Health-Conscious Optimization of Long-Term Operation for Hybrid PEMFC Ship Propulsion Systems

Abstract

The need to decarbonize the shipping sector is leading to a growing interest in fuel cell-based propulsion systems. While Polymer Electrolyte Membrane Fuel Cells (PEMFC) represent one of the most promising and mature technologies for onboard implementation, they are still prone to remarkable degradation. The same problem is also affecting Lithium-ion batteries (LIB), which are usually coupled with PEMFC in hybrid powertrains. By including the combined degradation effects in an optimization strategy, the best compromise between costs and PEMFC/LIB lifetime could be determined. However, this is still a challenging yet crucial aspect, rarely addressed in the literature and rarely yet explored. To fill this gap, a health-conscious optimization is here proposed for the long-term minimization of costs and PEMFC/LIB degradation. Results show that a holistic multi-objective optimization allows a 185% increase of PEMFC/LIB lifetime with respect to a fuel-consumption-minimization-only approach. With the progressive ageing of PEMFC/LIB, the hybrid propulsion system modifies the energy management strategy to limit the increase of the daily operation cost. Comparing the optimization results at the beginning and the end of the plant lifetime, daily operation costs are increased by 73% and hydrogen consumption by 29%. The proposed methodology is believed to be a useful tool, able to give insights into the effective costs involved in the long-term operation of this new type of propulsion system.

1. Introduction

In the last few decades, an increasing number of restrictions has been introduced in the transportation sector in order to reduce greenhouse gases and pollutant emissions. This has posed new technological challenges, and the need to explore new solutions is ever more urgent. In particular, growing efforts have been placed in reducing the emissions of the shipping sector, where the new and upcoming restrictions require a thorough revision of the whole ship building process. The urge to develop and upscale new technological solutions for decarbonizing the maritime transportation sector has promoted the research and development of different strategies. Among them, fuel cell-based propulsion has gained attention, as was demonstrated by several national and international initiatives on this topic [1,2,3,4]. As emerged from a previous study by the authors [3], one of the most promising solutions employs hydrogen-fuelled Polymer Electrolyte Membrane Fuel Cells (PEMFC), hybridized with a Battery Energy Storage System (BESS). In fact, it is well known that hydrogen-fueled PEMFC can potentially guarantee zero-emission propulsion: this feature makes their use fully compliant even to the most stringent regulations. As has already been demonstrated in several analyses, a small to medium sized vessel could benefit from the installation of hybrid PEMFC/BESS propulsion systems. In fact, the technical and regulatory barriers are currently limiting the installed power of PEMFC to few hundred kW, as highlighted in

Table 1. List of projects on hybrid PEMFC/BESS propulsion systems. Elaborated from [3].

Project Name	Start Date	End Date	PEMFC Power	Battery Type and Capacity	Ref.
Elektra and Elektra 2 (e4ships consortium)	2017	2024	300 kW	LIB, 2.5 kWh capacity	[6,7]
Energy observer	2017	2024	22 kW	LIB, 122 kWh propulsion, 18 kWh daily facilities	[8,9]
Aero 42	-	2023	2800 kW	LIB, 672 kWh	[10,11]
Race for water	2015	2021	60 kW	LIB, 745 kWh	[12]
Future Proof Shipping	-	2021	635 kW	LIB, 300 kWh	[13]
Kamine boat	2018	2020	60 kW	LIB, 60 kWh	[14,15]
Water Go Round Golden Gate Zero Emission	-	2020	360 kW	100 kWh, no data on the type	[16,17]
Zeff (part of Pilot-E scheme)	2018	2020	2200 kW	50 kWh estimated; type of batteries under evaluation	[18,19]
Zero V	-	2017	1800 kW	LIB provide for 200 kW power	[20,21]
Busan tourist boat	-	2016	56 kW	LIB, 47 kWh	[22]
ZEMSHIP FSC Alsterwasser	2006	2013	100 kW	Lead gel, 560 V (7 × 80 V) 360 Ah	[23,24,25]
Nemo H2	2008	2011	60–70 kW	Lead acid, 55 kW	[26]

and extracted from [3]. As for BESS, Table 1 also points out that the most recent projects use Lithium-Ion Batteries (LIB), as LIB are the commercial product with the best characteristics in terms of specific energy [5].

The main drawback of such hybrid PEMFC/LIB powertrains is that both PEMFC and BESS are prone to degradation, which is responsible for the overall reduction of the plant lifespan. Hence, a correct design and management of hybrid PEMFC/LIB powertrains is fundamental in order to achieve their cost effectiveness [27,28]. The management of hybrid PEMFC/LIB powertrains also includes the energy-efficient power allocation, usually addressed by the definition of an Energy Management Strategy (EMS). The latter can be defined following different approaches, among which rule-based and optimization-based ones are the most investigated in the literature on ship propulsion systems.

As for rule-based EMSs, an example is provided by Taccani et al. [29], who analyzed the implementation of a hybrid PEMFC/LIB power system for the propulsion of a small-sized ferry. The authors proposed a rule-based EMS in which PEMFC work at a constant load in order to cover the average power demand of the vessel, while LIB are used for peak shaving. Moreover, Han et al. [30] proposed a state-based EMS to control the energy flows between the PEMFC and the batteries of the hybrid ship Alsterwasser depending on the vessel power demand and the battery State of Charge (SOC) [23,24,25]. The same vessel was also studied by Bassam et al. [31], who proposed an improved state-based EMS for the hybrid PEMFC/battery system taking into account the PEMFC efficiency as input to the proportional-integral EMS. By adopting the proposed EMS, up to 3.5% of hydrogen daily consumption could be saved in comparison with a previously proposed strategy [30]. The main advantages of rule-based EMSs lie in their simplicity of use, which makes them practical for onboard applications. As a drawback, their operative principle is based on human expertise, and they could easily reflect human errors generated while defining the control strategy.

