Electric Power Systems Planning Using Linear Programming

JOEL BERGSMAN

Summary-A linear programming model can be used to plan newgeneration and transmission systems. The instance described is amodel for the Pacific Coast region of the United States for the years1965 through 1980. The method allows simultaneous optimization,at a high level of aggregation, of all decision variables relevant to thisgeographic space and time period. The results include marginal costestimates and analysis of the sensitivity of the solution to variousgiven data and parameters, as well as a description of the optimalsystem. The strengths and limitations of linear programming in thisuse are discussed.

I. INTRODUCTION

mqHIS PAPER describes a method of electric powerj systems planning which is more general than those

usually employed in the United States. Themethod, which uses linear programming, is illustratedby application to the Pacific coast region of the UnitedStates.' After a few general remarks, the applicationwill be described and the implications discussed.The combinations of natural resources, geography,

and existing electric power systems in the Pacific coastregion make the planning of future systems a complexproblem. Among the complicating factors are the com-bination of hydroelectric and thermal plants, the greatyear-to-year variation in stream flow in the ColumbiaRiver basin, and the differently-timed peak loads indifferent parts of the region (peak demand diversity).One of the novel ideas springing from these factors is anextra-high-voltage, very long transmission line con-necting the northwest (the systems in and around theColumbia River basin) to the southwest (California,Arizona, and Nevada). The latest published study2recommended two lines, with a combined capacity of1780 megawatts, lengths of 875 and 575 miles, and costsof $225 million. Some investor-owned utilities in thearea, at first suggesting that only about 200 megawattswere needed, later proposed building a 900-megawattline. (This northwest-southwest transmission line willhereinafter be called an "intertie.")The Special Task Force report, and all earlier studies

of an intertie, illustrate some shortcomings of typicalsystems planning methods. These methods are usually

Manuscript received September 16, 1963. The work describedherein was done at Stanford University, Stanford, Calif. Computa-tions were performed at the Stanford Computation Center.

The author is with the U. S. Agency for International Develop-ment, Washington, D. C.

I The first use of linear programming for electric power systemsplanning is described in P. Masse and R. Gibrat, "Application of lin-ear programming to investments in the electric industry," Manage-ment Sci., January, 1957; the latest developments in French electricpower systems planning in "Modele des Trois Plans," Electricite deFrance; April 15, 1962.

2 Special Task Force, U. S. Department of the Interior, "PacificNorthwest-Pacific Southwest Extra-High Voltage Common CarrierInterconnection," U. S. Bureau of Reclamation, Sacramento, Cali-fornia; 1961.

limited to planning only one, or at most a few, facilities,and consist essentially of forecasting a need, listing sev-eral alternative ways of satisfying the need, and choos-ing the best alternative among those listed. The choiceis made on the basis of highest benefits, net of costs.Prices at which to evaluate products of the facility (orequivalently, costs of the next best facility) must be as-sumed.The more general method to be described here over-

comes four shortcomings of the more typical partialmethods. First, partial analyses usually choose amongonly a few specific alternatives; one wonders whether thebest alternative has even been considered. Second, par-tial analyses must assume the effects that the facilitiesin question would have on the entire optimal system. Incomplex systems these effects are often far from obvious;one would prefer to choose the whole optimal system,rather than choose individual facilities one at a time,under possibly erroneous assumptions concerning the re-mainder. Third, partial analyses must assume prices forthe products of the facilities. Electric power is seldomsold in a competitive market, and one wonders whetherthese prices reflect the economic cost of supply. Fourth,partial analyses usually give no information as to whatdata are crucial to the results, and in what way. Predic-tions of future costs, technology, and demands are cer-tainly subject to error, and one wonders how the recom-mendations might change with a given change in a par-ticular type of data.

I I. AN APPLICATIONA. The ProblemThe work upon which the following discussion is based

is described more completely elsewhere.3 This paper isconcerned only with illustrating the above discussion,focusing on the intertie as an example of electric powersystems planning problems.Two models were used to perform the analysis; the

second is described here in detail. The difference be-tween this and the preliminary model will be mentionedfollowing the description. The subject of the linear pro-gramming model was the electric power system of thePacific coast region of the United States for a fifteenyear period starting in 1965. The activities (unknowns,variables) of the model represented various generationand transmission possibilities. The constraints (equa-tions) required satisfaction of predicted demands, andalso of physical and technological relationships. The

I J. Bergsman, "Economic Problems in Electric Power SystemsPlanning," Inst. Engrg.-Economic Systems, Stanford University,Stanford, Calif. Tech. Rept. No. EEP-6; 1963.

