Chapter 6 - Shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/44987/13/13...Chapter 6. Control of Shunt Active Filter 116 6.3 Synchronous Reference Frame Theory (dqtheory) The basic

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Chapter 6

6. Control of Shunt Active Filter

6.1 Introduction

The SAF generates a current equal and opposite to the load current

harmonics and reactive power component at fundamental frequency to

achieve balanced, sinusoidal supply current which is in phase with the

voltage at PCC. The critical task of the SAF controller is to generate reference

current. The accuracy of the SAF current reference determines performance

of SAF. Reference current generation methods utilized in SAF applications

can summarized in two groups as frequency domain methods and time

domain methods. Frequency domain methods are based on Fourier analysis.

The time domain methods are more preferred over the former due to

less number of calculations and fast response during transients. There exist

two well known time domain methods of harmonic extraction in SAF appli-

cation based on: 1) Instantaneous reactive power (IRP) theory 2) Synchronous

reference frame theory (SRF), briefly introduced in Chapter 1. A review of the

two techniques is presented in next section. A method to extract SAF refer-

ence current based on SSI (Sine Signal Integrator) for single phase SAF

application is presented in this chapter. This technique is particularly suited

for single phase applications as it simplifies the task of generating orthogonal

signal which is a main hurdle in extracting reference current for single phase

SAF using IRP and SRF techniques.

Chapter 6. Control of Shunt Active Filter

114

The current controller controls switching of the VSI switches so that the

current injected by the VSI is exactly equal to the reference current. A review

of different current control techniques and M-PWM (modified PWM) current

control technique is also explained.

In spite of compensation of harmonic currents of the load, SAF leaves

switching frequency ripple in the supply current if not filtered. This requires

a switching ripple filter at the input of SAF. Design of a broad-band tuned

type switching ripple filter is presented at the end.

6.2 Instantaneous Reactive Power Theory (p-q theory)

The p-q theory proposed by Akagi et. al. [H1] deals with instantaneous

powers. It is based on abc to 0 transformation to calculate instantaneous

powers defined in the time domain. As it is instantaneous, it assumes no

restrictions on the voltage and current waveforms. It is not only valid in

steady state but also in transient state. It can be applied to three phase sys-

tems with or without neutral wire. p-q theory has made it possible to design

very efficient and flexible controllers for power quality conditioners based on

power electronic devices. Fig. 6.1 illustrates basic idea of three phase shunt

current compensation. It is assumed that the compensator behaves like a

controlled current source that can draw arbitrary chosen current references*cai , *

cbi , *cci . Fig. 6.2 shows a general controller structure[H1,J5]. The calculated

real and imaginary powers p, q of the load can be separated into its average (

,p q ) and oscillating ( ,p q ) parts respectively. Then the undesired portions of

real and imaginary powers of the load that should be compensated are

selected and compensating powers * *,c cp q are derived. The negative sign

implies compensating current being exact inverse of the corresponding

undesired component of the load current. The inverse transformation from

0 to abc is applied to calculate instantaneous value of the compensating

current references *cai , *

cbi , *cci .


115

Figure 6-2 Control method for three phase shunt compensation based on p-q theory

The merits and drawbacks of p-q theory are reported as [H2] as: 1) easy

implementation 2) excellent steady state performance and the drawback

being sensitivity to harmonics and unbalances in the supply voltage [H17].

Figure 6-1: Basic shunt compensation principle


116

6.3 Synchronous Reference Frame Theory (dq theory)

The basic principle of synchronous reference frame controller [H2] for

the current reference generation in SAF is illustrated in Fig. 6.4. The meas-

ured load currents in abc reference frame are first transformed into frame

via abc- transformation and subsequently to synchronous reference frame

(dq) reference frame via to dq transformation. The to dq transformation

utilizes phase angle information ( t ) of the utility voltage for the transfor-

mation of the quantities to synchronous rotating frame. The and dq

frames are illustrated in Fig. 6.3. In synchronous rotating frame, once the

transformation is carried out, the positive sequence components at funda-

mental frequency appear as dc quantities whereas negative sequence

components at and other frequency components (harmonics) appear as ac.

By employing proper filtering technique in the dq reference frame, the dc and

ac quantities can be easily separated. Once the filtering in the dq reference is

carried out, the desired dq reference signals are obtained and transformed

back to . The three phase current references can be obtained via to dq

transformation.

One important characteristics of the SRF theory is that the reference

currents are directly obtained from the load current without considering the

source voltage. This is an important advantage since the generation of refer-

ence current is not affected by supply voltage unbalance or distortion.

