A novel adaptive control scheme for dynamic performance improvement of DFIG-Based wind turbines

Energy 38 (2012) 104e117

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A novel adaptive control scheme for dynamic performance improvement of DFIG-Based wind turbines Zhanfeng Song a, Tingna Shi a, *, Changliang Xia a, b, Wei Chen a a b

Department of Electrical Engineering & Automation, Tianjin University, Weijin Road 92, Tianjin 300072, China Department of Electrical Engineering & Automation, Polytechnic University, Tianjin 300160, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 March 2011 Received in revised form 20 December 2011 Accepted 22 December 2011 Available online 25 January 2012

A novel adaptive current controller for DFIG-based wind turbines is introduced in this paper. The attractiveness of the proposed strategy results from its ability to actively estimate and actively compensate for the plant dynamics and external disturbances in real time. Thus, the control strategy can successfully drive the rotor current to track the reference value, ensuring that the performance degradation caused by grid disturbances, cross-coupling terms and parameter uncertainties can be successfully suppressed. Besides, the two-parameter tuning feature makes the control strategy practical and easy to implement in commercial wind turbines. To quantify the controller performances, the transfer function description of the controller is derived. General disturbance rejection, robustness against parameter uncertainties, bandwidth and stability are also addressed. Simulation results, together with the time-domain responses, proved the stability and the strong robustness of the control system against parameter uncertainties and grid disturbances. Significant tracking and disturbance rejection performances are achieved. Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved.

Keywords: Doubly-fed induction generator Adaptive disturbance rejection control Current control Extended state observer

1. Introduction Wind energy today is becoming a global business [1e6]. This leads to increasing wind power penetration. Accordingly, wind turbine modeling and system control become research focus, resulting in rapid development of wind turbine technology [7e16]. Recently, the concern about the stability and reliability of power systems results in continuous reformulation of the grid connection requirements for wind turbines. This demands high-performance of connected plants with respect to power control ability [17e20], as well as the capability of remaining connected and providing necessary support to the grid during grid faults [21], leading industry to explore new high-performance control strategies. Wind turbines based on DFIGs (doubly-fed induction generators) with converters rated at around 30% of the generator rating dominate the world market. Compared with the wind turbines using fixed speed induction generators or fully-fed synchronous generators with full-sized converters, DFIG-based wind turbines offer not only the advantages of variable-speed operation and fourquadrant active and reactive power capabilities, but also lower converter cost and power losses [22]. Thus, design and control of DFIG-based wind turbines gain attentive focus. * Corresponding author. Tel./fax: þ86 22 27402325. E-mail address: [email protected] (T. Shi).

Dealing with control of DFIG-based wind turbines, a common way is to control the rotor current with stator flux orientation [23], or with stator voltage orientation [24], or with air-gap-flux orientation [25]. However, this technique is designed assuming the stator voltage to be ideal and the derivative of stator flux neglectable [26]. During grid disturbances which mean stator flux is not constant, conventional control methods usually cannot react directly and fast to reject these disturbances, although these control methods can finally suppress them through feedback regulation in a relatively slow way. This results in a degradation of system performance during grid disturbances. Moreover, the effect of the controller is dependent on accuracy information on system parameters, especially the rotor time constant. However, DFIG machine parameters are affected by temperature, saturation, and skin effects, which can deteriorate the controller performance when designed with nominal parameter values. Besides, DFIGbased wind turbines operate in a wide operating range during its whole life. Therefore, another concern regarding DFIG rotor current control is the robustness against variations in plant parameters. If the feed-forward terms do not entirely cancel the extra dynamics, the closed-loop poles are slower than intended and the damping factor deteriorates. The same effect was reported in [27], pointing out that variations of the system parameters compromise both the decoupling and the desired pole-zero cancellation.

0360-5442/$ e see front matter Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2011.12.029

Z. Song et al. / Energy 38 (2012) 104e117

Recently, with the rapid progress in power electronics, digital signal processors and modern control theories, nonlinear control methods gain attentive focus, and various algorithms have been proposed [28]. These algorithms have improved the control performance of DFIG systems from different aspects. In this paper, an adaptive disturbance rejection controller (ADRC) is developed for the rotor current control of DFIG-based wind turbines in which the grid disturbances, parameter variations, coupling terms and flux compensation are considered and compensated in real time. ADRC is a novel and unique design concept, proposed by Han [29] and further developed by Gao [30]. This method has achieved much success in some industrial control problems, e.g., electro-mechanical systems [31], machining processes [32], flexible-joint systems [33], robotic systems [34], manipulator systems [35], motor control systems [36,37] and power converters [38]. The principle of proposed control strategy for the DFIG-based wind turbines uses an extended state observer (ESO) to build a solid base for better performance and disturbances compensation via the provision of output and the real action component of internal dynamics and external disturbances. With accurate estimation of DFIG internal dynamics and external disturbances, the control strategy can successfully drive the rotor current to track the reference value, ensuring that the performance degradation caused by the grid disturbances and parameter uncertainties can be suppressed. More importantly, the two-parameter tuning feature makes the control strategy practical and easy to implement in commercial wind turbines. This paper is organized as follows. The dynamics of DFIG-based wind turbines is explained in Section 2. The control problems are described in Section 3. The proposed strategy design and its stability analysis are presented in Section 4 and Section 5, respectively. Performance tests, discussion, and results are shown in Section 6. Finally, concluding remarks are made in Section 7.

2. DFIG-based wind turbine system dynamics and modeling A DFIG-based wind turbine consists of a pitch controlled wind turbine and a DFIG. The DFIG is a wound-rotor induction machine with the stator windings directly connected to the three-phase grid and with the rotor windings connected to a partial scale frequency converter [1]. Direct control of the rotor currents allows for variable-speed operation and reactive power control. Consequently, DFIG-based wind turbines can operate at a higher efficiency over a wide range of wind speeds and help provide voltage support for the grid.

