Smart Power Systems: the Optimization Challenge

Transcription

1 Smart Power Systems: the Optimization Challenge Michel DE LARA Cermics, École des Ponts ParisTech, France and Pierre Carpentier, ENSTA ParisTech, France Jean-Philippe Chancelier, École des Ponts ParisTech, France Pierre Girardeau, post-doc EDF R & D Jean-Christophe Alais, École des Ponts and EDF R & D PhD student Vincent Leclère, École des Ponts PhD student Cermics, France December 19, 2012 Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

2 Outline of the talk In 2000, the Optimization and Systems team was created at École des Ponts ParisTech, with stochastic optimization as major theme. The idea was to extend decomposition-coordination optimization methods for large systems developed in the deterministic setting with Électricité de France (EDF R&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 from energy 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 grid perspective We shed light on the two main differences of stochastic control with respect to 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 December 19, / 112

3 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 December 19, / 112

4 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 December 19, / 112

5 Long term industry-academy cooperation Outline of the presentation École des Ponts ParisTech Cermics Optimization and Systems 1 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Industry partners 2 The smart grids seen from an optimizer perspective What is the smart grid? Optimization is challenged 3 Moving from deterministic to stochastic dynamic optimization Dam models The deterministic optimization problem is well posed In the uncertain framework, the optimization problem is not well posed There are two ways to express the information constraints 4 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Risk constraints in optimization 5 Conclusion: a need for training Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

6 Russian dolls Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

7 Long term industry-academy cooperation École des Ponts ParisTech École des Ponts ParisTech Cermics Optimization and Systems The École nationale des ponts et chaussées was founded in 1747 and is one of the world s oldest engineering institutes École des Ponts ParisTech is traditionally considered as belonging to the 5 leading engineering schools in France Young graduates find positions in professional sectors like transport and urban planning, banking, finance, consulting, civil works, industry, environnement, energy, etc. Faculty and staff 217 employees (including 50 subsidaries). 165 module leaders, including 68 professors students. École des Ponts ParisTech is part of University Paris-Est Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

8 Research at Long term industry-academy cooperation École des Ponts ParisTech École des Ponts ParisTech Cermics Optimization and Systems Figures on research Research personnel: 220 About 40 École des Ponts PhDs students graduate each year 10 research centers * CEREA (atmospheric environment), joint École 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 École des Ponts-EDF R&D * Paris School of Economic PSE Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

9 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems CERMICS, Centre d enseignement et de recherche en mathématiques et calcul scientifique The scientific activity of CERMICS covers several domains in scientific computing modelling optimization 14 senior researchers 14 PhD 11 habilitation à diriger des recherches Three missions Teaching and PhD training Scientific publications Contracts 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 December 19, / 112

10 Long term industry-academy cooperation Optimization and Systems Group École des Ponts ParisTech Cermics Optimization and Systems Three senior researchers J.-P. CHANCELIER M. DE LARA F. MEUNIER Five PhD students J.-C. ALAIS (CIFRE EDF) S. SEPULVEDA (foreign co-supervision) V. LECLÈRE (IPEF) T. PRADEAU P. 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 December 19, / 112

11 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Optimization and Systems Group research specialities Methods Stochastic optimal control (discrete-time) Large-scale systems Discretization and numerical methods Probability constraints Discrete mathematics; combinatorial optimization System control theory, viability and stochastic viability Numerical methods for fixed points computation Uncertainty and learning in economics Applications Optimized management of power systems under uncertainty (production scheduling, power grid operations, risk management) Transport modelling and management Natural resources management (fisheries, mining, epidemiology) Softwares Scicoslab, NSP Oadlibsim Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

12 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Publications since publications in peer-reviewed international journals 3 publications in collective works 3 books Modeling and Simulation in Scilab/Scicos with ScicosLab 4.4 (2e édition, Springer-Verlag) Introduction à SCILAB (2e édition, Springer-Verlag) Sustainable Management of Natural Resources. Mathematical Models and Methods (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 December 19, / 112

13 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Teaching Masters Mathématiques, Informatique et Applications Économie du Développement Durable, de l Environnement et de l Énergie Renewable Energy Science and Technology Master ParisTech École des Ponts ParisTech Optimisation et contrôle (J.-P. CHANCELIER) Modéliser l aléa (J.-P. CHANCELIER) Introduction à Scilab (J.-P. CHANCELIER, M. DE LARA) Modélisation pour la gestion durable des ressources naturelles (M. DE LARA) Économie du risque, effet de serre et biodiversité (M. DE LARA) Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