However, in order to transform to dq synchronous rotating frame, sine

and cosine signals synchronized with the respective phase-to-neutral supply

voltages are required. A phase locked loop (PLL) must be used to derive sine

and cosine signals corresponding to the supply voltage. A novel PLL and its

performance are also presented later in this chapter.


117

6.4 Single Phase IRP and SRF theory

The IRP (pq) and SRF (dq) theories are basically developed for three

phase systems. Due to their effectiveness, they can be also applied to single

phase systems. In three phase systems, three phase to two phase conversion

is carried out by using Clarke’s transformation. In single phase system, as

only one variable exists, the other fictitious variable has to be generated.

Literature presents different techniques to adapt IRP and SRF theories for

single phase systems [H8]-[H10], [H12]-[H16], [H18]. The basic concepts of

single phase IRP and SRF theories are discussed in the following section.

Figure 6-3: Relationship between space vectors stationary and rotating frames.

Figure 6-4: Control method for three phase shunt compensation based on SRF theory.


118

6.4.1 Single phase IRP theory

The single phase IRP theory is based on an instant2 lag or

2 lead of the

voltage and current [H15] to define the original system as a pseudo two

phase system. The overall system can be represented in frame. The single

phase voltage representation of voltage in can be written as,

( )( ) sin( )( ) cos( )( )

2

m

m

v tv t V tv t V tv t

(6.1)

( )( )( ) ( )

2

i ti ti t i t

(6.2)

The supply voltage in the above analysis is assumed to pure sinusoidal.

Once the single phase voltage and current are transformed into frame,

instantaneous powers can be calculated in the same way as they are for three

phase system.

( ) ( ) ( )( )( ) ( ) ( )( )

v t v t i tp tv t v t i tq t

(6.3)

( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )

v t i t v t i tp tv t i t v t i tq t

(6.4)

The instantaneous powers can be expressed in terms of dc and ac quantities

as,( ) ( ) ( )( ) ( ) ( )

p t p t p tq t q t q t

(6.5)

The instantaneous power ( )p t represents active component of the power

which is supposed to be supplied by the source and the remaining compo-

nents ( )p t , ( )q t , ( )q t are to be supplied by the SAF. In addition to that

SAF may need to absorb some additional active power for supplying its

losses and maintain dc bus voltage. The reference current for the SAF can be

calculated as,


119

*

* 2 2

( ) ( ) ( )( ) 1( ) ( ) ( )( ) ( ) ( )

c v t v t p ti tv t v t q ti t v t v t

(6.6)

The structure of the controller for single phase SAF based on pq theory is

shown in Fig. 6.5.

6.4.1.1 Single phase IRP theory case study

Assume single phase load current consisting of fundamental compo-

nent Im,1 and phase angle . Let the load current consist of 3rd and 5th

harmonic current components with peak values Im,3 and Im,5 respectively. The

supply voltage is assumed to be pure sinusoidal at frequency with peak

value Vm. The representation of voltage and current can be as obtained as

in (6.7) & (6.8).

( ) sin( )( ) cos( )

m

m

v t V tv t V t

(6.7)

,1 ,3 ,5

,1 ,3 ,5

sin( ) sin(3 ) sin(5 )( )( ) sin( ) sin(3( )) sin(5( ))

2 2 2

m m m

m m m

I t I t I ti ti t I t I t I t

(6.8)

,1 ,3 ,5

,1 ,3 ,5

sin( ) sin(3 ) sin(5 )cos( ) cos(3 ) cos(5 )

m m m

m m m

I t I t I tI t I t I t

(6.9)

By substituting in (6.3) we get the instantaneous powers as,

,1 ,3 ,5

,1 ,3 ,5

sin( ) sin( ) sin(3 ) sin(5 )( )

( cos( )) cos( ) cos(3 ) cos(5 )m m m m

m m m m

V t I t I t I tp wt

V t I t I t I t

(6.10)

,1 ,3 ,5

,1 ,3 ,5

( cos( )) sin( ) sin(3 ) sin(5 )( )

sin( ) cos( ) cos(3 ) cos(5 )m m m m

m m m m

V t I t I t I tq wt

V t I t I t I t

(6.11)

Figure 6-5: Control of single phase shunt compensation using IRP (pq) theory.


