Grid-forming control of doubly-fed induction generators based on the rotor flux orientation

Renewable Energy 207 (2023) 162–176

Available online 2 March 2023

0960-1481/© 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Grid-forming control of doubly-fed induction generators based on the rotor flux orientation

Jesús Castro Martínez

, Jos´e Luis Rodríguez Amenedo , Santiago Arnaltes G´omez , Jaime Alonso-Martínez

Electrical Engineering Department, University CARLOS III, 28911, Legan´es, Madrid, Spain

A R T I C L E I N F O Keywords:

Grid-forming power converter Wind energy

Doubly-fed induction generator Frequency response

A B S T R A C T

The increasing penetration of renewable energies in power systems demands new services from renewable generation plants. System operators are concerned about system stability since renewable generators behave as constant power sources. Therefore, new requirements have been imposed to grid-following generators to improve their contribution to system stability acting as grid-supporting generators. Nevertheless, grid-supporting control can still compromise system stability for high penetration of renewables, and grid-forming control has arised to ensure proper operation. This paper proposes a novel grid-forming control scheme for doubly-fed in- duction generators, so they behave as real voltage sources. The proposed grid-forming control is based on the rotor flux orientation to a reference axis obtained from the emulation of the synchronous generator swing equation. The rotor flux is oriented to the reference axis by means of a flux controller that also controls the flux magnitude. The flux orientation in turn allows to control the doubly-fed induction generator torque, while the flux magnitude control allows to regulate the generator reactive power or terminal voltage. The proposed control system has been validated through a comprehensive real-time simulation with hardware in the loop, assessing its grid-forming capability. Moreover, small signal analysis has also been performed to assess system stability.

1. Introduction

Doubly Fed Induction Generators (DFIGs) are widely used in wind power. In DFIGs, the stator is directly connected to the grid, while the rotor windings are fed at variable frequency through a back-to-back converter. This allows variable speed operation of the generator, while requiring a smaller power converter compared to full-converter gener- ators (FC). In DFIGs, the converter exchanges only the generator slip power, which usually does not exceed 30% of the nominal power of the wind turbine. This is the main advantage of this system. Classical DFIG vector-oriented control uses a grid-voltage-oriented or a stator-flux- oriented vector control to regulate the generator active and reactive power. Since the grid frequency and angle are estimated using a phase- locked loop (PLL), these systems are known as grid-following (GFL), operating as constant power sources under grid frequency disturbances.

Electrical grids are progressively being decarbonized through the addition of renewable and distributed generation, most of which use power electronic converter interfaces [1]. These technologies are dis- placing conventional synchronous generation (SG) that is key to ensure

stable operation regarding voltage and frequency control in electrical grids. Unless new requirements are established for non-synchronous generation to provide grid-supporting services similar to that of SG, the stability and robustness of the system may be compromised [1,2].

Recently, the feasibility of operating synchronous areas with high penetration of converter-based renewable generation (up to 100%) has been studied [3–5]. System stability and robust operation can be guar- anteed in the absence of SG, provided that enough power generation exhibits grid-forming (GFM) capabilities [6,7]. The assessment of the characteristics of a grid-forming converter are discussed in Ref. [8].

The first control approaches of grid-connected DFIGs were based on regulating the rotor current in a synchronous dq axes frame oriented along the stator flux [9]. A similar technique has been used in islanded systems [10,11], controlling the magnitude of the stator flux vector and aligning it with a system of dq axes rotating at the reference frequency.

Additional control alternatives have been suggested, such as in Ref. [12], where active and reactive stator powers are regulated through an internal rotor flux control loop, as opposed to classical methods based on rotor current control. In Ref. [13], a novel control system for the islanding operation of a DFIG is also proposed, which allows for the

* Corresponding author.

E-mail address: [email protected] (J.C. Martínez).

Contents lists available at ScienceDirect

Renewable Energy

journal homepage: www.elsevier.com/locate/renene

https://doi.org/10.1016/j.renene.2023.02.133

Received 9 September 2022; Received in revised form 22 February 2023; Accepted 27 February 2023

parallel operation of the multiple units in a wind farm through a fre- quency droop control but relying in a PLL. However, all these methods consider the grid voltage measured at the stator terminals. Conse- quently, using these methods the DFIG does not operate as a GFM. The same could be said about other control techniques, like direct torque/- power control (DTC/DPC) [14] or model predictive control (MPC) [15], although these control techniques do not use a PLL for measuring the grid voltage angle.

In the past few years, virtual synchronous machine (VSM) control techniques have been developed for wind power, emulating the char- acteristic swing equation of synchronous generators using an active power synchronization loop [16]. In Ref. [17] this active power syn- chronization loop is used to obtaining directly the rotor voltage angle.

However, controlling directly the rotor voltage does not lead to the control of a voltage source behind an impedance which is the principle of VSM.

