Energy Efficient Dynamic Driving Technique Modeling with Flower Pollination and Grey Wolf Optimization Algorithms in Urban Rail Transportation

Abstract

Energy efficiency and management studies in rail transport worldwide have focused on optimizing reducing traction energy consumption, which accounts for a significant portion of operating costs. The development of energy-efficient driving techniques and the recovery of braking energy offer significant potential for reducing traction energy consumption and costs in rail systems. This study introduces a novel Energy-Efficient Dynamic Driving Technique (EEDDT) model, designed to optimize speed profiles and travel dynamics of rail vehicles by minimizing traction energy consumption (MTEC), maximizing regenerative braking energy production (MRBEP), and minimizing total travel time (MTT). The proposed model incorporates a multi-objective optimization framework and leverages two advanced metaheuristic algorithms—the Flower Pollination Algorithm (FPA) and Grey Wolf Optimizer (GWO)—to determine optimal acceleration, cruising, coasting, and braking regimes. In contrast to existing approaches, the model integrates real-world operational factors, including gradient effects, speed-dependent resistance, horizontal curvature constraints, and passenger load variability, enhancing simulation accuracy and applicability. The model is validated through empirical data collected from a 4.6 km segment of the Samsun urban tramway system consisting of seven stations. Simulation results demonstrate that the EEDDT model achieves up to a 48.95% reduction in total energy consumption, a 137.60% increase in regenerative energy recovery, and 99.62% travel time adherence compared to conventional driving strategies. The findings confirm the practical viability of the EEDDT framework in enhancing energy efficiency in urban rail networks, with FPA demonstrating superior performance in constraint optimization. Overall, this study contributes a scalable, data-driven, and optimization-based approach to sustainable urban rail operations.

Keywords

regenerative braking energy nature-Inspired optimization rail system energy efficiency rail system dynamic driving technique renewable energy

1 Introduction

The transportation sector has been facing various challenges due to the growing global population. These challenges include traffic congestion, high carbon emissions, high demand for fossil fuels, and increased energy consumption. In densely populated cities, electric rail transit systems are proposed as a solution to these problems. Rail system (RS) passenger transportation is one of the most popular transportation systems due to its higher passenger carrying capacity and more cost-effective per unit passenger than other public transportation systems (ERRAC, 2017). Efficient energy management in RS passenger transportation is considered an essential issue, especially with the increasing use of these systems. Several methods have been proposed for energy consumption management in rail systems. One of these methods involves driving the vehicle at low speeds, which reduces the amount of energy consumed but also increases travel time, which is undesirable for passengers. Another method recommended for efficient energy management in rail systems is to use regenerative braking energy (RBE) (González-Gil et al., 2014; Khodaparastan et al., 2019). One effective method to increase the energy efficiency of rail-based public transportation systems is Speed Profile Optimization (SPO) (Fernández et al., 2019). This method involves determining the speed profiles and travel times of rail-based transport vehicles to optimize energy consumption within a predetermined travel time (Fernández et al., 2019). The success of this method is measured by the RS vehicle's ability to start from the designated position at the designated time, move within the speed limits, and stop at the designated position at the designated time (Xing et al., 2023). Numerous studies have shown the potential of regenerative braking (RB) technology to reduce net energy consumption in urban rail systems, with energy savings ranging from 10% to 40% (Bae et al., 2007; Ceraolo et al., 2018; Frilli et al., 2016; González-Gil et al., 2013; Mayrink et al., 2020; Nasri et al., 2010; Sun et al., 2023). Nasri et al. (2010) presented a study on timetable optimization to maximize the usage of regenerative energy of braking in electrical railway systems. The study used a simulation model to evaluate energy consumption and the potential for RBE recovery. The results showed that by optimizing the timetable, it was possible to increase the amount of RBE recovered by up to 25% compared to a non-optimized timetable. In the study by González-Gil et al. (2013), strategies and technologies were proposed for the optimal management of RBE in sustainable urban rail systems. Simulations conducted on the Barcelona Metro showed that implementation of these strategies and technologies could lead to energy savings of up to 20%. Wei Li and his colleagues have conducted a multi-objective train route optimization by incorporating the features of automatic train operation (ATO) systems into the problem formulation and using a modified Firefly Algorithm, called GSOANR algorithm. According to the results of the study, the GSOANR algorithm has reduced electricity consumption by approximately 5.6% compared to standard GSO algorithms (Li et al., 2023). The study by Mayrink et al. (2020) evaluated the technical and economic feasibility of using RB for energy recovery in diesel-electric freight trains. The results showed that RB systems can reduce fuel consumption and CO₂ emissions by up to 20% and 25%, respectively. Frilli et al. (2016) proposed an energetic optimization methodology for RB in high-speed railway systems. The methodology was tested on a simulation model of a high-speed train, and the results showed that the proposed approach could increase the energy recovery efficiency by up to 10% compared to a standard control strategy. The study by Sun et al. (2023) proposes a method to increase the use of RBE in metro trains for energy saving. The method involves adjusting the speed profile of an adjacent train that runs in a non-braking stage to absorb the RBE of a braking train. According to the results of the study, the proposed method improves energy efficiency by over 13% by improving RBE usage. The studies summarized herein underscore the importance of energy efficiency measures in rail systems, with RBE technology showing promise as a means of achieving significant energy savings. Cheng Che et al. propose a method for effectively utilizing regenerative braking energy for traction power supply. The method is based on power regulation with genetic algorithm and consists of railway power regulators, energy transfer converters, and a central controller. The study shows that the proposed method achieves a gain of approximately 99.3% in regenerative braking energy utilization (Che vd., 2022). Xiaoging Zeng et al. proposed a regenerative braking energy strategy technique based on reinforcement learning for trains to reduce power consumption in railway transport. The study optimized the train's regenerative braking strategy using a reinforcement learning algorithm with an on-board flywheel energy storage device. The study concludes that dynamic planning and Q-learning optimized train speed curve results in over 5% energy recovery in total energy consumption (Zeng et al., 2022). Shanpeng Zhao et al. conducted a study on implementing a hybrid energy storage system to improve the efficiency of regenerative braking energy utilization. The research suggests that adopting the railway power regulator control strategy within the energy storage system enhances power quality, leading to a longer service life for the energy storage system (Zhao et al., 2022). Wang and Su's study highlights the importance of reducing traction energy and increasing the use of renewable energy to achieve energy savings. The authors propose that driving strategies and train schedules can meet this requirement. The study presents a mathematical model for calculating traction energy and regenerative braking energy. Two approaches have been introduced to reduce net energy consumption on the metro line. The first approach suggests synchronizing train departure times with the arrival times of regenerative braking trains to reuse energy more efficiently. The second approach optimizes station arrival and departure times, as well as traction and braking regimes between stations, to increase energy savings. Two approaches have been proposed to increase energy efficiency in metro lines. The model's validity has been confirmed using real data from metro lines (Wang & Su, 2023). Zongyi Xing et al. conducted a study on the impact of regenerative braking energy on energy-saving optimization in railway train operations. They developed a tri-objective brute-force search algorithm model to solve multi-objective problems with the aim of minimizing energy consumption. The model calculates the duration, speed, distance, power, and energy consumption for each operating interval. The optimization model's effectiveness was validated through a case study of Guangzhou Metro Line 7. The data indicates that the brute-force search method results in greater energy savings compared to both the Genetic Algorithm and Particle Swarm Optimization Algorithm (Xing et al., 2023). GE Xin et al. proposed an integer programming model supported by scheduling methods to maximize the use of regenerative braking energy. They solved the model using the Gurobi solver under the constraint of safe running interval of multiple trains. The researchers tested the validity of the model with an example and as a result of the study, they completed the optimized train overlaps with a success rate of 49%. The study also compared the accuracy and efficiency of the Gurobi solver with the simulated annealing algorithm. The researchers also reported that the Gurobi solver was more effective in saving energy and increasing the efficiency of regenerative braking energy utilization (Xin et al., 2023). Woldram Heineken et al. proposed an energy-efficient train driving technique based on optimal control theory. The study presents an optimization approach using an algorithm by Khmelnitsky, which considers electric trains equipped with regenerative braking. Additionally, the paper introduces a strategy derived from a more general formulation of Khmelnitsky's maximum principle. The authors of the study assert that they were able to decrease the train's average energy consumption by 37% using the optimized strategy (Heineken et al., 2023).

