# Fuel Cell Design Considerations and Component Optimisation for Isolated Electric Energy Systems with Periodic Load Patterns

Bjarte Hoff

► To cite this version:

Bjarte Hoff. Fuel Cell Design Considerations and Component Optimisation for Isolated Electric Energy Systems with Periodic Load Patterns. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0047. hal-05077105

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Submitted on 21 May 2025

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# Fuel Cell Design Considerations and Component Optimisation for Isolated Electric Energy Systems with Periodic Load Patterns

Bjarte Hoff

UiT The Arctic University of Norway

Lodve Langes. gt. 2

Narvik, Norway

Tel.: +47 76 96 63 18.

E-Mail: bjarte.hoff@uit.no

URL: http://www.uit.no

## ACKNOWLEDGMENT

The authors acknowledge the funding support from Interreg Aurora's SAMHand project (20357984).

*Index Terms*—Design optimization, Fuel Cell system, Energy storage, Energy conversion, Fault tolerance.

*Abstract*—This paper presents design considerations for hydrogen-based fuel cell distributed generation units for electricity production in isolated energy systems and vehicles with periodic load pattern, replacing fossil fuel engines. Transient loads, inrush currents, fault currents that may occur in the power system is described. An optimization algorithm is used to find optimal component selection based on the load profile and cost parameters.

## I. INTRODUCTION

Hydrogen and fuel cells are considered as one of the alternatives to diesel for generating electricity in isolated power systems. Such systems can be found in remote areas without grid supply [1], feed barges for fish farming, diesel-electric ships and temporary electricity supply for e.g. construction. Larger vehicles for construction, foresting and mining can also be considered as small, isolated power systems. If green hydrogen is produced from renewable energy sources, this presents a fossil free alternative.

Preciously, fuel cells have been integrated in hybrid systems in combination with renewable and/or fossil energy sources, where one example can be found in [2]. For standalone systems, fuel cells are combined with a ultra capacitor (UC) battery in [3], where a UC bank was used for peak shaving. For replacing a diesel generator set with hydrogen fuel cell unit, the unit should be

made compatible with existing power systems, including its transient behaviour, fault situations and conventional protection devices.

The implementation of a fuel cell system will differ based on the application and load requirements. An auxiliary power unit for universal use has to consider all different types of load requirement, since the electrical loads will be unknown at the design stage and can change at any time. If information of the application is known, the system and energy storage can be optimised to known load patterns, which in some cases can allow the energy storage to be reduced. This will reduce the size, cost and possible simplifying the safety requirements for the energy storage. The performance, characteristics and recommended operation profiles of the fuel cell is one of key elements in designing the system. Transient behaviours of the fuel cell directly influences the required size of the energy storage, since a slow response will require larger energy storage to handle short termed mismatch between production and load. The transient behaviour of the fuel cells depends on its internal construction and operating recommendations to avoid degradation and reduced lifetime. Previously, a multi-objective optimisation problem for a fuel cell vehicle is proposed in [4] to determine fuel cell and battery net power, minimising fuel economy, durability, component cost and battery volume. In [5], a bi-level optimisation is implemented to optimize both component sizing and energy management. Optimisation based on the number of battery, capacitor and fuel cell modules is demonstrated in [6], but the authors does not consider minimum load constraints and potential cost for stopping and starting fuel cells.
This paper presents design considerations for a stand-alone fuel cell distributed generation unit (FCDG), with focus on the energy storage requirements to handle transient load changes, fault situations in the connected grid, and optimal selection of energy storage for periodic repeating loads. PEM (proton-exchange membrane) fuel cell stacks is considered as the main energy source and is assumed to able to produce the maximum load, where energy storage is limited to transient and dynamic behaviour in the system where the fuel cell is unable to respond sufficiently fast, and in low load situations. Unlike [5], this paper presents an optimization algorithm focusing on the component installation cost, incorporating the cost per installed module instead of per unit, including ultra capacitors as part of the energy system, and start-up costs for fuel cells. The objective is to investigate fuel cell as a direct replacement of an diesel generator, without introducing hybrid systems and minimising the cost and size of the energy storage systems. An aquaculture feed barge with a strong periodic load profile will be used as the target application and basis for the design considerations and discussion.

