Refuelling Nuclear Reactors – a multistage optimisation problem?
Richard Overton, British Energy, Gloucester
Mathematics 2010 – Saddlers’ Hall, London, April 22nd 2010
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Contents
1. A brief background to British Energy’s reactors and electricity production
2. A simplified description of the refuelling an Advanced Gas-cooled Reactor
3. A simple model for a stable equilibrium state
4. Formulating as an detailed time-dependent optimisation problem
5. The objective, constraints, and different approaches to optimising
6. Observations on obtaining an optimum (a personal view)
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Prologue
A youngster’s view of mathematics :
1. Logic
2. “A Right Answer” – no uncertainty
3. Precision
4. Perfection
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British Energy’s Reactors
> British Energy operates…
> a Pressurised Water Reactor (PWR) at Sizewell (Suffolk)
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British Energy’s Reactors (2)
> …. and 14 Advanced Gas-cooled Reactors (AGRs)
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British Energy’s Reactors (3)
> … that’s 15 nuclear reactors at 8 sites around the UK
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British Energy’s Reactors – scale of operations
> British Energy produces 15-20% of UK electricity from nuclear power
> Each AGR produces 500-600 MW of electricity (½ million “1 bar fires”).
> Together all our reactors save emission of around 35 million tonnes CO2
annually (equivalent to half the UK’s car emissions)
> Each reactor’s daily electricity production sells for around £½ m
> Fuel costs are around 20% of selling price
> Optimisation can potentially help
(a) maximise reactor output
(b) minimise fuel costs
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Generation of energy from nuclear fission
> Heat energy produced by splitting (fission) of uranium atoms by neutrons
Heat
(Uranium)
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Reactor schematic
> Enrichment : Proportion of U235 increased from natural (0.7%) to about 3.5% to assist
fission
> More neutrons produced by fission – chain reaction
Fuel assemblies
Reaction rate controlled
by neutron-absorbing
rods partially inserted
into reactor
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Gas Coolant
Steam
Reactor
Primary
Circuit
Turbine
Alternator
Boiler
Water
Water
Secondary
Circuit
Condenser
Continuous
Supply of
Cooling Water
Water
Pump
Pump
Pump
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Refuelling an Advanced Gas-cooled Reactor
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The AGR Fuel Element
Eight elements as a ‘fuel assembly’ loaded into each channel
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About 300 fuel channels per reactor….
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Typical Distribution of Channel Powers
332 fuel channels
New fuel
Old Fuel
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Typical distribution of Fuel ‘Age’
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The need to refuel…
Operation & Generation →
fuel becomes less reactive
control rods withdrawn slowly
Each channel needs refuelling every 6-7 years (generally the older fuel)
Every few weeks a batch of 8 or so channels are refuelled
Not a trivial operation …..
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AGR Fuelling Machine
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A simple refuelling model
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‘deadtime’ d
Shutdown period
Scaled Reactor Power
1
‘deadtime’ d
Refuelling batch 3
‘deadtime’ d
Refuelling batch 2
Refuelling batch 1
Simplified Refuelling Process
p
n channels
refuelled
r per day
n channels
refuelled
r per day
0
n channels
refuelled
r per day
0
n channels
refuelled
r per day
1095

N refuelling batches

  
   N  1n
d
1p
n/r

d70
dn/r
)11ppddnn
70
N
(p
i1
1165
1165

Nn
N
r
rr
Fuel demanded by reactor
1165
Time (days)
= Fuel supplied to reactor
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Number of batches
Equivalent days at full power
Fractional ‘load factor’
Daily income (£)
1095i  n
N
n  (1  p)i(d  n / r )
D  1095 N (1  p)(d  n / r )
 (1095i  n)(1  p)(d  n / r ) 
1095 

n

(
1

p
)
i
(
d

n
/
r
)