105

urd ¼ Lm

disd di  ðus  ur Þ Lm isq þ Rr ird þ Lr rd  ðus  ur ÞLr irq dt dt

(3)

urq ¼ Lm

disq dirq þ ðus  ur Þ Lm isd þ Rr irq þ Lr þ ðus  ur ÞLr ird dt dt

(4)

where Ls and Lr are the stator and rotor windings self-inductance, Lm is the mutual inductance between the stator and the rotor, us is the synchronous angular frequency and ur is the generator angular frequency. This model is able to represent rotor and stator transients correctly. The electromagnetic torque can be written in terms of stator flux linkages and currents as


 Te ¼ 1:5p jsd isq  jsq isd

(5)

where p represents the number of pole pairs, and jsd, jsq denote the d- and q-axis flux component, respectively. The stator reactive power expression, which is also the control objective of the rotor-side converter control as well as the electromagnetic torque, has the following form

  Qs ¼ 1:5 usq isd  usd isq

(6)

The equivalent circuit of DFIG is shown in Fig. 1. 2.2. Wind turbine and mechanical drive train model The turbine is made up of the three-bladed rotor and the hub. Through the turbine, wind energy is transformed into mechanical energy that turns the main shaft of the generator. The aerodynamic torque Tm captured by the wind turbine is given by

prr2 Cp v3 2 ut

Tm ¼

(7)

where r is the air density, r the radius of the turbine disk, v the wind speed, ut the turbine angular frequency and Cp represents the wind turbine power coefficient. The power coefficient Cp is a function of the tip speed ratio l as well as the blade pitch angle b in a DFIG-based wind turbine. The accurate computation of the power coefficient requires the use of Blade-Element Theory and the knowledge of the blade geometry. These complex issues are normally empirically considered. In this paper, the power coefficient is given by

Cp ðl; bÞ ¼ 0:22

 116

li

 12:5  0:4b  5 e li

(8)

where 2.1. Generator model

1

Dealing with the modeling of DFIG is not a trivial task and different levels of detail may be achieved doing some assumptions. In this work, it is assumed the stator and rotor windings to be placed sinusoidally and symmetrical, the magnetical saturation effects and the capacitance of all the windings neglectable. Using the motor convention, the relation between the voltages on the machine windings and the currents and its first derivative may be written in terms of a synchronous reference dq frame representation as

usd ¼ Rs isd þ Ls

disd di  us Ls isq þ Lm rd  us Lm irq dt dt

disq dirq usq ¼ Rs isq þ Ls $ þ us Ls isd þ Lm þ us Lm ird dt dt

(1) (2)

li

¼

1

l þ 0:08b



0:035

b3 þ 1

(9)

The tip speed ratio l is defined as

l ¼

um r V

(10)

The mechanical drive train of a DFIG-based wind turbine comprises the turbine shaft, the gearbox, and the generator’s rotor shaft. A common way to model the mechanical drive train is to treat it as a series of equivalent discrete masses connected together by springs and dampers with a multiplication ratio between them. When applications are limited to the impact of wind fluctuations, it is usually sufficient to consider the mechanical drive train as a single-mass shaft model because shaft oscillations of the wind turbines are not reflected to the grid due to the fast active power

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Z. Song et al. / Energy 38 (2012) 104e117

Jt

dut ¼ Tm  kqt þ cðut  ur Þ dt

(13)

where qt is the shaft twist angle. Variables and parameters of the low-speed shaft are referred to the high-speed side.

2.3. Back-to-back converter model

Fig. 1. DFIG equivalent circuits. (a) d-axis equivalent circuit. (b) q-axis equivalent circuit.

control. However, when the system response to grid disturbances is analyzed, the mechanical drive train should be approximated by at least a two-mass model [39]. This model treats the wind turbine as one inertia Jt and the gearbox and the generator’s rotor as the other inertia Jg connected through the elastic turbine shaft with an angular stiffness coefficient k and an angular damping coefficient c. Applying the Newton’s Laws, the dynamics of the resulting system can be described as

Jg

dur ¼ kqt þ cðut  ur Þ  Te dt

(11)

dqt ¼ ut  ur dt

(12)

A three-phase ACeAC converter is normally used to connect the rotor circuit of the DFIG to the grid, whereas the stator circuit is connected to the grid directly. The converter must be able to transfer energy in both directions, i.e. it must be able to work as a rectifier and as an inverter. When the generator operates in subsynchronous mode the converter will transfer energy to the rotor, while it is extracting energy from the rotor when the DFIG operates in super-synchronous mode. As graphically represented in Fig. 2, the most common topology of the converter of DFIG-based wind turbines is the IGBT voltage source back-to-back converter, consisting of a rotor-side converter, an intermediary DC-link capacitor and a grid-side converter in a series connection. The aim of the rotor-side converter is to control independently the active and reactive power on the grid, while the grid-side converter has to keep the DC-link capacitor voltage at a set value regardless of the magnitude and the direction of the rotor power and to guarantee a converter operation with unity power factor. The DC-link capacitor provides an intermediate energy storage, which decouples the rotor-side converter and the grid-side converter. The transfer of the active power from the generator into the grid can be realized via this DC link, only when the DC-link voltage is kept constant. Finally, in Fig. 2, a protection device called crowbar, is inserted between the rotor slip rings and the converter. Without this crowbar, the concern in DFIG is usually the fact that large grid disturbances lead to large fault currents in the stator due to the direct connection of its stator to the grid. Because of the magnetic coupling between the stator and the rotor and of the laws of flux conservation, the stator disturbance is further transmitted to the rotor. The results are both high rotor currents and voltages during the grid faults. The function of the protection device called crowbar as shown in Fig. 2, is to limit the rotor current. When the crowbar is triggered, the rotor-side converter is disabled and bypassed, and thereby protects the converter. Without a loss of generality, the back-to-back converter is modeled according to their circuit topologies, taking pulse width modulation switching of electronic devices into account. Dealing with the modeling of crowbar, phase A current is taken as an example. The current through the alternating-current side of the rotor converter can be described as

iac ¼ iar þ iaw

Rotor-side converter

Crowbar

DC link

(14)

Grid-side converter

Lf

Lg

vDC DFIG Lcb Rcb

Grid

Fig. 2. Topology of back-to-back converters.

