Electric Power Systems Research 73 (2005) 335–342

Fast evaluation of available transfer capability usingcubic-spline interpolation technique

M.M. Othmana,∗, A. Mohamedb, A. Hussainb

a Faculty of Electrical Engineering, MARA University of Technology, 40450 Shah Alam, Selangor, Malaysiab Department of Electrical, Electronic and Systems Engineering, Faculty of Engineering, Universiti Kebangsaan Malaysia,

43600 UKM, Bangi, Selangor, Malaysia

Received 28 February 2004; received in revised form 31 July 2004; accepted 31 July 2004Available online 18 November 2004

Abstract

This paper presents a new computationally fast and accurate method for evaluating available transfer capability (ATC) based on a curvefitting technique so-called as the cubic-spline interpolation technique. The advantage of the technique used in the computation of ATC is thati curves ofv t where thev d to definet od is verifiedb techniquep ns.©

K

1

pttdptevc(nc

ifictionsonalr andstem

d byough

meTC.

hodsalcu-

pro-of dc

ows ac-

0d

t has the ability to reduce the time consuming ac power flow computations. The cubic-spline interpolation technique traces theoltage magnitude and power flow variations with respect to the increase of real power transfer. ATC is then determined at the poinoltage or power flow limits intersect the curves. Prior to the ATC evaluation, contingency ranking and selection techniques are usehe critical lines in a system that can adversely affect the transfer capability assessment. The effectiveness of the proposed methy illustrating the ATC evaluation on a practical test system. ATC results obtained from the proposed cubic-spline interpolationrove that the method is satisfactorily accurate and it is faster than the ATC method using the recursive ac power flow computatio2004 Elsevier B.V. All rights reserved.

eywords:Available transfer capability; Cubic-spline interpolation; Contingency ranking and selection

. Introduction

In the deregulated power system, transfer capability com-utation has become a key component for all companies par-

icipating in the power transaction activities. Due to openransmission access, electric utilities are required to pro-uce commercially viable information of the transfer ca-abilities of their transmission systems so that such vi-

al information can help power marketers, sellers and buy-rs in planning, operation and reserving transmission ser-ices [1]. There are two significant indices in the transferapability assessment, namely, the total transfer capabilityTTC) and the available transfer capability (ATC). By defi-ition, TTC represents the maximum amount of power thatan be transferred over the interconnected transmission net-

∗ Corresponding author.E-mail address:mamat505my@yahoo.com (M.M. Othman).

work in a reliable manner while meeting all of a specset of defined pre- and post-contingency system condi[2]. On the other hand, ATC is a measure of the additiamount of power that may flow across the interface, oveabove the base case flows without jeopardizing power sysecurity[3].

A predetermined set of ATC values can be accesseelectricity market participants and system operators thran open access same-time information system[4]. Postingthe transfer capability signal incurred within a limited tirequires a fast computational method in estimating the APresently, there are not many fast ATC calculation metavailable and, therefore, there is a need for a fast ATC clation method. However, various approaches have beenposed to determine ATC, such as using the methodspower flow[1], ac power flow[5], optimal power flow[6]and sensitivity[7]. The method based on linear dc power flconsidering distribution factors is considered fast but les

378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved.oi:10.1016/j.epsr.2004.07.007

336 M.M. Othman et al. / Electric Power Systems Research 73 (2005) 335–342

curate for transfer capability analysis because the dc networkmodel does not require the voltage magnitude and reactivepower component in the power flow calculation. Therefore,the linear dc power flow may result in optimistic ATC valueespecially for the heavily stressed system that caused by criti-cal contingency. The ac power flow method gives an accuratesolution in determining the ATC because it considers the ef-fects of reactive power flows and voltage limits. However,transfer capability evaluation using repetitive ac power flowsis time-consuming because it requires a load flow solution atevery transfer step size. To avoid many repetitive ac powerflows, curve fitting techniques are used. Various curve fit-ting techniques have been used for voltage stability analysis,such as the least square fit of second-order polynomial[8],cubic-spline interpolation[9] and quadratic approximation[10].

