Lifetime Degradation Cost Analysis for Li-Ion Batteries in Capacity Markets using Accurate Physics-Based Models

: Capacity markets (CM) are energy markets created to ensure energy supply security. Energy storage devices provide services in the CMs. Li-ion batteries are a popular type of energy storage device used in CMs. The battery lifetime is a key factor in determining the economic viability of Li-ion batteries, and current approaches for estimating this are limited. This paper explores the potential of a lithium-ion battery to provide CM services with four de-rating factors (0.5 h, 1 h, 2 h, and 4 h). During the CM contract, the battery experiences both calendar and cycle degradation, which reduces the overall proﬁt. Physics-based battery and degradation models are used to quantify the degradation costs for batteries in the CM to enhance the previous research results. The degradation model quantiﬁes capacity losses related to the solid–electrolyte interphase (SEI) layer, active material loss, and SEI crack growth. The results show that the physics-based degradation model can accurately predict degradation costs under di ﬀ erent operating conditions, and thus can substantiate the business case for the batteries in the CM. The simulated CM proﬁts can be increased by 60% and 75% at 5 ◦ C and 25 ◦ C, respectively, compared to empirical and semiempirical degradation models. A sensitivity analysis for a range of parameters is performed to show the e ﬀ ects on the batteries’ overall proﬁt margins. The to obtain 0.5 h, 1 h, 2 h, and 4 h CM de-rating factors. During a one-year CM contract, the battery experienced cycle and calendar degradation, resulting in a degradation cost. At the same time, depending on a battery’s capacity obligation, a de-rated battery can generate revenue, overpayments, and penalties. The results illustrate that the 1 h de-rated battery had the highest proﬁt margin in the current CM design in all simulated scenarios. The results also show that batteries providing CM services should be stored at low temperatures, such as 5 ◦ C. However, during shortage periods when the battery is delivering power, the temperature should be increased to 25 ◦ C to avoid penalties. Moreover, the physics-based degradation model accurately predicted calendar and cycle degradation for a wider range of temperature conditions compared to empirical and semiempirical models. Due to the lower predicted battery capacity loss, the batteries received more capacity overpayment, thus increasing the overall revenue. As such, the proﬁt for the 1 h de-rated battery was increased by 59.6% and 75.5% for 5 ◦ C and 25 ◦ C, respectively, when compared to empirical and semiempirical models.


Introduction
The threat of climate change due to global warming has encouraged many countries to adopt policies to increase reliance on renewable energy sources (RES) in their electricity networks. The EU policies require European countries to increase the energy produced by RES by 20% by 2020 and 27% by 2030 [1]. The total world renewable energy generation capacity increased by 14.5% in 2019 [2]. The intermittent nature of RES [3], along with fast energy demand growth [4], raises considerable energy security concerns [5,6]. Driven by such concerns, several multidimensional approaches have been used to ensure adequate and cost-efficient power systems. These include enabling innovative technologies such as energy storage [7][8][9], improving market design [10,11], and enhancing system operation [12].
The introduction of the capacity markets (CMs) to improve the electricity market design is seen as an effective solution to enabling the integration of RES in electricity networks. As such, CMs have been implemented in many countries, including the US [13], Latin America [14], and Europe [15]. The aim of this is to adequately remunerate new electrical generators, reduce investment risks, and avoid electricity blackouts. Many of the new generators providing CM services use lithium-ion

