Eco-Driving Optimization with the Traffic Light Countdown Timer in Vehicle Navigation and Its Impact on Fuel Consumption

Abstract

For most drivers of fuel-powered vehicles who do not have specialized eco-driving knowledge, simple and practical strategies are the most effective way to encourage eco-driving habits. By incorporating traffic light countdown timers from vehicle navigation systems, this paper develops a 0–1 integer linear programming (ILP) model to determine the optimal speed curve and further provide actionable, easy-to-implement eco-driving recommendations. Specifically, time is discretized into one-second intervals, with speed and acceleration also discretized. Pre-calculating instantaneous fuel consumption under various speed and acceleration combinations ensures the linearity of the objective function. For a specified road and a given time duration, the optimal speed profile problem for approaching intersections is transformed into a series of speed and acceleration selections. Through the analysis of multiple application scenarios, this study proposes practical and easily adoptable eco-driving strategies, which can effectively reduce vehicle fuel consumption, thereby contributing to the sustainable development of urban traffic.

1. Introduction

With the continued advancement of electronic maps, vehicle navigation systems (VNSs) are becoming increasingly versatile and intelligent, offering greater convenience to drivers. Notably, the prompts and information provided by VNSs significantly influence driving behavior [1,2,3]. For example, the traffic light countdown timer (TLCT) feature, as illustrated in Figure 1, displays the status of traffic lights at upcoming intersections, sometimes extending to the second intersection ahead. By leveraging this information, drivers can adjust their driving strategies accordingly.

In urban settings, where intersections are abundant and vehicles frequently stop and start, the resulting high switching frequency imposes a significant burden on energy consumption and carbon emissions [4]. To address this issue, eco-driving strategies have been promoted globally for years [5,6,7,8]. Increasingly, individuals, researchers, and policymakers are recognizing the importance of eco-driving and expressing positive attitudes toward its adoption [9,10,11,12].

The concept of eco-driving aims to minimize energy consumption and carbon emissions by improving daily driving behavior. Studies indicate that adopting eco-driving habits can reduce energy consumption by 5% to 30% [13]. Additionally, an empirical study in Singapore demonstrated that eco-driving strategies not only maintain desirable journey speeds but also reduce fuel consumption and carbon emissions by more than 10% [14].

With rapid advancements in information and communication technologies, such as intelligent navigation systems and automated vehicles, the personal and societal benefits of eco-driving are becoming increasingly significant. It is well understood that a vehicle’s instantaneous energy consumption and carbon emissions are directly influenced by its speed and acceleration. Consequently, developing optimal driving strategies based on speed and acceleration has become a central focus of eco-driving research. For example, Ref. [15] developed an optimal control model for passenger vehicles that minimizes fuel consumption during specified traffic flows by coordinating speed and gear ratios. To balance data resolution with computational efficiency, Ref. [16] discretized speed, acceleration/deceleration, and road grade and formulated the optimal power-based vehicle longitudinal control problem for connected eco-driving as a 0-1 binary mixed-integer linear programming (MILP) problem. Similarly, Ref. [17] proposed an optimal control model for eco-driving in continuous time, which was then approximately transformed into discrete time using the Euler method. This approach allowed the model to be solved using a sequential quadratic programming algorithm. In addition to these methods, Ref. [18] incorporated driver comfort as a factor in the eco-driving model and employed a genetic algorithm (GA)-based approach to solve it. These advancements highlight the growing complexity and sophistication of eco-driving optimization strategies.

Advanced driver assistance systems and automated vehicle technologies are gradually becoming more prevalent, offering new opportunities for eco-driving applications [19,20]. For instance, to guide drivers in adjusting accelerator pedal usage to improve fuel efficiency, Ref. [21] conducted a driving simulator study that compared the designs of three in-vehicle eco-driving assistance systems. To simultaneously enhance energy efficiency and mobility, Ref. [22] proposed a hierarchical eco-driving strategy comprising a cloud-level controller and a vehicle-level controller. The cloud-level controller, based on dynamic programming, optimizes velocity and battery state-of-charge using global traffic information obtained from intelligent transportation systems. Meanwhile, Ref. [23] developed an eco-driving strategy leveraging constraint-enforced reinforcement learning for connected and automated vehicles (CAVs) navigating multiple signalized intersections.

