Chapter 3
Control of the Doubly-Fed Induction Generator based Wind Turbine Generator System
This chapter discusses the modelling and control design of a DFIG based WTGS in faultless or ‘healthy’ mode. A generator model that takes into account the internal faults of the machine is discussed in Chapter-4. The simulations have been carried out using Matlab/Simulink and Matlab/SimPowerSystems [Morren et al. 2003] [Ko et al. 2008] [Zhao et al. 2006] [Li & Chen 2006] [Serban et al.
2006] [Bouaouiche & Machmoum 2006] [Santos-Martin et al. 2006].
This chapter is organized such that an overview of the operation and description of the electrical system, presented in Section-3.1 and Section-3.2 respectively, leads to a brief discussion on the detail necessary in modelling the system components to study the phenomena of interest in Section-3.3. Models of the components used are then presented in Section-3.4 and controller design is discussed in Section-3.5. Section-3.6 tests the performance of the PI controllers designed for rated conditions at different wind speeds, for a 2 MW DFIG based WTGS. Two different control strategies namely the torque control strategy and the power control strategy are compared. The performance of the PI controller, for both strategies, is then compared to that of a Linear Quadratic Gaussian (LQG) controller at different wind speeds. Matlab/Simulink has been used for this purpose. Finally in Section-3.7, the start-up procedure of a DFIG based WTGS is explained with the help of simulations for a 15 kW rated system in Matlab/SimPowerSystems. Section-3.8 presents the conclusions.
3.1 Operating Regions of a DFIG based WTGS
The different regions of operation for a DFIG based WTGS are presented in Figure-3.1. The main distinction is between the Variable-Speed Region and the Constant-Speed Regions. In the variable-speed region MPPT is usually followed by maintaining the optimal TSR and keeping the pitch angle β = 0 [Li & Chen 2006]. Apart from the variable-speed region the speed of the system is kept at its minimum or maximum rated value. At wind speeds higher than the rated wind speed, here 12 m/s, the output power Pe of the system is limited to its rated value by controlling β. The Partial-Load Region is defined as the region of operation between wind speed at which the system rated speed occurs, here at 11 m/s, and the rated wind speed [Morales 2006]. In this region, the effect of drop in the value of the aerodynamic efficiency coefficient CP is over taken by the increase in the wind speed and there is a net gain in power since it is related to the cube of the wind speed; see Equation (2.5).
Figure 3.1 – Operating Regions of a DFIG based WTGS.
3.2 System Overview
An overview of a DFIG based WTGS is presented in Figure-3.2. The components necessary for simulation of the electrical part of the system are shown. The functional description of the components will now be given. The control laws are derived in the coming sections.
Figure 3.2 – System overview of a DFIG based WTGS.
The blocks marked as RSC and GSC, in Figure-3.2, refer to the Rotor-Side Converter and the Grid-Side Converter respectively. As can be judged from the wound-rotor induction generator symbol, marked as WRIG, the RSC is connected to the rotor windings while the GSC is connected via the inductance filter Lf and the transformer to the grid, hence the name. A three-winding transformer allows independent rated voltage on the stator and the rotor [Sorensen et al. 2006]. The converters share a DC-link formed by the capacitor CDC that allows bi-directional power flow between the machine’s rotor circuit and the grid since the converters are IGBT based. Together these three components namely the RSC, the DC-link and the GSC form a back-to-back converter. Since the DFIG is essentially a rotating transformer, the fundamental frequency of the current and the voltage in the rotor circuit changes with its speed. The DC-link thus provides the decoupling between the two AC sides, which are at different frequencies. The inductance filter mitigates the harmonics in the current, due to the switching converter, and provides the impedance over which energy can be exchanged. The stator windings are connected directly to the grid via the transformer.
Specific tasks are assigned to the two converters. The RSC has to control the machine response, depending on the control objective, via active power control while providing the required magnetization to the generator and reactive power support to the grid if required. The GSC, on the other hand, has to maintain the DC-link voltage and is usually reactive power neutral [Hansen & Michalke 2007]
[Lindholm 2003]. However, it can also be utilized for grid voltage support via reactive power exchange or grid harmonic compensation [Tremblay, Chandra &
Lagacé 2006] [Ko et al. 2008]. An alternate task for the RSC magnetization- current control can be to minimize the machine copper losses [Rabelo &
Hofmann 2005] [Li & Chen 2006] [El Aimani et al. 2003]. In other studies on the DFIG, the torque control of the RSC was modified to provide an inertia response by the wind turbine to support the grid frequency by releasing its kinetic energy when the frequency drops [Ekanayake & Jenkins 2004] [Morren, Pierik & de Haan 2006].
The vector control system based on Park’s transformation, discussed in Section- 3.4.1, provides two orthogonal channels d and q to each converter thereby allowing decoupled control of active and reactive power. This is equivalent to the control of a DC machine where this decoupling is achieved through machine construction [Mohan 2001].
3.3 A Note on Model Complexity
The detail used to model any system should be enough so as to allow accurate simulation of the phenomena of interest while not be computationally more
‘expensive’ than required. The time scale of events studied is important. For example, the effects of a change in wind speed represent a slow transient as compared to a voltage dip at the generator terminals [Morren et al. 2003].
For simulations focusing on the dynamics of the wind turbine itself, the frequency scale of 0 to 20 Hz is of interest [Iov 2003] [Wilkinson & Tavner 2006].
