Β© ABB Group 09 May, 2014 | Slide 1
Applications of Computational Mathematics to Industrial Products and ProcessesThe Case of ABB
L. Ghezzi, A. Sciacca, F. Rigamonti / ABB Italy β Mathematics for the Digital Factory, Berlin May 2014
Contents
ABB company overview
Applications to products
Applications to processes
Future research trends
Β© ABB Group 09 May, 2014 | Slide 2
Β© ABB Group
| Slide 4
A global leader in power and automation technologiesLeading market positions in main businesses
February 13, 2014
~150,000employees
Present
in
countries+100
Formed
in
1988merger of Swiss (BBC, 1891)
and Swedish (ASEA, 1883)
engineering companies
In revenue
(2013)
billion42$
Β© ABB Group
| Slide 5
Power and productivity for a better worldABBβs vision
February 13, 2014
As one of the worldβs leading engineering companies, we
help our customers to use electrical power efficiently, to
increase industrial productivity and to lower environmental
impact in a sustainable way.
Β© ABB Group
| Slide 6
Power and automation are all around usYou will find ABB technologyβ¦
February 13, 2014
crossing oceans and on the sea bed,
orbiting the earth and working beneath it,
on the trains we ride and in the facilities
that process our water,
in the fields that grow our crops and
packing the food we eat,
in the plants that generate our power and
in our homes, offices and factories
Β© ABB Group
Low Voltage Products division
09 May, 2014 | Slide 8
Low Voltage ProductsBusiness Units
MNS
Conventional
Switchgear
MNS Intelligent
Switchgear
MNS Integrated
Switchgear
Circuit Breakers
Switches
Fusegear and
Cable
Distribution
Cabinets
Modular DIN-Rail
Products
Intelligent
Building Control
KNX
Enclosures and
Cable Systems
Control and
Protection
Electronic
Products and
Relays
Connection
Wiring
Accessories
Industrial Plugs
and Sockets
Door Entry
Systems
Breakers
and SwitchesLV Systems
Enclosures and
DIN-Rail Products
Control
Products
Wiring
Accessories
Wire and cable
management
Cable protection
systems
Critical process
protection
People protection
Thomas &
Betts
Β© ABB Group
Low Voltage Products division
09 May, 2014 | Slide 9
Low Voltage ProductsChannels and Markets served
Distributors Panel Builders OEMs End-users and
Utilities
Industry Buildings and
infrastructuresRenewable energy E-mobility
System Integrators and
Contractors
Building Automation and
Energy Efficiency
Computational Magnetohydrodynamics (MHD)Numerical simulation of electric arcs
Β© ABB Group 09 May, 2014 | Slide 11
Navier-Stokes PDEβs
Maxwell PDEβs
Radiation PDEβs
Network DAEβs
Nonconstant media
Plasma-solid interface
Short circuits in
low voltage circuit breakers
numerical
One experiment
Experiment envelope
Short circuit prediction vs. measured
Computational Magnetohydrodynamics (MHD)Numerical simulation of electric arcs
Β© ABB Group 09 May, 2014 | Slide 18
Navier-Stokes PDEβs
Maxwell PDEβs
Radiation PDEβs
Network DAEβs
Nonconstant media
Plasma-solid interface
Stock sizingOverview
Β© ABB Group 09 May, 2014 | Slide 20
Time
Stock
level
Safety
stock
(re-order
point)
Re-order
issued
Supply
lead time
πΏ
π·
1
Demand
(average)
Re-order
lot
supplied
Re-o
rde
r lo
t
π = π·πΏ + ππ
π
Deterministic model
Ordered quantity
(average)πππ
Deterministic approach
Everything sharp
System may be mathematical, or black box, or else
Β© ABB Group 09 May, 2014 | Slide 21
System
π = πΏπ· + ππ
π
π
π·
πΏ
π
Stock sizingStochastic problem
Β© ABB Group 09 May, 2014 | Slide 23
Time
Stock
level
πΏ
π = π·πΏ + ππ
π
π· as a Stochastic variable
πππ
π·
1
Stock-out!
Stock sizingStochastic problem
Β© ABB Group 09 May, 2014 | Slide 24
Time
Stock
level
πΏ
π = π·πΏ + ππ
π
πΏ as a Stochastic variable
πππ
π·
1
Stock-out!
Stock sizingStochastic problem
Β© ABB Group 09 May, 2014 | Slide 25
Time
Stock
level
πΏ
π = π·πΏ + ππ
π
π as a Stochastic variable
πππ
π·
1
Stock-out!
