# Online Estimation of Peak Power Capability of Li-Ion Batteries in Electric Vehicles by a Hardware-in-Loop Approach

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## Abstract

**:**

## 1. Introduction

_{s}and neglects design limits like cell current, cell power or the state of charge (SoC) [8]. In order to overcome the drawbacks of the HPPC method, which is not suitable for continuous peak power capability prediction, and neglecting the SoC limits, the voltage-limited method was proposed by Plett [9]. However, the Rint model-based HPPC and voltage-limited method are not suitable for estimating the battery’s peak power capability due to the fact that it can hardly simulate the dynamic voltage performance.

_{2}O

_{4}lithium-ion cell as research subject. The paper is arranged as follows: Section 2 proposes an online parameter identification algorithm for the dynamic electrochemical-polarization (EP) battery model with the Simulink/xPC software; Section 3 proposes an online dynamic peak power capability estimation algorithm based on the dynamic EP model and designs a HIL test bench; Section 4 carries out the HIL test and evaluates the proposed method for hybrid electric vehicle (HEV) application; last is the conclusion of this paper.

## 2. Online Parameters Identification Method

#### 2.1. The EP Model

_{0}, K

_{1}and K

_{2}are three constants chosen to make the model well fit the test data, R

_{o}is the ohmic resistance and R

_{p}is the polarization resistance. The polarization capacitance C

_{p}is used to describe the transient dynamic voltage response during charging and discharging. U

_{p}is the polarization voltage across C

_{p}, U

_{t}is the terminal voltage.

#### 2.2. The Recursive Least Square Method with an Optimal Forgetting Factor

_{t}in this paper.

**φ**and

**θ**are the information matrix and the unknown parameter matrix, respectively. The parameters in

**θ**can either be constant or subject to infrequent jumps. ξ is a stochastic noise variable (random variable with normal distribution and zero mean), and k is a non-negative integer, which denotes the sample interval, k = 0,1,2... For the recursive function of (2), the system identification is realized as follows:

**K**

_{k}is the algorithm gain and

**P**

_{k}is the covariance matrix, $\lambda $ is the forgetting factor, typically $\lambda $ = [0.95, 1] and is very important to obtain a good estimated parameter, set with a small error.

#### 2.3. The Online Parameters Identification Method for the EP Model

_{s}is the sample intervals and is 1 second for this paper:

_{1}, p

_{2}, p

_{3}, p

_{4}, p

_{5}, p

_{6}, and as follows:

_{1}, p

_{2}, p

_{3}, p

_{4}, p

_{5}, p

_{6}.

## 3. Battery Online Peak Power Capability Estimation Method

#### 3.1. Online Peak Power Capability Estimation Method

#### 3.1.1. The HPPC Method

_{t}(t) is the terminal voltage at time t; U

_{oc}(s(t)) the open-circuit voltage at present SoC state (s(t)), and R

_{o}the charging or discharging internal resistance.

_{t,min}≤ U

_{t}(t) ≤ U

_{t,max}, the peak charge and discharge current under the voltage constraints are described as:

_{t,max}and U

_{t,min}are the maximum limit voltage when charging and the minimum voltage limit when discharging, respectively; ${I}_{\text{min}}^{\text{chg},\text{HPPC}}$, ${I}_{\text{max}}^{\text{dis},\text{HPPC}}$ are the minimum charge current and maximum discharge currents based on the HPPC method. Hence, the peak power capability of the lithium-ion cell can be described as:

#### 3.1.2. The SoC-Limited Method

_{i}is a coulomb efficiency factor for the current level I

_{L}(t), which is in function of the load current. C

_{max}is the present maximum available capacity.

_{max}, the minimum SoC s

_{min}and the current limit of cell can be expressed as follows:

#### 3.1.3. The EP Dynamic Model-Based Method

_{L}(t) and OCV is a nonlinear function of $s(t)$. Concerning this problem, the Taylor-series expansion is employed to linearize the equation and to solve the approximated values for the peak currents. The Taylor-series expansion equation is as follows:

_{p}is greater than zero when the battery is discharging and U

_{p}< 0 when charging, therefore the values computed by (21) are smaller in magnitude than those from (15) for the same ohmic resistance values. Meanwhile, ∂U

_{oc}/∂s is not constant within the entire SoC operation range, especially at the two extreme ends. Therefore the peak power capability estimates based on the EP-model method are more reasonable than those of the HPPC method regarding peak current calculations and thus safety issues. Once the current design limit is calculated, the peak currents with all limits enforced are calculated as:

_{max}, I

_{min}are the cell’s current design limits, I

_{max}denotes the maximum discharge current and I

_{min}denotes the minimum charge current of its design limits. The peak power capability may be calculated as follows:

_{max}, P

_{min}are the cell’s power design limits, P

_{max}denotes the peak discharge power and P

_{min}denotes the peak charge power of its design limits. Then we can build the Simulink model of the peak power capability estimation methods for the HPPC method and the EP dynamic model-based method, respectively, based on the equations above.

