Multi-objective optimization of electro-pneumatic braking process with fuzzy logic control for heavy haul railway applications

ABSTRACT

This paper presents a novel approach to multi-objective optimization in heavy haul trains’ electro-pneumatic braking systems using optimizable fuzzy control. Initial findings indicate that a straightforward replacement of pneumatically controlled valves with electro-pneumatic ones, without significant control schedule modifications, fails to substantially improve the dynamics behaviour of rolling-stock. Consequently, a more sophisticated strategy based on optimizable fuzzy logic control for the electro-pneumatic valves was developed. This approach utilizes multi-objective optimization to simultaneously minimize energy dissipation during braking and reduce force peaks in freight car draft gears. The optimal solution achieved demonstrates a 13.41% reduction in kinetic energy dissipation during downhill sections, along with a concurrent decrease in draft gear peak forces by 14.22% in traction and 3.16% in buff regimes compared to standard rolling-stock with pneumatically controlled valves. This optimized fuzzy braking controller outperforms standard procedures, highlighting its potential to enhance heavy haul train efficiency while adhering to safety standards.

KEYWORDS:

• Air brake
• freight railway
• train dynamics
• fuzzy logic control
• multi-objective optimization

1. Introduction

Heavy haul railways play a crucial role in developing the economy by offering a reliable and efficient mode of transporting large quantities of heavy and bulk commodities. These railways are the arteries of trade, especially for resource-rich countries, enabling the export of minerals, metals, and other raw materials to global markets. By moving goods in large volumes, heavy haul trains reduce the cost per ton, providing a competitive edge to industries and fostering economic growth. Furthermore, they alleviate pressure on road networks, reducing congestion and wear-and-tear. The infrastructural investment in heavy haul rail networks also generates employment, stimulates development in remote areas, and promotes the advancement of related technologies, which in turn catalyzes industrial innovation and productivity.

As a sustainable alternative to road transport, heavy haul railways also contribute to environmental conservation by lowering emissions and energy consumption, aligning economic progress with ecological responsibility.

Proper train braking operations are vital for the safety and efficiency of heavy haul train operations. Heavy haul trains, which carry massive loads over long distances, rely on scientific braking to maintain control, particularly when navigating varying terrains and gradients. Correct braking ensures steady deceleration and stopping, preventing derailments, and can also contribute to minimizing wear on the rail infrastructure. It also conserves energy, as smooth braking leads to more efficient fuel use and extends the life of braking systems. In the realm of heavy haulage, where cargo often includes hazardous materials, the importance of reliable braking cannot be overstated, as it safeguards both human life and the environment from potential accidents. Therefore, the correct application of braking systems is a critical component in the operational integrity of heavy haul railways.

The pneumatic-controlled braking system continues to be widely utilized in heavy haul rail transportation across various countries [1,2]. However, a notable challenge associated with pneumatic control is the delay of several seconds in activating brakes for the rear waggons of lengthy rolling-stocks. This delay results in longitudinal impacts at the couplers, where rear freight cars exert force on the frontal ones [3]. An effective alternative to address this braking delay involves replacing pneumatic-controlled valves with electro-pneumatic valves. Unlike their pneumatic counterparts, electro-pneumatic valves are electronically controlled, enabling simultaneous activation throughout the entire consist. Furthermore, these valves permit more frequent releases or increases in brake cylinder pressure, as they are not reliant on the brake pipe for control and, therefore, the brake pipe can continuously supply air pressure to the freight cars’ reservoirs. The adoption of electro-pneumatic systems also facilitates the implementation of more sophisticated control strategies, enhancing the overall braking process. This can result in minimized dissipated energy and improved safety for the consist, particularly during downhill sections of the railroad.

This paper focuses on the optimization of the train braking process in the context of heavy haul train driving control. There are still a range of research gaps in this research topic, as reviewed by Lin et al. [4]. Both Lin et al. [4] and Howlett [5] studied the energy issue for freight trains at downhill sections. Bai et al. [6] used the Quadratic Programming method to optimize train trajectory for energy efficiency and driving smoothness. Wang et al. [7] used the machine learning method to reduce in-train forces of heavy haul trains in downhill sections. Zhang et al. [8] used Model Predictive Control to achieve optimal heavy haul train operation in terms of both in-train forces and energy efficiency. Yi et al. [9] used a Bald Eagle search algorithm to optimize the freight train operation process.

It is noted that none of the published studies have included detailed fluid dynamic air brake models in the train braking related optimization process. As reviewed by Wu et al. [10], train air brake models can be classified into three categories: empirical [3,11,12] fluid dynamics [13–16] and fluid-empirical dynamics models [17–20]. Fluid dynamics and fluid-empirical dynamics models have the advantages of being high fidelity and adaptive to simulation scenarios. These advantages of greater values for brake related studies as this paper. This paper uses a detailed fluid dynamic air brake model [21] and a frictional draft gear model [22]. Different types of draft gear models [23–25] have significant influences on in-train force simulations. The detailed frictional draft gear model is then an important element for optimizations that aim to reduce in-train forces. This also represents another innovation of this paper.

As previously mentioned, electro-pneumatics enable the implementation of sophisticated control strategies that can be optimized. Among the existing optimizable control strategies, fuzzy logic stands out as a viable solution, demonstrating significant improvements after optimization [26,27]. Additionally, it proves to be robust when subjected to conditions different from those in which the controller was initially optimized [28]. Consequently, this paper employs a Genetic Algorithm to optimize a fuzzy-based train driving controller. Multiple objectives are considered to reduce in-train forces and energy consumption, thereby enhancing overall train efficiency.

Following the comprehensive literature review, it is evident that no analogous studies have employed optimization procedures in the context of fuzzy logic control for the electro-pneumatic-based braking process for heavy haul trains, particularly when considering detailed fluid dynamic air brake models in conjunction with the frictional draft gear model for the assessment of in-train forces. As a result of this identified research gap, the primary objective of this paper is to address and bridge this gap in the existing literature.

Section 2 will introduce the modelling method for train longitudinal dynamics. This includes the detailed fluid dynamic air brake model and frictional draft gear model. Section 3 presented the method of the brake process optimization. Section 4 presents the optimization results and discussions. Section 5 concludes this paper.

2. Train longitudinal dynamics

The simulation model is based on the train longitudinal dynamics present in [29], in which the sum of the forces applied in each freight car is used to define the resulting instantaneous acceleration ${\ddot{x}}_{\mathit{n}}$ [m/s${ }^2$] according to the car mass $m_n$ [kg]. The analysed rolling-stock is composed of two Dash9 locomotives located ahead of 168 ore freight cars as shown the Figure 1.

Figure 1. Kinetic diagram.

