Theory and Methodology

Optimizing the natural gas supply mix of local distribution

utilities
1
Jean-Michel Guldmann
a,
*
, Fahui Wang
b
a
Department of City and Regional Planning, The Ohio State University, 190 West 17th Avenue, Columbus, Ohio 43210, USA
b
Department of Geography, Northern Illinois University, DeKalb, Illinois 60115, USA
Received 1 January 1997; accepted 1 October 1997
Abstract
A large mixed-integer linear program (MILP) and a much smaller nonlinear programming (NLP) approximation of
the MILP, involving simulation and response surface estimation via regression analysis, are proposed to solve the
problem of the optimal selection of natural gas supply contracts by local gas distribution utilities. Each potential supply
source is characterized by several price and nonprice parameters. Weather variability is the basic stochastic factor that
drives the demand for gas by various market segments. The model minimizes the total cost of gas supply and market
curtailment, and thus determines the size of the interruptible market. A numerical application of the methodology il-
lustrates the excellent quality of the NLP approximation and the importance of the trade-os between contract char-
acteristics. A multi-temporal extension of the modeling methodology is outlined. 1999 Elsevier Science B.V. All
rights reserved.
Keywords: Energy; Purchasing; Optimization; Simulation; Regression
1. Introduction
During the past 15 years, there has been almost
continuous regulatory change in the US natural gas
industry, leading to signicant market changes. In
particular, Federal Energy Regulatory Commission
(FERC) Orders 436, 500 and 636 have transformed
the role of the transmission pipelines from mer-
chants, who owned the gas they transported from
the production areas to their customers, to nondis-
criminatory carriers. This has shifted the focus of
the industry to local gas distribution companies
(LDCs), which now face complex responsibilities in
making gas procurement and transportation deci-
sions. The unbundling of pipeline sales, transpor-
tation, and storage provides LDCs an opportunity
tosee more clearly the costs of each service, and thus
European Journal of Operational Research 112 (1999) 598612
*
Corresponding author. Fax: +1-614-292-7106; e-mail:
Guldmann.1@osu.edu.
1
This research has been, in part, supported by a grant from
the Urban Aairs and Urban Assistance Program, Ohio Board
of Regents. We are indebted to three anonymous referees for
valuable comments and suggestions on an earlier draft of this
paper.
0377-2217/99/$ see front matter 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 7 - 2 2 1 7 ( 9 7 ) 0 0 4 1 5 - 3
to achieve cost savings by building a least-cost
portfolio of these services. However, the sheer
number of these opportunities, and their complex
interactions with end-use gas demands, make such
contract portfolio selection much more complex, as
the LDCs must address not only cost issues, but also
supply reliability, price uncertainty, and various
operational factors. Comprehensive analytical tools
are therefore necessary to help LDCs make these
decisions. The purpose of this paper is to present
such a tool, with a focus on the optimal selection of
supply contracts, alsoknownas the ``optimal supply
mix'' problem.
The remainder of the paper is organized as fol-
lows. Section 2 expands the discussion and deni-
tion of the gas supply problem. Section 3 presents
the modeling approach. Section 4 consists in a
comprehensive numerical application of the meth-
odology. Section 5 presents a multi-period exten-
sion of the methodology. Section 6 concludes the
paper and outlines areas for further research.
2. The optimal gas supply mix problem
In order to clarify the gas supply mix problem,
and to set the methodology proposed in this paper
in proper perspective, we discuss the following
topics in this section: (1) gas demand variability, (2)
load balancing, (3) major characteristics of the new
supply situation, (4) objectives in building a port-
folio of contracts, (5) the focus of this paper, and
(6) a succinct literature review. For an extended
discussion of these issues, see also Duann (1991)
and EIA (1994, 1996).
A fundamental characteristics of the demand
for natural gas is its high variability. Over the
short-term, weather is the major factor, particu-
larly in service territories located in colder climates
and with a large share of space-heating customers.
Over the medium and longer terms, population
size, industrial activity, and the prices of gas and
alternative fuels, are of course major determinants.
Variations in industrial activities where gas is an
input to the production process, also have an eect
on short-term variations in gas demand. An overall
measure of this variability is the load factor, equal
to the ratio of the average day to the peak day
demands. The requirements of the residential and
commercial sectors display the highest seasonality.
These customers are generally classied as core (or
rm) customers, who have almost no alternatives
for fuel, and thus must be supplied on an uninter-
rupted basis. Noncore (or interruptible) customers
are those who have the ability to switch easily to
another fuel or to arrange gas purchases from other
sources (bypassing the LDC). The classication of
a given customer as core or noncore is somehow
relative, as any customer can technically establish a
fuel switching capacity, albeit maybe at a very high
cost. For instance, any residential customer could
maintain, together with a natural gas furnace, a
heating oil furnace or an electric heating system.
Hence, the costs of such alternatives, versus the
cost of receiving natural gas, should, to a large
extent, determine which market segments are core
and noncore. Such a consideration is a major fea-
ture of the methodology presented in Section 3.
A basic planning problem for an LDC is that of
supply-demand balance. On any given day with a
given potential demand, the LDC must decide how
much gas to take from its various supply sources,
and how much to allocate to the various customer
classes, possibly curtailing some of them if the
potential demand exceeds the potential aggregate
supply. This process is called load dispatching. In
the absence of storage, and when supply sources
are characterized only by unit commodity costs,
supplies are ordered by increasing unit cost, de-
mands are ordered by decreasing priority, and de-
mands are assigned to supplies until one or the
other runs out. However, this simple economic
dispatch becomes more complicated when supply
contracts are characterized by take-or-pay (T-O-P)
clauses, which specify a share (e.g., 60%) of the
maximum contract demand, that must be paid for,
even if the gas is not taken. In such a case, supply
sources are rst ranked in terms of decreasing unit
commodity cost, gas is taken in that order to avoid
all (or at least the most expensive) T-O-P decits,
and then the remaining volume (non-T-O-P) is
ordered from sources ranked by increasing unit
cost. Of course, this daily dispatching takes as
given the maximum daily supply available from
each source. However, this maximum daily supply
is a major decision variable in supply contracting,