#### 3.2. The Hardware-in-Loop Test Bench Design

## 4. Results and Discussion

Parameters | Value |
---|---|

Maximum load current /A | 350 |

Minimum load current /A | 175 |

Maximum terminal voltage /V | 4.2 |

Minimum terminal voltage /V | 3.0 |

Peak discharge power /W | 1500 |

Peak charge power /W | −700 |

SoC operation range for HEVs | 0.35–0.85 |

**θ**

_{k}are shown in Figure 4. The model’s parameters can be deduced by Equation (13) and the results are shown in Figure 5.

**Figure 4.**The online identification results for parameter matrix

**θ**

_{k}: (

**a**) p

_{1}~p

_{4}; (

**b**) p

_{5}and p

_{6}.

**Figure 5.**Online parameters identification results of the EP model and calculated SoC profiles: (

**a**) K

_{0}, K

_{1}, K

_{2}; (

**b**) Calculated SoC; (

**c**) U

_{oc}; (

**d**) R

_{o}; (

**e**) R

_{p}; (

**f**) C

_{p}.

**Figure 6.**The comparison profiles of the terminal voltages: (

**a**) The voltage error between the measured value and the online estimated value; (

**b**) Error’s statistical information.

_{oc}/∂s > 0 of the cell into account when making a prediction while the former doesn’t. For ∂U

_{oc}/∂s > 0, which leads to a lower peak discharge current value in contrast to the values of the HPPC method, the result is a more realistic representation. Further, the EP dynamic model can simulate the dynamic effect of lithium-ion batteries well, while the Rint model fails to do this. When discharging, U

_{p}> 0, it is obvious that the current based on the EP model is lower, especially in the ranges of high required power. The estimated peak current with the proposed peak power capability estimation decreases quickly because the polarization voltage increases quickly. Figure 7(b) shows the peak currents with the EP dynamic model-based method, which, considering the lithium-ion battery relaxation effect is well simulated, can meet the actual performance well. If the SoC value reaches the maximum design limit, the peak charge current will be very small while the peak discharge current will be very big. On the contrary, if the SoC value reaches the minimum design limit, the peak discharge current will be very small while the peak charge current will be very big. As a result, the method can optimize the operation range of batteries and extend their lifespan. At the same time, when the battery is being discharged with big currents, the peak discharge value is reduced significantly while the peak charge capability is significantly increased. This is conforming to the actual battery working characteristics and underlying optimization control.

**Figure 7.**Peak power capability and current online estimation results: (

**a**) Peak discharge currents; (

**b**) Peak charge currents; (

**c**) Peak discharge powers; (

**d**) Peak charge powers.

_{p}. The two methods are also compared with respect to charging power, as shown in Figure 7(d). Due to the ignorance of the SoC limits and the relaxation effect of the battery, the HPPC method is prone to overpredicting the charging power. With strong discharges at high required power ranges, the battery will allow a greater charging power while the HPPC method cannot give the accuracy peak power estimates quickly to match the driving cycle changes. However, the real-time peak power capability estimation results with the EP dynamic model-based method are not flat due to the polarization effects. Therefore the proposed method for peak power capability estimation gives satisfying results.

## 5. Conclusions

_{2}O

_{4}lithium-ion cell. Based on the above analysis, the following main concluding remarks can be made:

- (1)
- In order to avoid time-consuming, laborious and error-prone experiments for determining the tabulated OCV-SoC data, the EP model, which uses the Nernst model to define the open circuit voltage, is applied to model the lithium-ion battery.
- (2)
- For improving the dynamic performance of the EP model, the RLSF algorithm is applied to identify online the EP model’s parameters; the model’s accuracy is verified by the hardware-in-loop test, and the maximum error of the estimated terminal voltage is within 1% of its nominal voltage.
- (3)
- Compared with the HPPC method, the proposed peak power capability estimation method takes the cell voltage, current, SoC and power as its constraints; which can simulate the relaxation effect well. The evaluation results based on the HIL test show that the proposed method gives a more reliable estimation, especially when the load current changes suddenly or strongly. More importantly, when the SoC is high and low, the proposed method can give a more accurate estimate; avoiding overcharging or overdischarging.
- (4)
- The HIL test data has provided critical guidance for further development and improvement of the peak power capability estimation approach. This accelerates the overall system development process and reduces the cost of the EVs development.

## Acknowledgments

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**MDPI and ACS Style**

Xiong, R.; He, H.; Sun, F.; Zhao, K.
Online Estimation of Peak Power Capability of Li-Ion Batteries in Electric Vehicles by a Hardware-in-Loop Approach. *Energies* **2012**, *5*, 1455-1469.
https://doi.org/10.3390/en5051455

**AMA Style**

Xiong R, He H, Sun F, Zhao K.
Online Estimation of Peak Power Capability of Li-Ion Batteries in Electric Vehicles by a Hardware-in-Loop Approach. *Energies*. 2012; 5(5):1455-1469.
https://doi.org/10.3390/en5051455

**Chicago/Turabian Style**

Xiong, Rui, Hongwen He, Fengchun Sun, and Kai Zhao.
2012. "Online Estimation of Peak Power Capability of Li-Ion Batteries in Electric Vehicles by a Hardware-in-Loop Approach" *Energies* 5, no. 5: 1455-1469.
https://doi.org/10.3390/en5051455