ML19007A228

LTR-19-0006 Timothy Margulies, E-mail Turkey Point Severe Accident Modeling for Design Objectives Prevention or Mitigation

ML19007A228
Person / Time
Site:	Turkey Point
Issue date:	01/07/2019
From:	Margulies T Public Commenter
To:	Annette Vietti-Cook NRC/SECY
Shared Package
ML19007A229	List: ML19007A228
References
LTR-19-0006
Download: ML19007A228 (17)

Text

NRCExecSec Resource From:

Sent:

To:

Subject:

Attachments:

Timothy Margulies <t.margulies@gmail.com>

Monday, January 07, 2019 8:38 AM NRCExecSec Resource

[External_Sender] Report TurkeyPtUnitsSevere Accident.pdf Please find attached an investigation that may be of interest. It relates to an economic risk analysis applied to severe accidents at Turkey Point, FL Units.

Thank you for your attention to this matter.

Tim Margulies, Ph.D.

1

Severe Accident Modeling for Design Objectives: Prevention or Mitigation The purpose of this investigation is to provide an economic analysis for life extension of a nuclear generating station that is vulnerable to severe accidents. The Turkey Point, FL Units 3 and 4 are chosen for the calculation. The site consists of 3-loop pressurized water reactors with dry, ambient pressure, containments. They reside approximately twenty-five miles South of Miami (location is latitude 25° 26' 04" North and longitude 80° 19' 52" West). A renewed operating license allows an additional twenty years of operation to their licensed forty years (1972-2012) with extension to the year 2032.

A Code of Federal Regulation, Part 50, 10 CFR Appendix I, calculation for design back-fit for normal operation, or expected occurrences, would only use a surrogate $1000 per person-REM conversion factor for latent cancers. This was increased to $2000 in 1995 and to $ 5100 for cost-benefit analysis. A value of a life saved of $9.0 million, and the ICRP 103 risk coefficient of 5.7 x 10-4 per rem obtains $5100 per person-rem. The dollar per person-rem conversion factor is for stochastic effects only. Figure 1 displays a range of valuations.

The amounts for converting, or valuing person-REM to dollars ranges up to $25000 per person-REM averted as displayed in the figure considering 400 to 500 person-REM and 1 to 10 million dollars for a life saved as shown Figure 1. NRC adopted (1991) risk conversion factors of 4 x 10-4 (0.0004 latent cancer fatalities) per person-REM for workers and 5 x 10-4 (0.0005 latent cancer fatalities) per person-REM for the general public.

Do1'1ars per person REH a Stat1stica1 latent Cancer Fata11ty

[H1111on do11ars]

2000

$/person-REM 15000 10000 5000 0.01 person-REM/LCF rate

[Type text]

Figure 1: Regulatory Cost-Benefit Valuation

[Type text]

T. Margulies

Therefore, 10 9 expected person-REM.

$5,100 20 t

$ 1 1 M reac or years, or reactor-year person-REM Uncertainties or external event contributions would significantly increase the calculated results.

Data from the Turkey Point Units 3 & 4 submitted in a Life-Extension Licensing Application (Appendix F) were used to independently calculate the core-melt probability and expected off-site person-Sievert (1 Sv = 100 REM)). The reported core-melt probability and offsite person-REM [total collective effective dose equivalent] were verified.

Newer onsite and offsite costs were calculated for a range of interest rates (1 % to 7% per year). The expectation of core-melt probability was increased by a factor of three, as well as, the expected person-REM.

Offsite person-REM := 32.633 Probability of Core Melt:= 0.274 e-4 / Reactor-Year Averted Costs calculated assuming$ 5,000/Person-REM versus Interest Rates 1 through 7 %/year 7.715965012 107, 6.394434939 107, 5.334474691 107, 4.479472229 107, 3.785867476 107, 3.219995713 107, 2.755723629 107 Averted Costs calculated assuming$ 2000/Person-REM versus Interest Rates 1 through 7 %/year 7.444511433 107, 6.157579724 107, 5.126914452 107, 4.296804554 107, 3.624427058 107, 3.076721803 107, 2.628052579 107 The expected consequences reflect wind rose probabilities which are debatable, since during Three Mile Island incident the winds rotated around the compass. At Miami the wind direction, wind speeds, and frequencies are displayed in a wind rose figure. The large population of Miami is situated North. Projected population estimates for the year 2025 were used in the off-site calculations of total collective dose.

[Type text]

[Type text]

T. Margulies

700000 Turkey Point Units, FL Vear 2025 Population Estimates 600000 500000

=

0 *,-00000

'j.ooooo --.

