# Repeated game model to manage a collective self-consumption Smart grid with respect to the autonomy of prosumers *

Dominique Barth, Benjamin Cohen-Boulakia, Wilfried Ehounou

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Dominique Barth, Benjamin Cohen-Boulakia, Wilfried Ehounou. Repeated game model to manage a collective self-consumption Smart grid with respect to the autonomy of prosumers *. 2025. hal-05374732

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# Repeated game model to manage a collective self-consumption
## Smart grid with respect to the autonomy of prosumers *

Dominique Barth¹, Benjamin Cohen-Boulakia¹, Wilfried Ehounou¹,²

¹ Laboratoire DAVID, Université de Versailles - ST Quentin/Université Paris-Saclay Versailles, France

² Université Nangui Abrogoua, Abidjian, Côte d'Ivoire

## Abstract

This paper presents a repeated-game framework to manage a collective self-consumption Smart Grid in which prosumers retain autonomy over production, storage, and exchange decisions while preserving data confidentiality. The model removes the need for an internal market through a systematic comparison between the collective scenario enabled by the grid and the reference scenario corresponding to individual, uncoordinated operation by all actors. This comparative approach allows for a savings-sharing mechanism, by which aggregate cost reductions are dynamically allocated among participants to foster energy sharing. At each period, actors choose context-aware actions between drawing from the grid (and either prioritizing their own stock or not), disconnecting, or injecting surplus. These choices are guided by a short-horizon stock predictor. Three decision policies are evaluated: a systematic baseline that ignores grid effects (*SyA*), a selfish stock policy (*SSA*), and a centralized controller (*CSA*) used as a performance benchmark. Experiments on scenarios with varying storage capacities and linear buy/sell prices show that *SSA* often achieves the lowest grid cost in two instance families, while *CSA* leads on a third, with limited sensitivity to the prediction window. Analysis of purchases from and sales to the EPO indicates that, related to isolated strategies (*SyA*), grid coordination reduces bought energy and can increase sold energy. Overall, short-range prediction generally suffices to stabilize effective, autonomy-preserving policies that improve the grid's economic performance.

# 1 Introduction

This study focuses on the management method of a collective self-consumption grid that was carried out within LabCom HYPHEs of the French ANR, dedicated to the management of urban infrastructure networks, in partnership with Grand Paris Sud. The eco-responsible production and distribution of electrical energy at the local scale of an urban territory (a district, an activity zone for example) is nowadays an environmental and economic key objective in the development and planning of such territories. Indeed, urban governance trends ([19]), seeking to maintain or regain control over infrastructure networks (water, energy, mobility, waste), increasingly promote in the energy sector the deployment of Smart Grids linking private and public "prosumers" (e.g., companies, administrations, residential buildings, [5, 6, 12]), each potentially equipped with green electricity production and storage capabilities [17, 20]. In this context, collective self-consumption consists in sharing electricity production between several prosumers on the same territory. Thus, these prosumers are linked by the same legal entity which manages and maintains the distribution and consumption on a specific local smart grid: this Smart Grid is always connected to an external energy supplier to whom the territory can delegate technical management and maintenance (as is the case with water distribution networks in France [1]). The objective is threefold: to encourage and share eco-responsible local energy production; to control the cost of energy for users; and to ensure the territory control over its electricity production networks. It should be noted that such a self-consumption network is of interest if, occasionally, certain actors produce more energy than they consume. The legal entity linking electricity consumers and producers can take different forms, for example: co-ownership, association, local cooperative [10]. It is therefore a question of studying the shared operating model of such a collective self-consumption grid in

<footnote>*This study was carried out within the HYPHEs LabCom ANR-21-LCV1-0002 of the French ANR.</footnote>
a context where each participant maintains autonomy of decision regarding their own equipment and their consumption, and where the confidentiality of their data with respect to vis-à-vis other partners is guaranteed. Here, it is therefore this operating model, from both a control and economic point of view, which must encourage all partners to behave in an eco-responsible and financially profitable way for all, without any of these participants becoming a supplier of energy making a profit via the shared Smart Grid.

Since it is a question here of combining individual economic and environmental interests, autonomous decisions and virtuous collective behavior, our approach is to model such a system as a repeated collaborative game. The purpose is to propose and evaluate a decision process in a Smart Grid interconnecting prosumers, each one keeping the control of the quantity of energy it provides to or takes from the Smart Grid. Such a distributed multi-agent system is commonly modeled by a repeated game, in which each actor consumes energy in each period of time, is potentially capable of producing and storing electricity and has the ability to predict consumption and production for future periods, with a certain precision. Such hypotheses are not yet widely verified within urban areas, but their deployment in the near future is widely anticipated by energy stakeholders. The overarching goal is to identify management rules that respect individual interests while enabling the grid to maximize energy independence from external suppliers. In [3], we proposed and studied a first model for such grids by a repeated game which was based on a virtual economic model internal to the grid which, as the results showed, turned out to be complex to parameterize and which moreover only considered low predictive ability.

Building on the availability of more advanced predictive abilities, we therefore propose an alternative model that eliminates the need for an internal economic framework. This model relies solely on continuous comparison between collective integration within the Smart Grid and individual operation, which remains simple to deploy. We combine this game model with a distribution function that shares among the participants the total electricity cost savings, compared against the sum of individual costs incurred when dealing directly with the external supplier. This respect for the interests and independence of each actor naturally leads us not to seek a global optimum, but rather a virtuous equilibrium based on the policies of individual actors and beneficial to all. Furthermore, for such a collective Smart Grid model to be accepted, it must ensure that participation confers an advantage to every actor relative to purely selfish, individual behavior, which this work aims to address.

