Electric Power Systems Research 79 (2009) 1085–1096
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Advanced control of a doublyfed induction generator for wind energy conversion F. Poitiers ∗ , T. Bouaouiche, M. Machmoum Institut de Recherche en Electronique et Electrotechnique de Nantes Atlantique, rue Christian Pauc, 44306 Nantes, France
a r t i c l e
i n f o
Article history: Received 27 March 2007 Received in revised form 19 December 2008 Accepted 25 January 2009 Available online 27 February 2009 Keywords: Doublyfed induction generator Wind turbine Power generation PI, RST and LQG controller
a b s t r a c t The aim of this paper is to propose a control method for a doublyfed induction generator used in wind energy conversion systems. First, stator active and reactive powers are regulated by controlling the machine inverter with three different controllers: proportional–integral, polynomial RST based on pole placement theory and Linear Quadratic Gaussian. The machine is tested in association with a windturbine emulator. Secondly a control strategy for the gridconverter is proposed. Simulations results are presented and discussed for each converter control and for the whole system. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Squirrel cage induction machine is used in several wind energy conversion systems. This machine has proven its efﬁciency due to qualities such as robustness, low cost and simplicity when it is directly connected to the grid [1]. However, the wind turbine must be designed to keep the machine’s speed constant near the synchronous speed. This constraint reduces the possibility to increase the electrical energy produced for high wind speeds. A converter can be used between the stator of the machine and the grid but it is crossed by the full power and must be correctly cooled [5]. To overcome these disadvantages, a solution consists in using a doublyfed induction generator (DFIG) where the rotor is wounded and fed by slip rings (Fig. 1). It is then less robust and more expensive than the squirrel cage one. However, it allows putting a converter between the rotor and the grid which is designed only for a part of the full power of the machine (about 30 %) [2,7]. By controlling correctly this converter, variablespeed operation is allowed and electrical power can be produced from the stator to the grid and also from the rotor to the grid [6]. The control law can be used in order to extract maximum power of the wind turbine for different wind speeds. Many papers have been presented with different control schemes of DFIG. These control schemes are generally based on vector control concept (with stator ﬂux or voltage orientation) with
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classical PI controllers as proposed by Pena et al. in [1] and Poller in [10]. The same classical controllers are also used to achieve control techniques of DFIG when grid faults appear like unbalanced voltages [11,15] and voltage dips [17]. It has also been shown in [13,14] that ﬂicker problems could be solved with appropriate control strategies. Many of these studies conﬁrm that stator reactive power control can be an adapted solution to these different problems. This paper presents a control method for the machine inverter in order to regulate the active and reactive power exchanged between the machine and the grid. The active power is controlled in order to be adapted to the wind speed in a wind energy conversion system and the reactive power control allows to get a unitary power factor between the stator and the grid. Such an approach does not manage easily the compromise between dynamic performances and robustness or between dynamic performances and the generator energy cost. These compromises cannot easily be respected with classical PI controllers proposed in most DFIG control schemes. Moreover, if the controllers have bad performances in systems with DFIG such as wind energy conversion, the quality and the quantity of the generated power can be affected. It is then proposed to study the behaviour of a Polynomial RST controller and Linear Quadratic Gaussian (LQG) controller with state feedback and observer based on the pole placement theory [8]. The three controllers are compared and results are discussed, the objective is to show that complex controllers as LQG can improve performances of doublyfed induction generators in terms of reference tracking, sensibility to perturbations and parameters variations. After this comparison, a control law for the grid inverter is proposed in order
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This electrical model is completed by the mechanical equation: m = e + J
d˝ + f˝ dt
(3)
where the electromagnetic torque e can be written as a function of stator ﬂuxes and rotor currents: e = pp
M ( Ls
qs Idr
−
ds Iqr )
(4)
3. Control strategy of the DFIG
Fig. 1. Doublyfed induction generator connected to the grid.
