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Scientific Reports volume 16, Article number: 2655 (2026) Cite this article
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This paper demonstrates the enhanced performance of a novel optimization-based energy management strategy (EMS) for hybrid energy system (fuel cell/photovoltaic/battery/super-capacitor). A thorough comparison between several conventional and optimization-based EMSs is performed to define the robust approach. The energy management approaches are adapted to satisfy a highly volatile load demand by optimal power sharing from hybrid system components. The performance standards considered in this study are hydrogen fuel consumption, system efficiency, and components stress avoidance for increased lifetime. The simulation is performed in two scenarios; the first scenario assumes that photovoltaic power is available and the system is in charge of supplying the aforementioned load, the second scenario assumes the same conditions, but with a loss of photovoltaic power due to complete shading. The proposed EMS is designed based on recently presented metaheuristic algorithms, namely; coot bird optimizer (CBO) and improved grey wolf optimizer (IGWO). These algorithms are chosen upon recommendations for their reliability in handling similar engineering problems and fewer tunable parameters. The results of the proposed CBO-EMS surpassed the other competitors in the scale of hydrogen fuel saving, with values of 13.65 g for the first scenario and 17.51 g for the second scenario. The scale of system efficiency confirms the proposed strategy’s reliability, as it scored 92.65% and 89.65% for the two scenarios, respectively. In addition to the computational speed scale, the CBO-EMS achieved the lowest elapsed time of 0.0162 s, outperforming the other competitors.
Nowadays, the effects of climate change and energy market turmoil make it imperative for humanity to develop reliable and sustainable alternative energy solutions. In fact, the world has made great strides in replacing conventional energy sources with renewable ones, but they are still insufficient. This is because of the heavy reliance on fossil fuels as a major contributor to the current energy portfolio. Over the past few years, ambitious renewable energy technologies such as photovoltaics (PV), wind energy, biomass, hydropower, tidal energy and geothermal energy have achieved good penetration in the energy market¹. In addition to the utilization of green hydrogen produced by renewable sources in industrial, stationary and transportation applications. Focus on green transportation, including electric vehicles (EV) and fuel cell-electric vehicles (FCEV), which represent a key factor in the transition to a clean and sustainable energy future and the effective reduction of harmful emissions^2,3,4. Hydrogen fuel cell (FC) technology represents a ray of hope as pollutant-free electrochemical devices that efficiently generate electricity directly from hydrogen. It is distinguished from competing technologies by being eco-friendly, durable, light weight, no moving parts, silent, and low maintenance. FCs have different categories, the most commonly utilized are: alkaline FC (AFC), proton exchange membrane FC (PEMFC), solid oxide FC (SOFC), molten carbonate FC (MCFC), and phosphoric acid FC (PAFC)⁵. Notably, PEMFC is considered the promising contender among them due to its exceptional features of fast start-up time, higher efficacy, longer life time, compact size, and operation at low pressure and temperature. Those make it a prime candidate for integrated renewable systems and automotive applications⁶. Nothing is perfect; the main issue with FCs is their slow dynamic response to load fluctuations. A lot of attention has been paid by researchers to tackle FCs’ drawbacks and achieve effective integration in stationary energy and transportation applications⁷. Automotive applications typify fickle load conditions that vary between acceleration, braking, and start-stop states. A reliable performance for the fuel cell requires hybridization with such device that can compensate shortage in dynamic response. Configurations of fuel cell/batteries (FC/Bat), fuel cell/super-capacitor (FC/SC), and fuel cell/batteries/super-capacitor (FC/Bat/SC) were proposed to tackle the shortage of fuel cell performance⁸. This configuration also enhances the reliability and dependency of the system under transitional stages of fluctuated load demand. Additionally, they can operate as uninterruptible power supply in strategic applications like nuclear reactors, military command centers, continuous processes control, hospitals, and space-crafts⁹. The super-capacitors excel with high power density, long life time, very fast charging time in comparison with batteries and immediate response to demand Shocks¹⁰. Batteries have such a high energy density that they can meet the demands of load fluctuations at the average power supply^11,12. Significant hybridization of multiple energy sources requires a well-established energy management strategy (EMS) capable of coordinating its operations efficiently. In particular, systems integrating fuel cell devices, EMSs must give priority to the efficient use of hydrogen fuel and permissible limits of operation with regard to durability considerations. In the past few years, several EMS configurations have been proposed in pursuit of optimal operation of hybrid systems in stationary and automotive applications. From the point of view of the administrative level, EMSs can be categorized as: (i) rule-based management strategies (RMS), (ii) optimization-based management strategies (OMS), and learning-based management strategies (LMS)¹³. The first category depends on accumulated engineering experience, like; Deterministic rule-based EMSs and Fuzzy Logic-based EMSs. No prior knowledge about the load profile is required, so it excels in ease of implementation and less computational effort. They are widely utilized in automotive applications like FCEVs. The main disadvantages of RMSs are that they cannot ensure optimum operating conditions in all cases, and it has less adaptability. Recently, Researchers have paid more attention to OMSs. Although OMSs need prior knowledge of the load profile, they have the potential to obtain an optimal solution for such a hybrid system. Their capabilities of getting the hybrid systems optimality make them a benchmarking tool for other EMS types¹⁴. In the literature, several management criteria for multi-source systems, including FC devices, have been presented. The reported performance and results of competing EMSs are reviewed as follows. Wang et al. developed a power distribution RMS that can improve fuel economy and dynamic properties of the hybrid energy system. The suggested system included a PEMFC, a lithium-ion battery and a super-capacitor. The strategy makes regard to the grouped devices’ capabilities and the remaining stored energy¹⁵.