=iooooo..,__..---.,

100000

---

*::to;io:--

mi:

• --:11:2fi=3jt---*--

m-i-0 --

700000 600000 N

NNE NE Sector Direction Figure 2 Turkey Point Units, FL Vear 2025 Population Estimates 500000.... _______ ___

=

iooooo

)00000 200000 100000 0 _J::illl.-,&.----,-~-

____::::;::::=;;:::::=::;=:.:;::::::......,....---,-.......-.---,-1..:::::;:Jli:::.:::_-11::,

[Type text]

+#"+~~+~ ~~~ ~~~ ~~~~~ 'l-f++~

Sector Direction Figure 3

[Type text]

T. Margulies

[MIAJ MIAMI INTL AIRPORT Windrose Plot [All Year]

Period of Record: 01 Jan 1970

• 10 Nov 2018 N

6.1 N-W 4.8 3.6 w

s-w Generated: 10 Nov 2018 Wind Speed (mph!

IH Summary n:445571 Missing: 10196 Calm: 7.4%

Avg Speed: 8.8 mph 2*5 -

.7 7-10 C

J 10*1S -

15*20 -

20+

Figure 4: Wind rose packed with wind speeds, frequencies, and particular directions

[Iowa Environmental Mesoset, Iowa State University].

Potentfa1 Averted Costs 7, X 107

0. X 107 r

I

5. X 107 4, X 107 3, X 101 1

2 3

4 5

Interest I Year I

• 20000.

2000. I Figure 5: Onsite and Offsite Cost

[Type text]

[Type text]

I !

6 7

T. Margulies

The accident model scenarios account for containment barrier leaks (probability of 4.02

• 10 6

), ruptures (probability of 3.84

• 10 6

), or bypass of radioactivity to the Reactor-Year Reactor-Year 10-B atmosphere (probability of 7.95 *

).

Reactor-Year The technical probabilities are small; however, the accompanying disastrous consequences may not constitute an acceptable level of surety safety engineering for beyond the normal licensed service life. Prevention of core damage is generally achieved by stopping the fission chain reactions and by providing adequate coolant for heat removal requiring pumping, with its water and electrical supplies. The heat generated after shutdown would decay over a long period of time from accumulated fission products. The analysis would justify additional design hardware for achieving prevention or mitigation of accident consequences. To date, nuclear system safety is approached by safety goals, by aging management programs, by on-site inspections, by emergency preparedness, and by insurance. It is noted that Three Mile Island accident (March 28, 1979 ) liability claims cost: $ 70 M. and the Price-Anderson insurance for the nuclear industry is $ 450 M. per year. The final barrier to isolate large radioactive releases to the atmosphere remains vulnerable for these severe accidents beyond design basis assumptions.

Additions/Modifications

$ Million Subtotal Reactivity poison (boric acid) injection system (valve explosive bolts) 0.8 filtered hardened vent 15.0 15.8

( costs range from $ SM for sand/gravel to $ 30 M)

DC battery capacity 1.8 17.6 transfer & isolation switches

2.

19.6 diesel generator 10.8 30.4 Independent high pressure injection system 29.2 59.6

[Type text]

[Type text]

T. Margulies

Consider a discrete expected risk formulation, where the change in risk when a system modification is made is calculated by, A conversion factor for costs would also be included.

Two special cases may be obtained.

Case 1: Probabilities of core melt are essentially the same, Pi = p[11, so that the expected risk change is dominated by the change in consequences as in effective filtering mitigation of particulates.

l'!R = ~k Pi (ci - c[11)

Li=1 Case 2: Consequences remain essentially the same, ci = c[11, such that the change in expected risk is driven by a probability of core melt change, such as, from a preventative modification.

Federal regulation, Title 10 CFR 50.109, addresses that a backfit may only be imposed if the NRC determines that "there is a substantial increase in the overall protection of the public health and safety or the common defense and security to be derived from the backfit and that the direct and indirect costs of implementation for that facility are justified in view of this increased protection." This rule also provides the exceptions.

References:

H.P. Nourbakhsh, "Dealing with Beyond-Design-Basis Accidents in Nuclear Safety Decisions,", Presented at 12th International Probabilistic Safety Assessment & Management (PSAM 12) Conference, (June 22-27, 2014)

L. Rahm-Crites, How Health Risk from Radiation is Assessed, UCRL-ID-118487 (July 1994)

Decision Analysis Methodology for Assessing ALARA Collective Radiation Doses and Risks 30 May 2011 NUREG/BR-0184, Regulatory Analysis Technical Evaluation Handbook Final Report NUREG/BR-0058 Revision 5, US. Nuclear Regulatory Commission Guidance on Performing Benefit-Cost Analyses

[Type text]

[Type text]

T. Margulies

NUREG/BR-0058 Revision 4, Regulatory Analysis Guidelines of the US. Nuclear Regulatory Commission Margulies, T., "United States Nuclear Power Safety Risk", Presentation, Research Gate, DOI: 10.13140/RG.2.2.18082.68805 (2016).

