Wheeling and marginal wheeling rates: Theory and case study results

Electric Power Systems Research, 27 (1993) 11-26

11

Wheeling and marginal wheeling rates: theory and case study results K. L. Lo and S. P. Zhu Power Systems Research Group, Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow G1 1XW (UK) (Received November 17, 1992)

Abstract The theory of a novel approach to evaluate optimal wheeling rates for the case of bus to bus wheeling is developed in this paper. This approach does not assume the existence of a spot price based energy

market-place. It is based on the marginal cost theory which has been used for electricity pricing. On the basis of the equitable sharing of the benefits, arising from wheeling transactions, among the wheeler, the power seller and the buyer, this approach has the advantage over others that it avoids the direct evaluation of the network maintenance cost and the quality of supply cost components. Moreover, in the proposed approach, the additional power losses resulting from the wheeling transaction are modelled to be contributed by the seller and buyer, rather than by the wheeling utility. The numerical results of two case studies are presented in the paper. A comparison with the wheeling rates of other published approaches is also outlined. Keywords: wheeling, wheeling rates, transmission access, m a r g i n a l cost pricing, transmission service pricing.

1. I n t r o d u c t i o n W h e e l i n g is a term describing the s i t u a t i o n in which a given utility's t r a n s m i s s i o n / d i s t r i b u t i o n n e t w o r k is used to e x c h a n g e electric energy between two other parties. The i m p o r t a n t point is t h a t the two parties are not adjacent, so t h a t an i n t e r m e d i a r y u t i l i t y m u s t provide t r a n s m i s s i o n services, w h i c h is referred to as wheeling. Wheeling is becoming of i n c r e a s i n g i m p o r t a n c e in the o p e r a t i o n a n d p l a n n i n g of a multi-utility power system. Sometimes, the wheeling s i t u a t i o n occurs w h e n the wheeling u t i l i t y is simply being used to t r a n s m i t power from an electricity seller t h r o u g h this i n t e r m e d i a t e t r a n s m i s s i o n system to a buyer [1]. Of course, it is conceivable t h a t the transmiss i o n / d i s t r i b u t i o n c o m p a n y owns some limited g e n e r a t i n g power, the sole purpose of w h i c h is to accept a u t o m a t i c governor control and emergency response signals [2]. W h e e l i n g is receiving a lot of a t t e n t i o n because of a desire to i n t r o d u c e some competition into the electricity marketplace w i t h o u t giving up its basic r e g u l a t e d structure, t h a t is, p a r t i a l d e r e g u l a t i o n [3]. 0378-7796/93/$6.00

W h e n it happens, wheeling causes a v a r i e t y of physical and economic effects on the wheeling utilities. Therefore the following problems should be considered: • how to e v a l u a t e the a c t u a l economic costs and benefits of wheeling to the wheeling utility; • how to work out a set of wheeling rates which meet a v a r i e t y of criteria for an 'ideal' rates system. Wheeling rates determine p a y m e n t s by the electricity buyer and seller to the wheeling utility to compensate the wheeling u t i l i t y for the costs it incurs. Almost everyone agrees t h a t a wheeling u t i l i t y should get revenue to cover the effects of the wheeling on its o p e r a t i n g costs. These costs are incurred when facilitating generation redispatch (wheeling can force the u t i l i t y to redispatch its g e n e r a t i o n to m a i n t a i n acceptable line flows) and in the t r a n s a c t i o n itself (metering, billing, c o m m u n i c a t i o n s and c o m p u t a t i o n s [4]). The redispatch costs can sometimes be n e g a t i v e as a wheeling u t i l i t y m a y be able to r e d i s p a t c h genera t i o n more economically [5]. In the t h e o r y devel© 1993

Elsevier Sequoia. All rights reserved

12 oped in this paper, the additional power losses resulting from the t r a n s a c t i o n are modelled to be c o n t r i b u t e d by the seller and buyer, r a t h e r t h a n the wheeling utility. There are several different types of wheeling, depending on the relationships b e t w e e n the wheeling utility and the other two parties [4, 6]. Reference 4 categorizes three types of parties who may w a n t a utility to wheel energy for them: the external utility, the retail customer, and the requirements customer. The external utility is a n o t h e r r e g u l a t e d electrical utility t h a t is i n t e r c o n n e c t e d with the wheeling utility b u t does not necessarily t o u c h it. The retail customer is an individual customer l o c a t e d within the service t e r r i t o r y of the wheeling utility. This c u s t o m e r is likely to be a large industrial user or a small p o w e r producer. The requirements customer is a municipal dist r i b u t i o n c o m p a n y or other entity located within or touching the wheeling utility's service territory which p u r c h a s e s energy wholesale for subsequent resale to customers. The following are examples of wheeling transactions: from external utility to external utility; from retail customer to retail customer; from external utility to retail customer; from retail customer to external utility; from retail c u s t o m e r to requirements customer; from external utility to requirements customer. Some recent w o r k s on the pricing of wheeling [4, 6-9] focus on the marginal cost of pricing the service based on the operating cost of the p o w e r system (cost of losses and g e n e r a t i o n rescheduling and redispatch). References 10 and 11 describe a bidding process as a means of determining the value of the transmission service for the wheeling transaction. This paper is not m e a n t to discuss the economic issues related to the pricing of wheeling. Neither does it debate the a d v a n t a g e s or disadv a n t a g e s of the various pricing policies, such as embedded cost pricing, marginal cost pricing or value based pricing. It proposes an a p p r o a c h for e v a l u a t i n g the rates for a p o w e r transmission service. Based on the equitable sharing of the benefits arising from a wheeling transaction, among the wheeling, selling and buying utilities, this a p p r o a c h has the a d v a n t a g e over others t h a t it avoids the direct e v a l u a t i o n of the n e t w o r k m a i n t e n a n c e cost and the quality of supply cost components. This paper first describes the m e t h o d o l o g y w h e r e marginal wheeling rates f o r the bus to bus wheeling case are developed, and then presents

the numerical results of two h y p o t h e t i c a l case studies t h a t emphasize the differences of this a p p r o a c h in e v a l u a t i n g wheeling rates vis-~t-vis other existing approaches. Wheeling rate sensitivities to load levels, line flow constraints, etc., are discussed. A brief look at existing approaches is t a k e n and comparisons made with our approach. An assumed m o n e t a r y unit, ~ , is used throughout this paper ( m ~ = ~ × 10 3; in the Figures, millir -- m~).

