The Binomial Distribution

©2005 Dr. B. C. Paul

Scales for Data

Continuous Variables Numbers and anything in between over a set or infinite range

Ordered Number sets were one value is greater than another but there is

not a continuous distribution of numbers in-between Ordered category data

Category The numbers represent categories but one is not necessarily

greater than the other Example – my factories in three different locations

Some things are Category but they only have two states Yes or No

Two Category State Data Has a Binomial Distribution We see Binomial Data in engineering

In computer and microchip applicationsOften very useful in describing the operational

state of Equipment Our loader is working or not working Our truck is available or not available

Fundamentals of Binomial Probability Defining term in Binomial is P – the

probability of “success” (actually success can be arbitrary) If proportion of boys born is 50% then is P=.5

(someone else might have defined success as girls born)

Q is the opposite of P (ie prob of failure) Q+P = 1

N is the number of tries one makes

Illustrative Example

Weasel Nose power and light company is planning on building a new 1000 Megawattpower plant to meet baseload generation needs. Weasel Nose will build a coal plant because the cost of fuel will be low, but they have three different designs.

1- They could build a conventional pulverized coal power plant for $1400/KW capacityAll in cost

2- They could build an ultra super critical steam cycle plant for $1550/KW capacityAll in cost

3- They could build an IGCC for $1400/KW capacity all in cost

Things that influence cost to produce power Cost of Fuel Cost of Maintenance Environmental Cost General and Administrative Costs Capital Cost Amount of Power Produced

Amount of Power Produced

All plants need regular maintenance Operators schedule down time – hopefully during seasons of low

demand Weasel Nose will schedule their power plant to be running 97%

of the time – with 3% down time for maintenance Market may limit the amount of power the plant can sell

Demand for electricity varies by season and hour of the day In spring you probably don’t run your AC or your electric heat much

Power Plant Unscheduled Break-Downs Ratio of the power a plant does produce to what it could

on a 24/7/365 basis is called Capacity Factor Obviously a high capacity factor dilutes the fixed and investment

costs

Market Limitations on Power Sales(at least for this example)

Megawatts % Scheduled Time600 3750 7850 10

1000 80

Why Would they Build One Plant or the Other Conventional Steam Plant

It’s the cheapest to build It’s a well proven technology with large operating experience

base Ultra-Super Critical Steam Plant

Thermodynamics of fuel conversion to electricity are terrible Only about 28 - 34% of the fuel in a regular plant is ever converted

to electricity Ultra-Super Critical Steam Cycles convert 43% (maybe up to 45%) This can cut fuel costs and all environmental emissions by about

33% Design is simple but you do need to put more money into better

steel and other materials (higher pressures and corrosive tendency)

Why IGCC

Integrated Gas Combined Cycle converts the coal to gas and then burns the gas in a turbine and recovers waste heat in a steam boiler

Conversion efficiency is best But have a lot of parasitic processes so only net about 41 to

43% conversion efficiency (running an air separation plant takes a lot of energy)

IGCC can be more environmentally friendly SO2 becomes usable acid instead of tripling volume and

becoming a solid waste (scrubbers) IGCC may allow CO2 capture in the energy cycle at some future

date – Steam plants offer no reasonable hope of retrofit

Problems of IGCC

IGCC depends on a number of systems that all must work in succession First need an air separation plant to make oxygen

IGCC burns coal in a pure oxygen environment Next is a coal gasifier

Coal is reacted in a reducing environment to form carbon monoxide syngas

May have a shift gas reactor Third goes to a gas clean-up process

Ceramic candles bring down suspended ash Cryogenic- or Semi-Cryogenic processes are used to condense out

impure gases – need some real fancy heat exchangers Fourth gas goes to a gas turbine with a heat recovery boiler and

then a full steam generation system.

Binomial Probability for Processes in Succession Air Separation Plants run fairly reliably

Say they are 98% available Gasifiers have lots of mechanical parts and are hard on refractory

Say they are 85% available (if there is a maintenance schedule) Particulate Clean-Up

Ceramic candles are still being worked on – they like to break a lot Say 87% available

Cryogenic Gas Clean Up Complex but proven chemical process 97% available

Gas Combined Cycle Say 95% available

The Rule – Probability for Processes in Series is multiplied 0.98*0.85*0.87*0.97*0.95 = 0.6678 Has 66.78% Availability

Getting to the IGCC Capacity Factor Probability the things runs 66.78% Probability the things is scheduled to be

on line 97% Use Rule about Probability in Series

0.6678*0.97= 0.6478 or 64.78% Still Need to Deal with the Market

Limitation

Probability Rule – Mutually Exclusive Probabilities are Additive

Probability Market can take0.6 (600/1000 megawatts) is 3%0.75 (750/1000 megawatts) is 7%0.85 (850/1000 megawatts) is 10%1 all out 80%

Note that these probabilities are Mutually Exclusive and add up to 100%

Catching Suckers on Mutually Exclusive Suppose I have 5 haul trucks

Probability that a haul truck will start up and run is 85%

Not true that probability that one out of the 5 trucks will run is

0.85+0.85+0.85+0.85+0.85 = 4.45 or 450%

What went wrong What one haul truck does in fact has no effect on

whether other trucks will run Additivity works only for mutually exclusive events

