# Smart Charging of Electric Vehicles Considering SOC-Dependent Maximum Charging Powers

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## Abstract

**:**

## 1. Introduction

- a reformulation of the scheduling problem in terms of the control of charging energy, which facilitates the integration of SOC-dependent maximum charging power,
- a proposal of two transformations of SOC–power curves into SOC–energy curves, and
- a proposal and evaluation of two mixed integer linear programming based solution methods that consider SOC-dependent maximum charging powers.

## 2. Related Work

## 3. Problem Description

#### 3.1. Exact Maximum Energy

#### 3.2. Lower Bound for Maximum Energy

- If ${P}_{v}^{max}$ is a piecewise linear function, then ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ is piecewise linear as well. On the contrary, ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ might not be a piecewise linear function, even if ${P}_{v}^{max}$ is piecewise linear.
- If ${P}_{v}^{max}$ is a concave function, so are ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ and ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$.

#### 3.3. Converting Energy Back to Power

#### 3.4. Nonlinear Model

## 4. Problem Solving Approaches

#### 4.1. Concave Maximum Energy Functions

#### 4.2. General Piecewise Linear Maximum Energy Functions

## 5. Benchmark Instances

#### 5.1. Individual EVS-SOC Instances

#### 5.2. Rolling Horizon Benchmark Scenarios

#### Exemplary Solutions

## 6. Experimental Results

#### 6.1. EVS-SOC-LIN

#### 6.2. EVS-SOC-GLIN

#### Charging Cost Differences & Charging Errors

#### 6.3. Comparison of EVS-SOC-LIN and EVS-SOC-GLIN

#### 6.4. Model Based Predictive Control Simulations

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Comparison of ${P}_{v}^{max}$ curves with different numbers of segments. (

**a**) ${P}_{v}^{max}$ curves for BMW i3. (

**b**) ${P}_{v}^{max}$ curves for Energica Ego. (

**c**) ${P}_{v}^{max}$ curves for Mercedes-Benz EQC. (

**d**) ${P}_{v}^{max}$ curves for Audi e-tron. (

**e**) ${P}_{v}^{max}$ curves for Hyundai Kona Elektro. (

**f**) ${P}_{v}^{max}$ curves for Jaguar I-Pace. (

**g**) ${P}_{v}^{max}$ curves for MINI Cooper Electric. (

**h**) ${P}_{v}^{max}$ curves for Tesla Model 3 Long Range.

## Appendix B

**Table A1.**EVS-SOC-GLIN results for solving the static model versus B & C with ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ and ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ based on the original ${P}_{v}^{max}$ functions and ${P}^{\mathrm{gridmax}}=10n$.

n | $\mathbf{\Delta}\mathit{t}$ (min) | ${\mathit{n}}_{\mathbf{seg}}$ | ${\mathit{n}}_{\mathbf{opt}}$ | ${\mathit{n}}_{\mathbf{feas}}$ | Runtime (s) | ${\mathit{n}}_{\mathbf{cuts}}$ | %-gap | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Median | StdDev | Median | StdDev | Median | ||||||||||

Mean | Static | B & C | Static | B & C | Static | B & C | Static | B & C | B & C | Static | B & C | |||

${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ | ||||||||||||||

5 | 1 | 155 | 24 | 24 | 30 | 30 | 391.75 | 43.39 | 662.53 | 720.28 | 1038 | 1944 | 0.01 | 0.01 |

5 | 5 | 139 | 30 | 30 | 30 | 30 | 6.58 | 1.43 | 17.30 | 64.96 | 144 | 307 | 0.00 | 0.01 |

5 | 10 | 119 | 30 | 30 | 30 | 30 | 1.67 | 0.83 | 6.66 | 1.77 | 56 | 120 | 0.00 | 0.00 |

10 | 1 | 311 | 5 | 12 | 21 | 29 | 1800.00 | 1800.00 | 389.82 | 764.87 | 4068 | 2726 | 0.03 | 0.03 |

10 | 5 | 279 | 29 | 27 | 30 | 30 | 79.94 | 8.84 | 353.48 | 544.47 | 498 | 581 | 0.01 | 0.01 |

10 | 10 | 242 | 30 | 30 | 30 | 30 | 7.04 | 2.06 | 35.58 | 4.31 | 194 | 167 | 0.00 | 0.01 |

20 | 1 | 612 | 0 | 1 | 2 | 11 | 1800.00 | 1800.00 | 0.00 | 181.68 | 8974 | 1820 | 0.08 | 0.19 |

20 | 5 | 553 | 23 | 19 | 30 | 30 | 500.49 | 684.63 | 640.95 | 781.48 | 1846 | 1081 | 0.01 | 0.01 |

20 | 10 | 475 | 30 | 29 | 30 | 30 | 40.18 | 13.35 | 71.71 | 425.71 | 505 | 274 | 0.01 | 0.01 |

50 | 1 | 1544 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 15,910 | 2518 | - | - |

50 | 5 | 1393 | 2 | 2 | 26 | 30 | 1800.00 | 1800.00 | 174.76 | 351.60 | 6106 | 1249 | 0.05 | 0.05 |

50 | 10 | 1192 | 29 | 18 | 30 | 30 | 307.62 | 827.59 | 458.49 | 779.32 | 1930 | 594 | 0.01 | 0.01 |

100 | 1 | 3095 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 11,886 | 3940 | - | - |

100 | 5 | 2796 | 0 | 0 | 9 | 9 | 1800.00 | 1800.00 | 0.00 | 0.00 | 9961 | 1319 | 0.08 | 0.12 |

100 | 10 | 2399 | 11 | 4 | 30 | 30 | 1800.00 | 1800.00 | 418.13 | 452.22 | 4434 | 861 | 0.01 | 0.03 |

5 | 1 | 901 | 3 | 12 | 8 | 27 | 1800.00 | 1800.00 | 414.46 | 834.31 | 5304 | 6069 | 0.03 | 0.01 |

5 | 5 | 901 | 30 | 26 | 30 | 30 | 143.42 | 9.59 | 312.39 | 628.62 | 820 | 1553 | 0.00 | 0.00 |

5 | 10 | 901 | 30 | 30 | 30 | 30 | 34.53 | 2.60 | 106.85 | 48.02 | 319 | 623 | 0.00 | 0.00 |

10 | 1 | 1802 | 0 | 3 | 1 | 21 | 1800.00 | 1800.00 | 0.00 | 401.91 | 13,982 | 7431 | 0.04 | 0.08 |

10 | 5 | 1802 | 13 | 16 | 29 | 30 | 1800.00 | 725.37 | 568.29 | 872.54 | 2858 | 2600 | 0.01 | 0.01 |

