Journal of Electrical Engineering & Technology Vol. 6, No. 1, pp. 8~13, 2011

DOI:

10.5370/JEET.2011.6.1.008

8

A Study on Reliability Differentiated Pricing of

Long-Term Transactions

Balho H. Kim

†

Abstract – Reliability differentiated pricing (RDP) is known to improve the efficiency and benefits of

consumers and producers. Outage costs representing the economic and non-economic consequences of

service curtailments to customers can be used as an effective parameter of RDP in electricity markets.

This paper examines the theoretical aspects of an RDP scheme, and derives the optimal decisions of

consumers and electric utilities for long-term transactions. The proposed RDP is demonstrated through

a case study on a wheeling service.

Keywords: Reliability Differentiated Pricing (RDP), Outage Costs, Optimal Pricing

1. Introduction

Electricity consumers are often given the choice of dif-

ferent levels of service reliability [1]-[3]. Typically, cus-

tomers would opt for the level of reliability that best suits

their needs and processes. Such a choice is argued to “un-

bundle” the service, resulting in improved efficiency and

benefits to both consumers and producers. However, thus

far, these available priority pricing schemes have led to

economic inefficiencies; they even fall short in attaining

welfare maximization of the utility [4].

Real-time pricing or spot market pricing schemes have

failed to consider long-run costs entailed in electricity pro-

duction, such as capital costs of the added capacity. More-

over, real systems, in general, may not recover the suffi-

cient revenue for the utility. Thus far, the suggested

schemes to recover the required revenue, such as in the

Ramsey-type pricing schemes applied to real-time pricing,

result in welfare loss.

The pricing policy proposed in [5]-[8] could overcome

most of the shortcomings entailed in both priority pricing

and real-time pricing. In addition, under certain assump-

tions, the proposed pricing policy could recover revenues

without any welfare loss. It could also differentiate the

prices for varying levels of service reliability based on con-

sumer outage costs.

In the analysis that follows, outages are assumed to oc-

cur without prior notification and that they occur in “short”

duration. Outage costs are defined as costs incurred by

customers due to sudden shortage in power supply; this is

over and above the loss in relation to consumer benefits

due to the said shortage. One reason for this is the sudden

shortage of supply. Losses occur due to interruption of pro-

duction processes, which could have been avoided if re-

scheduling is carried out. Rescheduling could only be ac-

complished if prior notice of the outage is conducted.

2. Theory of Reliability Differentiated

Pricing (RDP) [4]-[7]

A single welfare-maximizing public utility is assumed to

own and operate the generating plants and the transmission

network of a given electric power system, which is sold

independently to customers. It is also assumed that the util-

ity could set and communicate the prices, as well as set a

different price range for each customer class, at each mo-

ment instantaneously. Supply outages occur randomly in

relation to the probability of a possibly known outage. In

effect, customers only learn of an outage after it occurs.

Outages are assumed to be of short duration, such that cus-

tomers who continue to access the utility service will not

find it feasible or profitable to reschedule their production

processes (i.e., change their electricity usage pattern) from

the high production prices during outage duration. It is

assumed that the utility could ration the supply shortages

among its targeted customer classes.

2.1 Consumer Behavior [5]

Individual customers act independently. Customers are

modeled as price-taking, profit-maximizing firms. Assume

that all customers, can be classified into a finite number of

classes,

iI

at I = {1,...,N}, then each class consisting

of customers would have the same outage cost at a particu-

lar location. Total time T consists of short discrete time

intervals

tT

. Customer i’s electricity demand at time t

is denoted by

it

Q

.

Consider a short-term rational customer whose benefit

from electricity usage at any time depends on consumption

at that time only. Hence, the customer maximizes consumer

surplus at each

t

independently. At any time

t

, let

i

F

be the value-added or benefit function for customer i’s use

of electricity

i

Q

. Then, in real-time, when there is no sup-

ply shortage, the customer would choose

i

Q

to maximize

† Corresponding Author: Department of Electrical and Electronic

Engeering, Hongik University, Korea. (bhkim@wow.hongik.ac.kr)

Received: February 5, 2010; Accepted: April 5, 2010

Balho H. Kim

9

profit

.

()

ii ii

F

QPQMax

(1)

i

i

i

dF

P

dQ

(2)

where

i

P

is the price of electricity.

Fig. 1 shows the expected short-run customer demand

curve DD, a very short-run customer demand curve D'E,

and supply curve SS of electricity.

e

i

Q

is the equilibrium

demand of customer

i at time

t

for the published ex-

pected price

e

i

P

. When supp1y is cut back to

0

i

Q

, the area

D’FE would represent the outage cost for the customer.

