# Considering the Life-Cycle Cost of Distributed Energy-Storage Planning in Distribution Grids
## 3 ESS Installation Site and Capacity Optimization
### The Second-Stage Optimization
#### 3.2.2 Capacity Fading Model  
At present, evaluating the lifetime of a battery is considered from the perspective of the degradation mechanisms, which cause the battery cell performance to deteriorate. When the degradation reaches a certain failure value, the end-of-life condition has been fulfilled. Conventionally, when the practical capacity of a battery has dropped to 80% of the battery's original capacity, it reaches the end-of-life condition. So, the number of cycles (length of use) of the ESS, when the capacity is reduced to 80% of its initial capacity, is defined as cycle life (calendar life). There are many factors that cause capacity degradation, including dissolution of cathode materials, phase change of cathode materials, decomposition of electrolyte, overcharging, self-discharging, interfacial film formation, and fluid-collecting corrosion, etc. [22]. The capacity degradation is mainly manifested by a decreased in-solution concentration and an increased internal resistance of the battery, and it is related to the charging and discharging power, depth of discharging, fluctuation of SOC, number of cycles, and the operation temperature [22]. [PERSON], and [PERSON], conducted a large number of tests and proposed a practical capacity fading model for Li-ion battery cells that can be especially used for irregular discharging, based on the experimental results [23]. The rest value of the Li-ion battery can be determined by the state-of-health (SOH) in this model. When the battery is at the initial capacity, the SOH is 1. If the SOH becomes 0, the battery reaches the end-of-life condition. The capacity fading rate is inseparable from the average SOC (\(SOC_{avg}\)) and the SOC deviation (\(SOC_{dev}\)), the higher the \(SOC_{avg}\) or \(SOC_{dev}\), the higher the capacity fading rate [23]. The SOC\({}_{avg}\) represents the average SOC over the cycle \(\tau\) (equal 24 h, in this study), and the \(SOC_{dev}\) represents the normalized deviation of the SOC from its mean, over a cycle period. According to the SOC, temperature, depth of discharging, fluctuation of SOC, and charging/discharging power, \(SOC_{avg}\) and \(SOC_{dev}\) can be calculated and the total capacity fading is a summation of the capacity fading that occurs under the experienced operating conditions, and can be calculated as:  
\[\Gamma=\sum_{I}^{\tau_{max}}\;((k_{s1}SOC_{dev,I}\cdot e^{(k_{s2}\cdot SOC_{ avg,I})}+k_{s3}e^{k_{s4}SOC_{dev,I}})e^{(-\frac{E_{d}}{R}(\frac{1}{I_{I}}- \frac{1}{\tau_{ref}}))})Ah_{I} \tag{22}\]  
where \(\Gamma\) is the total capacity fade, according to Reference [23] the parameter \(K\)s1 was set as \(-4.092\times 10^{-4}\), \(K\)s2 was set as \(-2.167\), \(K\)s3 was chosen as \(-1.408\times 10^{-5}\), and \(K\)s4 was set as 6.130. Parameters \(k\)s1 to \(k\)s4 were obtained through curve-fitting in the MATLAB Surface Fitting Tool, after analyzing the \(SOC_{dev}\) influence on the measured capacity fading rates with both \(SoC_{avg}\) and \(SoC_{dev}\) as variables. Equation (22) was obtained by combining the fitting-curve and the Arrhenius equation, which can analyze the temperature-dependence of the capacity fading rate. Ea is the activation energy which was 78.06 kmol/J and R is the gas constant which was 8.314 J/(mol * K). An event is represented by \(I\), which was a certain period \(\tau\), in this study, and \(\tau_{sum}\) is the total number of events. \(Ah_{I}\) represents the charging energy during event \(I\). \(SOC_{avg,I}\) and \(SOC_{dev,I}\) are the average SOC and the SOC deviation, during event \(I\). \(T_{ref}\) is the reference temperature in the operation process and is normally set to 25 \({}^{\circ}\)C. T\({}_{I}\) is the temperature of ESS in event \(I\) which can be formulated as Equation (23), according to the approximate linear relationship between the charging/discharging power and the temperature of Li-ion battery cells [24].