Modling Of Oil Refinery

Socro-Econ. Plan .Sci Vol. 19, No. I, PP. 35-40, Printed in the U.S.A.

1985 0

MODELING OIL REFINERY

GU384121/85 $3.00 + .oO 1985 Pergamon Press Ltd.

OPERATIONS

NOEL D. URIt Division of Antitrust, Bureau of Economics, Federal Trade Commission, Washington, DC 20580, U.S.A. (Received 29 May 1984)

Abstract-This paper develops a model to analyze the relative responsiveness of refined petroleum product output to changes in the relative prices of these products. A scheme is devised and implemented that shares out total production into motor gasoline, distillate fuel oil, kerojet fuel, and other refined products. A multinomial logit specification is used whereby the share of each of these products is a function of relative prices, a seasonal factor, and the relative amount of crude output to total product output. Finally, the structural stability of the estimated relationships are tested for and the null hypothesis of stability is not rejected.

INTRODUCIION

products, which in turn are determined by the interaction of the demand for the products and costs (i.e. marginal costs). Note that one cannot properly speak of supply in the neoclassical microeconomic sense, since supply exists only in that nonexistent theoretical world of perfect competition. Consider the operation of a petroleum refinery from a simplified perspective. A refinery produces output, as noted above, in the form of motor gasoline, distillate fuel oil (i.e. home heating oil), residual fuel oil, kerosene, kerojet fuel, etc. The typical refinery can use a number of different types of crude oil. For example, a typical refinery might be able to handle both light sweet and light sour$ crude that comes from, say, West Texas and Saudi Arabia [3]. It should be observed that the design of any specific refinery will depend upon, among other things, the specific characteristics of the crude oil to be processed and the output mix desired, within limits. There are different grades of crude oil and, as a result, there will be differences in the yield obtained from them. A light crude, for example, will yield a high percentage of useful products by distillation, whereas a heavy crude will produce significant amounts of crude-oil material that must be sold as heavy fuel oil or else processed further. As a result, there is some flexibility in switching a refinery to alternative crude streams, although this is limited [4]. In the refining process, some of the available crudes must be refined because of long-term minimum volume commitments or because of requirements for specialty products. These crudes are considered fixed, and yield constant motor gasoline and distillate fuel oil volumes. From the remaining crudes, and from those crudes which are available in volumes greater than their minimum volume commitment, are selected those which can supply the required refined products most economically. These are nominally referred to as the incremental crudes. The refinery problem becomes one of determining the minimum incremental cost of, say, distillate fuel oil as a function of incremental distillate fuel oil production, while

Modeling the activities of petroleum refineries with regard to altering the mix of refined petroleum product output has proven to be a most elusive proposition. The structural and data problems inherent in such analysis are difficult to resolve. It is imperative, however, that a framework be developed to enable such modeling efforts because of the overall importance of petroleum in a balanced energy network. For example, refined petroleum product consumption accounted for 42.6% of the total energy consumption in 1983 in the United States. Refinery operation is characterized by a phenomenon of being able to alter the relative amounts of output of refined petroleum products (e.g. motor gasoline, distillate fuel oil, residual fuel oil, kerosene, propane, and kerojet fuel) given a barrel of crude oil input, within technical limitations. There exists the possibility of output substitution in the refining of crude petroleum. In 1983, for example, the share of total output attributable to each of the various refined petroleum products continuously changed on a month-to-month basis [I]. Thus, in PADD Region III (this is defined below), for example, motor gasoline accounted for between 40.2 and 45.5% of total 1983 refined petroleum product output. Large output substitution elasticities of refined petroleum products are basic to the question of defining the extent of the market for petroleum products. That is, if refinery output shifted significantly between, say, kerosene and distillate fuel in response to changing relative prices, this would suggest that the market definition would be much broader than if no such shift was observed. See Stigler and Sherwin [2] for more on this. THE ECONOMIC ARGUMENT Given the unregulated nature of the market for petroleum products in the United States, the quantity supplied is a function of the prices of the various

t The views expressed are those of the author and do not necessarily represent the policies of the Federal Trade Commission or the views of other Federal Trade Commission staff members.