## II. ELECTRICAL LOAD ANALYSIS

A fuel cell acting as auxiliary electricity supply has different design consideration compared to specific applications such as vehicles and ship propulsion, since the load is unknown at the design stage. In vehicles, an energy storage is used to smooth out power variations, which also enables regenerative braking [7]. For an FCDG, regenerative braking is obviously not required, but there are other arguments for having an energy storage due to transient in power demand. The electrical load can vary between zero and full load at any time, inrush current during direct start of induction machines can result in temporary overload, and short circuits consumes a certain amount of let-through-energy before the fault is isolated by its protection devices.

This paper uses an aquaculture feed barge as the application for the FCDG. while similar load pattern can also be seen using fuel cells as mobile charging stations for battery powered vehicles. A feed barge is placed at the site for fish farming and acts as a distribution system for food to the fish. These barges have a very predictable load pattern. During feeding, large compressors is used to blow the food to the fish using compresses air and consumes high power from the electric system. The load when not feeding is very low and consist only of basic heating inside the office space, automation equipment, sensors and communication devices. The load therefore

<figure><img src="image_2.png" /><figcaption>Fig. 1. Short circuit current in an commercial converter [9]</figcaption></figure>

<table><caption>TABLE I
SYSTEM AND CONTROL PARAMETERS</caption><thead><tr><td>Description</td><td>Variable</td></tr></thead><tbody><tr><td>High load</td><td>P<sub>high</sub></td></tr><tr><td>Low load</td><td>P<sub>low</sub></td></tr><tr><td>High load duration</td><td>T<sub>high</sub></td></tr><tr><td>Low load duration</td><td>T<sub>low</sub></td></tr></tbody></table>

varies between high load and low load at a very predictable and predefined pattern as shown in Fig. 1.

Since fuel cells have a minimum load during operations, the duration of the low load period is actually more important than the transient phase from low to high load. It is not recommended to regularly turn on and off the fuel cell system, and the system should be able to be running also during low load in daily operation. During low load, the fuel cell is likely to produce more than is being consumed, such that the excess energy needs to be stored. Important design variables regarding the load are shown in Table I.

## III. DESIGN REQUIREMENTS

A simplified FCDG unit is made of a fuel cell stack, DC-DC converter, three phase inverter and an energy storage as shown in Fig. 2. The energy storage consists of capacitors and, if required, batteries. If batteries can be completely avoided, there is no need for a battery management system, temperature-controlled enclosures and fire safety regulations needed for lithium-ion batteries. Unless the load is controllable and can be adjusted to match the fuel cell, a battery of some size will in most cases be required. The startup procedure of the fuel cell system will also require some energy for a certain number of seconds before the production is started, which has to come from a battery or another available energy source in the installation. Similar for shutdown. If batteries cannot be omitted, by reducing the battery capacity, the requirements for battery rooms and other safety measures can in some cases be reduced.
<figure><img src="image_3.png" /><figcaption>Fig. 2. Overview of hydrogen power unit</figcaption></figure>

The energy storage is based on the following considerations:

1) Supply of energy when load change between zero and full load

2) Energy for short circuit currents

3) Energy when load is outside fuel cell operating range (higher and/or lower)

4) Energy for startup and shutdown

The two first design considerations are based on the assumption that dynamic properties in the fuel cell does not allow for instantaneous change in power. The last two considerations relate to the system design and rating of the components.

### A. Load changes

A sudden load change will occur every time a load is connected and disconnected. The change could be minor, but the worst case is between zero and full load:

$$P_L = \begin{cases} 0 & t \le 0 \\ P_N & t > 0 \end{cases}, \text{or} \quad (1)$$

$$P_L = \begin{cases} P_N & t \le 0 \\ 0 & t > 0 \end{cases}, \quad (2)$$

where $P_L$ is the load power and $P_N$ is the nominal power of the FCDG.

Due to the dynamic in the fuel cell and related components, the fuel cell are not able to increase or decrease its production instantaneously. An energy storage is therefore required to supply the temporary difference between the fuel cell and load power until the fuel cell has reached a new steady state:

$$E_{Load\_change} = \int_{t_0}^{t_1} (P_{FC} - P_L) dt, \quad (3)$$

where $t_0$ is the time just before the load change, $t_1$ is the time after reaching the new steady-state and $P_{FC}$ is power from the fuel cell.

