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Appl. Sci. 2021, 11, 1099. https://doi.org/10.3390/app11031099 www.mdpi.com/journal/applsci
Article
Comparison of Lead-Acid and Li-Ion Batteries Lifetime
Prediction Models in Stand-Alone Photovoltaic Systems
Rodolfo Dufo-López
1,
*, Tomás Cortés-Arcos
2
, Jesús Sergio Artal-Sevil
1
and José L. Bernal-Agustín
1
1
Departamento de Ingeniería Eléctrica, Universidad de Zaragoza, C/María de Luna, 3,
50018 Zaragoza, Spain; jsartal@unizar.es (J.S.A.-S.); jlbernal@unizar.es (J.L.B.-A.)
2
Escuela Universitaria Politécnica de La Almunia, EUPLA, Universidad de Zaragoza, C/Mayor, s/n,
50100 La Almunia, Zaragoza, Spain; tcortes@unizar.es
* Correspondence: rdufo@unizar.es
Abstract: Several models for estimating the lifetimes of lead-acid and Li-ion (LiFePO
4
) batteries are
analyzed and applied to a photovoltaic (PV)-battery standalone system. This kind of system usually
includes a battery bank sized for 2.5 autonomy days or more. The results obtained by each model
in different locations with very different average temperatures are compared. Two different loca-
tions have been considered: the Pyrenees mountains in Spain and Tindouf in Argelia. Classical bat-
tery aging models (equivalent full cycles model and rainflow cycle count model) generally used by
researchers and software tools are not adequate as they overestimate the battery life in all cases. For
OPzS lead-acid batteries, an advanced weighted Ah-throughput model is necessary to correctly es-
timate its lifetime, obtaining a battery life of roughly 12 years for the Pyrenees and around 5 years
for the case Tindouf. For Li-ion batteries, both the cycle and calendar aging must be considered,
obtaining more than 20 years of battery life estimation for the Pyrenees and 13 years for Tindouf. In
the cases studied, the lifetime of LiFePO
4
batteries is around two times the OPzS lifetime. As nowa-
days the cost of LiFePO
4
batteries is around two times the OPzS ones, Li-ion batteries can be com-
petitive with OPzS batteries in PV-battery standalone systems.
Keywords: lead-acid batteries; lithium batteries; photovoltaic; standalone systems; battery lifetime
1. Introduction
Renewable electricity generation is widely used in rural areas where the electrical
grid is weak or nonexistent. Stand-alone (off-grid) systems are typically powered by a
photovoltaic (PV) generator with battery storage. In this kind of system, the battery tech-
nology most widely used is lead-acid. In some cases, a hybrid PV–fossil fuel generator
(diesel or gasoline)–battery storage system can be optimal—that is, the system with lower
costs during the system’s lifetime. Standalone systems can be direct current (DC) or altern
current (AC) coupled [1]—that is, the bus where components are connected can be the DC
bus or the AC bus. DC coupled systems (Figure 1) are usual in low power systems (typi-
cally lower than 5 kW), while the AC-coupled system is commonly used in larger systems.
The charge controller is needed to avoid overcharge and over-discharge of the battery,
preventing premature failure. Additionally, an inverter (DC/AC) is needed when there is
AC load.
Citation: Dufo-López, R.; Cortés-
Arcos, T.; Artal-Sevil, J.S.; Bernal-
Agustín, J.L. Comparison of
Lead-Acid and Li-Ion Batteries
Lifetime Prediction Models in
Stand-Alone Photovoltaic Systems.
A
ppl. Sci. 2021, 11, 1099. https://
doi.org/10.3390/app11031099
Received: 17 November 2020
Accepted: 21 January 2021
Published: 25 January 2021
Publisher’s Note: MDPI stays neu-
tral with regard to jurisdictional
claims in published maps and insti-
tutional affiliations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license
(http://creativecommons.org/licenses
/by/4.0/).
Appl. Sci. 2021, 11, 1099 2 of 16
Figure 1. Direct current (DC) coupled standalone photovoltaic (PV) system.
In standalone systems, different types of batteries can be used [2]. Lead-acid batteries
(valve-regulated lead-acid type, VRLA) are the dominant technology for photovoltaic off-
grid applications [3] due to their affordable costs for large installed capacities. However,
lead-acid batteries are the overall weakness of the PV system and tend to be replaced by
new technologies such as Li-ion batteries [4], which can be competitive in some cases [5]
due to their higher cycle life, despite their higher cost.
