Optimal dispatch of Li-Ion battery energy storage, reviewing and considering cycling and calendar ageing models

Applied Thermal Engineering 265 (2025) 125597 Available online 17 January 2025 1359-4311/© 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Research Paper Optimal dispatch of Li-Ion battery energy storage, reviewing and considering cycling and calendar ageing models ☆ Andriy Vasylyev a,* , Alberto Vannoni a,b , Alessandro Sorce a a TPG, DIME, University of Genova, Italy b Materials and Generation Technologies Department, RSE, Milan, Italy ABSTRACT The growing share of renewable energy sources in the energy mix and the liberalisation of electricity markets has drastically affected the operation of electricity generators and grids. The transition from a fossil-fuel-based energy system to a renewable one has been significantly impacting the energy market, and energy storage systems have a pivotal role to play. In the coming years, a large amount of storage capacity is envisaged to be integrated into electricity grids to shave demand peaks, mitigate price volatility and face the growing demand for services to system operators. In such a situation, to properly manage these assets, and thus guarantee the economic viability of operating, it is essential to optimise their dispatch and define the best possible scheduling considering any hidden costs such as degradation. This paper focuses on Li-ion Battery Energy Storage (BES), as the fastest-deploying BES. A comprehensive literature study is carried out to provide a detailed review of ageing process modelling. Both the cycling and the calendar ageing processes are investigated considering the impacts of Depth of Discharge (DoD), State of Charge (SoC), temperature (T), C rate , number of cycles (N cycles ), and time (t). Moreover, the dependence of the efficiency of the charging and discharging phases on the current rate is remarkable. The dispatch optimisation is guaranteed by a proposed Mixed-Integer Linear Programming optimisation algorithm which considers the impact of degradation cost but is independent of the model selected from the literature. The literature review reveals that previous studies, dealing with battery dispatch optimisation, only include the cycling degradation explicitly in the objective function. In this work, the calendar degradation, impacting the battery capacity even when the battery is not in use, is also considered. This requires development of a strategy to properly weight the calendar contribution to fade, particularly when arbitrage opportunities are limited because of small daily price fluctuations. The proposed strategy involves the use of a factor R to adjust the cost associated with cycling. The optimal value of this factor is highly dependent on the economic conditions of the market, necessitating a sensitivity analysis to evaluate its impact. 1. Introduction The imperative shift from carbon-based to Renewable Energy Sources (RES) to mitigate global warming faces challenges. Despite the increasing use of RES, global greenhouse gas emissions are rising, as reported by the IEA [1]. Thus, more ambitious targets for carbon neutrality have been set, such as the EU’s ‘Fit for 55 ′ package [2]. Key targets include reaching 40 % of the total energy mix demand from RES by 2030. However, the stochastic and non–programmable nature of most renewable energy sources poses significant challenges in meeting instantaneous electricity generation, demand, and grid security requirements. In this scenario, the electricity grid is required to be more flexible, and large storage systems are considered essential to mitigate the variability of RES, temporarily shifting the load, at least on a daily basis, and providing services to the Transmission and Distribution System Operators. Depending on the storage technology, the potentiality for services varies, large pumped-hydro power plants have been traditionally employed for black start services, congestion relieving, replacement reserve, frequency restoration reserve, and even frequency containment reserve [3,4]. More recently, with the mothballing of large rotational inertia of spinning generators, the demand for ultrafast frequency regulation services and synthetic inertia has risen and Battery Energy Storage (BES) systems show a significant potential to play this role [5–7]. The BES potential is appreciated not only for its ability in fast frequency regulation. The independence of site requirements and the relevant round-trip efficiency are remarkable features. According to the Net Zero Emission by 2050 scenario of IEA, by 2030 680 GW of gridscale storage systems must be installed globally, while 16 GW had already been installed in 2021 [8], and most of this capacity is from Liion BES that is appreciated for its notable efficiency and reduced cost. This technology makes up 92 % of installed grid-scale BES in the US [9]. Besides lithium-ion batteries, flow batteries have emerged recently as a breakthrough technology for stationary storage as they do not show performance degradation for 25–30 years and are capable of being sized, allowing for a decoupling of investments in energy and power capacity, ☆ This article is part of a special issue entitled: ‘Sustainable Power Systems’published in Applied Thermal Engineering. * Corresponding author. E-mail addresses: [email protected] (A. Vasylyev), [email protected] (A. Vannoni). Contents lists available at ScienceDirect Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng https://doi.org/10.1016/j.applthermaleng.2025.125597 Received 6 March 2024; Received in revised form 19 December 2024; Accepted 15 January 2025 Applied Thermal Engineering 265 (2025) 125597 2 according to energy storage needs with limited investment [8]. Finally, BES systems are also playing an inescapable role in the mobility sector energy transition, including maritime, particularly for vessels with hybrid propulsion systems. The management of such energy systems presents many challenges, such as planning the use of the BES system to avoid partial loads on the engine [17]. Raggio M. et al. have discussed the use of Ni-Zn battery energy storage technologies to reduce peak demand by combining batteries with micro gas turbines [39]. However, for the most common BES types, battery life is a severe issue and, if BES is not optimally managed, the operational profits may not be enough to pay back the investment cost before the end of life. Indeed, lifespan is limited by the chemical degradation of the electrochemical cells of the battery pack. Many works in the literature investigate ageing of lithium-ion BES identifying two main ageing processes: calendar and cycling ageing; each of the processes is influenced by different idle and