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5.7.3 Numerical Example of Power Flow Control using
One HVDC-VSC

The ¬ve-bus system is used to illustrate the power ¬‚ow control performance of the HVDC-
VSC models. This power ¬‚ow controller may be used to regulate the amount of power ¬‚ow
at their points of connection or even to reverse the direction of power ¬‚owing through the
controller.



5.7.3.1 HVDC-VSC back-to-back model

The original network is modi¬ed to include one back-to-back (BTB) HVDC model to
regulate power ¬‚ow at the points of connection. Take, for instance, the case when the UPFC
is installed at the receiving end of line Lake“Main and is set to regulate active and reactive
powers ¬‚owing from Lake to Main at 40 MW and 2 MVAR, respectively. The voltage
magnitude at bus Lake is controlled at 1 p.u. The back-to-back HVDC model replaces
the UPFC used in the test case described in Section 5.6.3. As expected, the power ¬‚ow
results for both cases are exactly the same.



5.7.3.2 HVDC-VSC full model

A different situation arises when the full HVDC-VSC model replaces the combined UPFC“
transmission-line model connected between Lake and Main since the DC cable will contain
neither the inductance nor the capacitance of the transmission line. In this example, the
cable resistance in the DC system is taken to have the same value as the transmission-line
resistance in the AC system, which is 1 %. Figure 5.20 shows results for the case when the full
HVDC-VSC is used to control active power ¬‚ow at Lake at 40 MW, and Table 5.9 shows the
nodal voltages in the modi¬ed network.
The data given in function PowerFlowsData in Section 4.3.9 is modi¬ed to acco-
mmodate for the inclusion of the HVDC. For HVDC-BTB the modi¬cation is as in Section
5.4.3, and for the HVDC-VSC the transmission line originally connected between Lake and
Main is replaced by the HVDC-VSC. Function HVDCData is used to enter HVDC data:


%This function is used exclusively to enter data for:
%HIGH VOLTAGE DIRECT CURRENT (HVDC)

% NHVDC : Number of HVDC™s
% HVDCsend : Shunt converter™s sending bus
% HVDCrec : Shunt converter™s receiving bus
% Rcd : DC cable™s resistance for HVDC DC-Link model
% Psp : Target active power ¬‚ow (p.u.)
% VvrLo : Lower limit for voltage sources magnitudes (p.u.)
% VvrHi : Higher limit for voltage sources magnitudes (p.u.)
% SENDING BUS
226 POWER FLOW INCLUDING FACTS CONTROLLERS

% Xvr1 : Inductive reactance of transformer-sending
% TarVol1 : Target nodal voltage magnitude (pu.)
% VSta1 : control status for nodal voltage magnitude: 1 is on; 0 is off
% Qsp1 : Target reactive power ¬‚ow
% QSta1 : control status for reactive power: 1 is on; 0 is off
% Vvr1 : Initial condition for the shunt source voltage magnitude (p.u.)
% Tvr1 : Initial condition for the shunt source voltage phase angle (rad.)
% RECEIVING BUS
% Xvr2 : Inductive reactance of transformer-receiving
% TarVol2 : Target nodal voltage magnitude (pu.)
% VSta2 : Control status for nodal voltage magnitude: 1 is on; 0 is off
% Qsp2 : Target reactive power ¬‚ow
% QSta2 : Control status for reactive power: 1 is on; 0 is off
% Vvr2 : Initial condition for the shunt source voltage magnitude (p.u.)
% Tvr2 : Initial condition for the shunt source voltage angle (rad.)
NHVDC=1;
HVDCsend(1)=3; HVDCrec(1)=4; Xvr1(1)=0.1; Xvr2(1)=0.1; Rcd(1)=0.1;
TarVol1(1)=1.0; VSta1(1)=1; Qsp1(1)=0.02; QSta1(1)=0;
TarVol2(1)=1.0; VSta2(1)=0; Qsp2(1)=-0.02; QSta2(1)=1;
Psp(1)=0.4;
Vvr1(1)=1.0; Tvr1(1)=0.0; Vvr2(1)=1.0; Tvr2(1)=0.0;
VvrHi(1)=1.1; VvrLo(1)=0.9;


131.33 45 + j15
85.80 40 + j5


North Main
Lake 39.84
40.0
50.32 48.41

81.01 76.46 link
DC
17.8
18.4 8.93 3.49
0.15
36.58 13.49
11.72 13.51

2.85
78.71 75.92 13.62
12.97
37.50 0.77 13.34 4.21


South Elm
46.65
47.58

20 + j10 60 + j10
5.78
5.64
74.02
40

Figure 5.20 Power ¬‚ow results in the ¬ve-bus network with one full high-voltage direct-current-
based voltage source converter
227
EFFECTIVE INITIALISATION OF FACTS CONTROLLERS

Nodal voltages in the modi¬ed network
Table 5.9
Network bus
Nodal voltage North South Lake Main Elm
Magnitude (p.u.) 1.06 1 1 0.989 0.973
À 1.76 À 6.01 À 3.14 À 4.95
Phase angle (deg) 0




5.8 EFFECTIVE INITIALISATION OF FACTS CONTROLLERS

The modelling of FACTS controllers for application in power ¬‚ow analysis results in highly
nonlinear equations which should be suitably initialised to ensure quadratic convergent
solutions when using the Newton“Raphson method. This section addresses such a problem
and makes ¬rm recommendations for the use of simple and effective initialisation
procedures for all FACTS models in power ¬‚ow and related studies.


5.8.1 Controllers Represented by Shunt Synchronous Voltage Sources

Extensive use of FACTS models represented by shunt voltage sources indicates that
elements such as the STATCOM, the shunt source of the UPFC, and the two-shunt sources
representing the HVDC-VSC are suitably initialised by selecting 1 p.u. voltage magnitudes
and 0 phase angles.


5.8.2 Controllers Represented by Shunt Admittances

It has been found that the SVC is well initialised by selecting a ¬ring-angle value that cor-
responds to the reactance resonant peak; this value is calculated by using Equation (5.40).


5.8.3 Controllers Represented by Series Reactances

The TCSC can be represented as an equivalent variable reactance, the ability of which either
to generate or to absorb reactive power is a function of the thyristor ¬ring angle, TCSC . The
adjustable reactance representing the TCSC module shown in Figure 5.11 is well described
by Equations (5.31)“(5.35).
Normally, the active power ¬‚ow through the TCSC is chosen to be the control variable,
and TCSC is chosen to be the state variable. Hence, good initial values for TCSC become
mandatory in order to ensure robust iterative solutions. To this end, an approximation of
Equation (5.31) is used:

XTCSC°1Þ ° TCSC Þ % ÀC2 Á $ Á tan½$°p À TCSC ފ: °5:72Þ
228 POWER FLOW INCLUDING FACTS CONTROLLERS

Extensive testing carried out with a wide range of practical combination of values of
XC and XL con¬rm that Expression (5.72) represents the most signi¬cant term of
Equation (5.31) for the range of interest of TCSC operation.
Expression (5.72) is further altered to include the reactance of the compensated
transmission line (i.e. XTCSC °1Þ þ XTL ¼ XTCSC°1Þ ) and then solving for TCSC,
 
ÀXTCSC °1ÞÀTL
1
TCSC ¼ p À arctan : °5:73Þ
$ C2 $

It has been found that this expression yields very effective initialisations of TCSC when the
reactance contribution of the TCSC to XTCSC (1)“TL is assumed to be nil. Hence,
 
ÀXTL
1
TCSC ¼ p À arctan : °5:74Þ
$ C2 $



5.8.4 Controllers Represented by Series Synchronous
Voltage Sources

Suitable initialisation of series voltage sources in power ¬‚ow studies is mandatory to ensure
robust solutions. Examples of power electronic controllers that use one or more series
voltage sources are: the static synchronous series compensator (SSSC), UPFC, and the latest
addition to the family of FACTS controllers, the interline power ¬‚ow controller (IPFC).
Different equations exist for the purpose of initialising the series voltage source,
depending on the operating condition exhibited by the controller. For example, for the case
when active and reactive powers are speci¬ed at bus k, and assuming Vk ¼ Vm ¼ 1 p.u., and
k ¼ m ¼ 0 in Equations (5.50) and (5.51), leads to the following simple expressions:

VcR ¼ XcR °P2 sp þ Q2 sp Þ1=2 ; °5:75Þ
m m
 
Pm sp
cR ¼ arctan : °5:76Þ
Qm sp

These equations are used to initialise the parameters of series voltage sources within the
°0Þ °0Þ
Newton“Raphson power ¬‚ow solution. These parameters are referred to as VvR and vR .



