The capacity credit of wind power: A theoretical analysis

Solar Energy,Vol. 26, pp. 391--401, 1981 Prinled in Great Britain.

0038-.092X181/05039l-11502,00/0 Pergamon Press Ltd.

THE CAPACITY CREDIT OF WIND POWER: A THEORETICAL ANALYSIS JOHN HASLETTt Statistics Department, Trinity College, Dublin 2, Ireland and MARK DIESENDORF CSIRO Division of Mathematics and Statistics, P.O. Box 1965,Canberra City, A.C.T. 2601, Australia

(Received 20 June 1980; revision accepted 9 January 1981) A~traet--A probabilistic model of capacity credit (or value) is constructed for wind power in an electricity grid, The model-which is initially based on the assumptions that electricity supply, electricity demand and the output of a system of aerogenerators are Normally distributed and that the conventional power plant in the grid comprises units of identical capacity--provides simple analytical expressions for four different concepts of capacity credit and elucidates the qualitative differences between them. Two of these concepts of wind power capacity credit, the "equivalent conventional capacity" and the "equivalent firm capacity", are then studied in more detail by introducing a much more realistic probability distribution of wind power output than the Normal distribution and by calculating the Loss of Load Probability. For small penetrations of wind power into the grid, the capacity credit is approximatelyequal to the average wind power output, while for large penetrations the credit tends to a limit which is determined by the probabilityof zero wind power and the conventional plant characteristics. I. INTRODUCTION

This paper is stimulated by the considerable confusion, still existing even in the wind power literature, on the "capacity credit" or capacity value of wind power systems. By this is meant ultimately the extent to which the inclusion of wind power within an electricity grid can lead to savings in conventional power plant; or, more simply, we may ask whether a wind power system, because of its inherent unreliability, needs to have a complete back-up of conventional plant. A commonly expressed argument is that without storage, "the [wind power] system cannot be relied upon to deliver power at any given time, and, therefore, its economic value is only related to the amount of fuel saved, i.e. the wind power system receives no capacity credit"[1]. Others have argued that even without storage there may be a capacity credit of perhaps 10 per cent of the maximum wind power produced[2]. S~renson[3] gives a similar figure, expressed however as 25 per cent of the average wind power capacity. More thorough analyses have recently appeared[4,5] where it is shown that the credit is a function of many system parameters, including most importantly the degree of penetration of wind power into the grid. The intention of this paper is to contribute some relatively simple arguments and concepts to this debate. Simple probability models of the key variables (wind power, plant on outage and electricity load) will be used to obtain algebraic expressions for capacity credit. These expressions will then be used to illustrate analytically some of these concepts. For much of the work referred to above is empirical and/or numerical, and is often

expounded in the context of specific utilities; this inhibits the understanding of general principles. This paper will therefore stress analytic simplicity and generality at the expense, where necessary, of numerical accuracy. In the following therefore we first discuss, with the aid of simple models, the question of determining the required conventional generating plant, including static reserve, to meet a given load. We then extend the argument to include wind power. In the first stage of this extension (Section 3.1) we employ a grossly oversimplified model of wind power variability to introduce some alternative measures of "capacity credit". Subsequently (Section 3.2), we use a much more realistic model to analyse in some detail two of these measures, referred to as "equivalent conventional capacity" and "equivalent firm capacity", and their sensitivity to various system parameters. 2. GENERATION RESERVE STUDIES IN A CONVENTIONAL GRID

One crucial goal in planning the development of a conventional grid is to ensure that there is sufficient generating plant installed such that the available plant at any time will rarely exceed unconstrained load. The most common index of such reliability is the expected number of such events in a given period, measured via the Loss of Load Probability (LOLP); an alternative measure is in the Frequency and Duration approach. An easy introduction to LOLP, the method to be examined here, is given in [6]. This is the method examined in the context of wind power in most of the previous literature on capacity credit. 2.1 Loss of load probability If we denote by C~ the installed conventional plant, by

tWork performed at CSIRO Division of Mathematics and Statistics, Canberra. 391

392

J. HASLETr and M. DIESENDORF

A the available plant at any given time associated with C~ (A ~
p = ~(A < L).

Typically p is specified as part of utility policy, expressed perhaps as "1 day in 5 yr" or "8 hr in I yr". Note that A and L are both variable, the former due to outages (i.e. breakdowns of power plant and scheduled maintenance), the latter due to straightforward variations in demand. In the electricity industry the variation of A is modelled by a "capacity outage probability" table, constructed from an examination of the generating potential of the individual units, and of their "forced outage rates". The distribution of L is usually summarised in the Load Duration Curve. Suppose now that the Normal probability model is an adequate fit to each of these distributions; that is

2.2 Capacity implications of LOLP There is no single method for determining the desired installed capacity to meet unconstrained load with specific reliability; this is because of the large number of variables involved in the description of the plant. If all plant is taken to be identical, however (i.e. c~ = c, r~ = r for all i), then (6) and (4) may be solved for the single remaining parameter n. Although this is unrealistically simple it provides a basis for qualitative discussion. Suppose that po is the prescribed reliability according to the utility policy. Thus if Zo is such that ~(Zo) = Po = ~(Ao < L)

(7)

where Ao is the available plant corresponding to an installed capacity Co = noc, where no is to be found, we have (t~A0 -- ~[~L): Zo(O'Ao2 ~LO.L2)I/2