Urban environments pose additional challenges due to their complex traffic conditions, such as diverse road hierarchies, increased vehicle density, numerous intersections, and frequent traffic signals, all of which influence the effectiveness of eco-driving strategies [24]. Utilizing vehicle-to-infrastructure (V2I) and vehicle-to-vehicle (V2V) communication, Ref. [25] addressed the vehicle speed planning problem to minimize fuel consumption along signalized urban routes, employing a dynamic programming (DP) method to solve simplified models within prediction horizons. Considering regenerative braking efficiency, Ref. [26] formulated an eco-driving optimal control problem for electric vehicles (EVs) at urban signalized intersections, deriving a closed-form solution based on Pontryagin’s minimum principle. Additionally, Ref. [27] developed a deep reinforcement learning algorithm to optimize eco-driving strategies for EVs in urban environments.

As highlighted in this literature review, eco-driving has emerged as a prominent research focus in the road traffic domain over the past two decades, with advancements in relevant theories, technologies, and applications. However, despite the increasing presence of advanced vehicle technologies, the market share of conventional fuel-powered vehicles remains dominant, and manual driving is still the most prevalent mode of operation. For most drivers of fuel-powered vehicles who do not have specialized eco-driving knowledge, simple and practical strategies are the most effective way to encourage eco-driving habits. By incorporating traffic light countdown timers from vehicle navigation systems, this paper develops a 0-1 ILP model to determine the optimal speed curve and further provide actionable, easy-to-implement eco-driving recommendations. In this model, time is discretized into one-second intervals, and velocity and acceleration are similarly discretized. Instantaneous fuel consumption under varying conditions is pre-calculated to ensure the linearity of the objective function. For a given duration and distance, the optimal speed profile problem is reformulated as a series of velocity and acceleration selections.

Following this introduction, Section 2 outlines the modeling procedure. Using specific parameters, Section 3 analyzes and discusses several application scenarios. Finally, we conclude the study in Section 4 and provide some practical and easy-to-implement eco-driving recommendations for general drivers of fuel-powered vehicles.

2. Mathematical Formulation

We begin the mathematical formulation by introducing the necessary assumptions for this study. As the focus is on eco-driving strategies for general drivers of conventional fuel-powered vehicles, who are typically less concerned with factors such as air resistance and road slope, we make the following assumptions:

(1)

A conventional gasoline vehicle is the subject of this study;

(2)

The specified road is flat and straight;

(3)

Air resistance is not considered;

(4)

Urban roads and intersections impose speed limits.

(5)

The influence of other vehicles on the road is negligible.

Subsequently, we define the notations used for the subscripts, sets, parameters and decision variables, as detailed in

Table 1. Subscripts and parameters involved in the formulations.

Notation	Definition
$f$	Instantaneous fuel consumption (g/s) of a vehicle
$P_{tract}$	Total tractive power (kW) requirement at the wheels
$v$	Vehicle speed (m/s)
$v_{max}$	The maximum speed (m/s)
$I$	Set of indices of discretized speeds
$D^f$	Set of time-dependent freight demands
$v_i$	The $i$th discretized speed (m/s), $i\in I$
$a$ $a_{min}$	The vehicle acceleration (m/${\mathrm{s}}^2$) The minimum acceleration (m/${\mathrm{s}}^2$)
$a_{max}$	The maximum acceleration (m/${\mathrm{s}}^2$)
$J$	Set of indices of discretized accelerations
$a_j$	The $j$th discretized acceleration (m/${\mathrm{s}}^2$), $j\in J$
$f_{i,j}$	Instantaneous fuel consumption (g/s) when a vehicle is traveling at a speed of $v_i$and an acceleration of $a_j$
$A$	Rolling resistance term (kW/m/s)
$B$	Speed correction to rolling resistance term (kW/(m/${\mathrm{s}}^2$))
$C$	Air drag resistance term (kW/(m/${\mathrm{s}}^3$))
$M$	Vehicle mass (kg)
$g$	Gravitational constant (9.81 m/${\mathrm{s}}^2$)
$\theta$	Road grade (degrees)
$\alpha ,\beta ,\delta ,\zeta ,{\alpha }^{\prime }$	Parameters in the statistical model of fuel consumption
$L\left(0\right)$	A given traveling distance (m)
$\tau$	A given traveling time (s)
$T$	Set of discretized timestamps, $T=\left\{0, 1, 2, · · ·, \tau \right\}$
$t$	Index of discretized timestamps, $t\in T$
$\mu \left(0\right)$	A given initial speed (m/s) at timestamp 0
$\mu \left(\tau \right)$	A given final speed (m/s) at timestamp $\tau$
$y_{i,j}^t$	=1 if a vehicle adopts a speed of $v_i$and an acceleration of $a_j$at the timestamp $t$; =0 otherwise.

