# Field-Based Prediction Models for Stop Penalty in Traffic Signal Timing Optimization

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction and Background

_{S}is fuel consumed in a complete stop (gallon), F

_{I}is fuel consumed by 1-h idling (gal/hour), and 3600 is a conversion factor.

_{S}is fuel consumed in a complete stop (liter), F

_{I}is fuel consumed by 1-h idling (liter/hour), d

_{s}is stopped delay (hour), and d

_{h}is delay caused by deceleration-acceleration action (hour).

## 2. Methodology

#### 2.1. Overview of the Stop Penalty Derivation

_{CSSP}is total fuel consumed during a CSSP, FC

_{D}is fuel consumed during the deceleration mode, FC

_{I}is fuel consumed during the idling mode, and FC

_{A}is fuel consumed during the acceleration mode.

_{DA}(FC

_{D}+ FC

_{A}), from what is consumed during the stopped delay, represented as FC

_{I}. Thus, we can say that FC

_{DA}is equal to a constant (K

_{e}) multiplied by the FC

_{I}, as expressed in Equation (5).

_{e}) can be denoted as shown below:

_{I}is divided by the total idling time (T

_{I}) in seconds, as shown in Equation (7). This step is important to assign the number of seconds of stopped delay equivalent to a stopping event, which is the stop penalty (K-factor). The PI (Equation (1)) can then be called FC-PI (since it is derived to reduce FC) and is expressed as shown in Equation (8).

_{i}is stopped delay on link i (seconds), S

_{i}is total stops on link i, and n is number of links in the network or links included in the optimization process.

#### 2.2. Factors Impacting Stop Penalty

_{I}/T

_{I}) and deceleration duration (T

_{D})) was not examined in the previous studies. On the one hand, higher FC rates (FC

_{I}/T

_{I}) result in lower K values, as it can be concluded from Equation (7). On the other hand, a longer deceleration duration causes a higher K value because the excess FC during the deceleration phase depends on the duration of the deceleration process, which depends on several factors, including the driver’s behavior and the traffic dynamics of the vehicle(s) in front of the stopping vehicle. Thus, regardless of how small the FC (per unit of time) during deceleration is, longer deceleration times mean more fuel consumed.

#### 2.3. Collection of Field Data

#### 2.4. Data Preparation

#### 2.4.1. Vehicle Classification

#### 2.4.2. Instantaneous Fuel Consumption Rates

#### 2.4.3. Cruising Speeds and CSSPs

#### 2.4.4. Road Gradient

#### 2.5. Machine Learning (ML) Models

#### 2.5.1. Multigene Genetic Programming

_{2}, x

_{5}, and x

_{8}. Several functions can be used for the evolution process (e.g., ×, −, +, Log, and √). The model is linear in the parameters for the coefficients ${\beta}_{0}$, ${\beta}_{1}$, and ${\beta}_{2}$ despite using nonlinear terms. As it is seen from Figure 5, the evolved model is a linear combination of nonlinear transformations of the predictor variables. Two important MGGP parameters that need significant attention are the maximum allowable number of genes and maximum tree depth. Restricting the tree depth mainly results in generating more compact models. The products of MGGP are profoundly nonlinear equations, reached after forming millions of preliminary models through a complex evolutionary process [42]. As described in previous sections, field data is used to generate the MGGP models, consisting of thousands of K values for a wide range of operating condition scenarios.

#### 2.5.2. Development of MGGP Models

^{2}).

^{2}) and the root-mean-squared error (RMSE) were employed to judge the performance of the introduced models. RMSE and R

^{2}equations are displayed in Equations (18) and (19).

## 3. Results and Discussion

#### 3.1. Models Training, Testing, and Validation

^{2}values of more than 0.96. It is important to note that the same training datasets (for the seven-vehicle groups) were used to develop multivariate linear regression models. The obtained R

^{2}values were less than 0.35 for most of those regression models. Such poor performance of the conventional multivariate linear regression models can be explained by limitations of such statistical regression techniques. In most cases, the best linear or nonlinear models developed using the commonly used statistical approaches are obtained after controlling a few equations established in advance (30). Thus, such models cannot efficiently consider the interactions between the dependent and independent variables.