14 Long term industry-academy cooperation Industrial and public contracts since 2000 École des Ponts ParisTech Cermics Optimization and Systems Industrial contracts Conseil français de l énergie (CFE) SETEC Energy Solutions Électricité de France (EDF R&D) Thales Institut français de l énergie (IFE) Gaz de France (GDF) PSA Public contracts STIC-AmSud (CNRS-INRIA-Affaires étrangères) Centre d étude des tunnels CNRS ACI Écologie quantitative RTP CNRS Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

15 Long term industry-academy cooperation Outline of the presentation Industry partners 1 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Industry partners 2 The smart grids seen from an optimizer perspective What is the smart grid? Optimization is challenged 3 Moving from deterministic to stochastic dynamic optimization Dam models The deterministic optimization problem is well posed In the uncertain framework, the optimization problem is not well posed There are two ways to express the information constraints 4 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Risk constraints in optimization 5 Conclusion: a need for training Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

16 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 interacting with us, on subjects designed jointly developing methods, algorithms contribute to computer codes developed within the company training professional engineers in the company Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

17 Long term industry-academy cooperation Industry partners Électricité de France R & D / Département OSIRIS Électricité de France is the French electricity main producer collaborateurs dans le monde 37 millions de clients dans le monde 65,2 milliards d euros de chiffre d affaire 630,4 TWh produits dans le monde Électricité de France Research & Development 486 millions d euros de budget personnes Département OSIRIS Optimisation, simulation, risques et statistiques pour les marchés de l énergie Optimization, simulation, risks and statistics for the energy markets 145 salariés (dont 10 doctorants) 25 millions d euros de budget Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

18 Long term industry-academy cooperation Industry partners The Optimization and Systems Group has trained 8 PhD from 2004 to 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 * Eugénie LIORIS, former PhD student * Babacar SECK, former PhD student at EDF * Cyrille STRUGAREK, former PhD student at EDF, now with Munich-Ré * 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 December 19, / 112

19 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 December 19, / 112

20 Long term industry-academy cooperation Industry partners Other contacts with French small consulting companies ARTELYS is a company specializing in optimization, decision-making and modeling. Relying on their high level of expertise in quantitative methods, the consultants deliver efficient solutions to complex business problems. They provide services to diversified industries: Energy & Environment, Logistics & Transportation, Telecommunications, Finance and Defense. Créée en 2011, SETEC Energy Solutions est la filiale du groupe SETEC spécialisée dans les domaines de la production et de la maîtrise de l énergie en France et à l étranger. SETEC Energy Solutions apporte à ses clients la maîtrise des principaux process énergétiques pour la mise en œuvre de solutions innovantes depuis les phases initiales de définition d un projet jusqu à son exploitation. Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

21 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 December 19, / 112

22 The smart grids seen from an optimizer perspective Outline of the presentation 1 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Industry partners 2 The smart grids seen from an optimizer perspective What is the smart grid? Optimization is challenged 3 Moving from deterministic to stochastic dynamic optimization Dam models The deterministic optimization problem is well posed In the uncertain framework, the optimization problem is not well posed There are two ways to express the information constraints 4 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Risk constraints in optimization 5 Conclusion: a need for training Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

23 The smart grids seen from an optimizer perspective Outline of the presentation What is the smart grid? 1 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Industry partners 2 The smart grids seen from an optimizer perspective What is the smart grid? Optimization is challenged 3 Moving from deterministic to stochastic dynamic optimization Dam models The deterministic optimization problem is well posed In the uncertain framework, the optimization problem is not well posed There are two ways to express the information constraints 4 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Risk constraints in optimization 5 Conclusion: a need for training Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

24 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 December 19, / 112

25 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 December 19, / 112

26 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 management from centralized to more and more decentralized Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

27 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 technology due 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 be a new potential for optimized management Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

28 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 technologies Smart metering Storage Promises Quality, tariffs More safety More 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 December 19, / 112