Lets Get Our Market Capacity Limitation Now 0.6*0.03+0.75*0.07+0.85*.1+1*.8 = 0.955 And Finishing off our Capacity Factor

0.6478*0.955 = 0.619 or 61.9% Power Plant Generates

1000000*365*24*0.619= 5,422,082,551 kilowatt hours of electricy

How Capacity Factor Influences Cost Plant Cost $1,400,000,000 to build

Equivalent to $148,540,000 each year (assuming 30 year amortization at 10% interest – if

you had Engr. 361 you know how I did that – otherwise just believe)

Cost Per Kilowatt Hour to Build the Plant$148,540,000/ 5,422,082,551 = 0.0274

Ie 2.74 cents per kilowatt hour

Looking at the Alternative

Regular Power PlantTurbine Train 99% availableBoilers 97%0.97*0.99 = 0.9603 or 96.03% available

Put in Scheduled Hours and Market Limit0.9603*0.97*0.9555= 0.89Capacity Factor is 89%

What Does That Do to Cost

1000000*365*24*0.89=7,796,747,338 kilowatt hours

Same Cost for Regular Power Plant $148,540,000/7,796,747,338 = 1.91 cents per kilowatt

hour

The IGCC has a better “heat rate” (ie uses less fuel to do the same generation) If fuel and environmental costs are $1.25/million BTU

Getting Fuel and Environmental Costs Heat Rate represents the number of BTU

needed for 1 kilowatt hour of electricity If it were 100% conversion 3,467 BTU/kwh

Regular Steam Plant at 34% efficiency 3,467/0.34 = 10,197 BTU/kwh 1,000,000 BTU produces

1,000,000/10,197 = 98.07 kilowatt hours

$1.25/MMBTU / 98.07 kwh/MMBTU = 1.27 cents/Kwh

Now Checking the IGCC

3467/0.43 = 8063 heat rate 1,000,000/8063 = 124 kilowatt

hours/MMBTU $1.25/MMBTU/124kwh/MMBTU = 1 cent

per kwh

Compare and Line Up

Regular Steam Plant Fuel and

Environmental 1.27 c Capital Cost 1.91 c Total (less non fuel

OM and overhead) 3.18 cents/kwh

IGCC Fuel and

Environmental 1 cent Capital Cost 2.74 c Total (less non fuel

OM and overhead) 3.74 cent/kwh

Problem – the difference in capacity factor more than offsetsThe fuel and environmental savings

Can start to see why elegant processes that have to run inSeries can be the Binomial Kiss of Death

IGCC Plants are Actually a Little More Complicated than that Gas Turbines actually are not built larger than 250 megawatts

So our plant actually has 4 gas turbines – not 1 Also implies we have 4 HRSGs (Hertzigs – ok they are just heat

recovery boilers) We’ll assume we have more than one boiler on a steam header to

the steam turbines Two steam turbines on a header with 2 HRSGs each

We’ll also assume We can run the gas turbine with the steam generator down 1/3rd power from gas turbine (running on low BTU gas) and 2/3rd from

steam boilers We’ll add duct burners ahead of the HRSGs so we can bypass the gas

around the gas turbine and run the steam cycle if the gas turbine goes down (but can only run 2/3rds of gas or we’d have to much heat or flow for the boiler tube design)

We’ll add $50/kw of capacity for the extra flexability

More Complexities in the IGCC

Air Separation Plants larger than for 500 megawatts are really out of the realm of experience Will have two air sep plants not one

The cryogenic gas clean up trains are also limited in size Assume it will take 4 cryogenic trains.

Worst availability is in the gasifiers and ceramic candles Some designs compensate by adding spares Assume it takes 3 gasifiers per train We will put the gasifiers on a common header so we can route gas from

any gasifier to any turbine We will also add a couple of spare gasifiers with corresponding costs of

an extra $50/kw capacity (Our extras happen to now give us the same cost as an ultra-super

critical plant)

Now Lets Try to Get the Capacity Factor for Our More Complicated IGCC Start with the Air Sep Plants

Its now not all or nothing There are 4 possibilities

Plant #1 and Plant #2 could both be running Plant #1 could run and Plant #2 be down Plant #2 could run and Plant #1 be down Plants #1 and Plant #2 could be down

We probably know how to get probability for two independent events in series

Probability for both plants running 0.98*0.98 = 0.9604 or 96.04%

Probability for both plants down 0.02*0.02 = 0.0004 or 0.04%

By Elimination the probability for one running is 3.92%

For Our Air Separation Plant

96.04% we will have full capacity 3.92% we will have 50% capacity 0.04% we will be flat down Since we have headings to run oxygen

from any air separation plant to any gasifier each event at the air sep plant is equally likely to effect any gasifier

Moving on to those Gasifier Trains

I have 12 regular gasifiers plus 2 extra for a total of 14 gasifiers I have also have a really bad feeling about trying to

figure out how many possible combinations there are for that one

That’s why there is the Binomial Formula The formula says the probability for a particular

number of gasifiers being up is The probability that that event occurred * the number of ways

it could happen

Looking at My First Term

Consider the probability that I will have 12 gasifiers up and 2 down Probability is Pr*q(n-r)