_{10}value of the best RMSE and the mean RMSE achieved over the generations of a run. It is worth mentioning that the log

_{10}value of the RMSE is the error metric that GPTIPS attempts to minimize over the training data.

^{2}value on the training data. The final model for each vehicle group was selected based on two criteria, accuracy and model complexity. The developed models are validated with a fresh dataset to evaluate the generalization capability of the developed models. Figure 6(c1–c4) and Figure 7(c1–c3) show the acceptable performance of the models for the validation data.

#### 3.2. Parametric Analysis

#### 3.3. Comparison of Stop Penalties from Various Studies

- FC measurements were collected in the field, unlike Alshayeb et al. [24], whose stop penalties were simulation-based.
- Large number of LDVs and LDTs were included, whereas most previous studies used less than three vehicles.
- The tested fleet consisted of modern vehicles, whereas tested vehicles in the previous studies, except for Stevanovic et al. [23], are old for contemporary standards.
- Tested vehicles covered long distances, resulting in a significantly larger dataset than those used in the previous studies.
- The models cover multiple factors impacting the stop penalty (vehicle type, cruising speed, road gradient, FC idling rate, driving behavior, and decelerating duration), whereas most of the previous studies investigated only the impact of the cruising speed.

## 4. Conclusions and Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Dynamics and kinematics of a stopped vehicle [24].

**Figure 6.**Predicted versus computed stop penalty of light-duty vehicle (LDV) groups: (

**1**) LDV1, (

**2**) LDV2, (

**3**) LDV3, (

**4**) LDV4, (

**a**) training data, (

**b**) testing data, (

**c**) validation data.

**Figure 7.**Predicted versus computed stop penalty of light-duty truck (LDT) groups: (

**1**) LDT1, (

**2**) LDT2, (

**3**) LDT3, (

**a**) training data, (

**b**) testing data, (

**c**) validation data.

**Figure 10.**Parametric analysis of the developed models. (

**a**) Initial speed vs. stop penalty; (

**b**) Final speed vs. stop penalty; (

**c**) Accelerating grade vs. stop penalty; (

**d**) Idling FC rate vs. stop penalty; (

**e**) Deceleration duration vs. stop penalty; (

**f**) Acceleration vs. stop penalty.

**Figure 12.**Stop penalty vs. road gradient from the field and simulation [24].

Variable | Description |
---|---|

$C$ | Total cost function of the k-prototype algorithm |

$l$ | Number of clusters |

$i$ | Cluster |

${C}_{i}^{r}$ | Cost of assigning numerical objects in cluster i |

${C}_{i}^{c}$ | Cost of assigning categorical objects in cluster i |

$WCSS$ | Within-cluster sum of squares |

${x}_{ij}^{r}$ | Numerical object number j in cluster i |

${q}_{i}^{r}$ | Mean point of the centroid of cluster (i) |

${n}_{r}$ | Number of numerical objects in each cluster i |

${q}_{ij}^{c}$ | Categorical prototype number j in cluster i |

${n}_{c}$ | Number of categorical objects in cluster i |

${C}_{j}$ | Set of all unique values in the categorical attribute j |

LDV | Light-duty vehicle |

LDT | Light-duty truck |

Input Parameter | LDV1 Model | LDV2 Model | LDV3 Model | LDV4 Model | LDT1 Model | LDT2 Model | LDT3 Model | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Min | Max | Min | Max | Min | Max | Min | Max | Min | Max | Min | Max | Min | Max | |

${S}_{D}$ (mph) | 9.32 | 74.56 | 9.32 | 65.24 | 9.32 | 60.27 | 9.32 | 74.56 | 9.32 | 74.56 | 9.32 | 55.92 | 10.56 | 57.17 |

${S}_{A}$ (mph) | 9.32 | 75.19 | 9.32 | 70.21 | 9.32 | 55.92 | 9.32 | 72.7 | 9.32 | 66.49 | 9.32 | 59.65 | 9.32 | 59.65 |

${G}_{A}$ (%) | −13.67 | 13.16 | −13.43 | 12.08 | −13.29 | 13.69 | −12.8 | 15.28 | −9.54 | 8.55 | −10.48 | 8.65 | −6.36 | 11.49 |