2. M a r g i n a l w h e e l i n g rates: b u s t o b u s

2.1. Power balance Figure 1 shows the bus to bus type of wheeling, t h a t is, a seller of p o w e r S is located at one bus of the utility K, while a b u y e r B is located at a different bus. The power passed t h r o u g h will change the transmission losses incurred in the wheeling utility. The p o w e r b a l a n c e in the utility K w i t h o u t the wheeling t r a n s a c t i o n is gK=dK + L The power b a l a n c e in K with the wheeling transaction is

A W + g'K=dK + L + A L where AW denotes the net i n t e r c h a n g e of inflow from the seller S and outflow to the b u y e r B, and AL denotes the additional power losses b e c a u s e of wheeling. Generally speaking, AL is a function of the changes of p o w e r injections, which include tie line interchanges and g e n e r a t o r p o w e r outputs. It can be expressed as AL = f[A(Tie line injections), A(Generator node power injections)]

(1) (2)

~- f[Win , Wout, Ag K]

w h e r e Ag K = g,K _ gK. Usually, gK=g,K, b u t the associated cost Cgen(g K) is p r o b a b l y different from Cgen(g'K).

8 B' °o.r Bj Fig. 1. Bus to bus wheeling.

13

Hence, the b a l a n c e with the wheeling transaction can be r e w r i t t e n as

AW + gK=dK + L +AL t h a t is, AW = AL E q u a t i o n (2) also t u r n s o u t to be a function of the change of the tie line injections only, that is, AL = f[A(Tie line injections)] = f[Wm, Wout] AS far as the real o p e r a t i o n of electric p o w e r utilities is concerned, the power wheeling utilities normally m a k e a request t h a t b o t h p o w e r selling and buying utilities who require the wheeling (power transmission service) cont r i b u t e to the extra p o w e r losses due to wheeling. If all parties t a k i n g p a r t in a wheeling transaction agree t h a t the extra p o w e r losses due to wheeling from the seller S to the b u y e r B are supplied by inflow from S, and if the outflow to B is denoted by

Wout= W then we have

w~.= W + A w Therefore, the p o w e r losses AW resulting from wheeling are p a r t of the power i n t e r c h a n g e agreement, in which the A W will be supplied by the seller S. It should be noted t h a t A W can be either positive (wheeling increases losses), or negative (wheeling decreases losses). 2.2. Marginal wheeling rates W h e n a wheeling utility exercises a wheeling service by allowing its facilities to be used, it will d e m a n d a charge, the wheeling charge, to compensate for the use of transmission facilities and additional o p e r a t i n g costs. Suppose t h a t seller S, c o n n e c t e d at bus S inside a wheeling utility's territory, is wheeling to b u y e r B, c o n n e c t e d at bus B, t h r o u g h the wheeling utility K. As far as the wheeling utility K is concerned, this is a n a l o g o u s to b u y i n g energy at bus S (with i n c r e m e n t a l flow into K) and selling it at bus B ( i n c r e m e n t a l flow o u t of K). The short-run costs of wheeling are the marginal (incremental) costs of the last k W h of energy wheeled: Ideal wheeling rates = M a r g i n a l costs of wheeling Let ~5(t) be the marginal operating cost of wheeling at h o u r t ( m ~ / k W h ) . F o r simplicity and

clarity, the time dependence will be ignored for the moment. The basic formula for defining (5 is [6] (5 = 0(Operating costs of wheeling utility) 0(Amount of energy being wheeled)

(3)

The derivative of (3) is e v a l u a t e d s u b j e c t to constraints, such as the energy balance, Kirchhoff's law, and line flow limits. I f C (K) is defined as the total costs of utility K wheeling W k W h of electric energy (~/h), and B (B,s) the total benefits t h a t the wheeling parties (seller S and the b u y e r B) receive from the p o w e r being wheeled (~/h), then the social cost [12] can be defined as Social cost = C (K) - B (B' s)

(4)

where C (K) : Cwheeling (WOut)

and B(B. s) = RB(Wout)

_

Cs(Win ) __ (sWout

RB(Wout) is the r e v e n u e received by b u y e r B, b a s e d on the disposal of Wou t k W h of electric energy. If the b u y e r B supplies electricity to its customers, its gross (sales) r e v e n u e is equal to the price it charges its c u s t o m e r s multiplied by the demand. For simplicity it is assumed in this paper t h a t b u y e r B is a requirement customer, which is a p o w e r d i s t r i b u t o r who relies wholly on p o w e r from the supplier to serve its loads. Cs(Win) in the cost set by the seller S generating Win k W h of electric energy. If the marginal wheeling rates (5 are imposed, Wou t is chosen to maximize the difference between the wheeling gross r e v e n u e s and the wheeling costs, t h a t is, to maximize the net benefits of the wheeling utility [12] (~Wou t -- Cwheeling(Wout)

Thus Wo.t = Wout((5) is determined by the following condition: Cwheeling(Wout) = (5( Wout)

~Wout The above derivatives are b a s e d on the assumption t h a t only Wout depends directly on (5, and Win does not, which means t h a t b u y e r B will p u r c h a s e electricity t h r o u g h the wheeling utility K from seller S only w h e n it is profitable, and seller S will sell electricity as long as there is a demand.

14 S u b s t i t u t i n g Wout = Wout(05) into the social cost function equation (4) yields Social cost = Cwheeling[Wout(05)] - {RB[Wout(05)] -

cs(wio)

- 05Wout}

The following a u g m e n t e d formed:

Lagrangian

is now

+ F, ~Qs, i(zi - z maX)

i = Cwheeling(Wout) -- RB[Wout(05)] -~ Cs(Win ) -

Win]

B(K) = AC (K) + 05Wou t

~- 2 ~/QS,i(Zi -- zmax) i The L a g r a n g i a n multiplier 2 is i n t r o d u c e d to comply with the c o n s t r a i n t Win = W o u t - ~ - A W . If wheeling p o w e r Wou t (kWh) is t r a n s m i t t e d to a buyer, the line flow zi m a y r e a c h or exceed its capacity limit z}~ax, hence the K u h n - T u c k e r multipliers PQS.i are introduced: #QS, i = 0

when

z~ • z max

#QS, i > 0

when

z~>~z~ ax

N o w f] is a function of Win, Wout, 05, 2 and #VS, i (i = 1, 2 . . . . . NL). Setting O~/OWin = 0 yields

(~Cs (Win) (~iW ~Win -~-]~kOWin

)

~Zi

1 -t-2~QS'iSWin : O i

(5)