The objective of this study is to develop an Energy-Efficient Dynamic Driving Technique (EEDDT) model for urban rail transportation. Specifically, the study uses the FPA and GWO algorithms to optimize the selection of coasting position ranges that support driving speed, curve entry speed, and RBE formation and determine the operating boundary conditions and optimal operating regions in the model. The model considers a tramway line with 7 stations located in Samsun, Türkiye. To better approximate actual driving dynamics, the model includes dynamic weight dependent on passenger numbers, friction dependent on speed, gradient, and horizontal curve variables. This addition aims to achieve greater accuracy in simulating real-world conditions. Incorporating horizontal curvature dynamic driving techniques helps to increase the number of scenarios for driving strategy optimization, thereby expanding the selection range of coasting start points that support the generation of RBE. The model includes pre-curve and post-curve scenarios, encompassing the overall motion stages of the rail vehicle along a track, including a horizontal curve. The objective functions in the model consist of Minimum Traction Energy Consumption (MTEC), Maximum Regenerative Braking Energy Production (MRBEP) and Minimum Travel Time (MTT). The objective functions consist of MTEC, MRBEP, MTT single-stable and MTEC/MRBEP, MTEC/MRBEP, MTEC/MTT, MRBEP/MTT, MTEC/MRBEP/MTT multi-stable objective functions. The FPA and GWO algorithms are used to optimize the optimal values of the objective functions and the Pareto optimal operating boundary conditions. The developed EEDDT model enables speed/location stability in rail systems, creation of energy-efficient journey schedules, determination of travel time in advance, determination of the most efficient driving technique within constrained travel time, and stopping the rail system vehicle (RSV) at the destination station with minimum error. In order to clarify the research landscape and emphasize the contribution of this study, existing literature has been thematically and methodologically categorized in Table 1. The table presents an analytical comparison of prior works based on their methodology, focus, key contributions, limitations, and how each differs from the proposed approach. This classification allows for a clearer understanding of the innovative aspects of the present study in contrast to the state-of-the-art. Table 1 provides a comparative overview of recent studies focused on energy-efficient railway operation, highlighting the methodologies, optimization techniques, and key performance metrics adopted in each case. The comparison reveals the diversity of approaches employed in the literature, including regenerative braking energy utilization, timetable optimization, speed profile adjustment, and advanced control strategies using nature-inspired and intelligent algorithms. By summarizing the objective functions, algorithm types, and efficiency outcomes, the table offers a consolidated perspective on the current state of research in the field. This overview also underscores the novelty of the present study, which integrates multi-objective optimization with real-world dynamic driving scenarios to enhance energy efficiency in urban rail systems. The inclusion of both simulation-based and algorithm-driven methods in previous works validates the relevance of the metaheuristic techniques applied in this research.

Table 1.
Comparative Overview of Related Studies in Energy-Efficient Railway Operation.

Author/Study Method/Approach Focus Key Contribution Limitations Difference from This Study

Nasri et al. (2010) Timetable optimization RBE improvement 25% increase in RBE No consideration of dynamic driving scenarios Focused only on scheduling; coasting positions not optimized

González-Gil et al. (2013) Strategy proposal & simulation Energy management 20% energy saving Limited applicability; static geographical data Topographic variations (e.g., curves) not included

Li et al. (2023) GSOANR (modified Firefly Algorithm) Route optimization 5.6% energy saving Applicable only to ATO systems Coasting positions not explicitly modeled

Mayrink et al. (2020) Techno-economic analysis RB system evaluation 20% fuel saving Diesel-electric train model; non-electric system No coasting or drive strategy optimization; infrastructure-focused

Frilli et al. (2016) Energy optimization via simulation High-speed rail 10% improvement in energy efficiency Suitable for high-speed lines only Does not cover urban tram networks

Sun et al. (2023) Speed profile synchronization RBE alignment between trains 13% energy saving Limited to pairwise train scenarios No multi-stop routing or variable train loads

Che vd. (2022) GA & energy transfer system RBE utilization 99.3% RBE usage Requires additional infrastructure Hardware-focused; not an algorithmic driving technique

Zeng et al. (2022) Q-learning (RL-based) RBE scheduling strategy 5% energy saving Only tested in simulation Ignores dynamic slope, curvature, and load changes

Zhao et al. (2022) Hybrid energy storage + control Power quality & RBE Prolonged system lifespan Focused solely on hardware integration Does not perform algorithmic energy optimization

Wang and Su (2023) Timetable synchronization RBE overlap enhancement Real data model validation Static analysis; lacks dynamic scenarios Purely schedule-based; no driving strategy modeled

Xing et al. (2023) Brute-force optimization Energy reduction High potential energy savings Very high computational cost; not real-time applicable Simplistic speed profile; lacks intelligent adaptation

Xin et al. (2023) MILP + Gurobi solver RBE timing optimization 49% successful train overlap Single solution space; lacks driving strategy No consideration of road geometry, slope, or train weight

Heineken et al. (2023) Optimal control theory Energy-efficient driving 37% energy saving Omits passenger load, curves, and slopes No multi-objective optimization; lacks Pareto exploration

Author/Study	Method/Approach	Focus	Key Contribution	Limitations	Difference from This Study
Nasri et al. (2010)	Timetable optimization	RBE improvement	25% increase in RBE	No consideration of dynamic driving scenarios	Focused only on scheduling; coasting positions not optimized
González-Gil et al. (2013)	Strategy proposal & simulation	Energy management	20% energy saving	Limited applicability; static geographical data	Topographic variations (e.g., curves) not included
Li et al. (2023)	GSOANR (modified Firefly Algorithm)	Route optimization	5.6% energy saving	Applicable only to ATO systems	Coasting positions not explicitly modeled
Mayrink et al. (2020)	Techno-economic analysis	RB system evaluation	20% fuel saving	Diesel-electric train model; non-electric system	No coasting or drive strategy optimization; infrastructure-focused
Frilli et al. (2016)	Energy optimization via simulation	High-speed rail	10% improvement in energy efficiency	Suitable for high-speed lines only	Does not cover urban tram networks
Sun et al. (2023)	Speed profile synchronization	RBE alignment between trains	13% energy saving	Limited to pairwise train scenarios	No multi-stop routing or variable train loads
Che vd. (2022)	GA & energy transfer system	RBE utilization	99.3% RBE usage	Requires additional infrastructure	Hardware-focused; not an algorithmic driving technique
Zeng et al. (2022)	Q-learning (RL-based)	RBE scheduling strategy	5% energy saving	Only tested in simulation	Ignores dynamic slope, curvature, and load changes
Zhao et al. (2022)	Hybrid energy storage + control	Power quality & RBE	Prolonged system lifespan	Focused solely on hardware integration	Does not perform algorithmic energy optimization
Wang and Su (2023)	Timetable synchronization	RBE overlap enhancement	Real data model validation	Static analysis; lacks dynamic scenarios	Purely schedule-based; no driving strategy modeled
Xing et al. (2023)	Brute-force optimization	Energy reduction	High potential energy savings	Very high computational cost; not real-time applicable	Simplistic speed profile; lacks intelligent adaptation
Xin et al. (2023)	MILP + Gurobi solver	RBE timing optimization	49% successful train overlap	Single solution space; lacks driving strategy	No consideration of road geometry, slope, or train weight
Heineken et al. (2023)	Optimal control theory	Energy-efficient driving	37% energy saving	Omits passenger load, curves, and slopes	No multi-objective optimization; lacks Pareto exploration

Table 2.

Definitions and Values of Dynamic Driving Technique.

		Range definitions
Model Descrip	Explanation	Maximum	Minimum
V_driving _1,2	Cruising speed before and after the curve	70 km/h	15 km/h
V_braking _1,2	Safe stop braking speed before and after the curve	15 km/h
a,b	Travel resistance constant	2/2000
a_h	Speed up acceleration	1 m/s²	1.2 m/s²
a_g	Transition velocity acceleration	1 m/s²	1.2 m/s²
a_f	Braking acceleration	1 m/s²	1.4 m/s²
ξ	Constant	0.06
α	Coefficient depending on	0.7
β	the vehicle's brake type	0.9
γ	and brake elements	0.8

Table 3.

Rail System Vehicle Technical Specifications.

Vehicle Technical Specifications	Value	Vehicle Technical Specifications	Value
Number of Seats	50 Pieces	Empty Vehicle Weight	45.452 kg
Standing Passenger Capacity	240 Person	Fully Loaded (70 kg/person) Weight	65.752 kg
Total Passenger Capacity	290 Person	Velocity Acceleration	1.2 m/s²
Maximum speed	70 km/h	Braking Acceleration	2.8 m/s²
Average Operating Speed	25 km/h	Vehicle Traction Power Capacity	544 kW

2 Material and Method

This section provides an overview of RS motion dynamic calculations, traction system operating modes, and optimization algorithms used in the study.

2.1 Material

The objective of research on RSV driving techniques is to complete the journey from the starting station to the destination station with the shortest possible travel time and minimum energy consumption while considering speed and time constraints. The motion stages of an RSV is generally shown in Figure 1, where a high amount of energy is consumed during the acceleration stage, while lower energy consumption occurs during the cruising stage.

Figure 1.

Rail System Vehicle Speed Profile (Scheepmaker et al., 2017).

Vehicles equipped with RB capability generate energy during the coasting phase, and there is no energy consumption during the braking phase except for auxiliary loads (Scheepmaker et al., 2017). Vehicles equipped with RB feature can continuously repeat energy consumption and production stages during acceleration-deceleration cycles during the journey (Scheepmaker et al., 2017). By optimizing these motion phases of the rail vehicle, different driving techniques can be developed, and efficiency can be increased (Scheepmaker et al., 2017). The speed profile of an RSV is shown in Figure 1, where it accelerates with an a acceleration between 0-A for $Δ t_{1}$ . It continues to travel at Vmax speed for $Δ t_{2}$ between A-B, followed by coasting between B-C for $Δ t_{3}$ . Then, it comes to a stop by mechanical braking for $Δ t_{4}$ between C-D. The entire journey from the start position to the destination station is completed in $\sum Δ t = Δ t_{1} + Δ t_{2} + Δ t_{3} + Δ t_{4}$ .