Z. Song et al. / Energy 38 (2012) 104e117

where subscripts c, r and cb denote the converter output current, the generator input current and the crowbar input current, respectively. The dynamics of the current through the crowbar can be given by

diacb Rcb iacb va þ r ¼  dt Lcb Lcb

(15)

where Rcb and Lcb are the crowbar resistance and inductance, respectively. The value of Rcb and that of Lcb are both dependent on the generator data. 3. Problem description The equations of DFIG rotor currents within the stator flux orientation reference frame can be rewritten as

8 > > < dird ¼ urd  Rr ird þ ðus  ur ÞLm jsd þ ðus  ur Þ irq  sLr

dt

sLr

sLr Ls

dirq urq Rr irq Lm djsq > > : dt ¼ sLr  sLr  ðus  ur Þ us ird  Ls sLr dt

Lm djsd Ls sLr dt

(16) where s denotes the leakage coefficient. For small values of Rs, stator flux is mainly determined by stator voltage, which is practically constant, implying that the derivative of stator flux is close to zero, and can be neglected. Therefore, when dealing with the current controller design, the rate of change of stator flux linkage in (16) is often neglected which is equivalent to neglecting the stator electric transients. The reduced order model was derived by ignoring the differential term in (16). When the stator transient is neglected, the voltage equations given by (16) can be simplified to

8 > > < dird dt

di > > : dtrq

¼ ¼

urd

sLr



Rr ird ðus  ur ÞLm jsd þ þ ðus  ur Þirq sLr sLr Ls

(17)

urq Rr irq   ðus  ur Þird sLr sLr

Equation (17) shows that the dynamics of ird, for example, is not completely independent of irq. There are also additional terms in the current equations that depend on the generator parameters, flux magnitude and rotor angular frequency. Consequently, these terms cause a cross-coupling between the two current components ird and irq. To obtain control characteristics resembling that of the DC machine, where the field and armature currents are independent, the current controllers must provide decoupling as well as

107

regulation. Decoupling is traditionally achieved by using feedforward-voltage-control terms that attempt to cancel the coupling terms of the plant dynamics. The classic scheme for decoupled current control is shown in Fig. 3. The d-axis compensation voltage, for example, attempts to cancel the terms on the right-hand side of (17) that do not depend on ird. If the cancellation takes place, the overall dynamics is simplified, and it is easy to design suitable controllers. Reference voltages to obtain the desired currents can be written as

8 <

urd

ref

:u

rq ref

¼ u0rd 

ðus  ur ÞLm jsd  ðus  ur ÞsLr irq Ls

with

8 dird 0 > < urd ¼ Rr ird þ sLr dt > dirq : 0 urq ¼ Rr irq þ sLr dt

(19)

The ird and irq errors are processed by a proportional-integral controller to give u’rd and u’rq respectively. Treating cross-coupling terms as disturbances, the transfer function from the rotor voltage to the rotor current is given by

GðsÞ ¼

1

(20)

sLr s þ Rr

When proportional-integral controllers are used, the zero of the controller is usually selected to cancel the pole of the plant. The decoupled condition is always satisfied, but this technique can be applied assuming the stator voltage is ideal and the derivative of stator flux is neglected. Such system can provide good dynamic response during normal operation conditions, but the performance may be degraded during grid disturbances which mean stator flux is not constant. Furthermore, variations of DFIG parameters generally compromise both the decoupling and the desired pole-zero cancellation. The traditional decoupling controller assumes an exact knowledge of model parameters. In practical applications, however, situations may arise in which these parameters are not exactly known. Besides the nonlinearities related to measurement noise and harmonic pollution originating in the supply converters, electrical machine parameters are strongly affected by temperature, saturation, and skin effects, which can deteriorate the controller performance when designed with nominal parameter values. Besides, a problem related to rotor current control of DFIG is the tuning of the current loops. Proportional-integral controllers are

(

sd

-

)Lm/Ls

s

Active current regulator irq_ref

+

u'rq +

+

-

irq

urq_ref +

+

-

irq ird ird_ref

+

-

u'rd +

-

(18)

¼ u0rq þ ðus  ur ÞsLr ird

urd_ref

+ +

Reactive current regulator Fig. 3. Model structure and decoupled control scheme of DFIG.

ird

108

Z. Song et al. / Energy 38 (2012) 104e117

the most commonly used; however, selection of the controller gains is not easy and is usually subject to continuous adjustment. Very often, the gains obtained analytically or by simulation do not work well in implementation. There are several causes for this: floating of DFIG parameters with the change in external conditions, saturation, noise, delays, imperfections in signal acquisition, and improper field orientation. As a result, the system is often tuned by empirical methods or by trial and error. Meanwhile, the DFIG-based wind turbine operates in a wide operating range during its whole life. The proportional-integral control algorithm cannot assure a satisfying dynamic behavior in the entire operating range.

