Citation: Qian, G.; Wang, Q.; He, S.;

Dai, W.; Wei, N.; Zhou, N. Harmonic

Modeling and Analysis for Parallel

12-Pulse Rectifier under Unbalanced

Voltage Condition in Frequency-

Domain. Energies 2022, 15, 3946.

https://doi.org/10.3390/en15113946

Academic Editors: Dayi Li and Anna

Rita Di Fazio

Received: 14 March 2022

Accepted: 11 May 2022

Published: 26 May 2022

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energies

Article

Harmonic Modeling and Analysis for Parallel 12-Pulse Rectifierunder Unbalanced Voltage Condition in Frequency-DomainGuochao Qian 1, Qianggang Wang 2,*, Shun He 1, Weiju Dai 1, Nengqiao Wei 2 and Niancheng Zhou 2

1 Electric Power Research Institute, Yunnan Power Grid Co., Ltd., Kunming 650217, China;qianguochao@126.com (G.Q.); heshun@me.com (S.H.); weijudai@163.com (W.D.)

2 State Key Laboratory of Power Transmission Equipment & System Security and New Technology,Chongqing University, Chongqing 400044, China; weinengqiao@163.com (N.W.);cee_nczhou@cqu.edu.cn (N.Z.)

* Correspondence: qianggang1987@cqu.edu.cn; Tel.: +86-136-4055-8474

Abstract: This paper treats a parallel 12-pulse rectifier as the study subject. The influence of unbal-anced terminal voltage, the triggering angle, as well as the commutation angle of a thyristor-controlledrectifier circuit are considered in switching functions. Then, a frequency-domain harmonic coupledadmittance matrix under the effect of positive sequence and negative sequence voltages are proposed,respectively. With the integration of the above matrices, a harmonic coupled admittance matrix modelof a parallel 12-pulse rectifier under unbalanced voltage is proposed. On this basis, a simplifiedharmonic model of a 12-pulse rectifier is established, which only considers the effect of major elementsthrough analyzing the elements’ distribution and magnitude of the harmonic coupled admittance ma-trix, as well as the AC current harmonic producing mechanism. All the models proposed in this paperare verified with high accuracy through simulation in MATLAB/Simulink. The models proposed inthis paper can be used to guide the selection of harmonic suppression measures and equipment.

Keywords: 12-pulse rectification; unbalanced voltage; harmonic model; commutation overlap angle;coupled admittance matrix

1. Introduction

With the development of the power system, power electronic devices are widelyused [1–3]. As the interface of the AC grid and the DC load, thyristor rectifiers have becomethe most common converter [4]. However, with strong nonlinearity and time-variabilitycharacteristics, the rectifier causes serious distortion of the voltage and current on thegrid side. These harmonics may cause damage to the equipment of the power systemand loads [5,6]. There are two ways to suppress the harmonics generated by thyristorrectifiers. One is reducing the harmonics by installing a harmonic compensation device [7].This measure can significantly increase the size of the rectifier system, operating costs,losses, and reduce system reliability, but it is not applicable in many scenarios. The otherway is optimizing the rectifier [8–10] to reduce harmonics production. Typically, 12-pulseseries-type and parallel-type rectifiers are the optimal method. Compared with the 6-pulserectifier, the output current of the parallel 12-pulse rectifier on the DC side is doubled,which is very common in electrolysis, metallurgy, and other industrial occasions. In orderto accurately calculate the current harmonics generated by such devices and guide thedesign of multipulse rectifiers and the configuration of harmonic suppressing measures, itis necessary to establish a fast and practical harmonic analysis model.