2.2 Method

Trams are widely used in urban tramway transportation and typically have an average speed of 20 km/h to 30 km/h, taking into account their acceleration and deceleration capabilities. They usually accelerate with an acceleration value of approximately 1.1 m/s² and begin to decrease their speed with an acceleration of around 1.3 m/s² (Khodaparastan et al., 2019). From the moment an electric rail vehicle starts moving, it requires energy for traction. It reuses the kinetic energy gained during acceleration to overcome opposing forces such as rolling resistance, aerodynamic friction, and braking systems (Guo et al., 2017). To efficiently use the energy required, variables such as rolling resistance, maximum cruising speed, braking speed, and track gradient must be analyzed (Bae et al., 2007; Ceraolo et al., 2018; Frilli et al., 2016; González-Gil et al., 2013; Li et al., 2023; Mayrink et al., 2020; Nasri et al., 2010; Scheepmaker et al., 2017; Sun et al., 2023).

The dynamic movement calculations and optimization methods of the rail vehicle used in the developed EEDDT model are presented sequentially.

Accurate identification of the forces that drive the rail vehicle and those that resist its motion is essential in modeling the EEDDT. The vehicle's motion equation can be obtained by utilizing Newton's second law. According to this calculation, the net force can be expressed as given in Equation-1.

\begin{aligned} \sum F = Z - W = M a = M \frac{d v}{d v} (N) \end{aligned}

(1)

In Equation-1, $\sum F (N)$ represents the net force acting on the vehicle, $M (kg)$ represents the mass of the vehicle, a (m/s²) represents the acceleration, $Z (N)$ represents the traction force that accelerates the vehicle, and $W (N)$ represents the resistance force that makes the vehicle's motion difficult. The traction force expressions obtained $(Z_{a} + Z_{m})$ , as a function of speed for a locomotive with a specific adhesion weight and power, are shown in Figure 2.

Figure 2.

Attraction Forces and Sum Resistance/(Velocity -V) Relationship.

The graph provides some limit values for speed and traction force. The expressions for traction force are given in Equation-2 and Equation-3.

\begin{aligned} Z = Z_{a} & = μ g G_{a} (N); V \leq V_{t} \end{aligned}

(2)

\begin{aligned} Z = Z_{m} & = (3600 P) / V (N); V \geq V_{t} \end{aligned}

(3)

The expression Z_a defines the maximum adhesion force that can be transmitted between the locomotive wheels and the rail without slip. In this equation: Z_a (N) represents the adhesion force in newtons (N), μ (dimensionless) is the coefficient of adhesion between the wheel and the rail, g (m/s²) denotes the acceleration due to gravity (approximately 9.81 m/s²), G_a (kg) is the adhesion weight, referring to the portion of the locomotive's weight exerted on the driven axles. This relationship is derived from Newton's second law of motion and is fundamental in determining the limit of usable tractive effort before wheel slip occurs. The adhesion force increases with greater adhesion weight and higher friction conditions, but it is ultimately bounded by the coefficient of adhesion. The expression Z_m defines the theoretical tractive effort generated by the locomotive motor as a function of power and speed. In this equation: Z_m (N) represents the motor tractive force in newtons (N), P (kW) is the rated power of the locomotive in kilowatts (kW),V (km/h) denotes the locomotive speed in kilometers per hour (km/h).

The energy and power expressions in a RSV is given in Equation-4, Equation-5, and Equation-6.

\begin{aligned} Energy : E & = Z_{m} l (Joule = Nm) \end{aligned}

(4)

\begin{aligned} Power : P & = E / t (Watt = Nm/s) \end{aligned}

(5)

\begin{aligned} P & = (Z_{m} l) / t = Z_{m} v^{m / s} = Z_{m} V^{k m / h} / 3.6 \end{aligned}

(6)

These relationships allow for the detailed analysis of energy dynamics within the rail system, including both energy consumption and the energy recovered through regenerative braking.

The expression E defines the specific tractive effort, which represents the tractive force per unit mass of the locomotive or vehicle. In this equation: E (Joule) is the specific tractive effort, l (m) is the length of the track in meters (m). The consumed energy and the produced RBE in the general motion equation can be written as given in Equation-7 and Equation-8.

\begin{aligned} E_{c o n_{i}} (l) = Z_{i} Δ l_{i} (kJoule) \end{aligned}

(7)

\begin{aligned} E_{r e g_{i}} (l) = W_{i} Δ l_{i} (kJoule) \end{aligned}

(8)

Equations (7, and 8) model the energy consumption and regenerative braking energy production in rail vehicles. $E_{c o n_{i}}$ energy consumed at point i (kJ), $Z_{i}$ traction force at point i (N), $l_{i}$ distance traveled at point i (m). $E_{r e g_{i}}$ Regenerative energy produced at point i (kJ), $W_{i}$ Resistance force at point i (N).

The objective function equations can be written as given in between Equation-9 and Equation-15.

\begin{aligned} O F 1 & = min (\sum_{n = 1}^{i t e r} E_{c o n_{n}} (X)) \end{aligned}

(9)

\begin{aligned} O F 2 & = max (\sum_{n = 1}^{i t e r} E_{r e g_{n}} (X)) \end{aligned}

(10)

\begin{aligned} O F 3 & = min (\sum_{n = 1}^{i t e r} Δ t_{n} (X)) \end{aligned}

(11)

\begin{aligned} O F 4 & = min (\sum_{n = 1}^{i t e r} \sqrt[2]{\begin{array}{l} (O F 1 - E_{c o n_{n}} (X))^{2} \\ + (O F 3 - E_{r e g_{n}} (X))^{2} \end{array}}) \end{aligned}

(12)

\begin{aligned} O F 5 & = min (\sum_{n = 1}^{i t e r} \sqrt[2]{\begin{array}{l} (O F 2 - E_{r e g_{n}} (X))^{2} \\ + (O F 3 - Δ t_{n} (X))^{2} \end{array}}) \end{aligned}

(13)

\begin{aligned} O F 6 & = min (\sum_{n = 1}^{i t e r} \sqrt[2]{\begin{array}{l} (O F 1 - E_{c o n_{n}} (X))^{2} \\ + (O F 2 - E_{r e g_{n}} (X))^{2} \end{array}}) \end{aligned}

(14)

\begin{aligned} O F 7 & = min (\sum_{n = 1}^{i t e r} \sqrt[2]{\begin{array}{l} (O F 1 - E_{c o n_{n}} (X))^{2} \\ + (O F 2 - E_{r e g_{n}} (X))^{2} \\ + (O F 3 - Δ t_{n} (X))^{2} \end{array}}) \end{aligned}

(15)

Equations 9, 10, and 11 provide single- objective functions for solving MTEC, MRBEP, and MTT, respectively. Equations 12, 13, 14 and 15 provide multi- objective functions for solving MTEC / MRBEP, RBEP / MTT, MTEC / MRBEP and MTEC / MRBEP / MTT respectively. The single-objective functions (OF1–OF3) are designed to independently optimize three distinct performance metrics: minimizing traction energy consumption (MTEC) (Eq. 9), maximizing regenerative braking energy production (MRBEP) (Eq. 10), and minimizing total travel time (Eq. 11). Multi-objective functions (OF4–OF7) combine these aims via Pareto-optimal trade-offs, formulated as weighted Euclidean distances to ideal values (Eqs. 12–15). All objectives are subject to speed limits (V ≤ V_max) and time constraints (Δt_n≥Δt_min).

The proposed EEDDT model intrinsically embeds a rich optimization framework by addressing three conflicting objectives: MTEC, MRBEP, and minimizing travel time. These objectives are inherently contradictory in nature; for instance, reducing energy consumption through prolonged coasting phases may lead to increased travel time, while minimizing travel time often results in higher energy usage. To address this, the model formulates both single-objective (Equations 9–11) and multi-objective optimization problems (Equations 12–15), enabling the exploration of a Pareto-optimal trade-off space.

In this space, optimization does not occur in a fixed deterministic path, but rather as a dynamic selection process governed by the driving scenario, track profile, and performance requirements. The optimization process is guided by nature-inspired algorithms (FPA and GWO) which efficiently explore the multidimensional search space defined by decision variables such as acceleration profiles, coasting start positions, and braking thresholds.

The diversity in scenarios (e.g., pre-curve/post-curve strategies) further enriches the optimization landscape, allowing the model to identify optimal trade-offs under varying boundary conditions. The developed model thus explicitly defines the optimization space in terms of system states, speed constraints, and energy dynamics. The presented results (Tables 5–8, Figure 3) demonstrate that optimization occurs not only through equality constraints but within a well-characterised multi-objective framework, where decision-makers can prioritize among energy saving, travel time adherence, and regenerative energy maximization based on operational goals.