Setting ~f~/505 = 0, then using the chain rule 0f] OWo~t ~Wo~ &5

0

yields 0AW\ 0z~ s ( W o u t ) _ 2 l_t___~out)_~iitos.i~Wout 205 = ~ ROWo~t

(5 = 05o+ ACdispatch Wout

2.3. Net wheeling utility benefits The net benefit the wheeling utility receives from wheeling t r a n s a c t i o n B (K) ( ~ / h ) is defined as the additional operating costs the utility incurs b e c a u s e of wheeling plus the gross r e v e n u e it receives from wheeling:

f2 = C (K) - B (s's> + 2(Wout + AW - Win )

+ 05wout(05) + 2[Wou~(05) + n w

If we denote 05o and 05 as the marginal wheeling rates w i t h o u t and with consideration of the line flow limit c o n s t r a i n t s [13], respectively, then it can be shown t h a t

(6)

According to Kirchhoff's laws, z~ = z~ [Pc, Pc, n e t w o r k parameters, n e t w o r k structure]

~- Zi [Y] Here y = P g - Pc, which is the vector of nodal p o w e r injections. Wont is a sort of p o w e r demand, as far as the wheeling utility is concerned, therefore

Oz~ Ozi ~Wout ~Yout Yout, an element corresponding to bus B in vector y, is the net p o w e r injection at the b u y e r ' s bus B.

where AC (K) is the change in operating cost due to wheeling. It should be noted that some c i r c u m s t a n c e s w o u l d yield negative wheeling rates, when netw o r k line flows are affected f a v o u r a b l y by wheeling. Nevertheless, net wheeling benefits would always be positive.

2.4. Notes on implementation If we k n o w all the necessary data from the three parties such as the seller's cost function Cs, the buyer's r e v e n u e function RB, and the n e t w o r k parameters, the wheeling rates for the transmission service exercised by the wheeling utility can be e v a l u a t e d using a c o m p u t e r program based on the t h e o r y outlined above. The wheeling rate 05 is a c o n t i n u o u s function of the a m o u n t of electric energy wheeled, Wout. The gross revenue the wheeling utility receives is 05Wout. It is useful to u n d e r s t a n d t h a t when no energy is wheeled, t h a t is, Wout = 0, the wheeling utility's revenue, of course, is also zero, but the corresponding wheeling r a t e 05]Wout=0 is usually not zero. For example, let us suppose t h a t an electricity b u y e r B needs energy supplies and its marginal r e v e n u e from disposing of this electricity is 28 m ~ / k W h . If a seller S produces electricity at 25 m2~/kWh, and the wheeling rate w o r k e d out is 1 m ~ / k W h , then b u y e r B should at least pay an a m o u n t t h a t is a bit over 26 m ~ / k W h , b u t less t h a n its marginal r e v e n u e of 2 8 m ~ / k W h , of course. Further, suppose t h a t b u y e r B buys 100 MW wheeled energy in one h o u r at the price of 27 m ~ / k W h , it will pay 2700 ~ for this a m o u n t of energy, the wheeling utility K will receive 100 ~ for supplying the wheeling service, and seller S will usually produce a bit more t h a n 100 M W of the energy flowing into utility K while receiving 2600 ~.

15

3. Case s t u d i e s

B u y e r B's r e v e n u e function will be

Two examples are employed here to illustrate the b e h a v i o u r of optimal wheeling rates. In a two-bus example we look at the direction of the power flows, and show h o w p o w e r losses result from wheeling. In a three-bus example we add transmission line constraints. A time d u r a t i o n of only one hour is considered so the time dependence is dropped from the equations in the examples.

3.1. Two-bus example In this two-bus example, depicted in Fig. 2, the wheeling utility K is a p p r o x i m a t e d by a 'load centre' bus and a ' g e n e r a t i o n centre' bus connected by a single transmission line, which has a resistance of 0.025 p.u. on a 1000 M V A base. Load during a typical day is assumed to v a r y b e t w e e n 800 and 2200 M W according to the h o u r l y variation. Seller S's g e n e r a t i o n cost functions will be Cs(P) = 1500 + 21.18P + 0.00408P 2 ~ / h w h e r e P is in M W (Fig. 3).

RB(P) = 1500 + 23.76P + 0.004686P 2 ~ / h w h e r e P is also in M W (Fig. 3). Figure 4 shows the seller's marginal generation cost and the b u y e r ' s marginal revenue, corresponding to the h y p o t h e t i c a l curves in Fig. 3. The h a t c h e d area suggests the existence of benefits in p o w e r wheeling, if there are no nontechnical impediments to p o w e r transfer [3].

3.1.1. Forward flow wheeling In this case, the wheeling p o w e r flow is in the forward direction with r e s p e c t to the flow without wheeling. The seller S injects energy at the utility K's g e n e r a t i o n bus while the b u y e r B extracts the w h e e l e d a m o u n t of energy from K's load bus, t h e r e b y increasing losses. Wheeling rates corresponding to different load levels are listed in Table 1 for two different values of Wout, 200 and 400 MWh. These wheeling rates are plotted in Fig. 5. In this forward flow wheeling situation, wheeling increases power losses; there is a positive additional p o w e r loss AL caused by wheeling, as shown in Table 1 w h e r e Win is larger t h a n Wout.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

Utility K

W=

Seller S

,'i i

I

iW ~ t

( - ~ gl

:

d

L~J

Buyer B

I 6

',G~eradon C¢n~¢

Load Centre ',

'i. . . . . . Bus Bus . . . . . . . . . . . . . . . . . . . . . . . . . .

I

'

Fig. 2. T w o - b u s e x a m p l e d i a g r a m w i t h t h e s e l l e r c l o s e t o t h e generation centre.

2% V

o

501

3.1.2. Reversed flow wheeling In this case, the positions of seller and b u y e r are reversed (Fig. 6) so t h a t transmission losses decrease with wheeling, which means t h a t the energy injected at S will be smaller t h a n the w h e e l e d energy e x t r a c t e d from B. Wheeling rates corresponding to v a r i o u s load levels are listed in Table 2 for W o u t = 2 0 0 and 4 0 0 M W h . These wheeling rates are also plotted i n F i g . 7.

40

30

2@ >

20o

(.1

10-

0

I

0

200

I

4'00

I

I

600

800

I

1000

P (MW) Fig. 3. C o s t a n d r e v e n u e f u n c t i o n s : - -

, seller's cost; -

, buyer's revenue.

I

1200

I

1400

16

^ 4O

~" 35

30

i 2o

I

I

I

I

I

I

I

200

400

600

800

1000

1200

1 ~'00

P (MW) Fig. 4. Marginal cost and marginal revenue: - -

, seller's marginal cost; - - - , b u y e r ' s m a r g i n a l r e v e n u e .

