Electrical Modeling and Characterization of Electrochemical Impedance Spectroscopy-Based Energy Storage Systems

: This study presents the electrical modeling and characteristic analyses of energy storage systems (ESSs) based on the internal impedance characteristics of batteries to improve ESS stability. Frequencies ranging from 1 kHz to 0.1 Hz were injected into lithium-ion batteries, and the variation of the internal impedance of the batteries was obtained based on the reflected wave to determine the ESS state of charge (SoC) and temperature. The changes in the basic electrochemical impedance spectroscopy characteristics of the ESSs were observed. Specifically, the voltage, temperature, and SoC of an ESS that could be employed as a renewable ESS were analyzed. The impedance characteristics of the ESS were investigated via experimentation and simulation. The ESS comprised an electrically equivalent circuit of a series inductor ( L S ), series resistor ( R S ), parallel resistor ( R P ), and parallel capacitor ( C P ), as well as a MATLAB program based on its transfer function to generate energy. Furthermore, a method was developed for analyzing the frequency response of ESSs. The feasibility of the proposed electrical modeling was examined for a 58.4 V, 75 Ah, 4.4 kWh ESS.


Introduction
Global efforts to reduce global warming and carbon (CO 2 ) emissions are currently ongoing, and the demand for carbon neutrality, a state in which the amounts of carbon emitted and absorbed are equal, is increasing.Furthermore, the need for energy storage systems (ESSs) to replace existing fossil fuels and thereby reduce carbon emissions is increasing.Thus, renewable energy sources, including solar power, wind power, hydrogen, bio energy sources, NCM (Li[Ni,Co,Mn]O 2 ), NCA (Li[Ni,Co,Al]O 2 ), LFP (LiFePO 4 ), and LMO (LiMn 2 O 4 ), are being actively researched [1,2].ESSs constitute electrolyte-based lithium ions and have recently become popular, but due to their explosive nature, a constant risk of battery fire and explosion is present.Multifaceted research is being conducted on battery safety for ESSs [3,4].
To reduce global carbon emissions, the rapid status diagnosis of ESSs is essential, which can be quickly and accurately performed through electrochemical impedance spectroscopy (EIS).Based on the results, ESSs could be stably managed [3,4].This study comprehensively analyzes the relevance of EIS technology for ESSs of several kWh or more, improving the possibility and reliability of the online status diagnosis of ESSs.This study is expected to actively contribute to reducing carbon emissions.
Kong et al. [5] analyzed the fire hazards and safety strategies for lithium-ion batteries, while Kim et al. [6] assessed the explosion risk during lithium-ion battery fires.Furthermore, Troxler et al. [7] investigated the effect of thermal gradients on the performance of lithiumion batteries.Larsson et al. [8] studied the characteristics of lithium-ion batteries during fire tests.Thermal models of lithium-ion batteries have been studied, including Arora et al. [9], who proposed a neural network-based computational model for the thermal estimation of LFP pouch cells, and Cui et al. [10], who optimized the lumped parameter thermal model for hard-cased lithium-ion batteries.Kleiner et al. [11] experimentally studied the thermal modeling of prismatic lithium-ion batteries for electric vehicle batteries.Moreover, Wang et al. [12] developed a lithium-ion model based on high-speed impedance and presented an online estimation method for battery temperature.Peng and Jiang [13] investigated the surface temperature change and the solid electrolyte at 235 • C, 255 • C, 275 • C, 295 • C, and 315 • C for five lithium-ion batteries with varying cathode materials.Moreover, they examined the possibility of the thermal runaway of the lithium-ion batteries with varying cathode materials by analyzing their decomposition, negative solvent reaction, electrolyte decomposition, and positive solvent reaction.Li et al. [14] investigated the surface and battery temperatures during overcharging and simulated the impedance-based thermal coupling by measuring the resistance values for different states of charge (SoCs).
Studies have also investigated battery overcharging.Belov and Yang [15] reported the surface temperature, heat output, and voltage of 600 and 720 mAh batteries for overcharging at 1 C/12 V, 2 C/9 V, and 3 C/6 V and the time until thermal runaway for each overcharge.They proposed a method to improve the battery stability of lithium-ion batteries via analyses.
Battery modeling studies include Chan and Sutanto [16] and Schmid et al. [17], both of which summarized the (a) Rint model, (b) Thevenin model, (c) dual-polarization model, (d) N-RC Thevenin model, (e) RC model, and (f) PNGV model.Chen and Rincón-Mora [18] presented an accurate electrical battery model that can predict runtime and I-V performance.Bugryniec et al. [19] proposed the advanced abuse modeling of a lithium-ion battery model.Furthermore, Xiong et al. [20] afforded the data-driven multi-scale extended model based on Kalman filtering and the parameter or state estimation of lithium-ion pouch batteries for electric vehicles.Vergori et al. [21] performed battery modeling and simulation using programmable testing equipment.
Ziyad et al. [22,23] introduced a mathematical model of a lead-acid battery.Mahon et al. [24] measured and modeled the high-power performance of carbon-based supercapacitors.Gauchia et al. [25] reported on supercapacitor testing and dynamic modeling, while Buller et al. [26,27] modeled the dynamic behavior of supercapacitors using impedance spectroscopy.Moreover, Bae [28] investigated the electrical modeling and impedance spectra of lithium-ion batteries and supercapacitors.Karden et al. [29] developed the frequency-domain approach to dynamically model electrochemical power sources.
Stroe et al. [30] proposed different methods for measuring the impedance of lithiumion batteries during aging.Additionally, Varnosfaderani and Strickland [31] developed a method for the online impedance spectroscopy estimation of batteries.
Stroe et al. [32] diagnosed the state-of-health (SoH) of lithium-ion batteries using EIS.Maheshwari et al. [33] experimentally studied the cycle aging of lithium-nickel-manganesecobalt-oxide-based batteries through EIS.Buller et al. [34] presented impedance-based nonlinear dynamic battery modeling for automotive applications.Furthermore, Sihvo et al. [35] proposed a fast approach for battery impedance identification using pseudorandom sequence signals.Macdonald [36] interpreted applications for analyzing the material properties and electrode effects using impedance spectroscopy.Santoni et al. [37] developed a digital impedance emulator for battery measurement system calibration.Additionally, Gheem et al. [38] employed EIS in the presence of nonlinear distortions and nonstationary behaviors.
Multifaceted studies on EIS have been conducted.Stroe et al. [39] and Deng et al. [40] studied lithium-sulfur batteries, while Lee and Choi [41] studied lithium-polymer batteries.Moreover, Franke-Lang and Kowa [42] focused on zinc-air batteries, and Olarte et al. [43] analyzed the impedance spectra and electrical equivalent circuits of lead-acid batteries.
Oldenburger et al. [44] and Samuel and Paul [45] mathematically and electrically modeled the Warburg impedance (Z W ), which stems from diffusion in the extremely low-frequency region in lithium-ion batteries.Barreras et al. [46] presented a hardware-in-the-loop simulation battery model based on impedance spectroscopy.Liebhart et al. [47] proposed the use of passive impedance spectroscopy for monitoring lithium-ion battery cells during vehicle operations.
Kim and Kowal [48] presented a Matlab/Simulink model for monitoring cell SoH and SoC by examining the impedance of lithium-ion batteries.Guha and Patra [49] proposed the online estimation of EIS and the remaining useful life of lithium-ion batteries.Babaeiyazdi et al. [50] studied the SoC prediction of lithium-ion batteries for electric vehicles using EIS.Koleti et al. [51] developed an online method for lithium plating detection in lithium-ion batteries, and Crescentini et al. [52] proposed the online EIS and diagnostics for lithium-ion batteries via low-power integrated sensing and parametric modeling.Galeotti et al. [53] investigated the performance analysis and SoH evaluation of lithium-polymer batteries through EIS.
With regard to the online estimation of the SoC of lithium-ion batteries, Hasan and Scott [54] presented the impedance measurement of batteries under load, while Lee and Cho [55] and Wei et al. [56] conducted measurements using extended Kalman filtering.Furthermore, Kim et al. [57] studied the stable configuration of a lithium-ion series battery pack based on a screening process for improved voltage/SoC balancing, and Wang et al. [58] estimated the SoC of lithium-ion batteries from the fuse open-circuit voltage curve.
Chiang et al. [59] developed an online estimation method for the internal resistance and open-circuit voltage of lithium-ion batteries in electric vehicles.Wei et al. [60] studied the online model identification and SoC estimation for lithium-ion batteries with a recursive total least-squares-based observer.Rivera-Barrera et al. [61] also investigated the SoC estimation for lithium-ion batteries.Alvi et al. [62] examined a smart battery management system for electric vehicle applications, while Jang and Yoo [63] analyzed impedanceand circuit parameter-based battery models for hybrid electric vehicle power systems.Richardson et al. [64] and Gogona et al. [65] proposed methods for analyzing the power and drive systems and analyzed the battery impedance to estimate the internal temperature and lifespan of a battery cell.

