METHODS OF TECHNICAL PROGNOSTICS APPLICABLE TO EMBEDDED SYSTEMS

Transcription

1 VYSOKÉ UČENÍ TECHNICKÉ V BRNĚ BRNO UNIVERSITY OF TECHNOLOGY FAKULTA ELEKTROTECHNIKY A KOMUNIKAČNÍCH TECHNOLOGIÍ ÚSTAV AUTOMATIZACE A MĚŘICÍ TECHNIKY FACULTY OF ELECTRICAL ENGINEERING AND COMMUNICATION DEPARTMENT OF CONTROL AND INSTUMENTATION METHODS OF TECHNICAL PROGNOSTICS APPLICABLE TO EMBEDDED SYSTEMS METODY TECHNICKÉ PROGNOSTIKY APLIKOVATELNÉ V EMBEDDED SYSTÉMECH DOKTORSKÁ PRÁCE DOCTORAL THESIS AUTOR PRÁCE AUTHOR VEDOUCÍ PRÁCE SUPERVISOR Ing. MIROSLAV KRUPA doc. Ing. LUDVÍK BEJČEK, CSc. BRNO 2012

2 ABSTRACT The main aim of the thesis is to provide a comprehensive overview of technical prognostics, which is utilized in the condition based maintenance, based on continuous device monitoring and remaining useful life estimation, especially in the field of complex equipment and machinery. Nowadays technical prognostics is still evolving discipline with limited number of real applications and is not so well developed as technical diagnostics, which is fairly well mapped and deployed in real systems. Thesis provides an overview of basic methods applicable for prediction of remaining useful life, metrics, which can help to compare the different approaches both in terms of accuracy and in terms of computational/deployment cost. One of the research cores consists of recommendations and guide for selecting the appropriate prediction method with regard to the prognostic criteria. Second thesis research core provides description and applicability of particle filtering framework suitable for model-based forecasting. Verification of their implementation and comparison is provided. The main research topic of the thesis provides a case study for a very actual Li-Ion battery health monitoring and prognostics with respect to continuous monitoring. The case study demonstrates the prognostic process based on the model and compares the possible approaches for estimating both the runtime and capacity fade. Proposed methodology is verified on real measured data. KEYWORDS technical prognosis, prognostics, diagnostics, condition-based maintenance, embedded system, condition based maintenance, model-based prognosis, data-driven prognosis, remaining useful life, particle filters, model-based development, battery health management

3 ABSTRAKT Hlavní cílem dizertace je poskytnutí uceleného pohledu na problematiku technické prognostiky, která nachází uplatnění v tzv. prediktivní údržbě založené na trvalém monitorování zařízení a odhadu úrovně degradace systému či jeho zbývající životnosti a to zejména v oblasti komplexních zařízení a strojů. V současnosti je technická diagnostika poměrně dobře zmapovaná a reálně nasazená na rozdíl od technické prognostiky, která je stále rozvíjejícím se oborem, který ovšem postrádá větší množství reálných aplikaci a navíc ne všechny metody jsou dostatečně přesné a aplikovatelné pro embedded systémy. Dizertační práce přináší přehled základních metod použitelných pro účely predikce zbývající užitné životnosti, jsou zde popsány metriky pomocí, kterých je možné jednotlivé přístupy porovnávat ať už z pohledu přesnosti, ale také i z pohledu výpočetní náročnosti. Jedno z dizertačních jader tvoří doporučení a postup pro výběr vhodné prognostické metody s ohledem na prognostická kritéria. Dalším dizertačním jádrem je představení tzv. částicového filtrovaní (particle filtering) vhodné pro model-based prognostiku s ověřením jejich implementace a porovnáním. Hlavní dizertační jádro reprezentuje případovou studii pro velmi aktuální téma prognostiky Li-Ion baterii s ohledem na trvalé monitorování. Případová studie demonstruje proces prognostiky založené na modelu a srovnává možné přístupy jednak pro odhad doby před vybitím baterie, ale také sleduje možné vlivy na degradaci baterie. Součástí práce je základní ověření modelu Li-Ion baterie a návrh prognostického procesu. KLÍČOVÁ SLOVA Technická prognostika, diagnostika, údržba založená na stavu, prognostika založená na modelu, prognostika založena na datech, zbývající životnost, částicové filtry, vývoj založený na modelu, údržba a monitoring Li-Ion baterie

4 BIBLIOGRAPHIC REFERENCE KRUPA, M. Methods of Technical Prognostics Applicable to Embedded Systems, Brno: Brno University of Technology, Faculty of Electrical Engineering and Communication, pages. Supervisor: doc. Ing. Ludvík Bejček, CSc..

5 DECLARATION Prohlašuji, že svou disertační práci na téma Methods of Technical Prognostics Applicable to Embedded Systems jsem vypracoval samostatně pod vedením vedoucího disertační práce a s použitím odborné literatury a dalších informačních zdrojů, které jsou všechny citovány v práci a uvedeny v seznamu literatury na konci práce. Jako autor uvedené disertační práce dále prohlašuji, že v souvislosti s vytvořením této disertační práce jsem neporušil autorská práva třetích osob, zejména jsem nezasáhl nedovoleným způsobem do cizích autorských práv osobnostních a jsem si plně vědom následků porušení ustanovení 11 a následujících autorského zákona č. 121/2000 Sb., včetně možných trestněprávních důsledků vyplývajících z ustanovení 152 trestního zákona č. 140/1961 Sb. V Brně dne (podpis autora)

6 ACKNOWLEDGEMENT I would like to thank my wife Petra and my family for supporting me in this work and encouraging me to finish it in reasonable time. I would like to thank to my supervisor doc. Ing. Ludvík Bejček, Csc. for sharing his knowledge with me and for giving me the right direction. I would like to acknowledge the valuable contribution of other my colleagues (namely doc. Ing. Petr Beneš, PhD., Ing. Stanislav Klusáček, PhD.), who provided me feedback and helped me to improve the quality of the work.

7 TABLE OF CONTENTS 1 INTRODUCTION Overall Context and Motivation Prognosis Meaning in Thesis Context Diagnostics and Prognostics Coexistence STATE OF THE ART Current Research Topics Overview Metrics for RUL Estimation Prognostic Methods Classification Prognostics Methods and Applied Frameworks Including Use Cases RESEARCH TOPICS Comparison of Prognostics Methods from Embedded System Point of view Prognostics Utilizing Particle Filters Battery Life Prognostics - Use Case COMPARISON OF PROGNOSTIC METHODS WITH FOCUS ON EMBEDDED SYSTEMS Detailed Methods List with Pros and Cons Guide for Proper Method Selection How to Deploy Developed Prognostics Method into Embedded System Summary of Answers to Thesis Research Topic Number PROGNOSTICS BASED ON PARTICLE FILTERING FRAMEWORK Bayesian Framework and Methods Introduction Particle Filter Theory Applied to Dynamic System Particle Filter Implementation and Verification PF Prediction Capabilities Advantages and Disadvantages of PF Summary of Answers to Thesis Research Topic Number USE CASE - BATTERY HEALTH PROGNOSTICS Motivation Battery Health Monitoring and Prognostics Techniques Li-Ion Battery Model Considering Effects of Temperature and Capacity Fading

8 6.4 Battery Model Simulation RUL - Battery Runtime Estimation RUL Battery Capacity Fade Estimation Summary of Answers to Thesis Research Topic Number CONCLUSION REFERENCES SYMBOLS AND ABBREVIATIONS

9 LIST OF FIGURES Figure 1-1 Maintenance types overview per EN [18] Figure 1-2 Overview of categories of forecasting applications per [63] Figure 1-3 Relation between Diagnosis and Prognosis per [80] Figure 1-4 Informational data flow per ISO Figure 2-1 Explanation of RUL on Li-Ion battery capacity example simulated case Figure 2-2 Explanation of RUL on components failures in car simulated data Figure 2-3 Remaining useful life and associated attributes reproduced from [63] Figure 2-4 Classification of prognostic metrics as defined in [63] Figure 2-5 Taxonomy of prognostics methods according to [21] Figure 2-6 Taxonomy of prognostic methods proposed in [80] and reviewed in [3] Figure 2-7 Taxonomy of Prognostics methods as proposed in [68] with its sublevels Figure 2-8 Overall architecture of the WNN prognostic system introduced in [83] Figure 2-9 Model-Based Approach to Damage Identification reproduced from [11] Figure 2-10 Particle Filtering Framework for Battery Health Prognosis reproduced from [60] Figure 2-11 A model based approach to prognostics and health management for flight actuators reproduced from [11] Figure 2-12 Bathtub Curve derived from Weibull distribution adopted from [88] Figure 2-13 A framework of intelligent prognosis combining several prognostics methods proposed in [3] Figure 2-14 Graphical Comparison of prognostic methods adopted from [80] Figure 4-1 Onboard monitoring with offline prognostics

10 Figure 4-2 Graphical explanation of embedded online prognostics Figure 4-3 Guide for High Level Prognostics Approach Selection Figure 4-4 Model-Based Prognostics Process Steps Proposal Enhancing [23] and [48] Figure 4-5 Prognostics Algorithms Development flow in context of model based development Figure 5-1 Overview of Bayesian Methods and Particle Filters [82] Figure 5-2 Principle of particle filter re-sampling reproduced from [10] Figure 5-3 Particle Filtering Algorithm Steps reproduced from [10] Figure 5-4 Filtering of a system represented by Equation 5-42 for simple SIR, N s=60, Residual re-sampling, Resample every 2 samples Figure 5-5 Evolution of State Density Represented by Particles for SIR filter, N s=60, Residual re-sampling, Resample every 2 samples Figure 5-6 Filtering of a system represented by Equation 5-42 for simple Auxiliary PF, N s=60, Residual re-sampling, Resample every 2 samples Figure 5-7 Evolution of State Density Represented by Particles for Auxiliary filter, N s=60, Residual re-sampling, Resample every 2 samples Figure 5-8 Filtering of a system represented by Equation 5-42 for Regularized PF, N s=60, Residual re-sampling, Resample every 2 samples Figure 5-9 Evolution of State Density for Regularized PF, N s=60, Residual resampling, Resample every 2 samples Figure 5-10 Filtering of a system represented by Equation 5-42 for EKF-PF, N s=60, Residual re-sampling, Resample every 2 samples Figure 5-11 Evolution of State Density for EKF-PF, N s=60, Residual resampling, Resample every 2 samples Figure 5-12 Filtering of a system represented by Equation 5-42 for EKF filter Figure 5-13 RMSE of Estimation for Each PF algorithms one particular run Figure 5-14 RMS diff comparison of different particle filter types and their dependency on Number of particles Figure 5-15 Comparison of filtering methods for different re-sampling types for 150 simulation steps Figure 5-16 Filtering methods sorted based on RMSE

11 Figure 5-17 Execution time for different particle filter implementation Figure 5-18 Sorted execution time for different filter implementation Figure 5-19 Final Comparison of Bayesian framework based filters/estimators Figure 5-20 Testing of N-step prediction capability Figure 5-21 Simulation with 5 steps ahead prediction for Simple PF Figure 5-22 Simulation with 100 steps ahead prediction for Simple PF Figure 5-23 Particles and its density used for setting confidence limit of estimation (µ = , σ = ) Figure 6-1 Real Prognostics Process with focus on model-based prognosis Figure 6-2 Li-ion battery equivalent internal impedance (reproduced from [30]) Figure 6-3 Simulink dynamic model of lithium-ion battery Figure 6-4 Lithium-ion battery pulse charging process (C init=850mah) Figure 6-5 Lithium-ion battery pulse discharging process (C init=850 mah) for different current Figure 6-6 Battery charging for different cycle N (I charge=160 ma, C init=850 mah) Figure 6-7 Battery discharge for different cycle N (I disch=160 ma, C Cinit=850 mah) Figure 6-8 Charging process under different temperature condition and cycle number (N=100@293K, N=100@310K, N=500@293K, N=500@310K) Figure 6-9 Discharging process under different temperature condition and cycle number (N=100@293K, N=100@310K, N=500@293K, N=500@310K) Figure 6-10 Battery Measurement setup for basic model verification Figure 6-11 Measured data versus simulation data for charging at 100mA of initial capacity 50mA (during battery calibration the capacity was set to 35mAh) Figure 6-12 Measured data versus simulation data for discharging at 100mA (f sample= 5s) (during battery calibration the capacity was set to 35mAh) Figure 6-13 Difference between model curve and real measurement

12 Figure 6-14 Setup for measurement real data from laptop Figure 6-15 Discharging/ Charging Process for Li-Ion battery in Dell Precision M Figure 6-16 Current Discharging/Charging Process Figure 6-17 Battery current discharge rate histogram progression in time for Dell Precision (Time: t 1=1-1000s, t 2= s, t 2= s, t 1= s) Figure 6-18 Probability density function fit analysis Figure 6-19 Cumulative distribution function of current discharging rate Figure 6-20 Battery Run Time prediction per current SBS implementation with α = Figure 6-21 Root Mean Squared Percentage Error (RPSME) of RUL estimate Figure 6-22 Battery Run Time prediction per proposed PDF estimation approach with α limits set to Figure 6-23 Root Mean Squared Percentage Error (RPSME) of RUL estimate implemented per algorithm (6-9) Figure 6-24 Probability plot of RUL Estimate in times series Time: t 1=341s, t 2=3741s, t 3=5441s, t 4=8841s Figure 6-25 Runtime prediction based on PDF - boxplot defining mean and 25 th and 75 th percentile Figure 6-26 Battery Run Time prediction per proposed linear trend estimation approach with α limits set to 10% Figure 6-27 Root Mean Squared Percentage Error (RPSME) of RUL estimate implemented per algorithm (6-9) Figure 6-28 Battery Run Time prediction based on PF framework with α limits set to Figure 6-29 Root Mean Squared Percentage Error (RPSME) of RUL estimate implemented per algorithm (6-11) Figure 6-30 Runtime prediction based on PF framework - boxplot defining mean and 25 th and 75 th percentile Figure 6-31 Estimation of runtime with extreme current load and linear regression trend Figure 6-32 Regular battery usage measured on a laptop

13 Figure 6-33 Simulated capacity fade for operational temperature in range 273 K 313 K and 1-4 full discharge/charge cycles per week Figure 6-34 RUL Prediction for Capacity Fade by linear regression method Figure 6-35 RPSME of capacity fade prediction based on linear regression Figure 6-36 Particle Filter used for CCF tracking as an internal state Figure 6-37 RUL Prediction for Capacity Fade by Particle Filtering method Figure 6-38 RPSME of capacity fade prediction based on particle filtering Figure 6-39 Battery Health Monitoring and Prognostic Framework Figure 6-40 Battery Monitoring Solution for Hybrid Electrical Vehicles (HEV) from [74]

14 LIST OF TABLES Table 2-1 Accuracy based metrics collected from [80] Table 2-2 Precision based metrics reproduced from [64] Table 2-3 Robustness Based Metrics [63] and [80] Table 2-4 Computation Prognostics Metrics (mainly embedded application) - author s proposal Table 2-5 Cost/Benefit Prognostics Metrics reproduced from [29] and [63] Table 2-6 New Prognostics Metrics Proposed in [64] Table 2-7 Visualization Methods for RUL estimation reproduced from [60] Table 4-1 Probability Based Methods Pros and Cons composed from [60], [68] and [80] Table 4-2 Data-Driven Methods Pros and Cons composed from [60], [68] and [80] Table 4-3 Model Based Parameter Identification Methods Pros and Cons composed from [60] and [68] Table 5-1 Comparison of Different Implementation of Particle Filters Table 5-2 Comparison of Execution Time of Different Particle Filter Algorithms (Measured Time for 150 Steps) Table 5-3 Comparison of methods for Gamma distribution Table 5-4 Comparison of particles for different N step predictions Table 6-1 Comparison of SOC Estimation Technique merged from [55] and from [93] Table 6-2 Bantam BC6 Functionality Overview Table 6-3 Probability Distribution Parameters for Current rate Table 6-4 Runtime Estimation Parameters per SBS Spec [69] Table 6-5 Comparison of methods for RUL runtime estimation Table 6-6 Comparison of methods for RUL capacity fade Table 6-7 Overview of battery gauges and circuits for Smart Battery system

15 1 INTRODUCTION 1.1 Overall Context and Motivation Functionality and reliability of machines and equipments significantly affects operating expenses and safety of the modern systems over their life-cycle. Minimizing the impact of damages caused by equipment failure should be one of the key efforts for every organization/company. Diagnosis, monitoring and related maintenance occurs continuously around us in wide range of human activity from heavy machinery (mine equipment, port cranes), including aerospace, automotive to light but precise biotechnological equipments like pacemakers. Any equipment whose failure could cause damages to life or to property requires some way of monitoring, diagnostics, prognostics and maintenance in our modern safety oriented world. Conventional maintenance strategies could be split into two main groups, preventive maintenance and corrective maintenance. Corrective maintenance is based on a posteriori action, it means after fault occurrence (for example: replacing fully discharged batteries, fixing tire defect, replacement of the burst timing belt, replacement of functionless servomotor) [42]. Preventive maintenance (PM) is based on a priori action and its main goal is avoiding the problem before the fault occurs. PM is usually executed on predefined time period or alternatively based on objective indicators defined and created by reliability analysis and based on the empirical past data. This kind of maintenance is called predetermined. Battery replacement after specific amount of time, timing belt replacement after specific number of kilometers, jet engine compressor replacement after predetermined number of operation hours, all of those could be listed as an example of the predetermined maintenance. Preventive maintenance significantly increases availability, functionality and equipment uptime, but on the other hand it causes extra costs, because equipments or components are replaced during their useful life and long time before their real failure. Past studies showed [46] that most of critical failures were not aligned to operating time. For example the reliability of critical components such as aircraft actuators were estimated statistically and conservative safe life removal interval was determined. Historical evidence has indicated that the actual usage of military aircraft systems often differs greatly from the intended usage and operating environment. Usage also depends on the pilot and the flying style in manned systems. Furthermore, unanticipated and extreme operating scenarios are a major cause of unscheduled maintenance events [12]. This proves that preventive predetermined maintenance does not always bring expected results and is not really cost-effective as could be expected. It is obvious that conventional maintenance strategy do not cover needs for modern complex and expensive systems. Automotive and aerospace industry could be mentioned as a good example. Keeping minimal maintenance expenses and prolonging the systems operability is crucial in helping to survive in really competitive segment. Condition based maintenance (CBM) approach seems to be suitable alternative to the conventional methods [47] a [48] and is considered 15