A reduced-order induction machine model neglecting the stator flux dynamics can be used with both a one-mass and a two-mass model of the drive train.
However, neglecting both the stator and rotor flux dynamics and using a one- mass model of the drive train gives erroneous results while with a two-mass model correct results are obtained. For these aeroelastic simulations, the transfer function which is of interest is between the input wind turbine torque and the output electromagnetic torque [Iov 2003].
The DFIG based WTGS is more sensitive to grid disturbances compared to the full converter concept [Hansen & Michalke 2007] [Niiranen 2004] [Tsili, Patsiouras
& Papathanassiou 2008]. In order to correctly predict the currents in the machine, and therefore the design of the rotor converter protection, a full-order model that includes both the stator and rotor flux dynamics is to be used [Morren et al. 2003] [Hansen & Michalke 2007]. Furthermore, for fault-ride- through control performance analysis and its impact on the grid a two-mass model of the drive train is essential [Hansen & Michalke 2007] [Santos-Martin et al. 2006]. However, a simplified aerodynamic model based on Equation (2.5) is sufficient to illustrate the effect of turbine speed and pitch on the aerodynamic power during grid faults [Hansen & Michalke 2007] [Santos-Martin et al. 2006]
[Abo-khalil, Lee & Lee 2006]. In dynamic impact studies, the wind speed is usually assumed to be constant during the observed time frames [Hansen &
Michalke 2007]. The time duration of interest for voltage dip study is around one second compared to the wind dynamics of several seconds [Santos-Martin et al.
2006].
For events considered at the machine terminals, the transfer function of interest is from stator voltage to the electromagnetic torque. It has been shown that for stator connection to the grid and a three-phase grid fault notable difference exists between the full-order model and the reduced-order model which neglects the stator flux dynamics. While the results are somewhat better for the grid fault, the reduced-order model fails to predict the effect on electromagnetic torque and speed for stator connection to the grid [Iov 2003].
As for the converter, it suffices to model it without the Pulse-Width Modulation (PWM) switching. This can lead to 54 times faster simulation speed compared to the model which explicitly simulates switching [Morren et al. 2003]. The choice of a synchronously rotating reference frame means that all the currents, voltages and flux-linkages associated with the stator and rotor dq windings are DC in a balanced sinusoidal steady state [Mohan 2001]. Thus, a further boost by 100 times can be achieved if the three-phase system components, generator as well as the grid, are modelled in the dq reference frame. This is because, in steady- state, the quantities are not alternating and the solver can use larger time steps [Morren et al. 2003]. Such a model can be very useful for grid stability investigations since it provides accurate WTGS response with less computation effort.
In this work, a full-order model of the DFIG has been used i.e. both the stator and rotor flux dynamics have been taken into account. As for the drive train, a one- mass model is used since grid faults have not been studied. The aerodynamic model defined in Section-2.1 has been used in Section-3.6 [Galvez-Carillo 2011].
3.4 Electrical System Components Modelling
3.4.1 dq Reference Frame Transformation
Details of the dq reference frame transformation are presented in Appendix-A.
3.4.2 Induction Machine Model
The induction machine model can be implemented with motor convention on the rotor side and generator convention on the stator side [Zhao et al. 2006] [Li &
Chen 2006]. This is shown in Figure-3.3.
Figure 3.3 – Equivalent circuit of the induction machine in dq coordinates [Mohan 2001].
Equation (3.1) gives the model
sq d sd s sd
sd v R i
dt
d λ =− − +ω λ (3.1a)
sd d sq s sq
sq v R i
dt
d λ =− − −ω λ (3.1b)
rq dA rd r rd
rd v R i
dt
d λ = − +ω λ (3.1c)
rd dA rq r rq
rq v R i
dt
d λ = − −ω λ (3.1d)
+
= rd
r m sd s
sd L
L L
i 1 λ λ
σ (3.1e)
+
= rq
r m sq s
sq L
L L
i 1 λ λ
σ (3.1f)
+
= sd
s m rd r
rd L
L L
i 1 λ λ
σ (3.1g)
+
= sq
s m rq r
rq L
L L
i 1 λ λ
σ (3.1h)
(
rd sq rq sd)
r s
m
em L L
L 2
T p λ λ λ λ
σ −
= (3.1i)
(
m em eq m)
eq
m T T B
J 1 dt
d ω = − − ω (3.1j)
−
=
r s
2 m
L L 1 L
σ (3.1k)
λ, v and i signify flux-linkage, voltage and current respectively. Their subscripts s and r along with d and q represent a stator and rotor related quantity along the d and q axis of the reference frame respectively. Rs(Rr) and Ls(Lr) denote the resistance and inductance of the stator(rotor) respectively, while Lm refers to the magnetizing inductance. t represents time and σ is the leakage coefficient. ωd and ωdA are the electrical radian frequency on the stator and the rotor side respectively. ωdA = ωd - ωr where ωr is the electrical angular velocity of the rotor while ωm is its mechanical angular velocity. p and Tem represent the number of poles and the electromagnetic torque of the machine respectively whereas Tm, Jeq
and Beq represent the mechanical driving torque, the total inertia and the total viscous friction of the system when a lumped-mass model is used for the drive train.
Equation (3.1e) to Equation (3.1h) must be substituted in Equation (3.1a) to Equation (3.1d) before implementing in Simulink. This way, by proper initialization of the derivative terms, algebraic loops are avoided during simulations since the inputs to the model are not dependent on the outputs [Morren et al. 2003].
