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A new market clearing mechanism, based on comprehensive welfareallocation, considering participants’ optimality, ef ciency, and

extent of transmission use

Mohammadreza Baghayipour and Asghar Akbari Foroud*,†

Semnan University, Semnan, Iran

SUMMARY

This paper develops a new approach to fairly clear the market and assess the amount of each participant ’srevenue in a deregulated power system, through modifying the system nodal prices. This approach equalizes

the economic pro

t of each participant group to its fair and rational value. Transco behavior is modeled asan independent entity without direct supervision of an independent system operator, causing the possibilityof either making prot or incurring a loss for it. With this method applied, three main results for better system design and operation are produced:

(1) The rational and acceptable prot/loss values for all participant groups (producers, customers,

and Transco) are determined through fair apportionment of total system prot among them

based on their optimality and contribution of protability to the whole system.

(2) The prot/loss value for each of producers and consumers in the system is allocated according

to their proportional ef ciency and extent of transmission use.

(3) Transmission revenue assessment is performed through dening a criterion for evaluating the

transmission network optimality.

So this approach can correct wrong and unfair economic signals in the previous electricity pricingand participants’ revenue assessment methods, presenting such a fairer mechanism for it that each

participant ’s action will be oriented in maximizing the system

’s total economic ef

ciency. Copyright © 2012 John Wiley & Sons, Ltd.

key words: market clearing mechanism; social welfare allocation; revenue assessment; referencetransmission network; nodal prices

1. INTRODUCTION

In recent years, deregulation has resulted in some major changes in the economic behavior of the electricity

industry. These changes should be directed, such that the whole system act improves to attain the optimality

as well as nondiscriminatory use for all participants, namely producers, consumers, and a monopolistic

transmission company (Transco). In these systems, the private ownership manners of these entities and

their relevancy have special importance. These relations are usually controlled via an independent system operator (ISO). Here, three basic questions are posed:

(1) What are the rational revenue value and its evaluation mechanism for each participant group in

the system?

(2) How are the answered revenues between the constituents of each participant group in the system

(such as each of producers or consumers) shared, such that it is fair and nondiscriminatory from

their viewpoints?

*Correspondence to: Asghar Akbari Foroud, Semnan University, Semnan, Iran.†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

INTERNATIONAL TRANSACTIONS ON ELECTRICAL ENERGY SYSTEMS Int. Trans. Electr. Energ. Syst. 2013; 23:1335–1364Published online 21 June 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.1661


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(3) What is the appropriate manner and purpose of generation and transmission construction/expand-

ing, so that it reconciles all participants and improves the whole system operation to maximize its

economic ef ciency?

The answers to the rst and second questions are involved in the market clearing mechanism, and the

third broaches the generation expansion planning (GEP) and transmission expansion planning (TEP)

issues. It is clear that the answer to this question is directly affected by the answers from the previous

two queries. This is because, in the case of removing the ISO ’s force on each participant action, the

way of its revenue evaluation and recovery directly inuences its behavior. Hence, it is necessary to nd

a rational and fair mechanism for market clearing before broaching GEP and TEP issues.

The market clearing mechanism fundamentally includes two steps. The rst is to determine the

generation and consumption levels through solving the optimal power ow problem, with the purpose

of social welfare maximization (or total operational cost minimization). In fact, this step guarantees the

system economic ef ciency maximization during its operation and is completely rational. The second

step includes the determination of revenue and payment values for all participants. To this end, most of

previous papers have dened the locational marginal prices (LMPs) as the system nodal prices [1].

This way, which is usually affected by congestion, causes the total consumers’ payment to exceed total

generation revenue. As a result, the papers often have equated Transco ’s revenue with the resulted

network congestion surplus. That methodology has some drawbacks from the viewpoints of both

Transco and the other participants. However, most of the previous papers have discussed it from Transco’s point of view (summarized as follows). Moreover, the LMP-based pricing scheme will be

briey studied and criticized from the viewpoint of the other participants in Section 2.

(1) In the case that Transco’s act is optional and independent from ISO, value-based TEP is performed

with the purpose of congestion rent maximization, which consequently leads to the transmission

capacity withholding and more congestion in transmission lines, against the other participants’ inter-

ests and the system economic ef ciency [1].

(2) Alternatively, if Transco’s act is directly under ISO’s supervision, it must execute ISO’s approved

optimal transmission expansion plans. In this case, Transco earns from the network congestion rent,

exactly the same revenue as the variable part (lines capacity-dependent part) of the transmission

investment cost. However, the xed part (independent of lines capacity) of the aforementioned costs

is not recovered [1,2]. Hence, various approaches, reviewed in [3–5], have been presented to recover

the remaining xed part of the transmission costs and allocate it to the users. The postage stamp,contract pass, and megawatt-mile / megavoltampere-mile derived methods [1,3,4,6,7], transmission

capacity withholding [1,8], transmission lines’ extent-of-use evaluation-based methods [4,6,9–17],

and the evaluation of the participants’ prot/loss changes, due to the changes in the network topology

or line capacities [5,18,19] are in such group. Additionally, several papers [20–22] have used the nodal

prices or lines marginal capacity prices1 modication to equalize the congestion surplus to the total

required network revenue. Unlike the predecessors, those methods can perfectly recover both

xed and variable parts of the transmission investment costs through the modied congestion rent.

Nevertheless, all of the aforementioned methods have the principal dif culty that they recover only

the required net transmission costs as Transco’s revenue but not considering any rational prot or loss

value dependent on its optimality of act, as it is inherently incompatible with the direct supervision of

ISO. In this way, Transco will not have any incentive for ef cient transmission network constructing/

expanding (a detailed consideration of [20] and its additional drawbacks will be presented in Section 3).Rational transmission revenue assessment needs a benchmark for transmission optimality

evaluation. This benchmark should include the act of the united network, incorporating all its

lines, because evaluating the optimality of each line irrespective of the others is not so rational.

In this way, [1,23] have found the optimal transmission line capacities with the same network

topology, generations, and demands as the real one and named it the reference transmission

network (RTN). That is the network with minimum amount of total operational costs and

transmission investment costs (or long-run social welfare). Moreover, [24] denes the optimal

1Hourly operational cost reduction resulted for 1-MW increment in the capacity of each line [1].

M. BAGHAYIPOUR AND A. A. FOROUD1336

Copyright © 2012 John Wiley & Sons, Ltd. Int. Trans. Electr. Energ. Syst. 2013; 23:1335–1364

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transmission expansion plan as RTN. Nevertheless, they all neither evaluate the current network

optimality by using the resulted RTN nor determine the rational transmission revenue, on the ba-

sis of this benchmark.

