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Energy Procedia 18 (2012) 458 - 467

Control of Energy by the Technical Hysteresis fixed band

Y. Soufi a, S. Lekhchine b, T. Bahi b, A. Dekhane b, S. Ghoudelbourkc

a Department of Electrical Engineering, University Tebessa, Algeria b Department of Electrical Engineering, University of Annaba, Algeria cDepartment of Electrical Engineering, University of Skikda, Algeria

Abstract

Since its discovery, electrical energy represents one of the most decisive fields in any technology. In this paper, a comparative study between fixed hysteresis band current control (HBCC) and adaptive hysteresis band current control techniques in speed drive system is presented. These techniques control the variation of output voltage of the three phase rectifier. The theory and analysis of both the conventional fixed-HBCC and the adaptive HBCC which are the control techniques that operate at any condition with variable and constant switching frequency, are respectively developed. At the end, the performance of such control technique has been observed and discussed thanks to a simulation study carried with Matlab Simulink environnement.

© 2012 Published by Elsevier Ltd. Selection and/or peer review under responsibility ofThe TerraGreen Society. Keywords- Hysteresis current control, three-phase PWM rectifier, adaptive HBCC, switching frequency.

1. Introduction

The increasing consumption of electricity requires the use of continuous AC / DC converters and in particular the three-phase rectifier based on diodes to power some equipment in order to encourage the significant development of semiconductor power components that gives a large potential for the conversion of electrical energy. The control of electrical motors by static converters shows their speeds and improves their performances. So, the speed drive systems draw the attention of many researchers and

Y. Soufi. Tel.:00-213-696-392-576; fax: 00-213-553-429-898.

E-mail addres: y_soufi@yahoo.fr

1876-6102 © 2012 Published by Elsevier Ltd. Selection and/or peer review under responsibility of The TerraGreen Society. doi: 10. 1016/j .egypro .2012.05.057

industrials. In these systems, the three phase induction motors (IM) operating at variable speed can be powered by AC/DC/AC converters [1][2] [3]. For such dual power supplies, diode rectifiers( AC/DC) are widely employed in industrial fields and consumer products thanks to advantages of low cost, simple structure, robustness and absence of control. However, this type of converters results in only unidirectional power flow, low input power factor, high level of harmonic input currents, malfunction of sensitive electronic equipment and increased losses. It also contributes to inefficient use of electric energy [4]. Recently, many promising power factor correction techniques [5] [6] and various control techniques [7] [8] have been proposed for rectifiers. It is well known that in order to obtain better AC supply power quality and high performance of these converters, it is preferable to directly control the magnitude and phase angle of three phase supply currents. Among the control techniques, the hysteresis current control has retained much attention due to several advantages such as: the simplicity of its implementation, its independence to changes in parameters of the load and the power supply and ability on current limiting and fast current control response [9]. However, the control current by a fixed hysteresis band suffers from a major problem which lies in the absence of control of the switching frequency which shows high variation particularly in the case of three-phase isolated neutral systems where interactions between the phases occur during operation, This makes filtering out difficult [10]. So to improve the characteristics of the hysteresis current control to overcome these drawbacks, many researchers have been interested in developing efficient methods [4] [8]. The aim of this works is that the DC bus voltage must have a good adjustment to its desired value. The use of such converters has good industrial and economic results since its performance has improved significantly and the cost of operating the equipment is clearly reduced, which explains the wide use of these converters.

In this context, mathematical development of control technique operating at constant switching frequency, are studied. In this paper, we describe a control technique which leads to improvements in the quality of the waveforms of currents controlled with a variety of power converters. To this end, the problems of control technology for conventional hysteresis are discussed. In addition, improving the performance of the system obtained by setting the switching frequency is verified by numerical simulation. This method has the properties of robustness and simplicity of implementation as it has the speed control hysteresis and adds a limitation of variations on the switching frequency and greater accuracy. So, better benefits.

2. Modeling of studied system

In industry, the variable electric drives are widely used. Their development has been characterized primarily through the development of configurations and control techniques of power static converters.

Figure 1 shows the structure of such a system. There are basically two conversions (a rectifier and voltage inverter) between the power source and induction machine.

Fig. 1. Voltage fed PWM inverter system

The requirements of such a structure known as cascade, maintaining the output voltage of the rectifier has its required value. First, in the next section we present the modeling to implement a hysteresis control for adjusting the output voltage of the rectifier.

