An Adaptive Hysteresis Current Control Based on Unipolar
PWM for Active Power Filters
Sasan Zabihi
Firuz Zare
Department of Power
Mazandaran University
Babol, Mazandaran, Iran
School of Engineering Systems
Queensland University of Technology
Brisbane, GPO Box 2434,QLD, Australia
f.zare@qut.edu.au
Abstract:
When the load current exceeds the upper band, the
comparators generate control signals in such a way to
decrease the load current and keep it between the bands.
Switching frequency varies with respect to the band size,
the inverter and the grid parameters.
Current control techniques based on unipolar PWM
provide a better efficiency and less voltage stress across
loads compare to bipolar PWM technique. An important
issue in implementing the hysteresis control in power
converters is a variable switching pattern which causes
vast range of switching frequency variation. Although
variable switching frequency has been recognized as a
solution for motor drive systems to minimize mechanical
noise [7,8]; but it is not recommended for high power
system applications due to generating of sub-harmonics
and low order harmonics and difficulties to design low
pass filters. Different methods have been proposed to
control the switching frequency of three-phase inverters
with hysteresis current control methods [1]. Calculation
and implementation process of this technique with DSP
controller are described in [2,3,4].
Fig.1 shows a block diagram of an APF based on
hysteresis current control with bipolar and unipolar
modulations. Table I shows the load and the system
parameters which have been used for simulations.
This paper proposes and analyses a new adaptive current
control for a single-phase inverter with unipolar PWM
based on magnitude and time errors and simulation have
been carried out to evaluate the proposed method.
This paper presents a new adaptive hysteresis current
control based on unipolar pulse width modulation with
magnitude and time errors. Variable switching frequency
due to fixed hysteresis bands known as a drawback of
hysteresis method used in power system applications
which generates sub-harmonics and low order harmonics
and affects the quality of the power systems. Adaptive
hysteresis band is a classical method to control the
switching frequency. Several issues and solutions have
been discussed and proposed in this paper to keep the
switching frequency constant when the current error is
close to zero for the unipolar modulation. Magnitude and
time error signals are to control the load current and the
inverter voltage level change respectively; which are the
main issues to implement the adaptive current control
based on unipolar modulation. Simulations have been
carried out to verify the proposed adaptive hysteresis
band controller and the results are presented and
discussed in this paper.
1. Introduction:
Widespread utility capabilities, high efficiency and
suitable flexibility of power electronic equipments cause
vast applications of these instruments in industries and
power systems. Harmonics are main problems in power
network which cause power losses and heat in power
transformers and voltage distortion. Thus, harmonic
elimination seems to be vital and Active Power Filters
(APF) play an effective role in distortion recognition and
elimination [8,9]. These filters are classified with respect
to distortion determination strategy, inverter control
techniques, inverter topologies and their connection
types to the grid. Shunt filters are connected in parallel
with distribution networks. They recognize current
distortions by sampling the line current and compensate
distorted current components to maintain sinusoidal
source current.
Voltage source converters are the most useful power
converters and have wide rang of applications such as
electrical motor drives, uninterruptible power supply,
active power filters, dynamic voltage restore. Various
number of modulation techniques have been proposed
and implemented in these converters. Nowadays, a
hysteresis band Pulse Width Modulation (PWM) method
attracts researchers’ attention due to suitable stability,
fast transient response, simple implementation, high
accuracy and inherent current peak limitation [6].
The hysteresis band is used to control the load current
and determine the switching signals for inverters gates.
Fig.1: Hysteresis current control for an APF
Table I. System Parameters
Parameters
Appropriate Switching frequency of inverter under bipolar
or unipolar mode control
Fundamental frequency
Supply voltage AC
Inverter DC bus voltage
Rectifier load resistance
Inverter inductance
Cdc capacitor
Quantity
15 KHZ
50 Hz
150 V
200 V
5Ω
1 mH
1000 µF
2. Adaptive hysteresis current control based
on bipolar PWM
and substituting in Eq.8, the adaptive band value can be
determined by a controller as shown in Fig.3.
Different solutions have been proposed to provide a
constant switching frequency for three-phase systems
and mathematical analysis describes the relation between
the switching frequency and the system parameters [14]. The main concept is shown in Fig.2 where the
derivative of the load current and the reference current
determines the switching time and frequency. Ascendant
and descendant slope of the inverter current are
produced by imposing voltage stresses +Vdc & -Vdc on
an inductor which connects the inverter to the grid.