Figure 3.

Performance Comparison of Optimization Algorithms with Driving Technique Scenarios.

2.2.1 Optimization Models

Difficult optimization problems can be categorized as those with unclear mathematical models or those that require a significant amount of time to find the optimal solution (Mohamed vd., 2020). To address these challenges, intelligent optimization algorithms have been widely used in recent years to solve the Energy-Efficient Train Operation (EETO) problem.

The Energy Efficient Dynamic Driving Technique optimization problem is NP-hard due to its high-dimensional, nonlinear, and combinatorial nature:

Nonlinear and Discontinuous Dynamics: The motion equations (Eqs. 1–8) incorporate nonlinear resistance forces (e.g., speed-dependent friction, curve/gradient effects) and discrete regime switches (e.g., coasting → braking). Small changes in decision variables (e.g., coasting start position k_o) can lead to abrupt changes in energy consumption and regenerative braking, rendering the objective functions non-smooth and non-convex.

Combinatorial Explosion: For a tram line with N stations, the number of possible coasting sequences grows exponentially with N. This resembles classic NP-hard problems like the traveling salesman problem (TSP) with time windows, where exact methods (e.g., brute-force search) become computationally intractable for large N (Wang & Su, 2023; Xing et al., 2023).

Multi-Objective Pareto Optimization: The competing objectives (MTEC, MRBEP, MTT) require identifying a Pareto-optimal frontier (Figure 3), which scales poorly with traditional gradient-based or integer programming techniques. Given these challenges, conventional optimization methods fall short. Therefore, the adoption of evolutionary algorithms is justified.

Why Evolutionary Algorithms (FPA/GWO)? Nature-inspired metaheuristics, such as the Flower Pollination Algorithm (FPA) and the Grey Wolf Optimizer (GWO), offer a highly effective solution framework for this problem by leveraging several key advantages:

Global Optimization Capability: FPA's Lévy flight mechanism (Eq. 16) enables long jumps in the search space to escape local optima, while GWO's social hierarchy (alpha/beta wolves) balances exploration and exploitation. This is critical for navigating the non-convex objective landscape.

Constraint Handling Without Penalties: Speed limits, curve radii, and passenger loads are embedded as hard constraints via boundary checks (Table 1), thus avoiding the need for penalty functions that distort the objective space.

Computational Efficiency: Compared to brute-force search (used in [Xing et al., 2023] for small instances) or genetic algorithms (tested in [Li et al., 2023]), FPA and GWO achieve faster convergence (Tables 5–6) while maintaining solution quality. For instance, GWO reduces iteration completion time (ICT) by 50% compared to FPA (Table 6), critical for real-time applications. In addition, the problem exhibits sensitivity to initial conditions that resemble noisy landscapes.

Robustness to Noise-Like Discontinuities: While the objectives are deterministic, their sensitivity to initial conditions (e.g., passenger load variations) mimics “noisy” responses. Evolutionary algorithms inherently handle such landscapes via population-based sampling. These characteristics make FPA and GWO highly effective for solving the EEDDT problem, especially in real-time optimization scenarios.

These algorithms include genetic algorithms (Song et al., 2016; Tuyttens et al., 2013) differential evolutionary algorithms (Kim et al., 2013; Liu et al., 2019; Liu et al., 2020) ant colony optimization algorithms Fan et al., 2015; Cao et al., 2016; Naldini et al., 2021) particle swarm optimization algorithms (Wu et al., 2016), neural network algorithms (Chuang et al., 2008; Huang et al., 2016) and other meta-heuristic methods.

These algorithms mimic natural phenomena and processes or the intelligent behaviors of biological groups and are characterized by simplicity, generality, and ease of parallel processing. In this study, due to the problem's multidimensionality, intuitive methods such as the FPA and GWO were employed. Both algorithms rely on natural behaviors and interactions between individuals, making them highly effective in solving multidimensional optimization problems when appropriate parameters are selected.

The model we developed uses an optimization algorithm to determine the optimal speed and coasting positions based on the vehicle's driving characteristics. Within these limits, the algorithm identifies the situations in which the specified station distance can be completed in the shortest and longest time. These situations are showned in Figure 4, which are k_m₁, k_m₁’, k_x₁, k_y₁, k_m₂, k_m₂’, k_x₂, and k_{f_stop} points. These states also provide information on the speed, location, and time for the specified line. Figure 4 shows k_o(₁₋₂₎, which indicates the coasting starting position for generating RF energy. Table 5a presents scenarios for the pre-curve driving technique, including acceleration, cruising speed, and coasting starting position for each possibility where the driving speed (V_driving) is greater than the braking speed (V_braking). If the speed at the end of coasting is greater than V_curve, the RS vehicle mechanically brakes until V_curve. If it is smaller, it accelerates until V_curve. If the V_braking, V_curve, and V_driving speeds are equal, the RS vehicle accelerates up to the V_curve speed. After acceleration, it coasts to generate RF energy, supporting both safe entry into the curve region and RF energy generation.

Table 4.
Track Data for a Section of Samsun Tramway Line.

Starting Station	End Station	Inter-station Distance-m	Curve Entry Distance-m	Curve Exit Distance-m	Curve Distance-m	Curve Diameter-m	Curve Speed Limit-km/h	Slope-%
Dormitories	Faculty of Education	660	137	237	100	270	30	4.046
Faculty of Education	Life Centre	420	193	343	150	100	20	2.617
Life Centre	Health Sciences	600	40	167	127	100	20	2.281
Health Sciences	Dental Medicine	840	60	210	150	105	20	−2.030
Dental Medicine	Faculty of Medicine	720	190	640	450	105	20	−4.410
Faculty of Medicine	OMU Lodging	1420	450	1150	700	250	30	−2.460

Table 5a

Speed Limit Values for Pre-Curve Driving Technique Selection and Coasting Situation Comparison.

Before the Curve
Driving Technique Scenario-1		Driving Technique Scenario-2		Driving Technique Scenario-3
V_driving > V_curve > V_braking		V_driving > V_curve = V_braking		V_driving = V_curve > V_braking
COP	ADT	COP	ADT	COP	ADT
k_o ₁ = k_m ₁	AC + CO + BR/AC + CO/AC + CO + AC/ AC + CO + CR + AC	k_o ₁ = k_m ₁	AC + CO + BR AC + CO AC + CO + CR	k_o ₁ = k_m ₁	AC + CO + AC AC + CO + CR + AC
k_m ₁ < k_o ₁ < k_m ₁	AC + CR + CO + BR/ AC + CR + CO/AC + CR + CO + AC AC + CR + CO + CR + AC	k_m ₁ < k_o ₁ < k_m ₁	AC + CR + CO + BR AC + CR + CO AC + CR + CO + CR	k_m ₁ < k_o ₁ < k_m ₁	AC + CR + CO + AC AC + CR + CO + CR + AC
k_o ₁ = k_x ₁	AC + CR + BR	k_o ₁ = k_x ₁	AC + CR + BR	k_o ₁ = k_x ₁	AC + CR

Driving Technique Scenario-4		Driving Technique Scenario-5		Driving Technique Scenario-6
V_curve > V_driving > V_braking		V_driving =V_curve = V_braking		V_curve > V_driving = V_braking
COP	ADT	COP	ADT	COP	ADT
k_o ₁ = k_m ₁	AC + CO + AC AC + CO + CR + AC	k_o ₁ = k_m ₁	AC + AC + CO + BR AC + AC + CO AC + AC + CO + CR	k_o ₁ = k_m ₁	AC + AC + CO + AC AC + AC + CO + CR + AC
k_m ₁ < k_o ₁ < k_m ₁	AC + CR + CO + AC AC + CR + CO + CR + AC	k_m ₁ < k_o ₁ < k_m ₁	AC + CR + AC + BR AC + CR + AC + CO + BR AC + CR + AC + CO AC + CR + AC + CO + CR	k_m ₁ < k_o ₁ < k_m ₁	AC + CR + AC + CO + AC AC + CR + AC + CO + CR + AC
k_o ₁ = k_x ₁	AC + CR + AC	k_o ₁ = k_x ₁	AC + CR	k_o ₁ = k_x ₁	AC + CR + AC

Table 5b

Speed Limit Values for Post-Curve Driving Technique Selection and Coasting Situation Comparison.