16 as sub-group of the preventive maintenance [18] and [19]. Sometimes there is a term Predictive Maintenance used as an equivalent to CBM. Technically it is sub-group of CBM as defined in [18]. CBM deploys embedded diagnosis and prognosis to determine functional state of the equipment. Equipment degradation and prognosis of the potential failure is derived from the current and previously monitored state and is based on an evaluation of the operational conditions. Moreover this approach provides feedback and could help to prolong the uptime and system durability. Health and Usage Monitoring System (HUMS) could be mentioned as a typical example of the CBM system. HUMS could be used for monitoring helicopter vibrations respectively rotor blades vibration. Analysis of those vibrations and proposed corrective actions leads to minimizing the vibration levels for specific flight modes and helps to prolong life-cycle of the helicopter fuselage, because vibration causes mechanical stress potentially leading to material cracks. Deployment of similar equipment requires high initial investments but prolonged up time and system reliability will cover all beginning expenses within few years. Figure 1-1 Maintenance types overview per EN [18] Another example is the application of CBM strategy/system for wind power plants located in coastal waters [8]. There often occurs cracks and breaks in the rotor blades material (usually fiberglass) due to weather conditions (hail, lightning) or due to possible collisions with birds. Cracks, unless are not fixed by special techniques (fill in with an adhesive material patch), gradually expand over time and sooner or later the entire blade has to be replaced. Such replacement is relatively expensive and, moreover, can take up to 20 days, after which the plant produces electricity, and thus leads to further financial losses. However, in case a permanent monitoring system is put in place, in this case, infrared thermography and ultrasound diagnostics, then any new crack could be detected in a timely manner and crack growth could be monitored and its size could be predicted thus optimal maintenance could be scheduled. A potential crack could be consequently fixed before irreversible damage occurs to the blade, and therefore causes the necessary replacement of high costs. This type of monitoring is part of the whole field of interest called Structure Health Monitoring/Management - SHM. Typical areas of SHM application are airframe monitoring for cracks, hull of ships monitoring for cracks caused by corrosion etc. 16

17 CBM methodology is under extreme focus last decades. Older conventional maintenance methods are getting on background or are combined. Optimal maintenance of a multi unit system combines preventive predetermined maintenance and conditional-based maintenance is described in [71] or so called reliability centered maintenance RCE is an approach combining all maintenance strategies to guarantee maximal reliability [56]. Nowadays CBM could profit from expansion in model based development (MBD), offering significant reduced development time in any complex product. MBD is mostly deployed in automotive and aerospace engineering (GM, Boeing, Honeywell International, Lockheed Martin, Pratt and Whitney Industries). Increase in system reliability and robustness is related to the fact that system is modeled in beginning phases of the product development. Progress in this area simplifies model-based or data-driven prognosis and diagnosis into the system [47] as will be described later in this thesis. CBM requires continuous equipment monitoring and diagnosis as mentioned in previous paragraphs. Embedded systems used for monitoring range from simple 8-bit controllers up to high performance 32-bit multi-processors platforms including digital signal processors (DSP) and field programmable gate arrays (FPGA). 1.2 Prognosis Meaning in Thesis Context Prognosis is composition of Greek words pro and gnosis. Pro states for before and gnosis literally states for knowledge [80]. Prognostics is a wide range area trying to identify future behavior/state and we could see prognosis in finance, macro-economy, medicine, weather-forecasting, biology and of course in technical area machines, electronics etc. Vision of the future is one of the most attractive areas of interest for humans for a while. Unfortunately a possibility of precise future prediction is limited by huge uncertainties and wide range of external influences, which cannot be monitored or even are not known. More specific is the area of interest and more the external influences are minimized more precise the prediction could be. That is the reason why so many current prognostic applications are limited to only specific areas and to only specific number of conditions. We can see quite generic overview of forecasting applications with area of use in Figure 1-2. As could be seen from previous figure a prognosis is a wide area and only so called technical prognosis is the main and the only one area of our interest in this thesis. There are several definitions of technical prognosis term thus we will mention some of those most cited ones: Prognosis is an estimation of time to failure and risk for one or more existing and future failure modes [21]. Prognosis addressing the use of automated methods to detect and diagnose degradation of physical system performance, anticipate future failures and project the remaining life of physical systems in acceptable operating state before faults or unacceptable degradations of performance occurs [37]. Prognosis is the ability to predict accurately and precisely the remaining useful life of failing component and subsystem [80]. 17

18 What is common for all definition and is not exactly stated but will be considered in this work is that prognosis/prognostics is executed on component and sub-component level and includes predicting the time progression of a specific failure mode from it incipience to the time of component failure. Similarly to definition summary in [68]. Figure 1-2 Overview of categories of forecasting applications per [63] 1.3 Diagnostics and Prognostics Coexistence There is a long history of the technical diagnostics comparing to technical prognostics, which is quite new field of research interest. Prognostics and diagnostics are the key players in service planning, maintenance and in minimizing the down state of the equipment (aerospace is one of the critical area). Continuous increase of embedded system computation performance enables deployment of complex diagnostic algorithms in places, where it was not realistic several years ago. A huge number of data analyses are moved from specialized computation center directly into monitoring systems and enable us to evaluate conditions in real-time [84]. Diagnostics focuses on detection, isolation and identifies failure when they occur comparing to prognosis, which focuses to predict failure before they occur [2]. It means that technical prognostics could be understood as an extending/complementary element of technical diagnosis. We are able to determine not only the current state but we are able to predict future state with some relevance and level of probability based on the element and component degradation by using diagnosis and prognosis. The main goal of the technical prognosis is to make end of life (EOL) and remaining useful life (RUL) predictions that enable timely maintenance decision to be made [28]. Prognostics should be performed at the component of sub-component level and should involve predicting the time progression of a specific failure mode from its 18

19 incipience to the time of components failure. We can relation between diagnosis and prognosis in context of fault evolution at Figure 1-3. Figure 1-3 Relation between Diagnosis and Prognosis per [80] We can see in Figure 1-4 how blocks of diagnosis (Health Assessment) and prognosis (Prognostic Assessment) affect informational flow per ISO standard of condition based maintenance. Figure 1-4 Informational data flow per ISO Prognosis can be referred as the ability to predict how much time is left or remaining useful life (RUL) before a failure occurs given that an observed 19

20 machine condition variable and past operational profile. The observed condition can be attributed from physical characteristics or process performance to its failure. For instance vibration signature and oil analysis have been successfully used for monitoring the presence of failure in equipment. Other alternative condition parameters that can be used in prognostic are acoustic data, temperature, moisture, humidity, weather etc. [34]. 20

21 2 STATE OF THE ART 2.1 Current Research Topics Overview Technical prognosis, which is being considered as a part of the Prognostic Health Management (PHM) is quite new field of research and at the same time it is still considered as the weakest point in CBM processing chain. There are a lot of applications of prognostics method but the results and accuracy varies and are not always sufficient even researches claims so. Although a lot of patents have been registered and a lot of journal/conference papers have been published the area of technical prognosis is still quite new and not well researched, especially robust real system applications are still missing. Here are the main items of current research topics in area of technical prognosis: Metrics for RUL Estimation research area defining correct metrics, which enable us to compare different type of algorithms, methods and will help us in evaluation of prediction reliability. Definitions for prediction horizon, prediction confidence interval, algorithm performance accuracy, algorithm performance robustness, algorithm precision, algorithm performance trajectory, have been proposed [60], [63] and [64]. Prognostics Methods Classification there is a lot of different prognostics methods which are suitable for specific usage, depending on historical data availability, first principle model availability etc.. Researchers/Engineers who are not deeply familiar with could be lost in the number of different models and approaches [21], [66], [68]. Prognostics Frameworks there have been several prediction frameworks defining all steps needed for proper RUL prediction proposed during past years. All those frameworks are usually quite deployment area specific and are more use case studies. [3], [48]. Prognostics frameworks have been applied mainly to resolve following problems: Machinery/Materials (prognostics of flight actuators [11], crack growth monitoring in gearboxes of a helicopter [34]), Automotive where it is essential part of the OBD diagnostics. Electronics (electrical components failure, battery life prognosis [60], [85], [89] and [93]), Limited Sensing applications [28] applicable to all previous areas. All above mentioned items are just brief research topics overview and will be described in a detail within next sections. 2.2 Metrics for RUL Estimation A need for having metrics enabling us to compare different prognostics algorithm and helping us to evaluate prognostics results has been increasing last 21

22 years. This growing need is aligned to increased number of methods/prognostic framework. The paper [63] was quite important stimulus/progress on metrics evaluation. The main motivation to evaluate metrics was stated as in [63]: For end-of-life prediction of critical systems it becomes imperative to establish a fair amount of faith in the prognostic systems before incorporating their predictions into decision-making process. A maintainer needs to know how good the prognostic estimates are before he/she can optimize the maintenance schedule. Without any reasonable confidence bounds a prediction completely losses its significance. This statement is perfectly true since the main goal of prognostics is to provide the user/maintainer/technician relevant piece of information which could be appropriately assessed with making a right decision either to replace/fix components or sub-components. Remaining Useful Life (RUL) has been mentioned several times in previous text but this term has not been clearly defined even it could have different meaning across industries and thus could be interpreted differently. Here is the definition of RUL as will be used in this work: RUL is an amount of time before system health falls below a defined threshold i.e. the time interval beginning at a given instant of time and ending when the failure rate becomes unacceptable, or when the item is considered irreparable as a result of fault of for other relevant factors Definition composed from [18] and [64]. See example of RUL graphically represented on Li-ion Battery prognosis in Figure 2-1: Figure 2-1 Explanation of RUL on Li-Ion battery capacity example simulated case We can see that End of Life (EOL) is set once a battery capacity reached 80% of its initial capacity. Capacity percentage fade is considered as a threshold or as a failure. Such failure usually requires battery replacement (can be seen as a component replacement in general view). Threshold of 80% for battery capacity is quite commonly used because of significant operating performance decrease causing reduced operating time of equipment using the battery. Battery could be used next several weeks even with decreased capacity, thus it is clear that threshold level setting affects replacement costs, not always has to have technical value and threshold could be set by different users differently depending on user role in organization or user needs. Technician needs will be evidently different comparing to asset manager or financial manager who is 22

23 trying to keep minimal expenses. Consequently not only accuracy of RUL prediction but even the cost-benefits are important factors for evaluating performance metrics. As another example for understanding the RUL meaning could be occurrences of component failures in automotive industry, where tracking number of occurrences is used for reliability analysis but could be used at the same time for RUL prediction for specific cars and could be used in fleet management. Figure 2-2 Explanation of RUL on components failures in car simulated data RUL is estimated in discrete time manner and is usually computed with some period adequate to system dynamic and normal component/subcomponent life time. We get sequence of time series of RULs, which should track ground truth within predefined limit. See next Figure 2-3 describing graphically some of the most important attributes associated with RUL. Figure 2-3 Remaining useful life and associated attributes reproduced from [63] 23

24 Attributes like: mean, trajectory, prognostic horizon, confidential interval, computational demandingness, we define estimation tolerance and others are usually collected/calculated/monitored. It is obvious that in case we will use different predictions algorithms or different prognostics framework we will get different results. There has to be a provision or a way how to objectively compare algorithms to each other and to enable a user/organization to select appropriate one for demanded application. We have to consider not only accuracy/precision/robustness but even computational performance and cost benefit. There were several studies/papers proposing adequate metrics mainly [29], [63], [64] and [80]. Aerospace industry led in developing the metrics to evaluate prognostics algorithms, the main focus was accuracy and precision, but assessing business merits is getting more important as could be seen in recent years [80]. For example prognostics for electronics is not so well developed as prognostics for mechanical system, because electronics can profit from statistical reliability data, which are more precisely captured and maintained comparing to mechanical systems. Some of the current work in electronics prognostics emphasizes the potential cost savings provided and therefore relies on cost/benefit metrics such as life-cycle cost and mean time before failure MTBF [63]. Even a wide spectrum of prognostics metrics are used in domains like Aerospace, Electronics, Nuclear Power Stations, Automotive it still follows functional point of view as we can see Figure 2-4 representing basic functional classification of prognostics metrics: Figure 2-4 Classification of prognostic metrics as defined in [63] List of all most important metrics will be mentioned to provide a context and for later usage even a detailed list of all metrics could be found in above mentioned references. Description of each metric is described within the tables Table 2-1, Table 2-2, Table 2-3, Table 2-4, Table

25 Table 2-1 Accuracy based metrics collected from [80] Metric Name Definition Description Range Error defined the basic notion of deviation from desired output. Error is the ground truth, is, (2-1) the predicted RUL at time i; l PS = 0 represents l th Unit Under Tests (UUTs) Average Scale independent error Average Bias Mean Absolute Error (MAE) Root Mean Squared Percentage Error (RMSPE) 1 (2-2) "#$ $! %&' ' ( 1 (2-3) )% 1 +'),'% (2-4) (2-5) Weighs exponentially the errors in RUL predictions and averages over several UUTs; where D 0 is a normalizing constant, L represents number of UUT Averages the errors in predictions made at all subsequent times after prediction starts for the l th UUTs. It could be extended to average biases over all UUTs to establish overall bias Averages the absolute prediction error for multiple UUTs at the same prediction horizon Square root of the average of percentage error of the prediction from multiple UUTs. 0,1 PS = 1, PS = 0 *0, PS=0 *0, PS=0 Algorithmic performance can be measured by evaluating errors between the predicted and the actual RULs, other metrics use error to quantify other prediction characteristics such as statistical moments, robustness and convergence. Calculation of errors requires availability of ground truth data, which is quite rare case [64]. History data could be used instead in case those are available. Table 2-2 Precision based metrics reproduced from [64] Metric Name Definition Description Range Sample, 1 2 ) 0 Sample Standard deviation 3 measures the dispersion of the 4 1 error with respect to the sample *0, Standard where M is the sample mean mean of the error. This metrics is PS=0 Deviation of the error restricted to assumption of (2-6) normal distribution error. Mean Absolute Deviation (MAD) ) 3 where M=5674 and median is the 283 9: 0 ) 5674 ) (2-7) This is a resistant estimator of the dispersion of the prediction error. It is intended to be used in case small number of UTT. *0, PS=0 25

26 Robustness metrics deals mainly with prognostics sensitivity and utilize socalled reliability diagram. The profit of having robustness metrics is to understand how specific/single solution is our prognostics algorithm and how it could be affected by different set of input system data. We can recognize either if our system permanently over-estimate or under-estimate, which can help even in algorithms/methods fine tuning. Table 2-3 Robustness Based Metrics [63] and [80] Metric Name Definition Description Range, 1 Measures how sensitive prognostic algorithm is to input Sensitivity. ). *0, 2;<= changes or external disturbances. 3 PS=0 (2-8) Can be assessed against any performance metric of interest Reliability diagram Brier Score A > 1 The reliability diagram plots the observed frequency against the predicted probability of a random event. The deviation from diagonal gives the conditional bias. If the curve is below the line this indicates over-forecasting, if it is above it indicates under-forecasting (probabilities too low) *0 1 PS=0? (2-9) Assessing computational performance is not so critical in case of off-board system based on regular PC platform, where CPU with high computation power is available for computations but is getting more critical in case of real-time prognostics system, which will be usually embedded system with limited HW resources (usually limited size of NVM, CPU power). There is a specific domain in informatics dealing with comparing computational demandigness and usually profilers are used for this purpose. See Table 2-4 for some metrics which could be applied in case comparing computational performance. Table 2-4 Computation Prognostics Metrics (mainly embedded application) - author s proposal Metric Name Definition/Units Description Execution Time RAM Usage ms/seconds, Number of Instructions. ET = Algorithms(N), where N is number of inputs kbytes, Mbytes Collect this metric could be quite difficult in case of platform independent languages and could be specific for compiler settings and CPU usage. Execution time could have significant impact in case we acquire data and we need to perform real-time processing Size of hosted system RAM needed for algorithm execution. Usually could be in range of KBytes up to Mbytes. Some algorithms could require huge buffers. 26