3.1. Referenceframe to keep the DCbus voltage to a constant value and to have a unitary power factor between the inverter and the grid. Simulation results are presented and discussed and the whole system is then tested. The studied system can be modelled using a fully integrated model for general stability studies but as the aim is to analyze different control concepts, the componentwise model (model where the machine and the converters have their own model) is the best solution [16].
and the electromagnetic torque can then be expressed as follows:
2. Model of the doublyfed induction generator
e = −pp
2.1. Parameters of the machine
3.2. Control strategy
The studied machine is a 2 poles pairs LeroySomer woundedrotor induction machine. Its parameters are given below:
The torque and consequently the active power only depend on the qaxis rotor current component. If the per phase stator resistance is neglected, which is a realistic approximation for medium power machines used in wind energy conversion, the stator voltage vector is consequently in quadrature advance in comparison with the stator ﬂux vector. As shown in Fig. 2, stator voltages are
Electrical: Stator per phase resistance: Rs = 0.455 Rotor per phase resistance: Rr = 0.62 Stator leakage inductance: ls = 0.006 H Rotor leakage inductance: lr = 0.003 H Magnetizing inductance: M = 0.078 H Mechanical: Inertia: J = 0.3125 kg m2 Viscous friction: f = 6.73 × 10−3 N m s−1
⎪ ⎪ Vdr ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Vqr
=
and
s
M Iqr Ls
and
qs
=0
(5)
(6)
ds
Vqs = Vs = ωs
s
⎧ M ⎪ ⎨ P = Vs Iqs = −Vs Iqr Ls
ds
− ωs
qs
qs
+ ωs
ds
dr
− ωr
qr
+ ωr
⎧ ⎪ ⎪ ⎪ ⎨
ds
= Ls Ids + MIdr
qs
= Ls Iqs + MIqr
⎪ ⎪ ⎪ ⎩
dr
= Lr Idr + MIds ⎪ ⎪
qr
= Lr Iqr + MIqs
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ qr ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(1)
⎪ ⎩ Q = Vs Ids = Vs
s
Ls
−
Vs M I Ls dr
(8)
Field oriented control of the DFIG can then be applied with the active and reactive power considered as control variables. The block diagram of the system to be controlled is presented in Fig. 3. It takes into account the transfer function of the inverter used to feed the rotor: Kinv /(1 + pTinv ). We will consider that this inverter dynamic is very fast and that it can be reduced to a unitary coefﬁcient.
dr
⎫ ⎪ ⎪ ⎪ ⎬ (2)
⎪ ⎭
The stator and rotor angular velocities are linked by the following relation: ωS = ω + ωr
(7)
Using Eqs. (1), (2) and (5) the stator active and reactive power can then be expressed only versus these rotor currents as
The application of Concordia and Park’s transformation to the threephase model of the DFIG permits to write the dynamic voltages and ﬂuxes equations in an arbitrary d–q reference frame:
dt d = Rr Idr + dt d = Rr Iqr + dt
ds
Vds = 0
2.2. Equations
⎧ d ⎪ Vds = Rs Ids + ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Vqs = Rs Iqs + d
By choosing a reference frame linked to the stator ﬂux, rotor currents will be related directly to the stator active and reactive power. An adapted control of these currents will thus permit to control the power exchanged between the stator and the grid. If the stator ﬂux is linked to the daxis of the frame we have
Fig. 2. Voltage and ﬂux vectors settings.
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Fig. 5. System with PI controller.
The regulator terms are calculated with a polecompensation method. The time response of the controlled system will be ﬁxed at 10 ms. This value is sufﬁcient for our application and a lower value might involve transients with important overshoots. The calculated terms are
kp = Fig. 3. System to be controlled.
4. Controllers synthesis
ki =
In this section, we have chosen to compare the performances of the DFIG with three different linear controllers. The proportional–integral will be ﬁrst tested and will be the reference compared to the others: Polynomial RST controller and Linear Quadratic Gaussian controller. Before synthesizing these controllers, we will make two considerations:  The ﬁrst one will introduce two perturbations terms (εd and εq ) which represent residues of decoupling terms.  The second one will be to consider the constant term 4 as a perturbation which will have to be rejected by the control law. These simpliﬁcations allow us to consider the multivariable model as two monovariable models as shown in Fig. 4 where Rd and Rq represent the d and qaxis controllers. The perturbation 4 will be measured in order to improve the daxis controller’s behaviour. 4.1. PI regulator synthesis This controller is simple to elaborate. Fig. 5 shows the block diagram of the system implemented with this controller. The terms kp and ki represent respectively the proportional and integral gains. The quotient B/A represents the transfer function to be controlled, where A and B are presently deﬁned as follows:
A = Ls Rr + pLs
M2 Lr − Ls
and
B = MVs
1
Ls Lr −
5 × 10−3 1
M2 Ls
(10)
MVs Ls2 Rr
Lr −
M2 Ls
5 × 10−3 MV L L − s s r
M2 Ls
(11)
It is important to specify that the polecompensation is not the only method to calculate a PI regulator but it is simple to elaborate with a ﬁrstorder transferfunction and it is sufﬁcient in our case to compare with other regulators. We can also note that this regulator presents several disadvantages:  A zero is present in the numerator of the transferfunction.  The integrator introduces a phase difference which can induce instability.  The regulator is directly calculated with the parameters of the machine, if these parameters are varying, the robustness of the system can be affected.  The eventual perturbations are not taken into account and the system has few degrees of freedom to be tuned. 4.2. RST regulator synthesis The RST polynomial regulator seems to be an interesting alternative to the PI because it permits a best control of the compromise between speed and performances. It is based on the pole placement theory. The block diagram of a system with its RST controller is presented in Fig. 6. Our system deﬁned by the transferfunction B/A has Yref as a reference and as a disturbance. R, S and T are polynomials which constitute the controller. In our case, we have
(9) A = Ls Rr + pLs
Lr −
M2 Ls
and
B = MVs
(12)
The closedloop transferfunction of the controlled system is Y=
Fig. 4. Simpliﬁed model of the DFIG with controller.