J.-M. Guldmann, F. Wang / European Journal of Operational Research 112 (1999) 598612 599
and must be selected before any daily dispatching
takes place. The availability of seasonal under-
ground storage further complicates the supply
planning and dispatching problem. During the
summer season, when the demand is low, gas is
injected into storage; during the winter heating
season, that gas is withdrawn from storage to
complement the other supply sources. The use of
storage is driven in opposite directions by consid-
erations of reliability and economics. Reliability
suggests high storage inventories to allow for de-
mand peak forecasts that turn out to be low, and
unexpected supply interruptions. However, gas
inventory represents an investment of capital, and
is subject to carrying costs. Hence, economics
suggests low inventories. Also, storage introduces
complex intertemporal considerations: for in-
stance, interruptible customers can be curtailed
today instead of withdrawing gas from storage to
serve them. This gas can then be withdrawn later
on to serve core customers. Also, the volumes of
gas injection or withdrawal depend upon reservoir
pressure, hence upon the inventory, hence upon
past storage decisions. Finally, the dispatching
problem may be constrained by operational con-
siderations, such as the transmission capacity of
the LDC-owned pipelines that carry gas to the
various load centers.
Before the current restructuring of the gas in-
dustry, LDCs usedtobuy most of their gas fromone
or very few interstate pipelines under long-term
contracts (1020 y), under xed price conditions. In
this very stable environment, supply costs were
simply passed on to the LDC's customers, and there
was little attempt to minimize supply costs. Now
that pipelines act as carriers, and no longer as gas
merchants, LDCs purchase gas directly from pro-
ducers or marketers, arrange for transportation
services with one or more pipelines, and, more and
more, contract for seasonal storage services. Supply
contracts vary considerably in terms of duration
and pricing and nonpricing clauses. They include:
1. Short-term contracts, where xed daily volume
deliveries, at xed price, are arranged for one
month or less; they allow for short-term unbal-
ances in supply and/or demand to be corrected;
such contracts make up what is called the spot
market.
2. Mid-term contracts, for periods of up to 18
months, with variable prices indexed to some
future or spot price, and with xed reservation
and service fees, irrespective of volumes taken.
3. Long-term contracts, for periods of 18 months
to 1520 y, with reservation fees and minimum
take provisions; prices are indexed, and con-
tracts often include renegotiation and market-
out clauses.
In addition to contracting with gas suppliers,
LDCs must also contract for rm and/or inter-
ruptible transportation capacity with the pipelines
that will carry gas from the suppliers to the LDC.
Such contracts are further complicated by capacity
release programs, which enable an LDC with re-
served transportation capacity to release excess
capacity to a replacement shipper, with revenues
osetting some of the rm transportation reserva-
tion costs. Similar contracting decisions (e.g., rms
vs. interruptible) can be made with regard to stor-
age services. Finally, the LDC may use nancial
instruments to manage the risks in volatile gas
prices (e.g., futures contracts, options, swaps).
While these new contracting opportunities oer
savings potential for LDCs, they also carry signi-
cant risks. Reliance on short-term agreements
may render the LDC vulnerable to both price
shocks and gas shortages if, at some future time, a
current excess supply disappears and long-term
contracts are dicult to obtain. Short-term cost
savings may then be outweighed by higher long-
term costs, and reliable supply to core customers
may be jeopardized. On the other hand, too cau-
tious an approach, relying on long-term contracts
only, may deny customers signicant opportunities
for rate decreases. Thus, the LDC must build a
portfolio of contracts for gas purchases, transpor-
tation, and other services, that balances the con-
icting criteria of short-term cost, long-term costs,
price stability, supply reliability for core customers,
and overall gas transportation availability. To
build such a portfolio requires taking into account
all potential contracts characteristics (durations,
prices, volumes, minimum takes, etc.) and all
market segments characteristics (demand variabil-
ity, curtailment costs, expected growth, etc.),
within the framework of a single- or multi-objective
optimization model.
600 J.-M. Guldmann, F. Wang / European Journal of Operational Research 112 (1999) 598612
The purpose of this paper is to develop such a
model for a simplied portfolio problem where: (1)
simple supply contracts, of equal duration, are
considered, characterized by a peak deliverability, a
T-O-P minimum, and demand and commodity
charges; (2) transportation and storage services are
ignored; and (3) contract characteristics remain
constant. This model might constitute the kernel of
larger, more comprehensive models that would ac-
count for the other above-mentioned factors. An
example of suchextensions is presentedinSection 5.
In view of the practical importance of the LDC
supply portfolio problem, it is surprising to dis-
cover how small the literature on this subject is.
O'Neill et al. (1979) develop a network model that
allocates gas from supply nodes to demand nodes
while accounting for mass conservation and pres-
sure constraints across a pipeline network, but they
do not account for weather-related demand vari-
ability. Guldmann (1983) analyzes the interactions
between gas supply, storage, and service reliability,
using chance-constrained programming and ac-
counting for random weather variability from
month to month. While storage operational con-
straints are very detailed, only one supplier is
considered. In contrast, several suppliers and their
contract characteristics are considered in Guld-
mann (1986), but in a deterministic framework, i.e.,
only monthly expected demands are considered.
Also, the focus is primarily on pricing, and not on
supply mix optimization. A more comprehensive
approach is proposed in Avery et al. (1992), dealing
with both supply and transportation/storage con-
tracts, as well as with the network structure of the
pipeline companies considered, but still in a deter-
ministic framework. It is, however, clear that much
work is being carried on this issue within the in-
dustry (e.g., Jacobs, 1993) and its consultants (e.g.,
Planmetrics, 1988). Unfortunately, much of this
work is proprietary and remains unpublished.
3. The modeling approach
3.1. Denitions and assumptions
Consider a given market of potential end-use
gas consumers served by an LDC. This market
can be segmented into M (m = 1; . . . ; M) homo-
geneous submarkets (e.g., residential, commercial,
industrial, etc.). The basic factor that drives
variations in submarket gas demand is the
weather, as characterized by the number of daily
heating degree-days X, which is dened as the
dierence between the reference (thermal equi-
librium) average daily temperature of 65

