# A Real-Time Pricing Scheme for Energy Management in Integrated Energy Systems: A Stackelberg Game Approach

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## Abstract

**:**

## 1. Introduction

- A multi energy trading framework including one integrated energy provider and multiple integrated energy consumers, is proposed.
- A price-based energy management strategy is proposed to manage the electricity and gas trading between the integrated energy provider and smart energy hub operators.
- A Stackelberg game is proposed to capture the interactions between the integrated energy provider (leader) and smart energy hub operators (followers).
- A distributed algorithm between the integrated energy provider and energy hubs is proposed to derive the Stackelberg equilibrium, through which the optimal strategies for each player can be determined and the balance of energy supply and demand can be kept.

## 2. Smart Energy Hubs

## 3. The System Model

#### 3.1. Integrated Energy Provider Model

#### 3.2. The Smart Energy Hub Operator Model

#### 3.3. The Energy Pricing Mechanism: Stackelberg Game Approach

**Definition**

**1.**

**Theorem**

**1.**

#### 3.4. Distributed DR Algorithm and Implementation

Algorithm 1: An iterative algorithm executed by the EP and SEHOs |

1: Initialization: $k=0,\text{}\mathsf{\delta}={10}^{-5},\text{}{p}_{e}^{0},\text{}{p}_{g}^{0}$.2: While $t={\mathit{T}}_{T}$3: Repeat4: For each SEHO $n$5: Receive the new electricity price ${p}_{e,\text{}t}^{k}$ and gas price ${p}_{g,\text{}t}^{k}$ broadcasted by energy provider.6: Update the electricity consumption value ${E}_{n,\text{}t}^{in,k*}$ and gas consumption value ${G}_{n,\text{}t}^{in,k*}$ according to (12), and send them back to the energy provider.7: End for 8: Update the amount of electricity supply ${E}_{t}^{to,k*}$ and the natural gas supply ${G}_{t}^{to,k*}$ amount by solving (8)–(9).9: Update the electricity price ${p}_{e,t}^{k}$ and natural gas price ${p}_{g,t}^{k}$ according to (16).10: k= k+1.11: Until $\mathrm{max}\left(\left|{p}_{e,t}^{k+1}-{p}_{e,t}^{k}\right|,\left|{p}_{g,t}^{k+1}-{p}_{g,t}^{k}\right|\right)<\delta $.12: End while. |

