Flatness-Based Voltage-Oriented Control of Three-Phase PWM ...

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phase PWM-rectifier in synchronous reference frame. The. DC-link voltage and reactive current are shown to be flat outputs of the full-order system. Two different ...
Flatness-Based Voltage-Oriented Control of Three-Phase PWM Rectifiers J. Dannehl, F.W. Fuchs Institute of Power Electronics and Electrical Drives, Christian-Albrechts-University of Kiel, D-24143 Kiel, Germany, Phone: +49 (0) 431-880-6107, Email: [email protected]

Abstract— Flatness-based control is applied to the threephase PWM-rectifier in synchronous reference frame. The DC-link voltage and reactive current are shown to be flat outputs of the full-order system. Two different approaches are presented. At first the DC-link voltage is controlled directly. The second employs inner current loops. Feed forward design based on system flatness is shown and discussed. In many applications the DC-link voltage and reactive current are controlled to constant values. In this case the direct flatness-based approach offers no advantages compared to conventional voltage-oriented PI-control whereas the second approach outperforms it with respect to the obtained control dynamic. Keywords— Converter control, Non-linear control.

I. I NTRODUCTION Three-phase grid-connected PWM rectifiers are often applied in regenerative energy systems and in adjustable speed drives when regenerative braking is required. Besides power regeneration they offer the control of the power factor as well as the DC-link voltage while emitting less current harmonics to the grid compared to passive diode rectifier bridges. A cascaded control structure with an outer DC-link voltage control and inner current control loops are commonly used. For L-filter grid connections the current control is mostly done with PI controllers in line voltage-oriented coordinates [1]. For many applications this so-called voltage-oriented control (VOC) is suitable and well working but for special applications research still goes on. Line voltage distortions like harmonics and unsymmetries for example are challenging the control. As PI controller can not reject sinusoidal disturbances additional control concepts are often necessary in order to meet the standards. Stability problems due to interactions with the fundamental VOC can occur, especially in weak grid conditions [2]. Another issue which often requires additional concepts is the resonance damping if LCL-filters are used as grid connection. The different control subsystems are mostly designed separately and interactions are often neglected. As the conventional VOC is a cascaded control it requires different time constants of the different loops. Therefore the DC link control bandwidth is limited. Furthermore the outer loop is tuned assuming the DC link voltage near to its constant reference. The inherited nonlinearity can lead to instability if the voltage variations are too high. An approache for minimizing the DC capacitance

is presented in [3]. When PWM rectifiers with reduced DC capacitance are used, a load step will cause a DClink voltage dip. The smaller the capacitance the faster the controller has to react. In this case the time constants are getting closer to each other and the DC link voltage variations get higher. Stability problems may arise. The application of nonlinear control strategies does not require different time constants of the DC link and current dynamics and the control design can be done for the fullorder system without linearization around the constant DC voltage reference. Because of these reasons a faster control can be achieved which can be used for reducing the DC link capacitances. In [4], [5] and [6] the application of feedback linearization [7] for the PWM rectifier control yields faster control or smaller DC capacitors, respectively. Other nonlinear methods like Sliding Mode Control [8], passivity-based control [9] or the direct Lyapunov method [10] are also applied to the PWM rectifier control in order to improve the performance [11] [12] [10]. Another nonlinear method is the flatness-based control (FBC) [13] [14] which is successfully applied to motor control applications [15] [16] [17]. Recently, FBC is applied to the three-phase PWM rectifier [18] and [19] in order to achieve a higher control dynamic. In [18] the PWM rectifier was shown to be flat with the reactive current and the stored system energy as flat outputs. In [19] the experimental validation is shown. The stored energy depends on the active and reactive currents as well as the DC link voltage. For the flatness-based control the reference trajectories of the flat outputs have to be derived. As the basic control objective is the control of the DC link voltage and the reactive power in terms of the reactive current the above choice requires calculations in order to formulate the reference trajectories of the flat outputs. In particular, the trajectory of the active current component has to be derived. Because the application in [19] is a D-STATCOM without DC load the active current is zero is steady state. Therefore the stored system energy is mostly related to the DC link voltage and the reactive current. But for transients the trajectory of the active current has to be derived which remains unclear from [19]. For other applications like adjustable speed drives with regenerative braking or distributed energy generation the active current is mostly nonzero and time varying. In this paper the the DC link voltage and the reactive current are shown to be flat outputs itself. Its references

iL

Control

Fig. 1.

