# Pitch Controller Design of Wind Turbine Based on Nonlinear PI/PD Control

Authors: Feihang Zhou; Jun Liu
Journal: Shock and Vibration
 (2018)

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## Abstract

This paper analyzed and discussed the nonlinear PI/PD control for wind energy conversion system (WECS) pitch controller design. The nonlinear PI/PD controller consists of a classical PI/PD controller and a nonlinear gain table. In this paper, the detailed design procedures of the controller including not only the selection of control parameters but also the formulation of nonlinear gain table were given. In order to verify the effectiveness and correctness of the algorithm, a detailed 2 MW simulation test platform based on MATLAB/SIMULINK environment was established. Meanwhile, some valuable conclusions were also obtained. The presented contents of this article have the reference value and guiding significance for practical engineering application.


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## Body

## 1. Introduction

With the global climate problem increasingly prominent and nonrenewable energy sources being largely consumed, the exploration and utilization of renewable energy sources such as wind energy and solar energy have been paid to a high attention. As the fastest growing renewable energy sources in the recent years [1–4], wind generation is the most prevalent in coastal regions spanning temperate and boreal climates. Countries such as China, USA, Denmark, and Canada possess significant wind power potential due to their high average wind velocities [5–8]. Hence, the research and development about wind power have much practical value nowadays. Compared with the constant speed-constant frequency power system, the greatest advantage of variable speed-constant frequency wind power system widely used is the ability to get access to the maximum energy conversion. Mainstream variable speed-constant frequency power generation models include doubly-fed induction generator (DFIG) and permanent magnet synchronous generator (PMSG). Compared with DFIG, PMSG has many superior characteristics such as more efficient performance, higher reliability, and wider speed control range and is gradually becoming the first choice [9]. Therefore, the WECS based on PMSG was selected for research in this paper.As Figure1 shows, according to the wind speed, the working region of WECS typically can be divided into two regions, namely, partial-load region that has wind speed below the rated wind speed and full-load region that has wind speed above the rated wind speed. In the partial-load region, the control goal of the wind power system is generally to capture the maximum wind energy in order to achieve the maximum economic benefit. At this time, the partial-load region can also be called the maximum power point tracking (MPPT) region. In the full-load region, the variable propeller is particularly important. At this time, the generator-output power is limited at the rated value by the pitch control since the capacity of the generator and converter is limited [10, 11]. Furthermore, pitch control is also needed in some special working conditions such as the limited power control under rated wind speed, the low-voltage crossing in power grid, power grid needs to be injected into the inertia, and lighten the wind turbine loads. Meanwhile, it is pointed out that pitch control can also smoothen the power [12]. This paper focuses on the pitch control above the rated wind speed.Figure 1
WECS operation modes in different wind speeds.Figure2 illustrates the pitch-controlled system. Obviously, Figure 2(a) depicts the blades action under pitch-controlled, and Figure 2(b) shows the variable-pitch drive system in wind turbine hub. The variable-pitch drive system consists of pitch drive cabinet, variable-pitch drive, emergency power supply, pitch bearing and variable-pitch lubrication systems. The pitch control system and communication system are in the pitch drive cabinet. In general, the pitch controller and master controller are connected by the optical fiber communication. The PROFIBUS or CANopen protocols are generally used in this communication. Usually, the master controller sends the pitch instructions to the pitch controller and gets some important information from it at the same time. When the instructions are obtained, the pitch controller starts to control the variable-pitch drive-operating system.Figure 2
Pitch-controlled system. (a) Blades action under pitch-controlled. (b) Variable-pitch drive system in wind turbine hub.
(a)
(b)At present, the research on variable-pitch control mainly includes two main categories: linear control and nonlinear control. The research of the linear controller mainly has PI/PD control,H∞ controller with a linear matrix inequality approach [13], linear quadratic Gaussian LQR control [14], and generalized predictive control (GPC) [15]. The nonlinear control mainly contains LPV control [16] and some intelligent control methods such as fuzzy control [17]. Because wind turbine is a multi-input multi-output system containing strongly nonlinear dynamics, the effect of nonlinear control is better than the linear control. Taking into account the complexity of the algorithm, the most applied controllers in engineering is still PI/PD control. Therefore, the design of the variable propeller with the nonlinear PI/PD controller considering the nonlinear characteristics of the wind turbine has become the key. So, this article takes this as the focus of the study and summarizes some valuable conclusions. And this study can also provide the necessary reference and guidance for the follow-up engineering practice.The structure of this paper is as follows: the dynamic model and structure of WECS is briefly introduced in Section2. Section 3 explains the controller design in details, including the controller parameter tuning and the calculation of nonlinear gain. In Section 4, the validation of the nonlinear PI/PD controller is surveyed using the MATLAB/Simulink simulator. Finally, Section 5 provides some useful conclusions.


