Modeling Current-Mode-Controlled Three-Phase Converters for Simulating Multiple-Module InterConnected Power Supply Systems Runxin Wang, Student Member, IEEE, Jinjun Liu, Member, IEEE School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, CHINA E-mail: [email protected] Abstract- This paper extends the efforts on behaviorally modeling dc-dc power stages to modeling three-phase converters. By equating both the Y- and delta-connection of three-phase sources/loads to a new connection mode, which is suitable for calculating the generated/consumed power, this paper proposes the power transfer model respectively for three-phase boost rectifiers and three-phase voltage source inverters (VSI). The consistency between the proposed models and those previously published for dc-dc converters is discussed. The proposed models can find their applications in simulating multiple-module interconnected power supply systems.1

power stages, which include that of both three-phase boost rectifiers and three-phase voltage source inverters (VSI). Although more complex digital control technology may be used in three-phase converters, the power relationship in their power stages should hold as perfectly as that in dc-dc and PFC converters.

Index Terms- three-phase converter, rectifier, inverter, lowerfrequency power transfer model, computer simulation

I.

INTRODUCTION

Pursuing rapid simulation speed and requiring reflections on more circuit details are always contradictory in computer simulations, especially for those of complex circuits such as a distributed power system (DPS)[1,2]. Many studies have confirmed that we need developing simplified models to establish a balance between these two requirements[3]. At the instance of the power supply system that is formed by inter-connecting several individually well-designed and individually workable modules together onto the dc bus, as illustrated Fig. 1, the major attention during a simulation of the whole system is usually put on the potential interaction phenomena among modules rather than the operation details of each module. Following this viewpoint, the large-signal power transfer model of current-mode-controlled dc-dc and power factor correction (PFC) power stages was derived in literature [4, 5] by a novel methodology, of which the main idea is to treat the power stage as a power conservation unit and only externally present the averaged voltage-ampere relations at both the input- and output-port by arithmetic calculation. Literature [6, 7] went ahead to discuss the small-signal properties near any equilibrium and compare the results with those derived from conventional methods[8]. As a companion paper of literature [4, 5], this paper will extend the same idea to current-mode-controlled three-phase

In three-phase three-line power systems, which is most common in engineering and hence is considered in this paper, there only exist two independent variables in the currents flowing through each line (line-current). That is, one of the three line-currents is redundancy. With iA and iB been given, iC can be obtained by the KCL constraint in Eq. (1). iA  iB  iC 0 (1)

This work Foundation 50677053.

If arranging terminal C on the source lines as the common point in source side, one can reform to get another source connection that is equivalent to external circuits; where one line-current is associated one voltage source.

was of

supported by the China (NSFC)

National Nature Science under Grant Number

978-1-4244-1668-4/08/$25.00 ©2008 IEEE

Figure 1 A typical multiple-module inter-connected power supply system.

The rest of this paper is organized as follows. Section 2 gives methodology of transforming Ƹ -connection to Yconnection and making both the source and the load suitable for power calculations. Section 3 and section 4 derive the models of power stages respectively for three-phase boost rectifiers and three-phase voltage source inverters. Section 5 discusses the consistency between models in this paper and those previously published. Conclusion remarks are given in section 6. II. TRANSFORMING Ƹ-CONNECTION TO Y-CONNECTION

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A. Y-connection Sources Y-connection sources are shown in Fig. 2, where each of vA, vB and vC can take arbitrary values.

vA

vC

ª vAB  vBC º ª vC º « v »  «v » BC ¬ ¼ ¬ C¼

(4)

Arbitrarily letting vC=0 in Eq.(4), one can get Eq.(5), which leads to the connection shown in Fig. 5.

iA

vB

ªv A º «v » ¬ B¼

ªv A º «v » ¬ B¼

ª vAB  vBC º « v » BC ¬ ¼

iB

v AB  vBC

iC

vBC

(5)

iA

iB

Figure 2 Y-connection three-phase sources.

Moving vC from phase C line to phase A line and phase B line makes Fig. 3, by which the power generated by the three sources can be expressed as

 vA  vC iA   vB  vC iB

p(t )

v A  vC

(2)

Figure 5 Reformed Ƹ-connection three-phase sources. Fig. 5 shows that the power generated by the three sources can be expressed as

iA

vB  vC

p(t )

iC

one can also obtain the circuit in Fig. 5 directly by removing the redundancy voltage constraint vCA in Fig. 4 and moving vBC from phase C line to phase A line and phase B line.

