Maximizing Regenerative Braking Energy Harnessing in Electric Vehicles Using Machine Learning Techniques

Abstract

Innovations in electric vehicle technology have led to a need for maximum energy storage in the energy source to provide some extra kilometers. The size of electric vehicles limits the size of the batteries, thus limiting the amount of energy that can be stored. Range anxiety amongst the crowd prevents the entire population from shifting to a completely electric mode of transport. The extra energy harnessed from the kinetic energy produced due to braking during deceleration is sent back to the batteries to charge them, a process known as regenerative braking, providing a longer range to the vehicle. The work proposes efficient machine learning-based methods used to harness maximum braking energy from an electric vehicle to provide longer mileage. The methods are compared to the energy harnessed using fuzzy logic and artificial neural network techniques. These techniques take into consideration the state of charge (SOC) estimation of the battery, or the supercapacitor and the brake demand, to calculate the energy harnessed from the braking power. With the proposed machine learning techniques, there has been a 59% increase in energy extraction compared to fuzzy logic and artificial neural network methods used for regenerative energy extraction.

1. Introduction

Nowadays, transportation plays a crucial role in the economic growth and sustenance of countries [1]. Concerning global warming and the adverse effects of the emissions from automobiles on the environment, the electrification of vehicles has been in research for the past decade. Eco-friendly vehicles have been creating a demand in the market in the form of EVs and HEVs. Ferdinand Dudenhoeffer, the director of the Gelsenkirchen University of Applied Sciences’ Center of Automotive Research, stated in 2008 that “by 2025, all passenger vehicles produced in Europe would be electric or hybrid electric”. The reason for its rapid growth is its various advantages like eco-friendly, torque, and efficiency [2,3]. Bringing in the autonomous features in the vehicles leads to ease of transport, drive and safety, and comfort. These battery-operated vehicles are the new generation of transport that requires special control algorithms and electric power converters to run the motors on the wheels. However, the driving range of the electric vehicle is lagging when compared to fuel-powered vehicles—the main reason being the limitations of the energy source powering the car. The size of the energy source is directly proportional to the power it supplies. To extract excess power, larger battery packs are required, which is contradictory to the limited battery size based on the size of the vehicle.

A significant amount of research has been carried out in the field of batteries to last for a longer duration providing higher mileage. The driving range can be increased to an extent by harnessing the energy back into the storage system with the help of regenerative braking [4,5]. Various studies have proved that a fully electrified regenerative system is more efficient than conventional friction-based mechanical braking, which causes energy loss. Recent trends involve blended braking (hybrid-braking system) initially developed using fuzzy logic, which is faster and more effective in adverse situations and produces maximum efficiency [6,7]. Regenerative Braking harnesses the kinetic energy dissipated by the brakes into the powertrain and stores it in the energy source during deceleration. The mechanism involves numerous power converters to meet the battery voltage. But the battery takes a long time to recharge because of its lower power density. So, an additional storage device whose power density is higher is required to collect the regenerative braking faster. For the ideal application purpose, we need a storage device with power and energy densities higher which is not possible. The battery has a high energy density which can solely power the EV, but because of its low power density [8], it takes a longer time to harness the regenerative energy back. Whereas on the other side, the supercapacitor has high power density facilitating faster energy recovery through regenerative braking. Nevertheless, supercapacitors solely cannot support the EV.

A compound energy storage system (CESS) is developed to efficiently utilize the complementary features of supercapacitors and batteries [9]. With the CESS developed, it is required to ensure that optimal regenerative energy is being harnessed into the storage device. By increasing the amount of energy achieved through regenerative braking, the vehicle’s overall drive range and efficiency can be improved. It has been described that in the urban drive cycle, about 60% of the energy is utilized during braking; thus, to enhance the capacity of energy recovery, a composite braking system has been developed. The aim is to develop a system that can increase the energy during regeneration; a detailed understanding of the factors which limit the force has to be studied—especially the effect on the accelerator pedal [10]. A detailed understanding of battery and supercapacitor characteristics and the adhesive coefficient of the road is required to understand the factors that limit the regenerative braking energy capability.