$Sour crude contains significant amounts of sulfur and refineries must be capable of moving the sulfur. 35

36

N. D.

maintaining motor gasoline production and general refinery operation at a fixed level. The mechanics of solving this type of problem take us beyond the scope of the current concern. (The interested reader is referred to Garvin et al. [5] for its mathematical solution.) The element that needs to be highlighted, however, is that changing relative prices of refined products that precipitate the desire to alter the output mix can de facto change the relative shares of total output from a refinery. Given the technical flexibility that exists to change the relative output share of refined petroleum products, the question to be empirically examined is whether refineries actually do alter their behavior in response to changing relative prices and, if they do, what is the extent of this response. AN ANALYTICAL FRAMEWORK There are only a limited number of ways of addressing the question of refinery output response to changing relative prices of the various refined products. One approach is to use a linear programming framework along the lines, say, of the Refinery Evaluation Modeling System (REMS) of the Department of Energy [6]. This structure assumes that demands for the various types of refined petroleum products are given (i.e. quantities demanded are exogenously determined). Subsequently, the quantities demanded and an aggregate petroleum price determined. This allows one to relate price and output and hence to determine the relative responsiveness of output to variations in price. This approach does not really give any insights into the question at hand, however, since demand is assumed to be perfectly inelastic in such a setting (which has been repeatedly shown not to be the case [7]). This would suggest that the level of output is determined solely by demand while price is determined solely by the position of the cost curves. This is too unrealistic. An alternative approach in analyzing refinery output flexibility is to use an econometric approach. To date, unfortunately, this does not appear to have been tried. An appropriate structure will be suggested below. A major issue that needs resolution in considering a way to analyze refined petroleum product output is how to insure that the components of refined petroleum product output are exactly equal to total output. That is, if we are examining changes in, say, relative prices, and total output is predetermined, how can we be certain that the aggregation across the components precisely equals this total. To address this apparent anomaly, a scheme must be designed to share out total output based on some objective criteria. These shares as a proportion of total output will necessarily lie between zero and one. To guarantee that aggregation holds, it is only necessary to require that these shares across all refined petroleum products sum to one. Given that each of the shares is constrained by the interval (0, l), a linear model specification is not acceptable, due to the possibility that values may lie outside of it. The obvious solution is to have a transformation whereby for all possible values of the explanatory variables, the share values will be in the (0, 1) interval. One would also like the transformation to maintain the property that increases in the value

URI

of the explanatory variables are associated with increases (or decreases) in the value of the share. These requirements suggest that use of a cumulative function whose upper bound is one and whose lower bound is zero will provide a suitable transformation. There are a couple of such transformations that would prove acceptable [8]. The one adopted here is the logit specification because it has proven to be the most mathematically tractable. The logit model specification is based on the function &I = l/(

1 +

exp

(ai

+

C

bifl,rl,,))

(1)

I where S,, denotes the share of a given refined petroleum product produced in period t; x,, are the exogenous factors causing variations in output shares i in period t; and ai, b,, , . . . , bin are coefficients to be estimated. The specification provided in eqn (1) can be estimated directly by means of nonlinear least squares or, after a suitable transformation of the dependent variables S,,, by means of classical least squares. Such estimation techniques, however, would not use all of the information efficiently. Namely, the sum of the shares is equal to one given that the available choices are mutually exclusive (i.e. one gets motor gasoline or distillate fuel oil, not both). In what follows only four refined petroleum products are going to be considered. Consequently, when limiting the logit model to the four-choice case, one has log

log

$ 0 I

= a2, + C b,x,

: 0 I

= ad1 + C b,xj

1

I

where S, denotes the share of all refinery output besides motor gasoline, distillate fuel oil, and kerojet fuel, S2 denotes the share of motor gasoline output, S3 denotes the share of distillate fuel oil output, and S, denotes the share of kerojet fuel output; a*,, a3,, a4,, bll, . . , bdh are coefficients to be estimated, and x, are the exogenous factors causing variation in the relative outputs of the refined products. The subscript t, designating observations in each period, has been deleted to simplify the notation. That this is tantamount to the logit specification can easily be seen by solving each of the equations as a simple function of one specific share or by referring to McFadden [9]. Each of these equations presumes that the logarithm of the ratio of the share of one refined petroleum product to one minus the share of that product relative to a similar ratio for another refined petroleum product is a linear function of the set of explanatory factors. These values are dependent on the values associated with the remaining equations only to the extent that the system must be constrained so that the sum of the individual shares is one.

