Electrical Power and Energy Systems 44 (2013) 308–317

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

ATC determination with FACTS devices using PTDFs approach for multi-transactionsin competitive electricity markets

Ashwani Kumar ⇑, Jitendra KumarDepartment of Electrical Engineering, National Institute of Technology, Kurukshetra, Haryana, India

a r t i c l e i n f o

Article history:Received 10 February 2012Received in revised form 15 July 2012Accepted 25 July 2012Available online 26 September 2012

Keywords:Available transfer capabilityDC/AC power transfer distribution factorsPTDFs with FACTSFACTS devicesLine contingencyLine outage distribution factors

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.07.050

⇑ Corresponding author. Mobile: +91 9416366091;E-mail addresses: ashwa_ks@yahoo.co.in (A. Kum

(J. Kumar).

a b s t r a c t

Available transfer capability (ATC) is an important indicator for accommodating further transactions overand above already existing commitments. With flexible AC transmission system (FACTS) deployment in asystem for better utilization, the ATC information quantification is essential. ATC enhancement withthese devices can play an important role in an efficient and secure operation of competitive markets.The main objective of the paper is: (i) power transfer distribution factors determination with FACTSdevices, (ii) ATC determination for bilateral/multi-lateral transactions based on PTDFs with FACTSdevices, (iii) optimal location of FACTS devices based on power flow sensitivity corresponding to trans-actions, and (iv) comparison of ATC obtained with DC PTDF based approach. The results have been deter-mined for intact and line contingency cases without and with FACTS devices. The proposed method havebeen tested on IEEE 24 bus RTS.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

One of the key features of competitive electricity market is fairand open transmission access of the network to all users. This mayresult overloading of transmission system facilities more fre-quently. The assessment of available transfer capability for the eco-nomic utilization of the available system components with regardto system security plays a vital role in both operational planningand real time operation of a system. Therefore, secure and reliableoperation of transmission network requires the system operator(SO) to determine and update available transfer capability (ATC)at regular intervals for its optimal commercial use. North AmericanElectrical Reliability Council (NERC) in this regard established aframework for determining ATC of the interconnected transmis-sion networks for a commercially viable wholesale market [1,2].

With the introduction of competition in the power industry allover the world, the electricity supply industries are forced to uti-lize their network facilities in a more economic and secure manner[3]. Thus, the transfer capability determination of transmissionsystem has emerged as a new measure for secure and reliableoperation of a system. With established open access nondiscrimi-natory transmission services policy under FERC orders 888 and889, ATC is required to be posted on web to make competition rea-sonable and effective [4]. The information of ATC will help market

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elements to reserve transmission services well in advance for opti-mal commercial use of transmission network. Utilities are there-fore required to determine their ATC accurately to ensure secureand reliable operation of a system. ATC has to be continuouslyupdated and posted following changes in the system conditions.There are various sources of uncertainties involved in the ATC cal-culation that can be attributed to weather conditions, forced andscheduled transmission outages, and generation unavailability[5]. A number of software tools, such as continuation power flow(CPFLOW) [6], transmission and voltage limitation program(TVLIM) [7] and TRACE [8] have been developed for transfer capa-bility calculation. However these methods are time intensive for online implementation.

For fast computation of ATC, the power flow sensitivity basedmethods have been proposed by many authors. These methodsare based on Power Transfer Distribution Factors/Line Outage Fac-tors (PTDFs), (LODFs) using DC load flow approach. The DC loadflow based methods utilizing DC power transfer distributionfactors are well reported for ATC computation in [8]. The DC loadflow based approaches are fast however are based on DC load flowassumptions. More accurate methods based on AC load flowapproach for ATC determination using the sensitivity factors hasbeen reported in [9–14]. Ejebe et al. [9] presented a novel formula-tion of ATC problem based on full AC power flow solution to incor-porate the effects of reactive power flows, voltage limits as well asvoltage stability and line flow limits. Many authors utilized sensi-tivity based methods for computation of ATC [10–13]. Greene et al.[12] presented a computationally efficient formula for the first

Nomenclature

Pi real power injection at bus-iQi reactive power injection at bus-in total no of busesNl is the total number of lines in the systemPgi, Qgi real and reactive power generation at bus-iPdi, Qdi real and reactive power demand at bus-iVi, di voltage magnitude and voltage angle at bus-iYij = Gij + Bij i–jth element of Y-bus matrix|Yij|, hij magnitude and angle of Ybus elementsYsh shunt charging admittance of line-ijPij, Qij real power flow and reactive power flow in a line-ij

PSTATCOMi ;QSTATCOM

i real and reactive power injection at a particu-lar bus with STATCOM

PSSSCij ;QSSSC

ij real and reactive power injection at a particular buswith SSSC

PUPFCCij ;QUPFC

ij real and reactive power injection at a particular buswith UPFC

Vsh and dsh shunt voltage and angle for STATCOMVse and dse series injected voltage and angle for SSSCVsh, dsh Vse, dse shunt voltage and angle, Series injected voltage

and angle for UPFCPTDFij

mn;FACTS power transfer distribution factors with FACTS de-vices for transactions between seller bus m and buyerbus n

