Towards reactive power markets. Part 1: reactive power allocation

Towards reactive power markets. Part 1: reactivepower allocation

K.L. Lo and Y.A. Alturki

Abstract: Although real power is the main traded commodity in electricity markets, reactive powerplays a crucial role in power systems reliability and security. In competitive electricity markets, nomarket participant wants to subsidise others in any way. Market participants utilise the network indifferent ways to maximise their profits. It means that their effects on the system, such as losses, canalso be different. The development of a fair and accurate loss allocation scheme for real andreactive power is significant in avoiding cross subsidies and to have the correct charge for eachparticipant. A new method is introduced to allocate real and reactive power losses in bilateralmarkets. The basic idea of the method assumes that transactions have their own effects on thesystem as well as their interactive effects with each other. Each transaction share of losses is basedon its contribution to the system current flows. The proposed method determines these currentscontributions and adjusts them, due to system nonlinearity, using the introduced currentadjustment factors (CAFs). Unlike other approaches, the proposed method can effectively andeasily allocate both real and reactive power losses simultaneously, which saves time and effort. Italso determines the contribution of each transaction to every branch on the system. In addition,a new index, voltage participation index (VPI), is proposed to measure reactive power supportsparticipation. This index is based on modified Y-bus matrix method proposed previously. TheCAF method and VPI are illustrated using a simple three-bus system. Then they are verified onmany systems, but due to space limit, only the results of IEEE 14- and IEEE 300-bus systems arepresented. Results illustrate consistency with expectation. Part 2 will address the issue of how toprice reactive power based on the technique developed in Part 1.

1 Introduction

In electricity markets, the system operator assuressecurity of the network whether it is a pool-based marketor a bilateral-based market. The power system must bebalanced at every second, which means that generationequals loads plus losses all the time. Energy trading ofparticipants does not take into consideration system lossand the system operator is the entity who is in charge ofsecuring the system by providing the required real andreactive power [1]. Since the power network is not lossless,entities providing the network with the required lossesmust be compensated for their contribution, normally atthe pool marginal price in a pool based market, or at theirmarginal cost in bilateral markets [2]. The purpose of lossallocation is to assign each user of the network, whether agenerator or a load, its share of the cost of transmissionlosses based on the losses the user causes.

Network losses cost millions of dollars every year as theycan account for 5–10% of the total generation in the system[2]. So, fair allocation of the network losses has veryimportant impact on all users. This is so because unfairallocation causes cross subsidies and it gives wrongindicative signals to the network operator and users. A

user who causes more network losses must be charged morewhile a user who helps to reduce the losses, due to counterflow, must be rewarded.

Loss allocation methods that have been proposed so farfall into five categories: pro rata, marginal loss, proportionalsharing, circuit based, and different approaches for bilateralcontracts [1–3]. A short description of the first fourcategories is given here.

The pro-rata method is one of the most commontechniques. It is based on the generators’ injections or loadreal power level rather than on their relative locations in thenetwork. In other words, loads close to the ‘centre ofgravity’ of the generation subsidise remote loads andgenerators close to the ‘centre of gravity’ of the loadssubsidise remote generators.

The marginal loss method is based on incrementaltransmission loss (ITL) coefficients. This method dependson the location of the slack bus. It needs normalisation sincethe direct application of its coefficients results in overrecovery [1]. The ITL coefficients can be positive ornegative. Distributed slack bus is proposed in [3].

The proportional sharing technique [5–7] provides anefficient computational method for loss allocation startingfrom the output of a solved load flow. However, it only usesKirchhoff’s first law and it is based on the proportionalitysharing assumption, which is neither provable nor dis-provable. Furthermore, the technique traces power flowfrom generators to loads in which neither loads norgenerators have control on the price they would be chargedsince they do not have any control on how their powerreaches its destination and where that destination is.

Circuit-based loss allocation is proposed in [2]. The authorsuse the network Z-bus matrix without any simplifying

The authors are with the Power Systems Research Group, University ofStrathclyde, Glasgow, UK

E-mail: [email protected]

r IEE, 2006

IEE Proceedings online no. 20045266

doi:10.1049/ip-gtd:20045266

Paper first received 3rd December 2004 and in final revised form 10th August2005

IEE Proc.-Gener. Transm. Distrib., Vol. 153, No. 1, January 2006 59

assumptions. This method is based on a solved power flow,and all its computations are based on the admittancematrix. Similar to marginal loss method, the Z-bus techniquecan yield negative allocation to ‘reward’ those participantswho contribute to reduce network losses due to theirstrategically well positioned locations within the system.

The negative ITL coefficients are being interpreted ascross subsidies in [1] and [2]. The unsubsidised ITL (U-ITL)method has been proposed to avoid negative allocatedlosses. It was emphasised in [1] that the U-ITL method is toallocate non-negative losses costs and not to explainphysical facts. In [2], it is stated that the Z-bus method issimilar to the ITL method in which both methods can yieldnegative loss allocations. It is stated also that negative lossallocation using the Z-bus method is to ‘reward’ generatorsor loads that are strategically well positioned in the network.

The cross subsidy term means that a participant or agroup of participants are charged more than they should bewhile others are either not charged or rewarded. So, thereward of the second group comes actually from the chargesof the former group. For example, the case study conductedin [1] using the ITL method results in generators beingallocated 146% of losses and demands �46%.

The authors of this paper believe that a participant whocontributes to loss reduction should be rewarded by fewerallocation losses rather than by negative ones. By doing this,the possibility of cross subsidy is avoided while the incentiveto reduce losses is reflected by the small allocated losses.

