Optimal EMS Design for a 4-MW-Class Hydrogen Tugboat: A Comparative Analysis Using DP-Based Performance Evaluation

Abstract

In the current trend of hydrogen fuel cell-powered ships, batteries are used together with fuel cells to overcome the limitations of fuel cell technology. However, performance differences arise depending on fuel cell and battery configurations, load profiles, and energy management system (EMS) algorithms. We designed four hybrid controllers to optimize EMS algorithms for achieving maximum performance based on target profiles and hardware. The selected EMS is based on a State Machine, an Equivalent Consumption Minimization Strategy (ECMS), Economic Model Predictive Control (EMPC), and Dynamic Programming (DP). We used DP to evaluate the optimal design state and fuel efficiency of each controller. To evaluate controller performance, we obtained a 4-MW-class tug load profile as a reference and performed simulations based on Nedstack’s fuel cells and a lithium-ion battery model. The constraints were set according to the description of each equipment manual, and the optimal controller was derived based on the amount of hydrogen consumed by each EMS under the condition of completely tracking the load profile. As a result of simulating the hybrid fuel cell–battery system by applying the load profile of the tugboat, we found that the 4-MW EMPC, which requires more state variables and control inputs, is the most fuel-efficient controller.

1. Introduction

In response to challenges such as rising sea levels, abnormal climate patterns, and the acceleration of global warming attributed to the emission of greenhouse gases (GHGs), countries and international organizations are tightening regulations on GHG emissions [1]. With marine shipping accounting for more than 80% of global trade and contributing to approximately 3% of total global GHG emissions, it plays a significant role in this landscape. While the contribution of shipping to GHG emissions is lower than that of other methods of transportation, without further action, emissions are expected to escalate. Achieving carbon neutralization within the shipping industry requires technological advancements and shifts in operational practices, along with the adoption of low-carbon and zero-carbon fuels. However, this transition entails investments in ship design, engine modification, and the large-scale production of low-carbon and zero-carbon fuels, all of which are expected to inflate costs for ship owners, industry, trade, and consumers [2]. Consequently, the International Maritime Organization (IMO) has intensified regulations on ship emissions. Furthermore, the IMO has established a target of emission reduction of 50% by 2050 in relation to the 2008 levels [3]. The IMO is implementing regulations to reduce GHGs, focusing on technical regulations, the Energy Efficiency Existing Ship Index (EEXI), operational regulations, and the Carbon Intensity Indicator (CII). Regarding the EEXI regulations, it refers to a pre-determined value obtained by calculating and indexing the amount of carbon dioxide emitted per ton of cargo transported per mile, utilizing ship specifications, including the engine power and deadweight tonnage [4]. To meet the carbon emission regulations, pollution-reducing devices such as scrubbers or filters are often installed, alongside the ongoing development of various technologies. These include advancements in several areas: designing hulls that cut through water more easily, reducing friction between the hull and water, optimizing propellers for better thrust, streamlining operations to save fuel, exploring eco-friendly fuels such as LNG, ammonia, methanol, and hydrogen, and even developing electric propulsion systems that completely replace combustion engines. Among the array of eco-friendly ship technologies, electric propulsion ships primarily utilize electric energy to power the propulsion motor. The methods of supplying electric energy vary, encompassing the use of batteries, power generation through internal combustion engines, or harnessing electric energy from fuel cells employing hydrogen or ammonia [5]. Hydrogen fuel cell ships possess the advantage of extended range over battery-based electric ships and emit no carbon dioxide, thereby driving active research and development programs. By transitioning power sources for the propulsion motors and hotel services of ships to eco-friendly devices, not only can exhaust gas emissions be reduced but synergy can also be maximized by improving the equipment operation methods to enhance fuel efficiency. Most fuel cells installed in mobilities are operated using batteries or supercapacitors. By configuring the battery in a hybrid system, it is feasible to enhance fuel efficiency, performance, and reliability, in addition to using them to initiate the fuel cell stack [6].

The diversity in ship types, navigating routes, and retrofit structures, which influence the overall dynamic characteristics, give rise to various types of fuel cell and energy storage system (ESS) hybrids. Furthermore, to operate them at optimal efficiency, various algorithms applicable to energy management systems (EMSs) must be taken into account [7,8].

Several strategies have been reported for configuring fuel cells and batteries and optimizing them as hybrid systems, particularly in the automotive sector [9]. As an example of hybrid systems applied to ships, Ref. [10] applied a state-based EMS, equivalent fuel consumption minimization strategy (ECMS), charge-depleting–charge-sustaining (CDCS) EMS, and classical proportional–integral (PI) controller-based EMS to analyze the performance of the world’s first fuel cell–battery passenger ship, namely, the FCS Alsterwasser. Moreover, the authors of [11,12] analyzed the performance of the rule-based EMS of a ship for the high-efficiency zone of the Polymer Electrolyte Membrane Fuel Cell (PEMFC). In [6], an EMS based on a support vector machine (SVM) and a low-pass filter was proposed. Furthermore, a multi-objective optimization method for ESS sizing and EMS has been reported in the literature. In addition, Ref. [13] implemented an EMS using an ECMS filter based on a fuel cell ferry model.

This study aims to propose a method for selecting the optimal EMS among various candidates (State Machine (SM), ECMS, and Economic Model Predictive Control (EMPC)) for a tugboat equipped with a 4-MW PEMFC based on a given load profile. Specifically, we compared the optimization and performance of the EMS algorithm candidates by determining the hydrogen consumption through Dynamic Programming (DP)-derived optimal control inputs (demand power) into the PEMFC and Li-ion battery. Therefore, this paper explains the process of model and controller design and evaluation; Section 2 explains the configuration methods and models of the PEMFC and battery; Section 3 introduces and designs the EMS for hybrid control of the PEMFC and battery; Finally, Section 4 verifies the dynamic characteristics of each EMS and evaluates the performance of each controller based on DP.

2. Power Train Modeling of a Fuel Cell Ship

To implement the optimal control of the fuel cell–battery hybrid of a fuel cell-powered ship, we implemented a fuel cell power train model, which served as the control target. The fuel cell model—the core power generation source—was selected from commercially available products to closely emulate the performance of those installed and operational on actual ships, and the model was developed based on the datasheet provided by the company.

2.1. Power Train Structure

The power train structure of a fuel cell ship may vary depending on whether the main grid operates on AC or DC, with the addition of a converter or transformer contingent upon the configuration of the propulsion equipment or the voltage level between the battery voltage and the grid. Opting for a DC main grid yields several advantages, e.g., a less complex system requires fewer components, resulting in space savings. Moreover, the overall efficiency is improved owing to reduced power loss. Currently, the higher price of DC grid components for ships has created a demand for AC grid components.

In this study, we constructed a universal integration scheme to enhance the model utilization, as illustrated in the single-line diagram in Figure 1. For the power distribution, we entered the control input calculated using the EMS into the DC/DC converter to regulate the current generated from each converter. This strategy ensured that the ratio of power supplied by each power source could be freely controlled in response to the demand power, i.e., $P_{load}$ [14,15].

We configured the power grid based on the DC grid and omitted the modeling of equipment commonly found on electric ships but with minimal power consumption or negligible impact on the EMS design. Moreover, we focused on modeling fuel cells and batteries, which are key power sources.

To simplify our model, we focused solely on the propulsion motor’s load profile as we already had detailed data on its power usage. Including the hotel service load (power for lights, ventilation, etc.) was deemed unnecessary. We made this assumption because the power losses during the DC–AC or DC–DC conversion process by the converter are likely minimal and consistent across the different systems (fuel cells, batteries, and propulsion motors). These minor losses would not significantly impact the overall behavior of the power grid, particularly when evaluating the transient response performance of the EMS controller.

