CC BY
38
UDC 621.3
doi: 10.20998/2074-272X.2016.6.06
D.V. Tugay
THE PHASE REACTOR INDUCTANCE SELECTION TECHNIQUE FOR POWER ACTIVE FILTER
Purpose. The goal is to develop technique of the phase inductance power reactors selection for parallel active filter based on the account both low-frequency and high-frequency components of the electromagnetic processes in a power circuit. Methodology. We have applied concepts of the electrical circuits theory, vector analysis, mathematical simulation in Matlab package. Results. We have developed a new technique of the phase reactors inductance selection for parallel power active filter. It allows us to obtain the smallest possible value of THD network current. Originality. We have increased accuracy of methods of the phase reactor inductance selection for power active filter. Practical value. The proposed technique can be used in the design and manufacture of the active power filter for real objects of energy supply. References 12, figures 11.
Key words: power active filter, coefficient of harmonic distortion, phase reactors inductance, frequency modulation, Matlab-model of a three-phase energy supply system.
Цель. Целью статьи является разработка методики выбора индуктивности фазных реакторов параллельного силового активного фильтра, основанной на учете как низкочастотной так и высокочастотной составляющих электромагнитных процессов в силовой схеме. Методика. Для проведения исследований использовались положения теории электрических цепей, векторный анализ, математическое моделирование в пакете Matlab. Результаты. Разработана новая методика выбора индуктивности фазных реакторов параллельного силового активного фильтра, позволяющая получить минимально возможное значение коэффициента нелинейных искажений сетевого тока. Научная новизна. Повышена точность методов выбора индуктивности фазных реакторов силового активного фильтра. Практическое значение. Предложенная методика может использоваться при проектировании и изготовлении силовых активных фильтров для реальных объектов электроснабжения. Библ. 12, рис. 11.
Ключевые слова: силовой активный фильтр, коэффициент нелинейных искажений, индуктивность фазных реакторов, частота модуляции, Matlab-модель трехфазной системы электроснабжения.
Introduction. Electromagnetic compatibility of electrical consumers with the supply of industrial network is one of the key factors considered in the selection and installation of electrical equipment designed for a large installed capacity. Stringent standards for power quality [1, 2] impose obligations both producers and customers of electrical equipment to take additional measures to comply with them, which affects the characteristics of the end product, its cost, and certification opportunities.
In the industrial and municipal power systems, electric transport the main segment of powerful consumers of electricity consumers up with non-linear characteristics [1-3]. Joint connection to industrial consumers such network causes an increase in emission level of higher harmonic components of the current network, the distortion of the supply voltage circulating in conductors linear reactive currents and, as a consequence, leads to increased energy losses in the power supply system [1-3]. Exclude or partially reduce the listed effects it is possible with the help of modern filter-devices, one of which is power active filter (PAF). PAF of parallel type the simplest diagram of power circuits is shown in Fig. 1 have the greatest popularity.
The power part of the PAF identical scheme of autonomous voltage inverter and executed on the power transistor, diode modules, forming a three-phase bridge shoulders. DC converter unit included a storage capacitor C, the voltage at which the automatic control system is supported by PAF above the line supply voltage amplitude
U0 = k S •Usm , (1)
where Usm is the network phase voltage amplitude; k = 1.2-1.6.
Fig. 1. Diagram of parallel PAF power circuits
In the input circuit of the PAF a three-phase reactor is connected with the parameters of the inductance Lc and resistance Rc. To reduce high-frequency noise in the supply voltage at the input of the converter is connected in parallel with a three-phase filter of small capacity Cc.. Despite the simplicity of the power circuit and its power relative small element active filter is a complex dynamic device in which electromagnetic processes and parameters of circuit elements, depending on the mode and parameters of the load and the mains. Along with others, these factors complicate the calculation and selection of parameters of elements of power PAF scheme. The greatest difficulties arise when selecting the input inductance of Lc reactors, simplified method of calculation which is mainly linked to the taking into account the current growth rate through the reactor
© D.V. Tugay
winding commutations with transistor-diode keys, i.e. only the high-frequency component of the electromagnetic processes in a power scheme PAF [5-11]. As was shown in [12], this approach does not allow one to choose the parameters of the input reactor, so you need a new technique based on the specific operating conditions of the power compensator.