.

In the literature, numerous fuel consumption models for conventional fuel-powered vehicles have been proposed, ranging from overly simplistic to highly complex approaches. For this study, we adopt a widely cited statistical model developed by [28], which has been validated for its balance between accuracy and computational efficiency. Their study demonstrates that this model produces results that closely align with actual measurements, operates efficiently, and is relatively straightforward to calibrate. Specifically, the fuel consumption of a conventional fuel-powered vehicle is modeled as a bivariate function dependent on both speed and acceleration.

There are many fuel consumption models of fuel-powered vehicles in the literature, some of which are overly simple, while others are sufficiently complicated. In this paper, we adopt a widely cited statistical model, which has been studied in [28]. Their study indicates that this model not only outputs reasonable results compared to actual measurements but also runs fast and is relatively simple to calibrate. Specifically, the fuel consumption of a fuel-powered vehicle is a bivariate function with respect to speed and acceleration. Mathematically, it is expressed as

$$f\left(v,a\right)=\left\{\begin{array}{ll}\alpha +\beta \cdot v+\gamma \cdot v^2+\delta \cdot v^3+\zeta \cdot a\cdot v& \mathrm{if} P_{tract}>0;\\ {\alpha }^{\prime }& \mathrm{if} P_{tract}=0.\end{array}\right.$$

(1)

In the above expression, $P_{tract}$ represents the tractive power. When the tractive power is positive, it can be calculated as follows:

$$P_{tract}=A\cdot v+B\cdot v^2+C\cdot v^3+M\cdot a\cdot v+M\cdot g\cdot \sin \theta \cdot v$$

(2)

In Equations (1) and (2), $v$ and $a$ are the vehicle’s speed and acceleration, respectively, $M$ is the vehicle mass, $g$ is the gravitational constant, and $\theta$ is the road slope. According to the assumptions in this paper, $\theta =0$. Parameters $\alpha$, $\beta$, $\gamma$, $\delta$, $\zeta$ and ${\alpha }^{\prime }$ need to be calibrated using statistical methods, while parameters $A$, $B$ and $C$ need to be calibrated with physics, where $A$ is the rolling resistance term, $B$ is the speed correction to rolling resistance term and $C$ is the air drag resistance term.

For the given time range $\left[0,\tau \right]$, where $\tau$ can be understood as the time displayed by the traffic light countdown timer in vehicle navigation, we discretize it into intervals of one second. All discretized timestamps are collected in the set $T$, i.e., $T=\left\{0,1,2,···,\tau \right\}$. The vehicle’s potential speeds are discretized as $v_0$, $v_1$, $v_2$, ·····, $v_{max}$, with the subscripts of these discretized speeds grouped in the set $I$. Without loss of generality, for any $i, i^{\prime }\in I$ and $i<i^{\prime }$, we suppose that $v_i<v_{i^{\prime }}$. Similarly, the vehicle’s potential accelerations are discretized as $a_0$, $a_1$, $a_2$, ⋯, $a_{max}$, with the subscripts of these discretized accelerations grouped in the set $J$. For any $j, j^{\prime }\in J$ and $j<j^{\prime }$, we suppose that $a_j<a_{j^{\prime }}$. Using Equations (1) and (2), the fuel consumption per second can be precomputed for any specific combination of speed and acceleration. For simplicity, the fuel consumption at the speed $v_i$ and the acceleration $a_j$ is denoted as $f\left(v_i,a_j\right)$, or $f_{i,j}$ in abbreviated form. Let the binary decision variable $y_{i,j}^t$ indicate whether the driver adopts the speed $v_i$ and the acceleration $a_j$ at timestamp $t$. Based on the above definitions, the objective function for the eco-driving strategy that minimizes total fuel consumption in the time range $\left[0, \tau \right]$ can be expressed as follows:

$$min: \sum _{t\in T\backslash \left\{\tau \right\}}\sum _{i\in I}\sum _{j\in J}\left(f_{i,j} \cdot y_{i,j}^t\right).$$