Methods such as time-domain simulation [11–13] and physical experiments [14] aremainly used to analyze the harmonic characteristics of thyristor rectifiers. However, thesemethods have shortcomings such as long operation time and excessive calculation amount.In order to solve this problem, the literature [15,16] has established a harmonic coupling ad-mittance matrix model of a three-phase phase-controlled rectifier by the frequency-domain

Energies 2022, 15, 3946. https://doi.org/10.3390/en15113946 https://www.mdpi.com/journal/energies

Energies 2022, 15, 3946 2 of 9

analysis method. This model can illustrate the mechanism of harmonics generation directly.The literature [17] has applied the harmonic admittance matrix to the calculation of har-monic power flow and reduced the number of calculation iterations. The literature [18] hasestablished a three-phase uncontrolled rectifier based on the harmonic coupling admittancemodel. The above research does not consider the three-phase unbalance of the power grid.The negative sequence components generated by the unbalanced voltage of the powergrid have a great influence on the harmonic characteristics of the AC and DC sides. If theunbalanced condition is ignored, the calculation accuracy of the model will be reduced [19].The literature [20] has analyzed the harmonics characteristics of a 6-pulse rectifier underthe three-phase voltage unbalanced condition and calculates the harmonics with improvedswitching function. However, since the 12-pulse rectifier is composed of a phase-shiftingtransformer and two 3-phase rectifier bridges, there is a 30 phase difference betweenthe 3-phase voltages of the transformer windings, and part of the harmonics of the tworectifiers can cancel each other. The literature [19] has deduced the harmonic couplingadmittance matrix of a series of 12 pulse under positive sequence and negative sequencevoltages and established the frequency-domain harmonic model of a 12-pulse rectifierunder unbalanced voltage. However, the study of a frequency-domain harmonic model ofa parallel 12-pulse phase-controlled rectifier remains blank.

This paper analyzes the working principle of the parallel 12-pulse phase-controlledrectifier, establishes a balanced reactor decoupling equivalent circuit, and corrects theswitching function by considering the commutation overlap angle and the trigger angle.The AC current of the 12-pulse rectifier under the action of negative sequence voltageand negative sequence voltage is decomposed to obtain its harmonic coupling admittancematrix. Then, the frequency-domain harmonic analysis model of the parallel 12-pulserectifier under unbalanced voltage is established.

2. Operating Principle and Derivation of AC Current of Parallel 12-Pulse Rectifierunder Unbalanced Voltage

The structure of the parallel 12-pulse phase-controlled rectifier is shown in Figure 1a.The power supply side has Y/Y/4 three-winding transformers, two three-phase phase-controlled rectifier bridges with the same structure and parameters (referred to as bridge Iand bridge II), and a balanced reactor. After the power supply voltage is stepped down bythe star-shaped and delta-connected transformers through secondary winding, two sets ofAC voltages with the same size and a phase difference of 30 are generated, and after therectifier bridge, the DC voltages u1dc and u2dc are obtained. The average value of the twovoltages is the same, but the phase of the instantaneous waveform is 30 staggered, and theload must be powered simultaneously through a balancing reactor. The balancing reactor isa coupled inductance element that can withstand the difference between the line voltagesof the two rectifiers working at the same time to ensure that the two groups of rectifierbridge thyristors can be simultaneously on. Due to the action of the mutual inductance M12,the balancing reactor balances the DC currents i1dc and i2dc of the two groups of rectifierbridges by the induced electromotive force and synthesizes the load current idc = i1dc + i2dc,and then the rectifier bridge is then coupled with the AC power grid to cause harmonicdistortion of the AC current iabc of the 12-pulse rectifier.

In Figure 1a, the inductances on both sides of the balanced reactor are L1 and L2, andthe mutual inductance is M12. After decoupling the balanced reactor, the equivalent circuitof the parallel 12-pulse rectifier in Figure 1b is obtained. Compared with the derivationprocess of the series 12-pulse rectifier [19], Figure 2 shows the decoupling process of IPTand the calculation of the DC current of each rectifier bridge, which is required to analyzethe parallel 12-pulse rectifier.

