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978-1-4244-8542-0/10/$26.00 ©2010 IEEE

Abstract— In this paper, the nodal price for real and reactive power considering different reactive power price cost calculation methods for generator reactive power has been presented. The impact of FACTS controllers has also been considered taking their cost function into account. The comparison of fuel costs, reactive power cost, and cost of FACTS devices, and their impact on nodal price determination has been presented for different cases with reactive power cost models for determining reactive cost component for generators. The impact of SVC has also been considered on nodal price of real and reactive power. Mixed integer non-linear programming approach has been formulated for solution. The proposed approach has been tested on IEEE 24-bus Reliability Test System (RTS) for hybrid market model.

I. INTRODUCTION RANSMISSION price of real and reactive power in competitive electricity markets are among the most

important issues for better and economic operation of system. With the growing interest in determining the costs of ancillary services needed to maintain the quality of supply, the spot price for reactive power has also gained great importance. Spot pricing based on short run marginal cost (SRMC) has the potential to provide the economic signals for the power system operation from economic point of view. Various authors have proposed models and approaches for determining spot pricing [1-17]. Baughman and Siddiqi developed the model introducing reactive power pricing and revealed that the Lagrangian multipliers corresponding to node power balance equations in OPF represent the marginal costs of the node power injections [1]. Many investigations have been carried out for appropriate pricing of reactive power [10-16]. Recently Paucar and Rier [15-16] have presented approches for spot price determination of reactive power. The role of FACTS controllers on transmission pricing have been presented by many authors [17-23]. Recently Shrestha and Feng presented simulation studies on the effects of TCSC on the spot price of real and reactive power using heuristic method to determine the location of TCSC [19].

In a deregulated environment, the number of bilateral transactions has grown rapidly. It is required to help the system operator to evaluate their impacts on the system

A. Kumar is with the Department of Electrical Engineering, NIT Kurukshetra, INDIA(e-mail: ashwa_ks@yahoo.co.in).

P. Kumar is with the Department of Electrical Engineering, NIT Kurukshetra, INDIA and completed his masters in Power Systems. (e-mail: punit_pce007@yahoo.com).

operation. Bilateral transactions between sellers and buyers are deemed to be feasible, if these can be accommodated without the violations of system security limits [24]. References [25,26] discussed the secured bilateral transaction matrix determination in deregulated environment utilizing the approach of [27]. However, the secured transaction matrix was determined for the markets with only bilateral contracts.

In this paper, impact on nodal prices have been determined considering three different reactive cost model for generators’ reactive power cost calculations for hybrid electrcity markets where pool and bilateral transactions are occuring simultaneously. The impact of Static Var Compensator (SVC) and Unified Power Flow Controller (UPFC) has been incorporated with their cost model. GAMS and MATLAB interfacing has been utilized to model the system with non-linear equations considering DICOPT model in GAMS [28-29]. The comparison has been given for different reactive cost models for IEEE 24-bus reliability test system [30].

II. A BILATERAL TRANSACTION MATRIX-BETWEEN GENCOS AND DISCOS-T

The conceptual model of bilateral dispatch is that sellers and buyers enter in to transactions where the quantities traded and the trade prices are at the discretion of these parties and not a matter of ISO. These transactions are then brought to the ISO with a request that transmission facilities for the relevant amount of power be provided. If there is no violation of static and dynamic security, the ISO simply dispatches all requested transactions and charges for the service.

The bilateral concept can be generalized to the multi-node case where the seller, for example a generation company called Gencos, may inject power at several nodes and the buyer bus called Discos also draw load at several nodes. Unlike pool dispatch, there will be a transaction power balance in that the aggregate injection equals the aggregate draw off for each transaction.The bilateral contract model used in this paper is basically a subset of the full transaction matrix proposed in [6]. In its general form, the transaction matrix T as shown in (1) is a collection of all possible transactions between Generation (G), Demand (D), and any other trading Entities (E) such as the marketers and the brokers.

