1. Introduction
In recent decades, enhancements of information and communication technology have enabled the online collection of large quantities of driving data for the purpose of monitoring vehicle fleet operation for improved fleet management performance [
1]. An essential component of fleet management systems corresponds to solving vehicle routing problems (VRP) [
2], which is aimed at assigning vehicles and routes for accomplishing driving/delivery missions with a minimum number of vehicles and fuel cost. A VRP optimization algorithm requires the prediction of vehicle fuel consumption for the given route, period of day, driver, etc. [
3]. The fuel consumption prediction is based on a model, which should account for both driving behaviors and traffic conditions. Since some variables, which affect vehicle fuel consumption, are not measured on-board and/or are not broadcasted (e.g., vehicle weight and ambient conditions), significant efforts are made to model fuel consumption based only on a subset of key, standardly available variables, such as vehicle velocity and road slope [
4]. In general, fuel consumption models can be divided into two main categories: (i) first-principle (physics-based) models that describe vehicle dynamics at each time step using a set of mathematical equations corresponding to different vehicle subsystems and components [
5], and (ii) data-driven machine learning (ML) models that represent an abstract mapping of a set of input/explanatory variables into an output space defined by target variable(s) [
6].
The physics-based approaches can provide high prediction accuracy [
7,
8], but at the cost of low computing efficiency. Another disadvantage of physics-based models is that they require knowledge of many vehicle dynamics, powertrain parameters and multi-dimensional maps, which are usually not available. Therefore, instead of relying on a detailed microscopic model, there is a need to develop a fast, macroscopic model that predicts fuel consumption for an entire driving cycle at once to make VRP feasible. This problem can be efficiently solved by utilizing state-of-the-art ML modelling approaches [
9,
10,
11], which are characterized by automatic pattern learning from available data. Most commonly used models for these purposes are artificial neural networks (NNs), because they are universal approximators that can represent nonlinear characteristics of a complex system by using a nonlinear activation function [
12,
13]. The NNs can also readily be re-parameterized for different types of vehicle, which is not the case with physical models. The main challenge is to determine the NN architecture and inputs. The former is usually determined by the “trial and error” or “grid search” procedure [
14], while the latter varies between studies. For example, in [
15], the parameterization of models for the prediction of fuel consumption are realized on a subset of empirically selected statistical features, extracted from historical driving data. Another approach considers the application of a certain feature-selection technique such as a filter-based method (e.g., Pearson correlation) or an embedded method (e.g., LASSO regularization) to find the subset of the most relevant features for a given prediction task [
10,
16]. The disadvantages of these approaches are a need to extract most relevant input features instead of using full driving cycle information and a reliance on the automatic feature extraction property of NNs.
Recently, the success of NNs in regression and classification tasks has increased significantly with the development and application of more advanced NN types, such as convolutional neural networks (CNNs), recurrent neural networks (RNNs), etc., along with the utilization of deep learning (DL) techniques. A comprehensive analysis of different advanced ML models and DL techniques is given in [
17], where the prediction accuracies of corresponding models are examined and compared in the case of a short-term electric microgrid load forecasting problem. More complex prediction tasks should consider probabilistic model output, which, instead of point predictions (i.e., related to expected values), relies on the prediction of conditional probability distributions of target/dependent variables or related statistical indices (e.g., quantiles) [
18].
In this paper, we propose an NN model structured in two-stages, as outlined in
Figure 1, where the first NN predicts the driving cycle features based on the given route and traffic conditions (current, historical), while the second NN converts the driving cycle features into the fuel consumption prediction information. The main advantage of this two-stage model arrangement is that the traffic and energy modeling tasks are effectively separated.
This paper deals with the design of the second NN model based on experimentally acquired city bus GPS and CAN tracking data. By following the basic concept from [
19], the recorded driving cycles of different lengths are represented by a fixed-dimension histogram, whose axes correspond to vehicle and road states (velocity, acceleration and also road slope). For the purpose of well-conditioned NN training, which accounts for a limited resolution of fuel consumption measurement, the recorded driving cycles are pre-processed into a rich set of micro-cycles-based synthetic driving cycles. The fuel consumption prediction performance is examined in comparison with linear regression models, as well as the NN that does not account for road slope. The varying city-bus mass is not taken into account as the model input because (i) it is generally unknown and (ii) the NN model can implicitly extract a knowledge of mass from the driving cycle features [
20].
