzy Systemsrdquo IEEE Transactions on Fuzzy Systems pp 1-1
[32] S Bououden M Chadli and H R Karimi ldquoAn ant colonyoptimization-based fuzzy predictive control approach for non-linear processesrdquo Information Sciences vol 299 pp 143ndash1582015
[33] NWang S-F Su J Yin Z Zheng andM J Er ldquoGlobal Asymp-totic Model-Free Trajectory-Independent Tracking Control ofan Uncertain Marine Vehicle An Adaptive Universe-BasedFuzzy Control Approachrdquo IEEE Transactions on Fuzzy Systems2017
[34] R Saravanakumar andD Jena ldquoValidation of an integral slidingmode control for optimal control of a three blade variable speedvariable pitch wind turbinerdquo International Journal of ElectricalPower amp Energy Systems vol 69 pp 421ndash429 2015
[35] C Evangelista P Puleston F Valenciaga and L M FridmanldquoLyapunov-designed super-twisting sliding mode control forwind energy conversion optimizationrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 538ndash545 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 11
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
12 13 14 15
5 Local enlargeddrawing
1 2 3 4 5 6 7 8 90 11 12 13 14 15 16 17 18 19 2010t(s)
0
1
2
3
4
5
6
7
8
(d
eg)
(a) Pitch angles using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
8 9 10 11
1999
2
2001
Localenlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
19981999
2200120022003200420052006200720082009
2012011
w(r
ads
)
(b) Rotor speed using different controllers under random wind speed
PIDPID with feedback linearizationFuzzy PID in feedback loopThe proposed controller this paper
13 14 15 16
3
Local enlargeddrawing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200t(s)
25
3
35
4
P(M
W)
(c) Output power using different controllers under random wind speed
Figure 10 (a) Pitch angles using different controllers under random wind speed (b) rotor speed using different controllers under randomwind speed and (c) output power using different controllers under random wind speed
rated value pitch angles need to be adjusted to keep outputpower at constant
In Figure 9(a) the overshoot of the pitch angle is greaterwhile oscillation time is longer when adopting the conven-tional PID controller Although rotor speed also convergesat the rated value steady-state errors cannot be eliminatedwhich is approved in rotor speed chart depicted in Fig-ure 9(b) The rotor speeds of wind turbine are maintainedat constant when wind speed exceeds the rated value Dueto the steady-state errors of the pitch angle rotor speedshave deviations These results show that single linear PID
controller cannot track target well for wind turbine exhibitsstrongly nonlinear features
To address this problem we design a PID controllercombining with feedback linearization theory The PIDcontroller is feedback control while feedback linearizationis proposed as feedforward control Feedback control cankeep the system stable However the feedback control ishysteric Only when the deviation occurs the controller isimplemented to eliminate it Nevertheless feedforward canproduce a control variable in advanceThus the final outputsof the two controllers are proposed as the input of the wind
12 Mathematical Problems in Engineering
turbine system so that tracking accuracy of controller can beenhanced
