 Open Access
 Authors : Bisrat Gezahegn Lemma , Minfu Laio , Duan, Xiongying
 Paper ID : IJERTV9IS030362
 Volume & Issue : Volume 09, Issue 03 (March 2020)
 Published (First Online): 25032020
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
SecondOrder Odd Repetitive Control for ThreePhase FourWire Active Power Filter to Mitigation Current Harmonics, Unbalance and Neutral Current of Nonlinear Loads
Bisrat Gezahegn Lemma1 , Minfu Laio2* Duan Xiongying3
1,2,3, Institute of Electrical and electronics;
School of electrical engineering, Dalian University of Technology, Dalian 116024, China
Abstract:This paper describes a threephase fourwire (3P4W) active power filter (APF) for current harmonic, unbalance and neutral current compensation. The threephase fourwire system distribution system has been used for residential, industrial and commercial centers of single/ threephase loads. The singlephase nonlinear loads have been increasing into the low voltage distribution system. It causes a current harmonic, imbalance and excessive neutral current in the local grids. The shunt active power filter takes the upper hand among the other type of harmonics compensation techniques. The odd periodic frequencies nature of the power electronics harmonic maks suitable for repetitive control. Thus, the authors proposed a secondorder odd repetitive control for 3p4w APF for compensating current harmonics, unbalance and neutral current with generating high gain only at odd harmonic frequencies. The sensitivity magnitude response of second order odd RC has deeper and wider notches at the odd fundament frequencies. It compensated the total current harmonic distortion THDi from 87.42 % to about 3.52% and neutral current of unbalanced singlephase nonlinear loads. it reduced a total current harmonic from 49.45% to 3.43% for balanced and from 50.55% to 3.24% for unbalanced three phase nonlinear loads.
Keywords: Repetitive control (RC), secondorder repetitive controL; threephase fourwire (3p4w), Active power filter (APF), neutral current, current harmonic.
Introduction
Threephase fourwire (3P4W) distribution feeders have been used for residential, industrial and commercial centers of single/ threephase loads. The singlephase power electronicsbased devices have been increasing into the low voltage distribution systems [1]. This is due to its customer attraction features for house appliances, information technology [2] and electric traction [3], [4] [5]. However, the singlephase nonlinear loads contribute a current harmonic, imbalance and excessive neutral current in the local distribution system. The third (dominant zero sequence) current harmonic contributed by each single phase nonlinear loads summed up together and results in three times the zerosequence current of phase line [6]. Moreover, the singlephasing and abnormal phase change in
the industry aggravates the situation. The current harmonics and unbalances cause economical losses on the 3p4w distribution system. It increases the neutral current, circulation of current in the delta winding of star delta /Y distribution transformer. An excessive neutralline current causes overloading of distribution transformer and feeder, rise common mode noise, flat topping voltage waveform [7]. The excessive neutralline current may raise the potential of the neutral and may causes a safety concerns for custumers [8]. It's flattopping voltage waveform also affects the normal function of precision devices. On the other hand, the harmonic effects on the phase line(s) increase harmonic distortion and power losses.
Starting from the time of 1989 [9] there have been investigating different techniques such as passive filter [10], zigzag transformer [6], series APF [8], a combination of APF with zigzag transformer [11] etc for mitigating the excessive neutral current. The shunt APF are the most promising for mitigating current harmonics, reactive and unbalanced loads [8], [12],[13],[14],[15],[16] ,[17]. The authors in [12] introduced the concept of the 3p4w APF and made simulation compared with split capacitor 3p4w APF topology. In the following year, the same authors