Impedance Spectrum of Cylindrical and Pouch-Type ESS
Figure 1 depicts the complex plane diagram of the impedance spectrum of a cylindrical lithium-ion battery [27][28][29].The impedance spectrum comprises a series inductor (L S ) located in the high-frequency region ( f max ), and the point at which the trace of the impedance spectrum and the real axis meet provides the series resistor (R S ) of the cylindrical battery.In Figures 3 and 4, at SoCs of 100% and 50%, the  and  ,  corresponding to charge transfer arise from the SEI formed on the internal electrode of lithium-ion batteries.Furthermore,  and a double layer ( ,  ) representing the oxidation and reduction reactions of lithium ions, respectively, are disposed.
Thus, in the complex plane, the impedance spectrum of a cylindrical lithium-ion battery exhibits small and large semicircles that are associated with the SEI and the oxidation and reduction of lithium ions, respectively.The small semicircle is described using the film resistor (R SEI ) and capacitor (Q SEI , α SEI ) corresponding to charge transfer in the solid electrolyte interface (SEI) formed by the inner electrode.Z W is associated with the diffusion of solid-state lithium ions, and it increases with a 45 • slope in the complex plane [28,44,45].Z W can be expressed as follows [28,44,45]: Figure 2 displays the complex plane diagram of the impedance spectrum of a pouchtype lithium-ion battery.Unlike cylindrical lithium-ion batteries, pouch-type lithium-ion batteries do not exhibit diffusion-negative Z W but exhibit an impedance spectrum locus that moves parallel to the real axis with decreasing frequency at SoC 100% and 50%.In contrast, at 0% SoC, almost no charge transfer (R ct ) area is observed, and Z W increases with a 45 • slope.This is the most notable cause for the rapid increase in the internal impedance in pouch-type lithium-ion batteries.
and reduction of lithium ions, respectively.In Figures 3 and 4, at SoCs of 100% and 50%, the R SEI and Q SEI , α SEI corresponding to charge transfer arise from the SEI formed on the internal electrode of lithium-ion batteries.Thus, in the complex plane, the impedance spectrum of a cylindrical lithium-ion battery exhibits small and large semicircles that are associated with the SEI and the oxidation and reduction of lithium ions, respectively.
However, at 0% SoC, the R SEI and Q SEI , α SEI corresponding to charge transfer stem from the SEI formed on the internal electrode of lithium-ion batteries, leading to the appearance of a small semicircle.Since the oxidation or reduction of lithium ions does not occur, no R ct or double layer (Q dl , α dl ) is present.
As the temperature decreases below zero, R S and R P increase at 100% and 50% SoCs in ESSs, and the impedance spectrum shifts toward the imaginary axis.At 0% SoC, R S and R P increase and Z W tends to increase with a 45 • slope in the complex plane.
Figure 5 displays the electrical equivalent circuit of an ESS at SoC 100% and 50%, as follows: L S -R S -parallel connection of R SEI and Q SEI , α SEI -parallel connection of no R ct and Q dl , α dl .The equivalent impedance of the ESS at 100% and 50% SoC is as follows: As the temperature decreases below zero,  and  increase at 100% and 50% SoCs in ESSs, and the impedance spectrum shifts toward the imaginary axis.At 0% SoC,  and  increase and  tends to increase with a 45° slope in the complex plane.

