Smart Power Systems:the Optimization Challenge
Michel DE LARA
Cermics, Ecole des Ponts ParisTech, Franceand
Pierre Carpentier, ENSTA ParisTech, FranceJean-Philippe Chancelier, Ecole des Ponts ParisTech, France
Pierre Girardeau, post-doc EDF R & DJean-Christophe Alais, Ecole des Ponts and EDF R & D PhD student
Vincent Leclere, Ecole des Ponts PhD student
Cermics, France
December 19, 2012
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 1 / 112
Outline of the talk
In 2000, the Optimization and Systems team was created at Ecole des PontsParisTech, with stochastic optimization as major theme. The idea was toextend decomposition-coordination optimization methods for large systems
— developed in the deterministic setting with Electricite de France (EDFR&D) — to the stochastic case
During ten years, we have trained PhD students in stochastic optimization,mostly with EDF R&D. Since 2011, we witness a growing demand fromenergy firms for stochastic optimization, fueled by a deep and fast
transformation of power systems
We discuss to what extent optimization is challenged by the smart gridperspective
We shed light on the two main differences of stochastic control with respectto deterministic control: risk and information
We cast a glow on two snapshots highlighting ongoing research in the field
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 2 / 112
Outline of the presentation
1 Long term industry-academy cooperation
2 The “smart grids” seen from an optimizer perspective
3 Moving from deterministic to stochastic dynamic optimization
4 Two snapshots on ongoing research
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 3 / 112
Long term industry-academy cooperation
Outline of the presentation
1 Long term industry-academy cooperation
2 The “smart grids” seen from an optimizer perspective
3 Moving from deterministic to stochastic dynamic optimization
4 Two snapshots on ongoing research
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 4 / 112
Long term industry-academy cooperation Ecole des Ponts ParisTech–Cermics–Optimization and Systems
Outline of the presentation
1 Long term industry-academy cooperationEcole des Ponts ParisTech–Cermics–Optimization and SystemsIndustry partners
2 The “smart grids” seen from an optimizer perspectiveWhat is the “smart grid”?Optimization is challenged
3 Moving from deterministic to stochastic dynamic optimizationDam modelsThe deterministic optimization problem is well posedIn the uncertain framework, the optimization problem is not well posedThere are two ways to express the information constraints
4 Two snapshots on ongoing researchDecomposition-coordination optimization methods under uncertaintyRisk constraints in optimization
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 5 / 112
Long term industry-academy cooperation Ecole des Ponts ParisTech–Cermics–Optimization and Systems
Russian dolls
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 6 / 112
Long term industry-academy cooperation Ecole des Ponts ParisTech–Cermics–Optimization and Systems
Ecole des Ponts ParisTech
The Ecole nationale des ponts et chaussees was founded in 1747 and is oneof the world’s oldest engineering institutes
Ecole des Ponts ParisTech is traditionally considered as belonging to the5 leading engineering schools in France
Young graduates find positions in professional sectors like transport andurban planning, banking, finance, consulting, civil works, industry,environnement, energy, etc.
Faculty and staff
217 employees (including 50 subsidaries).165 module leaders, including 68 professors.1509 students.
Ecole des Ponts ParisTech is part of University Paris-Est
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 7 / 112
Long term industry-academy cooperation Ecole des Ponts ParisTech–Cermics–Optimization and Systems
Research at Ecole des Ponts ParisTech
Figures on research
Research personnel: 220About 40 Ecole des Ponts PhDs students graduate each year
10 research centers* CEREA (atmospheric environment), joint Ecole des Ponts-EDF R&D* CEREVE (water, urban and environment studies)* CERMICS (mathematics and scientific computing)* CERTIS (information technologies and systems)* CIRED (international environment and development)* LATTS (techniques, regional planning and society)* LVMT (city, mobility, transport)* UR Navier (mechanics, materials and structures of civil engineering, geotechnic)
* Saint-Venant laboratory (fluid mechanics), joint Ecole des Ponts-EDF R&D* Paris School of Economic PSE
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 8 / 112
Long term industry-academy cooperation Ecole des Ponts ParisTech–Cermics–Optimization and Systems
CERMICS, Centre d’enseignement et de recherche en
mathematiques et calcul scientifique
The scientific activity of CERMICS covers several domains in
scientific computingmodellingoptimization
14 senior researchers
14 PhD11 habilitation a diriger des recherches
Three missions
Teaching and PhD trainingScientific publicationsContracts
550 000 euros of contracts per year with
research and development centers of large industrial firms:CEA, CNES, EADS, EDF, Rio Tinto, etc.public research contracts
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 9 / 112
Long term industry-academy cooperation Ecole des Ponts ParisTech–Cermics–Optimization and Systems
Optimization and Systems Group
Three senior researchers
J.-P. CHANCELIERM. DE LARAF. MEUNIER
Five PhD students
J.-C. ALAIS (CIFRE EDF)S. SEPULVEDA (foreign co-supervision)V. LECLERE (IPEF)T. PRADEAUP. SARRABEZOLLES
Four associated researchers
P. CARPENTIER (ENSTA ParisTech)L. ANDRIEU (EDF R&D)K. BARTY (EDF R&D)A. DALLAGI (EDF R&D)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 10 / 112
Long term industry-academy cooperation Ecole des Ponts ParisTech–Cermics–Optimization and Systems
Optimization and Systems Group research specialities
MethodsStochastic optimal control (discrete-time)
Large-scale systemsDiscretization and numerical methodsProbability constraints
Discrete mathematics; combinatorial optimizationSystem control theory, viability and stochastic viabilityNumerical methods for fixed points computationUncertainty and learning in economics
Applications
Optimized management of power systems under uncertainty(production scheduling, power grid operations, risk management)Transport modelling and managementNatural resources management (fisheries, mining, epidemiology)
Softwares
Scicoslab, NSPOadlibsim
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 11 / 112
Long term industry-academy cooperation Ecole des Ponts ParisTech–Cermics–Optimization and Systems
Publications since 2000
24 publications in peer-reviewed international journals
3 publications in collective works
3 books
Modeling and Simulation in Scilab/Scicos with ScicosLab 4.4(2e edition, Springer-Verlag)Introduction a SCILAB(2e edition, Springer-Verlag)Sustainable Management of Natural Resources. Mathematical Models andMethods (Springer-Verlag)
2 books to be finished soon:
Stochastic Optimization. At the Crossroads between Stochastic Control and
Stochastic Programming
Control Theory for Engineers
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 12 / 112
Long term industry-academy cooperation Ecole des Ponts ParisTech–Cermics–Optimization and Systems
Teaching
Masters
Mathematiques, Informatique et Applications
Economie du Developpement Durable, de l’Environnement et de l’Energie
Renewable Energy Science and Technology Master ParisTech
Ecole des Ponts ParisTech
Optimisation et controle (J.-P. CHANCELIER)Modeliser l’alea (J.-P. CHANCELIER)Introduction a Scilab (J.-P. CHANCELIER, M. DE LARA)Modelisation pour la gestion durable des ressources naturelles (M. DE LARA)Economie du risque, effet de serre et biodiversite (M. DE LARA)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 13 / 112
Long term industry-academy cooperation Ecole des Ponts ParisTech–Cermics–Optimization and Systems
Industrial and public contracts since 2000
Industrial contracts
Conseil francais de l’energie (CFE)SETEC Energy SolutionsElectricite de France (EDF R&D)ThalesInstitut francais de l’energie (IFE)Gaz de France (GDF)PSA
Public contracts
STIC-AmSud (CNRS-INRIA-Affaires etrangeres)Centre d’etude des tunnelsCNRS ACI Ecologie quantitativeRTP CNRS
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 14 / 112
Long term industry-academy cooperation Industry partners
Outline of the presentation
1 Long term industry-academy cooperationEcole des Ponts ParisTech–Cermics–Optimization and SystemsIndustry partners
2 The “smart grids” seen from an optimizer perspectiveWhat is the “smart grid”?Optimization is challenged