Nodal Pricing with Different Reactive Power Cost Models in Hybrid Electricity Markets

Ashwani Kumar, Member, IEEE, and Punit Kumar

T

2

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

EEEDEGDEDDDGGEGDGG

T (1)

In present paper, all transactions are assumed between the suppliers (G) and consumers (D). It is also noted that the diagonal block matrices (GG and DD) are zero because it is assumed that there are no contracts made between two suppliers or two consumers. Neglecting transmission losses, the transaction matrix can be simplified as:

[ ] [ ]TDGGDT =≡ (2) Each element of T, namely tij, represents a bilateral

contract between a supplier (Pgi) of row i with a consumer (Pdj) of column j. Furthermore, the sum of row i represents the total power produced by generator i and the sum of column j represents the total power consumed at load j.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

≡

ndngng

nd

nd

tttttt

T

,1,

,21,2

,11,1

.........

(3)

where: ng = number of generators, and nd = number of loads.

In general, the conventional load flow variables, generation (Pg) and load (Pd) vectors, are now expanded into two dimensional transaction matrix T as given in (4).

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡

d

gT

g

d

u

u

TT

PP

00

(4)

Vector ug and ud are column vectors of ones with the

dimensions of ng and nd respectively. There are some intrinsic properties associated with this transaction matrix T [25]. These are column rule, row rule, range rule, and flow rule. These properties have been explained in [25]. Each contract has to range from zero to a maximum allowable value, tij

max. This maximum value is bounded by the value of corresponding Pgi

max or Pdj whichever is smaller. The range rule satisfies:

( )djgiijij PPtt ,min0 maxmax ≤≤≤ (5)

It is also possible for some contracts to be firm so that tij0

is equal to tijmax [25]. According to flow rule the line flows

of the network in AC model can be expressed as follows: [ ]dgline PPDFP −= (6)

The matrix DF is the distribution factors matrix [30]. If the representations of the Pg and Pd are substituted by using the definition of T as given in (4), the line flows can be expressed in an alternative as follows:

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

1

1T

line TTDFP (7)

),min( ,max

,max

bbDBsbGBsb PPGD = (8)

The values of the bilateral transactions considered in addition to the pool transactions are taken as:

Value of transaction between generator and load bus (p.u) T(1,1)=0.5 T(1,2)=0.3 T(1,3)=0.3 T(1,15)=0.1 T(1,18)=0.4

T(2,10)=0.2 T(2,13)=0.3 T(2,15)=0.4 T(2,18)=0.5 T(2,19)=0.2

T(7,9)=0.2 T(7,10)=0.2 T(7,13)=0.4 T(7,15)=0.5 T(7,18)=0.0

T(13,18)=1.5

III. COST MODEL OF FACTS DEVICES There are three basic kinds of FACTS devices. One kind

can be characterized as injection of current in shunt, the second kind as injection of voltage in series with the line and the third kind are a combination of current injection in shunt and voltage injection in series. Static model of FACTS devices viz. SVC has been well explained in [22-23].

A. Cost model of FACTS devices According to [21], the cost functions for SVC and TCSC

are taken as follows: SVC:

( ) 38.1273051.00003.0 2 +−= SSFCost $/KVAR (9) UPFC:

( ) 22.1882691.00003.0 2 +−= SSFCost $/KVAR (10) S is the operating range of the FACTS devices in MVar.

The unit for generation cost is US$/Hour and for the investment costs of FACTS devices are US$. They must be unified into US$/Hour. Normally, the FACTS devices will be in-service for many years. However, only a part of its lifetime is employed to regulate the power flow. In this paper, five years is applied to evaluate the cost function. Therefore the average value of the investment costs are calculated using the following equation:

( ) ( )5*87601

fcfc = $/hr (11)

where c(f) is the total investment costs of FACTS devices.

