Wim van AckooijÉlectricité de France (EDF) | EDF · R&D
Wim van Ackooij
HdR, Dr., Ir.
About
160
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Introduction
My research interests are in Stochastic programming and more specifically with probabilistic constraints.
Additional affiliations
December 2013 - March 2020
September 2003 - December 2013
Education
October 2017 - December 2017
February 2012 - December 2013
Ecole Centrale Paris
Field of study
- Applied Mathematics
September 1998 - June 2003
Publications
Publications (160)
Probabilistic constraints represent a major model of stochastic optimization. A possible approach for solving probabilistically constrained optimization problems consists in applying nonlinear programming methods. To do so, one has to provide sufficiently precise approximations for values and gradients of probability functions. For linear probabilist...
Probability constraints play a key role in optimization problems involving uncertainties. These constraints request that an inequality system depending on a random vector has to be satisfied with a high enough probability. In specific settings, copulæ can be used to model the probabilistic constraints with uncertainty on the left-hand side. In this...
We consider probability functions of parameter-dependent random inequality systems under Gaussian distribution. As a main result, we provide an upper estimate for the Clarke subdifferential of such probability functions without imposing compactness conditions. A constraint qualification ensuring continuous differentiability is formulated. Explicit...
Probability constraints are often employed to intuitively define safety of given decisions in optimization problems. They simply express that a given system of inequalities depending on a decision vector and a random vector is satisfied with high enough probability. It is known that, even if this system is convex in the decision vector, the associa...
Probability constraints are a popular modelling mechanism in applications. They help to model feasible decisions when the latter are taken prior to observing uncertainty and both decisions and uncertainty are involved in a constraint structure of an optimization problem. The popularity of this paradigm is attested by a vast literature using probabi...
We explore and construct an enlarged subdifferential for weakly convex functions. The resulting object turns out to be continuous with respect to both the function argument and the enlargement parameter. We carefully analyze connections with other constructs in the literature and particularize to the weakly convex setting well-known variational pri...
In this paper we investigate optimal control problems perturbed by random events. We assume that the control has to be decided prior to observing the outcome of the perturbed state equations. We investigate the use of probability functions in the objective function or constraints to define optimal or feasible controls. We provide an extension of di...
Chance constraints are a valuable tool for the design of safe decisions in uncertain environments; they are used to model satisfaction of a constraint with a target probability. However, because of possible non-convexity and non-smoothness, optimizing over a chance constrained set is challenging. In this paper, we consider chance constrained progra...
In this work we discuss the question of investment in power systems wherein the need for flexibility would be a driving force for investments. The investment cost itself consists of a capital investment cost and an operational cost, wherein the latter evaluates the cost of operating the system for a given installed capacity. We argue that the opera...
Probability functions appear in constraints of many optimization problems in practice and have become quite popular. Understanding their first-order properties has proven useful, not only theoretically but also in implementable algorithms, giving rise to competitive algorithms in several situations. Probability functions are built up from a random...
Handling cascading reservoir systems is an important energy management optimization problem. The difficulty of these problems stems, in part, from the modeling of the hydro-production function. Data are not always easy to come by. To remedy this issue, this paper proposes and describes a realistic instance generator, building instances of varying d...
Optimization problems with uncertainty in the constraints occur in many applications. Particularly, probability functions present a natural form to deal with this situation. Nevertheless, in some cases, the resulting probability functions are nonsmooth, which motivates us to propose a regularization employing the Moreau envelope of a scalar represe...
Probability functions appear in constraints of many optimization problems in practice and have become quite popular. Understanding their first-order properties has proven useful, not only theoretically but also in implementable algorithms, giving rise to competitive algorithms in several situations. Probability functions are built up from a random...
We consider a revenue maximization model, in which a company aims at designing a menu of contracts, given a population of customers. A standard approach consists in constructing an incentive-compatible continuum of contracts, i.e., a menu composed of an infinite number of contracts, where each contract is especially adapted to an infinitesimal cust...
Optimization problems with uncertainty in the constraints occur in many applications. Particularly, probability functions present a natural form to deal with this situation. Nevertheless, in some cases, the resulting probability functions are nonsmooth. This motivates us to propose a regularization employing the Moreau envelope of a scalar represen...
In many practical applications models exhibiting chance constraints play a role. Since, in practice one is also typically interesting in numerically solving the underlying optimization problems, an interest naturally arises in understanding analytical properties, such as differentiability, of probability functions. However in order to build nonline...
