# Collaborative Robust Optimization Strategy of Electric Vehicles and Other Distributed Energy Considering Load Flexibility

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Robust Optimal Control Framework

#### 2.2. EV Model and the Temporal Flexibility of EV Cluster

#### 2.2.1. Single EV Model under Demand Difference

#### 2.2.2. Temporal Flexibility for EV Clusters

_{1}c is the trajectory of the rated charging EV, and aa

_{1}c is the trajectory formed by charging to the desired SOC in the latest charging period $[{t}_{d},{t}_{l,out}]$. These constitute the upper and lower boundaries, respectively.

#### 2.3. Robust Optimization Model Considering Temporal Flexibility

#### 2.3.1. Robust Optimization Theory

#### 2.3.2. Construction of Robust Optimization Model

#### 2.3.3. The Description of Output Uncertainty Convex Set

#### 2.3.4. Decoupled Solving of Robust Optimization Models

**X**be the optimization vector in the whole $T$ period, then $\mathit{X}={[\mathit{u},{\mathit{P}}_{gi},{\mathit{P}}_{ev},{\mathit{P}}_{sw}]}^{T}$, and the dimension of

**X**is 2N

_{g}T + 2T. The matrix form of the min term is as follows:

**A**is the corresponding coefficient matrix; $H(\mathit{X})$ and $G(\mathit{X})$ respectively correspond to the inequalities and equality constraints of Equations (1)–(3), (10), (12), (14), (16) and (17); $\mathit{b}$ is a suitable constant vector; $0$ is a zero vector. Equation (21) is a typical mixed integer programming problem, which is solved efficiently by using the commercial solver CPLEX.

_{g}T + 2T; the dimension of

**Z**is 2T. The general form of the matrix of Equation (22) is:

**B**is the corresponding coefficient matrix; both

**M**and

**N**are matrices adapted to Formulas (7), (11), (12), (14)–(19);

**L**is a suitable constant vector; $0$ is a zero vector. Max-min is a two-level optimization problem. According to the duality theory, dual variables $\mu $ and $\rho $ are introduced to convert the two-level optimization into a single-level optimization. The dual form of Equation (24) is:

## 3. Results and Discussion

#### 3.1. Data Description and Parameter Setting

#### 3.2. Robust Optimization Results Analysis

^{4}, which is the compensation cost of Type 2 and Type 3 EVs. The power adjustment costs of the three units and EV cluster in the second stage in response to the fluctuation of renewable energy are CNY 2.467 × 10

^{4}, CNY 1.394 × 10

^{4}, CNY 0.352 × 10

^{4}and CNY 3.051 × 10

^{4}, respectively. DG

_{1}generates the highest cost, indicating that the power adjustment of DG

_{1}is the largest. Based on the model established in this paper, the second stage also ensures that the renewable energy is fully consumed.

^{4}and CNY 109.57 × 10

^{4}, respectively, under the prediction error of 5%. The solving time of Schemes 4 and 5 are 442.8 and 523.2 s, respectively, under the prediction error of 5%.

#### 3.3. Analysis of EV Charging Process

_{1}–A

_{4}is above 0.95 during their departure, which ensures the electricity demand of EV users. No matter when the ${l}_{1,k}$ is connected to a grid, it is charged to the desired SOC at the rated charging power; during the plugged-in periods of ${l}_{2,k}$, the actual charging power is less than the rated charging power, and the SOC growth is slower, but ${l}_{2,k}$ is not in the discharging state during the plugged-in periods; during the plugged-in periods of ${l}_{3,k}$, a few periods are in the discharging state, but it is guaranteed that the SOC is not lower than the 0.5 threshold during the discharging process, such as vehicle ${l}_{3,3}$. It should be noted that the charging and discharging states of EV in sets ${D}_{2,k}$ and ${D}_{3,k}$ are also constrained by the user′s plugged-in duration. However, in order to meet the user′s electricity demand, the EVs in sets ${D}_{2,k}$ and ${D}_{3,k}$ will still approach the rated charging power because of their shorter connection time, such as ${l}_{2,2}$ and ${l}_{3,4}$ in Figure 10.

#### 3.4. The Influence of Prediction Error and the Value of $\mathsf{\Gamma}$ on Robust Optimization

#### 3.5. The Impact of the Proportion of Three Types EV on Robust Optimization

^{5}; when all EVs are Type 2 or Type 3 EVs, the total robust cost is CNY 10.137 × 10

^{5}.