The goal of the paper is the development of a method of selecting values of phase power inductance of reactors of parallel power active filter based on the account as low-frequency and high-frequency components of the electromagnetic processes in the power circuit.
Features of the PAF operation. To analyze the characteristics of the PAF operation we will use a concrete example it is connected in parallel non-linear load, shown in Fig. 1. For the three-phase system with symmetrical sinusoidal voltage source usa, usb, usc whose parameters Ls and Rs are uniquely defined, connected six-pulse uncontrolled rectifier DR working on R-L load. To restrict network rectifier ripple current last networked via an inductive reactor LL, RL parameters. PAF is connected to terminals ABC, where there are voltages ua, ub, uc.
The principle of parallel PAF operation is based on generating compensation currents, which subtracts from the load current allows to obtain the desired shape of network currents. In this example, after connecting the PAF form of line current to be sinusoidal and instant network current should be in phase with the corresponding phase voltage.
Load current vector can be represented by a three-component vector, each component of which is a vector that stores information about the three current values of the respective phase currents
iL =[ha i1r iU J, (2)
where i1a is the vector of the fundamental harmonic active current three-phase system, which coinc es with the direction of voltage vector us (see Fig. 1); i1r is the vector of the fundamental wave -eac ive power three-phase system orthogonal to vector i1a ; ia is the vector of total current harmonic load three-phase system.
PAF current vector can also be represented by a three-component vector
C = [c1 -hr c2 ■ iU i2h(3) where is the vector total current harmonics whose frequencies are multiples of side-frequency PWM; c1 is the level of compensation of the primary harmonic of the reactive current (0<c1<1); c2 is the the level of compensation of the total load current of higher harmon cs (0<c2<1). The first two components of the vector iC describing the low-frequency component of the electromagnetic processes in PAF related to the characteristics of the load, and the third component - the high-frequency component, the appearance of which is associated with a feature of the compensator when switching of power semiconductor modules.
According to Fig. 1 and relations (2), (3) the line current vector at the working PAF may comprise four components
7s = \fla (l-clHr (l " c2 \U 1UT (4)
Quality compensation w l depend on the values of the coefficients cl, c2 vector iSh • Under ideal conditions
c1~1, c2~1, and |ish| ~0. To achieve full compensation is
virtually impossible in real installations. The quality of the PAF can be estimated value of the distortion factor THDi line current. In accordance with (4) THDt can be expressed as the sum of two components - a low-frequency (THDa) and high-frequency (THDih)
THDi = THDa + THDih =
ÊM+ïk
VIía+0 - c Y • 4
(5)
where I2n is the the sum of the squares of the rms values
of higher harmonic components of the load current; ijh is the sum of the squares of the rms values of higher harmonic components whose frequencies are multiples of side-frequency PWM.
With an increase in the inductance of the input reactor PAF THDa low-frequency component is increased, and high-frequency component THDh decreases. Thus, there is an optimal value of inductance Lopt corresponding to the minimum possible value THDi. To find the value Lopt necessary to examine separately the electromagnetic processes in the PAF on the low and high frequency.
Consider the work of one phase of the PAF in the circuit according to Fig. 1. In the circuit in the open state are alternately transistor diode and one group of the same group opposite phase. For example, with a negative reference current compensator ica* in working condition will be the transistor VT1 and diode VD4 circuit as shown in Fig. 2. The term repeatability mains voltage in accordance with the nature of the non-linear load are three possibilities of joint work with the transistor VT1 transistor other two phases: 1 - VT1, VT3, VT2; 2 - VT1, VT6, VT5; 3 -VT1, VT6, VT2. Fig. 1 illustrates the first option. Depending on the duration of the conduction intervals of transistors employed and the current values of the phase voltages on the PAF input for the described embodiment are possible 8 states of the circuit shown in Fig. 3. At the time interval when the transistor VT1, the energy stored in the capacitor C is transmitted through the reactor Lc in the network (Fig. 3,a,b) or stored in a capacitor through the reactors of the other two phases (Fig. 3,c). When transistor VT1 is closed in the open state is VD4 diode, and the energy stored in the reactor is transferred to the condenser, together with additional energy network (Fig. 3,e,g) is either returned to the network (Fig. 3f). The time intervals in a conducting state when there are three groups of one semiconductor device (Fig. 3,d and Fig. 3,h) the reactor is charged by the network.