The rest of this paper is organized as follows. First, Section 2 briey reviews the LMP-based pricing

methodology and its main drawbacks. Then, Section 3 formulates the participants’ revenue and payment

assessment, on the basis of old LMPs, and Section 4 reviews the approach presented by [20]. Afterwards,

in Section 5, an ef cient approach for optimally clearing the power market and correcting the previouslymentioned weaknesses is presented. This approach denes RTN as a benchmark and introduces the total

social welfare as an ef cient criterion for evaluating the present transmission network quality as compared

with RTN. With this criterion used, the participant groups’ rational revenues, including their net costs as

well as some excess prot/loss, are determined, so that it makes correct economic signals for optimizing

their performance. Finally, the determined revenues are fairly shared among the participant individuals

through modifying the system nodal prices, based on their optimality of act, ef ciency, and extent of trans-

mission use. The proposed method is applied on IEEE RTS 30-bus system in Section 6, and the numerical

results are analyzed. Finally, Section 7 concludes the paper.

2. A VIEW OF LOCATIONAL MARGINAL PRICE-BASED PRICING SCHEME FROM THE

VIEWPOINTS OF PRODUCERS AND CONSUMERS

Here, the main drawbacks of the old LMP-based pricing methodology from the viewpoints

of producers and consumers are studied. To this end, the variations of LMPs, as well as the

participants’ costs, revenues, and prots, versus different transmission capacities are analyzed

in Figure 1(A) for a simple two-bus example, with one producer and one consumer on each

bus. In this gure, the horizontal axis contains the variations of the local generations, and two

vertical axes represent the resulted LMPs at two buses, following either of marginal cost curves.

It is supposed that the loads on the rst and second buses are equal to q1,d and q2,d , respectively

(i.e., OF and O0F lengths), and that the dashed line ZY represents per-megawatt demand worth.

Let us start following the curves from point F, which represents the value of zero for the

transmission line capacity. In this case, the local generations are equal to the local demands,

and the LMPs at the rst and second buses are denoted by points E and L, respectively. As

the line capacity increases, the operating point moves to the right, and the generation on therst bus increases, whereas the second decreases. Finally, point R represents the case with

unconstrained economic power dispatch between two buses. In this case, both of the buses have

the same LMPs, denoted by point N.

The values of producers’ and consumers’ revenues, costs, and prots in each of the aforementioned

cases are observable from the areas of the corresponding polygons in Figure 1(A). The resultant areas

are summarized in Table I. Accordingly, the main drawbacks of the LMP-based pricing mechanism

can be categorized into four principal topics.

2.1. Unfair determination of pro t/loss values for some producers and consumers

Because of the inherent characteristics of power generation units, the generation cost function is

mostly modeled as a nonlinear parabolic function. The LMP on a bus with marginal generator isthus a linear function of its generation level. Therefore, the electricity price unlike many other

commodities will increase as its generation increases, causing the generation revenue to be

usually different from the net generation cost with a generation-dependent prot, regardless of

the producers’ ef ciencies. As illustrated in Figure 1 and Table I, if two generators supply the

same local demands without any power exchange, their nodal prices will equal their marginal

costs, and the producer with less ef ciency (more slope of marginal cost curve) earns more prot.

In the case of power exchange between two buses, this drawback can slightly be solved, because

the transmission line with unlimited capacity provides the unconstrained economic dispatch

applicability, more generation, and subsequently more prot for the more ef cient producer.

However, because of the transmission investment cost, the aforementioned case is not

A NEW MARKET CLEARING MECHANISM 1337


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economically reasonable. On the other hand, the line with the optimal capacity does not com-

pletely realize the unconstrained economic dispatch, and the less ef cient producer still may earn

the same or even more prot.

2.2. No fair and use-dependent allocation of the transmission payment among the participants

The LMP-based pricing mechanism does not guarantee fair and use-dependent allocation of the

transmission payment among the participants. For instance, if the rst producer sells a considerable

rate of electricity to the consumer on the second bus, the power price on the rst bus will increase,

and subsequently its consumers will sustain a loss, while they not only take no advantage from the

transmission line but also benet the system by reducing the line congestion. This problem can be

restated somewhat for the producers on the large demand buses, connected to a transmission line. In

this case, the line reduces the producers’ prots, and they decrease the line congestion, and so they

should not take part in transmission payment. Nevertheless, because their prot decrement is due to

their relatively less generation ef ciencies, their prots should still be less than the others’ (with more

Figure 1. (A) The variations of locational marginal prices, revenues, payments, and prots for producersand consumers in a two-bus system, when the line capacity grows and (B) the corresponding curves of

the aforementioned values versus different capacities of the line.



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T a b l e I . T h e p r o d u c e r s ’

a n d

c o n s u m e r s ’ e c o n o m i c m e a s u r e s f o r t h r e e d i f f e r e n t t r a n s m i s s i o n c a p a c i t i e s , a s

t h e a r e a s o f t h e c o r r e s p o n d i n g p o l y g o n s i n F i g u r e 1 ( A ) .

L i n e c a p a c i t y

0

T L i

U . E . D

o w

L i n e c a p a c i t y

0

T L i

U . E . D

o w

F i r s t p r o d u c e r ’ s r e v e n u e

O B

E F

O C G H

O D N R

S e c o n

d p r o d u c e r ’ s r e v e n u e

O 0

F L X

O 0

H M W

O 0

R N V

F i r s t p r o d u c e r ’ s c o s t

O A

E F

O A G H

O A N R

S e c o n

d p r o d u c e r ’ s c o s t

O 0

F L U

O 0

H M U

O 0

R N U

F i r s t p r o d u c e r ’ s p r o t

A B

E

A C G

A N D

S e c o n

d p r o d u c e r ’ s p r o t

U L X

U M W

U N

V

F i r s t c o n s u m e r ’

s r e v e n u e

O F

S Z

O F S Z

O F S Z

S e c o n

d c o n s u m e r ’

s r e v e n u e

O 0

F S Y

O 0

F S Y

O 0

F S Y


s p a y m e n t

O B

E F

O C I F

O D J F

S e c o n


s p a y m e n t

O 0

F L X

O 0

W K F = O 0

H M W + F G H + G M K I

O 0

V J F = O 0

R N V + R F J N


s p r o t

B E

S Z

Z C I S

Z D J S

S e c o n


s p r o t

S L X Y

W

K S Y

V J S Y

S y s t e m

c o n g e s t i o n s u r p l u s

0

G M K I

0

U . E . D . , U n c o n s t r a i n e d E c o n o m i c D i s p a t c h .



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ef ciency). In addition, another point that is observable from Figure 1(A) and Table I is that the system

congestion surplus is fully recovered via the second consumer, as the main transmission beneciary,

whereas the rst producer also takes advantage from the line, but it does not participate in its payment.

2.3. Improper pro le for the bulk generation pro t, as it directs the producers’ aggregate interests

against the system ef ciency

Figure 1(B) displays the variation curves of the quantities in Figure 1(A) versus different capacities of

the line. As depicted in this gure, although the economic power dispatch is perfectly compatible with

the bulk consumers’ interests, it does not satisfy the bulk producers’ desires, as their total revenue and

prot generally have descending proles as the line capacity increases. On the other hand, the line

optimum capacity is typically near its unconstraint ow [1]. Hence, although the different producers’

interests can be in diverse directions, their resultant bulk interest usually opposes the transmission

optimum capacity.