Experimental Equipment and Systems
Figure 7 presents the experimental setup for measuring the ESS impedance.

Experimental Equipment and Systems
Figure 7 presents the experimental setup for measuring the ESS impedance.Figure 8 depicts the voltage and current waveforms obtained when measuring the impedance spectra.The measurements were performed using the battery measuring equipment of BRS Messtechnik GmbH (Stuttgart, Germany).
The ESS's lithium-ion battery was manufactured by LG Chem.
(b) Energy storage system.The ESS's lithium-ion battery was manufactured by LG Chem.

Impedance Characteristic Analysis of a Cell of a Pouch-Type Lithium-Ion Battery (75 Ah)
The characteristics of a single cell of the pouch-type lithium-ion battery (75 Ah) were analyzed.
The following parameter values were considered in the analysis:

Impedance Characteristic Analysis of a Cell of a Pouch-Type Lithium-Ion Battery (75 Ah)
The characteristics of a single cell of the pouch-type lithium-ion battery (75 Ah) were analyzed.
The following parameter values were considered in the analysis: The cell voltage of the pouch-type lithium-ion battery (75 Ah) was 4.2 V at 100% SoC. Figure 9 presents the impedance spectrum measured for one cell of the pouch-type lithium-ion battery at 100% SoC under different temperatures.In Figure 9, the resistance value of the pouch-type lithium-ion battery (75 Ah) at 100% SoC ranges from 3.3083 to 7.9493 mΩ.
At 100% SoC,  ,  , parallel capacitor ( ), and  were as follows: In Figure 9, the resistance value of the pouch-type lithium-ion battery (75 Ah) at 100% SoC ranges from 3.3083 to 7.9493 mΩ.
At 100% SoC, R S , R P , parallel capacitor (C P ), and L S were as follows: (1) 100% SoC, 40  At 100% SoC, Z W was barely observed, but it appeared at the measurement frequency of around 0 Hz.
The cell voltage of the pouch-type lithium-ion battery (75 Ah) at 50% SoC was 3.65 V. Figure 10 exhibits the impedance spectrum measured for a single cell of the pouch-type lithium-ion battery at 50% SoC under varying temperatures.
The resistance of the pouch-type lithium-ion battery (75 Ah) at 50% SoC ranged from 5.1717 to 14.1187 mΩ.
At 50% SoC, R S , R P , C P , and L S were as follows.
(1) 50% SoC, 40  The cell voltage of the pouch-type lithium-ion battery (75 Ah) at 50% SoC was 3.65 V. Figure 10 exhibits the impedance spectrum measured for a single cell of the pouch-type lithium-ion battery at 50% SoC under varying temperatures.
The resistance of the pouch-type lithium-ion battery (75 Ah) at 50% SoC ranged from 5.1717 to 14.1187 mΩ.
At 50% SoC,  ,  ,  , and  were as follows.( As depicted in Figure 10,  ,  , and  decreased with increasing and decreasing temperatures.An analysis showed that it was increasing. was the largest at 10 °C, and it sharply decreased at −20 °C. Unlike the experimental results at 100% SoC, when the temperature decreased to −20 °C at 50% SoC,  and  increased, while  rapidly decreased.This resulted in the semicircle shape of the impedance spectrum (Figure 10).At 50% SoC, similar to the 100% SoC case,  almost did not appear, and it appeared near the measurement frequency of 0 Hz.
The cell voltage of the pouch-type lithium-ion battery (75 Ah) was 2.5 V at 0% SoC. Figure 11 depicts the impedance spectrum measured for a single cell of the pouch-type lithium-ion battery at 50% SoC under varying temperatures.
The battery resistance at 0% SoC ranged from 5.6889 to 29.6618 mΩ.Unlike the experimental results at 100% SoC, when the temperature decreased to −20 • C at 50% SoC, R S and R P increased, while C P rapidly decreased.This resulted in the semicircle shape of the impedance spectrum (Figure 10).At 50% SoC, similar to the 100% SoC case, Z W almost did not appear, and it appeared near the measurement frequency of 0 Hz.
The cell voltage of the pouch-type lithium-ion battery (75 Ah) was 2.5 V at 0% SoC. Figure 11 depicts the impedance spectrum measured for a single cell of the pouch-type lithium-ion battery at 50% SoC under varying temperatures.(  The battery resistance at 0% SoC ranged from 5.6889 to 29.6618 mΩ. (1) 0% SoC, 40       Figure 16 compares the impedance spectrum results for a single cell of the pouchtype lithium-ion battery (75 Ah) at 0%, 50%, and 100% SoC under 40 °C.The resistance value of the battery ranged from 3.212 to 14.6182 mΩ.
However, at 0% SoC,  rapidly increased compared to that at 100% and 50% SoC, denoting that  increased with  for 0% SoC.  Figure 16 compares the impedance spectrum results for a single cell of the pouchtype lithium-ion battery (75 Ah) at 0%, 50%, and 100% SoC under 40 °C.The resistance value of the battery ranged from 3.212 to 14.6182 mΩ.
However, at 0% SoC,  rapidly increased compared to that at 100% and 50% SoC, denoting that  increased with  for 0% SoC.At 100% SoC, R S and R P were small, and C P was also very small, verifying that the impedance spectrum curve progressed along the X-axis overall.
At 50% SoC, R S , R P , and C P were slightly higher than those at 100% SoC.At 0% SoC, R S , R P , and C P significantly increased along the Y-axis with a slope of about 45 • .However, at 0% SoC, C P rapidly increased compared to that at 100% and 50% SoC, denoting that C P increased with Z W for 0% SoC.
Figure 17     Figure 17 shows that at 25 °C,  ,  ,  , and  decreased with increasing SoC.Moreover, they increased with decreasing SoC.
However, at 0% SoC,  rapidly increased compared to that at 100% and 50% SoC.Thus, at 0% SoC,  increased with  .Figure 17 shows that at 25 • C, R S , R P , C P , and L S decreased with increasing SoC.Moreover, they increased with decreasing SoC.
However, at 0% SoC, C P rapidly increased compared to that at 100% and 50% SoC.Thus, at 0% SoC, C P increased with Z W .
Figure 18 compares the impedance spectrum results for a single cell of the pouch-type lithium-ion battery (75 Ah) at 0%, 50%, and 100% SoC under 10 • C.
The resistance value of the battery at 10 • C ranged from 3.4034 to 20.5187 mΩ.At 10 • C, R S , R P , C P , and L S were as follows: (1) 10   ,  ,  , and  decreased with increasing SoC, and they increased with decreasing SoC.
Figure 19 compares the impedance spectrum results for a single cell of the pouchtype lithium-ion battery (75 Ah) at 0%, 50%, and 100% SoC under −5 °C.The resistance value of the battery at −5 °C ranged from 3.4050 to 23.7239 mΩ. ,  ,  , and  decreased with increasing SoC, and they increased with decreasing SoC.
Figure 19 compares the impedance spectrum results for a single cell of the pouchtype lithium-ion battery (75 Ah) at 0%, 50%, and 100% SoC under −5 °C.The resistance value of the battery at −5 °C ranged from 3.4050 to 23.7239 mΩ.R S , R P , and L S decreased with increasing SoC.Additionally, they increased with decreasing SoC.
However, at 0% SoC, C P rapidly increased compared to that at 100% and 50% SoC.This signifies that at 0% SoC, C P increased with Z W .
Figure 20 compares the impedance spectrum results at 0%, 50%, and 100% SoC for a single cell of the pouch-type lithium-ion battery (75 Ah) under −20 • C. The resistance value of the battery at −20 • C ranged from 3.6055 to 30.6618 mΩ.
R S , R P , C P , and L S were as follows: (1) −20  At 0% SoC,  rapidly increased compared to that at 100% and 50% SoC.Therefore, at 0% SoC,  increased with  .

