# Research on Interval Optimal Scheduling Strategy of Virtual Power Plants with Electric Vehicles

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Structure and Principle of Virtual Power Plant with Electric Vehicles

#### 2.1. Structure of Virtual Power Plant with Electric Vehicles

#### 2.2. Principle of a Virtual Power Plant with Electric Vehicles

- (1)
- Wind power generation model

_{WPP}is the wind power output, kW, P

_{rated}is the rated wind power, kW/h, and v

_{in}, v

_{out}, and v

_{rated}are the cut-in wind speed, cut-out wind speed, and rated wind speed, m/s, respectively.

- (2)
- Photovoltaic power generation model

_{INC}and P

_{PV}are the actual radiation intensity and actual output power, respectively, k is the power temperature coefficient, T

_{r}is the standard test condition temperature, and T

_{e}is the actual temperature.

- (3)
- Gas turbine model

_{2}emission volume versus the power generated by the gas turbine is:

_{MT}and H

_{MT}are the electric and thermal power of the gas turbine, respectively,$\eta $

_{MT}is the gas turbine conversion efficiency, ${\eta}_{MT}^{re}$ is the waste heat recovery efficiency, K is the heat production coefficient, L

_{gas}is the low heating value of natural gas, Q

_{MT}is the natural gas consumption, Q

_{pai}is the total amount of CO

_{2}emissions, ${c}_{c{o}_{2}}$ is the CO

_{2}emission factor, and ${\rho}_{c{o}_{2}}$ is the density of the CO

_{2}.

- (4)
- Electric vehicle charging model

_{st}and the daily driving distance of the electric vehicle on the grid is:

_{p}is the length of the stopping time, ${\mu}_{\mathrm{p}}\mathrm{and}{\sigma}_{\mathrm{p}}$ are the expected value, and variance, of t, respectively, and the electric vehicle off-grid time is: t

_{q}= t + t

_{p}.

- (5)
- Power-to-gas equipment model

_{2}, m

^{3}, ${P}_{P2{H}_{2}}$ and ${P}_{P2{H}_{2},rated}$ are the electrical power consumed by the electrolyzer and the rated power, kW, respectively, ${\alpha}_{P2{H}_{2}}$ is the electrolytic efficiency factor, and V

_{ec}is the rated capacity, m

^{3}.

## 3. Optimized Scheduling Model

_{1}, c

_{2}, …, c

_{L})

^{T}, for which the i-th component c

_{i}, there are ${c}_{i}=[\underset{\_}{{c}_{i}},\overline{{c}_{i}}]$, g

_{j}(x, c) $\ge a$

_{j}is the interval inequality constraint, h

_{k}(x, c) = b

_{k}is the interval equation constraint [37], and ${f}_{i}^{I}(x,c)$ denotes the i-th objective function as follows:

#### 3.1. Objective Function

- (1)
- Economic objectives [38]

_{ev,t}denotes the electric vehicle charging power at time t, P

_{load,t}denotes the residential load power at time t, kW, $[{P}_{grid,t}^{ex}]$ and $[{V}_{gas,t}^{ex}]$ denote the energy interaction power between the virtual power plant and the grid and the natural gas network at time t, respectively, and ${c}_{gas,t}^{buy}$, ${c}_{gas,t}^{sell}$, ${c}_{grid,t}^{buy}$, and ${c}_{grid,t}^{sell}$ denote the purchase/sale price of gas from the virtual power plant to the natural gas network and the grid at time t, respectively.

- (2)
- Environmental objectives

_{2}emissions mainly considers the emissions corresponding to the power interaction with the grid [39], the emissions from gas turbine power generation, and the absorption of power-to-gas [40]. The environmental objective function is then:

- (3)
- Grid load smoothing target

#### 3.2. Constraints

- (1)
- Grid and gas networks need to maintain their power balance to operate properly; therefore, the power balance constraints are as follows:$$[{P}_{grid,t}^{ex}]=[{P}_{eload,t}]+{P}_{ev,t}+{P}_{P2G,t}+{P}_{es,t}-[{P}_{WPP,t}]-[{P}_{PV,t}]-{P}_{PMT,t}$$$$[{V}_{gas,t}^{ex}]=[{V}_{gload,t}]+{V}_{PMT,t}+{V}_{gs,t}-{V}_{P2G,t}$$
_{es,t}denotes the power of the energy storage system at time t. When P_{es,t}is positive, it represents discharge, and when it is negative, it represents charging. V_{gs,t}represents the output of the gas storage system at the moment t. When V_{gs,t}is positive, it represents gassing, and when it is negative, it represents gas storage. - (2)
- There are limits to the amount of electricity and natural gas that can be received by the grid and the gas grid from virtual power plants [42]; therefore, virtual power plants deliver power constraints to the grid and natural gas network as follows:$${P}_{grid,\mathrm{min}}^{ex}\le [{P}_{grid,t}^{ex}]\le {P}_{grid,\mathrm{max}}^{ex}$$$${V}_{gas,\mathrm{min}}^{ex}\le [{V}_{gas,t}^{ex}]\le {V}_{gas,\mathrm{max}}^{ex}$$
- (3)
- Electric vehicle charging power constraintDividing a day into 24 time periods, the total load demand for time period t is the sum of all EV charging loads, and the total charging load is:$${P}_{\mathrm{ev},t}={\displaystyle \sum _{m=1}^{M}{P}_{m,t}},t=1,2,\mathrm{...24}$$
_{ev,t}is the total charging power of all cars in the i-th time period. Therefore, the charging power should meet the following:$${P}_{ev,t,\mathrm{min}}\le {P}_{ev,t}\le {P}_{ev,t,\mathrm{max}}$$The charging power should also meet the dispatchable potential of the electric vehicle cluster. - (4)
- Total Electric Vehicle Power Balance Constraint$$\eta {\displaystyle \sum _{t=1}^{24}{P}_{ev,t}}={Q}_{ev}$$
- (5)
- Electricity storage equipment power constraints$$SOC(t)=SOC(t-1)+{\eta}_{es}\cdot {P}_{es,t}\cdot \Delta t$$
_{es}is the charging/discharging efficiency of the storage device. The charging and discharging behavior do not take place in the storage device simultaneously, so the storage state is:$${U}_{ech}+{U}_{edis}\le 1$$_{ech}and U_{edis}are for the storage equipment charging and discharging signs, respectively. - (6)
- Gas storage device gas storage capacity constraints$${V}_{gch}(t)={V}_{gch}(t-1)+({v}_{ch,t}-{v}_{dis,t})\cdot \Delta t$$
_{gch}(t) and V_{gch}(t–1) represent the volume of natural gas in the storage facility at the end of time periods t and t−1, respectively, m^{3}; and v_{ch,t}and v_{dis,t}are the storage and release rates of the storage facility, respectively, m^{3}/h. - (7)
- Each device has its own upper and lower limits of operating power, and can only operate properly if it is kept within these limits. The power constraints for each device are as follows:$$\left\{\begin{array}{l}\underset{\_}{{P}_{PMT}}\le {P}_{PMT,t}\le \overline{{P}_{PMT}}\\ \underset{\_}{{P}_{P2G}}\le {P}_{P2G,t}\le \overline{{P}_{P2G}}\\ \underset{\_}{{P}_{es}}\le {P}_{es,t}\le \overline{{P}_{es}}\\ \underset{\_}{{v}_{ch}}\le {v}_{ch,t}\le \overline{{v}_{ch}}\\ \underset{\_}{{v}_{dis}}\le {v}_{dis,t}\le \overline{{v}_{dis}}\end{array}\right\}$$

## 4. Scheduling Solution

#### 4.1. Outer Layer Optimization Model Solving

- (1)
- How to determine whether an individual satisfies the interval constraint.
- (2)
- How to define the predominance relation of feasible and infeasible solutions.
- (3)
- How to calculate the crowding distance of individuals when comparing individuals with the same ordinal value.