A conventional RMS is introduced in¹⁶ for a parallel hybrid EV. In the absence of a significant method for designing Boolean rules, it lacks an optimal solution under different conditions. Zhiming et al.¹⁷ demonstrated an EMS adapted to a regional integrated energy system. Whereas, a hesitant Fuzzy linguistic term set is utilized to express uncertainties of the evaluation criteria system. The proposed model excels in detecting tolerances between the alternatives, depending on its high level of uniqueness. A power distribution method based on double Q-learning EMS is developed in¹⁸. Since the method depends on the Markov approach for vehicle velocity prediction and the battery state of charge (SOC). A configuration of PV/FC/Bat is developed in¹⁹, in which the three sources generating power are conditioned through DC/DC converters. Uni-directional mode converters are utilized for PV and FC, whereas a bi-directional mode converter is utilized for Bat. The results showed fine performance, but the main issue in this configuration is that any fault in the converter may cause a system failure. Recently, Researchers have paid more attention to OMSs. Although OMSs need prior knowledge of the load profile, they have the potential to obtain an optimal solution for such a hybrid system. Their capabilities of getting the hybrid systems optimality make them a benchmarking tool to other EMS types^20,21. In this context, Bizon et al. proposed a real-time optimization technique (RTO) built on the global extremum seeking (GES) method to manage a stationary hybrid (PV/wind/Hydrogen storage) power system. The EMS main goal is to optimize the consumption of hydrogen fuel regardless of variable load conditions²². Three different energy management strategies (EMS) have been proposed to coordinate the output of the PV/FC/Bat system. Performance analysis was performed according to the battery state of charge (SOC) and the amount of hydrogen stored²³. A fuzzy logic controller (FLC) was adapted to a hybrid electric vehicle (HEV) to enhance its performance and the durability of its components. The integrated FC/Bat states and load demand are utilized by the FLC to derive the desired FC current. The FLC effectively distributed the power demand between the system components FC and Bat²⁴. A hybrid FC/SC electric vehicle (HEV) based on FC as a stand-alone renewable energy system is developed in²⁵. The proposed EMS for this system succeeded in achieving high efficiency level of 96.2% highly accurate DC-bus voltage. PEMFC/SC hybrid electric vehicle system is used with a highly volatile power demand, as presented in²⁶. Based on the passivity, the split power of PEMFC and SC is managed by a proportional integral (PI) controller. Various operating conditions are analyzed through the study. An equivalent consumption minimization strategy (ECMS) is adapted to HEV utilizes FC, SC and Bat. The adapted strategy is employed to increase the system components’ lifetime and decrease fuel consumption²⁷. Sulaiman et al.²⁸ studied various EMSs for hybrid electric vehicles. The study is concerned with factors affecting optimization algorithms’ performance, hydrogen sources, economic and environmental issues. It suggested guidelines for planning active EMSs suitable for FC hybrid vehicle applications. In ref.²⁹ Zhang et al. proposed an equivalent consumption minimization strategy (ECMS) to manage energy flow in a tram application powered by PEMFC, battery and super-capacitor. According to the reported results, the proposed EMS succeeded in improving tram performance and reducing hydrogen consumption. An energy dispatching strategy based on model predictive control (MPC) is proposed in ref.³⁰ to control an off-grid photovoltaic/wind/hydrogen/battery hybrid system, then it is compared to only a single strategy created on basis of state machine control technique, so it needs more investigations to ensure its effectiveness. Fathy et al.³¹ presented a comparative study for a hybrid hydrogen power system targeted for automotive applications. In which a new energy flow strategy is proposed based on meta-heuristic algorithms that can achieve an economic consumption of hydrogen fuel and enhance system’ overall efficiency. On the same side, Fletcher et al.³² presented an energy control strategy for hybrid electric vehicle applications. This strategy takes into consideration the fuel cell degradation factors (such as start-stop operation and cold start) as the core of the optimization target; a stochastic dynamic programming technique is utilized for this purpose. Bizon et al.³³ proposed an energy management algorithm implemented in Real-Time optimization loops for motorizing a hybrid renewable power system incorporated with a hydrogen energy storage system to satisfy a residential load demand of a dynamic profile. In ref.³⁴, an optimization based strategy is developed to manage a hybrid energy sources to minimize hydrogen use while maintaining system performance. Although it achieves high operational efficiency and significant hydrogen savings, the strategy suffers from drawbacks such as high data/computation requirements and real-time challenges. A technical–economic study of using a retired electric vehicle battery in grid-connected hybrid energy systems is presented in³⁵. Despite the reported Cost savings, sustainability, and battery reuse, the system is vulnerable to battery aging and performance uncertainty. The authors of³⁶ evaluated multiple system architectures to identify the most effective combinations of solar, wind, battery, and hydrogen storage for rural use. The proposed scheme optimizes the system design, improves feasibility and reliability, but lacks real-world validation. In³⁷, a study on modeling fuel cell devices using metaheuristic optimization algorithms is proposed. The study outputs are excellent model fit and rigorous validation. A metaheuristic based on the echolocation behavior of bats integrated in a hybrid form is presented in³⁸ to better navigate the complex solution space of timetable optimization. The study presented an efficient solution for complex, multi-track scenarios; demonstrated superior performance compared to state of the art method. It lacks real-world deployment or field validation; specifics on scaling or runtime are not detailed. Reference³⁹ focuses on transmission congestion management through real power rescheduling using the metaheuristic moth flame optimization algorithm. The hybrid Moth Flame Optimizer demonstrates high-quality solutions for complex scheduling problems. Details on the scalability, parameter tuning, or runtime performance under large network scenarios are not elaborated. In the same context, a bio-inspired metaheuristic algorithm called elephant herding optimization is deployed to real power rescheduling of generators⁴⁰. The study demonstrated the improved load flow performance through optimal generator rescheduling and mitigating the transmission congestion. An evolutionary optimizer called gravitational search algorithm is used to optimize economic power wheeling charges while ensuring fair allocation of transmission costs⁴¹. The study demonstrated the effect of efficient optimization in reducing congestion costs, improving electricity market efficiency and good convergence properties. The pelican optimization algorithm, a nature-inspired metaheuristic, is applied to solve the optimal power flow problem, which minimizes the cost of generation while satisfying load demand and system constraints⁴². The above-mentioned literature energy management studies deal with normal and low fluctuating load demands with definite available power, regardless of computational burden. In addition, there is a lack of a multi-objective energy management system that provides a comprehensive overview of the system, load, renewable energy availability conditions, and component constraints. This was a motive to conduct this study under the fickle nature of load and the uncertainty of available power from renewable PV source. This work presents a new effective metaheuristic optimization-based EMS for hybrid energy system. The proposed strategy has a holistic overview of the system components for minimizing hydrogen fuel minimized in comparison to other competitive schemes.
