Understanding the Doubly Fed Induction Generator ... - IEEE Xplore

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 27, NO. 2, JUNE 2012

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Understanding the Doubly Fed Induction Generator During Voltage Dips G. D. Marques, Member, IEEE, and Duarte M. Sousa, Member, IEEE

Abstract—The doubly fed induction generator (DFIG) is normally controlled with stator-flux orientation. The DFIG is a poorly damped system with a natural frequency near the grid frequency. This leads to natural oscillations on the stator flux and on other variables. When voltage dips occur, there are oscillations on the stator flux that produces electromotive forces and consequently disturbances on the rotor currents. This paper presents a comprehensive study of the DFIG during voltage dips. The method presented in this paper, valid for symmetric and for asymmetric voltage dips, is based on the classical model of the induction machine with stator flux orientation and neglects only a small part of the cross-coupling terms. The response depends on the design methodology of the proportional-integral inner controllers. Analytical, simulation, and experimental results are shown. The analysis and results can be divided into two sets depending if the voltage dip magnitude is smaller or deeper than 0.5 p.u. Index Terms—Doubly fed induction generators (DFIG), induction generators, ride through, voltage dips.

ψ Ls Lσ M Mem rs i u e Tp kp , ki kinv γs γm ωr f ω0 ωm ωc

NOMENCLATURE Stator or rotor flux linkage. Stator inductance. Leakage inductance. Mutual inductance. Electromagnetic torque. Stator resistance. Stator or rotor current. Stator voltage. Electromotive force. Small time constant. Parameters of the PI controller. Gain of the inverter. Position of the stator-flux vector. Electrical position of the rotor. Speed of the reference frame. Electrical frequency of the ac mains. Speed of the machine. Crossover frequency in closed loop.

Manuscript received August 16, 2011; revised October 27, 2011 and January 5, 2012; accepted February 22, 2012. Date of publication March 22, 2012; date of current version May 18, 2012. This work was supported in part by the Centre for Innovation in Electrical and Energy Engineering (CIEEE) of IST/TU Lisbon and Programa Operacional da Sociedade do Conhecimento (POSC). Paper no. TEC-00428-2011. The authors are with the Department of Electrical and Computer Engineering, TULisbon, 1049-001 Lisboa, Portugal (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TEC.2012.2189214

Fig. 1.

Block diagram of the DFIG.

Subscripts α, β d, q s, r R 0 Superscripts ∗ ∼

Usual αβ variables. Variables on a rotating reference frame. Stator or rotor quantities. Relative to Γ equivalent circuit. Steady state. Reference value. Variation around the operating point. I. INTRODUCTION

IND power is currently considered an alternative to the conventional energy sources. For turbines rated more than 2 MW, adjustable speed type are now prevailing because their performance is superior and the energy produced presents better quality. Within adjustable speed wind generators, the most widely used are those based on the doubly fed induction generator (DFIG) as they use a power converter rated to a fraction (about 30%) of the total power (see Fig. 1) [1]–[3]. The wind turbines based on the DFIG are very sensitive to grid disturbances, especially to voltage dips. The abrupt drop of the grid voltage causes a dc component on the stator flux resulting in high rotor-induced voltages [4]. In order to avoid losing the control of the rotor current, the converter should be sized to be able to generate a voltage equal, or even greater, than the maximum induced rotor voltage. Conversely, the rotor converter voltage is rated to only a fraction of the rated rotor voltage. This gives rise to high rotor currents, and so, the power electronics converter should be disconnected from the rotor terminals. A typical solution adopted by the manufactures, to protect the converter, was to use the so-called crowbar to short circuit the rotor windings and disconnect the turbine from the grid. More recent grid codes [5] impose that the turbine should be

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maintained connected during voltage dips and, in addition, should contribute to the voltage restoration injecting reactive current during the voltage dip. This subject has been of great interest during the last years. A considerable number of works were performed in order to study the ride through of the DFIG during voltage dips [6]–[20]. Many studies about the DFIG ride-through voltage dips, which propose different solutions, use simulations to predict the evolution of the electrical variables. These studies have the inherent difficulty to explain, from a theoretical and analytical point of view, the singular role played by each parameter during voltage dips. The methodology based on simulations makes possible to predict the behavior of the system, but has obvious limitations. A detailed theoretical and experimental study of the DFIG during a voltage dip is presented in [4]. This paper explains very well the behavior of the rotor voltage during a voltage dip. However, the influence of the rotor current controllers is not presented. When voltage dips occur, there are variations on the flux linkage, producing disturbance electromotive forces that introduce oscillations on the rotor currents. This paper presents a comprehensive study of the DFIG during voltage dips. It is based on the classic model of the induction machine [1], [22] and assumes stator-flux-oriented control. The analysis neglects a small part of the cross-coupling terms in the rotor equations leading to independent axis transfer functions that are easily evaluated. An earlier and very short presentation, considering only symmetric voltage dips, is introduced in [19]. Some analytical approximations are presented, discussed, and validated. The analysis is verified and complemented by simulation programs. Experimental validation based on a 3.2-kW prototype is also presented. Section III presents the closed-loop model of the system and the design of the rotor proportional-integral (PI) current controllers according to the integral of time multiplied by the absolute error (ITAE) and to the symmetrical optimum standard design procedures. It will be shown that the parameters of the controllers have an important role on the behavior of the DFIG during a voltage dip. Section IV presents the analytical work leading to the transfer functions that relate the rotor current and the disturbance electromotive forces. These are analyzed in Section V using only theoretical considerations. Section VI is devoted to the simulations. The experimental validation, based on a 3.2-kW prototype [15], is presented in Section VII. The conclusion is presented in Section VIII.