Energies 2022, 15, 3946 3 of 9Energies 2022, 15, x FOR PEER REVIEW 3 of 9

L

R

u1dc

u2dc

iabc

i1abc

i2abc

i1dc

Lag 30°

Ⅰ

ⅡY

Y

T1: 3k

T1 : k

i2dc

idc

L2

L1M12

(a)

L

R

u1dc

u2dc

iabc

i1abc

i2abc

i1dc

Lag 30°

Ⅰ

ⅡY

Y

T1: 3k

T1 : k

i2dc

idc

L1+M12

M12L2+M12

(b)

Figure 1. Circuits of a parallel 12‐pulse rectifier: (a) equivalent circuit of a parallel 12‐pulse rectifier;

(b) interphase transformer (IPT) decoupled equivalent circuit of a parallel 12‐pulse rectifier.

In Figure 1a, the inductances on both sides of the balanced reactor are L1 and L2, and

the mutual inductance is M12. After decoupling the balanced reactor, the equivalent cir‐

cuit of the parallel 12‐pulse rectifier in Figure 1b is obtained. Compared with the deriva‐

tion process of the series 12‐pulse rectifier [19], Figure 2 shows the decoupling process of

IPT and the calculation of the DC current of each rectifier bridge, which is required to

analyze the parallel 12‐pulse rectifier.

Figure 2. Harmonic analysis process of a parallel 12‐pulse rectifier.

The input voltages of bridges I and II are u1abc and u2abc, respectively. To analyze the

harmonics of a parallel 12‐pulse rectifier under unbalanced voltages, the voltage switch‐

ing functions Su1abc and Su2abc and the current switching functions Si1abc and Si2abc are cor‐

rected, which take the commutation overlap angle caused by the leakage inductance of

the transformer and the firing angle offset caused by the unbalanced voltage into consid‐

eration. Based on the input voltage and voltage modulation function of the parallel 12‐

pulse rectifier, the DC voltages u1dc and u2dc of bridges I and II are obtained:

1dc 1a u1a 1b u1b 1c u1c

1dc0 1dcn

2dc 2a u2a 2b u2b 2c u2c

2dc0 2dcn

( ) ( ) ( )+ ( ) ( )+ ( ) ( )

( )

( ) ( ) ( )+ ( ) ( )+ ( ) ( )

( )

u t u t S t u t S t u t S t

U u t

u t u t S t u t S t u t S t

U u t

(1)

where: U1dc0 and U2dc0 are the average values of DC voltages of bridges I and II, and u1dcn(t)

and u2dcn(t) are the DC voltage fluctuation components, respectively. In order to simplify the

Figure 1. Circuits of a parallel 12-pulse rectifier: (a) equivalent circuit of a parallel 12-pulse rectifier;(b) interphase transformer (IPT) decoupled equivalent circuit of a parallel 12-pulse rectifier.

Energies 2022, 15, x FOR PEER REVIEW 3 of 9

L

R

u1dc

u2dc

iabc

i1abc

i2abc

i1dc

Lag 30°

Ⅰ

ⅡY

Y

T1: 3k

T1 : k

i2dc

idc

L2

L1M12

(a)

L

R

u1dc

u2dc

iabc

i1abc

i2abc

i1dc

Lag 30°

Ⅰ

ⅡY

Y

T1: 3k

T1 : k

i2dc

idc

L1+M12

M12L2+M12

(b)

Figure 1. Circuits of a parallel 12‐pulse rectifier: (a) equivalent circuit of a parallel 12‐pulse rectifier;

(b) interphase transformer (IPT) decoupled equivalent circuit of a parallel 12‐pulse rectifier.

In Figure 1a, the inductances on both sides of the balanced reactor are L1 and L2, and

the mutual inductance is M12. After decoupling the balanced reactor, the equivalent cir‐

cuit of the parallel 12‐pulse rectifier in Figure 1b is obtained. Compared with the deriva‐

tion process of the series 12‐pulse rectifier [19], Figure 2 shows the decoupling process of

IPT and the calculation of the DC current of each rectifier bridge, which is required to

analyze the parallel 12‐pulse rectifier.