The total battery cost (including its replacement during the system lifetime) is the
highest in the system’s net present cost (NPC); thus, in the optimization process of
standalone systems, the accurate estimation of battery life is one of the most critical issues.
Significant errors in the battery lifetime prediction would lead to great errors in the esti-
mation of the NPC.
Lead-acid battery aging factors are charge and discharge rates, charge (Ah) through-
put, the time between full charge, time at a low state of charge (SOC), and partial cycling.
Several researchers have analyzed the lead-acid battery aging factors [6,7]. Classical mod-
els widely used by researchers and software tools to estimate the battery life are the
“equivalent full cycles model” and the “rainflow cycle counting model” [8].
The equivalent full cycles model counts the full charge (Ah throughput) cycled by
the battery since the start of its lifetime, without considering SOC, temperature, current,
or any other variable; when this value reaches the charge, the battery can cycle, consider-
ing the cycle life shown in the manufacturer datasheet obtained under standard tests, and
the end of the battery life is reached. The rainflow cycle count model includes the effect of
the depth of discharge (DOD). Nevertheless, real operating conditions (current rate, tem-
perature, DOD, SOC, etc.) are different from the laboratory conditions of the cycles shown
by the manufacturer datasheet, so a significant error in the lifetime prediction can be ob-
tained.
Most of the previously published studies of the simulation and/or optimization of
systems with battery systems do not use advanced models to estimate battery lifetime. In
many cases, the battery degradation is not considered or its lifetime is estimated in fixed
values based on the experience of the researcher [9–20]. In other cases, battery lifetime is
estimated by using the equivalent full cycles model [21–25]. In the best cases, it may be
estimated by using the rainflow cycle counting method [26–29]. However, a battery’s real
lifetime can differ from the estimated lifetime by many years using the mentioned meth-
ods, depending on the operating conditions. As previously mentioned, a high error in the
estimation of the battery life would imply a great error in the estimation of the total cost
of the batteries in the NPC of the system; therefore, the real levelized cost of Energy (LCE)
may be very different from the expectation.
Appl. Sci. 2021, 11, 1099 3 of 16
A much more accurate lead-acid aging model (and also more complex and with
higher computational difficulty) is the one described by Schiffer et al. [30], called
“weighted Ah throughput model” and used by iHOGA software [31]. The model is based
on applying weighting factors for the battery’s charge throughput to estimate the lost ca-
pacity (considering the different stress factors for cycling and corrosion). Schiffer et al.’s
model was used to estimate the battery life in PV systems [32,33]. This model obtained
results very similar to the real ones [33], while the equivalent full cycles model and the
cycle counting model obtained lifetime estimations that can be, in some cases (PV-battery
systems), two or three times higher than the real battery lifetime.
Li-ion batteries ([34–36]) have a higher cycle life, energy density, and energy effi-
ciency, and lower maintenance compared to lead-acid batteries. The LiFePO4 (LFP) type
is the most used in off-grid systems. Li-ion batteries’ most significant aging external fac-
tors are temperature, charge and discharge rates, and DOD [37]. In simulation and opti-
mization of standalone systems, Li-ion cell level aging models [38] are usually used due
to their simplicity. Electrochemical models are usually very complex, even the most sim-
plified ones [37,39,40], implying high calculation times [41]. “Calendar” aging occurs
when a battery is not being used while “cycle” aging occurs when the battery is under
charge or discharge current [41]. Cycle aging is affected by the total charge (Ah) through-
put from the start of battery lifetime, the current, the ambient temperature, and the SOC.
Calendar aging main factors are temperature and SOC [38]. A good example of Li-ion
aging model text matrix is shown in the work of Oyarbide et al. [42].
The rainflow cycle count model is also used for Li-ion batteries. Arrhenius kinetic-
based aging models [38] are the most used cycle aging models for Li-ion batteries. For
example, Wang et al.’s [43] model was obtained by performing many accelerated cycling
tests to commercial LiFePO4 cells, obtaining a capacity fade model that takes into account
the Ah-throughput and temperature for different charge/discharge rates. Li-ion battery
calendar aging has been modeled by different researchers [38]. For example, Petit et al.
[44] used an expression based on Arrhenius law, considering temperature, time, and SOC.
In this work, we compare the battery lifetime estimation of a PV-battery system used
to supply electricity to a household located in two different locations with very different
average temperatures, considering different models for the degradation of lead-acid or Li-
ion batteries. In Section 2, the models of the PV system components and the different bat-
tery lifetime models are shown. In Section 3, we show the comparison of the different
models applied in the PV-battery systems. Finally, the main findings and conclusions are
discussed in Section 4.