29 The smart grids seen from an optimizer perspective Outline of the presentation Optimization is challenged 1 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Industry partners 2 The smart grids seen from an optimizer perspective What is the smart grid? Optimization is challenged 3 Moving from deterministic to stochastic dynamic optimization Dam models The deterministic optimization problem is well posed In the uncertain framework, the optimization problem is not well posed There are two ways to express the information constraints 4 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Risk constraints in optimization 5 Conclusion: a need for training Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

30 The smart grids seen from an optimizer perspective Optimization is challenged A call from the industry to optimizers Every three years, Électricité de France (EDF) organizes an international Conference on Optimization and Practices in Industry (COPI) At the last COPI 11, Jean-François Faugeras from EDF R&D opened the conference with a plenary talk entitled Smart grids: a wind of change in power systems and new opportunities for optimization He claimed that power system players are facing high level problems to solve requiring 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 opérationnelle (PGMO) to support academic research in the field of optimization Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

31 The smart grids seen from an optimizer perspective Optimization is challenged What is optimization? Optimizing is obtaining the best compromise between needs and resources Resources: portfolio of assets production units costly/not costly: thermal/hydropower stock/flow, predictable/unpredictable: thermal/wind Marcel Boiteux (président d honneur d EDF) tariffs options, contracts Needs: energy, safety, environment energy uses safety, 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 December 19, / 112

32 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 the electrical load which unit? 0/1 variables which 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 the distribution companies and clears the market subject to more unpredictability, more players Long term planning subject 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 December 19, / 112

33 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 makes supply 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: smart meters, 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 December 19, / 112

34 The smart grids seen from an optimizer perspective Optimization is challenged A context of increasing complexity Multiple energy resources: photovoltaic, solar heating, heatpumps, wind, hydraulic power, 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 December 19, / 112

35 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 December 19, / 112

36 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 to more horizontal centralized unpredictible flow energies Ripples are touching the optimization community Specific optimization skills will be required, because optimal solution based on a nominal forecast (deterministic setting) might perform poorly under off-nominal conditions Roger Wets illuminating example: deterministic vs. robust of a furniture manufacturer deciding how many dressers of each type to make with man-hours and time as constraints; when they are uncertain, the stochastic optimal solution considers all 10 6 possibilities and provides a robust solution, whereas the deterministic solution does not and does no point in the right direction Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

37 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 December 19, / 112

38 Moving from deterministic to stochastic dynamic optimization Outline of the presentation Dam models 1 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Industry partners 2 The smart grids seen from an optimizer perspective What is the smart grid? Optimization is challenged 3 Moving from deterministic to stochastic dynamic optimization Dam models The deterministic optimization problem is well posed In the uncertain framework, the optimization problem is not well posed There are two ways to express the information constraints 4 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Risk constraints in optimization 5 Conclusion: a need for training Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

39 Moving from deterministic to stochastic dynamic optimization Dam models A t R t S t Qt Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

40 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) } {{ } = min{s, S(t) }{{} future volume 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 December 19, / 112

41 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) } {{ } = S(t) }{{} future volume volume q(t) }{{} turbined r(t) + }{{} a(t) }{{} spilled 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 December 19, / 112

42 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) } {{ } = S(t) }{{} future volume volume q(t) }{{} turbined r(t) + }{{} a(t) }{{} spilled 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 December 19, / 112

43 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 December 19, / 112

44 Moving from deterministic to stochastic dynamic optimization Typology of hydro-valleys Dam models (a) (b) (c) dams in cascade converging valleys pumping Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

45 Moving from deterministic to stochastic dynamic optimization Dam models Sketch of a cascade model with dams i = 1,..., N A 1,t S 1,t Dam 1 Q 1,t A 2,t R 1,t S 2,t Q 2,t A 3,t Dam 2 R 2,t S 3,t Q 3,t Dam 3 R 3,t a i,t : inflow into dam i at time t (rain, run off water) S i,t : volume in dam i at time t (water volume) q i,t : turbined from dam i at time t (valued at price p i,t ) r i,t : spilled volume from dam i at time t (irrigation...) Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

46 Moving from deterministic to stochastic dynamic optimization Dam models Water inflows historical scenarios Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