cycling parameters [10–12]. Calendar ageing is the progressive deterioration of a battery’s performance over time, regardless of its operational use. The main influencing factors are temperature and average State of Charge (SoC). Prolonged storage or operations at elevated temperatures accelerate calendar ageing favouring chemical reactions within lithium-ion batteries that compromise performance and cause capacity loss. Similarly, maintaining a lithium-ion battery at a high SoC for long periods contributes to the calendar ageing process, increasing the speed of the aforementioned chemical reactions within the battery. Long-term storage recommendations advocate keeping lithium-ion batteries at a low average SoC [10,12]. Cycle ageing is caused by the repetitive charge and discharge cycles inherent in the operational use of a battery. Each cycle causes a gradual loss of capacity and performance. The Depth of Discharge (DoD), i.e., the extent the ratio of battery capacity actually discharged in each cycle, has a significant impact on cycle ageing. Deeper discharge cycles generally cause accelerated capacity fade, whereas shallower discharge cycles, where the battery is not fully discharged, contribute to extending the life and the cumulative energy exchanged at the end of life. Charging parameters such as voltage and current also affect cycling ageing. Charging Li-ion batteries at higher voltages, or current rates, contributes to degradation over time. The total number of charge/discharge cycles, or cycling history, has a significant effect on wear [13–15]. Moreover, some authors propose non-linear models considering that the first cycles have a higher degradation impact [10]; however, even many linear formulations are proposed, according to which the degradation rate for each cycle is constant [11]. An effective model describing the real degradation process until the battery end-of-life is essential to develop a management strategy able to maximise the economic viability of operating the battery also considering the hidden costs of degradation [13]. For both cycling and calendar degradation processes, several models have been proposed in the literature, presenting different correlations that express the relationship between different parameters and battery degradation. Section 2 reports a detailed literature review of existing models, discussing how the impact of different parameters has been quantified and described through equations. In most countries, grid-scale batteries must operate in a liberalised electricity market design in which the most common business model consists of shifting the load if arbitrage opportunities subsist. The driving factor for arbitrage is electricity price fluctuation, which can be characterised by a period and a magnitude, consequently, the economic viability of storage strongly depends on the market scenario [16]. As mentioned before, the provision of services is also more and more promising as a source of revenue, however, it is characterised by considerable uncertainty and dependency on the local regulatory framework, and involves negligible amounts of energy if compared to arbitrage. Therefore, the focus is on lithium-ion BES, selected as the most common battery technology, while for arbitrage opportunities the fluctuations in the electricity day-ahead market, which is common to almost every market design, are considered as the base load for the business models. Some papers present optimisation models for the dayahead market, considering the influence of battery cycling as the main degradation effect and transforming it into a cost parameter to be included in the optimisation [18–20]. However, accounting also for calendar ageing is not investigated equally, to the best of the authors’ knowledge, the only paper integrating both cycle and calendar degradation costs into its optimisation model is by Bahloul M. et al. [21], presenting an approach translating degradation information into cost information for a hybrid PV-BES system. However, cycling and calendar costs have been linearised to reduce the complexity of the information in the optimiser. In addition, the cost functions are simplified as they do not fully account for various battery operating parameters such as DoD, SoC avg , t, T and C-rate. Amini A. et al. investigated the impact of both cyclic and calendar ageing during a battery life cycle [22]. They defined the effect of operating parameters on battery degradation, to estimate the battery’s State of Health (SoH). However, the cost-minimising objective function they proposed does not include the impact of dynamic battery degradation costs, as they considered a fixed cost for Nomenclature BES Battery Energy Storage CAPEX Capital Expenditures Crate Current Rate Crateref Reference Current Rate Cfade Capacity Fade Cfadecyclig Capacity fade due to Cycling Cfadeidling Capacity fade due to Idling Costcycle Cycle Cost function Costidele Idle Cost function Costcyclevirual Virtual Cycle Cost Costdegr Degradation Cost function DoD Depth of Discharge EoL End of Life iInterest rate LFP Lithium Iron Phosphate MILP Mixed Integer Linear Programming Ncycle Maximum Number of Cycles ncycle Number of Cycles η charge Charge efficiency η discharge Discharge efficiency OP Operational Profits OPnet Net Operational Profits Obji,j,tObjective Function Rfactor Virtual cycling cost factor SoC State of Charge SoCref Reference State of Charge ΔSoCiDiscretised State of Charge SoH State of Health TTemperature Tref Reference Temperature ttime xOptimised binary variable ZP Zonal Price A. Vasylyev et al. Applied Thermal Engineering 265 (2025) 125597 3 maintenance operations proportional to the BES nominal power. Instead, this work defines a comprehensive approach to evaluate the profitability of a BES system in various scenarios, simultaneously considering cycling and calendar degradation. Discretising the objective function, and optimising it through an MILP algorithm, also allows consideration of nonlinearities of degradation processes. The paper focuses on the point of view of a battery owner/manager operating in a liberalised electricity market, assessing the economic viability of energy arbitrage in the wholesale day-ahead market as a core business. The proposed method is designed to be independent of specific battery characteristics and versatile for any operational curves, i.e., degradation curves or efficiency curves. First, a comprehensive literature review is conducted in Section 2 to present the nature of degradation phenomena and the models proposed in past studies discussing the impact of key operating parameters on battery degradation. The keywords ‘number of cycles’, ‘capacity fade’, ‘cycling degradation’, and ‘calendar degradation’from 2010 