Electrochemical Impedance Spectrum Equivalent Circuit Parameter Identification Using a Deep Learning Technique

Citation: Zulueta, A.; Zulueta, E.; Olarte, J.; Fernandez-Gamiz, U.; Lopez-Guede, J.M.; Etxeberria, S. Electrochemical Impedance Spectrum Equivalent Circuit Parameter Identification Using a Deep Learning Technique. Electronics 2023,12, 5038. https://doi.org/10.3390/ electronics12245038 Academic Editors: Yi Xie, Dan Dan and Jiahao Liu Received: 26 September 2023 Revised: 24 November 2023 Accepted: 27 November 2023 Published: 18 December 2023 Copyright: © 2023 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 (https:// creativecommons.org/licenses/by/ 4.0/). electronics Article Electrochemical Impedance Spectrum Equivalent Circuit Parameter Identification Using a Deep Learning Technique Asier Zulueta 1, Ekaitz Zulueta 2, Javier Olarte 3, Unai Fernandez-Gamiz 1, Jose Manuel Lopez-Guede 2,* and Saioa Etxeberria 4 1Department of Energy Engineering, University of the Basque Country (UPV/EHU), 01006 Vitoria-Gasteiz, Spain; [email protected] (A.Z.); [email protected] (U.F.-G.) 2Department of System Engineering and Automatic Control, University of the Basque Country (UPV/EHU), 01006 Vitoria-Gasteiz, Spain; [email protected] 3Centre for Cooperative Research on Alternative Energies (CIC EnergiGUNE), Basque Research and Technology Alliance (BRTA), Alava Technology Park, Albert Einstein 48, 01510 Vitoria-Gasteiz, Spain; [email protected] 4Department of Mechanical Engineering, University of the Basque Country (UPV/EHU), 01006 Vitoria-Gasteiz, Spain; [email protected] *Correspondence: [email protected]; Tel.: +34-945014084 Abstract: Physical models are suitable for the development and optimization of materials and cell designs, whereas models based on experimental data and electrical equivalent circuits (EECs) are suitable for the development of operation estimators, both for cells and batteries. This research work develops an innovative unsupervised artificial neural network (ANN) training cost function for identifying equivalent circuit parameters using electrochemical impedance spectroscopy (EIS) to identify and monitor parameter variations associated with different physicochemical processes that can be related to the states or failure modes in batteries. Many techniques and algorithms are used to fit a predefined EEC parameter, many requiring high-human-expertise support work. However, once the appropriate EEC model is selected to model the different physicochemical processes associated with a given battery technology, the challenge is to implement algorithms that can automatically calculate parameter variations in real time to allow the implementation of estimators of capacity, health, safety, and other degradation modes. Based on previous studies using data augmentation techniques, the new ANN deep learning method introduced in this study yields better results than classical training algorithms. The data used in this work are based on an aging and characterization dataset for 80 Ah and 12 V lead–acid batteries. Keywords: neural networks; deep learning; electrical equivalent circuit; electrochemical impedance spectroscopy; model-based estimators; lead–acid batteries 1. Introduction The lecture notes and recordings in ECE4710/5710: Modeling, Simulation, and Identification of Battery Dynamics provide a 360 ◦ view of the modeling techniques used in batteries [ 1 ]. Research has been conducted to apply deep learning (DL) techniques to battery analyses, such as in [ 2 , 3 ], although there are also research studies based on more conventional neural networks, such as in [ 4 , 5 ]. Even more conventional algorithms such as the extended Kalman filter (EKF) are suitable for when battery state predictions must be calculated in real-time applications [ 6 ]. A detailed review of machine learning applications in battery