We also study here the impact of the actors’ ability to predict consumption and production, and on the precision of these predictions, on the effectiveness of the proposed model. Predicting electricity consumption, particularly its variations, is now crucial for the resilience and dynamics of Smart Grids. The prediction approaches generally proposed and analyzed use different Machine Learning concepts [9, 25]. Other studies have shown the possibility of predicting a few days in advance the renewable energy production capacity of prosumers, based in particular on meteorological prediction [5, 24]. One of our main objectives here is to evaluate the impact of such electricity consumption and green production prediction capabilities on the distributed management of a collective self-consumption grid.

## Related work

Some early works (as of the 2010’s) focus mainly on the use of game theory for energy supply to end consumers ([4, 14, 18, 27]). As renewable local energy production capabilities became more and more available, some work focused on considering those energy production capabilities in Smart Grid models, leading to include prosumers (i.e. producers/consumers) in those models. In [13], the authors propose a game theory approach to model the interactions among prosumers and distribution system operators for the control of electricity flows in real-time. A game theory based energy management is proposed in [26], based on the Stackelberg leadership model system which seeks for social optimality when each energy consumer focuses only on its own energy efficiency; the existence of a Nash equilibrium in the proposed game model is proved, and an optimal distributed deterministic algorithm is proposed that reaches this equilibrium.

Some other works seek to propose pricing policies for energy suppliers, considering various optimization objectives. In [22] is considered a smart power infrastructure, where several subscribers share a common energy source, Authors propose a distributed algorithm which automatically manages the interactions among subscribers and the energy provider. For example, [8] presents an equilibrium Se-
lection Multi-Agent Reinforcement Learning for consumer energy scheduling of residential microgrid, based on private negotiation between each consumer and the energy provider, and works from [15] include reliability parameters and energy fluctuations and propose a Deep Reinforcement Learning algorithm based on Q-Learning and Deep Neural Networks to determine a pricing policy for a power supplier that balances those two parameters. In [23] is also proposed a Vickrey-Clarke-Groves (VCG) based auction mechanism aiming to maximize the aggregate utility functions of all users minus the total energy cost. In [2], authors propose a game theory approach for selecting the best power subscription when several power suppliers are available, based on a two-stage process.

Taking into account the availability of several power suppliers led to various Demand Response Management (DRM) problems. Results of [7] focus on the real-time interactions among multiple utility companies and multiple users, and propose a distributed real-time demand response algorithm to determine each user's demand and each utility company's supply simultaneously. Contributions of [16] also focus on DRM, but solve electricity pricing questions, using a reinforcement learning-based decision system. The goal in this approach is to minimize the electricity payment and consumption dissatisfaction for end users. Some works focus on more distributed models, with autonomy goals. In [21] is proposed a coalitional game between a number of micro-grids (energy producers), each servicing a group of consumers (or an area), and in [11] is also considered autonomy goals for distributed set of micro-grids without support from a traditional centralized grid, and propose a distributed algorithm based on a cooperative game theory approach. This short review on related works leads to one observation: very few work focus on a general Smart Grid model, in which consumers may or may not have power production capabilities and power storage facilities, and consider learning approaches.

## 2 Smart Grid actors model

We consider that the activity time of the Smart Grid consists in a sequence of homogeneous periods (each can be, for example, a half-day or a day). In this discrete-time context, each energy indicator of each actor can be considered as a fixed value in each period. For each period, the system induced to represent the Smart Grid is considered here as a multi-agent system. Let $AG = \{a_1, ..., a_N\}$ the set of $N$ actors connected by the Smart Grid (SG), each actor $a_{i:1 \le i \le N}$ being potentially able to produce electricity, in particular with renewable energies, and to store it. Their production capacity and storage capacity are limited. The maximum storage capacity of the actor $a_i$ is denoted by $S_{max}(i)$.

For each actor $a_i$, we consider two integer vectors $C_i$ and $P_i$ where for each period $1 \le t \le T$ in the whole activity time considered, $C_i[t]$ is the energy consumption of $a_i$ in period $t$ and $P_i[t]$ the energy production of $a_i$ in this period. In an online approach, we consider that $a_i$ has an approximated (predicted) knowledge of $C_i[t']$ (resp. $P_i[t']$) for $t < t' \le t + \rho_i^c$ (resp. $t < t' \le t + \rho_i^p$), with $\rho_i^c \ge 1$ and $\rho_i^p \ge 1$ two input parameters representing the prediction capacity of $a_i$ in terms of consumption and production. In the following, we consider a unique value $\rho$ such that for any actor $i$ we have $\rho_i^c = \rho_i^p = \rho$,

Each actor $a_i, 1 \le i \le N$ in each period $t$ is also characterized by $S_i^t$ the amount of electricity stored by the actor at the beginning of the period. We denote by $S_i^{max}$ the maximum storage capacity and define the residual battery storage capacity $R_i^t = S_i^{max} - S_i^t$, that is, the maximum amount of energy the actor can store during the period. Note that during a period an actor cannot both use and provide its storage.

The Smart Grid is also connected to an external energy production operator (EPO) who can supply SG with electricity if necessary and to whom the Smart Grid can sell electricity unused produced by the actors. Note that the actors do not directly interact with this EPO.

