# Doubly-fed wind turbine generator control: A bond graph approach

## Doubly-fed wind turbine generator control: A bond graph approach

Simulation Modelling Practice and Theory 53 (2015) 149–166 Contents lists available at ScienceDirect Simulation Modelling Practice and Theory journa...

Simulation Modelling Practice and Theory 53 (2015) 149–166

Contents lists available at ScienceDirect

Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat

Doubly-fed wind turbine generator control: A bond graph approach Roberto Tapia ⇑, A. Medina Facultad de Ingeniería Eléctrica, División de Estudios de Posgrado, Universidad Michoacana de San Nicolás de Hidalgo, 58000 Morelia, Mexico

a r t i c l e

i n f o

Article history: Received 27 March 2014 Received in revised form 3 February 2015 Accepted 5 February 2015 Available online 28 February 2015 Keywords: Wind turbine Control Bicausality Bond graph

a b s t r a c t In this contribution, a bond graph doubly-fed wind turbine generator control is proposed. The control law is derived from the inverse model of the doubly-fed induction generator, which allows a different structure for the torque control to be obtained. The bond graph methodology and the concept of bicausality are applied to derive the control law. The robustness of the proposed control is veriﬁed and the simulation of the complete model is conducted for constant and variable wind speed operation conditions, respectively. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction The possibility of controlling active and reactive power, the capability of reducing stresses of the mechanical structure and acoustic noise are some of the advantages of using a doubly-fed induction generator (DFIG) in a wind turbine [1]. Also, losses in the power electronics converter are reduced, as compared to a direct-driver synchronous generator. This is due to the fact that the converter placed between the grid and the induction machine rotor handles only a fraction of the turbine rate power [2,3]. Control of a doubly-fed induction generator has been addressed in several works, e.g. [4–7]. In [4] a sliding mode control is used, and in [5] the power control of a doubly-fed induction machine via output feedback is presented. The behavior of such machines in large wind farms, along with the general active and reactive power control has been addressed in [6]. A novel simpliﬁed model of the DFIG appropriate for bulk power system studies is presented in [7]. In this paper, a different structure for the DFIG control, based on the bond graph methodology [8–10] is proposed. The wind turbine is a complex system in which different technical areas are involved (mechanics, aeronautics, electrical, among others). In order to analyze the system in the same reference frame, the bond-graph methodology can represent the whole structure. This methodology presents some proprieties that can be directly applied to the model reported in [11]. A bond graph consists of subsystems linked together by half arrows, representing power bonds [9]. They exchange instantaneous power at places called ports. The variables are forced to be identical when two ports are connected; the power variables are assumed to be functions of time. The different power variables are classiﬁed into an universal scheme, and are called either effort e(t) or ﬂow f(t). Their product P(t) = e(t)f(t) is the instantaneous power ﬂowing between the ports. The main advantages of the bond graph tool for modeling purposes is summarized through few keywords, which makes this approach quite speciﬁc and justiﬁes its use in the paper [12]. These are the following: ⇑ Corresponding author. E-mail addresses: [email protected]ﬁe.umich.mx (R. Tapia), [email protected] (A. Medina). http://dx.doi.org/10.1016/j.simpat.2015.02.005 1569-190X/Ó 2015 Elsevier B.V. All rights reserved.

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Nomenclature DFIG MSC NSC PLL P Q PI Md/q Ls Lm Lr Jm Rs Rr p

xs Tnom /s/r,d/q is/r,d/q

vrd/q vd/q br R b Lr k1,2 n Jhole Cp

doubly-fed Induction generator machine side converter network side converter phase lock-loop active power reactive power proportional integral controller I-ﬁeld magnetic coupling (axis d/q) stator self-inductance mutual inductance rotor self-inductance moment of inertia stator resistance rotor resistance number of pair of poles network angular frequency nominal torque ﬂuxes of stator/rotor in axis d/q currents of stator/rotor in axis d/q voltages of rotor in axis d/q voltages of stator in axis d/q estimated of Rr Estimated of Lr controllers gearbox ratio inertia power coefﬁcient