- (1)
- Interval credibility

_{j}(x, c) $\ge a$

_{j}, the confidence level of the constraint is:

_{j}(x, c) $\ge a$

_{j}, the confidence of the constraint is:

_{j}(x, c) $\ge a$

_{j}satisfies ${\delta}_{j}\ge {\delta}_{j}^{*}$, then x is said to be a feasible solution and vice versa. For two non-feasible solutions, ${x}_{1}$ with ${x}_{2}$, this paper will determine the dominance relation by comparing the sum of their violations in overall constraints, i.e., if there exists $\sum}_{j=1}^{N}{L}_{j$ (${x}_{1}$) < $\sum}_{j=1}^{N}{L}_{j$ (${x}_{2}$), then it is said that ${x}_{1}$ dominates ${x}_{2}$; if there exists $\sum}_{j=1}^{N}{L}_{j$ (${x}_{1}$) > $\sum}_{j=1}^{N}{L}_{j$ (${x}_{2}$), then it is said that ${x}_{2}$ dominates ${x}_{1}$; if there exists $\sum}_{j=1}^{N}{L}_{j$ (${x}_{1}$) = $\sum}_{j=1}^{N}{L}_{j$ (${x}_{2}$), then it is said that ${x}_{1}\text{}\mathrm{and}{x}_{2}$ are mutually exclusive.

- (2)
- Interval overlap degree

_{1}is V(${x}_{1}$), then the distance between these two evolved individuals with the same ordinal value is:

_{1}are${x}_{2}$ and ${x}_{3}$, then the larger D(${x}_{1}$,${x}_{2}$) and D(${x}_{1}$,${x}_{3}$) are, the less crowded the degree of${x}_{1}$ is; therefore, the individual crowding distance of ${x}_{1}$ is:

#### 4.2. Inner Layer Optimization Model Solving

- (1)
- Economic goals

- (2)
- Environmental Goals

- (3)
- Grid load smoothing target

#### 4.3. Nested Solutions of Inner and Outer Optimization Models

## 5. Example Analysis

#### 5.1. Parameter Setting

#### 5.2. Results of the Algorithm

#### 5.2.1. Scatter Plotting of Median Objective Function

#### 5.2.2. Distribution of Objective Function Values

#### 5.2.3. Typical Scenario Scheduling Scheme

- (1)
- The most economical scheduling scheme

- (2)
- The most environmentally friendly scheduling scheme

_{2}emissions.

- (3)
- Optimal scheduling scheme for grid smoothing objectives

#### 5.2.4. Optimal Solution

## 6. Conclusions

- (1)
- The most economical scenario, in which the gas turbine power is elevated during periods of low PV output and periods of high electricity prices, seeks to maximize the difference between the virtual plant’s revenue from electricity sales to the grid and the virtual plant’s own operating costs, resulting in the highest virtual plant revenue.
- (2)
- In the most environmentally friendly scenario, since conventional power plants still use coal-fired power generation and gas turbine operation also emits CO
_{2}, the only way to achieve a reduction in CO_{2}emissions is to reduce the share of coal-fired power generation, i.e., to reduce the power purchased from the grid and the power of gas turbine operation; conversely, the power-to-gas equipment absorbs CO_{2}and this scenario can appropriately increase the duration of power-to-gas equipment operation. - (3)
- The optimal scheme of grid load smoothing, in which load fluctuations are smoothed by orderly regulation of EV charging behavior, as well as regulation of each output power and virtual power plant power sales, to smooth out peak-to-valley differences.
- (4)
- The optimal solution using hierarchical analysis can better balance the three objectives and can effectively improve the economic benefits of virtual power plants and ensure environmental friendliness and grid load smoothing requirements are met.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations/Nomenclature

SET AND INDICES | |

V | wind speed |

V_{in} | Cut-in wind speed |

v_{out} | Cut out wind speed |

V_{rated} | Rated wind speed |

k | Power Temperature Coefficient |

T_{r} | Standard test conditions temperature |

T_{e} | Actual solar temperature |

$\eta $_{MT} | Gas turbine conversion efficiency |

${\eta}_{MT}^{re}$ | Waste heat recovery efficiency |

K | Heat production coefficient |

L_{gas} | Low calorific value of natural gas |

D | Distance traveled per day |

${d}_{R}$ | Maximum driving distance |

${\mu}_{L},{\sigma}_{L}$ | Expected value and variance of travel distance |

${\mu}_{s},{\sigma}_{s}$ | Expected value and variance of entry time |

${\mu}_{\mathrm{p}},{\sigma}_{\mathrm{p}}$ | Expected value and variance of parking duration |