Compared with traditional control theory, the proposed adaptive disturbance rejection control for DFIG has the unique characteristics of model independence and it actively rejects both internal and external disturbances. It uses an extended state observer to build a solid base for better performance and disturbances compensation via the provision of output and the real action component of disturbance. Since this strategy does not depend on an accurate model of the DFIG, it is robust against the structural uncertainties and disturbances. From equation (16), the coupled first-order plant can be rewritten as

  ¼ bvrd þ fd ird ; irq ; jsd

dt
 > di : rq ¼ bvrq þ fq irq ; ird ; jsq dt

(21)

sLr

sLr Ls

Rr irq Lm djsq > > : fq ¼  sLr  ðus  ur Þ ird  Ls sLr dt

> > > :

¼ x2d þ bvrd (24)

dx2d ¼ hd dt yd ¼ x1d

where hd denotes the time derivative of fd. The corresponding state space model is

dxd ¼ Ad xd þ Bd ud þ Ed hd dt y d ¼ Cd x d

(25)

where the matrixes are

 x2d T ; Ad ¼

xd ¼ ½ x1d ¼ ½1

0 ; Ed ¼ ½ 0

0 0

1 ; Bd ¼ ½ b 0

0  T ; Cd

1 T :

Based on (25), the ESO is constructed as follows

dzd ¼ Ad zd þ Bd ud þ Ld ðyd  Cd zd Þ dt ¼ ðAd  Ld Cd Þzd þ Bd ud þ Ld Cd xd

(26)

The states of this observer correspond to the estimated values of the quantities,

where b represents the controller coefficient (b ¼ 1/sLr). fd(ird, irq, jsd), fq(irq, ird, jsq) (or simply denoted as fd and fq, respectively) account for all the other factors excluding the control effort urd and urq, referred to as the general disturbance. fd and fq can be written as

8 > > < fd ¼ Rr ird þ ðus  ur ÞLm jsd þ ðus  ur Þ irq 

8 dx > > > dt1d <

(

4. Adaptive disturbance rejection controller design

8 > < dird

The extended state representation of the system dynamics can be derived directly from (17),

Lm djsd Ls sLr dt

(22)

As shown in (22), fd and fq are the combinations of the internal dynamics, grid disturbances, cross-coupling terms and flux compensation. The basic idea of the proposed strategy is to obtain the estimated fd and fq, and to compensate for them in the control law actively in real time. If the general disturbances can be estimated and canceled, the system is then reduced to a simple integral plant with a scaling factor, simplifying the control problem. The estimation problem of fd and fq leads to a unique state observer known as the ESO (Extended State Observer). Note that the control designs of the d- and q-axis are the same and are implemented in parallel. For clarity, the design process of the proposed controller is explained with the control of the d-axis in the following. The key to ESO design is to interpret the general disturbance fd as an additional state of the dynamical system. The inclusion of fd as an additional second state is what motivated the name Extended State. For this first-order system, the number of states would now increase from one to two. The states variables are chosen as



z1d ¼ ird z2d ¼ fd

(27)

which yields the desired estimate for the general disturbance x2d ¼ fd. The observer gain vector can be denoted as

Ld ¼ ½ b1d

b2d T

(28)

and the observer gains can be computed following the parameterization technique shown below. Combining (25) and (26), the error equation can be written as

ded ¼ Ae ed þ Ed hd dt

(29)

where

eid ¼ xid  zid ; i ¼ 1; 2

 A e ¼ A d  L d Cd ¼

b1d b2d

1 0

(30)

Obviously, the designed ESO is stable if the roots of the characteristic polynomial of Ae

lðsÞ ¼ s2 þ b1d s þ b2d

(31)

are all in the left half-plane. Suppose the observer poles are all placed at -u0

lðsÞ ¼ s2 þ b1d s þ b2d ¼ ðs þ u0 Þ2

(32)

then

8 < x1d ¼ ird

b1d ¼ 2u0 ; b2d ¼ u20

:x

This makes u0 the only tuning parameters for the observer, which is the observer bandwidth. Hence the implementation process of the observer is much simplified. The choice of u0 could be a trade-off

2d

Rr i ðus  ur ÞLm jsd Lm djsd ¼ fd ¼  rd þ þ ðus  ur Þ irq  sLr sLr Ls Ls sLr dt (23)

(33)

Z. Song et al. / Energy 38 (2012) 104e117

between how fast the observer tracks the states and how sensitive it is to the sensor noises [20]. Generally, the larger the observer bandwidth is, the more accurate the estimation will be. However, a large observer bandwidth will increase noise sensitivity. Therefore a proper observer bandwidth should be selected in a compromise between the tracking performance and the noise tolerance. With a well-tuned observer, the observer output of z1d, z2d closely track the states of the augmented plant (25), ird and fd, respectively. By canceling the effect of fd using z2d, the adaptive disturbance rejection controller actively compensates for fd in real time. It should be noted that the Ad and Bd matrices are constant and do not depend on a detailed knowledge of the plant. This feature enables one to choose the observer gains without such knowledge and is one of the reasons why the ESO is less sensitive to variations in plant parameters than classical observers. If the control law is designed as

urd

ref

¼

u0rd

ref

 z2d

(34)

b

the original plant will be reduced to a unit-gain single integral plant

dyd ¼ fd þ burd dt

ref zz2d

þb

u0rd

ref

b

 z2d

¼ u0rd

ref

(35)

as indicated in Fig. 4. This can easily be controlled by a simple proportional gain:

u0rd




ref

¼ kp ird

ref

 z1d



(36)

Note that simple proportional controllers are used to make the closed-loop transfer function of the current control problem pure first order without a zero:

Gcld ðsÞ ¼

kp s þ kp

(37)

The controller tuning reduces to tuning a single parameter kp, where kp has a specific meaning: it is the bandwidth of the controller.

kp ¼ uc

(38)

where uc represents the desired closed-loop natural frequency. The closed-loop natural frequency should be adjusted based on the

109

competing requirements of performance and stability margin, together with noise sensitivity. As shown above, the design is model independent. The only parameter needed is the approximate value of b in (24). This means that uc, the closed-loop natural frequency and u0, the bandwidth of the observer are the only two tuning parameters for the controller of the d- and q-axis, respectively. This process greatly simplifies the tuning of the adaptive disturbance rejection controller. The combined effects of internal dynamics, grid disturbances, cross-coupling terms as well as flux compensation are treated as a general disturbance. By augmenting the observer to include an extra state, it is actively estimated and canceled out, thereby achieving active disturbance rejection. Moreover, it should be noted that this scheme also compensates for the parameter variations that affect the time constants of the DFIG current loops, which will be shown later. More generally, the proposed current controller for DFIG-based wind turbines is unique in two aspects: one is that the controller is robust against external disturbances and parameter uncertainties of the DFIG systems; the other is that the two-parameter tuning feature makes the controller practical and easy to implement in commercial wind turbines. 5. Stability and robustness analysis 5.1. Transfer function representation development The transfer function representation enables stability analysis and evaluation of a steady-state performance of the closed-loop control system for the rotor current control of DFIG-based wind turbines. In this section, the Laplace transform of d-axis current controller will be developed. Then the frequency-domain analyses will be conducted on the basis of the transfer function representation. The Laplace transform of (16) for the d-axis is