5.9 SUMMARY

This chapter has covered the topic of power ¬‚ow models of FACTS controllers and assessed
their role in network-wide applications. Key aspects of modelling implementation in power
¬‚ow algorithms have received attention. Numerical examples have been included for each
one of the FACTS controllers presented.
The nonlinear power ¬‚ow equations of the various FACTS controllers have been
linearised and included in a Newton“Raphson power ¬‚ow algorithm. In this context, the state
variables corresponding to the controllable devices have been combined simultaneously
229
REFERENCES

with the state variables of the network in a single frame of reference for uni¬ed, iterative
solutions. The robustness of the method has been illustrated by numerical examples.
Coordinated strategies have been developed to handle cases when more than one
controller, either conventional or FACTS, regulates voltage magnitude at the same bus.
The starting values given to state variables of some FACTS controllers have proved to
have a determining effect as to whether or not the power ¬‚ow solution can be obtained. This
is an implementation aspect of paramount importance and has been duly addressed. A set of
analytical equations has been derived to give series synchronous voltage sources good initial
conditions. The case of shunt synchronous voltage sources is not a critical issue. The
variable series compensation representation based on ¬ring angle is a highly nonlinear
model, and use of the simple analytical equation presented in this chapter for initialisation
purposes should be used.



REFERENCES
Acha, E., 1993, ˜A Quasi-Newton Algorithm for the Load Flow Solution of Large Networks with FACTS
Controlled Branches™, in Proceedings of the 28th Universities Power Engineering Conference
1993, Staffordshire University, Volume 1, 21“23 September, Stafford, UK, pp. 153“156.
´
Ambriz-Perez, H., Acha, E., and Fuerte-Esquivel, C.R., 2000, ˜Advanced SVC Models for Newton“
Raphson Load Flow and Newton Optimal Power Flow Studies™, IEEE Trans. Power Systems 15(1)
129“136.
Arrillaga, J., Arnold, C.P., 1990, Computer Analysis of Power Systems, John Wiley & Sons, Chichester.
Chang, S.K., Brandwajn V., 1988, ˜Adjusted Solutions in Fast Decoupled Load Flow™, IEEE Trans.
Power Systems 3(2) 726“733.
Erinmez, I.A. (Ed.), 1986, ˜Static Var Compensators™, Working Group 38-01, Task Force No. 2 on SVC,
´
Conseil International des Grands Reseaux Electriques (CIGRE).
Fuerte-Esquivel, C.R., Acha, E., 1996, ˜Newton“Raphson Algorithm for the Reliable Solution of Large
Power Networks with Embedded FACTS Devices™, IEE Proceedings : Generation, Transmission
and Distribution 143(5) 447“ 454.
Fuerte-Esquivel, C.R., Acha, E., 1997, ˜The Uni¬ed Power Flow Controller: A Critical Comparison of
Newton“Raphson UPFC Algorithms in Power Flow Studies™ IEE Proceedings : Generation,
Transmission and Distribution 144(5) 437“ 444.
´
Fuerte-Esquivel, C.R., Acha, E., Ambriz-Perez, H., 2000a, ˜Integrated SVC and Step-down Transformer
Model for Newton“Raphson Load Flow Studies™, IEEE Power Engineering Review 20(2) 45“46.
´
Fuerte-Esquivel, C.R., Acha, E., Ambriz-Perez, H., 2000b, ˜A Thyristor Controlled Series Compensator
Model for the Power Flow Solution of Practical Power Networks™, IEEE Trans. Power Systems
15(1) 58“64.
´
Fuerte-Esquivel, C.R., Acha, E., Ambriz-Perez, H., 2000c, ˜A Comprehensive UPFC Model for the
Quadratic Load Flow Solution of Power Networks™, IEEE Trans. Power Systems 15(1) 102“109.
Fuerte-Esquivel, C.R., Acha, E., Tan, S.G., Rico, J.J., 1998, ˜Ef¬cient Object Oriented Power System
Software for the Analysis of Large-scale Networks Containing FACTS Controlled Branches™, IEEE
Trans. Power System 3(2) 464“472.
Grainger, J.J., Stevenson W.D. Jr., 1994, Power System Analysis, McGraw-Hill, New York.
Hingorani, N.G., Gyugyi, L., 2000, Understanding FACTS Concepts and Technology of Flexible AC
Transmission Systems, Institute of Electrical and Electronic Engineers, New York.
´
IEEE/CIGRE: (Institute of Electrical and Electronic Engineers/Conseil International des Grands
´
Reseaux Electriques), 1995, ˜FACTS Overview™, Special Issue, 95TP108, IEEE Service Center,
Piscataway, NJ.
230 POWER FLOW INCLUDING FACTS CONTROLLERS

IEEE SSCWG (Special Stability Controls Working Group), 1995, ˜Static Var Compensator Models for
Power Flow and Dynamic Performance Simulation™, IEEE Trans. Power Systems 9(1) 229“240.
Kundur P.P., 1994, Power System Stability and Control, McGraw-Hill, New York.
Nabavi-Niaki, A., Iravani, M.R., 1996, ˜Steady-state and Dynamic Models of Uni¬ed Power Flow
Controller (UPFC) for Power System Studies™, IEEE Trans. Power Delivery 11(4) 1937“1943.
Noroozian M., Andersson G., 1993, ˜Power Flow Control by Use of Controllable Series Components™,
IEEE Trans. on Power Delivery 8(3) 1420“1429.
Song, Y.H., Johns, A.T., 1999, Flexible AC Transmission Systems (FACTS), Institution of Electric
Engineers, London.
Tinney W.F., Hart C.E., 1967, ˜Power Flow Solution by Newton™s Method™, IEEE Trans. Power
Apparatus and Systems PAS-96(11) 1449“1460.
6
Three-phase Power Flow

6.1 INTRODUCTION

If no proper action is taken at the design stage, long-distance alternating-current (AC)
transmission circuits introduce a signi¬cant amount of geometric unbalance, which in turn
causes undesirable voltage, current, and power ¬‚ow imbalances (Wasley and Shlash, 1974a,
1947b). Over the years, a number of anomalies have been traced to the existence of power
system imbalances, such as increased power losses, heating of synchronous generators,
mis¬ring of power converters and ill-tripping of protective relays (Arrillaga and Harker,
1978; Harker and Arrillaga, 1979). Quite often, transmission lines are cited as the sole, most
important, reason for the existence of geometric imbalances (Hesse, 1966). In the past, line
transpositions were a popular resource for restoring geometric balance, but nowadays the
tendency is to avoid them on economic and design grounds. Under normal circumstances,
other power plant equipment such as transformers, generators, and shunt and series banks of
capacitors introduce little geometric unbalance and are no cause for concern. Moreover,
bulk transmission loads tend to be balanced.
In low-voltage distribution systems, the opposite situation exists. Three-phase transmis-
sion lines and cables are short and tend to be geometrically balanced, but urban loads are
mostly of the single-phase type, fed from single-phase feeders. In aggregate, at the
distribution substation, they result in three-phase loads exhibiting a high degree of
unbalance. The rapid growth of electri¬ed railroads has also been cited as a contributing
factor to distribution system imbalances (Zhang and Chen, 1994). In rural circuits,
continuity of supply has sometimes been maintained by using two of the three single-phase
transformers in the bank, following failure of one of the units. The resulting three-phase
transformer connection is termed ˜open delta™ and, although not recommended for normal
operation owing to its unbalanced nature, can be used as a last resort to maintain supply.
Positive sequence power ¬‚ows are not suitable for the study of power losses in systems
exhibiting signi¬cant transmission imbalances. The alternative solution approach is to use a
three-phase power ¬‚ow algorithm, with all the relevant power plant equipment modelled in
phase coordinates, as detailed in Chapter 3 (Chen and Dillon, 1974; Laughton, 1968).
Comprehensive assessments of the impact of unbalanced loading and equipment on system
operation are carried out with little effort using fully ¬‚edge three-phase power ¬‚ow solutions



