A fast interactive optimal load management algorithm under time-varying tariffs C S Chang Department of Electrical Engineering, Hong Kong Polytechnic, Hong Kong

A new approach to electricity tariff structuring, in which the tariff is varied continuously according to the actual production cost of electricity, has attracted many research workers. This paper begins by defining an analytical .framework, using which each consumer is able to maximize his returns. Under this .framework, a conceptually simple and yet computationally eJJ~cient algorithm has been developed to help each consumer in achieving his own goal. Based on the method of I.agrange multipliers, each method .liar impro~ing the consumption pattern is ranked according to its incremental cost of implementation. In the first m4merical example the algorithm provides a medium-size industrial consumer with the choice between se!f-generation a~M load sh~liin,q. In the second numerical example well- managed loads are shown to make good returns /br the supplier and the resulting load ,rite/or is improved, leading to savings in ./itel and capital costs.

Keywords. online control strategies, load .~low, optimal power Jlows

I. In t roduct ion This paper is concerned with a new electricity tariff proposal (spot pricing) which has come to prominence in the past few years. The proposal was first made by the Energy Laboratory at MIT ~'2 and has attracted many other workers 3 ~'. Experimental field trials 7 ~ " are also in progress in the USA and UK.

The essence of spot pricing is fhat the selling price of electricity to consumers will be updated continually throughout the day. The price will be the actual marginal cost of production, appropriately corrected for transmission and distribution losses. Thus when the production cost is high and less efficient plant has to be run, the selling price will be high. At times of low load, when high-efficiency base-load generators carry the bulk of the load, the selling price will be low.

The effect of spot pricing will be to enhance consumer interactions with the supplier. Sufficient price incentives will be given to move load away from high-load to less expensive low-load periods. Hence the peak demand for

Received: 10 September 1986; revised 26 January 1988

the same total energy consumption will be reduced and the load factor improved. Investment in installed capacity will be curtailed. The average fuel cost will be reduced. Both these savings will be reflected in a further reduction in consumer electricity bills.

The full benefits of the spot-pricing system cannot be realized unless consumers are able to respond to a varying tariff and to take full advantage of price incentives. 1-his paper begins by defining an analytical framework within which each consumer optimizes his consumption. Under this framework, it is assumed that electricity will be sold at its marginal cost of production, while each consumer will set his consumption to maximize his returns. The optimization is realized by setting marginal cost social welfare. An algorithm is developed that enables a consumer to shift his consumption from expensive periods to low-load periods and to run up self-generation should it become price competitive. This simple yet realistic algorithm makes use of the method of Lagrangian multipliers, and the incremental costs of all load management measures are equated with each other to maximize cost savings. It is ideally suited for near-real- time implementation on microprocessors at each consumer and for calculation of load control as well as interactions with the supplier.

II. Role of the opt imal load management a lgor i thm in a spot-pr ic ing env i ronment An investigation on tile potential for the application of spot pricing in Hong Kong was initiated in late 1984 and the work has been undertaken on consumer response 4. communication facilities ~' and the effects of spot pricing on the supply system 5. All these efforts will necessarily put together a consumer demand/cost model as shown in Figure 1. Given the new consumer load L(t) measured at time t, the load forecast module updates the predicted consumer loads over a period of time. The supplier module projects the price incentives which would bc prescribed by the supplier on the updated load h~recast. The optimal load management algorithm in tile feedback loop evaluates the optimal schedule for the consumer ill order to maximize his net welfare over the period of time.

Because of the simplicity of its formulation, computer overheads required to implement the basic algorithm arL"

260 0142 0615/88/040260-11/S03.00© Butterworth ~ Co (Publishers) Ltd Electrical Power ~ Energy Systems

