Master ParisTech REST
Renewable Energy Science and Technology
   
Graduate Degree STEEM
Energy Environment: Science Technology and Management

PHY661D 2017-2018

Stochastic and Decentralized Optimization
for the Management of Micro-Grids

Michel DE LARA, CERMICS-École des Ponts ParisTech


pdf version of this document


Eligibility/Pre-requisites.


Learning outcomes. After the course the student should be able to


Course main content. The course mixes theoretical sessions, modeling exercises and computer sessions.

In introduction, we present examples of micro-grid and virtual power plant management -- where the question of electrical storage is put, due to the need to answer a varying demand and to incorporate intermittent and highly variable renewable energies. During the course, we will present concepts and tools to formulate such problems as stochastic dynamic optimization problems. For this purpose, the first sessions are dedicated to mathematical recalls in probability and optimization, followed by an introduction to the scientific software Scicoslab.

Then, we turn to stochastic optimization. In a deterministic optimization problem, the values of all parameters are supposed known. What happens when this is no longer the case? And when some values are revealed during the stages of decision? We present stochastic optimization, at the same time as a frame to formulate problems under uncertainty, and as methods to solve them according to the formulation. More precisely, we present two-stage stochastic programming (and the resolution on scenario tree or by scenarios) and multi-stage stochastic control (and the resolution by stochastic dynamic programming). We finish with the Stochastic Dual Dynamic Programming (SDDP) algorithm (used in commercial software in the world of the energy), which mixes dynamic programming and cutting plane algorithm. Depending on time availability, we will try to shed light on decomposition methods that lead to decentralized optimization (especially adapted to micro-grid management).

Modeling exercises and computer sessions tackle issues like optimal economic dispatch of energy production units, storage/delivery optimization problem to buffer an intermittent and variable source of energy, dam optimal management with stochastic water inflows, battery optimal management with renewable energy inputs.


Examination and requirements for final grade. At the end of each computer session, the student produces a report, which receives a mark after evaluation. Mini-exams, presence and participation also contribute to the final grade.


Contact person.     Michel De Lara (Cermics--École des Ponts ParisTech)      professional webpage


Link course.    
http://cermics.enpc.fr/~delara/TEACHING/Graduate_Degree_STEEM_2017/      course webpage


Link Graduate Degree STEEM.    
https://portail.polytechnique.edu/graduatedegree/steem/      Graduate Degree STEEM               map of the courses rooms


Program


1 / Tuesday 9, January 2018

Introductory talks (14h00-15h00)

To introduce the course, we present examples of micro-grid and virtual power plant management (that can be solved using stochastic dynamic optimization):

Lecture (15h00-16h00)

Recalls on probability calculus: probability space, probability, random variables, law of a random variable, mathematical expectation, indicator function, independence of random variables, almost-sure convergence and law of large numbers. [Fel68,Bre93,Pit93]

Exercises (16h30-18h00)

Exercises on probability calculus.


2 / Tuesday 16, January 2018

Lecture and exercises (14h00-16h30)

Recalls and exercises on continuous optimization [Ber96].

Exercises (17h00-18h00)

We present, under the form of an exercise, an example of optimization problem under uncertainty: ``the newsvendor problem''.      slides


3 / Tuesday 23, January 2018

Computer session

Introduction to the scientific software Scicoslab. [CCN10]      computer session

Computer session

The newsvendor problem


4 / Tuesday 30, January 2018

Modeling session

``Day Ahead Energy Markets''     slides

Computer session

The newsvendor problem
You will send the results of the computer project The newsvendor problem
under the form of a pdf file TP1_STEEM_2017_MYNAME.pdf to delara@cermics.enpc.fr before .


5 / Tuesday 6, February 2018

Lecture

Two-stage stochastic programming on a scenario tree.

Non-anticipativity constraint along scenarios: tree representation.

[SDR09,KW12]

Computer session

     Sizing of reserves for the balancing on an electric market
(linear and quadratic optimization on a tree)


6 / Tuesday 13, February 2018

Lecture

Two-stage stochastic programming on a fan.

Non-anticipativity constraint along scenarios.