LODFijrs;FACTS line outage distribution factors with rs line outage

Limit tmaxij is thermal limit of any line i–j

A. Kumar, J. Kumar / Electrical Power and Energy Systems 44 (2013) 308–317 309

order sensitivity of the transfer capability with respect to the vari-ations in parameters such as operating conditions, other powertransfers, and system data and sensitivities can be used to estimatethe effect on the transfer capability of variations in parameters. Afast algorithm to incorporate the effect of reactive power flowsbased on circle equations and mega-Var corrected megawatt limitsfor ATC determination was presented in [13]. Authors in [14] pro-posed AC PTDFs based on AC load flow along with voltage sensitiv-ity factors for ATC determination. Othman [15] presented a newcomputationally fast and accurate method evaluating ATC basedon curve fitting technique so called cubic-spline interpolation tech-nique which traces the curves of voltage magnitude and powerflow variations with respect to increase of real power transfer. Liand Liu proposed maximum area concept based sensitivity basedmethod for computation of ATC in bilateral and simultaneoustransaction environment [16]. A novel method of contingencyATC computation using ac sensitivity factors and a sensitivity anal-ysis of system uncertainties was proposed in [17]. An approach fordetermination of TCSC reactance based on PTDFs for ATC enhance-ment is proposed in [18]. Comprehensive approach for ATC deter-mination in multi-transactions environment using DC PTDF basedapproach is presented in [19]. However, the method is based on DCload flow assumptions. The approach in [19] is extended with ACpower transfer distribution factors for multi-transactions environ-ment and comparison of results are presented with both DC and ACmethods [20]. However, the role of FACTS devices in ATC enhance-ment has not been considered in PTDF based approach. Based onthe literature available, it is observed that the power transfer dis-tribution factors based approaches which are proved fast can alsobe implemented with the incorporation of FACTS devices for ATCcomputation.

In the competitive environment, electricity supply industries allover the world are operating in a manner to utilize their existinginfrastructure in a best possible and efficient way. To utilize thepower system in a more secure and efficient manner, the FACTS de-vices have a great role to play. These devices have been installed ina system world wide for better power transfer capabilities of a sys-tem, security enhancement, voltage control, and transient and dy-namic stability improvements [21,22]. Thus, there is need in aderegulated electricity market to calculate ATC with FACTS con-trollers using PTDFs based approach. The FACTS technology pro-vides solutions for increasing transmission system capability[23–27]. Zhang and Handschin [23] presented mathematical mod-els of FACTS controllers such as the STATCOM, SSSC, UPFC, and thelatest FACTS devices GUPFC and IPFC using a non-linear optimiza-tion problem to determine ATC. Xiao and Song et al. presented anOPF based approach for ATC enhancement using FACTS device [24].

Harinder and Jeyasurya [25] presented the application of thirdgeneration FACTS controller, the unified power flow controller toimprove the transfer capability of the power system. Farahmandet al. described [26] the repeated power flow for enhancement ofATC with FACTS controllers by Genetic algorithm. Menniti et al.proposed a method to determine Static Synchronous Series Com-pensator (SSSC) best location that maximizes the power systemavailable transfer capability measured as the maximum systemload increase before any operating limit is reached [27]. Some ofthe authors have proposed fuzzy based contingency constrainedOPF and artificial intelligence based approach for ATC determina-tion in [28,29]. Sen transformer has emerged as one of the powerflow control devices. An analysis of comparison of UPFC and SENtransformer is presented recently in [30]. However, the authorshave utilized optimal power flow based methods for ATC enhance-ment with FACTS devices. The PTDFs with FACTS devices for ATCdetermination in multi-transaction market environment can beobtained for ATC determination as sensitivity based methods areproven faster.

In this paper, PTDFs with the incorporation of FACTS deviceshave been determined for ATC calculations. The location of FACTSdevices is decided based on the pattern of variation of PTDFs foreach line corresponding to different transactions. The line outagedistribution factors have been also been determined without andwith FACTS devices. The results have also been obtained for singleand multi-transactions between seller and buyer buses. The resultshave also been obtained using PTDFs based on DC load flow ap-proach for line intact and contingency cases for comparison. Theresults have been obtained for IEEE RTS 24 bus system [31].

2. Model of FACTS devices

2.1. Static compensator (STATCOM)

For the power flow analysis, STATCOM is represented by asynchronous voltage source with magnitude Vsh and angle dsh withits internal impedance Zse connected at any bus i, shown in Fig. 1.The real and reactive power injection at any bus i of the STATCOMare [22]:

PSTATCOMi ¼ V2

i Gsh þ ViVsh½Gsh cosðdi � dshÞ þ Bsh sinðdi � dshÞ� ð1Þ

QSTATCOMi ¼ �V2

i Bsh þ ViVsh½Gsh sinðdi � dshÞ � Bsh cosðdi � dshÞ� ð2Þ

where

1=Zsh ¼ Gsh þ jBsh

Vi Vj

Bus iVs

ZseBus j

cij

cij

jQP + + cji

cji jQP

IjiIij

Vsh

Zsh

Ish

0)**jiIseVshIshRe(V =-

Fig. 3. Equivalent circuit of UPFC.

Vi

Bus i

Iij

Vsh

Zsh

Pi +j Qi

I sh

0)*shIshRe(V =

Fig. 1. Equivalent circuit of STATCOM.

310 A. Kumar, J. Kumar / Electrical Power and Energy Systems 44 (2013) 308–317

The change in elements of Jacobian can be obtained appropriateplaces where STATCOM is connected. Vsh and dsh are the shunt volt-age and angle respectively.

With modified Jacobian and power flow sensitivity factors,PTDFs can be calculated with STATCOM.

2.2. Static Synchronous Series Compensator (SSSC)

Model of SSSC is shown in Fig. 2. If Vse is the compensating volt-age inserting in the transmission line with angle dse, then injectedreal and reactive power at bus i connected by line i–j where SSSC isplaced, can be written as [22]:

PSSSCij ¼ V2

i Gii þ ViVj½Gij cosðdijÞ þ Bij sinðdijÞ� þ ViVse½Gij cosðdi

� dseÞ þ Bij sinðdi � dseÞ� ð3Þ

Q SSSCij ¼ �V2

i Bij þ ViVj½Gij sinðdijÞ � Bij cosðdijÞ� þ ViVse½Gij sinðdi

� dseÞ � Bij cosðdi � dseÞ� ð4Þ

The injected real and reactive power at bus j can be written as:

PSSSCji ¼ V2

j Gjj þ ViVj½Gij cosðdijÞ þ Bij sinðdijÞ� þ VjVse½Gij cosðdj

� dseÞ þ Bij sinðdj � dseÞ� ð5Þ

Q SSSCij ¼ �V2

j Bij þ ViVj½Gij sinðdijÞ � Bij cosðdijÞ� þ VjVse½Gij sinðdj

� dseÞ � Bij cosðdj � dseÞ� ð6Þ

The modified Jacobian elements can be obtained with SSSC. Theprocedure for obtaining the Jacobian is explained in detail in [22].Obtaining power flow sensitivity with SSSC and Jacobian elements,PTDFs can be calculated for ATC determination.