onwards were used for the literature search. This review was essential to wisely model the BES, Section 3, and transform a degradation model into economic parameters, such as cycle cost and calendar degradation cost, used as inputs for the optimisation algorithm. The proposed MILP algorithm is described in Section 4, the objective function is defined using a discretised approach, enabling the BES system to operate according to predefined operating modes (i.e., net power output levels), thereby allowing holding non-linearities of complex objective functions and constraints [23], for example those following the proposed model of efficiency. Moreover, this approach well suits the need to enhance the versatility and adaptability of the proposed optimisation framework for any degradation or efficiency model and future purposes of integrated optimisation of energy arbitrage and grid service provision. The optimisation methodology is described in detail, along with the management of the degradation and efficiency curve models inside the optimisation approach. The model considers both cycling and calendar ageing dependencies on the state of charge and depth of discharge, and the dependency of efficiency on the current rate. Additionally, the progressive fade in capacity is accounted for to determine the actual amount of dispatchable energy. Section 5 concludes the methodology presentation by illustrating the scenarios investigated and the indicators assessed. Finally, Section 6 applies the presented optimiser to various market scenarios, examining the impact of average price and price variability on economic Key Performance Indicators (KPIs), such as capacity fade and annual specific Operational Profits (OP). This sensitivity analysis establishes a relationship between operating conditions and market scenarios. Under profitable conditions, battery management becomes more straightforward, ensuring that the battery operates only when operational profits exceed the hidden costs of degradation. This guarantees that, by the end of the battery’s life, the investment cost is recovered, and net profits are generated. Conversely, when arbitrage opportunities are diminished due to a flatter price profile, battery management becomes more challenging. If the battery remains idle unless potential operational profits outweigh degradation costs, calendar ageing will ultimately lead to higher economic losses by the end of its lifespan. 2. BES ageing processes review BES performance shows a decay of the nominal parameters over time, the fade of capacity (i.e., the maximum amount of energy that can be stored) due to the ageing processes has a major impact on the BES operating conditions, while power fade (i.e., the decrease of maximum power achievable during the charging and discharging phases) is considered negligible [24]. The rated capacity continuously decreases because of two major contributions, first the impact of each cycle, i.e., cycling ageing, and second in idle conditions a degradation occurs defined as calendar ageing. The State of Health (SoH), Eq. (1), is defined as the ratio of rated capacity on the nominal value, i.e., the complementary of fade. Typically for most lithium-ion BES and applications such as energy arbitrage on a daily basis, or integration in smart grids, the End-of-Life (EoL) is fixed at 20 % of fade, SoH =80 %. Under this threshold value, performance is too poor to keep BES operating. SoH =Crated Cnom •100% =100% −Cfade[%](1) Cfade =Cfadecycling +Cfadeidling [%](2) During both cycling and calendar degradation processes, different operating and idle parameters strongly impact the capacity degradation rate, which reduces the lifespan of a battery, the different dependencies are shown in Table 1. Eq. (3) and Eq. (4) report all the parameters impacting the calendar and cycling capacity fade, while the relationship (e.g., linear or exponential) between the fade and these parameters is reported in Table 1 according to different references. The model then selected for the analysis in Section 4 is highlighted in bold, likewise for Tables 2 and 3 in Sections 2.1 and 2.2. Cfadecycling =f(DoD,Crate,T,SoCavg,ncycle)[%](3) Cfadeidling =g(SoC,T,t)[%](4) EoL =∫Ncyclemax 0 f’(DoD,Crate,T,SoCavg)⋅dncycle[%](5) Ncyclemax =EoL fʹ(DoD,Crate,T,SoCavg)(6) EoL =∫tmax 0 g’(SoC,T)⋅dt[%](7) tmax =EoL gʹ(SoC,T)[h](8) DoD =SoCstart −SoCend (9) Equation (5) expresses the cumulative degradation of the battery before reaching end-of-life, if fis linearly dependent on cycles already performed, n cycle ,N cycle max can be explicated as in Eq. (6), where the function f’is derivative of fwith respect to n cyclce . EoL is the End-of-Life criterion, e.g., 20 % of capacity fade. In Section 3.1 the function fis presented specifically for the adopted model in the development of the optimisation algorithm. Analogously Eq. (7) defines t max , i.e., the calendar lifespan of the battery after which the EoL is reached even without performing any cycle. If function gis linearly dependent on the time t, t max can be explicated as in Eq. (8), where the function g’is derivative of gwith respect to t. In Section 3.1 the calculation is presented specifically for the model adopted in the developed optimisation algorithm. For f’ the unit of measurement is expressed in percentage points, pp, per cycle, while for g’pp per hour. Eq. (9) defines the Depth of Discharge (DoD) as the difference between the initial and the final State of Charge (SoC) for a cycle. 2.1. Cycling capacity ageing model As a battery undergoes charging and discharging cycles, its electrodes slowly degrade and become less effective at holding and releasing energy, causing cycling ageing. Many authors have analysed the factors influencing cycling ageing [10–15,24,25,27–29]. All of them agree that the DoD is of primary importance, some [10,27,28] describe the impact of temperature, nevertheless, this dependence is neglected by this paper since a temperature control system is assumed to be integrated with the BES and its impact as an auxiliary electrical consumption is considered on the charging and discharging efficiency model (Subsection 3.2). Since the models have different dependencies on the same operating parameters, e.g., polynomial or exponential, it was decided to implement a A. Vasylyev et al. Applied Thermal Engineering 265 (2025) 125597 4 uniform nomenclature for all fitting coefficients of the various degradation functions for better comparison and understanding of different degradation models. The subscripts of the fitting coefficients refer to the operational parameter, for example, a DoD refers to the fitting coefficient regarding the dependence on the depth of discharge. Only Eq. (12) has been presented with a specific discussion