47 Moving from deterministic to stochastic dynamic optimization Dam models Description of variables: a i,t Inflow water volume a i,t : amount of water inflowing without any control into the dam Inflow water volumes a i,t are part of the problem data, and may be either deterministic, in which case the values (a i,t0,...,a i,t ) are unique and are exactly known, for every dam i or uncertain, and many possibilities can be considered inflows chronicles may be available: (a s i,t 0,..., a s i,t) where s belongs to a set S of historical scenarios in addition, the probability π s of scenario s may be known more generally, the sequence (a i,t0,..., a i,t ) may be considered as a random process with known probability distribution Prices p i,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 December 19, / 112

48 Moving from deterministic to stochastic dynamic optimization Dam models Description of variables: q i,t Turbined water volume q i,t : amount of water turbined to produce electricity Turbined volumes q i,t are control variables, the possible values of which are in the hands of the decision-maker to achieve different goals In the deterministic framework, one looks for a single sequence of values (q i,t0,...,q i,t 1 ) for each dam i In the uncertain framework, the situation is more complex one has to specify the online available information upon which decisions are made the controls q i,t are also uncertain variables and they can be obtained by means of feedback policies feeding on online information Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

49 Moving from deterministic to stochastic dynamic optimization Dam models Description of variables: r i,t Spilled water volume r i,t : amount of water spilled from the dam with or without control The status of the spilled volumes depends on the context, and r i,t can represent an amount taken in the dam (for irrigation purposes, for instance), and can be 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 r i,t = [S i,t + a i,t + q i 1,t q i,t S i ] + and it is then an output variable which results from a calculation based upon other variables Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

50 Moving from deterministic to stochastic dynamic optimization Dam models The dynamics describes the temporal evolution of the water stock volume in the dam Q i 1,t A i,t The water volume in the dam evolves when time goes by and it is given by S i,t+1 = S i,t + a i,t + q i 1,t q i,t S i,t Dam i Q i,t A general form is S i,t+1 = Dyn i,t ( Si,t, q i,t, a i,t, q i 1,t ) Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

51 Moving from deterministic to stochastic dynamic optimization Dam models The payoffs are decomposed in instantaneous and final The payoff obtained by turbining the volume q i,t is p i,t q i,t We shall use the general form ( ) Util i,t Si,t, q i,t, a i,t, p i,t To avoid emptying the dam at the end of the optimization process, we introduce a water value term bearing on the final stock volume UtilFin i ( Si,T ) Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

52 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 ( T 1 ) ( ) ( ) Util i,t Si,t, q i,t, a i,t, p i,t +UtilFini Si,T t=t 0 i=1 This payoff must be optimized under constraints of dynamics S i,t+1 = Dyn i,t`si,t,q i,t,a i,t,q i 1,t, i 1,N, t t0,t 1 and under bounds constraints q i,t ˆq i,q i, i 1,N, t t0, T 1 S i,t ˆS i,s i, i 1,N, t t0, T Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

53 Moving from deterministic to stochastic dynamic optimization Outline of the presentation The deterministic optimization problem is well posed 1 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Industry partners 2 The smart grids seen from an optimizer perspective What is the smart grid? Optimization is challenged 3 Moving from deterministic to stochastic dynamic optimization Dam models The deterministic optimization problem is well posed In the uncertain framework, the optimization problem is not well posed There are two ways to express the information constraints 4 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Risk constraints in optimization 5 Conclusion: a need for training Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

54 Moving from deterministic to stochastic dynamic optimization The deterministic optimization problem The deterministic optimization problem is well posed In the deterministic case, inflows a i := ( a i,t0,...,a i,t ) and prices p i := ( p i,t0,..., p i,t ) are unique and are exactly known, for every dam i With q i := ( q i,t0,..., q i,t 1 ) and Si := ( S i,t0,..., S i,t ), the optimization problem is max (q i,s i) i=1,...,n N ( T 1 ) ( ) ( ) Util i,t Si,t, q i,t, a i,t, p i,t +UtilFini Si,T t=t 0 i=1 under constraints of dynamics S i,t0 given, i 1, N S i,t+1 = Dyn i,t ( Si,t, q i,t, a i,t, q i 1,t ), and under bounds constraints q i,t [ q i, q i ], i 1, N, t t0, T 1 S i,t [ S i, S i ], i 1, N, t t0, T i 1, N, t t0, T 1 Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

55 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 all variables to be scalar, the optimization must be 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 dam trajectories (qi,t 0,...,qi,T 1 ) and (Si,t 0,..., Si,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 December 19, / 112