In [18], a VSM control is compared to a standard PLL-based syn- chronization method regarding the inertial response of a DFIG wind turbine. In Ref. [19], a similar study assesses the provision of inertia by tuning the PLL dynamics. Furthermore [20], discusses a method for the provision of inertia with GFM controls. Since GFM behaves as a voltage source, some proposals lack internal current loops, which is a problem when a voltage dip occurs. In addition [21] discusses a DFIG-GFM control system considering voltage dips. Finally, in Ref. [22], the con- trol, operation, and stability of type III wind turbines (DFIG) are

obtained with this control configuration: first, it allows to improve the dynamics of the control loops given that the rotor flux is controlled directly through the rotor voltage which it is the physical variable applied to the rotor for controlling the machine; and second, this control allows the islanding operation of the generator, as the rotor flux gen- erates the machine internal emf, similarly to the excitation system of a synchronous generator. In this way, an internal emf behind an imped- ance model is obtained. The control scheme presents two regulators that control the rotor flux components in this dq reference frame. The advantage of this control system is that the measurement of the voltage angle at the output terminals is not used for maintaining the generator synchronized with the grid, which is a feature of GFM devices. The DFIG then behaves as voltage source.

The paper is structured as follows: Section II presents the system description. Section III presents the DFIG dynamic equations and the electromagnetic torque and reactive power expressions. Section IV dis- cusses the torque synchronization control loop. The RSC control scheme is presented in Section V. Section VI provides and discusses the experi- mental results. Finally, Section VII discusses the conclusions of the proposed grid forming control of the DFIG based on the rotor flux orientation.

2. System description

The DFIG grid forming control scheme is presented in Fig. 1 The DFIG stator winding is directly connected to the grid, while the rotor winding is connected through a back-to-back converter. One side of the converter is connected to the rotor and the other to the grid. Both windings operate at low voltage and are usually connected to a high voltage grid through a step-up three windings transformer.

The wind turbine controller provides the torque reference for the grid forming control. This reference is obtained following the well- known maximum power point tracking (MPPT) strategy at partial load, while a full load torque reference is kept constant and the pitch regulator controls the rotational speed.

For the proposed control system, the stator voltage vs,abc, and the stator and rotor currents is,abc and ir,abc are measured, and the electro- magnetic torque Te, and the dq components of the rotor flux vector λdr

and λqr are then calculated, using the control angle θ.

The RSC control is based on the regulation of the dq components of the rotor flux. The error between the reference values and the estimated values λdr and λqr, allows the flux regulators to calculate the rotor voltage components vdr and vqr. Then, the instantaneous rotor voltages vr,abc are calculated applying the coordinate transformation using the control angle θ and the rotor position θr measured from the encoder coupled to the DFIG shaft.

The grid side converter (GSC) is controlled using the well-known grid voltage- oriented control to keep a constant DC bus voltage Vdc, [23].

Fig. 1. Grid-forming DFIG control scheme.

3. DFIG dynamic model

The electrical equations of an induction machine in a reference frame, rotating at a speed of ω with regard to a stationary reference system, can be expressed as

d λ→

s

dt = − (1

σTs

+jω )

→λ

s+ 1

σTs

Lm

Lr

→λ

r+→v

s (1)

d λ→

r

dt = − (1

σTr

+jsω )

→λ

r+ 1

σTr

Lm

Ls

→λ

s+→v

r

where Ts=Ls/Rs and Tr=Lr/Rr are the stator and rotor time constants.

The slip is defined as s = (ω− ω_r)/ω and the leakage coefficient as

=1 − (L²_m/(LsLr)).

Separating (1) into its real and imaginary parts, the state-space model of the DFIG consists of four state variables and four input vari- ables, the dq components of the stator and rotor fluxes [x] = [λ_ds,λ_qs,λ_dr,λ_qr]^T and the dq components of the stator and rotor voltages [u] = [vds,vqs,vdr,vqr]^T, respectively.

3.1. Electromagnetic torque and reactive power

Neglecting the stator resistance and the stator flux derivative, v→s∼jω→λ

s, the DFIG stator active and reactive power can be expressed as follows,

Ps=3 2

ω σLs

λs

Lm

Lr

λrsin δ (2)

Qs=3 2

ω σLs

λs

(Lm

Lr

λrcos δ − λs

)

where δ is the angular difference between the stator and rotor flux vectors, the so-called torque angle.

Moreover, neglecting the stator resistance Rs, the stator power Ps is equal to the air gap power, and

Te= Ps

ω/p=3 2p

( Lm

σLsLr

)

λsλrsin δ. (3)

These expressions show that the DFIG torque can be controlled by controlling the torque angle, while the DFIG reactive power can be controlled by controlling the rotor flux magnitude. This is analogous to the operation of a SG.

This can also be shown by defining a DFIG internal electromotive force (emf)

→e

s=jω^Lm Lr

→λ

r. (4)

Then, the former equations can also be expressed as Te=3

2 p ω

1 σXs

vsessin δ (5)

Qs=3 2

1 σXs

vs(escos δ − vs)

showing the same control principles as the previous ones.

Fig. 2 shows the DFIG vector diagram. The DFIG is supplying active and reactive power to the grid, so the current vector i→

s lags the voltage vector v→s by the angle φ.

Moreover, the stator flux vector λ→

s lags v→s by 90^◦since v→s∼jω→λ

s, the rotor flux vector λ→

r can be obtained as

→λ

r=Lr

Lm

(

→λ

s+σLs→i

s

)

(6) Fig. 2. DFIG vector diagram.

Fig. 3. Torque synchronization loop (TSL).