Stock sizingStochastic problem
Β© ABB Group 09 May, 2014 | Slide 26
Time
Stock
level
πΏ
π = π·πΏ + ππ
π
π as a Stochastic variable
πππ
π·
1
Stock-out!
Stochastic approach
Β© ABB Group 09 May, 2014 | Slide 27
System
π = π·πΏ + ππ
Not necessarily Gaussian!
Uncorrelated
(for simplicity)
Stochastic approach
Β© ABB Group 09 May, 2014 | Slide 28
System
π = πΏπ· + πππ~ππ(ππ)
π·~ππ· ππ·
πΏ~ππΏ ππΏ
π~ππ ππ
π~ππ ππ
Stock sizingStochastic problem
Β© ABB Group 09 May, 2014 | Slide 30
Time
Stock
level
πΏ
π = π·πΏ + ππ
π
π as a Stochastic variable
πππ
π·
1
Stochastic approachTraditional
System linearization π:= πΈ πΏ πΈ π· + πΈ π πΈ π
Variance propagation ππ = πΏ2ππ·2 + π·2ππΏ
2 + π2ππ2 +π2 ππ
2
Output PDF assumed Gaussian, usually unproperly!
Β© ABB Group 09 May, 2014 | Slide 32
System
π = πΏπ· + ππ
πΈ π , ππ
πΈ π , ππ
πΈ π· , ππ·
πΈ πΏ , ππΏ
πΈ π , ππ
Stochastic approachMonte Carlo
Simple
Non-intrusive
Parallel
Large number of stochastic variables allowed
Slow, very slow(!): 106 (or even more) runs can be required
Β© ABB Group 09 May, 2014 | Slide 33
parameters
π
System
π¦π = π(π₯π)Input
π₯π
Output
π¦π
Output PDF
π~ππ(ππ)Input PDF
π~ππ(ππ)System
π¦π = π(π₯π)Input
π₯π
Output
π¦πSystem
π¦π = π(π₯π)Input
π₯π
Output
π¦πSystem
π¦π = π(π₯π)Input
π₯π
Output
π¦πSystem
π¦π = π(π₯π)Input
π₯π
Output
π¦πSystem
π¦π = π(π₯π)Input
π₯π
Output
π¦πSystem
π¦π = π(π₯π)Input
π₯π
Output
π¦πSystem
π¦π = π(π₯π)Input
π₯π
Output
π¦πSystem
π¦π = π(π₯π)Input
π₯π
Output
π¦πSystem
ππ(π)
= π(ππ(π)
, π)
Input
ππ(π)
Output
ππ(π)
Orthogonal Polynomial Sequences (OPS)
Probability density distribution ( metric)
π₯~ππ₯ π
Inner product in πΏ2(β)
π’ π , π£ ππ₯β πΈ π’π£ =
β
π’ π β π£ π β πππ₯ π ππ
Gram-Schmidt process applied to 1, π, π2, β¦ , ππ , β¦
polynomials πππ₯ π π=0
+β
Β© ABB Group 09 May, 2014 | Slide 34
Orthogonal Polynomial Sequences (OPS)
Polynomials are independent (degree argument)
Polynomals are orthogonal (or orthonormal, if desired)
πππ₯, ππ
π₯
π₯β πΈ ππ
π₯πππ₯ =
β
πππ₯ π β ππ
π₯ π β πππ₯ π ππ = πππ
π₯
π₯πππ
π₯
π₯
ππΏππ
Polynomials form a basis of πΏ2 β
π₯(π) =
π=1
+β
ππ β πππ₯(π)
Fourier series expansion
ππ =π₯ π , ππ
π₯ ππ₯
πππ₯ π , ππ
π₯ ππ₯
=πΈ[π₯ππ
π₯]
πππ₯
π₯
2
Β© ABB Group 09 May, 2014 | Slide 35
Stochastic approachPolynomial Chaos Expansion (PCE)
Β© ABB Group 09 May, 2014 | Slide 36
System
π = πΏπ· + πππ~ππ(ππ)
π·~ππ· ππ·
πΏ~ππΏ ππΏ
π~ππ ππ
π~ππ ππ
π· =
ππ·=0
+β
πππ· β πππ·π· (ππ·)
πΏ =
ππΏ=0
+β
πππΏ β πππΏπΏ (ππΏ)
π =
ππ=0
+β
πππ β ππππ (ππ)
π =
ππ=0
+β
πππ β πππ
π(ππ)
Stochastic approachPolynomial Chaos Expansion (PCE)
Β© ABB Group 09 May, 2014 | Slide 37
π~ππ ππ
π = π(π) =
π =0
+β
ππ β πππ(π)
π = (ππ·, ππΏ , ππ , ππ)
π = (ππ·, ππΏ , ππ , ππ)
πππ π β ππ
π ππ·, ππΏ , ππ , ππ = πππ·π· ππ· β πππΏ
πΏ ππΏ β ππππ ππ β πππ
πππ
π = ππ· + ππΏ + ππ + ππ
multi-index
to be truncated to π
πΈ πππππ
π = ππππ
ππππ
π
ππΏππ =
1 (π = π)0 (π β π)
(comput. by, e.g., fast MC)
πΈ π = ππ πππ
π
2
ππ =
π =0
π
ππ2 ππ
ππ
2
Surrogate model
By tensor product
Orthogonality inheritance