F and
the actual average daily temperature T. More
precisely, X = max(0; 65T), which implies that
X = 0 whenever T P65

F. We assume that a
statistical analysis of historical daily degree-day
data indicates that there are S (s = 1; . . . ; S)
possible dierent states for this random variable,
and we dene X
s
to be the number of daily
heating degree-days for state s, and P
s
to be the
probability of occurrence of state s.
The daily gas requirement of submarket m un-
der state s is a linear function of X
s
, with
Q
ms
= A
m
B
m
X
s
; (1)
where A
m
and B
m
are the base-load and heating-
load coecients for submarket m, respectively.
Some market segments may have a constant re-
quirement (e.g., industrial users that need gas in
production processes only), thus B
m
= 0. Other
segments may use gas for space-heating only (e.g.,
some residential users), thus A
m
= 0. However, in
general, both requirement components are pres-
ent in all submarkets, albeit in highly varying
degrees. We assume that A
m
and B
m
are exoge-
nously specied and remain constant within the
framework of this analysis, although it is clear
that they do vary over the long term as functions
of the prices of gas and alternative competing
energy sources, as well as of the costs of energy
conservation.
Curtailment of market m occurs if the demand
Q
ms
is not fully satised. Such curtailment entails a
cost, which may be related to a loss of production
for an industrial user, or to the use of a more ex-
pensive backup source of energy (e.g., oil burning
in dual-fuel furnaces), or to other forms of hard-
ship (e.g., school closings, health eects, etc.). We
assume that the total curtailment cost is propor-
tional to the amount of gas curtailed. This constant
J.-M. Guldmann, F. Wang / European Journal of Operational Research 112 (1999) 598612 601
unit cost assumption may not be fully satisfactory
if market m is substantially heterogeneous, that is,
made of sub-markets with dierent unit costs (e.g.,
industry). Further market segmentation would be
the simplest way to solve this problem, but would
increase the size of the model. We dene CR
m
to be
the unit curtailment cost for market segment m.
To satisfy its customers, the LDC can pur-
chase gas from N(i = 1; . . . ; N) suppliers, which
provide gas under specic contract characteristics.
In the following, we shall use the terms ``suppli-
er'', ``supply source'', and ``contract'' inter-
changeably. Each contract i is characterized by
three exogenous parameters: CC
i
is the com-
modity charge, i.e., cost per unit of gas actually
taken from source i; CD
i
the demand charge, i.e.,
cost per unit of maximum daily deliverability
(demand costs are strictly proportional to maxi-
mum deliverability); t
i
the take-or-pay (T-O-P)
rate, which species the minimum share of the
demand contract, below which gas must be paid
for, whether actually taken or not.
The basic initial decision that the LDC must
make with regard to contract i is the demand
contract D
i
, equal to the maximum daily deliver-
ability from source i. While some contracts do al-
low daily purchasing exceeding D
i
under some
circumstances, and with a cost penalty, we assume
that D
i
is the upper limit of daily supply from
source i on any day. The minimum purchase im-
plied by the T-O-P rate is then t
i
D
i
.
3.2. The optimal supply mix as a mixed-integer
linear program
In some ways, the gas supply mix problem is
similar to the generation mix and power dispatch-
ing problem in the electric utility industry. How-
ever, a distinct and complicating factor is the
minimum purchase, which implies some form of
discontinuity in the amount of gas actually taken,
and requires the use of integer 01 variables. Let
Q
s
=
X
M
m=1
Q
ms
=
X
M
m=1
(A
m
B
m
X
s
) (2)
be the total market requirement for gas under state
s. The maximum total amount of gas that can be
purchased on any day is:
P
i
D
i
. The total amount
of gas to be curtailed under state s is then
Q
Cs
= max 0; Q
s

X
i
D
i
!
: (3)
Next, we dene the total amount of gas required
by the market under state s beyond the minimum
total purchase,
P
i
t
i
D
i
, with:
Q
Ms
= max 0; Q
X
i
t
i
D
i
!
: (4)
The supply and curtailment variables under state s
are Y
is
, equal to the amount of gas taken from
source i in addition to the minimum take t
i
D
i
, and
Z
ms
, equal to the amount of gas curtailed for
market segment m.
The supply constraints are:
Y
is
6(1 t
i
)D
i
(i = 1; . . . ; N; s = 1; . . . ; S); (5)
i.e., the incremental take is limited by the demand
contract D
i
; and
X
N
i=1
Y
is

X
M
m=1
Z
ms
= Q
Ms
(s = 1; . . . ; S); (6)
i.e., the sum of the total incremental take and
market curtailment is equal to the total excess re-
quirement Q
Ms
.
The curtailment constraints are:
Z
ms
6Q
ms
(m = 1; . . . ; M; s = 1; . . . ; S); (7)
i.e., it is impossible to curtail more than the total
gas requirement Q
ms
; and
X
M
m=1
Z
ms
= Q
Cs
(s = 1; . . . ; S); (8)
i.e., the aggregate decit Q
Cs
must be allocated
among the market segments.
The total cost to minimize is the sum of supply
and curtailment costs. Supply costs include (1) the
minimum bills related to contract demand and T-
O-P penalties, and (2) the expected cost of the ac-
tual gas takes beyond the minimum levels. The
minimum bill (or xed) cost is
602 J.-M. Guldmann, F. Wang / European Journal of Operational Research 112 (1999) 598612
CF =
X
N
i=1
(CD
i
CC
i
t
i
)D
i
: (9)
The expected supply cost is
CS =
X
S
s=1
P
s
X
N
i=1
CC
i
Y
is
!
: (10)
The expected curtailment cost is
CR =
X
S
s=1
P
s
X
M
m=1
CR
m
Z
ms
!
: (11)
The optimization problem is to select the values
of the decision variables D
i
; Y
is
; Z
ms
[ i =1; . . . ; N; s
= 1; . . . ; S; m = 1; . . . ; M that minimize the total
cost
CT = CF CS CR (12)
subject to constraints (3)(8). Note that the model
can set D
i
= 0 for all supplies if curtailment costs
are much lower than supply costs, which would
mean that natural gas is not competitive with al-
ternative energy supplies or nonsupply strategies
such as energy conservation. At the other end of
the spectrum, the D
i
's may be set high enough,
when natural gas supply costs are low, to eliminate
the need for any curtailment.
Constraints (3), (4) cannot be used in their
present forms with standard algorithms. In order
to make the determination of the variables Q
Ms
and
Q
Cs
endogenous to the model, we transform the
max functions into sets of linear constraints via the
introduction of new 01 variables:
w
Ms
=
1 if Q
Ms
= 0;
0 if Q
Ms
= Q
s

P
i
t
i
D
i
> 0;

w
Cs
=
1 if Q
Cs
= 0;
0 if Q
Cs
= Q
s

P
i
D
i
> 0:

The following constraints guarantee the proper

determination of Q
Ms
and Q
Cs
, with M
0
being a
large given number:
Q
Ms
6(1 w
Ms
)M
0
; (13)
Q
Ms
PQ
s

X
i
t
i
D
i
; (14)
Q
Ms
6Q
s

X
i
t
i
D
i
w
Ms
M
0
; (15)
Q
Cs
6(1 w
Cs
)M
0
; (16)
Q
Cs
PQ
s

X
i
D
i
; (17)
Q
Cs
6Q
s

X
i
D
i
w
Cs
M
0
: (18)
The nal model, made of the objective function
(12) and the constraints (5)(8) and (13)(18), is a
mixed-integer linear program (MILP). The nal
set of decision variables is: D
i
; Y
is
; Z
ms
; Q
Ms
;
Q
Cs
; w
Ms
; w
Cs
. The total number of variables is
NV = NS MS N 4S, of which 2S are pure 0
1 variables. The number of constraints is
NC = (N M 8)S.
It is worth noting that the model is structurally a
two-stage stochastic programming problem with
recourse (Wets, 1966), where the D
i
's are the rst-
stage decision variables, and all the other variables
are the second-stage or recourse variables. How-
ever, in the standard model, only one of the S
scenarios does materialize, and the recourse vari-
ables are selected accordingly. In the present
model, all the S scenarios do take place sometime
during the year. The daily cost TC, when multiplied
by 365, represents the total annual cost of supply
and curtailment. Under the assumption that the
weather pattern (i.e., the distribution of X
s
) is the
same from year to year, this total annual cost is
deterministic.
The size of the MILP is highly dependent on S,
the number of possible values of daily heating de-
gree-days. If we assume that S = 100, the number
of market segments M = 10, and the number of
supply sources N = 10, then the model would in-
clude 2800 constraints, 2210 continuous variables,
and 200 integer variables. A value of S = 100
would correspond to an area where average daily
temperatures (rounded to the closest integer value)
take all values between 34

F and 65

F during the
year, a reasonable assumption for many northern
US states. While a model of this size can be solved
with available MILP algorithms, such computa-
tions might become cumbersome in the framework
J.-M. Guldmann, F. Wang / European Journal of Operational Research 112 (1999) 598612 603
of an extensive sensitivity analysis over the model
parameters or for much larger model sizes. An-
other approach would be to implement a resource
directive decomposition of the MILP, using
Benders' algorithm (see, for instance, Shapiro,
1979, Chapter 6, and Bienstock and Shapiro, 1988),
wherein the coordination problem would involve
the choice of D
i
's, and a subproblem would be
dened for each state s and would involve the
corresponding optimal dispatch and curtailment
for given D
i
's.
3.3. The optical supply mix as a combined simula-
tion/optimization
As an alternative to MILP branch-and-bound
algorithm or decomposition methods, we propose
a combined simulation/response curve estimation/
optimization approach to solve the supply mix
problem. Consider a given state measured by X
s
,
and assume that all the contract demands D
i
(i = 1; . . . ; N) are given exogenously. Under such
conditions, the optimal (least-cost) sequencing of
gas takes from the N sources by the gas dispatcher
is a straightforward task:
(1) If the total market demand Q
s
, (Eq. (2)) is
less than the sum of the T-O-P minimum purchases
(
P
i
t
i
D
i
), then the minimum total bill CF (Eq. (9))
applies, and there are no other costs. The dis-
patcher can take gas from any source i up to the
minimum level t
i
D
i
without any eect on CF.
(2) If
P
i
t
i
D
i
6Q
s
6
P
i
D
i
, then the dispatcher
will rst exhaust the minimum takes t
i
D
i
and then
will satisfy the residual gas requirement
(Q
s

P
i
t
i
D
i
) by purchasing gas from the N
sources ranked in terms of increasing commodity
charge CC
i
. The dispatcher starts purchasing from
source i only after having exhausted the available
supply from source i 1, that is: Y
is
> 0 i
Y
i1s
= (1 t
i
)D
i
. The total supply cost for state s is
then:
P
i
CC
i
Y
is
.
(3) If Q
s
>
P
i
D
i
, then curtailments are neces-
sary. The gas dispatcher purchases all the avail-
able gas (
P
i
D
i
), and the total supply cost is then:
P
i
(CD
i
CC
i
)D
i
. The total curtailment volume,
Q
s