2.4. The producers seldom sustain a loss, but the other participant groups may lose in many cases

With the old LMPs used, there are several cases that the consumers incur a loss from power

purchase. For example, if in Figure 1(B), the demand worth becomes slightly fewer than the

value of dashed line ZY, it causes a loss for consumers. Moreover, the transmission payment

based on the network congestion surplus causes a loss for Transco in many cases. In contrast,

producers only may incur a loss in one rare case, depicted in Figure 2. In this gure, LMP on

the rst bus follows line FD as long as its generation remains in allowable limit. But when its

generation ef ciency is so little that its optimal generation level decreases to the minimum and

if another producer can supply the unmet load on that bus with a lower price, LMP on the

bus will follow the line DC and declines to l1, below the rst generator minimum marginal price.

In this case, the rst producer ’s prot and generation cost are equal to the areas of rectangle

OBME and trapezoid OADE, respectively. Hence, the producer incurs a loss only when the trapezoid area

exceeds the rectangle area, that is, the area of triangle CMD exceeds the area of ABC.

3. LOCATIONAL MARGINAL PRICE-BASED REVENUE AND PAYMENT ASSESSMENTFORMULATION

Like in [20], the direct current optimal power ow (DCOPF) problem is recalled here, considering

given inelastic load vector QDx and the predened unit commitment schedule, in order to nd the

optimal generation levels as well as nodal voltage angles for each hour of the system operation

lifetime, with the goal of minimizing the total generation cost, subjected to the network constraints.

Figure 2. Variation of prot and loss values for a producer ’s maximum and minimum generation levels.



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The problem is expressed as in Equations (2)–(5), using the generation cost vector formulated as in

Equation (1). The two matrices GIM and CIM are mathematically dened according to Equations

(6) and (7), with symbols gimn,g and cimn,d denoting the elements of row n–column g and row n–col-

umn d of the matrices GIM and CIM , respectively.

GC QGð Þ ¼ a QG þ b QG^ 2 (1)

minQG;dX

GC QGð Þð Þn o

(2)

Subject to

GIM QG CIM QD ¼ P dð Þ ¼ B d; lð Þ (3)

QLowGx ≤QG≤QUpGx; s L;s Uð Þ (4)

t x≤H d≤t x; g L;g Uð Þ (5)

GIM ¼ gimn;g

¼ 1: if generation unit g connects to bus n

0: otherwise

;

n ¼ 1; 2; . . . ; N Bg ¼ 1; 2; . . . ; N G

(6)

CIM ¼ cimn;d

¼ 1: if demand d connects to bus n

0: otherwise

;

n ¼ 1; 2; . . . ; N Bd ¼ 1; 2; . . . ; N D

(7)

The main results of solving the DCOPF problem in the present power system x are the vectors QGxand dx, as well as lx, that is, the vector of LMPs at the system nodes. With these vectors used, the bulk

revenue and payment assessment for various participant groups can be formulated as Equations (8)–(13).

These values are suf xed with symbol x for representing their dependence with the vectors of

transmission line capacities (Equation (5)) as well as the generation of lower and upper limits

(Equation (4)) in the present power system x. Moreover, Equation (14) formulates the transmission

network per hour investment cost, as a function of its line capacities and lengths, supposing that Transco

has constructed all the transmission lines in the power system x in the same time, and its required invest-

ment cost can hourly be amortized with zero interest rate.

LR ¼X

d QD s QD^ 2 (8)

GC x ¼ XGC QGxð Þ (9)GRx ¼ lx

T GIMQGx (10)

LC x ¼ lxT CIMQD (11)

NRx ¼ LC x GRx (12)

¼ lxT GIMQGx CIMQDð Þ ¼ lx

T P dxð Þ ¼ lxT Bdx (13)

IC x ¼ k lT t x

8760 (14)

4. A VIEW OF [20]

In [20], the previous LMPs are modied, such that the required transmission cost can perfectly be re-covered via the resulted congestion surplus. To this end, that paper rst solves the DCOPF problem,

obtaining its main consequences, and then denes some additive penalties/rewards associated with

the system generations and nodal injections. These factors are obtained, by solving a special bi-level

optimization problem that applies the minimum nodal price deviations from the precalculated LMPs,

so that the following four groups of constraints can be satised:

(1) the constraint that ensures full recovery of the required network revenue from the congestion

surplus (Equation (16));

(2) the constraint that is associated with equal sharing of transmission revenue recovery between

producers and consumers (Equation (17));



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(3) the constraint that guarantees that the optimal precalculated generation levels and voltage angles

will not change after running the optimization problem (implicit in Equations (18)–(20)); and

(4) the constraints that are associated with the inner problem satisfaction that minimizes the sum-

mation of total generation cost and the additional payments for participants’ extents of transmis-

sion use, as some functions of the aforementioned penalties/rewards (Equations (18)–(21)).

These constraints have been formulated using the Lagrange representation of the internal opti-

mization problem, and they form the main one-level optimization problem (as in Equations(15)–(21)).

minr;l;s;g

l lxk k (15)

Subject to

lT P dxð Þ ¼ NRx (16)

lTCIMQD lxTCIMQD ¼ NRx-lx

TP dxð Þ

2

(17)

ddQGGC QGÞ QGx GIM

Tl þ rg sL þ sU ¼ 0. (18)

B l þ rdð Þ þ HT

gU gLð Þ ¼ 0 (19)sT L QGx Q

LowGx

þ sT U Q

UpGx QGx

þ gT L f x t xð Þð Þ þ g

T U t x f xð Þ ¼ 0 (20)

g L;g U≥0 ; s L;s U≥0 (21)

As illustrated in [20], its approach improves some drawbacks of the old LMPs. For example, it com-

pensates the resulted prot decrement for consumers on generation buses and producers on demand

buses. Nevertheless, it cannot correct a number of them, which can be summarized in three principal

issues:

(1) It does not provide any prot/loss for Transco and consequently any incentive for its better

performance.

(2) It modies the nodal prices, on the basis of minimizing their difference with the old LMPs. In

other words, that approach assumes the old LMPs as a basis and slightly changes them to equate

the resulted network congestion income with the required transmission costs. As illustrated in

Section 2, the old LMP-based mechanism inherently has some essential drawbacks and does

not produce correct economic signals in many cases. Hence, it cannot be a good basis for nodal

prices determination.

(3) In some cases, that approach even dispossesses the old LMP-based prot apportionment mechanism

from fairness. For instance, the prot decrement for less ef cient producers on the demand buses

should not be completely compensated, as it provides some incorrect economic signals for them.

