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. 11
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I I
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 2 12
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 q ca 7 6 7
qk
qk q SVC
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 q SVC q SVC
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 q ca 7 6 7
qk
qk q SVC
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 Á ab 7
0
qk q SVC 7 6
qk q SVC
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 Á bc 7
qk
qk q SVC q SVC
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 q ca
qk q SVC Á ca
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
q ab q ab
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
q ab q ab
SVC SVC


2h i
qBj j
¼À 1 þ cos 2 SVC : °6:44Þ
SVC
pXL
q ab
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 °Á j 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


qQ 
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  ފg À C2 cos2 °p À  Þ
‚ f! tan½!°p À  ފ 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
qQ 
¼ Qk BTCSC°1Þ ; °6:60Þ
k
q  q 
where

qXTCSC°1Þ  
¼ À2C1 ½1 þ cos°2  ފ þ C2 sin°2  Þf! tan½!°p À  ފ À tan  g
q 
 
cos2 °p À  Þ

þ C2 !2 À1 : °6:61Þ
cos2 ½!°p À  ފ
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
qQ
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
+


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

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