 It provides the analyst with a uniﬁed graphical language to represent power exchanges with a physical insight, energy dissipation and storage phenomena in dynamic systems of any physical domain.  It allows the visualization of causality properties before writing equations, according to selected modeling hypotheses.  Some software exists with a bond graph graphical editor, thus exempting the analyst from writing global equations. In this contribution, the bond graph is used to model the DFIG; with the concept of bicausality [13] being applied to obtain the control law. The outline of the paper is as follows: the wind turbine model is ﬁrst detailed. The induction machine model is recalled, and then simpliﬁed in Section 3; the proposed control is described in Section 4. The models of wind turbine and converter, respectively, are presented in Section 5, and the whole system simulated in Section 6. The global conclusions of the conducted investigation are drawn in Section 7. 2. System description From the generator point of view, a wind turbine has different conﬁgurations. Also, the wind turbine can operate with either under ﬁxed-speed or variable-speed mode. This operation depends directly on the generator connection. It means that for ﬁxed-speed wind turbines, the generator is directly connected to the power network, since the speed is closely tied to the grid frequency. Besides, for a variable-speed wind turbine, the generator is controlled through power electronics converters, which make possible to control the rotor speed. Each conﬁguration has its own advantages and is used under different operation conditions. Four basic conﬁgurations are often described in the literature: 1. 2. 3. 4.

Fixed-speed wind turbine with an induction generator. Variable-speed wind turbine with a cage-bar induction generator or synchronous generator. Variable-speed wind turbine with multiple-pole synchronous generator. Variable-speed wind turbine with doubly-fed induction generator.

For a ﬁxed-speed wind turbine, the rotor speed is in principle determined by a gearbox and the number of generator polepairs number. In this conﬁguration, the connection to the power network is directly made, as shown in Fig. 1. A two-windings generator having different ratings and pole-pairs is normally used in this conﬁguration.

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Fig. 2 shows a wind turbine connected to the power network via a power electronic converter. In this conﬁguration, the maximum rotor speed is rated to the generator speed with the use of a gearbox. The principal advantage of this conﬁguration is that the optimal power delivered by the wind can be extracted. The use of ‘‘back-to-back’’ power converter connections allows to appropriately handling power network variables and generator frequency. This conﬁguration is also used without the gearbox. For this purpose, it is necessary to use a synchronous generator with multiple poles. This allows to reduction of the friction losses and the vibrations introduced by the gearbox. As it was mentioned before, the doubly-fed induction generator (DFIG) offers an economical beneﬁt to the variable-speed wind turbines (Fig. 3). The stator is directly connected to the power network, while the rotor is connected through slip rings to a power electronic converter. The main advantage of this conﬁguration is the fact that the power electronic converter has to handle only a fraction (30%) of the total power [2]. Therefore, the losses in the power electronic converter can be reduced. Two power electronics converters, machine side converter (MSC) and network side converter (NSC), respectively, are used in order to have a DC-link between them, thus allowing the power transfer. With the MSC, it is possible to control the torque or speed in the DFIG and the power factor at the stator terminals, while with the NSC functions the DC-link voltage is kept constant. In the following section a model of the induction generator is developed. 3. Induction machine model 3.1. Induction machine model Induction machines have been addressed in many publications. Models can be represented in two general frameworks: one using a Park reference frame [14–16], and the other one using the natural reference frame (three sinusoidal waveforms) [17]. Fig. 4 shows a bond graph model of the induction machine in the Park 2-axis d–q reference frame [15].

Fig. 1. Fixed-speed wind turbine.

Fig. 2. Variable-speed wind turbine with an induction or synchronous generator.

Fig. 3. Variable-speed wind turbine with a doubly-fed induction generator (DFIG).

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Fig. 4. 2-axis reference frame induction machine.

The induction machine model is based on the following assumptions: – – – –

Magnetic hysteresis and magnetic saturation effects are neglected. The stator winding are sinusoidally distributed along the air–gap. The stator slots cause no appreciable variation of the rotor inductances with rotor position. An arbitrary dq-frame rotating around the homopolar 0-axis to the speed xs is chosen. The model was developed by using the equivalent electric circuit of the induction machine shown in Fig. 5 [15]. The equations that describe the bond graph model of Fig. 4 are given here after:

v d ¼ Rs isd þ d/dt  xs /sq v q ¼ Rs isq þ d/dt þ xs /sd v rd ¼ Rr ird þ d/dt  xs /rq þ pX/rq v rq ¼ Rr irq þ d/dt þ xs /rd  pX/rd sd

sq

rd

ð1Þ

rq

s11 ¼ pð/rq ird  /rd irq Þ s11 ¼ Jm ddtX  T m The inductance matrices for I-ﬁeld elements Md and Mq allow coupling the stator and rotor ﬂuxes by:

/sd=q /rd=q

!

