flexible ac transmission systems ( (5)

5 Modeling of FACTS in Three-Phase Power Flow and Three-Phase OPF AnalysisThree-phase power flow calculations are important tools to compute the realisticsystem operation states and eval

5 Modeling of FACTS in Three-Phase Power Flow and Three-Phase OPF Analysis

Three-phase power flow calculations are important tools to compute the realisticsystem operation states and evaluate the control performance of various controldevices such as transformer, synchronous machines and FACTS-devices, particu-larly because (a) there are unbalances of three-phase transmission lines in highvoltage transmission networks; (b) there are unbalanced three-phase loads; (c) inaddition, there are one-phase or two-phase lines in some distribution networks,etc Under these unbalanced operating conditions, three-phase power flow studiesare needed to assess the realistic operating conditions of the systems and analyzethe behavior and control performance of power system components includingFACTS-devices

A number of three-phase power flow methods such as Bus-Impedance Method[1], Newton-Raphson Method [2][3], Fast-Decoupled Method [4][6], Gauss-SeidelMethod [5], Hybrid Method [7], A Newton approach combining representation oflinear elements using linear nodal voltage equation and representation of nonlinearelements using injected currents and associated equality constraints [8], ImplicitBus-Impedance Method [9], Decoupling-Compensation Bus-Admittance Method[9], Fast Three-phase Load Flow Methods [10], and Newton power flow in currentinjection form [12] etc have been proposed since 1960s The Newton method

proposed in [8] is in particular interfaced with EMTP (Electro-Magnetic sients Program) and can be used to initialize the simulations The Fast Three-

Tran-phase Load Flow Methods proposed in [10] have been further implemented on aparallel processor [11]

In addition to the above three-phase power flow solution methods, specializedthree-phase power flow techniques [13]-[21] for distribution networks have alsobeen proposed with various success where the special structure of distributionnetworks is exploited and computational efficiency is improved Modeling ofpower system components can be found in [22][23][6]

An Optimal Power Flow (OPF) program can be used to determine the optimaloperation state of a power system by optimizing a particular objective while satis-fying specified physical and operating constraints Because of its capability of in-tegrating the economic and secure aspects of the system into one mathematicalmodel, the OPF can be applied not only in three-phase power system planning, butalso in real time operation optimisation of three-phase power systems With theincorporation of FACTS-devices into power systems, a three-phase optimal powerflow will be required In contrast to the research in three-phase power flow solu-

tion techniques, the research in optimal three-phase power flow methods has beenvery limited.

With the increasing installation of FACTS in power systems, modeling ofFACTS-devices into three-phase power flow and optimal three-phase power flowanalysis will be of great interest In recent years, three-phase FACTS models havebeen investigated for three-phase power flow analysis [24][25] Positive sequencemodels for FACTS-devices have been discussed in chapters 2, 3 and 4 However,three-phase FACTS models are more complex than those positive sequence onessince unbalanced conditions need to be considered This chapter introduces thefollowing aspects:

• review of three-phase power flow solution techniques;

• three-phase Newton power flow solution methods in polar and rectangular ordinates;

co-• three-phase FACTS models for SSSC and UPFC and their incorporation inthree-phase power flow analysis;

• formulation of optimal three-phase power flow problems

5.1 Three-Phase Newton Power Flow Methods in

Rectangular Coordinates

Modeling of power system components such as transmission lines, loads, etc havebeen discussed in [23][6] In the following, the formulation of three-phase Newtonpower flow in rectangular coordinates will be presented where the modeling ofsynchronous generator is discussed in detail

5.1.1 Classification of Buses

In three-phase power flow calculations, all buses may be classified into the lowing categories:

fol-Slack bus Similar to that in single-phase positive-sequence power flow

calcula-tions, a slack bus, which is usually one of the generator terminal buses, should beselected for three-phase power flow calculations At the slack bus, the positive-sequence voltage angle and magnitude are specified while the active and reactivepower injections at the generator terminal are unknown The voltage angle of theslack bus is taken as the reference for the angles of all other buses Usually there isonly one slack bus in a system However, in some production grade programs, itmay be possible to include more than one bus as distributed slack buses

