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## Demand-Side Management and Demand Response

The transportation of electricity from the generation location to the place of con-sumption has always been a challenge, starting from the very beginning. Originally, electricity networks were conceived in a unidirectional way: the electricity was pro-duced in large power plants (e.g. nuclear plants, which accounts for more than 70% of the total electricity production in France) and directly brought to the customers, in the aim to satisfy the demand at minimum cost. Such networks comprise three main parts (see Figure 2.1):

• The transmission network, directly connected to the large power plants, is able to transmit energy over long distances. To minimize the losses, the voltages applied to the transmission lines reach high values, between 200 kV and 800 kV [Garcia 2008].

• The sub-transmission network dispatches the energy to big consumers, such as heavy industries or railway companies, and to various substations that sup-ply distribution networks. The voltages on sub-transmission networks range between 20 kV and 275 kV [Puret 1991].

• The distribution network connects the end-users (such as residential cus-tomers) to the aforementioned substations. In France, a single-phase connec-tion has a voltage of 230 V, whereas a three-phase connection has a voltage of 400 V.

However, such networks have disadvantages. With the penetration of distributed generation (especially household photovoltaic (PV) panel installations) and their intermittence due to the unpredictability of the exact weather conditions, power injections into the network become more and more irregular [Barker 2000]. This can result in several issues: power outages, short-circuit current or islanding. Ensuring the supply-demand balance at any time is complex. To cope with the issue of supply-demand balance, two solutions coexist. Either the production can be adapted to the demand, or the demand can be adapted to the production. This second option is referred to as demand-side management (DSM). DSM is defined in [Gellings 1984] as « the planning, implementation and monitoring of those utility activities designed to influence customer use of electricity in ways that will pro-duce desired changes in the utility’s load shape, i.e., changes in the time pattern and magnitude of a utility’s load. Utility programs falling under the umbrella of demand-side management include: load management, new uses, strategic conservation, electrification, customer generation, and adjustments in market share. »

### Multi-leader-follower games

Multi-leader-follower games (MLFG) constitute another generalization of bilevel programming. In this case, the leader and the follower are not necessarily unique. The leaders as well as the followers are usually assumed to be in a competition situation. Therefore, the upper and the lower level are modeled as games. The direct consequence is that the followers react with a (generalized) Nash equilibrium (GNE) to the leaders’ decisions, and the optimal solution of the upper level is a GNE as well, though among the leaders. The following definition of GNE is taken from [Aussel 2018]. Let P := {1, . . . , p} be a finite set of p players. Each player ν ∈ P controls variables xν ∈ Rnν , also called strategy. The vector x := (x1, . . . , xp) ∈ Rn,

where n = n1 + · · · + np, represents the joint strategies of all players. To denote the strategies of all players but ν ∈ P , the notation x−ν is commonly used. By abuse of notation, x = (xν , x−ν ). In a generalized Nash equilibrium problem (GNEP), the variable xν is constrained to belong to the domain Xν (x−ν ), which is the set of feasible strategies, given the joint strategies of the other players. Furthermore, each player aims to minimize his objective function fν , which depends on the joint strategies of all players x. Given x−ν , the set Sν (x−ν ) := argminxν {fν (xν , x−ν ) | xν ∈ Xν (x−ν )} is the set of best responses of player ν ∈ P to the joint strategies x−ν of his oppo-nents.

#### Simplified problems and complexity

In this section, simplified versions of the problem (Pl) are defined and studied. In particular, they do not include DG or storage, nor do they consider multiple scenarios. The idea is to study the theoretical computational complexity of the problem: the problem having a specific structure, the fact that bilinear bilinear bilevel prob-lems are in general NP-hard does not guarantee the NP-hardness of (Pl). The complexity may arise from two diﬀerent features: the energy costs and/or the in-convenience costs. As for the rest of this chapter, the energy costs and inconvenience costs are assumed to be linear with respect to the energy that is produced, respec-tively consumed. However, the inconvenience factors can be arbitrarily chosen and do not depend on the delay.

**No energy cost – no inconvenience cost**

In this first case, we do not consider energy cost and inconvenience functions. It follows that:

• If the follower buys energy at time h, the price he pays will be equal to n o min xh, x¯h .

• As there is no inconvenience, the incentive for the follower to consume at a specific time slot is related to his billing cost, and thus to the price. Let k = bE/βc. The follower will consume β energy units during the k cheapest time slots, and E − kβ energy units during the (k + 1)th cheapest slot.

• There are no production costs for the leader, thus maximizing the leader’s objective requires to set the prices as high as possible. It results that an optimal solution for the leader is defined as ph = p¯h for all h ∈ H. This is naturally only true in the assumed optimistic setting.

**Table of contents :**

**1 Introduction **

**2 Literature review **

2.1 Demand-Side Management and Demand Response

2.1.1 Smart Grid

2.2 Bilevel programming

2.2.1 Definitions and theory

2.2.2 Multi-leader-follower games

**3 Single-Leader Single-Follower **

3.1 Stochastic bilevel problem

3.1.1 Scenario tree method

3.1.2 Notations

3.1.3 Follower’s problem

3.1.4 Leader’s problem

3.2 Simplified problems and complexity

3.2.1 One device

3.2.2 Multiple devices

3.3 Numerical resolution of (SBPP)

3.3.1 One-level formulation of (SBPP)

3.3.2 A single scenario case

3.3.3 Several scenarios

3.4 Rolling horizons

3.4.1 Toy examples

3.4.2 Tests on large instances

3.5 Conclusion

**4 Single-Leader Multi-Follower **

4.1 Introduction

4.2 Problem formulation

4.2.1 Local agents

4.2.2 Aggregators

4.2.3 End users

4.2.4 Electricity supplier

4.3 From a trilevel model to a bilevel model

4.4 Reformulations of the bilevel program and alternative solution concepts

4.4.1 First order formulation of the bilevel model

4.4.2 Revisited optimistic approach

4.4.3 Semi-optimistic approach

4.5 Numerical results

4.5.1 Competitive case

4.5.2 Noncompetitive case

4.6 Conclusion

**5 Multi-Leader Multi-Follower **

5.1 Definition of the model

5.1.1 Local agents

5.1.2 Electricity suppliers

5.2 Solution methods

5.2.1 The method of Leyffer and Munson

5.2.2 Formulation as a GNEP

5.3 Conclusion

**6 Conclusion **

**A Appendix to Chapter 4 **

A.1 KKT conditions

A.2 Instances parameters

**B Appendix to Chapter 5 **

B.1 Price, profit and demand graphs

**Bibliography**