Sliding-mode virtual flux oriented control of PWM

Michal KNAPCZYK1, Krzysztof PIENKOWSKI2 3Cap Technologies GmbH (1), Wroclaw University of Technology (2)

Sliding-mode virtual flux oriented control of PWM rectifier with DC-bus voltage and current sensors Abstract. The paper presents a line-side sensorless control of PWM rectifier with a virtual grid flux vector orientation and a sliding-mode approach. The grid currents and virtual flux are estimated using a sliding-mode current observer mixed with a nonlinear asymptotic observer and a virtual grid flux estimator. The only inputs of this nonlinear observer system are measured values of a DC-link current and voltage, as well as a PWM control pattern. The fast dynamics and robustness of the proposed sensorless control system has been presented in simulation and experimental results. Streszczenie. Artykuł prezentuje bezczujnikowy układ sterowania ślizgowego prostownikiem PWM z zastosowaniem nieliniowych obserwatorów wielkości sieci zasilającej. Układ sterowania ślizgowego przekształtnika oparty jest na orientacji wirującego układu odniesienia względem wektora wirtualnego strumienia sieci. Przedstawiono wyniki badań symulacyjnych i eksperymentalnych zaproponowanej metody sterowania ślizgowego. (Bezczujnikowy układ sterowania ślizgowego prostownikiem PWM z zastosowaniem nieliniowych obserwatorów wielkości sieci zasilającej).

Keywords: AC/DC line-side converter, PWM rectifier, virtual grid flux, sliding-mode control. Słowa kluczowe: Przekształtnik sieciowy AC/DC, prostownik PWM, wirtualny strumień sieci, sterowanie ślizgowe.

Introduction The most of modern electrical devices and electronic equipment requires the AC to DC power conversion. Referring to IEC 61000-3 – European Standards, modern line-side converters should draw sinusoidal currents from the mains without the flow of reactive power and have a possibility of injecting the electrical energy back into the mains. The high efficiency of the bidirectional power conversion can be provided by AC/DC line-side converters that are able to force sinusoidal grid current flow at the unity power factor condition while the DC-link voltage is kept constant in the wide range of load changes. A converter input voltage modulation method (a selected PWM technique) provides the desirable properties of the PWM rectifiers. In fact the sinusoidal grid currents are shaped via controlling the voltage drop over the line inductors (EMI filter) that are the integral part of the power circuit. Control strategies for the PWM rectifiers are adapted from the electrical drive applications and are still examined and developed. In most industrial applications with the PWM rectifiers the linear control is implemented [1]. The PI control performance is strictly dependent on the proper identification of the line chokes inductance and the DC-link capacitance. The values of the line and power circuit parameters may also vary under the influence of the electromagnetic interference generated by the converter itself. Under the closed-loop operation Sliding-Mode Control provides the robustness of the converter's control system within the limited range of the parameter mismatches and disturbances like grid voltage distortions and heavy load variations [4,6,7]. The sensorless control in power electronics and electrical drive has evolved from simple solutions based on the state simulators and estimators up to the advanced state observers using numerous approaches. The estimators are entirely based on the system model, having the same dynamics and inheriting the same sensitiveness to the parameter mismatch and outer disturbances. The state observers are adjustable reference system models tuned by feedback internal signals stemming from a controlled plant, so that the output estimates follow their real counterparts. The existence of these feedback signals provides robustness of the observer to the inaccuracies of the system parameter identification. However to some extent of the inaccuracies an observer may not provide the sufficient robustness towards them. The introduction of nonlinear methodologies into the state variable estimation

improves the observer insensitiveness to the outer disturbances and the parameter fluctuations. The slidingmode observers provide the high-quality estimation of all the components of the state vector by existence of disturbances and parameter mismatches [2,3,4,7]. AC/DC line-side converter and virtual flux vector The detailed circuit of the three-phase two-level PWM rectifier is presented in Fig.1. For a boost feature of the converter its input is required to be a current-source. Hence the three symmetrical chokes are interfacing the rectifier with the grid. The voltage-source output of the rectifier requires a capacitor to provide the step-up operation.

