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2 32 PROCEEDINGS OF THE IEEE, VOL. 71, NO. 2, FEBRUARY 1983

Spatial Electric Load Forecasting: A Tutorial Review

Abmuct-A spatial laad forecast is a forecast of the future locations and magnitudes of electric load within a utility's service territory. Such forecasts are a necespIly part of power deiive.ry system phning. Ihic paper presents a comprehensive review of spatial load f o h g techniques. A discussion of planning needs, uncertainty, spatial growth character, and forec+sting enor focuses attention on the essential qualities of a spatial f o r d The vnrious approaches to forecasting are then reviewed, compared, and evaluated on a uniform basis. Em- phasisisontheuseoftwo-dimeMionnlsignnltheoryrsaunifonn~e- work for analysis of load, growth, m, and forecast model behavior.

A I. INTRODUCTION AND OVERVIEW

N ELEClrRIC POWER system consists of a large number of interconnected components that work in concert to generate and deliver electrical power to many points

scattered over a wide geographical area. Generation equipment is generally concentrated at only a few large power stations. The power delivery system, however, is composed of many pieces of equipment scattered throughout the utility's service area, all sharing the burden of power distribution. Each unit of the delivery system must be properly located and sized to serve the demand in the vicinity. Planning the future expansion of this system involves determining both capacities and locations of future components. A first step in this planning is the forecast of the future electric demand with geographic detail sufficient to plan equipment sizes and locations. Such load forecasting is called spatial, or small area, load forecasting, and is the subject of this paper.

The objectives of power system planning are to provide an orderly and economic expansion to meet the electric utility's future electrical demand with an acceptable level of reliability. The components of the delivery system-subtransmission lines, substations, distribution feeders, and laterals-generally have capacities many orders of magnitude less than the load of the entire system. Planning of this system involves determining the correct sizes, locations, interconnections, and timing of future additions to this equipment. Such planning is a difficult task, compounded by recent trends of tightening design margins, longer equipment lead times, and increasing regulatory scrutiny. These trends have resulted in increasing power industry emphasis on delivery system planning. Correct planning of future delivery equipment requires a forecast of the future geographic distribution of electric demand in a manner sufficient to differentiate between available alternatives of capacity, location, and interconnection. The quality and accuracy of this forecast has a large influence on the quality of the subsequent delivery system planning. Therefore, methods to accomplish such forecasts have been the subject of increasing attention in

submission of this paper was encouraged after the review of an advance Manuscript received June 22, 1982;revised December 10, 1982. The

proposal. The authors are with the Advanced Systems Technology Division of

Westinghouse Electric Corporation, Pittsburgh, PA 15235.

the last few years. During the last decade, a number of practical and efficient automated computerized methods have been developed as an aid in delivery system planning. Many employ optimization to refine the system design and determine a minimum cost expansion plan [84]-[89], [95], [ 108 ]. These automatic procedures produce substantial savings over more traditional design methods, but require accurate projections of future small arealoads [49], [ 6 6 ] , 1831, [ S I , [94]. The desire t o gain maximum benefit from these procedures has fur- ther spurred development of improved forecasting methods.

In this paper we review the major aspects of spatial load forecasting, concentrating on the needs and methods used in modem power system planning. For this reason we concentrate almost exclusively on computerized techniques that utilize data available from modem utility load and customer data bases. We first briefly summarize planning and design of power delivery systems in order to identify the needs for and uses of a spatial load forecast. Then we discuss electric load and load growth, concentrating on those factors of load growth that are important in forecasting, particularly coincidence of load, spatial resolution, growth characteristics, time period, and forecasting error. In the final section, the bulk of this paper, we present the various types of forecast methods from the perspectives established earlier and summarize each technique, its application, and shortcomings. We have attempted to ob- jectively review and compare the various methods of forecasting with respect to one another and the planning needs of electric distribution systems.

In determining typical applications and usages, and in evaluating the state of the art, we are relying heavily on our own experience as well as four recent surveys of electric utility planning practices [ 901 - [ 93 ]. Where applicable, we have used the terminology developed under Electric Power Research Institute (EPRI) project RP-570 [49]. Where EPRI has not developed appropriate terminology, we use our own.

11. PLANNING AND FORECASTING NEEDS The primary purpose of power delivery planning is to deter-

mine an orderly expansion of the present system in order to meet future demand and to forecast budget needs for financial planning. Planning falls into the two broad categories of short- range and long-range analysis, which differ in both timespan and goals. Short-range planning has been the traditional approach for distribution systems and is done only far enough in- to the future to cover equipment delivery and construction lead times for the next series of system additions. The goals and motivations of short-range planning are to determine the single best plan for present construction commitments and to ensure that the required equipment is operational when needed.

By contrast, long-range planning is motivated by a desire to determine economic viability of the short-range commitments. Recognizing that any unit of equipment will have a useful life-

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WILLIS AND NORTHCOTE-GREEN: SPATIAL ELECTRIC LOAD FORECASTING 233

Fig. 1. A spatial load forecast locates electric demand geographically, as shown in th is load density grey scale plot of electric demand for a 25-by-25-mi part of a large metropolitan area.

Year 1 Year 10 Year 20 Fig. 2. Electric load grows in both magnitude and location as shown

here in displays of electric load (Z axis) for Panama City, Panama, for the years 1980, 1990, and 2000. In these diagrams the PaciTk Ocean is in the right portion of the plot, the Panama Canal runs along the lower left edge. Small area size is 4 k m 2 .

time of many years, long-range planning attempts to assure that all elements of the short-range plan are efficiently utilized throughout their lifetime, not just in the first few years after installation. Therefore, long-range planning concentrates on a time period beginning at equipment lead time and extending into the future. The uncertainty associated with long-range forecasting is minimized by using a number of individual forecasts, each a reasonable representation of future load growth. These forecasts are used in a multiple-scenario study of long- range needs. Short-range commitments can be studied to assure that they fit well into all reasonable long-range scenarios. There- fore, the forecasting for long-range planning needs a multiple scenario capability-the ability to produce a set of reasonable forecasts that cover the uncertainty of future load growth. The need for short-range planning is for a single most accurate forecast as a guide to determine construction commitments.

Spatial load forecasting involves prediction of both the magnitudes and locations of future electric load. The analysis of location must be made with sufficient geographic resolution to permit siting and sizing of delivery system components. A spatial forecast is therefore not only a forecast of a certain total magnitude of load but also a forecast of its geographic distribution. The Same total could be distributed differently in a variety of spatial forecasts-each a unique pattern of the

locations of load, and each perhaps requiring a different delivery system design.

The planning of most power systems is performed on an annual basis in anticipation of annual peak load. Spatial load analysis is accomplished by dividing a utility system into a number of small areas and forecasting the load in each. Loads for planning are typically referred to on an annual basis by small area

I&) = peak annual load for small area k, in year t ,

k = I to k small areas. h

Zk(t, t o ) = estimate of l k ( t ) , made from data from year to < t .

In some cases, the small areas used for geographic analysis are irregular in shape and size, corresponding to the service areas assigned to particular delivery system components such as substations or feeders. However, most modem techniques use an M X N grid of square cells, indexed by n , rn location, that cover the region to be studied.

I n , & ) = l k ( t ) , where k = n + (m - 1) * N,

where m = l , M and n = 1,N.

A cellular spatial load distribution is shown in Figs. 1 and 2.

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2 34 PROCEEDINGS OF THE IEEE, VOL. 71, NO. 2, FEBRUARY 1983

The planning of a power systems expansion always includes a forecast of the spatial load distribution because any plan for expansion (or no expansion) is based on some set of assumptions about future load growth, however poorly defied, cursory or implicit. Surveys indicate a fairly even split between use of a uniform grid and equipment-oriented area definitions. Area size varies depending on application. Currently, most U.S. utilities use 640-acre (mi2) or 160-acre (quarter-section) cells, or irregularly sized areas associated with distribution substations. The average short-range forecast period is four years ahead; that of long-range planning eight to twenty years ahead.

The first spatial forecast methods using a digital computer were developed and applied in the late 1950’s. These were done on an irregular, equipment area basis using multiple regression curve fitting to extrapolate historical load trends. Computerized land-use based methods were first developed in the early 1960’s, initially as methods in which the computer was used merely to store and output forecasts based totally on user intuition [ 1 1 J , [ 15 J , [ 17 J . Land-use based methods employ an analysis of the mix of customer types (land use) and density on an area by area basis. Not surprisingly, as the data- base management, analytical, and data-handling speed of computers has evolved, the methodology and quality of forecasting has kept pace. Development of computerized forecast methods has accelerated in the last five years, driven by the availability of optimization-based automatic distribution design programs [ 49 J , [ 841 - [ 891, [ 951, requiring a high-resolution spatial load forecast. The largest single impetus in this field was EPRI’s project RP-570 from 1975 to 1979 [49]. This investigated a wide range of forecast methods, established a uniform terminology, and identified many of the field’s major concepts and priorities.

Almost all modem spatial forecast methods are allocation methods that distribute on a small area basis a previously forecast total system load which is an input to the forecast procedure. There are two compelling reasons why this is done. First, considerable sophistication and effort is used by the average electric utility in projecting its total system load growth [ 75 J -[ 82 J . This attention to the forecast of the total load is motivated by the utility’s financial planning which needs a projection of electric power sales revenues. Also, forecasts of total system load are needed for the planning of generation. Consequently, the effort put into such forecasts is high, and the results from an engineering standpoint are excellent. While total system forecasts do not spatially locate the load growth, they typically stratify, or break down, the forecast on a rate or customer-class basis. Often they include very detailed specifics on the expected average per-customer use of electricity. There is no need for a spatial forecast procedure to repeat this analysis. Second, within most utilities the various engineering plans will be approved by management only if they are compatible with the company financial plan-a plan developed directly from the company’s revenue forecast. Plans developed from another, independent total system forecast would have a nearly hopeless uphill battle for approval.

111. LOAD AND LOAD GROWTH BEHAVIOR Coincidence of Load

The electric load on a power system is the composite demand of many electrical devices owned by many different customers. An important aspect of electric load is that customers do not simultaneously demand their peak power; there is a great diver- sity of when people, businesses, and devices demand power.

0.f 10 1W loo0 10000

NUMBER OF CUSTOMERS Fig. 3. A coincidence curve for sizing equipment gives the multiplier to

use in determining peak load based on the number of customers.

For this reason the peak total system load occurs when the combination of customer demands is at its highest. Some customers may be demanding only a part of their peak demand at that instant. The ratio of the peak total system demand and the sum of the individual customer peak is called the coincidence factor C,

peak system load sum of customer peak loads *

c, = (1)

For most power systems this ratio is in the range of 0.3-0.7. This apparently simple concept has subtle and far-reaching

consequences regarding the power system’s analysis [ 791 - [ 82 I , [ 84 J - [ 921. The peak load on any piece of equipment occurs when the combination of the demands of all the customers it serves is at a maximum. This peak may not occur at the same time as the total system peak, or as the peaks of any other equipment. For this reason, the sum of all feeder peak loads usually far exceeds the total system load.

Coincidence factors are used in much of the analysis and design of equipment loads because utility engineers normally will have timely, reliable load readings on only major equipment. Values used in the engineering and planning of much of the smaller equipment must be estimated in one manner or another from available data. Most electric utilities size distribution equipment with a demand coincidence curve similar to that shown in Fig. 3. The curve is based on observed behavior or probabilistic assumptions about the distribution of customer peak demands and off-peak demands. As a function of the number of customers involved, this curve gives a coincidence multiplier that is used to size equipment. For instance, if a planned feeder to a new subdivision were to serve 150 homes, it would be sized as follows:

total load = sum of individual estimated household peaks for 150 houses

feeder capacity = C( 150) * total load + margin (2)

where C( 150) is the coincidence value from the curve for 150 customers. The margin would be dictated by the utility’s planning policies and design standards. If ten laterals will distribute the feeder load among the 150 homes, each would be sized in a similar manner, using on the average C( 15) instead of C( 150). Note that since C( 15) > C( 1501, the sum of the lateral capacities would exceed the feeder capacity.