Fig.3: Block diagram of an adaptive controller
+
dica
1
= (Vdc − Vs )
dt
(1)
L
−
dica
1
= − (Vdc + Vs )
dt
L
Analysing the triangles in Fig.2 yields:
(2)
+
*
2HB = A − B = t1 ⋅ tan β − t1 ⋅ tanα = t1 ⋅ dica − t1 ⋅ dica
dt
dt
(3)
−
*
dt
dt
2 HB = C + D = −t2 ⋅ tan γ + t2 ⋅ tan α = −t2 ⋅ dica + t2 ⋅ dica (4)
The switching frequency is:
t1 + t 2 = TC =
1
fc
Simulations have been carried out using Simulink and
the switching frequency variations and the hysteresis
bands are shown in Fig.4 and Fig.5 respectively, where a
sinusoidal waveform is used as the reference current and
the switching frequency is selected at 15 kHz. Fig.5.a
shows that the switching frequency is kept around 15
kHz and the small error in keeping the switching
frequency constant is associated with the accuracy of the
simulation tool to calculate the distorted current by
several blocks in Simulink (Fig.1).
(5)
Adding Eq.3 & Eq.4 results:
di*
di +
di −
(6)
4 HB = (t2 − t1) ca + t1 ca − t2 ca
dt
dt
dt
Subtracting Eq.3 from Eq.4 results:
di*
di +
di −
(7)
(t1 + t2 ) ca − t1 ca − t2 ca = 0
dt
dt
dt
Eq.6 & Eq.7 show a relation between HB and (t1+t2)
which is completely independent from t1 & t2 & (t1-t2):
HB =
2
Vdc
− (Vs + Lm) 2
where m =
dica*
dt
(8)
4 f c ⋅Vdc ⋅ L
The slop of the reference current, DC link voltage, the
grid voltage and the inductance value are definite in this
equation. By choosing an arbitrary switching frequency
Fig.4: Variations of switching frequency for hysteresis
current control with constant band (HB=2.5A)
Fig.2: Current and voltage waveforms with hysteresis band current control based on bipolar PWM
voltage groups (+Vdc,0) & (0,-Vdc) are obtained as
follow for each group:
For the first half cycle when the reference current
ascending (+Vdc, 0):
+
dica
1
= (Vdc − Vs )
dt
L
(9)
dica−
1
= − Vs
dt
L
(10)
Analysing the two triangles yields:
2HB = A − B = t1 ⋅ tan β − t1 ⋅ tanα
(a)
dica+
di *
− t1 ⋅ ca
dt
dt
2 HB = C + D = −t2 ⋅ tan γ + t2 ⋅ tan α
di −
di*
= −t2 ⋅ ca + t2 ⋅ ca
dt
dt
And the switching frequency is as follow:
1
t1 + t2 = TC =
fc
Adding Eq.11 & Eq.12 results:
= t1 ⋅
di*
di +
di −
4 HB = (t2 − t1) ca + t1 ca − t2 ca
dt
dt
dt
(11)
(12)
(13)
(14)
Subtracting Eq.11 from Eq.12 results:
(b)
Fig.5: (a) Variations of switching frequency for an
adaptive hysteresis band (b) hysteresis bands and current
error for distorted current in an APF
3. Adaptive hysteresis current control based
on unipolar PWM
A hysteresis current control based on unipolar PWM has
advantages compare to bipolar PWM such as low
switching losses and voltage stress across a load.
As the output voltages of an inverter have three levels,
+Vdc, –Vdc and zero, the two switching states (similar
to bipolar) cannot control the load current sufficiently. In
this case more bands are required to achieve different
switching states corresponding to different output
voltages. For example, when the reference current has a
positive dic/dt, the load current can track the reference
current based on two voltage levels, +Vdc and zero
volts. But when the reference current has a negative
dic/dt, the output voltage of the inverter has to be
changed in such a case to generate negative dic/dt, thus
more band are required to change the voltage level from
+Vdc&0 to -Vdc&0.
Implementation of an adaptive hysteresis band current
control with unipolar PWM keeps the switching
frequency constant for a single-phase inverter which can
improve systems performance and quality. This paper
presents an adaptive hysteresis current control with
unipolar modulation and analyses problems associated
with the hysteresis band calculation.