To overcome this issue, some studies proposed optimization-based EMSs [32,33,34,35,36,37,38,39]. For example, Rivarolo et al. [32,33] developed two computational tools dedicated to the preliminary design and the thermo-economic analysis of different engineering solutions for ship power systems, including PEMFC and Solid Oxide Fuel Cells (SOFC). The proposed methodology provided the best energy system configuration in terms of environmental impact, cost, and overall dimensions. Bassam et al. [34] defined a multi-scheme EMS that minimizes the equivalent fuel consumption over one operation day of the hybrid PEMFC/BESS passenger ship Alsterwasser [23,24,25]. The developed EMS resulted in a 16.7% reduction of the total hydrogen consumption with respect to previously proposed EMSs. Zhang et al. [35] designed a real-time optimization EMS that took into account the degradation of PEMFC and used a high-pass filter to prolong battery life. A similar approach was followed by Chen et al. [36], who employed a whale optimization algorithm to optimize size and frequency for a hybrid battery/supercapacitor fuel cell ferry. Wu et al. [37] included the calculation of PEMFC degradation in the optimization of a hybrid plug-in PEMFC/BESS ferry for coastal navigation, defining the optimal size of the system by considering both greenhouse gas emissions and economic performances. For the same ferry, Wu et al. [38] recently proposed a reinforcement learning approach for the definition of an adaptive EMS. In both [37,38], the BESS degradation rate was set as a constant in each time step, independently from the operating conditions. Pivetta et al. [39] proposed a multi-objective design and operation optimization of three different ferries for coastal navigation, adopting a Mixed-Integer Linear Programming (MILP) approach based on a simultaneous minimization of PEMFC degradation, investment cost and operation cost. In this case, battery degradation was set as a constraint of the optimization model. As a main outcome, it was found that in order to limit both battery and PEMFC degradation, the battery capacity should be increased by up to 136%.

Although EMSs taking into account system degradation (also known as health-conscious EMSs) are of great interest for the shipping sector, the interplay between PEMFC and BESS degradation has been rarely addressed as of yet. Interesting contributions on this point can be found in the recent literature on hybrid PEMFC/BESS powertrains for other types of applications [40,41,42,43,44,45,46,47]. Some remarkable examples have been provided by Ҫinar et al. [43], Liang et al. [44], and Wang et al. [45]. The former group [43] used meta-heuristic algorithms to optimize hydrogen consumption in the aircraft sector. The second one [44] compared two EMSs (composite fuzzy control versus Pontryagin’s minimum principle) for hybrid PEMFC/LIB logistics, while the third one [45] proposed a health-conscious EMS for a hybrid PEMFC/LIB bus. Both PEMFC and LIB degradation were included as the objectives in a cost-optimization problem. In addition, Li et al. [46] studied an online adaptive EMS for a PEMFC hybrid vehicle, which takes into account the degradation of PEMFC, batteries, and supercapacitors. The authors concluded that the degradation of one of the power sources implied an increase in the degradation of the other ones. Moreover, they proved that the proposed EMS guaranteed normal plant operation also in case of plant source degradation. However, a holistic analysis throughout the whole plant lifetime was not addressed, and a quantitative plant lifetime estimation was not provided. Lastly, a comprehensive review on health-conscious EMSs for hybrid fuel cell vehicles was proposed by Yue et al. [47], who concluded that future trends in this research field should address the multi-objective optimization of a hybrid PEMFC/LIB system including degradation modeling and lifetime estimation methods of the power sources, possibly limiting the overall complexity of the problem.

Overall, the literature review pointed out the importance of considering the degradation of both PEMFC and LIB in optimizing the entire hybrid ship-propulsion system. However, it emerged that a holistic optimization procedure that takes into account the progressive worsening of both PEMFC and BESS performances with the progressive ageing over the plant lifetime has not been addressed yet. This indeed would be fundamental for a complete understanding of the costs involved in the operation of hybrid PEMFC/LIB propulsion systems. To fill this gap, the present study proposes an innovative methodology that (i) evaluates the degradation over time of both PEMFC and LIB, (ii) optimizes the operation of a hybrid PEMFC/LIB powertrain over the entire lifetime, and (iii) simultaneously minimizes the fuel consumption and the power source degradation rates. As is widely recognized, the MILP approach is a good candidate for optimizing complex energy systems because it allows a reduction in the computation effort with respect to other optimization techniques [48,49,50,51]. Therefore, in this study, MILP equations have been used to describe the hybrid PEMFC/LIB operation and a multi-objective optimization has been performed.

A small-size ferry for coastal navigation has been taken as case study to perform the optimization of operation over the entire power plant lifespan. The design of the system previously optimized in [39] has been taken as a reference for the operation optimization here proposed. Here, Lithium Iron Phosphate (LFP) chemistry has been chosen as LIB. Indeed, among different LIB chemistries Nickel Manganese Cobalt (NMC) and LFP batteries are addressed as the most mature technologies for shipping, but LFP are slightly better in terms of safety onboard [5].

The first part of the paper describes the methodology proposed for the analysis. Afterwards, the case study is described, and the results are presented and discussed.

2. Methodology

This section shows the main characteristics of the proposed method, with a particular focus on the equations describing the energy conversion and storage units. Given the complexity of the multi-objective optimization, a MILP approach has been adopted to limit the overall computational effort. Therefore, operation and performance degradation of the energy units, i.e., LFP battery and PEMFC, have been described with MILP equations. In the next paragraphs, a description of the procedure followed in the analysis is reported.

At first, the preliminary sizing of the hybrid PEMFC/LFP battery propulsion system has been determined by implementing the design and operation optimization model developed in [39]. The obtained sizes for PEMFC and LFP battery systems have been taken as input parameters of the new optimization approach for the long-term operation.

