Decoupled Double Synchronous Reference Frame PLL for Power Converters Control

Decoupled Double Synchronous Reference Frame Current Controller for Unbalanced Grid Voltage

Conditions Manuel Reyes1, Pedro Rodríguez1, Sergio Vázquez2, Alvaro Luna3, Juan Manuel Carrasco2, Remus

Teodorescu4

(1) Abengoa Research Seville, Spain

[email protected]

(2) Dpt. of Electronic Engineering

University of Seville Seville, Spain

(3) Dpt. of Electrical Engineering

Techinical University of Catalonia Terrasa, Spain

(4) Dpt. of Energy Technology

Aalborg University Aalborg, Denmark

Abstract— Due to the growing changes in the electrical network related to the new distributed generation scheme and the integration of renewable energy sources, new requirements for grid-connected power converters are being defined in the new grid codes. The injection of positive- and negative-sequence current components is becoming necessary for achieving new capabilities like the reactive power injection during a grid fault. This paper deals with a fundamental issue in this topic, i.e., the performance of the current controller. Classical dq controllers, which are extensively used in industrial applications, degrade its performance when both sequences are involved, as unavoidable oscillations in the dq axes appear. In this paper, a new scheme for controlling positive- and negative-sequence currents using dq controllers is proposed, based on an enhanced Double Synchronous Reference Frame (DSRF) controller. In this paper the DSRF controller is improved by adding a decoupling network which counteracts the oscillations caused by the presence of both sequences. Experimental results will demonstrate the validity of the proposed Decoupled DSRF (DDSRF) controller.

I. INTRODUCTION The connection of power converters to the electrical

network has become a crucial issue due to the fast penetration of renewable energy sources and distributed generation systems [1]-[4].

The integration of renewable energy sources into the electrical networks requires a good performance of the grid-side converters. The significant increase of these distributed generators in the energy market leads to the development of specific grid-connection codes by the Transmission System Operators (TSO) [5]-[6].

Under the presence of a grid fault the injected currents and the grid voltage may lose their sinusoidal and balanced appearances, giving rise to uncontrolled oscillations in the active and reactive power provided to the network. In the case of unbalanced faults positive- and negative-sequence

currents must be controlled independently under any grid voltage conditions to avoid the power oscillations.

Resonant controllers [7], [8], hysteresis current controllers [9], [10], direct power control methods [11], [12], and model based predictive controllers [13] can be used to control grid connected power converters operating under generic grid conditions.

However, the injection of positive- and negative-sequence currents into the grid cannot be accurately achieved by using most of the conventional current controllers implemented in the industry, based on implementing PI controllers working in rotational dq axis. However, industry is still interested in this kind of controllers, as they are a well-known and a can be easily tuned and controlled. This paper is focused on the improvement of synchronous frame based solutions oriented to improve the performance of classical controllers.

Under unbalanced conditions, two control loops, one for the positive-sequence and another one for the negative-sequence reference frames are needed. The DSRF regulator controls the positive- and negative-sequence current vectors by projecting the feedback currents on both reference frames by means of the Park’s transformation.

However, the interaction between current vectors and reference frames with different sequences gives rise to oscillations at twice the fundamental grid frequency (ω) in the dq signals obtained from the Park transformation.

Thus, authors in [14] proposed a double synchronous reference frame (DSRF) regulator using notch filters for reducing the oscillations. Likewise, other approaches for generalized unbalanced operating conditions were described in [15] and [16].

The decoupled DSRF (DDSRF) is proposed in this paper as a solution to eliminate the 2ω oscillations produced by the injection of positive- and negative-sequence currents to the

This work was supported in part by the Andalusian Government, Juntade Andalucía, under the project TIC-02991 and in part by the SpanishMinistry of Science and Innovation under the project ENE2011-29041-C02-01

grid. This new structure is based on a DSRF regulator with a cross-decoupling sequence network, which estimates the existing oscillations and compensates them. Finally, experimental tests have been performed in order to demonstrate the validity of the theoretical results.