II. MODELING AND CONTROL To simplify the analysis, the dynamic model correspondent to the Γ equivalent circuit of the induction machine, Fig. 2, in motor convention, is adopted in this paper for analytic calculations [3], [9]. In this case, the parameters and variables of the machine are multiplied by constant factors (1), (2) in order to simplify the model and to maintain its accuracy.

Fig. 2.

Γ equivalent circuit of the induction machine.

The relation between the variables of this circuit and the actual variables are the following: udR =

Ls udr M

ψdR =

Ls ψdr M

idR =

M idr Ls

uq R =

Ls uq r M

ψq R =

Ls ψq r M

iq R =

M iq r . Ls

(1)

For the parameters, the aforementioned relations are given by  2  2 Ls Ls rR = rr Lσ = Lr − Ls . (2) M M A. Basic Equations in Per Unit (p.u.) The two axis model of the induction machine in p.u. variables, using the time in seconds, and the speed in rad/s, in a common reference frame synchronous and aligned with the stator flux, is represented by [1], [3], [21], [22] 1 dψs ω0 dt ωr f + ψs ω0

uds = rs ids + uq s = rs iq s

udR = rR idR +

Lσ didR ωr f − ωm 1 dψs + − Lσ iq R ω0 dt ω0 dt ω0

uq R = rR iq R +

ωr f − ωm Lσ diq R + (ψs + Lσ idR ) ω0 dt ω0 Mem = ψs iq s .

(3) (4)

In this model, the system variables are the amplitude of the stator flux ψ s , the speed of the reference frame ω rf and the dq rotor currents. The input functions are the stator and rotor voltages. The mechanical speed ω m is considered a parameter, which is assumed constant during the analyzed transient as voltage dips are very fast and short when compared with mechanical transients. B. Simplified Rotor Model in p.u. The rotor equations of (3) can be represented by the block diagram of Fig. 3. This representation is appropriate for the analysis of some transients that are due to variations of the flux, like those that occur during voltage dips. In this model, the flux and its derivative, as well as ω rf and ω m , are input variables of the rotor equivalent model. The block diagram of Fig. 3 is characterized by crosscoupling terms that add a considerable complexity to the analysis. These crossing terms depend on the slip frequency and on the leakage inductance Lσ of the machine.

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Fig. 4. Model for the analysis of the transients associated with the control of rotor currents including flux variations. (a) d-axis model, (b) q-axis model.

frame can be obtained using (3) that leads to ωrf uq s − rs iq s = . ω0 ψs Fig. 3.

Block diagram of the rotor equations.

From (3), (5), and (6), the direct- and quadrature-axis disturbance electromotive forces, ed , and eq are defined as

The model can be considerably simplified if the crossing terms of Fig. 3 are neglected. In this case, two simple and decoupled models are obtained. Note that this simplification neglects only part of the crossing terms of the machine. The validity of this simplification was verified using simulation programs for the transients analyzed in this paper, i.e., voltage dips [19]. In these transients, and for usual parameters of the machine and of the PI controllers, it was verified that the rms values of these neglected terms are more than ten times smaller than the rms terms represented by eq and ed in Fig. 3. However, for the daxis and for voltage dips deeper than 0.5 p.u., this relation is not so small but the approximation is still good as will be seen in Fig. 12. This was considered appropriate for this study as will be confirmed by the results. The two components of the electromotive force defined in Fig. 3, can be interpreted as the derivative components of the stator-flux space vector that rotates at ω rf –ω m speed on the rotor reference frame, that is e¯ = (ed + jeq )ej (ω r f −ω m )t

1 dψs = −(uds − rs ids ) ≈ −uds ω0 dt   ωr f − ωm ωm eq = − ψs ≈ − u q s − ψs . ω0 ω0

ed = −

(7)