state analyses can be found in [7]. Physical models are suitable for the development and optimization of materials and cell designs, whereas models based on experimental data and electrical equivalent circuits (EECs) are suitable for the development of operation estimators, both for cells and batteries. A good example of physical models applied to battery design can be seen in [8]. Electronics 2023,12, 5038. https://doi.org/10.3390/electronics12245038 https://www.mdpi.com/journal/electronics Electronics 2023,12, 5038 2 of 13 The most basic EECs consist of a voltage source in series with a resistor, representing the open-circuit voltage and the internal resistance of the cell, respectively. Additional R-C-scale series are often added to simulate the charge and mass transfer phenomena occurring at and around the electrolyte–electrode interface. One method for selecting the equivalent circuit is to simulate different operating conditions. Then, the parameterization of the electrical components that constitute the circuit is usually performed using experimental voltage and current measurements [ 9 , 10 ]. Another option is using impedance spectroscopy [ 10 ]. Spectroscopy is easier to perform and more accessible. However, due to the variable nature of the R-C network components concerning the operating conditions (the state of charge (SOC), temperature, flow rate, and current), the parameterization process becomes a cumbersome task. The wider the operating window and the higher the desired accuracy, the greater the amount of experimental data required and the more complex the algorithms needed to perform the dynamic parameterization. The association or determination of an EEC for a spectrum allows to have a model of the behavior of the battery where different components of the circuit are identified at each frequency range, as detailed by [11]. Electrochemical impedance spectroscopy (EIS) can be used for real-time predictions by interpreting parameters from spectra [ 12 ]. Battery EIS data represent a very powerful tool for identifying battery EEC models, which help to evaluate different battery states and in operando conditions. However, initially, the electrochemical model is difficult to implement and is very technology-specific, as described in the study by Piller et al. [ 13 ]. Therefore, it is necessary to develop specific protocols for extracting EIS measurements for each type of electrochemical energy storage technology, as demonstrated by Meddings et al. [ 14 ]. The identification process for an EEC to be extracted from EIS measurements usually requires human expertise complementary to support-specific software such as Zview, which offers impedanceor gain-phase graphing tools and guided circuit selection advice, as seen in the study by Csomós et al. [ 15 ]. Good reviews of the remaining battery life prediction techniques are presented in [16,17]. Additionally, the continuous monitoring of EIS data allows for adjusting and improving the accuracy of EEC models based on continuous models by allowing the identification of the initial conditions, as identified by Olarte et al. in [12]. Artificial neural networks (ANNs) have been widely used in the development of battery models for diagnosis and prognosis, as reported by Lombardo et al. [ 8 ]. In their work, the authors applied NNs to parameter identification because of the NNs’ ability to fit non-linear relationships between inputs (EIS data) and outputs (equivalent circuit parameters), as demonstrated in the study by Chun et al. [ 18 ]. A related study can be found in [ 19 ]. Researchers have proposed a new loss function to train a neural network for EEC parameter predictions. Jimenez-Bermejo et al. [ 20 ] used NNs to obtain a more accurate estimation of the battery SoC in electric vehicles, and Yang et al. [ 21 ] used backpropagation (BP) NNs to estimate the battery