ª v AB º «v » « BC » «¬ vCA »¼ iA

C. Commonly Applicable Results Above reformed connections (Fig. 3 and Fig. 5) and power relation equations (Eq. (2) and Eq. (6)) can be uniformly illustrated or written as Fig. 6 and Eq. (8) respectively, where vAC and vBC stand for the two independent line-to-line voltages. vAC

iA

(3) vBC

v AB

iB iC

iB

vBC

(6)

iB

B. Ƹ-Connection Sources Ƹ-connection sources shown in Fig. 4 can not be uniquely reformed to Y-connection like that in Fig. 2, since equations in Eq. (3) are not mutually independent.

ª 1 1 0 º ª v A º « 0 1 1» « v » « »« B» «¬ 1 0 1 »¼ «¬vC »¼

 vAB  vBC iA  vBC iB

Actually, if noticing that each of the three voltage sources in Fig. 4 can not take arbitrary value due to the constraint presented by vAB  vBC  vCA 0 (7)

Figure 3 Reformed Y-connection three-phase sources.

vCA

iC

iC

Figure 4 Ƹ-connection three-phase sources. The solution set of Eq.(3), which tells that these equations have infinite number of solutions, is listed in Eq.(4).

Figure 6 Commonly applicable three-phase sources connection equivalent to external circuits.

p(t ) vAC iA  vBC iB

(8)

Fig. 6 and Eq. (8) are true for both Y-connection and Ƹconnection sources. They show that in both cases the power generated by the three-phase sources can be calculated out by detecting and only detecting the two independent line-to-line

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voltages (vAC and vBC) and the two independent line currents (iA and iB). Actually this result is consistent to the practical way that measure three-phase power by two watt-meters. This result is also suitable to measure the power consumed by the three-phase loads, as will be mentioned in section 4. During modeling process that will be presented in following sections, all voltage and current variables should be denoted as LJ x(t) Lj Ts defined by Eq. (9), where Ts is the switching period[3,4].

x(t )

1 Ts

Ts

³

t Ts

t

x(W )dW

To make following things clear, let us divide the power distribution through the rectifier into three parts: psource: power generated by the sources; pinductor: power stored in inductors; pload: power consumed by the dc load. Above items have the following values respectively: psource v AC iA  vBC iB (10)

pinductor diA  iB LB dt di iA LA A  iB LB dt

iA LA

(9)

However, to make figures and equations seem neat, this paper will intentionally omit all angle bracket symbols, writing LJx(t)Lj Ts as x. Attentions should be paid to avoid confusions.

    iA  iB LC

dt di di · § iA ¨  LA  LC A  LC B ¸ dt dt ¹ © di di · § iB ¨  LB  LC B  LC A ¸ dt dt ¹ © pload iout vout

III. MODELING THREE-PHASE BOOST RECTIFIERS Three-phase boost rectifiers shown in Fig. 7 transform threephase ac voltage at the source side to dc voltage at the load side. It is current-mode-controlled means that the averaged values of input currents to the switching network, iA and iB, are always tracking external command signals well, by the work of the quick-responding current loop, hence these current values can be regarded as independent signals during analysis. iout

vout

b

LC

iC

c

C

DC LOAD

C

pinductor  pload

(13)

iout vout

a

LB

B iB

(12)

So, inserting Eq. (10), (11) and (12) into Eq. (13) makes Eq. (14), which can be used to establish the model shown in Fig. 8.

LA

iA

(11)

Eq.(13) is true due to the power balance relation,

psource A

di diB  iC LC C dt dt diB dt d    iA  iB

vAC iA  vBC iB  diA diB · · § § ¨ iA ¨  LA  LC dt  LC dt ¸ ¸ ¹ ¸ ¨ © ¨ diB di · ¸ §  LC A ¸ ¸ ¨ iB ¨  LB  LC dt dt ¹ ¹ © ©

Figure 7 Circuit of three-phase boost rectifiers.

v AC

LA

 LC



diA dt

LC



d iB dt

LC

(14)

The power source in Fig. 8 can be implemented as that in [4, 5].

di B dt

iout

A iA  LB

v BC

 LC

di A dt

¦

B

C vout

iB iB

iA

C Figure 8 Model of three-phase boost rectifiers.