Conversely to ed , eq depends on the speed of the rotor. C. Introduction of the Rotor Current Controllers The model obtained, when PI controllers are introduced and when the crossing-coupling terms are neglected, is represented in Fig. 4. The inverter is modeled using the first-order block diagram with the kinv gain and the small time constant Tp , which represents the natural delay introduced by the pulsewidth modulation (PWM). The d and q block diagrams, in Fig. 4, have the disturbance electromotive forces as disturbance inputs. D. Design of the PI Controllers Using ITAE Criterion The classic design of the PI controllers uses the ITAE performance criterion [2], [3] where the zero of the PI transfer function is located over the dominant pole in the block diagrams of Fig. 4. In this case, the design of the PI controller is done in order to have a determined and specified crossover frequency ω c in closed loop. The synthesis gives rise to [3]

1 dψ¯s =− ω0 dt 

1 d  ψs ej (ω r f −ω m )t ω0 dt   1 dψs + j(ωrf − ωm )ψs ej (ω r f −ω m )t . =− ω0 dt

=−

(6)

(5)

In steady state, the speed of the reference frame is constant and equal to 1 p.u. However, to analyze the transients that can occur on the DFIG, this simplification is not always valid. As will be shown in Sections V and VI, during these transients, a better approach should be considered. The speed of the reference

kp =

ωc Lσ , ω0 kinv

ki =

ωc rR . ω0 kinv

(8)

The closed-loop transfer function will be [3] Gcl =

1 . 1 + (s/ωc )

(9)

It is common to adjust the PI controllers to have ω c = 7ω 0 [3]. In this case, this transfer function can be considered almost a

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unitary gain. Effectively, for ω = ω 0 , the gain is 0.99 and the phase is 8 degrees. For asymmetric voltage dips, the presence of an inverse component gives rise to flux oscillations of ω = 2ω 0 leading to a less precise approximation. This result can be used to simplify the analysis presented in Section IV. E. Design of the PI Controllers Using the Symmetrical Optimum For high-power rated machines, the rotor resistance is small and consequently using (8) a very small ki parameter results. In this case, the PI symmetrical optimum design procedure can be applied alternatively to the ITAE criterion. Defining a α parameter that will be chosen later, the tuning of the PI controller can be performed considering the following parameters [1]: T =

Lσ , kinv ω0

Ti = α 2 T p .

(10)

Therefore, the PI parameters are [1] kp s y m =

T , αTp

ki s y m =

kp s y m ω0 Ti

(11)

Fig. 5. Model for the analysis of variations around an operating point during transients associated with stator-flux variations. (a) Original block diagram, (b) first manipulation with the block diagram, and (c) final block diagram.

the crossover frequency is given by ωc s y m

1 = αTp

(12)

and the damping factor is given by ξsym =

α−1 . 2

(13)

For a given Tp , the bandwidth and the damping factor in closed loop are dependent of the α parameter adopted. Normally, to obtain good bandwidths and appropriate damping factors, α = 2 or α = 3 are used. III. ANALYSIS OF THE DISTURBANCE TRANSFER FUNCTION A. d-Axis Disturbance Model For transients with origin on the stator, like the produced during voltage dips when the reference rotor currents are maintained constants and there are important variations of the stator flux, the model of Fig. 4(a) can be used in terms of variations leading to Fig. 5, where ˜idR = idR − idR 0 ,

u ˜dR = udR − udR 0 .

(14)

These variations will have different waveforms depending on the magnitude of the voltage dip. The frequency depends also on the voltage dip type. If the voltage dip is symmetrical, the frequency is ω = ω 0 . A negative sequence is presented in unsymmetrical voltage dips that leads to flux oscillations of ω = 2ω 0 . In general, in unsymmetrical voltage dips, there are the frequencies ω 0 and 2ω 0 . Performing simple block manipulations, the model of Fig. 5(a) is transformed into the model of Fig. 5(b). In this figure, the first transfer function computes an equivalent rotor reference current using the disturbance electromotive force and the inverse of the product of two transfer functions.

The closed-loop block diagram of Fig. 5(b) represents the system in closed loop. Replacing this system by its equivalent (9), the model of Fig. 5(b) is transformed into the block diagram of Fig. 5(c) where the time constant Tp was neglected in the first transfer function. Considering, as mentioned before, that for ω = ω 0 or ω = 2ω 0 , the second transfer function is unitary, the equivalent transfer function is ˜idR s/ω0 s 1 . (15) = = e˜d (kp s + ki ω0 )kinv kp kinv (s/ω0 ) + (ki /kp ) When the ITAE criterion is used, introducing the tuning (8), the transfer function (15) becomes ˜idR s/ω0 1 . = G(s) = e˜d kp kinv (s/ω0 ) + (rR /Lσ )

(16)