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criterion used to define the optimal system was mini-mum total cost, on a present value basis. The modelhad the following features:

Geographic Regions: The total area was divided intothree regions. The "northern" region consisted of thestates of Washington, Oregon, Idaho and western Mon-tana. The "central" region, northern California andnorthwestern Nevada. The "southern" region, southernCalifornia, the rest of Nevada and Arizona.As of 1962, almost all electric power facilities within

each of the three regions could be operated on a co-ordinated basis. Little coordination was possible, how-ever, between the regions. Because of the intraregionalcoordination that did exist, the regions were treated asdimensionless points (i.e., the geographic distribution ofloads and resources within each region was ignored) toa limited extent. Distribution systems within each re-gion were not explicitly considered; costs of facilities fortransmission of power from generators to loads were in-cluded in costs of generation facilities. The descriptionof the interregional intertie, however, recognized thatits length must increase as its capacity increases, inorder to connect the actual facilities and loads served.

Time Periods: The planning period of the model was1965 to 1980. This time was divided into the followingfour periods, of unequal length:

Period 1 1965-67 3 yearsPeriod 2 1968-70 3 yearsPeriod 3 1971-73 3 yearsPeriod 4 1974-79 6 years.Demand Requirements: The requirements which an

electric power system must satisfy are of two types,capacity and production. At every instant of time, thesystem must meet a certain power demand; enoughcapacity must have been constructed to meet this de-mand. In meeting this demand over time, the systemmust furnish the energy to be used during the period;enough coal must be burned, water passed through tur-bines, etc. One usually assumes (except when consider-ing several centuries) that mineral fuels are available inunlimited quantities at certain (not necessarily con-stant) prices. Therefore, satisfaction of demand by afuel-fired system is assured by constructing capacitysufficient to meet the highest demand expected, and bypurchasing fuel sufficient to generate the energy de-manded during the period. One cannot assume, how-ever, that water is available in unlimited quantities. Itmay be that a hydroelectric system can meet the peakdemand, but is unable to generate all the energy neededduring a dry season. Such a system requires more capac-ity; either more hydroelectric plants to use the samewater more times, or fuel-fired plants. Thus, two capac-ity requirements must be satisfied to insure satisfactionof demand by a hydroelectric system; one on its peakingcapability, which is limited only by the installed gener-ation capacity, and another on what may be called its"firm" capability, which may be further limited bystreamflow.

The model, then, had three types of demand require-ments. The peak and firm requirements expressedcapacity, i.e., power, needs; the average requirementsexpressed production, i.e., energy use. The peak andaverage requirements were imposed on all three geo-graphic regions; however, conditions in the central andsouthern regions insure that the firm requirement wouldbe redundant, so it was imposed on the northern regiononly.Each demand requirement was imposed for each time

period. The peak requirement was the maximum ex-pected load during the period. The average requirementwas the average expected load during the last year ofthe period, which was further divided into four "sea-sons": January-March, April-June, July-September,and October-December. The firm requirement was theaverage expected load during the Coumbia River stor-age drawdown period, in the northern region.

Activities: The activities represented various ways ofsatisfying the demand requirements. Just as the re-quirements on the system are of two types, capacity andproduction; so the activities were of two types, buildingnew capacity, and generating energy at (new and old)existing plants. The technological scope of these activi-ties included conventional steam plants, hydroelectricplants, and the intertie. Technological detail was de-scribed by variations in costs and unit contributions tothe requirements.

Constraints: The constraints were of two types. De-mand constraints expressed the satisfaction of demandrequirements; for each requirement there was a con-straint of the form

ti unitve

all Vcontribution}activities

(activity > demandlevel - requirement

A second set of constraints expressed technological orgeographical limitations on certain activities; genera-tion cannot exceed generating capacity; new hydro-electric capacity cannot exceed a limit set by availabilityof sites, etc.

Preliminary Model: A preliminary model was usedmainly for sensitivity analysis, i.e., for deciding whatfactors were crucial to certain features of the optimalsolution. These important factors were then further ana-lysed in the model described above. In the preliminarymodel, the central and southern regions were consideredas one. The planning period was 1965 to 2015, dividedinto five time periods. Nuclear-powered steam plantswere among the activities considered.

B. Results

Two types of results will be discussed here. The firstconcerns the sensitivity of the optimal level of intertiecapacity to various types of data and parameters; thesecond concerns marginal cost estimates from the analy-sis and rate structure implications. These results illus-trate the meaning and importance of the points men-tioned in Section I.