Capacity Market Fundamentals
The reliability of liberalized electricity markets has been questioned due to increasing energy demands, the decommissioning of conventional power plants (i.e., coal), and the steady growth of RES. In particular, many policymakers argue that the current energy-only markets may not ensure resource adequacy [49]. One issue is that the energy-only market neglects the energy adequacy problem, because it assumes that the energy demand and supply are always balanced (quantity supplied = quantity needed). Thus, when the supply side becomes scarce, there must be a load reduction from the demand side to ensure market clearance. However, due to the inelastic nature of the demand side and rational customer responses, electricity markets do not guarantee a demand response or market clearance [50].
Another reason for the energy-only market failure is its inefficiency during electricity blackouts [51]. If there is a blackout similar to the one that happened in the UK in 2019, which affected over 1 million customers [52], then there is at least one supplier that does not have power to sell at any price, as illustrated in Figure 1a. Despite the scarce capacity and the peak demand, generators do not earn money in the blackout events [48].
RES. In particular, many policymakers argue that the current energy-only markets may not ensure resource adequacy [49]. One issue is that the energy-only market neglects the energy adequacy problem, because it assumes that the energy demand and supply are always balanced (quantity supplied = quantity needed). Thus, when the supply side becomes scarce, there must be a load reduction from the demand side to ensure market clearance. However, due to the inelastic nature of the demand side and rational customer responses, electricity markets do not guarantee a demand response or market clearance [50].
Another reason for the energy-only market failure is its inefficiency during electricity blackouts [51]. If there is a blackout similar to the one that happened in the UK in 2019, which affected over 1 million customers [52], then there is at least one supplier that does not have power to sell at any price, as illustrated in Figure 1a. Despite the scarce capacity and the peak demand, generators do not earn money in the blackout events [48].
Nevertheless, even if there are no generation adequacy issues, a "missing money" problem will exist in both normal operating conditions and scarcity periods. In normal operating conditions, where the demand quantity is below the peak available capacity (shown in Figure 1b) and the market is competitive, some generators, such as the peakers (plants operating only at high demand), cannot earn sufficient revenue beyond covering their operating costs [49]. Therefore, they may not cover their capital investment cost. In scarcity periods, as shown in Figure1c, all generators may charge high scarcity prices (red dot), which in real cases can be 367 times higher than the average price [53]. However, market power can be exercised in scarcity periods, even by small generators (see [54]), therefore triggering regulatory intervention. Regulators usually set the scarcity prices low to mitigate market power abuse, thus creating the missing money problem depicted in Figure 1c. Since the investment in new capacity to increase the reliability of the supply side depends on the scarcity prices, energy-only markets, therefore, do not provide the incentives needed to increase capacity.  Nevertheless, even if there are no generation adequacy issues, a "missing money" problem will exist in both normal operating conditions and scarcity periods. In normal operating conditions, where the demand quantity is below the peak available capacity (shown in Figure 1b) and the market is competitive, some generators, such as the peakers (plants operating only at high demand), cannot earn sufficient revenue beyond covering their operating costs [49]. Therefore, they may not cover their capital investment cost. In scarcity periods, as shown in Figure 1c, all generators may charge high scarcity prices (red dot), which in real cases can be 367 times higher than the average price [53]. However, market power can be exercised in scarcity periods, even by small generators (see [54]), Energies 2020, 13, 2816 4 of 21 therefore triggering regulatory intervention. Regulators usually set the scarcity prices low to mitigate market power abuse, thus creating the missing money problem depicted in Figure 1c. Since the investment in new capacity to increase the reliability of the supply side depends on the scarcity prices, energy-only markets, therefore, do not provide the incentives needed to increase capacity.
A well-designed CM can mitigate the issues of market power and missing money by determining the adequate level of supply capacity needed. This can be done by designing a capacity auction for generators to determine the scarcity price required to secure an adequate level of capacity, as set by the regulator, in order to reduce the number of shortage hours. The market is open to new and existing generators, considering the investment level needed in the new generators. The result is that the market discovers the true value of the scarcity price corresponding to the optimal level of capacity, while the regulatory intervention is limited to only control the level of the capacity needed.
For several reasons, energy storage devices may have limited discharge capacity (for example, degradation). Therefore, CMs (in the UK, Germany, France, Italy, Ireland, and Denmark) have introduced de-rating methodologies to account for the percentage of firm capacity they can supply during shortage periods [55]. Fraunholz et al. [56] found that the choice of a suitable de-rating factor is challenging and may affect batteries' market competitiveness. Figure 2 depicts the number of batteries providing CM services in the current UK CM with different de-rating factors. The de-rating factors are CM design parameters that are used to measure the net contribution of energy-limited devices such as energy storage and energy security. It can be seen that there is an increased interest in obtaining higher de-rating factors, such as 3 h or 4 h durations. This study considers a real-life scenario, in which LIBs are utilized to obtain 0.5 h, 1 h, 2 h, and 4 h de-rating factors. A well-designed CM can mitigate the issues of market power and missing money by determining the adequate level of supply capacity needed. This can be done by designing a capacity auction for generators to determine the scarcity price required to secure an adequate level of capacity, as set by the regulator, in order to reduce the number of shortage hours. The market is open to new and existing generators, considering the investment level needed in the new generators. The result is that the market discovers the true value of the scarcity price corresponding to the optimal level of capacity, while the regulatory intervention is limited to only control the level of the capacity needed.
For several reasons, energy storage devices may have limited discharge capacity (for example, degradation). Therefore, CMs (in the UK, Germany, France, Italy, Ireland, and Denmark) have introduced de-rating methodologies to account for the percentage of firm capacity they can supply during shortage periods [55]. Fraunholz et al. [56] found that the choice of a suitable de-rating factor is challenging and may affect batteries' market competitiveness. Figure 2 depicts the number of batteries providing CM services in the current UK CM with different de-rating factors. The de-rating factors are CM design parameters that are used to measure the net contribution of energy-limited devices such as energy storage and energy security. It can be seen that there is an increased interest in obtaining higher de-rating factors, such as 3 h or 4 h durations. This study considers a real-life scenario, in which LIBs are utilized to obtain 0.5 h, 1 h, 2 h, and 4 h de-rating factors.

Problem Setup
This study models battery degradation for LIBs used to provide a CM service to the UK grid operator over a one-year contract using a physics-based model. During the contract period, many simultaneous degradation mechanisms can affect the battery performance, resulting in a degradation cost. As illustrated in Figure 3, when the CM's contract begins, the battery should provide reserve services to the grid and be ready to respond to electricity SEs. Therefore, in the absence of SEs, the calendar battery degradation can be quantified. During SEs, and depending on their duration, the obligation capacity will be calculated for each de-rated battery. The duration of the event and the obligation capacity amount is updated at each settlement period (30 min). Then, a physics-based battery model is used to discharge the required capacity. The operating conditions of the battery, such as the temperature, are then fed to the degradation model to quantify the cycle degradation and update the initial capacity in the model. Afterwards, depending on the amount of generated power, the penalties and overpayment can be quantified to obtain the overall revenue. The battery capacity and the de-rating factors used in this study are shown in Table 1, which are in line with the current batteries participating in the CM [57]. The CM parameters used in this study are given in an earlier study [48].