(analytical, by
orthogonality)
ππ =π π ,ππ
π ππ
πππ π ,ππ
π ππ
=πΈ[π(π)ππ
π(π)]
πππ
π
2 = π=1π π€ππ ππ ππ
π(ππ)
πππ
π
2 = π=1π π€π π· ππ·,π πΏ ππΏ,π + π ππ,π π(ππ,π) ππ
π(ππ)
πππ
π
2
Stochastic approachPolynomial Chaos Expansion (PCE)Non-intrusive approach (collocation)
Β© ABB Group 09 May, 2014 | Slide 38
π = π(π) =
π =0
π
ππ β πππ(π)
Inner products integrals Gaussian quadrature π model evaluations
Must not be too large
(tensorization!)
Contain stochastic variables!
System
π = πΏπ· + ππ
Stochastic approachPolynomial Chaos Expansion (PCE)Intrusive approach (Galerkin)
Β© ABB Group 09 May, 2014 | Slide 39
π = π(π) =
π =0
π
ππ β πππ(π)
Differential system
β(π) = π
β π ,πππ
π= π, ππ
π
π
β
π =0
π
ππ β πππ , ππ
π
π
= π,πππ
π
π =0
π
ππ β β πππ , ππ
π
π= π, ππ
π
π
Strong form
Weak form
In case of inear differential operators
In case ofβ¦
Stock sizingStochastic problem
Β© ABB Group 09 May, 2014 | Slide 40
Not Gaussian at all!
Co
st
Stocked quantity
Se
rvic
e le
ve
l (n
on
-sto
ck
-ou
t p
rob
ab
ilit
y)
Stock sizingPareto Front, per single item
Β© ABB Group 09 May, 2014 | Slide 41
COST
PE
RF
OR
MA
NC
E
Same performance
higher cost
inferiorSame cost
lower performance
inferior
Each point on the front is Β«optimalΒ» in the senseβ¦
Infeasible
Stock sizingPareto Front, per single item
Β© ABB Group 09 May, 2014 | Slide 42
COST
PE
RF
OR
MA
NC
E
Concave hereOperational
excellence
Where to locate
the optimal choice,
item by item?
?
Stock sizingConstrained optimization problem
Β© ABB Group 09 May, 2014 | Slide 43
π₯π
π¦π
COST
PE
RF
OR
MA
NC
E
π-th item
π1πππ₯π + π2πππ¦π = ππ
π1πππ₯π + π2πππ¦π > ππ
π1πππ₯π + π2πππ¦π β€ ππ
Stock sizingConstrained optimization problem
Β© ABB Group 09 May, 2014 | Slide 44
π¨π β π₯π , π¦π β€ ππ
π¦ππ£π
Same cost, lower performance inferior
π₯π
π¦π
COST
PE
RF
OR
MA
NC
E
Stock sizingConstrained optimization problem
Β© ABB Group 09 May, 2014 | Slide 45
π¨π β π₯π , π¦π β€ ππ
Same performance, higher cost inferior
π₯π
π¦π
COST
PE
RF
OR
MA
NC
E
π₯ππ€π
Stock sizingConstrained optimization problem
Β© ABB Group 09 May, 2014 | Slide 46
max π =
π=1
π
π¦ππ£π
π=1
π
π₯ππ€π β€ π,
subject to
ππ β€ π₯π β€ π’π , βπ β {1,β¦ , π}
π¨π β π₯π , π¦π β€ ππ , βπ β {1,β¦ , π}
Nonlinear knapsack problem
All variables real
Many exact methods known
(easy problem)
Linear Programming (LP)
Simplex methodBudget constraint
Other possible bounds
System laws constraint
Goal function:
performance
Stock sizingConstrained combinatorial optimization problem
Non-concave case
Made piecewise concave
Dinsjunctive choice variables
Integer: π§π β β€2, π β {1,2,3}
π§1 + π§2 + π§3 = 1
Exactly one equals 1, others vanish
Difficult problem
Β© ABB Group 09 May, 2014 | Slide 47
π§1 π§2 π§3
Stock sizingConstrained combinatorial optimization problem
Β© ABB Group 09 May, 2014 | Slide 48
max π =
π=1
π
π¦ππ£π + π§πππ β 1 β π§π ππ(ππ)π£π
π=1
π
π₯ππ€π + π§πππ β 1 β π§π πππ€π β€ π,
subject to
ππ β€ π₯π β€ π’π , βπ β {1,β¦ , π}
π§π β 0,1 , βπ β {1,β¦ , π}
π¨π β π₯π , π¦π β€ ππ , βπ β {1,β¦ , π}
πβπΎπ