2.2. Fuel Cell Plant

Our fuel cell model is based on the Low-Temperature Proton Exchange Membrane (LT-PEMFC), the current leader in mobility applications. Unlike other fuel cell types, the temperature can be readily controlled during power generation, the weight and volume of the Balance of Plant (BoP) for running fuel cells are relatively small, and it is relatively economical [16]. In this study, we performed the modeling based on FT-FCPI-500 and FCS 13-XXL, which are maritime fuel cell power installations of Nedstack, a Dutch company [17,18,19,20]. These systems are modular, with units capable of generating up to 4 MW stored in a single 40-foot container. Additionally, separate 20-foot containers house the hazardous BoP (hydrogen supply) and non-hazardous BoP (air supply) for each fuel cell unit.

Nedstack’s PEMFC systems feature a modular design, allowing them to scale up or down to meet various power requirements. The basic building block is the individual cell. Multiple cells are grouped together to form a stack (typically 96 cells delivering 13.6 kW). These stacks are further combined into strings (12 stacks producing 100 kW) and, finally, housed within containers for larger-scale applications (500 kW or more). This modularity offers a significant advantage: the required power capacity for a ship can be easily achieved by adjusting the number of strings incorporated into the system. Thus, the ship’s specific power needs do not significantly impact the dynamics of individual fuel cells. Instead, the key factors influencing fuel cell behavior are the amount of power delivered over time and the resulting temperature changes.

To capture the key dynamics of the fuel cell system, we focused on modeling the modular fuel cell stack and its BoP. These components have the most significant influence on the overall system’s behavior. Additionally, we prioritized readability, usability, and scalability when analyzing the load profile of a high-capacity ship. This ensures the model can be easily adapted to various ship capacities using Nedstack’s modular products.

We employed MATLAB/Simulink(2023b) as the primary modeling platform for this project. We decided to exclude the Fuel Gas Storage System (FGSS) and hydrogen tanks from the model as their impact on the power system is minimal. Similarly, the motor and hotel service loads were incorporated into the overall load profile instead of being modeled separately. This approach allowed us to focus on the core components directly interacting with the EMS: fuel cells, batteries, and the BoP.

Figure 2 illustrates the MATLAB/Simulink-based PEMFC–Battery model. It represents the load profile, expressed as the sum of motor power and general service load, and the power consumption (loss) generated at the BoP of the PEMFC, combined as $P_{load}$, which is fed into the EMS. The EMS determines the power to be supplied to the PEMFC plant and Li-ion battery models. Subsequently, dynamic computations within the PEMFC plant and Li-ion battery model yield results such as temperature changes, demi-water production, hydrogen/air consumption, etc. $P_{FC}$ denotes the power supplied by the PEMFC, $P_{BAT}$ represents the power supplied by the battery, and $P_{FC-BAT}$ indicates the power supplied from the PEMFC to the battery for charging purposes. The relationships between $P_{load},P_{FC},P_{BAT}$, and $P_{FC-BAT}$ will be detailed in Section 3.1.

2.2.1. Fuel Cell Mathematical Model

The model of the fuel cells serves two key control requirements: the model allows us to calculate the voltage $V_{stack}\left[\mathrm{V}\right]$ generated by the current $I_{stack}\left[\mathrm{A}\right]$ to the load. Thus, this strategy allowed us to derive the I–V curve and generate the efficiency map required for control. The relationship between the fuel cells’ $I_{stack}\left[\mathrm{A}\right]$ and $V_{stack}$ exhibits nonlinear characteristics. This nonlinearity arises from the physical and chemical processes occurring within the fuel cell when hydrogen and oxygen react in the presence of an electrolyte. Based on the electromotive force generated from the fuel cell $E_{Nerst}\left[V\right]$ and using the Nerst Potential Equation [21], we determined the final generated voltage based on the three major losses generated during the reaction, as expressed in Equation (1).

$$V_{stack}=N_{cell}\times \left(E_{Nerst}-V_{act}-V_{\Omega }-V_{con}\right),$$

(1)

where $N_{cell}$ is the number of cells containing a membrane electrode assembly (MEA), $V_{act}$ denotes the activation loss, $V_{\Omega }$ denotes the Ohmic loss, and $V_{con}$ denotes the concentration loss. The electromotive force and each loss can be obtained using Equations (2)–(7), respectively, and we referred to existing studies to derive these equations [20,21].

Open-Circuit Voltage:

$$E_{Nerst}=1.23-8.5\times {10}^{-4}\left(T_{stack}-298.15\right)+4.31\times {10}^{-5}\times T_{stack}\times \ln \left(P_{H_2}\sqrt{P_{O_2}}\right)$$

(2)

Activation Loss:

$$V_{act}=-\left[{\mathsf{\gamma }}_1+{\mathsf{\gamma }}_2\times T_{stack}+{\mathsf{\gamma }}_3\times T_{stack}\times ln\left(C_{O_2}\right)+{\mathsf{\gamma }}_4\times T_{stack}\times ln\left(I_{stack}\right)\right]$$

(3)

Ohmic Loss:

$$V_{\Omega }=I_{stack}\times \left(R_m+R_C\right)$$

(4)

$$R_m={\mathsf{\rho }}_{m}l{S}^{-1}$$

(5)

$${\mathsf{\rho }}_m=\frac{181.6\left[0.062{\left(\frac{T_{stack}}{303}\right)}^2{\left(\frac{I_{stack}}{S}\right)}^{2.5}+0.03\left(\frac{I_{stack}}{S}+1\right)\right]}{\left[\mathsf{\lambda }-0.063-3\left(\frac{I_{stack}}{S}\right)\right]\times e^{\frac{T_{stack}-303}{T_{stack}}}}$$

(6)

Concentration Loss:

$$V_{con}=-\beta \ln \left(1-\frac{J}{J_{max}}\right)$$

(7)

In the equation for determining $E_{Nerst}$, $T_{stack}\left[\mathrm{K}\right]$ denotes the temperature of the stack, and $P_{H_2}\mathrm{a}\mathrm{n}\mathrm{d} P_{O_2}$ denote the partial pressures of hydrogen and oxygen, respectively. The constant parameter ${\gamma }_1\dots {\gamma }_4$ of the activation loss is applied as shown in

Table 1. Polymer electrolyte membrane fuel cell parameters for mathematical models.

Parameter	Description	Value	Unit
$N_{cell}$	Number of cells	* 78	EA
$N_{stk}$	Number of sub-stacks	* 37	EA
S	Membrane area	500	cm2
$J_{\mathrm{m}\mathrm{a}\mathrm{x}}$	Maximum current density	1200	mA/cm2
l	Membrane thickness	178	µm
$\lambda$	Tunable parameter	4	-
$P_{H_2}$	Partial pressure of hydrogen	1.2520	bar. a
$P_{O_2}$	Partial pressure of oxygen	0.2792	bar. a
$\beta$	$\mathrm{Constant} \mathrm{associated} \mathrm{with} {\mathrm{T}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{k}}$	0.15	-
${\gamma }_1$	Activation loss constant	−0.92	-
${\gamma }_2$	Activation loss constant	3.53 × 10−3	-
${\gamma }_3$	Activation loss constant	7.59 × 10−5	-
${\gamma }_4$	Activation loss constant	−9.62E × 10−5	-
$C_{O_2}$	Concentration of dissolved oxygen	1.7685 × 10−7	mol/cm3
$C_{H_2}$	Concentration of dissolved hydrogen	9.0021 × 10−7	mol/cm3
$R_m$	Resistance to the flow in the membrane	0.260608	$\Omega$
$R_c$	Contact resistance of the electrodes	3.00 × 10−4	$\Omega$

* The stack and cells of the PEMFC equipment are expected to have sufficient margins, and the number of cells and stacks is adjusted to approximate the hydrogen consumption in [18] as closely as possible.