In this addendum to our paper published in [Optim. Meth. Softw. 34(1) (2019), pp. 890–920], we provide more details concerning Proposition 2.1 figuring therein. The original proposition implicitly assumed that the constraint mapping was Clarke-regular. Here, not only do we fix some inaccuracies and confusing statements in Section 2 of that paper, b...
We propose a generic multistage stochastic model for the Alternating Current Optimal Power Flow (AC OPF) problem for radial distribution networks, to account for the random electricity production of renewable energy sources and dynamic constraints of storage systems. We consider single-phase radial networks. Radial three-phase balanced networks (me...
We consider a control problem for a heterogeneous population composed of customers able to switch at any time between different contracts, depending not only on the tariff conditions but also on the characteristics of each individual. A provider aims to maximize an average gain per time unit, supposing that the population is of infinite size. This...
![Function fξ\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/358219490/figure/fig2/AS:1146390155661312@1650332120193/Function-fxdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
In this work we consider probability functions working on parameter dependent sets that are given as an intersection of convex sets and their complements. Such an underlying structure naturally arises when having to handle bilateral inequality systems in various applications, such as energy. We provide conditions under which the probability functio...
In decision-making problems under uncertainty, probabilistic constraints are a valuable tool to express safety of decisions. They result from taking the probability measure of a given set of random inequalities depending on the decision vector. Even if the original set of inequalities is convex, this favourable property is not immediately transferr...
We consider a profit-maximizing model for pricing contracts as an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation price and multiple price coefficients (attributes). Classical approaches suppose that the customers have deterministic utilities; then, the respo...
Probability functions measure the degree of satisfaction of certain constraints that are impacted by decisions and uncertainty. Such functions appear in probability or chance constraints ensuring that the degree of satisfaction is sufficiently high. These constraints have become a very popular modelling tool and are indeed intuitively easy to under...
In this paper, we tackle the hydro unit commitment problem and scheduling in a hydro valley. We first decompose the problem into several simpler subproblems, one for each reservoir/plant. Then, we model each of them as an optimization problem on graphs with or without resource constraints. We compare our method with a commercial solver for mixed in...
We propose a generic multistage stochastic model for the Alternating Current Optimal Power Flow (AC OPF) problem for radial distribution networks, to account for the random electricity production of renewable energy sources and dynamic constraints of storage systems. We consider single-phase radial networks. Radial three-phase balanced networks (me...
Chance constraints are a valuable tool for the design of safe decisions in uncertain environments; they are used to model satisfaction of a constraint with a target probability. However, because of possible non-convexity and non-smoothness, optimizing over a chance constrained set is challenging. In this paper, we establish an exact reformulation o...
This work considers nonsmooth and nonconvex optimization problems whose objective and constraint functions are defined by difference-of-convex (DC) functions. We consider an infeasible bundle method based on the so-called improvement functions to compute critical points for problems of this class. Our algorithm neither employs penalization techniqu...
In decision-making problems under uncertainty, probabilistic constraints are a valuable tool to express safety of decisions. They result from taking the probability measure of a given set of random inequalities depending on the decision vector. Even if the original set of inequalities is convex, this favourable property is not immediately transferr...
Intermittent sources of energy represent a challenge for electrical networks, particularly regarding demand satisfaction at peak times. Energy management tools such as load shaving or storage systems can be used to mitigate intermittency. In this work, the value of different mechanisms to move energy through time is examined through a multi-objecti...
The valuation of a real option is preferably done with the inclusion of uncertainties in the model, since the value depends on future costs and revenues, which are not perfectly known today. The usual value of the option is defined as the maximal expected (discounted) profit one may achieve under optimal management of the operation. However, also t...
The original version of this article unfortunately contained a mistake in equation (1). The correct expression is given below:
A significant share of stochastic optimization problems in practice can be cast as two-stage stochastic programs. If uncertainty is available through a finite set of scenarios, which frequently occurs, and we are interested in accounting for risk aversion, the expectation in the recourse cost can be replaced with a worst-case function (i.e., robust...
This works investigates a conceptual algorithm for computing critical points of nonsmooth nonconvex optimization problems whose objective function is the sum of two locally Lipschitzian (component) functions. We show that upon additional assumptions on the component functions and or feasible set, the given algorithm extends several well-known optim...