All rolling-stock cars are submitted to their respective movement resistance force defined in Equation 1(1) $F_{\mathit{r}\left(\mathit{n}\right)}=m_n\mathit{g}\left(A_{1(\mathit{n})}+\frac{{\mathit{A}}_{2(\mathit{n})}{N}_{\mathit{a}(\mathit{n})}}{m_n}+B_n{\dot{x}}_{\mathit{n}}+C_n{\dot{x}}_{\mathit{n}}^2+\mathit{sin}{\alpha }_n+\frac{6.116}{g{R}_n}\right)$(1) , that englobe the wheel – rail rolling resistance and the aerodynamic drag, defined according to the empirical coefficients presented in the Table 1. Moreover, the railroad geometry influence is also considered according to the railroad slope angle ${\alpha }_n$ [rad] and the curves’ radii $R_n$ [rad].

(1) $F_{\mathit{r}\left(\mathit{n}\right)}=m_n\mathit{g}\left(A_{1(\mathit{n})}+\frac{{\mathit{A}}_{2(\mathit{n})}{N}_{\mathit{a}(\mathit{n})}}{m_n}+B_n{\dot{x}}_{\mathit{n}}+C_n{\dot{x}}_{\mathit{n}}^2+\mathit{sin}{\alpha }_n+\frac{6.116}{g{R}_n}\right)$(1)

Table 1. Locomotives and freight cars parameter.

Longitudinal train dynamics parameters
	Locomotives		Freight cars
Parameters	${\dot{x}}_{\mathit{n}}\le 0.1$	${\dot{x}}_{\mathit{n}}>0.1$	${\dot{x}}_{\mathit{n}}\le 0.1$	${\dot{x}}_{\mathit{n}}>0.1$
$A_1$	5	0.1	15	0.15
$A_2$	425	8.5	425	4.25
$\mathit{B}$	0	0.00938	0	0.007
$\mathit{C}$	0	0.0046	0	0.000325
Mass $m_n$	$162\times {10}^3$ kg		$106.5\times {10}^3$ kg
Number of axis $N_a$	5		4

The first and second locomotive accelerations are defined by Equations 2(2) ${\ddot{x}}_1=\frac{{\mathit{F}}_{\mathit{t}1}-F_{\mathit{d}1}-F_{\mathit{r}1}-F_{\mathit{dg}1}}{m_1}$(2) and 3(3) ${\ddot{x}}_2=\frac{{\mathit{F}}_{\mathit{t}2}-F_{\mathit{d}2}-F_{\mathit{r}2}+F_{\mathit{dg}1}-F_{\mathit{dg}2}}{m_2}$(3) respectively, as a function of the train control schedule, that defines the applied traction $F_{\mathit{t}(\mathit{n})}$ [N] among the 8 available notches ($P_1$ to $P_8$) curves for the Dash9 locomotives according to Figure 2(a), or dynamic braking $F_{\mathit{d}(\mathit{n})}$ [N] forces, which convert a portion of the kinetic energy into heat to be dissipated according to the 4 available stages ($D_1$ to $D_4$), as shown in Figure 2(b).

(2) ${\ddot{x}}_1=\frac{{\mathit{F}}_{\mathit{t}1}-F_{\mathit{d}1}-F_{\mathit{r}1}-F_{\mathit{dg}1}}{m_1}$(2)

(3) ${\ddot{x}}_2=\frac{{\mathit{F}}_{\mathit{t}2}-F_{\mathit{d}2}-F_{\mathit{r}2}+F_{\mathit{dg}1}-F_{\mathit{dg}2}}{m_2}$(3)

Figure 2. Locomotive traction and dynamic brake curves.

Once the locomotive traction or dynamic braking forces are defined, the resulting acceleration is defined as a function of its respective draft gear forces $F_{\mathit{dg}(\mathit{n})}$ [N] that represent the interaction between two connected cars and will be properly discussed in Section 2.1.

The intermediary freight cars accelerations ${\ddot{x}}_{\mathit{n}}$ are defined by the Equation 4(4) ${\ddot{x}}_{\mathit{n}}=\frac{-{\mathit{F}}_{\mathit{r}(\mathit{n})}-F_{\mathit{bn}}+F_{\mathit{dg}(\mathit{n}-1)}-F_{\mathit{dg}(\mathit{n})}}{m_n}$(4) , as the sum of the frontal ($F_{\mathit{dg}(\mathit{n}-1)}$) and rear ($F_{\mathit{dg}(\mathit{n})}$) draft gear forces, the respective rolling resistance ($F_{\mathit{r}(\mathit{n})}$) and the braking force $F_{\mathit{b}(\mathit{n})}$ [N] applied by the wagons pneumatic braking system, that will be presented in section 2.2. Finally, the last freight car acceleration ${\ddot{x}}_{\mathit{f}}$ is calculated by Equation 5.(5) ${\ddot{x}}_{\mathit{f}}=\frac{-{\mathit{F}}_{\mathit{r}(\mathit{f})}-F_{\mathit{b}\left(\mathit{f}\right)}+F_{\mathit{dg}(\mathit{f}-1)}}{m_f}$(5)

(4) ${\ddot{x}}_{\mathit{n}}=\frac{-{\mathit{F}}_{\mathit{r}(\mathit{n})}-F_{\mathit{bn}}+F_{\mathit{dg}(\mathit{n}-1)}-F_{\mathit{dg}(\mathit{n})}}{m_n}$(4)

(5) ${\ddot{x}}_{\mathit{f}}=\frac{-{\mathit{F}}_{\mathit{r}(\mathit{f})}-F_{\mathit{b}\left(\mathit{f}\right)}+F_{\mathit{dg}(\mathit{f}-1)}}{m_f}$(5)

Once all instantaneous accelerations ${\ddot{x}}_{\mathit{n}}$ [m/s${ }^2$] are defined they are numerically integrated by the Matlab/Simulink ODE1 (Euler) integrator, to define each car speed ${\dot{x}}_{\mathit{n}}$ [m/s] and displacement $x_n$[m]. The use of ODE1 to solve the train dynamics is allowed due to the small-time steps (${\mathrm{\Delta }}_t\le 0.0005$ s) required by the brake pipe airflow simulation. Furthermore, as shown in [30], the ODE1 showed similar results when compared to high order integrators such as ODE4 (Runge – Kutta) and ODE8 (Dormand – Prince).

2.1. Draft gear model

The draft gear forces $F_{\mathit{dg}(\mathit{n})}$ are defined by the model presented by Wu et al. [31], combined with the dissipated energy approximation method [2,30] applied in the transition between the loading and unloading stages, to enhance the computational efficiency. The draft gear model has considered all key components and their geometries (Figure 3). The force-displacement characteristics are expressed as:

Figure 3. Friction draft gear model: (a) loading stage 1, (b) loading stage 2, (c) unloading stage 1 and (d) unloading stage 2; 1- follower, 2- central wedge, 3- wedge shoe, 4- release spring, 5- outer stationary plate, 6- movable plate 7- lubricating metal, 8- inner stationary plate, 9- spring seat, 10- main springs, and 11- housing.