5. MARKET CLEARING STEPS, BASED ON THE PRESENT PAPER APPROACH (AFTER

SOLVING DIRECT CURRENT OPTIMAL POWER FLOW PROBLEM)

5.1. De nition and calculation of “total social welfare,” as a basis for correcting the previous defects

In order that the revenue and payment assessment process can fairly be accomplished, at rst, a bench-

mark for evaluating the whole system ef ciency affected by different participants’ behaviors is re-

quired. This benchmark, named here as “total social welfare,” is dened as the total prot of all the

participants according to Equations (22) and (23). These equations demonstrate that the mentioned

value equals the resulted net prot (or the negated net cost) of the system, considering the generation

and transmission costs as well as the obtained consumption revenue. This consequence can be inter-

preted by noting the fact that the three main participant categories indeed are some copartners in a

communal participation to produce some commodities or services, and their total prots equal the

net participation prot, that is, the net revenue minus the net cost. It is noticed that this denition



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can be extended to a more general formula, which is the difference between the net system revenue and

the net system cost. The net system revenue includes all the revenues resulted from the electricity

consumption, and the net system cost comprises all the costs required to produce and transmit the

electricity, such as generation investment and operational costs, generation emission costs, transmis-

sion investment cost, and cost of environmental impacts. From this extended denition, the economic

measure of all the advantages and disadvantages resulted from the power system operation and

planning can be considered, and the total social welfare can be obtained as the difference betweenthe equivalent economic values of the total advantages and disadvantages. The equivalent economic

evaluation of different advantages/disadvantages is usually performed by the related entities. For

example, Environmental Safety Organization assesses the emission taxes for generation companies

or the taxes of environmental impacts for both the generation and transmission companies.

SWt x ¼ Gprofit x þ Lprofit x þ Nprofit x¼ GRx GC x þ LR LC x þ NRx IC x¼ NRx GC x þ LR þ NRx IC x

(22)

¼ LR GC x IC x (23)

5.2. Bulk competence measure: the criterion for fair apportionment of total social welfare among the

participant groups and modifying Equations (16) and (17)

The resulted total social welfare in the system should be fairly apportioned among the main three bulk

participant entities, namely generation, consumption, and the monopolistic Transco. As the essence of

the bulk entities’ act in the system is completely different, this apportionment should be performed

considering their own effects on the value of total social welfare and in such a manner that their

long-term behaviors can be oriented in the total social welfare maximization. The mechanism proposed

here to attain this goal is to apportion the bulk prot shares among the three main participant groups in

proportion to a mathematical criterion named here as “bulk competence measure.” This criterion is de-

ned here as the mathematical product of two main criteria, namely bulk optimality measure and bulk

protability measure, which are introduced as follows.

5.2.1. Bulk optimality measure. Bulk optimality measure presents a criterion for evaluating the opti-mality of act for each participant group in comparison with its own best performance. To this end,

the present behavior of each group as well as its optimal case and the case of removing it from the sys-

tem should be analyzed, and the bulk optimality criterion for each participant group can be dened as

in Equation (24). In this equation, symbol BOM Co,x denotes the bulk optimality measure for the bulk

entity, suf xed by Co index in present power system x, and SWt x, SWt absence,Co, and SWt best,Co repre-

sent the resulted total social welfare supposing the present system x, the system without the entity, and

the system with its best possible behavior, respectively.

BOM Co;x ¼ SWt x SWt absence;Co

SWt best ;Co SWt absence;Co(24)

Because the participants’ behaviors can generally be evaluated from the viewpoints of either their

operation or their planning, the term for the best total social welfare in the aforementioned criterioncan be obtained in different ways, dependent on each group’s work nature. However, for the sake of

simplicity, the operation manners of the bulk entities are ignored here, and the bulk optimality measure

for both transmission and bulk generation entities are evaluated by considering their planning beha-

viors. Furthermore, as the consumers do not take part in the electrical planning processes (i.e., TEP

and GEP); their best possible manner is equated to their present manner, and so the bulk optimality

measure for them is assumed equal to 1. This assumption seems to be rational, as in essence, the con-

sumers are those who are mainly extracting the net prot or welfare from electricity consumption.

However, the discussion on how to determine the bulk optimality measure for the bulk consumption

entity in a more accurate way may still be an open problem and requires more analysis, which is out

of the scope of this paper.



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According to Equation (24), the bulk optimality measure for Transco can be obtained in this way:

(1) The total social welfare that resulted from the present transmission network (x) is denoted here

as SWt x and can be obtained by running the DCOPF problem (Equations (2)–(5) in the present

system and by using the formulation of Equation (23).

(2) The optimum case of transmission design can be attained through constructing the RTN. RTN is

dened here as the network with the same generations, demands, and topology as the present

one but with different line capacities such that the DCOPF-resulted total social welfare can be

maximized (denoted here as SWt RTN). RTN can be obtained by solving a special bi-level

optimization problem as in Equations (25) and (26). This problem can be solved by using a stochas-

tic optimization algorithm that executes the DCOPF subproblem, once for each set of the line

capacities given from the external subproblem, and that minimizes its resulted total generation cost.

SWt RTN ¼ maxt x

SWt xð Þ ¼ maxt x

LR GC x IC xð Þ (25)

Subject to

t x ≥0 (26)

as well as to

DC OPF 2 5ð Þh i

(3) The system total social welfare in the case of removing all its transmission lines is denoted here

as SWt 0. This quantity can be calculated by solving a DCOPF problem, which takes into account

variable load vector as a new decision variable in addition to the formers and consequently

considers the consumption revenue decrement due to the required load curtailment and the

generation decrement to satisfy the power balance constraint.

(4) Finally, the bulk optimality measure for Transco can be formulated in a way similar to Equation

(24), as seen in Equation (27), where symbol BOM Transco, x denotes the bulk optimality measure

for Transco in the present power system x.

BOM Transco;x ¼

SWt x SWt 0

SWt RTN SWt 0 (27)

In a similar way, the bulk optimality measure for the bulk generation entity can be obtained as

follows:

(1) The total social welfare that resulted from the present system (x) has been calculated previously

as SWt x.

(2) The optimum case of generation capacity design can be attained through constructing the refer-

ence generation capacities (RGCs). RGC is dened here as the network with the same demands,

transmission lines, and topology as the present one but with different generation capacities such

that the DCOPF-resulted total social welfare can be maximized (denoted here as SWt RGC). In

principle, RGC should be obtained by using a bi-level optimization problem similar to that of

Equations (25) and (26) and considering the generation investment costs (beside the transmissioncosts). However, for the sake of simplicity, here, it is assumed that the generation cost functions

can include both the fuel costs and the generation investment costs; thus, there is no need to con-

sider the generation total investment cost as a separate variable. Consequently, the proposed

problem for nding the RGC can be simplied to an ordinary DCOPF as in Equations (2)–(5),

with the difference that the generation upper limits are set to innity. Then, the resulted

unbounded generation levels can introduce the best possible generation capacities, and SWt RGCcan easily be calculated by applying Equation (23) on the outcomes obtained from the mentioned

DCOPF solution.

(3) Provided that the whole generation entity can be removed from the system, there will be neither

load revenues nor generation costs. Then, the total social welfare will decrease to the negated



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total transmission investment cost. On this condition, it seems rational to neglect the transmis-

sion investment costs in the total social welfare calculation, because if this case were actually

realizable, Transco, indeed, would not construct the transmission network at all. On the other

hand, according to Equation (24), considering a negative value of total social welfare in this

case results in an overestimated bulk optimality measure for the bulk generation entity. Hence,

in order to avoid such inequities, the value of total social welfare in the case of removing the

whole generation entity is assumed equal to 0. It should be noted that, because of similar rea-soning, the value of total social welfare in the case of removing the whole consumption entity

(which is required in the next section) is also equated to 0.