3 Moving from deterministic to stochastic dynamic optimizationDam modelsThe deterministic optimization problem is well posedIn the uncertain framework, the optimization problem is not well posedThere are two ways to express the information constraints
4 Two snapshots on ongoing researchDecomposition-coordination optimization methods under uncertaintyRisk constraints in optimization
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 15 / 112
Long term industry-academy cooperation Industry partners
We cooperate with industry partners
As academics, we cooperate with industry partners,looking for longlasting close relations
We are not consultants working for clients
Our job consists mainly in
training Master and PhD students, working in the company and interactingwith us, on subjects designed jointlydeveloping methods, algorithmscontribute to computer codes developed within the companytraining professional engineers in the company
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 16 / 112
Long term industry-academy cooperation Industry partners
Electricite de France R & D / Departement OSIRIS
Electricite de France is the French electricity main producer
159 000 collaborateurs dans le monde37 millions de clients dans le monde65,2 milliards d’euros de chiffre d’affaire630,4 TWh produits dans le monde
Electricite de France Research & Development
486 millions d’euros de budget2 000 personnes
Departement OSIRISOptimisation, simulation, risques et statistiques pour les marches de l’energieOptimization, simulation, risks and statistics for the energy markets
145 salaries (dont 10 doctorants)25 millions d’euros de budget
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 17 / 112
Long term industry-academy cooperation Industry partners
The Optimization and Systems Group has trained
8 PhD from 2004 to 2011 + 2 PhD students,the majority related with EDF and energy management
* Laetitia ANDRIEU, former PhD student at EDF, now researcher EDF* Kengy BARTY, former PhD student at EDF, now researcher EDF* Daniel CHEMLA, former PhD student* Anes DALLAGI, former PhD student at EDF, now researcher EDF* Laurent GILOTTE, former PhD student with IFE, researcher EDF* Pierre GIRARDEAU, former PhD student at EDF, now with ARTELYS* Eugenie LIORIS, former PhD student* Babacar SECK, former PhD student at EDF* Cyrille STRUGAREK, former PhD student at EDF, now with Munich-Re* Jean-Christophe ALAIS, PhD student at EDF* Vincent LECLERE, PhD student (partly at EDF)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 18 / 112
Long term industry-academy cooperation Industry partners
PhD subjects
Contributions to the Discretization of Measurability Constraints for Stochastic
Optimization Problems,
Optimization under Probability Constraint,
Uncertainty, Inertia and Optimal Decision. Optimal Control Models Applied to Greenhouse
Gas Abatment Policies Selection,
Variational Approaches and other Contributions in Stochastic Optimization,
Particular Methods in Stochastic Optimal Control,
From Risk Constraints in Stochastic Optimization Problems to Utility Functions,
Resolution of Large Size Problems in Dynamic Stochastic Optimization and Synthesis of
Control Laws,
Evaluation and Optimization of Collective Taxis Systems,
Risk and Optimization for Energies Management,
Risk, Optimization, Large Systems,
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 19 / 112
Long term industry-academy cooperation Industry partners
Other contacts with French small consulting companies
ARTELYS is a company specializing in optimization, decision-making andmodeling. Relying on their high level of expertise in quantitative methods,the consultants deliver efficient solutions to complex business problems. Theyprovide services to diversified industries: Energy & Environment, Logistics &Transportation, Telecommunications, Finance and Defense.
Creee en 2011, SETEC Energy Solutions est la filiale du groupe SETECspecialisee dans les domaines de la production et de la maıtrise de l’energieen France et a l’etranger. SETEC Energy Solutions apporte a ses clients lamaıtrise des principaux process energetiques pour la mise en œuvre desolutions innovantes depuis les phases initiales de definition d’un projetjusqu’a son exploitation.
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 20 / 112
Long term industry-academy cooperation Industry partners
Summary
The following slides on “smart grids” express a viewpoint
from an optimizer perspective
working in an optimization research group
in an applied mathematics research center
in a French engineering institute
having contributed to train students now working in energy
having contacts and contracts with energy/environment firms
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 21 / 112
The “smart grids” seen from an optimizer perspective
Outline of the presentation
1 Long term industry-academy cooperationEcole des Ponts ParisTech–Cermics–Optimization and SystemsIndustry partners
2 The “smart grids” seen from an optimizer perspectiveWhat is the “smart grid”?Optimization is challenged
3 Moving from deterministic to stochastic dynamic optimizationDam modelsThe deterministic optimization problem is well posedIn the uncertain framework, the optimization problem is not well posedThere are two ways to express the information constraints
4 Two snapshots on ongoing researchDecomposition-coordination optimization methods under uncertaintyRisk constraints in optimization
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 22 / 112
The “smart grids” seen from an optimizer perspective What is the “smart grid”?
Outline of the presentation
1 Long term industry-academy cooperationEcole des Ponts ParisTech–Cermics–Optimization and SystemsIndustry partners
2 The “smart grids” seen from an optimizer perspectiveWhat is the “smart grid”?Optimization is challenged
3 Moving from deterministic to stochastic dynamic optimizationDam modelsThe deterministic optimization problem is well posedIn the uncertain framework, the optimization problem is not well posedThere are two ways to express the information constraints
4 Two snapshots on ongoing researchDecomposition-coordination optimization methods under uncertaintyRisk constraints in optimization
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 23 / 112
The “smart grids” seen from an optimizer perspective What is the “smart grid”?
Three key drivers for the “smart grid” concept
Environment
Markets
Technology
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 24 / 112
The “smart grids” seen from an optimizer perspective What is the “smart grid”?
Key driver: environmental concern
Strong impulse towards a decarbonized energy system
Expansion of renewable energy sources
⇒ Successfully integrating renewable energy sources has become critical,and made especially difficult due to their inpredictable and highly variable nature
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 25 / 112
The “smart grids” seen from an optimizer perspective What is the “smart grid”?
Key driver: deregulation
A power system (generation/transmission/distribution)
less and less vertical (deregulation of energy markets)hence with many players with their own goals
with some new players
industry (electric vehicle)regional public authorities (autonomy, efficiency)
with a network in horizontal expansion(the Pan European system counts 10,000 buses, 15,000 power lines,2,500 transformers, 3,000 generators, 5,000 loads)
with more and more exchanges (trade of commodities)
⇒ A change of paradigm for managementfrom centralized to more and more decentralized
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 26 / 112
The “smart grids” seen from an optimizer perspective What is the “smart grid”?
Key driver: telecommunication, metering and computing
technology
A power system with more and more technologydue to evolutions in the fields of computing and telecoms
smart meters
sensors
controllers
grid communication devices, etc.
⇒ A huge amount of data which, one day, will bea new potential for optimized management
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 27 / 112
The “smart grids” seen from an optimizer perspective What is the “smart grid”?
The “smart grid”? An infrastructure project with promises
to be fulfilled by a “smart power system”
Hardware / infrastructures / smart technologies
Renewable energies technologiesSmart meteringStorage
Promises
Quality, tariffsMore safetyMore renewables (environmentally friendly)
Software / smart management(energy supply being less flexible, make the demand more flexible)smart management, smart operation, smart meter management, smart distributed generation, load management, advanced distribution
management systems, active demand management, diffuse effacement, distribution management systems, storage management, smart home,
demand side management, etc.