(3)

The driver must select a speed and an acceleration at each timestamp, except for the final one. Therefore, for any $t\in T$, decision variables $y_{i,j}^t\left(i\in I, j\in J\right)$ must satisfy the following equation:

$$\sum _{i\in I}\sum _{j\in J}{y}_{i,j}^t=1,\forall t\in T\backslash \left\{\tau \right\}.$$

(4)

Except timestamps 0 and $\tau$, the vehicle’s speed at each timestamp is determined by both the speed and the acceleration at the previous timestamp. Thus,

$$\sum _{i\in I}\sum _{j\in J}\left(v_i\cdot y_{i,j}^t\right)=\sum _{i\in I}\sum _{j\in J} \left[\left(v_i+a_j\right)\cdot y_{i,j}^{t-1}\right],\forall t\in T\backslash \left\{\tau \right\}$$

(5)

The vehicle has an initial speed at timestamp 0, denoted by $\mu \left(0\right)$, and a final speed at timestamp $\tau$, denoted by $\mu \left(\tau \right)$. In addition to Equation (5), the vehicle’s motion must be subject to two boundary conditions, namely

$$\sum _{i\in I}\sum _{j\in J}\left(v_i\cdot y_{i,j}^0\right)=\mu \left(0\right).$$

(6)

$$\sum _{i\in I}\sum _{j\in J} \left[\left(v_i+a_j\right)\cdot y_{i,j}^{t-1}\right]=\mu \left(\tau \right).$$

(7)

Urban roads typically impose speed limits to ensure safety. For the specified road in this study, the speed limit is denoted as $v_{max}$. Therefore, the vehicle’s speed must satisfy the following constraint at all timestamps:

$$0\le \sum _{i\in I}\sum _{j\in J}\left(v_i\cdot y_{i,j}^t\right)\le v_{max}, \forall t\in T.$$

(8)

Due to the discretization of time, speed, and acceleration, the distance traveled by the vehicle each second should approximately equal the average of its instantaneous speeds at the start and end of that second, i.e., ($v_i$ + $v_{i+1}$)/2. Over the time range $\left[0, \tau \right]$, the vehicle is required to traverse the specified road exactly. However, the discretization means that this requirement can only be approximated as follows, where $L\left(0\right)$ is the length of the specified road and $\mathsf{\Delta }$ is the allowable error:

$$L\left(0\right)-\mathsf{\Delta }\le \sum _{t\in T\backslash \left\{\tau \right\}}\sum _{i\in I}\sum _{j\in J}\left(v_i\cdot y_{i,j}^t\right)+{\displaystyle \frac{\mu \left(0\right)+\mu \left(\tau \right)}{2}}\le L\left(0\right)+\mathsf{\Delta }.$$

(9)

Finally, the possible values of the decision variables are defined as follows:

$$y_{i,j}^t\in \left\{0,1\right\}, \forall i\in I, j\in J, t\in T.$$

(10)

From (3) to (10), we formulate an eco-driving optimization model that incorporates information from the traffic light countdown timer into vehicle navigation. This model is particularly designed to optimize the eco-driving strategy for a vehicle approaching a signalized intersection on a specified road. This model is expressed as a 0-1 ILP model, making it well suited for obtaining exact solutions with commercial solvers like Gurobi.