SoH in electric vehicles, using a large amount of experimental data to feed the NNs. ANNs can identify complex non-linear relationships between inputs (experimental EIS data, in our case) and outputs (EIS parameters, in our case). However, ANNs require extensive amounts of data to identify complex relationships because the architecture and number of weights are very large in these cases. Obtaining experimental data for all battery operating conditions is very time-consuming; therefore, we propose that an NN be trained with an initial set of real EIS data augmented with synthetic data corresponding to normal values for the equivalent circuit model. We emphasize the possibility of reducing the amount of experimental data by more efficiently generating synthetic data. This data augmentation process must be carefully carried out, since the EEC parameters are very dependent on each other, meaning it is not possible to achieve complete stochastic data augmentation. As such, we propose that a feasible data augmentation process is conducted via an experimental data comparison. Electronics 2023,12, 5038 3 of 13 A two-step parameter identification algorithm to monitor EEC parameter variations was reported in a publication by Olarte et al. [ 22 ]. The authors’ goal with the proposed method was to automatically monitor the variations in the EEC parameters in operation with impedance sensors to infer the variations in the battery health state or failure modes from the electrochemical interpretation of these values, as in the study by Chun et al. [18]. In this work, we propose a new deep learning (DL) technique for identifying an equivalent circuit’s parameters from a series of EIS data using a trained ANN. Once the ANN is trained, algorithms of very low computational complexity can be implemented, allowing the design of systems for economically monitoring the evolution of the EEC parameters associated with the battery states. The structure of this paper is as follows. Section 2poses the target problem, providing the foundations and addressing the solution. The method proposed to directly learn the model from the EIS data is provided in Section 3. Section 4reports the obtained results, while Section 5discusses the results. Finally, Section 6presents our conclusions. 2. EEC Parameter Identification Problem Here, we propose a new DL NN-based algorithm that predicts an equivalent circuit’s parameters via EIS, as defined in Section 2.1. To achieve this objective, we designed a new training loss function that forces the ANN to propose an equivalent circuit’s parameter set (see Tables 1and 2) that explains the EIS data as well as possible (see Section 2.2). The variables defined in the mathematical model are presented in Table 3. Table 1. Parameters extracted from EIS spectra by adjusting the Zview software (initial raw data for the batteries in one of six rounds of essays). SoC R1 CPE1T CPE1P R2 CPE2T CPE2P R3 100% 0.0027176 7.17 0.85729 0.0092174 87.18 0.65421 - 80% 0.0027953 9.21 0.77865 0.0039696 184.13 0.61221 0.21606 60% 0.0031349 11.21 0.75909 0.0021683 218.8 0.56847 0.088716 40% 0.0033452 18.01 0.62091 0.0020905 229.5 0.5006 0.066692 20% 0.0039584 14.92 0.65745 0.0020599 199.4 0.38122 0.12304 0% 0.0046775 10.12 0.70804 0.0025044 152.2 0.29418 - Table 2. EEC parameter proposal. Parameter Unit Range 1 R1Ω[0.00271760, 0.00467750] L1H[10−6, 10−3] R2Ω[0.00205990, 0.00921740] T1secp1.Ω−1[7.17, 18.01] p1- [0.62091000, 0.85729000] R3Ω[0.06669200, ∞) T2secp1.Ω−1[87.18, 229.50] p2- [0.29418000, 0.65421000] 1 In this table, we list the proposed EEC parameters. The range for each parameter is a value span within which the parameter can reasonably be found. The maximum and the minimum values of these ranges are fixed by the experimental data. To compare the results of this new algorithm, we propose using the same equivalent circuit and tests presented in the work by Olarte et al. [22]. 