120

Solving further,

,1 ,3 ,5( ) cos( ) cos(4 ) cos(4 )m m m m m mp wt V I V I t V I t (6.12)

,1 ,3 ,5( ) sin( ) sin(4 ) sin(4 )m m m m m mq wt V I V I t V I t (6.13)

The oscillating active power can be represented as,

,3 ,5( ) cos(4 ) cos(4 )m m m mp wt V I t V I t (6.14)

*,3 ,5( ) 1( cos(4 ) cos(4 ))c m m m mp wt V I t V I t (6.15)

*,1 ,3 ,5( ) 1( sin( ) sin(4 ) sin(4 ))c m m m m m mq wt V I V I t V I t (6.16)

The SAF reference current can be derived from (6.6) above,

,3 ,5*2

,1 ,3 ,5

sin( ) cos(4 ) cos(4 )1( )cos( ) ( sin( ) sin(4 ) sin(4 ))

m m m m m

m m m m m m m m

V t V I t V I ti t

V V t V I V I t V I t

(6.17)

*,1 ,3 ,5( ) 1 sin( )cos( ) sin(3 ) sin(5 )m m mi t I t I t I t (6.18)

The SAF reference current derived from application of IRP theory is such that

the resulting instantaneous active power is constant as seen by the supply

side. This means that the SAF will compensate for the oscillating component

of the load active power, p wt( ) and reactive power component, q wt( ) . As

stated in [H17], the SAF controlled according to the IRP theory, in the pres-

ence of supply voltage harmonics “attempts” to compensate even an ideal,

resistive, unity-power factor load and the compensator injects a distorted

current into the supply system unnecessarily. In the present work, a method

based on SSI is used to extract fundamental component of supply voltage

before calculating instantaneous active and reactive powers as per IRP

theory.

6.4.2 Single phase SRF theory

In order to apply the concept of dq transformation for single phase

systems, the basic requirement is representation of single phase system in

reference frame. As stated previously, it requires a phase shift of2 in the

original variable to generate the component. The resulting equations in dq

reference frame can be stated as,


121

( ) ( )sin( ) cos( )( ) ( )cos( ) sin( )

d

q

i t i tt ti t i tt t

(6.19)

( ) ( )( ) ( )sin( ) ( )cos( )( ) ( )cos( ) ( )sin( )( ) ( )

d dd

q q q

i t i ti t i t t i t ti t i t t i t ti t i t

(6.20)

The dc terms in (6.20), ( )di t and ( )qi t represents fundamental active and

reactive current components of the load and the non-dc terms, ( )di t and

( )qi t represents harmonic components of active and reactive current. The dc

terms from ( )di t and ( )qi t can be easily extracted by using a low pass filter.

The current reference in dq domain is generated by subtracting the dc com-

ponents, ( )di t and ( )qi t from ( )di t and ( )qi t respectively. The structure of

the controller for single phase SAF based on SRF theory is shown in Fig. 6.6

6.4.2.1 Single phase SRF theory case study

The instantaneous load currents in frame are already defined in

(6.8). The dq components of the load current can be calculated by using (6.19)

as,

,1 ,3 ,5

,1 ,3 ,5

sin( ) sin( ) sin(3 ) sin(5 )( )

cos( ) sin( ) sin(3( )) sin(5( ))2 2 2

m m m

dm m m

t I t I t I ti t

t I t I t I t

(6.21)

Figure 6-6: Control of single phase shunt compensation based on dq theory


122

Solving further,

,1 ,3 ,5

,1 ,3 ,5

sin( ) sin( ) sin(3 ) sin(5 )( )

cos( ) cos( ) cos(3 ) cos(5 )m m m

dm m m

t I t I t I ti t

t I t I t I t

(6.22)

Similarly,

,1 ,3 ,5

,1 ,3 ,5

cos( ) sin( ) sin(3 ) sin(5 )( )

sin( ) cos( ) cos(3 ) cos(5 )m m m

qm m m

t I t I t I ti t

t I t I t I t

(6.23)

Applying trigonometric identities,

,1 ,5 ,3( ) cos( ) ( )cos(4 )d m m mi t I I I t (6.24)

,1 ,3 ,5( ) sin( ) ( )sin(4 )q m m mi t I I I t (6.25)

It is evident from (6.25) that the 3rd and 5th harmonics are mapped to 4th

harmonic frequency in dq reference frame. The analytical calculations are

verified by simulation. The simulated waveforms of load current and its

harmonic spectra is shown in Fig. 6.7. It is seen that the load current has 3rd

5th harmonics. The dq currents are shown in Fig. 6.8. The dq currents are

shown in Fig. 6.8a. The harmonic spectra of the dq currents, id and iq is shown

in Fig. 6.8b and 6.8c respectively. It is seen that the 3rd and 5th harmonics of

the load current are mapped to 4th harmonic in dq domain and the fundamen-

tal components are mapped to dc.

