# Synthesis and Feature Selection-Supported Validation of Multidimensional Driving Cycles

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Research Aim and Contributions

## 2. Recording of Driving Cycle Data

#### 2.1. Data Collection

- timestamp,
- geographical coordinates (latitude, longitude),
- elevation (altitude),
- vehicle speed,
- travelled distance (from odometer),
- cumulative fuel consumption.

#### 2.2. Data Processing

## 3. Synthesis of Multidimensional Driving Cycles

#### 3.1. Markov Chain-Based Synthesis Method

- The vehicle velocity is discretized with resolution of 0.1 km/h in the range from 0 km/h to 90 km/h, which is equal to the velocity limit of considered buses (a total of 901 velocity discrete values).
- The vehicle acceleration is reconstructed by means of recorded vehicle velocity differentiation, with the sampling time of one second. The acceleration range is set from –2 m/s
^{2}to 2 m/s^{2}, because the majority of accelerations observed in recorded driving cycles fall in this range [25], while the acceleration resolution is set to 0.15 m/s^{2}(a total of 28 acceleration discrete values). - The road slope is discretized with the resolution of 0.1° in the full range from −6° to 6° (a total of 121 road slope discrete values; see Figure 4b).
- The road slope time derivative range is set from −0.75°/s to 0.75°/s with the resolution of 0.25°/s (a total of 7 road slope derivative discrete values).

#### 3.2. Computing of Transition Probability Matrix

#### 3.3. Generating Synthetic Driving Cycles

## 4. Driving Cycle Feature Selection

#### 4.1. Preparing Driving Cycle Data for Regression Modelling

#### 4.2. Nominating Initial Set of Statistical Features

- velocity $\mathit{v}$, acceleration $\mathit{a}$, and road slope $\mathit{\theta}$,
- total power on the wheels ${\mathit{P}}_{\mathit{w}}={\mathit{F}}_{\mathit{w}}\circ \mathit{v}$ (where the operator $\circ $ denotes element-wise multiplication),
- horizontal velocity component ${\mathit{\phi}}_{\mathit{h}}=\mathit{v}\circ \mathrm{cos}\mathit{\theta}$,
- vertical velocity component ${\mathit{\phi}}_{\mathit{v}}=\mathit{v}\circ \mathrm{sin}\mathit{\theta}$,
- and power-to-mass ratio (i.e., specific power) ${\mathit{P}}_{\mathit{M}\mathit{R}}=\mathit{v}\circ \mathit{a}$,
- specific driving cycle energy per kilometre ${\mathit{E}}_{\mathit{d}\mathit{c}}={\mathit{P}}_{\mathit{w}}\xb7\Delta T/d$ (sampling time $\Delta T$ is 1 s and $d$ is a total distance travelled),

^{−1}for the first mode, and [0.0025, 0.005] m

^{−1}for the second mode). Then, the ratio of maximum and mean amplitude value is calculated for each of these two dominant, low-mid frequency bands.

#### 4.3. Selection of Significant Statistical Features Based on Linear Regression Models

#### 4.4. Neural Network Model for Predicting Fuel Consumption for Synthetic Cycles

#### 4.5. Performance Analysis of Regression Models

^{2}score, mean value ${\mu}_{res}$, and standard deviation ${\sigma}_{res}$ of model prediction residuals/errors. The R

^{2}indicator represents the proportion of variance in response variable which can be explained by the predictor variables. The value R

^{2}= 1 corresponds to the ideal fit, while R

^{2}= 0 means no correlation, i.e., it corresponds to the case when the model output is constant and equal to the mean value of recorded fuel consumptions.

^{2}of 0.942 and lowest ${\sigma}_{res}$ of 2.47 L/100 km, thus representing the ultimate prediction accuracy. Although being significantly simpler, both LR and LASSO models follow the NN model very closely, with R

^{2}equal to 0.935 and 0.939, respectively, and ${\sigma}_{res}$ being around 5% larger. It should also be noted that the predictions are well balanced, i.e., the mean error ${\mu}_{res}$ is close to zero.

