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2009 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2009), October 4-6, 2009, Kuala Lumpur, Malaysia

Dynamic Evolution Controller for Single Phase

Inverter Application

Ahmad Saudi SamosirDepartment of Electrical Engineering

Lampung University

Bandar Lampung, [email protected], [email protected]

Abdul Halim Mohd YatimDepartment of Energy Conversion

Universiti Teknologi Malaysia

Johor Bahru, [email protected], [email protected]

Abstract — This paper proposes a new control method based

on Dynamic Evolution Control for single phase inverter. A

new approach for synthesis of inverter controller based on

dynamic evolution control theory presented. In order to

synthesize the converter controllers, this method uses a

simple analysis of non-linear equation models of the

converter. Synthesis of single phase inverter controller

based on the proposed control method is discussed in detail.

Performance of the proposed dynamic evolution control is

verifying through simulation under step load condition.

Simulation results show that the proposed techniques are

suitable for controlling power inverter.

Keywords — Dynamic evolution control; ew control

method; Single phase inverter; Inverter control.

I. I NTRODUCTION

Power Inverter is the most crucial component to manydc to ac conversion equipment such as the uninterruptible power supply (UPS), induction motor drive and inductionheating. The main feature of a well-designed inverter is itsability to provide clean and stable ac voltage regardless ofa type of load connected to it. It must also have the abilityto recover from transients caused by external disturbancesas quickly as possible [1].

During the past several years, various instantaneousfeedback controls on a linear model have been proposed[1-4]. In designing of linear model control theory, e.g. PIand PID controller, the small signal linear approximationshave applied to the non-linear system. This approachenables the designer to use a simple linear controller to

keep the system stable. However, the main disadvantageof PI and PID controller are only applicable to near aspecified operating point.

For the past few decades, many controller techniqueshave been proposed for power inverter applications.Since power inverters are non-linear time-varyingsystems, so, the design of controllers must have capabilityto cover up the non-linearity and time-varying propertiesof the inverter system.

The power inverter operated with parameter variation,non-linearity, load disturbance, etc. Therefore, the needfor good controller design to perform tight regulationunder large unpredictable load variation is increased.

In this paper, a new approach for synthesis of convertercontroller based on dynamic evolution control theory is

presented. The dynamic evolution control exploits thenon-linearity and time-varying properties of the system tomake it a superior controller. This control method tries toovercome the mentioned problem of linear control byexplicitly using the dynamic equation model of theconverter for control synthesis.

The dynamic evolution control is expected to obtainseveral advantages such as zero steady state error, widerange stability, and robust. The synthesis process is simpleand requires a quite low bandwidth, simple calculation,and it is easy to be realized in digital. So, this method issuitable for digital control implementation. Moreover, thedynamic evolution control is operated at constantswitching frequency. Therefore, the complexity of powerfiltering problems is reduced.

II.

DYNAMIC EVOLUTION CONTROL[7,8,9] A. Basic Idea

Feedback control refers to an operation that, in the presence of disturbances, tends to reduce the difference between the output of a system and the reference inputand that it does so on the basis of the difference. [11]

In order to generalize this definition, we can assumethat: “The difference between the output system and thereference input, which is denoted by error state, must bereduced to zero all the time, regardless, whether thedisturbance is present or not.” This assumption is thefundamental based of Dynamic Evolution Control.

The basic idea of the dynamic evolution control is to

reduce the error state by forcing the error state to followthe specific path, that ensure the error state goes to zero inincrease of time.

B. Evolution Path and Dynamic Evolution Function

The objective of the dynamic evolution controller is tocontrol the dynamic characteristic of the system to operateon the target equation, Y = 0. In dynamic evolutioncontroller, the dynamic characteristic of the convertersystem is forced to make evolution by following anevolution path. The selected evolution path is anexponential function as shown in Fig. 1. The equation ofthis exponential function can be written as:

mt Y Ce−= (1)

978-1-4244-4683-4/09/$25.00 ©2009 IEEE 530

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Figure 1. Dynamic evolution path

Figure 2. PWM Signal Generator

where C is the initial value of Y , and m is proportional tothe initial decrease rate of Y .

In this exponential function, the value of Y is decreaseexponentially to zero as a function of time. The decreasespeed of Y is proportional to the decrease rate m.

Let Y represent the state error function of the converter.Then, Y is forced to follow the evolution path as show infig.1, so the dynamic evolution of the state error function(Y ), with initial value Y O, can be represented as:

. mt

OY Y e−

= (2)

It means that the state error function (Y ) is drivendecrease to zero exponentially with decrease rate m.