The paper structure is detailed as follows: Section II describes the configuration of the hybrid energy system. In Section III an explanation of the comparative energy management schemes under study is presented, whereas Section IV illustrates the development of the proposed optimization-based EMS. Section V demonstrates the results and discussions, and finally, the conclusions and main findings are presented Section VI.
The main contributions of this work are summarized as follows:
An efficient optimization-based EMS is proposed for a hybrid (PEMFC/PV/Bat/SC) energy system under the condition of fluctuated load.
It is the first time to adapt metaheuristic CBO and IGWO algorithms to this energy management application.
The impact of hydrogen fuel consumption, overall efficiency, devices’ stress avoidance and computational burden are considered the key factors of assessment.
The proposed technique performance is compared to conventional reputable strategies.
The reliability of the developed CBO-EMS is verified according to its best performance indicators compared to other competitors under different scenarios.
The multi-source system under study consists of PEM fuel cell, PV array, battery, and super capacitor. This hybridization aims to tackle the drawbacks of each source. The scheme of the hybrid system is depicted in Fig. 1. In which the available renewable power generated by the PV array plays the main role of satisfying the load demand. The load demand is compared with the available PV power to calculate the net capacity. In case of the existence of surplus PV power, it will be used for charging the Bat/SC or operating the polymer electrolyte membrane electrolyzer (PEMEZ) to generate the hydrogen fuel. The hydrogen fuel is of great importance to operate the FC during times of renewable power insufficiency or to be used to fuel automotive applications. The Bat is utilized to share the loading power with FC for the low-frequency demand. The SC is utilized to satisfy the transient load demands or sudden load changes. In the following subsections, a brief description and modeling background of the system components are presented.
The hybrid (FC/PV/Bat/SC) energy system scheme.
FC is an electrochemical device that efficiently generates electrical energy through internal chemical reactions. The reactions of hydrogen and oxygen concurrent inside the membrane sandwiched by anode and cathode electrodes. The reactions start at the catalyst coated anode by splitting the hydrogen to protons and electrons. The electrons flow to the external load circuit, generating voltage difference between the cell terminals. The protons migrate to the cathode through the membrane to unite with the oxygen forming water. The reaction product is electric energy, and the byproduct is water vapor. Especially, the polymer electrolyte membrane (PEM) fuel cell type has several advantages, like; fast starting up due to low operating temperature, longer life time, clean (no exhaust just water vapor), silent, high energy density and compact size³¹. These capabilities make it a potential contender for hybrid energy systems²¹. The FC terminal voltage can be expressed as follows^46,48,49.
where ({N}_{Cells}) is the number of series cells of PEMFC stack, ({E}_{rev}) is the reversible or Nernst voltage which expresses the max terminal voltage given by the cell expressed by Eq. (2). The last three parts of Eq. (1) are the main potential loss of the cell defined as; ({V}_{Act}) is the activation loss to initiate the reaction, ({V}_{Ohmic}) is the ohmic loss, and ({V}_{Conc}) is the concentration loss occurs due to protons mass transfer through the membrane. The mathematical expressions of potential loss are described as follows:
where (T) is the FC operating temperature, ({p}_{{H}_{2}}) is the hydrogen partial pressure, and ({p}_{{O}_{2}}) is the oxygen partial pressure.
where ({xi }_{1}), ({xi }_{2}), ({xi }_{3}), and ({xi }_{4}) represent parametric coefficients of the cell, ({I}_{FC}) is the cell operating current in (A), and ({C}_{{O}_{2}}) is the concentration of oxygen in (mol/{cm}^{3}) at the electrodes defined by:
The Ohmic loss ({V}_{Ohmic}) represents two parts of voltage drop one due to membrane resistance taking the symbol ({R}_{m}) and the other is the contact resistances at the interface of the electrodes and the membrane taking the symbol ({R}_{c}).
where (B) is a parametric coefficient in (Volts), (i) is the current density driven from the cell and ({i}_{L}) is The maximum current density in ({(text{mA}/text{cm}}^{2})).
The PEMFC utilization or consumption rate of hydrogen fuel in (mol/s) is given by:
where (F) is Faraday constant (96,487 C/mol).
A PV system is one or a group of solar cells connected in series and parallel configuration, which directly converts solar radiation into electric power. PV cells have two common manufacturing technologies: mono-crystalline and poly-crystalline⁵⁰. The commonly used single diode model presented by Park et al.⁵¹ is utilized in this study. The model represents the PV cell physics. Considering the equivalent circuit shown in Fig. 2 to represent the performance of PV cell consisting of a current source, a diode, a shunt resistor ({R}_{Sh}), and a series resistor ({R}_{S}), which is solar radiation and temperature dependent^23,52. Since the voltage and current generated by the photovoltaic cell are proportional to the solar radiation, according to the diode model, the voltage and current drop to zero in the dark. As the output of PV array depends on the changing weather conditions, such a technique is needed to overcome this problem by searching and adjusting the operating point at maximum power yield. The maximum power point tracking (MPPT) technique is commonly utilized for maximizing the energy yield from PV array³⁰. Many MPPT approaches were presented in the literature, such as perturbation and observation, incremental conductance, fuzzy logic control, and ripple correlation. The reported results of MPPT-based fuzzy logic control make it the best candidate for fluctuating weather conditions⁵². The PV cell behavior can be calculated as follows:
where ({I}_{L}) is the cell light-generated current by the photo-effect, ({I}_{SH}) is the shunt leakage current and ({I}_{D}) is the diode current given by:
where ({V}_{PV}) is the PV cell output current, ({I}_{0}) is the diode saturation current, q is the electron charge, (K) is the Boltzmann constant (1.3806*10^–23 J/K), ({T}_{PV}) the PV cell temperature in Kelvin, ({A}_{PV}) is the ideality factor of the PV cells and the shunt leakage current is given by:
The Equivalent circuit of a single PV cell.