PV Buses PV buses in three-phase power flow calculations are usually generator

terminal buses For these buses, the total active power injections and sequence voltage magnitudes are specified

positive-5.1 Three-Phase Newton Power Flow Methods in Rectangular Coordinates 141

PQ Buses PQ buses are usually load buses in the network For these buses, the

active and reactive power injections of their three-phases are specified

5.1.2 Representation of Synchronous Machines

A synchronous machine may be represented by a set of three-phase balanced age sources in series with a 3 by 3 impedance matrix Such a synchronous ma-chine model is shown in Fig 5.1 The impedance matrix Zg i may be determined

volt-by positive-, negative-, and zero-sequence impedance parameters of a nous machine Zg i is defined in Appendix A of this chapter

synchro-It is assumed that the synchronous generator in Fig 5.1 has a round rotor ture, and saturation of the synchronous generator is not considered in the presentmodel However, in principle, there is no difficulty to take into account the satura-tion

struc-In Fig 5.1, V i a =E i a+ jF i a,V i b =E i b+ jF i b,V i c=E i c+ jF i c, which are thethree-phase voltages at the generator terminal bus, are expressed in phasors in rec-tangular coordinates Similarly, the voltages at the generator internal bus may begiven by E i a =Eg i a + jFg i a, E i b =Eg i b+ jFg i b, E i c=Eg i c+ jEg i c In fact thevoltages at the generator internal bus are balanced, that is:

3 /

2 π

j a i b

i = E e−

3 /

2 π

j a i c

a i a

i b

Eg

2

32

1+

−

a i a i b

Fg

2

12

i c

Eg

2

32

Fg

2

12

5.1.3.1 Power Mismatch Equations at Network Buses

The network buses include all buses of the network except the internal buses ofgenerators The power mismatch equations of phase p at the network bus i aregiven by:

m j pm ij m j pm ij p i m j pm ij m j pm ij p i p

m j pm ij m j pm ij p i m j pm ij m j pm ij p i p

PQ Machines.For a PQ machine, the total three-phase active and reactive powers

at the terminal bus of the machine are specified:

b

m i pm i m i pm i p i m i pm i m i pm i p i

Spec

i

i

Eg Bg Fg Gg F Fg Bg Eg Gg E

E Bg F Gg F F Bg E Gg E

[

)]

()(

b

m i pm i m i pm i p i m i pm i m i pm i p i

Spec

i

i

Eg Bg Fg Gg E Fg Bg Eg Gg F

E Bg F Gg E F Bg E Gg F

)]

()(

(5.10)

5.1 Three-Phase Newton Power Flow Methods in Rectangular Coordinates 143

where Pg i Spec and Qg i Spec are the specified active and reactive powers of the

gen-erator at bus I, which are in the direction of terminal bus i.

PV Machines.For a PV machine, the total three-phase active power flow and the

positive sequence voltage magnitude at its terminal bus i are specified The active

power flow mismatch equation is given by (5.9) while the voltage mismatch

equa-tion at bus i is given by:

2 1 2 1 1

)()

Spec i i Spec i

where V i1 is the positive-sequence voltage magnitude voltage at the generator

terminal bus i e and1i f are the real and imaginary parts of the positive-sequence i1

voltage phasor at bus i and they are given by:

3/)

i j b i a i

3/)

i j b i a i

1 i Spec ( i) ( i)

i Spec i

5.1.4 Formulation of Newton Equations in Rectangular Coordinates

Combining the power mismatch equations of network buses and generator activepower and voltage control constraints for the case of PV machines, the followingNewton equation in rectangular coordinates can be obtained:

)

(X

F X

where ∆X=[∆X ,∆X ]T

T a i a

i

gen=[∆Eg ,∆Fg ]

∆X

T c j c j b j b j a j a j c i c i b i b i a i a

i

sys =[∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ]

∆X

T sys

gen

gen =[f1 , f2 ]

F

T c j c j b j b j a j a j c i c i b i b i a i a

)(

)or,()

