Discrete Sliding Mode Current Control of Grid-Connected ... - Uni Kiel

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Discrete Sliding Mode Current Control of Grid-Connected Three-Phase PWM Converters with LCL Filter F. Fuchs, J. Dannehl*, F.W. Fuchs** *Student Member, IEEE, ** Senior Member, IEEE Inst. for Power Electronics and Electrical Drives, Christian-Albrechts-University of Kiel D-24143 Kiel, Germany [email protected], * [email protected], ** [email protected]

Abstract—The development of discrete sliding mode control for three-phase PWM converters with LCL filter is presented. This robust control method is investigated for the current control loop. Done in a discrete state space system, a feedforward feedback structure is utilized. A reference state trajectory and feedforward control in αβ coordinates, discrete integral SMC in dq coordinates are designed and analysed. Simulation and experimental results are shown.

x2 System I a

x1

x2 System I

978-1-4244-6391-6/10/$26.00 ©2010 IEEE

779

System II

b

I. I NTRODUCTION PWM converters are used in many industry applications. Here, a system for connecting a machine or generator to the three-phase grid is utilized. The work deals with the control of the grid-connected converter, which is equipped with turn-off power semiconductors, namely IGBTs (Insulate Gate Bipolar Transistor). In combination with a grid filter, this configuration offers the possibility to adjust the grid current with a unity power factor and bidirectional power flow. Beyond the DC link capacitor, a three-phase PWM converter supplying an induction machine with load is used. For switching frequency filtering, a three-phase grid-side LCL filter is chosen. An LCL filter offers economic advantages in comparison to an equivalent L filter [1], but needs a more complex control structure for handling the sensible resonance [2]. Different control theories have been investigated and are under investigation (f.e. [3]). In this context, sliding mode control (SMC) with its well known robustness is an attractive option. SMC as part of variable structure control (VSC) has its origin in the former Soviet Union [4]. Its most important property is the so-called invariance [5], which guarantees absolute independence of model uncertainties and very good disturbance rejection. As VSC, SMC switches abruptly between system structures, but in extension to VSC, it utilizes this to keep the state trajectory on a so-called switching surface. This shall be explained with an example (Fig. 1, from [5]). Subplot (a) shows trajectory lines for a second order differential equation I, subplot (b) for another second order differential equation II (with x2 = dx1 /dt). Both systems are unstable. The nonlinear feedback in SMC switches between these two systems (differential equations) with the aim of driving the trajectory into the origin. This is successfully done in subplot

x2

c

x1

System II System II

Sliding mode control of a converter

d

A0

Direct approach

Indirect approach

xx11 Continuous model System II System I

A analogue

B digital

C D analogue digital

Discrete model

E digital

Fig. 1. (a) State space trajectories of unstable system I (b) State space trajectories of unstable system II (c) Combination of (a) & (b) : SMC system of second order in state space (d) Overview of SMC strategies for a converter

(c). Validity areas for the two systems are defined, separated by two straight lines. The vertical switching line guarantees the reaching of the angulate second switching line during the socalled reaching mode. All trajectories in the neighbourhood of the second switching line are directed to itself, so the trajectory is kept on the line and the switching leads to the desired motion into the origin. This switching motion is called sliding mode. It should be shown for a concrete starting point A0 . At the beginning in the reaching mode the trajectory follows the lines of system I. Crossing the x2 axis, the system changes and following the trajectory lines of system II, the second switching line is reached. Here the trajectory is held on the line and ’slides’ into the origin where it stays. This is the sliding mode, where the trajectory moves on a line, which none of the origin systems contains: absolute invariance in sliding mode. With the progress in digital computing, discrete SMC (DSMC)

2

PWM converter UDC + a) Grid

Filter ugabc

Rr,Lr

ig

Rg,Lg

Lc,Rc

uc

ic

C Load

C

ucap -

b) iq* UDC*

+ -

UDC PI Control DC link voltage control

id*

dq Įȕ

Feedforward control

ĭ

+

uĮȕ *

Įȕ reference system

Įȕ dq

ĭ

uĮȕ

+ ¨uĮȕ Įȕ dq

Įȕ abc

Plant PWM

ĭ ĭ

Discrete sliding mode control

PLL

abc dq

ĭ

¨xdq xdq * -

+ xdq

Current Control Loop

Fig. 2.

