Electr Eng (2009) 91:27–34DOI 10.1007/s0020200901132
ORIGINAL PAPER
A new approach for cost allocation and reactive power pricingin a deregulated environment
S. Hasanpour
·
R. Ghazi
·
M. H. Javidi
Received: 29 June 2008 / Accepted: 14 April 2009 / Published online: 7 May 2009© SpringerVerlag 2009
Abstract
Power industry has been facing restructuringproblems during the past decade. Appropriate managementof reactive power is very essential for supporting power system security. Reactive power has dominant effects on realenergy transfer. Furthermore, it can support the secure operation of the system as an ancillary service. However, mostresearches have been focused on active power as the maingood transacted in electricity markets. On the other hand,while reactive power production cost is highly dependent onrealpoweroutput,itismainlyconﬁnedtolocalconsumption.Asaresult,toavoidmarketpower andtomaintainthesecureoperation of the system, a fair cost allocation method seemsto be very essential. Appropriate pricing of reactive poweras an ancillary service has been a challenging problem during the past decade. However, most methods proposed so farfor reactive power pricing are essentially based on empirical approximations. In this paper, a new method for reactivepower cost allocation is proposed. The method is based oncalculation of the accurate cost which will be imposed ongenerators due to supporting reactive power. The proposedmethod isfair,accurate andrealisticanditcanbeformulatedvery easily. Furthermore, a new approach based on tracingalgorithm is proposed for pricing of reactive power whichconsidersthecostofbothactiveandreactivelossesallocatedto each generator. Application of the proposed method onIEEE 9bus standard network conﬁrms its validity and effectiveness.
Keywords
Reactive power pricing
·
Cost allocation
·
Tracing algorithm
·
Restructuring
S. Hasanpour (
B
)
·
R. Ghazi
·
M. H. JavidiDepartment of Electrical Engineering,Ferdowsi University of Mashhad, Mashhad, Iranemail: so_ha73@stumail.um.ac.ir; hasanpour_s@yahoo.com
1 Introduction
Provision of reactive power is very essential for secure andreliable operation of power systems. In vertically integratedelectricity industry, reactive power support is considered aspart of system operator’s activities and its cost which shouldbe recovered is usually calculated based on approximatemethods. Insome systems,reactive power costisincluded inthe price of active power. In some other systems, the powerfactor is used for calculation of a penalty factor for the priceto compensate the cost.In a restructured environment, in spite of the fact that thecost of reactive power may be dominantly linked with theprice of active energy as well as other services, it is considered as an ancillary service which is priced separately. Onthe other hand, it is well known that a fair pricing of such aservice can lead to market liquidity which in turn results inapproaching the optimal condition.Many investigations have been carried out for appropriate pricing of reactive power [1–10]. Some of these meth
ods utilize various search techniques such as genetic andant colony algorithms for pricing [4], others have focused
on formulating reactive power pricing [5,6]. Muchayi [7]
havepresentedasurveyonsomeofthereactivepricingalgorithms. Dona and Paredes [8] have proposed a pricing tech
nique based on minimization of the operation cost as well asthe transmission losses using decoupled OPF. Cost allocation of reactive power using modiﬁed Ybus matrix methodhas been reported by Chu and Chen [9]. Ro [10] has pre
sentedthereactivechargingschemecomposedofrecoveringcapital cost and operational cost. Pricing of real and reactive power as bundled products in synchronous machine hasbeen investigated in [11]. Rider and Paucar [12] have pro
posed a nonlinear reactive power pricing method. They havepresented the total cost of reactive power production as a
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28 Electr Eng (2009) 91:27–34
nonlinear model which is solved by modiﬁed predictorcorrector interiorpoint method. Active and reactive pricing using interior point nonlinear optimization method hasbeen demonstrated by Xie [13]. Chung et al. [14] have pre
sented a method for costbased reactive power pricing inwhich the cost of reactive power production by generatorsand capacitors are minimized. Also a methodology for calculationofcostofreactivepowerbygenerators,synchronouscondenserandstaticreactivepowersourceshasbeenreportedby Deksnys and Staniulis [15].
The cost of reactive power produced by a generator isessentially composed of two components: ﬁxed costs orinvestment costs and variable costs. Variable costs, in turn,consist of operating costs (including fuel and maintenancecost) and the opportunity cost which is imposed on the generatorresultingfromreductionofitsactivepowergeneration.Some of the abovementioned methods consider only theopportunitycost[4,14]whileothersconsideronlytheopera
tionandinvestmentcosts[5,13].Therefore,suchapproaches
lead to approximate reactive pricing techniques.Inthispaper,we have proposed anew method for reactivepowerpricinginapoolbasedpowermarket.Ourmethodutilizes the accurate relation between active and reactive powerto assign an accurate function for the cost of reactive powerproduction. Various components of reactive power cost havebeen considered in our proposed approach.While,insomeofreactivepowerpricingmethods,thecostofactivelossesisattributedonlytooneplant(theoneatslack bus) in the optimization problem [14], our method not only
considers the cost of active power losses but also considersthe cost of reactive power losses. Furthermore, the contribution of each generator in covering the active and reactivelossesisdeterminedusingtracingmethod.Toshowthecredibility of the proposed approach, it has been applied to IEEE9bus system.The presented paper is organized in ﬁve sections. Theprocedure of the proposed reactive cost allocation method isdiscussed in Sect. 2. In Sect. 3, the analysis of cost for reac
