PART II: ECONOMIC MODELING AND IMPLICATIONS OF THE DECISION ON
10. EQUATIONS FOR ORIGINAL DESIGN PLANT LIFE OPERATION
10.2. Operations period
The cost equations applicable during the operations period are similar to the basic cost equations derived in section 9 above, except that we need to be more explicit about the definition of each cost component. To recapitulate, the basic annual net cash flow equation is:
NCF(t) = REV(t) – OM(t) – FL(t) - DD(t) – I(t-1) × {(N – t + 1) -1 + r} (12) This is similar to equation (8) and to its graphic representation in Figure 4 above, with the specific cost item defined following equation (1). The remaining investment in year t during the operations period to be recovered and to provide a return on is defined as
I(t) = I(t-1) × {(N-t) / (N-t+1)} + CAPADD(t) (13) This is identical to equation (5) above, and cost item definitions as given following Equation (4). The representation of equation (5) is shown in Figure 3. Note that in equation (13) when we set t = 1 we get I (0) which represents the discounted cumulative NPP Investment carried over from the end of the construction period to the start of the operations period. The numerical value of I(0) is given by the first (left) part of equation (11).
Finally, the discounted cumulative net cash flow, which accounts for all the negative cash flow incurred during the construction period (discounted), and the repayment of that negative cash flow overhang by year t from incoming revenues (minus operations costs), is given by the following:
t t
CUMCF(t) = CUMCF(t = 0) +
Σ
{NCF(k) × (1+r)-K = - I(t=0) +Σ
{NCF(k) × (1+r)-K} (14)k=1 k=1
This is similar to equation (3) above, with definitions following discussions related to equations (3) and equation (11) above.
Having defined the basic equations governing plant economics during its nominal operations period, we will now provide more explicit definitions for the various terms in the above equations. The more detailed equations eventually introduce NPP-specific elements into the hitherto generalized discussion, as seen below. The incoming revenues in equation (12) can be defined as
REV(t) = ELCPRC(t) × CAP × 8.76 × 10-3 × CF(t) (15) Where
REV(t) – Annual incoming revenues obtained from sale of electricity in year t, [M$/year]
ELECPR(t) – Unit electricity sales price in year t, [$/MWh]
CAP – Plant Capacity [MWe]
CF(t) – NPP Annual capacity factor in year t, [%]
This revenue stream is the only source from which the operating costs can be repaid, and the negative investment overhang can be depreciated. Annual schedules of expected electricity prices and plant capacity factors should be provided in order to compute the value of the incoming revenue stream by year t. In general, over the operations lifetime of the plant’s unit electricity prices are expected to increase with the general rise of the Producers’ Price Index above inflation. Plant capacity factors are expected to rise above early-in-life low values, achieve a high plateau value and maintain it throughout most of the operations life, and possibly decline a bit prior to the scheduled end of life.
The plant’s annual O&M expenditures could be defined as
OM(t) = NSTAF(t) × SALAV(t) × 10-3 + FOM(t) × CAP × 10-3 + VOM(t) × CAP × 8.76 ×
10-3 × CF(t) (16)
Where
OM(t) – Annual O&M expenditures in year t, excluding capitalized capital addition costs [M$/year]
NSTAF(t) – Number of plant staff in year t, [persons]
SALAV(t) – Average loaded salary for the mix of plant employees defined by NSTAF(t), in year t, [K$/year]
FOM(t) – Annual fixed O&M costs in year t, excluding personnel costs, and including small capital addition costs and interim replacement costs which are expensed in the year they accrue [$/KWe –year]
VOM(t) – Annual variable O&M costs in year t, [$/MWh]
In this analysis we break the total annual O&M costs into four components: personnel salaries and benefits costs, fixed O&M costs excluding salaries and including expensed capital additions, variable O&M costs which include mostly plant consumables determined by the plant’s power generation rate, and large capitalized capital addition costs CAPADD(t), discussed in section 9 above. We find it useful to break out personnel costs separately, since both the staffs mix and the average salaries will change during the transition from the operations to the decommissioning phases. Thus, an explicit schedule of plant staffing mix and the average salary levels of those mixes are useful ways of capturing changes in this important O&M cost category which, specifically in the NPP case, contributes 50 – 70 percent of overall annual O&M expenses. We further break out the remaining fixed O&M costs from the variable operations costs, again, bearing in mind that variable O&M costs disappear during the plant decommissioning phase. In the nuclear-specific case, with NPP O&M costs comprising about two thirds of all annual operations costs, this is an important cost category, and the more we can break it into time-related or fixed components, data permitting, the better we can describe it during the analysis.
