Three-phase multi-pulse converter StatCom analysis

Ricardo Davalos M., Juan M. Ramırez*, Ruben Tapia O.

CINVESTAV-I.P.N.-Unidad Guadalajara, Guadalajara, Jal., C.P. 45090, Mexico

Received 21 February 2002; accepted 11 August 2004

Abstract

This paper is aimed to present the analysis of operating characteristics of the static synchronous compensator (Statcom) based on a multi-

pulse voltage-source inverter (VSI). A detailed analysis of voltages and currents on both 6- and 12-pulses is presented and equations

describing the performance of each circuit are derived. Finally a comparison between digital simulations using the EMTDC/PSCAD and

analytical results is shown, verifying the validity of the derived expressions. The device is now being implemented in the laboratory, based on

IGBT’s.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: FACTS; StatCom; VSI

1. Introduction

In recent years, voltage stability and control are

increasingly becoming a limiting factor in the power

systems planning and operation, mainly in longitudinal

ones. However, a variety of factors constrain the construc-

tion of new transmission lines. This has been reflected in the

necessity of maximizing the use of existing transmission

facilities. On steady state, bus voltages must be controlled

within a short interval. A suitable voltage and reactive

power control allows to get important benefits in the power

systems operation such as the reduction of voltage

gradients, the efficient transmission capacity’s utilization

and the increase of stability margins. By different control

means and operating techniques the voltage control task in

transmission levels can be got. Some solution technologies

can involve a series voltage injection, or a shunt reactive

current injection in strategic sites of the power system.

When a disturbance occurs, changes in the voltage system

are presented and its restoration to the reference values

0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijepes.2004.08.007

* Corresponding author. Address: Priv. Josefa Ortiz de Dominguez 101,

Col. Emiliano Zapata, 64390 Monterrey, NL, Mexico. Fax: C52 3 684

1580.

E-mail addresses: rdavalos@gdl.cinvestav.mx (Ricardo D.M.), jramir-

ez@gdl.cinvestav.mx (J.M. Ramırez), rtapia@gdl.cinvestav.mx (Ruben

T.O.).

depends on the dynamic response of the excitation systems

and the employed control devices.

In the last decade commercial availability of Gate Turn-

Off thyristor devices (GTO) with high power handling

capabilities, and the advancement of other types of power-

semiconductor devices have led to the development of

controllable reactive power sources utilizing electronic

switching converter technology [1]. Additionally, these

technologies offer considerable advantages over the existing

ones in terms of space reductions and performance. The

GTOs enable the design of a solid-state shunt reactive

compensation equipment based upon switching converter

technology. This concept is used to create a flexible shunt

reactive compensation device named Static Synchronous

Compensator (StatCom) due to its operating characteristics

similar to that of a synchronous compensator although

without the mechanical inertia.

The advent of Flexible AC Transmission Systems

(FACTS) is giving rise to a new family of power electronic

equipment emerging for controlling and optimizing the

performance of power system, e.g. StatCom, SSSC and

UPFC. The use of voltage-source inverter (VSI) has been

widely accepted as the next generation of reactive power

controllers of power system to replace the conventional

VAR compensation, such as a thyristor-switched capacitor

(TSC) and thyristor controlled reactors (TCR).

Electrical Power and Energy Systems 27 (2005) 39–51

www.elsevier.com/locate/ijepes

Ricardo D.M. et al. / Electrical Power and Energy Systems 27 (2005) 39–5140

Some researchers are aiming their efforts to apply

FACTS in different ways to enhance the power systems

operation. The major applications are: voltage stability

enhancement, damping torsional oscillations, power system

voltage control, and power system stability improvement.

These applications can be implemented with a suitable

control (voltage magnitude and phase angle control) [2–4].