(4) Finally, the bulk optimality measure for the whole generation entity can be formulated as in

Equation (28), where symbol BOM Generation,x represents the bulk optimality measure for the

whole generation entity in the present power system x.

BOM Generation; x ¼ SWt x

SWt RGC(28)

5.2.2. Bulk pro tability measure. The bulk protability measure for each participant category

introduces a criterion to evaluate how much its present performance participates in attaining the pres-

ent system total social welfare, relative to the case without it. In other words, this criterion determines

the percentage increase in the total social welfare in the present system compared with the case of

removing each bulk entity, as formulated in Equation (29) with symbol BPM Co,x denoting the bulk

protability measure for the bulk entity indexed with Co in the present power system x.

BPM Co; x ¼ SWt x SWt absence;Co

SWt x(29)

Because, as described in Section 5.2.1, the value of SWt absence,Co for both the bulk generation and

consumption entities is equal to 0, according to Equation (29), the bulk protability measure for both

of them is equal to 1. This means that both the bulk generation and consumption entities have the same

protability measures. This consequence seems to be rational, as with either of them removed, the total

social welfare declines to zero.

The bulk protability measure of Transco in the present power system x, denoted here as BPM Transco,x,

can also be derived from Equation (29) as formulated in Equation (30). According to Equation (30), thisvalue for Transco is typically lower than two other bulk entities, or in other words, its act is typically of

lower importance. This outcome is also rational, because with all the transmission lines removed, the

remaining network may still be able to supply some loads (which are connected to local generations)

and thereby be protable.

BPM Transco;x ¼ SWt x SWt 0

SWt x(30)

5.2.3. Bulk competence measure. The bulk competence measure value is denoted here as BCM Co,x for

the bulk entity, indexed with Co in the present power system x, and is dened as the product of the two

aforementioned criteria (bulk optimality measure and bulk protability measure), according to

Equation (31).

BCM Co; x ¼ BOM Co; x BPM Co; x ¼ SWt x SWt absence;Co

SWt best ;Co SWt absence;Co

SWt x SWt absence;CoSWt x

(31)

In the view of the comments pointed out in the previous two sections and of Equations (27)–(31),

the bulk competence measure for each of the three bulk participant entities can be formulated as in

Equations (32)–(34). In these equations, symbols BCM Transco,x, BCM Generation,x, and BCM Consumption,xrespectively represent the bulk competence measure for Transco, entire generation entity, and entire

consumption entity in the present power system x.

BCM Transco;x ¼ SWt x SWt 0SWt RTN SWt 0

SWt x SWt 0

SWt x(32)



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BCM Generation;x ¼ SWt x

SWt RGC(33)

BCM Consumption;x ¼ 1 (34)

5.2.4. Determination of the nal bulk pro t shares for each participant group. Finally, the bulk prot

shares for the three bulk participant entities are obtained using Equations (35)–(37). Two optional

equations from these equations will be replaced later, instead of Equations (16) and (17) in the optimi-

zation problem, formulated by [20].

Nprofit x ¼ BCM Transco;x

BCM Transco;x þ BCM Generation;x þ BCM Consumption;x:SWt x (35)

Gprofit x ¼ BCM Generation;x


Lprofit x ¼ BCM Consumption;x


5.2.5. The long-term investment signals of the proposed bulk pro t apportionment scheme. The mech-

anism presented in Sections 5.2.1,5.2.2,5.2.3,5.2.4 for sharing the system ’s total social welfare among

the three bulk participant entities can produce some positive long-term investment and planning sig-

nals for both the generation and transmission entities and even for the bulk consumption entity, which

can be listed as follows:

(1) According to Equations (32) and (35), the proposed mechanism encourages the Transco (which,

as supposed here, has an active role in the system with variable prot/loss) to determine and

construct better transmission expansion plans and line capacities, which liken and close the entire

transmission network to the RTN and increase the total social welfare as much as possible.

(2) According to Equations (33) and (36), the mechanism also encourages the producers to make bet-

ter generation capacities, which are as close as possible to the RGCs. Thereby, the more ef cient

producers (with lower costs) usually have the incentive to increase their presented generation ca-

pacities, as the RGC usually introduces more capacities for more ef cient producers.

(3) As is obvious, more values of load shedding in the system with no transmission lines correspond tomore dependence between supplying the required demand and the existence of transmission

lines because of far distances between generation units and load centers. According to Equations

(32) and (35), this leads to a more decrement in SWt 0, as well as to a more prot share for

Transco. Because this consequently results in reducing the other participants’ prots, it

encourages them to decrease the farness between generation and demand centers while locating

at the design time (for example, by using the distributed generations) and leads to the system

long-term economic ef ciency increment.

5.3. Modi cation of the objective function (Equation (15)) to affect the pro t sharing mechanism

among individual producers and consumers directly by their ef ciency and extent of transmission use

In the previous sections, the way to fairly modify the constraints of Equations (16) and (17) is

explained. However, this method is ef cient to determine the bulk rational prot shares for each of

the three participant groups, and it does not modify the prot sharing mechanism for their individuals

(i.e., each producer or consumer). This means that, for example, the bulk-determined prot for the ent-

ire generation entity should be apportioned among the individual producers such that the more ef cient

producer earns more prot, although its prot should decrease, depending on its relative transmission

use, in comparison with the others. This target is attained in this section via determining a new set of

nodal power prices such that the predened entire producers and consumers’ bulk prots can be

allocated to their individuals, on the basis of their ef ciencies and extents of transmission use. In this

direction, the system nodal prices will be determined regardless of the previous LMPs but with the

purpose of causing the minimum possible differences between participants’ monetary quantities



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(revenues and payments) and their rational values, such that the network constraints as well as two

additional constraints for bulk participants’ prots can be satised. To this end, rstly, it is necessary

to dene several vectors relating to either participants’ ef ciencies or their relative extents of transmis-

sion use in the system as follows.

5.3.1. Relative extents of transmission use for producers. The aim of this section is to dene a

criterion for evaluating the producers’

relative extents of transmission use. As described in Section1, various approaches have been presented so far to attain the aforementioned goal. Those methods

mostly consider the act of each transmission line individually, and only a few of them regard the

use of the whole transmission network. For the sake of summarization, this paper avoids broaching

the aforementioned problem and simply assumes that the vector of relative extents of transmission

use for producers can be represented with symbol UGx, which, in the simplest form, is dened as in

Equation (38), with symbol uiG, representing its ith elements. In addition, in this equation, symbols

pi and gni respectively denote the ith elements of the nodal injections vector (P(dx) = Bdx) and vector

GN, that is, the vector of the numbers of generation units connected to each of the system buses

according to the denition of Equation (39). As will be seen later, the role of the recent vector in

the denominator of the fraction is to equally share the transmission use for the buses containing several

generation units among the units.