The area under the short-run demand curve DD gives the

loss in customer benefit.

0

i

P

is the price that the customer

is willing to pay in the very short-run period for the contin-

ued supp1y of e1ectricity. The very short-run demand

curve is initially very steep (vertical), as it represents the

price the customer (end use) is willing to pay to avoid any

service interruption.

Consider the scenario wherein a shortage of supply oc-

curs. In real-time, the customer would choose

i

Q

to

maximize profits:

() ()

e

i

i

Q

ii ii ii

Q

M

ax F Q MOC Q PQ

(3)

=>

i

ii

i

dF

M

OC P

dQ

(4)

where

i

M

OC

is the marginal outage cost of customer i

as a function of usage

i

Q

. At

0

ii

QQ

,

i

M

OC

is equal

to D’F (Fig. 1). The benefit that the customer obtains for

electricity usage

0

i

Q

is highlighted in case of supply

shortage; this is not the area under the very short-run demand

Fig. 1. Customer Demand and the Supply Curves of Elec-

tricity.

curve integrated from 0 to

0

i

Q

. In contrast, the customer

acts along with the very short-run demand curve to mini-

mize losses caused by the shortage. If the duration of the

outage is sufficiently long, or if the outage and subsequent

rise in price are known to the customer in advance, then the

customer would adjust in electricity consumption and set

0

i

i

i

dF

P

dQ

(i.e., to remain on demand curve DD). However,

the customer could not plan and/or adjust to the outage

condition during unexpected outages of short duration; thus,

the surplus of the consumer for customer

i

is expressed

by Equation (3).

Consider the case of a long-term rational customer

i

benefitting at time

t

.

it

F

depends not only on the electricity

usage of customer

i

at a certain time, but also on the usage

of customer

i

at other times, and possibly, on the usage

by other customers (end uses); that is,

()

it it

FFQ

where

{,}

it

QQiItT

. Assuming that the customer

seeks to maximize the total expected surplus of consumers

(i.e.,

{() }

it

t

E

FQ Costs

) for time T, where E is the

expectation operator. Thus, the customer would choose

it

Q

to maximize the expected profit,

() ()

it it it

t

M

ax E F Q OC Q P Q

(5)

it it

TT

it

it it

FOC

P

QQ

(6)

where

it

P

is the expected price of electricity. The outage

cost function includes loss in customer benefit (the bold

letters represent expected values). Thus, the customer

would equate the expected net marginal benefit to the ex-

pected price of electricity.

In the analysis that follows, the value-added function of

the customer is extended to include both real and reactive

power demands at the fundamental frequency of 60 Hz:

((), ())

dd

it

it i i i i

F

FP p Qq

(7)

where

d

i

P

: vector of real power demands of customer

i

,

d

i

Q

: vector of reactive power demands of customer

i

,

i

p

: vector of prices of real power for customer

i

, and

i

q

: vector of prices of reactive power for customer

i

.

A Study on Reliability Differentiated Pricing of Long-Term Transactions

10

2.2 Objective

The objective of the short-run operating and optimal

pricing problem in a welfare-maximizing utility using the

criterion of consumers plus producers surplus can be stated

as

(, ) (,)

dd uu

it i i it i i

ti i

WE FPQ OCPQMax

(, ) (,)

gg ss

it it it it it it

ii

PC P Q EC P Q

(8)

where

[]E

: expected value of the argument,

(, )

dd

it i i

F

PQ

: short-run va1ue-added function of customer

i

,

d

i

P

: real power demand of customer

i

,

d

i

Q

: reactive power demand of customer

i

,

u

i

P

: outage of real power for customer

i

,

u

i

Q

: outage of reactive power for customer

i

,

g

i

P

: generation of real power by generator

i

,

g

i

Q

: generation of reactive power by generator

i

,

s

i

P

: emergency purchase of real power at bus

i

,

s

i

Q

: emergency purchase of reactive power at bus

i

,

(, )

uu

it i i

OC P Q

: outage cost function of customer

i

,

(, )

g

g

it it it

P

CPQ

: cost of producing real and reactive power at

bus

i

, and

(, )

s

s

it it it

EC P Q

: cost of purchasing real and reactive power

at bus

i

.

All decision variables, with the exception of those in-

volving reactive power, are nonnegative. . The objective

function is assumed as a differentiable with a continuous

first derivative. For optimal operating and pricing strate-

gies, the objective is to maximize the subject for the oper-

ating and network constraints inherent in the system.

