# A Novel Distributed Large-Scale Demand Response Scheme in High Proportion Renewable Energy Sources Integration Power Systems

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## Abstract

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## 1. Introduction

- (1)
- The bias of CBL cannot be avoided, and it will result in welfare losses to customers or LSEs [41]. In addition, DR event frequency will be much higher in large-scale DR to integrate a higher proportion of RES, which causes obstacles to establishing accurate CBL, as it is calculated by historical load data in the absence of DR events.
- (2)

- (1)
- This paper proposes a novel DR performance measurement method called customer directrix load (CDL) to replace CBL. Due to the global uniqueness of CDL, it is more suitable for distributed schemes and the moral hazard problem can be avoided.
- (2)
- The process of allocating DR resources in this paper is designed as a distributed method, which is similar to the red envelop allocation in many social networks. The LSE only needs to publish a total DR rebate and corresponding DR regulation goal to the demand side through social networks, and customer claims for an ideal rebate are performed in a distributed manner. This process is modeled as a non-cooperative game, and the Nash equilibrium is proven to exist.
- (3)
- The Gossip algorithm is used to realize distributed iterations in solving the non-cooperative game. To match factors of socially connected network, the Gossip algorithm is also improved in this paper.

## 2. Timeline of the Proposed DR Scheme

## 3. CDL-Based DR Scheme

#### 3.1. CDL

#### 3.2. Broadcast Total Rebate

## 4. Non-Cooperative Game of Customers

#### 4.1. Customer Model

#### 4.2. Formulation of Non-Cooperative Game

- Players: customer $i\in \mathcal{I}$;
- Strategy: the ideal rebate value ratio ${k}_{i}(t)$ for customer i;
- Utility: the welfare function of customer given in (10) and limited to (11).

- The player set is finite.
- The strategy sets are closed, bounded, and convex.
- The utility functions are continuous and quasi-concave in the strategy space.

- If $\widehat{k}(t)>{\overline{k}}_{i}(t)$,$$\frac{\partial {w}_{i}^{DR}(t)}{\partial {k}_{i}(t)}>0\text{\hspace{0.05em}}$$
- If $\widehat{k}(t)<{\underset{\_}{k}}_{i}(t)$,$$\frac{\partial {w}_{i}^{DR}(t)}{\partial {k}_{i}(t)}<0$$
- If ${\underset{\_}{k}}_{i}(t)\le \widehat{k}(t)\le {\overline{k}}_{i}(t)$$$\frac{\partial {w}_{i}^{DR}(t)}{\partial {k}_{i}(t)}\{\begin{array}{l}>0\text{\hspace{1em}}{\underset{\_}{k}}_{i}(t)\le {k}_{i}(t)<\widehat{k}\\ =0\text{\hspace{1em}}{k}_{i}(t)=\widehat{k}\\ <0\text{\hspace{1em}}\widehat{k}<\text{\hspace{0.05em}}{k}_{i}(t)<{\overline{k}}_{i}(t)\end{array}$$

Algorithm 1: Non-cooperative process executed by DR customer $i\in \mathcal{I}$ | |

1: | for each $i\in \mathcal{I}$ do |

2: | Obtain global configure information from LSE day-ahead; |

3: | m = 0; |

4: | Keep initial estimation value of total rebate ratio ${\widehat{K}}_{i}^{(m=0)}=1$ |

5: | repeat |

6: | Calculate optimal ideal rebate ratio ${k}_{i}^{opt(m)}$ according to (20); |

7: | m = m + 1; |

8: | Execute Algorithm 2 to update the estimation value of total rebate ratio ${\widehat{K}}_{i}^{(m)}$; |

9: | until (21) is satisfied; |

10: | Report the optimal rebate ratio ${k}_{i}^{opt(m)}$ to LSE; |

11: | end for |

#### 4.3. Iterative Distributed Scheme Based on Gossip Algorithm

**A**and degree matrix

**D**, and their entries ${a}_{ij}$ and ${d}_{ij}$ are:

**L**, which is defined as:

- (1)
- Initial ${k}_{\mathrm{ave}.i}^{(r=0)}={k}_{i}^{opt(m)}$, and calculate weights;
- (2)
- Exchange information with others and update the estimating average value of ${\widehat{K}}_{i}(t)$ according to (24).

**L**is hard to obtain not only for customers but also for LSE. In addition, different from [49,50], the network in our scheme is not physically connected but is socially connected, and customers only know the connecting information regarding themselves. That is, there is no one who can gather and calculate an adjacent matrix

**A**and degree matrix

**D**. To solve this problem, this paper improves the approach to calculating the heights to be adjusted to our DR scheme.

- Each graph has a certain degree ${d}^{*}$ for every node, and ${d}^{*}=\{1,2,3,\cdots ,20\}$;
- The total nodes number $I$ range from 100 to 1000, i.e., $I=\{100,200,\cdots ,1000\}$;
- In a graph with a certain degree and total nodes number, randomly change $\mathcal{E}$ 10 times, namely, choose the link ways randomly for 10 times.

**A**, which ensures the distributedly integrative process. The process for each customer to estimate ${\widehat{K}}_{i}^{}$ can be characterized by the Algorithm 2.