$F{C}_{I}$ (gram/sec) | 0.09 | 1.2 | 0.09 | 1.314 | 0.1 | 1.18 | 0.1 | 1.17 | 0.09 | 1.21 | 0.09 | 0.98 | 0.1 | 1.04 |

${T}_{D}$ (sec) | 1.7 | 30 | 1.4 | 30 | 3.1 | 30 | 0.6 | 30 | 1.3 | 30 | 2.3 | 30 | 2.6 | 33.5 |

${A}_{r}$ (ft/sec^{2}) | 0.28 | 37.97 | 0.39 | 36.45 | 0.1 | 22.02 | 0.16 | 36.45 | 0.24 | 30.38 | 0.59 | 22.78 | 0.3 | 9.11 |

Attribute * | Options/Value |
---|---|

Function set | +, −, x, /, log, sqrt, square |

Population size | 800 |

Number of generations | 500 |

Maximum number of genes allowed in an individual | 6 |

Maximum tree depth | 4 |

Tournament size | 80 |

Tournament type | Pareto (probability = 1) |

Elite fraction | 0.7 |

Number of inputs | 8 |

Constants range | [−10, 10] |

Complexity measure | Node count |

Model | Equation # |
---|---|

LDV1 | |

${K}_{LDV1}=\frac{1.321e-2\xb7{S}_{D}{}^{2}\xb7{T}_{D}+0.3979\xb7{T}_{D}{}^{2}-5.102\xb7F{C}_{I}\xb7{T}_{D}}{F{C}_{I}\xb7{T}_{D}}+\frac{1.608\xb7{S}_{D}+0.2311\xb7{S}_{D}\xb7{T}_{D}+4.966e-3\xb7{S}_{D}{}^{2}\xb7{G}_{A}\xb7{T}_{D}+3.073e-2\xb7{S}_{D}{}^{2}\xb7F{C}_{I}\xb7{T}_{D}+7.796e-3\xb7{S}_{A}\xb7{S}_{D}\xb7{T}_{D}}{F{C}_{I}\xb7{T}_{D}\xb7A}$ | (20) |

LDV2 | |

${K}_{LDV2}=\frac{{S}_{D}{}^{2}\xb7\left[8.426\times {10}^{15}\xb7\left({G}_{A}+F{C}_{I}\right)+4.229\times {10}^{16}\right]\xb74.337\times {10}^{-19}}{F{C}_{I}\xb7A}+$$\frac{4.235\times {10}^{-22}[F{C}_{I}\xb7{S}_{A}{}^{2}\xb7{T}_{D}{}^{2}\xb76.245\times {10}^{19}\xb7{S}_{D}{}^{2}+8.126\times {10}^{20}\xb7{T}_{D}+2.781\times {10}^{22}\xb7F{C}_{I}-1.44\times {10}^{22}\xb7F{C}_{I}\xb7\mathrm{log}\left(\left|A\right|\right)]}{F{C}_{I}\xb7A}$ | (21) |

LDV3 | |

${K}_{LDV3}=\frac{3.341e-4\xb7F{C}_{I}\xb7\left({S}_{A}+{S}_{D}\right)}{{G}_{A}{}^{2}}-$ $\frac{1.11\times {10}^{-15}\left[3.904\times {10}^{15}\xb7{T}_{D}-2.643\times {10}^{15}\xb7{S}_{D}+2.972\times {10}^{14}\xb7{S}_{D}\xb7{T}_{D}\right]}{{T}_{D}}+$ $\frac{8.674\times {10}^{-19}\xb7\left[4.58\times {10}^{17}\xb7F{C}_{I}\xb7{T}_{D}+6.994\times {10}^{15}\xb7{S}_{D}{}^{2}\xb7{G}_{D}+2.46\times {10}^{16}\xb7{S}_{D}{}^{2}\xb7{T}_{D}-6.994\times {10}^{15}\xb7{S}_{D}\xb7{G}_{D}\xb7{T}_{D}\right]}{{G}_{A}\xb7{T}_{D}}$ | (22) |