BT BS Y + AS + BR ref AS + BR
Fig. 6. System with RST controller.
(13)
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By applying the Besout equation, we have D = AS + BR = CF
(14)
where C is the command polynomial and F is the ﬁltering polynomial. In order to have a good adjustment accuracy, we choose a strictly proper regulator. So if A is a polynomial of n degree (deg(A) = n) we must have deg(D) = 2 n + 1 deg(S) = deg(A) + 1 deg(R) = deg(A) To ﬁnd the coefﬁcients of polynomials R and S, the robust pole placement method is adopted with Tc as control horizon and Tf as ﬁltering horizon. We have pc = −
1 Tc
and pf = −
1 Tf
where pc is the pole of C and pf the double pole of F. The pole pc must accelerate the system and is generally chosen 2–5 times greater than the pole of A pa . pf is generally chosen 3–5 times smaller than pc . In our case:
Tc =
1 1 = T =− 4 f 4pa
Ls Lr −
M2 Ls
(16)
4Ls Rr
Perturbations are generally considered as piecewise constant. can then be modelled by a step input. To obtain good disturbance rejections, the ﬁnal value theorem indicates that the term BS/(AS + BR) must tend towards zero: lim p
p→0
S =0 Dp
(17)
To obtain a good stability in steady state, we must have D(0) = / 0 and respect relation (17). The Bezout equation leads to four equations with four unknown terms where the coefﬁcients of D are related to the coefﬁcients of the polynomials R and S by the Sylvester Matrix:
⎛
⎞
⎛
d3 a1 ⎜ d2 ⎟ ⎜ 0 = ⎝d ⎠ ⎝0 1 d0 0
0 a1 a0 0
0 0 b0 0
Fig. 7. Block diagram of the daxis of the system with LQG controller.
(15)
⎞⎛
⎞
0 s2 0 ⎟ ⎜ s1 ⎟ 0 ⎠ ⎝ r1 ⎠ r0 b0
(18)
(19)
Because of S(0) = 0, we can conclude that T = R(0). In order to separate regulation and reference tracking, we try to make the term BT/(AS + BR) only dependent on C. We then consider T = hF (where h is real) and we can write BT BT BhF Bh = = = AS + BR D CF C

y: measurements, e: error output to be regulated, u: control input, w: exogenous input (disturbances and references).
The aim is to ﬁnd an observerbased controller allowing us to derive the control needed (u) to keep nil all control errors (e) from the measurement (y) whatever are the exogenous inputs (w). All exogenous signals (Qref , 3 , 4 ) are considered constant. The LQG controller synthesis is presented below only for the qaxis because the methodology for the daxis is similar. The ﬁrst step in the synthesis of the observer is to write the system as a standard problem:
˙ Pref = 0
(21)
ε˙ q = 0
In order to determine the coefﬁcients of T, we consider that Y must be equal to Yref in steady state. In consequence, BT =1 lim p→0 AS + BR
LQG controller. The control problem is described by the way of four vectors:
(20)
As T = R(0), we conclude that h = R(0)/F(0). 4.3. LQG regulator synthesis It is important to precise that the designed LQG controller has been obtained using the methodology proposed by De Larminat [3]. This approach is interesting because it allows us to reduce to only two the freedom degrees. It makes it easy to handle the tuning of such a control law even if the control problem is multivariable. Fig. 7 shows a block diagram of the daxis of the system controlled by an
⎧
x˙ 1q A11q A12q x1q B1q ⎪ ⎪ + uq + B˛q ˛q ⎪ ⎪ x˙ 2q = 0 A22q x2q 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Aq ⎨
x1 ⎪ eq = ( Ce1q Ce2q ) ⎪ ⎪ x2 ⎪ ⎪ ⎪
⎪ ⎪ ⎪ x ⎪ ⎩ yq = Cyq 1q + Dˇq ˇq
(22)
x2q
where
⎧ ⎪ ⎨ x1q = i qr ⎪ ⎩ x2q =
Pref εq
and
⎧ ⎨ eq = P − Pref
⎩ yq =
P Pref
(23)
The different matrices used in Eqs. (22) and (23) are detailed in Appendix B. The terms ˛ and ˇ (respectively state noise and measurement noise) are taken into account only for the synthesis of the observer (in our case, a Kalman ﬁlter). The second step consists in determining an asymptotical control that must be applied to the system in steady state to ensure reference tracking and disturbance rejections. This control is obtained by resolving Eq. (24). This equation gives the two parameters Taq
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(asymptotical trajectory) and Gaq (asymptotical gain).