of the degradation parameters, as it contains coefficients that are not easily generalisable like those in other models. This is because the proposed model considers combining the contributions of the two degradation mechanisms into a single degradation function. To the best of the authors’knowledge, the first work in the open literature that defines a degradation curve, describing the relationship between the DoD and the admissible number of cycles N cycle max , was provided in 2011 by Zhou et al. [25] for V2G (Vehicle to Grid) applications. In this work, the impact of some parameters (temperature and DoD) was investigated for different battery energy storage types (lithium-ion, lead-acid, and nickel metal hydride). Concerning the temperature, Zhou suggests using a hyperbolic correlation, resulting from the Arrhenius equation [30], meanwhile, the DoD dependence is a logarithmic, thus the number of cycles exponentially increases as the DoD decreases. Mallon et al. [26], proposed new coefficients for the equation reported by Zhou to describe the DoD influence on the number of cycles. In their work, the temperature dependence was not mentioned. The proposed equations were used in heavy-duty electric powertrain applications, assessing the impact of the installation of solar panels on the bus’s roof and sides on the battery’s lifespan. Eq. (10) describes the ageing relationship previously suggested by Zhou et al. and later adopted and revised by Mallon. Cfadecycling =aDoD⋅DoDbDoD ⋅e(cDoD⋅(1−DoD) )⋅ncycle (10) Stroe et al. [10,14,24] analysed in detail the influence of more, compared to Zhou, operating and idling parameters, developing models not only for battery capacity fade but also for battery power degradation, for specific lithium-ion phosphate/graphite batteries (LiFePO 4 or LFP). They adopted an equivalent-electrical circuit performancedegradation modelling approach to develop the lifetime models, considering both effects of cycling and calendar processes; in Section 2.1.2 the calendar ageing model will be discussed in more detail. The first application in which they had to develop the degradation model relates to an integrated wind power plant to compensate for the stochastic nature of wind energy [10]. The dependence on the number of cycles is polynomial (degree 0.5), while to quantify the dependence on the DoD the authors used a power-law function and the temperature impact on capacity fade is described by an exponential function. Finally, the last parameter considered by Stroe and colleagues is the average State of Charge (SoC avg ) during the cycle and it has a decreasing exponential impact: a high average SoC generally has a positive effect on the degradation rate increasing the lifespan. This model with minor variations is used in other works [14,24,31] to perform techno-economic optimisation of a Virtual Power Plant (VPP) dispatch, frequency regulation in the Danish electricity market or lifetime estimation for fully electric vehicles. In the next equation, Eq. (11), the relationship described by Stroe et al. is explicated. Cfadecycling =a⋅(aDoD⋅DoDbDoD )⋅e(aSoC⋅SoCavg)⋅nan cycle⋅(aT⋅ebT⋅T)(11) Xu et al. [12] in their first work analyse both ageing processes, considering all the operating and idling parameters. This model integrates both theoretical considerations and empirical evidence in its analysis. The degradation model focuses on the two primary stress functions: calendar ageing and cycling ageing. It is used for a technoeconomic analysis of Primary Frequency Regulation (PFR) and Table 1 Cycling and idling parameters in different degradation models. Cycling Ageing Calendar Ageing Ref No. DoD n Cycle SoC avg T C rate SoC T time [25] exp linear −hyperbolic − − − − [26] exp linear − − − − − − [10,14,24] power root-square exp exp −exp exp exp [12,27] exp exp exp exp exp exp exp exp [11,28] poly linear ¡poly ¡poly poly linear [15] poly linear − − − poly −linear [13] exp linear − − − − − − Table 2 The fitting coefficient for each Cycling ageing model. Reference Coefficient [25] [26] [10,14,24] [31] [12,27,29] [11,28] [15] [13] a− − 2.64e-2 2.4e-4 −¡− − aDoD 2.88e-1 7.3e-1 1.23e-2 2.982e-2 8.95e4 ¡4.72e-5 4.83e-4 5.2e-5 bDoD 7.95e-1 6.79e-1 7.162e-1 4.904e-1 −4.86e-1 9.62e-5 2.38e-5 6.8e-1 cDoD 0−1.61e1 − − − 7.28e-4 ¡− − 1.6e1 aSoC − − − 1.943e-2 −1.04e1 ¡− − acrate − − − − 2.63e-1 ¡− − aT− − 4e-3 −6.93e-2 3.62e-3 − − bT− − 1.705e-2 2.717e-2 −¡1.05e-1 − − cT− − − −1.93 − − an− − 0.5e-1 0.5e-1 −¡− − Table 3 The fitting coefficient for each calendar ageing model. Reference Coefficient [12,27,29] [10,14,24] [31] [11,28] [15] aSoC 1.04e1 1.639e-1 1.9e-2 6.02e-6 4.34e-6 bSoC −7.38e-3 8.23e-1 1.35e-5 2.73e-5 cSoC − − 5.195e-1 1.85e-5 1.45e-5 aT6.93e-2 1.977e-11 3.258e-9 2.31e-3 − bT−7.51e-2 5.087 ¡4.01e-2 − cT− − 2.95e-1 ¡1.21 − at1 8e-1 8e-1 11 A. Vasylyev et al. Applied Thermal Engineering 265 (2025) 125597 5 Dynamic Frequency Regulation (DFR) provision by storage. The models’ fitting parameters are defined for Lithium Manganese Oxide (LMO). Unlike the other proposed models, the calendar and cycling ageing does not impact linearly on the faded capacity that is expressed as in Eq. (12). Cfade =1−psei⋅e−rsei⋅fd+ (1−psei)⋅e−fd(12) Where: p SEI and r SEI are two constants characterising different BES models and correlated with the formation of a combination of stress functions, While f d is a combined degradation function consisting of two terms: f cycling is the contribution due to the cycling ageing process, and f calendar is the contribution due to the calendar ageing process. In this subsection the cycling contribution is analysed, while the parameters impacting f calendar will be discussed in subsection 2.2. f cycling , Eq. (10) is the product of 4 functions describing the impact of the DoD, SoC, C rate , and temperature. f cycling has a polynomial dependence on the DoD that plays the major role, Eq. (11). The other parameters have an exponential impact on f cycling when they deviate from the reference values Eq. (13–17). For Stroe et al. [10], during cycling, a high state of charge has a negative effect on life consumption, while a low state of charge has a positive effect, while for Xu et al. [12] there is a symmetrical trend with respect to the 50 % state of charge, increasing the impact of extreme SoC. This article is the only study reporting the dependence on the C rate (i.e., the current rate at which the charge and discharge phases occur, defined as a ratio between the charging/discharging power and the nominal capacity) fcycling =fDoD(DoD)⋅fSoC(SoCavg)⋅fCrate (Crate)⋅fT(T)⋅ncycle (13) fDoD(DoD) = (aDoD⋅DoDbDoD +cDoD)−1(14) fSoC(SoC) = eaSoC⋅(SoC−SoCrefavg )2 (15) fCrate (Crate) = eaCrate ⋅(Crate−Crateref )(16) fT(T) = eaT⋅(T−Tref )⋅Tref T(17) In