2.3 Procedure for Calculating Fuel-Cycle Costs

To calculate fuel-cycle costs, it is necessary to focus attention on individual fuel sublots and determine:

1. The amount and composition of each sublot when charged to the reactor

2. The amount of electricity generated by each sublot in each period in which electricity is paid for

3. The amount and composition of each sublot when discharged from the reactor

4. The cost incurred in each step for preparing fuel before it is charged to the reactor

5. The cost or credit incurred in each step for recovering fuel after it is discharged from the reactor

6. The time at which each cost is paid or each credit is received, and the time at which revenue is received for each increment of electricity generated by each lot of fuel

A somewhat simplified, approximate procedure for calculating fuel-cycle costs will be illustrated by the example of sublot 2A of the PWR whose fuel management was described in

Center­ line |AB C d E F G H 6FG 8AH 7GB 8FG 8FF 7AH 7BG 9AH 27815 10500 22820 8023 12011 20926 23442 0 35977 20927 31825 18967 22722 29455 32594 10506 0.72 0,94 0.81 1.03 1.03 0.82 0.93 1.12 8HA 8BG 8CH 7DH 8DH 7EG 9BG 9BH 10500 12820 9879 19252 8086 21115 0 0 20926 23452 20532 28625 19246 30121 12827 10847 0.94 0.97 0.98 0.86 1.08 0.88 1.36 1.16 7GB 8HC 8GB 7FG 8EG 7CH 7DG 9CH 22820 9879 12818 18969 10517 20537 23395 0 31821 20527 22820 27841 21118 29352 32494 9884 0.81 0.98 0.91 0,81 1.02 0.86 0.92 1,03 8FG 7HD 7GF 7FF 8BH 8DG 9DG 9DH 8023 19240 19382 22713 10843 12829 0 0 18957 28597 28183 31175 21656 23409 12829 8086 1.02 0.86 0.80 0.79 1.07 1.07 1.37* 0.83 8FF 8HD 8GE 8HB 8GF 7BH 9EG 12011 8085 10516 10843 8023 21663 0 22715 19236 21099 21651 19382 31076 10514 1.03 1.08 1.02 1.07 1.15 0.95 1.10 7HA 7GE 7HC 8GD 7HB 9FF 9FG 20925 21094 20532 12828 21659 0 0 29453 30101 29344 23406 31072 12007 8021 0.82 0.88 0.86 1.07 0.95 1.25 0.82 78G 9GB 7GD 9GD 9GE 9GF 23442 0 23393 0 0 0 32594 12827 32492 12829 10514 8021 0.93 1.36 0.92 1.37* 1.10 0.82 9HA 9HB 9ИС 9HD 0 0 0 o 10506 10847 9884 8086 • = Maximum Relative Power 1.12 1.16 1.03 0.83 Fuel Lots 6,7, 8, 9 Initially 3.20 w/o U-235 Key

Cycle Average Burnup * 10,081 MWd/MT Cycle Thermal Energy = 896.8 GWd

Assembly Number BOC Burnup, MWd/MT EOC Burnup, MWd/MT BOC Relative Power (Assent)!у /Average)

Center­

line

7FG	9AH	8GB	9FG	9FF	8AH	8BG	10AH
27841	10506	22820	8021	12007	20927	23452	0
35997	20929	31824	18967	22721	29456	32601	10505
0.72	0.94	0.81	1.03	1.03	0.82	0.93	1.12
9HA	9BG	9CH	8DH	9DH	8EG	10BG	10BH
10506	12827	9884	19247	8086	21118	0	0
20928	23454	20535	28621	19249	30125	12826	10847
0.94	0.97	0.98	0.86	1.08	0.88	1.36	1.16
8GB	9HC	9GB	8FG	9 EG	8CH	8DG	10CH
22820	9884	12827	18968	10514	20532	23409	0
31820	20530	22827	27840	21117	29348	32506	9883
0.81	0.98	0.91	0.81	1.02	0.86	0.92	1.03
9FG	8HD	8GF	8FF	98H	90G	10DG	100H
8021	19236	19382	22722	10847	12829	0	0
18956	28594	28183	31183	21660	23410	12830	8086
1.02	0.86	0.80	0.79	1.07	1.07	1.37*	0.83
9FF	9HD	9GE	9HB	9GF	8BH	10EG
12007	8086	10514	10847	8021	21656	0
22714	19238	21098	21656	19382	31071	10516
1.03	1.08	1.02	1.07	1.15	0.95	1.10
8HA	8GE	8HC	9GD	8HB	10FF	10FG
20926	21099	20527	12829	21651	0	0
29454	30105	29340	23408	31067	12010	8022
0.82	0.88	0.86	1.07	0.95	1.25	0.82
8BG	10GB	8GD	10GD	10GE	10GF
23452	0	23407	0	0	0
32601	12825	32504	12829	10516	8022
0.93	1.36	0.92	1.37*	1.10	0.82
ЮНА	10HB	10HC	10HD
0	0	0	0
10505	10847	9883	8086		= Maximum Relative Power
1.12	1.16	1.03	0.83