Probability functions appearing in chance constraints are an ingredient of many practical applications. Understanding differentiability, and providing explicit formulae for gradients, allow us to build nonlinear programming methods for solving these optimization problems from practice. Unfortunately, differentiability of probability functions canno...
Multi stage stochastic programs arise in many applications from engineering whenever a set of inventories or stocks has to be valued. Such is the case in seasonal storage valuation of a set of cascaded reservoir chains in hydro management. A popular method is Stochastic Dual Dynamic Programming (SDDP), especially when the dimensionality of the prob...
![Solution progress for instance T=25\documentclass[12pt]{minimal}...](publication/332821397/figure/fig2/AS:961388600111122@1606224312613/Solution-progress-for-instance-T25documentclass12ptminimal-usepackageamsmath_Q320.jpg)
![Solution progress for instance T=61\documentclass[12pt]{minimal}...](publication/332821397/figure/fig3/AS:961388600102921@1606224312700/Solution-progress-for-instance-T61documentclass12ptminimal-usepackageamsmath_Q320.jpg)
![Instance T=97\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/332821397/figure/fig4/AS:961388600107011@1606224312915/Instance-T97documentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
We consider well-known decomposition techniques for multistage stochastic programming and a new scheme based on normal solutions for stabilizing iterates during the solution process. The given algorithms combine ideas from finite perturbation of convex programs and level bundle methods to regularize the so-called forward step of these decomposition...
![Graphs of functions and fρ\documentclass[12pt]{minimal}...](publication/331872438/figure/fig2/AS:779422978281499@1562840327832/Graphs-of-functions-and-frdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
We propose an optimization technique for computing stationary points of a broad
class of nonsmooth and nonconvex programming problems. The proposed approach
(approximately) decomposes the objective function as the difference of two convex
functions and performs inexact optimization of the resulting (convex) subproblems.
We prove global convergence...
This work concerns the study of a constraint qualification for nonsmooth DC-constrained optimization problems, as well as the design and convergence analysis of minimizing algorithms to address the task of computing a stationary/critical point for problems of this class. Specialized algorithms for DC programming approximate the nonconvex optimizati...
Probability functions are a powerful modelling tool when seeking to account for uncertainty in optimization problems. In practice, such uncertainty may result from different sources for which unequal information is available. A convenient combination with ideas from robust optimization then leads to probust functions, i.e. probability functions act...
In this paper we investigate probability functions acting on nonlinear systems wherein the random vector can follow an elliptically symmetric distribution. We provide first and second order differentiability results as well as readily implementable formulæ. We also demonstrate that these formulæ can be readily employed within standard non-linear pr...
The Unit Commitment problem in energy management aims at finding the optimal production schedule of a set of generation units, while meeting various system-wide constraints. It has always been a large-scale, non-convex, difficult problem, especially in view of the fact that, due to operational requirements, it has to be solved in an unreasonably sm...
The ever-growing insertion of intermittent energy sources requires to account for this uncertainty by precise models. Probabilistic constraints are an interesting technique to deal with the high fluctuations of such energy resources, while accounting for power flow and unit generating constraints. In the context of hydrothermal unit commitment prob...
In this paper we consider energy management optimization problems in a future wherein an interaction with micro-grids has to be accounted for. We will model this interaction through a set of contracts between the generation companies owning centralized assets and the micro-grids. We will formulate a general stylized model that can, in principle, ac...
Increasing the share of non-synchronous sources in the energy mix leads to a decline in the dynamic response of electrical systems. In particular, the performance of the primary frequency control may drop, increasing the risk of Under Frequency Load Shedding (UFLS) as a consequence of large power imbalances. This fact has drawn attention to the sim...
We solve probability maximization problems using an approximation scheme that is analogous to the classic approach of p-efficient points, proposed by Prékopa to handle chance constraints. But while p-efficient points yield an approximation of a level set of the probabilistic function, we approximate the epigraph. The present scheme is easy to imple...
We present a computational study of several strategies to solve two-stage stochastic linear programs by integrating the adaptive partition-based approach with level decomposition. A partition-based formulation is a relaxation of the original stochastic program, obtained by aggregating variables and constraints according to a scenario partition. Par...
We propose a family of proximal bundle methods for minimizing sum-structured convex nondifferentiable functions which require two slightly uncommon assumptions that are satisfied in many relevant applications: Lipschitz continuity of the functions and oracles which also produce upper estimates on the function values. In exchange, the methods: (i) u...
