# Model Predictive Control-Based Fast Charging for Vehicular Batteries

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## Abstract

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## 1. Introduction

#### 1.1. Literature Review

^{2}[5], and so forth. A smooth control circuit (SCC) is proposed to ensure the stable transition from the constant-current (CC) to the constant-voltage (CV) stage [6]. A further review of CCCV family is given in [7].

#### 1.2. Overview of Proposed Charging Controller

- A SOC predictor, predicting the SOC of battery when fed by a sequence of future charging current;
- A temperature predictor, predicting the future battery temperature under a sequence of charging current;
- A fitness evaluator, evaluating the performance of the sequence of charging current;
- An optimizer, finding the best sequence of charging current using genetic algorithm (GA).

## 2. Predictive Models

#### 2.1. RC Model for SOC Prediction

#### 2.2. Thermal Model in Simulation

#### 2.3. Thermal Model in Experiment

_{4}cells during charge processes with different charging rates. The increase in temperature agrees with the two results reported in the literature above.

#### 2.4. Model Based Prediction

## 3. Formulation under MPC Framework

#### 3.1. Performance Indexes

#### 3.2. Constraints

## 4. Optimization Using Genetic Algorithm

- l.
- Coding. The standard GA generally codes a candidate solution as a string of characters which are usually binary digits, referred to as a chromosome. The candidate solution is termed an individual. Accordingly, the set made up of a number of individuals is termed a population. In this paper, we apply a real-value coding method, which codes a candidate solution as a set of floating decision variables. The real-value coding method is proven to have superior performance to the binary-coded method in control optimization problems [35].
- 2.
- Initialization. The standard GA starts with an initial population. Usually, individuals in the initial population are produced randomly. In MPC, the initialization process is executed at each control time to start the GA. Since the best control sequence optimized at time $k$ contains good candidates from $k+1$ to $k+p-1$, one initial individual is specially designed by shifting it one time step and filling the last charging current with the same value as $\widehat{I}(k+p-1)$, as shown in Figure 6. This individual introduces historical best charging sequence into the current optimization process, thus it is helpful to improve optimization performance to be at least very similar with the previous optimized performance.

**Figure 6.**Initialization of one special individual by introducing the best control sequence optimized in previous step into the present step.

- 3.
- Fitness evaluation. We evaluate the fitness of each individual in each generation according to the Equation (25). The smaller the fitness, the better the individual. However, to facilitate the following selection step, the raw fitness is usually scaled to assign suitable selection pressure to each individual. In this work, the scale function is expressed as follows:$${F}_{scale}=\frac{1}{1+{F}^{r}}$$
- 4.
- Selection. Individuals are selected from the previous generation to the current generation based on the scaled fitness ${F}_{scale}$, following the survival of the fittest rule. Many selection methods have been developed to avoid genetic drift and premature phenomena. In this work, the roulette wheel selection method is adopted [36]. The elitism strategy is also applied in selection to assure that the best solution will never be lost.
- 5.
- Crossover. In the crossover step, the standard GA exchanges information between two parent individuals and produces two child individuals. In this work, the arithmetical crossover method is used. Given two parents ${x}_{1}$ and ${x}_{2}$, the children ${y}_{1}$ and ${y}_{2}$ are produced by linear combinations of parents with a random coefficient $\lambda $:$${y}_{1}=\lambda {x}_{1}+\left(1-\lambda \right){x}_{2}$$$${y}_{2}=\lambda {x}_{2}+\left(1-\lambda \right){x}_{1}$$
- 6.
- Mutation. After the crossover step, a subset of individuals is selected with a mutation probability of ${p}_{m}$. To explore the search space, we use Gaussian mutation, which adds a random value from a Gaussian distribution with variance $\sigma $ to each item of the selected individual.
- 7.
- Termination. Many terminating conditions have been proposed to stop the iteration process. For example, when the distances among individuals are smaller than a predetermined value, an individual satisfies a minimum criterion, or the maximum number of generations is reached. The last method is applied here.

## 5. Performance Demonstration

#### 5.1. Settings

Symbol | Description | Value | Unit | |
---|---|---|---|---|

Battery (simulation) | C | Battery nominal capacity | 7 | Ah |

m_{bat} | Battery mass | 0.37824 | kg | |

C_{bat} | Battery heat capacity | 795 | J/kgK | |

R_{eff} | Effective thermal resistance | 7.8146 | K/W | |

T_{amb} | Ambient temperature | 20 | ^{o}C | |

${\dot{m}}_{air}$ | Airflow rate | 5.8333 | g/s | |

C_{air} | Air heat capacity | 1009 | J/kgK | |

MPC | T_{s} | Control period | 30 | s |

p | Prediction horizon | 5 | -- | |

${\omega}_{1}$ | Weight of SOC tracking
J_{1} | 100 | -- | |

${\omega}_{2}$ | Weight of termperature rising
J_{2} | 1 | -- | |

GA | MaxGen | Maximum generation number | 30 | -- |

Popsize | Population size | 50 | -- | |

r | Power of raw fitness in scaling | 2 | -- | |

${R}_{\lambda}$ | The range of crossover coefficient | [0.1, 0.9] | -- | |

p_{m} | Mutation probability | 0.2 | -- | |

$\sigma $ | Variance of Gaussian mutation | 1 | -- |

_{1}is around 0.08 while that of J

_{2}is around 2. Therefore, the real weight ratio of J

_{1}to J

_{2}is around 4:1.

#### 5.2. Evaluation Method

**Figure 8.**Pareto fronts of CCCV and MPC charging methods in simulation. The expected trajectories of MPC are modified from CCCV by multiplying 1.05.

#### 5.3. Simulation Results

#### 5.4. Experimental Results

_{4}cell has good charging acceptance at low SOC. Therefore, a high charging current is utilized to increase the charging speed. Before reaching around 80% SOC, another high charging period occurs. This is because MPC predicts the high temperature increase beyond 80%, so it applies a high charging current at this stage and switches to a low charging current when SOC is close to 1.

**Figure 10.**Pareto fronts of CCCV and MPC charging methods in experiments. The expected trajectories of MPC are modified from CCCV by multiplying by 1.10.

**Figure 11.**Experimental curves during the CCCV and MPC charging processes using a modified 2 C profile.

## 6. Conclusions

## Acknowledgments

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**MDPI and ACS Style**

Yan, J.; Xu, G.; Qian, H.; Xu, Y.; Song, Z. Model Predictive Control-Based Fast Charging for Vehicular Batteries. *Energies* **2011**, *4*, 1178-1196.
https://doi.org/10.3390/en4081178

**AMA Style**

Yan J, Xu G, Qian H, Xu Y, Song Z. Model Predictive Control-Based Fast Charging for Vehicular Batteries. *Energies*. 2011; 4(8):1178-1196.
https://doi.org/10.3390/en4081178

**Chicago/Turabian Style**

Yan, Jingyu, Guoqing Xu, Huihuan Qian, Yangsheng Xu, and Zhibin Song. 2011. "Model Predictive Control-Based Fast Charging for Vehicular Batteries" *Energies* 4, no. 8: 1178-1196.
https://doi.org/10.3390/en4081178