flexible ac transmission systems ( (4)

The Optimal Power Flow OPF problem was defined in early 1960’sCar-as an expansion of conventional economic dispatch to determine the optimal tings for control variables in a power networ

4 Modeling of FACTS-Devices in Optimal Power Flow Analysis

In recent years, energy, environment, deregulation of power utilities have delayedthe construction of both generation facilities and new transmission lines Betterutilisation of existing power system capacities by installing new FACTS-deviceshas become imperative FACTS-devices are able to change, in a fast and effectiveway, the network parameters in order to achieve a better system performance.FACTS-devices, such as phase shifter, shunt or series compensation and the mostrecent developed converter-based power electronic devices, make it possible tocontrol circuit impedance, voltage angle and power flow for optimal operation ofpower systems, facilitate the development of competitive electric energy markets,and stipulate the unbundling the power generation from transmission and mandateopen access to transmission services, etc

However, in contrast to the practical applications of the STATCOM, SSSC andUPFC in power systems, very few publications have been focused on the mathe-matical modeling of these converter based FACTS-devices in optimal power flowanalysis This chapter covers

• Review of optimal power flow (OPF) solution techniques

• Introduction of OPF solution by the nonlinear interior point methods

• Mathematical modeling of FACTS-devices including STATCOM, SSSC,UPFC, IPFC, GUPFC, and VSC HVDC

• The detailed models of the multi-converter FACTS-devices GUPFC, and VSCHVDC and their implementation into the nonlinear interior point OPF

• Comparison of UPFC and VSC HVDC, and GUPFC and multiterminal VSCHVDC

• Numerical examples for demonstration of FACTS controls

4.1 Optimal Power Flow Analysis

4.1.1 Brief History of Optimal Power Flow

The Optimal Power Flow (OPF) problem was initiated by the desire to minimizethe operating cost of the supply of electric power when load is given [1][2] In

1962 a generalized nonlinear programming formulation of the economic dispatch

problem including voltage and other operating constraints was proposed by pentier [3] The Optimal Power Flow (OPF) problem was defined in early 1960’s

Car-as an expansion of conventional economic dispatch to determine the optimal tings for control variables in a power network considering various operating andcontrol constraints [4] The OPF method proposed in [4] has been known as thereduced gradient method, which can be formulated by eliminating the dependentvariables based on a solved load flow Since the concept of the reduced gradientmethod for the solution of the OPF problem was proposed, continuous efforts inthe developments of new OPF methods have been found Several review paperswere published [5]-[13] Among the various OPF methods proposed, it has beenrecognized that the main techniques for solving the OPF problems are the gradientmethod [4], linear programming (LP) method [15][16], successive sparse quad-ratic programming (QP) method [18], successive non-sparse quadratic program-ming (QP) method [20], Newton’s method [21] and Interior Point Methods [27]-[32] Each method has its own advantages and disadvantages These algorithmshave been employed with varied success

set-4.1.2 Comparison of Optimal Power Flow Techniques

It has been well recognised that the OPF problems are very complex mathematicalprogramming problems In the past, numerous papers on the numerical solution ofthe OPF problems have been published [7][10][11][13] In this section, a review

of several OPF methods is given

4.1.2.1 Gradient Methods

The widely used gradient methods for the OPF problems include the reduced dient method [4] and the generalised gradient method [14] Gradient methods ba-sically exhibit slow convergence characteristics near the optimal solution In addi-tion, the methods are difficult to solve in the presence of inequality constraints

gra-4.1.2.2 Linear Programming Methods

LP methods have been widely used in the OPF problems The main strengths of

LP based OPF methods are summarised as follows:

1 Efficient handling of inequalities and detection of infeasible solutions;

2 Dealing with local controls;

3 Incorporation of contingencies

Noting the fact that it is quite common in the OPF problems, the nonlinear ties and inequalities and objective function need to be handled In this situation, allthe nonlinear constraints and objective function should be linearized around thecurrent operating point such that LP methods can be applied to solve the linear op-timal problems For a typical LP based OPF, the solution can be found through theiterations between load flow and linearized LP subproblem The LP based OPF

equali-4.1 Optimal Power Flow Analysis 103

methods have been shown to be effective for problems where the objectives areseparable and convex However, the LP based OPF methods may not be effectivewhere the objective functions are non-separable, for instance in the minimization