Margulies, T., "Simple Cost Risk-Benefit Calculation: Nuclear Plant Back-fit Analysis,"

Reliability Engineering and System Safety, vol. 86/13, 2004.

[Type text]

[Type text]

T. Margulies

restart: with(Student) : with(LinearAlgebra) : with(plots) : Ml := Matrix( 47, 7, [ [Al, 5.46 X lQ/'1, 5.49 X 10"1, 5.46 X 10"1, 1.18 X 10"1, 1.19 X 10"4, 1.18 X 10"7], [A.2,2.48 X 10"4,2.48 X 10"4,2.48 X 10"4,4.48 X 10"9,4.47 X 10"9,4.48 X 10"9 ], [Bl,8.46 X 10 "2,8.47 x 10"2,8.46 x 10"2, 1.49 x 10"7,3.06 x 10"6, 1.49 x 10"7 ], [B2-L,2.64 x 10 14,2.65 x 10"4,2.64 x 10"4,5.ll x 10"9,5.10 x 10"9,5.ll x 10"9], [ B2-R,3.73 x 10 "4,3.74 X 10"4,3.73 X 10"4,9.37 X 10"9,9.36 X 10"9,9.37 X 10"9 ], [B3-L,8.46 X 10 "2,8.47 X 10"2,8.46 X 10"2, 1.49 X 10"7,3.06 X 10"6, 1.49 X 10"7], [B3-R, 1.92 X 10 "3,1.92 x 10"3, 1.92 x 10"3,5.12 x 10"7,3.93 x 10"7,5.12 x 10"7],[B4-L,2.64 x 10 "4,2.65 x 10"4,2.64 x 10"4,5.ll x 10"9,5.10 x 10"9,5.11 x 109],[B4-R,3.78 x 10 "4,3.78 x 10"4,3.78 x 10"4,9.38 x 10"9,9.37 x 10"9,9.38 x 10"9 ], [B5-L, 1.54 x 10 "3, 1.54 x 10"3, 1.54 x 10"3,2.74 x 10"7, 1.56 x 10"7,2.74 x 10"7 ], [B5-R,3.28 x 10 "3, 3.28 X 10"3, 3.28 X 10"3, 1.23 X 10"8, 1.12 X 10"8, 1.23 X 10"8 ], [B6-L, 2.29 X 10 "4, 2.29 x 10"4, 2.29 x 10"4, 3.84 x 10"9, 3.83 x 10"9, 3.84 x 10"9], [B6-R, 2.74 x 10 "4,2.74 X 10"4,2.74 X 10"4,5.10 X 10"9,5.09 X 10"9,5.10 X 10"9], [BP-V,4.46 X 10 "4,4.86 x 10"4,4.46 x 10"4, 1.27 x 10"10, 1.27 x 10"10, 1.27 x 10"10], [BY-SGTR, 8.07 X 10"3,8.08 X 10"3,8.07 X 10"3, 7.99 X 10"8, 7.93 X 10"8, 7.99 X 10"8], [Cl-L, 8.46 X 10"2, 8.47 X 10"2, 8.46 X 10"2, 1.49 X 10"7, 3.06 X 10"6, 1.49 X 10"7 ], [ Cl - R, 1.92 x 10"3, 1.92 x 10"3, 1.92 x 10"3,5.12 x 10"7,3.93 x 10"7,5.12 x 10"7], [C2-L, 2.65 x 10"4,2.65 x 10"4,2.65 x 10"4,5.ll x 10"9,5.10 x 10"9,5.ll x 10"9], [C2-R, 3.79 x 10"4,3.80 x 10"4,2.48 x 10"4,9.38 x 10"9,9.37 x 10"9,4.37 x 10"9),[ C3-L, 8.46 x 10"2,8.47 x 10"2,8.46 x 10"2, 1.49 x 10"7,3.06 x 10"6, 1.49 x 10"7 ], [C3-R, 1.92 X 10"3, 1.92 X 10"3, 1.92 X 10"3,5.12' X 10"7,3.93 X 10"7,5.12 X 10"7 ], [C4-L, 2.65 x 10"4, 2.65 x 10"4, 2.65 x 10"4, 5.11 x 10"9, 5.10 x 10"9, 5.11 x 10"9 ], [ C4-R, 3.79 x 10"4,3.80 x 10"4, 2.48 x 10"4, 9.38 x 10"9, 9.37 x 10"9,4.37 x 10"9 ], [ C5-L, 1.54 x 10"3, 1.54 x 10"3, 1.54 x 10"3,2.74 x 10"7, 1.56 x 10"7,2.74 x 107 ], [C5-R, 3.28 x 10"3,3.28 x 10"3, 3.28 x 10"3, 1.23 x 10"8, 1.12 x 10"8, 1.23 x 10"8], [ C6-L, 2.29 x 10"4,2.30 x 10"4,2.29 x 10"4,3.84 x 10"9,3.83 x 10"9,3.84 x 10"9 ], [C6-R, 3.27 X 10"4,3.27 X 10"4, 3.27 X 10"4, 7.56 X 10"9, 7.55 X 10"9, 7.56 X 10"9 ], [Dl-L, 5.63 X 10"3,5.64 X 10"3,5.63 X 10"3,3.18 X 10"8,3.07 X 10"8,3.18 X 10"8], [Dl-R, 8.20 X 10"3, 8.83 X 10"3, 8.20 X 10"3, 8.01 X 10"8, 7.95 X 10"8, 8.01 X 10"8 ], [D2