Figure 6-7: Load current and harmonic spectra for single phase SRF case


122

Solving further,

,1 ,3 ,5

,1 ,3 ,5

sin( ) sin( ) sin(3 ) sin(5 )( )


dm m m

t I t I t I ti t

t I t I t I t

(6.22)

Similarly,

,1 ,3 ,5

,1 ,3 ,5

cos( ) sin( ) sin(3 ) sin(5 )( )


qm m m

t I t I t I ti t

t I t I t I t

(6.23)















122

Solving further,

,1 ,3 ,5

,1 ,3 ,5

sin( ) sin( ) sin(3 ) sin(5 )( )


dm m m

t I t I t I ti t

t I t I t I t

(6.22)

Similarly,

,1 ,3 ,5

,1 ,3 ,5

cos( ) sin( ) sin(3 ) sin(5 )( )


qm m m

t I t I t I ti t

t I t I t I t

(6.23)















123

6.5 Reference current generation using SSI

In a single-phase system, the use of a rotating frame is not possible

unless a virtual system is coupled to the real one in order to simulate a two-

axis environment. One common requirement of single phase IRP and SRF

reference current generation technique is necessity to create a fictitious

orthogonal or imaginary signal in which all frequency components are phase

shifted through 900 electrical degrees with respect to original variable. Most

common techniques to create such orthogonal component are to use Hilbert

Transformation or to use FIR filter. The drawback of Hilbert transformation

is that it leads to non-causal system and cannot be implemented directly. FIR

filter may cause some phase delays in the orthogonal variable [H16].

(a)

(b) (c)Figure 6-8(a) id, iq waveforms (b) Harmonic spectra of id (c) Harmonic spectra of iq


124

Alternatively, computation of the SAF reference current can be performed

using Sine Signal Integrator (SSI) with less computational burden. This

approach enables delay less orthogonal signal generation. Moreover, refer-

ence current generation is insensitive to grid voltage distortion. The transfer

function of the basic Second Order Generalized Integrator(SOGI) is defined

by (6.26) which correspond to the block diagram depicted in Fig. 6.9. It

presents two poles located at 0s jw and a zero at 0s .

2 20

( )SOGIsG s

s w

(6.26)

Taking into account Laplace transform of a sine and cosine signal are:

sine 2 20

sG (s)s w

(6.27)

0cosine 2 2

0

G (s)s

(6.28)

Figure 6-9: Block diagram of SOGI

Figure 6-10: Bode Diagram of the basic SOGI


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Bode diagram of the continuous time domain transfer function of the SOGI

tuned to 0 2 50rad/sec is shown in Fig. 6.10. Its phase diagram

represents zero phase at 0 and theoretically infinite gain at that frequen-

cy. In rest of the frequencies, the phase takes the values from to2 2 rad.

One of the main characteristics of SOGI is that it presents very narrow band-

width around the resonant frequency, 0 i.e. it is very selective. It also rejects

the dc component, since it has a zero at s=0, thus it shown infinite attenuation

at 0 . The basic SOGI has the disadvantage of having infinite gain at

resonant frequency, which can make the system unstable. To avoid stability

problems associated with the infinite gain and to increase bandwidth, a

modified version of SOGI i.e. SSI can be used as shown in Fig. 6.11. The SSI

presents finite gain at the desired frequencies and adjustable bandwidth

through the gain ka.

Figure 6-11: Block diagram of the SSI

The input signal is xi(t), while the output signal in phase with the input signal

is xo(t). The output quadrature signal is xoq(t). It can be seen that SSI consist of

SOGI plus a gain ka and a negative feedback path from xo(t). The transfer

function for each output correspond to (6.29) and (6.30) respectively and their

bode diagrams are as shown in Fig.6.12. In this diagram, a difference of

phases of c

2is observed at 0 . Besides that, ( )

( )o

i

x sx s

behaves as a band pass


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filter and the bandwidth is determined by value of ka and( )( )

oq

i

x sx s

behaves as a

low pass filter.