According to comparisons among different battery technologies such as Lead acid, Metal Hydride (NiMH), Nickle cadmium (NiCd), and Lithium Ion (Li-ion). The last one is considered the good candidate for hybrid renewable technologies due its high energy density capabilities and reliability⁵³. The Li-ion battery can be mathematically modeled taking in account the SOC of the battery, the voltage polarization, and the filtered battery current is used instead of the actual current for purpose of simulation stability enhancement. The Li-ion model is shown in Fig. 3. and the battery voltage is given by the following equations⁷.
where; (E_{0}) is the battery constant voltage in volts, ({K}_{p}) is the polarization constant (V/Ah), Q is the battery capacity (Ah), (it) is the actual battery charge (Ah), ({R}_{b}) is the battery internal resistance in ohms, ({A}_{b}) is the exponential zone amplitude in volts, B is the exponential zone time constant inverse (Ah)^-1, and ({i}^{*}) is the battery filtered current (A). The term ({K}_{p}frac{Q}{Q-it}) represents the polarization resistance, in case of fully charged battery (only at charging stage) the output voltage is suddenly stepped up, so this term is modified to express this phenomena as follows:
The Li-ion battery model.
A fast-response supplementary storage device is needed to compensate for high fluctuations in DC bus voltage due to transient load requirements. The most effective device to accomplish this task is the Super-Capacitor (SC), which has high reliability, long life and very low maintenance. The extremely fast power response and high specific capacity of the SC complement the relatively slower response of other system components. It can reduce peak current in the connected battery on fluctuating load demand⁵⁴.
Considering a SC of a capacitance C in Farads with initial voltage ({V}_{SC,i}) in volts before discharge process and final ({V}_{SC,f}) in volts after discharge process, the net electrical energy ({E}_{SC}) extracted or stored in the super-capacitor in W.s can be calculated as follows:
As well known a SC bank consists of a configuration of series and parallel capacitors called a bank to produce a desired rated terminal voltage and capacity, The SC can be represented by an equivalent electrical circuit as shown in Fig. 4, Where C is the equivalent capacitance of the SC in Farads, according to the SC Stern model that integrates the Helmholtz and Gouy-Chapman models⁵⁵, it can be given by:
where ({N}_{e}) is the number of electrode layers, (varepsilon) is the electrolyte permittivity in (F/m) of the material, (varepsilon_{0}) is the space permittivity in (F/m), ({A}_{i}) is the joint area between electrodes and electrolyte (m²), d is the Helmholtz layer length (or molecular radius) (m), ({Q}_{C}) is the cell electric charge (C), and c is the molar concentration (mol/m³).
SC equivalent electrical circuit.
For a bank super-capacitor the total equivalent capacitance ({C}_{total}) and the total equivalent series resistance ({ESR}_{total}) can be calculated as follows³¹.
where ({n}_{sc,s}) and ({n}_{sc,p}) represents the number of series and parallel super-capacitors in the bank, respectively. ESR is the equivalent series resistance in (ohm) which represents the charging/discharging resistance and EPR is the equivalent parallel resistance in (ohm) which represents the self-discharging losses.
EMS’s primal role is to control and divide the load demand among the hybrid system components PV, FC, Bat, and SC. In a certain way, reliable EMS works to minimize the hydrogen fuel consumption rate by maintaining the SOC of the auxiliary storage (Bat and SC) inside their desired operating limits.
In the following subsections a brief explanation of the comparative energy management schemes is addressed.
SMC is used to avoid FC output power drift during load fluctuations by taking account of the minimum, maximum and optimum to run the FC at high efficacy. It considers three predetermined levels of the battery SOC in addition to the charge/discharge power⁵⁶. SMC inputs are load requirement, and SOC of the battery, FC output power can be calculated as illustrated in Fig. 5³¹.
The SMC strategy.
PID control technique is one of the most familiar management schemes in industrial and power systems fields. Despite that PI control technique still has a similar popularity as it satisfies the performance requirements with less implementation capabilities, which means less cost. Additionally, the D component in PID techniques gives raid actions for system error variations that can cause high oscillations in system response. So PI control technique is mostly preferred for noisy systems to minimize the effects of noise in system response. According to the desired limits of Battery SOC, the PI control technique can optimize its power⁵⁷. In this work, the PI control scheme extracts the Bat full power when its SOC is higher than the mean value (SOC*) and the FC output power is low. On the other hand, the FC produces its full load in case of the SOC of the Bat is lower than its mean value. The transfer function of the PI control scheme is described as follows:
where ({K}_{pc}) and ({K}_{ic}) are the controller gain constants, their values are extracted from ref.⁵⁴. The generated current of the FC I_FC is computed depending on its power, obtained from the controller, and the FC output voltage V_FC. The PI control strategy is illustrated in Fig. 6.
The PI control strategy.
The ECMS is a well-known real time cost function aims at reducing the fuel consumption in the FC. In this strategy, a variable equivalence factor that is based on the battery’s SOC is used. It works to maintain interconnected battery and/or super-capacitor SOC within limits. Moreover, considering the equivalence factor as a part of the objective function, which is required to be optimized, makes the ECMS less sensible to the SOC balance coefficient⁷. In this case, the objective function of the optimization problem can be formulated as follows.
Optimal solution values are given by Eq. (18):
where ({P}_{FC}, {alpha }_{p},) and ({P}_{Bat}) are fuel cell output power, penalty coefficient, and battery output power, respectively.
Optimization goal is to minimize:
Constrained by:
And their conditional boundaries are:
where (Delta T) is the sampling time and (mu) is the SOC balance coefficient of the battery. ({P}_{Bat}) and ({P}_{FC}) are the battery and FC powers, respectively, while P_load is the load demand. ({P}_{FC_min}) and ({P}_{FC_max}) are the operating boundaries of the FC. ({P}_{Bat_min}) and ({P}_{Bat_max}) are the operating boundaries of the Bat.
In the ECMS, the DC bus voltage is controlled via the Bat converters. Therefore, in the optimization process, SC power is not taken into consideration. Once, the SC is discharged, it is recharged via the Bat. Accordingly, in each load cycle, only the FC and the battery can handle the total energy of the load. The ECMS Scheme is depicted in Fig. 7.
The ECMS strategy.
FDFLCS forces the FC to provide a nearly constant power, the other energy sources deal with the high-frequency demands. The essential advantage of FDFLCS is the fact that the average energy of the battery closes to zero, as observed, this scheme provides the lowest use of the battery energy (SOC between 70 and 59%). This guarantees a narrow scope of the battery SOC to avoid stress and increase its life time. However, FLC is needed for controlling the battery SOC nearby the smallest limit. Figure 8 shows the scheme of FDFLCS. Considering a low pass filter is employed for frequency decoupling. The fuzzy logic controller rules are tabulated in Table 1⁵⁸.
