# Coevolution and Evaluation of Electric Vehicles and Power Grids Based on Complex Networks

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## Abstract

**:**

## 1. Introduction

## 2. Regional Evolution Model of a Power Grid Based on a Complex Network

#### Evolution of Power Systems

- (1)
- Growth: The scale of the network is constantly increasing. If there is a connected network containing M
_{0}nodes, each new node will be connected to the existing M nodes, and m ≤ M_{0}. - (2)
- Priority connection: the probability of a new node connecting to an existing node
**i**is, and the connection probability of node**i**is related to the degree of that node,**k**:_{i}$${\mathbf{\Pi}}_{\mathit{i}}\left({\mathit{k}}_{\mathit{i}}\right)=\frac{{\mathit{k}}_{\mathit{i}}}{{\sum}_{\mathit{j}}{\mathit{k}}_{\mathit{j}}}$$**Σ**is the sum of node degrees in a network, assuming that the network evolves to step t. The network produces a network with N = t + m_{j}k_{j}_{0}nodes and m_{t}+ M_{0}edges. When t = 0, m_{0}links between M_{0}nodes. The relationship between the continuous variable and node number is 0 < M_{0}≤ 1/2 (m_{0}(m_{0}− 1)).

- (3)
- Local world: Select M nodes (N > M) from the network of N nodes as the local world of newly added nodes.

- Step 1: The components and connection modes in the network are calibrated, and the substation, transformer, and EV charging piles are specified as nodes. If there is a physical connection between nodes, it is marked as 1, and if there is no connection, it is marked as 0. The topology matrix, X, without electric vehicles is established. Suppose there are N nodes in the grid.
- Step 2: Set M as the size of the local area network (N > M), and randomly select a submatrix X′ with size M from the topology matrix.
- Step 3: Given probability P, the newly added node is connected to the nodes in the local world with probability P, and the nodes that meet the characteristics of the scale-free network are preferentially connected. Its probability for priority connection is:$${\mathbf{\Pi}}_{\mathbf{local}}\left({\mathit{k}}_{\mathit{i}}\right)={\mathbf{\Pi}}^{\mathbf{\prime}}\left(\mathit{i}\in \mathbf{localword}\right)\frac{{\mathit{k}}_{\mathit{i}}}{{\sum}_{\mathbf{jlocal}}{\mathit{k}}_{\mathit{i}}}$$
- Step 4: Reconstruct the network for the newly added node.

## 3. Distribution Network Stability Evaluation Model Based on the Coevolution of Vehicle Networks

#### 3.1. Fault Types

- (1)
- Distribution wire failure: The distribution wire is a link connecting two nodes. Each distribution wire connecting two nodes has a probability of failure. The failure of links at different locations will affect different areas.
- (2)
- Circuit breaker failure: The circuit breaker isolates the power grid in a section of the region, protecting the regional power grid as a whole. The position of the circuit breaker in the power grid is relatively fixed, and the affected region is fixed.
- (3)
- Transformer failure: the transformer is located at the power grid transmission terminal, and power passes through the transformer power supply to the user. The failure of the transformer will affect the power consumption of the connected users, and the affected area is constant.

#### 3.2. Component Fault Sampling

**TTF**is the normal operation time before component failure (h),

**TTR**is the component failure time, namely, the component repair time (h),

**λ**is the component failure rate (number of failures/year),

**u**is the component repair rate (number of repairs/year), and

**R**and

_{1}**R**are uniformly distributed random numbers between [0, 1].

_{2}#### 3.3. User Load Model

**P**is the annual maximum load,

_{Lmax}**P**(

_{week}**t**) is the percentage of weekly peak load,

**P**(

_{day}**t**) is the percentage of daily load peak to weekly load peak, and

**P**(

_{hour}**t**) is the he percentage of peak hourly load to peak daily load.

#### 3.4. Charging and Discharging Power Sequence Model of Electric Vehicles

**P**is the discharge power of electric vehicles,

_{discharge}**P**refers to the charging power of electric vehicles, and

_{charge}**D**(

**t**) is the time gate function.

#### 3.5. Fault Mode Analysis Method

_{F}= [S

_{F}, E

_{F}, T

_{F}, N

_{F}, L

_{F}], where M

_{F}is the failure mode analysis table, S

_{F}is the failure type, E

_{F}is the failure component, T

_{F}is the failure time of the component, and T

_{F}is the repair time. N

_{F}is the number of affected users, and L

_{F}is the load loss after failure.

_{F}can be directly determined. When the circuit breaker fails, its position in the power grid is fixed, so its affected area is relatively fixed. When the circuit breaker fails, it can determine whether there is output of electric vehicles in the isolated island formed by the circuit breaker fault and then determine whether the isolated island has power failure, the number and duration of power failures, and other parameters. In the case of wire distribution failure, since every wire distribution in the power grid is equally likely to fail, the location of the fault is uncertain, and the affected area is also uncertain. Therefore, the traversal of the fault is carried out by establishing a topology matrix and by using forward search and back search [33]. The power grid after failure can be divided into two areas: the unaffected area L and the affected area I. It can be determined whether there are EV nodes in the affected area I, and it can be determined whether there is power failure in area I and the number and duration of power failure and other parameters.

_{E,}with the load, Q

_{L,}at the time of power grid failure. When Q

_{E}> Q

_{L}, the network is not affected; when Q

_{E}< Q

_{L}, the network is still out of operation. The outage time is the difference between the component maintenance time and the electric vehicle power supply time. The load loss is equal to the load loss at the fault point minus the battery input of the electric vehicle.

