APPENDIX BPer-Unit Values for VSC Systems

B.1 INTRODUCTION

It is often more convenient to express a power-electronic converter system in a per-unitterm. This can be achieved based on the following per-unitization system.

B.1.1 Base Values for AC-Side Quantities

The base values for a VSC system AC-side quantities are given in Table B.1. Asshown in the table, the base voltage for a VSC system is chosen as the peak value ofthe line-to-neutral voltage of the point of common coupling (PCC); this is in contrastto the conventional power system for which the rms line-to-neutral voltage representsthe base voltage. The rated three-phase power is selected as the base power.

B.1.2 Base Values for DC-Side Quantities

The DC-side base values are determined based on those of the AC side. The basepower is the same for both DC and AC sides. However, the DC-side base voltage isdefined to be two times the AC-side base voltage. This is to obtain the AC-side voltageof 1.0 pu from the DC-side voltage of 1.0 pu, at unity modulation index. The basevalues for DC-side quantities are summarized in Table B.2.

EXAMPLE B.1 Model of Three-Phase VSC-Based Rectifier

Figure B.1 illustrates a schematic diagram of a three-phase VSC system thatoperates in the rectifying mode of operation and supplies a DC RL load. Theopen-loop model of the VSC system is described by

Ldid

dt= −Rid + Lωiq + 1

2mdVDC − vsd,

Ldiq

dt= −Riq − Lωid + 1

2mqVDC − vsq,

Voltage-Sourced Converters in Power Systems, by Amirnaser Yazdani and Reza IravaniCopyright © 2010 John Wiley & Sons, Inc.

426

INTRODUCTION 427

TABLE B.1 Based Values for VSC AC-Side Quantities

Quantity Symbol and Expression Description

Power Pb = 3

2VbIb VA rating of the VSC

Voltage Vb = V̂s Amplitude of the line-to-neutral nominal voltage

Current Ib = 2Pb

3Vb

Amplitude of the nominal line current

Impedance Zb = Vb

Ib

Capacitance Cb = 1

Zbωb

Inductance Lb = Zb

ωb

Frequency ωb = ω0 Usually the power system nominal frequency

TABLE B.2 Based Values for VSC DC-Side Quantities

Quantity Symbol and Expression Description

Power Pb−dc = Vb−dcIb−dc = Pb Same as the AC-side base powerVoltage Vb−dc = 2Vb

Current Ib−dc = 3

4Ib

Impedance Rb−dc = 8

3Zb

Capacitance Cb−dc = 3

8Cb

Inductance Lb−dc = 8

3Lb

CdVDC

dt= −il − iDC = −il − 3

4

(mdid + mqiq

),

Ll

dil

dt= −Rlil + VDC. (B.1)

Let us signify a per-unitized value by the underline. Thus, for the AC-sidequantities we have

L = LbL,

R = ZbR,

vsd = Vbvsd,

428 APPENDIX B: PER-UNIT VALUES FOR VSC SYSTEMS

Pt

+

−

PCC

VSC

+ −

Ps

PDC

il

Ll C VDC

Rl

R L iabcυs-abc

FIGURE B.1 Schematic diagram of a three-phase VSC-based rectifier.

vsq = Vbvsq,

id = Ibid,

iq = Ibiq,

ω = ωbω, (B.2)

and for the DC-side quantities, we obtain

C = 3

8CbC,

Ll = 8

3LbLl,

Rl = 8

3ZbRl,

VDC = 2VbVDC,

il = 3

4Ibil. (B.3)

Substituting from (B.2) and (B.3) in (B.1), and using the relationships betweenthe base values as given in Table B.1, one deduces the following per-unitizedset of equations:

1

ωb

Ldid

dt= −Rid + Lωiq + mdVDC − vsd,

1

ωb

Ldiq

dt= −Riq − Lωid + mqVDC − vsq,

1

ωb

CdVDC

dt= −il − (mdid + mqiq),

INTRODUCTION 429

1

ωb

Ll

dil

dt= −Rlil + VDC. (B.4)

It should be noted that based on the foregoing per-unit system, we do notexpress the modulating signals md and mq in per-unit terms. The reason is thatthe absolute values of the modulating signals are between zero and unity, andthus expressing them in per-unit terms does not yield more insight. Equation(B.4) indicates that each derivative term of an original equation is premultipliedby the factor 1/ωb in the corresponding per-unit counterpart. This factor canbe avoided if the time is also expressed in per-unit terms using the base valuetb = 1/ωb. Based on such per-unitization of time, (B.4) assumes the form

Ldid

dt= −Rid + Lωiq + mdVDC − vsd,

Ldiq

dt= −Riq − Lωid + mqVDC − vsq,

CdVDC

dt= −il −

(mdid + mqiq

),

Ll

dil

dt= −Rlil + VDC. (B.5)