C C

Lab

k

Ckb

Ckab Ckbc

Lbc

k

Cka Ckc

Lca

ca k

C k

Three-phase model of the static VAR compensator (SVC) in phase coordinates

Figure 6.3

250 THREE-PHASE POWER FLOW

Equations (2.10) and (2.14):

2 3 2 32 3

ÀjBab ÀjBac

a a

jBaa

Ik Vk

SVC SVC SVC

6 b 7 16 76 b 7

4 Ik 5 ¼ 4 ÀjBba ÀjBbc 54 Vk 5: °6:30Þ

jBbb

SVC SVC SVC

3

ÀjBca ÀjBcb

c c

jBcc

Ik Vk

SVC SVC SVC

In this matrix expression, two different kinds of terms exist, namely, self and mutual terms:

8 1 X X 1 j

> BC

> ÁB BjC À BTCR ; for 1 ¼ 1;

>C

<

B1 2 ¼

j¼a;b;c j¼a;b;c

°6:31Þ

j6¼1 j6¼1

>

SVC

> B1 B2

>

: À C C þ B1 2 ; for 1 6¼ 1;

ÁBC TCR

where

2 p À 12 À sin 212

TCR TCR

B12 ¼ ; °6:32Þ

p!L12

TCR

TCR

B1 ¼ !CC ;

1

°6:33Þ

C

X j

ÁBC ¼ BC : °6:34Þ

j¼a;b;c

The superscripts 1, 2, and j take values a, b, and c. Note that parameters with double

superscripts, 1 and 2, correspond to branch parameters connected between phases 1

and 2.

6.3.1 Variable Susceptance Model

The three-phase power ¬‚ow equations for the SVC may be derived with reference to the

equivalent circuit in Figure 6.3 and using the variable susceptances B12 as state variables.

SVC

The three susceptance values are adjusted automatically by the iterative algorithm in order

to constrain the nodal voltage magnitude at the speci¬ed value. The ¬nal values of

susceptance represent the line susceptances in the delta-connected SVC equivalent circuit.

With reference to Figure 6.3, and using the SVC transfer admittance matrix of Equa-

tion (6.30), the three-phase power ¬‚ow equations for the SVC are as follows:

X À Á

P ¼ ÀVk

Vk Bj sin À jk ;

j

°6:35Þ

SVC

k k

j¼a;b;c

j6¼

X

À Á2 À Á

Q Vk B

Vk Bj cos À jk ;

j

¼À þ °6:36Þ

Vk

SVC SVC

k k

j¼a;b;c

j6¼

where the variables and j take values a, b, and c.

Taking the partial derivatives of the power ¬‚ow Equations (6.35) and (6.36), with

respect to the equivalent susceptances (state variables), we arrive at the following

251

STATIC VAR COMPENSATOR

linearised equation:

2 3°iÞ 2 a 3°iÞ 2 3

Áa °iÞ

qPk qPa qPa qPa ab qPa ca

ÁPa k k k k k

B 0 BSVC

k

6 7 6 qa 76 7

qc qBca

qb qBab SVC

6 7 6k 76 7

SVC

k SVC

k

6 7 6b 76 7

6 7 6 qPk 7 6 Áb 7

qPb qPb qPb ab qPb bc

6 ÁPb 7 6 76 k7

k k k k

B B 0

6 k7 6 qa 76 7

qc

qk qBSVC SVC qBSVC SVC

b ab bc

6 7 6k 76 7

k

6 7 6 76 7

6 7 6 qPc qPc ca 7 6 Ác 7

qPc qPc qPc bc

6 c7 6k 76 k7

k k k k

6 ÁPk 7 6a ca BSVC 7 6 7

0 B

6 q qk qBSVC

qk qBSVC SVC

6 7 76 7

c

b bc

6 7 ¼6 k 76 7:

6 7 6 qQa qQa ca 7 6 ÁBab 7

qQa qQa qQa ab

6 7 6k 76 SVC 7

6 ÁQk a7 6a ca BSVC 7 6 Bab 7

k k k k

B 0

6 7 6 qk 7 6 SVC 7

qk qBSVC

qk qBSVC SVC

c

b ab

6 7 6 76 7

6 7 6b 76 bc 7

6 7 6 qQk 7 6 ÁBSVC 7

qQb qQb qQb ab qQb bc

6 b7 6 76 7

k k k k

B B 0

6 ÁQk 7 6 qa 7 6 Bbc 7

qk

qk qBSVC SVC qBSVC SVC

c

b ab bc

6 7 6k 7 6 SVC 7

6 7 6c 76 7

4 5 4 qQ qQc ca 5 4 ÁBca 5

qQc qQc qQc bc SVC

k k k k k

0 B ca BSVC

ÁQc qk qk qBSVC

qk qBSVC SVC ca

a c

b bc BSVC

k

°6:37Þ

The new Jacobian entries in the linearised expression have the following form:

qP À Á

Bj ¼ ÀVk Vk Bj sin À jk ;

j

°6:38Þ

k

qBj

SVC SVC k

SVC

qQ À Á2 À Á

Bj ¼ À2 Vk Bj þ Vk Vk Bj cos À jk :

j

°6:39Þ

k

qBj

SVC SVC SVC k

SVC

The terms corresponding to partial derivatives of active and reactive powers with respect

to nodal voltage phase angles have the same form as Equations (6.18), (6.20), (6.22), (6.24),

(6.26), and (6.28), respectively.