I need a translation to even understand that formula P is probability that any one gasifier is available 85% (ie – 0.85) n is the number of gasifiers 14 q is the probability that any one gasifier is down 15% (ie-0.15) r is the number of units I propose to have available 12

Plug and Chug 0.8512*0.15(14-12) = 0.0032 or 0.32% probability that any one

configuration with 12 gasifiers running and 2 down will occur

Now for the Second Part

How many configurations are there with 12 gasifiers running and 2 down As you can see you’ll probably go nuts trying to

figure that out by hand

Time for a Formula – The Binomial Coefficient (or how many different configurations are there that does that)

!)!(

!

rrn

n

r

n

Note that the explanation mark means

factorial

Lovely – Whats a Factorial

Let n = 14, then n! is (14*13*12*11*10*9*8*7*6*5*4*3*2*1) Or its also a key you push on your calculator or the

=fact(n) function in excel

I kind of like the last couple options Using Excel I find

479001600208717829120

91

Thus there are 91 different configurations with 12 gasifiers running and twoDown – anyone want to list them or shall we just believe the formula

I Thought We Were in a Believing Mood Finishing

91*0.0032 = 0.29124 or 29.124% probability that there will be 12 gasifiers fit to run

Now we are ready for the probability for 14 gasifiers, 13 gasifiers, 11 gasifiers, 10 gasifiers, 9

gasifiers, 8 gasifiers, 7 gasifiers, 6 gasifiers, 5 gasifiers, 4 gasifiers, 3 gasifiers, 2 gasifiers, 1 gasifier and 0 gasifiers running

I bet if I offer to just give you the results off an Excel Spreadsheet you’ll be willing to believe again.

Lets Throw in One More Issue

Ceramic Candles are used to collect particulate matter out of the hot reducing syngas coming off the gasifier They have only 87% availability and each gasifier is paired with

a single set of candles Thus the probability that a gasifier is actually going to be

producing particulate free syngas is 0.85*0.87 = 0.7395

Now we’ll plug that into the formulas 14 running 1.33%, 13 running 6.7311%, 12 running 15.8151%,

11 running 22.8668%, 10 running 22.7306%, 9 running 16.4329%, 8 running 8.91%, 7 running 3.681%, 6 running 1.1642%, 5 running 0.2805%, 4 running 0.0507%, 3 running 6.66X10-3%, 2 running 6.02X10-4%, 1 running 3.35X10-5%, everything down 8.65X10-7%

Effect of Gasifiers on Availability

Remember we had 2 extra gasifiers – we would actually only run 12 for full capacity even if 14 were available

We apply the probability additivity rule here Prob of 14 + Prob of 13 + Prob 12 = Prob that we will

have full capacity on the gasifiers Note the events are mutually exclusive if you have 12

gasifiers that are fit to run you don’t have 14, 13, or 10

Prob of Full Capacity is 23.88%

That was Probably the Worst – Lets Look at the Cryogenic Systems

Any one train has 97% probability of being up

Probability of all 4 up 88.53% Probability of 3 up 10.95% Probability of 2 up 0.5081% Probability of 1 up 0.0105% Probability of a total bust 8.1X10-5%

Now for the Combined Cycle

We rigged it so we could run the gas turbines without the HRSGs or vs. versa Gas Turbines have 1/3rd of our capacity and would

need 100% of syngas to run Heat Recovery Boilers

Can run either with the gas turbine If Gas Turbine runs 97% of the time and the HRSGs

run 98% and the steam turbines run 99% we get just under 95% availability for full gas combined cycle

Of course we are rigged to be able to get partial capacity even when break-downs occur

Combined Cycle Probabilities

Gas Turbines have same probability as the cryogenic cycle gas clean-up Have 4 trains with 97% availability on each unit

HRSG 4 units but 98% probability of being available Prob of all 4 HRSGs is 92.24% Prob of 3 HRSGs is 7.53% Prob of 2 HRSGs is 0.23% Prob of 1 HRSG is 3.14X10-3% Prob of no HRSG running is 1.6X10-5%

Steam Turbine Trains (2 available) Prob of both running is 98.01% Prob of one running is 1.98% Prob of bust is 0.01%

Now We Have to Combine all these probabilities to get a capacity factor We will use cross multiplication tables with

capacity and Probability If you don’t see what I’m doing the first type – take a

deep breath (we’ll be doing it more than once)

Starting with Air Separation Prob of 100% capacity is 96.04% Prob of 50% is 3.92% Prob of 0% is 0.04%

Probability Table Starting with Air Separation Plant Data

Gasifier ProbAir Sep Prob

96.04%3.92%0.04%

We are starting to set the probabilities into a cross multiplicationTable

Now Add the Gasifier Probability Data

Gasifier ProbAir Sep Prob 23.88% 22.87% 22.73% 16.43% 8.91% 3.68% 1.16% 0.28% 0.05% 0.01% 0.00% 0.00% 0.00%

96.04%3.92%0.04%

We will now cross multiply to get the probability of every possibleCombination of gasifiers being operational with air separation plantsBeing available.