A vehicle braking system is now incorporated with artificial intelligence to measure the intensity of the braking required based on real-time scenarios [11,12]. With the leap to digital transformation, machine learning and artificial intelligence has been an interdisciplinary area applied to various domains, including electric vehicles [13,14]. The main focus is regenerative braking during deceleration, especially at traffic signals or turnings. A special algorithm for regenerative braking has been designed that optimizes the power harnessing during decelerations at the traffic signals [15,16]. When it is impossible to represent the system explicitly and accurately, machine learning paves its path to understanding the data and making appropriate best decisions accordingly. A nested learning framework has been developed to manage the energy in an HEV and minimizes the cost of power sources using machine learning techniques [17,18]. Deep learning networks are now used to estimate the SOC of the lithium-ion batteries that can self-learn the behavior in real-time and change their activation functions to provide the best-optimized method for SOC calculation [19]. The use of regenerative braking is not limited to electric vehicles or four-wheelers but has moved to every segment of vehicles that can be powered by batteries [20].

From the abovementioned literature, the key motivation for this research can be derived as modeling the braking force/energy estimation with the help of machine learning models, and comparing its performance with statistical models to find out if machine learning models can be used in the braking scenarios. The idea is to check regenerative energy maximized with machine learning models and the methods to further leverage these models in real time.

The work proposes a machine learning model using two different algorithms, i.e., multi-variate polynomial regression and random forest regression, for maximizing the regenerative braking energy in an electric vehicle. The driving range of a vehicle is compared between a vehicle with regenerative braking and one without regenerative braking. The work also presents a study comparing the proposed model with that of the fuzzy, artificial neural network and real-time optimization, to maximize the energy harnessing from the braking power. The work is described in detail in various sections, where Section 2 gives an overview of the system designed to harnessing the braking power and send it back to the battery. Section 3 and 4 introduces the polynomial regression and random forest methods used in the regenerative braking energy extraction, different training technique, and their unique characteristics to the work. A detailed comparison is presented, and the most suitable algorithm is identified that extracts maximum power due to deceleration providing extra kilometers to the electric vehicle.

2. System Description

Brake Blending is often ruled by deterministic rules that are based on physical considerations that are performed on relatively simple models of vehicle behavior [21,22]. Prior work on brake blending states different lane time simulations, braking conditions, and the amount of braking torque required. A vehicle in motion can have various drive cycles wherein sometimes sudden braking is required, very fast change is needed, as well as panic braking conditions. In such situations, energy harnessed from regenerative braking will vary. The work focuses on a comparison model using machine learning techniques subjected to various driving conditions and finding the optimal algorithm that provides maximum regenerative energy on an average across all driving conditions. A complete understanding of vehicle dynamics and the effect on braking force among the front and rear axles are required to design the system. Figure 1 represents the block diagram of the estimation of the regenerative force required to provide the regenerative braking facility to an electric vehicle for range efficiency. The considered system is a part of a front-wheel drive electric vehicle where the motor is connected to the front axle. The braking force has two parts one is the force due to electrical braking at the motor and the force due to friction. The performance of the braking effect plays a major role in vehicle safety systems. In electrified vehicles, the kinetic energy lost due to braking force can be recovered and sent back to the energy source, thus charging it back.

The state of charge (SOC) of the energy source, such as the battery or supercapacitors, is dependent on determining the braking force required during regenerative braking. The machine learning algorithms take the applied brake force as the input and calculate the regenerative force based on multiple logic functions. The maximum braking energy is harnessed using the optimal logic function and the energy is sent back to the storage element. This increases the SOC of the battery by charging it and thus provides a longer range to the vehicle. Various machine-learning algorithms used to harness regenerative energy have been discussed in further sections. A comparative analysis using polynomial regression, random forest regression, fuzzy logic, artificial neural network, and real-time optimization was performed. The latency in braking is assumed to be 0 for simulation purposes.

3. Polynomial Regression

A dependent variable can be expressed as a linear combination of independent variables, but the coefficients are unknown, which can be determined by the regression analysis. It is one of the essential statistical tools used in diverse fields used in economics and business to investigate the causal link between two or more variables. To generate a regression equation, a theoretical relationship model is established, and parameter values are calculated. The model is subsequently subjected to a set of tests to gauge its suitability. Model validation is a crucial stage in the modeling process that determines the model’s trustworthiness before it can be utilized in decision-making.