Ut u,
Ua "f Uc
Fig. 2. Phase A operation of the circuit with PAF
Account of the low-frequency component of the electromagnetic processes in the selection phase inductance reactors. In accordance with Fig. 1, 2, and by (3) a low-frequency component of the PAF current can be represented as the sum of the reactive component of the first harmonic current, and harmonics of the order 6n±1 due to the spectral composition of the input current six-pulse uncontrolled rectifier.
Let us evaluate the effectiveness of the PAF at the frequency of the mains voltage. Fig. 4 shows an equivalent circuit of one phase of the PAF for the fundamental harmonic voltage and current.
The first harmonic voltage, the current at the input rectifier bridge PAF is shown in the equivalent circuit source uca1 and the load impedance phase - equivalent resistance ZL1.
Fig. 5,a and shows a vector diagram of the voltages and currents corresponding to the equivalent circuit of Fig. 4. The diagram shows that the instantaneous value of the first harmonic voltage at the input rectifier bridge uca1 practically in phase with the instantaneous mains voltage usa. The amplitude of the fundamental harmonic voltage at the input rectifier bridge (Umca1) PAF will always be larger than the amplitude-phase mains voltage (Umcai) on the value of the voltage drop across the phase reactorUmcLa1
Umca1 ~ Uma UmcLa1. (6)
If we taking into account the orthogonal of vectors of current fundamental harmonic of the network Isa1 and fundamental harmonic PAF Ica1 , the parameters of the source and the load to take the same, it is possible to prove the existence of the optimal value of inductance phase reactor Lopt1 ensuring the fulfillment of the above conditions.
In the vector diagram in Fig. 5,b shows that by increasing the inductance of the phase of the reactor there is a shift of the fundamental harmonic line current with respect to the supply voltage on the light angle y thereby deteriorating the quality of the PAF compensation.
Lc 3 L? ri pi-
re jiU \ia
T2
a, +
Ma Ur
1 c
g h Fig. 3. Possible states of the PAF circuit at transistor VT1 and diode VD4 operation
Using a superposition method according to the equivalent circuit of Fig. 4 write ratio to calculate an integrated value of the fundamental harmonic load current.
b
a
Fig. 4. Equivalent circuit of the PAF one the phase on the network voltage frequency
Zr
1 1
+ -
Liai ~ Uma
Z11 -(¿L + ZC ) Z11 Z
22
1 - A-Z 22
(7)
where zc = Rc+jmLc1; Zl = RL+jmLL+ZL1; Z11 = Zs+Zl'Zc/(Zl+Zc); z22 = Zc+Zl-Zs/(Zl+Zs); Zs = r^+jffll^.
C/cial
Ua
Isa 1 opi
b
Fig. 5. Vector diagrams of voltages and currents of the equivalent circuit on the network voltage frequency
The real part of the complex values of the fundamental harmonic load current Re(LLa1) corresponds to the optimum value of the fundamental harmonic current network Lsa1 opt, and the imaginary part Im(LLa1) - optimal value of the fundamental harmonic current PAF Ica1 opt.
Complex value of the fundamental harmonic voltage at the input of the rectifier bridge PAF
Uca1 = Ua + j * Im(Lla1) - Zc . (8)
From the equivalent circuit (Fig. 4) we express the imaginary part of the complex values of the fundamental harmonic current PAF
Lca1 = Im
' gcan - U(ZL ^ Z 22 Z11 -(Ll + ZC )
., ,, ., (9) -22 Z11' ' '
Fig. 6,a and shows the dependence of the fundamental harmonic current PAF on the relative values
(in the total inductance of the load shares L{L = = LL+Im(ZL1)/ffl) phase inductance PAF reactor at different values of relative total inductance of the load and a fixed value of the network relative inductance LJL{Z = 0.065. Dependencies are constructed with the following parameters of the equivalent circuit elements: Uma = 311.13 V, Rc = Rl = 0.002 Q, Rs = 0.00036 Q, LL = = 75 ^H, ZL1 = 0.438+j0.358 Q.