1 2 20

( ) 2( )( ) 2

o a

i a

x s k sH sx s s k s

(6.29)

02 2 2

0

( ) 2( )( ) 2

oq a

i a

x s kH sx s s k s

(6.30)

In steady state operation the relationship between the phases of the transfer

functions H s1( )and H s2( ) in frequency domain is,

H s H s1 1( ) ( )2

(6.31)

The inherent capability to produce quadrature component is utilized in

generating instantaneous fundamental reactive power component of the

load.

(a) (b)

Figure 6-12:Bode Diagrams of (a) H s1( )and (b) H s2( )


127

6.6 Reference current generation for direct control of SAF

The block diagram for SAF reference current generation based on SSI

is depicted in Fig. 6.13. There are two SSI blocks to extract fundamental

components of supply voltage and load current respectively. The SSIv block

generates sv 1 and sv 1 components of the supply voltage sv . As the SSIv and

SSII blocks are sharply tuned to fundamental frequency, sv 1 and sv 1 outputs

are insensitive to distortion present in sv .The harmonic component of the

load current L hi*1 is calculated by subtracting component of load current

Li 1 from the actual load current Li . The output of the SSII block is the

fundamental component of the load current (including active and reactive

components). To extract the fundamental reactive current, single phase IRP

theory as discussed in section 6.4.1 has been used. The instantaneous reactive

power reference q* can be calculated by using (6.4). Equating real power to

zero, the fundamental reactive current reference can be calculated using (6.6).

The resulting equation reactive current reference is,

v t v ti tv t v t q ti t v t v t

*

* 2 2

( ) ( ) 0( ) 1( ) ( ) ( )( ) ( ) ( )

(6.32)

In (6.32), the reactive power component is fundamental i.e. q t( ) as the

calculation of q(wt) performed on fundamental terms. The resulting

expression for i t* ( ) becomes

s s L s Lv v t i t v t i ti t

v t v t1 1 1 1 1*

2 2

( ) ( ) ( ) ( )( )

( ) ( )

(6.33)

As the system is single phase, i* component can be neglected and the i*

component acts as the reactive current reference. The necessary fundamental

active current, dci*1 needed to be absorbed by the SAF to maintain dc bus

voltage is generated by a DC voltage controller and added to i* . The resulting

fundamental current i *1 and L hi*

1 are added to generate the SAF reference

current ci* .


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6.7 Reference current generation for indirect control of SAF

In indirect control of SAF, the supply current is sensed and the SAF

switches are controlled in such a manner so that the supply current is sinu-

soidal and in phase with the supply voltage. A This means the SAF current is

indirectly controlled. As the sensed current is sinusoidal at fundamental

frequency, this offers an advantage of selecting a low bandwidth current

transducer. Ideally, the supply should generate only fundamental active

component of the load current. In attempt to maintain sinusoidal and in

phase supply current, the SAF indirectly supplies the fundamental reactive

and harmonic current. The block diagram of the SAF reference current

extraction for indirect control is shown in Fig. 6.14. The SSI structure is as

same as shown for direct current control technique. As both of the SSI units

extract fundamental components of supply voltage and load current

respectively, the instantaneous fundamental active power p t( ) can be

calculated by using (6.4). The component i *1 can be calculated as,

v t v t i t v t i ti t

v t v t1 1 1 1 1*

2 2

( ) ( ) ( ) ( ) ( )( )

( ) ( )

(6.34)

The desired supply current is addition of i t* ( ) and dci*1 . The use of SSI

eliminates the need of additional low pass filter to extract fundamental

powers and a2 lag/lead for supply voltage and load current signals to

generate fictitious signal to apply IRP theory as proposed by [H15]. The

resultant structure of the SAF controller is greatly simplified and eliminates

the need of phase lag elements and filters.


129

Figure 6-13: Block diagram of single phase SAF direct control using SSI

Figure 6-14: Block diagram of SAF controller -indirect control using SSI


130

6.8 Phase synchronization

From the Bode diagrams shown in Fig. 6.12, it can observed the SSI is

sharply tuned to a particular frequency i.e. nominal frequency of the funda-

mental supply voltage. If the supply frequency deviates from its nominal

value, the gain of the SSI will drop sharply and consequently it will affect

output. It requires automatic adaptation to grid frequency which will main-

tain the same response characteristics in spite of small variations in grid

frequency. SSI can be made frequency adaptive by using a PLL. Various

single phase PLL techniques are reported in literature [D1]-[D5]. The block

diagram of the conventional park-PLL[D1] is shown in Fig 6.15. As it can be

seen from the block diagram, it is a single phase version of the three phase

synchronous reference frame PLL. The utility voltage signal vs is given as an

input to the Park transformation block. The component v of the stationary

reference frame is obtained by inverse Park transforming the filtered syn-

chronous components 'dv and '

qv . Thus, a balanced three phase system is

emulated. By using a single SSI block, component v can be easily generated

and this will reduce the complexity of the PLL. The block diagram of the

proposed PLL structure is shown in Fig. 6.16. SSIV block generates com-

ponents of the supply voltage. The components are further transformed

to dq frame by using appropriate transformation. The closed loop control

system tries to maintain qV =0. The dV represents peak of the fundamental

supply voltage. Performance of the PLL is tested by giving step change in

frequency as well as by giving frequency ramp.