Thus, the Smart Grid (SG) itself can be considered as an energy container connecting all actors $a_{i:1 \le i \le N}$; each actor $a_i$ in each period $t$ provides a quantity of electricity $Prod_i^t$ to SG and uses a quantity of electricity $Cons_i^t$ from SG (note that both $Prod_i^t$ and $Cons_i^t$ cannot be greater than 0).
We denote $In_{SG}^t = \sum_{i=1}^{N} Prod_i^t$ and $Out_{SG}^t = \sum_{i=1}^{N} Cons_i^t$. Then, if $In_{SG}^t < Out_{SG}^t$, the energy production operator EPO provides a quantity of energy $q = Out_{SG}^t - In_{SG}^t$ to SG from a given linear or concave increasing cost function $\phi_{EPO}^{-}(q)$; else if $Out_{SG}^t \le In_{SG}^t$, SG sells a quantity of energy $q' = In_{SG}^t - Out_{SG}^t$ from a given linear or convex increasing benefit function $\phi_{EPO}^{+}(q')$. Definitions of
$\bar{\phi}_{EPO}$ and $\phi_{EPO}^{+}$ are inputs of the model.

We define several parameters for each actor $a_i$ at each period $t$:

* $ValNoSG_i^t = \phi_{EPO}^{-}(Cons_i[t]) - \phi_{EPO}^{+}(Prod_i[t])$ and $ValNoSG_i^t = \sum_i ValNoSG_i^t$. These metrics indicate the price that actors would pay based on their choices over time if SG did not exist and they dealt directly with EPO.

* $ValSG^t = \phi_{EPO}^{-}(|Out_{SG}^t - In_{SG}^t|^{+}) - \phi_{EPO}^{+}(|In_{SG}^t - Out_{SG}^t|^{+})$. These quantities represent the overall price that actors must collectively pay through SG.

* $QttEPO^{-} = \sum_{1 \le t \le T} |Out_{SG}^t - In_{SG}^t|^{+}$ (resp. $QttEPO^{+} = \sum_{1 \le t \le T} |In_{SG}^t - Out_{SG}^t|^{+}$) giving at period $t$ the quantity of energy provided by (resp. sold to) EPO to actors of SG during all the periods.

The objective of all actors in SG for all the periods is to minimize $Obj_{SG} = \sum_{1 \le t \le T} ValSG^t$.

### 3 Repeated game modelisation

We consider here a repeated game model and not just the repetition of a simultaneous game, because the strategy chosen by each actor during a period has an influence on the situation of the game in the future, in terms of stock of energy; this is where the ability to predict comes into play. The sum of the costs of each actor $Obj_{a_i}$ is therefore not necessarily minimal when in each period $t$, each actor minimizes its cost $Price_i^t$ (which would consist, in the event of overproduction of energy, in always preferring to sell rather than store).

Thus we consider the whole system during the whole activity time as a repeated game in which at each period, players play a unitary simultaneous game consisting for each one to make a decision about managing its production and storage activity for this period. We first define this unitary game played at each period $t$, in which the player set is AG. Each actor $a_{i:1 \le i \le N}$ has to choose a strategy among four:

* **CONS+**: in the case where an actor consumes more than they produce, this strategy consists in satisfying energy demand using its own production and stored energy, and supplementing demand with energy supplied by the grid.

* **CONS-**: In the case where an actor consumes more than he produces, in this strategy an actor only uses its own production and supplements demand with energy supplied by the grid, leaving its own stored energy untouched.

* **DIS**: In the case where an actor produces enough in relation to what he consumes, this strategy disconnects an actor from the grid, relying only on its own energy production and storage. Note that depending on its state, an actor applying this strategy may still inject energy to the grid if it is unable to store it. This point is developed below.

* **PROD**: In the case where an actor produces more than he consumes, this strategy consists in supplying all the produced energy that is not used to the grid.

Thus, at each period $t$, each actor can only choose between two of these four strategies, depending on his situation regarding consumption and production values. These situations are considered four possible states of each actor implied by $P_i[t]$, $S_i^t$, $C_i[t]$. The variation of the amount of energy stored by an actor depends on both its state and its strategy:

* **State Deficit**: If $P_i[t] + S_i^t \le C_i[t]$ then actor $a_i$ needs energy from SG for his consumption; It can choose to also use its storage or not. Thus, in this state, $Cons_i^t$ and $S_i^{t+1}$ can have two values corresponding to two strategies:

$$ - \text{Strategy CONS+}: Cons_i^t = C_i[t] - (P_i[t] + S_i^t), S_i^{t+1} = 0 $$
– Strategy CONS-: $\text{Cons}_i^t = C_i[t] - P_i[t]$, $S_i^{t+1} = S_i^t$

In both cases, $\text{Prod}_i^t = 0$.

* **State Self:** if $P_i[t] + S_i^t > C_i[t]$ and $P_i[t] < C_i[t]$, the actor $a_i$ can choose to use either the grid or its own stock to meet their energy consumption needs. The two corresponding strategies are:

– Strategy CONS-: $\text{Cons}_i^t = C_i[t] - P_i[t]$, $S_i^{t+1} = S_i^t$

– Strategy DIS: $\text{Cons}_i^t = 0$, $S_i^{t+1} = S_i^t - (C_i[t] - P_i[t])$

In either case, $\text{Prod}_i^t = 0$.

* **State Surplus:** if $P_i[t] > C_i[t]$, $\text{Cons}_i^t = 0$, the actor $a_i$ produces more energy than what is needed, and it can choose to inject it in the grid, or store it for future use. The two possible strategies are:

– Strategy PROD: $\text{Prod}_i^t = P_i[t] - C_i[t]$, $S_i^{t+1} = S_i^t$

– Strategy DIS: $\text{Prod}_i^t = |P_i[t] - C_i[t]| - R_i^t|^+$ with $R_i^t = S_i^{\text{max}} - S_i^t$, $S_i^{t+1} = \min \text{in}(S_i^{\text{max}}, S_i^t + (P_i[t] - C_i[t]))$. This means that the actor favors the storage of his remaining produced energy, but if its storage capacity is reached, the remaining energy is still injected to the grid. Note that if $S_i^t = S_i^{\text{max}}$, this strategy is forbidden, since it is equivalent to PROD.