2.1. Electrical Equivalent Circuit Proposal We propose the EEC in Figure 1a because the real data the battery essays produced, as described in Section 3, fit well with two pairs of constant phase elements in parallel with a resistor. The outputs of the real experiments (in black) with the theoretical model predictions obtained with Zview are shown in Figure 1b (in green). The model obtained Electronics 2023,12, 5038 4 of 13 using Zview requires expert human supervision. The physical meanings of the EEC parameters are described in [ 23 ]. Here, R1 is the inner resistance, R2 is the charge transfer resistance for the negative electrode, CPE1 is associated with a distribution of relaxation times, and CPE2 is an impedance related to higher time constants than CPE1 . Finally, R3 is the resistance associated with the charge transfer resistance for the positive electrode. Table 3. EEC variables. Variable 2Unit Definition j - Imaginary unit number wrad/s Frequency EIS(jw)ΩElectrochemical impedance Spectrum of the battery CPE1(jw)ΩFirst constant-phase element CPE2(jw)ΩSecond constant-phase element ZCPE1,R2Ω Equivalent parallel impedance of first constant element and the second resistance ZCPE2,R3ΩEquivalent parallel impedance of second constant element and third resistance 2In this table, we list the proposed EIS equivalent circuit’s main variables. Electronics 2023, 12, x FOR PEER REVIEW 5 of 13 Figure 1. (a) Proposed EEC circuit model and (b) adjustment of EIS spectra of 12 V block at 100% of the state of charge (SoC) using Zview (version 3.5i) manual software. Figure 1a shows the selected electrical equivalent circuit. The inductive (see 𝐿) parts do not contain valuable information about the battery degradation process because they are very dependent on the electrical setup, so they are not considered. Conceptually, it is good practice to derive all expressions of impedance in the real and imaginary parts because the current automatic differentiation algorithms do not support complex variable operations: 𝑟𝑒𝑎𝑙(𝐶𝑃𝐸)=󰇡     󰇢     (6) 𝑖𝑚𝑎𝑔(𝐶𝑃𝐸)=󰇡     󰇢     (7) 𝑟𝑒𝑎𝑙(𝐶𝑃𝐸)=󰇡     󰇢     (8) 𝑖𝑚𝑎𝑔(𝐶𝑃𝐸)=󰇡     󰇢     (9) 𝑟𝑒𝑎𝑙𝑍  ,  =  ∙(  )(  (  ))  (  ) (  (  ))    (  ) (10) 𝑖𝑚𝑎𝑔𝑍  ,  =  ∙(  )(  (  ))(  ) (  (  ))    (  ) (11) 𝑟𝑒𝑎𝑙𝑍  ,  =  ∙(  )(  (  ))  (  ) (  (  ))    (  ) (12) 𝑖𝑚𝑎𝑔𝑍  ,  =  ∙(  )(  (  ))(  ) (  (  ))    (  ) (13) Figure 1. (a) Proposed EEC circuit model and (b) adjustment of EIS spectra of 12 V block at 100% of the state of charge (SoC) using Zview (version 3.5i) manual software. Table 1shows the reference parameters extracted from the EIS spectra of a lead–acid battery analyzed by adjusting the Zview manual software. These reference parameters are considered to generate the synthetic data. Electronics 2023,12, 5038 5 of 13 The mathematical model is defined via Equations (1)–(3): EIS(jw)=R1+jL1w+ZCPE1,R2+ZCPE2,R3(1) ZCPE1,R2=R2CPE1(jw) CPE1(jw)+R2(2) ZCPE2,R3=R3CPE2(jw) CPE2(jw)+R3(3) The constant phase elements’ impedances are defined in Equations (4) and (5): CPE1(jw)=1 T1(jw)p1(4) CPE2(jw)=1 T2(jw)p2(5) Figure 1a shows the selected electrical equivalent circuit. The inductive (see L1 ) parts do not contain valuable information about the battery degradation process because they are very dependent on the electrical setup, so they are not considered. Conceptually, it is good practice to derive all expressions of impedance in the real and imaginary parts because the current automatic differentiation algorithms do not support complex variable operations: real(CPE1) = cosπ 2p1 T1wp1(6) imag(CPE1) = sinπ 2p1 T1wp1(7) real(CPE2) = cosπ 2p2 T2wp2(8) imag(CPE2) = sinπ 2p2 T2wp2(9) realZCPE1,R2=R2·real(CPE1)(R2+real(CPE1))+imag2(CPE1) (R2+real(CPE1))2+imag2(CPE1)(10) imagZCPE1,R2=R2·imag(CPE1)((R2+real(CPE1))−real(CPE1)) (R2+real(CPE1))2+imag2(CPE1)(11) realZCPE2,R3=R3·real(CPE2)(R3+real(CPE2))+imag2(CPE2) (R3+real(CPE2))2+imag2(CPE2)(12) imagZCPE2,R3=R3·imag(CPE2)((R3+real(CPE2))−real(CPE2)) (R3+real(CPE2))2+imag2(CPE2)(13) The