We consider the Unit Commitment subproblem dedicated to hydro valley management, also known as Hydro Unit Commitment Problem (HUCP). The problem consists in finding an optimal hydro schedule for hydro valleys composed of head-dependent reservoirs for a short term period in which the electricity prices and the inflows are forecasted. We propose a Mi...
This Thesis was presented to obtain a Habilitation (HdR) at Université Paris 1 Panthéon Sorbonne
![Example of a penalization function ψt\documentclass[12pt]{minimal}...](publication/317065446/figure/fig6/AS:960357917028365@1605978578212/Example-of-a-penalization-function-pstdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
We consider a general robust block-structured optimization problem, coming from applications in network and energy optimization. We propose and study an iterative cutting-plane algorithm, generic for a large class of uncertainty sets, able to tackle large-scale instances by leveraging on their specific structure. This algorithm combines known techn...
In energy management, the unit-commitment problem deals with computing the most cost
efficient production schedule that meets customer load, while satisfying the operational constraints of
the units. When the problem is large-scale and/or much modelling detail is required decomposition
approaches are vital for solving this problem. The recent stron...
http://paperity.org/p/77751146/editorial-for-the-special-issue-optimization-in-energy
In this paper, we study second-order differentiability properties of probability functions. We present conditions under which probability functions involving nonlinear systems and Gaussian (or Student) multi-variate random vectors are twice continuously differentiable. We provide an expression for their Hessian that can be useful in numerical metho...
For problems when decisions are taken prior to observing the realization of underlying random events, probabilistic constraints are an important modelling tool if reliability is a concern. A key concept to numerically dealing with probabilistic constraints is that of p-efficient points. By adopting a dual point of view, we develop a solution framew...
We explore modifications of the standard cutting-plane approach for minimizing a convex nondifferentiable function, given by an oracle, over a combinatorial set, which is the basis of the celebrated (generalized) Benders’ decomposition approach. Specifically, we combine stabilization—in two ways: via a trust region in the L1 norm, or via a level co...
The aim of the project is to study a crucial problem in energy management: the Unit Commitment (UC) sub-problem dedicated to hydro valley management, see [3]. The problem consists in finding an optimal short-term hydro scheduling for a hydro valley composed of head-dependent reservoirs [4]. When continuous, such a problem is easily solved to optima...
This poster describes a model for designing offshore wind turbines while using probability constraints
L’augmentation de la part des sources nonsynchrones dans le mix énergétique entraîne une dégradation de la réponse dynamique d’un système électrique. En particulier, une baisse de performance de la régulation primaire de fréquence pourrait augmenter le risque de délestage de charge suite à de forts déséquilibres de puissance. Ceci remet en cause le...
In energy management, the unit-commitment problem deals with computing the most cost efficient production
schedule that meets customer load, while satisfying the operational constraints of the units. When the problem
is large-scale and/or much modelling detail is required decomposition approaches are vital for solving this
problem. The recent stron...
Everyday, electricity generation companies submit a generation schedule to the grid operator for the coming day; computing an optimal schedule is called the unit-commitment problem. Generation companies can also occasionally submit changes to the schedule, that can be seen as intra-daily incomplete recourse actions. In this paper, we propose a two-...
A key procedure in proximal bundle methods for convex minimization problems is the defini-tion of stability centers, which are points generated by the iterative process that successfully decrease the objective function. In this paper we study a different stability-center classifica-tion rule for proximal bundle methods. We show that the proposed bu...
This project deals with theoretical and numerical studies aiming at the improvement of unit-commitment problems subject to uncertainty. Unit-commitment refers to the operation of a physical system - the power mix - for which it is becoming more and more important to incorporate stochastic effects, due to the increased use of renewable energy source...
For optimization problems involving probabilistic constraints it is important to be able to compute gradients of these constraints efficiently. For distribution functions of Gaussian random variables with positive definite covariance matrices, computing gradients can be done efficiently. Indeed, a formula, linking components of the partial derivati...
The Unit Commitment problem in energy management aims at finding the optimal productions schedule of a set of generation units while meeting various system-wide constraints. It has always been a large-scale, non-convex difficult problem, especially in view of the fact that operational requirements imply that it has to be solved in an unreasonably s...