(

,

p m i j

F B E G F B E G

p m i j F

B E G

E

P

i

p i pp ii p i pp ii c

b m

m j pm ij m j pm ij

p i pm ij p i pm

)(

)or,(

,

p m i j

F G E B E B F G

p m i j F

G E

a m

m j pm ij m j pm ij

p i pm ij p i pm

)(

)or,(

,

p m i j

F G E B E B F G

p m i j F

G E

a m

m j pm ij m j pm ij

p i pm ij p i pm

)(

)or,(

,

p m i j

F B E G F B E G

p m i j F

B E

G

F

Q

c a m

p i pp ii p i pp ii m j pm ij m j pm ij i

j

p i pm ij p i pm

m i

5.1 Three-Phase Newton Power Flow Methods in Rectangular Coordinates 145

p i pm i p i pm i m i

p i

F Bg E Gg Fg

Assuming Pg i p and Qg i p are the active and reactive generator output of phase

=a c p

p i

p i

Qg

∂

∆

∂,

Qg

∂

∆

∂

The differentials of the synchronous machine

power mismatches with respect to the internal voltage variables Eg , i m Fg (m = i m

a, b, c) are given by:

i

i

Eg

Pg Eg

i

i

Fg

Pg Fg

i

i

E

Pg E

i

i

F

Pg F

i

i

Eg

Qg Eg

i

i

Fg

Qg Fg

i

i

E

Qg E

i

i

F

Qg F

Qg

where m = a, b, c.

As mentioned, actually in the three-phase power flow equations of the tor, Eg and i a Fg i a can be considered as independent state variables of the internalgenerator bus while Eg b i and Fg , and b i Eg i c and Eg i c are dependent state vari-ables and can be represented by Eg i a and Eg We have: i a

c i c i

p i c

a

c i c i

p i

c a

b i b i

p i c

a

b i b i

p i

c a

p i a

i

i

Eg

Fg Fg

Pg Eg

Eg Eg Pg

Eg

Fg Fg

Pg Eg

Eg Eg Pg Eg

Pg Eg

Pg

, ,

, ,

c i c i

p i c

a

c i c i

p i

c a

b i b i

p i a

i

b i b i

p i

c a

p i a

i

i

Fg

Fg Fg

Pg Fg

Eg Eg Pg

Fg

Fg Fg

Pg Fg

Eg Eg Pg Fg

Pg Fg

Pg

, ,

c i c i

p i c

a

c i c i

p i

c a

b i b i

p i c

a

b i b i

p i

c a

p i a

i

i

Eg

Fg Fg

Qg Eg

Eg Eg Qg

Eg

Fg Fg

Qg Eg

Eg Eg Qg Eg

Qg Eg

Qg

, ,

, ,

c i c i

p i c

a

c i c i

p i

c a

b i b i

p i c

a

b i b i

p i

c a

p i a

i

i

Fg

Fg Fg

Qg Fg

Eg Eg Qg

Fg

Fg Fg

Qg Fg

Eg Eg Qg Fg

Qg Fg

Qg

, ,

, ,

,

(5.36)

5.1 Three-Phase Newton Power Flow Methods in Rectangular Coordinates 147

Using the relationships in (5.3)-(5.6), (5.25)-(5.36) can be simplified as:

p i c

a

p i

c a

p i c

a

p i

c a

p i a

i

i

Fg

Pg Eg

Pg

Fg

Pg Eg

Pg Eg

Pg Eg

Pg

, ,

, ,

,

2

32

1

2

32

p i c

a

p i

c a

p i b

i

p i

c a

p i a

i

i

Fg

Pg Eg

Pg

Fg

Pg Eg

Pg Fg

Pg Fg

Pg

, ,

,

2

12

3

2

12

p i c

a

p i

c a

p i c

a

p i

c a

p i a

i

i

Eg

Qg Eg

Qg

Fg

Qg Eg

Qg Eg

Qg Eg

Qg

, ,

, ,

,

2

32

1

2

32

p i c

a

p i

c a

p i c

a

p i

c a

p i a

i

i

Fg

Qg Eg

Qg

Fg

Qg Eg

Qg Fg

Qg Fg

Qg

, ,

, ,

,

2

12

3

2

12

3

(5.40)