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IV. MODELING THREE-PHASE VOLTAGE SOURCE INVERTERS (VSI)

pinductor

Three-phase voltage source inverters shown in Fig. 9 transform dc voltage at the source side to three-phase ac voltage at the load side. It is current-mode-controlled means that the averaged values of output currents to the load, ia and ib, are always tracking external command signals well, by the work of the quick-responding current loop, hence these current values can be regarded as independent signals during analysis. What is different from dc cases is that sinusoidal currents injected into the load do not always make sinusoidal voltages at the load terminals if the load is non-linear. iin

ia LA

ia

DC SOURCE

a

vin

b

c

LA

ib

LB

ic

LC

dia  ib LB dt di ia LA a  ib LB dt

    ia  ib LC

dib di  ic LC c dt dt dib dt d    ia  ib dt

di di · § ia ¨  LA  LC a  LC b ¸ dt dt ¹ © di di · § ib ¨  LB  LC b  LC a ¸ dt dt ¹ © pload v AC ia  vBC ib

A

(16)

(17)

Inserting Eq. (15), Eq. (16) and Eq. (17) into the power balance relation equation Eq. (18) leads to Eq. (19). psource pinductor  pload (18)

B C

vin iin vAC ia  vBC ib di di · § ia ¨  LA  LC a  LC b ¸ dt dt ¹ © di di · § ib ¨  LB  LC b  LC a ¸ dt dt ¹ ©

Figure 9 Circuit of three-phase voltage source inverters (VSI).

Again, we define the following three variables, which satisfy the equations listed below: psource: power generated by the dc sources; pinductor: power stored in inductors; pload: power consumed by the three-phase ac loads. psource vin iin (15)

(19)

The model established based on Eq. (19) is shown in Fig. 10. With the filter capacitors including in the load network, the power source in Fig. 10 can also be implemented as that in [4, 5]. LC

iin

d ib dt

LA

 LC



d ia dt

A

vAC _ Load

ia

¦

LC

d ia dt

 LB

 LC



d ib dt

B

v in

vBC _ Load

ib ia

ib

C Figure 10 Model of three-phase voltage source inverters (VSI).

V. DISCUSSION ON THE MODELING RESULTS The models in Fig. 8 and Fig. 10 seem more complex than that in [4, 5] and lose the two-port property on the surface.

However, things will seem familiar if defining the power as the dot product between voltage vector and current vector. Eq. (14) and Eq. (19) can be expressed as Eq. (20) and Eq. (21) respectively.

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diA diB ª «¬v AC   LA  LC dt  LC dt

iout vout

vBC   LB  LC

diB di º ªi º  LC A » « A » dt dt ¼ ¬iB ¼

T

ª v º ªi º  « 11 » « A »  V1T I1 ¬ v12 ¼ ¬iB ¼ vin iin

V1
dia dib ª «v AC   LA  LC dt  LC dt ¬

vBC   LB  LC

dib di º ªi º  LC a » « a » dt dt ¼ ¬ib ¼

T

ª v º ªi º  « 21 » « a »  V2T I 2 ¬v22 ¼ ¬ib ¼

V1 v 2

REFERENCES [1]

[2] [3]

i2

[4]

Figure 11 Vector arithmetic operation unit for model of the threephase boost rectifiers in Fig. 8.

v1

[5]

V2

[6]

i1

(21)

V2
With the definitions in Eq. (20) and Eq. (21), the vector arithmetic operation unit shown in Fig. 11 and Fig. 12 can take the place of the adder, multipliers and power source in the model respectively shown in Fig. 8 and Fig. 10.

I1

(20)

[7]

I2

[8]

Figure 12 Vector arithmetic operation unit for model of three-phase voltage source inverters (VSI) in Fig.10.

VI. CONCLUSION This paper finishes following work to establish simplified large-signal model of three-phase three-line current-modecontrolled power stages: 1. Deriving a connection mode of three-phase three-line sources/loads suitable to calculating their generated/ consumed power; 2. Proposing the power transfer model for three-phase boost rectifiers and three-phase voltage source inverters (VSI); 3. Making clear that the modeling results are consistent to those for dc-dc converters previously published. The models proposed in this paper can find their applications in simulating multiple-module inter-connected power supply systems, by taking the places of power stages. ACKNOWLEDGMENT The authors thank Mr. Qinsan Hou, for his constructive work to the research project related to this paper.

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