56 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 the framework of mathematical programming, that is, the maximization of a function over R n under equality and inequality constraints The volumes S i,t are intermediary variables, completely determined by the choice of the q i,t (however, such intermediary variables may be used to compute gradients by means of an adjoint state) The optimization problem is often solved after formulating it as a linear problem and with softwares such as CPLEX In the nonlinear case, the optimization problem can be difficult to solve, due to its large size, but it is known how to apply decomposition and coordination techniques Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

57 Moving from deterministic to stochastic dynamic optimization Outline of the presentation In the uncertain framework, the optimization problem is not well posed 1 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Industry partners 2 The smart grids seen from an optimizer perspective What is the smart grid? Optimization is challenged 3 Moving from deterministic to stochastic dynamic optimization Dam models The deterministic optimization problem is well posed In the uncertain framework, the optimization problem is not well posed There are two ways to express the information constraints 4 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Risk constraints in optimization 5 Conclusion: a need for training Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

58 Moving from deterministic to stochastic dynamic optimization In the uncertain framework, the optimization problem is not well posed Stock volumes in a dam following an optimal strategywith a final value of water (volume) (time) Figure: Stock volumes in a dam following an optimal strategy, with a final value of water Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

59 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 account in 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 December 19, / 112

60 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) ( N ( T 1 ( ) ( E Util i,t Si,t, q i,t, a i,t, p ))) i,t +UtilFini Si,T t=t 0 i=1 and to satisfy all (physical) constrainsts almost surely that is, practically, for all possible issues of the uncertainties (robust approach) ( q i,t [ ] ) q P i, q i, i 1, N, t t 0, T 1 S i,t [ ] = 1 S i, S i, i 1, N, t t 0, T Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

61 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 advance what will happen for all times: clairvoyance is impossible The in-between is non anticipativity constraint In our case, the decision q i,t will be looked after as a fonction of past uncertainties, that is, of the water inflows and the prices ( a j,τ, p j,τ ) for j 1, N and τ t 0, t Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

62 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 q i,t is looked after under the form q i,t = φ i,t`a1,t0,p 1,t0,...,a N,t0,p N,t0,..., a 1,τ, p 1,τ,...,a N,τ, p N,τ,... a 1,t 1, p 1,t 1,...,a N,t 1, p N,t 1 Denote the uncertainties at time t by w t = ( a 1,t, p 1,t,..., a N,t, p N,t ) When uncertainties are considered as random variables (measurable mappings), the above formula for q i,t expresses the measurability of the control variable q i,t with respect to the past uncertainties, also written as σ(q i,t ) σ ( w t0,..., w t 1 ) qi,t σ ( w t0,...,w t 1 ) Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

63 Moving from deterministic to stochastic dynamic optimization On-line information structure can reflect decentralized information In the uncertain framework, the optimization problem is not well posed When the control q i,t is looked after under the form q i,t = φ i,t ( ai,t0, p i,t0,..., a i,τ, p i,τ,...,a i,t 1, p i,t 1 ) this expresses that each dam is managed with local information Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

64 Moving from deterministic to stochastic dynamic optimization The stochastic optimization problem In the uncertain framework, the optimization problem is not well posed In a probabilistic setting, where uncertainties are random variables and where a probability P is given, a possible formulation is max (q i,s i ) i=1,...,n E N X i=1 T 1 X under constraints of dynamics t=t 0 Util i,t`si,t,q i,t,a i,t, p i,t +UtilFini `Si,T «S i,t0 given, i 1, N S i,t+1 = Dyn i,t`si,t,q i,t,a i,t,q i 1,t, i 1, N, t t0,t 1 under bounds constraints q i,t ˆq P i,q i S i,t ˆS i, S i, i 1, N, t t 0,T 1, i 1, N, t t 0,T! = 1 and under measurability constraints q i,t σ`w t0,...,w t 1, i 1, N, t t0,t 1 Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

65 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 Histogram of the optimal payoff with a final value of water 1.0e e e e e e e e e e e Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

66 Moving from deterministic to stochastic dynamic optimization Outline of the presentation There are two ways to express the information constraints 1 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Industry partners 2 The smart grids seen from an optimizer perspective What is the smart grid? Optimization is challenged 3 Moving from deterministic to stochastic dynamic optimization Dam models The deterministic optimization problem is well posed In the uncertain framework, the optimization problem is not well posed There are two ways to express the information constraints 4 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Risk constraints in optimization 5 Conclusion: a need for training Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