whole impedance can be derived via Equations (14) and (15). real(EIS)=R1+realZCPE1,R2+real(ZCPE2,R3)(14) imag(EIS)=L1w+imagZCPE1,R2+imag(ZCPE2,R3)(15) Electronics 2023,12, 5038 6 of 13 2.2. Supervised Versus Unsupervised Strategies We propose a new unsupervised training algorithm. The usual approach is to train the artificial neural network with EIS empirical data as inputs and EEC parameters as outputs, as seen in Figure 2. This approach has the disadvantages of the high costs and time periods required to obtain a sufficient dataset for training. In this research, we produced essays for 5 months. Each month, the batteries were charged and discharged at different SoCs. The costs of these essays were for the research personnel, the battery sets, the laboratory spaces, and the measuring equipment and refrigeration systems’ amortization. The process produced only 36 essays, including six different SoC levels (100% to 20%) and six different times (from the start to the fifth month). Therefore, the essay measurements were expensive and time-consuming. Each impedance measurement produced an EIS essay. Due to these time and budget limitations, the amount of data was limited to 36 essays. As seen in Equation (17), PANN(EISq(jw , θ) is an artificial neural network that predicts the EEC parameters: → P=[R1L1R2T1p1R3T2p2] t(16) → PANN =→ PANN(EISq(jw,→ θ)(17) MSELoss→ θ=∑q=Npatterns q=1∑µ=Nfrequency µ=1 → Pq−PANN(EISq(jwµ,→ θ)) 2(18) EISmodeljw,→ P=R1+jL1w+ZCPE1,R2+ZCPE2,R3(19) MSELoss→ θ=∑q=Npatterns q=1∑µ=Nfrequency µ=1realEISqjwµ− realEISmodeljwµ,→ P 2 + ∑q=Npatterns q=1∑µ=Nfrequency µ=1 imagEISqjwµ−imagEISmodeljwµ,→ P 2(20) Electronics 2023, 12, x FOR PEER REVIEW 6 of 13 The whole impedance can be derived via Equations (14) and (15). 𝑟𝑒𝑎𝑙(𝐸𝐼𝑆)=R+𝑟𝑒𝑎𝑙𝑍  ,  +𝑟𝑒𝑎𝑙(𝑍  ,  ) (14) 𝑖𝑚𝑎𝑔(𝐸𝐼𝑆)=L 𝑤+𝑖𝑚𝑎𝑔𝑍  ,  +𝑖𝑚𝑎𝑔(𝑍  ,  ) (15) 2.2. Supervised Versus Unsupervised Strategies We propose a new unsupervised training algorithm. The usual approach is to train the artificial neural network with EIS empirical data as inputs and EEC parameters as outputs, as seen in Figure 2. This approach has the disadvantages of the high costs and time periods required to obtain a sufficient dataset for training. In this research, we produced essays for 5 months. Each month, the batteries were charged and discharged at different SoCs. The costs of these essays were for the research personnel, the battery sets, the laboratory spaces, and the measuring equipment and refrigeration systems’ amortization. The process produced only 36 essays, including six different SoC levels (100% to 20%) and six different times (from the start to the fifth month). Therefore, the essay measurements were expensive and time-consuming. Each impedance measurement produced an EIS essay. Due to these time and budget limitations, the amount of data was limited to 36 essays. As seen in Equation (17), 𝑃(𝐸𝐼𝑆(𝑗𝑤,𝜃) is an artificial neural network that predicts the EEC parameters: 𝑃 󰇍  =󰇟𝑅 𝐿𝑅𝑇𝑝𝑅𝑇𝑝󰇠 (16) 𝑃 󰇍   =𝑃 󰇍  (𝐸𝐼𝑆( 𝑗 𝑤,𝜃 ) (17) 𝑀𝑆𝐸  󰇡 𝜃 󰇍 󰇍  󰇢=∑∑ 󰇻𝑃 󰇍   −𝑃  (𝐸𝐼𝑆  (𝑗𝑤  , 𝜃 󰇍 󰇍  ))󰇻        (18) 𝐸𝐼𝑆 𝑗 𝑤,𝑃 󰇍  =R+ 𝑗 L𝑤+𝑍  ,  +𝑍  ,  (19) 𝑀𝑆𝐸𝜃 =∑∑ 󰇻𝑟𝑒𝑎𝑙󰇡𝐸𝐼𝑆 𝑗 𝑤󰇢−       𝑟𝑒𝑎𝑙󰇡𝐸𝐼𝑆 𝑗 𝑤,𝑃 󰇍  󰇢󰇻+ ∑∑ 󰇻𝑖𝑚𝑎𝑔󰇡𝐸𝐼𝑆 𝑗 𝑤󰇢−𝑖𝑚𝑎𝑔󰇡𝐸𝐼𝑆 𝑗 𝑤,𝑃 󰇍  󰇢󰇻       (20) Figure 2. Supervised training model. Figure 2. Supervised training model. We propose a different mean loss function (an unsupervised training method) to reduce the prediction error, as shown in Figure 3. Table 4defines the main variables of the EEC neural network predictor, including the inputs and outputs. Electronics 2023,12, 5038 7 of 13 Electronics 2023, 12, x FOR PEER REVIEW 7 of 13 We propose a different mean loss function (an unsupervised training method) to reduce the prediction error, as shown in Figure 3. Table 4 defines the main variables of the EEC neural network predictor, including the inputs and outputs. Figure 3. Unsupervised training model. Table 4. The parameters of the proposed neural network for EEC parameter predictions. These parameters are defined in the electrical scheme in Figure 1a. Parameter Unit Range Definition 𝑟𝑒𝑎𝑙󰇡𝐸𝐼𝑆 𝑗 𝑤󰇢 - [0, 1] It is a 121-term input vector. It contains the real part of 𝐸𝐼𝑆 𝑗 𝑤, and these inputs are normalized. 