# Estimating Degradation Costs for Non-Cyclic Usage of Lithium-Ion Batteries
## 1 Introduction
Furthermore, they defined_ PF _as a function that depends on ohmic resistance of the cell (_R\({}_{0}\)_), charge transfer resistance of the cell (_R\({}_{1}\)_), the ohmic resistance of the cell at the beginning of life (_R\({}_{0,BOL}\)_), charge transfer resistance of the cell at the beginning of life (_R\({}_{1,BOL}\)_), and the cell total resistance factor at the end of life compared to the beginning of life (_\(\mu_{PF}\)_)._  
\[CF\ =\ 1-\frac{C(t)-\mu_{CF}C_{BOL}}{C_{BOL}-\mu_{CF}C_{BOL}}\ =\ \frac{C_{BOL}-C(t)}{C_{ BOL}-\mu_{CF}C_{BOL}}, \tag{1}\]  
\[PF\ =\ \frac{1}{\mu_{PF}-1}\Bigg{(}\frac{R_{0}(t)+R_{1}(t)}{R_{0,BOL}+R_{1,BOL}} -1\Bigg{)}, \tag{2}\]  
\[Cost_{dgg}\ =\ \max(CF\text{-}Cost_{bat},PF\text{-}Cost_{bat}), \tag{3}\]  
_where_ \(\mu_{CF}\) _is 0.8, as the capacity degradation at the end of life is 20%. Moreover,_ \(\mu_{PF}\) _is 2, as the increase in total resistance at the end of life is 100%._  
_In Equations (1) and (2), the terms_ \(C\)_,_ \(R_{0,BOL}\)_, and_ \(R_{1,BOL}\) _are expressed as_ \(C(t)\)_,_ \(R_{0}(t)\)_, and_ \(R_{1}(t)\) _to reinforce the idea that they change over time, and the other terms are constants._  
_Other authors applied different methodologies. [PERSON] et al. [33] multiplied the_ Cost\({}_{bat}\) _by the ratio of magnitude involved in actual use to the same magnitude over the battery lifetime. For the calendar cost, the magnitude used is time, and for cyclic cost, it is Ah-throughput. [PERSON] et al. [34] used battery degradation to achieve better strategies in charging batteries, applying a multi-objective approach in a market with different hourly energy prices. They obtained degradation cost as a fraction of battery replacement cost, proportional to the used Ah-throughput._  
_Sometimes calculating degradation costs requires a dynamic simulation of the cell during the charge, discharge, and rest periods. Using a simulation with the current applied and ambient temperature as inputs, the output values of voltage and cell temperature can be obtained. For this purpose, the most commonly used electric model is the Thevenin model. It consists of a voltage source named open circuit voltage (_OCV_), a serial resistance, and an RC circuit, as seen in Figure 1._  
This model is extensively described in the literature; for example, by [PERSON] et al. [35], where the Thevenin model uses hysteresis. A more traditional formulation can be found in [PERSON] et al. [36], which uses it for the state of charge (_SoC_) for NMC estimation with adaptive filters, [PERSON] et al. [37], or [PERSON] et al. [38]. A similar formulation to that of the above studies was developed by [PERSON] et al. [39]. Figure 1: Thevenin cell model. Cell temperature depends on cell use and ambient temperature. Models that evaluate the cell temperature take into account that heat generation affects temperature variation. Thus, [PERSON] et al. [40] differentiate reversible heat generation, which is due mainly to entropy change, from irreversible heat generation, which depends on internal resistance and current. The temperature model applied by [PERSON] et al. [10] uses the thermal heat generated by internal resistance. It models heat transference from/to the battery, ambient atmosphere, and cabin, based on the work of [PERSON] et al. [41]. They provide values for the heat transfer coefficient and thermal mass of the battery. Another study on heat generation is [PERSON] et al. [42], which tests the temperature model on an LFP cell. [PERSON] et al. [31] use entropy variation, based on a [PERSON] et al. [43] study that calculates the internal temperature of an LFP/graphite 26,650 cell depending on surface temperature and ambient temperature. [PERSON] et al. [39] developed a thermal model with air convection. [PERSON] et al. [24] present a complete thermal model with coefficients for Panasonic 17,500 cells. Furthermore, [PERSON] et al. [32] report a thermal model applied to 18,650 cells. [PERSON] et al.