Rg , Lg

Ka

Kb

Kc

Ka

Kb

Kc

Fig.1. Power circuit of PWM rectifier with symmetrical DC-link

The dynamical model of the converter in three-phase system can be described by the following matrix equations [7]:

(1)

u d i g  e g  R g i g  dc J  K abc dt 6 i d 1 u dc  K Tabc  i g  load dt Cd 2C d

where

 i gA   egA      i g  i gB  , e g   egB  i gC  egC     

and

 2  1  1 K a    J   1 2  1 , K abc   K b  .  1  1 2   K c 

Lg

PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 2/2010

253

where Ka, Kb, Kc are the conduction states {1,-1}, Rg, Lg – grid resistance [] and inductance [H]; Cd – DC-link capacitance [F]; igA, igB, igC – grid three-phase currents [A]; egA, egB, egC – grid three-phase voltages [V]; udc – DC-link voltage [V]; idc – DC-link current [A], iload – load current [A]. The issue of the vector control for the PWM rectifiers is the decomposition of each state variable vector into the two perpendicular components from the natural three-phase coordinates into a stationary (-) frame oriented with the A phase axis and than into a synchronous (d-q) frame oriented with a state variable vector rotating in the (-) coordinates with an angle . Here the amplitude-invariant transformations are assumed:

(2)

(3)

 x  2 1 x       3 0

 xd  cos  x     q   sin 

x  sin    x  αβ     x   Pdq  x  .  cos       

xA   x   3 C ABCT P αβ T  xd  .  dq   B  2 dq  xq   xC 

where

igd  egd  i gdq    , e gdq    , i  gq  e gq  K d  K dq    , Kq 

f dq

Sliding-Mode Virtual Flux Oriented Control The principles of the Sliding-Mode Virtual Flux Oriented Control (SM-VFOC) for the PWM rectifier is based on the decomposition of the grid current vector ig into the two components igd and igq in the (d-q) coordinate frame oriented with the virtual grid flux vector as depicted in Fig.2.

b

u dc 2 Lg









 1   L e gd  Rg igd  i gq  . e gdq , i gdq ,   g  1 e  R i  i  g gq gd   L gq  g 





 - grid angular frequency. For the Sliding Mode Control the following switching functions for the grid current sliding-mode control are formulated [7]: (7)

s dq  i gdqref  i gdq  0

where

 sd  igdref  igd  s dq       0.  sq  i gqref  i gq 

The goal of calculations is defining the sliding-mode conditions. A domain in the system space has to be found from which any state trajectory converges to the sliding line described by (7). Substituting (6) into the time-derivative of (7) results in the following matrix expression:





s dq  i gdqref  f dq e gdq ,i gdq ,  bK dq 

(8)

ψ g   e g dt  ψ g0



i gdq  f dq e gdq , i gdq ,  bK dq

xA  xA   1 2   ABC  x B   Cdq  x B  ,   3 2  3 2  xC   xC 

The distorted voltage drops in the distribution line are mostly caused by the pulsed currents of the nonlinear loads. The effective approach to the problem of the line voltage disturbances is the introduction of the virtual grid flux vector [5,7]. The virtual flux vector defined in (5) is far less sensitive to the line disturbances and maintains near sinusoidal shape even in case of the low-harmonic pollution in the supply voltage: (5)



(6)

1 2

The back transformation to the natural three-phase frame is given by:

(4)

For a control design purpose it is more convenient to transform the mathematical model of the converter (1) into the (d-q) coordinates using the formulas (2) and (3) as presented in the following generalized matrix form [4,6,7]:

 Fdq  Τ  K abc

where





Fdq  i gdqref  f dq e gdq , i gdq , ,

αβ ABC T  bPdq Cdq .