400
leal, A
Ls* = 0.065
200
0.125 0.25 / \ 0.375 ô!i Lcloptmm* Lcl opt mar*
210
205
200
195
190
lea 1, A
a
1
/ / Ls - 0.65
/ r 0.02 y
If, 0.25
leal opt I —7/ i/f Ls• - 0.05 Ls* — 0.065
—L-L»- / Ls = 0.1 Lei*
t 0.1 L,c 1 opt mill*
0.2 0,3
, Lcl opt max* b
0.4
Fig. 6. Dependencies Lca1 = f(Lc1*)
Fig. 6,b shows the dependence of the fundamental harmonic current SAF on the relative values of the inductance of the phase reactors SAF for different values of the network inductance and fixed value relative total inductance load Lis* = 1.
a
At the intersection built in Fig. 6 curves with curves corresponding optimal value of current fundamental harmonic PAF Ica1 opt are point abscissa are equal to the optimum values of the phase inductance reactors Lopt1.
From a comparison of Fig. 6,a and Fig. 6,b it can be concluded that the optimum value of the phase inductance reactors PAF largely depends on the inductance of the network than the load inductance. If the total inductance of the load changes from 0.2Lls» to 3Lls» optimum value of the phase inductance reactors PAF decreased from 0.275Lls» to 0.261Lls*. If you change the network inductance of 0.02Lls» to 0.65Lls» range of the optimum load inductance is much wider from 0.083Lls» to 0.273Lls*. Moreover, by increasing the inductance of first optimum network inductance increases, and after a certain Ls* value decreases.
From the vector diagram in Fig. 6 we obtain the ratio to calculate the angle of the shift of the fundamental harmonic phase current with respect to the appropriate voltage
( Im(/Lal)~ Im(/ca1 ) ) . Re(/Lal) ,
Fig. 7 shows the dependence of the angle ç of the relative phase inductance reactors Lc1* for various values of a combination of the relative inductance of the network and relative total inductance of the load.
p _ arctan
(10)
Account of the high-frequency component of the electromagnetic processes in the selection phase inductance reactors. We consider the work of one PAF when alternately switched transistor VT1 and diode VD4. According to Fig. 2 and Fig. 3 in the first working interval VT1 and VD4 alternate conductivity four options PAF circuit state (Fig. 3,a, Fig. 3,d, Fig. 3f Fig. 3,h). Fig. 8 shows SAF current shape in the first valve VT1 and VD4 working conduction interval corresponding to one period of modulation Tmod.
i <','., i
Ton Toff
I ¿1 Y J M tb \
! Imi ■ m ■ ' / ■ - m i ■ CÔ5.' ' t VTl' , \Ti- ■ VTÏ. . ■ vt* ■■■ ; ¡73 'i ■ m , ,. ; ' V1H. : t. №6 1 172' j i', ' vm. •. ;■. m ' ;
■/■4 :. .....yil-y.... k i
/in . ' 1 \ ^^^ :
Ah
.............IfJM
:
■ : .
Fig. 8. Current shape of the PAF in the conduction interval VTl and VD4 corresponding to one period of modulation Tmod
0.5
0.4
0.3
0.2
0.1
\ Ls* = 0.
IA" Ll = 0.65v :•= 1 j \Ls- \ Lli Luz*-•= 1
¡Ls* \ = 0.065
/ Lll = 1N ^Ls* — ( ).065~"~ 5\___
1/. Ls* = 0. Ll*-A Lc\*
The short time intervals tl and t3 (see. Fig. 3,a) when in the open condition of the transistors VTl, VT3, VT2 the energy stored in the capacitor is transmitted into the network. The time intervals t2 and t5 when in a conducting state at the same time the valves are three phases of one group (see. Fig. 3,d and Fig. 3,h) Lc reactor stores energy from the network. At time intervals t4 and t6 along with transistors VT3, VT2 involved diode VD4 (Fig. 3, f) the energy stored in the reactor is returned to the network.