^{2}score on test dataset. Finally, it is checked which of the models result in a prediction accuracy close to the original model (cf. Table 1), while containing a low number of features. The obtained results indicate that in the case of LR model the first 18 features are sufficient for a given prediction task (see Figure 10c and also green plot in Figure 10a), while the first six features are sufficient for LASSO model (see Figure 10d and also blue plot in Figure 10a). Since LASSO model results in better performance and less features than LR model (R

^{2}value of 0.940 vs. 0.937 for the case of 6 vs. 18 features), it is adopted as a referent model for further analyses. The main reason for choosing LASSO instead of NN model is because it uses a limited set of features rather than the complete driving cycle information as input, and it also performs automatic feature selection through L1 regularization (see Section 4.3). Therefore, the values of the regression coefficients ${\widehat{\beta}}_{j}$ for the corresponding subset of six most significant features (Table 2) are utilized for further validation of synthetic driving cycles. Note that the small excess or shortfall in resulting R

^{2}values for the case of reduced order models when compared to full order models (cf. the above R

^{2}values with those given in Table 1) are the result of better or worse generalization on unseen/testing data.

## 5. Validation of Synthetic Driving Cycles

#### 5.1. Lumped Performance Indicators

#### 5.2. Correlation Analysis of Lumped Performance Indicators with Respect to Fuel Consumption Deviation

#### 5.3. Selection of Representative Driving Cycles

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

A/C | Air conditioning (system) |

ADAM | Adaptive Moment Estimation |

CAN | Controller Area Network |

CMC | Combined micro-cycles |

Comb | combined recorded driving cycles |

ED | Euclidean distance |

FCD | Fuel consumption deviation |

FSM | Finite state machine |

GPRS | General Packet Radio Service |

GPS | Global Positioning System |

H3D | Three-dimensional histogram |

LASSO | Least Absolute Shrinkage and Selection Operator |

LR | Linear Regression |

MSDE | Mean specific driving energy |

MSE | Mean square error |

NEDC | New European Driving Cycle |

NN | Neural network |

RI | Regression Index |

SAFD | Speed-acceleration frequency distribution |

TPM | Transition probability matrix |

VRP | Vehicle routing problem |

WLTC | World-wide harmonized Light duty Test Cycles |

## Appendix A

Algorithm A1. Pseudo Code for Parameterization of TPM |

Step 1: Initialize set containing all recorded driving cycles data $\mathsf{\Psi}=\left\{{\mathbf{\psi}}_{1},{\mathbf{\psi}}_{2},\dots ,{\mathbf{\psi}}_{N}\right\}$, where |

${\mathbf{\psi}}_{i}=\left[\mathbf{v}\left(k\right)\mathbf{a}\left(k\right)\mathbf{\theta}\left(k\right)\dot{\mathbf{\theta}}\left(k\right)\right]{}^{T};i=1,2,\dots ,N$ |

Step 2: Initialize 8-D TPM as empty dictionary ${\Pi}_{8D}$ |

Step 3: Initialize discrete Markov state values |

${\mathit{v}}_{states}=\left[{v}_{\mathrm{min}},{v}_{\mathrm{min}}+\Delta v,{v}_{\mathrm{min}}+2\xb7\Delta v,\dots ,{v}_{\mathrm{max}}\right]$ |

${\mathit{a}}_{states}=\left[{a}_{\mathrm{min}},{a}_{\mathrm{min}}+\Delta a,{a}_{\mathrm{min}}+2\xb7\Delta a,\dots ,{a}_{\mathrm{max}}\right]$ |

${\mathit{\theta}}_{states}=\left[{\theta}_{\mathrm{min}},{\theta}_{\mathrm{min}}+\Delta \theta ,{\theta}_{\mathrm{min}}+2\xb7\Delta \theta ,\dots ,{\theta}_{\mathrm{max}}\right]$ |

${\dot{\mathit{\theta}}}_{states}=\left[{\dot{\theta}}_{\mathrm{min}},{\dot{\theta}}_{\mathrm{min}}+\Delta \dot{\theta},{\dot{\theta}}_{\mathrm{min}}+2\xb7\Delta \dot{\theta},\dots ,{\dot{\theta}}_{\mathrm{max}}\right]$ |

Step 4: Iterate through $\mathsf{\Psi}$ and update TPM |

for$i=1,2,\dots ,N$ do |

Step 4.1: Get number of recorded samples $M$ in ${\mathit{\psi}}_{i}$ |

Step 4.2: Iterate through ${\mathit{\psi}}_{i}$ |

for$k=1,2,\dots ,M-1$do |

Step 4.2.1: Assign indices for current discrete Markov state values |

$\hspace{1em}\hspace{1em}\hspace{1em}q=\mathrm{argmin}\left(\left|{\mathit{v}}_{states}-{v}_{i}\left(k\right)\right|\right);r=\mathrm{argmin}\left(\left|{\mathit{a}}_{states}-{a}_{i}\left(k\right)\right|\right);$ |

