Article citation information:Elmaihy, A., Abd El-Latif, M. Performance and sizing of PEM fuel cells under variable operating conditions for an autonomous underwater vehicle. Scientific Journal of Silesian University of Technology. Series Transport. 2026, 131, 63-87. ISSN: 0209-3324. DOI: https://doi.org/10.20858/sjsutst.2026.131.4Ali ELMAIHY, Mohamed ABD El-LATIFPERFORMANCE AND SIZING OF PEM FUEL CELLS UNDER VARIABLE OPERATING CONDITIONS FOR AN AUTONOMOUS UNDERWATER VEHICLESummary. Ensuring a reliable power supply is a key challenge for autonomous underwater vehicles (AUVs), especially when deeper diving, longer endurance, and extended mission range are required. Proton exchange membrane (PEM) fuel cells offer a promising solution due to their high-energy density and quiet operation. This study conducted a detailed PEM fuel cell model in MATLAB/Simulink to evaluate how operating conditions affect single-cell electrochemical behavior and stack-level sizing. The model incorporates activation and ohmic losses to quantify variations in cell voltage under different temperatures and reactant supply pressures. Based on these analyses, performance and sizing indices are derived for a practical AUV power system. Results indicate that increasing operating temperature from 55°C to 85°C, together with increasing oxygen and hydrogen supply pressures from 1 to 3 atm, enables a 20% reduction in the required number of cells, directly translating into similar reductions in stack mass and volume. Moreover, reactant consumption decreases by 9.5%, improving overall energy efficiency by 9.6%. These findings demonstrate the strong influence of operating conditions on PEM fuel cell performance and provide guidelines for optimized sizing of AUV fuel cell power systems.Keywords: fuel cell performance, activation over potential, ohmic losses, operating temperature, supply pressures, AUV1.INTRODUCTIONUnderwater vehicles, including miniature submarines and autonomous underwater vehicles (AUVs), are increasingly used in the military and commercial fields. Similar to AUVs, the power system of AUVs has high requirements for stealth (without vibration and noise) and long endurance [1]. Because of their low pollution and energy efficiency, fuel cell vehicles (FCVs) have gained popularity in response to growing worries about global warming, pollution, and energy consumption [2]. PEMFC and solid oxide fuel cells (SOFC) can be utilized for AUVs [3,4]. PEMFC is the recommended option since it performs better and runs at lower temperatures (below 100°C). Many AUVs currently run on batteries [5], but AUV endurance will increase with the adoption of fuel cell systems [6,7]. Because the storage media in fuel cells have a far higher energy density than batteries, they can outperform them in terms of energy density; nonetheless, the issues arise from their weights and system sizes [8]. Recent research indicates that commercial AUVs use batteries to store power. Battery-powered AUVs with greater endurance and range are significantly bigger than regular ones, which reduces their utility and increases their space requirements [5,9]. So, fuel cells are used in the AUV industry. According to [10], the properties of suitable fuel cell systems are:The size of fuel cell systems must be as small and as light as possible.The power output of fuel cell systems must be in the range of 1 kW to 2 kW.Fuel cell systems should be removable, exchangeable, and replaceable.The integration of fuel cell technology into Autonomous Underwater Vehicles for naval applications presents a compelling avenue for enhanced operational capabilities, characterized by extended endurance, reduced acoustic signatures, and improved energy efficiency [11]. However, the realization of these benefits hinges critically on a comprehensive understanding of the intricate relationship between fuel cell operating conditions, system size, and overall performance within the unique constraints imposed by the underwater environment. AUVs represent a paradigm shift in underwater exploration and data acquisition, offering unparalleled maneuverability and the ability to access remote and complex underwater environments without direct human intervention [12]. The evolution of AUV technology has been driven by advancements in control systems, communication protocols, and sensory technologies, enabling increasingly sophisticated missions in challenging underwater settings [13]. The operational demands of AUVs in naval contexts are particularly stringent, necessitating robust performance under varying environmental conditions, including temperature fluctuations, pressure variations, and exposure to corrosive seawater [14].The performance of fuel cells is intricately linked to various operating conditions, encompassing temperature, pressure, humidity, and reactant stoichiometry, all of which exert a significant influence on the electrochemical reaction kinetics, mass transport phenomena, and membrane properties within the fuel cell stack. Hydrogen energy systems, particularly those employing PEM fuel cells, have been reviewed for underwater applications, emphasizing the importance of storage methods and operating parameters in system design [9]. The literature