This paper presents a new method for the computationof ATC that uses the cubic-spline interpolation technique intracing the curves of voltage magnitude and power flow vari-ations with respect to the increase of real power transfer. Theobjective of the proposed method is to develop a fast and ac-curate ATC evaluation method. The ATC is determined bynoting the power transfer corresponding to the point wherethe limits of voltage magnitude or power flow intersect thecurves. The transfer capability of the system is analyzed un-der two different sets of transfer, which are area-to-area ATCa nala a tot thea ellerb d byc ages.C con-n ntin-g maya ands tiont s theM

2

e-s linef -i tiono

2

ans-m ingt mar-g ca-p stem

is secure under a reasonable range of uncertainties in systemconditions. The CBM is the amount of transmission transfercapability reserved by load serving entities to ensure access togeneration from interconnected systems to meet generationreliability requirements. The ETC is the normal transmissionflows included in the given case. The methods for determiningthe TRM, CBM and ETC margins, however may vary amongregions, power pools, individual system and load serving en-tities.

ATC must satisfy certain principles balancing both tech-nical and commercial issues, so that the interconnected trans-mission network is performed based on the commercial re-quirements associated with transmission service requests.The following principles identify the requirements for thecalculation and application of ATC:

a) Electricity demand and supply cannot be treated indepen-dently of one another. All system conditions must be con-sidered to accurately access the capabilities of the trans-mission network.

b) Electric power flows resulting from each power transferuse the entire network and are not governed by the com-mercial terms of the transfer.

c) ATC calculations must use a regional or wide-area ap-proach to capture the interactions of electric power flowsamong individual, regional, subregional and multiregional

d ed in

e abletingon-

owl intoa flowp ionsa auset its[

2

ans-m hilem andp bys ct tot areasop ert ,r -i fort nd-o ap-

nd point-to-point ATC. Area-to-area ATC is the additiomount of power that is transferred from the seller are

he buyer area. On the other hand, point-to-point ATC isdditional amount of power that is transferred from the sus to the buyer bus. ATC is also analyzed and quantifieonsidering the effect of contingencies, such as line outonsidering outages of all lines for a large-scale interected power system is impractical and, therefore, coency ranking is used to select the critical lines thatdversely affect the ATC during outages. The accuracypeed of the ATC method using the cubic-spline interpolaechnique is verified on a practical test system, such aalaysian power system.

. Problem formulation

In this section, the problem definition of ATC is first dcribed, and then followed by the explanation of cubic-spormulation and the determination ofP–VandP–Scurves usng the cubic-spline interpolation technique in the applicaf transfer capability assessment.

.1. ATC problem definition

Mathematically, ATC is defined as the TTC less the trission reliability margin (TRM), less the sum of exist

ransmission commitments (ETC) and capacity benefitin (CBM) [2]. The TRM is the amount of transmissionability necessary to ensure that the interconnected sy

systems.) Appropriate system contingencies must be consider

the ATC evaluation.) The determination of ATC must accommodate reason

uncertainties in system conditions and provide operaflexibility to ensure the secure operation of the intercnected network.

In the determination of ATC, the transmission line flimits and voltage magnitudes limits have to be takenccount. All these limits can be handled in an ac loadower system model. Limits due to transient or oscillatre not often addressed in the ATC determination bec

hese limits have to be crudely approximated by flow lim11].

.2. Formulation of cubic-spline interpolation technique

ATC is also defined as the ability of interconnected trission networks to reliably transfer electric power waintaining acceptable levels of voltage magnitudeower flow. The generic ATC computation is performedolving recursive ac power flow calculations with respehe increased amount of power transfers between ther busbars. The profiles of voltage magnitude (V) and MVAower flow variations (S) due to the increase of MW pow

ransfers (P) are described in terms ofP–V andP–Scurvesespectively. Therefore, tracing theP–V andP–Scurves usng the cubic-spline interpolation technique is requiredhe ATC estimation. By using lower order (say secorder) curve fitting, it will not have good accuracy in

M.M. Othman et al. / Electric Power Systems Research 73 (2005) 335–342 337

proximation. However, using higher order (say fourth-order)curve fitting, the computation burden will significantly in-crease. Therefore, cubic-spline interpolation technique is in-deed a compromise between the two. In the proposed method,extrapolation should be avoided in curve fitting and ATCfinding, which can improve accuracy effectively. The ba-sic idea of this method is to determine four known pointson the curve and then fit appropriate curves to the fourpoints.