A11q Taq − Taq A22q + B1q Gaq = A12q
(24)
Ce1q Taq = Ce2q
The system is reduced to its controllable part q which represents the difference between iqr and its asymptotical trajectory −Ta x2q :
˙ q = A11q q + B1q u˜ q
(25)
e = Ce1 q with u˜ q = uq − uaq
and uaq = −Gaq x2q
(26)
The gain of the feedback loop: u˜ q = −klq,q q is calculated in order to minimize the quadratic criterion:
∞ (qt Qcq q + Uqt Rcq Uq )dt
J=
(27)
0
where is the vector regulation error and v the control input vector. Qcq and Rcq are weighting matrices allowing tuning control loop performances. The term Tc (control horizon) is introduced to adjust Qcq and Rcq :
⎡ Rcq = 1,
Qcq = ⎣Tc
⎤−1
Tc t eA11q t B1q B1q e
At
11q
dt ⎦
(28)
0
After solving the Riccati equation: (29)
t P =0 At11q P + PA11q + Qcq − PB1q Rc−1 B1q
the gain klq,q is given by −1 t Rcq B1q P
(30)
In the same way the observer is designed to insure the minimum of variance for the estimation error in respect to the state and measurement noise variances (Qf and Rf ). Such matrixes can also be considered as tuning parameters to adjust the observer bandwidth and/or the robustness of the closedloop. The observer structure is given below:
⎧ ⎨ xˆ˙ q = A11q A12q xˆ q + B1q uq + Lq (yq − yˆ q ) ⎩
0
structure is given by Eq. (34):
⎧ x1d ⎪ ⎪ ud = −( klq,d klq,d Tad + Gad ) ⎪ ⎪ x2d ⎪ ⎨
(Rd ) : A11d A12d B1d xˆ˙ d = ud + Lq (yd − yˆ d ) xˆ d + ⎪ ⎪ 0 A 0 ⎪ 22q ⎪ ⎪ ⎩
A22q
0
(31)
where
⎧ x1d = id ⎪ ⎨ ⎪ ⎩ x2d =
Qref εd 4
and
⎧ eq = Q − Qref ⎪ ⎨ ⎪ ⎩ yq =
Q Qref 4
5.1. Direct and indirect control The aim of the control is to have measured active and reactive powers equal to the reference values. These powers must then be collected. In order to measure only the rotor current, we can use
After solving the Ricatti equation applied to the observer the gain Lq is given by Lq = Qf Cyt Rf−1
(32)
Finally the qaxis controller is given by Eq. (33):
⎧ x1q ⎪ ⎪ uq = −( klq,q klq,q Taq + Gaq ) ⎪ ⎪ x2q ⎪ ⎨
xˆ˙ q = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
A11q 0
A12q A22q
xˆ q +
B1q 0
uq + Lq (yq − yˆ q )
(33)
yˆ q = Cyq xˆ q
The same methodology is used for synthesis of the daxis controller but the perturbation 4 must be taken into account. The
(35)
5. Performances of the controllers
yˆ q = Cyq xˆ q
(Rq ) :
(34)
yˆ d = Cyd xˆ d
∃P = P t > 0 as :
klq,q =
Fig. 8. Principle of direct and indirect control.
Fig. 9. Feedback loops for the two control methods.
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Fig. 10. Reference tracking (indirect control).
Fig. 11. Effect of a speed variation (indirect control).