vDC

Load

CDC vL

iload

iDC

Lf

vC Gate Signals

PWM rectifier with L-Filter connected to the grid TABLE I S YSTEM PARAMETERS

Symbol VL ω Lf Rf fc CDC

Quantity Line voltage (phase-to-phase, rms) Line angular frequency Filter inductance Resistance of filter inductor Switching/ control frequency DC link capacitance

value 400 V 2 π 50 Hz 6 mH 100 mΩ 2 kHz 2200 μF

can be used directly for the feed forward without precalculating other trajectories. Here two different flatnessbased approaches are shown. First a direct control of the DC link voltage and reactive current without an inner active current loop is designed and discussed. The second approach is a cascade structure with an inner active current loop. In section II the system description and modeling is shown which is followed by a discussion of the system flatness in section III. For the purpose of comparison the conventional VOC is shown in section IV. Both flatnessbased control approaches are shown and analyzed in section V. Simulation results are presented and analyzed in section VI. Finally, a conclusion is given. II. S YSTEM M ODEL The analyzed system is shown in Fig. 1. A threephase IGBT voltage source converter is connected to the grid through a line-side filter. Here, the grid is modeled as an ideal, sinusoidal three-phase voltage source without line impedances and distortions like harmonics and unbalances. The DC link voltage and the line currents are measured for control purpose. The line voltages are measured for synchronizing with the grid. The system parameters can be found in Tab. I. Thereby the copper losses of the inductors are taken into account and modeled by Rf whereas its iron losses are neglected. Here, the space vector notation is used [20]. The threephase values are transformed into a two-phase stationary αβ-reference frame. Applying Kirchhoffs laws gives: d αβ αβ i = v→ αβ − Rf · → i αβ (1) →C L −v L dt → L Transformation of (1) into the dq-reference frame rotating with the line voltage vector yields: Lf

Lf

d dq dq i = v→ dq − Rf · → i dq i dq →C L −v L − jωLf · → L dt → L

(2)

The DC link voltage dynamic can be written as dvDC = iDC − iload (3) CDC · dt In order to get a direct relation between (3) and (2) a power relation of the line side and the DC side can be used. The active line power of a three-phase system is given by (φI : phase angle between voltage and current):   ∗ 3ˆ vLˆiL 3 PAC = · √ √ · cos(φI ) = · Re v→ αβ i αβ L ·→ L 2 2 2 (4)   3 3 dq dq ∗ = · vLd iLd · Re v→ L · → = iL 2 2 Neglecting the filter and converter losses the active power balance of the line side (PAC ) and DC side (PDC = vDC iDC ) gives: 3 vLd iLd iDC = · (5) 2 vDC Finally, the DC link voltage dynamic can be written as 3 vLd iLd dvDC = · − iload (6) CDC · dt 2 vDC A similar derivation gives the reactive line side power: 3 (7) QAC = − · vLd iLq 2 From (6) and (7) it becomes clear that the DC link voltage can be controlled by the d component of the line current and the reactive power by the q component. Note that iLq is commonly directly regulated to zero. Defining the system state x = [iLd , iLq , vDC ]T and the control input vector u = [vCd , vCq ]T the following state space representation can be derived out of (2) and (6): ⎤ ⎡ v ⎤ ⎡ Ld −Rf vCd ⎢ Lf iLd + ωiLq − Lf ⎥ ⎢ Lf ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ d x = ⎢ ⎢−Rf i − ωi − vCq ⎥+⎢ 0 ⎥ (8) Lq Ld ⎥ ⎢ ⎥ ⎢ dt Lf ⎥ ⎢ i ⎥ ⎢ Lf load ⎦ ⎣− ⎦ ⎣ 3 vLd iLd · CDC 2 CDC vDC III. F LATNESS OF THE S YSTEM Consider the following system with the system state vector x = [x1 , ..., xn ]T of order n and the system input vector u = [u1 , ..., um ]T of the order m: x˙ = f (x, u), x(0) = x0