ugabc

LCL states

¨udq

UDC

Grid-connected PWM converter with LCL filter a) and control structure b)

became of interest [6]–[11]. Newer publications investigate the discretization behaviour of SMC systems [12], [13]. Here DSMC is the method of choice. When implementing SMC for a power converter, there are several possibilities (Fig. 1 d)): 1. Direct implementation where the IGBT is the switching part of the SMC with analogue (A) or digital (B) implementation; 2. Indirect approach with continuous model with analogue (C) implementation or digital (D) implementation; 3. Indirect approach with discrete model and digital implementation (E). A is done in [14]. [15] shows a continuous approach with simulation results. D is done in [16], [17] and E in [17]–[20], but was until now not investigated with LCL filter (to the author’s knowledge) and will be developed here. An investigation of the current control for a grid-connected PWM converter by means of sliding mode with discrete model and digital implementation is presented here. In section II, the system and control structure is described. Modeling of the system is shown in section III. In section IV the developed current control strategy is shown. Section V gives an analysis and results. The paper is finished by a conclusion (section VI).

capacitor voltage and converter current and the DC voltage are measured. An outer DC link PI controller gives the id reference for the current control, which is described in the following. The reference for iq is adjusted manually. The phase angle φ of the grid voltage is computed by a PLL. III. M ODELING There exist several approaches for designing the SMC. The first task is to get a description/model of the system with switching feedback (no linear representation is possible). Gao summarizes these descriptions in [21]. Here, the state space description with a separation into switching and linear system (described in [7]) is chosen. Assuming symmetry for the three-phase LCL filter, the state space equations in αβ space vector coordinates can be formed with differential equations, defining the converter voltage as input and the grid voltage as distortion:      R i g,αβ (t) i g,αβ (t) − Lgg 0 − L1g − → − → d     i 1 c i c,αβ (t)  = 0  →  − −R →c,αβ (t)  + Lc Lc dt − 1 u cap,αβ (t) u cap,αβ (t) − C1 0 − → − → C {z } | Ac

II. S YSTEM DESCRIPTION AND C ONTROL OVERVIEW Fig. 2 a) shows the system topology. The grid with the threephase LCL filter is shown. The filter states (ig , ic , ucap ) are marked. The converter feeds the DC link capacitor. The load is for example a converter with induction motor. In Fig. 2 b), the control structure can be seen. Grid voltage, grid current, 780



 1  0 Lg  − 1  u c,αβ (t) +  0  u g,αβ (t) Lc − → − → 0 0 | {z } | {z } 

bc

dc

(1)

3

Transforming them to the dq rotating frame aligned to the grid voltage vector and neglecting couplings leads to the same dynamics in d and q (used for DSMC feedback design). Hence, Eq. (2)-(5) are valid for the αβ and dq system. The discrete representation (solution for one time step) is now obtained by the following equations [22]:   a11 a12 a13 ! Ad = a21 a22 a23  = L−1 {(sI − Ac )−1 }(t = Tc ) (2) a31 a32 a33   b1 ! bd =  b2  = (Ad − I)A−1 (3) d bc b3   d1 ! dd =  d2  = (Ad − I)A−1 (4) d dc d3 (1/Tc control frequency (Tab. I), L Laplace transformation) Due to complexity reasons, no symbolic representation of Ad , bd and dd can be given. Considering the converter, a onestep delay of the control frequency is introduced. In the first step for control design, the grid voltage is neglected. It will be considered later. The state space system results in:       ig (k + 1) 0  ig (k)   ic (k + 1)   ic (k)   0  A b d d        ucap (k + 1)  = 000 0  ucap (k)  +  0  u(k) | {z } u (k) 1 uc (k + 1) c Adc | {z } | {z } x(k)

bdc

(5)