- L, 3.04 x 10"4, 3.07 x 10"4, 3.04 x 10"4, 6.98 x 10"9, 6.97 x 10"9, 6.98 x 10"9 ],

[D2-R,2.87 X 10"4,3.92 X 10"4,2.87 X 10"4,8.78 X 10"9,8.77 X 10"9,8.78 X 10"9],

[D3-L, 1.63 x 10"4, 1.64 x 10"4, 1.63 x 10"4, 1.77 x 10"9, 1.76 x 10"9, 1.77 x 10"9 ],

[D3-R, 1.89 X 10"4,2.28 X 10"4, 1.89 X 10"4,3.41 X 10"9, 3.41 X 10"9,3.41 X 10"9],

[D4-L, 2.73 X 10"4, 2.76 X 10"4, 2.73 X 10"4, 5.23 X 10"9, 5.22 X 10"9, 5.23 X 10"9],

[ D4-R,2.69 X 10"4, 3.56 X 10"4,2.69 X 10"4, 7.14 X 10"9, 7.14 X 10"9, 7.14 X 10"9],

[ El-L,5.70 X 10"3,5.72 X 10"3,5.70 X 10"3,3.18 X 10"8,3.07 X 10"8,3.18 X 10"8],

[ El-R, 8.38 x 10"3, 9.05 x 10"3, 8.38 x 10"3, 8.02 x 10"8, 7.95 x 10"8, 8.02 x 10"8],

[ E2-L,3.20 x 10"4,3.23 x 10"4,3.20 x 10"4,6.98 x 10"9,6.97 x 10"9,6.98 x 10"9),

[ E2 - R, 3.05 x 10"4, 4.22 x 10"4, 3.05 x 10"4, 8. 78 x 10"9, 8. 78 x 10"9, 8. 78 x 10 "9], [E3-L,5.70 X 10"3,5.72 X 10"3,5.70 X 10"3,3.18 X 10"8,3.07 X 10"8,3.18 X 10 "8], [£3-R,8.38 X 10"3,9.05 X 10"3,8.38 X 10"3,8.02 X 10"8,7.95 X 10"8,8.02 X 10 "8], [ E4 - L, 3.20 x 10"4, 3.23 x 10"4, 3.20 x 10"4, 6.98 x 10"9, 6.97 x 10"9, 6.98 X 10"9], [ £4-R,3.05 X 10"4,4.22 X 10"4,3.05 X 10"4,8.78 X 10"9,8.78 X 10"9, 8.78 x 10"9], [ E5-L, 1.63 x 10"4, 1.64 x 10"4, 1.63 x 10"4, 1.77 x 10"9, 1.76 x 10"9, 1.77 X 10"9], [ E5-R, 1.89 X 10"4,2.29 X 10"4, 1.89 X 10"4,3.41 X 10"9,3.41 X 10"9,3.41 X 10"9], [ £6-L, 2.83 X 10"4, 2.86 X 10"4, 2.83 X 10"4, 5.26 X 10"9, 5.25 X 10"9, 5.26 x 10"9], [ E6-R,2.83 x 10"4,3.80 x 10"4,2.83 x 10"4, 7.15 x 10"9, 7.14 x 10"9, 7.15 X 10"9)))

[Type text]

[Type text]