27 NVM Usage Algorithm Program Size kbytes, Mbytes kbytes, Mbytes Size of Non-Volatile memory needed for memory storage (EEPROM, FLASH), This memory is usually needed for storing long term data, to perform back-up in case of power-down event. Even in case run-time RAM usage will be low the program size still could be huge in e.g. highly optimized switches. Cost/Benefit metrics are key driving aspects in considering deployment of any particular prognostics approach and framework even it is not often consider in technical discipline. Only safety-oriented application like aerospace or nuclear power plants could ignore Return on Investments (ROI) less than 1 because those fields are oriented more towards reliability. In Table 2-5 the important metrics are mentioned. Table 2-5 Cost/Benefit Prognostics Metrics reproduced from [29] and [63] Metric Name Definition Description Metric measures the ratio between how long component lasts and how long it is used MTBF/MTBUR (mean time before replacing it. Prognostics should MTBF/MTB between failure/meantime enable to prolong component usage until UR ratio between unit replacement) near future failure occurrence. This should save money and is related to condition based maintenance approach Life Cycle Cost Return on Investments (ROI) Technical Value (TV) Acquisition cost + operations cost Gain/Investment ' B..C ( D.E 1 ' F. ' G.H ( ' I.J (2-10) As a metric compare the life cycle cost of the system (which includes the cost of building it or acquiring it and the cost of operating it) with and without prognostics. An investment in prognostics is expected to save money on a maintenance and possibly prevention of downtime or lost hardware over the life of the system. The gain is amount of money saved as a result of using prognostics (reduction in life-cycle cost) and the investment is the cost of developing, installing and maintaining the prognostics system. The benefits achieved through accurate detection, fault isolation and prediction of critical failure modes and weighed against the costs associated with false alarms. ' B - probability of failure mode D - overall detection confidence metric C savings realized by fault detection in advance l overall isolation confidence metric E - savings realized by isolation a fault in advance ' G false positive detection metric H - cost of false positive detection ' I false positive isolation metric J - cost of false positive isolation All metrics mentioned above reflects current status quo and metrics, which are used in other disciplines (diagnostics, metrology). Problem is that those are 27

28 not well spread and commonly used in researches related to technical prognostics and only a small number of research papers use at least some metrics but those are usually limited to error estimation. Independently on the lack of having valuable metrics in current research a new additional metrics has been proposed in [64]. Some of those metrics are completely new and the main reason for introducing them was fact that some of older accuracy metrics make assumptions about the probability distribution of the error and use standard estimators like mean and variance, which requires always knowledge of ground truth either from measured data or from historic data. See Table 2-6 for some of newly proposed prognostics metrics. Table 2-6 New Prognostics Metrics Proposed in [64] Metric Name Definition Description Range Prognostic PH EoL j Where: Q 54 %@ C T T %@ C! is the time index when predictions satisfy α- bound; is the set of all time indexes when a prediction is *0, Horizon (2-11) made; D is the index for D FU unit under test (UUT); is the ground truth, is the predicted RUL at time i; EoL is the ground truth End-Of-Life (actual failure). PS=0 Data Frame Size Horizon/ Precision Ratio DFS n; n 1, HPR \ Pi PHi Indicates how many consecutive sets of feature values data frame size are required to be known at any given time for the algorithm to function within the constraint set on nominal performance. Metrics calculates the ratio of the precision over the horizon. It quantifies the spread as function of distance to EOL. Prediction accuracy at specific time instances, C - accuracy modifier; _- window modifier. *0, PS=ε 0 α-λ Performance *1 C. 9 T 9 T *1 ( C. 9 where, 9^ ' ( _%& ' (2-12) True/ False 28

29 Metric Name Definition Description Range Relative Accuracy + 1 `a F b cad F b `, a F b where, 9^ ' ( _%& ' (2-13) *0,1 PS=1 RUL estimate is a sequence of predictions till the EoL is reached for each particular RUL(i) estimate in time index i. Could be represented either graphically for better user understanding. The most important statistical attributes (mean, variance) are aligned to RUL. See graphical visualization methods proposed in [60]. Table 2-7 Visualization Methods for RUL estimation reproduced from [60] Normal Distribution Mixture Of Gaussians Location (central tendency) Spread (variability) Visualization Mean (µ) Sample standard deviation (σ), IQR (inter quantile range) Confidence Interval (CI), box plot with mean Means - µ 1, µ 2,,.. µ n Weights - ω 1, ω 2,,.. ω n Sample standard deviation - σ 1, σ 2,,.. σ n Multiple CIs with varying box plots As has been clearly described metrics are important part of the technical prognostics and we can expect that those will be used in broader range and will be part of each prognostics frameworks analysis. The problem is in the number of different metrics and not all of those will be always applicable thus the reduced set should be proposed in near future for specific technical areas. Of course metrics should be formalized and should become part of international standards. RUL definition and the problem related to RUL estimation are clearly identified in this section and main current research topics are described. 2.3 Prognostic Methods Classification RUL and its attributes are the outcome of the prognostics and are used in prognostic assessment by applying appropriate metrics and additional criterions. 29

30 There is a wide range of methods dealing with RUL computation and calculation. A significant amount of research has been undertaken to develop prognostics models over a recent years. By design, models are subject to specific assumption and approximations, some of which are mathematical, while others relate to practical implementation issues such as the amount of data required to validate and verify a proposed models as stated in [68]. Selection of an appropriate method is predominant to success in condition based program deployment and is related to already mentioned return on investment (ROI) attribute. Adequate model selection requires of course mathematical understanding of each model type and its basic pros and cons. Research activity in this area (like [6], [66], [68]) deals with managing and organizing prognostics methods, mainly for business purposes. Even we can consider methods grouping as a background activity it is still valid effort. Three classifications will be mentioned for clarity and to provide context of problematic. Official ISO could be mentioned as reference even it is not commonly used in papers/journals, because of its granularity, when grouping prognostics methods into twelve different types (Mathematical/Life usage models, Behavioral, Statistical, Probabilistic, Artificial Neural Networks, Life expectancy models, Reliability based, deterioration based, knowledge based models, rule based, causal tree models (ISO 13379), case-based reasoning) see [21] for details and overview in Figure 2-5. Figure 2-5 Taxonomy of prognostics methods according to [21] Even it is somehow regulatory classification approach a different grouping is used in praxis. Classification seems to converge into less granular approach consisting of three basic groups; model-based prognostics, data-driven prognostics and experience-based prognostics [34], [46] and [65]. We can see prognostic taxonomy proposed in [46], [48], [80] in Figure 2-6. Taxonomy of prognostics method inherited from [80] will be used in thesis as a default one. 30

31 Figure 2-6 Taxonomy of prognostic methods proposed in [80] and reviewed in [3] Another classification approach has been proposed in [68] and it has to be mentioned in thesis just to demonstrate how complex is the problem of prognostics method classification and to provide complete list. Figure 2-7 Taxonomy of Prognostics methods as proposed in [68] with its sublevels Of course part of the classification a detailed methods description need to be provided - see next section where classification proposed in [80] is described in detail including description of real application and proposed frameworks either model-based, data-driven or probability based. In a summary this area of research is still open and will not be closed soon because of different industrial views and needs. Whenever new frameworks and prognostics approaches will be provided all of those need to be classified. Independently how solid the grouping it still, there is a lack of basic grouping either methods are online/real-time applicable or not. 31

32 2.4 Prognostics Methods and Applied Frameworks Including Use Cases Each of the prognostics methods and approached independently on their classification has its own pros and cons and sometimes the hybrid methodology is used, which profits from pros of all methods. It is quite common that prognostics framework is part of the diagnostics framework and cannot be always isolated. Several prognostic frameworks has been developed and described but before the analysis is performed we need to describe each prognostic approach in detail including its state of the art review Data Driven Prognostics A data-driven approaches use the ordinarily observed operating data (power, vibration and acoustic signals, temperature, pressure, oil debris, currents, voltages, calorimetric data, frequency response) to track, approximate and forecast the system degradation behavior [48]. Measured input/output data is the major source for getting a better understanding of the system degradation behavior. The data-driven (DD) approaches rely on assumption that the statistical data are relatively unchanged unless a failure occurs in the system. The common cause variations are entirely due to uncertainties and random noise and special cause variations (e.g. due to degradations) account for not attributed to common cause [66] The data driven prognosis is based on statistical and learning techniques from the theory of pattern recognition. These range from multivariate statistical methods (static and dynamic principle component, linear and quadratic discriminants, partial least squares and canonical variance analysis) to black-box methods based on artificial neural networks (probabilistic neural networks, multi-layer perceptrons, radial basis functions), graphical models (Bayesian networks, hidden Markov model), self-organizing feature maps, signal analysis (filters, auto-regressive models, FFT etc.) decisions trees) and fuzzy rule based systems [46]. Most of the work in data-driven prognostics has been for structural prognostics. Many of those systems use vibration sensors to monitor the health of rotating machinery such as helicopter gearboxes. Some systems monitor the exhaust gases or the oil stream from the engine for contamination that could indicate a fault [66]. As a particular example we can mention applying the dynamic neural networks (DNN) on accurate jet engine life prediction [53]. New NNRAX (Neural Network Auto Regressive Model) has been developed, that enables authors to compute the stresses and temperatures at critical locations of gas turbine, in orders of less computation time than required by more detailed thermal and stress non-linear models. Real engine flight data are used as an input data set for the neural network training needs. Authors demonstrated model reduction technique for computing critical component parameters for RUL. Dynamic neural network model reduces the original thermal model of a turbo-machinery component and the temperatures could be computed on the fly if needed. The results show that such data driven prognostic techniques can be applied with minimal error in RUL estimation while taking into account the actual operating conditions [53]. 32

33 Dynamic Wavelet Neural Network (DWNN) utilization and RUL estimation of bearings could be mentioned as another example of the current research in this area. Neural network was trained by using vibrations signals from the damaged bearings with different level and signs of wear. This approach seems to be accurate enough for the diagnostic and prognostic purposes [83]. See Figure 2-8 for graphical representation. Figure 2-8 Overall architecture of the WNN prognostic system introduced in [83] The ability to transform and to reduce large amount of noisy data into smaller amount of valid and meaningful data set is the big advantage of the data driven approaches. The big disadvantage is the dependency on quality and quantity of operating data, which is driving key element of the prognostic accuracy and reliability. Sometimes there is a problem especially in aerospace area that we are missing faulty data. In summary, the data driven approaches are preferred in case large amount of run-to failure data set is available in required operational range and in case system models are not available (e.g. model is not known, too complex or not shared because of intellectual property) Model Based Prognostics Model-based approaches or so called physics-based are applicable, when relatively accurate mathematical model could be developed from first principle of system s failure modes [1] and [73]. Model is often represented by differential equation in a form: ef9 ge9 ( hij ( k9 l9 me9 ( ni9 ( o9 (2-14) where e p 2 is vector of internal states connected with fast/slow dynamic changes, i p 2 is vector of system inputs,; l p q is vector of outputs it is usually measurable quantity and thus called as a measurement vector, ω is input noise, v is measurement noise; C represents relational between internal states and outputs, D is matrix of direct relations between inputs and outputs. Models could be classified as a qualitative or quantitative (alternatively called parametric/non-parametric). The quantitative model represents 33

34 mathematical and functional relationship between the inputs and outputs of a system, while the qualitative models represent these relationships in terms of qualitative functions centered on different units in the system [80]. The qualitative models can be developed either as qualitative causal models or abstraction hierarchies, such as diagraphs based causal models and fault trees, but quantitative models are used in most cases [49]. Figure 2-9 Model-Based Approach to Damage Identification reproduced from [11] Model based approaches are based on analytical redundancy. A process contains analytical redundancy if an input or output can be calculated by using only other inputs or outputs [46]. The analytical redundancy is utilized by comparing the outputs from the real process and outputs from a process model, which is fed by the same inputs as the real process in the simplest case. Inconsistencies between the model and the real process are represented as residuals. In case of no fault the residual should be close to zero (considering the model accuracy, signal noise etc.) and in the case of a fault the residual should be significantly non zero if it is sensitive to that particular fault. A number of residuals are used and they are made sensitive to different faults to achieve the fault isolation [51]. There is limited number of real application in this area and it could be considered as a most complex and accurate approach. Use cases defining model based approach has been created for suspension system in [48], where simple state-space model was defined and degradation measurement was involved as a slowly changing feature. Degradation measurements was connected to potential crack in suspension system and based on the system load (Palmer-Miner Law). Simulations proving this approach were performed. New term called degradation measurement has been introduced in [47] and system model is represented as: ef ve,tr,w, rf xye,r, l me ( ni ( o 34 (2-15) where, r p s is a vector of states related to slow dynamics changes, respectively represents vector of system degradation, t p u is parametric

35 vector of function r, 0 z x 1 is constant representing time separation between dynamic events and slow drifts. Similar model-based approach has been applied in diagnosis/prognosis of wheeled mobile robots, where six subsystems were under diagnostics/prognostics inspection and monitoring: power supply subsystem, driving subsystem, steering subsystem, suspensions, communication, sensors [49]. Quantitative physics - based model were developed for each subsystem and Kalman filtering methods were used for model parameter estimation. Another use case was demonstrated on battery health prognosis, where Li- Ion battery degradation model was developed and based on changes in internal resistances (features), received from electro-impedance spectroscopy (EIS) the degradation in battery capacity e.g. RUL was estimated with acceptable accuracy (5-10%). Features (internal battery resistance) from EIS were extracted by using Relevance Vector Machine (RVM) approach. Technique for estimation was based on Bayesian network and particularly on particle filter approach [60]. This approach seems to be more promising especially because dealing with measurement noise and is able to track internal non-measureable changes. Figure 2-10 Particle Filtering Framework for Battery Health Prognosis reproduced from [60] Impact Technologies has developed a diagnostic and prognostic framework for flight actuators. This model-based approach to prognostics and health management (PHM) deploys physical modeling and advanced parametric identification techniques, along with fault detection and failure prediction algorithms, in order to predict the time-to-failure for each of the critical, competitive failure modes within the system. This approach for condition-based maintenance seems to overcome black-box based health-monitoring [11]. The big advantage of model-based approach is the possibility to take into account the physical knowledge of the system into the monitoring process, in other wording it means that we can reduce the amount of sensed parameters or we could determine some parameters directly from a model we can limit number of real sensors and replace those by virtual ones. Model - based prognostics under limited sensing was researched in [23]. Model adaptation to system degradation is another advantage of these methods, because it helps to keep the prognostics accuracy at demanded level. 35

36 One of the disadvantages could be quite computational exhaustive model implementation/developing into embedded systems for online/real-time deployment. Figure 2-11 A model based approach to prognostics and health management for flight actuators reproduced from [11] The simplest form of prognostic RUL prediction is based on trend analysis of a single monotonic parameter correlated with remaining life. This parameter may have been originated from single sensor of from a number of sensors aggregated into a single variable. This one parameter is than plotted as a function of time [68] Probability Based Prognostics Determining a precise dynamic model in terms of differential equations that relates the inputs and outputs of the system being may be impractical or even nearly impossible [80]. Statistical/Probability models estimate the damage initiation and progression based on previous inspections results on similar machines. Forecasting of future deterioration is often undertaken by comparing these results from with models representing good behavior [68]. Probability based methods have the longest history, comparing to other previous approaches does not require too much detailed data and utilize different kinds of probability distribution functions - PDFs, which were parameterized for individual systems / subsystems / components based on production parameters, operational data, statistical data from history. The most commonly used distribution functions are normal, Weibull and exponential distribution. The probability density function of a Weibull random variable x is [72]: 36

37 ;_,E} ~^ =^?c3 c b ~, 0ƒ (2-16) 0,, z 0 where β > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential function. A typical distribution describing the failure rate versus time is called a "bathtub curve", which was first time published in 1965 and still has its merit [88]. The prognostics approach also provides confidence level in which we operate and we can rely on. It is important for determining the probability and accuracy of our estimate. Probability-based prognostic approach is still most common and is very often applied in the electrical industry. This research area is well described. Figure 2-12 Bathtub Curve derived from Weibull distribution adopted from [88] Hybrid approaches and Intelligent Maintenance Frameworks Hybrid approach usually combines the data-driven approach with one or more of the other approaches and it possibly offers a more reliable and accurate prognostic results per [66]. By using the data-driven approach, a framework for implementing of an intelligent maintenance prognosis tool was introduced in [3]. The framework utilizes the existing equipment operating performance data in the industry for prognosis process. Consequently the framework combines the ability of prognosis in estimating remaining useful life (RUL) of equipment with the maintenance action knowledge to generate a well-received maintenance plan [3]. See Figure The first step of the framework is data acquisition process. The process is important to obtain the observed condition data. Once the observed performance dataset stated that there is a fault based on the level unacceptable control limit of behavior threshold model, next, the research looks at the prognosis process stage. Since the complete historical failure data is available in the existing database, the prognosis model parameter can easily be estimated [3]. 37

38 Figure 2-13 A framework of intelligent prognosis combining several prognostics methods proposed in [3] Summary Three basic approaches of prognostics were described. All approaches utilize similar mathematical apparatus and for example border between datadriven and model-based approach could be quite fuzzy. Applicability of approaches could vary and usually probability based approach has the biggest application probability, but the accuracy is not always good enough. Model based prognostics is quite specific for area of its deployment and approach cannot be deployed for another system even the development cost could be quite high, but on the other hand the prediction accuracy overcomes other methods. Model-based approach could profit of emerging model-based development, where number of physics based model is increasing. Availability of tools enabling mathematical model conversion to C/C++ language and consequent targeting to embedded platform is another key factor why model based prognostics is getting into foreground in recent years. There is a progress in hardware in the loop simulation, which again together with model based development could provide new area of deployment of model-based prognostics. See comparison of each prognostic approaches in Figure