10 | 10 | 1802 | 30 | 28 | 30 | 30 | 201.32 | 10.29 | 327.78 | 451.48 | 680 | 845 | 0.00 | 0.01 |

20 | 1 | 3605 | 0 | 0 | 0 | 10 | 1800.00 | 1800.00 | 0.00 | 0.00 | 23,449 | 6856 | - | 0.14 |

20 | 5 | 3605 | 2 | 6 | 14 | 30 | 1800.00 | 1800.00 | 85.05 | 629.36 | 6479 | 4009 | 0.07 | 0.05 |

20 | 10 | 3605 | 22 | 18 | 30 | 30 | 1038.91 | 116.59 | 569.76 | 862.58 | 1507 | 1708 | 0.01 | 0.01 |

50 | 1 | 9041 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 6856 | 4528 | - | - |

50 | 5 | 9041 | 0 | 1 | 0 | 23 | 1800.00 | 1800.00 | 0.00 | 308.85 | 15,048 | 3971 | - | 0.11 |

50 | 10 | 9041 | 1 | 7 | 4 | 30 | 1800.00 | 1800.00 | 123.65 | 585.67 | 6160 | 3202 | 0.18 | 0.03 |

100 | 1 | 18,078 | 0 | 0 | 0 | 0 | - | 1800.00 | - | 0.00 | 0 | 5698 | - | - |

100 | 5 | 18,086 | 0 | 0 | 0 | 10 | 1800.00 | 1800.00 | 0.00 | 0.00 | 18,944 | 5630 | - | 0.08 |

100 | 10 | 18,086 | 0 | 2 | 0 | 25 | 1800.00 | 1800.00 | 0.00 | 393.05 | 10,750 | 3536 | - | 0.06 |

**Table A2.**EVS-SOC-GLIN results for solving the static model versus B & C with ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ and ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ based on the original ${P}_{v}^{max}$ functions and ${P}^{\mathrm{gridmax}}=25n$.

n | $\mathbf{\Delta}\mathit{t}$ (min) | ${\mathit{n}}_{\mathbf{seg}}$ | ${\mathit{n}}_{\mathbf{opt}}$ | ${\mathit{n}}_{\mathbf{feas}}$ | Runtime (s) | ${\mathit{n}}_{\mathbf{cuts}}$ | %-gap | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Median | StdDev | Median | StdDev | Median | ||||||||||

Mean | Static | B & C | Static | B & C | Static | B & C | Static | B & C | B & C | Static | B & C | |||

${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ | ||||||||||||||

5 | 1 | 155 | 12 | 5 | 29 | 30 | 1800.00 | 1800.00 | 690.82 | 636.49 | 3184 | 3153 | 0.02 | 0.06 |

5 | 5 | 139 | 28 | 26 | 30 | 30 | 25.52 | 6.68 | 450.62 | 616.54 | 422 | 461 | 0.01 | 0.01 |

5 | 10 | 119 | 30 | 29 | 30 | 30 | 1.27 | 1.62 | 10.37 | 329.09 | 153 | 154 | 0.01 | 0.01 |

10 | 1 | 312 | 1 | 0 | 20 | 23 | 1800.00 | 1800.00 | 29.78 | 0.00 | 7298 | 2829 | 0.10 | 0.12 |

10 | 5 | 279 | 29 | 19 | 30 | 30 | 183.39 | 770.59 | 445.94 | 815.87 | 1132 | 771 | 0.01 | 0.01 |

10 | 10 | 242 | 30 | 29 | 30 | 30 | 17.87 | 11.88 | 163.75 | 437.28 | 452 | 223 | 0.01 | 0.01 |

20 | 1 | 612 | 0 | 0 | 4 | 3 | 1800.00 | 1800.00 | 0.00 | 0.00 | 11,938 | 2372 | 0.26 | 0.28 |

20 | 5 | 553 | 14 | 6 | 30 | 30 | 1800.00 | 1800.00 | 608.45 | 637.22 | 2702 | 936 | 0.01 | 0.05 |

20 | 10 | 475 | 29 | 22 | 30 | 30 | 60.59 | 201.06 | 422.28 | 755.21 | 967 | 359 | 0.01 | 0.01 |

50 | 1 | 1544 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 22,034 | 4220 | - | - |

50 | 5 | 1393 | 2 | 1 | 29 | 30 | 1800.00 | 1800.00 | 160.23 | 280.55 | 6997 | 1257 | 0.08 | 0.11 |

50 | 10 | 1192 | 20 | 5 | 30 | 30 | 902.21 | 1800.00 | 698.40 | 604.91 | 2575 | 653 | 0.01 | 0.03 |

100 | 1 | 3095 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 29,193 | 6236 | - | - |

100 | 5 | 2796 | 0 | 0 | 14 | 7 | 1800.00 | 1800.00 | 0.00 | 0.00 | 11,737 | 1494 | 0.12 | 0.18 |

100 | 10 | 2399 | 6 | 0 | 30 | 30 | 1800.00 | 1800.00 | 482.70 | 0.00 | 5340 | 1077 | 0.03 | 0.06 |

${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ | ||||||||||||||

5 | 1 | 901 | 1 | 1 | 9 | 25 | 1800.00 | 1800.00 | 274.48 | 138.10 | 15,258 | 9382 | 0.21 | 0.20 |

5 | 5 | 901 | 26 | 17 | 30 | 30 | 448.47 | 761.59 | 644.73 | 831.42 | 2153 | 2330 | 0.01 | 0.01 |

5 | 10 | 901 | 29 | 26 | 30 | 30 | 56.12 | 16.43 | 321.42 | 610.48 | 866 | 990 | 0.00 | 0.01 |

10 | 1 | 1802 | 0 | 0 | 1 | 18 | 1800.00 | 1800.00 | 0.00 | 0.00 | 23,328 | 9977 | 0.23 | 0.33 |

10 | 5 | 1802 | 12 | 7 | 26 | 30 | 1800.00 | 1800.00 | 580.44 | 699.52 | 5220 | 3467 | 0.04 | 0.06 |

10 | 10 | 1802 | 29 | 22 | 30 | 30 | 204.26 | 233.60 | 417.05 | 757.12 | 2063 | 1389 | 0.01 | 0.01 |

20 | 1 | 3605 | 0 | 0 | 0 | 2 | 1800.00 | 1800.00 | 0.00 | 0.00 | 17,970 | 9466 | - | 0.32 |

20 | 5 | 3605 | 1 | 1 | 15 | 29 | 1800.00 | 1800.00 | 113.34 | 318.65 | 10,784 | 4058 | 0.08 | 0.12 |