#### 3.6. Reliability Indicators Based on the Monte Carlo Method

**λ**(times/year) is the average failure rate of the system load point,

_{S}**λ**(times/year) is the failure rate of component

_{i}**i**,

**R**(hour/year) is the fault repair time of component

_{i}**i**,

**R**(hour/year) is the equivalent repair time of each failure of the system load point, and

_{S}**Us**(hour/year) is the average annual power failure time of the load point.

- System Average Interruption Frequency Index (
**SAIFI**)

- Customer Average Interruption Duration Index (
**CAIDI**)

- System Average Interruption Duration Index (
**SAIDI**)

- Average Service Availability Index (
**ASAI**)

- Energy Not Supplied (
**ENS**)

**λ**is the load failure outage rate numbered

_{j}**j**,

**N**is the number of load users numbered

_{j}**j**, and

**n**is the total load.

**U**is the average power outage duration of load users numbered

_{j}**j**,

**L**indicates the power required by the user in the case of a fault,

_{i}**U**indicates the power supply duration required by the user in the case of a fault,

_{i}**L**represents the discharge power of EV under the condition of failure, and

_{EV_dis}**U**represents the discharge time of power from electric vehicles during power grid failure.

_{EV_dis}#### 3.7. Reliability Evaluation of the Power Grid Based on the Sequential Monte Carlo Method

- Step 1: Initialize the system clock and statistics and set t = 0.
- Step 2: The fault probability of each component is generated as a value between 0 and 1, the TTF of each component in the distribution network is generated using Equation (3), and t = t + min (TTF). Check whether t is less than the simulation time. If so, go to Step 3; otherwise, go to Step 9.
- Step 3: Select the smallest TTF component and generate the TTR of the component according to Equation (3).
- Step 4: Identify 1 power grid failure. Options are: 1 for cable failure, 2 for circuit breaker failure, and 3 for transformer failure. Go to Step 5 for option 1, Step 6 for Option 2, and Step 7 for Option 3.
- Step 5: Perform a traversal search for the distribution wire fault to find the point set I affected by the fault and determine whether the point set contains the main network node. If it does not, go to Step 6; if it does, go to Step 7.
- Step 6: Determine whether or not there are electric vehicles in the regional power grid and whether they participate in V2G. If there are no electric vehicles, go to Step 7; if there are no electric vehicles, go to Step 8.
- Step 7: Read the fault impact table, record the blackout duration, affected users, and lost loads, and then go to Step 2.
- Step 8: Calculate the electric vehicle discharge power, modify the fault mode analysis table, record the blackout duration, affected users and lost load value, and go to Step 2.
- Step 9: Calculate the reliability indicators of each node and the system.
- Step 10: Output the calculation results.

## 4. Power Grid Evolution and Evaluation

#### 4.1. Evolution of the Power Grid

#### 4.2. Monte Carlo Grid Reliability Evaluation

- (1)
- Transformer failure

- (2)
- Circuit breaker failure

- (3)
- Distribution lines

**M**= [

_{F}**S**,

_{F}**E**,

_{F}**T**,

_{F}**N**,

_{F}**L**]. Finally, the reliability index can be calculated.

_{F}#### 4.3. Impact of the Coevolution of EVs on the Reliability of Distribution Networks

- Working Condition 1: there are no electric vehicles in the system,
- Working Condition 2: An EV node is connected to node 12 and participates in V2G,
- Working Condition 3: An EV node is connected to node 6 and participates in V2G,
- Working Condition 4: An EV node is connected to node 12 and does not participate in V2G.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Fault type and traversal search: (

**a**) transformer failure chart, (

**b**) circuit breaker failure chart, (

**c**) distribution lines chart, and (

**d**) back/forward searches flow chart.

Label | Node Degrees | Connection Probability |
---|---|---|

1 | 2 | 0.04 |

2 | 2 | 0.04 |

3 | 2 | 0.04 |

4 | 2 | 0.04 |

5 | 2 | 0.04 |

6 | 2 | 0.04 |

7 | 2 | 0.04 |

8 | 3 | 0.06 |

9 | 2 | 0.04 |

10 | 2 | 0.04 |

11 | 2 | 0.04 |

12 | 3 | 0.06 |

13 | 2 | 0.04 |

14 | 2 | 0.04 |

15 | 1 | 0.02 |

16 | 3 | 0.06 |

17 | 2 | 0.04 |

Indicators | Working Condition 1 | Working Condition 2 | Working Condition 3 | Working Condition 4 |
---|---|---|---|---|

SAIFI | 1.1021 | 1.0764 | 1.0983 | 1.1154 |

SAIDI | 4.8137 | 4.6974 | 4.7826 | 4.8869 |

CAIDI | 4.3628 | 4.3462 | 4.3594 | 4.3709 |

ASAI | 0.99945 | 0.99946 | 0.99946 | 0.99944 |

EENS | 31.5372 | 30.8659 | 31.0832 | 32.0276 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, D.; Kang, Y.; Ji, L.; Shi, R.; Jia, L.
Coevolution and Evaluation of Electric Vehicles and Power Grids Based on Complex Networks. *Sustainability* **2022**, *14*, 7052.
https://doi.org/10.3390/su14127052

**AMA Style**

Zhang D, Kang Y, Ji L, Shi R, Jia L.
Coevolution and Evaluation of Electric Vehicles and Power Grids Based on Complex Networks. *Sustainability*. 2022; 14(12):7052.
https://doi.org/10.3390/su14127052

**Chicago/Turabian Style**

Zhang, Di, Yaxiong Kang, Li Ji, Ruifeng Shi, and Limin Jia.
2022. "Coevolution and Evaluation of Electric Vehicles and Power Grids Based on Complex Networks" *Sustainability* 14, no. 12: 7052.
https://doi.org/10.3390/su14127052