Once the SVC linearised equation has been evaluated at a given iteration, (i), it is then

combined with the linearised expression representing the overall external system “ Equa-

tion (6.9) “ and a new set of state variables is obtained. The SVC susceptances are updated

by using the following expression:

!°iÞ

ÁBj

j °iÞ j °iÀ1Þ j °iÀ1Þ

BSVC ¼ BSVC þ BSVC : °6:40Þ

k

j

BSVC

This calculation completes iteration (i), and the three-phase mismatch power equations are

calculated and checked for convergence. If the convergence criterion has not been satis¬ed,

a new iteration is carried out.

6.3.2 Firing-angle Model

An alternative SVC model is realised by using the ¬ring angles of the thyristors as state

variables, rather than equivalent susceptances. In this situation, the new SVC linearised

252 THREE-PHASE POWER FLOW

equation takes the form:

2 3 2a 32 3°iÞ

ÁPa °iÞ qPa °iÞ Áa

qPk qPa qPa qPa

k k

k k k k

0

6 7 6 qa qca 7 6 7

qk

qk qSVC

c

b ab

6 7 6k SVC 7 6 7

6 7 6b 76 7

6 ÁPb 7 6 qPk 7 6 Áb 7

qPb qPb qPb qPb

6 k7 6 0 76 k7

k k k k

6 7 6 qa 76 7

qk

qk qSVC qSVC

c

b ab bc

6 7 6k 76 7

6 7 6 qPc c76 7

qPc qPc qPc qPk 7 6

6 ÁPc 7 6k c7

7 6 Ák 7

6 k7 6a k k k

0

6 7 6 q qca 7 6 7

qk

qk qSVC

c

b bc

6 7 ¼6 k SVC 7 6 7: °6:41Þ

6 7 6 qQa qQa 7 6 7

qQa qQa qQa

6 a7 6k k76 7

k k k

6 ÁQk 7 6 qa ca 7 6 Áab 7

0

qk qSVC 7 6

qk qSVC

6 7 6k SVC 7

c

b ab

6 7 6b 76 7

6 7 6 qQ 76 7

qQb qQb qQb qQb

6 7 6k 0 76 7

k k k k

6 ÁQ b7 6 qa 7 6 Ábc 7

qk

qk qSVC qSVC

c

b ab bc

6 k7 6k 76 SVC 7

6 7 6c c76 7

4 5 4 qQk qQk 5 4 5

qQc qQc qQc

k k k

0

qa qk qca

qk qSVC Áca

c

b bc

ÁQc SVC

k SVC

k

The new Jacobian entries in the linearised expression have the following form:

qP j

À j Á qBSVC

j

¼ ÀVk Vk sin k À k ; °6:42Þ

k

qab qab

SVC SVC

hÀ Á i j

qQ À j Á qBSVC

2 j

¼ À Vk ÀVk Vk cos k À k ; °6:43Þ

k

qab qab

SVC SVC

2h i

qBj j

¼À 1 þ cos 2SVC : °6:44Þ

SVC

pXL

qab

SVC

The terms corresponding to partial derivatives of active and reactive powers with respect to

nodal voltage phase angles are the same terms referred to in Equation (6.37).

Upon solution of the combined Equations (6.9) and (6.41), a new set of state variable

increments is obtained. The increments are used to update the state variable values “ among

them the SVC ¬ring angles “ using the following expression:

°j Þ°iÞ ¼ °j Þ°iÀ1Þ þ °Áj Þ°iÞ : °6:45Þ

SVC SVC SVC

This calculation completes iteration (i), and the three-phase mismatch power equations are

calculated and checked for convergence.

6.3.3 Numerical Example: Static VAR Compensator Voltage

Magnitude Balancing Capability

A three-phase SVC is added to the unbalanced ¬ve-bus network in Section 6.2.4 in order to

explore the capability of the SVC to restore geometric balance at the point of connection

while at the same time providing effective voltage magnitude regulation. The SVC is

assumed to be connected at Elm, and the voltage magnitude is set at 0.98 p.u. The study is

conducted using the SVC reactance model, and convergence is achieved in 5 iterations to

satisfy a power mismatch tolerance of 1 e À 12.