demonstrates its prototype for the practical test.
The instantaneous reactive power theory ( ) method proposed by Akagi [18], the synchronous reference frame ( ) method [19], are the control strategy applied for the 3p4w APFs. In the PQ method, the source only demanded to supply the load dc power. All the remaining power of loads (ac power and zero sequences components power) must compensate by the active power filter. The (id iq) method, works to accomplish the source only supplies the mean value of the direct axes component current of the loads. The comparison of the 3p4w APF control indicates that perfect harmonic cancellation (PHC) eliminates the imbalance in the source currents [20]. The 3p4w APF can compensate harmonics and unbalance caused by singlephase nonlinear loads [16]. The performance of the APF control depends on the accuracy of signal conditioning, the accuracy of the reference generation method and gating signal stages. Among these three stages, the reference generation stage is
the most important and has a diversified approach for APF application. So far different inner control approaches have been investigated for APF such as PI control, hysteresis control [21], repetitive control [22], [23],[24], predictive control [16], deadbeat control [25] etc. The conventional PI controller is unable to react frequency and phase angle variation at steadystate or under transient conditions. The hysteresis controls cause power loss and acoustic resonance [21].
Fortunately, the power electronic harmonics are odd periodic harmonic frequencies [26], i.e. the 6 Â±1, = 1,2,3, of the fundamental frequencies for threephase threewire system and odd harmonics for the 3p4w system. An odd repetitive control (RC) has a promising control for tracking and disturbance rejection on the odd period harmonic without introducing high gain at even frequencies [27]. However, the gain of the repetitive control drastically fails for deviation from the center frequency. The higher order RC (HORC) can be achieved a higher robustness for frequency variation and uncertainties in the design of active power filter [28]. The secondorder repetitive control (RC) is used to compensate for the current harmonics and neutral current . The authors proposed the higher second
order odd RC for compensation, current harmonics, reactive and neutral current in the 3p4w system. The secondorder odd RC provides a wide and deep notch at the odd fundamental frequencies which are the targeted frequencies of the 3p4w system. The rest of the paper is organized as follows: Section 2 describes of 3p4w active power filter and its mathematical model, section 3 companies the different orders of odd harmonic repetitive control and its tracking performance, section 4 presents analysis the simulation results, and finally, the conclusion was drowned in section 5.

System description and mathematical model
Fig.1 presents a threephase fourwire (3p4w) active power filter (APF) with a fourleg IGBTs based voltage source inverter (VSI) and a common capacitor for dc source. The loads are modeled with three singlephase diode rectifiers (Singlephase type1 and type2 ), threephase diode rectifier RC diode (threephase type3) and a linear resistive load as shown in Table 1. Hereafter we call it by its load type for this paper. The conventional PI controllers are used to regulate the dcbus voltage variation. The output of the PI regulator used for the magnitude reference of the source current.
source
id
P
G G G5
Isa
Isb
Isc
Isn
g
I fa
G
G
Lfa
7
ILa
ILb
ILc
ILn
I fn
L
nonlinear load balanced or unbalanced
1 3 fn
b
b
a
VDC 0 c
n
N G4
G6 G2 G8
Fig. 1 Circuit diagram of the threephase fourwire active power filter.
D1
v
v
LacD 3
m
CLm
RLm
D2 D 4

b)
Fig. 2 Diode rectifier with RC loads a) circuit diagram b) power triangle of the load.
The power capacity of the loads is also indicated in a three dimensional (3D) power triangle diagram [29] using the load parameters given in Table 1. The design system has a voltage rating of voltage is 230 V and a capacity of 5 kVA.
The zerosequence current of the load was varied by adjusting the load resistor and the load fundamental current [29].
Table 1 The load parameters for singlephase and threephase loads.
Load types
(5%)
Singlephase type1
3.2 m
1100
75
Singlephase type2
3.2 m
1100
100
Threephase type3
3.2 m
1100
100
Resistive Linear load
50
Decreasing the load resistor causes an increase in the fundamental current or system load power (i.e. the system is overloaded) and viceversa. This provides a variable zerosequence current without affecting other power variables.
The 3p4w APF is connected in parallel to the nonlinear load as seen from the PCC. The first order output ripple filters connected at the PCC with the 4leg VSI. The governing dynamic equation of the active power filter following the current direction given in Fig. 1 can be written as [29], [16]:
= 0
(1)
Where = , , for the phases and the voltage from source ground to point n of the 4LVSI is given as 0 =
0+0+0+0. Thus, taking the source voltage as
1
= +
(2)
1
= +
(2)
4
disturbance the final firstorder model of the inner plant can be obtained as (2):