Few spatial load models explicitly address coincidence of



40001 TOTAL SYSTEM LOAD

....... ...... . .

........ ..... . . ’““I SEVERAL SQUARE MILES

/ . . . . ..... I----

........

’“7 . . .

Fig. 4. As a typical electric utility system is subdivided into smaller and smaller areas, its load growth behavior becomes less and less a smooth continuous process, and more of the “S”-curve behavior, characterized by a sharp, brief period of growth.

load in the manner that we have discussed above. Coincidence is acknowledged in one of two ways. First, it may simply be ignored; the load-value used for a small area is the best avaii- able data on the annual peak demand in that area. Ignoring coincidence has no adverse consequences if the small area size is large enough that a typical small area contains at least several hundred customers. The coincidence factor for that number of customers will be very close to C,, thus a small area’s peak load will be approximately its “share” of system peak, and very little error is introduced by ignoring coincidence.

Alternatively, the values used may include an explicit adjustment to account for coincidence.

Z&) used for area k = (C(e)/C(a)) * (estimated peak load in cell k) (3)

where a is the average number of customers in a cell, e is the average number of customers in a substation or feeder service area (whichever equipment level is being planned from the load forecast).

This is the most common manner of addressing load coincidence in modem spatial load models. The recent trend toward much higher spatial resolution has made the effects of load coincidence significant. For instance, 10-acre cells often contain only 10 or 20 customers. The coincidence factor for numbers of customers in this range is not close to C,. As noted earlier, the crux of distribution planning is the overall system design and the sizing of its major components. An estimate of the peak load in any one small area is not as important as that area’s role in the overall system. By using such loads on a small area basis, any number of Ik( t ) can be directly added to form load estimates for any arbitrary area. If the sole purpose of the planning is to plan for large equipment (that with over 500 customers), this is valid because the coincidence curve (Fig. 3) is flat for the large numbers of customers served by major units of equipment.

Load Growth Character

Load growth of a power system is due to two simultaneous processes. First, growth is driven by increases in the number of customers in the utility service area. Second, it can be caused by increases in the average demand usage per customer. New customers are added to a system due to migration into an area (population growth) or the electrification of previously nonelec-

tric households. Growth of the number of customers causes the spread of electric load into areas that previously had no electric load- cells that were “vacant” from the power system’s standpoint. Changes in usage per customer occur simultaneously and largely independently of any change in the number of customers. Except for areas of urban renewal, over the past decade, there have been only slight increases in usage per customer in most US. electric utilities. This trend is a result of increasing prices, conservation, and improvements in the efficiency of electrical usage. A number of comprehensive methods exist to assess the changes in customer usage due to such factors. These methods are adequately covered in current literature [ 75 ] -[ 821 and will be discussed in this paper only to the extent that they impact spatial forecast models. Here the character of spatial load growth will be examined in two ways-analysis of temporal growth trends and analysis of load level “transitions.”

When viewed from a system total basis, a growing power system generally exhibits a smooth, continuous trend of annual peak load growth. Slight variations are caused by weather and minor factors that vary from one year to the next. By contrast, the behavior of growth in a typical small area is not a smooth curve but is more like that shown in Fig. 4-a sharp Ogive, or Gompertz curve, commonly referred to as an “S” curve [29] , [42], [49], [ 581. The majority of the small area load growth occurs during only a few years, during which load goes from a near zero to a value close to a final “saturated” peak load. The smooth constant trend of the total system load is due to it being the aggregate of many small areas whose “bursts” of growth occur at different times.

Fig. 4 shows growth curve averages for a large North Ameri- can city. As the cell size used for analysis is reduced, the typical growth curve shape gradually becomes less a smooth constant trend and more a sharper “S” curve shape. Typical growth curve shape is a function of spatial resolution. Of course, each small area will differ-most slightly, and a few significantly-from the average growth behavior. The quantitative aspects-the average rise time and its changes as a function of the resolution-differ from one utility system to another. This basic behavior is observable in nearly every power system [49], [ 581, [64].

A second perspective of electric load growth can be studied with a matrix approach to transitions among various levels of load [ 571. Define a set of load levels that partition the small area load levels, the following sixteen-level set defined on the last year t 2 of a period t l , t2


236 PROCEEDINGS OF THE IEEE, VOL. 71, NO. 2, FEBRUARY 1983

(4 (f) (9) 01)

Fig. 5. Transition matrices for electric load growth as observed at various resolutions and time periods in a large city. (a) 2560 acres-2 yn. (b) 2560 acres-4 yn . (c) 2560 acres4 yrs. (d) 2560 acres-8 yrs. (e) 640 acres-4 yrs. (f) 160 acres-4 yrs. (g) 40 acres-4 p. (h) 40 acres- 8 y n .

916 = load level exceeded by of all cells Similarly, the total growth due to all cells that ended the period in level i is the row sum

For any year t , all small areas in a utility system can be classi- fied by their load l k ( t ) into one of these 16 classes. A transition matrix of movement of small areas from one level to others during any particular period of time can be formed as a Q by Q matrix with elements

T i j = number of small areas that moved from level i to level j during the period from year tl to year t z .

From this, a matrix useful in the diagnosis of the forecast method applicability can be formed as

G ~ J = Ti,i * (2, - Qi) = number of cells that moved from level i to level j , weighted by the difference in average load between the two levels (Qi - ai). (5)

The G matrix has element values that, neglecting load-level quantization errors, are the portion of load growth due to transitions among the various levels. Total system load growth from year t l to t 2 is

The growth over the period, due only to growth of all cells that started the period at level j is a column sum

16 growth from leveli = Ggj . (6)

i = l

16 growth from level i = G g j . ( 7)

j=l

Fig. 5 shows G matrices for a large metropolitan area with a fairly high growth rate. These matrices are displayed as grey scale plots of the element values to facilitate the easy recognition of their essential qualities. The form of these matrices varies considerably depending on both the cell size and time period, reflecting the change in the character of growth as a function of both cell size and time period. They highlight an interesting quality of spatial load growth; from the standpoint or the character of observed load growth, reducing cell size and lengthening time period are equivalent.

Fig. 5(a) shows G matrices of growth in four-square mile areas over a two-year period. This matrix is diffuse, indicating that the majority of growth occurred due to transitions of cells from among many different load levels-particularly in the intermediate levels. Column one-growth due to development of load in vacant areas-is nearly empty, indicating little load growth in previously vacant areas. Little growth occurs in the rows indicating growth to the higher load levels, the final or “saturation” levels. If the time period is lengthened, as shown in Fig. 5(b), (c), and (d), the form of the matrix changes. Less of the total growth is due to transitions among the intermediate levels; more is due to transitions of cells into the higher load levels. There is significant growth due to transitions of cells that began the period with no load, as evidenced by the first column of the matrix.

A similar change in character occurs if the time period is held constant but the cell size is reduced. Fig. 5(a), (e), (f), and (g) show, respectively, four-year G matrices for 2560,640, 160, and 40 acre resolutions. If cell size is reduced while period length is held constant, more growth occurs as transitions to the final load levels, and column 1-growth of cells


WILLIS AND NORTHCOTE-GREEN: SPATIAL ELECTRIC LOAD FORECASTING 2 3 1

N

t LLW 1 mile u-. 1 km. WL + ++-+

-- -. .++ +- - -- - - + - + -+ -.- --+. .

(a) (b) Fig. 6. The spatial distribution of error in three different forecasts of

the same group of forty-acre small areas. Flus and minus signs indicate over or under forecasts in each area; the size of the sign gives an indication of magnitude. Broad lines indicate approximate boundaries of substation service areas; thin lines indicate feeder areas. All three forecasts were produced with Houston Lighting and Power Company’s ELUFANT forecast model, with variations to the parameter setup of its high-frequency suitability model [ 331.

that began the period with no load-becomes significant. The TABLE I predominant growth is in column 1 -growth of previously EVALUATION OF ERROR IMPACT BY THREE DIFFERENT MEASURES VERSUS

vacant areas. reveals that from the perspective Of A N A L Y S ~ OF THREE DIFFERENT FORECASTS AND THE POWER S Y ~ M I M P A C ~ ON SUBSTATION AND FEEDER PLANNING, MEASURED BY ACTUAL

observed behavior, increasing spatial resolution or increasing time period has much the same effect. ERROR FORECAST

These two different ways to study load growth-curve shapes and transition matrices-reveal the basic change in the character R.M.S. 13.44 14.4 14.0

of the load growth as the resolution and time period are varied. Average Absol. Value 12.3 11.3 12.3

At a fairly low spatial resolution load growth is generally a Impact on Substations

gradual growth of each area, lasting many years, with few pre- measured by examinatiol, 9.8 2.4 2.6 measured by eq.9 10.0 2 .5 2 .6

UEASURE - - - A B C

viously “vacant” areas developing any load. At higher resolu- -ct on Feeder iaads tions growth behavior is much sharper and briefer, with many cases of previously vacant cells developing load growth. If the

measured by examination 1 2 . 0 3.5 8.2 measured by eq. 9 1 2 . 1 3.7 8 .2

time period of observation is long compared to the average cellular rise time, then most of the growth will appear to be transitions to the final load levels, and will be mostly due to development of previously vacant areas.

The two methods given above allow the applicability of various forecast methods to be studied quantitatively for any particular situation. For example, traditional load forecast methods such as curve fitting and other trending methods are incapable of forecasting development in presently vacant areas. The column 1 sum of the transition matrix for a particular problem is the amount of “vacant area growth” that such trending methods cannot predict. Some modem simulation methods work with a simplifying assumption that growth occurs as transitions directly from the vacant to a final level [ 361, [ 431. In cases where most growth appears in column 1 and the higher load level rows, such an assumption causes little error.

Analysis of Spatial Forecast Error Before turning to the forecast methods themselves, we wish

to briefly examine the quantitative measurement of error in a load forecast and its relationship to the planning of a power system. Generally, forecast methods are tested by predicting present small area peak loads, using data from a past year, perhaps ten years before. The accuracy in forecasting present loads is taken as an indication of probable accuracy in forecasting future loads over an equivalent future time period [ 231 - [ 271, [291, [331, [371-[421. When a system expansion plan is

developed based on an actual forecast, the planner is of course unaware of the magnitudes and locations of the errors in the forecast. Define the error as the difference between the estimated and actual load on a small area basis

= error in forecast load for area n, m . (8)

A number of traditional error measures have been applied to the en,m including the rms, average absolute value, and Mean and Variance and the “68 percent Rule” developed under EPRI sponsorship [49]. These various measures have given conflicting, inconsistent rankings of different forecast methods applied to the same problem I441. More importantly, none correspond to the error impact when assessed by detailed study of the actual impact on planning [471, [ 63 1 . Tradi- tional statistical measures do not properly assess error measure because they are not sensitive to the spatial qualities of the errors, while a power system is.

Fig. 6 shows the forecast errors for the same portion of a utility system, from three different forecasts. Each forecast has roughly 13.5-percent rms error, as shown in Table I. In- spection of the three forecasts will show that they obviously have a far different impact on planning. The forecast shown in Fig. 6(b) has errors distributed in a manner such that positive and negative errors usually cancel almost entirely within any



predicted - ’ FEEDERS actual 0

1

vl W

0 iL

vl Y

n

a

0 0 1 0 5 1

FREPUENCY - Cyclewmi.