UGx ¼ uiGf g ¼ pi= P

Pj jgnið Þif pi > 0

0 if pi≤0

8<:

9=; ; i ¼ 1; 2; . . . ; N B (38)

GN ¼ gnif g ¼ GIMEg (39)

Equation (38) supposes that the relative extents of transmission use for the producers connected to

each bus are directly related to the net injected power from that bus to the system. Thereby, if the total

generation on the node equals its local demand, the relative extent of transmission use for its both

producers and consumers can be equated to 0, as they do not exchange electrical power with the

network. In the same way, positive net injected power at the bus corresponds to the transmission

use for only the producers connected to the bus and the negative value for only the consumers. How-

ever, this method does not consider the extents of use for individual transmission lines, although someparticipants may use shorter and cheaper lines, but it is the simplest way for evaluating the participants’

extent of transmission use. It should be noted that the main goal of this paper is not to determine the

aforementioned criterion, but in contrast, it attempts to modify the prot allotment mechanism with the

use of the criterion. Hence, the criterion of Equation (38) can be replaced here with any other appro-

priate equation according to the papers discussed in Section 1 or any other related paper.

5.3.2. Relative generation ef ciency vector. Generation ef ciency vector can be dened as the vec-

tor containing the ratios of the producers’ generations to their costs, according to Equation (40),

with symbol l x denoting the average electricity price in the system as dened in Equation (41).

This value presents a criterion for evaluating the worth of the generated power. It is used in Equa-

tion (40) in order to homogenize the numerator and denominator of the fraction, both in monetary

units. The costs remarked in Equation (40) may indeed include any costs or disadvantages requiredfor electricity generation or anything imposed by generation units, such as generation fuel and in-

vestment costs, emission costs, and costs of environmental impacts due to construction of power

plants, as well as equivalent transmission use costs, that is, the costs indicating the producers’

shares of transmission use. However, for the sake of simplicity, here, it is supposed that all such

costs except the equivalent transmission use cost can be included in the hourly generation cost

functions (as practically may be quite correct). Hence, in Equation (40), the denominator in the

right side only contains the vectors of hourly generation cost functions as well as the term IC xGIMUGx,indicating the vector of hourly equivalent transmission use costs for the generation units. In this term,

the extents of transmission use for the producers are indeed homogenized with their generation costs,

both as pecuniary quantities, so that both their effects can easily be compared and analyzed. To this



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end, the equivalent transmission use cost for each producer is dened here as the previously dened rel-

ative transmission use criterion multiplied by the net required transmission investment cost and is as-

sumed as the generation cost increment, superimposing to the previous generation cost. The role of

the product of two vectors GIM and UGx is also to extract the transmission use extents for individual

generation units from those calculated for the system buses. Thereby, the vector of relative generation

ef ciencies for producers considering their relative extents of transmission use can nally be formulated

as hGx=PhGxaccording to the denition of Equation (40) for hGx.

hGx ¼ lxQGx= GC QGxð Þ þ IC xGIMUGxð Þ (40)

lx ¼

Plx

N B(41)

5.3.3. Relative extents of transmission use for consumers. In the same way as with Equation (38),

Equation (42) formulates the relative extents of transmission use for consumers, with symbols uiDand dni denoting the ith elements of vectors UDx and DN, respectively. In a similar way to

Equation (39), vector DN contains the numbers of separate demands connected to each node and

can be formulated as Equation (43).

UDx ¼ uiDf g ¼ pi= P

Pj jdnið Þ if pi > 0

0 if pi≤0

8<:

9=; ; i ¼ 1; 2; . . . ; N B (42)

DN ¼ dnif g ¼ CIMEd (43)

5.3.4. Relative consumption ef ciency vector. Similar to the previously dened relative generation

ef ciency vector, the relative consumption ef ciency vector can be formulated as in Equation (44).

In this equation, vector CIM, as a multiplier, plays the role of obtaining the relative extents of

transmission use for the individual demands from those previously calculated for the system nodes.

hDx ¼ LR QDð Þ IC xCIMUDxð Þ= lxQD

(44)

5.3.5. Final modi ed optimization problem formulation. The nal modied objective function isdened here in such a manner that each producer ’s revenue and each consumer ’s payment can be matched

as close as possible with their rational values. This goal is achieved considering the fact that the products

of the power price on each bus and its connected producers ’ generation levels equal their resulted

revenues. In addition, the power price on each node multiplied by its connected consumers’ demands will

equal their obligated payments. On the other hand, each individual producer ’s rational revenue can be

dened as the summation of its generation cost and the rational prot share from the bulk-determined

prot for the entire generation entity (determined in Equation (36)) according to its relative generation

ef ciency. Similarly, the rational individual consumers’ payments can be obtained as the differences

between their load revenues (known as demand worth) and their rational prot shares from that predened

for the whole consumption entity (Equation (37)) according to their relative consumption ef ciencies.

Therefore, the modied objective function, as well as the constraints, can nally be formulated as in

Equations (45)–

(51). As is obvious, the two norm operators in Equation (45) are respectively de

ned over the sets of individual producers and consumers, even if there are two or several producers or consumers

connected to a bus. Herein, the roles of the vectors GIMT and CIMT multiplied byl is to obtain the nodal

prices that belong to the individual producers and consumers, in accordance with the aforementioned

goal. In addition, it should again be noted that the real meaning of the generation cost functions in

Equation (45) might practically include all the costs related to the electricity generation by the producers,

such as hourly generation fuel and investment costs, hourly emission costs, and hourly costs of the

environmental impacts imposed by the generation units. However, the equivalent transmission use cost

should obviously be ignored in the generation costs inclusion, because it is not actually a part of the costs

that producers are directly necessitated and only reduces their prot through separately being considered

in the generation ef ciency vector according to Equation (40).



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minr;l;s;g

GIMT l QGx GC QGxð Þ þ Gprofit xhGx=PhGx

þCIMT l QD LR QDð Þ Lprofit x

hDx=PhDx

8<:

9=; (45)

Subject to

l

T

GIMQGx GC x ¼ Gprofit x ¼

BCM Generation;x

BCM Transco;x þ BCM Generation;x þ BCM Consumption;x :SWt x (46)

LRx lTCIMQD ¼ Lprofit x ¼

BCM Consumption;x


ddQGGC QGÞjQGx GIM

Tl þ rg sL þ sU ¼ 0.

(48)

B l þ rdð Þ þ HT gU gLð Þ ¼ 0 (49)

sT L QGx QLowG

þ sT U Q

UpG QGx

þ gT L f x t xð Þð Þ þ g

T U t x f xð Þ ¼ 0 (50)

g L;g U≥0 ; s L;s U≥0 (51)

6. NUMERICAL ANALYSIS

The proposed algorithm (Equations (45)–(51)) has been applied on modied IEEE 30-bus system and

has been solved using MATLAB fmincon general-purpose optimization function in several different

cases. As previously mentioned, the DCOPF problem is formulated here, considering inelastic loads

(except in the case of removing all the transmission lines, which vast load shedding is required),

and so its decision variable is just the generation vector. Hence, the applied demand schedule is con-

sidered the same as those presented in the IEEE 30-bus system, listed in Table II beside the

corresponding coef cients of the consumption revenue functions appended to the mentioned test case.

