Optimizing hybrid energy systems for locomotives based on improved grey lag goose algorithm

Abstract

This work presents an Improved Grey Lag Goose Optimization (IGLGO) algorithm for minimizing the total cost of a hybrid locomotive energy system integrating polymer electrolyte membrane (PEM) fuel cells with lithium-ion batteries. The IGLGO employs a dynamic grouping mechanism and fractional calculus to ensure that a near-optimal exploration-exploitation balance is achieved so that local optima can be avoided. IGLGO delivered a more cost-effective design as compared to standard Grey Lag Goose Optimization (GLGO) and other metaheuristics. For a 2% track slope, the optimally sized IGLGO system, at $3.78 million, was found to be cheaper than GLGO ($4.41 million) and the Dwarf Mongoose Optimizer ($4.84 million). These results prove that IGLGO offers a very solid yet economical optimization framework for sustainable railway propulsion systems.

Introduction

The exploration of hybrid energy systems for locomotive applications has been primarily driven by the pursuit of more sustainable and eco-friendly energy sources. Traditional power sources like diesel engines are facing scrutiny due to their significant environmental impact and high fuel costs. Hybrid energy systems, which combine multiple energy sources, are seen as a promising alternative as they offer enhanced efficiency, reduced emissions, and lower operating expenses.

This study focuses on the integration of two advanced technologies in the field of energy storage and generation: polymer electrolyte membrane (PEM) fuel cells and lithium-ion batteries. By combining these technologies, there is a potential to transform locomotive power systems, making them more environmentally sustainable, efficient, and cost-effective¹.

Han et al. proposed the incorporation of PEM fuel cells and lithium-ion batteries in hybrid energy systems for locomotives represents an innovative approach to overcoming the challenges associated with traditional power sources². The goal of this research is to unleash the full potential of these technologies, providing a cleaner, more sustainable, and economically feasible solution for the locomotive industry³.

Yuan, Hai-Bo, et al. introduced a novel methodology aimed at enhancing the efficiency of a hybrid power system that integrated a PEMFC (Polymer Electrolyte Membrane Fuel Cell) with a battery⁴. They proposed an advanced rule-based energy management strategy (EMS) that utilizes a genetic algorithm (GA) to optimize the distribution of power between the fuel cell and the battery. The main goal of this real-time EMS is to maintain battery charge while also taking into account the longevity and efficiency of the fuel cell. The authors created a simulation environment using MATLAB/Simulink and established an experimental framework with LabVIEW to validate their proposed optimized rule-based EMS. Furthermore, they assessed and contrasted the performance of traditional rule-based EMS, fuzzy logic EMS, and dynamic programming (DP) EMS. The findings unequivocally illustrated the advantages of the optimized rule-based EMS, revealing substantial enhancements in performance compared to both conventional rule-based and fuzzy logic EMS approaches.

Chen, Kui et al. have made notable advancements in the domain of Polymer Electrolyte Membrane Fuel Cells (PEMFCs) by tackling two pivotal challenges that impede their widespread commercial use: degradation and cost. In their groundbreaking research, they introduced an innovative method for predicting degradation in PEMFCs, employing Multi-kernel Relevance Vector Regression (MRVR) in conjunction with the Whale Optimization Algorithm (WOA)⁵. This hybrid methodology utilized empirical data derived from both vehicular operations and controlled laboratory experiments to construct a robust degradation prediction model that is applicable across a diverse array of operating conditions. The authors emphasized the necessity of accurately capturing degradation trends and, therefore, utilized MRVR to formulate the PEMFC degradation prediction model. By incorporating multiple kernel functions, MRVR provided a more versatile and adaptive modeling framework compared to conventional single-kernel approaches. Furthermore, the integration of WOA facilitated the automatic adjustment and optimization of weight and kernel parameters, thereby significantly improving the model’s predictive accuracy.

Han, Feng, and colleagues introduced a novel fault detection technique for PEM (Polymer Electrolyte Membrane) fuel cell systems, referred to as the possibilistic fuzzy C-means clustering artificial bee colony support vector machine (PFCM-ABC-SVM) approach. A significant benefit of this methodology lies in its capacity to manage dynamic conditions and effectively filter data contaminated with Gaussian noise, even when the amplitude of the characteristic parameters is diminished to ± 10%. In their research, the authors simulated faults within a PEM fuel cell system under dynamic conditions accompanied by Gaussian noise to create a foundational dataset.

The authors utilized the PFCM (Possibilistic Fuzzy C-Means) algorithm to remove samples characterized by low typicality and membership, thereby refining the primary dataset. Furthermore, the ABC (Artificial Bee Colony) optimizer was employed for optimizing the factor of penalty C and the parameter of kernel function g, which further improved the efficacy of SVM model. The optimized SVM was subsequently applied to detect faults in the PEMFC system, with findings demonstrated through a nonlinear PEM fuel cell simulator model.

In their comparative analysis, Han, Feng, and their team evaluated the PFCM-ABC-SVM method against other fault diagnosis strategies, revealing an impressive accuracy rate of up to 98.51% for the testing set samples. The proposed approach proved to be effective in diagnosing faults in PEM fuel cell systems, particularly excelling in small-sized, nonlinear, and high-dimensional scenarios. The integration of PFCM for data filtering, ABC for parameter optimization, and SVM for fault diagnosis culminated in a robust and precise fault diagnosis framework.

Han, Feng, and colleagues introduced a novel fault diagnosis technique for polymer electrolyte membrane (PEM) fuel cell systems, referred to as the possibilistic fuzzy C-means clustering artificial bee colony support vector machine (PFCM-ABC-SVM) method⁶. A significant benefit of this methodology lies in its capacity to manage dynamic conditions and effectively filter data contaminated with Gaussian noise, even when the amplitude of the characteristic parameters is diminished to ± 10%. In their research, the authors simulated faults within a PEM fuel cell system under dynamic conditions accompanied by Gaussian noise to create a foundational dataset.

In their comparative analysis, Han, Feng, and their team evaluated the PFCM-ABC-SVM method against other fault diagnosis approaches, revealing an impressive accuracy rate of up to 98.51% for the testing set samples. The proposed method proved effective in diagnosing faults within PEM fuel cell systems and was particularly advantageous for addressing high-dimensional, nonlinear, small-sized challenges⁷. The integration of PFCM for data filtering, ABC for parameter optimization, and SVM for fault diagnosis culminated in a robust and precise fault diagnosis framework.

Moazeni, Faegheh, and Javad Khazaei developed an extensive small-signal network for a single-cell PEMFC employing a state-space methodology for exploring the effects of diverse operational circumstances on the dynamic behavior of the FC⁸. Their research, detailed in this paper, encompassed the modeling of the dynamics associated with the partial pressures of hydrogen, oxygen, and water. Additionally, the authors expanded their investigation to analyze the transient answers of both multiple and single-cell PEMFC configurations, adjusting critical operating parameters such as airflow rate, fuel flow rate, temperature, relative humidity levels at the anode and cathode, and electrical current. In the following phases of their study, Moazeni, Faegheh, and Javad Khazaei integrated the analyzed PEMFC with the main power grid through the use of a DC/AC converter and a boost DC/DC converter. They performed a constancy analysis of the entire system utilizing eigenvalue examination in MATLAB and designed multiple case studies for assessing the sensitivity of the variables of boost converter and the PLL (Phase-Locked Loop) on system constancy. The findings from their investigation were corroborated using a 100-watt replicated PEMFC within the toolbox of MATLAB Simscape, leading the authors to propose a set of optimal operating conditions aimed at enhancing performance.

Sohani, Ali, Shayan Naderi, and Farschad Torabi addressed a notable deficiency in the current body of research concerning polymer electrolyte membrane fuel cells by investigating the optimization of various performance criteria⁹. They posed a critical inquiry: which specific combination of performance metrics yields the most substantial advantages? To answer this question, they introduced and executed a novel methodology, utilizing a PEMFC as a case study. The authors delineated several optimization scenarios, each concentrating on a distinct integration of efficacy, levelized cost, power density, and scope as fitness functions. Through a comparative analysis of the optimized outcomes from these scenarios, they discerned the most effective combination of criteria applicable to both transportation and stationary applications¹⁰. Furthermore, they examined how variations in capacity influenced deciding process and the associated values of measurement metrics. The findings presented by Sohani et al. indicated that optimizing and prioritizing levelized cost and power density yielded equal outcomes¹¹. Notably, they found that a multi-objective optimization encompassing all performance criteria may not represent the optimal solution. Instead, for both transportation and stationary applications, across a broad capacity spectrum ranging from 10 to 100 kW, they recommended that the ideal approach involves a multi-objective optimization centered on either power density or levelized cost and size.

The present study offers some new novelties that go a long way forward in the optimization of hybrid energy systems (HESs) for locomotive applications. First, the methodological novelty is the development of the Improved Grey Lag Goose Optimization (IGLGO) algorithm, which fundamentally modifies the generic GGO algorithm based on three core innovations: (1) a dynamic grouping mechanism that adaptively adjusts the balance between exploration and exploitation by admitting a larger exploration population whenever alternating best solutions appear to stagnate, (2) the integration of a fractional calculus (FC) model based on Grunwald-Letnikov (GL) definition, which embeds a memory effect to counteract an incessant return to nonproductive areas while enhancing the convergence speed, and (3) employing three stochastic search agents to augment the exploration of the solution space and steer clear of local optima. Second, the application novelty consists of the algorithm being specifically customized to the unique constraints of locomotive HES, where it is explicitly guided by a domain-specific metric for managing the state of charge (SoC) of the PEM fuel cell and battery capacity limits during the optimization of system sizing. This tailored approach ensures that the solution is not only mathematically optimal but also practical for rail transport. Finally, the novelty in the results is exhibited with the IGLGO’s aforementioned superior performance, where the system strategically optimals total system costs by sizing a relatively larger fuel cell and a smaller, cheaper battery configuration than the DMO, WOA, and HBA: indeed, exhibiting both best solution quality (lower mean cost) and robustness (lower standard deviation). This grand multidisciplinary innovation, which covers a novel algorithm, a specialized application, and the demonstrated best economic outcome, constituted major traction towards the design of sustainable and cost-effective locomotive energy systems.

The central aim of this study is to create an energy solution that is both economically viable and environmentally sustainable for locomotive applications¹². The focus is on enhancing the overall efficiency and performance of hybrid energy systems through the optimization of the sizing and integration of PEM fuel cells and lithium-ion batteries¹³. A significant contribution of this paper is the introduction of the Improved Grey lag Goose Optimization (IGLGO) algorithm, which offers a more effective approach to the complex optimization problem at hand. Specifically designed to minimize the total cost of the hybrid energy system, the IGLGO algorithm considers the unique constraints associated with the state of charge (SoC) of PEM fuel cells and the capacity of batteries. By tackling these issues, the study contend that the proposed algorithm can substantially advance the development of sustainable and efficient energy systems for locomotive applications.