IV. ACTIVE AND REACTIVE NODAL PRICE DETERMINATION FOR HYBRID ELECTRICITY MARKET MODEL

The real and reactive power nodal prices, fuel cost, cost components of reactive power with different FACTS devices have been obtained solving an optimization problem of minimizing total cost subject to equality and inequality constraints for a pool model based electricity market. The objective function can be represented as:

(a) Objective function ( ) ( ) ( )iii FCostQCostPCost ++∑ (12)

The objective function consist three cost components as cost of real power, cost of reactive power, and cost of FACTS devices. These can be represented as:

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Where i ε NG (no. of generator) Cost(Pi) = Cost function of real power Cost(Qi) = Cost function of reactive power Cost(Fi) = Cost function of FACTS devices. Where:

pipipi cPbPaPCost ++= 2)( $/h (13)

The cost of reactive power produced by a generator is essentially composed of two components: fixed costs or investment costs and variable costs. Variable cost in turn consists of operating costs (including fuel and maintenance cost) and the opportunity cost which is imposed on the generator resulting from reduction of its active power generation. Three methods have been considered to evaluate the cost of reactive power of generators.

A. Method-1: Triangular approach [20] This method of reactive power cost calculation is

essentially based on the formulation for active power cost, in which the active power is replaced by reactive power using the triangular relationship.

( ) cQbQaQCost ′′+′′+′′= 2 $/hr (14) where, a’’, b’’, c’’ are constants depending on power factor (cos θ) and are calculated as follows from power triangle. a’’ = ap sin2 θ b’’ = bp sin θ c’’ = cp

B. Method-2: Maximum real power based approach If a generator produces its maximum active power (Pmax),

then its cost for generating active power equals cost (Pmax). In such a situation, no reactive power is produced and therefore, S equals Pmax. Reactive power production by a generator will reduce its capability to produce active power. Hence, provision of reactive power by generator will result in reduction of its active power production. To generate reactive power Qi by generator i which has been operating at its nominal power (Pmax), it is required to reduce its active power to Pi such that:

iii PPPQPP −=Δ−= max22

max , (15) ΔP represents the amount of active power that will be

reduced as a result of generating reactive power. To accurately calculate the cost of reactive power Qi, we should include all the costs imposed on generator as below: Cost(Pmax) : cost of producing active power equal to Pmax in one hour. Cost(Pmax −ΔP) : cost of generator when producing both active and reactive power with the amounts Pi and Qi , respectively. Cost(Pmax) − Cost(Pmax −ΔP) : Reduction in the cost of active power due to compulsory reduction in active power generation (ΔP) which happens due to generating reactive power with the amount of Qi . This represents the cost of reactive power production while the operating point of generator is moved from point 1 to point 2 as below:

( ) ( ) ( )ii PPCostPCostP

PPQCost −−Δ−= maxmaxmax

max $/hr (16)

C. Method-3: Maximum apparent power based approach The reactive power production cost of generator is called

opportunity cost reactive power output may reduce active power output capacity of generators which can at least serve as spinning reserve, therefore causes implicit financial loss to generators. Actually, opportunity cost depends on the real-time balance between demand and supply in the market, so it is difficult to determine the real value. For simplicity, we consider the opportunity cost approximately as:

( ) ( ) ( ) kQSCostSCostQCostGiGGGi *22

max max−−= ($/h)(17)

where SGi,max is the nominal apparent power of the generator at bus i; QGi is the reactive power output of the generator at bus i; K is the profit rate of active power generation, usually between 5% and 10%. (b) Equality constraints: Power flow equation The real and reactive power flow equations from bus-i to bus-j can be written as:

( )ijijijijjiijiij BGVVGVP δδ sincos2 +−= (18)

( ) ( )ijijijijjishijiij BGVVBBVQ δδ cossin2 −−+−= (19)

The real and reactive power flow equations from bus-j to bus-i can be written as:

( )ijijijijjiijjji BGVVGVP δδ sincos2 −−=

(20)

( ) ( )ijijijijjishijjji BGVVBBVQ δδ cossin2 +++−=

(21)(c) Power Injection at buses

Subject to following set of constraints

idigi PPP =− NBi ∈∀ (22)

idiSVCiSVCigi QQQQ =−+ *ξ NBi ∈∀ (23)

∑−∑=∈∈ TBiij

jiFBiij

iji PPP NBi ∈∀ (24)

∑−∑=∈∈ TBiij

jiFBiij

iji QQQ NBi ∈∀ (25)