, and $C_{O_2}\left[\mathrm{m}\mathrm{o}\mathrm{l}/{\mathrm{m}}^3\right]$ is the concentration of dissolved oxygen in liquids, which is defined according to Henry’s law. In the equation for Ohmic loss, $R_m[\Omega ]$ is the resistance to the flow in the membrane and $R_C[\Omega ]$ is the contact resistance of the electrodes. $\lambda$ denotes the tunable parameter, and $l\left[\mathrm{m}\right] \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{S}\left[{\mathrm{m}}^2\right]$ denote the thickness and stack area of the membrane, respectively. In the equation for concentration loss, $\beta \left[\mathrm{J}/\mathrm{C}\mathrm{ }\mathrm{o}\mathrm{r}\mathrm{ }\mathrm{V}\right]$ is a constant associated with $T_{stack}$, the ideal gas constant, and the Faraday constant. $J \mathrm{a}\mathrm{n}\mathrm{d} J_{max}$ denote the current density and the maximum current density $\left[\mathrm{A}/{\mathrm{m}}^2\right]$, respectively.

The I–V curve of the fuel cell disclosed by Nedstack was fitted based on the aforementioned mathematical model and the parameters listed in Table 1. The similarity between the two models was quantified using the correlation coefficient, the coefficient for the I–V curve was 0.8314, indicating a strong correlation. Moreover, the correlation coefficient for the I–P curve was 1.000, indicating perfect collinearity. In Figure 3, $V_{ref} \mathrm{a}\mathrm{n}\mathrm{d} P_{ref}$ are the curves of the Nedstack product, and $V_{mdl} \mathrm{a}\mathrm{n}\mathrm{d} P_{mdl}$ are mathematical models based on the parameters above.

A fuel cell’s final response to changes in power demand (transient response) is influenced by three major factors, each occurring at different timescales.

• Electrochemical reaction (fast): This response happens very quickly due to the nature of the chemical reactions within the fuel cell.
• Fluid Movement (slow): The movement of fluids within the fuel cell stack takes longer to respond to changes, impacting the overall response time.
• BoP Dynamics (slowest): The BoP, which includes components like pumps and valves, has the slowest response time among the three factors. Additionally, temperature, humidity, and the fuel-to-oxidant ratio (stoichiometry) can further influence these response times [22]. The specific transient response of a fuel cell system can vary depending on the equipment used. However, these variations typically occur within a timeframe of tens to hundreds of seconds. Understanding the transient response is crucial for optimizing the performance of the EMS controller.

For example, when the step input ranges from 100 [A] to 200 [A], the voltage is delayed and requires time to reach the steady state [23]. There are three main factors behind the transient response: the relatively fast electrochemical reaction, the moderate mass transport process, and the relatively slow system equipment. We can estimate a model that integrates these three factors through system identification based on data to analyze their transient responses.

The manufacturer (Nedstack) does not disclose data on transient responses. To address this challenge, we relied on research data to illustrate the general characteristics of this response [23]. In this simplified model, we assumed constant humidity and fuel-to-oxidant ratio (stoichiometry) to isolate the impact of temperature on the transient response. However, we could not validate the mathematical correlation between the transient response characteristics and temperature change. Thus, this part of the model remains a black box. To estimate this black box model, we performed system identification based on MATLAB Toolbox(2023b) and data. The difference in gain between input and output can be determined using an electrochemical relationship, and min–max normalization was performed for all input and output data to obtain only the transient response characteristics. The derived model is based on multi-input single-output with input $\mathbf{r}\left(t\right)={\left[\begin{array}{cc}{I}_{stack}\left(t\right)& T_{stack}\left(t\right)\end{array}\right]}^{\prime }$ and output $y\left(t\right) = V_{stack}\left(t\right)$, and when $\mathit{x}\in {\mathbb{R}}^3$, it was realized with a state space model, as shown in Equations (8)–(13).

$$\dot{\mathit{x}}\left(t\right)=\mathrm{A}\mathit{x}\left(t\right)+\mathrm{B}\mathbf{r}\left(t\right)$$

(8)

$$y\left(t\right)=\mathrm{C}\mathit{x}\left(t\right)+\mathrm{D}\mathbf{r}\left(t\right)$$

(9)

$$\mathrm{A}=\left[\begin{array}{ccc}-0.5663& 1.388& -5.427\\ -1.422& 0.1839& 4.759\\ 14.83& -6.979& -26.73\end{array}\right]$$

(10)

$$\mathrm{B}=\left[\begin{array}{cc}-0.4496& 4.049e-5\\ 1.221& -3e-5\\ -9.947& 1.226e-4\end{array}\right]$$

(11)

$$\mathrm{C}=\left[\begin{array}{ccc}3.572& 1.006& -0.08484\end{array}\right]$$

(12)

$$\mathrm{D}=\left[\begin{array}{cc}0& 0\end{array}\right]$$

(13)

Creating a more complex model can improve its accuracy (fidelity) in reflecting real-world behavior. However, overly complex models can become too sensitive to minor variations in the data (noise). Therefore, it is crucial to find the right balance between complexity and robustness. The current model was modeled as a 3D model based on the Hankel Singular Value, with all eigenvalues exhibiting negative real values, ensuring asymptotic stability. We validated the output derived based on the aforementioned model with the original data, as depicted in Figure 4 and Figure 5.

2.2.2. BoP Model of the Fuel Cell

The main devices of the BoP of a fuel cell system installed on a hydrogen ship include the hydrogen tank, humidifier, air blower, compressor, heat exchanger, coolant pump, and condenser (Figure 6). These devices are responsible for balancing fuel supply, pressure, humidity, and temperature, ensuring stable power generation through fuel cells.

When hydrogen and oxygen are supplied to the PEMFC stack, power is generated; however, it also leads to internal losses within the PEMFC and BoP operation. Therefore, establishing an appropriate standard is essential for operating the PEMFC-based power generation system at peak efficiency. This standard can be estimated based on the PEMFC’s I–V curve (Figure 4) and the approximate loss range of the BoP. The loss in the PEMFC stack is expressed as follows:

$$P_{stk,loss}=P_{net}-P_{FC},$$

(14)

where $p_{net}$ is the net energy gain in stack [W] and $p_{FC}$ is the produced electrical power of the PEMFC [W]. $P_{net}$ and $P_{FC}$ can be represented as follows:

$$P_{net}=P_{rxn}+P_{∆std}+P_{trans},$$

(15)

$$P_{FC}={{\eta }_{cnv}I}_{stack}{V}_{stack},$$

(16)

where $P_{rxn}$ is the reaction energy at the standard temperature [W] and the total energy [W] gained by the stack, $P_{∆std}$ is the energy gain in the membrane by bringing reactants and products to the standard temperature [W], $P_{trans}$ is the energy gain in the membrane due to water transport [W], and ${\eta }_{cnv}$ is the converter efficiency [24,25]. Therefore, the PEMFC stack efficiency is given by:

$${\eta }_{stk}=\frac{P_{FC}}{P_{FC}+P_{stk,loss}}.$$

(17)

However, to generate power from the PEMFC stack, power is inevitably consumed to drive the BoP. Therefore, the overall efficiency of the PEMFC power generation system is expressed as follows:

$${\eta }_{FC}=\frac{P_{FC}}{P_{FC}+P_{BoP}+P_{stk,loss}}.$$

(18)

where $P_{BoP}$ is the power consumption [W] of the BoP. Based on the aforementioned equation, ${\eta }_{stk}$ and ${\eta }_{FC}$ can be represented in a 3D efficiency map correlating the temperature and produced power, as illustrated in Figure 7. The optimal control was performed based on this map.