3.4.3 Converter Model
A simplified converter model has been used in simulations. As mentioned previously, the switching events need not be simulated. However, the converter delay has to be taken into account [Serban 2003] [Soens, Driesen & Belmans 2003]. This can be done, as a first approximation, by delaying the reference to the converter block by multiples of the sampling time determined by the ratio of the sampling frequency to the switching frequency. The only difference observed, with this way of modelling, in the converter current is the lack of switching harmonics.
A model of the converter based on power balance on the AC and DC sides is used [Morales 2006] [Morren et al. 2003] [Serban 2003]. This neglects the losses in the converter. The current on the DC side iDC is then given in terms of power on the AC side and the DC-link voltage vDC.
(
d d q q)
DC
DC v i v i
v
i = 1 + (3.2)
vd and id are the d components of voltage and current on the AC side respectively, while vq and iq are the q components of the same. The expression of AC power in terms of dq voltages and currents is explained in Appendix-A.
3.4.4 DC-link Model
The DC-link is modelled here as a pure capacitance in series with a resistance [Iov et al. 2004]. The equivalent circuit of a capacitor actually has four components: a pure capacitance, an equivalent series resistance (ESR) and an equivalent series inductance (ESL) all in series while the fourth is a resistance parallel to the capacitance [Gebbia 2001] [Evox Rifa n.d.]. All the components except the pure resistance, parallel to the capacitance, are frequency dependent.
This pure resistance represents the leakage current path and depends on the quality of the dielectric [Evox Rifa n.d.].
The ESR value is normally quoted at 100 Hz, 20 ˚C as base values and corresponds to an ESR factor value of 1. Since it is inversely proportional to capacitance, frequency and hot-spot temperature, therefore, from the ESR tables the factor value at the correct hot spot temperature and frequency should be taken. For example, the quoted ESR value for a 2200μF, 400 V electrolytic capacitor from EPCOS at base values is 0.072Ω [EPCOS 2008]. Assuming a realistic switching frequency of 2 kHz, the ESR factor value can be 0.45 and 0.43 for a hot spot temperature of 40˚C and 60˚C respectively [Evox Rifa n.d.]. This gives the ESR value of 0.03 Ω. The ESL is ignored here due to small values of around 20 nH [EPCOS 2008] [Evox Rifa n.d.]. The DC-link model is given in Equation (3.3).
0 DC t t
0 net DC net DC
DC i dt v
C i 1 R
v = +
∫
+ = (3.3)RDC, CDC and inet are the equivalent series resistance, the capacitance and the current flowing through this series branch respectively.
3.4.5 Inductance Filter Model
As a first-order filter, the inductance used at the output of the GSC provides an attenuation of -20 dB/decade [Roufi & Lamchich 2004] [Lindgren & Svensson 1995]. Equation (3.4) gives its mathematical representation.
(
d,grid fd f fd d f fq)
f
fd v v R i L i
L i 1 dt
d = − − +ω (3.4a)
(
q,grid fq f fq d f fd)
f
fq v v R i L i
L i 1 dt
d = − − −ω (3.4b)
vd,grid, vq,grid and vfd, vfq are the grid voltage and the converter output voltage components respectively. Rf, Lf are the filter resistance and inductance respectively. ifd and ifq are the filter current components.
3.4.6 Controller Model
The classical technique based on Proportional Integral control has been used for the system as is usually the case for the WTGS [Ko et al. 2008] [Hansen &
Michalke 2007] [Zhao et al. 2006] [Rabelo & Hofmann 2005] [Li & Chen 2006]
[Morales 2006]. The transfer function for the continuous-time controller is given in Equation (3.5).
s K K ) s (
C = p+ i
(3.5)
Kp and Ki are the proportional and the integral gain respectively. For implementation on a computer system the controller must be discretized.
Forward Euler method is used which approximates 1/s as Ts/(z-1), where Ts is the sampling time [Matlab/Simulink R2008b]. The output of the discrete-time integrator block is generated as follows
) ( ) (k x k
y = (3.6a)
) ( . ) ( ) 1
(k x k T u k
x + = + s (3.6b)
y, x and u represent the output, the state and the input to the block respectively while k is the sample number at time t = kTs. y(0) = x(0) is equal to the initial condition specified for the block. The discrete-time controller is given as
z 1
K T K ) z (
C p i s
+ −
= (3.7)
For this discretization to be stable, the poles of the continuous-time controller must lie in the left-half plane within a circle with its origin at (-1/Ts, 0) and of radius 1/Ts [Petersson 2005].
3.4.7 Phase-Locked Loop
The Phase-Locked Loop (PLL) provided by the Wind Turbine Block-set is modified for line (phase-phase) voltages and used [Iov et al. 2004]. It gives the position θPLL of the grid voltage space vector and the frequency fPLL of the grid voltage at its output. The block diagram for the PLL is presented in Figure-3.4. It aligns the q-axis of the dq reference frame with the grid voltage space vector, thus vd,grid = 0.
Figure 3.4 – Phase-locked loop [Iov et al. 2004].
Two phase-phase voltages vac and vbc can be measured, in case the neutral is not available, and transformed into three phase-neutral voltages va, vb and vc using the transformation matrix given as Equation (3.8). Then using Clarke’s transformation, Equation (3.9), the voltage space vector formed by these three voltages is converted into two orthogonal components vα and vβ taking phase a as reference in the transformation [Morren et al. 2003] [Soens, Driesen &
Belmans 2003]. The orthogonal components are then used to find the cosine and sine function values of angle γ of the voltage space vector.