(8)

(2)

A ~ N(~A, o'ff)

and hence using (4) and solving for no, that

and L ~ N(#L, o¥2)

(3)

where ~ and cr2 denote mean and variance respectively. In practice (3) is often quite a good approximation but (2) is not. Hence the detailed numerical results of our model should be regarded with caution; however, the assumptions suffice for the largely qualitative analysis performed here. If the conventional plant consists of n independent units with capacity c~ (i.e. C~ =2c~) and forced outage rate r, (i = 1, 2. . . . . n) then it follows from the theory of independent binary events that =

( 1 - r,)c,,

=

r,)c,

(4)

The peak load Lo in a given year may be computed from /~L and aL, given the Normal distribution, in the following sense. For the load duration curve for that year is but an empirical distribution function based on 8760 observations, and 1/8760 is approximately equal to the probability of a standard Normal random variable exceeding 3.7. Consequently we may write for the peak load Lp = m- + 3.7~rL. Lp, defined in this manner, is such that the expected number of hourly loads exceeding Lp in a year is one. This expression however should only be regarded as indicative. In practice, the value of the peak load in a given period, which is subject to many unpredictable factors, could not be deduced accurately in the above manner. If A and L are independent, it follows from (2) and (3) that A

-

L

~ N(I, tA -/~L,

O'A2 + O'L2 )

(5)

and that p = • (A < L) = ~{(#A -- ~L)/(~Az + O'L2)1/2}

(6)

where t}(z) = (2¢r)-"/2~ f f e -(1/2y2dy, is widely tabulated.

no={c(l-r)}-l(tzL +½Zo2Cr+ Zoao)

(9)

ao = (~L cr + O'L2 .~1 Zo2C2r2)II2.

(10)

where

Note that ~Ao = noc(1- r) and that if we write ~ = ( ~ - / z L ) and ~r2= ~ Z + ~ L ~ (as will be convenient later) then by (8) and (9) /z = Zoa = zo(½zocr + ao).

(11)

Note also that if ~L and c are written as proportional to ~L, then # is also proportional to/ZL. For example, let ~L = IGW and ~rL--0.2GW (corresponding to Lo = 1.74 GW). Further let c = 0.2 GW and r--0.1. Then it follows for po = 0.001 (corresponding to 8.76 hr on average per yr when load would have to be constrained), that zo=3.1 and hence that no = 10.35. Thus Co--2.069 GW is the installed plant (of this type) needed to meet a load characterised by an average value of 1 GW and peak load of 1.74 GW at this level of reliability. The static reserve (Co-Lp) is 0.329GW, or more than 1.5 plants of size 0.2 GW. This is the penalty associated with non-zero forced outage rates or with the choice of 8.76 hr/yr, rather than a greater number, as the definition of acceptable system reliability. We now use these concepts to examine the capacity implications of wind power. 3. CAPACITY "IMPLICATIONS OF' WIND POWER

The wind power W sent out to the grid by a large array of wind turbines is variable. This variability is partly due to mechanical outages of individual turbines; but, since there will typically be several hundred machines in such an array, the proportion on outage due to breakdown will he essentially constant. Most of the variation is due to variations in wind speed.

The capacity credit of wind power Many factors can influence the resulting variability of W and its effects. These include the choice of the rated speed of the wind turbines (usually related to the average wind speed at the site) and the extent to which the sites are dispersed (thus exploiting the lack of perfect correlation between hourly wind speeds at separated locations). The capacity implications of this variation are clearly also a function of the correlation between W and L. For a recent Irish study[7], correlation coefficients of 0.150.20 were recorded. However, in this paper we shall ignore correlations which, although interesting, would prematurely complicate the discussion with second order effects. To an electricity grid with installed conventional capacity Co and LOLP Po prescribed by utility policy, let us now add a maximum (or rated) wind power capacity Wr. Then the LOLP decreases to p , = ~(Ao + W < L),

(12)

where W is the available wind power and Zo increases to zw which is given by t~(zw) = p,. Two models of wind power variability will be presented below to examine p, and its implications for capacity credit. The first (Section 3.1) is grossly oversimplified, but will allow an easy introduction to the concepts. The second (Section 3.2) is much more realistic and will allow us to examine the effects on capacity credit of a wide range of system parameters.

3.1 A simple model of wind power variability Suppose simply that wind power is Normally distributed, i.e. that W ~ N(gw, aw2). Then A + W - L ~ N(ga + ~w - gL, O'a2 + O'w2 + O'L") (13)

393

ing zo. Thus C~ = G w - C o

-- (1 - r)-l{½ cr(zw 2 - 202) + aO(ZW - 2o)}

(16)

since aw -'- ao for small values of cr. This measure of capacity credit is essentially the one proposed by Melton [8] and followed by I)¢shmuida[5]. In Section 3.2 it is calculated under more general conditions than determined by the Normal model of wind power. However, under this latter assumption, C, can be obtained directly, without evaluating pw, by simply equating the two expressions for pw: (17)

• (Ao+ W < L ) = ~ ( A o + A c < L )

where Ac is the available conventional plant corresponding to Co Hence equating the arguments of the corresponding ~ functions, we obtain zw = (tz + tzw)(o,2 + a ~ ) - " m (18)

= (]Z + l£Ac)(or2 + O,Ac2) -(112).