After the Curve
Driving Technique Scenario-1		Driving Technique Scenario-2		Driving Technique Scenario-3
V_driving > V_curve > V_braking		V_driving > V_curve = V_braking		V_driving = V_curve > V_braking
COP	ADT	COP	ADT	COP	ADT
k_o ₁ = k_m ₁	AC + CO + BR AC + CO + CR + BR	k_o ₁ = k_m ₁	AC + CO + BR AC + CO + CR + BR	k_o ₁ = k_m ₁	CO + BR CO + CR + BR
k_m ₁ < k_o ₁ < k_m ₁	AC + CR + CO + BR AC + CR + CO + CR + BR	k_m ₁ < k_o ₁ < k_m ₁	AC + CR + CO + BR AC + CR + CO + CR + BR	k_m ₁ < k_o ₁ < k_m ₁	CR + CO + BR CR + CO + CR + BR
k_o ₁ = k_x ₁	AC + CR + BR	k_o ₁ = k_x ₁	AC + CR + BR	k_o ₁ = k_x ₁	CR + BR

Driving Technique Scenario-4		Driving Technique Scenario-5		Driving Technique Scenario-6
V_curve > V_driving > V_braking		V_driving =V_curve = V_braking		V_curve > V_driving = V_braking
COP	ADT	COP	ADT	COP	ADT
k_o ₁ = k_m ₁	BR + CO + BR/BR + CO + CR + BR	k_o ₁ = k_m ₁	AC + CO + BR AC + CO + CR + BR	k_o ₁ = k_m ₁	BR + AC + BR/BR + AC + CO + BR BR + AC + CO + CR + BR
k_m ₁ < k_o ₁ < k_m ₁	BR + CR + CO + BR BR + CR + CO + CR + BR	k_m ₁ < k_o ₁ < k_m ₁	CR + AC + BR CR + AC + CO + BR C + AC + CO + CR + BR	k_m ₁ < k_o ₁ < k_m ₁	BR + CR + AC + BR BR + CR + AC + CO + BR BR + CR + AC + CO + CR + BR
k_o ₁ = k_x ₁	BR + CR + BR	k_o ₁ = k_x ₁	CR + BR	k_o ₁ = k_x ₁	BR + CR + BR

Figure 4.

Optimization Points for Cruising Speeds, Dynamic Driving Technique and Transition Between States.

The model also includes a possible braking situation after the curve. After mechanical braking at the exit of the curve, the RS vehicle maintains a cruising speed until reaching the k_o₂’ point, as determined by the optimization algorithm. It then accelerates to the speed of the curve, where safe stopping is once again possible, before continuing with the coasting event. If the coasting event ends at position k_b₂, the RS vehicle will maintain a cruising speed until reaching the k_{f_stop} position, where safe braking will commence. The model allows for the ko position to be located anywhere between the k_m and k_x positions, resulting in the most suitable driving technique being achieved according to the desired objective function. Additionally, the model includes comparisons of state values such as k_o₁ = k_m₁, k_o₁ > k_m₁ / k_o₁ < k_x₁, and k_m₁ = k_x₁, along with V_driving, V_curve, and V_braking speed values. The flow chart of the proposed EEDDT driving technique is shown in Figure 5.

Figure 5.

EEDDT Driving Technique Flow Chart.

To enhance the reproducibility, transparency, and technical depth of the proposed optimization approach, this section provides detailed pseudocode, algorithmic workflow, and the integration scheme of the FPA and GWO within the EEDDT framework. Both algorithms were employed independently to solve the single-objective and multi-objective functions (Equations 9–15) in the developed model and to compare their performance. In this study, the primary reason for employing evolutionary optimization algorithms such as the FPA and Grey Wolf Optimization (GWO) for the optimization of Energy-Efficient Dynamic Driving Techniques (EEDDT) lies in the inherent complexities of the problem and the limitations of traditional methods.

2.2.1.1 Flower Pollination Algorithm (FPA)

Approximately 80% of plant species on Earth are classified as flowering plants. The primary function of a flowering plant is to reproduce by transferring pollen, which is produced through its reproductive organs, to recipients with the aid of biotic or abiotic agents, thereby facilitating the continuation of the species life cycle. In some instances, abiotic carriers such as wind and floodwaters can also facilitate pollination without the involvement of biotic agents (Kar, 2016). This natural pollination process has existed for millions of years and has been successfully employed to address engineering problems (Yang, 2012). The FPA operates on four fundamental principles that are explained as follows:

Rule 1: A predetermined selection probability threshold (p ∈ [0, 1]) is used in the selection of the method to be applied when determining the search space in the pollination process (global and local search).Rule 2: When performing the global operation to represent the biotic process, the step space is determined using the Levy distribution (Levy Flight). Rule 3: When performing local search representing the abiotic process, the step space is determined using a uniform distribution. Rule 4: To increase the probability of re-selection, pollinators used in biotic pollination change the flower density as given in Equation-16.

\begin{aligned} x_{i}^{t + 1} = x_{i}^{t} + γ L (λ) (g_{*} - x_{i}^{t}) \end{aligned}

(16)

In Equation-16,

x_{i}^{t}

represents the i'th solution at the t'th iteration,

g_{*}

the current best solution,

L (λ)

the search space step size corresponding to the lambda value and,

γ

the scaling factor. Before proceeding to the implementation phase of the algorithm, parameters such as the lower upper bounds of the objective function search space (U_b and L_b), the number of critical driving points (k_1,2…n), the driving constants and variables, the maximum number of iterations (MaxGen) and the probability threshold value (p) are determined. Before the iteration process starts, these values are randomly generated between the boundary values of the search space. After the initial assignment, the objective function is run according to these values and the objective value is calculated, these values are assigned as the best value and the optimization process starts. In each step, a random number (rand ∈ [0, 1]) is generated at the beginning of the iteration and this generated number is checked against the threshold value. When this randomly generated number is greater than the threshold value, the search process is a biotic process and global search rules are applied to organize the variable step space according to Levy's flight, following the equation given in rule 4. Otherwise, the search space is determined with the help of the coefficients obtained with a uniform distribution, using values in the range 0–1.The new solution space is applied to the objective function to calculate the performance values and if there is an improvement, the best value is updated. This process continues until the number of iterations reaches a predetermined maximum. When the iterations are completed, the best value is declared.

The following pseudocode outlines the step-by-step procedure of the FPA, which is employed in this study to solve single-objective and multi-objective optimization problems by mimicking the global and local pollination behaviors observed in nature.

Input: Objective function f(x), population size n, switch probability p ∈ [0,1], max iterations MaxGen, search bounds [L_b, U_b]

Initialize a population of n flowers x_i ∈ [L_b, U_b]

Evaluate fitness f(x_i) for each i

Determine the global best solution g*

for t = 1 to MaxGen do

for each flower i do

Generate random number rand ∈ [0,1]

if rand < p then

Global pollination (Levy flight):

x_i(t + 1) = x_i (t) + L * (g*−x_i (t))

else

Local pollination:

Select x_j and x_k randomly from population

x_i(t + 1) = x_i(t) + ε * (x_j−x_k), where ε ∈ [0,1]

end if

Enforce boundary constraints and handle feasibility

Evaluate f(x_i(t + 1)) and update g* if improved

end for

Return best solution g*

Initialization: A population of candidate solutions (flowers) is randomly initialized within the predefined lower and upper bounds of the search space. Each individual encodes decision variables of the EEDDT model.

Fitness Evaluation: The fitness of each flower is evaluated by calculating the value of the selected objective function (e.g., MTEC, MRBEP, or MTT). The best-performing individual is assigned as the global best solution g*.

Pollination Mechanism: A switching probability p is used to determine whether global or local pollination is performed: Global Pollination simulates long-distance pollination via Lévy flights, facilitating global exploration of the search space. This step helps to avoid local optima by using the current best solution to guide the search. Local Pollination represents neighborhood-based search using two randomly selected individuals from the population. It enables exploitation by refining existing solutions.

Boundary and Constraint Handling: If a new candidate solution exceeds the defined search bounds or violates any problem-specific constraints (e.g., speed, safety), it is corrected through reinitialization or boundary clipping.

Solution Update: If the new candidate yields a better fitness value than the current best, the global best solution is updated.

Termination: The algorithm iterates until the maximum number of generations is reached or convergence is achieved based on the stability of the best solution.

2.2.1.2 Grey Wolf Algorithm (GWO)

GWO is a nature-inspired optimization algorithm that was introduced in 2014 by Mirjalili et al. (2014). GWO is one of the promising metaheuristic algorithms, which are general- objective optimization algorithms that are inspired by natural phenomena such as the behavior of animals, plants, and other natural systems. The algorithm is based on the social hierarchy and hunting behavior of grey wolves. Alpha wolf is the most powerful of the entire pack and leads in hunting ana feeding. In case of disappearing of alpha, beta wolf leads the pack. Also, whether in needed, delta and omega wolves lead the pack, respectivel (Faris et al., 2018). The GWO algorithm starts with an initial population of randomly generated solutions, represented by a set of wolves. The fitness of each wolf is evaluated based on its ability to solve the problem, and the wolves are ranked according to their fitness. The alpha, beta, and delta wolves are then selected based on their fitness, and a search is conducted in the search space using their positions. Wolf pack generally follows steps which are chasing or encircling, harrasing and attacking in hunting a prey. In the encircling step is stocasting process and each wolf of the pack changes location depends on the position of the prey.

\begin{aligned} X (t + 1) & = X (t) - A D \end{aligned}

(17)

\begin{aligned} D & = C X_{p} (t) - X (t) \end{aligned}

(18)

In encircling phase, $X (t)$ existing position and $X (t + 1)$ next position of the wolf and $X_{p} (t)$ is the position of the prey. D is a vector matrix chaning depends on position of the prey, A and C are coefficient values which are generated randomly. In harrasing and hunting phase positions of the wolves $(X_{1}, X_{2} and X_{3})$ get closer to the prey and circle surrounding the prey is decreasing relatively quickly.

\begin{aligned} X_{1} = X_{α} (t) - A_{1} D_{α} \end{aligned}

(19)

\begin{aligned} X_{2} = X_{β} (t) - A_{2} D_{β} \end{aligned}

(20)

\begin{aligned} X_{3} = X_{δ} (t) - A_{3} D_{δ} \end{aligned}

(21)

Algorithm ends when local search error untill not change siginificantly or iteration limit is exceeded.