COMPARISION OF THE FIRST, SECOND, AND THIRDORDER REPETITIVE CONTROL
According to the internal model principle [30], zero error signal tracking and the disturbance rejection for a periodic signal can be achieved if a representative signal generator is included in the stable closedloop system. The fullRC system has an infinite loop gain at the integer multiple of the fundamental frequencies of the disturbance. Fortunately, the harmonic of the power electronics are the odd periodic signal. The repetitive control is effective for perfect asymptotic tracking and rejection of periodic signals. However, the convention first order repetitive control loses it's gain drastically when the frequency deviates from the harmonic frequency. This due to the structure of the periodic signal generator. Steinbuch [31] proposed a general way of the periodic signal generator to achieve higher gain around the targeted frequency and its neighborhood.

z N
z N
z N
z N M
3
2
I (z)HORC 1
Fig. 3 Structure of a general Morder periodic signal generator [31].
Fig. 3 is generalized MHORC controller having M delay function weighted by M to form Mmemory loops. Thus HORC offers an enlargement of the interval around the harmonic at which IM provides a peak gain. This provides to HORC to have better performance robustness under a varying or uncertain frequency of the disturbance/reference signals. The general internal model (IM) for first, second and thirdorder odd repetitive control can be defined as [28]:
()()
() = 1+ ()()
(3)
where
Fig. 4 Magnitude response of the internal model of odd repetitive control with () = for order 1,2 and 3.
() =
/ ,
+ ( + /), . and H(z) is the FIR filter. Assuming that H(z) = 1, the poles of (3)
{ + ( + )
,
are generated infinite gain at frequencies of = ( ) /, for k = 1,2,.., N/21 for first order odd repetitive control at frequencies of = /, for k = 0, 1,2,.., N/21. These poles are evenly distributed at the circle at a point of z =
( /) for the first order and mthe ultiplicity of the higher order ,m

Bode plot for first, second, and thirdorder Repetitive control
The bode plot in Fig. 4 showed that the secondorder odd RC has a higher magnitude followed by the third order and firstorder RC. The numerical value of the maximum magnitude in dB for the first, second and thirdorder odd RCs are 4052.18, 261931.72 and 4114.20 respectively. This implies that the secondorder odd RC the most robust than the first and third order odd RCs. Of course, it is not only evaluated at a single point but also for other points in the graph showed that secondorder RC has a higher magnitude for all fundamental frequencies. The third order followed the secondorder in magnitude response. The first order odd RC has the smallest magnitude response. Therefore, the secondorder odd RC is the most robust RC than the third and the first order odd RC. This implies that the firstorder RC is unsuitable for frequency variations around the fundamental frequencies. Moreover, for higher frequencies, the thirdorder RC has some magnitude variation. Its practical implementation needs a conservation filter design for higher frequencies.
Fig. 5 Sensitivity magnitude response of first, second and thirdorder odd RC.

Sensitivity for first, second, and thirdorder Repetitive control
The sensitive magnitude response of the first, second, and thirdorder odd RC are depicted in Fig. 5. The thirdorder odd RC has higher magnitude between the fundamental frequencies. This indicates that the thirdorder RC amplifies the interharmonic between two fundamental harmonics. In this regard, the firstorder RC has the smallest magnitude response between two fundamental frequencies. However, the firstorder sensitivity magnitude response has a shallow and narrow notch at the fundamental frequencies. This characteristic also indicates
that the firstorder odd RC may not effective for frequency deviation or uncertainties. On the other hand, the second order odd RC has moderate magnitude between two fundamental frequencies. Moreover, the secondorder odd RC has deeper and wider notch at the harmonic frequencies. The thirdorder follows the secondorder odd RC as shown in the magnified plot of Fig. 5. The more the deeper notch means the more attenuate the error signals. In this aspect, the secondorder odd RC has better characteristics for frequencies variation and interharmonic frequency. The sensitivety function is computed as (4):
1 ()()
() = 1 ()()( () () 1)
(4)
Repetitive controller
IHORC ( z)
D(z)
R(z)
E(z)
Whorc (z)
H (z)
Gx (z)
Ci (z)
U (z)
Gp( z)
Y (z)
Fig. 6 pluge in Highorder Repetitive controller
Fig. 6 is the control diagram of the plugin high order RC consists of the internal model of the, IHORC(z) , lowpass filter H(z) and the stabilizing controller Gx(z). The low pass filter guarantees the stability of the contrl by the attention of high order signals beyond the targeted frequency. A constant gain low than unity can be used
instead of the lowpass filter. But it provides a lowers gain at the required points and will not suppress effectively at
higher frequencies as well. Finally, the learning controller, () and the plant model () are connected inseries to the HORC. The design procedure of HORC follows the same procedure as the other RC. The controller
() must meet a sufficient margin for the loop without RC. The transform function of the system without HORC can be written as
()()
() = 1 + () ()
(5)