2. Materials and Methods
In this section, the components’ models are described, emphasizing the different bat-
tery lifetime models used. The system’s simulation was performed over the course of a
whole year in hourly intervals (t = 0 …. 8760 h), and the results were extrapolated for the
remaining years of the system’s lifetime.
2.1. Photovoltaic Generator
If there was no maximum power point tracking in the controller (typical in DC-cou-
pled systems), the PV output current during time t, 𝐼(𝑡) (A), was calculated as follows
([33]), where the effect of the module temperature is negligible:
𝐼(𝑡)=𝐼 ·𝐺(𝑡) (1)
where 𝐼 is the shortcut current (A) of the PV module and G(t) (kW/m2) is the irradiance
over the module surface at time t.
The output power of the PV generator (W) of 𝑁_ strings in parallel was obtained
by using Equation (2):
Appl. Sci. 2021, 11, 1099 4 of 16
𝑃(𝑡)=𝑁· 𝐼(𝑡)·𝑉(𝑡)·
𝑓
_ (2)
where VDC(t) (V) is the battery bank voltage in DC-coupled systems and fPV_loss is the loss
factor (PV module mismatch or power tolerance, losses due to dirt in the PV modules, and
losses in the wires).
2.2. Charge Controller
It prevents overcharge and over-discharge of the battery. In lead-acid batteries, over-
charge is avoided by charging batteries in three stages (bulk, absorption, or boost and float
stages, Figure 2). Absorption includes battery equalization in some cycles, periodically.
During the second and third stages, the current is limited using the pulse with modulation
(PWM) technique. The limits between stages are voltage setpoints, while advanced con-
trollers compare the minimum battery state of charge (SOC) of the previous discharge
with SOC setpoints to determine the charge stages to be applied. In this work, SOC con-
trollers performed the boosting stage if the SOC during the last discharge was lower than
a specific value (usually 70%), and equalization was performed if battery SOC during dis-
charge was lower than another specific value (usually 40%).
Figure 2. Charge controller charging stages.
Lithium batteries require a specific charge controller, as there is a very little voltage
difference for a high SOC difference (state of function is estimated in Li-ion batteries), and
they cannot accept overcharge; therefore, there is no equalization stage and no float stage,
just a constant voltage/constant current (CV/CC) charge algorithm. Some controller mod-
els are programmable and can be used for lead-acid or for lithium batteries.
The charge controller prevents over-discharge by disconnecting the battery when a
specific setpoint voltage (or SOC) is reached and reconnecting after partial recharging
when another higher specific setpoint voltage (or SOC) is reached.
2.3. Inverter
Although charge controllers include battery over-discharge protection, standalone
inverters also include this feature (in some cases, they are directly connected to the bat-
tery). The inverter efficiency depends on its output power, as shown in Figure 3. Many
researchers and software tools use a constant value for the inverter efficiency, leading to
significant errors when the AC load profile has great peaks and valleys.
Appl. Sci. 2021, 11, 1099 5 of 16
Figure 3. Typical inverter efficiency [45].
2.4. Battery
During each time step, the state of charge SOC (t) (per unit) was calculated from the
previous time step SOC, adding or subtracting the charge of the battery current:
𝑆𝑂𝐶(𝑡)=𝑆𝑂𝐶(𝑡−Δ𝑡)𝐼
(𝜏)
𝐶
𝑑𝜏 (3)
where Δ𝑡 is the length of the time step (h), 𝐼
(𝑡) (Equation (4)) is the current that effec-
tively affects the battery charge, C
N
is the nominal capacity of the battery (Ah), and 𝜏 is
the time bewteen t-Δ𝑡 and t.
𝐼
(𝑡)=𝐼
(𝑡)·
_
; 𝐼
(𝑡)0 (𝑐ℎ𝑎𝑟𝑔𝑒)
𝐼
(𝑡)=𝐼
(𝑡)
_
; 𝐼
(𝑡)0 (𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒) (4)
where 𝐼
(𝑡) (positive charging, negative discharging) is the battery current,
_
is
the charging efficiency, and
_
is the discharging efficiency. Usually both efficiencies
are considered to be the same, equal to the square root of the roundtrip efficiency.
Battery degradation models are shown in the next subsections. In all the models, if
the battery lifetime estimation in years was higher than the floating life (shown in the
manufacturer datasheet), the floating life was used as the battery lifetime. In the upcoming
subsections, the battery voltage will be modelled.