L
1165
 G.L
Can now determine sensitivity to n,p,d,r and
help define long term refuelling strategy
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Results from a simple model
Annual Reactor Output
Demand exceeds supply
Supply exceeds demand
Load low enrichment fuel
Fuel supply capability (elements/year)
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Results from a simple model
Annual Reactor Output
Demand exceeds supply
Supply fulfills demand
Load higher
enrichment fuel
Fuel supply capability (fuel assemblies/year)
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A more detailed refuelling model
[
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Control rod position
Reactor Power
Timeline of repeated refuelling & operation
Time
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Channel Power
Effect of age on channel power
5
Age of fuel (yrs)
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Effect
Effectofofhigher
age on
enrichment
channel power
on channel power
Channel Power
Maximum channel power allowed
Higher enrichment
again
Higher enrichment
5
6
Age of fuel (yrs)
7
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Effect of gadolinium poison on channel power
Channel Power
Maximum channel power allowed
Effect of gadolinium ‘poison’
1.5
5
6
Age of fuel (yrs)
7
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Optimisation of the detailed model
[
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Objective Function & Constraints
> Objective :
Maximise fuel life, or generation, or ‘profit’ over a long period (years), in which
there are k refuelling batches
i.e. find max ΣFi
> Constraints:
Keep power profile across reactor as flat as possible i.e. limit peak & spread
‘Shutdown requirement’ ensures reactor can be shutdown safely
Max {Pi} ≤ P’
maximum power limit
Max {Pi} - Min {Pi} ≤ S’
limit spread in channel powers
Use 2 enrichments
(usually fixed zones in reactor)
> … and many other constraints not mentioned here!
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Mathematical Model
> Choices (“controls”) at refuelling batch k :
No. of channels to refuel nk
Size of refuelling batch k
(k choices)
Channels for refuelling
(Ii)k = 0 ……………..…….. do not refuel channel i
= 1 ……………………. refuel channel i
300C choices (~1015)
…. for i = 1,2,… nk
nk
Fuel type to use
(ei)K ε { E1, E2 }
………… Enrichment choice (of 2)
(qi)K ε { Q1, Q2,…QL } ……... Poison choice (typically 2-3)
…. for i = 1,2,… nk
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Mathematical Model (2)
> Constraints
Let Pik = power in channel i after refuelling batch k
Pik ≤ Pmax i = 1, 300
Power limit for each channel
(300 constraints)
Maxi {Pik} – Mini {Pik} ≤ S
Range of powers limited
(1 constraint)
∫ Pik(t) dt ≤ A
Energy output (Age) limit on fuel assembly
(300 constraints)
SDk (nk, (Ik,ek,qk)) ≥ SDmin
Shutdown capability
(1 constraint)
Typically 20000 constraints to cover a 4 year period (30 batches)
Constraint functions are nonlinear, evaluated by ‘black box’ physics model (PANTHER code)
Importantly
(1) Refuelling a channel affects neighbouring channels
(2) Refuelling choice affects next choice for neighbouring channels
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Optimisation options (1)
> Option 1 : Exhaustive evaluation
> Choosing 8 channels for one refuelling batch
> In practice, will choose from oldest 10% of fuel
300
15
C8 ~ 10 choices
30
7
C8 ~ 10 choices
8
> Must also choose refuelling batch size 4-12 channels → 10 choices
> To be useful, need to evaluate over k = 20 refuelling batches (3 years) :
6000
C160 choices
> ….and also need to choose fuel types for each refuelled channel
Each choice requires one evaluation of the objective
…and evaluation of all constraints (PANTHER physics model)
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Optimisation options (2)
> Option 2 : Heuristic approach
> Choose oldest 8 channels for refuelling
> Quickly leads to violation of range of power constraint
> Have programmed in a set of heuristic rules, based on knowledge/experience, but
still performs poorly
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Optimisation options (3)
> Option 3 : Formal ‘textbook’ methods
> Iterative hillclimbing approach?
> Newton-Raphson?
…with penalty functions to handle constraints
x n 1  x n 
f '(xn )
f "(xn )
> Far too high dimensionality, discrete choices, ‘black box’ model (no derivatives)
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Optimisation options (4)
> Option 4 : Linear/integer programming
> Piecewise linear approximation to physics model
> Integer 0/1 variables
> Refuelling choice (Ii)k (0/1) variables
> Again, far too high dimensionality, beyond computational
feasibility
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Optimisation options (5)
> Option 5 : Dynamic programming
> Exploit multistage structure to subdivide problem into k subproblems
> Use Bellman’s recursive Principle of Optimality
> Fi(Si-1) = Maxui { fi(ui, Si-1) + Fi+1(Si (ui, Si-1)) } for i = 1, 2, 3 …k
> Relies on state Si at stage i depending only on state at stage i-1 (Si-1) and
choices made at stage i (ui)
8
> Still ~ 10 choices of channels to refuel at each stage
> But unfortunately state Si depends on choices ui, ui-1, ui-2,… made at several