20 | 10 | 3605 | 20 | 10 | 29 | 30 | 1097.26 | 1800.00 | 573.95 | 709.09 | 4647 | 2500 | 0.01 | 0.03 |

50 | 1 | 9041 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 23,986 | 9245 | - | - |

50 | 5 | 9041 | 0 | 0 | 0 | 17 | 1800.00 | 1800.00 | 0.00 | 0.00 | 23,708 | 5721 | - | 0.18 |

50 | 10 | 9041 | 0 | 3 | 16 | 28 | 1800.00 | 1800.00 | 0.00 | 439.63 | 12,160 | 4186 | 0.04 | 0.08 |

100 | 1 | 18,086 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 0 | 4697 | - | - |

100 | 5 | 18,086 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 25,754 | 8585 | - | - |

100 | 10 | 18,086 | 0 | 0 | 0 | 19 | 1800.00 | 1800.00 | 0.00 | 0.00 | 19,752 | 5121 | - | 0.09 |

**Table A3.**EVS-SOC-GLIN results for solving the static model versus B & C with ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ and ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ based on the original ${P}_{v}^{max}$ functions and ${P}^{\mathrm{gridmax}}=40n$.

n | $\mathbf{\Delta}\mathit{t}$ (min) | ${\mathit{n}}_{\mathbf{seg}}$ | ${\mathit{n}}_{\mathbf{opt}}$ | ${\mathit{n}}_{\mathbf{feas}}$ | Runtime (s) | ${\mathit{n}}_{\mathbf{cuts}}$ | %-gap | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Median | StdDev | Median | StdDev | Median | ||||||||||

Mean | Static | B & C | Static | B & C | Static | B & C | Static | B & C | B & C | Static | B & C | |||

${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ | ||||||||||||||

5 | 1 | 155 | 11 | 2 | 29 | 29 | 1800.00 | 1800.00 | 542.57 | 57.30 | 4476 | 2923 | 0.04 | 0.15 |

5 | 5 | 139 | 28 | 24 | 30 | 30 | 31.04 | 55.93 | 513.40 | 715.78 | 619 | 492 | 0.01 | 0.01 |

5 | 10 | 119 | 30 | 29 | 30 | 30 | 2.49 | 4.05 | 53.44 | 371.17 | 247 | 153 | 0.01 | 0.01 |

10 | 1 | 311 | 0 | 0 | 20 | 20 | 1800.00 | 1800.00 | 0.00 | 0.00 | 8161 | 3130 | 0.21 | 0.17 |

10 | 5 | 279 | 21 | 8 | 30 | 30 | 301.14 | 1800.00 | 745.75 | 677.68 | 1410 | 676 | 0.01 | 0.03 |

10 | 10 | 242 | 28 | 26 | 30 | 30 | 27.80 | 36.06 | 450.92 | 660.00 | 456 | 201 | 0.01 | 0.01 |

20 | 1 | 612 | 0 | 0 | 2 | 1 | 1800.00 | 1800.00 | 0.00 | 0.00 | 13,361 | 2440 | 0.27 | 0.48 |

20 | 5 | 553 | 5 | 0 | 30 | 30 | 1800.00 | 1800.00 | 365.20 | 0.00 | 2863 | 884 | 0.04 | 0.10 |

20 | 10 | 475 | 28 | 19 | 30 | 30 | 69.51 | 571.16 | 479.77 | 745.04 | 1078 | 327 | 0.01 | 0.01 |

50 | 1 | 1544 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 25,908 | 3569 | - | - |

50 | 5 | 1393 | 0 | 0 | 28 | 28 | 1800.00 | 1800.00 | 0.00 | 0.00 | 7110 | 1096 | 0.12 | 0.21 |

50 | 10 | 1192 | 18 | 1 | 30 | 30 | 1097.80 | 1800.00 | 640.13 | 183.90 | 2748 | 520 | 0.01 | 0.05 |

100 | 1 | 3095 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 29,066 | 6072 | - | - |

100 | 5 | 2796 | 0 | 0 | 7 | 2 | 1800.00 | 1800.00 | 0.00 | 0.00 | 11,782 | 1239 | 0.22 | 0.21 |

100 | 10 | 2399 | 1 | 0 | 29 | 30 | 1800.00 | 1800.00 | 121.93 | 0.00 | 5650 | 808 | 0.06 | 0.10 |

${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ | ||||||||||||||

5 | 1 | 901 | 2 | 0 | 9 | 24 | 1800.00 | 1800.00 | 261.72 | 0.00 | 20,190 | 9588 | 0.23 | 0.44 |

5 | 5 | 901 | 25 | 9 | 30 | 30 | 582.18 | 1800.00 | 651.87 | 643.80 | 3180 | 2231 | 0.01 | 0.07 |

5 | 10 | 901 | 30 | 23 | 30 | 30 | 80.12 | 34.07 | 160.32 | 753.56 | 1228 | 955 | 0.00 | 0.01 |

10 | 1 | 1802 | 0 | 0 | 1 | 13 | 1800.00 | 1800.00 | 0.00 | 0.00 | 24,450 | 8643 | 0.49 | 0.77 |

10 | 5 | 1802 | 12 | 0 | 26 | 30 | 1800.00 | 1800.00 | 598.34 | 0.00 | 6026 | 3161 | 0.02 | 0.17 |

10 | 10 | 1802 | 29 | 17 | 30 | 30 | 245.17 | 1147.26 | 375.49 | 837.79 | 2161 | 1553 | 0.01 | 0.01 |

20 | 1 | 3605 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 17,460 | 9716 | - | - |

20 | 5 | 3605 | 0 | 0 | 15 | 29 | 1800.00 | 1800.00 | 0.00 | 0.00 | 13,276 | 3457 | 0.14 | 0.22 |

20 | 10 | 3605 | 19 | 3 | 29 | 30 | 1437.18 | 1800.00 | 550.74 | 447.72 | 5692 | 2190 | 0.01 | 0.08 |

50 | 1 | 9041 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 12,253 | 7961 | - | - |

50 | 5 | 9041 | 0 | 0 | 0 | 11 | 1800.00 | 1800.00 | 0.00 | 0.00 | 27,617 | 4805 | - | 0.21 |

50 | 10 | 9041 | 0 | 0 | 14 | 27 | 1800.00 | 1800.00 | 0.00 | 0.00 | 13,538 | 2670 | 0.10 | 0.12 |

100 | 1 | 18,083 | 0 | 0 | 0 | 0 | - | 1800.00 | - | 0.00 | 0 | 9122 | - | - |

100 | 5 | 18,086 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 31,692 | 13,113 | - | - |