another work by the same authors [27], they again perform a techno-economic analysis for PFR and DFR, but they consider different lithium-ion battery technologies such as Lithium Iron Phosphate (LFP), Lithium Nickel Manganese Cobalt Oxide (NMC), and Lithium Manganese Oxide (LMO). Sayfutdinov et al. [11,28] presented a polynomial model to describe the dependence on both cycling and calendar ageing mechanisms. The model was used to optimise the size of the energy storage system considering the impact of degradation over the years. Four different lithium-ions were compared considering different fitting coefficients for the models: Lithium Iron Phosphate (LFP), Lithium Nickel Manganese Cobalt Oxide (NMC), Lithium Manganese Oxide (LMO), and LithiumTitanium-Oxide (LTO). The only two parameters that were taken into consideration were the DoD and the temperature, for both a quadratic function was utilised to describe the impact on the capacity degradation. Fallahifar et al. [15] utilise Sayfutdinov et al. models to characterise the capacity degradation function for cycling and calendar fade, while defining the fitting coefficients for their specific batteries. The degradation model is applied to establish the optimal scheduling in microgrid applications, taking into account capacity fade. Various lithium-ion technologies are considered, including Lithium Iron Phosphate (LFP), Lithium Nickel Manganese Cobalt Oxide (NMC), Lithium Manganese Oxide (LMO), and Lithium-Titanium-Oxide (LTO). The next equation, Eq. (18), explicates the relationship for Sayfutdinov et al. and Fallahifar et al. as described above. For this case study, the second-order polynomial used to describe the dependence on Tis equal to 1, neglecting the temperature effect. Cfadecycling =(aDoD⋅DoD2+bDoD⋅DoD)⋅(aT⋅T2+bT⋅T+cT)⋅ncycle (18) Lee et al. [13] considered only the dependence of DoD on capacity degradation, expressed by an exponential relationship. The proposed model was used to define the optimum way to manage the cycling cost during scheduling problems. In the next equation, Eq. (19), the relationship described earlier for the Lee et al. models can be seen. Cfadecycling =aDoD⋅DoDbDoD ⋅ecDoD⋅(1−DoD)⋅ncycle (19) Table 2 shows the coefficients for the models discussed above, thus making it easier for the reader to compare different ones. It is important to consider whether in the different works the coefficients are fitted for percentage DoD or dimensional values. In Fig. 1 the reviewed cycling ageing models are compared calculating the maximum number of cycles N cycle max that a battery can perform before reaching the EoL criterion, all models’fitting coefficients for different batteries are considered for Li-ion technology. The main models presented in this paragraph are represented with continuous lines with a different marker from the others [11–14,25], while those based on these models but suggesting different fitting coefficients have the same marker but a dotted line [15,26]. The logarithmic scale adopted on the y-axis is significant for the considerable prediction gap between some models. Only for the Stroe et al. models [10,14,24] does the progressive number of cycles have a nonlinear impact on the degradation; this means that the first cycles have a higher degradation rate than the last cycles, this information is not directly representable in the figure. To plot the relative curve, N cycle max is calculated numerically, through Eq. (5), for different DoD assuming C rate equal to 1, temperature equal to 25 ◦C, and an intermediate value of SoC avg equal to 60 %. Xu et al., Stroe et al., Sayfutdinov et al., and Fallahifar et al. [11,15,28] also report in the same articles an analysis of the calendar degradation model; these references allow for consistency of models as they refer to the same lithium-ion battery energy storage technology and are reviewed in detail in the subsequent section. 2.2. Calendar capacity fade model Calendar ageing is a natural degradation behaviour of the chemical component inside the battery energy storage, it is due to the occurrence of collateral reactions generated by the thermodynamic instability of constituent materials [10]. Furthermore, the thermodynamic stability of Fig. 1. Maximum number of cycles at fixed operating conditions, in function of main operating parameter DoD. A. Vasylyev et al. Applied Thermal Engineering 265 (2025) 125597 6 the negative electrode is pivotal since graphite is not electrochemically stable when used with most electrolyte types. Calendar ageing is generally less investigated than cycling in the open literature [15,24,27,28], nevertheless a proper model, also including this effect, is essential if the BES is forced to long idling periods, e.g., because the market is not profitable enough to perform arbitrage. Stroe et al., Sayfutdinov et al., and Fallahifar et al. point out that an idling SoC is the key parameter and agree on the detrimental effect of a high SoC. As summarised in Table 1, according to Stroe et al. [10,14,24], Sayfutdinov et al. [11,28] and Fallahifar et al. [15] a second-order polynomial dependence is defined while for Xu et al. [12,27] the dependence of capacity fade on SoC is a symmetrical exponential trend with respect to 50 %, so describing a non-monotonic trend that implies an optimal SoC (50 %) that minimises calendar ageing during idling periods. Even when it concerns the influence of temperature on the calendar capacity fade Stroe et al. [10,14,24] and Xu et al. [12,27] maintain the same exponential dependence, while Sayfutdinov et al. [11,28] and Fallahifar et al. [15] maintain a second-order dependence. The proposed models differ in the kind of dependency on time, if [15,27,28] propose a linear relationship, the dependence indicated by Stroe [10] reports a power law dependence with exponent atequal to 0.8 affirming that calendar degradation progressively slows down over time. Eq. (20) describes Stroe et al.’s model; then Eq. (21) describes Xu et al.’s model for the linearised time degradation function [12]; and, finally, Eq. (22) explicates Sayfutdinov et al.’s and Fallahifar et al.’s models [11,15]. Cfadeidling = (aT⋅ebT⋅T+cT)⋅(aSoC⋅ebSoC⋅SoC +cSoC)⋅tat(20) fcalendar =at⋅eaSoC⋅(SoC−SoCref )2 ⋅eaT⋅(T−Tref )⋅Tref T⋅t(21) Cfadeidling =(aSoC⋅SoC2+bSoC⋅SoC +cSoC)⋅(aT⋅T2+bT⋅T+cT)⋅t(22) Fig. 2 shows the trends associated with the revised calendar ageing models by calculating, fixing temperature at 25 ◦C, the maximum lifetime t max that can be achieved before the battery reaches the EoL criterion. In Eq. (5) the x-axis shows the SoC at which the battery is maintained over the years, and it can be seen that Xu et al. [12,27] have proposed a non-monotonic model, with a maximum for average SoC equal to 50 %. Xu’s model is