Apart from proposing the two-stage structure of fuel consumption prediction NN model, as shown in
Figure 1 and elaborated above, the main contributions of the paper include: (i) a method of synthesizing a rich set of representative and well-conditioned driving cycles by randomly concatenating a varying number of recorded micro-cycles corresponding to fuel consumption measurement resolution, and (ii) a systematic examination of the benefits of using the proposed NN fuel prediction consumption model when compared to simpler, linear regression and NN models, where the emphasis is on analyzing the influence of the effect of road slope.
In addition to fuel consumption prediction, the proposed NN model can be used (after retraining) to predict time-at-destination and electric vehicle (EV) battery state-of-charge (SoC), which represent crucial information for developing optimal EV charging management strategies [
21]. Additionally, the NN model can be employed as a key component in the validation of synthetic driving cycles [
22], which are typically used for the purpose of optimal design of EV configurations and energy management strategies, and in general fleet simulations [
23].
The paper is organized as follows. Driving cycle data acquisition for a city bus is described in
Section 2. The considered linear and neural network-based regression models are presented in
Section 3. A detailed comparative analysis of different models in terms of fuel consumption prediction accuracy is provided in
Section 4. Concluding remarks are given in
Section 5.
4. Comparative Results
The performance of fuel consumption prediction based on the four models established in
Section 3 has been examined by using the test dataset, comprised of augmented experimental data for two different resolutions of fuel consumption measurements, separately (rough and fine; see
Section 2).
Figure 8 shows the corresponding plots of predicted vs. real/recorded fuel consumption. Evidently, the simplest linear model
Poly1D exhibits the largest errors regarding fuel consumption predictions (i.e., deviation with respect to an ideal 1:1 straight line), while the H3D input-based NN model provides the most accurate predictions.
The respective distributions of absolute and relative prediction errors/residuals are shown in
Figure 9. Interestingly, the more complex linear regression model
Poly2D has comparable error distribution to the simpler NN model (NN-H2D). Specifically, accounting for the road slope information can make the regression model competitive with the NN model with no road slope input. The ultimate NN-H3D model significantly outperforms other models, with its prediction error distributed mostly within ±0.5 L (i.e., within ±10% of relative error). It should also be noted that the predictions are well balanced, with their mean error close to zero.
The model’s performance is further quantified by an
R2 value which is determined on the test dataset as:
The R2 value is often interpreted as an amount of variance in recorded data, which can be explained by the considered model. It takes values between 0 and 1, where R2 = 1 corresponds to an ideal fit, while R2 = 0 corresponds to the case in which the model output is constant and equal to the mean value of recorded fuel consumptions (here ). Additionally, R2 is expressed with respect to Poly1D model fuel consumption predictions, where is used instead of in (4). This indicator is denoted as R2corr, and can be interpreted as an additional variance in recorded data that can be explained by the considered model and that is not explained by the Poly1D model.
Table 1 provides standard deviations of prediction residuals, corresponding
R2 and
R2corr indicators for each model and average execution times
(for a single prediction), all calculated on the test dataset. The standard deviation of residuals of the NN-H3D model equals 0.19 L, and is reduced approximately by the factors of 4 and 2.5 when compared to the
Poly1D model and the
Poly2D and NN-H2D models, respectively. Note that an interval which contains 95% of model predictions can be calculated as approximately ±2∙
σres, which in the case of the NN-H3D model equals to around 0.35 L. All models have large
R2 values (>0.9), where the NN-H3D model approaches the almost ideal value of 1. The relative relations of different models are better described when using the
R2corr indicator, which shows that the
Poly2D and NN-H2D models explain around 60% more of the variance in the recorded data compared to the
Poly1D model, while in the case of the NN-H3D model this share exceeds 90% (i.e., again nearing the ideal value of 1). The average execution time
for a single prediction falls in the range from 1.5 to 1420 μs, depending on the model complexity. Moreover, the average execution time
for the NN-H2D model is 35% lower with respect to the NN-H3D model, in spite of significant differences in input size (i.e., vector with 13,325 elements for the NN-H3D model vs. 533 elements for the H2D model; see
Section 3).