P
i
D
i
, is then allocated to the various market
segments in terms of increasing unit curtailment
cost. The dispatcher starts curtailing segment m
only if segment m 1 has been completely cur-
tailed, that is: Z
ms
> 0 i Z
m1s
= Q
m1s
. Note,
however, that the total curtailment of market m
under all states,
P
s
Z
ms
, would, generally, only
represent a partial curtailment of its total re-
quirement,
P
s
Q
ms
. The total curtailment cost for
state s is then:
P
m
CR
m
Z
ms
. It must be added to
the xed and supply costs to obtain the total cost
CT
s
.
The above simulation is then repeated for all
states s, leading to a set of costs CT
s
(s = 1; . . . ; S).
The next step is to weigh these costs by their
probabilities, with:
CT =
X
S
s=1
P
s
CT
s
: (19)
CT is the minimum total cost for given contract
demands D
i
(i = 1; . . . ; N). Thus:
CT = CT(D
1
; . . . ; D
i
; . . . ; D
N
): (20)
The optimization problem now is to select the D
i
's
that minimize CT(D); D = (D
1
; . . . ; D
i
; . . . ; D
N
).
As there is no closed-form expression of CT(D),
the only possible approach is to approximate this
function by (1) generating a response surface by
varying the D
i
's extensively over a grid of values
and computing CT by simulation for every com-
bination of the D
i
's and (2) estimating the response
surface CT(D) with curve-tting regression tech-
niques. While the problem
Minimize
(D
i
)
CT(D
1
; . . . ; D
i
; . . . D
N
) (21)
is considerably simpler than the earlier MILP (N
continuous variables, no constraints), one could
argue that this simplication does not justify the
necessary large number of simulations and the
regression curve-tting. However, the approach
can be expanded to include, among the arguments
of CT(:), not only the decision variables D, but
also the various parameters of the model. Consider
the following vectors of parameters: CC = CC
i
;
CD = CD
i
; CR = CR
m
; T = t
i
; A = A
m
;
B = B
m
. By varying these parameters, together
with D = D
i
, over grids of values, we could es-
timate a more general function
604 J.-M. Guldmann, F. Wang / European Journal of Operational Research 112 (1999) 598612
CT = CT(D; CC; CR; T; A; B); (22)
which would provide a convenient approach to
sensitivity analyses over the parameters. Such an-
alyses may be critical to gas supply planners when
the parameters are uncertain or may be random
variables. For instance, the base and heating load
coecients A and B may change in the future due
to energy conservation and/or competition from
other energy sources. Analyzing the impacts of
such possible changes on the optimal demand
contracts D may help the supply planner select a
robust contract portfolio. Similar arguments may
be made with regard to the other parameters. Of
course, the number of simulation runs would in-
crease with an increase in the number of parame-
ters, and, as a result, the accurate estimation of the
function (22) would become more dicult. How-
ever, as illustrated in the application presented in
Section 4.2, this diculty may be alleviated by a
pre-estimation analysis of the specic roles of the
parameters in the dierent cost components.
4. Application
4.1. Data
The data used to illustrate the modeling ap-
proach are, in part, hypothetical, yet realistic, and,
in part, based on actual data characterizing an
LDC, the National Fuel Gas Distribution Com-
pany (NFGDC), which serves primarily the west-
ern parts of the states of New York and
Pennsylvania. The availability of usable data was
the primary criterion in selecting this LDC.
Gas requirement data were drawn from the
Uniform Statistical Report (USR) led by
NFGDC with the American Gas Association
(AGA). This report provides monthly gas sales to
various customer classes (sectors), and the corre-
sponding numbers of monthly heating degree-days.
NFGDC market is segmented into four sectors:
residential, commercial, industrial, and public au-
thorities (noncommercial institutional buildings).
Using the 1982 USR, we have divided each
monthly gas volume and the number of degree-
days by the number of days in that month. Next,
we have regressed, for each sector, the average
daily gas requirement on the average daily number
of degree-days. The regression results, with the
corresponding R
2
and t-values, are presented in
Appendix A. The base and heating load coecients
(A
m
and B
m
) to be used here are based on these
estimated values, with some rounding, and are
presented in Table 1.
Daily heating degree-day data were available
for Bualo, NY (located in NFGDC service terri-
tory) for 4 yr (19761979). A frequency analysis of
these data shows that X
s
ranges from 0 to 70, with
70 distinct values (i.e., states), the only value
missing in the range [0,70] is 69. The average daily
heating degree-day is X = 19:26. About 25% of the
days of the typical year (here assumed to be the
average of 19761979) have zero heating degree-
days, and thus no space-heating requirements.
Using the selected load coecients, we have com-
puted the average and peak daily requirements for
each sector (with X = 19:26 and X
p
= 70). Their
ratio, the load factor, is a measure of the variability
of the daily requirements (e.g., a 100% load factor
Table 1
Gas markets load and curtailment cost data
Market segment Load coecients (MCF)
a
Average load
(MCF)
Peak load
(MCF)
Load factor Curtailment unit
cost ($/MCF)
Base Space-heating
Residential 55,000 11,000 266,838 825,000 0.323 12.00
Commercial 15,000 3000 72,774 225,000 0.323 8.00
Industrial 100,000 3000 157,774 310,000 0.509 6.00
Public authorities 5000 1200 28,110 89,000 0.316 9.00
Total 175,000 18,200 525,496 1,449,000 0.363 )
a
MCF: thousand cubic feet.
J.-M. Guldmann, F. Wang / European Journal of Operational Research 112 (1999) 598612 605
indicates a completely constant demand). These
loads and load factors are also presented in Ta-
ble 1.
We consider ve hypothetical gas suppliers,
ranked (1; . . . ; 5) in terms of their increasing com-
modity charge: $2:00; $2:50; $3:00; $3:50; $4:00; per
MCF (thousand cubic feet). To these charges
(CC
i
), which are assumed constant throughout the
analysis, are associated demand charges (CD
i
) and
T-O-P rates (t
i
). We assume that the demand
charge may vary between $0:20/MCF and $0:80/
MCF, and the T-O-P rate between 0.40 and 0.80. A
primary goal of the analysis reported in Section 4.3
will be to uncover the trade-os between these
contract parameters.
The selection of the sectoral unit curtailment
costs was based, in part, on the end-use prices of oil
products in 1990 in the state of New York, as
drawn from EIA (1992). The average price of dis-
tillate oil in the industrial sector was $6:20 per
MMBtu (Million British Thermal Units). In the
commercial and residential sectors, we have com-
puted the average of the prices of distillate fuel and
liqueed petroleum gas (LPG), with $8:50/MMBtu
and $11:00/MMBtu for commercial and residential
end-users, respectively. The selected curtailment
costs, presented in Table 1, are close to these val-
ues. No data were available for the Public Au-
thorities sector, for which we selected a unit cost of
$9:00/MCF, close to but higher than the commer-
cial unit cost.
4.2. Response curve estimation
The average daily cost CT is the sum of three
components: (1) the xed cost, CF, equal to the
sum of the demand and T-O-P costs, (2) the ex-
pected supply cost, CS, and (3) the expected cur-
tailment cost, CR. Eq. (9) in Section 3.2 provides
an explicit formulation of CF, allowing the esti-
mation process to focus on CS and CR. Consider
rst the case of CR. As the unit curtailment costs
CR
m
are xed, it is clear that the distribution of the
total daily volumes of gas curtailed, and thus the
distribution of the resulting costs, CR
s
, only de-
pend upon the aggregate contract demand
D
T
=
P
i
D
i
, but not upon the individual demands
D
i
's. Hence, a general formulation of the expected
curtailment cost function is
CR = f (D
T
): (23)
We have simulated the curtailment process by
unit-incrementing D
T
from 0 to 1500, and for each
value of D
T
we have computed CR. We have next
used a third-order polynomial
CR = a
0
a
1
D
T
a
2
D
2
T
a
3
D
3
T
(24)
to approximate the true cost function (23). An
excellent t was obtained, with R
2
= 0:999. Also,
the coecient a
0
was estimated at 9.365, which is
very close to the true average curtailment cost of
9.484 when D
T
= 0. The case of the supply cost CS
is, however, more complicated. We dene the
minimum take for contract i as: Z
i
= t
i
D
i
. We
know that supply dispatching takes place when-
ever
P
i
Z
i
< Q
s
, and that the total amount dis-
patched is limited by D
T
=
P
i
D
i
. The amount of
gas dispatched from each source i will depend
upon the ranges [Z
i
D
i
[ and the commodity costs
CC
i
. Depending upon the state s, this amount
varies between 0 and (D
i
Z
i
), and the corre-
sponding cost between 0 and CC
i
(D
i
Z
i
). How-
ever, the distribution of the supply cost for
contract i clearly depends upon the values of D
j
and Z
j
for all the other contracts j, hence a com-
plex interaction between the variables D
k
and Z
k
and the cost parameters CC
k
(k = 1; . . . ; 5). As the
CC
k
's are assumed xed, we focus on the varia-
tions of D
k
and t
k
, and hence on those of Z
k
. Let
Z = (Z
1
; . . . ; Z
5
) and D = (D
1
; . . . ; D
5
). We can
then posit a general cost relationship
CS = g(Z; D): (25)
We have simulated the dispatching model de-
scribed in Section 3.3 over a grid of values for (1)
the demand contracts D
i
, and (2) the T-O-P rates
t
i
(i = 1; . . . ; 5). Each D
i
may take six values (0,
300, 600, 900, 1200, and 1500 MMCF), but all
combinations such that
P
i
D
i
> 1500 were dis-
carded, because of their irrelevance with a peak
daily load of 1449 MMCF (see Table 1). Each t
i
may take the following values: 0.4, 0.5, 0.6, 0.7,
0.8. The total number of combinations in the input
606 J.-M. Guldmann, F. Wang / European Journal of Operational Research 112 (1999) 598612
space (D; T) or (D; Z) is 787,500. We have next
used a third-order polynomial
CS = a
0