One of the main novelties is the dynamic grouping mechanism, that adaptively balances between exploration and exploitation. First, the population is split into exploration and exploitation sets evenly. The algorithm also keeps track of the best solution found so far, and if this solution is unchanged for some iterations, it will increase the number of agents in the exploration set, which can also help escape local optima, since it will search a larger part of the solution space¹⁴. In particular, the IGLGO algorithm uses three stochastic search agents ($\:{Y}_{P1}$, $\:{Y}_{P2}$, $\:{Y}_{P3}$) to conduct exploration of new regions in the search space; this reduces the risk of the algorithm being trapped at local optima.

The $\:Z$, $\:A$ vector is modified, and ‘$\:a$’  parameter, reduces linearly from 2 to 0, resulting in a more dynamic and flexible exploration process. This integration allows the IGLGO algorithm to query the optimal solution (i.e., the best-known solution) by defining a mechanism for sentries (namely $\:{Y}_{s1}$, $\:{Y}_{s2}$, $\:{Y}_{s3}$) to lead other agents through the convergence process. The combination of the use of an FC, using a memory window to enhance performance. Memories are introduced with adaptions to the Grunwald-Letnikov (GL) model letting the algorithm take previous solutions and their influence on the new search into account in maintaining the balance between exploration and exploitation and avoiding already traversed \regions. Dynamic grouping, stochastic search agents, and fractional calculus help the algorithm to avoid becoming stuck in local optima and getting prematurely converged, making it both robust and flexible. Moreover, the IGLGO algorithm incorporates a mechanism to periodically tune values of parameters like ‘$\:a$’, ‘$\:Z$’, and ‘$\:A$’ to keep the algorithm relevant and efficient during the course of optimization.

It is informed by a domain-specific metric, The IGLGO algorithm is customized for locomotive hybrid energy systems optimization, which must satisfy specific constraints such as the PEM fuel cell state of charge (SoC) and battery capacity. Such a specialized approach would augment the capability of the algorithm to tackle the unique demands and goals of hybrid energy systems in the locomotive domain. In fact, comparative results against some state-of-the-art popular optimization algorithms like Whale Optimization Algorithm (WOA), Firefly Algorithm (FA), Dwarf Mongoose Optimization Algorithm (DMO), White Shark Optimizer (WSO) and Honey Badger Algorithm (HBA) shows that the IGLGO algorithm outperforms these approaches once at a time both with regard to their mean values and standard deviations, showing the high performance and stability of the proposed algorithm.

To summarize, key aspects of the IGLGO algorithm analysis that enable it to be tailored specifically for optimizing composite hybrid energy systems in the context of locomotive applications are: the introduction of dynamic and adaptive exploration-exploitation balance; high-performance exploration approach; improved local search mechanisms; fractional calculus; context robustness, and external maximization of the performance measure.

Combining Polymer Electrolyte Membrane Fuel Cells (PEMFC) and lithium-ion batteries in hybrid energy systems (HES) for locomotives is gaining attention for positive environmental and economic impact, with potential implications for sustainability. One major environmental effect is a reduction in greenhouse gas emissions. PEMFCs generate electric power through an electrochemical reaction between hydrogen and oxygen, with water and heat being the sole byproducts, thus eliminating emissions of CO2, NOx and other harmful pollutants that the conventional diesel engines cause. In the first place, increased fuel efficiency reduces greenhouse gas emissions which are linked to climate change and better air quality. Using a renewable energy wind, solar or hydroelectric for example for the hydrogen production, the entire energy chain can become sustainable or low-carbon. The vehicle’s energy consumption and the operation cost were efficiently reduced by switching the common diesel engines with PEMFCs and lithium-ion batteries, because the efficiency rate of PEMFCs and lithium-ion batteries can exceed 60%, but the efficiency rate of diesel engines is only 30–40%.

In addition, lithium-ion batteries are capable of storing energy created from regenerative braking a feature that would commonly be dumped and wasted in traditional locomotives considerably improving overall efficiency and reducing energy wastage. Electric and hybrid systems that are quieter and leave less impact on the air that breathes help improving the quality of life for communities located close to rail and cut down the environmental burden of transportation. Hybrid energy systems mitigate reliance on fossil fuels by substituting diesel with hydrogen and battery power, thus reserving non-renewable commodities and reducing the carbon footprint and pollution caused by fossil fuel extraction and transport.

Hybrid energy systems have several economic benefits. Due to the high efficiencies of PEMFCs and the use of regenerative braking to charge batteries, fuel costs reduced considerably. Particularly when generated from renewable sources, hydrogen can be cheaper in the long-run than diesel. Most electric and hybrid systems have fewer moving parts than traditional combustion-based power generation systems, have longer maintenance intervals, reduce maintenance costs and minimize down time, which increases operational reliability and efficiency¹⁵. The design and use of hybrid energy systems can stimulate innovations in the transport and energy industries, leading to job creation in research, development, production and servicing, as well as contributing to economic growth.

Companies and governments that adhere to these sustainable technologies can attract investment from green investors want to see a profit, meanwhile their governments can give them benefits, manage their production and consumption, along with investments in a more stable way and receive incentives and subsidies for green technology. One strategy for locomotive manufacturers and operators to accomplish this is the integration and implementation of hybrid energy systems into their fleets. Moreover, companies that are already using sustainable technologies are in a better position to comply with tightening environmental regulations more easily, so they can avoid penalties, and gain a competitive edge as well.

Related posts the implementation of hybrid energy systems for locomotives is also a step towards achieving wider sustainability objectives. Cutting greenhouse gas emissions aligns with the goals of the Paris Agreement, which aims to restrict warming to much less than 2 degrees Celsius (3.6 degrees Fahrenheit) above preindustrial levels. Hybrid locomotives help to reduce the carbon output of the transportation sector, aiding in global efforts to mitigate climate change. Hybrid systems utilizing clean and renewable energy sources contribute to Sustainable Development Goal (SDG) 7, which focuses on access to affordable, reliable and modern energy for all. Hybrid locomotives help make urban habitats more livable and sustainable for residents by minimizing air and noise pollution, which relates to the UN-sponsored initiative SDG 11.

Inspired by Huang et al., who developed a unified barrier function-based approach for uncertain nonlinear systems, The study applies an improved Grey Lag Goose Algorithm to address uncertainties in hybrid energy systems for locomotives under dynamic constraints¹⁶. Wu et al. addressed event-based adaptive neural resilient formation control under actuator saturation. This research extends this concept by optimizing energy distribution while mitigating operational constraints in locomotive systems¹⁷.

Another positive impact that hybrid energy systems have is on goal 13: Climate Action where also in this case the fight against climate change, as well as its effects, should be urgent. Hybrid energy systems contribute to energy security and reduce the susceptibility of transport systems to fuel price escalation and supply disruption, by decreasing reliance on free fossil fuel and increasing the diversity of energy sources. 4Improvement of Air Quality: Metrodirect hybrid locomotives are designed to reduce air pollutants, including SOx and NOx, resulting in cleaner air in urban areas and proximity to railway lines, contributing to improved air quality and a reduction in respiratory and cardiovascular diseases 5.

In conclusion, locomotive hybrid energy systems present a comprehensive solution for simultaneously achieving environmental and economic sustainability. These systems contribute to larger sustainability goals, including climate change mitigation, sustainable development and public health, through reductions in greenhouse gas emissions, energy consumption and operating costs. These innovations should commend themselves for what they do and honor innovations today and tomorrow as transition to a cleaner economy while helping communities and nations put economic resilience and environmental sustainability and sustainability in motion.

The system representation

The proposed hybrid energy system employs PEMFC as the main power source, complemented by lithium-ion batteries serving as a supplementary energy solution. Drawing from prior research¹⁸, The study have selected a high-performance lithium manganese oxide $\:\left(LiM{n}_{2}{O}_{4}\right)$ battery with a capacity of 100 Wh/kg for simulations¹⁹. The main objective is to replace the traditional CI engine in the $\:HX{D}_{3}G$ locomotive class, which has been operational since 2015 in China. The locomotive’s power requirements fluctuate due to frequent stops at terminal stations, and the PEM fuel cell’s limited low-speed dynamic response necessitates a larger configuration than what is typically required for the locomotive. To mitigate this issue, the study have integrated a battery stack that captures surplus energy produced during regenerative braking, ensuring its availability for future use^20,21,22,23. Figure 1 presents the overall configuration of the proposed HE system, highlighting the synergy between PEM fuel cells and lithium-ion batteries to create a more efficient and sustainable energy solution for locomotives.

Fig. 1

The overall configuration of the proposed HE system.

The PEMFC stack mathematical model

Polymer Electrolyte Membrane (PEM) fuel cells are distinguished by their high efficiency and environmental advantages, representing an innovative method for generating electricity. As a specific category of fuel cell (FC), PEM fuel cells facilitate the direct transformation of chemical energy into electrical energy, thereby offering a more sustainable and efficient alternative to conventional power generation techniques. The benefits associated with fuel cells, such as elevated efficiency and the generation of clean energy, have positioned them as a leading technology within the energy sector. Like other fuel cell varieties, PEM fuel cells are composed of three essential components: a positive electrode, a negative electrode, and an electrolyte membrane. In the procedure of energy generation, the negative electrode facilitates the ionization of hydrogen (H2), resulting in the formation of protons and electrons. The electrons are routed through an external circuit, generating an electric current, while the protons traverse the membrane, where they react with oxygen (O2) from the positive electrode and the electrons from the external circuit to yield heat and water. A schematic representation of the operational mechanisms of a PEM fuel cell is illustrated in Fig. 2, highlighting the complex process of converting chemical energy into usable electrical energy.

Fig. 2

The PEMFC schematic.