ðus  ur ÞLm jsd ðsÞ Lm þ ðus  ur Þ irq ðsÞ  sj ðsÞ s Lr Ls Ls sLr sd ird ðsÞ ¼ s þ bRr b þ u ðsÞ ¼ fd ðsÞ þ Gp ðsÞurd ðsÞ s þ bRr rd

Fig. 4. Design process of the rotor current controller for DFIG. (a) Detailed scheme. (b) Simplified scheme.

(39)

110

Z. Song et al. / Energy 38 (2012) 104e117

where Gp(s) denotes the transfer representation of the d-axis. The Laplace transform of the controller given by (34) and (36) is

1 kp i b rd

urd ðsÞ ¼

ref ðsÞ



1 kp b

1 zd ðsÞ

(40)

The Laplace transform of the ESO represented by (25) and (26) is

szd ðsÞ ¼ ðAd  Ld Cd Þzd ðsÞ þ Bd vrd ðsÞ þ Ld Cd xd ðsÞ

uc ðs þ u0 Þ2 i bsðs þ 2u0 þ uc Þ rd

urd ðsÞ ¼

ref ðsÞ





u20 þ 2u0 uc s þ u20 uc ird ðsÞ bsðs þ 2u0 þ uc Þ (42)

Let

 Gc ðsÞ ¼

HðsÞ ¼ 



u20 þ 2u0 uc s þ u20 uc bsðs þ 2u0 þ uc Þ

(43)

uc ðs þ u0 Þ2  2 u0 þ 2u0 uc s þ u20 uc

(44)

Equation (42) can be rewritten as

h urd ðsÞ ¼ Gc ðsÞ HðsÞird

ref ðsÞ

 ird ðsÞ

i

(45)

The closed-loop control system for d-axis is shown in Fig. 5, which is in the form of a two-degree-of-freedom closed-loop system. From the figure, it can be seen that the closed-loop d-axis current control system of DFIG-based wind turbines can be represented by a unity feedback loop with a pre-filter. From Fig. 5, the open-loop transfer function is

G0 ðsÞ ¼ Gc ðsÞGp ðsÞ

(46)

The closed-loop transfer function is

Gcl ðsÞ ¼

HðsÞGc ðsÞGp ðsÞ ird ðsÞ ¼ 1 þ Gc ðsÞGp ðsÞ ird ref ðsÞ

Gp ðsÞ ird ðsÞ ¼ 1 þ Gc ðsÞGp ðsÞ fd ðsÞ

Gc ðsÞGp ðsÞ 1 þ Gc ðsÞGp ðsÞ

(48)

From these transfer functions, the frequency-domain analysis will proceed.

5.2. Bandwidth and stability The closed-loop stability is the main concern for rotor current control of DFIG-based wind turbines. Since the transfer function of the system is known, it is easy to investigate its stability by

(49)

The characteristic polynomial is

i h
s3 þ ð2u0 þ uc þ bRr Þs2 þ bRr ð2u0 þ uc Þ þ u20 þ 2u0 uc s þ u20 uc ¼ 0

(50)

For maintaining the stability of the closed-loop system, it is required that

bRr >  2u0  uc bRr ð2u0 þ uc Þ>  u20  2u0 uc    ð2u0 þ uc þ bRr Þ bRr ð2u0 þ uc Þ þ u20 þ 2u0 uc >u20 uc

(51)

If inequality (51) is satisfied, the system dynamics are locally stable. Considering (51), it is clear that the stability of closed-loop system depends on the coefficient b ¼ 1/sLr and rotor resistance Rr. Also it depends on the controller bandwidth uc and ESO bandwidth u0. Since parameters b and Rr are both positive, inequality (51) is obviously satisfied regarding rotor current controller of DFIG-based wind turbines. 5.3. General disturbance rejection This section will show how the general disturbance, including the internal dynamics, grid disturbances, flux compensation and cross-coupling terms between d- and q-axis, is rejected by the proposed controller. Taking d-axis for example, the disturbance rejection capability of the proposed controller can be characterized by the transfer function between the general disturbance fd given by (22) and the d-axis rotor current ird. Substituting the Gp(s) in (39) and the Gc(s) in (43) into (48), the transfer function is obtained

(47)

Moreover, the transfer function from the general disturbance fd(s) to the output is

Gd ðsÞ ¼

G0cl ðsÞ ¼

(41)

Replacing the zd(s) in (40) with (41) gives



analyzing pole locations. It is noted that both coefficients in the denominator of H(s) are positive and, therefore, H(s) is guaranteed stable. Consequently, the closed-loop stability can be determined by the pole locations of

Gd ðsÞ ¼ ¼

bsðs þ 2u0 þ uc Þ   sðs þ 2u0 þ uc Þðs þ bRr Þ þ u20 þ 2u0 uc s þ u20 uc

sðs þ 2u0 þ uc Þ   sðs þ 2u0 þ uc ÞðsLr s þ Rr Þ þ u20 þ 2u0 uc sLr s þ sLr u20 uc (52)

From (52), it can be seen that, as the frequency u converges to zero or infinity, the Gd(ju) will go to zero. This suggests that the disturbance will be attenuated to zero with the increase of system bandwidth. The Bode plots of Gd(s) are shown in Figs. 6 and 7, respectively, in which the mutual inductance Lm, and rotor resistance Rr are varying. The results show the disturbance rejection ability of the designed controller since the magnitude responses are under 0 dB at any given frequency. Meanwhile, these figures demonstrates a desirable disturbance rejection property which is unaffected by the plant parametric uncertainties. 5.4. Robustness against parameter uncertainties

Fig. 5. Block diagram of the adaptive disturbance rejection control system for DFIG rotor current in transfer function form (d-axis).