2.1. Rectifier

Unlike conventional rectifiers, PWM rectifiers are made of opening and closing controlled semiconductor. The structure of three-phase PWM rectifier is shown in Fig. 2. The rectifier is composed of a three-phase converter with six IGBT or Mosfet elements. The opening order possibility allows full control of the converter, because the switches can be switched as required, as well as closing the opening [11]. The main objective of this structure is to generate three-phase sinusoidal input currents in the input phase voltages. The voltage rectifier works by keeping the DC bus voltage to a desired reference, using a closed loop control where a proportional-integral PI controller is used to regulate the DC output voltage and provide the input of this controller is the error between the square of the reference DC voltage and the DC bus capacitors voltage, the output is considered as the magnitude of the desired supply currents and the reference currents are estimated by multiplying this magnitude with the unit vectors in phase with the AC main source voltages provided by the phase-locked-loop (PLL). After having found the reference currents, the supply current is sensed and compared with the sinusoidal reference current for each phase. The comparison result is introduced in hysteresis comparator to generate the PWM signal of the rectifier.

Fig.2. Control block diagram of three-phase PWM rectifier.

The rectifier is modelled by a set of ideal switches i.e., no resistance to the state, infinite resistance to blocking state. The switch in the same arms are complementary, their condition is defined by the following function [12] [13].

1 +1, = -1

Si = \ ~ j Y i Sj=+1

Where: j=a,b and c

Then the input phase voltages and the output currents can be written in function of Sj, VDC and input currents whose sum is zero for balanced three-phase system without connecting the neutral (ia + ib + ic = 0).The input voltage between phases of the PWM rectifier can be described by:

UU sab = ( Sa - Sb )Vc

USbC II - Sc )Vc

UU sca = ( Sc - Sa )Vdc

The voltage equations are the following:

~ ea ' ~ia ' t d + L.— dt ~ia ' ~ Usa _

eb = R. ib ib + Usb

_ec _ _ic _ _ic _ Uc _

Where:

sa - V ' DC

U sb 3

2 -1 -1 -1 2 -1 -1 -1 2

Finally, we deduce the equation of coupling between the input side and the alternative output continues

CVDC = (S a ia + Sb ib + Sc ic ) - 11

Where I is the load current.

2.2. Inverter\Jnduction machine

Fig. 3 shows a three-phase PWM inverter feeding an induction machine. Where each phase of the machine is represented by an electromotive force (ei) against in series with an inductance (Lm) and resistance (Rm).

Fig.3. Voltage inverter fed induction machine

The equations of stator and rotor voltages of the IM can be written as following matrix form [14]:

Vj ]=k Vj hi

Where, j =s and j=r respectively for the stator and rotor .

[vj ]=h Vjb Vjc ]t; [lj ]=[lja j j J [Rmj \ (T) is the index exponent that indicates the transpose.

[LAlj ]+[Lr ][ijJ

RmJa 0 0

0 Rmb 0

0 0 RmJc

rJl L JJJL JJ L srJL J

The mutual inductance is written:

Lm sraa Lm srab Lm srac

Lm srba Lm srb Lmsrbc

Lm srca Lm srcb Lmsrcc

Lmsr : Mutual inductance between stator phase i and rotor phase j. [Lmrs ]=[Lmsr ]T

The dynamic equation is described by the following law:

Q = T - T - fn

em l J

Tem : electromagnetic torque; f : friction coefficient; a : rotation speed; T : load torque; J : inertia moment.

Since AC motors are normally connected in a delta configuration or with an insulated neutral, the previous analysis must be extended to unity vector. Considering the three phase IM with an insulated neutral motor load as show in figure 2. Where the neutral point of winding is no longer connected to supply neutral and the electromotive force voltage can mutually interact. Now, expression (6) will become:

u = Li+ Ri + e + u,I

Where the input voltage of the machine can take the values ±VDC/2 depending on the state of the semiconductor switching of the inverter.

Where I and uo are respectively the unity vector and the motor neutral voltage referred to the supply neutral. they are defined as:

1 = [1 1 1]T (12)

Since the motor neutral is insulate, the instantaneous sum of the three currents is ( ia+ ib+ic =0) and the instantaneous sum of the three back Electromotive force is (ea + eb+ec=0); the expression of uo is:

Uo = (ua + ub + uc )/3

3. Hysteresis current control

The technique of the hysteresis band switch allows switching of the rectifier, where the error between the signal exceeds a set point and amplitude determined by the hysteresis band. This technique requires only a comparator with hysteresis in phase.

The hysteresis comparator operates on the principle explained in Figure 6, the switch opens if the error falls below-H / 2 and it closes behind if that is greater than + H / 2, where H represents the range (or larger) of hysteresis. If the error is now between -l>V 2 and +l>V 2 (in the band), the switch does not switch [15].

When the midpoint HDD of dc capacitors is connected to ground, the PWM rectifier is symmetrical, and each phase can be regarded as independent. Referring to the topology, the relationship for phase a voltage and current can be formulated as:

u = Li+ Ri+e (14)

If i* is the load reference current, consequently, the phase reference voltages may be defined as:

* T * Tk*

u =Li +Ri +e (15)

Where u* the reference voltage that should be applied to obtain the current i*. Now, the instantaneous current errors are the deviations between the reference and actual current, and his defined as:

< =1 -l (16) Subtracting (11) from (14) and substituting from (16) gives:

L<+R<=u-u

The instantaneous output phase voltage take ± VDC / 2 with duration Ton of the positive pulse and Toff of the negative pulse for a total switching period T. Usually, for a reasonably high switching frequency, the effects of load resistances R can be neglected [2][16] , and the term u-u* can be considered constant during a modulation period. So, it results that the current error has a triangular behaviour:

L < = u - u

Fig. 4 illustrates the operation of HBCC technique over one switching cycle.