Unfortunately, transmission losses are nonlinear func-tions of line flows. This nonlinearity characteristic makes itimpossible to divide system losses to unique separate parts;i.e. each part is uniquely assigned to a generator or a load.So, any loss allocation approach has a certain degree ofarbitrariness. Nonetheless, there are characteristics that aloss allocation scheme must have to be equitable, or at leastaccepted as a reasonable approach. The scheme shouldhave most of the following characteristics:

1 It should reflect the amount produced or consumed by auser.

2 It should take into consideration the relative location of auser within the network.

3 The sum of all loss allocated terms should be consistentwith the results of the power flow.

4 It should avoid volatility.

5 It should be easy to understand and to implement.

2 Proposed current adjustment factor (CAF)method

In a deregulated energy system, users should be responsiblefor the system losses that they cause. System losses arecaused by individual users and their interactions, whichmake loss allocation more difficult. To illustrate thisproblem, let us consider the following branch that carriestwo power flows: PA and PB, as shown in Fig. 1. Realpower losses can be easily calculated as follows:

PLoss ¼ I2

t

�� R ¼ IA þ IB� �2�� R

¼ I2

A þ I2

B þ 2IAIB

�� R ð1Þ

where �I ¼ current vector, I�� ¼ magnitude of I , and

R¼ line resistance. But if the power losses of these flowsare calculated individually, then

PLoss;A ¼ I2

A

�� R and PLoss;B ¼ I2

B

�� R ð2Þ

The problem comes from

PLoss 6¼ PLoss;A þ PLoss;B ð3ÞThere is a cross term difference, ð2� IA � IBÞ � R, insidethe absolute term. If one traces the �Is of A transaction andthose of B transaction, then

PLoss ¼ IA þ IB

�� 2�R

¼ ðIAx þ IBxÞ þ jðIAy þ IByÞ�� 2�R ð4Þ

where IAx, IBx are the real parts of �IA and �IB, respectively,and IAy, IBy are the imaginary parts of �IA and �IB,respectively. But the squared absolute term of a vector isequal to the dot product of that vector, so

PLoss ¼ ðIAx þ IBxÞ þ jðIAy þ IByÞ�� 2�R

¼½ðIAx þ IBxÞ2 þ ðIAy þ IByÞ2� � R

¼½I2Ax þ I2Ay þ 2� IAx � IBx þ 2� IAy

� IBy þ I2Bx þ I2By � � R ð5ÞThen it is fair enough to assign each contributor its share ofthe losses as follows

PLoss;A ¼I2Ax þ I2Ay þ 2� IAx � IBx �

ðIAxÞ2

ðIAxÞ2 þ ðIBxÞ2

þ2� IAy � IBy �ðIAyÞ2

ðIAyÞ2 þ ðIByÞ2

266664

377775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}MA

�R

ð6Þ

PLoss;B ¼I2Bx þ I2By þ 2� IAx � IBx �

ðIBxÞ2

ðIAxÞ2 þ ðIBxÞ2

þ2� IAy � IBy �ðIByÞ2

ðIAyÞ2 þ ðIByÞ2

266664

377775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}MB

�R

ð7ÞIn contrast to other methods proposed to allocate realpower losses only, this proposed method can be appliedeffectively to reactive power loss allocation because it isbased on current flows rather than power flows:

Qloss ¼½I2Ax þ I2Ay þ 2� IAx � IBx þ 2� IAy � IBy

þ I2Bx þ I2By � � X ; X¼ line reactanceð8Þ

Qloss;A ¼ MA � X ð9Þ

Qloss;B ¼ MB � X ð10Þ

More generally, for any system, real and reactive powerloss allocation can be determined through the followingprocedure:

1 From a solved load flow (base case) where all transac-tions deliver their shares of the market, i.e. each generatorinjects its contracted real power and each counterpart

PA

PB

Fig. 1 Branch that carries two traded real power flows PA and PB

60 IEE Proc.-Gener. Transm. Distrib., Vol. 153, No. 1, January 2006

load consumes it, calculate all currents in all branches of thenetwork:

�Ik ¼ Ikx þ jIky ; k ¼ 1; 2; . . . ; NB ð11Þ

where NB¼ total number of branches, Ikx¼ real part of thecomplex current, and Iky¼ imaginary part of the complexcurrent

2 With the transaction of interest Ti inactivated, run powerflow program again and calculate resulting currents in allbranches:

�ITik ¼ ITi

kx þ jITiky ; k ¼ 1; 2; . . . ; NB; i ¼ 1; 2; . . . ; NT ð12Þ

where NT ¼ total number of transactions. In this step,generator i (or groups of generators under the same entity i)is kept active with zero real power output. This ensures aconvergent solution.

3 The contribution of each transaction Ti in a branch isequal to the corresponding current flow difference betweenthe base case and that when Ti is inactive;

ITik; cont ¼ Ik � I

Tik ð13Þ

4 The nonlinearity of the network due to the interactionbetween loads and generators when they are run at the sametime makes the sum of currents obtained in step 3 notmatch that obtained in step 1, i.e.

�Ik 6¼XNT

i¼1I

Tik; cont ð14Þ

So, current adjustment factors (CAFs) are introduced toadjust the obtained currents in step 3 as follows:

�Ik ¼ CAFk

XNT

i¼1

�ITik; cont ð15Þ

where CAF¼ current adjustment factors. CAF is generallya complex matrix, which is expected since the nonlinearityof the system is due to real and imaginary factorinteractions. Note that each branch on the system has itscorresponding CAF, which means that CAF is a (1� k)vector.