## 4. Conclusions

^{4}and 442.8 s, respectively. Compared with other methods, the robust total cost and computational efficiency are reduced about by 3.8% and improved by about 15.4%, respectively.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

EVs | Electric vehicles |

EVAs | Electric vehicle aggregators |

SOC | State of charge |

V2G | Vehicle to grid |

## Mathematical Notation Explanations

${P}_{c,l}^{r}$, ${\eta}_{c,l}$ | the rated charging power and charging efficiency of vehicle $l$, respectively |

${P}_{l,t}$, ${S}_{l,t}$ | the actual charging/discharging power and actual SOC of vehicle $l$ at period $t$, respectively |

${P}_{d,l}^{r}$, ${\eta}_{d,l}$ | the rated discharging power and discharging efficiency of vehicle $l$, respectively |

${S}_{l,in}$, ${S}_{l,ex}$ | the initial SOC and expected SOC of vehicle $l$, respectively |

${S}_{l,thr}$ | the discharging SOC threshold of vehicle $l$ |

${E}_{l,in}$, ${E}_{l,ex}$, ${E}_{d,thr}$ | the initial electric quantity, the expected electric quantity of departure time and the discharging electric quantity threshold of vehicle $l$, respectively |

$\Delta {E}_{l,max}$ | the maximum electric quantity reduction of vehicle $l$ |

${E}_{d,l}$ | the discharging electric quantity of vehicle $l$ |

${P}_{ev,t}$ | the power of an EV cluster at period $t$ |

${N}_{A}$, ${N}_{g}$ | the number of EVAs, unit |

${N}_{x}$, ${N}_{y}$, ${N}_{z}$ | the number of rated charging power EVs, adjustable charging EVs and flexible charging–discharging EVs, respectively |

$\Delta {E}_{2,t}^{o}$, $\Delta {E}_{3,t}^{o}$ | the electric quantity reduction of adjustable charging EVs and flexible charging–discharging EVs, respectively |

$\Delta {E}_{s,t}$ | the electric quantity reduction of an EV cluster |

${C}_{s}$ | the total cost of the system |

${P}_{g}$, ${P}_{b}$, ${E}_{s}$, $W$ | the unit power vector, standby power vector, EV cluster power vector and abandoning renewable power vector, respectively |

${a}_{i}$, ${b}_{i}$, ${c}_{i}$ | the cost coefficient of the $i$-$\mathrm{th}$ unit |

${u}_{i,t}$ | boolean variable that indicates the startup and shutdown of the $i$-$\mathrm{th}$ at period $t$ |

${s}_{i}$ | the startup and shutdown cost of the $i$-$\mathrm{th}$ unit |

${P}_{gi,t}^{up}$, ${P}_{gi,t}^{down}$ | the up-regulated and the down-regulated power of the $i$-$\mathrm{th}$ unit, respectively |

${\gamma}_{i}^{up}$, ${\gamma}_{i}^{down}$ | the price of up-regulated power and price of down-regulated power of $i$-$\mathrm{th}$ unit, respectively |

${C}_{a}$ | the compensation cost of the EV cluster |

${\beta}_{t}$ | the compensation factor of the EV cluster |

${s}_{l,\mathrm{max}}$, ${c}_{l,\mathrm{max}}$ | the charging benefit and charging cost corresponding to the upper boundary, respectively |

${s}_{l}$ | the charging benefit corresponding to the broken line $abc$ |

$\phi $ | the discharging compensation coefficient |

${E}_{l}^{\text{'}}$ | the total discharging electric quantity of the EV cluster |

$\Delta W$ | the power of abandoning renewable energy |

${\tilde{P}}_{s,t}$, ${\tilde{P}}_{s,t}$ | the forecast solar energy output and wind energy output, respectively |

${\theta}_{s,t}$, ${\theta}_{w,t}$ | the forecast error of solar energy and wind energy, respectively |

${P}_{sw,t}$ | the actual consumption of renewable energy |

${P}_{gi}^{\mathrm{min}}$, ${P}_{gi}^{\mathrm{max}}$, ${P}_{gi}^{umax}$, ${P}_{gi}^{dmax}$ | the maximum and minimum power, the increased power limit and the decreased power limit of the $i$-$th$ unit, respectively |

${t}_{i,on}$, ${t}_{i,off}$, ${T}_{i,on}$, ${T}_{i,off}$ | the continuous start-up and shutdown time, and the minimum start-up and shutdown time of the $i$-$\mathrm{th}$ unit, respectively |