indicates that hydrogen/oxygen storage strategies and fuel cell operating conditions are interconnected factors that determine the overall system size and performance. Proper management of these parameters can lead to more compact and efficient fuel cell systems suitable for AUV deployment [9]. Knyazhev and  HYPERLINK "https://ieeexplore.ieee.org/author/193458371244297"Loshchenkov [15] review various fuel cell types, emphasizing their performance under different operating conditions. It highlights how these conditions affect size and efficiency, making them suitable for naval AUV applications, ensuring extended operational duration and enhanced cruising range. Mendez et al. [16,17] discussed that fuel cell operating conditions significantly influence size and performance, emphasizing the need for compact, efficient systems. It highlights the importance of optimizing reactant storage and system configurations to enhance endurance and efficiency in AUV applications. Pielecha [18] analyzed fuel cell performance under various driving conditions, highlighting that urban, rural, and motorway scenarios significantly affect power output and losses. It emphasizes the importance of optimizing operating conditions to enhance fuel cell efficiency and longevity in applications like AUVs. Tsukioka et al. [19] discussed that fuel cell size and performance in AUVs are influenced by operating conditions such as temperature, pressure, and the number of cells in series, which determine output voltage and efficiency, impacting overall energy generation and operational capability. Challenges associated with hydrogen fuel cells in transportation and underwater environments further underscore the necessity of optimizing operating conditions. These include issues related to durability, efficiency, and system integration, which are directly impacted by temperature, humidity, and load fluctuations [20]. Addressing these challenges through advanced energy management strategies, such as high-fidelity simulation models combined with experimental data, has shown promise in enhancing performance and reducing system size [21].Because PEMFC systems are complex, multimodal, and intricately linked, modeling their properties for design and performance assessment is difficult. Over the years, significant progress has been made in comprehending the behavior of PEMFC features using mathematical modeling [22-34]. Models of integrated cars with fuel cell stacks are necessary to examine the system behavior under varied load, reagent gas pressure, and temperature conditions in order to maximize the utilization of PEMFCs [35]. In the meantime, modeling is becoming more and more significant in design, analysis, and performance evaluation. Qasem and Abdulrahman [36] reported that, despite several modeling approaches used in literature for the prediction of PEMFCs performance, Amphlett et al.'s mathematical modeling [37,38] was recognized for its prediction accuracy under different operating conditions. The majority of models in the literature are supported by experimental work. Evaluating and comparing the polarization curve derived from both mathematical and experimental investigations is the most popular validation method. If the findings of two polarization curves are the same or nearly identical, the model is reliable [39]. The objective of this paper is to use MATLAB Simulink to build a fuel cell model according to [37,38]. Then, select a well-known AUV to ascertain its power needs. The final result was a block diagram that forms the basis for forecasting the fuel cell's performance under various operating circumstances. The second part of the paper investigates the impact of operating conditions on the sizing of the fuel cell system, as well as the consumption rates of hydrogen and oxygen, which play a crucial role in determining the overall endurance of the AUV.2. FUEL CELL MODEL DESCRIPTIONAn electrochemical model that can be used to predict the behavior of PEMFC is presented in this section. The optimal simulation results depend on the definition of a set of parameters used in this mathematical model. The following expression can be used to define the output voltage of a single cell, Vcell [22,37,38].Vcell=ENernst+ηOhmic+ηAct(1)Where ENernst is the Nernst equation, which is an expression for the electromotive force (EMF) for a given product; QUOTE is the activation overvoltage, which is the amount of voltage used to drive the reaction; QUOTE is the ohmic overvoltage, which is the amount of voltage lost to the resistance to electron flow in the electrodes and the resistance to ion flow in the electrolyte; the terms of Eq. 1 are discussed in the following sections.2.1. Nernst VoltageThe Nernst equation for the reaction described above:ENernst=1.229-0.85 x 10-3 x T-298.15+4.3085x10-5 x T x (logPH2*+0.5 x logPO2*) (2)Where T is the stack temperature in (K), and PH2* and PO2* are the hydrogen and oxygen partial gas pressures in (atm) at the surface of the catalyst at the anode and cathode, respectively. At each electrode, two distinct cases need to be taken into account (for example, the first has an inert diluent in the input flows, either carbon dioxide in the anode flow or nitrogen in the cathode