Intelligent Estimation of LiFePO4 Battery State-of-Charge (SOC) in Electric Vehicles

Transcription

1 Intelligent Estimation of LiFePO4 Battery State-of-Charge (SOC) in Electric Vehicles Igor Filipe Nunes Montes Thesis to obtain the Master of Science Degree in Mechanical Engineering Supervisors: Prof. Paulo José da Costa Branco Prof. João Miguel da Costa Sousa Examination Committee Chairperson: Prof. João Rogério Caldas Pinto Supervisor: Prof. Paulo José da Costa Branco Member of the Committee: Profª Susana Margarida da Silva Vieira November 2016

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3 Acknowledgements First of all, I would like to express my gratitude towards my supervisors, Prof. Paulo Branco and Prof. João Sousa, not only for their guidance in this thesis, but also for the experiences exchanged outside university. I want to express my gratitude to both my parents and my sister Alexandra, for their undying faith in my work and also for being so patient with me, no matter the situation. I really love you. To my friends from Carcavelos who I still meet regularly, whether it is a reunion to have dinner, to play games or some other special event. You are all like my second family. I want to thank all my friends from university, whom I not only shared amazing experiences, but also because they always supported me in my university decisions, whether they were from mechanical engineering or my companions at the machine laboratory. I also want to show my appreciation for all people from Zeugma, who helped me in my first industrial experience. I also want to thank Mónica Brioso. Although we are no longer together, it is true that I have learned a lot from all years we have been together and it would be unfair not to mention her, since she was really close to my work. I really wish you all the best. I really, really want to give my kindest regards to Filipa Diogo. She really helped me through the last year, since a sequence of bad things started to happen all of a sudden. If it was not for you, I would probably not have found the path I am starting to carve on my own. I hope everything goes well for you from now on, as well. Finally, to three very important people. Grandmother and grandfather, I really hope you are proud of me for all I tried to accomplish. Luis Huang Xu, you may no longer be here and unfortunately, you could not become a mechanical engineer. So, I will gladly carry your wishes within me. I love you all three. iii

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5 Resumo Atualmente, as baterias de lítio ferro fosfato (LiFePO4) têm sido cada vez mais utilizadas, nomeadamente em veículos elétricos. Como tal, um dos parâmetros mais importantes para saber as condições físicas da bateria é o estado de carga. Uma aquisição precisa do seu valor permite manter a bateria em boas condições e consequentemente, o aumento da vida útil da bateria. Neste contexto, dados provenientes da unidade de Tecnologia de Energia da empresa VITO, situada na Bélgica, foram utilizados. Estes dados são retirados através de uma espetroscopia de impedância eletroquímica a uma bateria de LiFePO4 e são destinados à indústria automóvel. Esta coleção de dados contem informação física relativa à gama de frequências aplicadas à bateria, assim como a sua temperatura e vida útil. Estes dados foram estudados previamente numa dissertação de mestrado, em que se procurou fazer um primeiro estudo para obter um sistema com lógica fuzzy, que permitia a estimação do valor do estado de carga de uma bateria de LiFePO4. Além do mais, nessa dissertação, uma técnica para gerar perfis de curvas de resposta em frequência cujo nível de carga não foi divulgado nos dados fornecidos pela VITO foi criada. Nesse trabalho, o sistema fuzzy usa somente as impedâncias obtidas nas curvas de resposta em frequência. As variáveis físicas apresentadas nos dados não têm em conta efeitos físicos como a corrente a que as baterias foram carregadas ou o tempo de carga da bateria. Para além disso, a relação entre a curva de resposta em frequência e o estado de carga não é linear. Como tal, a solução sugerida nesta dissertação utiliza um sistema adaptativo baseado em lógica fuzzy, cujas variáveis de entrada são parâmetros geométricos da curva de resposta em frequência. Alguns dos dados são apresentados ao modelo neuro-fuzzy adaptativo para que as características da bateria sejam extraídas e associadas a um certo estado de carga. Com essas características, é possível inferir outros valores de estados de carga que não sejam conhecidos previamente pel o modelo neurofuzzy adaptativo. Nessa mesma dissertação, o autor desenvolveu um método que utiliza interpolação bilinear para criar novas curvas de resposta em frequência, que foi usado por não ser possível realizar resultados experimentais na bateria. Nesta dissertação de mestrado, é assumido que a evolução do valor do estado de carga não utiliza uma distribuição bilinear, mas sim cúbica, para ter em conta as suas nãolinearidades. Desta forma, pretende-se aperfeiçoar o modelo neuro-fuzzy adaptativo, através da introdução de características que possam não ter sido extraídas inicialmente. Para todos os modelos, utilizou-se o erro quadrático médio como critério de escolha. Palavras-Chave: baterias de LiFePO4; estado de carga; interpolação; modelos neuro-fuzzy adaptativos; parâmetros geométricos. v

