Stochastic methods and models for optimising dynamic multi-objective problems

Stochastic methods and models
for multi-objective dynamic
optimisation problems
Eric S Fraga
Department of Chemical Engineering, UCL
University College London (UCL)
20 October 2017
Universidad de Salamanca
v1.39 23rd November 2017

2 Dynamics http://www.ucl.ac.uk/~ucecesf/
Acknowledgements
The following did the actual work for the case studies
presented:
Dr Oluwamayowa Amusat, formerly Chemical
Engineering, UCL.
Mr Alistair Rodman, University of Edinburgh.
with inputs from Dr Dimitrios Gerogiorgis (Edinburgh IMP)
and Dr Paul Shearing (UCL Chemical Engineering).
Any errors, omissions or bloopers are mine and mine alone.

Examples
Pressure Swing Adsorption (PDE)
∂
∂z
(uci )+ b
∂ci
∂t
+(1− b)ρs
∂qi
∂t
= DL
∂2
ci
∂z2
, ∀z ∈ (0, L]
Dwelling energy systems (Delay DE)
C
d
dt
Tr (t) = Whp(t − λ2) − ∆T(t − λ1) · α + γ∆T2
Beer fermentation (Nonlinear, control)
d
dt
[XA] = µx ·[XA]−µDT ·[XA]+µL·[XL] , µX =
µx0 (T) · [S]
kx + [EtOH]

Challenges

Deﬁnition (Stochastic)
adjective, Statistics.
1 of or relating to a process involving a randomly
determined sequence of observations each of which is
considered as a sample of one element from a probability
distribution.

Optimisation methods
Optimisation
methods
Deterministic Stochastic
Gradient
based
Direct
search
Steepest
descent
Dynamic
programming
Nelder
Mead
Hooke &
Jeeves
ESF & McKinnon, 2003
Whole
process
synthesis
ESF & Zilinskas, 2006
Integrated
process
with HENS
Nature
inspired
Genetic
algorithm
Ant
colony
Plant
propagation
Garrard & ESF, 1998
MENS
ESF & Rowe 2003
HENS
ESF & Amusat, 2016
Energy systems
(Strawberry)
Rodman, ESF & Gerogiorgis, 2017
Beer fermentation
(Strawberry)

Nature inspired optimisation
Source

Strawberry plant propagation algorithm
Given: f (x), ng , np, nr .
Output: z, set representing approximation to Pareto front.
1: p ← initial random population of size np
2: for ng generations do
3: prune population p, removing similar solutions
4: N ← ﬁtness(p) rank based
5: ˜p ← φ
6: for i ← 1, . . . , np do
7: j ← select(p, N) ﬁtness based
8: for k ← 1, . . . , nr ∝ Nj do runners
9: ˜p ← ˜p ∪ neighbour(xj , 1 − Nj ) propagation
10: p ← ˜p ∪ {xi |xi ∈ p ∧ f (xi ) nondominated} elitism
11: z ← {xi |xi ∈ p ∧ f (xi ) nondominated} Pareto set
A Salhi & ESF (2011), Proc. IceMATH K2–1–8.

9 Fermentation http://www.ucl.ac.uk/~ucecesf/
Fermentation
essential step in manufacture of
alcoholic beverages.
responsible for alcohol content
and taste of final product.
process is temperature
controlled.
aim is to enhance alcohol
production, reduce time and
control flavours.
Source
Problem
What are the trade-offs between time and alcohol content
while constraining flavour affecting chemicals?

Optimisation problem
Solve
max
T(t)
z =
[EtOH]tf
−tf
, t ∈ [0, tf ]
where
T(t) temperature control profile,
tf fermentation time, and
[EtOH]tf
final ethanol concentration
subject to end-point constraints on concentration of
flavour affecting chemicals.
Concentrations of all species modelled by set of nonlinear
ordinary differential equations.

