### SOP estimation

**1. SOP Estimation**

The peak power state SOP is the maximum power that the power battery can release or absorb in a predetermined time interval. The peak power estimation of power battery can evaluate the charging and discharging power limit capacity of power battery under different SOC and SOH, and optimize the relationship between power battery system and vehicle power performance, so as to meet the acceleration and climbing performance of electric vehicle and maximize the regenerative braking performance of motor. Due to the influence of electrochemical kinetics and thermodynamics, the real-time peak power of power battery is restricted by its voltage, current, temperature, available capacity and SOC. In this section, the continuous peak power prediction method will be described in detail.

**2. Continuous SOP forecasting method**

Aiming at the problem of SOP prediction which needs continuous sampling interval, this section describes the continuous SOP prediction method for multi sampling interval in detail.

**Continuous peak current estimation**

In Thevenin model, OCV can be expressed as a function of SOC

(1) |

The state space expression equation of the model is

(2) |

In the formula, the definition of each matrix is

(3) |

Assuming that the system output is a constant current during the sampling interval from *k* to *k+L*, and L is the duration of the sampling time, that is, the Thevenin model can be used to estimate the voltage at the k+L sampling time, the expression is

(4) |

For the convenience of calculation, the power battery prediction model for the duration of is regarded as a steady system, that is, the model parameters remain unchanged and can be further derived:

(5) |

In order to solve the maximum discharge current and the minimum charge current , the output of the battery model should satisfy the following two equations:

(6) | |

(7) |

The estimating equation of the continuous peak current of the power battery in time is

(8) |

Note that if *L* is 1, the above formula can be degenerated into a peak current calculation formula.

**Continuous SOP estimation**

When the SOC is close to the lower limit z_{min}, the maximum discharge current should be restricted, and the charging current should be maximized, otherwise the power battery system will face the danger of over-discharge. On the contrary, when the SOC is close to the upper limit z_{max}, the maximum charging current should be restricted, and the discharge current should be maximized, otherwise it will face the danger of overcharging. Adopting SOC constraint is a key principle of peak current estimation, and its equation is expressed as

(9) |

In the formula, and are the minimum charging current and the maximum discharging current in the sampling interval , respectively. Once the constraint current is obtained, the peak current expression is

(10) |

In the formula, Imax and Imin are the maximum allowable discharge current restriction value and the minimum allowable charge current restriction value of the power battery, respectively. and are the minimum charge current and maximum discharge current under consideration of all constraints, respectively. The continuous power capability based on Thevenin model can be estimated by the following formula:

(11) |

Then derive the estimation results of the peak charge and discharge power and :

(12) |

In the formula, Pmax and Pmin are power battery discharge power design constraints and charging power design constraints, respectively.

**3. References**

[1] R. Xiong. Core Algorithms of Battery Management System. Beijing：China Machine Press，2018. (Chinese) (Chapter Four)

[2] R. Xiong*, F. Sun, H. He and T. Nguyen, “A data-driven adaptive state of charge and power capability joint estimator of lithium-ion polymer battery used in electric vehicles”, Energy, vol. 63, pp. 295–308, Dec 2013.（Download）

[3] F. Sun; R. Xiong* and H. He, “Estimation of state-of-charge and state-of-power capability of lithium-ion battery considering varying health conditions”, J. Power Sources, vol.259, pp.166–176, Aug 2014.（Download）

[4] R. Xiong*, Y. Duan, J. Cao and Q. Yu*, “Battery and ultracapacitor in-the-loop approach to validate a real-time power management method for an all-climate electric vehicle”, Applied energy, vol. 217, pp. 153-165, Feb 2018.（Download）

[5] R. Xiong*, H. He; F. Sun and K. Zhao, “Online Estimation of Peak Power Capability of Li-Ion Batteries in Electric Vehicles by a Hardware-in-Loop Approach”, Energies, vol. 5, no. 5, pp. 1455-1469, May 2012.（Download）

**4. Available Resources**

(1) SOP estimation algorithm information: click to download (PDF watermark textbook)

(2) Model example:Resource Application Form.pdf

(3) Lecture notes:Resource Application Form.pdf