The pseudocode presented below illustrates the implementation logic of the GWO (GWO), which simulates the leadership hierarchy and cooperative hunting behavior of grey wolves to explore and exploit the solution space efficiently within the context of the proposed EEDDT model.

Input: Objective function f(x), number of wolves N, max iterations MaxGen, search bounds [L_b, U_b]

Initialize a population of N wolves X_i ∈ [L_b, U_b]

Evaluate fitness of each X_i

Identify the top three solutions as α (best), β (second), δ (third)

for t = 1 to MaxGen do

for each wolf i do

Compute distance vectors:

D_α = |C₁ * X_α−X_i|, D_β = |C₂ * X_β−X_i|, D_δ = |C₃ * X_δ−X_i|

Compute candidate positions:

X₁ = X_α−A₁ * Dα

X₂ = X_β−A₂ * D_β

X₃ = X_δ−A₃ * D_δ

Update position:

X_i(t + 1) = (X₁ + X₂ + X₃) / 3

Enforce boundary constraints and handle feasibility

end for

Recalculate fitness and update α, β, and δ

end for

Return α as the best solution

Initialization: Population of wolves is initialized randomly within the decision space. Each wolf represents a potential solution composed of EEDDT parameters such as coasting distance and curve speed.

Role Assignment: The best, second-best, and third-best individuals are assigned as the α, β, and δ wolves, respectively. These act as leaders in guiding the search process.

Position Update/Encircling Mechanism: Each wolf updates its position based on the relative position to the α, β, and δ wolves. Coefficients A and C are dynamically adjusted to simulate the encircling behavior observed in nature. This mechanism balances exploration and exploitation.

Averaging Strategy: The final updated position of each wolf is the average of its positions relative to α, β, and δ. This strategy enhances the convergence rate toward optimal solutions by considering multiple leaders simultaneously.

Constraint Management: As in FPA, solutions that fall outside the feasible region are corrected or penalized. Domain-specific constraints related to safety (e.g., braking limits, coasting zone) are enforced.

Selection and Update: After updating positions, fitness values are recalculated. The roles of α, β, and δ are reassigned based on the new fitness values.

Termination: The process continues until a predefined maximum iteration count is reached, or solution convergence is observed.

3 Simulation Results and Discussion

Energy-efficient dynamic driving techniques supported by accelerating and decelerating cycles can minimize energy consumption while maximizing energy production, thereby achieving efficient driving. Efficient and optimized decision-making models, such as the developed EEDDT model, can achieve this objective.

The EEDDT model proposed in this study is formulated under the assumption that tram vehicles operate exclusively on dedicated tracks, fully segregated from other road traffic. This assumption eliminates the influence of external and unpredictable factors such as traffic signals, vehicle congestion, and mixed-use interactions, which are typically present in shared corridors. Consequently, the model concentrates on optimizing controllable parameters, including speed profiles, delay minimization, and energy consumption, thereby enabling a more precise evaluation of system performance. The definitions and values of the dynamic driving technique in the model are given in Table 2 and the vehicle characteristics used in the model are given in Table 3.

During optimization, both the FPA and the GWO iteratively modify speed profiles and coasting strategies in response to load-dependent resistance forces. Since increased passenger mass leads to higher rolling friction and inertia, the optimization algorithms are designed to adaptively adjust acceleration and braking parameters to maintain energy efficiency across different loading scenarios (Figure 6).

Figure 6.

Simulated Section of Samsun Tramway Line.

The application subject to the model consists of a tramway line in Samsun, Türkiye. A particular segment of Samsun Urban Rail Network was taken into account for the case study and a real-like tram model was created with characteristics which are given in Table 4. The total length of the line is approximately 4600 meters. There are 7 stations on the line as given in Table 4. The driving phases consist of pre-curve and post-curve dynamic driving phases shown in Figure 4.

Table 6.

Single Stable Objective Function Optimization Comparison Values.

Description	OF1	OF2	OF3
FPA TEC (kWh)	4.550	7.442	19.186
FPA RBEP (kWh)	0.000	1.800	0.022
FPA DTR(kWh)	4.550	5.642	19.164
FPA TT (s)	1141.317	802.438	522.386
FPA ICT (s)	6754.961	4063.678	7051.664
FPA SPSD (m)	4659.950	4660.121	4660.000
FPA NP (person)	0.000	1740.000	297.000
GWO TEC (kWh)	4.607	6.948	19.503
GWO RBEP (kWh)	0.000	1.856	0.001
GWO DTR(kWh)	4.607	5.092	19.502
GWO TT (s)	1141.317	795.122	521.379
GWO ICT (s)	5072.121	2546.135	5780.892
GWO SPSD (m)	4660.000	4660.125	4660.000
GWO NP (person)	0.000	1740.000	1119.000

TEC: Traction Energy Consumed, RBEP: Regenerative Braking Energy Produced, DTR: Difference TEC and RBEP, TT: Travel Time, ICT: Iteration Completion Time, SPSD: Station Platform Stop Distance, OF1: MEC, OF2: MRBE, OF3:MTT, NP: Number of Passenger.

The driving technique for the shortest and longest travel time was obtained within the specified limits, and k_m₁, k_m₁’, k_x₁, k_y₁, k_m₂, k_m₂’, k_x₂ and k_{f_stop} shown in Figure 4 were calculated as a result of these driving techniques. The locations where RBE begins to occur were indicated by k_o₁, k_c₁, k_o₂ and k_c₂.

The vehicle performs acceleration, cruising, coasting, regenerative braking and air braking movements in all scenarios, and the coasting points were determined by the optimization algorithm in accordance with the objective function and scenario. At the same time, the optimization algorithm decides which of the dynamic driving stra egies should be implemented in accordance with the specified objective function and scenario. Speed and position comparisons were separately handled for pre-curve and post-curve driving techniques. If there are similar complex structures in a station interval, the driving technique is determined according to the objective function.

The comparison of speed limit values, V_driving, V_curve, and V_braking, was performed to determine the driving dynamics scenarios in the developed EEDDT driving technique model and based on this comparison, pre and post curve scenarios were created for each station interval and given in Tables 5a and 5b.

The optimization results using single-stable objective functions for the modeled tramway line are given in Table 6. The single-stable objective functions consist of MTEC, MRBEP, and MTT.

In the proposed EEDDT model, key decision variables are optimized using the FPA and GWO to MTEC, MRBEP, and reduce MTT. The explicit decision variables include:

Coasting start positions (k_o_1, k_o₂): Transition points from cruising to coasting, optimized within defined positional bounds to enhance braking energy recovery.

Driving speeds (V_driving₁, V_driving₂) : Cruising speeds before and after curves, selected between 15–70 km/h to balance energy efficiency and travel time.

Braking speeds (V_braking₁, V_braking₂): Predefined deceleration speeds (15 km/h) used before stations and curves to support safe operation and regenerative recovery.

Acceleration/deceleration rates (a_h, a_g, a_f): Including acceleration (1.0–1.2 m/s²), transition acceleration (1.0–1.2 m/s²), and braking deceleration (1.0–1.4 m/s²), optimized for dynamic consistency and passenger comfort.

In addition, implicit variables such as the duration of each motion phase and curve entry/exit speeds ( V_curve = 20–30 km/h) are derived from the speed profiles.

These variables are evaluated under operational constraints—track geometry, speed limits, and vehicle dynamics (Eq. 1–8)—to produce Pareto-optimal solutions for both single- and multi-objective formulations (see Tables 6–7). The comprehensive optimization of these variables forms the foundation of our model's ability to generate practical, energy-efficient driving strategies while meeting operational constraints.

Table 7.

Multi-Stable Objective Function Optimization Comparison Values.

Description	OF4	OF5	OF6	OF7
FPA TEC (kWh)	6.289	13.365	5.325	6.330
FPA RBEP (kWh)	1.066	1.454	1.221	1.201
FPA DTR(kWh)	5.223	11.911	4.102	5.129
FPA TT (s)	701.807	552.999	926.075	699.888
FPA ICT (s)	51092.816	36181.432	54726.541	58097.631
FPA SPSD (m)	4659.973	4660.189	4660.049	4659.968
FPA NP (person)	37.000	0.000	1740.000	0.000
GWO TEC (kWh)	6.336	13.282	5.375	6.374
GWO RBEP (kWh)	1.101	1.490	1.271	1.228
GWO DTR(kWh)	5.235	11.872	4.104	5.146
GWO TT (s)	700.944	553.480	917.808	699.114
GWO ICT (s)	25335.642	18886.214	24658.697	26410.490
GWO SPSD (m)	4659.977	4660.192	4660.009	4659.977
GWO NP (person)	50.000	0.000	1740.000	0.000

TEC: Traction Energy Consumed, RBEP: Regenerative Braking Energy Produced, DTR: Difference TEC and RBEP, TT: Travel Time, ICT: Iteration Completion Time, SPSD: Station Platform Stop Distance, OF4: MEC/MTT, OF5:MRBE/MTT, OF6:MEC/MRBE, OF7:MEC/MRBE/MTT, NP:Number of Passenger.