F. Poitiers et al. / Electric Power Systems Research 79 (2009) 1085–1096
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Fig. 12. Effect of machine’s parameters variation (indirect control).
an indirect control method by comparing Pref and Qref respectively to −Vs (M/Ls )Iqr and (Vs s /Ls ) − (Vs M/Ls ) in accordance with Eq. (8). Fig. 8 shows the two methods of control for the qaxis and Fig. 9 shows the feedback loops for the two methods. The three controllers have been tested using the two direct and indirect control method. The indirect control mode is based on the rotor currents measurements and gives then best results for the DFIG control (in spite of small static error). In order to minimize the number and complexity of the presented results, the differences between the three controllers will be then showed only for the indirect control. 5.2. Reference tracking The ﬁrst test investigated to compare the three controllers is reference tracking by applying stator active and reactive power steps
(respectively −5 kW and 2 kVAR) to the DFIG, while the machine’s speed is maintained constant at 1450 rpm. The machine is considered as working over ideal conditions (no perturbations and no parameters variations), The results for the indirect control mode are presented in Fig. 10. In this control mode, transient oscillations due to the coupling terms between the two axes and static error appear on active and reactive powers. This is inherent to the adopted control mode: the feedback signal of the controller is calculated using rotor currents and considering that stator resistance is neglected. It can be seen that transient oscillations amplitude are minimized with LQG controller witch have better rejection of perturbations. However, it is important to note that RST and LQG controllers are very sensitive to their tuning parameters (ﬁltering and control horizon). Therefore we can consider that the three controllers have almost equivalent behaviour for this test.
Fig. 13. Device studied for test with turbine emulator.
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Fig. 14. Active and reactive power control with turbine emulator.
5.3. Sensibility to perturbations
stability and quality of the generated power when the speed is varying.
The aim of this test is to analyze the inﬂuence of a speed variation of the DFIG on active and reactive powers. The active and reactive power references are maintained to −5 kW and 2 kVAR and at time = 2 s the speed varies from 1350 rpm to 1450 rpm. The results are shown in Fig. 11. This ﬁgure shows the limits of the PI controller which is only based on the machine’s parameters and does not take into account any disturbances. Indeed, for this controller, a speed variation induces an important variation of the active and reactive powers (20% for active power and 10% for reactive power). The RST controller includes the presence of perturbations in its synthesis, so it shows better disturbance rejection than PI controller (11% of active power variation and 6% of reactive power variation). This rejection is still not perfect because the synthesis of the controller includes parameters which have no explicit inﬂuence on its behaviour and are difﬁcult to be tuned. The LQG controller has a nearly perfect speed disturbance rejection, indeed, only small power variations can be observed (less than 1% for active and reactive powers). This result is interesting for wind energy applications to ensure
5.4. Robustness The aim of this test is to analyze the inﬂuence of the DFIG’s parameters variations on the controller’s performances. The machines’ model parameters have been deliberately modiﬁed with excessive variations: the values of the stator and the rotor resistances RS and Rr are doubled and the values of inductances Ls , Lr and M are divided by 2. The DFIG is running at 1350 rpm, active and reactive power steps of respectively −5 kW and 2 kVAR are applied at 2 s and 2.5 s and the speed is increased to 1450 rpm at t = 2.8 s. The obtained results are presented in Fig. 12. These results show that parameters variations of the DFIG increase the timeresponse of the PI and RST controllers but not the LQG controller’s one. The transient oscillations due to the coupling terms between the two axes are always present for all controllers but their amplitudes have not been increased compared to the test with no parameters variation. However, a static error on reactive power appears when the value of the active power is changed
Fig. 15. Block diagram of the front end converter control.
F. Poitiers et al. / Electric Power Systems Research 79 (2009) 1085–1096
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Fig. 16. Behaviour of the DCbus voltage with speed and power variations.