ABCOHFQH

Key

Fuel Lots 7, 8. 9. 10 Initially 3.20 w/o U-235

Assembly Number BOC Burnup, MWd/MT EOC Burnup, MWd/MT BOC Relative Power (Assembly/Average)

Figure 3.24 PWR, assembly power and burnup distribution, cycle 8.

Sec. 4. Information on material quantities and energy production from sublots 1A through 4B in cycles 1, 2, and 3, applicable to this example calculation, is given in Table 3.5.

The 63 assemblies of sublot 2A contain 28,171.8 kg of uranium enriched to 2.8 w/o aU. The average bumup experienced by this sublet is 16,448 MWd/MT in cycle 1 and 8700 MWd/MT in cycle 2, for a total bumup B2a of 25,148 MWd/MT. The total thermal energy produced by this sublot is

(25,148 MWd/MTX28.1718MTX24,000 kWh/MWd)= 17,003 X 106 kWh Calculations by computer code LEOPARD made by Rieck [Rl] predict that this fuel wnen

Table 3.3 Fuel-cycle performance of PWR in successive cycles Cycle number Fuel lot number w/o 233 U charged Burnup, MWd/MT Peak radial power ratio Cycle average Fuel discharged Charged Discharged Max. value Position 1 1 1 2.25 15,535 16,943 1.24 EE 2* 2.8 3+ 3.3 2 4 2* 3.2 9,652 25,115 1.34 DG 3 5 3* 3.2 9,894 32,076 1.36 DG 4 6 4* 3.2 10,284 30,306 1.34 DG 5 7 5* 3.2 10,038 30,401 1.38 DG 6 8 6* 3.2 10,084 30,419 1.37 DG 7 9 7* 3.2 10,081 30,399 1.37 DG 8 10 8* 3.2 10,081 30,400 1.37 DG ^Contains burnable poison. * Sixty-three assemblies from this lot and one from previous lot.

exposed to burnup of 25,148 MWd/MT will contain 27,991 kg of uranium enriched to 0.920 w/o 235 U and 161.330 kg of the fissile isotopes of plutonium 239Pu and 241 Pu.

To obtain the durations of cycles 1 and 2, and from them the times at which payments are made, credits received, and revenue obtained from the sale of electricity, the total thermal energy per cycle Ht is required and is calculated from burnup increments in the next to the last row of the table. The duration of irradiation during the ith cycle, r,-‘ — /’■, in years is obtained from Hi in the last row of the table, using a rated thermal output of 3.250 GW.

Figure 3.25 is a schematic flow sheet for lot 2A fuel cycle. This shows the fuel-cycle steps to be considered and defines notation for the material and service quantities involved in each step, the total cost or credit associated with each step, and the timing of all transactions (dashed arrows).

Table 3.6 gives numerical values for material quantities, unit costs or credits, and total direct costs or credits involved in each fuel-cycle step and calculates the overall net direct cost for lot 2A as $26.4 million. A total of $27.8 million is paid out for UF6 and fabrication in transactions 1 and 2 before any revenue is received from the sale of electricity. Because of this delay in receiving revenue, the total fuel-cycle cost includes also charges for carrying the $27.8 million advanced several years before it is recovered through revenue from the sale of electricity. Similarly, there is a financing charge on the net credit of $1.4 million in steps 3 through 6, delayed until after revenue is received from the sale of electricity.