At this stage, $\text{Cons}_i^t = 0$.

Thus, at each period $t$, each actor $a_i$ chooses a strategy $\text{Strat}_i^t \in \{0,1\}$ with in State Deficit $\{0 = \text{CONS}+, 1 = \text{CONS}-\}$, in State Self $\{0 = \text{DIS}, 1 = \text{CONS}-\}$ and in State Surplus $\{0 = \text{DIS}, 1 = \text{PROD}\}$. Thus in any case, an actor has always two possible strategies. Note that an actor's stock can only be used for its own consumption, which ensures that an actor will not behave as an independent energy supplier in SG.

These strategies ensure that the production of the actor $a_i$ is intended primarily for its consumption and the stored energy of $a_i$ is never imported from SG but is only filled by its own production. In fact, each actor $a_i$ consumes the total amount of production (and eventually his current storage) before importing electricity from SG. The choice of actors therefore only concerns the policy of supply or use of the stock.

The behavior of the grid over the sequence of $T$ periods can be modeled by three matrices $C$, $P$ and $\pi$ of dimension $N \times T$ and a vector $St$ of dimension $N$, where for each $1 \le i \le N$ and $1 \le t \le T$

* are given $C[i, t] = C_i^t$, $P[i, t] = P_i^t$ and $St[i] = St_i^0$,

* must be decided $\pi[i, t] = \text{Strat}_i^t \in \{0,1\}$.

The objective is, given $C$, $P$ and $St$, computing $\pi$ to minimize $\text{Obj}_{SG}$, i.e., the price paid by SG to EPO over all periods. Computing $\pi$ consists in a repeated game since each actor $a_i$ has to decide $\text{Strat}_i^t$ at each period $t$, knowing only columns 1 to $t-1$ in line $i$ of matrices $C$, $P$ and $\pi$, value $St_i^{t-1}$ and eventually (predicted) values of columns $t+1$ to $t+\rho$ of matrices $C$ and $P$. Note that the evolution of stock market values over time makes each period dependent on the others, which defines $\pi$ as a repeated game and not just a repetition of independent simultaneous games.

Consider now a strategy profile $\pi^t = \text{Strat}_1^t, \dots, \text{Strat}_N^t$ of the unitary game at a period $t$ (i.e., column $t$ of $\pi$). This profile $\pi^t$ induces values of $\text{Prod}_i^t$, $\text{Cons}_i^t$ of each actor $a_i$ and thus values of $\text{In}_i^t$ and $\text{Out}_i^t$. It also determines the value of each $S_i^{t+1}$.

# 4 Unitary game

In this section, we deal with the processes used by each actor to select one strategy of the unitary game in each period. We first define in the following subsection a parameter used to predict the impact of prediction on current stock.
## 4.1 Stock prediction

To implement in each unitary game the fact that it could be important to stock energy for the next (predicted) periods, we have to take it into consideration the gain of a player in the game.

For each $0 \le h \le \rho$, we consider $\tau_{i,t,h}^{-} = |C_{i}^{t+h} - P_{i}^{t+h}|^{+}$ and $\tau_{i,t,h}^{+} = |P_{i}^{t+h} - C_{i}^{t+h}|^{+}$ which represent the predicted energy surplus and deficit.

For any $1 \le h \le \rho$, we can recursively compute a stock prediction by considering its energy surplus or deficit and the stock prediction from the previous period $t+h-1$:

$$SP_{i}^{t+h} = \begin{cases} \min(SP_{i}^{t+h-1} + \tau_{i,t,h-1}^{+}, S_{max}) & \text{if } \tau_{i,t,h-1}^{+} > 0 \\ \max(SP_{i}^{t+h-1} - \tau_{i,t,h-1}, 0) & \text{if } \tau_{i,t,h-1}^{+} > 0 \\ SP_{i}^{t+h-1} & \text{if } \tau_{i,t,h-1}^{+} = \tau_{i,t,h-1}^{-} = 0 \end{cases}$$

We consider that $SP_{i}^{t} = S_{i}^{t}$.

Then, for any $1 \le h \le \rho$, we define $\gamma_{i}^{t} = \min \{1 \le h \le \rho : (SP_{i}^{t+h} = S_{max} \text{ or } (h = \rho)\} - 1$; if $\gamma_{i}^{t} = \rho - 1$, and $SP_{i}^{t+\rho} < S_{max}$ then set $\gamma_{i}^{t} \leftarrow \rho$. Then, for any $1 \le h \le \gamma_{i}^{t}$ we define

* The amount of free energy injected in the grid $\mathcal{G}^h$. This energy is necessarily re-injected as it cannot be saved due to saturated storage capacities.

$$Free^h = \sum_{1 \le i \le N} |P_i^{t+h} - C_i^{t+h} - (S_{max} - SP_i^{t+h})|^+$$

* The needs in energy $Needs_{i}^{t+h}$ of prosumer $i$, based on predicted storage, consumption, and production; the total needs for all prosumers.

$$Needs_{i}^{t+h} = |\tau_{i,t,h}^{-} - SP_{i}^{t+h}|^{+} \text{ and } GridNeeds_{i}^{t+h} = \sum_{1 \le i \le N} Needs_{i}^{t+h}$$

* The amount of free energy $Help_{i}^{t+h}$ the grid can provide to the prosumer. This is computed as a proportion of the free energy $Free^h$, without exceeding the total amount of the energy actually needed by the prosumer.

$$Help_{i}^{t+h} = \min \left( \frac{Needs_{i}^{t+h}}{GridNeeds_{i}^{t+h}} \times Free^h, Needs_{i}^{t+h} \right)$$

* The energy $Rq_{i}^{t+h} = Needs_{i}^{t+h} - Help_{i}^{t+h}$ that the grid cannot provide for free to a prosumer, which is the remaining energy once the grid has provided a quantity of energy $Help_{i}^{t+h}$ to the prosumer.

