Passivity Based Control of PWM Current Source Inverters Based Control of PWM... · Description and...

International Electrical Engineering Journal (IEEJ)

Vol. 5 (2014) No. 1, pp. 1216-1228

ISSN 2078-2365

http://www.ieejournal.com/

1216

Padmabeaula et. al., Passivity Based Control of PWM Current Source Inverters

Passivity Based Control of PWM Current

Source Inverters

Padmabeaula Aa, Josephine R L

b, Suja S

b Helen Catherine

c, Dhayal Raj

d

a Department of Electronics & Communication Engineering,PSNA College of Engineering & Technology,

Dindigul,India b Department of Electrical and Electronics Engineering, Coimbatore Institute of Technology, Coimbatore, India c Department of Electrical and Electronics Engineering, ,Coimbatore ,Sri Krishna College of Engineering and

Technology, India d Department of Physics, Sacred Heart College ,Vellore,India.

Email: [email protected]

Abstract— AC power is generally used for various

practical applications. Inverters are capable of

producing AC output from DC input. Current

source inverters produce variable voltage output. If

the output voltage is regulated at constant level, it

can be used as an uninterruptible power supply. In

this paper, passivity-based control (PBC) strategy is

employed for single-phase pulse width-modulated

current-source inverters feeding a resistive load. In

this strategy, it is required to estimate the load

resistance and the inductor-current reference.

However, the estimation accuracy depends on the

inductance value when the load resistance and the

inductor-current reference are estimated using an

existing adaptive algorithm. The inductor-current

reference estimation is possible at the expense of a

constant reference function for the inductor

current. In addition, the design of inductor and the

sensitivity of the output voltage to the value of

output capacitor used in the controller are studied

extensively through the steady-state analysis of the

control method. It is also shown that even very

large values of the estimation error do not give rise

to significant change in the output-voltage

amplitude.

Keywords – Output voltage; passivity control; load; pulse width modulation; current source inverters.

1. Introduction

Generating clean AC power for household and

small business use is on the rise. Systems such as

Uninterruptible Power Supplies (UPS), AC power

generators, and active filter line conditioners are just

a few examples of power electronic systems that

make use of some type of inverter circuit. As with

any consumer application, economic concerns play a

large factor in system design. Power electronics is a

relatively new and fast-growing electronics, with

wide practical application. It is concerned with

efficient conversion of electrical power from one

form to another. To achieve high efficiency power

conversion, the active semiconductor devices

(thyristors, power transistors) are always used in a

switching mode in combination with passive

components (power diodes, inductors, capacitors and

transformers).

Power converters exhibit a wealth of non-linear

phenomena. The Pulse Width Modulated (PWM)

Current Source Inverters (CSIs) possess attractive

features such as regeneration capability when

supplied from a thyristor rectifier, short-circuit

protection provided by the inductor, and a load

voltage with low total harmonic distortion (THD).

As a result, they received a considerable attention by

the researchers [1]-[7]. In addition, the CSI drive has


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Fig.1.Schematic diagram of current source

inverter the ability to boost output voltages[25]. This brings

the possibility of another application for the CSI,

namely, as an uninterruptible power supply. Thus,

AC voltages required in certain utility applications

can be obtained from a low battery voltage in one

single power stage. However, the control of CSI is

more difficult compared to that of voltage-source

inverters. The prime source of non-linearity is the

presence of switching element present in all power

electronic circuits. Non-linear components (e.g., the

power diodes) and control methods (e.g., pulse-

width modulation) are further sources of non-

linearity. So a non-linear control technique should

be employed to stabilize the system. 2. Problem Description An inverter is an electrical device that converts

direct current (DC) to alternating current (AC).

There are three main types of inverters namely

Voltage source inverters, Variable voltage inverters

and Current source inverters.

In voltage source inverter, output voltage is

maintained constant and current is forced to vary.