Fig. 7. Results of tests comparing actual impact of error on the estima-

signal analysis of system error sensitivity. Spatial frequencies repre- tion of feeder and substation loads, with the impact as estimated by

senting error were added to small area load data and the consequent impact measured and recorded 88 shown. “Predicted” response is the normalized spectrum of the Pdd) fdter.

areas the size of feeder or substation areas. Thus errors in forecast (b) would have little adverse impact on planning feeder and substation capacities. When its loads are added manually or by computer to form the equipment load estimate for any reasonable set of feeder or substation areas (not just those shown), most of the errors cancel leaving a net error close to zero; hence, errors of this type have little adverse impact on planning. As shown in Table I, these errors cause less than 2-3 percent error in estimating substation loads, as determined by detailed examination.

Forecast errors distributed as shown in Fig. 6(a) do not similarly cancel. Positive and negative errors tend to “clump” together in large contiguous areas, whose size in general is equal to or larger than equipment service areas. The net effect is that a substantial amount of error accumulates in feeder and substation areas. A forecast method that spatially distributed error in this manner could be very damaging to planning. The forecast shown in Fig. 6(c) is an example of a forecast that has strong impact on feeders and little impact on substations. In this forecast the errors are distributed so that they largely cancel within substation areas, but on the average only partially cancel over areas the size of feeder areas. This forecast would mislead the planning of feeder capacities but generally lead to correct sizing of substations.

The difference in impacts of the three forecast error sets discussed above reveals an interesting aspect of system planning. From the standpoint of spatial error impact, the power system planning is a spatial low-pass filter. High spatial frequencies of the load forecast error, as in Fig. 6(b), have no impact, but lower frequencies, as in Fig. 6(a), do have an impact. Small area electric load can be viewed as a signal function of space. When the cells used in the small area analysis are the cells of a uniform grid the Zn, m(t ) values are uniform discrete samples of the spatial electric load signal. These values are not strictly samples of the load level at the particular points, but the aggregate of electric load in each cell. This type of data is roughly equivalent to that obtained by passing the spatial load signal Z(t) through a spatial low-pass filter, and sampling the output of the filter [ 1001. A set of two-dimensional fiiters, whose derivation is fully explained elsewhere can measure impact on the particular level of a power system. For instance, for substations the filter’s space domain representation is

P,(d) = probability that a randomly selected cell a distance d away from a particular point is served by the substation serving that point. PJx, y ) is a twodimen- sional function P,(~W).

This function, P,(x, y ) , is a circularly symmetric function and embodies the overall design characteristics of the substation design of the system. Actual impact of the cell’s errors on substation load estimates is closely approximated by

N M

(10)

where

P , ( d ) = O , forall d > P .

The Us value usually corresponds closely to the actual impact of the errors determined by examination, as shown in Table 1. The P,(d) filter derived above from the substation area structure, and others similarly derived for transmission, subtransmission, and feeders give the “response” to spatial forecast errors of the planning of equipment capacities. Agreement with observed behavior is usually excellent and has been veri- fied on a number of systems as shown in Fig. 7 [63]. All such filters are low pass; specifics vary from one system to another depending on equipment specifications, design standards, and

A load forecast must contain an accurate representation of all the spatial frequencies to which the design is sensitive. The spatial sampling of the electric load must be done with a rate sufficient to represent the frequencies to which the design is sensitive. The filter’s spectra can be used along with the Nyquist criteria to determine the required forecast cell size, the sampling rate, needed to accurately represent the loads. Table I1 shows sampling rates calculated for a large metropolitan system. As would be expected, the design of the smaller parts of the system (feeders) requires substantially more resolution than that required for larger equipments.

The preceding discussion of error shows that the forecast of the individual loads is not as important to proper power system planning as the assessment of some of the overall spatial aspects of the load distribution. For this reason, the authors refer to forecasting of loads for distribution planning asspatial, rather

geography.



I I

Fig. 8. Two scenarios of electric load growth for a major city. Both represent a forecast of the same total amount of load but differ in the pattern of its geographic location (most of the difference is in the lower right portions of the patterns). The difference in the two patterns is due to modeling of a new highway as along either of two proposed alternate routes.

TABLE I1 CELL SIZES NEEDED FOR ADEQUATE LOAD MODELING FOR THE VARIOUS

LEVEU OF A LARGE METROPOLITAN POWER SYSTEM

Lcvel of System Area Sizes Needed

PIV hansmission 4 sq. miles (10 kn 1 2

Transmission 1 sq. mile (2.5 km 1 2

S U b S a t i O n S 160 acres (.63 kni 1 2

2 Feeders 71 acres (.28 km )

than just small area, forecasting. This highlights the need for an integrated assessment of the entire load pattern.

The discussion above is a workable definition of error assessment, or quality, etc., when only one forecast is compared to actual loads. How does one measure the “error” in a set of multiple scenario forecasts? If one of these alternative forecasts were absolutely correct, all the others would by their very nature be “incorrect.” Criteria for multi-scenario forecasts are much harder to define. A common misconception is that planning can be done based on the small area expectations. These expectations are obtained by probabilistically weighting the various scenario loads for each cell according to the perceived likelihood of each scenario, as

I E(lk( t ) ) = ( P i ’ zk(i, t ) ) p j

/ I (1 1)

i=1 j = 1

where there are I scenarios and pi = probability of scenario i , Ik(i, t ) = forecast load for year t in cell k, for scenario i . Using such E(Zk(t)) is invalid, as demonstrated by a simple example. Suppose there are two scenarios, each representing theaddition to the future loads of a major factory, which could go at either of two widely separated but equally likely sites. Each of the two scenarios generated from assumptions that the factory would locate at either site is a valid, reasonable forecast of what might happen. Combining the loads probabilistically results in a “scenario” with half a factory at either site-something that is quite unlikely. In general, combining small area loads in a probabilistic manner results in a geographic pattern that does

not sufficiently identify problems caused by localized areas of high density load. Using expectations gives load patterns much “smoother” than normal; essentially low-pass spatial-filtered patterns. The various scenario load sets must be carried through the planning process individually, as shown in Fig. 8. Multiple scenarios will be discussed in more detail later in this article.

IV. LOAD FORECASTING METHODS

In this section the defiitions and concepts developed earlier will be used in a discussion of the various spatial forecasting procedures. Spatial load forecast methods can be categorized by several characteristics. These categories are: 1) whether or not they analyze historical data in the course of the forecast; 2) the type of grid or area distinction; and 3) the type of forecast algorithm. AnaZytic forecast methods perform analysis of past or present data, identifying trends and patterns that are then used to project future load growth. By contrast, nonanalytic methods are approaches that use a computer only for the record keeping and report generation of an intuitively generated forecast. Most of the newer forecast techniques use a rectangular grid of uniformly sized small areas as the basis of the forecast. Many methods, particularly the traditional approaches, use irregularly sized areas defined by the service areas assigned to different members of some equipment level, such as substations or feeders.

Fig. 9 shows the overall relationship of the spatial forecast algorithms which we will discuss in this section. Analytic methods fal l into several categories. The simplest in terms of data and method are the various algorithms that extrapolate past small area load growth, generally called trending methods [SI, [91, [ 101, 1141, [251, [291, W I , [@I, 1581, [MI, [691. Some are quite novel. Multivariate methods work with other data in addition to small area load data [ 271, [ 281, [49]. The types of possible algorithms vary considerably, but the term “multivariate” alone has generally been used for methods that extrapolate simultaneously a number of variables on a small area basis. A very successful class of multivariable methods are Zand-use based methods that employ one or another form of urban system land-use simulation to predict future electric load ~ ~ l , ~ ~ I , ~ ~ ~ l , ~ ~ ~ l , ~ ~ ~ l , ~ ~ ~ l , ~ ~ ~ l , ~ ~ ~ l , ~ ~ ~ l , ~ ~ ~ l - ~ ~ ~ l ,



NON-ANALYTIC METHODS THAT

WORK ONLY WITH METHODS THAT

ANALYTIC WORK WITH MORE

MULTIVARIATE

FILTERING 2 3-3 OR 3-3-3

Fig. 9. The different classes of spatial electric load forecast method are shown here, plotted by general level of forecast quality (accuracy, flexibility) vertically, and complexity (ease of use, data needs, resource

ship of various methods, following the classification of methodology requirements) horizontally. Tree structure show the family relation-

described in the text. Year given for each method is the authors’ esti- - mate of its fvst use.

[431-[511, [601, [651, [661. Although these are technically multivariate methods, they are generally regarded as a separate class because they often do not extrapolate in the strictest sense.

Nonunalytic Methods

Nonumlytic methods for spatial load forecasting rely almost completely on user intuition. A popular method, dubbed the “coloring book” approach, will be described here as typical of all such methods [ 71, [ 101, [ 181. A map of the utility’s service area is divided by a series of grid lines into 160 acre cells. The amount of existing land use in several classes, for example, residential, apartments, commercial, and industrial, is placed on the map by literally coloring each cell with colored pencils, a different color indicating each class. Future land- use, obtained from user intuition, is similarly coded onto the map. An average per-acre load density for each land use class is developed from experience, survey data, and the utility’s overall system load forecast. These densities are used to manually convert the colored land uses into a load for each cell. Despite the apparent crudeness, this method or a variation has been widely used as a long-range planning tool by over one fourth of all utilities surveyed.

It is a natural step to employ a computer as the data manipu- lation and reporting medium for such a scheme. A number of techniques doing just this were developed in the 1960’s and early 1970’s [8] , [ 111, [ 161, [ 171, [45]. Use of a computer allowed expansion of the methods in a number of ways. More classes of load, up to twenty, could be used. A common fea- ture was the ability to scale or adjust the total growth to meet

specified system total load leveL Results from such methods are generally poor, but required resources are minimal.

Trending

Trending is a generic term that encompasses all methods that extrapolate load based on past load values. Such methods have two advantages; they employ simple, direct algorithms, and they require a minimum of data. Nearly every utility maintains records of major equipment loadings, so the required data are always available at least on an equipment area basis. The most obvious trending method involves curve fitting to the historical loads on an area-by-area basis. A K area forecast is done as K independent extrapolations, using multiple regression to fit a function to each cell (area), typically

A

I k ( t ) = f k ( t , t2 , t3 , r-l) (12)

or h

=fk(log t , (log o’, (log 0 3 ) . (13)

Applied on a low-resolution, short-term forecast basis (typically on l-square mile or larger areas for three-to-five year forecast periods) multiple regression gives useful results. When applied to areas smaller than 1 square mile, curve fit and extrapolation methods have problems dealing with the sharp “S”Curve shapes typical of small areas 1421, [49], [58]. Equipment service areas-the usual source of a nonuniform area basis-are time varying due to transfer of loads from one feeder t o another that occur when operating personnel reswitch loads and feeder segments, or when new construction is done.



These transfers cause variations in recorded loads, creating substantial error in trend identification.

Most applications of trending use a “horizon year” load estimate for each area

A

l&) = estimated eventual load level for cell k, in year h, a year well beyond end of forecast period.

The function f k ( t ) is fitted to both the historical data values and this horizon year load. The contribution of such horizon year loads t o improved forecasts has been well documented [ 291 , [ 491 , [ 581. Short-term, low-resolution regression-based forecasts are fairly robust as regards these horizon year estimates; results are not highly sensitive to reasonable levels of error in the horizon year estimates [ 581, [64].

A considerable amount of research has been devoted to regression-based trending. Meinke [ 421 performed extensive tests of historical data period, function type, and forecast period. Similar tests of extrapolation sponsored by EPRI [251, [29] reached roughly the same conclusions: trending is applicable to short-term ( 5 years or less) forecasting, and the best extrapolation for most applications is a cubic logarithmic function (( 13) above) fitted to only the most recent five or six years of data. These studies were all at relatively low spatial resolutions of 640 or 520 acres. Whether these conclusions are valid at higher resolutions has not been determined.

The industry surveys discussed earlier indicate a wide variety of slight variations in applications of regression-based methods. Variations have included heuristic rules for ignoring “bad” data points and numerous adjustment schemes for scaling the sum of all trends to equal a global control total. Most adjustment methods fall into one of the three categories covered in EPRI sponsored research: 1) adjustment proportional to the area loads themselves; 2) adjustment proportional t o the small area growth or; 3) adjustment proportional to the curve fit residuals (49).