FACTS: Modelling and Simulation in Power Networks.
´ ´
Enrique Acha, Claudio R. Fuerte-Esquivel, Hugo Ambriz-Perez and Cesar Angeles-Camacho
# 2004 John Wiley & Sons, Ltd ISBN: 0-470-85271-2
232 THREE-PHASE POWER FLOW

(Birt, Graffy, and McDonald, 1976; Chen et al., 1990; Laughton and Saleh, 1985; Smith and
Arrillaga, 1998).
Line transpositions are no longer regarded as the preferred option for keeping geometric
imbalances under control. Instead, a new solution is emerging, based on the use of power
electronics. If a thyristor-controlled series compensator (TCSC) is already available for the
purpose of impedance compensation then the idea would be to operate it in an unbalanced
manner so that geometric balance can be restored at the point of connection. The
applicability of an static VAR compensator (SVC) to restore voltage balance, in addition to
achieving its primary function of providing reactive power support, has been established at
the simulation level. However, this is at the expense of injecting a substantial amount of
third harmonic current into the AC system. An alternative solution is to use a static
compensator (STATCOM) for which the harmonic generation pattern is not signi¬cantly
in¬‚uenced by terminal AC voltage conditions.
To carry out comprehensive studies of active and reactive power ¬‚ows in unbalanced
transmission systems, and to determine the role that FACTS controllers may play in
reducing transmission imbalances, it is mandatory to have a three-phase power ¬‚ow
computer program with FACTS equipment modelling capability (Angeles-Camacho, 2000;
Venegas and Fuerte-Esquivel, 2001). This is the object of this chapter, where the theory of
three-phase power ¬‚ow is presented. It builds on the strength of the material presented in
Chapters 2“5. Chapters 2 and 3 addressed the modelling of FACTS controllers and
conventional power systems plant in phase coordinates, respectively. Chapters 4 and 5
studied the theory of conventional and FACTS power ¬‚ow using the Newton“Raphson
method, respectively.


6.2 POWER FLOW IN THE PHASE FRAME OF REFERENCE

The starting point for developing nodal power equations suitable for three-phase power ¬‚ow
solutions using the Newton“Raphson method is to establish a relationship between injected
bus currents and bus voltages. This may be achieved by using an approach similar to that
followed in Section 4.2.1 for the case of positive sequence power ¬‚ows.
With reference to the three-phase transmission circuit shown in Figure 3.2, and redrawn
for convenience in Figure 6.1 in a slightly modi¬ed form, the three-phase currents and

Bus k Bus m
La
Ra
I ka a
Im

Eka a
Em
Lab
Rb Lb
I kb b
Im

Ekb b
Em
Lac
Lbc
Rc Lc
c c
I Im
k


Ekc c
Em

Three-phase branch
Figure 6.1
233
POWER FLOW IN THE PHASE FRAME OF REFERENCE

voltages are related by the transfer admittance matrix of the branch:
" #" #" #
Iabc Yabc Yabc Eabc
¼ ; °6:1Þ
k kk km k
abc abc abc
Eabc
Im Ymk Ymm m

where
2 3À1
Raa--g þ j!Lkm
aa--g ab--g ab--g
Rac--g þ j!Lkm
ac--g
Rkm þ j!Lkm
6 7
km km
Yabc ¼ Gabc þ jBabc ¼ 6 Rba--g þ j!Lkm Rbc--g þ j!Lkm 7 ;
ba--g bb--g bb--g bc--g
Rkm þ j!Lkm
4 km 5
kk kk kk km
Rca--g þ j!Lkm
ca--g cb--g cb--g
Rkm þ j!Lcc--g
cc--g
Rkm þ j!Lkm
km km
°6:2Þ
Yabc ¼ Yabc ¼ ÀYkm ¼ ÀYmk ;
abc abc
kk mm

Eabc ¼ ½Ek Ek Ek Št ¼ ½Vk ek Vk ek Vk ek Št ;
a b c
abc a b c
k
°6:3Þ
Eabc ¼ ½Em Em Em Št ¼ ½Vm em Vm em Vm em Št ;
a b c
a b c a b c
m
Iabc ¼ ½Ik Ik Ik Št ;
abc
k
Iabc ¼ ½Im Im Im Št ;
abc
m

and where t is the transpose of the matrix or vector. Notice that the impedance parameters in
Equation (6.2) are assumed to include the impedance contribution due to ground return loops.


6.2.1 Power Flow Equations

Expressions for active and reactive power injected at the three-phase buses k and m of
Figure 6.1 may be derived from the following complex power expression:
" #" #" #
Pabc þ jQabc
Sabc Eabc IabcÃ
¼ ¼ : °6:4Þ
k k
k k k
Pm þ jQm
abc abc abc abcÃ
abc
Em Im
Sm
After some arduous algebra, the expressions for active and reactive powers injected at
phases a, b, and c of bus k are arrived at:
( )
X X j j
P ¼ Vk
Vi ½Gki cos° À i j Þ þ Bj sin° À i j ފ ; °6:5Þ
k k ki k
i ¼ k;m j ¼ a;b;c
( )
X X
Q ¼ Vk

Vij ½Gj sin° À i j Þ À Bj cos° À i j ފ ; °6:6Þ
k ki k ki k
i ¼ k;m j ¼ a;b;c

where the superscript  is used to denote phases a, b, and c.
As expected, the expressions for calculating the active and reactive powers injected at bus
m are of the same form as Equations (6.5) and (6.6), with the subscript m replacing k, and
vice versa:
( )
X X j j
Vi ½Gmi cos° À i j Þ þ Bj sin° À i j ފ ;
P ¼ V m

°6:7Þ
mi
m m m
i ¼ m;k j ¼ a;b;c
( )
X X
Vij ½Gj sin° À i j Þ À Bj cos° À i j ފ :
Q ¼ Vm

°6:8Þ
mi mi
m m m
i ¼ m;k j ¼ a;b;c
234 THREE-PHASE POWER FLOW

6.2.2 Newton“Raphson Power Flow Algorithm

Solution of the positive sequence nodal power equations using the Newton“Raphson method
has shown strong reliability towards convergence. Building on experience, the Newton“
Raphson technique has been adopted to solve the three-phase nodal power equations.
The power expressions Equations (6.5)“(6.8) are linearised around a base operating point,
as illustrated in Section 4.3.2 for the case of positive sequence power ¬‚ow. In the three-
phase application, mismatch powers and state variables terms become vectors of order
3 ‚ 1, and individual Jacobian terms become matrices of order 3 ‚ 3. The resulting
linearised equation, suitable for iterative solutions, becomes:
2 3°iÞ 2 3°iÞ
qP˜ qP  Áh
2  3°iÞ ˜
ÁP˜ j
6 qh qV Vj 7 6 7
6 7 6j 76 7
j
5 ¼6  7 6 ÁV 7 ; °6:9Þ
4 
4 qQ˜ qQ˜  5 4 j5