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 28 / 112
The “smart grids” seen from an optimizer perspective Optimization is challenged
Outline of the presentation
1 Long term industry-academy cooperationEcole des Ponts ParisTech–Cermics–Optimization and SystemsIndustry partners
2 The “smart grids” seen from an optimizer perspectiveWhat is the “smart grid”?Optimization is challenged
3 Moving from deterministic to stochastic dynamic optimizationDam modelsThe deterministic optimization problem is well posedIn the uncertain framework, the optimization problem is not well posedThere are two ways to express the information constraints
4 Two snapshots on ongoing researchDecomposition-coordination optimization methods under uncertaintyRisk constraints in optimization
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 29 / 112
The “smart grids” seen from an optimizer perspective Optimization is challenged
A call from the industry to optimizers
Every three years, Electricite de France (EDF) organizes an internationalConference on Optimization and Practices in Industry (COPI)
At the last COPI’11, Jean-Francois Faugeras from EDF R&D opened theconference with a plenary talk entitled “Smart grids: a wind of change inpower systems and new opportunities for optimization”
He claimed that “power system players are facing high level problems to solverequiring new optimization methods and tools”,with “not only a ’smart(er)’ grid but a ’smart(er)’ power system”and called on the optimizers to develop new methods
EDF R&D has just created a new program Gaspard Monge pour
l’Optimisation et la recherche operationnelle (PGMO)to support academic research in the field of optimization
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 30 / 112
The “smart grids” seen from an optimizer perspective Optimization is challenged
What is “optimization”?
Optimizing is obtaining the best compromise between needs and
resources
Marcel Boiteux (president d’honneur d’EDF)
Resources: portfolio of assetsproduction units
costly/not costly: thermal/hydropowerstock/flow, predictable/unpredictable: thermal/wind
tariffs options, contracts
Needs: energy, safety, environment
energy usessafety, quality, resilience (breakdowns, blackout)environment protection (pollution) and alternative uses (dam water)
Best compromise: minimize socio-economic costs (including externalities)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 31 / 112
The “smart grids” seen from an optimizer perspective Optimization is challenged
Electrical engineers metiers and skills are evolving
Unit commitment, optimal dispatch of generating units:finding the least-cost dispatch of available generation resources to meet theelectrical load
which unit? 0/1 variableswhich power level? continuous variables
subject to more unpredictable energy flows (solar, wind)and demand (electrical devices, cars)
Markets: day-ahead, intra-day (balancing market):dispatcher takes bids from the generators, demand forecasts from thedistribution companies and clears the marketsubject to more unpredictability, more players
Long term planningsubject to more unpredictability (technologies, climate), more players
. . . Without even speaking of voltage, frequency and phase control
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 32 / 112
The “smart grids” seen from an optimizer perspective Optimization is challenged
Optimization skills will follow the power system evolution
We focus on generation and trading, not on transmission and distribution
Less base production and more wind and photovoltaic fatal generation makessupply more unpredictible→ stochastic optimization
Hence more storage (batteries, pumping stations)→ dynamical optimization, reserves dimensioning
The shape of the load is changing due to electric vehicle penetration→ demand-side management, “peak shaving”, adaptive tariffs
New subsystems emerge with local information and means of action: smartmeters, new producers, micro-grid, virtual power plant→ agregation, coordination, decentralized optimization
Markets (day-ahead, intra-day)→ optimization under uncertainty
Environmental constraints on production (CO2) and resources usages (water)→ risk constraints
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 33 / 112
The “smart grids” seen from an optimizer perspective Optimization is challenged
A context of increasing complexity
Multiple energy resources: photovoltaic,solar heating, heatpumps, wind, hydraulicpower, combined heat and power
Spatially distributed energy resources(onshore and offshore windpower,solarfarms), producers, consumers
Strongly variable production: wind, solar
Intermittent demand: electrical vehicles
Two-ways flows in the electrical network
Environmental and risk constraints (CO2,nuclear risk, land use)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 34 / 112
The “smart grids” seen from an optimizer perspective Optimization is challenged
An optimization birdview
Complex power systems
stocks: water reservoirs, storage,accumulators, fuels, etc.
uncertain demand and production
multiple constraints: costs, environment,blackout risk, etc.
large interconnected networks
multiple producers and consumers
information is not shared by all the actors
Complex decision problems
dynamics
uncertainties
multiple objectives
large scale
multiple decision-makers
decentralized information
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 35 / 112
The “smart grids” seen from an optimizer perspective Optimization is challenged
Summary
Three major key factors — environmental concern, deregulation,telecommunication, metering and computing technology —drive the power systems mutation
This induces a change of paradigm for management:from vertical centralized predictible “stock” energies tomore horizontal centralized unpredictible “flow” energies
Ripples are touching the optimization community
Specific optimization skills will be required, because optimal solution basedon a nominal forecast (deterministic setting) might perform poorly underoff-nominal conditions
Roger Wets’ illuminating example: deterministic vs. robust
of a furniture manufacturer deciding how many dressers of each type to make withman-hours and time as constraints; when they are uncertain, the stochasticoptimal solution considers all 106 possibilities and provides a robust solution,whereas the deterministic solution does not and does no point in the rightdirection
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 36 / 112
Moving from deterministic to stochastic dynamic optimization
Outline of the presentation
1 Long term industry-academy cooperation
2 The “smart grids” seen from an optimizer perspective
3 Moving from deterministic to stochastic dynamic optimization
4 Two snapshots on ongoing research
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 37 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
Outline of the presentation
1 Long term industry-academy cooperationEcole des Ponts ParisTech–Cermics–Optimization and SystemsIndustry partners
2 The “smart grids” seen from an optimizer perspectiveWhat is the “smart grid”?Optimization is challenged
3 Moving from deterministic to stochastic dynamic optimizationDam modelsThe deterministic optimization problem is well posedIn the uncertain framework, the optimization problem is not well posedThere are two ways to express the information constraints
4 Two snapshots on ongoing researchDecomposition-coordination optimization methods under uncertaintyRisk constraints in optimization
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 38 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
At
St
Qt
Rt
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 39 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
A single dam nonlinear dynamical model where turbinating
decisions are made before knowing the water inflows
We can model the dynamics of the water volume in a dam by
S(t + 1)︸ ︷︷ ︸
future volume
= minS♯, S(t)︸︷︷︸
volume
− q(t)︸︷︷︸
turbined
+ a(t)︸︷︷︸
inflow volume
S(t) volume (stock) of water at the beginning of period [t, t + 1[
a(t) inflow water volume (rain, etc.) during [t, t + 1[
decision-hazard:a(t) is not available at the beginning of period [t, t + 1[
q(t) turbined outflow volume during [t, t + 1[
decided at the beginning of period [t, t + 1[supposed to depend on the stock S(t) but not on the inflow water a(t)chosen such that 0 ≤ q(t) ≤ S(t)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 40 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
A single dam nonlinear dynamical model in decision-hazard:
equivalent formulation
We can model the dynamics of the water volume in a dam by
S(t + 1)︸ ︷︷ ︸
future volume
= S(t)︸︷︷︸
volume
− q(t)︸︷︷︸
turbined
− r(t)︸︷︷︸
spilled
+ a(t)︸︷︷︸
inflow volume
S(t) volume (stock) of water at the beginning of period [t, t + 1[
a(t) inflow water volume (rain, etc.) during [t, t + 1[
q(t) turbined outflow volume during [t, t + 1[
decided at the beginning of period [t, t + 1[supposed to depend on the stock S(t) but not on the inflow water a(t)chosen such that 0 ≤ q(t) ≤ S(t)
r(t) =[S(t) − q(t) + a(t) − S♯
]
+the spilled volume
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 41 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
When turbinating decisions are made after knowing the
water inflows, we obtain a linear dynamical model
We can model the dynamics of the water volume in a dam by
S(t + 1)︸ ︷︷ ︸
future volume
= S(t)︸︷︷︸
volume
− q(t)︸︷︷︸
turbined
− r(t)︸︷︷︸
spilled
+ a(t)︸︷︷︸
inflow volume
S(t) volume (stock) of water at the beginning of period [t, t + 1[
a(t), inflow water volume (rain, etc.) during [t, t + 1[;
hazard-decision:a(t) is known and available at the beginning of period [t, t + 1[
q(t) turbined outflow volume and r(t) spilled volume