Problem Setup
This study models battery degradation for LIBs used to provide a CM service to the UK grid operator over a one-year contract using a physics-based model. During the contract period, many simultaneous degradation mechanisms can affect the battery performance, resulting in a degradation cost. As illustrated in Figure 3, when the CM's contract begins, the battery should provide reserve services to the grid and be ready to respond to electricity SEs. Therefore, in the absence of SEs, the calendar battery degradation can be quantified. During SEs, and depending on their duration, the obligation capacity will be calculated for each de-rated battery. The duration of the event and the obligation capacity amount is updated at each settlement period (30 min). Then, a physics-based battery model is used to discharge the required capacity. The operating conditions of the battery, such as the temperature, are then fed to the degradation model to quantify the cycle degradation and update the initial capacity in the model. Afterwards, depending on the amount of generated power, the penalties and overpayment can be quantified to obtain the overall revenue. The battery capacity Energies 2020, 13, 2816 5 of 21 and the de-rating factors used in this study are shown in Table 1, which are in line with the current batteries participating in the CM [57]. The CM parameters used in this study are given in an earlier study [48].  The total revenue of battery represents the revenue from the CM contract in addition to overpayments minus potential penalties , as given in Equation (1), where is the de-rated capacity, is the de-rating factor, is the CM auction clearing price, and is a factor used to reward slightly more payment in peak demand months: The de-rated capacity depends on the battery's output power, as in Equation (2), where ( ) is the battery current, ( ) is the battery voltage, and is the total number of battery cells.
The capacity obligation is calculated at each settlement period (i), as in Equations (3) and (4), where is the peak electricity demand during the SE divided by the total CM contracted capacity through the CM auction , and is the capacity offered by the battery to other grid services.  The total revenue of battery R represents the revenue from the CM contract in addition to overpayments R ov minus potential penalties p, as given in Equation (1), where C de is the de-rated capacity, k de is the de-rating factor, λ cl is the CM auction clearing price, and f is a factor used to reward slightly more payment in peak demand months: The de-rated capacity C de depends on the battery's output power, as in Equation (2), where I b (t) is the battery current, V b (t) is the battery voltage, and N is the total number of battery cells.
Energies 2020, 13, 2816 6 of 21 The capacity obligation C o is calculated at each settlement period (i), as in Equations (3) and (4), where D p is the peak electricity demand during the SE D se p divided by the total CM contracted capacity through the CM auction C auc , and C b is the capacity offered by the battery to other grid services.
The penalties and the overpayments are obtained by calculating the amount of undelivered or overdelivered capacity over the CM contract at each settlement period, as in Equations (5) and (6): By multiplying the lost battery capacity C lost (t), which is obtained from the battery degradation model by a degradation price λ degr , the total energy degradation cost E lost can be roughly estimated, as in Equation (7):

Battery Cycling Profile
The battery cycles in the CM are calculated according to the SE period. The expected number of these SEs is based on the loss of the lead expectation (LOLE) reliability metric [58]. Since accurately predicting the LOLE is difficult, as it is a function of complex processes such as generator availability, blackouts, and environmental factors, many studies deem LOLE to be unreliable [59,60]. In the presence of a high share of RESs, it is found that the LOLE can reach 62 h per year [61]. Another study found that LOLE can reach 83 h per year, considering the current CM scarcity prices [62]. Therefore, by considering the difficulty in estimating realistic LOLE values and the previous studies, this work assumes that the total number of SEs of different durations is 20 in the first year of the CM's contract, while increasing to 90 in the following year (from months 13 to 24), as shown in Figure 4. This distribution of SEs considers the electricity peak demand periods in most parts of Europe [63]. Furthermore, the distribution reflects the probability of the duration of SEs; for instance, events lasting 1 h and 2 h are more probable than events lasting 4 h [64].
Energies 2020, 13, x FOR PEER REVIEW 6 of 21 The penalties and the overpayments are obtained by calculating the amount of undelivered or overdelivered capacity over the CM contract at each settlement period, as in Equations (5) and (6): By multiplying the lost battery capacity ( ), which is obtained from the battery degradation model by a degradation price , the total energy degradation cost can be roughly estimated, as in Equation (7):

Battery Cycling Profile
The battery cycles in the CM are calculated according to the SE period. The expected number of these SEs is based on the loss of the lead expectation (LOLE) reliability metric [58]. Since accurately predicting the LOLE is difficult, as it is a function of complex processes such as generator availability, blackouts, and environmental factors, many studies deem LOLE to be unreliable [59,60]. In the presence of a high share of RESs, it is found that the LOLE can reach 62 h per year [61]. Another study found that LOLE can reach 83 h per year, considering the current CM scarcity prices [62]. Therefore, by considering the difficulty in estimating realistic LOLE values and the previous studies, this work assumes that the total number of SEs of different durations is 20 in the first year of the CM's contract, while increasing to 90 in the following year (from months 13 to 24), as shown in Figure  4. This distribution of SEs considers the electricity peak demand periods in most parts of Europe [63]. Furthermore, the distribution reflects the probability of the duration of SEs; for instance, events lasting 1 h and 2 h are more probable than events lasting 4 h [64].

