# Measurement-Based Current-Harmonics Modeling of Aggregated Electric-Vehicle Loads Using Power-Exponential Functions

## Abstract

**:**

## 1. Introduction

_{2}[1]. In order to meet the goals of the Paris Declaration on Electromobility and Climate Change that set the requirements that at least 20% of all road transport vehicles globally need to be electrically driven by 2030, EVs play a major role in the future of sustainable transport systems, and the global proliferation of EVs is necessary [2].

_{v}) would be very low. A method for predicting harmonic currents generated by a concentration of EVs was presented in [19]. The authors’ analysis showed that the net produced harmonic current was not proportional to the number of EVs being charged, that is, the current-harmonics magnitude sum was far less than the total value of the current harmonics emitted by each charger. In [20], the integration of EVs in the Portuguese distribution grid was analyzed and showed that the simultaneous operation of multiple one-phase chargers in the same LV feeder could significantly raise current-harmonic levels. Simulations made in [21], where an EV charging station was modeled, showed that permissible current-harmonic limits were exceeded when six or more vehicles were simultaneously charging. In [22,23], it was shown that distortion limits would be exceeded when charging in relatively weak grids. The combination of four different EVs, and what their cancellation effect does to phase-angle diversity, was analyzed in [24]. Analysis was made during constant-charging mode, during which the phase angles were assumed to be constant. The results show significant harmonic cancellation. The same authors simulated a distribution network supplied by EV charging load, and the result indicate a minor increase in harmonic currents due to EV charging [13]. Kim et al. [25] showed that the low penetration and slow charging rate of EVs slightly impacted network harmonic distortion. However, the high penetration of EVs and fast charging rates may result in considerable voltage- and current-harmonic distortion [25,26]. Deilami et al. [27] established that randomly charging EVs could violate the standard level of voltage harmonics.

_{i}(total harmonic distortion of current), but had low impact on THD

_{v}, while moderate-to-high EV penetration was shown to have an adverse impact on both THD

_{i}and THD

_{v}. With wind generation included in the simulations, THD could either be reduced or cause an increase; hence, the size and placement of wind generation is essential in order to manage THD levels. In [30], the interaction between EVs and other loads (foremost heat pumps) was analyzed. The study showed that the level of cancellation depends on the considered harmonic order, EV operating state, the phase angles of the background harmonic voltages, and the harmonic currents emitted by other loads.

_{n}= 230 V); hence, on the assumption of nonvarying voltage supply. This was justified by the measured voltage levels and individual voltage harmonics. This is, of course, a simplification, since harmonics are affected by nearby loads and distortion in the supply voltage. It can, therefore, be argued that the modeling is only applicable in certain grid configurations. However, as an initial work that will be followed by a more exhaustive model (i.e., coupled Norton model) where these factors are considered, it is still justified because of the originality of the model construction.

## 2. Theoretical Background

#### 2.1. Electric Vehicles and Harmonics

_{h}is the RMS value of the current h-th harmonic component, and I

_{1}is the current RMS value of the main frequency. THD

_{v}can be similarly calculated, but with the current values in Equation (1) replaced with respective voltage values. THD

_{i}does not always reflects the actual harmonic content. For example, if the fundamental current decreases while the harmonic content remains the same, the THD

_{i}value is higher, indicating higher emissions, but not reflecting the adverse impact on the grid. A more suitable way to express distortion in terms of current harmonics is the total harmonic current (THC):

_{h}is the RMS value of each harmonic order, and h is the harmonic order.

#### 2.2. Harmonic Phasor Aggregation

## 3. Measurement Setup

_{C,k}and phase angle φ

_{k}. In a 50 Hz network, harmonic component Y

_{H,h}is identical to spectral component Y

_{C,k}, with k = h × 10, and is represented by harmonic phasor

## 4. Modeling Harmonic Currents Using Power Exponential Functions

_{n}marginally improves the model, but at the expense of computational effort.

_{1}are set to zero, that is, before the vehicle starts to charge. In many cases, choosing suitable intervals for one individual harmonic gives a set of suitable points for most or all other harmonics. The second step is then separately fitting the model in each interval. For this application, a least-squares fitting procedure was applied, and, since the model was nonlinear with respect to the parameters, a numerical procedure for finding the fitting was used. This was conducted with software [54] that uses an interior reflective algorithm based on [55,56].

- inspection of measured value (imaginary and real part is done separately);
- manually choosing moments in time when peaks occur;
- choosing initial values for p
_{1}, p_{2}, and p_{3}(or q_{1}, q_{2}and q_{3}); - applying Function (7) or (8) between peaks;
- least-squares fitting procedure (MATLAB) to find values of p
_{1}, p_{2}and p_{3}(or q_{1}, q_{2}and q_{3}) for a function that best fits measured values between peaks; - procedure is repeated until all peaks are handled. Functions between each peak are acquired, and a piecewise combination of the functions is scaled and translated, so that the resulting function is continuous.

#### Aggregation of Multiple Electric-Vehicle Loads

## 5. Results

## 6. Discussion and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Measurement of fifth harmonic-current phase angle (I5 leads Up1 α5 > 0) [39].