Figure 2. Harmonic analysis process of a parallel 12‐pulse rectifier.

The input voltages of bridges I and II are u1abc and u2abc, respectively. To analyze the

harmonics of a parallel 12‐pulse rectifier under unbalanced voltages, the voltage switch‐

ing functions Su1abc and Su2abc and the current switching functions Si1abc and Si2abc are cor‐

rected, which take the commutation overlap angle caused by the leakage inductance of

the transformer and the firing angle offset caused by the unbalanced voltage into consid‐

eration. Based on the input voltage and voltage modulation function of the parallel 12‐

pulse rectifier, the DC voltages u1dc and u2dc of bridges I and II are obtained:

1dc 1a u1a 1b u1b 1c u1c

1dc0 1dcn

2dc 2a u2a 2b u2b 2c u2c

2dc0 2dcn

( ) ( ) ( )+ ( ) ( )+ ( ) ( )

( )

( ) ( ) ( )+ ( ) ( )+ ( ) ( )

( )

u t u t S t u t S t u t S t

U u t

u t u t S t u t S t u t S t

U u t

(1)

where: U1dc0 and U2dc0 are the average values of DC voltages of bridges I and II, and u1dcn(t)

and u2dcn(t) are the DC voltage fluctuation components, respectively. In order to simplify the

Figure 2. Harmonic analysis process of a parallel 12-pulse rectifier.

The input voltages of bridges I and II are u1abc and u2abc, respectively. To analyze theharmonics of a parallel 12-pulse rectifier under unbalanced voltages, the voltage switchingfunctions Su1abc and Su2abc and the current switching functions Si1abc and Si2abc are cor-rected, which take the commutation overlap angle caused by the leakage inductance of thetransformer and the firing angle offset caused by the unbalanced voltage into consideration.Based on the input voltage and voltage modulation function of the parallel 12-pulse rectifier,the DC voltages u1dc and u2dc of bridges I and II are obtained:

u1dc(t) = u1a(t)Su1a(t) + u1b(t)Su1b(t) + u1c(t)Su1c(t)= U1dc0 + u1dcn(t)

u2dc(t) = u2a(t)Su2a(t) + u2b(t)Su2b(t) + u2c(t)Su2c(t)= U2dc0 + u2dcn(t)

(1)

where: U1dc0 and U2dc0 are the average values of DC voltages of bridges I and II, andu1dcn(t) and u2dcn(t) are the DC voltage fluctuation components, respectively. In order tosimplify the analysis, the DC current generated by the two DC voltages is divided into twoparts—the average components and the fluctuation components—that satisfy:

i1dc(t) = I1dc0 + i1dcn(t) (2)

i2dc(t) = I2dc0 + i2dcn(t) (3)

Energies 2022, 15, 3946 4 of 9

In Figure 1a, R is the DC load resistance, and L is the DC load inductance. Let Ls1 =L1 + M12, Ls2= L2 + M12, Ls = L −M12, the average components of each DC current can beobtained as,

I1dc0 = I2dc0 =U1dc0 + U2dc0

4R(4)

The DC current fluctuation component can be transformed into a plurality of fre-quency components, which are generated by the DC voltage fluctuation component ofthe corresponding frequency. To calculate the h order frequency phasor of the DC currentfluctuation component, the frequency-domain method is used:

.I1dch = U1dch∠ϕ1dch

jhωLs1+jhωLs2//(R+jhωLs)

− U2dch∠ϕ2dchjhωLs2+jhωLs1//(R+jhωLs)

R+jhωLsjhωLs1

(5)

.I2dch = −U1dch∠ϕ1dch

jhωLs1+jhωLs2//(R+jhωLs)R+jhωLs

jhωLs2

+ U2dch∠ϕ2dchjhωLs2+jhωLs1//(R+jhωLs)