39 Figure 2-14 Graphical Comparison of prognostic methods adopted from [80] 39

40 3 RESEARCH TOPICS 3.1 Comparison of Prognostics Methods from Embedded System Point of view Questions related to this topic which should be answered in thesis: What are the conditions when to select model-based approach? What are the most promising methods for embedded systems? How to deploy developed prognostic method into embedded system? As has been described in previous sections there are several methods and approaches for prognostics and RUL estimation. This research topic should find the most appropriate techniques for embedded system representing an online onboard system or at least identify steps, which need to be done to properly select the right technique/method. This research area should focus on prognostics only and should not deal with diagnosis, which is much more developed. The aim of this research part is to provide list of applicable methods for embedded system with simple guide which method and when need to be selected. 3.2 Prognostics Utilizing Particle Filters Questions related to this topic which should be answered in thesis: What are the different possible implementations of particle filters? What are the performance differences for each of those? How could be particle filters used for n step prediction? Model-based approaches and some of hybrid approaches utilize Bayesian framework, which especially particle filters can help with system non-linearity. The main aim of this thesis area is to fully identify all features and different modifications of particle filters as a one of the most promising method in technical prognostics. The outcome should provide clear overview of performance details and list of all potential roadblocks in utilizing particle filters in model based prognostics. 3.3 Battery Life Prognostics - Use Case Questions related to this topic which should be answered in thesis: 40

41 What are the current battery life prognostics approaches? What are the pros and cons of model-based approach and simple statistics/trending approach? How should look optimal battery health prognostics framework? A demonstration of real case where prognostics could be applied is the main motivation of this research topic. It is interesting that even so many products rely on battery source, the battery health monitoring and prognostics is still under exploration and it is still vivid area of research and interest. A brief overview of state of the art of lithium-ion battery health monitoring techniques (State-of-life estimation, state-of-capacity estimation, run-time estimation, internal resistance estimation, capacity fade estimation) will be provided. Verification and utilization of the dynamic lithium-ion model in estimation of the remaining useful life (RUL) is discussed as well. This research part of thesis aims to provide a novel framework utilizing battery embedded monitoring and continues RUL estimation both discharging runtime and capacity fade. This Use Case should be main research goal and should clearly demonstrate how particular methods could be used in real world and how complex could be the prognosis. 41

42 4 COMPARISON OF PROGNOSTIC METHODS WITH FOCUS ON EMBEDDED SYSTEMS The main aim of this research topic is to provide prognostics methods comparison from embedded system point of view and to provide guide for proper method selection. Diagnostics and prognostics could be performed either on off-board system as an offline process or onboard as an online process providing near real-time. Both have its own pros and cons and in some cases only a hybrid approach is suitable. Performing diagnostics/prognostics only on off-board system could be done only in case of slow fault propagation since we cannot react immediately and usually degradation measurement requires offline measurement done per inspect intervals (typically oil debris in engine, structure health analysis utilizing roentgen scan etc.). This approach is against CBM, which philosophy automatically prefers online permanent monitoring utilizing onboard/embedded systems. Of course embedded system could be used only for data collection and the rest of processing chain (health assessment, prognostics assessment) could be done online/offline for different assets/systems. This is approach which could be helpful in case huge number of data is available, huge computational power is needed and automated way of data collection (automated data connectivity) is provided not to rely on manual data download. See graphical representation of system with offline prognosis. Figure 4-1 Onboard monitoring with offline prognostics 42

43 On the other hand embedded systems enable us not to only monitor all relevant signals but even track and evaluate all changes/degradation. The potential roadblock of moving prognostics and diagnostics to embedded system could be limited hardware resources like RAM memory, non-volatile memory, CPU power, OS and additional software. Those limitations are getting less significant nowadays, when there is nearly no difference in embedded system computation power and regular laptop/pc. Some embedded systems utilize even PC architecture and could be called embedded PC with running regular of lightweight version of OS (Embedded Linux, Embedded Windows). Having online prognostics significantly simplifies maintenance and provides always specific prognostics data for particular assets (cars, aircrafts etc.) Real deployment target could be important in selection of proper prognostics method besides other prognostics metric mentioned in section 2.2. There will not be only one method or one approach to select for project/system needs. Before any selection guide will be proposed all possible approaches with its detailed pros and cons need to be listed/compared and selection criteria need to be identified. Figure 4-2 Graphical explanation of embedded online prognostics 4.1 Detailed Methods List with Pros and Cons This section could be considered as continuation of Section 2.4, where brief prognostics methods overview is introduced including real application and where methods are classified upon data/model availability criteria. See next overview with detailed list of methods and its advantages and disadvantages. It would make sense to provide comparison of methods utilizing metrics introduced in 43

44 section 2.2 but such comparison could be performed only in specific real deployment and would be out of scope of this thesis Probability Based Methods Probability Based Methods (summary and continuation from ): Aggregate Reliability Functions: Reliability analysis of aggregated failure data is one of the most common methods. This approach involves analyzing times to failure of a population of equipments/assets and determining probability density functions [70]. It provides information when failures are typically expected to occur. Various distributions are used Normal, Exponential, Weibull, Gamma (typically used for component life testing), Poisson, Rayilegh, Student etc.. RUL Probability Density Functions: one of the simplest Bayesian approaches it is an extension to aggregate reliability analysis. Prediction for new state could be improved by more advanced state estimation techniques like Kalman and Particle filters. The accuracy and precision of remaining life estimation improves with approached ground truth of EOL. [68] Static Bayesian Network: Bayesian network are probabilistic graphical models that represents set of random variables and their probabilistic independence. Sometimes those methods are called as Bayesian Belief Network or Causal Probabilistic methods [23]. Semi-Markov Models: Markov models assume that a system of component can be in only one of a finite number of states. Steady states probabilities are one of the most preferred. These methods are suitable for one step failure prediction and for diagnostics Hidden Markov Models: There are several types of Markov model. The most common is ergodic model, in which every state can be reached from every other states in a finite number of steps. These methods are computationally extensive and could be used only for known faults [68] Table 4-1 Probability Based Methods Pros and Cons composed from [60], [68] and [80] Method Advantages Disadvantages Online Aggregate Reliability functions RUL PDF Confidence limits are available for RUL prediction Could be used in case sample size is statistically significant and representative Software toolboxes/packages available Simple and easy adaptation of basic reliability approaches Confidence limits are available for RUL prediction Accuracy and precision 44 Failure must be statistically independent and identically distributed We need statistically significant sample size linking each failure mode to RUL predictions Accuracy and precision is dependant of forecasting interval Need statistically important sample size We should avoid this No No

45 Method Advantages Disadvantages Online increases as RUL decreases resulting in ability to set useful warning limits Should be consider in case condition monitoring data are not available method in case past operating conditions are not representative for current environment and usage Do not provide high level of accuracy and precision in case long prediction horizon. Static Bayesian Network Suitable in case incomplete data sets, captures and integrates expert knowledge Computer software modeling available Confidence limits are provided Computational difficulty Results sensitive to selection of prior knowledge Modeling experts required in addition to domain knowledge No Markov, Semi- Markov models Hidden Markov, semi-markov models Can handle incomplete data sets Well established approach and able to model numerous system design Computationally efficient once developed Provide confidence limits as part of the their RUL prediction Can model different stages of degradation so failure trend does not need to be monotonic Can model temporal data Can handle incomplete data sets Large volume of data required for training Assumes single monotonic, non temporal failure degradation Results may be sensitive to selection of prior distribution Not appropriate for repairable systems Large volume of data required for training Computationally extensive No No Data Driven Methods Data Driven Methods (summary and continuation from 2.4.1): Trend Extrapolation: Is the simplest technique used for RUL prediction. It is based on simple trend analysis of a single monotonic parameter correlated with remaining life. Parameter could originate from single sensor data or could be calculated value. Parameter is considered and plot as a function of time. Trend equations could be different but usually it is linear trend, polynomial, exponential and different regression techniques are used like least squares etc. Neural Networks: Neural networks are popular approach for solving problems at black box level. Neural networks have different structure for example - feedforward layout, perceptrons, wavelet neural networks, and others. Important part of usage NN is its training on sample data, which could be complicated and not only 45

46 those data are available. Neural networks are popular in forecasting applications as an N step predictor. Dynamic Wavelet Neural Networks: The wavelet neural network is a static model in the sense it provides static relation between it inputs and outputs. All signals flow in forward direction with this configuration. The predictor uses the discrete wavelet transform and recurrent neural networks [66]. Fuzzy Systems: Fuzzy systems implements rule base and algorithms for applying logic. Fuzzy logic is suitable in case mathematical model is not available. Deployment of fuzzy system for RUL prediction is limited Table 4-2 Data-Driven Methods Pros and Cons composed from [60], [68] and [80] Method Advantages Disadvantages Online Trend Extrapolation RUL Forecasting with Artificial neural networks DWWN (Dynamic Wavelet Neural Network) Fuzzy systems Simplest technique to apply and explain Easy to set alarms No special software tools needed Complex, multi dimensional, non-linear systems can be modeled Physical understanding of the system does not required Dynamic Neural Networks seems to be promising Physical understanding of the system does not required Input vector could consist of different type of signals even mixed time signals and frequency spectra Successfully deployed for bearing crack size prediction problem Inputs can be imprecise, noisy or incomplete Confidence limits could be provided in some types of models Could be used in case mathematical model is not available or not feasible to implement Few failures have a well defined monotonic, singleparameter trend Not handling variations in operating conditions Definition of proper trend function requires huge number of samples Requires significant amount of data for training data that needs to be representative of true data range and its variability Most networks cannot provide confidence limits Determining to proper neural network is a trial and could be time consuming Requires significant amount of data for training data that needs to be representative of true data range and its variability Setting proper parameters is not trivial Domain experts required to develop rules Better for diagnostics Yes Yes Yes Yes 46

47 4.1.3 Model Based Methods Model Based Methods (summary and continuation from 2.4.2) are one of the most complex approaches to prognostics and accuracy should significantly overcome other methods since first principle of fault generation and propagation is introduced. Development of model could not be easy and requires several trials and approximating using empirical data. If we are mentioning model its definition could vary and we can consider even simple degradation model defining only several internal states with slow dynamic as a model. System model does not always require complete physical description and definition all internal system states represented by differential equation. Model parameter could not be always precisely defined and thus estimation of parameters is required as a part of model identification. See list of parameter estimation technique: ARMA Models: the ARMA model is a tool for understanding and, perhaps, predicting future values in this series. The model consists of two parts, an autoregressive (AR) part and a moving average (MA) part. The model is usually then referred to as the ARMA(p,q) model where p is the order of the autoregressive part and q is the order of the moving average part (as defined below). Sometimes it is called as Box-Jenkins model [86]. ARMA and ARMAX models can remove temporal trends they should be used only for stationary data. Artificial Neural Networks (ANN): Estimates parameter as black box and do not incorporate system model. Suitable in case of large models and huge number of state variables. Kalman Estimation: Is one method inherited from Bayesian network and is quite commonly used in control theory. It is applicable only for linear system with Gaussian noise. Some non-linear variant exists. See detailed description in [7] and [85]. Estimation based on Particle Filter: this is one of the most promising methods inherited from Bayesian network. Particle filter are suitable for non-linear system and in case non-gaussian noise occurs. Particle filters are used in computer graphics because of it tracking capability and ability of prediction. System model need to be always defined. See more details in [2] or in section 5. Table 4-3 Model Based Parameter Identification Methods Pros and Cons composed from [60] and [68] Method Advantages Disadvantages Online ARMA models Simple and easy adaptation of basic reliability approaches Confidence limits are available for RUL prediction Accuracy and precision increases as RUL decreases resulting in ability to set useful warning limits Consider only in case shortterm prediction needed Accuracy and precision is dependant of forecasting interval Need statistically important sample size Avoid in case long term prediction required Yes 47

48 Method Advantages Disadvantages Online Parameter Estimated by ANN Parameter Estimated by Kalman Filter Parameter Estimated by Particle Filters Confidence limits available from underlying model (with estimated parameters) Can be used to model multivariate, dynamic process Basic KF is computationally efficient particularly with systems with a large number of states Can be used to model multivariate, dynamic process Noise does not need to be either linear of Gaussian More accurate than Kalman filter variant for non-linear systems Parameters are estimated as black box with minimal first-principle knowledge Process and measurement noise must be Gaussian Measurement data required A large number of particles are required to avoid degeneracy problem Can be more computationally intensive than Kalman filters All methods/algorithms/approaches have a lot of different implementation and different aspect of deployment. Even methods description is quite brief it is still at the level, which should provide enough background information needed to proper understanding of their advantages and disadvantages. Summary and prognostic guide will be proposed in next section. Yes Yes Yes 4.2 Guide for Proper Method Selection As all potential methods have been layout we can defines steps to select the right and appropriate method. As a first step of prognostics selection should be identification a requirement for precision and accuracy. In case high precision and accuracy is required it always leads to selection of either model-based or data-driven approach. Second important step which need to be answered is what are the return on investments (ROI) cost in case we have no system model, and we are limited by ROI it would be recommended to select data-driven approach, where will not be high cost in model development and verification - of course this is limited by case when we have enough training data - in other wording our product already exists and is deployed and valid failures already occurs.. Criteria of ROI and Accuracy need to be always balanced and for some applications it could be opposite way. See Figure 4-3 for detailed description of each step in appropriate method selection. It is obvious that model-based approach is the most precise and accurate, because it follows physics on failure and thus helps precisely identify the problem. Model could be developed already in phase of design and could be actually used even for system control or for system diagnosis. 48

49 Figure 4-3 Guide for High Level Prognostics Approach Selection In case model based prognostics process has been selected, which should be preferred because of its accuracy, the first step will be system model identification. We should consider number of system states, type of system load, what is the process noise either it is Gaussian or non-gaussian. In a next step we should test system under different type of loads and conditions and should verify how precisely we are able to track the system degradation. Feature estimation is next step which is important to identify/estimate internal variables/states, which are not directly measureable. It could be a degradation measure or wear parameter hidden in state variable. If estimation process is set properly we can start tracking of degradation. Tracking in this case means proper estimation for several successive steps. Once tracking process is set-up we can proceed to predicting of RUL with confidence interval. The prediction will be as much accurate as accurate will be estimate of system load. In this case we can excite our system by deterministic operational sequence, probabilistic operational sequence or online sequence estimation [46]. The remaining useful life depends on the current damage state/wear parameter as well as on the future system usage. If the operation of a system is known a priori, the remaining life could be estimated using this known/estimated knowledge, which were already enumerated above and are detailed below according to [46]: Deterministic Operational Sequence: In this case we assume that system will be operated under known sequence of mode and mode durations. (Like in case of aircraft: taxi, takeoff, flight, and landing). In this case we can define sequence 5,, <! 3, where is sequence start, < is sequence end time, mode 5 represent operational sequence mode defined by specific operations condition, N represent number of different possible sequences. 49

50 Probabilistic Operational Sequence: this is the case where we expect that the system operates under J operational sequences, 5,, ˆ <! 3, where, could occur with known probability p Sj. If is the estimate of RUL based on ˆ sequence, then the overall RUL is given by 3 Š. Online Sequence Estimation: This approach estimates the operational sequence based on measured data. We could use particle filters for this purposes or so called IMM (Interactive Multiple Models) based on Monte-Carlo simulation as demonstrated in [47]. We can use either probability density function for estimation of the sequence or system load. In case of electrical circuit it could be for example PDF of current load, temperature load etc. Final step should always be prognostics metrics evaluation at least following should be always provided: RPMSE (Accuracy) Sample Standard Deviation SSD (Precision) α-λ Performance See Figure 4-4 representing steps of model based-development prognostics, which is novel proposal utilizing approaches drawn in [23] and [48]. Figure 4-4 Model-Based Prognostics Process Steps Proposal Enhancing [23] and [48] 50

51 4.3 How to Deploy Developed Prognostics Method into Embedded System Even methods and algorithms are mentioned, most of them will be developed in special environment either in most common MATLAB/Simulink or in LabView, which both are perfect tools for analysis, algorithms evaluation and simple prototyping. Once algorithms are defined and basically verified in simulated conditions it is time to verify it on real HW platform. Code could be generated into C/C++ code, which is common language for embedded system because of its high efficiency and support of different compilers and platforms. There are options for targeting floating- or fixed-point processors, code could be generated either from MATLAB and Simulink, could be optimized for specific processor architectures, could be integrated with hand-written software. Another important support is profiling and verifying embedded code directly on microntrollers [77]. This code generation approach is really popular nowadays and a lot of technological company prefers automatic code generation (see Figure 4-5 for development steps), because is simplifies product delivery and enable engineers to focus on system level work and on final integration and validation and it is actually part of the model based design, mentioned couple times before. Figure 4-5 Prognostics Algorithms Development flow in context of model based development It used to be quite complicated to develop the prognostics algorithms in C language with limited way for verification and testing. With current tools support, the developers and system engineers could focus on functionality and the process of coding and targeting is significantly simplified, consequently the effort/cost of algorithms development is decreasing. It is significant especially for MATLAB environment, which becomes industry standard for model based development, even there is a lack in instrumentation support, the algorithms library and support of toolboxes overcomes other software tools. MATLAB enables to generate code not only to C/C++ but even to Java and VHDL. More details of code generation steps could be found in [77]. Part of the verification of code generation process, SIMULINK model of Li-Ion battery detailed described in section 6.3, has been generated and deployed for real on-line execution. The same approach was verified for M-code generation and MATLAB Run-Time environment execution was tested as well. 51