2.3. Battery Model

A mathematical model was created to represent the behavior of the batteries configured with fuel cells as a hybrid and to establish constraints for the stable operation of the battery model. The dynamics of the state of charge (SOC) of the battery can be expressed as follows using the battery’s internal resistance model ($R_{eq})$:

$$I_{BAT}\left(t\right)=\frac{V_{OC}\left(SOC\right)-\sqrt{{V_{OC}\left(SOC\right)}^2-4R_{eq}{P}_{BAT}\left(t\right)}}{2R_{eq}},$$

(19)

$$\dot{SOC}\left(t\right)=\frac{I_{BAT}\left(t\right)}{C_{BAT}},$$

(20)

$$C_{BAT}=N_P{C}_{max}$$

(21)

Here, $I_{BAT}\left(t\right)$ is the battery current [A], $V_{OC}$ is the open-circuit voltage [V], $R_{eq}$ is the equivalent resistance [$\Omega$], $C_{BAT}$ is the battery capacity [Ah], $C_{max}$ is the maximum capacity for a parallel line [Ah], and $N_P$ is the number of parallel lines [-]. The voltage increases proportionally as more batteries are connected in series, and the total internal resistance of the battery also increases proportionally. The real-time behavior of the $SOC$ can be derived by integrating Equation (15) with respect to time. The power $P_{BAT}$ supplied to the battery module can be expressed as the power $P_{BAT,grid}$ flowing to the grid and the converter efficiency ${\eta }_{cnv}$, as follows:

$$P_{BAT}=\left\{\begin{array}{c}{\eta }_{inv}{P}_{BAT,grid}, P_{BAT,grid}\ge 0\\ \frac{1}{{\eta }_{inv}}{P}_{BAT,grid}, P_{BAT,grid}<0\end{array}\right.$$

(22)

To provide a specific amount of power to the grid, the battery must furnish additional power, equivalent to the reciprocal of the inverter efficiency.

2.4. Power Distribution

To optimize the formation of a hybrid system integrating fuel cells and batteries, hardware that can control the power supply freely according to the control input is required. This power distributor can be implemented using an EMS and converters. The DC/DC or DC/AC converter to be installed at each power source and the load can control the desired amount of power in real time using internal switching components. In [15], two types of converters are introduced: a non-isolated converter type and an isolated converter type. In non-isolated converters, the input side and the load side are electrically connected. Although the volume and cost are minimal, the noise generated from the input can be transmitted to the load. In contrast, isolated converters utilize a transformer that insulates the input from the load, thereby mitigating the noise transmission. While this design facilitates high step-up/step-down conversion ratios, it comes with increased costs, weight, and volume due to the inclusion of transformers and additional power components like DC–AC converting circuits.

Given the dynamic nature of fuel cell power generation, where the output varies with the supplied amount of hydrogen and oxygen, the DC/DC converter must accommodate a wide input power range and track the reference voltage of the main grid. Moreover, as batteries require both charging and discharging capabilities, a bidirectional DC/DC converter is preferable. However, in cases where the grid voltage aligns closely with the battery voltage, direct connection of batteries to the grid without a DC/DC converter may be feasible, depending on the specific circumstances [26].

In this study, it is assumed that the power equipment is configured based on the scheme in Figure 1b and that all DC/AC inverter efficiencies (${\eta }_{inv}$) are the same. Additionally, it is assumed that the proportion of the ship’s demand power to be supplied by PEMFC and batteries can be freely controlled at the DC/AC inverter by receiving EMS control inputs.

3. PEMFC–Battery EMS Design

Configuring a hybrid system with the PEMFC and batteries has inherent advantages, including improved fuel efficiency and reliability. For example, If the system’s power demand exceeds the power generated by the PEMFC, the battery can supplement the energy to improve output (peak shaving). When traveling for a short distance, the system can operate only with energy from the batteries, with the PEMFC on standby mode (spinning reserve), or fast-response batteries can be used to compensate for the shortcomings of the slow-response PEMFC (enhanced dynamic performance) [27]. Various EMS algorithms are being explored to ensure the timely application of these functionalities. Moreover, by leveraging multiple power consumption profiles of the ship, it becomes feasible to design an optimal controller capable of achieving performance levels close to those observed in real-world scenarios, which can potentially reduce design costs.

Various algorithms have been studied to maximize the characteristics of hybrids, and in general, the types of algorithms for optimal hybrid control of fuel cell-ESS can be largely divided into rule-based, optimization-based, and learning-based control strategies. In this study, optimal controllers are designed based on the State Machine (SM, rule-based), Model Predictive Control (MPC, optimization-based: online), Equivalent Consumption Minimization Strategy (ECMS, optimization-based: online), and Dynamic Programming (DP, optimization-based: offline), and the optimal controller is selected by comparing the performance.

The rationale for selecting each controller was based on their frequent appearance in literature reviews of EMS algorithms applied not only in ships but also in other mobility applications. These controllers are prominently featured in optimal control theory as representative algorithms. The core focus of this study was to introduce a strategy that evaluates the optimization states and performance of other controllers based on the global optimal algorithm, i.e., DP. Therefore, specialized controllers optimized for specific purposes were excluded from the evaluation criteria. Additionally, in the context of a learning-based control strategy, the reliability of the controller design often correlates with the amount of available data. However, in this study, due to the limited availability of load profiles for 4-MW-class tugboats, the performance analysis of the learning-based control strategy has been omitted.

The implementation methods and characteristics of the selected controllers in this study are as follows:

• State Machine (rule-based): It transitions to predefined states when specific conditions are met and performs predefined roles. It determines the energy to be released from the fuel cell and battery based on the current states of the fuel cell, battery, load profile, etc. In this study, we designed the control logic based on MATLAB’s Stateflow, aiming to tailor it for the fuel cell–battery ship structure by referencing the MAX SOC from automotive hybrid rule-based EMS [28].
• ECMS (Equivalent Consumption Minimization Strategy): In a hybrid system that includes the battery, the strategy converts the energy stored in the battery into fuel and designs a cost function to minimize its use, finding the optimal control input based on the optimal algorithm. In this study, we utilized the Sequential Quadratic Programming algorithm, which finds the solution with the minimum cost under constraints.
• EMPC (Economic Model Predictive Control): Based on the model, it estimates the state of the system up to a specified time horizon and finds the control input with the lowest cost based on the designed cost function. In general MPC, control inputs are minimized and the system is controlled to track the reference input. However, considering that the PEMFC should charge the battery in a timely manner and that the efficiency of the fuel cell changes in real time according to the load, we designed the economic cost function for fuel use, including the energy flow from the PEMFC to the battery, in the form of EMPC (nonlinear).

For the controller design, we actively used MATLAB’s Optimization Toolbox(2023b) and Model Predictive Control Toolbox(2023b).

The characteristics of each controller are illustrated in Figure 8. Transient behavior is categorized into Robust and Sensitive based on the degree of performance variation relative to the ship’s environment or operating conditions. General rule-based control involves sequential actions according to the current status and operating conditions of the ship, resulting in consistent performance across different ship types. Optimal control is classified by the extent to which fuel efficiency is enhanced by operating fuel cells and batteries efficiently. Specifically, MPC predicts the operation method with the least fuel consumption based on the fuel cell and battery model (prediction horizon) to generate the optimal input (control horizon). Real-time property is an indicator of real-time control performance. Unlike other controllers, DP requires extensive computational time and maximizes performance only when provided with information about the load profile in advance. It is utilized to derive the globally optimal input for the total load profile. Therefore, it is classified as offline.

As reported in the following sections, we designed an appropriate controller based on the aforementioned controllers to optimally control a fuel cell–battery hybrid for the 4-MW tugboat load profile.