flow), which will not be considered here. The flows in the second scenario don't contain any diluent. So PH2* and PO2* will be described as [37].The first case: diluted H2 and O2:PH2*=PAnode×1-0.5×xH2OAnode-ΧCO2channel×e0.183×iT0.832(3)PO2*=PCathode×1-xH2OCathode-ΧN2channel×e0.291×iT0.882 (4)Where i is the current density i=IAcell Am2. PAnode and PCathode are the total humid gas pressures at both the anode and the cathode, and are given by:PAnode=PAnodeDry+0.5×PH2Osat(5)PCathode=PCathodeDry+PH2Osat(6)Where PAnodeDry and PCathodeDry are the total dry gas pressures at both the anode and the cathode. ΧH2Oanode and ΧH2OCathode are the molar fractions of water in a gas stream at anode and cathode, respectively, at saturation condition for a given temperature, and are given by: ΧH2OAnode=0.5×PH2OsatPAnode(7)ΧH2OCathode=PH2OsatPCathode(8)The PH2Osat term is the saturation pressure of water at the operating temperature T. It is determined in a fuel cell by the following empirical equation:PH2Osat=e70.434643-7362.6981T+0.006952085×T-9×logT(9)ΧN2channel and ΧCO2channel are the average molar fractions of nitrogen in the air stream and carbon dioxide in the hydrogen stream, respectively, and are given by:ΧN2Channel=ΧN2,Humidout-ΧN2,HumidinlogΧN2,HumidoutΧN2,Humidin (10)ΧCO2Channel=ΧCO2in-ΧCO2outlogΧCO2inΧCO2out(11)ΧN2,Humidout and ΧN2, Humidin are the mole fractions of inlet and outlet N2 in the mixture of oxygen and nitrogen, saturated with water vapor, given by:ΧN2,Humidout=1-ΧH2OCathodet1+SR1-1SR1×ΧO21-ΧO2(12)ΧN2,Humidin=1-ΧH2OCathodet×ΧN2,Dryin (13)ΧCO2in and ΧCO2out are the mole fractions of inlet and outlet CO2 in the mixture of Hydrogen and carbon dioxide, and given by:ΧCO2in=1-ΧH2×1-ΧH2OAnode(14)ΧCO2out=1-ΧH2OAnode1+SR2-1SR2×ΧH21-ΧH2(15)SR1 and SR2 are the stoichiometric ratios of oxygen and hydrogen, respectively. ΧO2 and ΧH2 are the mole fractions of O2 and H2 on the cathode and anode streams, respectively. The second case: Pure inlet streams of H2 and O2:PH2*=PH2Osat×1exp4.192×iT1.334.xH2OAnode-1(16)PO2*=0.5PH2Osat×1exp1.6532×iT1.334.xH2OCathode-1(17)2.2. Activation overvoltageActivation losses (also called activation overpotentials) arise from the energy barrier that must be overcome for the electrochemical reactions to occur at the electrodes. These losses are more significant at low current densities. The loss can be decreased if steps are taken to increase the reaction rates by increasing the temperature or pressure in the fuel cell, which operates in a temperature range of approximately 55°C to 85°C. The semi-empirical equation for the activation overvoltage is given by [37,38]:ηact=-0.9514+0.00312×T+7.4×10-5×T×logCO2*-0.00187×TlogI(18)Where the concentration of dissolved oxygen at the gas/liquid interface(CO2*) can be defined by a Henry’s law expression of the form:CO2*=PO2*5.08×106×e-489T (19)Where the expression for PO2*is given by Eq. 2.2.3. Ohmic OvervoltageAnother portion of the voltage generated by the electrochemical reaction is lost due to the resistance to electron flow in the electrodes and graphite collector plates, and the resistance to ion flow in the electrolyte. The ohmic losses can be reduced by using thin, well-hydrated membranes with high proton conductivity, improving electrical contact, and using materials with low resistivity, and maintaining proper membrane hydration (dry membranes increase resistance). The ohmic overvoltage can be expressed following Ohm’s law as:ηohmic=-Rinternal×I (20)Where Rinternal is the resistance to electron transfer and is calculated by:Rinternal=0.01605-3.5×10-5×T+8×10-5×I (21)2.4. Fuel Cell Model ImplementationIn this paper, MATLAB/Simulink is used to set up the PEMFC system model. Simulink is a software package for modeling and simulation, which can be used to evaluate the system performance and optimize the system design. (Figure 1) shows the Simulink model of the simulation diagram of the PEMFC dynamic model based on Eqs (1) to (21).3. MODEL VALIDATIONTo validate the model, a single cell, the Ballard Mark IV model, was simulated and fed with gases H2 and O2. For oxygen, the dry-basis mole fraction, ΧO2, settings were 0.21, 0.46, and 1.00; for hydrogen, ΧH2, they were 0.65, 0.81, and 1.00. By keeping the total dry gas pressure at the cathode and anode PAnodeDry and PCathodeDryconstant at 3.06 atm abs, the partial pressure settings were accomplished. The stoichiometries of the input flows were maintained at 1.73 cm3 of fuel (anode gas) for every cm3 of hydrogen consumed and 8.33 cm3 of oxidant (cathode gas) per cm3 of oxygen used throughout all runs. The stoichiometric ratios SR1 and SR2 are calculated for validation by: SR1=8.33×ΧO2(22)SR2=1.73×ΧH2(23)The inlet dry partial pressure:PH2in, Dry=ΧH2PAnodeDry(24)PO2in, Dry=ΧO2PCathodeDry(25)The Mark IV employs a Nafion membrane (0.18 mm in thickness), with an active surface area of 50.56 cm2. Table 1 displays the experimental settings of 28 runs with the corresponding measured parameters and corresponding predicted results. Comparing diagrams of the total output voltage, fuel cell resistance, and activation overvoltage between the model predictions and the experimental data [38] are presented in Figures 2 to 4, respectively. Calculated output voltage response to a range of independent operating current, temperature, and oxygen partial pressure variations, as shown in the experimental settings shown in Table 1.It is demonstrated that the model accurately predicts fuel cell voltage for current densities as high as 0.33 A/cm2 in the temperature range of 55 to 85 degrees Celsius and correlates fuel cell voltage over the experimental range of current densities. In conclusion, the modeling strategy is found to be successful, efficient, and simple to use with a limited quantity of data.2171705270500Fig. 1. The block diagram of the MATLAB/SIMULINK model of PEMFCsTab. 