Finally the matrix including control signals can be obtained as follows: (9)

K abc   signs abc 

where

s abc 

3 2 T b Τ s dq . 2

In a real application based on a digital microcontroller and a power module the control signals (9) must be separated into six gate logic signals of {1, 0} for each IGBT transistor according to the following relations:

1 1  K a  2 1 K b  1  K b  2 1 K c  1  K c  2

K a  (10) Fig.2. Diagram of virtual grid flux vector orientation method

254

K a  1  K a K b  1  K b K c  1  K c


The AC/DC line-side converter can be characterized with the fast dynamics associated with the grid current control loop and the relatively slow dynamics of the DC-link voltage control loop. Due to the fact that digd/dt, digq/dt >> dudc/dt the current differential equations from (1) become algebraic ones by making their left-hand sides equal to zero. Moreover, when the system is already in sliding-mode (7) than igdqref=igdq and the following modified equations are true: (11)

egA Rload

Grid Current Preliminary Reconstruction Sliding-Mode Virtual Grid Flux Observer

uconvd  2egd  Lg  igqref  / K d

gA

uconvq  2egq  Lg  igdref  / K q

idc

Virtual Grid Flux Estimator & NA Observer

igA

gB

ABC

igB

K a Kb K c SlidingSliding-Mode Mode Controller

ABC

udc

Controller

Similarly the DC-link voltage differential equation gets modified into the form: (12)

EMI filter (Rg,Lg)

egB egC

g

g

ig

sa sb sc

ig

ABC

g

d-q

K d igdref  K q igqref iload d u dc   dt 2Cd Cd

d-q

igd gd

ref

The adjustable power factor condition requires the input current phase angle  to trace its reference value:

igq

ref

udcref

PI

sd

igdref

gq

d-q

sq

udcref i gqref iload

udc

(13)

 ref

 i gdref  tan 1   i gqref 

   

Fig.3. Sensorless Sliding-Mode Virtual Flux Oriented Control

An important requirement for the proper operation of the PWM rectifier is the power balance condition. The instantaneous values of the electrical power at the converter input must be equal to their corresponding quantities at the converter output assuming that the power conversion process is lossless and the pure sinusoidal grid voltages and grid currents are transformed according to (2) and (3), which can be expressed in the following manner: (14)

3    gd  igdref  gq  igqref   udcref  iload 2

Solving equations (13) and (14) with respect to igdref, igqref yields the following formulas for the reference grid currents in the (d-q) coordinate frame:

(19)

igdref 

   gd   gq tan  ref 

(16)

igqref 

   gd   gq tan  ref 

Table 1. DC to AC currents relationship idc Ka , Kb , Kc igA 1,-1,-1 -igC 1,1,-1 igB -1,1,-1 -igA -1,1,1 igC -1,-1,1 -igB 1,-1,1

u dcref iload

The linear dynamics of the DC-link voltage closed loop at pure resistive load (edc=0) is calculated by substituting the reference grid currents (15) and (16) into the equation (12):





udcref  udc  iload udcref  udc d , udc   dt udcCd Rload Cd

The starting point of the design of a robust nonlinear observer for virtual grid flux is the following sliding-mode current observer for the source grid voltage [4]:



(20)





u dcref  u dc  e dc   iload u dcref  u dc  e dc  d  u dc  u dc  edc   C d Rcable C d dt

According to the above mathematical background the Sliding-Mode Virtual Flux Oriented Control system for the PWM rectifier has been designed and presented in Fig.3.





d 1 i gA  e g 0  sign î gA  i gA  dt Lg  R g i gA 

and for the DC-power regeneration (edc > udc, Rload=Rcable): (18)

idc  K a  igA  K b  i gB  K c  i gC

According to the equation (19) the partial information about the grid phase currents can be extracted using the logical conditions presented in Table 1.

u dcref iload tan  ref

(15)

(17)

Sliding-mode-based observer design The strict knowledge of the virtual grid flux and grid current is necessary to determine the virtual grid flux vector position and the line current deviations. Their instantaneous values may be successfully estimated based only on the information from the DC-link voltage sensor, DC-link current sensor and the actual PWM pattern. The mathematical relationship between the DC-link current and the line currents is based on the following equation involving the PWM pattern:

u dc 2 K a  K b  K c  6 

where eg0 is an observer gain, î gA is an observed grid current described as follows:

(21)

