# Stochastic Modeling of the Charging Behavior of Electromobility

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## Abstract

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## 1. Introduction

_{2}emissions in Europe, causing dramatic environmental issues [1]. As the Earth’s climate shows signs of change, more and more states and organizations across the globe start to consider the electrification of the mobility sector as an opportunity to reduce the emission of greenhouse gases. This means a shift from conventional oil-based transportation towards electric grid as well as off-grid or H2-based supplied transportation.

_{2}emissions within Europe [1]. It is worth noting at this point that electrical drives in electric vehicles have an efficiency of up to 96%, no transmission at all and low noise emission. An internal combustion engine (ICE), on the other hand, only shows a 35% maximum efficiency in general [2]. However, to increase public trust in electric vehicles, battery technologies and charging infrastructure including fast charging stations must be further researched and developed.

- The charging pattern of a charging station is described by a stochastic process implementing the Markov chain.
- Based on the stochastic process, an algorithm designed for high performance and scalability is developed.
- A case study, simulating a 22 kW charging station in Vienna, Austria considering weekdays and weekends to show typical occupation together with load patterns is conducted.
- Finally, the charging pattern, the parameters’ variation, and the charging station operator’s (CSO) revenues are illustrated.

## 2. Methods

#### 2.1. Markov Chain

- “Unoccupied” ${S}_{u}$: No PEV is connected to the charger.
- “Charging” ${S}_{c}$: A PEV is plugged-in to the station, and its battery is being charged (SOC < 100%).
- “Plugged-in but not charging” ${S}_{n}$: A PEV is still plugged-in to the station, however, its battery has already been fully charged (SOC = 100%).

#### 2.2. Algorithm for Describing the Charging Process

#### 2.3. Revenues of the Charging Station Operator

**Tariff on energy consumption:**The simplest tariff charges energy consumption only (e.g., $\mathrm{EUR}/\mathrm{kWh}$). This kind of tariffs is often used, calculating the revenues in a given interval with the equation:

**Tariff on plug-in duration:**Consumption is only charged on a time basis (e.g., $\mathrm{EUR}/\mathrm{min}$). Such a tariff helps to prevent PEV of being plugged but not charged. Equation

## 3. Case Study

## 4. Results and Discussions

#### 4.1. Verification of the Model

#### 4.2. Sensitivity Analysis

- the mean and the deviation of the charging distribution;
- times of the PEV being plugged in; and
- the probabilities ${p}_{cn,t}$ and ${p}_{nu,t}$ within and beyond the average charging duration.

#### 4.2.1. Sensitivity of the Charging Duration

#### 4.2.2. Sensitivity of the Plug-in Time

#### 4.2.3. Sensitivity of the Probabilities ${p}_{cn,t}$, ${p}_{nu,t}$

#### 4.3. Monetary Impact of Different Tariff Designs to the Charging Station Operator

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AC | Alternating Current |

CSO | Charging Station Operator |

EV | Electric vehicle |

PEV | Plug-in electric vehicles |

PHEV | Plug-in hybrid electric vehicles |

SOC | State of charge |

ICE | Internal combustion engine |

NHTS | National Household Travel Survey |

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**Figure 3.**Example for roulette wheel selection: generated number (gray dot) falls on “plugged in but not charging”.

**Figure 6.**Average energy not charged due to plugged-in fully charged PEVs in each interval for a $N=1000$.

**Figure 8.**Variation of the plug-in time distributions in Figure 4.

Witgin Average Duration | Overnight | Morning Rush | Working Hours | Evening Commute | Evening | Weekend |

${p}_{cc,t}$ | 0.6 | 0.45 | 0.5 | 0.6 | 0.6 | 0.6 |

${p}_{cu,t}$ | 0.1 | 0.45 | 0.4 | 0.3 | 0.1 | 0.3 |

${p}_{cn,t}$ | 0.3 | 0.1 | 0.1 | 0.1 | 0.3 | 0.1 |

${p}_{nn,t}$ | 0.7 | 0.6 | 0.6 | 0.6 | 0.7 | 0.6 |

${p}_{nu,t}$ | 0.3 | 0.4 | 0.4 | 0.4 | 0.3 | 0.4 |

Beyond Average Duration | Overnight | Morning Rush | Working Hours | Evening Commute | Evening | Weekend |

${p}_{cc,t}$ | 0.3 | 0.2 | 0.2 | 0.2 | 0.3 | 0.2 |

${p}_{cu,t}$ | 0.1 | 0.5 | 0.3 | 0.3 | 0.1 | 0.3 |

${p}_{cn,t}$ | 0.6 | 0.3 | 0.5 | 0.5 | 0.6 | 0.5 |

${p}_{nn,t}$ | 0.7 | 0.3 | 0.7 | 0.4 | 0.4 | 0.4 |

${p}_{nu,t}$ | 0.3 | 0.7 | 0.3 | 0.6 | 0.6 | 0.6 |

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**MDPI and ACS Style**

Sokorai, P.; Fleischhacker, A.; Lettner, G.; Auer, H.
Stochastic Modeling of the Charging Behavior of Electromobility. *World Electr. Veh. J.* **2018**, *9*, 44.
https://doi.org/10.3390/wevj9030044

**AMA Style**

Sokorai P, Fleischhacker A, Lettner G, Auer H.
Stochastic Modeling of the Charging Behavior of Electromobility. *World Electric Vehicle Journal*. 2018; 9(3):44.
https://doi.org/10.3390/wevj9030044

**Chicago/Turabian Style**

Sokorai, Peter, Andreas Fleischhacker, Georg Lettner, and Hans Auer.
2018. "Stochastic Modeling of the Charging Behavior of Electromobility" *World Electric Vehicle Journal* 9, no. 3: 44.
https://doi.org/10.3390/wevj9030044