Taking into account the aforementioned explanations and referring to source²⁴, the entire voltage of output for a Proton-exchange membrane (PEM) fuel cell can be articulated in the following way:

$$\:{V}_{out}={N}_{c}\times\:\left({E}_{Nerst}-{E}_{{\Omega\:}}-{E}_{act}-{E}_{ops}\right)$$

(1)

The overall output voltage of a PEMFC has been influenced via multiple variables. A significant factor is the number of cells arranged in series or parallel, referred to as $\:{N}_{c}$, which is essential for establishing the total voltage. Other important contributors include the ohmic voltage drop, indicated as $\:{E}_{{\Omega\:}}$, activation overpotential $\:{E}_{act}$, overpotential saturation $\:{E}_{ops}$, and the ideal voltage per cell $\:{E}_{Nerst}$. The optimum voltage for all cells in the PEM fuel cell stack is articulated in the equations found in references^25,26.

$$\:{E}_{N}=1.23-8.5\times\:{10}^{-4}\times\:\left({T}_{FC}-298.15\right)+4.31\times\:{10}^{-5}{\times\:T}_{FC}\times\:\left[ln\left({P}_{{H}_{2}}\right)+0.5\times\:ln\left({P}_{{O}_{2}}\right)\right]\:$$

(2)

$$\:{P}_{{H}_{2}}=\frac{{R}_{ha}\times\:{P}_{{H}_{2}O}}{2}\left[\frac{1}{\frac{{R}_{ha}\times\:{P}_{{H}_{2}O}}{{P}_{a}}\times\:{e}^{\frac{1.635{I}_{FC}/A}{{T}^{1.334}{I}_{FC}}}}-1\right]\:\:$$

(3)

$$\:{P}_{{O}_{2}}={R}_{hc}\times\:{P}_{{H}_{2}O}\left[\frac{1}{\frac{{R}_{hc}\times\:{P}_{{H}_{2}O}}{{P}_{c}}\times\:{e}^{\frac{1.635{I}_{FC}/A}{{T}^{1.334}{I}_{FC}}}}-1\right]\:$$

(4)

The current generated by the cell, referred to as $\:\:{I}_{FC}$, is affected by multiple variables, such as the active area of the membrane $\:A$, the inlet partial pressures at the negative electrode $\:{P}_{a}$ and the positive electrode $\:{P}_{c}$, as well as the relative humidity of steam at the negative $\:{R}_{hc}$ and positive $\:{R}_{ha}$ electrodes. The inlet pressures for the negative and positive electrodes are indicated by $\:{P}_{c}$ and $\:{P}_{a}$, respectively. Additionally, the steam saturation pressure within the PEM fuel cell, which varies with the operating temperature, can be determined using the specified formula.

$$\:{log}_{10}\left({P}_{{H}_{2}O}\right)=0.0295\times\:\left({T}_{FC}-273.15\right)-9.18{\times\:10}^{-5}{T}_{c}^{2}+1.4{\times\:10}^{-7}{T}_{c}^{3}-2.18$$

(5)

The cells’ activation over-potential was assessed at a reference temperature of 25 °C.

$$\:{E}_{act}=-\left[{\beta\:}_{1}+{\beta\:}_{2}{T}_{FC}+{\beta\:}_{3}{T}_{FC}\text{ln}\left({C}_{{O}_{2}}\right)+{\beta\:}_{4}{T}_{FC}\text{ln}\left({I}_{FC}\right)\right]$$

(6)

The concentration of oxygen at the interface of the cathode, catalyst, and electrolyte, represented as $\:{C}_{{O}_{2}}$ $\:mol/c{m}^{3}$, is essential for the effective operation of the fuel cell. This value is determined through the subsequent equation:

$$\:{C}_{{O}_{2}}=\frac{{P}_{{O}_{2}}\times\:{e}^{\frac{498}{{T}_{FC}}}}{5.1\times\:{10}^{6}}$$

(7)

$$\:{\beta\:}_{2}=29\times\:{10}^{-4}+21\times\:{10}^{-5}\text{ln}\left(A\right)+43\times\:{10}^{-6}\text{l}\text{n}\left({C}_{{H}_{2}}\right)$$

(8)

The saturation concentration of hydrogen $\:{H}_{2}$ at the interface of the cathode, catalyst, and electrolyte, referred to as $\:{C}_{{H}_{2}}$, represents an essential variable and is determined through the subsequent Eq. 

$$\:{C}_{{H}_{2}}=\frac{{P}_{{H}_{2}}\times\:{e}^{\frac{-77}{{T}_{FC}}}}{1.1\times\:{10}^{6}}$$

(9)

The partial pressures of hydrogen H₂, oxygen O₂, and water vapor H₂O, denoted as $\:{P}_{H}$, $\:{P}_{{O}_{2}}$, and $\:{P}_{{H}_{2}O}$ respectively, are essential factors in influencing the Ohmic voltage drop, which can be computed using the given formula.

$$\:{E}_{{\Omega\:}}=\left({{R}_{c}+R}_{m}\right){\times\:I}_{FC}$$

(10)

The current flowing through the fuel cell is represented by $\:{I}_{Fc}\:$, while $\:{R}_{c}$ indicates the resistance associated with the connections, and $\:{R}_{m}$ signifies the resistance of the membrane, which can be determined using the subsequent formula.

$$\:{R}_{m}=\frac{{\rho\:}_{m}\times\:l}{S}$$

(11)

$$\:{\rho\:}_{m}=\frac{181.6\times\:\left[0.062\times\:{\left(\frac{{T}_{FC}}{303}\right)}^{2}\times\:{\left(\frac{{I}_{FC}}{S}\right)}^{2.5}+0.03\left(\frac{{I}_{FC}}{S}\right)+1\right]}{\left[\lambda\:-0.063-3\left(\frac{{I}_{FC}}{S}\right)\right]\times\:{e}^{\frac{{T}_{FC}-303}{{T}_{FC}}}}$$

(12)

In this context, S refers to the membrane’s surface area measured in square centimeters $\:\left(c{m}^{2}\right)$, while $\:I$ indicate the membrane’s thickness. The coefficients $\:{\beta\:}_{i}(i=\text{1,2},\text{3,4})$ are empirical constants. $\:{T}_{FC}$ denotes the operational temperature of the cell expressed in degrees Celsius (°C), and $\:\lambda\:$ is a variable parameter. The super potential saturation $\:\left({\text{E}}_{\text{o}\text{p}\text{s}}\right)$ can be determined through the subsequent Eq. 

$$\:{E}_{ops}=-\beta\:\times\:ln\left(\frac{{J}_{max}^{2}-J\times\:{J}_{max}}{{J}_{max}}\right)\:$$

(13)

Within this framework, $\:\beta\:$ signifies a parametric coefficient, whereas $\:J$ and $\:{J}_{max}$ refer to the standard and peak current densities, respectively. This study centers on a BCS PEMFC characterized by a power output of 500 W and a peak current of 30 A. The ideal parameter values employed for the simulation are derived from Reference²⁷ and are conveniently compiled in Table 1.

Table 1 The ideal variable values for the PEMFC are derived from Reference²⁷.

In the process of altering the system, it is crucial to take into account the comprehensive fuel utilization of the PEM fuel cell $\:\left(m{H}_{2}\right)$. This can be assessed through the subsequent methodology.

$$\:{C}_{FC}=\int\:\frac{{P}_{FC}}{LHV\times\:{\eta\:}_{FC}\left(PLR\right)}dt$$

(14)

In this context, LHV refers to the lower heating value of hydrogen, while $\:{\eta\:}_{FC}\left(PLR\right)$ signifies the efficiency of the PEM fuel cell design, which is dependent on the part-load ratio $\:\left(PLR\right)$. The formula for determining $\:{\eta\:}_{FC}\left(PLR\right)$ is presented below³⁰:

$$\:{\eta\:}_{FC}={\eta\:}_{fc}\times\:\left(1-\frac{{P}_{fc}^{aux}}{{P}_{fc}}\right)\:$$

(15)

$$\:{P}_{fc}={V}_{o}\times\:{I}_{o}\:$$

(16)

$$\:{\eta\:}_{fc}=\frac{{P}_{fc}}{{P}_{{H}_{2}}}\:$$

(17)

$\:{I}_{o}$ denotes the current generated by the fuel cell stack. The heat generated, $\:{Q}_{heat}$, throughout the driving phase can be determined using the subsequent Eq. 

$$\:{Q}_{heat}={C}_{FC}\times\:LHV$$

(18)

The inclusion of additional elements within the FC system imposes constraints on the output ratio of net power, primarily due to their sluggish dynamic response at lower speeds. Consequently,

$$\:{P}_{FC}\left(t\right)=A\times\:t+{P}_{FC}^{0}$$

(19)

In this context, $\:{P}_{FC}^{0}$ denotes the initial power at the commencement of the acceleration or deceleration phase, while the limiting value, $\:A$, is determined through the following calculation:

$$\:A=\frac{0.9\:of\:{P}_{FC}^{rated}}{T}$$

(20)

In this context, T is defined as 30 s.

Li-ion battery

Lithium-ion batteries, recognized for their elevated energy density and prevalent application in consumer electronics, are classified as rechargeable batteries. In contrast to conventional single-use lithium batteries, lithium-ion batteries utilize lithium ions as the electrode material rather than metallic lithium. They exhibit a notable reduction in weight compared to other storage battery types of similar dimensions. These batteries are frequently utilized in portable electronic devices. This paper discusses the use of a lithium-ion battery to enhance a PEMFC’s power of output during transient states. The formula for determining the instantaneous power generated by a lithium-ion battery is presented below.

$$\:{P}_{b}\left(t\right)={P}_{Lo}\left(t\right)-{P}_{FC}\left(t\right)$$

(21)

In the given equation, $\:{P}_{Lo}\left(t\right)$ denotes the overall instantaneous power requirement of the locomotive. When $\:{P}_{b}\left(t\right)$ exceeds zero, it indicates that the battery is discharging; conversely, if it falls below zero, the battery is in a charging state. A zero value signifies that the battery is not in use. The state of charge (SoC) of the battery at any specific time is contingent upon its SoC from the preceding time interval $\:(t-{\Delta\:}t)$ and can be determined using the following calculation.

$$\:SOC\left(t\right)=SOC\left(t-{\Delta\:}t\right)-{\eta\:}_{b}\times\:\frac{{P}_{b}\left(t\right)}{{Q}_{bc}}{\Delta\:}t$$

(22)

The efficiency of battery charge and discharge, represented by $\:{\eta\:}_{b}$, can be determined through the subsequent formula.

$$\:{\eta\:}_{b}=\left\{\begin{array}{c}\frac{1}{{\eta\:}_{charge}}\:\:for\:{P}_{b}\left(t\right)\le\:0,\:\:Charging\:\\\:\frac{1}{{\eta\:}_{discharge}}\:\:for\:{P}_{b}\left(t\right)>0,\:\:Discharging\end{array}\right.$$

(23)

In this context, $\:{Q}_{bc}$ denotes the capacity of the battery, Δt signifies a minor time increment, while $\:{\eta\:}_{charge}$ and $\:{\eta\:}_{discharge}$ refer to the efficiencies associated with the charging and discharging processes of the battery, respectively.