* î  i gA gA  when i gA


i*gA  0

255

In order to achieve a smooth estimate of the virtual grid flux the discontinuous sign term from (18) should be directly integrated as follows: (22)

 gA  e g 0  signî gA  i gA dt   gA0

The total virtual grid flux can be obtained using the following estimator: (23)

  Lg î gA  igA     gA  gA

The virtual grid flux signal is next submitted to the nonlinear asymptotic observer (NAO) that filters out the input signal giving the smoothed virtual grid flux: (24)

where

ˆx1  ˆx2  L1W1 x1 ˆx 2   2 ˆx1  L2W1 x1 x1  ˆx1   gA , is the grid angular frequency

[rad/s], L1, L2 are positive observer gains and W1 is the "window" signal for the phase A that is defined by the following formula: (25)

1 W1   0

i*gA  0

if

Selected simulation results The proposed control method has been tested using Simulink incorporating PLECS. The simulation model has been divided into three subsystems which have been computed with different rates: circuital model of the PWM rectifier as well as EMI filter and supply grid in PLECS, Sliding-Mode Virtual Flux Oriented Control and slidingmode-based AC-side observer [7]. The parameters of the simulation model are presented in Table 2. Table 2. Parameters of simulation model Phase grid voltage eg: Grid voltage frequency fg: Choke resistance RgN: Choke inductance LgN: DC-link capacitance CdN: DC-link nominal voltage udc: Load resistance Rload: Sliding-mode observer gain eg0 NAO observer gain L1 NAO observer gain L2 Power module sample time Tp1: SM-VFOC rate Tp2: SM-based observer rate Tp3:

230 Vrms 50 Hz 100 m 10 mH 1000 F 600 V 100  500 1e3 2e6 1e-6 s 30e-6 s 15e-6 s

Fig.5 shows the measured phase grid voltage egA and the transients of the measured grid currents igA, igB, igC at applying the full load resistance Rload.

otherwise

The estimated virtual grid flux signal from NAO ˆ gA is then the input signal of the grid current simulator that is described by the following equation:

2.0

easier later to move or change the dimensions of the 1.5 figures. 1.0 0.5

(26)

  igA



1 ˆ gA   convA Lg



0 -0.5 -1.0

where (27)

-1.5

 convA 

1 u dc 2 K a  K b  K c dt   convA0 6

Fig.4 presents the single-phase block diagram of the proposed sliding-mode observer for the virtual grid flux and grid current. i

i dc

 gA

 igB  igC

Ka Kb Kc

ˆ ψconvA ψ gA L-1g

udc

~ uconvA

ψgA

~

 igA

îgA Lg

egA

uLA

L-1g

u RA Fig.4. Phase representation of sliding-mode-based observer mixed with nonlinear asymptotic observer and linear estimator for virtual grid flux and grid current

256

-2.0 0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

Fig.5. Phase grid voltage and three-phase grid currents

It can be noticed that for the unity power factor operation of the PWM rectifier the phase shift îgA between the grid phase currents and the respective grid phase voltages has been reduced to zero. Fig.6 presents ˆ ψ ψ gA gA the transient of the DC-link voltage and îgA current under the step change of the rectifier load. In sliding-mode due to igA the equation (17) the DC-link voltage tends to its reference value udcref with the time constant =RloadCd irrespective of the values of the parameters of the control system and the AC grid.


The time function of the DC-link current corresponds to the active component igq of the line current. (1) udc

(2) idc

[p.u.]

Fig.9 presents the robustness of the control system to changes of output capacitance values (40%200% of CdN). 1.01

udc [p.u.]

1.0

1.02

0.99 1.01

40%CdN

0.98

(1)

1.0

1.01

udc [p.u.]

1.0

0.99

0.99

0.98

0.98

2.0

1.01

80%CdN udc [p.u.]

1.0

1.5 1.0

(2)

0.99

0.5

0.98

0

1.01

-0.5

1.0

-1.0

0.99

0.1

0.12

0.14

0.16

0.18

0.2

t [s]

120%CdN udc [p.u.]