67 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 q i,t = φ i,t ( wt0,..., w t 1 ) In the special case of the Markovian framework with white noise, it can be shown that there is no loss of optimality to look for solutions under the form q i,t = φ i,t ( S1,t,..., S N,t ) } {{ } state Measurability constraints q i,t σ ( w t0,..., w t 1 ) In the special case of the scenario tree approach, uncertainties are supposed to be observed, and all possible scenarios `w s t 0,..., w s T, s S, are organized in a tree, and controls q i,t are indexed by nodes on the tree Equivalently, q i,t are indexed by s S with the constraint that if two scenarios coincide up to time t, so must do the controls at time t `ws t0,..., wt 1 s = `ws t 0,..., wt 1 s q s i,t = qi,t s Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

68 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 memory a decision/control is chosen, and the system evolves Our aim is to minimize some objective function that depends on the evolution of system Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

69 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 x t of stock at each time step t using decision u t for each time step t while dealing with stochastic data w t, such as the power demand, unit breakdowns, market prices All random variables are written using bold letters Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

70 Moving from deterministic to stochastic dynamic optimization Mathematical formulation There are two ways to express the information constraints Standard problem min x,u C t (x t,u t,w t+1 ) + K (x T )), T 1 E( t=0 s.t. x t+1 = f t (x t,u t,w t+1 ), t = 0,...,T 1, x 0 = w 0, where F t := σ{w 0,...,w t } P-a.-s. constraints (e.g. bounds on x t and u t ), u t is F t -measurable, Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

71 Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints Classical methods Stochastic Programming Discretize random inputs of the problem using scenario trees Write the constraints and objective function on this structure (time has become arborescent) Use standard numerical methods from mathematical programming (linear/convex programming to overcome the combinatorial explosion of the tree) Carpentier/Cohen/Culioli (1995); Birge/Louveaux (1997); Pflug (2001); Bacaud/Lemaréchal/Renaud/Sagastizábal (2001); Heitsch/Römisch (2003); Dupačová/Gröwe-Kuska/Römisch (2003); Barty (2004); Shapiro/Dentcheva/Ruszczyński (2009)... Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

72 Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints Classical methods Stochastic Programming Discretize random inputs of the problem using scenario trees Write the constraints and objective function on this structure (time has become arborescent) Use standard numerical methods from mathematical programming (linear/convex programming to overcome the combinatorial explosion of the tree) Complexity of multistage stochastic programs The amount of scenarios needed to achieve a given precision grows exponentially w.r.t. the number of time steps See Shapiro (2006); Girardeau (2010) Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

73 Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints Classical methods Dynamic Programming (1) Markovian framework Noise variables w t are independent over time Cost-to-go function, Bellman function, value function! TX 1 V t (x) := min E C s (x u s, u s,w s+1) + K (x T ) x t = x, x X, s=t subject to constraints of the original problem Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

74 Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints Classical methods Dynamic Programming (2) The Dynamic Programming (DP) principle Bellman (1957) There is no loss of optimality in looking for the optimal strategy as feedback functions on x t only. Moreover, DP provides a way to compute Bellman functions backwards: V T (x) = K (x), x X T, and, for all t = t 0,..., T 1: V t (x) = min u U t E(C t (x,u,w t+1) + V t+1 (f t (x,u,w t+1))), x X t Reference textbooks: Bertsekas (2000); Puterman (1994); Quadrat/Viot (1999) Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

75 Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints Classical methods Dynamic 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 discretization grows exponentially w.r.t. the dimension of the state space Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

76 Moving from deterministic to stochastic dynamic optimization There are two ways to express the information constraints Classical methods Dynamic 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 discretization grows 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 December 19, / 112

77 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 of online information Many open problems, because many ways to represent risk (criterion, constraints) many information structures tremendous numerical obstacles to overcome Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

78 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 December 19, / 112

79 Two snapshots on ongoing research Outline of the presentation Decomposition-coordination optimization methods under uncertainty 1 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Industry partners 2 The smart grids seen from an optimizer perspective What is the smart grid? Optimization is challenged 3 Moving from deterministic to stochastic dynamic optimization Dam models The deterministic optimization problem is well posed In the uncertain framework, the optimization problem is not well posed There are two ways to express the information constraints 4 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Risk constraints in optimization 5 Conclusion: a need for training Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