253

THYRISTOR-CONTROLLED SERIES COMPENSATOR

Table 6.3 Nodal voltage in the three-phase unbalanced network with a static VAR compensator:

(a) phase voltages and (b) sequence voltages

Network bus

”””

””””””””””””””””””””””””

”

Voltage Phase North South Lake Main Elm

(a) Phase voltages

Magnitude (p.u.) a 1.06 1.00 0.9822 0.9810 0.98

b 1.06 1.00 0.9888 0.9974 0.98

c 1.06 1.00 0.9947 0.9923 0.98

À 2.04 À 4.64 À 4.79 À 5.76

Phase angle (deg) a 0

b 240 238.16 235.17 234.84 234.81

c 120 117.60 115.37 114.86 113.09

(b) Sequence voltages

Magnitude (p.u.) Zero 0.00 0.0028 0.0047 0.0050 0.0087

Positive 1.06 1.0000 0.9886 0.9859 0.9799

Negative 0.00 0.0028 0.0025 0.0022 0.0086

The three-phase nodal voltage magnitudes and phase angles are given in Table 6.3(a), in

which it is shown that the SVC is effective in regulating and balancing nodal voltage

magnitude at Elm. As expected, the phase angles at that bus are still unbalanced. It should

be mentioned that power losses now stand at 1.01 %, a result that compares favourably with

the unbalanced case where no SVC is used and where power losses stand at 2 %. Note that

negative sequence voltages have also reduced in magnitude [Table 6.3(b)].

6.4 THYRISTOR-CONTROLLED SERIES COMPENSATOR

Based on the nodal admittance representation of the TCSC, derived in Chapter 2, two quite

useful positive sequence power ¬‚ow models were developed in Section 5.4. One model

uses an adjustable reactance as the state variable and the other uses the thyristor ¬ring angle.

The same idea is now extended to the case of the TCSC power ¬‚ow models in phase

coordinates.

6.4.1 Variable Susceptance Model

The three-phase TCSC representation is simply obtained by using three independent TCSC

modules, as shown in Figure 2.12. The changing susceptance, shown in Figure 6.4,

represents the fundamental frequency equivalent susceptance of each series module making

up the three-phase TCSC. The value of BTCSC is determined by using the Newton“Raphson

method to regulate active power ¬‚ow through the three branches to a speci¬ed value.

254 THREE-PHASE POWER FLOW

a a

Preg BTCSC

Vka Vm

I ka a

Im

b b

BTCSC

Vkb Preg Vm

I kb b

Im

c c

Preg BTCSC

Vkc Vm

I kc c

Im

Three-phase variable series susceptance

Figure 6.4

The transfer admittance of the TCSC may be derived from visual inspection of the

equivalent circuit shown in Figure 6.4. Assuming that takes the values a, b, c:

! ! !

jB jB

I Vk

¼ : °6:46Þ

kk km

k

jBmk jBmm V

Im m

In Equation (6.46) the terms B ; B ; B , and B are diagonal matrices since the three

mm

kk km mk

TCSC modules are electromagnetically decoupled:

1

B ¼ B ¼ B

TCSC ¼ À ; °6:47Þ

X

mm

kk

1

B ¼ B ¼ ÀB

TCSC ¼ ; °6:48Þ

X

km mk

where X represents the fundamental frequency equivalent reactance of the th series

modules making up the TCSC.

With reference to Figure 6.4 and using the transfer admittance matrix in Equation (6.46),

the three-phase nodal power injections at bus k are:

À Á

P ¼ Vk Vm B sin À ;

°6:49Þ

m

k km k

À Á2 À Á

Q ¼ À Vk B À Vk Vm B cos À :

°6:50Þ

m

k kk km k

Power equations at bus m are obtained by replacing the subscript k with m, and vice versa, in

Equations (6.49) and (6.50).

The ¬rst partial derivatives of the power equations with respect to X are:

qP

X ¼ ÀP ; °6:51Þ

k

qX k

X ¼ ÀQ : °6:52Þ

k

qX k

255

THYRISTOR-CONTROLLED SERIES COMPENSATOR

When the TCSC is controlling active power ¬‚owing from k to m, at speci¬ed value, the set

of linearised power ¬‚ow equations is

2 3°iÞ 2 3°iÞ 2 3°iÞ

ÁP qP qP qP qP qP Á

k k k k k k k

V V X76

7 7

6 6 q q qX

qVk k qVm m

7 76 7

6 6 m

k

6 7 6 qP 7 6 Á 7

qPm qPm qPm qPm 7 6

6 ÁP 7 6 m7

m7 X76 7

6 6 m

Vk Vm

7 76 7

6 6 qk qX

qm qVk qVm

6 7 6 76 7

6 7 6 qQ qQ 7 6 ÁVk 7

qQ qQ qQ

6 ÁQ 7 6 X 7 6 V 7 ; °6:53Þ

k 7 ¼6

k k k k k

6 76 k7

Vk Vm

6 qk qm qVk qVm qX

6 7 76 7

6 7 6 76 7

6 7 6 qQm qQm qQm qQm qQm 7 6 ÁVm 7 7

6 7 6 X76 7

6 ÁQ 7 V V

6 q 7 6V

q qX

qVk k qVm m

6 m7 6 76 m7

m

k

6 7 6 ;X 76 7

;X ;X ;X ;X

4 5 4 qPkm qPkm qPkm qPkm qPkm 5 4 ÁX 5

V V X

q q

ÁP;X qVk k qVm m qX X

m

k

km

where ÁP;X , given by

km

ÁP;X ¼ P;X;Re g À P;X;cal ;

km km km

is the active power ¬‚ow mismatch for the TCSC. ÁX is the incremental change in the total

series TCSC reactance, and the superscript (i) indicates iteration number.