The Scheme of FDFLCS.
The cost function commonly used in many real-time hydrogen consumption optimization strategies including ECMS consists of two parts, the first part is the hydrogen consumed by the FC and the second is the equivalent hydrogen consumption of the connected battery/super-capacitor. Equivalent hydrogen fuel is consumed to keep the state of charge (SOC) of the Bat/SC within limits, depending on the load profile, which is challenging to determine precisely. This may lead to poor strategy performance. Rather than minimizing fuel consumption, which requires the evaluation of the equivalent fuel consumption, EEMS is designed to maximize the battery and super-capacitor energies at any given instant, while keeping the battery SOC and DC bus voltage (or super-capacitor SOC) within their operating limits. The optimization problem is formulated as follows⁵⁴:
Optimal solution values are given by:
where ({P}_{Bat}) is the battery output power and (Delta V) is the super-capacitor charge/discharge voltage.
Optimization goal is to maximize:
Constrained by:
And their conditional boundaries are:
where ({C}_{R}) is the rated value of the SC, (Delta V) is the SC charge/discharge voltage, ({V}_{Bat_n}) is the battery nominal voltage, (Q) is the battery rated capacity ({V}_{DC_min}) is the minimum permissible DC bus voltage and ({V}_{DC_max}) is the maximum permissible DC bus voltage.
As shown in Fig. 9, the outputs of the EEMS algorithm are the battery reference power and the SC charge/discharge voltage. The battery reference power is afterwards removed from the load power to get the FC reference power. The SC charge/discharge voltage is added to the DC bus voltage reference to force the SC system to charge or discharge. Similar to the ECMS, the DC bus voltage is controlled by the battery converters.
The Scheme of (a) EEMS, and (b) the interconnected DC bus voltage regulator.
The metaheuristic algorithms are designed to optimize the outputs of the FC and the battery. The CBO and the IGWO and other metaheuristic algorithms are adapted to the proposed EMS as shown in Fig. 10. The chosen algorithms are good candidates due to their balanced exploration and exploitation capabilities which helps in avoiding local minima entrapment. The chosen techniques showed reliable performance in dealing with similar optimization problems^46,47,48. In the proposed EMS, the metaheuristic optimizers are adopted to maximize the output power of the Bat and SC according to the objective function presented in Eq. (24). The optimization variables are the FC’s output power, P_FC, the battery’s output power, P_Bat, and the battery’s state of charge (SOC). The lower and upper bounds of the variables under consideration are selected as follows:
The proposed metaheuristic algorithms based EMS.
The objective of the proposed EMS is to satisfy the load demand by defining the shared power ratio from FC and Bat at times of renewable source shortages or lose. It considers for overall system efficiency and avoiding stresses on system components. The SC works to compensate load demands at transient periods to enhance system response.
The outputs of the proposed EMS are I_FC* which is fed to the FC’s DC-DC converter reference input to define the maximum value of the FC current, and I_{Bat_conv}* which is fed to the Bat charging/discharging converters to define the maximum currents of the Bat during these two states. The proposed control strategy is implemented in an S-function block named Algorithm. The load profile and the battery’s SOC are the inputs of the strategy while the outputs are the reference fuel cell current and the reference current fed to the battery’s converter. The main objective of the proposed metaheuristic based EMS is to obtain the minimum hydrogen consumption and improve the overall system efficiency. In the following subsection a brief description of mechanisms of the CBO and the IGWO algorithms is demonstrated.
The CBO algorithm is a metaheuristic approach designed to mimic the swarm of water coot birds. It has been presented by Naruei et. al.⁴⁴. The amazing swarming behavior of coots tends to save energy and increase the flock speed of surfing the water in search of food. This includes replacing weak coot leaders by stronger coots to achieve the flock goal. The nature-inspired CBO algorithm mechanism is mathematically expressed as follows. The total flock population (({N}_{Pop})) equals flock leader plus subordinate coots (({N}_{Pop}={text{N}}_{text{leader}}+{text{N}}_{text{coot}})). The replacement behavior of stronger subordinate coots with less performance leaders is expressed by their positions (({Pos}_{text{coot}})) and (({Pos}_{text{leader}})), respectively. The algorithm at start gives initial random position values to coots by⁴⁶:
where ({U}_{text{b}}) and ({L}_{text{b}}) are the problem boundaries. Then, the fitness of every coot can be calculated by:
Since ({F}_{text{obj}}) represents the fitness objective function and i takes values from 1 to ({N}_{coot}). The optimal score and optimal position can be given by:
Similarly, the fitness of coot leader, and both optimal score and position can be calculated by:
At the start the CBO nominates a subordinate for each leader coot randomly, after that their positions are updated with each iteration. And the position boundaries are checked as follows:
where ({rand}_{text{coot}}) and ({rand}_{text{leader}}) are random operators. In order to replace weaker leader by a stronger subordinate coot the fitness of each coot is calculated by:
The leaders’ positions are randomly upgraded by:
where the (It) represents to the current iteration and ({It}_{text{max}}) is the maximum iterations limit. The optimal score (({text{Optim}}_{text{score}})) and its corresponding positions (({text{Optim}}_{text{pos}})) are given by:
The flowchart of the CBO algorithm which describes its operation sequence is depicted in Fig. 11.
Flowchart of the CBO algorithm.
The IGWO algorithm basically is designed to mimic the hunting behavior of grey wolf predators. The Pack of wolves consists of three categories namely; alpha, beta, delta and omega. The alpha category is the grey wolf pack leader, which is responsible for attacking decisions. The beta category is the second level of Pack hierarchy, they follow the leader instructions and rule the rest of the pack and ensure obeying the leader. The beta wolves are the future candidate for leader position. At the third level of the pack hierarchy are the delta wolves. Which follow the instructions of the alpha and beta wolves, and take control of the lowest category in the pack, the omega wolves⁴⁸. A new improvement is added depending on dimension learning-based hunting strategy, which can enhance the balance between the exploration and exploitation phases, and avoid the premature convergence of the GWO algorithm. The grey wolves’ movement strategy is enhanced during hunting sessions using dimension learning-based hunting (DLH) search strategy. In DLH strategy each wolf has a neighborhood to support information sharing, this can support the search activity and ensure diversity of population to avoid local optima entrapment. The IGWO algorithm stages are initialization, movement, selection, and upgrade. This stages can be represented mathematically by the following equations⁴⁵.