𝑖𝑚𝑎𝑔󰇡𝐸𝐼𝑆 𝑗 𝑤󰇢 - [0, 1] It is a 121-term input vector. It contains the imaginary part of 𝐸𝐼𝑆 𝑗 𝑤, and these inputs are normalized. 𝑅 Ω [0, 1] The first resistance. The value is normalized. 𝐿 H [0, 1] The serial inductance. The value is normalized. 𝑅 Ω [0, 1] The second resistance. It is in parallel with the first constant-phase element. 𝑇 sec p1. Ω −1 [0, 1] The inverse gain of the first constant-phase element. It is a normalized value. 𝑝 - [0, 1] The power index parameter of the first constant-phase element. It is a normalized value. 𝑅 Ω [0, 1] The third resistance. It is in parallel with the second constant-phase element. It is a normalized value. 𝑇 sec p1. Ω −1 [0, 1] The inverse gain of the second constant-phase element. It is a normalized value. 𝑝 - [0, 1] The power index parameter of the second constantphase element. It is a normalized value. 𝑤 rad/sec [0.0628, 63323] with an exponential step of Δ𝑤 The frequency sample. There are 121 different frequency samples. Δ𝑤 rad/sec Δ𝑤=𝑒     The frequency sampling step. 𝜇∈0,1,2,..,𝑁  2.3. Data Augmentation Proposal We have proposed a data augmentation algorithm to generate synthetic training from five essays with different states of charge (SoCs) from 20% to 100% (see Table 1 for the whole set of essays used). The generation process is defined in Algorithm 1. The objective is to generate three different datasets: a training dataset with 20,000 synthetic equivalent circuits, a validation dataset of 2500 synthetic equivalent circuits, and a test dataset of 500 synthetic equivalent circuits. The difference between the EIS values of a selected experimental essay and the synthetic equivalent circuit is defined as the absolute value of the difference between these two EIS values, which is divided by the absolute value of the selected experimental essay’s Figure 3. Unsupervised training model. Table 4. The parameters of the proposed neural network for EEC parameter predictions. These parameters are defined in the electrical scheme in Figure 1a. Parameter Unit Range Definition realEISjwµ - [0, 1] It is a 121-term input vector. It contains the real part of EISjwµ, and these inputs are normalized. imagEISjwµ - [0, 1] It is a 121-term input vector. It contains the imaginary part of EISjwµ, and these inputs are normalized. R1Ω[0, 1] The first resistance. The value is normalized. L1H [0, 1] The serial inductance. The value is normalized. R2Ω[0, 1] The second resistance. It is in parallel with the first constant-phase element. T1 sec p1.Ω−1[0, 1] The inverse gain of the first constant-phase element. It is a normalized value. p1- [0, 1] The power index parameter of the first constant-phase element. It is a normalized value. R3Ω[0, 1] The third resistance. It is in parallel with the second constant-phase element. It is a normalized value. T2 sec p1.Ω−1[0, 1] The inverse gain of the second constant-phase element. It is a normalized value. p2- [0, 1] The power index parameter of the second constant-phase element. It is a normalized value. wµrad/sec [0.0628, 63323] with an exponential step of ∆wµThe frequency sample. There are 121 different frequency samples. ∆wµrad/sec ∆wµ=e 15µ Nfrequency −5The frequency sampling step. µ∈n0,1,2, . . . , Nfrequencyo 2.3. Data Augmentation Proposal We have proposed a data augmentation algorithm to generate synthetic training from five essays with different states of charge (SoCs) from 20% to 