Temperature control profile
Strawberry requires a discrete representation of potential
solutions.
Representation must balance cover and efficiency for the
search.
Consider representation based on piecewise linear
temperature control profiles by specifying
1 initial temperature,
2 sequence of δt, ∆T values, and
3 final time interval duration.
with
δt ∈ [δtmin, δtmax] and δtmin > 0
∆T ∈ [∆Tmin, ∆Tmax]
based on engineering insight.

Example
x =


T0
11 40 3
δt1,∆T1
δt2,∆T2
60 −1 40 −1
δt3,∆T3
δt4
20


results in this piecewise linear temperature control proﬁle:
Control proﬁle
t T
0 11
40 14
100 13
140 12
160 12

First attempt

Evolution of population

Better discretisation

and proﬁles

Polynomial spline control vector
11 design variables describing cubic, quintic & cubic
polynomials with matching slopes.

Pareto trade-oﬀ curve
A Rodman, ESF, D Gerogiorgis (2018), CACE.

19 Oﬀ-grid mining operations http://www.ucl.ac.uk/~ucecesf/
Energy systems
Current energy systems
distributed geographically
but managed centrally
large scale power
generation
national and
trans-national power and
fuel transmission networks
distribution to users via
single branching networks.
Photo courtesy: Benkid77 Puddington-Shotwick footpath 27 110809

Local generation and distribution
Motivation
transmission losses
improved micro-generation
technologies
energy storage advances
need for oﬀ-grid operation
peak demand management
Examples include single buildings, neighbourhoods and large
scale possibly oﬀ-grid industrial operations.

Technologies I
Generation
µ-CHP
solar photovoltaic (PV)
solar thermal
wind turbines
for power and heat.

Technologies II
Storage
batteries
compressed air
hydraulic
molten salts

Energy network design
Must handle variability of generation and of demand:
dynamic modelling is necessary;
need to consider cyclic behaviour, e.g. periodicity of
demand or generation;
multiple technologies may be required;
is grid connection (electricity, gas) necessary?
⇒ models are diﬀerential, combinatorial and nonlinear.

Continuous mining operations
Olympic Dam, BHP Billiton, South Australia

Oﬀ-grid

Sustainable oﬀ-grid operation
Large scale, both thermal
and power.
Continuous operation for
many years.
Currently meet demands
using transported diesel
fuel.
Use renewable energy
Space not a constraint!
Question
What is the best integrated renewable energy system design
including energy storage?

Variability

Large scale storage – Molten salts
Google Maps

Large scale storage – Compressed air

Large scale storage – Pumped hydraulic

Integrated system
Photo-
voltaics
Heat
Power
Power
Tower
Pumped Hydro Energy
Storage (PHES)
Advanced Adiabatic
Compressed Air Energy
Storage (AA-CAES)
Molten salt Tank Storage
(MTS)
Heat
Electricity direct to plant
GENERATION
DEMAND
Wind
Turbine
Power
block
Electricity
Heat
Vanadium Redox Flow
Battery System (VRFBS)

Energy generation
Sample equations:
photovoltaic
.
E
gen
PV (t) = ηpv (t)ηinv Ap
.
G
tot
(t)
wind
.
E
gen
wind (t) =
1
2
ηwtρAwtν3
(t)
storage ρcpVTES
d
dt
TTES
(t) = . . .
Resulting model is an initial value diﬀerential algebraic
system for t ∈ [0, 8760] h.
Quantities in red are variable and unpredictable.

Variability & Predictability
Climate is what you expect, weather is what you get.
Robert A. Heinlein (1973), Time Enough For Love.

Data modelling: collection
For each variable input (solar irradiance, wind speed):
1 Given hourly data for every day of every month over a set
of years, Dy,m,d,h, y = 1, . . . , ny , combine these data into
12 × 24 sets, one for each hour of a mythical day for each
month:
Dm,h ≡
ny
y=1
nd,m
d=1
Dy,m,d,h
for m = 1, . . . , 12
h = 1, . . . , 24
where Dy,m,d,h is a data point for year y, month m, day d
and hour h, and nd,m is number of days in month m.