${z}_{t}^{+}$, ${z}_{t}^{-}$ | the 0–1 binary auxiliary variable |

$\mathsf{\Gamma}$ | conservative degree control parameter of the robust model |

## Appendix A

Time Periods | Electricity Price/CNY | φ | β |
---|---|---|---|

Peak Time (9:00–12:00, 19:00–24:00) | 1.85 | 1.85 | 2.41 |

Usual time (7:00–9:00, 12:00–19:00) | 1.4 | 1.4 | 1.82 |

Valley Time (0:00–7:00) | 0.95 | 0.95 | 1.23 |

Parameters | DG1 | DG2 | DG3 |
---|---|---|---|

${P}_{\mathrm{min}}/\mathrm{MW}$ | 3 | 6 | 4 |

${P}_{\mathrm{max}}/\mathrm{MW}$ | 15 | 30 | 15 |

${P}^{\mathrm{d}\mathrm{m}\mathrm{a}\mathrm{x}}/(\mathrm{M}\mathrm{W}\cdot 15\mathrm{m}\mathrm{i}{\mathrm{n}}^{-1})$ | 500 | 600 | 700 |

${P}^{\mathrm{u}\mathrm{m}\mathrm{a}\mathrm{x}}/(\mathrm{M}\mathrm{W}\cdot 15\mathrm{m}\mathrm{i}{\mathrm{n}}^{-1})$ | 380 | 450 | 500 |

${\gamma}_{}^{up}/\mathrm{CNY}\cdot \mathrm{KW}$ | 1.25 | 1.25 | 1.25 |

${\gamma}_{}^{down}/\mathrm{CNY}\cdot \mathrm{KW}$ | 0.45 | 0.45 | 0.45 |

a | 300 | 373 | 300 |

b | 170 | 200 | 150 |

c | 0.251 | 0.082 | 0.452 |

$s/\mathrm{CNY}$ | 600 | 900 | 600 |

${T}_{on},{T}_{off}/h$ | 1 | 1.5 | 1 |

EV Parameters | Value |
---|---|

${P}_{c}/\mathrm{KWh}$ | 20 |

${P}_{d}/\mathrm{KWh}$ | 15 |

${\eta}_{c}$ | 0.95 |

${\eta}_{d}$ | 0.95 |

$E/\mathrm{KWh}$ | 65 |

${S}_{ex}$ | 1 |

${S}_{thr}$ | 0.5 |

${S}_{l,ex}$ | 0.95 |

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**Figure 2.**The schedulable capability of Type 2 and Type 3 EVs. (

**a**) Adjustable charging EV. (

**b**) Flexible charging–discharging EV.

**Figure 10.**Three SOC change curves of three EV types in four EVAs. (

**a**) SOC changes of three EV types in A

_{1}; (

**b**) SOC changes of three EV types in A

_{2}; (

**c**) SOC changes of three EV types in A

_{3}; (

**d**) SOC changes of three EV types in A

_{4}.

Cost/CNY | First Stage/CNY | Second Stage/CNY | Total Cost/CNY |
---|---|---|---|

Unit 1 | 210,080 | 24,670 | 234,750 |

Unit 2 | 331,050 | 13,940 | 344,990 |

Unit 3 | 151,050 | 3520 | 154,570 |

EV cluster | 289,280 | 30,510 | 319,790 |

Abandoning renewable energy | 0 | 0 | 0 |

Total cost/CNY | 981,460 | 72,640 | 1,054,100 |

The Solving Time/s | Scheme 1 | Scheme 2 | Scheme 3 | Scheme 4 | Scheme 5 |
---|---|---|---|---|---|

5% | 491.5 | 473.2 | 469.5 | 442.8 | 523.2 |

10% | 493.5 | 475.6 | 487.3 | 462.4 | 518.7 |

15% | 499.4 | 472.1 | 483.4 | 459.8 | 529.6 |

20% | 489.3 | 478.3 | 484.8 | 451.7 | 532.4 |

25% | 501.4 | 481.3 | 489.3 | 461.3 | 529.5 |

30% | 497.5 | 482.1 | 477.6 | 459.2 | 525.6 |

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**MDPI and ACS Style**

Wang, Y.; Zhang, B.; Li, C.; Huang, Y.
Collaborative Robust Optimization Strategy of Electric Vehicles and Other Distributed Energy Considering Load Flexibility. *Energies* **2022**, *15*, 2947.
https://doi.org/10.3390/en15082947

**AMA Style**

Wang Y, Zhang B, Li C, Huang Y.
Collaborative Robust Optimization Strategy of Electric Vehicles and Other Distributed Energy Considering Load Flexibility. *Energies*. 2022; 15(8):2947.
https://doi.org/10.3390/en15082947

**Chicago/Turabian Style**

Wang, Yuxuan, Bingxu Zhang, Chenyang Li, and Yongzhang Huang.
2022. "Collaborative Robust Optimization Strategy of Electric Vehicles and Other Distributed Energy Considering Load Flexibility" *Energies* 15, no. 8: 2947.
https://doi.org/10.3390/en15082947