## 2. Mathematical Model of WECS

WECS is mainly constitutive of wind turbine, transmission chain, permanent magnet synchronous generator, generator-side converter, DC bus, and the grid-side inverter. Its structure is shown in Figure3.Figure 3
Structure diagram of WECS.


### 2.1. Wind Turbine Model

Based on the Baez theory, the aerodynamic power,Pt and torque Tt of the wind turbine are [18–22](1)Pt=0.5πρR2CPβ,λv3,Tt=Ptωt=0.5πρR3CPβ,λv2λ,where ρ is the air density, R is the radius of wind wheel, v is the wind speed, β is the pitch angle, ωt is the speed of wind turbine, λ is the tip speed ratio (TSR), and CP is defined as the wind energy conversion coefficient. A 2 MW baseline wind turbine-power coefficient is shown in Figure 4. Obviously, CP is the function of β and λ.Figure 4
2-MW baseline wind turbine power coefficient.The tip-speed ratioλ is given by(2)λ=Rωtv.


### 2.2. Dynamic Model of PMSG

If PMSG was supposed to be an ideal motor, it should meet the following criteria:(1)
three-phase stators are symmetry.(2)
induced electromotive force is sinusoidal.(3)
there is no damping winding on the rotor(4)
magnetic saturation of iron core, vortex, and hysteresis loss can be neglected.(5)
electronic conductivity of permanent magnet material is zero.Under this assumption, the mathematical model of PMSG [23, 24] is(3)Ld00Lqi˙di˙q=−RsnpωgLq−npωgLd−Rsidiq+uduq+0npωgΨ.Electromagnetic torque is given by(4)Tg=1.5npLd−Lqidiq+Ψiq,where Ld and Lq are the d-axis and q-axis stator reluctance, Rs is the stator resistance, Ψ is the permanent flux, np is pole pairs, ωg is the speed of PMSG, id and iq are d-axis and q-axis currents, respectively, and ud and uq are d-axis and q-axis voltages, respectively.Because nonsalient PMSG meetsLd=Lq=L, the mathematical model of PMSG has coupling terms such as npωgeniq,−npωgenid+Ψ/L, according to Equation (3).Given(5)vdvq=0npωgL−npωgL0idiq+0−npωgΨ+uduq.The mathematical model of PMSG could be rewritten as(6)i˙di˙q=−RsL00−RsLidiq+1L001Lvdvq.The above formula also can be rewritten into a transfer functionG0s.(7)G0s=idsvds=iqsvqs=1Ls+Rs.Figure5 shows the inner control of current. The closed loop transfer function of the current or torque can be obtained by Figure 5. And the closed loop transfer function also could be taken as a first-order inertial system.(8)φs=ididref=iqiqref=TgTg_ref=kiPs+kiILs2+Rs+kiPs+kiI=1τTs+1,(9)τT=RskiI=LkiP,where τT is the inertial time constant and Tg_ref is the command signal of Tg.Figure 5
Inner control of current (remark: PI parameters are denoted askiP and kiI). (a) Current feedforward decoupling control. (b) First-order tuning for PI parameters.
(a)
(b)Equation (8) shows that the dynamic model of PMSG could be equivalent to a first-order inertial system by current feedforward decoupling control and first-order tuning for PI parameters. Usually, when τT≤0.01, we can leave out the inertial time delay of PMSG.


### 2.3. Dynamics of Blade Pitch Actuator

There are two kinds of drives in the blade pitch actuator at present. They are the motor drive and hydraulic drive. Compared with the hydraulic drive, motor drive has the characteristics of lower cost and higher reliability and has been widely used. Actual motor drive is shown in Figure6.Figure 6
Motor drive system in pitch actuator.The collecting variable-pitch strategy was used. According to Figure7, the blade dynamics is expressed as(10)JBladeβ¨=TDrive−μ+fβ˙.Figure 7
Blade (1, 2, 3) model (note:JBlade is the moment of blade inertia. TDrive is the motor-driving torque. Tv is the resistance torque of wind. Tf is the resistance torque of blade root friction. f and μ are the coefficient of friction of blade root and wind drag, respectively).By leaving out the inertia time delay of the motor drive, the drive torque of motor is assumed as(11)TDrive=KβDs+KβPβref−β.The blade pitch actuator dynamic can be given by Figure8 where βref is the command of β and the PD parameters of the blade pitch actuator are KβP and KβD.Figure 8
Blade pitch actuator dynamic.If the delay coefficient meets(12)τβ=μ+fKβP=JBladeKβD,the blade pitch actuator dynamic model can be simplified as(13)Gβs=βsβrefs=1τβs+1.Obviously, the inertia delay coefficient,τβ in Equations (12) and (13) can be reduced by increasing PD parameters. When KβP and KβD take the appropriate value, the inertia delay can also be left out. Actually, because the value of the parameters JBlade, μ, and f is unknown, the value of the control parameters KβP and KβD can only depend on experience.