Table 8.

Comparison of Driving Techniques of Samsun Tramway Line.

Driving Technique	Description	Between Station 1–2	Between Station 2–3	Between Station 3–4	Between Station 4–5	Between Station 5–6	Between Station 6–7
Operational Driving Technique	Traction Energy Consumption (kWh)	2.707	2.741	1.654	2.201	3.142	4.636
	Regenerative Energy Production (kWh)	0.200	0.020	0.113	0.141	0.001	0.001
	Travel Time (s)	95.150	60.487	86.402	120.991	114.300	204.481
	Station Distance (m)	660.000	420.049	600.036	840.029	720.00	1420.001
	Average Driving Speed (km/h)	25.000	25.000	25.000	25.000	22.677	25.000
EEDDT Driving Technique (FPA)	Traction Energy Consumption (kWh)	1.048	1.858	1.152	1.054	2.363	1.245
	Regenerative Energy Production (kWh)	0.410	0.081	0.223	0.169	0.030	0.196
	Travel Time (s)	95.082	61.993	86.513	121.092	115.134	204.578
	Station Distance (m)	660.020	420.037	600.031	840.031	720.041	1419.994
	Average Driving Speed (km/h)	24.990	24.392	24.969	24.974	22.514	24.988
EEDDT Driving Technique (GWO)	Traction Energy Consumption (kWh)	1.050	1.864	1.155	1.057	2.368	1.254
	Regenerative Energy Production (kWh)	0.418	0.083	0.227	0.172	0.031	0.200
	Travel Time (s)	95.078	61.983	86.513	121.092	115.147	204.585
	Station Distance (m)	660.021	420.037	600.031	840.031	720.040	1420.019
	Average Driving Speed (km/h)	24.991	24.396	24.969	24.974	22.512	24.988

To accurately represent operational dynamics, the model incorporates variable loading conditions by adjusting the total vehicle mass m in Equation-22. The total mass is calculated as:

\begin{aligned} m = m_{e m p t y} + n_{p} . m_{p} \end{aligned}

(22)

Where m_empty is the empty (curb) weight of the tram (45.452 kg, see Table 3), n_p is the number of passengers, m_p is the average passenger mass (assumed to be 70 kg/person).

Passenger load is treated as a time-varying parameter, updated at each station based on boarding and alighting behavior derived from empirical data.

This dynamic update enables the model to account for variations in traction force, rolling resistance, and braking energy recovery throughout the journey.

The traction energy consumption of the vehicle moving on the tramway line occurs only during the acceleration (AC) and cruising (CR) phases.

No energy is consumed during the coasting (CO) phase; instead, regenerative braking energy is generated. Increasing the duration of the CO phase in any strategy leads to a decrease in energy consumption since no energy is consumed during this phase.

However, this extension results in an increase in the total travel time. Punctuality is a crucial aspect of the total travel time and should be considered in energy efficiency studies. To address this, separate objective functions were created: TEC, RBEP, and TT, and their results are given in Table 6.

The shortest journey completion time is achieved in the driving technique specified by OF3. Indeed, this driving technique has the highest TEC value.

There is an approximate 61% reduction in traction energy consumption in the OF2 driving technique compared to OF3. Despite the significant reduction in traction energy consumption, the OF2 driving technique results in an approximate 54% increase in journey duration.

OF1 driving techniques has the minimum TEC. Despite achieving the minimum TEC, the OF1 driving technique has the highest TT among the considered strategies.

Additionally, the OF1 driving technique does not result in any RBEP. The driving techniques that focus on single-stable objective functions are not sufficient in terms of energy efficiency.

For this purpose, multi-objective functions whose result values are given in Table 7 were created. These objective functions were tested separately using FPA and GWO algorithms in 7 station intervals.

At this stage, the decision variables have been compared both pairwise and ternary to evaluate their effectiveness. OF4, OF5, and OF6 consist of pairwise comparisons of decision variables, while OF7 consists of simultaneous comparisons. OF4, OF5, OF6, and OF7 objective functions are formed by jointly optimizing the decision variables MEC/MTT, MRBE/MTT, MEC/MRBE, and MEC/MRBE/MTT, respectively.

Figure 3 shows the convergence behavior of the FPA and GWO algorithms across multiple objective functions (OF1–OF7), based on normalized median best fitness values over iterations. The results reveal that GWO achieves faster initial convergence due to its hierarchical search mechanism, while FPA demonstrates superior refinement in later iterations, particularly for multi-objective problems (OF4–OF7). Both algorithms converge stably within 100–150 iterations, with FPA attaining marginally better final fitness values. Therefore, FPA is preferable for high-precision optimization tasks, whereas GWO is more suitable for real-time applications requiring rapid, near-optimal solutions. To ensure solution reliability, both algorithms were executed independently 30 times under identical initial conditions.The Table 7 gives the information that OF6 achieves the lowest DTR value. However, despite this achievement, OF6 has the highest TT value among the considered driving techniques. This result indicates that despite achieving energy efficiency in terms of the lowest DTR value, OF6 is not sufficient as an energy-efficient driving technique considering the importance of punctuality. Other factors such as reducing TT need to be taken into account to achieve a more balanced and efficient driving technique. OF7 is essential for achieving EEDDT, where all decision variables converge to their optimal values. Indeed, in addition to considering the DTR and TT, OF7 also takes into account the RBEP efficiency. This comprehensive objective function aims to balance energy efficiency, decision coverage, and punctuality in the driving technique. Furthermore, the developed model contributes to predicting the travel by utilizing the driving technique that aligns with the specified objective function. The FPA and GWO algorithms used in the model were additionally tested and compared with the OF7 function between dormitories and education faculty stations, resulting in a Pareto frontier comparison. The results of this comparison can be seen in Figure 7.a and Figure 7.b.

Figure 7.

Pareto Frontier Comparison OF7 (FPA and GWO - Between Dormitories and Faculty of Education). a.FPA; b.GWO.

To evaluate the performance of the proposed optimization framework, the FPA and GWO were applied to solve the OF7 objective function, representing the transit route between dormitories and the education faculty stations. The outcomes were benchmarked against Pareto-optimal boundary conditions derived using the MTEC/MRBEP/MTT multi-objective function, as defined in Equation (15) and illustrated in Figures 7.a and 7.b.

Figures 7.a and 7.b clearly demonstrate that the local search space explored by the GWO algorithm is notably narrower than that of the FPA. This observation suggests that GWO adopts a more focused convergence approach, likely due to its hierarchical leadership-based search strategy. This behavior results in more rapid convergence toward local optima, which is crucial in scenarios requiring timely suboptimal solutions, such as real-time or near-real-time railway trajectory adjustments. This trend is further corroborated by the computational metrics presented in Tables 6 and 7, where GWO consistently achieves convergence within fewer iterations compared to FPA. Consequently, GWO exhibits superior convergence speed and efficiency, reinforcing its suitability for applications where execution time is a critical constraint. (Table 9).

Table 9.

Comparison of Result Values of Driving Techniques Applied to Samsun Tramway Line.

Description	Operational Driving Technique	EEDDT Driving Technique-FPA	EEDDT Driving Technique-GWO	Efficiency-FPA	Efficiency-GWO
Total Traction Energy Consumed	17.081 (kWh)	8.72 (kWh)	8.748 (kWh)	48.950%	48.785%
Total Regenerative Braking Energy Produced	0.476 (kWh)	1.109 (kWh)	1.131 (kWh)	132.983%	137.605%
Difference Between Energy Consumed and Energy Produced	16.605 (kWh)	7.611 (kWh)	7.617 (kWh)	54.164%	54.129%
Total Travel Time	681.811 (s)	684.392 (s)	684.398 (s)	99.621%	99.620%
Total Track Lenght	4660.119 (m)	4660.154 (m)	4660.178 (m)	99.999%	99.998%

However, despite its slower convergence, the FPA demonstrates higher accuracy in identifying near-optimal solutions, particularly when energy efficiency and travel time precision are of paramount importance. While GWO tends to prioritize convergence speed and local search effectiveness, FPA excels in fine-tuning solutions during later iterations, making it a favorable choice in applications that demand high-resolution optimization and global solution accuracy.

Furthermore, Figure 7 showcases the combined performance of both algorithms in capturing the trade-off surface across the three objectives MTEC, MRBEP, and MTT. The resulting Pareto fronts exhibit uniform distribution, and the spacing metrics and hypervolume indicators confirm that the objective space is well-covered, without significant clustering or voids.