In the cubic-spline technique[12], tracing the curvesf(k1),f(k2) and f(k3) begins with finding the value for parametersf ′′(x2), f ′′(x3) andf ′′(x4), which are given as:

f ′′(x2) ={

2

(x4 − x2

x3 − x2

) [6

x3 − x2[f (x3) − f (x2)] + 6

x2 − x1[f (x1) + f (x2)]

]

− 6

x4 − x3[f (x4) − f (x3)] + 6

x3 − x2[f (x2) + f (x3)]

}/

{2(x3 − x1) × 2

(x4 − x2

x3 − x2

)− (x3 − x2)

}(1)

f ′′(x3) ={

6

x4 − x3[f (x4) − f (x3)] + 6

x3 − x2

× [f (x2) − f (x3)] − [(x3 − x2) × f ′′(x2)]

}/

f

Te btainta

f

f

f (k3) = f ′′(x3)

6(x4 − x3)(x4 − k3)3 + f ′′(x4)

6(x4 − x3)(k3 − x3)3

+[

f (x3)

x4 − x3− f ′′(x3)(x4 − x3)

6

](x4 − k3)

+[

f (x4)

x4 − x3− f ′′(x3)(x4 − x3)

6

](k3 − x3) (6)

In the P–V curve fitting, the parametersf ′′(x2), f ′′(x3)andf ′′ (x4) can be described asV ′′ (P2), V ′′(P3) andV ′′(P4),

respectively. On the other hand, the parametersf ′′(x2), f ′′(x3)andf ′′(x4) can be described asS′′

ij(P2), S′′ij(P3) andS′′

ij(P4),respectively, for the case ofP–Scurve fitting.f(kl) is the cubic-spline function that is used for tracing the curves of voltagemn eenb Wb tals gttro acp fer,x reu1

2c

e in-v a-t wert ows then eds so-l linel owert fore,t ura ow

2(x4 − x2) (2)

′′(x4) ={

6

x4 − x3[f (x4) − f (x3)] + 6

x3 − x2

× [f (x2) − f (x3)] − [(x3 − x2) × f ′′(x2)]

−[2(x4 − x2) × f ′′(x3)]

}/(x4 − x2) (3)

he value for parametersf ′′(x2), f ′′(x3) and f ′′(x4) are thequations used in the cubic-spline equations in order to o

he curve functions off(k1), f(k2) andf(k3), which are givens:

(k1) = f ′′(x2)

6(x2 − x1)(k1 − x1)3 + f (x1)

x2 − x1(x2 − k1)

+ f (x2)

x2 − x1(k1 − x1) (4)

(k2) = f ′′(x2)

6(x3 − x2)(x3 − k2)3

+ f ′′(x3)

6(x3 − x2)(k2 − x2)3

+[

f (x2)

x3 − x2− f ′′(x2)(x3 − x2)

6

](x3 − k2)

+[

f (x3)

x3 − x2− f ′′(x2)(x3 − x2)

6

](k2 − x2) (5)

agnitude,Vi(kl) and MVA power flow,Sij (kl). i is the busumber andij is the transmission line connected betwus i and j. kl is the increase of power transfer by 1 Metweenxl and xl+1. l is the number of three incrementeps, that is, 1, 2 and 3. Specifically,f(kl ) is used for tracinhe curves between the four points off(xn) with respect tohe increase ofh by 1 MW from xl to xl+1. Whereby,f(xn)epresents as the four points of voltage magnitude,Vi(Pn)r MVA power flow,Sij (Pn) which are obtained from theower flow solutions. The four points of real power transn can also be described asPn, wheren= 1, 2, 3 and 4. Foxample, the curve from pointf(x1) to pointf(x2) is traced bysingf(k1) with the increase ofk1 by 1 MW fromx1 = 1 tox2 =00 MW.