Analysis of Impedance Characteristics of 16 Cells of 4.4 kWh ESS
To analyze the stability of an impedance-based ESS, a 4.4 kWh ESS with a pouch-type lithium-ion battery and 16 cells was studied, which could be used as an ESS for renewable power generation.
For the 4.4 kWh EES, the voltage was 67.2 V at 100% SoC.

Analysis of Impedance Characteristics of 16 Cells of 4.4 kWh ESS
To analyze the stability of an impedance-based ESS, a 4.4 kWh ESS with a pouch-type lithium-ion battery and 16 cells was studied, which could be used as an ESS for renewable power generation.
For the 4.4 kWh EES, the voltage was 67.2 V at 100% SoC. Figure 21 presents the impedance spectrum for the 4.      and  decreased with increasing temperature, and they increased with decreasing temperature.
At 50% SoC, the voltage of the 4.4 kWh ESS was 59.2 V. Figure 22 illustrates the impedance spectrum of the 4.4 kWh ESS at 50% SoC under varying temperatures.At 50% SoC, the ESS resistance value ranged from 89.020 to 195.796 mΩ.R S , R P , C P , and L S were as follows: (1) 50% SoC, 40    At 0% SoC,  ,  ,  , and  were as follows: (  The resistance value of the 4.4 kWh ESS at 0% SoC ranged from 121.639 to 279.797 mΩ.At 0% SoC, R S , R P , C P , and L S were as follows: (1) 0% SoC, 40 At 0% SoC, C P rapidly increased compared to that at 100% and 50% SoC.Thus, C P increased with Z W .
Figures 24-27 exhibit the impedance spectra of the 16-cell ESS at 100%, 50%, and 0% SoCs under varying temperatures, unifying the ranges of the X and Y axes.
At 100% SoC, R S , R P , and C P were relatively small compared to those at SoC 50% and 0% SoC.
At 50% SoC, R S , R P , and C P were slightly higher than those at 100% SoC.Particularly, at −20 • C, R P very rapidly increased at both 100% and 50% SoC.At 0% SoC, R S , R P , and C P became very large and increased along the Y-axis with a slope of about 45 At 0% SoC,  ,  , and  became very large and increased along the Y-axis with a slope of about 45°.Specifically,  ,  , and  decreased with increasing SoC and increased with decreasing SoC.
At 50% SoC,  exhibited the minimum value, and at 0% SoC,  rapidly increased compared to that at 100% and 50% SoC. increased with  .
Figure 29 shows that the resistance value of the 4.4 kWh 16-cell ESS at 25 °C ranged from 52.991 to 180.482 mΩ.At 25 °C,  ,  ,  , and  were as follows: (1) 25  Specifically, R S , R P , and L S decreased with increasing SoC and increased with decreasing SoC.
At 50% SoC, C P exhibited the minimum value, and at 0% SoC, C P rapidly increased compared to that at 100% and 50% SoC.C P increased with Z W .
Figure 29 shows that the resistance value of the 4.  Specifically,  ,  , and  decreased with increasing SoC and increased with decreasing SoC.
At 50% SoC,  exhibited the minimum value, and at 0% SoC,  rapidly increased compared to that at 100% and 50% SoC. increased with  .
Figure 29 shows that the resistance value of the 4.4 kWh 16-cell ESS at 25 °C ranged from 52.991 to 180.482 mΩ.At 25 °C,  ,  ,  , and  were as follows:       ,  , and  decreased with increasing SoC and increased with decreasing SoC.At 50% SoC,  exhibited the minimum value, and at 0% SoC,  rapidly increased compared to that at 100% and 50% SoC.Additionally,  was greatly influenced by the structure or shape (cylindrical or straight shaped) of the electrode throughout the battery.The  value of the diffusion area rapidly increased and decreased with decreasing temperature and SoC, respectively.
Further research is required to determine whether Ls,  ,  , and  are completely related to the performance of lithium-ion batteries.
However, beyond a certain range, the sharp increases in  and  could stem from the decrease in the ESS performance.
Since the occurrence of  had the greatest influence on the increase in the internal resistance of lithium-ion batteries, a battery management system (BMS) should be incorporated.
This study employed the Thevenin model, which is the most widely used model, and Figures 9-32 highlight that R S is more significantly affected by SoC than temperature.Furthermore, R P is more significantly affected by temperature than SoC.
Specifically, R P rapidly increased at temperatures below −20 • C, and Z W rapidly increased at 0% SoC.
Additionally, L S was greatly influenced by the structure or shape (cylindrical or straight shaped) of the electrode throughout the battery.The R P value of the diffusion area rapidly increased and decreased with decreasing temperature and SoC, respectively.