T A B L E 1. Two-bus example: seller close to generation centre

Load

05 ( m ~ / k W h )

Win ( M W h )

Benefits(B. s) ( ~ / h )

( MW)

800 1000 1200 1400 1600 1800 2000 2200

Wout = 2 0 0

Wout = 4 0 0

Wout = 2 0 0

Wout=400

Wout = 200

Wo~t = 400

0.78760 0.66220 0.53540 0.40730 0.27780 0.14690 0.01454 --0.32190

0.69390 0.55350 0.41170 0.26820 0.12310 -0.02356 -0.17190 -0.11930

204.5 205.5 206.5 207.5 208.5 209.5 210.5 211.5

410.0 412.0 414.0 416.0 418.0 420.0 422.0 424.0

276.2139 278.4082 280.8626 283.5816 286.5688 289.8297 293.3676 297.1884

591.4257 598.3472 603.8557 613.9612 622.6707 631.9973 641.9504 652.5396

1.0

•

0.5

0.0

l

1000

.

i

1200

I

1400

I

1600

~

0

T

-0,5

LocLd L e v e l (BIW) Fig. 5. Two-bus example wheeling rates with the seller close to the generation centre for Wout = 0 ([~), 100 ( + ) , 200 (.) a n d 400 ( A ) MWh.

In this reversed flow wheeling situation, wheeling decreases power losses, therefore the additional power loss AL caused by wheeling is negative, as demonstrated in Table 2 where Win is smaller than Wout.

3.1.3. Illustration of the implementation We illustrate the case w h e n the load level is 1000 MW and Wout= 200 M W h (see Table 1). The corresponding 05 =0.6622 m ~ / k W h and Win = 205.5 MWh. Then

17 I

I

z 1

Buyer B

I f t h e b u y e r B b u y s 200 M W o f e l e c t r i c e n e r g y in 1 hour at 24m~/kWh, dividing the Benefits (B's) = 278.4082 ~ e v e n l y b e t w e e n t h e b u y e r and the seller, then we have

Utility K

w m l t J,

m,

i

i 0°' ,GenerauonCentre : Bus

U t i l i t y B ' s p a y m e n t = 24.00 × 200 = 4800

LoadC~ B~

U t i l i t y K ' s g r o s s r e v e n u e = (5 × 200 = 132.44 2~

Fig. 6. Two-bus example diagram with the seller close to the load centre.

Utility S's revenue (receipt)=

4800-

132.44

= 4667.55 3Cs(P) 6~ - 3 × (7.06 + 2 ×

Therefore,

O.O0139P)lP=2O~.Suw

B ' s n e t b e n e f i t = 4 9 3 9 . 4 4 - 4800 = + 139.44 ----22.894 m # / k W h S ' s n e t b e n e f i t = 4 6 6 7 . 5 5 - 4528.59 = + 138.97

ORs(P)

_ 3 × (7.92 + 2 × O.O01562P)lp=2oo u w

They are roughly equal.

= 25.706 m ~ / k W h

3.2. Three-bus example

aP

and

RB[Wout=2OOMWh=

1500 + 23.76Wout + 0.004686Wout 2

= 1500 + 4939.44

Cslwin=2os.sMWh =

1500 + 21.18Win ÷ 0.00417Win 2

= 1500 + 4528.59

T h e t h r e e - b u s e x a m p l e is e m p l o y e d h e r e t o illustrate the effect of transmission line capacity constraints on optimal wheeling rates. The threebus-three-line model used to approximate a w h e e l i n g u t i l i t y is s h o w n i n F i g . 8. B u s e s ( a ) a n d (c) a r e g e n e r a t i o n c e n t r e s w i t h v a r i a b l e g e n e r a tion costs under normal outage conditions rang-

TABLE 2. Two-bus example: seller close to load centre Load (MW)

Win (MWh)

(5 (m~/kWh)

800 1000 1200 1400 1600 1800 2000 2200

Benefits(B. s) (~/h)

Wout= 200

Wout =400

Wo~t =200

Wout=400

Wout=200

Wout=400

1.0581 0.9434 0.8277 0.7107 0.5925 0.4731 0.3524 0.2305

1.2703 1.1520 1.0324 0.9116 0.7895 0.6662 0.5416 0.4157

196.5519 196.5885 194.6345 193.6898 192.7543 191.8277 190.9101 190.0012

394.0598 392.1190 390.1973 388.2943 386.4100 384.5439 382.6959 380.8656

403.7629 448.6647 493.5797 538.5135 583.4738 628.4676 673.5029 718.5850

751.9102 846.7290 941.5300 1036.3310 1131.1480 1225.9940 1320.8870 1415.8410

1.5

1.0

0.5

0.0

[

800

1000

I

1200

I

1 'tOO Load L e v e l

I GO0 (MW)

1800

I

|

2000

2200

Fig. 7. Two-bus example wheeling rates with the seller close to the load centre for Wo, t = 0 ([3), 100 ( + ) , 200 (*) and 400 ( A ) MWh.

18

r - - - q w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iw ,r___n

3.2.1. Wheeling flow direction from seller at bus (c) to buyer at bus (a) This situation is illustrated in Fig. 8 and different cases are d e m o n s t r a t e d below. Case 1. Assuming that, prior to wheeling, utility K operates in the state shown in Table 4, and

G 1'°' , .

.

.

.

.

.

.

.

.

.

.

0>

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

i

Operating costs = 90.12 k ~ / h

Fig. 8. T h r e e - b u s e x a m p l e (seller at b u s (c) to b u y e r at b u s (a)).

.,c:

140

E

130 120 110 1 O0

~

8o

o -~ ._~

70 60

:~

us (b) Generation

~

j,,,.,=

Bus (c) Generation

50 4O 30

2O

•

0

I

•

400

I

•

800

I

•

1200

1

1600

Generation (MW) Fig. 9. M a r g i n a l g e n e r a t i o n cost at utility K.