Starting from time $t_i$ = 0, corresponding to a novel installation of both PEMFC and LFP battery, consecutive optimizations have been run in a loop until either PEMFC or LFP battery reached the end of their life. In order to limit the computational effort, it has been assumed that the first day of the month i is representative for the operation of all the days of the month (assuming all months have 30 days). Operation optimization has been performed for the day representative of the entire month i. The cumulative degradation of PEMFC and LIB calculated at the end of month i has been taken as an input parameter for the operation optimization of month i + 1.

The optimization problem has been developed in Python [52] and solved with the MILP optimizer Gurobi Optimization [53]. The general formulation of the MILP optimization problem has been set to find the optimal value of the generic continuous variables $x^{*}\left(t\right)$ and binary variables ${\delta }^{*}\left(t\right)$ that maximize or minimize the set objective function(s) $Z$ (Equation (1)). The MILP problem is subject to equality constraints $g\left(t\right)$ and inequality constraints $h\left(t\right)$ (Equations (2) and (3)), which include the description of the modeled ship energy system. The continuous and binary variables represent the decision variables of the optimization problem. In particular, binary variables have been implemented to decide about activation/de-activation of each energy unit during operation.

$$Z=f\left(x^{*}\left(t\right), {\delta }^{*}\left(t\right)\right)$$

(1)

$$g\left(x^{*}\left(t\right), {\delta }^{*}\left(t\right)\right)=0$$

(2)

$$h\left(x^{*}\left(t\right), {\delta }^{*}\left(t\right)\right)\le 0$$

(3)

In the following subsections, the main constraints describing the energy system operation and the degradation models of PEMFC and LFP battery are presented, i.e., the relationships $g\left(t\right)$ and $h\left(t\right)$ in Equations (2) and (3). Subsequently, the objective functions of the optimization problem, i.e., the objective functions $Z$ in Equation (1), and the solving procedure are illustrated and discussed.

2.1. Energy System Constraints

It has been assumed that at each time step t the ship energy demand $P_{demand}\left(t\right)$ (propulsion and auxiliaries power demand) is fulfilled by the power output of the $n$ fuel cell stacks $P_{F{C}_j}\left(t\right)$ and by the discharging power of battery $P_{batt}^{+}\left(t\right)$. $P_{batt}^{-}\left(t\right)$ indicates the charging power of battery, which is charged by fuel cells whenever $P_{F{C}_i}\left(t\right)$ is higher than $P_{demand}\left(t\right)$. The overall power balance has been set as (Equation (4)):

$${\displaystyle \sum }_{j=1}^n{P}_{F{C}_j}\left(t\right)+P_{batt}^{+}\left(t\right)=P_{demand}\left(t\right)+P_{batt}^{-}\left(t\right)$$

(4)

Equations (5) and (6) are the constraints adopted to limit the proposed energy system volume and weight, set to be lower than the values of volume $V_{max}$ and weight $w_{max}$ of the ferry existing energy system (i.e., internal combustion engines powered by marine gasoil). A correction factor ($c_{ov})$ has been introduced to consider the current limited development state of PEMFC and LFP battery technologies.

$${\displaystyle \sum }_{j=1}^n{\int }_{t_{in}}^{t_{fin}}\left(F_{F{C}_j}\left(t\right)·dt\right)·V_{H_2}+E_{batter{y}_{max}}·V_{batt}+n·P_{F{C}_{max}}·V_{FC}\le V_{max}·\left(1+c_{ov}\right)$$

(5)

$${\displaystyle \sum }_{j=1}^n{\int }_{t_{in}}^{t_{fin}}\left(F_{F{C}_j}\left(t\right)·dt\right)·w_{H_2}+E_{batter{y}_{max}}·w_{batt}+n·P_{F{C}_{max}}·w_{FC}\le w_{max}·\left(1+c_{ov}\right)$$

(6)

where $t_{in}$ and $t_{fin}$ identify the time interval considered for the operation of ferry without refueling. With reference to Equation (5), the volume occupied by hydrogen is expressed by ${\int }_{t_{in}}^{t_{fin}}\left(F_{F{C}_i}\left(t\right)·dt\right)·V_{H_2}$, where $F_{F{C}_j}\left(t\right)$ is the fuel consumption of the stack j at time t and $V_{H_2}$ is the volume occupied by hydrogen and its storage system. The volume occupied by LIB is expressed as $E_{batter{y}_{max}}·V_{batt}$, where $E_{batter{y}_{max}}$ is the energy capacity of the battery and $V_{batt}$ is the specific volume. The volume required by PEMFC is determined as $n·P_{F{C}_{max}}·V_{FC}$, with $n$ being the number of installed PEMFC stacks, $P_{F{C}_{max}}$ the fuel cell stack rated power and $V_{FC}$ the specific volume of a PEMFC stack. Similarly, in Equation (6) the weight of hydrogen is expressed as ${\int }_{t_{in}}^{t_{fin}}\left(F_{F{C}_j}\left(t\right)·dt\right)·w_{H_2}$, being $w_{H_2}$ the weight of bunkered hydrogen and its storage system. Weight of LIB is calculated as $E_{batter{y}_{max}}·w_{batt}$, where $w_{batt}$ is the specific weight of LIB, and weight of PEMFC is $n·P_{F{C}_{max}}·w_{FC}$, where $w_{FC}$ is the specific weight of PEMFC stack.

2.2. Polymer Electrolyte Membrane Fuel Cells Model

It has been considered that separate PEMFC stacks with fixed rated power are installed on board. Each stack has been assumed to operate independently from the others. Stacks with different rated power or with different performances could be included in the model by modifying the stack operation parameters (see Section 3).