(9)

A system is said to be flat if there exists a so-called flat output vector y of the same order as the input vector u (dim(y) = dim(u)) that can be expressed as functions of the system states (x ), the system inputs (u ) and a finite number of time derivatives (which are denoted by superscripts in brackets)

(α ) m) (10) y = φ x , u1 , ..., u1 1 , ..., um , ..., u(α m and which fulfills the following two conditions [21]:

(β ) (βm ) x = ψ 1 y1 , ..., y1 1 , ..., ym , ..., ym (11)

(β +1) (βm +1) u = ψ 2 y1 , ..., y1 1 , ..., ym , ..., ym (12)

Note that the above is not the most general definition of flatness. For a detailed mathematical definition the reader is referred to [21] or [13]. For a flat system there exist many different outputs that can be chosen as flat outputs. As the DC link voltage and the reactive current are the quantities to control, here, it is shown that the these are flat outputs itself. Its references can be used directly for the feed forward without precalculating other trajectories. For the proof the disturbances (second term of the right hand side of (8) are neglected and the line voltage amplitude is assumed constant. For sake of simplicity in the following derivatives are denoted by dots (x˙ instead of dx/dt for example). Consider as output vector y = [y1 , y2 ]T = [iLq , vDC ]T = φ(x1 , x3 )

vDC*

VDC PI control

current control

PWM

plant

vC,abc

vC,dq* gL

idq* idq

PLL

vDC iabc

vL,abc

dq a,b,c

Fig. 2. control

Cascaded control structure of conventional voltage-oriented Nonlinear Feed forward *

v C,ff ~* y

y

Reference Generation

*

+ -

Linear Feed back

v C,fb

*

+ +

vC

*

y

vC PWM

System

(13)

First the system state vector x = [iLd , iLq , vDC ] will be expressed as function of the flat outputs. Obviously two states are flat outputs themselves and as dvDC 3 vLd iLd = v˙ DC = y˙ 2 = · (14) dt 2 CDC y2 the d current component can be written as: 2 CDC y2 y˙ 2 iLd = · (15) 3 vLd By using (13) and (15) the system state vector can be expressed as ⎡ ⎤ 2 CDC y2 y˙ 2 · ⎢ 3 ⎥ vLd ⎥ = ψ (y1 , y2 , y˙ 2 ) x˙ = ⎢ (16) ⎣ ⎦ 1 y1 y2 T

The second step is proving (12) whereas u = [vCd , vCq ]T . Rearranging the first line of (8) gives

diLd Rf + vCd = −Lf · iLd − ω iLq (17) dt Lf Substituting iLd by (15) and rearranging gives 2 CDC Lf · (y˙ 2 · y˙ 2 + y2 · y¨2 ) vCd = 3 vLd 2 Rf CDC · · y2 y˙ 2 + ωLf y1 − 3 vLd = ψ 2,1 (y1 , y2 , y˙ 2 , y¨2 )

Fig. 3. Control structure of direct flatness-based DC link voltage and the reactive current control

IV. C ONVENTIONAL VOLTAGE - ORIENTED CONTROL The conventional voltage-oriented control consists of an outer DC link voltage control loop and inner current control loops [1]. The cascaded control structure is shown in Fig. 2. The outer loop regulates the DC link voltage ∗ by applying PI control. In to its constant reference VDC order to prevent wind up problems in case of limitation of the current references an anti-wind up mechanism is used here [22]. For the design of the PI controller parameters (kDC ,TDC ) the inner active current loop is modeled as PT1 -lag element with the delay time of Tinner = 4Tc . Assuming the DC-link voltage near to its constant reference the PI controller can be tuned with the symmetrical optimum [23]: kDC =

∗ 2 CDC VDC · ; TDC = 4 Tinner 3 2 Tinner vLd

(22)

For the voltage-oriented current control PI controllers are used. In order to decouple the d and q current dynamics a decoupling by feedback is inserted [1]: (18)