X
i
a
1
i
Z
i

X
i;j
a
2
ij
Z
i
Z
j

X
i;j;k
a
3
ijk
Z
i
Z
j
Z
k

X
i
b
1
i
D
i

X
i;j
b
2
ij
D
i
D
j

X
i;j;k
b
3
ijk
D
i
D
j
D
k
(26)
to approximate the true cost function (25). An
excellent t was obtained, with R
2
= 0:994.
Prior to the estimations of Eqs. (24) and (26),
the costs CR and CS have been divided by the
average total daily gas requirement TD =
P
s
P
s
Q
s
,
or 525,496 MCF (see Table 1). The average daily
cost AC ( =CT/TD) has, as arguments, the vectors
D; CD, and T, and is summarized as follows:
AC = (a
0
a
0
)
X
5
i=1
[(CD
i
CC
i
t
i
)=TED
a
1
a
1
i
t
i
b
1
i
[D
i

X
i;j
(a
2
ij
t
i
t
j
b
2
ij
)D
i
D
j
a
2
X
i
D
i
!
2

X
i;j;k
a
3
ijk
t
i
t
j
t
k
b
3
ijk

D
i
D
j
D
k
a
3
X
i
D
i
!
3
: (27)
4.3. Optimal contract selection
The MILP presented in Section 3.2 includes
1190 constraints, 140 integer 01 variables, and 915
continuous variables, when applied with the data
presented in Section 4.1 (S = 70; N = 5; M = 4).
The nonlinear program (NLP) involved in mini-
mizing the cost function (27) involves only ve
continuous variables (D
i
's). The MILP and NLP
have been solved using the OSL and MINOS
solvers of GAMS (General Algebraic Modeling
System). For further details, see Brooke et al.
(1996). In the following, we rst assess the quality
of the NLP approximation by comparing its solu-
tion to the true optimum generated by the MILP.
Next, we analyze under what parameter values do
contracts 25 become competitive with contract 1.
4.3.1. Assessment of the NLP approximation
We assume here that the parameters CD
i
and T
i
are the same for all ve contracts. Contract 1 (with
CC= $2:00=MCF) is then the only one selected. Of
course, such a situation could not take place in
reality, because suppliers 25 would always lose,
and thus be out of business. It is simply a conve-
nient way to analyze the case of one sole supplier
and its contract. We vary T
i
from 0.4 to 0.8 by in-
crements of 0.1, and CD
i
from 0.2 to 0.8 by in-
crements of 0.1. The resulting minimum average
costs and optimal demand contracts for supplier 1,
as produced by the MILP, are presented in Ta-
bles 2 and 3. As could be expected the minimum
average cost AC decreases and the demand con-
tract D
1
increases with decreasing T-O-P rates and
decreasing demand charges. The lowest
AC=2.781 (T
i
= 0:4 and CD
i
= 0:2) represents
about 65% of the highest AC = 4:308, and the
lowest D
1
= 684:6 (T
i
= CD
i
= 0:8) represents
about 70% of the highest D
1
= 971:8 (T
i
= 0:4 and
CD
i
= 0:2), clearly underlining the strong impact
that the contract parameters T and CD have on the
overall supply cost and quantity. As the average
daily load is 525.5 MMCF, and the peak one 1449
MMCF, it is clear that the optimal values of D
1
are
always larger than the average load, and always
smaller than the peak one. Thus, curtailments oc-
cur in all cases, as it is never optimal to contract for
the peak load. The LDC's customer set must in-
clude interruptible customers, which will be cur-
tailed when the actual load exceeds the contract
demand. While the desirability of having inter-
ruptible customers is well known and well accepted
by LDCs, the optimal mix of rm and interruptible
loads is generally not easy to determine. The
present model does just that.
The minimum average costs and optimal de-
mand contracts for supplier 1, as produced by the
NLP, are presented in Tables 4 and 5, together
with the percentage deviations from the exact
MILP solutions. The relative cost deviation varies
between )1.99% (NLP underestimation) to +0.95%
(NLP overestimation). In absolute terms, the cost
deviation varies between 0.41% and 1.99%, with a
mean of 1.33%. The relative demand deviation
varies between )1.55% and 3.43%, and, in absolute
terms, between 0.017% and 3.43%, with a mean of
J.-M. Guldmann, F. Wang / European Journal of Operational Research 112 (1999) 598612 607
1.06%. These results suggest that the NLP solution
is, generally, a very close approximation of the
exact MILP solution. Using the NLP approxima-
tion is further warranted by the dierence in com-
putation time. The average time for solving the
MILP on a Pentium-based 100 MHz personal
computer is 54.12 CPU seconds, while the time for
the NLP is 0.133 CPU seconds, implying that
solving the NLP is 407 times faster than solving the
MILP. While this ratio cannot be extrapolated as
such to larger problems, it suggests that large
computational gains may be achieved by using the
simulation/estimation/NLP approach.
4.3.2. Contract trade-o analysis
Because of its relatively small size, we continue
using the MILP formulation of the model to assess
under what conditions contracts 25 become
competitive with or preferable to contract 1.
Throughout, we keep contract 1 parameters con-
stant, with T
1
= 0:8 and CD
1
= 0:8 (the most un-
favorable conditions). We rst vary the parameters
T
2
and CD
2
of contract 2 over the previous ranges,
while keeping the parameters for contracts 35 at
the same levels as contract 1. Hence contracts 35
cannot become active, and the choice is between