Figures 9(a)ndash9(c) demonstrate that the control perfor-mance of feedforward feedback controller based on com-bining PID algorithm with feedback linearization algorithmis clearly superior to conventional PID controller in whichthe overshoot of pitch angle is smaller and adjusting timeof oscillating is shorter and furthermore the steady-statedeviation is eliminated which is also shown both in therotor speed diagram and in the output power diagram Tofurther improve control performance sliding mode controlis combined with feedback linear algorithm in feedforwardloop while the fuzzy self-tuning algorithm is used to adjustthe parameters of the PID controller in feedback loopFurthermore in order to decrease the effect when inputvariable and output variable change a lot variable universeapproach is proposed to optimize fuzzy rules Known fromthe results the deficiencies of controller produced by systemconstruct and parameters variation can be overcome FromFigures 9(a)ndash9(c) it is shown that the performance of thecontroller we propose in this paper is the best among thefour controllers whose overshoot is the least while it rapidlyconverges to constant without steady-state errors Althoughcontinuous step speed can evaluate controller performanceit still exhibits limitation
In order to further verify the effectiveness and universal-ity of the proposed controller in this paper we further adoptrandom wind speed Figure 8(a) depicts random wind speedFigures 10(a)ndash10(c) show the control results of the pitch anglerotor speed of turbine and output power produced by fourdifferent control algorithms under random wind speed Theresults are reasonable which are the same as those producedby step wind speed Therefore the control algorithm wepropose in this paper is the best among the described fouralgorithms and it can keep the output power at the rated valuewhen wind speed exceeds the nominal velocity
5 Conclusions
In this research a novel pitch control strategy for largevariable speed variable pitch wind turbines is studied aimingat maintaining the output power at constant so that theenergy quality is improved Known from the results as thesliding mode theory is combined with feedback linearizationalgorithm in feedforward loop the uncertainty of modeland time-varying of system parameters are overcome Fur-thermore with setting a control variable in advance theovershoot can be decreased when wind speed is varyingOn the other hand in feedback loop as variable universefuzzy algorithm is employed to optimize the parameters ofthe conventional PID controller the tracking performanceof the feedback controller is enhanced Meanwhile insteadof sign function saturation function is implemented todecrease oscillation of sliding mode algorithm and thus thefluctuations of output power and rotor speed of wind turbineare mitigated Both control accuracy and robustness usingthe proposed algorithm are the best among four differentcontrollers Another in future we plan to supply fuzzy
predictive adaptive algorithm in power control to furtherimprove system performance
Data Availability
The data used to support the findings of this study arecurrently under embargo while the research findings arecommercialized Requests for data 6 months after publica-tion of this article will be considered by the correspondingauthor
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National Natural ScienceFund of China under Grant no 51405052
References
[1] C-SWang andM-H Chiang ldquoA novel dynamic co-simulationanalysis for overall closed loop operation control of a large windturbinerdquo Energies vol 9 no 8 2016
[2] E Van Solingen J Beerens S P Mulders R De Breuker andJ W Van Wingerden ldquoControl design for a two-bladed down-wind teeterless damped free-yaw wind turbinerdquo Mechatronicsvol 36 pp 77ndash96 2016
[3] Y-J Chen and Y C Shiah ldquoExperiments on the performance ofsmall horizontal axis wind turbine with passive pitch control bydisk pulleyrdquo Energies vol 9 no 5 2016
[4] A Choudhry M Arjomandi and R Kelso ldquoMethods tocontrol dynamic stall for wind turbine applicationsrdquo Journal ofRenewable Energy vol 86 pp 26ndash37 2016
[5] H Li C Yang Y Hu X Liao Z Zeng and C Zhe ldquoAnimproved reduced-ordermodel of an electric pitch drive systemfor wind turbine control system design and simulationrdquo Journalof Renewable Energy vol 93 pp 188ndash200 2016