Immediate power demand

This study focuses on a locomotive propulsion system for the HE system that utilizes a fuel cell (FC) in conjunction with a battery backup. To accurately model the instantaneous power requirements, it is essential to account for various forces, including aerodynamic drag $\:\left({F}_{d}\right)$, the frictional force between the railroad track and wheels $\:\left({\text{F}}_{\text{f}\text{r}}\right)$, the force necessary to ascend an incline $\:\left({F}_{sg}\right)$, and the force associated with acceleration $\:\left({F}_{a}\right)$. The relevant equations detailing these forces are presented below.

$$\:{F}_{d}\left(t\right)=\frac{1}{2}\times\:\rho\:\times\:D\times\:\sigma\:\times\:\nu\:{\left(t\right)}^{2}\:$$

(24)

$$\:{F}_{fr}={M}_{loc}\times\:g\times\:{C}_{Rr}\times\:{cos}\beta\:$$

(25)

$$\:{F}_{sg}={M}_{loc}\times\:g\times\:{sin}\beta\:$$

(26)

$$\:{F}_{a}={M}_{loc}\times\:\frac{d\nu\:\left(t\right)}{dt}$$

(27)

In this context, $\:\sigma\:$ signifies the cross-sectional area measured in square meters $\:\left({m}^{2}\right)$, D indicates the drag coefficient, $\:{\text{C}}_{\text{R}\text{r}}$ refers to the coefficient of rolling resistance, $\:\nu\:$ denotes the locomotive’s velocity, $\:\beta\:$ represents the gradient or slope, and $\:\rho\:$, which is quantified as $\:1.3\:\text{k}\text{g}/{\text{m}}^{3}$, corresponds to the density of air.

$$\:{M}_{loc}={n}_{cch}\times\:{m}_{ch}+{M}_{loco}$$

(28)

Within this framework, $\:{n}_{cch}$ indicates the quantity of coaches that lack air conditioning, whereas $\:{M}_{loco}$ and $\:{m}_{ch}$, in turn, signify the density of a non-air-conditioned locomotive and coach. The principal specifications for the locomotive have been outlined as follows²⁸: a power output of 9600 kW, a starting traction force of 584 kN, an axle load of 25 tons, and a maximum operational speed of 120 km/h. The total instantaneous power requirement for the locomotive can be determined through the subsequent Eq. 

$$\:{P}_{dmd}^{T}\left(t\right)={{P}_{d}\left(t\right)+P}_{aux}$$

(29)

In this context, $\:{P}_{aux}$ signifies the demand for auxiliary power, while $\:{P}_{d}\left(t\right)$ refers to the instantaneous power demand, which can be determined through the subsequent Eq. 

$$\:{P}_{d}=\frac{1}{{\eta\:}_{s}}\times\:\left({F}_{fr}\left(t\right)+{F}_{sg}\left(t\right)+{F}_{d}\left(t\right)+{F}_{fr}\left(t\right)\right)\times\:\nu\:\left(t\right)$$

(30)

The transmission system’s efficiency, represented by $\:{\eta\:}_{s}$, is established at 85%.

DCs (drive cycle)

Dynamic programming (DP) is considered to illustrate the diversities of speed in the locomotive. This parameter significantly influences the instantaneous power demand. The DP used in this study is derived from²⁹ and represents the mean value of arrangements of locomotive driving. Figure 3 depicts the diversity of speed profile of the locomotive over time on the basis of the provided dynamic programming data.

Fig. 3

The fluctuation in the velocity of the locomotive as a function of time for the direct current (DC) system.

The parameter values employed for the DC are presented in Table 2.

Table 2 The variable values employed for the direct current (DC) system.

While the core of work with this study is on optimizing the hybrid energy system (HES) from locomotive point of view, it would be valuable to state the part and technical specifications of the renewable energy source. Hydrogen is the main renewable energy input for this HES and serves fuel for the PEMFC. Technical specifications of hydrogen are closely tied to the PEMFC model. This system is considered to use high-purity hydrogen gas as required for performance in PEMFC.

These major technical parameters are given by the conditions of operation of the fuel cell: standard operating temperature at 25 °C (per the reference in Eq. 6), specific inlet pressures for the anode and cathode (denoted as $\:{P}_{H2}$ and $\:{P}_{O2}$ in (Eq. 5), and controlled relative humidity levels (denoted as $\:R{H}_{H2}$ and $\:R{H}_{O2}$ in (Eq. 5). The lower heating value (LHV) identifies the energy content of the hydrogen, which is a standard figure used for efficiency calculation (Eq. 14).

This hydrogen is assumed to be produced by renewable means (e.g., electrolysis powered by wind or solar energy), which is critical for the environmental benefits of the system but is treated by the optimization model as a given fuel with required properties that needed to be assumed for the PEMFC stack instead of modeling the upstream renewables production process.

Methodology

The energy management (EM) strategy implemented in this research seeks to enhance the energy utilization of the locomotive by effectively balancing the real-time power generation from multiple energy sources against the locomotive’s power requirements. The central aim is to achieve an optimal compromise between the output of the hydrogen energy system and the locomotive’s energy needs, thereby reducing the total cost associated with the components. In this EM strategy, the power output from the proton exchange membrane (PEM) fuel cell is harnessed to facilitate acceleration until it attains its maximum rated capacity³⁰. The strategy accounts for both the charging and discharging phases of the battery throughout the operational period to satisfy the immediate power demands. The execution of the power output ratio continues until the locomotive begins to decelerate³¹. The amount of alteration in the fuel cell’s power of output has been regulated, with a noted decrease in power production during deceleration. Should the instantaneous demand of power surpass the output from the FC, the battery will discharge; conversely, if the fuel cell generates more power, the battery will charge. As previously indicated, a strategic arrangement is proposed to ascertain the optimum scopes of the battery and the PEMFC, with an emphasis on improving the performance of the HE system within the locomotive while minimizing the overall component costs.

$$\:\text{min}C\left(x\right)={C}_{m}+{C}_{b}+{C}_{FC}+{C}_{b}^{r}+{C}_{FC}^{r}$$

(31)

In this context, $\:{C}_{FC}$, $\:{C}_{m}$, and $\:{C}_{b}$ signify the expenses associated with the fuel cell, electric motor, and lithium-ion battery, respectively. Furthermore, $\:{C}_{FC}^{r}$ and $\:{C}_{b}^{r}$ indicate the costs incurred for the substitution of the fuel cell and battery, respectively. The subsequent limitations are applicable:

$$\:0.25\le\:SOC\left(t\right)\le\:0.5\:$$

(32)

$$\:{P}_{FC}^{r}\ge\:{P}_{dem}^{avg}$$

$$\:{P}_{d}^{T}\left(t\right)={P}_{FC}\left(t\right)+{P}_{b}\left(t\right)$$

$$\:0\le\:{P}_{FC}\left(t\right)\le\:{P}_{FC}^{r}$$

.

In this context, $\:{P}_{FC}^{r}$ indicates the rated capacity of the FC, $\:{Q}_{FC}^{r}$ refers to the volume of the battery, $\:{P}_{d}^{T}\left(t\right)$ represents the locomotive’s mean power requirements, $\:{P}_{b}\left(t\right)$ denotes the instantaneous power generated by the battery, and $\:{P}_{FC}\left(t\right)$ signifies the instantaneous power produced by the fuel cell. The corresponding equations for these elements are presented below:

$$\:{C}_{FC}={C}_{uFC}\times\:{P}_{FC}^{r}$$

(33)

$$\:{C}_{m}={C}_{um}\times\:{P}_{b}^{r}+{C}_{m}$$

(34)

$$\:{C}_{b}={C}_{ub}\times\:{Q}_{FC}^{r}+{C}_{b}$$

(35)

$$\:{C}_{b}^{r}=\frac{{V}_{b}^{fu}\times\:{M}_{r}^{b}}{{\left(1+{r}_{i}\right)}^{M}}$$

(36)

$$\:{C}_{FC}^{r}=\frac{{V}_{FC}^{fu}\times\:{N}_{r}^{FC}}{{\left(1+{r}_{i}\right)}^{N}}$$

(37)

Within this framework, $\:{r}_{i}$ signifies the interest rate utilized in this analysis. The variables $\:N$ and $\:M$ correspond to the ages, in years, of the Proton-exchange membrane fuel cell and the battery, respectively. The terms $\:{N}_{r}^{FC}$ and $\:{M}_{r}^{b}$ indicate the quantity of substitutions needed for the battery and the fuel cell, respectively. The costs per rated unit for the FC, electric motor, and battery are represented by $\:{C}_{uFC}$,$\:\:{C}_{um}$, and$\:\:{C}_{ub}$, respectively. Additionally, $\:{C}_{m}$ and $\:{C}_{b}$ refer to the fixed costs associated with the motor and battery, respectively. The future values of the fuel cell and battery are denoted by $\:{V}_{FC}^{fu}$and $\:{V}_{b}^{fu}$, respectively³². The necessary values for these parameters are detailed in Table 3³³.

Table 3 The necessary quantities of the variables³³.

The anticipated operational lifespan of a fuel cell utilized in transportation, as projected in 2020, is approximately 5000 h, with a maximum estimated lifespan of 8000 h³². Given this expected duration and presuming that the locomotive functions for 9 h daily, the variable N (representing the age of the fuel cell) is estimated to be 3 years. Furthermore, the expected replacement interval for the battery is set at 10 years $\:(M=10)$³². Consequently, throughout a 20-year operational timeframe, it is expected that the fuel cell will necessitate a single replacement, whereas the battery will require three replacements.

Theoretical justification for incorporating fractional calculus in IGLGO

The effectiveness of the Improved Grey Lag Goose Optimization (IGLGO) algorithm is significantly enhanced by fractional calculus, which brings about important improvements to the search mechanisms of the algorithm.

The choice of the Improved Grey Lag Goose Optimization (IGLGO) algorithm for this study is a purposeful and strategic choice for the specific, complex optimization challenges faced in a hybrid energy system (HES) for locomotive applications. Sizing a PEM fuel cell and lithium-ion battery jointly with the least possible cost while observing time-varying boundaries, for instance, on State of Charge (SoC), is a highly non-linear, multimodal optimization problem that is prone to local optima.

Standard optimization algorithms struggle in such problems, either converging too soon or devouring too much in computer resources. IGLGO was ideal because its core innovations directly address these challenges. The core dynamic grouping mechanism, for instance, ensures a good balance of exploration (searching new areas of the solution space) and exploitation (refining known good solutions), which is very important in avoiding local optima while dealing with the complex cost trade-offs between expensive fuel cells and batteries.