The StatCom basically consists of a step-down transfor-

mer with a leakage reactance, a three-phase GTO and

PWM-based voltage source inverter (VSI) or current source

inverter (CSI), and a DC capacitor. The AC voltage

difference across the leakage reactance produces reactive

power exchange between the StatCom and the power

system, such that the AC voltage at the bus bar can be

regulated so as to improve the voltage profile of the power

system, which is the primary duty of the StatCom. However,

for instance, a secondary damping function can be added

into the StatCom for enhancing power system oscillation

stability [4].

The basic principle of StatCom operation is as follows.

The VSI generates a controllable AC voltage source behind

the leakage reactance. This voltage is compared with the AC

bus voltage system; when the AC bus voltage magnitude is

above that of the VSI voltage magnitude, the AC system sees

the StatCom as an inductance connected to its terminals.

Otherwise, if the VSI voltage magnitude is above that of the

AC bus voltage magnitude, the AC system sees the StatCom

as a capacitance connected to its terminals. If the voltage

magnitudes are equal, the reactive power exchange is zero. If

the StatCom has a DC source or energy storage device on its

DC side, it can supply/absorb real power to/from power

system. This can be achieved adjusting the phase angle of the

StatCom’s terminals and the phase angle of the AC power

system. When the phase angle of the AC power system leads

Fig. 1. Six-pulse VSI w

the VSI phase angle, the StatCom absorbs active power from

the AC system; if the phase angle of the AC power system

lags the VSI phase angle, the StatCom supplies active power

to AC system [5–7].

In this paper, an analysis of the major characteristics and

basic operation principles of StatCom are carried out and is

organized as follows. Sections 2 and 3 present a detailed

signal analysis on a six-pulse VSI, including AC and DC

currents and voltages. Simulations are carried out employ-

ing the derived expressions. Section 4 exhibits the detailed

analysis on a 12-pulse StatCom. Section 5 verifies the

analysis through PSCAD/EMTDC simulations.

2. DC–AC voltage analysis

The major aim of a VSI is to generate an AC voltage

from a DC one; that is because of it is often referred to as a

DC–AC converter or inverter. It is able to generate a

symmetric AC voltage with a desired magnitude and

frequency, which can be fixed or varied according to the

application. The voltage-source inverter is the building

block of a StatCom and other FACTS devices such as the

SSSC and the UPFC.

The three-phase basic configuration is called six-pulse

inverter, consisting of six asymmetric turn-off devices, such

as a GTO or IGBT, with reverse-parallel diodes connected as

a six-pulse Graetz bridge [6] (Fig. 1). The applied transistors’

firing control signals gi is controlled in such a way that each

one conducts during 1808 when the inverter is connected to a

resistive load. The inverter can be seen as set up by three

single-phase inverters, where each phase produces a voltage

output phase-shifted by G1208 with respect to the other two

legs outputs. The firing control signals are shifted by 608 from

ith Y-connected.

Fig. 2. Phase-to-phase vab(t) and phase-to-neutral van(t) voltage waveforms.

Ricardo D.M. et al. / Electrical Power and Energy Systems 27 (2005) 39–51 41

each other. The phase-to-phase voltage vab(t) and the phase-

to-neutral van(t) are depicted in Fig. 2; it is noticed that the

phase-to-phase voltage has 1208 pulse widths with peak

voltage magnitude VDC.

Carrying out Fourier’s analysis to the waveforms of

Fig. 2, the harmonic content of voltages vab(t) and van(t) are

attained. The instantaneous values of vab(t) and van(t) based

on such analysis are given by,

vabðtÞ ZXN

mZ1

Vabmsin mut C

p

6m

� �(1)

vanðtÞ ZXN

mZ1

VanmsinðmutÞ (2)

where

T Z 2p; VabmZ

4

mpVDC cos

p

6m

� �;

VanmZ

4

3mpVDC cos

p

3m

� �C1

� �cm Z 6rG1;

r Z 0; 1; 2;.

Fig. 3 shows that the index harmonic terms are mZ6rG1,

r being any positive integer; that is, mZ1, 5, 7, 11, 13..