3. Optimal Long-Term Prices

Section 2 deals with the short-run problem of a welfare-

maximizing utility from which optimal short-term reliabil-

ity-differentiated prices are derived from. This section ex-

amines the long-run social welfare maximization problem

of the utility developed from the previous section. Based

on these, we derive the differentiated prices of long-term

reliability, which considers cost of capital or other fixed

costs of production. Long-term transactions, such as firm

capacity purchased from or by the utility, or purchased

from individual generators in the case of a network utility,

the access to transmission services and wheeling transac-

tions would require the provision of long-term prices in

order for the participants to enter into contracts and to ac-

commodate optimal investment decisions.

Consider the long-run resource planning problem of a

welfare-maximizing utility. To obtain the given values of

customer demands, the welfare-maximizing utility must

first solve the following problem of minimizing long-run

expected societal costs in order to determine the optimal

resource plan,

{(,) (,) (,)

dd uu gg

it i i it i i it it it

ti i i

M

axW E FPQ OCPQ PCPQ

(, )} ( , )

s

spq

it it it i i i

ii

E

CPQ CCK K

(9)

max

(,) ( )

pq

ii i K

k

ik

SC S S TC T

which is subject to the following constraints:

()0

gsud

it it it it it jt ij ij jt it

j

PPPP VVVCos

(10-1)

()0

gsud

it it it it it jt ij ij jt it

j

QQQQ VVVSin

(10-2)

0

gp

it i it

P

Ka

(10-3)

qgq

iit it iit

K

aQ Ka

(10-4)

max

ijt ij ijt

TTb

(10-5)

min max

iiti

VVV

(10-6)

(, , , ,,)0

ddggss

tt t t t t t

gP Q P Q PQ

(10-7)

0

s

p

it i

P

S

(10-8)

0

s

q

it i

QS

(10-9)

ng n n

iit it it

eP E A

(10-10)

ng n n

iit t t

i

eP E A

(10-11)

hhd

it i it

I dP

h=2,3,4,… (10-12)

hhh

it ij jt

j

VZI

h=2,3,4,… (10-13)

2

h

it

2

2

l

it

h

t

i

V

THD

NB V

(10-14)

maxt

THD D

(10-15)

0

u

it

P

(10-16)

0

u

it

Q

(10-17)

where

p

i

K

: real power generation capacity of generator i,

q

i

K

: reactive power generation capacity of generator i,

p

i

S

: real power spinning reserve capacity at bus i,

Balho H. Kim

11

q

i

S

: reactive power spinning reserve capacity at bus i,

and

max

k

T

: volt-ampere transmission capacity of line k

are the decision variables. In addition,

(,)

pq

ii i

CC K K

: capital cost of capacity for generator i,

(,)

pq

ii i

SC S S

: purchase price of spinning reserve at bus i,

and

max

()

kk

TC T

: capital cost of capacity for transmission line k.

This problem can be solved accordingly. By taking

d

it

P

and

d

it

Q

as the parameters of the problem, in order to ob-

tain the optimal values of all variables as the functions of

the parameters (i.e.,

d

it

P

and

d

it

Q

(sensitivity theorem), the

objective function can be written as

(,) (,) (,) (,)

dd dd dd dd

tt ttt ttt

it i i it

ti i

opt

F PQ OC P Q PCP Q ECP QW

(, ) (, ) (, )

dd dd dd

iik

iik

CCPQ SCPQ TCPQ

(11)

where

;,

dd

it

PPiItT

,

;,

dd

it

QQiItT

.

The customer sets the expected net marginal benefit

from the electricity consumption equal to the expected

price of electricity. Then, the respective optimal long-term

prices for real and reactive power inducing customers to

behave in a social welfare-maximizing manner are given

by:

L

it t t

jt

ddd

ij

j

tjtjt

OC PC EC

P

PPP

(12)

L

it t t

jt

ddd

ij

j

tjtjt

OC PC EC

q

QQQ

(13)

Thus, by employing socially optimal investment deci-

sion-making, long-term prices of real and reactive power

could coincide with short-term reliability-differentiated

prices. However, optimal investments require the evalua-

tion of future uncertainty, which is often difficult to carry

out. Another difficulty is the unavailability of proper de-

mand models that could capture the interdependence of

intertemporal demands. In principle, one should utilize

benefit functions that depend on the entire time stream of

demand rather than a single point in time.

4. Case Study

4.1 Implications for Pricing Firm Capacity

Firm capacity refers to the generation of capacity that is

purchased by the utility from neighboring utilities, namely,

independent power producers (IPP) and cogenerators, in

order to provide customers with long-run higher levels of

service reliability. The price paid for firm capacity should

equal the value or marginal benefit that the customers de-

rive from the added capacity. Purchasing firm capacity is a

long-term contract forged to protect against the loss of cus-

tomer load due to unplanned outage of generating units.