 ¼

Ls

Lm

Lm

Lr



isd=q ird=q

 ð2Þ

where Ls, Lm and Lr are the stator self-inductance, mutual inductance between the stator and rotor, and rotor self-inductance, respectively. I-ﬁeld element Jm represents the shaft and rotor moment of inertia; Rs and Rr are stator and rotor resistances, p is the number of pole pairs and x corresponds to 2pf, with f being the network frequency. It is important to remark that the four modulated sources (dependent on xs) represent virtual sources (not physical sources), since their power sum is zero and are only a mathematical consequence of the model, as demonstrated in [15]. They are considered by the arbitrary framework assumption and can be removed if the stationary frame is chosen (setting axis-d with stator phase a) since xs = 0. As the machine model is related to the rotor, the stator equations are not inﬂuenced by the rotor speed. It is known that causal path and causal loops allow getting the mathematical expression directly from the bond graph. In this case, the induction machine model shows four virtual MSe sources which are related to magnetic ﬂuxes (/rd, /rq, /sd, /sq). Causal paths need to be taken into account in order to formulate the complete causal path model. Since the bond graph machine model has I-ﬁeld multiports, the state variables are obtained by using the methodology proposed in [18], which

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Fig. 5. Induction machine electrical circuit in dq0 arbitrary frame.

Rr

Ls Lm

R

6

Lm

1

m11 m 21

Lr

Id

5

ird 1 8

p

R

2

4

1

1isd 3

M Se

9

MGY

2

Rs

m12 m22

M Se

36 35

23

Jm I

1

33

10

Se Tm 11

sq

rq

1

34

32

37

30

s

s s

22 12

MGY

p

-1

39

38

31

MSe

13

15

-1

MSe 21

16

irq1

s sd

rd

Iq

2

17

1

Rr R

18

i

1 sq 19

R

Rs

Fig. 6. Causal loops and paths stator axis-d.

allows obtaining the causal loops and paths when a multiport is used. Fig. 6 shows some causal loops and paths for the axis-d stator. The rest of causal loops (cli) and causal paths (cpi) for this axis are given in Table 1. Since symmetrical induction machine model is used (from the graphical point of view), the trajectories shown in Table 1 are the same for the q-axis.

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Table 1 Trajectories and gains of some causal loops and paths. Elements

Trajectories

Gain

Elements

Trajectories

Gain

I1d I1d I1d I1d I2d I2d

4–2–2–4 4–5–6–6–5 30–31–21–18 4–5–9–10–11–22 5–6–6–5 5–4–2–2–4

cl1 ¼ Rs  m11 cp1 ¼ Rr  m21 cp2 ¼ xS cp3 ¼ /rq  p  m21

I2d ! I2q I2d ! I2q I2d ! Jm I2d ! Jm J m ! I2d J m ! I2q

5–37–39–13–17 5–37–38–16–17 5–9–10–11–22 5–37–39–12–11–22 22–11–10–9–5 22–11–12–13–17