Figure 6-15: Block diagram of Improved single phase Park PLL


131

Dynamic performance of the PLL is tested by giving step input of input

frequency as shown in Fig. 6.17. The performance of the PLL under distorted

utility conditions is shown in Fig. 6.18. From the responses, it can be con-

cluded that the SSI based PLL has satisfactory dynamic response and has

ability to extract instantaneous phase angle information even in presence of

distorted supply.

Figure 6-18: Frequency tracking performance of PLL

Figure 6-16: Block diagram of single phase PLL based on SSI

Figure 6-17: PLL performance under distorted supply voltage conditions


132

6.9 SAF current control

The generated SAF reference current is sent to the current controller.

The current controller takes the instantaneous feedback of the SAF current at

its input ( ci ) and the reference current ( ci* ) and then creates the switching

signals to the individual switches of VSI for regulation of SAF current. Per-

formance of current controller can be characterized by following

requirements [G8]:

Non sinusoidal multi frequency tracking.

Ability to operate with low VSI output inductance (Lac < 5%).

High di/dt reference current tracking and high current control-

ler. bandwidth for tracking of high frequency harmonics.

Minimization of low as well as high frequency errors in current.

Desired implementation by constant switching frequency PWM

scheme.

Maintain predictable ripple bounds.

Different current control strategies are reported in literature for SAF

applications. Hysteresis and sliding mode controls allow both a direct control

of the current but at the price of a time-varying switching frequency. Hyste-

resis current regulators are on-off type current regulators and widely utilized

for tracking of non-sinusoidal, multiple frequency and high di/dt current

references. When analog implementation is utilized, hysteresis current

regulators exhibit high bandwidth and superior tracking capability of the

reference currents with high di/dt. Various techniques to implement hystere-

sis control are discussed in [I2]-[I7]. The well known drawback of hysteresis

control technique is variable switching frequency. Hysteresis current regula-

tors also generate an undesirable ‘white noise’ current spectrum as a result of

variable inverter switching frequency. The design of the output filter is also

difficult for these methods as well as control of noise level. [G8].

Linear and on-off current controllers also form a viable solution for

SAF applications. The basic principle of the linear current regulators is


133

shown in Fig.6.19. The difference between the current reference and the

current feedback constitutes the current error signal which is converted to the

inverter voltage reference via the linear current regulator. The inverter

voltage reference is synthesized by carrier based PWM modulator. The

modulator creates the switching signals for the VSI to form the desired

current through the filter inductor which is connected between the VSI

output terminals and AC utility grid. The advantage of the linear current

regulators involves the carrier based PWM modulator implementation. In the

carrier based implementation, the switching frequency fsw can be adjusted to

a fixed value. The fixed fsw results in well defined switching current harmon-

ics on the current waveform. For a particular value of fsw, harmonics appear

at the frequency of fsw and multiple frequencies of fsw. Also, the fixed fsw is

kept high to achieve high bandwidth in SAF application. Therefore, since the

frequency of the switching harmonics is known and high, the elimination of

them is simple by a simple passive switching ripple filter structure at the

input terminals of the SAF. The drawback of PWM implementation is that a

sudden change of the reference signal results in unavoidable delay nearly

equal to one or one-half carrier period (PWM period = Tsw =1/fSW) depending

on the PWM implementation technique. Therefore, the switching frequency

is kept as high as possible to overcome this drawback while considering

thermal stability and capability limits of semiconductor switches .

Figure 6-19: Basic block diagram of linear current regulator

The modulation methods for the PWM are divided into two main groups;

scalar PWM and space vector PWM. In the scalar PWM method, the voltage

reference wave (modulation wave) is compared with a triangular carrier

wave and the intersections define the switching instants for the switches of

VSI. In the space vector PWM method, the switch on-state durations are


134

calculated from the complex number from volt-seconds balance equation for

the inverter voltage and the switch pulse pattern is programmed via digital

PWM.