Original Modified consumerloadL(t) ~ ~ IoadL'(t) Load =, forecast

module

Predicted l Projected loads__. ~ I SUopp~ il g r I _ _ _ _ i .SpO_t prices

/

"1

~ Communication links

Optimal load management measures Optimal load ~_

management module

Figure 1. Functional block diagram of a consumer demand/cost model in a spot-pr ic ing environment

very low. This is an important advantage, since in practical realization the algorithm has to be run as often as possible on an updated load forecast, etc. Other features should also be taken into account and the resulting computer overheads for an inefficient algorithm would increase exponentially. For example, industrial consumers would like to consider effects due to labour and capacity limits, inventory, process flexibility, etc. In addition, most consumers would consider switching to other forms of energy should the electricity price become too high, or would simply curtail its use if it became prohibitively expensive. The method of Lagrangian multipliers, as adopted here, is formulated to accommodate various inequality constraints. Varying electricity prices may change the market shares of competing fuels. Self-generation may become operative. Self-generation is a standard feature in the formation of the demand/price elasticity matrix (Section IV). Energy storage and forward contract of electricity ~3 may be the other two options open to consumers in maximizing their cost savings. In the first option energy is purchased and stored at low prices and then utilized when the price is high. In the second option a forward contract is agreed between the consumer and the supplier to buy or sell a specific quantity of electricity at a stated future time and price. [n this manner financial and operational risks due to unexpected tariff changes as a r~ult of the weather, breakdowns in the supply system, etc. can be reduced. Both these energy options may be modelled by discrete- time equations and are ideally suited for incorporation into the algorithm.

Although originally designed for consumers of all sectors, the algorithm should find more applications in the industrial/commercial sectors. Domestic consumers cannot respond optimally unless technology makes available sufficiently simple to use and understand devices by which consumers can regulate electricity use. The project at the Hong Kong Polytechnic completed a three-way survey of domestic consumers 4 to ascertain the response to flexible tariffs, and the results obtained have been moderately encouraging.

III. O p t i m a l load m a n a g e m e n t a lgor i thm based on a quadra t ic cost func t ion The following formulation provides a method of ranking each energy option for the consumer against the others by comparing their incremental costs of implementation. For the sake of illustrating the salient features of this formulation, only load shifting and private generation are presented in this paper. Formulations for energy storage and forward contract have, however, been completed. At the outset, the algorithm is provided with a set of updated load forecasts and a set of updated electricity prices on an hourly basis (or in any other interval as appropriate) for the next 24 hours.

During the hours when electricity prices are high, the consumer is faced with the options of either shifting part of his load to cheaper time intervals or reducing the intake from the supplier by making it up with self-generation. Thus consider the potential cost saving ACp for generating the consumer's own power Ag and for shifting a load Am. Further, let ACp, be derived from the marginal cost of production of the public supply Di and the price incentive for load regulation lz (Section IV). Both Di and li are updated in every hour i.

ACpi = D i Agi + li A m i (l)

where li is the price incentive in hour i for load regulation (Section IV).

Assume that both the private generation cost ACg and load management cost ACm are approximated by quadratic functions of A 9 and Am respectively. Then in every hour i

A C g i = Ag A g 2 -I- Ug Agi 4- Cg (2)

ACmi = A m Am 2 + B m Ami + C m (3)

Thus in every hour i the total cost saving ACi is expressed by

A C i = ACpi + ACgi + ACre i (4)

Vol 10 No 4 October 1988 261

The total daily saving AC is calculated by summing over all the 24 hours:

2 4

AC = E ACi i - 1

2 4 2 4 2 4

= E l iAm,+ ~, D i A g i - ~, A~Ag 2 i - 1 i = 1 i 1

2 4 2 4 2 4

-- E AgAg i - E Cg- E AmAm2i i 1 i = 1 i 1

2 4 24

- y, & Z Um (5) i 1 i - 1

The sign convention is that Ami is positive when it adds to the load in the hour i, and Ag~ is either positive (generating) or zero (on standby).

The net contribution of load shift and private generation to the consumer load is

2 4

(Ami + Agi)= O (6) i - 1

To generalize the above expression, let

2 4

~" (Ami+Agi)=PR (7) i 1

where PR is the residual load gain (positive if PR increases the consumer load).

The private generation is constrained by

,qei~Agi~Oui for i= 1, 2 . . . . . 24 (8)

where g¢~ is the lower private generation limit in hour i and g~ is the upper private generation limit in hour i.