cp5 ¼ /rd  p  f 12 cp6 ¼ xs cp7 ¼ /rq  p

\$ Rs ! I2d ! I1q ! Jm \$ Rr ! I1d

cl2 ¼ Rr  m22 cp4 ¼ m12  Rs

cp8 ¼ /rd  p  f 13 cp9 ¼ /rd  p=Jm cp10 ¼ /rq  p=J m

As the numerical value of ﬂuxes /rq, /rd, variable f12 (speed), f17 (rotor current q-axis), and f4 (rotor current d-axis) are required in order to analyze the dynamics of the loop gains, the linearization process around an equilibrium point is conducted. However, a close approximation of stator and rotor dynamics in the machine can be achieved. From the bond graph of Fig. 6, the state variables can be decomposed into slow and fast variables. This is achieved by using the causal loops concept, which gives an estimation of the time constants and natural frequencies present in the model. Table 2, shows the d-axis variables. Values of Table 2 have been calculated by taking into account the machine parameters of Table 3. It can be observed from Table 2 that the stator presents the fastest dynamic (loop 1). It is important to notice that the difference between loops 1 and 2 is not very signiﬁcant. 3.2. Model simpliﬁcation The dynamic model can be simpliﬁed through an algebraic equivalent. This can be achieved if the dynamics are eliminated, the transient is cancelled and the order of the model is reduced. Thus, by setting /_ sq and /_ sd equal to zero, the model order is reduced from 5 to 3. The methodology shown in [19] can be applied to the bond graph model of the induction machine, in order to reduce the order model from the energy metric point of view. However, since the application of the induction machine in the wind turbine (as a generator) does not involve a high rotor speed, it is assumed that the neglected mode cannot be excited. Regarding Fig. 4, the causality change is required in bonds 4 and 19. Fig. 7 shows the stator d–q-axis simpliﬁed model of the induction machine. Effort in bond 4 is zero, since the stator dynamics are eliminated. As the current isd is used to calculate the ﬂux /sd (necessary for the MSe source along q-axis), the same structure is kept, and only the causality is changed. A similar process is made in bond 19. Causality changes require a modiﬁcation of (2). Matrix expressions in the I-ﬁeld elements are changed to:



/sd;q ird;q



0 ¼@

2

Ls  LLmr

Lm Lr

 LLmr

1 Lr

1 A

isd;q

! ð3Þ

/rd;q

Causality changes reduce from 5 to 3 the number of independent state variables in the model. The dynamic equation in the stator part is changed for a static one, and the effort is set equal to zero in these bonds.

Table 2 Time constant of causal loops. No.

Elements involved in the causal loop

1 2 3

Ls Lr Jind

Rs Rr Lr or Ls

Gain (s)

Time constant (s)

Rs/Ls ⁄ s Rr/Lr ⁄ s /2r,s/Jind ⁄ Lr,s ⁄ s2

Ls/Rs = 11.44 Lr/Rr = 13.66 (2p ⁄ /r,s)/sqrt(Jind ⁄ Lr,s) = 7.058

Table 3 Induction machine parameters. Power Poles numbers Voltage Stator resistance Rotor resistance

2 MW 2 690 V 0.0022 O 0.0018 O

Stator inductance Rotor inductance Mutual inductance Inertia Frequency

25.16 mH 24.58 mH 2.9 mH 53.87 kgm2 60 Hz

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Fig. 7. Stator d, q-axis simpliﬁed induction machine.

In order to compare the behavior of the simpliﬁed model against the complete one, a simulation is carried-out by considering the induction machine as a motor. A squirrel cage machine conﬁguration is used in both models. It means that voltages vrd and vrq (Fig. 4) are considered with a zero value. Also, the Park transformation [20] is added. The machine is started at instant t = 0 s, then at t = 5.5 s the nominal torque is applied (Tnom = 10,685 Nm). After that, at t = 6.0 s a voltage reduction of 50% during 200 ms is applied. Fig. 8 shows the obtained results. As observed from Fig. 8, the response obtained with both models is in close agreement. It is important to observe that oscillations are eliminated when the simpliﬁed model is used. For example, the speed in the complete model (speed C) oscillates at the instant voltage falls; these are eliminated in the simpliﬁed model (speed S). A more pronounced difference between models is observed at the machine starting. If nominal values around the equilibrium point are assumed, pole location analysis can be performed. Table 4 shows the pole values for the two models. These poles are calculated using the data of Table 2 and (2). By using these values, it is veriﬁed again that the change of causality (Fig. 7) and dynamics elimination (/_ sq ¼ /_ sd ¼ 0) allow to obtain a model without oscillations.

Fig. 8. Comparison responses of the simpliﬁed and complete induction model.

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R. Tapia, A. Medina / Simulation Modelling Practice and Theory 53 (2015) 149–166 Table 4 Numerical poles values. Complete machine model

Simpliﬁed machine model

P1 = 6.944 + 38.73i P2 = 6.944 – 38.73i P3 = 12.09 P4 = 15.69 + 376.5i P5 = 15.69–376.5i