Load changes may also occur from inrush currents from direct start of induction machines. These high but temporary currents are considered to stay within the nominal power of the converter. Larger machines today are assumed to be equipped with frequency converters, thereby avoiding inrush currents during startup. If the converter current and power ratings are selected based on the nominal fuel cell values, the energy requirement if covered by (3).

An overshoot or undershoot is expected in the fuel cell, since the energy storage needs to be charged or discharged back to its initial reference value.

### B. Short circuit currents

A short circuit in the public AC grid is essentially a large load limited only by the grid impedance and transient behaviour of the generator. A circuit breaker will allow a certain amount of energy to pass before the fault can be isolated, known as let-through-energy, $I^2t$. In addition, the fault current has to exceed a threshold known as $I_5$ to instantaneous trigger the protection device. In a FCDG, all load current is supplied through the inverter and a fault current of multiple kA is normally not available. Additional fault current can be supplied from a separate device as suggested in [8] if needed. Due to compatibility with existing electrical power systems, advanced protection devices will not be considered.

A grid inverter can implement a short circuit functionality, where the inverter itself can be programmed to temporary supply a higher current than its nominal rating [9]. The principle is shown in Fig. 3, where the current is controlled at a predefined limit and duration once a short circuit is detected. The converter parameters has to be set according to the system ratings and current requirements for tripping the available circuit breakers. The required energy storage for this functionality will be given by

$$E_{SC} = I_{SCC}^2 R_{loop} t_{SC}. \quad (4)$$

where $I_{SCC}^2$ is the fault current supplied from the converter, $R_{loop}$ is the resistance in the fault loop and $t_{SC}$ is the duration of the short circuit.

### C. Load outside fuel cell operating range

If load outside the operating range of the fuel cell is expected, sufficient energy storage has to be available for peak shaving or ensuring minimum load for the fuel cell system.

In the case of feed barges, the minimum load is very low and expected to be lower than the allowed minimum fuel cell load. In that case, a battery needs
<figure><img src="image_4.png" /><figcaption>Fig. 3. Short circuit current in an commercial converter [9]</figcaption></figure>

to have enough available capacity to store the surplus energy from the production, at least in normal daily operation. If the feed barge is taken out for maintenance or feeding is suspended for a longer time period, the fuel cell obviously has to be stopped at some point when there are no storage capacity left.

Energy outside fuel cell operating range can be described as

$$E_{OR} = \int_{t_0}^{t_1} (P_{FC} - P_L) dt, \quad (5)$$

, where the time-range from $t_0$ and $t_1$ now refers to the total high or low load duration, depending on what is relevant for the application.

If feed barges are using battery to store energy during low load, the stored energy can be used to reduce the peak load, discharging the battery back down and ready for a new cycle.

### D. Startup and shutdown

The fuel cell system will have some auxiliary energy demand for running its control systems, circulating pumps and cooling systems in the short period between giving the start command and until the fuel cell starts to produce. Similarly, there will be a shutdown procedure and a energy need between when the fuel cell stops producing and until the whole systems has safely shut down.

This energy requirement depends on the fuel cell and how it is integrated, and such information must be acquired from the fuel cell system manufacturer or integrator. If the feed barge, or a similar application, has its own onboard emergency energy system, it could simplify the system if this could provide energy during startup and shutdown, avoiding additional battery capacity in the fuel cell system.

## IV. DYNAMIC SYSTEM CONSIDERATIONS

The dynamic properties during step changes in load are important to evaluate the required energy storage in