Further research is required to determine whether Ls, R S , R P , and C P are completely related to the performance of lithium-ion batteries.
However, beyond a certain range, the sharp increases in R S and R P could stem from the decrease in the ESS performance.
Since the occurrence of Z W had the greatest influence on the increase in the internal resistance of lithium-ion batteries, a battery management system (BMS) should be incorporated.
This study employed the Thevenin model, which is the most widely used model, and performed analysis by adding an L S .
Figure 34 illustrates the simplified electrical equivalent circuit of an ESS based on lithium-ion batteries.
Based on the  ,  ,  , and  values obtained in the previous experimental re sults, the gain margin (GM) and phase margin (PM) were calculated using MATLAB Ver sion 9.4.
Figure 35 displays the Bode plot for a single cell of the pouch-type lithium-ion battery (75 Ah) at 0%, 50%, and 100% SoC under 40 °C and −20 °C.When voltage V batt (t) is generated in the ESS and a current i(t) is flowing through it, the voltages of L S , R S , and R P can be converted using Laplace transform as follows: V per (s) = R P 1 + R P C P s I(s) Therefore, the Laplace transform of the ESS voltage is as follows: The ESS transfer function can be written as follows: Based on the R S , R P , C P , and L S values obtained in the previous experimental results, the gain margin (GM) and phase margin (PM) were calculated using MATLAB Version 9.4.In the cell, the GM decreased when the angular frequency (ω = 2 π f) was above 10 rad/s (frequency: 1500 Hz), and it decreased as the angular frequency approached 0 rad/s (frequency: 0 Hz), representing the characteristics of a typical low-pass filter (LPF).
In general, GM increased and decreased with the temperature.At 0% SoC, the angular frequency at which PM was maximum exceeded 50 rad/s ▷ 1.6-2.2rad/s, and the lowfrequency GM sharply decreased.
The GM at 0 rad/s (frequency: 0 Hz) for the single cell of a pouch-type lithium-ion battery (75 Ah) was calculated using Equation ( 8), and the  values at 40 °C and −20 °C were as follows: (1) 1 Cell, 40 °C, 100% SoC:  44.8664 dB; (   In the cell, the GM decreased when the angular frequency (ω = 2 π f) was above 10 4 rad/s (frequency: 1500 Hz), and it decreased as the angular frequency approached 0 rad/s (frequency: 0 Hz), representing the characteristics of a typical low-pass filter (LPF).
In general, GM increased and decreased with the temperature.At 0% SoC, the angular frequency at which PM was maximum exceeded 50 rad/s ▷ 1.6-2.2rad/s, and the lowfrequency GM sharply decreased.
The GM at 0 rad/s (frequency: 0 Hz) for the single cell of a pouch-type lithium-ion battery (75 Ah) was calculated using Equation ( 8  In the 16 cells, the GM decreased when the angular frequency (ω = 2π f ) exceeded 2000 rad/s (frequency: 300 Hz), and it decreased when the angular frequency approached 0 rad/s (frequency: 0 Hz).These trends represent the characteristics of a typical LPF.
At 50% or 100% SoC, PM was maximum at the angular frequency of ≥200 rad/s (frequency: 30 Hz), but at 0% SoC, PM was maximum at the angular frequency of ≥21 rad/s (frequency: 2.4 Hz).PM was minimum at the maximum angular frequency of 15-21 rad/s (frequency: 2.4 Hz), signifying that the gain was reduced at 0 rad/s (frequency: 0 Hz).In the 16 cells, the GM decreased when the angular frequency (ω 2 π ) exceeded 2000 rad/s (frequency: 300 Hz), and it decreased when the angular frequency approached 0 rad/s (frequency: 0 Hz).These trends represent the characteristics of a typical LPF.
At 50% or 100% SoC, PM was maximum at the angular frequency of ≥200 rad/s (frequency: 30 Hz), but at 0% SoC, PM was maximum at the angular frequency of ≥21 rad/s (frequency: 2.4 Hz).PM was minimum at the maximum angular frequency of 15-21 rad/s (frequency: 2.4 Hz), signifying that the gain was reduced at 0 rad/s (frequency: 0 Hz).
In general, GM increased with temperature, and it decreased with temperature.At 0% SoC, the angular frequency at which the PM was at its maximum exceeded 200 rad/s ▷ 15 to 21 rad/s, and the low-frequency GM sharply decreased.
The GM at 0 rad/s (frequency: 0 Hz) for the 16 cells of the pouch-type lithium-ion battery (75 Ah) was calculated using Equation (9).The  values at 40 °C and −20 °C were as follows: (

