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 Þ: °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 t ¼ ½Vk ek Vk ek Vk ek t ;

a b c

abc a b c

k

°6:3Þ

Eabc ¼ ½Em Em Em t ¼ ½Vm em Vm em Vm em t ;

a b c

a b c a b c

m

Iabc ¼ ½Ik Ik Ik t ;

abc

k

Iabc ¼ ½Im Im Im t ;

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 Þ ; °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 Þ ; °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 Þ ;

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 Þ :

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 t ; °6:10Þ

˜ k k k m m m

ÁQ ¼ ½ÁQa ÁQb ÁQc ÁQa ÁQb ÁQc t ; °6:11Þ

˜ k k k m m m

D ¼ ½Áa Áb Ác Áa Áb Ác t ; °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