37

Modeling oil refinery operations One is now in a position ~21,

~31,

~41,

b2,,

. . . ,

to estimate

the parameters

b4,,. If the set of explanatory

factors (i.e. xj) in each equation are not identical, efficiency can be gained by using the iterative seemingly unrelated technique of Zellner [lo]. This is just a generalized least squares estimation procedure used to reflect the correlation among the error terms associated with each equation in the multinomial model. The iterative Zellner technique is used here.

(3) Technical characteristics. As discussed above, not all crude oils yield the same output mix, nor the same volume of total output of refined products. As crude input increases, given that there is a quality change in the crude and hence more or less refined products produced, relative output shares will change. This affect is reflected by including a variable defined as the ratio of crude input to total refined product output. These basic data were taken from the Petroleum Supply Monthly.

DATA Before proceeding to a presentation of the estimation results, a discussion of the data is in order. The data employed in the estimation are monthly observations over the period 1981-1983. The year 1981 was used as the line of delineation since that is when decontrol became effective, ostensibly permitting the market mechanism to work, at least as far as refined petroleum products are concerned. Monthly refined petroleum product data were obtained from the Department of Energy’s Petroleum Supply Monthly [ 1 I]. Given the limited time period for which data are available, practical econometric consideration suggested that looking at a large number of individual refined petroleum products (data are collected on 18 separate items by the Department of Energy) would present problems. Consequently, the investigation was limited to just four products: motor gasoline, distillate fuel oil (these two combined account for about 65% of refinery output), kerosene-jet fuel type (which accounts for about 6% of refinery output) and all other refined petroleum products. The output data were collected on a Petroleum Administration for Defense District (PADD) level (see the Monthly Energy Review [ 1] for a characterization of the five PADD regions). This is the most disaggregated basis publicly available for these data. Regionally disaggregated data were selected over national aggregates since there are some differences in refinery vintages by region and some differences in the technical characteristics of the refineries in the various regions. The set of exogenous variables used in the estimation consisted of the following: (1) Prices. The price of motor gasoline relative to the price of all other refined petroleum products (i.e. of motor gasoline, distillate fuel oil, and kerojet fuel), the price of distillate fuel oil relative to the price of all other refined petroleum products, and the price of kerojet fuel relative to the price of all unleaded refined petroleum products. The price of other refined products is simply measured as a weighted average. All of the price data were taken from Platt ‘s Oil Price

Handbook [ 121. (2) Seasonal variation.

To capture the seasonal variations in the output shares, crude oil input into the refining process is used. This variable really is a proxy for changing refined product requirements (i.e. more motor gasoline in the summer and more distillate fuel oil in the winter) over the year. The data are taken from the Petroleum Supply Monthly. Note that this method of reflecting seasonal variation was adopted in difference to using dummy variables since the latter method would use up eleven degrees of freedom. This was judged unacceptable given the limited number of observations to begin with.

Other variables were considered in preliminary analyses and proved to be statistically insignificant. In particular, a weather variable (measured by heating degree days), the relative price of imported refined petroleum products, and electric utility generation (residual fuel oil is a significant boiler fuel) all failed to demonstrate any impact on the output shares of the products. What one is left with now is a mechanism for analyzing refined petroleum product output shares based on relative prices, technical refinery characteristics, and crude oil input (as a proxy for seasonal variation). Empirical estimates of the impact of these factors on changing shares is the subject of the following section. EMPIRICAL

RESULTS

Using the data and the estimating technique previously discussed, the empirical results were obtained. Preliminary analyses suggested that the imposition of symmetry constraints might be appropriate (that is, b, = b,, on the relative price terms). To test this hypothesis, a Quandt test is employed. The test consists of computing the determinants of the unrestricted and restricted disturbance covariance matrices. The logarithm of this times minus the number of observations used in the estimation has a chi-square distribution with degrees of freedom equal to the number of independent restrictions being imposed [ 12l.t The test was performed for each of the regions considered with the null hypothesis being that symmetry holds. In all instances, the null hypothesis could not be rejected. That is, it is not possible to statistically reject the contention that a relative increase in the price of refined petroleum product j will affect the logarithm of the relative product share i to the same extent that a relative price increase in the price of refined petroleum product i will affect the logarithm of the relative product share j. The estimation results for each of the PADD regions are given in Tables l-5. The standard errors of the estimates are in parentheses. All of the reported estimated coefficients are statistically significant at the 95% level or better. The Durbin-Watson statistics are either inconclusive or suggest the absence of serial correlation. In interpreting the results, note that they relate to the change in the share of refined petroleum product output of either motor gasoline, distillate fuel oil, or

t This result is valid only for maximum likelihood estimates. Kmenta and Gilbert [ 141, however, have shown that iteration of the Zellner estimation procedure until convergence (as is done here) results in maximum likelihood estimates.