P1 = 7.01 + 38.7i P2 = 7.01 – 39.7i P3 = 12.08

4. DFIM control As previously shown, the induction machine is a coupled 5th or 3th order system (depending on the considered model) with four inputs. Actually, the simpliﬁcation made in the previous section allows to intuitively know that stator dynamics are not directly involved in the control law. For that reason, the control inputs which are present in the doubly fed induction machine are those of rotor. The last assumption is commonly made when a generator control law is deduced, but here, the bond graph model supports this assumption. When a wind turbine generator is used, the objective is to control the output torque. Also, the torque can be controlled indirectly by the speed. This selection will be dependent on the particular application. As the paper is focused in the general behavior of the wind turbine, it is important to remark that the proposed control does not consider the Sommerfeld effect [21,22], since unbalance rotor conditions are not considered. In order to control the output torque, it is necessary to control the d-axis rotor ﬂux supplied by the induction machine. To this aim, a speciﬁc algorithm is designed, based on bicausal bond graph [13]. For the formulation of the inverse bond graph, it is necessary to change the effort detectors De (assumed ideal), which will be placed in bond 5 (Fig. 4) of the original bond graph, by a source named SS, (which impose zero ﬂow but non-zero effort to the inverse model), then propagate bicausality (in only one line of power transfer) from this source (SS: /_ rd ) to the input source of the original bond graph which becomes a detector (i.e. SS: vrd) in the inverse bond graph [23]. The structure of the control in open loop is designed with the inverse bond graph. The decoupling actions are deﬁned (inverse matrix and disturbance compensation). The open loop structure is then extended to a closed loop control by ﬁxing the dynamics of errors. Fig. 9 shows the preferred derivative causality assigned in the inverse bond graph, which allows to deduce the open loop control laws. The rotor ﬂux and torque sensors are simultaneously inverted via bicausal bonds; two disjoint bicausal paths are drawn to the two desired inverse model outputs, corresponding to the two control signals (MSe: vrd, MSe: vrq), which demonstrates that the model is invertible [23]. It is important to notice that the Jm inertia is not considered in the inverse bond graph of Fig. 9. This is because non-unbalanced rotor operation has been considered and because this inertia will represent the whole wind turbine inertia.

Fig. 9. Inverse bond graph for calculation of the controls laws.

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For the rotor ﬂux control law, no change is presented in the I-ﬁeld element Md. Otherwise, the causality change in bond 4 is due to the simpliﬁed model consideration (this bond has not a dynamic behavior). Eq. (4) is derived from the inverse bond graph of Fig. 9, and this relationship corresponds to the 1-junction placed between bonds 5, 6, 7, 8 and 9, i.e.

e6 þ e7 þ e8  e9 ¼ e5

ð4Þ

It is assumed that the numerical value of the elements (resistances and inductances) is the same for 2 axis. These values b r of the actual system parameters, in order to consider a slight error. By replacing have been taken as estimated values e.g. R these values in (4) it gives,

v rd ¼

d b r ird þ p/ X  x/ / þR rq rq dt rd

ð5Þ

To establish the closed loop control law, the dynamics of the error (e ¼ /rq represents the controller to be used. Expression (5) becomes (6) as:

v rd ¼

ref

 /rq ) are set in (5), as e_ þ k1 e ¼ 0; where k1

d b r ird þ p/ X  x/ ð/  eÞ þ R rq rq dt rdref

ð6Þ

d b r ird þ p/ X  x/ / þ k1 ð/rdref  /rd Þ þ R rq rq dt rdref

ð7Þ

Finally,

v rd ¼

For the last expression, the rotor ﬂux /rq is assumed to be zero. This is justiﬁed since /rq, compared to /rd has not an important impact on the voltage vrd. Actually, his value is very small compared to the d-axis ﬂux. Taking the last consideration, (7) is represented in the Laplace domain as,

b r ðird Þ ðv rd ÞðsÞ ¼ sð/rdref ÞðsÞ þ k1 ð/rdref  /rd ÞðsÞ þ R ðsÞ

ð8Þ

In the same manner as for the determination of the rotor ﬂux control, the torque control law is calculated. Eq. (9) is derived from the bond graph of Fig. 9 as.