### 2.4. Dynamic Model of Transmission Chain

Figure9 shows the generator cabin internal components of high-speed permanent magnet wind turbine. It is clear that its transmission chain mainly consists of spindle (or low-speed shaft), gearbox, and high-speed shaft. The dynamics of the transmission chain is characterized by Figure 10.Figure 9
Internal components of generator cabin.Figure 10
Dynamic model of transmission chain (remark: Sun and Sate represent the sun wheel and planetary gear of gearbox).According to Figure10, the dynamics of the transmission chain is derived as(14)Jtω˙t=Tt−Dtωt−TSun,Jgω˙g=TSate−Dgωg−Tg.The gearbox ratio is defined as(15)n=ωgωt=TSunTSate.From Equations (14) and (15), the dynamic model of the transmission chain can be simplified as(16)Jeqω˙t=Tt−Deqωt−nTg,where(17)Jeq=Jt+n2Jg,Deq=Dt+n2Dg.


## 2.1. Wind Turbine Model

Based on the Baez theory, the aerodynamic power,Pt and torque Tt of the wind turbine are [18–22](1)Pt=0.5πρR2CPβ,λv3,Tt=Ptωt=0.5πρR3CPβ,λv2λ,where ρ is the air density, R is the radius of wind wheel, v is the wind speed, β is the pitch angle, ωt is the speed of wind turbine, λ is the tip speed ratio (TSR), and CP is defined as the wind energy conversion coefficient. A 2 MW baseline wind turbine-power coefficient is shown in Figure 4. Obviously, CP is the function of β and λ.Figure 4
2-MW baseline wind turbine power coefficient.The tip-speed ratioλ is given by(2)λ=Rωtv.


## 2.2. Dynamic Model of PMSG

If PMSG was supposed to be an ideal motor, it should meet the following criteria:(1)
three-phase stators are symmetry.(2)
induced electromotive force is sinusoidal.(3)
there is no damping winding on the rotor(4)
magnetic saturation of iron core, vortex, and hysteresis loss can be neglected.(5)
electronic conductivity of permanent magnet material is zero.Under this assumption, the mathematical model of PMSG [23, 24] is(3)Ld00Lqi˙di˙q=−RsnpωgLq−npωgLd−Rsidiq+uduq+0npωgΨ.Electromagnetic torque is given by(4)Tg=1.5npLd−Lqidiq+Ψiq,where Ld and Lq are the d-axis and q-axis stator reluctance, Rs is the stator resistance, Ψ is the permanent flux, np is pole pairs, ωg is the speed of PMSG, id and iq are d-axis and q-axis currents, respectively, and ud and uq are d-axis and q-axis voltages, respectively.Because nonsalient PMSG meetsLd=Lq=L, the mathematical model of PMSG has coupling terms such as npωgeniq,−npωgenid+Ψ/L, according to Equation (3).Given(5)vdvq=0npωgL−npωgL0idiq+0−npωgΨ+uduq.The mathematical model of PMSG could be rewritten as(6)i˙di˙q=−RsL00−RsLidiq+1L001Lvdvq.The above formula also can be rewritten into a transfer functionG0s.(7)G0s=idsvds=iqsvqs=1Ls+Rs.Figure5 shows the inner control of current. The closed loop transfer function of the current or torque can be obtained by Figure 5. And the closed loop transfer function also could be taken as a first-order inertial system.(8)φs=ididref=iqiqref=TgTg_ref=kiPs+kiILs2+Rs+kiPs+kiI=1τTs+1,(9)τT=RskiI=LkiP,where τT is the inertial time constant and Tg_ref is the command signal of Tg.Figure 5
Inner control of current (remark: PI parameters are denoted askiP and kiI). (a) Current feedforward decoupling control. (b) First-order tuning for PI parameters.
(a)
(b)Equation (8) shows that the dynamic model of PMSG could be equivalent to a first-order inertial system by current feedforward decoupling control and first-order tuning for PI parameters. Usually, when τT≤0.01, we can leave out the inertial time delay of PMSG.