52 4.4 Summary of Answers to Thesis Research Topic Number 1 What are the conditions when to select model-based approach? Detailed lists of all methods advantages and disadvantages are presented mainly in sections 2.4.1, 2.4.2, 2.4.3, 2.4.4, 2.4.5, 4.1.1, and At the same time identification whether the method is suitable for online evaluation is done in Table 4-1,Table 4-2 and Table 4-3. It is clearly identified that model based prognostic approach should be selected anytime the precise and accurate prognostics is demanded and in case we are not strictly limited by financial aspects, because of development cost. Model-based approach will be used especially in more complex and expensive systems and in case of high reliable system is required. Guide realized by flow chart was proposed in Figure 1-1. Next advantage is the progress in model-based development and increased availability of precise physical model as are presented on conferences and are published as a demo cases in MATLAB environment. We could expect wider and wider model-based prognostics deployment in future. What are the most promising methods for embedded systems? Embedded system and its limitations are described in section 4.1. Based on the analysis and list of disadvantages the most promising online method suitable for embedded system are simple linear trending and mainly particle filters utilizing Bayesian framework because of its capability to handle non-linear problems with higher accuracy than Kalman filters and to deal with non- Gaussian noise. Particle filters can be used to model multivariate, dynamic process. Particle filters are used in tracking and navigation applications which is similar problem used in prognostics when we want to track and predict potential failure. Consequently particle filters should be deeply investigated in next thesis research areas, where not only advantages but even disadvantages will be described. How to deploy developed prognostic method into embedded system? The answer could be found in Section 4.3. Deployment of prognostic algorithms with modern software tools and ability to convert system model to C/C++ language is getting easier. The most difficult part remains the final validation and verification of deployed embedded system especially system initialization, adaptation and run-time execution. This research question could be considered more as a technical solution research the main intent was to demonstrate that with currently available tools the problem of algorithms development and it verification has been significantly reduced. 52

53 5 PROGNOSTICS BASED ON PARTICLE FILTERING FRAMEWORK 5.1 Bayesian Framework and Methods Introduction Fault or fault indicator prediction is a process with high uncertainty and it has been proven that Bayesian estimation techniques provides framework which can deal with such uncertainties [80], consequently it is not surprising that those techniques are finding application domains in machinery fault diagnosis and prognosis of the remaining useful life of a failing component/subsystem. Bayesian estimation with particle filters is alternative to Kalman filters for estimating the posteriori in Bayesian framework model not limited by either linearity or Gauss noise assumption. They are also known as sequential Monte Carlo simulation methods and are particularly useful for situations where the posterior distribution is multivariate and or non-standard [68]. Particle filters are methods based on point mass (or particle ) representations of probability densities, which can be applied to any state-space model and which generalize the traditional Kalman filtering methods. Several variants of the particle filter such as Sequential Importance Re-sampling (SIR), Adaptive Sample Importance Re-sampling (ASIR) and Regularized Particle Filter (RPF) exists within a generic framework of the sequential importance sampling (SIS) algorithm [2]. See overview of Bayesian Network methods in Figure 5-1. Figure 5-1 Overview of Bayesian Methods and Particle Filters [82] 53

54 5.2 Particle Filter Theory Applied to Dynamic System As has been mentioned in text above the particle filtering is a technique for implementing recursive Bayesian filter by Monte Carlo sampling. The main idea is to represent the posterior density by a set of random particles with associated weights. Estimate is computed based on these samples and weights. Particle filters use weighted set of samples (particles) for approximating the filtering distributions. Dynamic system could be described as state sequence represented by Markov random process. State equation (is quite commonly available but equations are collected from [2], [10] and [80] ): e v e e cœ,k (5-1) Where x k is state vector at time instant k, f x is state transition function, ω k is process noise with known distribution. Observation equation: l e e,o (5-2) Where y k is observations at time instant k, h x is observation function, v k is observation noise with known distribution. Similar equation as mentioned in Section The alternative representation of the system by density function is conditional probability formulated for state equation: e? e?c3 (5-3) And for observation (measurement) equation could be represented as: Žl e (5-4) The main Bayesian framework objective is to estimate unknown state x k, based on a sequence of observations y k, k=0,1. In other wording find posterior distribution: Že : l Œ: (5-5) Sequential importance sampling (SIS) is a very commonly used PF algorithm that approximates the filtering distribution denoted in (5-5) by a set of P weighted particles {(w (i) k, x (i) k ): i = 1,, P}. The importance normalized (i) weights w k are approximations to the relative posterior probabilities of the particles such that: e? e? :? 6?? e?, $ 3? The weights update is given by: $ 3 1 (5-6)??c3 ql e qe e š e e œ: š, l š: (5-7) 54

55 Where the importance distribution πe? e :?c3, l 3:? is approximated as e? e?c3 and? is particle weight before normalization. Basically particle filter based on SIS technique consists of three main steps: 1. Particle Generation:? ~ e? e?c3 (5-8) 2. Weight Processing: a. Weight Computation equation: b. Weight Normalization:??c3 l? e? (5-9)? Ÿ (5-10) š 3. Estimate Computation is expressed by equation 5-6. A common problem with the SIS particle filter is the degeneracy phenomenon, where after a few iterations, all but one particle will have negligible weight[2]. There is a way how suitable measure degeneracy of weights. An effective sample size approach has been introduced in [7] is defined as: Ÿ <BB $ 38 (5-11) Where <BB cannot be evaluated and thus is estimated by <BB and relation is defined as: <BB 3 Ÿ š (5-12) Where <BB T ' indicates severe particles degeneracy. One of the approaches how to deal with particle degeneracy is having huge number of particles (P or N s ) but this is computational too expensive and we can consider this as a really brute force. There are to reasonable options how to solve it either by good choice of importance density function or by resampling method utilizing. Re-sampling is used to avoid whenever severe particle degeneracy is indicated by <BB T ', the main goal as mentioned above is avoiding cases in which all but one of the importance weights are close to zero, in other wording the basic idea of is to eliminate particles that have small weights and to focus on particles with large weights. Re-sampling is performed by drawing P particles from the current set with probabilities proportional to their weights and then simply replacing the current set with the new one an assigning the same weight 1/P to all [61], [62]. Graphically re-sampling process is showed in Figure

56 Figure 5-2 Principle of particle filter re-sampling reproduced from [10] There is a way how suitable measure degeneracy of weights. An effective sample size approach has been introduced in [7] is defined as: <BB $ Ÿ š (5-13) Where <BB is referred as a true weight and in case <BB T ' it indicates severe degeneracy. There are several technique for re-sampling like multinomial, residual, stratified, systematic in detailed described in [22]. Example of systematic resampling algorithm: e? l :? $ 3?. e.6? (5-14) Where e.6? denotes the Dirac delta function located at e? [2]. Although the resampling step reduces the effects of the degeneracy problem, it causes other practical problem - the particles that have high weights are statistically selected many times. This leads to a loss of diversity among the particles as the resultant sample will contain many repeated points. This problem, which is known as sample impoverishment, is severe in the case of small process noise. In fact, for the case of very small process noise, all particles will collapse to a single point within a few iterations [2]. See [2] and [7] for more details. Particle filter algorithm with re-sampling based on [10] could be graphically then represented as: 56

57 Figure 5-3 Particle Filtering Algorithm Steps reproduced from [10] There are several re-sampling algorithms multinomial, systematic, residual, stratified [22]. The simplest one so called multinomial re-sampling algorithm could be expressed as: e?,? ª 3 «+%,)'% e?,? ª 3 «INITIALIZE CDF c 1 = 0 FOR i = 2 : N s o Construct CDF: c3 (? END FOR Start at the bottom of the CDF: i=1 Draw a starting point w 3 ~*0, c3 FOR j = 1 : N o Move along the CDF w w 3 ( c3 o WHILE w : ( 1 o END WHILE o Assign sample e? e? o Assign weight c3? o Assign parent END FOR (5-15) Generic Particle Filter Algorithm Generic particle filter represents implementation, which includes the observations into the proposal density. This is basis implementation of SIS filter 57

58 including re-sampling. Although the re-sampling step reduces the effects of the degeneracy problem, it introduces other practical problems. First, it limits the opportunity to parallelize since all the particles must be combined. Second, the particles that have high weights are statistically selected many times [52]. e?,?! 3 '± e?c3,?c3! 3,l? FOR i = 1 : N o Draw:? ~ e? e?c3 o Calculate:? i? e?c3 END FOR Calculate total weight: ² 3? FOR i = 1 : N o END FOR Calculate <BB Normalize:? Ÿ ³ (5-16) z FUa< U ; IF <BB o RESAMPLE using algorithm o e?,?,! 3 +%,)'% e?c3 END IF,?c3! 3,i? SIR Particle Filter Algorithm As the importance sampling density for the SIR filter is independent on measurement, the state space is explored without any knowledge of the observations [2]. Therefore, this filter can be inefficient and is sensitive to outliers. Furthermore, as re-sampling is applied to every iteration, this can result in rapid loss of diversity in particles. However, the SIR method does have the advantage that the importance weights are easily evaluated and that the importance density can be easily sampled [2]. e?,?! 3,µ+ e?c3,?c3! 3,l? FOR i = 1 : N o Draw:? ~ e? e?c3 o Calculate:? l? e?c3 END FOR Calculate total weight: ² 3? FOR i = 1 : N o Normalize:? Ÿ END FOR RESAMPLE using algorithm o e?,?,! 3 +%,)'% e?c3 ³,?c3! 3,l? (5-17) Regularized Particle Filter Regularized particle filter (RPF) implementation focuses on avoiding the problem when all particles will collapse to a single point within a specified time period [7]. The RPF is identical to the SIR filter, except for the resampling stage. The RPF resamples from a continuous approximation of the posterior 58

59 density e? l :?, meantime the SIR re-samples from the discrete approximation. Where: $ e? l :??.> U.?? 3 > U 3 (5-18) U > = U (5-19) Where is the rescaled Kernel density >, : is the Kernel bandwidth (a scalar parameter), 4 = is the dimension of the state vector. As a Kernel Epachnikov kernel is used: e 0, e z 1 > #$ } 0 ƒ (5-20) 0 e 1 Where 2 is the volume of the unit hypersphere. The optimal choice for bandwidth : is: And consequently A is defined as: š : #$ ' º» (5-21) 8. c3 2 4 = ( 42 À 2 º» (5-22) š When RPF written in algorithm way [16]: e?,?! 3 +'± e?c3,?c3! 3,l? FOR i = 1 : N s o Draw:? ~ e? e?c3 o Calculate:? l? e?c3 END FOR Calculate total weight: ² Á 3? FOR i = 1 : N s o END FOR Calculate <BB Normalize:? Ÿ z FUa< U ; ³ IF <BB o Calculate Empirical covariance matrix S k of e?,?! 3 o Compute n? n?,? o Resample using e?,?, ª 3 «+%,)'% e?c3,?c3 ª,l? 3 «o FOR i = 1 : N s Draw  ~ > from Epachnikov Kernel e? e? ( : #$.n?.â o END FOR END IF (5-23) 59

60 5.2.4 Auxiliary Sampling Importance Re-sampling Filter ASIR filter is a variant of the standard SIR filter. This filter can be derived from the SIS framework by introducing an importance density Ãe?, l 3:?, which samples the pair e?,! 3, where refers to the index of the particle at 1. Algorithm steps are defined as per [52]: e?,?! 3,µ+ e?c3,?c3! 3,l? FOR i = 1 : N s o Calculate: w? o Calculate:? Ã `l 1:.C.l w?.?c3 END FOR Calculate total weight: ² 3? FOR i = 1 : N s o END FOR Normalize:? Ÿ Resample using e?,?, ª 3 FOR i = 1 : N s END FOR ³ Á «+%,)'% e?c3 o Draw:? ~ Ãe?`,l? e? Ä?c3 o Assign weight:? ql e Å ql Æ Å Calculate total weight: ² 3? FOR i = 1 : N s o END FOR Normalize:? Ÿ ³ Á,?c3 ª,i? 3 «(5-24) Rao-Blackwellized Particle Filter If the model is of certain form the efficiency of SIR can be enhanced by using the theorem of Rao-Blackwell. In the case of fault models with large scale vectors, re-sampling may not be sufficient in reducing the variance of particle errors. In such cases, if a part of i? of the state space x È can be used to analytically compute the remaining part, then e? É %e? i?, which is known as Rao-Blackewellized version of x È, can be used as the state estimator with the same mean as x È. E denotes the statistical operator expectations [61]. It is actually combination of PF and Kalman filter and sometimes it is called hybrid particle filter. In summary [62] this RBPF consist of following steps: * eê? +'± e?c3,?c3! 3,l? Use Monte Carlo sampling to the values Ë? Given these values, compute distribution of? with Kalman filter equations Result is a Mixture Gaussian distribution, where each particle consist of value Ë?, associated weight? and the mean and covariance conditional to the history Ë 3:? (5-25) 60

61 5.2.6 N Step Prediction with Particle Filters for RUL As has been described in sections above the first step of particle generation and weight computation is part of the prediction step. In the prediction phase we wish to compute at time k p, %&?Ì :?Ì and +Í?Ì :?Ì, we can approximate a prediction distribution n steps forward as [23]: $ e (4Äl :. e (4.e (4 e 82c3 3 (5-26) For a particle i propagated n steps forward (without new data), we can simply take its weight as. Similarly we can approximate EOL as: $ %& Äl :. "#.6%& 3 (5-27) The pseudo-code for the prediction procedure is given as in as [23], which requires hypothesizing future inputs of the system iê? : %&?Ì,?Ì! 3 «%&'6 e?ì,?ì! 3,l? «FOR i = 1 : N s o Assign: k = k p o Assign: e e?ì o WHILE Î "# e? 0 do Predict iê??83 ~?83?,iÊ? Assign: k = k + 1??83 END FOR (5-28) n step prediction is expressed as: %&?Ì,?Ì! 3 «'6 e?ì,?ì! 3,l? «FOR i = 1 : N s o Assign: k = k p o Assign: e e?ì o FOR j = 1:n Predict iê??83 ~?83?,iÊ? Assign: k = k + 1??83 o END FOR END FOR (5-29) Extended Kalman Filter Even it is not particle filter we should mention Extended Kalman filter (EKF) as an option for non-linear solution. EKF is the nonlinear version of the Kalman filter which linearizes about an estimate of the current mean and covariance. To estimate a process with non-linear difference and measurement relationships, we can define governing equations that linearize an estimate about system represented by 5-1 and 5-2 according to [85]: 61

62 e?83 eï?83 (.e? eï? ( ².Ð? l? lï È ( Ñ.e? eï? ( Ò.υ? (5-30) Where eï?83 and lï È are the approximate state and measurement/output vector, eï? is an a posteriori estimate of the state at time step k. A is the Jacobian matrix of partial derivatives of f x ( ) with respect to x represented as: *, ÔB * Ô= *Å eï?,i?,0 (5-31) W is the Jacobian matrix of partial derivatives of f x ( ) with respect to ω represented as: ² *, ÔB * ÔÕ *Å eï?,i?,0 (5-32) H is the Jacobian matrix of partial derivatives of h x ( ) with respect to x represented as: Ñ *, ÔU * Ô= *Å eï?,0 (5-33) V is the Jacobian matrix of partial derivatives of h x ( ) with respect to ω represented as: Ò *, ÔU * ÔÕ *Å eï?,0 (5-34) As another step we define a priori and a posteriori prediction error as: Ö c c = e? eï È? Ö = e? eï? (5-35) and similarly and a priori and a posteriori observation error also known as measurement residual: Ö c c Ø l? lï? Ö Ø l? lï? (5-36) A priori estimate error covariance ' È c is defined as: And a posteriori estimate error covariance ' È is: ' c È E*Ö c c = Ö Ù = (5-37) ' È E*Ö =.Ö = Ù (5-38) To measure update in Kalman filter we need to define: Ú? eï? ( >?.Ö Ø (5-39) 62

63 Where >? is Kalman filter gain nûm matrix. EKF time update is then performed as: c Ú?83 = Ú?,i?,0 c '?83? '?? ( ²? Ü? ²? (5-40) Where Ü? is a process noise covariance at time k. EKF measurement update equations are following: >? Ñ? Ñ? '? c Ñ? ( +? eê? eê? c ( >l? ÑeÊ? c '? I >? Ñ? '? c (5-41) Where +? is measurement noise covariance matrix. See next algorithm steps written as pseudo-code (summarized from [85]): * eê? %>±*Ú c?,l? c with initial estimate for '? Time Update Predict c o Project the state ahead: eê?83 = eê?,i?,0 o Project the error covariance ahead: c? '?? ( ²? Ü? ²? '?83 Measurement Update ( Correct ) o Compute the Kalman gain: >? Ñ? Ñ? ' c? Ñ? ( +? o Update Estimate with measurement: eê? eê c? ( >l? ÑeÊ c? c o Update the error covariance matrix: '? I >? Ñ? '? (5-42) 5.3 Particle Filter Implementation and Verification In previous section detailed background theory has been explained. Next step is particle filter implementation before any real implementation a detailed research of available libraries and toolboxes was performed with success. Mathworks do not offer any standard toolbox for particle filters, even there is System Identification Toolbox with Kalman filters and ARMA models. There is only limited number of real examples of PF implementation but no solid and mature approach. Finally PFLib - An Object Oriented MATLAB Toolbox for Particle Filtering developed by [16] was selected for all simulation and system verification. This toolbox is product of Scientific Systems Company Inc. and University of North Carolina at Chapel Hill, developed under a United States Army Small Business Technology Transfer (STTR) project called Advanced Computational Algorithms for Nonlinear Filtering for Real Time Environment". Toolbox could be used under GNU License, which is followed in this thesis project The implementation of a filtering algorithm and its use are separated by an object oriented approach. Major algorithms and options have been implemented by authors [16]: Simple Particle Filter Auxiliary Particle Filter Regularized Particle Filter, 63