* $Val_{i}^{t+h}$ represents the estimated energy that the prosumer will need to acquire or store over the time horizon $h$. This amount is computed from $Rq_{i}^{t+h}$ without exceeding the remaining available storage.

$$Val_{i}^{t+h} = \min \left( Rq_{i}^{t+h}, S_{max} - SP_{i}^{t+h}, \min_{1 \le x \le h-1} \left| S_{max} - SP_{i}^{t+x} - \sum_{x \le y \le h-1} Val_{i}^{t+y} \right|^{+} \right)$$

Then, the exact estimation of the amount of energy the prosumer should store in the near future is

$$EstimStock_{i}^{t} = \sum_{1 \le h \le \gamma_{i}^{t}} \frac{Val_{i}^{t+h}}{h}$$

To measure the impact of an inaccuracy in stock estimation (due to prediction errors of $C_{i}^{h}$ and $P_{i}^{h}$) on performances of the strategy choice algorithms defined in the next section, we consider $0 \le \epsilon \le 1$
a prediction error rate used as follows. For any period *t* and any prosumer *a*<sub>*i*</sub>, let *K*<sub>*i*</sub><sup>*t*</sup> follow a Poisson distribution with parameter *λ* = 50, truncated to {0, ..., 100} and renormalized; we then define

$$u_i^t = \frac{K_i^t}{100},$$

which yields a discrete variable on [0, 1] with centesimal resolution, a mode near 0.5, and decreasing probability mass as one moves away from 0.5. The stock quantity considered to be needed then by each actor *a*<sub>*i*</sub> is then:

$$QTStock_i^t = (1 + 2 \times (u_i^t - 0.5) \times \epsilon) \times EstimStock_i^t$$

Note that if $\epsilon = 0$, we assume no prediction error occurs. We will thus make experimentations with $\epsilon \in \{0, 0.02, 0.05\}$.

## 4.2 Unit game algorithms

The purpose of this section is to define how each actor *a*<sub>*i*</sub> can use at each period *t* of the repeated game to choose *Strat*<sub>*i*</sub><sup>*t*</sup>. Note that all the algorithms defined hereafter consider that each actor has the control of its own situation in SG (except CSA used for comparison with other algorithms).

We will compare the perfomance of three algorithms, described in Table ??, to choose at each period *t* the action *Strat*<sub>*i*</sub><sup>*t*</sup> of each actor *a*<sub>*i*</sub>. These algorithms are:

### Systematic Algorithm (SyA)

This algorithm consists of choosing the best decision that an actor should take if there were no SG, i.e. giving priority to the stock, without taking into account the prediction of future consumption/production;

### Centralized SG Algorithm (CSA)

This algorithm considers a model where all the actors' decisions are taken by SG with the objective of minimizing the quantity of electricity purchased from the operator at each period (see Table ??). Note that this algorithm does not respect the autonomy of choice of each actor and is only considered for performance comparison.

### Selfish stock algorithm (SSA)

This algorithm consists of choosing the most selfish a priori decision that an actor can take within a SG, i.e. always giving priority to the stock, taking into account the prediction of future consumption/production (which means that each agent will share its energy with the other agents via SG if its own stock plus its future production is sufficient to cover its future consumption, see Section 4.1 and Table ??).

<table><caption>Table 1: Definition of *SyA*, *SSA* and *CSA* algorithms.</caption><thead><tr><th>State</th><th>Strat<sub>i</sub><sup>t</sup> in SyA</th><th>Strat<sub>i</sub><sup>t</sup> in SSA</th><th>Strat<sub>i</sub><sup>t</sup> in CSA</th></tr></thead><tbody><tr><th>Deficit</th><td>CONS+</td><td>CONS+</td><td>CONS-</td></tr><tr><th>Self</th><td>DIS</td><td>DIS</td><td>CONS+</td></tr><tr><th>Surplus</th><td>DIS</td><td>PROD if S<sub>i</sub><sup>t</sup> ≥ QTStock<sub>i</sub><sup>t</sup><br>else DIS</td><td>PROD</td></tr></tbody></table>

# 5 Experimentation and performance evaluation

In order to assess the proposed management model and the impact of predictive abilities on collective self-consumption Smart Grids, this section details the experimental framework and performance evaluation procedures. The experimentation is designed to compare the strategies introduced in the
<figure><img src="image_2.png" /><figcaption>Figure 1: Markov chains used in the instance generation model.</figcaption></figure>

previous sections, under varied scenarios that consider several actor profiles with heterogeneous storage capacities ($S^t$ values) and consumption/production patterns ($C^t$ and $P^t$ values). Instances are generated to represent heterogeneous population mixes and fluctuating energy demands. The evaluation focuses on several key criteria: aggregate grid cost, individual incentives, energy exchanges with the external supplier EPO, and sensitivity to prediction horizon.

## 5.1 Instances generation

We generate instances by distinguishing three generation schemes for prosumers and two temporal profiles. At each period $t$, a prosumer can be in one of three assignment schemes that impact $P_i^t$ and $C_i^t$ as follows:

* Scheme A: $P_i^t = 0$ and $C_i^t$ uniformly taken between 5 and 20.

* Scheme B: $C_i^t = 24$ and, following the Markov chain in Figure 1(a) with states 1 and 2, $P_i^t$ is uniformly taken between 5 and 10 in State 1 and between 25 and 35 in State 2; when Scheme B applies at consecutive periods, the current state is the final state reached at the previous application of Scheme B.