## 2.3. Dynamics of Blade Pitch Actuator

There are two kinds of drives in the blade pitch actuator at present. They are the motor drive and hydraulic drive. Compared with the hydraulic drive, motor drive has the characteristics of lower cost and higher reliability and has been widely used. Actual motor drive is shown in Figure6.Figure 6
Motor drive system in pitch actuator.The collecting variable-pitch strategy was used. According to Figure7, the blade dynamics is expressed as(10)JBladeβ¨=TDrive−μ+fβ˙.Figure 7
Blade (1, 2, 3) model (note:JBlade is the moment of blade inertia. TDrive is the motor-driving torque. Tv is the resistance torque of wind. Tf is the resistance torque of blade root friction. f and μ are the coefficient of friction of blade root and wind drag, respectively).By leaving out the inertia time delay of the motor drive, the drive torque of motor is assumed as(11)TDrive=KβDs+KβPβref−β.The blade pitch actuator dynamic can be given by Figure8 where βref is the command of β and the PD parameters of the blade pitch actuator are KβP and KβD.Figure 8
Blade pitch actuator dynamic.If the delay coefficient meets(12)τβ=μ+fKβP=JBladeKβD,the blade pitch actuator dynamic model can be simplified as(13)Gβs=βsβrefs=1τβs+1.Obviously, the inertia delay coefficient,τβ in Equations (12) and (13) can be reduced by increasing PD parameters. When KβP and KβD take the appropriate value, the inertia delay can also be left out. Actually, because the value of the parameters JBlade, μ, and f is unknown, the value of the control parameters KβP and KβD can only depend on experience.


## 2.4. Dynamic Model of Transmission Chain

Figure9 shows the generator cabin internal components of high-speed permanent magnet wind turbine. It is clear that its transmission chain mainly consists of spindle (or low-speed shaft), gearbox, and high-speed shaft. The dynamics of the transmission chain is characterized by Figure 10.Figure 9
Internal components of generator cabin.Figure 10
Dynamic model of transmission chain (remark: Sun and Sate represent the sun wheel and planetary gear of gearbox).According to Figure10, the dynamics of the transmission chain is derived as(14)Jtω˙t=Tt−Dtωt−TSun,Jgω˙g=TSate−Dgωg−Tg.The gearbox ratio is defined as(15)n=ωgωt=TSunTSate.From Equations (14) and (15), the dynamic model of the transmission chain can be simplified as(16)Jeqω˙t=Tt−Deqωt−nTg,where(17)Jeq=Jt+n2Jg,Deq=Dt+n2Dg.


## 3. Design of Controller

When the wind speedv is above the rated wind vrate, the system needs pitch control to unload the power and enable the speed ωt to approach the rated speed ωrate. If the inertia delay τT and τβ are neglected by adjusting control parameters, the electromagnetic torque Tg and pitch angle β can be approximated as their command values Tg_ref and βref.(18)Tg≈Tg_ref=Prateωg=Pratenωt,β≈βref=kDs+kP+kIsωt−ωrate,where s is the Laplace factor.By linearizing Equations (1) and (18), the small signal values are(19)T˜t=aβ˜+bω˜t+cv˜+oβ˜,ω˜t,v˜,T˜g≈−Prateω¯g2ω˜g=−Pratenω¯t2ω˜t,β˜≈kDs+kP+kIsω˜t,where(20)a=∂Tt∂βop=ρπR2v¯32ω¯t⋅∂CP∂βop,b=∂Tt∂ωtop=ρπR3v¯22ω¯t∂CP∂λop−C¯Pλ¯,c=∂Tt∂vop=ρπR3v¯23C¯Pλ¯−∂CP∂λop,Superscripts ∼ and — refer to the small signal value and mean value, respectively. o is high-order infinitesimal.Meanwhile, Equation (16) can also be rewritten as(21)Jeqω¯˙t+ω˜˙t=T¯t+T˜t−Deqω¯t+ω˜t−nT¯g+T˜g.By eliminating the mean value of the upper equation and Equations (19) and (22) can be obtained.(22)Jeq−akDs2+Deq−akP−b−Prate/ω¯t2s−akIsω˜t=cv˜.Due toa<0, kD can increase the inertia of the system. In general, kD and kI are not used at the same time because they have the opposite effect on the system, namely, either PI control or PD control can be used in pitch control.