but it keeps reasonable values (no more than 10% for all controllers). 5.5. Test with a turbine emulator The study carried out herein consists in integrating the model of the doublyfed induction generator in a complete wind energy generating system [9]. The ﬁrst step is to model the wind speed variations starting from its spectral decomposition [4]. This wind model is then applied to a small 10 kW wind turbine. The resulting torque thus generated is applied to the DFIG. The active power produced by the DFIG is controlled in order to keep an optimal rotation speed for the wind turbine characteristics and the reactive power reference is maintained to zero. The complete synopsis of the studied system is presented in Fig. 13 and the results are presented in Fig. 14. The behaviour of the active power variations is similar for the three controllers because the machine has no parameters variations and the
speed as well as the active power reference, have slow variations. However, it appears clearly that the reactive power has limited variations around zero while using LQG controller. The power factor is then maintained around 1, which improves the quality of the energy produced. 6. Control of the grid inverter In order to test the behaviour of the whole device presented in Fig. 1, the control of the grid inverter has been integrated in the system. The block diagram of the converter with its controllers [12] is presented in Fig. 15. This system can be described by the following equation:
Va Vb Vc
=R
ia ib ic
d +L dt
ia ib ic
+
Vainv Vbinv Vcinv
(36)
which can be written in a rotating reference frame by using Park transformation:
⎧ ⎪ ⎨ Vd = RId + L dId − ωs LIq + Vdinv dt
⎪ ⎩ Vq = RIq + L dIq + ωs LId + Vq inv
(37)
dt
The direct axis current is used to regulate the reactive power and the quadrature axis current is used to regulate the DCbus voltage. The reference frame is considered oriented along the stator voltage vector. This method gives possibility to make independent control of the active and reactive powers between the front end converter and the supply side. The voltage components in the Park frame Vq and Vd are given at the output of the PI regulator:
⎧ ⎪ ⎨ Vd = RId + L dId dt
Fig. 17. Block diagram of the whole system.
⎪ ⎩ Vq = RIq + L dIq dt
(38)
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Fig. 18. Whole system simulation: blade’s pitch angle, wind speed and relative speed.
Fig. 19. Whole system simulation: statoric active and reactive power, DFIG’s rotor current, DFIG’s speed and DCbus voltage.
F. Poitiers et al. / Electric Power Systems Research 79 (2009) 1085–1096
In the Park reference frame, the voltage source components are Vd = 0 and Vq = Vs and the powers can be written as P = Vq Iq
and
Q = Vq Id
(39)
By neglecting the converter losses, we have Vdc Idc = Vq Iq Vdc C
and
C
dVdc = Idc − Im dt
dVdc = P − Pm dt
(40)
Va Vb Vc
1 = 3
2 −1 −1 2 −1 −1
speeds (synchronous speed is 1500 rpm with a speed range from 670 rpm to 1670 rpm). The rotor current phase is consequently modiﬁed during these variations. Besides, the DCbus voltage is kept around 700 V with less than 5% variations. The stator active power produced by the DFIG is controlled according to the MPPT strategy and is limited to 7.5 kW which represents the nominal power of the DFIG while the stator reactive power is maintained to zero.
(41) 8. Conclusion
The machine inverter is still controlled in order to regulate the both active and reactive stator powers with the PI controller and the two inverters are included in the simulations by using the following models:
1095
−1 −1 2
F1 F2 F3
VDC
(42)
where F1 , F2 , F3 are the logical functions which traduce the states of the inverter switches (1 when the high switch of an inverter arm is closed, 0 when it is opened) and VDC the voltage of the DC bus. This DCbus voltage is linked to the RMS value of the inverted voltage V through the relationship: E V =r √ 2 2
(43)
where r is the modulation index. The value of the capacitance C is ﬁxed to 1500 F and the DCbus voltage is regulated to 600 V. The behaviour of the DCbus voltage is presented in Fig. 16 together with the active and reactive stator powers and the speed variations. We can notice that these steps have a poor inﬂuence on the DCbus voltage variations due to the high value of the capacitance C and the DCbus controller. The oscillations on the active and reactive power measurements are due to the inverters commutations which are taken into account in this simulation. 7. Simulation of the whole system Simulation of the whole system has been realized using Matlab Simulink. The block diagram of this simulation is presented in Fig. 17. The DFIG model is the same as the previously used one. It is associated with the turbine emulator which is now controlled with MPPT (Maximum Power Point Tracking) strategy. In this control mode, three running areas can be distinguished:  For low wind speeds, the DFIG is controlled at variable speed in order to maintain the power coefﬁcient of the wind turbine to its optimal value.  For medium wind speeds, the DFIG is controlled in order to maintain the rotational speed to its nominal value.  