N. D.

38

URI

Table I. PADD 1 refinery share equation estimates”) Explanatory Variables c2) Equation

Constant

log ( s2 ) -

-6.9229 (0.9177)

Pmg’Po 1.5933 (0.7551)

pd’po

~2(3) pkj’po

Crude/Total Output

Crude

D. W.c4)

(5)

(5)

3.8799 (1.2328)

0.0167 (0.0014)

0.6722

1.57

0.0171 (0.0028)

0.7154

1.75

0.7078

1.79

Sl log ( s3 ) G log

( s4 ) G

0.1719 (0.0308)

(5)

0.5111 (0.2155)

0.3310 (0.1518)

-0.7564 (0.3138)

-2.4013 (0.7120)

(5)

0.3310 0.1518)

1.4729 (0.6449)

-0.0404 (o.ooao)

(5)

(1)

Standard errors of estimates in parentheses.

(2)

Pk. : price of kerojet, Pw : price of motor gasoline, P : price of other nfined petroleum products, crdde : crude oil input to refinery, and total output = ?otal refinery petroleum output.

(3)

Coefficient

(4)

Durbin-Watson statistic.

(5)

Not statistically

of determination.

significant.

kerojet fuel relative to all other refined petroleum products. It is consistently the case that month-tomonth variations in the relative prices of the own fuels are significant in explaining relative variations in the relative amounts produced of the various refined petroleum products. Of equal or more significance is the relative shift between refined products in response to changing relative fuel prices. Thus, for example, in region PADD 3, an increase in the relative price of kerojet fuel will result in a reduction in the relative share of distillate fuel oil output. In this latter regard, just what is the mechanism through which this substitution can take place? Take kerojet fuel as an illustration. Kerojet production can be manipulated in the refining process via a number of alternatives. One way is to divert kerojet quality

kerosene from other uses (e.g. kerosene blended with heavier gas oil to produce distillate fuel oil). Another way is for refiners to technically adjust the refining process by changing the boiling point ranges on the distillation towers to increase or decrease the percentage of crude oil extracted as kerosene (at the expense of other light refined products such as either motor gasoline or distillate fuel oil). Finally, it is possible to change crude oil types to those with a higher percentage of kerosene or for which the kerosene is kerojet quality. There are two other inferences to be drawn from these results. First, as the ratio of crude input to total output increases, there is a general indication that the relative share of motor gasoline increases at the expenses of distillate fuel oil and kerojet fuel. This is

Table 2. PADD 2 refinery share equation estimates”’ Explanrrtory Variables (2)

g2(3)

Equation

constant

Pmg’Po

pd’po

pkj’po

Crude/Total Output

Crude

log ( S2 ) -

-6.1965 (2.4393)

0.9005 (0.4589)

-0.4051 (0.2023)

1.1442 (0.5811)

3.7031 (1.921)

0.0104 (0.0044)

0.3676 (0.0526)

-0.4051 (0.2033)

0.5119 (0.1420)

-2.3297 (0.7770)

1.1442 (0.5811)

Sl log

( “3 1 Sl

log ( S4 ) <

(5)

(5)

1.7197 (0.4994)

0.6524

1.62

-0.4153 (0.1700)

0.0129 (0.0063)

0.6937

1.99

-0.0397 (0.0075)

(5)

0.6845

2.20

(1)

Standard errors of estimates in parentheses.

(2)

= price of kerojet, Pw = price of motor gasoline, P : price of other refined petrolem products, CFa de = crude oil input to refinery, and total output : ?otal refinery petroletan output. Pk.

(3)

Coefficient

(4)

Lurbin-Watson statistic.

(5)

Not statistically

of determination.

significant.