The operation of the PEMFC stack j is described by the following MILP equations (Equations (7)–(11)):

$$F_{F{C}_j}\left(t\right)=\left(k_{1_F}·I_{F{C}_j}\left(t\right)+k_{2_F}\right)·{\delta }_{F{C}_j}\left(t\right)+{\delta }_{st,u{p}_j}\left(t\right)·F_{start}·P_{F{C}_{max}}$$

(7)

$$P_{F{C}_j}=\left(k_{1_P}·I_{F{C}_j}\left(t\right)+k_{2_P}\right)·{\delta }_{F{C}_j}\left(t\right)$$

(8)

$$I_{F{C}_{min}}·{\delta }_{FC}{}_j\left(t\right)\le I_{F{C}_j}\left(t\right)\le I_{F{C}_{max}}·{\delta }_{F{C}_j}\left(t\right)$$

(9)

$$\left|P_{F{C}_j}\left(t\right)-P_{F{C}_j}\left(t-1\right)\right|\le {\mathsf{\Delta }}_{P_{FC}}$$

(10)

$$0\le {\delta }_{F{C}_j} \left(t\right)-{\delta }_{F{C}_j} \left(t+1\right)+{\delta }_{st,u{p}_j}\left(t\right)$$

(11)

where $I_{F{C}_j}$ is the current density (in A/cm²), $k_1{}_F$, $k_2{}_F$, $k_1{}_P$ and $k_2{}_P$ are the linearization coefficients, ${\delta }_{F{C}_j}$ is the binary variable defining on/off status, ${\delta }_{st,u{p}_j}$ is the binary variable defining the occurrence of a start-up phase (if ${\delta }_{st,u{p}_j}$ = 1, it will be counted as one start/stop cycle), $F_{start}$ is the fuel consumption in the start-up phase, $I_{F{C}_{min}}$ and $I_{F{C}_{max}}$ are the limiting values for the current density, and ${\mathsf{\Delta }}_{P_{FC}}$ is the allowed load variation.

The PEMFC stack degradation is a complex phenomenon that affects the performances of several stack components (e.g., electrodes, membranes, bipolar plats, gas diffusion layers). Degradation effects may be different from one cell to another, e.g., near the edges, cells are subject to a faster degradation [54]. Besides, at the component level, the degradation mechanisms are hard to estimate. As reported in [47], the ageing mechanisms in mobility application could be analyzed through three possible approaches: impedance estimation based on electrochemical impedance spectrometry, remaining useful life estimation, and a stack voltage degradation model. Even though the latter approach is less accurate with respect to other degradation models and strongly depends on experimental data, voltage degradation can be expressed by MILP equations, allowing a sensible reduction of the computational effort required by an optimization with respect to, for example, Mixed-Integer Quadratic Programming (MIQP).

Therefore, the stack voltage degradation model has been adopted in this study to describe the PEMFC ageing. PEMFC voltage loss has been considered to depend mainly on three operating conditions: the stack load variation, the start/stop cycle and the idling/high-current operation of the stack [55]. The voltage loss caused by the stack degradation has been expressed as the voltage reduction of a single cell at an equal current density output, as proposed in other studies available in the literature [37,56,57,58].

Performance degradation of the PEMFC stack j at time t is described by Equations (12)–(15).

$$d{V}_{loa{d}_j}\left(t+1\right)=\left|P_{F{C}_j}\left(t\right)-P_{F{C}_j}\left(t+1\right)\right|·\mathsf{\Delta }{v}_{load}$$

(12)

$$d{V}_{st,u{p}_j}\left(t\right)={\delta }_{st,u{p}_j}\left(t\right)·\mathsf{\Delta }{v}_{st,up}$$

(13)

$$d{V}_{P_{F{C}_j}}\left(t\right)=k_{1_{dv}}·I_{F{C}_j}\left(t\right)+k_{2_{dv}}$$

(14)

$$d{V}_j\left(t\right)=d{V}_{loa{d}_j}\left(t\right)+d{V}_{st,u{p}_j}\left(t\right)+d{V}_{P_{F{C}_j}}\left(t\right)$$

(15)

where $d{V}_{loa{d}_j}\left(t\right)$ is the voltage reduction due to load variation, $\mathsf{\Delta }{v}_{load}$ and $\mathsf{\Delta }{v}_{st,up}$ are the proportionality constants, $d{V}_{st,up,j}\left(t\right)$ is the voltage reduction due to start-up, $d{V}_{P_{FC,j}}\left(t\right)$ is the voltage reduction depending on the power load, $k_{1_{dV}}$ and $k_{2_{dV}}$ are the linearization coefficients, $d{V}_j\left(t\right)$ is the total loss of voltage at time t.

The cumulative voltage degradation evaluated for the single cell modifies the characteristic curves of the entire stack. In particular, the relation between the cumulative loss of voltage (Equation (16)) and power has been approximated by a linear curve, which relates voltage loss to the angular coefficient of the power curve ($k_{1_P}$), as described in Equation (17). The SOH for PEMFC stack ($SO{H}_{PEMFCj}$) has been defined as in Equation (18).

$$d{V}_{TOT,j}=\int d{V}_j\left(t\right)·dt$$

(16)

$$k_{1_P}=k_{1_{deg}}·d{V}_{TOT,j}+k_{2_{deg}}$$

(17)

$$SO{H}_{PEMF{C}_j}=\frac{V_{ref,FC}-d{V}_{TOT,j}}{V_{ref,FC}}$$

(18)

where $d{V}_{TOT,j}$ is the cumulative loss of voltage, $k_{1_{deg}}$ and $k_{2_{deg}}$ are the linearization coefficients, $V_{ref,FC}$ is the reference maximum voltage for a novel single cell (voltage at the minimum current density $I_{F{C}_{min}}$).