Initializing the positions of grey wolves of number ({N}_{Pop}) to randomly explore the search space of range [l_i, u_j] is given by:
where D is the dimension number of the problem. At each iteration, the position vector of the i-th wolf, then the corresponding fitness value f (X_i (t)) of each wolf position is calculated.
Movement strategy based on DLH strategy included the addition of individual hunting behavior of wolves besides group hunting behavior. This improves the original algorithm exploration phase by developing two neighbor wolves’ candidate.
Encircling the prey behavior is expressed by:
As (mathop{X}limits^{rightharpoonup}) is the position vector of a grey wolf, (It) is the current iteration index, (mathop{A}limits^{rightharpoonup}) and (mathop{C}limits^{rightharpoonup}) are coefficient vectors, (mathop{X}limits^{rightharpoonup} _{p}) is the prey position vector.
The vectors (mathop{A}limits^{rightharpoonup}) and (mathop{C}limits^{rightharpoonup}) are evaluated as follows:
The random vector (mathop{A}limits^{rightharpoonup}) takes the values in the range of [-2a, 2a] representing the wolves attack towards the prey. Where (mathop{r}limits^{rightharpoonup} _{1}), (mathop{r}limits^{rightharpoonup} _{2}) are random vectors ranged of [0, 1] and (mathop{a}limits^{rightharpoonup}) is a linearly descending vector start from 2 to 0 during the iteration session, which controls the exploration and exploitation till ending the hunting session.
The positions of grey wolves are updated randomly in the search space according to Eqs. (41) and (42) to enhance their positions around the prey.
At such iteration, the alpha grey wolf is considered the best candidate solution, as it has the best fitness. Beta and delta wolves are the following best fitness in order (search agents), so they are qualified to update the pack leader. The least fitness values are considered as omegas. The replacement sequence is repeated at each iteration step till reaching the maximum iteration limit ({It}_{text{max}}). This behavior is represented as follows:
The DLH strategy mechanism depends on the learning from neighboring of each individual wolf position (mathop{X}limits^{rightharpoonup} _{i} left( {It} right)) to support the search space information. The strategy gives new position (mathop{X}limits^{rightharpoonup} _{IGWO} left( {It + 1} right)) for the developed neighbor. This is expressed mathematically using Euclidean distance between and its neighbor wolf X_IGWO (It + 1) to calculate the radius R_i(It) as follows:
The X_i(It) neighborhood are given the symbol ({N}_{i}left(Itright)) and calculated by:
where ({D}_{i}) is Euclidean distance between (mathop{X}limits^{rightharpoonup} _{i} left( {It} right)) and (mathop{X}limits^{rightharpoonup} _{j} left( {It} right)).
For position (mathop{X}limits^{rightharpoonup} _{i} left( {It} right)) neighborhood, there are several neighbor learning wolves are working in the search space. Considering d-th dimension, the corresponding random neighbor (mathop{X}limits^{rightharpoonup} _{n,d} left( {It} right)) (belongs to ({N}_{i}left(Itright))) and Population random wolf position (mathop{X}limits^{rightharpoonup} _{r,d} left( {It} right)) are used to compute the d-th dimension (mathop{X}limits^{rightharpoonup} _{i – DLH,d} left( {It + 1} right)) as follows:
Subsequently, the best candidate is nominated based on the evaluation of fitness values of two wolves (mathop{X}limits^{rightharpoonup} _{IGWO} left( {It + 1} right)) and (mathop{X}limits^{rightharpoonup} _{i – DLH,d} left( {It + 1} right)) according to following equation.
Then the fitness value of (mathop{X}limits^{rightharpoonup} _{i} left( {It + 1} right)) is computed and compared to this of (mathop{X}limits^{rightharpoonup} _{i} left( {It} right)) to decide replacing the old position by new one or keep it. This is done for all individuals and repeated at each iteration till reaching ({It}_{text{max}}).
The flowchart of the IGWO algorithm which describes its operation sequence is shown in Fig. 12.
Flowchart of the IGWO algorithm.
The characteristics of the energy management strategies (SMC, FDFLC, PI strategy, ECMS, EEMS, CBO-EMS, IGWO-EMS) under study are summarized in Table 2. The objectives, advantages, disadvantages, and input/output variables of each technique are included.
The aforementioned system components are modeled and integrated under the environment of MATLAB-simulink software version: R2020a (9.8.0.1323502) 64-bit⁵⁹ running on an Intel® core™ i5-5200U CPU, 2.7 GHz, 6 GB RAM Laptop. The system components (PV array/PEMFC/Bat/SC) and conditioning circuits specifications are depicted in Table 3^7,31,52,54. The study is designed to compare the performance of the proposed metaheuristic based EMS with conventional strategies (PI strategy, SMC, FDFLCS, ECMS, and EEMS) at two scenarios. At the first scenario, The EMSs work to satisfy a highly fluctuated load demand profile based on actual measured data extracted from ref.⁷ as shown in Fig. 13. The duration of the system simulation is 350 s under the condition of the availability of renewable PV power. The second scenario is designed to operate the system for the same load in case of lack of PV power due to complete shading or darkness, which puts the system in a severe evaluation test to satisfy the same fluctuated load.
The fluctuated load demand.
For fair assessment, the sampling time is set at 0.0001 s and the utilized solver is ode23tb (stiff/TR-BDF2) for all the strategies under test. The controlling parameters of the adopted metaheuristic optimizers are set the same as the number of search agents (population) is 20 and the maximum number of iterations is 500. Additionally, the initial values of the system components are set the same for all compared strategies. The hybrid renewable energy system Simulink model is shown in Fig. 14. The proposed metaheuristic based EMS Simulink diagram is depicted in Fig. 15.
The hybrid renewable energy system Simulink model.
The metaheuristic CBO based-EMS Simulink diagram.