${G}_{{H}_{2}}$ | Outputs of H_{2} |

${\alpha}_{P2{H}_{2}}$ | Electrolytic efficiency factor |

x, y | Decision variables in the outer inner layer |

$\mathsf{\Omega}$ | Decision Space |

c | Interval vector |

${f}_{i}^{I}(x,c)$ | The i-th objective function |

M | Total number of electric vehicles |

Q_{ev} | Total electric vehicle charging capacity |

$\eta $ | Electric vehicle charging efficiency |

$\eta $_{es} | Charging/discharging efficiency of power storage devices |

U_{ech}, U_{edis} | Charging and discharging signs for power storage equipment |

C_{WPP}, C_{PV}, C_{PMT}, C_{es}, C_{P2G} | Cost of each piece of equipment |

${c}_{gas,t}^{buy}$${c}_{gas,t}^{sell}$, ${c}_{grid,t}^{buy}$, ${c}_{grid,t}^{sell}$ | t-time virtual power plants to the natural gas network, grid purchase/sale prices |

$\lambda $_{grid} | Emission factors for electricity purchased from the grid |

PARAMETERS | |

P_{WPP} | Wind power output |

P_{rated} | Wind power rating |

G_{INC} | Actual solar radiation intensity |

P_{PV} | Photovoltaic actual output power |

P_{MT} | Electrical power of the gas turbine |

H_{MT} | Thermal power of the gas turbine |

Q_{MT} | Natural gas consumption |

Q_{pai} | CO_{2} Total Emissions |

SOC_{st} | Electric Vehicle On-grid Initial Power |

${P}_{P2{H}_{2}}$, ${P}_{P2{H}_{2},rated}$ | Electrical power consumed by the electrolytic bath vs. rated power |

P_{PMT,t} | t time period gas turbine output electric power |

P_{P2G,t} | t time period electricity to gas equipment power |

P_{ev,t} | t time period electric vehicle charging power |

P_{es,t} | t time period power of power storage equipment |

V_{gs,t} | t time period gas storage equipment gas storage capacity |

$[{C}_{sell}^{energy}]$$[{C}_{buy}^{energy}]$, $[{C}_{op}^{DGs}]$ | Virtual power plant revenue, acquisition cost, maintenance cost of each unit |

P_{load,t} | Residential load power at time t |

$[{P}_{grid,t}^{ex}]$,$[{V}_{gas,t}^{ex}]$ | t time virtual power plant with grid and natural gas network energy interaction power |

${P}_{grid,\mathrm{min}}^{ex}$,${P}_{grid,\mathrm{max}}^{ex}$ | Upper and lower power limits for energy interaction between the virtual power plant and the grid |

${V}_{g\mathrm{as},\mathrm{min}}^{ex}$,${V}_{gas,\mathrm{max}}^{ex}$ | Upper and lower power limits for energy interaction between virtual power plant and natural gas network |

SOC(t) | Storage device charge at time t |

V_{gch} (t), V_{gch} (t–1) | Volume of natural gas in the gas storage facility at the end of time periods t and t–1 |

v_{ch,t}, v_{dis,t} | Storage and release rates of gas storage devices |

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**Figure 1.**Structure diagram of gas-electric interconnection virtual power plant incorporating an electric vehicle.

**Figure 15.**Economically optimal solution. (

**a**) Scheduling results of each power supply; (

**b**) Interactive power dispatch results with the grid.

**Figure 16.**Optimal solution for environmental friendliness. (

**a**) Scheduling results of each power supply; (

**b**) Interactive power dispatch results with the grid.

**Figure 17.**Optimal scheme for grid load smoothing. (

**a**) Scheduling results of each power supply; (

**b**) Interactive power dispatch results with the grid.

**Figure 18.**Optimal solution. (

**a**) Scheduling results of each power supply; (

**b**) Interactive power dispatch results with the grid.

Name | Interval Number | Probability-Based | Time-Domain Rolling and Multiple Time Scales | Multiple Scenarios | Combinatorial Optimization |
---|---|---|---|---|---|

advantage | simple practicality and efficient | effective | better results | good robustness | high accuracy |

disadvantage | low accuracy | difficult to obtain engineering practice | complex process | high dependence | complex process and difficult |