Recall that earlier the discussion of the growth transition matrices showed that for small cell sizes, or long forecast periods, a significant portion of growth occurs in previously vacant areas (the column 1 sum of the G matrices in Fig. 5). Trending on an individual cell basis cannot forecast development of load where the historical load is zero. A nonzero horizon year load for a vacant cell may force a nonzero extrapolation, but it hardly provides a reasonable forecast. For this reason curve fit techniques are limited to short-term, low- resolution applications, where the amount of load occurring in “vacant” areas is only a small fraction of total growth.

An interesting variation in extrapolation uses a clustering- template matching method instead of regression [61] , [64]. This method uses a set of about six typical curve shapes, called templates, t o forecast load on a cell-by-cell basis. Each cell’s recent 6-year load history is shifted back and forth against each template. The best combination of template and shift determines the cell’s forecast, as shown in Fig. 10. A simple clustering algorithm is used to derive the set of templates from the historical small area data. The calculations are simple and amenable to 16-bit integer arithmetic, making this method adaptable to the smallest computers. Due to the use of clustering, the method works only on a uniform a d basis.

Multiple Cell Trending The most successful trending methods work not on a cell-

by-cell basis, but on groups of cells. A simple method of multiple cell trending, called vacant area inference (VAI) is

RESENT

4 . . . . . . . . . . . . . . . . , FUTUlE

YEAR

Fig. 10. Clustet curve method fits a “template,” selected from a set of typical growth curve shapes, to each small area’s load history, shifting the template t o minimize mismatch with the load history. The template then provides the forecast. The method f M solves for the templates based on a cluster analysis of past histories of all cells.

partially successful at forecasting load growth in cells with no past or present electric load. Future growth in a vacant area can often be correctly inferred by observing that a region’s smooth, continuous load growth can continue only if vacant areas within the region then begin to grow. Fig. 11 shows a square mile divided into four areas, three of which have a load. These three subareas can be trended-they each have a load history. The fourth area cannot be trended-its load history is zero. The entire square mile can be trended-using a horizon year load that is the sum of the four suburea horizon year loads. The trend for the square mile can be compared to the sum of the trends of the three subareas that were individually trended. The difference is the load attributable to the fourth (vacant) area. Computer programs applying this VAI concept have been applied to both regression-based and cluster/ template trending [58], [67]. Extensive testing has proven that it is much more accurate than cell-by-cell regression curve fit, with typically only 3 to 3 the average absolute error.

Since the early 19603, a number of trending methods have used a threedimensional function to fit to the small area loads

load = f(x, y) , where x, y index the small area locations.

(14)

Generally, these methods severely restrict the form of f by u priori assumptions about its character. Recognizing a strong tendency for higher load density in the central core of a city, several methods use a function that is the sum of one or more radially symmetric, monotonically decreasing functions of distance from an urban pole, as shown in Fig. 12. Lazzari f i s t used this approach in the mid 1960’s [ 1 1 ] , [ 131. Many similar approaches are now in use [8], [ 161, [34].

There are three features to such functions: 1) the location of the urban pole, ( x p ( t ) , y p ( t ) ) ; 2) the central load density, or pole “height,” D ( t ) ; and 3) a per-unit function of distance, d, from the center, g(d, t ) with g(0, t ) 1 .O. As usually applied, g(d, t ) is a monotonically decreasing function of distance, whose “radius” or point where g(d, t ) = 6 increases with time. As applied, the volume under the curve expands in concert with the previously projected system total, the central height expands as necessary to accommodate increases in load density, and the diameter increases to accommodate an expanding city.

For the cell at the center (x&), y p ( t ) )



difference is vacant area

trend of sum of ....... histories

0

trends

Y E A R

/ l / i l / l i / l Fig. 11. Vacant Area Inference (VAI) techniques trend the load in small

areas that have no load or load history. This is done by comparing the sum of the individual area trends of a block of areas to the trend of the sum of their histories, as shown. The difference is the inferred trend, assigned to the vacant area. The critical element is the horizon

trends and the trend of the sum. year estimates, w h i c h causes the difference between the sum of the

h as applied, perspective

DISTANCE FROM CITY CENTER

Fig. 12. An “urban pole” function models the tendency of electric load density to increase in the center of a city. These simple functions are

load, which can then be refrned by other means. often very successful at modeling the overall spatial distribution of

and

l i j = D(t) - g ( d ( x p - i)’ + ( y p - j ) 2 , t). (16)

The volume of the function is the total load, on a discrete cellular basis.

i = j j= l

An important and easy addition often made to these methods is to restrict the load in certain cells t o zero, where u priori information indicates no load growth is possible. Cells representing such areas as cemeteries, parks, swamps, and land with sharp gradients, etc., can all be defined as I,.,,,(t) = 0 for all t . This results in a much improved forecast [ 341, [651.

A forecast of load using this technique is straightforward. The function is fitted to load values for the most recent year, and also for the loads for several years before, finding in each year the center D(t ) and the form of g ( d , t ) . The forecast sys-

\ actu

functwn nd r u n t bnd data

--. ,_I *j P

y- -.::.: ..... ‘2.,r.‘...‘’ ............ -hnctiM-C 1

0 FREOUENCV - CYCLES/YILE .5

Fig. 13. Spectrum of the spatial load distribution for a small city (pop. 220 000), and an urban pole function carefully fitted to the load distribution. As shown, addition of information on the restricted vacant areas such as parks and cemetaries improves the match to actual distribution. Note that most of the serious mismatch is in the higher frequencies to w h i c h planning is not highly sensitive (see Fig. 7).

tem total (obtained from the utility’s revenue forecast) defines the forecast total volume of the function. The change in the location of the center over the “calibration period” is extrapolated linearly, as is any change in o(t). For any future year these three extrapolated changes can be used to determine a new diameter, completely defining the function’s values. Implicitly, this method forecasts the growth of previously “vacant” areas.

When fitted to past data and extrapolated into the future, functions of this type typically produce “S”-curve shapes for each cell’s forecast load growth. The “S” curves obtained from such a method can be compared to the observed range of curve shapes, as shown earlier in Fig. 4. They will be a reasonable match to the “S”-curve behavior at some (usually large) area size. In a sense, that area size is the resolution limit of the method; it cannot generate local growth curves that are any “sharper.” Even though it may be applied on a higher resolution basis, the method is really modeling each cell as a part of a larger area growth curve, and has limited resolution.

Analysis of the spatial frequency content of this function also provides insight into its applicability. Fig. 13 shows the function’s spectrum and that of actual load density for a small city to which it has been carefully fitted. The control total de f i e s the spatial dc component of load. Use of u priori data on known vacant cells such as cemeteries provides some of the


high frequencies in the method’s spectrum. Most of the error is in the band of spatial frequencies to which substation planning (see Fig. 7) is not sensitive.

As the total system load (the volume under the function) increases, the resolution of this method viewed by either of the two methods described here, decreases. When used to cover a larger city, the function’s limit of “accurate” spatial resolution decreases. The same number of parameters and functional complexity are being used to cover a much larger geographic area; “resolution” suffers. Since this method loses its resolution when its size is expanded, there is a largest city or region size for which it is really suited, a function of the resolution needed. Recognizing that many larger cities were far from circularly symmetric, several forecast methods use a number of different “urban pole” functions [ 161, [ 341, [ 651. A more complex geographic distribution of load can then be modeled, and a large city can be modeled while maintaining higher resolution.

The authors feel that urban pole trending methods and variations on them to be discussed later may be the most undenated form of spatial load forecasting. Their simplicity causes many people to dismiss them without adequate testing. They have proven particularly suitable for forecasting in small cities with a well-defined “center of town.”

Other Trending Techniques An early form of multi-area trending involved disaggregation

of the total load forecast, by “trending” area loads against the system total load, rather than in time

I k ( t ) =Fk(Z(f), other data) (17)

where L ( t ) is the total system load (aggregate) forecast for year t . There were a great many variations of this theme [ lo] , [ 161 , [ 67 I . Results are roughly equivalent to regression-based trending, and depend on the form of solution of the equations Fk( ) from historical data. Methods that solve for all Fk simultaneously, rather than one at a time, are more accurate and can be improved by using generalized, or Markov, regression [35] .

There are entire bodies of apparently applicable trending methodology that have to the authors’ knowledge not been widely applied to spatial load forecasting. These and other methods of trending have not begn vigorously pursued for several reasons. First, trending methods cannot relate changes in load to “causal factors,” such as the completion of a new highway, changes in the economy, etc. This means trend methods are not amenable to multiple scenario forecasting. Secondly, a once-important virtue, economy of data needs, is not as critical as it was in the past. Many utilities now have computerized transformer load management and customer information system data bases that can provide detailed, rele- vant data [ 891 -[ 941. Third, the continuing increase in economical computational power is allowing practical implemen- tation of more involved analysis techniques. These reasons have focused most development and research along other lines.

Multrivariate Extrapolation

A

Multivariate methods of spatial load forecasting make use of more than just the I&) peak annual load values, using instead from one to as many as sixty additional data measure- ments on each small area. These variables form a “data vector” for each area.

W t ) = (Ul ,k( t ) , U 2 , k ( t ) - - - U&k(t)) .

uik(f) is variable i in year f, for cell k. I is the total number of variables. Some of the data may be load data.

The nature of these additional data variables varies considerably among approaches. All multivariate methods differ from trending methods, in that they work as a series of iterations, extending data in time

Vn(t + a ) = G( Vn(t), C ( t + a ) ) (18)

when V( t ) is a vector composed of all the u i J t ) and C(f + a) denotes a set of “control” totals and data. a is the forecast iteration period, typically 1 to 3 years. G is the multivariate forecast function. Multiyear forecasts for periods more than a year long are done as several iterations.

The definitive computer program for multivariate forecasting was developed by Wilreker, Strintsis, and Long under EPRI sponsorship and is now widely known within the electric power industry as “Multivariate” [491. This program is flexible and can be set up to perform most types of multivariate load analysis. The mathematical formulation is covered elsewhere in great detail [27], [281. ’The computer program itself is available from EPRI [491. The concept behind this method is simple.

1) The space is not the I data dimensions. Multivariate small area data are usually highly collinear. A principal components rotation transforms each cell’s vector into P dimensional space, P < I . These P dimensions are linearly independent and statis- tically normalized, allowing the next steps to be performed validly [ 271.

2) The cells are not individually trended. They are clustered into sets. Each cluster is used to develop a linear function of movement within the cluster, the average “trajectory” of its cells. The resulting model is several piecewise (cluster) linear approximations of a nonlinear process [ 281.

3) Once established, these trajectories could be projected for many years into the future. Instead, the forecast trends each a short distance in the space, representing change over only a few years. The forecast proceeds iteratively, recalculating the entire process every few years. The reason is that as cells move in the space, they could change clusters and hence trajectory characteristics.

4) At any point in the process the cell’s vector in P space can be interpreted into electric load using the principal components loading matrix and a simple linear function [491.

This procedure produced reasonable results in a series of tests sponsored by EPRI [44], [ 491 when “driven” by control variables (the C(t + n) in (18)) from a land-use projection program of the type to be discussed later. This land-use projection provided the information on change from vacant to positive counts of residential, etc., customer type in year t + a. Viewed from the perspectives of load growth dynamics discussed earlier, the method can handle extreme types of ‘ Y c u r v e behavior because of the piecewise linear model based on the clusters. By itself, it cannot accommodate vacant area growth, a major shortcoming. The burden of forecasting vacant areas is shifted onto the “control variable” forecast, which almost totally determines the quality of the procedure’s long-range, high-resolution accuracy,

As small area size is reduced, the number of customer types in an average cell decreases. Viewed at a one-mile resolution, most cells in a city or town will have a great many types of load; usually each will have some fraction of load for about 50 percent of all the customer classes but will vary greatly in the mixture in each cell. The clustering method used by “Multi-



Fig. 14. Overall structure of a land-use based spatial load forecast method.

variate” will f i id a rich statistical mix on which to work and will distinguish a number of clusters. As cell size is reduced to 160,40, or 10 acres, an average cell will be composed of fewer, at most a few, customer types. Thus at high resolutions, the individual VK(t) vectors used will be sparse. Consequently, clustering will not work. During a test at 25 acres cell size by the authors, the cluster definitions in Multivariate “collapsed” to little more than the customer class definitions. Cells were best distinguished only by land-use type and density-distinctions that are clear without the use of a clustering method.