In likewise fashion we can define annual fuel costs, specific to a NPP, as
FL(t) = FFL(t) × CAP × 10-3 + {VFL(t) + WMC(t)} × CAP × 8.76 × 10-3 × CF(t) (17)
Where
FL(t) – Annual fuel costs including waste management charge in year t, [M$/year]
FFL(t) – Fixed portion of fuel cycle costs in year t, including first core costs payable in year t (if any), or inventory charges on fresh fuel supplies held in storage for more than a year in year t, [M$/year]
VFL(t) – Variable portion of annual fuel cycle cost in year t, representing mostly costs incurred in the front end of the fuel cycle [M$/year]
WMC(t) – Annual payment towards the ultimate spent fuel disposal by the appropriate agency during the NPP Shutdown phase [M$/year]
In general, the annual costs of storing spent fuel in the pool at the NPP fuel storage building are accounted by the FOM(t) cost defined above. Construction of an on-site spent fuel dry cask storage facility is accounted for as a CAPADD(t) cost item. The annual spent fuel control and monitoring expenses during the post shutdown phase could be included in the FOMDD(t) term during the decommissioning period as discussed in Sub-section 10.3. The WMC(t) term represents the annual cost in paid year t of the operations period, for the ultimate disposal of the NPP’s spent fuel, during the post shutdown phase. We included here a time dependency in the waste management charge WMC(t), to account for the possibility that this annual charge might be changed, sometime during the NPP operations period.
The FFL(t) term – the NPP fixed fuel cost component is relatively small and is included here for completion only. Typical cost numbers are 3–5 [$/MWh] for VFL(t), 1 [$/MWh] for WMC(t), and a fraction of [$/MWh] for FFL(t). This breakdown would be different for other power plant types.
We can now define the last term in equation (12), not yet discussed – the decommissioning cost. Decommissioning cost and the significant attention paid to their estimation and collection are unique features of a nuclear plant operation. While fossil-fired plants now start incorporating some (yet small value) decommissioning cost items in their cost evaluations, the discussions here, and its later implications, are NPP-specific.
DD(t) = CDD × SFFDD = CDD × {iDD × [(1 + iDD)N -1]-1} (18) Where
DD(t) – Annual decommissioning fund outlay in Year t, [M$/year]
CDD – Final value of the decommissioning fund at the end the operations phase, expressed in today’s constant currency value [M$]
SFFDD – Sinking fund factor – the fraction of the final decommissioning fund value that needs to be set aside annually and deposited into an interest bearing fund that will accumulate to the desired value after N years [1/year]
iDD – The interest rate on the sinking fund accumulating annual decommissioning outlays [%/year]
Thus, if we would like to accumulate 400 [M$] in constant dollars of today at the end of the NPP nominal operations period of 30 years, and we deposit annual decommissioning outlays into a sinking fund invested in Government bonds bearing 4 [%/year] interest, the sinking
fund factor computed for that outlay will be 0.04/ (1.0430 -1) or 0.0178 [1/year], and the annual decommissioning outlay will be 7.13 [M$/year].
Getting back to the more generalized plant economics discussion we must first define end of operations period cost factors, before discussing issues related to the decommissioning period.
Two important changes occur at the point of end of the original design plant life. Firstly, all initial investments in the plant have been completely depreciated, since depreciation payments were set to be completed during an operations period of N years. Most of the capitalized annual capital additions costs have also been depreciated in like manner save for the last one.
Thus
I(t = N) = CAPADD(t = N) (19) We have assumed for simplicity sake that the book life period over which depreciation occurs is identical to the nominal plant operations period. If we assume shorter book life than operations life, then the modified investment recovery equations will become more complicated having to address two different time intervals, however the principle embodied in equations (13) and (19) would still be valid and equation (19) would remain unchanged.
Assuming that the initial investment overhang except for equation (19) above has been depreciated, this implies that the investment related recursive process that has driven the net cash flow equations during the operations period is no longer necessary during the decommissioning period.
The second change occurring at the end of the plant operations period is that the decommissioning fund has been fully accumulated, and it becomes available to the plant to pay for decommissioning period expenses. Thus CDD is added to the cumulative discounted net cash flow at the end of life, and these two sources of funds are available to pay all decommissioning period costs. Once we get into a discussion of a large decommissioning fund and its disposition, we do however imply that we address a NPP-specific situation. Thus Sub-section 10.3 below represents a nuclear-specific section.