Active p o w e r losses = 56.10 M W There is a wheeling d e m a n d Wont from utility B c o n n e c t e d at bus (a). Wheeling rates for v a r i o u s a m o u n t s of power wheeled and the r e l e v a n t results are illustrated in Table 5. If the real p o w e r demand at bus (b) varies, that is, the load level changes, the curves of wheeling rates with respect to different load levels are as presented in Fig. 10. Case 2. Assuming implementation of optimal power dispatch at the wheeling utility K [13], the load flow results are given in Table 6. Correspondingly, Operating costs = 86.95 k ~ / h

ing, respectively, from 55 and 35 m ~ / k W h at low generation levels to 80 and 60 m ~ / k W h at generation levels of 1000 MW, as s h o w n in Fig. 9. Bus (b) is modelled as having some expensive generating p o w e r available and all of utility K's load. All three transmission lines in Fig. 8 are assumed to be indentical. Here, the seller's g e n e r a t o r cost function and the b u y e r ' s gross r e v e n u e are the same as those in the two-bus example. Wheeling utility K's marginal variable generation costs are s h o w n in Fig. 9. Table 3 gives the n e t w o r k data of utility K and Table 4 gives the load flow data when the load level is 1600 MW. T A B L E 3. N e t w o r k p a r a m e t e r s of u t i l i t y K Branch

r ÷ jx

Susceptance a b

S m"x (MVA)

a-b b c c-a

0.04 ÷ j0.20 0.04 + j0.20 0.04 + j0.20

0.015 0.015 0.015

1000.0

aLine c h a r g i n g is h a l f t o t a l line c h a r g i n g , a n d d a t a are on t h e b a s e SB = 1000 MVA.

and Active p o w e r losses = 48.95 MW Utility K is operating in an optimal operating state prior to wheeling. Then t h e r e is a wheeling r e q u i r e m e n t Wout from utility B c o n n e c t e d at bus

T A B L E 5. B u s to b u s w h e e l i n g r a t e s (seller at b u s (c) to b u y e r at b u s (a))

(MWh)

~ (m~/kWh)

Line b c flow P b - c (MW)

Win (MWh)

Benefits(B, s) (:~/h)

0 200 400 600 800 1000 1200 1400

0.8376 0.8953 0.9435 0.9818 1.0100 1.0277 1.0347 1.0306

915.97 983.25 1051.31 1120.20 a 1189.96 a 1260.63 a 1332.26 a 1404.92 a

0.00 202.53 407.24 614.22 823.48 1035.09 t249.11 1465.60

0.00 299.60 559.34 771.46 930.04 1027.38 1055.74 1007.31

Wout

aLine flows are over t h e c a p a c i t y c o n s t r a i n t s .

T A B L E 4. G e n e r a t i o n a n d d e m a n d at utility K

T A B L E 6. O p t i m a l g e n e r a t i o n d i s p a t c h at u t i l i t y K

Bus

Voltage (p.u.)

Generation (MW, MVAR)

Demand (MW, MVAR)

Bus

Voltage (p.u.)

Generation (MW, MVAR)

Demand (MW, MVAR)

(a) (b) (c)

1.00 1.00 1.05

600.00 - j341.779 0.00 +j372.547 1056.10 + j356.670

1600.0 + j200.0

(a) (b) (c)

1.0206 1.0072 1.0500

367.14 - j141.78 103.45 + j300.30 1178.36 + j191.46

1600.0 + j200.0

19

I .25

?. .-~

1.oo

cg

c~ 0.75 *.4

0.50

J 800

1000

1200

1400 1600 L o a d L e v e L (MW)

1800

2000

2200

Fig. 10. T h r e e - b u s e x a m p l e (seller at b u s (c) to b u y e r at b u s (a)): Wou t = 0 ( [ ~ ) , 100 ( + ) , 200 (*) a n d 400 (A) M W h .

T A B L E 7. B u s to b u s w h e e l i n g r a t e s w i t h o p t i m a l power disp a t c h (seller at b u s (c) to b u y e r at B u s (a)) Wo, t (MWh)

~5 (m~/kWh)

L i n e b - c flow Pb~c ( M W )

Win (MWh)

Benefits(B, s) (~/h)

0 200 400 600 800 1000 1200 1400

0.8760 0.9253 0.9648 0.9941 1.0131 1.0212 1.0183 1.0038

913.08 981.00 1049.73 1119.30 a 1189.76 a 1261.16 a 1333.56 a 1407.01 a

0.00 204.80 411.79 621.02 832.56 1046.45 1262.78 1481.60

0.00 241.89 459.10 585.01 672.61 694.48 642.81 709.13

aLine flows a r e over t h e c a p a c i t y c o n s t r a i n t s .

(a). Wheeling rates for various wheeling a m o u n t s and the r e l e v a n t results are given in Table 7, and the curves of wheeling rates with respect to the a m o u n t of e n e r g y wheeled are illustrated in Fig. 11. Case 3. Wheeled e n e r g y W o u t = 2 0 0 M W h is used t h r o u g h o u t all the different load levels at bus (b) in utility K. All the wheeling rates (5 in Table 8 are obtained on the premise t h a t before wheeling happens utility K a l r e a d y operates at the optimal power dispatch state for each individual load level at bus (b), which means the lowest operating cost while the load is supplied by the utility K's g e n e r a t i o n [13]. All inductive

I .04

1.00

%

5 "~, v

o.ss

~'

0.92

*,,4 o

z:

0.88

0.84 0

I 200

I 400

I BOO

I 800

I ! 000

I 1200

I 1400

Wout. ( MW )

Fig. 11. B u s to b u s w h e e l i n g r a t e s w i t h o p t i m a l p o w e r d i s p a t c h (seller at b u s (c) to b u y e r at b u s (a)).

20 T A B L E 8. W h e e l i n g r a t e s w i t h line c a p a c i t y c o n s t r a i n t s L o a d at b u s (b) (MW, MVAR)

(5 (m~/kWh)

Wi n (MWh)

Benefits(B. S) (~/h)

800+j100 1000 + jl00 1200 + jl00 1400 ÷ jl00 1600 + jl00 1800a+jl00 2000a+jl00 2200 ÷ jl00

1.1674 1.1067 1.0424 0.9830 0.9201 7.9275 (0.86655) 10.1931 (0.8391 b) 0.8886

204.1143 204.1840 204.8631 204.2329 205.4850 204.7510 205.4020 202.2654

209.0840 219.6320 116.9580 243.2606 227.1719 -1157.5081 -1525.5175 307.1382

ACrodi~p~tch (:F/h)

L i n e b c flow w i t h o u t r e d i s p a t c h , P5~c (MW) 571.66 674.76 800.55 881.10 981.68 1077.24 1099.71 961.50

1412.2 1870.8

aLoad levels a r e over t h e m a x i m u m flow c o n s t r a i n t . G e n e r a t i o n m u s t be r e d i s p a t c h e d in o r d e r to m e e t t h e t r a n s m i s s i o n line b - c m a x i m u m flow c o n s t r a i n t of 1000 M W . T h e e x t r a cost is g i v e n by ACredispatch. b W h e e l i n g r a t e s before r e d i s p a t c h .