100 | 10 | 18,086 | 0 | 0 | 0 | 11 | 1800.00 | 1800.00 | 0.00 | 0.00 | 23,081 | 4035 | - | 0.14 |

**Table A4.**EVS-SOC-GLIN results for solving the static model versus B & C with ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ based on five-segment piecewise linear approximations of the original ${P}_{v}^{max}$ functions, ${P}^{\mathrm{gridmax}}=25n$.

n | $\mathbf{\Delta}\mathit{t}$ (min) | ${\mathit{n}}_{\mathbf{seg}}$ | ${\mathit{n}}_{\mathbf{opt}}$ | ${\mathit{n}}_{\mathbf{feas}}$ | Runtime (s) | ${\mathit{n}}_{\mathbf{cuts}}$ | %-gap | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Median | StdDev | Median | StdDev | Median | ||||||||||

Mean | Static | B & C | Static | B & C | Static | B & C | Static | B & C | B & C | Static | B & C | |||

${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ | ||||||||||||||

5 | 1 | 40 | 29 | 22 | 30 | 30 | 60.14 | 19.63 | 394.28 | 791.01 | 387 | 485 | 0.01 | 0.01 |

5 | 5 | 46 | 30 | 30 | 30 | 30 | 2.40 | 1.98 | 5.97 | 263.17 | 88 | 102 | 0.01 | 0.01 |

5 | 10 | 43 | 30 | 30 | 30 | 30 | 0.64 | 1.13 | 1.37 | 1.21 | 42 | 50 | 0.00 | 0.01 |

10 | 1 | 80 | 27 | 13 | 30 | 30 | 509.28 | 1800.00 | 582.27 | 830.23 | 1162 | 639 | 0.01 | 0.02 |

10 | 5 | 92 | 30 | 30 | 30 | 30 | 11.01 | 8.34 | 28.13 | 224.13 | 232 | 136 | 0.01 | 0.01 |

10 | 10 | 87 | 30 | 30 | 30 | 30 | 1.49 | 2.68 | 1.78 | 8.36 | 118 | 62 | 0.01 | 0.01 |

20 | 1 | 160 | 5 | 2 | 12 | 30 | 1800.00 | 1800.00 | 193.77 | 407.09 | 2488 | 722 | 0.03 | 0.06 |

20 | 5 | 185 | 30 | 25 | 30 | 30 | 54.58 | 61.09 | 199.06 | 659.96 | 516 | 192 | 0.01 | 0.01 |

20 | 10 | 174 | 30 | 30 | 30 | 30 | 5.03 | 7.45 | 13.02 | 37.35 | 217 | 79 | 0.01 | 0.01 |

50 | 1 | 398 | 0 | 0 | 0 | 12 | 1800.00 | 1800.00 | 0.00 | 0.00 | 5598 | 796 | - | 0.24 |

50 | 5 | 459 | 28 | 10 | 30 | 30 | 640.74 | 1800.00 | 516.17 | 754.54 | 1556 | 363 | 0.01 | 0.02 |

50 | 10 | 433 | 30 | 29 | 30 | 30 | 37.23 | 36.95 | 54.09 | 379.05 | 624 | 160 | 0.01 | 0.01 |

100 | 1 | 798 | 0 | 0 | 0 | 0 | 1800.00 | 1800.00 | 0.00 | 0.00 | 9312 | 1458 | - | - |

100 | 5 | 921 | 12 | 3 | 30 | 30 | 1800.00 | 1800.00 | 466.38 | 464.39 | 3237 | 568 | 0.01 | 0.06 |

100 | 10 | 871 | 30 | 25 | 30 | 30 | 112.16 | 84.83 | 156.15 | 652.92 | 1360 | 259 | 0.01 | 0.01 |

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**Figure 1.**Maximum charging power of a Hyundai Kona Elektro in dependence of the EV’s SOC; data obtained from Fastned [15].

**Figure 5.**Optimal solution for an instance with $n=5$, $\Delta t=5$ min, ${P}^{\mathrm{gridmax}}\in \{50,125,200\}$ kW using EVS-SOC-GLIN. (

**a**) ${P}^{\mathrm{gridmax}}=200$ kW; total charging costs: 290.42 cent. (

**b**) ${P}^{\mathrm{gridmax}}=125$ kW; total charging costs: 296.91 cent. (

**c**) ${P}^{\mathrm{gridmax}}=50$ kW; total charging costs: 330.10 cent.

**Figure 6.**EVS-SOC-LIN runtime comparison for directly solving the LP problem versus the cutting plane approach corresponding to the results of Table 2.

**Figure 7.**Visualization of EVS-SOC-GLIN results for solving the static model versus B & C with ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ and ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ based on five-segment piecewise linear approximations of the original ${P}_{v}^{max}$ functions, ${P}^{\mathrm{gridmax}}=10n$. (

**a**) Number of feasible solutions found for ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$. (

**b**) Number of feasible solutions found for ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$. (

**c**) Median runtimes for ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$. (

**d**) Median runtimes for ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$. (

**e**) Median number of cuts added within B & C for ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ as well as ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$.

**Figure 8.**Visualization of EVS-SOC-GLIN results for solving the static model versus B & C with ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ and ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ based on five-segment piecewise linear approximations of the original ${P}_{v}^{max}$ functions, ${P}^{\mathrm{gridmax}}=25n$. (

**a**) Number of feasible solutions found for ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$. (

**b**) Number of feasible solutions found for ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$. (

**c**) Median runtimes for ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$. (

**d**) Median runtimes for ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$. (

**e**) Median number of cuts added within B & C for ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ as well as ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$.

**Figure 9.**Visualization of EVS-SOC-GLIN results for solving the static model versus B & C with ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ and ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ based on five-segment piecewise linear approximations of the original ${P}_{v}^{max}$ functions, ${P}^{\mathrm{gridmax}}=40n$. (

**a**) Number of feasible solutions found for ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$. (

**b**) Number of feasible solutions found for ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$. (

**c**) Median runtimes for ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$. (

**d**) Median runtimes for ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$. (

**e**) Median number of cuts added within B & C for ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ as well as ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$.

**Figure 10.**Mean charging cost gaps of EVS-SOC-LIN and EVS-SOC-GLIN with ${P}^{\mathrm{gridmax}}=40n$. Whiskers indicate the standard deviations. Note that for $n=20$ and $\Delta t=1$ only a single instance was solved to optimality, and therefore the corresponding standard deviation is zero.

**Figure 11.**Mean charging error when scheduling with convex ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ and realizing the plan with nonconvex ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ using ${P}^{\mathrm{gridmax}}=40n$. Whiskers indicate the standard deviations.