(6)

where h = 1, 5, 7, 11, . . . , ω is the fundamental angular frequency, and U1dch, ϕ1dch,U2dch,and ϕ2dch are the effective values and phases of the h order harmonics of the DCvoltage fluctuation. By multiplying the DC current by the switching function of eachphase current, the AC side currents i1abc(t) and i2abc(t) of bridges I and II can be calculated.Taking phase A as an example, according to the transformation ratio of the three-windingtransformer, the time-domain expression of the AC current of the parallel 12-pulse rectifieris obtained.

ia(t) = i1a(t) + i2a(t)= kTi1dc(t)Si1a(t) +

kT√3

i2dc(t)[Si2a(t)− Si2b(t)](7)

In a similar way, the AC currents of phases B and C of the parallel 12-pulse rectifierscan be calculated.

3. Frequency-Domain Harmonic Model of Parallel 12-Pulse Rectifier3.1. Frequency-Domain Harmonic Model under Unbalanced Voltage

Through the derivation process mentioned above, the frequency-domain harmonicmodel is as follows. The relationship between AC harmonic current and terminal harmonicvoltage satisfies:

I+ = Y+U+ + Y+∗U+∗

I− = Y−U− + Y−∗U−∗(8)

where Y+, Y+∗ and Y−, Y−∗ are harmonic coupled admittance matrices under positivesequence voltage and negative sequence voltage, respectively; U+, U+∗ and U−, U−∗ arethe positive sequence voltage and the negative sequence voltage as well as their conjugatevector, respectively; I+ and I− are the positive sequence current vector and the negativesequence current vector of the AC side.

The frequency-domain harmonic model of the parallel 12-pulse rectifier can be ob-tained by adding each harmonic current together. The k order harmonic current is,

.Ik =

.I+

k +.I−k

= M(y+k U+ + y+∗k U+∗) + y−k U− + y−∗k U−∗(9)

where yk+, yk+∗ and yk−, yk−∗ are k-th roe vector of Y+, Y+∗ and Y−, Y−∗.

3.2. Simplification of Frequency-Domain Harmonic Model

The harmonic model can be simplified by analyzing the distribution and magnitudeof the harmonic coupled admittance matrix elements, as shown in Figures 3 and 4 (themagnitude of element y+

1,1 is considered 1).

Energies 2022, 15, 3946 5 of 9

Energies 2022, 15, x FOR PEER REVIEW 5 of 9

The frequency‐domain harmonic model of the parallel 12‐pulse rectifier can be ob‐

tained by adding each harmonic current together. The k order harmonic current is,

( )

k k k

k k k k

I I I

M y U y U y U y U

(9)

where yk+, yk+* and yk−, yk−* are k‐th roe vector of Y+, Y+* and Y−, Y−*.

3.2. Simplification of Frequency‐Domain Harmonic Model

The harmonic model can be simplified by analyzing the distribution and magnitude

of the harmonic coupled admittance matrix elements, as shown in Figures 3 and 4 (the

magnitude of element y+1,1 is considered 1). Hence, the simplified harmonic model, which only considers the effect of the main

elements, is as follows:

1 1 1 11 1

* *'' ''

h h h h

k kk k kk k kk k kk k

I y U y U y U

I y U y U y U y U

(10)

voltage harmonic order

ampl

itud

e/pu

current harmonic order

voltage harmonic order

ampl

itud

e/pu

current harmonic order

(a) (b)

Figure 3. Positive sequence harmonic coupled admittance matrix elements of parallel 12‐pulse rec‐

tifier. (a) Matrix Y+; (b) Matrix Y+*.

1020

3010

20

30

0

0.5

1

Voltage harmonic order

Am

plit

ude/

pu

Current harmonic order

1020

3010

20

30

0

0.5

1

Voltage harmonic order

Am

plit

ude/

pu

Current harmonic order

(a) (b)

Figure 4. Negative sequence harmonic coupled admittance matrix elements of parallel 12‐pulse rec‐

tifier. (a) Matrix Y−; (b) Matrix Y−*.

