**1. Introduction**

Energy is the backbone of modern society. It provides the means to support everyday infrastructure, such as hospitals, schools, and homes. In the case of homes, for most of the 90 percent of the global population with access, it is difficult to imagine living without electric energy [1]; it powers our essential needs, such as water pumping, lighting, cooling, and very often also cooking, among others. Additionally, electric energy provides comfort and entertainment, and although these are not essential needs, they can also improve quality of life.

Throughout the years, with an increasing energy demand, managing energy consumption has become important. Demand Side Management (DSM) encapsulates those strategies that change the main power consumption to better match the power supply. Through DSM methodologies, one of the purposes is to create an energy demand scheduling to benefit a household.

Many DSM studies focus on minimizing energy cost, achieving utility stability, and shifting peak demand. Wu et al. [2] proposed a mixed-integer linear programming (MILP) model for the energy system optimization to reduce the annual cost in a building distributed heating network. Similarly, Tang et al. [3] proposed a game theoretic method to maximize net profit and reduced demand fluctuation using real data of building on a campus in

**Citation:** Ortiz, S.; Ndoye, M.; Castro-Sitiriche, M. Satisfaction-Based Energy Allocation with Energy Constraint Applying Cooperative Game Theory. *Energies* **2021**, *14*, 1485. https://doi.org/10.3390/en14051485

Received: 4 January 2021 Accepted: 22 February 2021 Published: 9 March 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Hong Kong. Conversely, Lokeshgupta et al. [4] proposed a mathematical model of an intelligent multi-objective home energy managemen<sup>t</sup> (HEM) to simultaneously minimize the consumer's bill and system peak demand.

Some authors have aimed at the time of device usage instead of the costs while ensuring a level of satisfaction. Yang et al. [5] implemented a Nash-based game theoretic approach to optimize time-of-use (ToU) pricing strategies considering the costs of fluctuating demands to the utility company and the satisfactions costs of user. Additonally, Marzband et al. [6] introduced a satisfaction function in a bi-level model to maximize and allocate the profit. The authors included a satisfaction function as part of the payoff function. The function is calculated at the end of each time slot and depends on the amount of energy generation.

Among the studies that include satisfaction as part of the objectives, it is defined in terms on how much their expectations are met [7,8]. The aforementioned methodologies tackle the comfort/satisfaction/welfare part as an indirect measure. It is derived from another variable taking this into account. Ogunjuyigbe et al. [8], on the other hand, developed a cost per unit satisfaction index. Their model considered individual devices at each time of the day. The index was maximized by using a genetic algorithm.

Game theory approaches have become one of the tools adopted for modeling and analyzing energy consumption, due to its effectiveness to capture complex interactions between multiple players. The Stackelberg game is one of the most used game strategies for demand response problems [3,9–12]. Another largely used non-cooperative approach, as a solution for DSM, is the Nash equilibrium strategy [3,13–16].

Summarizing the aforementioned studies, even though research efforts are starting to emerge in the point of confluence of analyzing energy systems while considering quality of life, they are still less prevalent. Two knowledge gaps in these studies are that satisfaction is not computed directly, neither is satisfaction considered in most cases from the point as time dependent. This study provides a motivation for such a granular level of smart meter data with distinct energy uses. The proposed study explores more ways to harness smart meters data to improve people's wellbeing. Detailed and granular information on energy consumption is expected to be broadly available in most households in the near future. This study is centered on the specific power usages at the household level. The benefits they bring to the household, as captured by human satisfaction, is also studied. The user satisfaction is not studied as a posteriori parameter to test the model but as a key part of the problem.

DSM requires the processing of a high amount of data to coherently use consumption patterns and manage demand. Smart metering infrastructure (SMI) provides the means to gather this high amount of electrical consumption information. However, it is still a challenge to consider people´s wellbeing while using smart meter data. Buchanan et al. [17] studied how smart meters can affect consumer's wellbeing. Under the 'five ways to wellbeing framework' [18], they explored other areas that may be found with the consumer acceptance and engagemen<sup>t</sup> with smart meter enabled services (SMES). To address this gap, this work is also contributing to a new platform to insert smart meter research directly into the exploration of wellbeing and the human impact of energy socio-technical systems. The present research offers a Shapley Value (SV) game-theory approach to solve the multiobjective optimization problem (MOO) to optimize energy consumption. The hours of the day for which energy should be allocated are found. Quantifiable user satisfaction metric is used through the concepts of power satisfaction (PS) and energy satisfaction (ES). PS and ES were recently developed by the authors [19]. PS and ES were computed hourly and incorporated the detrimental impact that excess consumption can have in the quality of life. Although the state of art may offer other traditional [20] and metaheuristics [21] multi-objective based approaches, the present novel SV-based game-theoretic approach, as seen in mentioned research, offers a simpler and more intuitive way to tackle the problem.

Chambers [22] proposed responsible wellbeing to combine the concept of wellbeing with personal responsibility. Castro-Sitiriche and Ozik [23] delved into the matter when

considering responsible wellbeing in terms of energy consumption. The energy threshold hypothesis is defined. It was previously presented by Max-Neef [24] in terms of economic growth and quality of life. The proposed MOO consists of responsibly fulfilling user's needs by maximizing satisfaction while minimizing the power consumption. A novel model is proposed to include customer's satisfaction in an optimization problem to minimize the energy consumption. To summarize, the contributions of this paper can be highlighted as follows:


#### **2. Cooperative Game Theory**

#### *2.1. Overview of Game Theory*

Game theory provides a series of analytical tools, which allows us to understand what is observed in decision-making interactions. The foundation of the theory is formed by two basic assumptions: decision-makers are rational, and reason strategically by considering the expectations of other decision-makers' behaviors. Real-life situations are modeled by game theory through highly abstract representations, thus, allowing their use to study problems in many fields [26].

#### *2.2. Types of Games*

There are noncooperative games and cooperative (or coalitional) games. In the former, each action is taken by a single player in response to the other players [27]. In cooperative games, the model consists of the set of joint actions that each group of players (or coalition) can take in response to the other players. Cooperative games are concerned with the interactions among players, the value of each coalition and how the value can be distributed to the participating players.