Similarly, ifEg and i a Fg i a can be considered as independent state variables ofthe internal generator bus while Eg i b and Fg , and i b Eg i c and Eg i c are dependentstate variables and can be represented by Eg i a and Eg , then we have i a

)(

2

3)(

21

)(

2

3)(

21

)(

p i pc i p i pc i p

i pc i p i pc i

p i pb i p i pb i p

i pb i p i pb i

p i pa i p i pa i a

i

p

i

F Gg E Bg F

Bg E Gg

F Gg E Bg F

Bg E Gg

F Bg E Gg

−

+

−

−+

2

1)(

23

)(

2

1)(

2

3

p i pc i p i pc i p

i pc i p i pc i

p i pb i p i pb i p

i pb i p i pb i

p i pa i p i pa i a

i

p

i

F Gg E Bg F

Bg E Gg

F Gg E Bg F

Bg E Gg

F Gg E Bg

−

+

−

−+

2

3)(

2

1

)(

2

3)(

2

1

p i pc i p i pc i p

i pc i p i pc i

p i pb i p i pb i p

i pb i p i pb i

p i pa i p i pa i a

i

p

i

F Bg E Gg F

Gg E Bg

F Bg E Gg F

Gg E Bg

F Gg E Bg

2

1)(

2

3

)(

2

1)(

2

3

p i pc i p i pc i p

i pc i p i pc i

p i pb i p i pb i p

i pb i p i pb i

p i pa i p i pa i a

i

p

i

F Bg E Gg F

Gg E Bg

F Bg E Gg F

Gg E Bg

F Bg E Gg

−+

5.2 Three-Phase Newton Power Flow Methods in Polar Coordinates 149

5.2 Three-Phase Newton Power Flow Methods in Polar Coordinates

5.2.1 Representation of Generators

In Fig 5.1,V i a =V i a∠θi a,V i b =V i b∠θi b,V i c=V i c∠θi c, which are the three-phasevoltages at the generator terminal bus, are expressed in phasors in rectangular co-ordinates Similarly, the voltages at the generator internal bus may be given by

2 π

j a i b

i = E e−

3 /

2 π

j a i c

3

2πδ

δ = a−

i b

a i c

3

2πδ

δ = a+

i c

5.2.2 Power and Voltage Mismatch Equations in Polar Coordinates

5.2.2.1 Power Mismatch Equations at Network Buses

The network buses include all buses of the network except the internal buses of

generators The power mismatch equation of phase p at the network bus i are

j

p i p i p

P

,

)sincos

p i p i p

Q

,

)cossin

PQ Machines For a PQ machine, the total three-phase active and reactive powers

at the terminal bus of the machine are specified:

c

a

pm i pm i pm i pm i m i p i

Spec i i

Bg Gg

E V

Bg Gg

V V Pg

Pg

))sin(

)cos(

([

)sincos

([

δθδ

θ

θθ

c

a

pm i pm i pm i pm i m i p i

Spec i i

Bg Gg

E V

Bg Gg

V V Qg

Qg

))cos(

)sin(

([

)cossin

([

δθδ

θ

θθ

(5.54)

where Pg i Spec andQg i Spec are the specified active and reactive powers of the

gen-erator at bus I, which are in the direction of into terminal bus i.

PV Machines For a PV machine, the total three-phase active power flow and the

positive sequence voltage magnitude at its terminal bus i are specified The active

power flow mismatch equation is given by (5.9) while the voltage mismatch

equa-tion at bus i is given by:

2 1 2 1 1

)()

Spec i i Spec i

where V i1 is the positive-sequence voltage magnitude voltage at the generator

terminal bus i e1i and f i1are the real and imaginary parts of the positive-sequence

voltage phasor at bus i and they are given by (5.11) and (5.12)

Slack Machine At the terminal bus of the Slack machine, the positive-sequence

voltage magnitude is specified and the positive-sequence voltage angle is taken asthe system reference We have:

0

1=

=

5.2 Three-Phase Newton Power Flow Methods in Polar Coordinates 151

2 1 2 1

1 i Spec ( i) ( i)

i Spec i

5.2.3 Formulation of Newton Equations in Polar Coordinates

Combining the power mismatch equations of network buses and generator activepower and voltage control constraints for the case of PV machines, the followingNewton equation in polar coordinates can be obtained:

)

(X

F X

where∆X=[∆Xgen,∆Xsys]T

T a i a

i

gen=[∆ ,∆E ]

T c j c j b j b j a j a j c i c i b i b i a i a

i

sys =[∆ ,∆V ,∆ ,∆V ,∆ ,∆V ,∆ ,∆V ,∆ ,∆V ,∆ ,∆V ]

T sys

gen

gen=[f1 , f2 ]

F

T c j c j b j b j a j a j c i c i b i b i a i a

(

),()cossin

(

V Q

p m i j B

G V V

P

pp ii p i p

i

pm ij pm ij pm ij pm ij m j p i m

j

p

i

θθ

),()

sincos

(

p m i j V

P B

V

p m i j B

G V

V

P

p i p i pp ii

p

i

pm ij pm ij pm ij pm ij p

(5.60)

≠

≠+

(

),()

sincos

(

2

p m i j G

V P

p m i j B

G V

V

Q

pp ii p i p i

pm ij pm ij pm ij pm ij m j p

),()

cossin

(

p m i j V

Q B

V

p m i j B

G V

V

Q

p i p i pp ii

p

i

pm ij pm ij pm ij pm ij p

In comparison to positive sequence model of SSSC, the three-phase SSSCmodels should consider:

• The differences between three-phase and positive sequence SSSC models Thethree-phase SSSC models are basically different from the positive sequenceSSSC models, which are able to give realistic results of power system operationwith presence of unbalances of networks and loads while the positive sequenceSSSC models can provide meaningful results only if both networks and loadsare balanced

• The transformer connection types In the three-phase SSSC models, it is sary to consider how the SSSC is connected with the transformer while, in thepositive sequence SSSC models for conventional power flow calculations, suchconsiderations are not needed

neces-• The similarity between three-phase models and positive sequence models Inprinciple, the three-phase models should be identical to the positive sequencemodels when both networks and loads are balanced

5.3 SSSC Modeling in Three-Phase Power Flow in Rectangular Coordinates 153

5.3.1 Three-Phase SSSC Model with Delta/Wye Connected

Transformer

5.3.1.1 Basic Operation Principles

Fig 5.2 shows the basic operation principles of a three-phase SSSC The SSSCconsists of three converters, which are series connected with a three-phase trans-mission line via three single-phase transformers with Delta/Wye connections Theprimary sides of the three single-phase transformers are delta-connected It is as-

sumed here that the transmission line is series connected with the SSSC bus j With such an assumption, the active and reactive power flows entering the bus j

are equal to the sending-end active and reactive power flows of the transmissionline, respectively In principle, the SSSC can generate and insert three-phase seriesvoltage sources, which can be regulated to change the three-phase impedances(more precisely reactance) of the transmission line In this way, the power of thetransmission line, which the SSSC is connected with, can be controlled

5.3.1.2 Equivalent Circuit of Three-Phase SSSC

The equivalent circuit of the three-phase SSSC is given in Fig 5.3 The SSSC isrepresented by an ideal fundamental frequency three-phase voltage source vector

abc

se

V in series with an impedance matrix Z se abc Z se abc represents the impedancematrix of the three series transformers The switching losses of SSSC may be in-cluded directly in Z abc