of transmission losses

4.1.2.3 Quadratic Programming Methods

QP based OPF methods [17]-[20] are efficient for some OPF problems, especiallyfor the minimization of power network losses In [20], the non-sparse implementa-tion of the QP based OPF was proposed while in [17][18][19], the sparse imple-mentation of the QP based OPF algorithm for large-scale power systems was pre-sented In [17][18], the successive QP based OPF problems are solved through asequence of linearly constrained subproblems using a quasi-Newton search direc-tion The QP formulation can always find a feasible solution by adding extra shuntcompensation In [19], the QP method, which is a direct solution method, solves aset of linear equations involving the Hessian matrix and the Jacobian matrix byconverting the inequality constrained quadratic program (IQP) into the equalityconstrained quadratic program (EQP) with an initial guess at the correct active set.The computational speed of the QP method in [19] has been much improved incomparison to those in [17][18] The QP methods in [17]-[19] are solved usingMINOS developed by Stanford University

4.1.2.4 Newton’s Methods

The development of the OPF algorithm by Newton’s method [21]-[24], is based

on the success of the Newton’s method for the power flow calculations Sparsematrix techniques applied to the Newton power flow calculations are directly ap-plicable to the Newton OPF calculations The major idea is that the OPF problemsare solved by the sequence of the linearized Newton equations where inequalitiesare being treated as equalities when they are binding However, most critical as-pect of the Newton’s algorithm is that the active inequalities are not known prior

to the solution and the efficient implementations of the Newton’s method usuallyadopt the so-called trial iteration scheme where heuristic constraints enforce-ment/release is iteratively performed until acceptable convergence is achieved In[22][25], alternative approaches using linear programming techniques have beenproposed to identify the active set efficiently in the Newton’s OPF

In principle, the successive QP methods and Newton’s method both using thesecond derivatives, which are considered as second order optimization method, aretheoretically equivalent

4.1.2.5 Interior Point Methods

Since Karmarkar published his paper on an interior point method for linear gramming in 1984 [26], a great interest on the subject has arisen Interior pointmethods have proven to be a promising alternative for the solution of power sys-tem optimization problems In [27] and [28], a Security-Constrained Economic

pro-Dispatch (SCED) is solved by sequential linear programming and the IP Affine Scaling (DAS) In [29], a modified IP DAS algorithm was proposed In[30], an interior point method was proposed for linear and convex quadratic pro-gramming It is used to solve power system optimization problems such as eco-nomic dispatch and reactive power planning In [31]-[36], nonlinear primal-dualinterior point methods for power system optimization problems were developed.The nonlinear primal-dual methods proposed can be used to solve the nonlinearpower system OPF problems efficiently The theory of nonlinear primal-dual inte-rior point methods has been established based on three achievements: Fiacco &McCormick’s barrier method for optimization with inequalities, Lagrange’smethod for optimization with equalities and Newton’s method for solving nonlin-ear equations [37] Experience with application of interior point methods to powersystem optimization problems has been quite positive.

u - the set of control variables

x - the set of dependent variables

The objectives, controls and constraints of the OPF problems are summarized

in Table 4.1 The limits of the inequalities in Table 4.1 can be classified into twocategories: (a) physical limits of control variables; (b) operating limits of powersystem In principle, physical limits on control variables can not be violated whileoperating limits representing security requirements can be violated or relaxedtemporarily

In addition to the steady state power flow constraints, for the OPF formulation,stability constraints, which are described by differential equations, may be consid-ered and incorporated into the OPF In recent years, stability constrained OPFproblems have been proposed [38]-[42]

4.2 Nonlinear Interior Point Optimal Power Flow Methods 105

Table 4.1 Objectives, Constraints and Control Variables of the OPF Problems

Objectives • Minimum cost of generation and transactions

• Minimum transmission losses

• Minimum shift of controls

• Minimum number of controls shifted

• Mininum number of controls rescheduled

• Minimum cost of VAr investmentEqualitiy constraints • Power flow constraints

• Other balance constraintsInequalitiy constraints • Limits on all control variables

• Branch flow limits (amps, MVA, MW, MVAr)

• Bus voltage variables

• Transmission interface limits

• Active/reactive power reserve limitsControls • Real and reative power generation

i i i

(

N

j j i

(

N

j j i

=

(4.7)

where V and i θ are the magnitude and angle of the voltage at bus i , respec- i

tively; Y ij =G ij+jB ij is the system admittance element whileθij=θi−θj N is the total number of system buses.