Battery and Degradation Models
The electrochemical battery cell model is shown in Figure 1 and based on the seminal work by

Battery Model
The electrochemical battery cell model is shown in Figure 1 and based on the seminal work by Newman et al. [65]. The input to the model is the load current, material properties, geometry design parameters, and the operating temperature. The output is the cell voltage and the SoC. The model parameters values used in this study are given in Appendix A. The five model states are the lithium concentration in the solid (c s ) and electrolyte (c e ) phases, the electric potential in the solid (φ s ) and electrolyte (φ e ) phases, along with the rate of lithium movement between the phases ( j Li ) [66]. These variables can be found by solving five coupled differential equations, along with their boundary conditions, as described below.
The mass conservation of lithium (assuming the concentration of lithium within the particles is spherically symmetric) in the solid phase can be described using Fick's second law in Equation (8), where (D s ) is the solid phase diffusion constant and (r) is the radial (pseudo) dimension. The first boundary condition of Equation (8) is given by Equation (9), indicating that there is no diffusion in the center of the particle. The second boundary condition is given by Equation (10), indicating that the transfer of charges occur at the outer boundary of the particle, where (a s ) is the specific interfacial area between the solid and the electrolyte, R p is the particle radius, and (F) is Faraday's constant: The lithium concentration in the electrolyte phase is the result of diffusion (first term) and due to charge transfer between the solid and the electrolyte (second term), as in Equation (11), where (ε e ) is the porosity and (t + 0 ) is the transference number of the cation with respect to the solvent. Since there must be no electrolyte flux at the cell boundaries, the boundary conditions of Equation (11) are given in Equation (12): The solid-phase charge conservation follows Ohm's law, since (φ s ) depends on the current passing through the solid, as in Equation (13), where (σ e f f ) is the effective conductivity. The current only flows at the collector-solid interface, as in the boundary conditions in Equation (14), where (A) is the electrode area: −σ e f f ∂φ s ∂x x=0 The electrolyte-phase charge conservation follows Ohm's law in a liquid electrolyte (first term) and the local concentration of lithium (second term), as in Equation (15), where (κ e f f ) is the effective conductivity of the electrolyte. At the boundary of the electrode-current collector interphase, the ionic current must be zero, as in Equation (16), where: Equations (8)- (16) are coupled through the Butler-Volmer equation in Equation (17), where (i o ) is the exchange current density; (R) is the universal gas constant; (T) is the temperature; (α a , α c ) are anode and cathode symmetry factors, respectively; and (η) is the reaction overpotential.
After solving the above equations, the cell voltage and SoC are given in Equations (18) and (19):

Degradation Model
Several degradation mechanisms for LIBs are presented in the literature and various models are reported to describe these mechanisms, with more than one model often used to describe a single mechanism [36]. Here, three dominant aging mechanisms are included and are shown in Figure 1: SEI layer growth, active material (AM) loss, and SEI layer fracture. For a complete derivation of these models, please refer to [67][68][69]. The degradation model used in this work is dependent on the P2D battery model described in [70].
The total degradation equation that represents all the three mechanisms is given in Equation (20). The SEI layer growth (Q SEI ) is directly proportional to the side reaction current density i s . The governing equations related to this are given in Equations (21)- (24). During the charge-discharge process, the mechanical stress generated inside the active material could result in particle fracture, which in turns may isolate the active material. As such, the amount of lithium is reduced, leading to capacity loss (Q AM ), as given in Equation (25). SEI fracture is observed in [71], where the SEI layer experiences tensile stress as the active material expands. This results in the SEI layer stretching and cracking (Q SEI,crack ), which exacerbates the battery cell's harmful side reaction. The governing equations of (Q SEI,crack ) are given in Equations (26) and (27). All the parameters used in this degradation model are given in Appendix B.

