Battery degradation: Impact on economic dispatch

Batteries are crucial to manage the rising share of intermittent energy sources and variability in demand. Most technoeconomic models in the literature oversimplify battery degradation representation. Accounting properly for battery degradation allows for better cost tradeoffs and optimal battery usage, especially in dynamic settings. We propose a highly accurate and scalable formulation for battery degradation that considers the combined impact of cycle depth (CD) and state of charge on calendar and cycle aging. This includes a novel way to track charge‐discharge cycles. We test the consequences of battery degradation in a stylized price arbitrage model on battery operation and solution times. When ignoring battery degradation, ex post calculations reveal hidden degradation costs that exceed revenues and hence turn seemingly profitable trades into losing trades. Considering battery degradation leads to smaller CDs and lower average states of charge. Overall, we show that a much‐improved representation of battery degradation is possible at modest computational cost, allowing better decisions and higher profits.


| INTRODUCTION
Batteries have favorable characteristics such as high power and energy density, flexibility in spatial placement and sizing, and fast response times.2][3] Intensive use of batteries accelerates their aging due to physical and chemical stresses, leading to reduced performance and reduced safety. 4,5Batteries are expensive, and their degradation should be minimized to maximize their lifetime.However, in settings with very high and low electricity prices, minimizing battery degradation may mean charging or discharging at unfavorable prices and forgoing opportunities to charge very cheaply.Arbitraging price differences can allow for steep financial gains; however, at the expense of increased battery degradation.Models for smart grids and other dispatch models should explicitly trade off profit opportunities with the added battery stress and battery life reduction.
This paper addresses two questions: How do different battery degradation mechanisms affect battery operation in an economic dispatch setting?, and How can we balance accurate battery degradation formulations with model scalability?
In the following, we discuss battery structure and aging mechanisms, and their relevance in economic dispatch modeling.We formulate an economic dispatch model for price arbitrage with adequate, scalable functional forms for three main battery degradation mechanisms: Cycle Depth (CD), average cycle state of charge (SOC), and SOC-based calendar aging.We also contribute a new cycle counting formulation, and perform a sensitivity analysis on battery replacement costs.
We assess the impact of the degradation on optimal battery dispatch, capturing over 95% of battery degradation and solution times short enough for practical applications.

| LITERATURE
Lithium-ion batteries consist of a carbonaceous anode, a metal oxide cathode, a lithium salt electrolyte, and a separator.Each of these four components experiences degradation, causing decreasing power output and reducing the maximum stored energy, thus affecting overall battery lifetime.Battery lifetime is related to a battery's purpose.Typically, end of life means that the battery's capacity is reduced so much that it can no longer adequately perform its intended purposes.
Battery life has two components: calendar life, and cycle life, which are additive, so we minimize both. 6,7alendar life corresponds to the time before the battery reaches a purpose-specific capacity threshold (eg, for electric vehicles [EVs] this is often considered to be 80%).Cycle life is connected to the number of charge and discharge cycles a battery can experience before the battery reaches its end of life.In frequently used batteries, the cycle life is the decisive component of lifetime. 8Stationary, grid-connected, batteries are often used frequently; hence, for these, cycle life tends to be decisive for their life.In contrast, EV batteries are typically idle most of the time, with few charge cycles.For EVs calendar aging is very often the main factor impacting battery life.

| Calendar degradation mechanisms
Calendar aging is mostly driven by time, ambient temperature, and the SOC.][10] Prolonged, high SOC levels are devastating to batteries. 9,11ost batteries operate optimally at a cell temperature of approximately 25 C. 4 While cell temperature has a very significant impact, with about doubled battery aging for every 10 C to 15 C increase, it is an easily controllable parameter in stationary conditions.Furthermore, it is difficult to assess the influence of the ambient temperature on cell temperature, especially for actively used batteries in low ambient temperature conditions.3][14] Finally, while low ambient temperatures are beneficial for the calendar lifetime, they can be harmful to the cycle lifetime. 15