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7 Abstract Nowadays, lithium iron phosphate batteries (LiFePO4) are very popular in electrical vehicles (EV). One of the most important parameters is the state-of-charge (SOC), since it is an indicator of the battery physical conditions. A precise acquisition of the SOC value allows to check if the battery is in good working conditions and therefore, it is possible to increase its lifetime. In this context, datasets provided from the unit of Energy Technology at VITO company, in Belgium, were used. These datasets were obtained using the electrochemical impedance spectroscopy (EIS) method in a LiFePO4 battery and were used for the automotive industry. These datasets contain physical information regarding the range of frequencies at which the battery was subjected, as well as the room temperature and its battery lifetime. These datasets were previously studied in a master thesis, where a first approach to attain a fuzzy system capable of estimating and predicting a LiFePO4 battery SOC value was studied. Also, a technique to generate EIS profiles for SOC values which are not provided in VITO datasets was developed. In this work, some improvements based on that study were implemented. In the abovementioned work, the studied fuzzy system only uses the complex impedance contained in the frequency curve response. The physical variables presented in the datasets do not have into account the battery charging current or the battery charging time. Also, the relation between the battery frequency response and its SOC is not linear. Therefore, the solution suggested in this thesis uses an adaptive system based in fuzzy logic, whose input variables are geometrical parameters of the frequency curve response. Some of the VITO datasets were used as input data to the adaptive neuro-fuzzy inference system (ANFIS) model to extract battery features and map them to a certain SOC value. With these features, it is to understand the battery dynamics and therefore, it is possible to estimate other SOC values which were not previous acquired by the ANFIS model. Also, in the mentioned work, a bilinear method to interpolate new EIS curves when the SOC value is not provided was developed, based on the SOC values previously known, which was used when it is not possible to perform experiments on the battery. In this master thesis, it is assumed that the SOC evolution is not bilinear, but cubic, which has into account its non-linear evolution. With this, it is intended to extract new features which the ANFIS model may have not taken into account and therefore, improve its inference abilities. For all of the obtained models, the root mean squared error () was used as a choosing criteria. Keywords: ANFIS; EIS; interpolation, geometrical parameters, LiFePO4 battery, SOC estimation vii

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9 Contents Acknowledgements... iii Resumo... v Abstract... vii Contents... ix List of figures... xii List of tables... xiv List of Abbreviations... xvi List of symbols... xvii 1. Introduction The advent of the electrical vehicles Objectives Thesis outline Contributions Lithium-ion batteries Li-ion functional analysis Typical Li-ion batteries for EV State-of-charge (SOC) of a battery: definition Review of methodologies for SOC estimation Open circuit voltage (OCV) Coulomb counting method Kalman filters Electrochemical Impedance Spectroscopy (EIS) Soft computing techniques Implemented solution Adaptive Neuro-Fuzzy Inference Systems (ANFIS) Principles of fuzzy modelling If-then fuzzy rules and membership functions ix

10 Takagi-Sugeno-Kang fuzzy model ANFIS Hybrid learning Characteristic features of battery EIS profile Physical variables Sequential pattern-based variables for EIS parameterization Polynomial models Frequency based models Principal component analysis (PCA) hybrid models SOC estimation of LiFePO4 batteries Polynomial based model: linear approximation Case study: without temperature (T) and lifetime (L) physical variables Case study: with temperature (T) and without lifetime (L) physical variables Case study: with lifetime (L) and without temperature (T) physical variables Case study: with temperature (T) and lifetime (L) physical variables Polynomial based model: cubic approximation Case study: without temperature (T) and lifetime (L) physical variables Case study: with temperature (T) and without lifetime (L) physical variables Case study: with lifetime (L) and without temperature (T) physical variables Case study: with temperature (T) and lifetime (L) physical variables Case study: frequency based models Case study: without temperature (T) and lifetime (L) physical variables Case study: with temperature (T) and without lifetime (L) physical variables Case study: with lifetime (L) and without temperature (T) physical variables Case study: with temperature (T) and lifetime (L) physical variables PCA hybrid models Case study: without temperature (T) and lifetime (L) physical variables Case study: with temperature (T) and without lifetime (L) physical variables Case study: with lifetime (L) and without temperature (T) physical variables Case study: with temperature (T) and lifetime (L) physical variables Discussion SOC interpolation with VITO datasets Case study: 25% to 50% range x

11 % to 75% range % to 100% range Discussion Conclusions Achievements Future work Bibliography Appendix A A.1. Chapter a. 25% to 50% range a. Case study: without temperature (T) and lifetime (L) physical variables b. Case study: with temperature (T) and without lifetime (L) physical variables c. Case study: with lifetime (L) and without temperature (T) physical variables b. 50% to 75% range a. Case study: without temperature (T) and lifetime (L) physical variables b. Case study: with temperature (T) and without lifetime (L) physical variables c. Case study: with lifetime (L) and without temperature (T) physical variables c. 75% to 100% range a. Case study: without temperature (T) and lifetime (L) physical variables b. Case study: with temperature (T) and without lifetime (L) physical variables c. Case study: with lifetime (L) and without temperature (T) physical variables xi

