# An Integer Non-Cooperative Game Approach for the Transactive Control of Thermal Appliances in Energy Communities

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## Abstract

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## 1. Introduction

- We formulate a game with binary variables representing the on/off control of TCLs and explicitly model the consumers’ comfort constraints;
- We discuss the theoretical foundations of this integer TCLs game, and we show that local or global optimal equilibria can be reached;
- We generalize the paper’s results to two types of cost functions often applied to energy communities: quadratic and peak pricing.

## 2. System Model

#### 2.1. System Description

#### 2.2. Load Modeling

#### 2.3. Community Costs

#### 2.3.1. Quadratic Cost Function

#### 2.3.2. Peak Pricing Function

## 3. Thermal Loads Scheduling as a Game

- $\mathcal{N}=\{1,2,\dots ,N\}$ is the set of consumers living in the community and participating in the scheduling game (i.e., players);
- ${\mathcal{S}}_{n}={\left\{{\mathbf{l}}_{n}\right\}}_{n\in \mathcal{N}}$ denotes the action space for consumer $n\in \mathcal{N}$. This set is composed by feasible scheduling vectors, ${\mathbf{l}}_{n}$, that respect users’ comfort constraints defined in Equation (6);
- ${u}_{n}:\mathcal{S}\mapsto \mathbb{R}$ is the utility function user $n\in \mathcal{N}$ receives. It is defined as the negative of consumers’ bill, which is a share of the community’s total cost (either quadratic or peak pricing). The utility function is shown in Equation (12), in which $C({\mathbf{l}}_{n},{\mathbf{l}}_{-n})$ can be Equation (8) or (10). The share ${f}_{n}$ depends on the billing mechanism, and we use a version that bills each consumer according to his/her energy consumption during the scheduling horizon—see Equation (13). This is a popular billing model in the literature of non-cooperative scheduling games [7,8,9,10,11,19,36,37], and we analyze its advantages and limitations when applied to the game with TCLs.$$\begin{array}{c}\hfill {u}_{n}({\mathbf{l}}_{n},{\mathbf{l}}_{-n})=-{f}_{n}C\left({\mathbf{l}}_{n},{\mathbf{l}}_{-n}\right)\end{array}$$$$\begin{array}{c}\hfill {f}_{n}=\frac{{\sum}_{t\in \mathcal{T}}{l}_{n,t}}{{\sum}_{t\in \mathcal{T}}{L}_{t}}\end{array}$$

#### 3.1. Model Assumptions

#### 3.2. Equilibrium Points of the Integer Game

#### 3.3. Advantages of the Game Model

#### 3.4. On Market and Price Formation

#### 3.5. Best Response Algorithm

- Consumers receive the total load vector and calculate the load of opponents ${\mathbf{L}}_{-n}^{k}$ using their previous best strategies;
- They update their strategies ${\mathbf{l}}_{n}^{k+1}$ as a response to ${\mathbf{L}}_{-n}^{k}$ by solving the local MIQP (15) with $C({\mathbf{q}}_{n},{\mathbf{L}}_{-n}^{k})$ defined by (8), if the quadratic total cost function is applied, or the local MILP (15) with $C({\mathbf{q}}_{n},{\mathbf{L}}_{-n}^{k})$ defined by (10), if the peak pricing total cost function is applied;$$\begin{array}{cc}\hfill {\mathbf{l}}_{n}^{k+1}& =B{R}_{n}\left({\mathbf{L}}_{-n}^{k}\right)=\underset{{\mathbf{q}}_{n}\in {\mathcal{S}}_{n}}{\mathrm{argmax}}{u}_{n}({\mathbf{q}}_{n},{\mathbf{L}}_{-n}^{k})=\underset{{\mathbf{q}}_{n}\in {\mathcal{S}}_{n}}{\mathrm{argmin}}C({\mathbf{q}}_{n},{\mathbf{L}}_{-n}^{k})\hfill \end{array}$$
- Consumers add their local strategy ${\mathbf{l}}_{n}^{k+1}$ to the opponents’ total consumption vector ${\mathbf{L}}_{-n}^{k}$ and send this new aggregated consumption profile ${\mathbf{L}}^{k+1}$ to the next player.