This transfer function (16) has a zero on the origin and a corner frequency located at rR /Lσ . The Bode diagrams for typical 2-MW and 3.2-kW rated machines are shown in Fig. 6(a). It is clear that there are considerable differences due to very different rotor resistance values. In Fig. 6, the closed-loop transfer function (9) is also considered. Small circles and crosses are used to identify the points of ω = ω 0 and ω = 2ω 0 , respectively. For these two frequencies, the phase angles of the transfer function are small. Being the range where the disturbances normally occur, ω = ω 0 and ω = 2ω 0 , higher than the corner frequency, a simplified transfer function (17) can be adopted ˜idR 1 = G(s) ≈ e˜d kp kinv

(17)

representing a gain, which is known. The problem of disturbance currents is now reduced to the determination of disturbance electromotive forces. Using (8), the transfer function (17) takes

MARQUES AND SOUSA: UNDERSTANDING THE DOUBLY FED INDUCTION GENERATOR DURING VOLTAGE DIPS

Fig. 7.

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Stator-flux trajectories for several magnitudes of voltage dips.

B. q-Axis Disturbance Model Being the d- and q-axis models similar, the work performed on the previous section can be also applied to this axis leading to similar results. IV. THEORETICAL ANALYSIS OF THE DISTURBANCE ELECTROMOTIVE FORCES

Fig. 6. Disturbance transfer functions. (a) Bode diagram using ITAE criterion and (b) bode diagram using the symmetry criterion.

the form ˜idR ω0 = G(s) ≈ . e˜d ωc Lσ

(18)

Neglecting the stator resistive voltage drop, the stator linkage flux is computed by the integral of the stator voltage. When a voltage dip occurs, the forced component of the stator-flux linkage ψ sf decreases proportionally to the new value of the steady-state stator voltage. As the flux is a state variable, it should remain continuous during the transient, and so a dc natural flux component ψ sn appears. This natural flux is proportional to the voltage variation and slowly decreases exponentially to zero. At the very beginning, the damping is very small, meaning that the natural flux is almost constant. The following analysis considers also that the voltage dip occurs in a particular instant corresponding to a null β flux component. So, the flux components in the first instants are ψsα = ψsf cos (ω0 t) + ψsn

In conclusion, the amplitude of the rotor current oscillations is inversely proportional to the closed-loop bandwidth of the current controller. When the symmetry criterion is used to tune the PI controller, similar calculations can be performed. Fig. 6(b) shows the Bode diagram obtained for two different adopted α values (α = 2 and α = 3) for both machines. Because this criterion uses only the leakage inductance and this value in p.u. is similar for both machines, the resultant transfer functions are almost undistinguishable. It is clear that there is a higher crossover frequency obtained for the symmetry criterion leading to different gains and phases for ω = ω 0 and ω = 2ω 0 . When the symmetry criterion is used, the gain and phase depend on the α parameter adopted. For α = 2, the system leads to smaller gains, which give rise to a higher reduction of the disturbance rotor currents.

ψsβ = ψsf sin (ω0 t) .

(19)

Defining ψ sf0 as the forced flux before the voltage dip, the natural flux component will be ψsn = ψsf 0 − ψsf .

(20)

The stator-flux space vector defined as (ψ sα , ψ sβ ) by (19) describes trajectories as shown in Fig. 7. This figure shows the stator-flux trajectories in the αβ plane for several values of the voltage dip magnitude. As an example, several stator-flux positions are presented, in gray, for a voltage dip of 0.25-p.u. magnitude. Fig. 7 is the key to understand the disturbance electromotive force waveforms. For voltage dips of magnitude smaller than 0.5 p.u., ψ sf is greater than ψ sn and the trajectory encircles the origin: this means that the angle representing the flux phase varies between −π and π. The consequence is that ω rf is

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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 27, NO. 2, JUNE 2012

Fig. 8. Waveforms for voltage dips (0.1, 0.25, 0.4) smaller than 0.5 p.u. (deeper voltage dips are represented by less imprinted gray).

not constant but its average is ω 0 . Conversely, for voltage dip magnitudes deeper than 0.5 p.u., ψ sf is smaller than ψ sn and the trajectory does not encircle the origin: the average value of ω rf is null. The speed ω rf takes the constant value of 0.5 p.u. for the particular amplitude voltage dip of 0.5 p.u. Using (19), it is possible to obtain the amplitude and phase of the stator-flux space vector  2 + ψ2 ψs = ψsα sβ θ = arctan 2(ψsβ , ψsα ).