Voltages vbc(t), vbn(t), vca(t), vcn(t), exhibit a similar pattern,

except phase shifted by 120 and 2408, respectively. Thus

vabðtÞ Z4

pVDC

XN

m

1

mcos

p

6m

� �sin mut C

p

6m

� �(3)

vanðtÞ Z4

3pVDC

XN

m

1

mcos

p

3m

� �C1

� �sinðmutÞ (4)

It is worth noting that the fundamental component and

the harmonic components of the phase-to-phase voltages

and the phase-to-neutral voltages are phase-shifted by

308 from each other. The amplitude of the phase-to-

phase voltages isffiffiffi3

ptimes the phase-to-neutral ones; the

harmonic components not included within the set

mZ12rG1, are in phase opposition. This is emphasized

by the following expression

VabmZ ðK1Þr

ffiffiffi3

pVanm

(5)

where mZ6rG1 and rZ0, 1, 2,..

If the VSI of Fig. 1 is resistively loaded (a unitary power

factor) the diodes do not conduct at any time, but if the load

is inductive, the transistors’ conduction period would be

between 90 and 1808, and that of the diodes is between 0 and

908. For a pure reactive load (zero power factor) the

transistors’ and diodes’ conduction period is 908.

Fig. 3. vab(t) harmonic content.

Ricardo D.M. et al. / Electrical Power and Energy Systems 27 (2005) 39–5142

3. Basic VSI StatCom

The static compensators are devices with the ability to

both generate and absorb reactive and active power, though

the most common applications are in reactive power

exchange between the AC system and the compensator.

The static synchronous compensator (StatCom) based on

six-pulse VSI is the basic building block of a high power

static VAR compensator. Hence, it exhibits a high harmonic

content; that is, in high power applications the six-pulse

configuration does not have an appropriate performance.

The reactive power exchange between the AC system

and the StatCom is controlled varying the magnitude of the

fundamental component of the inverter voltage above and

below that of the AC system; this is achieved by small

variations in the switching angle of the semiconductor

devices. When the voltage’s fundamental component

produced by the inverter is forced to lag or lead the AC

system voltage by a few degrees, an active power may flow

into or out of the inverter modifying the DC capacitor

voltage value. Otherwise, if the compensator supplies only

reactive power, the active power provided by the capacitor

is zero, therefore the DC capacitor does not modify its

voltage. One could say then that the capacitor plays not any

role in the active power generation [5–8].

Fig. 4. Phase relationship between the inductor voltage VL and the

fundamental current Ia1.

3.1. AC current signals

The current flow between the compensator and the AC

system is determined by the voltage across the tie inductor

L. Let the AC bus voltage be a pure sinusoidal signal,

vANðtÞZVm sinðutÞ; from Eq. (2) the fundamental voltage’s

compensator is vanðtÞZ0:6366VDC sinðutÞ, then the magni-

tude of the fundamental and harmonics current components

are given by equations below:

Ia1Z

Vm K0:6366 VDC

uL(6)

IanZ

0:6366VDC

n2uL; cnO1 (7)

The fundamental current will be leading when

Vm!0.6366VDC; that is, if the amplitude of the inverter

voltage is increased above that of the AC bus (the current

flows from the converter into the AC system); in this case

the compensator is seen as a capacitor by the AC bus

(Fig. 4). The fundamental current will be lagging when

VmO0.6366VDC; that is, if the amplitude of the inverter

voltage is decreased below that of the AC bus (the current

flows from the AC system into the compensator); in this

Fig. 5. AC current waveform, ia(t).

Ricardo D.M. et al. / Electrical Power and Energy Systems 27 (2005) 39–51 43

case the compensator is seen as an inductor by the AC

bus. The equation that describes the AC current ia(t) is

derived taking into account the voltage across the tie

inductor over each 608 conduction period (Fig. 5). The

another two phase currents, ib(t) and ic(t), will be

identical to ia(t), except phase-shifted by 120 and 2408,

respectively. At any time, vLðtÞZL ddt

iaðtÞ, where vL(t) is

the instantaneous coupling transformer voltage; which is

the instantaneous difference between the AC system

voltage, vAN(t), and the compensator one, van(t). Con-

sidering the waveforms displayed in Fig. 5, the following

differential equations can be written for each 608

conduction interval.