It has been assumed that $k_{1_F}$ and $k_{2_F}$ are not affected by the stack degradation $d{V}_{TOT,j}$. Therefore, when the power loss caused by degradation increases, the efficiency of the stack tends to decrease. The stack efficiency ${\eta }_{FC}$ has not been included as a decisional variable in the optimization problem, but it has been evaluated as a result (Equation (19)).

$${\eta }_{FC}=P_{F{C}_j}/F_{F{C}_j}$$

(19)

2.3. Lithium Iron Phosphate Battery Model

At each time step t, the energy stored in the battery $E_{batt}\left(t\right)$ has been defined according to the charging/discharging power of the battery $P_{batt}^{\pm }\left(t\right)$ and the relative efficiency ${\eta }_{batt}$, assumed constant and equal in the charging/discharging phase (Equation (20)). The SOC of the battery at time t has been calculated as the ratio between $E_{batt}\left(t\right)$ and the energy capacity of the battery $E_{batter{y}_{max}}$ (Equation (21)). The battery C-rate has been determined as the ratio between the charging/discharging power of the battery and the battery capacity (Equation (22)), and the C-rate has been constrained in order not to exceed the maximum C-rate ($Crat{e}_{max}$) achievable by the battery (Equation (23)). A constraint has been set on the SOC in the first-time step of the day $t_0$ to make it correspond to the SOC at the end of the day $t_{fin}$ (Equation (24)).

$$E_{batt}\left(t\right)=E_{batt}\left(t-1\right)+\left({\eta }_{batt}·P_{batt}^{-}\left(t\right)-\left(1/{\eta }_{batt}\right)· P_{batt}^{+}\left(t\right)\right)·\mathsf{\Delta }t$$

(20)

$$SOC\left(t\right)=\frac{E_{batt}\left(t\right)}{E_{batter{y}_{max}}}$$

(21)

$$Crat{e}^{\pm }\left(t\right)=\frac{P_{batt}^{\pm }\left(t\right)}{E_{batter{y}_{max}}}$$

(22)

$$\left|Crat{e}^{\pm }\left(t\right)\right|\le Crat{e}_{max}$$

(23)

$$SOC\left(t_0\right)=SOC\left(t_{fin}\right)$$

(24)

As for battery degradation, the most dominant ageing phenomenon of LIB is the formation of a Solid Electrolyte Interface (SEI) at the electrode/electrolyte interface. The thickness of SEI increases over time, leading to a progressive increase in the impedance of the battery and a consequent decrease in the battery capacity. Different degradation models are available in the literature, based either on electrochemical/equivalent circuit methods (e.g., SEI film thickness models and internal resistance models), or on empirical/semi-empirical methods (e.g., capacity fade models) [47,59]. In this study, it has been chosen to model the battery degradation over time with a capacity fade model. In fact, such an approach allows to limit the overall computational burden with respect to electrochemical models, allowing a comprehensive analysis of the entire system.

Commonly, battery ageing can be classified into two categories: calendar ageing and cycle ageing. Calendar ageing occurs when the battery is at rest condition, i.e., when no current is flowing through the battery. Calendar degradation rates depend on SOC and temperature. Cycle ageing occurs when the battery is charged or discharged, and depends on battery C-rate, SOC, temperature, number of performed equivalent cycles, and Depth of Discharge (DOD) [60,61].

In the proposed study, the hybrid propulsion system has been considered to power a small-size ferry for public transport, which operates 24 h, seven days a week. Therefore, it is reasonable to assume that only cycle ageing occurs on the battery. Nevertheless, to prevent an excessive calendar ageing of the battery in case of a prolonged stop of the vessel it has been decided to set a constraint on the SOC at the end of each day of operation (Equation (25)). The reference $SO{C}_{cal}$ has been set equal to 60%, according to the data available in the literature [62].

$$SOC\left(t_{fin}\right)=SO{C}_{cal}$$

(25)

Cycle ageing has been evaluated following the experimental studies available in [61] for a single LFP cell. As done for PEMFC (see Section 2.2), in this case it has also been considered in terms of the degradation of a single cell, representative for the whole battery. At each time step t of an operative day, $T_{batt}\left(t\right)$ has been expressed as linear function of $Crate\left(t\right)$, through the linearization coefficients $a_{1T,x}$ and $a_{2T,x}$ that depend on the ambient temperature (Equations (26) and (27)). The capacity fade of battery $Q_{loss}\left(t\right)$ has been calculated as a linear function of battery temperature $T_{batt}\left(t\right)$ (Equation (28)).

$$T_{batt}\left(t\right)=a_{1T,x}·Crate\left(t\right)+a_{2T, x}$$

(26)

$$a_{1T,x}=f\left(T_{amb,x}\right); a_{2T,x}=f\left(T_{amb,x}\right)$$

(27)

$$Q_{loss}{}_i\left(t\right)=a_{1QT,i}·T_{batt}\left(t\right)+a_{2QT,i}$$

(28)

The time dependency of $Q_{loss}\left(t\right)$ in the long-term operation has been taken into account by determining the linearization coefficients $a_{1QT,x}$ and $a_{2QT,x}$ in Equation (27) at each month of the plant lifetime. In particular, $a_{1QT,x}$ and $a_{2QT,x}$ at the month i have been evaluated as a function of the energy throughput $A{h}_i$ at month I, as expressed in Equation (29).

$$a_{1QT,i}=k_{11,Ah}·A{h}_i^2+k_{21,Ah}·A{h}_{i }+k_{31,Ah}$$

(29)

$$a_{2QT,i}=k_{12,Ah}·A{h}_i^2+k_{22,Ah}·A{h}_{i }+k_{32,Ah}$$

(30)

where $k_{11,Ah}$, $k_{21,Ah}$, $k_{31,Ah}$, $k_{12,Ah}$, $k_{22,Ah}$, and $k_{32,Ah}$. Note that Equations (29) and (30) have been evaluated at the beginning of the optimization and hence have not been formulated as MILP. Equation (31) expresses the energy throughput $A{h}_i$ at month i.