4.2.2 Transmission Line Limits

The transmission MVA limit may be represented by:

2 max 2

(2

ij ij ij ij j i ij i

)cossin

(2

ij ij ij ij j i ii i

where b ii=−B ij+bc ij/2 bc ijis the shunt admittance of transmission lineij.

4.2.3 Formulation of the Nonlinear Interior Point OPF

Mathematically, as an example the objective function of an OPF may minimizethe total operating cost as follows:

Pg Pg

x

while being subject to the following constraints:

Nonlinear equality constraints:

0)()

0 ) ( )

Nonlinear inequality constraints

max min

4.2 Nonlinear Interior Point Optimal Power Flow Methods 107

)

(x

h functional inequality constraints including line flow and voltage

magnitude constraints, simple inequality constraints of ables such as generator active power, generator reactive power,transformer tap ratio

vari-Pg the vector of active power generation

Qg the vector of reactive power generation

t the vector of transformer tap ratios

θ the vector of bus voltage magnitude

V the vector of bus voltage angle

By applying Fiacco and McCormick’s barrier method, the OPF problem tions (4.11)-(4.14) can be transformed into the following equivalent OPF problem:

j

sl x

j

j j j j

N

i

i i N

i

i i M

j

M

j j j

h su h u h

sl h

ʌl

Q Ȝq P Ȝp su

min

1 1

)(

)(

)(ln)

(ln

π

(4.20)

where Ȝp i , Ȝq i , ŋl j , ŋu jare Langrage multipliers for the constraints of equations

(4.16)-(4.19), respectively N represents the number of buses and M the number of

inequality constraints Note thatȝ>0 The Karush-Kuhn-Tucker (KKT) first orderconditions for the Lagrangian function shown in equation (4.20) are as follows:

0)

where Sl=diag(sl j), Su=diag(su j), Πl=diag(πl j), Πu=diag(πu j)

As suggested in [31], the above equations can be decomposed into the ing three sets of equations:

−

∇

−

∇Π

∇

−Π

µ µ

π

µ µ

π

λ

ππ

L L

L u L

L l L

x u l

J

J H h h

h Su u

h Sl

l

x

Su u

Sl l

T T

T

1

1 1

1

00

0

00

00

(4.28)

)(

1

l Sl L l

sl=Π ∇sl µ − ∆π

(4.29))

(

1

u Su L u

su=Π −∇su µ− ∆π

(4.30)where H(x,λ,πl,πu) =∇2f(x)−¦λ∇2g(x)−¦(πl+πu)∇2h(x) ,

)()(

x Q

x P x

The elements corresponding to the slack variables sl and su have been

elimi-nated from equation (4.28) using analytical Gaussian elimination By solvingequation (4.28), ∆ŋl, ∆ŋu, ∆x, ∆λ can be obtained, then by solving equations(4.29) and (4.30), respectively,∆sl, ∆su can be obtained With ∆ŋl, ∆ŋu, ∆x, ∆Ȝ,

∆sl, ∆su known, the OPF solution can be updated using the following equations:

4.2 Nonlinear Interior Point Optimal Power Flow Methods 109

l

for those sl<0,∆su<0, ∆ŋl<0 and ∆ŋu>0

The barrier parameter µ can be evaluated by:

β

whereȕ∈ [0.01-0.2] and Cgap is the complementary gap for the nonlinear interior

point OPF and can be determined using:

1

)

4.2.4 Implementation of the Nonlinear Interior Point OPF

Equations (4.28)-(4.30) are the basic formulation of the nonlinear interior pointOPF that has been well reported in [31][51] Equation (4.28) is the reduced equa-tion with respect to the original OPF problem However, equation (4.28) can befurther reduced by eliminating all the dual variables of the inequalities, generatoroutput variables and transformer tap ratios The elimination will result in new fill-

in elements For example, eliminating a transformer tap ratio will result in sixteennew elements in the reduced Newton equation Such a significant reduction meansthat the reduced Newton equation only involves the state variables of θi,V , i λp i,

i

q

λ The details will be discussed in the next section

4.2.4.1 Eliminating Dual Variables ŋl, ŋu of the Inequalities

In order to obtain the final reduced Newton equation consisting of only the ablesθ, V , λp, λq, the following Gaussian elimination steps can be applied

vari-The dimension of the Newton equation (4.28) can be reduced using analyticalGaussian elimination techniques Basically, the dual variablesŋl, ŋu in equation

(4.28) can be eliminated to obtain:

µλ

'