#### 3.6. Computational Complexity and Communication Overhead

#### 3.7. General Applicability of the Model

## 4. Results

#### 4.1. Case Study

#### 4.2. Solutions of the Non-Cooperative Game with TCL

#### 4.3. Convergence Process of the Algorithm

#### 4.4. Scalability and Solution Times

#### 4.5. Consumers’ Comfort and Savings

#### 4.6. Discussion and Future Challenges

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$n,m$ | Indexes for consumers |

t | Indexes for time slots |

k | Indexes for game stages |

$\mathcal{N}$ | Set of consumers |

$\mathcal{T}$ | Set of time slots |

${\mathcal{S}}_{n}$ | Set of strategies of n |

${x}_{n,t}$ | Binary scheduling variable: is 1 if consumer n’s TCL is on at t and 0 otherwise |

${l}_{n,t},{q}_{n,t}$ | Total consumption of n at t |

${L}_{t}$ | Community’s total load at t |

$C\left(\mathbf{L}\right)$ | Community’s total cost when its load curve is $\mathbf{L}$ |

${u}_{n}({\mathbf{l}}_{n},{\mathbf{l}}_{-n})$ | Utility of consumer n |

$\delta $ | Size of time slot (in hours) |

T | Last slot of time horizon |

N | Number of participants/consumers |

${w}_{n,t}$ | Inflexible load of n at t |

${\theta}_{n,t}$ | Internal temperature of n’s household at t |

${\theta}_{n,t}^{min}$ | Minimum accepted internal temperature of n at t |

${\theta}_{n,t}^{max}$ | Maximum accepted internal temperature of n at t |

${\theta}_{t}^{et}$ | External temperature at t |

$T{H}_{n}$ | Thermal capacity of n’s AC |

${R}_{n}$ | Thermal resistance of n’s AC |

${\eta}_{n}$ | Performance of n’s AC |

${E}_{n}$ | Power rate of n’s AC |

${a}_{t}$ | Quadratic cost at time slot t |

${b}_{t}$ | Linear cost at time slot t |

${c}_{t}$ | Fixed cost at time slot t |

${d}_{t}$ | TOU tariff at time slot t |

e | Peak load charge |

${f}_{n}$ | Energy share of consumer n |

AC | Air Conditioning |

BAS | Base Scenario |

BRD | Best Response Dynamics |

CEN | Centralized Scenario |

DSO | Distribution System Operator |

ECC | Energy Consumption Controller |

EV | Electric Vehicle |

HEMS | Home Energy Management System |

HVAC | Heating, Ventilation and Air Conditioning |

LAN | Local Area Network |

MIQP | Mixed-Integer Quadratic Program |

MILP | Mixed-Integer Linear Program |

NE | Nash Equilibrium |

PoA | Price-of-Anarchy |

PP | Peak Price |

Q | Quadratic |

TCL | Thermostatically Controlled Load |

TC | Transactive Control |

TOU | Time-of-Use |

TSO | Transmission System Operator |

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**Figure 1.**System model considered in this study. Each consumer has TCLs controlled locally by his/her ECC, which is part of a HEMS that communicates with the other HEMSs and with the aggregation platform.

**Figure 2.**Flowchart of the scheduling process using the non-cooperative game model and the Best Response Dynamics (BRD).

**Figure 3.**Solution algorithm to solve the proposed game model: consumers communicate the total load of the community, which guarantees information privacy.

**Figure 5.**Total energy consumption of the community with ACs in the base scenario (without energy management).

**Figure 6.**Final load of the community in each scenario: centralized optimization (blue), game solved with BRD (orange), and base scenario (green).

**Figure 8.**Final load curve with the dispatch of one consumer’s AC. Theta, tmin, and tmax are the temperature inside the room, the minimum, and the maximum comfort constraints, respectively.

**Table 1.**Computational complexity and communication overhead of the best response algorithm if compared to the centralized model (worst-case analysis).