(21)

Using (19), (21), and (5) to (7), it is possible to obtain ψ sα , ψ sβ , ψ s , θ, ed , eq , and ω r f for voltage dips smaller and deeper than 50%, as shown in Figs. 8 and 9, respectively. When transformed to a constant speed synchronous reference frame, the forced flux component gives rise to constant signals and the natural flux to sinusoidal components. Due to the presence of the natural flux component that translates the trajectory out of the center of the αβ plane, the speed of the flux vector is not constant. The consequence is to have oscillations that are not sinusoidal. This is clearly seen in Figs. 8 and 9. The d and q disturbance electromotive forces are due to the stator-flux oscillations on the field reference frame that occur during voltage dips. For small magnitude voltage dips, the variation of the statorflux waveform in the field reference frame is almost sinusoidal, resulting in ed and eq almost sinusoidal, as shown in Fig. 8 (black traces). For deeper voltage dips, the flux is no more sinusoidal, resulting in d and q nonsinusoidal electromotive forces. This is shown in Figs. 8 (in gray) and 9. For very deep voltage dips, the natural flux is determinant giving rise to almost sinu-

Fig. 9. Waveforms for voltage dips (0.55, 0.7, 0.85) deeper than 0.5 p.u. (deeper voltage dips are represented by less imprinted gray).

soidal oscillations (soft gray in Fig. 9). Eq is in this case almost constant. A. Analysis of the d-Axis Disturbance Electromotive Force The ed electromotive force is given by (7). This electromotive force is due to oscillations of the stator-flux amplitude. If the stator-flux waveform is known, this can be computed using its derivative. Comparing Figs. 8 and 9, it is possible to conclude that the ed component is always smaller than 0.5 p.u. In the first stage, its amplitude increases with the voltage dip magnitude, takes the maximum when it is 0.5 p.u. and then starts to decrease for deeper voltage dips. It is also possible to note that the stator flux and ed waveforms have some kind of symmetry because these waveforms for 0.6 p.u. are equal to the 0.4-p.u. waveforms and so forth. B. Analysis of the q-Axis Disturbance Electromotive Force This disturbance electromotive force is due to oscillations on the speed of the reference frame and on the amplitude of the stator flux. The q electromotive force is given by (7). The simplified (22) can be adopted for analytical calculus eq ≈ −uq s + ωm ψs .

(22)

The terms, uq s and ω m ψ s assume different relevance depending on the amplitude of the voltage dip. For small magnitude voltage dips, smaller than 50%, and after the voltage dip, uq s can be

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considered constant and equal to the applied stator voltage. In terms of variations, (22) leads to e˜q ≈ ωm ψ˜s .

(23)

This concludes that the variation of the q electromotive force depends on the amplitude of the flux variation and is proportional to the speed of the machine. For deeper voltage dips, deeper than 50%, uq s and ψ s have opposite oscillations that almost cancel each other giving rise to oscillations of small amplitude on eq . This can be observed in industrial machines and will be analyzed in the next sections. The rotor current disturbance can now be computed using ed and eq and the relations (15). V. SIMULATION RESULTS This section presents simulation results obtained with a detailed model using the MATLAB/Simulink environment for a machine of 2 MW of rated power. The wound induction machine was simulated using the classical model. Conversely to (1), where a reference frame synchronous and aligned with the stator flux was adopted, the equations are written in a common and general reference frame. The stator and rotor linkage fluxes were chosen as state variables. That is ωr f 1 dψds = uds − rs ids + ψq s ω0 dt ω0 1 dψq s ωr f = uq s − rs iq s − ψds ω0 dt ω0 1 dψdr ωr f − ωm = udr − rr idr + ψq r ω0 dt ω0 1 dψq r ωr f − ωm = uq r − rr iq r − ψdr . ω0 dt ω0

(24)

The torque is given by Mem = ψds iq s − ψq s ids .

(25)

The relation between the d-axis linkage fluxes and the currents are





ψds Ls M ids = . (26) ψdr idr M Lr Similar equations apply for the q-axis. As the program integrates directly the linkage fluxes, the currents are obtained by the inverse of the inductance matrix (26) that is computed only once. More detailed information about this representation can be found in [23]. The inverter connected to the rotor is represented using 2Sa − Sb − Sc Udc 3 2Sb − Sa − Sc Udc = 3 2Sc − Sb − Sa Udc . = 3

uar = ubr ucr

(27)

Fig. 10.

Voltage dip down to 0.6 p.u.

Here Sa , Sb , and Sc are the gate signals that for each inverter leg are 0 or 1 depending if the upper leg is OFF or ON. These gate signals are given by a space vector modulation system that generates the gate signals using the output of the PI current controllers. The transformations of the abc to dq variables are performed in the usual way. This program takes into account other aspects, as for instance, the detection of the stator flux. The PI controllers with antiwindup were synthesized as described previously. The ac grid and the model of insulated gate bipolar transistors are considered ideal. The dc link was simulated using a constant voltage source. Figs. 10 and 11 present the stator-flux waveforms in the stator reference frame αβ and in a constant speed reference frame dq, the stator-flux amplitude ψ s and the speed of the reference frame ω rf for two different magnitudes of voltage dip: down to 0.6 p.u. and to 0.25 p.u., respectively. The disturbance electromotive forces are also presented. The dc components on the αβ reference frame lead to sinusoidal waveforms in the dq reference frame at constant speed. These are almost sinusoidal because the reference frame is at constant speed and there is small damping. Due to the dc flux components on the αβ reference frame, the resultant speed of the stator-flux vector is not constant. As the system is controlled in a reference frame synchronous with the stator-flux vector, the speed of this reference frame is not constant leading to a deformation of the stator-flux magnitude waveform. In Fig. 10, the comparison between the eq and its approximation (23) leads to good results. The speed of the reference frame is now oscillating around the synchronous speed. High speeds as ω rf = 3ω 0 can be observed.