L q

J p J x

T Q T

P∆ − ∆ =−∇

−

µ λ

J P∆ =−∇ p

µ λ

µ µ

L u L Su

h

L l L Sl

h L L

u su

T x

x

∇Π+

∇

∇+

∇Π+

1 '

(4.47)

x

x P

00

'

Jq Jp

Jq Jp

µ λ µ

L L L

q p x

equa-4.2 Nonlinear Interior Point Optimal Power Flow Methods 111

4.2.4.2 Eliminating Generator Variables P g and Q g

In equation (4.50), generator variables Pg, Qg can be further eliminated The

equation (4.50) may be written in the following form, in which only the relevant

major diagonal block of bus i is displayed:

µ λ µ

µ θ µ µ

λλθ

θθθ

θθθ

θ

L L L L L L

q p V Qg Pg

V Jq Jq

V Jp Jp

V Jq V Jp V H V H

Jq Jp V H H

Qg Qg

q p V

Qg Pg

i i i i i i

i i i i

i i i i

i i i i i i

i i i i i i

i i

i

i

' ' ' '

' '

' ' '

'

00,,

00,,

,,

,,

01000

µ λ µ

µ θ

λλθ

λθ

λθ

θ

θθ

θθ

θ

' ' ' ' '

'

' '

0,,

0,

,

,,

,,

L L L L

q p V

q J V

Jq Jq

p J V Jp Jp

V Jq V Jp V V H V

H

Jq Jp

V H H

i i i i

q p V

i i i i

i i

i i

i

i i

i i

i

i i i i i i i

i

i i i i i i i

Jλ

µ µ

λ µ

λp i L' =∇ p i L +(H'Pg i Pg i) 1∇Pg i L'

µ µ

λ µ

)(H'Pg Pg 1 λp L'µ

Pg

i

Pg i i

i

)(

)(H'Qg Qg 1 λq L'µ

Qg

i

Qg i i

4.2.5 Solution Procedure for the Nonlinear Interior Point OPF

The solution of the nonlinear interior point OPF may be summarised as follows:

Step 0: Formulation of equation (4.28)

Step 1: Forward substitution

(a) Eliminating the dual variables ŋl, ŋu of the inequalities from

equation (4.28), obtain equation (4.50);

(b) Eliminating generator variables Pg, Qg from equation (4.50),

ob-tain equation (4.54);

(c) Eliminating the transformer tap ratio t from equation (4.54),

ob-tain highly reduced Newton matrix equation

Step 2: Solution of the final highly reduced Newton equation by sparse matrix

techniques

(a) The highly reduced system matrix has a dimension of 4Nwhere

N is the total number of buses.

(b) Having been grouped into 4 by 4 blocks, solution to the final trix is produced by sparse matrix techniques

ma-Step 3: Back substitution

(a) First substitution for transformer tap ratio After solving the finalmatrix equation, ∆θ, ∆V ∆λp and ∆λq are known, then ∆t

can be found by back substitution

(b) Second substitution for generator output variables: ∆Pg and

Qg

∆ can be found from equations (4.55) and (4.56);

(c) Third substitution for all the dual variables of the inequalities:The dual variables ŋl, ŋu of the inequalities can be found by

equation (4.51) and equation (4.52);

(d) Fourth substitution for all slack variables: All slack variables can

be found by equations (4.29) and (4.30)

4.3 Modeling of FACTS in OPF Analysis

Very few publications have been focused on the mathematical modeling ofFACTS-devices in optimal power flow analysis In [44][45], a UPFC model hasbeen proposed, and the model has been implemented in a Successive QP In [46],mathematical models for TCSC, IPC and UPFC have been established, and theOPF problem with these FACTS-devices is solved by Newton’s method In [47], aversatile model for UPFC in OPF analysis has been proposed and the model hasbeen implemented into the nonlinear interior point methods In this model, explicitcontrols such local voltage and power flow controls can be explicitly represented.Furthermore, in this model, global controls of UPFC can be achieved without ex-plicit controls In [48], the modeling techniques in [47] have been extended to

