# Real-Time Pricing Scheme in Smart Grid Considering Time Preference: Game Theoretic Approach

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## Abstract

**:**

## 1. Introduction

## 2. System Model

#### 2.1. Real-Time Pricing Model

#### 2.1.1. Supplier-Side Model

**Property**

**1.**

**Property**

**2.**

**Property**

**3.**

#### 2.1.2. User-Side Model

**Property**

**4.**

**Property**

**5.**

**Property**

**6.**

**Property**

**7.**

**Proof of Property**

**4.**

**Proof for Property**

**5.**

**Proof for Property**

**6.**

**Proof for Property**

**7.**

#### 2.2. Non-Real-Time Pricing Model

## 3. Equilibrium

#### 3.1. Equilibrium of Real-Time-Pricing Model

**Proposition**

**1.**

- A.
- If${\sum}_{t=1}^{T}\left({\alpha}_{i}^{t}-{p}^{t}{x}_{i}\right)\le 1$, and$\frac{{\alpha}_{i}^{t}}{{x}_{i}}\ge {p}^{t}$:
- i.
- If${\sum}_{i=1}^{N}{y}_{i}^{t}}<K$, the optimal energy price and consumption are given by$${p}^{t*}=\frac{{\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}{x}_{i}\right)+{d}_{2}^{t}{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}}}{2{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}},{y}_{i}^{t*}={\alpha}_{i}^{t}{x}_{i}-\frac{{\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}{x}_{i}\right)+{d}_{2}^{t}{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}}}{2{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}}{\left({x}_{i}\right)}^{2}$$
- ii.
- If${\sum}_{i=1}^{N}{y}_{i}^{t}}=K$, the optimal energy price and consumption are given by$${p}^{t*}=\frac{{\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}{x}_{i}\right)-K}}{2{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}},{y}_{i}^{t*}={\alpha}_{i}^{t}{x}_{i}-\frac{{\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}{x}_{i}\right)-K}}{2{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}}{\left({x}_{i}\right)}^{2}$$

- B.
- If${\sum}_{t=1}^{T}\left({\alpha}_{i}^{t}-{p}^{t}{x}_{i}\right)>1$, and$\frac{{\alpha}_{i}^{t}}{{x}_{i}}\ge {p}^{t}$, the optimal energy price and consumption are given by${\sum}_{t=1}^{T}{y}_{i}^{t*}}={x}_{i$,${p}^{t*}\to +\infty $
- C.
- If$\frac{{\alpha}_{i}^{t}}{{x}_{i}}<{p}^{t}$, the optimal energy price and consumption is given by${p}^{t*}=0$, and${y}^{t*}=0$.

#### 3.2. Comparative Statics

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

#### 3.3. Equilibrium of Non-Real-Time Pricing Model

**Proposition**

**6.**

- A.
- If${\sum}_{t=1}^{T}\left({\alpha}_{i}^{t}-p{x}_{i}\right)\le 1$, and$\frac{{\alpha}_{i}^{t}}{{x}_{i}}\ge p$:
- i.
- If${\sum}_{i=1}^{N}{y}_{i}^{t}}<K$, the optimal energy price and consumption are given by$${p}^{*}=\frac{{\displaystyle \sum _{t=1}^{T}{\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}{x}_{i}\right)}}+{\displaystyle \sum _{t=1}^{T}{\displaystyle \sum _{i=1}^{N}{d}_{2}^{t}{\left({x}_{i}\right)}^{2}}}}{2T{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}},{y}_{i}^{t*}={\alpha}_{i}^{t}{x}_{i}-\frac{{\displaystyle \sum _{t=1}^{T}{\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}{x}_{i}\right)}}+{\displaystyle \sum _{t=1}^{T}{\displaystyle \sum _{i=1}^{N}{d}_{2}^{t}{\left({x}_{i}\right)}^{2}}}}{2T{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}}{\left({x}_{i}\right)}^{2}$$
- ii.
- If${\sum}_{i=1}^{N}{y}_{i}^{t}}=K$, the optimal energy price and consumption are given by$${p}^{*}=\frac{{\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}{x}_{i}\right)-K}}{{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}},{y}_{i}^{t*}={\alpha}_{i}^{t}{x}_{i}-\frac{{\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}{x}_{i}\right)-K}}{{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}}{\left({x}_{i}\right)}^{2}$$

- B.
- If${\sum}_{t=1}^{T}\left({\alpha}_{i}^{t}-p{x}_{i}\right)>1$, and$\frac{{\alpha}_{i}^{t}}{{x}_{i}}\ge p$, the optimal energy price and consumption are given by${\sum}_{t=1}^{T}{y}_{i}^{t*}}={x}_{i$,${p}^{*}\to +\infty $
- C.
- If$\frac{{\alpha}_{i}^{t}}{{x}_{i}}<p$, the optimal energy price and consumption are given by${p}^{*}=0$, and${y}^{t*}=0$.