Table 1 also provides the results related to the dataset with a rough fuel consumption resolution (equal to 0.5 L; see
Section 2). These results confirm that a similar level of performance can be achieved regardless of whether low-resolution or high-resolution fuel consumption tracking data are available.
5. Conclusions
Different approaches to fuel consumption prediction based on driving cycle data-fed linear regression and NN models have been presented. For the needs of well-conditioned model training, an originally recorded driving cycle dataset has been augmented by generating a rich set of combined driving cycles in terms of randomly concatenating a different number of recorded micro-cycles, where each micro-cycle is determined by cumulative fuel consumption equal to the measurement resolution of 0.5 L. In the case of the NN model, the driving cycles of different lengths are faithfully represented by a fixed-dimension histogram/matrix, where each cell value is obtained by counting the driving cycle discrete-amplitude state information related to vehicle velocity and acceleration, as well as road slope.
It has been shown that adding the road slope input significantly increases the prediction accuracy, i.e., the standard deviation of residuals reduces from 0.8 L to 0.5 L in the case of linear regression models, and from 0.5 L to 0.2 L in the case of neural network models. The ultimate performance of the NN model based on 3D histogram matrix input is manifested in its ability to capture variance in the recorded data to an extent that is 95% higher when compared to the basic linear regression model. By proper synthesis of driving cycles used for NN training, the favorable accuracy of NN-based fuel consumption prediction is preserved in the case of using more common, low-resolution fuel consumption measurement ( of 0.534 and 0.964 for the NN-H2D and NN-H3D models, respectively). The NN models have also shown to be rather computationally efficient ( between 920 and 1420 μs), thus confirming their suitability for vehicle routing problem (VRP) applications.
The main advantage of the proposed NN-based fuel consumption prediction approach when compared to the use of physical model-based microsimulations is its high accuracy with no demand on a number of physical city bus parameters. However, the approach can have deteriorated accuracy in extrapolation regions, i.e., when facing unseen input data. An additional disadvantage of the proposed NN model is that it does not take into account the impact of auxiliary devices such as heating, ventilation and air conditioning (HVAC) systems, which are not directly related to the driving cycle. Similarly, although implicitly captured through the driving cycle, the absence of an explicit passenger mass input can affect the prediction accuracy. Additionally, the presented study relates to a city bus fleet consisting of the same vehicle/engine type, and the NN designed may be inaccurate in the case of a mixed-vehicle fleet. Finally, the NN is not directly applicable in real-time applications, because it requires knowledge of the driving cycle in advance, which is usually not available.
Therefore, the main effort of future research should be directed towards the design of the first NN from
Figure 1, which would provide a driving cycle histogram for the NN designed in this paper, based on the knowledge of vehicle route and current and historical traffic condition data, as well as vehicle-specific features (e.g., curb weight, vehicle category, engine displacement or similar). In addition, the impact of HVAC consumption should be modeled through a separate regression model trained on the same experimental dataset, which would take into account the weather condition data (ambient air temperature and solar irradiance) as primary inputs, as well as vehicle velocity (reflecting the HVAC radiator inlet airmass flow) and passenger load (related to passengers’ metabolic heat). To further improve the prediction accuracy, vehicle mass data could be added as an input to the NN model. A possible practical realization of such a prediction model could be based on increasing each element of the flattened input histogram/matrix by the amount of vehicle mass averaged over the entire driving cycle. Similarly, information on vehicle or engine type should be fed to the NN to account for mixed fleet conditions. Finally, using a geographically and chronologically broader input dataset and/or applying certain regularization techniques should be considered to overcome possible extrapolation issues.