2.4.1. Equivalent Full Cycles Model
This method estimates the end of the battery lifetime when a specified number of full
charge–discharge cycles (regardless of the operating conditions) are reached (Z
IEC
), de-
fined by the IEC standard [46].
During the simulation, for every time step, the equivalent number of full cycles since
the beginning (Z
N
) was calculated as follows:
𝑍
(𝑡𝑡)=𝑍
(𝑡)|𝐼
(𝑡)|·𝑡
𝐶
(5)
where |𝐼
(𝑡)| is the absolute value of the discharge current. When Z
N
(t) = Z
IEC
, the end of
the lifetime of the battery has been reached.
2.4.2. Rainflow Cycles Counting Model
This model is more complex and precise as it considers the depth of discharge (DOD)
of the charge/discharge cycle, and is based on Downing’s algorithm [47]. This method is
0
20
40
60
80
100
120
0 20406080100
EFFICIENCY (%)
OUTPUT POWER (%)
Appl. Sci. 2021, 11, 1099 6 of 16
based on counting, during one year of the system’s simulation, the charge/discharge cy-
cles Zi corresponding to each range of the DOD (split into m intervals). There are a number
of cycles to failure (CFi) obtained from the manufacturer datasheet (Figure 4).
Battery duration, in years, can be calculated as follows:
𝐿𝑖𝑓𝑒=1
∑𝑍
𝐶𝐹
(6)
Figure 4. Lead-acid battery: cycles to failure vs. depth of discharge (DOD) [48].
2.4.3. Lead-Acid Batteries: Schiffer et al.’s Weighted Ah-Throughput Model
The weighted Ah-throughput model presented by Schiffer et al. [30] assumes that
operating conditions are typically more severe than those used in standard tests of cycling
and float lifetime.
This model uses weighting factors for the charge throughput over the battery life to
model the lost capacity due to the different aging mechanisms. These weights depend on
the DOD, the current rate, the acid stratification, and the time since the last full charging.
Battery voltage at each time step was calculated depending on if the battery was
𝐼(𝑡)>0 (charging), Equation (7), or 𝐼(𝑡)(< 0 (discharging), Equation (8), using the
Shepherd model [49]:
𝑉(𝑡)=𝑉−𝑔𝐷𝑂𝐷(𝑡)+𝜌(𝑡)𝐼(𝑡)
𝐶+𝜌(𝑡)𝑀𝐼(𝑡)
𝐶 𝑆𝑂𝐶(𝑡)
𝐶−𝑆𝑂𝐶(𝑡) (7)
𝑉()=𝑉−𝑔𝐷𝑂𝐷(𝑡)+𝜌(𝑡)𝐼(𝑡)
𝐶+𝜌(𝑡)𝑀𝐼(𝑡)
𝐶 𝐷𝑂𝐷(𝑡)
𝐶(𝑡)−𝐷𝑂𝐷(𝑡) (8)
where 𝑉 (V) is the open-circuit equilibrium cell voltage at the fully charged state, 𝑔 (V)
is an electrolyte proportionality constant, 𝜌(𝑡) and 𝜌(𝑡) (ΩAh) represent the aggre-
gated internal resistance during c harge or dischar ge, a nd 𝐶 and 𝐶(𝑡) represent the nor-
malized capacity of the battery during charge or discharge.
This model estimates the capacity loss by corrosion, 𝛥𝐶(𝑡) and the capacity loss by
cycling (degradation), 𝛥𝐶(𝑡). Each hour, the remaining battery capacity, 𝐶(𝑡), could
be estimated as the normalised initial battery capacity 𝐶(0) minus the capacity loss by
corrosion and degradation. The end of the battery life was considered to be when 𝐶(𝑡) =
0.8 𝐶.
Appl. Sci. 2021, 11, 1099 7 of 16
𝐶(𝑡)=𝐶(0) −𝛥𝐶(𝑡)−𝛥𝐶(𝑡) (9)
Capacity loss by degradation 𝛥𝐶(𝑡) was calculated counting the weighted number
of cycles, with the impact of the SOC, the discharge current, and the acid stratification. At
the same time, the capacity loss by corrosion was estimated, which is proportional to the
effective corrosion layer thickness, which grows during the lifetime of the battery depend-
ing on the corrosion voltage of the positive electrode and temperature. This is a complex
model with many equations; further details can be found in [30].