However, the generation schedule has been slightly modied through inserting a new generation unit

into it, connected to bus 2, so that the effectiveness of the proposed approach in different cases can bebetter demonstrated. The default generation schedule containing the default generation cost function

coef cients and default generation capacities (i.e., the upper limits, as the lower limits are constantly

assumed equal to 0) for all the generation units in the present system is available in Table III. However,

it should be noted that the sets of generation capacities and generation cost function coef cients can

completely be changed in different cases under study, and the sets listed in Table III represent the

default generation schedule. In addition, the default transmission schedule including the two line

ending nodes, their lengths, default capacities, and required investment costs are available in Table IV.

However, the transmission topology and the line lengths are the same in all the cases under study, the

line capacities and their investment costs can also be varied in different cases, and those available in

Table IV only represent their default values. The annualized per-kilometer per-megawatt marginal

investment costs of building the transmission lines and the annualized per-megawatt investment costs

of building the power transformers and the generation units are supposed to be equal to $35/MW.km.

year, $250/MW.year, and zero, respectively. According to Equation (14), it is supposed here that therequired investment cost of a transmission line is linearly proportional to the product of its length and

capacity, whereas the corresponding cost for a transformer branch is only in linear proportion to its

capacity. In addition, for the sake of simplicity, the required investment costs of the generation units

are assumed equal to 0 (as in reality, such costs can be included in their generation cost functions).

Here, for better demonstration, the presented mechanism has been applied on such power system for

each of the several different cases described in the following remarks:

(1) To illustrate the effectiveness of the proposed approach even though two or several generation

units are simultaneously connected to one node, two cases containing two different sets of

generation cost function coef cients for both the generation units at bus 2 are applied by xing



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T a b l e I I . T h e c o e f c i e n t s o f t h e d e m a n d r e v e n u e f u n c t i o n s a n d t h e g i v e n d e

m a n d l e v e l s i n t h e s y s t e m .

D e m a n d #

B u s #

d

s

Q D

D e m a n d #

B u s #

d

s

Q D

D e m a n d #

B u s #

d

s

Q D

1

2

5 0 . 0 5

0 . 0 1 1

2 1 . 7 0

8

1 2

5 0 . 4 0

0 . 0 1 8

1 1 . 2 0

1 5

2 0

5 0 . 7 5

0 . 0 2 5

2 . 2 0

2

3

5 0 . 1 0

0 . 0 1 2

2 . 4 0

9

1 4

5 0 . 4 5

0 . 0 1 9

6 . 2 0

1 6

2 1

5 0 . 8 0

0 . 0 2 6

1 7 . 5 0

3

4

5 0 . 1 5

0 . 0 1 3

7 . 6 0

1 0

1 5

5 0 . 5 0

0 . 0 2 0

8 . 2 0

1 7

2 3

5 0 . 8 5

0 . 0 2 7

3 . 2 0

4

5

5 0 . 2 0

0 . 0 1 4

9 4 . 2 0

1 1

1 6

5 0 . 5 5

0 . 0 2 1

3 . 5 0

1 8

2 4

5 0 . 9 0

0 . 0 2 8

8 . 7 0

5

7

5 0 . 2 5

0 . 0 1 5

2 2 . 8 0

1 2

1 7

5 0 . 6 0

0 . 0 2 2

9 . 0 0

1 9

2 6

5 0 . 9 5

0 . 0 2 9

3 . 5 0

6

8

5 0 . 3 0

0 . 0 1 6

3 0 . 0 0

1 3

1 8

5 0 . 6 5

0 . 0 2 3

3 . 2 0

2 0

2 9

5 1

0 . 0 3 0

2 . 4 0

7

1 0

5 0 . 3 5

0 . 0 1 7

5 . 8 0

1 4

1 9

5 0 . 7 0

0 . 0 2 4

9 . 5 0

2 1

3 0

5 1 . 0 5

0 . 0 3 1

1 0 . 6

Σ

2 8 3 . 4



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the other producers’ coef cients in either cases to be unchanged. Hence, the applied two sets for

all producers are namely the default set of the generation cost function coef cients (listed in

Table III) and a new set available in Table V. The applied generation capacities for both the cases

are the same as the default set presented in Table III. According to the proposed algorithm, each

case may result in distinct sets of the best transmission and generation capacities (RTN and

RGC). However, because the two considered sets of the generation cost function coef cients

are completely symmetrical, the resulted RTNs in both cases are the same, which is scheduled

in Table VI. In addition, Tables VII and VIII respectively contain the resulted RGCs for the

two systems, namely the default generation cost function coef cients and the new coef cientsof Table V. In both cases, the RTN is calculated using the bi-level optimization problem of

Equations (25) and (26) and supposing the presented (default or new) generation cost function

coef cients and the default generation capacities (according to Table III), whereas the RGC is

Table III. Default generation cost function coef cients as well as default generation capacities in thesystem.

Generation unit # 1 2 3 4 5 6 7

Bus # 1 2 2 5 8 11 13a 0.038432 0.01 0.01 0.01 0.01 0.01 0.01b 20 20 10 40 40 40 40

QGxup 360.2 140 100 100 100 100 100

Table IV. Default transmission schedule including two ending nodes, lengths, default capacities, and re-quired investment costs of the transmission lines in the system. The zero lengths correspond to the trans-

former branches, of which their investment costs are only in proportion to their capacities.

From bus

Tobus

Length(km)

Capacity(MW)

Investment cost ($/h)

From bus

Tobus

Length(km)

Capacity(MW)

Investment cost ($/h)

1 2 112 114.38 51.18 15 18 62 6.41 1.591 3 156 57.21 35.66 18 19 44 3.21 0.562 4 182 28.51 20.73 19 20 26 6.31 0.663 4 59 54.81 12.92 10 20 88 8.51 2.99

2 5 322 64.06 82.42 10 17 71 4.62 1.312 6 320 40.27 51.49 10 21 71 15.90 4.514 6 149 51.86 30.87 10 22 68 7.70 2.095 7 49 18.87 3.69 21 22 35 1.62 0.236 7 45 41.67 7.49 15 23 72 5.76 1.666 8 0 12.85 0.37 22 24 73 6.09 1.786 9 0 12.02 0.34 23 24 77 2.56 0.796 10 42 10.78 1.81 24 25 37 0.09 0.019 11 0 19.72 0.56 25 26 73 3.51 1.029 10 0 31.72 0.91 25 27 54 3.59 0.774 12 28 23.87 2.67 28 27 0 16.59 0.4712 13 0 21.79 0.62 27 29 48 6.07 1.1612 14 97 8.03 3.11 27 30 110 6.95 3.0512 15 86 18.54 6.37 29 30 90 3.67 1.3212 16 60 7.90 1.89 8 28 252 1.76 1.77

14 15 64 1.83 0.47 6 28 248 14.84 14.7016 17 58 4.40 1.02 Σ 359.06

Table V. A new set of generation cost function coef cients in the system.


a 0.038432 0.01 0.01 0.01 0.01 0.01 0.01b 20 10 20 40 40 40 40



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T a b l e V I . T h e r e f e r e n c e t r a n s m i s s i o n n e t w o r k t h a t r e s u l t e d i n t h e c a s e o f t h e d e f a u l t g e n e r a t i o n c o s t f u n c t i o n c o e f c i e n t s a n d g e n e r a t i o n c a p a c i t i e s i n t h e s y s t e m .