12 List of figures Figure 2.1 Discharging (a) and charging (b) mechanisms of Li-ion rechargeable batteries, adapted from [6]...3 Figure 2.2 Relation between SOC and OCV, adapted from [16]...5 Figure 2.3 Equivalent circuit model of a LiFePO4 battery...6 Figure 2.4 Effect of the temperature T in a LiFePO4 battery EIS, second month, SOC = 25%...8 Figure 2.5 Effect of the battery lifetime L in a LiFePO4 battery EIS, T = 00C, SOC = 25%...8 Figure 3.1 Graphical representation of y = (x 2) Figure 3.2 Discretized graphical representation of y = (x 2) Figure 3.3 Application of membership functions to y = (x 2) Figure 3.4 Output after rule application Figure 3.5 Triangular fuzzy model rules and respective singletons imposed on the domain Figure 3.6 Takagi-Sugeno first-order system with two rules and two inputs Figure 3.7 Equivalent ANFIS architecture of a Takagi -Sugeno model Figure 4.1 Linear approximation with temperature T = 100C and battery lifetime L = 3, for different SOC values Figure 4.2 Case with four extrema, for temperature = 100C, battery lifetime L = 3 and SOC = 25% 24 Figure 4.3 Application of the algorithm steps, for T = 100C, L = 3 and SOC = 25% Figure 5.1 Graphical representation of the selected model from table 5.1: r = 0.3, S1T,S1C Figure 5.2 Graphical representation of the selected model from table 5.2: r = 0.3, S1T,S1C Figure 5.3 Graphical representation of the selected model from table 5.3: r = 0.3, S2T,S2C Figure 5.4 Graphical representation of the selected model from table 5.4: r = 0.3, S2T,S2C Figure 5.5 Graphical representation of the selected model from table 5.5: r = 0.3, S1T,S1C Figure 5.6 Graphical representation of the selected model from table 5.6: r = 0.3, S2T,S2C Figure 5.7 Graphical representation of the selected model from table 5.7: r = 0.3, S2T,S2C Figure 5.8 Graphical representation of the selected model from table 5.8: r = 0.7, S2T,S2C Figure 5.9 Graphical representation of the selected model from table 5.9: r = 0.4, S2T,S2C Figure 5.10 Graphical representation of the selected model from table 5.10: r = 0.4, S1T, S1C Figure 5.11 Graphical representation of the selected model from table 5.11: r = 0.4, S1T, S1C Figure 5.12 Graphical representation of the selected model from table 5.12: r = 0.3, S2T, S2C xii

13 Figure 5.13 Graphical representation of the selected model from table 5.13: r = 0.8, S2T, S2C Figure 5.14 Graphical representation of the selected model from table 5.14: r = 0.8, S1T, S1C Figure 5.15 Graphical representation of the selected model from table 5.15: r = 0.9, S1T, S1C Figure 5.16 Graphical representation of the selected model from table 5.16: r = 0.9, S2T, S2C Figure 6.1 Representation of the selected model from table 6.4: frequency based model, r = Figure 6.2 Graphical representation of the selected model from table 6.8: linear approximation model, r = Figure 6.3 Graphical representation of the selected model from table 6.12: linear approximation model, r = Figure 0.1 Graphical representation of the selected model from table 0.1: frequency based model, r = Figure 0.2 Graphical representation of the selected model from table 0.2: cubic approximation model, r = Figure 0.3 Graphical representation of the selected model from table 0.3: cubic approximation model, r = Figure 0.4 Graphical representation of the selected model from table 0.4: cubic approximation model, r = Figure 0.5 Graphical representation of the selected model from table 0.5: linear approximation model, r = Figure 0.6 Graphical representation of the selected model from table 0.6: frequency based model, r = Figure 0.7 Graphical representation of the selected model from table 0.7: cubic approximation model, r = Figure 0.8 Graphical representation of the selected model from table 0.8: cubic approximation model, r = Figure 0.9 Graphical representation of the selected model from table 0.9: cubic approximation model, r = xiii

14 List of tables Table 3.1 Description of the passes used in hybrid learning [22] Table 4.1 Variables provided in VITO datasets Table 5.1 for several radius without using T and L as input variables Table 5.2 for several radius using only T as an input variable Table 5.3 for several radius using only L as an input variable Table 5.4 for several radius using T and L as input variables Table 5.5 for several radius without using T and L as input variables Table 5.6 for several radius using only T as an input variable Table 5.7 for several radius using only L as an input variable Table 5.8 for several radius using T and L as input variables Table 5.9 for several radius without using T and L as input variables Table 5.10 for several radius using only T as an input variable Table 5.11 for several radius using only L as an input variable Table 5.12 for several radius using T and L as input variables Table 5.13 for several radius without using T and L as input variables Table 5.14 for several radius using only T as an input variable Table 5.15 for several radius using only L as an input variable Table 5.16 for several radius using T and L as input variables Table 6.1 for several radius using T and L as input variables Table 6.2 for several radius using T and L as input variables Table 6.3 for several radius using T and L as input variables Table 0.1 for several radius without using T and L as input variables Table 0.2 for several radius using only T as an input variable Table 0.3 for several radius using only L as an input variable Table 0.4 for several radius without using T and L as input variables Table 0.5 for several radius using only T as an input variable Table 0.6 for several radius using only L as an input variable Table 0.7 for several radius without using T and L as input variables xiv

15 Table 0.8 for several radius using only T as an input variable Table 0.9 for several radius using only L as an input variable xv

16 List of Abbreviations ANFIS Adaptive neuro-fuzzy inference system EIS Electrochemical impedance spectroscopy EV Electric vehicle HEV Hybrid electric vehicle LCO Lithium cobalt oxide battery LFP Lithium iron phosphate battery LMO Lithium manganese oxide battery MF Membership function NCA Lithium nickel cobalt aluminium battery PCA Principal component analysis PCHIP Piecewise cubic Hermit interpolant polynomial Root mean square value SOC State-of-charge VAF Variance accounted for xvi