2.4.4. LFP Li-Ion Models
In Li-ion batteries, during each time step, the capacity loss 𝑄(𝑡) (percentage capacity
fade) can be calculated as the sum of cycle capacity fade and calendar capacity fade:
𝑄(𝑡) = 𝑄(𝑡)+𝑄(𝑡) (10)
Cycle capacity fade is affected by the charge cycled, which depends on the number
of cycles N and DOD, and it is also affected by other factors such as SOC, current and
ambient temperature T. Calendar aging depends on the temperature, SOC, and time t [38].
Different models can calculate the cycling capacity fade. Electrochemical models are
usually very complex. One of the most simplified electrochemical models is Astaneh et
al.’s Li-ion battery lifetime prediction model [37,39], which integrates the simplified single
particle model (SSPM) and reduced-order model (ROM) to predict solid electrolyte inter-
phase growth (SEI), giving good results for moderate cycling currents; however, it in-
cludes many variables that are dependent on the specific chemistry of the battery and are
difficult to estimate.
It must be settled that each aging model is intrinsically related to each cell type and,
therefore, the same aging model cannot be valid for all the Li-ion models.
Wang et al.’s Cycle Aging Model
Wang et al. [43] obtained the cycle capacity fade for commercially available 2.2 Ah
cells from A123 Systems, for different C-rates and temperatures:
𝑄(𝑡)=𝐵·exp [
−31700 + 370.3 · 𝐶𝑟𝑎𝑡𝑒
𝑅𝑇 ](𝐴ℎ). (11)
where B is the pre-exponential factor that depends on Crate (h−1) (current through the bat-
tery in A divided by its nominal rated capacity in Ah), R is the gas constant (8.314 J/mol-
K), and T is the ambient temperature (K). 𝐴ℎ =𝑁·𝐷𝑂𝐷·𝐶 is the total Ampere-hour
(Ah) throughput of the battery.
Groot et al.’s Cycle Aging Model
Groot et al. [50] model was obtained after testing commercial LFP cells for different
currents I (A) and temperatures T (°C):
𝑄(𝑡)=𝑎·𝑒··𝑇·· +
𝑓
(12)
where a, b, c, d, e, and f are fit parameters.
Petit et al.’s Calendar Aging Model
Calendar capacity fade is modelled by Petit et al. [44] using an expression based on
Arrhenius law, dependent on temperature, time, and SOC:
𝑄(𝑡)=𝐵(𝑆𝑂𝐶)·exp [
−𝐸_(𝑆𝑂𝐶)
𝑅𝑇 ]𝑡() (13)
where 𝐵(𝑆𝑂𝐶) (Ah/sZcal) is the pre-exponential factor depending on SOC, 𝐸(𝑆𝑂𝐶)
is the activation energy (Jmol−1), and 𝑧(𝑆𝑂𝐶) is a dimensionless constant (assumed to
be a value of 0.5).
Appl. Sci. 2021, 11, 1099 8 of 16
Wang et al.’s Calendar Aging Model Combined with Petit et al.’s Calendar Model
Cycle aging mainly occurs when the battery is charging. Petit et al. [44] postulates
that it only happens when the current is above a given limit 𝐼 (A), using in this case
the Wang model, while when current is lower than the limit calendar aging occurs, using,
in said case, Equation (13). The limit current depends on the battery capacity to tackle high
charging rates.
3. Results
An off-grid household AC load profile from a previous publication [33] has been
considered to compare the battery lifetime estimation of the PV-battery system. The aver-
age measured AC load was 3.61 kWh/day. The system is located in the Pyrenees moun-
tains, in Aragon, Spain (latitude 42.772°, longitude −0.334°) with an average outdoor an-
nual temperature of 5.1 °C. For comparison, the same system is considered to be in a
desertic place (Tinduf, Argelia, latitude 27.669°, longitude −8.144°), with an average out-
door annual temperature of 23.1 °C.
Different battery lifetime estimation models will be used.
For lead-acid batteries:
• Equivalent full cycles model;
• Rainflow cycle count model;
• Schiffer et al.’s weighted Ah-throughput model.
For LiFePO4 batteries:
• Equivalent full cycles model;
• Groot et al.’s cycle aging model.
Wang et al.’s calendar aging model combined with Petit et al.’s calendar model con-
sidered an 𝐼 of 5% of the battery bank’s nominal capacity (that is, current at C-rate of
20 h, C20).
The nominal voltages are 48 VDC and 230 VAC.
The PV modules considered have a peak power of 100 W, 12 V nominal voltage, and
short-circuit current of 6.79 A. In the Pyrenees, the PV generator is composed of four serial
× seven parallel PV modules (total 2800 W), while in Tindouf (higher irradiation) it is com-
posed of four serial × six parallel PV modules (total 2400 W). A loss factor of 𝑓_ = 0.8
was considered.