For high wind speeds, the DFIG is controlled in order to produce its nominal power and the power extracted from the wind turbine is limited by adjusting the blade’s pitch angle. This way of control is illustrated by Fig. 18 where the pitch angle of the turbine blades ˇ is changing only when the wind speed v is greater than 8 m s−1 . Before this speed the relative speed of the wind turbine is maintained to a constant value corresponding to the optimal power coefﬁcient of the wind turbine. The rotor converter of the DFIG is controlled by the same way as presented previously. The grid inverter is controlled with proportional–integral controllers in order to maintain the DCbus voltage to 700 V and to get a unitary powerfactor. The results of the whole system simulation are shown in Fig. 19 where the DFIG is used at subsynchronous and supersynchronous
In this paper, we have presented a complete system to produce electrical energy with a doublyfed induction generator by the way of a wind turbine. The studied device is constituted of a DFIG with the stator directly connected to the grid and the rotor connected to the grid by the way of two converters (machine inverter and grid inverter). The control of the machine inverter has been presented ﬁrst in order to regulate the active and reactive powers exchanged between the machine and the grid. Two methods of control are described, the ﬁrst one is based on the measured active and reactive powers compared to their references (direct control) and the second one, which is used in the simulations, is based on the calculated active and reactive powers from the rotor currents measurements (indirect control). Three different controllers are synthesized and compared. In term of power reference tracking with the DFIG in ideal conditions (no parameters variations and no disturbances), the performances of the three controllers are similar. When the machine’s speed is modiﬁed (witch represents a perturbation for the system), the impact on the active and reactive powers values is important for PI and RST controllers whereas it is almost nonexistent for LQG one. A robustness test has also been investigated where the machine’s parameters have been modiﬁed. These changes induce timeresponse variations with PI and RST controllers but not with LQG controller. The static error of about 10% appears on the reactive power when the active power is modiﬁed but it is due to the indirect control mode and it can be numerically compensated in future works. The grid inverter has also been controlled with a PI controller in order to maintain the DCbus voltage to a constant value. A test has been investigated with speed, active and reactive power variations showing negligible variations of the voltage. The last part of the paper shows the possibility to simulate the whole system behaviour with a turbine emulator controlled with the MPPT strategy. The presented results show that robust control method as LQG can be a very attractive solution for devices using DFIG such as wind energy conversion systems. Indeed, most of the studied DFIG control schemes use classical PI controllers but the comparison done in this paper show that the limits of this type of controller can have negative effects on the quality and the quantity of the generated power. Using a polynomial RST can be a solution but its tuning parameters are difﬁcult to adjust and disturbances can be not perfectly rejected. The LQG controller is no more complex than the RST one for a numerical implementation and it is more attractive in terms of tuning parameters and provided results. All the simulations have been elaborated with a ﬁxedstep size of 0.1 ms in order to consider digital implementation in future works. Appendix A. List of symbols
Vds , Vqs , Vdr , Vqr stator and rotor dq voltages Ids , Iqs , Idr , Iqr stator and rotor dq currents Rs , Rr stator and rotor per phase resistance qs , qr stator and rotor dq ﬂuxes dr , ds , stator ﬂux position
s
electrical angle
1096
F. Poitiers et al. / Electric Power Systems Research 79 (2009) 1085–1096
Ls , Lr M pp m J f p g ωs , ωr
cyclic stator and rotor inductances magnetizing inductance number of poles pairs generator mechanical torque generator inertia viscous friction coefﬁcient Laplace variable induction machine’s slip Stator and rotor angular velocities
Appendix B Matrixes used in LQG standard problem:
Aq =
Bq =
A11q 0
B1q 0
Cyq =
Ad =
Bd =
⎛ =⎝
−MVs Ls 0
Ceq = ( Ce1q
A12q A22q
=⎝
−
Rr Ls Ls Lr − M 2 0 0
Ls Ls Lr − M 2 0 0
0 0 0
Ls Ls Lr − M 2 0 0
⎞ ⎠
⎞ ⎠ ; B˛q = I3
0
0
Dˇq = I2
;
1 0
Ce2q ) =
A11d 0
B1d 0
⎛
A12d A22d
⎛ ⎜ ⎝
=⎜
+MVs Ls
⎛ ⎜ ⎝
=⎜
−
1
0
Rr Ls Ls Lr − M 2 0 0 0
Ls Ls Lr − M 2 0 0 0
0 0 0 0
Rr Ls Ls Lr − M 2 0 0 0
⎞ 0
⎟
0⎟ ⎠ 0 0
⎞ ⎟ ⎟ ; B˛d = I4 ⎠
References [1] R. Pena, J.C. Clare, G.M. Asher, A doubly fed induction generator using back to back converters supplying an isolated load from a variable speed wind turbine, IEE Proceeding on Electrical Power Applications 143 (September (5)) (1996). [2] S. Heier, Grid Integration of Wind Energy Conversion Systems, John Wiley & Sons Ltd, England, 1998. [3] P. De Larminat, Le contrôle d’états standard, Collection pédagogique d’automatique, Hermes, 2000.