This is the used description in the αβ as well in the dq frame (negligence of couplings). IV. D ISCRETE S LIDING M ODE C URRENT C ONTROL Applying SMC with the aim of reference tracking requires some modifications of the control structure. Here, a feedforward feedback structure [7] is used (Fig. 2 b), current control loop). The incoming reference i∗d,q is transformed to αβ coordinates (reason explained in section IV-B). The feedforward controller (described in section IV-A) computes the input signal u∗αβ (converter voltage) which leads to perfect tracking of the current references in the state space system (αβ reference system in Fig. 2). Due to modeling simplifications, the (real) plant does not track these references perfectly. SMC feedback has the function to eliminate this deviation. It is done in dq coordinates. Receiving as input ∆x, the difference of the measured and the reference states (Eq. (6)), it computes a voltage ∆udq , that must be added to u∗αβ .      ∗  ig (k) ∆ig (k) ig (k)  ∆ic (k)   ic (k)   i∗c (k)       ∆x =   ∆ucap (k)  =  ucap (k)  −  u∗cap (k)  (6) ∆uc (k) uc (k) u∗c (k) ∆udq guarantees compensation of the deviations of the plant states. With this control structure SMC can take place in a transformed state space model with states ∆x. In this model, reaching the origin (aim of SMC) as illustrated in Fig. 1 means that all states reach their reference value. 781

Fig. 3.

Principle of feedforward control; left: feedforward, right: plant

A. Feedforward Control Here the inversion principle is applied [7]. The idea is to use the inverted transfer function of the plant for the feedforward controller (Fig. 3). In this case, the overall transfer function becomes one. Two (well known) problems occur with this principle: 1) The entrance current reference has to be known some steps in advance (if the degree of the denominator is higher than that of the numerator). 2) The converter voltage reference U (z) can be beyond the converter output limit. 1) is solved by delaying the current reference. 2) can be solved by limiting the current reference step hight (because high current steps entail high converter voltage levels). The feedforward control and parallel reference model could run in dq frame. As already mentioned, in this case also model Eq. (5) would be valid (couplings neglected). But here a non tolerable phase angle error is introduced. This can be illustrated. If calculating in dq frame with neglecting the couplings, the circuit in Fig. 4 a) would be valid for dqDC values. In this case in steady state and for example zero converter voltage, the phase angle of the dq state space vector of ig equals that of ic . This error would not be tolerable. A solution is to work in the dq frame without neglecting the couplings or in the αβ frame. In dq frame without neglecting the couplings the state space system becomes a MIMO system. Feedforward control design for a MIMO system is complex. The second solution, running feedforward control and the parallel model in αβ frame, is taken. Here, no negligences are made and the references have correct phases. Forming the transfer function of the plant, it has to be defined which current shall be controlled. The incoming reference i∗d,q is normally a reference for the converter-side current ic . Analysis showed, that taking feedforward to control ic leads to a unstable overall transfer function to ig . A conjugate complex pole pair appears and effects oscillations in ig . Hence, ig is taken as current to be controlled. In this case the LCL filter is an additional part in the plant of the UDC controller and must be considered there. The ig output equation of the state space system Eq. (5) is: y = ig (k) = [1000] x(k) | {z }

(7)

cT d

So the (discrete) plant transfer function from Eq. (5) written in terms of poles and zeros with output ig can be calculated: U (z) = cTd (zI − Ad )−1 bd + 0 Ig (z) (z − z01ig )(z − z02ig ) (8) = kig (z − z∞1 )(z − z∗∞1 )(z − z∞2 )(z − z∞3 )

Gp (z) =

4

For the used setting (Table (I)), the zeros and poles ∗ =−0.3are z01ig =−2.92; z02ig =−0.33; z∞1 =−0.3+0.94i; z∞1 0.94i; z∞2 =0.99; z∞3 =0. They are plotted in red in the pole zero map in Fig. 6. z∞ =1 is an additional pole, which appears later in SMC design. The resonance pole pair at the resonance frequency (≈ 1, 2kHz), the delay pole at z∞ 3=0 and another LCL filter pole z∞ 2=0.99 are marked. It can be seen that one zero is outside the unit circle. This non minimum phase system would be unstable, if inverted. In [23], possibilities to circumvent this problem are shown: the non minimum phase zero is neglected in Gf eedf (z), but phase and gain must be adjusted to compensate the resulting error. The feedforward transfer function becomes (with complex adjusting gain factor γ and already mentioned delays):

a)