12.5

2 .,¢ %

~

/+

10.0

/t

:,.s

/

J

/ /

0

e,,

/

/

5.0

/

O~ ¢-

/ /

r.

/

2.5

/ I

0.0

I

800

1000

!

I

1200

I

4t

I

I

1400

1600

I

1000

i

2000

Load L e v e l ( MY) Fig. 12. W h e e l i n g r a t e s w i t h line c a p a c i t y c o n s t r a i n t s : - c o n s t r a i n t on line b - c .

loads c o n t a i n the same a m o u n t of reactive power, as given in the first column of Table 8. Table 8 shows optimal wheeling rates for different load levels in utility K, t h a t is, different real power demands at bus (b). In this Table, it is indicated t h a t flow on line b - c is sometimes over its c o n s t r a i n t limit of 1000 MW. Therefore utility K has to redispatch its power in order to reduce the line flow, which is achieved at an additional operating c o s t ACredispatch, as shown in the fifth column of the Table. While transmission line c a p a c i t y c o n s t r a i n t s are not binding, wheeling rates do not exceed 1.2 m ~ / k W h . W h e n they become effective the wheeling rate increases to as much as 10.2 m~t/kWh (Fig. 12). This reflects the high operating costs incurred by utility K to r e a r r a n g e its g e n e r a t i o n p a t t e r n t o w a r d higher generation at bus (a) and, especially, at bus (c),

, w i t h o u t c a p a c i t y c o n s t r a i n t on line b - c ; - -

-, with capacity

and lower generation at bus (b), in order to meet the c a p a c i t y c o n s t r a i n t on transmission line b - c . The extent to which the wheeling utility has to r e a r r a n g e its generation p a t t e r n to meet flow c o n s t r a i n t s can be a p p r e c i a t e d by observing the last column of Table 8, w h i c h shows flow on b - c w i t h o u t g e n e r a t i o n redispatch. Certainly, there is a limited capability for wheeling utility K to satisfy its line c o n s t r a i n t s by r e a r r a n g i n g its g e n e r a t i o n pattern. Moreover, it is w o r t h noting t h a t wheeling m a y f a v o u r a b l y affect lines flows. In the last case of Table 8 (load at bus (b) --2200 + j l 0 0 MVA), w h e n the load is further increased the loading on line b - c actually decreases to within it maximum limit. In this case it also happens to be the optimal loading for the wheeling utility, and hence no redispatch is needed.

21

3.2.2 Wheeling flow direction: from seller at bus (a) to buyer at bus (c) N o w we assume the n e w s i t u a t i o n w h e r e the location of the selling and b u y i n g utilities are reversed, t h a t is, the seller is l o c a t e d at bus (a) and the b u y e r at bus (c), as illustrated in Fig. 13. Case 1. Prior to wheeling, we assume t h a t utility K operates at the state s h o w n in Table 4. Therefore, the operating cost and p o w e r losses are the same as those w h e r e the wheeling flow is from the seller at bus (c) to the b u y e r at bus

(a).

w

~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

: . .w-.

!

Fig. 13. Three-bus example (seller at bus (a) to b u y e r at b u s (c)).

There is a wheeling d e m a n d Wout from utility B c o n n e c t e d at bus (c). The wheeling rates for various a m o u n t s of p o w e r w h e e l e d and the r e l e v a n t results are listed in Table 9. The curves of wheeling rates with respect to different load levels, t h a t is, different real p o w e r demands at bus (b), are p r e s e n t e d in Fig. 14. Case 2. Assuming t h a t optimal p o w e r dispatch is implemented at wheeling utility K, t h e n after dispatch from utility K the costs and losses are the same as those for the wheeling direction from the seller at bus (c) to the b u y e r at bus (a). N o w there is a wheeling r e q u i r e m e n t WOUt from utility B c o n n e c t e d at bus (c). The wheeling rates for various wheeling r e q u i r e m e n t s from the b u y e r located at bus (c) and the r e l e v a n t results are listed in Table 10. These wheeling rates with respect to the power w h e e l e d are plotted in Fig. 15. Case 3. Wheeled energy W o u t = 2 0 0 M W h is used t h r o u g h o u t all the different load levels at bus (b) in utility K.

TABLE 9. Bus to bus w h e e l i n g r a t e s (seller at bus (a) to b u y e r at bus (c)) Wout (MWh)

(J~ (m~/kWh)

Line b c flow Pb~c (MW)

Line b - c Ib~ c (p.u.)

Line c - a Ic~ a (p.u.)

Win (MWh)

Benefits(B, s) (~t/h)

0 200 400 600 800 1000 1200 1400

0.8420 0.9051 0.9575 0.9990 1.0294 1.0481 1.0551 1.0498

715.34 777.52 840.03 902.90 966.14 1029.78 a 1093.84 a 1158.37 a

0.8847 0.8228 0.7612 0.7002 0.6398 0.5802 0.5218 0.4650

0.2608 0.2490 0.3026 0.3965 0.5097 0.6325 0.7608 0.8927

0.00 199.63 401.41 605.39 811.58 1020.07 1230.87 1444.07

0.00 364.08 696.98 993.23 1247.66 1453.82 1606.10 1697.67

aLine flows are over the capacity c o n s t r a i n t s .

I .25

0.75

i 0.~0

'

800

I

I

1000

1200

I

I

1400 1600 Load L e v e l (I'IV)

---T'-

1800

I

I

2000

2200

Fig. 14. Three-bus example (seller at b u s (a) to b u y e r at bus (c)): Wout = 0 ([~), 100 ( + ) , 200 (*) and 400 ( A ) MWh.

22 TABLE 10. Bus to bus wheeling r a t e s with optimal power dispatch (seller at bus (a) to buyer at bus (c)) Wout (MWh)

~5 (m,~/kWh)

Line b c flow Pa~b (MW)

Line b c Ib_ c (p.u.)

Line c a Ic_ ~ (p.u.)

W~,1 (MWh)

Benefits(B, s) (~/h)

0 200 400 600 800 1000 1200 1400

0.8840 0.9624 1.0309 1.0893 1.1375 1.1750 1.2017 1.2172

614.23 676.30 738.68 801.38 864.43 927.86 991.68 1055.92 ~

0.8768 0.8145 0.7525 0.6909 0.6298 0.5692 0.5095 0.4511

0.2724 0.1755 0.1475 0.2181 0.3299 0.4541 0.5834 0.7157

0.00 197.39 396.85 598.41 802.09 1007.96 1216.05 1426.41

0.00 403.82 779.47 1121.93 1426.06 1685.82 1895.44 2048.69

aLine flows are over the capacity c o n s t r a i n t s . 1,3

r

1,2

i

/

L J=

/

1,1

v VI &, ,.,p

"~

1,0

¢. .,,,-,i

r

3~

0.9

0,8

I

I

I

]

200

400

600

800

I

1000

I

I

1200

1400

Wout ( MWh ) Fig. 15. Bus to bus rates with optimal p o w e r d i s p a t c h (seller at bus (a) to b u y e r at bus (c)).

