PART II: ECONOMIC MODELING AND IMPLICATIONS OF THE DECISION ON
9. BASIC EQUATIONS
The basic equation used throughout the computations is that of the plant’s net annual cash flow.
NCF(t) = REV(t) – OM(t) – FL(t) - DD(t) – RETDebt(t) – RETEq(t) (1) Where
NCF(t) – Net annual cash flow in year t, [M$/year]
REV(t) – Annual revenues obtained from sale of electricity in year t, [M$/year]
OM(t) – Annual operations and maintenance (O&M) expenditures in year t, excluding capital additions [M$/year]
FL(t) – Annual fuel costs including waste management charge in year t, [M$/year]
DD(t) – Decommissioning fund outlay in year t, [M$/year]
RETDebt(t) – Annual payments for the return of and on the remaining plant debt in year t, [M$/year]
RETEq (t) – Annual payments for the return of and on the remaining equity in the plant in year t, [M$/year]
Equation (1) denotes that the plant’s net annual cash flow in year t is equal to the revenues coming in from the sale of electricity to the grid minus all the costs incurred during plant operations, and minus expenses paid to provide a return on and of the remaining plant investment.
All costs incurred in year t are discounted to the start of commercial operation of the power plant using the discount rate, or the average cost of money r [%/year], based of the following standard equation
r = d × id + (1-d) × ie (2) Where
r – Annual discount rate or the average cost of money (assumed here to be the same) [%/year]
id – interest rate on plant debt [%/year]
ie – Return on equity invested in the plant [%/year]
d – Debt fraction of total plant investment 1-d – Equity fraction of total plant investment
We assume that all investments made in the plant include combinations of debt and equity only. We further assume that the debt/equity investments ratios are kept constant throughout
the entire plant life span and equally apply to both the initial investment and all annual capital additions expenditures treated as investments.
It is now possible to derive the basic equation for the discounted cumulative net cash flow in the year t during the plant operations period, as
t
CUMCF(t) = - I(t=0) +
Σ
{NCF(k) × (1+r)-K} (3)k=1
Where
CUMCF(t) – Discounted cumulative net cash flow by year t, [M$]
I(t) – Remaining un-depreciated investment, yet to be recovered by the year t, [M$]
k – Running index from first plant operation year to the year t
By definition, CUMCF(t = 0) = - I(t = 0), implying that the initial value of the discounted cumulative net cash flow at the beginning of plant operations period is the transition discounted cumulative cash flow from the end of the previous construction period representing a negative-value accumulated construction investment of – I(t = 0).
– I(t = 0) represents the un-depreciated investment in the plant at the commercial operation start point – sum cumulative of all plant investments during the construction period, including base construction cost, contingency, owners costs and accumulated interest during construction (IDC). At the end of the construction period the net cash flow is negative, since only construction related e×penditures have occurred, with no offsetting revenues received.
Hence the minus sign in front of the I(t=0) term. This negative cash flow is being offset during the operations period by the accumulation of discounted annual net cash flows from the start point of commercial operation until the year t. Not all annual cash flows might turn out positive, and some might add to the large negative cash flow overhang accumulated during the construction period. Yet, year-by-year, assuming mostly positive net annual cash flows, the large initial negative cash overhang is being reduced, and in most cases, after fifteen to twenty five years depending on the specific plant cost data, the cumulative cash flow turns positive and the plant provides increasing larger net positive cash flows (profits).
The issue of extended operation versus early shutdown relates to the question of has the discounted cumulative net cash flow at the decision point turned positive or not. This issue is covered in more detail later.
It is now possible to introduce the recursive element into the basic equations, as it relates to the net remaining un-depreciated plant investment
I(t) = I(t-1) × {1 – (N-t+1) -1} + CAPADD(t) (4) Where
I(t-1) – Remaining un-depreciated plant investment in year (t-1) [M$]
CAPADD(t) –Annual capital additions to the plant treated as investment [M$/year]
N – Nominal expected plant lifetime (original design plant life) [years]
Equation (4) denotes the change in remaining un-depreciated plant investment between the year (t-1) and the year t. A portion of the existing investment at the end of the (t-1) period is
depreciated, assuming a straight-line depreciation procedure. The remaining lifetime over which the investment I(t-1) can be straight-line depreciated is N – (t-1) or N – t + 1 [Years] . The portion of the investment I(t-1) depreciated during year t is then I(t-1) × (N – t + 1) -1. Thus the remaining un-depreciated investment carried over from period (t-1) to the end of period t, is then {I(t-1) - I(t-1) × (N – t + 1) -1}, which upon rearrangement, yields the first term on the right hand side of equation (4).