Scenario decomposition by Lagrangian relaxation. Progressive Hedging [RW91].

Computer session

     Sizing of reserves for the balancing on an electric market
(linear and quadratic optimization on a fan)
You will send the results of the computer project Sizing of reserves for the balancing on an electric market
under the form of a pdf file TP2_STEEM_2017_MYNAME.pdf to delara@cermics.enpc.fr before .


7 / Tuesday 27, February 2018

Exercises (14h00-16h00)

Exercises on two-stage stochastic programming.

Exam (16h30-18h00)

Exam on two-stage stochastic programming.


8 / Tuesday 6, March 2018

Lecture and exercises

    
Dynamical models of storage (battery models, dam models).
Dynamical sequential systems with control.
Optimal control of dynamical sequential systems.      slides
Dynamic programming. Curse of dimensionality.
Exercises on dynamic programming.

Dynamical sequential systems with control and noise.
Optimal control of stochastic dynamical sequential systems.      slides
Stochastic dynamic programming. Curse of dimensionality.      slides
Exercises on stochastic dynamic programming.

[Bel57,Put94,Ber00,Whi82,CCCD15]


9 / Tuesday 13, March 2018

Computer session

Code the dynamic programming algorithm.

     A Numerical Toy Stochastic Control Problem Solved by Dynamic Programming

Computer session

    Dam optimal management under uncertainty
You will send the results of the computer project Dam optimal management under uncertainty
under the form of a pdf file TP3_STEEM_2017_MYNAME.pdf to delara@cermics.enpc.fr before .


10 / Tuesday 20, March 2018

Lecture (14h00-16h00)

    

Inventory problems. Optimal storage management in the hazard-decision framework, with linear costs.

Stochastic optimal control with convex costs and linear dynamics.

Presentation of the Stochastic Dual Dynamic Programming (SDDP) algorithm.      slides


Exam (16h30-18h00)

Exam on stochastic dynamic programming.


Bibliography

Bel57
R. E. Bellman.
Dynamic Programming.
Princeton University Press, Princeton, N.J., 1957.

Ber96
D. P. Bertsekas.
Constrained Optimization and Lagrange Multiplier Methods.
Athena Scientific, Belmont, Massachusets, 1996.

Ber00
D. P. Bertsekas.
Dynamic Programming and Optimal Control.
Athena Scientific, Belmont, Massachusets, second edition, 2000.
Volumes 1 and 2.

Bre93
L. Breiman.
Probability.
Classics in applied mathematics. SIAM, Philadelphia, second edition, 1993.

CCCD15
P. Carpentier, J.-P. Chancelier, G. Cohen, and M. De Lara.
Stochastic Multi-Stage Optimization. At the Crossroads between Discrete Time Stochastic Control and Stochastic Programming.
Springer-Verlag, Berlin, 2015.

CCN10
Stephen Campbell, Jean-Philippe Chancelier, and Ramine Nikoukhah.
Modeling and Simulation in Scilab/Scicos with ScicosLab 4.4.
Springer-Verlag, New York, 2 edition, 2010.

Fel68
W. Feller.
An Introduction to Probability Theory and its Applications, volume 1.
Wiley, New York, third edition, 1968.

KW12
Alan J. King and Stein W. Wallace.
Modeling with Stochastic Programming.
Springer Series in Operations Research and Financial Engineering. Springer New York, 2012.

Pit93
J. Pitman.
Probability.
Springer-Verlag, New-York, 1993.

Put94
M. L. Puterman.
Markov Decision Processes.
Wiley, New York, 1994.

RW91
R.T. Rockafellar and R. J-B. Wets.
Scenarios and policy aggregation in optimization under uncertainty.
Mathematics of operations research, 16(1):119-147, 1991.

SDR09
A. Shapiro, D. Dentcheva, and A. Ruszczynski.
Lectures on stochastic programming: modeling and theory.
The society for industrial and applied mathematics and the mathematical programming society, Philadelphia, USA, 2009.

Whi82
P. Whittle.
Optimization over Time: Dynamic Programming and Stochastic Control, volume 1 and 2.
John Wiley & Sons, New York, 1982.