3.1. Multiple Regression Model

If output can be expressed as a linear/non-linear combination of multiple independent variables, then this type of problem is called multi-variate regression. To solve a multiple regression model, a technique called polynomial regression is used where the dependent variable is expressed in the powers of independent variables.

Model:

Consider a basic multiple regression model of a dependent variable (i.e., output variable$\overrightarrow{Y}$) on the set of features (i.e., predictors) X₁, X₂, X₃, …., X_k can be expressed as (1).

$$\left.\begin{array}{c}{y}_1={\theta }_o+{\theta }_1{x}_{11}+\dots +{\theta }_k{x}_{1k}\\ \begin{array}{c}{y}_2={\theta }_o+{\theta }_1{x}_{21}+\dots +{\theta }_k{x}_{2k}\\ ⋮\end{array}\\ y_n={\theta }_o+{\theta }_1{x}_{n1}+\dots +{\theta }_k{x}_{nk}\end{array}\right\}$$

(1)

i.e.,

$$y_i={\theta }_o+{\theta }_1{x}_{i1}+{\theta }_2{x}_{i2}+\dots +{\theta }_k{x}_{ik},for i=1,2,\dots , n$$

where $y_i$ is the value of the dependent variable $\overrightarrow{Y}$ for the $i_{th}$ case, $x_{ij}$ is the value of the $j_{th}$ independent variable $X_j$ for the i^th case, ${\theta }_o$ is the Y-intercept of the regression surface (for multidimensional features), each ${\theta }_j$, $j=1,2,\dots , k,$ is the slope of the regression surface for variable $X_j$. In Equation (1), there are n examples and k features. In regression analysis, the variables $X_j$ are considered fixed quantities, although, in correlation analysis, they are random variables. $X_j$ are independent of the error term for any case considered. In matrix notation, the model can be rewritten as (2).

$$\overrightarrow{Y}=x\theta$$

(2)

where the output vector $\overrightarrow{Y}$, the parameter vector $\theta$ is the column vector of length k + 1, and the feature matrix X is $n\times \left(k+1\right)$ (with all first elements equal to 1, and the second columns are feature values of $X_1$, etc. We have to estimate the unknown values of $\theta$. One of the basic common methods used to identify the unknown value is the gradient descent algorithm.

3.2. Gradient Descent Algorithm

Gradient descent is mainly of three kinds, each of which uses different sizes of data to compute the gradient of a function. Sometimes where the data is huge, it takes a long time for the gradient descent algorithm to converge. To decrease the time complexity, data will be divided into chunks of data. Based on the volume of the chunk received, a trade-off can be established between maintaining the accuracy of the parameter intended to update and the time taken to update.

Batch Gradient Descent

Vanilla gradient descent shown in (3), often referred to as batch gradient descent, is used to calculate the gradient of the cost function about the parameters for the whole training dataset.

$$\theta =\theta -\alpha .{\nabla }_{\theta }J\left(\theta \right)$$

(3)

where $\alpha$ is the learning rate, ${\nabla }_{\theta }J\left(\theta \right)$ is the gradient of the cost function for the parameter $\theta$. If there is a huge amount of data, computation of the gradient will be carried out on the whole dataset, which is challenging and sometimes infeasible for storage. Batch gradient descent also prevents us from updating our model in real-time, i.e., on the go. The derivative can be computed by either using state-of-art libraries in MATLAB or Python. Manual derivation requires gradient checking to be performed. The size of the update we can make is determined by the learning rate, after which the parameters are modified in the gradient’s direction. With batch gradient descent, a global minimum is obtained for the convex optimization function and a local minimum for the non-convex optimization function. It might be challenging to select an appropriate learning pace. A trade-off has to be achieved for the learning rate. If the learning rate is high, the loss function diverges from the minimum point, whereas if the learning rate is small, the loss function converges to the minimum very slowly.

3.3. Cost Function

The cost function for the polynomial regression is defined by the mean squared error (i.e., the square of the difference between the expected output and the actual output), as shown in (4).

$$J\left(\theta \right)=\frac{1}{2m}\left[{{\displaystyle \sum }}_{i=1}^m{\left(\left(h_{\theta }\left(x^{\left(i\right)}\right)\right)-y^{\left(i\right)}\right)}^2+\lambda {{\displaystyle \sum }}_{j=1}^n{\theta }_j^2\right]$$

(4)

In Equation (4), $\lambda$ is the regularization parameter. The term is added to prevent overfitting. $\lambda$ values have to be chosen based on whether it is a bias or variance problem. For cost function in the (4) gradient descent algorithm will be as given in (5).

