22.812
Nuclear Energy Economics and Policy Analysis
S’04
Classnote – March 1, 2004
The Levelized Cost of Production and the Annual Carrying Charge Factor
First, define levelized cash flows:
1. Discrete cash flows
Consider the non-uniform cash flow series:
Aj-1
Aj
Aj+1
We can define an ‘equivalent levelized’ cash flow, AL, such that the uniform
series PW is equal to the PW of the actual series:
N
N
ÂA
L
n=1
(P/F,i,n) = Â An (P/F,i,n)
n=1
N
 A (P/F,i,n)
n
AL =
n =1
N
 (P/F,i,n)
n =1
1
2. Continuous cash flow rate
A(t)
We obtain, by analogy,
T
- rt
ÚA e
dt
o
AL =
0
T
Úe
-rt
dt
0
For the special case of an exponential increase in A
A(t) = Aoeyt
T
(y -r)t
ÚA e
o
AL =
dt
0
= Ao
T
Úe
-rt
dt
r È 1- e(y- r)T ˘
r - y ÍÎ 1 - e- rT ˙˚
0
And expanding the exponentials as Taylor series and retaining terms through
second order, yielding, to first order,
2
(r - y)
T + ......
2
rT
1+ ....
2
ª 1 + yT
2
AL 1 ª
Ao
Levelized Unit Cost of Product
The lifetime levelized cost, the constant cost that is equivalent in a present worth
sense to the relevant time-varying cost, is a useful benchmark for comparisons of
facilities which might otherwise be difficult to compare (e.g., windmills versus gas
turbines.)
Example – manufacturing facility
Consider a factory with initial investment cost Io at t=0, which operates for N
years after which it is salvaged at IN.
Suppose that during this period the factory produces Qj units per year at an
annual operating cost of Mj dollars per year.
Qj
Mj
What is the levelized cost of a unit of product – i.e., the uniform cost which, if
recovered on every unit produced, will provide lifetime revenues just sufficient to
cover all capital and operating costs?
Case I: No Taxes
Write the levelized unit cost, c, as the sum of operating and capital
components:
3
c = cM + cI
1. Operating cost component, cm
N
N
j=1
j=1
 cMQ j(P/F,i, j) =  Mj (P/F,i.j)
N
 M (P/F,i.j)
j
cM =
j=1
N
 Q (P/F,i, j)
j
j=1
2. Capital cost component, cI
cIQj
4
Io -
N
IN
=
 cIQ j (P/F,i, j)
(1 + i)N j=1
(1)
Define : Average (levelized) productionrate QL
N
N
 QL (P/F,i,j) =
j=1
 Q (P/F,i, j)
j
j=1
N
 Q (P/F,i, j)
j
QL =
j=1
(P/A,i,N)
andsubstituting for QL in(1)
cI =
1
I - I (P/F,i,N)
QL (P/A,i,N) o N
=
1
[I (A/P,i,N) - IN (A/F,i,N)]
QL o
[
]
i.e.,
levelizedunit cost =
1
I ¥ capital recoveryfactor - IN ¥ sinking fund factor
levelized productionrate o
[
Case II: With Taxes
Qjc
Tj
Mj
5
]
As before, write c = cm + cI
Next, transform the cash flow problem into an equivalent tax-implicit problem
Qj(cI+cM)(1-t)
IN
tDj
Mj(1-t)
Io
And, decomposing into capital and operating components,
CIQj(1-t)
cMQj(1-t)
tDj
IN
+
Mj(1-t)
Io
Then solve separately for cI and cM.
6
a. cM
N
N
j =1
j =1
(1- t )Â cMQ j (P / F,x, j ) = (1 - t )Â Mj (P / F, x, j )
N
 M (P / F,x, j )
j
cM =
j =1
N
 Q (P / F, x, j )
j
j =1
b. cI
N
N
(1- t ) cIQ j (P / F,x, j ) = Io - IN (P / F, x,N) - t  Dj (P / F,x, j )
j =1
j =1
For the case of straight line depreciation:
Dj =
Io - IN
N
and
È
˘
(Io - IN )
I
I
(P
/
F,
x,N)
t
(P
/
A,x,
N)
˙
1 Ío N
N
cI =
N
˙
1- t Í
Q
(P
/
F,x,
j
)
 j
Í
˙
Î
j =1
˚
as before,define a levelized production rate,QL
N
QL =
N
 Q j (P / F, x, j )
j =1
N
 (P / F, x, j )
 Q (P / F, x, j )
j
=
j =1
(P / A,x,N)
j =1
And substituting in (2) above
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(2)
cI =
=
cI =
1
(1- t )QL
Io
QL
È
I -I ˘
I (A / P, x,N) - IN (A / F, x,N) - t o N
ÍÎ o
N ˙˚
È 1 Ï
¸˘
t ÊÁ IN ˆ˜ IN
(A
/
P,
x,N)
1
(A
/
F,
x,N)
Ì
˝˙
Í 1- t
N Ë Io ¯ Io
Ó
˛˚
Î
(3)
Io
f
QL
where f, the term in square brackets, is the annual carrying charge factor (with
units of yr-1)
Notes
1. Io is the PW of the initial investment at the start of operation.
2. In a tax-free environment (t=0), the annual carrying charge factor f reduces
to the capital recovery factor, adjusted for NSV.
3. In the limit of large N (N‡ •)
(A / P, x,N) =
x(1 + x)N
Æx
(1+ x)N - 1
(A / F, x,N) =
x
Æ0
(1+ x)N - 1
f Æ f• =
x
1- t
This is a good approximation for large N.
4. The form of the annual capital charge factor in equation (3) applies to the
case of straight-line depreciation. Equivalent expressions can be derived for
other depreciation schedules.
8