ÁQ˜ V
V
qh qV j j
j j
where ˜ ¼ k; m; j ¼ k; m, and (i) is the iteration number.
The vector terms take the following form:
ÁP ¼ ½ÁPa ÁPb ÁPc ÁPa ÁPb Ác Št ; °6:10Þ
˜ k k k m m m

ÁQ ¼ ½ÁQa ÁQb ÁQc ÁQa ÁQb ÁQc Št ; °6:11Þ
˜ k k k m m m

D ¼ ½Áa Áb Ác Áa Áb Ác Št ; °6:12Þ
j k k k m m m
!
ÁV ct
ÁVk ÁVk ÁVk ÁVm ÁVm ÁVm
a b c a b
j
¼ : °6:13Þ
V a c
b a b c
Vk V k Vk V m Vm Vm
j
The Jacobian terms are:
2 3
q P a q Pa q Pa
˜ ˜ ˜
6 q a q b q c 7
6j j7
j
6b 7
q P 6 q P ˜ q Pb q Pb 7
¼6 a ˜7
˜
c 7; °6:14Þ
l
6 q

6 j q j q j 7
q hj b
6c 7
4 q P˜ q P c q Pc 5
˜ ˜
q a q b q c
j j
j
2a 3
q P˜ a q Pa b q Pa c
˜ ˜
6 q Vja Vj q V b Vj q Vjc Vj 7
6 7
j
6b 7
q P  6 q P˜ a q P˜ b q P˜ c 7
b b
¼6 7
˜
6 q V a Vj q V b Vj q V c Vj 7; °6:15Þ
V
q V j 6j 7
j
j
j
6 7
4 q P˜ a q P˜ b q P ˜ c 5
c c c
V V V
q Vja j q Vjb j q Vjc j
2 3
q Qa q Qa q Qa
˜ ˜ ˜
6 q a q b q c 7
6j j7
j
6 7
q Q 6 q Qb q Qb q Qb 7
¼6 a ˜7
˜ ˜
˜
; °6:16Þ
6 q q b q c 7

q hj 6j j7
j
6 7
4 q Q˜ q Q˜ q Qc 5
c c
˜
q a q b q c
j j
j
235
POWER FLOW IN THE PHASE FRAME OF REFERENCE

2 3
q Qa a q Qa b q Qa c
˜ ˜ ˜
6 q Vja Vj V V
q Vjc j 7
q Vjb j
6 7
6 7

q Q˜  6 q Qb a q Q˜ c 7
q Qb b b
6 V 7:
˜ ˜
 Vj ¼ 6 °6:17Þ
a Vj V
q Vjc j 7
6 q Vj q Vj
q Vj bj
7
6 7
4 q Qc a q Q˜ c 5
q Qc b c
˜ ˜
a Vj V V
q Vj q Vjc j
q Vj
bj


It should be noted that the linearised Equation (6.9) applies to only one three-phase
transmission line connected between buses k and m. However, the result may be readily
extended to the more practical case, involving nl transmission lines connected between nb
generic buses ˜ and j, where ˜ ¼ 1; . . . ; k; m; . . . ; nb À 1, and j ¼ 1; . . . ; k; m; . . . ; nb À 1.
Note that only nb À 1 buses are considered since the contribution of the slack bus is not
explicitly represented in the linearised system of equations.
Consider the ˜th element connected between buses k and m in Equation (6.9), for which
the self and mutual Jacobian terms are explicitly given below, with the help of two phase
superscripts 1 and 2 used to denote a, b, and c, respectively.
For k ¼ m, and 1 ¼ 2:
 2
qP1 1 cal 1
Bkk 1 ;
1
k;l
¼ ÀQk À Vk °6:18Þ
1
qk;l
 2
qP1 1 1 cal 1
Gkk 1 ;
1
k;l
V ¼ Pk þ Vk °6:19Þ
1 k;l
qVk;l
qQ1 À  Á2
¼ P1 cal À Vk 1 G1 1 ;
k;l
°6:20Þ
q1
k kk
k;l
 2
qQ1
V 1 Q1 cal 1
B1 1 :
k;l
¼ À °6:21Þ
Vk
1 k;l k kk
qVk;l
For k ¼ m, and 1 6¼ 2:
h    i
qP1 1 2 12 1 2 1 2 1 2
k;l
¼ sin k À k À Bkk cos k À k ; °6:22Þ
V k Vk Gkk
q2
k;l
h    i
qP1
V 2 ¼ Vk Vk G1 2 cos 1 À 2 þ B1 2 sin 1 À 2 ;
1 2
k;l
°6:23Þ
2 k;l kk k k kk k k
qVk;l
h    i
qQ1 1 2 1 2 1 2 1 2 1 2
k;l
¼ ÀVk Vk Gkk cos k À k þ Bkk sin k À k ; °6:24Þ
q2
k;l
h    i
qQ1 2 1 2 1 2 1 2 1 2 1 2
k;l
V ¼ Vk Vk Gkk sin k À k À Bkk cos k À k : °6:25Þ
2 k;l
qVk;l

For k 6¼ m:
h    i
qP1
¼ Vk Vm G1 2 sin 1 À 2 À B1 2 cos 1 À 2 ;
1 2
k;l
°6:26Þ
q2
m m
km k km k
m;l
236 THREE-PHASE POWER FLOW


h    i
qP1
V 2 1 2
G1 2 1 1 2 1
2 2
k;l
¼ cos k À m þ Bkm sin k À m ; °6:27Þ
V k Vm
2 m; l km
qVm;l

h    i
qQ1 1 2
G1 2 1 1 2 1
2 2
k;l
¼ ÀVk Vm cos k À m þ Bkm sin k À m ; °6:28Þ
q2
km
m;l

h    i
qQ1
V 2 1 2 1 2 1 1 2 1
2 2
k;l
¼ Gkm sin k À m À Bkm cos k À m : °6:29Þ
Vk Vm
2 m;l
qVm;l

The iterative solution of the three-phase power ¬‚ow equations using the Newton“Raphson
method requires similar considerations to those applied in the case of positive sequence
solutions regarding state variable initialisation and generator reactive power limit checking,
as presented in Sections 4.3.3 and 4.3.4, respectively. However, note that in the three-phase
application the voltage phase angles of phases a, b, and c are initialised at values of 0,
À2p=3, and 2p=3, respectively.


6.2.3 Matlab1 Code of a Power Flow Program in the
Phase Frame of Reference

The Matlab1 computer program, given in Section 4.3.6, has been extended to cater for the
power ¬‚ow solution of three-phase networks. The function TLParameters is used to
furnish transmission-line data in phase coordinates, starting from positive, negative, and
zero sequence information. This information is widely available in utility data bases since it
is used for the purpose of short-circuit current calculations, even though its usefulness is
of somewhat limited applicability in three-phase power ¬‚ow studies, as it assumes that
transmission lines are geometrically balanced. The relevant theory is covered in
Section 3.2.13. If more realistic representation of transmission lines are required then the
function Longline given in Sections 3.2.7 and function TransmissionLineData given
in Section 3.2.11 can be used, and the ensuing transmission-line parameters supplied to the
three-phase power ¬‚ow application.
Generators are represented as three-phase active power injections and adjustable reactive
power injections to meet speci¬ed nodal voltage magnitudes at their terminals. If a more
realistic synchronous generator representation is required then the three-phase model
derived in Section 3.4 should be implemented in the power ¬‚ow program. Note that this is a
very comprehensive model which caters for saliency and has explicit representation of the
generator load angle.
To keep the length and complexity of the current program (Program 6.1) within bounds, it
does not contain provisions for three-phase transformer representation. However,
implementation of three-phase transformer banks with a wide range of connections can
be implemented with ease. The most popular transformer connections are detailed in
Section 3.3.4, where emphasis is placed on transformer complex tap modelling.
Loads are taken to be constant sinks of active and reactive power in the program but,
again, voltage dependency can be incorporated by using the relevant models provided in
Section 3.5.
237
POWER FLOW IN THE PHASE FRAME OF REFERENCE