Repeat

$$\begin{gathered} \{ \\ \left.\begin{array}{c}{\theta }_o:≔{\theta }_o-\alpha \frac{1}{m}{{\displaystyle \sum }}_{i=1}^m\left(h_{\theta }\left(x^{\left(i\right)}\right)-y^{\left(i\right)}\right)x_o^{\left(i\right)}\\ {\theta }_j:≔{\theta }_j-\alpha \left[\frac{1}{m}{{\displaystyle \sum }}_{i=1}^m{h}_{\theta }\left(x^{\left(i\right)}\right)-y^{\left(i\right)})x_j^{\left(i\right)}+\frac{\lambda }{m}{\theta }_m\right]\\ for j=1,2,3,\dots ,n\end{array}\right\} \\ \} \end{gathered}$$

(5)

3.4. Principal Component Analysis

For data exploration and analysis, principal component analysis (PCA) is the most widely recommended algorithm across all domains of research when the data is vast, consisting of multiple variables and incredibly correlated with a couple of observations in keeping with the variable. To cope with high-dimensional statistics, a smaller set of trends should be replicated in the authentic statistics in a lower-dimensional subspace with the least diploma of statistics loss is wished to be found.

PCA is one of the most effective dimension reduction methods, as it identifies new dimensions from the linear combinations of the original ones, due to the simplicity with which these additional dimensions may be generated, as shown in Figure 2, PCA can be used for data exploration and data pretreatment.

3.4.1. Method

The major component of computation is rather simple. It is built on the principles of linear algebra, including eigenvectors and eigenvalues as well as vectors, matrices, their operations, and properties. A variety of mathematical concepts, such as the Karhunene–Loeve theorem, demonstrate that the space represented by a data matrix Y of size n x p may be precisely partitioned into rank-1 vectors U and V, which are the foundation of PCA. The data matrix Y of size n x p may be divided perfectly into rank-1 vectors U and V like, as given in (6).

$$Y=U S V^T$$

(6)

where U, S, and V are obtained using a method known as singular value decomposition (SVD). S is a diagonal matrix, with all off-diagonal elements being zeros, and the diagonal elements, known as singular values, are ordered as given in (7).

$${\lambda }_1>{\lambda }_2>\dots >{\lambda }_p$$

$$S=\left[\begin{array}{cccc}{\lambda }_1& 0& \dots & 0\\ 0& {\lambda }_2& \dots & 0\\ \dots & \dots & :& \dots \\ 0& 0& \dots & {\lambda }_p\end{array}\right]$$

(7)

where U is a n x p orthonormal matrix. Each of the p columns of U represents one of the eigenvectors or major axes of Y, and they are all orthogonal to one another. i.e., $U{U}^T=U^{T}U=I (I=Identity matrix$ and $U^{-1}=U^T)$. V is an orthonormal matrix of type p. The eigenvectors or primary axes of Y^T are represented by each of the $\mathrm{p}\times \mathrm{p}$ columns of V, and they are all orthogonal to one another, i.e., $V{V}^T=V^{T}V=I$ and $V^{-1}=V^T$. The decomposition of the matrix Y into a set of rank-1 vectors U and V is depicted graphically in Figure 1. The number of singular values that are non-null is equal to the number of vectors. This number is the same as the rank of matric Y with $Rank\left(Y\right)\le \min \left(n,p\right)$. These notations are significant since the primary axes are represented by the eigenvectors of the data matrix. There is an associated eigenvalue for each eigenvector. When the data are projected onto the eigenvector, the eigenvalue is a scalar that indicates how much variation is explained by it.