(3). A dynamic model of a fuel cell can be modelled analytical as shown in [10], or by using an empirical approach as in [11]. However, the performance should not be evaluated on the basis of only its maximum dynamic performance, since sudden load changes can result in improper water management with reduced lifetime and performance as result [12]. The DC-DC boost converter that connects the fuel cell to the DC link is therefore used to slow down the dynamic response. Instead of a detailed fuel cell model, the boost converter control algorithm will be given a time constant that dominates the response in supplying current from the fuel cell. Since the fuel cell dynamics and changes in cell voltage is not considered visible to the DC-link, the fuel cell is modelled as a constant voltage source, where the dynamic is dominated by the boost converter feedback control loops. From a system perspective, the design therefore does not require any of the fuel cell or boost converter parameters except for its safe operation region, where the fuel cell and boost converter combined is modelled as

$$H_{FC}(s) = \frac{i_{FC}(s)}{i_{Boost}^{*}(s)} = 1. \quad (6)$$

and the energy storage directly connected to the DC-link is modelled as a capacitor

$$H_{C}(s) = \frac{i_{C}(s)}{v_{DC}(s)} = \frac{1}{Cs}, \quad (7)$$

where $i_{FC}$ is the current from the fuel cell, $i_{Boost}^{*}$ is the current reference for the boost converter, $C$ is the capacitance of the energy storage in the DC-link, $i_{C}$ is the capacitor current and $v_{DC}$ is the DC-link voltage.

The whole control loop is shown in Fig. 4, $i_{G}$ is the current that flows to the grid inverter, $V_{DC}^{*}$ is the DC-link reference voltage, and PI-controllers are defined as the transfer functions

$$H_{PI}(s) = k_p + \frac{k_i}{s} = \frac{k_p s + k_i}{s}, \quad (8)$$

where $k_p$ and $k_i$ is the proportional and integral gain.

If the DC-link voltage reference is considered constant, the closed loop transfer function for the whole system with $i_{G}$ as input is shown in (9).

Based on the design considerations and the transfer function in (9), the expected voltage deviation in the DC-link can be calculated. Sudden load changes can be seen as a step response, while short circuit currents can be seen as an impulse with energy equal to the let-through energy of the circuit breaker under consideration.

As an example, the control parameters in Table II have been chosen such that the inner current loop has a
$$M(s) = \frac{v_{DC}(s)}{i_g(s)} = \frac{(k_{pi} + 1)s^2 + k_{ii}s}{(Ck_{pi} + C)s^3 + (Ck_{ii} + k_{pi}k_{pv})s^2 + (k_{ii}k_{pv} + k_{iv}k_{pi})s + k_{ii}k_{iv}} \quad (9)$$

<figure><img src="image_5.png" /><figcaption>Fig. 4. System control loop</figcaption></figure>

<table><caption>TABLE II
SYSTEM AND CONTROL PARAMETERS</caption><thead><tr><th>Description</th><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>Current control proportional gain</td><td>k<sub>pi</sub></td><td>10</td></tr><tr><td>Current control integral gain</td><td>k<sub>ii</sub></td><td>1</td></tr><tr><td>Voltage control proportional gain</td><td>k<sub>pv</sub></td><td>1</td></tr><tr><td>Voltage control integral gain</td><td>k<sub>iv</sub></td><td>0.5</td></tr><tr><td>DC-link capacitor</td><td>C</td><td>0.1 F</td></tr></tbody></table>

time constant of 10 seconds, while the outer voltage loop is made stable and significantly slower. Fig. 5 shows the voltage deviation with a load increase of 10 A and 100 A. Depending on the chosen components, it is shown that minor load changes can be handled by just a capacitor as energy storage, while larger load changes such as the 100 A case, needs additional energy storage as long as the load change cannot be controlled as a gradient. This additional storage could be a battery connected directly of through a DC-DC converter to the DC-link. Note that a direct connection would change the dynamic in the system, where (7) should include the battery dynamics as well.

For short circuits, Fig. 6 shows the impulse response when subjected to a short circuit of 100 A with a duration of 0.01 seconds, sufficient to trigger a 10 A miniature circuit breaker with C characteristic. Note that the current from the DC-link during a short circuit in most cases is lower to the current conversion then grid inverter, depending on the grid fault loop impedance. Due to the quick clearing time in short circuits, a capacitor should be large enough to handle short circuits without depending on an additional storage in form of batteries.

The simplified system perspective and transfer function in (9) can also be used in reverse direction to design the fuel cell control loops and energy storage

<figure><img src="image_6.png" /><figcaption>Fig. 5. Voltage deviation after a 10 A and 100 A increase in load.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. Voltage deviation after a short circuit of 100 A and duration of 0.01 seconds.</figcaption></figure>
requirements based on maximum voltage deviation limits and requirement for sudden load changes and short circuit current capabilities.