†

Interval 0!ut%p/3

vLðtÞ Z Vm sinðutÞK1

3VDC Z L

d

dtiaðtÞ (8)

so that,

iaðtÞZKVm

uLðcosðutÞK1ÞK

1

3LVDCtCI0 I0Ziað0Þ (9)

†

Interval p/3!ut%2p/3

vLðtÞZVmsinðutÞK2

3VDCZL

d

dtiaðtÞ

iaðtÞZKVm

uLðcosðutÞK0:5ÞK

2

3LtK

2p

9uL

� �VDCCI1

I1Zia2p

3u

� �

†

Interval 2p/3!ut%p

vLðtÞZVmsinðutÞK1

3VDCZL

d

dtiaðtÞ

iaðtÞZKVm

uLðcosðutÞK0:5ÞK

1

3LtK

2p

9uL

� �VDCCI2

I2Zia2p

3u

� �

It is assumed that in steady state the AC current

waveform is symmetric, therefore

ia

p

2u

� �Z 0

Then the initial condition I0 can be calculated as follows

ia

p

2u

� �ZK

Vm

uLcos

p

2

� �K0:5

� �

K2

3L

p

2uK

2p

9uL

� �VDC C I1 Z 0

I1 ZKVm

uL0:5 C

p

9uLVDC

I1 Z iap

3u

� �ZK

Vm

uLcos

p

3

� �K1

� �K

1

3L

p

3uVDC C I0

I1 ZVm

uL0:5 K

p

9uLVDC C I0

where I0 is the initial condition at tZ0, ia(0)ZI0

I0 ZKVm

uLC

2p

9uLVDC (10)

Ricardo D.M. et al. / Electrical Power and Energy Systems 27 (2005) 39–5144

Therefore, the steady state half-period equations result

iaðtÞ ZKVm

uLcosðutÞK

1

3Lt K

2p

9uL

� �VDC;

0!ut%p=3

(11)

iaðtÞ ZKVm

uLcosðutÞK

2

3Lt K

p

3uL

� �VDC;

p=3!ut%2p=3

(12)

iaðtÞ ZKVm

uLcosðutÞK

1

3Lt K

p

9uL

� �VDC;

2p=3!ut%p

(13)

Within the interval p%ut!2p the AC current wave-

form is negative respect to that described in the above

equations. In this paper, the results of examples have

academic purposes only and are obtained with the following

values: leading case VDCZ6 V; VmZ2.5 V and LZ3 mH;

and for the lagging case VDCZ6 V; VmZ4.5 V and

LZ3 mH, respectively.

3.1.1. Transistors and diodes conduction period

Now, with the precedent AC current equations, it is

possible to predict the transistor’s and diode’s conduction

period. When only reactive power is generated (zero power

factor) the transistors and diodes of the inverter circuit

conduct for 908. If the compensator is absorbing reactive

power, the transistors naturally turn off at zero current. On

the other hand, when it is generating reactive power, the

transistors turn off at the peak of the AC current waveform.

The following sequence proposes a way of determining the

conduction period of each device at any power factor based

on the AC current waveforms and the firing control signals

(Fig. 1).

Leg 1

Pulse g1 on: Pulse g4 on:

If ia(t) is positive, D1 conducts. If ia(t) is positive, Q4 conducts.

If ia(t) is negative, Q1 conducts. If ia(t) is negative, D4 conducts.