decided at the beginning of period [t, t + 1[supposed to depend on the stock S(t) and on the inflow water a(t)chosen such that
0 ≤ S(t) − q(t) + a(t) − r(t) ≤ S♯
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 42 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
Complexity increases with interconnected dams
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 43 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
Typology of hydro-valleys
(c)(a) (b)
dams in cascade converging valleys pumping
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 44 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
Sketch of a cascade model with dams i = 1, . . . , N
Dam 2
Dam 3
Dam 1
S1,t
S2,t
S3,t
Q1,t
R1,t Q2,t
R2,t Q3,t
R3,t
A1,t
A2,t
A3,t
ai ,t : inflow into dam i at time t (rain, run off water)Si ,t : volume in dam i at time t (water volume)qi ,t : turbined from dam i at time t (valued at price pi ,t)ri ,t : spilled volume from dam i at time t (irrigation. . . )
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 45 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
Water inflows historical scenarios
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 46 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
Description of variables: ai ,t
Inflow water volume
ai ,t : amount of water inflowing without any control into the dam
Inflow water volumes ai ,t are part of the problem data, and may be
either deterministic, in which case the values (ai ,t0 , . . . , ai ,T ) are unique andare exactly known, for every dam i
or uncertain, and many possibilities can be considered
inflows chronicles may be available:(as
i,t0, . . . , as
i,T ) where s belongs to a set S of historical scenariosin addition, the probability πs of scenario s may be knownmore generally, the sequence (ai,t0 , . . . , ai,T ) may be considered as arandom process with known probability distribution
Prices pi ,t can also be part of the problem data
In the same way, the prices of turbined volumes can be part of the problem data,with an uncertain or deterministic status
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 47 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
Description of variables: qi ,t
Turbined water volume
qi ,t : amount of water turbined to produce electricity
Turbined volumes qi ,t are control variables, the possible values of which arein the hands of the decision-maker to achieve different goals
In the deterministic framework, one looks fora single sequence of values (qi ,t0 , . . . , qi ,T−1) for each dam i
In the uncertain framework, the situation is more complex
one has to specify the online available informationupon which decisions are madethe controls qi,t are also uncertain variablesand they can be obtained by means of feedback policies feeding on onlineinformation
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 48 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
Description of variables: ri ,t
Spilled water volume
ri ,t : amount of water spilled from the dam with or without control
The status of the spilled volumes depends on the context,and ri ,t can represent
an amount taken in the dam (for irrigation purposes, for instance), and canbe a data (deterministic or uncertain)
a volume deliberately taken in the dam and released without being turbined,and it is then a control variable
the amount of water which overflows in case of excess
ri ,t = [Si ,t + ai ,t + qi−1,t − qi ,t − S♯i ]+
and it is then an output variable which results from a calculation based uponother variables
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 49 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
The dynamics describes the temporal evolution
of the water stock volume in the dam
Dam i
Qi−1,t
Si,tQi,t
Ai,t
The water volume in the dam evolves whentime goes by and it is given by
Si ,t+1 = Si ,t + ai ,t + qi−1,t − qi ,t
A general form is
Si ,t+1 = Dyni ,t
(Si ,t , qi ,t, ai ,t , qi−1,t
)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 50 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
The payoffs are decomposed in instantaneous and final
The payoff obtained by turbining the volume qi ,t is
pi ,tqi ,t
We shall use the general form
Utili ,t
(Si ,t, qi ,t , ai ,t , pi ,t
)
To avoid emptying the dam at the end of the optimization process, weintroduce a “water value” term bearing on the final stock volume
UtilFini
(Si ,T
)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 51 / 112
Moving from deterministic to stochastic dynamic optimization Dam models
The management of the dams cascade can be formulated
as an intertemporal optimization problem
The payoff attached to the dam cascade is
N∑
i=1
( T−1∑
t=t0
Utili ,t
(Si ,t , qi ,t, ai ,t , pi ,t
)+ UtilFini
(Si ,T
))
This payoff must be optimized
under constraints of dynamics
Si,t+1 = Dyni,t
`
Si,t , qi,t , ai,t , qi−1,t
´
, i ∈ J1, NK, t ∈ Jt0, T−1K
and under bounds constraints
qi,t ∈ˆ
qi, qi
˜
, i ∈ J1, NK, t ∈ Jt0, T−1K
Si,t ∈ˆ
S i , S i
˜
, i ∈ J1, NK, t ∈ Jt0, T K
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 52 / 112
Moving from deterministic to stochastic dynamic optimization The deterministic optimization problem is well posed
Outline of the presentation
1 Long term industry-academy cooperationEcole des Ponts ParisTech–Cermics–Optimization and SystemsIndustry partners
2 The “smart grids” seen from an optimizer perspectiveWhat is the “smart grid”?Optimization is challenged
3 Moving from deterministic to stochastic dynamic optimizationDam modelsThe deterministic optimization problem is well posedIn the uncertain framework, the optimization problem is not well posedThere are two ways to express the information constraints
4 Two snapshots on ongoing researchDecomposition-coordination optimization methods under uncertaintyRisk constraints in optimization
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 53 / 112
Moving from deterministic to stochastic dynamic optimization The deterministic optimization problem is well posed
The deterministic optimization problem
In the deterministic case, inflows ai :=(ai ,t0 , . . . , ai ,T
)and prices
pi :=(pi ,t0 , . . . , pi ,T
)are unique and are exactly known, for every dam i
With qi :=(qi ,t0 , . . . , qi ,T−1
)and Si :=
(Si ,t0 , . . . , Si ,T
),
the optimization problem is
max(qi ,Si )i=1,...,N
N∑
i=1
( T−1∑
t=t0
Utili ,t
(Si ,t , qi ,t , ai ,t, pi ,t
)+ UtilFini
(Si ,T
))
under constraints of dynamics
Si ,t0 given, i ∈ J1, NK
Si ,t+1 = Dyni ,t
(Si ,t , qi ,t, ai ,t , qi−1,t
), i ∈ J1, NK, t ∈ Jt0, T−1K
and under bounds constraints
qi ,t ∈[q
i, qi
], i ∈ J1, NK, t ∈ Jt0, T−1K
Si ,t ∈[S i , S i
], i ∈ J1, NK, t ∈ Jt0, T K
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 54 / 112
Moving from deterministic to stochastic dynamic optimization The deterministic optimization problem is well posed
The deterministic optimization problem is well posed
If, to make things simple, we assume allvariables to be scalar, the optimization mustbe made with respect to all control variables(∈ R
N(T−t0)) and to all “state” variables(∈ R
N(T−t0+1))
The optimization yields for every damtrajectories (q⋆
i ,t0, . . . , q⋆
i ,T−1) and(S⋆
i ,t0, . . . , S⋆
i ,T ) of the controls and of the“states”, as well as the optimal payoff
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 55 / 112
Moving from deterministic to stochastic dynamic optimization The deterministic optimization problem is well posed
Here are some characteristics
of the deterministic optimization problem
The resolution of the deterministic optimization problem can be made in theframework of mathematical programming, that is, the maximization of afunction over R
n under equality and inequality constraints
The volumes Si ,t are intermediary variables,completely determined by the choice of the qi ,t
(however, such intermediary variables may be used to compute gradients bymeans of an adjoint state)
The optimization problem is often solved after formulating it as alinear problem and with softwares such as CPLEX
In the nonlinear case, the optimization problem can be difficult to solve, dueto its large size, but it is known how to apply decomposition and coordinationtechniques
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 56 / 112
Moving from deterministic to stochastic dynamic optimization In the uncertain framework, the optimization problem is not well posed
Outline of the presentation
1 Long term industry-academy cooperationEcole des Ponts ParisTech–Cermics–Optimization and SystemsIndustry partners
2 The “smart grids” seen from an optimizer perspectiveWhat is the “smart grid”?Optimization is challenged
3 Moving from deterministic to stochastic dynamic optimizationDam modelsThe deterministic optimization problem is well posedIn the uncertain framework, the optimization problem is not well posedThere are two ways to express the information constraints
4 Two snapshots on ongoing researchDecomposition-coordination optimization methods under uncertaintyRisk constraints in optimization
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 57 / 112
Moving from deterministic to stochastic dynamic optimization In the uncertain framework, the optimization problem is not well posed
0 50 100 150 200 250 300 350 4000
10
20
30
40
50
60
70
80
90
100
Stock volumes in a dam following an optimal strategywith a final value of water
(time)
(vol
ume)
Figure: Stock volumes in a dam following an optimal strategy, with a final value of waterMichel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 58 / 112
Moving from deterministic to stochastic dynamic optimization In the uncertain framework, the optimization problem is not well posed
In the uncertain framework,
two additional questions must be answered
Question (expliciting risk attitudes)
How are the uncertainties taken into accountin the payoff criterion and in the constraints?