4.3 Modeling of FACTS in OPF Analysis 113

general mathematical models for the converter based FACTS-devices such asSTATCOM, SSSC, and UPFC suitable for optimal power flow analysis Applyingthe techniques in chapter 3, the Thyristor controlled FACTS-devices such as SVCand TCSC can be modeled in OPF analysis The detailed models of STATCOM,SSSC, UPFC are referred to [47][48] In the next sections, novel models for IPFC,GUPFC, multi-terminal VSC HVDC will be discussed, the modeling techniques

of which are applicable to STATCOM, SSSC and UPFC

4.3.1 IPFC and GUPFC in Optimal Voltage and Power Flow Control

An innovative approach to utilization of FACTS-devices providing a tional power flow management device was proposed in [43] There are severalpossibilities of operating configurations by combing two or more converter blockswith flexibility Among them, there are two novel operating configurations,namely the Interline Power Flow Controller (IPFC) and the Generalized UnifiedPower Flow Controller (GUPFC) [43][49][50], which are significantly extended tocontrol power flows of multi-lines or a sub-network rather than controlling thepower flow of a single line by a UPFC or SSSC

multifunc-In contrast to the practical applications of the GUPFC in power systems, veryfew publications have been focused on the mathematical modeling of this newFACTS-device in power system analysis A fundamental frequency model of theGUPFC consisting of one shunt converter and two series converters for EMTPstudy was proposed quite recently in [50] The modeling of IPFC and GUPFC inpower flow and optimal power flow (OPF) analysis has been reported [51][52] Inthe next, novel model for GUPFC will be proposed, which are very convenient toconsider various control constraints and control modes The model for IPFC can

be very easily derived once the model for GUPFC has been established

4.3.2 Operating and Control Constraints of GUPFC

As discussed in chapter 3, the GUPFC combining three or more converters ing together extends the concepts of voltage and power flow control beyond what

work-is achievable with the known two-converter UPFC The simplest GUPFC conswork-ists

of three converters, one connected in shunt and the other two in series with twotransmission lines in a substation It can control total five power system quantitiessuch as a bus voltage and independent active and reactive power flows of twolines The equivalent circuit of such a GUPFC, which is shown in Fig 4.1, is used

to show the basic operation principle for the sake of simplicity However, themathematical derivation is applicable to a GUPFC with an arbitrary number of se-ries converters

In the steady state operation, the main objective of the GUPFC is to controlvoltage and power flow Real power can be exchanged among these shunt and se-ries converters via the common DC link The sum of the real power exchangeshould be zero if we neglect the losses of the converter circuits

For the GUPFC shown in Fig 4.1, it has total 5 degrees of control freedom, thatmeans it can control five power system quantities such as one bus voltage, and 4active and reactive power flows of two lines It can be seen that with more seriesconverters included within the GUPFC, more degrees of control freedom can beintroduced and hence more control objectives can be achieved.

In Fig 4.1 Zsh i and Zse in are the shunt and series transformer impedances,respectively.Vsh i=Vsh i∠șsh i and Vse in =Vse in∠șse in ( n = j, k) are the control-

lable injected shunt and series voltage sources PEsh and i PEse in are the powerexchange of the shunt converter and series converter, respectively, via the com-mon DC link

4.3.2.1 Power Flow Constraints of GUPFC

The power flow constraints of the GUPFC are summarized as follows:

Shunt power flows:

))sin(

)cos(

(

2

i i i i i i i i i i

))cos(

)sin(

(

2

i i i i i i i i i i

Series power flows:

)sincos

(

2

in in in in n i in i

))sin(

)cos(

(

2

in in in in n i in i

))cos(

)sin(

4.3 Modeling of FACTS in OPF Analysis 115

))sin(

)cos(

(

2

i n in i n in n i in n

))sin(

)cos(

)sin(

(

2

i n in i n in n i nn n

))cos(

)sin(

tively, P , ni Q (n = j, k) are equal to the active and reactive power flows at the ni

sending-end of the transmission lines, respectively

The operating constraint representing the active power exchange among verters via the common DC link is:

Re

i i i

0)(

Re

ni in in

4.3.2.2 Operating Control Equalities of GUPFC

The GUPFC shown in Fig 4.1 can control both active and reactive power flows of

the two transmission lines The active and reactive power flow control constraints

of the GUPFC are given by (3.50) and (3.51)

The GUPFC has additional capability to control the voltage magnitude of bus i:

i

where V is the voltage magnitude at bus i i V i Specis the specified bus voltage

con-trol reference at bus i In the point of view of the implementation, the inequality is

preferred since incorporation of the simple variable inequality is very easy

4.3.2.3 Operating Inequalities of GUPFC

For the operation of the GUPFC, the injected voltage sources should be withintheir operating ratings while the currents through the converters should be withinthe current ratings:

Shunt converter

max min

i i

max min

i i

i Vsh Vsh

max max

i i

in in

max min

in in

in Vse Vse

max max

in in

I is the current rating of the series converter

4.3.3 Incorporation of GUPFC into Nonlinear Interior Point OPF

4.3.3.1 Constraints of GUPFC

The GUPFC consists of the power flow constraints (4.57)-(4.62), the internalpower exchange balance constraint (4.63), the operating inequality constraints(4.67)-(4.74), and the power flow control constraints and voltage control con-straint (4.66) In the formulation of the nonlinear interior point OPF algorithm, thepower flow constraints (4.57)-(4.62) can be directly incorporated into the power

mismatch equations at bus i, j and k By introducing slack variables and barrier

parameter, the inequalities (4.67)-(4.74) can be converted into equalities Then allthe transformed equalities of the GUPFC can be incorporated into the Lagrangianfunction of the OPF problem

4.3.3.2 Variables of GUPFC

The state variables of the GUPFC areθsh i,Vsh , i PEsh , i θse in, Vse , in PEse in

With incorporation of PEsh , i PEse into the state variables of the GUPFC, the in

formulation of the Newton OPF equation and the implementation of multi-control

4.3 Modeling of FACTS in OPF Analysis 117

functional model become simple and straightforward In addition to the state ables, dual variables should be introduced for all the equalities and inequalitieswhile slack variables should be introduced for all the inequalities In the imple-mentation, the angle constraints (4.67) and (4.71) are optional since they are usu-ally allowed to move around 360 For the simplicity of presentation, the angle$constraints (4.67) and (4.71) are not discussed here

vari-The dual variables of the GUPFC inequality constraints are defined as follows:

In the above equations, all the variables that start with ‘S’ are slack variablesand they are positive values while all the variables that start with ‘π’ are dualvariables

The dual variables of the equalities of the GUPFC are defined as follows:

m

q

where ∆P m and ∆Q m are power mismatch equations at bus m.

4.3.3.3 Augmented Lagrangian Function of GUPFC in Nonlinear Interior OPF

The augmented Lagrangian function of the equalities (4.75) -(4.84) is as follows:

)ln(

m

Spec ni ni ni Spec ni ni

ni

* in in in

in

* i i i

i k

j

i

Q q P

p

Q Q Q P

)(

)(

))Re(

(

))Re(

()

(

,

Ise Vse

Ish Vsh

(n = j, k; m = i, j, k)

(4.93)

4.3.3.4 Newton Equation of Nonlinear Interior OPF with GUPFC

With the incorporation of the augmented Lagrangian functions above into the OPFproblem in section 4.2, a reduced Newton equation can be derived:

x B C

C A

sys T

[

Xgupfc= ∆ ik gufc ∆ ij gupfc ∆ i gupfc

variables, and

T

] ,

, ,

, , [

Xin gupfc= ∆πuIse ni ∆șse in ∆Vse in ∆PEse in λPEse in ∆λPse ni ∆λQse ni

incremental vector of the variables of the GUPFC series branch in.

T

] ,

, , , [

Xi gupfc= ∆πuIsh i ∆șsh i ∆Vsh i ∆PEsh i ∆λPEsh i ∆λPEx

vector of the variables of the GUPFC shunt branch i.

T

] X , X ,

X

[

Xsys= ∆ sys i ∆ sys j ∆ sys k

∆ - the incremental vector of the variables of the tem buses

[

Xm sys= ∆θm ∆V m ∆λp m ∆λq m

variables of system bus m.

4.3 Modeling of FACTS in OPF Analysis 119

In (4.94), all the slack and dual variables of the simple variable inequalities

have been eliminated from the formulation B and b are the system matrix and

right hand vector, which have similar structure to the system matrix and right hand

of (4.54), respectively except that in calculating the former, the contributions fromthe GUPFC should be considered ain and a are given byi

∇

−

−+

µ λ

µ λ µ

µ

µ θ

µ µ

π

µµπ

L L L

SuPEse SlPEse

L

SuVse SlVse

L

L

L uIse

L

in in in in

in

in

in in

Qse Pse PEse

in in

PEse

in in

Vse

se

SuIse in

uIse

)/

1/

1(

+

∇

−

−+

µ λ µ

µ

µ θ

µ µ

π

µµπ

L L

SuPEsh SlPEsh

L

SuVsh SlVsh

L

L

L uIsh

L

PEx PEsh

i i

PEsh

i i

Vsh

sh

SuIsh i

uIsh

i

i i

i

i

i i

)/

1/

1(

)/1/

1(

in in

in in

in in

SuVse uVse

SlVse lVse

d

(n = j,k)

(4.100)