comparable to the other models for a high average SoC idling period, while for low SoC it defers significantly, being more cautionary in terms of lifespan expectancy. Analogously to cycling ageing, to plot the Xu curve t max was assessed for different SoC numerically, through Eq. (7), assuming T equal to 25 ◦C. Table 3 shows the coefficients for the models discussed above, thus making it easier for the reader to compare different ones. It is important to consider in different works if the coefficients are fitted for percentage SoC or dimensional values. According to the other models, it is always advisable to keep the battery at a low SoC to reduce the rate of degradation over time. As for the cycling models, the same logic is used to visualise the models mentioned: continuous lines for original models [10–13,25], while dotted lines for the same models but revised with different coefficients [15,26]. It is important to underline that each author defines models considering different time units, this can lead to difficulty in comparing different models. In this work, in order to make a better comparison, the calendar capacity fade models are reported with an hourly time unit, to define the idle cost per hour in accordance with time discretisation of the optimisation carried out in Section 4, as the Day Ahead Market (DAM) electricity market is based on hourly time resolution. 3. BES model This section describes the BES model. The MILP algorithm is independent of the BES model. Indeed, the proposed approach can be replicated for different batteries just by modifying the assumption presented in this section. 3.1. Adopted cycle and calendar fade model For the analysis carried out in this paper, a model is selected that consistently describes the calendar and cycling ageing; among the models reviewed in Section 2, Stroe [10,14,24], Xu [12,27,29], and Sayfutdinov [11,28] are eligible options. Since the impact of the average SoC on the cycling agreeing is not certain, controversial, and in any case of secondary importance, the model proposed by Sayfutdinov [11] was selected, because of its independence from this parameter. Consequently, Eq. (6), 8) indicate the theoretical limits of cycles and idling time independently of the considered ageing model, considering an EoL criterion of 20 % of capacity fade. Adopting the coefficients proposed by Sayfutdinov et al. [11] and reported in Table 2, the cycling and calendar limits are quantified respectively in 4109 cycles with DoD =80 % and 2.3•10 4 h (approximately 25 years) of idling at SoC =20 %. Table 2 reports the coefficients of the Eq. (23), 24). Ncyclemax(DoD) = EoL aDoD⋅DoD2+bDoD⋅DoD (23) tmax(SoC) = EoL (aSoCav ⋅SoC2+bSoC⋅SoC +SoC)(24) 3.2. Efficiency model When considering a BES operating on the electricity grid, it is important to consider the global efficiency, from alternate current to alternate current (AC-AC) which is lower than the value provided by some manufacturers concerning the battery itself (DC-DC). Rancilio et al. [32,33] identify the C rate and the SoC as the factors with the highest impact on AC-AC efficiency. However, the results reported show that the effect of SoC is negligible if compared to the C rate contribution. The temperature at which the battery operates during its lifetime has a strong effect on its operating parameters, lifetime, and on its fade rate. Yang T. et al. conducted an experimental campaign with 21,700Li-ion battery packs, under several operating conditions, e.g., C rate and layout pack configuration [34]. Their work pointed out that a high C rate can lead to excessive battery heating, so to keep the temperature within optimal operating limits it is necessary to consider integrating Battery Thermal Management Systems (BTMS). Several solutions for BTMS were Fig. 2. Lifespan at fixed idling conditions for the battery energy system, in function of the idling State of Charge (SoC). A. Vasylyev et al. Applied Thermal Engineering 265 (2025) 125597 7 reviewed in the literature by Kumar Thakur A. et al. and Haosong H. et al. [35,36]. According to the reviews, an air-conditioned system may only be sufficient for quasi-static applications, C rate limit equal to 1, for fast charge and discharge applications it is necessary to consider higherperformance systems such as solid–liquid phase transition. As the proposed system has to operate in the DAM electricity market, there is no real advantage in installing batteries with high C rate , as the market time unit is one hour. The air conditioning in the battery pack is therefore considered sufficient to maintain the temperature limits. The effect of the auxiliaries consumption was successively integrated into the efficiency curve between 0 ◦C and 1 ◦C. The dependence of efficiency on the C rate is then considered interpolating the reported experimental data [32,33]. Consistently with the reference, discharge, and charge efficiencies are assumed to be equal. The associated trend reports a maximum approximatively for C rate =0.5, moving toward higher current causes a slight decrease because of the parasitic current losses and greater effort required to maintain the cell design temperature. Decreasing C rate below 0.2 forces the energy conversion auxiliary systems to operate in strong off-design conditions. Consequently, a significant drop in charging and discharging overall efficiency occurs at reduced C rate .Table 4 reports the efficiency values as a function of the discretisation adopted and C rate . 4. MILP optimiser accounting for degradation costs 4.1. From an ageing model to a cost quantification The present section focuses on the optimisation algorithm. The optimiser implementation can be generalised regardless of the specific model selected to describe the ageing process or the factors impacting the BES efficiency. In the previous sections 3.1 and 3.2, the degradation and efficiency models were selected to conduct a sensitivity analysis for the MILP model under varying market scenarios. The purpose of this section is to illustrate how ageing, and efficiency are considered within the optimisation stage. To consider the degradation cost associated with the capacity fade due to cycling and idling, ad hoc cost functions are defined. These functions, Eq. (25–26), express the cost [ € ] associated with each unit, i.e., a cycle for cycling ageing and an idling hour for calendar ageing. Costcycle(DoD) = CAPEX Nmax(DoD)(25) Costcalendar(SoC) = CAPEX tmax(SoC)(26) Fig. 3(a) expresses the cycle cost according to the model reviewed in Section 2.1 with the same assumptions made for plotting Fig. 1. Analogously, in Fig. 3(b), the calendar cost is expressed, for the models reviewed in Section 2.2, as a function idling SoC with the same assumptions made to