6.4.2 Firing-angle Model

The TCSC structure shown in Figure 2.12 is extended to account for the three phases as

shown in Figure 6.5. This is used as the basis for deriving the three-phase power ¬‚ow model,

a

XC a

Vka Vm

ILOOP

I ka a

Im

a

XL

a a

b

XC b

Vkb Vm

ILOOP

I kb b

Im

b

XL

ab c

XC c

Vkc Vm

ILOOP c

I kc Im

c

XL

ac

Figure 6.5 Three thyristor-controlled series compensator modules

256 THREE-PHASE POWER FLOW

where the TCSC ¬ring angle is used as state variable (Venegas and Fuerte-Esquivel, 2001).

The three TCSC branches are assumed to be electrically and magnetically decoupled.

The fundamental frequency TCSC equivalent reactance, as a function of TCSC ¬ring

angle, is given by Equation (2.52). It follows that the extension to three phases is quite

straightforward, owing to the decoupled nature of the three modules:

XTCSC°1Þ ¼ ÀXC þ C1 f2°p À Þ þ sin½2°p À Þg À C2 cos2 °p À Þ

‚ f! tan½!°p À Þ À tan°p À Þg; °6:54Þ

where

XC X L

¼ ; °6:55Þ

XLC

XC À X L

XC þ XLC

¼ ; °6:56Þ

C1

p

À Á2

4 XLC

C2 ¼ : °6:57Þ

XL p

The transfer admittance matrices of both TCSC representations are identical, given by

Equations (6.46)“(6.48). Moreover, the TCSC nodal power equations also coincide. The

TCSC power equations with respect to the ¬ring angle are:

qB 2 qX

TCSC°1Þ TCSC°1Þ

¼ BTCSC°1Þ ; °6:58Þ

q

q

qXTCSC°1Þ

qP

¼ Pk BTCSC°1Þ ; °6:59Þ

k

q q

qXTCSC°1Þ

¼ Qk BTCSC°1Þ ; °6:60Þ

k

q q

where

qXTCSC°1Þ

¼ À2C1 ½1 þ cos°2 Þ þ C2 sin°2 Þf! tan½!°p À Þ À tan g

q

cos2 °p À Þ

þ C2 !2 À1 : °6:61Þ

cos2 ½!°p À Þ

When the TCSC module is controlling the active power ¬‚owing from buses k to m, at a

speci¬ed value, the set of linearised power ¬‚ow equations is:

2 3°iÞ 2 3°iÞ 2 3°iÞ

ÁP qP qP qP qP qP Á

k k k k k k k

Vk Vm

6 7 6 q 7 6 7

qm qVk qVm q 7 6

6 7 6 7

k

6 7 6 7 6 Á 7

6 qPm qPm qPm qPm qPm 7 6

6 ÁP 7 m7

6 m7 6 76 7

Vk Vm

6 7 6 qk q 7 6 7

qm qVk qVm

6 7 6 76 7

6 7 6 qQ qQ 7 6 ÁVk 7

qQ qQ qQ

6 ÁQ 7 6 k7 6 7

m 7 ¼6 ; °6:62Þ

6 7 6 Vk 7

k k k k

Vk Vm

6 7 6 qk q 7 6 7

qm qVk qVm

6 7 6 76 7

6 7 6 qQ qQ qQ qQ 7 6 7

6 7 6 m 7 6 ÁV 7

m m m m

6 ÁQm 7 6 q q 7 6 7

Vk Vm

q m

qVk qVm

6 7 6 7 6 Vm 7

m

k

6 7 6 ; 76 7

4 qPkm qP; qP; qP; qP; 5 4

4 5 5

km km km km

Vk Vm

Á

q q

ÁP; q

qVk qVm

m

k

km

257

STATIC COMPENSATOR

where ÁP;

km

ÁP; ¼ Pkm

;;Reg

À P;;cal ;

km km

and P;;cal ; P , are the active power ¬‚ow mismatches for the three-phase TCSC module, and

km k

°iÞ , given by

°iÞ ¼ °iÀ1Þ þ Á°iÞ ;

is the incremental change in the TCSC ¬ring angle. The superscript (i) indicates iteration

number.

6.4.3 Numerical Example: Power Flow Control using One

Thyristor-controlled Series Compensator

In order to show the ¬‚exibility of the TCSC model in the phase domain, a three-phase power

¬‚ow study is carried out. The TCSC is added to the unbalanced ¬ve-bus network, connected

between buses Lake and Main. The aim of this example is to balance out the amount of

active power through the TCSC at 21 MW. The increase of active power in phase a is almost

50 % and in phase b 5 %; in phase c it decreases by 14 % with respect to the unbalanced

case in Section 6.2.4. The three-phase nodal voltage magnitudes and phase angles are given

in Table 6.4(a), whereas Table 6.4(b) gives the nodal voltage magnitudes in the sequence

domain. The power ¬‚ows are shown in Figure 6.6.