T. Margulies

proby:= Matrix(47,2, ((Al, 2.49 x 10"(-7) ], [A2, 1.18 x 10"(-7)], [BJ,8.45 x 10"(

-7) ], [B2 - L, 1.48 x 10" ( -7) ], [B2 - R, 1.47 x 10"( -7) ], [B3 - L, 2.90 x 10"(

[Type text]

-7) ], [B3 - R,2.90 x 10" ( -7) ], [B4 - L, 1.01 x 10" ( -7) ], [B4 - R, 1.01 x 10" (

-7) ], [B5 - L, 1.44 x 10"( -10) ], [B5 - R, 1.44 x 10" ( - IO} ], [B6 - L, 6.57 x 10" (

-11) ], [B6 - R, 6.56 x IO"( -11) ], [BP - V, 6.24 x 10"( -8) ], [BP - SGTR, 1.71 x 10"(-8 )], [CJ -L, 1.06 x 10"(-6) ], [Cl -R, 1.06 x 10"(-6) ], [C2 -L, 7.02 x 10"(-7) ], [C2-R,5.96 x 10"(-7) ], [C3 -L, 1.07 x 10"(-6) ], [C3 -R, 1.07 x 10"(-6) ], [C4-L,6.46 x 10"(-7) ], [C4-R,5.59 x 10"(-7) ], [C5-L,2.02 x IC

"(-10) ], [ C5 - R, 2.02 x 10"( -IO)], [ C6 - L, 1.30 x 10"( -10)], [ C6 -R, 1.01 x IC

"(-10)],[DJ -L, 3.25 x 10"(-10)], [DJ -R,3.25*10- 10], [D2-L, 1.74 x 10"(

[

-IO]

-10) ], D2 - R, 1.74* 10

, [ D3 - L, 0], [ D3-R, 3.32 x 10"( -12) ], [D4 -L, O],

[D4-R, 1.55 x 10" ( -12)], [El - L, 0 ], [ El-R, 6.36 x 10" ( -9} ], [E2 - L, 0 ], [E2-R, 3.13 x 10" ( -10) ], [E3 - L, 0 ], [E3-R,4.73 x 10" ( -9) ], [E4 - L, 0 ], [E4-R, 2.35 x I<

"( -10) ], [E5 - L, 0 ], [E5 - R, 2.68 x 10"( -11) ], [E6 -L, O], [E6-R, 4.79 x 10"(

-13) ))); Ml[ 1, 7]; proby[ 1, 2]; n== 47;prob== [seq(proby[i, 2], i = 1.. n) ];probt==

sort(prob ); probs := add(prob); pSvO := [seq(MJ[i, 2] *proby[i, 2], i = I.. n) ]; pSvOt :=

sort(pSvO); pSvl := add(pSvO); case:= [seq( l.O*i, i = 1.. n) ]; ptsp== [seq( [case[}],

probt[j)),j = 1.. n) ];plp := plot(ptsp, style= point, symbol= solidcircle, symbolsize= 12, color= blue, gridlines, title ='core melt probability per RY', titlefont

= [ COURIER, Bold, 14 ], labels= ["bin", "Prob."],labelfont= [COURJER, BOLDITALIC, 12], axesfont= [COURIER, 12]) ;ptsrm := [seq( [case[J],pSvOt[j)),j = 1

.. n) ]; plpr== plot(ptsrm, style = poin~ symbol= solidbox, symbolsize = 12, color= green, gridlines, title ='expected population dose per RY', titlefont = [ COURIER, Bold, 14 ], labels= ["bin", "person-Sv"],labelfont= [COURIER,BOLDITALIC, 12],

axesfont= [COURIER, 12]);

[Type text]

T. Margulies

Severe Accident Cost Analysis Notes Ofex = f

• prem

• LV

• Ct (Offsite exposure cost after discounting)

Onst = f

• DoeA

• LV

• Ct (Onsite exposure cost, short term)

Ontot = Onst + Ofex (Total Onsite cost=short-term + long-term costs)

OnLT = f

• 80 ee

• LV

• Ct* Sm (Onsite exposure cost, long-term)

Dev = f

• Cev *Ct* Sm (expected present value of decontamination & clean-up costs)

Rp = f

• CRP *Cf* PL (Power level factor (per 910MWe))

(replacement power cost summation of single event costs over the facility service life)

Total1nt = Ontot + Dev + Rp (Total Internal costs=Onsite total +Clean-up & decontamination total+ replacement power)

TotalExt = e

• Total1nt (External event costs obtained by a multiplier of the internal event cost total)

Total = Total1nt + TotalExt = (1 + e)