Vi Vj

Bus iVs

eZseBus j

0)*jiIseRe(V =

cij

cij jQP + + c

jicji jQP

IjiIij

Fig. 2. Equivalent circuit of SSSC.

2.3. Unified Power Flow Controller (UPFC)

In the steady state operation, the main objective of an UPFC is tocontrol voltage and power flow. The equivalent circuit of an UPFCis shown in Fig. 3.

The injected active and reactive power equations at bus i andbus j can be written as [22]:

PUPFCij ¼ V2

i ðGii þ GshÞ þ ViVj½Gij cosðdijÞ þ Bij sinðdijÞ� þ ViVse½Gij

� cosðdi � dseÞ þ Bij sinðdi � dseÞ� þ ViVsh½Gsh cosðdi

� dshÞ þ Bsh sinðdi � dshÞ� ð7Þ

QUPFCij ¼ �V2

i ðBij þ BshÞ þ ViVj½Gij sinðdijÞ � Bij cosðdijÞ�þ ViVse½Gij sinðdi � dseÞ � Bij cosðdi � dseÞ� þ ViVsh½Gsh

� sinðdi � dshÞ � Bsh cosðdi � dshÞ� ð8Þ

PUPFCji ¼ V2

j Gjj þ ViVj½Gij cosðdijÞ þ Bij sinðdijÞ� þ VjVse½Gij cosðdj

� dseÞ þ Bij sinðdj � dseÞ� ð9Þ

QUPFCji ¼ �V2

j Bij þ ViVj½Gij sinðdijÞ � Bij cosðdijÞ� þ VjVse½Gij

� sinðdj � dseÞ � Bij cosðdj � dseÞ� ð10Þ

where 1/Zsh = Gsh + jBsh; Gij and Bij are taken from Ybus.The Jacobian elements according to the power injection model

with UPFC can be obtained modifying the base case N–R Jacobianelements as explained in [22]. With the power flow sensitivityand Jacobian with UPFC, PTDFs can be obtained for ATCdetermination.

3. PTDFs determination with FACTS devices

For a change in the transmission line quantity DPij for a transac-tion Pmn among the buyer and seller buses with FACTS devices, theAC power transfer distribution factors can be defined as,

PTDFijmn;FACTS ¼

DPFACTSij

Pmnð11Þ

For PTDF calculations with FACTS devices, the power flow sen-sitivity and N–R load flow Jacobian can be calculated. The powerflow equations without FACTS devices in polar form can be repre-sented as:

Pi ¼Xn

j¼1

jVikVjkYijj cosðhij � di þ djÞ ð12Þ

Qi ¼Xn

j¼1

jVikVjkYijj sinðhij � di þ djÞ ð13Þ

A. Kumar, J. Kumar / Electrical Power and Energy Systems 44 (2013) 308–317 311

These equations are modified with FACTS devices as given inSection 3 (1)–(10). Using Taylor series expansion, the change inpower flows at any bus i can be formulated in terms of Jacobian as:

DP

DQ

� �¼

J1;FACTS J2;FACTS

J3;FACTS J4;FACTS

" #DdFACTS

DjV jFACTS

" #ð14Þ

where

½J1�FACTS ¼@P@d

; ½J2�FACTS ¼@P@jV j ; ½J3�FACTS ¼

@Q@d

; ½J4�FACTS ¼@Q@jV j ð15Þ

The change in the angle and voltage magnitude can be deter-mined as:

DdFACTS

DjV jFACTS

" #¼

J1;FACTS J2;FACTS

J3;FACTS J4;FACTS

" #�1DP

DQ

� �ð16Þ

The Jacobian elements with FACTS devices can be determinedas:

With STATCOM:With real power injection Pi,statcom = 0, there is reactive power

injection at a bus i. therefore, the sub matrices J3 and J4 of JacobianJ on with STATCOM can be modified. The change in J3 and J4 at theappropriate places can be represented as:

J3;statcomði; iÞ ¼ J3ði; iÞ þ ViVsh½Gsh cosðdi � dshÞ þ Bsh sinðdi

� dshÞ� ð17Þ

J4;statcomði; iÞ ¼ J4ði; iÞ þ Vsh½Gsh sinðdi � dshÞ � Bsh cosðdi � dshÞ� ð18Þ

With SSSC:The change in sub Jacobian elements J1 and J2 can be obtained

as:

J1;ssscði; iÞ ¼ J1ði; iÞ þ ViVse½�Gij sinðdi � dseÞ þ Bij cosðdi � dseÞ� ð19Þ

J2;ssscði; iÞ ¼ J2ði; iÞ þ Vse½Gij cosðdi � dseÞ þ Bij sinðdi � dseÞ� ð20Þ

J1;sssscðj; jÞ ¼ J1ðj; jÞ þ VjVse½�Gij sinðdj � dseÞ þ Bij cosðdj � dseÞ� ð21Þ

J2;ssscðj; jÞ ¼ J2ðj; jÞ þ Vse½Gij cosðdj � dseÞ þ Bij sinðdj � dseÞ� ð22Þ

The change in sub Jacobian elements J3 and J4 can be obtained as:

J3;ssscði; iÞ ¼ J3ði; iÞ þ ViVse½Gij cosðdi � dseÞ þ Bij sinðdi � dseÞ� ð23Þ

J4;ssscði; iÞ ¼ J4ði; iÞ þ Vse½Gij sinðdi � dseÞ � Bij cosðdi � dseÞ� ð24Þ

J3;ssscðj; jÞ ¼ J3ðj:jÞ þ VjVse½Gij cosðdj � dseÞ þ Bij sinðdj � dseÞ� ð25Þ

J4;ssscðj; jÞ ¼ J4;ssscðj; jÞ þ Vse½Gij sinðdj � dseÞ � Bij cosðdj � dseÞ� ð26Þ

With UPFC:The change in the Jacobian elements can be obtained as:

J1;upfcði; iÞ ¼ J1ði; iÞ þ ViVse½� sinðdi � dseÞ þ Bij cosðdi � dseÞ�þ ViVsh½�ðGij þ GshÞ sinðdi � dshÞ þ Bij cosðdi

� dshÞ ð27Þ

J2;upfcði; iÞ ¼ J2ði; iÞ þ Vse½Gij cosðdi � dseÞ þ Bij sinðdi � dseÞ�þ Vsh½Gsh cosðdi � dshÞ þ Bij sinðdi � dshÞ� ð28Þ

J1;upfcðj; jÞ ¼ J1ðj; jÞ þ VjVse½�Gij sinðdj � dseÞ þ Bij cosðdj � dseÞ� ð29Þ

J2;upfcðj; jÞ ¼ J2ðj; jÞ þ Vse½Gij cosðdj � dseÞ þ Bij sinðdj � dseÞ� ð30Þ

J3;upfcði; iÞ ¼ J3ði; iÞ þ ViVse½Gij cosðdi � dseÞ þ Bij sinðdi � dseÞ�þ ViVsh½Gsh cosðdi � dshÞ þ Bsh sinðdi � dshÞ� ð31Þ

J4;upfcði; iÞ ¼ J4ði; iÞ � 2ViðBij þ BshÞ þ Vj½Gij sinðdijÞ � Bij

� cosðdijÞ� þ Vse½Gij sinðdi � dseÞ � Bij cosðdi � dseÞ�þ Vsh½Gsh sinðdi � dshÞ � Bsh cosðdi � dshÞ� ð32Þ

J3;upfcðj; jÞ ¼ J3ðj; jÞ þ VjVse½Gij cosðdj � dseÞ þ Bij sinðdj � dseÞ� ð33Þ

J4;upfcðj; jÞ ¼ J4ðj; jÞ þ Vse½Gij sinðdj � dseÞ � Bij cosðdj � dseÞ� ð34Þ

Using N–R load flow analysis, bus voltage magnitudes and an-gles can thus be obtained with FACTS devices. For calculation ofPTDFs, the power flow sensitivity can be determined using thepower flow equations with FACTS devices. Using Taylor’s seriesapproximation and ignoring higher order terms, change in realpower flow can be written as:

DPFACTSij ¼

@PFACTSij

@dFACTSi

DdFACTSi þ

@PFACTSij

@dFACTSi

DdFACTSj þ

@PFACTSij

@VFACTSi

DVFACTSi

þ@PFACTS

ij

@VFACTSi

DVFACTSj ð35Þ

The sensitivity coefficients in (35) can be obtained taking thepartial derivatives of real power flows. The sensitivity of powerflow equation can be written in the compact matrix form as:

DPFACTSij ¼

@PFACTSij

@dFACTS2

; . . . ;@PFACTS

ij

@dFACTSn

@PFACTSij

@VFACTS2

; . . . ;@PFACTS

ij

@VFACTSn

" #DdFACTS

2

..

.

DdFACTSn

DjVFACTS2 j...

DjVFACTSn j

2666666666664

3777777777775ð36Þ

In (36), the change in angles and voltages can be substitutedfrom (16) and the change in power flows can be obtained corre-sponding to power injection vector. For a bilateral transactionamount Pt between seller bus m and buyer bus n, and substitutingin power injection column vector at its respective position,DPm = +Pt, DPn = �Pt, the change in power flows can be obtained as,

DPFACTSij ¼

@PFACTSij

@dFACTS2

; . . . ;@PFACTS

ij

@dFACTSn

@PFACTSij

@VFACTS2

; . . . ;@PFACTS

ij

@VFACTSn

" #J1;FACTS J2;FACTS

J3;FACTS J4;FACTS

" #�1

0...

þPt

0...

�Pt

0

26666666666664

37777777777775

¼ PTDFijmn;FACTS � Pt ð37Þ

Therefore, the PTDFs with FACTS devices for bilateral transac-tion can be represented as:

PTDFijmn;FACTS ¼

@PFACTSij

@dFACTS2

; . . . ;@PFACTS

ij

@dFACTSn

@PFACTSij

@VFACTS2

; . . . ;@PFACTS

ij

@VFACTSn

" #J1;FACTS J2;FACTS

J3;FACTS J4;FACTS

" #�1

0...

þ10...

�10

26666666666664

37777777777775ð38Þ

Bus-sBus- r

Srs Ssr

Srs Ssr

Fig. 5. Post-outage state of the power system.

0.40.6

T1 T2 T3

312 A. Kumar, J. Kumar / Electrical Power and Energy Systems 44 (2013) 308–317

3.1. PTDFs determination for simultaneous/multi-lateral transactionswith FACTS devices

The sellers and buyers can have simultaneous/multi-lateraltransactions during trading of power in a hybrid markets. WhenATC is determined for more than one transactions occurring simul-taneously in a system, ATC in such a case is called as simultaneousor multi-transaction ATC. The procedure for simultaneous ATC issimilar as discussed for single transactions case with a change inthe power injection matrix. In the simultaneous ATC case, thepower injection matrix can be modified based on the transactionsoccurring between many sellers m, p and buyers n, q respectively.The change in power injection vector with multi-lateral transac-tions can be represented as:

DP ¼

0ðmÞ þ Pt

..