LDV4 | |

${K}_{LDV4}=\frac{0.01948\xb7{S}_{D}{}^{2}\xb7{A}^{3}-F{C}_{I}\xb7{S}_{D}\xb70.01757}{F{C}_{I}\xb7{A}^{4}}-$ $\frac{6.345\xb7F{C}_{I}+0.2576\xb7{T}_{D}+0.008612\xb7{S}_{D}{}^{2}+4.518\times {10}^{-6}\xb7{A}^{2}\xb7F{C}_{I}\xb7{T}_{D}{}^{2}+\frac{0.0025\xb7{S}_{D}{}^{2}\xb7{G}_{A}}{\sqrt{\left(\left|A\right|\right)}}}{F{C}_{I}}$ | (23) |

LDT1 | |

${K}_{LDT1}=\frac{8.674e-19\xb7{S}_{D}\xb7\left(1.574e14\xb7{S}_{A}\xb7{G}_{A}{}^{2}+1.574e14\xb7{S}_{A}\xb7{S}_{D}\xb7{G}_{A}+7.553e17\right)}{\left(F{C}_{I}\xb7A\right)}-\frac{\left(8.314e-3\xb7{S}_{D}{}^{2}\xb7F{C}_{I}{}^{2}\xb7{T}_{D}{}^{2}+8.977\xb7{S}_{D}{}^{2}\xb7F{C}_{I}+5.725e3\xb7F{C}_{I}{}^{2}+2.983\xb7\left({S}_{A}\xb7F{C}_{I}\xb7{T}_{D}\right)+4.946\xb7{T}_{D}\right)\xb71.694e-3}{F{C}_{I}{}^{2}}$ | (24) |

LDT2 | |

${K}_{LDT2}=\frac{1.059e-3\xb7{S}_{A}\xb7{S}_{D}{}^{2}\xb7F{C}_{I}{}^{3}+{G}_{A}\xb7\mathrm{log}\left(\left|{T}_{D}\right|\right)\xb7{S}_{D}{}^{2}\xb7F{C}_{I}{}^{2}\xb70.002112+0.5789\xb7{S}_{D}\xb7F{C}_{I}{}^{2}}{{\left(F{C}_{I}\right)}^{3}\xb7A}-\frac{0.0002673\xb7{S}_{D}-0.01809\xb7{S}_{D}{}^{2}\xb7F{C}_{I}{}^{2}-2.792\xb7F{C}_{I}{}^{2}\xb7\sqrt{\left(\left|{T}_{D}\right|\right)}+9.811\xb7F{C}_{I}{}^{3}}{F{C}_{I}{}^{3}}$ | (25) |

LDT3 | |

${K}_{LDT3}=\frac{3.006e-2\xb7{S}_{D}{}^{2}}{{G}_{A}}+0.4344\xb7F{C}_{I}\xb7{T}_{D}-0.003328\xb7{\left({S}_{A}+{S}_{D}+F{C}_{I}+{T}_{D}\right)}^{2}+\frac{13.16}{{G}_{A}}+\frac{\left(5.023\times {10}^{15}\left({S}_{A}+{S}_{D}+{G}_{D}\right)\right)\xb74.441\times {10}^{-16}}{{T}_{D}}+\frac{({G}_{D}\left(-1\xb7{G}_{D}{}^{2}+\frac{{S}_{D}}{{G}_{A}}\right)\xb70.2658}{{T}_{D}}-39.4$ | (26) |

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**MDPI and ACS Style**

Alshayeb, S.; Stevanovic, A.; Park, B.B. Field-Based Prediction Models for Stop Penalty in Traffic Signal Timing Optimization. *Energies* **2021**, *14*, 7431.
https://doi.org/10.3390/en14217431

**AMA Style**

Alshayeb S, Stevanovic A, Park BB. Field-Based Prediction Models for Stop Penalty in Traffic Signal Timing Optimization. *Energies*. 2021; 14(21):7431.
https://doi.org/10.3390/en14217431

**Chicago/Turabian Style**

Alshayeb, Suhaib, Aleksandar Stevanovic, and B. Brian Park. 2021. "Field-Based Prediction Models for Stop Penalty in Traffic Signal Timing Optimization" *Energies* 14, no. 21: 7431.
https://doi.org/10.3390/en14217431