F r o m

b u s

T o b u s

C a p a c i t y

( M W )

I n v e s t m e n t c o s t

( $ / h )

F r o m b u s

T o

b

u s

C a p a c i t y

( M W )


( $ / h )

F r o m b u s

T o b u s

C a p a c i t y

( M W )


( $ / h )

1

2

7 . 5 0

3 . 3 6

4

1 2

4 2 . 5 6

4 . 7 6

2 1

2 2

4 . 1 3

0 . 5 8

1

3

5 0 . 9 0

3 1 . 7 3

1 2

1 3

2 . 0 0

0 . 0 6

1 5

2 3

5 . 8 5

1 . 6 8

2

4

6 0 . 4 8

4 3 . 9 8

1 2

1 4

9 . 3 5

3 . 6 3

2 2

2 4

7 . 2 1

2 . 1 0

3

4

4 8 . 5 0

1 1 . 4 3

1 2

1 5

1 8 . 2 0

6 . 2 5

2 3

2 4

2 . 6 5

0 . 8 2

2

5

8 5 . 3 5

1 0 9 . 8 1

1 2

1 6

7 . 8 1

1 . 8 7

2 4

2 5

4 . 8 4

0 . 7 2

2

6

7 2 . 9 6

9 3 . 2 9

1 4

1 5

3 . 1 5

0 . 8 1

2 5

2 6

5 . 5 0

1 . 6 0

4

6

5 8 . 8 3

3 5 . 0 2

1 6

1 7

4 . 3 1

1 . 0 0

2 5

2 7

8 . 3 4

1 . 8 0

5

7

1 2 . 8 5

2 . 5 2

1 5

1 8

7 . 3 0

1 . 8 1

2 8

2 7

2 1 . 3 4

0 . 6 1

6

7

3 5 . 6 5

6 . 4 1

1 8

1 9

4 . 1 0

0 . 7 2

2 7

2 9

8 . 0 6

1 . 5 5

6

8

3 1 . 6 6

0 . 9 0

1 9

2 0

9 . 4 0

0 . 9 8

2 7

3 0

8 . 9 4

3 . 9 3

6

9

3 0 . 5 0

0 . 8 7

1 0

2 0

1 1 . 6 0

4 . 0 8

2 9

3 0

5 . 6 6

2 . 0 4

6

1 0

1 8 . 3 0

3 . 0 7

1 0

1 7

8 . 6 9

2 . 4 7

8

2 8

2 . 3 4

2 . 3 5

9

1 1

2 . 0 0

0 . 0 6

1 0

2 1

1 7 . 3 7

4 . 9 3

6

2 8

2 1 . 6 8

2 1 . 4 8

9

1 0

3 0 . 5 0

0 . 8 7

1 0

2 2

9 . 3 4

2 . 5 4

Σ

4 2 0 . 4 5



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calculated supposing the presented generation cost function coef cients as well as the default

transmission capacities (according to Table IV).

(2) Furthermore, in order to further illustrate the capability of the proposed mechanism in rationally appor-

tioning the total social welfare among the main three participant entities, one additional case namely

the one containing the default generation cost function coef cients (as in Table III) with a new set

of generation capacities, listed in Table IX, is also analyzed. Then, this case is compared with the

previously analyzed case of the default generation cost function coef cients and generation capacities,

from the viewpoint of the bulk generation prot, to demonstrate the resulted positive long-term invest-

ment signals in the generation planning processes. In the recent new case, the RTN and RGC are

respectively obtained supposing the new set of generation capacities and the default transmission ca-

pacities.It is obvious that the RGC does not change from that of the previous default system, presented

in Table VII, whereas the new resulted RTN can be completely different, as scheduled in Table X.

(3) The nal analyzed case is the case with the default generation cost function coef cients and generation

capacities but with the present transmission line capacities equal to those resulted as the RTN in sim-

ilar condition. It is obvious that, in this case, the RTN is the same as the present transmission network,

whereas the RGC should again be recalculated, as displayed in Table XI. The resulted bulk Transco’s

prot can be compared with that of the similar case with default transmission line capacities, in order that its resulted positive long-term investment signals in the TEP processes can be demonstrated.

In brief, according to the aforementioned remarks, the systems under analysis here can be summa-

rized as follows:

(1) the system with the default generation cost function coef cients and generation capacities

(Table III), as well as the default transmission capacities (Table IV);

(2) the system with the new generation cost function coef cients (Table V) and the default generation

capacities (Table III), as well as the default transmission capacities (Table IV);

(3) the system with the default generation cost function coef cients (Table III) and the new generation

capacities (Table IX), as well as the default transmission capacities (Table IV); and

(4) the system with the default generation cost function coef cients and generation capacities

(Table III) but with the best possible transmission capacities (i.e., RTN in Table VI).Here, two different kinds of analysis on the numerical results obtained from the aforementioned four

systems, including the analyses on the monetary quantities for both the bulk participant entities and

their individuals, are presented as follows.

Table VII. The resulted reference generation capacities as well as current generation levels in the system with the default generation cost functions coef cients and default transmission line capacities.


QG,RGC,xUp 169.96 0.00 40.97 11.98 19.00 19.70 21.79

QGx 169.96 0.00 40.97 11.98 19.00 19.70 21.79

Table VIII. The resulted reference generation capacities as well as current generation levels in the system with the new generation cost functions coef cients and default transmission line capacities.


QG,RGC,xUp 169.96 40.97 0.00 11.98 19.00 19.70 21.79

QGx 169.96 40.97 0.00 11.98 19.00 19.70 21.79

Table IX. The new applied set of generation capacities in the system.


QGxup 200 20 20 100 100 100 100



DOI: 10.1002/etep


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T a b l e X . T h e r e f e r e n c e t r a n s m i s s i o n n e t w o r k t h a t r e s u l t e d i n t h e s y s t e

m w i t h t h e d e f a u l t g e n e r a t i o n c o s t f u n c t i o n c o e f c i e n t s a n d n e w g e n e r a t i o n c

a p a c i t i e s ( T a b l e I X ) .

F r o m

b u s

T o b u s

C a p a c i t y

( M W )


( $ / h )

F r o m b u s

T o

b

u s

C a p a c i t y

( M W )


( $ / h )

F r o m b u s

T o b u s

C a p a c i t y

( M W )


( $ / h )

1

2

1 3 0 . 8 8

5 8 . 5 7

4

1 2

4 0 . 8 4

4 . 5 7

2 1

2 2

3 . 9 8

0 . 5 6

1

3

7 1 . 9 8

4 4 . 8 6

1 2

1 3

1 0 . 7 3

0 . 3 1

1 5

2 3

5 . 7 6

1 . 6 6

2

4

4 0 . 1 2

2 9 . �

Transmission Pricing 38

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