Question (expliciting available online information)
Upon which online information are turbinating decisions made?
In practice, one assumes
more or less precise knowledge of the past (online information)
statistical assumptions about the future (offline information)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 59 / 112
Moving from deterministic to stochastic dynamic optimization In the uncertain framework, the optimization problem is not well posed
How are the uncertainties taken into account
in the payoff criterion and in the constraints?
In a probabilistic setting, where uncertainties are random variables,a classical answer is
to take the mathematical expectation of the payoff (risk-neutral approach)
E
( N∑
i=1
( T−1∑
t=t0
Utili ,t
(Si ,t, qi ,t , ai ,t, pi ,t
)+ UtilFini
(Si ,T
)))
and to satisfy all (physical) constrainsts almost surely that is, practically,for all possible issues of the uncertainties (robust approach)
P
(
qi ,t ∈[q
i, qi
], i ∈ J1, NK, t ∈ Jt0, T−1K
Si ,t ∈[S i , S i
], i ∈ J1, NK, t ∈ Jt0, T K
)
= 1
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 60 / 112
Moving from deterministic to stochastic dynamic optimization In the uncertain framework, the optimization problem is not well posed
Upon which online information
are turbinating decisions made?
On the one hand, it is suboptimal to restrict oneself,as in the deterministic case, to open-loop controls depending only upon time
On the other hand, it is impossible to suppose that we know in advancewhat will happen for all times: clairvoyance is impossible
The in-between is non anticipativity constraint
In our case, the decision qi ,t will be looked after as a
fonction of past uncertainties,
that is, of the water inflows and the prices(aj,τ , pj,τ
)
for j ∈ J1, NK and τ ∈ Jt0, tK
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 61 / 112
Moving from deterministic to stochastic dynamic optimization In the uncertain framework, the optimization problem is not well posed
On-line information feeds turbinating decisions:
the non anticipativity constraint
As a consequence, the control qi ,t is looked after under the form
qi,t = φi,t
`
a1,t0 , p1,t0 , . . . , aN,t0 , pN,t0 , . . . ,
a1,τ , p1,τ , . . . , aN,τ , pN,τ , . . . a1,t−1, p1,t−1, . . . , aN,t−1, pN,t−1
´
Denote the uncertainties at time t by
wt =(a1,t , p1,t , . . . , aN,t, pN,t
)
When uncertainties are considered as random variables (measurablemappings), the above formula for qi ,t expresses the measurability of thecontrol variable qi ,t with respect to the past uncertainties, also written as
σ(qi ,t) ⊂ σ(wt0 , . . . , wt−1
)⇐⇒ qi ,t σ
(wt0 , . . . , wt−1
)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 62 / 112
Moving from deterministic to stochastic dynamic optimization In the uncertain framework, the optimization problem is not well posed
On-line information structure can reflect
decentralized information
When the control qi ,t is looked after under the form
qi ,t = φi ,t
(ai ,t0 , pi ,t0 , . . . , ai ,τ , pi ,τ , . . . , ai ,t−1, pi ,t−1
)
this expresses that each dam is managed with local information
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 63 / 112
Moving from deterministic to stochastic dynamic optimization In the uncertain framework, the optimization problem is not well posed
The stochastic optimization problem
In a probabilistic setting, where uncertainties are random variablesand where a probability P is given, a possible formulation is
max(qi ,Si )i=1,...,N
E
„ NX
i=1
“
T−1X
t=t0
Utili,t
`
Si,t , qi,t , ai,t , pi,t
´
+ UtilFini
`
Si,T
´
”
«
under constraints of dynamics
Si,t0 given, i ∈ J1, NK
Si,t+1 = Dyni,t
`
Si,t , qi,t , ai,t , qi−1,t
´
, i ∈ J1, NK, t ∈ Jt0,T−1K
under bounds constraints
P
qi,t ∈ˆ
qi, qi
˜
, i ∈ J1, NK, t ∈ Jt0,T−1K
Si,t ∈ˆ
S i , S i
˜
, i ∈ J1, NK, t ∈ Jt0,T K
!
= 1
and under measurability constraints
qi,t σ`
wt0 , . . . , wt−1
´
, i ∈ J1, NK, t ∈ Jt0,T−1K
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 64 / 112
Moving from deterministic to stochastic dynamic optimization In the uncertain framework, the optimization problem is not well posed
The outputs of a stochastic optimization problem
are random variables
210000 215000 220000 225000 230000 235000 240000 245000 2500000.0e+00
1.0e−05
2.0e−05
3.0e−05
4.0e−05
5.0e−05
6.0e−05
7.0e−05
8.0e−05
9.0e−05
1.0e−04
Histogram of the optimal payoff with a final value of water
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 65 / 112
Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints
Outline of the presentation
1 Long term industry-academy cooperationEcole des Ponts ParisTech–Cermics–Optimization and SystemsIndustry partners
2 The “smart grids” seen from an optimizer perspectiveWhat is the “smart grid”?Optimization is challenged
3 Moving from deterministic to stochastic dynamic optimizationDam modelsThe deterministic optimization problem is well posedIn the uncertain framework, the optimization problem is not well posedThere are two ways to express the information constraints
4 Two snapshots on ongoing researchDecomposition-coordination optimization methods under uncertaintyRisk constraints in optimization
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 66 / 112
Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints
There are two ways to express the information constraints
Functional approach qi ,t = φi ,t
(wt0 , . . . , wt−1
)
In the special case of the Markovian framework with white noise, it can beshown that there is no loss of optimality to look for solutions under the form
qi ,t = φi ,t
(S1,t , . . . , SN,t
)
︸ ︷︷ ︸
state
Measurability constraints qi ,t σ(wt0 , . . . , wt−1
)
In the special case of the scenario tree approach, uncertainties are supposed tobe observed, and all possible scenarios
`
w st0
, . . . , w sT
´
, s ∈ S , are organized in atree, and controls qi,t are indexed by nodes on the treeEquivalently, qi,t are indexed by s ∈ S with the constraint that if two scenarioscoincide up to time t, so must do the controls at time t
`
wst0
, . . . , wst−1
´
=`
ws′
t0, . . . , w
s′
t−1
´
⇒ qsi,t = q
s′
i,t
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 67 / 112
Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints
General framework
Stochastic Optimal Control (SOC)
Consider a dynamical system in discrete time influenced by exogenous noise
At each time step:
observations are made on the system and kept in memorya decision/control is chosen, and the system evolves
Our aim is to minimize some objective function that depends on the evolution ofsystem
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 68 / 112
Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints
A motivating example
Typical application of SOC: portfolio management
Consider a power producer which has to
manage an amount xt of stock at each time step t
using decision ut for each time step t
while dealing with stochastic data wt , such as the power demand, unitbreakdowns, market prices
All random variables are written using bold letters
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 69 / 112
Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints
Mathematical formulation
Standard problem
minx,u
E(
T−1∑
t=0
Ct (xt ,ut ,wt+1) + K (xT )),
s.t. xt+1 = ft (xt ,ut ,wt+1) , ∀t = 0, . . . , T − 1,
x0 = w0,
P-a.-s. constraints (e.g. bounds on xt and ut),
ut is Ft-measurable,
where Ft := σw0, . . . ,wt
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 70 / 112
Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints
Classical methodsStochastic Programming
Discretize random inputs of the problemusing scenario trees
Write the constraints and objective functionon this structure (time has becomearborescent)
Use standard numerical methods frommathematical programming (linear/convexprogramming to overcome the combinatorialexplosion of the tree)
Carpentier/Cohen/Culioli (1995); Birge/Louveaux (1997); Pflug (2001);
Bacaud/Lemarechal/Renaud/Sagastizabal (2001); Heitsch/Romisch (2003);
Dupacova/Growe-Kuska/Romisch (2003); Barty (2004); Shapiro/Dentcheva/Ruszczynski (2009)
...