Two types of battery banks were considered for both locations. The battery bank size
was selected, considering that about 2.5 days of autonomy are required.
• The lead-acid battery bank, which consists of 24 × 2 V OPzS [30] (flooded, tubular-
plated, deep cycle) commercial batteries in serial, CN = 270 Ah (total 12.96 kWh), 1258
equivalent full cycles (CF vs. DOD curve shown in Figure 4), float life of 20 years at
20 °C (manufacturer datasheet) and roundtrip efficiency of 85%. SOC to disconnect
load (SOCmin) 20%. The parameters for the Schiffer model were the ones used in [30].
• Li-ion LFP battery consists of a commercial 48 V pack of CN = 213.3 Ah (total 10.24
kWh), 5600 cycles at 80% DOD, 4022 equivalent full cycles, float life of 20 years at 20
°C (manufacturer datasheet does not show float life, but operating life of 20 years for
stationary battery systems is usually considered [51]) and roundtrip efficiency of
90%. SOCmin is 10%.
The nominal inverter power is 600 W, and its efficiency is shown in Figure 3.
The annual average ambient temperature in the battery room in the Pyrenees location
was estimated to be higher than the outdoor temperature. We considered two values as the
average: 8 or 12 °C. In the desertic place of Tindouf, we considered that in the battery room,
the average temperature can be similar to the average outdoor temperature, 23.1 °C.
Considering Arrhenius law, corrected float life at the different temperatures consid-
ered are shown in Table 1.
Appl. Sci. 2021, 11, 1099 9 of 16
Table 1. Float life at different temperatures considering Arrhenius law.
Standard Pyrenees Tinduf
Average temperature (°C) 20 8 12 23.1
Battery float life (years) 20 46.1 35.1 16.1
Irradiation over the optimal slope (south orientation) for both locations is shown in
Table 2.
Table 2. Monthly average irradiation data from Photovoltaic Geographical Information System (PVGIS) [52], year 2015.
Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.
Pyrenees, 65° slope irradiation
(kWh/m2/day) 3.31 2.69 3.92 4.71 4.76 4.49 5.03 5.17 4.93 3.87 3.68 3.87
Tinduf, 35° slope irradiation
(kWh/m2/day) 6.75 7.54 7.7 7.41 7.36 7.02 6.65 6.5 6.64 5.85 7.07 6.43
Table 3 shows the results of the simulation during a whole year for the two locations
considered. It can be seen that the battery bank charge/discharge energy is very low
(roughly 700 kWh/yr), which implies around 60 equivalent full cycles. The charge/dis-
charge rates are very low, as is usual in PV-battery stand-alone systems: the average
charge rate is roughly 4% of the battery’s nominal capacity, while the average discharge
rate is around 1%.
Table 3. Results of the simulation during a whole year for the different systems.
Pyrenees Tindouf
Battery type Lead-acid Li-ion Lead-acid Li-ion
Load (kWh/yr) 1318 1318 1318 1318
PV generation (kWh/yr) 2804 2804 3944 3944
Battery charge/discharge energy
(kWh/yr) 740 717 767 669
Equivalent full cycles per year 57.1 70 53.6 65.3
Hours of battery charge per year 2261 2151 3102 2399
Hours of battery discharge per year 5757 5792 5050 5050
Average charge rate (% of CN) 3.77 4.67 4.15 4.88
Average discharge rate (% of CN) 0.97 1.23 1.03 1.30
The hourly simulation of a whole year is shown in Figure 5 (Pyrenees, lead-acid) and
6 (Tinduf, Li-ion). For Tindouf, the battery bank is almost all the time at SOC higher than
80% (Figure 6). For the Pyrenees, the battery bank is also most of the time at higher SOCs
than 80%; however, in winter, there are periods with lower SOCs (Figure 5), reaching 30%
during short periods.
Appl. Sci. 2021, 11, 1099 10 of 16
(a)
(b)
(c)
(d)
0
500
1000
1500
2000
2500
PV generation (W)
Hour of the year
0 2000 4000 6000 8000
0
2
4
6
8
10
12
14
16
Battery bank charge rate (% of Cn)
Hour of the year
0 2000 4000 6000 8000
0
0.5
1
1.5
2
2.5
3
3.5
4
Battery bank discharge rate (% of Cn)
Hour o f the y ear
0 2000 4000 6000 8000
Appl. Sci. 2021, 11, 1099 11 of 16
(e)
Figure 5. Hourly simulation of the Pyrenees—lead-acid battery: load (W) (a), PV generation (W)
(b), battery bank discharge rate (% of Cn) (c), charge rate (% of Cn) (d) and state of charge (SOC)
(%) (e).