3.1. Control Target Constraints

Considering the practical operation of the power train, it becomes imperative to devise a controller that accommodates the constraints imposed by both the PEMFC and batteries. Additionally, for the EMS, it is essential to determine the optimal control inputs that adhere to these constraints. The limitations of the PEMFC and batteries, pivotal components of the power train, can be expressed as follows:

$$P_{load}=P_{FC}+P_{BAT},$$

(23)

$$P_{FC}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le P_{FC}\left(\mathrm{t}\right)+P_{FC-BAT}\left(\mathrm{t}\right)\le P_{FC}^{\mathrm{m}\mathrm{a}\mathrm{x}},$$

(24)

$${dP}_{FC}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le {dP}_{FC}\left(\mathrm{t}\right)+{dP}_{FC-BAT}\left(\mathrm{t}\right)\le {dP}_{FC}^{\mathrm{m}\mathrm{a}\mathrm{x}},$$

(25)

$$P_{BAT}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le P_{BAT}\left(\mathrm{t}\right)-P_{FC-BAT}\left(\mathrm{t}\right)\le P_{BAT}^{\mathrm{m}\mathrm{a}\mathrm{x}},$$

(26)

$${dP}_{BAT}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le {dP}_{BAT}\left(\mathrm{t}\right)-{dP}_{FC-BAT}\left(\mathrm{t}\right)\le {dP}_{BAT}^{\mathrm{m}\mathrm{a}\mathrm{x}},$$

(27)

$${SOC}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le \mathrm{S}\mathrm{O}\mathrm{C}\left(\mathrm{t}\right)\le {SOC}^{\mathrm{m}\mathrm{a}\mathrm{x}},$$

(28)

$$P_{FC}\left(\mathrm{t}\right),P_{FC-BAT}\left(\mathrm{t}\right)\ge 0,$$

(29)

where min and max denote minimum and maximum values and $P_{FC-BAT}$ refers to the power supplied from the fuel cell to the battery for charging the battery. The power generated from each power source is expressed as a positive number; nevertheless, it can be expressed as a negative number if the energy flows toward the storage, such as a battery. Therefore, $P_{FC-BAT}$ is added to $P_{FC}\left(\mathrm{t}\right)$, because it is the power generated from the PEMFC, and it is added as a negative number to $P_{BAT}\left(\mathrm{t}\right)$. These constraints are used in the constrained nonlinear optimization algorithm [29,30].

3.2. SM Rule-Based Control

The primary objective of the controller is to ensure continuous electricity generation over an extended period, particularly in high-efficiency regimes for fuel cells, thereby minimizing hydrogen consumption and mitigating ship downtime. The rule-based EMS for hydrogen ships was designed by referring to the rule-based EMS (MAX. SOC) of hybrid electric vehicles (HEVs) [28]. A notable distinction between HEVs and hybrid ships lies in regenerative braking, a feature seldom found in ships. Thus, the controller could be designed with a relatively simple structure. Accordingly, we designed an SM control system, as shown in Figure 9.

According to the battery SOC and fuel cell state, the power required by the load is designed to transition between four SMs based on the deterministic rule-based control to maximize the efficient state. One of the main purposes of a hybrid is to operate the fuel cell only in the maximum-efficiency section to improve fuel efficiency.

In the event of the battery being in a state that requires charging, if its charge level falls below the predefined SOC, the vessel can be operated while the batteries are being replenished via the fuel cells’ high-efficiency range. However, if the batteries are depleted, simultaneous operation of the fuel cells and battery power to achieve maximum output becomes unfeasible. When the vessel is in the battery propulsion state, it operates solely on battery power as utilizing the fuel cells during periods of low power demand is inefficient. Conversely, the fuel cell propulsion state is engaged when the load’s power demand aligns with the fuel cells’ highest efficiency range, ideal for sustained-speed cruising. The hybrid propulsion state is activated when the load’s power demand surpasses the fuel cells’ rated power, necessitating supplementary power from the batteries.

The designed controller operates based on predefined conditions without requiring complex calculations. Thus, it can be seamlessly applied across different ship types as it does not involve intricate computational processes. However, it operates on a control method that disregards the ship’s specific load profile and the dynamic behavior of the fuel cells, thereby deviating from optimal control strategies.

3.3. ECMS

The ECMS assumes that the electrical energy consumption of the batteries in a hydrogen hybrid ship can be equated to fuel consumption. Furthermore, the power generation of the fuel cells and the batteries is distributed by finding a solution that minimizes the hydrogen consumption of the fuel cells and the hydrogen consumption of the batteries. Therefore, the equivalent fuel consumption rate of the fuel cell and the battery power can be represented as follows:

$${\dot{m}}_{eqv}\left(\mathrm{t}\right)={\dot{m}}_{FC}\left(\mathrm{t}\right)+{\dot{m}}_{BAT}\left(\mathrm{t}\right),$$

(30)

where ${\dot{m}}_{eqv}$ is the total equivalent hydrogen consumption (kg/s), ${\dot{m}}_{FC}$ is the fuel cell hydrogen consumption (kg/s), and ${\dot{m}}_{BAT}$ is the battery-equivalent fuel consumption [kg/s]. According to [31], ${\dot{m}}_{FC}\left(\mathrm{t}\right)$ and ${\dot{m}}_{BAT}\left(\mathrm{t}\right)$ can be represented as follows:

$${\dot{m}}_{FC}\left(\mathrm{t}\right)=\frac{P_{FC}\left(t\right)}{{\eta }_{FC}(T,P_{FC})Q_{LHV}},$$

(31)

$${\dot{m}}_{BAT}\left(\mathrm{t}\right)=\mathrm{s}\left(\mathrm{t}\right)P_{BAT}\left(\mathrm{t}\right).$$

(32)

where ${\eta }_{FC}$ is the fuel cell efficiency [-], $Q_{LHV}$ is the hydrogen lower heating value [MJ/kg], and $\mathrm{s}$ is the virtual specific fuel consumption [kg/kW]. $s$ can reflect the characteristics of the converter, efficiency during charging and discharging, and the SOC of the batteries, but this study considers the coulombic efficiency and the fuel cell efficiency because the power generated from the fuel cell is stored $(\mathrm{s}\left(\mathrm{t}\right)=1/{(\mu }_C·\overline{{\eta }_{FC}}))$.

The constraints (23)–(29) of the batteries and fuel cells must be considered to actually apply the ECMS to hardware. In particular, to maintain a stable battery state, we set the SOC target to 0.6 and applied two penalty functions to maintain the SOC [31]. The penalty functions applied here are as follows:

$$p\left(SOC\right)=1-{\left(\frac{SOC\left(t\right)-{SOC}_{target}}{\left({SOC}^{\mathrm{m}\mathrm{a}\mathrm{x}}-{SOC}^{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}\right)}^a,$$

(33)

$$w\left(SOC\right)=\left\{\begin{array}{c}\begin{array}{cc}K& SOC\left(t\right)<{SOC}^{\mathrm{m}\mathrm{i}\mathrm{n}}\end{array}\\ \begin{array}{cc}-K& {SOC\left(t\right)>SOC}^{\mathrm{m}\mathrm{a}\mathrm{x}}\end{array}.\end{array}\right.$$

(34)

Penalty $p\left(SOC\right)$ of (33) varies with changes in $a$, as shown in Figure 10. The constant $K$ of (34), which determines the charging and discharging of the batteries, is determined by trial and error through repeated simulations. Given that the penalty function is applied and the actual controller is a discrete system, the cost function for the ECMS controller design is as follows:

$$J_{ECMS}\left[k\right]={\dot{m}}_{FC}\left[\mathrm{k}\right]+(\mathrm{p}\left(\mathrm{S}\mathrm{O}\mathrm{C}\right)+\mathrm{w}\left(\mathrm{S}\mathrm{O}\mathrm{C}\right))·{\dot{m}}_{BAT}\left[\mathrm{k}\right].$$

(35)

Since it is a discrete system, we used $k$ instead of $t$ to represent the current control interval. Based on Equation (35), we applied Successive Quadratic Programming (SQP) based on the Newton–Raphson method to determine the solution of nonlinear constrained cost function minimization [32].

