# Estimating Degradation Costs for Non-Cyclic Usage of Lithium-Ion Batteries
## 1 Introduction  
This study aims to determine which cell technology models are most suitable for estimating batteries' degradation costs to determine the most appropriate model for use in optimization problems. It is common to consider two types of battery degradation: calendar and cyclic. Calendar degradation occurs when a battery is at rest, i.e., when it is neither being charged nor discharged. Cyclic degradation occurs when the battery is being charged or discharged. In lithium-ion batteries, the main reason for calendar degradation is an increase in the solid electrolyte interface layer on the negative (graphite) electrode. In cyclic degradation, the most crucial reason is the accumulation of lithium on the negative electrode [1]. In this work, we reviewed and applied the most suitable models for estimating calendar degradation and for evaluating cyclic degradation. Battery degradation costs can be an essential economic variable when designing an installation with battery power storage [2]. Several models in the technical literature can estimate batteries' degradation costs, sometimes including thermal and electrical aspects [3; 4; 5; 6; 7]. Depending on the significance of the degradation costs, the best solutions may or may not include extensive use of the batteries. These solutions often include random battery use that differs from standard cyclic usage. It is necessary to estimate degradation costs accurately. While selecting the models studied, the fact that the feedback of values in real-time is not available when mathematical models are used for optimization problems were taken into account [6; 7]. There are several papers with literature reviews of degradation models. [PERSON] et al. [3] analyzed several models, including those of [PERSON] et al. [8; 9; 10]; [PERSON] et al. [11; 12; 13; 14]; [PERSON] et al. [15]; [PERSON] et al. [16], and; [PERSON] et al. [17]. In other reviews, [PERSON] [4] studied and identified three main models: NREL, [PERSON], and MOBICUS; and [PERSON] et al. [5] reviewed adaptive aging models, which are ofinterest for on-board systems but not for our objective. The review carried out by [PERSON] et al. [6] is useful for optimization problems; this review presented deterministic models that are easy to apply, as they provide functions fitted to real data. Finally, a review by [PERSON] et al. [7] is of interest, in which several classifications of mathematical model studies related to battery degradation were presented. Several degradation models have been applied to optimization problems. [PERSON] et al. [2] applied several battery aging models developed for the optimization of lead-acid batteries, and [PERSON] et al. [18] also applied various battery life models to the optimization of off-grid systems, considering both lead-acid and lithium-ion batteries. Due to their increasing use, a detailed study of lithium-ion battery life models is necessary in order to be able to apply them to this type of problem. The degradation of lithium nickel cobalt aluminum oxide (NCA) cells is treated by [PERSON] et al. [15] with a simplified lifetime model applied to the optimization of electric vehicle charging. The battery life model estimates both energy capacity fading and power fading due to temperature, state of charge profile, and daily depth of discharge (_DoD_). In another study of lithium-ion batteries, [PERSON] et al. [19] fitted functions for capacity and internal resistance based on data obtained from [PERSON] et al. [20]. They modeled calendar loss and cyclic loss. Capacity fade and resistance growth due to calendar degradation depend on cell voltage (_V_), absolute surface cell temperature (_T_), and the actual time of cell life (_t_). Capacity fade and resistance growth due to cyclic usage depend on average voltage, _DoD_ or depth of cycle for NMC (_DoD_), and total Ah-throughput from the beginning of cell life (_Q_).