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Sensitivity Analysis: Both models were solved manytimes under different parameter values, cost and de-mand assumptions, structures, etc. Among the factorsshown not to affect the optimal level of intertie capacitywere the choice of discount rate, cost and technologypredictions for conventional and for nuclear-fired steamplants, peak demand diversity, and variations in anyparameter or estimate more than fifteen or twentyyears in the future. The discount rate was varied be-tween 2.5 per cent and 7.5 per cent; cost and technologypredictions were varied between very high and very lowestimates, encompassing what in 1962 seemed all butvery unlikely ranges; and peak demand diversity wasvaried between zero and twice the expected values.The intertie and Columbia River hydroelectric plants

were shown to be highly complementary. Building theintertie speeds up the optimal rate of installation of thishydroelectric capacity so much that the present valueof total optimal system generation capacity costs isactually increased if the intertie is constructed. Theseincreased generation capacity costs, and the cost of theintertie itself, are more than offset by savings in energygeneration at steam plants in the central and southernregions made possible by the transmission of inexpen-sive energy from the north.The value of the marginal units (i.e., the last units

that are worth at least as much as they cost) of intertiecapacity comes from the transmission of inexpensiveenergy from the northern region to the central andsouthern regions. Capacity savings from peak diversityare important sources of benefits, but are completelyexploited at a level of intertie capacity far below theoptimum. Thus peak diversity has no effect on theoptimal level of intertie capacity.

Engineering cost estimates for the intertie show anincrease in marginal cost somewhere in the 1600- to1800-megawatt capacity range. The optimal level ofintertie capacity was shown to be at the upper limit ofthe lower-cost range. The range over which the costestimates may vary without affecting this conclusionare set by the marginal values of the intertie at differentcapacity levels, as given in the following model solution:

(megw)1600 or less1800 or more

($/kw)$196 or more$168 or less

($/kw)$160 or less$225 or more.

Thus, the solution shows that at least 1600 megawattscapacity should be built as long as the marginal cost at1600 megawatts is less than $196 per kilowatt, and thatno more than 1800 megawatts capacity should be builtunless the marginal cost at 1800 megawatts is less than$168 per kilowatt.

This information about what data do not affect a

certain recommendation, and about the allowable varia-tion in the data that do affect it, is of considerable value

to the planner, both in pointing out where to focusfurther effort in more refined studies, and in defendingthe completed analysis.

Costs and Rates: In a linear programming model, themarginal cost of satisfying each requirement appears aspart of the solution. The way in which the "real world"requirements are expressed in the model is of course thedecision of the model builder. Demands for electricpower were described in this analysis as follows:

For each of the four time periods:

peak demand, each geographic region;firm demand, northern region;average (i.e., energy) demand, each region, each four-month "season."

These marginal costs represent the economic costs ofsupplying the last units of each of the services described.The most interesting result of this part of the presentanalysis was the great discrepancy between the marginalcost of meeting peak demand, as estimated by the linearprogramming analysis, and traditional evaluations ofthis cost. The estimated cost, until about 1973, is about$0.50 per kilowatt per year. Bonneville Power Adminis-tration charges $17.50 per kilowatt per year for thisservice, and traditional evaluations in the Pacific south-west are around $10.00 to $15.00 per kilowatt per year.This discrepancy partially explains the large number ofunused sites for inexpensive peaking equipment in theBonneville system, existing contemporaneously withmore expensive peaking equipment which Bonnevillecustomers have built for themselves. The discrepancyalso confirms our earlier suspicion that the priceswhich are assumed in partial analyses to evaluate bene-fits may not always reflect the true cost of supply.

III. CONCLUDING DISCUSSIONA. General or Partial Analysis4

Four shortcomings of the more usual partial analysiswere mentioned early in this paper. The more generalmethod described here illustrated both that these short-comings are significant and that they can largely beovercome. First, a linear programming analysis can con-sider several hundred possible activities, and thus opti-mize decision variables which represent virtually all pos-sible components of a large system for many years inthe future. Furthermore, the possible levels of each ac-tivity (i.e., the sizes of all facilities) need not be speci-fied, but can be continuously varied and allowed toreach the optimum level rather than just the best levelamong a few possibilities. In this analysis, for instance,the intertie could have been any size; the results cannotbe challenged because of failure to consider, say, a 200-megawatt or a 3000-megawatt intertie.

I For an excellent general discussion of this topic, see M. Boiteuxand F. Bessiere, "Sur l'emploi des methodes globale et marginale dansle choix des investissements," Rev. Francaise de Recherche Opera-tionelle, vol. 5, 1960.

