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A Decomposition based on Disaggregation
As stated in the introduction, we consider a centralized entity (e.g. an energy operator) in-terested in minimizing a possibly nonconvex cost function p 7!f (p), where p 2 RT is the aggregate allocation of T dimensional resources (for example power production over T time periods). This resource allocation p is to be shared between a set N of N individual agents, each agent obtaining a part xn 2 Xn, where Xn denotes the individual feasibility set of agent n. The global problem the operator wants to solve is described in (1.1). The idea behind the results of this chapter is that, in problem (1.1), the constraints set Xn and individual profile xn are confidential to agent n and should not be disclosed to the central operator or to another agent. Let us define the set PD of feasible aggregate allocations that are disaggregeable as: def p 2 P j 9x 2 X ; p = ån xn . (1.4) PD = Feasibility of problem (1.1) is equivalent to having PD not empty. Constraints for each agent are composed of a global demand over the resources and lower and upper bounds over each resource, as given below: Assumption 1.1. For each n 2 N , there exists En > 0, xn 2 RT, xn 2 RT such that : Xn = fxn 2 RT : åt2T xn,t = En and x n,t 6 xn,t 6 n,tg 6= ˘. (1.5) x In particular, Xn is convex and compact. Given an allocation p, the structure obtained on the matrix (xn,t)n,t, where sums of coefficients along columns and along rows are fixed, is often referred to as the transportation problem. These problems found various applications (see e.g. [AN79; Mun57]). We focus on this case in Sections 1.2 and 1.3, while in Section 1.4, we shall give a generalization of some of our results in the general case where Xn is a poly-hedron. Given a particular allocation p 2 P, the operator will be interested to know if this allo-cation is disaggregeable, that is, if there exists individual profiles (xn)n2N 2 Õn X n summing to p, or equivalently if the disaggregation problem (1.2) has a solution. Following (1.2), the disaggregate profile refers to x, while the aggregate profile refers to the allocation p. Problem (1.2) may not always be feasible. Some necessary conditions for a disaggregation to exist, obtained by summing the individual constraints on N , are the following aggregate constraints: p>1T = E>1N (1.6a) and x >1N 6 p 6 >1N . (1.6b) x Those conditions are not sufficient in general, as explained in the following section.
An equivalent flow problem and Hoffman conditions
The particular structure of the problem we consider implies that we can write it as a flow problem in a graph, as stated in Proposition 1.1. We refer the reader to the book [Coo+09, Chapter 3] for the terminology. Definition 1.1. Consider a directed graph G = (V and demands d : V ! R (where dv < 0 means capacities ‘ : E ! R+ and upper capacities u , E) with vertices V and edges E V V, that v is a production node), edge lower : E ! R+. A flow on G is a function.
Generation of Hoffman’s constraints with APM
In this section, we propose an algorithm that solves (1.1) while preserving the privacy of each agent constraints Xn and individual profile xn 2 RT. To do this, the proposed algorithm is implemented in a decentralized fashion and relies on the alternate projections method (APM) to solve the disaggregation problem (1.2). Let us consider the polyhedron enforcing the agents constraints: X def= X1 XN , where Xn = nxn 2 R+ j åt2T xn,t = En and 8t, xn,t 6 xn,t 6 xn,to . (1.10) def T Besides, given an allocation p 2 P, we consider the set of profiles aggregating to p : Yp = x 2 RNT j 8t 2 T , ån2N xn,t = pt . def Note that Yp is an affine subspace of R NT (to be distinguished from P which is a subset of RT), and that Yp \ X is empty iff p 2/ PD, according to the definition of PD in (1.4). The idea of the proposed algorithm is to build a finite sequence of decreasing subsets (P(s))06s6S such that: = P(0) P(1)P(S) PD .
At each iteration, a new aggregate resource allocation p(s) is obtained by solving an instance of the master problem introduced in (1.3) with Q = P(s): min f (p) (1.11a) p2RT s.t. p 2 P(s) . (1.11b) In the remaining of the chapter, we will refer to (1.11) as an instance of master problem. Our procedure relies on the following immediate observation: Proposition 1.3. If p(s) is a solution of (1.11), and Yp(s) \ X 6= ˘ and x 2 Yp(s) \ X , then (p(s), x) is an optimal solution of the initial problem (1.1). Having in hands a solution p(s), we can check if Yp(s) \ X 6= ˘ using APM on X and p(s) , as described in Algorithm 1.1 below (where Y = Yp).
Table of contents :
General Introduction
1 From a Centralized to a Decentralized Electric System
2 Mathematical framework: from Optimization to Games
3 Contents of the manuscript
4 Contributions of the Chapters
Notation and Conventions
I Decentralized Management of Flexibilities and Optimization
1 Privacy-preserving Disaggregation for Optimal Resource Allocation
1.1 Introduction
1.2 Resource Allocation and Transportation Structure
1.3 Disaggregation based on APM
1.4 Generalization to Polyhedral Agents Constraints
1.5 Numerical examples
1.6 Conclusion
Appendices
1.C Fast Projection on a Boxed Simplex
II Decentralized Management of Flexibilities and Game Theory
2 Two billing mechanisms for Demand Response: Efficiency and Fairness
2.1 Introduction
2.2 Energy Consumption Game
2.3 Measuring efficiency and fairness
2.4 Daily Proportional Billing: social optimality
2.5 Hourly Proportional Billing: Fairness
2.6 Numerical Experiments
2.7 Conclusion
3 Analysis of an Hourly Billing Mechanism for Demand Response
3.1 Introduction
3.2 Consumption Game with Hourly Billing
3.3 Fast Computation of the Nash Equilibrium
3.4 Simulation of Online Demand Response
3.5 Conclusion
Appendices
4 Impact of Consumers Temporal Preferences in Demand Response
4.1 Introduction
4.2 Context and Energy Consumption Game
4.3 Social cost versus System cost
4.4 Properties
4.5 Numerical experiments
4.6 Conclusion
III Efficient Estimation of Equilibria in Large Games: from Nash to Wardrop
5 Estimation of Equilibria of Large Heterogeneous Congestion Games
5.1 Introduction
5.2 Congestion Games with Coupling Constraints
5.3 Approximating VNEs of a Large Game
5.4 Application to Demand Response and Electricity Flexibilities
5.5 Conclusion
Appendices
6 Nonatomic Aggregative Games with Infinitely Many Types
6.1 Introduction
6.2 Monotonicity, Coupling Constraints and Symmetric Equilibrium
6.3 Approximating Nonatomic Aggregative Games with an Infinity of types
6.4 Illustration on a Smart Grid Example
6.5 Conclusion
IV Decentralized Energy Exchanges in a Peer to Peer Framework
7 A p2p Electricity Market Analysis based on Generalized Nash Equilibrium
7.1 Introduction
7.2 Prosumers and Local Communities
7.3 Centralized Market Design
7.4 Peer-to-Peer Market Design
7.5 Test Cases
7.6 Conclusion
Conclusion and Perspectives
A Acronyms and Abbreviations
B Introduction (Français)
B.1 Système Électrique : du Centralisé vers le Décentralisé
B.2 Cadre Mathématique : de l’Optimisation aux Jeux
B.3 Organisation du Manuscrit
B.4 Contributions des Chapitres
Bibliography