π§π β€ 1 ,
πβπ β€ π§π , βπ β 1,β¦ , π
π₯π β π’π β ππ π§π β€ ππ , βπ β {1,β¦ , π}
πβπΎπ
π§π = 1 ,
Dependency chain of non Β«must-haveΒ» items
Dependency chain of Β«must-haveΒ» items
Future research trendsIncreasing size
Industrial needs
Large scale problems
Multi-processor HW is very cheap
Only marginal improvements to current
transistor technology
Mathematical challenges
Efficient and robust preconditioners
Parallel algorithms
Hierarchy and multi-level attacks
Β© ABB Group 09 May, 2014 | Slide 77
Future research trendsIncreasing complexity
Industrial needs
Nonlinear problems
Including strongly nonlinear!
Mathematical challenges
Robust formulations
Necessary and/or sufficient conditions for
convergence
Β© ABB Group 09 May, 2014 | Slide 78
linear weakly nonlinear
strongly nonlinear
Future research trendsIncreasing specificity
Industrial needs
From general purpose to specific technologies
From general, Β«textbookΒ» problems & solutions
To specific, taylorized problems & solutions
Mathematical challenges
Exploit the mathematical structure of the operators
E.g., Whitney forms & algebraic topology
geometrically conformal comput. electromag.
Find out closest standard problem
Develop original approach to add specific features
Standard ideas exist, a lot of work to do
Innovative ideas welcome!
Β© ABB Group 09 May, 2014 | Slide 79
Whitney 0-forms
Whitney 1-forms
Whitney 2-forms
Whitney 3-forms = FVM
Future research trendsIntegration of systems
Industrial needs
Multi-phisical problems
Multi-scale problems
PDEβs + DAEβs
Multi-disciplinary combinatorial optimization
Mathematical challenges
Robust formulations
Necessary and/or sufficient conditions for convergence
From weak coupling to strong coupling?
Original attacks to nonstandard discrete problems
Β© ABB Group 09 May, 2014 | Slide 80
CFD
MECH
EMAG Planning
Logistics
Production
Future research trendsStochastics
Industrial needs
Stochastic differential equations (SDE)
Non-deterministic problems and/or boundary conditions
Either intrinsically (finance, markets, etc.)
Or de facto (e.g., deterministic PDE with non
completely known material properties or forcing terms)
Mathematical challenges
Beyond MonteCarlo & non-intrusive, black-box attacks
Β«Stochastic dimensionsΒ» added to physical dimensions
Preconditioners, robustness, convergence, etc.
Pure mathematics as well!
Β«LargeΒ» number (> 10) of stochastic variables
Β© ABB Group 09 May, 2014 | Slide 81
Future research trendsCombinatorics
Industrial needs
Discrete, combinatorial optimization ( processes)
Sub-optimal solutions frequently adequate
A good solution to a realistic problem is betterβ¦
β¦than the mathematically best solution to an
oversimplistic problem
Mathematical challenges
Large scale
From pure (meta-)heuristics
To Β«ancillaryΒ» heuristics in exact methods
As a fast albeit sharp bounding tool
Branching + Bounding / Pricing / β¦
Β© ABB Group 09 May, 2014 | Slide 82
if z==1 then
x >= LB;
else
x = 0;
end
πΏπ΅ β π§ β€ π₯ β€ π β π§
(π β« πΏπ΅)