As mentioned in the previous section, DFIG machine parameters are affected by temperature, saturation, and skin effects, which can deteriorate the controller performance when designed with nominal parameter values. Moreover, DFIG-based wind turbine operates in a wide operating range during its whole life. Therefore, another concern regarding DFIG rotor current control is the robustness against variations in plant parameters. The robustness

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111

It should be noted that the controller design process is depen^ dent on the control coefficient b, as presented in (34). Normally b, the estimation of b, can be obtained. The influence r, the ratio ^ on the between real plant parameter b and its estimation ðr ¼ b=bÞ system stability and disturbance rejection property when it does not equal to 1 is discussed. Accounting for r, the open-loop transfer function can be rewritten as



G00 ðsÞ





r u20 þ 2u0 uc s þ u20 uc ¼ ^ þ 2u þ u Þs2 þ rbR ^ ð2u þ u Þs s3 þ ðrbR r c r c 0 0

(53)

The transfer function from the input disturbance fd(s) to the output is given by

G0d ðsÞ ¼

^ þ 2u þ uc Þ rbsðs 0 ^ Þ þ u2 þ 2u u s þ u2 u sðs þ 2u0 þ uc Þðs þ rbR r 0 c 0 0 c (54)

Fig. 6. Bode plots of Gd(s) with different Lm.

of the proposed controller against parameter uncertainties is assessed in this section. The bode diagrams of the loop gain transfer function given by (46) with varying parameters are shown in Figs. 8 and 9. These figures demonstrate the robustness of the proposed controller in the presence of variations in plant parameters, or when the assumed parameter values differ from the actual plant parameters. Also it is shown that the closed-loop control system is stable with reasonably large stability margins for the chosen uc (uc ¼ 50 rad/s) and u0 (u0 ¼ 300 rad/s).

Fig. 7. Bode plots of Gd(s) with different Rr.

Figs. 10 and 11 show the Bode plots of G00 (s) and Gd’(s) with large variations of r. It is demonstrated that the controller is a little sensitive to large variations of r, which is proportional to the uncertainties in the plant parameter b. However, it should be noted that the normal variation of DFIG parameters and the reasonable differences between the assumed parameter values and the actual plant parameters only causes a small variation of b. Therefore, the controller can reject the general disturbance when DFIG parameters vary, as shown previously in Figs. 6 and 7.

6. Controller performance evaluation The simulation studies are conducted to demonstrate the feasibility and the performance of the proposed controller for a DFIG-based wind turbine rated at 1.75 MVA, with parameters and specifications listed in Appendix. The DC-link voltage is rated at 1200 V, with a capacitor of 145000 mF. The control algorithm, including the space-vector pulsewidth modulation technique, is

Fig. 8. Bode plots of G0(s) with different Lm.

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Fig. 9. Bode plots of G0(s) with different Rr.

Fig. 11. Bode plots of G00 (s) with different r.

executed in one 200-ms interrupt and the switching frequency of the converter is 5 kHz.

strategy. Without a loss of generality, the DFIG system is subject to a change in q-axis reference current to 1580 A at t ¼ 6 s. Meanwhile, in order to show the controller flexibility for achieving power system operator requirements, a step change in d-axis reference current is applied to the system at t ¼ 6.5 s. From the figure, it can be demonstrated that by using the proposed control strategy, the qaxis current response shows no overshoot and settles down quickly to a steady-state without any steady-state error. The d-axis current response, corresponding to the reactive power step response, also exhibits excellent dynamic performance. In the proposed control scheme, the ESO estimated the system state and the summation of the system’s general disturbance. Figs. 13 and 14 show the actual ird and irq in comparison with the value estimated by d- and q-axis ESO, respectively. Close agreement is observed with the estimation catching up the actual response. Fig. 15 demonstrates that the extended state output by d-axis ESO correspond to the compensation voltages that would be generated by the decoupling compensator. Close agreement is also observed for q-axis components, as shown in Fig. 16. This confirms that the ESO has the same effect as the compensation voltages and these estimations can be used to compensate for the system control.

6.1. Reference current tracking The performance of the proposed control method is first evaluated by reference current tracking responses during normal operation when subject to change of first q-axis and then D-axis current. Initially the wind speed is 8.5 m/s, and the d-axis reference current is set to 1000A, corresponding to the common situation of a DFIG-based wind turbine operating in the maximum wind energy capture region. Fig. 12 shows the dynamic response of the proposed

Fig. 10. Bode plots of G0d (s) with different r.

Fig. 12. Current tracking performance for the proposed adaptive disturbance rejection controller.

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Fig. 15. Actual and observed value of the general disturbance fd (d-axis).

6.2. Fault mitigation

Fig. 13. ird and z1 output of d-axis ESO.

Fig. 14. irq and z1 output of q-axis ESO.

Fault mitigation is a challenging problem for wind power plants to provide uninterrupted electric power. Grid voltage disturbances could induce a large transient current into the rotor circuit, which may damage the power electronic converters and result in disconnection of DFIG wind turbines from the power grid. To evaluate transient performance of the system, a voltage profile should be selected. However, it should not suddenly drop or rise before and after the fault conditions. Subsequently, in this paper, voltage profile is selected according to the E.ON code, as shown in Fig. 17, in which the variation curve of grid voltage magnitude is marked with lines for clarity. At the moment of voltage dip at t ¼ 2 s, the generator is operated at unity power factor. The transient response with the proposed control strategy is shown in Fig. 18(a). The transient behavior of DFIG with traditional controllers is presented in Fig. 18(b) for comparison. It is clear how the currents at the rotor winding are reduced, avoiding the tripping of the rotor-side converter. In addition, Fig. 19 presents the fundamental component of the phase voltage at the rotor of DFIG with different controllers, and Figs. 20e22 show the responses of stator output active power along with stator current and voltage. With regard to reactive power response, Fig. 23 demonstrates transient responses of the reactive power output when 1.2 MVar reactive power compensation is required under the grid fault. It is noted that the main concern for fault ride through of wind turbines is to decrease the peak current. From Fig. 18, it is demonstrated that the proposed control scheme is able to suppress

Fig. 16. Actual and observed value of the general disturbance fq for q-axis.