### 3.1. Design of Nonlinear PI Controller

WhenkD=0, Equation (22) could be rewritten as(23)dsω˜t=cJeqsv˜,ds=s2+Deq−b−Prate/ω¯t2−akPJeqs+−akIJeq.By comparingds with the typical second-order system, we get(24)ds=s2+2ζωn+ωn2.Equation (22) can be obtained.(25)ωn=−akIJeq,ζ=Deq−b−Prate/ω¯t2−akP2−akIJeq.Figure11 shows the torque characteristics of WECS. Obviously, there are two equilibrium points A and B in the system. For equilibrium point B, if there is a perturbation increasing the rotor speed, then PMSG torque will be over wind turbine torque. After the perturbation is eliminated, rotor speed can decrease. By the same method, we can also confirm the system will be back to point B by the assumption that a little perturbation reduced the rotor speed. Hence, the working point B is stable. It could be confirmed that the equilibrium point A is unstable by this perturbation observation method. Therefore, only the point B is a stable equilibrium point, and the range of stability is ωt≥ωA. The slope of wind turbine torque characteristic, Tt is b and the slope of PMSG torque characteristic, nTg is −Prate/ω¯t2. When the system works at the point B, we can get Equation (26) by Figure 11.(26)0>−Prateω¯t2>b.Figure 11
Torque characteristics of WECS.Generally, the dampingDeq is unknown. From Equation (26), the damping ratio ζ in Equation (25) meets(27)ζ>−akP2−akIJeq.Given−akP/2−akIJeq=ε≥1 and(28)−akP=KP,−akI=KI,the damping ratio ζ and PI parameters will meet ζ>ε and KP2=4KIJeqε2. Therefore, the parameters KP and KI are given by(29)KP=2Jeqωnε,KI=Jeqωn2.By the assumption ofKP=ηIKI, Equation (29) could be rewritten as(30)ωn=2εηI,KP=4Jeqε2ηI,KI=4Jeqε2ηI2.


### 3.2. Design of Nonlinear PD Controller

Inertia has the ability to prevent frequency mutation in the system. Low frequency oscillation and rotor oscillation of the system can be suppressed by increasing inertia. Therefore, PD control-providing the virtual inertia for the system, is often applied to the small inertia systems and situations that need to be injected into inertia.Similarly, whenkI=0, Equation (22) also could be rewritten as(31)Jeq−akDs+Deq−b−Prateω¯t2−akPω˜t=cv˜.IfDeq−b−Prate/ω¯t2 in Equation (31) is neglected, the transfer function Gvω is given by(32)Gvωs=v˜sω˜s≈c−akPG0s,G0s=1Jeq−akD/−akPs+1.Given(33)−akP=KP,−akD=KD,and KD=ηDJeq, the parameter KP is(34)KP=Jeq+KDBW=1+ηDJeqBW,where BW is the bandwidth of G0s.In general, we keepKD and KP (or KI and KP) fixed and make −1/a as a nonlinear gain table. Meanwhile, the limit of the variable pitch rate and angle is taken into account, and pitch control based on nonlinear PI or PD controller is shown in Figure 12.Figure 12
Pitch control based on nonlinear PI or PD controller.