D. W.(‘I)

39

Modeling oil refinery operations Table 3. PADD 3 refinery share equation estimates”’ Eicplanatory Variables (*I Equation

constant

$(3)

Pmg'Po

pd'po

pkj'po

Crude/Total Output

Crude

D. W.(4)

)

-2.5924 (0.6233)

0.8130 (0.3847)

-1.5168 (0.5847)

-1.5237 (0.4640)

1.6474 (0.6235)

3.0163 (0.0059)

0.7862

1.55

log ( S3 ) G

0.4086 (0.0732)

-1.5168 (0.5847)

0.7838 (0.3734)

-0.3944 (0.1510)

-0.9271

(0.4298)

0.0476 (0.0059)

0.8115

1.91

log ( S4 ) Sl

-2.9126 (0.9373)

-1.5237 (0.4640)

-0.3944 (0.1510)

1.9371 (0.6678)

-0.6713 (0.0987)

(5)

0.7527

1.93

log (2

s1

(1)

(2)

Standard errors of estimates

in parentheses.

Pk. z price of kernjet,Pm = price of motor gasoline,P : price of other refined petroleumproducts, crdde = crude oil input,to refinery,and total output : total refinerypetrolem output.

(3)

Coefficientsof determination.

(4)

Durbin-Watsonstatistic.

(5)

Not statisticallysignificant.

consistent with the results of Copp [ 151. Second, both the relative production of motor gasoline and the relative production of distillate fuel oil are sensitive to seasonal variations. This is hardly a surprising result [ 161. A final useful result would be obtained if estimates of the actual share elasticities were available. This would provide information on the percentage changes in the various refined petroleum product shares in response to changes in the various explanatory factors. Unfortunately, as specified, there are no ready machinations available to allow for deriving such estimates in an easily interpretable form. Thus, for example, to obtain a refined petroleum product share for motor gasoline, one needs to take the exponential of the right-hand-side of eqn (2b) and multiply it by the all

other refined petroleum share. This latter term is equal to one over one plus the sum of the exponentials on the right-hand-side of each of the eqns (2a)-(2c). Needless to say, such an expression is quite complex and not subject to easy interpretation. Thus, share elasticities are not readily available. TESTING FOR STABILITY A final concern in the analysis is with the structural stability of the estimated relationships in the sense that the estimated coefficients on the explanatory variables have remained constant over time. The test is carried out using the test statistic reported in Uri [ 171, where the test statistic equals the difference between the sum of squared residuals of the entire sample less the cumulative sum of the squared resid-

Table 4. PADD 4 refinery share equation estimates”’ ExplanatoryVariables (2) Equation

Constant

log ( s2 )

-3.0426

Plllg’PO

0.9328 (0.4320)

$(3)

D. W.c4)

'd"o

pkj'po

Crude/TotalOutput

(5)

-1.3212 (0.51731

1.8353 (0.9255)

0.0428 (0.0206)

0.6151

1.87

Crude

< log ("3 I % log ( s4 )

0.1278 (0.0423)

(5)

0.3158 (0.1356)

(5)

-0.6352 (0.2206)

-0.0397 (0.0129)

0.5924

1.97

-1.9122 (0.3812)

-1.3212 (0.5173)

(5)

0.5697 (0.1951)

-0.3606 (0.1784)

(5)

0.6544

1.96

s1

(1)

Standarderrom of estimatesin parentheses.

(2)

Pk. = price of kerojet, Pw = price of motor gasoline,P = price of other refined petroleumproducts, crdde : crude oil input to refinery,and total output : ?otal refinerypetrolem output.

(3)

Coefficientof determination.

(4)

Durbin-Watsonstatistic.

(5)

Not statisticallysignificant.

40

N. D.

URI

Table 5. PADD 5 refinery share equation estimates”) ExplanatoryVariables 12f Equation log

csz ) s1

log (s3 1 Sl log

(2 $1

)

~2(3)

D. W.t4)

Crude

Constant

Pnlg'Po

pd'po

pkj'po

Crude/TotalOutput

-2.5749 (a.83841

1.0646 (0.3702)

(5)

-0.9337 (0.4105)

0.1525 (0.0716)

0.0218 f0.0117)

0.6982

1.82

0.2701 (0.0548)

(5)

0.3465 (0.1528)

-0.1447 (0.0625)

0.3466 (0.1528)

0.0556 (0.0052)

0.7599

2.14

-2.7998 (0.7646)

0.9337 (0.4105)

-0.1447 (0.0625)

0.9455 (0.3023)

-0.2381 10.0761)

(5)

0.7505

2.05

(11

Standarderrom of estimatesin parentheses.