e14  e18 þ e15 þ e13  e17 ¼ 0

ð9Þ

For (9), the effort e17 needs to be calculated from (3), by assuming that /rq = Se17. Making the appropriate substitutions in (9), it yields,

v rq ¼ bL r

d b r irq  p/ X þ x/ irq þ R rd rd dt

ð10Þ

In (10), the current isq of (3) has been assumed zero, since the balanced condition of the Park transformation has been considered. However, the mutual inductance is not present in (10). The voltage vrq and torque relation is not explicitly shown in (10). Thus, in order to relate (10) with the torque, another expression is needed. Two different procedures are visualized, i.e. the ﬁrst is to replace the speed O for his equivalent relation with the torque, and the second one is to establish an internal control loop for the irq current, and cascade the torque relationship in the previously internal control loop. The second solution is chosen, since it provides a control law structure similar to the traditional vector control [24] applied to this machine. By using the estimated values and establishing the closed control loop (for irq variable) as previously described, (10) becomes,

v rq ¼ Rb r irq  p/rd X þ x/rd þ bL r

d L r k2 ðirqref  irq Þ irqref þ b dt

ð11Þ

where k2 represents the controller used for this loop. The other necessary expression is taken from the original model (Fig. 4); speciﬁcally, in bond 11 the torque is given by,

T e11 ¼ pð/rq ird  /rd irq Þ

ð12Þ

As /rq = 0, expression (12) becomes T e11 ¼ p/rd irq ; and can be expressed as T ref ¼ p/rd irqref . This expression is replaced in (11), to obtain the control law (12).

b r ðirq Þ  p/ X þ x/ þ b ðv rq ÞðsÞ ¼ R Lr s rd rd ðsÞ



T ref p/rd

 ðsÞ

þb L r k2



T ref  irq p/rd

 ð13Þ ðsÞ

The controller block diagram (Eqs. (8) and (13)) is shown in Fig. 10. It is important to notice that the structure of the control law contains a feed-forward with a derivative action on the reference signals. The signals are constants, so that they can be removed, and by taking into consideration that the wind does not have a sudden change, they are kept in this proposed control law. In addition, the control law needs an estimation of /rd, then, the expression (Ls ⁄ ird + Lm ⁄ isd) [25] is used.

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Fig. 10. Torque control law schema.

Fig. 10 shows that the proposed gains of each control loop are proportional + integral; the estimate values are considered in the control law. In order to verify the robustness of the proposed control law, a simulation is carried-out, by considering that the generator has a primary governor to emulate the torque provided by the turbine mechanical part. Also, as it is considered that the converter has not an important impact in the machine behavior, the machine side converter is not used. This allows to connect the vrd and vrq voltages directly via the MSe sources. Fig. 11 shows the simulation scheme. The stator machine power is regulated by the rotor. To verify the robustness of the proposed control, a three phase load is connected at machine terminals. Park’s transformation is used in order to transform the dq reference frame to the abc framework.

Fig. 11. Constant speed generator schema.

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The simpliﬁed induction machine model is used in the simulation, i.e. this model represents the behavior of the machine, without the transient oscillations due to fast dynamics. The results presented in this paper have been veriﬁed against the complete induction machine model, obtaining the same results; the only difference being that the oscillations due to the fast dynamics are present in the complete model. A speed governor is used to maintain the nominal speed in the generator. As the objective is to test the behavior of the control law; this stage has been represented by using an ideal Governor (PI controller). The pole-plot is taken directly from the simulator. In order to verify the robustness of the proposed model different values of power factor in the load (0.5–1) are used. The nominal values of torque and the magnetic ﬂux (Tref and phiref in Fig. 11) have been considered by taking into account the parameters of Table 3. Fig. 12 shows the 3D pole-plot evolution. Poles which have a signiﬁcant movement correspond to the load poles (red and orange colors). Unlike the generator and control poles, these do not have a signiﬁcant movement when the power factor is varied. This shows the robustness of the proposed control in this system. It is important to mention that in order to take into account the uncertainties due to parameter estimation the simulation is carried-out by assuming a slight error in the parameters, i.e. 10% between the model and the controller parameters. The numerical parameters of the controller are given in Table 5. 5. Turbine and converter models In order to simulate the complete wind turbine model with a doubly-fed induction machine, it is necessary to introduce two more elements; i.e. the mechanical part of turbine (gearbox, blade, etc.) and the converter models. 5.1. Converter model For the network side converter (NSC), the model is taken from [26,27]. This model corresponds to a three phase converter using a LC ﬁlter connection. The ﬁlter allows to give a voltage source behavior for the converter, and also to eliminate the harmonic distortion introduced by the converter. Fig. 13 shows the NSC converter used. In contrast to the model presented in [26,27], this model shows explicitly the three phases (3 MTF elements), due to the fact that single phase voltages are assumed here, instead of phase to phase voltages. The control law of this converter is also derived from the inverse bond graph model (using the bicausality concept); the robustness and accurate performance of the proposed control has been demonstrated for voltage regulation [27]. The control is composed by three principal stages:

Fig. 12. 3D pole-plot evolution.