## V. STATIONARY DESIGN CONSIDERATIONS

The sizing of energy storage in stationary conditions is proposed to be optimised using a mixed integer linear programming (MILP) together with knowledge of requirements and component cost. The advantage of using MILP is to include integer actions from starting up the fuel cell and initial costs of including one or more batteries in the system. The algorithm will therefore consider if it is optimal to install batteries or not.

A minimisation problem is formulated in (10) with the objective to find the optimal selection of energy storage for a given load and fuel cell ratings. Both batteries and ultra capacitors are considered, where the algorithm will use one or both of them to minimise the cost. An overview of parameters and variables are presented in Table III and IV, where estimated parameter values are included for reference. The number of installed fuel cell units and their total installed power is here considered to be known and matched to the high load part of the load profile as illustrated in Figure 1. This algorithm differs from similar work such as [6] that does not consider minimum fuel cell load and the cost of startup.

The cost function in (11) is defined as the sum of battery capacity cost per kWh, ultra capacitor cost per kWh, the cost of starting the fuel cell, and the initial cost of installing a battery energy storage systems and ultra capacitor packs. The cost for capacities is calculated based on the number of installed modules, not on the usage of them. The algorithm can choose to install additional battery and capacitor modules, but are not allowed to uninstall modules since $s_{cap}$ and $s_{bat}$ are restricted to positive values. Since the fuel cell eventually will supply all the required energy, it is not relevant to add cost of the hydrogen to the problem. While this algorithm only implements one type of battery module and one capacitor module, it can easily be extended to include more module sizes to optimise among.

The last six optimisation variables in Table IV are restricted to integer numbers in the MILP algorithm, since it describes the number and initialization of battery and capacitor modules, in addition to startup of fuel cells.

There are a total of 24 constraints in (10), including both inequality and equality constraints. Inequality constraints set the upper and lower limits of the optimisation variables, while equality constraints implement the behaviour of the system. Through these, the battery charge is constrained to predefined values at beginning

and end of the periodic pattern. Ultra capacitors is considered short time energy storage and is required to be charge/discharged back to its initial point of zero before the end of the period. The last equality constraint ensures the energy balance according to the defined load pattern.

<table><caption>TABLE III
OPTIMISATION PARAMETERS</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th></tr></thead><tbody><tr><td>C<sub>bat</sub></td><td>150</td><td>€/kWh</td><td>Cost of battery capacity</td></tr><tr><td>C<sub>bat init</sub></td><td>10000</td><td>€</td><td>Initial cost of battery</td></tr><tr><td>C<sub>cap</sub></td><td>7500</td><td>€/kWh</td><td>Cost of ultra capacitor capacity</td></tr><tr><td>C<sub>cap init</sub></td><td>1</td><td>€</td><td>Initial cost of ultra capacitor</td></tr><tr><td>C<sub>FC start</sub></td><td>1000</td><td>€</td><td>Cost for fuel cell startup</td></tr><tr><td>E<sub>load,t</sub></td><td></td><td>kWh</td><td>Load energy per div.</td></tr><tr><td>E<sub>bat max</sub></td><td>100</td><td>kWh</td><td>Max capacity per battery pack</td></tr><tr><td>E<sub>bat T,min</sub></td><td>50</td><td>kWh</td><td>Min. end of period battery charge</td></tr><tr><td>E<sub>bat T,max</sub></td><td>50</td><td>kWh</td><td>Max. end of period battery charge</td></tr><tr><td>E<sub>bat init</sub></td><td>50</td><td>kWh</td><td>Initial battery charge</td></tr><tr><td>E<sub>cap max</sub></td><td>1</td><td>kWh</td><td>Max capacity per capacitor pack</td></tr><tr><td>E<sub>FC min</sub></td><td>20</td><td>kWh</td><td>Minimum FC production</td></tr><tr><td>E<sub>FC max</sub></td><td>200</td><td>kWh</td><td>Maximum FC production</td></tr><tr><td>M<sub>bat</sub></td><td>1/0.175</td><td>kg/kWh</td><td>Battery pack mass</td></tr><tr><td>M<sub>cap</sub></td><td>1/0.035</td><td>kg/kWh</td><td>Ultra capacitor mass</td></tr><tr><td>N<sub>bat</sub></td><td>10</td><td></td><td>Max number of battery packs</td></tr><tr><td>N<sub>cap</sub></td><td>10</td><td></td><td>Max number of capacitor packs</td></tr><tr><td>N<sub>FC</sub></td><td>2</td><td></td><td>Max number of FC units</td></tr><tr><td>N<sub>FC T,min</sub></td><td>1</td><td></td><td>Min. FC units running at end of period</td></tr><tr><td>V<sub>bat</sub></td><td>1/300</td><td>m<sup>3</sup>/kWh</td><td>Battery pack volume</td></tr><tr><td>V<sub>cap</sub></td><td>1/10</td><td>m<sup>3</sup>/kWh</td><td>Ultra capacitor volume</td></tr><tr><td>ω<sub>bat M</sub></td><td>1</td><td></td><td>Weighting factor battery mass</td></tr><tr><td>ω<sub>bat V</sub></td><td>1</td><td></td><td>Weighting factor battery volume</td></tr><tr><td>ω<sub>cap M</sub></td><td>1</td><td></td><td>Weighting factor capacitor mass</td></tr><tr><td>ω<sub>cap V</sub></td><td>1</td><td></td><td>Weighting factor capacitor volume</td></tr></tbody></table>