Voltage source inverters are generally used to

regulate the speed of three-phase squirrel cage

motors by changing the frequency and the voltage. It

consists of input rectifier, DC link and output

converter. They are available for low voltage range

and medium voltage range applications.

The variable voltage inverter (VVI) uses an SCR

converter bridge to convert the incoming AC voltage

into DC. The SCRs provide a means of controlling

the value of the rectified DC voltage from 0 to

approximately 600 VDC. The inverter section

consists of six switching devices. Various devices

can be used such as thyristors, bipolar transistors,

MOSFETS, and IGBTs. The above schematic

shows an inverter that utilizes bipolar transistors.

Control logic uses a microprocessor to switch the

transistors on and off providing a variable voltage

and frequency to the motor.

In current source inverter, output current is

maintained constant and voltage will be forced to

vary. The harmonics present in the output is a

major challenge for designing the control technique

[14]-[18]. A current source inverter has a constant

DC current intermediate circuit. The current is kept

constant with a controlled rectifier and high

inductance reactors. The output frequency is

controlled by the inverter. The schematic diagram of

current source inverter is shown in Fig.1.

Current spikes, caused by switching, can be seen in

the output. At low speeds current pulses can cause

the motor to cog. Voltage and current waveforms of

CSI are shown in Fig.2.

Fig.2. Voltage and current waveforms of CSI

Control of CSI is difficult because the control

variable appears in both inductor-current and

capacitor-voltage equations. The voltage control of

such systems is possible at the expense of a lower

bandwidth than the current control. The lower

Current

Voltage

Converter DC Link

Inverter



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bandwidth, however, slows down the dynamic

response of the system. The dynamic response of the

CSI system can be faster if a current controller is

employed.

Discrete time control strategies are based on one-

step-ahead control [3] and sliding-mode control

approaches [8]-[9]. However, response times of

discrete-time-based control approaches are limited

by microcontroller speed, which leads to

considerable distortion in the output voltage. It is

well known that the continuous-time control

strategies are much faster, leading to much less

distorted output voltage. A passivity-based control

(PBC) proposed in continuous time offers fast

dynamic response and exhibits an ability to regulate

the output-voltage variation caused by the load

perturbations. In this project, passivity based control

of current source inverter is achieved and the design

of inductor and sensitivity of the output voltage to

the value of output capacitor used in the controller

are studied extensively through the steady-state

analysis of the control method.

3. Description and operation of single-phase CSI

The main objectives of the CSI control system are

to produce an output voltage with low THD and to

force the inductor current to follow its reference.

The direct control of output voltage may lead to an

unstable system due to the non-minimum phase

feature of the CSI. Hence the output voltage is

controlled by tracking input current.

3.1 Single phase CSI circuit

Single phase CSI circuit is shown in Fig.3. It is

composed of a DC voltage source (Vs), an inductor

(L), a single-phase PWM CSI using four

unidirectional switches, and a capacitive output filter

(C). The load is assumed to be purely resistive (RL).

Fig.3. Single phase CSI circuit

The equations describing the operation of the CSI

can be written as

iss vVRidt

dyK

(1)

L

oo

R

vi

dt

dvD 0

(2)

where R is the resistance of the inductor, K denotes

inductance and D denotes capacitance. vi and io are

the DC input voltage and the AC output current

which are produced by PWM operation of the CSI,

respectively. The switching function is denoted as d

which takes values in the finite set {−1, 0, +1} and

acts as a control input variable.

The inductor absorbs the voltage harmonics

produced by the CSI and behaves as a current source

to the CSI. On the other hand, the capacitor absorbs

the current harmonics generated by the CSI and

defines a sinusoidal voltage across the load.

The control objective is to design a control law for

the control input d such that the output voltage is

regulated to the desired reference once the inductor-

current tracking is achieved. At the same time,



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harmonics should also be reduced to the minimum

extent.