Substituting Zw from (18) and ao=~r-½zocr = a - ½gcrhr from (11) into (16) we obtain (1 - r)C~ = {(/z + gw)(1 + (O'w/O')2)- " m - g} × [1 + ½(crzo2/g) x (1 + ttw/tt)(1 + (trw/~)2)-"12q.

(19)

(b) Equivalent firm capacity. A disadvantage of the previous measure is that it gives a definition of capacity credit in terms of an arbitrary currency. Hence it is also useful to compare wind power (and conventional plant) with idealised plant having zero forced outage rate. The "equivalent firm capacity" CF of a grid, which has conventional capacity Co and rated wind power capacity Wr, is again given by equating two expressions for Pw:

and pw = ~(Ao + W < L)= ~{Ao + CF < L}.

(20)

p~ = '~{(pA + g w - m.)/(o'~ 2 + o ' J + o-Le)'n}. (14)

There are various ways in which we can assess the capacity implications of pw. We shall discuss four of these. (a) Equivalent conventional capacity. Instead of adding wind power capacity Wr to Co, we can also obtain the new LOLP pw by adding some conventional capacity Cc(Wr) to Co. We call Cc the "equivalent conventional capacity" corresponding to W,. Clearly we would expect C, < W~ in general. Writing Q w = C,(W,) + Co as the total conventional plant corresponding to a grid with conventional capacity Co and rated windpower capacity W,, we have as an analogue to (9), C1w =(1-r)-l{IZL +½z2cr+zwaw},

(15)

where pw = ~(zw), and aw is as in (10), with zw replac-

We have, therefore, for a general wind power distribution (21)

C~=tr(zw-zo).

For the special case of Normally distributed W we know zw and may simplify to C~ = (g + gw){1 + ( o ' w / o t ) 2 }

-(112) -

It.

(22)

(c) Effective load carrying capability. An alternative measure, motivated like the "equivalent firm capacity" by the desire to measure credit in terms of a well-defined standard, is determined as follows: Given that 9(Ao + W < L) = Pw, by what constant value may we increase L before LOLP returns to po? That is, we.find the value of CL such that ~(Ao+ W < L + CL)= Po.

(23)

394

J. HASLETTand M. DI~ENDORF

In the Normal case being considered here we have simply that (see eqn 6) CL = # + #w -/z (I + (O'w/O')2) I/2.

Ignoring terms in (rc) 2 we find for small values of wind power penetration

(24)

(I - r) Csh, ---(~,.I~,)[I + ½ rzoclo~o- ½ ZoyC(~w/C~o)l

The algebra here does not generalise quite so easily to more realistic cases. It does however have the advantage that G. is not tied to a particular type of plant. CL is essentially the measure proposed by Kahn[4] and is utilised by the recent EPRI[9] study. (d) Capacity saving (CS). Perhaps the most direct measure of capacity credit is related to the total amount of conventional plant Cow which together with W , provides a grid with precisely the specified LOLP Pp. Therefore

(I - r)Cl. ---(~wh,)[1+ l rzo2Ch, _ ½ zo2yC(~wl~)]

Po = ~(Ao < L) = 9~(Ao w + W < L)

CFI~ = CLI# = (#wl#)[l - ~ Zo~yJ(#wl.)]

where we have introduced the coefficient of variation of wind power, yw = ~w/#w. Note that under all measures the capacity credit at low wind penetration is of the same order as the average power output, and that this is, to first order, independent of the variability of this output. Equation (27) also reemphasises that Cc and Cs are tied closely to the forced outage rate of their arbitrarily adopted basic unit, which is conventional power plant. As expected, it demonstrates that wind power has a higher capacity credit when compared with units with higher forced outage rates. In particular, when 100 per cent firm capacity or firm load is used as a standard, the capacity credit is always lower, as Table 1 demonstrates. The anomalous behavior of Cc and CF for large wind penetration is an indication of a basic inadequacy of the Normal model of wind power. For the argument of ~ in (14) may be written as Z w = Zo(1 + ~w/~){l + (Zoyw)2(~w/#)2} -"/2) for conventional plant Co, and it is easy to show that Zw increases, and hence pw decreases, with (#w/#), only when (/~w/#)< 1/(Zoyw). Thus, under the Normal model, increasing the wind penetration could ultimately increase the LOLP. This is a consequence of the fact that the Normal distribution attributes non-zero probabilities to negative values of W, and that these will be significant for large values of yw. Thus this section is certainly invalid for values of #w > (½zocr+ ao)yw. For the given example this corresponds to #w/#L = 0.8. Further, the value of yw used in Table 1 (yw = 0.3) was chosen to give small probability to negative W under the Normal model.

(25)

where the unknown Ao w is the available conventional capacity corresponding to CoTM. Then the "capacity saving" C~ = Co- Cow becomes (1 - r)Cs = # w + zoao- Zo(Ow2 - rclzw + Or02)1/2 = # w + # - ½Zo2re

- Zo[~C + ~ 2 _ rc(m~ - ~ ) + ¼(zorC)2] 1/2

(26)

using (10) and (11). Although the main purpose of this section is to introduce the basic concepts, with the aid of a grossly oversimplified model of W, it is of some relevance to make a few limited comparisons. Table 1 below compares the numerical values of the four measures of capacity credit for varying #w and for O'w taken arbitrarily to be 0.3ttw. The other parameters are r=O.l, c=0.2, GW, I~L = 1.0GW, trL = 0.2 GW, Zo= 3.1, as in the example of Section 2. These results are not changed significantly by reducing the unit size to 0.1 GW.