tive power support and reactive power pricing are discussed.The simulationresultsaswellastheircomparison withothermethods are presented in Sect. 4. The conclusions that canbe drawn from this paper are presented in Sect. 5.
2 Reactive support cost allocation
Conventionalcostmethodsforactiveandreactivepowersupport are based on empirical methods in which active andreactive costs of generators are deﬁned in quadratic forms asbelowCost
(
P
)
=
a
p
P
2
+
b
p
P
+
c
p
(1)Cost
(
Q
)
=
0
.
05
b
p
Q
2
(2)It should be noted that the active and reactive power generated by each generator are essentially bundled with eachother. This bundling property highly depends on the operating point of generator. Therefore, while Eq. 1 provides analmost accurate value for the cost of active power, Eq. 2results in a rough estimation for the value of the cost of reactivepower.Furthermore,Eq.2considersonlytheoperationalcost of reactive power generation and the cost for coveringthe investment for reactive power generation is not includedin this equation.To overcome inaccuracies associated with conventionalcost methods, Song, Irving and Zhau proposed a method forcost evaluation of reactive power which is based on the triangular relationship between active and reactive power [5].In this triangular approach, the cost of reactive power is formulated as below:Cost
(
Q
)
=
a
Q
2
+
b
Q
+
c
(3)where,
a
,
b
,
c
are constants depending on power factor
(
cos
θ)
and are calculated as follows:
a
=
a
p
sin
2
θ
b
=
b
p
sin
θ
(4)
c
=
c
p
This method of reactive power cost calculation is essentiallybased on the formulation for active power cost, in which theactive power is replaced by reactive power using the triangular relationship. However, as the investment for generatorsis essentially based on the optimal solution for active powergeneration, employing the same formula for the cost of reactive power will lead to calculation of wrong ﬁxed costs forreactive power. Xie [13] in another approach, have used asecondorder polynomial for the cost of reactive power inwhich a, b and c constants are approximated to be onetenthof those for the cost of active power. Furthermore, such anequation normally is valid for a special range of reactivepower production.The present paper proposes a new method for the formulation of reactive cost allocation. In the proposed approach,all the investment, operation and opportunity costs due toreactive power support are taken into account. Attempt hasbeen made to formulate the equation by a quadratic functionas below.In generator, the relationship between active and reactivepower is:
S
2
=
P
2
+
Q
2
(5)If a generator produces its maximum active power
(
P
max
)
,then its cost for generating active power equals cost
(
P
max
)
.In such a situation, no reactive power is produced and therefore,Sequals
P
max
.Reactivepowerproductionitselfdoesnotseem to impose any fuel cost on generator except for the cost
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Electr Eng (2009) 91:27–34 29
imposedforlosses.However,reactivepowerproductionbyagenerator will reduce its capability to produce active power.Hence, provision of reactive power by generator will resultin reduction of its active power production.To generate reactive power
Q
i
by generator
i
which hasbeen operating at its nominal power
(
P
max
)
, it is required toreduce its active power to
P
i
such that
P
i
=
P
2max
−
Q
2
i
P
=
P
max
−
P
i
(6)where,
P
represents the amount of active power that willbe reduced as a result of generating reactive power.To accurately calculate the cost of reactive power
Q
i
, weshould include all the costs imposed on generator as below:Cost
(
P
max
)
:
cost of producing active power equal to
P
max
in one hour.Cost
(
P
max
−
P
)
:
cost of generator when producing bothactiveandreactivepowerwiththeamounts
P
i
and
Q
i
,respectively.Cost
(
P
max
)
−
Cost
(
P
max
−
P
)
:
Reduction in the cost of active power due to compulsory reduction in active powergeneration
(
P
)
which happens due to generating reactivepower with the amount of
Q
i
.Thisrepresentsthecostofreactivepowerproductionwhilethe operating point of generator is moved from point 1 topoint 2 (Fig. 1) as below:Cost
(
P
max
)
−
Cost
(
P
max
−
P
)
=
Cost
(
Q
i
)
+
PP
max
Cost
(
P
max
)
(7)where,
PP
max
Cost
(
P
max
)
is related to the change of operatingpoint (In fact this represents the cost of
P
Mwh energywhen the generator is generating its nominal power). There
i
P
i
Q
max
P
S
P Q
Field limit
(2)
Armature limit
(1)
Fig. 1
Capability curve of generator
fore, from the above equation it can be concludedCost
(
Q
i
)
=
Cost
(
P
max
)
−
Cost
(
P
max
−
P
i
)
−
PP
max
Cost
(
P
max
)
Cost
(
Q
i
)
=
P
max
−
PP
max
Cost
(
P
max
)
−
Cost
(
P
max
−
P
i
)
(8)Now,weshouldexpressCost
(
Q
)
asafunctionof
Q
.Assuming we will use the full potential of generator capability, wemay conclude that its operating point will always be suchthat its current will be equal to its nominal value and we willbe able to write
Q
as a function of
P
(Eq. 6). Therefore, considering
Q
as variable and using Eqs. 6 and 8, its production
cost can be calculated for different values of
Q
. The results,interpolated by using Newton–Gregory polynomial, conﬁrmthat they can accurately be ﬁtted into a quadratic polynomialform as below:Cost
(
Q
)
=
a
q
Q
2
+
b
q
Q
+
c
q
(9)This equation is very simple and as it is extracted from thepower cost function of the generator, it is more realistic andcanprovideaccurateresultsinreactivepowerpricingascomparedwithconventionalempiricalapproximatemethod.Theproposed cost function, as compared with previously usedmethods, not only considers the operational cost imposed tothe system due to reactive power support, but also the opportunity cost is taken into account. Furthermore, investmentcost in this equation is accurately included.Figure 2 shows the plotted cost curves for active powerand the proposed reactive power formulation. From the ﬁgure, it can be observed that both cost curves show similarcharacteristics. However, as it should be, the reactive cost ismuch smaller than the active cost.
0 20 40 60 80 100 120 140 160 180 200050010001500200025003000350040004500