(a)
(b)
0
10
20
30
40
50
60
70
80
90
100
Battery bank SOC (% )
Hour of the year
0 2000 4000 6000 8000
0
200
400
600
800
1000
1200
1400
1600
1800
2000
PV generation (W)
Hour of the year
0 2000 4000 6000 8000
Appl. Sci. 2021, 11, 1099 12 of 16
(c)
(d)
(e)
Figure 6. Hourly simulation of Tindouf—Li-ion battery: load (W) (a), PV generation (W) (b), bat-
tery bank discharge rate (% of Cn) (c), charge rate (% of Cn) (d) and SOC (%) (e).
The estimated battery lifetimes obtained by the different models are shown in Tables
4 (lead-acid) and 5 (Li-ion).
For the Pyrenees and lead-acid batteries (Table 4), lifetime estimation using the
equivalent full cycles model or rainflow model is higher than float life at 20° (20 years) but
lower than expected float lifetime considering temperature (Table 1). For Tindouf and
le ad-ac id, both m ode ls obt ain ed val ues much h igher t han th e expec ted fl oat life. Howeve r,
the more accurate Schiffer model obtained values much lower than the expected float life,
and these results are the selected ones (marked in green in Table 4).
0
2
4
6
8
10
12
14
Battery bank c harge rate ( % of Cn)
Hour o f the y ear
0 2000 4000 6000 8000
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5Battery bank disc harge rate ( % of Cn)
Hour of the year
0 2000 4000 6000 8000
0
10
20
30
40
50
60
70
80
90
100
Battery bank SO C (%)
Hour of the year
0 2000 4000 6000 8000
Appl. Sci. 2021, 11, 1099 13 of 16
Table 4. Results. Battery lifetime estimation (years) for lead-acid batteries.
Eq. Full Cycles Rainflow Schiffer Value Selected
Pyrenees 8 °C 22.7 24.2 12.2 12.2
Pyrenees 12 °C 22.7 24.2 11.8 11.8
Tinduf 23.1 °C 23.0 26.8 4.8 4.8
For the Pyrenees and Li-ion batteries (Table 5), all the models obtained values much
higher than the float life at the average temperature considered. The selected values are
20 years, marked in green in Table 4, as it is the value usually considered for stationary
battery systems, although at low temperatures than 20 °C higher lifetimes could be
achieved [51]. For Tindouf and Li-ion, the equivalent full cycles model and Groot model
obtained overestimated values, as they do not consider the calendar aging. The only
model that is considered correct is Wang–Petit, as it is the only one that considers calendar
aging, which has great importance in standalone systems, with very low charge/discharge
rates.
The results were not verified, as extensive time and resources would have been
needed to conduct Li-ion aging tests and to verify the aging models. In this work, we
wanted to estimate the lifetime of the battery using several available models. To obtain
the tuning parameters of the different models, Li-ion batteries should be tested under sim-
ilar conditions as the working conditions, during several years, as working conditions in
stand-alone PV systems are low current (usually lower than C/20) and take many hours
duri ng the da y at floatin g stage. Thi s was not poss ible if w e wanted to make t he predi ction
now, so the available models were used. Usually, researchers and engineers use the equiv-
alent full cycles model, but the results show that in many cases (most of the typical stand-
alone PV systems) it leads to overestimation of the battery lifetime.
Table 5. Results. Battery lifetime estimation (years) for Li-ion batteries.
Eq. Full Cycles Groot Wang + Petit Value Selected
Pyrenees 8 °C 67.2 162.1 66.5 20
Pyrenees 12 °C 67.2 141.1 41.4 20
Tinduf 23.1 °C 68.9 118.2 13.7 13.7
4. Discussion
Considering the typical PV-battery standalone systems, with large battery banks
with enough energy to supply more than 2 days of autonomy (in many cases 3 or 4 days,
or even more), the charge/discharge rates were very low, typically around C20 (current in
A roughly 5% of the nominal capacity in Ah).
In these cases, for lead-acid batteries, the equivalent full cycles model or the rainflow
cycle counting model overestimated the battery lifetime, being necessary to use Schiffer
et al.’s [30] model, obtaining in the case studied a lifetime of roughly 12 years for the Pyr-
enees and 5 years for Tindouf.