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 71 / 112
Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints
Classical methodsStochastic Programming
Discretize random inputs of the problemusing scenario trees
Write the constraints and objective functionon this structure (time has becomearborescent)
Use standard numerical methods frommathematical programming (linear/convexprogramming to overcome the combinatorialexplosion of the tree)
Complexity of multistage stochastic programs
The amount of scenarios needed to achieve a given precision grows exponentially w.r.t. thenumber of time steps
See Shapiro (2006); Girardeau (2010)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 71 / 112
Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints
Classical methodsDynamic Programming (1)
Markovian framework
Noise variables wt are independent over time
Cost-to-go function, Bellman function, value function
Vt (x) := minu
E
T−1X
s=t
Cs (xs , us ,ws+1) + K (xT ) | xt = x
!
, ∀x ∈ X,
subject to constraints of the original problem
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 72 / 112
Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints
Classical methodsDynamic Programming (2)
The Dynamic Programming (DP) principle Bellman (1957)
There is no loss of optimality in looking for the optimal strategy as feedback functionson xt only.Moreover, DP provides a way to compute Bellman functions backwards:
VT (x) = K (x) , ∀x ∈ XT ,
and, for all t = t0, . . . , T − 1:
Vt (x) = minu∈Ut
E(Ct (x , u,wt+1) + Vt+1 (ft (x , u,wt+1))), ∀x ∈ Xt
Reference textbooks: Bertsekas (2000); Puterman (1994); Quadrat/Viot (1999)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 73 / 112
Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints
Classical methodsDynamic Programming (3)
Most of the time, the DP equation cannot be solved analytically
Numerical methods have to be used
Curse of dimensionality
The complexity associated with the resolution of the DP equations by discretizationgrows exponentially w.r.t. the dimension of the state space
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 74 / 112
Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints
Classical methodsDynamic Programming (3)
Most of the time, the DP equation cannot be solved analytically
Numerical methods have to be used
Curse of dimensionality
The complexity associated with the resolution of the DP equations by discretizationgrows exponentially w.r.t. the dimension of the state space
Some attempts of breaking the curse are through functional approximations:
Approximate Dynamic Programming Bellman/Dreyfus (1959); Gal (1989);de Farias/Van Roy (2003); Longstaff/Schwartz (2001); Bertsekas/Tsitsiklis (1996)...
Stochastic Dual Dynamic Programming Pereira/Pinto (1991); Philpott/Guan (2008);Shapiro (2010)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 74 / 112
Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints
Summary
Stochastic optimization highlights risk attitudes tackling
Stochastic dynamic optimization emphasizes the handling ofonline information
Many open problems, because
many ways to represent risk (criterion, constraints)many information structurestremendous numerical obstacles to overcome
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 75 / 112
Two snapshots on ongoing research
Outline of the presentation
1 Long term industry-academy cooperation
2 The “smart grids” seen from an optimizer perspective
3 Moving from deterministic to stochastic dynamic optimization
4 Two snapshots on ongoing research
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 76 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
Outline of the presentation
1 Long term industry-academy cooperationEcole des Ponts ParisTech–Cermics–Optimization and SystemsIndustry partners
2 The “smart grids” seen from an optimizer perspectiveWhat is the “smart grid”?Optimization is challenged
3 Moving from deterministic to stochastic dynamic optimizationDam modelsThe deterministic optimization problem is well posedIn the uncertain framework, the optimization problem is not well posedThere are two ways to express the information constraints
4 Two snapshots on ongoing researchDecomposition-coordination optimization methods under uncertaintyRisk constraints in optimization
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 77 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
Decomposition-coordination: divide and conquer
Spatial decomposition
multiple players with their local informationlocal / regional / national /supranational
Temporal decomposition
A state is an information summaryTime coordination realized through Dynamic Programming, by value functionsHard nonanticipativity constraints
Scenario decomposition
Along each scenario, sub-problems are deterministic (powerful algorithms)Scenario coordination realized through Progressive Hedging, by updatingnonanticipativity multipliersSoft nonanticipativity constraints
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 78 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
Problems and methods: a very tentative tableFeel free to comment and criticize! ;-)
Decompose and coordinateSpace Time Scenarios Scenarios Trees
DP PH 2SSP
decentralized units OKstorage OKunit commitment OKshort term market OK
Dynamic Programming: DPProgressive Hedging: PHTwo and Multi-Stage Stochastic Programming : 2SSP
Use specific analytical features
Stochastic Programming: linearity, convexityStochastic Dual Dynamic Programming: linear dynamics, convex Bellmanfunction approximated by “cuts”
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 79 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
We have a nice decomposed problem but. . .Flower structure
We are almost in the case where
units could be driven independently
one from anotherUnit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 80 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
We have a nice decomposed problem but. . .Flower structure
Unfortunately...Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Couplingconstraint
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 80 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
The associated optimization problem may be written as
min(u1,...,uN)
N∑
i=1
Ji (ui )
︸ ︷︷ ︸
costs minimization
underN∑
i=1
Θi(ui ) = D
︸ ︷︷ ︸
offer = demand
where
ui is the decision of each unit i
Ji (ui ) is the cost of making decision ui for unit i
Θi(ui ) is the production induced by making decision ui for unit i
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 81 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
Proper prices allow decentralization of the optimum
min(u1,...,uN)
N∑
i=1
Ji (ui ) underN∑
i=1
Θi(ui ) = D
The simplest decomposition/coordination scheme consists in
buying the production of each unit at a price λ(k)
and letting each unit minimizing its costs
minui
Ji (ui ) + λ(k)︸︷︷︸
price
Θi(ui )
then, modifying the price depending on the coupling constraint
λ(k+1) = λ(k) + ρ( N∑
i=1
Θi(ui ) − D)
(like in the “tatonnement de Walras” in economics)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 82 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
Extension to dynamics and stochasticity
If we explicitely take into account time and uncertainties, a classical approchconsists in writing the new problem
minui,t
t∈0,T−1
i∈1,N
E
( N∑
i=1
( T−1∑
t=0
Li ,t
(xi ,t,ui ,t,wi ,t
)+ Ki
(xi ,T
)))
under the constraints
N∑
i=1
Θi ,t
(xi ,t,ui ,t ,wi ,t
)− dt = 0, t ∈ J0, T−1K
xi ,t+1 = fi ,t(xi ,t,ui ,t ,wi ,t
), i ∈ J1, NK, t ∈ J0, T−1K
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 83 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
We need to specify information constraints
The optimization problem is not well posed, because we have not specifiedupon what depends the control ui ,t of each unit i at each time t!