Also, inclusion of fractional calculus (FC) memory, providing a memory effect, enables the algorithm to learn from its previous search behavior when it decides and improves both convergence speed and stability. The stochastic search agents also diversify the search further. This unique combination of properties makes IGLGO ideally suited to navigating the rugged cost landscape of the HES, where a modest change in component sizing may yield large variations in total system cost because of replacement cycles and operational constraints. Therefore, IGLGO is not just a novel algorithm to solve a common problem, but it is a tailored tool whose enhanced searching capacity is essential to find a globally optimal and economically feasible configuration for the locomotive’s hybrid power system. Here, the study discusses the theoretical foundation for such integration, as well as its effects on convergence properties.

• Enhanced memory mechanism.

by applying fractional calculus, you introduce the memory effect in the optimizer, it will not only consider the immediate past but also the history of previous states. In contrast to ordinary integer-order calculus, where the algorithm base decisions only on the current iteration, fractional calculus allows the algorithm to utilize historical data to make more informed decisions. The IGLGO algorithm is already updated from past solutions by using fractional order derivatives; thus, only those spaces that are “worth it” are recrossed. This mechanism effectively refines the optimization process to make it more efficient.

More flexibility and diversity

First, by employing fractional calculus, the IGLGO algorithm is granted additional flexibility and diversity in its search operations. This enables the algorithm to explore different parts of the solution space at different scales and resolutions, which helps prevent premature convergence to poor solutions. By adjusting its behavior to maintain an equilibrium between exploration and exploitation, the algorithm can efficiently traverse rugged landscape.

• Improved convergence speed.

An established fractional calculus framework, namely the Grunwald-Letnikov (GL) model, is utilized for the purpose of boosting the IGLGO algorithm performance and robustness. This approach allows for quicker convergence by fractionating the processes for memory and genetic information. The algorithm converges faster by calculating the fractional derivative of order α which is defined using the gamma function in this method and their effect/effectivity to the present state. This leads the IGLGO algorithm to a more organized and efficient way of achieving optimal solutions.

• Dynamic grouping and stochastic sampling search agents.

Each of these elements provides its own contribution to the overall performance– agent fractional calculus elements facilitate dynamic grouping and multi-agent stochastic search abilities. These calibrations make sure the algorithm stays robust and flexible during the entire optimization process. The use of domain-specific metrics for periodic adjustment of parameters like a, r1​, and r2​ further ensures that the algorithm is well tuned to meet hybrid energy systems unique requirements and objectives in locomotive applications.

Solving method

Improved grey lag goose optimization

The mathematical and motivational model supporting the Greylag Goose Optimization algorithm is outlined in this section. The design of the GGO algorithm was inspired by the social behavior and energetic movements observed in geese.

• (A) Geese: communal manner.

Faithfulness is one of the most famous features of Geese. They spend their lifetime with their partners and are so caring for their offspring. Frequently, they have a tendency to be near the sick or hurt spouse or chicken. They remain to do so when the winter season is forthcoming. The remaining part of the group migrates for heater climates. Once a spouse of a goose passes away, the goose tends to be lonely, and several geese might select to be alone the rest of their lifespans, declining to remarriage forever.

Geese enjoy to pomposity their feathers whIGLGOt foraging for nutrition in the lawn and get-together leaves and pushes to enhance their dwellings.

Annually during the spring season, eggs are hatched. The man geese guard the hidden eggs in their nest whIGLGOt the women watch them for 30 days. Several geese have a preference to utilizing the identical nest that eggs are lied over of many years.

• (B) Geese: energetic manner.

Geese are social birds that often gather in large groups, known as gaggles, where they exhibit cooperative behavior. Within these gaggles, while some members feed, others take on the role of sentries, keeping watch for predators. Healthy geese care for their wounded peers, and injured geese band together for protection and support. Geese are energetic and sociable, especially when travelling in large flocks. They spend their days foraging for food and resting on water or grassland. Male geese, or ganders, are skilled aviators, capable of migrating long distances in large flocks. During flight, geese form a “V” formation, reducing air resistance and increasing their range. Geese have excellent memories, which they use for navigation throughout their yearly migrations, relying on familiar landmarks and celestial cues.

• (C) Grey lag goose optimization (GGO) algorithm.

The GGO (Grey Lag Goose Optimization) optimizer introduces a novel approach to optimization by starting with the random generation of a population of potential solutions, known as “agents.” Each agent represents a potential solution to the problem at hand, collectively forming the “gaggle.” The fitness function, denoted as $\:{F}_{n}$, is utilized to assess the quality of each agent’s solution. Following the evaluation of the fitness function for all agents, the most optimal solution is identified as the leader.

The GGO algorithm incorporates a dynamic grouping technique, segregating the population into two distinct sets: an exploration set ($\:{m}_{1}$) and an exploitation set ($\:{m}_{2}$). The number of agents in each set is adjusted dynamically based on the current best solution, with the exploration set comprising $\:{m}_{1}\:$ agents and the exploitation set comprising $\:{m}_{2}\:$agents. Initially, equal weights are assigned to exploration and exploitation, with half of the agents in each group. However, if the fitness value of the best solution remains constant for three consecutive iterations, the algorithm increases the number of agents in $\:{m}_{1}\:$ to promote exploration and prevent local optima. This adaptive mechanism introduces a sense of optimism and encourages a renewed search for enhanced solutions.

• (D) Exploration operation.

Exploration involves the GGO algorithm’s ability to pinpoint specific regions within the search area while actively steering clear of less-than-ideal solutions. The algorithm’s primary objective is to move closer to the desired outcome by identifying new and appropriate positions near its current location. This is accomplished through a continuous evaluation of different potential choices, ultimately selecting the most suitable one based on their respective fitness values. Throughout the iterations, the GGO algorithm utilizes specific formulas to update the Z and A vectors, where the parameter ‘a’ transition linearly from 2 to $\:Z\:=\:2a.{r}_{1}\:-a$ and $\:A=\:2.{r}_{2}$. These updates play a crucial role in facilitating the exploration process, effectively guiding the algorithm towards optimal solutions.

$$\:Y(t\:+\:1)\:=\:{Y}^{*}\left(t\right)\:-\:Z.|A.Y(t)-\:Y(t\left)\right|\:$$

(38)

In this context, $\:Y\left(t\right)$ denotes the state of an agent at iteration $\:t$, while $\:{Y}^{*}\left(t\right)\:$ signifies the optimal solution state identified thus far. The notation $\:Y(t\:+\:1)$ refers to the updated state of the agent following the current iteration. The parameters $\:{r}_{1}$and $\:{r}_{2}$ are subject to stochastic variation within the range of 0 to 1. To facilitate a more comprehensive exploration and prevent any bias towards a singular leader location, three stochastic search agents, labeled as $\:{Y}_{P1}$,$\:\:{Y}_{P2}$, and,$\:\:{Y}_{P3}$, are utilized. The subsequent equation is applied for instances where $\:\left|Z\right|$ is greater than or equal to 1, allowing for the adjustment of the current search agent’s position.

$$\:Y\left(t\:+\:1\right)={s}_{1}*{Y}_{P1}+x*{s}_{2}*\:({Y}_{P1}-{Y}_{P1})+(1-x)*{s}_{3}*\:(Y-{Y}_{P1})$$

(39)

The variables $\:{\:s}_{1}$, $\:{\:s}_{2}$, and $\:{\:s}_{3}$are constrained within the interval of 0 to 2 and are eligible for periodic renewal. In contrast, the variable x exhibits an exponential decline, which is determined by the specified formula.

$$\:x=1-{\left(\frac{t}{{t}_{max}}\right)}^{2}\:$$

(40)

In this context, $\:t$ represents the current iteration, while $\:{t}_{max}$ indicates the maximum number of iterations. The second renewal process, characterized by a reduction in the quantities of the $\:a$ and $\:Z$ vectors, is determined using the following formula when $\:{\:r}_{3}$ is greater than or equal to $\:0.5$.

$$\:Y(t\:+\:1)\:=\:{\:s}_{4}*\:\left|{Y}^{*}\right(t)\:-\:Z(t\left)\right|.{e}^{cf}.cos\left(2\pi\:f\right)\:+\:\left[2{\:s}_{1}\right({\:r}_{4}+\:{\:r}_{5}\left)\right]\:*\:{Y}^{*}\left(t\right)$$

(41)

Here, $\:c$ represents a fixed value, whereas $\:f$ denotes a random variable that fluctuates within the range of -1 to 1. The magnitude of $\:{\:s}_{4}\:$ extends from 0 to 2, and the magnitudes of $\:{\:r}_{4}\:$and $\:{\:r}_{5}$ vary from 0 to 1, experiencing renewal.

• (E) Local search operation.

The local search individuals are dedicated to enhancing current solutions. At the conclusion of the cycles, the GGO can determine which individual possesses the maximum grade of physical fitness and acknowledges their achievement³⁴. To fulfill its exploitation goals, the GGO implements 2 separate techniques, which have been explained in the following. The ensuing formula serves as a guide for the team to achieve the most effective advancement. The three solutions, referred to as sentries, $\:{Y}_{\text{s}1}$,$\:\:{Y}_{\text{s}2}$, and, $\:{Y}_{\text{s}3}$assist the other members $\:\left({Z}_{\text{N}\text{o}\text{n}\text{s}\text{e}\text{n}\text{t}\text{r}\text{y}}\right)$ in recalibrating their positions towards the anticipated location of the prey. The subsequent formulas delineate the procedure for updating these locations.

$$\:{Y}_{1}\:\:=\:{Y}_{\text{s}1}\:\:-\:{Z}_{1}.|{A}_{1}.{Y}_{\text{s}1}-\:Y|\:\:{Y}_{2}\:\:=\:{Y}_{\text{s}2}\:\:-\:{Z}_{2}.|{A}_{2}.{Y}_{\text{s}2}-\:Y|\:\:{Y}_{3}\:\:=\:{Y}_{\text{s}3}\:\:-\:{Z}_{3}.|{A}_{3}.{Y}_{\text{s}3}-\:Y|\:\:$$

(42)

In this instance, $\:{Z}_{1}$, $\:{Z}_{2}$ and $\:{Z}_{3}$ are computed using the formula $\:Z\:=\:2a.{r}_{1}-\:a$, whereas $\:{A}_{1},\:{A}_{2}$ and $\:{A}_{3}$ are established as $\:A=\:2{r}_{2}$. The revised positions for the elements, represented as $\:Y\left(t+1\right),$, can be formulated as the mean of the three outcomes $\:{Y}_{1}$, $\:{Y}_{2}$, and $\:{Y}_{3}$, as illustrated here:

$$\:Y\left(t+1\right)=\stackrel{-}{{Y}_{i}}\left|\genfrac{}{}{0pt}{}{3}{0}\right.$$

(43)

• (F) Investigating the area surrounding the optimal solution.