3. Numerical Experiments

In this section, we discuss optimal eco-driving strategies under different parameter settings. We use the calibrated parameters of the experimental car in [27], i.e., $M=1325 \mathrm{k}\mathrm{g}$, $A={\displaystyle \frac{0.1326 \mathrm{k}\mathrm{W}}{\mathrm{m}/\mathrm{s}}}$, $B=2.7384\times {\displaystyle \frac{{10}^{-3} \mathrm{k}\mathrm{W}}{{\left(\mathrm{m}/\mathrm{s}\right)}^2}}$, $C=1.0843\times {\displaystyle \frac{{10}^{-3} \mathrm{k}\mathrm{W}}{{\left(\mathrm{m}/\mathrm{s}\right)}^3}}$, $\alpha =0.365$, $\beta =1.4\times {10}^{-3}$, $\gamma =0$, $\beta =9.65\times {10}^{-7}$, $\zeta =9.43\times {10}^{-2}$, and ${\alpha }^{\prime }=0.299$. For convenience, the maximum and minimum values of acceleration are approximately set to 3.6 m/s² and −4 m/s², respectively. The speed limit for vehicles on some urban roads is 60 km/h, and the speed limit when vehicles pass intersections is 40 km/h. So the maximum speed on roads and the maximum speed of passing intersections are approximately converted to 16.6 m/s and 11.1 m/s, respectively. In the experiments, speed and acceleration are discretized into intervals of 0.1 m/s and 0.1 m/s², respectively. Using the calibrated parameters mentioned above, the fuel consumption per second of the experimental vehicle can be calculated for any specific combination of speed and acceleration. Due to the vast number of possible combinations resulting from the discretization, the corresponding fuel consumption data set becomes extremely large, so only a representative subset of the data is presented in Appendix A.

(1)

The optimal eco-driving strategies with respect to the initial distance under the scenario that the traffic light at the approaching intersection is green with a countdown timer of 35 s.

In this set of experiments, we suppose that vehicle navigation shows that the traffic light at the approaching intersection is green with a countdown timer of 35 s. By means of the proposed model, we need to identify the optimal eco-driving strategies for different distances. For this purpose, we set the initial distance (i.e., $L\left(0\right)$ in the model) to range from 100 m to 500 m, increasing in increments of 50 m. Considering the speed limits on urban roads and intersections mentioned above, we set $\mu \left(0\right)=16.6 \mathrm{m}$/s and $\mu \left(\tau \right)=11.1 \mathrm{m}$/s. For any given distance, the model determines the most fuel-efficient speed profile and calculates the corresponding fuel consumption by adjusting the parameter τ second by second. This ensures that the experimental car reaches the intersection before the green light turns red. Figure 2 illustrates the experimental results. For example, when the car is 100 m from the traffic light, its speed needs to drop quickly from 16.6 m/s to 11.1 m/s, as shown in the leftmost curve of Figure 2. In this process, the experimental car takes only seven seconds and consumes only 2.09 g of fuel. From all speed curves in Figure 2, it is clear that different deceleration strategies are used in the early and later stages. Generally, in the early stage, the driver should take full advantage of the car’s inertia to keep it moving fast. However, in the later stage, especially in the final second, a more dramatic deceleration strategy is required to satisfy the final speed condition.

In fact, the first five eco-driving strategies shown in Figure 2, i.e., the optimal solutions for initial distances of 100 m, 150 m, 200 m, 250 m and 300 m, are also the most time-saving driving strategies. However, for initial distances of 350 m, 400 m, 450 m and 500 m, the corresponding optimal eco-driving strategies are not the most time-saving. As shown in Figure 3, when the initial distance is 350 m or 400 m, the most time-saving driving strategies save 1 s compared to the corresponding optimal eco-driving strategies. For initial distances of 450 m or 500 m, the most time-saving strategies save 2 s. However, these time-saving strategies result in approximately 3.77%, 4.52%, 5.91% and 6.11% higher fuel consumption, respectively, compared to the corresponding optimal eco-driving strategies.

The optimal eco-driving strategies and the most time-saving driving strategies shown in Figure 2 and Figure 3 can be applied in more scenarios. For instance, when the initial distance is 100 m and the traffic light at the approaching intersection is red with a countdown timer of less than 7 s, the leftmost curve in Figure 2 remains both the optimal eco-driving strategy and the most time-saving strategy. Similarly, when the initial distance is 500 m, and the traffic light is red with a countdown timer of less than 33 s, the rightmost curve in Figure 2 represents the optimal eco-driving strategy for this situation. Furthermore, if the initial distance is 500 m and the traffic light is green with a countdown timer of just 31 s, the driving strategy corresponding to the rightmost dotted curve in Figure 3 is preferable. This strategy is not only the most time-saving but also the optimal eco-driving strategy, as the car would otherwise have to stop and wait for the red light if this strategy is not adopted.