From Fig. 8 it can be seen that in the period of modulation Tmod greatest slope of the curve corresponds to the current instant network time intervals t1 and t3, when open transistors VTl, VT3, VT2. For Fig. 3,a the following relation is correct
dica L _ u , £ç_
l^c — u Sa _
dt 3
and for Fig. 3,d
dic
dt
Lc _ usa .
(ll)
(12)
Fig. 7. Dependence ç = f(Lc1*)
The optimal values of the inductance of the phase reactors correspond to points of intersection of plotting with the horizontal axis (ç = 0). If the inductance of the network is unknown or tends to zero, the point of intersection of the graph with the x-axis will be absent (see. Fig. 7 at Ls* = 0, Lls» = l). In such a case it is necessary to limit the range of variation of inductance values PAF phase reactors, in which the angle 9 will not vary by more than 1°. In this example, the corner ç = 1° value corresponds to the inductance of the relative phase reactors Lc1* = 0.15.
Using auxiliary constructions shown in Fig. 8, it is easily to determine the maximum angle of PAF current [6]
Aic
2 ■AIc
_ 4 ■AIc ■ ./mod :
(13)
At Tmod 12
where AIc is the maximum deviation of the instantaneous current of the current PAF set, fmod is the frequency pulse width modulation. Let us consider the case when
fmod = const.
Substituting the relation (l), (13) in (ll) we obtain the equation for determining the minimum inductance phase reactor PAF
min 2
"m 'I' ^
4 • /mod -A/c* • '
where
A/c
I
sal
Re(/Lai)-
(14)
(15)
LE
1 -
A V3
Lc2* -
L
c min2 + Lc max2
15--L
s
7. If the value AIc» more than 5 % from the first harmonic amplitude of the current network, we make adjustments of parameters. In the example, you can slightly increase the modulation frequency to the value /mod = 7045 Hz, and Mc* = 0.05.
With a high degree of accuracy (14) can be replaced by a simplified relationship
Lcmin2 -Tj— . (16)
4 ' /mod '^c*
where RLE = RL + Re^l) is the total resistance of the load phase.
From Fig. 3,d in a similar manner to obtain the ratio of the maximum value of PAF inductance phase reactor, substituting (1), (13) in (12) and bearing in mind that the maximum voltage at the time of switching transistor reaches half the phase voltage amplitude usa~0.5Um (see Fig. 2)
L _ RLE (17)
cmax2 _ „ t at . ( )
8 ' /mod 'A1c*
The relative value of the average inductance of the reactor phase
(18)
2 ' LLE 8 ' /mod ' ^c* ' TL ,
where tl = LlE/RlE is the time constant of the load phase.
Fig. 9 shows the dependence of the relative values of the inductance of the phase reactors by PAF parameter AIc* for various values of the modulation frequency /mod for these parameters earlier scheme.
Method of choosing PAF inductance. On the basis of the material presented above describe the steps to select inductance phase reactors PAF. Consider a specific example of the application of the scheme PAF with the previous settings of its elements.
1. We determine total load inductance LLE = LL+Im(ZL1)/ffl). In the example LLE = 0.433 mH.
2. From the known parameters of the scheme, using the relation (7) expect to complete the value of the first harmonic of the load current. In the example ILa1 = = 645.4—199.5 A.
3. Using (8) we obtain the relation for the calculation of the first harmonic current SAF in the form (9).
4. By (10) we build dependence y = /Lc1*) (see Fig. 7).
5. At the intersection of the curve y = /Lc1*) with the x-axis determines the point coordinate which corresponds to the optimum value of the relative phase of the reactor inductance PAF. In the example Lc1 op* = 0.249 (Lc1 opt = = 0.108 mH).
6. The obtained value of inductance is substituted in equation (18) and determines the relative value of the maximum deviation of the instantaneous current of the given current PAF AIc*. In the example AIc* = 0.059.
