# Energy Trading and Pricing in Microgrids with Uncertain Energy Supply: A Three-Stage Hierarchical Game Approach

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

- Leader: the energy provider determines the energy purchase and the pricing strategy to maximize its profit.
- Followers: the consumers determine the energy demands to maximize their payoffs.

## 3. Wind Power Generation Model

## 4. Scenario A: The Three-Stage Game for Price-Taking Consumers

#### 4.1. Consumer’s Energy Demands in Stage III

#### 4.2. Optimal Pricing Strategy in Stage II

- ${S}_{1}(p)$ (excessive supply): ${S}_{1}(p)$ doesn’t intersect with $D(p)$, ${p}^{*}={p}_{d}$;
- ${S}_{2}(p)$ (excessive supply): ${S}_{2}(p)$ has one intersection with $D(p)$, where $D(p)$ has a non-negative slope, ${p}^{*}={p}_{d}$;
- ${S}_{3}(p)$ (conservative supply): ${S}_{3}(p)$ has one intersection with $D(p)$, where $D(p)$ has a negative slope, ${p}^{*}={p}_{h}$, where ${p}_{h}$ is the intersection point of $D(p)$ and $S(p)$ and ${p}^{*}$ is the optimal price announced by the energy provider.

#### 4.3. Energy Supply Strategy in Stage I

- (1)
- Interval I: ${p}_{s}\in [0,\frac{Q}{2}]$. In this interval, the energy provider’s profit function is:$$\begin{array}{cc}\hfill {W}_{II}^{1}({p}_{s})& ={E}_{P\in [{P}_{\mathrm{min}},{P}_{\mathrm{max}}]}\left[{W}_{II}^{CS}(P)\right]\hfill \\ & ={\int}_{{P}_{\mathrm{min}}}^{{P}_{\mathrm{max}}}{W}_{II}^{CS}(P){f}_{WP}(P)\mathrm{d}P.\hfill \end{array}$$
- (2)
- Interval II: ${p}_{s}\in [\frac{Q}{2},\infty ]$. The energy provider’s profit function is:$$\begin{array}{cc}\hfill {W}_{II}^{2}({p}_{s})=& {E}_{P\in [{P}_{\mathrm{min}},\frac{Q}{2}]}\left[{W}_{II}^{CS}(P)\right]+{E}_{P\in [\frac{Q}{2},{P}_{\mathrm{max}}]}\left[{W}_{II}^{ES}(P)\right]\hfill \\ \hfill =& {\int}_{{P}_{\mathrm{min}}}^{\frac{Q}{2}}{W}_{II}^{CS}(P){f}_{WP}(P)\mathrm{d}P+{\int}_{\frac{Q}{2}}^{{P}_{\mathrm{max}}}{W}_{II}^{ES}(P){f}_{WP}(P)\mathrm{d}P.\hfill \end{array}$$

## 5. Scenario B: The Three-Stage Game for Price-Anticipating Consumers

#### 5.1. Consumer’s Energy Demands in Stage III

#### 5.2. Optimal Pricing Strategy in Stage II

- ${S}_{1}(\omega )$ (excessive supply): ${p}_{0}(\beta +{\beta}_{0}){p}_{s}\ge {p}_{0}G/2h$, ${\omega}^{*}=0$,
- ${S}_{2}(\omega )$ (conservative supply): ${p}_{0}(\beta +{\beta}_{0}){p}_{s}<{p}_{0}G/2h$, ${\omega}^{*}={\omega}_{p}$,

- ${S}_{3}(\omega )$ (excessive supply): ${S}_{3}(\omega )$ has one intersection with $D(\omega )$, where $D(\omega )$ has a non-negative slope, ${\omega}^{*}={\omega}_{0}$,
- ${S}_{4}(\omega )$ (conservative supply): ${S}_{4}(\omega )$ has three intersections with $D(\omega )$, ${\omega}^{*}={\omega}_{p}$,
- ${S}_{5}(\omega )$ (conservative supply): ${S}_{5}(\omega )$ has one intersection with $D(\omega )$, where $D(\omega )$ has a negative slope, ${\omega}^{*}={\omega}_{p}$.

- ${S}_{6}(\omega )$ (excessive supply): ${S}_{6}(\omega )$ doesn’t intersect with $D(\omega )$, ${\omega}^{*}={\omega}_{0}$,
- ${S}_{7}(\omega )$ (excessive supply): ${S}_{7}(\omega )$ has one or two intersections with $D(\omega )$, where both intersections are located in the increasing interval of $D(\omega )$, ${\omega}^{*}={\omega}_{0}$,
- ${S}_{8}(\omega )$ (conservative supply): ${S}_{8}(\omega )$ has two intersections with $D(\omega )$, where both intersections are located in the both sides of ${\omega}_{0}$, respectively, ${\omega}^{*}={\omega}_{p}$.