(2)

The optimal eco-driving strategies with respect to the initial distance under the scenario that the traffic light at the approaching intersection is red with a countdown timer of 35 s.

In this set of experiments, we suppose that the vehicle’s navigation indicates a red traffic light at the approaching intersection with a countdown timer of 35 s. Since the light is red, the vehicle must stop and wait if it reaches the intersection within this period, continuing its journey only when the light turns green.

Using the proposed model, we determine the optimal eco-driving strategies for initial distances of 100 m, 150 m, 200 m, 250 m, 300 m, 350 m, 400 m, 450 m and 500 m. Without loss of generality, we set $\mu \left(0\right)=16.6 \mathrm{m}$/s and $\mu \left(\tau \right)=0 \mathrm{m}$/s. After the red light turns green, we further assume that the vehicle crosses the intersection in the most fuel-efficient manner. The intersection is considered to have a length of 25 m. In other words, we solve the model again with $L\left(0\right)=25 \mathrm{m}$, $\mu \left(0\right)=0 \mathrm{m}$/s and $\mu \left(\tau \right)=11.1 \mathrm{m}$/s. Under these parameter settings, the most fuel-efficient speed curves and the corresponding fuel consumptions are illustrated in Figure 4. In this figure, the green curve on the right represents the vehicle’s speed while crossing the intersection. The blue portion of each bar in the subplot represents the fuel consumption before reaching the intersection, the red portion corresponds to the fuel consumed while waiting at the red light and the green portion represents the fuel consumption during the intersection crossing.

From Figure 4, the fuel consumption before reaching the intersection increases as the distance from the intersection grows, while the fuel consumption during the red light wait decreases as the waiting time shortens. However, due to the car’s relatively high initial speed (i.e., 16.6 m/s) and its ability to coast a significant distance at idle speed, the total fuel consumption across these nine scenarios remains nearly the same. The only exception is the 500 m scenario, where the fuel consumption is 0.1 g higher than in the others. For example, in the 100 m scenario, the car consumes a total of 15.86 g of fuel (i.e., 2.69 + 7.77 + 5.40) from the start of the 35 s countdown until it departs at the intersection, while the total fuel consumption in the 500 m scenario is 15.96 g (i.e., 10.56 + 5.40). Additionally, the speed curves for all scenarios in Figure 4 exhibit a gentle downward trend in the early stage, followed by a steep decline in the later stage.

When the waiting time at the red light is relatively short, e.g., 7 s in the 400 m scenario shown in Figure 4, an eco-driving strategy that avoids stopping (Strategy II) is often preferable. To demonstrate the advantages of this approach, we further solve the model with $\mu \left(0\right)=16.6 \mathrm{m}$/s $\mathrm{a}\mathrm{n}\mathrm{d}$ $\mu \left(35\right)=11.1 \mathrm{m}$/s for the 400 m, 450 m and 500 m scenarios. The resulting speed curves (under Strategy II) are illustrated in Figure 5. For comparison, the corresponding speed curves from Strategy I (shown in Figure 4) are also included in Figure 5.

In the first subplot, the speed curve of Strategy II shows a steep decline during the first 10 s, followed by a gradual decrease before rising to 11.1 m/s at the 35th second. This final speed, 11.1 m/s, matches the speed limit for passing through the intersection, which the car maintains while crossing. Compared to Strategy I, the fuel consumption during the 35 s red light period increases (from 10.46 g to 12.38 g), but the fuel consumption for crossing the intersection is significantly reduced (from 5.40 g to 1.15 g). As a result, Strategy II achieves total fuel savings of 14.69% under the 400 m scenario. For the 450 m and 500 m scenarios, the savings increase to 26.80% and 27.26%, respectively. Moreover, under Strategy I, the car requires 4 s to cross the intersection, while under Strategy II, it only takes 3 s. These results demonstrate that the eco-driving strategy (Strategy II) can also function as a time-saving driving strategy when the remaining red light duration is within an appropriate range.