The load shift is constrained by

l 0 <~ Am~ s <~ u~ s (9)

where I u is the lower shift limit between hour i and hour j, U 0. is the upper shift limit between hour i and hour j, and Am~i is the load shift between hour i and hour j.

Other shift constraints are

Li <.Ami~ u i (10)

where Li is the lower physical shift limit in hour i and Ui is the upper physical shift limit in hour i. L i and U~ are related to the capacity of the eqtripment used for shifting the load.

Am~ is also restrained economically by

L,i~ AmiE U~i (11)

where L~ is the lower economical shift limit in hour i and U~ is the upper economical shift limit in hour i. L~ ~and U~ ensure that loads are shifted in the correct direction and by a reasonable amount. They are updated hourly in accordance with the consumer's demand/price charac- teristic and the supplier's marginal cost of production as well as cross-elasticity across time to other time periods. Section 1V presents the key equations used to evaluate L~ and U~i.

In addition, lower and upper load limits are used to restrict each load after shifts.

Augment C in (5) with a Lagrange multiplier term ( "

24-

AC'=AC+)~ y" (Am~+Ag3 (12) i = 1

Then, to minimize A C ' subjected to all the above constraints,

(?AC'/(3Ami=O i = 1 , 2 . . . . . 24 (13)

~?AC'/'#Agi=O i = 1 , 2 . . . . . 24 114)

~0 AC 7'82 = 0 ( 15 )

Solving for 2, Am~ and kg~ gives

I N N "~ ) . . . . . +___ " ~2Am 2Ag)

x E i-2Am 2 i + ~ m m + ~ - F P R . ) (16~

Arni = (). -Bm + li)/ 2Am 117)

Ag i = (£ -- Bg + Di),/2Ag (1~)

Ami and Agi are the optimal private generation and load shift in every hour to maximize the total cost saving in (5). In cases where zXgi in (8), Am u in (9) or kmi in (10), (i 1) is outside its limits, it must be set at the corresponding limit. The additional Arn~ or A9~ must then be absorbed by other hours. Equation (7) is used to accumulate all the additional Agi and Ami. Equations (16F(18) are used to distribute the accumulated Ag and Am to other hours.

IV. Der ivat ion of economical load shift l imits In Figure 2a the marginal cost £ against the generating plant capacity characteristic G is derived from the incremental fuel cost of the lowest order of merit unit currently loaded in the system, appropriately corrected for transmission and distribution losses. Generation start-up and shut-down costs are neglected and not contrained from one period to the next. The marginal generation cost becomes infinite at the point of maximum generation Gma x and the area under the curve at that point is the maximum total variable generation cost.

Figure 2b shows the total consumer welfare modelled by the economist's demand/price curve, which assumes consumers to be welfare-maximizing with constant marginal utility of income. The total consumer welfare is the area under the demand/price curve for any price demand.

Since consumers react to price information according to how they measure benefits obtained from electricity. i.e. to benefit their welfare, they will set their demand at a level (the optimal demand Yi) where the price of electricity equals the marginal value of total production per k w h of elecgricity for industrial consumers, and the total welfare for commercial and domestic consumers, i.e. where -;-,,,i- h,,pi (Figure 2c). The total net consumer welfare is thus the shaded area in Figure 2c between the consumer demand curve and the marginal cost curve. The objective for bol h

262 Electrical Power 8 Energy Systems

e<

"En Area = Dly)

Generation G

'U- Q -

=

Consumption Y

a b c

,4

Xi = h~

i i

yi

Figure 2. (a) Product: D ( y ) = c o s t of product ion. (b) Consumer: C ( y ) = b e n e f i t from use of electricity. (c) Equil ibrium: ()., or h i, y i )=opt ima l operating point; shaded area=maximum net benefi t

the supplier and the consumer is to maximize this net welfare

With an objective of maximizing his net welfare in every hour i, the consumer should adjust his present demand Pi toward the optimal demand ii,.. The discrepancy between Pi and Yi thus represents the target value of each load shift mi that the consumer is aiming for:

O<-.Ami <~- Y i - P i Pi < Yi (19)

Yi-Pi<~Ami<~O Pi> Yi (20)

In actual implementation it would be impossible for the consumer to achieve such a target since each load shift will be constrained by a set of demand/price elasticity coeffÉcients as follows.