previous refuellings (the gadolinium poison effect persists over several
refuelling batches)
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Optimisation options (6)
> Option 6 : Genetic Algorithms
> Collaborative work with Imperial College
> “Breed” from initial population of possible solutions
> Random selection of characteristics (refuelling channels, fuel type) from 2
individual solutions to create new solution.
> Evaluate constraints
> Evaluate objective
> If improvement, keep in population with pre-determined probability
> Does need many tens of thousands of evaluations to get good solutions
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Objective Function (1)
> First attempts with Genetic Algorithm
> Objective : maximise Age of fuel
> Tends to
> (a) minimise batch size (take each fuel assembly to its age limit)
> (b) create more refuelling batches => loss of generation
> Instead, need a financially based objective function….
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Objective Function (2)
Let
G
= value of 1 full power day generation
F1
= cost of 1 assembly of fuel type 1
F2
= cost of 1 assembly of fuel type 2
fpd (i) = number of full powers days generation after refuelling batch i
Then in N refuelling ‘outages’ and generation runs, the value of generation is :
N
G   fpd (i )
i 1
The cost of the fuel for N refuelling batches is :
N
 (F n ( i )  F n ( i ))
i 1
1 1
2 2
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Objective Function (3)
And a suitable objective function to maximise is then:
Z
N 1
N 1
i 1
N 1
i 1
G. fpd (i )   [ F1n1 (i )  F2 n2 (i )]  G. fpd ' ( N )  F1n'1 ( N )  F2 n' 2 ( N ))
N
 fpd(i )  fpd' ( N )   (a  (n (i )  n (i ))b(1  p)  m(i )c(1  p))
i 1
i 1
1
2
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Objective Function (4)
> Success with ~50000 to 100000 evaluations of objective function,
giving choice of different refuelling batches to refuelling planner
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> However, in practice :
> Other constraints appear –
Early refuelling of a channel required
Occasional breakdown of fuelling machine – smaller refuelling batch
> Objective function changes (selling price, fuel cost) Step change, contractual change, government intervention
> Improvements in safety –
Additional restrictions on fuel age or power : new or changed constraints
> Solution needs to be robust to such changes
i.e. avoid significant losses if rules change
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UK Baseload Power Prices to July 2008 (annuals)
90
85
Electricity
Pool
BETTA
Market
NETA
Market
Tight supply/demand
balance in the oil and coal
markets push prices to
record levels
80
75
70
65
Station Gate
Market Prices
60
Market Price - £/MWh
Further Oil &
Gas price rises
NBP
Market Prices
Cold winter &
high gas prices
Rising oil,
coal & gas
prices
Oct '02
TXU collapses E.on
55
buys generation &
supply business
50
45
40
Sept '02
BE announces
Apr '02
state-aid
1.5GW plant is
mothballed
35
30
25
Oil & Gas
price rises
Apr '03
2 GW of plant closes
2.3GW is mothballed
Increasing security
over winter gas
supplies
Fear of Winter
05 gas supply
Effect of LCPD
& high carbon
shortage
20
15
Oct '03
3GW of plant returns
10
prices
Weak gas &
mild winter
weather
Dec '03
EU ETS enters
front year
5
0
Aug-99
Feb-00
Aug-00
Feb-01
Aug-01
Feb-02
Aug-02
Feb-03
Aug-03
Feb-04
Aug-04
Trade Date
Feb-05
Aug-05
Feb-06
Aug-06
Feb-07
Aug-07
Feb-08
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Uranium Ore prices
Historic Uranium Spot Prices
150
140
130
120
110
90
80
70
60
50
40
30
20
2006 $
2008
2006
2004
2002
2000
1998
1996
1994
1992
1990
1988
1986
1984
1982
1980
1978
1976
1974
0
1972
10
1970
US$ / lb U3O8
100
Outturn $
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Conclusion
> For complex systems such as AGR refuelling:
> Not worth the effort in optimising too finely
> Optimisation can be undermined by step change in objective or
constraints
> System managers do not necessarily seek optimum
> Develop training on amelioration techniques and know when to stop!
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References
D. Barrable, J. H. Kershaw, R. S. Overton, K. Brearley and D. R. Gray
“Increasing fuel irradiations in advanced gas-cooled reactors in the UK”
Nuclear Energy, 37
A. K. Ziver , C. C Pain, C. R. E. de Oliveira, A. J. H. Goddard and R. S. Overton
“Genetic algorithms and artificial neural networks for loading pattern
optimisation of advanced gas-cooled reactors.”
Annals of Nuclear Energy 31, Issue 4
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Acknowledgements
> Grateful thanks to:
Vanessa Strong, John Foulkes, Andy Williams, Nigel Rhodes,
Chris Hall, Rik Bahia (colleagues at British Energy)
… and Kay & Liz Overton
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Epilogue
Trainee mathematician
Precision
Perfection
A right answer
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Metamorphosis in the attic
Mature industrial
Trainee
mathematician
mathematician/physicist
Precision
Perfection
Puts
up with uncertainty
A right answer
Tolerates imperfection
Faced with intractable problems
With apologies to Oscar Wilde
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Questions
Any questions?
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