plot Fig. 2. Additionally, to the previously assumed values, CAPEX has been fixed at 150 k € /MWh [37]. In the previous figures, all the models analysed are presented for a general comparison. The model by Sayfutdinov et al. is selected for the optimisation phase and is highlighted with a thicker red line. 4.2. Definition of virtual cycle cost Because of calendar ageing, the lifetime is not exclusively related to the BES operation, in order to maximise the net operating profits over the whole lifetime it is advantageous to introduce a virtual cost increasing or decreasing artificially the actual cycle cost during the optimisation process adopting different operational strategies. Costcyclevirtual(DoD,Rfactor)=Costcycle(DoD)⋅Rfactor (27) The Cycle Cost defined in Eq. (27) is multiplied by R factor which can increase or reduce the cost associated with one cycle at fixed DoD creating a virtual cost. R factor is defined as non-negative real and the sensitivity to this parameter is investigated in this paper in the range between 0 and 2 with a discretisation step of 0.1. R factor equal to zero means performing the analysis with the cycle cost equal to zero, so the battery will perform all the cycles that pay off the inefficiencies of the system, while R factor equal to 2 means that the virtual cycle cost is double the real cost. The objective of a variable virtual cycle cost is to evaluate whether the overall BES performance can be enhanced by increasing the number of operating cycles during the low profitability market, reducing the operational losses, avoiding that, with the BES idling, the EoL is reached due to calendar ageing. On the other hand, in market periods characterised by high profitability, an artificial increase in cycle costs can lead to maximising final net OP by selecting just the more profitable cycles. It must be stressed that the change in cycling cost is just a matter of fine tuning one of the main optimisation parameters, while the actual degradation cost is computed without considering the Rfactor to correctly evaluate the economic performance. 4.3. Optimisation algorithm description The MILP algorithm is implemented through a discretised approach function since it is demonstrated that it is capable of handling the complexity of the problem without losing efficiency in finding the optimal solution [23]. Moreover, it can be adopted in future works for integrated optimisation of the day-ahead and ancillary service markets [38]. Eq. (28) illustrates a general definition of the MILP optimisation problem. Looking at the terms in the equation: f T is the vector coefficient for the objective function, and xis the optimised variable; in this work it is binary. To describe the problem considering technical and economic constraints, matrix Ais used for the linear constraints definition and lb and ub for the lower and upper bounds. The problem of an optimal arbitrage strategy is described by the objective function, i.e., the net operational profits defined by Eq. (29), that account for the Cost cycle and the Cost idle , as defined by Eq. (25–26) and represented in Fig. 3. In Eq. (29) two terms can be distinguished, the first concerns the operational profits (OP t ) that are exclusively related to the charging and discharging phases, and the second quantifies the degradation cost (Cost degrt ), determined as the sum of the contributions of both cycling and idling ageing. Eq. (25) and (26) respectively for each time, t. In this way, the algorithm defines the economic optimum that also pays back the cycle cost associated with degradation. Indeed, capacity fade is a cost since, once the EoL criterion is reached, the battery must be replaced. While OP represents the actual cash flow for the operator, what is worth maximising is the OP net to avoid performing cycles characterised by too low OP to justify the lifetime consumption of the BES. Such a strategy guarantees maximising earnings, net of CAPEX, over a long period. On a yearly basis, the summation of Cost degr can be considered as a provision for new investment at the end of BES life or amortisation of already paid CAPEX. Eq. (29) shows how the operating phases of the battery are divided and how the objective function coefficients (f T ) are defined. In the matrix representation, it can be seen that the elements on the diagonal represent the idling phase (i=j), while the upper matrix represents the charging phase Table 4 Efficiency values for the discretised C rate , for MILP optimisation [32,33]. C rate 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 η 0.09 45.79 74.40 86.75 93.83 94.49 93.13 91.96 90.94 A. Vasylyev et al. Applied Thermal Engineering 265 (2025) 125597 8 (i < j), and finally the lower matrix represents the discharging phase (i >j). In (30), there are: Ithe final state of charge reached by the battery, j the initial state of charge, and finally toptimised hour. Since the optimisation is carried out considering the day ahead market (DAM), the time discretisation is set to one hour in accordance with the market time resolution. Optimisation of dispatch is performed subsequently day by day, considering a forecasting rolling horizon, H, of 36 h as the best trade-off between computational time and global optimum identification [23]. min xfTxs.t.{A⋅x≤b lb ≤x≤ub (28) OPNet =∑ H=36 t=1(OPt−Costdegrt) =∑ H=36 t=1((Revenuedisch −Costch) − (Costcyclevirtual +Costcalendar))t (29) The problem formulation for the MILP is presented in Eq. (29), While the mentioned discretised character of the problem can be appreciated through the matrix approach of Fig. 4 that provides a visualisation of the optimised binary variable x, which multiplies the coefficient matrix defined in Eq. (30). For each time step t, a squared matrix is defined, columns indicate the initial SoC while rows determine the final. At each time step, one and only one element of the matrix must be set to 1 selecting a univocal operational mode (i.e., a SoC variation). In Fig. 4, rows are characterised by index iand columns by index j. The matrix dimension depends on the discretisation resolution of SoC; by adopting a finer step the number of columns and rows of N SoC increases and the computational effort rises. As a mere example, in Fig. 4, SoC is discretised with a step of 20 percentage points, as the selected mode is i =1 and j =1, which means that the battery is maintained at 100 % SoC. Analogously to x, the Obj array and its terms, shown in Eq. (30), can be defined with 3D matrix shape. For each position in the matrix, Cost cycle is determined by the associated ΔSoC,Cost idle will be defined by a diagonal matrix determined by the SoC associated with the index, and OP is computed from the ΔSoC of each operational mode, the electricity