Table 1. Comparison of capacity credit measures

Wind Power Penetration

•04

(27)

Capacity Credit

(MW)

CC

CF

CL

CS

56

49

49

55

•09

108

94

95

107

.17

203

176

180

203

•26

281

243

256

287

.43

390

337

383

426

•86

469

404

594

644

I. 72

396

342

813

852

395

The capacity credit of wind power Table 2. Coefficientsof variation "rw of wind power NO. of sites

Rated Speed/Mean Speed 1.5

2.0

1

I.ii

1.62

2

.99

1.45

3

.84

1.23

4

.84

1.23

However, a recent study[7] found significantly higher values of yw, as shown in Table 2. For this reason a more realistic model of the distribution of wind power W is introduced in the next section. 3.2 A more realistic model of wind power variability The model we now propose for G(w), the distribution function of W, is G(w)

=

~(w <~W)

w<0j

=

I!

when - q exp{-

)[l(W/Wr)--l~2(W]Wr)2}

where A, =(Ir/(2mZ)){Vo(V,Vo)} and ~z= (A/(4mZ)) {(v,- re)Z}. Clearly ~(W<~ W,)= 1. Typically qz may be ignored. The distribution (28) is therefore entirely realistic in this case. In more general cases also it exhibits the required characteristics of a wind power distribution. Given the distribution G(w) we now compute p~ = • (Ao + W - L < 0). For notational simplicity we write as previously /~ = ~U'A0 -- /ff'L and tr 2 = 0 " 2 0 + O'L 2 . Then L - Ao ~ N(-/z, ~r2). Now, for a given value, say y, of L - Ao, ~( W < L - Ao) = G(y). Consequently, integrating over all values of L - Ao, we have

O<~w~ Wr

~

(28) where q, A, and ,~2 are dimensionless parameters which will typically be fitted to the disbribution. They may also be derived in an important special case, as we now show. The variability of wind power W derives from the variability in V, the wind speed. A common model[10] of the distribution of V is the Rayleigh model

• (V ~
p,, : (2~ro'2)- ° m

(33)

which by a change of integration becomes, since G(y) = 0 for y
(34)

I? if V ~
using (11). Now since ( 1 - q ) ~ 0, we have (1 - q)po <-Pw <~Po. This provides useful bounds for later examination. Equations (34) and (28) now yield

-]

pw =

W = I W'(v - Vo)(V,.- re) -~, if Vo~< V < v,, the rated speed | W , , if Vr <<- V < v2, the furling speed

[_0, if V >/v2.

It should be noted that W~ can represent the rated power of a single turbine exposed to wind speed V, or equivalently the maximum power of a set of identical turbines located at the same site. Then

f zo(l+v) (l-qexp{-A,[(Z/Zo)-l]/v J zO - ~2[(zlzo) - 1]2/v~})x e -(m~= dz/(21r) m + ~{Zo(1 + v)}

(30)

(35)

where v = Wr/~ measures the penetration of wind power into the grid; this is equal to #w/# divided by the capacity factor (typically 0.2-0.4) of the wind plant. A little algebra then yields pw = p o - ( q e"l~)[~{~o-'(Zo + A,/(~Zo)}

~ ( W = O)= ~(V~< Vo)+ ~ ( V > v2)

- ~{~-' (Zo+ ,h/(VZo)) + ZoV}]

(36)

(31)

= 1-qo+qz

where q, = exp {- (Tr/4)(vJm)2}, i = 0, 2. Further, 9 ( W <~w)= 9 { V <~Vo+ (WlWr)(V,-+ 9 ( V < v2),

G(y) exp {--~ (y + #)2/o'2} dy

(29)

where m is the mean wind speed. A simple model of the relationship between V and W is

"

f

where to2 = 1 + 2,~2/u2Zo2

vo)} 0<<- w < W,

= 1 + q2 - qo exp {- Al(w[ W,) - A2(w[W,)2} (32)

and /3 = {A,2/u2Zo2 _ 2h2/v2 + 2h,/v}/(2J).

(37)

J. HASLETTand M. DIESENDORF

396

This represents an exact expression for pw in this very realistic case. In many important cases (36) may be approximated by much simpler expressions as discussed below. Before doing so however we note from (11) that ~,= W,[zo(½zocr+ no)]-~ indicating that the penetration parameter is a fairly complex function of local and conventional plant characteristics. It will be increased by increasing W, or by decreasing/~L and trL2, as we should expect, but it will also be reduced by increasing Po and hence Zo and increased by reducing c or r. However if c is proportional to ~L, ~' is proportional to W,/I~L or to ~w/~L. LOLP: Small wind power penetration. A Taylor expansion of (36) in terms of small ~, yields, with a little algebra, Pw

=

Po - I/7oe-°/2)z°~{(q/(2A~)u~)exp

plications of this reduction for capacity credit through the concepts of "equivalent conventional capacity" and "equivalent firm capacity" (Section 3.1) and eqns (16) and (21). The other measures of Section 3.1 could also be used here of course; we do not pursue them simply because the algebra is less insightful. In the following section we shall discuss fairly fully some numerical applications. Some analytic examination of the limiting cases is possible however and we consider this here. In (16) put z, = Zo+ 8z and expand for small values of 8z. Thus for & ~ 1 (1 - r)Cc = [Cno+ Oto]8z where we ignore terms in (cr) 2. Now by definition pw - p o - 8z@(zo)

(~.12/4~2)

x [~{,~./(2A2) ~,=} - ~{,h/(2A2) m + (2,~=)'/~}]}.