II. CURRENT CONTROLLERS FOR UNBALANCED CURRENTS INJECTION

A. Double Synchronous Reference Frame (DSRF) The dq synchronous current controller is based on a

rotating reference frame synchronized with the frequency which is desired to be compensated. If not only fundamental positive-sequence, but also negative-sequence is involved, two dq synchronous controllers synchronized with the respective angular position of each sequence θ +, θ − are needed. This is the so-called DSRF structure that can be seen in Figure 1.

The dq current components in the positive and negative reference frame must be calculated by applying the Park´s transformation over iαβ with the suitable positive-sequence, θ +, and negative-sequence angle, θ −, which have been obtained from the grid-voltage:

( ) ( )( ) ( )

( )DC term AC term

DC termAC term

cos sini i

sin cos

+

+ −

− + − − + −++ −

+ − + − − + −

− −+ −

⎡ ⎤⎡ ⎤ − + −⎢ ⎥⎡ ⎤ ⎢ ⎥= ⋅ = + =⎣ ⎦ ⎢ ⎥⎢ ⎥ − − + −⎣ ⎦ ⎣ ⎦

⎡ ⎤+ ⋅⎢ ⎥⎣ ⎦

d qdJdq

q d q

Jdq dq

i iie

i i i

i e i

θαβ

θ θ

θ θ θ θ

θ θ θ θ (1)

( ) ( )( ) ( )

( )DC term AC term

DC termAC term

cos sini i

sin cos

−

− +

+ − + + − +−− −

− + − + + − +

− −− +

⎡ ⎤⎡ ⎤ − + −⎢ ⎥⎡ ⎤ ⎢ ⎥= ⋅ = + =⎣ ⎦ ⎢ ⎥⎢ ⎥ − − + −⎣ ⎦ ⎣ ⎦

⎡ ⎤+ ⋅⎢ ⎥⎣ ⎦

d qdJdq

q d q

Jdq dq

i iie

i i i

i e i

θαβ

θ θ

θ θ θ θ

θ θ θ θ (2)

The Park’s transformation in (1) and (2) is given by:

cos( ) sin( ).

sin( ) cos( )− ⎡ ⎤⎡ ⎤ = ⎢ ⎥⎣ ⎦ −⎣ ⎦

Je θ θ θθ θ

(3)

The main drawback of the DSRF controller is the cross-coupling between the dq axis signals of both synchronous reference frames that can be checked in (1) and (2). This coupling effect is manifested by a 2ω oscillation overlapping the dc signals on the dq axes, determined by the difference between the angular positions of each synchronous frame:

22 ,

−+

− +

− =− =

+ Δ− − Δ

tt

ωω

θ φθθ φθ

(4)

where Δφ is the difference of initial phase between the positive- and negative-sequence voltage vector.

The positive-sequence frame is rotating counter clockwise while the negative-sequence one is rotating

clockwise. From the positive-sequence reference frame point of view, the negative-sequence is rotating at double frequency and vice versa.

To illustrate the effect of the cross-coupling between the positive and negative reference frames, simulations of power converter injecting positive- and negative-sequence currents into the grid through an inductor have been performed using Matlab.

Considering the current references provided to the controller in this simulation test, which are showed on the positive- and negative-sequence reference frames, dq+ and dq- in the Figure 2, the measured currents that are obtained with a DSRF current controller are shown in Figure 3. From the figures it can be appreciated how the injection of any positive-sequence current gives rise to oscillations at 2ω on the negative reference frame, and vice versa, degrading thus the performance of the current controller.

a)

b) Figure 2. Current references for a grid-connected converter in the synchronous frames: a) positive-sequence (id

*+ and iq*+); b) negative-

sequence (id*-and iq

*-)

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4-5

0

5

i d*+, i

q*+ [A

]