Rg

UgĮȕ

Lg

IgĮȕ,noUc

C

Lc

Rc

UcĮȕ=0

b)

UĮȕ*

IĮȕ* + feedf

IgĮȕ,noUc Feedforward controller

Fig. 4. a) One-phase equivalent circuit of LCL filter b) Principle of feedforward controller

(with grid voltage angle determined by a PLL) and taken as reference states for the SMC (Fig. 2). In this way, only from the incoming reference for the grid current i∗dq dynamic 1 (z − z∞1 )(z − z∗∞1 )(z − z∞2 )(z − z∞3 ) references x∗ for all filter states are computed. Gf eedf (z) = γ kig z 3 (z − z02ig ) (9) C. Discrete Sliding Mode Feedback Controller The whole transfer function is not one, but includes the added Now the sliding mode controller, which eliminates the delays: deviation from reference and real states, has to be designed. It is done in dq coordinates to make the extension to integral Gsys (z) = Gf eedf (z)Gp (z) SMC easier. Two methods will be presented: equivalent control γz01ig γ(z − z01ig ) γ = 2− (10) and Gao’s reaching law [7]. As it has been shown in [18] = z3 z z3 an implementation as described in [7] leads to steady state γ is chosen, such as Gsys equals unity at 50 Hz (because errors. For this reason, the theory is expanded by an additional feedforward control and reference model run in the 50 Hz integral state w(k) [18], [24], which sums all other states. The alternating αβ frame): discrete system results in (valid for d and q system):      ! ∆ig (k) a11 a12 a13 b1 0 ∆ig (k + 1) Gsys (z = ej2π50HzTC ) = 1 (11)  ∆ic (k + 1)   a21 a22 a23 b2 0   ∆ic (k)  1       ∆ucap (k + 1)  =  a31 a32 a33 b3 0   ∆ucap (k)  (12) ⇒ γ = −j4π50tc      − z01ig e−j6π50tc e  ∆uc (k + 1)   0 0 0 0 0   ∆uc (k)  In the state space model Eq. (5), the grid voltage is neglected. w(k) ki1 ki2 ki3 ki4 1 w(k + 1) {z } | {z }| {z } Until here, the developed feedforward control can adjust a | Ai ∆xi,dq (k+1) ∆xi,dq (k) current in a LCL filter without grid voltage. To include the grid     d1 0 voltage in the approach, the grid current caused by the grid  d2   0  voltage is computed and subtracted from the actual current      d3  ∆ug,dq (k)   0 ∆u (k) + + dq reference. In that way, the grid voltage is ’masked’. With      0   1  complex AC calculation (ω = 2π50Hz), the current caused 0 0 by the grid voltage with zero converter voltage becomes (see | {z } | {z } Fig. 4 a)): bi di (14) Ugαβ (13) With ki1 -ki4 , the weightings of the states can be adjusted. Igαβ,noUc = 1 Rg + jωLg + jωC ||(Rc + jωLc ) As only the grid current steady state error is important, the || means parallel circuit. This current is subtracted from the other gains ki2 -ki4 are set to zero. Now SMC feedback shall input current references and guarantees compensation of the be developed for this system. In [7], a comprehensive design grid voltage. Eq. (9) is the transfer function of the feedforward procedure can be found. Here a short description is given: The switching lines in Fig. 1 are defined by switching functions s. controller. It is shown in Fig. 4 b). The sign of s decides which system is valid. s=0 means that the trajectories are located on the switching line. B. Reference Generation 1) Integral Equivalent Control: In equivalent control, the To extract the correct references for all states of the LCL switching function s, which is defined as linear combination filter, the constructed feedforward is given to the parallel state of the states, is set to zero in the next time step (grid voltage space system (see Fig. 2,’αβ reference model’). The states of neglected): this model are indexed with ∗ , because they are references s(k + 1) = 0 = Ci ∆xi,dq (k + 1) for the SMC. Every control time step the actual references = Ci (Ai ∆xi,dq (k) + bi ∆udq,eq (k) + di ∆ug,dq (k)) are computed with Eq. (5), including the grid voltage. The (15) states of this system are transformed back to the dq frame 782