| Cycle degradation mechanisms
Besides the intended exchange of electrons, battery charging and discharging causes side reactions that promote battery degradation by increasing the internal resistance and reducing the storage capacity.Physical aging refers to the loss of active material (eg, lithium oxide) in the electrodes, and is affected by operating decisions CD and SOC.In contrast, chemical aging refers to the loss of material for transport between electrodes (eg, lithium inventory), and is affected by calendar time, cell temperatures, and current rate (C-rate) when charging and discharging. 4,8attery cycling induces physical stresses in the form of volume changes through the intercalation and deintercalation of lithium ions in the anode and the cathode.These volume changes lead to particle fractures at the electrodes, thereby exposing additional electrode surface to the electrolyte, which leads to a growth of the solidelectrolyte interphase (SEI) layer.This in turn results in a permanent drop in cell capacity and thus overall battery capacity. 11he four main cycling-related degradation drivers are: CD, C-rate, temperature, and SOC.
Deeper discharge cycles result in faster battery aging. 10,16In the literature, depth of discharge (DOD) is used for both the absolute discharge level of the battery (such that SOC + DOD = 100%), and for the depth of a discharge compared to a starting SOC that may be different from 100%.We rather use CD for the latter meaning.
Operating a battery with 10% CD compared to 100% CD allows 100 times more cycles and 10 times larger total energy throughput. 17The pronounced nonlinear relation between CD and aging of Li-ion batteries is typically not accounted for in economic dispatch models.
The second important cycling-related aging driver is the C-rate.It is defined as the (dis)charging current divided by the rated battery storage capacity.Lower C-rates tends to result in lower battery aging. 6In grid applications, the (dis)charge voltage is considered constant; hence, we express the C-rate relative to a full (dis) charge in 1 h. 2 Batteries have three degradation phases.A new battery experiences rapid aging due to the initial formation of the SEI layer, resulting in up to 5% capacity loss.In the second phase, the battery is more stable and ages at a slower rate than in the other phases.Batteries spend most cycles in this stage.After approximately 12% capacity loss (including phase 1) phase 3 follows with accelerated degradation. 5hile low C-rate are generally good for battery longevity, in low-temperature conditions the discharge efficiency is worse with low discharge rates.This is due to a lower solid-state diffusivity of the Li ions, low ionic conductivity of the electrolyte and much higher interfacial charge transfer resistance. 18egardless of the battery type, C-rates below 1C have modest impact on battery capacity, 8,19 for lithium iron phosphate (LFP) batteries this continues even up to 4C.For EVs battery management systems prevent the occurrence of damaging high C-rates. 20For batteries in grid applications, the power ratings are usually lower than the energy rating, which prevents C-rates >1C from happening.Consequently, degradation due to C-rate can often be ignored, except in high-power applications such as frequency regulation. 2 The third important cycling-related aging driver is cell temperature.High cell temperatures accelerate chemical reactions and thus harmful side reactions.Compared to the ambient temperature, the temperature gradient in the battery depends primarily on the C-rate: high C-rate will result in a high cell temperature.It is difficult to separate cell temperature impact from the C-rate stress, and at room temperature the C-rate is the main driver, therefore cell temperature effects can be captured by a C-rate degradation mechanism.
The fourth driver, SOC, is often considered for calendar aging, but it affects cycle aging too.The optimal average SOC for battery cycling is 50%: cycles passing symmetrically through SOC = 50% cause the least damage. 16,21| BATTERY DEGRADATION Battery degradation mechanisms have complex nonconvex behavior.This complexity makes integrating battery degradation in economic dispatch models difficult because of numerical tractability.Electrochemical models are the most accurate, but are highly nonconvex and need much data. 22They do not scale to the longer time horizons considered in economic dispatch models.Many technoeconomic studies consider battery degradation only through fixed upper and lower bounds for SOC and a limit on C-rate. 23,24From a financialeconomic perspective such limits unnecessarily disallow potentially profitable operations.Ignoring other degradation mechanisms, typically leads to high resting SOC level and deep cycles, which are both harmful to battery life. 25We desire a degradation model usable in economic dispatch optimization that considers factors affected by operational decisions SOC, CD, and C-rate.Hence, we formulate degradation based on measurements of a specific battery cell. 21,26Total battery degradation equals the sum of calendar and cycle aging 8,27 :

| Calendar degradation
Arrhenius law is the basis for electrochemical calendar aging models, wherein temperature is the main driver. 28s we lack data to apply Arrhenius law and rather avoid representing temperature, we opt for an empirically based expression wherein calendar aging increases when longer time is spent at high SOC levels.Several polynomial curve fits have been used to define this aging relationship for battery types LFP 28 and nickel managanese cobalt (NMC). 29Linear fits are imprecise as they disregard aging plateaus. 7Higher order polynomial functions are more accurate, and these can be represented using piecewise linear functions.