Conclusions
This study discussed the electrical modeling and characteristic analysis of an ESS based on a pouch-type lithium-ion battery according to the internal impedance characteristics.The characteristics of the 58.4 V, 75 Ah, 4.4 kWh ESS, which could be used in renewable energy and mobility fields, were comprehensively analyzed by varying the SoC and temperature.∎ 100% SoC of One Cell (75 Ah) In general, GM increased with temperature, and it decreased with temperature.At 0% SoC, the angular frequency at which the PM was at its maximum exceeded 200 rad/s ▷ 15 to 21 rad/s, and the low-frequency GM sharply decreased.
The GM at 0 rad/s (frequency: 0 Hz) for the 16 cells of the pouch-type lithium-ion battery (75 Ah) was calculated using Equation ( 9).The GM 0Hz values at 40

Conclusions
This study discussed the electrical modeling and characteristic analysis of an ESS based on a pouch-type lithium-ion battery according to the internal impedance characteristics.The characteristics of the 58.4 V, 75 Ah, 4.4 kWh ESS, which could be used in renewable energy and mobility fields, were comprehensively analyzed by varying the SoC and temperature.Through the above research, the spread of renewable energy and ESS could be promoted, and carbon emissions could be reduced by improving the reliability and safety of impedance-based ESSs.

Figure 1 .
Figure 1.Complex plane diagram of the impedance spectrum of a cylindrical lithium-ion battery [27-29].

Figure 1 .
Figure 1.Complex plane diagram of the impedance spectrum of a cylindrical lithium-ion battery [27-29].

Figure 2 .
Figure 2. Complex plane diagram of the impedance spectrum of a pouch-type lithium-ion battery.

Figure 2 .
Figure 2. Complex plane diagram of the impedance spectrum of a pouch-type lithium-ion battery.

Figure 3 .
Figure 3. Impedance spectrum in a cell according to the SoC.

Figure 4 .
Figure 4. Impedance spectrum of the ESS based on 16 cells according to the SoC.

Figure 3 .Figure 2 .
Figure 3. Impedance spectrum in a cell according to the SoC.

Figure 3 .
Figure 3. Impedance spectrum in a cell according to the SoC.

Figure 4 .
Figure 4. Impedance spectrum of the ESS based on 16 cells according to the SoC.

Figure 4 .
Figure 4. Impedance spectrum of the ESS based on 16 cells according to the SoC.

Figure 5 .
Figure 5. Electrical equivalent circuit of the ESS at 100% SoC and 50% SoC.

Figure 6 .
Figure 6.Electrical equivalent circuit of an ESS at SoC 0%.

Figure 5 .
Figure 5. Electrical equivalent circuit of the ESS at 100% SoC and 50% SoC.

Figure 6
Figure 6 exhibits the electrical equivalent circuit of an ESS at 0% SoC, as follows: L S -R S -parallel connection of R SEI and Q SEI , α SEI -Z W .The equivalent impedance of the ESS at 0% SoC is as follows:

Figure 5 Figure 6 Figure 5 .
Figure 5 displays the electrical equivalent circuit of an ESS at SoC 100% and 50%, as follows:  - -parallel connection of  and  ,  -parallel connection of no  and  ,  .The equivalent impedance of the ESS at 100% and 50% SoC is as follows:      1  •

Figure 6 .
Figure 6.Electrical equivalent circuit of an ESS at SoC 0%.

Figure 6 .
Figure 6.Electrical equivalent circuit of an ESS at SoC 0%.

Figure 7 .
Figure 7. Experimental apparatus for measuring the ESS impedance.

Figure 8 .
Figure 8. Voltage and current waveforms when measuring the impedance spectrum.

Figure 7 .
Figure 7. Experimental apparatus for measuring the ESS impedance.

Figure 8
Figure 8 depicts the voltage and current waveforms obtained when measuring the impedance spectra.The measurements were performed using the battery measuring equipment of BRS Messtechnik GmbH (Stuttgart, Germany).

Figure 7 .
Figure 7. Experimental apparatus for measuring the ESS impedance.

Figure 8 .
Figure 8. Voltage and current waveforms when measuring the impedance spectrum.Figure 8. Voltage and current waveforms when measuring the impedance spectrum.

Figure 8 .
Figure 8. Voltage and current waveforms when measuring the impedance spectrum.Figure 8. Voltage and current waveforms when measuring the impedance spectrum.

Figure 9 .
Figure 9. Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 100% SoC under different temperatures.

Figure 9 .
Figure 9. Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 100% SoC under different temperatures.

Figure 10 .
Figure 10.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 50% SoC under different temperatures.

Figure 10 .
Figure 10.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 50% SoC under different temperatures.

31 Figure 11 .
Figure 11.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 0% SoC under different temperatures.

Figure 11 .
Figure 11.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 0% SoC under different temperatures.

31 Figure 12 .
Figure 12.Impedance spectrum of a single cell of the pouch-type lithium-ion battery under different SoCs and temperatures.

Figure 13 .
Figure 13.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 100% SoC under different temperatures.

Figure 12 . 31 Figure 12 .
Figure 12.Impedance spectrum of a single cell of the pouch-type lithium-ion battery under different SoCs and temperatures.

Figure 13 .
Figure 13.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 100% SoC under different temperatures.

Figure 13 .
Figure 13.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 100% SoC under different temperatures.

Figure 14 .
Figure 14.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 50% SoC under different temperatures.

Figure 15 .
Figure 15.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 0% SoC under different temperatures.

Figure 14 . 31 Figure 14 .
Figure 14.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 50% SoC under different temperatures.

Figure 15 .
Figure 15.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 0% SoC under different temperatures.