(38)

Now the term in square brackets may readily be identified as follows. For given the distribution function (28), the mean wind power is ~w = fo w" {1 - G(w)} dw

(39)

= - Sz(2~r)-"/2> e -"/2)~,

8z ~ I,

(47)

and comparing (47) with (42) we find that &

=

Zo#~l~

=

~1~,

and after substituting for ~ from (11), (46) becomes C~ = (1 - r)-~/zw[1 + ½crzo2/l~].

=

(46)

(48)

W,(q(2¢r)'121(232)'/2) exp (A,214A2) (40)

[~{,~ ,/(2,~2'/2)} _ 4;{,h/(2~2)'/2 + (2,~),=}]. Thus (38) becomes P,, = Po - (l.twll~)(21r)-~/Z)zo e -°/:)'~

(41)

= po[1 - Zo21~Jlz]

(42)

where the last step has utilized the large Zoexpansion for

For small values of rc this is as in (27). A small wind power penetration expansion for the "equivalent firm capacity" is obtained from pw - Po = $(Zo + Cvlo')- 6(zo) ---(C~4o')~'(Zo) = (CFlcr(2~')~/2)e-"/2>~' Equating this to (47) yields

6(Zo):

C~ = ~. [I + o(aw/~)]

~(Zo) = (Zo(2~r)V2)-~ e-~/2)~{l - Zo-2 + 0(~7o-4)}. (43)

Thus the impact of a small wind penetration on the LOLP is determined by the average wind power output. LOLP: Large wind power penetration. As v ~ , :P{Zo+a~/Zo)hO + ~,Zo}~0 in (36). Performing a Taylor expansion of the remaining ~ function we obtain pw = po(1 - q) - (q,~,/~')LOo- (Zo(21r)'/2)-~ e -(~/2)'d} (44)

and utilising (43) we obtain the further approximation p. = po{l - q(l + ,h/Zo2)}.

(45)

This limiting case emphasises that capacity credit cannot continue to increase with W, but rather is asymptotic to a value determined by the probability q of non-zero wind power. The parameter A2 does not appear in this result. Capacity credit. The foregoing expressions quantify the reduction in LOLP as a consequence of adding wind power onto an existing grid plan. We examine the im-

(49)

(50)

also as (27). Equations (48) and (50) emphasise that the capacity credit for small windpower penetration is slightly greater than (in the case of Cc) or equal to (in the case of CF) the average windpower output, confirming the results derived in Section 3.1 for much simpler models of wind power variability. The similarity of these results to those obtained in that section is not coincidental. It may in fact be shown that the expansion (27) is valid for any distribution of W[I1]. For large penetration we may write p= = lim pw = po(1-q). Then Cc(~) and CF(~) corresponding to p= may be obtained from (16) and (21) in terms of z~, where @(z®)= p®. These represent upper bounds to the possible capacity credit. For more useful approximations we may write zw = z = - 8z, yielding from (16) that Cc(v) = Cc(~) - 8z[crzo + a= + (crzo)2/2a=] (51)

and CF0,) = CF(~) - 8z~ and a® corresponds to ao in (10).

(52)

The capacity credit of wind power

397

16o]

Now expanding the definition of pw about z~ and equating to (43) we obtain

~40I

~z

=

(q,hh, zfl){exp - ½(Zo2 - z~2)}.

(53)

~//fEcc _

/

. . . . . .

. .- / /

w~

• 0.3

These then provide the basis for approximations to the capacity credit for large wind penetrations. 3.3 Applications The foregoing provides the basic theoretical development for the capacity credit of windpower and its approximate computation. In the following we show how this may be applied to provide approximate numerical results in three cases: (a) a system consisting of identical wind turbines experiencing the same wind regime as modelled by (29) and (30), (b) a system involving the use of dedicated storage to even out the fluctuation in wind power, and (c) a conventional generating unit. (a) Single site. Consider the model described in (29) and (30). We now have expressions for q, A, and A2 in terms of the characteristics of the wind turbines and the wind regime. In Table 3 and Fig. I below the relationship between ECC and ECF and various system parameters is shown. The behaviour of the measures of capacity credit with increasing penetration is as foreshadowed in Section 3.2; it is initially roughly equal to the mean wind power output and asymptotes to a final value about 10 per cent or so greater than the credit at about 50 per cent penetration by energy. EFC is seen, as already explained, to be less than ECC. Their difference is roughly (1- 3r/2) over the parameter range considered. Changes in the parameters of the conventional system increase or decrease the credit of wind power according as they decrease or increase respectively the reliability of the system. Thus increasing r and po increases the unreliability of the system; decreasing c, and hence increasing the number of conventional generating units, increases the reliability of this system. Changes in the wind system parameters increase the credit with decreasing coefficient of variation of wind; the values of 3'w for the three wind systems are 1.03 (basic), 1.51 (v~ = 2m) and 0.69 (vo = 0.3v~). The approximate formulae in (27) provide reasonably accurate estimates of capacity credit up to about a 2 per cent penetration by energy for the parameters considered, and well beyond this (up to about 5 per cent) when the coefficient of variation of wind is low. The expansions (51) and (52) seem adequate for these parameters down to a penetration of about 60 per cent, and slightly further in the case of high coefficient of variation. This is useful for this would certainly represent the highest penetration ever likely to be considered in practice, as opposed to the purely theoretical interpretation that must be placed on Cp(~) and C~(~). (b) Dedicated storage. The results of Section 3.2 apply not only to the situation in (a) above, but to any wind power distribution which may be adequately modelled as in (28). Further our analysis has shown that the parameters of real importance for large penetration are those which control the distribution for low values of w. This then leads to the possibility of using (28) as a means