Accuracy of Battery and Degradation Models
The output voltage against the battery capacity is shown in Figure 5a,b for different current rates and temperatures during discharge. The model results are in good agreement with the experimental data presented in [72] for the same 53 Ah nickel (N) manganese (M) cobalt (C) LIB cell. In Figure 5a, as expected, the higher the current rating, the lower the battery capacity, which is a common feature for many other LIB types [73]. One reason for the lower capacity retention at higher current rates is due to the induced overpotential, which reduces the cell capacity and the cell voltage, especially at higher DoD [74].The battery capacity at the 4C rate, for instance, is 49.7 Ah in the first cycles, meaning its effective capacity is 93.7% or lower compared to 56Ah at the 0.25C rate.
Energies 2020, 13, x FOR PEER REVIEW 9 of 21 due to the induced overpotential, which reduces the cell capacity and the cell voltage, especially at higher DoD [74].The battery capacity at the 4C rate, for instance, is 49.7 Ah in the first cycles, meaning its effective capacity is 93.7% or lower compared to 56Ah at the 0.25C rate. Figure 5b shows the battery capacity and voltage for different temperatures at the 1C rate plotted against their experimental data. It can be seen that at lower temperatures, such as 5 °C, the battery capacity predicted by the model is 48 Ah compared to the 47.5 Ah value obtained experimentally. This suggests that without any degradation, this LIB cell capacity is 89.6% at 5 °C because of the increased battery resistance at lower temperatures [75]. The model's average root mean square error is 1.1%. It should be noted that the cycle life performance of the NMC battery chemistry is highly dependent on different temperatures and electrochemical conditions, which may affect the total capacity presented in this study. For further information on the different NMC compositions and their capacity retention levels under different operation conditions, please see [76]. The LIB cell degradation as predicted by Equation (20) and the corresponding experimental data are presented in Figure 6a for calendar degradation and Figure 6b for cycle degradation. The calendar experimental data were taken from [77] and the cycle experimental data were taken from [78]. The model results show good agreement with experiment data, except for the results at 45 °C at 100% SoC. In Figure 6a, high capacity loss is evident at higher temperatures and SoCs, and vice versa. For instance, at 5 °C, the LIB's state of health is over 96% after 500 days, indicating that there is minimal capacity loss. Figure 6b shows the cycling results for different C-rates at the same temperature of 35 °C, except for one result at 5 °C. The results indicated that the higher the DoD, the higher the expected capacity loss at the same C-rate. Moreover, it can be seen that the cycling result at 5 °C is as expected based on the battery model results obtained in Figure 5b, in which the battery's state of health is 89% without any cycling, reaching 85% after 2000 cycles. This mitigates the limitations of other models, which are unable to show that calendar degradation is minimal at lower temperatures, while it is maximal when cycling at the same lower temperatures [48]. This is important in applications where calendar and cycle degradation quantification are needed, such as in the CM. In Figure 6c, a magnified version of the 1C rate (100% DoD) is depicted to relate it to the three degradation mechanisms predicted using Equations (21), (25), and (26). It can be seen that at higher DoD levels, the , is in high agreement with [70].  Figure 5b shows the battery capacity and voltage for different temperatures at the 1C rate plotted against their experimental data. It can be seen that at lower temperatures, such as 5 • C, the battery capacity predicted by the model is 48 Ah compared to the 47.5 Ah value obtained experimentally. This suggests that without any degradation, this LIB cell capacity is 89.6% at 5 • C because of the increased battery resistance at lower temperatures [75]. The model's average root mean square error is 1.1%. It should be noted that the cycle life performance of the NMC battery chemistry is highly dependent on different temperatures and electrochemical conditions, which may affect the total capacity presented in this study. For further information on the different NMC compositions and their capacity retention levels under different operation conditions, please see [76].
The LIB cell degradation as predicted by Equation (20) and the corresponding experimental data are presented in Figure 6a for calendar degradation and Figure 6b for cycle degradation. The calendar experimental data were taken from [77] and the cycle experimental data were taken from [78]. The model results show good agreement with experiment data, except for the results at 45 • C at 100% SoC. In Figure 6a, high capacity loss is evident at higher temperatures and SoCs, and vice versa. For instance, at 5 • C, the LIB's state of health is over 96% after 500 days, indicating that there is minimal capacity loss.

Revenue and Degradation Costs in the Capacity Market
The collected revenues over a 12 months CM contract and any incurred degradation costs are depicted in Figures 7 and 8 for the four de-rated batteries in different conditions. In Figure 7, the batteries were kept at room temperature (25 °C) and the SoC was varied. This was to account for a real case scenario, whereby the battery could be at a different SoC level in a thermally controlled environment until it is ready to respond to SEs. First, it can be seen that the 1 h de-rated battery has the highest revenue compared to the others due to the relatively high compared to its . The modeling results are confirmed, with many battery asset owners opting for 1 h de-rating factors in the latest CM auction [79]. Second, the degradation cost is lower at low SoC levels, as the calendar degradation is generally low. Third, compared with the same cases presented in [48], the physics-based degradation model predicts lower capacity losses under the same conditions. Therefore, it offers the opportunity for more overpayment, represented by Equation (6), as well as lowers the exposure to penalties (Equation (5)) for the 4 h de-rated battery. As such, the total collected revenue per CM contract is higher compared to the results presented in [48]. (b) cycle aging results with different depth of discharge ranges and C-rates at T = 35 • C with experimental data taken from [78], except for the 1C result at 5 • C (90-10) taken from [77]; (c) detailed version of 1C (100-0) degradation based on the physics-based degradation model mechanisms. Figure 6b shows the cycling results for different C-rates at the same temperature of 35 • C, except for one result at 5 • C. The results indicated that the higher the DoD, the higher the expected capacity loss at the same C-rate. Moreover, it can be seen that the cycling result at 5 • C is as expected based on the battery model results obtained in Figure 5b, in which the battery's state of health is 89% without any cycling, reaching 85% after 2000 cycles. This mitigates the limitations of other models, which are unable to show that calendar degradation is minimal at lower temperatures, while it is maximal when cycling at the same lower temperatures [48]. This is important in applications where calendar and cycle degradation quantification are needed, such as in the CM. In Figure 6c, a magnified version of the 1C rate (100% DoD) is depicted to relate it to the three degradation mechanisms predicted using Equations (21), (25), and (26). It can be seen that at higher DoD levels, the Q SEI,crack is in high agreement with [70].