**Table 1.**Used EV types with battery capacity ${C}_{v}$, ${P}_{v}^{max}$ domain $[{s}_{v}^{\mathrm{min}},{s}_{v}^{\mathrm{max}}]$ and the number of linear pieces of ${P}_{v}^{max}$.

EV Name | ${\mathit{C}}_{\mathit{v}}$ (kWh) | ${\mathit{s}}_{\mathit{v}}^{\mathbf{min}}$ | ${\mathit{s}}_{\mathit{v}}^{\mathbf{max}}$ | #${\mathit{P}}_{\mathit{v}}^{max}$-lin. Pieces |
---|---|---|---|---|

Energica Ego | 21.5 | 1.1 | 99.9 | 53 |

MINI Cooper Electric | 32.6 | 12.1 | 93.8 | 34 |

BMW i3 | 42.2 | 15.1 | 96.0 | 26 |

Hyundai Kona Elektro | 67.5 | 10.1 | 94.9 | 28 |

Tesla Model 3 Long Range | 82.0 | 11.1 | 99.0 | 35 |

Mercedes-Benz EQC | 85.0 | 2.1 | 97.8 | 24 |

Jaguar I-Pace | 90.0 | 8.0 | 100.0 | 29 |

Audi e-tron | 95.0 | 3.1 | 99.8 | 44 |

**Table 2.**EVS-SOC-LIN runtime comparison for concave maximum power functions and ${P}^{\mathrm{gridmax}}=25n$: solving the static MILP versus the cutting plane method.

n | $\mathbf{\Delta}\mathit{t}$ (min) | ${\mathit{n}}_{\mathbf{seg}}$ | Runtime (s) | ${\mathit{n}}_{\mathbf{cuts}}$ | ||||
---|---|---|---|---|---|---|---|---|

Static | Cutting Plane | Cutting Plane | ||||||

Mean | Median | StdDev | Median | StdDev | Mean | StdDev | ||

${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ | ||||||||

5 | 1 | 49 | 0.07 | 0.04 | 1.34 | 0.26 | 10,423 | 5209 |

5 | 5 | 46 | 0.01 | 0.00 | 1.04 | 0.23 | 574 | 269 |

5 | 10 | 43 | 0.01 | 0.00 | 1.03 | 0.24 | 190 | 85 |

10 | 1 | 99 | 0.18 | 0.15 | 1.52 | 0.38 | 15,949 | 5580 |

10 | 5 | 93 | 0.02 | 0.01 | 1.05 | 0.28 | 1243 | 520 |

10 | 10 | 86 | 0.01 | 0.00 | 1.03 | 0.26 | 416 | 159 |

20 | 1 | 199 | 0.60 | 0.30 | 2.09 | 0.49 | 25,549 | 6715 |

20 | 5 | 187 | 0.05 | 0.02 | 1.10 | 0.25 | 2593 | 747 |

20 | 10 | 172 | 0.02 | 0.01 | 1.05 | 0.25 | 862 | 245 |

50 | 1 | 495 | 2.78 | 1.02 | 6.72 | 2.07 | 87,375 | 19,749 |

50 | 5 | 464 | 0.16 | 0.06 | 1.28 | 0.31 | 6499 | 1167 |

50 | 10 | 427 | 0.06 | 0.02 | 1.10 | 0.23 | 2157 | 335 |

100 | 1 | 994 | 9.34 | 2.60 | 12.84 | 3.99 | 193,069 | 27,979 |

100 | 5 | 931 | 0.56 | 0.22 | 1.68 | 0.32 | 13,502 | 1664 |

100 | 10 | 858 | 0.13 | 0.05 | 1.25 | 0.27 | 4367 | 475 |

${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ | ||||||||

5 | 1 | 901 | 1.19 | 1.02 | 1.31 | 0.38 | 12,800 | 5986 |

5 | 5 | 901 | 0.23 | 0.11 | 0.90 | 0.25 | 1102 | 542 |

5 | 10 | 901 | 0.08 | 0.07 | 0.97 | 0.26 | 322 | 205 |

10 | 1 | 1802 | 4.98 | 3.27 | 1.65 | 0.52 | 25,271 | 9541 |

10 | 5 | 1802 | 0.59 | 0.22 | 1.06 | 0.24 | 2341 | 950 |

10 | 10 | 1802 | 0.22 | 0.10 | 1.01 | 0.20 | 757 | 387 |

20 | 1 | 3605 | 14.33 | 8.48 | 3.29 | 0.83 | 60,778 | 18,725 |

20 | 5 | 3605 | 1.21 | 0.45 | 1.16 | 0.27 | 5117 | 1547 |

20 | 10 | 3605 | 0.68 | 0.20 | 1.07 | 0.21 | 1585 | 516 |

50 | 1 | 9041 | 70.69 | 31.89 | 9.11 | 2.66 | 175,979 | 28,195 |

50 | 5 | 9041 | 4.17 | 1.58 | 1.57 | 0.33 | 13,737 | 2329 |

50 | 10 | 9041 | 1.57 | 0.54 | 1.15 | 0.21 | 3989 | 858 |

100 | 1 | 18,086 | 280.22 | 100.87 | 25.45 | 9.66 | 390,873 | 44,162 |

100 | 5 | 18,086 | 13.11 | 4.73 | 2.11 | 0.51 | 27,920 | 3515 |

100 | 10 | 18,086 | 3.80 | 1.35 | 1.32 | 0.34 | 8126 | 1419 |

**Table 3.**EVS-SOC-GLIN results for solving the static model versus B & C with ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ and ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ based on the original ${P}_{v}^{max}$ functions and ${P}^{\mathrm{gridmax}}=10n$.

n | $\mathbf{\Delta}\mathit{t}$ (min) | ${\mathit{n}}_{\mathbf{seg}}$ | ${\mathit{n}}_{\mathbf{feas}}$ | Runtime (s) | ${\mathit{n}}_{\mathbf{cuts}}$ | %-gap | |||
---|---|---|---|---|---|---|---|---|---|