a
p
=.11 , b
p
=2 , c
p
=150
P  Q (MWMVar)
c o s t ( $ / h )
active cost curvereactive cost curve
Fig. 2
Active cost curve and proposed reactive cost curve
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30 Electr Eng (2009) 91:27–34
0 20 40 60 80 100 120 140 160 180 200050010001500200025003000350040004500
a
p
=.11 , b
p
=2 , c
p
=150
P Q (MW  MVar)
c o s t ( $ / h )
active cost curve proposed method conventional method triangular method
Fig. 3
Active and reactive power cost function in three methods
0 20 40 60 80 100 120 140 160 180 2000200400600800100012001400160018002000
a
p
=.0025 , b
p
=8.4 , c
p
=225
P  Q (MW  MVar)
c o s t ( $ / h )
active cost curve proposed method conventional method triangular method
Fig. 4
Comparison of the new method with conventional and triangular methods
In Figs. 3 and 4, the cost allocated to reactive power,
obtained by using the proposed method is compared withthose obtained using conventional and triangular methodsfor two different cases in which
a
p
,
b
p
and
c
p
parametersare different. While, for both cases, the triangular method isalmostcompatiblewithourproposedmethod,itcanbeeasilyobserved that the conventional cost method may not be reasonable(Fig.4).Thisismainlyduetothefactthatinvestmentcost is not at all included in the pricing of reactive power forthe conventional method. Therefore, depending on the values of
a
p
,
b
p
and
c
p
parameters for the generator, the resultsobtainedbytheconventionalmethodmaydiffersignificantlyfrom the actual cost of reactive power production imposedon generator.
3 Reactive power pricing
Active and reactive marginal prices are normally obtainedthroughsolvingtheoptimalpowerﬂowinwhichanobjectivefunctionsubjecttoasetofequalityandinequalityconstraintsis minimized. In this paper, we also propose a new frame forreactive power cost allocation which covers all costs associated with reactive power generation in objective function of optimization problem. In our approach, both active and reactive losses allocated to each generator are included in theproposed objective function. Therefore, it guarantees a moreaccurate and nondiscriminative pricing scheme for activeandreactivepower.Inthissection,ﬁrsttheproposedcostformulationforreactive power isusedintheproposed objectivefunction of OPF. Then, to specify the cost allocated to eachgenerator,theresultsareappliedtotracingalgorithm.Inorderto demonstrate inaccuracies of the conventional method, theobtained results are compared with the results of proposedapproach.3.1 Objective functionSo far, different objective functions have been used for OPF;the following formulations (a and b) are more common:(a) Summation of active and reactive power productioncosts,
C
total
=
N
g
i
=
1
[
Cost
(
P
G
i
)
+
Cost
(
Q
G
i
)
]
(10)where,Cost
(
P
G
i
)
:
Active power cost function of generator
i
,Cost
(
Q
G
i
)
:
Reactive power cost function of generator
i
,
N
g
:
Number of generators.While the abovementioned method considers the totalcosts of active and reactive power produced by generators, it may not be accurate, because all active andreactive losses are assumed to be generated at slack bus.Therefore,thisapproachmaynotbefair,especiallywhenthemarginalcostatslackbussignificantlydiffersfrom those in other buses.(b) Summation of active and reactive power productioncosts and the cost of active and reactive losses,
C
total
=
N
g
i
=
1
[
Cost
(
P
G
i
)
+
Cost
(
Q
G
i
)
]+
P
loss
α
+
Q
loss
β
(11)
1 3
Electr Eng (2009) 91:27–34 31
where,
P
loss
:
Active power losses,
Q
loss
:
Reactive power losses,
α
:
Price of active power losses,
β
:
Price of reactive power losses,where,
α
and
β
aremeanmarginalpricevaluesforactiveandreactive power generated by different generators. Therefore,we can write
α
=
N
g
i
=
1
λ
p
i
/
N
g
β
=
N
g
i
=
1
λ
Q
i
/
N
g
(12)where,
λ
P
i
:
Active power price in generator
i
,
λ
Q
i
:
Reactive power price in generator
i
.Whilethisapproachconsiderstheeffectofmarginalpriceof various generators, losses are accounted for twice in thisformulation.Infact,lossesaretakenintoaccountbyadditionof third and fourth terms, while they are also included in theﬁrst and second terms.To overcome the deﬁciencies in previously proposedmethods, we have proposed a new formulation for the objective function, in which all generators contribute in active andreactivepowerlosses.Toaccuratelyincludetheeffectofmarginal cost of different generator on the total cost imposedby losses, we should ﬁrst evaluate the amount of active andreactive power losses attributed to each generator. This isachieved by using a wellknown tracing algorithm. As aresult, the cost of losses assigned to each generator can befairly calculated. To do this, we have formulated the objective function as the summation of cost functions for pureconsumed active and reactive power as well as the cost functions for losses. In fact, we have formulated the total costimposed on each generator in four different terms includingcostsforactiveandreactivepowersuppliedtocustomersandcosts for active and reactive losses. Therefore, we can writethe objective function as below:
C
total
=
N
g
i
=
1
[
Cost
(
P
G
i
−
P
G
i
)
+
Cost
(
Q
G
i
−
Q
G
h
)
+
P
G
i
.λ
P
i
+
Q
G
i
.λ
Q
i
]
(13)where,
P
G
i
:
Active power losses allocated to generator
i
,
Q
G
i
:
Reactive power losses allocated to generator
i
,
P
G
i
−
P
G
i
:
Activepowerproductionbygenerator
i
withoutconsidering active losses,
Q
G
i
−
Q
G
i
:
Reactive power production by generator
i
without considering reactive losses.In the above formulation,
P
G
i
and
Q
G
i
are calculatedusing a tracing based loss allocation algorithm [16].
3.2 ConstraintsThe constraints, considered in this problem, are the standardset of equality and inequality constraints which are normallyconsidered in OPF. In fact, the set of equality constraintsrepresent the standard power ﬂow equations for active andreactive power and the set of inequality constraints representthe physical and security limits of the system as below:
N
g
i
=
1
P
G
i
−
N
i
=
1
P
D
i
−
P
loss
=
0
N
g
i
=
1
Q
G
i
−
N
i
=
1
Q
D
i
−
Q
loss
=
0where,
P
loss
=
N
i
=
1