100% (see Table 1for the whole set of essays used). The generation process is defined in Algorithm 1. The objective is to generate three different datasets: a training dataset with 20,000 synthetic equivalent circuits, a validation dataset of 2500 synthetic equivalent circuits, and a test dataset of 500 synthetic equivalent circuits. The difference between the EIS values of a selected experimental essay and the synthetic equivalent circuit is defined as the absolute value of the difference between these two EIS values, which is divided by the absolute value of the selected experimental essay’s EIS value (see line 9 in Algorithm 1). The maximum accepted difference level is equal to 30%. The records of each dataset were randomly selected. Electronics 2023,12, 5038 8 of 13 Algorithm 1: Data augmentation process for synthetic essay generation Inputs: The set of essays with each parameter set. There are 5 different essays, with different SoCs. •EISµ(jw),µ∈{20%, 40%, 60%, 80%, 100%} •EquivalentCircuitParameterµ= [R1µ, R2µ, R3µ, T1µ, p1µ, T2µ, p2µ, L1µ] The number of essays generated. •NgeneratedEIS The maximum difference level. •Jmax Outputs: The set of synthetic essays. There are NgeneratedEIS different essays. •EISsynthetic,q(jw), q ∈{1, NgeneratedEIS} •EquivalentCircuitParameterq= [R1q, R2q, R3q, T1q, p1q, T2q, p2q, L1q] 1: procedure 2: Obtain the maximum and the minimum values of each EEC parameter. These values are calculated from EquivalentCircuitParameterµ. ◦R1max, R2max, R3max, T1max, p1max, T2max, p2max, L1max ◦R1min, R2min, R3min, T1min, p1min, T2min, p2min, L1min 3: for qfrom 1to NgeneratedEIS 4: NextCircuit = False 5: while not NextCircuit 6: µselected = mod(µ,Cardinal(µ)) 7: Generate an equivalent parameter set (CircuitParameterq) based on EquivalentCircuitParameterµselectced, The parameters are calculated with a pseudorandom uniform value between the maximum and the minimum values calculated in line 2. 8: Generate synthetic EISsynthetic,q(jw). See Equation (1). 9: Calculate difference level as follows: J=100µ=Nfrequency ∑ µ=1 |EISsynthetic,q(jwµ)−EISµselected(jwµ)| |EISµselected(jwµ)| 10: if J<Jmax 11: NextCircuit=True 12: end if 13: Store EISsynthetic,q(jw), and Store Equivalent CircuitParameterµselectced 14: end while 15: end for 16: return EISsynthetic, q(jw), CircuitParameterq, q ∈{1, NgeneratedEIS} 3. Materials and Methods The data used in this work were based on the aging and measurements of a set of 80–100 Ah and 12 V lead–acid batteries. The testing protocol included a CC charge stage and a CV charge stage. Periodic impedance measurements were performed at different state-of-charge levels, at each ∆ h% = 20 in the discharge process. At each SOC level, a 12 h relaxation time was established before performing the impedance measurement, which was conducted under an excitation current of 50 mA and in the frequency range of 10 mHz to 10 kHz. As already outlined in Section 2, the EIS spectra extracted from the test were adjusted to a specific EEC to identify an EEC parameter. The proposed neural network’s architecture has two inputs, where the first input is the real part of the EIS spectra and the second input is the imaginary part of the EIS spectra, with each input having 121 frequency samples. These two inputs are concatenated in one feature vector. After this concatenation, there is a fully connected layer with 100 neurons and a ReLU layer. After this, there are three blocks of fully connected layers with a ReLU layer of 10 neurons. Finally, there is a fully connected layer and a sigmoid layer of eight neurons. The output layer has eight neurons Electronics 2023,12, 5038 9 of 13 because the artificial neuron network has to predict the parameters of the equivalent circuit (see Equation (16)). The architecture has 25,600 learnable parameters. In Figures 4and 5, we have attached a description of the whole