• Total1nt T

[ -rt]T l

-rT A present value factor useful in the costs calculation yields, fi0 e-rt dt = _e -

=....:.:..__,

-r o r

n n -t the limit definition of e, limitn->oo ( 1 + ;;:) ~ e, limitn-+oo [ ( 1 + ;;:) ]

~ e-t Note Define, 2:.. = ~. n = !:!., where, r = the interest rate per year, N = the number of compounding N

n r

intervals per year, and, t = term in years. This yields a continuous compounding interest formulation, l

-n*rt

= limit~00 [ ( 1 + ;) ]

~ e-rt

[Type text]

[Type text]

T. Margulies

[Type te><

restart: with(Student) : with(plots) : ir := 0.01; Ooca := 3300.; oocc := 2.* 104 ; Ced:= 1.5

• 109 ; Crp := l.2* 108; Lvalue := 20000.; E := evalf( O~OOOOOOOOi::l } Onst := 3.* 109 ;

Co VJ := 2.5; nmes := "Turkey Pts.";

yrsr

= 20.; nt := 7; Ct := seq

,j = 1.. nt ; cmp2 := [ cmp*yrsr ];

(

( 1 - exp( -ir*j *yrsr))

)

lf*J rep! := 10* Sm := seq

• = 1 nt

• pwrl := 2300* PL := --

(

(1-exp(-ir* i*repl))

)

1 rep! *ir*j 910

• pwrl; offprem := 3* 10.87769; pcm:= 3*9.14522* 10- 6;forjfrom 1 to ntdo Qffex[J] :=

Lvalue*pcm*offprem*Ct[j]; OnLT[j]== pcm*oocc*Lvalue*Ct[j]*Sm[j]; Onst[j]==

pcm* Lvalue*ooca*Ct[j]; Ontot[j]== Onst[j] + Offex[j] + OnL T[j]; Ocd[j]== pcm

• Crp*Ct[j]2.PL;Rp[j] := pcm*Crp*Ct[Jf*PL; Totlnt[j] := Ontot[j] + Ocd[j]

+ Rp[J]; Totext[j] := E

• Totlnt[j] : Total[j]== Totlnt[j] + Co VJ* Totext[j] :end do:

pty := seq( Total[}],}= l.. nt); ptx := seq( l.O* i, i = 1.. nt); pts== [ seq( [ptx[ i],pty[ i] ], i

= 1.. nt) ]; pltl== plot(pts, style= point, symbol= solidbox, color= blue, legend

= [Lvalue],gridlines, title ='Potential Averted Costs', titlefont = [ COURIER, Bold, 14 ], labels= ["Interest/ Year","$"], labelfont= [ COURIER,BOLDITALIC, 12],

axesfont= [COURIER, 12]); Lvaluel := 2000.;forjfrom 1 tontdo Offexl[j] := Lvaluel

• pcm*offprem*Ct[j]; OnLTJ[j]==pcm* oocc*Lvaluel *Ct[j] *Sm[}]; Onstl[j] := pcm

• Lvaluel *ooca*Ct[j]; OntotJ[j]== Onstl[j] + Offexl [j] + OnLTl [j]; OcdJ[j] :=

pcm* Crp*Ct[j]2.PL; Rpl[j] := pcm*Crp*Ct[Jf PL; Totlntl[j] := Ontotl[j] + Ocdl[f

+ Rpl [J]; TotextJ[J] := E*Totlntl[j] : Totall [j] := Totlntl[j] + Co VJ* TotextJ[j]:

end do:ptyJ := seq( Totall[j],j = 1.. nt);ptx := seq( 1.0* i, i = l.. nt);ptsl==

[seq( [ptx[i],ptyJ[i] ], i = I.. nt) ]; plt2 := plot(ptsl, style= point, symbol= solidbox, color

= green, legend= [Lvaluel], gridlines, title ='Potential Averted Costs', titlefont

[ COURIER, Bold, 14 ], labels= ["Interest/ Year", "$"],labelfont= [COURIER, BOLDITALIC, 12], axesfont= [COURIER, 12]); display( {pltl,plt2} }; offpremg :

278.; pcm== 9.14522*10- 6;forjfrom 1 tontdo Offex[j]== Lvalue*pcm*offpremg

• Ct[j]; OnLT[J]==pcm* oocc*Lvalue*Ct[j] *Sm[}]; Onst[j]==pcm* Lvalue*OOca

• Ct[j]; Ontot[j] := Onst[j] + Offex[j] + OnLT[j]; Ocd[j]== pcm* Crp*Ct[J]2.PL; Rp[j] := pcm*Crp*Ct[J]2.PL; Totlnt[j] := Ontot[j] + Ocd[J] + Rp[J]; Totext[j] := E