Now We Will Cross Multiply

Gasifier ProbAir Sep Prob 23.88% 22.87% 22.73% 16.43% 8.91% 3.68% 1.16% 0.28% 0.05% 0.01% 0.00% 0.00% 0.00%

96.04% 22.93% 21.96% 21.83% 15.78% 8.56% 3.54% 1.12% 0.27% 0.05% 0.01% 0.00% 0.00% 0.00%3.92% 0.94% 0.90% 0.89% 0.64% 0.35% 0.14% 0.05% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00%0.04% 0.01% 0.01% 0.01% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

We will set up the same type of table for capacity

Gasifier CapacityAir Sep Cap 100.00% 91.67% 83.33% 75.00% 66.67% 58.33% 50.00% 41.67% 33.33% 25.00% 16.67% 8.33% 0.00%

100.00%50.00%

0.00%

We are interested in capacity to produce departicalized syn-gas

We don’t get this by cross multiplyingIf you have 50% air sep and gasifiers for 66.67% you don’t get33.33% capacity – you will get 50%

Trick here is to select the most limiting number in our cross-comparison table

Gasifier CapacityAir Sep Cap 100.00% 91.67% 83.33% 75.00% 66.67% 58.33% 50.00% 41.67% 33.33% 25.00% 16.67% 8.33% 0.00%

100.00% 100.00% 91.67% 83.33% 75.00% 66.67% 58.33% 50.00% 41.67% 33.33% 25.00% 16.67% 8.33% 0.00%50.00% 50.00% 50.00% 50.00% 50.00% 50.00% 50.00% 50.00% 41.67% 33.33% 25.00% 16.67% 8.33% 0.00%

0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

Now I’ll Transpose All This Into Columns

Gasifier CapacityAir Sep Cap 100.00% 91.67% 83.33% 75.00% 66.67% 58.33% 50.00% 41.67% 33.33% 25.00% 16.67% 8.33% 0.00%

100.00% 100.00% 91.67% 83.33% 75.00% 66.67% 58.33% 50.00% 41.67% 33.33% 25.00% 16.67% 8.33% 0.00%50.00% 50.00% 50.00% 50.00% 50.00% 50.00% 50.00% 50.00% 41.67% 33.33% 25.00% 16.67% 8.33% 0.00%0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

Capacity Probability100.00000%

50.00000%0.00000%

91.66667%50.00000%

0.00000%83.33333%50.00000%

0.00000%75.00000%50.00000%

0.00000%66.66667%50.00000%

0.00000%58.33333%50.00000%

0.00000%50.00000%50.00000%

0.00000%41.66667%41.66667%

0.00000%33.33333%33.33333%

0.00000%25.00000%25.00000%

0.00000%16.66667%16.66667%

0.00000%8.33333%8.33333%0.00000%0.00000%0.00000%0.00000%

I’ll do the same to lining up the correspondingProbabilities in the next column of the table IAm making.

Now the Columns are Completed I Will use the Sort Option and arrange the results by capacity

Capacity Probability100.00000% 0.229308

50.00000% 0.00935950.00000% 9.551E-05

91.66667% 0.219612450.00000% 0.0089638

0.00000% 9.147E-0583.33333% 0.218304350.00000% 0.0089104

0.00000% 9.092E-0575.00000% 0.157821150.00000% 0.0064417

0.00000% 6.573E-0566.66667% 0.085571550.00000% 0.0034927

0.00000% 3.564E-0558.33333% 0.035350350.00000% 0.0014429

0.00000% 1.472E-0550.00000% 0.011180850.00000% 0.0004564

0.00000% 4.657E-0641.66667% 0.002694441.66667% 0.00011

0.00000% 1.122E-0633.33333% 0.00048733.33333% 1.988E-05

0.00000% 2.028E-0725.00000% 6.401E-0525.00000% 2.613E-06

0.00000% 2.666E-0816.66667% 5.784E-0616.66667% 2.361E-07

0.00000% 2.409E-098.33333% 3.217E-078.33333% 1.313E-080.00000% 1.34E-100.00000% 8.305E-090.00000% 3.39E-100.00000% 3.459E-12

This will be my sort control column but I will have itSort the one by the side as well

Capacity Probability100.00000% 0.229308

50.00000% 0.00935950.00000% 9.551E-05

91.66667% 0.219612450.00000% 0.0089638

0.00000% 9.147E-0583.33333% 0.218304350.00000% 0.0089104

Enlarged view ofPart of columns

Looking at Our Sorted ResultCapacity Probability

0.00000% 9.551E-050.00000% 9.147E-050.00000% 9.092E-050.00000% 6.573E-050.00000% 3.564E-050.00000% 1.472E-050.00000% 4.657E-060.00000% 1.122E-060.00000% 2.028E-070.00000% 2.666E-080.00000% 2.409E-090.00000% 1.34E-100.00000% 8.305E-090.00000% 3.39E-100.00000% 3.459E-128.33333% 3.217E-078.33333% 1.313E-08

16.66667% 5.784E-0616.66667% 2.361E-0725.00000% 6.401E-0525.00000% 2.613E-0633.33333% 0.00048733.33333% 1.988E-0541.66667% 0.002694441.66667% 0.0001150.00000% 0.009359550.00000% 0.008963850.00000% 0.008910450.00000% 0.006441750.00000% 0.003492750.00000% 0.001442950.00000% 0.011180850.00000% 0.000456458.33333% 0.035350366.66667% 0.085571575.00000% 0.157821183.33333% 0.218304391.66667% 0.2196124

100.00000% 0.229308

Capacity Probability0.00000% 9.551E-050.00000% 9.147E-050.00000% 9.092E-050.00000% 6.573E-050.00000% 3.564E-050.00000% 1.472E-050.00000% 4.657E-060.00000% 1.122E-06

Enlarged view ofcolumn

I will next add all the probabilities for the sameCapacity to simplify the table before going on.