contracts 1 and 2. The results are presented in
Tables 6 and 7. The indierence curve for the two
contracts, that is, the set of combinations (CD
2
; T
2
)
for which the two contracts are equally attractive,
is clearly suggested by Tables 6 and 7 and repre-
sents, roughly, a sub-diagonal of the tables. For
any combination (CD
2
; T
2
) below this curve, con-
tract 1 is always selected. Contract 2 is always se-
lected for any combination (CD
2
; T
2
) above the
curve. Note that both contracts are active
(D
1
> 0; D
2
> 0) in six cases (see footnote of Ta-
ble 7).
Next, we carry the same sensitivity analysis over
the parameters of contract 3 (CD
3
; T
3
), leaving the
parameters of all other contracts at CD
i
= 0:8 and
T
i
= 0:8. The choice is thus here between contracts
1 and 3. The results are presented in Table 8. As
could be expected, lower values for CD
3
and T
3
Table 3
MILP optimal contract demand (MMCF) contract #1
Demand charge ($/MCF) Take-or-pay rate
0.400 0.500 0.600 0.700 0.800
0.2 971.80 903.00 865.40 812.00 775.60
0.3 921.20 880.60 835.00 793.80 757.40
0.4 895.80 850.20 812.00 775.60 742.00
0.5 866.60 830.20 793.80 757.40 728.60
0.6 847.00 812.00 775.60 743.80 719.25
0.7 812.00 789.40 757.40 728.60 702.80
0.8 793.80 774.20 739.20 718.00 684.60
Table 2
MILP minimum average costs ($/MCF) contract #1
Demand charge ($MCF) Take-or-pay rate
0.400 0.500 0.600 0.700 0.800
0.2 2.781 2.943 3.115 3.292 3.474
0.3 2.960 3.113 3.276 3.445 3.620
0.4 3.133 3.278 3.433 3.595 3.763
0.5 3.301 3.438 3.585 3.742 3.904
0.6 3.463 3.593 3.735 3.885 4.041
0.7 3.621 3.745 3.881 4.026 4.176
0.8 3.774 3.893 4.024 4.163 4.308
608 J.-M. Guldmann, F. Wang / European Journal of Operational Research 112 (1999) 598612
Table 4
NLP minimum average costs ($/MCF) and deviations (%) from the MILP optimum contract #1
Demand charge ($/MCF) Take-or-pay rate
0.400 0.500 0.600 0.700 0.800
0.2 2.800 2.916 3.062 3.228 3.405
(0.68)
a
()0.92) ()1.70) ()1.94) ()1.99)
0.3 2.979 3.085 3.222 3.379 3.549
(0.64) ()0.90) ()1.65) ()1.92) ()1.96)
0.4 3.154 3.251 3.379 3.528 3.691
(0.67) ()0.82) ()1.57) ()1.86) ()1.91)
0.5 3.324 3.413 3.533 3.674 3.829
(0.70) ()0.73) ()1.45) ()1.82) ()1.92)
0.6 3.490 3.571 3.683 3.817 3.965
(0.78) ()0.61) ()1.39) ()1.75) ()1.88)
0.7 3.652 3.726 3.831 3.958 4.099
(0.86) ()0.51) ()1.29) ()1.69) ()1.84)
0.8 3.810 3.877 3.975 4.095 4.230
(0.95) ()0.41) ()1.22) ()1.63) ()1.81)
a
The percent deviation of the NLP value from the MILP value is indicated in parentheses.
Table 5
NLP optimal contract demand (MMCF) and deviations (%) from the MILP optimum contract #1
Demand charge ($/MCF) Take-or-pay rate
0.400 0.500 0.600 0.700 0.800
0.2 957.57 902.85 852.39 806.10 763.60
()1.46)
a
()0.02) ()1.50) ()0.73) ()1.55)
0.3 930.45 880.94 833.90 789.99 749.24
(1.00) (0.04) ()0.13) ()0.48) ()1.08)
0.4 905.52 860.26 816.20 774.45 735.31
(1.08) (1.18) (0.52) ()0.15) ()0.90)
0.5 882.31 840.63 799.19 759.40 721.76
(1.81) (1.26) (0.68) (0.26) ()0.94)
0.6 860.51 821.89 782.80 744.81 708.57
(1.59) (1.22) (0.93) (0.14) ()1.49)
0.7 839.89 803.94 766.97 730.63 695.71
(3.43) (1.84) (1.26) (0.28) ()1.01)
0.8 820.29 786.69 751.63 716.84 683.16
(3.34) (1.61) (1.68) ()0.16) ()0.21)
a
The percent deviation of the NLP value from the MILP value is indicated in parentheses.
Table 6
MILP minimum average costs ($/MCF) contract #2
Demand charge ($/MCF) Take-or-pay rate
0.400 0.500 0.600 0.700 0.800
0.2 3.323 3.505 3.695 3.891 4.089
0.3 3.492 3.664 3.846 4.033 4.225
0.4 3.656 3.819 3.993 4.173 )
a
0.5 3.815 3.969 4.136 4.303 )
0.6 3.968 4.117 4.274 ) )
0.7 4.119 4.252 ) ) )
0.8 4.238 4.307 ) ) )
a
Benchmark solution (AC=4.308, D
1
=684.60, D
2
=0).
J.-M. Guldmann, F. Wang / European Journal of Operational Research 112 (1999) 598612 609
than in the case of contract 2 are necessary to make
contract 3 competitive with contract 1, because
these lower values must compensate for a higher
commodity charge (CC
3
= $3:00=MCF vs :CC
2
= $2:50=MCF). The sub-diagonal indierence
curve moves therefore upward and leftward.
Whenever CD
3
P0:7 or T
3
P0:7, contract 3 is
never competitive. Also, note that both contracts
are active (D
1
> 0; D
3
> 0) in six cases (see foot-
note of Table 8).
Similar sensitivity analyses are then carried for
contracts 4 and 5. Contract 5 is never competitive
with contract 1. Contract 4 is selected in only two
cases (CD
4
= 0:2; 0:3; T
4
= 0:4), where both con-
tracts are active (D
1
> 0; D
4
> 0). When CD
4
=
0:2; AC = 4:256; D
1
= 408:10, and D
4
= 349:30;
when CD
4
= 0:3; AC = 4:300; D
1
= 578:93; and
D
4
= 127:67.
5. Multi-period extension
In order to illustrate the potential of the com-
bined simulation/optimization approach, we out-
Table 8
MILP minimum average cost and optimal contract demand contract #3
Demand charge ($/MCF) Minimum average cost ($/MCF) Contract demand of supplier 3 (MMCF)
T-O-P rate T-O-P rate
0.400 0.500 0.600 0.400 0.500 0.600
0.2 3.848 4.044 4.247 850.20 793.80 648.50
b
0.3 4.008 4.192 830.20 692.53
c
0
a
0.4 4.152 4.286 640.80
d
291.20
e
0