* Scheme C: $C_i^t$ uniformly taken between 15 and 40 and $P_i^t$ uniformly taken between 30 and 50.

We then consider two temporal profiles that determine, at each period, which generation scheme is instantiated for a prosumer and thus drive the evolution of $P_i^t$ and $C_i^t$: Profile $P_1$ follows the Markov chain in Figure 1(b) with states A and B, and Profile $P_2$ follows the Markov chain in Figure 1(c) with states B and C, where state B is common to both chains.

We generate according to this model three sets $I_1, I_2$ and $I_3$ of 15 instances each, with storage capacity $S_{max}$ equal respectively to 10, 20 and 30. Each instance concerns $N = 10$ actors with $T = 50$ periods. The values of the parameters are $\epsilon = 0, \mu = 0.1, \phi_{EPO}^+(x) = 10 \times x$ and $\phi_{EPO}^-(x) = 2 \times x$.

## 5.2 Performance evaluation

For each generated instance, we run *SyA*, *CSA*, *SSA* algorithms once per instance to obtain average performances for this last algorithm.

Let us define obtained values parameters $ValSG(a)$ and $ValNoSG(a)$, $QTTEpo^-(a)$ and $QTTEpo^+(a)$ obtained with any algorithm $\alpha \in A = \{SyA, SSA, CSA\}$. For each instance and each $\alpha \in A$, we define

* $VSG(\alpha) = \frac{ValSG(\alpha)}{Min_{a \in A} ValSG(a)}$ and $VNSG(\alpha) = \frac{ValNoSG(\alpha)}{Min_{a \in A} ValNoSG(a)}$, to evaluate the performances of each algorithm when it does not give the best value of $ValSG$ and $ValNoSG$ on each generated instance.

* $QT^-(\alpha) = \frac{QTTEpo^-(\alpha)}{Min_{a \in A} QTTEpo^-(a)}$ and $QT^+(\alpha) = \frac{QTTEpo^+(\alpha)}{Max_{a \in A} QTTEpo^+(a)}$ to evaluate the energy quantity bought and sold to EPO in comparison with the less quantity bought and the maximum quantity sold by any method for each instance.
<table><caption>Table 2: Number of best performances of each method, considering 15 instances for $I_1$, $I_2$ and $I_3$.</caption><thead><tr><th>I<sub>1</sub>/I<sub>2</sub>/I<sub>3</sub></th><th>#instances</th><th>#instances</th><th>#instances</th><th>#instances</th></tr><tr><th>Algorithm α</th><th>VSG(α) = 1</th><th>VNSG(α) = 1</th><th>QT<sup>-</sup>(α) = 1</th><th>QT<sup>+</sup>(α) = 1</th></tr></thead><tbody><tr><td>SyA</td><td>0/0/0</td><td>15/14/9</td><td>0/0/0</td><td>0/0/0</td></tr><tr><td>CSA</td><td>6/10/3</td><td>0/0/0</td><td>5/8/1</td><td>13/13/15</td></tr><tr><td>SSA (ρ = 5)</td><td>9/5/10</td><td>0/2/5</td><td>9/7/12</td><td>2/3/0</td></tr><tr><td>SSA (ρ = 10)</td><td>9/5/11</td><td>0/1/4</td><td>11/7/13</td><td>2/3/0</td></tr><tr><td>SSA (ρ = 20)</td><td>9/5/12</td><td>0/1/5</td><td>11/7/11</td><td>2/4/0</td></tr></tbody></table>

<table><caption>Table 3: Average performance (value/best) ratio when not the best performances in $I_1$.</caption><thead><tr><th>I<sub>1</sub></th><th>Average</th><th>Average</th><th>Average</th><th>Average</th></tr><tr><th>Algorithm α</th><th>VSG(α) > 1</th><th>VNSG(α) > 1</th><th>QT<sup>-</sup>(α) > 1</th><th>QT<sup>+</sup>(α) < 1</th></tr></thead><tbody><tr><td>SyA</td><td>1.575</td><td>—</td><td>1.284</td><td>0.653</td></tr><tr><td>CSA</td><td>1.204</td><td>1.107</td><td>1.069</td><td>0.938</td></tr><tr><td>SSA (ρ = 5)</td><td>1.034</td><td>1.051</td><td>1.040</td><td>0.658</td></tr><tr><td>SSA (ρ = 10)</td><td>1.036</td><td>1.050</td><td>1.035</td><td>0.907</td></tr><tr><td>SSA (ρ = 20)</td><td>1.034</td><td>1.051</td><td>1.035</td><td>0.837</td></tr></tbody></table>

<table><caption>Table 4: Average performance (value/best) ratio when not the best performances in $I_2$.</caption><thead><tr><th>I<sub>2</sub></th><th>Average</th><th>Average</th><th>Average</th><th>Average</th></tr><tr><th>Algorithm α</th><th>VSG(α) > 1</th><th>VNSG(α) > 1</th><th>QT<sup>-</sup>(α) > 1</th><th>QT<sup>+</sup>(α) < 1</th></tr></thead><tbody><tr><td>SyA</td><td>2.421</td><td>1.010</td><td>1.706</td><td>0.678</td></tr><tr><td>CSA</td><td>1.071</td><td>1.148</td><td>1.054</td><td>0.982</td></tr><tr><td>SSA (ρ = 5)</td><td>1.176</td><td>1.047</td><td>1.059</td><td>0.910</td></tr><tr><td>SSA (ρ = 10)</td><td>1.130</td><td>1.047</td><td>1.059</td><td>0.910</td></tr><tr><td>SSA (ρ = 20)</td><td>1.130</td><td>1.047</td><td>1.074</td><td>0.905</td></tr></tbody></table>