Program 6.1 Program written in Matlab1 to carry out power ¬‚ow calculations in the three-
phase frame of reference using the Newton“Raphson method



PowerFlowsData3Ph; % read threephasedata


[TLImpedInv,TLAdmit] = TLParameters(ntl,tlresisp,tlreacp,tlcondp,...
tlsuscepp,tlresisz,tlreacz,tlcondz,tlsuscepz);


[YR,YI] = YBus3Ph(nbb,ntl,tlsend,tlrec,TLImpedInv,TLAdmit,nsh,...
shbus, shresis,shreac);


[VM,VA,it] = NewtonRaphson3Ph(nmax,tol,itmax,ngn,ntl,tlsend,tlrec,...
nld, nbb,bustype,genbus,loadbus,PGEN,QGEN,QMAX,QMIN,PLOAD,QLOAD,...
YR,YI, TLImpedInv,TLAdmit,VM,VA);


[PQsend,PQrec,PQloss] = PowerFlows3Ph(nbb,ntl,tlsend,tlrec,VM,VA,...
TLImpedInv,TLAdmit);


% END of main three-phase program



function [TLImpedInv,TLAdmit] = TLParameters(ntl,tlresisp,tlreacp,...
tlcondp, tlsuscepp, tlresisz,tlreacz,tlcondz,tlsuscepz)
% Transmission line parameters
TLImpedInv = zeros(3,3,ntl);
TLAdmit = zeros(3,3,ntl);
for kk = 1 : ntl
Zself = ((tlresisz(kk)+ tlreacz(kk)*i) + 2*(tlresisp(kk) + tlreacp...
(kk)*i))/3;
Zmutual = ((tlresisz(kk) + tlreacz(kk)*i)-(tlresisp(kk) + tlreacp...
(kk)*i))/3;
Yself = ((tlcondz(kk) + tlsuscepz(kk)*i) + 2*(tlcondp(kk) + ...
tlsuscepp (kk)*i))/3;
Ymutual = ((tlcondz(kk) + tlsuscepz(kk)*i)-(tlcondp(kk) + tlsuscepp...
(kk)*i))/3;
for ii = 1 : 3
for jj = 1 : 3
if ii == jj
TLImpedInv(ii,jj,kk) = Zself;
TLAdmit(ii,jj,kk) = Yself;
else
TLImpedInv(ii,jj,kk) = Zmutual;
238 THREE-PHASE POWER FLOW

TLAdmit(ii,jj,kk) = Ymutual;
end
end
end
imped = TLImpedInv(1:3,1:3,kk);
imped2 = inv(imped);
TLImpedInv(1:3,1:3,kk) = imped2;
end



function [YR,YI] = YBus3Ph(nbb,ntl,tlsend,tlrec,TLImpedInv,...
TLAdmit,nsh,shbus,shresis,shreac)
% Set up YY
YY = zeros(nbb*3,nbb*3);
% Transmission lines conribution
for kk = 1 : ntl
ii = (tlsend(kk)-1)*3 + 1;
jj = (tlrec(kk)-1)*3 + 1;
YY(ii:ii + 2,ii:ii + 2) = YY(ii:ii + 2,ii:ii + 2) + ...
TLImpedInv(:,:,kk) + 0.5*TLAdmit(:,:,kk);
YY(ii:ii + 2,jj:jj + 2) = YY(ii:ii + 2,jj:jj + 2) - TLImpedInv(:,:,kk);
YY(jj:jj + 2,ii:ii + 2) = YY(jj:jj + 2,ii:ii + 2) - TLImpedInv(:,:,kk);
YY(jj:jj + 2,jj:jj + 2) = YY(jj:jj + 2,jj:jj + 2) + ...
TLImpedInv(:,:,kk) + 0.5*TLAdmit(:,:,kk);
end
% Shunt elements conribution
for kk = 1 : nsh
SHAdmit = zeros(3,3);
jj = shbus(kk)*3;
for ii = 1 : 3
SHAdmit(ii,ii) = 1/(shresis(kk,ii) + shreac(kk,ii)*i);
end
YY(jj-2:jj,jj-2:jj) = YY(jj-2:jj,jj-2:jj) + SHAdmit(:,:);
end
YR = real(YY);
YI = imag(YY);


function [VM,VA,it] = NewtonRaphson3Ph(nmax,tol,itmax,ngn,ntl,...
tlsend, tlrec,nld,nbb,bustype,genbus,loadbus,PGEN,QGEN,QMAX,QMIN,...
PLOAD,QLOAD,YR, YI,TLImpedInv,TLAdmit, VM,VA)
% GENERAL SETTINGS
D = zeros(1,nmax*3);
¬‚ag = 0;
it = 1;
% CALCULATE NET POWERS
239
POWER FLOW IN THE PHASE FRAME OF REFERENCE

[PNET,QNET] = NetPowers3Ph(nbb,ngn,nld,genbus,loadbus,PGEN,QGEN,...
PLOAD,QLOAD);

while ( it <= itmax & ¬‚ag==0 )
% CALCULATED POWERS
[PCAL,QCAL] = CalculatedPowers3Ph(nbb,ntl, tlsend,tlrec,VM,VA,...
TLImpedInv,TLAdmit);
% POWER MISMATCHES
[DPQ,¬‚ag] = PowerMismatches3Ph(nmax,nbb,tol,bustype,¬‚ag,PNET,...
QNET, PCAL,QCAL);
if ¬‚ag == 1;
break;
end


% JACOBIAN FORMATION
[JAC] = NewtonRaphsonJacobian3Ph(nmax,nbb,bustype,PCAL,QCAL,VM,...
VA,YR, YI);

% SOLVE FOR THE STATE VARIABLES VECTOR
D = JAC\DPQ™;


% UPDATE STATE VARIABLES
[VA,VM] = StateVariablesUpdates3Ph(nbb,D,VA,VM);

it = it + 1;
end



function [PNET,QNET] = NetPowers3Ph(nbb,ngn,nld,genbus,loadbus,...
PGEN,QGEN, PLOAD,QLOAD);
% CALCULATE NET POWERS
PNET = zeros(1,nbb*3);
QNET = zeros(1,nbb*3);
for ii = 1 : ngn
for jj = 1 : 3
PNET((genbus(ii)-1)*3 + jj) = PNET((genbus(ii)-1)*3 + jj) + ...
PGEN(ii,jj);
QNET((genbus(ii)-1)*3 + jj) = QNET((genbus(ii)-1)*3 + jj) + ...
QGEN(ii,jj);
end
end
for ii = 1 : nld
for jj = 1 : 3
PNET((loadbus(ii)-1)*3 + jj) = PNET((loadbus(ii)-1)*3 + jj) - ...
PLOAD(ii,jj);
240 THREE-PHASE POWER FLOW