In the causal and perfect memory case, we say that the control ui ,t dependsof all past noises up to time t
ui ,t = φi ,t
(w1,0, . . . ,wN,0,d0 . . . . . .w1,t , . . . ,wN,t,dt
)
When the control ui ,t is looked after under the form
ui ,t = φi ,t
(wi ,0,d0 . . . . . .wi ,t ,dt
)
this expresses that each unit is managed with local information,which is a way to handle decentralized information
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 84 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
Looking after decentralizing prices models
Going on with the previous scheme, each unit i solves
minui,0,...,ui,T−1
E
„ T−1X
t=0
“
Li,t
`
xi,t , ui,t ,wi,t
´
+ λ(k)i,t Θi,t
`
xi,t , ui,t ,wi,t
´
”
+ Ki
`
xi,T
´
«
under the constraints
xi ,t+1 = fi ,t(xi ,t,ui ,t,wi ,t
), t ∈ J0, T−1K
In all rigor, the optimal controls u⋆i ,t of this problem depend upon xi ,t and
upon all past prices (λ(k)i ,0 , . . . , λ
(k)i ,t )
Research axis: find an approximate dynamical model for the prices, driven by
proper information (for instance, replace λ(k)i ,t by E
(
λ(k)i ,t | yt
)
, where the
information variable yt is a short time memory process)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 85 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
Dual Approximate Dynamic Programming
λk
Samples/scenarios of
dual variable
at iteration k
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 86 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
Dual Approximate Dynamic Programming
λk
Samples/scenarios of
dual variable
at iteration k
Subproblem iSubproblem 1 Subproblem N
We solve subproblems
using E(λk |y)
by Dynamic Programming
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 86 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
Dual Approximate Dynamic Programming
λk
Samples/scenarios of
dual variable
at iteration k
Subproblem iSubproblem 1 Subproblem N
We solve subproblems
using E(λk |y)
by Dynamic Programming
Policy Φ1“
x1, y”
Policy Φi“
xi , y”
Policy ΦN“
xN , y”
We obtain policies
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 86 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
Dual Approximate Dynamic Programming
λk
Samples/scenarios of
dual variable
at iteration k
Subproblem iSubproblem 1 Subproblem N
We solve subproblems
using E(λk |y)
by Dynamic Programming
Policy Φ1“
x1, y”
Policy Φi“
xi , y”
Policy ΦN“
xN , y”
We obtain policies
We simulate scenarios of strategies and prices and
update prices using a gradient step
λk+1t = λk
t + ρ ×PN
i=1 g it
“
xi,kt , u
i,kt , wt
”
We update prices
using a gradient step
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 86 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
Dual Approximate Dynamic Programming
λkAt iteration k + 1
Subproblem iSubproblem 1 Subproblem N
We solve subproblems
using E(λk |y)
by Dynamic Programming
Policy Φ1“
x1, y”
Policy Φi“
xi , y”
Policy ΦN“
xN , y”
We obtain policies
We simulate scenarios of strategies and prices and
update prices using a gradient step
λk+1t = λk
t + ρ ×PN
i=1 g it
“
xi,kt , u
i,kt , wt
”
We update prices
using a gradient step
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 86 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
Extension to interconnected dams
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 87 / 112
Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty
Contribution to dynamic tariffs
Spatial decomposition of a dynamic stochastic optimization problem
Lagrange multipliers attached to spatial coupling constraints are stochasticprocesses (prices)
By projecting these prices, one expects to identify approximate dynamicmodels
Such prices dynamic models are interpreted as dynamic tariffs
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 88 / 112
Two snapshots on ongoing research Risk constraints in optimization
Outline of the presentation
1 Long term industry-academy cooperationEcole des Ponts ParisTech–Cermics–Optimization and SystemsIndustry partners
2 The “smart grids” seen from an optimizer perspectiveWhat is the “smart grid”?Optimization is challenged
3 Moving from deterministic to stochastic dynamic optimizationDam modelsThe deterministic optimization problem is well posedIn the uncertain framework, the optimization problem is not well posedThere are two ways to express the information constraints
4 Two snapshots on ongoing researchDecomposition-coordination optimization methods under uncertaintyRisk constraints in optimization
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 89 / 112
Two snapshots on ongoing research Risk constraints in optimization
Dam management under “tourism” constraint
Maximizing the revenuefrom turbinated water
under a tourism constraintof having enough waterin July and August
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 90 / 112
Two snapshots on ongoing research Risk constraints in optimization
Stock trajectories
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 91 / 112
Two snapshots on ongoing research Risk constraints in optimization
A single dam nonlinear dynamical model in decision-hazard
We can model the dynamics of the water volume in a dam by
S(t + 1)︸ ︷︷ ︸
future volume
= minS♯, S(t)︸︷︷︸
volume
− q(t)︸︷︷︸
turbined
+ a(t)︸︷︷︸
inflow volume
S(t) volume (stock) of water at the beginning of period [t, t + 1[
a(t) inflow water volume (rain, etc.) during [t, t + 1[
decision-hazard:a(t) is not available at the beginning of period [t, t + 1[
q(t) turbined outflow volume during [t, t + 1[
decided at the beginning of period [t, t + 1[supposed to depend on S(t) but not on a(t)chosen such that 0 ≤ q(t) ≤ S(t)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 92 / 112
Two snapshots on ongoing research Risk constraints in optimization
The traditional economic problem is
maximizing the expected payoff
Suppose that
a probability P is given on the set Ω = RT−t0 of water inflows scenarios
`
a(t0), . . . , a(T − 1)´
turbined water q(t) is sold at price p(t), related to the price at which energycan be sold at time t
at the horizon, the final volume S(T ) has a value given by UtilFin`
S(T )´
,the “final value of water”
The traditional (risk-neutral) economic problem is to maximize theintertemporal payoff (without discounting if the horizon is short)
max E
T−1∑
t=t0
price︷︸︸︷
p(t)turbined
q(t)︸︷︷︸
+
final volume utility︷ ︸︸ ︷
UtilFin(S(T )
)
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 93 / 112
Two snapshots on ongoing research Risk constraints in optimization
Dam management under “tourism” constraint
Traditional cost minimization/payoff maximization
max E
T−1∑
t=t0
turbined water payoff︷ ︸︸ ︷
p(t)q(t) +
final volume utility︷ ︸︸ ︷
UtilFin(S(T )
)
For “tourism” reasons:
volume S(t) ≥ S , ∀t ∈ July, August
In what sense should we consider this inequality which involves therandom variables S(t) for t ∈ July, August ?
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 94 / 112
Two snapshots on ongoing research Risk constraints in optimization
Robust / almost sure / probability constraint
Robust constraints: for all the scenarios in a subset Ω ⊂ Ω
S(t) ≥ S , ∀t ∈ July, August
Almost sure constraints
Probability
water inflow scenarios along whichthe volumes S(t) are above the
threshold S for periods t in summer
= 1
Probability constraints, with “confidence” level p ∈ [0, 1]
Probability
water inflow scenarios along whichthe volumes S(t) are above thethreshold S for periods t in summer
≥ p
and also by penalization (to be discussed later), or in the mean, etc.