Algorithm 2: Gossip algorithm for each DR customer $i\in \mathcal{I}$ to estimate ${\widehat{K}}_{i}^{(m)}$ | |

1: | for each $i\in \mathcal{I}$ do |

2: | Obtain global configure information from LSE day-ahead; |

3: | Calculate heights ${\omega}_{ii}$ and ${\omega}_{ij}$ according to (27) and (29); |

4: | r = 0; |

5: | Keep initial estimation average value of total rebate ratio ${k}_{\mathrm{ave}.i}^{(r=0)}={k}_{i}^{opt(m)}$; |

6: | repeat |

9: | r = r + 1; |

10: | update ${k}_{\mathrm{ave}.i}^{(r)}$ according to (24); |

11: | until (28) is satisfied; |

12: | Calculate the estimation value of total rebate ratio ${\widehat{K}}_{i}^{(m)}=I\cdot {k}_{\mathrm{ave}.i}^{(r)}$; |

13: | end for |

#### 4.4 Flowchart of the Whole DR Program

## 5. Case Study

#### 5.1. Simulation Results about Consuming RES

#### 5.2. Simulation Results of the Non-Cooperative Game of Customers

#### 5.3. Impacts of RES and DR Percentage and Rebate Rate Factor in the DR Scheme

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## Nomenclature

$\mathcal{T}$ | Set of time, index by t |

$\mathcal{I}$ | Set of DR customers, index by i |

$\mathcal{G}$ | Graph of social network |

$\mathcal{V}$ | Set of all nodes in the social network |

$\mathcal{E}$ | Set of logical links between nodes |

${\mathcal{N}}_{i}$ | Set of all the customer nodes who can exchange information with customer i, |

${P}_{c}(t)$ | Total power output of conventional generators at time slot t |

${D}_{i}(t)$ | Power consumption DR of customer i at time slot t; |

${P}_{CDL}(t)$ | The calculated CDL at time slot t |

${D}^{*}(t)$ | The forecasted total load of DR customers at time slot t |

$\Delta {P}_{\mathrm{up}.\mathrm{max}}$ | The maximal increasing capacity of total conventional generations |

$\Delta {P}_{\mathrm{down}.\mathrm{max}}$ | The maximal decreasing capacity of total conventional generations |

${k}_{i}$ | Claiming rebate ratio of customer i at time slot t |

${k}_{i}^{opt}(t)$ | Optimal decision value of ${k}_{i}$ |

${k}_{i}^{*}(t)$ | Actual value of rebate ratio for customer i calculated by LSE |

K(t) | Total rebate ratio of all the DR customers |

${\widehat{K}}_{i}(t)$ | Estimation value of K(t) by customer i |

${\overline{k}}_{i}(t)$, ${\underset{\_}{k}}_{i}(t)$ | Maximal and minimal rebate ratio for customer i at time slot t |

${\pi}_{r}$ | Retail rate of electricity |

$\sigma $ | Rebate rate factor, a positive constant |

${u}_{i}(t)$ | Utility of consumer i at time slot t |

${\mathrm{w}}_{i}(t)$ | Welfare of consumer i at time slot t |

$\rho $ | A index to evaluate two time series’ similarity |

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**Figure 6.**Conventional generation curve and RES curve through the proposed DR scheme to get balance.

**Figure 8.**The dynamic process of the non-cooperative process for every customers’ ideal rebate ratio.

**Figure 9.**The dynamic process of the non-cooperative process for every customers’ actual rebate value.

**Figure 10.**The dynamic process for customer 1 to estimate ${\widehat{K}}_{i=1}(t=7)$ based on the proposed Gossip algorithm.

Number of DR Customers | Iteration | Number of DR Customers | Iteration |
---|---|---|---|

150 | 16 | 750 | 8 |

300 | 15 | 900 | 7 |

450 | 12 | 1050 | 7 |

600 | 10 | 1200 | 6 |

Iteration | Estimation Value | Actual Value | Iteration | Estimation Value | Actual Value |
---|---|---|---|---|---|

1 | 2.76 | 2.94 | 6 | 411.98 | 412.45 |

2 | 7.90 | 7.78 | 7 | 569.36 | 569.67 |

3 | 21.53 | 21.66 | 8 | 591.64 | 591.76 |

4 | 59.68 | 59.89 | 9 | 593.13 | 593.27 |

5 | 164.87 | 165.26 | 10 | 593.23 | 593.36 |

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## Share and Cite

**MDPI and ACS Style**

Fan, S.; He, G.; Jia, K.; Wang, Z.
A Novel Distributed Large-Scale Demand Response Scheme in High Proportion Renewable Energy Sources Integration Power Systems. *Appl. Sci.* **2018**, *8*, 452.
https://doi.org/10.3390/app8030452

**AMA Style**

Fan S, He G, Jia K, Wang Z.
A Novel Distributed Large-Scale Demand Response Scheme in High Proportion Renewable Energy Sources Integration Power Systems. *Applied Sciences*. 2018; 8(3):452.
https://doi.org/10.3390/app8030452

**Chicago/Turabian Style**

Fan, Shuai, Guangyu He, Kunqi Jia, and Zhihua Wang.
2018. "A Novel Distributed Large-Scale Demand Response Scheme in High Proportion Renewable Energy Sources Integration Power Systems" *Applied Sciences* 8, no. 3: 452.
https://doi.org/10.3390/app8030452