Table 6.4 Nodal voltages in the three-phase unbalanced network with a thyristor-controlled series

compensator (TCSC): (a) phase voltages and (b) sequence voltages

Network bus

””””””””””””””””” ”””””””””””””

Voltage Phase North South Lake Main Elm LakeTCSC

(a) Phase voltages

Magnitude (p.u.) a 1.06 1.00 0.9839 0.9797 0.9788 0.9800

b 1.06 1.00 0.9884 0.9823 0.9751 0.9871

c 1.06 1.00 0.9912 0.9866 0.9593 0.9902

À 1.87 À 5.28 À 3.86 À 5.54 À 3.48

Phase angle (deg) a 0.00

b 240.00 238.14 235.36 234.80 235.21 235.07

c 120.00 117.52 115.58 114.51 113.05 114.94

(b) Sequence voltages

Magnitude (p.u.) Zero 0.00 0.0035 0.0046 0.0097 0.0147 0.0098

Positive 1.06 1.0000 0.9878 0.9828 0.9710 0.9857

Negative 0.00 0.0035 0.0052 0.0079 0.0092 0.0086

6.5 STATIC COMPENSATOR

With reference to the single-phase equivalent circuit shown in Figure 2.18(b), and assuming

´

that the three Thevenin equivalents representing a three-phase STATCOM are decoupled,

the equivalent circuit is shown in Figure 6.7.

258 THREE-PHASE POWER FLOW

34.78 + j4.35

51.75 + j17.25

94.81

132.38

21.00

North Lake

45.46 Main

21.02

43.65

86.92 78.31 16.37

16.50 2.35

3.02 0.57

29.10 0.86

8.70

22.46

29.77 2.14 4.06

4.38

77.35

84.49

8.77

22.79 3.74

0.96

South Elm

51.23

51.93

20 + j10

60 + j10

6.71 6.26

40.0 64.06

(a)

46 + j5.75

45 + j15

89.49

126.52

21.00

North Lake

41.76 Main

20.89

40.35

84.76 18.33

71.84

17.65 5.18

5.29 6.97

0.48

25.65 3.31

28.40

25.92 1.96 0.74

0.74

71.30

82.24

29.03 3.31 5.20

2.98

South Elm

48.86

49.89

17.4

+ j8.7

52.17 + j8.7

3.20 3.50

40.0 63.12

(b)

40 + j5

39.13 + j13.04

86.57

138.11

21.00

Lake Main

North 40.96 20.93

39.49

97.15 71.66 16.01

14.91 1.80

2.03 3.62

0.76

20.64 9.39

28.46

20.91 0.94 2.14

4.21

69.76

94.60

5.16

9.13

28.83

0.33

Elm

South 59.87

61.86

23 + j11.5

69 + j11.5

8.55 6.34

40.0 64.25

(c)

Figure 6.6 Five-bus network with thyristor-controlled series compensator: (a) phase a, (b) phase b,

and (c) phase c

259

STATIC COMPENSATOR

Vka q ka I ka g ka

aa

YvR

Vkb q kb I kb g k +

b

bb

YvR

VvR δ vR

a a

_

+

Vkc q kc I kc g kc

cc

YvR VvR δ vR

b b

_

+

VvR δ vR

c c

k _

Figure 6.7 Three-phase static compensator equivalent circuit

Based on this equivalent circuit, and the three-phase transfer admittance Equation (2.65),

the following expressions for active and reactive power injections at bus k may be

written:

À Á2 ‚ À Á À ÁÃ

P ¼ Vk G þ Vk VvR G cos À vR þ B sin À vR ; °6:63Þ

vR vR vR

k k k

À Á2 ‚ À Á À ÁÃ

Q ¼ À Vk B þ Vk VvR G sin À vR À B cos À vR : °6:64Þ

vR vR vR

k k k

The corresponding expressions for the three sources are:

‚ À Á À ÁÃ

P ¼ °VvR Þ G þ VvR Vk G cos vR À þ B sin vR À ;

2

°6:65Þ

vR vR vR vR

k k

‚ À Á À ÁÃ

Q ¼ À°VvR Þ B þ VvR Vk G sin vR À À B cos vR À ;

2

°6:66Þ

vR vR vR vR

k k

where refers to phases a, b, and c at bus k and at the terminals of the source.