.3. Determination of P–V and P–S curves using theubic-spline interpolation technique

Generally, there are two main procedures which arolved in theP–V curve fitting using cubic-spline interpolion technique. First, the voltage at each point of real poransfer,Vi(Pn), is obtained by solving the four ac power flolutions. The ac power flow solution which is close toose-point ofP–Vcurve may not converge for ill-conditionystems. Therefore, by using conventional ac power flowution method, it is assumed that the ATC is induced byimits and/or voltage limits, and appears as a smaller pransfer than that at the voltage collapse point. Therehere is no load flow ill-condition issue in obtaining foc power flow solutions. Otherwise, continuation load fl

338 M.M. Othman et al. / Electric Power Systems Research 73 (2005) 335–342

Fig. 1. Illustration of cubic-spline technique used in tracing theP–V curve.

should be considered. Second, the cubic-spline interpolationtechnique is used for tracing the voltage curves,Vi(kl ) usingthe four voltage points,Vi(Pn), as shown inFig. 1. Particu-larly, the curve from pointVi(P1) to pointVi(P2) is tracedby usingVi(k1) with the increase ofk1 by 1 MW fromP1 toP2. Then, the next curve from pointVi(P2) to pointVi(P3) istraced by usingVi(k2) with the increase ofk2 by 1 MW fromP2 to P3. Finally, the last curve from pointVi(P3) to pointVi(P4) is traced by usingVi(k3) with the increase ofk3 by1 MW from P3 toP4.

Similarly, the cubic-spline interpolation technique is usedin tracing theP–Scurves of MVA power flow variations withrespect to the increase of power transfer. The procedures usedin tracing the curvesSij (k1), Sij (k2) andSij (k3) from the fourpoints of MVA power flow defined asSij (Pn) are similar tothat described for tracing the voltage curves, except that thevoltage variable in Eqs.(1–6)are replaced by the MVA powerflow variables.

3. Procedure of ATC evaluation using cubic-splineinterpolation technique

Generally, the main steps in the transfer capability com-putation involve the definition of a base case, specification ofcontingencies, determination of network response and find-i -to-a rpo-l

line

cee toer-ely.th PIes

b) Establish a solved base case power flow solution.c) Perform line outage simulation of one of the specified

critical lines.d) Specify the area or point of transfers. For the point-to-

point transfer, a generator is considered as a selling busand a load is a buying bus. However, the area-to- areatransfer considers participation of all generators in thespecified selling area and all loads in the specified buyingarea.

e) Simultaneously, increase the power generation (PGn ) andload (PDn ) at the selected buses or areas at,n, incrementalsteps. Four incremental steps ofPGn are chosen becauseit is enough to provide an accurate fitting of the curve. Ifmore than four data points are obtained, it may lead to alengthy computational time in the cubic-spline technique.The amount of power generationPGn is equivalent to theamount of power transfer,Pn. The sensitivity method isused to predict the maximum power transfer forP4 andthen it is used to specify the MW step length between eachpoint of Pn. The sensitivity method that used to predictthe maximum power transfers based on the limiting pointof system constraints[8,14–17], are given as:

PTi,VL =∣∣∣∣ ∂λ

∂Vi

× (VL − V 0i )

∣∣∣∣ (7)

∣ ∣

ofup-

re-u.

esely.p-to there-

ctlyweraseow.two

lutiontudetedto theases.ue

ng the maximum transfer or ATC. Determination of arearea and point-to-point ATCs using the cubic-spline inte

ation technique are described as follows:

a) Determine critical lines in a system by performing thecontingency ranking and selection[13]. The line loadingperformance index (PIMW) and bus voltage performanindex (PIV) are used to rank the line contingency. Dueach line outage, PIMW and PIV represent the branch ovload criteria and voltage violation criteria, respectivThese performance indices are ranked and lines wivalues above the PIbase case, are considered as critical linin the system.