64 Particle Filter with an EKF-type proposal distribution Extended Kalman Filter, Resampling schemes includes - None, Simple (Multinomial), Residual, Branch-and-Kill (not well documented) and Systematic. Other available parameter choices include sampling, frequency, number of particles, and specification of Jacobians for EKF needs. Part of the library is graphical interface enabling user to initially set particle filters, but it is only for simple systems Comparison of PF algorithms As a part of particle filter verification and as part of the familiarization with new filtering technique a simple one dimensional system model has been selected in a form:? 0.5?c3 ( 25?c3 1 (?c3 0? = 0 (5-43) Simulation has been performed with following variances: Different Filter Types (Simple, Auxiliary, Regularized, EKF-PF, EKF) Different parameters ω process noise distribution mean, variance Different re-sampling technique Different number of particles See results of simulation separately for each particle type and comparison of RMS at the end of this section. Figure 5-4 Filtering of a system represented by Equation 5-42 for simple SIR, N s =60, Residual re-sampling, Resample every 2 samples 64

65 Figure 5-5 Evolution of State Density Represented by Particles for SIR filter, N s =60, Residual re-sampling, Resample every 2 samples Figure 5-6 Filtering of a system represented by Equation 5-42 for simple Auxiliary PF, N s =60, Residual re-sampling, Resample every 2 samples 65

66 Figure 5-7 Evolution of State Density Represented by Particles for Auxiliary filter, N s =60, Residual re-sampling, Resample every 2 samples Figure 5-8 Filtering of a system represented by Equation 5-42 for Regularized PF, N s =60, Residual re-sampling, Resample every 2 samples 66

67 Figure 5-9 Evolution of State Density for Regularized PF, N s =60, Residual re-sampling, Resample every 2 samples Figure 5-10 Filtering of a system represented by Equation 5-42 for EKF-PF, N s =60, Residual re-sampling, Resample every 2 samples 67

68 Figure 5-11 Evolution of State Density for EKF-PF, N s =60, Residual re-sampling, Resample every 2 samples Figure 5-12 Filtering of a system represented by Equation 5-42 for EKF filter 68

69 As we can see in previous figures there is no big difference in particle filter response. Method of comparing RMS has been chosen to numerically compare each PF methods. Figure 5-13 RMSE of Estimation for Each PF algorithms one particular run In next table you can see comparison of root mean square error (RMSE) for different number of particles (EKF is mentioned just for completeness). RMSE or sometimes called as a Root Mean Square Deviation (RMSD) is quite common metric for comparison estimators and is usually defined as: +),% ß à = š, c =, š (5-44) where 3, is simulated system state and 0, is estimated value. Table 5-1 Comparison of Different Implementation of Particle Filters Filter Type Simple PF RMSE [-] AUX PF RMSE [-] EKF-PF RMSE [-] Number of Particles 69 Regularized PF RMSE [-] EKF RMSE [-] Residual resampling every 3 steps N s = N s = N s = N s = Average Residual resampling every 1 step N s =15 N s = N s = N s = Average

70 Filter Type Simple PF RMSE [-] AUX PF RMSE [-] Regularized PF RMSE [-] EKF-PF RMSE [-] Simple resampling every 3steps N s = N s = N s = N s = EKF RMSE [-] Systematic resampling every 1step N s = N s = N s = N s = In next figure you can see graphical representation of received RMSE results. Figure 5-14 RMS diff comparison of different particle filter types and their dependency on Number of particles As we can see in Table 2-1 and Figure 5-14 above the best RMS performance offers EKF, which seems to be more suitable for non-linear application with Gaussian noise. EKF-PF seems to be nearly as good as EKF with the advantage of applicability to even non-gaussian noise. We can see that there is no so big impact of particle numbers in some cases the performance was better with less number of particles, consequently the bigger number of particles is not always better solution. 70

71 Figure 5-15 Comparison of filtering methods for different re-sampling types for 150 simulation steps Figure 5-16 Filtering methods sorted based on RMSE MATLAB software provides the Profiler functionality, which enables to evaluate execution time of each algorithm including its sub-functions. This tool is quite helpful in algorithms development and optimizing. It provides information not only about execution time but even about the number of calls/hits. See results of profiling for different particle filters. 71

72 Table 5-2 Comparison of Execution Time of Different Particle Filter Algorithms (Measured Time for 150 Steps) Filter Type Simple PF Exec Time[ms] AUX PF Exec Time[ms] Regularized PF Exec Time[ms] EKF-PF Exec Time[ms] EKF Exec Time[ms] Number of Particles Residual re-sampling every 3steps N s = N s = N s = N s = Avg. per part Number of Particles Residual re-sampling every 1 step N s = N s = N s = N s = Avg. per part Number of Particles Simple re-sampling every 3steps N s = N s = N s = N s = Avg. per part Number of Particles Systematic re-sampling every 1 steps N s = N s = N s = N s = Avg. per part As we can see from results in Table 5-2 the most time consuming execution is for implemented EKF-PF algorithm. The rest of the execution seems to be proportionally dependent on number of particles. When dealing with details and root cause of slow performance of EKX-PF most of the time consumes repmat function which is often up to 15646time in case of simulation length 150steps and N s =15. EKF significantly overcomes other methods with its execution time this is because of PF needs some time for stabilization. In next figure you can see graphical representation of execution time results. 72

73 Figure 5-17 Execution time for different particle filter implementation Figure 5-18 Sorted execution time for different filter implementation We can from RMSE results and execution time results that even EKF-PF seems to be promising regarding the RMSE value, it has disadvantage in its execution ecution time. To compare different implementation a simple normalized 73

74 RMSE and Execution time is merged together to easily identify each algorithms advantages and disadvantages. See Figure 5-19: Figure 5-19 Final Comparison of Bayesian framework based filters/estimators Just for completeness a testing for Non-Gaussian distribution has been performed. Gamma distribution (a = 2, b = 1) was used for simulation of internal states and the results were similar to previous utilizing Gaussian PDF, except the EKF-PF where it was not possible to properly tune all parameters and consequently was not used. See next table with brief overview of results, where could be noticed that in this case a number of particles has impact on RMSE unlike the run with Gaussian noise. Table 5-3 Filter Type Number of Particles Comparison of methods for Gamma distribution Simple PF AUX PF RMSE[-] RMSE[-] Regularized PF RMSE[-] EKF RMSE[-] N s =15 N s =30 N s =60 N s =120 Average Residual re-sampling every step In next work Simple PF, AUX PF and Regularized PF with Residual retesting approaches. From embedded point of view there is a room for algorithms optimization especially in sampling technique will be used to limit number of EKF-PF filter, which seems to be promising because of its RMSE estimation capability. As well EKF is still promising and is considered as a default approach. What has to be mentioned in support of promising PF is fact that 74

75 initialization portion of the internal state estimates takes several steps, which cause worse RMSE. It has been verified than in longer simulation run the results overcomes EKF. To compare all algorithms in detail a more complex simulation system should have been chosen (multi-dimensional arrays, different noise mean and variance etc.) but this is not the intent of this section. The main aim was to prove that implementation of particle filters is ready for use and could be utilize for prognostics approaches. 5.4 PF Prediction Capabilities Particle filters could be used not only for tracking the internal states, which could be used in technical diagnostics but at the same tide it could be used as an N-step predictor. This is crucial functionality, which is used for RUL estimation. We can choose either an estimation for next N steps or we can modify prediction till specific condition is reached threshold etc. To verify prediction capabilities of a current PFLib mentioned in section 5.3 was updated by adding function of predict. Simulation steps consists of particle filtering, when all particles where drawn and weights were set as it could be seen in Figure Figure 5-20 Testing of N-step prediction capability Table 5-4 Comparison of particles for different N step predictions Filter Type N steps Prediction Simple PF RMSE[-] AUX PF RMSE[-] Regularized PF RMSE[-] As we can see from results captured in Figure 5-21 and Figure 5-22, it is possible to make N step prediction with particle filters. In any prediction step we automatically rely on the same distribution function with unchanged attributes (mean, variance), which cannot be always guaranteed. With prediction we are getting probability of estimate, which is represented by particle distribution density. See Figure 5-23 for plor of probability density function. 75

76 Figure 5-21 Simulation with 5 steps ahead prediction for Simple PF Figure 5-22 Simulation with 100 steps ahead prediction for Simple PF As per [80] long term predictions can be used to estimate the probability of failure in a system, given by hazard zone that is defined lower and upper limits. The prognostic confidence interval as well as expected time to failure (TTF) can be defined from the TTF PDF [80]: á ± 3 ' Ñ T á T Ñ Æ! Ï á (5-45) The uncertainty usually increases with the prognostic horizon and prediction farther to future, to reduce the uncertainty in the particle-filter-based 76

77 failure prognostics an additional learning method for correction of estimated TTF is used which is similar to continues linear regression performed above specific set of prognostics estimation [80]. In next figure Figure 5-23 we can how distribution of particles providing us mean and confidence limit once particle weights are applied. Figure 5-23 Particles and its density used for setting confidence limit of estimation (µ = , σ = ) Prediction utilizing particle filtering technique is quite unique and only limited number of real use cases were demonstrated. Based on the research work it seems that mainly academicians from Georgia Institute of Technology, NASA Ames Research Center and engineers from Impact Technology are pioneers of this promising technology. Thesis demonstrates prediction capability on real system in section of Battery Health Management for runtime estimation and for capacity fade degradation. Currently known application of particle filtering prediction capability could be found in [61] where authors deal with battery health prognostics and Rao-Blackwellized version of particle filtering is used but no precise metrics for comparison are mentioned. Another area, where PF prediction capability was demonstrated is Model-Based Prognostics under Limited Sensing [23], where authors provides α-λ metrics for spring damage prediction, it was proven that even with limited number of sensing still in case accurate model is and accurate prediction could be performed. 5.5 Advantages and Disadvantages of PF PF tuning could be quite complex task and need several trials. The most complicated step is to select the right prior density function as well there is no clear suggestion what the right number of particles is 77

78 All pros and cons were collected during simulation/evaluation of different algorithms implementation and from available literature [2], [4] and [7]. It could be summarized as follows: Advantages: Applicable even to non-linear systems Adaptive focusing on probable regions of state-space Working for non-gaussian noise Ability to represent arbitrary densities The framework allows for including multiple models (tracking maneuvering targets) Disadvantages: Difficult to determine optimal number of particles High computational complexity (depending on number of system states) Number of particles increase with increasing model dimension Potential problems: degeneracy and loss of diversity The choice of importance density is the most important Minimal real applications Difficult to develop accurate model and deep knowledge of particle filtering technique is needed 5.6 Summary of Answers to Thesis Research Topic Number 2 What are the different possible implementations of particle filters? Particle filters and the theoretical background of Bayesian framework has been described in section 5.1 and 5.2. Several important implementation of particle filters are described in section with description of generic particle filter without re-sampling mechanism, Sample Importance Re-sampling PF is introduced in section 5.2.2, Regularized PF implementation and Auxiliary SIR PF implementation details are described in section and Rao- Blackwellized version of PF is mentioned in section Alternatives to particle filters represents Extended Kalman filter for non-linear systems are described in section All sections contain description of algorithms and detailed steps needed for implementation including different types of re-sampling mechanism. Algorithms are verified and compared by utilizing PFLib toolbox developed in under GNU license. Most precise in case of used Gaussian noise seems to be Regularized PF and Simple PF which has low RMSE and good performance at the same time. In case of Gaussian noise EKF overcomes particle filters but this was caused mainly because on simple simulation system. Similar comparison of particle filters implementation is unique and has not been performed on similar basis in known literature references and this was the main reason of this effort. 78

79 What are the performance differences for each of those? There is no big difference in performance differences. Algorithms are compared not only from accuracy point of view but even from performance point of view, which could affect final deployment to embedded system. MATLAB profiler was used for this purpose. All results are clearly described and evaluated in Table 5-2 Comparison of Execution Time of Different Particle Filter Algorithms (Measured Time for 150 Steps) and in figures Figure 5-17 and Figure It cannot be clearly stated, which implementation is better because problems, where PF will be deployed will vary but in metric considering evaluation time and accuracy the regularized particle filter with systematic resampling could be considered as the best one. How could be particle filters used for n step prediction? Algorithm steps for prediction are defined in Section in Equations (5-28) and (5-29). N-step prediction is verified and tested in Section 5.4. Based on the results it could be assumed that PF are suitable framework for estimation application mainly because of its capability to provide estimation confidence interval, to handle non-linear systems (even not fully tested in this thesis) and non-gaussian noise, which is quite common case. Thesis demonstrates prediction capability on real system in section of Battery Health Management for runtime estimation and for capacity fade degradation. Correctness of prediction algorithms has not been fully verified and this capability needs to be deeply investigated in future research work. N-step prediction would not be always needed in case we are able to precisely track the system degradation we can still use some more common extrapolation/regression technique like linear, polynomial fit. 79

80 6 USE CASE - BATTERY HEALTH PROGNOSTICS 6.1 Motivation Technical prognosis is a research field which has limited number of real deployment. Model development, load identification, algorithms definition and different prognostics approach comparison simple but real and easy to understand use case has been chosen to demonstrate all activities/steps related to any prognostics process. See next Figure 6-1 with all steps, which follows prognostics process proposed in Section 4.2 and is applicable for Li-Ion battery health prognosis. Figure 6-1 Real Prognostics Process with focus on model-based prognosis Rechargeable battery cells mainly lithium-ion batteries are one of the most important electrical energy accumulation technology. Utilizing battery sources varies in a huge range from small portable devices like cell phones, PDAs, notebook, laptops, over universal power sources (UPS) up to bigger devices/machines/systems like pure electrical/hybrid automobiles, solar power plants accumulating energy etc. Popularity of rechargeable lithium-ion batteries is growing even thanks to climate changing and green thinking of the population. As usual there are some cons to any used technology and in case of lithium-ion battery it is the 80

81 problem of battery ageing and degradation during regular life-cycle. Battery capacity decreases just in case it is not used and stored only because its selfdischarge. Environment temperature affects it as well (approx. capacity loss 8% at 21 C, 31% at 60 C per [87]). Normal life-cycle of lithium-ion accumulator is about charge/discharge cycles of course depending on operational condition and manufacture technology [76]. Pieces of information about battery health status including current state of charge, capacity wear out, remaining useful life are essential for every system running on batteries. Cost of battery failure increase with cost and complexity of the equipment for example in case a cell phones run out the battery it is quite simple to recharge it and user just losses connectivity (small business impact), but in case of battery failure of electric car or solar power plants it is getting more complicated and causes significant extra expenses to fix it. UPS is another system utilizing battery and where battery failure could cause data loss and moreover could affect human safety. Availability and reliability of any battery powered system is a main driving factor for monitoring battery health and for estimating RUL. It is quite important to know not only when we will run out from energy source in case of one charging cycle called as a battery run-time, but knowing the progression of capacity fade and knowing the predicted time to replace battery pack is important as well, especially in mission readiness applications. It is surprising that in commercial sector (e.g. laptops, PDAs) there is not available precise and accurate battery health monitoring solution including battery prognostics. Current solutions/applications focus mainly on basic battery status indication (simple battery bar) with limited accuracy of runtime estimation. 6.2 Battery Health Monitoring and Prognostics Techniques In this section the most important research areas in lithium-battery prognostics and health monitoring will be briefly described. We will follow basic research topics as grouped and described in [93] State of charge (SOC) Estimation SOC is percentage of the maximum possible charge that is present inside rechargeable battery as defined in [55]. SOC is a parameter providing us the information about the remaining useful energy and the remaining useful time could be inherited, but it is not directly measurable parameter. SOC computation from charge counting or current integration is one of the most commonly applied method [76]:,&Î 1 âûã;f ä *Ë;:,5: (6-1) Where i is the current; η is Coulombic efficiency defined as the ration between charging and discharging energy required to restore original capacity; C n is the nominal capacity; t is the time. Having Coulombic efficiency is quite important because errors in terminal measurements could cause large SOC 81

82 errors [85]. Other techniques to estimate SOC are represented by Open Circuit Voltage (OCV) look up table, electromotive force (EMF) determination, applying fuzzy logic and neural networks etc. Overview of different SOC Estimation techniques is presented in Table 6-1. Table 6-1 Comparison of SOC Estimation Technique merged from [55] and from [93] SOC Technique Field of application Pros Cons Discharge test Coulomb counting Coulomb Counting with Impedance tracking Opend Circuit Voltage (OCV) Electromotive Force (EMF) Used for capacitydetermination at the beginning of life All battery systems, most applications Li-Ion, patented by Texas Instruments Lead, Lithium, Zn/Br, Lead, Lithium Easy and accurate; independent of SoH Accurate if enough re-calibration points are available and with good current measurements Precise, Online, real ICs avaliable Online, cheap, OCV prediction Online, cheap, EMF, prediction Offline, time intensive, modifies the battery state, loss of energy Sensitive to parasite reactions; needs regular recalibration points Needs long rest time (current = 0) Needs long rest time, (current = 0) Linear model Lead Photovoltaic Online, easy Needs reference data, for fitting parameter Electrochemical Impedance Spectroscopy (EIS) [43] All systems Gives information on, SoH and quality Temperature sensitive, cost intensive Artificial Neural Networks [40] All battery systems Online Needs training data of a similar battery, expensive to implement Fuzzy logic [40] All battery systems Online Need a lot of memory in real-word application Kalman filters Particle Filters Support Vector Machine All battery systems, PV, dynamic application All battery systems, PV, dynamic application Lithium Online, Dynamic Online Dynamic Precise parameter estimation Difficult to implement the filtering algorithm that considers all features as e.g. nonnormalities and nonlinearities Difficult to implement the filtering algorithm considering all features, model dependency Offline, Training data needed only method for feature identification Voltage Estimation The main goal is to establish a battery model enabling us to simulate/estimate OCV. An empirical battery model which is capable of 82