Similarly, the second line of (8) combined with (15) gives

2 ω CDC Rf vCq = −Lf y˙ 1 + · y2 y˙ 2 + y1 (19) 3 vLd Lf (20) = ψ 2,2 (y1 , y˙ 1 , y2 , y˙ 2 ) Combining (18) and (19) gives u = ψ 2 = [ψ 2,1 , ψ 2,2 ]T = ψ 2 (y1 , y˙ 1 , y2 , y˙ 2 , y¨2 ) (21) It can be seen from (16) and (21) that y = [iLq , vDC ]T fulfils the conditions (11) and (12) and consequently is a flat output vector (with β1 = 0 and β2 = 1). If a cascade control structure with inner current loops is used only the current dynamics of (8) are of interest for the current control. Obviously, in this case both current components are directly flat outputs and therefore y = [iLd , iLq ]T is a flat output vector.

∗ vCd

∗ vCq

= ω Lf iLq + kI (iLd − i∗Ld )  kI + (iLd − i∗Ld ) dt TI   = −ω Lf iLd + kI iLq − i∗Lq    kI + iLq − i∗Lq dt TI

(23)

(24)

Note that the line voltage can be compensated as well but this is omitted here. For the controller parameter design the converter is modeled as a delay of one switching period (Tc = 1/fc ). Tuning with symmetrical optimum gives kI =

−Lf ; TI = a2I Tc ; aI = 3 aI Tc

(25)

As the available converter output voltage is limited by the DC link voltage an anti-wind up mechanism is also used [22]. Here, the determination of the line voltage phase angle is done by a PLL algorithm. A survey of different synchronization solutions can be found in [24].

V. F LATNESS - BASED CONTROL The flatness property can effectively be used for designing control algorithms. Basically, the control structure consists of a feed forward and a feedback part [14]. The key idea of the flatness-based control is to drive the system towards the reference trajectory by feed forward. The feedback part is inserted only in order to eliminate the deviations caused by disturbances and other nonidealities like model uncertainties, actuator limitations and other perturbations. Flatness allows to formulate the control input vector as function of only the flat output vector even for nonlinear systems, see (12). This expression is used for designing the feed forward. Under ideal conditions the feed forward can track the (time varying) reference if it is smooth enough. Otherwise due to derivatives in the references and limitations in the control inputs tracking errors would appear. Even if the references are smooth enough deviations from perfect tracking will appear due to disturbances, model uncertainties and other perturbations. Therefore feedback is inserted. As the system is near to the reference trajectory via feed forward, thus linearizable around it, the feedback can be designed with linear methods even for nonlinear systems [14]. As already mentioned the references should be smooth enough in order to make effectively use of the feed forward [25]. Therefore the references are smoothed in the block called reference trajectory generation. In the following two different approaches are shown and analyzed. At first the DC link voltage and the reactive current are controlled directly and afterwards a cascaded control structure with an inner active current control loop will be used. A. Direct Approach In this section the direct control of the DC link voltage and the reactive current is designed and analyzed (y = [y1 , y2 ]T = [iLq , vDC ]T ). The control structure is shown in Fig. 3. The feed forward can be designed directly by using (18) and (19): 2 CDC Lf ∗ ∗ · (y˙ 2 · y˙ 2 + y2∗ · y¨2∗ ) 3 vLd (26) 2 Rf CDC − · · y2∗ y˙ 2∗ + ωLf y1∗ 3 vLd

2 ω CDC ∗ ∗ Rf ∗ ∗ = −Lf y˙ 1 + y2 y˙ 2 + y (27) 3 vLd Lf 1

∗ vCd,f f =

∗ vCq,f f

Note that the load current iload is not included in the feed forward as it is assumed unknown. If it was measured or estimated it could be included for improving the DC link voltage control. It is common practice to control the DC link voltage to a constant value and the reactive current constantly to zero. Therefore no real tracking is necessary with respect to the control variables. In this case the feed forward from (26) and (27) reduces to the following simple expressions ∗ vCd,f f ∗ vCq,f f

= ω Lf y1∗ = Rf y1∗

(28) (29)

