# Multi-Objective Demand Response Model Considering the Probabilistic Characteristic of Price Elastic Load

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Stochastic DR Characteristic

#### 2.1. Demand Response to Balance Wind Power Fluctuations

_{Gi}and ΔP

_{l}

_{,i}are the active power and active load at node i respectively, N is the number of buses, and ΔP

_{Loss}is the change of the transmission loss in the system. The grid change is usually small in a short time. Therefore, $\Delta {P}_{Loss}$ is negligible. Thus, the interaction benefit of all PELs can be calculated as follows:

#### 2.2. Probabilistic Characterization of PELs

_{0}, c

_{0}are, respectively, the initial electricity demand and initial price.

_{l}

_{,i}) and its price (c

_{i}) can be defined as follows:

_{l}

_{,i}) and the price (c

_{i}) is linear and can be expressed as follows:

_{i}< 0 and β

_{i}< 0.

_{i}| > |α

_{j}|.

_{i}> k

_{j}. Thus, the uncertain demand response range of PEL i is larger than that of PEL j (Figure 1).

## 3. PEL Interaction Benefit Model

_{i}) of PEL and its amount of demand response (ΔP

_{l}

_{,i}) can be defined as follows:

_{i}) can be defined as the changes in the electricity costs of customers.

_{l}

_{,i}and ΔP

_{l}

_{,i}can be expressed as follows:

_{i}and ΔP

_{l}

_{,i}can be described as Equation (12) by substituting Equation (11) into Equation (10):

## 4. Stochastic DR Model of PEL

#### 4.1. Demand Response Satisfaction of PEL

_{i}represents the ECS of PEL i, and ECS submits to ${\mathsf{\eta}}_{i}\in (0,1]$. ECS reaches the maximum (its value is 1.0) when the electricity consumption manner of PEL is not changed.

#### 4.2. Probabilistic Demand Response Model

#### 4.2.1. Objective Function

_{i}

_{1}represents the weight of IBS and λ

_{i}

_{2}represents the weight of ECS. With regard to different PELs, weights can be set as different values to reflect that the degrees of attention to IBS and ECS are different for various PELs.

#### 4.2.2. Equality Constraints

- Power balance constraints

- Stochastic Constraint

#### 4.2.3. Inequality Constraints

#### 4.3. Solution Methodology

## 5. Case Study

#### 5.1. Data and Assumptions

PEL | α_{i} | β_{i} | λ_{i}_{1} | λ_{i}_{2} |
---|---|---|---|---|

1 | −0.559 | 9.3403 | 0.7 | 0.3 |

2 | −0.509 | 8.3973 | 0.7 | 0.3 |

3 | −0.506 | 5.3461 | 0.6 | 0.4 |

4 | −0.556 | 4.1522 | 0.6 | 0.4 |

5 | −0.107 | 3.2877 | 0.5 | 0.5 |

6 | −0.117 | 3.2574 | 0.5 | 0.5 |

7 | −0.306 | 2.9123 | 0.4 | 0.6 |

8 | −0.336 | 2.9642 | 0.4 | 0.6 |

_{i}and β

_{i}of each PEL bus can be calculated as Table 1. We then set the corresponding distribution coefficient of probabilistic parameter k

_{i}.

_{i}decreases. On the contrary, a larger $\left|\mathsf{\alpha}\right|$ corresponds to the greater uncertainty of demand response. Thus, k

_{i}increases. Therefore, ${k}_{4}>{k}_{1}>{k}_{3}>{k}_{2}>{k}_{8}>{k}_{7}>{k}_{6}>{k}_{5}$. In this paper, the following are set: ${k}_{4}=0.25$, ${k}_{1}=0.2$, ${k}_{3}=0.18$, ${k}_{2}=0.15$, ${k}_{8}=0.12$, ${k}_{7}=0.1$, ${k}_{6}=0.08$, and ${k}_{5}=0.05$.

#### 5.2. Relationship between Wind Power Fluctuation and Demand Response Amount

#### 5.3. Price Elasticity Affecting Demand Response Amount

#### 5.4. Effect of PEL Probabilistic Characterization on Demand Response Amount

#### 5.5. Computing Performance

## 6. Conclusions

- (1)
- The output of the uncertain model contains abundant probability information. It provides practical information on how PELs actively respond to the power grid integrated with wind power, thus decreasing the effects caused by the response deviation;
- (2)
- The relationship between the elasticity coefficient of PELs and the distribution coefficient of the stochastic demand response is elaborated. The increasing elasticity coefficient of PELs, i.e., decreasing flexibility, leads to a large distribution coefficient of stochastic demand;
- (3)
- Choosing PELs with small sensitivity to the price elasticity coefficient into the interaction with the power grid reduce the uncertainty and enhance reliability,
- (4)
- Proper choice of the distribution coefficient of ECS for PELs increases the comprehensive satisfaction of demand responses,
- (5)
- The proposed model enables demand response resources to respond to wind power variability. It also contributes to mitigating power imbalance, and consideration of the interaction profit with the power grid is presented; and
- (6)
- This approach is applicable in hourly real-time pricing models, and also in day-ahead pricing models.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

DR | Demand response |

PEL | Price elastic load |

ECS | Electricity consumption satisfaction |

IBS | Interaction benefit satisfaction |

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**MDPI and ACS Style**

Yang, S.; Zeng, D.; Ding, H.; Yao, J.; Wang, K.; Li, Y.
Multi-Objective Demand Response Model Considering the Probabilistic Characteristic of Price Elastic Load. *Energies* **2016**, *9*, 80.
https://doi.org/10.3390/en9020080

**AMA Style**

Yang S, Zeng D, Ding H, Yao J, Wang K, Li Y.
Multi-Objective Demand Response Model Considering the Probabilistic Characteristic of Price Elastic Load. *Energies*. 2016; 9(2):80.
https://doi.org/10.3390/en9020080

**Chicago/Turabian Style**

Yang, Shengchun, Dan Zeng, Hongfa Ding, Jianguo Yao, Ke Wang, and Yaping Li.
2016. "Multi-Objective Demand Response Model Considering the Probabilistic Characteristic of Price Elastic Load" *Energies* 9, no. 2: 80.
https://doi.org/10.3390/en9020080