3.2 Non-minimum phase feature

Direct control of output voltage may lead to

unstable operation due to the non-minimum phase

feature of CSI. Current source inverter (CSI) has a

right-half-plane (RHP) zero in its control-to-output

transfer function [24].

This RHP zero causes the inverter output to fall

before rising when a step increase in command

reference is required (commonly known as non-

minimum-phase effect)

[13].In order to avoid the

unstable operation and eliminate the non-minimum

phase obstacle, the output voltage should be

indirectly controlled by tracking the following

inductor-current reference:

)2sin( tIIi HDD

(3)

where ID is a DC reference and IH is the harmonic-

ripple current.

4. Passivity based control

Passivity Based Control (PBC) is a controller

design approach with ‘energy shaping’. PBC yields

a closed-loop energy that is equal to the difference

between the stored and the supplied energies,

namely energy-balancing [10]-[12] .When a

passivity-based controller is designed a desired

energy function is first selected. Then the controller

can be designed to ensure this objective. This

energy-balancing property is clearly a universal

property of passive physical systems, including

nonlinear and time-varying ones.

4.1 Passivity

Passivity is a fundamental property of many

physical systems which may be roughly defined in

terms of energy dissipation and transformation. It is

an inherent Input-Output property in the sense that it

quantifies and qualifies the energy balance of a

system when stimulated by external inputs to

generate some output. Passivity is therefore related

to the property of stability in an input-output sense,

that is, we say that the system is stable if bounded

“input energy” supplied to the system, yields

bounded output energy. This is in contrast to

Lyapunov stability which concerns the internal

stability of a system, that is, how “far” the state of a

system is from a desired value.

Passivity based control is a methodology which

consists in controlling a system with the aim at

making the closed loop system, passive.

4.2 Passive system

From an energetic point of view, we can define a

passive system as a system which cannot store more

energy than is supplied by some “source”, with the

difference between stored energy and supplied

energy, being the dissipated energy. Hence, it shall

be clear that passivity is closely related to the

stability of a system, in the input-output viewpoint.

In PBC achieving stability from this viewpoint is the

first goal.

4.2.1 Fundamental property of passive system

A fundamental property of passive systems is that,

the feedback interconnection of two passive systems

yields a passive system. Thus, if the overall energy

balance is positive, in the sense that the energy

generated by one subsystem is dissipated by the

other one, the closed loop will be stable in an input-

output viewpoint.[23] This property constitutes the

basis of passivity-based control (PBC).

4.2.2 Energy shaping and Damping Injection

In terms of energy dissipation, the PBC approach

may be viewed as an extension of the so-called

energy-shaping plus damping injection technique.

An energy shaping stage consists of modifying the

potential energy of the system in such a way that the

“new” potential energy function has a global and

unique minimum at the desired equilibrium.

Damping injection stage consists of modifying the

dissipation properties of the system, to render it

strictly passive.

The energy shaping stage accomplishes a

passivation objective with a desired storage function

that consists of the original kinetic energy and the

new desired potential energy. The damping injection

reinforces this property to output strict passivity.

4.3 Passivity Based Control of CSI



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The basic idea behind the PBC is to shape the

closed-loop energy function to a desired energy

function. After energy shaping, a damping injection

is needed to achieve an asymptotic convergence.

Let the error variables of the CSI system be

defined as,

Ds iiy 1

(4)

Do vvy 2

(5)

where iD and )sin( tVv mD are the internally

generated references for is and vo respectively.

The error dynamics including the damping terms,

can be written as

11

.

11211

.

ydviRiKVydyRyyK DDDs

(6)

22

.

22212

. 11yv

RidvDyy

RdyyD D

L

DD

L

(7)

where 1 > 0 and 2 > 0 are constants providing a

suitable damping injection and “ ˙ ”. If the right-

hand sides of equations (6) and (7) are set to zero,

we can obtain,

11 ydvRiKiV DDDs

(8)

22

.

0 yR

vdivD

L

DDD

(9)

11211 ydyRyyK

(10)

.