5 Calculate the new adjusted currents:

�ITik;Adj ¼ CAFk � �ITi

k; cont ð16Þ

6 Calculate the real and reactive power losses allocationsfor each transaction, PLosses

Ti and QLossesTi , respectively, as

follows:

P Tilosses ¼

XNB

k¼1

"ðITi

kx; adjÞ2 þ ðITi

ky; adjÞ2 þ CRe

k

(

�ðITi

kx; adjÞ2

IRe; sumk

þ CImk �

ðITiky; adjÞ

2

I Im; sumk

#� Rk

)ð17Þ

QTilosses ¼

XNB

k¼1

"ðITi

kx; adjÞ2 þ ðITi

ky; adjÞ2 þ CRe

k

(

�ðITi

kx; adjÞ2

IRe; sumk

þ CImk �

ðITiky; adjÞ

2

I Im; sumk

#� Xk

)ð18Þ

where

IRe; sumk ¼

XNT

i¼1ðITi

kx; adjÞ2; I Im; sum

k ¼XNT

i¼1ðITi

ky; adjÞ2

CRek ¼

XNT

i¼1

XNT

j¼1i 6¼j

ITikx; adj � ITj

kx; adj; CImk ¼

XNT

i¼1

XNT

j¼1i 6¼j

ITiky; adj � ITj

ky; adj

Rk¼ the resistance of branch k, Xk¼ the reactance ofbranch k

Ikx,adjTi ¼ the real part of �ITi

k; adj, and Iky,adjTi ¼ the imaginary

part of �ITik; adj

It is worth noting here that all loss allocations are greaterthan or equal to zero, which ensures eliminating any crosssubsidies, while at the same time it does consider counterflow on branches, which can be looked at as how much atransaction utilises a branch. A counter flow transactionreduces the flow on the branch, yet actually it utilises thebranch. Because of its good effect on flow reduction, itshould be rewarded by allocating less loss at that branchthan other transactions that increase the flow. In contrast topower factor schemes, the proposed method has the abilityto charge individual transactions even when they trade unitypower factor loads due to the real and reactive losses theycause on the system.

3 Choice of slack bus

The choice of a slack bus is always arbitrary in powersystem analysis. Different locations of the slack bus result indifferent total system losses and branch current flows.Because the proposed method is based on a solved base casewith prespecified slack bus, changing the location of theslack bus does affect the results. However, several casestudies, including case studies presented in this paper, havebeen conducted to see how significant the effect is, in whichit was discovered that the location of slack bus has a minoreffect on the final loss allocations. This subject will bereviewed in a separate paper.

4 Proposed voltage participation index (VPI)

The authors in [7] consider reactive power allocation whenpurchased in a competitive market. Their proposed methodis straightforward without any assumptions. It is derivedfrom basic circuit theory by partitioning the Y-bus matrixinto four parts: gen–gen, gen–load, load–gen, and load–loadparts. Then each load is replaced by its correspondingadmittance element using power flow results. This meansthat the diagonal elements of the load–load part are theonly modified ones. The derivation of the method is asfollows:

YGG YGL

YLG YLL

� �VG

VL

� �¼ IG

IL

� �ð19Þ

Note here that the Y-bus matrix is built in such a way thatall generator buses are inserted in the admittance matrixbefore load buses. From load flow calculations, all loadvoltages are known, so each load is replaced by its


corresponding admittance element, which can be calculatedusing the following:

YLj ¼1

VLj

SLj

VLj

� ��ð20Þ

where (*) means conjugate. This will modify the diagonalelements of YLL. So, the new equation will be

YGG YGL

YLG Y mLL

� �VG

VL

� �¼ IG

0

� �ð21Þ

where YLLm is the modified Y-bus.

Then the load voltages can be expressed in terms ofgenerator voltages using the second part of (21) where loadcurrents become zeros:

VL½ � ¼ � Y mLL

�1YLG½ �|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

½YA�

VG½ � ¼ YA½ � VG½ � ð22Þ

The voltage of each load bus, consisting of voltagecontributions received from generator buses, is expandedas follows:

VLj¼XNG

i¼1YAji � VGi ð23Þ

and it is assumed that

DVLi; j ¼ YAj; i � VGi ð24Þwhere DVLi, j is the voltage contribution that load bus jacquires from generator bus i. Equation (23) may also beexpressed as

VLj¼XNG

i¼1DVLi; j ð25Þ

where NG equals the total number of generator buses.Equation (25) indicates that the load voltage VLj is asummation of the contributions of voltages DVLi,j fromgenerator buses i¼ 1,y,NG.

Then the authors used this formula to determine reactivepower that each generator contributes to a load bymultiplying the voltage contribution by the load current.This approach is similar to proportional sharing methodsfrom the point of view that neither generators nor loads canchoose their counterpart contractors. In this paper, ameasure index of reactive power support participation using(25) has been developed.

For the purpose of a competitive market, it is desirablethat load buses have many voltage contributions frommany voltage sources (voltage source is a reactive powersource). This means less opportunity for market power involtage support. The proposed index to evaluate the systemvoltage profile according to the above criteria needs toreflect the number of contributions on each load bus andhow much each contribution is. Equation (26) is used todevelop voltage participation index (VPI). It is defined asfollows:

VPI ¼ 1

NL

XNL

i¼1

XNG

j¼1

DVLi; j

Vi

��2

ð26Þ

where NL¼ number of load buses and NG¼ number ofreactive resources buses.

Similar to real power participation indices, such as HHI(Herfindahl-Hirshman index [Note 1]), which give informa-tion about how concentrated the real power generation is,

the purpose of VPI is to give a quick picture of how diversethe reactive power support is. VPI has a maximum value of104 (as percentage or unity as per unit), which represents afull monopoly with one reactive support provider within thenetwork. As the number of effective participants in reactivesupport increases, the VPI value decreases. This is similar tothose indices dealing with market power in a power market.A system operator can define threshold levels for compe-titive and less competitive market power ranges for VPI.Since reactive power cannot travel far, VPI can serve as anindicator of comparison between different areas of thenetwork to provide signals about how much reactive poweris available within those areas. The larger the value of VPIthe more important it is for new reactive providers to enterthe market. Note that VPI takes into consideration thenature of the locality of reactive power, which means thatadding many resources that are far away from wherereactive power is needed has little effect on VPI value. Thiscrucial phenomenon will be explored later in case studies.