The complete equation (4) indicates that two plant investment-related processes happen in year t:
1. A portion of the remaining investment from the previous period is straight-line-depreciated, and
2. A new annual capital addition, if it has occurred and if it has been recognized by the regulatory agencies as an investment increment, is capitalized and added to the books, rather than being treated as an O&M cost item and expensed in the year it occurs.
Thus the remaining investment at the end of period t, is the remaining investment from the previous period after deducting depreciation during period t, plus the capital additions investment increment which might have incurred during the year t. This process is depicted graphically in Figure 3. We should mention here that in some years during the plant operations period no capital additions might have occurred, or those that have occurred might be relatively small and are directly expensed rather than capitalized. Capital additions that qualify as investments in a NPP case could include large projects that can not be expensed in one year, such as the construction of a dry cask spent fuel storage facility on site, the construction (and equipment) of a new computer building, a new training center and simulator addition, steam generator replacement, plant primary pumps replacements and up-rates programmes, up-rating the turbine generator for higher plant output, core shroud replacement in a BWR, or CANDU reactor re-tubing. In all cases the decision on whether to treat capital addition expenditure as an investment or as an annual O&M expense will depend on the tax treatment for either option afforded by the regulatory agencies in that year. Annual capital additions not treated as investments will be considered here as a part of the fixed O&M annual expenditure in that year, as discussed later.
Equation (4) can be rearranged to yield
I(t) = I(t-1) × {(N-t) / (N-t+1)} + CAPADD(t) (5) Equation (5) is the basic recursive equation of this analysis, and it is graphically presented in Figure 3. As indicated above this treatment equally applies to all types of power plants.
It is now possible to simplify equation (1) based on elements of the above discussion, by re-defining the last two components of that equation. Thus
RETDebt(t) = I(t-1) × d × (N – t + 1) -1 + I(t-1) × d × id
RETEq(t) = I(t-1) × (1-d) × (N – t + 1) -1 + I(t-1) × (1-d) × ie
Year (t - 1) Year (t) Year (t+1)
Depreciable Investment [M$]
Capital Additions
0.0
Capital Additions
Capital Additions Investment
(t-2)
Investment (t-1)
Investment (t)
Year
Depreciation Depreciation Depreciation
Fig. 3 Annual change in plant depreciable investment*
* Axes and bars not drawn to scale
The annual return in the year t on and of the debt fraction of the total investment, and the annual return on and of the equity fraction can each be expressed in a two terms equation. The first term represents the amount straight-line-depreciated in the year t, as discussed above, i.e.
the return of the investment expended on the plant. The second term represents the annual interest payment on the plant debt and the return on plant investment (ROI) during the year t on the yet un-depreciated plant investment. The second term thus represents the overall return on the investment in the plant, including both its debt and equity fractions. The above two equations can be re-arranged as follows:
RETDebt (t) = I(t-1) × {d × (N – t + 1) -1 + d × id} RETEq (t) = I(t-1) × {(1-d) × (N – t + 1) -1 + (1-d) × ie} Combining the above two equations and rearranging we get
RETDebt(t) + RETEq(t) = I(t-1) × (N – t + 1)-1 × {d + (1-d)} + I(t-1) × { d × id + (1-d) × ie} (6) The bracket in the second term on the right hand side of the above equation is however the average cost of money r, as defined in Equation (2). Thus upon further rearrangement, we get RETDebt (t) + RETEq (t) = I(t-1) × {(N – t + 1) -1 + r} (7)
And substituting back into Equation (1) we get
NCF(t) = REV(t) – OM(t) – FL(t) - DD(t) – I(t-1) × {(N – t + 1) -1 + r} (8) Equations (8), (3), and (5), are the basic equations of the recursive process, used throughout this analysis. A graphic representation of equation (8) is shown in Figure 4.
Net Annual Cash Flow [M$/Year]
Year
O&M Expenses
Return On and Of the Investment
Net Annual Cash Flow Annual Revenues
from
Electricity Sales
Fuel Cost Decomm.
fund
RESTRUCTUREDUTILITY
Fig. 4 Components of plant net annual cash flow*
• Axes and bars not drawn to scale
10. EQUATIONS FOR ORIGINAL DESIGN PLANT LIFE OPERATION