In this scenario the hybrid energy system simulation is performed for 350 s with the availability of the renewable PV power. The seven aforementioned EMSs (Classic PI, SMC, FDFLCS, ECMS, EEMS, IGWO-EMS, and CBO-EMS) are investigated. Figure 16 depicts the hydrogen fuel consumption for each strategy over the simulation period. The resulted data of hydrogen fuel consumption and system efficiency are tabulated in Table 4. Hence, the overall system efficiency is computed by dividing the load power over the summation of shared power of each source in the system. According to hydrogen consumption data, the conventional strategies are ranked in descending order as: PI strategy, EEMS, FDFLCS, SMC, and ECMS. The comparison data shows a superiority for metaheuristic based EMSs (CBO-EMS, and IGWO-EMS) over other conventional ones in the scale of hydrogen fuel consumption saving. Their promising results surpassed the nearest conventional competitor (PI strategy) by decrement in hydrogen consumption of 24.7 and 14.2%, respectively.
First scenario; Hydrogen fuel consumption for the competing EMSs under study.
It is notable that the CBO-EMS achieved the best result at the scale of hydrogen fuel consumption of 13.65 g and the worst result is for ECMS of 28.89 g. At the right of the table, the overall system efficiency deeply expresses the goodness of the CBO-EMS with a value of 92.65%. This means a superior management capability for the shared power from each source to meet the variable load demand. The highly fluctuated load demand and shared generated powers managed by the CBO-EMS versus time are depicted in Fig. 17. The simulation starts with zero load demand and the hybrid system operates to charge the energy storing devices (Bat/SC). At time of 40 s the load demand steps up so the fast response device SC begins to compensate this load demands then the bat discharge amount of energy in order to regulate the DC bus voltage till the FC changes its operating point. The strategy gives the responsibility of DC bus regulation to the battery. This sequence is repeated at time of 60 s and the following sudden increase along the simulation period. The SC is recharged at times of low load demands by surplus power from the PV array or the FC.
First scenario; Load demand and shared generated powers by CBO-based EMS versus time.
A close look on Fig. 17, there is a mismatch between the peak of available PV power and the load demand at the period from 170 to 190 s (which represents a common problem of renewable energy systems). It is obvious that proposed EMS succeeded to manage the system by storing the surplus PV power in storage devices (Bat/SC), then it is used later to overcome the shortage at the period from 215 to 242 s and efficiently satisfy the load demand. The fast response of SC and Bat, respectively is clear at times of sudden load changes and epochs, which assure the effectiveness of the proposed CBO-EMS and system configuration in satisfying the load demand with minimum stresses on the FC and the Bat. From Fig. 18, it is remarkable that the more efficient EMSs give the priority to extract more power from the Bat and the SC to meet the load demand. This is obvious from the proposed CBO-EMS Bat SOC (%) curve which has the higher reduction in SOC within limits, that is why the proposed strategy achieved the best hydrogen fuel saving. The figure also clarifies the adherence of the proposed strategy with system components constraints as SOC doesn’t violate the 60% limit, so increased component life time.
The battery SOC versus time for the competing EMSs under study.
Further evaluations are considered, the proposed strategy CBO-EMS is tested for 100 independent runs versus popular state-of-the art competitors namely; multi-trial vector- differential evolution-EMS (MTDE-EMS)⁴⁶, Genetic algorithm-EMS (GA-EMS)³¹, and Particle swarm optimizer-EMS (PSO-EMS)⁴⁶ in addition to SMC, PI strategy, FDFLCS, ECMS and EEMS. The frequent results statistics of strategies under test (efficacy and average execution time) are displayed in Table 5. The repetitive results of each scheme is statistically analyzed in means of average efficacy of the system and its corresponding standard deviation besides the average elapsed time. It is obvious that CBO-EMS and MTDE-EMS has achieved the best results on the scale of efficacy, but the average execution time of MTDE-EMS is disappointing of 3.8 s. This means heavy computational burden and need costive processing hardware and limits the real-time implementation. As one of the main objective for this study is proposing efficient optimization-based EMS with minimal computational burden to suite real-time application. It is clear that CBO-EMS and IGWO-EMS excels with high speed of convergence achieving average time of 0.0162 and 0.0171 s, that qualifies them for more validation in next scenarios. In Fig. 19, A Boxplot describing the efficacy results’ distribution of the state-of-the-art competing optimization-based EMS’s. A closed look at the figure, The MTDE-EMS shows very concentrated results reflecting minimum deviation values. On the other hand, the CBO-EMS surpassed all the competitors with the higher efficacy values, supporting previous findings. Subsequently, the well-known non-parametric Wilcoxon test is used to compare between the medians of resulted efficacy of the 100 runs of CBO-EMS and the closest competitors at the scale of minimum computational burden; IGWO-EMS and GA-EMS. With a significant level of 0.05, Table 6 depicts the outputs of the analysis carried out under the IBM-SPSS statistical tool⁶⁰. The outputs (positive rank, negative rank, and probability values) are shown for the relevant pairwise comparisons of CBO-EMS results, taking into account its superiority over IGWO-EMS and GA-EMS results. The analysis outputs show statistically significant differences between the repeated results of the compared techniques. This indicates the contrasting performance of each. Furthermore, a statistical t-test is conducted to the clarify significant difference between the means of results groups of each strategy. The results of the test are tabulated in Table 7. Considering 95% Confidence Interval of the Difference, the larger absolute t-value indicates a greater difference between the groups or between a sample mean and a hypothesized value, providing stronger evidence against the null hypothesis and assure the Wilcoxon test results of the existence of a statistically significant difference. Additional to the values of the Sig. (2-tailed) parameter is less than the chosen significance level (0.05), so the null hypothesis is rejected and there is a statistically significant difference between the means the grouped results of each technique.
Boxplot describing the results’ distribution of the state-of-the-art competing algorithms.
As the impact of computational speed is a crucial factor for real-time implementation requirements, convergence curves of the competing optimization-based EMSs are depicted in Fig. 20. It is obvious that the CBO-EMS achieved the best convergence speed followed by the IGWO-EMS as shown in Fig. 20a,b. The convergence speed of the remaining optimizers is in doubt as shown in Fig. 20c–e, respectively. It can be concluded that all the above statistics confirm the robustness of the CBO-EMS for motorizing hybrid energy systems.
Convergence curves of the competing optimization-based EMSs.