Mean | Median | Median | Median | ||||||

Static | B & C | Static | B & C | B & C | Static | B & C | |||

${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ | |||||||||

5 | 1 | 155 | 30 | 30 | 391.75 | 43.39 | 1038 | 0.01 | 0.01 |

5 | 5 | 139 | 30 | 30 | 6.58 | 1.43 | 144 | 0.00 | 0.01 |

5 | 10 | 119 | 30 | 30 | 1.67 | 0.83 | 56 | 0.00 | 0.00 |

10 | 1 | 311 | 21 | 29 | 1800.00 | 1800.00 | 4068 | 0.03 | 0.03 |

10 | 5 | 279 | 30 | 30 | 79.94 | 8.84 | 498 | 0.01 | 0.01 |

10 | 10 | 242 | 30 | 30 | 7.04 | 2.06 | 194 | 0.00 | 0.01 |

20 | 1 | 612 | 2 | 11 | 1800.00 | 1800.00 | 8974 | 0.08 | 0.19 |

20 | 5 | 553 | 30 | 30 | 500.49 | 684.63 | 1846 | 0.01 | 0.01 |

20 | 10 | 475 | 30 | 30 | 40.18 | 13.35 | 505 | 0.01 | 0.01 |

50 | 1 | 1544 | 0 | 0 | 1800.00 | 1800.00 | 15,910 | - | - |

50 | 5 | 1393 | 26 | 30 | 1800.00 | 1800.00 | 6106 | 0.05 | 0.05 |

50 | 10 | 1192 | 30 | 30 | 307.62 | 827.59 | 1930 | 0.01 | 0.01 |

100 | 1 | 3095 | 0 | 0 | 1800.00 | 1800.00 | 11,886 | - | - |

100 | 5 | 2796 | 9 | 9 | 1800.00 | 1800.00 | 9961 | 0.08 | 0.12 |

100 | 10 | 2399 | 30 | 30 | 1800.00 | 1800.00 | 4434 | 0.01 | 0.03 |

${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ | |||||||||

5 | 1 | 901 | 8 | 27 | 1800.00 | 1800.00 | 5304 | 0.03 | 0.01 |

5 | 5 | 901 | 30 | 30 | 143.42 | 9.59 | 820 | 0.00 | 0.00 |

5 | 10 | 901 | 30 | 30 | 34.53 | 2.60 | 319 | 0.00 | 0.00 |

10 | 1 | 1802 | 1 | 21 | 1800.00 | 1800.00 | 13,982 | 0.04 | 0.08 |

10 | 5 | 1802 | 29 | 30 | 1800.00 | 725.37 | 2858 | 0.01 | 0.01 |

10 | 10 | 1802 | 30 | 30 | 201.32 | 10.29 | 680 | 0.00 | 0.01 |

20 | 1 | 3605 | 0 | 10 | 1800.00 | 1800.00 | 23,449 | - | 0.14 |

20 | 5 | 3605 | 14 | 30 | 1800.00 | 1800.00 | 6479 | 0.07 | 0.05 |

20 | 10 | 3605 | 30 | 30 | 1038.91 | 116.59 | 1507 | 0.01 | 0.01 |

50 | 1 | 9041 | 0 | 0 | 1800.00 | 1800.00 | 6856 | - | - |

50 | 5 | 9041 | 0 | 23 | 1800.00 | 1800.00 | 15,048 | - | 0.11 |

50 | 10 | 9041 | 4 | 30 | 1800.00 | 1800.00 | 6160 | 0.18 | 0.03 |

100 | 1 | 18,078 | 0 | 0 | - | 1800.00 | 0 | - | - |

100 | 5 | 18,086 | 0 | 10 | 1800.00 | 1800.00 | 18,944 | - | 0.08 |

100 | 10 | 18,086 | 0 | 25 | 1800.00 | 1800.00 | 10,750 | - | 0.06 |

**Table 4.**EVS-SOC-GLIN results for solving the static model versus B & C with ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ and ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ based on the original ${P}_{v}^{max}$ functions and ${P}^{\mathrm{gridmax}}=25n$.

n | $\mathbf{\Delta}\mathit{t}$ (min) | ${\mathit{n}}_{\mathbf{seg}}$ | ${\mathit{n}}_{\mathbf{feas}}$ | Runtime (s) | ${\mathit{n}}_{\mathbf{cuts}}$ | %-gap | |||
---|---|---|---|---|---|---|---|---|---|