From (2), the derivative of Y is given as:

. . mt

O

dY mY e

dt

−

= −

.dY

mY dt

= −

As a result, the dynamic evolution function can bewritten as equation (3).

0; 0dY

mY m

dt

+ = > (3)

where, m is a design parameter specifying the rate ofevolution.

C. Synthesis Process

The main objective of the synthesis process is to obtainthe control law that guarantees the state error function (Y)of converter decrease to zero by following the evolution

path. For dc-dc power converter, this control lawrepresents the duty cycle equation of the converter. Thisduty cycle equation α(VO,Vg,iL), represents α as a functionof the state VO, Vg, and iL.

In order to obtain the duty cycle equation α(VO,Vg,iL),

the dynamic equation of the converter system is analyzedand substituted into the dynamic evolution function (3).

The duty cycle equation is used to calculate the desiredvalue of signal level control Vcontrol. The pwm signal isgenerated by comparing a signal level control Vcontrol witha repetitive waveform as shown in Fig. 2. The frequencyof the repetitive waveform with a constant peak, which isa sawtooth Vst, establishes the switching frequency. Thisfrequency is kept constant. Therefore, the dynamicevolution control is operated at constant switchingfrequency.

III. SYNTHESIS OF THE POWER I NVERTER

CONTROLLER

Many applications of power electronic used powerinverter as their basic topology. Power inverter is used toconvert a DC input voltage to an AC output voltage. Inthis paper, a single phase power inverter is controlled byusing the proposed control method to produce a sinusoidaloutput voltage. Single phase inverter scheme is depictedas Fig. 3.

As already mentioned, to synthesize the control law ofthe dynamic evolution controller, the dynamic equation of

the converter system must be analyzed and substitutedinto the dynamic evolution function (3).

The operation condition of inverter can be divided totwo conditions as follows:

First: When S1, S2 on and S3, S4 off. The voltageequation of inverter can write written as:

Ldc O

diV L V

dt = + (4)

Second: When S3, S4 on and S1, S2 off. The voltageequation of inverter can write written as:

Ldc O

diV L V

dt − = + (5)

Let say the duration time of first condition is tON andduration time of second condition is tOFF. Thus, the volt-time equations on the first and second condition are

( ) Ldc O O O

diV t L V t

dt ⋅ = + ⋅ (6)

( )

L

dc OFF O OFF

di

V t L V t dt − ⋅ = + ⋅

(7)

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Figure 3. Single Phase Inverter scheme

By summing (6) and (7), and divided the result with theswitching period, thus the dynamic equation of single

phase inverter is obtained as follows:

(2 1) ; 0< <1 Ldc O

diV L V

dt α α ⋅ − = + (8)

Where Vdc is the dc input voltage, α = tON/T is dutycycle, iL is inductor current, VO is output voltage, L is

inductor inductance and T is switching period.

Rearranging (8), the output voltage of inverter can bewritten as:

(2 1) LO dc

diV V L

dt α = − − (9)

The dynamic evolution synthesis of the controller begins by defining the state error function (Y ) as follows

. err Y k V = (10)

Where k is a positive coefficient and Verr is error

voltage err ref OV V V = − .

The derivative of Y is given by:

. err dV dY k

dt dt = (11)

Substitution (10) and (11) into (3), yields

. . . 0err err

dV k m k V

dt + =

. ( . 1)err err ref O

dV k m k V V V

dt + − + = (12)

Directly substituting the converter voltage output VO from (9) into (12) we can get:

. ( . 1) (2 1)err Lerr ref dc

dV dik m k V V V L

dt dt

α + − + = − −

(13)

Solving for duty cycle α, the obtained control law isgiven by:

. ( . 1)

2.

err Lerr ref dc

dc

dV dik m k V L V V

dt dt

V

α

+ − + + +

= (14)

The expression for duty cycle α is the control action forthe converter controller.

Duty cycle (14) forces the state error function (Y) tosatisfy the dynamic evolution function (3). Consequently,the state error function (Y ) is forced to make evolution byfollowing (2) and decrease to zero (Y = 0) with a decreaserate m. so, the state error function (Y ) satisfy the equation

. 0err Y k V = =

Thus the state error of the converter will converge tozero.

0err V = (15)

Substituting err ref OV V V = − into (15), we can see

that the voltage output of converter converges to theconverters steady state:

O ref V V = (16)

From the synthesis procedure, it is clear that thedynamic evolution controller works on the full non-linear

system and does not need any linearization orsimplification on the system model at all as is necessaryfor application of traditional control theory.