|

.~ 8oF

[ ,',/

o

I:I//

[

ECC

/ ,/ :/

o

~----

/! / "

200

o

4~o

. . . . . . . . . . . . . ~/t4. -O5 EFC ° ' "

660

~

,800

' 1200

1400

Mean wind po~er /zw,MW

Fig. 1. Capacitycredit as a functionof penetrationfor two different starting speeds and two measures of credit. System parameters with the exception of vo]v, are defined in footnote(2) to Table 3. for studying, for example, the impact on capacity credit of the use of dedicated storage, as described by Serensen[12], who has published power duration curves for storages of various size. Clearly an appropriate model for such power duration curves, and one which could be straightforwardly included in the analysis of Section 3.2, is as follows G,(w) =

Ii

when - q , exp{-Al,(W/W~)-A,2(w/W,)Z}, q2exp{- A2~(w/W,)-AE2(W/W,)2},

w<0 1 0~
w>-W,.] (54) If we wish to concentrate on estimating meaningful upper limits to the capacity credit in such circumstances, then it is sufficient to concentrate on the region 0 < w < /~w.Further for the purpose of demonstrating the procedure we may use the simpler model t~, ( w )

-~ l - q exp {-A(w/W,)}, for small

w.

(55) Serensen's Fig. 8 allows estimates of (1- q) directly as the availability of non-zero wind power, and a through (1-qm), the availability of the mean wind power as 1 -q~

=

(56).

1-qexp{-A(~JWr)]

whence ,~ = ( W r l ~ w )

In {qo/q,.,. )}.

(57)

Table 4 below gives the values of q and ,~ as obtained by crude fitting to Sorensen's data, together with the value of capacity credit at the asymptote and at 50 per cent

83.5

5,120

17.3

69.9

69.4

68.3

66.3

62.6

55.7

44.4

29.8

5.3

70.3

69.8

68.9

67.0

63.4

57.0

46.2

31.6

18.7

i0.I

64.4

64.2

63.8

61.2

58.0

52.1

42.3

29.0

17.1

9.3

4.8

EFC

i00.i

99.2

97.5

94.1

87.7

76.4

58.3

37.2

20.9

ii.i

5.7

ECC

.38 for wind system.

85.3

84.9

82.7

79.9

74.5

65.0

49.7

31.7

17.9

9.5

4.9

EFC

po =. 01

aL=2~L;

Capacity

Capacity

(4)

factor

factor

.58

.20

64.2

63.8

62.9

61.2

58.0

52.1

42.3

29.0

17.1

9.3

4.8

EFC

50.1

49.9

49.3

48.3

46.2

42.5

36.2

27.3

17.8

10.2

5.5

ECC

42.0

41.8

41.4

40.5

38.8

35.7

30.4

22.9

14.9

8.6

4.6

EFC

Vr/m=2.0 (3)

c=.l~L , r=.10, po=.001,

In other cases one parameter

at a time is changed.

4.9 9.7

125.1

123.2

119.6

113.0

101.8

84.3

59.0

34.8

18.7

EFC

for large ~w"

150.2

147.9

143.6

135.6

122.0

100.9

70.4

41.4

22.3

11.5

5.9

ECC

Vo/Vr=0.3 (4)

Vr/m=l.5 , Vo/Vr=.5 , and hence capacity

clearly not all of ~w is useful

74.2

73.7

72.7

70.7

67.0

60.2

48.8

33.4

19.7

10.7

5.5

ECC

c=50 ( M W )

Thus the credit associated with a mean wind oower XHw in a system with

~L=IGW,

mean load x~ L is x times the tabulated value.

factor

Basic system parameters

(3)

(2)

(1)

4.8

9.3

ECC

r=. 05

~w is the average energy production of the wind turbines;

82.9

NOTES

81.7

1,280

2,560

66.5

160

74.7

52.9

80

79.3

35.4

20

40

320

20.6

i0

640

5.7

ii.i

5

ECC

~w (MW)

EFC

Basic System (2)

Energy (i) Penetration

Table 3. Capacity credit for various system parameters

The capacity credit of wind power

399

Table 4. Effect of dedicated storage on capacity credit Storage Size

(hrs)

(i)

1 (2)

q

ECC (~) (3)

EFC (~) (3)

EFC (50%) (4

ECC (50),4)t

(sw)

(MW)

(MW)

(MW)

0

.69

1.64

95.3

79.7

84.4

70.6

3

.71

1.09

100.5

84.0

92.2

77.1

I0

.74

.48

109.0

91.0

104.6

87.4

24

.80

.35

129.3

107.8

124.7

104.1

200

.92

.ii

197.7

164.0

193.9

160.9

NOTES (1)

Number of hours for which average power may be delivered.