In summary, although GWO outperforms FPA in terms of local convergence efficiency and ICT, FPA proves to be more effective in solving the objective functions within the developed model framework. The choice between these two algorithms should be made based on the specific priorities of the application: GWO for computational speed and localized optimization, and FPA for comprehensive exploration and solution refinement.

In the proposed model, the balanced distribution of the Pareto Optimal Front across the objective space is achieved through the inherent diversity mechanisms of both the FPA and GWO algorithms. FPA ensures broad exploration via global and local pollination strategies, while GWO maintains a structured search through its hierarchical leadership mechanism. The quality of distribution is quantitatively validated using spacing metrics and hypervolume indicators, which confirm that the generated solution set is uniformly spread without significant gaps or clustering. As illustrated in Figure 7, the resulting Pareto front effectively captures the trade-off surface among the objectives, indicating a well-distributed and robust optimization outcome.

Tables 8 and 9 present a comparison of the EEDDT driving technique along a section of the Samsun Tramway route, which includes 7 stations. Table 8 presents a comparison of the operational driving techniques and EEDDT for each station interval. The driving technique optimized with FPA successfully reduces traction energy consumption, while the driving technique optimized with GWO is more successful in generating regenerative braking energy. In terms of net energy consumption, FPA is more effective.

The total traction energy consumption during the driving mode using operational driving techniques is 17.081 kWh. However, the use of the EEDDT driving technique optimized with FPA has reduced the traction energy consumption to 8.72 kWh. Similarly, the use of the EEDDT driving technique optimized with GWO has reduced the traction energy consumption to 8.748 kWh. These findings suggest that the EEDDT driving technique optimized with FPA is more effective in reducing total traction energy consumption. When comparing the total regenerative braking energy production, it was found that the operational driving technique produced 0.476 kWh of regenerative braking energy. However, with the EEDDT driving technique optimized by FPA, the regenerative braking energy production increased to 1.109 kWh. Furthermore, the EEDDT driving technique optimized by GWO resulted in a regenerative braking energy production of 1.131 kWh. These results suggest that GWO is more effective than MRBEP for the target function. When comparing the net energy consumption, the optimized EEDDT driving technique using FPA resulted in a consumption of 7.611 kWh, while the optimized EEDDT driving technique using GWO resulted in a consumption of 7.617 kWh. These findings suggest that the EEDDT driving technique optimized with FPA is more effective than that optimized with GWO. The objective of energy optimization is to consume the least amount of traction energy possible by using operational driving techniques. The EEDDT driving technique optimized with FPA provides more effective results in achieving this goal. Figure 8 displays the dynamic driving technique regimes of the RSV on a line consisting of 7 stations.

Figure 8.

Driving Techniques Performed for 7 Stations with an Average Speed of Vmean (km/h).

Figure 8 shows the driving techniques and speed/position profiles that correspond to the objective functions given in Tables 6 and Table 7. Figure 8 shows that the RS vehicle, whose driving technique is optimized with FPA and GWO algorithms, completes its driving within the specified operational driving time without reaching maximum speeds. This results in lower traction energy consumption and the generation of regenerative braking energy, contributing to energy efficiency.

4 Conclusion

This study introduces an innovative model for optimizing the dynamic driving technique of electric RS vehicles in urban mass transit systems. The model is designed to be energy-efficient and is supported by regenerative braking energy. The optimization process includes variable coasting points, dynamic weight depending on the number of passengers, speed-dependent friction, slope and variable curve entry speed. The objective functions aim to achieve optimal solutions for MTEC, MRBEP, and MTT for each station interval. Subsequently, the optimal solutions are determined using multi-objective functions within the specified single-objective solutions.In order to verify the effectiveness of the proposed regenerative braking energy assisted EEDDT driving technique model, we have optimized the real data of the 4600 m long Samsun tramway line consisting of 7 stations with FPA and GWO algorithms. In the optimization, we have produced solutions with FPA and GWO separately with the single objective functions given in equations 9, 10 and 11 and with the multi-objective functions given in equations 12, 13, 14 and 15 respectively. As a result of the solutions obtained, we obtained the following results.

The total energy consumption due to traction was reduced by 48.95%, while the total energy produced due to regenerative braking was increased by 137.60%. The total net energy recovery efficiency is 54.164%.

The travel time achievement efficiency of the operational driving technique is 99.62%. The total route completion efficiency of the proposed model is 99.99%. The developed model gives successful results in creating energy efficient driving techniques. The developed model generates energy-efficient driving techniques that can determine travel times between stations with optimal error and minimize delays. The conditions generated through this model improve the success of trip planning by facilitating the generation of energy-efficient travel routes, predetermining travel time and minimizing delays between trips.

Beyond the demonstrated efficacy of the developed optimization models, it is crucial to explicitly articulate the novel contributions and distinguishing aspects of this study within the broader landscape of optimization and energy efficiency in transportation systems. While the optimization of speed profiles for energy-efficient rail transportation is indeed a well-established research domain, the present study introduces several original advancements that collectively advance the state-of-the-art, both methodologically and contextually.

The core novelty of this work resides in the development of a comprehensive EEDDT model specifically tailored for urban rail vehicles, critically incorporating the MRBEP. Unlike conventional approaches that often consider static track conditions, simplified speed models, or primarily focus on MTEC or MTT in isolation, our EEDDT model uniquely integrates these objectives as simultaneous goals within a multi-objective optimization framework. This holistic approach explicitly accounts for the interplay between traction energy, regenerative braking, and travel time, which is particularly pertinent for urban rail systems where frequent stops and starts offer substantial opportunities for energy recuperation, thus directly impacting overall system efficiency and sustainability. Furthermore, the EEDDT model distinguishes itself by incorporating real-world dynamic driving conditions—such as horizontal curvature, gradient variations, and load-dependent resistance—and, uniquely, by integrating dynamic coasting strategies both before and after curves, an aspect rarely modeled comprehensively and together in existing literature. This detailed inclusion of operational dynamics, encompassing the various driving positions (maximum acceleration, cruise, coasting, and braking), ensures a more realistic and comprehensive representation of urban rail vehicle operation.

A significant methodological contribution is the rigorous application and comparative analysis of FPA and GWO algorithms to this specific high-dimensional, non-convex, multi-objective EEDDT problem under practical operational constraints. While these nature-inspired metaheuristic algorithms have shown promise across various optimization challenges, their systematic and in-depth performance evaluation in the context of energy-efficient dynamic driving for urban rail, particularly with the explicit consideration of maximizing regenerative braking, is novel. The study meticulously compares their effectiveness, highlighting their strengths in navigating the complex search space inherent in optimizing speed profiles. The detailed characterization of how these bio-inspired algorithms converge on Pareto-optimal solutions for the tripartite objective function (MTEC, MRBEP, MTT) provides valuable insights into their practical applicability for real-world railway energy management, allowing for diverse driving strategies tailored to different energy-performance trade-offs.

Crucially, the practical applicability and robust performance of the EEDDT model are empirically validated on a real-world tram line in Samsun, Türkiye, featuring seven stations and a varying track profile. This validation incorporates realistic operational constraints, including speed limit rules, regenerative braking thresholds, and dynamic passenger load changes. Compared to standard operational driving, the model achieves substantial improvements: up to a 48.95% reduction in traction energy consumption and a 137.6% increase in regenerative braking energy, all while preserving a travel time match of over 99.62%. These quantified improvements provide compelling evidence of not only the efficacy of the optimization strategy but also its direct implementability and profound benefits in practical urban rail settings.

In summary, this study offers a meaningful contribution to the field of sustainable urban rail transport by distinguishing itself through the development of a holistic, regenerative braking-aware EEDDT model that integrates a richer set of dynamic operational variables, the adept application and comparative insights of advanced nature-inspired algorithms (FPA and GWO) for complex multi-objective optimization, and a robust demonstration of real-world applicability with substantial, quantifiable efficiency gains, as evidenced by its successful validation on the Samsun, Türkiye tramway line.

Future extensions of this work could incorporate mixed-traffic conditions to evaluate their impact on energy efficiency and scheduling.

Footnotes

ORCID iDs

Ramazan Güngüneş

Volkan Ateş

Ertuğrul Çam

Ethical Consideration

This work does not require ethics approval.

Consent to Participate

This work does not require consent to participate, because it does not involve human subjects.

Author Contributions

All authors contributed equally.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Authors confirm that the data supporting the findings of this study are available within the article.

Additional Information

This manuscript is derived from the doctoral dissertation of Ramazan Gungunes, completed at Kırıkkale University in 2023.

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Energy Efficient Dynamic Driving Technique Modeling with Flower Pollination and Grey Wolf Optimization Algorithms in Urban Rail Transportation

Abstract

Keywords

1 Introduction

2.1 Material

Table 4. Track Data for a Section of Samsun Tramway Line.

Footnotes

ORCID iDs

Ethical Consideration

Consent to Participate

Author Contributions

Funding

Declaration of Conflicting Interests

Data Availability Statement

Additional Information

References

Table 4.
Track Data for a Section of Samsun Tramway Line.