[6] E B Muhando T Senjyu A Uehara and T Funabashi ldquoGain-scheduledHinfin control forWECS via LMI techniques and para-metrically dependent feedback part II Controller design andimplementationrdquo IEEE Transactions on Industrial Electronicsvol 58 no 1 pp 57ndash65 2011
[7] F D Bianchi H De Battista and R JMantz ldquoRobustmultivari-able gain scheduled control of wind turbines for variable powerproductionrdquo International Journal of Systems Control vol 1 no3 pp 103ndash112 2010
[8] I Soslashnderby andMH Hansen ldquoOpen-loop frequency responseanalysis of a wind turbine using a high-order linear aeroelasticmodelrdquoWind Energy vol 17 no 8 pp 1147ndash1167 2014
[9] S Bououden M Chadli S Filali and A El Hajjaji ldquoFuzzymodel basedmultivariable predictive control of a variable speedwind turbine LMI approachrdquo Journal of Renewable Energy vol37 no 1 pp 434ndash439 2012
[10] J Chen T Lin CWen and Y Song ldquoDesign of a Unified PowerController for Variable-Speed Fixed-Pitch Wind Energy Con-version Systemrdquo IEEETransactions on Industrial Electronics vol63 no 8 pp 4899ndash4908 2016
Mathematical Problems in Engineering 13
[11] Y Ren L Li J Brindley and L Jiang ldquoNonlinear PI control forvariable pitch wind turbinerdquo Control Engineering Practice 2015
[12] Y Yuan and J Tang ldquoAdaptive pitch control of wind turbinefor load mitigation under structural uncertaintiesrdquo Journal ofRenewable Energy vol 105 pp 483ndash494 2017
[13] C-S Wang and M-H Chiang ldquoA novel pitch control systemof a large wind turbine using two-degree-of-freedom motioncontrol with feedback linearization controlrdquo Energies vol 9 no10 article no 791 2016
[14] Z Civelek E Cam M Luy and H Mamur ldquoProportional-integral-derivative parameter optimisation of blade pitch con-troller in wind turbines by a new intelligent genetic algorithmrdquoIET Renewable Power Generation vol 10 no 8 pp 1220ndash12282016
[15] A Dahbi N Nait-Said andM-S Nait-Said ldquoA novel combinedMPPT-pitch angle control for wide range variable speed windturbine based on neural networkrdquo International Journal ofHydrogen Energy vol 41 no 22 pp 9427ndash9442 2016
[16] M L Corradini G Ippoliti and G Orlando ldquoAn observer-based blade-pitch controller of wind turbines in high windspeedsrdquo Control Engineering Practice vol 58 pp 186ndash192 2017
[17] M Zhu J Liu Z Lin and H Meng ldquoMixed H2 Hinfin pitchcontrol of wind turbine generator system based on globalexact linearization and regional pole placementrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 5 pp921ndash930 2016
[18] R Gao and Z Gao ldquoPitch control for wind turbine systemsusing optimization estimation and compensationrdquo Journal ofRenewable Energy vol 91 pp 501ndash515 2016
[19] X-X Yin Y-G Lin W Li H-W Liu and Y-J Gu ldquoAdaptivesliding mode back-stepping pitch angle control of a variable-displacement pump controlled pitch system for wind turbinesrdquoISA Transactions vol 58 pp 629ndash634 2015
[20] M-H Chiang ldquoA novel pitch control system for a wind turbinedriven by a variable-speed pump-controlled hydraulic servosystemrdquoMechatronics vol 21 no 4 pp 753ndash761 2011
[21] H Camblong S Nourdine I Vechiu and G Tapia ldquoCom-parison of an island wind turbine collective and individualpitch LQG controllers designed to alleviate fatigue loadsrdquo IETRenewable Power Generation vol 6 no 4 p 267 2012
[22] F A InthamoussouHDe Battista andR JMantz ldquoLPV-basedactive power control of wind turbines covering the completewind speed rangerdquo Journal of Renewable Energy vol 99 pp996ndash1007 2016
[23] F D Bianchi R J Mantz and C F Christiansen ldquoGainscheduling control of variable-speed wind energy conversionsystems using quasi-LPVmodelsrdquoControl Engineering Practicevol 13 no 2 pp 247ndash255 2005