Fig. 4. Stator and rotor flux vector diagram referred to dq axes.

and, according to (4), the emf e→s leads λ→

r by 90^◦. Therefore, as shown in Fig. 2, the torque angle δ between the stator and rotor flux vectors is equal to the angle between the stator voltage and the emf.

4. Torque synchronization control loop

As illustrated in Fig. 1, the synchronization loop sets the control angle θ that determines the position of the dq-axes, relative to a

stationary reference system, based on the difference between the refer- ence torque T_e^∗, and the actual torque Te. The rotational speed of the dq- axes is ω (as shown in Fig. 2) and it is calculated as follows

ω=1 J

∫ʀ

T_e^∗− Te− D(ω− ω0))

dt (7)

where J is the inertia constant of the VSM, in seconds, D is the damping constant, being its inverse R = 1/D the droop constant, which is the ratio between the normalized frequency variation, Δf/fn, and the normalized power deviation, ΔP/Pn. The control angle θ is then obtained by the integration of ω, Fig. 3. As it will be shown in the next paragraph, rotor flux will be oriented to this angular position, while θ is continuedly adjusted by the torque synchronization loop so that the angular differ- ence with the stator flux is the required torque angle.

Fig. 5. RSC flux control.

Fig. 6. Voltage and reactive power control.

Fig. 7. DFIG eigenvalues loci (s = − 0.2, Te0=1 p.u., us=1 p.u.).

Table 1

Eigenvalues, damping ratio and natural frequency.

Variable state Eigenvalue (rad/sec) Damping ratio

ξ (p.u.). Frequency

ω_{n (rad/sec)}

Δλds,Δλqs − 14.7 ± j322 0.045 322

Δλ_dr,Δλqr − 315 ± j65.6 0.98 322

Δxdr,Δxqr − 14.8 ± j3.7 0.98 15.3

Δδ − 18.5 1.00 18.5

Fig. 8. Power system scheme implemented in the RTDS.

5. Rotor side converter control

The conventional RSC control aims to regulate the electromagnetic torque and reactive power at the stator output. The reference electro- magnetic torque T^∗_e is calculated based on the wind turbine MPPT strategy (see Fig. 1). This reference is compared to the actual DFIG torque, and as specified in the previous section, the synchronization loop calculates the control angle θ.

Torque can be easily estimated from the direct measurement of the stator power based on equation (3), using the measurements of the instantaneous stator voltage and current vectors, vs,abc and is,abc. Te=3

2 p ω

ʀuαsiαs+uβsiβs

) (8)

Rotor flux can also be easily calculated from the direct measurements of the stator and rotor currents, is,abc and ir,abc. Then, rotor flux vector in a

stationary reference frame is calculated as

→λ^αβ

r =λαr+jλβr= − Lm→i

s+Lr→i

re^jθr (9)

where θr is the rotor angular position, as measured by the encoder attached to the rotor shaft.

The rotor flux vector λdr+jλqr in a rotating reference system is found by multiplying the vector λαr+jλβr by e^−jθ.

λdr= +λαrcosθ + λβrsin θ (10)

λqr= − λαrsin θ + λβrcos θ

After calculating the rotor flux components on dq-axes, the RSC control is applied to maintain the rotor flux vector oriented along the d- axis of the rotating reference frame. In Fig. 4, the position of stator and rotor flux vectors, the dq-axes rotating at speed ω, as well as the torque Fig. 9. Testbench implementation in RSCAD software (top) and RSC control implementation in Simulink environment (bottom).

angle δ and the control angle θ are shown.

When the rotor flux vector, λ→

r, is aligned with the d-axis, the DFIG is operating in synchronism. To produce this alignment, it is necessary to control λ→

r by means of the rotor voltage, v→r. The relationship between both vectors, based on (1) and disregarding the voltage drop at Rr, is as follows:

→v

r=d λ→

r

dt +j (ω− ωr)→λ

r (11)

This means that, in the rotating reference frame, disregarding the voltage drop at Rr, the dq components of the rotor flux λ→

r can be Fig. 10. RTDS (left) and dSpace (right) HIL testing setup.

Fig. 11. Wind turbine and DFIG rotational speed and torque.

Fig. 12. DFIG rotor currents.

Fig. 13. Active and reactive power response to a reactive power step.

Fig. 14. Rotor flux components response to a reactive power step.

Fig. 15. Detail of instantaneous voltage and current in phase a of DFIG stator.

controlled directly by the corresponding dq components of the rotor voltage vector v→r, after compensating the cross-coupling terms in (11).

The control loops for the rotor flux are represented in Fig. 5. On the q-axis, the setpoint value is maintained at λ^∗_qr=0 to align λ→

r with the d- axis, and by comparing it with λqr , the rotor voltage component vqr is obtained as the sum of the PI regulator output and the cross-coupling term + sωλ_dr. In the same way, the component vdr is obtained after calculating the difference between λ^∗_dr, which is equal to the reference magnitude of the rotor flux, λ^∗_r, and λdr. In this case the cross-coupling term added to the regulator output is − sωλ_qr. Hence, using both regu- lators, and adding the cross-coupling terms, the dq components of the rotor voltage vector v→r are obtained

By applying the inverse Park transformation to such components, using the slip angle θslip, the reference phase components var,vbr,vcr are calculated. From these values, and using pulse width modulation (PWM), the trigger signals S1…6 of the RSC switches are obtained. The slip angle is calculated based on the difference between the control angle, θ, and the rotor angle, θr. θslip =θ − θ_r.