TABLE 11. Wheeling r a t e s w i t h line capacity c o n s t r a i n t s (seller at bus (a) to b u y e r at bus (c)) Load at bus (b) (MW, MVAR)

800÷j100 1000+ jl00 1200+j100 1400+j100 1600+j100 1800+j100 2000÷j100 2200 + jl00

~5 (m~/kWh)

1.1891 1.1340 1.0807 1.0117 0.9731 0.9201 1.0156 1.0274

Win (MWh)

198.3135 197.7376 197.1102 197.5546 198.1824 197,6141 195.0356 195.1409

Benefits (B,s) (:~/h)

337.2984 361.5118 386.4929 390.1462 383.5330 407.1010 446.8510 442.0910

Table 11 shows optimal wheeling rates for various load levels at bus (b) in utility K. All the wheeling rates 05 are o b t a i n e d on the premise t h a t before wheeling utility K a l r e a d y operates at the optimal p o w e r dispatch state for each

Line flow (MW) Pa ~ b

Pb ~ c

Pc ~ a

314.47 394.83 470.79 592.31 668.98 747.54 548.11 529.70

432.67 553.38 673.15 767.81 884.09 1003.41 984.87 962.30

99.42 134.94 174.04 142.30 150.82 185.13 371.19 366.66

individual load level at bus (b), which means the lowest operating cost while the load is supplied by utility K's generation. All inductive loads c o n t a i n the same reactive power, as given in the first column of Table 11.

23

We may r e m a r k t h a t the results in the fifth line of Table 8 (corresponding to the case w h e n the load at bus (b) = 1600 ÷ jl00 MVA) are very close to the results in the second line of Table 7 (Wo, t = 200 MW). There are differences b e c a u s e in case 1 of the three-bus example the load at bus (b) = 1 6 0 0 + j 2 0 0 M V A (Table 4), the reactive part of w h i c h is different from t h a t in case 2 of the three-bus example, despite the same real p o w e r load. Likewise, the results in the fifth line of Table 11 are close to those in the second line of Table 10. Indeed, reactive power flows will change the effects of the wheeling transaction, as we m a y expect.

4. W h e e l i n g r a t e s e n s i t i v i t i e s

4.1. Effect of changes in load levels The effects on wheeling rates at different load levels are illustrated in Figs. 5, 7, 10, 12, 14 and 16. H i g h e r load levels p r o d u c e lower wheeling rates, provided t h a t n e t w o r k transmission limits are not reached. It is interesting to note t h a t there is an almost linear relationship b e t w e e n load level and wheeling rate. A l t h o u g h p o w e r systems are by n a t u r e not linear, in this case they exhibit a near-linear behaviour. P r o b a b l y this is partly b e c a u s e the changes in load were minor and the n e t w o r k was not stressed. It is possible t h a t u n d e r certain c i r c u m s t a n c e s wheeling t u r n s o u t not to be 'cost effective' ('bad' wheeling), for instance, w h e n wheeling rates are

~ ,00

negative as loading reaches high levels. This means t h a t wheeling will n e i t h e r improve the economic efficiency, nor r e c o v e r fully the extra costs of the wheeling utility. Generally speaking, the rates e v a l u a t e d by this a p p r o a c h should be positive. Otherwise, wheeling utility K would be made worse off by wheeling.

4.2. Effect of power losses due to wheeling The additional p o w e r losses caused by wheeling, which are modelled to be c o n t r i b u t e d by seller and buyer, can be either positive (wheeling increases losses) or negative (wheeling decreases losses). U n d e r different circumstances, the effects are different. Therefore, this effect needs to be observed case by case. For instance, in Tables 9 and 10, w h e n the a m o u n t of p o w e r wheeled, Wout, is at certain levels, the additional p o w e r loss A W is negative; b u t when Wout reaches o t h e r levels, AW is positive. As far as the wheeling rates evaluated by the a p p r o a c h developed in this paper are concerned, this is an interesting observation.

4.3. Effect of line flow constraints W h e n wheeling causes flows to exceed transmission limits, corrective action must be taken, at some cost, which is c h a r g e d to the wheeling transaction. U n d e r some circumstances, wheeling flows affect transmission line limits favourably, and enable the wheeling utility to redispatch its g e n e r a t i o n and reduce its operating costs.

--

3.75 t_ ,-$

v

2.50 o iv om ¢.

1.25 -~L

0.00

e

I

000

1000

E)--------'~----B---- --

a

1200

i

i

I 'tOO Load

0

Level.

1800

~..B---~

--------0

i

1000

l

2000

(MY)

Fig. 16. Wheeling r a t e s w i t h line capacity c o n s t r a i n t s (seller at bus (a) to b u y e r at bus (c)).

--~EJ

I

2200

24

TABLE 12. Line flow comparison (seller at bus (c) to buyer at bus (a)) Load at bus (b) (MW, MVAR)

800 + 1000 + 1200 + 1400 + 1600 + 1800 + 2000 + 2200 +

jl00 jl00 jl00 jl00 jl00 jl00 jl00 jl00

Line flow during wheeling (MW)

Line flow without wheeling (MW) P~-b

Pb~c

Pc~a

P~-b

Pb~o

P¢~

241.21 337.09 417.20 537.32 654.22 726.54 861.05 912.66

504.10 607.04 732.48 813.16 914.37 1009.94 1032.56 894.96

245.83 247.91 288.40 245.69 279.59 263.58 239.63 32.34

175.01 270.77 350.76 470.80 588.72 660.83 795.70 846.88

571.66 674.76 800.55 881.10 981.68 1077.25 a 1099.71 a 961.50

382.38 384.37 425.20 381.98 417.77 401.02 377.88 168.07

aLine flows are over the capacity constraints.