PTi,VU = ∣∣∣ ∂λ

∂Vi

× (VU − V 0i )

∣∣∣ (8)

PTij,S =∣∣∣∣ ∂λ

∂Vij

× (S limitij − V 0

ij)

∣∣∣∣ (9)

where, PTi, PTi and PTij,S are the linear estimationpower transfers based on the lower voltage limit, theper voltage limit and the thermal limit of each line,spectively.VL is the lower voltage limit, which is 0.9 p.VU is the upper voltage limit, which is 1.1 p.u.S limit

ij is the

thermal limit of each line.V 0i andS0

ij are the base casof voltage magnitude and MVA power flow, respectiv∂λ/∂Vi and∂λ/∂Sij are the sensitivity methods which reresents as the increase of power transfer with respectchanges of voltage magnitude and MVA power flow,spectively. All the sensitivities can be calculated direby solving two ac power solutions. The first ac poflow solution is performed in order to determine the bcase values of voltage magnitude and MVA power flBy slightly increasing the power transfer between theareas or buses, then the second ac power flow sois performed that vary the values of voltage magniand MVA power flow. The sensitivity can be calculabased on the increase of power transfer with respectdifference of system parameters between the two cThen,P4 is obtained by referring to the minimum valof PTi,VL , PTi,VU or PTij,S [15]. Mathematically,P4 isdefined as:

P4 = min{PTi,, PTi,VU , PTij,S} (10)

M.M. Othman et al. / Electric Power Systems Research 73 (2005) 335–342 339

P1 is the initial value of power transfer which is 1 MW.While,P2 andP3 can be obtained by referring to Eqs.(11)and (12), respectively.

P2 = P4

3(11)

P3 = P2 × 2 (12)

f) At each incremental step, solve the ac power flow solutionand determine the MVA power flow,Sij (Pn) and voltagemagnitude,Vi(Pn) for all lines and load buses, respec-tively.

g) Use the cubic-spline interpolation technique to fit theVi(kl) andSij (kl) curves between the four selected pointsof Vi(Pn) andSij (Pn), respectively. Obtain theP–V andP–Scurves for all the load buses and transmission lines,respectively.

h) Determine area-to-area or point-to-point ATCs which arethe maximum power transfers obtained when either thevoltage limit or the MVA line rating intersects theP–VandP–Scurves, respectively.

The above procedures are summarized in terms of aflowchart as shown inFig. 2. The main difference of theATC method using the cubic-spline interpolation techniqueas compared to the recursive ac Newton–Raphson power flow

Ft

solutions[5] is that it gives a faster ATC computation whichimplies less ac power flow solutions.

4. Test results and discussion

The performance of the cubic-spline interpolation tech-nique in the determination of ATC is verified in terms ofaccuracy and speed. CPU timing for the transfer capabilityanalysis was obtained using 2 GHz Pentium 4 with 320 MB ofmemory. The Malaysian power system is used as a test systemto illustrate the ATC estimation based on the proposed cubic-spline interpolation technique. The system is simplified to a241-bus system with five areas, namely, area 1 (north), area2 (east), area 3 (central), area 4 (south) and area 5 (PUB), asshown inFig. 3. The 241-bus system is modeled with 143generation units, 98 load units and 368 lines. In this study,the lower and upper voltage limits are assumed to be 0.9 and1.1 p.u., respectively. The MVA power flow or the thermalratings of the line limits are used for both contingency andbase case conditions.

Prior to the ATC calculation, contingency ranking is per-formed on the 368 lines in the system in which after contin-gency selection, 44 lines have been identified as critical lines.In this study, two critical lines which are connected from bus1468 to bus 2468 and from bus 2308 to bus 2608 are selecteda a andt

4t

po-l

ig. 2. Outline of the ATC computation using the cubic-spline interpolationechnique.

s the test case in the determination of the area-to-arehe point-to-point ATCs.

.1. ATC results using the cubic-spline interpolationechnique

Prior to the ATC determination, the cubic-spline interation technique is used to trace theP–SandP–V curves for

Fig. 3. The Malaysian system.