Resulting Simplified Table(Represents probabilities for different amounts of syngas with the particulate matter removed)

Capacity Probability Percent0% 0.000400009 0.040001

8.33% 3.348E-07 3.35E-0516.67% 6.0204E-06 0.000602

25% 6.66213E-05 0.00666233.33% 0.000506842 0.05068441.67% 0.002804336 0.280434

50% 0.050248129 5.02481358.33% 0.035350349 3.53503566.67% 0.085571479 8.55714875.00% 0.157821095 15.7821183.33% 0.218304321 21.8304391.67% 0.219612428 21.96124

100.00% 0.229308036 22.9308

So What Did I Just Do

I took probability for the air separation plant running and probability for the gasifier and ceramic candles to be running I did a cross multiplication table to get all possible combinations

I figured the percent capacity I would have in each case in another table

I transposed all the figures from the two tables into two long columns

I sorted the columns by capacity I added all the probabilities together for the same

capacity to simplify the table

What Will I Do Next?

I will take the probability of having particle free syngas and repeat the process with the cryogenic gas clean-up systemThis will give me table of clean syn-gas

capacities that will be available for me to burn in my combined cycle.

Lets Set Up My Cross Multiplication and Capacity Tables

Probability TableCryogenic Prob

Syngas Prob 0.8852928 0.109521 0.005081 0.00010476 8.1E-070.00040001 0.0003541 4.38E-05 2.03E-06 4.19049E-08 3.24E-10

3.348E-07 2.964E-07 3.67E-08 1.7E-09 3.50736E-11 2.71E-136.0204E-06 5.33E-06 6.59E-07 3.06E-08 6.30697E-10 4.88E-126.6621E-05 5.898E-05 7.3E-06 3.38E-07 6.97924E-09 5.4E-110.00050684 0.0004487 5.55E-05 2.58E-06 5.30968E-08 4.11E-100.00280434 0.0024827 0.000307 1.42E-05 2.93782E-07 2.27E-090.05024813 0.0444843 0.005503 0.000255 5.26399E-06 4.07E-080.03535035 0.0312954 0.003872 0.00018 3.7033E-06 2.86E-080.08557148 0.0757558 0.009372 0.000435 8.96447E-06 6.93E-08

0.1578211 0.1397179 0.017285 0.000802 1.65333E-05 1.28E-070.21830432 0.1932632 0.023909 0.001109 2.28696E-05 1.77E-070.21961243 0.1944213 0.024052 0.001116 2.30066E-05 1.78E-070.22930804 0.2030048 0.025114 0.001165 2.40223E-05 1.86E-07

This table was produced by multiplying numbers at the edge of theTable to get numbers in green in the center

Lets Get the Capacity TableCapacity Table

Cryogenic CapSyngas Cap 100.00% 75.00% 50.00% 25.00% 0.00%

0.00% 0.00% 0.00% 0.00% 0.00% 0.00%8.33% 8.33% 8.33% 8.33% 8.33% 0.00%

16.67% 16.67% 16.67% 16.67% 16.67% 0.00%25.00% 25.00% 25.00% 25.00% 25.00% 0.00%33.33% 33.33% 33.33% 33.33% 25.00% 0.00%41.67% 41.67% 41.67% 41.67% 25.00% 0.00%50.00% 50.00% 50.00% 50.00% 25.00% 0.00%58.33% 58.33% 58.33% 50.00% 25.00% 0.00%66.67% 66.67% 66.67% 50.00% 25.00% 0.00%75.00% 75.00% 75.00% 50.00% 25.00% 0.00%83.33% 83.33% 75.00% 50.00% 25.00% 0.00%91.67% 91.67% 75.00% 50.00% 25.00% 0.00%

100.00% 100.00% 75.00% 50.00% 25.00% 0.00%

The values at the center come from selecting the most limiting value in the yellowCells at the edge.

Now We Will Put Our Data Into Columns

Capacity Probability0.00% 0.00035418.33% 2.964E-07

16.67% 5.33E-0625.00% 5.898E-0533.33% 0.000448741.67% 0.002482750.00% 0.044484358.33% 0.031295466.67% 0.075755875.00% 0.139717983.33% 0.193263291.67% 0.1944213

100.00% 0.20300480.00% 4.381E-058.33% 3.667E-08

16.67% 6.594E-0725.00% 7.296E-0633.33% 5.551E-0541.67% 0.000307150.00% 0.005503258.33% 0.003871666.67% 0.009371975.00% 0.017284775.00% 0.023908975.00% 0.024052175.00% 0.025114