Fig. 9. Dependencies Lc2* = f(A7s*)
Experimental verification of the theoretical results. To test the proposed methodology has been developed Matlab-model of a three-phase power supply system with a non-linear load and parallel active filter shown in Fig. 10.
Matlab-model consists of 5 groups of blocks:
1. Power circuit (Us - three-phase voltage source with definable parameters; DR - three-phase uncontrolled rectifier; Load - rectifier load; PAF - inverter bridge of PAF; C - DC link capacitor of PAF; Rc - charging resistor; LL - load current smoothing reactor; Lpafl, Lpaf2 - PAF phase reactor divided into two sections; Cpaf - PAF input capacitor filter; SA_PAF, SA_C - three-phase contactors of PAF and capacitor filter, respectively).
2. Currents and voltages sensors (blocks Is, IL, Ipaf, U, UL, Upaf, UC).
3. Control system (Control System - PAF control system; Gate_Paf, Gate_G, Gate_C, Gate_R - power circuit control blocks).
4. Measuring subsystem (Measurements - subsystem for calculating total harmonic distortion and operating currents).
5. Virtual instrumentation (Multimeters - multimeters; Us_Is, Upaf_Ipaf, Uc - oscilloscopes).
For computer simulation the same parameters elements of the power circuit, as for the calculation of the phase inductance reactors PAF were chosen. In the experiment, the input capacitor filter PAF has been disconnected.
A a B b
C g
Discrete, Ts = 2e-006 s. powergui
Load
Us
< ta o
Gate_PAF
ŒK
SA PAF
L
IlLlsj) ^
Uabs A R □
IL
Us Is
Is A R z£ □
IL C
Control System "+Ï V'''
Upaflpaf
THDJ > [ 0.34561
THD_U 1 0 039461
THD^Load » 1 034581
THD_SAF 1 0.04051
lc_rms | 8.044e-007|
leaf mns —► 1 0.00021671
Gate_C
Gate G
',A_A
Cpaf
■ c
SA C
Lpafl ■—"B b
Lpaf2 Upaf
A a B b
C c
< CÜ <J
Ipaf
J
Rc
C=f
LP
□
Uc
uc
PAF
I Cpaf
Measurements
Multimeters
Fig. 10. Matlab-model of the three-phase power supply system with PAF
Fig. 11 shows the current waveform of phase A for three values of the relative phase inductance reactors PAF: Lc* = Lopt = 0.249 - Fig. 11,a; L* = 0.577 - Fig. 11,è; Lc* = 0.093 - Fig. 11,c. In the figures also indicate the values of the line current THD. The experiment confirmed that the minimum possible value of the line current THD for given circuit parameters corresponding to the selected value of the inductance of the phase reactors PAF. Further studies have shown that when the input PAF C-filter (see Fig. 1) THD line current can be reduced by 1 % (i.e., up to 5 %).
Fig. 11. Waveforms of the network current of the phase A at three values of the PAF phase reactors inductance
Conclusions.
1. The correct choice of inductance of PAF should be based on the account of the two components of the electromagnetic processes in the power circuit - the low-
frequency one related with compensation of reactive power and harmonic load current and high frequency one associated with the processes of «pumping» of energy in phase reactor of PAF.
2. It is proved that the optimal value of inductance of phase reactors corresponds to the smallest possible THD of the line current.
3. The method of selecting inductance of PAF phase reactors taking into account the parameters of elements of the supply network and the load, i.e., the specific operating conditions of the PAF in the power supply system is developed. The proposed method can be used in the design and manufacture of PAF for installation on concrete objects.
REFERENCES
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Received 27.10.2016
D.V. Tugay, Candidate of Technical Science, Associate Professor,
O.M. Beketov National University of Urban Economy in Kharkiv,
12, Revolution Str., Kharkiv, 61002, Ukraine. phone +38 057 7073111, e-mail: [email protected]
How to cite this article:
Tugay D.V. The phase reactor inductance selection technique for power active filter. Electrical engineering & electro-mechanics, 2016, no.6, pp. 31-38. doi: 10.20998/2074-272X.2016.6.06.
