a sliding mode controller in single phase voltage source inverters

8/12/2019 A Sliding Mode Controller in Single Phase Voltage Source Inverters

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A Sliding Mode Controller in Single Phase Voltage

Source Inverters

H.

Pmheiro,

A.S. Martins,

J.R. plnheiro

Universidade Federal de Santa Maria, CT-DELC-NUPEDEE-DESP, 97119-900, Santa Maria, RS, Brazil

Fax:O5

226

1

975, e-mail

:

[email protected].

Abstract- This

paper

deals with a

sliding

mode controller

for single phase inverter used in UPS applications. The

proposed system provides

overload

and short Circuit

protection. It c n

operate in

constant

or

variable

frequency. The

use

of areduced order observerelhinates

the requirement of the load current measurement

and

improves the noise immunity Experimental results

obtained

in

laboratoryarepresented and they confirm the

simulation

and theoretical

analysis.

JNTRODUCIION

Static converters are intrinsically variable structure

systems, so the slidmg mode control

SMC)

approach is a

strong candidate method for the inverter controller design.

Many papers have been published in the field of

slidmg mode control[2]-[10]. It is claimed th the

use

of the

sliding mode control result in order reduction, disturbance

rejection, insensitivity to parameter variations ancl simple

implementation.

Implementation of sliding mode control implies hgh

frequency discontinuous signals. This is not a problem static

converters, because d~scontinuous ignals is always present.

Furthermore, SMC can provide a possibility to solve the

closed loop system design and to create the switching signals

(PWM pattem) in the same framework.

Some paper have been published about

shdmg

mode

control applied to DC/AC inverters. In [8] a three level

PWM

inverter with fixed frequency is presented, the proposed

method is complex, because it needs

two

frequency closed

loops and a midpoint control. The proposed current limiter

degrades

the

transients response during large perhx-bations. In

[9] is presented a two level PWM inverter with fixed

switching frequency and current limiter. The overall

perFormance is good, but it needs two current measurement,

namely, the load and filter indqctor current. In

[lo] is

presented a comparison of three and

two

levelsP W M mverter.

In the

above mentioned works the output voltage posses

steady-state error.

l b s paper is concemed on a

slidmg

mode controller

implementation for

U P S

inverters. The suggested system

posses the following features: first, it does not need

measurement of the load current; second, it is possible to

obtain zero steady state error to step input; thnd, it can operate

with constant frequency; fourth, it provides overload protection

for the power semiconductors.

SYSTEM DESCRIPTION

Figure 1 shows the basic circuit of single phase

inverter with a LC output filter whch is often used in single

phase U P S .

T

,L

L

P P

Fig.1: Single phase voltage source inverter.

The control goal is to make the output voltage v, be

equal to a reference input v4 (sinusoidal) while

the

inductor

current absolute value

ir

is kept lower

than

a maximum

value

ik.

SLIDINGMODE

ROLLER

T&ng into account

that

the output voltage

v,

must

track the reference voltage

vmf

and the inductor current must

be limited, the sliding mode control

in

the voltage and current

error coordinates is formulated. The dynamic behaviour of the

inverter, shown in Fig.1, is govemed by the

state

space

equation 1).

h 2

=

- X I

-

ref

t L L L d t

Where x , is the voltage error

(

vref-vc

,

x, is the current

error (i&) and L and C re inductance and capacitance of

0-7803-1328 -3/94 03.000 1994 IEEE 394


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the output filter respectively.

defined as:

The sliding surface and the control action

U

are

s=

C g x , czx,

o

and

U=E sign(s)

Average Model

The equation of the motion along

the

surface

s=O

is

given by

In order to achieve the invariance property in slidmg

mode the inductor reference current

is

made to

be:

Substituting

(3)

into

2)

yelds:

(3)

4)

So,

once the system is

in

slidmg mode it behaves

as

a stable first order system and the voltage error decays

exponentially to zero

C,IC,>O).

Existence Co ndition

The state space equation l),when the i s given by

(3),

is:

U

dt L L

- - + g ( t )

or

Where

and

g( t )

is given

by:

g ( t )

=

v4 lL

diJdt +

C&vJ&.

If s

and its time denvatwe have opposite signs the

state trajectory reaches the slidmg

surface s=O

after finite time

interval. This constraint can

be

achieved

if

the following

inequality is true:

possible to construct a reduced dimension observer to obtain

io. The state space equation of a Luenberger reduced-order

observer, which has

been

implemented, is:

l

c z

+Ck0vc

,

where w s the natural frequency of the output filter and z

is

an

auxiliary variable.