The assumptions going into the calculation of direct costs in Table 3.6 will be described first. Then the procedure for calculating financing charges will be described, and finally a value will be given for the complete fuel-cycle cost.

Direct costs. The unit costs used in the examples of this chapter are those anticipated in 1975 for the year 1980. Because of changes since 1975, readers are cautioned to regard these costs more as examples than as firm numbers.

The unit cost of enriched uranium in the form of UF6 depends on the 235 U content of the uranium, the price paid for the natural uranium from which the uranium was enriched, the cost of the separative work expended in enriching the uranium, and the composition of the tails stream containing depleted uranium leaving the uranium enrichment plant. The procedure for calculating the cost of enriched uranium is described in Chap. 12. The unit costs

Сц[6] = $848.66/kg U for uranium enriched to 2.8 w/o 235 U fed to fabrication and cV" = $152.83/kg U for uranium containing 0.920 w/o 235 U recovered from reprocessing are based on the following assumptions:

Price of natural uranium ore concentrates, $31.55ДЬ U308

Price of natural UF6, $89.11/kg U

Cost of separative work, $100/separative work unit

235 U content of enrichment plant tails, 0.3 w/o

Table 3.4 Zion reactor, cycle 8: Relative power and bumup Assembly location DA DB DC DD DE DF DC DH Number of previous cycles 1 2 2 2 1 1 0 0 Axial position Relative power at beginning 12 (top) 0.72 0.64 0.60 0.57 0.68 0.62 0.61 0.35 11 1.03 0.87 0.81 0.78 1.00 0.94 1.07 0.64 10 1.13 0.95 0.89 0.86 1.13 1.10 1.32 0.81 9 1.17 0.98 0.92 0.89 1.20 1.18 1.48 0.91 8 1.18 0.99 0.92 0.90 1.23 1.23 1.58 0.97 7 Mid- 1.17 0.98 0.92 0.90 1.24 1.26 1.64 1.01 6 plane 1.15 0.96 0.90 0.89 1.23 1.26 1.68 1.04 5 1.11 0.93 0.87 0.86 1.21 1.25 1.70* 1.04 4 1.07 0.89 0.83 0.82 1.16 1.21 1.67 1.02 3 1.00 0.83 0.78 0.77 1.09 1.13 1.57 0.96 2 0.91 0.76 0.70 0.69 0.97 0.99 1.32 0.79 1 (bottom) 0.64 0.56 0.52 0.51 0.68 0.66 0.77 0.44 Average 1.02 0.86 0.80 0.79 1.07 1.07 1.37* 0.83 Assembly location AA AB AC AD AE AF AG AH Number of previous cycles 4 2 3 2 2 3 3 1 Axial position Bumup, MWd/MT, at end of cycle 12 27,179 14,426 23,474 13,087 15,589 21,108 22,962 6,100 11 35,525 19,961 31,028 18,154 21,530 28,394 31,095 9,347 10 37,482 21,536 32,909 19,575 23,259 30,387 33,421 10,543 9 37,850 21,992 33,348 19,972 23,794 30,932 34,124 11,018 8 37,893 22,174 33,478 20,120 24,028 31,129 34,421 11,267 7 37,897 22,303 33,564 20,221 24,199 31,256 34,631 11,454 6 37,913 22,430 33,653 20,320 24,364 31,378 34,833 11,629 5 37,952 22,568 33,762 20,431 24,537 31,513 35,042 11,803 4 38,028 22,710 33,894 20,546 24,710 31,653 35,247 11,956 3 38,052* 22,740 33,952 20,566 24,760 31,661 35,281 11,957 2 37,078 21,905 33,049 19,808 23,889 30,614 34,090 11,236 1 29,045 16,317 25,703 14,715 17,904 23,379 25,999 7,671 Average 35,991* 20,922 31,818 18,959 22,714 29,450 32,595 10,498 ^Maximum local value. * Maximum assembly average.