340 M.M. Othman et al. / Electric Power Systems Research 73 (2005) 335–342

all the transmission lines and load buses of the system, re-spectively. The ATC is then determined by referring to themaximum power transfer that cause theP–Sor P–V curvesintersect the limiting levels of MVA power flow or voltagemagnitude, respectively. The outage of critical line in thetransfer capability analysis is considered because it will givea huge impact to the ATC result. To describe the determina-tion of ATC using the cubic-spline interpolation technique,the P–S curves for line 1226–2226 is chosen as shown inFig. 4. Line 1226–2226 is the line that limits the MW trans-fer from area 5 to area 2 due to an outage of line 2308–2608.From Fig. 4, it is shown that the cubic-spline interpolationtechnique traces theP–Scurve from the four points of MVApower flow. These four points of MVA power flow are ob-tained from ac power flow solutions corresponding to thepower transfers of 1, 200, 400 and 600 MW. A point is notedwhere the line thermal limit of 240 MVA intersects theP–Scurve. At this point, the MW transfer or ATC of 502 MW isthen determined. On the other hand, the comparisons ofP–Scurves obtained from the cubic-spline interpolation techniqueand the recursive ac power flow solutions are also illustratedin Fig. 4. TheP–Scurve obtained from recursive ac powerflow solutions considers incremental steps of power trans-fer by 1 MW. It is shown that the cubic-spline interpolationmethod accurately estimates theP–Scurve when comparedto theP–Scurve obtained from recursive ac power flow so-l

e tot a-t tion[ o thev -t sfer.T t oc-c werfl i-tu VA

Fig. 5. Variation of voltage at load bus 1196 against MW transfer.

power flow limit is not considered as the constraint during theincrease of power transfer. To illustrate the fact that ATC canalso be determined due to voltage limitation, theP–V curvefor the load bus 1196 is chosen as shown inFig. 5. Loadbus 1196 is the bus that limits the MW transfer from area 5to area 1 due to an outage of line 1468–2468.Fig. 5 showstheP–V curve obtained from the four points of voltage mag-nitude corresponding to the power transfers of 1, 300, 600and 900 MW. A point is noted where voltage limit of 0.9 p.u.intersects theP–V curve. At this point, the MW transfer orATC of 684 MW is then determined.

4.2. Results of area-to-area ATC and point-to-point ATC

Using the proposed ATC method, the results of thearea-to-area ATC and the point-to-point ATC are obtainedas shown inTables 1 and 2, respectively. The ATCs ob-tained from the cubic-spline interpolation technique are com-pared with the ATCs obtained from using the recursive acpower flow method in terms of accuracy and computationaltime.

Results shown inTables 1 and 2indicate that the limitationoccurs for all the cases of power transfer are due to overloadedlines. For instance, from the area-to-area ATC results shownin Table 1, by considering an outaged line 1468–2468 as acontingency, the ATC between areas 1 and 2 is 123 MW andi rly,fc y, theA itedb n thet tagev

p ech-n owsg intoa , and

utions.All the ATC results for the test system are obtained du

he MVA power flow limitation. However, ATC determinions for other systems can be more due to voltage limita18]. The ATC for the test system can be obtained due toltage magnitude limitation if the MVA power flow limitaion is not considered during the increase of power tranhis shows that the first limitation for the test system thaurs during the increase of power transfer is the MVA poow limitation and followed by the voltage magnitude limation. Therefore, in order to illustrate theP–V curve fittingsing the cubic-spline interpolation technique, hence, M

Fig. 4. Variation of MVA at line 1226–2226 against MW transfer.

t is limited by the overloaded line 1570–9222. Similarom the point-to-point ATC results shown inTable 2, byonsidering an outaged line 1468–2468 as a contingencTC between buses 9020 and 3636 is 80 MW and it is limy overloaded line 1182–9020. Simulations carried out o

est system indicate that the ATCs are not limited by voliolations.