1Experimental settings and comparison between measured values and predicted resultsExperimental settingMeasured ResultsPredicted results% ErrorRun No.T[K]I[Amp]ΧO2ΧH2Vcell[volt]Rint[Ω]Vcell[volt]Rint[Ω]VcellRint13582.720.460.810.910.003880.9160.00374-0.663.6123286.660.210.810.7730.004890.7750.00510-0.26-4.2933436.660.460.810.820.004550.8230.00458-0.37-0.6643586.6610.810.8670.00410.8690.00405-0.231.2253436.66110.8510.004530.8560.00458-0.59-1.1063436.660.210.650.7810.004550.7840.00458-0.38-0.6673286.660.460.650.8020.004890.8000.005100.25-4.2983432.720.210.810.8630.00420.8670.00426-0.46-1.4393436.660.460.810.8240.004460.8230.004580.12-2.69103436.660.460.810.8270.004410.8230.004580.48-3.85113282.720.460.810.8880.004950.8820.004790.683.23123286.6610.810.8350.005050.8330.005100.24-0.99133436.660.460.810.8260.004370.8230.004580.36-4.81143586.660.210.810.8090.004060.8050.004050.490.25153586.660.4610.8460.003930.8440.004050.24-3.05163586.660.460.650.8360.003990.8330.004050.36-1.501734316.330.4610.7150.004770.7130.005350.28-12.161834316.3310.810.730.004930.7360.00535-0.82-8.52193436.660.460.810.8230.004560.8230.004580.00-0.44203436.660.210.810.7920.004410.7900.004580.25-3.852135816.330.460.810.7290.004420.7270.004830.27-9.28223432.720.4610.9010.004280.9040.00426-0.330.47233436.6610.650.8470.004440.8440.004580.35-3.15243432.720.460.650.8980.004280.8930.004260.560.472534316.330.210.810.6620.004770.6760.00535-2.11-12.16263436.660.460.810.8220.004560.8230.00458-0.12-0.44273286.660.4610.8070.005020.8120.00510-0.62-1.592834316.330.460.650.70.00490.7020.00535-0.29-9.1811880854191000Fig. 2. Parity plot of the model prediction and the measured values for the total output voltage109283511779200Fig. 3. Parity plot of the model prediction and the measured values for the FC internal resistance1189430508000Fig. 4. Parity plot of the model prediction and the measured values for the FC activation overpotential4. METHODOLOGYThis section aims to present a procedure for obtaining the most suitable operating conditions for the fuel cell to power an AUV. The inferred requirements of FCs for AUVs are mentioned in [5]. The proposed procedure is summarized as follows:Selection of the AUV.Estimation of the required power for the selected AUV.Prediction of the performance of one cell of Ballard IV FC (cell output voltage and power) at different operating conditions, namely, Cell temperature, H2 supply pressure, and O2 supply pressure at different current densities.Recording the maximum power and the corresponding current density for each operating condition (PO2in , PH2in and T).Calculations of the number of cells needed to produce the AUV's required power at each operating condition. Calculation of the H2 and O2 consumption, Calculation of the FC efficiency.Analysis of the obtained results.Figure 5 presents a flowchart illustrating the previously described procedures. 4.1. AUV Selection and Required Power EstimationThe AUV ECA A27-E [5,40] is a Classical rear propeller with control surface architectures, such as in large conventional submarines equipped with rudders, which take an important place in the AUV market since it has a great performance following straight lines due to its torpedo-shaped hull. The main characteristics of A27-E AUV are shown in Table 2.The viscous resistance coefficient CV, the friction resistance coefficient CF, and the shape factor (1+ k) are used to calculate the power consumption of propulsion engines.CF can be calculated using the International Towing Tank Conference 57 correlation line (ITTC, 1957) [41]:CF=0.075log10Re-22(26)Where Re is the Reynolds Number based on the AUV length and speed:Re=vL μ (27)Where: μ is the kinematic viscosity of seawater [m2/s]; L is the length of ROV [m]; v is the calculated speed of ROV [m/s]. For AUVs with a circular cross-section, Droblenkov's curve can be used to determine the form coefficient:1+k=1+0.2DL+8DL2-10DL3(28)Thus, viscous resistance is calculated by the expression:Rv=12ρAv2Cf(1+K)(29)Where: ρ = 1025 [kg/m3]- density of seawater; A is the wet surface area of the ROV, calculated as a cylinder of diameter D and length L. The required thrust power is calculated by:WAUV=1ηDρACf(1+K)v3(30)Where ηD is the propeller thrust performance.Figure 6 shows the dependence of AUV resistance and required thrust power on the AUV speed. At the maximum AUV speed, the required thrust power is 1523 W.3048001270EndCalculation of the No. of cells (Eq. 31) as shown in Table 5Calculation of H2 and O2 consumptions for the FC (Eq. 32 to 35) as shown in Table 5Calculation of the FC efficiency (Eq. 36) as shown in Table 5Analysis of the obtained results (Sec. 5.2)StartCalculation of power requirements of the selected AUV (Eq. 26 to Eq. 30 Sec. 4.1)Prediction of one cell performance of Ballard IV FC at different operating conditions as shown in Table 3Selection of the AUV (Sec 4.1)Recording the maximum power and corresponding current as shown in Table 500EndCalculation of the No. of cells (Eq. 31) as shown in Table 5Calculation of H2 and O2 consumptions for the FC (Eq. 32 to 35) as shown in Table 5Calculation of the FC efficiency (Eq. 36) as shown in Table 5Analysis of the obtained results (Sec. 5.2)StartCalculation of power requirements of the selected AUV (Eq. 26 to Eq. 30 Sec. 4.1)Prediction of one cell performance of Ballard IV FC at different operating conditions as shown in Table 3Selection of the AUV (Sec 4.1)Recording the maximum power and corresponding current as shown in Table 5Fig. 5. Flow diagram of the method presented for sizing the AUV fuel cell and calculating the gas consumption at different operating conditionsTab. 2Main Characteristics of A27-E AUV [5]ParametersValue Length4.5 mDiameter0.73 mWeight 850 kgMaximum diving depth300 mMaximum speed6 knotes-3.09 m/sUnderwater endurance at speed 6 knots30 hr115190710644000Fig. 6. Viscous resistance power requirement at different AUV speeds4.2. Single FC at different operating conditionsThe mathematical model described in Section II is used to predict the performance of one cell of Ballard IV FC at different operating conditions. Pure oxygen systems are usually exclusively utilized in situations when air is unavailable, such as submarines and other vessels, due to technical challenges, the additional bulk and weight of air storage, and associated safety considerations [42].The operating conditions selected in this study are consistent with those commonly reported for PEM fuel cells used in AUV and underwater applications. Specifically, the temperature range of 55-85°C aligns with the operating range of PEM fuel cells [10,11, 27] and the typical operating window of Nafion-based PEM fuel cells (60-80°C), where high proton conductivity and stable membrane performance are achieved [25]. Similarly, the selected hydrogen and oxygen inlet pressures (1-3 atm) fall within the commonly adopted range in the literature [27], where moderate pressurization improves reactant availability and reduces mass transport losses without imposing excessive compression penalties or safety risks, as discussed in [43]. Therefore, these ranges represent a practical compromise between performance, efficiency, system compactness, and operational safety, making them suitable for AUV applications and consistent with established literature.Table 3 shows 36 combinations of different operating conditions with a special designation that is developed to enable straightforward identification and correlation with the operating conditions. The designation is OIHJTM where O is for oxygen, I is the inlet dry pressure of O2 [atom], H is for hydrogen, J is inlet dry pressure of H2 [atom], T is for operating cell temperature and M is the value of the temperature [oC] so O1H3T60 means a case in which the inlet oxygen dry pressure is 1 atom, the inlet hydrogen dry pressure is 3 atoms operating at 60oC. The stoichiometric ratios SR1 and SR2 are fixed at 1.4 and 1.25 for oxygen and hydrogen, respectively, for all runs as recommended by [42].Tab. 3Different Calculations Operating ConditionsCasePO2inPH2inT[K]CasePO2inPH2inT[K]O1H1T551155O1H1T751175O1H2T551255O1H2T751275O1H3T551355O1H3T751375O2H1T552155O2H1T752175O2H2T552255O2H2T752275O2H3T552355O2H3T752375O3H1T553155O3H1T753175O3H2T553255O3H2T753275O3H3T553355O3H3T753375O1H1T651165O1H1T851185O1H2T651265O1H2T851285O1H3T651365O1H3T851385O2H1T652165O2H1T852185O2H2T652265O2H2T852285O2H3T652365O2H3T852385O3H1T653165O3H1T853185O3H2T653265O3H2T853285O3H3T653365O3H3T8533854.3. Fuel Cell Stack for AUVUsing the polarization curve V-I and the power curve WFC-I for each operating condition, the maximum power for the single FC, WFC, max, is determined with the corresponding current density and output voltage.The number of cells, NFC, is determined by:NFC=WAUVWFC, max(31)According to the electrochemical reactions of the PEMFC, the molar consumption of H2 and O2 can be determined as follows. nH2=SR2NFCI2F(32)nO2=SR1NFCI4F(33)The inlet mass flow rate of H2 and O2 in gm/s is calculated by:mH2=nH2.MH2(34)mO2=nO2.MO2(35)Where MH2 and MO2 are the molar masses of hydrogen and oxygen.The fuel cell energy efficiency is calculated by [44]:ηFC=NFC×WFC, maxnH2×HHVH2(36)5. RESULTS and DISCUSSIONS5.1. Single FC at Different Operating ConditionsThe impact of operating supply pressure of H2 and O2 on PEMFC performance is assessed and displayed in Figures 7 to 11. Anode and cathode gas supply conditions are examined under various combinations of PO2in and PH2in as shown in Table 3. The maximum power for all cases occurs at a current range of 36.5 A and 41.75 A, corresponding to a current density of 0.72 A/cm2 and 0.83 A/cm2, respectively. The output voltage and power at a current of 40 A are displayed in Table 4 and will be used for comparisons. Table 4 demonstrates the output cell voltage and power at 40 A and operating temperatures of 55°C and 85°C.Tab. 4The output cell voltage