This scenario is designed to be a verification test for the proposed strategy by assuming unavailability of PV array due to complete shading or darkness. This mimics the conditions of real operation shaping a severe real test the system should tackle. The same load profile in Fig. 13 and conditions of first scenario are applied. The resulted hydrogen fuel consumption profiles of competing EMSs are depicted in Fig. 21 and the extracted data are listed in Table 8. According to hydrogen consumption data, the competing strategies results are ranked starting from the best as follows: CBO-EMS, IGWO-EMS, EEMS, FDFLCS, PI strategy, SMC, and ECMS. The time response of hydrogen fuel consumption supports the validity of metaheuristic based EMSs. The minimum consumption values were for CBO-EMS and IGWO-EMS with values of 17.51, and 17.68, respectively, which surpassed the results of nearest conventional competitor (EEMS) by decrement of 38.5, 37.9%, respectively. At the other side of Table 8 reports percentage efficiency of 89.65% for CBO-EMS and 86.43% for IGWO-EMS surpassing the results of conventional strategies. The higher efficiency gives a special advantage to the proposed CBO-EMS, which means tight management for the shared powers from the hybrid sources with respect to specified constraints. The CBO-EMS succeeded in satisfying the load demand despite the lack of PV power as shown in Fig. 22, which shows higher delivered levels of power from FC and Bat sources in comparison with first scenario to meet the load requirements. This explains the higher hydrogen fuel consumption and the relative decrease in system efficiency. The SC performed very well at transient events at instants of 40, 60, 130, 180, 245, and 260 s as it compensates the shortage of delivered power from the FC and the Bat.
Second scenario; hydrogen fuel consumption for the competing EMSs under study.
Second scenario; load demand and shared generated powers by CBO-based EMS versus time.
It is concluded that the proposed CBO-EMS has the capability to perform well under the conditions of uncertainty of renewable PV power. It has succeeded to manage the available power from FC/Bat/SC which represent storage devices to meet the fickle load demands. This expresses its effectiveness and reliability of managing universal hybrid systems such as stationary or automotive applications.
A new metaheuristic optimization-based EMS is proposed to efficiently manage a hybrid renewable energy system. The study included comprehensive comparisons between the proposed CBO-EMS and other conventional and metaheuristic-based strategies, namely: PI strategy, SMC, FDFLCS, ECMS, EEMS, and IGWO-EMS. The simulation is performed for two scenarios, the first considered the availability of solar PV power and the second considered complete PV array shading, which mimics the uncertainty of available renewable PV power. In the first scenario of simulation, the conventional strategies are ranked in descending order as: PI strategy, EEMS, FDFLCS, SMC, and ECMS according to the indicators of hydrogen consumption and efficiency. The metaheuristic optimization-based strategies (CBO-EMS and IGWO-EMS) showed promising performance; their results surpassed the nearest conventional competitor (PI strategy) by a decrement in hydrogen consumption of 24.7% and 14.2%, respectively. The additional statistical analyses confirmed the robustness of the proposed CBO-EMS results in comparison with the state-of-the-art (GA-PSO-MTDE-IGWO) for 100 independent runs. In the second scenario of simulation, the competing strategies’ results are ranked starting from the best in descending order as follows: CBO-EMS, IGWO-EMS, EEMS, FDFLCS, PI strategy, SMC, and ECMS. The proposed CBO-EMS and IGWO-EMS surpassed the results of the nearest conventional competitor by a decrement in hydrogen consumption of 38.5%, 37.9%. Another unique aspect is the computational speed of CBO-EMS, which achieves a minimum elapsed time of 0.0162 s, qualifying it for real-time implementation. From both scenarios results, the proposed CBO-EMS confirmed its superiority over the other competing metaheuristic and conventional strategies. In the scale of hydrogen fuel saving, it recorded the values of 13.65 g for the first scenario and 17.51 g for the second scenario. The other scale of system efficiency supports its reliability, as it scored the highest values of 92.65% and 89.65% for the two scenarios, respectively, outperforming the rest of the competitors. In addition to constrained operation to avoid stress on system components to increase their life-time. All of this confirms the validity and reliability of the proposed CBO-EMS to manage universal hybrid energy systems such as stationary or automotive applications. In the future, this study can be developed to include automotive applications such as the complete driving cycle of the vehicle.
The authors confirm that the data supporting the findings of this study are available within the article.
Alternating current
Alkaline fuel cell
Battery
Coot bird optimizer
Dimension learning-based hunting
Direct current
Equivalent consumption minimization strategy
Electrochemical double layer capacitors
External energy maximization strategy
Energy management strategy
Equivalent parallel resistance of super-capacitor
Equivalent series resistance of super-capacitor
Electric vehicle
Electrolyzer
Fuel cell
Fuel cell electric vehicles
Frequency decoupling- fuzzy logic control strategy
Fuzzy logic controller
Global extremum seeking
Hybrid renewable system
Hybrid electric vehicle
The improved grey wolf optimizer
Learning-based management strategies
Lithium ion
Molten carbonate fuel cell
Metal hydride
Model predictive control
Maximum power point tracking
Nickle cadmium
Optimization-based management strategies
Phosphoric acid fuel cell
Polymer electrolyte membrane
Polymer electrolyte membrane electrolyzer
Polymer electrolyte membrane fuel cell
Proportional integral control strategy
Photovoltaic
Rule-based management strategies
Real time optimization technique
Super-capacitor
State of charge
Solid oxide fuel cell
Statistical package for social science
State machine strategy
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Ministry of Transport, Egyptian National Railways Authority (ENR), Cairo, 11794, Egypt
Mohamed Ahmed Ali
Electrical Power and Machines Department, Faculty of Engineering, Zagazig University, Zagazig, 44519, Egypt
Mohey Eldin Mandour & Mohammed Elsayed Lotfy
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M.A. and M.L. wrote the main manuscript text and M.A. and M.M. prepared figures. All authors reviewed the manuscript.
Correspondence to Mohamed Ahmed Ali or Mohammed Elsayed Lotfy.
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Ali, M.A., Mandour, M.E. & Lotfy, M.E. Efficient coordination of hybrid energy system (fuel cell/photovoltaic/battery/supercapacitor) under the condition of fluctuated load using optimization based energy management strategy. Sci Rep 16, 2655 (2026). https://doi.org/10.1038/s41598-025-27685-4
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Received: 02 April 2025
Accepted: 05 November 2025
Published: 21 January 2026
Version of record: 21 January 2026
DOI: https://doi.org/10.1038/s41598-025-27685-4
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