Tracking capacity of the first, second, and thirdorder Repetitive control
The performance of the tracking ability if the three odd RCs were evaluated using a mixed harmonic signal composed of the fundamental, third, fifth and seventh order harmonic frequencies
v(s) = v1(s) + v3(s) + v5(s) + v7(s)
(6)
The numerical value of the harmonic signal where v1(s) = 10 sin(wt + 1 ) + 3.33 sin(3wt + 3) + 0.2 sin(5wt + 5) +
1.43 sin (7wt + 7 ).
Fig. 7 Comparisons of tracking capacity of first, second, and thirdorder odd RCs.
The tranking performance of the three odd RC harmonic are good as shown in Fig. 7 and Fig. 8. Both odd RCs have showen a good traking performance for a typical high distorted haronic input signal.
Fig. 8 Comparisons of error for tracking capacity of first, second and thirdorder odd RCs.
TABLE 2. SYSTEM DATA FOR SIMULATION
Description
Symbol
unit
Total capacity
S
5090 VA
Source voltage
VLL
400 V
System frequency
f
50Hz
source resistance
0.1 (0.4%)
Source inductance
0.3 mH (0.5%)
Feeder reactance
3.2 mH (5%)
Load Power Factor
Pf
0.77
Dc bus voltage
Vdc
680V
Load capacitance
Cdc
2660F
Filter reactance
Lf
3 mH
APF Rating = 2.4kVA,



SIMULATION RESULTS AND DISCUSSION The threephase fourwire system was modeled in

MATLAB/Simulink toolboxes with three singlephase nonlinear loads, threephase nonlinear load and a linear resistive load. Two singlephase diode rectifiers with RC were used to simulate and design the proposed system. The two singlephase diode rectifiers with RC nonlinear loads
are differed only by load resistor (i.e. 75 and 100 and keeping the other parameter of the load constant) to modifying only the fundamental current without changing the active and reactive power for both loads. This helps to take in to account the performance of the proposed control for the zerosequence harmonics only [29].
A fourleg IGBT based threephase converter was used for circuits modeling with a common capacitor at the dc side. The simulation incorporates balanced and unbalanced loads of threephase and/or singlephase nonlinear loads.

Balanced three singlephase nonlinear loads without compensation
A balanced three singlephase nonlinear RC loads are connected to the threephase source system to collect the
harmonic spectrum of the RCs loads. The waveforms in Fig. 9 showed that the source current is highly distorted for singlephase type1. Moreover, the current magnitude of the neutral current is the same as the current of the phase lines but the neutral line with zerosequence harmonic distortion. The total source current harmonic distortion THDi was found to be 80.32 %. Fig. 9 demonstrates that the system current harmonic for singlephase type1 before the proposed control has been connected to the system.
Fig. 9 Waveform of the source phase and neutral current with balanced three singlephase type1 loads.
The neutral and phase current have the same magnitude of 7.52 A as demonstrated in Fig. 9. However, the phase lines have distorted with odd harmonics (i.e. 3 = 71.53% 5 = 33.63% 7 = 9.71% and 9 = 7.99% )
and a total current harmonic distortion of 80.32% as shown in Fig. 10. The neutral line only distorted with the zero sequence current. Unfortunately, all the zero sequence
currents of the phase lines were not canceled out at the junction of neutral line instead they summed up and create burden for the system. Notice that the wellknown theory of a balanced threephase system has zero neutral current works only for linear loads. Therefore, balancing the three phase loads will not guarantee zero neutral currents.
Fig. 10 Harmonic spectrum for the balanced three Singlephase type1 type nonlinear loads.