3.4. EMPC

In MPCs, a model for the control target is included, and the cost incurred for the prediction step size $p$ (prediction horizon) is predicted based on the model to determine the control input that minimizes the cost for the control step size $m$. Unlike the ECMS, it used a model, which is its primary distinction. The key difference between the EMPC and a typical MPC lies in the cost function. A typical MPC is based on a quadratic cost function that penalizes the deviation between the input and the state variable for the corresponding steady-state value. However, it is a cost function considering process economics, and it is a linear/nonlinear cost function that may not be positive definite. Hence, its purpose is to obtain a control input that results in the most economical fuel economy through the integrated operation of the fuel cells and the batteries.

For the EMPC, the mathematical model Equations (8), (9), and (20) describing the fuel cells and batteries are used. Given the vector ${\mathit{u}}_k={\left[\begin{array}{ccc}{P}_{k,FC}& P_{k,BAT}& \begin{array}{cc}{P}_{k,FC-BAT}& P_{k,load}\end{array}\end{array}\right]}^{\prime }$, the equation for determining the optimal cost function $J^{*}\left(P_k,\dots ,P_{k+p}\right)$ for the prediction horizon $p$ to find the optimal manipulated input ${\mathit{u}}_k^{*}$ is as follows:

$$J^{*}\left({\mathit{u}}_k,\dots ,{\mathit{u}}_{k+p}\right)=\underset{P_{\mathit{FC}},{\mathit{P}}_{\mathit{BAT}}}{\mathit{min}}{J}_{EMPC}\left[k\right],$$

(36)

$$J_{EMPC}\left[k\right]=J_{ref}\left[k\right]+J_{FC}\left[k\right]+J_{BAT}\left[k\right]+J_{SOC}\left[k\right],$$

(37)

$$J_{ref}\left[k\right]=\sum _{i=1}^p{\left\{w_{i,ref}\left(P_{i,load}\left[k\right]-P_{i,FC}\left[k\right]-P_{i,BAT}\left[k\right]\right)\right\}}^2,$$

(38)

$$J_{FC}\left[k\right]=\sum _{i=1}^p\left\{\frac{w_{i,FC}}{{\eta }_{FC}\left[k\right]}\left(P_{i,FC}\left[k\right]+P_{i,FC-BAT}\left[k\right]\right)\right\},$$

(39)

$$J_{BAT}\left[k\right]=\sum _{i=1}^p\left(\frac{w_{i,BAT}}{{\eta }_c·\overline{{\eta }_{FC}}}{P}_{i,BAT}\left[k\right]\right),$$

(40)

$$J_{SOC}\left[k\right]=\sum _{i=1}^p{\left\{w_{i,P}\left(SOC\left[k\right]+∆SOC[k,P_{i,FC-BAT}])-{SOC}_{target}\right)\right\}}^2.$$

(41)

where $w_i$ denotes the weight for each cost function. The $J_{ref}$ term is designed as a quadratic equation so that $P_{load}$ is tracked by $P_{FC}{ \mathrm{a}\mathrm{n}\mathrm{d} P}_{BAT}$. The $J_{FC}$ term is the cost function for the power use of the PEMFC, and $J_{BAT}$ is the cost function for battery power use. For $J_{SOC}\left[k\right]$, the cost function is designed to charge power while tracking the target SOC [33]. The results of all cost functions are positive, and for the EMPC, the solution of the cost function is calculated using SQP to obtain the solution of the nonlinear constructed cost function minimization.

3.5. DP

DP has been used to solve finite horizon multi-stage deterministic decision problems, minimizing the total cost incurred [34]. Even if the system complexity is high, the optimal result can be derived as long as the computational capacities are supported. However, DP is noncausal, that is, it requires the ship’s load profile for the entire optimization horizon. For that reason, it is primarily used in simulation. DP in control theory is designing optimal control input $\mathit{u}\left(t\right)$ to follow a state $\mathit{x}\left(t\right)$ to minimize a cost function $J\left(x_0,u_0,\dots ,u_{N-1}\right)$ for system ${\mathit{x}}_{k+1}\left(t\right)=f({\mathit{x}}_{\mathit{k}},{\mathit{u}}_{\mathit{k}})$ on $k=\mathrm{0,1},\dots ,N-1$. The cost function used to implement DP is as follows:

$$J_{DP}\left({\mathit{x}}_0,{\mathit{u}}_0,\dots ,{\mathit{u}}_{N-1}\right)=g_N\left({\mathit{x}}_N\right)+\sum _{k=0}^{N-1}{g(\mathit{x}}_k,{\mathit{u}}_k).$$

(42)

where ${g(x}_k,u_k)$ is the stage cost for DP, that is, the cost incurred for each stage $k$ and $g_N\left(x_N\right)$ is the terminal cost. Obtaining the optimal solution in this optimization problem is necessary to obtain the control inputs at the minimum total cost, as follows:

$$J_{DP}^{*}\left({\mathit{x}}_0\right)=\underset{{\mathit{u}}_k,k=0,\dots ,N-1}{\min }{J}_{DP}\left({\mathit{x}}_0,{\mathit{u}}_0,\dots ,{\mathit{u}}_{N-1}\right),$$

(43)

$${\mathit{u}}_0^{\mathrm{*}},\dots ,{\mathit{u}}_{N-1}^{\mathrm{*}} =\underset{{\mathit{u}}_k,k=0,\dots ,N-1}{\arg \mathit{min}}{J}_{DP}\left({\mathit{x}}_0,{\mathit{u}}_0,\dots ,{\mathit{u}}_{N-1}\right).$$

(44)

The optimal control sequence ${\mathit{u}}_0^{\mathrm{*}},\dots ,{\mathit{u}}_{N-1}^{\mathrm{*}}$ minimizes cost. The optimization is divided into two phases: a “backward” phase and a “forward” phase. The procedure of the backward and forward phases was referred in [34].

The optimal stage cost function for each stage $k$ is as follows:

$$g_k\left({\mathit{u}}_k\right)=g_{FC}\left[k\right]+g_{BAT}\left[k\right],$$

(45)

$$g_{FC}\left[k\right]=P_{k,FC}/{\eta }_{FC}\left[k\right]+P_{k,FC-BAT}/\overline{{\eta }_{FC}}\},$$

(46)

$$g_{BAT}\left[k\right]={{\eta }_{c}P}_{k,BAT}.$$

(47)

Based on DP, the cost for the entire optimal horizon can be analyzed, and as a terminal cost is involved, we can adjust the terminal cost ${\mathit{x}}_N$ to obtain ${\mathit{u}}_0^{\mathrm{*}},\dots ,{\mathit{u}}_{N-1}^{\mathrm{*}}$, thereby ensuring the desired state for the SOC. Therefore, unlike other controllers, our approach does not require a penalty function to maintain the target SOC as it prepares for a load profile where outcomes are uncertain.

As shown in the analysis in [31,35], DP is used for a global optimization algorithm. However, its real-time application is hindered by its high consumption of computational resources and time. Nevertheless, it proves valuable in selecting an EMS for a ship and analyzing the optimal state for the chosen EMS. While a real-time EMS poses challenges in predicting the final SOC state based on the load profile and comparing fuel efficiency based on the stored SOC, DP allows for the determination of the terminal state, facilitating EMS performance comparison. In the subsequent section, we compare the operation and effectiveness of each controller based on DP and describe the process of selecting the optimal controller.