Time [s]

iq*+

id*+

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4-5

0

5

i d*-, i

q*- [A

]

Time [s]

id*-

iq*-

Figure 1. Double synchronous reference frame (DSRF) current controller.

qv+

dv+

qi+

PIdi+

*qi+

*di+

+

+

+−

di+Δ

qi+Δ

+

+

−

−

qv−

Je θ −

dv−

qi−

di−

*qi−

+

+

+

−

+

+

−

−

Je θ −−

Je θ +− Je θ +

vαβ+

vαβ−

vαβiαβ

idq−

idq+

*di−

+

+

qi−Δ

di−Δ

PI

PI

PI

Lω

Lω

Lω

Lω

B. Decoupled DSRF

The dq current in the DSRF under unbalanced conditions results from the addition of a dc value and an overlapped 2ω perturbation. This oscillation must be avoided in order to achieve full capability for injecting the desired active and reactive power during a grid fault. The DDSRF introduced in this paper is based on the estimation of the oscillation for minimizing this undesirable effect.

As it was evidenced in (1) and (2), the amplitude of the oscillation in the positive-sequence measured current matches the dc value of the dq negative-sequence current component and vice versa; therefore it can be easily stated that there exists a cross-coupling effect between both sequences. For example, in Figure 3, from t=1s to t=1.1s, the amplitude of the oscillation in the negative-sequence is equal to the positive-sequence injected current. However, the PI controller achieves the tracking of the dc value due to its infinite dc gain but cannot avoid the 2ω ripple. In the proposed DDSRF case, not only the PI will track the dc value, but also the decoupling-network will estimate and eliminate the oscillation from the measured current.

( ) ( )'

DC term DC term DC term

AC term Cross Coupling Term

i+ −+ − − −− −+ + − −⎡ ⎤⎡ ⎤= + ⋅ − ⋅⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

JJdq dq dq dqi e i e i

θ θθ θ (5)

( ) ( )'

DC term DC term DC term

AC term Cross Coupling Term

i− +− + − −− −− − + +⎡ ⎤⎡ ⎤= + ⋅ − ⋅⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

JJdq dq dq dqi e i e i

θ θθ θ (6)

The DDSRF equations in (5) and (6) are directly obtained from equations (1) and (2) by subtracting the undesirable ac term using the decoupling-network. As a result, the estimated current is free from the 2ω oscillating ac term.

The proposed DDSRF current controller scheme is shown in Figure 4.

The dc value of one sequence is the value for the opposite sequence oscillation´s amplitude. On the other hand, the Park´s transformation is applied over the angular

position difference of both frames with the aim of obtaining the estimated oscillation waveform.

The DDSRF current controller filters the corresponding opposite sequence current to obtain its dc value and decelerate (if the sequence to be compensated is the positive one) or accelerate (if the sequence to be compensated is the negative one) by two times the rotating frequency (considering also the initial voltage phase shift) in order to estimate the undesirable oscillations.

The low-pass (LP) filter for obtaining the mean value does not have a high selectivity because in steady-state conditions, the decoupled current is free from oscillations and matches its dc value. The cut-off frequency of the LP filter is set to:

/ 2fω ω= (7)

The performance of the DDSRF under the current references in Figure 2 is shown in Figure 5.

The cross-decoupling network allows a perfect estimation of the oscillations in the measured currents, achieving the elimination of the 2ω oscillations after a short transient error. This transient error is produced by the

a)

b) Figure 5 Decoupled currents using the DDSRF: a) in the positive-sequence reference frame (id

+ and iq+); b) in the negative-sequence reference frame

(id- and iq

-).