5

1

fc=4kHz 1e3 1.2e3 800 0.1 1.4e3 600 0.2 Resonance 0.3 pole pair 1.6e3 400 0.4 0.5 Delay pole 0.6 0.7 1.8e3 200 0.8 0.9 2e3 2e3

100

0.8 90

0.6 80

Imag Axis

degree of dominance [%]

0.4 70 60 50

0.2 0

-0.2

Dominant pole pair

1.8e3

40

-0.4 30

1.6e3

-0.6 20

1.4e3

-0.8 -1 -3 0.8

0.6 0.4 absolute value

0.2

0

0

50

150 100 angle [degree]

-2

600

-1.5 -1 Real Axis

1

200

Fig. 6. Pole zero map of open (red) and closed loop (green) with integral equivalent control

Fig. 5. Degree of dominance under variation of absolute value and angle of the moved resonance pole pair, delay pole kept at z=0 (setting from Tab. I)

Ci is the linear combination matrix for the switching function. Here s is a scalar, so Ci is a five-dimensional vector: Ci = [c1 c2 c3 c4 c5 ]

-2.5

400

1.2e3 800 1e3 -0.5 0 0.5

10 0 1

200

(16)

In accordance with [18], c5 is set to one. This leads to the equivalent feedback (control law):
! ∆udq,eq (k) = −(Ci bi )−1 Ci Ai ∆xi,dq (k) + Ci di ∆ug,dq (k) . (17) The grid voltage is assumed to be estimated perfectly, so ∆ug,dq (k) is set to zero. The parameters to adjust this controller are the switching function vector Ci and ki1 . Following [7], they should be determined by pole placement for the sliding mode and reaching mode. But choosing equivalent control, the feedback (Eq. (17)) is consistent, and linear control design is possible. This is done by pole placement for the closed loop. The poles of the system are reduced to two dominant poles. For these two poles, parameters in the time domain are defined. The rise time tr and percentage overshoot ∆h for the step response are set to tr = 6Tc and ∆h = 1%. In Fig. 6, the created dominant pole pair is marked (green). In addition to the poles mentioned in feedforward control design, the integrator pole z=1 appears. The aim is now to place the three other poles (resonance pole pair and delay pole) in a way, that they do not affect the adjusted time behaviour. The delay pole is chosen to rest in the origin. The resonance poles are placed, provided that • they are not too far from the original poles, • a minimum distance to the unit circle is kept (stability), • the impact of the two dominant poles is not decreased too much. Therefore, the degree of dominance (DOD) [25] is used. The DOD gives a percentage which describes the dominance of a pole opposite to other existing poles and zeros. The DOD of the dominant pole pair is computed under variation of the resonance pole pair. Hereby, only locations within the unit circle are considered (stability). The result is shown in Fig. 783

5. A dominance of the adjusted pole pair of over 85% is given for all possible poles with a phase angle higher than 75◦ . 85% is considered as a sufficient value for the DOD. So the resonance poles are chosen to be pulled to a radius of 0.7 in direction to the origin as a trade-off between stability and not being too far from the original location. The result can be seen in Fig. 6. With these poles, the parameters ki1 and c1 -c4 can be computed by comparing the characteristic equation of the desired and given system. The system is underdetermined, because the delay pole is not being moved. This is circumvented by choosing c4 = 1. For the integrator an anti wind-up is used. 2) Gao’s Integral Reaching Law Control: Gao’s reaching law is described in [7], [8]. Here the system structure in accordance with [7] chapter 2.1.1, is obtained by interchanging the order of the last two states in Eq. (14). Now the design progress can be devided in designing sliding mode and reaching mode dynamics. As in reaching mode, only the delay pole is considered, the pole placement strategy similar to the one for equivalent control is suitable. Following the nomenclature of [7] leads to: 

a11  a21 =  a31 ki1  = 0 0

a12 a22 a32 ki2

 s(k) = c1 c2  S¯1 = c1 c2   S¯2 = c4

c3

A¯11 A¯21

0 0

a13 a23 a33 ki3 

  0 b1  b2 0   ; A¯12 =   b3 0  1 ki4   ¯ ; A22 = 0

   