QNET((loadbus(ii)-1)*3 + jj) = QNET((loadbus(ii)-1)*3 + jj) - ...
QLOAD(ii,jj);
end
end


function [PCAL,QCAL] = CalculatedPowers3Ph(nbb,ntl,tlsend,tlrec,...
VM,VA, TLImpedInv,TLAdmit);
% Include all entries
PQsend = zeros(ntl,3);
PQrec = zeros(ntl,3);
PQloss = zeros(ntl,3);
for iii = 1 : ntl
Vsend = ( VM(tlsend(iii),:).*cos(VA(tlsend(iii),:)) + ...
VM(tlsend(iii),:).*sin(VA(tlsend(iii),:))*i );
Vrec = ( VM(tlrec(iii),:).*cos(VA(tlrec(iii),:))...
VM(tlrec(iii),:).*sin(VA(tlrec(iii),:))*i );
for jj = 1 : 5
if jj < 4
PQsend(iii,jj) = Vsend(1,jj)*(conj(-TLImpedInv(jj,:,iii)) *...
(Vrec(1,:))™ + conj(TLImpedInv(jj,:,iii) + 0.5*...
TLAdmit(jj,:,iii))*(Vsend(1,:)™));
PQrec(iii,jj) = Vrec(1,jj)*(-conj(TLImpedInv(jj,:,iii))* ...
(Vsend(1,:))™ + conj(TLImpedInv(jj,:,iii) + 0.5*...
TLAdmit(jj,:,iii))* (Vrec(1,:)™));
elseif jj==4
PQsend(iii,jj) = tlsend(iii);
PQrec(iii,jj) = tlrec(iii);
else
PQsend(iii,jj) = tlrec(iii);
PQrec(iii,jj) = tlsend(iii);
end
PQloss = PQsend - PQrec;
end
end
PCAL1=zeros(nbb,3);
for ii = 1 : nbb
for jj = 1:ntl
if PQsend(jj,4) == ii
PCAL1(ii,:) = PCAL1(ii,:) + PQsend(jj,1:3);
end
if PQrec(jj,4) == ii
PCAL1(ii,:) = PCAL1(ii,:) + PQrec(jj,1:3);
end
end
end
for ii = 1 : nbb
241
POWER FLOW IN THE PHASE FRAME OF REFERENCE

PCAL2(1,3*ii-2:ii*3)=PCAL1(ii,:);
end
PCAL = real(PCAL2);
QCAL = imag(PCAL2);


function [DPQ,¬‚ag] = PowerMismatches3Ph(nmax,nbb,tol,bustype,...
¬‚ag,PNET,QNET,PCAL,QCAL);
% POWER MISMATCHES
DPQ = zeros(1,nmax);
DP = PNET - PCAL;
DQ = QNET - QCAL;
% To remove the active and reactive powers contributions of the slack
% bus and reactive power of all PV buses
kk = 1;
for ii = 1 : nbb
for jj = 1 : 3
if (bustype(ii) == 1 )
DP(kk) = 0;
DQ(kk) = 0;
elseif (bustype(ii) == 2 )
DQ(kk) = 0;
end
kk = kk 1;
end
end
% Re-arrange mismatch entries
kk = 1;
for ii = 1 : nbb
for jj = 1 : 3
DPQ((ii-1)*3 + kk) = DP(kk);
DPQ((ii-1)*3 + kk + 3) = DQ(kk);
kk = kk + 1;
end
end
% Check for convergence
for ii = 1 : nbb*6
if (abs(DPQ) < tol)
¬‚ag = 1;
end
end



function [JAC] = NewtonRaphsonJacobian(nmax,nbb,bustype,PCAL,QCAL,...
VM,VA, YR,YI);
% JACOBIAN FORMATION - Include all entries
JAC = zeros(nmax,nmax);
242 THREE-PHASE POWER FLOW

iii = 1;
for ii = 1 : nbb
kk = (ii-1)*3 + 1;
jjj = 1;
for jj = 1 : nbb
ll = (jj-1)*3 + 1;
if ii == jj
for mm=1:3;
for nn=1:3;
if nn==mm
JAC(iii + mm-1,jjj + nn-1) = - QCAL(kk + mm-1) - VM(ii,mm)...
^2*YI(kk + mm-1,kk + mm-1);
JAC(iii + mm-1,3 + jjj + nn-1) = PCAL(kk + mm-1) + ...
VM(ii,mm)^2*YR(kk + mm-1,kk + mm-1);
JAC(iii + 3 + mm-1,jjj + nn-1) = PCAL(kk + mm-1) - VM(ii,mm)...
^2*YR(kk + mm-1,kk + mm-1);
JAC(iii + 3 + mm-1,jjj + 3 + nn-1) = QCAL(kk + mm-1) - ...
VM(ii,mm)^2*YI(kk + mm-1,kk + mm-1);
else
JAC(iii + mm-1,jjj + nn-1) = VM(ii,mm)*VM(ii,nn)*(YR(kk + ...
mm-1,kk + nn-1)*sin(VA(ii,mm)-VA(ii,nn))-YI(kk + mm-1,...
kk + nn-1)*cos(VA(ii,mm)-VA(ii,nn)));
JAC(iii + mm-1,3 + jjj + nn-1) = VM(ii,mm)*VM(ii,nn)*...
(YR(kk + mm-1,kk + nn-1)*cos(VA(ii,mm)-VA(ii,nn)) + YI(kk + ...
mm-1,kk + nn-1)*sin(VA(ii,mm)-VA(ii,nn)));
JAC(iii + 3 + mm-1,jjj + nn-1) = -VM(ii,mm)*VM(ii,nn)*...
(YR(kk + mm-1,kk + nn-1)*cos(VA(ii,mm)-VA(ii,nn)) + YI(kk + ...
mm-1,kk + nn-1)*sin(VA(ii,mm)-VA(ii,nn)));
JAC(iii + 3 + mm-1,jjj + 3 + nn-1) = VM(ii,mm)*VM(ii,nn)...
*(YR(kk + mm-1,kk + nn-1)*sin(VA(ii,mm)-VA(ii,nn))-YI...
(kk + mm-1,kk + nn-1)*cos(VA(ii,mm)-VA(ii,nn)));
end
end
end
else
for mm=1:3;
for nn=1:3;
JAC(iii + mm-1,jjj + nn-1) = VM(ii,mm)*VM(jj,nn)*(YR(kk + ...
mm-1,ll + nn-1)*sin(VA(ii,mm)-VA(jj,nn)) - YI(kk + mm-1,ll + ...
nn-1)*cos(VA(ii,mm)-VA(jj,nn)));
JAC(iii + mm-1,3 + jjj + nn-1) = VM(ii,mm)*VM(jj,nn)*(YR(kk + ...
mm-1,ll + nn-1)*cos(VA(ii,mm)-VA(jj,nn)) + YI(kk + mm-1,ll + ...
nn-1)*sin(VA(ii,mm)-VA(jj,nn)));
JAC(iii + 3 + mm-1,jjj + nn-1) = -VM(ii,mm)*VM(jj,nn)*(YR(kk + ...
mm-1,ll + nn-1)*cos(VA(ii,mm)-VA(jj,nn)) + YI(kk + mm-1,ll + ...
nn-1)*sin(VA(ii,mm)-VA(jj,nn)));
JAC(iii + 3 + mm-1,jjj + 3 + nn-1) = VM(ii,mm)*VM(jj,nn)...
243
POWER FLOW IN THE PHASE FRAME OF REFERENCE