4. Simulation Results

We applied the same 4-MW-class tugboat load profile to each EMS to compare the performance. Subsequently, for the prepared four types of EMSs (SM, ECMS, EMPC, and DP), we compared the performance based on the load profile-tracking performance and hydrogen usage (fuel efficiency). Moreover, we examined the behavior of the SOC of the battery model based on the 4-MW-class load profile. These data will aid in determining the required fuel cell/battery output and battery capacity based on the specified load profile, fuel cell, and battery model.

To ensure compatibility with the target fuel cells and batteries, we compiled the constraints based on the installation guidelines, outlined in

Table 2. Fuel cell and battery constraints.

Item		MIN	MAX	Unit
PEMFC $\left(P_{FC}+P_{FC-BAT}\right)$	Power	0	4	MW
	* Power Rate	−4	0.4	MW/s
Battery $\left(P_{BAT}-P_{FC-BAT}\right)$	Power	−4	4	MW
	Power Rate	−4	4	MW/s
	SOC	0.1	0.9	-

* 10%/s ramp-up speed considered.

. The PEMFC power rating reflects that the maximum ramp-up speed of the Nedstack 4-MW-Class PEMFC is 10% per second. For the battery constraints, the performance margin is set at the maximum level because the batteries should accommodate all the power that the fuel cells cannot handle. However, when selecting the lithium-battery ESS used for the ship, it is important to consider that the battery is maintained at a proper C-rate (maximum 3[C], [36]) in most cases due to safety issues. The constraints listed in Table 2 are applied to all four controllers being designed.

4.1. Load Profile

A load profile of the tugboat is required to design an optimal EMS for the target ship. Therefore, we obtained the load profile of the 4-MW-class tugboat from Korea Maritime Consultants Co., Ltd.(Busan, South Korea), a South Korean ship design company (Figure 11). The general service of this profile refers to the power consumed other than propulsion power, which does not fluctuate much in relation to the propulsion power. As shown in Figure 11, some fluctuation occurs when maneuvering a stopped ship; however, during normal sea travel, the ship cruises at a constant speed, leading to minimal fluctuations. After that, there are instances when the load temporarily increases during towing operations to berth a large ship. After completing the towing operation, the ship maintains a constant speed toward the dock for berthing (normal sea-going and maneuvering.

4.2. Controller Performance Review

4.2.1. Simulation Configuration

In the case of the ECMS and EMPC, the input solution is determined using the SQP algorithm. Determining the appropriate initial point is crucial for selecting the appropriate optimal solution. This is because, in some situations, the solution of $\mathit{u}\left[k\right]$ may not be found or may be a locally optimal solution depending on the constraints and the model’s state variable $\mathit{x}\left(t\right)$. The SQP algorithm is not designed to find a global optimal solution, and it requires a lot of time and computing resources depending on the method used to obtain the global optimal solution. Hence, this algorithm is difficult to apply for real-time controllers. The solution to the global optimal for the entire load profile was verified using DP, and its performance was compared to that of DP. Therefore, the initial point of the input at $k$ of the ECMS and EMPC was specified identically as ${\mathit{u}}_0\left[k\right]=\left[\begin{array}{ccc}{(P_{FC}^{\mathrm{m}\mathrm{i}\mathrm{n}}+P}_{FC}^{\mathrm{m}\mathrm{a}\mathrm{x}})/2& {(P_{BAT}^{\mathrm{m}\mathrm{i}\mathrm{n}}+P}_{BAT}^{\mathrm{m}\mathrm{a}\mathrm{x}})/2& \begin{array}{cc}0& P_{k,load}\end{array}\end{array}\right]$. The last column of ${\mathit{u}}_0\left[k\right]$ is not a term that affects the solution, because it is a reference that the large PEMFC and batteries must track. For the simulations, we used MATLAB/Simulink. The optimal algorithm was implemented using MATLAB’s Optimization Toolbox.

Regarding the capacity of the lithium-ion batteries selected as the ship’s Energy Storage System (ESS), we deliberately chose a relatively small capacity, distinct from that of typical hybrid and fuel cell ships, to analyze controller dynamics and evaluate EMS performance effectively through SOC behavior.

Table 3. Simple Li-ion battery model specification.

Parameter	Value	Unit
$\mathrm{Open}-\mathrm{circuit} \mathrm{voltage} {(V}_{OC})$	3.5057–4.1928	V
$\mathrm{Cell} \mathrm{capacity} {(C}_{max})$	4.7357	Ah
$\mathrm{Equivalent} \mathrm{resistance} {(R}_{eq})$	0.0392–0.0456	$\Omega$
$\mathrm{Number} \mathrm{of} \mathrm{cells} \left(N_P\right)$	7500	EA
$\mathrm{Total} \mathrm{capacity} {(C}_{BAT})$	133,192	Wh

lists the specifications of the Li-ion battery model applied in this study, along with the parameters used in Equations (19)–(21).

The initial SOC of the Li-ion batteries is 0.5, the simulation time is 180 min, and the step size is 1 s for all the SM, ECMS, EMPC, and DP simulation models.

Table 4. Parameters for SM, ECMS, and EMPC.

Controller	Parameter	Value	Unit
SM	$P_{FC,UL}$	2.21	MW
	$P_{FC,LL}$	1.03	MW
ECMS	${SOC}_{target}$	$0.6$	-
	$a$	3	-
	$K$	0.5	-
EMPC	$w_{i,ref}$	${[1, 0, 0]}_{i=\mathrm{1,2},3}$	-
	$w_{i,FC}$	${[0, 0, 1]}_{i=\mathrm{1,2},3}$	-
	$w_{i,BAT}$	${[0, 0, 1]}_{i=\mathrm{1,2},3}$	-
	$w_{i,P}$	${\left[0, 0, 1300000\right]}_{i=\mathrm{1,2},3}$	-
	$* \overline{{\eta }_{FC}}$	0.6541	-
Common	${SOC}^{\mathrm{m}\mathrm{i}\mathrm{n}}$	0.1	-
	${SOC}^{\mathrm{m}\mathrm{a}\mathrm{x}}$	0.9	-
	${SOC}_{init}$	0.5	-
	${SOC}_{target}$	0.6	-
	${\eta }_{inv}$	0.9	-

* PEMFC efficiency for $P_{FC}^{\mathrm{m}\mathrm{a}\mathrm{x}}/2.$

shows the parameters applied to the SM, ECMS, and EMPC. $w_i, K, \mathrm{a}\mathrm{n}\mathrm{d} a$ are experimentally obtained coefficients.

4.2.2. Controller Behavior Analysis

The SM is designed to maximally use the section (1.03–2.21 [MW]) where the efficiency of the PEMFC is 65% or more. As shown in Figure 12, the part that exceeds 2.21 MW in the SM, $P_{FC}+P_{FC-BAT}$, PEMFC supplies more power over the maximum efficiency section only when the SOC is in the lower limit for battery charging. When the load is less than 1.03 MW, the battery power is used. In the case of the ECMS and EMPC, optimal control is achieved based on the designed cost function. In the case of the ECMS, the optimal control input ${\mathit{u}}_k^{*}$ is obtained only for the specified unit step size. In the case of EMPC, the prediction horizon is set to three step size; thus, the optimal control input for the three step size derived for supplying the appropriate power is distributed to each ESS and PEMFC. $P_{load}$ shows that the load profile is tracked through the sum of $P_{FC}$ and $P_{BAT}$. By comparing the results based on the simulations, we revealed that the SM, ECMS, and EMPC can effectively track the load profile. $P_{FC}$ exceeding the load profile and the charging of $P_{BAT}$ (minus power) indicate the energy flow of $P_{FC-BAT}$.