(6) $F_{\mathit{j},\mathit{i}}={\mathrm{\Psi }}_{\mathrm{i}}{F}_{\mathit{sm}}-\left({\mathrm{\Psi }}_{\mathrm{i}}-1\right)F_{\mathit{sr}},\left(\mathit{i}=1,4\right)$(6)

(7) ${\mathrm{\Psi }}_1=\frac{1+tan\left(\beta +a\mathrm{rctan}\left({\mu }_3\right)\right)\mathit{tan}\left(\mathit{\gamma }+\mathit{arctan}\left({\mu }_1\right)\right)}{1-\mathit{tan}\left(\mathit{\alpha }+\mathit{arctan}\left({\mu }_2\right)\right)\mathit{tan}\left(\mathit{\gamma }+\mathit{arctan}\left({\mu }_1\right)\right)}$(7)

(8) ${\mathrm{\Psi }}_2={\mathrm{\Psi }}_1+\frac{2\left(1-{\mu }_1\mathit{tan}\left(\mathit{\gamma }\right)\right){\mu }_4\left({\mathrm{\Psi }}_1-1\right)}{{\mu }_1+\mathit{tan}(\mathit{\gamma })}$(8)

(9) ${\mathrm{\Psi }}_3=\frac{1+tan\left(\beta -a\mathrm{rctan}\left({\mu }_3\right)\right)\mathit{tan}\left(\mathit{\gamma }-\mathit{arctan}\left({\mu }_1\right)\right)}{1-\mathit{tan}\left(\mathit{\alpha }-\mathit{arctan}\left({\mu }_2\right)\right)\mathit{tan}\left(\mathit{\gamma }-\mathit{arctan}\left({\mu }_1\right)\right)}$(9)

(10) ${\mathrm{\Psi }}_4=\frac{\left(tan(\gamma )-{\mu }_1\right){\mathrm{\Psi }}_3}{\mathit{tan}(\mathit{\gamma })\left(1-2{\mu }_1{\mu }_4+2{\mu }_1{\mu }_4{\mathrm{\Psi }}_3\right)+2{\mu }_4{\mathrm{\Psi }}_3-2{\mu }_4-{\mu }_1}$(10)

where $F_{\mathit{j},\mathit{i}}$ is the draft gear force; $\mathit{i}\left(\mathit{i}=1,4\right)$ indicates different stage of the draft gear working process; ${\mathrm{\Psi }}_{\mathrm{i}}$ is the force amplifying factor; $F_{\mathit{sm}}$ is the main spring force; $F_{\mathit{sr}}$ is the release spring force; $\mathit{\alpha }$, $\mathit{\beta }$, and $\mathit{\gamma }$ are wedge angles as shown in Figure 3; and ${\mu }_1-{\mu }_4$ are coefficients of friction that are also shown in Figure 3. It can be seen that the model considered four stages of the working process: two loading stages and two unloading stages. The first loading stage describes the period before the follower touches the movable plates. The second loading stage describes the period corresponding to the downward movements of the movable plates. The first unloading stage represents the period before the spring seat touches the movable plates. The second unloading stage represents the period when movable plates are moving upwards. The simulation utilized coefficients derived from the MARK50 draft gear, following the methodology outlined in a prior study [2].

To handle the transitions between any loading and unloading stage, a Dissipated Energy Model [2,30] was used:

(11) $F_c=\left\{\begin{array}{cc}0& \mathrm{within}\mathrm{couplers}\mathrm{lack}\\ \frac{F_c\left(\mathit{t}-\mathit{dt}\right)+F_j(\mathit{t})}{2}& \mathrm{transition}\\ F_j& \mathrm{otherwise}\end{array}\right.$(11)

where $F_c$ is the current step of coupler force; $\mathit{t}$ is time; $\mathit{dt}$ is the simulation time step-size; and $F_j$ is the previously calculated draft gear force. More details about the draft gear model can be found in Wu et al. [31] and Eckert et al. [2,30].

2.2. Pneumatic brake system

The pneumatic brake system can be separated into two categories, automatic pneumatic brake system and electro-pneumatic brake systems. The automatic pneumatic brake system is activated by the pressure variation in the brake pipeline, which connects all brake components through the composition. Whenever the pressure drops, a control valve (ABDX or DB-60, names of the commonly used control valves) identifies this pressure change and applies brake force, connecting a reservoir located in the waggon to its brake cylinders. To release the brake force, an increase in the pipeline pressure activates the control valve, and it connects the brake cylinder to the environment to completely depressurize it and connects the pipeline to the waggon reservoirs to pressurize them. This type of brake has a delay in the application signal due to the pressure drop propagation speed through the pipeline, and it does not support several sequences of brake applications and release, since it needs time to pressurize the reservoirs in each waggon to the maximum work pressure. Models to represent this type of brake system are presented by Pugi et al. [32] and Teodoro et al. [33].

The electro-pneumatic brake system uses electrical signals to apply or release brakes in railway composition. The system evaluated in this work uses an EP60, which is composed of a segregated module of electro-pneumatic valves connected to a commonly used pneumatic brake control valve, such as ABDX or DB-60. This module is responsible for connecting the emergency reservoir to its responsible brake cylinder and the brake cylinder to the environment. Figure 4 presents the electro-pneumatic brake model implemented based on Teodoro et al. [33] brake schematic and model.

Figure 4. Brake model schematic.

The driver controls the brake system, setting up the desirable pressure in the locomotive control. This procedure sends electrical signals from the locomotive to all waggons through a wired connection between the electro valves without any delay. The signal is read by the EP60 control module, which changes the connections in the pneumatic components of each waggon to apply or release the brake force. When a signal to apply brake force is sent, the EP60 module connects the emergency reservoir to the brake cylinder, pressurizing it until the cylinder reaches the desirable pressure. As for the release brake force signal, the EP60 module connects the brake cylinder to the environment, depressurizing it until its pressure drops to the value established by the signal. In parallel, the brake pipeline continuously feeds the auxiliary and emergency reservoir, maintaining its pressure close to the maximum work pressure. Also, it is possible to execute partial releases and execute brake applications and releases in sequence, without losing the ability to brake efficiently.

The electro-pneumatic brake system is modelled according to the model proposed by Pugi et al. [32]. The model is represented by orifices that represent the pressure losses between two connected pneumatic components, and by chambers that store the compressed air of each component and represent the pressure in each one. Besides, the brake pipeline is modelled using several chambers and orifices connected. This model shows satisfactory results, with lower computational cost when compared with other methods as evaluated [33].

3. Electro-pneumatic valves fuzzy logic control optimization

3.1. Train braking process

In this paper, a real rail downhill section of the EFVM (Vitória-Minas railroad) in Brazil is evaluated. The standard braking procedure for a heavy-haul train with two locomotives followed by 168 ore freight cars is shown in Figure 5. At the beginning of the section, the locomotives are in the traction stage using notch 8 ($N_8$); when the consist reaches the 1 km mark, the traction is turned off and the consist runs in the neutral regime until it reaches the 2 km mark. The dynamic brake of the locomotives is used at maximum capacity ($D_4$), simultaneously with the freight cars braking by the pneumatic system between the 3 km and 8 km marks.