Fig. 4. Dperation of HBCC

For the interval 0 < t <To,

- u* = L A = L

¿(Ton) -¿(0)

Ton - 0

For the interval Ton < t <T

^ - u* = L A£ = L

¿(T) -¿(Tc

From (19) and (20), the switching period expression is given by:

DC - u *2

The constant o" is the band of accepted error between the maximum positive current error and the minimum negative current error. From (21), we note that if the bandwith o" is constant, a variable modulation frequency is produced et par consequent , the switching period (T ) is variable..

To obtain a constant switching frequency, the hysteresis band o" has to be dynamically modified according to this equation [17].

ß = 4fr (1 - 2u'/Vdc )

Where fd is the desired switching frequency control.

The controller maintains its analog structure, but an adaptive bandwidth digital control is added which ensures constant switching frequency. Fig .5 shows the adaptation of the hysteresis band for two consecutive modulation periods [6][17][18][19].

Fig. 5. Principle of hysteresis adaptive band

From Fig.5, we deduce the following equations:

A = S+Ton = S ~Toff

on oJJ (93)

T = Ton + TOff

Where S+ and S- are the positive and negative slopes of the non interacting error ¿"in modulation period. For i switching period corresponding to k, we have:

For i switching period corresponding to k +1, the equation (24) is written:

T (k +') = A (k) (25)

For two consecutive switching periods, we have the following simplifying:

iS *(k) = S *(k +1) (26)

[S- (k) = S - (k +1)

From (24), (25) and (26), we can derive the control equation:

A(k +1) = A (k)

T(k) (27)

The principle of control given by equation (27) is to keep the switching period constant, where T(k + 1) is the desired switching period which we want to impose and T(k) is the measured one, i.e. to define the switching period T(k + 1) we predict at time k the bandwidth ui(k+1) and hence this reasoning is intended to evaluate the proper bandwidth to achieve the error between the real switching period and the desired switching period equal to zero. This reasoning leads to an algorithm which is equivalent to a first order dead-beat control of the switching period.

The control algorithm is very simple and it is able to ensure a good switching frequency regulation with any knowledge on the system parameters, but it cannot control the position of modulation pulses. This means that the distribution of the modulation pulses inside the modulation period is random.

4. Simulation results and discussions

To verify the performances of fixed f hysteresis -band current control (HBCC) and adaptive hysteresis-band current control techniques, the simulation results are shown in Fig. 5 and 6.

Fig. 5a shows the voltage obtained at the output of the rectifier and its desired value. Fig 5b and 5c respectively show the waveforms of the input current of the rectifier and the load current of phase an for fixed-band hysteresis current control. These two currents have oscillations and deformations causing non-negligible distortion and consequently significant losses. This is a consequence of the control technology used. In fact, it uses the variation of the error between the actual current and reference in a band in prefixed order. Therefore, the result of the simulation of the variation of the error is shown in Figure 5d. We

note the same quantities but for the adaptive hysteresis-band control, current control techniques are shown in Fig. 6. We notice a good set of output voltage to its set value (see Fig 6a.). The 6b and Fig 6c shows waveforms of the input current of the rectifier and the load of phase QD In this case the supply current ripples are minimized. Consequently, low levels of distortion and therefore low losses. Finally, Figure 6c shows the variation of the error within the band variable that is a function of constant frequency

1 T I .Real DC voltage

.Reference DC voltage

0.41 0.4! 0.43 0.« 0.45 0.46 0.47

fme(s)

0.4 0.4! 0.42 0.43 0.44 0.45 0.46 0.41

030 0.4 0.41 0.43 0.43 0.44 0.45 0.46 0.4Ï

0.M3 0.044 0.046 d.m

Fig. 5 Results de simulation with fixed f hysteresis band

— —1

I -I ? - i

LJ .Real DC voltage

\ Reference DC

0.45 046

i :... ~dpr\

036 6.38

Fig. 6. Simulation results with adaptive hysteresis-band

5. Conclusion

This paper has presented the advantages and the control strategies based on fixed hysteresis band current control and adaptive hysteresis band current control techniques of the three phase voltage source PWM rectifier. The comparison of the performances of voltage regulating in the output of the rectifier in speed drive system is discussed and has convincing results which requires adjustment as such. The adaptive hysteresis band current control techniques ensure a good regulation of switching frequency. The simulation results show a good performance of the studied technique, providing a good regulation of output DC voltage and a sinusoidal input AC current to come closer to a sinusoid.

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