Let hi be the present price of electricity on the consumer's price/demand curve and let hop i be the optimal price of electricity on the same curve, where }oopi=hopl. The discrepancy between hi and hopi is thus the price incentive for the consumer to regulate his demand. The consumer should increase his load Pi if hi>hop i or he should decrease Pi if hi< hopi.

Next, let the consumer's demand/price elasticity be available and be represented in the form

E =

I C1 --C1. 2

--C2,1 C 2

--C24,1 -- C24,2

•. --C1,2 4

•. __ C2v.24

•. C24

(21)

where E is the consumer's demand/price elasticity matrix, eij (= %i, i #:j) is the cross-elasticity coefficient across time from hour i to hour.j,

24. Ci=Cii~- Z Cij

i=l iCj

and eii is the self-elasticity coefficient for hour i only - the amount by which the consumer would vary his load with a change of electricity price.

Let H be a vector containing a set of price incentives for load regulation:

H = [ h l -hopl, h 2 -hop2 . . . . . h24- hop24]* (22)

Multiply E by H:

B = E H (23)

where

a = [ b l , b 2 . . . . . b24]*

is a vector of economic load shift limits• The elements of B are related to the economic load shift limits in (11) by

i f b i > 0 i = 1 , 2 , 3 . . . . . 24

U~i=bi Lei--- 0 (24)

i f b i < 0 i = 1 , 2 , 3 . . . . . 24

Uei=0 gei=b i (25)

Uei and Le,-are used in the optimal load shift algorithm to limit each Am/j and thus Ami. Figure 3 shows a simple graphical representation of the relationships among U~i, Lci, H and E.

V. S i m u l a t i o n o f t h e s u p p l i e r m o d u l e In the proposed consumer demand/cost model shown in Figure 1 the consumer receives a set of advance information from the supplier likely future spot prices and/or future generating plant conditions. The more advanced this information made available to the consumer, the more efficiently will the consumer demand/ cost model work to the benefit of all concerned.

At the present stage of work being undertaken in the Hong Kong Polytechnic, the supplier module is simulated with a well-established unit commitment algorithm which makes use of the techniques of dynamic programming and dynamic priority ranking 14. The optimal load management algorithm and the unit

Vol 10 No 4 October 1988 263

~N~, h°ur2 I i To houri

~ ~ ei'j / ~ OhOur23

ei'~ hi-h°Pi

believed to be a reasonable figure for industrial consumers in Hong Kong.

The upper and lower physical sh(ti limits (referring to equation (10)) are

Li= 4 MW { J i = 7 MW

The load management cost co£[ficient is assumed to be quadratic and has the value Am = 15, where the cost is expressed in S/hour and the load in MW.

Private generation has a rating in MW and a heat rate equation of

A~ = 4.968 Bg = 130.7

where the cost is expressed in S/hour and the load in MW. The large-size consumer data project the electric vehicle

(EV) loads in Hong Kong to the year 2000. Figure I1 shows the projected electricity consumption for recharging all those EVs when no price incentives are

Figure 3. Electrical analogue of consumer's demand/price elasticity

commitment algorithm are executed one at a time in successive approximations ~s. Figure 4 shows the flow chart of the scheme of successive approximations. Demand changes calculated from the optimal load management algorithm and price changes calculated from the unit commitment algorithm are used continually to update information in the iterative loops.

Vl. Case studies - data Based on the formulation and algorithm discussed in the previous sections, two cases of optimal load management to maximize the social welfare were studied.

VI.1 Supplier data The supply system data project the Hong Kong supply system to the year 2000 ~6. The data include generator ratings, the system demand curve and the optimal plant loading and show the start-up times and close-down times of various units (Figure 5).

VI.2 Consumer data The medium-size industrial conxumer data include the consumer consumption curve (Figure 6), the consumer price/demand curve, the consumption/price elasticity coefficient, thc upper and lower shift limits, the load management cost coefficient, the rating of private generation as well as the heat rate.