A marginal VPI (MVPI) can be derived as an indicationabout the available marginal participation of each reactivesource if a marginal change of a load bus voltage is needed.While VPI gives information about the current situation ofthe system, MVPI provides information about the marginalreactive support availability if a marginal reactive powersupport is needed. In other words, the network can have alow VPI value due to a large number of effective reactivepower sources while it has high MVPI because one or moreof these sources will not be able to increase their support tosatisfy any system marginal needs for reactive power. From(21), the rate of change of load voltages with respect togenerator buses voltages can be expressed as follows (seeAppendix 1 for full derivation):

@VL

@VG

� �¼ � Y m

LL þ Y m0LL VL

h i�1|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

D

YLG½ � ¼ VS½ � ð27Þ

@VL; i

@VG; j¼ VSij ð28Þ

where VSij is the voltage sensitivity of load bus i with respectto a change on the set voltage of voltage bus j. A factor, Wj,

can be used to reflect whether a reactive source is availablefor further support or not, i.e. Wj can take binary valueszero or one (zero when a source is not available for furtherreactive support and one when it is available):

VSsumi ¼

XNG

j

VSij ð29Þ

Then MVPI is defined as

MVPI ¼ 1

NL

Xi

Xj

VSij

VSsumi

��2

ð30Þ

MVPI, like VPI, has a maximum value of 104 (if percentagevalues are used or unity if per unit values are used) and asystem operator can define intervals of secured operationaccording to the value of MVPI. VPI and MVPI needadditional research work before they can be adopted inpractice.

5 Case studies

5.1 Simple three-bus system for illustrationpurposesThis system is illustrated in Fig. 2. There are twotransactions: transaction one is between generator 1 and

Note 1: This index is defined as follows: HHI ¼PN

iu s2i where the summation isover all N participants in the real market and si refers to the market share ofeach participant.


load 3a and transaction two is between generator 2 andload 3b. Three scenarios have been studied:

(a) Since the slack bus location affects system lossesgenerally, a rotating slack bus is used between buses 1and 2. Then the average is calculated. With unity powerfactor loads, traded power of transaction 2 increases fromsingle MW to 200MW in steps of 1MW while keepingtransaction one’s at 100MW. Figure 3 shows that astransaction 2 increases, its allocated losses increase while theallocated losses of transaction 1 have slightly increased. Thelittle increase of loss allocated to transaction 1 is due to theinteraction between transactions when they are appliedsimultaneously, as shown in (17) and (18).

It is worth noting that real and reactive power lossallocations given above have the same pattern except that they-axis values are different. This is expected since the methodis based on branch currents that are the same for real andreactive loss allocations, as shown in (17) and (18). They aredifferent only according to line resistance and reactance andthis is actually the strength of the proposed method.

Since the electric distance for both transactions is thesame, the loss allocations must be the same when thetraded powers of both transactions are 100MW, whichcan be seen from the results above. For VPI, it changesslightly over the range, as shown in Fig. 4. Both identicalsources have exactly the same relative location within

G2 G1

L3aL3b

Z = 0.013 + 0.1j p.u.

Z = 0.012 + 0.1j p.u. Z = 0.012 + 0.1j p.u.

bus 2

bus 3

100 MW 102 MW

100 MW 0 Mvar

100 MW 0 Mvar

11.4 Mvar9.2 Mvar

bus 1

Fig. 2 Simple three-bus system

0 50 100 150 2000

1

2

3

4

5

transaction 2 traded MW

allo

cate

d M

W lo

sses

0 50 100 150 2000

5

10

15

20

25

30

35

40


allo

cate

d M

var

loss

es

transaction 1

transaction 2

transaction 1

transaction 2

Fig. 3 Real and reactive power loss allocation to both transactionsin case study 1(a)

0 50 100 150 2005000.0

5000.5

5001.0

5001.5


VP

I and

MV

PI

0 50 100 150 200−15

−10

−5

0

5

10


Q1−Q

2 Q2>Q1

Q1>Q2

VPI

MVPI

Fig. 4 VPI and MVPI results of case study 1(a)


the network. The small changes of VPI are due to thesmall difference in the generator reactive power outputs.The reactive power output of generator 2 is slightlyless than the reactive power output of generator 1 whentransaction 1 is smaller than transaction 2, equal to reactivepower output of generator one when both transactionsare 100MW, and larger than reactive power output ofgenerator 1 when transaction 2 is smaller than transactionone. That is why VPI is equal to 0.5p.u. at 100MWand slightly bigger than 0.5 elsewhere. This means thatboth generators participate by half in maintaining the loadbus voltage when real power outputs of generator 1 and 2are 100MW and one bigger than the other elsewhere,as shown in Fig. 4. For MVPI, it is constant at 5000 sinceboth generators have the same relative positions withinthe network and same operating conditions. This is nottrue if they have different reactive power participationcharacteristics.

(b) In this case, the traded power of transaction 2 has a fixednon-unity power factor (0.89 lagging). This means that thistransaction burdens the system more than it does in theprevious case. The results shown in Fig. 5 are consistent

with expectation as transaction 2 is now allocated morelosses compared to the previous case for the same MWoutput.