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Fig. 11.

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 27, NO. 2, JUNE 2012

Fig. 12.

Effects of the simplification used (ap: approximate).

Fig. 13.

Simulation result of a voltage dip down to 0.6 p.u.

Voltage dip down to 0.25 p.u.

Fig. 11 presents a similar result when a deeper voltage dip occurs. In this case, the forced components of the stator flux are smaller than the natural dc components, leading to different waveforms. Although being this voltage dip much bigger in magnitude, it produces similar oscillations on the stator-flux amplitude. The major difference is observed in the speed of the stator flux. Now, this variable oscillates around the zero speed. The approximation given by (23), presented in Fig. 11 is not so accurate. The term uq s has now oscillations that almost cancel the oscillations of the flux. For voltage dips of small amplitude, less than 50%, the waveforms obtained are similar to the results shown in Fig. 10. For deeper voltage dips, the waveforms obtained are similar to those presented in Fig. 11. Fig. 12 shows a comparison between the rotor currents computed using the complete model of the machine and the results obtained using the simplified model that neglects the crossing terms of Fig. 3. This was obtained for a voltage dip down to 0.25 p.u. It can be seen that the differences are small. Fig. 13 shows two simulation results during a voltage dip of 150 ms down to 0.6 p.u. using the ITAE and the symmetry criterion (α = 2) methodologies. The stator flux has the same waveform in both cases. The ITAE criterion gives the oscillations of the idr in phase with the stator flux and the iqr in quadrature and delayed. Conversely, the symmetry criterion gives the iqr in phase and the idr in quadrature and leading the stator flux. The oscillations of the active power transferred in the stator circuits are much more important if the ITAE method is adopted. This is in agreement with the results stated previously.

VI. EXPERIMENTAL RESULTS This section presents some experimental results obtained with a prototype using a 3.2-kW wound induction machine [15], [21]. The control algorithms are implemented in a Microchip dsPIC30F4011. Two dq PI current controllers were implemented as described before. To obtain experimental results in real time, four PWM output channels with simple RC filters were used. The actual rotor position was measured using an incremental encoder and the quadrature encoder interface of the dsPIC used. The voltage dips were obtained by a mechanical commutator and a transformer. The voltage applied to the stator circuits can be switched OFF from the mains and switched ON from the secondary of the transformer applying a lower voltage. There is a considerable time between the original point to the final point when the lower voltage is applied. This can be clearly seen in Fig. 14 (between t = 50 ms and t = 100 ms).

MARQUES AND SOUSA: UNDERSTANDING THE DOUBLY FED INDUCTION GENERATOR DURING VOLTAGE DIPS

Fig. 14. Stator flux, dq rotor currents and eq (ITAE criterion). Voltage dip down to 0.7 p.u.

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Fig. 16. Stator flux, dq rotor currents and eq (symmetry criterion, α = 3). Voltage dip down to 0.7 p.u.

Fig. 17. Stator flux, dq rotor currents and eq (ITAE criterion). Voltage dip down to 30%. Fig. 15. Stator flux, dq rotor currents and eq (symmetry criterion, α = 2). Voltage dip down to 0.7 p.u.

During this period, the DFIG is fed by the mains with a resistor in series eliminating, in this way, previous oscillations. Only the second transient is considered in this study. Figs. 14–20 show only the second transient. The direct rotor current reference was set close to zero while the quadrature rotor current was set close to −0.5 p.u. A. Response to Voltage Dips of Magnitude Smaller that 50% The responses to a small symmetrical voltage dip down to 70%, using the different tuning methodologies of the PI controllers previously analyzed in Section II, are presented in Figs. 14–16. The frequency of the oscillations is f0 = 50 Hz. Fig. 14 shows that the Iqr is slightly in advance with the stator flux, as expected. In Fig. 15, the response using the PI synthesized with the symmetry criterion (α = 2) is shown. As was predicted, the oscillations of the rotor currents are now much smaller.