−
−
−
−
=
bc ac
c b a
v v
3 1 3
1 3 2 3
1 3
1 3 2
v v v
(3.8)
−
−
−
=
c b a
v v v
3 1 3 0 1
3 1 3
1 3 2 v
v
β
α (3.9)
) sin(
sin cos cos
sin
e= γ θPLL − γ θPLL = γ −θPLL (3.10) If the PLL output angle θPLL is equal to γ in Equation (3.10) the input error e to the PI controller is zero. If the frequency of the grid voltage changes from 50 Hz, there is a non-zero error input to the PI controller and it adds a positive or negative output to the nominal radian frequency of the grid voltage. This leads to a zero input error and thus the PLL is locked again to the grid voltage frequency and phase angle. Other PLL types are discussed in [Timbus et al. 2005].
3.4.8 Measurement Filter Model
A 2nd order low-pass Butterworth filter with a cut-off frequency of 500 Hz is used.
The characteristic of a Butterworth filter is a flat response in the pass-band and an adequate rate of roll-off. Designing the same filter in Matlab using the code [zro,pol,kn]=butter(2,2*pi*500,'s') yields the poles pol as -2.2214 ± j2.2214, the gain kn as 9.8696e6 and no zeros zro. If a quadratic equation has poles a ± jb, then it has roots of (s – a – jb) and (s – a + jb). When these roots are multiplied together, the resulting quadratic polynomial is s2 – 2as + a2 + b2. Thus
the transfer function Hf of the filter is given in Equation (3.11) while the Bode plot is presented in Figure-3.5.
( )
s 4442.8s 9869600 9869600s
Hf 2
+
= + (3.11)
Figure 3.5 – Bode plot of the 2nd order Butterworth filter.
3.4.9 Pulse-Width Modulation Strategy
In order to carry out simulations with a switching converter in SimPowerSystems, two blocks were developed for the Space Vector Modulation (SVM) and Sinusoidal PWM following the methods described in references [Mohan 2001] [Iov et al. 2004] [Kim & Sul 2004] [Perruchoud & Pinewski 1996].
The blocks are presented in Figure-3.6 and Figure-3.7. These blocks provide the required duty-cycles as outputs and can be readily used in experimentation.
Figure 3.6 – Duty-cycle generation for Space Vector Modulation.
Figure 3.7 – Duty-cycle generation for Sinusoidal PWM.
In Figure-3.7, -1 ≤ ma ≤ 1 gives 0 ≤ DCa ≤ 1. Same is true for mb and mc. The reference voltages to the converter with SVM do not have the same purely sinusoidal nature as for Sinusoidal PWM [Mohan 2001]. In order to generate a reference voltage vreference,a from the control voltage v*a given by the control system, for phase a when using SVM, the following expressions are to be used [Mohan 2001].
( )
pktri DC
k a a ,
reference .v
2 / v
v
v v −
=
∗
(3.12)
( ) ( )
2
v , v , v min v
, v , v
vk max a b c a b c
∗
∗
∗
∗
∗
∗ +
= (3.13)
The reference voltages are of the same form as the duty-cycles. The duty-cycles at the output of the blocks are compared in Figure-3.8(a). The reference voltages are compared with a triangular wave vtri, -1 ≤ vtri ≤ 1, to generate the gate pulses for the IGBTs as shown in Figure-3.8(b). IGBTTop and IGBTBot refer to the switching signals for IGBTs in the same branch of a two-level converter.
(a) (b)
Figure 3.8 – Outputs of blocks for simulating a switching converter.
(a) Duty-cycles. (b) Gate pulses for IGBTs with dead time.
Figure 3.9 – IGBT gate pulse generation.
The block to generate gate pulses with the inclusion of dead-time, to avoid turning both the IGBTs in one arm ON at the same time, is presented in Figure- 3.9 where Tdt and vpktri refer to the dead time and the peak value of the triangular wave respectively. The active-low PWM Enable function is provided to replicate the provision in the dSPACE hardware DS1104, used in experimentation, since it has to be explicitly enabled.
3.5 Control Design for the DFIG System
A cascade control structure has been used to command the RSC and GSC [Morales 2006] [Zhao et al. 2006] [Da Silva et al. 2008] [Forchetti et al. 2007].
The control objectives must be achieved by the manipulation of currents, thus the inner control loops are current loops. They receive the reference from the outer control loop, which relates to the process variable defined in the control objective.
3.5.1 Selecting the Angular Velocity and Alignment of the dq Reference Frame
As already mentioned, the choice of a synchronously rotating reference frame has the advantage that all the currents, voltages and flux-linkages associated with the stator and rotor dq windings are DC in a balanced sinusoidal steady state [Mohan 2001].
While the GSC control scheme always uses grid voltage space vector alignment, two possibilities exist for the alignment of the RSC reference frame. It can be aligned with either the stator flux space vector [Serban et al. 2006] [Zhao et al.