17 List of symbols LiFePO4 Lithium iron phosphate battery LiCoO2 Lithium cobalt oxide battery C N Battery nominal capacity Q b Net discharged current η Coulombic efficiency i(t) Current [A] ω Frequency [rad/s] f Frequency [Hz] V(t) Voltage [V] φ Phase shift [ 0 ] Z Complex impedance Z Re Real part impedance Z Im Imaginary part impedance j Imaginary unit μ A(x) Membership function A(x) Linguistic values set Y(x) Takagi-Sugeno crisp output p i Parameter from Takagi-Sugeno rule q i Parameter from Takagi-Sugeno rule r i Parameter from Takagi-Sugeno rule t i Parameter from Takagi-Sugeno rule xvii

18 w Output B(x) Linguistic values set O i Node output w Normalized firing strength T Temperature L Battery lifetime m Slope from first order equation b Y-intercept from first order equation (a, b, c, d) Third order polynomial coefficients α Slope angle from first order equation R Rotation matrix f i Frequency evaluated at extrema points a i Concavity evaluated at extrema points φ Linear transformation z i S it S ic r Principal component Training set Checking set Cluster radius xviii

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20 1. Introduction 1.1. The advent of the electrical vehicles With the constant need to move quickly from one city to another, or even another countries, easy mobility became a necessity. This is especially true in the automotive industry, where parents need to leave their children at school or move from home to work. However, with the crescent number of vehicles existent on the street, it is necessary to increase the crude exploration. This creates two problems. The first issue is that fossil sources of energy are non-renewable, which means other energy sources such as solar energy. The second issue is the fact that energy deriving from fossil sources pollute the atmosphere, creating health problems. Measures have been taken to reduce the greenhouse effect, mostly coming from the Kyoto Protocol and more recently, the Paris Agreement [1] [2]. To overcome these problems, hybrid electrical vehicles (HEV) and electrical vehicles (EV) were used as an alternative to oil-based vehicles. HEV and EV use lithium-ion batteries because they are environment friendly and comparing to lead-acid batteries, they have a longer cycle-life, provide higher power and higher energy density [3]. However, EV vehicles are mostly driven in urban circuits, since the Li-ion battery storage cells only have enough energy for few kilometres and they take too long to fully charge. Also, most existing battery models are impractical to provide the remaining energy left in the battery. This value is called state-of-charge (SOC). This parameter is important because with it, it is possible to monitor the battery anomalies, such as deep discharges, overcharges or self-discharges, and extend the Li-ion battery lifetime. Also, knowing a battery SOC helps to detect what caused the anomaly, by analysing the temperature at which the battery is subjected or the user driving style. Regarding this subject, some authors [4] [5] used soft computing techniques and fuzzy logic to compute the battery SOC. In this thesis, it is intended to further develop the work developed in [4], since this is a work which attempted to find a first approach to compute the battery SOC with datasets provided from the belgium company VITO. These datasets are obtained after testing LiFePO4 batteries, which are starting to be widely used in the automotive industry because of their thermal safety. In this work, some non-linear components and behaviours were introduced in the models, to further study the behaviour of this kind of batteries. This work aims to create a model which can predict a battery s SOC in an accurate way, having into account different temperature conditions and its age. 1

21 1.2. Objectives With this work, it is intended to achieve two main goals: Development of a model which can extract features from impedance profiles with known SOC values. With this model, it is possible to compare the stored information with new impedance profiles and therefore, predict its SOC. Development of an interpolation technique which generates new impedance profiles, based on already known profiles, for different SOC values than the ones provided in VITO datasets. This allows to create new profiles when information is limited to few SOC values and it is not possible to perform experimental setups Thesis outline This work has the following structure: Chapter 2 explains the basic physical properties of a battery and also explains the importance of using LiFePO4 batteries. The definition of SOC is presented, as well as a review of SOC estimation methods and an introduction to the proposed SOC prediction method. In chapter 3, the principles of fuzzy logic are presented, as an introduction to adaptive neurofuzzy inference systems (ANFIS), which are applied in this work. Chapter 4 presents the variables existent in VITO datasets and methodologies to extract alternative variables which are characteristic of the battery dynamics at different working conditions. In chapter 5, ANFIS models are created with the methodologies developed in chapter 4. Its results are also analysed. Chapter 6, a nouvelle technique is developed to generate new datasets with different SOC values than the ones provided by VITO. ANFIS is then used to validate the technique. Chapter 7 summarizes the main conclusions and some approaches that may be studied in future works are presented Contributions In this thesis, batteries made of lithium iron phosphate (LiFePO4) were used in EV and the originated datasets were obtained having in mind automotive applications. In this context, the proposed system uses the battery impedance profiles to infer its SOC values, using adaptive neuro-fuzzy models. This system must be capable of estimating the battery SOC based on the impedance profiles used for the learning process. 2