0.00% 2.032E-068.33% 1.701E-09

16.67% 3.059E-0825.00% 3.385E-0733.33% 2.575E-0641.67% 1.425E-0550.00% 0.000255350.00% 0.000179650.00% 0.000434850.00% 0.000801950.00% 0.001109250.00% 0.001115850.00% 0.0011651

0.00% 4.19E-088.33% 3.507E-11

16.67% 6.307E-1025.00% 6.979E-0925.00% 5.31E-0825.00% 2.938E-0725.00% 5.264E-0625.00% 3.703E-0625.00% 8.964E-0625.00% 1.653E-0525.00% 2.287E-0525.00% 2.301E-0525.00% 2.402E-05

0.00% 3.24E-100.00% 2.712E-130.00% 4.877E-120.00% 5.396E-110.00% 4.105E-100.00% 2.272E-090.00% 4.07E-080.00% 2.863E-080.00% 6.931E-080.00% 1.278E-070.00% 1.768E-070.00% 1.779E-070.00% 1.857E-07

Capacity Probability0.00% 0.00035418.33% 2.964E-07

16.67% 5.33E-0625.00% 5.898E-0533.33% 0.000448741.67% 0.002482750.00% 0.044484358.33% 0.031295466.67% 0.0757558

Enlarged view of tabledata

I will now sort by capacity so I can add all theProbabilities for exactly the same capacity togetherAnd simplify the final table.

Simplified Table for Clean Sygas

Clean Syngas Capacity and ProbabilityCapacity Probability Percent

0% 0.000400818 0.0400828.33% 3.34799E-07 3.35E-05

16.67% 6.0204E-06 0.00060225% 0.000171332 0.017133

33.33% 0.000506789 0.05067941.67% 0.00280404 0.280404

50% 0.055049154 5.50491558.33% 0.035167007 3.51670167.67% 0.085127668 8.512767

75% 0.230077533 23.0077583% 0.193263246 19.3263292% 0.194421303 19.44213

100% 0.203004755 20.30048

This represents all the stagesOf the power plant thatProduced a clean syngas thatIs now available to burnIn the combined cycle.

Assumed Plant Features

Combined Cycle uses a gas turbine and then the waste heat goes into a HRSG to produce steam for a steam turbine

This design would allow either the steam or gas turbines to run without the otherAlthough obviously there is an efficiency

penalty We’ll go after the Gas Turbines first

Gas Turbine is 97% AvailableGas Turbine Prob

Syngas Prob 0.885293 0.109521 0.00508086 0.00010476 8.1E-070.0004008 0.000355 4.39E-05 2.0365E-06 4.19897E-08 3.25E-103.348E-07 2.96E-07 3.67E-08 1.70107E-09 3.50736E-11 2.71E-13

6.02E-06 5.33E-06 6.59E-07 3.05888E-08 6.30697E-10 4.88E-120.0001713 0.000152 1.88E-05 8.70512E-07 1.79487E-08 1.39E-100.0005068 0.000449 5.55E-05 2.57492E-06 5.30912E-08 4.1E-10

0.002804 0.002482 0.000307 1.42469E-05 2.93751E-07 2.27E-090.0550492 0.048735 0.006029 0.000279697 5.76695E-06 4.46E-08

0.035167 0.031133 0.003852 0.000178679 3.6841E-06 2.85E-080.0851277 0.075363 0.009323 0.000432522 8.91797E-06 6.9E-080.2300775 0.203686 0.025198 0.001168992 2.41029E-05 1.86E-070.1932632 0.171095 0.021166 0.000981943 2.02463E-05 1.57E-070.1944213 0.17212 0.021293 0.000987827 2.03676E-05 1.57E-070.2030048 0.179719 0.022233 0.001031439 2.12668E-05 1.64E-07

Gas Turbine CapSyngas Cap 100.00% 75.00% 50.00% 25.00% 0.00%

0.00 0.00% 0.00% 0.00% 0.00% 0.00%8.33% 8.33% 8.33% 8.33% 8.33% 0.00%

16.67% 16.67% 16.67% 16.67% 16.67% 0.00%25.00% 25.00% 25.00% 25.00% 25.00% 0.00%33.33% 33.33% 33.33% 33.33% 25.00% 0.00%41.67% 41.67% 41.67% 41.67% 25.00% 0.00%50.00% 50.00% 50.00% 50.00% 25.00% 0.00%58.33% 58.33% 58.33% 50.00% 25.00% 0.00%67.67% 67.67% 67.67% 50.00% 25.00% 0.00%75.00% 75.00% 75.00% 50.00% 25.00% 0.00%83.33% 83.33% 75.00% 50.00% 25.00% 0.00%91.67% 91.67% 75.00% 50.00% 25.00% 0.00%

100.00% 100.00% 75.00% 50.00% 25.00% 0.00%

Set up the cross tables with the probability of gas turbines available withThe probability for various amounts of syngas

We will take these results and put them in a column and sort them by capacity

Finishing the Capacity Factor for the Gas Turbines

Capacity Prob0.00% 0.0003550.00% 4.39E-050.00% 2.04E-060.00% 4.2E-080.00% 3.25E-100.00% 2.71E-130.00% 4.88E-120.00% 1.39E-100.00% 4.1E-100.00% 2.27E-090.00% 4.46E-080.00% 2.85E-080.00% 6.9E-080.00% 1.86E-070.00% 1.57E-070.00% 1.57E-070.00% 1.64E-078.33% 2.96E-078.33% 3.67E-088.33% 1.7E-098.33% 3.51E-11