Leg 2

Pulse g3 on: Pulse g6 on:

If ib(t) is positive, D3 conducts If ib(t) is positive, Q6 conducts

If ib(t) is negative, Q3 conducts If ib(t) is negative, D6 conducts

Leg 3

Pulse g5 on: Pulse g2 on:

If ic(t) is positive, D5 conducts If ic(t) is positive, Q2 conducts

If ic(t) is negative, Q5 conducts If ic(t) is negative, D2 conducts

3.1.2. Capacitor current

The capacitor current is made up of segments of the

three AC phase currents and is dependent on which

semiconductor devices are conducting over each 608

period. The capacitor current IC(t) can be described in

the following fashion:

ICðtÞ Z iaðtÞC icðtÞ; 0%ut!p=3

ICðtÞ Z iaðtÞ; p=3%ut!2p=3

ICðtÞ Z iaðtÞC ibðtÞ 2p=3%ut!p

where

ICðtÞ ZVm

uLsin ut K

p

6

� �K

2

3Lt K

p

9uL

� �VDC;

0%ut!p=3

(14)

The capacitor current waveform over each of the

remaining five conduction periods is identical to that

described by Eq. (14) and yields a repetitive waveform,

with six times the corresponding AC power system

frequency. Fig. 6 shows the capacitor current, correspond-

ing to the AC current depicted in Fig. 5.

3.1.3. DC capacitor voltage

The previous analysis assumes a constant DC voltage.

This is tantamount to consider an infinite capacitor and

consequently zero DC ripple voltage. If a finite capacitor is

dealt with, a DC voltage ripple exists which is dependent on

the capacitor value and the capacitor current. It is assumed

that the capacitor current remains unchanged from that of

Eq. (14), the capacitor voltage can be estimated; under such

condition a minimum DC ripple voltage is obtained. The

capacitor voltage vC(t) over the first 608 period results

vCðtÞ Z1

C

ðt

0ICðtÞdt CV0 (15)

where V0 is the initial condition at tZ0, V0ZvC(0).

Substituting Eq. (14) into (15),

vCðtÞ ZKVm

u2LCcos ut K

p

6

� �K

1

3LCVDCt2

Cp

9uLCVDCt C

ffiffiffi3

pVm

2u2LCCV0 (16)

Making use of the DC voltage level VDC, the value of V0

can be calculated from the average component of Eq. (15)

with a period TZp/3u,

VDC Z1

T

ðp=3u

0vCðtÞdt (17)

V0 Z 0:0889Vm

u2LCK0:0609

1

u2LCVDC CVDC (18)

The capacitor voltage’s waveforms obtained by

expressions (14) and (16) are displayed in Fig. 7. This

figure illustrates that the peak capacitor voltage Vpk, occurs

at utZ308 when the compensator is generating reactive

power, that is, leading current. The peak voltage results in

Vpk ZK0:0451Vm

u2LCC0:0305

1

u2LCVDC CVDC (19)

Fig. 6. Capacitor current; (a) generating reactive power; (b) absorbing reactive power.

Ricardo D.M. et al. / Electrical Power and Energy Systems 27 (2005) 39–51 45

When the compensator exchanges reactive power only,

the DC component of the capacitor current is zero and the

capacitor voltage is made up only by sinusoidal functions

with fixed amplitude and a DC offset. This results in zero

DC power on the DC side; thereby the capacitor voltage

does not vary. If a phase shift f exists between the AC bus

and the fundamental compensator voltage, both reactive and

active power is exchanged. It is noteworthy that if the angle

f increases the capacitor current would have a greater DC

Fig. 7. DC capacitor voltage; (a) generating reac

level; this effect is illustrated in Fig. 8 where IC(t) is

exhibited assuming fZ2 and 158.

Under this condition the instantaneous capacitor current

and voltage, based on Fourier analysis are expressed by

ICðtÞ Z IC_DC CXN

nZ1

ICn sinðnutÞ (20)

vCðtÞ Z VDC CXN

nZ1

VCn cosðnutÞ (21)

tive power; (b) absorbing reactive power.

Fig. 8. Instantaneous capacitor current.

Fig. 9. Instantaneous capacitor voltage.