#### 5.3. Energy Supply Strategy in Stage I

- (1)
- Interval I: ${p}_{s}\in [0,\frac{A}{4h(N+1)}]$. In this interval, the energy provider’s profit function is:$$\begin{array}{cc}\hfill {W}_{II}^{{1}^{\prime}}({p}_{s})=& \hfill {E}_{P\in [{P}_{\mathrm{min}},{P}_{\mathrm{max}}]}\left[{W}_{II}^{CS}(P)\right]\\ \hfill =& \hfill {\int}_{{P}_{\mathrm{min}}}^{{P}_{\mathrm{max}}}{W}_{II}^{CS}(P){f}_{WP}(P)\mathrm{d}P.\end{array}$$
- (2)
- Interval II: ${p}_{s}\in [\frac{A}{4h(N+1)},\infty ]$. The energy provider’s profit function is:$$\begin{array}{cc}\hfill {W}_{II}^{{2}^{\prime}}({p}_{s})& ={E}_{P\in [{P}_{\mathrm{min}},\frac{A}{4h(N+1)}]}\left[{W}_{II}^{CS}(P)\right]+{E}_{P\in [\frac{A}{4h(N+1)},{P}_{\mathrm{max}}]}\left[{W}_{II}^{ES}(P)\right]\hfill \\ & ={\int}_{{P}_{\mathrm{min}}}^{\frac{A}{4h(N+1)}}{W}_{II}^{CS}(P){f}_{WP}(P)\mathrm{d}P+{\int}_{\frac{A}{4h(N+1)}}^{{P}_{\mathrm{max}}}{W}_{II}^{ES}(P){f}_{WP}(P)\mathrm{d}P.\hfill \end{array}$$

## 6. Simulation Results

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Total Energy Obtained | Optimal Price | Optimal Profit |
---|---|---|

in Stages I and II | ${\mathit{p}}^{*}({\mathit{p}}_{\mathbf{s}},\mathit{\beta})$ | ${\mathit{W}}_{\mathbf{II}}({\mathit{p}}_{\mathbf{s}},\mathit{\beta})$ |

Excessive Supply Regime: ${p}_{s}\ge \frac{Q}{2}$ | ${p}^{ES}={p}_{d}$ | ${W}_{II}^{ES}({p}_{s},\beta )$ in Equation (17) |

Conservative Supply Regime: ${p}_{s}<\frac{Q}{2}$ | ${p}^{CS}={p}_{h}$ | ${W}_{II}^{CS}({p}_{s},\beta )$ in Equation (18) |

Total Energy Obtained | Optimal Parameter | Optimal Profit |
---|---|---|

in Stages I and II | ${\mathit{p}}^{*}({\mathit{p}}_{\mathbf{s}},\mathit{\beta})$ | ${\mathit{W}}_{\mathbf{II}}({\mathit{p}}_{\mathbf{s}},\mathit{\beta})$ |

Excessive Supply Regime: ${p}_{s}\ge \frac{A}{4h(N+1)}$ | ${\omega}^{ES}={\omega}_{0}$ | ${W}_{II}^{ES}({p}_{s},\beta )$ in Equation (43) |

Conservative Supply Regime: ${p}_{s}<\frac{A}{4h(N+1)}$ | ${\omega}^{CS}={\omega}_{p}$ | ${W}_{II}^{CS}({p}_{s},\beta )$ in Equation (44) |

Scenario A | Scenario B | |||
---|---|---|---|---|

${\mathit{\beta}}_{0}$ | ${\mathit{p}}_{\mathbf{s}}$ | Profit | ${\mathit{p}}_{\mathbf{s}}$ | Profit |

0.1 | 349 | 33.79 | 399.6 | 31.52 |

0.3 | 246 | 34.61 | 288.6 | 33.27 |

0.6 | 186 | 35.2 | 209 | 35.13 |

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**MDPI and ACS Style**

Ma, K.; Hu, S.; Yang, J.; Dou, C.; Guerrero, J.M.
Energy Trading and Pricing in Microgrids with Uncertain Energy Supply: A Three-Stage Hierarchical Game Approach. *Energies* **2017**, *10*, 670.
https://doi.org/10.3390/en10050670

**AMA Style**

Ma K, Hu S, Yang J, Dou C, Guerrero JM.
Energy Trading and Pricing in Microgrids with Uncertain Energy Supply: A Three-Stage Hierarchical Game Approach. *Energies*. 2017; 10(5):670.
https://doi.org/10.3390/en10050670

**Chicago/Turabian Style**

Ma, Kai, Shubing Hu, Jie Yang, Chunxia Dou, and Josep M. Guerrero.
2017. "Energy Trading and Pricing in Microgrids with Uncertain Energy Supply: A Three-Stage Hierarchical Game Approach" *Energies* 10, no. 5: 670.
https://doi.org/10.3390/en10050670