In this work, modified PWM (M-PWM) [I8] method has been used for

control of current. The technique is based on two comparisons between a

high frequency triangular carrier signal on one side and load harmonic and

its opposite on other side. As compared to standard PWM technique, use of

M-PWM pushes the first significant switching frequency harmonics in the

source current to twice the switching frequency. In addition, it eliminates

group of rays that are centered on odd multiples of switching frequency.

Principle of MPWM is illustrated in Fig. 6.20 and the timing diagram is

shown in Fig. 6.21. It is based on two comparisons: 1) between control signal,

and a high frequency triangular carrier and 2) between opposite of control

signal, and the same carrier. The control signal is assumed to be slow

varying signal and practically constant within each half period of triangular

carrier. The control signal is delivered by a closed loop PI controller which

tries to minimize the error ( c ci i* ). The effectiveness of the M-PWM method

can be studied by comparing its performance with the standard PWM tech-

nique [I8]. To validate the findings, an inverter operating at fsw =10 kHz is

simulated and the harmonic spectrums are studied. The harmonic spectrums

of both the cases are given in Fig.6.22 (a) and (b).

Figure 6-20: Basic block diagram of MPWM current controller


135

Figure 6-21: Timing diagram of MPWM current control technique

(a)


136

(b)

Figure 6-22: Comparison of vc spectrum: a) with S-PWM b) with M-PWM

6.10 Switching ripple filter for the SAF

The current ripple generated by the VSI of the SAF power circuitry can

spread to the power line through the PCC where the PAF system is con-

nected to the power system. High frequency switching harmonics create

noise problems for other loads connected to the same PCC. To filter the high

frequency switching ripple currents due to the switching of the VSI, passive

switching ripple filters are placed at the PCC as an integral part of the SAF.

Linear current regulators generate regular PWM ripples around the switch-

ing frequency and its multiples and sidebands over the frequency spectrum.

The M-PWM switching technique causes the switching ripple to be at 2fsw as

shown in the previous discussed above. This section presents overview of

different switching ripple filter topologies and design of broad band switch-

ing ripple filter topology to reduce switching ripples.

6.10.1 Switching Ripple Filter Topologies

The choice of SRF topology is directly related to the ripple currents

frequency spectrum, hence that is related to the current regulator type uti-

lized in the PAF control system. The switching ripple currents of linear

current regulators are observed at switching frequency (fsw) and its multiples

(2fsw, 3fsw...) i.e. at specific frequency components which depend upon the

switching frequency. Therefore, a tuned type switching ripple filter which is


137

tuned to fsw should be utilized to filter the switching ripple currents of the

PAF when utilizing linear current regulators. Various filter topologies are

already shown in Fig. 3.4 of Chapter 3. Broad band tuned filter is the most

suitable choice considering the specific harmonic frequencies present in the

current.

6.10.2 Broad-band Tuned Type switching ripple filter

The broad-band tuned type switching ripple filter (RF) illustrated in

Figure 3.4.b is utilized to filter the switching ripple currents created by the

linear current regulators in the SAF applications and also for providing

passive damping of the parallel resonance for a wide frequency range [F7].

The filter involves two separate paths; one for the attenuation of the switch-

ing ripple currents and the other for damping of the load and/or source side

induced resonances. The LF, CF branch is a series resonant branch which is

tuned to frequency fsw. The impedances of CF and LF cancel each other at fsw

and the total impedance of filter is equal to filter resistance RF at fsw. To sink

the switching ripple currents, RF should be low. RF is mainly constituted by

the internal resistance of LF. By tuning resonant frequency of the LF CF RF

filter to fsw, the most dominant switching ripple currents at fsw are filtered

through the low impedance path. The value of the filter damping resistor Rd

is selected such that it will damp out the resonances for a wide frequency

range with the capacitor C connected in series with the two parallel con-

nected arms. The series resonant frequency fs of the broad-band tuned type

filter, considering the inclusion of C to the filter structure, is defined as,

sF

FF

fC CL

C C

1

2

(6.35)

For the design simplicity, the approximate value of fs can be expressed in

terms of LF and CF values, as,


138

sF F

fL C1

2 (6.36)

The size of C should be such that, C >> CF. As suggested in [F5], the value of

CF should be such that,

FC C10 (6.37)

However, the larger C sizes are not preferred since they result in excessive

reactive power at the fundamental frequency. The equivalent circuit of broad

band switching ripple filter at harmonic frequencies is shown in Fig. 6.23.