Table 3.5 Example of material quantities and energy production by lot and cycle Cycle i ——————— Fuel discharged Fuel sublot number к Number of assemblies Fuel charged 1 2 3 Average burnup per lot, MWd/MT, Bk = 2/ ABik Thermal energy per lot, GWh, Hk = 24 X 10’6 w/o 235 у kg U, w* kg fissile Pu, Pk Average burnup increment of sublot к in cycle i, MWd/MT, ДBlk w/o 235 и kg U, ui 1A 64 2.25 29,168.2 16,943 _ _ 16,943 11,861 0.944 28,457.6 141.736 IB 1 (#1GC) 2.25 455.8 15,950 7,210 — 23,160 253.35 0.663 441.1 2.497 2A 63 2.8 28,171.8 16,448 8,700 — 25,148 17,003 0.920 27,199.1 161.330 2B 1 (#2FE) 2.8 447.2 14,934 8,721 8,040 31,695 340.17 0.653 428.2 2.719 ЗА 63 3.3 27,499.5 13,280 10,132 8,671 32,083 21,175 0.921 26,320.8 171.258 3B 1 (#3HD) 3.3 436.5 10,194 8,553 8,601 t t t t t 4A 63 3.2 29,039.1 — 10,242 10,743 t t t t t 4B 1 (#4FG) 3.2 460.9 — 6,828 11,051 t І * t t

Thermal energy per cycle, GWh

Я/ = 24 X 10~6 Ujf ABjic 32,188 20,044 §

Duration of irrad.,yr 8766XJ250X a9 1-2553 0.7817 8 1

Receive Revenue from Sole of Electricity

Zu-	ZFa"				ZSh =	Zr =
о C X					cShU	CRU’	CpPf’	c0.Uf’f"
Times — ty.	lFa		t Ґ	t2 t2t2	lSh	Ir	V=tp

Figure 3.25 Schematic flow sheet for lot 2A fuel cycle, showing material and service quantities and timing of transactions.

It is assumed that /’ = 0.99 fraction of uranium charged in the form of UF6 will be recovered as fabricated fuel. Hence, to provide U’= 28,171.8 kg of fabricated uranium, 28,171.8/0.99 = 28,456.364 kg uranium in the form of UF6 must be purchased. The direct cost of this UF6 is 28,456.364 X 848.66 = $24,149,778.

A similar procedure is used to calculate the other components of the direct cost shown in Table 3.6. Other assumptions are as follows:

Fraction of uranium and plutonium recovered in reprocessing, f" = 0.99.

Fraction of recovered uranium converted to UF6,/’" = 0.995.

The fabrication unit cost of $130/kg includes cost of converting UF6 to U02 and packaging U02 in fuel assemblies.

The shipping cost of $30/kg includes storage charges at the reactor for around 150 days to permit fuel radioactivity to decrease.

The reprocessing and conversion cost of $180/kg includes charges by the government for perpetual storage of radioactive wastes.

Financing charges. A company generating electricity that pays out Z dollars for fuel-cycle costs t years before it receives revenue from generation of electricity from that fuel must pay to the bondholders and stockholders who advanced the funds for the fuel the return they require on their investment, and must also pay income taxes to the government on the profits from which the stockholders’ return is obtained. It is possible to represent all of these financing charges as the product yZt, where у is known as the annual cost of money before income taxes. For a privately owned U. S. electric company, a value of у = 0.151 per year is representative.

To find the total fuel financing charge, it is necessary to find the amount of money advanced for fuel as a function of time. Figure 3.26 is a schematic plot of the amount of

Figure 3.26 Plot of dollars invested in fuel versus time.
money invested in fuel cycle as a function of time for an example like sublot 2A, in which revenue is assumed received from the sale of electricity at two times during irradiation, from sale of Ex kWh at time r, in cycle 1 and from sale of E2 kWh at time t2 in cycle 2. The generalization to a more realistic case, in which revenue is received at more times, should be clear.

At time fu’, Zxj’ dollars are invested in enriched uranium. This amount of money is invested for tFa — tV’ years, until tFm when ZFa more dollars are paid out to fabricate fuel. The financing cost of carrying the initial investment of Zy> dollars for tFa — ty’ years is the product of the cost of money before income taxes, y, and the area of region A, Zy(tFa — fu’)-

Between tFtt, when fabrication is paid for, and f1; when revenue is received from production of Ex kWh of electricity, the dollars invested in fuel is Zy’ + ZFm and financing costs are the product of у and area B, (Zu’ + ZFa)(tx ~ tFa).