| Cycle degradation
To represent CD-induced degradation the rainflow algorithm is commonly used to count the occurring cycles.2][33][34][35][36] The latter allows the penalization of discharges more than proportional to the CD.The basis is the material stress function, which determines capacity loss per cycle as a function of the CD.Two approaches to this stress function exist.First, it can be based on the Arrhenius equation, but this results, however, in a concave stress function, 32 so we disregard this option.Second, it can be based on an amplitude function for physical stress (eg, Equation 1). 9,16Parameter a displays the maximum capacity loss per cycle, while m is the fatigue strength exponent.
This stress function is the input for the segmented cost function Equation (2), wherein r is the battery replacement cost €=kWh ð Þ , e RAT battery capacity, and N the number of virtual battery segments. 9Since it more easily allows implementation of other aging mechanisms compared to other alternatives in the literature, we base Equation (2) for CD-based damage in virtual segment j on 33,34 : Naturally, there is a tradeoff in the number and sizes of the segments, and thereby the accuracy of a piecewise linear approximation and the complexity of the resulting model.For the battery type used in our case study, with 16 segments the relative error is approximately 2% only for an NMC battery cell. 9We try to minimize this error further by considering a nonuniform segmentation in Section 5.2.

| MODEL FORMULATION
The model is set up to determine the maximum profit obtainable from charging and discharging the battery, given hourly varying prices, while considering battery degradation costs.We list variable names in Table 1.Parameter values related to the battery can be found in Tables 2 and 3 in Section 5.For reference, we provide sets, variables and parameter names in Tables 6-8 in Nomenclature at the end of the paper.
Equation (3) minimizes the battery degradation cost K over all time steps.We treat the arbitrage profit from charges C j,h (purchases) and discharges D j,h (sales) as negative costs, both multiplied by the hourly electricity price, parameter p h .MIN : We include constraints for operation and for degradation.

| Battery operation constraints
As indicated previously, we propose a nonuniform virtual battery segmentation.Equation ( 4) computes the storage level in a segment at the start of a time step as the previous storage level modified by loss-corrected charges or discharges in the previous time step.Equation ( 5) restricts stored energy in each segment.Equations ( 6) and ( 7) restrict charges and discharges to their maximum values.
Equations ( 8) and ( 9) compute binary variables for the mutually exclusive battery modes charging, discharging and steady state, which are used to count cycles and determine battery degradation.The reader may notice that Equation ( 8) enforces a minimum charge or discharge of 1.The scaling of the data is such that this is never binding.It is computationally cheaper to use continues variables in the right-hand side of Equation ( 8), however, a more general formulation would use a bigM.

| Battery degradation constraints
We adopt the widely used rainflow algorithm-based model for CD.In addition, we consider the influence of the SOC on both calendar and cycle aging.Thus, the model considers three aging factors: CD, average cycling SOC and SOC-based calendar aging.
The degradation costs K consist of the sum of three individual capacity losses Q (%) multiplied by the replacement cost r in €/kWh and e RAT in kWh.

| CD-based degradation
To reflect the nonuniform virtual segmentation for the Q CD , in Equation (11) we multiply the rated energy e RAT with the length of each segment.Here, o CD j is the SOC value at the segment breakpoint.
To measure CD degradation, we keep track of accumulative discharges in consecutive time periods, not interrupted by periods with charges or without activity.Equation (11) computes the (accumulative) CD-based capacity loss after each discharging time period.To do so, the first term locates all segments that the discharge in the current time step passes through and how much of each segment is discharged.The second term is the degradation value of specific segments computed with piecewise linear function

| Average cycle SOC degradation
We introduce auxiliary variables to track battery cycles and SOC levels to compute the average cycle SOC degradation at the end of each discharge cycle.First, Equation ( 12) computes the SOC at the end of every hour.
Equation ( 19) uses B ST h to save the SOC at the start of a cycle in auxiliary variable A ST h .In the following timesteps, as long as the cycle continues, the first term becomes zero and the second term copies the value of the previous timestep.One time step after the cycle end, A ST becomes zero due to B END hÀ1 .
Equations ( 20) and ( 21) compute the deviation between the average SOC and 50%.
Finally, Equation ( 22) computes the cycle degradation, at the end of each cycle.
Note that Equations ( 19) and ( 22) are bilinear constraints.The Gurobi solver can handle such constraints efficiently.