2006] [Chowdhury & Chellapilla 2006] [Morren et al. 2003] [Abo-Khalil, Lee &
Lee 2006] [Bouaouiche & Machmoum 2006] or the stator voltage space vector [Rabelo & Hofmann 2005] [Zhang et al. 2006] [Santos-Martin et al. 2006] [Soens, Driesen & Belmans 2003] [Da Silva et al. 2008]. The obvious advantage is the simplification of either the torque or the power expression; see Equation (3.1i) and Equation (3.42) [Morren et al. 2003] [Soens, Driesen & Belmans 2003]. A decoupling is therefore obtained to control the respective quantity through either the d or the q axis. To simplify the design of the controllers two assumptions are generally made which seem to make the above alignment choice immaterial, since the stator voltage and stator flux end up being orthogonal to one another. These assumptions are a negligible stator resistance [Zhao et al.
2006] [Li & Chen 2006] [El Aimani et al. 2003] and neglecting the stator flux transients due to the assumption of a stiff grid [Iov 2003] [Yuan, Chai & Li 2004]
[Morren et al. 2003] [Morales 2006].
However, from an operational point of view there is a difference. The dynamics of the DFIG have two poorly damped eigenvalues (poles) that have an oscillation frequency close to the line frequency [Iov 2003] [Petersson 2005] [Zhan &
Barker 2006]. These reflect as a decaying 50 Hz variation on the synchronously rotating reference frame [Iov 2003] [Morales 2006]. In case of a grid disturbance these poles will cause an oscillation in the stator flux, which leads to oscillations in the torque. A high slip is produced which leads to high rotor voltages and currents and over speed of the rotor. If the rotor gets accelerated too much then to bring it back to normal speed requires large voltages and currents, which can damage the rotor circuit [Zhan & Barker 2006]. Once the voltage recovers, the control system will damp the flux-linkage oscillations more rapidly if the reference frame is oriented towards the stator voltage rather than the stator flux [Naess et al. 2005]. The system is more stable for stator voltage oriented reference frame and the space vector angle can be directly measured with a PLL while for the stator flux orientation it has to be estimated from the stator currents, rotor currents and rotor position [Petersson 2005].
Grid voltage alignment has been used throughout this thesis where the q axis of the dq reference frame is aligned to the grid voltage space vector, thus vd,grid = 0.
3.5.2 Selecting the Sampling and Switching Frequency
It is recommended in literature to select an angular sampling frequency at least 10 times the closed-loop bandwidth and the angular switching frequency no less than half the angular sampling frequency, when using SVM of a voltage source converter [Harnefors & Nee 1998]. In order to assure that the discrete-time controller can be made to closely match the performance of the continuous-time controller, the angular sampling frequency should be greater than 30 times the bandwidth of the system [Franklin, Powell & Workman 1998] [Serban 2003].
The switching frequency in modern WTGS is between 2 – 5 kHz [Schulz et al.
2002] and below 2 kHz for the MW class [Rabelo & Hofmann 2005]. The achievable bandwidth of current control loops in electric drives is approximately 10 – 30 times smaller than the angular switching frequency [Blasko, Kaura &
Niewiadomski 1998]. Sampling the signal twice in a switching period increases the bandwidth to about 1/10 of the angular switching frequency, while synchronous sampling once every cycle leads the bandwidth to be 1/20 of the angular switching frequency [Lindholm 2003]. The unit of bandwidth and angular frequency is rad/s.
The sampling frequency chosen in this thesis is twice the switching frequency selected, unless stated otherwise.
3.5.3 Grid-Side Converter Control
The GSC as mentioned earlier has two tasks. It has to maintain the DC-link voltage and exchange reactive power with the grid if required. At the start-up stage of the DFIG system, it is the GSC that has to bring the DC-link voltage to the
level required for the operation of the system after which the RSC control can be enacted [Da Silva et al. 2008] [Bouaouiche & Machmoum 2006] [Abo-khalil, Lee
& Lee 2006]. During normal operation, the GSC maintains the DC-link voltage by exchanging the active power of the RSC with the grid. The active power pg and the reactive power qg on the grid side are given in Equation (3.14) and Equation (3.15) respectively with vd,grid = 0.
fq grid , q
g v i
p = (3.14)
fd grid , q
g v i
q = (3.15)
Thus the corresponding filter current component has to be manipulated to control the powers.
3.5.3.1 The Control Scheme
The cascaded control loop structure for the GSC is given in Figure-3.10. As mentioned before the control of the DC-link voltage is related to the flow of active power, thus the inner control loop is for ifq component of the filter current.
The converter and measurement filter delays are included as Tcd and Tfd
respectively. The control loop structure for ifd has a similar form.
If the system modelling is carried out completely in the dq domain, as discussed in Section-3.3, then the filter of Section-3.4.8 cannot be directly used on the measurement of the ifd and ifq currents since they are normally DC values.
Therefore, the phase delay introduced is not the same, due to a different frequency, as for the corresponding AC currents for which the filter is originally intended. The measurement delay is then included as an equivalent time delay since there is no attenuation at the frequency of interest.
Figure 3.10 – Cascaded control structure for the grid-side converter.
The outer loop has the DC current iDC,RSC from the RSC as the disturbance while the output voltage measurement is fed back after filtering with the 2nd order filter presented in Section-3.4.8. The delays due to the converter and the measurement filter have not been included in the design process.