zonal price (ZP), which depends on the time, and the charging or discharging efficiency, depending on C rate and so on ΔSoC. Eq. (30) reports in detail how each element of Obj is computed according to the objective function. Obj is then reshaped as a 1-D array before running the MILP algorithm. The constraints of consistency between the final SoC of time t and the initial SoC at the time t +1 is expressed by Eq. (31) while Eq. (32) imposes selecting one, and only one, operational mode at every time step, finally, Eq. (33) imposes the SoC at the first time step at the first day and guarantees that the initial SoC, i=1, at the first hour of the day n is equal to the final SoC at the 24th hour, j=24, of the day n–1 , a detailed description of the rolling horizon optimisation logic is provided in [23]. xis imposed as a binary variable. Fig. 3. Cycle cost associated with each specific occurred cycle (a). Calendar cost associated with each specific idle hour (b). The CAPEX (Capital Expenditure) cost was assumed to be € 150 k/MWh, to calculate cycle and calendar function costs [37]. Fig. 4. Matrix visualisation of optimisation variable x. Obji,j,t= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ΔSoCi,j 100 ⋅Cnom⋅ η disch(ΔSoCi,j)⋅ZPt−Costcyclevirtual(DoDi,j,Rfactor) − Costcalendar(SoCmeani,j),ΔSoCii>j −Costcalendar(SoCi,j),ΔSoCii=j (ΔSoCi,j 100⋅ η ch(ΔSoCi,j)⋅Cnom⋅ZPt−Costcalendar(SoCmeani,j))ΔSoCii<j (30) A. Vasylyev et al. Applied Thermal Engineering 265 (2025) 125597 9 ∑ NSoC j=1 xn,j,t=∑ NSoC i=1 xi,n,t+1;∀t∈T∧ ∀n∈NSoC (31) ∑ NSoC i=1∑ NSoC j=1 xi,j,t=1;∀t∈T(32) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ∑NSoC j=1x1,j,1=1,dayn=1 (∑NSoC j=1xn,j,1)dayn =(∑NSoC i=1xi,n,24)dayn−1 ∀n∈NSoC,dayn>1(33) 5. Investigated market scenarios and KPIs This section defines the input and boundaries for the optimiser described in Section 3, the ageing model proposed by Sayfutdinov [11] in Subsection 4.1, and the efficiency formulation presented in Subsection 4.2 to a 1 MWh/1 MW Li-ion BES in different real market scenarios. First, the Italy NORD electricity market zone is assumed as a case study, and different scenarios are created considering day-ahead electricity prices for years 2022 and 2023, each repeated several times until the EoL. The dispatch of BES is optimised on a daily basis, with a 36 h time horizon and perfect price knowledge. The BES-rated capacity is continuously updated consistently with the adopted ageing model and once the EoL (20 %) is reached, the process stops. Concerning the methodology presented in Section 3.2, the SoC discretisation is limited between 20 % and 100 % with 10-percentage-point steps. This analysis aims to assess how the costs associated with degradation can affect the economic performance of the battery. Subsequently, it is therefore essential to carry out a sensitivity analysis considering different market scenarios because it is important to check if there is any relationship between the most important parameter in the market when it comes to arbitrage, i.e., the price fluctuation magnitude, and the optimum R factor . For this analysis, different market scenarios were evaluated, characterised by both years and different market zones: 4 years from 2019 to 2022 and 7 different market zones (the north of Italy, Ita-NORD, Sicily, Ita-SICI, Portugal, PT, Poland, PL, Greece, GR, Norway NO1, and Sweden SE3). For further analysis, different KPIs were considered: lifespan, optimised R factor , cycling and calendar percentage fade, charged and discharge energy, yearly OP and finally the Present Value for Operation Profits PV OP as defined in Eq. (34) for different yearly interest rates i(0 %, 5 % and 10 %). PVOP =∑ lifespan−1 year=0 OPyear (1+i)year (34) 6. Results First, the Italian electricity market, NORD bidding zone, is considered to focus on the price scenarios defined by the years 2022 and 2023, representative of profitable and non-profitable conditions for battery operations respectively. At this stage, the R factor is maintained equal to 1. Fig. 5 shows how battery degradation is divided between cycle and calendar fade before reaching the EoL criterium and associated Operational Profits OP. Fig. 5 shows the cumulative operational profits Eq. (34), plotted as lines on the right y-axis, differently coloured according to the interest rate, and the degradation, represented with blue and red areas for cycling and calendar contribution respectively, on the left y-axis the cumulative capacity fade can be read Eq. (2) while on the right the corresponding cumulative degradation costs are reported which are equal to CAPEX as fade reaches the EoL. Subfigures (a) and (b) report the results for two different scenarios in which the 2022 and 2023 hourly market prices are repeated until the BES EoL. 2022 was characterised by high prices, which are commonly associated with increased variability [16], i.e., the main driving factor for energy arbitrage [16]. In 2022 the average daily variability of electricity price, i.e., the difference between the maximum and minimum daily prices, scored on average 161.52 € / MWh, while in 2023 it was limited to 78.42 € /MWh. Consequently, BES operates much more frequently in the 2022 scenario performing 0.84 equivalent cycles per day against 0.49 if 2023 prices are adopted. This is immediately reflected in the ageing process; in the first case, the lifespan is estimated at 8.0 years with a relevant impact of cycling (63.8 %) on the overall capacity fade. In the second scenario, BES lasts longer (11.4 years), and cycling ageing has a lower impact on the total fade, but the percentage is reduced to 52.3 %. However, what is important is to look at the cumulative sum of OP at the end of life: 234.9 k € and 126.2 k € for the two scenarios respectively. Considering that the CAPEX for a 1 MWh BES is assumed to be 150 k € [37], it is possible to conclude that the optimisation of operations guarantees 10.63 k € /year of net profits in the 2022 scenario. Conversely, considering the prices of 2023, it minimises the losses to 2.09 k € /year. The optimisation approach presented in the previous section guarantees that BES performs a cycle only if the arbitrage revenues (OP) overcome the associated degradation cost. Indeed, in Fig. 5, the cumulative operational profit lines are always above the blue area. However, the presence of calendar ageing jeopardises the viability of this approach and tuning of R factor is needed. Fig. 5. Cycling and calendar contribution to capacity fade (left y-axis) and cumulative OP for 2022 (a) and 2023 (b) scenarios. Dashed horizontal lines indicate the CAPEX (150 k € [37]) and the cumulative OP at the EoL, thus the gap between these highlights the net profits or economic loss. A. Vasylyev et al.