TABLE 13. Generation redispatch due to network c o n s t r a i n t s Load at bus (b) (MW, MVAR)

Generation unit

No wheeling

Wheeling Wout = 200 MW

1800 + jl00

Gen. at bus (a) Gen. at bus (b) Gen. at bus (c)

466.00 + 122.23 + 1273.52100.02 628.56 + 172.97 + 1272.19 114.94

510.75 + 174.35 + 1170.34101.43 767.73 + 192.99 + 1107.94 116.81

Cost ( k ~ / h ) 2000 + j 100

Gen. at bus (a) Gen. at bus (b) Gen. at bus (c)

Cost ( k ~ / h )

j93.15 j129.80 j278.70 j723.13 j47.80 j404.38

j280.21 j24.64 j0.77 j763.67 j52.48 j448.36

TABLE 14. Line flow comparison (seller at bus (a) to buyer at bus (c)) Load at bus (b) (MW, MVAR)

800 + jl00 1000 + jl00 1200+j100 1400 + jl00 1600 + jl00 1800 + jl00 2000 + jl00 2200 + jl00

Line flow during wheeling (MW)

Line flow without wheeling (MW) Pa~b

Pb-c

Pc~a

Pa~b

Pb-o

Pc~a

248.86 329.28 405.35 526.65 603.46 682.31 483.25 464.58

498.86 619.68 739.57 834.30 950.22 1069.44 1051.41 1029.06

233.24 268.65 307.63 275.82 284.69 319.10 504.65 499.90

314.47 394.83 470.79 592.31 668.98 747.54 548.11 529.70

432.67 553.38 673.15 767.81 884.09 1003.41 984.87 962.30

99.42 134.94 174.04 142.30 150.82 185.13 371.19 366.66

In Fig. 12, we show the wheeling rates for a three-bus system wheeling 200 M W p o w e r to the b u y e r at bus (a) from the seller at bus (c) with the flow on line b - c in excess of its c a p a c i t y w h e n the load at bus (b) is 1800 and 2000 MW. As expected, the c o n s t r a i n t s cause the wheeling rates to increase substantially. Table 12 lists the line flows in the system b o t h w i t h o u t wheeling and during wheeling, and also indicates the fine flows t h a t are over the limit. P o w e r redispatch is applied to m a i n t a i n system

o p e r a t i o n security, and the results of redispatch are given in Table 13. Similar to Table 12, Table 14 gives the comparison w h e n the wheeling direction is from the seller at bus (a) to the b u y e r at bus (c).

4.4. Effects of reactive power flow It is certain that reactive power flow will change the effects of the wheeling transaction. Reactive power flow is another complex and difficult aspect of wheeling and wheeling rate

25

evaluation. I n s t e a d of using the c o n v e n t i o n a l p o w e r factor penalties, the a u t h o r s believe t h a t this aspect should be developed s i m u l t a n e o u s l y with active p o w e r wheeling for the sake of smooth o p e r a t i o n of the electricity market-place [14].

5. Comparison with existing approaches Comparing the p r e s e n t a p p r o a c h with the existing a p p r o a c h e s in refs. 4, 6 and 7, the major differences are as follows. • Our a p p r o a c h e v a l u a t e s optimal wheeling rates w i t h o u t any a s s u m p t i o n of the existence of a spot price based energy market-place. • We model the additional p o w e r losses resulting from wheeling to be c o n t r i b u t e d by the selling and buying utilities. Therefore, while wheeling happens, inflow into the wheeling utility and outflow out of the wheeling utility are different. Different o p e r a t i n g conditions of the wheeling utility result in different A W, the difference bet w e e n Win and Wout, which is to cover the additional p o w e r losses. • On the basis of the equal sharing of benefits arising from the wheeling t r a n s a c t i o n among the wheeling, selling and buying utilities, our approach avoids the direct e v a l u a t i o n of the netw o r k m a i n t e n a n c e cost and the quality of supply cost components. The following wheeling rates [6] follow a variety of different approaches. M a n y are b a s e d on case specific n e g o t i a t i o n s and there is a lot of room for i n t e r p r e t a t i o n and 'fair play' in implem e n t a t i o n [7]. Postage stamp: a c o n s t a n t wheeling rate t h a t is often used t o d a y to r e m u n e r a t e utilities affected by the activity of wheeling parties. Contract path~red line: simple models b a s e d on the a s s u m p t i o n t h a t w h e e l e d energy travels pred o m i n a n t l y over a specific transmission line (red line) have often been used to date to determine wheeling costs; a l t h o u g h the red line model m a y be i n a d e q u a t e in some cases, it m a y prove a good a p p r o x i m a t i o n in others• Megawatt mile: this a p p r o a c h also considers the physics of n e t w o r k flow, b u t in terms of DC load flow. Some illustrative features of setting the wheeling rates by the t h r e e a p p r o a c h e s [7] listed above are as follows. • The wheeling rates are flat and do not depend on the a m o u n t w h e e l e d or o t h e r transmission flows. They often involve d e m a n d as well as energy charges.

• The wheeling rates are often 'postage stamp', t h a t is, the rate is the same regardless of distance wheeled. Voltage levels are u s u a l l y a factor. • Some rates are for firm wheeling [15], t h a t is, wheeling utility K can refuse to wheel only under emergency conditions. In these cases the rate is u s u a l l y per peak kW wheeled during some period, or per c o n t r a c t e d kW. • Rates are often determined by examining embedded capital costs for some p o r t i o n of the wheeling utility's transmission system [15], and sometimes with a m e a s u r e of a v e r a g e losses added. One method is to draw a line on a map of the transmission system from the wheeling utility K's border with the seller of p o w e r S to K's border with b u y e r B. • Some a r r a n g e m e n t s allow wheeling utility K to i n t e r r u p t the wheeling at its sole discretion. In these cases the rates are often per kWh.

6. Summary We have p r e s e n t e d a solid t h e o r e t i c a l foundation for wheeling rates in a multi-utility system. Additional problems which we will address in future papers are the following. (1) The formulae for area to area wheeling are more complicated t h a n the bus to bus formulae as many buses and lines will be affected. However, the insights gained here still apply. (2) Like o t h e r methods published in the literature, the ideal wheeling rates (5(t) e v a l u a t e d by this t h e o r y could over- or under-recover the real wheeling costs. Hence, r e v e n u e reconciliation m a y need to be adopted. W h e n the wheeling rates of this p a p e r are used, the buying and selling utilities automatically t a k e the costs which t h e y impose on o t h e r utilities into c o n s i d e r a t i o n when deciding w h e t h e r to b u y and.sell to each other• Wheeling becomes a 'no loss' proposition for all utilities. The effect of our a p p r o a c h is to allow closer coordination among spatially s e p a r a t e d utilities. The n u m b e r of potential p a r t n e r s for interchanges increases, since they need n e i t h e r be members of the same pool nor be a d j a c e n t to each other.

Acknowledgements S. P. Zhu wishes to extend his heartfelt gratitude to the British Council and the Chinese Embassy for their sponsorship.

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