and power at 40 A and operating temperatures of 55°C and 85°CCasePO2inPH2inT[K]V [V]WFC[W]CasePO2inPH2inT[K]Vcell[V]WFC[W]O1H1T5511550.40116.043O1H1T8511850.45818.308O1H2T5512550.41116.424O1H2T8512850.46718.696O1H3T5513550.41616.650O1H3T8513850.47318.933O2H1T5521550.42316.912O2H1T8521850.48119.257O2H2T5522550.43217.293O2H2T8522850.49119.645O2H3T5523550.43817.519O2H3T8523850.49719.881O3H1T5531550.43617.421O3H1T8531850.49519.812O3H2T5532550.44517.802O3H2T8532850.50520.200O3H3T5533550.45118.027O3H3T8533850.51120.436Examining how voltage loss responds could help explain the improvement in cell performance. The activation overpotential does not exhibit a pressure term. Activation losses are the voltage losses due to the finite kinetics of the electrochemical reactions at both electrodes, namely the Oxygen Reduction Reaction (ORR) at the cathode and Hydrogen Oxidation Reaction (HOR) at the anode. They reflect the energy barrier that must be overcome for the reactions to proceed at a given current density. The change in activation overpotential is influenced by the O2 concentration, which is a function of PO2* as shown in Eq. 19. At a current of 40 A, the reduction percent in activation overpotential approximately attains 5.4% and 6.4% when PO2in increases from 1 atm to 3 atm, at 55°C and 85°C, respectively. Gerling et al. [45] showed that under controlled O₂ partial pressure conditions, the exchange current density for ORR increased significantly, causing lower activation overpotential at the cathode. Butori et al. [25] decoupled the effects of O₂ partial pressure, humidity, and temperature and found that raising O₂ partial pressure improved voltage stability across current densities by maintaining adequate cathode kinetics even at higher temperatures. On the other hand, increasing PH2in does not affect the activation overpotential as shown in Figures 7a and 7b at temperatures 55°C and 85°C, respectively. The negligible effect of overpotential with the partial pressure of H2 is often true in well-optimized PEMFCs under moderate to high H2 pressures and moderate current densities, where mass-transport losses are not dominant. In other regimes (low H2 pressure, high current density, degraded diffusion layers, transient operation), the effect can be noticeable [46]. The HOR is often neglected in full cell studies for the sake of simplicity, justified by the high rates of the kinetics [45].Operating temperature has a significant effect on the activation energy and activation losses of PEMFC. As operating temperature increases, the reaction kinetics improve, mainly due to faster charge transfer reactions, increased proton conductivity in the membrane, and reduced activation energy barrier for the electrochemical reactions. The effect can be understood using the Arrhenius equation, which shows that increasing operating temperature results in higher exchange current density and reduced activation overpotential. The effect of operating temperature is shown in Figures 8a and 8b. When the operating temperature increases from 55°C to 85°C, the percentage decrease of activation overpotential is 7.84% at both O2 and H2 supply pressures of 1 atm and 8.82% at both O2 and H2 supply pressures of 3 atm a) b)Fig. 7. Effect of PO2in and PH2inon the activation overpotential at operating temperatures of 55°C and 85°C a) b)Fig. 8. Effect of different operating temperatures on the FC ohmic losses at PO2in and PH2in of 1 and 3 atomsThe ohmic overpotentials do not exhibit a pressure term. At higher operating temperatures, water activity and water content of the membrane, and as a result, membrane conductivity increases and ohmic loss decreases [47,48]. Figure 9 indicates that the ohmic losses decrease as the operating temperature increases at all O2 and H2 supply pressures. When the operating temperature increases from 55°C to 85°C, the percentage decrease of ohmic losses is 13.5% at all O2 and H2 supply pressures.Fig. 9. Effect of different operating temperatures on the FC ohmic losses at all values of PO2in and PH2inThe cell output voltage depends on the open circuit voltage and overpotentials. The open circuit voltage of a fuel cell depends directly on the reactants' partial pressures, as shown in Eq. 2. When O2 and H2 partial pressures increase, the logarithmic term becomes larger, raising the equilibrium cell potential. This reduces the relative magnitude of the overpotential (difference between actual voltage and theoretical voltage). The percentage increase of the Nernst voltage is about 1.3% when PH2in increases from 1atm to 3 atm at all values of Po2in. On the other hand, the percentage increase of Nernst voltage is about 0.65 % when PO2in increases from 1atm to 3 atm at all values of PH2in. Increasing either the inlet pressure of H2 or O2 can result in a high output voltage, as seen in Figures 10a, 10c,10e, and 10f, and high-power output as shown in Figures 10b, 10d, 10f, and 10g. The increase in the output voltage at a temperature of 55°C when PH2in Increases from 1 atm to 3 atm approximately attain 3.78%, 3.58%, and 3.48% for Po2in equal 1 atm, 2 atm, and 3 atm, respectively. These percentages become 3.41%, 3.24%, and 3.15% at an operating temperature of 85°C. The increase in the