• Totlnt[j] : Total[j] := Totlnt[j] + Co VJ* Totext[j] :end do: pty3 := seq( Total[}],}= 1

.. nt); ptx := seq( 1.0* i, i = l.. nt}; pts3== [seq( [ptx[i],pty3[i)), i = 1.. nt) ]; plt3==

plot(pts3, style= point, symbol= solidbox, color= purple, legend= [Lvalue ], gridlines, title ='Potential Averted Costs', titlefont = [ COURIER, Bold, 14 ], labels

= ["Interest/Year", "$"], labelfont = [ COURIER, BOLDIT ALIC, 12 ], axesfont

= [COURIER, 12]); Lvaluel := 2000.;forjfrom 1 tontdo Offexl[j] := Lvaluel

• pcm

• offpremg*Ct[J]; OnLTJ[j] := pcm* oocc*Lvaluel *Ct[j]-Sm[j]; Onstl[j] := pcm

• Lvaluel *Ooca*Ct[j]; Ontotl[J] := Onstl[j] + Offexl [j] + OnLTl [J]; OcdJ[j] :=

pcm* Crp*Ct[j]2.PL; Rpl [J] := pcm*Crp*Ct[J]2.PL; Totlntl[j] := Ontotl[j] + Ocdl[f

+ Rpl [J]; TotextJ[j] := E*Totlntl[j] : Totall [j]== Totlntl[j] + Co VJ* Totextl[j]:

end do: pty4 := seq(Totall[j],j = 1.. nt);ptx := seq( 1.0* i, i = 1.. nt);pts4==

[ seq( [ptx[ i],pty4[ i] ], i = 1.. nt) ]; plt4== plot(pts4, style= point, symbol= solid box, color

= red, legend= [ Lvaluel ], gridlines, title = 'Potential Averted Costs', titlefont

= [ COURIER, Bold, 14], labels= ["Interest/ Year","$"], labelfont= [COURIER, BOLDITALIC, 12 ], axesfont= [ COURIER, 12]}; display( {plt3, plt4}};

plot3d( dollr* lO;O, dollr = 1.JO, carate = 4.. 5, title ='Dollars per person REM carafe* I

• ( a Statistical Latent Cancer Fatality Saved $M) ', titlefont

= [ COURIER, Bold, 14], labels= [ "$ M dollar", "person-REM/LCF rate",

"],labelfont= [COURIER,BOLDITALIC, 12],axesfont= [COURIER, 1argulies person -REM 12]}

[Type text]

Reliability System Model Basics Series system of independent R

1denttca1 components 1.0 0.9 0.8

0. 7 0.6

0. 5

0. 4 0.3 0.2 0.1,.__ __ ~-....---.-~~.....-~-......-~~

1 2

3 4

5 6

number of components

[-

0.1 0.01 0,001 -

0.0001 1 0.00001 -

0.000001 Parallel system oF independent identical components 1.0~=----?"-----:::;:;:=..... --~~~~~

0. 9 R

0. 8

0. 7 1

2 3

4 5

6 number of components l

~---0-.1-----0-.0-1___

0.001 -

0.0001 1 0.00001 -- 0.000001

[Type text]

T. Margulies

[Type text]

6 C0111b1nator1a1 system Probab11 ity number failed III of components m

1 2

3 4

5 6

-1

-2 1og system prob - 3

-4

-5

-6

• 1

• 2 3

• 4

• 5

• 61 Standby system Rel iabfl ity 1

0.8 0.6 system prob 0.4 0.2 0.j___~

.;:::::,._:::=::;;;;a.;;:-......

0 50000 100000 150000 t

[Type text]

T. Margulies

[Type text]

Standby system Re11abi1ity di fference 0.3 0.2 system prob delta 0.1 0.wt............. ~:.,..:~

0 100000 t

1-2 3 -

4 -

5-6 !

[Type text]

T. Margulies

[Type tex restart: with(Student): with(plots): with(CurveFitting): with(plottools): digits== 20; nt== 6;prob== [seq(.lk, k= l.. nt) ]; A.:= l.; a== 4.;probf== [ seq( ( 1 - exp(

b["] )ex)

)]

- pro/\\.