In decision-making problems where uncertainty plays a key role and decisions have to be taken prior to observing uncertainty, chance constraints are a strong modelling tool for defining safety of decisions. These constraints request that a random inequality system depending on a decision vector has to be satisfied with a high probability. The chara...
Ce document, à but pédagogique, illustre comment la prise en compte des incertitudes au sein d'un
problème d'optimisation entraîne une multiplicité de questions de modélisation. Les propos sont
illustrés à l'aide du problème de la gestion de production court-terme. Nous rappelons également les
travaux menées en interne autour de l'optimisation sous...
The Unit Commitment problem in energy management aims at finding the optimal productions schedule of a set of generation units while meeting various system-wide constraints. It has always been a large-scale, non-convex difficult problem, especially in view of the fact that operational requirements imply that it has to be solved in an unreasonably s...
The unit commitment problem, aims at computing the production schedule that satisfies the offer-demand equilibrium at minimal cost. Often such problems are considered in a deterministic framework. However uncertainty is present and non-negligible. Robustness of the production schedule is therefore a key question. In this paper, we will investigate...
In this paper, we deal with a cascaded reservoir optimization problem with
uncertainty on inflows in a joint chance constrained programming setting. In particular,
we will consider inflows with a persistency effect, following a causal time series
model with Gaussian innovations. We present an iterative algorithm for solving
similarly structured joi...
We propose restricted memory level bundle methods for minimizing constrained convex nonsmooth optimization problems whose objective and constraint functions are known through oracles (black-boxes) that might provide inexact information. Our approach is general and covers many instances of inexact oracles, such as upper, lower and on-demand accuracy...
Joint chance constrained problems give rise to many algorithmic challenges. Even in the convex case, i.e., when an appropriate transformation of the probabilistic constraint is a convex function, its cutting-plane linearization is just an approximation, produced by an oracle providing subgradient and function values that can only be evaluated inexa...
In optimization problems involving uncertainty, probabilistic constraints are an important tool for defining safety of decisions. In Energy management, many optimization problems have some underlying uncertainty. In particular this is the case of unit commitment problems. In this Thesis, we will investigate probabilistic constraints from a theoreti...
In this paper, we will consider a midterm offer demand equilibrium model for electricity. This kind of model can be used for investment opportunity studies, wherein new assets are valued against obtained marginal costs on a restricted set of uncertainty scenarios. In order to correctly value peak-load assets, realistic marginal costs are required a...
This paper deals with a Chance-Constrained Programming formulation and approximate resolution of an offer-demand equilibrium
problem in the context of electricity markets. First, we state the probabilistic model. Computing the coefficients of the
problem matrix is easy for financial assets, but a challenging task for physical assets. By introducing...
Questions
Questions (3)
My question is as follows:
Let us consider an energy system (hydro thermal say) and the task of coming up with a feasible production schedule today for tomorrow. This will typically engage some commitment decisions and due to technical constraints some of these may be binding. Now over the course of the day, uncertainty on load, perhaps partially due to the presence of intermittent renewable generation may lead to situations where it is hard to recover (e.g., the presence of a duck curve inducing a heavy constraint on ramping).
Typically however unit commitment is solved in "deterministic" mode. Perhaps with various tweaks to account for uncertainty without saying this (power margins, spinning reserve requirements etc...). As an alternative one could consider methods from stochastic programming and set up a stochastic unit-commitment model (perhaps 2-stage).
Frequently decision makers believe that it is sufficient to tweak the deterministic model and for instance, provided that sufficient power margin is set aside, everything will work out fine. I would personally disagree, but what is your opinion on the topic?
My question is as follows:
a) Let M be a C^r manifold in R^m of dimension k (assume say k < m)
b) Let us consider the set R_+M = { rz : r >= 0, z in M } equally as a subset of R^m ;
c) Can we conclude that R_+M is a C^r manifold too, and if so how is the dimension impacted. (I am guessing it is not larger than k + 1). If true how would one show this, or what reference to a (positive or negative) result could you provide?
I am interested in the following question:
1) Is given a continuous set-valued mapping M : R^n => R^m ;
2) I have a fixed vector z in R^m and am interested in D := {x in R^n : z \in int co M(x) },
where int = the interior, co the convex hull.
Now the question is : can we conclude that for a given x1, x2 in D that there exist points
z1, ..., zk in M(x1) \cap M(x2) and weights \lambda_1, ..., \lambda_k such that z = \sum_{i=1}^k \lambda_i z_i.
If so any reference to a work where this c