Table 5 Control parameters. bs R br R

0.00242

p

3

Proportional gain torque Proportional gain ﬂux

100 15

0.00198

b Ls b Lr

0.132

b M Integral gain torque Integral gain ﬂux

3.19

0.055 0.01 0.2

160

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Fig. 13. Network side converter with LC ﬁlter.

1. DC bus regulation. Based on a traditional PI (proportional + integral) controller. It provides the active power reference. 2. Active and reactive power regulation. Inspired from the power ﬂow concept between two sources, connected through a line impedance. 3. Voltage and current regulation. A resonant and proportional controller for voltage and current control, respectively. The machine side converter is modeled by using the three MTF elements without the ﬁlter. 5.2. Turbine model For the turbine model, different bond graph models can be formulated. Some of they have been reported in the literature e.g. [28–30]. In this contribution the model shown in Fig. 14 will be used. The characteristics of this model are:     

Only hub and generator inertias are considered. They are placed together and represent the Jhole inertia. Gearbox ratio n is considered. Wind speed is represented by a Sf source. Aerodynamic force conversion is considered. Rigid structure is assumed (no stiffness is used).

Aerodynamic conversion is represented by a MGY element. Within this element, the traditional mathematical expressions are introduced [25]. This means that the MGY element is modiﬁed in order to convert the wind speed input (Sf source) into the aerodynamic force (bond 2), which will be applied to the mechanical turbine part. Also, the MGY element has the external modulated input beta, which represents the pitch angle of blades.

Fig. 14. Turbine model.

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Two control loops are related with the turbine, i.e. the pitch angle and the angular speed control, respectively. These control laws are taken from [25] and used here, in order to allow the complete wind turbine simulation. Parameters of the turbine are given in Table 6. In the next section the complete wind turbine model is presented. 6. Complete system The generator, power converters, and the turbine models are associated as shown in Fig. 15 to assemble the complete wind turbine model. Fig. 15 shows the whole control law involved in the wind turbine. The block called ‘‘Wind control’’ corresponds to the wind turbine angular speed control. This control provides the reference torque, which is introduced in the proposed torque control for the doubly-fed generator. The active power is sensed in order to calculate the maximum power available in the mechanical part of the turbine. Then, this power is regulated via the pitch angle, given by the block ‘‘Pitch control’’. The block NSC control, groups the continuous voltage (DC bus), power, voltage and current controls for the converter. Additional elements, such as power sensors, Park’s transformation, and the phase lock-loop (PLL) are required to simulate the system. The turbine, the generator and the power converter require sensors. For the turbine a power and angular speed sensors are used, while for the generator the rotor and stator currents sensors are needed to estimate the magnetic ﬂuxes. These current sensors are also used in the power converter control. In order to show the behavior of the complete bond graph model, two different simulations are conducted. First, a constant wind proﬁle with a sensor noise is considered, and then real wind proﬁle data are used. These simulations have been performed using the 20Sim software. The scenario for simulation is as follows: a constant wind of 5 m/s is applied at the simulation start; then, at t = 10 s, a wind ramp is applied. This wind ramp increases from 5 m/s, at t = 10 s to 14 m/s, at t = 55 s; this value is maintained until the simulation concludes. A Gaussian distribution with variance 1 is used as the noise added into the sensors. Fig. 16 (from a to b) shows the simulation responses for the wind speed, DFIM speed, the tip speed, the power coefﬁcient and the pitch angle, respectively.

Table 6 Turbine parameters. Power Blade number Gearbox ratio Hub

2M 3 92.6 90,000

Density of the air Optimal power coefﬁcient Optimal non tips Radio of the blade

Fig. 15. Complete wind turbine model simulation.

1.22 0.47 8 40 m

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Fig. 16. Responses for a constant wind proﬁle.