**Figure 4.**Third current harmonic during electric-vehicle (EV) charging of (

**a**) EV1, (

**b**) EV2, (

**c**) EV3, and (

**d**) EV4.

**Figure 5.**Illustration of how steepness and slopes of power exponential function varies with p [51].

**Figure 6.**Some examples of what $g\left(t;{t}_{1},{t}_{2},{p}_{1},{p}_{2},{p}_{3}\right)$ function can look like in interval $\left[{t}_{1},{t}_{2}\right]$.

**Figure 7.**Example of third current harmonic during (blue) charging of EV1 with (orange) fitted function.

**Figure 8.**Third current-harmonic simulation of aggregated electric vehicle loads (100,000 random simulations). Current magnitude and phase angle during circumstances when maximal current-harmonic magnitude was obtained. Maximal number of simultaneous charging vehicles was set to six. Polar plot visualizes how magnitude and phase angle continuously changed over time.

**Table 1.**Technical and measurement data of vehicles included in the study. Note: THC, total harmonic current.

EV 1 | EV 2 | EV 3 | EV 4 | ||
---|---|---|---|---|---|

Battery capacity (kWh) | 18.8 | 24 | 24.2 | 18.7 | |

Motor engine power (kW) | 125 | 80 | 85 | 60 | |

Maximum charging power (kW) | 7.3 | 6.7 | 3.6 | 3.6 | |

Charging current (A) | Mean | 20.9 | 16.7 | 13.3 | 14.5 |

THC (A) | Mean | 1.13 | 0.79 | 0.56 | 0.58 |

**Table 2.**Uncertainty requirements for current measurements (CLASS I instruments) [49]. I

_{nom}, nominal range of power-quality (PQ) instrument; I

_{m}, measured value.

Measurement | Conditions | Maximum Error (±ε) |
---|---|---|

Current | I_{m} ≥ 3% of I_{nom} | ±5% of I_{m} |

I_{m} < 3% of I_{nom} | ±0.15% of I_{nom} |

Harmonic Order | 3 | 5 | 7 | 9 | 11 | |||||
---|---|---|---|---|---|---|---|---|---|---|

Imax (A) | 6.89 | 2.52 | 1.61 | 1.88 | 1.26 | |||||

Time (min) at Imax | 400 | 437 | 439 | 598 | 354 | |||||

Phase-angle location (Quadrant) | Second and third | Second and third | First and second | Third | Second | |||||

Charging start time (min) | SOC (%) | Charging start time (min) | SOC (%) | Charging start time (min) | SOC (%) | Charging start time (min) | SOC (%) | Charging start time (min) | SOC (%) | |

EV1 (1) | 229 | 30–40 | 356 | 70–80 | 397 | 70–80 | 508 | 30–40 | 289 | 50–60 |

EV1 (2) | 281 | 70–80 | 329 | 30–40 | 329 | 70–80 | Not charging | 349 | 50–60 | |

EV1 (3) | 288 | 50–60 | 280 | 50–60 | 316 | 30–40 | Not charging | 283 | 60–70 | |

EV1 (4) | 277 | 60–70 | 427 | 60–70 | 395 | 80–90 | 517 | 50–60 | 331 | 80–90 |

EV2 (1) | Not charging | 309 | 36–40 | Not charging | Not charging | 736 | 60–70 | |||

EV2 (2) | 357 | 40–50 | 302 | 90–100 | 586 | 90–100 | Not charging | Not charging | ||

EV2 (3) | 251 | 36–40 | Not charging | 315 | 90–100 | Not charging | 51 | 40–50 | ||

EV2 (4) | Not charging | Not charging | Not charging | not charging | Not charging | |||||

EV3 (1) | Not charging | Not charging | Not charging | 373 | 40–50 | Not charging | ||||

EV3 (2) | Not charging | Not charging | Not charging | 528 | 70–80 | Not charging | ||||

EV3 (3) | Not charging | Not charging | Not charging | Not charging | Not charging | |||||

EV3 (4) | Not charging | Not charging | Not charging | Not charging | Not charging | |||||

EV4 (1) | Not charging | Not charging | Not charging | 472 | 80–90 | Not charging | ||||

EV4 (2) | Not charging | Not charging | Not charging | 587 | 50–60 | Not charging | ||||

EV4 (3) | Not charging | Not charging | Not charging | Not charging | Not charging | |||||

EV4 (4) | Not charging | Not charging | Not charging | Not charging | Not charging |

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**MDPI and ACS Style**

Foskolos, G.
Measurement-Based Current-Harmonics Modeling of Aggregated Electric-Vehicle Loads Using Power-Exponential Functions. *World Electr. Veh. J.* **2020**, *11*, 51.
https://doi.org/10.3390/wevj11030051

**AMA Style**

Foskolos G.
Measurement-Based Current-Harmonics Modeling of Aggregated Electric-Vehicle Loads Using Power-Exponential Functions. *World Electric Vehicle Journal*. 2020; 11(3):51.
https://doi.org/10.3390/wevj11030051

**Chicago/Turabian Style**

Foskolos, Georgios.
2020. "Measurement-Based Current-Harmonics Modeling of Aggregated Electric-Vehicle Loads Using Power-Exponential Functions" *World Electric Vehicle Journal* 11, no. 3: 51.
https://doi.org/10.3390/wevj11030051