Ensemble Gradient Boosted Tree for SoH Estimation Based on Diagnostic Features
Abstract
:1. Introduction
2. Proposed SoH Estimation’s Framework
2.1. Experimental Data
2.2. Data Preparation
2.3. Feature Engineering
2.3.1. Feature Exploration
Statistical Features
Distortion Metrics
Spectral Features
2.3.2. Prognostic Feature Ranking
2.4. Ensemble Gradient Boosted Tree
 $\left\{\left({x}_{1},{y}_{1}\right),\dots ,\left({x}_{n},{y}_{n}\right)\right\}$ is the training set;
 $L$ is the number of leaves;
 ${R}_{1},\text{}{R}_{2},\dots ,\text{}{R}_{L}$ are the disjoint regions constitute from input data;
 ${P}_{i}$ is the output of region ${R}_{i},\text{}i=1\text{}to\text{}L$;
 $H$ is a decision tree and its output is calculated as: $H={\displaystyle \sum}_{i=1}^{L}{P}_{i}{1}_{{R}_{i}}\left(x\right)$;
 $\widehat{F}\left(x\right)=\mathrm{arg}minF\left(x\right)\text{}\left[L\left(y,F\left(x\right)\right)\right]$ is a function that maps $x$ to $y$ in a way that reduces the loss function $L\left(y,F\left(x\right)\right)$ over the joint distribution of all ensembles;
 The pseudoresidual is determined as: ${g}_{i}\left(x\right)=\left[\frac{\partial L\left({y}_{j},{F}_{i1}\left({x}_{i}\right)\right)}{\partial {F}_{i1}\left({x}_{i}\right)}\right],\text{}j=1\dots N.\text{}$
 Crossvalidate a set of ensembles. Exponentially increase the treecomplexity level for subsequent ensembles from decision stump (one split) to at most n  1 splits. n is the sample size. In addition, vary the learning rate for each ensemble between 0.05 to 0.2.
 Vary the maximum number of leaves using the values in the sequence {${2}^{0}$,$\text{}{2}^{1}$,…, ${2}^{m}$}. m is such that 2 m is no greater than n − 1.
 For each variant, adjust the learning rate using each value in the set {0.05, 0.1, 0.15, 0.2}.
 Estimate the RMSE for each ensemble.
 Identify the number of trees (N), maximum leaves number of (L), and learning rate (R) that yields the lowest RMSE overall.
3. Results and Discussion
3.1. Experimental Results
3.2. Discussion
4. Conclusions and Outlooks
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Positive Electrode Material  Nickel Manganese Cobalt Oxid 

Negative electrode material  Graphite 
Cell wight  428 $\mathrm{g}$ 
Nominal voltage  3.65 $\mathrm{V}$ 
Nominal capacity  20 $\mathrm{Ah}$ 
Voltage range  3 to 4.2 $\mathrm{V}$ 
Power density  2300 $\mathrm{W}/\mathrm{Kg}$ 
Specific energy  174 $\mathrm{W}/\text{}\mathrm{Kg}$ 
Conditions  Cell 1  Cell 2  Cell 3 

DoD (%)  75  60  75 
Middle SoC %  50  50  50 
Temperature  45  45  10 
Number of full charges and discharge cycles per day  6  12  6 
Number of WLTC trips in each cycle  5  4  5 
Features  Description 

Basic Features  
MAV  $MAV=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}\left{s}_{i}\right$ 
SD  Measures data spreadation around mean value: $SD=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{N}{\left\left({s}_{i}mean\left(s\right)\right)\right}^{2}}{N1}}$ 
RMS  Root mean square of an input signal: $RMS=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{N}{s}_{n}{}^{2}}{N}}$ 
SF  Calculated by dividing RMS by MAV. It is dependent on the signal shape. $SF=\frac{RMS}{MAV}$ 
Impulsive features  
Peak values  The maximum absolute value of a signal. The basic parameter for computation of other impulsive features. 
Impulse parameter  The height of a peak divided by to the signal’s mean absolute level: $\frac{Peakvalue}{MAV}$ 
Crest parameter  Calculated by dividing peak value over RMS: $C=\frac{\leftPeak\right}{RMS}$ 
Highorder features  
Skewness  Describes the symmetry of a distribution: $Sk=\frac{1}{N1}\left(\frac{{{\displaystyle \sum}}_{i=1}^{N}{\left({s}_{i}mean\left(s\right)\right)}^{3}}{S{D}^{3}}\right)$ 
Kurtosis  Characterizes the difference between a distribution and a normal distribution: $\frac{1}{N1}\left(\frac{{{\displaystyle \sum}}_{i=1}^{N}{\left({s}_{i}mean\left(s\right)\right)}^{4}}{S{D}^{4}}\right)$ 
EGBT Algorithm 


Metrics  Definition 

Mean absolute error (MAE)  $MAE=\frac{1}{N}{\displaystyle {\displaystyle \sum}_{i=1}^{N}}\left\left({\widehat{Y}}_{i}{Y}_{i}\right)\right$ 
Mean absolute percentage error (MAPE)  $MAPE=\frac{1}{N}{\displaystyle {\displaystyle \sum}_{i=1}^{N}}\left\frac{({\widehat{Y}}_{i}{Y}_{i}}{{Y}_{i}})\right\times 100\%$ 
Root mean squared error (RMSE)  $RSME=\sqrt{\frac{1}{N}{\displaystyle {\displaystyle \sum}_{i=1}^{N}}{\left({\widehat{Y}}_{i}{Y}_{i}\right)}^{2}}$ 
Model  MAPE  Computational Cost (s) 

Model 1  0.45  11.8 
Model 2  0.21  8.1 
Model 3  0.20  6.9 
Error Evaluation Metric  Cell 1  Cell 3 

MAE  0.53  0.64 
RMSE  0.69  0.70 
MAPE  0.58  0.63 
Training  Validation on Cell 2  Validation on Cell 3  

Error Evaluation Metric  EGBT  Decision Tree  EGBT  Decision Tree  EGBT  Decision Tree 
RMSE  0.29  0.34  0.69  1.18  0.70  2.67 
MAE  0.083  0.081  0.53  1.05  0.64  2.34 
MAPE  0.20  0.21  0.58  1.14  0.63  2.35 
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Khaleghi, S.; Firouz, Y.; Berecibar, M.; Mierlo, J.V.; Bossche, P.V.D. Ensemble Gradient Boosted Tree for SoH Estimation Based on Diagnostic Features. Energies 2020, 13, 1262. https://doi.org/10.3390/en13051262
Khaleghi S, Firouz Y, Berecibar M, Mierlo JV, Bossche PVD. Ensemble Gradient Boosted Tree for SoH Estimation Based on Diagnostic Features. Energies. 2020; 13(5):1262. https://doi.org/10.3390/en13051262
Chicago/Turabian StyleKhaleghi, Sahar, Yousef Firouz, Maitane Berecibar, Joeri Van Mierlo, and Peter Van Den Bossche. 2020. "Ensemble Gradient Boosted Tree for SoH Estimation Based on Diagnostic Features" Energies 13, no. 5: 1262. https://doi.org/10.3390/en13051262