(c) If generator 2 is far away from the load centre, theshipping of its energy burdens the system much more thanthat of transaction 1. To study this, the impedance of thebranch connecting generator 2 to bus 3 is increased toz¼ 0.03+j0.2p.u. It is expected that transaction 2 will beallocated the same losses as those of transaction 1 whengenerator 2 has a lower output than that of generator 1. Inother words, for 100MW from both generators, transaction2 will be allocated more losses than generator 1 due to itslarger electrical distance. This is shown in Fig. 6.

For VPI and MVPI, Fig. 7 shows that both indices havelarger values compared to those of case 1(a). This reflectsthat one participant, generator 1, seizes a bigger share ofreactive power support than the other participant, generator2. This is to be expected since the former has a betterrelative location within the network. In other words,generator 2 now has longer electrical distance to the loadbus than generator 1 does. This phenomenon is one of themost important differences between real and reactive power.As transaction 2 increases, more reactive power support is

0 50 100 150 2000

1

2

3

4

5

6

7


allo

cate

d M

W lo

sses

0 50 100 150 2000

10

20

30

40

50

60


allo

cate

d M

var

loss

es

transaction 1

transaction 2

transaction 1

transaction 2

Fig. 5 Real and reactive power loss allocation to both transactionsin case study 1(b) when transaction 2 has 0.89 lagging power factor

0 50 100 150 2000

1

2

3

4

5

6

7

8

9


allo

cate

d M

W lo

sses

0 50 100 150 2000

10

20

30

40

50

60


allo

cate

d M

var

loss

es

transaction 1

transaction 2

transaction 1

transaction 2

Fig. 6 Real and reactive power loss allocation to both transactionsin case study 1(c) when transaction 2 has longer electrical distance


needed from generator 1, which leads to a bigger value ofVPI reflecting the market power of generator 1. For MVPI,it is bigger than that in case 1(a) for the same reason.

It is found from the above VPI results that the ranges ofchanges between Figs. 4 and 7 are very small. This happensbecause, as mentioned earlier, VPI incorporates two factors.The first one depends on the number of reactive sourcesthat are available for voltage support and the second factoris the magnitude that each reactive source contributes to thesystem, which depends on its location within the network.Network configuration and location of sources play animportant role in reactive power service, so it shouldbe emphasised here that the second factor has a larger effecton VPI than the first one does because of the localityphenomenon of reactive power. So, the range of changes inFigs. 4 and 7 are small while the actual values of the indexesdiffer greatly between Fig. 4 (about 5001) and that of Fig. 7(about 5568), where the location of generator 2 has beenchanged. So, changing one branch on the system (thebranch between bus 2 and 3) has increased VPI significantlyby about 570.

5.2 IEEE 14-bus systemThe one-line diagram of the system is shown in Fig. 8and transaction data is given in Table 1. Table 2 showsthe results for different scenarios. Case (a) is the base case.The locations of the five reactive power sources withinthe network as well as the operating conditions determinethe reactive outputs of each source and the voltage source

participations delivered to each load bus. So, if thesesources were assumed to have exactly the same relativelocations within the network, VPI would be 2000 as a result

0 50 100 150 2005560

5562

5564

5566

5568

5570

5572

5574

5576


VP

I and

MV

PI

VPI

MVPI

Fig. 7 VPI and MVPI results of case study 1(c)

generators

synchronouscondensers

12

13

14

1011

96

54

78

87

9

4

transformer equivalentthree-winding

3

AEP 14-bus test system bus code diagram

2

1

C

G

C

C

C

G

G

C

Fig. 8 One-line diagram of IEEE 14-bus system

Table 1: Bilateral contracts between generators and loadsof IEEE 14-bus system

Bus Transactions quantities in MVA

Generator 1 Generator 2

1 0 0

2 21+12.7j 0

3 94+19j 0

4 0 47�3.9j

5 0 10+1.6j

6 0 15+7.5j

7 0 0

8 0 0

9 0 30+16.6j

10 0 9+5.8j

11 0 15+1.8j

12 0 35+1.6j

13 0 30+5.8j

14 0 20+5j

Total 115+31.7j 211+41.8j

Table 2: Results of case study 2

Case Voltage Indices Generator Load Allocated losses

VPI MVPI P (MW) Q (Mvar) P (MW) Q (Mvar)

a 4702.2 4644.7 G1 115 31.7 3.8081 15.155

G2 211 41.8 12.501 64.90

b 4994.2 4925.9 G1 115 31.7 3.82 15.20

G2 211 41.8 12.79 65.22

c 3089.8 3016 G1 115 31.7 3.85 15.30

G2 211 41.8 13.28 71.26

d 3950.8 3905.8 G1 115 31.7 3.807 15.153

G2 211 41.8 12.43 64.465


of one-fifth share of each source. But in our case, VPI equals4702.2, which indicates that some sources seize more voltagesupports to load bus voltages than other sources do. This isnoticed from the voltage share of each source on each loadbus (24), as shown in Table 3. Synchronous condensers atbuses 6 and 8 provide more voltage shares than othersources do, especially for buses 9, 10, 11, 12, 13 and 14 (loadcentre). This causes VPI to be higher than 2000.

Case (b) studies the system when the source at bus 8 isdisconnected. VPI and MVPI now are higher, as shown inTable 2, which reflect that there are fewer voltage sourcesthan the previous case, emphasising that the source at bus 6now seizes more of the load bus voltages (especially loadcentre area). On the other hand, total system losses arehigher, as expected, and most of the increase is allocated togenerator 2 (96.67 and 86.49% of real and reactive lossesincrease, respectively) reflecting the fact that all its contractdestinations are in the load centre, so it is expected to beaffected more for losing this source.