Balanced three singlephase nonlinear loads with compensation
Fig. 11 depicted that a proposed control connected to the system with balanced three Singlephase type1 The waveforms in Fig. 11 showed that the source current is composted from the current distortion caused by the
nonlinear load of singlephase type1. Moreover, the current magnitude of the phases was reduced from 7.52 to
7.49 A and that of the neutral current reduced to almost zero current. The total source current harmonic distortion THDi was reduced from 80.32 % to 4.20% as shown in Fig. 12.
Fig. 11 The waveform of the source current with a balanced three Singlephase type1 type nonlinear loads with RC compensation.
Fig. 12 Harmonic spectrum of the balanced three Singlephase type1 type nonlinear loads with RC compensator.
Fig. 13 The waveform of the source current with unbalanced three singlephase RCs nonlinear loads with secondorder RC compensation connected to the system at 0.3 seconds. From top to down: (a) source voltage (V), (b) compensation current (A), (C) Load current (A) and (d) source current (A).

Unbalanced load with three singlephase nonlinear and a resistive linear loads
An unbalanced nonlinear system was formed by connecting phaseA with singlephase type2, phaseB with Single phase type1 and phaseC with Singlephase type1 with a resistive linear load at its ac side. The singlephase type2 load changes the zerosequence current of the phaseA and the linear resistive load changes the current of phaseC. The proposed RC compensator was connected to the system at 0.3 seconds as shown in Fig. 13. The waveform in Fig. 13 demonstrates from top to down that the source voltage (V), compensation current (A), the load current (A) and (d) source current (A). The unbalance and distortion of
the threephase current was collected without the RC compensation (i.e. before 0.3 seconds). The total current harmonic distortion THDi were phase A, phase B and phase C 87.42%, 87.42%, and 47.91% respectively. However, after the RC compensator was connected (i.e. after 0.3 seconds) the system load become balanced and the total current harmonic THDi compensated phase A from 87.42% to 3.51%, phase B from 87.42% to 3.52% and phase C from 47.91% to 4.48% as shown in Table 3. These achieved a compensaton of 83.89%, 83.9% and 43.43% in phase A, phase B, and phase C respectively. At the same time the neutral current reduced from 7.95 A (before compensation) and 0.03 A(after compensation).
Table 3 Current harmonic in an unbalance three singlephase nonlinear loads.
Unbalanced load with three singlephase nonlinear and a resistive linear loads
A
B
C
w/out
RC
Achieved
w/out
RC
Achieved
w/out
RC
Achieved
1
5.85
9.6
*
7.707
9.6
*
14.01
9.72
*
3
78.16
0.59
77.57
75.41
0.69
74.72
41.33
0.72
40.61
5
46.67
1.01
45.66
40.38
1.12
39.26
22.13
0.68
21.45
7
27.86
0.42
27.44
13.50
0.41
13.09
7.40
0.48
6.92
8.
7.43
0.18
7.25
7.96
0.13
7.83
4.37
0.06
4.31
11
6.28
0.18
6.10
6.28
0.07
6.21
3.44
0.12
3.32
THD
87.42%
3.51
83.89%
87.42%
3.52
83.9%
47.91%
4.48
43.43%
Neutral current reduce from 7.95A 0.03 A

Balanced Threephase nonlinear loads.
A balanced threephase system is created with a three phase type3 connecting with the threephase source. The current harmonic spectrum of the system and the waveforms are shown in Fig. 14. The total source current harmonic distortion THDi was found to be 49.45% before the compensation is connected to the system. However, after
the compensator is connected to the system at 0.3 seconds the system total harmonic reduced from 49.45% to 3.435 for the threephases. The individual nonzerosequence odd current harmonics are presented in Table 3. However, the zerosequence currents are almost zero in balanced three phase nonlinear loads.
.
Fig. 14 The waveform of the source current with balanced Threephase type3 nonlinear loads with RC compensation connected to the system at 0.3 seconds.
From top to down: (a) source voltage (V), (b) compensation current (A), (c) Load current (A) and (d) source current (A).