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4

-5

0

5

i d+',

i q+' [

A]

Time [s]

id+'

iq+'

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4

-5

0

5

i d- ', i q- ' [

A]

Time [s]

id- '

iq- '

Figure 4. DDSRF current controller based on the measured current.

qv+

dv+

'qi+

PILω

'di+

*qi+

*di+

+

+

+

−

di+Δ

qi+Δ

+

+

−

−

qv−

Je θ −

dv−

'qi−

'di−

*qi−

+

+

+

−

+

+

−

−

Je θ −−

Je θ +− Je θ +vαβ

+

vαβ−

vαβiαβ

idq−

idq+

*di−

'idq−

'idq+

+

+

( )Je θ θ+ −− −

−

−

f

fsω

ω+

( )Je θ θ− +− −

qi−Δ

di−Δ

Lω

PI

PILω

Lω

PI

f

fsω

ω+

a)

b) Figure 3. Measured currents using the DSRF: a) in the positive-sequence reference frame (id

+ and iq+); b) in the negative-sequence reference frame

(id- and iq

-).

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4

-5

0

5

i d+, i

q+ [A

]

Time [s]

id+

iq+

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4

-5

0

5

i d- , iq- [A

]

Time [s]

id-

iq-

deviation in the estimation of the dc value by the decoupling-network under a current reference step.

The cross-decoupling network concept for the current controller is similar that the one that is applied for a different problem like voltage synchronization with a PLL, but with different constraints [18]. Actually, the decoupled output magnitudes are not obtained from the same point in the structure, so that the processing paths followed by the signals are different.

III. EXPERIMENTAL RESULTS

A. Experimental set-up Several experimental tests have been performed in the

aim of demonstrating the validity of the proposed controllers in a real system.

In a test bench, a three-phase three-wire inverter has been connected to the grid through an inductor L. A capacitor C and a damping resistor R are also added. The main parameters of the set-up are summarized in Table I. The inverter also includes a filter tuned at 10Khz (Lf,, Cf, Rf) for avoiding the switching frequency effects and a 1:1 wye-wye isolation transformer. A dc voltage external source fixed the dc-link voltage to 750V.

The control algorithm has been programmed in a DSpace ACE1103 setup. The experimental measurements shown in this section have been taken with an oscilloscope. Moreover, the internal dq currents measured by the dSpace are also shown for the sake of clarity.

TABLE I. EXPERIMENTAL PARAMETERS

Parameter Description Value

fs Switching frequency 10kHz

fsamp Sampling frequency 10kHz

f Grid frequency 50Hz

L Grid connection inductor 6mH

C Filter capacitor 5μF

R Filter resistor 4.7Ω

Lf 10kHz filter inductor 100μH

Cf 10kHz filter capacitor 2.5μF

Rf 10kHz filter resistor 0.5Ω

Cdc Dc-link capacitance 4700μF

Vdc dc-link voltage 750V

Kp Proportional constant 0.797p.u.

Ki Integral constant 277.22p.u.

S Nominal power of the converter 5.5kVA

Vg Nominal grid phase voltage 230Vrms

Three different tests are presented in this paper. The objective of the first one is to illustrate experimentally the concept of the cross-coupling between the sequences. A phase-to-neutral short circuit is directly produced at the converter terminals.

The second and third tests are performed over a more complex scheme. The objective in these cases is to test the performance of the DDSRF under different grid voltage conditions.

With that objective, a maneuver block allows the creation of short-circuits over a multi-tap transformer, dropping the voltage from its nominal value to the half. Thus, a voltage sag at any phase can be forced.

The set-up is connected to this mechanism for creating voltage sags through a delta-wye transformer, so that the sags are changed after its propagation over this transformer having different amplitudes and phase values at the converter’s terminals than the ones measured in the primary windings [19], as it can be deducted from Figure 6.

B. One-phase to ground fault

As it has been stated, the purpose of this first test is to show experimentally the relation between the amplitude of one sequence with the oscillation of the opposite one, and also the performance of the DDSRF proposed in this paper.

The unbalanced conditions are provoked by a phase-to-neutral short circuit directly produced at the converter terminals. Then, a step current reference command id

+ = 4A is sent to the control hardware.