.

ðpÞ � Pt

0ðnÞ þ Pt

..

.

ðqÞ � Pt

0

26666666666666666664

37777777777777777775

ð39Þ

Depending on the number of transactions, the entry at the corre-sponding seller and buyer buses in (39) can be added. Using (38),the PTDFs with simultaneous transactions can be calculated as:

PTDFijmn;FACTS ¼

@PFACTSij

@dFACTS2

; . . . ;@PFACTS

ij

@dFACTSn

@PFACTSij

@VFACTS2

; . . . ;@PFACTS

ij

@VFACTSn

" #

J1;FACTS J2;FACTS

J3;FACTS J4;FACTS

" #�1

0þ1

..

.

�10þ1

..

.

�10

2666666666666666664

3777777777777777775

ð40Þ

Thus, PTDFs with FACTS devices can be determined for simulta-neous transactions/multi-transactions case using (40).

4. Line outage distribution factors with FACTS devices

The line outage distributions (LODFs) with FACTS devices can bedefined as the change in the line flows due to the outage of any linebetween bus r and bus s to the pre outage flow in the respectiveline. This can be mathematically defined as:

Psr

Qsr

Prs

Qrs

Bus- r Bus-s

Fig. 4. Pre-outage state of the power system.

LODFijrs;FACTS ¼

DPijrs;FACTS

P0rs

ð41Þ

The pre-outage state of a part of an interconnected power systemnetwork, where a line-l connected between bus-r and bus-s isshown in Fig. 4. Fig. 5 shows the post outage state of the power sys-tem network with a line-l to be considered as out of service. Thesimulation of a line outage will require modification of [YBus]parameters to exclude the parameters of the line-l, which changesthe Jacobian matrix. This involves a time intensive process. A lineoutage has been approximately simulated by considering two ficti-tious generators at bus-r and bus-s and a fictitious line between thebuses having the same parameters as the original line to retain theoriginal [YBus] and also the elements of Jacobian and power flowsensitivity matrix [14]. Thus, retaining a fictitious line with thesame parameters as that of an original line, [YBus] remains un-changed. The power flow in this fictitious line is considered as thepre-outage power flow in the actual line.

The power injected due to the fictitious sources has been takensame as the line flows at the two ends in order to make the netpower flow to be zero thus, simulating the line outage condition.The changes in the bus power from pre-outage to post outage stateat bus-r and bus-s for outage of the line-l are represented as:

DPr ¼ P0rs;DPs ¼ P0

sr ð42Þ

DQr ¼ Q 0rs;DQ s ¼ Q0

sr ð43Þ

These are the injections of real and reactive power flows to bus sand bus r. Using these injections at appropriate places in the powerinjection matrix, the change in the voltage and angle can be ob-tained due to outaged line rs as:

Ddrs;FACTS

DjV jrs;FACTS

" #¼

J1 J2

J3 J4

� ��1

FACTS

DP

DQ

� �ð44Þ

1 4 710 13 16 19 22 25 28 31 34 37 T1

-1.2-1

-0.8-0.6-0.4-0.2

00.2

PTDFs

Lines

Tr...

Fig. 6. Variation of PTDFs obtained with AC power flow method.

A. Kumar, J. Kumar / Electrical Power and Energy Systems 44 (2013) 308–317 313

Obtaining changes in voltage and angles, new values of voltage andangles with line outage can thus be obtained adding by thesechanges to voltage and angles obtained at base case.

dFACTSj;rs ¼ d0

j þ Dd;FACTSj;rs ; dFACTS

i;rs;S ¼ d0i þ DdFACTS

i;rs ð45Þ

VFACTSi;rs ¼ V0

i þ DVi;rs;FACTS;VFACTSj;rs ¼ V0

j þ DVFACTSj;rs ð46Þ

Using these new voltage and angles, the change in the power flowfor all lines with outage line-r–s can be obtained using power flowequations with FACTS devices. Knowing the change in the powerflow due to line-r–s outage, the line outage distribution factorscan be calculated. Thus, ATC can be evaluated for any lines r–s usingthese LODFs.

start

Read system dataBus data(initial Pd, V, Angles, PV, PQ

bus..etc)Line data(R,X, P flow limit..etc)

Select the line or bus for FACTS Devices

Select the transactions (bilateral or simultaneous)

Set the transaction countKK=0

Form the Ybus

Form N-R Jacobian without FACTS

Any contingency occurs?

Form N-R Jacobian with FACTS

Calculate line flow sensitivity factors and and ACPTDFs with and without FACTS

Calculate ATC with FACTS and without FACTS

Is all transactions completed?

Calculate Voltage magnitudes and angles using N-R load flow and line flows

Stop

Yes

Yes

KK=KK+1No

No

Fig. 7. Flow chart for A

5. ATC determination with FACTS devices

With the determination of PTDFs and LODFs with FACTS de-vices, the ATC can be determined for any number of transactions.Now Pmax

ij�mn;FACTS for any transaction between seller bus m to buyerbus n can be obtained as:

Pmaxij�mn;FACTS ¼

Limit tmaxij�Pij

PTDFijmn;FACTS

; PTDFijmn;FACTS > 0

1ðinfiniteÞ; PTDFijmn;FACTS ¼ 0

�Limit tmaxij�Pij

PTDFijmn;FACTS

; PTDFijmn;FACTS < 0

8>>>>><>>>>>:

ð47Þ

where Pij is the real power flow through any line i–j.

Set the contingency countTT=0

Select outaged line based on contingency screening

calculate Voltage magnitudes and angles(contingency case)

Calculate line flow sensitivity and LODFs with line outage

Is all contingency taken?