0.5 4.250 371.67
f
0
a
0
0.6 4.300 121.53
g
0 0
a
Benchmark solution (AC=4.308, D
1
=684.60).
b
D
1
=95.30.
c
D
1
=81.67.
d
D
1
=148.60.
e
D
1
=429.80.
f
D
1
=372.13.
g
D
1
=581.47.
Table 7
MILP optimal contract demand (MMCF) contract #2
Demand charge ($/MCF) Take-or-pay rate
0.400 0.500 0.600 0.700 0.800
0.2 903.00 848.40 804.60 757.40 719.25
0.3 877.40 823.20 777.00 739.20 702.80
0.4 848.40 804.60 759.00 721.00 0
a
0.5 819.80 786.80 743.80 348.64
b
0
0.6 793.80 759.00 553.00
c
0 0
0.7 758.33
e
546.00
d
0 0 0
0.8 485.33
f
28.93
g
0 0 0
a
D
1
=684.60.
b
D
1
=349.53.
c
D
1
=168.00.
d
D
1
=17.27.
e
D
1
=193.20.
f
D
1
=253.87.
g
D
1
=655.67.
610 J.-M. Guldmann, F. Wang / European Journal of Operational Research 112 (1999) 598612
line here how it could be used to deal with contracts
of unequal durations and with characteristics that
vary over time. Consider a time horizon made of T
periods (t = 1; . . . ; T), usually years, and a set of N
contracts (i = 1; . . . ; N) that may be selected at
various times over this horizon. Let t
1i
and t
2i
be the
rst and last periods of the time interval over which
contract i can be active. Let d
it
= 1 if contract i can
be active at t, d
it
= 0 if not (d
it
= 1 if t [t
1i
t
2i
[).
Let D
i
be the contract demand decision variable,
which we assume constant throughout the contract
period. The decision variable vector for period t
can then be dened as: D
t
= (d
1t
D
1
; . . . ;
d
it
D
i
; . . . ; d
Nt
D
N
). Next, consider the following
vectors of parameters: CC
t
= d
it
CC
it
; CD
t
= d
it
CD
it
; TOP
t
= d
it
TOP
it
, where CC
it
;
CD
it
, and TOP
it
are the commodity, demand, and
take-or-pay rates for contract i in period t. If r is the
interest rate, then the supply mix problem is to nd
the values of the variables D
i
that minimize the
present value of all supply and curtailment costs
over the period [1 ) T], with:
Min
X
T
t=1
[CT
t
(D
t
; CC
t
; CD
t
; TOP
t
)=(1 r)
t
[: (28)
For each period t, the dispatch simulation would
be implemented while considering only those
contracts that could be active at t. Hence, a dif-
ferent cost function CT
t
would be estimated for
each period t. The above model requires, as inputs,
forecasts of the contract characteristics CC
it
; CD
it
,
and TOP
it
. Various scenarios for these parameters
could be easily considered in the above model,
extending to a multi-period framework the sensi-
tivity analyses presented in Section 4.3.
6. Conclusions and areas for further research
We have presented a modeling methodology
designed to help LDCs select the optimal mix of
supply contracts, all characterized by dierent
commodity and demand charges, as well as dier-
ent take-or-pay rates. Weather variability is the
basic stochastic factor that drives the demand for
gas by the various market segments. The model
minimizes the total cost of gas supply and market
curtailment, and thus determines the optimal
sharing of the total demand between rm and in-
terruptible customers. The methodology is illus-
trated, numerically, with actual and synthetic data.
The application underlines the critical trade-os
between contract characteristics, and demonstrates
that the solution of the combined simulation, re-
sponse curve estimation, and simplied optimiza-
tion approach is a very good approximation of the
exact solution produced by a large mixed-integer
linear program. An extension of the approach to a
multi-period framework with variable contract
characteristics and durations has been outlined.
Further extensions of the methodology might in-
volve storage capacity and injections/withdrawals
decisions. However, as demonstrated in Guldmann
(1983), this would require introducing the intra-
annual time dimension into the model to account
for the intertemporal interactions of storage oper-
ating decisions, and for time-linked stochastic
weather patterns. This time dimension would allow
for seasonally changing prices. The access to
transportation also needs to be considered. There
are clearly interactions between the selection of
transportation contracts (which pipelines, whether
rm or interruptible) and the selection of the sup-
ply contracts. The secondary market for capacity
resale, as well as the impacts of operating con-
straints (the physical conguration of the LDC
network often imposes limitations on its ability to
take possession of gas from suppliers and deliver it
to customers) should also be considered in this
extended framework.
Appendix A. Daily gas requirements functions for
NFGDC
Residential sector:
Q
R
= 54; 739 11; 103HDD; R
2
= 0:99
(6:93) (36:26)
Commercial sector:
Q
C
= 14; 978 2; 841HDD; R
2
= 0:99
(6:46) (31:61)
J.-M. Guldmann, F. Wang / European Journal of Operational Research 112 (1999) 598612 611
Industrial sector:
Q
I
= 98; 209 2; 851HDD; R
2
= 0:95
(17:76) (13:30)
Public authorities sector:
Q
P
= 4; 403 1; 218HDD; R
2
= 0:99
(6:09) (43:46)
Here, HDD=daily heating degree-days,
Q
:
=daily gas requirement (MCF), with t-values in
parentheses.
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