According to Eq. (2), the DFIG rotor flux magnitude λ^∗_r can be ob- tained from a reactive power controller or directly from a voltage controller, as shown in Fig. 6. From (2), it can be stated that reactive power increases with λr.

6. Small signal stability analysis

This section aims to analyze the stability of the proposed control system. By adding the control equations to the dynamic model of the machine, given by (1), the dq components of the rotor voltage that were inputs in (1) are obtained as outputs of the rotor flux control represented in Fig. 5.

→v

r=kp

(

→λ^∗

r− →λ

r

) +ki

∫ (

→λ^∗

r− →λ

r

)

dt (12)

where kp and ki are proportional and integral gains of the PI regulators.

Separating (12) into its real and imaginary parts, vdr and vqr are obtained as follows:

vdr=kp

ʀλ^∗_r− λdr

)+ki

∫ʀ λ^∗_r− λdr

)dt (13)

vqr= − kpλqr− ki

∫ λqrdt

The d-axis reference of the rotor flux is equal to the flux magnitude (λ^∗dr =λ^∗_r) when the control action is completed, while the q-axis refer- ence of the rotor flux is zero (λ^∗qr =0) when the rotor flux vector is aligned with the d-axis.

Two additional state variables, xdr and xqr, are introduced as dxdr

dt =ki

ʀλ^∗_r− λdr

) (14)

dxqr

dt = − kiλqr

The equations for vdr and vqr in (12) using xdr and xqr are as follows:

vdr=kp

ʀλ^∗_r− λdr

)+xdr (15)

vqr= − kpλqr+xqr

Substituting the dq components of the rotor voltage (13) in (1), the following dynamic expressions are obtained for the rotor fluxes:

dλdr

dt = 1 σTr

Lm

Ls

λds− (1

σTr

+kp

)

λdr+sωλqr+xdr+kpλ^∗_r (16)

dλqr

dt = 1 σTr

Lm

Ls

λqs− sωλdr− ( 1

σTr

+kp

) λqr+xqr

Considering a constant slip, these equations are linear. On the other hand, the dynamic equations of the stator flux do not depend on vdr and vqr so that

dλds

dt = − 1 σTs

λds+ωλqs+ 1 σTs

Lm

Lr

λdr+vssin δ (17)

dλqs

dt = − ωλds− 1 σTs

λqs+ 1 σTs

Lm

Lr

λqr+vscos δ

where vds=vssin δ and vqs =vscos δ. In contrast to (16), these equa- tions are not linear because the dq components of the stator voltage are obtained as the product of vs and a trigonometric function of δ, which is a state variable on the controlled system. In (17), the stator voltage ap- pears as a disturbance of the controlled DFIG system.

The equation of the torque angle δ in the synchronization loop is dδ

dt=ω0

D (

T_e^∗− 3 2p

(1 σLs

Lm

Lr

) λsλrsin δ

)

(18) This is a non-linear equation since the electromagnetic torque is the product of three variables. In (18), torque synchronization considers only the droop constant 1/D term to simplify the stability study.

Taking a small variation around an equilibrium point, the state Fig. 16. System frequency and DFIG active and reactive power after a

load step.

Fig. 17. Comparison of system frequency and active power of three different wind power models simulated after a load step.

Table 2

Frequency nadir and ROCOF values of the cases depicted in Fig. 17.

PARAMETER WT WTx20 (GFL) WTx20 (GFM)

Frequency nadir 49.57 Hz 49.66 Hz 49.71 Hz

ROCOF at t = 1 s − 0.39 Hz/s −0.34 Hz/s −0.26 Hz/s

variables of the DFIG linearized model, including the proposed control, are then:

[Δx] =[

Δλds,Δλqs,Δλdr,Δλqr,Δxdr,Δxqr,Δδ]_T

(19) Three new state variables are added to the dq components of stator and rotor fluxes. These state variables are Δxdr,Δxqr and the torque angle Δδ. The inputs are now the stator voltage magnitude and the rotor flux and electromagnetic torque references [Δu] = [Δvs,Δλ^∗_r,ΔT^∗e]^T. The output vector includes the rotor flux magnitude and the electromagnetic torque [Δy] = [Δλr,ΔTe]^T. Therefore, the set of linearized equations at the point of equilibrium of a controlled DFIG can be expressed as follows d[Δx]

dt =A[Δx] + B[Δu] (20)

[Δy] = C[Δx] + D[Δu]

The elements of the A, B, C, and D matrices are given in the Appendix.

Fig. 7 shows the eigenvalues loci of the controlled DFIG, whose pa- rameters are given in the Appendix. The point of equilibrium is Te0= − 1 p.u., vs=1 p.u., and s = − 0.2 p.u. The PI regulator parameters are the proportional gain kp=1 p.u. and the integral gain ki =kp/ (ωTr), where the regulator time constant is Tc =σTr =0.059 s, so that ki = 0.054 kp. And the synchronization droop constant is 1/D = 0.05 p.u.