(2)

- price of motor gasoline,P : price of other refined petroleumproducts, "J = price of kerojet, P cr de = crude oil input t?&finery, and total output = ?otal refinerypetroleumoutput.

(3) Coefficientof determination. (4) Durbin-Watsonstatistic. 151 Not statisticallysignificant.

uals over the non-overlapping segments divided by the cumulative sum of squared residuals of the nonoverlapping segments. The null hypothesis that a regression relationship is constant over time implies that the value of the test statistic is distributed as F. The data upon which the equations were estimated were divided into three equal length intervals. Since they do not provide much information in and of themselves, the tabulated value of the test statistic for each equation in each region is not reported. None of the estimated relationships, however, demonstrates any instability over the period 198 i-1983. The implications of these results in estimating the response of the relative refined petroleum product shares is clear. Events since i 98 1 have left virtually unchanged the responsiveness of the relative shares to changes in relative prices, crude input, etc. That is, the relative importance of the various explanatory factors has not changed. One must be careful, however, to avoid inferring that relative shares have not changed. The estimation results clearly demonstrate that, say, the relative price of motor gasoline to other refined petroleum products influences the relative share of motor gasoline produced. The magnitude of the associated responses in the aggregate, however, have remained unaltered over the sample period. CONCLUSION The objective of the foregoing analysis has been to develop a model to analyze the relative responsiveness of refined petroleum product output to changes in the relative prices of these products. A scheme is devised and implemented that shares out total production into motor gasoline, distillate fuel oil, kerojet fuel, and the remaining refined products. A multinomial logit specification is used whereby the share of each of these products is a function of relative prices, a seasonal factor, and the relative amount of crude input to total product output. Finally, the structural stability of the estimated relationships is

tested for and the null hypothesis rejected.

of stability

is not

REFERENCES 1. Department of Energy, Monthly Energ_v Rev., U.S. Government Printing Office, Washington, DC., monthly. 2. G. Stigier and R. A. Sherwin, The extent of the market. Unpublished manuscript (1983). 3. R. 0. Jones, Vertical integration, cartel coordination and the petroleum industry. Unpublished Ph.D. dissertation, University of Maryland (1983). 4. D. Teece, Vertical integration in the U.S. oil industry. In Verticaf Integration in the Oil Industry (Edited by E. J. Mitchell), The American Enterprise Institute, Washington, D.C. (1976). 5. W. W. Garvin, H. W. Crandall, J. B. John and R. A. Spellman, Application of linear pro~mming in the oil industry. Mgmt. Sci. 3, 407-430 (I 957). 6. Department of Energy, Annual Energ_p Outlook, U.S. Government Printing Office, Washington, DC. (1984). 7. N. D. Uri, The Demand ,jbr Energy and Conservation in fhe United States, JAI Press, Greenwich, Connecticut ( 1983). 8. R. Pindyck and D. Rubinfeld, Econometric Models and Methods. McGraw-Hill, New York (1976). 9. D. McFadden, Multinomial models. In Frontiers in Econometrics (Edited by P. Zarembka). Academic Press. New York ( 1974). 10. A. Zellner, An efficient method of estimating seemingly unrelated regression and tests for aggregation bias, J. of Am. Stat. Ass. 57, 348-368 (1962). 11. Department of Energy, Petroieum Supply ~~~nt~iy, U.S. Government PrintingOffice, Washington, D.C., monthly. 12. R. Pindyck and D. Rubinfeld, P(att’s Oil Price Handbook. McGraw-Hill, New York, annual. 13. S. Goldfeld and R. Quandt, Nonlinear Methods in Economeirics. North Holland, Amsterdam (197 1). 14. J. Kmenta and R. Gilbert, Small sample properties of alternative estimaters of seemingly unrelated regression, J. Am. Stat. Ass. 63, 1180-1200 (1968). 15. E. A. Copp, Regulation Competition in Oil. Texas A and M University Press, College Station (1976). 16. A. Bopp and G. M. Lady, Conservation and U.S. petroleum consumption. Energy 3, 533-538 (1978). 17. N. D. Uri, The demand for energy in Korean industry. App. Energy 14, 295-3 16 (1983).