222

12

.

yR

ydyyD

L

(11)

The total energy of the system and its derivative are

given by,

2

2

2

12

1

2

1)( DyKytG

(12)

.

22

.

11

.

)( yDyxKytH

(13)

Substituting (10) and (11) into (13) and simplifying

gives,

2

2

1

2

2

1

2

11

.

)()( yRRyytG L

(14)

Since it is negative for all values of y1 and y2, then it

can be concluded that the error dynamics are

asymptotically stable. The controller can be obtained

from equation (9) by solving for d as

))((

1)(

2

1.~

DoLDDD vvRvvDi

td

(15)

Where ~

D is the estimated value of D.

The implementation of (3) requires the estimation

of Is sin(2ωt + θ) which increases the complexity of

the controller. In addition, in the case of unknown

load, the knowledge of RL is needed not only in (3)

but also for the computation of isd. Nevertheless, the

adaptive approach can be used to estimate 1/RL,

provided that its upper and lower bounds are known

[2]. However, since the value of L is directly

proportional to the estimation accuracy and

inversely proportional to the dynamic response of

the proposed controller, a compromise between

dynamic response and correct estimation for isd is

essential when L is being chosen. A remedy to this



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problem is to ignore the harmonic-ripple component

and hence,

222

222

L

mss

DsRR

V

R

V

R

VIi

(16)

Therefore, once the estimate of 1/RL is obtained,

isd can be computed easily. The division by iD in (15)

can easily be accomplished by modulating the

amplitude of the carrier with iD.

6. SIMULATION RESULTS

In this section, the simulink diagram of current

source inverter and the obtained results are

presented. The effect of passivity based control on

current source inverter is clearly shown. The

response of current source inverter with and without

controller is shown. The existing control methods

and the passivity based control methods [19]-[22]

are compared.

6.1 Simulink diagram of current source inverter

The Simulink model of current source inverter has

been developed and it is shown in Fig.4. It is

simulated to obtain the response of current source

inverter and the response is shown in Fig.5. Then

passivity based control is implemented in the circuit

and it is shown in Fig.6.and the response obtained is

shown in Fig.7. It can be seen that the controller

makes the system output to be maintained at the

desired reference value. The controller switching

response is shown in Fig.8. The load current

waveform is shown in Fig.9.



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Fig 4. Simulink diagram of CSI

Fig.5. Output Voltage of CSI without controller

The non-linear response of CSI circuit is controlled

by employing Passivity based controller. The output

voltage is controlled and regulated at a constant

level by switching the input voltage accordingly

based on the variations in the output voltage. The

switching function maintains the input voltage and

output current in such a way that the output voltage

is maintained at the desired reference value. Thus it

is observed that the passivity based control provides

faster dynamic response and maintains the output

voltage at the required reference value.



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Fig.6. Simulink diagram of passivity based controlled CSI



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Fig.7. Output Voltage of CSI with controller



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Fig.8. Switching function



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Fig.9. Load current

Thus this method in comparison with the existing

methods has more advantages which are discussed

as follows: In adaptive digital and sliding mode

control methods, discrete control is employed but

the passivity based control employs continuous

control technique. The parameters sensitivity in this

technique is negligibly small and the dynamic

response of the system is found to be good.

Realization complexity is less and the load

regulation of the inverter is better than the discrete

control technique.

7. CONCLUSION

A PBC strategy has been proposed for single-phase

PWM CSIs. In this strategy, it is required to estimate

the load resistance and the inductor-current

reference. However, the estimation accuracy

depends on the inductance value when the load

resistance and the inductor-current reference are

estimated using an existing adaptive algorithm. It is

shown that the inductor current reference estimation

is possible at the expense of a constant reference

function for the inductor current. Moreover, the

design of inductor and the sensitivity of the output

voltage to the value of output capacitor used in the

controller are studied and it is shown that this

method regulates the output voltage efficiently.

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