Fig 5.2 Operating principles of three-phase SSSC with a Delta/Wye transformer

[ c]T

se b se a

In the practical operation of the SSSC, the equivalent injected voltage tude of each phase should be within a specific voltage limit We define:

magni-p se p

j b j a j abc

k b

ij = Q ,Q ,Q

vectors of branchijleaving busi [ c]T

ji b ji a ji abc

ji = P ,P ,P

ji b ji a ji abc

ji = Q ,Q ,Q

Q

are active and reactive power flow vectors of branchijleaving busj

5.3.1.3 Power Equations of the Three-Phase SSSC

Assume thatV m p =E m p+ jF m p (m=i,j,k and p=a,b,c), the following power tions of the SSSC branch are derived according to the equivalent circuit shown inFig 5.3:

equa-Fig 5.3 Equivalent circuit of three-phase SSSC

5.3 SSSC Modeling in Three-Phase Power Flow in Rectangular Coordinates 155

m se pm ij m se pm ij p

i c

a

m

m se pm ij m se pm ij p

i

c a m

m j pm ij m j pm ij p

i c

a

m

m j pm ij m j pm ij p

i

c a m

m i pm ii m i pm ii p i c

a

m

m i pm ii m i pm ii p

i

p

ij

E B F G F

F B E G E

E B F G F

F B E G E

E B F G F

F B E G E

P

, ,

, ,

, ,

m se pm ij m se pm ij p

i c

a

m

m se pm ij m se pm ij p

i

c a m

m j pm ij m j pm ij p

i c

a

m

m j pm ij m j pm ij p

i

c a m

m i pm ii m i pm ii p i c

a

m

m i pm ii m i pm ii p

i

p

ij

F B E G F

E B F G E

F B E G F

E B F G E

F B E G F

E B F G E

Q

, ,

, ,

, ,

m se pm jj m se pm jj p

j c

a

m

m se pm jj m se pm jj p

j

c a m

m i pm ji m i pm ji p

j c

a

m

m i pm ji m i pm ji p

j

c a m

m j pm jj m j pm jj p

j c

a

m

m j pm jj m j pm jj p

j

p

ji

E B F G F

F B E G E

E B F G F

F B E G E

E B F G F

F B E G E

P

, ,

, ,

, ,

m se pm jj m se pm jj p

j c

a

m

m se pm jj m se pm jj p

j

c a m

m i pm ji m i pm ji p

j c

a

m

m i pm ji m i pm ji p

j

c a m

m j pm jj m j pm jj p

j c

a

m

m j pm jj m j pm jj p

j

p

ji

F B E G F

E B F G E

F B E G F

E B F G E

F B E G F

E B F G E

Q

, ,

, ,

, ,

p se

m i pm ji m i pm ji p

se c

a

m

m i pm ji m i pm ji p

se

c a m

m j pm jj m j pm jj p

se c

a

m

m j pm jj m j pm jj p

se

c a m

m se pm jj m se pm jj p

se c

a

m

m se pm jj m se pm jj p

se

p

se

E B F G F

F B E G E

E B F G F

F B E G E

E B F G F

F B E G E

P

, ,

, ,

, ,

(5.69)

where G ij pm+ jB ij pm =G pm ji + jB ji pm =−y se pm, G ii pm+ jB ii pm =y se pm,

pm se pm

00

00

bb se

aa se abc

se cc se cb se ca se

bc se bb se ba se

aa se ab se aa se abc

se

z z z y

y y

y y y

y y y

5.3.1.4 Three-Phase SSSC Model with Independent Phase Power Control

In the operation of the three-phase SSSC, the active power exchange of the threeconverters with the DC link should be zero Such a constraint is described by(5.68) Besides, due to the fact that the SSSC is delta-connected with the threesingle-phase series transformers, the zero sequence component of the equivalentinjected voltage vectorV se abcshould be zero In other words, the following con-straints should hold,

0)

Re(

, ,

p se c

a p

p se

0)

Im(

, ,

p se c

a p

p se

Since the SSSC steady model has six state variables such

as E se a,F se a,E se b,F se b,E se c,F se c, it still has three control degrees of freedom Hereassuming that the three three-phase transmission line phase power flows can becontrolled, we have:

ji p

ji p

Q

(5.73)

where p=a, b, c Pspec , p ji Qspec p ji are the control references of the active and

re-active power flows, respectively, of phase p.