Figure 15 . 31 Figure 16 .
Figure 15.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at 0% SoC under different temperatures.Batteries 2024, 10, x FOR PEER REVIEW 14 of 31

Figure 16 .
Figure 16.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at different SoCs under 40 • C.
compares the impedance spectrum results for a single cell of the pouch-type lithium-ion battery (75 Ah) at 0%, 50%, and 100% SoC under 25 • C. The resistance value of the battery ranged from 3.3083 to 16.3872 mΩ.

Figure 16 .
Figure 16.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at different SoCs under 40 °C.

Figure 17 .
Figure 17.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at different SoCs under 25 °C.

Figure 17 .
Figure 17.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at different SoCs under 25 • C. At 25 • C, R S , R P , C P , and L S were as follows: (1) 25 • C, 100% SoC: R S = 3.4660 mΩ, R P = 2.2553 mΩ, C P = 3.5815 F, L S = 526.532nH; (2) 25 • C, 50% SoC: R S = 5.4694 mΩ, R P = 4.0496 mΩ, C P = 3.1778 F, L S = 803.796nH; (3) 25 • C, 0% SoC: R S = 5.7159 mΩ, R P = 9.6713 mΩ, C P = 103.4333F, L S = 805.117nH.Figure17shows that at 25 • C, R S , R P , C P , and L S decreased with increasing SoC.Moreover, they increased with decreasing SoC.However, at 0% SoC, C P rapidly increased compared to that at 100% and 50% SoC.Thus, at 0% SoC, C P increased with Z W .Figure18compares the impedance spectrum results for a single cell of the pouch-type lithium-ion battery (75 Ah) at 0%, 50%, and 100% SoC under 10 • C.The resistance value of the battery at 10 • C ranged from 3.4034 to 20.5187 mΩ.At 10 • C, R S , R P , C P , and L S were as follows:

Figure 18 .
Figure 18.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at different SoCs under 10 °C.

Figure 19 .
Figure 19.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at different SoCs under −5 °C.

Figure 18 .
Figure 18.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at different SoCs under 10 • C.

Figure 19
Figure 19 compares the impedance spectrum results for a single cell of the pouch-type lithium-ion battery (75 Ah) at 0%, 50%, and 100% SoC under −5 • C. The resistance value of the battery at −5 • C ranged from 3.4050 to 23.7239 mΩ.

Figure 18 .
Figure 18.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at different SoCs under 10 °C.

Figure 19 .
Figure 19.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at different SoCs under −5 °C.

Figure 19 .
Figure 19.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at different SoCs under −5 • C. R S , R P , C P , and L S were as follows: (1) −5 • C, 100% SoC: R S = 3.8573 mΩ, R P = 2.6260 mΩ, C P = 4.2824 F, L S = 541.923nH; (2) −5 • C, 50% SoC: R S = 6.3150 mΩ, R P = 4.6714 mΩ, C P = 2.5950 F, L S = 920.122nH; (3) −5 • C, 0% SoC: R S = 9.0026 mΩ, R P = 13.7213mΩ, C P = 70.0385F, L S = 1326.620nH.R S , R P , and L S decreased with increasing SoC.Additionally, they increased with decreasing SoC.However, at 0% SoC, C P rapidly increased compared to that at 100% and 50% SoC.This signifies that at 0% SoC, C P increased with Z W .Figure20compares the impedance spectrum results at 0%, 50%, and 100% SoC for a single cell of the pouch-type lithium-ion battery (75 Ah) under −20 • C. The resistance value of the battery at −20 • C ranged from 3.6055 to 30.6618 mΩ.R S , R P , C P , and L S were as follows:

Figure 20 .
Figure 20.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at different SoCs under −20 °C.Specifically,  ,  ,  , and  decreased with increasing SoC.Additionally,  ,  , and  increased with decreasing SoC.At 0% SoC,  rapidly increased compared to that at 100% and 50% SoC.Therefore, at 0% SoC,  increased with  .

Figure 20 .
Figure 20.Impedance spectrum of a single cell of the pouch-type lithium-ion battery at different SoCs under −20 • C.

Figure 21 .
Figure 21.Impedance spectrum of the 16-cell ESS at 100% SoC under different temperatures.

𝑅
and  decreased with increasing temperature, and they increased with decreasing temperature.At 50% SoC, the voltage of the 4.4 kWh ESS was 59.2 V. Figure22illustrates the impedance spectrum of the 4.4 kWh ESS at 50% SoC under varying temperatures.At 50% SoC, the ESS resistance value ranged from 89.020 to 195.796 mΩ.

Figure 21 .
Figure 21.Impedance spectrum of the 16-cell ESS at 100% SoC under different temperatures.
temperature. exhibited the largest value at 10 °C, and it rapidly decreased at −20 °C.Moreover,  was absent.Particularly,  significantly increased at −20 °C, and the battery performance significantly deteriorated as the temperature decreased.At 0% SoC, the voltage of the 4.4 kWh ESS was 40.0 V. Figure 23 depicts the impedance spectrum of the 4.4 kWh ESS at 0% SoC under varying temperatures.The resistance value of the 4.4 kWh ESS at 0% SoC ranged from 121.639 to 279.797 mΩ.

Figure 23 .
Figure 23.Impedance spectrum of the 16-cell ESS at different temperatures at 0% SoC.

Figure 23 .
Figure 23.Impedance spectrum of the 16-cell ESS at different temperatures at 0% SoC.

Figure 24 .
Figure 24.Impedance spectrum of the 16-cell ESS under different SoCs and temperatures.

Figure 25 .
Figure 25.Impedance spectrum of the 16-cell ESS at 100% SoC under different temperatures.

Figure 24 .
Figure 24.Impedance spectrum of the 16-cell ESS under different SoCs and temperatures.

Figure 24 .
Figure 24.Impedance spectrum of the 16-cell ESS under different SoCs and temperatures.

Figure 25 .
Figure 25.Impedance spectrum of the 16-cell ESS at 100% SoC under different temperatures.