(d) Power generating limits maxmin gigigi PPP ≤≤ (26)

maxmin gigigi QQQ ≤≤ (27)

(e) Power Balance equation

01

=−−∑=

lossdi

Ng

igi PPP (28)

01

=−−∑=

lossdi

Ng

igi QQQ (29)

(e) Transmission limits maxmin jijij PiPP ≤≤ (30)

(f) Voltage limits maxmin iii VVV ≤≤ (31)

4

(g) Phase angle limits maxmin iii δδδ ≤≤ (32)

(h) Reactive Power Capability Curves limit ( )222

atgg IVQP ≤+ (33)

(i) Equality constraints for bilateral transactions ∑

∈

=sbi

ijDB GDP (34)

∑∈

=bbj

ijGB GDP (35)

GPGB PPPg += (36)

DPDB PPPd += (37)

)( DBGBDF PPPfb −= (38)

)( DPGPDF PPPfp −= (39)

fpfbf PPP += (40)

),min( ,max

,max

bbDBsbGBsb PPGD = (41)

(j) Limit on FACTS controller UPFC: maxmax *.*. TTT φφφ uu ≤≤− (42)

maxmax *.*. qqq III uu ≤≤− (43) max.0 TT VuV ∗≤≤ (44)

u is the vector of binary variable (‘0’s and ‘1’s) representing the presence or absence of UPFC with ‘1’s represent presence and ‘0’s represent absence of FACTS devices. UPFC can be incorporated in the model replacing the power flow equations with UPFC.

More than one FACTS controller can be incorporated in the problem using the constraint on number of FACTS devices and can be represented as: However, in the present work, impact of only one FACTS controller has been studied. (k) Limit on FACTS controller

maxFACTSi

FACTSiFACTSi N≤∑ξ (45)

Where GD = bilateral transaction matrix PDB = vector of bilateral demand PDP = vector of pool demand PGB = vector of bilateral generation PGP = vector of pool generation s and b are the sets of the seller and buyer buses, respectively.

The optimization problem is solved using the GAMS 21.3 / DICOPT solver [28-29] and utilizing interfacing with MATLAB 6.5.

V. RESULTS AND DISCUSSIONS

A. Results for IEEE-24 Bus Test System The results have been determined for IEEE-24 bus system

for a pool model. The results have been determined without

and with FACTS devices considering the different methods for the incorporation of reactive power cost model. The results have been obtained for different cases as follows- Case 1: Results without FACTS devices for all methods Case 2: Results with FACTS device (SVC) for all methods Case 3: Results with FACTS device (UPFC) for all methods (Case-1): The results of the marginal cost for real and reactive power for Case 1 using different methods of reactive power model is shown in Figs. 1 to 3.

Fig. 1. Marginal Cost of real and reactive power without any FACTS device (Method-1)

Fig. 2. Marginal Cost of real and reactive power without any FACTS device (Method-2)

Fig. 3. Marginal Cost of real and reactive power without any FACTS device (Method-3)

It is observed from Fig. 1 to Fig. 3 that the marginal cost of real power is minimum at bus 22 is 34.86 $/MWh. At each bus the marginal cost of real power is different with a maximum value 52.49 $/MWh. It can be observed that the marginal cost of reactive is both positive and negative. The negative reactive power marginal cost represents that the Lagrange multiplier is negative at that bus. It is observed that marginal cost of real power is minimum using Method-

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3. As given in Table I, cost component of generators’ reactive power is minimum using method 2.

TABLE I

RESULTS FOR IEEE-24 BUS SYSTEM FOR CASE-1

Method-1 Method -2 Method -3

Fuel Cost (US $/h) 118771.58 120357.76 118438.99 Q cost (US $/h) 5324.99 663.62 1582.31

Cost of FACTS (US $/h) 0.0 0.0 0.0 Total cost (US $/h) 124096.57 121021.38 120021.30

(Case-2): The results of the marginal cost for real and reactive power for Case 2 is shown in Fig. 4 to 6.