<table><caption>TABLE IV
OPTIMISATION VARIABLES</caption><thead><tr><th>Variable</th><th>Unit</th><th>Description</th></tr></thead><tbody><tr><td>c<sub>bat,t</sub></td><td>kWh</td><td>Battery state of charge</td></tr><tr><td>c<sub>cap,t</sub></td><td>kWh</td><td>Capacitor state of charge</td></tr><tr><td>e<sub>bat,t</sub></td><td>kWh</td><td>Battery energy per div.</td></tr><tr><td>e<sub>cap,t</sub></td><td>kWh</td><td>Capacitor energy per div.</td></tr><tr><td>e<sub>FC,t</sub></td><td>kWh</td><td>Fuel cell energy per div.</td></tr><tr><td>s<sub>bat,t</sub></td><td></td><td>Battery packs initialised</td></tr><tr><td>s<sub>cap,t</sub></td><td></td><td>Capacitor banks initialised</td></tr><tr><td>s<sub>FC,t</sub></td><td></td><td>Fuel cells started up</td></tr><tr><td>u<sub>bat,t</sub></td><td></td><td>Number of battery packs in operation</td></tr><tr><td>u<sub>cap,t</sub></td><td></td><td>Number of ultra capacitors in operation</td></tr><tr><td>u<sub>FC,t</sub></td><td></td><td>Number of fuel cells in operation</td></tr></tbody></table>
$$
\begin{array}{ll}
\text{minimise} & f \\
\text{subject to} & e_{FC,t} \ge E_{FC_{min}} u_{FC,t} \\
& e_{FC,t} \le E_{FC_{max}} u_{FC,t} \\
& s_{FC,t} \ge u_{FC,t} - u_{FC,t-1} \\
& s_{FC,t} \ge 0 \\
& s_{bat,t} = u_{bat,t} - u_{bat,t-1} \\
& s_{bat,t} \ge 0 \\
& s_{cap,t} = u_{cap,t} - u_{cap,t-1} \\
& s_{cap,t} \ge 0 \\
& u_{FC,t} \le N_{FC} \\
& u_{FC,t} \ge 0 \\
& u_{FC,t} \ge N_{FC_{T,min}}, t = T \\
& u_{bat,t} \le N_{bat} \\
& u_{bat,t} \ge 0 \\
& u_{cap,t} \le N_{cap} \\
& u_{cap,t} \ge 0 \\
& c_{bat,t} \ge 0 \\
& c_{bat,t} \ge E_{bat_{T,min}}, t = T \\
& c_{bat,t} \le E_{bat_{T,max}}, t = T \\
& c_{bat,t} \le E_{bat_{max}} u_{bat,t} \\
& c_{cap,t} \le E_{cap_{max}} u_{cap,t} \\
& c_{bat,t} = c_{bat,t-1} - \frac{e_{bat,t}}{\Delta_t} \\
& c_{cap,t} = c_{cap,t-1} - \frac{e_{cap,t}}{\Delta_t} \\
& c_{cap,t} = 0, t = T \\
& e_{FC,t} + e_{bat,t} + e_{cap,t} = e_{load,t},
\end{array}
\qquad (10)
$$