(2)

Roughly fitted,

(3)

The credit in MW for a system with parameters

as described,

to Figure 8 of S~rensen's paper. as in the basic system of Table 3 for indefinitely

large

wind penetration. (4)

The credit for a 50% penetration by energy.

penetration by energy, as calculated using (51) and (52). It can be seen that the credit increase declines with increasing storage size. These figures should however be regarded as indicative only, and the implementation of a model such as (54) awaits a more thorough study. (c) Conventional plant. To a certain extent the model (28) may be applied to a conventional plant of different size and forced outage rate from the basic plant used heretofore. Consider a plant with two states producing power Pr with probability q and power zero with probability (1- q). This corresponds to (28) with A, = he = 0. From (36) we now have that p .... the LOLP with this new conventional plant added, is poo° = po(l - q ) + q4,{Zo(1 + ~)}

where i, now represents the penetration of additional conventional plant into the grid. For large values of z,, p¢o, tends to the same limit as pw. In other words, the same upper limit applies to capacity credit (as measured by "equivalent conventional" or "firm capacity") with equal probability of zero power output of conventional and wind plants.

Table 5 below shows the value of capacity credit of such plant for increasing q and v, other parameters being as previously. These figures provide yet another means of interpreting capacity credit of wind power, by cross reference to Table 3. In some ways it is a more satisfactory extension of the "equivalent firm capacity" measure. For example, it will be noted that if the additional unit is of size 100 MW and forced outage rate 0.1, it is indistinguishable from the generating units already in the system. The fact that ECC is not however precisely equal to 100 MW is a reflection of a slight inconsistency in methodology. In Section 3.1 the distribution of the available plant (basic and additional) was assumed to be Normal; in the above only the former is assumed to be Normally distributed. It seems difficult to treat this measure other than numerically (see [13]). 4. DISCUSSION This paper has been concerned with the complex question of capacity credit of wind power in combination with conventional units on an electricity grid. Its objectire has been to provide an analytic treatment of the

Table 5. Capacity credit for added conventional plant, MW Forced Outage Rate

Rated Power (MW)

.2 ECC

EFC

ECC

.02

.05

.I EFC

ECC

EFC

ECC

EFC

20

18.4

15.5

21.0

17.7

22.3

18.

23.1

19.5

50

43.9

37.0

51.2

43.2

55.2

46.5

57.7

48.6

200

117.3

98.4

157.7

131.9

189.8

158.4

218.4

182.0

NOTE:

other system parameters

are as in the basic system of table 3.

400

J. HASLETTand M. DIESENDORF

subject in order to focus on the important concepts suitable for general discussion. As such, we have been forced to use some relatively simple mathematical models of the grid, at the loss (it must be admitted) of some numerical accuracy. The main assumptions made are: (i) that the conventional plant is composed of a large enough number of identical machines for the Central Limit Theorem to allow an adequate Normal approximation to the distribution of the available conventional plant; (ii) that the demand for electricity is Normally distributed; and (iii) that wind power is statistically independent of demand. The latter is probably the weakest point conceptually, although its numerical importance should not be exaggerated. For even where there is an appreciable correlation between wind and load on an annual basis, for the purposes of computation only the large values of load are really crucial, and the correlation between these loads and their associated wind power will typically be small. Thus, for example, in the Irish study referred to previously, the correlation coefficient, on an annual basis, was 0.15, but the correlation between the top 100 values of L and their associated values of W was in fact -0.03, i.e. essentially zero. Thus, provided the computations proceed on the basis of the statistical properties of these conditioned variables, the correlation assumption is not crucial. A more general analytic treatment is readily available within the simpler analysis of Section 3.1. Indeed, the extension is almost trivial as only the variance of (A + W - L ) is altered by correlation, and trivially so, and the resulting analysis is also unchanged. No simple mechanism has yet been found for including correlation within the analysis of Section 3.2. The requirement that the conventional plants be identical is a necessary consequence of adopting a single criterion, LOLP, as the basis for specifying the composition of the conventional deck. In practice, of course, economic considerations will also be present, and will dictate a mix of plant types. These considerations are too complex to raise here, and so we cannot expect the present models to give a good account of effect of the size c of conventional units on the capacity credit of wind power. The assumption that the available plant follows a Normal distribution is not a good approximation in practice, and is the most important single source of numerical errors in practical applications of the above. For a limited comparison with a numerical evaluation of the above, the reader is referred to [13]. The main findings of the paper, within the foregoing assumptions, are as follows: (i) The use of very simple models of the variation of load, wind power and plant availability allows helpful comparisons to be made between various alternative measures of "capacity credit". (ii) For small penetrations into the grid, the capacity credit of wind power is approximately equal to the average wind power. (iii) There is a definite upper limit to the credit that may be earned, no matter how large the wind system, as long as there is a non-zero probability of no wind power.