Ricardo D.M. et al. / Electrical Power and Energy Systems 27 (2005) 39–5146

The above expressions show that a DC power will flow

through the DC side. Thus, the capacitor voltage will rise or

down, depending upon the signal IC_DC is positive or

negative. A capacitive circuit is employed to clarify this

effect. The circuit is initially at steady state with fZ08; at

time tZ0.0333 s the angle f is suddenly changed to K0.58.

Fig. 9 shows the capacitor voltage and emphasizes that the

phase shift f affects the capacitor voltage; it is due to the DC

component in IC(t), that is, the active power exchange. This

proves that the mechanism of phase angle adjustment can be

used to control the VAR generation or absorption by

increasing or decreasing the capacitor voltage, and thereby

the amplitude of the converter output voltage.

For an ideal converter (without losses) supplying reactive

power only, the AC bus voltage and the fundamental

converter voltage are in phase. In practical converters, the

semiconductor switches are not lossless, and therefore

the energy stored in the DC capacitor would be required by

the internal losses. However, these losses can be supplied

from the AC bus by lagging the converter’s output voltage

with respect to the AC bus voltage by a small angle; this

allows an active power flow from the AC system to the

converter which compensates for the losses, keeping the

capacitor voltage at the desired level. Such concept is

illustrated in Fig. 10. The system is operating in steady state

with a fixed DC source at tZ0.1 s. The DC source is then

changed by a DC capacitor. In Fig. 10(a) both the AC bus

voltage and the compensator one have the same phase; in this

case the energy stored in the DC capacitor is used by the

internal losses. In Fig. 10(b) with a small angular difference

between the AC bus voltage and the compensator one,

the losses are supplied from the AC system and the capacitor

voltage is kept at the desired level. Fig. 10 was obtained by

PSCAD/EMTDC.

4. Improving the six-pulse VSI

The six-pulse StatCom is the simplest arrangement used

in this kind of devices; however, due to the high harmonic

content, in high power applications it does not offer a good

performance. A better one is reached combining two six-

pulse converters. Such configurations are named 12-pulse

StatCom. The 12-pulse circuit is the lowest practical pulse-

numbered circuit for power system application to achieve a

satisfactory harmonic performance. The six-pulse output

Fig. 10. Instantaneous AC current with: (a) fZ08; (b) fZK0.58.

Ricardo D.M. et al. / Electrical Power and Energy Systems 27 (2005) 39–51 47

voltages are: vab(t), vbc(t) and vca(t), where vab(t) is given by

Eq. (1). If this compensator is connected to a wye–wye

transformer with a 1:1 turn ratio, the line-to-neutral voltage,

van(t), can be expressed by

vanðtÞ Z1ffiffiffi3

pXN

mZ1

Vabm

ðK1ÞrsinðmutÞ; cm Z 6rG1;

r Z 0; 1; 2;.;

(22)

Suppose that the second six-pulse converter produces

phase-to-phase voltages lagged 308 with respect to the other

converter and with the same magnitude. That is,

vabðtÞ2 ZXN

mZ1

VabmsinðmutÞ (23)

If this second converter is connected to a delta–wye

transformer with a 1 : 1=ffiffiffi3

pturn ratio, the line-to-neutral

voltage in the wye-connected secondary would be:

vanY ðtÞ2 Z1ffiffiffi3

pX

VabmsinðmutÞ; cm Z 6rG1;

r Z 0; 1; 2;.

(24)

then the line-to-line voltage is

vabY ðtÞ2 ZXN

mZ1

Vabm

ðK1Þrsin mut C

p

6m

� �;

cm Z 6rG1; r Z 0; 1; 2;.