[4] C. Nichita, D. Luca, B. Dakyo, E. Ceanga, N.A. Cutululis, Modelling nonstationary wind speed for renewable energy systems control, The Annals of “Dunarea de Jos” University of Galati Fascicle III, 2000, pp. 29–34. [5] D. Schreiber, State of the art of variable speed wind turbines, in: 11th International Symposium on Power Electronics, Novi Sad, Yugoslavia, 31 October–2 November 2001. [6] F. Poitiers, M. Machmoum, R. Le Doeuff, M.E. Zaim, Control of a doublyfed induction generator for wind energy conversion systems, International Journal of Renewable Energy Engineering 3 (December (3)) (2001) 373–378. [7] R. Datta, V.T. Ranganathan, Variablespeed wind power generation using doubly fed wound rotor induction machine—a comparison with alternative schemes, IEEE Transactions on Energy Conversion 17 (September (3)) (2002) 414–421. [8] C. Darengosse, F. Poitiers, M. Machmoum, Advanced control of a doublyfed induction machine for variablespeed wind energy generation, in: EPE 2003, Toulouse, France, 2–4 September 2003, CDROM Proceedings. [9] F. Poitiers, M. Machmoum, R. Le Doeuff, Simulation of a wind energy conversion system based on a doublyfed induction generator, in: EPE 2003, Toulouse, France, 2–4 September 2003, CDROM Proceedings. [10] M.A. Poller, Doublyfed induction machine models for stability assessment of wind farms, in: Power Tech Conference Proceedings, 2003, IEEE, Bologna, vol. 3, 23–26 June 2003. [11] T. Brekken, N. Mohan, A novel doublyfed induction wind generator control scheme for reactive power control and torque pulsation compensation under unbalanced grid voltage conditions, in: IEEE 34th Annual Power Electronics Specialist Conference, 2003, PESC ‘03, 15–19 June 2003, vol. 2, pp. 760–764. [12] T. Bouaouiche, M. Machmoum, F. Poitiers, Doubly fed induction generator with active ﬁltering function for wind energy conversion system, in: EPE 2005, Dresden, Germany, 11–14 September 2005, CDROM Proceedings. [13] T. Sun, Z. Chen, F. Blaabjerg, Flicker study on variable speed wind turbines with doubly fed induction generators, IEEE Transactions on Energy Conversion 20 (December (4)) (2005) 896–905. [14] L. Piegari, R. Rizzo, A control technique for doubly fed induction generators to solve ﬂicker problems in wind power generation, in: International Power and Energy Conference, Putrajaya, Malaysia, 28 and 29 November 2006, pp. 19–23. [15] T.K.A. Brekken, N. Mohan, Control of a doubly fed induction wind generator under unbalanced grid voltage conditions, IEEE Transaction on Energy Conversion 22 (March (1)) (2007) 129–135. [16] CIGRE Working group C4.601, Modeling and dynamic behavior of wind generation as it relates to power System Control and Dynamic Performance, August 2007. [17] J. Lopez, P. Sanchis, X. Roboam, L. Marroyo, Dynamic behavior of the doubly fed induction generator during threephase voltage dips, IEEE Transaction on Energy Conversion 22 (September (3)) (2007) 709–717. F. Poitiers was born in Nantes, France, on 1973. He received the master degree of science in 1997 from Nantes University. and his Ph.D. degree at the Polytechnic Institute of Nantes University in Electrical Engineering in 2003. He joined the Technological University Institut of Nantes in 2004 as “Maître de Conférences”. He studies generator and converters associations for wind energy conversion systems. T. Bouaouiche was born in Hussein Dey, Alger, Algeria in 1979. He received the engineering degree from University of Science and Technology Houarie Boumedienne of Alger in 2001 and he received the master degree of science from Ecole Polytechnique of Nantes University in 2003. He is currently a Ph.D. student in electrical engineering at the Ecole Polytechnique de l’Université de Nantes. His research interests are wind energy conversion systems and power quality. M. Machmoum was born in Casablanca, Morocco, on November 29, 1961. He received the Diplo. Ing. Degree from the Institut Supérieur Industriel of Liège, Belgium, in 1984, and the Ph.D. degree from the Institut National Polytechnique of Lorraine (INPL), France, in 1989, all in electrical engineering. In 1991 he joined l’Ecole Polytechnique de l’Université de Nantes as a “Maître de Conférences” and he is now “Professeur des Universités”. His main area of interest includes power quality applications, tidal marine and wind energy conversion systems and power line communications.