$$A{h}_i=n_{days}·N_{cycle}{}_{i-1}·DOD·A{h}_{cell}$$

(31)

where $n_{days}$ is the number of days per month (assumed all equal to 30 days), $N_{cycl{e}_{i-1}}$ is the number of equivalent cycles performed by the battery during one operation day, representative for the month i − 1. $N_{cycle}$ has been determined via a rainflow algorithm. $A{h}_{cell}$ indicates the capacity of the single battery cell. DOD effect on the capacity fade has not been directly taken into account in the evaluation of $Q_{loss}\left(t\right)$, since its effect on the capacity fade is negligible if compared to the cycling effect [61]. As for the SOC effect, a constraint has been set (Equation (32)) to limit the SOC-window of operation and hence further limit the degradation, as proposed for example in [63].

$$SO{C}_{min }\le SOC\left(t\right)\le SO{C}_{max}$$

(32)

At the end of each month i, the SOH of the battery ($SO{H}_{batter{y}_i}$) has been determined, as in Equation (32).

$$SO{H}_{batter{y}_i}=\frac{E_{battery,max}-Q_{los{s}_i}}{E_{battery,max}}$$

(33)

2.4. Objective Functions

A multi-objective optimization has been performed to determine the optimal long-term operation of the proposed FC/LFP battery hybrid system, taking into account the fuel consumption and the degradation of both PEMFC and LFP battery over their lifetime.

Three objective functions have been specified, namely fuel consumption ($f_1$ in Equation (34)), PEMFC performance degradation ($f_2$ in Equation (35)), and LFP battery performance degradation ($f_3$ in Equation (36)). For the multi-objective optimization, a “Blended Objectives” method has been adopted, considering a linear combination of the objective functions, each with a fixed weight [53]. In this case, a cost-related weight has been assumed for each objective function. The weight of $f_1$ ($w_1$ in Equation (38)) depends on the cost of hydrogen. Weights of $f_2$ and $f_3$ ($w_2$ and $w_3$ in Equations (39) and (40)) derive from the cost of components and their lifetime. The multi-objective optimization has been performed by minimizing the linear combination of the three objective functions, defined as $f_{MO}$ (Equation (37)).

A 1% deviation from the optimal value of the $f_{MO}$ function has been allowed.

$$f_1={{\displaystyle \sum }}_{j=1}^n{\int }_0^{t_{fin}}{F}_{F{C}_j}\left(t\right) dt$$

(34)

$$f_2={{\displaystyle \sum }}_{j=1}^n{\int }_0^{t_{fin}}d{V}_j\left(t\right)dt$$

(35)

$$f_3={\int }_0^{t_{fin}}{Q}_{loss battery}\left(t\right)dt$$

(36)

$$f_{MO}=Minimize \left(w_1·f_1+w_2·f_2+w_3·f_3\right)$$

(37)

$w_i$ parameters have been defined as in Equations (38)–(40).

$$w_1=c_{H_2}$$

(38)

$$w_2=c_{FC}·\frac{P_{F{C}_{max}}}{V_o}$$

(39)

$$w_3=c_{batt}·\frac{E_{battery,max}}{Q_{los{s}_{max}}}$$

(40)

where $c_{FC}$ is the cost of a PEMFC stack (€/kW), and $V_o$ is the total voltage loss permitted for the single cell of the PEMFC stacks (set equal to 20% of $V_{ref,FC}$). Similarly, $c_{batt}$ is the cost of LFP battery (€/kWh), and $Q_{los{s}_{max}}$ is the maximum capacity fade of the battery (set equal to 20%).

3. Case Study

A typical small size RO-Pax (Roll-on/roll-off Passengers) ferry operating in coastal areas has been chosen as a case study.

Table 2. The main characteristics of the RO-Pax ferry proposed as a case study. Power system volume and weight have been elaborated from data available on diesel storage on board and a diesel-electric generator for marine propulsion, scaled by the installed power of the ferry.

	Unit	Value
Vessel Type	-	RO-Pax ferry
Length Overall	m	42
Breadth Extreme	m	9
Gross Tonnage	t	280
Propulsion engines	kW	2 × 206
Auxiliary engines	kW	2 × 28
Power system volume	m3	15.5
Power system weight	kg	7185

reports the main characteristics of the ferry. The proposed RO-Pax is similar to the represented ferry in Figure 1. Power system volume and weight in Table 2 have been elaborated from data available on diesel storage on board and diesel-electric generator for marine propulsion, scaled by the installed power of the ferry.

The ferry power demands (propulsion and auxiliaries demands) over a typical operation day have been evaluated in [39]. Ferry power demands rated per minute enter as input in the proposed long-term operation optimization. More details on power profile and the typical sailing cycle of the ferry chosen as a case study are reported in Appendix A.

A simplified schematic of the propulsion system for the proposed case study is presented in Figure 2. Resulting from the design optimization performed in [39], the proposed energy system is composed of two 100-kW PEMFC stacks fueled by hydrogen, an energy management system, and a 286-kWh LFP battery. Hydrogen has been assumed to be stored in liquid form in a cryogenic tank, with the aim of reducing the volume and weight for the storage system in comparison to other types of hydrogen storage systems (e.g., compressed hydrogen tanks). It has been assumed that the ferry has enough fuel for a daily operation.

The parameters that have been used for the long-term operation optimization are reported in

Table 3. Input parameters of the multi-objective optimization.