Figure 5. Train traction and braking schedule according to the railroad position.

The evaluated braking procedure is based on the conventional pneumatic controlled valves, in which the braking cylinder pressure varies according to the pressure drop in the main pipe (brake handle input) which controls the ABDX valve. At the beginning of the rail section, the main pipe is fully pressurized (90 psi), which corresponds to a completely released freight car’s brake system. When the locomotives reach the 2 km mark, the main pipe pressure is dropped to 84 psi, which triggers the ABDX valve to pressurize the brake cylinder, generating braking force. When the locomotives reach the 4 km point, the main pipe pressure is decreased once more to 78 psi resulting in another increase in the brake cylinder pressure, therefore, amplifying the applied braking force. Finally, at the end of the downhill section (8 km mark), the main pipe pressure is restored to 90 psi, releasing the brakes.

To provide a standard result to compare the differences between the conventional pneumatic controlled valves and electro-pneumatic ones, the standard braking procedure presented in Figure 5 is simulated. After that, the braking procedure is adapted to the electro-pneumatic valves, that control directly the braking cylinder pressure, without any interference of the main pipe pressure.

The initial results show that using the electro-pneumatic system to replicate a braking schedule developed for the conventional valves, does not present expressive gains. Despite the elimination of the braking delay characteristic of the pneumatically controlled valves, the in-train forces and impacts among the freight cars during the braking do not improve. However, the automatic pneumatic brake system is constantly pressurized by the locomotives, which allows more frequent changes in the braking cylinder pressure, without the risk of depressurizing the braking system. Therefore, this paper proposes the implementation of an automatic braking control based on fuzzy logic to improve the electro-pneumatic braking schedule.

3.2. The method of fuzzy logic control optimization

Fuzzy logic methodology has found extensive application across various engineering domains in academic and technical literature [26]. These controllers are well known for their robustness and anti-disturbance properties. Moreover, fuzzy logic control can be numerically tuned employing optimization algorithms, such as the Interactive Adaptive-Weight Genetic Algorithm (i-AWGA) [34,35] or multi-objective particle swarm optimization [36], which avoids possible biases in control formulation, as seen in traditional fuzzy logic controllers developed based on specialist knowledge [34].

The developed controller for the electro-pneumatic valves system is based on the Matlab ^TM fuzzy logic toolbox, employing the Mamdani inference method as successfully implemented in previous works [26,34,35]. Additionally, the centroid defuzzification method was chosen to prevent sudden changes [37] in the brake control output, which could potentially result in undesirable high impacts on the draft gears within the system.

In this work, the control is configured as a multiple input, multiple output (MIMO) system. Figure 6 shows the implemented control structure, with the inputs being the consist speed and acceleration, and the outputs being the brake cylinder pressure and the locomotive dynamic brake mode (Figure 2(b)). To ensure optimal control precision with feasible computational cost in the optimization process, after initial evaluations, each membership function was defined with five adjustable levels, which can be adjusted following the procedure outlined below. Moreover, the selection of triangular linear levels for the membership functions was also based on its low computational cost [38].

Figure 6. Fuzzy logic optimizable membership functions.

The optimization algorithm allows for numerical tuning of the membership function levels. Each membership function comprises 5 levels, established by 13 points within the valid range of each function membership, as shown in Figure 6, the speed input membership function triangular levels are defined by the $P_{\mathit{S}1}$ to $P_{\mathit{S}13}$ points. Once, the initial and final points are predetermined, directly corresponding to the range of each membership function, there are 11 changeable points per membership function ($P_{\mathit{S}2}$ to $P_{\mathit{S}12}$, $P_{\mathit{A}2}$ to $P_{\mathit{A}12}$, $B_{\mathit{P}2}$ to $B_{\mathit{P}12}$ and $B_{\mathit{D}2}$ to $B_{\mathit{D}12}$) that serve as design variables to be optimized by the algorithm. The vectors $\left[\mathbf{M}{\mathbf{F}}_{\mathbf{in}}\right]$ (Equation 12(12) ${\left[\mathbf{M}{\mathbf{F}}_{\mathbf{in}}\right]}_{1\times 22}=\left[P_{\mathit{S}2}{P}_{\mathit{S}3}\dots P_{\mathit{S}11}{P}_{\mathit{S}12}{P}_{\mathit{A}2}{P}_{\mathit{A}3}\dots P_{\mathit{A}11}{P}_{\mathit{A}12}\right]$(12) ) and $\left[\mathbf{M}{\mathbf{F}}_{\mathbf{out}}\right]$ (Equation 13(13) ${\left[\mathbf{M}{\mathbf{F}}_{\mathbf{out}}\right]}_{1\times 22}=\left[B_{\mathit{P}2}{B}_{\mathit{P}3}\dots B_{\mathit{P}11}{B}_{\mathit{P}12}{B}_{\mathit{D}2}{B}_{\mathit{D}3}\dots B_{\mathit{D}11}{B}_{\mathit{D}12}\right]$(13) ) are used to store these design variables for the input and output membership functions, respectively.

(12) ${\left[\mathbf{M}{\mathbf{F}}_{\mathbf{in}}\right]}_{1\times 22}=\left[P_{\mathit{S}2}{P}_{\mathit{S}3}\dots P_{\mathit{S}11}{P}_{\mathit{S}12}{P}_{\mathit{A}2}{P}_{\mathit{A}3}\dots P_{\mathit{A}11}{P}_{\mathit{A}12}\right]$(12)

(13) ${\left[\mathbf{M}{\mathbf{F}}_{\mathbf{out}}\right]}_{1\times 22}=\left[B_{\mathit{P}2}{B}_{\mathit{P}3}\dots B_{\mathit{P}11}{B}_{\mathit{P}12}{B}_{\mathit{D}2}{B}_{\mathit{D}3}\dots B_{\mathit{D}11}{B}_{\mathit{D}12}\right]$(13)

To prevent any potential control gaps, the rules are created by combining all levels of the two input functions (see Figure 6), resulting in 25 $\mathit{n}$ rules for optimization. For each combination of inputs, the optimization method selects one level of the brake pressure output ($L_{\mathit{P}1}$ to $L_{\mathit{P}5}$) referred to as $R_{\mathit{P}(\mathit{n})}$ and one level of the dynamic braking output ($L_{\mathit{D}1}$ to $L_{\mathit{D}5}$) designated as $R_{\mathit{D}(\mathit{n})}$. Additionally, each $\mathit{n}$ rule has its corresponding weight, denoted as $W_{\mathit{PD}(\mathit{n})}$. Table 2 displays the combinations of rules and control weights.

Table 2. Fuzzy control rules.