Thus, in order to determine the optimal price of firm capac-

ity purchases, the long-run problem of welfare-maximizing

utility given in Equations (9)–(10) must be considered.

Again, this problem could be solved when obtaining the

optimal values of all variab1es as functions of the parame-

ters of the problem, in particular,

p

i

K

and

q

i

K

(sensitivity

theorem).

The shadow price of added capacity (i.e., the Lagrange

multiplier of the capacity constraint) represents the will-

ingness of customers to pay for the obtained added capacity.

Based on Equations (12) and (13), the respective optimal

firm capacity purchase prices at bus j for real and reactive

power capacity are expressed by

P

ittt

j

PPP

tT i

jjj

OC PC EC

KKK

(14)

q

ittt

j

qqq

tT i

jjj

OC PC EC

KKK

(15)

4.2 Numerical Example: Calculation of Long-Term

Rates

Consider the simple four-bus system in Fig. 2. In the

figure, Gi and Li represent generating unit and load at Bus

i

, while Ti represents transmission line

i

. The character-

istics of the system are as follows:

G1: Rating = 200 MW, FOR= 0.00, Generation cost =

$30/MWH

G2: Rating = 250 MW, FOR= 0.10, Generation cost =

$0/MWH

G3: Rating = 110 MW, FOR = 0.00, Generation cost =

$40/MWH

G4: Rating = 250 MW, FOR = 0.20, Generation cost =

$20/MWH

L1: Customer load = 180 MW, Outage cost = $600/

MWH

L2: Customer load = 150 MW, Outage cost = $200/

MWH

L3: Customer load = 100 MW, Outage cost = $600/

MWH

L4: Customer load = 200 MW, Outage cost = $400/

MWH

T1: Rating = 100 MW, FOR = 0.00

T2: Rating = 120 MW, FOR = 0.00

T3: Rating = 80 MW, FOR = 0.30

T4: Rating = 80 MW, FOR = 0.00

A Study on Reliability Differentiated Pricing of Long-Term Transactions

12

Fig. 2. Four-Bus Power System for Sample Rate Calcula-

tions.

In the model, there are two generating units (G2 and G4)

and one transmission line (T3) with non-zero forced outage

rates. Thus, at each instance over time, there are eight pos-

sible system configurations or states that the system can

utilize. The load at each bus is assumed consisting of 1

MW of individual customer load for those that belong to

the same class (i.e., classes that have the same outage cost.)

Assuming that the system is operated optimally by a sin-

gle welfare-maximizing utility, then the expected reliabil-

ity-differentiated price for customers of class j is given by

i

j

ij

jj

OC

P

C

P

L

L

Thus, the expected prices for the customer classes of the

system are

1

P

= $72.96/MWH,

2

P

= $52.45/MWH

3

P

= $72.96/MWH,

4

P

= $87.72/MWH

The probability of service interruption (POSI) for cus-

tomers of each class, which measures the level of reliabil-

ity in which each customer class is served, are

POSI

1

= 0.0000, POSI

2

= 0.0825,

POSI

3

= 0.0000, POSI

4

= 0.0746.

The total expected cost of unserved energy for this linear

system is

$8, 444.00

i

ii

ii

i

OC

OC MOC PU

L

where

i

M

OC

is the marginal outage cost of customer

i

(e.g.,

2

M

OC

= $200/MWH).

The total expected revenue from the sale of electric

power is

$45,848.80

ii

i

TR PL

The total expected cost of generation is

$9,612.00

ii

i

PC MPC PG

where

i

M

PC

is the marginal production cost of generator

i

(e.g.,

1

M

PC

= $30/MWH).

The price of capacity purchase from generator

i

is

given by

j

i

j

ii

OC

P

C

K

K

Thus, the prices of capacity purchase from the genera-

tors of the system are

1

= $42.96/MW,

2

= $48.96/MW

3

= $40.16/MW,

4

= $21.60/MW.

The total payment to the generators for capacity pur-

chase is

$30,649.60

ii

i

CP K

The price of capacity purchase from transmission line k

is given by

max max

j

k

j

kk

OC

P

C

TT

Thus, the prices of capacity purchase from the transmis-

sion lines are

1

= $0.00/MW,

2

= $0.00/MW

3

= $25.20/MW,

4

= $44.64/MW

The total payment to the transmission lines for capacity

purchase is

max

$5,587.20

kk

k

TP T

The sum total of all expected costs/payments for the sys-

tem is

TC = PC + CP + TP =$45,848.80