output voltage at a temperature of 55°C when PO2in increases from 1 atm to 3 atm approximately attains 8.58%, 8.39%, and 8.27% for PH2in equal to 1 atm, 2 atm, and 3 atm, respectively. These percentages become 8.21%, 8.04%, and 7.94% at an operating temperature of 85°C.The effect of increasing PO2in is more pronounced than that of the H2. The percentage increase in cell voltage when increasing both PO2in and PH2in from 1 atm to 3 atm is 12.37% and 11.63 at operating temperatures of 55°C and 85oC respectively. The same percentage increase applies to the ell output power at all corresponding operating conditions.Generally speaking, raising the operating pressure could enhance cell performance. Nevertheless, high inlet gas pressure requires more energy from the gas compressor; via onboard compression, the associated parasitic power can offset these gains beyond an optimal compression level. In the current work, the increase in reactant inlet conditions is achieved by supplying H₂ and O₂ from one of the different storage options, including compressed gas storage and cryogenic storage for H₂ and O₂, metal hydrides for H₂ storage, and chemical O2 sources [5], which is a commonly adopted configuration for autonomous underwater vehicles (AUVs). Under this assumption, no onboard compressors are required during operation, and therefore, compressor power does not directly influence the instantaneous system performance or the reported efficiency. This energy input occurs off-board during refueling and is not part of the onboard energy balance considered in this study. Each storage option introduces different system-level trade-offs in terms of energy density, thermal management, system complexity, and auxiliary power requirements. A comprehensive evaluation of these storage strategies and their impact on overall system performance is beyond the scope of the present paper. However, this aspect represents an important direction for future research, and it is intended to be addressed in detail in subsequent work.The effect of operating temperature enhances the fuel cell output voltage and power for all inlet oxygen and hydrogen supply pressures, as shown in Figures 11a to d. When the operating temperature increases from 55°C to 85°C, the increase in the cell output voltage is 14.12% and 13.36% at both PO2in and PH2in of 1 and 3 atm, respectively. On the other hand, open circuit voltage decreases with the increase in operating temperature. As the operating temperature increases, the Gibbs free energy of the reaction decreases (1.229 for standard conditions), leading to a lower Nernst voltage. At the same time, higher temperatures increase the value of RT/2F, which increases the Nernst voltage. This can partially offset the drop caused by the decrease in Gibbs free energy, especially in pressurized operation. In general, the Nernst voltage slightly decreases with temperature under constant pressure. Results show that when the operating temperature increases from 55°C to 85°C, the Nernst voltage decreases 2% and 1.85% at both values of PO2in and PH2in of 1 atm, 3 atm, respectively. Finally, decreasing ohmic loss and activation overpotential dominate the Nernst voltage; consequently, the fuel cell performance enhances.Increases in operating temperature and pressure can improve cell performance, according to experimental research on the impact of operating circumstances on cell performance [30]. Therefore, lowering the irreversibility rate and improving the cell's performance can lower the cost and aid in the fuel cell's commercialization.5.2. Fuel Cell Stack for AUVAll calculations shown by Eqs. 31 to 36 at maximum predicted cell power under different operating conditions are shown in Table 5. The stack power for the considered AUV is 1523W.Tab. 5Different Calculations at Different Operating ConditionsCaseWFC, max[W]IWFC, max[Ampere]NFCO2 Cons.[Mole/s]O2 Cons.[gm/s]H2 Cons.[Mole/s]H2 Cons.[gm/s]ηEnergyO1H1T5516.2136.5940.01160.3700.0220.0450.283O1H2T5516.5637920.01150.3670.0220.0450.286O1H3T5516.7737.25910.01140.3650.0220.0440.287O2H1T5517.0137.25900.01120.3590.0220.0440.291O2H2T5517.3737.75880.01110.3570.0210.0430.294O2H3T5517.5838870.01110.3550.0210.0430.295O3H1T5517.4937.75880.01110.3540.0210.0430.296O3H2T5517.8538.25860.01100.3520.0210.0430.298O3H3T5518.0738.25850.01090.3480.0210.0420.301O1H1T6516.8737.5910.01140.3650.0220.0440.287O1H2T6517.2438890.01130.3620.0220.0440.289O1H3T6517.4638.25880.01120.3600.0220.0440.291O2H1T6517.7238.5860.01110.3570.0210.0430.294O2H2T6518.1038.75850.01100.3520.0210.0430.298O2H3T6518.3239840.01090.3490.0210.0420.300O3H1T6518.2338.75840.01090.3490.0210.0420.300O3H2T6518.6139.25820.01080.3460.0210.0420.302O3H3T6518.8339.5810.01080.3440.0210.0420.304O1H1T7517.5738.75870.01130.3620.0220.0440.289O1H2T7517.9539850.01110.3570.0210.0430.294O1H3T7518.1839.25840.01110.3540.0210.0430.295O2H1T7518.4739.5830.01100.3510.0210.0430.298O2H2T7518.8640810.01090.3480.0210.0420.301O2H3T7519.0940800.01070.3440.0210.0420.304O3H1T7519.0140810.01080.3450.0210.0420.303O3H2T7519.4040.25790.01060.3410.0200.0410.307O3H3T7519.6340.5780.01060.3390.0200.0410.309O1H1T8518.3139.75840.01110.3560.0210.0430.294O1H2T8518.7040820.01100.3510.0210.0430.298O1H3T8518.9340.25810.01090.3490.0210.0420.300O2H1T8519.2640.5800.01080.3450.0210.0420.303O2H2T8519.6641780.01070.3420.0210.0420.306O2H3T8519.9041.25770.01060.3400.0200.0410.308O3H1T8519.8341770.01060.3390.0200.0410.309O3H2T8520.2341.5760.01050.3370.0200.0410.311O3H3T8520.4841.75750.01050.3350.0200.0410.313The