Recently Thompson and Gonen have developed a novel multivariate approach to distribution load forecasting that employs tables of production rules, similar in concept to the developmental Lindenmayer systems, automata, and grammars used to describe the growth of biological systems [ 521, [62]. This multivariate model is novel in that it simultaneously pre- dicts the growth of both load and the delivery system network configuration. It has given promising results in limited tests for forecasting rural electric load growth.

Land-Use Based Forecasting Land-use based methods involve an intermediate forecast of

the growth of land-use type and density as a f i t step in the forecast of electric load (Fig. 14). This type of analysis has a number of practical and operational advantages over other approaches. Chief among these is the complete decoupling of the two causes of load growth. Changes in the electrical usage on a per-customer basis is tallied by a usage modeL Change in the number and location of customers, which drives the geographic spread of electric load, is handled by the landuse forecast. Other forecast methods do not explicitly make this distinction and consequently suffer in their ability to infer valid trends and patterns from historical data. Another advantage of land-use methods is that they can use the utility’s total system forecasts to provide the bulk of the data needed for an electric usage model that is related to land-use classes. As mentioned earlier, the techniques used to project total system load provide extremely sophisticated projections of the expected average behavior by customer class. A third advantage is that land-use methods are particularly well suited to multiple scenario studies. They relate to the basic causal aspects of the growth. By contrast, the trending and multivariate methods discussed earlier cannot realistically vary growth modeling. The concept of land-use/customer class appeals to intuition. Therefore, subjective evaluation of a land-use based forecast

is easy, improving the credibility of the results over that of other methods, an important point when the forecast is to be used to justify major system expansion. The largest advantage is that analytic land-use techniques are the most accurate and useful spatial forecast methods available. Their advantage over other techniques increases with a lengthened forecast period and increasing spatial resolution [451, 1491, [541, [651.

All modem methods work on a square-grid basis, with data on the land-use type and density (amount) in each small area. An- cillary data on geography and other factors are often included. There are C classes of land use, c = 1, C , typically 9 < C < 20. These classes include such distinctions as residential, commercial, and industrial (usually three or four subclasses within each).

Defiie

&(t) = amount of land-use type c in cell k for year t .

These a i ( t ) are a subset of the Uk(t) vector for cell k. The additional data vary from one approach to another but usually include zoning and data on the transportation network.

The units of measurement of the ag(t) are usually one of three types: 1) the actual number of customers of type c in cell k; 2) the number of acres of land devoted to class c in cell k; or 3) the percent of land area in cell k devoted to class c.

All land-use methods work in the same basic manner, differing only in the method and sophistication of each of the several steps shown in Fig. 14. A land-use based method is first used in a calibration phase in which present and historical data are analyzed to determine trends and locational patterns. .These trends and patterns are then used to forecast the cellular data on land-use type and density on a cell-bycell basis. The resulting projected land use is then transformed into electric load with the application of the usage model which separately models load of each land-use type.

There are three approaches to the conversion of the land-use data to electric load: 1) a single factor per class; 2) a daily load profiie (24 hourly loads) for each class; or 3) use of several 24-h curves for each class, each curve representing a part of the class’s load.

1) The first method uses a single factor of load density per unit of each land-use type, and may be a function of the year

L,( t ) = electric demand of one unit of land-use type c in year t .

The load in cell k, in year t is

2) The second method uses 24 hourly load readings rather than a single value

L J h , t ) E electric demand of one unit of land-use type c, in hour h of the peak day, in year t

and the 24 hourly loads for the day, for the cell become

c load in cell k at h hour of the day = (ag(t) * L C @ , r)).

c = 1

(20) This second method’s advantage is that the 24-h curves explicitly address the fact that class’s loads peak at different times of day.



3) The most complete load models used in spatial load analysis are end-use type models, in which a set of up to 200 load types are used. Each is a 24-h daily curve that describes the load behavior of a particular subclass of load. For example, residential might be represented by subfunctions for lighting, cooking, water heating, space heating, and air conditioning loads, etc. Multiple-scenariovariations to a particular subfunc- tion, representing alternate scenarios of future usage, can be included in the spatial loads by merely varying and recalculating the load functions, without having to recalculate the land-use forecast.

Land-Use Forecasting Computerized analysis of land usage on a spatial basis has

grown into an established field of socioeconomic studies that has at one time or another looked at man’s use of land from nearly every imaginable perspective. Three concepts of analysis, borrowed from urban modeling, are key ingredients in many spatial urban models: 1) the concept of basic causal growth and balance; 2) the “gravity” model of influence; and 3) the concept of small area growth suitability. In the early 1960’s, Lowry developed a model of a metropolitan area’s growth as driven by “basic industry”-that portion of the local employment sector that marketed outside the region. The expansion of the basic industry would increase employment, leading to expansion of the residential sector, leading to an increase in the retail (commercial) sector of the local economy [ 961 . This concept can be interpreted as several basic rules for “balancing” the forecast growth of any city, town, or rural area: a) a region must provide sufficient room and services to house and sustain the populace; b) there must be a market area to deliver food and other necessary items of sustenance to the populace; c) land, activity, and electric load must be devoted to industries that provide the local economy with basic employment; and d) these needs must be balanced in ratio. One does not find a residential area of small size in a city with a tremendous industrial base; the industrial workers must live somewhere close to their employment. Similarly, the residential segment of the city necessitates a market segment large enough to feed, clothe, entertain, and otherwise sustain it. Therefore, any city, town, or region will display reasonable and somewhat predictable ratios of land use that grow in this coupled, driven manner.

The second concept, the “gravity model,” assumes the growth influence (coupling) of two areas is proportional to magnitude of the influencing factors and inversely proportional to a function of distance (not necessarily inverse squared) [ 95 1, [ 961. In Lowry’s original model of Pittsburgh, the expansion of residential growth was assumed to be driven by a monotonically decreasing function of distance centered in the industrial areas; there was more demand for housing close to these employment centers than far away from them. “Distance” meant travel time by road and commuter methods, not straight line distance. Much of the spatial distribution of a city can be understood by analysis of the street, freeway, rail, and water transportation systems [96], [98], [ loll , [102]. Insomecases,thesimpli- fying assumption that travel time is proportional to straight- line distance can be used with good effect, particularly on small cities lacking dominant geographical barriers.

The third concept of urban modeling is land parcel suitability. Individual parcels of land are considered suitable only for certain land uses. Suitability as used here can be thought of as a numerical measure of a particular parcei of land meeting spe-

XIC landuse needs. For example, a parcel of land near a railroad track may be ideal for industrial purposes and yet marginal at best-due to noise, etc.-for residential use. A parcel of land only a half mile away may have opposite suitabilities.

A spatial load forecast procedure can use these three concepts of balance, “gravity” influence, and suitability to refiie estimates of the likelihood of future land-use class develop ment on a small area basis. These concepts are very amenable to computerization, particularly in simulation approaches. In some cases spatial load growth models have addressed these three concepts by directly applying urban models developed for transportation and city planning [ 221, [ 231 , [ 481, most of which forecast land-use growth on a census tract basis. Relat- ing census tract output of such models to the cells of a uniform grid or to the nonuniform equipment areas can become extremely complicated in itself [ 3 3 ] , [35]. Urban modeling concepts applied in techniques specifically designed for electric load forecasting have performed much better than existing urban models applied to electric utility planning [ 1 11, 1231, [261, [381, [491 , [601, [651,[661 , [941.

Spatial Frequency Ranges of DemandlSupply All land-use based spatial forecast methods, both analytic

and nonanalytic, can be viewed from a spatial land demand and land-use supply modeling perspective. Many have not been so identified in the literature, but for consistency and illumina- tion of similarities we will use this perspective. In order for growth or change to occur, there must be a simultaneous spatial match of both demand and supply. For example, if residential growth occurs in a cell, it is because both a demand for housing and a supply of suitable housing land was simultaneously present in that cell. A fascinating aspect of landuse modeling as applied to electric load forecasting is that if demand and supply are appropriately defined, they are respectively the low and high spatial frequency aspects of growth. This concept was f i t explicitly used for electric load modeling by one of the authors in the late 1970’s [ 301, but it is clearly an implicit part of most land-use approaches.

In order to fit into this perspective, demand must be viewed as the restrictions on where someone seeking expansion for a particular land-use type searches to find land matching his needs. Examples would be the individual seeking to find a new home, or a business desirous of a new location. Both have a specific, exact, defintion of w h r is suitable for their needs (good neighborhood, close to schools, not close to competing industry, etc.). However, in addition to these suitability needs, each restricts his search to a certain, perhaps fuzzily defined, region. Regardless of what suitable land might be available outside of this “search area,” the search for a new site is restricted to this region. The prospective home buyer may examine only a quadrant of a city due to a desire to be relatively close to his employment. A business may likewise desire to locate a new retail outlet only in an expanding quadrant of the city. Each is willing to accept almost any parcel of land in a wide area if it matches his requirements better than any other parcel in that area, but will not accept a parcel outside the area even if it is better. In a large city, the homeowner is usually indifferent to a change in exact location of one or two miles if he finds a particular parcel of land that better matches what he considers suitable. But he would not accept a change in location of twenty miles for even a larger improvement. The key concept is that demand, defined as the locational restrictions on growth due to where people desire to



F I L T E R C E R E C l D C N T I I L

. . . . . . .... . . _ . . . . . . ..

Fig. 15. Spatial diatribution of 1980 residential electric load on a. 40- acre small area basis for Calgary, Canada. (a) Actual load data. (b) Data after filtering that attenuates frequencies modeled as due to land suitability effects. These same data are plotted in Fig. 16.

live and work in rehtion to the city or region, is indifferent to small changes in distance. So defined, a quantitative measure of "demand" varies slowly on a spatial basis; it is composed of low and intermediate spatial frequencies.

The homeowner and the business have specific suitability definitions of the qualities that would make a parcel of land suitable for their purposes. Each land,use type has certain characteristics that are critical, or at least important, to its suitability. Commercial retail generally wants to find a parcel of land that is close to a major traffic corridor. Residential needs to be close to other residential (good neighborhood), convenient to transportation, near amenities (schools, etc.) but not near industry or railroads. These characteristics tend to change rapidly in space. One particular parcel of land may be ideal for a land-use purpose, and yet a parcel only mile away may be very marginal for that purpose. Therefore, quantitative measures of land suitability for specific usages are of relatively high spatial frequency. If one views the set of all land parcels that are suitable for a particular land-use class as the supply of land for that purpose, then supply is spatially of high frequency.

Regardless of how suitable any site is, it will develop only if there is also demand for development there. If demand for housing is restricted, for whatever reason, to only the East region of a city, little development will occur in the West. The excellent sites in the West nonetheless rate high as a supply of land suitable for that purpose. A parcel of land may be suitable for a particular land-use class, but growth is not guaran- teed just because it is suitable. Likewise, demand alone does not guarantee growth; there must also be a supply of suitable land. It is the combination of both that results in growth. Be- cause demand is mostly low frequency, and supply is high spatial frequency, the authors interpret them respectively as the low- and high-frequency models of growth.