80 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Decomposition-coordination: divide and conquer Spatial decomposition multiple players with their local information local / regional / national /supranational Temporal decomposition A state is an information summary Time coordination realized through Dynamic Programming, by value functions Hard nonanticipativity constraints Scenario decomposition Along each scenario, sub-problems are deterministic (powerful algorithms) Scenario coordination realized through Progressive Hedging, by updating nonanticipativity multipliers Soft nonanticipativity constraints Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

81 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Problems and methods: a very tentative table Feel free to comment and criticize! ;-) Decompose and coordinate Space Time Scenarios Scenarios Trees DP PH 2SSP decentralized units OK storage OK unit commitment OK short term market OK Dynamic Programming: DP Progressive Hedging: PH Two and Multi-Stage Stochastic Programming : 2SSP Use specific analytical features Stochastic Programming: linearity, convexity Stochastic Dual Dynamic Programming: linear dynamics, convex Bellman function approximated by cuts Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

82 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty We have a nice decomposed problem but... Flower structure Unit 3 Unit 4 Unit 2 We are almost in the case where units could be driven independently one from another Unit 5 Unit 1 Unit 6 Unit 8 Unit 7 Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

83 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty We have a nice decomposed problem but... Flower structure Unit 3 Unit 4 Unit 2 Unfortunately... Unit 5 Coupling constraint Unit 1 Unit 6 Unit 8 Unit 7 Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

84 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty The associated optimization problem may be written as where N min J i (u i ) (u 1,...,u N ) i=1 } {{ } costs minimization u i is the decision of each unit i under N Θ i (u i ) = D i=1 J i (u i ) is the cost of making decision u i for unit i } {{ } offer = demand Θ i (u i ) is the production induced by making decision u i for unit i Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

85 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Proper prices allow decentralization of the optimum min (u 1,...,u N ) N J i (u i ) under i=1 N Θ i (u i ) = D The simplest decomposition/coordination scheme consists in i=1 buying the production of each unit at a price λ (k) and letting each unit minimizing its costs min u i J i (u i ) + λ (k) Θ i (u i ) }{{} price then, modifying the price depending on the coupling constraint ( N ) λ (k+1) = λ (k) + ρ Θ i (u i ) D i=1 (like in the tâtonnement de Walras in economics) Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

86 Two snapshots on ongoing research Extension to dynamics and stochasticity Decomposition-coordination optimization methods under uncertainty If we explicitely take into account time and uncertainties, a classical approch consists in writing the new problem ( N ( T 1 ( ) ( ) )) min E L i,t xi,t,u i,t,w i,t + Ki xi,t {u i,t} t {0,T 1} i {1,N} under the constraints i=1 t=0 N ( ) Θ i,t xi,t,u i,t,w i,t dt = 0, i=1 x i,t+1 = f i,t ( xi,t,u i,t,w i,t ), t 0, T 1 i 1, N, t 0, T 1 Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

87 Two snapshots on ongoing research We need to specify information constraints Decomposition-coordination optimization methods under uncertainty The optimization problem is not well posed, because we have not specified upon what depends the control u i,t of each unit i at each time t! In the causal and perfect memory case, we say that the control u i,t depends of all past noises up to time t u i,t = φ i,t ( w1,0,...,w N,0,d w 1,t,...,w N,t,d t ) When the control u i,t is looked after under the form u i,t = φ i,t ( wi,0,d w i,t,d t ) 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 December 19, / 112

88 Two snapshots on ongoing research Looking after decentralizing prices models Decomposition-coordination optimization methods under uncertainty Going on with the previous scheme, each unit i solves TX 1 (k) «min E L u i,0,...,u i,t`xi,t,u i,t,w i,t + λ i,t Θi,t`xi,t,u i,t,w i,t + K i `xi,t i,t 1 t=0 under the constraints x i,t+1 = f i,t ( xi,t,u i,t,w i,t ), t 0, T 1 In all rigor, the optimal controls u i,t of this problem depend upon x i,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 y t, where the information variable y t is a short time memory process) Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

89 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Dual Approximate Dynamic Programming Samples/scenarios of dual variable at iteration k λ k Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

90 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Dual Approximate Dynamic Programming Samples/scenarios of dual variable at iteration k λ k We solve subproblems using E(λ k y) Subproblem 1 Subproblem i Subproblem N by Dynamic Programming Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