Derivation of these power equations with respect to the STATCOM state variables VvR

and vR yields the following linearised equation:

2 3 2 32 3

ÁP qP qP qP qP Á

k k k k k k

V V

6 7 6 q qVvR vR 76 7

qVk k qvR

6 76k 76 7

6 7 6 76 ÁV 7

qQ qQ

6 ÁQk 7 6 qQk qQk 76 k7

6 76 VvR 76 V 7

k k

Vk

6 7 6 qk 76 k 7

qVk qvR qVvR

6 7¼6 76 7: °6:67Þ

6 76 76 7

qP qP

6 ÁP 7 6 qPvR qPvR 76 ÁvR 7

vR vR

6 vR 7 6 76 7

V V

qVvR vR 76

7 6 qk qVk k qvR

6 7

6 76 76 7

4 5 4 qQvR qQvR 54 ÁVvR 5

qQ qQ

vR vR

ÁQvR V V

q

qVvR vR

qVk k qvR VvR

k

260 THREE-PHASE POWER FLOW

The Jacobian elements created for this application are as follows:

qP À Á2

¼ ÀQk À Vk GvR ; °6:68Þ

k

qk

qP À Á2

Vk ¼ Pk þ Vk GvR ; °6:69Þ

k

qVk

qP qQ ‚ À Á À ÁÃ

Vk ¼ VvR Vk G cos vR À þ B sin vR À ;

vR

¼ °6:70Þ

k

vR vR

qvR qVk k k

qP qQ ‚ À Á À ÁÃ

VvR ¼ À vR ¼ Vk VvR G cos À vR þ B sin À vR ; °6:71Þ

k

q vR vR

qVvR k k

k

qQ À Á2

¼ Pk À Vk GvR ; °6:72Þ

k

qk

qQ À Á2

Vk ¼ Q À Vk B ; °6:73Þ

k

vR

qVk k

qQ qP ‚ À Á À ÁÃ

¼ À vR Vk ¼ ÀVk VvR G cos À vR þ B sin À vR ; °6:74Þ

k

vR vR

qvR qVk k k

qQ qP ‚ À Á À ÁÃ

vR

VvR ¼ ¼ Vk VvR GvR sin k À vR À BvR cos k À vR ; °6:75Þ

k

qVvR qk

qP 2

vR

¼ ÀQvR À °VvR Þ BvR ; °6:76Þ

qvR

qP 2

vR

VvR ¼ PvR þ °VvR Þ GvR ; °6:77Þ

qVvR

qQ 2

vR

¼ PvR À °VvR Þ GvR ; °6:78Þ

qvR

qQ 2

vR

VvR ¼ QvR À °VvR Þ BvR : °6:79Þ

qVvR

Solution of the linearised Equation (6.67) yields information on the state variable

increments. The increments are, in turn, used to update the state variables. Voltage mag-

nitude limits are checked at the end of each iterative step and if one or more limits are

violated the voltage magnitude is ¬xed at the violated limit.

6.5.1 Static Compensator Three-phase Numerical Example

The three-phase STATCOM model is used to balance voltage magnitude at Elm at 0.98 p.u.

This is, in essence, the same case study carried out with the SVC model in Section 6.3.3.

261

UNIFIED POWER FLOW CONTROLLER

The source impedances are XvR ¼ 0:1 p.u. per phase. The power ¬‚ow results indicate that

the STATCOM generates 4.81 MVAR, 8.47 MVAR and 15.25 MVAR in phases a, b and c,

respectively, in order to achieve the three-phase voltage magnitude target. The STATCOM

parameters associated with this amount of reactive power generation are: VvR ¼ 0:9849,

0.9886 and 0.9955 p.u. for phases a, b and c, respectively. As expected power ¬‚ows results

coincide with those obtained using the SVC model in Section 6.3.3. Nodal voltage

magnitudes and phase angles are given in Table 6.3(a), and sequence domain voltage

magnitudes are given in Table 6.3(b).

6.6 UNIFIED POWER FLOW CONTROLLER

The UPFC schematic representation and its operational control were presented in Sec-

tion 2.4.4, and a positive sequence power ¬‚ow model was developed in Section 5.6.

However, in order to assess the role of the UPFC operating under unbalanced conditions it is

necessary to develop the model in phase coordinates.

Assuming that the equivalent circuit of a three-phase UPFC consist of three single-phase

UPFC equivalent circuits, with no couplings between them, as shown in Figure 6.8, the

m

k a a

VcR d cR

aa

Ig YcR

a a

1 1

Vka q ka Vm q m

a a

+

I vR g vR

a a

b b

VcR d cR

bb

I1b g 1

b

YcR

Vq _

b b

Vm q m

b b

+

k k

cc

I vR g vR

b b c c

VcR d cR

YcR

Igc c

_

Vkc q kc 1 1

Vm q m

c c

+

I vR g vR

c c

aa bb cc

YvR YvR YvR

+

Re ’VvR IvR* + VcR IcR* = 0

{ }

pp pp

Vδa a

vR vR

+

’

Vδ

b b

vR vR

’ +

VvR δ vR

c c

’

Three-phase uni¬ed power ¬‚ow controller equivalent circuit

Figure 6.8

262 THREE-PHASE POWER FLOW

three-phase power ¬‚ow equations are as follows:

À Á2 ‚ À Á À ÁÃ

P ¼ Vk G þ Vk Vm G cos À þ B sin À

‚ À km Á k m À km ÁÃk m

k kk

þ Vk VcR Gkm cos k À cR þ Bkm sin k À cR

‚ À Á À ÁÃ

þ Vk VvR G cos À vR þ B sin À vR ;

°6:80Þ

vR vR

k k

À Á2 ‚ À Á À ÁÃ

Q ¼ À Vk B þ Vk Vm G sin À À B cos À

‚ À Á À ÁÃ

m m

k kk km k km k

þ Vk VcR G sin À cR À B cos À cR

‚ Àk Á À ÁÃ

km km k

þ Vk VvR GvR sin k À vR À BvR cos k À vR : °6:81Þ

Equations (6.80) and (6.81) are the three-phase counterparts of Equations (5.50) and (5.51).