Using Li-ion batteries, models that just consider cycle aging are not correct—it is nec-
essary to consider both cycle and calendar aging, as the model of Wang et al. [43] com-
bined with the calendar aging model of Petit et al. does [44]. For the studied standalone
PV-battery system with Li-ion batteries and low temperatures (much lower than 20 °C),
the typical value of 20 years for stationary battery systems can be considered as the battery
lifetime. However, if the average temperature is higher than 20 °C (as in Tindouf), the
battery life is significantly reduced to 13.7 years.
Summarizing, comparing a similar battery bank size in a PV-battery standalone sys-
tem, the LiFePO4 battery life is expected to be around two times the OpzS lead-acid one.
As the LiFePO4 battery cost at the end of 2020 can be around two times the OPzS cost, this
means that economically LiFePO4 batteries can be competitive with the OPzS technology.
Appl. Sci. 2021, 11, 1099 14 of 16
Considering the expected reduction in Li-ion battery cost, we can expect that Li-ion bat-
teries will be widely installed in PV-battery standalone systems in a few years.
Author Contributions: Conceptualization, R.D.-L.; methodology, R.D.-L. and T.C.-A.; software,
R.D.-L.; validation, J.S.A.-S., T.C.-A. and R.D.-L.; formal analysis, J.L.B.-A. and R.D.-L.; investiga-
tion, R.D.-L. and T.C.-A.; resources, J.S.A.-S. and J.L.B.-A.; data curation, T.C.-A. and R.D.-L.; writ-
ing—original draft preparation, R.D.-L. and T.C.-A.; writing—review and editing, J.S.A.-S., R.D.-L.
and J.L.B.-A.; visualization, R.D.-L. and T.C.-A.; supervision, R.D.-L. and J.L.B.-A. All authors have
read and agreed to the published version of the manuscript.
Funding: This work was supported by the Universidad de Zaragoza programme “Proyectos de in-
vestigación–proyectos puente–convocatoria 2019”, project “Modelos de envejecimiento de baterías
de litio para su aplicación en simulación y optimización de sistemas aislados de la red eléctrica”
[grant number: UZ2020-TEC03].
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data available on request.
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
𝐵(𝑆𝑂𝐶) Pre-exponential factor depending on SOC (Ah/sZcal)
CN Nominal capacity of the battery (Ah)
𝐶 Normalized capacity of the battery during charge (per unit)
𝐶(0) Normalized initial battery capacity (Ah)
𝐶(𝑡) Normalized capacity of the battery during discharge (per unit)
Crate Current through the battery divided by its nominal rated capacity (h-1)
𝐶(𝑡) Remaining battery capacity (Ah)
DOD Depth of discharge (%)
𝐸(𝑆𝑂𝐶) Activation energy depending on SOC (Jmol-1)
fPV_loss PV loss factor
𝑔 Electrolyte proportionality constant (V)
G(t) Irradiance over the module surface at time t (kW/m2)
𝐼(𝑡) Current that effectively affects the battery charge (A)
𝐼(𝑡) Battery current (positive charging, negative discharging)
𝐼 Limit current above which only cycling degradation is considered (A)
𝐼(𝑡) PV output current during time t (A)
𝐼 Shortcut current of the PV module (A)
N Number of cycles
𝑁_ Number of PV strings in parallel
𝑃(𝑡) Output power of the PV generator (W)
𝑄(𝑡) Capacity loss (%)
𝑄(𝑡) Calendar capacity fade (%)
𝑄(𝑡) Cycle capacity fade (%)
R Gas constant (8.314 J/mol-K)
SOC (t) State of Charge of the battery (per unit)
t Time (hour)
T
Ambient temperature (K)
VDC(t) DC bus voltage (V)
𝑉 Open-circuit equilibrium cell voltage at the fully charged state (V)
𝑧(𝑆𝑂𝐶) Dimensionless constant (assumed a value of 0.5).
ZIEC Number of full charge-discharge cycles defined by the IEC standard
ZN Equivalent number of full cycles since the beginning
𝛥𝐶(𝑡) Capacity loss by corrosion (Ah)
𝛥𝐶(𝑡) Capacity loss by cycling (degradation) (Ah)
𝛥𝑡 Length of the time step (h)
_
Battery charging efficiency
Appl. Sci. 2021, 11, 1099 15 of 16
_
Battery discharging efficiency
𝜌(𝑡) Aggregated internal resistance during charge (ΩAh)
𝜌(𝑡) Aggregated internal resistance during discharge (ΩAh)
𝜏 Time between t-𝛥𝑡 and t
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