### 3.3. Nonlinear Gain Table

Obviously, all of the wind speedv¯, rotor speed of wind turbine ω¯t, and pitch angle β¯ can have effect on the nonlinear gain a by Equation (20). But, the sensitivity of the system to the pitch angle is far greater than other factors generally. When the system works at the rated working point ω¯t=ωrate and T¯t=Trate, the relationship between v¯ and β¯ is(35)πρR2CPβ¯,Rv¯/ωratev¯32ωrate=Trate.In general, the nonlinear gaina only depends on the pitch angle β¯. The calculation process of nonlinear gain is shown in Figure 13, and the parameters of WECS are shown in Table 1.Figure 13
The calculation process of nonlinear gaina (remark: Δβ is the precision value of β, and the initial value of Δβ is 0.01°).Table 1
WECS parameters.
Parameters
Value
Equivalent moment of inertiaJeq (kg·m2)
2 × 104
Equivalent damping coefficientDeq
100
Blade lengthR (m)
28
Gear box ration
100
Air densityρ (kg/m3)
1.225
Cut-out wind speedvOut (m/s)
25
Inertia delay coefficient of blade pitch actuatorτβ
0.01
Inertia delay coefficient of torque controlτT
0.01
Rated powerPrate (MW)
2
Rated torqueTrate (kN·m)
103
Rate rotor speedωrate (rad/s)
2
Rated wind speedvrate (m/s)
12
Generator pole logarithm
3
Permanent flux (Wb)
1.25
Maximum power coefficientCP_max
0.48In practice, the functionCPβ,λ can only be obtained by data fitting, and these data for fitting are generated by Bladed Software. In order to verify the correctness of the algorithm shown in Figure 8, an empirical formula of CP is used.(36)CPβ,λ=0.5176116λm−0.4β−5e−21/λm+0.0068λ,1λm=1λ+0.08β−0.035β3+1.The partial derivative of the upper formula is(37)∂CP∂β=0.5176e−21/λm116∂1/λm∂β−0.4−21∂1/λm∂β116λm−0.4β−5,∂1/λm∂β=−0.08λ+0.08β2+0.105β2β3+12.The truth value ofa could be calculated by Equations (20) and (37). Table 2 shows the calculated and truth values of nonlinear gain at different wind speed. From Table 2, the calculated value of a is very close to the truth value of a. Therefore, the algorithm shown in Figure 13 is valid. The minimum of β is 2.21° above the rated wind, and the relationship between −1/a and β is shown in Figure 14,. Nonlinear gain −1/a is(38)−1aβ=9.37×10−6β+5.79×10−7,2.21°≤β<4.2°,3.50×10−8β2−2.54×10−6β+4.99×10−5,4.2°≤β≤33.298°.Table 2
Nonlinear gaina.
Wind speedv¯ (m/s)
Pitch angleβ¯ (°)
Nonlinear gaina1
Nonlinear gaina2
Calculated value ofa
Truth value ofa
11
0.769
−131258
−185437
−158348
−158348
12
2.21
−50628
−52895
−51761
−51760
12.1
2.53
−25142
−54975
−40058
−40057
12.2
2.94
−25702
−38593
−32147
−32147
12.3
3.43
−24971
−30670
−27821
−27821
12.4
3.98
−35384
−16564
−25974
−25974
12.5
4.55
−28240
−22884
−25562
−25562
13
7.3495
−29605
−29670
−29638
−29638
14
12.0014
−24736
−58744
−41740
−41740
15
15.6949
−44415
−65296
−54856
−54856
16
18.7385
−87893
−50143
−69018
−69018
17
21.309
−94964
−73747
−84355
−84355
18
23.518
−97211
−104702
−100956
−100956
19
25.44
−91846
−145894
−118870
−118870
20
27.136
−90789
−185643
−138216
−138216
21
28.64
−143466
−174484
−158975
−158975
22
29.9874
−230630
−131854
−181242
−181242
23
31.1999
−148312
−261735
−205023
−205023
24
32.2981
−244074
−216678
−230376
−230376
25
33.298
−357896
−156795
−257346
−257346Figure 14
Fitting curve and data.


## 3.1. Design of Nonlinear PI Controller

WhenkD=0, Equation (22) could be rewritten as(23)dsω˜t=cJeqsv˜,ds=s2+Deq−b−Prate/ω¯t2−akPJeqs+−akIJeq.By comparingds with the typical second-order system, we get(24)ds=s2+2ζωn+ωn2.Equation (22) can be obtained.(25)ωn=−akIJeq,ζ=Deq−b−Prate/ω¯t2−akP2−akIJeq.Figure11 shows the torque characteristics of WECS. Obviously, there are two equilibrium points A and B in the system. For equilibrium point B, if there is a perturbation increasing the rotor speed, then PMSG torque will be over wind turbine torque. After the perturbation is eliminated, rotor speed can decrease. By the same method, we can also confirm the system will be back to point B by the assumption that a little perturbation reduced the rotor speed. Hence, the working point B is stable. It could be confirmed that the equilibrium point A is unstable by this perturbation observation method. Therefore, only the point B is a stable equilibrium point, and the range of stability is ωt≥ωA. The slope of wind turbine torque characteristic, Tt is b and the slope of PMSG torque characteristic, nTg is −Prate/ω¯t2. When the system works at the point B, we can get Equation (26) by Figure 11.(26)0>−Prateω¯t2>b.Figure 11
Torque characteristics of WECS.Generally, the dampingDeq is unknown. From Equation (26), the damping ratio ζ in Equation (25) meets(27)ζ>−akP2−akIJeq.Given−akP/2−akIJeq=ε≥1 and(28)−akP=KP,−akI=KI,the damping ratio ζ and PI parameters will meet ζ>ε and KP2=4KIJeqε2. Therefore, the parameters KP and KI are given by(29)KP=2Jeqωnε,KI=Jeqωn2.By the assumption ofKP=ηIKI, Equation (29) could be rewritten as(30)ωn=2εηI,KP=4Jeqε2ηI,KI=4Jeqε2ηI2.