Parameters	Unit	Value	Ref.
$c_{H_2}$	€/kWh	0.3	[64,65]
$c_{batt}$	€/kWh	818	[5,66]
$c_{FC}$	€/kW	3750	[64]
$c_{ov}$	%	20	Assumed
$\mathsf{\Delta }{v}_{load}$	µV/kW	0.0441	[37,55,56,57,58]
${\mathsf{\Delta }}_{P_{FC}}$	%	10	Assumed
$\mathsf{\Delta }{v}_{st,up}$	µV/cycle	23.91	[37,55,56,57,58]
$\mathsf{\Delta }t$	min	1	Assumed
${\eta }_{batt}$	%	90	Assumed
$F_{start}$	%	10	Assumed
$I_{F{C}_{max}}$	A/cm2	600	[67]
$I_{F{C}_{min}}$	A/cm2	75	[67]
$k_{1_{deg}}$	µVcm2/A	0.243	[37,55,56,57,58]
$k_{2_{deg}}$	µV	0.159	[37,55,56,57,58]
$k_{1_F}$	kWcm2/A	0.354	[57,58,67]
$k_{2_F}$	kW	3.837	[57,58,67]
$k_{1_P}$	kWcm2/A	0.159	[57,58,67]
$k_{2_P}$	kWcm2/A	7.251	[57,58,67]
$k_{11,Ah}$	1/(KAh2)	$-7.1·{10}^{-9}$	[61,62,63]
$k_{21,Ah}$	1/(KAh)	$2.2·{10}^{-4}$	[61,62,63]
$k_{31,Ah}$	1/K	0.033	[61,62,63]
$k_{12,Ah}$	1/(KAh2)	$2.1·{10}^{-6}$	[61,62,63]
$k_{22,Ah}$	1/(KAh)	−0.48	[61,62,63]
$k_{32,Ah}$	1/K	−8.885	[61,62,63]
$SO{C}_{min}$	%	20	[63]
$SO{C}_{max}$	%	90	[63]
$V_{batt}$	m3/kWh	0.0091	[5]
$V_{FC}$	m3/kW	0.0312	[64]
$V_{H_2}$	m3/kgH2	0.025	[64,65]
$V_{max}$	m3	15.5	Assumed
$V_o$	V	0.19	Assumed
$w_{batt}$	kg/kWh	8	[5]
$w_{FC}$	kg/kW	20	[64]
$w_{H_2}$	kg/kgH2	2.5	[64,65]
$w_{max}$	kg	7185	Assumed

. These model parameters represent the main characteristics of the considered ferry and of the current market scenario. In particular, the PEMFC and battery characteristics have been evaluated on the basis of the existing technologies.

4. Results

In this section, the main results of the optimization are presented and discussed. In the first part, a comparison of the plant lifetime obtained with the proposed optimization approach and the one obtained without considering energy units degradation (i.e., non-health-conscious optimization) is presented. Afterwards, the optimal operation profile during one day of operation along the entire lifetime operation is presented and compared with the one obtained through the non-health-conscious optimization. Lastly, the increase/decrease path of different optimization variables during the plant lifetime is analyzed.

4.1. Plant Lifetime Estimation

In this subsection, an evaluation of the energy system lifetime obtained with the proposed long-term optimization approach is compared with a non-health-conscious optimization approach. The latter considers the minimization of fuel consumption only (i.e., the minimization of the objective function $f_1$ in Equation (34)), and will be later referred to as FM (Fuel consumption Minimization). Figure 3 shows the SOH of both LFP batteries and PEMFC during the plant lifetime, expressed in the number of operative days. The dots indicate the SOH evaluated at the month i. Figure 3a shows the results of the FM optimization, while Figure 3b illustrates the results obtained with the proposed health-conscious multi-objective optimization.

From Figure 3a it can be noticed that with an FM approach the plant lifetime results to be about 210 days. After this time, $SO{H}_{PEMFC}$ reaches 80%, while $SO{H}_{battery}$ is still about 93%. This is due to the fact that the design of the system is fixed following the methodology explained in [39], where battery degradation was anyway included as a constraint in the methodology. Therefore, if PEMFC degradation is not minimized the battery results to be oversized. This implies low C-rates and hence lower degradation rates of LIB. On the other hand, PEMFC stacks tend to follow the power demand. Such an operation implies that PEMFC operates at a high current density and performs large load variations, causing high degradation rates of PEMFC and hence limited lifetime.

The optimization approach proposed here allows to increase the plant lifetime from 210 to 600 operation days, as shown in Figure 3b. In this case, the first energy unit to reach the threshold value of SOH is the LFP battery, while the $SO{H}_{PEMFC},$ at this time is still about 83% (so theoretically PEMFC could work for another 60–90 days). This aspect is also due to the current higher cost of PEMFC with respect to battery (see Table 3). As a consequence, the degradation of PEMFC ($f_2$) weighs more on the overall objective function ($f_{MO}$), and hence the optimal EMS advantages the minimization of PEMFC degradation instead of the LFP battery one.

4.2. Health-Conscious Optimal Daily Operation over the Entire Lifetime

The proposed optimization approach allows also to determine the optimal daily operation of the ferry, representative for the entire month i, during the whole lifetime of the energy system.

Figure 4 shows the health-conscious optimal operation of the ferry during one day in the first month of operation (Figure 4a) and the last month of operation (Figure 4b). Here, at both the beginning and the end of the plant lifetime, the PEMFC tends to avoid the operation at rated power, as this would result in a higher degradation rate and a lower energy conversion efficiency, and tends also to avoid large load variations and start-up phases, which would also cause higher degradation rates. As for battery, it is possible to see that the operation changes from the beginning till the end of the plant lifetime. As mentioned before, the cost of battery is lower than that of PEMFC, hence the minimization of battery $f_3$ (Equation (36)) weighs less on the overall objective function than the minimization of PEMFC degradation $f_2$ (Equation (35)). As a consequence, at the beginning of the plant lifetime when the degradation rate of the battery is low, $f_3$ slightly affects $f_{MO}$ (Equation (37)), and battery operates to fulfill the power demand of the ferry not covered by PEMFC. Differently, at the last month of operation, the degradation rate of the battery is higher and hence $f_3$ weighs more on $f_{MO}$. Consequently, the optimal EMS finds a compromise between the battery degradation and that of the PEMFC. For example, at the beginning of the day in the last month (Figure 4b) the PEMFC stacks operated at a higher load with respect to the first month to avoid high C-rates of the battery. To better understand the advantages of the proposed health-conscious EMS, the results of the long-term operation optimization have been compared with the results of FM optimization. Figure 5 shows the optimal operation of the ferry during one day in the first month of operation (Figure 5a) and the last month of operation (Figure 5b) if the FM approach is followed. It can be noticed that at the first month of operation (Figure 5a) the PEMFC operated to follow the highest possible value of the total energy conversion efficiency. Indeed, storing energy in the LIB would imply a lower overall efficiency, as LIB is subject to a charging/discharging efficiency. In the first month, LIB operates in a range of SOC (about 35% to 65%) smaller than in the health-conscious optimization (about 20% to 65%), and hence the LIB degradation rate is only partially affected by the non-health-conscious EMS at the beginning of the plant lifetime. In the last month of operation (Figure 5b), the situation was different. The non-health-conscious EMS would still advantage the PEMFC operation instead of LIB in order not to decrease the efficiency. However, at the end of the plant lifetime there is an increased use of LIB to overcome the voltage loss experienced by PEMFC stacks.