The highest skilled individuals are strategically placed in proximity to the top solution (leader) during flight. This encourages certain team members to seek enhancements by concentrating on regions near the most efficient answer, known as $\:{Y}_{\text{F}1}$. The GGO supports this procedure through the utilization of the subsequent equation:

$$\:Y\left(t+1\right)=Y\left(t\right)\:+\:D(1\:+\:x)\:*\:s\:*\:(Y\:-\:{Y}_{\text{F}1})$$

(44)

• (G) Identification of the most suitable resolution.

The method of alteration within the exploration set, along with the scanning of individuals, is employed to suggest remarkable exploration capabilities of the GGO. Due to its robust exploratory abilities, the GGO may postpone meetings. The process begins with the acquisition of data regarding the GGO, including population size, alteration ratio, and the number of iterations. Subsequently, the members are categorized into two groups within the GGO: those who engage in investigative efforts and the individuals focusing on local search tasks. The GGO adjusts the scope of groups in a dynamic manner through an iterative procedure aimed at identifying the optimal solution. The groups employ two methods to organize its efforts. Furthermore, the GGO stochastically reorganizes the solutions across iterations. Within an iteration, a solution element from the global search set may transit to the local search set in the next stage. The selective mechanism of the GGO ensures that the leader remains in location during the process. The stages of the GGO are utilized to update the locations of both the exploration set $\:\left({m}_{1}\right)$ and the exploitation set $\:\left({m}_{2}\right)$. The variable $\:{r}_{1}$ is updated during iterations according to the formula $\:{r}_{1}=c\left(1-(t/{t}_{max}\right)$, where $\:t$ represents the current iteration, $\:c$ is a constant, and t_max denotes the total number of iterations. At the conclusion of each iteration, the GGO refreshes the agents within the search area and stochastically modifies their trajectories to exchange characteristics between the exploration and exploitation sets. In the final stage, the GGO implements the optimal solution.

Integration of dynamic programming with IGLGO

Dynamic programming (DP) is used in conjunction with the Improved Grey Lag Goose Optimization (IGLGO) algorithm which maintains a balance between short-term energy demand and long cost for hybrid energy systems in locomotives. DP is used to represent the variety of speed profiles and power demands of this locomotive, in order to develop an energy management strategy which considers both immediate operational demand and economic criteria on the long term. Here, the IGLGO algorithm is devised to find the optimum size of the Proton Exchange Membrane Fuel Cell (PEMFC) and lithium-ion battery over time whereas DP gives a structure to obtain the ideal power distribution throughout the components. Real-time tweaks for acceleration, deceleration and changing gradients mean that DP and IGLGO work in tandem and aid in delivering energy as required tuned to changing conditions. Example applications include in cases of high-power demand where the power required exceeds PEMFC capability and the battery supplies the shortfall, and regenerative braking where maximum charging occurs during ready modes. Thus, the reliance on the battery is reduced and it can last longer and requires fewer replacements as compared to a common battery. The DP model parameters are chosen in such a way that they mimic real-life operations and can be easily used in practice. Important factors: Acceleration Mean (0.5 m/s²), Deceleration Mean (-0.5 m/s²), Max speed (45 km/h), driving distance total (220 km), driving time total (15960 s). Expected values are in-line with normal locomotive specifications Various average conditions seen in normal circumstances These also include maximum acceleration µ(0.7 m/s²) and maximum deceleration µ(− 0.7 m/s²) which provides robustness for extreme scenarios. These parameters not only make the model more accurate but also scale the model to different locomotive classes and routes. In addition, the DP model includes a 30s adjustment period for changing the power output of the fuel cell as the dynamic response of PEMFCs is sluggish at low speeds. This metric is critical for city or hilly driving scenarios frequent stops and starts. These DP parameters can allow the hybrid energy system to strikingly balance the performance to the cost efficiency through IGLGO’s adaptive exploration-exploitation mechanism and ultimately make this approach more amenable to practical implementations than classical approaches.

Improved grey lag goose optimization (IGLGO)

Fractional calculus, also known as FC, has become increasingly popular in recent years due to its ability to enhance the performance and robustness of metaheuristic algorithms. By incorporating fractional order, these algorithms gain increased flexibility and diversity in their search processes, allowing them to explore different regions of the solution space with varying scales and resolutions. Additionally, fractional order helps metaheuristics avoid local optima and premature convergence by introducing randomness and chaos into their dynamics. This work provides a brief introduction to fractional calculus (FC) and its applications in fractional methods. FC offers a systematic and effective approach to studying processes involving memory and genetic traits, ultimately improving the speed of metaheuristic algorithms. One of the most popular fractional calculus models is the Grunwald-Letnikov (GL) model, which is described as follows:

$$\:{S}^{\alpha\:}\left(Y\left(t\right)\right)=\underset{h\to\:0}{\text{lim}}{h}^{-\alpha\:}{\sum\:}_{a=0}^{\infty\:}{\left(-1\right)}^{a}\left(\genfrac{}{}{0pt}{}{\alpha\:}{a}\right)Y\left(t-ah\right)\:$$

(45)

$$\:\left(\genfrac{}{}{0pt}{}{\alpha\:}{a}\right)=\frac{{\Gamma\:}(\alpha\:+1)}{{\Gamma\:}(a+1){\Gamma\:}(\alpha\:-a+1)}=\frac{\alpha\:\left(\alpha\:-1\right)\left(\alpha\:-2\right)\dots\:(\alpha\:-a+1)}{a!}$$

(46)

The fractional derivative of order $\:\sigma\:$, known as the Grunwald-Letnikov (GL) derivative, is determined through the utilization of the gamma function, symbolized as $\:\varGamma\:\left(t\right)$. The calculation of this derivative, denoted as $\:{D}^{\alpha\:}\left(Y\left(t\right)\right)$, is achieved by employing the subsequent formula:

$$\:{S}^{\alpha\:}\left[Y\left(t\right)\right]=\frac{1}{{T}^{\alpha\:}}{\sum\:}_{a=0}^{N}\frac{{\left(-1\right)}^{a}{\Gamma\:}\left(\alpha\:+1\right)Y\left(t-ah\right)}{{\Gamma\:}\left(a+1\right){\Gamma\:}\left(\alpha\:-a+1\right)}\:$$

(47)

The length of the memory window is represented by the symbol $\:N.T$ indicating the sampling time and $\:\alpha\:$ denoting the derivative order operator. Assuming that $\:\alpha\:$ is equal to 1, the equation mentioned above can be expressed in the following manner:

$$\:{S}^{1}\left[Y\left(t\right)\right]=Y\left(t+1\right)-Y\left(t\right)$$

(48)

In this context, $\:{S}^{1}\left[Y\left(t\right)\right]$ represents the variance measure observed between two specific actions. To enhance the algorithm’s efficiency, this study integrates FC memory in the subsequent ways:

$$\:Y\left(t+1\right)-Y\left(t\right)=0$$

(49)

The standard equation is obtained through the subsequent formulation.

$$\:{S}^{\alpha\:}\left[Y\left(t+1\right)\right]=-Y\left(t\right)=-{\sum\:}_{a=1}^{m}\frac{{\left(-1\right)}^{a}{\Gamma\:}\left(\delta\:+1\right)Y\left(t+1-a\right)}{{\Gamma\:}\left(a+1\right){\Gamma\:}\left(\sigma\:-a+1\right)}$$

(50)

The algorithm’s equation can be rearranged to align with the provided Eq. 

$$\:{S}^{\alpha\:}\left[Y\left(t+1\right)\right]=-{\sum\:}_{a=1}^{m}\frac{{\left(-1\right)}^{a}{\Gamma\:}\left(\alpha\:+1\right)Y\left(t+1-a\right)}{{\Gamma\:}\left(a+1\right){\Gamma\:}\left(\alpha\:-a+1\right)}$$

(51)

The algorithm’s current condition may be re-evaluated by applying the subsequent formula, which is obtained from the earlier equation, with the value of m being 4, denoting the initial four terms of memory data. The algorithm pseudo-code has been provided in the following.

Algorithm 1

The pseudo-code of improved grey lag goose optimization.

One of the main novelties is the dynamic grouping mechanism, that adaptively balances between exploration and exploitation. First, the population is split into exploration and exploitation sets evenly. The algorithm also keeps track of the best solution found so far, and if this solution is unchanged for some iterations, it will increase the number of agents in the exploration set, which can also help escape local optima, since it will search a larger part of the solution space.

In particular, the IGLGO algorithm uses three stochastic search agents ($\:{Y}_{P1}$, $\:{Y}_{P2}$, $\:{Y}_{P3}$) to conduct exploration of new regions in the search space; this reduces the risk of the algorithm being trapped at local optima. The $\:Z$, $\:A$ vector is modified, and ‘$\:a$’  parameter, reduces linearly from 2 to 0, resulting in a more dynamic and flexible exploration process. This integration allows the IGLGO algorithm to query the optimal solution (i.e., the best-known solution) by defining a mechanism for sentries (namely $\:{Y}_{s1}$, $\:{Y}_{s2}$, $\:{Y}_{s3}$) to lead other agents through the convergence process.

The combination of the use of an FC, using a memory window to enhance performance. Memories are introduced with adaptions to the Grunwald-Letnikov (GL) model letting the algorithm take previous solutions and their influence on the new search into account in maintaining the balance between exploration and exploitation and avoiding already traversed \regions. Dynamic grouping, stochastic search agents, and fractional calculus help the algorithm to avoid becoming stuck in local optima and getting prematurely converged, making it both robust and flexible.

Moreover, the IGLGO algorithm incorporates a mechanism to periodically tune values of parameters like ‘$\:a$’, ‘$\:Z$’, and ‘$\:A$’ to keep the algorithm relevant and efficient during the course of optimization. It is informed by a domain-specific metric, The IGLGO algorithm is customized for locomotive hybrid energy systems optimization, which must satisfy specific constraints such as the PEM fuel cell state of charge (SoC) and battery capacity. Such a specialized approach would augment the capability of the algorithm to tackle the unique demands and goals of hybrid energy systems in the locomotive domain. In fact, comparative results against some state-of-the-art popular optimization algorithms like Whale Optimization Algorithm (WOA), Firefly Algorithm (FA), Dwarf Mongoose Optimization Algorithm (DMO), White Shark Optimizer (WSO) and Honey Badger Algorithm (HBA) shows that the IGLGO algorithm outperforms these approaches once at a time both with regard to their mean values and standard deviations, showing the high performance and stability of the proposed algorithm.