Equations for bus m, and the series and shunt converters are also obtained by direct

extensions of Equations (5.52)“(5.57) into phase coordinates.

Ã

In this situation, the active power supplied to the shunt converter, RefVvR IvR g satis¬es the

È Ã É

active power demanded by the series converter, Re VcR Im . The impedance of the series

and shunt transformers, ZcR and ZvR , are included explicitly in the model.

The UPFC power equations, in linearised form, are combined with those of the AC

network. For the case when the UPFC controls the following parameters: (1) voltage

magnitude at the shunt converter terminal (bus k), (2) active power ¬‚owing from bus m to

bus k, and (3) reactive power injected at bus m, and taking bus m to be a PQ bus, the

linearised system of equations is as follows:

3 32 3

2 2

Á

ÁP qP qP qP qP qP qP qP k

k k k k k k k k

7 6 q qvR 76 7

6 V V V

q

qVk k qVm m qcR qVcR cR

76 76 7

6 m

76 76 Ám

7

6 k

7 76 7

6 ÁPm 6 qP qP qP qP qP qP

76 0 76 7

6 m m m m m m

V V VcR

7 6 q 76 7

6

qm qVk k qVm m qcR qVcR

76 76 ÁV 7

6 k

76 76 7

6

qQ qQ qQ qQ qQ qQ 76 k 7

6 ÁQ 7 6 qQk

k7

k 76 V 7

6 6 k k k k k

Vk Vm VcR

7 6 q

qvR 76 k 7

6 qm qVk qVm qcR qVcR

76 76 7

6 k

7 6 qQ 76 ÁV 7

6 qQ qQ qQ qQ qQ

76 0 76 m 7:

6

6 ÁQ 7 ¼ 6

m m m m m m

76 7

V V V

q q

qVk k qVm m qcR qVcR cR

76 76 Vm 7

6 m

m

7 6 k 76 7

6

7 6 qP 76 7

6

qP qP qP qP

qPmk

7 0 76 ÁcR 7

6 6 mk

mk mk mk mk

6 ÁPmk 7 6 q 76 7

V V V

q

qVk k qVm m qcR qVcR cR

76 76 7

6 m

k

76 76 7

6

qQ qQ qQ qQ

7 6 qQmk 76 7

6 qQmk

76 0 76 ÁV 7

6 mk mk mk mk

V V V

6 ÁQ 7 6 q 76 7

q

qVk k qVm m qcR qVcR cR

mk 7 76 cR 7

6 6 m

k

76 76 7

6

qP qP qP qP qP 54 VcR 5

5 4 qPbb

4 qPbb bb bb bb bb bb

V V V

q q

qVk k qVm m qcR qVcR cR qvR

ÁP Á

m

k vR

bb

°6:82Þ

The Jacobian entries, which are 3 ‚ 3 matrices, are derived in a similar way to those of

the STATCOM in Section 6.5. The linearised equation (6.82) is solved for the vector of state

variables increments, and this information is used to update the state variables. If the

convergence criterion has not been satis¬ed then a new iteration is started using the latest,

state variables information available.

263

UNIFIED POWER FLOW CONTROLLER

34.78 + j4.35

51.75 + j17.25

89.06

132.46

30.00

North Lake

49.20 Main

29.96

47.37

83.26 79.49 9.22

9.57 20.22 5.50

7.13

34.38 2.41

11.66

16.48

35.37 12.19 7.44

13.58

78.77

80.90

11.67

16.65 2.05

3.99

Elm

South 48.33

48.89

20 + j10

60 + j10

8.05 7.95

40.0 70.31

(a)

46 + j5.75

45 + j15

84.64

126.34

North 45.43 Lake Main

30.00 29.87

43.94

80.91 72.92 12.30

11.72 9.24 0.86

2.47

1.99

31.06 6.22

22.35

31.50 6.54 4.62

8.64

72.63

78.47

22.74 6.28 2.67

1.77

Elm

South 45.89

46.83

17.4

+ j8.7

52.17 + j 8.7

5.39 6.03

40.0 65.41

(b)

40 + j5

39.13 + j13.04

82.02

138.39

Lake Main

North 44.77 30.00

43.12 29.87

93.62 72.64 10.12

9.38 11.89 4.45

6.05

1.78

26.01 12.27

22.40

26.53 8.97 7.67

11.66

71.02

91.17

2.45

11.92

22.68

4.77

Elm

South 57.08

58.96

23 + j11.5

69 + j11.5

10.96 9.05

40.0 55.45

(c)