Figure 3. Positive sequence harmonic coupled admittance matrix elements of parallel 12-pulserectifier. (a) Matrix Y+; (b) Matrix Y+∗.

Energies 2022, 15, x FOR PEER REVIEW 5 of 9

The frequency‐domain harmonic model of the parallel 12‐pulse rectifier can be ob‐

tained by adding each harmonic current together. The k order harmonic current is,

( )

k k k

k k k k

I I I

M y U y U y U y U

(9)

where yk+, yk+* and yk−, yk−* are k‐th roe vector of Y+, Y+* and Y−, Y−*.

3.2. Simplification of Frequency‐Domain Harmonic Model

The harmonic model can be simplified by analyzing the distribution and magnitude

of the harmonic coupled admittance matrix elements, as shown in Figures 3 and 4 (the

magnitude of element y+1,1 is considered 1). Hence, the simplified harmonic model, which only considers the effect of the main

elements, is as follows:

1 1 1 11 1

* *'' ''

h h h h

k kk k kk k kk k kk k

I y U y U y U

I y U y U y U y U

(10)

voltage harmonic order

ampl

itud

e/pu

current harmonic order

voltage harmonic order

ampl

itud

e/pu

current harmonic order

(a) (b)

Figure 3. Positive sequence harmonic coupled admittance matrix elements of parallel 12‐pulse rec‐

tifier. (a) Matrix Y+; (b) Matrix Y+*.

1020

3010

20

30

0

0.5

1

Voltage harmonic order

Am

plit

ude/

pu

Current harmonic order

1020

3010

20

30

0

0.5

1

Voltage harmonic order

Am

plit

ude/

pu

Current harmonic order

(a) (b)

Figure 4. Negative sequence harmonic coupled admittance matrix elements of parallel 12‐pulse rec‐

tifier. (a) Matrix Y−; (b) Matrix Y−*.

Figure 4. Negative sequence harmonic coupled admittance matrix elements of parallel 12-pulserectifier. (a) Matrix Y−; (b) Matrix Y−∗.

Hence, the simplified harmonic model, which only considers the effect of the mainelements, is as follows:

.I1 ≈ y+1hUh

+ + y+∗1hUh+∗ + y−∗11U1

−∗.Ik ≈ y+kk

.U

+

k + y+kk′.

U+

k′ + y−∗kk.

U−∗k + y−∗kk′′Uk′′

−∗(10)

4. Simulation Verification

An electromagnetic transient model is boiled in MATLAB/SIMULINK to verify thefrequency-domain harmonic model of the parallel 12-pulse rectifier proposed in this paper.The simulation parameters are shown in Table 1.

Table 1. Simulation parameters of 12-pulse rectifier.

Case Fire Angle α/ THDu/% γ Overlap Angle µ/ Ua1/V R/Ω L1,L2/H L/H H

1 10 1.94 0.1 0 380 5 0.42 0.02 302 30 4.79 0.20 10 380 10 0.5 0.2 30

Figure 5 shows the waveforms of the AC current of the 12-pulse rectifier, ia(t), ia1(t),and ia2(t), respectively, with different parameters. The current calculated by the modelproposed in this paper, which considers the influence of the negative sequence voltage(complete model), is matched with the simulation results in cases 1 and 2. However, thewaveform that does not consider the influence of the negative sequence voltage (half model)

Energies 2022, 15, 3946 6 of 9

is obviously different from the complete model and simulation result in case 2 (which hasa bigger negative sequence). The peak value of the simulation, complete model, and halfmodel of ia(t) of case 2 are 83.5A, 83A, and 70A, respectively.

Energies 2022, 15, x FOR PEER REVIEW 6 of 9

4. Simulation Verification

An electromagnetic transient model is boiled in MATLAB/SIMULINK to verify the

frequency‐domain harmonic model of the parallel 12‐pulse rectifier proposed in this pa‐

per. The simulation parameters are shown in Table 1.