(iv) A realistic probability distribution of the power output of a system of aerogenerators has been put forward. Within the limits of the assumptions above, certain of the formulae presented for determining capacity credit, particularly those for large and small wind systems, are very easy to compute given this wind power distribution. The distribution may be computed to take account of effects such as the use of dedicated storage. Thus the expressions provided should be useful and usable as "first-cut" tools. Finally, it should be emphasised that this paper does not deal with the wider, and ultimately more important, question of the economic value of the capacity impact of wind power. Recent work[ll, 14] on the so called "reoptimised mix" indicates that, at least in all-thermal grids, the credit above, computed in MW terms, should be given a value no less than that of highly capital intensive base generating plant. This capacity value is, in turn, a small but by no means negligible part of the overall economic value of wind power to the electricity grid.

Acknowledgements--Thework of one of us (MOD) is supported by a grant from the National Energy Research, Development and Demonstration Council (NERDDC). We acknowledge the stimulation and computing assistance of Brian Martin. NOMENCLATURE A, Ao, Ao Aw, Aow available power from conventional plant (MW) is different circumstances ao, a~, a® expressions defined in (I0), following (15), and following (52) resp. algebraic expression defined in (37) c, ci capacity of individual conventional unit, MW Co, CI, Cm, Cow total installed conventional capacity, MW Cs capacity saving CL effective load carring capability (ELCC) Measures of CF equivalent firm capacity (EFC) capacity Cc equivalent conventional capacity (ECC) credit, MW Yw coefficientof variation of wind power 8z small change in z ~(z) the probability that a standard normal random deviate exceeds z G(w) probability distribution function for wind power Gs(w),Gs(w) probability distribution for wind power output from system incorporating dedicated storage, and an approximation to it L electricity load at a given time, MW Lp peak load, MW AI,A2 parameters of G(w) All, Ai2,,t21,A22 parameters of GAw) m mean wind speed at a site, msec-l # defined following (10) ~A, ~L, #W mean of available conventional plant, load and wind power, (resp.), MW v expression measuring penetration of wind power, defined following (35) N Normal distribution n, no numbers of conventional generating units probability Pr rated power of a conventional unit, MW P, Po, pw LOLP in general, for conventional grid only, for wind-augumented conventional grid, and for an infinitely augmented conventional grid, (resp.)

The capacity credit of wind power Pco, LOLP for basic grid augmented by a further conventional unit q,, the probability that the wind power exceeds /zw q the probability that wind power is non-zero ql, q2 parameters of model G(s) r, ri forced outage rate of conventional generating units 0.w2, 0.a2, 0.L2 variance, for wind power, for available plant and for load, (resp.), MW2 0.2 variance defined following (10) v, V wind speeds, msec -~ Vo,V, v2 cut-in, rated and furling speeds, (resp.), msec -| ~o defined following (36) W, w wind power, MW W, maximum wind power, MW z, Zo,z,, zs quantiles of • such that O(z)= p, Po, pw, (resp.)

REFERENCES

1. H. M. Bae and M. D. Devine, Optimisation models for the economic design of wind power systems. Solar Energy 20, 469-48 i (1978). 2. Danish Academy of Technical Sciences Working Group-Wind Power in the Electricity Supply System, Copenhagen 1977. 3. B. S~rensen, Renewable Energy. Academic Press, London

(1980). 4. E. Kahn, Reliability of wind power for dispersed sites: a preliminary assessment. Lawrence Berkeley Lab. Energy and Environmental Div., LBL-6889 (1978).

401

5. R. G. Deshmukh, A probabilistic study of wind electric conversion systems from the point of view of reliability and capacity credit. Ph.D. Thesis, Oklahoma State Univ. (May 1979). 6. L. L. Garver, The electric utilities. In Handbook of Operations Research (Edited by J. J. Moder and S. E. EImaghraby), Vol. 2, Chap. II.6. Van Nostrand, New York (1978). 7. T. G. Gibbons, J. Haslett, E. Kelledy and M. O'Rathaille, The potential contribution of wind power to the Irish electricity grid. Rep. 7902 of Statistics and Operations Research Laboratory, Trinity College, Dublin (1979). 8. W. D. Melton, Loss of load probability and capacity credit calculations for WECS. Proc. 3rd Biennial Conf. and Workshop on Wind Energy Conversion Systems, CONF-770921, Washington, DC, Vol. 2, pp. 728-741 (Sept. 1977). 9. EPRI, Requirements assessment of wind power plants in electric utility systems. Electric Power Research Institute, Summary Rep. ER-978-SY, Vol. 1 (Jan. 1979). 10. R. B. Corotis, A. B. Sigl and J. Klein, Probability models of wind velocity and persistence. Solar Energy 20, 483-493 (1978). 11. J. Haslett, On a general theory for modelling the interaction between wind energy conversion systems and the electricity grid. J. Opl. Res. J. Submitted. 12. B. Sorensen, On the fluctuating power generation of large wind energy converters, with and without storage. Solar Energy 20, 321-331 (1978). 13. B. Martin and M. O. Diesendorf, The capacity credit of wind power: a numerical model. 3rd Int. Syrup. Wind Energy Systems, Copenhagen, Denmark, Paper L3, pp. 555-564. BHRA Fluid Engineering, Cranfield, Bedford, England, 2629 Aug. 1980. 14. H. Darvition, Wind power and electric utilities--a review of the problems and prospects. Wind Engng 2, 234-255 (1978).