TT=TT+1No

Calculate ATC with line contingency cases using ACPTDFs

and LODFs

Yes

TC determination.

Table 1PTDFs AC methods for transactions T1–T3.

Lines AC method

From To T1 T2 T3

1 2 �0.0152 �0.1102 �0.12541 3 0.0337 0.3225 0.35611 5 �0.0148 �0.2464 �0.26132 4 �0.0048 0.0178 0.013082 6 �0.0089 �0.1662 �0.17513 9 �0.075 �0.4341 �0.50923 24 0.0991 �0.2731 �0.173404 9 �0.0047 0.0135 0.00885 10 �0.012 �0.2626 �0.27476 10 �0.0049 �0.2064 �0.21147 8 0.0012 �0.0292 �0.02808 9 0.0028 0.1076 0.11058 10 �0.0027 �0.1062 �0.10909 11 0.0038 �0.1518 �0.14809 12 �0.0622 �0.1788 �0.2410

10 11 0.0231 0.3097 0.332810 12 �0.0449 0.2768 0.231911 13 �0.1538 0.022 �0.131811 14 0.183 0.1698 0.352912 13 �0.0066 0.0623 0.055712 23 �0.1393 0.0868 �0.052513 23 �0.1592 0.0615 �0.097714 16 0.1791 0.1587 0.337815 16 �0.7393 �0.2637 �0.993015 21 �0.0818 �0.0284 �0.110215 21 �0.0818 �0.0284 �0.110215 24 �0.1592 0.5064 0.347316 17 0.1565 0.054 0.210516 19 �0.6857 �0.141 �0.826817 18 0.1457 0.0508 0.196517 22 0.0229 0.0081 0.031018 21 0.0728 0.0253 0.098118 21 0.0728 0.0253 0.098119 20 �0.3366 �0.0678 �0.4044719 20 �0.3366 �0.0678 �0.4044720 23 �0.3452 �0.0688 �0.4139920 23 �0.3452 �0.0688 �0.4139921 22 �0.0223 �0.0077 �0.03002

314 A. Kumar, J. Kumar / Electrical Power and Energy Systems 44 (2013) 308–317

Pmaxij;mn is the maximum allowable transaction (bilateral, simulta-

neous/multi-lateral transaction) amount from bus m to bus n con-strained by the line flow limit from bus i to bus j. For the giventransaction, the ATC can be defined and obtained as:

ATCmn;FACTS ¼min Pmaxij;mn;FACTSij 2 Nl

n oð48Þ

5.1. ATC determination with line contingency case

According to ATC principles, reasonable level of uncertaintiesshould be accommodated in ATC calculations [1]. Thus the impactof line outage as contingency shall be considered for ATC calcula-tions. Line contingencies have been taken based on the contin-gency analysis to obtained LODFs. Thus, the LODFs and PTDFscan be combined together to calculate available transfer capability.This is the maximum increase in the transaction amount from agiven bus to another bus under (n – 1) contingency condition. Con-sider a transaction from bus m to bus n and the outage of the linefrom bus r to bus s (line-rs). The change in the flow on the line-rsdue to the given transaction is:

DPnewrs;FACTS ¼ PTDFrs

mn;FACTS � Pnewmn ð49Þ

When the outage of line-rs is considered, the part of the flow ap-pears on any line ij. Thus, the change in flow in the line ij resultingfrom outage of the line-rs along with a new transaction from bus mto bus n is given by:

DPnewij;rs;FACTS ¼ ðPTDFij

mn;FACTS þ LODFijrs;FACTS � PTDFrs

mn;FACTSÞ � Pnewmn ð50Þ

The ATC from bus/zone m to bus/zone n, with outage of line rs isgiven as [18]:

ATCmn;rs;FACTS ¼ minPmax

ij � P0ij

PTDFijmn;FACTS þ LODFij

rs;FACTS � PTDFrsmn;FACTS

ij 2 Nl

( )

ð51Þ

All possible combinations of lines outages and limiting lines canbe checked. Then, ATC can be evaluated as:

ATCFACTS ¼minðATCmn;FACTS;ATCmn;rs;FACTSÞ ð52Þ

The steps for calculation of ATC can be summarized in the flowchart shown in Fig. 7 with FACTS devices and can be utilized for alltypes of FACTS devices.

6. Location of FACTS devices

The ATC is a function of power flow sensitivity to any transac-tions occurring between any seller and buyer buses correspondingto those transactions. These sensitivities provide information aboutthe power flow pattern change corresponding to the transactionbetween any seller bus and buyer bus. The location of power flowcontrollers based on their pattern of change corresponding to eachtransaction can be obtained based on the variation pattern ofPTDFs. The calculations of PTDFs without FACTS devices are wellexplained in [14,20]. The sensitivity of power flow to any transac-tions can be plotted and the slope of the curve can be observed cor-responding to each transactions. The lines connected between thebuses where slope variation is higher can be observed for all trans-actions critically. These lines with higher variation can be taken aspotential candidates for optimal location of FACTS devices as withsmall change in the power injection at a particular bus will bringlarger change in the power flows. Thus, based on the pattern ofPTDFs, the lines between the buses can be identified where powerflow sensitivity variations are of lower magnitude but with highervariations. Although the slope of the curve may be higher at someother lines connected between the respective buses but the PTDFsvalues may be higher for those lines. These lines with higher PTDFsmay not result considerable increase in ATC.

Therefore, for obtaining enhancement of ATC, lines with lowervalues of PTDFs with higher slope of sensitivity curves are bettercandidates and shall be chosen for placement of FACTS devices.In sensitivity based approaches, the judicious choice plays animportant role for obtaining better results. Thus, the lines withlower magnitude of power flow sensitivities and higher slope ofpower flow sensitivity curve are identified as the candidate linesfor installation of power flow devices for enhancement of ATC.