The consumer price/demand curve is assumed to be quadratic and the second-order term is neglected for convenience:

price in $/MWh = 406.2 - 0.11367 x demand in MW

(26)

The consumption/price elasticity coe/ficient has the value of 0.1 p.u. MW consumption/p u. electricity price and is

Start )

Optimal t plant loading

l Store marginal production costs changes

Optimal load management

Store demand changes

Print output results

Figure 4. Schematic diagram of hourly schedule of generation and optimal load management

264 Electrical Power ~t Energy Systems

8250

g Q.

g

I l l C g -o" C c~

C3

7500

6750

6000

5250

4!500

3750

3000

2250

7 .... i .... I

Plant load ing

I I

"•---. Base system demand

a=

150C I L I I t 1 ~ I t I L I 1 I 1

Hours

Figure 5. EVIoaddis t r ibu t ion and corresponding plant loading before shift

1 1 1 I ,1 1

I ! I I I

i I

given (fiat tariffs). The EV consumption has a peak demand of 67 MW.

El~" conslonptio¢,price dasticity coefficient. EV loads are believe to be manageable ~ s and can be shifted to fill in late-night valleys on the system demand curve if sufficient price incentives are given.

V I I . P e r f o r m a n c e o f t h e c o n s u m e r

o p t i m a l l o a d s h i f t a l g o r i t h m Using the categorization proposed in Reference 4, industrial load may be classified in the following categories:

{a) simple loads of variable marginal utility, which include a large variety of loads,

(b) sequential loads,

(el on off loads.

I-he method of equal incremental cost is formulated here primarily for the load category (a). If other load

categories were considered, it would be difficult to maintain all load management measures on a common incremental cost since some limits could be exceeded. It would be easier, however, to operate each type of load management measure on an individual equal incremental cost. Orders of merits would then be set up to rank various types of measures, each aiming for a certain load category. Although such a scheme would seem "sub- optimal' , it has the advantage of being computationally efficient and may be implemented on a microprocessor in a near-real-time mode. An algorithm was developed for the load category (a) to rank self-generation against load shifting when the limits are exceeded. The key equations of this algorithm are presented in the Appendix.

Figure 6 shows the consumption curve of the medium- size industrial consumer. Other information on this consumer has been presented in Section VI. It is further assumed that only the shaded portion of the consumption in Figure 6 belongs to the category (a). Both load shifting and self-generation were considered and are presented in the following case studies. Optimal load management results showing characteristics of both measures are presented in Figures 7 10.

Vol 10 No 4 October 1988 265

O J

20

15

10

iiiiii!iiii!i!!!iiiiiii!ii!i]i]i!!i!iiii!i!ili!!!i

12 2

i I Iz~,~,,~,~i,i~i~i~i,ii,i!iiiiii~i!i~ii:i!iiiiiiii~i~iii .................. iii!!~:ili!ii!i Ii i iii iiiiiiiiiiiiiiliiiiiiiiiiiiiiiiiiil

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300 #,

~, 200

.~ 1 O0

I

I I I

Price of electricity

Noon I t I I I ] I I I I [ I I I I I I 1

12 2 4 6 8 10 12 2 4 8 10

Figure 6. Electricity consumpt ion and price of electricity

1

12

o J

v

o

c

co

20

15,

I0i

~ ~:::::::::::::::::::::::::::::::::::::::: Shaded area = load category (a)

, : : : : : : : : : : : : : : : : : : : : : : : :

I 1 5 Modified electricity consumption

] N o o n 0 , , , , , , , , , , , , I , , , , l

12 2 4 6 8 10 12 2 4 6 8

5

2.5

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"4

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J Noon l i I I I I L I L L I I I L I

2 4 6 8 10 12 2 4 6 l _ _ i i

t I

Optimal self-generation

i I I I I I I i I I I i I I I ] I i I i

10 t2 l

10 12

Figure 7. Optimal load management case study (i)

266 Electrical Power ~t Energy Systems

0 . J

2 0 -

101

0~::::::::::::::::::::::::::::::::::::::::::::::: 12 2

2.5 o 12

2.5

0 12

Shaded area = load category (a)

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i3 iiiiii~i~i~i~;Eiiiiiiiiiiiiii I ~ 1