22 2. Lithium-ion batteries In this chapter, an overview regarding the battery charging and discharging mechanisms is given. The charging conditions are very important when analysing the battery performance and its maintenance conditions, since they can prolong the battery lifetime. Also, an important factor to analyse the battery condition is also the composition of the battery. LiFePO4 are starting to be widely used in the automotive industry, so the main advantages and drawbacks of its use will be pointed out in this chapter. Finally, it is intended to use a method which allows to estimate a LiFePO4 battery state-of-charge (SOC). This physical variable measures the remaining electrical energy in the battery and it is a key parameter to evaluate the battery working conditions, since it is affected by the external temperature and the own battery lifetime. A precise estimation of its value can manage and optimize the battery resources. Therefore, the propulsion system can be adjusted to the new driving conditions Li-ion functional analysis A battery is a device consisting of one or more electrochemical cells which uses the chemical energy stored in them and converts it into electrical energy, resorting to an oxidation-reduction reaction. Each battery cell has the following components: an anode usually composed of carbon or graphite, a cathode whose composition depends on the type of battery, a membrane which prevents contact between the anode and the cathode and a non-aqueous liquid electrolyte. As it was already mentioned in chapter 1, Li-ion batteries are used nowadays in the automotive industry, therefore Li-based molecules compose the cathode. In the discharging process, the anode releases its excessive electrons, which pass through the membrane and are accepted by the cathode. In rechargeable batteries, this process is inverted when the battery is being charged. Both processes are exemplified in figure 2.1. a) b) Figure 2.1 Discharging (a) and charging (b) mechanisms of Li-ion rechargeable batteries, adapted from [6] 3

23 2.2. Typical Li-ion batteries for EV Nowadays, there are several types of Li-ion batteries available on the market. According to [7], it is possible to find lithium nickel cobalt aluminium oxide (NCA), lithium manganese oxide (LMO), lithium cobalt oxide (LCO) and lithium iron phosphate (LFP), as well as many others. In the case of electrical vehicles (EV) and hybrid electric vehicles (HEV), the choice tends to be between LMO, NCA and LFP batteries [8]. However, looking at other industries, there are several applications which use LCO batteries, namely in the avionic industry. It is also commonly used in cameras and laptops. One of the main issues of using LiCoO2 is that it tends to release greater amounts of energy than the other Li-ion batteries. Also, the temperature at which this occurs is the lowest of the available Li-ion batteries, around 170 ºC, against a 300ºC peak in LiFePO4 batteries, according to [9]. This shows that even out of the automotive industry, it is interesting to further study and develop LiFePO 4 batteries. They are preferred because they are environment friendly, FePO4 is easy to obtain, so it is cheaper and these batteries are thermally more stable [7] [10] [11] [12]. There are also some issues, such as the volume energy density being lower than LiCoO2 and the low electronic conductivity. However, the last problem can be overcome with substance doping [10] [11], so in general, it is safer and acceptable to use LiFePO4 batteries State-of-charge (SOC) of a battery: definition The state-of-charge (SOC) of a battery is a measure of the remaining electrical energy stored in a cell and can be defined in the following way, SOC = C N Q b C N (2.1) where Q b is the net discharged charge from the last time the battery was fully charged and C N is the nominal capacity. However, this is assuming it is always possible to fully charge the battery. For example, the battery s age deteriorates the quality of the cathode and after some time, it is not possible to achieve its nominal capacity. Therefore, it is important to not only know the initial Q b but also the battery state of usage, which leads to other SOC definitions [13]. An accurate SOC determination in LiFePO4 batteries is hard to attain. First of all, the number of charge and discharge cycles the battery can handle is empirical. Secondly, most of the SOC determination techniques are off-line, which means it is rarely possible to monitor the battery characteristics in real time. Lastly, the batteries are sealed, which implies there is a need to use non-invasive methods. There are also several non-linear effects which cannot be easily accounted for in SOC determination, such as self-discharge or thermal differences [14]. If these problems are solved, it is possible to extend the life of the battery, instantaneously compute the SOC level and if possible, predict the SOC in a HEV based on the user driving style. 4

24 OCV [V] 2.4. Review of methodologies for SOC estimation The following is a list of techniques used to compute a battery SOC. Usually, they are combined to give a more accurate estimation of its measurement and are called hybrid methods [15] Open circuit voltage (OCV) OCV is the difference of potential when the battery is not connected to any circuit. A decrease in SOC implies a drop in the OCV in a LiFePO4 battery, as it can be seen in figure ,4 3,3 3,2 3,1 3 2,9 2, SOC [%] Figure 2.2 Relation between SOC and OCV, adapted from [16] Typically, there are tables where the known OCV-SOC relations are stored and compared with the OCV measured in the battery. This method is not very accurate because of two reasons. It is visible in figure 2.2 that there are some regions where the battery voltage is almost constant, independently of its SOC. If there is noise in the measurement, it is harder to get a precise value. The second reason is the influence of temperature and the number of cycles of the battery, which may change the initial OCV - SOC relation. 5