[24] F A Shirazi K M Grigoriadis and D Viassolo ldquoWind turbineintegrated structural and LPV control design for improvedclosed-loop performancerdquo International Journal of Control vol85 no 8 pp 1178ndash1196 2012
[25] F A Inthamoussou F D Bianchi H De Battista and R JMantz ldquoLPV wind turbine control with anti-windup featurescovering the complete wind speed rangerdquo IEEE Transactions onEnergy Conversion vol 29 no 1 pp 259ndash266 2014
[26] K Z Oslashstegaard J Stoustrup and P Brath ldquoLinear parametervarying control of wind turbines covering both partial loadand full load conditionsrdquo International Journal of Robust andNonlinear Control vol 19 no 1 pp 92ndash116 2009
[27] Y Lin L Tu H Liu and W Li ldquoHybrid power transmissiontechnology in a wind turbine generation systemrdquo IEEEASMETransactions onMechatronics vol 20 no 3 pp 1218ndash1225 2015
[28] S Bououden M Chadli F Allouani and S Filali ldquoA newapproach for fuzzy predictive adaptive controller design usingparticle swarmoptimization algorithmrdquo International Journal ofInnovative Computing Information and Control vol 9 no 9 pp3741ndash3758 2013
[29] S H Li and H Gu ldquoFuzzy adaptive internal model controlschemes for PMSM speed-regulation systemrdquo IEEE Transac-tions on Industrial Informatics vol 8 no 4 pp 767ndash779 2012
[30] A B Asghar and X Liu ldquoAdaptive neuro-fuzzy algorithmto estimate effective wind speed and optimal rotor speed forvariable-speed wind turbinerdquo Neurocomputing vol 272 pp495ndash504 2018
[31] Y Wang H Shen H R Karimi and D Duan ldquoDissipativity-based Fuzzy Integral SlidingModeControl ofContinuous-TimeT-S Fuzzy Systemsrdquo IEEE Transactions on Fuzzy Systems pp 1-1
[32] S Bououden M Chadli and H R Karimi ldquoAn ant colonyoptimization-based fuzzy predictive control approach for non-linear processesrdquo Information Sciences vol 299 pp 143ndash1582015
[33] NWang S-F Su J Yin Z Zheng andM J Er ldquoGlobal Asymp-totic Model-Free Trajectory-Independent Tracking Control ofan Uncertain Marine Vehicle An Adaptive Universe-BasedFuzzy Control Approachrdquo IEEE Transactions on Fuzzy Systems2017
[34] R Saravanakumar andD Jena ldquoValidation of an integral slidingmode control for optimal control of a three blade variable speedvariable pitch wind turbinerdquo International Journal of ElectricalPower amp Energy Systems vol 69 pp 421ndash429 2015
[35] C Evangelista P Puleston F Valenciaga and L M FridmanldquoLyapunov-designed super-twisting sliding mode control forwind energy conversion optimizationrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 538ndash545 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
12 Mathematical Problems in Engineering
turbine system so that tracking accuracy of controller can beenhanced
Figures 9(a)ndash9(c) demonstrate that the control perfor-mance of feedforward feedback controller based on com-bining PID algorithm with feedback linearization algorithmis clearly superior to conventional PID controller in whichthe overshoot of pitch angle is smaller and adjusting timeof oscillating is shorter and furthermore the steady-statedeviation is eliminated which is also shown both in therotor speed diagram and in the output power diagram Tofurther improve control performance sliding mode controlis combined with feedback linear algorithm in feedforwardloop while the fuzzy self-tuning algorithm is used to adjustthe parameters of the PID controller in feedback loopFurthermore in order to decrease the effect when inputvariable and output variable change a lot variable universeapproach is proposed to optimize fuzzy rules Known fromthe results the deficiencies of controller produced by systemconstruct and parameters variation can be overcome FromFigures 9(a)ndash9(c) it is shown that the performance of thecontroller we propose in this paper is the best among thefour controllers whose overshoot is the least while it rapidlyconverges to constant without steady-state errors Althoughcontinuous step speed can evaluate controller performanceit still exhibits limitation