Speed	Acceleration input
input	$L_{\mathit{A}1}$	$L_{\mathit{A}2}$	$L_{\mathit{A}3}$	$L_{\mathit{A}4}$	$L_{\mathit{A}5}$
$L_{\mathit{S}1}$	$R_{\mathit{P}1}$	$R_{\mathit{P}2}$	$R_{\mathit{P}3}$	$R_{\mathit{P}4}$	$R_{\mathit{P}5}$
	$R_{\mathit{D}1}$	$R_{\mathit{D}2}$	$R_{\mathit{D}3}$	$R_{\mathit{D}4}$	$R_{\mathit{D}5}$
	$W_{\mathit{PD}1}$	$W_{\mathit{PD}2}$	$W_{\mathit{PD}3}$	$W_{\mathit{PD}4}$	$W_{\mathit{PD}5}$
$L_{\mathit{S}2}$	$R_{\mathit{P}6}$	$R_{\mathit{P}7}$	$R_{\mathit{P}8}$	$R_{\mathit{P}9}$	$R_{\mathit{P}10}$
	$R_{\mathit{D}6}$	$R_{\mathit{D}7}$	$R_{\mathit{D}8}$	$R_{\mathit{D}9}$	$R_{\mathit{D}10}$
	$W_{\mathit{PD}6}$	$W_{\mathit{PD}7}$	$W_{\mathit{PD}8}$	$W_{\mathit{PD}9}$	$W_{\mathit{PD}10}$
$L_{\mathit{S}3}$	$R_{\mathit{P}11}$	$R_{\mathit{P}12}$	$R_{\mathit{P}13}$	$R_{\mathit{P}14}$	$R_{\mathit{P}15}$
	$R_{\mathit{D}11}$	$R_{\mathit{D}12}$	$R_{\mathit{D}13}$	$R_{\mathit{D}14}$	$R_{\mathit{D}15}$
	$W_{\mathit{PD}11}$	$W_{\mathit{PD}12}$	$W_{\mathit{PD}13}$	$W_{\mathit{PD}14}$	$W_{\mathit{PD}15}$
$L_{\mathit{S}4}$	$R_{\mathit{P}16}$	$R_{\mathit{P}17}$	$R_{\mathit{P}18}$	$R_{\mathit{P}19}$	$R_{\mathit{P}20}$
	$R_{\mathit{D}16}$	$R_{\mathit{D}17}$	$R_{\mathit{D}18}$	$R_{\mathit{D}19}$	$R_{\mathit{D}20}$
	$W_{\mathit{PD}16}$	$W_{\mathit{PD}17}$	$W_{\mathit{PD}18}$	$W_{\mathit{PD}19}$	$W_{\mathit{PD}20}$
$L_{\mathit{S}5}$	$R_{\mathit{P}21}$	$R_{\mathit{P}22}$	$R_{\mathit{P}23}$	$R_{\mathit{P}24}$	$R_{\mathit{P}25}$
	$R_{\mathit{D}21}$	$R_{\mathit{D}22}$	$R_{\mathit{D}23}$	$R_{\mathit{D}24}$	$R_{\mathit{D}25}$
	$W_{\mathit{PD}21}$	$W_{\mathit{PD}22}$	$W_{\mathit{PD}23}$	$W_{\mathit{PD}24}$	$W_{\mathit{PD}25}$

The selected output levels $R_{\mathit{P}(\mathit{n})}$ and $R_{\mathit{D}(\mathit{n})}$ and their respective weights $W_{\mathit{PD}(\mathit{n})}$ are also defined as design variables to be optimized and are stored in the vector $\left[\mathbf{R}{\mathbf{W}}_{\mathbf{PS}}\right]$ as shown in Equation 14.(14) ${\left[\mathbf{R}{\mathbf{W}}_{\mathbf{PS}}\right]}_{1\times 75}=\left[R_{\mathit{P}1}{R}_{\mathit{D}1}{R}_{\mathit{P}2}{R}_{\mathit{D}2}\dots R_{\mathit{P}25}{R}_{\mathit{D}25}{W}_{\mathit{PD}1}{W}_{\mathit{PD}2}\dots W_{\mathit{PD}25}\right]$(14)

(14) ${\left[\mathbf{R}{\mathbf{W}}_{\mathbf{PS}}\right]}_{1\times 75}=\left[R_{\mathit{P}1}{R}_{\mathit{D}1}{R}_{\mathit{P}2}{R}_{\mathit{D}2}\dots R_{\mathit{P}25}{R}_{\mathit{D}25}{W}_{\mathit{PD}1}{W}_{\mathit{PD}2}\dots W_{\mathit{PD}25}\right]$(14)

The design variables subject to optimization are governed by the set $\mathit{C}$ presented in Equation 15.(15) $\mathit{C}=\left\{\begin{array}{cc}{P}_{\mathit{G}1}<P_{\mathit{G}(\mathit{k})}<P_{\mathit{G}13}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}1}<P_{\mathit{G}3}<P_{\mathit{G}2}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}12}<P_{\mathit{G}11}<P_{\mathit{G}13}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}3}<P_{\mathit{G}4}<P_{\mathit{G}5}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}6}<P_{\mathit{G}7}<P_{\mathit{G}8}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}9}<P_{\mathit{G}10}<P_{\mathit{G}11}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}6}<P_{\mathit{G}5}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}9}<P_{\mathit{G}8}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}1}<P_{\mathit{G}4}<P_{\mathit{G}7}<P_{\mathit{G}10}<P_{\mathit{G}13}& \mathit{\hspace{0.17em}}\\ 1\le R_{\mathit{P}(\mathit{n})}\le 5(\mathrm{integer}\mathrm{value})& \mathit{\hspace{0.17em}}\\ 1\le R_{\mathit{D}(\mathit{n})}\le 5(\mathrm{integer}\mathrm{value})& \mathit{\hspace{0.17em}}\\ 0\le W_{\mathit{PD}(\mathit{n})}\le 1& \end{array}\right.$(15) The membership function levels cannot establish gaps within their respective ranges. Therefore, each triangular level must intersect at least the triangle immediately preceding and following its position, excluding the first and last triangles, which are constrained by the membership function range (Figure 6). When considering a generic membership function consisting of 11 level points, denoted as $P_{\mathit{G}(\mathit{k})}$, the same principle can be applied analogously to the input levels, denoted as $P_{\mathit{S}(\mathit{k})}$ and $P_{\mathit{A}(\mathit{k})}$, as well as the output points, denoted as $B_{\mathit{P}(\mathit{k})}$ and $B_{\mathit{D}(\mathit{k})}$.