$\hspace{1em}\hspace{1em}\hspace{1em}w=\mathrm{argmin}\left(\left|{\mathit{\theta}}_{states}-{\theta}_{i}\left(k\right)\right|\right);h=\mathrm{argmin}\left(\left|{\dot{\mathit{\theta}}}_{states}-{\dot{\theta}}_{i}\left(k\right)\right|\right);$ |

Step 4.2.2: Assign indices for next discrete Markov state values |

$\hspace{1em}\hspace{1em}\hspace{1em}x=\mathrm{argmin}\left(\left|{\mathit{v}}_{states}-{v}_{i}\left(k+1\right)\right|\right);y=\mathrm{argmin}\left(\left|{\mathit{a}}_{states}-{a}_{i}\left(k+1\right)\right|\right);$ |

$\hspace{1em}\hspace{1em}\hspace{1em}z=\mathrm{argmin}\left(\left|{\mathit{\theta}}_{states}-{\theta}_{i}\left(k+1\right)\right|\right);t=\mathrm{argmin}\left(\left|{\dot{\mathit{\theta}}}_{states}-{\dot{\theta}}_{i}\left(k+1\right)\right|\right);$ |

Step 4.2.3: Increment count of registered transitions from state ${Z}_{k}$ to ${Z}_{k+1}$ stored in TPM |

if${\Pi}_{8D}$ do not have any registered transition for input keys $\left(q,r,w,h\right)$ then |

Assign empty dictionary to ${\Pi}_{8D,qrwh}$ |

else |

if${\Pi}_{8D,qrwh}$ do not have registered transition to $\left(x,y,z,t\right)$ then |

Initialize counter to zero, ${\Pi}_{8D,qrwh,xyzt}=0$ |

else |

Increment counter, ${\Pi}_{8D,qrwh,xyzt}={\Pi}_{8D,qrwh,xyzt}+1$ |

end for |

Step 5: Scale ${\Pi}_{8D}$ so that $\sum}_{x}{\displaystyle \sum}_{y}{\displaystyle \sum}_{z}{\displaystyle \sum}_{t}{p}_{qrwh,xyzt}=1,\forall \left(q,r,w,h\right)$ |

for each primary key denoted with $\left(q,r,w,h\right)$ and contained in ${\Pi}_{8D}$ do |

for each secondary key denoted with $\left(x,y,z,t\right)$ and contained in ${\Pi}_{8D,qrwh}$ do |

Calculate transition probabilities as ${\Pi}_{8D,qrwh,xyzt}=\frac{{\Pi}_{8D,qrwh,xyzt}}{\sum {\Pi}_{8D,qrwh}}\equiv {p}_{qrwh,xyzt}$ |

where $0<{p}_{qrwh,xyzt}\le 1$ |

end for |

end for |

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**Figure 3.**Histograms of (

**a**) distance travelled and (

**b**) travel duration for circular bus route from Figure 2.

**Figure 4.**Examples of recorded driving cycles for both driving directions: (

**a**) vehicle speed vs. time profile, and (

**b**) road slope vs. travelled distance profile.

**Figure 5.**Histograms of recorded and synthesized profiles of (

**a**) velocity, (

**b**) acceleration, and (

**c**) road slope.

**Figure 6.**Overview of data augmentation process based on splitting recorded driving cycles into micro-cycles and combining micro-cycles into artificial driving cycles.

**Figure 8.**Neural network architecture for fuel consumption prediction based on 3D histogram input derived from (synthetic) driving cycle.

**Figure 9.**Plots of (

**a**) predicted vs. real fuel consumption and (

**b**) probability distribution of prediction errors/residuals for different models and test dataset.

**Figure 10.**Results of feature selection method for full-order and reduced-order LR model (

**a**), and LASSO model (

**b**); and dependence of model’s performance in terms of R

^{2}score on number of features of LR model (

**c**) and LASSO model (

**d**).

**Figure 11.**Dependence of FCD on (

**a**) ED and (

**b**) RI performance indicators for the case of 3000 synthetic driving cycles, along with belonging correlation coefficient values (${\rho}_{ED,FCD}$, ${\rho}_{RI,FCD}$).

**Figure 12.**Synthesized velocity vs. time profiles (

**a**,

**b**) and road slope profiles (

**c**,

**d**) according to ED (

**a**,

**c**) and RI (

**b**,

**d**), along with recorded road slope profiles in (

**c**,

**d**).

**Table 1.**Performance indicators calculated for different regression models aimed at predicting fuel consumption (test dataset used).