*(YR(kk + mm-1,ll + nn-1)*sin(VA(ii,mm)-VA(jj,nn)) - YI(kk + ...
mm-1,ll + nn-1)*cos(VA(ii,mm)-VA(jj,nn)));
end
end
end
jjj = jjj + 6;
end
iii = iii + 6;
end
% Delete the voltage magnitude and phase angle equations of the slack
% bus and voltage magnitude equations corresponding to PV buses
for kk = 1 : nbb
if (bustype(kk) == 1)
ll = (kk-1)*6 + 1;
for ii = ll : ll + 2
for jj = 1 : 6*nbb
if ii == jj
JAC(ii,ii) = 1;
else
JAC(ii,jj) = 0;
JAC(jj,ii) = 0;
end
end
end
end
if (bustype(kk) == 1) j (bustype(kk) == 2)
ll = (kk-1)*6 + 1;
for ii = ll + 3 : ll + 5
for jj = 1 : 6*nbb
if ii == jj
JAC(ii,ii) = 1;
else
JAC(ii,jj) = 0;
JAC(jj,ii) = 0;
end
end
end
end
end



function [VA,VM] = StateVariablesUpdates3Ph(nbb,D,VA,VM)
for ii = 1 : nbb
iii = (ii-1)*6 + 1;
for jj = 1 : 3
VA(ii,jj) = VA(ii,jj) + D(iii);
VM(ii,jj) = VM(ii,jj) + D(iii + 3)*VM(ii,jj);
244 THREE-PHASE POWER FLOW

iii = iii + 1;
end
end


function [PQsend,PQrec,PQloss] = PowerFlows3Ph(nbb,ntl,tlsend,...
tlrec,VM, VA,TLImpedInv,TLAdmit);
% Include all entries
PQsend = zeros(ntl,3);
PQrec = zeros(ntl,3);
PQloss = zeros(ntl,3);
for iii = 1 : ntl
Vsend = ( VM(tlsend(iii),:).*cos(VA(tlsend(iii),:)) + ...
VM(tlsend(iii),:).*sin(VA(tlsend(iii),:))*i );
Vrec = ( VM(tlrec(iii),:).*cos(VA(tlrec(iii),:)) + ...
VM(tlrec(iii),:).*sin(VA(tlrec(iii),:))*i );
for jj = 1 : 5
if jj < 4
PQsend(iii,jj) = Vsend(1,jj)*(conj(-TLImpedInv(jj,:,iii))...
*(Vrec(1,:))™ + conj(TLImpedInv(jj,:,iii) + ...
0.5*TLAdmit(jj,:,iii)) *(Vsend(1,:)™));
PQrec(iii,jj) = Vrec(1,jj)*(-conj(TLImpedInv(jj,:,iii))* ...
(Vsend(1,:))™ + conj(TLImpedInv(jj,:,iii) + ...
0.5*TLAdmit(jj,:,iii)) *(Vrec(1,:)™));
elseif jj == 4
PQsend(iii,jj) = tlsend(iii);
PQrec(iii,jj) = tlrec(iii);
else
PQsend(iii,jj) = tlrec(iii);
PQrec(iii,jj) = tlsend(iii);
end
PQloss = PQsend - PQrec;
end
end



6.2.4 Numerical Example of a Three-phase Network

The ¬ve-bus network shown in Section 4.3.9 is used as the basis for illustrating how the
three-phase power ¬‚ow performs under balanced and unbalanced operating conditions. The
¬le threephasedata contains all the required data for the power ¬‚ow solution. Notice that
voltage information is provided explicitly for the three phases, where a balanced set of
three-phase voltages means equal voltage magnitude and phase angles between adjacent
phases separated by 2p=3 radians, with the following rotation: 0, À2p=3, 2p=3.
In this application, transmission lines require zero sequence information for resistance,
reactance, susceptance, and conductance, in addition to the corresponding positive sequence
parameters. Negative sequence parameters are not explicitly required since they are equal to
245
POWER FLOW IN THE PHASE FRAME OF REFERENCE

positive sequence parameters in transmission lines. It should be mentioned that in the
original ¬ve-bus network, aimed at the testing of positive sequence power ¬‚ow algorithms,
no information exists for zero sequence transmission-line parameters. For the purpose of the
current exercise, zero sequence transmission-line parameters have been taken to be three
times the positive sequence values.

The function threephasedata for the balanced test case is as follows:

% Bubars data
nbb=5;
bustype(1)=1; VM(1,1)=1.06; VA(1,1)=0*pi/180;
VM(1,2)=1.06; VA(1,2)=240*pi/180; VM(1,3)=1.06; VA(1,3)= 120*pi/180;
bustype(2)=2; VM(2,1)=1.00; VA(2,1)=0*pi/180;
VM(2,2)=1.00; VA(2,2)=240*pi/180; VM(2,3)=1; VA(2,3)=120*pi/180;
bustype(3)=3; VM(3,1)=1.00; VA(3,1)=0*pi/180;
VM(3,2)=1.00; VA(3,2)=240*pi/180; VM(3,3)=1; VA(3,3)=120*pi/180;
bustype(4)=3; VM(4,1)=1.00; VA(4,1)=0*pi/180;
VM(4,2)=1.00; VA(4,2)=240*pi/180; VM(4,3)=1; VA(4,3)= 120*pi/180;
bustype(5)=3; VM(5,1)=1.00; VA(5,1)=0*pi/180;
VM(5,2)=1.00; VA(5,2)=240*pi/180; VM(5,3)=1; VA(5,3)= 120*pi/180;

% Generators data
ngn=2;
genbus(1)=1; PGEN(1,1)=0.0; QGEN(1,1)=0; PGEN(1,2)=0.0;
QGEN(1,2)=0; PGEN(1,3)=0.0; QGEN(1,3)=0; QMAX(1)=9; QMIN(1)=-9;
genbus(2)=2; PGEN(2,1)=0.4; QGEN(2,1)=0.0; PGEN(2,2)=0.4;
QGEN(2,2)=0.0; PGEN(2,3)=0.4; QGEN(2,3)=0.0; QMAX(2)=9;
QMIN(2)=-9;

% Transmission lines data
ntl=7;
tlsend(1)=1; tlrec(1)=2; tlresisp(1)=0.02; tlreacp(1)=0.06;
tlcondp(1)=0; tlsuscepp(1)=0.060; tlresisz(1)=0.06;
tlreacz(1)=0.18; tlcondz(1)=0; tlsuscepz(1)=0.18;
tlsend(2)=1; tlrec(2)=3; tlresisp(2)=0.08; tlreacp(2)=0.24;
tlcondp(2)=0; tlsuscepp(2)=0.050; tlresisz(2)=0.24;
tlreacz(2)=0.72; tlcondz(2)=0; tlsuscepz(2)=0.15;
tlsend(3)=2; tlrec(3)=3; tlresisp(3)=0.06; tlreacp(3)=0.18;
tlcondp(3)=0; tlsuscepp(3)=0.040; tlresisz(3)=0.18;
tlreacz(3)=0.54; tlcondz(3)=0; tlsuscepz(3)=0.12;
tlsend(4)=2; tlrec(4)=4; tlresisp(4)=0.06; tlreacp(4)=0.18;
tlcondp(4)=0; tlsuscepp(4)=0.040; tlresisz(4)=0.18;
tlreacz(4)=0.54; tlcondz(4)=0; tlsuscepz(4)=0.12;
tlsend(5)=2; tlrec(5)=5; tlresisp(5)=0.04; tlreacp(5)=0.12;
tlcondp(5)=0; tlsuscepp(5)=0.030; tlresisz(5)=0.12;
tlreacz(5)=0.36; tlcondz(5)=0; tlsuscepz(5)=0.09;
tlsend(6)=3; tlrec(6)=4; tlresisp(6)=0.01; tlreacp(6)=0.03;
246 THREE-PHASE POWER FLOW

tlcondp(6)=0; tlsuscepp(6)=0.020; tlresisz(6)=0.03;
tlreacz(6)=0.09; tlcondz(6)=0; tlsuscepz(6)=0.06;
tlsend(7)=4; tlrec(7)=5; tlresisp(7)=0.08; tlreacp(7)=0.24;
tlcondp(7)=0; tlsuscepp(7)=0.050; tlresisz(7)=0.24;
tlreacz(7)=0.72; tlcondz(7)=0; tlsuscepz(7)=0.15;