91 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Dual Approximate Dynamic Programming Samples/scenarios of dual variable at iteration k λ k We solve subproblems using E(λ k y) Subproblem 1 Subproblem i Subproblem N by Dynamic Programming We obtain policies Policy Φ 1 x 1, y Policy Φ i x i, y Policy Φ N x N, y Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

92 Two snapshots on ongoing research Dual Approximate Dynamic Programming Decomposition-coordination optimization methods under uncertainty Samples/scenarios of dual variable at iteration k λ k We solve subproblems using E(λ k y) Subproblem 1 Subproblem i Subproblem N by Dynamic Programming We obtain policies Policy Φ 1 x 1, y Policy Φ i x i, y Policy Φ N x N, y We update prices using a gradient step We simulate scenarios of strategies and prices and update prices using a gradient step λ k+1 t = λ k t + ρ P N i=1 g t i x i,k t, u i,k t, wt Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

93 Two snapshots on ongoing research Dual Approximate Dynamic Programming Decomposition-coordination optimization methods under uncertainty At iteration k + 1 λ k We solve subproblems using E(λ k y) Subproblem 1 Subproblem i Subproblem N by Dynamic Programming We obtain policies Policy Φ 1 x 1, y Policy Φ i x i, y Policy Φ N x N, y We update prices using a gradient step We simulate scenarios of strategies and prices and update prices using a gradient step λ k+1 t = λ k t + ρ P N i=1 g t i x i,k t, u i,k t, wt Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

94 Two snapshots on ongoing research Extension to interconnected dams Decomposition-coordination optimization methods under uncertainty Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

95 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 stochastic processes (prices) By projecting these prices, one expects to identify approximate dynamic models Such prices dynamic models are interpreted as dynamic tariffs Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

96 Two snapshots on ongoing research Outline of the presentation Risk constraints in optimization 1 Long term industry-academy cooperation École des Ponts ParisTech Cermics Optimization and Systems Industry partners 2 The smart grids seen from an optimizer perspective What is the smart grid? Optimization is challenged 3 Moving from deterministic to stochastic dynamic optimization Dam models The deterministic optimization problem is well posed In the uncertain framework, the optimization problem is not well posed There are two ways to express the information constraints 4 Two snapshots on ongoing research Decomposition-coordination optimization methods under uncertainty Risk constraints in optimization 5 Conclusion: a need for training Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

97 Two snapshots on ongoing research Risk constraints in optimization Dam management under tourism constraint Maximizing the revenue from turbinated water under a tourism constraint of having enough water in July and August Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

98 Two snapshots on ongoing research Risk constraints in optimization Stock trajectories Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

99 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) } {{ } = min{s, S(t) }{{} future volume 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 December 19, / 112

100 Two snapshots on ongoing research The traditional economic problem is maximizing the expected payoff Risk constraints in optimization Suppose that a probability P is given on the set Ω = R T t 0 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 energy can 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 the intertemporal payoff (without discounting if the horizon is short) T 1 price final volume utility {}}{ max E p(t) turbined { }} { q(t) }{{} + UtilFin ( S(T) ) t=t 0 Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

101 Two snapshots on ongoing research Risk constraints in optimization Dam management under tourism constraint Traditional cost minimization/payoff maximization T 1 turbined water payoff final volume utility { }} { { }} { max E p(t)q(t) + UtilFin ( S(T) ) For tourism reasons: t=t 0 volume S(t) S, t { July, August } In what sense should we consider this inequality which involves the random variables S(t) for t { July, August }? Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

102 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 water inflow scenarios along which Probability the volumes S(t) are above the threshold S = 1 for periods t in summer Probability constraints, with confidence level p [0, 1] water inflow scenarios along which Probability the volumes S(t) are above the threshold S p for periods t in summer and also by penalization (to be discussed later), or in the mean, etc. Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

103 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=t 0 turbined water payoff { }} { p(t)q(t) + final volume utility { }} { UtilFin ( S(T) ) and a failure tolerance is accepted water inflow scenarios along which Probability the volumes S(t) S for periods t in July and August 99% Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112

104 Stock trajectories Two snapshots on ongoing research Risk constraints in optimization Michel DE LARA (Cermics, France) Horizon Maths 2012, Paris, 19 Dec December 19, / 112