(15) $\mathit{C}=\left\{\begin{array}{cc}{P}_{\mathit{G}1}<P_{\mathit{G}(\mathit{k})}<P_{\mathit{G}13}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}1}<P_{\mathit{G}3}<P_{\mathit{G}2}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}12}<P_{\mathit{G}11}<P_{\mathit{G}13}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}3}<P_{\mathit{G}4}<P_{\mathit{G}5}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}6}<P_{\mathit{G}7}<P_{\mathit{G}8}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}9}<P_{\mathit{G}10}<P_{\mathit{G}11}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}6}<P_{\mathit{G}5}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}9}<P_{\mathit{G}8}& \mathit{\hspace{0.17em}}\\ P_{\mathit{G}1}<P_{\mathit{G}4}<P_{\mathit{G}7}<P_{\mathit{G}10}<P_{\mathit{G}13}& \mathit{\hspace{0.17em}}\\ 1\le R_{\mathit{P}(\mathit{n})}\le 5(\mathrm{integer}\mathrm{value})& \mathit{\hspace{0.17em}}\\ 1\le R_{\mathit{D}(\mathit{n})}\le 5(\mathrm{integer}\mathrm{value})& \mathit{\hspace{0.17em}}\\ 0\le W_{\mathit{PD}(\mathit{n})}\le 1& \end{array}\right.$(15)

Finally, the resulting 119 design variables are stored in a vector named chromosome $\mathbf{X}$, which will be used in the applied i-AWGA optimization method.

(16) ${\left[\mathbf{X}\right]}_{1\times 119}=\left[\mathbf{M}{\mathbf{F}}_{\mathbf{in}}\mathbf{M}{\mathbf{F}}_{\mathbf{out}}\mathbf{R}{\mathbf{W}}_{\mathbf{PS}}\right]$(16)

3.3. Problem formulation

Once the design variables of the fuzzy controller are defined, it becomes necessary to formulate an optimization problem. This problem will be used to evaluate the obtained results and classify each potential novel control solution according to the applied optimization criteria.

The first optimization criterion is to minimize the energy dissipated by the braking system, denoted as $\mathit{ED}$ [J], as shown in Equation 17.(17) $f_1\left(\mathbf{X}\right)=min(\mathit{ED})$(17) The primary goal of this criterion is to enhance the speed of the rolling-stock on downhill sections, aligning it more closely with the safety-prescribed speed limits of the railroad. By minimizing energy dissipation through the use of brakes, the rolling-stock concludes downhill segments with higher speed. Consequently, locomotives require less traction power to maintain the consist speed, leading to fuel savings.

(17) $f_1\left(\mathbf{X}\right)=min(\mathit{ED})$(17)

The $\mathit{ED}$ energy is comprised of two components: the braking force applied to the freight car’s wheels, denoted as $F_b(\mathit{n})$ [N], and the energy dissipated by the dynamic braking system of the locomotives, represented by the force $F_d(\mathit{n})$ [N]. As reported by Wu, Luo and Cole [39], the energy dissipated by the draft gears is insignificant as compared to the traction and braking energy.

The force applied to each locomotive/waggon in the rolling-stock ($F_b(\mathit{n})$ and/or $F_d(\mathit{n})$) is multiplied by the respective velocity of each car, denoted as $\dot{x}(\mathit{n})$ [m/s]. This multiplication yields the instantaneous power dissipated in each waggon/locomotive.

The instantaneous power of all the cars is then aggregated, resulting in the instantaneous power of the entire consist. This total instantaneous power is then numerically integrated over time $\mathit{t}$ [s] to calculate the total energy dissipated in the simulated section, as illustrated in Equation 18.(18) $\mathit{ED}={\int }_0^t\sum _1^n\left(F_b\left(\mathit{n}\right)+F_d\left(\mathit{n}\right)\right)\dot{x}\left(\mathit{n}\right)\mathit{dt}$(18)

(18) $\mathit{ED}={\int }_0^t\sum _1^n\left(F_b\left(\mathit{n}\right)+F_d\left(\mathit{n}\right)\right)\dot{x}\left(\mathit{n}\right)\mathit{dt}$(18)

The second and third optimization criteria refer, respectively, to the minimization of the peak forces in the draft gears, both in traction and compression regimes. These criteria are defined as a counterpart of the minimization of the braking energy dissipation ($f_1\left(\mathbf{X}\right)$), resulting in higher speeds and, consequently, more demanding operational conditions for the draft gears, which could lead to premature damages of the components. Moreover, it is well-known that force peaks occur in some of the draft gears within the consist, especially in the last wagons of the rolling-stock [2]. Therefore, the formulated optimization problem aims to minimize the highest force peak in both the traction regime (Equation 19(19) $f_2\left(\mathbf{X}\right)=min\left(max\left[F_{\mathit{dg}1}{F}_{\mathit{dg}2}\cdots F_{\mathit{dg}(\mathit{n}-1)}{F}_{\mathit{dg}(\mathit{n})}\right]\right)$(19) ) and the compression regime (Equation 20(20) $f_3\left(\mathbf{X}\right)=min\left(\left|min\left[F_{\mathit{dg}1}{F}_{\mathit{dg}2}\cdots F_{\mathit{dg}(\mathit{n}-1)}{F}_{\mathit{dg}(\mathit{n})}\right]\right|\right)$(20) ).

(19) $f_2\left(\mathbf{X}\right)=min\left(max\left[F_{\mathit{dg}1}{F}_{\mathit{dg}2}\cdots F_{\mathit{dg}(\mathit{n}-1)}{F}_{\mathit{dg}(\mathit{n})}\right]\right)$(19)

(20) $f_3\left(\mathbf{X}\right)=min\left(\left|min\left[F_{\mathit{dg}1}{F}_{\mathit{dg}2}\cdots F_{\mathit{dg}(\mathit{n}-1)}{F}_{\mathit{dg}(\mathit{n})}\right]\right|\right)$(20)

Additionally, some extra constraints have been included to prevent unfeasible solutions. First, the rolling-stock must meet the defined railroad section requirements (see Figure 5) without making any unexpected stops. Furthermore, the travel time of the consist should not exceed 10% of the time taken when the standard braking schedule developed for the ABDX system (refer to Figure 5). On the other hand, the rolling-stock must not exceed a speed of 65 km/h, which corresponds to the speed limit of the analysed rail section. If any of the simulated control configurations fail to comply with these mentioned restrictions, that solution will be eliminated from the optimization process.

3.4. Interactive adaptive-weight genetic algorithm

To identify the optimal solutions for the proposed control system, the Interactive Adaptive-Weight Genetic Algorithm (i-AWGA) technique [40], was employed. This technique was selected due to its efficiency in eliminating the need to define appropriate optimization weights for each optimization criterion. Determining such weights can be particularly challenging, especially in problems featuring multiple conflicting optimization criteria.

Initially, fuzzy controller configurations with randomly generated membership functions are simulated, while adhering to the constructive system constraints, such as the intersections between triangular functions. During this stage, the output parameters and weights of the 25 controller rules are also randomly generated. The results obtained are evaluated, and if they satisfy the constraints, they are included in the population database. This process is iterated until 100 valid solutions are reached, representing the initial population.