7. Results and discussions

The results are obtained for available transfer capability withbilateral and simultaneous/multi-transactions occurring betweensellers and buyer buses. ATC has been obtained for line intactand contingency cases for IEEE 24 bus RTS without and with FACTSdevices. The results have also been obtained with DC method forcomparison. These various transactions are categorized as:

Bilateral transaction (T1): transaction between seller bus 23 tobuyer bus 15.

Bilateral transaction (T2): transaction between seller bus 10 tobuyer bus 3.

Simultaneous/multilateral transactions (T3): transaction betweenseller buses 23 and 10 to buyer bus 15 bus 3.

Simultaneous/multilateral transactions (T4): transaction betweenseller buses 23, 10, 21 to 15, 3, 6.

Table 2PTDFs with FACTS devices for transactions T1–T3.

STATCOM SSSC UPFC

T1 T2 T3 T1 T2 T3 T1 T2 T3

�0.015 �0.107 �0.1265 �0.00601 �0.0688 �0.08982 �0.00609 �0.07653 �0.085720.0339 0.3229 0.3552 0.03796 0.338498 0.369891 0.038287 0.338823 0.375241�0.0146 �0.2124 �0.2642 �0.00665 �0.16394 �0.19041 �0.00627 �0.17734 �0.18338�0.0008 0.0221 0.0109 0.00078 0.037151 0.029663 0.000833 0.039511 0.038081�0.0088 �0.1397 �0.1773 �0.00381 �0.12901 �0.14313 �0.00343 �0.11696 �0.12102�0.0738 �0.4238 �0.5124 �0.07256 �0.42055 �0.49747 �0.07146 �0.41644 �0.48929

0.0989 �0.2764 �0.1737 0.101042 �0.26745 �0.16911 0.100914 �0.26867 �0.16854�0.004 0.0212 0.0056 0.00078 0.034184 0.026566 0.001166 0.037857 0.036634�0.0117 �0.2079 �0.279 �0.00383 �0.21215 �0.23114 �0.00275 �0.17722 �0.17958�0.0045 �0.1425 �0.2162 0.000711 �0.15209 �0.16451 0.002305 �0.10082 �0.09407

0.0017 �0.0083 �0.048 0.001721 �0.01811 �0.01843 0.002629 �0.01002 �0.007460.0031 0.0961 0.1171 0.00259 0.097415 0.101657 0.002828 0.092822 0.09417�0.003 �0.0947 �0.1025 �0.00251 �0.09607 �0.10021 �0.00275 �0.09148 �0.09276

0.0037 �0.1484 �0.148 0.005234 �0.14107 �0.13881 0.004829 �0.13944 �0.13378�0.0622 �0.1722 �0.2413 �0.05668 �0.16582 �0.22982 �0.05297 �0.16299 �0.22306

0.0232 0.3007 0.3333 0.024273 0.291861 0.317452 0.025503 0.288078 0.308356�0.0447 0.2703 0.2321 �0.03956 0.261224 0.218549 �0.03416 0.258363 0.210914�0.1542 0.016 �0.1311 �0.15104 0.018409 �0.13494 �0.15035 0.015963 �0.1385

0.1833 0.1659 0.3535 0.183254 0.163057 0.347054 0.183337 0.161687 0.344016�0.0067 0.0564 0.0564 �0.01024 0.057601 0.051677 �0.01244 0.055317 0.048133�0.1396 0.0811 �0.0517 �0.14158 0.081537 �0.05704 �0.14327 0.079222 �0.06094�0.1593 0.0614 �0.0976 �0.15968 0.059689 �0.09924 �0.1598 0.059518 �0.10011

0.1866 0.1593 0.3379 0.186548 0.154493 0.334207 0.186813 0.159403 0.345266�0.7186 �0.2663 �0.9933 �0.71663 �0.25747 �0.99767 �0.71668 �0.24531 �0.96279�0.082 �0.0287 �0.1102 �0.08182 �0.02775 �0.10964 �0.08183 �0.02801 �0.10992�0.082 �0.0287 �0.1102 �0.08182 �0.02775 �0.10964 �0.08183 �0.02801 �0.10992�0.1584 0.5153 0.3457 �0.16203 0.499581 0.341412 �0.16146 0.502936 0.342499

0.1584 0.0545 0.2105 0.157991 0.052708 0.209425 0.158004 0.054082 0.212261�0.6762 �0.1434 �0.8266 �0.67395 �0.13818 �0.82432 �0.67374 �0.13304 �0.80871

0.1456 0.0513 0.1965 0.145194 0.049589 0.195446 0.145205 0.049701 0.1950680.0227 0.0081 0.031 0.022641 0.007862 0.030813 0.022643 0.00775 0.0304180.0729 0.0255 0.0981 0.072749 0.024699 0.097545 0.072755 0.024903 0.0977380.0729 0.0255 0.0981 0.072749 0.024699 0.097545 0.072755 0.024903 0.097738�0.3351 �0.069 �0.4044 �0.33399 �0.06647 �0.40331 �0.33389 �0.06593 �0.40077�0.3351 �0.069 �0.4044 �0.33399 �0.06647 �0.40331 �0.33389 �0.06593 �0.40077�0.3453 �0.07 �0.4139 �0.34412 �0.06744 �0.41281 �0.34402 �0.06793 �0.41294�0.3453 �0.07 �0.4139 �0.34412 �0.06744 �0.41281 �0.34402 �0.06793 �0.41294�0.0224 �0.0078 �0.03 �0.02238 �0.00754 �0.02986 �0.02238 �0.00766 �0.03007

Table 3ATC (p.u) with and without FACTS devices.