Variable | Description |
---|---|

P_{PMT,t} | Gas turbine output electric power |

P_{P2G,t} | Electric to gas equipment power |

P_{ev,t} | Electric vehicle charging power |

P_{es,t} | Power of power storage device (both positive and negative) |

V_{gs,t} | Gas storage equipment storage capacity (both positive and negative) |

Variable | Description | Values |
---|---|---|

C_{WPP} | Wind turbine equipment cost | 0.11 (yuan/kW) |

C_{PV} | Photovoltaic unit equipment cost | 0.08 (yuan/kW) |

C_{PMT} | Gas turbine equipment costs | 0.10 (yuan/kW) |

C_{es} | Cost of power storage equipment | 0.01 (yuan/kW) |

C_{P2G} | Power-to-gas equipment cost | 0.01 (yuan/kW) |

${c}_{gas,t}^{buy}$ | Price of gas purchased from the natural gas network during t hours | 2.5 (Yuan/m^{3}) |

${c}_{gas,t}^{sell}$ | Price of gas sold to the natural gas grid during t hours | 2.0 (Yuan/m^{3}) |

$\lambda $_{grid} | Emission factors for electricity purchased from the grid | 330.644 (kg/kWh) |

$\lambda $_{gas} | Emission factors for natural gas-fired power generation | 203.953 (kg/kWh) |

L_{gas} | Natural gas low calorific value | 9.70 (kWh/m^{3}) |

$\eta $ | Electric vehicle charging efficiency | 0.90 |

$\eta $_{es} | Charging and discharging efficiency of power storage equipment | 0.90 |

$\eta $_{MT} | Gas turbine conversion efficiency | 0.32 |

${\eta}_{MT}^{re}$ | Waste heat recovery efficiency | 0.54 |

K | Heat production coefficient | 2 |

${c}_{c{o}_{2}}$ | Natural gas-fired power generation emissions CO_{2} coefficients | 0.20374 (kg/kWh) |

${\rho}_{c{o}_{2}}$ | Carbon dioxide density | 1.977 (kg/m^{3}) |

ξ | Satisfy the constraint likelihood threshold | 0.80 |

Name | Population Size | Number of Maximum Iteration | Crossover Algorithm | Variation Probability | Reliability Threshold |
---|---|---|---|---|---|

Value | 100 | 3000 | 2 | 0.8 | 0.8 |

Economical Target (Yuan) Median of the Interval | Environmental Target (kg) Median of the Interval | Peak-Shaving Target (kW) Median of the Interval |
---|---|---|

6503.16 | 9821.03 | 2402.56 |

Economical Target (Yuan) Median of the Interval | Environmental Target (kg) Median of the Interval | Peak-Shaving Target (kW) Median of the Interval |
---|---|---|

6250.47 | 9797.43 | 2158.7 |

Economical Target (Yuan) Median of the Interval | Environmental Target (kg) Median of the Interval | Peak-Shaving Target (kW) Median of the Interval |
---|---|---|

4122.54 | 10,490.6 | 2475.7 |

Name | Economical | Environmental Protection | Peak-to-Valley Difference |
---|---|---|---|

Economical | 1 | 5 | 5 |

Environmental Protection | 1/5 | 1 | 5 |

Peak-to-valley difference | 1/5 | 1/5 | 1 |

Economical Target (Yuan) Median of the Interval | Environmental Target (kg) Median of the Interval | Peak-Shaving Target (kW) Median of the Interval |
---|---|---|

6456.11 | 9860.01 | 2402.56 |

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## Share and Cite

**MDPI and ACS Style**

Li, T.; An, J.; Zhang, D.; Diao, X.; Liu, C.; Liu, W.
Research on Interval Optimal Scheduling Strategy of Virtual Power Plants with Electric Vehicles. *World Electr. Veh. J.* **2022**, *13*, 235.
https://doi.org/10.3390/wevj13120235

**AMA Style**

Li T, An J, Zhang D, Diao X, Liu C, Liu W.
Research on Interval Optimal Scheduling Strategy of Virtual Power Plants with Electric Vehicles. *World Electric Vehicle Journal*. 2022; 13(12):235.
https://doi.org/10.3390/wevj13120235

**Chicago/Turabian Style**

Li, Taoyong, Jinjin An, Dongmei Zhang, Xiaohong Diao, Changliang Liu, and Weiliang Liu.
2022. "Research on Interval Optimal Scheduling Strategy of Virtual Power Plants with Electric Vehicles" *World Electric Vehicle Journal* 13, no. 12: 235.
https://doi.org/10.3390/wevj13120235