The conventional DSRF controller will be able to track the dc positive-sequence current, but 4A oscillations will appear in the negative-sequence. The dq currents measured by the dSpace are shown in Figure 7. As long as there is no dc current to track in the negative-sequence, there are no oscillations in the positive-sequence controller in steady-state conditions.

However, if the proposed DDSRF controller is implemented, this oscillation in the negative-sequence reference frame disappears, as it can be seen in Figure 8.

The actual currents injected with the DDSRF and the measured voltages at the inverter terminals captured with the oscilloscope are shown in Figure 9.

Figure 6. Experimental set-up for the 2nd and 3rd test

C. One-phase to ground fault through a delta-wye transformer In this second experiment, the performance of the

DDSRF is tested under different voltage conditions. In this way, the voltage sag has been generated over one of the phases and then propagated through a delta-wye transformer.

According to [20], the B type voltage sag, which is originated by the one-phase fault, gives rise to a C type at the secondary of the delta-wye transformer, and consequently at the converter’s terminal.

If the DSRF controller is chosen, the PI controller cannot avoid the appearance of the 2ω oscillations, as it is demonstrated in Figure 10. In this case, the converter is injecting initially positive-sequence d and q currents, when a command step is applied to the d and q negative-sequence currents.

Despite these voltage conditions, if the DDSRF is used, the power converter compensates the 2ω oscillations in the dq frame as it can be seen in Figure 11, where the dq decoupled currents used for the controller by the DSpace are shown.

It can be observed that there is a cross-coupling effect during the transient response because of the estimation error of the dc value of the current during the transient. This transient performance can be modified by adjusting the LP filter.

The real currents injected with the DDSRF are shown in the bottom part of Figure 12. It should be noted that they are unbalanced currents because positive- and negative-sequence are injected. The contribution of the DDSRF lays on the capability to control the dq currents in order to provide positive or negative active and reactive power in order to modify the grid conditions according to decisions taken in a higher hierarchical control level, which is out of the scope of this paper.

a)

b) Figure 10. Measured dq currents with the DSRF: a) in the dq positive reference frame (id

+ and iq+) under id

*+=4A and iq*+=-2A constant current

commands; b) in the dq negative reference frame (id - and iq

-) where a step command from 0A to 2A is provided to id

*- at t=0s and another step from 0A to -2A is also applied to iq

*- at t=0s. Current references are representedby dashdot lines.

-0.01 0 0.01 0.02 0.03 0.04 0.05

-5

0

5

i d+, i

q+ [A

]

Time [s]

id+

iq+

-0.01 0 0.01 0.02 0.03 0.04 0.05

-5

0

5

i d- , iq- [A

]

Time [s]

id -

iq -

Figure 9. Grid voltage (vga, vgb and vgc), with a 100V/div scale and injectedcurrents by the DDSRF (ia, ib and ic) with a 2A/div scale in the abcstationary frame corresponding to a constant id

*+=4A (time axis limits arerepresented from -40ms to 60ms with a 10ms/div scale).

a)

b) Figure 8. Decoupled dq currents with the DDSRF: a) in the dq positive reference frame (id

+ and iq+) under id

*+=4A step current at t=0s; b) in the dqnegative reference frame (id

- and iq-) where no current reference command

was provided. Current references are represented by dashdot lines.

-0.01 0 0.01 0.02 0.03 0.04 0.05

-5

0

5

i d+',

i q+' [

A]

Time [s]

id+'

iq+'

-0.01 0 0.01 0.02 0.03 0.04 0.05

-5

0

5

i d- ', i q- ' [

A]

Time [s]

id- '

iq- '

a)

b) Figure 7. Injected dq currents with the DSRF: a) in the dq positive reference frame (id

+ and iq+) under id

*+ 4A step current at t=0s; b) in the dqnegative reference frame (id

- and iq-) where no current reference command

was provided. Current references are represented by dashdot lines.