In 1979 one of the authors (Willis) tested these frequency

concepts at a 25-acre basis for the western half of Houston, TX, based on growth from 1973 to 1979 (a period where population roughly doubled). Supply was determined as the best fit to the small area growth characteristics using the ELUFANT spatial land-use model to be discussed later [26], [30], [33]. Eighty-one percent of all growth occurred in areas designated as a highly suitable supply (most highly rated 12 percent of cells). The supply was calculated with a suitability model that had spatial frequency content of only 4-mi wavelengths and shorter. Only 43 percent of such highly rated cells displayed any growth. The demand was 0 in the others. Demand in this experiment was calculated by a transportation- type urban model that modeled growth demand with functions that were in the spatial frequency range dc-to-4-mi wavelength. When combined, demand and supply calculations explained a total of 93 percent of the growth locations and magnitudes of electric load, verifying the basic concept of high frequency supply and low frequency demand.

Often, when small area load data are filtered, rejecting those frequencies in the suitability-supply band, an obvious urban pole structure of low frequency emerges. Figs. 15 and 16 show the correlation between amount of residential load on a per-cell basis and distance from the major employment centers for Calgary, Canada. When the cellular residential development is filtered with a spatial filter that attenuates frequencies in the "suitability band," an obvious correlation emerges from what was otherwise an apparent random scatter. The result clearly shows the simple, almost linear tendency for residential location to be near the employment center. The fdter rejected spatial distribution caused by suitability, leaving only that caused by demand.

Calculation of Spatial Demand and Supply As discussed above, demand is largely transportation ori-

ented on the basis of convenient travel to employment. The



UNFILTERED DATA FILTERED DATA

MILES

(a)

MILES

(b) Fig. 16. Residential electric load versus distance to downtown on a 40-

acre basis, is plotted here in scatter diagrams. (a) No particular pattern emerges from the actual data, due to cell-by-ceJl variations in load density caused by factors not related to distance from the urban core. (b) Filtering of the data results in data values which show the expected correlation, attributable to land-use demand factors.

better transportation urban models, such as Putman’s model of transportation influence [ 1011, work with zones of about ten square miles or larger. Hence, they are low-frequency models of growth. Little if any modification of these concepts, procedures, or calculation methods is needed for application in spatial load models. Analysis can be performed on a large area basis and the low-frequency results interpolated for a high resolution [ 4 3 ] .

In our discussion of trending methods given earlier, we showed that for many situations, the simple methods of the urban pole-type model all the low-frequency components of spatial growth. These are the same frequencies that are the “demand band.” Therefore, such urban pole models applied in a land-use method often sufficiently fulfill the needs of a demand model. Because urban pole functions are low frequency, in the authors’ viewpoint they are land-use-demand models even if implemented as part of the trending approaches discussed earlier.

By contrast with the low-frequency models, the high-frequency models used in spatial load modeling have not applied the quantitative methods used in urban modeling, where there is a perceived difficulty in calculating a measure at high spatial resolution [ 10 1 1 . Define suitability for land-use type c, in cell k as

p i = suitability of type c in cell k, a measure of how cell k matches the needs of land-use type c.

The exact manner of calculation of the pk varies among present forecasting methods. Usually, & is a function of both the cell’s data vector V k ( t ) and the data vectors of other small areas next to or near to it. Many modern spatial load models determine suitabilities using very extensive calculations on the basis of two types of factors-“proximity” and “ s ~ r r o ~ n d ” factors. A proximity factor is an “inverse distance” to the nearest item of some set of amenities. An example from Houston Lighting and Power’s ELUFANT model [36] is used in industrial suitability analysis.

Dk = 5 if there is a railroad in cell k

{ (5 - distance in cell widths to the nearest railroad),

D~ = if it is less than 5 cell widths away

0, otherwise.

The distance weighted sum of all of one class or item in the area is called a surround factor.

S:,j = sum of class c development near cell i, j , weighted by a function of distance W,(d).

x=1 y=1

Here, W,(d) is monotonically decreasing with distance and W ( d ) = 0 for d >8. Both W,(d) and 8 will vary depending on the factor being calculated.

Combinations and functions of several proximity and surround factors are used in many land-use based models to explain suitability on a high-resolution basis. For instance, the suitability for residential development may be a function of proximity t o schools, proximity to commercial (shopping), amount of surrounding residential development-all positive factors. Negative factors may include proximity t o railroads, amount of surrounding industry, and proximity t o airports. Calculation of the suitabilities based on these factors is straightforward. It is the determination of the proximity and surround factors that is difficult. Direct calculation proved very lengthy in some of the first electric load models in which it was applied [ 3 0 ] , [ 3 6 ] .

Factors such as Dn and Wn above can be calculated rapidly in the spatial frequency domain. Note that since W(d) is of finite duration, monotonically decreasing t o 0 beyond some distance b (consistent with the gravity concept) ( 16) may be rewritten as

a convolution of the u,“ with a radially symmetric FIR oper-



ator. Calculation can be done in three steps: 1) Transform both the as “image” and W(d) into their Discrete Fourier Transforms (DFT’s) by use of a Fast Fourier Transform (FFT) method; 2) multiply them point by point; and 3) inverse transform the products using the FFT. The result is the Sg. This procedure is fast, making such calculations practical [ 361, [47], [ 711. The authors have recently implemented a similar method that calculates Dk factors within 8 percent using the FFT, a logarithmic operator and a “trick” interpretation [ 1061,

Land-Use Based Forecast Models

As discussed above, all land-use based spatial load forecast models consist of three basic components: 1) demand model; 2) supply or suitability model; and 3) land-use to load model, varying only in the level of sophistication of each.

Demand models have been implemented in three basic ways:

1) no spatial demand model other than a control total by

2) “urban poles” spatial model [ 131 , [ 341, [ 651 ; 3) urbanltransportation model [ 301, [66].

class [171, [SO];

The third type is normally done on a large area (zonal) basis. The second is composed of simple functions of inherently low spatial frequency but normally calculated at high resolution for the sake of simplicity.

Supply models are high resolution and also fall into three basic categories:

1) manual input suitability values based on user intuition

2) combination of zoning and manual input suitability [ 341 ; 3) suitability measures calculated from detailed spatial fac-

tors, usually based on proximity and surround factors [361, i 1051.

[ I l l , [ 1 7 1 ;

The load model can be done in any of three forms as discussed earlier:

1) a single load value for each class [ 171, [ 341 ; 2) a 24-h curve for each class [45], [65] ; 3) a 24-h curve for each of several end-use subclasses of

each class [65] , [66] .

The utility’s total system load forecast provides much of the information needed for any of these three load models.

Because all land-use based methods employ some form of analysis in each of the three categories, it is convenient to refer to any land-use based forecast method with three numbers corresponding respectively to the basic type of demand, supply, and load models. Thus a 2-1-2 model is a technique that uses urban pole demand functions, manually input small area suitabilities, and 24-h class-by-class load curves. We will use this categorization in the discussion of models that have been used in the utility industry.

A number of electric utilities have obtained the output of low resolution, census-tract based urban models used for municipal planning and combined this with manually determined suitabilities in a semi-analytic 3-1-1 method [8 ] , [ 151, [ 201, [48]. This approach gives better results than purely manual, coloring book methods because the urban model balances growth as discussed earlier. Problems in credibility and accuracy have been reported [ 161, [22], [46], [65], due to the dependence on an urban model external to the utility.

Most important, multiple-scenario variations cannot be performed due to lack of access to the urban model itself.

Fischer developed a unique land-use method for Dallas Power and Light that used preferences based on zoning, road, and highway networks [SO]. I t is the only 1-3-1 method the authors have encountered. The model has an extensive and very interesting method of tying historical load data on an equipment area basis, to the small area land use. It was recently replaced with a 2-3-3 method developed by the authors [ 1051.

The EPRI sponsored Multivariate model used EMPIRIC, a census-tract based urban analysis model whose census tract projections of growth, allocated manually among the small areas in each tract, were the “control variables” referred to in the earlier discussion of Multivariate. EMPIRIC performed the spatial land-use demand analysis, Multivariate accomplished both high-resolution suitability analysis and the load model in one step. This method was basically a 3-2-1 model and gave good results in tests on Phoenix, AZ [ 491.

Brooks developed a now widely used 2-2-1 model using two urban pole functions which are coupled by different ratios to each of 10 land-use or customer classes. Suitabilities for each land-use class are calculated from two input data factors. The combination of these two factors with the demand function yields a growth allocation value for each cell for each land-use class. The resulting values are the relative probability of development in a “Monte Carlo” allocation of the total system growth by land-use class among all K cells. A type-1 load model is used. This method has given excellent results in forecasting for small metropolitan areas [ 341 , [ 1071.

Willis developed a nonanalytic land-use method during the early 1970’s that was subsequently evolved into a 3-3-1 model called ELUFANT [301 , [33], [361,[461, [471, used at Hous- ton Lighting and Power from 1974 to 198 1. It operated on a grid of over 250 000 cells of 25.6 acres (& square mile). A detailed suitability model operating at this resolution calculated suitabilities based on 20 proximity and surround factors. A Lowry urban model, operating on 16-miz square zones, was used to allocate the total system growth (an input) to these large zones. Once calculated, the suitability factors were matched with demand factors using a complex clustering-pattern recognition analysis borrowed from the EPRI multivariate model [30], [36]. A type-1 load model was used, with load values estimated from historical substation load data using a novel method developed by Sadler [35].

ELUFANT, one of the f i t attempts at large-scale high- resolution land-use simulation, explicitly modeled demand and supply as low- and high-frequency processes. The original model (1975) had a “gap” between the demand model’s 16-miz resolution and those frequencies easily and efficiently handled by the 25-acre high-resolution model. Gregg subsequently developed an improved low-frequency model for ELUFANT to correct this problem [431.

A 2-3-3 land-use based method has recently been completed by the authors’ organization, under sponsorship of the Canadian Electric Association [65]. This model uses a demand model with multiple urban poles d e f i e d as the center of growth of each class. A suitability model combines manually input zoning with ten proximity and surround factors calculated in the frequency domain. For the demand model, the center of each urban pole, its diameter and its slope are found by a statisti- cally guided iterative procedure utilizing filtering of the type demonstrated in Fig. 15. Calibration of the high-frequency model is basically heuristic, but straightforward if carried out



with the aid of: 1) data fitering that attenuates low frequencies; and 2) display maps of calculated surround and proximity factors. A type-3 load model uses 96-point (1 5-min) daily load curves, similar to those used in load management studies [ 771 - [ 821. This model, called CEALUS, has proven extremely easy to use and accurate in recent tests on Calgary, Canada.

The authors recently developed a very flexible computer program for land-use based spatial forecasting. This program, called SLF-2, is controlled by a user-specified “macro” program, of instructions for the basic forecast steps [ 1051 . This macro program capability allows it to be quickly tailored to a particular problem/data/situation. SLF-2 is now in use on a dozen utility systems, implementing 1-1-3, 2-1-2, 2-2-2, 2-3-3, and 3-3-3 forecast models. Procedures used in SLF are similar to those of other land-use simulation methods with three ex- ceptions. First, calculation of suitability, not just the surround factors, is done in the frequency domain. Suitability is modeled as linear functions of the surround and proximity factors, and the suitability function’s spectrum can be directly calculated in the frequency domain. Second, calculation of low-frequency effects is done at low resolution and interpolated as needed. Third, the program will automatically analyze multiple scenarios in the manner to be discussed later. Several high-speed fitering procedures, which reduce two-dimensional symmetri- cal filters to one-dimensional approximations, are used to increase speed [ 711, [ 1041.