architecture. Electronics 2023, 12, x FOR PEER REVIEW 9 of 13 was conducted under an excitation current of 50 mA and in the frequency range of 10 mHz to 10 kHz. As already outlined in Section 2, the EIS spectra extracted from the test were adjusted to a specific EEC to identify an EEC parameter. The proposed neural network’s architecture has two inputs, where the first input is the real part of the EIS spectra and the second input is the imaginary part of the EIS spectra, with each input having 121 frequency samples. These two inputs are concatenated in one feature vector. After this concatenation, there is a fully connected layer with 100 neurons and a ReLU layer. After this, there are three blocks of fully connected layers with a ReLU layer of 10 neurons. Finally, there is a fully connected layer and a sigmoid layer of eight neurons. The output layer has eight neurons because the artificial neuron network has to predict the parameters of the equivalent circuit (see Equation (16)). The architecture has 25,600 learnable parameters. In Figures 4 and 5, we have attached a description of the whole architecture. Figure 4. The proposed architecture with a directional acyclical graph of the proposed neural network. Figure 5. Summary information of each layer of the proposed neural network. Figure 4. The proposed architecture with a directional acyclical graph of the proposed neural network. Electronics 2023, 12, x FOR PEER REVIEW 9 of 13 was conducted under an excitation current of 50 mA and in the frequency range of 10 mHz to 10 kHz. As already outlined in Section 2, the EIS spectra extracted from the test were adjusted to a specific EEC to identify an EEC parameter. The proposed neural network’s architecture has two inputs, where the first input is the real part of the EIS spectra and the second input is the imaginary part of the EIS spectra, with each input having 121 frequency samples. These two inputs are concatenated in one feature vector. After this concatenation, there is a fully connected layer with 100 neurons and a ReLU layer. After this, there are three blocks of fully connected layers with a ReLU layer of 10 neurons. Finally, there is a fully connected layer and a sigmoid layer of eight neurons. The output layer has eight neurons because the artificial neuron network has to predict the parameters of the equivalent circuit (see Equation (16)). The architecture has 25,600 learnable parameters. In Figures 4 and 5, we have attached a description of the whole architecture. Figure 4. The proposed architecture with a directional acyclical graph of the proposed neural network. Figure 5. Summary information of each layer of the proposed neural network. Figure 5. Summary information of each layer of the proposed neural network. The training process is an ADAM algorithm with 60 epochs, with a mini-batch size of 100 synthetic circuits. The initial learning ratio is equal to 10 −3 . The gradient decay factor is set to 0.9 and the squared decay factor is set to 0.999. The epsilon value (small constant) is set to 10−8. 4. Results We compared the normalized theoretical EIS data with the EIS data generated using the parameters predicted by the neural network. We plotted the theoretical EIS data (in red) calculated using Equations (1)–(5) with the EEC parameter predicted by the neural network. When the neural network takes an experimental EIS datum as an input, it predicts the EEC parameter set (see Table 1). We trained the neural network to predict parameters to match the experimental EIS data via Equations (1)–(5), not the EEC parameters. In Figure 6, the axes are the normalized real parts of the EIS data (x-axis) and the imaginary parts of the EIS data (y-axis). The figures do not have units because the real parts were normalized from [0 to 1] and the imaginary parts from [ − 1 to 0]. The scales were normalized with the