Case (c) examines the system when the reactive source atbus 6 is disconnected. Total real and reactive losses increasesince required reactive power needs to travel a longerdistance than it does in the base case. VPI and MVPIdecrease even though the system has lost a well positionedreactive source. Although this seems in contradiction tocommon sense and expectation, the results are correct.Looking in depth at the results shows that voltage sharesfrom reactive sources that build load buses voltages aremore evenly distributed when synchronous condenser atbus 6 is disconnected (assuming no voltage securityviolation occurs). In other words, the reactive support ismore competitive without having source 6. This can be seenby comparing the results of Fig. 9 with those of Fig. 10,where the voltage participations are calculated using (24).Regarding system losses, generator 2 is allocated most ofthe loss increases (95.12 and 97.7% of real and reactive loss

increases, respectively) for the same reason as mentioned incase (b).

To decrease the dependence on the voltage source at bus6, another voltage source is added at the area where source6 is most needed, i.e. load the centre. Case (d) studies thiswhen a voltage provider is added at bus 14. As seen inTable 2, VPI and MVPI decrease and so do total real andreactive losses. Again, this installation benefits the loadcentre, which has a contract with generator 2. This isreflected by the fact that more percentage of loss reductionis given to transaction 2 (98.47 and 98.42% of real andreactive loss reduction, respectively). Among the examinedcases, case (c) has the best value of VPI (minimum), whichmeans that more voltage sources participate in buildingeach load bus with more evenly distributed shares. This canbe concluded from Figs. 9–11.

Table 3: Voltage source participations to load buses voltages [case 2(a)]

Generator Participation of a generator on a load bus (voltage magnitude)

4 5 7 9 10 11 12 13 14

1 0.3884 0.17311 0.39361 0.16488 0.13528 0.068946 0.011047 0.022929 0.1009

2 0.12065 0.053775 0.19941 0.051218 0.042022 0.021417 0.0034317 0.0071227 0.031344

3 0.23374 0.10418 0.14363 0.099226 0.08141 0.041492 0.0066482 0.013799 0.060723

6 0.17795 0.23108 0.23255 0.40458 0.51502 0.77915 0.98121 0.94616 0.62573

8 0.11724 0.5013 0.072041 0.34282 0.28127 0.14335 0.022969 0.047674 0.2098

0

0.1

0.2

0.3

0.4

0.5

0.6

4 5 6 7 9 10 11 12 13 14

load bus number

volta

ge s

hare

s in

mag

nitu

des

G1

G2

G3

G8

Fig. 10 Generators participation to load buses voltages in casestudy 2(c)

0

0.2

0.4

0.6

0.8

1.0

1.2

4 5 7 9 10 11 12 13 14load bus number

volta

ge s

hare

s in

mag

nitu

des

G1

G2

G3

G6

G8

Fig. 9 Generators participation to load buses voltages in casestudy 2(a)

0

1.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

4 5 7 9 10 11 12 13

load bus number

volta

ge s

hare

s in

mag

nitu

des

G1G2G3G6G8G14

Fig. 11 Generators participation to load buses voltages in casestudy 2(d)


5.3 IEEE 300-bus systemThe introduced loss allocation method and voltageparticipation indices are applied to an IEEE 300-bussystem. Results shown in Table 4 (in Appendix 2) areconsistent with the analysis outputs of previous cases. Eachloss allocation reflects two factors: the amount of contractedMVA and the electric distance between source and sink.VPI and MVPI are 3558.6 and 3522.4, respectively. Tostudy the effect of adding a new reactive source, asynchronous condenser is added at bus 225 whichaffects loss allocations, as shown in Table 5 (in Appendix3), and reduces VPI and MVPI values to 3519.4 and3485.9, respectively. The contract loss allocations changedifferently according to the relative locations of eachbilateral contract’s source and sink with respect to thenew added reactive power source. Most transactionsbenefit from this addition, which appears in lower lossallocations. Generator 191 is most helped by the installa-tion. This is because the installed synchronous condenseris closer to many destinations of generator 191 contracts.This stresses the well known fact about the reactive powerlocality characteristic.

6 Conclusions

Reactive power is important for system security and lossescompensation, yet no one is charged for this cost becausethe whole traded commodity in electricity markets untilnow is real power. Since all participants benefit fromreactive power support and each pure MW transportationis accompanied by reactive power support, the requiredreactive power should be allocated to all participants fairlyaccording to their real use of the network. The real use ofthe system depends mainly on two factors: the nature of theindividual contracts and the relative locations of partiesinvolved in the contract. This paper contributes towards acompetitive reactive power market by allocating eachcontract its share of both real and reactive power lossesusing the current adjustment factors (CAFs) method. It alsocontributes to the current real power markets, wherereal power is still the main traded commodity, by proposinga new method for real power loss allocation. In addition,the paper has introduced two new measure indices ofvoltage source participations on system reactive powerneeds: voltage participation index (VPI) and marginal VPI.The proposed method and indices have shown consistencywith intuitive expectation through many test systems whereonly a few of these cases are presented in this paper dueto space limitation. The cases included a simple three-bussystem, an IEEE 14-bus system, and an IEEE 300-bussystem.