In order to further verify the effectiveness and universal-ity of the proposed controller in this paper we further adoptrandom wind speed Figure 8(a) depicts random wind speedFigures 10(a)ndash10(c) show the control results of the pitch anglerotor speed of turbine and output power produced by fourdifferent control algorithms under random wind speed Theresults are reasonable which are the same as those producedby step wind speed Therefore the control algorithm wepropose in this paper is the best among the described fouralgorithms and it can keep the output power at the rated valuewhen wind speed exceeds the nominal velocity
5 Conclusions
In this research a novel pitch control strategy for largevariable speed variable pitch wind turbines is studied aimingat maintaining the output power at constant so that theenergy quality is improved Known from the results as thesliding mode theory is combined with feedback linearizationalgorithm in feedforward loop the uncertainty of modeland time-varying of system parameters are overcome Fur-thermore with setting a control variable in advance theovershoot can be decreased when wind speed is varyingOn the other hand in feedback loop as variable universefuzzy algorithm is employed to optimize the parameters ofthe conventional PID controller the tracking performanceof the feedback controller is enhanced Meanwhile insteadof sign function saturation function is implemented todecrease oscillation of sliding mode algorithm and thus thefluctuations of output power and rotor speed of wind turbineare mitigated Both control accuracy and robustness usingthe proposed algorithm are the best among four differentcontrollers Another in future we plan to supply fuzzy
predictive adaptive algorithm in power control to furtherimprove system performance
Data Availability
The data used to support the findings of this study arecurrently under embargo while the research findings arecommercialized Requests for data 6 months after publica-tion of this article will be considered by the correspondingauthor
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was supported by the National Natural ScienceFund of China under Grant no 51405052
References
[1] C-SWang andM-H Chiang ldquoA novel dynamic co-simulationanalysis for overall closed loop operation control of a large windturbinerdquo Energies vol 9 no 8 2016
[2] E Van Solingen J Beerens S P Mulders R De Breuker andJ W Van Wingerden ldquoControl design for a two-bladed down-wind teeterless damped free-yaw wind turbinerdquo Mechatronicsvol 36 pp 77ndash96 2016
[3] Y-J Chen and Y C Shiah ldquoExperiments on the performance ofsmall horizontal axis wind turbine with passive pitch control bydisk pulleyrdquo Energies vol 9 no 5 2016
[4] A Choudhry M Arjomandi and R Kelso ldquoMethods tocontrol dynamic stall for wind turbine applicationsrdquo Journal ofRenewable Energy vol 86 pp 26ndash37 2016
[5] H Li C Yang Y Hu X Liao Z Zeng and C Zhe ldquoAnimproved reduced-ordermodel of an electric pitch drive systemfor wind turbine control system design and simulationrdquo Journalof Renewable Energy vol 93 pp 188ndash200 2016
[6] E B Muhando T Senjyu A Uehara and T Funabashi ldquoGain-scheduledHinfin control forWECS via LMI techniques and para-metrically dependent feedback part II Controller design andimplementationrdquo IEEE Transactions on Industrial Electronicsvol 58 no 1 pp 57ndash65 2011