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Noon I I I I I I I I I 1 I I I I I I 4 6 8 10 12 2 4 6

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I 10 ~

i I I I I I I I I I I I I l l I I l

Optimal self-generation (fuel cost reduced by 66,7%)

Noon I I I I 1 I I I I I I I I I I I I

12 2 4 6 8 10 12 2 4 6

O p t i m a l l oad m a n a g e m e n t - case s t u d y ( i i i )

12

t

10 12

Vol 1 0 No 4 October 1 988 267

VII.1 Case s tudy (i) - l oad shi f t ing only At the outset it is assumed that the cost of self-generation is prohibitively high. Thus by giving Ag in (16) a high value which forces out all Ag~, load management will contain load shifting only. The resulting optimal load shifts are shown in Figure 7.

VII.2 Case (ii) - l o a d sh i f t s o u t s i d e l imits In contrast to the above unconstrained case, reductions of the upper shift limits in (9) and (10) would bring in self- generation, which operates at an incremental cost above that of load shifting (Figure 8).

VII.3 Case (iii) - s e r f - g en er a t i o n on l y Figure 9 shows a case where self-generation is very cheap and has a capacity large enough to supply all the shaded load in Figure 6.

VII.4 Case (iv) - s e r f -genera t i on o u t s i d e l imits Figure 10 shows both self-generation and load shifting operating together, and the former is at an incremental cost lower than that of the latter.

The final case study was performed to evaluate the potential impacts of EV loads in Hong Kong ~ ~. Well- managed EV loads are seen to have favourable feedback effects on the supplier (Figures 11 and ! 2) by improving the load factor. In a typical case here, the supplier would save about l ~',~, on the fuel cost. As for the EV owner, he could earn as much as $600 per annum should he take full advantage of price incentives offered to him.

VII. C o n c l u s i o n s An algorithm for optimizing consumption patterns as a result of load management in a spot-pricing environment

20

15

-o lO 8

. d ~ilili~i !iiii!ii:!~!i~ili~i i:!~i:i!i!i!!!i~

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5

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2.5 c

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10o

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> W

o 12

F igure 1 1.

Shaded area = load category (a)

I I ]

I

Modified electricity consumption (from public supply)

Noon I L i I I I i I I I L I I L I I i I

4 6 8 10 12 2 4 6 8

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I Noon L I i t I I i I I I L 1 I I I I 1 I i

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I I 10 12

10 ]2

I 8 10

Optimal self generation (capacity reduced from 5 MW to 2 MW)

I Noon I I I I I I I I I I 1 I I I 1 I I I

2 4 6 8 10 12 2 4 6

Optimal load managemen t - case study (iv)

T2

J

Power requirement under time-varying tariffs (see figure 6)

2 4 6 8 10 12 2 4 6 8 10 12

P r o j e c t e d EV p o w e r r e q u i r e m e n t s u n d e r f la t a n d t i m e - v a r y i n g tar i f fs

268 Electrical Power ~t Energy Svstem~

>.

tm

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7500

6750

6000

5250

4500

3750

3000

2250

12

Figure 1 2.

~' Peak / looping

F - - ~ / I not n e e d e d I 1

Plant loading

l__i STBF

EV load after shift

\ ~ / / / /

" l / l i l l l j

Base system demand

- - - - , plant loading before shift STBF, starting time brought forward CTBF, closing time brought forward CTBB, closing time brought backward

Noon I l I I I I I I L I I I L L

2 4 6 8 10 12 2

EV load distribution and corresponding plant loading after shift

. . . . . I I I

CTBF I I I

L

CTBB I

C T B B - - - ~

L_

Z22~

½

1 I I 1 I I I I 4 6 8 10 12

has been presented. Expressions for calculating incremental costs of implementing various methods of improving consumption patterns have been developed. These expressions are used to rank various methods of improving consumption patterns. The methodology should find more applications in other load categories. Limits on Ami and Agi due to capac~y and labour limits, inventory, process flexibility, etc. may be included as inequality limits in the basic formulation. The execution of the algorithm is very fast since Am and A# are calculated by employing sensitivity analysis as suggested in Section III. The disadvantages of this formulation centre on its gross simplicity. Cost functions are assumed to be quadratic, stationary and deterministic. Load shift costs relating to interchange Amij between hours i and j are not individually represented and are lumped together for each hour i or j. For purposes of interactive load control in a near-real-time mode, this simple formulation is probably adequate. Load shift costs relating to interchange Amq may be expressed in terms of the lumped load shift Am~. The resulting 'loss' coefficients may be

updated by performing a detailed simulation if and when necessary.