## 4. Case Studies

#### 4.1. Basic Parameters

#### 4.2. Simulation Results and Discussions

#### 4.2.1. Real-Time Energy Prices and Load Profiles

#### 4.2.2. Comparisons of the Peak-to-Average Ratio (PAR)

#### 4.2.3. The Convergence of the Demand Response Algorithm

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

SEHO | Smart energy hub operator |

EP | Energy provider |

SEH | Smart energy hub |

SEMS | Smart energy management system |

DR | Demand response |

T | Transformer |

EHP | Electric heat pump |

GT | Gas turbine |

GB | Gas boiler |

$n$ | Index of SEH |

$t$ | Index of time slot |

$x$ | Index of load type |

$e/h$ | Electricity/heating load |

$X\in \left\{E,G\right\}$ | Index of electricity or gas |

${T}_{T}$ | The set of time slots |

${a}_{t}^{X}/{b}_{t}^{X}/\text{}{c}_{t}^{X}$ | Cost coefficients of EP |

${a}_{n,t}^{X}/{\beta}_{n}^{X}$ | Preference parameter of SEHO |

${E}_{n,t}^{in}/{G}_{n,t}^{in}$ | Electricity/gas purchased by SEH$n$ |

${E}_{n,t}^{T}/{E}_{n,t}^{EHP}$ | Power consumed by T/EHP |

${G}_{n,t}^{GT}/{G}_{n,t}^{GB}$ | Gas consumed by GT, GB |

${L}_{n,t}^{e}/{L}_{n,t}^{h}$ | power/heating load |

${\eta}_{n}^{T}/{\eta}_{n}^{GB}$ | Efficiency of T/GB |

${\eta}_{n}^{GT,e}/{\eta}_{n}^{GT,h}$ | Electrical/heating efficiency of GT |

$CO{P}_{n}^{EHP}$ | Coefficient of performance of EHP |

${p}_{e,t}/{p}_{g,t}$ | Real time power/gas price |

${R}_{x}$ | Load shifting ratio |

${L}_{n,x}^{dr,t}/{L}_{n,x}^{0,t}$ | Load $x$ after/before DR |

${E}_{t}^{to}/{G}_{t}^{to}$ | Total electricity/natural gas supplied by EP |

${E}_{t,max}^{to}/{G}_{t,max}^{to}$ | Maximum electricity/natural gas supplied by EP |

$\gamma $ | Step size |

## Appendix A

**Proof of Theorem 1:**

**Proof**

**1.**

**Proof**

**2.**

**Proof**

**3.**

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**Figure 2.**The schematic diagram of a smart integrated energy system consisting of one energy provider and N smart energy hub operators.

**Figure 10.**The convergence processes of energy prices: (

**a**) The convergence processes of electricity prices without DR; (

**b**) The convergence processes of electricity prices with DR; (

**c**) The convergence processes of gas prices without DR; (

**d**) The convergence processes of gas prices with DR.

Smart EH Type | ${\mathit{\eta}}_{\mathit{n},\mathit{T}}$ | ${\mathit{\eta}}_{\mathit{n},\mathit{G}\mathit{T}}^{\mathit{e}}$ | ${\mathit{\eta}}_{\mathit{n},\mathit{G}\mathit{T}}^{\mathit{h}}$ | $\mathit{C}\mathit{O}{\mathit{P}}_{\mathit{n},\mathit{E}\mathit{H}\mathit{P}}$ | ${\mathit{\eta}}_{\mathit{n},\mathit{G}\mathit{B}}$ |
---|---|---|---|---|---|

Type I | [0.95, 0.98]* | [0.3, 0.38] | [0.4, 0.48] | [2.5, 3.5] | — |

Type II | [0.92, 0.96] | [0.28, 0.33] | [0.44, 0.5] | — | [0.85, 0.95] |

^{1}[a, b]* represents that the parameter obeys the uniform distribution of the range [a, b].

Type I | $\overline{{E}_{n}^{T}}$ | $\overline{{E}_{n}^{EHP}}$ | $\overline{{G}_{n}^{GT}}$ |

1500 kW | 500 kWh | 4000 kWh | |

Type II | $\overline{{E}_{n}^{T}}$ | $\overline{{G}_{n}^{in}}$ | $\overline{{G}_{n}^{GB}}$ |

2000 kW | 5000 kW | 1000 kW |

Items | Before DR | After DR |
---|---|---|

PAR of Electricity | 1.370 | 1.143 |

PAR of Natural gas | 1.214 | 1.077 |

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**MDPI and ACS Style**

Ma, T.; Wu, J.; Hao, L.; Yan, H.; Li, D. A Real-Time Pricing Scheme for Energy Management in Integrated Energy Systems: A Stackelberg Game Approach. *Energies* **2018**, *11*, 2858.
https://doi.org/10.3390/en11102858

**AMA Style**

Ma T, Wu J, Hao L, Yan H, Li D. A Real-Time Pricing Scheme for Energy Management in Integrated Energy Systems: A Stackelberg Game Approach. *Energies*. 2018; 11(10):2858.
https://doi.org/10.3390/en11102858

**Chicago/Turabian Style**

Ma, Tengfei, Junyong Wu, Liangliang Hao, Huaguang Yan, and Dezhi Li. 2018. "A Real-Time Pricing Scheme for Energy Management in Integrated Energy Systems: A Stackelberg Game Approach" *Energies* 11, no. 10: 2858.
https://doi.org/10.3390/en11102858