Revenue and Degradation Costs in the Capacity Market
The collected revenues over a 12 months CM contract and any incurred degradation costs are depicted in Figures 7 and 8 for the four de-rated batteries in different conditions. Energies 2020, 13, x FOR PEER REVIEW 11 of 21 In Figure 8, the impact of the temperature change on the revenue is huge, necessitating a battery thermal management system to control the temperature at lower temperatures. However, when discharge occurs during SEs, it is necessary to increase the battery's temperature to respond effectively and avoid penalties. Since the cycling here is low during the first 12 months of the CM contract, as shown in Figure 4, the average revenue stays the same compared to Figure 7.  Table 2 summarizes the accuracy of different degradation models, which have been used in the same CM application to predict the degradation cost. These are the empirical and semiempirical models reported in earlier study [48] and the physics-based degradation model used in this study. The physics-based degradation model is more accurate than the other two in predicting calendar and cycle degradation under different operating conditions. This is due to the feedback mechanism, which uses an accurate pseudo 2-dimensional (P2D) battery cell model, in which most of the parameters used account for the temperature-C rate change, as given in Appendix A. Additionally, although empirical and semiempirical modes that are tied to equivalent circuit models can be useful and computationally fast, the process of determining the parameter values in these models uses empirical system identification. Therefore, changing the operating conditions necessitates the use of a different fitting, which is a time-consuming and unreliable process. In Figure 7, the batteries were kept at room temperature (25 • C) and the SoC was varied. This was to account for a real case scenario, whereby the battery could be at a different SoC level in a thermally controlled environment until it is ready to respond to SEs. First, it can be seen that the 1 h de-rated battery has the highest revenue compared to the others due to the relatively high k de compared to its C o . The modeling results are confirmed, with many battery asset owners opting for 1 h de-rating factors in the latest CM auction [79]. Second, the degradation cost is lower at low SoC levels, as the calendar degradation is generally low. Third, compared with the same cases presented in [48], the physics-based degradation model predicts lower capacity losses under the same conditions. Therefore, it offers the opportunity for more overpayment, represented by Equation (6), as well as lowers the exposure to penalties (Equation (5)) for the 4 h de-rated battery. As such, the total collected revenue per CM contract is higher compared to the results presented in [48].
In Figure 8, the impact of the temperature change on the revenue is huge, necessitating a battery thermal management system to control the temperature at lower temperatures. However, when discharge occurs during SEs, it is necessary to increase the battery's temperature to respond effectively and avoid penalties. Since the cycling here is low during the first 12 months of the CM contract, as shown in Figure 4, the average revenue stays the same compared to Figure 7.  In Table 3, the profit (revenue minus degradation cost) for the 1 h de-rated battery is calculated using the three different degradation model approaches (empirical, semiempirical, and physics). By using a physics-based degradation model when accounting for the degradation cost, the profit can be increased by 59.6% and 75.5% for 5 °C and 25 °C, respectively, when compared to both empirical and semiempirical models. This is due to the lower predicted degradation cost, which allows more overpayment to be collected. At higher temperatures such as 45 °C, the physics model predicts higher losses compared to the other two temperatures.  Table 2 summarizes the accuracy of different degradation models, which have been used in the same CM application to predict the degradation cost. These are the empirical and semiempirical models reported in earlier study [48] and the physics-based degradation model used in this study. The physics-based degradation model is more accurate than the other two in predicting calendar and cycle degradation under different operating conditions. This is due to the feedback mechanism, which uses an accurate pseudo 2-dimensional (P2D) battery cell model, in which most of the parameters used account for the temperature-C rate change, as given in Appendix A. Additionally, although empirical and semiempirical modes that are tied to equivalent circuit models can be useful and computationally fast, the process of determining the parameter values in these models uses empirical system identification. Therefore, changing the operating conditions necessitates the use of a different fitting, which is a time-consuming and unreliable process. In Table 3, the profit (revenue minus degradation cost) for the 1 h de-rated battery is calculated using the three different degradation model approaches (empirical, semiempirical, and physics). By using a physics-based degradation model when accounting for the degradation cost, the profit can be increased by 59.6% and 75.5% for 5 • C and 25 • C, respectively, when compared to both empirical and semiempirical models. This is due to the lower predicted degradation cost, which allows more overpayment to be collected. At higher temperatures such as 45 • C, the physics model predicts higher losses compared to the other two temperatures. Table 3. Profit at the end of a 1-year capacity market contract fora battery with a de-rating factor of 1 h when using several degradation models.

Sensitivity Analysis
This section investigates the effects of changing important parameters on the profitability of the batteries participating in the CM. This include changes to the CM clearance price, battery degradation cost, de-rating factors, and increased SEs as a result of a predicted increase in energy demand. It should be noted that all the sensitivity analysis results are conducted at a standard temperature of 25 • C, except for in Section 4.3.4, where the temperature was set to 5 • C to study the effects of CM penalties.