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Fig. 17. Grid voltage under grid fault.

the rotor current overshoot to only around 4000 A, during and after clearance of the fault, while the value for traditional controls is about 6100 A. This indicates that the transient rotor current is notably reduced with the proposed scheme. Consequently, it is generally easier to ride through the fault of the same severity with the proposed strategy. This is due to that with estimation and actively compensation of grid disturbances in real time, the control strategy ensures that the performance degradation caused by the grid disturbances can be suppressed. Thus, under the same

Fig. 19. DFIG rotor voltage transient response under grid fault. (a) Rotor voltage response with proposed controller. (b) Rotor voltage response with traditional controller.

conditions, the peak values of the rotor current is decreased by using the proposed controller, indicating that the rotor current may not exceed their acceptable limits and the protective devices are not triggered. This means that the proposed controller provides more opportunity for the wind turbine to remain connected during grid faults and contribute to system stability after fault clearance. It should be noted that voltage profile of DFIG during grid faults is also an important aspect. In Fig. 17, the grid voltage is shown, where it can be clearly noticed how the voltages at DFIG stator

Fig. 18. DFIG rotor current transient response under grid fault. (a) Rotor current response with proposed controller. (b) Rotor current response with traditional controller.

Fig. 20. DFIG stator current transient response under grid fault.

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Fig. 21. DFIG stator voltage under grid fault.

Fig. 23. Transient response of DFIG stator output reactive power when reactive power compensation is required under grid fault.

windings varies during the fault. With regard to rotor voltage, Fig. 19 demonstrates the voltage profile during this fault with two different control schemes. Compared to the conventional controller, a little additional rotor voltage is required in this new control scheme, as indicated in Fig. 19. However, in this way, as it has been shown above, it is possible to prevent generator windings from suffering over currents. Furthermore, the maximum instantaneous value of rotor voltage is much lower than the rotor-side converter maximum output voltage, indicating that the voltage stay within acceptable limit. Therefore, the proposed scheme successfully suppressed the transient current in the rotor circuit, apart from acceptable voltage peaks during the transients. This minimizes the occurrence of crowbar interruptions, and thus enable DFIG provide uninterrupted active and reactive power output during the whole grid fault period. Besides, in Fig. 19, changes in rotor current frequency are observed after fault clearance. The reason is that during the voltage dip, the electromagnetic torque of DFIG drops accordingly and therefore, the generator starts to accelerate. When the fault is cleared, the rotor speed will reach higher value than the prefault speed [40e42]. The increase of machine speed further results in changes in rotor current frequency. As the machine speed in the simulated condition is at a level lower than the synchronous speed, the rotor current frequency decreases accordingly, as demonstrated in Fig. 19. However, it is clearly demonstrated in Figs. 17e19 that compared with traditional controller, the current transient period of the proposed scheme lasts longer and current transient components decay slower. This indicates that the main advantage of the proposed controller is reduction in rotor current and not in

transient response. It is worth noting that one of major challenges must be overcome in achieving the ride-through requirements of DFIG-based wind turbines during the voltage dip is the peak rotor fault current that may exceed its limit. Therefore, the main concern for the controller design is to decrease the peak current. Therefore, reduction in rotor peak current presented by the proposed controller benefits fault ride-through capability of wind turbines.

Fig. 22. Transient response of stator output active power under grid fault.

6.3. Parameter uncertainties It should be noted that the realization of the proposed control law requires little knowledge of the DFIG parameters. To examine robust performance of the controller against parameter

Fig. 24. DFIG response with Rr uncertainties. (a) Response with traditional controller. (b) Response with proposed controller.

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Fig. 25. Calculated and observed value of the general disturbance fd with traditional controller and proposed controller (100% overestimation in Rr.)

uncertainties, in the following, the system transient performance as the rotor resistance and the mutual inductance are severely altered. Fig. 24 shows the transient behavior of the DFIG system, for the rotor current tracking performance under uncertainties in rotor resistance. The time constants of the plant current loops are detuned under such condition. In Fig. 24(a), the rotor resistance is unknown and two cases are considered. In the first case, 60% underestimation, ^ r ¼ 0:4Rr , and in the second case, 100% overestimation, i.e. R ^ r ¼ 2Rr is considered. To see the influence of the proposed R controller, the performances of the adaptive disturbance rejection controller and conventional proportional-integral-differential controller are compared under same conditions. From Fig. 24(b), it is shown that under uncertainties in rotor resistance, the adaptive disturbance rejection controller can still maintain its excellent dynamic performance. Furthermore, the transient time of about 100 ms confirms that the time constant of the rotor current is 20 ms, and this correspond to the desired closed-loop pole at 50. This demonstrates that the proposed scheme also compensates for the parameter variations that affect the time constants of the DFIG current loops, as pointed out previously in Section 4. In Figs. 25 and 26, the values of the ESO output is shown in comparison with the compensation voltages that would be applied by the traditional decoupling compensator e note that they do not match exactly (due to the detuning of the plant). It can be observed that the output of ESO can track the actual value in real time. Also the proposed controller shows good results for 50% error in estimation of mutual inductance. With the changed mutual inductance, the output of ESO and the calculation of the compensation voltages that would be applied by the traditional decoupling

Fig. 27. Calculated and observed value of the general disturbance fd withith traditional controller and proposed controller (50% underestimation in Lm.)