Data modelling: model creation
2 Identify suitable statistical parameters, mean, standard
deviation, skewness and kurtosis, for best ﬁtting
stochastic models:
pm,h = probability distribution function(Dm,h)
for m = 1, . . . , 12
h = 1, . . . , 24

Data modelling: scenarios
3 A scenario is set of hourly data, {vm,d,h}, for a complete
year:
1: for m ← 1, . . . , 12 do
2: for h ← 1, . . . , 24 do
3: vm,1,h ← pearsrnd(pm,h)
4: for d ← 2, . . . , nm,d do
5: ˆv ← pearsrnd(pm,h)
6: vm,d,h ← wd vm,d−1,h + (1 − wd )ˆv
where wd ∈ [0, 1] provides problem speciﬁc balance
between climate and weather.

Stochastic model

Optimisation problem
Photo-
voltaics
Heat
Power
Power
Tower
Pumped Hydro Energy
Storage (PHES)
Advanced Adiabatic
Compressed Air Energy
Storage (AA-CAES)
Molten salt Tank Storage
(MTS)
Heat
Electricity direct to plant
GENERATION
DEMAND
Wind
Turbine
Power
block
Electricity
Heat
Vanadium Redox Flow
Battery System (VRFBS)
Solve
min
x,y
z =
C(x, y)
−R(x, y)
,
an MI(D)NLP problem, where
y are choices of technologies and operating scheme,
x are the sizes and capacities of the generation and
storage units,
C is the capital cost (generation, storage) and
R is the measure of the reliability of the system in
meeting the given demands for a number of randomly
generated scenarios.
subject to thermo-physical constraints.

Reliability estimation
Deﬁnition (Loss of power supply probability, LPSPm)
Given a set of scenarios, S = {si , i = 1, . . . , ns},
LPSPm =
|{si : Ri < R , i = 1, . . . , ns}|
ns
where R is a lower bound on the acceptable reliability.

Solver
Use NSGA-II1
:
Solar & wind
Population size npop 100 100
Generations ngen 300 300
Scenarios ns 300 1200
Process cores nproc 8 12
Wall clock time tcpu 108 299 h
with binary tournament selection, intermediate crossover
(0.75), and Gaussian mutation (0.1).
1
Deb (2000), Comput Methods Appl Mech Engng 186:311-338.

Results
1 . 0 0 . 8 0 . 6 0 . 4 0 . 2 0 . 0
3 0 0 0
4 0 0 0
5 0 0 0
6 0 0 0
R u n 1
R u n 2
R u n 3
Capitalcost(€million)
L P S P m
Amusat, Shearing & ESF (2017). Computers & Chemical Engineering 103:103–115.

A design
WT
Th.
Demand
El.
Demand
PT
PHES
MTS
R3
R4
R2
VRFB
R1
R5
R6
MTS molten salts
PHES pumped
hydro
PT power tower
Rn operating
dispatch
priorities
VRFB Vanadium
redox ﬂow
battery
WT wind
turbine

Reliability versus cost
Stochastic data modelling enables a second criterion for the
design of large scale oﬀ-grid mining operations with renewable
energy.

44 Conclusions http://www.ucl.ac.uk/~ucecesf/
Summary
Optimisation
methods
Deterministic Stochastic
Gradient
based
Direct
search
Steepest
descent
Dynamic
programming
Nelder
Mead
Hooke &
Jeeves
Nature
inspired
Genetic
algorithm
Ant
colony
Plant
propagation
Thanks to Dr Dimitrios Gerogiorgis, Mr Alistair Rodman (U of
Edinburgh); Professor Abdel Salhi (U of Essex); Dr Mayowa Amusat, Dr
Paul Shearing (UCL), Birse Charitable Trust Fellowship (UK), PRESSID
(Nigeria).
www.ucl.ac.uk/~ucecesf

45 Conclusions http://www.ucl.ac.uk/~ucecesf/
Fresa
An implementation of a plant propagation algorithm in Julia.
http://www.ucl.ac.uk/~ucecesf/fresa.html
http://julialang.org/

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