Capacity Market Price Change Effects
The CM clearing price λ cl is an important parameter that can be changed with different CM auction results. As such, λ cl has been changed twice to reflect the maximum and minimum auction prices obtained in the UK's CM auctions from the start of the CM until now. The original λ cl = £19.4/kW/year values were changed to λ cl = £27.5/kW/year and λ cl = £6/kW/year [80,81]. The profit (revenue-degradation cost) for a one-year CM contract is depicted in Figures 9 and 10, respectively, for the four de-rated batteries. In Figures 9 and 10, it can be seen that λ cl can hugely affect the batteries' profitability. In Figure 9, nearly all the four batteries are profitable at the end of the CM contract, as shown by an average increase of 167% compared to the normal case. In Figure 10, all the four batteries incurred huge losses due to low clearing prices and high degradation costs, as shown by a decrease of 170% compared to the normal case.
Energies 2020, 13, x FOR PEER REVIEW 13 of 21 Table 3. Profit at the end of a 1-year capacity market contract fora battery with a de-rating factor of 1 h when using several degradation models.

Sensitivity Analysis
This section investigates the effects of changing important parameters on the profitability of the batteries participating in the CM. This include changes to the CM clearance price, battery degradation cost, de-rating factors, and increased SEs as a result of a predicted increase in energy demand. It should be noted that all the sensitivity analysis results are conducted at a standard temperature of 25 °C, except for in Section 4.3.4, where the temperature was set to 5 °C to study the effects of CM penalties.

Capacity Market Price Change Effects
The CM clearing price is an important parameter that can be changed with different CM auction results. As such, has been changed twice to reflect the maximum and minimum auction prices obtained in the UK's CM auctions from the start of the CM until now. The original = £19.4/kW/year values were changed to = £27.5/kW/year and = £6/kW/year [80,81]. The profit (revenue-degradation cost) for a one-year CM contract is depicted in Figures 9 and 10, respectively, for the four de-rated batteries. In Figures 9 and 10, it can be seen that can hugely affect the batteries' profitability. In Figure 9, nearly all the four batteries are profitable at the end of the CM contract, as shown by an average increase of 167% compared to the normal case. In Figure  10, all the four batteries incurred huge losses due to low clearing prices and high degradation costs, as shown by a decrease of 170% compared to the normal case.

Degradation Cost Effects
The degradation cost is changed from 176 $/kWh or 0.5 £/Ah to an optimistic 100 $/kWh (0.29 £/Ah), which is regarded as the ultimate goal for battery pack cost reduction in the future [82]. It can be seen in Figure 11 that all the batteries are profitable, with an average increase of nearly 50% over the standard case.

De-rating Factor Effects
The presumed de-rating factors for all four batteries (0.5 h to 4 h) are projected to decrease in the future to allow for new-generation entries in the CM, as stated in [64]. Therefore, the de-rating factors have been changed accordingly to 17.80% for the 0.5 h battery, 36.44% for the 1 h battery, 64.79% for the 2 h battery, and 96.11% for the 4 h battery. As shown in Figure 12, the overall profitability decreased for the 0.5 h and 4 h batteries, meaning they are no longer profitable.

Degradation Cost Effects
The degradation cost is changed from 176 $/kWh or 0.5 £/Ah to an optimistic 100 $/kWh (0.29 £/Ah), which is regarded as the ultimate goal for battery pack cost reduction in the future [82]. It can be seen in Figure 11 that all the batteries are profitable, with an average increase of nearly 50% over the standard case.

Degradation Cost Effects
The degradation cost is changed from 176 $/kWh or 0.5 £/Ah to an optimistic 100 $/kWh (0.29 £/Ah), which is regarded as the ultimate goal for battery pack cost reduction in the future [82]. It can be seen in Figure 11 that all the batteries are profitable, with an average increase of nearly 50% over the standard case.

De-rating Factor Effects
The presumed de-rating factors for all four batteries (0.5 h to 4 h) are projected to decrease in the future to allow for new-generation entries in the CM, as stated in [64]. Therefore, the de-rating factors have been changed accordingly to 17.80% for the 0.5 h battery, 36.44% for the 1 h battery, 64.79% for the 2 h battery, and 96.11% for the 4 h battery. As shown in Figure 12, the overall profitability decreased for the 0.5 h and 4 h batteries, meaning they are no longer profitable.

De-rating Factor Effects
The presumed de-rating factors for all four batteries (0.5 h to 4 h) are projected to decrease in the future to allow for new-generation entries in the CM, as stated in [64]. Therefore, the de-rating factors have been changed accordingly to 17.80% for the 0.5 h battery, 36.44% for the 1 h battery, 64.79% for the 2 h battery, and 96.11% for the 4 h battery. As shown in Figure 12, the overall profitability decreased for the 0.5 h and 4 h batteries, meaning they are no longer profitable.

Degradation Cost Effects
The degradation cost is changed from 176 $/kWh or 0.5 £/Ah to an optimistic 100 $/kWh (0.29 £/Ah), which is regarded as the ultimate goal for battery pack cost reduction in the future [82]. It can be seen in Figure 11 that all the batteries are profitable, with an average increase of nearly 50% over the standard case.

De-rating Factor Effects
The presumed de-rating factors for all four batteries (0.5 h to 4 h) are projected to decrease in the future to allow for new-generation entries in the CM, as stated in [64]. Therefore, the de-rating factors have been changed accordingly to 17.80% for the 0.5 h battery, 36.44% for the 1 h battery, 64.79% for the 2 h battery, and 96.11% for the 4 h battery. As shown in Figure 12, the overall profitability decreased for the 0.5 h and 4 h batteries, meaning they are no longer profitable.