83 estimating OCV under a range of discharge rates and ambient temperatures has been presented in [31] and will be described in section 6.3. Another area is an effort to improve the effectiveness of voltage monitoring in protection circuit so that severe failures such as overcharging and overheating can be prevented [93] Capacity Estimation The loss of capacity as a result of increased impedance, mainly on a battery s cathode, will cause reduced performance when electrical devices cannot operate at a satisfactory level and functional level when the battery fails to supply the required energy and power [93]. Capacity estimation is closely related to SOC estimation and could be considered by researches as an interchangeable area Remaining-useful-life (RUL) prediction Remaining-useful-life represents available run time left before a battery degrades bellow an acceptable level. RUL is important information and is crucial for battery operating systems and at the same time for mission planning and mission readiness planning, this fact is supported by increasing number of research effort and projects sponsored by NASA [60] and as stated in [93]. In [61] prognostics method for battery health monitoring based on a Bayesian framework has been introduced. Proposed method combines Relevance Vector Machine RVM regression/classification technique for features extraction from EIS data and particle filtering (PF) technique providing features (electrode resistance) prediction on first-principle battery model. Prognostics framework has been split to offline part applying RVM technique and to online part running particle filters predictions. This approach seems to be really promising regarding the accuracy of RUL estimation. At the same time correlation between battery performance capacity and battery internal resistance was exploited. Particle filters used in RUL prediction has significant advantage because of its capability to provide confidence interval of estimation and ability to decide on prognostic horizon to keep prediction more accurate. Using Rao- Blackwellized type of particle filters seems to be suitable for battery prognostics application [61]. The big disadvantage of previously mentioned method is dependency on periodical EIS data measurement to track battery capacity degradation. This could be time consuming and cannot be considered as a standalone prognostics framework, because battery needs to be removed from equipment for each measurement. We can see that there is no mature approach for battery prognostics. Determining the optimal approach for battery prognostics will be the main goal of next sections. 6.3 Li-Ion Battery Model Considering Effects of Temperature and Capacity Fading Model-based prognostic is one of the most precise approach as has been stated in sections above. Any first principle model provides us deep knowledge of the system and of course could help in development of prognostics algorithm. 83

84 Numerous lithium-ion battery models have been developed for purpose of battery performance simulation, complex hybrid system simulation, for I-V characteristic generation etc.. Despite many models have been presented in last years, most of them have limitations because of ignoring transient behavior or simulating only steady state. A dynamic lithium-ion battery model considering the effects of temperature and capacity fading has been introduced in [24]. The proposed model has been inherited from [17] and modified by adding temperature and capacity fading effects on battery dynamics according to [58]. Parameters used in model are as follows [17], [24], [31]: CCF Capacity correction factor [-] C init Initial battery capacity [Ah] long-time transient capacitance [F] C transient_l C transient_s C usable I bat k 1 short-time transient capacitance [F] Usable battery capacity [Ah] Battery current [A] Coefficient for the change in battery in SOC of battery negative electrode [cycle -2 ] k 2 Coefficient for the change in battery in SOC of battery negative electrode [cycle -1 ] k 3 Coefficient for the change in R cycle [Ω/cycle 1/2 ] N Cycle number [-] Qn Change in state-of-charge of battery negative electrode [-] R cycle battery resistance increased by cycling [Ω] resistor causing instantaneous voltage drop [Ω] R series R transient_l R transient_s long-time transient resistance [Ω] short-time transient resistance [Ω] SOC State of charge [-] SOC init Initial state of charge [-] T t storage V bat V oc Z eq ΔE(T) Temperature [ C- K] storage time [s] Battery output voltage [V] Battery open circuit [V] Battery equivalent internal impedance [Ω] Temperature correction of the potential [V] Even most of the common equations are mentioned in previously listed references we will state only the most important ones for providing the context of problematic. The battery final output voltage could be calculated from battery open circuit voltage, voltage drop from the battery equivalent internal impedance and the temperature correction of the battery potential as defined in [31]: Ò åæf Ò #ä åæf ç <u ( % *Ò (6-2) 84

85 Battery open circuit voltage could be calculated as defined in [17]: Ò #ä,&î céêûš#ä ( ( ,&Î ,&Î 0 ( ,&Î é *Ò (6-3) Capacity correction factor can be calculated as defined in [93]: Îα 1 Î7D467 D D@ËËË ( Î D D D@ËËË (6-4) And usable capacity is then expressed as [93]: Î Æ æå< Î 2Fæ Îα (6-5) Correlation for variation of SOC with cycling was developed based on experimental data in [58]. The rate of change of state of charge of electrode material is expressed as: ;í ; 3 ( 0 (6-6) The values of coefficients k 1 and k 2 vary with temperature and this variations need to be consider in case proper cycle life losses will be determined. The problem is that coefficients are empirical values and currently only reduced set of values for specific temperatures is available (25 C and 50 C). Battery dynamic and voltage drops are caused due to battery internal impedance. See battery equivalent internal impedance model in Figure 6-2: Figure 6-2 Li-ion battery equivalent internal impedance (reproduced from [30]) The battery equivalent internal impedance consists of a series resistor composed of R cycle, R series, R transient_l, R transient_s, C transient_l, C transient_s and values could be calculated dynamically as function of SOC as defined from empirical data in [17]: + <a<,&î c0î.éïûš#ä ( Faæ2 <2F_Š,&Î c0ñ.3îûš#ä ( Î Faæ2 <2F_Š,&Î c3é.ê3ûš#ä ( Faæ2 <2F_,&Î c3êê.0ûš#ä ( Î Faæ2 <2F_,&Î 6056 c0ï.30ûš#ä ( 4475 (6-7) Only R cycle seems to be SOC independent (based on research in [30]) and could be calculated as: 85

86 + Ø < é 3/0 (6-8) ΔE(T) calculation could be found in [24] and [31]. This model considers all important degradation attributes and could be considered as a suitable for next simulations. 6.4 Battery Model Simulation Model is implemented per equations describe in section 6.3. As a development environment MATLAB/Simulink environment is chosen. Model has been verified for functional correctness by running following simulations: Charging for different current rates (80 ma, 120 ma, 160 ma) Discharging for different current rates (80 ma, 120 ma, 160 ma) Discharging for different cycle numbers N (100, 500, 800) Model was verified by comparing data collected from test bed described in next sections. Created model consists of battery internal logic and configurable parameters affecting model behavior like SOC init, charging/discharging current I bat, temperature T, storage time t storage and cycle number N. Similar as described in [30]. Figure 6-3 Simulink dynamic model of lithium-ion battery Following simulation condition were used for pulse charging/discharging process 95% duty 2500 seconds pulse period 125 seconds relax time Battery capacity was set to 850 mah 86

87 Based on the received results a charging time is adequate to battery capacity (t = C(SOC)/i). And even pulse dynamics of the system is reflected as you can see in Figure 6-4 and Figure 6-5. SOC was set in range 20% -100%. Figure 6-4 Lithium-ion battery pulse charging process (C init =850mAh) Figure 6-5 Lithium-ion battery pulse discharging process (C init =850 mah) for different current Capacity degradation simulation is the most important area of interest, enabling us to identify the best approach for RUL estimate as will be discussed in Section 6.5. Results could be seen in Figure 6-6 and Figure

88 Figure 6-6 Battery charging for different cycle N (I charge =160 ma, C init =850 mah) Figure 6-7 Battery discharge for different cycle N (I disch =160 ma, C Cinit =850 mah) Charge/discharge curve on capacity fade was simulated for N= 1, N=100, N=500, N=800. Simulation of temperature change was added to map behavior of the model for different number of cycles as well. Cycle of N=100 at temperature of 293 K (19.85 C), N=100 at temperature of 310 K (36.85 C), N=300 at temperature of 293 K (19.85 C)and N=300 at temperature of 310 K (36.85 C) was selected 88

89 Figure 6-8 Charging process under different temperature condition and cycle number Figure 6-9 Discharging process under different temperature condition and cycle number In a summary battery model provides reasonable results for basic system simulation and is sufficient for model based development, but in case of real deployment as reference model - a parametric estimation logic should be added. 89

90 6.4.1 Model verification with e-station Balanced Charger/Discharger Having a Simulink model developed is only initial part of the real problem. A simple test bed utilizing commercial charger/discharger with connectivity to PC was created to verify the real functionality. Verification environment consisted of following items: Li-Ion battery Silverlit 3.7 V with original 50 mah capacity and after calibration it was set to 35 mah ( ) e-station Bantam BC6 Charger/Discharger with USB convertor (s.n ) Laptop Dell Precision M4500 Set of connectors Figure 6-10 Battery Measurement setup for basic model verification Bantam BC6 is a charger used mainly in RC application and supports several multi-chemistry battery types, different current charge and different modes. Table 6-2 Bantam BC6 Functionality Overview Functionality Description Operating Voltage DC: V, AC: V Power Max 50/5 W 90

91 Functionality Description NiCd 1-15 cells NiMh 1-15 cells Li-Ion/Li-Po/LiFe 1-6 cells Pb 2-20 V Charge Current 0.1 to 5.0 A Discharge Current 0.1 to 1.0 A Modes Fast Charging, Charge, Discharge, Storage Mode for charging and discharging was selected for data measurement. Temperature was not specially controlled or measured but system was operated in an ambient temperature in range 20 C 22 C. No special measurement technique was used because of effort to align testing environment to real non specialized condition. BC6 charger/discharger has not been deeply calibrated but for basic model verification is sufficient. The main intent is not to verify measurement setup itself but mainly focus on getting real and meaningful data. We have to keep in mind that limited instrumentation will be offered in real application especially in case of laptops, mobile phones and other mass commercial products. Two modes were tested first charging with current of 100mA and discharging for the same 100 ma. Figure 6-11 Measured data versus simulation data for charging at 100mA of initial capacity 50mA (during battery calibration the capacity was set to 35mAh) 91

92 Figure 6-12 Measured data versus simulation data for discharging at 100mA (f sample = 5s) (during battery calibration the capacity was set to 35mAh) In case of the charging curve the model behaves more properly (RMSE = 0.028V). We can see that discharging curve is not identical (RMSE = 0.262V) and parameters from exponential equation (3) have to be slightly modified or reestimated. The main reason why battery model has been introduced is to demonstrate how current load, temperature, cycling and storage time affects battery life either capacity fade but even the battery runtime and how important is to monitor those data and to verify that at the same time model is appropriate enough. Figure 6-13 Difference between model curve and real measurement 92

93 6.4.2 Measurement of real battery characteristics in laptop Basic model measurement was described in previous section but measurement was done only for static current load of 100mA and for one cell. These conditions cannot be always guaranteed and consequently real current load and I-V characteristics would make sense to obtain to help in understanding of battery load process. The reasonable idea is to measure load in any real equipment like cell phones, laptops. The instrumentation for measurement of voltage and current could be potential roadblock but in case of laptop the situation is easier. Most of the nowadays laptops use so called Smart Battery System (SBS) with its detailed specification described in [69]. SBS system consists of battery pack and electronics with logic enabling to collect and evaluate environmental conditions (voltage, current, temperature, N cycles, calculate SOC) and communicates with external system. There is a memory containing all battery cell specific data and information related to calibration. SBS is technology independent i.e. it should be applicable even to NiCd or NiMh batteries and we could consider SBS as a true embedded condition based maintenance. SBS serves real time data on Smart bus and those data are propagated via OS kernel up to application layer and could be used by host applications. Even there are couples of commercial products called as a battery monitor a new application was implemented in C++ accessing all battery parameters like: Number of Cycles, Charged Capacity, Voltage, Current Rate, Life Percent, Remaining Time, Power State charging/discharging/ac, Battery Info. Application was written in Visual Studio 2008 and implemented to work with Windows based operating system and based on testing it is compatible with Windows XP, Windows Vista and Windows 7. Application BatteryMonitor was implemented to log data every 1 second. BatteryMonitor application could run on any laptops and in case detailed measurement and wide spectrum data collection would be requested a monitoring network could be created and logged data would be uploaded via network on any FTP server. This could help in capturing more relevant data. But this is only potential nice to have approach and is not developed as part of this theses. Figure 6-14 Setup for measurement real data from laptop 93

94 Data from battery were collected from different laptops for comparison (Dell Precision M4500, Aspire AspireOne 752 Netbook). Operational conditions were specific for the user and applications running on a machine. Data in next figures were measured on Dell Precision M4500 with 3-cells battery and original designed capacity 94500mWh (7.5Ah). Figure 6-15 Discharging/ Charging Process for Li-Ion battery in Dell Precision M4500 Figure 6-16 Current Discharging/Charging Process We can see that current discharging rate in opposite to charging rate is quite noisy and has several multiple peaks. A histogram sequence was created to capture current discharge rate progression in time (it represents real battery load). 94

95 Figure 6-17 Battery current discharge rate histogram progression in time for Dell Precision (Time: t 1= s, t 2= s, t 2= s, t 1= s) Knowledge of discharge current distribution could be quite helpful as a consumption indicator and could be used for trending/prediction purposes. The entire time series data were analyzed and fit for the right probability distribution. Figure 6-18 Probability density function fit analysis 95

96 Figure 6-19 Cumulative distribution function of current discharging rate We can see that the best fitted distribution is a Generalized Extreme Value (GEV) distribution and that there is not big difference between Gamma distribution and Normal distribution. See parameters of all distribution in next table. Table 6-3 Probability Distribution Parameters for Current rate Distribution Type Generalized Extreme Value Parameters µ = Variance = k = , σ = , ν = Normal µ = Variance = Gamma µ = Variance = a = , b = Data received from real measurement will be used in next RUL estimates and prognostics approach analysis. Measuring on real laptop will be our final verification deployment. 6.5 RUL - Battery Runtime Estimation Remaining useful life could be represented either by battery run-time or useful capacity capacity fade, which falls below specified threshold. Both are the most desired attributes in case of battery health prognosis especially from user perspective and both cases could be represents estimation of time to failure. Run-time is defined as a time for which system could operate before recharging 96

97 is needed. Usually the threshold is set to either SOC level (usually 10%) or battery voltage (3.0 V in case of one lithium-ion cell [85]). RUL is defined in as a time when C usable is less than 80% of C init [55]. Capacity loss during battery service is not linear and depends on many factors as mentioned before. A confidence interval and an estimation probability should be provided as recommended in when RUL is estimated. Let focus on quite simple but easy to compare process of run-time estimate as a typical RUL estimation problem. In next evaluation we expect that SOC value is available to us Estimation Based on Average Current Load Nowadays modern laptops/netbooks provides information about remaining time at least in Windows/Linux based operating system. Mobile phones are limited in runtime estimation and provide usually simple 3-5 bars indication only. Laptops already utilize one of the SBS function RunTimeToEmpty providing information about remaining time to empty in minutes defined and computed per SBS specification and described in Table 6-4. Accuracy for SBS is not exactly stated and could vary in range of 0-100%, which is not sufficient. Table 6-4 Runtime Estimation Parameters per SBS Spec [69] Parameter Units: Range: Granularity: Accuracy: Attributes minutes 0 to 65,534 min 2 min or better )7%@ ±wddî:7ôî77 9 õ7ôîw49 where MaxError is set by manufacture and could vary from (0%-100%) We can evaluate a precision of estimate from our laptops measurement using BatteryMonitor application. Figure 6-20 Battery Run Time prediction per current SBS implementation with α =

98 Figure 6-21 Root Mean Squared Percentage Error (RPSME) of RUL estimate It could be stated that approach implemented nowadays by SBS and Win operating system is not accurate enough especially at the begging of prediction the error reaches up to 50% and at the end the values reaches nearly 100% and prediction is highly unreliable. At the same time no confidence interval is provided. Relative percentage error of such prediction is about 28% and about 86% of prediction is outside of α boundary limit Prediction Based on Evaluating PDF of Current Load As we can see from previous SBS approach there are no valid information regarding confidence of prediction. Knowledge of current rate PDF is important to enhance the prediction of runtime because in previous example only current average is used in case we will use mean calculated from PDF. Current rate PDF fits to General Extreme Value distribution as has been proved in section 6.5.1, thus this type of distribution will be used in this approach. Prediction algorithm for battery runtime is defined as +Í_%Ë9579_'±,&Î ö,µ_79 c :, w Ëø Calculate current load PDF for (i-n, i) FOR j = 1 : N o Calculate,Q 3600,&Î ù /µ_79 END FOR Calculate r(i) mean and variance from '±,Q: Provide r(i) histogram Provide r(i) probability plot (6-9) Buffer size affects the number of samples and could affect PDF parameters in could be considered as way of simple filtering. In our case of MATLAB simulation we will use N=300 and overlap of buffers will be set to N/2 it represents our processing window. 98

99 Figure 6-22 Battery Run Time prediction per proposed PDF estimation approach with α limits set to 0.1 We can see that prediction trajectory in Figure 6-22 is more accurate and nearly within α boundary limit. There is a huge drop in time t=3000 s, which causes RPSME error up to 45%. Similar results were obtained from another test machine Acer Aspire One. In next Figure 5-23 we can see plot of estimation errors and RPSME time series. Relative error mean was 11.57% which is still quite high and outside 10% limit. Prediction error was in 44% above limit α-λ limit. Figure 6-23 Root Mean Squared Percentage Error (RPSME) of RUL estimate implemented per algorithm (6-9) 99

