# A New Cooperative Game—Theoretic Approach for Customer-Owned Energy Storage

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## Abstract

**:**

## 1. Introduction

- Building a fundamental game with multiple energy storage technologies which can be easily expanded and adapted. In this game, multiple modes of operation were modeled: normal and emergency conditions. The players’ utility functions to reflect their satisfaction level with the outcomes were also developed for each player, ensuring that they were not modeled in a simplified linear structure, but rather the risk-averse human nature was considered in the modeling of utility functions. Additionally, the utility functions were modeled for each mode, normal and emergency, considering realistic consequences that each player would face under emergency circumstances and their realistic response to these conditions.
- Studying a cooperative and non-cooperative approach for the energy storage hub problem. While energy storage hubs are a unique concept, the main focus is usually energy allocation modeled as optimization problems. However, the interaction of the owner with the consumers in a realistic manner has not been tackled in the literature. The cooperative approach applied was the regret matching algorithm, which allowed for the improvement of the social welfare of every player involved in the game. Additionally, the non-cooperative approach utilized was the ascending price-clinching auction. Both methodologies produced pricing schemes as well as decisions on which consumers’ needs will be served by the hub.

## 2. Game Model

#### 2.1. System Overview

#### 2.2. Players and Actions

#### 2.2.1. Energy Hub Owner

#### 2.2.2. Energy Hub Customers: The Electric Grid

#### 2.2.3. Energy Hub Customers: Residential

#### 2.2.4. Energy Hub Customers: Industrial

#### 2.3. Utility Functions

#### 2.3.1. Energy Hub Owner

#### 2.3.2. Energy Hub Customers: The Electric Grid

#### 2.3.3. Energy Hub Customers: Residential

#### 2.3.4. Energy Hub Customers: Industrial

_{j,i}to the equation indicates that if there is some kind of disconnection from the grid, which leads to a failure in the industrial customer’s plant, the customer now values the energy resource even more than they did before. As an example, loss of service in the telecommunications industry is critical; therefore, their factor would be a high value, indicating the significance of the energy received to the continuity of service.

- Players: The hub’s owner and the hub’s customers. The hub’s customers are the electric grid, aggregated residential customers, and industrial customers.
- Strategies: The hub’s owner selects how much energy it is supplying to each customer. Each consumer selects its energy consumption in either a cooperative or a greedy manner and that is decided based on the type of game they will play, as discussed in the following sections.
- Utility or payoff functions: ${\mathrm{u}}_{\mathrm{j}}$, where all the players’ utilities are shown in Equations (1)–(3).

## 3. Cooperative Correlated Equilibrium Game Approach

#### Regret Matching Algorithm

_{1}and w

_{2}were chosen to be of equal weight, 0.5 each, to emphasize both each player’s own utility as well as the overall welfare of all other players. As soon as all this is carried out, the game is now ready to be played to achieve a correlated equilibrium. As mentioned earlier, this game is modified at every iteration as the players learn from their previous actions and observe how to improve their own happiness as well as that of others. To achieve this, first, a value called the difference value is calculated. This difference value can be described as such: if player 1 plays action 1, while all other players play their randomly selected actions, player 1 must calculate the difference in their utility if they had played any other action in her action set other than action 1, while all other players did not change their actions, as seen in (8). The current iteration is denoted as n, ${\mathrm{S}}_{\mathrm{j}}$ describes player j’s chosen action at iteration k, while all other actions that player j did not choose at iteration k are denoted as ${\mathrm{S}}_{\mathrm{j}}^{\prime}$. Finally, all other player’s actions at iteration k are represented as ${\mathrm{S}}_{-\mathrm{j}}^{\mathrm{k}}$.

## 4. Non-Cooperative Correlated Equilibrium Game Approach

#### Auction Framework

## 5. Results

#### 5.1. Cooperative Correlated Equilibrium Approach

#### 5.2. Non-Cooperative Ascending Price Clinching Auction Approach

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Parameter | Parameter Description |
---|---|

i | $\mathrm{Energy}\mathrm{resource}\mathrm{type}.I=\{1,\dots ,\mathrm{i},\dots ,{\mathrm{N}}_{\mathrm{i}}$$\},|I|={\mathrm{N}}_{\mathrm{i}}=$3 |

j | $\mathrm{Game}\mathrm{player},\mathrm{where}\mathrm{player}1\mathrm{is}\mathrm{the}\mathrm{ESH}\mathrm{owner}\mathrm{and}\mathrm{the}\mathrm{remaining}\mathrm{players}\mathrm{are}\mathrm{energy}\mathrm{consumers}.J=\{1,\dots ,\mathrm{j},\dots ,{\mathrm{N}}_{\mathrm{j}}\},\left|J\right|={\mathrm{N}}_{\mathrm{j}}=4$ |

${\mathrm{p}}_{\mathrm{i}}$ | Price of one unit of energy type i, in cents per kilowatt hour. |

${\mathrm{e}}_{\mathrm{j},\mathrm{i}}^{\mathrm{max}}$ | Maximum possible demand for energy based on historical data. |

${\mathrm{B}}_{\mathrm{j},\mathrm{i}}$ | Price sensitivity coefficient of player j, in cents per kilo-Watt hour to energy type i. |

${\mathrm{cf}}_{\mathrm{j},\mathrm{i}}$ | Continuity of service coefficient for player, j, and energy resource, i. |

Energy Type i | i = 1 | i = 2 | i = 3 |
---|---|---|---|

Maximum Supply Available | 40 kWh | 50 kWh | 30 kWh |

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**MDPI and ACS Style**

Hanna, M.O.; Shaaban, M.F.; Salama, M.M.A.
A New Cooperative Game—Theoretic Approach for Customer-Owned Energy Storage. *Sustainability* **2022**, *14*, 3676.
https://doi.org/10.3390/su14063676

**AMA Style**

Hanna MO, Shaaban MF, Salama MMA.
A New Cooperative Game—Theoretic Approach for Customer-Owned Energy Storage. *Sustainability*. 2022; 14(6):3676.
https://doi.org/10.3390/su14063676

**Chicago/Turabian Style**

Hanna, Maria O., Mostafa F. Shaaban, and Magdy M. A. Salama.
2022. "A New Cooperative Game—Theoretic Approach for Customer-Owned Energy Storage" *Sustainability* 14, no. 6: 3676.
https://doi.org/10.3390/su14063676