It becomes clear from (28) that the coupling from the q component into the d path is decoupled. The voltage drop of the q path across the parasitic resistance of the inductor is compensated by (29). In case of constant references the core of the flatness-based structure, that is the feed forward, does not contribute remarkably to the system behavior. Most of the control has to be done of the feedback controller. As no tracking performance is required the direct flatness-based control is not suitable if the DC link voltage and reactive current is kept constant. Actually, the quantity which has to be tracked is the active current. Depending on the load current it has to be controlled to a certain value in order to stabilize the DC link voltage. Therefore a cascaded structure with an inner flatness-based active current control loop will be designed and analyzed in the next section. As the direct approach is not well suited the design of the feedback and the reference smoothing is omitted here. The reader is referred to [19] for approaches. The direct approach is not further treated in this paper. B. Cascaded Control The cascaded control structure is shown in Fig. 4. From the first line of (16) the outer current feed forward can be directly derived: ∗ ∗ v˙ DC 2 CDC vDC · (30) 3 vLd Note again that the load current is not included in the feed forward because it is assumed unknown. In case of constant DC link voltage reference the feed forward does not contribute to the active current reference. As this is common practice there is no difference to the conventional voltage-oriented control since feedback is performing the tracking as well. For the same reason the PI controller already used for VOC can be used, see (22) for its parameters. The DC link voltage reference smoothing is not treated in this paper since it is constant. For the design of the inner voltage feed forward the first two lines of (8) are used: d ∗ = −Lf i∗Ld − Rf i∗Ld + ωLf i∗Lq (31) vCd,f f dt d ∗ vCq,f = −Lf i∗Lq − Rf i∗Lq − ωLf i∗Ld (32) f dt The line voltage compensation is omitted here again. The active current reference depends on the load current which is time varying. For constant reactive current reference (32) reduces to: Rf ∗ ∗ vCq,f = − i − ωLf i∗Ld (33) f Lf Lq

i∗Ld,f f =

It can be seen that the flatness-based feed forward design yields a decoupling by feed forward whereas for the conventional VOC this is done by feedback, see (23) and (24). Comparing (31) and (33) with the control law of the conventional voltage-oriented control, see (23) and (24), shows that the FBC is additionally compensating the voltage drop across the parasitic filter resistances. As the FBC is using reference signals instead of measured

*

vCq,ff Current Feedforward

Voltage Feedforward *

* ~ vDC

iLd,ff Reference Generation

vDC

*

iLd,fb

+ -

*

+ DC link voltage Controller

+

~* iLd

vCd,fb +

vDC

+

vCq,fb Current Controller

iLd

vS,123

*

*

iLd + Reference ~* iLq Gener- iLq* + ation

*

vCd,ff

iLq

PWM

*

System

+ +

vL1

gL

iL,123

PLL

dq a,b,c

Fig. 4.

Cascaded control structure of flatness-based control

ones the sensitivity against measurement noise and errors is lower compared to VOC. Another improvement of the FBC is the derivative part in (31) which increases the control bandwidth as will be shown later by simulations. Combining (31) with the first line of (8) and (32) with the second line of (8) and neglecting the computation and PWM delay yields the dynamics of the current tracking errors ΔiLd = (i∗Ld − iLd ) and ΔiLq = (i∗Lq − iLq ): Rf vCd,f b vLd dΔiLd =− ΔiLd + ω ΔiLq + − (34) dt Lf Lf Lf dΔiLq Rf vCq,f b =− ΔiLq − ω ΔiLd + (35) dt Lf Lf For the feedback control design the couplings are neglected as the currents are assumed near to its reference due to the feed forward. Additionally, the line voltage is treated as disturbance. Finally, two decoupled first order P T1 -dynamics are achieved which can be controlled with PI controller as already used for the conventional voltage-oriented control. The difference to VOC is that (34) and (35) are the error dynamics, thus its references are zero. This makes clear again that the tracking is not done by the feedback but by the feed forward. Even if optimization for disturbance rejection instead of tracking behavior was possible, here the same PI controller as for VOC are used, see (25) for its parameters. Note that the computation and PWM delay is taken into account for the feedback design. So, the feed forward of (31) and (32) can be used as plug-in solution in order to improve the VOC. The anti-windup is included as well. As already mentioned smooth references are important for the effective use of the feed forward. Here simple first order lag filters are used. See [19] [25] for improved methods and [17] for different simple ones. At best, the voltage limitations are directly included into the reference trajectory generation. VI. S IMULATION R ESULTS In this section simulation results are shown and analyzed which are obtained with Matlab/Simulink. As emphasis is put on cascaded control structures and the DC link voltage control is the same for all approaches only the