Fig. 4. Marginal Cost of real and reactive power with SVC (Method-1)

Fig. 5. Marginal Cost of real and reactive power with SVC (Method-2)

Fig. 6: Marginal Cost of real and reactive power with SVC (Method-3)

It is observed from Fig. 4 to 6 that the marginal cost of

real power is minimum at bus 22 is 34.74 $/MWh. At each

bus the marginal cost of real power is different with a maximum value 53.85 $/MWh. It can be observed that the marginal cost of reactive is both positive and negative. It is observed that marginal cost of reactive power is almost constant because SVC is a source of reactive power. In this study SVC is optimally connected at bus 9. The minimum cost is obtained for method 3 as shown in Table II. SVC is optimally connected at bus 3. Cost component for SVC is obtained minimum for method 1 as reactive power support is obtained minimum for this case. The cost comparison for different methods of reactive power cost component calculation for generators’ reactive support with SVC is given in Fig. 7. It is found that without and with SVC cost is found minimum for method 3. The total fuel cost slightly increases with SVC compared to without case of SVC as cost of SVC is also added to the total fuel cost due to cost component of SVC.

TABLE II RESULTS FOR IEEE-24 BUS SYSTEM (CASE-2)

Method-1 Method -2 Method -3

Fuel cost (US $/h) 118847.65 120338.186 118442.76

Q-cost (US $/h) 5430.03 650.405 408.84 Cost of SVC (US $/h) 1.452 3.666 5.785

Total cost (US $/h) 124279.14 120992.26 118857.38

QSVC (MVAR) 0.5 1.265 2.0

Fig. 7. Comparison of costs for different methods without and with SVC Case (3) The results of the marginal cost for real and reactive power with UPFC is shown in Figs. 8 to 10.

Fig. 8. Marginal Cost of real and reactive power with UPFC (Method-1)

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Fig. 9. Marginal Cost of real and reactive power with UPFC (Method-2)

Fig. 10. Marginal Cost of real and reactive power with UPFC (Method-3)

UPFC is optimally connected between bus 3 and 24 and the maximum injection voltage is 0.5 p.u., maximum current injection is 1.0 p.u. and maximum limit of phase angle is п. In this case also, we are getting most negative reactive power marginal cost at bus no 6, while the reactive power marginal cost is almost same at all the other buses. It is observed from Fig. 8 to 10 that the marginal cost of real power is found minimum at bus 22 and highest at bus 1. With method 2, the marginal prices are found higher compared to other methods. At each bus, the marginal cost of real power is different with a maximum value at bus 1. In comparison to case without FACTS devices, the maximum value of real power marginal cost is found to increase at all buses due to the cost of UPFC.

TABLE III

RESULTS FOR IEEE-24 BUS SYSTEM WITH UPFC

Method-1 Method -2 Method -3

Fuel cost (US $/h) 114617.244 114670.600 113216.570

Q-cost (US $/h) 6837.512 369.590 100.407 Cost of UPFC (US $/h) 15.761 34.386 19.514 Total cost (US $/h) 121,470.52 115074.576 113,336.491 QUPFC (MVAR) 3.651 8.099 4.573 The above table represents the total Fuel Cost, the reactive

power cost, and cost of FACTS Device (UPFC) with all the

three methods of reactive power cost model. We can observe from the table that the fuel cost is found to be minimum for Method-3. Cost component for FACTS device UPFC is found lowest for Method 1 as reactive power supplied by UPFC is lower for method 1.With UPFC, total cost is found to be more compared to costs obtained without FACTS devices for all the methods. The comparison of marginal prices for all methods for bilateral model is given in Table II.

VI. CONCLUSIONS In this paper work, nodal price for real and reactive power

have been obtained including the reactive power cost model for generators’ reactive support with three different approaches. FACTS devices viz. SVC and UPFC with its cost model have been incorporated to find their impact on real and reactive power management and its nodal price at each node. The total cost have been obtained minimum with method 3. The cost component of reactive power is obtained minimum using method 3. It is observed that reactive power cost component is important to consider for nodal price determination and overall cost computation. The FACTS devices can be remunerated based on the cost component obtained for their reactive support in the system.

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