Transactions ATC (p.u)

DC method AC method without FACTS STATCOM SSSC UPFC

T1 7.8362 7.5785 7.7968 7.8183 7.8276T2 3.6876 2.8342 3.1800 3.2331 3.2903T3 3.5024 2.7629 2.9368 3.0036 3.2449T4 0.9211 1.1183 1.1886 1.1920 1.2006

A. Kumar, J. Kumar / Electrical Power and Energy Systems 44 (2013) 308–317 315

For optimal location of FACTS devices, the pattern of power flowchange corresponding to all transactions is obtained and is shownin Fig. 6. Based on the pattern, it is observed that sensitivity varia-tions are of lower magnitude but with higher variations betweenlines 3 and 10. Although the slope of the curve is also quite highfor lines 22–28 but the PTDFs values are higher. Therefore, forobtaining enhancement of ATC, lower values of PTDFs with higherslope of sensitivity curves are chosen for placement. The FACTS de-vices location is thus based on the trend of PTDFs variations and istaken at bus 12 for location of STATCOM, line 5–10 for SSSC, andUPFC has been placed on line 6–10. ATC has been obtained for lineintact and contingency cases for IEEE 24 bus RTS. The flow chart ofATC computation is shown in Fig. 7.

The PTDFs have been obtained without and with FACTS devicesand are given in Tables 1 and 2 and are given for transactions T1 toT3. ATCs obtained without and with FACTS devices are given inTable 3. The ATC obtained with DC method is also given in

Table 3. From Table 3, it is observed that ATC enhances with FACTSdevices for all transactions respectively. Comparing ATC valuesgiven in Table 3, the ATC enhancement is higher with UPFCcompared to STATCOM and SSSC for all transactions. For multi-transactions cases, the ATC is less compared to the bilateraltransactions. With DC method, ATC estimate is more optimisticcompard to AC method due to the assumptions involved in DCmethod.

7.1. ATC under line contingency cases

The ATCs have been obtained for few contingency cases. Thelines taken for study are: 9–12, 16–19, 19–20, and 20–23 basedon contingency analysis. The ATC determined without and withFACTS devices given in Table 4. The ATC values obtained withoutand with FACTS devices are also shown in Fig. 8. The ATC obtainedfor all line contingency cases enhances with FACTS devices and is

Table 4ATCs (p.u) with FACTS devices with line outage case.

ATC (p.u.) with STATCOM ATC (p.u.) with SSSC ATC (p.u.) with UPFC

T1 T2 T3 T4 T1 T2 T3 T4 T1 T2 T3 T4

9–12 6.0078 0.4314 0.4012 0.8457 5.7895 0.4687 0.4291 0.9021 5.8598 0.4741 0.4307 0.888116–19 6.3244 3.2219 2.9607 1.2045 6.4996 3.1947 3.0262 1.2046 6.4801 3.3571 3.3068 1.233619–20 6.4193 3.1621 2.9099 1.1868 6.4533 3.1172 2.9773 1.1910 6.4574 3.2730 3.2159 1.200520–23 4.0688 3.1601 2.9091 1.1873 4.0708 3.1143 2.9764 1.1914 4.0760 3.2699 3.2136 1.2005

0

1

2

3

4

5

6

7

ATC

(p.u

.)

T1 T2 T3 T4 T1 T2 T3 T4 T1 T2 T3 T4 T1 T2 T3 T4

AC method STATCOM SSSC UPFC

L(9-12) L(16-19) L(19-20) L(20-23)

Fig. 8. Comparison of ATC obtained with AC method and with FACTS devices.

Table 5ATCs (p.u) using DC and AC method with line outage case.

Outagedline

ATC(p.u)

DC method AC method

T1 T2 T3 T4 T1 T2 T3 T4

9–12 6.63 1.03 0.91 0.76 5.2978 0.4683 0.4303 0.562616–19 6.49 3.64 3.27 0.93 6.3310 2.8714 2.7236 1.120319–20 5.22 3.68 3.46 0.92 6.4196 2.8199 2.7236 1.116720–23 4.09 3.68 3.39 0.92 4.0590 2.8180 2.7236 1.1175

00.010.020.030.040.050.060.070.08

DC AC(JN-R) STATCOM SSSC UPFC

Tim

e (s

ec.)

T1 T2 T3 T4

Fig. 9. Time for ATC calculations.

316 A. Kumar, J. Kumar / Electrical Power and Energy Systems 44 (2013) 308–317

found higher with UPFC compared to all other FACTS devices. Forbilateral transaction cases, ATC obtained is higher than the ATC ob-tained for multi-transaction cases. ATC obtained with DC/AC meth-od is also given in Table 5 for comparison with all line contingencycases. With DC methods, results of ATC with all line contingenciesare optimistic compared to AC method.

The time in sec. for ATC computation with and without FACTSdevices is shown in Fig. 9. The time in seconds for ATC calculationis also given for DC method. The time taken for AC method is morecompared to DC method and with FACTS devices, the time furtherincreases due to more number of iterations. The results have been

obtained using MATLAB 7.0.4 version on 2.4 GHz core 2 processorwith 2 GB ram.

8. Conclusions

In this paper, PTDFs have been obtained with FACTS devicesusing N–R load flow approach. ATC has been determined for bilat-eral and simultaneous/multi-transaction cases without and withline contingency cases. The results have been also obtained withDC power transfer distribution factors for intact and line outagecases for comparison. The ATC obtained with FACTS devices en-hances for all transactions. The ATC for bilateral transactions arehigher and for multi-transaction cases, the ATC reduces. For linecontingency cases, ATC is found to decreases for all transactions.With FACTS devices, the ATC enhances for all transaction caseswith all line contingencies. The proposed method can accommo-date any number of transactions in a competitive electricity mar-ket environment for ATC determination. This study is importantfor the ISO to quantify ATC with the incorporation of FACTS devicesfor its declaration well in time for its optimal commercial use.

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