To initiate the evolutionary process of the population, it is classified according to a fitness function, $\mathit{Ft}(\mathbf{X})$, as presented in Equation 21.(21) $\mathit{Ft}(\mathbf{X})=\frac{f_{1_{pop}^{max}}-f_1\left(\mathbf{X}\right)}{f_{1_{pop}^{max}}-f_{1_{pop}^{min}}}+\frac{f_{2_{pop}^{max}}-f_2\left(\mathbf{X}\right)}{f_{2_{pop}^{max}}-f_{2_{pop}^{min}}}+\frac{f_{3_{pop}^{max}}-f_3\left(\mathbf{X}\right)}{f_{3_{pop}^{max}}-f_{3_{pop}^{min}}}+P_r\left(\mathbf{X}\right)$(21) This function is used to define a numerical evaluation parameter based on the maximum and minimum values $f_{i_{pop}^{max}}$ and $f_{i_{pop}^{min}}$ present in the population for each optimization criterion. Additionally, a penalty factor $P_r$ is added to favour non-dominated solutions present on the Pareto frontier.

(21) $\mathit{Ft}(\mathbf{X})=\frac{f_{1_{pop}^{max}}-f_1\left(\mathbf{X}\right)}{f_{1_{pop}^{max}}-f_{1_{pop}^{min}}}+\frac{f_{2_{pop}^{max}}-f_2\left(\mathbf{X}\right)}{f_{2_{pop}^{max}}-f_{2_{pop}^{min}}}+\frac{f_{3_{pop}^{max}}-f_3\left(\mathbf{X}\right)}{f_{3_{pop}^{max}}-f_{3_{pop}^{min}}}+P_r\left(\mathbf{X}\right)$(21)

After calculating the values $\mathit{Ft}(\mathbf{X})$ for all members of the population, they are sorted to enhance the selection probability of configurations with the highest $\mathit{Ft}(\mathbf{X})$ values. Equation 22(22) $S_P(\mathbf{X})=\frac{Ft\left(\mathbf{X}\right)}{{\sum }_{k=1}^{PS}\mathit{Ft}\left({\mathbf{X}}_k\right)}$(22) defines the selection probability $S_P(\mathbf{X})$ for each population member, which is determined by their respective $\mathit{Ft}(\mathbf{X})$ value and the total sum of all $\mathit{Ft}(\mathbf{X})$ values within the population.

(22) $S_P(\mathbf{X})=\frac{Ft\left(\mathbf{X}\right)}{{\sum }_{k=1}^{PS}\mathit{Ft}\left({\mathbf{X}}_k\right)}$(22)

After the classification of the population, 2 solutions are selected for the crossover and mutation processes, which combine/modify the existing parameters.

The crossover process randomly combines the parameters of the selected members. In this process, the 4 membership functions and the 25 rules with their respective weights are combined, generating a new control configuration. This configuration is then simulated, and if it meets the imposed constraints, it is added to the population.

To promote diversity among the values of the existing parameters in the population, the mutation process randomly modifies some of the parameters of the solution generated by the crossover process and the two selected members of the population. This generates 3 new configurations, which are simulated and added to the population if they meet the constraints.

The selection, crossover, and mutation processes are repeated until the algorithm converges, which is characterized by the stagnation of the evolutionary process over a high number of generations [41,42]. The population size is controlled by setting a maximum member count $P_{\mathit{limit}}=100$. If the population size exceeds $P_{\mathit{limit}}$, the worst solutions, which present the highest Pareto ranking, are eliminated from the population database. In case of a population composed only of non-dominated solutions (First Pareto ranking) that reaches the $P_{\mathit{limit}}$ value, the limit is increased ($P_{\mathit{limit}}=P_{\mathit{limit}}+50$) in order to avoid the elimination of the whole population [26].

4. Results

After the algorithm’s convergence, the optimized configurations that are not dominated by other solutions in all the evaluated criteria (Pareto frontier) are presented in Figure 7. The optimization procedure results in 61 feasible optimum control configurations. However, only the solutions that achieved the best results in the three optimized criteria are presented, along with the solution with the highest fitness value $\mathit{Ft}(\mathbf{X})$, representing the result with the best trade-off among the three criteria. Additionally, the results obtained using the default configuration (ABDX valve and the standard braking schedule) are highlighted, as well as the result obtained with an electro-pneumatic system while maintaining the default operating mode. Table 3 summarizes the results of the analysed solutions.

Figure 7. Optimized solutions.

Table 3. Standard and optimized results.

	Results
	Dissipated	Draft gear peak force		Expended	Average
	energy	Traction	Buff	time	speed
Configurations	[MJ]	[kN]	[kN]	[s]	[km/h]
Standard ABDX	7835	1090	−1139	614.6	50.15
Equivalent	7956	1090	−1139	625.7	49.65
Electro-pneumatic
Min. $f_1(\mathbf{X})$	6246	1031	−1130	544.4	53.58
Min. $f_2(\mathbf{X})$	7301	840	−1122	615.8	50.10
Min. $f_3(\mathbf{X})$	7705	1060	−1073	627.6	49.57
Max. $\mathit{Ft}(\mathbf{X})$	6784	935	−1103	613.9	50.18

As depicted in Figure 7, all optimized solutions (the Pareto frontier) outperform the results obtained by simulating the conventional ABDX valve. However, solutions that exhibit minimum values for a specific criterion usually do not yield the best performance concerning the other criteria. Finally, the results obtained by the standard ABDX valve and the electro-pneumatic system applying a control equivalent to the default one will serve as benchmarks for comparison with the optimized solutions.

As depicted in Table 3 and Figure 7, for the scenario under evaluation for this specific consist configuration, merely substituting conventional ABDX valves with electro-pneumatic counterparts, without altering the braking schedule control, does not lead to a significative reduction in in-train forces. This happens even though the electro-pneumatic system effectively eliminates the well-known braking delay issue inherent in the conventional system. The in-train force peaks for these solutions are presented in Figure 8.

Figure 8. Draft gear peak forces – standard schedule.

Figure 9 illustrates the differences in rolling-stock speed among the analysed solutions, taking into account the travel time of the analysed rail section and the consist speed based on the distance travelled. As shown in Figure 5, the standard braking schedule applies an initial braking load when the leading locomotive reaches the 2 km point, which decreases the rolling-stock speed acceleration. Afterwards, the rolling-stock speed continues to increase, and the braking force is increased at the 3 km mark to prevent the consist from exceeding the maximum allowed railroad speed of 65 km/h, which is reached when the rolling-stock passes the 4.5 km mark. Due to safety reasons, it is not allowed to release the brakes of the ABDX system until the end of the downhill section to avoid high accelerations and brake failures due to the long time required to repressurize the standard braking system. In the context of the electromechanical system employing a braking schedule adapted from the one presented in Figure 9, the behaviour remains nearly identical, except for a minor reduction in speed attributable to the elimination of braking delays.

Figure 9. Rolling-stock speed and braking cylinder pressure in the downhill section.