[7] F D Bianchi H De Battista and R JMantz ldquoRobustmultivari-able gain scheduled control of wind turbines for variable powerproductionrdquo International Journal of Systems Control vol 1 no3 pp 103ndash112 2010
[8] I Soslashnderby andMH Hansen ldquoOpen-loop frequency responseanalysis of a wind turbine using a high-order linear aeroelasticmodelrdquoWind Energy vol 17 no 8 pp 1147ndash1167 2014
[9] S Bououden M Chadli S Filali and A El Hajjaji ldquoFuzzymodel basedmultivariable predictive control of a variable speedwind turbine LMI approachrdquo Journal of Renewable Energy vol37 no 1 pp 434ndash439 2012
[10] J Chen T Lin CWen and Y Song ldquoDesign of a Unified PowerController for Variable-Speed Fixed-Pitch Wind Energy Con-version Systemrdquo IEEETransactions on Industrial Electronics vol63 no 8 pp 4899ndash4908 2016
Mathematical Problems in Engineering 13
[11] Y Ren L Li J Brindley and L Jiang ldquoNonlinear PI control forvariable pitch wind turbinerdquo Control Engineering Practice 2015
[12] Y Yuan and J Tang ldquoAdaptive pitch control of wind turbinefor load mitigation under structural uncertaintiesrdquo Journal ofRenewable Energy vol 105 pp 483ndash494 2017
[13] C-S Wang and M-H Chiang ldquoA novel pitch control systemof a large wind turbine using two-degree-of-freedom motioncontrol with feedback linearization controlrdquo Energies vol 9 no10 article no 791 2016
[14] Z Civelek E Cam M Luy and H Mamur ldquoProportional-integral-derivative parameter optimisation of blade pitch con-troller in wind turbines by a new intelligent genetic algorithmrdquoIET Renewable Power Generation vol 10 no 8 pp 1220ndash12282016
[15] A Dahbi N Nait-Said andM-S Nait-Said ldquoA novel combinedMPPT-pitch angle control for wide range variable speed windturbine based on neural networkrdquo International Journal ofHydrogen Energy vol 41 no 22 pp 9427ndash9442 2016
[16] M L Corradini G Ippoliti and G Orlando ldquoAn observer-based blade-pitch controller of wind turbines in high windspeedsrdquo Control Engineering Practice vol 58 pp 186ndash192 2017
[17] M Zhu J Liu Z Lin and H Meng ldquoMixed H2 Hinfin pitchcontrol of wind turbine generator system based on globalexact linearization and regional pole placementrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 5 pp921ndash930 2016
[18] R Gao and Z Gao ldquoPitch control for wind turbine systemsusing optimization estimation and compensationrdquo Journal ofRenewable Energy vol 91 pp 501ndash515 2016
[19] X-X Yin Y-G Lin W Li H-W Liu and Y-J Gu ldquoAdaptivesliding mode back-stepping pitch angle control of a variable-displacement pump controlled pitch system for wind turbinesrdquoISA Transactions vol 58 pp 629ndash634 2015
[20] M-H Chiang ldquoA novel pitch control system for a wind turbinedriven by a variable-speed pump-controlled hydraulic servosystemrdquoMechatronics vol 21 no 4 pp 753ndash761 2011
[21] H Camblong S Nourdine I Vechiu and G Tapia ldquoCom-parison of an island wind turbine collective and individualpitch LQG controllers designed to alleviate fatigue loadsrdquo IETRenewable Power Generation vol 6 no 4 p 267 2012
[22] F A InthamoussouHDe Battista andR JMantz ldquoLPV-basedactive power control of wind turbines covering the completewind speed rangerdquo Journal of Renewable Energy vol 99 pp996ndash1007 2016
[23] F D Bianchi R J Mantz and C F Christiansen ldquoGainscheduling control of variable-speed wind energy conversionsystems using quasi-LPVmodelsrdquoControl Engineering Practicevol 13 no 2 pp 247ndash255 2005
[24] F A Shirazi K M Grigoriadis and D Viassolo ldquoWind turbineintegrated structural and LPV control design for improvedclosed-loop performancerdquo International Journal of Control vol85 no 8 pp 1178ndash1196 2012
[25] F A Inthamoussou F D Bianchi H De Battista and R JMantz ldquoLPV wind turbine control with anti-windup featurescovering the complete wind speed rangerdquo IEEE Transactions onEnergy Conversion vol 29 no 1 pp 259ndash266 2014