3.5.3.2 Inductance Filter Current Control
The current control loop is based on the model of the inductance filter from Section-3.4.5. Equation (3.4) is rearranged and presented as Equation (3.16).
vfd(s)=−
(
Rf +Lfs)
ifd(s)+vd,grid(s)+Lfωdifq(s) (3.16a) vfq(s)=−(
Rf +Lfs)
ifq(s)+vq,grid(s)−Lfωdifd(s) (3.16b) This gives the process model GLf and the compensation terms, for the cross coupling between the d and q axis current control loops, as given in Equation (3.17) and Equation (3.18) respectively. v’fd, v’fq and vfdcomp, vfqcomp are the controller output voltage and the compensation voltage respectively.(
R 1L s)
) s ( ' v
) s ( i ) s ( ' v
) s ( ) i s ( G
f f fq
fq fd
fd
Lf = = =− + (3.17)
vcompfd (s)=vd,grid(s)+Lfωdifq(s) (3.18a) vcompfq (s)=vq,grid(s)−Lfωdifd(s) (3.18b) The controller gains are determined by the pole-zero cancellation method using Equation (3.17) [Forchetti et al. 2007]. Same gains are used to control the current in each axis [Iov 2003]. The proportional gain Kp,if and the integral gain Ki,if are thus given as
f if , cl if
,
p L
K =−ω (3.19a)
f if , cl if
,
i R
K =−ω (3.19b)
ωcl,if is the closed-loop bandwidth selected. The resulting closed-loop transfer function Gcl,if is given in Equation (3.20), with the time period τcl,if being the inverse of ωcl,if, with a pole at s = -1/ τcl,if . Since this is a first order system, the rise time trise,if is given by Equation (3.21).
1 s ) 1
s ( G
if , cl if
,
cl = +
τ (3.20)
( )
if , cl if , rise
9 t ln
=ω (3.21)
3.5.3.3 DC-link Voltage Control
The DC-link voltage control loop is based on the model of the capacitor given in Section-3.4.4, for which the equivalent Laplace form is given in Equation (3.22).
s C R 1 ) s ( i
) s ( ) v s ( G
DC DC net
DC
CDC = = + (3.22)
Neglecting the converter losses, the DC current iDC,GSC on the GSC side can be related to the current ifq, by power equivalence, with the power on the AC side given as
fq fq fd fd
f v i v i
p = + (3.23)
This is non-linear since ifq itself is controlled by vfq. If we assume that the inductance filter has low losses then the active power at the grid side and at the output of the GSC is equal [Serban 2003] [Lindholm 2003]. Thus
fq DC
grid , q GSC ,
DC i
v
i = v (3.24)
Since vq,grid is constant for an ideal grid and if the resultant vDC will not change much then the ratio of vq,grid to vDC is approximately constant. In order to get the value for this ratio, Equation (3.4) has to be rearranged for ifd and ifq considering steady-state conditions and Rf ≈ 0. Substituting these currents in Equation (3.23) gives
f d
grid , q fd
f L
v p v
= ω (3.25)
Thus, the transfer of active power is dependent upon the phase difference between the grid voltage and the converter voltage rather than the magnitude difference. This means that the required converter voltage magnitude will not be much larger than that of the grid. With vd,grid = 0 and the power-invariant transformation used, vq,grid is equal to the grid line (phase-phase) RMS voltage magnitude and therefore the required converter output voltage, see Appendix-A.
DC ma grid
,
q m v
2 2
v = 3 (3.26a)
DC ma grid
,
q m v
2
v = 1 (3.26b)
Equation (3.26a) and (3.26b) are related to Sinusoidal PWM and SVM respectively [Morales 2006] [Mohan 2001] [Mohan, Undeland & Robbins 2003].
An amplitude modulation index mma equal to 1 gives the linear relationship between iDC,GSC and ifq as
fq ma GSC ,
DC k i
i = (3.27)
kma is 0.612 or 0.707 depending on which modulation strategy is used.
If the inner current loop is very fast compared to the outer loop then its transfer function can be taken as unity otherwise it can be explicitly included in the
design. With the inclusion of the current loop transfer function, the process model for the voltage control loop becomes
( )
(
s 1)
C s1 s C R k ) s ( i
) s ( ) v s ( G
DC if
, cl
DC DC ma fq
DC
vDC +
= +
= τ (3.28)
The PI controller can be designed by specifying phase margin as a design parameter along with the required bandwidth [Astrom & Hagglund 2006].
Separating the process model into real and imaginary parts gives
( )
( )
( )
(
cl2,if)
2 DC
if , cl 2 DC DC ma 2
if , cl 2 DC
if , cl DC DC ma
vDC C 1
1 C
R jk 1
C C R ) k
j (
G ω ω τ
τ ω τ
ω ω τ
− + +
−
= − (3.29)
Similarly, the transfer function of the PI controller with the proportional gain Kp,vDC and integral gain Ki,vDC becomes
ω p,vDC iω,vDC
vDC
jK K
) j (
C = − (3.30)
Let ωcl,vDC be the gain cross-over frequency and фm be the required phase margin.