In reference to the optimized solutions, the solution minimizing dissipated energy ($min{f}_1$) was successful in reducing braking energy by 20.28% when compared to the ABDX system. Furthermore, draft gear forces decreased by 5.41% during the traction stage and only by 0.79% during the buff stage. A comparison of the peak forces differences between the ABDX systems and the $min{f}_1$ solution is possible by analysing Figures 8(a) and 10(a).

Figure 10. Draft gear peak forces – best of each criterion.

The solution for $min{f}_1$ initiates the downhill rail section by applying an initial braking effort (Figure 9(c)). This effort slightly reduces the rolling-stock speed (Figure 9(a)) in comparison to the standard solution, which does not apply braking in the first km of the section (Figure 9(b)). Afterwards, the braking effort gradually increases to compensate for the load effect of the freight cars entering the downhill section, keeping the consist acceleration controlled. It continues until it approaches the 65 km/h speed limit, at which point the brake pressure reaches its peak ($3.7\times {10}^5$ Pa) to prevent exceeding the speed limit. During this stage, the brake control stabilizes the rolling-stock speed and then gradually decreases the pressure until the system is completely released. This proposed solution reduces the overall mechanical work of braking efforts, maintaining the pressure below that of the other analysed solutions for the majority of the time (Figure 9(c)). Consequently, it effectively reduces the dissipated kinetic energy and, therefore, the higher average rolling-stock speed in the analysed section (Table 3).

The solution for $min{f}_2$ seeks to reduce the peak traction forces in the draft gear. During the initial phases of braking control development, the optimization procedures solely focus on minimizing the impact during the buff stage of draft gears while also reducing the dissipated energy. However, the simultaneous pursuit of these objectives results in an increase in traction forces due to intermittent braking applications, which, in turn, reduces the dissipated energy but generates an accordion effect among freight cars.

To address the mentioned issues, we incorporated the minimization of peak traction forces into the optimization procedures. The $min{f}_2$ solution successfully reduced the peak traction forces by 22.93%, reaching a maximum value of 840 kN. This improvement came with a 1.49% decrease in the buff peak force and a 6.81% reduction in dissipated energy compared to the standard ABDX version. The comparison of peak forces is illustrated in Figures 8(a) and 10(a).

The solution represented by $min{f}_2$ commences the downhill rail section by applying a braking effort higher than that applied by the $min{f}_1$ solution (Figure 9(c)), which reduces the rolling-stock speed to nearly 35 km/h (Figure 9(a)). Subsequently, the brake effort diminishes, allowing the consist to accelerate and increase speed until it reaches the 65 km/h speed limit. During this acceleration phase, the braking pressure gradually increases to compensate for the load effect of the freight cars entering the downhill section. In contrast to the $min{f}_1$ solution, the $min{f}_2$ solution applies a smooth increase in braking pressure to prevent exceeding the 65 km/h speed limit, thereby avoiding collisions between the waggons. Finally, this solution gradually reduces the brake pressure until the complete system is released.

The solution for minimizing $f_3$ aims to reduce the forces exerted on the draft gears during the buff stage. This braking control strategy operates similarly to the minimization of $f_2$ solution (refer to Figure 9), but it involves applying slightly higher pressure to the braking cylinder. This increased pressure effectively decreases the impact forces to −1073 kN, representing a 5.94% improvement compared to the ABDX version. Moreover, this solution also minimizes the traction forces by 2.75% and the dissipated energy by 1.66% as compared to the ABDX valve with the standard braking schedule. The difference in the in-train peak forces can be analysed by comparing Figures 8(a) and 10(a).

Finally, the best trade-off solution, considering the three analysed criteria, is defined by the high fitness value $max\mathit{Ft}$. This solution combines the advantages of the other three mentioned solutions, reducing the dissipated energy by 13.41% while minimizing the draft gear forces by 14.22% and 3.16% in the traction and buff stages, respectively, as compared to the ABDX control. Figure 11 show the peak forces of the $max\mathit{Ft}$ solution.

Figure 11. Draft gear peak forces for the best trade-off solution.

This solution initiates the application of brake load at the beginning of the downhill section, akin to the $min{f}_2$ and $min{f}_3$ solutions (see Figure 9). However, it extends the application of braking until the train reaches a speed of nearly 34 km/h. Subsequently, it maintains a braking pressure that exceeds that of the $min{f}_2$ and $min{f}_3$ solutions, facilitating a slower acceleration of the rolling-stock and affording better speed control with a smoother increase in braking pressure. This, in turn, prevents a sudden pressure peak when this solution approaches the 65 km/h speed limit. Upon reaching a speed of approximately 63 km/h, the train maintains this velocity until the end of the analysed section. Ultimately, the optimal trade-off solution successfully enhances all three criteria, completing the analysed railroad section within the same timeframe as the standard braking schedule while employing the ABDX valve. This approach mitigates component wear and ensures the train schedule remains unaffected by any delays.

5. Conclusion

This paper proposes a novel multi-objective optimization approach for electro-pneumatic braking in heavy haul trains, using the Interactive Adaptive-Weight Genetic Algorithm (i-AWGA). The goal is to minimize energy dissipation and force peaks in freight cars’ draft gears during braking.

The optimization converged, resulting in 61 fuzzy logic controllers, each with unique trade-offs. Specific solutions were chosen for in-depth analysis, focusing on the one with the highest fitness value. This refers to the optimal fuzzy braking controller, that offers the most favourable balance between kinetic energy conservation and draft gear peak force reduction.

As compared to the standard braking schedule procedure, the best trade-off solution was able to minimize the 13.41% of the dissipated kinetic energy during the analysed railroad downhill section, at the same time that the draft gear peak forces were reduced by 14.22% at the traction stage and 3.16% as buff regime, without any delay in the train operation.

Incorporating fuzzy logic in an optimization framework enhances heavy haul train performance with electro-pneumatic braking control valves, ensuring rolling-stock safety. These procedures, with precise simulations, refine train control, reducing costs and preventing premature wear. The outlined optimization methodology extends to broader applications, including train traction control for fuel savings and travel time reduction. Fuzzy logic integration introduces a versatile and effective approach to improving heavy haul train efficiency.

The optimal controllers proposed in this paper necessitate additional analyses before feasible real-world applications. The assessment of the entire rail section through which the consist operates is crucial when considering the developed controllers. The optimization process was restricted to a limited section of the railroad due to the high computational cost associated with the applied simulation/optimization models. Additionally, it is important to account for extra constraints related to safety and unforeseen operational conditions before implementing these controllers in practical scenarios. Restrictions for ore transportation, including the unloading process, can prevent the use of cables to transmit the information. Maintenance issues must also be evaluated since the harsh environment can attack the electrical connections and internal valve systems. Notwithstanding, more studies can provide adequate economic and safety reasons for the adoption of electro-pneumatic controls in heavy haul trains.

Acknowledgements

This work was conducted with scholarships and financial support from VALE S.A, National Council for Scientific and Technological Development) grant 315304/2018-9, Universidade Estadual de Campinas (UNICAMP), Brazil and Central Queensland University, Australia.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico [315304/2018-9].