| Calendar degradation
To calculate the calendar life loss Q CAL we use SOS2 variables via Equations ( 23)- (25).Equation ( 26) selects the appropriate segment and computes weighting factors Z i,h .Equation ( 27) determines the calendar aging.

| CASE STUDY
We apply our model to assess the impact of the three considered degradation mechanisms and conduct a sensitivity analysis on battery replacement costs.The optimization is implemented in Pyomo, using Gurobi solver 9.1.2on an Intel Core i5-6200U CPU @ 2.30 GHz, 8 GB RAM.Solution times for specific instances vary from a few seconds to about an hour.

| Input parameters
The data that support the findings of this study are openly available in Zenodo at https://zenodo.org/doi/10.5281/zenodo.10255080. 37We consider a 2-day period, 22 to 24 April 2019 with hourly German spot prices, 38 to which we add a grid fee of 7.39 ct 39 and 19% value-added tax.Any negative hourly prices based on this are replaced by the value 0.1 ct/kWh.We fix the start and end SOC level to zero.We consider an NMC battery, values ares given in Table 2.The maximum C-rate, or power to energy ratio, is 0.6.Values for a, m, f are estimated based on Laresgoiti et al 16 (c.f. Figure 1).In contrast to Laresgoiti et al, 16 we disregard the positive constant (the function value at 50% SOC) for parameter f as it is CD damage and we reflect that mechanisms separately.Also, we scale the value by the maximum deviation of 50%.
For the calendar degradation, we compute hourly degradation values from the 10-month NMC data at 25 C from the piecewise linear curves with plateaus in the study of Keil et al. 7 As several studies give significantly lower annual calendar degradation values at full SOC than 8%, we divide the values by 4. Accordingly, a permanent 100% SOC then causes 2% capacity loss per year.Table 3 shows the five chosen breakpoints, and Figure 2 presents the piecewise linear function.
We parameterize the model with values for hourly prices and battery degradation mechanisms.We perform several analyses to validate the model and to analyze the impacts of different types of battery degradation on battery charging and discharging behavior, and profitability of price arbitrage.

| Model calibration and validation
We verify model outcomes for degradation with this formula: Relative error = (model value/measured value) À 1, wherein measured value may be an interpolation between two actual measurement points from a literature source.For convex functions, piecewise linear segments are always above the curve, and Relative error ≥0.
First, we compare a uniform segmentation and a nonuniform segmentation optimized for minimal average distance between segments and the original curve (c.f.Imamoto and Tang 40 ).The segment breakpoint locations turn out to be very similar.In the nonuniform approach, the breakpoints shift somewhat toward the steeper upper end of the stress function, making that part more accurate, and the lower segments are larger and slightly less accurate.We perform a model run considering Q CD only, and find that the battery cycles most often in the lower segments, which are better represented in the uniform segmentation.The nonuniform segmentation performs slightly worse.However, the difference in relative error is small: 5.000% for uniform segmentation and 5.007% for the nonuniform.We continue the analyses using a uniform segmentation.
Next, we consider all three degradation mechanisms.Table 4 shows that the relative error for each component is between 1.6% and 6.6% only.
Figure 1, shows the quadratic and piecewise linear curve fits for the five original data points, and the model output data.As the quadratic curve fit is very bad for high SOC levels, we use the piecewise linear function to find the relative error for Q CD of 3.24%.
Considering Q CAL with 16 data measurement points (in green) in Figure 2, 1 we get a relative error of 1.62% for a piecewise linear fit with four segments (five breakpoints).Adding a sixth breakpoint at 90% SOC, improves accuracy but at the expense of an increase in solution time from 2 min to over an hour.Considering scalability, we disregard this option.The total relative error is 3.32% only.For a model run disregarding degradation, the left column in Figure 3 shows the ex post calculated degradation costs.In this case, the battery cycles to the full extent for every price arbitrage opportunity, even the smallest.
In this 2-day planning horizon, the disregarded degradation costs of €27 are significantly higher than the revenues obtained, €20.When degradation is considered, revenues are about 40% lower, but degradation reduces by more than 75%.Consequently, a €7 loss turns into a €6 profit.Furthermore, in agreement with literature on grid-connected batteries, CD degradation (the red areas in the columns) is the primary degradation mechanism.The following section analyzes the impact of the individual degradation mechanisms on the SOC profile of the battery.