Most land-use methods use only a single year of data history (the most recent) as the base for the entire forecast. Contrasted with this are multiple-year methods that base the forecast on several years of land-use data. These two types of method differ in how they are calibrated against past data. In single- year methods, the demand and supply models are calibrated or fitted to best “explain” all land-use distribution in the only year of data used. In multiyear methods the model’s functions are adjusted to explain the changes between two or more years. The single-year methods are surprisingly effective and currently outperform multiple-year methods by quite large margins [ 591, [ 651 . The critical factor limiting multiyear methods is data error in year-to-year land-use data [45] , [66]. As an example, data from 1977 and 1981 for a utility with a customer growth rate of 4 percent annually, would represent a 17-percent increase in total land-use density (measured as number of customers). If each land-use cellular data base had 5-percent random data error-about the best level economically attainable- differences in the two year’s data due only to error are roughly 10 percent [ 701. Total difference-error plus actual changes caused by growth-is in the order of 27 percent, of which over f is error. Calibrating a multiyear model to this difference in- troduces substantial error into the forecast. At the present time, the quality of available data usually gives better results if single year models are fitted to all land-use location in only the most recent year of data [59], [ 701. The desire to improve this situation has led the authors to research LANDSAT data and automatic interpretation as a source of historical land-use data [371, 1551. It is hoped that interpretation errors from automatic procedures would be highly correlated from year to year, minimizing the problem discussed above. So far, results have been disappointing [ 591.

Applicability of Land-Use Based Methods The applicability of the demand and supply models of land-

use growth is interesting when viewed from a spatial frequency standpoint. The high-frequency suitability model’s resolution

(cell size required) is set by the requirement of spatial frequencies needed by the planning, as described earlier in the discussion of error. The type of demand model is really dictated by the complexity of the metropolitan area. Large cities such as Los Angeles or Houston are so extensive that a simple urban pole model is generally not sufficient. A model with two, three, or even four urban poles is fairly easy to calibrate and apply. An 11 urban pole demand model of Houston gave results equivalent to the transportation analysis urban model portion of ELUFANT, but was no easier to use because determining the factors for eleven urban poles was more difficult than calibrating a transportation model. The choice of demand model type is really based on which is easiest to use in the application at hand.

Land-use methods are ideally adapted to multiple-scenario forecasting. Earlier, we discussed how a type-3 load model could be used for scenarios involving the application of load management and other changes in usage. In addition, land-use simulation models have data and parameters in both the demand and supply models adaptable to represent scenarios of significant shifts in social, economic, and transportation parameters [231, [431, [491, [65). In this respect type-2 demand models are somewhat harder to use than type 3, but have been made to perform very well [33], [45], [ 1071.

It was shown earlier that errors-differences between forecast and actual loads-had an impact, or “difference of design” that was assessible by a filtering analysis. The same basic assessment of impact can be done to compare two or more alternate scenario forecasts. The cellular differences in loads of two forecasts can be filtered in the same manner used to evaluate error impact. If a significant portion of their differences is passed, it means the scenarios will have an impact on the planning. The authors’ spatial load forecasting program, SLF, uses this concept in two ways. First, the differences among scenarios are filtered and the scenarios are ranked by expected “impact” or design difference for each level of the system-a guide for the engineer in designing systems for each. Second, significant input and model parameters are pertubated in fl0-percent in- crements and the resulting changes in loads are filtered. Only those parameters that result in significant filtered impact are varied, singly and in combination, to establish some of the model’s scenarios [66]. Seldom do changes to the high- frequency (suitability) model of growth result in filtered impact, since these changes are high frequency, and the P,(d )- type filters are low pass. Therefore, most scenarios of importance represent variations in the land-use demand and electric load (usage) model.

Spatial frequency analysis of spatial load distribution itself is interesting and a useful step in determining applicability of forecast procedures. Fig. 17 shows the spectra, in one dimen- sion, of four cities in North America. The high and intermediate frequency components vary among these examples largely due to the differing terrain and geographic restrictions in the various cities. Our last figure, Fig. 18, shows spatial frequency ranges of a number of concepts we have talked about in this and the previous section, the averages of data and parameters on four cities in North America. It is presented in no more quantitative form because we are not certain the results are quantitatively general. A broad range of high spatial frequencies are of vacant land types. The availability, or lack of availability, of land for growth is critical to high-resolution forecasting.

Analysis of the frequency content of the electric load, and



~RE&r- C~CLES/MILE .5 5

Fig. 17. Spectra of spatial load distribution of four cities in North America. Two-dimensional spectra have been reduced to one dimen- sion using a method based on an inverse to the McClellan Transform [ 1061. Each spectrum is normalized to dc component equal to 1 .O. Low-frequency differences are due mwtly to the size and ove-rall lay- out of the cities. Mid- and high-frequency differences are largely a

region that is geographically rather featureless, has lower medium- function of terrain and geographic restrictions. Houston, built in a

small mountains and a flood zone, or Syracuse, a city that is built and high-frequency content than Phoenix, a city built around s e v a a l

around a lake and several other geographic restrictions.

I I I I I DC 0 5 1 0 1 5 2 0 2 5

SPATIAL FREOUENCY CYCLESIMILE

Ti. 18. Qualitative range of spatial frequencies of needs, effects, and data involved in spatial load forecasting.

the various data sources and available forecast models, can be used to tailor and calibrate a land-use approach to best fit a particular situation. As mentioned earlier, evaluation of spatial resolution needs using the P,(d) and other filters provides insight on the required small area size, and implicitly, on the limit of spatial frequencies that must be accurately covered by the forecast model. The spatial frequency content of the electric load itself gives an indication of the degree of importance of the low- and high-frequency models of growth. If the high- frequency content of the spatial load pattern is low, the error introduced into a forecast by ignoring high frequencies may be tolerable. Alternately, it may be possible t o establish that the high frequencies of interest can be tied to one particular data source, such as vacant land (Fig. 13), or to a very simplified model. Considerable attention to the high-frequency modeling is justified because cost in terms of data, computer and human resources, and complexity increases rapidly as spatial resolution is increased.

Applicability of a forecasting method to a particular problem is dependent on a number of factors, only some of which are related to the forecasting. It is important not to lose sight of the ultimate goal, which is not the forecast but the planning of a well-designed, economical power delivery system. There fore, the load forecast should be selected to provide the type

of analysis consistent with the planning goals, planning period, and planning method. The forecast method must also be compatible with both the available data base and computer and human resources.

Applicability of any forecast method, as well as accuracy and quality, varies greatly depending on data, planning goals, and resources. For this reason quantitative comparison of the performance of all the forecasting methods covered in this paper is difficult. In Fig. 9 we attempted to plot the various algorithms by both forecast quality and overall complexity of method, data, and resources. Table 111 gives the relative error levels we use in our own work when selecting from among forecast methods for any particular study. It was developed based on our own work over the past 12 years. Within each category of performance measure, we have defined trending with regression based curve fitting as 1 .O.

We admit to a very strong preference for the land-use based approach, largely because of its intuitive appeal, its multiple- scenario capabilities, and its compatibility with the growth characteristics of electric load. The higher costs, mostly due to data needs, are easily justified by the performance advantage over other techniques. Although present industry applications still make heavy use of load extrapolation, the trend is clearly toward land-use based methods [ 901 -[ 941. The reasons we



TABLE 111 RELATIVE PERFORMANCE OF GENERIC CLASSES OF FORECASTING METHOD

Planning Impact as Cost of Cost of

MS Measured Required Required Method T m Error by Eq. 8’ Data Resources

TRENDING - -

Regression)(ref. 29) 1.0 1.0 1.0 1.0

VAI-TRENDING (581 .6 .55 1.0 1.3

URBAN POLE LOAD FIT (13) .5 .40 1.5 2.0

NON-A)(ALYTIC LAND USE (17) .72 .55 10.0 10.0

ANALYTIC LAND USE 2-2-1 .30 .30 10.0 (34)

8.0

ANALYTIC LAND USE 3/2-3-3 2 0 .15 12.0 10.0 ( 3 6 )

ANALYTIC LAND USE 3/2-3-3 .15 .075 12.0 15.0 w i t h m u l t i p l e f i l t e r e d scenarios based on p a r m t e r p e r t u b a t i o n s (66) ‘AS measured on feeders using the best application o f the forecasting nethod.

have given above provide most of the motivation for this shift. Another and equally important reason is that a land-use based method can be heavily tied to a detailed, customer class-oriented forecast of a utility’s total system load. Therefore, use of a land-use based method improves the coordination between the power delivery planning forecast and the overall total system forecast.

The advantages of land-use based methods have focused most present research activity on them, almost to the complete exclusion of all other approaches. These techniques will un- doubtedly become the power industry’s standard approach during the remainder of this decade.

V. CONCLUSION

Spatial load forecasting is currently undergoing a rapid change as more traditional methods are replaced by newer techniques that properly mesh with modem optimization-based planning and design techniques. The apparent characteristics of load growth depend on time period and spatial resolution of the data and planning. Therefore, application of particular forecast methods is best evaluated by relating their qualities to the characteristic dynamics of load growth at the resolution and time period being studied. Signal theory is invaluable in spatial load analysis as it provides a comprehensive framework for study of planning sensitivity, load growth character, model structure and performance, and error. Accuracy in a load forecast is best judged by evaluating errors in a spatial f i ter manner to estimate impact on planning. The planning of the various levels of a power system differ in the sensitivity to spatial error distribution.

Spatial load forecasting methods operate with one of three approaches-trending of historical load growth, multivariate extrapolation, or developing a simulation of landuse growth as an intermediate forecast. The land-use based methods are suited to long-range study needs, both because they can perform valid multi-scenario studies and also because they are well suited to high-resolution, long-range growth dynamics. The various methods of land-use based load forecasting differ only in the degree of sophistication of three basic components:

land demand model, land supply model, and electric load model.

Spatial load forecasting is not an end in itself but only the first step in the planning of a power system. Most of the con- straints, goals, and priorities that have shaped the form of these forecasting methods and their application are due to needs of other planning tools. Spatial load forecast methods and optimization-based techniques are dependent on one another, neither can produce large improvements without the other. The recent recognition that these two together can cut delivery system costs by 10 percent has spurred development of both. We believe that the historical development rate and capabilities of spatial load forecast methods looks much like the “S” curves of small area growth itself, with the present state of the art midway up the high slope part of the curve. We expect intense development and increasing application of such methods during the remainder of the 1980’s.

ACKNOWLEDGMENT The authors wish to thank A. B. Cummings, J. Gregg, S. D.

Havemann, T. W. Parks, C. R. Sadler, D. L. Wall, and C. L. Brooks for their many helpful suggestions and comments.

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[ 331 H. L. Willis, J. Gregg, Y. Chambers, “Spatial load forecasting for system planning,” presented at the American Power Conf.

[ 341 C. L. Brooks and J. E. D. Northcote-Green, “A stochastic-prefer- (Chicago, IL, Apr. 1978).

ence technique for allocation of customer growth using small area modeling,” in Proc. American Power Conf. (Univ. Illinois,

[ 35) C. R. Sadler and H. L. Willis, “Boundary location errors in spatial Chicago, IL, Apr. 1978).

Simulation Cons (Univ. Pittsburgh, Pittsburgh, PA, Apr. 22, allocation models,” in Roc. 9th AnnuaZPittsburghModelingand

[ 36 1 H. L. Willis, Y. Chambers, and J. Gregg. “The design of a spatial 1978, SA).

energy model for electric utility system design,” in Proc. 9th Annual Modeling and Simulation Conf. (Pittsburgh, PA, Apr.

[ 371 H. L. Willis and V. F. Wilreker, “An improved method for spa- 1978, ISA).

tialiy related load forecasts for power system planning,” presented at the 1st Annual Lawrence Symposium (Berkeley, CA, Sept.