(25)

The waveforms given by Eqs. (1) and (25) are added

using a summing transformer to give the third waveform

Vab12(t), closer to being a sine wave,

vab12ðtÞ Z vabðtÞCvabY ðtÞ2 (26)

thus, vab12(t) is the phase-to-phase voltage of a 12-pulse

converter. These waveforms are shown in Fig. 11, where the

two six-pulse converters are connected in parallel on the

same DC bus, and work together as a 12-pulse VSI

StatCom. The 12-pulse voltage vab12(t) holds only harmo-

nics of order mZ12rG1, where r is any positive integer,

that is, mZ1, 11, 13, 23, 25,., with amplitudes 1/11, 1/13,

1/23, 1/25,., respectively. The 12-pulse voltages given by

Eq. (26) can be expressed as,

vab12ðtÞ Z 2XN

mZ1

Vab12msin mut C

p

6m

� �(27)

where

Vab12mZ

4

mpVDC

ffiffiffi3

p; cm Z 12rG1; r Z 0; 1; 2;.

4.1. AC current signals

Applying a similar procedure to that presented in the

previous section, the AC current analysis is carried out. In such

case the fundamental voltage is vanðtÞZ1:2732VDC sinðutÞ,

then the magnitude of the fundamental and harmonics current

components are given by:

Ia1Z

Vm K1:2732VDC

uL(28)

Iam Z1:2732VDC

m2uL;cmO1 (29)

Fig. 11. (a) vab(t) and vabY(t); (b) 12-pulse voltage.

Ricardo D.M. et al. / Electrical Power and Energy Systems 27 (2005) 39–5148

The fundamental current will be leading when Vm!1.2732VDC; thus, the compensator is seen as a capacitor by

the AC system and the current flows from the compensator

into the AC system; the fundamental current will be

lagging when VmO1.2732VDC, thus the compensator

behaves as an inductor by the AC system and the current

flows from the AC system into the compensator. To obtain

the AC current equations, the procedure is carried out in a

similar way as that of the six-pulse circuit, taking into

account that the width over each conduction period is 308

(Fig. 11). The steady state current ia(t) results

iaðtÞ ZKVm

uLcosðutÞK

1

3Lt K

7:4641p

18uL

� �VDC;

0%ut!p=6

(30)

iaðtÞ ZKVm

uLcosðutÞK

0:9107

Lt K

3:0654p

6uL

� �VDC;

p=6%ut!p=3 ð31Þ

iaðtÞ ZKVm

uLcosðutÞK

1:2440

Lt K

3:7320p

6uL

� �VDC;

p=3%ut!p=2 ð32Þ

The current flowing into each wye-connected secondary

and the wye-connected primary is equal to the AC line

current, and the current flowing into the delta-connected

primary is phase-shifted by 308 with respect to the wye-

connected primary. Fig. 12(c) depicts the AC current

waveform, assuming for the leading case, VDCZ3 V;

VmZ2.5 V and LZ3 mH were used; and for the lagging

case VDCZ3 V; VmZ4.5 V and LZ3 mH.

4.2. Capacitor current

The capacitor current is made up of the DC currents

contributed by each six-pulse converter; thus the capacitor

current is given by

iDC12ðtÞ Z iDC1ðtÞC iDC2ðtÞ (33)

where iCD12(t) is the 12-pulse capacitor current; iCD1(t) is

the first compensator’s capacitor current; iCD2(t) is the

second compensator’s capacitor current, where

iDC1ðtÞ ZKVm

uLcos ut C

p

3

� �K

1:2440

Lt K

p

6u

� �VDC;

0%ut!p=6 ð34Þ

iDC2ðtÞ ZKVm

uLcos ut C

p

6

� �K

1:2440

Lt K

p

3u

� �VDC;

p=6%ut!p=2 ð35Þ

From Eqs. (34) and (35),

iDC12ðtÞ ZK1:9319Vm

uLcos ut C

5p

12

� �

K2:4880

Lt K

p

12u

� �VDC; 0%ut!p=6 ð36Þ

The capacitor current waveform over the remaining

11 conduction periods is identical to that described by

Eq. (36); for the previous example, Fig. 12(a) and (b) depict

the 12-pulse capacitor current.

Fig. 13. DC capacitor voltage; (a) generating rea

Fig. 12. (a) capacitor current: generating reactive power; (b) capacitor

current: absorbing reactive power; (c) AC current waveform, ia(t).