The observed load current is given by:

Q ? .

1 =1 I

o c l

9)

By proper choice of the gain

k, the

desired

convergence rate of

I

to ic and, consequently, to &may

be

provided.

Current Limiter

A mod~fied

eference current may be obtained by

rearranging the sliding surface equation

as

shown

in l0).

L

s c , x , c , x ,

=C, ( r , +x , )

C,

L

s=c,( X,

rt,- l

=C, i

Lf i t )

C

L

It is possible to limit the inverter output current

through the limitation i , assuring, in

this

way, overload and

short-circuit protection, as shown in Fig.2.

0 a b i 8.h B B3 s)

-Me

Fig.2

:

Output voltage and inductor current when a

temporary short-circuit in the load occurs.

Elimination of the Steady State Error

This inequality also

detennines

the

minimum

inverter

DC

voltage needed for enforcing

the

sliding mode.

The Use of Reduced-Order Observer

As

the state variables

v,

and

il

are available, it is

Whenever the steady-state specification

is

not

satisfied, due to the use of hysteresis comparator leadmg to

finite

switching frequency, it is possible

to

use extra state

variables in order to cancel the voltage errors. For instance,

the steady-state error to a constant reference input is

395


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eliminated by introducing an extra state variable, as shown

below.

= x ,

r

dt

=

- - - - + g ( t )

I U

dt

L L

t B i

The slidmg surface is gven by:

Fig.4: State trajectories in the vicinity of slidmg surface

when the hysteresis comparator

is

used.

s=

CdE, +c ,x ,

Q2 =o (12)

~=f x8,g t>);f=f x,-E,g t)).

The average dynamic behaviour is govemed by:

The maximum switchmg frequency is obtained setting

v , ~ ~o zero, which results in the following equation:

(13)

'x,

C,

dx

CO

- - - r l = O

d t2 C2C t C,C

Freq- =-CP 15)

4AL

Figure 3 shows the output voltage error for a step

Constant Frequency O peration

nput.

It is easily

seen

from the equation (14), that the

switchmg frequency depends on the reference and

DC

nput

voltages. It can result in a very wide switching frequency

range. This is a disadvantage from the point of view of the

output filter design and implementation There

are

basically

two methods to achieve constant frequency operation:

a) Variable Hysteresis Band with feed fo mad or feed

back techmque. In t h ~ s ase some points must

be

carefully

investigatedsuch

as:

stability problems, transient

perform nce

limitations, complexity of implementation[121 131.

b) Disturbance Method:

it is possible to obtain

1.t-3

2.t-3

3 t - 3

4.1 3

constant frequency operation adding an

adequate

constant

Fig.3: Output voltage error for a step input.

SWITCHING

FREQUENCY

Variable Frequency O peration

The hysteresis is responsible for the fact th t once the

state trajectories hit the dscontinuity surface s=U, the

describing points do not move exactly along

the

surface but

wll oscillate in its vicinity with width equal to

2A

(Fig.4).

Under the assumption th t the states trajectories are constant

near

the

surface s=U the switchmg frequency is given by:

frequency signal to

s

variable, whch is comparable to the

approach proposed in

[9].

Figure

5

shows athree evel disturbance signal, whch

with an apropriate design provides constant switching

frequency operation. 6t determines the maximum duty-cycle

at steady state. The amplitude D must be greater

than

maximum s value. Disturbance

signal

frequency

Fd

must obey

the folowing inequality:

d t

Fig5 Three level dlsturbance signal.

396


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THREE LEVEL PWM

The three level P W M method proposed in [8]

presents a good performance, but it is complex. A simpler way

to implement a three levelP W M s presented in [lo]. The later

treats the three level

P W M

s a double two level

PWM,

ne

of them is active when

ueq

s positive and the other when

ueq

is negative

(ueq

s defined in [l]). However, it was used vref

instead of ueq.

This

practice resulted in degradation of the

transient response and &stortion around the zero

(ueq

may have

opposite sign of

vref

, as illustrated in Fig.6.

Fig.6:

Three

Level P W M nverter with 0.7 inductive load.

INSULATINGRANSFORMER

Usually it is required galvanic insulation of the load.