Figure 25 .
Figure 25.Impedance spectrum of the 16-cell ESS at 100% SoC under different temperatures.

Figure 26 .
Figure 26.Impedance spectrum of the 16-cell ESS at 50% SoC under different temperatures.

Figure 27 .
Figure 27.Impedance spectrum of the 16-cell ESS at 0% SoC under different temperatures.

Figure 28
Figure 28 displays the impedance spectra of the 16-cell ESS at 0%, 50%, and 100% SoC under 40 °C.The resistance value of the 4.4 kWh 16-cell ESS at 40 °C ranged from 53.603 to 169.168 mΩ. ,  ,  , and  were as follows:

Figure 27 .
Figure 27.Impedance spectrum of the 16-cell ESS at 0% SoC under different temperatures.

Figure 28 Figure 27 .
Figure 28 displays the impedance spectra of the 16-cell ESS at 0%, 50%, and 100% SoC under 40 °C.The resistance value of the 4.4 kWh 16-cell ESS at 40 °C ranged from 53.603 to 169.168 mΩ. ,  ,  , and  were as follows: Figure 27.Impedance spectrum of the 16-cell ESS at 0% SoC under different temperatures.

Figure 28 .
Figure 28.Impedance spectrum of the ESS at different SoCs for 40 °C.

Figure 29 .
Figure 29.Impedance spectrum of the 16-cell ESS at different SoCs for 25 °C.

Figure 30
Figure 30 denotes that the resistance value of the 4.4 kWh ESS under 10 °C ranged from 54.557 to 199.100 mΩ.The  ,  ,  , and  values were as follows:

Figure 30 .Figure 30 .
Figure 30.Impedance spectrum of the 16-cell ESS at different SoCs for 10 °C. ,  , and  decreased with increasing SoC.Furthermore, they increased with decreasing SoC.At 0% SoC,  rapidly increased compared to that at 100% and 50% SoC.Therefore, at 0% SoC,  increased with  .The resistance value of the 4.4 kWh 16-cell ESS was 56.682-227.427mΩ under −5 °C

Figure 31 .
Figure 31.Impedance spectrum of the 16-cell ESS at different SoCs for −5 °C.

Figure 31 .Figure 32 .
Figure 31.Impedance spectrum of the 16-cell ESS at different SoCs for −5 • C.R S , R P , and L S decreased with increasing SoC and increased with decreasing SoC.At 50% SoC, C P exhibited the minimum value, and at 0% SoC, C P rapidly increased compared to that at 100% and 50% SoC.Hence, at 0% SoC, C P increased with Z W . Figure32denotes that the resistance value of the 4.4 kWh 16-cell ESS at −20 • C ranged from 62.375 to 279.797 mΩ.R S , R P , C P , and L S were as follows:

Figure 32 .
Figure 32.Impedance spectrum of the ESS at different SoCs for −20 • C.

Figure 33 .
Figure 33.Electrical equivalent circuit corresponding to the lithium-ion battery.

Figures 9 -
Figures 9-32 highlight that  is more significantly affected by SoC than temperature.Furthermore,  is more significantly affected by temperature than SoC.Specifically,  rapidly increased at temperatures below −20 °C, and  rapidly increased at 0% SoC.Additionally,  was greatly influenced by the structure or shape (cylindrical or straight shaped) of the electrode throughout the battery.The  value of the diffusion area rapidly increased and decreased with decreasing temperature and SoC, respectively.Further research is required to determine whether Ls,  ,  , and  are completely related to the performance of lithium-ion batteries.However, beyond a certain range, the sharp increases in  and  could stem from the decrease in the ESS performance.Since the occurrence of  had the greatest influence on the increase in the internal resistance of lithium-ion batteries, a battery management system (BMS) should be incorporated.

Figure 33 .
Figure 33.Electrical equivalent circuit corresponding to the lithium-ion battery.

Figure 34 .
Figure 34.Electrical equivalent circuit of a simplified ESS.

Figure 35 displays
the Bode plot for a single cell of the pouch-type lithium-ion battery (75 Ah) at 0%, 50%, and 100% SoC under 40 • C and −20 • C.

Figure 35 .
Figure 35.Bode plot of a single cell of a pouch-type lithium-ion battery (75 Ah).

Figure 35 .
Figure 35.Bode plot of a single cell of a pouch-type lithium-ion battery (75 Ah).

Figure 36 presents
Figure 36 presents the Bode plot for the 16 cells of a pouch-type lithium-ion battery (75 Ah) at 0%, 50%, and 100% SoC under 40 • C and −20 • C.In the 16 cells, the GM decreased when the angular frequency (ω = 2π f ) exceeded 2000 rad/s (frequency: 300 Hz), and it decreased when the angular frequency approached 0 rad/s (frequency: 0 Hz).These trends represent the characteristics of a typical LPF.At 50% or 100% SoC, PM was maximum at the angular frequency of ≥200 rad/s (frequency: 30 Hz), but at 0% SoC, PM was maximum at the angular frequency of ≥21 rad/s (frequency: 2.4 Hz).PM was minimum at the maximum angular frequency of 15-21 rad/s (frequency: 2.4 Hz), signifying that the gain was reduced at 0 rad/s (frequency: 0 Hz).

Figure 36 .
Figure 36.Bode plot of the 16 cells of the pouch-type lithium-ion battery (75 Ah).

Figure 36 .
Figure 36.Bode plot of the 16 cells of the pouch-type lithium-ion battery (75 Ah).
4 kWh ESS at 100% SoC under varying temperatures.The ESS resistance value at 100% SoC ranged from 53.603 to 119.909 mΩ.
, the Laplace transform of the ESS voltage is as follows: Figure 34.Electrical equivalent circuit of a simplified ESS.The ESS transfer function can be written as follows:( 1) 16 Cell, 40 °C, 100% SoC:  23.0223 dB;