current control behavior is analyzed here. Instead of a DC link capacitor and its voltage control here a constant DC voltage source is used. By that the influence of the outer DC link voltage control is canceled and the current control behavior can be pointed out more clearly. The threephase IGBT converter, filter inductors and line voltage sources as well as the DC voltage source are modeled in PLECS, see Tab. I for the system parameters. The signals are sampled in the middle of each switching period and afterwards the control algorithm is executed. The PLL algorithm is executed with 20 kHz. The reference currents are smoothed with first-order lag filters which are discretized by the Tustin approximation with prewarping [22]. In Fig. 5 and 6 the response of the current control is shown for the different approaches. The reactive current reference is zero whereas a step from zero to 30 A in the active current reference is applied to the control system. Fig. 5(a) illustrates good tracking behavior of VOC even if only PI controllers are used and the references are not filtered. After 5 Tc the reference is reached the first time and an overshoot of 9 A can be seen. Couplings between the d and q components are clearly visible in Fig. 6(a). Filtering the reference yields only slightly less overshoot and a longer response time but the couplings are reduced. As can be seen in (23) and (24), in the conventional VOC an additional decoupling by feedback is used. Its effect can be seen in Fig. (5)(b) and (6)(b). Instead of improving the response a more oscillatory behavior in both current components is obtained with decreasing the response time. One reason can be the quite low control frequency. A compensation of the delay could improve the response. As the FBC in (31) and (32) shows a decoupling by feed forward another approach is shown here. Instead of using the measured current values the reference currents are used for the decoupling of VOC in (23) and (24). This approach is referred as VOC with feed forward decoupling whereas the other is called VOC with feedback decoupling. As can be seen in Fig. 5(c) and 6(c) the feed forward decoupling considerably improves the step response. The couplings are effectively reduced and a faster response is obtained as well. With smoother references the coupling are decreased but the response time is increased as well.

VOC without decoupling

a)

VOC without decoupling 400

vCd/V

iLd/A

40 20

200

without filter f =f /8

0

Filter

f 0

0.05

0.055

0.06

0.065

0.07

0.075

-200

0.08

0.05

VOC with feed back decoupling

b)

0.055

0.06

0.065

c

= f / 20

Filter

0.07

c

0.075

0.08

VOC with feed back decoupling 400

vCd/V

iLd/A

40 20

200

without filter f =f /8

0

Filter

f 0

0.05

c)

0.055

0.06

0.065

0.07

0.075

-200

0.08

0.05

VOC with feed forward decoupling

0.055

0.06

0.065

c

= f / 20

Filter

0.07

c

0.075

0.08

VOC with feed forward decoupling 400

vCd/V

iLd/A

40 20

200

without filter f =f /8

0

Filter

c

fFilter= fc / 20 0

0.05

0.055

0.06

0.065

0.07

0.075

-200

0.08

0.05

0.055

0.06

FBC

d)

0.065

0.07

0.075

0.08

FBC 400

vCd/V

iLd/A

40 20

200

without filter f =f /8

0

Filter

c

fFilter= fc / 20 0

0.05

0.055

0.06

0.065 t/s

0.07

0.075

-200

0.08

0.05

0.055

0.06

0.065 t/s

0.07

0.075

0.08

Fig. 5. Simulated responses to an active current reference step (applied at t = 0.05s) with different reference filters (i∗Lq = 0A, VDC = const. = 700V). Left: active current component iLd , right: d component of the converter output reference voltage uLd . a) Voltage-oriented control without decoupling, b) Voltage-oriented control with feedback decoupling, c) Voltage-oriented control with feed forward decoupling, d) Flatness-based control

VOC without decoupling

VOC without decoupling

20

100

vCq/V

Lq

i /A

a)