The overall variance of the data is represented by the sum of all the eigenvalues. Using the eigenvectors of the covariance matrices $C_p\left(C_p=Y^{T}Y\right)$ and $C_n\left(C_n=Y{Y}^T\right)$ provides an alternate method of seeing the main components. These two square symmetric matrices’ respective eigenvectors, U and V, are what we previously discovered. The square roots of the eigenvalues are represented by singular values S. In actuality, we are related to the following people, as given by (8);

$$\left.\begin{array}{c}Y=US V^T\\ Y^{T}YV=V{S}^2\to V=eig\left(Y^{T}Y\right)=eig\left(C_p\right)\\ Y{Y}^{T}U=U{S}^2\to U=eig\left(Y{Y}^T\right)=eig\left(C_n\right)\end{array}\right\}$$

(8)

3.4.2. Dimension Reduction and Variance

We create the new data Y with fewer components using the Eigen decomposition, or SVD technique after computing the eigenvectors U and V and the eigenvalue. Only the principal components up to rank r—specifically, the r principal components with the largest variance explained—are used to rebuild Y. We presume that the remaining factors are noise and that they only slightly affect the variance. The reconstructed Y is given by (9),

$$Y_{rec}=U\left(:,1:r\right)diag\left(S_{\left[1\dots r\right]}\right)V{\left(:,1:r\right)}^T$$

(9)

There are no straightforward statistical tests to determine how many major components there are since PCA is a data-pushed technique that does not make any assumptions on the approximate noise distribution. Calculating the cumulative variance defined and selecting a sure threshold, inclusive of eighty percent, is one approach to choosing what number of additives to keep. Each component’s proportion of variation is explained and the total variance explained is given by (10):

$$\left.\begin{array}{c}\left(percent\right)Va{r}_i=\frac{{\lambda }_i}{\sum {\lambda }_i}*100\\ \left(cumulative\right)Va{r}_i={\displaystyle \sum }_1^{r}perent Va{r}_i\end{array}\right\}$$

(10)

4. Random Forest

Polynomial regression and principal component analysis were used to extract regenerative energy, and the results are compared with another regression method known as random forest regressor, as it is generally recommended as one of the most accurate regression algorithms for many applications.

As illustrated in Figure 3, random forest ensembles B trees (T 1 (X),...,T B (X), where X = x₁,...,x_p is a p-dimensional vector of molecular descriptors or properties linked with a molecule. The ensemble produces B outputs $\left\{\widehat{Y_2}=T_1\left(X\right),\dots , {\widehat{Y}}_B=T_B\left(X\right)\right\}$ where ${\widehat{Y}}_b, b=1, \dots , B$, is the $b^{th}$ tree’s forecast for a molecule. The effects of all trees are mixed to generate a single, very last forecast, Y. In type issues, the elegance forecasted with the aid of using maximum trees is Y. It is the average of the individual tree predictions in the regression.

4.1. Procedure for Training

The training procedure goes as follows, given activity data on a set of n molecules, D = (X₁, Y₁),...,(Xn, Yn), where Xi,i = 1,..., n is a vector of descriptors and Y_i is either the matching category label (for example, active/inactive) or training of interest.

• Make a bootstrap sample using the n-molecule training set (i.e., sampled randomly, with replacement, n molecules)
• Assemble a tree with more modifications at each bootstrap instance by choosing the most beneficial split from an m to try a descriptor subset (rather than all of them) at each node. The single tuning parameter of the algorithm, in this case, is an m try. The tree is not chopped back until it has reached its maximum capacity (i.e., until no further splits are imaginable).
• Until the B identical trees are put together, repeat the previous steps. When mtry = p, the random forest approach is the same as bagging since the best split at each node is chosen among all descriptors. Although other alternatives are taken into consideration, CART is the tree-growing algorithm employed in random forests.