There are basically two ways of implementing the galvanic

insulation: the first one is by using a low frequency

transformer and the second one by using hgh frequency

transformer.

Low Frequency Insulating Tramformer

To improve the performance of the system in low

frequency it is possible to substitute the derivative of the

reference voltage by the integral of the output voltage in the

equation (3) [81, it because the reference signal is sinusoidal

in

U P S

applications.

High Frequency Insulating Transformer

Several works have been published on inverters using

hgh

frequency transformer link [11,14,15].

The

DC/AC

converter proposed in [ l l ] is shown in Fig.7. The phase

control used there can be repalced by the sliding mode

controller previously presented here.

Fig.7: DC/AC converter with high frequency transformer.

The switches S1 and S2 are commanded with fixed

frequency and duty-cycle equal to

0.5,

so producing a

high

frequency square wave v,, at the input of the cycleconverter.

On

the other hand, the switches S 3 and S4 are commanded by

the following law:

S4= sign(s) sign@,)

S3=

-sign(s)

.

sign(v,)

,where -1 is for switch opened and 1 for switch closed. The

inverter and the clycleconverter topology can be implemented

using other configurations, for instance, half and full-bridge

respectively.

EXPERIMENTALESULTS

It has

been

implemented a prototype of the single

phase inverter shown in Fig.1. The implemented inverter

controller block &gram with reduced order observer and

current limiter is shown in Fig.8.

%-

Fig.8: Block diagram of the implemented controller.

The controller and output filter parameters have been

derived from the following input datas:

Rated output power Po=600VA

DC input voltage E =lOOV

Maximum switchmg frequency F,,,,,= 15KHz

Current Ripple(limiting current mode) AM.4

Current transducer gain

k,,,=. 1V/A

2=1/1oooo

s.

RMS fundamental output voltage v0=55v

Output voltage Ripple v0,=4v

Desired time constant of system

The calculated parameter values are:

Hysteresis width AS.65

Output filter inductance

M . 2 5

mH

Output filter capacitance C=85.5

JJF

Gain C,

C,=0.0855

Mmimum switchmg frequency Fm,,=2.85KHZ

Gain c c;=o. 1

397


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The experimental results shown in Figures 9, lO and

11 validate previous analysis and simulation.

Fig.9: Output voltage v, wth variable frequency operation

for step input.Voltage Scale: 20 V/div.

Time scale:0.2ms/dlv.

Fig.10: Output voltage v, and inductor current il ( variable

frequency operation mode) with sinusoidal input and when a

temporary output short-circuit occurs.Voltage Scale:

20

V/div. Current Scale:lONdw. Time scale: Sms/dv.

Fig. 11: Output voltage

v,

and

PWM

signal with constant

switching frequency operation (15 Khz) .Voltage Scales:

upper 50 V/dlv,lower 100 V/div. Time scale:0.2ms/&v.

CONCLUSIONS

The s l i m mode control techmque is a powerfull tool

for the controller design of static converters. The resulting

system is robust, simple and posses

h h

perfomance.

Furthermore, it can create the switching signals in same

framework.

The suggested controller has an observer, whch

becomes the system less sensitive to noise when compared

with the one that use hfferentiator and avoids the

measurment of the load current. The use of the observer does

not increase significantly the circuitry implementation.

The prototype has been sucessfully implemented with

both variable and constant frequency operation. The power

semiconductor currents

are

limited during overload

and

short-

circuit on the load.

REFERENCES

[l] V.I.Utkin,

Sliding Modes

and

Their Applications

in

Variable Structure Systems.

Mmcow:Mir-Publisher, 1978.

[2] J.P.Karunadasa, A.C.Renfrew, Design and Implementation

of Microprocessor Based Slidmg Mode Controler for Brushless

Servomotor ,

IEE Proceedings-B,

vol. 138, n0.6, Nov./1991.

[3] J.Y.Hung,W.GaoJ.C.Hung, IVariabletructure contro1:A

Survey JEEE

Trans.Ind.Electron. vo1.4O7o.1,February 1993.

[4] S.Bolognani,E.Ognibeni,M.Zigliotto,Sliding Mode

Control of the Energy Recovery Chopper in a C-Dump

Switched Reluctance Motor Drive ,

IEEE T rans.Ind.Electron.,

vo1.29, no.1, February 1993.

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Automatica,

vo1.5, 1969.

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a sliding mode controller in single phase voltage source inverters

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