7 References

1 Conejo, A.J., Arroyo, J.M., and Alguacil, N.: ‘Transmission lossallocation: a comparison of different practical algorithms’, IEEE Trans.Power Syst., 2002, 17, (3)

2 Conejo, A.J., Galiana, F.D., and Kockar, I.: ‘Z-bus loss allocation’,IEEE Trans. Power Syst., 2001, 16, (1)

3 Galiana, F.D., Conejo, A.J., and Kockar, I.: ‘Incremental transmissionloss allocation under pool dispatch’, IEEE Trans. Power Syst., 2002, 17,(1)

4 Kirschen, D., Allan, R., and Strbac, G.: ‘Contributions ofindividual generators to loads and flows’, IEEE Trans. Power Syst.,1997, 12, (1)

5 Strbac, G., Kirschen, D., and Ahmad, S.: ‘Allocating transmissionsystem usage on the basis of traceable contributions of generators andloads to flows’, IEEE Trans. Power Syst., 1998, 13, (2)

6 Kirschen, D., and Strbac, G.: ‘Tracing active and reactive powerbetween generators and loads using real and imaginary currents’, IEEETrans. Power Syst., 1999, 14, (4)

7 Chu, W., Chen, B., and Liao, C.: ‘Allocating the costs of reactive powerpurchased in an ancillary service market by modified Y-bus matrixmethod’, IEEE Trans. Power Syst., 2004, 19, (1)

8 Schewppe, F.C., Caramanis, M.C., Tabors, R.D., and Bohn, R.E.:‘Spot pricing of electricity’ (Kluwer Academic Publishers, Boston, MA,USA, 1988)

8 Appendixes

8.1 Appendix 1MVPI derivationDerivation of (30) is as follows:The second equation of (21) is

½YLG�½VG� þ ½Y mLL�½VL� ¼ 0 ð31Þ

which can be written in more detail as

Y 11LG � � � Y 1Ng

LG

..

. ...

Y Nl1LG � � � Y NlNg

LG

2664

3775

VG1

..

.

VGNg

2664

3775

þY 11

LL � � � Y 1NlLL

..

. ...

Y Nl1LL � � � Y NlNl

LL

2664

3775

VL1

..

.

VLNl

2664

3775 ¼ 0 ð32Þ

So, there are Nl equations, where Nl is the number of loadbuses on the system:

Y 11LGVG1 þ Y 12

LGVG2 þ � � � þ Y 1Ng

LG VGNg

þ Y 11LL VL1 þ Y 12

LL VL2 þ � � � þ Y 1NlLL VLNl ¼ 0

..

.

ð33Þ

Y Nl1LG VG1 þ Y Nl2

LG VG2 þ � � � þ Y NlNg

LG VGNg

þ Y Nl1LL VL1 þ Y Nl2

LL VL2 þ � � � þ Y NlNlLL VLNl ¼ 0 ð34Þ

The partial derivatives of (33) and (34), @VL@VG

, yield to

(Nl�Ng) equations as follows:

@

@VGf½YLG�½VG� þ ½Y m

LL�½VL�g ¼ 0 ð35Þ

Y 11LG þ Y 11

LL �2S�1V 2

L1

� �@VL1

@VG1þ Y 12

LL@VL2

@VG1þ � � � þ Y 1Nl

LL@VLNl

@VG1¼ 0 ð36Þ

Y 12LG þ Y 11

LL �2S�1V 2

L1

� �@VL1

@VG2þ Y 12

LL@VL2

@VG2þ � � � þ Y 1Nl

LL@VLNl

@VG2¼ 0 ð37Þ

^

Y Nl1LG þ Y Nl1

LL@VL1

@VG1þ Y Nl2

LL@VL2

@VG1þ � � � þ Y NlNl

LL �2S�Nl

V 2LNl

!@VLNl

@VG1¼ 0 ð38Þ

Y NlNg

LG þ Y Nl1LL

@VL1

@VGNg

þ Y Nl2LL

@VL2

@VG2þ � � � þ Y NlNl

LL �2S�Nl

V 2LNl

!@VLNl

@VGNg

¼ 0 ð39Þ

Note that all YLGij and YLL

ij elements, except YLLii , are

constant values. YLLii is a function of VLi given by

Y iiLL ¼

1

VLi

S�

V �Li

� �ð40Þ

So,

@Y iiLL

@VGj¼ @Y ii

LL

@VLi

@VLi

@VGj¼ �2S�

V 3Li

@VLi

@VGjð41Þ


Equations (36)–(39) can be written in a matrix form asfollows:

Y 11LL �

2S�1V 2

L1

� �Y 12

LL � � � Y 1NlLL

Y 21LL

. ..

Y 21LL

..

. . .. ..

.

Y Nl1LL � � � � � � Y NlNl

LL �2S�Nl

V 2LNl

!

26666666664

37777777775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}D

�

@VL1

@VG1� � � @VL1

@VGNg

..

. ...

@VLNl

@VG1� � � @VLNl

@VGNg

2666666664

3777777775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}@VL@VG

h i

¼ �

Y 11LG � � � Y 1Ng

LG

..

. ...

Y Nl1LG � � � Y NlNg

LG

266666664

377777775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}YLG½ �

ð42Þ

@VL

@VG

� �¼ �D�1 YLG½ � ¼ VS½ � ð43Þ

where [VS] is the voltage sensitivity of load buses withrespect to a change in the set voltages of the voltage buses,i.e. VSij is the voltage change of load bus i with respect to achange in set voltage of the voltage bus j:

@VL; i

@VG; j¼ VSij ð44Þ

VSsumi ¼

Xj

VSij; j ¼ 1; . . . ; Ng ð45Þ

Then MVPI is defined as

MVPI ¼ 1

Nl

Xi

Xj

VSij

VSsumi

��2

; i ¼ 1; . . . ; Nl ð46Þ

8.2 Appendix 2Table 4: Results of case study 3 (slack bus is 7049)

Generator ContractedMW

ContractedMvar

AllocatedMW

AllocatedMvar

Actualoutput

84 375 194 12.89 134.38 375

91 155 0 2.957 13.81 155

92 290 98 10.37 60.508 290

98 68 33 0.54647 5.1844 68

108 117 23 1.744 7.9824 117

119 1930 793.2 5.2913 570.44 1930

124 240 161.5 1.2624 4.75 240

125 0 260.6 0.12906 6.5984 0

138 0 400 0.03846 0.33179 0

141 281 38.3 4.5524 46.568 281

143 696 262.7 19.746 198.55 696

146 84 �24.3 0.1982 1.5321 84

147 217 145.7 3.4843 26.674 217

149 103 139 0.61904 4.92 103

152 372 104.9 7.6128 54.449 372

153 216 0 3.8972 35.016 216

156 0 9 0.023241 0.067937 0

170 205 83.3 2.7346 33.479 205

176 228 0 11.949 48.452 228

177 84 219 1.4754 7.3122 84

185 200 0 3.5185 31.919 200

186 1200 198.54 6.3337 142.42 1200

187 1200 288.3 19.123 201.21 1200

190 475 100 1.1036 7.6647 475

191 1973 674.6 35.858 703.81 1973

198 424 382 5.3571 34.905 424

213 272 119 3.187 47.577 272

220 100 0 0.03361 0.8472 100

221 450 0 0.54572 6.8746 450


Table 4 (continued )


ContractedMvar

AllocatedMW

AllocatedMvar

Actualoutput

222 250 0 0.12237 2.4765 250

227 303 143 0.58289 14.732 303

230 345 348 1.9297 44.238 345

233 300 188 0.49882 9.2137 300

236 600 151 2.4926 63.409 600

238 250 212 1.5204 11.461 250

239 550 152 2.2893 47.785 550

241 575.43 139.7 7.452 85.457 575.43

242 170 79.3 2.2929 19.966 170

243 84 29 1.9496 11.44 84

7001 467 161 6.5252 83.239 467

7002 623 210 24.746 163.59 623

7003 1210 169 17.703 343.14 1210

7011 234 29 2.7305 38.64 234

7012 372 103.8 8.881 131.97 372

7017 330 127 13.067 95.746 330

7023 185 110 5.8562 33.329 185

7024 410 137 17.552 180.51 410

7039 500 190 11.097 167.5 500

7044 37 20 0.97235 7.3926 37

7049 46.42 24.53 4.2582 52.518 454.58

7055 45 0 0.51856 5.3578 45

7057 165 0 3.5972 29.004 165

7061 400 0 15.342 131.16 400

7062 400 0 14.721 132.97 400

7071 116 0 10.061 74.515 116

7130 1292 185.8 38.526 643.01 1292

7139 700 140 4.4149 117.22 700

7166 553 35.5 19.643 314.22 553

9054 50 0 0.21524 8.9129 50

9055 8 0 0.0080366 0.69448 8

Total 23525.85 7787.97 408.15 5503.05 23934.01

8.3 Appendix 3Table 5: Results of case study 3 when synchronous condenser is added at bus 225


ContractedMvar

AllocatedMW

AllocatedMvar

Actualoutput

84 375 194 12.628 130.76 375

91 155 0 2.951 13.782 155

92 290 98 10.343 60.378 290

98 68 33 0.54525 5.1849 68

108 117 23 1.7429 7.9856 117

119 1930 793.2 5.2904 570.92 1930

124 240 161.5 1.2666 4.7684 240

125 0 260.6 0.12969 6.5746 0

138 0 400 0.03787 0.32696 0

141 281 38.3 4.5549 46.585 281

143 696 262.7 19.719 198.52 696

146 84 �24.3 0.19778 1.5281 84

147 217 145.7 3.4786 26.637 217

149 103 139 0.6199 4.9255 103


Table 5 (continued )


ContractedMvar

AllocatedMW

AllocatedMvar

Actualoutput

152 372 104.9 7.61 54.436 372

153 216 0 3.8961 35.005 216

156 0 9 0.023582 0.06876 0

170 205 83.3 3.1551 36.558 205

176 228 0 11.949 48.448 228

177 84 219 1.4738 7.305 84

185 200 0 3.5267 31.984 200

186 1200 198.54 6.3403 142.63 1200

187 1200 288.3 19.131 201.37 1200

190 475 100 1.1545 8.1034 475

191 1973 674.6 30.177 605.26 1973

198 424 382 5.3254 34.759 424

213 272 119 3.1963 47.763 272

220 100 0 0.035706 0.85656 100

221 450 0 0.49392 6.5099 450

222 250 0 0.1221 2.5012 250

227 303 143 0.7153 15.729 303

230 345 348 1.9368 44.437 345

233 300 188 0.49715 9.2799 300

236 600 151 2.4814 63.107 600

238 250 212 1.511 11.334 250

239 550 152 2.2794 47.777 550

241 575.43 139.7 7.3953 85.226 575.43

242 170 79.3 2.2627 19.557 170

243 84 29 1.9422 11.307 84

7001 467 161 6.5407 83.376 467

7002 623 210 24.727 163.56 623

7003 1210 169 17.662 342.89 1210

7011 234 29 2.7015 38.509 234

7012 372 103.8 8.7328 130.98 372

7017 330 127 12.937 94.767 330

7023 185 110 5.837 33.181 185

7024 410 137 17.427 179.83 410

7039 500 190 10.944 166.58 500

7044 37 20 0.98621 7.3797 37

7049 46.42 24.53 4.3059 52.215 448.27

7055 45 0 0.5183 5.3554 45

7057 165 0 3.5796 28.9 165

7061 400 0 15.258 130.8 400

7062 400 0 14.671 132.72 400

7071 116 0 10.107 74.701 116

7130 1292 185.8 38.454 642.45 1292

7139 700 140 4.4477 117.51 700

7166 553 35.5 19.639 313.96 553

9054 50 0 0.21467 8.8949 50

9055 8 0 0.0079894 0.69251 8

Total 23525.85 7787.97 401.84 5399.44 23927.70


Towards reactive power markets. Part 1: reactive power allocation

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