Table 1. Simulation parameters of 12‐pulse rectifier.

Case Fire Angle α/° THDu/% γ Overlap Angle

μ/° Ua1/V R/Ω L1,L2/H L/H H

1 10° 1.94 0.1 0° 380 5 0.42 0.02 30

2 30° 4.79 0.20 10° 380 10 0.5 0.2 30

Figure 5 shows the waveforms of the AC current of the 12‐pulse rectifier, ia(t), ia1(t),

and ia2(t), respectively, with different parameters. The current calculated by the model

proposed in this paper, which considers the influence of the negative sequence voltage

(complete model), is matched with the simulation results in cases 1 and 2. However, the

waveform that does not consider the influence of the negative sequence voltage (half

model) is obviously different from the complete model and simulation result in case 2

(which has a bigger negative sequence). The peak value of the simulation, complete

model, and half model of ia(t) of case 2 are 83.5A, 83A, and 70A, respectively.

50

0

-50

50

0

-50

80

0

-80

(a)

Energies 2022, 15, x FOR PEER REVIEW 7 of 9

50

0

-50

50

0

-50

80

0

-80

(b)

Figure 5. Time‐domain current comparison of rectifier under different conditions. (a) Case 1; (b)

Case 2.

The amplitude and phase of the harmonic current of the 12‐pulse rectifier are shown

in Figure 6. The current main contains 12 k ± 1th harmonic current. With the influence of

negative sequence, 3rd, 9th, 15th…, the harmonic current is observed in the simulation

and complete model image.

0 5 10 15 20 25 300

40

80

0 5 10 15 20 25 30

-3

0

3

Pha

se /

rad

Am

plit

ude

/A

(a)

Figure 5. Time-domain current comparison of rectifier under different conditions. (a) Case 1; (b) Case 2.

The amplitude and phase of the harmonic current of the 12-pulse rectifier are shownin Figure 6. The current main contains 12 k ± 1th harmonic current. With the influence ofnegative sequence, 3rd, 9th, 15th . . . , the harmonic current is observed in the simulationand complete model image.

Energies 2022, 15, 3946 7 of 9

Energies 2022, 15, x FOR PEER REVIEW 7 of 9

50

0

-50

50

0

-50

80

0

-80

(b)

Figure 5. Time‐domain current comparison of rectifier under different conditions. (a) Case 1; (b)

Case 2.

The amplitude and phase of the harmonic current of the 12‐pulse rectifier are shown

in Figure 6. The current main contains 12 k ± 1th harmonic current. With the influence of

negative sequence, 3rd, 9th, 15th…, the harmonic current is observed in the simulation

and complete model image.

0 5 10 15 20 25 300

40

80

0 5 10 15 20 25 30

-3

0

3

Pha

se /

rad

Am

plit

ude

/A

(a)

Energies 2022, 15, x FOR PEER REVIEW 8 of 9

0 5 10 15 20 25 300

40

80

0 5 10 15 20 25 30-4

-2

0

2

4

Pha

se /r

adA

mpl

itude

/A

(b)

Figure 6. AC current harmonic spectrum of 12‐pulse rectifier under different conditions. (a) Case

1; (b) Case 2.

Figure 7 shows the error analysis results with case 1′s parameters, and the results of

the complete model are considered the real value. It is obvious that the simplified model

is much more accurate than the half model.

20 40 60 800

5

10

15 Half model Simplified model

Rel

ativ

e er

ror/

%

Triggering angle /rad

Figure 7. Error analysis of models.

5. Conclusions

This paper established a frequency‐domain harmonic coupled admittance matrix

model of a parallel 12‐pulse rectifier. This model can efficiently calculate the harmonic

current produced by the rectifier under different terminal voltage conditions, including

negative sequence voltage, and it has obvious accuracy advantages when the unbalanced

factor of the terminal voltage is large. To adapt to different demands, a simplification

model is proposed. The models proposed in this paper can be used to guide the selection

and optimization of harmonic suppression measures and equipment.