## 3.2. Design of Nonlinear PD Controller

Inertia has the ability to prevent frequency mutation in the system. Low frequency oscillation and rotor oscillation of the system can be suppressed by increasing inertia. Therefore, PD control-providing the virtual inertia for the system, is often applied to the small inertia systems and situations that need to be injected into inertia.Similarly, whenkI=0, Equation (22) also could be rewritten as(31)Jeq−akDs+Deq−b−Prateω¯t2−akPω˜t=cv˜.IfDeq−b−Prate/ω¯t2 in Equation (31) is neglected, the transfer function Gvω is given by(32)Gvωs=v˜sω˜s≈c−akPG0s,G0s=1Jeq−akD/−akPs+1.Given(33)−akP=KP,−akD=KD,and KD=ηDJeq, the parameter KP is(34)KP=Jeq+KDBW=1+ηDJeqBW,where BW is the bandwidth of G0s.In general, we keepKD and KP (or KI and KP) fixed and make −1/a as a nonlinear gain table. Meanwhile, the limit of the variable pitch rate and angle is taken into account, and pitch control based on nonlinear PI or PD controller is shown in Figure 12.Figure 12
Pitch control based on nonlinear PI or PD controller.


## 3.3. Nonlinear Gain Table

Obviously, all of the wind speedv¯, rotor speed of wind turbine ω¯t, and pitch angle β¯ can have effect on the nonlinear gain a by Equation (20). But, the sensitivity of the system to the pitch angle is far greater than other factors generally. When the system works at the rated working point ω¯t=ωrate and T¯t=Trate, the relationship between v¯ and β¯ is(35)πρR2CPβ¯,Rv¯/ωratev¯32ωrate=Trate.In general, the nonlinear gaina only depends on the pitch angle β¯. The calculation process of nonlinear gain is shown in Figure 13, and the parameters of WECS are shown in Table 1.Figure 13
The calculation process of nonlinear gaina (remark: Δβ is the precision value of β, and the initial value of Δβ is 0.01°).Table 1
WECS parameters.
Parameters
Value
Equivalent moment of inertiaJeq (kg·m2)
2 × 104
Equivalent damping coefficientDeq
100
Blade lengthR (m)
28
Gear box ration
100
Air densityρ (kg/m3)
1.225
Cut-out wind speedvOut (m/s)
25
Inertia delay coefficient of blade pitch actuatorτβ
0.01
Inertia delay coefficient of torque controlτT
0.01
Rated powerPrate (MW)
2
Rated torqueTrate (kN·m)
103
Rate rotor speedωrate (rad/s)
2
Rated wind speedvrate (m/s)
12
Generator pole logarithm
3
Permanent flux (Wb)
1.25
Maximum power coefficientCP_max
0.48In practice, the functionCPβ,λ can only be obtained by data fitting, and these data for fitting are generated by Bladed Software. In order to verify the correctness of the algorithm shown in Figure 8, an empirical formula of CP is used.(36)CPβ,λ=0.5176116λm−0.4β−5e−21/λm+0.0068λ,1λm=1λ+0.08β−0.035β3+1.The partial derivative of the upper formula is(37)∂CP∂β=0.5176e−21/λm116∂1/λm∂β−0.4−21∂1/λm∂β116λm−0.4β−5,∂1/λm∂β=−0.08λ+0.08β2+0.105β2β3+12.The truth value ofa could be calculated by Equations (20) and (37). Table 2 shows the calculated and truth values of nonlinear gain at different wind speed. From Table 2, the calculated value of a is very close to the truth value of a. Therefore, the algorithm shown in Figure 13 is valid. The minimum of β is 2.21° above the rated wind, and the relationship between −1/a and β is shown in Figure 14,. Nonlinear gain −1/a is(38)−1aβ=9.37×10−6β+5.79×10−7,2.21°≤β<4.2°,3.50×10−8β2−2.54×10−6β+4.99×10−5,4.2°≤β≤33.298°.Table 2
Nonlinear gaina.
Wind speedv¯ (m/s)
Pitch angleβ¯ (°)
Nonlinear gaina1
Nonlinear gaina2
Calculated value ofa
Truth value ofa
11
0.769
−131258
−185437
−158348
−158348
12
2.21
−50628
−52895
−51761
−51760
12.1
2.53
−25142
−54975
−40058
−40057
12.2
2.94
−25702
−38593
−32147
−32147
12.3
3.43
−24971
−30670
−27821
−27821
12.4
3.98
−35384
−16564
−25974
−25974
12.5
4.55
−28240
−22884
−25562
−25562
13
7.3495
−29605
−29670
−29638
−29638
14
12.0014
−24736
−58744
−41740
−41740
15
15.6949
−44415
−65296
−54856
−54856
16
18.7385
−87893
−50143
−69018
−69018
17
21.309
−94964
−73747
−84355
−84355
18
23.518
−97211
−104702
−100956
−100956
19
25.44
−91846
−145894
−118870
−118870
20
27.136
−90789
−185643
−138216
−138216
21
28.64
−143466
−174484
−158975
−158975
22
29.9874
−230630
−131854
−181242
−181242
23
31.1999
−148312
−261735
−205023
−205023
24
32.2981
−244074
−216678
−230376
−230376
25
33.298
−357896
−156795
−257346
−257346Figure 14
Fitting curve and data.