unbalanced threephase nonlinear loads.
The unbalanced threephase system is created with a three phase type3 and a linear load connecting with phaseC to the threephase source. The unbalance nonlinear load creates an unbalance current of 5.64, 6.71, and 11.86 A for
phaseA, phaseB, and phaseC respectively. The zero sequence current before the compensation connected to the system has a very neutral current of 6.48 A and reduced to 0.01A after compensation. The reduction of neutral current in the threephase fourwire system has direct practical
implication in the lives of sensitive electronic devices, circulating current in Y distribution transformer etc. On the other hand, the current harmonic distortions before compensation in phase A, phase B and phase C were 50.55%, 49.46%, and 26.62 %. When the proposed compensator was connected to the system at 0.3 seconds,
the threephase source current harmonic distortion reduced to from 50.55%, to 3.24% for phase A, from 49.46% to 3.17% for phase B and from 26.62%to 2.91 % for phase C. At the same time, the magnitude of the phase C current reduce from 11.86 to 10.29 A after compensation.
Table 3 Current harmonic in a balance and unbalance threephase rectifier loads.
Balanced threephase nonlinear load 
Unbalanced threephase nonlinear loads 

without APF 
With APF 
Without APF 
With APF 

A 
A 
A 
B 
C 
A 
B 
C 

1 
5.95 
10.27 
5.64 
6.71 
11.86 
10.28 
10.29 
10.29 
3 
* 
* 
14.97 
5.26 
8.26 

5 
46.08 
0.28 
42.62 
40.47 
22.93 
0.15 
0.14 
0.06 
7 
19.74 
0.37 
20.00 
18.82 
9.01 
0.24 
0.10 
0.21 
9 
* 
* 
4.13 
0.52 
1.86 

11. 
7.97 
0.06 
7.21 
6.68 
4.34 
0.03 
0.03 
0.02 
13 
4.47 
0.13 
4.08 
6.70 
1.76 
0.13 
0.04 
0.09 
THD 
49.45% 
3.43% 
50.55% 
49.46 
26.62% 
3.24% 
3.17% 
2.91% 
Neutral 
The neutral current reduced from 6.48 to 0.01. 
Fig. 15 The waveform of the source current with unbalanced threephase RCs nonlinear loads with secondorder RC compensation connected to the system at
0.3 seconds. From top to down: (a) source voltage (V), (b) compensation current (A), (C) Load current (A) and (d) source current (A).
CONCLUSION
The repetitive control can be implemented in the first, second and thirdorder harmonics. The first order odd repetitive drastically fails for any frequency variation from the center frequency. The thirdorder odd harmonics have wider and higher gains around the fundamental frequency. However, it is unsuitable at higher frequencies. The secondorder fond to be the optimal odd repetitive control for odd harmonic frequencies. Its sensitivity magnitude response has deeper and wider notches at the fundament frequencies. Moreover, it has a high gain for lower dominant frequencies and decreases its gain evenly for unwanted higher odd fundamental frequencies. Therefore, the performance of the secondorder odd RC was intensively investigated for a threephase fourwire active power filter against the current harmonics, unbalanced and
neutral current compensation. The result indicates that the secondorder odd RC compensates the total current harmonic distortion THDi from 87.42 % to about 3.52% for unbalanced singlephase nonlinear loads. At the same time, it also compensated the neutral current from the phase current level to a negligible current level. In the case of the balanced threephase nonlinear loads, it achieved a total current harmonic compensatio of 49.45% to 3.43% and from 50.55% to 3.24% for unbalanced threephase nonlinear loads.
ACKNOWLEDGMENTS
The project is supported by the National Natural Science Foundation of China (No.51777025); and the Fundamental Research Funds for the Central Universities (No.DUT19ZD219).
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