Degradation mechanism
Relative error [%] Combined degradation 3.32 Note: Relative errors are not additive; the combined relative error is 3.32% only.difference.Due to the overlap, no degradation is more clearly visible in Figure 5.This results in six cycles, three of which have a 100% CD depth that causes high battery degradation.At 00:00 on the second day, CAL charges with a 1-h delay compared to no degradation.Thereby, the battery charges at a slightly higher price, but avoids 1 h in 100% SOC, and thus some calendar aging.
In contrast with the two cases discussed so far, in hours five-six of the first day for CD, there is one, small, 10% cycle instead of a 60% cycle.This is due to the small price difference in the charge and discharge hours, which reduces profitability of larger discharges because of the discharge degradation costs.The charging in the early afternoon is more gradual, and not completely full at 93%.In the first evening, there is a much lower discharge at 18:00 of 25% only, and at midnight we see no discharge at all.The next morning at 6:00, there is a smaller discharge again, but at 18:00 there is a full discharge.The battery charged almost for free at 12:00 of the first day and now discharges for more than 13 ct/kWh.A full battery discharge is typically disallowed by lower and upper storage limits in optimization models, but here we show the benefit of directly trading off degradation costs and revenues.
In addition, consider full cycle degradation consisting of both Q CD and Q SOC .In this scenario, the cycle at 05:00 to 06:00 does not happen at all.Once charging starts at noon, there is a lower average SOC in the next periods.Until 18:00 the SOC is 75% (the blue and yellow lines overlap), and we see a deeper discharge at 18:00.The deeper discharge brings the average SOC during cycling closer to 50%, which is beneficial from a degradation perspective.
Finally, we observe the combination of all three mechanisms in full degradation line.The first day, the SOC profile is similar to that of CD and SOC.However, the calendar aging punishes high SOC levels, which induce the battery to charge 2 h later.This becomes more pronounced starting at 06:00 on the second day as a deeper discharge occurs (the yellow line dives below the blue line).This has two effects: calendar aging is reduced due to lower SOC levels, and Q SOC is reduced (to zero) because the cycle goes from SOC 75% to SOC 25% (50% on average).In contrast, during the last charge-discharge cycle, calendar aging causes a later charge hour, a lower SOC level and lower degradation in the period 11:00 to 17:00; however, this gain is partially offset as the discharge to 0 cannot be symmetrically around 50%, thereby causing a higher Q SOC .Overall, the impact of battery degradation on optimal charging and discharging is pronounced.Considering CD-based degradation alone may result in long-lasting high resting SOC levels, as well as cycles in the higher and lower SOC spectrum, which may accelerate calendar aging.Thus it is advisable to consider multiple mechanisms.5 present a range of battery replacement costs.The range from 50 to 500 €/kWh captures uncertainty in battery price forecasts 41 as well as (installation) cost differences for different battery pack sizes (note that No degradation corresponds with r ¼ 0).

| Sensitivity on battery replacement cost
Generally, the maximum SOC level decreases and cycle amplitudes become smaller with rising replacement cost r, due to higher calendar aging costs and disproportionally high discharge costs for deep cycles.For r ¼ 500 the highest SOC level is 31%, whereas for r ¼ 50 and r ¼ 100 it is 100%.
If batteries would become significantly cheaper, for instance, r ¼ 100, or even r ¼ 50, the annualized CD degradation could become very high with respective values 12.1% and 18.7%.As the profit values show, these high degradation values are more than offset by the added revenues from trading.This clearly shows the advantage of trading off degradation costs explicitly in the optimization rather than imposing hard bounds on SOC or CD.If replacement costs are lower, we can benefit more often Note that for r ¼ 500 there is an operating loss, but this loss would have been significantly larger if no trades had occurred.The pay-back period indications provide evidence that large-scale battery use for intra-day price arbitrage should be (close to) commercially viable.Considering future battery cost reductions, our sensitivity analysis shows that cheaper batteries lead to spikier chargedischarge cycles.Increasing price volatility has the same effect as lower battery costs.More frequent and larger price differences will more quickly offset degradation costs, and therefore also result in spikier charge-discharge cycles.

