22.812 Nuclear Energy Economics and Policy Analysis S’04
Classnot e – Marc h 1 , 2004
Th e Levelize d Cos t o f Productio n an d th e Annua l Carryin g Charg e Factor
First , define levelized cash flows:
1. Discret e cas h flows
Consider the non-uniform cash flow series:
A j-1 A j A j+1
We can define an ‘equivalent levelized’ cash flow, A L , such that the uniform series PW is equal to the PW of the actual series:
N N
A L (P/F,i,n) A n (P/F,i,n)
n 1 n 1
N
A n (P/F,i,n)
N
A L n 1
(P/F,i,n)
n 1
2. Continuou s cas h flo w rate
A(t)
We obtain, by analogy,
T
o
A e rt dt
A
0
L T
e rt dt
0
For the special case of an exponential increase in A
A (t) A o e
yt
T
o
A e (y r)t dt
r 1 e (y r)T ˘
A 0 A
L T
e rt dt
0
o r y 1 e rT
And expanding the exponentials as Taylor series and retaining terms through second order, yielding, to first order,
A L A o
1 ( r y ) T ......
2
1 rT ....
2
1 yT
2
Levelize d Uni t Cos t o f 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.)
Exampl e – manufacturin g facility
Consider a factory with initial investment cost I o at t=0, which operates for N years after which it is salvaged at I N .
Suppose that during this period the factory produces Q j units per year at an annual operating cost of M j dollars per year.
Q j
M j
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?
Cas e I : N o Taxes
Write the levelized unit cost, c, as the sum of operating and capital components:
c = c M + c I
1. Operating cost component, c m
N N
c M Q j (P/F,i, j) M j (P/F,i.j)
j 1 j 1
N
M j (P/F, i.j)
c
j 1 M N
Q j (P/F,i, j)
j 1
2. Capital cost component, c I
c I Q j
I o
I N (1 i) N
N
c I Q j (P/F,i, j) (1)
j 1
Define : Average (levelized) production rate Q L
N N
Q L (P/F,i, j) Q j (P/F,i, j)
j 1 j 1
N
Q j (P/F,i, j)
Q j 1 L (P/A, i, N)
and substituting for Q L in (1)
1
I
o
N
c I
I (P/F,i,N)
Q L (P/A,i,N)
1
Q
[I o (A/P,i, N) I N (A/F,i,N)]
L
i.e. ,
levelizedunit cost 1 I capital recovery factor I
sinking fund fa
levelized production rate o N
Cas e II : Wit h Taxes
Q j c
T j
M j
As before, write c = c m + c I
Next, transform the cash flow problem into an equivalent tax-implicit problem
D j
M j (1- )
I o
Q j (c I +c M )(1- )
I N
And, decomposing into capital and operating components,
C I Q j (1- )
D j
I N
c M Q j (1- )
+
M j (1- )
I o
Then solve separately for c I and c M .
a. c M
N N
( 1 ) c M Q j ( P / F , x , j ) ( 1 ) M j ( P / F , x , j )
j 1 j 1
N
M j ( P / F , x , j )
c
j 1
M N
Q j ( P / F , x , j )
j 1
b. c I
N N
( 1 ) c I Q j ( P / F , x , j ) I o I N ( P / F , x , N ) D j ( P / F , x , j )
j 1 j 1
For the case of straight line depreciation:
I o I N
D j N
and
I ( I o I N ) ˘
1 o I N ( P / F , x , N ) N ( P / A , x , N )
c I 1
N
Q j ( P / F , x , j )
j 1
(2 )
a s b efore , d efin e a l e vel i zed p rod uction rate , Q L
N N
Q j ( P / F , x , j ) Q j ( P / F , x , j )
Q j 1 j 1
L N
( P / F , x , j )
j 1
( P / A , x , N )
And substituting in (2) above
I
c 1 I ( A / P , x , N ) I ( A / F , x , N ) I o I N ˘
( 1 ) Q L
o N
N
I o 1 I N I N ˘
Q 1 ( A / P , x , N ) N 1 I I ( A / F , x , N ) (3)
L
o o
I o
Q
c I
L
where , the term in square brackets, is the annua l carryin g charg e factor (with units of yr -1 )
Notes
1. I o is the PW of the initial investment a t th e star t o f operation .
2. In a tax-free environment ( =0), the annual carrying charge factor reduces to the capital recovery factor, adjusted for NSV.
3. In the limit of large N (N )
x ( 1 x ) N
( A / P , x , N ) ( 1 x ) N 1 x
( A / F , x , N ) x 0
( 1 x ) N 1
x
1
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.