As shown in Fig. 7, the rotor flux eigenvalues Δλdr,Δλ_qr are shifted toward the left side when compared to the non-controlled model, because of the effect of the rotor flux PI regulators, and the imaginary part of these eigenvalues is the same as in the non-controlled model, so they have the same oscillation frequency in both models. Eigenvalues Δλ_ds,Δλ_qs do not change significantly compared to the non-controlled

system, because the stator flux is non-controlled. And the eigenvalue of the torque angle Δδ is also on the negative side of the real axis.

In Table 1, the eigenvalues of the state variables of the dynamic system are presented. All eigenvalues are in the negative half plane which guarantees system stability. Rotor flux modes are very close to the critical damping (ξ = 0.98), as well as auxiliary variables Δxdr,Δxqr, but the stator flux modes are very poorly damped (ξ = 0.045).

The comparison of the small signal stability of the proposed control system with that of a conventional control implementing droop control and inertia emulation, as presented in Ref. [24], shows that the stability of the DFIG has significantly improved when referring to the most critical eigenvalues, i.e., those related to the stator flux and synchro- nizing angle, which are the closest to the positive half plane.

7. Experimental results

The assessment of capabilities of the proposed grid forming control of the DFIG has been done through an experimental setup using real time simulation with hardware in the loop (HIL). A comprehensive model of the DFIG and the testing grid has been implemented in the Real Time Digital Simulator (RTDS) platform, while the proposed control system has been implemented in a dSpace control board, running the control in real time from a control model developed in Simulink, implemented in the target hardware through its Real-Time Interface. Fig. 8 denotes the scheme of the power system implemented in the RTDS. For the imple- mentation of power system model, the real time simulator manufacturer provides a dedicated software called RSCAD. Moreover, the simulated test bench follows the specifications of the Spanish normative for the assessment of the requirements for generators [25]. The benchmark Fig. 18. Response to a grid fault during the voltage drop.

parameters are given in the Appendix.

The RSC control system presented in this paper is implemented in the dSpace board using Simulink environment, see Fig. 9. The dSpace reads the DFIG phase currents, phase voltages, DFIG rotational speed and rotor angular position through its analog input board (DS2004). These signals are previously generated in the testbench running in the real-time simulator processor and passed through the FPGA. Then, they are sent to the RTDS analog output board (GTAO). On the other hand, the control system outputs the PWM duty cycle signals, which are converted into pulses through a high-speed pulse board (DS5101). These pulses are then sent to the RTDS high speed digital I/O card (GTDI) and used as input signals for the commutation of the IGBTs bridge simulated in the

RSCAD model. The whole process can be monitored in real time using both ControlDesk and RSCAD runtime, Fig. 10 shows the testing setup.

7.1. DFIG response to a wind gust

The first test shows the DFIG response to a wind gust. At t = 1 s, wind velocity changes from 5 to 9 m/s. The wind turbine is initially operating at the MPP for 5 m/s (2.1 kNm and 1083 rpm, at the generator side).

Then, because of the wind gust, the wind turbine torque increases, producing a rotor acceleration.

The increasing rotational speed produces increasing DFIG torque commands following the MPPT strategy. Finally, a new steady state point is reached at the MPP corresponding to 9 m/s (6.9 kNm and 1950 rpm, at the generator side). All the above is illustrated in Fig. 11.

This test demonstrates the capability of the proposed control system to track the input torque command while maintaining the synchronism.

This is achieved by keeping the rotor flux oriented to the reference angle θ, whilst this angle changes following the increasing generator torque command.

Fig. 12 denotes the rotor currents at the transition around the syn- chronous speed of the DFIG. The figure clearly illustrates the change in the rotor current phase sequence when changing from sub-synchronous to super-synchronous rotational speed, i.e., from positive to negative slip.

7.2. DFIG response to a reactive power step

In this subsection, the DFIG response to a reactive power step com- mand is obtained when the rotor flux magnitude is given by a reactive power controller as the one depicted in Fig. 6. Initially, the reactive power command is set at 0 MVAr and at t = 0.3 s a 1000 MVAr command is set. The time response of the active and reactive powers is given in Fig. 19. Response to a grid fault during the voltage recovery.

Fig. 20. Active and reactive power response.

Fig. 13.

The wind turbine is operating at a wind velocity of 10 m/s and the power shown in the figure is the active power of the DFIG stator. The reactive power reaches the commanded value in a rapid and well damped manner, while the active power suffers a slight disturbance due to transient misalignment produced while increasing the rotor flux through the voltage applied by the RSC.

Fig. 14 denotes the rotor flux components. The direct component increases as demanded by the reactive power controller, while during the transient a slight misalignment can be observed in the transversal rotor flux component, producing the active power disturbance noted before.

Finally, Fig. 15 denotes the stator voltage and current in one phase of the stator. Initially, voltage and current pulse in phase because the commanded reactive power is zero. Then, at t = 0.3 s when the reactive

power command is changed, the current wave immediately changes to lag the voltage wave as required by commanded reactive power.

7.3. DFIG response to a system load step

The DFIG response to a system load increase is shown in this sub- section. The main purpose of this simulation is to demonstrate that the proposed control system behaves as a voltage source. So, in opposition to the classical vector-oriented control, where the DFIG behaves as a cur- rent source providing constant active and reactive powers, here the generator can “see” the load change. So, at t = 1 s the system load is increased by 0.05 p.u. Initially, the wind turbine is supplying 6.9 kNm at 1950 rpm at 9 m/s wind velocity. After the load step, the system Fig. 21. Stator voltage and frequency, rotor flux components, stator instantaneous voltage and current, and rotor voltage and current.