Rearranging the control law equation (14), the controllaw can be written as:

( . 1)

2. 2. 2. 2.

err L

ref dc err

dc dc dc dc

dV dik LV V m k V dt dt

V V V V α

+ −= + + + (17)

It is interesting to note that the control law in (17)consists of four distinct parts. The first part is thefeedforward term, (V

ref +V

dc)/2V

dc, which is calculated

based on the duty cycle at the previous sampling instant.This term compensates for variations in the input voltages.The second and third terms consist of proportional andderivative terms of the perturbations in the output voltagerespectively. The last term consists of the derivative termsof the inductor current.

From (17), it is also seen that the input voltage, outputvoltage and inductor current are involved in controloutput. The advantage is the dynamic evolution controlcan compensate all of variation in the input and outputvoltages also the change of inductor current. It contributesto the better dynamic performance of the controlledsystem.

The dynamic evolution control law theoretically doesnot require precise knowledge of the model parameters.The required parameter is the inductor inductance (L).

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Figure 4. Single phase inverter Simulation Model

This requirement places a small limitation on thecontrol system for a reason: sometimes it is difficult to getthe fixed value of L. This limitation can be solved by usethe nominal value of inductor inductance (L).

IV. SIMULATION R ESULTS

A comprehensive simulation analysis was conducted toverify the controller performance. The performance ofdynamic evolution control under step load variationcondition is tested through simulation by using MATLABSIMULINK. The single phase inverter scheme and thedynamic evolution control law, which is described by(14), are modeled in Simulink as shown in Fig. 4. Themodel parameters are listed in Table I. The control goal isto produce 240V, 50 Hz sinusoidal output voltage. Thereference of the output voltage is specified based on thedesired output voltage, which means that

240 2 sinref V t ω = (18)

where 100ω π = .

These reference values are not varied during theoperation. The selected parameters of the controller arek = 0.01 and m = 5000. The simulation waveform of

output voltage and load current are shown in Fig. 5 – 10.The simulation waveform shown the output voltage is notdisturbed by the step load variation. The controllersaccomplish to regulate the converter output voltage keepon steady-state at 240V, 50Hz reference. This conditionindicates that the state error of the converter goesconverge to zero.

A. Steady-state Performance

Figure 5 and 6 show the result for steady-state performance of the proposed dynamic evolution control

for no-load and full-load condition. Full load condition isachieved with a resistor load of 10 ohm. The results give asatisfactory performance which indicates that the

proposed dynamic evolution control is capable to avoidvoltage output level from dropping when a load isconnected.

B. Small-signal PerturbationPerformance

To show the effectiveness of the proposed dynamicevolution control in handling small-signal disturbance, asudden small load disturbance (from 20% to 50% load) isimposed to the system. Fig. 7 shows the results for

proposed controller performance. Fig. 8 shows the zoomof inverter output voltage under sudden small load

disturbance condition. The controller accomplishes tomaintain output voltage close to reference voltage.

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C. Large-signal disturbance PerformanceThe superiority of the proposed controller can be

judged by evaluating its performance under large signaldisturbance. Fig. 9 shows the performance of the

proposed dynamic evolution control when a large loaddisturbance (from no-load to full load) is imposed. Fig. 10shows the zoom of inverter output voltage under suddenlarge load disturbance condition. From the figures, it can

be observed that the proposed controller exhibits goodtransient performance: fast rising time and settling time(0.6 ms).

CONCLUSION

This paper presents a dynamic evolution control for single phase power inverter. The performance of dynamicevolution control under small and large load disturbancecondition has been tested.

From Simulation that have been done, the dynamicevolution control shown many advantages such as fastresponse and good transient performance. The controlleraccomplishes to regulate the converter output voltagekeep on steady-state at 240 V, 50 Hz reference.

ACKNOWLEDGMENT

The authors would like to thank the Ministry ofScience, Technology and Innovation (MOSTI) of the

Malaysian government for providing the funding for thisresearch.

Figure 10. Zoom of inverter output voltage under Sudden Large

Load Disturbance condition

Figure 9. Performance of inverter output voltage under SuddenLarge Load Disturbance condition

Figure 8. Zoom of inverter output voltage under Sudden SmallLoad Disturbance condition

Figure 5. Steady-state performance in no-load condition

Figure 7. Performance of proposed controller when subjected to

small load disturbance

Figure 6. Steady-state performance in full-load condition

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R EFERENCES

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