V
i

V
j

Y
ij

cos
(θ
ij
+
δ
j
−
δ
i
)
Q
loss
=
N
i
=
1

V
i

V
j

Y
ij

sin
(θ
ij
+
δ
j
−
δ
i
)
and,
P
G
min
≤
P
G
i
≤
P
G
max
Q
G
min
≤
Q
G
i
≤
Q
G
max
(14)
P
2
G
i
+
Q
2
G
i
≤
S
G
i
,
max
i
=
1
,...,
N
g

V
i

min
≤ 
V
i
 ≤ 
V
i

max
i
=
1
,...,
N
Intheaboveformulasdifferentvariablesaredeﬁnedasbelow:
N
: Number of buses of the network
P
G
i
,
Q
G
i
:
Supply of active and reactive power in
i
th bus
P
D
i
,
Q
D
i
:
Active and reactive demand in
i
th bus
S
G
i
,
max
:
Maximum apparent power in bus
iV
i
= 
V
i

δ
i
:
Voltage phasor in bus
iY
ij
θ
ij
:
The
ij
th element of admittance matrix
4 Case study
To investigate the validity of the proposed algorithm, it hasbeen applied to IEEE 9bus system (Fig. 5) with a typicaldailyloadasshowninFig.6.Tables1and2showtheparam
eters of this system.To be able to make an analytical comparison betweenthe proposed method and previous algorithms, two different
1 3