Figure 13 shows the state of the SM. State 0 represents the battery charging state, state 1 represents the battery propulsion state, state 2 represents the fuel cell propulsion state, and state 3 represents the hybrid propulsion state. The state transition occurs according to the load profile.

The SOC trajectory of each controller is plotted in Figure 14. SM represents the SM. Charging starts from the moment the SOC drops to below 0.3, and the charging continues until the target SOC is achieved. Thus, compared to other controllers, it shows the largest fluctuation. As the SOC behavior of the SM is dominated by the specified ${SOC}_{FC,UL} \mathrm{a}\mathrm{n}\mathrm{d} {SOC}_{FC,LL}$, it is difficult to select the optimal capacity of the batteries. In the case of the optimum algorithm-based controllers, i.e., ECMS and EMPC, it seems that they can be used to select an appropriate capacity of the battery that does not interfere with the SOC behavior when performing the optimal control for the 4-MW tugboat load profile based on SOC behavior profile. The SOC behavior of the ECMS in Figure 14 shows that the SOC moves only between 0.47 and 0.78 (gap: 0.31) to track the load profile. This means that there may be no problem with the behavior if the capacity required for optimal EMS operation is only 31% (41,290 Wh) of the total battery capacity. However, as the ship will not solely operate with this load profile, an appropriate margin is necessary, which can be enhanced to some extent with various load profiles.

In the case of EMPC, the SOC was observed to fluctuate only between 0.5 and 0.59.

Table 5. Start and end SOCs for SM, ECMS, and EMPC.

EMS	$\mathbf{Start} \left({\mathit{S}\mathit{O}\mathit{C}}_{\mathit{i}\mathit{n}\mathit{i}\mathit{t}}\right)$	$\mathbf{End} \left({\mathit{S}\mathit{O}\mathit{C}}_{\mathit{t}\mathit{e}\mathit{r}\mathit{m}}\right)$
SM	0.5	0.403985
ECMS	0.5	0.777764
EMPC	0.5	0.593971

provides details on the initial and final SOC for each controller. However, comparing the performance of each controller proves challenging due to the disparity in the final SOC. This issue will be thoroughly addressed in Section 4.2.3 to identify a solution.

Figure 15 demonstrates the cumulative hydrogen consumption for each controller: 157.664 [kg] for the SM, 161.028 [kg] for the ECMS, and 157.008 [kg] for the EMPC. Although the EMPC consumed the least amount of hydrogen in terms of cumulative consumption. However, we cannot say that the EMPC has the best fuel efficiency EMS when the energy stored in the SOC is considered. Therefore, in the next section, we evaluated the optimality and performance of each controller at the same time by comparing the amount of fuel consumption(hydrogen) using DP, which can obtain the global optimal solution.

4.2.3. DP-Based Performance Comparison

Depending on the operation method of each controller, some fuel consumed was stored in the SOC, making it difficult to compare fuel efficiency accurately. Therefore, the ending point of the SOC was specified through the terminal cost $g_N\left({\mathit{x}}_N\right)$, and the optimality of each controller was examined based on DP, which is a global optimal algorithm. Furthermore, the fuel efficiency and performance were compared based on the difference in hydrogen consumption in DP. The optimal input derived through DP was designed based on the battery model introduced in Section 2.2, not the MATLAB/Simulink model. To perform DP computations and secure the computation speed, we divided the state of the SOC and the input that distributes power into 0.01 units discretely. The ${SOC}_{term}$ values obtained from the SM, ECMS, and EMPC were entered into DP(Table. 5), and the obtained global optimal input ${\mathit{u}}_k^{*}$ was entered into the EMS in the simulations. The simulation results are shown in Figure 16, Figure 17 and Figure 18.

There is a subtle model difference between DP and MATLAB/Simulink, and there may be no solution to ensure that ${SOC}_{term}$ obtained from each controller simulation is accurately reached. For that reason, we allowed an error of about 0 to 0.01, but the results of each EMS show that the error in ${SOC}_{term}$ is less than 0.001 (Figure 18,

Table 6. The actual DP ${SOC}_{term}$ results after aligning DP’s ${SOC}_{term}$ with each EMS ${SOC}_{term}$(simulation error).

Target EMS	$\mathbf{Target} {\mathit{S}\mathit{O}\mathit{C}}_{\mathit{t}\mathit{e}\mathit{r}\mathit{m}}$	$\mathbf{DP} {\mathit{S}\mathit{O}\mathit{C}}_{\mathit{t}\mathit{e}\mathit{r}\mathit{m}}$
SM	0.403985	0.400441
ECMS	0.777764	0.773405
EMPC	0.593971	0.598299

).

Figure 18 and

Table 7. Tracking results of ${SOC}_{term}$ for each EMS based on DP.

EMS	H2 Consumption [kg]	H2 Consumption-DP [kg]	Deviation [kg]	${\mathit{S}\mathit{O}\mathit{C}}_{\mathit{t}\mathit{e}\mathit{r}\mathit{m}}$ [-]
SM	157.664	152.933	4.731	0.40
ECMS	161.028	161.17	−0.142	0.77
EMPC	157.008	157.086	−0.78	0.59

present the hydrogen consumption for the three real-time controllers based on DP, satisfying the ${SOC}_{term}$ of each EMS. Theoretically, the hydrogen consumption of DP should be the lowest, but the real-time optimization-based algorithm outperformed DP due to differences in the model and cost function structure. It was found that the SM, a rule-based EMS, has the largest deviation in hydrogen consumption compared to DP. On the other hand, it was found that the EMPC, which has the most state information for driving compared to the other controllers, has the largest minus deviation compared to DP. This means that the EMPC is well-optimized and exhibits the best performance for the current profile considering that DP is a global optimization algorithm.

5. Conclusions

We designed the SM, ECMS, EMPC and DP to select and apply an EMS algorithm to a 4-MW-class tugboat. The PEMFC and Li-ion batteries were mathematically modeled, and each EMS were applied to evaluate performance in MATLAB simulation. Based on the given tugboat’s load profile and model, simulation was conducted. The goal of the simulation was to evaluate the fuel efficiency performance of each EMS under conditions where the load profile is fully tracked and to select the EMS with the highest fuel efficiency.

Comparing the performance of each EMS solely based on fuel consumption proved challenging due to the need to account for the energy stored in the batteries. Consequently, to facilitate a comprehensive comparison of EMS performance, we employed DP, a non-real-time global optimization algorithm. In this approach, we inputted the initial and terminal states of charge of DP, which were determined based on the behavior of the derived SOC, to obtain a global optimal solution for the entire load profile duration. Subsequently, by inputting this solution into the EMS, we compared the optimal performance states and fuel consumption rates within each EMS. Our analysis revealed that EMPC, equipped with relatively more information to steer the controller, exhibited the most optimal performance. Despite the real-time optimization algorithm outperforming DP owing to differences in cost function structures and model designs, DP remains a valuable tool for comparing the performance of the SM, ECMS, and EMPC.

Given that our study relied on only one load profile, extrapolating results to other load profiles presents challenges. Hence, there is a need for further improvement, achievable through performance comparisons and validations using diverse profiles. Additionally, it is essential to acknowledge that the EMS employed in actual ships operates based on signals from the ship’s control room rather than real-time demand power input.

Validating the performance of hydrogen fuel cell-powered ships through empirical testing is extremely challenging due to the high costs associated with equipment and fuel. Through this study, we aim to enhance the simulation technology and reliability of EMS for fuel cell-battery ships, thereby reducing development costs for actual implementation and promoting the wider adoption of hydrogen-powered vessels in the future.