### SOS Estimation

**1. SOS Estimation**

As lithium-ion batteries are used in electric vehicles and stationary storage applications, more battery cells and higher energy densities increase the risk of possible catastrophic events. It is necessary to estimate the state of safety(SOS) of the battery.

Based on the concept that security is inversely proportional to abuse, a definition and method for calculating the security status of energy storage systems are given. As the latter increases, the former decreases to zero. The previous description is qualitative in nature, but does not provide a digital quantification of the security of the storage system. For battery test standards, they only define pass or fail criteria. The proposed SOS state uses the same range as other commonly used state quantities (such as SOC, SOH), with a value between 0 (completely unsafe) and 1 (completely safe). The developed function combines the effects of any number of sub-functions in one or more variables (such as voltage, temperature or mechanical deformation). Each sub-function describes a specific situation of abuse. These variables can be detected by sensors or by other technical estimation. By adding new sub-functions or improving existing sub-functions, the security status definition can be made more general.

Use and abuse to define safety in a way that is inversely proportional:

(1) |

Among them, is an abuse state, and is a safe state, *x *represents all types of state and control variables that describe battery behavior, such as voltage, temperature, charging and discharging current, internal impedance, battery expansion, and battery deformation. In a given time, as the abuse status increases, the security status will decrease accordingly. In order to limit SOS to a reasonable working value, we believe that it has the same value range as SOC, from 0 to 1. As the absolute value of abuse becomes infinite, the safety value should tend to 0 or completely unsafe. When the abuse does not exist or is zero, the safety value should be limited to 1 or completely safe.

In general, the abuse function *h(x)* considers all variables that may affect the ESS. We can assume that there is only one variable, and the abuse increases with that variable. Therefore, the is a sub-function, when combined with other such sub-functions, the characteristic value of SOS will be obtained. According to the definition of SOS as a probability function, the characteristic value can be calculated as the multiplication and integral distribution of each sub-function.

**2. Mathematical equations**

Define the SOS value as:

(2) |

Where *g(x)* is the abuse function defined for the value.

Use 2 times representation:

(3) |

Where *h(x)* can take any value, including zero and negative numbers, and *m* and *d *are constants that allow us to control the rate of decline as needed.

In many practical uses of storage systems, when its capacity (power or energy) can only reach 80% or 0.8 of the same capacity at the beginning of life (BOL), it is considered to have reached the end of life (EOL), and the capacity at the beginning of life is standardized 100% or 1.0. Therefore, we will also use the familiar value 0.8 as a boundary to indicate when the battery is at an acceptable abuse value.

(4) | |

(5) | |

(6) |

We can get parameters:

(7) | |

(8) |

Among them, and are combinations of battery variables that describe 100% and 80% safety respectively at the original time t0. You can get:

(9) |

You can also use the number *ζ *between zero and one instead of 0.8 to summarize the end of life.

(10) | |

(11) | |

(12) |

In general, the abuse function *h(x)* considers all possible variables. We can assume that there is only one variable, and the abuse increases with that variable, which is:

(13) | |

(14) | |

(15) | |

(16) |

The is a sub-function, when combined with other such sub-functions, the characteristic value of SOS will be obtained. According to the definition of SOS as a probability function, the characteristic value can be calculated as the multiplication and integral distribution of each sub-function.

(17) |

Where n is the number of sub-functions. Since each sub-function has a lower value* ζ* (to ensure safe behavior), the maximum value is 1, so we have important values:

Figure 1 Suggests the operating area of the SOS function, where ζ is the minimum value, between 0 and 1 to ensure safety, and n is the number of sub-functions

**3. Main results**

Since SOS is a multi-dimensional quantity, it is usually impossible to plot SOS in a three-dimensional space to observe the dependence of each variable. For this reason, only two variables need to be considered at a time. Figure 7 shows the impact of low pressure, high pressure and high temperature on SOS, so other sub-functions are not considered. This is equivalent to assuming that the value of any other sub-function is 1, so it is completely safe.

Figure 7 The surface map of SOS, considering only the two variables V and T in the sub-functions fv and ft

**4. References**

[1] Eliud Cabrera-Castillo,Florian Niedermeier,Andreas Jossen. Calculation of the state of safety (SOS) for lithium ion batteries[J]. Journal of Power Sources,2016,324.（Download）

[2] Ashtiani C. Analysis of Battery Safety and Hazards' Risk Mitigation[J]. Ecs Transactions, 2008, 11(19).（Download）

[3] L.L.C. Chrysler, Ford Motor Company, General Motors Corporation, Potential Failure Mode and Effects Analysis (FMEA)[S], AIAG, 2008.（Download）

[4] Languang Lu,Xuebing Han,Jianqiu Li,Jianfeng Hua,Minggao Ouyang. A review on the key issues for lithium-ion battery management in electric vehicles[J]. Journal of Power Sources,2013,226.（Download）

[5] S. Abada,G. Marlair,A. Lecocq,M. Petit,V. Sauvant-Moynot,F. Huet. Safety focused modeling of lithium-ion batteries: A review[J]. Journal of Power Sources,2016,306.（Download）