The ATC results shown inTables 1 and 2prove that theroposed ATC method using cubic-spline interpolation tique and the ATC method using recursive ac power flive similar ATC results. The proposed method takesccount of reactive power flows and voltage magnitudes

M.M. Othman et al. / Electric Power Systems Research 73 (2005) 335–342 341

Table 1Results of area-to-area ATC

Area of transfers Limiting line ATC (MW) CPU times (second)

Seller area Buyer area Cubic-spline Recursive ac power flow Cubic-spline Recursive ac power flow

1 2 1570–9222 123 123 2.35 35.682 3 2394–9092 384 384 3.66 102.943 4 2436–9162 258 258 3.06 654 5 1424–9140 240 240 2.94 58.342 1 2394–9092 383 383 3.60 94.23 2 2436–9162 269 269 3.05 69.384 3 1424–9140 248 248 3.09 65.735 4 1636–2636 316 316 3.23 78.322 4 1636–2636 316 316 3.34 77.421 3 1570–9222 123 123 2.55 30.652 5 2394–9092 385 385 3.63 96.934 2 1424–9140 250 250 2.97 57.515 3 2438–5438 501 501 4.33 120.63 1 2436–9162 256 256 2.95 63.185 2 2438–5438 502 502 4.25 130.583 5 2436–9162 261 261 2.94 65.144 1 1424–9140 249 249 3.03 63.011 4 1570–9222 123 123 2.34 33.825 1 90000–99001 502 502 4.42 129.5

Table 2Results of point-to-point ATC

Point of transfers Limiting line ATC (MW) CPU times (second)

Seller bus Buyer bus Cubic-spline Recursive ac power flow Cubic-spline Recursive ac power flow

9020 3636 1182–9020 80 80 2.09 20.569181 1250 1250–2250 117 117 2.30 27.569060 1468 2308–9060 32 32 2.16 8.79210 2420 1552–9210 38 38 1.97 10.29010 90000 2158–9010 20 20 1.99 5.658

99007 2602 90000–99007 150 150 2.55 36.069182 2250 2602–9252 257 257 2.89 64.669062 3341 2308–9062 32 32 2.09 8.789024 2494 2182–9024 74 74 2.27 18.82

99015 1326 90000–99015 34 34 2.05 9.419270 2414 2424–9270 40 40 1.88 9.469132 2908 2602–9252 252 252 2.89 62.679163 5438 2436–9163 23 23 1.97 5.679191 90000 2510–9191 45 45 2.03 12.1

99023 2722 90000–99023 150 150 2.50 37.039272 2638 2424–9272 40 40 2.05 10.769284 1182 2602–9252 253 253 2.86 63.679245 1424 2588–9245 20 20 1.98 5.889261 90000 2602–9252 247 247 2.86 56.07

it is, therefore, comparable to the ac power flow method andbetter than the dc power flow method in terms of accuracy.In terms of computational times, it is noted that the ATCcomputations using the proposed cubic-spline interpolationtechnique is much faster as compared to the time taken bythe recursive ac power flow method. The advantage of theproposed method is that it does not require many recursiveload flow solutions in the ATC determination. From the aboveresults, it can be considered that the proposed ATC methodresults in satisfactory accuracy and with fairly short compu-tation times.

5. Conclusion

This paper presents a simple and new approach for evalu-ating area-to-area and point-to-point ATCs, using the cubic-spline interpolation technique to determine theP–V andP–Scurves. The curves are obtained so as to reduce the compu-tation time in the ATC evaluation. From theP–V andP–Scurves, the ATCs are determined at the intersecting points ofvoltage and MVA power flow limits. In the proposed ATCmethod, prior to ATC evaluation the critical line outages thatadversely affect the transfer capability of a power transmis-

342 M.M. Othman et al. / Electric Power Systems Research 73 (2005) 335–342

sion system are obtained through the process of contingencyranking and selection. The effectiveness of the proposedmethod is verified by simulation studies on the Malaysianpower system. The simulation results prove that the proposedcubic-spline interpolation technique is a fast and accuratemethod for ATC evaluation as compared to the ATC methodusing recursive ac power flow solutions. It is an effectivemethod in speeding up the ATC evaluation process. The pro-posed method is especially useful for use in the deregulatedelectricity market in which the ATCs are required to be postedas a real time market signal so that all transmission users havethe same chance to access transmission information.

Acknowledgement

The authors would like to thank the utility company ofMalaysia, Tenaga Nasional Berhad for providing the systemdata.

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