# Electric Vehicle Fast Charging: A Congestion-Dependent Stochastic Model Predictive Control under Uncertain Reference

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## Abstract

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## 1. Introduction

- The reduction of the power flow at the POC of the service area with the main grid, which is essential in order to reduce operation costs of the service area;
- The tracking of the charging power demand for each charging PEV (i.e., the controller must strive to assign to each PEV the power it requires), which is needed in order to assure minimum charging times for the drivers;
- A third requirement is related with the operation of the ESS. It is desirable to avoid the ESS to become fully depleted during operation, since in this case, it cannot contribute to balance future charging demand peaks.

#### 1.1. Literature Review

#### 1.2. Paper Contributions

- We provide a stochastic MPC formulation for the service area control problem, assuming knowledge of the expected value of the charging demand, which is a realistic assumption, since it can be estimated by the service area operator from the historical data. The proposed formulation overcomes the drawback of deterministic MPC ones, which rely on the accurate knowledge of the demand profiles, which cannot be assumed in the fast-charging use case.
- The performances of two possible hardware configurations for the service area are compared, highlighting the peculiarities of each one.
- The proposed controller jointly manages the ESS control and also the charging station control, allowing to optimize the performance of the service area, while maximizing the experience of the users, by lowering the charging power to the PEVs only in periods of extreme congestion in the service area, by the means of a state-dependent weight in the MPC objective function.
- Finally, the proposed MPC objective function adapts (through a congestion-dependent weight) to the congestion state of the service area, which further improves the performance in terms of peak reduction at the POC with the grid.

#### 1.3. Paper Structure

## 2. Problem Formalization

#### 2.1. Case 1—BUS Configuration

**Problem**

**1**

#### 2.2. Case 2—UPS Configuration

**Problem**

**2**

## 3. Simulation Results

#### 3.1. Simulation Setup

#### 3.2. Simulation 1: Comparison between Fixed and Variable Weight for the Charging Power Tracking Term

#### 3.3. Simulation 2: Effects of Power Losses

#### 3.4. Simulation 3: Comparison between BUS and UPS Configuration in Case of Uncertainties

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ESS | Energy Storage System |

MPC | Model Predictive Control |

PEV | Plug-in Electric Vehicles |

POC | Point of Connection |

PMP | Pontryagin Minimum Principle |

RES | Renewable Energy Sources |

SOC | State-of-Charge |

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**Figure 3.**The curve $w\left(t\right)$ and ${\widehat{u}}^{ev}\left(t\right)$ chosen for the simulations. The black line represents the expected values of $\mathbb{E}\left[w\right(t\left)\right]$ and $\mathbb{E}\left[{\widehat{u}}^{ev}\left(t\right)\right]$ in (

**a**) and (

**b**), respectively; the red line represents a realization of the actual curve $w\left(t\right)$ and ${\widehat{u}}^{ev}\left(t\right)$ in (

**a**) and (

**b**), respectively. (

**a**) Curve $w\left(t\right)$ chosen for the simulations; (

**b**) The curve ${\widehat{u}}^{ev}\left(t\right)$ chosen for the simulations.

**Figure 5.**Evolution of mismatch between power demand for charging PEVs (${\widehat{u}}^{ev}\left(t\right)$) and power delivered to PEVs (${u}^{ev}\left(t\right)$). ${\xb7}^{1}$ is computed with fixed $c=15$; ${\xb7}^{2}$ is computed with $c\left(t\right)$, as in (15).

**Figure 6.**Power evolution at POC. $p{\left(t\right)}^{1}$ is computed with fixed c = 15; $p{\left(t\right)}^{2}$ is computed with c(t), as in (15).

**Figure 7.**ESS SOC evolution. $x{\left(t\right)}^{1}$ is computed with fixed c = 15; $x{\left(t\right)}^{2}$ is computed with c(t), as in (15).

**Figure 8.**Power evolution at POC with different power conversion losses factors $\eta \in \{0.5,0.7,0.98,1\}$.

**Figure 9.**ESS SOC evolution with different power conversion losses factors $\eta \in \{0.5,0.7,0.98,1\}$.

**Figure 10.**Evolution of power at POC in case of uncertainties for BUS (red) and UPS (black) configurations.

**Figure 11.**ESS SOC evolution in case of uncertainties for BUS (red) and UPS (solid black) configurations.

**Figure 12.**Comparison between power delivered to PEVs (bottom) and variable cost $c\left(t\right)$ (top) for BUS and UPS configurations.

Symbol | Explanation |
---|---|

g | Constant used to define weight c in (15). |

h | Constant used to define weight c in (15). |

${h}_{0}$ | Constant used to define weight c in (15). |

N | Length [number of sampling intervals] of the MPC prediction horizon. |

$p\left(t\right)$ | Power [kW] flowing at the POC during time interval t. |

${p}_{max}^{ev}$ | Maximum power [kW] of a high-power charging station. |

q | Weight of the state term in the objective function. |

r | Weight of the power term in the objective function. |

s | Weight of the control term in the objective function. |

t | Generic time interval. |

${t}_{0}$ | Initial time interval. |

T | Sampling time. |

${u}^{ess}\left(t\right)$ | ESS charging/discharging power [kW] during interval t. |

${u}_{max}^{ess}$ | Maximum ESS charging power [kW]. |

${u}_{min}^{ess}$ | Maximum ESS discharging power [kW]. |

${u}^{ev}\left(t\right)$ | Total power [kW] delivered to the PEVs during interval t. |

${\widehat{u}}^{ev}\left(t\right)$ | Total power [kW] demand from the PEVs during interval t. |

${\widehat{u}}_{max}^{ev}$ | Maximum aggregated power demand [kW] for charging stations. |

$w\left(t\right)$ | RES power [kW] produced during interval t. |

$x\left(t\right)$ | ESS SOC [kWh] at the beginning of time interval t. |

${x}_{0}$ | ESS initial SOC [kWh]. |

${x}_{min},{x}_{max}$ | respectively, minimum and maximum possible ESS SOC [kWh]. |

${x}^{ref}$ | Reference ESS SOC [kWh]. |

${\alpha}_{t}$ | Boolean variable equal to one if the ESS recharges during time interval t. |

${\beta}_{t}$ | Boolean variable equal to one if the ESS discharges during time interval t. |

${\eta}_{ch},{\eta}_{dis}$ | The ESS charging and discharging conversion losses, respectively. |

$\rho \left(t\right)$ | Total power [kW] entering or exiting the ESS at time t in the UPS configuration. |

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## Share and Cite

**MDPI and ACS Style**

Di Giorgio, A.; De Santis, E.; Frettoni, L.; Felli, S.; Liberati, F.
Electric Vehicle Fast Charging: A Congestion-Dependent Stochastic Model Predictive Control under Uncertain Reference. *Energies* **2023**, *16*, 1348.
https://doi.org/10.3390/en16031348

**AMA Style**

Di Giorgio A, De Santis E, Frettoni L, Felli S, Liberati F.
Electric Vehicle Fast Charging: A Congestion-Dependent Stochastic Model Predictive Control under Uncertain Reference. *Energies*. 2023; 16(3):1348.
https://doi.org/10.3390/en16031348

**Chicago/Turabian Style**

Di Giorgio, Alessandro, Emanuele De Santis, Lucia Frettoni, Stefano Felli, and Francesco Liberati.
2023. "Electric Vehicle Fast Charging: A Congestion-Dependent Stochastic Model Predictive Control under Uncertain Reference" *Energies* 16, no. 3: 1348.
https://doi.org/10.3390/en16031348