## 4. Validation of Algorithm by Simulation

In this section, simulations are carried out with MATLAB/Simulink environment. And the block diagram of WECS is shown in Figure15. In order to verify the correctness and effectiveness of nonlinear PI and PD pitch control, a detailed 2 MW simulation test platform of WECS was constructed. The simulation parameters of the system are shown in Table 1, and the function CPβ,λ is the empirical formula.Figure 15
Block diagram of WECS.The step-change wind shown in Figure16(a) is used to simulate the actual wind condition. And the response of PI control is shown in Figure 17. Given ηI=ε=2×103, the PI parameters KP and KI were calculated by Equation (30). Meanwhile, simulation results of PD control under step-change wind shown in Figure 16(b) are illustrated in Figure 18. Given ηD=0.7 and BW=50, the PD parameters KP and KD were calculated by Equation (34).Figure 16
Step-change wind speed.
(a)
(b)Figure 17
Responses under PI control. (a) Electromagnetic power curve. (b) Generator rotor speed curve. (c) Pitch angle curve. (d) Generator stator three-phase current under PI nonlinear gain.
(a)
(b)
(c)
(d)Figure 18
Responses under PD control. (a) Electromagnetic power curve. (b) Generator rotor speed curve. (c) Pitch angle curve. (d) Generator stator three-phase current under PI nonlinear gain.
(a)
(b)
(c)
(d)Figures17(a) and 18(a) show the electromagnetic power curves. It is clear that the output power of WECS is maintained at 2 MW, and the power fluctuation with nonlinear gain control is obviously less than the one with fixed parameters control. The generator rotor speed curves are shown in Figures 17(b) and 18(b). Obviously, the speed steady-state errors of PI control is less than the speed steady-state errors of PD control. Figures 17(c) and 18(c) describe the pitch angle curve. Compared with the PI control, the actual value of the pitch angle under PD control failed to achieve the value in Table 2. However, the pitch control is generally used as a coarse control. Therefore, the small pitch angle errors and rotor speed errors of PD control are acceptable. By Figures 17(d) and 18(d), the stator three-phase currents could be observed. The simulation results also indicate that the power of WECS is more sensitive to pitch control than the rotor speed, when the wind speed changes.In Figure19(a), the wind speed decays at step change. Under this wind speed, the responses of WECS are shown in Figures 19(b)–19(f). Meanwhile, simulation results under gradual change wind speed shown in Figure 20(a) are given by Figures 20(b)–20(f). Different from other responses curves such as power, torque, and rotor speed, the changes in the pitch angle are not smooth. This is because the rate of pitch angle change is limited at ±7°/s.Figure 19
Responses under step-change wind speed. (a) Wind speed curve. (b) Electromagnetic power curve. (c) Electromagnetic torque curve. (d) Generator rotor speed curve. (e) Pitch angle curve. (f) Wind turbine power coefficient curve.
(a)
(b)
(c)
(d)
(e)
(f)Figure 20
Responses under gradual change wind speed. (a) Wind speed curve. (b) Electromagnetic power curve. (c) Electromagnetic torque curve. (d) Generator rotor speed curve. (e) Pitch angle curve. (f) Wind turbine power coefficient curve.
(a)
(b)
(c)
(d)
(e)
(f)By thesis comparisons among classical PI control algorithm, classical PD control algorithm, nonlinear PI control algorithm, and nonlinear PD control algorithm, the similar conclusions could be summed up. Compared with the PD control, PI control has high tracking accuracy and the pitch angle has changed into the right place in Table2. Under the nonlinear PI/PD control, the output power fluctuate is small. And the output power is very sensitive to the change in pitch angle under different wind conditions. Furthermore, the merits of nonlinear PI/PD control algorithms have been clearly revealed in Figures 19 and 20.


## 5. Conclusion

In this research, the design procedure of a pitch controller with nonlinear PI/PD control is given in detail, including the selection of PI/PD parameters and the calculation of nonlinear gain. After that, a simulation test platform for a two-megawatt WECS was built. Finally, the effectiveness and correctness of the algorithm was verified by this test platform, and some useful conclusions were summed up:(i)
The output power of WECS is the most sensitive to the change in the pitch angle.(ii)
Compared with the fixed parameters control, the nonlinear control with nonlinear gain can make the power fluctuate smaller.(iii)
The precision of speed tracking is high, and the pitch angle can change into the right place under the PI control, compared with the PD control.



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*Source: 7859510-2018-10-11.xml*