(18) (19)

and

c3

c5 c4 



˙ igao ∆xgao ∆xgao i,dq (k)=C i,dq (k) (20)

Indexed with gao means, that the equivalent from system Eq. (14) with interchanged state order is taken. The sliding mode matrix [7], which describes the dynamics in the sliding mode,

6 (a)

(b)

10 0

0.3

0.305 (c)

0.31

(b) 20 ig,q/[A]

ig,d/[A]

10 0

10

0.305 (d)

0

0.31

0.3

0.305 (c)

0.31

0.3

20

0.31

0.3

0.305 (f)

0.3

0.31

0.305 (e)

0.31

0.3

0.305 (h)

0.31

0.3

0.305 time [s]

0.31

0.3 (f) 40

-20

300

-40

0.3

0.305 (g)

0.31

-60

350

0.3

0.305 (h)

uc,d /[V]

uc,q /[V]

-100 0.3

0.305 time [s]

0.31

0.3

0.305 (g)

0.31

0.305 time [s]

0

-50

250

-100 0.3

0.31

Fig. 7. States of LCL filter for d- and q-step of grid current in simulation (equivalent control) (a) & (b) grid current (c) & (d) converter current (e) & (f) capacitor voltage (g) & (h) converter voltage each d- & q-component with reference (red)

-60

50

300

200

0.3

0

-40

350

0

20

-20

300

0.31

-50

250

320

280

50

300

340

ucap,q /[V]

0

uc,q /[V]

320

20

ucap,d /[V]

340

u cap,q /[V]

u cap,d /[V]

0.31

ic,q /[A]

ic,d /[A]

i c,q /[A] 0.305 (e)

40

uc,d /[V]

0.305

0 0

0.3

200

0.31

10

10

0 0

280

0.305 (d)

20

20

10

10

10

0

0.3

20 i c,d /[A]

(a) 20

20 i g,q /[A]

i g,d /[A]

20

0.305 time [s]

0.31

Fig. 8. States of the LCL Filter for d- and q-step of grid current in simulation (Gao’s reaching law) (a) & (b) grid current (c) & (d) converter current (e) & (f) capacitor voltage (g) & (h) converter voltage TABLE I S YSTEM PARAMETERS

results in: Asm = A¯11 − A¯12 S¯2−1 S¯1  (a11 − cb14 c1 ) (a12 − cb14 c2 )  (a21 − b2 c1 ) (a22 − b2 c2 ) c4 c4 =  (a31 − b3 c1 ) (a32 − b3 c2 ) c4 c4 ki1 0

(a13 − cb14 c3 ) (a23 − cb24 c3 ) (a33 − cb34 c3 ) 0

 (− cb14 ) (− cb24 )   (− cb34 )  0 (21)

For the pole placement for this matrix, the same poles as for equivalent control, except the delay pole (one dimension lower), are taken. In the second step, the feedback has to be designed. [8] defines it as: −1 ··· ∆udq (k) = −(Cigao bgao i )   gao gao gao (22) · · · [Ci Ai − ks Ci ]∆xgao i,dq (k) + qs sign(s(k))

From simulation experience, ks = 0.1 and qs = 5 is chosen. V. A NALYSIS AND R ESULTS