Mean | Median | Median | Median | ||||||

Static | B & C | Static | B & C | B & C | Static | B & C | |||

${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ | |||||||||

5 | 1 | 155 | 29 | 30 | 1800.00 | 1800.00 | 3184 | 0.02 | 0.06 |

5 | 5 | 139 | 30 | 30 | 25.52 | 6.68 | 422 | 0.01 | 0.01 |

5 | 10 | 119 | 30 | 30 | 1.27 | 1.62 | 153 | 0.01 | 0.01 |

10 | 1 | 312 | 20 | 23 | 1800.00 | 1800.00 | 7298 | 0.10 | 0.12 |

10 | 5 | 279 | 30 | 30 | 183.39 | 770.59 | 1132 | 0.01 | 0.01 |

10 | 10 | 242 | 30 | 30 | 17.87 | 11.88 | 452 | 0.01 | 0.01 |

20 | 1 | 612 | 4 | 3 | 1800.00 | 1800.00 | 11,938 | 0.26 | 0.28 |

20 | 5 | 553 | 30 | 30 | 1800.00 | 1800.00 | 2702 | 0.01 | 0.05 |

20 | 10 | 475 | 30 | 30 | 60.59 | 201.06 | 967 | 0.01 | 0.01 |

50 | 1 | 1544 | 0 | 0 | 1800.00 | 1800.00 | 22,034 | - | - |

50 | 5 | 1393 | 29 | 30 | 1800.00 | 1800.00 | 6997 | 0.08 | 0.11 |

50 | 10 | 1192 | 30 | 30 | 902.21 | 1800.00 | 2575 | 0.01 | 0.03 |

100 | 1 | 3095 | 0 | 0 | 1800.00 | 1800.00 | 29,193 | - | - |

100 | 5 | 2796 | 14 | 7 | 1800.00 | 1800.00 | 11,737 | 0.12 | 0.18 |

100 | 10 | 2399 | 30 | 30 | 1800.00 | 1800.00 | 5340 | 0.03 | 0.06 |

${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ | |||||||||

5 | 1 | 901 | 9 | 25 | 1800.00 | 1800.00 | 15,258 | 0.21 | 0.20 |

5 | 5 | 901 | 30 | 30 | 448.47 | 761.59 | 2153 | 0.01 | 0.01 |

5 | 10 | 901 | 30 | 30 | 56.12 | 16.43 | 866 | 0.00 | 0.01 |

10 | 1 | 1802 | 1 | 18 | 1800.00 | 1800.00 | 23,328 | 0.23 | 0.33 |

10 | 5 | 1802 | 26 | 30 | 1800.00 | 1800.00 | 5220 | 0.04 | 0.06 |

10 | 10 | 1802 | 30 | 30 | 204.26 | 233.60 | 2063 | 0.01 | 0.01 |

20 | 1 | 3605 | 0 | 2 | 1800.00 | 1800.00 | 17,970 | - | 0.32 |

20 | 5 | 3605 | 15 | 29 | 1800.00 | 1800.00 | 10,784 | 0.08 | 0.12 |

20 | 10 | 3605 | 29 | 30 | 1097.26 | 1800.00 | 4647 | 0.01 | 0.03 |

50 | 1 | 9041 | 0 | 0 | 1800.00 | 1800.00 | 23,986 | - | - |

50 | 5 | 9041 | 0 | 17 | 1800.00 | 1800.00 | 23,708 | - | 0.18 |

50 | 10 | 9041 | 16 | 28 | 1800.00 | 1800.00 | 12,160 | 0.04 | 0.08 |

100 | 1 | 18,086 | 0 | 0 | 1800.00 | 1800.00 | 0 | - | - |

100 | 5 | 18,086 | 0 | 0 | 1800.00 | 1800.00 | 25,754 | - | - |

100 | 10 | 18,086 | 0 | 19 | 1800.00 | 1800.00 | 19,752 | - | 0.09 |

**Table 5.**EVS-SOC-GLIN results for solving the static model versus B & C with ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ and ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ based on the original ${P}_{v}^{max}$ functions and ${P}^{\mathrm{gridmax}}=40n$.

n | $\mathbf{\Delta}\mathit{t}$ (min) | ${\mathit{n}}_{\mathbf{seg}}$ | ${\mathit{n}}_{\mathbf{feas}}$ | Runtime (s) | ${\mathit{n}}_{\mathbf{cuts}}$ | %-gap | |||
---|---|---|---|---|---|---|---|---|---|

Mean | Median | Median | Median | ||||||

Static | B & C | Static | B & C | B & C | Static | B & C | |||

${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ | |||||||||

5 | 1 | 155 | 29 | 29 | 1800.00 | 1800.00 | 4476 | 0.04 | 0.15 |

5 | 5 | 139 | 30 | 30 | 31.04 | 55.93 | 619 | 0.01 | 0.01 |

5 | 10 | 119 | 30 | 30 | 2.49 | 4.05 | 247 | 0.01 | 0.01 |

10 | 1 | 311 | 20 | 20 | 1800.00 | 1800.00 | 8161 | 0.21 | 0.17 |

10 | 5 | 279 | 30 | 30 | 301.14 | 1800.00 | 1410 | 0.01 | 0.03 |

10 | 10 | 242 | 30 | 30 | 27.80 | 36.06 | 456 | 0.01 | 0.01 |

20 | 1 | 612 | 2 | 1 | 1800.00 | 1800.00 | 13,361 | 0.27 | 0.48 |

20 | 5 | 553 | 30 | 30 | 1800.00 | 1800.00 | 2863 | 0.04 | 0.10 |

20 | 10 | 475 | 30 | 30 | 69.51 | 571.16 | 1078 | 0.01 | 0.01 |

50 | 1 | 1544 | 0 | 0 | 1800.00 | 1800.00 | 25,908 | - | - |

50 | 5 | 1393 | 28 | 28 | 1800.00 | 1800.00 | 7110 | 0.12 | 0.21 |

50 | 10 | 1192 | 30 | 30 | 1097.80 | 1800.00 | 2748 | 0.01 | 0.05 |

100 | 1 | 3095 | 0 | 0 | 1800.00 | 1800.00 | 29,066 | - | - |

100 | 5 | 2796 | 7 | 2 | 1800.00 | 1800.00 | 11,782 | 0.22 | 0.21 |

100 | 10 | 2399 | 29 | 30 | 1800.00 | 1800.00 | 5650 | 0.06 | 0.10 |

${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ | |||||||||

5 | 1 | 901 | 9 | 24 | 1800.00 | 1800.00 | 20,190 | 0.23 | 0.44 |

5 | 5 | 901 | 30 | 30 | 582.18 | 1800.00 | 3180 | 0.01 | 0.07 |

5 | 10 | 901 | 30 | 30 | 80.12 | 34.07 | 1228 | 0.00 | 0.01 |

10 | 1 | 1802 | 1 | 13 | 1800.00 | 1800.00 | 24,450 | 0.49 | 0.77 |

10 | 5 | 1802 | 26 | 30 | 1800.00 | 1800.00 | 6026 | 0.02 | 0.17 |

10 | 10 | 1802 | 30 | 30 | 245.17 | 1147.26 | 2161 | 0.01 | 0.01 |

20 | 1 | 3605 | 0 | 0 | 1800.00 | 1800.00 | 17,460 | - | - |

20 | 5 | 3605 | 15 | 29 | 1800.00 | 1800.00 | 13,276 | 0.14 | 0.22 |

20 | 10 | 3605 | 29 | 30 | 1437.18 | 1800.00 | 5692 | 0.01 | 0.08 |

50 | 1 | 9041 | 0 | 0 | 1800.00 | 1800.00 | 12,253 | - | - |

50 | 5 | 9041 | 0 | 11 | 1800.00 | 1800.00 | 27,617 | - | 0.21 |

50 | 10 | 9041 | 14 | 27 | 1800.00 | 1800.00 | 13,538 | 0.10 | 0.12 |

100 | 1 | 18,083 | 0 | 0 | - | 1800.00 | 0 | - | - |

100 | 5 | 18,086 | 0 | 0 | 1800.00 | 1800.00 | 31,692 | - | - |

100 | 10 | 18,086 | 0 | 11 | 1800.00 | 1800.00 | 23,081 | - | 0.14 |

**Table 6.**EVS-SOC-GLIN results for solving the static model versus B & C with ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ based on five-segment piecewise linear approximations of the original ${P}_{v}^{max}$ functions, ${P}^{\mathrm{gridmax}}=25n$.

n | $\mathbf{\Delta}\mathit{t}$ (min) | ${\mathit{n}}_{\mathbf{seg}}$ | ${\mathit{n}}_{\mathbf{feas}}$ | Runtime (s) | ${\mathit{n}}_{\mathbf{cuts}}$ | %-gap | |||
---|---|---|---|---|---|---|---|---|---|