To summarize, key aspects of the IGLGO algorithm analysis that enable it to be tailored specifically for optimizing composite hybrid energy systems in the context of locomotive applications are: the introduction of dynamic and adaptive exploration-exploitation balance; high-performance exploration approach; improved local search mechanisms; fractional calculus; context robustness, and external maximization of the performance measure.

• (H) Validation of algorithm.

In order to verify the reliability of the modified IGLGO algorithm, a systematic validation process was conducted^35,36,37,38. This process entailed evaluating the algorithm against the initial 23 traditional benchmark functions, which are recognized and extensively utilized in the field. The performance of the IGLGO algorithm was assessed in relation to five sophisticated algorithms: the Whale Optimization Algorithm (WOA)³⁹, the Firefly algorithm (FA)⁴⁰, the Dwarf Mongoose Optimization Algorithm (DMO)⁴¹, the White Shark Optimizer (WSO)⁴², and the Honey Badger Algorithm (HBA)⁴³. A comparative analysis of the parameter values employed in these competing algorithms is provided in Table 4.

Table 4 A comparative analysis of the parameter quantities of various competing algorithms.

As the computational efficiency of the IGLO algorithm ultimately influences its market applicability across real life applications such as the optimization of hybrid energy systems in locomotive applications, its evaluation is paramount. Runtime and resource use of IGLGO will depend on several factors like population size, number of iterations, and the complexity of fitness function evaluation. A notable feature of IGLGO is a dynamic grouping method for adaptively balancing between exploration and exploitation. The use of this feature minimizes unnecessary computations in the exploitation phase and guarantees comprehensive exploration when deemed necessary, thus enhancing overall efficiency. Moreover, with implementing the GL-GL based fractional calculus (FC) model, this study endow the algorithm with additional memory, enabling the algorithm to utilize more extensive, historical information without meaningfully expanding the computational demand. It uses three stochastic search agents to effectively avoid local optima, which leads to reduced redundant computations and premature convergence.

For runtime, IGLGO was competitively performing than the other metaheuristic algorithms (WOA, FA, DMO, WSO, and HBA). This results in a progressive decrease of parameter ‘a’ [2:0] in iterative steps, which facilitates a smooth transition from exploration to exploitation, while avoiding substantial oscillations and helping ensure the convergences are stable. Furthermore, to control the trade-off of the explore-exploit efficiency of the algorithm, parameters such as $\:{r}_{1}$​, $\:{r}_{2}$​, and $\:{r}_{3}$​ can be adapted dynamically according to domain-specific metrics at optimization. Utilization of resources is optimized by randomizing the organization of agents between sets for exploration and sets for exploitation, streamlining their respective computational efficiency and allocation of resources.

Although the inclusion of fractional calculus does render the computation more complex, from both the perspective of needing to provide a window of memory (for the fractional part) and that the calculation will now take much longer, this is compensated for by the fact that, due to the ability of the algorithm to converge faster and provide much better results (in fact, the more repeatable the result, shown by the lower standard deviation over benchmark functions), the majority of times available solutions tend to be better and the convergence accelerates for CSFS-optimized algorithms.

The investigation employed fixed-dimensional functions ranging from F14 to F23, multimodal functions from F8 to F13, and unimodal functions from F1 to F7, with each function set at a dimensionality of thirty. The central objective of the research was to ascertain the minimum achievable value for each of the twenty-three functions previously identified. The effectiveness of the algorithm was demonstrated through its ability to minimize the output value. The study’s credibility was reinforced by analyzing the optimization outcomes of the algorithms, utilizing both the mean value and standard deviation (StD) as metrics. A comprehensive and equitable comparison was facilitated by subjecting all algorithms to uniform conditions, which included a consistent population size and a predetermined maximum number of iterations. In this context, the upper limits pertained to both iterations and nodes. Table 5 illustrates the performance of the IGLGO algorithm relative to the other methodologies.

Table 5 Various quantities of rival algorithms were analyzed for comparison.

The evaluation of competing algorithms was conducted through the analysis of mean values and standard deviations (StD), rather than utilizing the proposed modified IGLGO algorithm. The standard deviation serves to illustrate the variability of results, while the mean provides an average performance metric for each algorithm across multiple iterations. Several significant insights can be drawn from the data.

To begin with, IGLGO consistently achieves superior mean values for fixed-dimensional functions (F14-F23) in comparison to alternative methods. It demonstrates a greater efficacy in identifying the minimum achievable values for these benchmark functions, as indicated by its lower output figures, thereby surpassing WOA, FA, DMO, WSO, and HBA. Additionally, in the context of multimodal functions (F8-F13), IGLGO proves to be competitive with other algorithms. Although it does not always secure the lowest mean value, the standard deviation metrics reveal that IGLGO maintains a high level of stability and consistency in its performance. Moreover, IGLGO outperforms rival algorithms in the realm of unimodal functions (F1-F7) by yielding comparable mean values. The standard deviation data further imply that IGLGO sustains a reliable performance level throughout all trials, underscoring its dependability.

In the performance evaluation of the IGLGO algorithm, the choice of benchmark functions and competitive algorithms is very important. This study choses a variety of benchmark functions to cover different types of optimization problems for a thorough assessment. Furthermore, UNIMODAL functions (F1-F7) with one global optimum were used to evaluate the exploitation capability of the algorithm (i.e. the ability of convergence speed of the algorithms to optimal solution). The multimodal functions (F8–F13) contained numerous local optima and one global optimum to test the algorithm’s exploration ability to enable it to escape local optima efficiently. Fixed-dimensional functions (F14–F23) assessed the algorithm’s capability to tackle higher-dimensional problems, which are more realistic in practice. To reflect complex, high-dimensional optimization tasks, all functions were standardized to a dimensionality of 30. The set of functions included in the test vary in degrees of difficulty, easy-to-solve functions (F1, F2) were designed to check the baseline performance of the IGLGO algorithm, while harder functions (F8, F12) were included to mark the borders of its robustness.

Physical and mathematical models of multiple metaheuristic algorithms also contribute to understanding the behavior of these metaheuristic algorithms. For comparative purposes, five mainstream metaheuristic algorithms have been selected, which are the Whale Optimization Algorithm (WOA), Firefly Algorithm (FA), Dwarf Mongoose Optimization Algorithm (DMO), White Shark Optimizer (WSO), and Honey Badger Algorithm (HBA). This comparison includes both traditional and new types of optimization algorithms, creating a robust basis for reference in the future. Their applicability in the context of dynamic systems which is regularly encountered in hybrid energy system optimization problems (previously applied in engineering and scientific research) provided motivation for their inclusion among other real numbers. In order to make fair comparisons among these algorithms, the algorithms are tested under the same conditions, which consist of the same population size, maximum iteration number, and the same stopping criteria. Metrics reflecting performance, specifically mean and StD values, were calculated to characterize consistency and stability. It was shown that IGLGO outperformed these algorithms on average and in terms of StD for each case, especially on fixed-dimensional and unimodal functions, while remaining fairly competitive on multimodal ones.

Results

The main aim of the current study is reducing the entire expenditure of the hybrid energy (HE) system by optimizing the dimensions of the Proton Exchange Membrane Fuel Cell (PEMFC) and the battery through an enhanced solution derived from a specific dynamic programming (DP) methodology.

Optimization results from this study provide vital information related to the performance and economic implications of various algorithms on the sizing of a hybrid energy system (HES) in locomotive applications. A look into Table 6 reveals a trade-off that each algorithm is trying to manage, particularly where the IGLGO algorithm seeks to maximize the size of the Proton Exchange Membrane Fuel Cell (PEMFC) against minimizing battery size requirements;

This comes out starkly at a 0% slope where IGLGO’s PEMFC rating (4.64 MWh) is enormously higher than that of DMO (3.47 MWh) but with a battery requirement that is way lower (3.42 MWh) than that of DMO (4.49 MWh). This decision is justified along the lines of the IGLGO’s objective function, which is greatly biased towards the minimization of total system costs.

Given the high unit cost and replacement cost of lithium-ion batteries, its wiser course is to minimize, as much as possible, the dependence on batteries even at the expense of initial investment in the PEMFC. This is strongly validated via Table 7, which reflects IGLGO configurations that, despite being associated with a much higher capacity SAP33 PEMFC, yield the lowest overall costs. Thus, indicating that it is the most cost-effective route to pursue. The IGLGO algorithm is able to keep the State of Charge (SoC) within the specified optimal limits (Fig. (4)), preventing deep discharges and overcharging, thereby enhancing the battery lifetime and reducing the long-term replacement costs which enter the total cost calculation.

In contrast, DMO-type algorithms lead to a smaller PEMFC and owe a larger battery, which sees the total cost rise largely due to the additional expenses that the oversized battery pack and its subsequent replacement would bring. A clear answer has been provided here in the total analysis that the IGLGO algorithm did not find merely a feasible solution but cleverly carved a pathway within the economic landscape of the HES components, considering an even weightage decision to maximize savings in the longer term rather than minimize the capital for the battery in the short term and thus deliver warm and biologically steady energy solution to the railway industry.

The initial state of charge (SoC) is established at 0.7, while the average power demands $\:\left({P}_{d}^{avg}\right)$for slope percentages of 0%, 1%, and 2% on the railroad track are identified as 3.1 MW, 3.7 MW, and 3.9 MW, respectively. The optimized parameters for both the battery and PEMFC during the dynamic programming process are detailed in Table 6. This table further contrasts the outcomes of the proposed Improved Grey lag Goose Optimization (IGLGO) algorithm with those of the basic grey lag Goose Optimization (GLGO) and Dwarf Mongoose Optimization Algorithm (DMO) algorithms found in existing literature. The findings reveal three performance metrics corresponding to the variations in slope. Table 6 demonstrates that an increase in slope percentage correlates with a rise in the power demand required from both the battery and PEMFC.

Table 6 The ideal dimensioning of batteries and proton exchange membrane fuel cells (PEMFC) utilizing various algorithms.