Fig. 22. WT rotational speed and pitch angle.

Fig. 23. DFIG torque and WT torque in isolated mode.

frequency drops as depicted in Fig. 16. Also, in this figure, the DFIG active power increases following the load increment, demonstrating that the DFIG behaves as a voltage source.

Obviously, the DFIG supplies the load increment only temporally because the incoming power of the wind turbine does not change. In fact, the generated power increment is obtained from the kinetic energy of the rotor, producing the deceleration of the rotor, due to the DFIG torque increment.

7.4. Wind farm response to a load step

In this subsection, the system frequency response is obtained when a wind farm, made of 20 WTs, is connected in the scheme of Fig. 8, instead of a single wind turbine. This simulation test pursues to demonstrate the advantages of grid-forming DFIG in weak grids. For simulation, a single WT, rated 20 × 1.5 MW, represents the aggregate model of the WF. Wind velocity, system load and load increment are the same as those used in subsection 7.3. The objective of this simulation is to prove the contri- bution of the proposed control system to the frequency stability under a high penetration of wind power. For comparison purposes, the system frequency obtained in subsection 7.3 is also represented here. Thus, upper part of Fig. 17 shows in blue the system frequency when only one wind turbine was connected to the system and in red when the wind farm is connected. As shown in the figure, frequency response improves in both ROCOF and frequency nadir. The reason for this is that the droop control response of the DFIG, given by the synchronization loop of Fig. 3

and the fast flux control of Fig. 5, is much faster than that of the syn- chronous generator, because the inherent the response delay of the governor and turbine. This is what in the literature is being called fast frequency response, although the proposed control system provides it naturally. Finally, Fig. 17 also shows the wind farm and the single wind turbine cases power response. A higher ripple can be observed in the power response in the case with high wind power penetration because of the higher power electronics-based generation in the system. As in subsection 7.3, the wind farm responses to the load increment by increasing its output, alleviating in this way the frequency deviation of the synchronous generator. Finally, a comparison with the frequency response of the same wind farm embodying now a classical grid- following control with frequency support based on droop frequency regulation [24] has been included. The frequency response showed in Fig. 17 shows that both frequency nadir and ROCOF are higher in this case, see Table 2. This was expected due to the inherent delays in fre- quency measurement and filtering of this control strategy, whilst the grid-forming provides its response in a natural and immediate way.

7.5. DFIG response to a grid fault

The response of the DGIG under the proposed grid-forming control scheme to a symmetrical grid fault is obtained in this subsection. This simulation test pursues to demonstrate the low voltage ride through (LVRT) capability of the proposed control system. The fault is applied at Fig. 24. Terminal voltages at DFIG stator and at POC.

Fig. 25. Motor rotational speed and torque.

Fig. 26. Active and reactive power generated by the DFIG.

the POC and it is correctly cleared after 500 ms. As a consequence, a 20%

voltage dip is seeing at the DFIG terminals. Fig. 18 shows the DFIG response to the voltage dip. This figure shows, from top to bottom, POC voltages, rotor currents, stator currents, stator active and reactive powers and rotor flux components. Initially, after the fault is detected, the crow-bar protection is activated for 50 ms. in order to protect the rotor side converter against the short-circuit currents [26]. After that period, the DFIG control takes over, active power is taken to zero and reactive power is increased to support the voltage. Note that the rotor flux is being oriented to the reference angle, by maintaining the rotor flux quadrature component to zero, and the rotor flux magnitude is increased as much as possible considering the rotor current limitation, due to the rotor side converter current capability.

Fig. 19 shows the DFIG response during the dip recovery. As before, this figure shows, from top to bottom, POC voltages, rotor currents, stator currents, stator active and reactive powers and rotor flux com- ponents. After the fault is cleared, voltage is recovered and, after the transient, the control system takes the active and reactive power to the initial values of 1500 kW and 0 kVAr, respectively. The active power reference is ramped from 0 to the reference value in 0.5 s. The rotor continues being oriented by maintaining the quadrature rotor flux component to zero, while the direct rotor flux component is increased to the value commanded by the reactive power controller, because this value can now (after the stator voltage recovery) be achieved without increasing the rotor currents over the allowed limits. It is worth mentioning that this control is also compatible with another LVRT control strategies as the one proposed in Ref. [27], in order to improve the reactive power support.

7.6. DFIG response to a transition from grid connected to isolated In this subsection, according to the scheme of Fig. 8, the DFIG will supply a local system load in isolated mode after the sudden trigger of the interconnection line. It has to be noted that the DFIG switches from grid-connected to isolated mode automatically, i.e., there is no needed to detect the islanded operation. After the transition, the DFIG automati- cally changes its output, due to its voltage source feature, until the power demanded by the local load is met with a certain frequency higher than nominal because the DFIG reduces its output.

Initially, the DIFG is connected to the grid supplying the maximum power, 1500 kW, for the actual wind velocity, 10 m/s, at 1938 rpm.

Also, the reactive power command is set at 400 kVAr (see Fig. 20).