## 4. Numerical Analysis

#### 4.1. User-Side Result

#### 4.2. Supplier-Side Result

#### 4.3. Social Welfare

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proof of Proposition 1

- ①
- ${\mathcal{L}}_{{y}_{i}^{t}}\left({y}_{i},{\mathsf{\mu}}_{i}\right)=-\frac{{\alpha}_{i}^{t}\beta {y}_{i}^{t}}{2{\left({x}_{i}\right)}^{2}}+\beta \frac{{\alpha}_{i}^{t}}{{x}_{i}}-{p}^{t}-{\displaystyle \sum _{i=1}^{N}{\mu}_{i}=0}$
- ②
- $\sum _{t=1}^{T}{y}_{i}^{t}\le {x}_{i}},\forall i\in \left[1,N\right]$
- ③
- ${\mu}_{i}\left({\displaystyle \sum _{t=1}^{T}{y}_{i}^{t}-{x}_{i}}\right)=0,\forall i\in \left[1,N\right]$
- ④
- ${\mu}_{i}\ge 0,\forall i\in \left[1,N\right]$

- Case A: $\sum _{i=1}^{N}{\mu}_{i}^{t}}=0,{y}_{i}^{t*}=\left({\alpha}_{i}^{t}-\frac{{p}^{t}}{\beta}{x}_{i}\right){x}_{i$
- Case B: $\sum _{i=1}^{N}{\mu}_{i}^{t}}>0,{\displaystyle \sum _{t=1}^{T}{y}_{i}^{t}}={x}_{i$

- ①
- $\begin{array}{ll}{\mathcal{L}}_{{p}^{t}}\left(p,\lambda \right)& =\left[-\frac{1}{\beta}{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}\right]\left[{p}^{t}-{c}_{2}^{t}-{c}_{1}^{t}{\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}-\frac{{p}^{t}}{\beta}{x}_{i}\right){x}_{i}-{\displaystyle \sum _{t=1}^{T}{\lambda}^{t}}}\right]+\left[{\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}-\frac{{p}^{t}}{\beta}{x}_{i}\right){x}_{i}}\right]\left[1+\frac{{c}_{1}^{t}}{\beta}{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}\right]\\ & =\left[-\frac{1}{\beta}{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}\right]\left({p}^{t}-{c}_{2}^{t}-{\displaystyle \sum _{t=1}^{T}{\lambda}^{t}}\right)+{\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}-\frac{{p}^{t}}{\beta}{x}_{i}\right){x}_{i}}=0\end{array}$
- ②
- $\sum _{i=1}^{N}{y}_{i}^{t}\le K},\forall t\in \left[1,T\right]$
- ③
- ${\lambda}^{t}\left({\displaystyle \sum _{i=1}^{N}{y}_{i}^{t}-K}\right)=0,\forall t\in \left[1,T\right]$
- ④
- ${\lambda}^{t}\ge 0,\forall t\in \left[1,T\right]$

- ①
- If $\sum _{t=1}^{T}{\lambda}^{t}}=0$, equilibrium energy price and consumptions are$${p}^{t*}=\frac{\beta {\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}{x}_{i}\right)+\beta {d}_{2}^{t}{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}}}{2{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}},{y}_{i}^{t*}={\alpha}_{i}^{t}{x}_{i}-\frac{{\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}{x}_{i}\right)+{d}_{2}^{t}{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}}}{2{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}}{\left({x}_{i}\right)}^{2}$$
- ②
- If $\sum _{t=1}^{T}{\lambda}^{t}}>0$, equilibrium energy price and consumptions are$${p}^{t*}=\frac{\beta {\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}{x}_{i}\right)-\beta K}}{2{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}},{y}_{i}^{t*}={\alpha}_{i}^{t}{x}_{i}-\frac{{\displaystyle \sum _{i=1}^{N}\left({\alpha}_{i}^{t}{x}_{i}\right)-K}}{2{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}\right)}^{2}}}{\left({x}_{i}\right)}^{2}$$

- ①
- ${\mathcal{L}}_{{p}^{t}}\left(p,\lambda \right)={\displaystyle \sum _{i=1}^{N}{x}_{i}}>0$
- ②
- $\sum _{i=1}^{N}{y}_{i}^{t}\le K},\forall t\in \left[1,T\right]$
- ③
- ${\lambda}^{t}\left({\displaystyle \sum _{i=1}^{N}{y}_{i}^{t}-K}\right)=0,\forall t\in \left[1,T\right]$
- ④
- ${\lambda}^{t}\ge 0,\forall t\in \left[1,T\right]$

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

${p}^{t}$ | Energy price at time $t$ |

${x}_{i}$ | Maximum daily requirement of user $i$ |

${\alpha}_{i}^{t}$ | Time preference (weight) for user $i$ at time $t$ |

$T$ | Number of time slots |

$N$ | Number of users |

${y}_{i}^{t}$ | Actual consumption of user $i$ at time $t$ |

$y$ | Matrix of users’ energy consumption schedule |

$p$ | Vector of supplier’s energy supplement schedule |

${Y}^{t}$ | Total amount of energy consumption at time $t$ |

$\mathsf{\Pi}$ | Profit obtained by the energy supplier during time period $[1,T]$ |

${C}^{t}$ | Cost function of energy generation |

${d}_{i}^{t}$ | Coefficients of cost function ($i=1,2,3$) |

$K$ | Production capacity for each time slot |

${u}_{i}^{t}$ | Utility of user $i$ at time $t$ |

${W}_{i}$ | Payoff of user $i$ for using energy |

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**MDPI and ACS Style**

Piao, R.; Lee, D.-J.; Kim, T.
Real-Time Pricing Scheme in Smart Grid Considering Time Preference: Game Theoretic Approach. *Energies* **2020**, *13*, 6138.
https://doi.org/10.3390/en13226138

**AMA Style**

Piao R, Lee D-J, Kim T.
Real-Time Pricing Scheme in Smart Grid Considering Time Preference: Game Theoretic Approach. *Energies*. 2020; 13(22):6138.
https://doi.org/10.3390/en13226138

**Chicago/Turabian Style**

Piao, Ri, Deok-Joo Lee, and Taegu Kim.
2020. "Real-Time Pricing Scheme in Smart Grid Considering Time Preference: Game Theoretic Approach" *Energies* 13, no. 22: 6138.
https://doi.org/10.3390/en13226138