The method’s effectiveness is due to two differences between the two techniques compared to building a single decision tree. The traditional tree-growing method examines all descriptors for splitting efficiency at every node, whereas Random Forest tests the m_try of the descriptors. The analysis is fairly fast since “m” is often quite little (customary within the program is that the root of the quantity of descriptors is used for categorization). Second, they often would like some pruning, employing a single call tree to induce the optimum model quality and forecast strength. This can be frequently accomplished by cross-validation and may fritter away a large amount of the estimations. On the other side, random forest undertakes absolutely no pruning. We found that random forest can be taught faster than a single decision tree when there are many descriptors. The two essential ensemble learning techniques, in addition to random forest, are bagging and boosting, both of which significantly outperform a single tree in terms of efficiency. Bagging may be characterized as a random forest with mtry = p, as was previously mentioned. A series of trees are trained on all of the data in boosting, but each tree’s weighting of the data depends on whether the one before it correctly classified the data. Each tree in the regression is generated on the residuals of the preceding trees. The ensemble outputs are predicted using a unit vote (in classification) or a sample mean (in the regression). Different ensemble tactics—random FIRM15 and decision forest—are communicated. In random FIRM, an ensemble of timber is constructed for the usage of all the schooling data, however, at every split, a descriptor is selected at random primarily based totally on eventualities associated with the split’s statistical importance as decided with the aid of using the classifier’s relevance. Using descriptors that the other trees do not have access to, each tree in the ensemble of decision forest is built to maintain prediction accuracy beyond a certain threshold.

4.2. Importance of Descriptor

The significance of descriptors in a decision tree is well-known for its potential to choose “essential” descriptors from a large set while ignoring (often irrelevant) others. Furthermore, the decision tree provides a specific model that describes the link among these descriptors as well as predictions, making model interpretation easier. Random forest learns the ability to pick “important” descriptions as an ensemble of trees; it does not generate an effective model. The link between descriptors and activities of concern, on the other hand, is buried behind a “black box.” However, during training, an assessment of how each descriptor contributed to Random Forest’s predictive performance may be derived. As the performance descriptor “noise up” (e.g., substituted with random noise), the accuracy rate should reduce at a considerable rate. However, if a descriptor is unimportant, “noising” should not influence performance. In the classified situation, it turns out that the difference in predictive performance is frequently a less accurate metric than the changes in margins. The margin is determined by comparing the total number of votes cast for the correct class to the group of votes cast most frequently for the incorrect class. The margin is 0.6—max (0.3, 0.1) = 0.3 if a molecule in class 1 obtains 60, 30, and 10% of the options in categories 1, 2, and 3, respectively. The ensemble forecast is accurate when the range is positive; anytime the range is negative, it is incorrect. To determine a descriptor’s importance in classification, one uses the likelihood that the margin of error will become smaller (more negatively or even less positively) when it is “noised” up.

The next approach is used to estimate the relevancy of descriptors in detail. Based on the OOB data for each tree as it grows, complete predictions are made. The tree predicts each such updated data set while simultaneously randomly permuting each descriptor in the OOB data. Let M be the mean margins depending on the OOB forecast and the predictive model with the j the descriptor permuted. Then the j^th descriptor’s significance is just $M-M_j$. Margin values are efficiently substituted with squared forecast errors in regress situations. $M_j$ reflects the average margin depending on the OOB. The dataset from the required model is gathered and simulated in MATLAB to predict the maximum energy that is discussed further. The analysis from prior work of ANN and real-time optimization are obtained and compared with the current research on random forest regression and the polynomial regression model to extract energy from regenerative braking.

5. Dataset and Simulation

The whole system was designed, verified, and implemented in MATLAB/Simulink.

Table 1. System parameters used for the simulation.

S. No	Parameter	Value
1	Mass (M)	1704 kg
2	Air density (ρ)	1.2 kg/m2
3	Height of the centroid of the EV	0.4 m
4	Wheelbase (L)	2.63 m
5	The centroid of the vehicle to the front axle (La)	1.24 m
6	The centroid of the vehicle to the rear axle (Lb)	1.39 m

represents the system parameters used for the simulation that has been taken from the previous work done in [12] by the author. A real-time dataset was collected from the open-loop analysis of the system in MATLAB/Simulink a part of which is shown in Figure 2. Initially, the data was reconfigured by rounding off the decimal values and verifying the data results theoretically. The Verified results were then normalized to ensure the minimum time of convergence and also to make sure that the result did not diverge. From this data, 60% of it was used for training the system using polynomial regression. A hypothesis function that predicted the proportion of the regenerative braking force to the total force on the front axle was produced using the polynomial regression. The ideal regenerative braking force was then produced by multiplying the regenerative ratio by the sum of the frontal braking forces.