At t = 1 s, the line is triggered and the DFIG supplies the local load in isolated mode. In isolated operation the DFIG has to supply exactly the active and reactive powers demanded by the load. In this case, a con- stant impedance load demanding 1000 kW and 500 kVAr has been used.

During this simulation, the external loop setting the λrd command is the voltage control mode, as depicted in Fig. 6. Fig. 16 shows this transition from grid connected to isolated operation. The control system enables a fast and stable response.

The synchronizing loop of Fig. 3 detects a reduction in the DFIG torque, as a consequence of the reduction of the electrical load (see Fig. 23). Therefore, the droop control produces a positive frequency increment, as shown in Fig. 21. The internal frequency is increased, and the synchronizing control loop operates with a permanent torque error, as it does not follow the reference torque anymore, but produces the torque required for the actual electrical load. On the other hand, the voltage control of Fig. 6 automatically obtains the rotor flux magnitude required for supplying the isolated load at the commanded voltage.

Fig. 21 shows the voltage magnitude and frequency, the rotor flux components, the instantaneous stator voltage and current and rotor voltage during the transition. The rotor flux q-axis component is kept as its reference value of zero, so the reference frequency is achieved. While the d-axis component produces the desired rotor flux magnitude to obtain the reference voltage at the DFIG terminals. Also, it has to be pointed out that if the operation at nominal frequency was desired, a secondary frequency controller could have been implemented that would modify the reference frequency ω₀ in Fig. 3, so that the nominal frequency is obtained for the internal frequency ω. Moreover, Fig. 21 also shows the detail of the instantaneous stator voltage and current and rotor voltage during the transition. A higher ripple can be observed in the voltage at the POC after the transition, because initially voltage was imposed by the synchronous generator and in isolated mode is imposed by the DFIG being generated through the induced emf produced by the modulated voltage applied to the rotor.

Moreover, Fig. 22 shows the wind turbine rotational speed, at the DFIG side, and the pitch angle, while Fig. 23 shows the wind turbine torque, also at the high-speed shaft. Here, after the transition, the wind turbine accelerates as a consequence of the reduction of the DFIG torque.

When the wind turbine reaches its maximum rotational speed, the speed control based on the pitch control starts increasing the pitch angle, which reduces the wind turbine driving torque to maintain constant Fig. 27. Frequency at POC.

rotational speed. By maintaining constant rotational speed, it can be assured that the power delivered by the wind turbine equals the power demanded by the DFIG, which in turn it is equal to the power demanded by the electrical load.

These results demonstrate the ability of the proposed control system to switch from grid connected to isolated operation, without having to detect the islanded mode and without changing the control mode.

Therefore, these results also prove the operation of the DFIG as a true voltage source using the proposed control system.

7.7. Start-up of an induction motor from a blackout situation

In this subsection, the black start capability of the control proposed will be tested when feeding a dynamic load (induction motor) from a de- energized system. Attending to Fig. 8, the motor is located at the POC and the DFIG is working in isolated operation as the line connecting it to the SG is tripped during the simulation. The system will be energized by ramping up the voltage reference, in order to smooth the inrush currents generated during the system energization. Wind turbines are receiving a wind speed above nominal.

Voltage magnitudes measured at the DFIG stator and at the POC are shown in Fig. 24. The voltage reference of the DFIG voltage control loop is ramped from 0 to 1 in 3 s. At t = 4s, the nominal mechanical load is connected to the motor and the voltage drops consequently. Motor rotational speed and torque are shown in Fig. 25. The motor rotational speed increases from zero to the synchronous speed as no load is applied to the motor initially. At t = 4 s the mechanical load is connected and the rotational speed drops. This figure also shows the motor electrical tor- que, where the motor starting-up and loading can be noticed.

Fig. 26 shows the active and reactive powers delivered by the DFIG.

Note that the active power provided by the generator practically matches the active power consumed by the motor, given that the rest of the elements of the system are mainly inductive. The active power provided initially accelerates the motor until it reaches its synchronous speed, dropping practically to zero afterwards; at t = 4 s the mechanical load is applied to the motor and the active power increases to feed it.

Attending to the reactive power, the effect of the inrush currents is prevented due to the voltage ramping. At t = 4 s the mechanical load is applied and the reactive power increases with the slip.

The frequency at the POC is depicted in Fig. 27. Due to the fact that the DFIG is operating in an isolated system, the frequency is determined by the synchronization loop, as a result of the active power error and the droop constant. The power reference is set to zero, so the frequency starts at 50 Hz. As the system starts being loaded, the frequency deviates below 50 Hz and comes back to practically 50 Hz when the motor rea- ches its synchronous speed, given that no active power is demanded. At t = 4 s the mechanical load is applied and the frequency drops again.

8. Conclusions

This paper has proposed a novel grid-forming control strategy for a DFIG based on the orientation of the rotor flux. By aligning the rotor flux to the reference axis, two control objectives are met: controlling the synchronism of the DFIG and controlling the DFIG torque. Moreover, the flux controller also allows to modify the rotor flux magnitude, which in turn allows to control the DFIG reactive power.

The dynamic model of the DFIG is first presented in the paper and then, using the definition of the proposed emf, an equivalent circuit analogous to that of the synchronous generator can be obtained in steady state. Based on this dynamic model, the DFIG synchronizing

torque is obtained, which is used in the synchronization control loop that emulates the swing equation of the synchronous generator. It is demonstrated that this synchronizing loop allows the generator to follow changes in the grid load, in opposition to the classical vector- oriented control.

