# Estimating Degradation Costs for Non-Cyclic Usage of Lithium-Ion Batteries
## 1 Introduction
Considering another cell technology, the Zabala model provides cyclic and calendar degradation of an LFP/graphite 26650-type cell of 2.3 Ah and 3.3 V. It is described in four studies by [PERSON] et al. First, calendar degradation and cyclic degradation models were introduced [11], then only the calendar model was studied [12]. The cyclic degradation model was adjusted again [13], and finally, the two degradations were revisited [14]. Modeling of total degradation was achieved, including cyclic and calendar types. Electrochemical models are usually very complex, so they are unsuitable for optimization problems because there are many possible solutions, a high number of evaluations, and therefore excessive calculation times. The models of [PERSON] et al. [21; 22], [PERSON] et al. [23], [PERSON] et al. [24], [PERSON] et al. [25; 26], and [PERSON] et al. [27; 28] are examples. A simplified electrochemical model is provided by [PERSON] et al. [29]. They proposed a hybridization of the single-particle model and an electrical equivalent model, with better computation time than a single-particle model and more accuracy than an electrical equivalent model. [PERSON] et al. [3], to include variations in operating conditions, determined that based on the work of [PERSON] et al. [12; 13], the heterogeneous use of a cell can be taken into account by considering the conditions of a specific use of the cell at an initial degradation state resulting from the end of its previous use. Other studies, on the health prognosis of lithium batteries, achieved very satisfactory results predicting batteries' end of life, such as one carried out by [PERSON] et al. [30]. In the present work, however, we have not considered that study, or similar ones, because no studies about the prognostics and health management applicable to our purpose are founded. Previous studies use some parameters adjusted to replicate experimental data, with expressions that depend on full equivalent cycles and on other abstract parameters. It is not clear the relation with cell properties and with usage conditions like _DoD_, _CR_; or temperature can be used to parametrize cell charges/discharges and rest time. The revised works are based on experiments performed that considered little cyclic usage and a reduced variety of conditions. In addition to the availability of degradation models, the cost associated with the reduction of battery life needs to be appropriately assessed. Several authors have applied various methodologies in order to assess this cost. [PERSON] et al. [17] used a ratio of total energy transferred over the complete lifetime of the battery. [PERSON] et al. [31] used the derivative of capacity loss over time and applied it at the current point in time of cell life. [PERSON] et al. [10] applied a similar strategy; they used the derivative of capacity loss over time and multiplied it by the time of use. Two indicators commonly used to measure battery degradation are the capacity fade (_CF_), which relates to capacity loss, and the power fade (_PF_), which relates to the increase of internal resistance. [PERSON] et al. [32] obtained the degradation cost (_Cost\({}_{dgg}\)_) from _CF_, _PF_, and battery replacement cost (_Cost\({}_{bat}\)_). _CF is_ a function that depends on several parameters of the equivalent circuit model: cell remaining capacity at a given time (_C_), capacity at the beginning of life (_C\({}_{BOL}\)_), and the cell capacity factor at the end of life compared to the beginning of life (_\(\mu_{CF}\)_).