<table><caption>Table 5: Average performance (value/best) ratio when not the best performances in $I_3$.</caption><thead><tr><th>I<sub>3</sub></th><th>Average</th><th>Average</th><th>Average</th><th>Average</th></tr><tr><th>Algorithm α</th><th>VSG(α) > 1</th><th>VNSG(α) > 1</th><th>QT<sup>-</sup>(α) > 1</th><th>QT<sup>+</sup>(α) < 1</th></tr></thead><tbody><tr><td>SyA</td><td>3.289</td><td>1.020</td><td>2.115</td><td>0.402</td></tr><tr><td>CSA</td><td>1.458</td><td>1.168</td><td>1.256</td><td>—</td></tr><tr><td>SSA (ρ = 5)</td><td>1.112</td><td>1.053</td><td>1.085</td><td>0.794</td></tr><tr><td>SSA (ρ = 10)</td><td>1.073</td><td>1.048</td><td>1.111</td><td>0.782</td></tr><tr><td>SSA (ρ = 20)</td><td>1.096</td><td>1.050</td><td>1.118</td><td>0.799</td></tr></tbody></table>

<table><caption>Table 6: Minimum, maximum and average quantities of energy bought from EPO or sold to EPO for all periods and all methods in $I_1$, $I_2$ and $I_3$.</caption><thead><tr><th>In/out EPO</th><th>I<sub>1</sub></th><th>I<sub>2</sub></th><th>I<sub>3</sub></th></tr></thead><tbody><tr><td>Min QTTEpo<sup>-</sup></td><td>41</td><td>54</td><td>41</td></tr><tr><td>Max QTTEpo<sup>-</sup></td><td>297</td><td>194</td><td>210</td></tr><tr><td>Avg QTTEpo<sup>-</sup></td><td>134</td><td>102</td><td>109</td></tr><tr><td>Min QTTEpo<sup>+</sup></td><td>27</td><td>35</td><td>69</td></tr><tr><td>Max QTTEpo<sup>+</sup></td><td>169</td><td>206</td><td>259</td></tr><tr><td>Avg QTTEpo<sup>+</sup></td><td>107</td><td>138</td><td>149</td></tr></tbody></table>
First, Table 2 clearly shows that the best strategies in the case where all actors are associated with SG are different from those when they are all independent and directly connected to EPO. Indeed, the *SyA* algorithm almost always has the best performance regarding *ValNoSG*, and never regarding *ValSG*. In the case of instance sets $I_1$ and $I_3$, the ability to predict the required stock allows *SSA* to obtain the best results regarding *ValSG* (9 times out of 15 for $I_1$ and from 10 to 12 times out of 15 for $I_3$). The results are in favor of *CSA* for $I_2$, even if in comparison the performances of *SSA* remain close to *CSA* as shown by the values of *VSG* (see Tables 3, 4, 5), which is less the case for *CSA* when it is not the best (*VSG* equal to 1.2). Finally, no notable difference appears in terms of performances between the three values of $\rho$, despite the three different values of maximum stock considered. It therefore seems that a short prediction window is sufficient for the efficiency of *SSA*.

The different behaviors and performance differences between the methods are clearly apparent when considering the quantities of energy purchased and sold by SG to EPO, analyzed by the values of $QT^-$ and $QT^+$ (see Tables 2 to 5). The values of these two parameters obtained by *SyA* are a reference to analyze the contribution of SG using *SSA* and *CSA*: the SG grid allows buying less and sell more energy to EPO. As noted in the previous paragraph, we can see that *CSA* is the most efficient for the set $I_2$ since on average, it is the algorithm that both buys the least and sells the most to EPO, unlike $I_1$ and $I_3$ where *SSA* has better performances, mainly for what concerns the energy bought from EPO. It should also be noted that *CSA*'s performance, particularly for $I_2$, is likely related to the fact that it is a centralized algorithm at the SG level, thus not taking into account the autonomy of choice of each actor. In all cases, Table 6 shows that the average quantities bought and sold by the set of methods remain similar between $I_1$, $I_2$ and $I_3$, which shows that there is no notable difference between the energy requirements between these three sets of instances.

In conclusion, these first experiments show on the one hand that the proposed grid model provides an economically profitable use of green energy shared between actors while preserving their decision-making independence, and on the other hand that it is not necessary for actors to resort to complex algorithmic strategies to benefit from belonging to such a collective self-consumption network. The performance comparisons between *SyA* on the one hand and *SSA* and *CSA* on the other hand clearly show the differences in policies to be applied with or without grids: favoring your stock when you are not on the grid, favoring the grid and other players otherwise. Indeed, the savings made by each consist of reducing the quantity of energy purchased for others. The grid therefore encourages the collective use of local energy whose production is eco-responsible. Finally, these initial experiments with the *SSA* algorithm seem to show that taking into account a consumption and production forecast of only a few days in advance improves inventory management performance, even if this is not necessarily systematic, as demonstrated by the comparative performance of *SSA* and *CSA* (centralized algorithm) for $I_2$.

# 6 Actor prices computation model

Whatever the chosen unit game algorithm, the value of *ValSG* represents the amount that all players will have to pay to the EPO if it is positive, and the profit that all actors will make if it is negative. The aim is therefore to propose and evaluate a distribution model that allows each actor to be shown their individual price or profit, knowing that the SG status excludes direct remuneration between actors. The principle proposed here is based on the impact of each actor on the difference between *ValNoSG* and *ValSG*, knowing that if *ValSG* < *ValNoSG*, each actor will benefit from a price reduction.