[26] K Z Oslashstegaard J Stoustrup and P Brath ldquoLinear parametervarying control of wind turbines covering both partial loadand full load conditionsrdquo International Journal of Robust andNonlinear Control vol 19 no 1 pp 92ndash116 2009
[27] Y Lin L Tu H Liu and W Li ldquoHybrid power transmissiontechnology in a wind turbine generation systemrdquo IEEEASMETransactions onMechatronics vol 20 no 3 pp 1218ndash1225 2015
[28] S Bououden M Chadli F Allouani and S Filali ldquoA newapproach for fuzzy predictive adaptive controller design usingparticle swarmoptimization algorithmrdquo International Journal ofInnovative Computing Information and Control vol 9 no 9 pp3741ndash3758 2013
[29] S H Li and H Gu ldquoFuzzy adaptive internal model controlschemes for PMSM speed-regulation systemrdquo IEEE Transac-tions on Industrial Informatics vol 8 no 4 pp 767ndash779 2012
[30] A B Asghar and X Liu ldquoAdaptive neuro-fuzzy algorithmto estimate effective wind speed and optimal rotor speed forvariable-speed wind turbinerdquo Neurocomputing vol 272 pp495ndash504 2018
[31] Y Wang H Shen H R Karimi and D Duan ldquoDissipativity-based Fuzzy Integral SlidingModeControl ofContinuous-TimeT-S Fuzzy Systemsrdquo IEEE Transactions on Fuzzy Systems pp 1-1
[32] S Bououden M Chadli and H R Karimi ldquoAn ant colonyoptimization-based fuzzy predictive control approach for non-linear processesrdquo Information Sciences vol 299 pp 143ndash1582015
[33] NWang S-F Su J Yin Z Zheng andM J Er ldquoGlobal Asymp-totic Model-Free Trajectory-Independent Tracking Control ofan Uncertain Marine Vehicle An Adaptive Universe-BasedFuzzy Control Approachrdquo IEEE Transactions on Fuzzy Systems2017
[34] R Saravanakumar andD Jena ldquoValidation of an integral slidingmode control for optimal control of a three blade variable speedvariable pitch wind turbinerdquo International Journal of ElectricalPower amp Energy Systems vol 69 pp 421ndash429 2015
[35] C Evangelista P Puleston F Valenciaga and L M FridmanldquoLyapunov-designed super-twisting sliding mode control forwind energy conversion optimizationrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 538ndash545 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 13
[11] Y Ren L Li J Brindley and L Jiang ldquoNonlinear PI control forvariable pitch wind turbinerdquo Control Engineering Practice 2015
[12] Y Yuan and J Tang ldquoAdaptive pitch control of wind turbinefor load mitigation under structural uncertaintiesrdquo Journal ofRenewable Energy vol 105 pp 483ndash494 2017
[13] C-S Wang and M-H Chiang ldquoA novel pitch control systemof a large wind turbine using two-degree-of-freedom motioncontrol with feedback linearization controlrdquo Energies vol 9 no10 article no 791 2016
[14] Z Civelek E Cam M Luy and H Mamur ldquoProportional-integral-derivative parameter optimisation of blade pitch con-troller in wind turbines by a new intelligent genetic algorithmrdquoIET Renewable Power Generation vol 10 no 8 pp 1220ndash12282016
[15] A Dahbi N Nait-Said andM-S Nait-Said ldquoA novel combinedMPPT-pitch angle control for wide range variable speed windturbine based on neural networkrdquo International Journal ofHydrogen Energy vol 41 no 22 pp 9427ndash9442 2016
[16] M L Corradini G Ippoliti and G Orlando ldquoAn observer-based blade-pitch controller of wind turbines in high windspeedsrdquo Control Engineering Practice vol 58 pp 186ndash192 2017
[17] M Zhu J Liu Z Lin and H Meng ldquoMixed H2 Hinfin pitchcontrol of wind turbine generator system based on globalexact linearization and regional pole placementrdquo InternationalJournal of Machine Learning and Cybernetics vol 7 no 5 pp921ndash930 2016