% Loads data
nld=4;
loadbus(1)=2; PLOAD(1,1)=0.20; QLOAD(1,1)=0.10; PLOAD(1,2)=0.20;
QLOAD(1,2)=0.10; PLOAD(1,3)=0.20; QLOAD(1,3)=0.10;
loadbus(2)=3; PLOAD(2,1)=0.45; QLOAD(2,1)=0.15; PLOAD(2,2)=0.45;
QLOAD(2,2)=0.15; PLOAD(2,3)=0.45; QLOAD(2,3)=0.15;
loadbus(3)=4; PLOAD(3,1)=0.40; QLOAD(3,1)=0.05; PLOAD(3,2)=0.40;
QLOAD(3,2)=0.05; PLOAD(3,3)=0.40; QLOAD(3,3)=0.05;
loadbus(4)=5; PLOAD(4,1)=0.60; QLOAD(4,1)=0.10; PLOAD(4,2)=0.60;
QLOAD(4,2)=0.10; PLOAD(4,3)=0.60; QLOAD(4,3)=0.10;

% General parameters
itmax=10;
tol=1e-12;
nmax=6*nbb;


As expected, the solution given by the three-phase program essentially agrees with that
provided by the positive sequence power ¬‚ow program, given in Table 4.1. More
speci¬cally, the nodal voltage magnitudes and phase angles for phase a of the network
coincide with those for the positive sequence. The voltage magnitude for phases a, b, and c
have equal values, with the phase angles for phases b and c displaced by 240 and 120 ,
respectively, with respect to those of phase a. Table 6.1 summarises the results for the
balanced three-phase solution. Convergence was achieved in 5 iterations to a power
mismatch tolerance of 1e - 12.
Since this is a case of balanced operation and design parameters “ all loads are taken to be
balanced “ neither negative nor zero sequence voltages exist.



Table 6.1 Three-phase nodal voltages for the balanced case
Network bus
”””
””””””””””” ”
””””””””””””””
Voltage Phase North South Lake Main Elm
Magnitude (p.u.) a 1.06 1.00 0.9872 0.9841 0.9717
b 1.06 1.00 0.9872 0.9841 0.9717
c 1.06 1.00 0.9872 0.9841 0.9717
À 2.06 À 4.63 À 4.95 À 5.76
Phase angle (deg) a 0
b 240 237.93 235.36 235.04 234.23
c 120 117.93 115.36 115.04 114.23
247
POWER FLOW IN THE PHASE FRAME OF REFERENCE

An altogether different situation arises if imbalances are introduced into the test network,
say in the system load. This requires only a straightforward change in the data ¬le, with the
¬le unbalthreephasedata re¬‚ecting these changes “ at each bus, active and reactive
power loads have been altered arbitrarily by Æ15 % with respect to the base, balanced case:

%Loads data with 15 unbalance

nld=4;

loadbus(1)=2; PLOAD(1,1)=0.20; QLOAD(1,1)=0.10;
PLOAD(1,2)=0.1739; QLOAD(1,2)=0.08695; PLOAD(1,3)=0.23;
QLOAD(1,3)=0.115; loadbus(2)=3;
PLOAD(2,1)=0.5175; QLOAD(2,1)=0.1725;
PLOAD(2,2)=0.45; QLOAD(2,2)=0.15; PLOAD(2,3)=0.3913;
QLOAD(2,3)=0.1304; loadbus(3)=4;
PLOAD(3,1)=0.3478; QLOAD(3,1)=0.0435;
LOAD(3,2)=0.46; QLOAD(3,2)=0.0575; PLOAD(3,3)=0.40;
QLOAD(3,3)=0.05; loadbus(4)=5;
PLOAD(4,1)=0.60; QLOAD(4,1)=0.10;
PLOAD(4,2)=0.5217; QLOAD(4,2)=0.087; PLOAD(4,3)=0.69;
QLOAD(4,3)=0.115;


Table 6.2(a) shows the three-phase voltage solution for unbalanced loading. The solution
was achieved in 5 iterations to a power mismatch tolerance of 1 e À 12.
The impact of unbalanced loading on system performance can be appreciated by
comparing the results given in Table 6.2(b), where small amounts of negative and zero



Table 6.2 Three-phase nodal voltages in the unbalanced network: (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 0.9820 0.9811 0.9789
b 1.06 1.00 0.9881 0.9831 0.9755
c 1.06 1.00 0.9908 0.9872 0.9599
À 2.02 À 4.67 À 4.84 À 5.96
Phase angle (deg) a 0
b 240 238.16 235.26 234.95 235.26
c 120 117.58 115.38 114.88 113.23
(b) Sequence voltages
Magnitude (p.u.) Negative 0.00 0.0030 0.0032 0.0027 0.0148
Positive 1.06 1.0000 0.9870 0.9838 0.9713
Zero 0.00 0.0030 0.0020 0.0017 0.0070
248 THREE-PHASE POWER FLOW

51.75 + j17.25
95.25
132.45 34.78 + j4.35
Lake
North Main
42.60 14.02
40.91 14.02

89.85 77.25 18.39
18.00 1.83 3.70
24.86
27.18 0.43
6.42
25.34 0.69
1.08
2.09
76.03
87.40
6.45 5.17
27.70
1.51
Elm
South 53.55
54.36
20 + j10
60 + j10
5.67 4.83
40.0 63.96
(a )


46 + j5.75
45 + j15
89.78
126.45

Main
Lake
North 41.67 20.49
40.20 20.59

84.78 73.05 17.39
16.73 2.78 4.60
25.39
28.67 0.31
3.17
25.75 0.39
0.84
2.27
72.49
82.27
3.19 4.47
29.18
1.46
Elm
South 48.98
49.95
17.39+
j8.70
52.17 + j8.7
3.92 4.23
40.0 63.60
(b)

40 + j5
39.13 + j13.04
86.84
138.05

Main
Lake
North 42.10 40.64 24.03 23.98

95.95 70.97 16.69
15.87 3.92 5.66
22.52
26.41 1.50
10.39
22.78 0.26
0.84
2.84
69.25
93.37
10.16 5.80
26.78
1.86
Elm
South 58.84
60.81
23 + j11.5
69 + j11.5
7.69 5.70
40.0 55.11
(c)

Three-phase power ¬‚ows: (a) phase a, (b) phase b, and (c) phase c
Figure 6.2
249
STATIC VAR COMPENSATOR

sequence voltages are now evident. Power system loss increased by nearly 2 % with respect
to the balanced case. It can be seen from the power ¬‚ow results in Figures 6.2(a)“6.2(c) that
the power ¬‚ows in all three phases are unbalanced.
It has been stated in the introduction of this chapter that FACTS controllers intended for
nodal voltage control could perform the role of restoring voltage magnitude balance at the
point of connection. It was also argued that a series compensator could provide a useful role
in balancing out power ¬‚ows at the point of compensation. Such use of FACTS controllers is
assessed in the following sections.



6.3 STATIC VAR COMPENSATOR

In order to assess the role of SVC operation in unbalanced three-phase power systems
it is necessary to develop a more detailed model of the SVC than the one developed in
Section 5.3 for the case of positive sequence power ¬‚ows. The new SVC power ¬‚ow model
is developed in the frame of reference afforded by the phases, building on its admittance
matrix representation derived in Section 2.3.2.
The model corresponds to a three-phase, delta-connected thyristor-controlled reactor
(TCR) placed in parallel with a three-phase bank of capacitors connected in star
con¬guration, with its star point ¬‚oating. Figure 6.3 shows the SVC equivalent circuit
used to derive the three-phase power ¬‚ow equations. The individual branches are adjusted
individually, by controlling the ¬ring angles of the thyristors, in order to achieve speci¬ed
nodal voltage magnitudes while satisfying the constraint power equations. Two distinct SVC
power ¬‚ow models are described in this section: one uses controllable susceptances as state
variables whereas the other uses the ¬ring angles of the thyristors.
It is illustrated in Figure 6.3 that the three-phase, star-connected capacitor bank has an
alternative representation in the form of a delta-connected equivalent circuit. Equation (6.30)
describes the three-phase SVC model, which is obtained by the simple addition of



Vka q ka Vkb q kb Vkc q kc Vka q ka Vkb q kb Vkc q kc
b
I TCR
b
IC c
a
I TCR
I TCR
a c

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