This is due to the value of the gain obtained with this synthesis as described in Section IV. Being the oscillations of the rotor current so small, there is no possibility to verify the correctness of the phase. The results obtained for α = 3 are presented in Fig. 16. In this case, because the phase value of the transfer function is almost 80◦ , Idr and the flux are almost in phase. B. Response to Voltage Dips Deeper than 50% For a deeper voltage dip, down to 30%, the responses obtained are presented in Figs. 17–19. Because the machine is small, the characteristics differ from an industrial machine: it is possible to observe the decreasing of the natural flux along the time in those figures. These figures can be divided, for analysis, into two periods. Before t ≈ 120 ms, the natural flux is greater than the forced flux. In this case, eq has grid frequency oscillations of considerable amplitude. In the second period, after t ≈ 120 ms, the forced flux is greater than the natural flux and the oscillations of eq are reduced. Note that eq has similar waveforms for the three cases.

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Fig. 18. Stator flux, dq rotor currents and eq (symmetry criterion, α = 2) voltage dip down to 30%. Fig. 20. Stator flux, dq rotor currents and eq for an unsymmetrical voltage dip (symmetry criterion, α = 3).

Fig. 19. Stator flux, dq rotor currents and eq (symmetry criterion, α = 3) voltage dip down to 30%.

However, the rotor current oscillations have different waveforms as was expected. This is due to the different transfer functions (see Fig. 6). C. Response to Unsymmetrical Voltage Dips An unsymmetrical voltage dip was obtained in a similar way as in the previous section. In this case, the main experimental difference is that phase 3 was maintained unchanged. The results are presented in Figs. 20 and 21. For this case, there are two frequencies present, ω 0 and 2ω 0 . The component related with ω 0 is due to the natural flux and extinguishes exponentially to zero. Conversely, the component related to 2ω 0 is due to the inverse sequence component, and so, it is a forced type component. From the Bode plot of Fig. 6, it is possible to conclude that the phase angle is now 60◦ . VII. CONCLUSION This paper presents a comprehensive study of the DFIG behavior during voltage dips. Based on a single simplification, the paper shows that it is possible to obtain simple transfer functions relating the rotor flux transients and their electromotive forces.

Fig. 21.

Zoom of Fig. 19 in steady state (ω = 2ω 0 ).

Simple expressions were obtained for the electromotive forces and their oscillations. The analytical results were confirmed with the help of simulation and experimental results. APPENDIX PARAMETERS OF THE WOUND INDUCTION MACHINES 2-MW rated machine in the simulations: Ls = 3.1 p.u., M = 3 p.u., rs = 0.01 p.u, rr = 0.01 p.u. Lr = 3.1 p.u. Laboratory prototype; Induction Machine: stator 380 V, 8.1 A, rotor 110 V, 19 A, 3.2 kW, four poles, 1,400 rpm, Ls = 1.62 p.u., Lσ = 0.26 p.u., rs = 0.06 p.u., rR = 0.084 p.u. REFERENCES [1] W. Leonhard, Control of Electrical Drives. Berlin, Germany: SpringerVerlag, 1990. [2] R. Pena, J. C. Clare, and G. M. Asher, “Doubly fed induction generator using back-to-back PWM converters and its application to variable-speed wind-energy generation,” IEE Proc. Electr. Power Appl., vol. 143, no. 3, pp. 231–241, May 1996.