Increased Shortage Events in the CM
In case of an expected increase in energy demand, the batteries in the CM are required to cycle more times per year, as shown in Figure 4, from month 13 to 24. In Figure 13, it can be seen that the 4 h battery is totally unprofitable due to the incurred penalties when cycling because of the high amount of capacity needed, which the battery cannot meet.

Increased Shortage Events in the CM
In case of an expected increase in energy demand, the batteries in the CM are required to cycle more times per year, as shown in Figure 4, from month 13 to 24. In Figure 13, it can be seen that the 4 h battery is totally unprofitable due to the incurred penalties when cycling because of the high amount of capacity needed, which the battery cannot meet. Figure 13. Profitability of the four de-rated batteries over 1 year with an increase in shortage events.

Conclusions and Future Work
This paper presented physics-based battery and degradation models that are used to inform a degradation cost analysis for lithium-ion batteries in the CM. The degradation model considers the SEI layer growth, active material loss, and SEI crack growth. The battery is utilized to obtain 0.5 h, 1 h, 2 h, and 4 h CM de-rating factors. During a one-year CM contract, the battery experienced cycle and calendar degradation, resulting in a degradation cost. At the same time, depending on a battery's capacity obligation, a de-rated battery can generate revenue, overpayments, and penalties.
The results illustrate that the 1 h de-rated battery had the highest profit margin in the current CM design in all simulated scenarios. The results also show that batteries providing CM services should be stored at low temperatures, such as 5 °C. However, during shortage periods when the battery is delivering power, the temperature should be increased to 25 °C to avoid penalties. Moreover, the physics-based degradation model accurately predicted calendar and cycle degradation for a wider range of temperature conditions compared to empirical and semiempirical models. Due to the lower predicted battery capacity loss, the batteries received more capacity overpayment, thus increasing the overall revenue. As such, the profit for the 1 h de-rated battery was increased by 59.6% and 75.5% for 5 °C and 25 °C, respectively, when compared to empirical and semiempirical models.
A sensitivity analysis for a range of parameters used in this study revealed that the CM profit can be affected in several ways. First, the profit for batteries is highly sensitive to the CM auction price. For instance, increasing the CM auction price by 30% can increase the profit by nearly 170% for the 1 h de-rated battery. Second, decreasing the degradation cost to the optimal battery pack price of $100/kWh can increase the profit by 50% for the 1 h de-rated battery. Third, decreasing the derating reduced the profitability for the 0.5 h and 4 h de-rated batteries. Fourth, increased cycling in the case of frequent SEs hugely decreased the profitability of the 4 h de-rated battery. Future work will include investigating how degradation of lithium plating can affect the overall CM design, considering different regions and energy storage technologies.

Conclusions and Future Work
This paper presented physics-based battery and degradation models that are used to inform a degradation cost analysis for lithium-ion batteries in the CM. The degradation model considers the SEI layer growth, active material loss, and SEI crack growth. The battery is utilized to obtain 0.5 h, 1 h, 2 h, and 4 h CM de-rating factors. During a one-year CM contract, the battery experienced cycle and calendar degradation, resulting in a degradation cost. At the same time, depending on a battery's capacity obligation, a de-rated battery can generate revenue, overpayments, and penalties.
The results illustrate that the 1 h de-rated battery had the highest profit margin in the current CM design in all simulated scenarios. The results also show that batteries providing CM services should be stored at low temperatures, such as 5 • C. However, during shortage periods when the battery is delivering power, the temperature should be increased to 25 • C to avoid penalties. Moreover, the physics-based degradation model accurately predicted calendar and cycle degradation for a wider range of temperature conditions compared to empirical and semiempirical models. Due to the lower predicted battery capacity loss, the batteries received more capacity overpayment, thus increasing the overall revenue. As such, the profit for the 1 h de-rated battery was increased by 59.6% and 75.5% for 5 • C and 25 • C, respectively, when compared to empirical and semiempirical models.
A sensitivity analysis for a range of parameters used in this study revealed that the CM profit can be affected in several ways. First, the profit for batteries is highly sensitive to the CM auction price. For instance, increasing the CM auction price by 30% can increase the profit by nearly 170% for the 1 h de-rated battery. Second, decreasing the degradation cost to the optimal battery pack price of $100/kWh can increase the profit by 50% for the 1 h de-rated battery. Third, decreasing the de-rating reduced the profitability for the 0.5 h and 4 h de-rated batteries. Fourth, increased cycling in the case of frequent SEs hugely decreased the profitability of the 4 h de-rated battery. Future work will include investigating how degradation of lithium plating can affect the overall CM design, considering different regions and energy storage technologies.  [72] Lithium diffusion in the electrolyte, D e 3.8037 × 10 −10 × exp (−0.792C e ) × exp ( −10000 R ( 1 T − 1 Tre f )) [72] Lithium diffusion in the positive electrode D s,pos 3 × 10 −14 × exp ( −35000 R ( 1 T − 1 Tre f )) 2 [72] Lithium diffusion in the negative electrode D s,neg 3 × 10 −14 × exp ( −35000 R ( 1 T − 1 Tre f )) 2 [72] Reaction rate in the positive electrode, k pos k dis 0,pos × exp (−5x) × exp ( −20000 [72] Reaction rate in the negative electrode, k neg k dis 0,neg × exp ( −20000