1

, i = l.. nt

coll== [green, orange,yellow, blue,purple, black] :

mult== l.;yl== mult;for ifrom l tontdo p[i]; sys[i]== [seq( ( 1 - probf[i])i,j = l..nt)] :

syps[i]== [seq( 1-(probf[i])J,j = l.. nt)] :end do:forjfrom 1 to ntdo ;ptsl[j] :=

[seq( [ i, mult* (sys[i][j]) ], i = I.. nt)]: ptslp[j] := [seq( [ i, mult* (syps[i][j]) ], i = l

.. nt)] :end do:forj from 1 to ntdo pll[j]== plot(ptsl [j], style= line, thickness= 2, color

coll[j], gridlines, title

' Series system of independent identical components', titlefont

= [ COURIER, Bold, 14], labels= ["number ofcomponents",

11 R"], labelfont

= [ COURIER, BOLDITALIC, 12 ], axesfont= [ COURIER, 12 ], legend= [prob[j)));

pllp[j]== plot(pts Jp[j], style= line, thickness= 2, color= coll[j], gridlines, title =

' Parallel system of independent identical components', titlefont

= [ COURIER, Bold, 14 ], labels= ["number of components"," R 11 ], labelfont

= [ COURIER, BOLDIT ALIC, 12 ], axesfont = [ COURIER, 12 ], legend= [ prob[j)));

pll 2[j]== Polynomiallnterpolation(ptsl [j], x); plsp2[j] := plot( {pll 2[j],ptsl [j]}, x

= I.. 6,y= 0.. yl, gridlines, color= coll[j]);end do: display(seq(pll[j],j = I.. nt) );;

display(seq(pllp[j],j = 1.. nt)); nc== [ seq( 1. i, i = I.. nt) ]; n== 6;for k from 2 to nt nt dopcmbl[k,2]== m2=J:)*(l -probf[2])m*(probf[2])1't-m :pcmbl[k,3]==

m m

m;;k(:) * ( 1 - prob.1[3])

111

• ( prob.1[3])1' 1-m: pcmbl[k, 4] := JJ:) *{ I - prob.I[ 4Jr

• ( prob.I[ 4 ]t 1 -m :end do: probe==.l;pcn := [ seq( m!* (n~t~ m) ! *probc!'

1*(l nt

- probc)nt-m, m= 1.. nt)] :forjfrom 1 tontdopsysc[j]== Lpcn[m] :end do:fork m=1 from 1 to nt do pie[ k]== plot( [ [ k, loglO (psysc[ k]) ] ], style= point, symbol= solidcircle, color= coll [ k], symbolsize = 12, gridlines, title =

' Combinatorial system Probability number failed m of n components',

titlefont = [ COURIER, Bold, 14 ], labels= [

11m 11," log system prob"], labelfont

= [COURIER,BOLDITALIC, 12],axesfont= [COURIER, 12], legend= [kl) ;end do:

display(seq(plc[ k], k= l.. nt)); AS==.0001; tot== 1.5

• 105 ;for kfrom 1 k-1

~ (AS t1f tontdo Rsb[k]== tl-exp(-t.s*tl)* L

plr[k]==

g=O g!

plot( Rsb [ k] ( tl), tl = 0.. tot, thickness= 2, color= coll [ k], gridlines, title=

' Standby system Reliability ', titlefont = [ COURIER, Bold, 14], labels

= [

11t",

11 system prob"], labelfont= [ COURIER, BOLDITALIC, 12 ], axesfont

= [COURIER, 12]) : co/2== [green, orange,yellow, blue,pwple, black];end do:forj from2 tontdosbdl[J] := t2-exp(-t.s*t2) * [

1i (AS*t2f g'

g=O 1I (AS*t2)g ] ;plrd[j] := plot(sbdl [J] ( t2), t2 = 0.. tot, thickness= 2, g=O g!

color= col2[i], gridlines, title=' Standby system Reliability difference',

titlefont = [ COURIER, Bold, 14 ], labels= [

11t",

11 system prob delta 11],labelfont

= [COURIER,BOLDITALIC, 12],axesfont= [COURIER, 12], legend= [j]) :end do:

display(seq(plr[ k], k= I.. nt) ) ; display(seq(plrd[j],j = 2.. nt) );

1argulies

About The Author Timothy S. Margulies enjoys presenting a collection of investigations and explorations for introducing and for teaching quantitative studies. He earned both Masters and Doctorate degrees at The Johns Hopkins University in Baltimore, Maryland. His career path is comprised of ten years at The Applied Physics Laboratory of Johns Hopkins, nine years at the US NRC, and eight at the US EPA. He was a Lecturer at Johns Hopkins on Risk Mathematics, Management, and System Safety. He was awarded a Civil Service Special Achievement Award and Civil Service Peer Award, Civil Service Bronze Medal.

[Type text]

[Type text]

T. Margulies