[18] R Gao and Z Gao ldquoPitch control for wind turbine systemsusing optimization estimation and compensationrdquo Journal ofRenewable Energy vol 91 pp 501ndash515 2016
[19] X-X Yin Y-G Lin W Li H-W Liu and Y-J Gu ldquoAdaptivesliding mode back-stepping pitch angle control of a variable-displacement pump controlled pitch system for wind turbinesrdquoISA Transactions vol 58 pp 629ndash634 2015
[20] M-H Chiang ldquoA novel pitch control system for a wind turbinedriven by a variable-speed pump-controlled hydraulic servosystemrdquoMechatronics vol 21 no 4 pp 753ndash761 2011
[21] H Camblong S Nourdine I Vechiu and G Tapia ldquoCom-parison of an island wind turbine collective and individualpitch LQG controllers designed to alleviate fatigue loadsrdquo IETRenewable Power Generation vol 6 no 4 p 267 2012
[22] F A InthamoussouHDe Battista andR JMantz ldquoLPV-basedactive power control of wind turbines covering the completewind speed rangerdquo Journal of Renewable Energy vol 99 pp996ndash1007 2016
[23] F D Bianchi R J Mantz and C F Christiansen ldquoGainscheduling control of variable-speed wind energy conversionsystems using quasi-LPVmodelsrdquoControl Engineering Practicevol 13 no 2 pp 247ndash255 2005
[24] F A Shirazi K M Grigoriadis and D Viassolo ldquoWind turbineintegrated structural and LPV control design for improvedclosed-loop performancerdquo International Journal of Control vol85 no 8 pp 1178ndash1196 2012
[25] F A Inthamoussou F D Bianchi H De Battista and R JMantz ldquoLPV wind turbine control with anti-windup featurescovering the complete wind speed rangerdquo IEEE Transactions onEnergy Conversion vol 29 no 1 pp 259ndash266 2014
[26] K Z Oslashstegaard J Stoustrup and P Brath ldquoLinear parametervarying control of wind turbines covering both partial loadand full load conditionsrdquo International Journal of Robust andNonlinear Control vol 19 no 1 pp 92ndash116 2009
[27] Y Lin L Tu H Liu and W Li ldquoHybrid power transmissiontechnology in a wind turbine generation systemrdquo IEEEASMETransactions onMechatronics vol 20 no 3 pp 1218ndash1225 2015
[28] S Bououden M Chadli F Allouani and S Filali ldquoA newapproach for fuzzy predictive adaptive controller design usingparticle swarmoptimization algorithmrdquo International Journal ofInnovative Computing Information and Control vol 9 no 9 pp3741ndash3758 2013
[29] S H Li and H Gu ldquoFuzzy adaptive internal model controlschemes for PMSM speed-regulation systemrdquo IEEE Transac-tions on Industrial Informatics vol 8 no 4 pp 767ndash779 2012
[30] A B Asghar and X Liu ldquoAdaptive neuro-fuzzy algorithmto estimate effective wind speed and optimal rotor speed forvariable-speed wind turbinerdquo Neurocomputing vol 272 pp495ndash504 2018
[31] Y Wang H Shen H R Karimi and D Duan ldquoDissipativity-based Fuzzy Integral SlidingModeControl ofContinuous-TimeT-S Fuzzy Systemsrdquo IEEE Transactions on Fuzzy Systems pp 1-1
[32] S Bououden M Chadli and H R Karimi ldquoAn ant colonyoptimization-based fuzzy predictive control approach for non-linear processesrdquo Information Sciences vol 299 pp 143ndash1582015
[33] NWang S-F Su J Yin Z Zheng andM J Er ldquoGlobal Asymp-totic Model-Free Trajectory-Independent Tracking Control ofan Uncertain Marine Vehicle An Adaptive Universe-BasedFuzzy Control Approachrdquo IEEE Transactions on Fuzzy Systems2017
[34] R Saravanakumar andD Jena ldquoValidation of an integral slidingmode control for optimal control of a three blade variable speedvariable pitch wind turbinerdquo International Journal of ElectricalPower amp Energy Systems vol 69 pp 421ndash429 2015
[35] C Evangelista P Puleston F Valenciaga and L M FridmanldquoLyapunov-designed super-twisting sliding mode control forwind energy conversion optimizationrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 538ndash545 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