Similar to bipolar modulation, the relation between the
load and the reference current changes are found
according to Fig.6 where ascendant and descendant
slope of the inverter current generated by two different
di*
di +
di −
(t1 + t2 ) ca − t1 ca − t2 ca = 0
dt
dt
dt
(15)
These two Eq14 & Eq.15 show a relation between HB
and (t1+t2) which is completely independent from t1 &
t2 & (t1-t2):
*
Vdc (Vs + Lm) − (Vs + Lm) 2 , m = dica (16)
dt
2 f c ⋅Vdc ⋅ L
For the second half cycle when the reference current
descending (0,-Vdc):
HB =
dica+
1
= − Vs
dt
L
(17)
−
dica
1
= − (Vdc + Vs )
dt
L
(18)
By assuming:
tb −ta =t3
&
tc −tb =t4
(19)
Geometric analyses of two right triangles gives:
2 HB = E + F = t3 ⋅ tan θ − t3 ⋅ tan ω
= t3 ⋅
di ca+
di *
− t 3 ⋅ ca
dt
dt
(20)
dica−
di *
+ t 4 ⋅ ca
dt
dt
(21)
2 HB = H − G = −t 4 ⋅ tan φ + t 4 ⋅ tan ω
= −t 4 ⋅
As:
1
fc
Adding Eq.20 & Eq.21 results:
t 3 + t 4 = TC =
4 HB = (t 4 − t 3 )
*
dica
di +
di −
+ t 3 ca − t 4 ca
dt
dt
dt
(22)
(23)
Subtracting Eq.20 from Eq.21 results:
(t 3 + t 4 )
*
ca
+
ca
−
ca
di
di
di
− t3
− t4
=0
dt
dt
dt
(24)
These two Eq.23 & Eq.24 results in a relation between
HB and (t1+t2) which is completely independent from
(t1 & t2) & (t1-t2):
HB = −
Vdc (Vs + Lm) + (Vs + Lm) 2
di *
m = ca (25)
dt
2 f c ⋅ Vdc ⋅ L
Switching frequency variations for the fixed and the
adaptive bands are shown in Fig.8 and Fig.9,
respectively; with considering a sinusoidal reference
current and 15 kHz switching frequency. As shown in
Fig.8 the switching frequency is not constant (15 kHz)
for a traditional (fixed band) hysteresis current control
based on unipolar modulation; and also there are
problems in the adaptive band hysteresis current control
where the current error is close to zero, the switching
frequency drops as shown in Fig.9. In the hysteresis
current control based unipolar PWM with magnitude
error, there are two upper and lower bands in order to
change the voltage level (0 & Vdc) or (-Vdc & 0). As
shown in Fig.9.b, when the current error is close to zero,
the load current exceeds the first band and the switching
time happens when the load current crosses the second
band in order to change the voltage level. In this case,
the switching time depends on the size of the second
band and other parameters such as the voltage across the
load, the grid voltage and the reference current.
If the output voltage is zero when the load current
exceeds the first band, the only voltage to change the
load current is the grid voltage which can affect the
switching frequency. This is one of the problems in
implementing the adaptive band current control based on
unipolar modulation which can be solved by including a
time error in parallel with the magnitude current error.
The time error can be define based on 15kHz switching
frequency where the load current exceeds the first band,
the controller waits until t=1/(15kHz) and then changes
the output voltage one level.
Fig.8: Variations of switching frequency with constant
hysteresis bands (HB1=1A&HB2=1.5A)
Fig.6: Current and voltage waveforms with hysteresis band current control based on unipolar PWM
Fig.7: block diagram (1st & 2nd half cycles) of an adaptive controller
The switching frequency variation around 15 kHz shown
in Fig.10.a is due to two main reasons:
1: When the load current crosses the first band, the load
current derivative may not be changed significantly and
for a short period of time it is kept between the first
upper and the lower bands, and after that, the current
exceeds the first band again and the controller changes
the voltage level based on the time error. This issue can
change the switching time and the switching frequency
(for example at t=.0.42 S)
2: Due to implementing several blocks in Simulink,
calculation time changes the accuracy of the simulation
results.
(a)
(a)
(b)
Fig.9: (a) Variations of switching frequency with
adaptive hysteresis bands (b) hysteresis bands limitations
and first band current error egression
The hysteresis band convergence in the unipolar
modulation happens according to Eq.16 & Eq.25 which
are described as follow:
V dc (V S + Lm ) − (V S + Lm ) 2 = 0
(26)
Or:
− V dc (V S + Lm ) − (V S + Lm ) 2 = 0
(27)
This means:
(VS + Lm)(±Vdc − VS − Lm) = 0
(28)
The last term of the above equation is never zero while
the first term may be zero. As shown in Fig.9.b, this
issue can be controlled by considering a minimum value
for the first band and applying the time error when the
load current is between the first and the second bands in
order to control the switching frequency.
The controller has been modified based on the time and
the magnitude errors and the simulation result is shown
in Fig.10 where the switching frequency is kept constant
around 15 kHz for the unipolar modulation. In this case,
the first band never becomes less than 0.07A and the
time error is applied based on 15 kHz switching
frequency and the simulation result confirm the
proposed method.
(b)
Fig.10: (a) Variations of switching frequency for an
adaptive hysteresis band limiter with the time error
control (b) current error and the first upper and lower
bands
4. Conclusions:
Hysteresis current control techniques based unipolar
PWM has lower switching losses and voltage stress
compare to bipolar modulation. Variable switching
frequency due to fixed hysteresis bands known as a
drawback of the hysteresis method for power system
applications which causes sub-harmonics and low order
harmonics and affects the quality of active power filters.
Adaptive hysteresis band is a classical method to control
hysteresis PWM switching frequency. This paper
presents a new adaptive hysteresis current control based
on time and band errors for unipolar pulse width
modulation to control the switching frequency for high
power applications. Simulation results verify the
proposed adaptive hysteresis band controller with
unipolar modulation based on time and magnitude
errors.
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