The transfer function T(s) defined as the ratio of the line current IHS to har-

monic current source IH expressed in terms of filter circuit components CF, LF,

C, and Rd as follows,

F F d F F F d

S F F F S F d F F S F d

s L C CR s L C s C C RT ss L L C C s L L C CR s L C L C s C C R

3 2

4 3 2

( ) 1( )( ) ( ) ( ) ( ) 1

(6.38)

It can be seen that, very low value of Rd there can exist a condition for paral-

lel resonance between switching ripple filter capacitance C and supply side

inductance Ls, at frequency fp1,

ps

fL C1

12 (6.39)

As Rd increases, there is a chance of parallel resonance which occurs between

filter capacitor CF, filter inductor LF and line inductance LS at a frequency fp2

given as,

ps F F

fL L C2

12 ( )

(6.40)


139

Therefore, the Rd value should be selected such that harmonics between the

two resonant frequencies are damped effectively. Selection of various para-

meters of switching ripple filter is illustrated with an example as shown

below. Various ac supply system parameters are as shown in Table 6.1. The

supply current without switching ripple filter, after load harmonic compen-

sation by SAF, is shown in Fig. 6.24. The harmonic spectrum of the supply

current is shown in Fig. 6.25.

Table 6-1 AC supply system parameters

From the harmonic spectrum of the supply current, it can be seen that there is

a switching frequency ripple component at 10kHz. The LF CF can be tuned to

filter the 10kHz ripple component.

Figure 6-23: Equivalent circuit of broad band tuned LCR type switching ripple filter andAC supply side at harmonic frequencies

Parameter Value

sV , Supply voltage 240V

f , Fundamental supply frequency 50Hz

sL , Supply side inductance 100 H

sR , Supply side resistance 100 m


140

Selecting the value of CF as F2 , the LF can be calculated from (6.36) as

FL mH0.1266

The internal resistance of LF is assumed to be 50 m . According to (6.37), the

value of the capacitor C is selected to be

C = F20

The value of Rd can be selected such that parallel resonance frequencies fp1

and fp2 are damped out. Frequency response of the transfer function T(s) for

different values of Rd are shown in Fig. 6.26. The magnitude of T(s) is desired

to be as close as possible to unity for low frequencies. Effect of various Rd

values on the frequency response of T(s) is shown in Fig. 6.26. It can be

Figure 6-24: Source current after load harmonic compensation without switching ripplefilter

Figure 6-25: Source current harmonic spectrum without switching ripple filter


141

observed that for low value of Rd (Rd=1 ), the gain of T(s) is increased at

parallel resonant frequency fp1. Similarly, for a higher value of Rd (Rd=10 ),

the gain of T(s), is increased around fp2. The responses for Rd = 4 indicate

that it effectively damps out the frequencies between fp1 and fp2. The optimum

value of Rd is selected to be 4 . The various filter parameters are tabulated in

Table 6.2.

Table 6-2: Various parameters of switching ripple filter

Parameter Value

LF, Filter inductance LCR branch 0.1266 mH

RF, Internal resistance of LF 50 m

CF, Filter capacitance of LCR brach 2 F

C, filter capacitance 20 F

Rd Damping resistance 4

Figure 6-26: T(s) magnitude versus frequency for various values of Rd


142

The effectiveness of the filter is tested by simulating the ripple filter along

with the SAF and the responses of the source current and the harmonic

spectrum are presented in Fig. 6.27 and Fig. 6.28. From the harmonic spec-

trum of the supply current, it is clear that the broad band type switching

ripple filter effectively filters out the switching frequency ripple component

and the band of frequencies above 2fsw. The THD of the supply current

without connection of ripple filter is 6%. The effect of switching ripple filter

can be clearly seen in the THD of the supply current which reduces to 3.1%.

This can be attributed to the attenuation of ripple frequencies near 10kHz

because of the tuned LCR branch and attenuation of the higher frequency

components because of the broadband nature of the filter.

Figure 6-27: Supply current with switching ripple filter at the input of SAF

Figure 6-28: Harmonic spectrum of the supply current with switching ripple filter at theinput of SAF.


143

6.11 Summary

In this chapter, a method based on SSI to extract reference current of

single phase SAF is presented. Two models to extract SAF reference current

for direct and indirect controlled SAFs have been put forth. A single phase

PLL based on SSI is also proposed. The current controller is based on M-

PWM technique. The design of broad-band tuned LCR filter to minimize

switching frequency ripple is also discussed along with its simulation results.

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