In Figure 5, a noticeable difference between the speed curve of Strategy II in the first subplot and those in the other two subplots is the upward trend observed in the 35th second for the former. This observation prompts us to explore whether a more optimal eco-driving strategy exists between $\mu \left(35\right)=11.1$ and $\mu \left(35\right)=0$ when $L\left(0\right)=$400, $\mu \left(0\right)=16.6$, and the red light countdown timer shows exactly 35 s. To investigate this, we conducted two additional experiments for $\mu \left(35\right)=9.6$ and $\mu \left(35\right)=8.1$, corresponding to Strategies III and IV, respectively. For comparison, the resulting speed curves, along with those from the first subplot in Figure 5, are presented in Figure 6. From the fuel consumption subplot in Figure 6, it can be seen that Strategy III demonstrates the best overall performance. Compared to Strategy II ($\mu \left(35\right)=11.1$), the eco-driving strategy of Strategy III ($\mu \left(35\right)=9.6$) achieves total fuel savings of 4.18% under the 400 m scenario. These results highlight the importance of considering eco-driving strategies holistically when vehicles cross intersections with traffic lights.

(3)

The optimal eco-driving strategies with respect to the initial speed under the scenario of being a given distance away from the approaching intersection with a given final speed.

In this set of experiments, we assume that the initial distance between the car and the approaching intersection is 300 m, and the traffic light remains green for a sufficiently long duration. In the model, the above conditions are represented as $L\left(0\right)=300$ and $\mu \left(\tau \right)=11.1$. Using the proposed model, we determine the optimal eco-driving strategies for initial speeds of 16.6 m/s, 15.5 m/s, 14.4 m/s, 13.3 m/s, 12.2 m/s, 11.1 m/s, 10.0 m/s and 5.0 m/s. The experimental results are shown in Figure 7, where all subplots, except the last, correspond to specific initial speeds and include multiple eco-driving speed profiles for varying travel times, and the last subplot Figure 7ix aggregates the most fuel-efficient profiles for each of these initial speeds.

In Figure 7, the first subplot shows that the optimal eco-driving strategy for an initial speed of 16.6 m/s is also the most time-saving, requiring 19 s. The second and third subplots reveal that the optimal eco-driving strategies for initial speeds of 15.5 m/s and 14.4 m/s take 21 s and 22 s, respectively, which are not the most time-saving. From subplot (iv) to subplot (viii), the optimal eco-driving strategies require a longer time. For instance, the optimal eco-driving strategy for an initial speed of 13.3 m/s takes 5 s longer than the most time-saving strategy. From subplot (ix), the lower the car’s initial speed, the more fuel it consumes. For instance, the optimal eco-driving strategy for an initial speed of 10.0 m/s consumes 9.9% more fuel compared to the optimal eco-driving strategy for an initial speed of 11.1 m/s.

4. Conclusions

In this study, we developed a discrete model for the eco-driving problem, taking into account the traffic light countdown timer in vehicle navigation as vehicles approach intersections. This model was used to investigate the impact of various eco-driving strategies on fuel consumption across different scenarios. The analysis of different initial speeds, different distances to the approaching intersection and their corresponding eco-driving profiles provides a comprehensive understanding of eco-driving strategies. Although vehicles and drivers exhibit significant variability in real-world conditions, several common patterns identified in this study can be applied generally.

First, the optimal eco-driving strategy tends to be the most time-efficient when the vehicle has a relatively high initial speed, the distance to the approaching intersection is relatively short and the traffic light countdown timer indicates a relatively long green light. Second, an eco-driving strategy that avoids stopping may be preferable when the vehicle is relatively far from the intersection and the countdown timer shows a relatively short red light. Finally, in most scenarios, drivers must balance the trade-off between time savings and fuel efficiency. It is important to note that an excessive focus on eco-driving could negatively impact traffic flow, as not all eco-driving behaviors are the most time-efficient.

This study does not account for the influence of multiple intersections and other vehicles. Therefore, future research could focus on incorporating more dynamic traffic conditions to further refine eco-driving strategies. Additionally, further experiments are needed to validate these results in real-world driving scenarios. With the growing application of autonomous vehicles [20,29], studying ecological autonomous driving strategies has become an important research direction.