Figure 4 illustrates how the predicted value was assessed for the NEDC: ECE R15 drive cycle, a European driving cycle with four urban and two extra-urban vehicle modes. Figure 4 depicts the polynomial regression distribution of braking forces for the NEDC: ECE R15 driving cycle. As only braking was considered in the scope of this paper, the force distribution was considered only during the deceleration curve. From the ECE R15 drive cycle, it was observed that the acceleration demand and deceleration demand required were less since it was urban-driven. For the whole drive cycle, the range of the brake demand was between 0 and −0.1. Since the vehicle was front-driven, i.e., the motor is on the front side, the front braking force will be split into regenerative braking force and frictional braking force on the front wheels. From Figure 4, it is observed that the regenerative braking force was higher than the front braking force, thereby reducing the total amount of frictional braking force applied to the vehicle. In this way, the regenerative braking force was maximized using polynomial regression.

The power harvested for the cycle using polynomial regression is shown in Figure 5. The fact that the power is negative means that the voltage at the DC-Link will also be negative, making it possible to recover the energy and store it in a battery or supercapacitor. The motor power is proportional to the car’s speed and the driver’s brake demand. The same analysis has been performed using an ensemble method, bootstrap aggregation, like the random forest, to evaluate its efficiency. With ensemble methods, a lot of other analysis has been carried out to reduce its computation time. Initially, all the descriptors/parameters were classified and plotted to know their homogeneity. From Figure 6, the number of levels for the state of charge of the battery and supercapacitor are the same. The highest number of levels can be seen for the speed variable, considering it is non-categorical. Therefore, homogeneity is not maintained. So, to understand the predictor’s importance, the standard CART algorithm was not used. Instead, the curvature test or interaction test was used for the accurate predictor importance estimates.

With the curvature test, the predictor importance estimate was calculated and is shown in Figure 7. It can be seen that the result is highly dependent on the brake demand of the vehicle, i.e., the force driver applied to the brake. There is void importance for the speed variable. Following the brake demand, the next priority was equally shared by the SOC of the battery and supercapacitor.

After analyzing that there was high importance of the brake demand of the vehicle, it could be possible that a correlation could be identified among the other parameters and club them into one. From Figure 8, it was observed that there was no correlation among the parameters. First, speed had an association estimate of 1, i.e., fully correlated, with the speed parameter itself. With the rest of the parameters, the association estimate was 0, indicating there was no correlation at all. Thus, there was no possibility of combining the other three parameters, so that computation time could be reduced or data visualization was possible with the approach of ensemble methods.

For data visualization purposes, principal component analysis was carried out on 4 parameters and reduced to 3 parameters. The data was visualized in three-dimensional using the scatter and surface plots that are shown in Figure 9, where z_1, z_2, and z_3 were the principal vectors projected onto the 3D plane. The graph depicts the data scattered upon the projection of data onto a 3D plane. The input had 4 attributes, and as PCA was used for visualization purposes only, the visualization did not indicate anything significant about how the data was represented. The scatter plot showed discontinuity in data and hence, the derivative-based models were ruled out; this also ruled out the importance of PCA in work.

From the analysis done, it was identified that the selection of the split predictors at each tree could not be reduced because of their low correlation. Therefore, computing the ratio of the regenerative braking force to that of the total front braking force was carried out using a random forest regression algorithm. The distribution of the forces is shown in Figure 10.

From Figure 10, it can be observed that the regenerative braking force was equal to the total front braking force. The regenerative braking force during the time interval of 150–200 s is observed in Figure 4, where the force was around −1000 N, whereas in Figure 10, the force was around −700 N. From this analysis, polynomial regression produced better results in maximizing the regenerative braking force than compared to that of random forest regression. The corresponding regenerative powers are plotted in Figure 11. As a result of the force not being maximized, there was low power harnessed by the regenerative braking with the random forest regression methodology.

From Figure 11, it can be observed that the regenerative power was −7000 W (approximately) during the time interval 150–200 s, whereas from Figure 5, it can be observed that regenerative power was around −8000 W during the time interval 150–200 s. Therefore, polynomial regression performed better in terms of maximizing the regenerative braking energy.

Apart from the NEDC drive cycle, other drive cycles such as FTP, HWFET, and WLTP class 3 drive cycles were considered to analyze the effect of different driving conditions. The Federal Test Procedure (FTP) drive cycle was created for an urban commute that includes frequent stops and a part of highway driving by the US Environmental Protection Agency. The Highway Fuel Economy Test (HWFET) is used to measure the energy efficiency of a vehicle that will be produced while driving on highways. However, a Class 3 cycle is made of four-speed zones to mimic the real-time drive scenario; one representative of urban driving, suburban driving, extra-urban driving, and a highway zone.