MARQUES AND SOUSA: UNDERSTANDING THE DOUBLY FED INDUCTION GENERATOR DURING VOLTAGE DIPS

[3] A. Petersson, “Analysis, modeling and control of doubly-fed induction generators for wind turbines,” Ph.D. dissertation, Chalmers Univ. Technol. Gothenburg, Sweden, 2005. [4] J. Lopez, P. Sanchis, X. Roboam, and L. Marroyo, “Dynamic behavior of the double-fed induction generator during three-phase voltage dips,” IEEE Trans. Energy Convers., vol. 22, no. 3, pp. 709–717, Sep. 2007. [5] E.ON Netz GmbH, Bayreuth, Germany (2003, Aug.). Grid Code, High and Extra High Voltage, [Online]. Available: http://www.eon-netz.com [6] T. Thiringer, A. Petersson, and T. Petru, “Grid disturbance response of wind turbines equipped with induction generator and doubly-fed induction generator,” in Proc. IEEE Power Eng. Soc. General Meeting, 2003, vol. 3, pp. 1542–1547. [7] A. Perdana, O. Carlson, and J. Persson, “Dynamic response of grid connected wind turbine with doubly fed induction generator during disturbances,” presented at the Nordic Workshop Power Ind. Electron., Trondheim, Norway, 2004. [8] T. Sun, Z. Chen, and F. Blaabjerg, “Transient analysis of grid-connected wind turbines with DFIG after an external short-circuit fault,” presented at the Nordic Wind Power Conf. Chalmers Univ. Technol, Gothenburg, Sweden, Mar. 2004. [9] A. Petersson, T. Thiringer, L. Harnefors, and T. Petru, “Modeling and experimental verification of grid interaction of a DFIG wind turbine,” IEEE Trans. Energy Convers., vol. 20, no. 4, pp. 878–886, Dec. 2005. [10] J. Morrent and S. W. H. de Haan, “Ride-through of wind turbines with doubly-fed induction generator during a voltage dip,” IEEE Trans. Energy Convers., vol. 20, no. 2, pp. 435–441, Jun. 2005. [11] A. Petersson, L. Harnefors, and T. Thiringer, “Evaluation of current control methods for wind turbines using doubly-fed induction machines,” IEEE Trans. Power Electron., vol. 20, no. 1, pp. 227–235, Jan. 2005. [12] D. Xiang, L. Ran, P. J. Tavner, and S. Yang, “Control of a doubly fed induction generator in a wind turbine during grid fault ride-through,” IEEE Trans. Energy Convers., vol. 21, no. 3, pp. 652–662, Sep. 2006. [13] S. Seman, J. Niiranen, and A. Arkkio, “Ride-through analysis of doubly fed induction wind-power generator under unsymmetrical network disturbance,” IEEE Trans. Power Syst., vol. 21, no. 4, pp. 1782–1789, Nov. 2006. [14] J. Lopez, E. Gubia, E. Olea, J. Ruiz, and L. Marroyo, “Ride through of wind turbines with doubly fed induction generator under symmetrical voltage dips,” IEEE Trans. Ind. Electron., vol. 56, no. 10, pp. 4246–4254, Oct. 2009. [15] G. D. Marques, V. F. Pires, S. Sousa, and D. M. Sousa, “A DFIG sensorless rotor position detector based on a hysteresis controller,” IEEE Trans. Energy Convers., vol. 26, no. 1, pp. 9–17, Mar. 2011. [16] J. Hu, H. Nian, H. Xu, and Y. He, “Dynamic modeling and improved control of DFIG under distorted grid voltage conditions,” IEEE Trans. Energy Convers., vol. 26, no. 1, pp. 163–175, Mar. 2011. [17] O. Abdel-Baqi and A. Nasiri, “Series voltage compensation for DFIG wind turbine low-voltage ride-trough solution,” IEEE Trans. Energy Convers., vol. 26, no. 1, pp. 272–280, Mar. 2011. [18] L. Yang, Z. Xu, J. Ostergaard, Z. Y. Dong, K. P. Wong, and X. Ma, “Oscillatory stability and eigenvalue sensitivity analysis of a DFIG wind turbine system,” IEEE Trans. Energy Convers., vol. 26, no. 1, pp. 328– 339, Mar. 2011.

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[19] G. D. Marques and D. M. Sousa, “Analysis of the DFIG during a voltage dip,” presented at the IEEE Region 8 Eurocon-2011, Lisbon, Portugal, Apr. 27–29. [20] M. Heller and W. Schumacher, “Stability analysis of doubly-fed induction machines in stator flux reference frame,” in Proc. 7th Eur. Conf. Power Electron. Appl., Trondheim, Norway, Sep.1997, vol. 2, pp. 707–710. [21] G. D. Marques and D. M. Sousa, “Air-gap power vector based sensorless method for DFIG control without flux estimator,” IEEE Trans. Ind. Electron., vol. 58, no. 10, pp. 4717–4726, Oct. 2011. [22] P. C. Krause, Analysis of Electrical Machinery. Piscataway, NJ: IEEE Press, 1995. [23] G. D. Marques, “A computer application for teaching and learning on the induction motor dynamics,” Enhancement of Education in Electrical and Information Engineering through Industry Co-operation and Research, pp. 173–178, 1998.

G. D. Marques (M’95) was born in Benedita, Portugal, on March 24, 1958. He received the Dipl. Ing. and Ph.D. degrees in electrical engineering from the Technical University of Lisbon, Lisbon, Portugal, in 1981 and 1988, respectively. He joined the Instituto Superior T´ecnico, Technical University of Lisbon, in 1981, where he teaches power systems in the Department of Electrical and Computer Engineering. He has been an Associate Professor since 2000. He is also a Researcher at the Center for Innovation in Electrical and Energy Engineering. His research interests include electrical machines, static power conversion, variable-speed drive and generator systems, harmonic compensation systems, and distribution systems.

Duarte M. Sousa (M’09) was born in Viana do Castelo, Portugal, in 1970. He received the Dipl. Ing., M.S., and Ph.D. degrees in electrical and computer engineering from the Instituto Superior T´ecnico, Technical University of Lisbon, Lisbon, Portugal, in 1993, 1996, and 2003, respectively. He joined the Instituto Superior T´ecnico, Technical University of Lisbon, in 1993. Since 2003, he has been an Assistant Professor. He is also a Researcher at the Center for Innovation in Electrical and Energy Engineering. His research interests include electrical drives, power electronics, nuclear magnetic resonance equipment, and electric vehicles.