16.67% 5.33E-0616.67% 6.59E-0716.67% 3.06E-0816.67% 6.31E-1025.00% 0.00015225.00% 1.88E-0525.00% 8.71E-0725.00% 1.79E-0825.00% 5.31E-0825.00% 2.94E-0725.00% 5.77E-0625.00% 3.68E-0625.00% 8.92E-0625.00% 2.41E-0525.00% 2.02E-0525.00% 2.04E-0525.00% 2.13E-0533.33% 0.00044933.33% 5.55E-0533.33% 2.57E-0641.67% 0.00248241.67% 0.00030741.67% 1.42E-0550.00% 0.04873550.00% 0.00602950.00% 0.0002850.00% 0.00017950.00% 0.00043350.00% 0.00116950.00% 0.00098250.00% 0.00098850.00% 0.00103158.33% 0.03113358.33% 0.00385267.67% 0.07536367.67% 0.00932375.00% 0.20368675.00% 0.02519875.00% 0.02116675.00% 0.02129375.00% 0.02223383.33% 0.17109591.67% 0.17212

100.00% 0.179719

Capacity Prob Percent0.00% 0.000401628 0.0401638.33% 3.34799E-07 3.35E-05

16.67% 6.02039E-06 0.00060225.00% 0.000276031 0.02760333.33% 0.000506735 0.05067441.67% 0.002803744 0.28037450.00% 0.059824744 5.98247458.33% 0.034984616 3.49846267.67% 0.08468616 8.46861675.00% 0.293576993 29.357783.33% 0.171094562 17.1094691.67% 0.172119782 17.21198

100.00% 0.17971865 17.97187

Weighted Average Capacity Factor80.93%

Sorted ColumnOf results fromThe cross tablesGives the sumOf the probabilityOf each capacity Multiply

EachCapacity byThe probability of its occurrence

Now Need the Steam Cycle Part

This will be a two step process We have 4 HRSGs that generate the steam for 2

steam turbines We will need to do the cross multiplication tables

to get the probability that the steam cycle equipment will be running

Then we can do another cross mulitplication exercise to get the capacity factor for the steam cycle

Getting Probabilities for the Steam Cycle Running

Steam Turbine ProbHRSG Prob 0.9801 0.0198 0.0001

0.92236816 0.904013 0.018263 9.22E-050.07529536 0.073797 0.001491 7.53E-060.00230496 0.0022591 4.56E-05 2.3E-07

3.136E-05 3.074E-05 6.21E-07 3.14E-091.6E-07 1.568E-07 3.17E-09 1.6E-11

Steam Turbine CapacityHRSG Capacity 100% 50% 0%

100% 100% 50% 0%75% 75% 50% 0%50% 50% 50% 0%25% 25% 25% 0%

0% 0% 0% 0%

We do our probability and capacity tables

Compiling Results for Availability of the Steam Cycle

Capacity Probability0% 1.568E-070% 3.168E-090% 9.224E-050% 7.53E-060% 2.305E-070% 3.136E-090% 1.6E-11

25% 3.074E-0525% 6.209E-0750% 0.002259150% 0.018262950% 0.001490850% 4.564E-0575% 0.073797

100% 0.904013

We take our capacity and probability results and sort them

Capacity Probability Percent0% 0.00010016 0.010015998

25% 3.13569E-05 0.00313568650% 0.022058467 2.2058467275% 0.073796982 7.379698234

100% 0.904013034 90.40130336Sum up the results(We will next do tables with theProbability of syngas being available)

Now Do the Cross Tables of Steam Cycle and Syngas Availability

Steam Cycle ProbSyngas Prob 0.0001002 3.14E-05 0.022058 0.073796982 0.904013034

0.00040082 4.015E-08 1.26E-08 8.84E-06 2.95792E-05 0.0003623453.348E-07 3.353E-11 1.05E-11 7.39E-09 2.47072E-08 3.02663E-07

6.0204E-06 6.03E-10 1.89E-10 1.33E-07 4.44287E-07 5.44252E-060.00017133 1.716E-08 5.37E-09 3.78E-06 1.26438E-05 0.0001548860.00050679 5.076E-08 1.59E-08 1.12E-05 3.73995E-05 0.0004581440.00280404 2.809E-07 8.79E-08 6.19E-05 0.00020693 0.0025348890.05504915 5.514E-06 1.73E-06 0.001214 0.004062461 0.0497651530.03516701 3.522E-06 1.1E-06 0.000776 0.002595219 0.0317914330.08512767 8.526E-06 2.67E-06 0.001878 0.006282165 0.0769565220.23007753 2.304E-05 7.21E-06 0.005075 0.016979028 0.2079930880.19326325 1.936E-05 6.06E-06 0.004263 0.014262244 0.174712493

0.1944213 1.947E-05 6.1E-06 0.004289 0.014347705 0.1757593920.20300476 2.033E-05 6.37E-06 0.004478 0.014981138 0.183518945