25 Coulomb counting method This methodology takes into account the initial SOC and integrates the battery current, independently of being charged or discharged. It is expressed as it follows: SOC(t) = SOC(0) 0 t η. i(t) dt C n (2.2) where SOC(t) is the SOC at a given time, SOC(0) is the initial SOC, i(t) is the current, C n is the battery estimated capacity and η is the coulombic efficiency. There are several problems when applying this technique, which the most relevant are the low accuracy of the initial SOC measurement, losses during charging cycles and also self-discharging [17]. Also, these effects tend to accumulate during the battery lifetime, which further degrades the quality of SOC measurements Kalman filters Usually, this technique is used when it is required to obtain a physical model of a battery, that is, to model the battery as an equivalent circuit with resistance, capacitors and coils. An example is provided in figure 2.3. Figure 2.3 Equivalent circuit model of a LiFePO4 battery After obtaining the physical model with state-space theory, Kalman filters are applied. This filter is an estimator which assumes the battery possesses internal noise and it is independent from the sensor measurements. The battery is observed during a certain time, allowing the filter to predict the next time step state and its uncertainty. After the next step is measured, the estimation is corrected and the filter parameters are updated. If the changes are too drastic, for example due to abrupt temperature changes, the model may not follow properly the system behavior. Some variants of Kalman filters are used to attenuate such effects, such as extended Kalman filter, fading Kalman filter or a combination with the OCV technique [18] [19] [20] 6

26 Electrochemical Impedance Spectroscopy (EIS) EIS is a technique where a small voltage, typically around 5mV to 10mV, is applied to the battery terminals to measure its impedance over a range of frequencies [21]. The battery is inserted in a Faraday cage, to avoid parasitic currents and typically it has three electrodes connected to it: one in each terminal and a reference electrode [22]. To measure the impedance, either potentiostatic mode can be used, where an AC potential is imposed to a cell, or galvanostatic mode can be used, where the excitation current is imposed. In the case of the galvanostatic mode, the imposed current is expressed as: I(t) = I max sin(ωt) = I max sin(2πft) (2.3) where I(t) is the current at the specified time t, I max is the maximum current, ω is the angular frequency and f is the frequency in hertz. As for the voltage, the expression is written as: V(t) = V max sin(ωt + φ) = V max sin(2πft + φ) (2.4) where V(t) is the voltage at the specified time t, V max is the maximum voltage and φ is the phase shift between the current and the voltage. Rewriting equations 2.3 and 2.4 in polar form, the impedance can be computed as it follows: Z = V j(ωt +φ) max. e = Ze jφ I max. ej (ωt) (2.5) where Z is the complex impedance, in ohm and j is the imaginary unit. Each point in the complex plane is associated to a certain frequency, so it is usual to plot the impedance spectra in a Nyquist plot, as it is shown in figures 2.4 and 2.5. The curve frequency responses displayed in these figures are attained using VITO datasets, studied in this work. 7

27 Figure 2.4 Effect of the temperature T in a LiFePO4 battery EIS, second month, SOC = 25% Figure 2.5 Effect of the battery lifetime L in a LiFePO 4 battery EIS, T = 0 0 C, SOC = 25% The spectra is then compared with an existing spectra database and therefore, the SOC can be estimated. The datasets provided by VITO were created with this technique. One thing that can be noted in the datasets is that Z is closer to 0 in the battery inductive zone, comparing with the capacitive zone. It is not common to find equivalent circuits possessing coils, so the inductive effect can be explained by the 8

28 experimental setup connections, since a small change in the electrodes position affects the EIS appearance and adds noise in the measurements. Therefore, only the battery capacitive zone was studied Soft computing techniques Soft computing techniques are data based systems which use datasets as inputs and extracts relations between its variables, relating them directly with the output. These kind of systems are capable of selfdesign and adjust themselves to match the uncertainty of the battery parameters [15]. This happens because they are modelled to categorize data the same way a human does. For battery applications, it is common to find one of the following techniques: Fuzzy logic: this technique uses a rule based system to distinguish each physical phenomena different working conditions. These rules are based on the human interpretation and therefore, they are often created with linguistic variables. Also, data is categorized with some uncertainty and in a broad sense. For example, the battery SOC can be defined as high within the range of 75% and 100%, but its value can be medium within the range of 30% and 80%. As it is seen, there is an overlap between the battery SOC ranges, so the concept of degree of membership and membership function appear. These concepts allow to measure the degree to which the variables satisfy the linguistic property, based on its behaviour. With these rules, the fuzzy model extracts the battery features to build a mathematical model which is capable of inferring the output variables values based on the examples previously learned. For example, if information regarding the battery discharging condition, the corresponding voltage, as the author in [23] used, or the temperature at which the battery EIS profiles were acquired is used as input parameters, it is possible to build a fuzzy model capable of predicting the battery SOC. Neural networks: this technique works based on the mechanism of information transmission present in the human brain. Unlike fuzzy logic, no rules are used. Instead, every input is crossed against each other in the existing nodes, independently of the relation between the variables. If there is a relation, the system launches an output response. Typically, there are three zones in a neural network. The first one is the input layer where raw data is inserted as it is, such as the extracted electric charge and the no-load battery voltage [24], with as many nodes as input variables. The second one is the hidden layer, which may have one or more layers. This zone performs the input combination, giving different weights t o the input variables. Each node propagates the output signal to the output layer, where the incoming signals are gathered and evaluated. One of the advantages of using neural networks is that the system can be iteratively trained, until it reaches an error goal. This is done by propagating the error between the output and the generated signal backwards, to update each input variable weight, since the first iteration may 9