-0.01 0 0.01 0.02 0.03 0.04 0.05

-5

0

5

i d+,i q+

[A

]

Time [s]

id+

iq+

-0.01 0 0.01 0.02 0.03 0.04 0.05

-5

0

5

i d- , iq- [A

]

Time [s]

id -iq -

D. Two-phase to ground fault through a delta-wye

transformer In the final test, a two-phase to ground fault has been

applied for generating an E type voltage sag, which is converted into an F type at the power converter’s terminal by its propagation through the delta-wye transformer.

Figure 13 shows the successful decoupling of the dq positive-sequence and negative-sequence currents by the DDSRF controller.

This type of sag entails a different phase shift between

the phases of the voltage, as can be seen in the top part of Figure 14.

In the bottom part of Figure 14, the injected currents are shown. As in the previous cases the controller is also able to provide the desired unbalanced currents under these unfavorable conditions.

IV. CONCLUSIONS Unbalanced grid voltage conditions degrade the

performance of dq synchronous controllers. Undesirable 2ω oscillations appear due to the cross-coupling between the

Figure 14. Grid voltage (vga, vgb and vgc), with a 100V/div scale and injected currents by the DDSRF (ia, ib and ic) with a 2A/div scale in the abcstationary frame corresponding to a constant id

*+=4A and iq*+=-2A current

reference together with a id*- step from 0A to 2A and a iq

*- step from 0A to -2A at t=0s (time axis limits are represented from -40ms to 60ms with a 10ms/div scale).

a)

b) Figure 13. Decoupled dq currents with the DDSRF: a) in the dq positive reference frame (id

+ and iq+) under id

*+=4A and iq*+=-2A constant current

commands; b) in the dq negative reference frame (id - and iq

-) where a step command from 0A to 2A is provided to id

*- at t=0s and another step from 0A to -2A is also applied to iq

*- at t=0s. Current references are represented by dashdot lines.

-0.01 0 0.01 0.02 0.03 0.04 0.05

-5

0

5

i d+',

i q+' [

A]

Time [s]

id+'

iq+'

-0.01 0 0.01 0.02 0.03 0.04 0.05

-5

0

5

i d- ', i q- ' [

A]

Time [s]

id -'

iq -'

Figure 12. Grid voltage (vga, vgb and vgc), with a 100V/div scale and injectedcurrents by the DDSRF (ia, ib and ic) with a 2A/div scale in the abcstationary frame corresponding to a constant id

*+=4A and iq*+=-2A current

reference together with a id*- step from 0A to 2A and a iq

*- step from 0A to-2A at t=0s (time axis limits are represented from -40ms to 60ms with a10ms/div scale).

a)

b) Figure 11. Decoupled dq currents with the DDSRF: a) in the dq positive reference frame (id

+ and iq+) under id

*+=4A and iq*+=-2A constant current

commands; b) in the dq negative reference frame (id - and iq

-) where a step command from 0A to 2A is provided to id

*- at t=0s and another step from0A to -2A is also applied to iq

*- at t=0s. Current references are representedby dashdot lines.

-0.01 0 0.01 0.02 0.03 0.04 0.05

-5

0

5

i d+',

i q+' [

A]

Time [s]

id+'

iq+'

-0.01 0 0.01 0.02 0.03 0.04 0.05

-5

0

5

i d- ', i q- ' [

A]

Time [s]

iq -'

id -'

positive- and negative-sequence, and they cannot be removed by the PI in the dq synchronous frame.

In this paper, the DDSRF current controller for grid connected inverter under unbalanced conditions is proposed. This dq current controller decouples the performance of each positive and negative reference frame by estimating the cross-coupling oscillations. The significance of this controller is that positive- and negative-sequence active and reactive power can be controlled independently.

Experimental results under different grid conditions demonstrate the capability of the controller for compensating the oscillations and the good performance that is obtained has been compared to the classical DSRF solution.

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