6. Result Analysis

A comparative analysis was made between polynomial regression (PR), random forest regression (RFR), artificial neural networks (ANN), and real-time optimization (RTO) to understand which was the better performer in terms of maximizing the regenerative braking force/energy. The analysis has not been limited to only one drive cycle but has been extended over four different drive cycles with four different types of driving conditions at a particular brake demand of 0.05. The analysis covers city driving conditions, highway driving conditions, different terrain conditions, and congested road conditions. The results for fuzzy, ANN, and RTO are taken from the papers cited [12,23,24].

Table 2. Comparative analysis between polynomial regression (PR), random forest regression (RFR), fuzzy, artificial neural networks (ANN), and real-time optimization (RTO).

Brake Demand = 0.05 (between 155–165 s)
	Parameter	PR	RFR	Fuzzy	RTO	ANN
FTP	Max Regen Power (W)	16,000	15,770	8160	22,050	8643
	Max Regen Force (N)	1409	1453	490	1550	910
HWFET	Max Regen Power (W)	17,600	13,470	10,360	21,470	8500
	Max Regen Force (N)	860	850	612	1300	450
NEDC	Max Regen Power (W)	8100	8850	7850	11,000	7663
	Max Regen Force (N)	970	1000	810	1130	620
WLTP class 3	Max Regen Power (W)	11,700	11,442	7990	13,510	7775
	Max Regen Force (N)	1343	1358	949	1574	903

* All Values are negative.

represents a complete comparison of the analysis of maximum regenerative power generation using the proposed machine learning techniques—polynomial regression and random forest method with the already analyzed techniques such as artificial neural networks, real-time optimization, and fuzzy logic. However, it is proved that fuzzy logic is just the basic algorithm that yields a low power. It is also observed that real-time optimization maximizes the regenerative braking energy as much as possible and fully leverages the regenerative force by making the frictional braking force as low as possible.

From the proposed algorithms, the yield of energy from polynomial regression and Ensemble methods are close by, but in the case of urban driving and congested roads, random forest outperforms polynomial regression. When compared to real-time optimization, the latter is also the best algorithm that yields maximum regenerative energy.

The accuracy metrics taken into consideration for polynomial regression and random forest regression are RMSE and R2 Squared. Comparatively, random forest regression produces more accurate results than polynomial regression as per

Table 3. Accuracy metrics for polynomial regression (PR) and random forest regression (RFR).

Method	RMSE	R2 Squared
PR	0.042588	1
RFR	5.16E-04	1

, however, polynomial regression has better standards.

Overall, polynomial regression performs better in terms of maximizing the regenerative braking force at different driving conditions when compared with random forest regression; the RTO performs far better than polynomial regression as well. Therefore, the RTO can be used to yield maximum regenerative energy, but it also takes a longer time to process. RTO used in [12] performs better compared to the machine learning models. In addition, RTO is a blatant optimization problem, it might not work for out-of-sample or real-time data. The machine learning models presented will work for the out-of-sample data collected in real-time and will follow the standard brake distribution curve. Reference [3] considers critical situations or heavily congested driving situations, machine learning algorithms with lesser processing time should be used, and a polynomial regression algorithm can be the best alternative for real-time optimization as it can harness the maximum energy.

7. Conclusions and Future Scope

The regenerative braking force is estimated using two different machine learning algorithms, i.e., random forest regression, and polynomial regression. Among these two, polynomial regression performs better in terms of maximizing the regenerative braking force under varied driving conditions. The paper also presents a comparison with other methodologies, and it is proved that real-time optimization yields the most amount of energy followed by the polynomial regression method; polynomial regression can be the best alternative to yield very high regenerative energy faster. All the methodologies are performed and implemented in MATLAB/Simulink and have considered the same metrics and constraints while designing the algorithm.

The paper can be extended to include battery control, SOC, and SOH estimation of battery and supercapacitors. The latency of the braking is also not considered in the computation of regenerative energy and can be taken up in the future.