29 not provide an acceptable weight. The goal is the same of fuzzy models, which is to build a model capable of predicting the battery SOC based on the EIS profile There are also some variants that combine properties of several techniques, such as adaptive neurofuzzy inference systems (ANFIS), which was previously studied by some authors [5] [25], where physical properties such as voltage, current and voltage change rate were used as input variables. This technique was applied in this work and will be described in chapter Implemented solution In this work, EIS tests were performed monthly in a set of LiFePO4 batteries, between the second and the sixth testing month. Also, the batteries were studied at several temperatures, which were -10ºC, 0ºC, 10ºC, 25ºC and 40ºC. The impedance profiles were then stored in a database and then the SOC value was computed with a known relation between SOC and OCV from VITO. The SOC values provided in the databases are 25%, 50%, 75% and 100%. With the initial datasets, interpolation was used to generate intermediate EIS profiles, with a SOC of 37.5%, 62.5% and 87.5%. Then, a fuzzy inference system was developed to automatically estimate the battery SOC, based on the information contained in some of the EIS measurements. This database needs to be updated periodically, since there may be a need to estimate the SOC of a battery which is older or studied at a higher temperature than the ones used in this study and therefore, keep track of the battery physical conditions and usage. 10

30 3. Adaptive Neuro-Fuzzy Inference Systems (ANFIS) Nowadays, several important engineering problems which are too complex to model with analytic models, since either some phenomena cannot be included or have not been included in those models. One current example is to develop an analytic model that can accurately characterize the behaviour of the new LiFePO4 battery and with it, obtain its SOC. Although there are studies in this field [23] [5], there is still no consensus regarding a solid mathematical background to model this type of batteries. Regarding this problem, fuzzy logic is used to extract relationships between its physical phenomena and therefore, estimate the battery SOC, since it generates a mathematical model based solely on experimental data, without prior knowledge of the physical system. In this chapter, the main characteristics of fuzzy modelling are explained, as an introduction to the adaptive neuro-fuzzy inference system (ANFIS) modelling. In this case, the information existent in the EIS profiles provided in VITO datasets is used to build these models Principles of fuzzy modelling To understand how fuzzy inference works, it is necessary to explain how inputs are mapped in functional relations and how it is different from fuzzy mapping. As an example, figure 3.1 shows the graphical representation of the equation y = (x 2) This equation was used because as it was seen in chapter 2, subsection 2.4.4, figures 2.4 and 2.5, the EIS curves of a LiFePO4 battery have a wave-like behaviour when the electric frequency varies and a cubic equation can emulate this property. Figure 3.1 Graphical representation of y = (x 2)

31 This function can be also turned into a piecewise function. Limiting our domain in x [0, 10], as it is suggested in figure 3.1, with the discretization values x = 1, 3, 5, 7 and 9, it is possible to define crisp relations. Figure 3.2 represents the result after applying the restrictions defined in equation 3.1. If x [0, 4[ then y = 20 If x [4, 6[ then y = 50 If x [6, 8[ then y = 155 (3.1) If x [8, 10] then y = 380 Figure 3.2 Discretized graphical representation of y = (x 2) If-then fuzzy rules and membership functions The situation abovementioned can be applied in the batteries context. As example, let us assume that x is the battery usage and y is the battery discharge rate. In human language, the following set of linguistic rules can be formulated: Rule 1: If x is low, then y is low Rule 2: If x is moderate, then y is high (3.2) Rule 3: If x is high, then y is very high The formulation of equation 3.2 uses linguistic values instead of numerical values. For the input variable x, the possible values are low, moderate and high, whereas the output variable y can be assigned 12

32 Low High Very high with the values low, high and very high. These sets of linguistic values are called the universe of discourse, that is, the possible values of the variable domain. One of the main drawbacks of this classification is its subjectivity. For example, for electric vehicles (EV), the battery usage may be high because the EV is constantly used, but its discharge rate can still be low, if the battery maintenance conditions are ideal. Also, the rules are based on linguistic values, so there is no certain corresponding numerical value. To overcome these problems, the concept of membership function (MF) is used. A MF is a mathematical object whose characteristic function is defined as it follows, μ A(x) : X [0,1] (3.3) where A(x) is the set containing the linguistic values defining X and μ A(x) is the membership function, ranged between 0 and 1, which allows the variables to have a certain degree of membership. Also, each linguistic value is has a certain MF shape, which tries to emulate as close as possible the physical behaviour inherent to the linguistic values. An application of the rules written in equation 3.2 can be found in figure 3.3. Region 3 Region 2 Region 1 Low Moderate High Figure 3.3 Application of membership functions to y = (x 2) In this example, triangular MF are used, but it is also common to find gaussian or trapezoidal MF. Also, the rules written in equation 3.2 can be re-written in the following way: 13

33 Rule 1: If x is low, then y is low A (1) B 1 (1) Rule 2: If x is moderate A (2), then y is high B (2) (3.4) Rule 3: If x is high, then y is very high A (3) B (3) where A (i) = μ A (i), B (i) = μ B (i) and i is the i-th rule. The application of these restrictions give a fuzzy output. An example of such an output can be found in figure 3.4. Figure 3.4 Output after rule application Takagi-Sugeno-Kang fuzzy model This kind of modelling was proposed by Takagi, Sugeno and Kang to develop a systematic approach to generating fuzzy rules from a given input-output data set [26]. The rules in this type of modelling are written in the following form, If x 1 is A and x 2 is B then w = g(x 1,x 2 ) (3.5) where x and y are system inputs, g(x, y) is typically a zero-order or a first-order polynomial and w is the output crisp value. w can be written generically as n (3.6) w = t i x i i =0 14