Steam Cycle CapacitySyngas Cap 0.00% 25.00% 50.00% 75.00% 100.00%

0.00% 0.00% 0.00% 0.00% 0.00% 0.00%8.33% 0.00% 8.33% 8.33% 8.33% 8.33%

16.67% 0.00% 16.67% 16.67% 16.67% 16.67%25.00% 0.00% 25.00% 25.00% 25.00% 25.00%33.33% 0.00% 25.00% 33.33% 33.33% 33.33%41.67% 0.00% 25.00% 41.67% 41.67% 41.67%50.00% 0.00% 25.00% 50.00% 50.00% 50.00%58.33% 0.00% 25.00% 50.00% 58.33% 58.33%67.67% 0.00% 25.00% 50.00% 67.67% 67.67%75.00% 0.00% 25.00% 50.00% 75.00% 75.00%83.33% 0.00% 25.00% 50.00% 75.00% 83.33%91.67% 0.00% 25.00% 50.00% 75.00% 91.67%

100.00% 0.00% 25.00% 50.00% 75.00% 100.00%

Now Put Results in Columns and Sort by Capacity

Capacity Probability0.00% 4.015E-080.00% 3.353E-110.00% 6.03E-100.00% 1.716E-080.00% 5.076E-080.00% 2.809E-070.00% 5.514E-060.00% 3.522E-060.00% 8.526E-060.00% 2.304E-050.00% 1.936E-050.00% 1.947E-050.00% 2.033E-050.00% 1.257E-080.00% 8.841E-060.00% 2.958E-050.00% 0.00036238.33% 1.05E-118.33% 7.385E-098.33% 2.471E-088.33% 3.027E-07

16.67% 1.888E-1016.67% 1.328E-0716.67% 4.443E-0716.67% 5.443E-0625.00% 5.372E-0925.00% 1.589E-0825.00% 8.793E-0825.00% 1.726E-0625.00% 1.103E-0625.00% 2.669E-0625.00% 7.215E-0625.00% 6.06E-0625.00% 6.096E-0625.00% 6.366E-0625.00% 3.779E-0625.00% 1.264E-0525.00% 0.000154933.33% 1.118E-0533.33% 3.74E-0533.33% 0.000458141.67% 6.185E-0541.67% 0.000206941.67% 0.002534950.00% 0.001214350.00% 0.000775750.00% 0.001877850.00% 0.005075250.00% 0.004263150.00% 0.004288650.00% 0.00447850.00% 0.004062550.00% 0.049765258.33% 0.002595258.33% 0.031791467.67% 0.006282267.67% 0.076956575.00% 0.01697975.00% 0.014262275.00% 0.014347775.00% 0.014981175.00% 0.207993183.33% 0.174712591.67% 0.1757594

100.00% 0.1835189

Capacity Prob Percent0% 0.000500938 0.050094

8.33% 3.34766E-07 3.35E-0516.67% 6.01979E-06 0.00060225.00% 0.000202653 0.02026533.33% 0.000506722 0.05067241.67% 0.002803671 0.28036750.00% 0.075800289 7.58002958.33% 0.052360372 5.23603767.67% 0.083238687 8.32386975.00% 0.268563204 26.8563283.33% 0.174712493 17.4712591.67% 0.175759392 17.57594

100.00% 0.183518945 18.35189101.7974

Weighted Average Capacity Factor81.78%

Figuring the Plants Over-All Capacity

Overall Capacity Factor Fraction of CapacityCapacity for Gas Turbines 80.93% 0.333333333Capacity for Steam Cycle 81.78% 0.666666667

81.50%

With a 97% schedule the Capacity Factor will be79.05%

Some Observations

We went from 65% to 79% How Did We Do That

Avoided Big Single Train Processes (we had to) Also helps chances with not having everything down at the same

time Set up to Allow Things to Run as Independent as Possible

Any air sep plant could supply any gasifier Any gasifier could supply any cryogenic unit Any syngas supply could go to any turbine Gas turbines could run without the HRSGs Steam turbines could run with the gas turbines down

Added Spare Units in the Achilles Heal Areas that just can’t be down

Issues Remaining

79% Capacity Factor is still behind 89% Issue of Peak Demand

When whether is hot (or you have a hot opportunity to sell power for a good price) what is the chance your plant will be ready to run full out?

IGCC unit was around 18% chance This has implications for needing to build additional system or back-

up capacity

Problem – Even if IGCC could run at the same price the process complexity increases the need for back-up systems that cost money and limits capacity factor

The Comparison

Ultra-Super Critical Power Plant probably has Two trains A boiler burns fuel to make steam The steam turns the turbines

Our IGCC has Two air separation plants 14 gasifiers with ceramic candle precipitators 4 cryogenic gas clean-up systems 4 gas turbines 4 HRSGs 2 steam turbine trains

Which one would you rather try to keep running

Binomial Summary

Binomial Distributions Describe Yes/No types of data Probabilities in Series multiply each other Only mutually exclusive probabilities can be added The Binomial Probability Formula gives the probabilities

that any number of multiple processes will be running at any one time

Probabilities for two events in series each with multiple outcomes can be dealt with by cross multiplying individual probabilities

(Side Note – Can do this kind of thing on games of Black Jack)