5.3 SSSC Modeling in Three-Phase Power Flow in Rectangular Coordinates 157

Combining the six operation and control constraint equations (5.68),

(5.71)-(5.73) and six power mismatch equations at buses i, j together, the Newton power

flow equation including the SSSC in rectangular coordinates may be given by:

)(X F X

where

T sys

se

sssc =[∆E ,∆F ,∆E ,∆F ,∆E ,∆F ]

∆X

T c j c j b j b j a j a j c i c i b i b i a i a

i

sys =[∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ]

∆X

T sys

sssc =[∆PΣ ,∆E¦,∆F¦,∆P ,∆P ,∆P ]

F

T c j c j b j b j a j a j c i c i b i b i a i a

a p

p ji

p ji

Combining the six operation and control constraint equations (5.71), (5.72),

(5.75) and (5.76) and six power mismatch equations at buses i, j together, the

Newton power flow equation including the SSSC in rectangular coordinates may

be given by:

)(X F X

where

T

],

∆

T c se c se b se b se a se a

se

sssc =[∆E ,∆F ,∆E ,∆F ,∆E ,∆F ]

∆X

T c j c j b j b j a j a j c i c i b i b i a i a

i

sys =[∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ]

∆X

T sys

$

se j

b se a

A set of symmetrical or balanced three-phase voltage phasors are equal in nitude while their phase angles have120 displacement among them For the sake$

mag-of computation, equation (5.78) may be replaced by the following four equations

in real and imaginary parts:

02

32

1)

se b

se a

se j

b se a se

02

12

3)

se b

se a

se j

b se a se

02

32

1)

se c

se a

se j

c se a se

02

12

3)

se c se a

se j

c se a se

When (5.79)-(5.82) hold, equations (5.71) and (5.72) will be satisfied more, the active power exchange constraint (5.68) should be balanced at any in-stant So the SSSC with symmetrical control has only one control degree of free-dom Assuming that the total active power or reactive power of the three-phasetransmission line is controlled, we have the following power flow control con-straint:

p ji

p ji

Q

(5.83)

5.3 SSSC Modeling in Three-Phase Power Flow in Rectangular Coordinates 159

where p=a,b,c PSpec¦ji and QSpec are the specified total three-phase active¦ji

and reactive power flow control references, respectively

The SSSC model with symmetrical voltage control has six operation and trol constraint equations (5.68), (5.79)-(5.82) Combining the six operation andcontrol constraint equations (5.68), (5.79)-(5.82) and six power mismatch equa-

con-tions at buses i, j together, the Newton power flow equation including the SSSC in

rectangular coordinates may be given by:

)

(X

F X

where∆X=[∆Xsssc,∆Xsys]T

T c se c se b se b se a se a

se

sssc =[∆E ,∆F ,∆E ,∆F ,∆E ,∆F ]

∆X

T c j c j b j b j a j a j c i c i b i b i a i a

i

sys =[∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ]

∆X

T sys

sssc =[∆PΣ ,∆VRe1 ,∆VIm1 ,∆VRe2 ,∆VIm2 ,∆P¦]

F

T c j c j b j b j a j a j c i c i b i b i a i a

5.3.2.1 Basic Operating Principles of Single Phase SSSC

In Fig 5.4, three single-phase SSSCs are series connected with phase a, b, c of a

transmission line, respectively The three SSSCs have neither electrical nor netic connections between them Each single phase SSSC is series-connected withthe transmission line via a single-phase transformer Each can independently con-trol the phase power flow of the transmission line The single-phase SSSC is at-tractive and practical when there are unbalanced loads and one or two phase linesexisting in the systems

mag-5.3.2.2 Equivalent Circuit of Single Phase SSSC

Due to the fact that there are no electrical and magnetic couplings between thethree single-phase SSSCs, each SSSC branch can be represented by an equivalentcircuit shown in Fig 5.5 Such an equivalent circuit is exactly the same to that ofthe SSSC for the positive sequence power flow calculations However, the physi-cal meaning of the single-phase equivalent circuit here is quite different from that

of the positive sequence SSSC in the positive sequence power flow calculations

5.3.2.3 Single-Phase SSSC

The power flow equations (5.64)-(5.67) for the three-phase SSSC are still

applica-ble to the system with the three separate single-phase SSSCs installed on phase a,

b , c of the transmission line in Fig 5.4, respectively.