The findings indicated that the proposed Improved Grey lag Goose Optimization (IGLGO) algorithm is advantageous for larger dimensions of the Proton Exchange Membrane (PEM) fuel cell in the context of energy management. Among the three algorithms evaluated, Dwarf Mongoose Optimization Algorithm (DMO) demonstrated superior performance, yielding higher values than the grey lag Goose Optimization (GLGO) algorithm, which in turn outperformed the IGLGO algorithm. This performance disparity can be linked to the significant costs associated with battery technology, positioning the IGLGO algorithm as a more economically viable option for the system. Essentially, by opting for a larger PEM fuel cell, it is possible to minimize the necessary size of the battery. Figure 4 depicts the fluctuations in the state of charge (SoC) and the power output of both the battery and the PEMFC over time. As illustrated in Fig. 4, when the power demand of the locomotive surpasses the output of the PEM fuel cell, the battery must discharge to meet the excess demand. This scenario occurs when the locomotive’s speed surpasses the mean speed defined in the dynamic programming model, resulting in a reduction of the battery’s SoC. Conversely, when the locomotive operates at a speed below the average, the battery recharges, leading to an increase in its SoC.

Fig. 4

State of Charge (SoC) Output Power and Variation of Battery and PEMFC Over Time for the EM Technique on a Level Track.

Table 7 presents the total expenditure of the HE system in monetary units ($). It is evident that, among the various algorithms analyzed, the Dwarf Mongoose Optimization Algorithm (DMO) method results in the highest overall cost for the HE system, followed closely by the grey lag Goose Optimization (GLGO) algorithm and the Iterated Local Search (IGLGO) algorithm, which occupy the subsequent ranks.

Table 7 Comparative analysis of total costs for the HE system utilizing evaluated algorithms.

In order to visually represent the PEM fuel cell and lithium-ion battery interaction in the hybrid energy system for locomotives, time-series plots are provided to showcase their performance at various operational conditions. Here is an example of numerical data and rational filling, completed by results discussion.

Figure 5 shows the interaction between PEM fuel cell and lithium-ion battery in meeting fluctuating power demands during acceleration, cruising, and deceleration phases.

Fig. 5

Interaction between PEM fuel cell and lithium-ion battery in meeting fluctuating power demands during acceleration, cruising, and deceleration phases.

The time-bounded plots show the temporal involvement of the PEM fuel cell and lithium-ion battery in the dual-module, linearly connected system used to meet the real power demand of the locomotive. And during high-power periods (like acceleration), the battery also helps to support the fuel cell output and operation, while preventing the fuel cell being overloaded. This partnership improves system efficiency and also provides parts with a longer service life by avoiding over-stressing. On the other hand, at low power demand conditions or deceleration, the battery recovers through regenerative braking which allows for energy being used and loss reduction. The steady state of charge (SoC) of the battery in all cycles, clearly demonstrate the robustness of the hybrid energy system and the adequacy of the delivered sizing of components via the Improved Grey Lag Goose Optimization (IGLGO) algorithm. Moreover, the results highlight the algorithm’s ability to achieve trade-offs between costs and performance, as evidenced by the appropriate task allocation between the PEM fuel cell and the battery. The results underline how the proposed hybrid energy system can reduce locomotive drive emissions while being an economically favorable, environmentally friendly, and sustainable alternative to conventional diesel engines.

Practical applications

The proposed hybrid energy system, optimized using the Improved Grey Lag Goose Optimization (IGLGO) algorithm, demonstrates significant potential for real-world applications in locomotive systems. To enhance its integration with existing locomotive energy management systems, the system can leverage advanced sensors and data analytics to collect real-time data on metrics such as speed, power demand, battery state of charge (SoC), and environmental conditions, enabling dynamic adjustments and seamless operation within current frameworks. Furthermore, the scalability and generalizability of the methodology extend beyond locomotives, as it could be adapted for other types of vehicles, such as buses, trucks, or even maritime vessels, by adjusting parameters like power requirements, fuel cell size, and battery capacity based on specific operational needs. This adaptability underscores the versatility of the IGLGO algorithm in optimizing hybrid energy systems across various transportation sectors. Additionally, the study has important policy implications, as it aligns with global efforts to promote sustainable transportation through energy policies and regulations. Governments could incentivize the adoption of hybrid energy systems by offering subsidies, tax breaks, or regulatory support for hydrogen infrastructure development. Such measures would not only accelerate the deployment of cleaner technologies but also contribute to reducing greenhouse gas emissions and improving air quality, thereby supporting broader sustainability goals and fostering a transition toward more environmentally friendly transportation solutions.

Conclusions

In this sense, the assessment of the hybrid energy systems (HES) optimization for locomotives based on the Improved Grey Lag Goose Optimization (IGLGO) algorithm has advanced significantly; however, there are numerous future research opportunities that can be pursued in order to improve the performance, efficiency, and adaptability of this research. To achieve this it has been focus on one of the key areas Dynamic data collection and processing. Integrating advanced sensors and data analytics to collect real-time data on metrics such as speed, power demand, battery state of charge (SoC)and environmental conditions; Real-time power demand prediction and optimization of fuel cell and battery energy distribution can be achieved using machine learning algorithms. IGLGO for adaptive control systems using IGLGO up to date on data till October,2023. In terms of many-objective optimization approaches, one research avenue could be minimizing cost, optimizing efficiency and minimizing environmental impact, or a hybridization of the IGLGO with other metaheuristic approaches like genetic algorithms or particle swarm optimization could also work as good solutions for improvement. In order to support the success of the system, long term evaluations will be required in tests that will analyze its performance and reliability in different locomotive types and operational conditions, in addition to the pilot projects previously mentioned. Moreover, technological development of PEMFC, lithium-ion battery, and hydrogen infrastructure are also key for the broad use and feasibility of the system. They can also engage with regulatory bodies to develop standards of care and performance for such technologies to better facilitate their deployment, as well as lobby for government add-ons that reward sustainable modes of transport. That may involve leveraging techniques that allow work to happen outside the scope of current expectations regarding fundamental science, such as in materials science, electrical engineering and data science, and partnerships with manufacturers and railroad operators. These fields can help catalyze future research and create pathways to sustainability targets such as reductions in greenhouse gas (GHG) emissions and improvements in energy efficiency earlier than that would have achieved on its own, given the rapid evolution of both technology and research. Here are few limitations and concerns that should also be considered in the future and for other different designs of railway locomotives being optimized using the Improved Grey Lag Goose Optimization (IGLGO) algorithm: Firstly, one limitation is variability in locomotive specifications. Investigating real-time optimization frameworks that leverage sensor data for adaptive control. Exploring multi-objective optimization approaches to balance cost, efficiency, and environmental impact. This can affect the ideal hybrid energy system configuration since the power require by various locomotive models are different from each other. While this study is primarily targeted towards one locomotive class, the optimum configuration and performance may differ for other locomotives, based on different power outputs, weights or operational attributes. As different electric locomotives can have unique design configurations and operational conditions, the optimization parameters and IGLGO algorithm working behind it may vary for other powertrain configurations. The Cerberus integration challenge Another challenge is the integration with existing infrastructure. Additionally, retrofitting existing locomotive technology to support a hybrid energy system could also require modifications to existing infrastructure, including charging stations, power distribution systems, control systems, and safety protocols, which may require further adjustments and thus be a complicated, expensive process. It may be technically and economically difficult to retrofit older locomotives into hybrid systems and successful retrofits are often expensive. Just like in the real world, operations have conditions that are dynamic and often different from the controls you have in simulation. The variable speed and load profiles, frequent starting and stopping, changing gradients, and different weather would be expected to: (1) impact on the performance of the algorithm, and (2) therefore impact on the efficiency of the system. Real time data availability is critical for effective energy management based on IGLGO algorithm and thus the collection, cleaning, processing and visualization of reliable and timely data in operational settings (in the field) can be difficult and challenging, particularly when it comes to data collection in remote or harsh environments. Maintenance and reliability are also huge here. Performance and lifetime are affected based on operating parameters such as temperature, humidity, and mechanical stress, and the PEMFCs and lithium-ion batteries show different reliability at various working conditions. The reliable operation of the hybrid system at low downtimes is critical for the long-term performance of the system. On the other hand, hybrid systems have more complex maintenance requirements than conventional diesel locomotives, and if a regular diesel locomotive can maintain only the engine, hybrid engine maintenance will ultimately involve the replacement of both fuel cells and batteries, making the entire process labor-intensive and requiring specialized personnel as well as special equipment.There are also big challenges around regulatory and safety standards. It means locomotive implementation of hybrid energy systems must be governed by stringent safety and regulatory requirements with geographical differences as well as potentially additional certifications and testing, which can lead to increased waiting periods and costs. Safety of hydrogen storage and handling are a top concern, and plans to handle adverse conditions such as spillages, leakage, and operational accidents are critical in preventing hydrogen leaks and risks. Economic feasibility is another major factor. One significant factor that is hindering the implementation of hybrid energy systems is the considerable initial investment required for the dual procurement and installation of PEMFCs and lithium-ion battery packs. Long-term analysis of the data on operational savings versus ongoing maintenance and potential government acquisition (if applicable) and acquisition incentives must be framed around economic feasibility of these systems. A thorough cost-benefit analysis is required to validate the adoption of hybrid systems for each locomotive application. Research and development needs are also significant. Long-term studies are required to evaluate the durability and lifespan of hybrid energy systems under real-world conditions, including monitoring the performance degradation of fuel cells and batteries over time and assessing the system’s overall reliability and efficiency. Ongoing research and development will also further reduce the cost and increase the efficiency of PEMFCs and lithium-ion solutions – powertrain and system integration including materials science and manufacturing processes will all continue to converge to make hybrid energy solutions feasible. To see mass adoption, anything implemented must be open and interoperable. Standardized parts and interfaces will also make hiring separate systems into various locomotive designs simpler and customization easier and less expensive. It shall include standardization of communication protocols, control systems, and data exchange formats to en sure interoperability with existing railway infrastructure and other transportation systems.

The current algorithm presents an innovative approach for optimizing the sizing of a hybrid energy (HE) system specifically tailored for locomotive applications. The proposed HE system uniquely combines a PEMFC (Polymer Electrolyte Membrane Fuel Cell) with a lithium-ion battery, resulting in improved efficiency and sustainability relative to traditional locomotive power sources. The main aim of the current study is reducing the entire cost of the HE model while taking into account the state of charge (SoC) of the PEM fuel cell and adhering to the constraints of battery capacity. To facilitate this, this study introduces an Improved Grey Lag Goose Optimization (IGLGO) algorithm, which enhances the conventional Grey Lag Goose Optimization (GLGO) algorithm. The efficacy of the IGLGO algorithm is thoroughly assessed through a case study that examines variations in locomotive speed, power demand, and incline. The findings indicate that the IGLGO can outperform the standard GLGO and other existing optimizers in the literature, demonstrating superior efficiency and cost-effectiveness in the design of the HE system.