200%CdN

0.98 0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

t [s]

Fig.6. Transients of DC-link voltage and current

Fig.9. DC-link voltage at different values of output capacitance

Fig.7 presents the three steps of the grid current reconstruction and estimation in the proposed sliding-modebased observer for AC-side variables.

Fig.10 presents the robustness of SM-VFOC with proposed sliding-mode observer to changes of values of the EMI filter inductance (60%140% of LgN).

îgA

 igA

 igA

(1) 2egA 2.0

1.0

(2) igA

(3) igB

(4) igC

[p.u.]

60%LgN

0

1.0

(4)

-1.0

0

(3) (2)

-1.0 1.0

(1)

-2.0

0

2.0

-1.0

140%LgN

1.0

(4)

1.0

0

0

-1.0

(3) (2)

-1.0 0.11

(1)

-2.0 0.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

0.155

0.11

0.12

0.13

0.14

0.15

0.16

0.17

t [s]

Fig.7. Phase grid current reconstruction and estimation

Fig.10. Grid currents at different values of choke inductance

Fig.8 presents the three steps of the virtual grid flux estimation in the proposed sliding-mode-based observer for AC-side variables.

Experimental setup and test outcomes Fig.11 shows the experimental test-bench of the PWM rectifier with the TMX320R2812 DSP-based control board [7].

 (2) ψ gA

(1) ψgA

(3) ψˆ gA

[p.u.]

1.0 0.5 0 -0.5 -1.0

(1)

1.0 0.5 0 -0.5 -1.0

(2)

1.0 0.5 0 -0.5 -1.0

(3)

0.11

0.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

0.155

t [s]

Fig.8. Phase virtual grid flux estimation

Fig.11. Prototype of AC/DC line-side converter

Figures 9 and 10 show the robustness of SM-VFOC to the parameter mismatches of the power circuit components.

The power unit is based on the 3.3 kW IGBT module by EUPEC with EiceDRIVER™ 6ED003E06-F interface.


257

Fig.12 shows the phase grid voltage and the respective three-phase currents at the unity power factor. Despite the voltage distortions the grid currents are sinusoidal.

igC igB igA egA Fig.12. Measured grid voltage and three-phase grid currents

Fig.13 presents the transient of the DC-link voltage at the application of the full load resistance. Additional frequencies stem from the grid voltage distortions.

Conclusions The paper presents the design of the sliding-mode current control technique for the PWM rectifier based on the virtual grid flux vector orientation. The SM-VFOC technique with the sliding-mode grid current controllers provides an excellent dynamics of the inner control loop and robustness of the control system against the parameter mismatches and load variations. Throughout the property of the sinusoidal current shaping the inconvenient influence of the rectifier on the grid is thus eliminated. The introduction of the virtual grid flux vector enhances the insensitiveness of SM-VFOC to the limited grid voltage distortions and brief voltage collapses. The special stress has been put on the designing of the sliding-mode observer for the virtual grid flux and grid current to provide a sensorless operation of the PWM rectifier without the AC-side transducers. The maximal limitation of the number of the necessary sensors has become the subject of many researches in area of the PWM rectifiers. There are several solutions based on the same idea of the grid current reconstruction that consist in the DC-link current sampling in particular times of the PWM signal period. However they take into account only the SVPWM technique that provides the control pattern every fixed time period. Thus the DC-link current sampling is technically more convenient as in case of the -modulation where the switching frequency varies. Then according to the NyquistShannon sampling theorem the DC-link current should be sampled at the rate at least twice higher that maximal PWM frequency. The sliding-mode-based observer of the AC-side variables proposed in this paper has been designed with help of the Sliding-Mode Control methodology mixed with the design assumptions of the classical nonlinear asymptotic observers. The proposed sliding-mode observer for the virtual grid flux and grid current provides the excellent operation not only under the nominal conditions but also within a relatively wide range of the parameter fluctuations. REFERENCES

Fig.13. Transient of DC-link voltage

Fig.14 shows the estimation of the virtual grid flux. It can be noticed that the virtual flux is phase shifted with the grid voltage signal of an angle /2.

g

gA

egA

Fig.14. Phase virtual grid flux and position angle of its vector

258

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