Given one selected algorithm above, namely here *SSA*, let us define at each period *t* the reduction cost value by $Reduc^t = ValNoSG^t - ValSG^t$, with positive or zero value since by definition $\bar{\phi}_{EPO}(x) > \phi_{EPO}^+(x)$ for any $x > 0$, giving a reduction cost obtained by using SG in comparison with *ValNoSG*.

The price paid for all the *T* periods by each actor $a_i$ is then $Price_i = \sum_{1 \le t \le T} ValNoSG_i^t - Repart_i^t$ where $Repart_i^t$ is a part of the reduction $Reduc^t$ given to $a_i$ according to its action in SG at period *t*. Note that $Price_i$ could be negative (the actor $a_i$ thus receive a revenue from SG).

More precisely, $Repart_i^t$ consists for each actor in the sum of $\frac{Reduc \times \mu}{N}$ plus a repartition of $Reduc \times$
$(1 - \mu)$ over all actors such that $Prod_i[t] > 0$, with $0 \le \mu \le 1$ an input parameter.

The definition of function Repart we propose to focus on is the following one:

$$Repart_i^t = \mu \times \left( \frac{Reduc^t}{N} \right) + (1 - \mu) \times \left( Reduc^t \times \frac{Prod_i^t}{Max(In_{SG}^t, 1)} \right)$$

Note that if $In_{SG}^t = 0$ then $Reduc^t = 0$. It is also easy to see that $\sum_i Price_i^t = ValSG^t$. This is the most intuitive method of distributing the reduction, which must guarantee a minimum reduction for each. Thus, the comparisons between the behaviors of the algorithms studied subsequently will not be impacted by a distribution function that is too irregular.

For each actor $a_i$, we consider

* its impact defined by $Imp_i = \frac{\sum_{1 \le t \le T} \frac{Prod_i^t - Cons_i^t}{|In_{SG}^t - Out_{SG}^t| + |Out_{SG}^t|}}{T}$,

* its interest $Inter_i = \frac{\sum_{1 \le t \le T} \frac{Repart_i^t}{Max(1, Reduc^t)}}{T}$.

The impact $Imp_i$ measures the average impact of each actor $a_i$ over all periods on its difference between energy production to the grid and energy consumption supplied by the grid compared to the totality of what the grid has produced, whether for all actors or sold to EPO. The interest $Inter_i$ of each actor $a_i$ the average share over the $T$ periods of the cost reduction from which it benefited.

The experiment carried out to evaluate the proposed economic model consists of running *SyA*, *CSA* and *SSA* algorithms, with $\rho = 5$ for *SSA*, for 20 instances of 10 actors each, with $T = 50$ periods, generated as in Section 5.1. This will therefore provide for each algorithm 200 pairs $(Imp_i, Inter_i)$ represented in the point clouds of Figure XX.

Each of the three figures presents the set of points $(Imp_i, Inter_i)$. The three sets of points clearly show that, regardless of the methods, the datasets are realistically composed of two types of actors: those who produce and are mainly autonomous, or even overproduce (blue points called producers) and those who always require external energy (red points called consumers). There are no intermediate situations. As this is the objective for which we designed instance generation (see Section 5.1), these curves therefore validate the relevance of the two metrics $Imp_i$ and $Inter_i$.

Each figure also presents the linear regression line for its point set, in dotted lines, and the regression lines for the two sets of red (consumers) and blue (producers) points, the equations of which are given below each figure. By comparing in Figure 1 the slope of these two lines with that of the dotted line, we see that the *SyA* method, which is very greedy in terms of energy purchased from EPO (see Section 5.2), is also the most unfavorable both in terms of cost reduction for consumers and the most unequal in terms of profits for producers, which is reinforced by the fact that the maximum values of $Imp_i$ and $Inter_i$ are maximum for *SyA* as y is the minimum value of $Imp_i$. Figure 2 shows that on similar criteria, the algorithm most favorable to the economy of consumers and the least unequal between the profits of producers is *CSA*, which we recall does not respect the constraints of independence of choice by each actor. In comparison, Figure 3 shows that the *SSA* algorithm, much more favorable to consumers and much more egalitarian for producers than *SyA* does not reach the performances of *CSA*. We can conclude that compared to *CSA*, the *SSA* algorithm is more favorable to the most powerful producers, without this having a significant impact on the overall economy of SG compared to energy purchased from EPO (see Section 5.2). We can hypothesize that this is an impact of the autonomy of choice of each actor, a specific form of "price of anarchy" [?].
<figure><img src="image_3.png" /><figcaption>Figure 2: Cloud of points (*Imp*<sub>*i*</sub>, *Inter*<sub>*i*</sub>) by *SyA*.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3: Cloud of points (*Imp*<sub>*i*</sub>, *Inter*<sub>*i*</sub>) by *CSA*.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 4: Cloud of points (*Imp*<sub>*i*</sub>, *Inter*<sub>*i*</sub>) by SSA.</figcaption></figure>

# 7 Conclusion

The aim of the study presented in this article is to define and evaluate the relevance of a management model for a collective self-consumption grid, interconnecting prosumers, each retaining its independence of choice in the management of the energy it produces and stores. In this model, it is the grid, as a territorial entity, which is seen as a customer of an energy production operator, and no longer each prosumer. In a repeated game model, the results obtained, based on the use of a first simple algorithm by each actor, *SSA*, show the economic relevance (for each actor and for the grid) and environmental relevance of the proposed approach, based on the ability to predict short-term consumption and production. By considering a distribution function of the reductions in energy costs induced by the grid, we noted that a gain in performance was possible in terms of the savings made by consumers and the equality of distribution between producers. The objective is now to improve the performance of such a collective self-consumption grid while maintaining the independence of each actor, by using distributed reinforcement learning methods, whose reward function integrates both the price paid by SG and the distribution of savings.

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