Mean | Median | Median | Median | ||||||

Static | B & C | Static | B & C | B & C | Static | B & C | |||

${E}_{v}^{\mathrm{max}-\mathrm{lb}}$ | |||||||||

5 | 1 | 40 | 30 | 30 | 60.14 | 19.63 | 387 | 0.01 | 0.01 |

5 | 5 | 46 | 30 | 30 | 2.40 | 1.98 | 88 | 0.01 | 0.01 |

5 | 10 | 43 | 30 | 30 | 0.64 | 1.13 | 42 | 0.00 | 0.01 |

10 | 1 | 80 | 30 | 30 | 509.28 | 1800.00 | 1162 | 0.01 | 0.02 |

10 | 5 | 92 | 30 | 30 | 11.01 | 8.34 | 232 | 0.01 | 0.01 |

10 | 10 | 87 | 30 | 30 | 1.49 | 2.68 | 118 | 0.01 | 0.01 |

20 | 1 | 160 | 12 | 30 | 1800.00 | 1800.00 | 2488 | 0.03 | 0.06 |

20 | 5 | 185 | 30 | 30 | 54.58 | 61.09 | 516 | 0.01 | 0.01 |

20 | 10 | 174 | 30 | 30 | 5.03 | 7.45 | 217 | 0.01 | 0.01 |

50 | 1 | 398 | 0 | 12 | 1800.00 | 1800.00 | 5598 | - | 0.24 |

50 | 5 | 459 | 30 | 30 | 640.74 | 1800.00 | 1556 | 0.01 | 0.02 |

50 | 10 | 433 | 30 | 30 | 37.23 | 36.95 | 624 | 0.01 | 0.01 |

100 | 1 | 798 | 0 | 0 | 1800.00 | 1800.00 | 9312 | - | - |

100 | 5 | 921 | 30 | 30 | 1800.00 | 1800.00 | 3237 | 0.01 | 0.06 |

100 | 10 | 871 | 30 | 30 | 112.16 | 84.83 | 1360 | 0.01 | 0.01 |

**Table 7.**Objective value comparison using EVS-SOC-GLIN and different ${E}_{v}^{max}$ functions based on the five-segment ${P}_{v}^{max}$ approximation and the original ${P}_{v}^{max}$; ${P}^{\mathrm{gridmax}}=40n$.

n | $\mathbf{\Delta}\mathit{t}$ (min) | ${\mathit{n}}_{\mathbf{opt}}$ | Charging Costs | |||
---|---|---|---|---|---|---|

${\mathit{E}}_{\mathit{v}}^{\mathbf{max}-\mathbf{lb}}$ | ${\mathit{E}}_{\mathit{v}}^{\mathbf{max}-\mathbf{ex}}$ | %-gap | ||||

Mean | Mean | Mean | StdDev | |||

Original ${P}_{v}^{max}$ | ||||||

5 | 1 | 2 | 109.08 | 108.97 | 0.10 | 0.01 |

5 | 5 | 25 | 209.40 | 208.83 | 0.29 | 0.18 |

5 | 10 | 30 | 227.10 | 225.78 | 0.64 | 0.40 |

10 | 5 | 11 | 374.24 | 372.98 | 0.34 | 0.13 |

10 | 10 | 28 | 447.51 | 445.05 | 0.59 | 0.35 |

20 | 10 | 19 | 882.53 | 877.33 | 0.60 | 0.30 |

5-segment approx. ${P}_{v}^{max}$ | ||||||

5 | 1 | 2 | 109.10 | 108.98 | 0.10 | 0.01 |

5 | 5 | 25 | 209.38 | 208.82 | 0.29 | 0.17 |

5 | 10 | 30 | 227.11 | 225.77 | 0.64 | 0.41 |

10 | 5 | 11 | 374.14 | 372.92 | 0.33 | 0.13 |

10 | 10 | 28 | 447.44 | 445.04 | 0.57 | 0.32 |

20 | 10 | 19 | 882.39 | 877.26 | 0.60 | 0.30 |

**Table 8.**Charging error comparison when scheduling with ${E}_{v}^{\mathrm{max}-\mathrm{ex}}$ using EVS-SOC-GLIN and realizing the schedule with ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$; ${P}^{\mathrm{gridmax}}=40n$.

n | $\mathbf{\Delta}\mathit{t}$ (min) | ${\mathit{n}}_{\mathbf{opt}}$ | Mean Charging Error (% SOC) | |||
---|---|---|---|---|---|---|

Original ${\mathit{P}}_{\mathit{v}}^{max}$ | 5-seg. Approx. ${\mathit{P}}_{\mathit{v}}^{max}$ | |||||

Mean | StdDev | Mean | StdDev | |||

5 | 1 | 3 | 0.23 | 0.08 | 0.21 | 0.08 |

5 | 5 | 25 | 1.14 | 0.26 | 1.06 | 0.28 |

5 | 10 | 30 | 2.01 | 0.58 | 1.94 | 0.60 |

10 | 5 | 12 | 1.14 | 0.16 | 1.18 | 0.18 |

10 | 10 | 29 | 2.03 | 0.45 | 2.03 | 0.46 |

20 | 10 | 20 | 2.01 | 0.29 | 1.97 | 0.34 |

**Table 9.**Rolling horizon charging cost difference for EVS-SOC-LIN vs. EVS-SOC-GLIN using ${E}_{v}^{\mathrm{max}-\mathrm{lb}}$; ${P}^{\mathrm{gridmax}}=40n$.

n | $\mathbf{\Delta}\mathit{t}$ (min) | Charging Cost Difference | |||
---|---|---|---|---|---|

Absolute (Cent) | Relative (%) | ||||

Mean | StdDev | Mean | StdDev | ||

5 | 5 | 0.97 | 0.73 | 0.22 | 0.16 |

5 | 10 | 0.91 | 0.60 | 0.20 | 0.12 |

10 | 5 | 1.75 | 0.99 | 0.20 | 0.11 |

10 | 10 | 1.78 | 0.77 | 0.20 | 0.08 |

20 | 5 | 3.78 | 1.34 | 0.21 | 0.08 |

20 | 10 | 3.80 | 1.03 | 0.21 | 0.06 |

50 | 5 | 9.14 | 2.42 | 0.20 | 0.05 |

50 | 10 | 9.39 | 2.64 | 0.21 | 0.06 |

100 | 5 | 24.42 | 2.40 | 0.27 | 0.03 |

100 | 10 | 19.96 | 4.82 | 0.22 | 0.05 |

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**MDPI and ACS Style**

Schaden, B.; Jatschka, T.; Limmer, S.; Raidl, G.R.
Smart Charging of Electric Vehicles Considering SOC-Dependent Maximum Charging Powers. *Energies* **2021**, *14*, 7755.
https://doi.org/10.3390/en14227755

**AMA Style**

Schaden B, Jatschka T, Limmer S, Raidl GR.
Smart Charging of Electric Vehicles Considering SOC-Dependent Maximum Charging Powers. *Energies*. 2021; 14(22):7755.
https://doi.org/10.3390/en14227755

**Chicago/Turabian Style**

Schaden, Benjamin, Thomas Jatschka, Steffen Limmer, and Günther Robert Raidl.
2021. "Smart Charging of Electric Vehicles Considering SOC-Dependent Maximum Charging Powers" *Energies* 14, no. 22: 7755.
https://doi.org/10.3390/en14227755