Since each single phase SSSC can neither generate nor absorb active power, thepower exchange of each SSSC with the system should be zero Mathematically,such a constraint may be represented by:

Fig 5.4 Operation principle of single phase SSSC for three-phase power flow analysis

Fig 5.5 Equivalent circuit of single phase SSSC for three-phase power flow analysis

5.3 SSSC Modeling in Three-Phase Power Flow in Rectangular Coordinates 161

0

=

p se

where p=a, b, c P , which is given by (5.69), is the active power exchange of se p

SSSC with the DC link or the system

Assuming that each SSSC independently controls the phase active or reactivepower flow of the transmission line, the power flow control constraint may be rep-resented by:

0

=

ji p

ji p

where p=a, b, c Pspec ji p andQspec p ji are the specified active and reactive power

flow control references of phase p, respectively.

Combining the six operation and control equations (5.85) and (5.86) of the

three single phase SSSCs and six power mismatch equations of buses i and j, the

three-phase Newton equation may be given by:

)

(X

F X

where∆X=[∆Xsssc,∆Xsys]T

T c se c se b se b se a se a

se

sssc =[∆E ,∆F ,∆E ,∆F ,∆E ,∆F ]

∆X

T c j c j b j b j a j a j c i c i b i b i a i a

i

sys =[∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ,∆E ,∆F ]

∆X

T sys

5.3.3 Numerical Examples

A 5-bus system and the IEEE 118 bus system have been used to test the phase Newton power flow algorithm with modeling of the SSSC The 5 bus three-phase system is shown in Fig 5.8 in the Appendix of this chapter while the systemparameters are listed in Table 5.11 - Table 5.14 In the tests, a convergence toler-ance of 1.0e-12 p.u is used For the sake of convenience, the three-phase SSSCmodel with independent phase power control, three-phase SSSC model with totalthree-phase power control and three-phase SSSC model with symmetrical injectedvoltage control in Section 2 are denoted as Model 1, Model 2 and Model 3, re-spectively, while the three-phase and single phase SSSC models in Section 3 arereferred to Model 4 and Model 5, respectively

three-5.3.3.1 Test Results for the 5-Bus System

Based on the 5-bus system, tests under the following conditions have been carriedout:

load

Case 2: Non-transposed transmission lines and unbalanced load at bus 3 as given

by Table 5.13 and Table 5.14

Case 3: As for case 1, but a SSSC is installed at the sending-end of the

transmis-sion line 1-3

Case 4: As for case 2, but a SSSC is installed at the sending-end of the

transmis-sion line 1-3

Case 5: As for case 3, but the whole system is represented by the positive

se-quence network only

The number of iterations of the three-phase power flow algorithm on cases 1-4 aresummarized in Table 5.1 For cases 3 and 4, the control references of the SSSCModels 1, 4 and 5 are Pspec a ji =Pspec b ji =Pspec c ji =7.0p.u.while the control ref-erence for the SSSC Models 2 and 3 isPspec¦ji =21.0p.u.In order to verify thevalidity of the three-phase power flow algorithm and the SSSC models, case 5 hasbeen carried out, in which the whole system is represented only by the positive se-quence network since the system is balanced The power flow solution of case 5 isobtained by a positive sequence power flow program The detailed power flow so-lutions of case 3 and case 5 are given by Table 5.2

5.3 SSSC Modeling in Three-Phase Power Flow in Rectangular Coordinates 163

Table 5.1 Number of iterations of three-phase power flow algorithm for the 5-bus system

Number

of tions

itera-1 P13a=P13b=P13c =4.94 u.Q13a =Q13b =Q13c = 2 01 u.

82 14

P a= P13b = 5 17 u. P13c = 4 69 u.

96 1

Q a = Q13b = 1 46 u.Q13c = 2 33 u.

82 14

V V

= 43 83

c se

θ

SSSC Model 2:

1933

V V

= 43 83

c se

= 43 83

c se

V V

= 43 83

c se

θ

SSSC Model 1:

1781

= 48 22

c se

θ

SSSC Model 2:

1437

V V

= 43 95

c se