0

0 -100 without filter

-20

0.055

0.06

0.065

0.07

0.075

0.08

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0

0.055

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0.055

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0.08

c

0.075

0.08

f

Filter

=f /8

0.065

c

0.07

f

Filter

= f / 20

0.075

c

0.08

0.05

0.055

0.06

fFilter= fc / 8 0.065

0.07

fFilter= fc / 20 0.075

0.08

FBC 100

vCq/V

i /A

Lq

0.05

-200

FBC

0 -100 without filter

-20

0.07

= f / 20

0

without filter

0

Filter

100

0

20

0.065

f

VOC with feed forward decoupling

20

d)

c

0

VOC with feed forward decoupling

-20

=f /8

100

vCq/V

Lq

i /A

c)

Filter

0.06

without filter -20

f

VOC with feed back decoupling

vCq/V

20

i /A

VOC with feed back decoupling

b) Lq

0.05

-200

0.05

0.055

0.06

0.065 t/s

0.07

0.075

0.08

-200

0.05

0.055

0.06

fFilter= fc / 8 0.065 t/s

0.07

fFilter= fc / 20 0.075

0.08

Fig. 6. Simulated responses to an active current reference step (applied at t = 0.05s) with different reference filters (i∗Lq = 0A, VDC = const. = 700V). Left: reactive current component iLq , right: q component of the converter output reference voltage uLq . a) Voltage-oriented control without decoupling, b) Voltage-oriented control with feedback decoupling, c) Voltage-oriented control with feed forward decoupling, d) Flatness-based control

Finally, the results achieved with FBC are shown in Fig. 5(d) and 6(d). In addition to the feed forward decoupling the derivative of the active current reference is included and the voltage drop across the parasitic resistances are compensated, see (31) and (32). Mostly the derivative part further decreases the response time. Without filtering the reference the d-current already exceeds the reference to 40 A. Thus, the fastest response is achieved but the overshoot is around 20 A which is more than 60 % overshoot. Applying a reference filter with a cut-off frequency of fc /8 yields a lower overshoot but compared to the other approaches still the fastest response time is obtained. Compared to VOC with feed forward decoupling higher couplings are obtained but compared to the other approaches the couplings are still reduced. VII. C ONCLUSION Flatness-based control of three-phase PWM rectifiers in synchronous reference frame is designed and analyzed. The DC link voltage and the reactive current are shown to be be flat outputs of the PWM rectifier system. Two different flatness-based control approaches are presented and analyzed. First the DC link voltage and the reactive current are controlled directly. The second approach employs an inner active current loop. The feed forward design based on the system flatness is shown and discussed. For the cascade structure simulations are carried out as well. For purpose of comparison the conventional voltageoriented control is also shown and simulation results are presented. In many applications the DC link voltage and the reactive current are controlled to constant values. In this case the direct flatness-based approach offers no advantages compared to the conventional voltage-oriented control. The second approach outperforms it as a faster step response is achieved with less coupling between both current components. In this case the good tracking behavior can be effectively used for the active current control. Filtering the reference currents yields a less oscillatory but still faster performance compared to the conventional control. The difference between decoupling by feed forward and feedback is shown by simulations. VIII. ACKNOWLEDGMENT This work has been funded by German Research Foundation (DFG). R EFERENCES [1] Kazmierkowski, M.P., Krishnan, R., and Blaabjerg, F., Control in Power Electronics: Selected Problems. Oxford: Academic Press, 2002. [2] Liserre, M., Teodorescu, R., and Blaabjerg, F., “Stability of photovoltaic and wind turbine grid-connected inverters for a large set of grid impedance values,” IEEE Transactions on Power Electronics, vol. 21, no. 1, pp. 263– 272, January 2006. [3] Winkelnkemper, M. and Bernet, S., “Impact of control model deviations on the dc link capacitor minimization in ac-dc-ac converters,” in IEEE Industrial Electronics, IECON 2005 - 31st Annual Conference on, 2005, CD-ROM paper. [4] Burgos, R.P., Wiechmann, E.P., and Holtz, J., “Complex statespace modeling and nonlinear control of active front-end converters,” IEEE Transactions on Industrial Electronics, vol. 52, no. 2, pp. 363– 377, April 2005.

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