100 Figure 6-24 Probability plot of RUL Estimate in times series Time: t 1 =341s, t 2 =3741s, t 3 =5441s, t 4 =8841s Figure 6-25 Runtime prediction based on PDF - boxplot defining mean and 25 th and 75 th percentile Runtime prediction based on PDF could be used either as input for other algorithms or as additional info for particle filtering approach. Algorithms 100

101 precision could be modified by generic density distribution function (Epanechnikov kernel) which will more precisely describe current load and thus mean estimate will be more accurate. Computation of probability density function could be time consuming and will not be applicable to simple embedded system Prediction Based on Linear Trending The simplest form of prognostics RUL prediction could be based on simple trend analysis of a single monotonic parameter correlated with remaining life it is SOC in our case. The disadvantage of trending is low ability to reflect changes in battery usage. Another difficulty could be selection of proper trending function e.g. regression methods it will be either linear or polynomial function. Resulting trend need to be extrapolated. The problem of extrapolation is in considering only history data and cannot handle extra battery loads including environmental temperature changes. The proposed approach defined in this section computes linear trend of SOC and computes runtime based on the received linear equation it means extrapolation of linear function is performed. +Í_%Ë9579_47 46,&Î ö, w Ëø Calculate SOC linear trend parameters a, b for (i-n, i) in a form,&î 7 95 ( Calculate,&Î af æ,&î /7, where SOC critical could is usually set to 5% of original SOC Provide r(i) probability plot (6-10) Buffer size was set to N=240 during our simulations, it means RUL r (i) is calculated every 240s=4 minutes, which should be enough as a refresh rate for laptop user, of course it will depend on battery capacity and total possible runtime in case battery runtime is just couple minutes then estimation with 4 minutes refresh rate will be quite low. Figure 6-26 Battery Run Time prediction per proposed linear trend estimation approach with α limits set to 10% 101

102 Figure 6-27 Root Mean Squared Percentage Error (RPSME) of RUL estimate implemented per algorithm (6-9) Linear trending or so called time series linear regression is the easiest approach for real implementation and has the lowest computational requirements only RMS computation for linear trend equation is needed. It seems that in case of the battery run-time prognostics it could be valid approach with sufficient accuracy and precision overcoming original SBS runtime prediction and even PDF estimation based prognostics. Independently on this fact RSMPE is still out of 10% boundary in significant amount of cases (42%). Linear trending provides confidence interval, which could be calculated from regression residuals Prediction Based on Particle Filtering PF framework could be used for runtime estimation in case no SOC state is available and only discharge current and battery voltage is available. SOC is then considered as unknown internal state, which has to be estimated. Discharge current is our system input variable and a battery voltage is our observations in this case. This approach would require more precise physical model for multi-cell Li-ion battery, but independently on this fact the Li-Ion battery model introduced in section 6.3 is simplified and multiplication by number of cells is performed. It is obvious that this approach would be computationally inefficient comparing to previously mentioned methods and would make sense to use it only in case no SBS is deployed and no SOC info is available, which is always preferred approach. See description of algorithm we used for estimation +Í_%Ë9579_'±,&Î ö, Use system simulation model described in section 6.3 modified for 3 cells voltage:,&î? : 3,? 3,?c3 w? ( Ð 3,?, : w? Ò? : 3,? Ò&Î 3,? w? ç <u 3,? ( ú 3,? Track internal variable SOC (6-11) 102

103 WHILE(SOC > 0.05*SOC initial ) o Make N-step prediction with updated particles o Use computed current PDF END WHILE Provide r(i) histogram Provide r(i) probability plot Gaussian noise was used for measurement noise simulation and for density function calculation. Regularized PF was used with number of 30 particles and systematic re-sampling algorithms. Figure 6-28 Battery Run Time prediction based on PF framework with α limits set to 0.1 Figure 6-29 Root Mean Squared Percentage Error (RPSME) of RUL estimate implemented per algorithm (6-11) 103

104 Figure 6-30 Runtime prediction based on PF framework - boxplot defining mean and 25 th and 75 th percentile Runtime RUL Estimation Methods Comparison Prediction of battery runtime is demonstrated in previous section including different approaches. Four methods were selected: SBS original approach based on average current (representing simple data-driven approach), Estimation based on PDF current load (representing probability based approach) Simple linear regression trending (representing data-driven approach) Particle Filtering approach based on simplified battery model (representing complex model-based approach) Most precise seems to be approach based on particle filtering, but system model need to be well parameterized and estimation of density is relatively difficult model was adapted manually to fit real data measured in laptop and modified to handle 3-cell Li-Ion battery. PF technique is not recommended because of high computational demandingness and difficulty to adjust initial values and to have more generic solution. See Table 6-5 where comparison of methods is mentioned. Table 6-5 Comparison of methods for RUL runtime estimation Method Mean RPSME[%] Computational Efficiency SBS Original Approach Estimation based on PDF Simple Linear Regression Particle Filtering α-λ Performance (α = 0.1, λ = 0) False/True ratio [%] 104 Estimation Resolution [min] for 7.5Ah battery Refresh Rate [s] High ± Medium ± High ± Low ±17 360

105 Computational efficiency was determined based on MATLAB profiler capability and enumeration instead of parametric values is used in range High, Medium, and Low. Simple linear trending seems to be most suitable for battery runtime estimation purposes even more complex and robust methods are available mainly particle filtering approach considering Li-Ion battery model and its open voltage characteristics depending on SOC and current load. Estimation of runtime was done on 20 samples with similar results. Approach combining several methods could be considered we can use PDF estimation which will be collected continuously during battery life time and will be used as initial estimate at the beginning of runtime estimation then linear regression of SOC with appropriate samples averaging method could be used including applying time window for buffer. Runtime estimation results described in Table 6-5 are compared with respect to one particular measurement of discharge cycle. Algorithms were verified on multiple runs (30 full discharge cycles) with minimal outputs variances. It could be clearly stated that prediction could be as precise as precisely current load is estimated in case of extreme current drop occurs the prediction is highly unreliable as could be seen from another experimental discharge cycle shown in Figure Figure 6-31 Estimation of runtime with extreme current load and linear regression trend 6.6 RUL Battery Capacity Fade Estimation Battery capacity fade estimation respectively RUL prediction is more complex task comparing to runtime estimation. Current peaks, number of cycles, operational temperature and temperature peaks, storage time, battery life time and other attributes affects capacity fade over the battery life. In next section we will describe two basic approaches to predict capacity fade the first will be simple trending based on linear regression and utilizing SBS data like number of wear coefficient (C current /C initial ), second approach is represented by particle filtering technique utilizing accurate system degradation model. In both cases mainly simulation data from capacity fade Simulink model are used, even real operational data were collected from several laptops using BatteryMonitor application. In all tested machines (Dell Precision M4500, S/N: NH237866, S/N: JT457856) the battery degradation/wear out was only about 2% percentage for period of 4months and those results are not suitable for real algorithms 105

106 verification. Accelerated degradation could be used in laboratory conditions but this was already performed in developing of capacity fade model [54], [65] and [70] and there is no advantage to repeat the same experiments. It is outside the thesis scope. The main intent is to define and propose novel functional framework, which could be developed in near future and could be used in wide areas, where Li-Ion batteries are utilized and offers potential for embedded application. As has been described in model-based development prognostics approach - once model is verified and features are defined then we can proceed with degradation tracking and usage monitoring. In case of capacity fade the most important features we should track are number of battery charge/discharge cycles and its time dependency ΔN/t [-/s], battery life (time from production date till now) and degradation of capacity during time ΔC/t [-/s] Capacity Fade Estimation Based on Linear Regression SBS provides information about charging and discharging states and SOC states. This piece of information could be used for counting N cycles and create relation to time. In this case we will define our RUL as a number of cycles, for which battery could be used then we can collect usage data and extrapolate our linear function represented as: N üýþ t(n (6-12) Where N üýþ is threshold set for battery useful life and N is number of cycles, from production, it should be set to 0. Collecting N cycles values could be quite difficult especially for cases when N is computed only for full charge/discharge cycles. But this is usually not the real case. For example rain-flow counting method developed for mechanical systems could be used to determine proper number of full/half cycles. Figure 6-32 Regular battery usage measured on a laptop 106

107 Cycle counting complicates the estimation and is applicable only in case RUL is defined in a number of cycles N RUL. More suitable feature to track is capacity degradation itself. SBS interface provides original capacity, which is set by manufacture and actual capacity is calculated during charging and discharging process. This value could oscillate because of Coulombic counting method inaccuracy and because of missing full charge status is reached, but newer SBS HW provides more precise and stable capacity measurement called Impedance Track patented by Texas Instruments. The advantage of capacity tracking approach is that even we are not monitoring temperature and current peaks, affecting capacity significantly, we are still getting actual and relatively accurate capacity and its evolution over time this could be considered as a true embedded monitoring solution. Linear regression equation would be defined as: C % ΔCCF t(c (6-13) Capacity degradation represented as ΔCCF should be monotonic decreasing function of time with slope estimated by linear regression. Estimation accuracy/precision will depend, as mentioned couple times, on unknown variable operational sequence. Equations 6-4, 6-5, 6-6 were used to Monte-Carlo simulation of capacity fade by setting random temperature in range of 273K 313K and random N cycles rate in range 1-4 cycles per week, which was based on real laptop measurements (Figure 6-32). Both values define simulated operational sequence and represents battery usage. Simulated capacity fade process utilizing equations 6-4, 6-5, 6-6 Figure 6-33 Simulated capacity fade for operational temperature in range 273 K 313 K and 1-4 full discharge/charge cycles per week Linear regression is computed for specific number of measured capacity samples in a similar way as it is done for runtime estimation. Different regression function could be selected for situations, when linear regression has 107

108 big residuals. There are options like exponential, polynomial and others. Generic method testing the best fit could be recommended as a future enhancement. Figure 6-34 RUL Prediction for Capacity Fade by linear regression method Figure 6-35 RPSME of capacity fade prediction based on linear regression We can derive based on results shown in Figure 6-34 and Figure 6-35 that capacity fade estimation is less accurate and relative error mean of RPSME is about (11±5) % it means that accuracy of our prediction for simulated case at prognostic horizon of 300 days is ±33 days and ±6 days in case of 50 days prognostic horizon, which could be considered as not sufficient resolution, depending on criterions set by users. There are still areas for prediction improvement similar to runtime ones prediction filtering, buffer window etc. 108

109 6.6.2 Capacity Fade Estimation Based on Particle Filtering Another approach is to use parameter estimation and n-step prediction by using Bayesian framework specifically particle filters as was already mentioned in section 5.2. We can use equations 6-4, 6-5, 6-6 and use k 1 and k 2 and ΔCCF as internal parameter or states. The system model is following: x È k 3 x 3,È x 3,Èc3 ( ω 3,È k 0 x 0,È x 0,Èc3 ( ω 0,È CCF x é,è x 3,È N È N Èc3 ( x 0,È ( ω é,è ƒ y È C y 3,È y 3,Èc3 x é,è ( υ 3.È (6-14) ΔCCF is cycle life capacity loss per k and in our case, we have omitted calendar life loss for simplicity, C usable could be measured and is obtained either by SOC battery recalibration or could be calculated from accumulated charge under condition of real-time current measurement is available. Parameters k 1 and k 2 are function of temperature and in this case effect of temperature on capacity fade will tracked as well. Number of cycles could be calculated based on charging/discharging rate or as defined in previous section by using rain-flow algorithm. Algorithms will be executed every time full charge/discharge cycle is executed on battery. Particle filter with N particle = 50 was implemented and normalized re-sampling method was used. Results of simulation demonstrating ability to track ΔCCF are shown in Figure Figure 6-36 Particle Filter used for ΔCCF tracking as an internal state In case we will be able to track ΔCCF during battery usage we are able to estimate rate of capacity fade and battery RUL at the same time. Simulations show that particle filtering approach enables us to estimate RUL with precision 109

110 of days depending on prognostic horizon. This needs to be proven by real measurement in normal environment and on more samples. Degradation measurements are time consuming and require bigger number of samples to get statistical reasonable values. Precision and estimation process need to be deeply investigated. This section aims to provide basic concept and idea of RUL estimate in broader context. Figure 6-37 RUL Prediction for Capacity Fade by Particle Filtering method Figure 6-38 RPSME of capacity fade prediction based on particle filtering Again it could be assumed based on the results shown in Figure 6-37 and Figure 6-38 that capacity fade estimation is more accurate than linear regression approach and relative error mean of RPSME is about (9±4)% it means that accuracy of our prediction at prognostic horizon of 300 days is ±27 days and ±5 days in case of 50 days prognostic horizon, which is nearly sufficient but future work needs to focus on better performance or on possible model improvement. 110

111 Table 6-6 Comparison of methods for RUL capacity fade Method Mean RPSME[%] Computational Efficiency Simple Linear Regression Particle Filtering α-λ Performance (α = 0.1, λ = 0) False/True ratio [%] Estimation Resolution for PH = 300 days 11±5 34 High ±33 9±4 35 Low ±27 Comparison of two proposed methods is captured in Table 6-6. Particle filtering methods overcomes linear trending but both are not precise and are outside of α limit boundary. Even it could be seen this estimation as not accurate enough it could be still considered as a functional approach ready for deployment in embedded system. Another valid approach demonstrated in runtime estimation could be PDF estimation of temperature and number of cycles but this has not been verified since it is not applicable when simulation data and no real data are used especially when we use Gaussian distribution used with specific mean and variance and there is no value added for our tests but definitely for future enhancements Battery RUL Prognostics Framework As has been described above all attributes affecting capacity fade are measurable quantities, without any extra instrumentation needs and served by SBS system. Disadvantage of SBS is that even it could produce a valuable runtime history data there is a lack of prognostics information like cycling rate and temperature history data. Only sum N is provided and wear-out coefficient with not well defined accuracy. In case SBS approach and battery model knowledge will be merged we can get quite robust battery prognostics framework. It will consists of lithium battery cells, smart battery electronics physically connected to battery cells and prognostics evaluation module, realized either by extra microcontroller ( bit) running simplified prognostics algorithms (camera battery, electric car system, solar power plants system) or it could be high level program OS specific utilizing system resources (laptops/notebooks, cell phones). Figure 6-39 Battery Health Monitoring and Prognostic Framework 111

112 Prognostics evaluation module will provide RUL estimate and precise runtime estimate, which is a weak part of current SBS system. Several metrics like current load represented not by average current but by probability density function could be computed. Framework is more concept specification and need to be further evaluated as part of next research. There are following areas, which need to be explored to confirm/prove framework added value: Define RUL/Run-time requirements Accuracy, Precision (minutes in case of runtime, days in case of RUL) Availability of HW resources. In case simple µc will be used only simple algorithms could be deployed and accuracy and precision has to be reduced Particle filtering method and its re-sampling types (simple, normalized, Rao-Blackwellized) need to be tested. Other valuable methods of prediction need to be investigated. Functional prototype need to be fully tested and (currently only BatteryMonitor application has been prepared for Window OS platform) Prognostics Evaluation as algorithms plug-in Battery Prognostics Framework HW Consideration This section describes real HW applicable to battery health management and currently available for commercial use. There are several manufactures producing integrated circuits (IC) for smart battery system and battery fuel gauges. The most known ones are MAXIM, Dallas Semiconductors (subsidiary of Maxim) and Texas Instruments Inc., which is member of Smart Battery Implementers Forum. Texas Instruments produces battery fuel gauges as a part of TI s battery management portfolio. Fuel gauges measure battery current, voltage and temperature to find state-of-charge. Their patented battery measurement method allows the gauge to calculate remaining battery capacity with high precision [74]. This technology is called Impedance Track and is advertised as more precise than simple Coulombic counting. Brief overview of available fuel gauges and smart battery circuits is listed in Table 6-7: Table 6-7 Overview of battery gauges and circuits for Smart Battery system Company Type Chemistr y Dallas Semiconductor Dallas Semiconductor Texas Instruments Texas Instruments DS2751 Li-Ion, NiMh Max Number of cells Approx. Capacity [mah] Commun ication Protocol 1-3 N/A 1-Wire DS2782 Li-Ion 1 N/A 1-Wire bq2750 Li-Ion 1 bq3060 Li-Ion 2, 3, to to I 2 C SMBus 112

113 Hybrid electrical vehicles (HEV) are emerging area on automobiles and it seems that this will be one the main path for future automotive development. Electro mobiles of course utilizes huge number of battery cells and their permanent monitoring and potential degradation tracking will be value added for any companies deploying condition based maintenance. Collection of degradation data could help even to producers to optimize their parts replacement cost. TI provides quite complex battery usage monitoring for HEV, where proposed prognostic framework could be deployed as well and enhance potential condition based functionality. Figure 6-40 Battery Monitoring Solution for Hybrid Electrical Vehicles (HEV) from [74] 6.7 Summary of Answers to Thesis Research Topic Number 3 What are the current battery life prognostics approaches? In this research topic a brief overview of state of the art of lithium-ion battery health monitoring techniques (State-of-life estimation, state-of-capacity estimation, run-time estimation, internal resistance estimation, capacity fade estimation) is described to provide a problem context see section 6.2.1, and Dynamic lithium-ion battery model with capacity fade modeling is introduced and different kinds of simulations are performed (different temperature, different number of cycles, and different number of charging currents) in Section 6.3 and in Section 6.4. Results are discussed and real verification is performed using one-cell Li-Ion battery. Demonstrated model contains capacity fade feature and could be used either in battery runtime 113