compensator are shown in Fig. 27 respectively for comparison. It can be observed that, similar to the condition under resistance variation, there exist differences and the output of ESO can track the actual value in real time. Thus, the performance of the proposed controller, under a wide range of uncertainty in estimation of rotor/ stator resistances and mutual inductance, is robust. In conclusion, these results verify that by using the adaptive disturbance rejection controller, the estimation and compensation of the impact of general disturbance is implemented efficiently and no integration operation is needed. 7. Conclusion In this paper, an adaptive disturbance rejection controller for DFIG-based wind turbines has been presented. The combined effects of internal dynamics, grid disturbances, cross-coupling terms as well as flux compensation are treated as a general disturbance. By augmenting the observer to include an extra state, it is actively estimated and canceled out, thereby achieving active disturbance rejection. Moreover, it should be noted that this scheme also compensates for the parameter variations that affect the time constants of the DFIG current loops. The time-domain design-based controller is analyzed and evaluated using frequency response techniques, resulting in better explanation and examination of the control concept. The drawbacks of traditional controller, viz., the degradation of performance in the presence of modeling uncertainties, have been removed and demonstrated through frequency analysis as well as simulation validation. Besides, it is demonstrated that the proposed control scheme constrains the rotor currents during grid disturbances, indicating that the rotor current may not exceed their acceptable limits and the protective devices are not triggered. These results, together with the frequency-domain responses, proved the stability and the strong robustness of the control system against parameter uncertainties and external disturbances. Acknowledgment

Fig. 26. Calculated and observed value of the general disturbance fd with traditional controller and proposed controller (60% underestimation in Rr.)

The authors would like to express their sincere gratitude to Prof. Z. Gao at the Cleveland State University and Prof. D. Wu at the Department of Precision Instruments and Mechanology, Tsinghua University, for their excellent guidance regarding the frequencydomain analysis of the controller. The authors gratefully acknowledge the support of National Science Fund for Distinguished Young Scholars (no: 50825701), National Natural Science Foundation of China (no: 51107084 &

Z. Song et al. / Energy 38 (2012) 104e117

51077097), Research Fund for the Doctoral Program of Higher Education of China (no. 20100032120081). Appendix A. Parameters of the 1.75-MVA, 575-V, 50-Hz DFIG WT Ratings: Sn ¼ 1.75 MVA, Un ¼ 575 V (lineeline, rms), Winding connection (stator/rotor): YdY Stator resistance: Rs ¼ 0.003428 U Stator leakage inductance: Lls ¼ 0.000103052 H Mutual inductance: Lm ¼ 0.0033967 H Rotor resistance: Rr ¼ 0.00384 U Rotor leakage inductance: Llr ¼ 0.000111347 H Number of pole pairs: p ¼ 2 DC-link voltage: uDC ¼ 1200 V DC-link capacitor: C ¼ 145,000 mF Turbine inertia: Jt ¼ 303.96 kg m2 Generator inertia: Jg ¼ 60.79 kg m2 Shaft stiffness coefficient: k ¼ 2.17  105 N m/rad Shaft damping coefficient: c ¼ 3.48  105 N m s/rad B. Controller Parameters Desired closed-loop natural frequency: uc ¼ 50 rad/s Observer bandwidth: u0 ¼ 300 rad/s C. Operating Conditions Interrupt period: 200 ms Converter switching frequency: 5 kHz References [1] Melicio R, Mendes VMF, Catalao JPS. Comparative study of power converter topologies and control strategies for the harmonic performance of variablespeed wind turbine generator systems. Energy January 2011;36(1):520e9. [2] Ozbek M, Rixen DJ, Erne O, Sanow G. Feasibility of monitoring large wind turbines using photogrammetry. Energy December 2010;35(12):4802e11. [3] Liu HP, Shi J, Erdem E. Prediction of wind speed time series using modified Taylor Kriging method. Energy December 2010;35(12):4870e9. [4] Sawetsakulanond B, Kinnares V. Design, analysis, and construction of a small scale self-excited induction generator for a wind energy application. Energy December 2010;35(12):4975e85. [5] Mabel MC, Raj RE, Fernandez E. Adequacy evaluation of wind power generation systems. Energy December 2010;35(12):5217e22. [6] Saheb-Koussa D, Haddadi M, Belhamel M, Hadji S, Nouredine S. Modeling and simulation of the fixed-speed WECS (wind energy conversion system): application to the Algerian Sahara area. Energy October 2010;35(10):4116e25. [7] Lin WM, Hong CM. Intelligent approach to maximum power point tracking control strategy for variable-speed wind turbine generation system. Energy June 2010;35(6):2440e7. [8] Hocine L, Mounira M. Effect of nonlinear energy on wind farm generators connected to a distribution grid. Energy May 2011;36(5):3255e61. [9] Mikel PG, Oriol GB, Andreas S, Joan BJ. Analysis of a multi turbine offshore wind farm connected to a single large power converter operated with variable frequency. Energy May 2011;36(5):3272e81. [10] Kamel RM, Chaouachi A, Nagasaka K. Wind power smoothing using fuzzy logic pitch controller and energy capacitor system for improvement microgrid performance in islanding mode. Energy May 2010;35(5):2119e29. [11] Connolly D, Lund H, Mathiesen BV, Leahy M. Modelling the existing Irish energy-system to identify future energy costs and the maximum wind penetration feasible. Energy May 2010;35(5):2164e73. [12] Kusiak A, Zheng HY. Optimization of wind turbine energy and power factor with an evolutionary computation algorithm. Energy March 2010;35(3):1324e32. [13] Lanzafame R, Messina M. Power curve control in micro wind turbine design. Energy February 2010;35(2):556e61. [14] Migoya E, Crespo A, García J, Moreno F, Manuel F, Jiménez Á, et al. Comparative study of the behavior of wind-turbines in a wind farm. Energy October 2007;32(10):1871e85.

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