| Model runtime
The model takes 5 min 46 s to solve with r ¼ 150 and considering all three degradation mechanisms.Because we wish to implement the degradation formulations in larger models, this may not be scalable.Table 5 shows that the contribution of each mechanism to the total battery degradation is very different.When the battery is very actively used, with many, and deep, cycles, r ¼ 50, CD degradation accounts for 93.5% of the total degradation, Q CAL 5% and Q SOC 1.5% only.For the moderate, realistic r ¼ 150 these values change to Q CD 83.1%, Q CAL 11.1% and Q SOC 5.8%.Given that Q CD has the most impact, we execute two additional runs, one without Q SOC and one without Q CAL , to assess the runtime influence of each mechanism.Combination Q CD -Q SOC solves in 17 s, while Q CD -Q CAL solves in just under a second.We conclude that Q SOC has the least impact on the model accuracy but by far the biggest impact on the run time.
To maintain scalability, one may remove the Q SOC degradation from the optimization with little impact on model accuracy for larger systems, longer planning horizons, and in stochastic models.

| CONCLUSION
In this paper, we develop scalable, accurate formulations for battery degradation to allow better tradeoffs in smart grid and other technoeconomic electricity dispatch models.These formulations consider CD, average cycle SOC, and SOC-based calendar aging.We present and discuss how different degradation measures affect optimal battery operation, generally resulting in less deep chargedischarge cycles and moderated SOC levels.In many models, these damage-reducing effects are commonly enforced by merely imposing hard lower and upper bounds on SOC levels, but we emphasize this unnecessarily reduces the potential benefit from price arbitrage opportunities when price variations are very large.In a stylized, realistic setting we demonstrate profitability, and estimate a degradation model accuracy of within 3.32%.This estimate is based on linearly interpolated measurement data; real battery tests could further validate functional forms and parameter values.Two aspects are not considered in our formulation are ambient and battery cell temperature, and very high charge rates, opening up for future work.In addition, it would be interesting to apply the degradation models considering demand and generation assets in a smart grid setting.The figure shows that for low and high SOC levels, capacity degradation is higher compared to levels in between.Capacity degradation is linked to the anode potential at specific SOC levels, caused by the correlation of anode potential to the loss of cyclable lithium (c.f.Keil et al 7,11 ).High capacity degradation occurs during the lowest anode potential, from around 60% SOC and up for NMC cells.Degradation is more moderate in SOC range about 30% to 60% for medium anode potentials.Below 30% SOC, the anode potential is highest, and therefore capacity degradation lowest.

Figure 4 1
Figure4shows the SOC over 2 days, with and without the different degradation mechanisms considered.First, we consider no degradation and calendar aging CAL, then the cycle degradation mechanisms, before considering all three.For reference, the hourly spot price is indicated by the dashed, blue line.No degradation and CAL have almost the same SOC profile characterized as very volatile (spiky) cycles that exploit every price Calendar degradation in %/hF I G U R E 2 Piecewise linear approximation of calendar aging.The red line is with a sixth breakpoint at 90% SOC.T A B L E 4 Relative model degradation errors.

3
Revenues and degradation cost calculation.
State of Charge in % Price in ct/kWh F I G U R E 5 SOC for no or all three degradation mechanisms for varying battery replacement cost.

Figure 5
Figure 5 and Table5present a range of battery replacement costs.The range from 50 to 500 €/kWh captures uncertainty in battery price forecasts41 as well as (installation) cost differences for different battery pack sizes (note that No degradation corresponds with r ¼ 0).Generally, the maximum SOC level decreases and cycle amplitudes become smaller with rising replacement cost r, due to higher calendar aging costs and disproportionally high discharge costs for deep cycles.For r ¼ 500 the highest SOC level is 31%, whereas for r ¼ 50 and r ¼ 100 it is 100%.
Note: Subscripts j and h in various variables indicate virtual battery segment and hour, respectively.T A B L E 2 Battery input parameters.
T A B L E 3 Break point calendar aging values(10 À7).
T A B L E 5 Battery replacement costs sensitivity results (annualized by multiplying model results with 182).