The developed control algorithm is simulated under Matlab/SimuLink/PLECS and is implemented on a 22 kW induction-motor-based test drive with dSpace control board. The system parameters can be found in Tab. I. Fig. 7 shows simulation results with equivalent control without the DC controller assuming a constant DC link voltage. All states of the LCL filter are plotted in dq coordinates. Thereby, the states of the PLECS model are marked blue (x in Fig. 2), and the reference states of the parallel αβ state space system are marked red (x∗ in Fig. 2). In (g) & (h), the feedforward control (red, u∗αβ in Fig. 2) and corrected feedforward control (blue, uαβ in Fig. 2) are plotted (dq transformed). 784

Symbol

Quantity

Value

UL ω UDC P fs fc = CDC Lg Lc Cf fres

Line voltage (phase-to-phase, rms) Line angular frequency DC link voltage Nominal power (grid-side) Switching frequency Control frequency DC link capacitance Grid-side inductivity Converter-side inductivity Filter capacitance Resonance frequency of filter

400 V 2 π 50 Hz 700 V 25 kW 2 kHz 4kHz 2710 µF 0.75 mH 2 mH 32 µF 1.2 kHz

1 Tc

In (a), (b), it can be seen, that the grid-side current, which is to be controlled, follows the references i∗d , i∗q (green, see also Fig. 2) according to the step response of the transfer function Eq. (10). After two delays and one step, the final value of 20 A is reached without overshoot in the d- as well as in the q-component. The reference system (red) and the PLECS system (blue, with PWM etc.) react quite similarly. For the converter current (c),(d) and capacitor voltage (e),(f), steady state references show slight deviations (not significant, because their integrator gains are set to zero), while the important dynamic behaviour is quite similar. Aliasing effects cause oscillations in the measured capacitor voltage ((e),(f)). They lead to oscillations in all states, which can be avoided by filtering the measured capacitor voltage. (g),(h) show that the SMC is active to compensate these oscillations of the PLECS system. Fig. 8 shows the same system with Gao’s reaching law. It can be seen that the introduction of the switching part offers no advantages. Especially in d-component, the switching of uc,d

7

(a)

(b)

10

10

0

0 5 (c)

10

20

0

5 (f)

10

0

5 (h)

10

0 0

5 (e)

10 50 ucap,q /[V]

360 ucap,d /[V]

10

10

0

340 320

0

-50

300 0

5 (g)

10

-100 100 uc,q /[V]

400 uc,d /[V]

5 (d)

20

10

300

200

0

ic,q /[A]

ic,d /[A]

0

280

R EFERENCES

20 ig,q/[A]

ig,d/[A]

20

0

5 time [ms]

10

0

-100

0

5 time [ms]

10

Fig. 9. Laboratory: Reference states for grid current step (a) & (b) grid current (c) & (d) converter current (e) & (f) capacitor voltage (g) & (h) converter voltage each with d&q-component

in (g) causes chattering in the converter current (b). For that reason, only equivalent control is shown in lab (Fig. 9). The control is tested with UDC controller under load in the reactive part. Due to the distorted grid voltage (high 5. and 7. harmonic), the states of the reference model contain oscillations. Equivalent control can reach the desired 20 A in short time with no overshoot (b). The reference for ic,d shows slight deviations, which does not effect the control, because its integrator gain is set to zero. In (b), (d) it can be seen, that the step response of the references and the real states is quite similar. (g), (h) show, that SMC is active to compensate the deviations from the references. VI. CONCLUSION A successful design, implementation and test of discrete sliding mode current control for PWM rectifiers with LCL filter is shown. Using the discrete state space description, a feedforward feedback structure is investigated. For feedforward, the inversion approach is utilized. Reference generation for the LCL filter by an αβ model running in parallel, is implemented. Two sliding mode feedback designs are analysed: Integral equivalent control and Gao’s integral reaching law. Hereby, a pole placement strategie for the LCL filter is presented. Results from simulation and laboratory show a fast, robust control algorithm, where no advantage for the extension to Gao’s reaching law can be seen. More research activities would be interesting: There exist several other control laws (f.e. chapter 7 of [26], [27]). They could be investigated with this approach. In the feedforward control, a rather simple approach with neglecting the non minimum phase zero is taken. In [23], more advanced approaches are shown, which would be interesting to be developed. Other pole placement strategies would be of interest, too. 785

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