34 B (1) B (2) B (3) Low High Very high where t i are the coefficients relating the n input variables and the output crisp value w. With this expression, the rules stated in equation 3.4 can be rewritten as: Rule 1: If x is A (1), then y is w (1) Rule 2: If x is A (2), then y is w (2) (3.7) Rule 3: If x is A (3), then y is w (3) Applying this to the initial example, y = (x 2) , figure 3.5 is obtained: w (3) w (2) w (1) Low Moderate High A (1) A (2) A (3) Figure 3.5 Triangular fuzzy model rules and respective singletons imposed on the domain After evaluating each input with all MF, the inputs can be combined using the weighted average of all rule outputs, which is computed as it follows, c l=1 Y(x) = (l) μ Al c (l) l=1 μ Al (x).ω (l) (x) (3.8) where x is the input numerical value, c is the number of rules and Y(x) is the crisp output after all input values are inferred through the MF. 15

35 3.2. ANFIS As it was written in the beginning of this chapter, ANFIS uses the fundaments of fuzzy logic, namely the ones used in Takagi-Sugeno fuzzy systems. Under certain circumstances [26] [27], ANFIS is an adaptive network functionally equivalent to Takagi-Sugeno models, only changing the system weights computation method. Re-writing equation 3.6 to have two input variables, the following rule set is obtained: Rule 1: If x is A 1 and y is B 1, then w 1 = p 1 x + q 1 y + r 1 (3.9) Rule 2: If x is A 2 and y is B 2, then w 2 = p 2 x + q 2 y + r 2 A graphical representation of a regular Takagi-Sugeno model using these rules can be found in figure 3.6. A 1 B 1 w 1 A 2 B 2 w 2 Figure 3.6 Takagi-Sugeno first-order system with two rules and two inputs If the Takagi-Sugeno fuzzy model is mapped into a neural network, the equivalent ANFIS model is created. Figure 3.7 represents one of the most common ANFIS architectures. 16

36 Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Figure 3.7 Equivalent ANFIS architecture of a Takagi-Sugeno model Each layer has several nodes whose meaning is described next. Layer 1: this layer is composed only of adaptive nodes, defined by: O 1,i = μ i (x i ) (3.10) where O 1,i is the output of the i-th node of the first layer. The parameters generated in this layer are called premise parameters. Layer 2: in this layer, all nodes are fixed. Every output is the product of all outputs coming from layer 1: O 2,i = w i = μ j j (3.11) Each node output corresponds to the firing strength of the rule. Layer 3: here, every node is fixed and each output is the normalized firing strength: O 3,i = w i = w i (3.12) j w j 17

37 Layer 4: all nodes are adaptive, where the normalized firing strengths are multiplied by the consequents of the if-then rules. Using the notation of equation 4.9, the expression for the output can be written as it follows: O 4,i = O 3,i. w i = w i. (p i x + q i y + r i ) (3.13) As it is implied, the output contains the consequent parameters. Layer 5: finally, this layer contains a single node, which is the overall output. This is obtained summing the outputs of the previous layer: O 5,1 = O 4,i i i w i w i = i w i (3.14) Hybrid learning ANFIS has two possible types of learning, backpropagation and hybrid learning. Typically, hybrid learning is implemented because it does not use the same space dimensions of the backpropagation method. It combines the original backpropagation method with least-squares method and it is based on the decomposition of the ANFIS output: O 5,1 = w i (p i x + q i y + r i ) i (3.15) = (w i x). p i + (w i y). q i + (w i ). r i i Assuming the premise parameters are fixed, all outputs are passed until layer 4. Then, the consequent parameters are computed using least-squares estimation, to guarantee these values are optimal. After this, the error signals are propagated backwards until they reach layer 1 by gradient descent, to update the premise parameters. A summary of the two passes used in ANFIS hybrid learning is given in table 3.1. Table 3.1 Description of the passes used in hybrid learning [26] Forward pass Backward pass Premise parameters Fixed Gradient descent Consequent parameters Least-squares estimator Fixed Signals Node outputs Error signals 18

38 This procedure is repeated in each iteration (epoch, as it is mentioned in the literature) until the final output is close to the intended result. This procedure cannot be performed for an unlimited amount of epochs. In the particular case of LiFePO4 batteries, there are physical phenomena which have not been studied yet, so there are some features which are not taken into account in the ANFIS model. If the model matches all of the provided battery EIS curve characteristics, it will result in loss of inference capability, i.e., a small change in the battery EIS profile shape originated by a noisy acquisition may be inferred as an EIS profile associated to a different SOC value. This effect is called overfitting. The ideal number of epochs depends on the data quality and quantity. In the ANFIS learning process, the datasets are partitioned into two sets: a training set which contains the battery EIS curve features necessary to create the premise and the consequent parameters for SOC prediction, and a checking set which is used to confirm if the ANFIS model parameters can be used to correctly infer any SOC value based on the previous knowledge. When the checking root mean square mean value () reaches a minimum, the training process is completed and that model is chosen, otherwise overfitting will occur. 19