The speed in the DFIM increases in the same ratio as the wind input, i.e. up to the maximum value (226 rad/s) when the wind speed exceeds 12 m/s (see Fig. 16b). This is due to the fact that this value has been assumed to be the maximum set point. The curve called ‘‘lambda’’ corresponds to the tip speed ratio, which presents its nominal value before the ramp proﬁle starts (Fig. 16c). When machine speed increases to its maximum value, the tip speed decreases from 8 to 7 m/s. A similar case is shown for the power coefﬁcient Cp (Fig. 16d). Fig. 16e also shows the pitch angle applied to the blades, which starts at the instant when the wind speed exceeds 12 m/s. As this investigation centers on the control of the DFIM, the reference torque provided by the angular velocity control law is compared to the actual torque in the generator (see Fig. 17). The curves of the DFIM and the reference torques closely agree, only a small difference has been identiﬁed during the transient state. It is important to notice that the noise added does not present a signiﬁcant impact on the control law behavior. This noise effect is present in all responses. The power transferred to the network is the most important parameter to take care in a wind turbine. Fig. 18a and b shows the active (P) and reactive power (Q), respectively, supplied to the power network. Zero reactive power has been set as reference, with the whole active power (2MW) being supplied. The regulation of the continuous bus is also an important parameter to take into consideration. Fig. 18c shows the DC bus capacitor voltage, which follows the voltage reference (1200 volts). It is important to mention that an initial condition is used in order to allow the simulation to be performed.

Fig. 17. Reference and DFIM torques for a constant wind proﬁle.

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Fig. 18. (a) Active power (P), (b) reactive power (Q), and (c) voltage of the DC bus (Vbus) – constant wind proﬁle.

Fig. 19. Responses for a real wind proﬁle.

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Fig. 20. Reference and DFIM torque for a real wind data.

The last simulation results allow the veriﬁcation of theoretical concepts related to the wind turbine dynamic operation. Practical wind behavior has a varying proﬁle instead of being constant. The performance of the proposed control law can be conﬁrmed if a real wind proﬁle is used. Data are introduced in the simulator with a ‘‘data from ﬁle’’ block, which takes the numerical values from a table each time step. Fig. 19 shows the simulation response for the selected wind turbine variables. The wind changes (from 9 to 13 m/s) are reﬂected in the DFIM speed with the same ratio. When the wind crosses the limit of 12 m/s for small periods of time, i.e. from 10 to 60 s, the DFIM reaches its maximum value (226 rad/s). Also, this behavior is presented in the lambda, the Cp and the pitch angle responses. In order to verify that the reference torque (given by the angular velocity control) is the same as the actual DFIM torque, Fig. 20 shows the comparison between the reference and DFIM torque. Similarly to the simulation with constant wind speed, the curves of Fig. 20 are in close agreement; only a very small difference can be appreciated. Finally, the DC bus voltage and the active and reactive power are presented in Fig. 21. The active power is limited to the maximum power that can be extracted from the wind turbine (2 MW). The reactive power reference has been selected to be zero. For the DC bus voltage, some oscillations of small amplitude (around 1V) can be appreciated.

Fig. 21. (a) Active power (P), (b) reactive power (Q), and (c) DC voltage (Vbus) – real wind proﬁle.

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The simulation results can verify that the proposed control law applied to the DFIM presents a good performance, either for constant or variable wind speed. Nevertheless, some improvements can be suggested, as mentioned in the section to follow. 7. Conclusions and future work The control of a DFIM has been proposed. The causal loop concept applied to the bond graph model of the induction machine allowed to identify the dynamics (slow and fast) of the model approximation. Then, by choosing the elimination of the fast dynamics, in order to reduce the order of the model, the simpliﬁed machine model has been used. By applying the bicausality concept, the simpliﬁed model has been used in order to formulate the inverse bond graph. The control laws were intuitively obtained by considering the simpliﬁed model. As the proposed control law strongly depends on the estimated DFIM parameters, a traditional pole-place analysis has been conducted in order to verify the robustness of the control. The small error considered in the parameters did not vary the pole-place diagram. A complete wind turbine model with a doubly-fed induction generator has been proposed. The wind turbine behavior was simulated in order to verify the dynamic performance of the whole system. This dynamic behavior has been obtained by assuming either a constant or variable wind speed. It is suggested in future research work to test the proposed control law in a real-time simulator, in order to make a validation still closer to reality, before being tested in a real wind turbine. Also, the control law has been synthetized from the inverse bond graph model, which has been successfully obtained by considering the DFIM in its natural reference frame. This means that the dq model can be replaced by an abc model. This allows to formulate a control law in the natural reference frame, without resorting to the Park´s transformation. 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