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