Furthermore, the paper introduces later the inner flux control loops.

The rotor voltage components applied by the RSC are used to control of the rotor flux components, based on the dynamic equations of the rotor circuit. Moreover, also based on the proposed dynamic model of the DFIG, the rotor flux magnitude is used to control the DFIG reactive power, whereas torque is controlled by the TSL, by means of the load angle.

Finally, comprehensive simulation models, including the wind tur- bine and a test grid, have been implemented in the RTDS real-time platform to obtain the response of the proposed control system, which interacts with the RTDS using HIL. The test setup is based on the Spanish normative for the assessment of the requirements for generators.

Experimental results demonstrate the capability of the proposed control system to follow torque and reactive power commands, being torque commands a consequence of the wind speed variations following the MPPT, and also to demonstrate the grid-forming capability, acting as a voltage source under changes in the grid load. Moreover, the response to a grid fault has also been obtained, demonstrating the low voltage ride through capability of the proposed control system. The transition from grid-connected to isolated operation has also been tested to prove the capability of the proposed system to act as a voltage source even in isolated operation. Finally, the start-up of and induction motor from a non-energized grid situation has been proved feasible with this control strategy. The experimental results also prove the excellent dynamic response of the proposed control system.

CRediT authorship contribution statement

Jesús Castro Martínez: Investigation, Resources, Conceptualiza- tion, Methodology, Software, Validation, Formal analysis, Writing – original draft, Writing – review & editing, Visualization. Jos´e Luis Rodríguez Amenedo: Investigation, Resources, Conceptualization, Methodology, Software, Validation, Formal analysis, Writing – original draft, Writing – review & editing, Visualization, Supervision, Project administration, Funding acquisition. Santiago Arnaltes G´omez:

Investigation, Resources, Conceptualization, Methodology, Software, Validation, Formal analysis, Writing – original draft, Writing – review &

editing, Visualization, Supervision, Project administration, Funding acquisition. Jaime Alonso-Martínez: Investigation, Resources, Conceptualization, Methodology, Software, Validation, Formal analysis, Writing – original draft, Writing – review & editing, Visualization, Su- pervision, Project administration, Funding acquisition.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was supported by the Spanish Research Agency (Agencia Estatal de Investigaci´on) under Project PID2019-106028RB-I00/AEI/

10.13039/501100011033.

Appendix A. State space model matrix elements Matrix A

B11=cos δ0B21= − sin δ0B52=kiB73= D Matrix C

C13=1 C21= − A71C22= − A72C23= − A73C27= − A77

Matrix D

D11=D12=D21=D22=0

Appendix B. Wind turbine and doubly-fed induction generator parameters Table 3

DFIG parameters

PARAMETER SYMBOL VALUE UNITS

Rated power PN 1500 kW

Rated stator voltage Vs 690 V

Stator frequency fs 50 Hz

Poles pair number p 2 –

Slip range s ±1/3 p.u.

Stator resistance Rs 3.46 m Ω

Rotor resistance R^′r 3.87 m Ω

Magnetization inductance Lm 3.33 mH

Stator leakage inductance Lσ_s 0.116 mH

Rotor leakage inductance L^′σr 0.116 mH

Moment of inertia J 650 kg/m²

Appendix C. Simulation benchmark parameters Table 4

synchronous generator parameter

PARAMETER SYMBOL VALUE UNITS

Rated power SN 200 MVA

Rated line-line voltage VN 5 kV

Frequency fs 50 Hz

Poles pair number p 2 –

Stator winding resistance Rs 0.020 p.u.

Stator leakage inductance Ls 0.112 p.u.

d-axis magnetizing inductance Ldm 1.79 p.u.

q-axis magnetizing inductance Lqm 1.60 p.u.

Field winding resistance (referred to stator) Rf 1.21 p.u.

Field winding leakage inductance (referred to stator) Lfl 0.117 p.u.

Rotor damping cage d-axis resistance (referred to stator) Rdr 0.03 p.u.

Rotor damping cage d-axis leakage inductance (referred to stator) Ldrl 0.375 p.u.

Rotor damping cage q-axis resistance (referred to stator) Rqr 0.004 p.u.

Rotor damping cage d-axis leakage inductance (referred to stator) Lqrl 0.200 p.u.

Inertia constant H 2.5 s

Stator and rotor windings turn ratio N_s/N_r 1/3

Table 5

Line and transformers parameters

PARAMETER SYMBOL VALUE UNITS

T1: Rated power SN 220 MVA

T1: Rated line-line voltage primary V1N 20 kV

T1: Rated line-line voltage secondary V2N 5 kV

T1: Short-circuit ratio Zsc 10 %

T1: X/R ratio X/R 30 p.u.

T1: Frequency f 50 Hz

T2: Rated power SN 2.5 MVA

T2: Rated line-line voltage primary V1N 20 kV

T2: Rated line-line voltage secondary V2N 0.69 kV

T2: Short-circuit ratio Zsc 8 %

T2: X/R ratio X/R 25 p.u.

T2: Frequency f 50 Hz

LINE: Inductance Ll 3.4 mH

LINE: Resistance Rl 0.5 Ω

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