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Second, a very large system can be optimized at once,rather than being built up a little at a time under pos-sibly incorrect assumptions about the total system. Inthis case, for example, the optimal size of the intertiewas shown to be closely dependent on the rate of instal-lation of Pacific northwest hydrolectric capacity. Partialanalysis would probably not even discover this depend-ency, and is incapable of optimizing the two simultane-ously except by coincidence.

Third, no values of products had to be assumed, andthe marginal costs which were part of the linear pro-gramming solution showed that the traditional assump-tions of the value of peak power are nowhere near therelevant economic cost of supply in the western UnitedStates.

Fourth, those data which influenced certain featuresof the optimal solution were distinguished from thosewhich had no effect, and the effects of variations in theinfluential data were studied. Contrary to strong a prioribeliefs, peak demand diversity predictions had no ef-fect whatsoever on the optimal intertie capacity, nordid the choice of discount rate. Thus the analysis cannotbe challenged on its diversity predictions, even thoughthey may be subject to large error, or on its use of theparticular, somewhat arbitrarily chosen discount rateof 4 per cent.

All this does not mean that linear programmingmodels should supplant all other methods of electricpower systems planning. A general model, to be useful,must be of so great a scope that computational limita-tions preclude detail sufficient to specify each individualfacility precisely. Moreover, assumptions which are nec-essary in a linear programming model (see Section II I-B)may be accurate only for the large scale of variables inthe general model, and not at the smaller scale of indi-vidual facilities. General models such as the one de-scribed here can only give a rough outline of the optimalsystem and its cost structure; given this information,partial analyses must be used to fill in the plant-by-plant details.

B. Pertinent Characteristics of Linear ProgrammingThis section is not intended to be a complete outline

even of linear programming assumptions and methods,but only to mention the more important effects of theseassumptions on the utility of the method for electricpower systems planning.A linear programming problem must be stated as the

minimization (or maximization) of a certain function,subject to certain fixed constraints on the variables. Theelectric power systems planning problem seems to fitthis necessity rather easily; it is stated as the minimiza-tion of total costs, subject to the satisfaction of pre-dicted demands. If, however, the necessary data wereavailable, one might prefer to generalize further and

determine both demand and supply, given only futuretechnological possibilities (data on which are availableand were used in the present analysis) and future de-mand-price relationships (which are not, and may neverbe, known in sufficient detail). This more general prob-lem does not seem to be susceptible to the linear pro-gramming formulation.A more significant limitation is the necessary assump-

tion that marginal costs must be nondecreasing func-tions of capacity; i.e., returns to scale must be eitherconstant or decreasing. (Decreasing returns to scalemust be approximated by a cost curve composed oflinear segments, but this is not a serious problem.) Sinceincreasing returns to scale appear significant in steamplants up to about 500 megawatts capacity, the scaleof the model must be large enough so that each activityrepresenting steam plants actually represents severallarge plants. If this is impossible, recourse to integer pro-gramming, or to some method of searching among localoptima can be useful.5 Returns to scale appear to beconstant or decreasing in other activities.A third limitation is the assumption that the produc-

tion per unit of a given activity is independent of thelevels of all other activities. A hydroelectric facilitywhich can store a significant amount of water may in-crease the production capability of all downstream facil-ities. This effect is significant in an area like the Pacificnorthwest, where in wet years the streamflow exceedsboth current load requirements and storage facilities,and is therefore spilled over dams and never used. If newfacilities are always built upstream from existing ones,there is no problem, for the unit production coefficientof the activity can include the increased downstreamcapability. This strict upstream progression of new damsis of course not realized, and assumptions about down-stream facilities must be made.The costs of a complete study such as the one partially

described in this paper would be a few man-years, orabout $50,000, plus perhaps eight hours use of an elec-tronic computer such as the IBM 7090, or about $5000.The present value of costs to be minimized is of theorder of tens of billions of dollars. The possible savingsseem to justify the costs.

ACKNOWLEDGMENT

The author would like to acknowledge the helpfulguidance of A. S. Manne, W. K. Linvill, and J. E.Howell, and the financial support of the Ford Founda-tion and Resources for the Future.

5 For examples of two different approaches to this problem, seeA. S. Manne, "Plant Location Under Economies of Scale, Centraliza-tion, and Computation," unpublished notes, August, 1963; and R.Frisch, "The Non-plex Method," presented at Conf. on Nat'l Plan-ning, University of Pittsburg, Pa., March, 1964.

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