[ 381 A. B. Cumminkp and J.E.D. Northcote-Green, “Method for fore- 1978).

casting loads in small areas,” presented at the Pacific Coast Elec- trical Assoc. Meeting (Salt River Project, Tempe. AZ, Mar. 16,

[39] J.E.D. Northcote-Green, “Spatial effects on feeder optimization 1979).

in distribution planning,” in Proc 10th Modeling andSimulation

[40] C. R. Sadler e t al., “Multi-scenario spatial forecast modeling,” Conf. (Pittsburgh, PA, Apr. 1979, ISA).

in Proc. 10th Annual Pittsburgh.Con$ on Modeling and Simuh-

[41] H. L. Willis, “Spatial load fdtering for power distribution plan- tion (Pittsburgh, PA, Apr. 1979, ISA).

n+g.” in Proc. 10th Annual Pittsburgh Con$ on Modeling and

(421 J. R. Meinke, “Sensitivity analysis of small area load forecasting Stmulation (Pittsburgh, PA, Apr. 1979, ISA).

models,” in Proc. 10th Annual Pinsburgh Modeling and Simula-

[43] J. Gregg et al., “Improvements in spatial load forecasting,’’ in tion Conf. (Pittsburgh, PA, Apr. 1979, ISA).

Proc. 10th Annual Pittsburgh Conf. on Modeling and Simuhtion (Pittsburgh, PA, Apr. 1979, EA).

I 4 4 1 C. L. Brooks, “The sensitivity of small area load forecasting to

and Simuhtion Conf. (Pittsburgh, PA, Apr. 1979, ISA). customer class variant input,’’ in Proc. 10th Annual Modeling

[45] “Summary of six years experience with ELUFANT forecast and data base program,” Houston Lighting & Power Co., Houston, TX, Tech. Rep. EP-79-3, 1979.

[ 46 1 H. L. Willis and J. Gregg, “Computerized spatial load fomxsting.” TM”n and Distribution, p. 48, May 1979.

[47] H. L. Willis and J. V. Aanstoos, “Some unique signal proceging applications in power system planning.” ZEEE Tmm Acoust.,

[48] M. Munasinghe, The Economics of Power System ReliobUityand Speech, Signal Process., vd. ASSP-27, p. 685, Dec. 1979.

Planning. Washington, D.C.: The World Bank, 1979.

[ 49 1 “Research into load forecasting and distribution planning,” Elec- tric Power Research Institute, F’alo Alto, CA, EPRI Rep. EL-1 198,

[ S O ] R. L. F i h e r , “LANDUSE User’s Guide,” Doctor of Electrical 1979.

[ 5 1 1 H. L. Willis, “Microcomputer based simulation of small area Engineering Internship Rep. Texas A&M Univ., May 1980.

urban electric demand growth,” in Roc 12th Annual Pittsburgh Con$ on Modeling and Simulation (Pittsburgh, PA, Apr. 1981, ISA).

521 J. C. Thompson and I. Gonen, “A developmental system approach to modeling electric energy demand growth in radial network,” in Proc. 12th Modeling and Simulation Conf. (Pittsburgh, PA,

531 R. F.Wolff,“ComputeninT&D,”Ekc World,p.67,Apr. 1981. Apr. 1981, SA).

541 H. L. Willis e t al., “Computerized small area load forecasting,” in Proc 7th Power Systems Computcrtion Conf. (Lausanne,

551 H. L. Willis, “LANDSAT applications to spatial electric demand Switzerland, July 1981).

forecasting,” in Proc. Western Remote Sensing Con$ (Monterey,

[ 561 M. A. Abu-El Magd and N. K. S i b a , “Two new algorithms for CA, May 1981, NASA).

.~

node power system,” ZEEE Trans Power App. Syst., p. 3246, on-line modeling and forecasting of the load demand of a multi-

July 1981. [ 57) H. L. Willis and T. D. Vismor, “Discrete transition matrix analysis

of spatial electric demand time series,” presented at the 5th

[58] H. L. Willis and J.E.D. Northcote-Green, “A hierarchical re- Annual Joint Time Series Conf. (Houston, TX, Aug. 1981).

cursive method for substantially improving trending of small area load forecasts,’’ ZEEE Tram Power App. Syst., p. 1776,

[ 591 T. D. Vismor et al., “Landsat data applications to small area June 1982.

Workshop on Remote Sensing Applications to Electric Utitities electric demand forecasting,” presented at the 13th EEI/APA

[60] C. L. Brooks, “Overview of load forecasting,” in Proc. 13th (Palo Alto, CA, Sept. 24,1981).

[ 61 1 A. E. Schauer et al., “A new load forecasting method for distribu- Modeling and Simulation Conf. (Pittsburgh, PA, Apr. 22,1982).

tion planning.” presented at the 13th Annual Modeling and Simulation Conf. (Pittsburgh, PA, Apr. 22, 1982).

[62] J. C. Thompson and T. Gonen, “Simulation of load growth developmental system models for comparison with field data on radial networks,” in Proc. 13th Modeling and Simulation Conf.

631 H. L. Willis, “Load forecasting for distribution planning-Error (Pittsburgh, PA, Apr. 22, 1982, ISA).

and impact on design,” to be presented at the 1982 IEEE/PES Summer Meeting (San Francisco, CA, July 18-23, 1982, paper

641 H. L. Willis et al., “Forecasting distribution system loads using curve shape clustering,” to be presented at the 1982 lEEE/PES Summer Meeting (San Francisco, CA, July 18-23,1982), paper

BZSM470-3.

B2SM385-3. [ 651 “Final report on CEA Project 079DlS6-Urban distribution load

forecasting-Canadian Electric Association,” 1982. [66] R. W. Powell e t al., “Advances in distribution planning tech-

niques,” presented at the 1983 CEPSI Conf. (Bangkok, Thailand, Nov. 30, 1982).

[67] E. P. Cody, “Load forecasting method cuts time, cost,” EIec.

(681 F. Y. Murad and J. A. Perry, “Computerized distribution load World, p. 87, Nov., 1982.

projection in planning,” presented a t the Southeastern Electric Exchange Meeting (Hot Spring, AR, Apr. 13, 1970).

[ 691 H. L. Willis and H. N. Tram “A cluster based V.A.I. method for distribution load forecasting,” to be presented at the 1983 IEEE PES Winter Power Meeting (New York, NY, Jan. 31,1983).

701 H. N. Tram et al., “Load forecasting data and data base development for distribution planning,” submitted to the 1983 Power Industry Computer Applications Conf. (Houston, TX, May 19, 1983).

71 ] H. L. Willis and T. W. Parks, “Fast algorithms for small area load forecasting,” submitted to the 1983 Power Industry Computer

721 “A Guide to Distribution Load Forecasting,’’ Canadian Electric Applications Conf. (Houston, TX, May 19,1983).

731 S. E. Collier, “A comparison of load research project results with Association, Sept. 1977.

the REA demand tables,” in Proc. 1982 Rural Electric Power Conf. (Oklahoma City, OK, Am. 25,1981). IEEE CH 1733-5-D4.

[ 74) J. &r& e t al., “A Markov ~ & ~ e s s appged to forecasting,” presented at the IEEE PES Summer Meeting (Vancouver, B.C., C m d a . July, 1973), C73 475-1.

Non-Spatial Load Forecasting [ 751 “Load forecasting bibiliography: Part 1,”ZEEE Tram PowerApp.

(761 “Load forecasting bibiliogrpphy: part 11,” ZEEE Trans Power Syst., p. 53, Jan. 1980.

App. Syst., p. 3217, July 1981.



[ 771 J. H. Broehl, “An end-use approach to demand forecasting,” IEEE Trans Power App. Syst., p. 2714. June 1981.

[ 781 “Bibliography on Load Management-IEEE Committee Report,” IEEE Trans. PowerApp. Syst., p. 2597, May 1981.

[ 791 R. B. Adler et al., “A distribution system/endvser cost model,” IEEE Trans. PowerApp. Syst., p. 3590, July 1981.

[ 801 R. F. Hamilton, “The summation of load curves,” AIEE Trans., vol. 61, p. 369, 1942.

[ 81 ] “Modeling and analysis of electricity demand by time of day,” EPRI Roj. EA-1304, Palo Alto, CA, Dec. 1979.

[ 821 “Energy Demand and Consumption by Time of Use, Electric

CA, Dec. 1979. Power Research Institute,” Rep. Project EA-1294, Palo Alto,

Distribution Planning [ 831 C. J. Baldwin et al., “Techniques for the simulation of subtrans-

Industry Computer Applications Conf. (May 1969), p. 575. mission and distribution system expansion,”in h o c . IEEEPower

[ 841 E. Masud, “An interactive procedure for sizing and timing distribution substations using optimization techniques,” presented at the IEEE PES Winter Meeting (New York, NY, 1974), paper T74142.142-6.

[85] D. M. Crawford and S. B. Holt, “A mathematical optimization technique for locating and sizing distribution substations,” IEEE

[ 861 L. S. Neal, “An interactive computer approach to distribution Trans. PowerApp. Syst., vol. 94, no. 2, p. 230, Mar. 1975.

circuit modeling,” presented at the Power Distribution Conf. (Austin, TX, Oct. 1977).

[ 87 1 Y. Backlund and J. A. Babenko, “Distibution system design using computer graphics technique,” presented at the 1979 IEEE

[ 88) D. L. Wall et aZ., “An optimization model for planning radial Power Industry Computer Applications Conf.

distribution networks,” presented at the IEEE PES Summer Meeting (July 1978), paper F78653-8.

[ 891 T. Gonen, “A research progress report for electrical energy distribution systems,” Elec. Eng. Dep. Rep., Univ. of Oklahoma,

[ 901 “Survey of industry distribution planning needs,” in EPRI Roj. Norman, OK, Sept. 1978.

[91] “Survey of Canadian distribution load forecasting,” CEA Roj. RP-570, EPRI Rep. EL 1198, Palo Alto, 1979.

[ 921 ‘‘Survey of industry distribution planning needs and variations,” 079D136, 1981 (project report forthcoming).

Westinghouse AST Rep. AST-1667, 1982 (proprietary).

[93] “Industry distribution planning survey,” Houston Lighting and Power Co., Houston. TX, tech. rep. EP-80-8, 1980.

[ 941 R. F. Wolff, “The new electronic frontier-Distribution design,” Elcc. World, p. 65, May 1982.

(951 “System design using dispersed generation and load management,” Final Rep. Roj. DEDS1941, U.S. Dep. Energy (in preparation).

Other References

I. Walter, Location and Space Economy. Cambridge, MA: The MIT Res, 1956. Calibrating and Testing a Gravity Model f o r A n y Size Urban Area, D.O.C., Bureau of Roads, U.S. Gov. Printing Office, 1963. I. S . Lowry, A Model of Mezropolir; Rand Corp., Santa Monica, CA, Memo. RM-403541964. ‘‘Final report: EMPIRIC land-use forecasting model,” Final Rep. for Eastern Massachusetts Regional Planning Project, 1970.

Proc. 9th Modeling and Simuiation Conf. (Pittsburgh, PA, Apr. C. R. Sadler, “Practical aspects of urban model data error,’’ in

1977, HA).

portation and location,” U.S. Dept. of Transportation, DOT S . H. Putman, “Integrated policy analysis of metropolitan trans-

S . H. Putman, Urban Residential Location Models Boston: Mirtinus Nijhoff, 1979. R. J. Bennett, Spatial Time SeriesAnalysi& London: Pion Ltd.,

J . H. McClellan, “Design of 2-dimensional digital fdters by trans- 1979.

Science (Princeton, NJ, 1973), p. 247. forms,” in h o c . 7th Annual Princeton Conf. on Information

house Advanced Systems Technology, Pittsburgh, PA, Rep. “Computer aided distribution planning and design,” Westing

H. L. Wiltis, “Rapid calculation of proximities with FFT’s,” submitted to IEEE Trans. S y s t , Man, Cybernetics. C. Silva etal., “Advanced distribution planning for Panama City,” presented at the 1981 IEEE Central America Power Conf. (Pan- ama City, Panama, July 9, 1981).

ning-A unified approach,” in hoc. 7th IEEEIPES T t D Conf. J.E.D. Northcote-Green et al., “Long range distribution plan-

(Atlanta, GA, Apr. 1979), IEEE Publ. 79CH1399-5-PWR.

P-30-80-32, Aug. 1980.

DB52-235, J a n . 1982.


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