This means that the open-loop transfer function is given as
( )
m( )
mvDC , cl vDC vDC , cl vDC vDC
, cl vDC ,
ol (j ) G ( j )C ( j ) cos jsin
G ω = ω ω =− φ − φ
(3.31) Solving for Kp,vDC and Ki,vDC gives
(
( ))
)cos (
K 1 m cl,vDC
vDC , cl vDC
,
p φ ψ ω
ω
χ −
−
= (3.32a)
(
( ))
)sin
K ( m cl,vDC
vDC , cl
vDC , cl vDC
,
i φ ψ ω
ω χ
ω −
= (3.32b)
2 vDC , cl vDC vDC
,
cl ) G ( )
(ω ω
χ = (3.32c)
) (
G )
(ωcl,vDC vDC ωcl,vDC
ψ =∠ (3.32d)
3.5.4 Rotor-Side Converter Control
The RSC control has different objectives depending on which stage of operation the WTGS is in. Before any power can be produced, the generator stator voltage has to be synchronized and connected to the grid much like a conventional generator. Therefore, the three modes of RSC control are to generate voltage at the stator terminals, to synchronize with the grid, connect and produce the required power or control the electromagnetic torque as required. Four controllers need to be designed, the inner loop current control, the outer loop voltage control and the power or torque control.
3.5.4.1 The Control Scheme
The control loop structure for the RSC is given in Figure-3.11.
Figure 3.11 – Cascaded control structure for the rotor-side converter.
The control structure for the d axis quantities is similar to the presented one for the q axis. The current control loop has a filter but the delay introduced is relatively less compared to the GSC current loop due to lower fundamental frequencies in the rotor. The reference to the outer loop can be the stator voltage vsd, stator active power ps or the electromagnetic torque Tem. Normally to switch between control objectives, and therefore controllers, requires some method to avoid sudden change in references. Here this ‘bumpless transfer’ is not needed since the references to the current loop do not change when switching controllers, as will be shown in the coming sections [Zhang et al. 2006]. The gains kps and kvs, in Figure-3.11, correspond to the stator power or stator voltage control objective selected for the outer loop respectively. The delays due to the converter and the measurement filter have not been included in the design process.
3.5.4.2 Rotor Current Control
The current control loop is based on the rotor circuit equations of the induction machine from Section-3.4.2. Before the connection of the stator to the grid, both isd and isq are zero which leads to the rotor circuit equations in Equation (3.33).
( )
( ) ( ))
(s R L s i s L i s
vrd = r + r rd − rωdA rq (3.33a)
( )
( ) ( ))
(s R L si s L i s
vrq = r + r rq + rωdA rd (3.33b) This gives the process model Gr and the compensation terms, for the cross coupling between the d and q axis rotor current control loops, as given in Equation (3.34) and Equation (3.35) respectively. v’rd, v’rq and vrdcomp, vrqcomp are the controller output voltage and the compensation voltage respectively.
(
R L s)
1 )
s ( ' v
) s ( i ) s ( ' v
) s ( ) i s ( G
r r rq
rq rd
rd
r = = = + (3.34)
vrdcomp(s)=−LrωdAirq(s) (3.35a) vrqcomp(s)= LrωdAird(s) (3.35b)
The controller gains are determined by pole-zero cancellation using Equation (3.34) [Forchetti et al. 2007]. Same gains are used to control the current in each axis [Iov 2003]. The proportional gain Kp,ir and the integral gain Ki,ir are thus given as
r ir , cl ir ,
p L
K =ω (3.36a)
r ir , cl ir ,
i R
K =ω (3.36b)
ωcl,ir is the closed-loop bandwidth selected. The resulting closed-loop transfer function Gcl,ir is given in Equation (3.37), with the time period τcl,ir being the inverse of ωcl,ir, with a pole at s = -1/ τcl,ir . Since this is a first order system, the rise time trise,ir is given by Equation (3.38).
1 s ) 1
s ( G
ir , cl ir
,
cl = +
τ (3.37)
( )
ir , cl ir , rise
9 t ln
=ω (3.38)
3.5.4.3 Stator Voltage Control
Stator voltage control is based on the stator circuit equations of the induction machine from Section-3.4.2. Before the connection of the stator to the grid both isd and isq are zero, which leads to the stator circuit equations given in Equation (3.39).
vsd(s)=−Lmωdirq(s)+Lmsird(s) (3.39a) vsq(s)=Lmωdird(s)+Lmsirq(s) (3.39b) In steady state, the differential terms will also be zero. This means that transfer function Gvs for the stator voltage control loop is given as
m d
rd sq rq
sd
vs L
s i
s v s i
s s v
G − = = ω
= ( )
) ( )
( ) ) (
( (3.40)
If the grid is ideal then the stator can be synchronized and connected using only the current control loop. However, this is not usually the case. Therefore, a voltage control loop must be designed to achieve perfect synchronization and avoid large stator currents on connection.
The transfer function Gvs can be represented by a constant kvs for control design.
Then using pole-zero cancellation, with the inclusion of the transfer function of the current loop, and a closed-loop bandwidth ωcl,vs, the proportional gain Kp,vs
and integral gain Ki,vs are found to be
vs ir , cl
vs , cl vs ,
p k
K ω
= ω (3.41a)
vs vs , cl vs ,
i k
K ω
= (3.41b)
The signs of the gains for the d axis stator voltage controller have to be opposite to that of the q axis since its transfer function is negative, see Equation (3.40).
3.5.4.4 Stator Power Control
The stator active power ps and reactive power qs are given as
sq sq sd sd
s v i v i
p = + (3.42)
sq sd sd sq
s v i v i
q = − (3.43)
The active and reactive power thus has to be controlled through stator currents which can be manipulated through rotor currents. The stator power control is based on the stator circuit equations of the induction machine, see Section-3.4.2.