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 95 / 112
Two snapshots on ongoing research Risk constraints in optimization
Environmental/tourism issue leads to dam management
under probability constraint
The traditional economic problem is max E [P(T )]where the payoff/utility criterion is
P(T ) =
T−1∑
t=t0
turbined water payoff︷ ︸︸ ︷
p(t)q(t) +
final volume utility︷ ︸︸ ︷
UtilFin(S(T )
)
and a failure tolerance is accepted
Probability
water inflow scenarios along whichthe volumes S(t) ≥ S
for periods t in July and August
≥ 99%
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 96 / 112
Two snapshots on ongoing research Risk constraints in optimization
Stock trajectories
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 97 / 112
Two snapshots on ongoing research Risk constraints in optimization
An improvement compared to standard procedures
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 98 / 112
Two snapshots on ongoing research Risk constraints in optimization
However, though the mean is optimal,
the payoff effectively realized can be far from it
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 99 / 112
Two snapshots on ongoing research Risk constraints in optimization
We propose a stochastic viability formulation
to treat symmetrically andto guarantee both environmental and economic objectives
Given two thresholds to be guaranteed
a volume S (measured in cubic hectometers hm3)a payoff P (measured in numeraire $)
we look after policies achieving the maximal viability probability
Π(S, P) = max Proba
water inflow scenarios along which
the volumesS(t) ≥ S
for all time t ∈ July, August
and the final payoff P(T ) ≥ P
The maximal viability probability Π(S, P) is the maximal probability toguarantee to be above the thresholds S and P
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 100 / 112
Two snapshots on ongoing research Risk constraints in optimization
A stochastic viability formulationState, control and dynamic
The state is the couple x(t) =(S(t), P(t)
)volume/payoff
The control u(t) = q(t) is the turbined water
The dynamics is
S(t + 1)︸ ︷︷ ︸
future volume
= minS♯, S(t)︸︷︷︸
volume
− q(t)︸︷︷︸
turbined
+ a(t)︸︷︷︸
inflow volume
,
t = t0, . . . , T − 1
P(t + 1)︸ ︷︷ ︸
future payoff
= P(t)︸︷︷︸
payoff
+ p(t)q(t)︸ ︷︷ ︸
turbined water payoff
, t = t0, . . . , T − 2
P(T ) = P(T − 1) + UtilFin(S(T )
)
︸ ︷︷ ︸
final volume utility
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 101 / 112
Two snapshots on ongoing research Risk constraints in optimization
A stochastic viability formulationState and control constraints
The control constraints are
u(t) ∈ B(t, x(t)
)⇐⇒ 0 ≤ q(t) ≤ S(t)
The state constraints are
x(t) ∈ A(t) ⇐⇒
S(t) ≥ S , ∀t ∈ July, August
P(T ) ≥ P
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 102 / 112
Two snapshots on ongoing research Risk constraints in optimization
Dynamic programming equation
Abstract version
V (T , x) = 1A(T )(x)
V (t, x) = 1A(t)(x) maxu∈B(t,x)
Ew(t)
h
V“
t + 1, Dyn`
t, x , u, w(t)´
”i
Specific version
V (T ,S, P) = 1P≥P
V (T − 1,S, P) = max0≤q≤S
Ea(t)
h
V“
t + 1, S − q + a(t), P + UtilFin`
S´
”i
V (t,S, P) = max0≤q≤S
Ea(t)
h
V“
t + 1, S − q + a(t), P + pq”i
, t 6∈ July, August
V (t,S, P) = 1S≥S max0≤q≤S
Ea(t)
h
V“
t + 1,S − q + a(t), P + pq”i
,
t ∈ July, August
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 103 / 112
Two snapshots on ongoing research Risk constraints in optimization
Maximal viability probability Π(S , P)
as a function of guaranteed thresholds S and P
For example, the probabilityto guarantee
a final payoff aboveP = 1 Meuros
and a volume aboveS = 40 hm3 in Julyand August
is about 90%
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 104 / 112
Two snapshots on ongoing research Risk constraints in optimization
Iso-values for the maximal viability probability
as a function of guaranteed thresholds S and P
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 105 / 112
Two snapshots on ongoing research Risk constraints in optimization
Contribution to quantitative sustainable management
Conceptual framework forquantitative sustainablemanagement
Managing ecological and economicconflicting objectives
Displaying tradeoffs betweenecology and economy sustainabilitythresholds and risk
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 106 / 112
Conclusion: a need for training
Outline of the presentation
1 Long term industry-academy cooperation
2 The “smart grids” seen from an optimizer perspective
3 Moving from deterministic to stochastic dynamic optimization
4 Two snapshots on ongoing research
5 Conclusion: a need for training
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 107 / 112
Conclusion: a need for training
Trends are favorable to statistics and optimization
More telecom technology→ more data
More data, more unpredicability→ more statistics
More unpredicability → more storage→ more dynamic optimization
More unpredicability→ more stochastic dynamic optimization
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 108 / 112
Conclusion: a need for training
Opportunities for higher education
Masters
Mathematiques, Informatique et Applications
Economie du Developpement Durable, de l’Environnement et de l’Energie
Renewable Energy Science and Technology Master ParisTech
and others ;-)
Research centers
CERMICS (mathematics and scientific computing)Electricite de France Research & Development Departement OSIRISOptimization, simulation, risks and statistics for the energy marketsand others ;-)
Consulting positions for Master and PhD graduates: ARTELYS, SETEC, etc.
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 109 / 112
Conclusion: a need for training
Pedagogical material
CEA / EDF / INRIA Summer School on Stochastic Optimization slideshttp://www-hpc.cea.fr/SummerSchools2012-SO.htm
Advanced Course on Sustainable Optimization slideshttp://cermics.enpc.fr/~delara/ENSEIGNEMENT/
Sustainable_Optimization/Sustainable_Optimization.html
Scicoslab practical computer works http://cermics.enpc.fr/~scilab
More on the Optimization and Systems web pagehttp://cermics.enpc.fr/equipes/optimisation.html
More on my web pagehttp://cermics.enpc.fr/~delara/
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 110 / 112
Conclusion: a need for training
Some related initiatives
Gaspard Monge Program for Optimization and operations researchhttp://www.fondation-hadamard.fr/pgmo
From January 7 to April 5, 2013, Institut Henri Poincare (IHP) thematicquarter Mathematics of Bio-Economics as a contribution to Mathematics ofPlanet Earth - MPE2013http://www.ihp.fr/en/ceb/mabies
From January 7 to March 1, 2013, Master Course on Stochastic Optimization(MMMEF)http://cermics.enpc.fr/equipes/optimisation.html
From April 8 to April 12, 2013, CIRM School on Stochastic Control for theManagement of Renewable Energies as a contribution to Mathematics ofPlanet Earth - MPE2013http://cermics.enpc.fr/~delara/ENSEIGNEMENT/
CIRM2013/Optim_ENR_CIRM2013/
More on my web page http://cermics.enpc.fr/~delara/
Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 111 / 112
Conclusion: a need for training
References
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Barty, K. (2004). Contributions a la discretisation des contraintes de mesurabilite pour les problemes
d’optimisation stochastique. These de doctorat, Ecole Nationale des Ponts et Chaussees.Barty, K., Carpentier, P., and Girardeau, P. (2010). Decomposition of large-scale stochastic optimal control
problems. RAIRO Operations Research, 44(3):167–183.Bellman, R. (1957). Dynamic Programming. Princeton University Press, New Jersey.Bellman, R. and Dreyfus, S. E. (1959). Functional approximations and dynamic programming. Math tables
and other aides to computation, 13:247–251.Bertsekas, D. P. (2000). Dynamic Programming and Optimal Control. Athena Scientific, 2 edition.Bertsekas, D. P. and Tsitsiklis, J. N. (1996). Neuro-Dynamic Programming. Athena Scientific.Birge, J. R. and Louveaux, F. V. (1997). Introduction to Stochastic Programming. Springer Series in
Operations Research. Springer Verlag, New York.Carpentier, P., Cohen, G., and Culioli, J.-C. (1995). Stochastic optimal control and decomposition-coordination
methods. In Durier, R. and Michelot, C., editors, Recent Developments in Optimization. Lecture Notes inEconomics and Mathematical Systems, 429, pages 72–103. Springer Verlag, Berlin.
de Farias, D. P. and Van Roy, B. (2003). The Linear Programming Approach to Approximate DynamicProgramming. Oper. Res., 51(6):850–856.
Dupacova, J., Growe-Kuska, N., and Romisch, W. (2003). Scenario reduction in stochastic programming. Anapproach using probability metrics. Mathematical Programming, 95(3):493–511.
Gal, S. (1989). The Parameter Iteration Method in Dynamic Programming Management Science,35(6):675–684.
Girardeau, P. (2010). A comparison of sample-based Stochastic Optimal Control methods. StochasticProgramming E-Print Series, http://www.speps.org. submitted to Optimization and Engineering.
Heitsch, H. and Romisch, W. (2003). Scenario Reduction Algorithms in Stochastic Programming.Computational Optimization and Applications, 24:187–206.
Longstaff, F. A. and Schwartz, E. S. (2001). Valuing american options by simulation: A simple least squaresapproach. Review of Financial Studies, 14(1):113–147.Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec. 2012 December 19, 2012 112 / 112