where

$$
\begin{aligned}
f = & (C_{bat}E_{bat_{max}} + V_{bat}\omega_{bat_{V}} + M_{bat}\omega_{bat_{M}})u_{bat,T} \\
& + (C_{cap}E_{cap_{max}} + V_{cap}\omega_{cap_{V}} + M_{cap}\omega_{cap_{M}})u_{cap,T} \\
& + \sum_{t=1}^{T} C_{FC_{start}} s_{FC,t} + \sum_{t=1}^{T} C_{bat_{init}} s_{bat,t} \\
& + \sum_{t=1}^{T} C_{cap_{init}} s_{cap,t}.
\end{aligned}
\qquad (11)
$$

VI. RESULTS

The optimisation algorithm in section VI is applied
to a aquaculture feed barge case, which is known for
a strong periodic load pattern. The load is alternating
between high load during feeding of the fish, and low
load. This profile is very similar to the use of a fuel

cell generating unit for charging construction vehicles at
remote locations without grid connection.

MATLAB R2024a is used to solve the optimisation problem through the "intlinprog" command. The result of the optimisation is shown in Fig. 7, where one fuel cell and one battery is chosen to supply the load within the fuel cell maximum and minimum power limits. For such pattern, batteries are considered sufficient, where ultra capacitors are more suitable for short peak load. This is also reflected on the results from the optimisation that only included one battery. It is also worth noting that while the fuel cell initially was set to run, the algorithm found it to have lower cost to stop two times than installing another battery module to receive the minimum fuel cell energy during low load periods. Note that the optimisation is based on the energy within each time division with kWh as unit as shown in Fig. 7. In this particular case, the time division is one hour and the average power within the hour will therefore equal the energy.

A second case is presented in Fig. 8, where the cost for starting the fuel cell has been increased to € 100 000 to avoid shutting it down during a normal working day. Note that this cost must represent the cost over the lifetime of the system, not for each individual start/stop since this is primarily a component selection optimisation, not an energy management optimisation. In this case, three battery modules are used instead of one. This also increases the desired stored energy in the battery packs at $t = T$ to three times its initial value.

A third case introduces a short peak power during startup of the high load period. The results are shown in Fig. 9, where a capacitor is used to handle the additional load, instead of installing another battery pack.

VII. CONCLUSION

Design considerations for using a fuel cell to supply
a isolated grid is presented together with the transfer
function of the control loops for transient behaviour and
an energy storage optimisation algorithm for stationary
considerations. It is shown that by using a simplified
system approach to the control loops, it is possible to
determine the required dynamic behaviour and energy
storage requirement to account for sudden load changes
and short circuits. Stationary considerations including
the additional cost for starting a fuel cell, and instal-
lation costs for batteries and ultra capacitors are pro-
posed solved using a mixed integer linear programming
optimisation algorithm, where the number of battery
and capacitor modules are designed based on a set of
parameters and a predefined periodic load pattern.
<figure><img src="image_8.png" /><figcaption>Fig. 7. Optimal use of fuel cell and battery for a feed barge case with $C_{FCstart} = 1000$. Maximum and minimum limits shown as dashed lines.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 9. Optimal use of fuel cell and battery for a feed barge case with $C_{FCstart} = 100000$ and a high load at $t = 14$. Maximum and minimum limits shown as dashed lines.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. Optimal use of fuel cell and battery for a feed barge case with $C_{FCstart} = 100000$. Maximum and minimum limits shown as dashed lines.</figcaption></figure>

While this work uses a predefined fuel cell size and optimized its use and the attached energy storage system, the method can be extended to include a catalogue of fuel cells, capacitors and batteries, designing the optimal combination of number of units and their sizes. Alternative methods such as the cost for a backup diesel generator could be integrated for comparison, although the fuel cost would then be different.

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