When focusing attention on H₂ consumption, the results show that H₂ daily consumptions obtained with the FM approach (155 kg_H2/day at the first month and 208 kg_H2/day at the last month) are similar to the ones obtained with the health-conscious approach (157 kg_H2/day at the first month and 204 kg_H2/day at the last month). This indeed demonstrates that energy conversion efficiency has not been affected considerably by limiting the degradation of PEMFC.

4.3. Behaviour of the Energy System Variables during the Plant Lifetime

As seen in Section 4.2, the proposed methodology allows to optimally define the daily operation of the ferry during the entire lifetime, and hence it is possible to analyze the behavior of the energy system variables in the long-term operation. Figure 6 shows the daily H₂ consumption (Figure 6a), the PEMFC average efficiency (Figure 6b), the volume and weight of the plant (Figure 6c), and the daily cost (Figure 6d) over time. The PEMFC average efficiency is defined as the average conversion efficiency of PEMFC over an operation day. It can be noticed that the hydrogen consumption at the end of the plant lifetime (204 kg/day) is about 29% higher than the hydrogen consumption at the beginning of the plant lifetime (157 kg/day). While this increase could be partially caused by the change in the optimal operation over time (see Section 4.2), the rise of hydrogen daily consumption is mainly due to the decrease of PEMFC average efficiency (Figure 6b). Indeed, with the progressive degradation of PEMFC, the efficiency is decreased and hence a higher amount of hydrogen is needed to reach the same power output.

An important drawback of the increase of hydrogen daily consumption over time is the increase of volume and weight required by the storage system onboard. Figure 6c reports the increase of volume and the weight of the main components of the system (i.e., LFP battery, PEMFC stacks, hydrogen tank). It is possible to observe that the system volume and weight at the end of the plant lifetime increased by up to 8.9% and 1.7%, respectively, with respect to the volume and weight at the beginning of the plant lifetime. While this does not exceed volume and weight constraints of the plant currently installed on board (Equations (5) and (6)), such aspects should be carefully taken into account when designing hybrid PEMFC propulsion systems, as it could affect the space available for the total payload of the vessel.

Lastly, from Figure 6d it is possible to see that the daily cost of the plant, i.e., the value of the $f_{MO}$ (in Equation (37)), increases from 3183 €/day to 5518 €/day. While this increase is partially due to the increase in hydrogen daily consumption (Figure 6a), it can be mainly related to the ongoing PEMFC and LFP battery degradation over time. The average daily cost over the entire plant lifetime was calculated as being equal to 4273 €/day.

5. Conclusions

The present study proposes a new methodology for the definition of the optimal EMS over the long-term operation of hybrid PEMFC/LFP battery propulsion by developing and applying a holistic MILP optimization methodology to a small-size ferry for coastal navigation.

The proposed EMS allows to increase the overall lifetime of the plant from 210 to 600 typical operation days with respect to a FM optimization. Given the current higher cost of PEMFC in comparison with LFP battery, the first component to reach the end of life is the LFP battery system, as its degradation rate weighs less in the overall objective function of the optimization problem. The results also show that the usage of LIB helps in reducing the degradation rate of PEMFC, and plant operation changes with the passing of time to allow the optimal economic performance of the plant. With the progressive ageing of the components, the decrease in the average PEMFC efficiency causes an increase in hydrogen daily consumption (from 157 kg/day to 204 kg/day at the end of the plant lifetime). This reflects on the need to oversize the hydrogen storage system in order to ensure the daily autonomy of the ferry over the entire lifespan. Such an implication is thus jeopardizing the technical feasibility of directly installing on board the system, as well as on the cost and possible total loss of payload. Moreover, the components ageing also affects the plant daily cost, which was found to increases by up to 73% at the end of lifetime with respect to the daily cost at the beginning of the plant lifetime.

The proposed methodology is a useful tool, which can be easily adapted to other types and designs of hybrid PEMFC/LFP battery propulsion systems, as well as to other types of vessels and to other market scenarios. Moreover, the present study points out that changes in the costs of PEMFC, LFP battery and liquid hydrogen could noticeably affect the optimal long-term operation of the energy system. These results can provide some interesting inputs for future research on this topic, such as the analysis of how the optimization results are affected by uncertainties on hydrogen, PEMC and LIB costs. In addition, the uncertainty on the mission profile, caused by variations on hull resistance (due to, e.g., fouling of ship’s hull) or seasonal variability of power will also be considered in future works.

Future studies will be also addressed to the improvement of hybrid PEMFC/LIB powertrains with the addition of onboard solar energy plants and to the possible connection to a battery charging station when ferries are staying at the quay for a long time. Lastly, it should be stressed that safety and regulatory aspects still play a key role in limiting the onboard installation of such systems and hence further analyses should address these points to provide information about the technical feasibility of the plant.