Regression Model | ${\mathit{\mu}}_{\mathit{r}\mathit{e}\mathit{s}}[\mathbf{L}/100\mathbf{km}]$ | ${\mathit{\sigma}}_{\mathit{r}\mathit{e}\mathit{s}}[\mathbf{L}/100\mathbf{km}]$ | R^{2} [-] |
---|---|---|---|

LR | 0.24 | 2.59 | 0.935 |

LASSO | 0.24 | 2.51 | 0.939 |

NN | 0.12 | 2.47 | 0.942 |

**Table 2.**Finally adopted subset of most significant features of Lasso regression model, including the corresponding regression coefficient values ${\widehat{\beta}}_{j}$ and minimum/maximum feature values (calculated before applying normalization).

No. | Feature | Min/Max Values | Unit | Regression Coefficient Value * |
---|---|---|---|---|

1. | Mean specific driving energy per kilometre travelled $\left({\overline{E}}_{dc}^{+}={\displaystyle \sum}_{k=1}^{N}{E}_{dc,k};\forall {E}_{dc,k}0\right)$ | 1.23/14.31 | MJ/km | 98.11 |

2. | Standard deviation of vertical velocity component | 0.02/1.82 | km/h | −16.35 |

3. | Mean velocity | 0.05/41.69 | km/h | −16.34 |

4. | Root mean square of amplitude of road slope frequency response | 0.39/5.23 | - | 7.93 |

5. | Number of vehicle stops per kilometre | 0/8.37 | km^{−1} | 7.17 |

6. | Mean positive acceleration | 0.22/1.12 | m/s^{2} | 3.91 |

7. | Intercept | 33.52 |

**Table 3.**Values of selected statistical features for combined recorded cycle (Comb) and most representative synthetic driving cycles selected according to RI and ED criteria.

No. | Statistical Feature | Unit | Comb | Most Representative Driving Cycle | |
---|---|---|---|---|---|

${\mathit{\psi}}_{\mathbf{ED}}^{*}$ | ${\mathit{\psi}}_{\mathbf{RI}}^{*}$ | ||||

1. | Mean velocity | km/h | 23.03 | 23.25 (+1.0%) ^{1} | 22.18 (−3.7%) |

2. | Mean positive acceleration | m/s^{2} | 0.45 | 0.48 (+5.6%) | 0.47 (+3.6%) |

3. | Number of stops per kilometre | km^{−1} | 1.87 | 2.15 (+15.0%) | 1.95 (+4.5%) |

4. | Mean specific driving energy per kilometre travelled | MJ/km | 4.32 | 4.45 (+3.2%) | 4.40 (+2.0%) |

5. | Standard deviation of vertical velocity component | km/h | 1.34 | 1.32 (−1.5%) | 1.16 (−13.5%) |

6. | Root mean square of amplitude of road slope frequency response | - | 1.33 | 1.25 (−6.0%) | 1.08 (−19.1%) |

7. | Standard deviation of road slope | ° | 2.59 | 2.41 (−6.6%) | 2.44 (−5.5%) |

8. | Standard deviation of the total power on the wheels | kW | 67.46 | 65.55 (−2.8%) | 64.33 (−4.6%) |

9. | Standard deviation of power-to-mass ratio | W/kg | 3.61 | 3.64 (+0.8%) | 3.47 (−3.9%) |

10. | Fuel consumption | L/100 km | 39.84 | 42.87 (+7.6%) | 41.05 (+3.0%) |

^{1}Values in parentheses represent relative differences of statistical features from the ones related to Comb driving cycle.

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**MDPI and ACS Style**

Topić, J.; Škugor, B.; Deur, J. Synthesis and Feature Selection-Supported Validation of Multidimensional Driving Cycles. *Sustainability* **2021**, *13*, 4704.
https://doi.org/10.3390/su13094704

**AMA Style**

Topić J, Škugor B, Deur J. Synthesis and Feature Selection-Supported Validation of Multidimensional Driving Cycles. *Sustainability*. 2021; 13(9):4704.
https://doi.org/10.3390/su13094704

**Chicago/Turabian Style**

Topić, Jakov, Branimir Škugor, and Joško Deur. 2021. "Synthesis and Feature Selection-Supported Validation of Multidimensional Driving Cycles" *Sustainability* 13, no. 9: 4704.
https://doi.org/10.3390/su13094704