Author Contributions: Conceptualization, G.Q. and Q.W.; methodology, G.Q. and Q.W.; software,

G.Q. and Q.W.; validation, S.H., W.D. and N.W.; formal analysis, G.Q. and Q.W.; investigation, G.Q.

and Q.W.; resources, N.Z.; data curation, S.H.; writing—original draft preparation, W.D.; writing—

review and editing, N.W.; visualization, S.H.; supervision, N.Z.; project administration, G.Q.; fund‐

ing acquisition, G.Q. All authors have read and agreed to the published version of the manuscript.

Figure 6. AC current harmonic spectrum of 12-pulse rectifier under different conditions. (a) Case 1;(b) Case 2.

Figure 7 shows the error analysis results with case 1’s parameters, and the results ofthe complete model are considered the real value. It is obvious that the simplified model ismuch more accurate than the half model.

Energies 2022, 15, x FOR PEER REVIEW 8 of 9

0 5 10 15 20 25 300

40

80

0 5 10 15 20 25 30-4

-2

0

2

4

Pha

se /r

adA

mpl

itude

/A

(b)

Figure 6. AC current harmonic spectrum of 12‐pulse rectifier under different conditions. (a) Case

1; (b) Case 2.

Figure 7 shows the error analysis results with case 1′s parameters, and the results of

the complete model are considered the real value. It is obvious that the simplified model

is much more accurate than the half model.

20 40 60 800

5

10

15 Half model Simplified model

Rel

ativ

e er

ror/

%

Triggering angle /rad

Figure 7. Error analysis of models.

5. Conclusions

This paper established a frequency‐domain harmonic coupled admittance matrix

model of a parallel 12‐pulse rectifier. This model can efficiently calculate the harmonic

current produced by the rectifier under different terminal voltage conditions, including

negative sequence voltage, and it has obvious accuracy advantages when the unbalanced

factor of the terminal voltage is large. To adapt to different demands, a simplification

model is proposed. The models proposed in this paper can be used to guide the selection

and optimization of harmonic suppression measures and equipment.

Author Contributions: Conceptualization, G.Q. and Q.W.; methodology, G.Q. and Q.W.; software,

G.Q. and Q.W.; validation, S.H., W.D. and N.W.; formal analysis, G.Q. and Q.W.; investigation, G.Q.

and Q.W.; resources, N.Z.; data curation, S.H.; writing—original draft preparation, W.D.; writing—

review and editing, N.W.; visualization, S.H.; supervision, N.Z.; project administration, G.Q.; fund‐

ing acquisition, G.Q. All authors have read and agreed to the published version of the manuscript.

Figure 7. Error analysis of models.

Energies 2022, 15, 3946 8 of 9

5. Conclusions

This paper established a frequency-domain harmonic coupled admittance matrixmodel of a parallel 12-pulse rectifier. This model can efficiently calculate the harmoniccurrent produced by the rectifier under different terminal voltage conditions, includingnegative sequence voltage, and it has obvious accuracy advantages when the unbalancedfactor of the terminal voltage is large. To adapt to different demands, a simplification modelis proposed. The models proposed in this paper can be used to guide the selection andoptimization of harmonic suppression measures and equipment.

Author Contributions: Conceptualization, G.Q. and Q.W.; methodology, G.Q. and Q.W.; software,G.Q. and Q.W.; validation, S.H., W.D. and N.W.; formal analysis, G.Q. and Q.W.; investigation, G.Q.and Q.W.; resources, N.Z.; data curation, S.H.; writing—original draft preparation, W.D.; writing—review and editing, N.W.; visualization, S.H.; supervision, N.Z.; project administration, G.Q.; fundingacquisition, G.Q. All authors have read and agreed to the published version of the manuscript.

Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Conflicts of Interest: The authors declare no conflict of interest.

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