If the total electricity production of the lot of fuel is I, mEm (Ex + E2 for sublot 2A) and the net direct fuel-cycle cost is Z/Z/=Z (where Zj is the direct cost of the /’th fuel-cycle step and ZjZj = $26,439,162 for sublot 2A), the amount of money invested in fuel after tx should be reduced by ZExl(Ex + E2 ). Between fj and t2, when the second (and in this case, the last) revenue is received from production of E2 kWh of electricity, the dollars invested in fuel is Zy + ZFa — ZExl(Ex + E2), and financing costs are the product of у and area C, [Zy< + ZFa-ZEi/(Ex +E2)} (t2-tx).

The amount of electricity generated by sublot 2A in cycle 1, Et, is obtained from the thermal efficiency of the power plant

77 = ilr = 032615 (ЗЛ4)

the average burnup increment of sublot 2A in cycle 1, AB12a = 16,448 MWd/MT from Table 3.5, and the mass of uranium, U2a =28,171.8 kg. Hence

Ex = 24n ABxzaU’ia = (24)(0.32615)(16,448)(28,171.8) = 3.6271 X 109 kWhe (3.15) Similarly, the amount of electricity generated by sublot 2A in cycle 2 is

E2 = (24)(0.32615) (8700)(28,171.8) = 1.9185 X 109 kWhe (3.16)

The total electric generation is

E = Ex +E2 = (3.6271 + 1.9185) X 109 = 5.5456 X 109 kWhe (3.17)

Between t2 and t$h when payment is made for shipping fuel, the dollars invested in fuel is the difference between the initial outlay Zy’ + ZFa and the direct fuel-cycle cost Z, which is equivalent to

Zy’+ ZFa — Z = Zy" + Z? — Zsh — ZR (3.18)

The financing cost for this time interval is the product of у and area D, (Zy" + ZP — Zr —ZSh)(tsh ~h)-

Between tgh and tR, when payment is made for reprocessing fuel and converting uranium to UF6, the dollars invested in fuel is Zy" + Zp — ZR. The financing cost for this time interval is the product of у and area E, (Zy" + ZP — ZR)(tR — tsh).

If credit for plutonium and uranium is received at a time ty“ later than tR, an additional financing charge is incurred on the value of this uranium and plutonium, Zy" + ZP, for the time interval ty" — tR, equal to the product of у and the area F, (Zy" + ZPX? u" — tR).

[*r

The sum of areas А, В, C, D, E, and F is

+ (Zu" + Zp — ZR —Zsh)(fsh + Zp ~ZR)(tR — tsf,)

+ (Zu — + Zp)(tV" — tR) = Zu — — tl)j

_l *7 (Eiti+E2t2 . ±T {Eiti +E2t2 „

Zf‘ VT^T* Fa) ^{еГ+е, tsh)

* — ‘*) — (z“-+z-> fihrir — —

Thus, the area under the curve is the sum of the product of each expenditure or credit times the difference between the mean time for receipt of revenue t

i= Eiti + E2t2

Ei +E2

and the time when the expenditure or credit is paid. This result generalizes to an expression for the area of

(3.21)

where Zj is the outlay for fuel-cycle step / (negative if a credit) at f;- and tm is the time at which revenue is received, for production of Em kWh of electricity. The fuel-cycle cost e in mills per kilowatt-hour then is

The assumptions of Table 3.7 are made to obtain the times needed to calculate the carrying-charge term for sublot 2A:

{EttT ~'<) й E’- <3-22>

A time basis of zero is taken for the start of cycle 1 (t = 0).

The mean time for receipt of revenue from sale of electricity, t, is

r= EiU +E2t2 _ (3.6271)(0.7920)+ (1.9185)(1.9355) _ , ,0„._

1- ~ЁГПГ~~————————— ШпТШБ 1 1876 yr (3-23)

From the foregoing transaction times and the direct fuel-cycle costs or credits of Table 3.6, the carrying charges and total fuel-cycle costs may be calculated, as shown in Table 3.8 for sublot 2A. Division of the total fuel-cycle cost of $33,173,168 by the electricity generated by sublot 2A, 5.5456 X 109 kWhe, and conversion to mills per kilowatt-hour of electricity by Eq.

(3.21) gives 5.9819 mills/kWhe for the unit fuel-cycle cost of lot 2A.

Table 3.8 shows that more than 20 percent of the fuel-cycle cost arises from carrying charges.