Ricardo D.M. et al. / Electrical Power and Energy Systems 27 (2005) 39–51 49

4.3. DC capacitor voltage

The capacitor voltage over the first 308 conduction period

is

vCðtÞ ZK1:9319Vm

u2LCsin ut C

5

12p

� �

C1:8681Vm

u2LCK

1:2440

LCVDCt2

C1:2440p

6uLCVDCt CV0 (37)

where

V0 Z 0:04181

u2LCVm K0:0568

1

u2LCVDC CVDC (38)

Fig. 13 exhibits the 12-pulse DC-capacitor voltage. This

figure also shows that the peak capacitor voltage occurs at

utZ158 when the compensator is generating reactive

power. The peak voltage is given by

Vpk ZK0:02201

u2LCVm C0:0284

1

u2LCVDC CVDC

(39)

5. PSCAD/EMTDC simulations

The previous analysis has been verified by comparing the

equations above with simulations made in PSCA-

D/EMTDC. Figs. 14 and 15 exhibit the phase-to-phase

voltage, vab(t), for a 12-pulse StatCom when the compen-

sator is operated with a finite capacitor as a DC source.

These figures illustrate the capacitor ripple voltage effect.

The capacitor voltage and the current waveforms are

ctive power; (b) absorbing reactive power.

Fig. 14. (a) 12-pulse AC current ia(t); (b) 12-pulse phase-to-phase voltage vab(t).

Ricardo D.M. et al. / Electrical Power and Energy Systems 27 (2005) 39–5150

depicted too. The considered parameters are: CZ220 mF,

LZ3 mH, VmZ2.5 V and VDCZ3 V.

6. Conclusions

A detailed circuit based analysis of a six-pulse and

12-pulse VSI-StatCom is presented; the analytical results

Fig. 15. Capacitor current and DC

are validated employing EMTDC/PSCAD simulations;

the equations obtained allow to choose the DC

capacitance size and a predefined design. The 12-pulse

VSI-StatCom is the lowest practical pulse-numbered

circuit for power applications to achieve a satisfactory

harmonic performance. The most important difference,

regarding to the six-pulse VSI-StatCom, is on the DC

capacitor: the 12-pulse VSI-StatCom requires a lower DC

capacitor voltage waveform.

Ricardo D.M. et al. / Electrical Power and Energy Systems 27 (2005) 39–51 51

rating capacitor than that required for the six-pulse VSI-

StatCom in order to obtain a comparable capacitor ripple

voltage. The DC capacitor voltage required to achieve a

specified reactive current is half that needed for the six-

pulse VSI-StatCom. Such analysis is the first stage of the

design procedure; in a future paper, results on an actual

device will be exhibited.

References

[1] CIGRE, Static Synchronous Compensator. Working Group 14.19,

September 1998.

[2] Yang Z, Shen C, Zang L. Integration of a StatCom and battery energy

storage. IEEE Trans Power Syst 2001;16(2):254–60.

[3] Chun L, Qirong J, Jianxin X, Investigation of voltage regulation

stability of static synchronous compensator in power system. IEEE

Power Engineering Society, Proceedings of the Winter Meeting 2000,

IEEE vol. 4, 2642–7.

[4] Wang HF. Applications of damping torque analysis to StatCom control.

Elsevier, Electric Power Energy Syst 2000;22:197–204.

[5] Hingorani NG, Gyugyi L. Understanding FACTS.: IEEE Press; 2000.

[6] Song YH, Johns AT. Flexible ac transmission systems FACTS. IEE

Power Energy Ser 1999;30.

[7] Yang Z. Integration of battery energy storage with flexible AC

transmission system devices. PhD Thesis, University of Missouri-

Rolla; 2000.

[8] Mohaddes M, Gole AM, Elez S. Steady state frequency response of

StatCom. IEEE Trans Power Deliv 2001;16(1):2–23.