Computational Complexity | Communication Overhead | |
---|---|---|

Best Response Dynamics | $r\times N\times {2}^{\frac{24}{\delta}}$ | $r\times N$ |

Centralized Model | ${2}^{N\frac{24}{\delta}}$ | - |

System Parameters | Market Parameters | ||||
---|---|---|---|---|---|

N | 201 | ${\eta}_{n}$ | [3.0, 3.2] | ${a}_{t}$ ($\xa2/{\mathrm{kWh}}^{2}$) | 0.081 |

T | 96 | ${\theta}_{n,t}^{min}$ (${}^{\circ}$C) | [18, 22] | ${b}_{t}$ ($\xa2/\mathrm{kWh}$) | 12.605 |

$\delta $ | 0.25 (15 min) | ${\theta}_{n,t}^{max}$ (${}^{\circ}$C) | [20, 26] | ${c}_{t}$ ($\$/\mathrm{kWh}$) | 1.701 |

${R}_{n}$ (${}^{\circ}$C/kW) | [5.0, 8.0] | ${\theta}_{n,0}$ (${}^{\circ}$C) | [${\theta}_{n,t}^{min}$, ${\theta}_{n,t}^{max}$] | ${d}_{t}$ ($/kWh) | 0.12 [0–17] |

${E}_{n}$ (kW) | [1.5, 3.5] | ${\theta}_{t}^{et}$ (${}^{\circ}$C) | 35 | 0.20 [17–0] | |

${C}_{n}$ (kWh/${}^{\circ}$C) | [0.5, 4.2] | e ($/kW) | 1.00 |

**Table 3.**Solution of each scenario: community’s total cost and peak-to-average ratio of the group’s final load curve.

Scenario | Quadratic Function | Peak Pricing | ||
---|---|---|---|---|

Total Cost ($) | PAR | Total Cost ($) | PAR | |

Base | 618.32 | 1.492 | 625.20 | 1.492 |

Centralized | 617.42 | 1.285 | 580.84 | 1.141 |

Transactive Control | 617.42 | 1.285 | 585.49 | 1.188 |

Total Cost Function | Aspect | N = 5 | N = 10 | N = 50 | N = 100 | N = 150 | N = 201 |
---|---|---|---|---|---|---|---|

Quadratic | Time (s) | 211.60 | 254.53 | 465.23 | 1426.71 | 592.06 | 10,295.66 |

Rounds | 3 | 3 | 8 | 12 | 8 | 12 | |

Peak Pricing | Time (s) | 251.47 | 317.16 | 2603.32 | 1778.08 | 1877.39 | 2299.58 |

Rounds | 4 | 4 | 10 | 9 | 9 | 10 |

**Table 5.**BRD solution times when changing the complexity of each consumers’ problem (all instances have 10 consumers).

Total Cost Function | Aspect | Random 01 | Random 02 | Random 03 | Highest Load | Highest AC | Smallest Load |
---|---|---|---|---|---|---|---|

Quadratic | Time (s) | 254.53 | 600.14 | 1137.92 | 1536.55 | 739.21 | 430.03 |

Rounds | 3 | 4 | 4 | 4 | 6 | 4 | |

Peak Pricing | Time (s) | 317.16 | 687.64 | 378.49 | 5616.34 | 871.75 | 308.71 |

Rounds | 4 | 5 | 5 | 10 | 4 | 6 |

Total Cost Function | Aspect | $\mathit{\delta}=1.00$ | $\mathit{\delta}=0.50$ | $\mathit{\delta}=0.25$ |
---|---|---|---|---|

Quadratic | Time (s) | 135.03 | 1632.21 | 10,295.66 |

Rounds | 4 | 6 | 12 | |

Peak Pricing | Time (s) | 64.88 | 1042.80 | 2229.58 |

Rounds | 2 | 4 | 10 |

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## Share and Cite

**MDPI and ACS Style**

Marques, L.; Uturbey, W.; Heleno, M.
An Integer Non-Cooperative Game Approach for the Transactive Control of Thermal Appliances in Energy Communities. *Energies* **2021**, *14*, 6971.
https://doi.org/10.3390/en14216971

**AMA Style**

Marques L, Uturbey W, Heleno M.
An Integer Non-Cooperative Game Approach for the Transactive Control of Thermal Appliances in Energy Communities. *Energies*. 2021; 14(21):6971.
https://doi.org/10.3390/en14216971

**Chicago/Turabian Style**

Marques, Luciana, Wadaed Uturbey, and Miguel Heleno.
2021. "An Integer Non-Cooperative Game Approach for the Transactive Control of Thermal Appliances in Energy Communities" *Energies* 14, no. 21: 6971.
https://doi.org/10.3390/en14216971