results indicate a consistent improvement in fuel cell performance with increasing operating temperature and reactant pressures, which is well aligned with findings reported by [30,38,49]. As the temperature rises from 55°C to 85°C, the required number of cells (NFC) decreases significantly (from 94 to 75), reflecting enhanced reaction kinetics and reduced activation losses, as discussed in [49]. A similar trend is observed with increasing O₂ and H₂ pressures, where higher pressurization reduces both oxygen and hydrogen consumption rates due to improved reactant availability and reduced mass transport limitations, consistent with observations by [5]. Moreover, the gradual decrease in reactant consumption indicates better fuel utilization efficiency, which agrees with classical PEMFC performance behavior reported in [50]. The energy efficiency (ηEnergy) increases from about 0.283 to 0.313, confirming that elevated temperature and pressure enhance overall system efficiency, as also highlighted by [42]. Notably, the best performance was obtained at high temperature (85°C) and high pressures (O3H3). Overall, the present trends are in strong agreement with established literature on PEM fuel cell operation. a) b) c) d) e) f) g) h)Fig. 10. Effect of PO2in and PH2in on the FC output voltage and power at different operating temperatures163119335210274623-74220 a) b)163621-14030274623-15730 c) d)Fig. 11. Effect of different operating temperatures on the FC output voltage and power at PO2in and PH2in of 1 and 3 atomsResults presented in Table 5 show that increasing operating temperature from 55°C to 85°C and increasing both O2 and H2 supply pressures from 1 to 3 atm result in:Reduction of the No. of cells from 94 cells at 1 atm to 75 cells for 3 atm, with a decreasing percentage of 20%. The percentage is close to the percentage decrease in fuel cell stack weight and volume. Reduction of the O2 and H2 consumptions by 9.5%Increasing the energy efficiency increases by 9.6%6. CONCLUSIONSThe research methodology entails creating a fuel cell model to predict how different operating conditions affect the performance of a single fuel cell. The impact of operating conditions on cell voltage losses is investigated through a comprehensive investigation. Based on these investigations, size and performance indices are calculated for a fuel cell stack to derive a commercial AUV under varying operating conditions. The following conclusions can be made in light of this study:The higher H2 and O2 supply pressures increase cell performance mainly due to the decrease in activation overpotential at all current densities and operating temperatures. On the other hand, they do not affect the ohmic losses.Increasing the cell operating temperature results in better fuel cell performance, mainly due to the decrease in both ohmic and activation losses at all current densities and gas supply pressures.The effect of increasing PO2in on the FC performance is more pronounced than that of and PH2in.As PO2in increases from 1 atm to 3 atm at a current density of 40 A, the reduction in activation overpotential approximately attains 5.4% and 6.4%, at 55°C and 85°C, respectively.The ohmic losses decrease as the operating temperature increases at both values of PO2in and PH2in of 1 and 3 atm respectively (ohmic losses is 13.5% at operating temperature increases from 55°C to 85°C).Concerning the FC stack for the assigned AUV, the increase of operating temperature from 55°C to 85°C and the increase of both O2 and H2 supply pressures from 1 to 3 atm result in a reduction of the number of cells by 20%. 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Zhang. 2013. “PEM Fuel Cell Testing and Diagnosis”. Elsevier B.V. ISBN: 978-0-444-53688-4. DOI: https://doi.org/10.1016/B978-0-444-53688-4.00004-8.EG&G Technical Services Inc. 2004. Fuel Cell Handbook. 7th ed. U.S. Department of Energy. Under Contract No. DE-AM26-99FT40575.Dicks A.L., D.A.J. Rand. 2018. Fuel Cell Systems Explained. 3rd ed. John Wiley & Sons Ltd. ISBN: 9781118706978.Received 09.01.2026; accepted in revised form 04.05.2026Scientific Journal of Silesian University of Technology. Series Transport is licensed under a Creative Commons Attribution 4.0 International LicenseScientific Journal of Silesian University of Technology. Series TransportZeszyty Naukowe Politechniki Śląskiej. Seria TransportVolume 1312026p-ISSN: 0209-3324e-ISSN: 2450-1549DOI: https://doi.org/10.20858/sjsutst.2026.131.4Journal homepage: http://sjsutst.polsl.plPerformance and sizing of PEM fuel cells under variable operating conditions for… PAGE 17. PAGE 16A. Elmaihy, M. Abd El-Latif Faculty Department of Ships and Submarines, Military Technical College, Egypt. Email: ali.elmaihy@mtc.edu.eg. ORCID: https://orcid.org/0000-0002-3917-9533 Faculty of Mechanical Design and Production Department, Military Technical College, Egypt. Email: abdelmenim@mtc.edu.eg. 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