To simplify the controller design, it is normally assumed that the voltage drop on the stator resistance is negligible due to its small value [Zhao et al. 2006] [Li &
Chen 2006] [El Aimani et al. 2003] [Yuan, Chai & Li 2004]. For an ideal grid, the stator flux dynamics can be neglected since the stator is directly connected to the grid and the stator flux is almost constant [Iov 2003] [Yuan, Chai & Li 2004]
[Morren et al. 2003] [Morales 2006]. With the stator connected to the grid, vsd = 0, this leads to
sq =0
λ (3.44a)
d sq sd
v
λ =−ω (3.44b)
Using Equation (3.44a) in Equation (3.1f) and simplifying gives
rq s m
sq i
L
i = L (3.45a)
Substituting the value of λsd from Equation (3.44b) into Equation (3.1e) and simplifying gives
rd s m s d
sq
sd i
L L L i =− v +
ω (3.45b)
Using Equation (3.45) in the stator power equations, Equation (3.42) and Equation (3.43), and rearranging for the rotor currents as references leads to [Zhang et al. 2006] [Abo-khalil, Lee & Lee 2006] [Bouaouiche & Machmoum 2006]
s m sq
s
rq p
L v
i = L (3.46a)
m d
sq s m sq
s
rd L
q v L v i L
+ω
= (3.46b)
When power control is activated, the reference values for the stator active and reactive powers must be zero. Therefore, Equation (3.46) will give the same references to the rotor current loop as required in the stator voltage loop, Equation (3.40), and therefore bumpless controller switching is achieved.
The transfer function for the stator active and reactive power is therefore [Bouaouiche & Machmoum 2006] [Zhao et al. 2006]
sq
s m rd
s rq
s
ps v
L L s i
s q s i
s s p
G = = =
) (
) ( ) (
) ) (
( (3.47)
For an ideal grid, the transfer function Gps can be represented by a constant kps
for control design.Using pole-zero cancellation, with the inclusion of the transfer function of the current loop, and a closed-loop bandwidth ωcl,ps, the proportional gain Kp,ps and integral gain Ki,ps are found to be
ps ir , cl
ps , cl ps
,
p k
K ω
= ω (3.48a)
ps ps , cl ps ,
i k
K ω
= (3.48b)
3.5.4.5 Electromagnetic Torque Control
The electromagnetic torque control is based on Equation (3.1i) for the induction machine from Section-3.4.2. Making the same assumptions as for stator power control, and thereby using Equation (3.44), the transfer function between the electromagnetic torque Tem and rotor current component irq is found to be [Zhao et al. 2006]
d sq s m rq
em Te
v L pL s i
s s T
G ( ) 2 ω
) ) (
( = = (3.49)
For an ideal grid, the transfer function GTe can be represented by a constant kTe
for control design.Using pole-zero cancellation, with the inclusion of the transfer
function of the current loop, and a closed-loop bandwidth ωcl,Te, the proportional gain Kp,Te and integral gain Ki,Te are found to be
Te ir , cl
Te , cl Te ,
p k
K ω
= ω (3.50a)
Te Te , cl Te ,
i k
K ω
= (3.50b)
3.6 Comparison of Control Techniques
A performance comparison between the classical PI and the modern state-space LQG control technique is carried out for a 2 MW DFIG based WTGS. The torque and power control strategies are implemented and the comparison is carried out at different wind speeds for the variable speed region and the rated wind speed.
A detailed analysis is provided for the PI controller design. The details of the LQG technique are not discussed here but can be found in Appendix-C, Paper-1 [Zafar et al. 2010] [Galvez-Carillo 2011].
The modelling of the system and the simulations are carried out in the dq domain using Matlab/Simulink. This provides the possibility for fast simulations and the models can be modified as required. On the other hand, SimPowerSystems does not provide the possibility to calculate the initial conditions for a wound-rotor induction machine by load flow, using the ‘Powergui’ utility [Matlab/SimPowerSystems R2008b]. Furthermore, the initial conditions have to be specified for each phase rather than as dq quantities.
The parameter values for the simulated 2 MW, 690 V, 50 Hz, 4-pole, 1950 rpm DFIG based WTGS are given in Table-3.1 [Ackermann 2005] [Zhi & Xu 2007]
[Galvez-Carillo 2011]. The rated values are the base values for per-unit conversion.
Table 3.1 – Model parameter values for the 2 MW DFIG based WTGS [Ackermann 2005] [Zhi & Xu 2007] [Galvez-Carillo 2011].
DFIG
Slip s (at Rated Wind Speed) = -0.30 Stator Resistance Rs = 0.048 p.u.
Stator Leakage Reactance Xls = 0.075 p.u.
Magnetizing Reactance Xm = 3.80 p.u.
Rotor Leakage Reactance Xlr = 0.12 p.u.
Rotor Resistance Rr = 0.018 p.u.
Generator Rotor Inertia Hg = 0.5 s Generator Rotor Damping Dg = 0.02 p.u.
Wind Turbine Rated Wind Speed = 12 m/s Blade Radius Rblade = 37.5 m
Gearbox Ratio = 108 Air Density ρ = 1.225 kg/m3 Turbine Rotor Inertia Ht = 2.5 s Turbine Rotor Damping Dt = 0.02 p.u.
Efficiency Coefficient CP(λTSR,β):
Equation (2.7) DC-link
Capacitor Reactance XCDC = 1.34 p.u.
Capacitor Resistance RDC = 0.004 p.u.
Grid Filter
Filter Reactance Xf = 0.33 p.u.
Filter Resistance Rf = 0.025 p.u.