Uni¬ed power ¬‚ow controller upgraded test network and power ¬‚ow

Figure 6.9

264 THREE-PHASE POWER FLOW

6.6.1 Numerical Example of Power Flow Control using

One Uni¬ed Power Flow Controller

In order to assess the effectiveness of UPFC controllers to regulate active and reactive power

¬‚ow and to control voltage magnitude in one of the UPFC connecting buses, the ¬ve-bus

network is modi¬ed to include a three-phase UPFC model to compensate and to balance the

transmission line linking buses Lake and Main. The modi¬ed network is shown in

Figure 6.9. The UPFC is used to maintain active power leaving the UPFC, towards Main, at

30 MW in each phase; reactive power towards Main is selected in such a manner that

balanced voltage magnitudes of 0.98 p.u. are obtained at the bus connecting the UPFC and

compensated transmission line; the reactive power injections are set at 7.13 MVAR, 2.47

MVAR, and 6.05 MVAR for phases a, b, and c, respectively; voltage magnitudes at bus Lake

are ¬xed at 1 p.u. The three-phase nodal voltage magnitudes and phase angles are given in

Table 6.5(a), where it is shown that the UPFC is effective in regulating voltage magnitude in

both of the connecting buses. Figure 6.9 shows the power ¬‚ow results when the UPFC

regulates reactive power at the above values. It is clear that the UPFC is an effective device

for restoring power balance. Table 6.5(b) shows the nodal voltage magnitudes in the

sequence domain.

Table 6.5 Nodal voltages in the three-phase unbalanced network with a uni¬ed power ¬‚ow

controller: (a) phase voltages and (b) sequence voltages

Network bus

”””””””””””””

””” ””””””””””””””””

Voltage Phase North South Lake Main Elm LakeUPFC

(a) Phase voltages

Magnitude (p.u.) a 1.06 1.00 1.00 0.980 0.979 0.98

b 1.06 1.00 1.00 0.977 0.973 0.98

c 1.06 1.00 1.00 0.978 0.957 0.98

À 1.74 À 6.05 À 3.04 À 5.19 À 2.49

Phase angle (deg) a 0.00

b 240.00 238.27 234.80 235.90 235.67 236.41

c 120.00 117.65 114.83 115.36 113.39 115.98

(b) Sequence voltages

Magnitude (p.u.) Zero 0.00 0.0036 0.0050 0.0087 0.0161 0.0077

Positive 1.06 1.0000 1.0000 0.9782 0.9695 0.9799

Negative 0.00 0.0036 0.0050 0.0073 0.0090 0.0078

6.7 SUMMARY

In the ¬rst part of this chapter we presented the theory of power ¬‚ow in phase coordinates

using the Newton“Raphson method. This enables the reliable solution of three-phase power

systems exhibiting any degree of geometric and operational imbalance. To illustrate the

265

REFERENCES

additional ¬‚exibility introduced by the phase coordinates modelling, the ¬ve-bus network

was solved for cases of balanced and unbalanced system load, and comparisons were drawn.

A three-phase power ¬‚ow function written in Matlab1 code was provided to enable a hands-

on application of the theory. It used the Newton“Raphson method and is suitable for solving

small and medium-sized networks with balanced and unbalanced system loads. The

function is quite general, but modelling capability has been kept at a relatively low level

to avoid cumbersome and lengthy code. Nevertheless, incorporation of advanced power

plant models, such as those studied in Chapter 3, is quite a straightforward programming

exercise.

The second part of the chapter focused on developing three-phase models of key FACTS

controllers, such as the SVC, STATCOM, TCSC, and UPFC. The ¬rst two controllers are

shunt-connected, and the test cases presented emphasise the fact that, at least in principle,

these controllers are capable of restoring voltage magnitude at the point of connection, in

addition to ful¬lling their basic function of providing reactive power support. The TCSC and

UPFC are series-connected controllers and, in their respective numerical examples, they are

set to enable balanced active power ¬‚ows in the compensated transmission line. This is

achieved by contributing unbalanced compensation, a fact that does not dent the quadratic

convergence characteristic of the three-phase power ¬‚ow Newton“Raphson method.

Incorporation of these FACTS controller models into the Matlab1 function given in this

chapter is a more cumbersome exercise than the incorporation of conventional power plant

models.

REFERENCES

Angeles-Camacho, C., 2000, Modelado en Estado Estacionario del Controlador Uni¬cado de Flujo de

´

´ ´ ´

Potencia para el Analisis Trifasico de Sistemas Electricos, MSc thesis, Instituto Tecnologico de

´

Morelia, Mexico.

Arrillaga, J., Harker B.J., 1978, ˜Fast Decoupled Three-phase Load Flow™, Proceedings of the IEE

125(8) 734“740.

Birt, K.A., Graffy, J.J., McDonald J.D., 1976, ˜Three-phase Load Flow Program™, IEEE Trans. on Power