IX. References 1 C a r a m a n i s , M C , B o h n , R E a n d S c h w e p p e , F C 'Optimal

spot pricing: practice and theory' IEEE PES Winter Meeting, New York (31 January-5 February 1982)

2 Schweppe , F, Tabors, R, K i r t ley , J, Onthred, H, Pickel, F a n d Cox, A 'Homeostatic utility control' IEEE Trans. Power Appar. 6" Syst. Vol PAS-99 No 3 (May/June 1 980) pp 11 51- 1163

3 B e r r i e , T W 'Interactive load control (Parts 1-6) ' Elec. Rev. (1982) part 1, Sept 81, pp 2-4; part 2, Oct 1 981, pp 5-7; part 3, Nov 1981, pp 8-9; part 4, Dec 1981, pp 10-14; part 5, Jan 1982, pp 13-14; part 6, Feb 1982, pp 15-16

4 David, A K, Nut t , D J, Chang, C S a n d L e e , Y 'The variation of electricity prices in response to supply-demand conditions and devices for consumer interaction' Elec. Power &" Energy Syst. Vol 8 No 2 (April 1 986)

5 David , A K 'Estimation of load pattern modification and short range load forecasting under time varying tariffs' Proc. Fifth Int. Conf. on Meeting Apparatus and Tariffs for Electricity Supply, Edinburgh (14-16 April 1 987)

Vol 10 No 4 October 1988 269

6 Chang, C $, Nutt , D J and Lee, Y C 'Design and Ioaboratory implementation of a low cost meter tariff control system' Int. J. Proc Sixth CEPAI Conference, Jakarta, 3-7 Nov, 1986, Paper 1-12

7 Geier, D L and Samaniego, G M 'Evaluation of load management as an electric system resource' IEEE Summer Power Meeting, Vancouver (July 1 985) Paper No 85 SM475- 9

8 Peddle, R A and Fielden, J S 'Credit and load management systems for an electricity supply utility' CIRED ConL lEE 197 (June 1981) pp 231-235

9 Caldwel I , JE, Crawley, DF, Hensrnan, GOandJames , A L 'Energy management field trials in U K' CIRED ConL, Brighton (May 1985) pp 261-265

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Appendix. Ranking self-generation against load shifting Recall (5), which evaluates the total potential cost saving for load management, and split it into two parts:

Saving due to load shifting

2 4 2 4

ACm= 2 liAmi- 2 AmAmZi i = 1 i= 1

24 2 4

- -~ /~m Am, Z Cm (27) i = 1 i = l

saving due to self-generation

2 4 2 4

A 2 a G : Y. Diao,- Z Ag g, i = ! i ~ 1

2 4 2,4

- E B g a g , - E G <2sl i = 1 i = 1

The total cost saving is

A C = AC m + ACg (29)

" 24- Pro) A C ' = A C + Z m ( ~, Ami- 1

/ 2 4 ' t

+),g( ~ A g i - i Pg ], (30)

where 2m is the incremental cost of load shifting, Pm is the total load shifted (i= 1, 2 . . . . . 24), 2g iS the incremental cost of self-generation and Pg is the total self-generation (i = l, 2 . . . . . 24).

By differentiating (30) w.r.t. Am i, Ag i, ).m and ).~,

Z/,N ';'m :~ ( Pro+ 2 A m - 2Am/,2Am ~31~

' - 1321

Self-generation is ranked higher than load shifting if 2g<;%. The former is ranked lower than the latter if 2m > 2 e- For the c a s e /~m =";~g, (31), (32) and (16) arc equivalent to each other if no limits on self-generation and load shifting are exceeded.

270 Electrical Power ~t Energy System~