Effect of Load Priority Modeling on the Size of Fuel Cell as an Emergency Power Unit in a More-Electric Aircraft

Featured Application

Investigation of electric load demand management upon the hydrogen consumption and the optimum power size of fuel cell for the aircraft industry is the specific application of this research work.

Abstract

The proton exchange membrane fuel cell as a green power source is a suitable replacement of the engine mounted generators in the emergency power unit of a more-electric aircraft. Most existing energy management methods for operation of fuel cells in the more-electric aircraft refer to the hydrogen consumption minimization. But due to the increasing number of electrical components and hence electrical demand in the aircraft, demand-side management should be considered in these methods. In order to determine the effect of demand-side management on the fuel cell operation and size, an efficient load priority model is presented and integrated into an optimization framework. The proposed optimization framework is formulated as mixed-integer quadratic programming using Karush–Kuhn–Tucker optimality condition and is solved by CPLEX optimization tool. The Boeing 787 electrical distribution system is considered as a single-bus case study to evaluate the performance of the proposed optimization framework. Numerical results show that the size of fuel cell as an emergency power unit resource depends on the type and importance of the system’s loads in different emergency conditions. Also, with an efficient priority model, both hydrogen consumption and load shedding can be decreased.

1. Introduction

1.1. Motivation

The more-electric aircraft (MEA) uses the electric power to supply the aircraft’s non-propulsive systems like hydraulic and pneumatic which have been replaced by the electrical systems. Currently, most of the electrical power required by an MEA is provided by the engine mounted generators which have the maximum efficiency of 40% in flight. This efficiency decreases to 20% on the ground using the emergency power unit and also with noise pollution and gaseous emissions [1]. Now in the era of artificial intelligence, deep learning has achieved great success [2,3,4], because it requires very little engineering by hand, so it can easily use the increase of available computation and data. In order to monitor and track [5,6,7] the surrounding objects in real time, the intelligent algorithm used in aircraft will consume a lot of power. Intelligent aircraft needs stable and reliable electric supply, and the electric consumption is increasing. Although battery can replenish from environment [8], the size of these generators should be increased due to increasing electric demand in the advanced MEAs, which lead to the occupation of larger available space and also increase the system’s weight.

The hybrid fuel cell/battery system is a suitable power resource in transportation systems [9]. Nowadays, fuel cells are being used by airplane manufacturers, especially Boeing and Airbus, because of their high power efficiency, low noise, and near-zero pollution. Hybridizing the fuel cells with supercapacitors or batteries as the energy storage systems (ESS) can improve their performance and efficiency [10]. The feasibility of using fuel cells as the emergency power unit of aircraft has been considered in many research studies [1], and [11,12,13,14,15]. Therefore, the replacement of the emergency power unit based on the conventional generation system like ram air turbine by the hybrid fuel cell system is a proper solution to get closer to the advancement of electric aircraft [16]. Also, this makes the emergency power unit more efficient at low aircraft’s altitude and speed.

The energy management, including management of generation, conversion, and distribution plays an important role in an MEA. The energy management system should control different units, including generators, storage devices and loads. Generally, the electrical system in an aircraft may operate in three modes: normal, abnormal or emergency where main generators in these modes work ideally, isolated or with faults, respectively [16]. In all modes, the energy management system is responsible for controlling various units. Load shedding is one way of controlling systems in abnormal or emergency conditions.

1.2. Related Works

The energy management strategies for different transportation systems have been studied in some research works. In [17], a hybrid configuration for operating a tram without connecting to the main grid including proton exchange membrane fuel cell system (PEMFCS) and ESS was proposed. The equivalent consumption minimization strategy (ECMS) was used to optimize the hydrogen consumption. Different energy management for a fuel cell emergency power unit in an MEA sas investigated in [18]. As a result, ECMS method had a better performance in the case of fuel consumption minimization. The specific energy of a fuel cell emergency power unit in an aircraft was maximized in [19] by the proposed optimization of the ratio energy/weight. Also, the feasibility of using PEMFCS in an aircraft was analyzed. Optimizing the size and the hydrogen consumption in a powertrain system was investigated in [20]. This system included typical PEMFCS and their durability was also considered. First, the model was solved by dynamic programming method which had some results for multi-objective optimization as the rules. These rules were named soft-run strategy which helps the fuel cell being stable. A convex model was proposed in [21] to simultaneously optimize the hybrid ESS (battery and supercapacitor) size and the power allocation between hybrid ESS and fuel cell in an electric bus. As a result, the hybrid ESS has less hydrogen consumption rather than battery-only ESS. The stochastic dynamic programming method was used in [22] to minimize costs of the fuel cell, including fuel consumption and degradation. A new hydrogen consumption minimization strategy is proposed in [23] and was compared with ECMS. The proposed model was more robust to change in the load profile in an MEA but had high stress on the ESS.

The load management in the MEA system often happens by interrupting the loads or load shedding in order to prevent the main generators from overloading. In this regard, prioritization of different loads has been used repeatedly in some references. A load with a higher priority will be disconnected later. Variable priority method was used in [24] and also in [25] in which each load sends a variable request to a control unit. A load with high request level will be connected earlier. The proposed load management in [26] prevents loads from being reconnected instead of load shedding. Some studies used other methods for load management like the power-sharing, time-sharing and peak-compression which were used in [27] to exploit long responding times of MEA like heater and galleys. However, in some studies, the authors suggest a learning method such as in [28,29]. Also the authors gave a novel method to optimize the system performance using clustering [30,31,32], heuristic [33], genetic [34,35,36], evolution [37], and optimization [38] approaches. Other robust methods have also been used for uncertainty removal [39,40,41,42,43,44].

Regarding the review of past references, no article has addressed the joint optimization of fuel cell hydrogen consumption (hydrogen economy) and demand-side management in an MEA system. Most of the articles have examined this topic separately. The purpose of this paper is to provide a convex multi-objective optimization model for both hydrogen consumption and also load management in an MEA and examine the effect of load priority during emergency condition in a flight on the hydrogen consumption. Choosing the large size for PEMFCS will increase the hydrogen consumption as well as aircraft’s weight, but the probability of load interruption during the emergency condition will decrease (the purpose of optimal size in this paper is to determine the optimal electric power and weight). On the other hand, selecting a small size for the PEMFCS reduces the weight and the hydrogen consumption of aircraft but it also causes more load shedding during emergency conditions. By prioritizing loads in the system, both size of PEMFCS and optimal load shedding can be achieved (Figure 1). The formulation of the problem in the form of mixed-integer quadratic programming (MIQP) can effectively examine tradeoffs between the load shedding and hydrogen consumption. The logic constraints in the proposed model are transformed into mixed-integer constraints using Karush–Kuhn–Tucker (K.K.T) optimality condition. Hence, the proposed convex model can be solved by the strong CPLEX solver.

The paper is organized as follows. System modeling including distribution system, PEMFCS, and ESS is presented in Section 2. The proposed optimization formulation is presented in Section 3. Section 4 obtains the simulation results and finally, Section 5 is dedicated to the conclusion.

2. Materials and Methods

2.1. Distribution System Model

The electrical distribution system of the Boeing 787 is shown in Figure 2. It has four main generators, each with a capacity of 250 kVA and an emergency power unit. Each 230 VAC main bus is fed by its connected main generator. Its nominal voltage is 230 VAC, 115 VAC and 28 VDC [45]. The emergency power unit has two generators, each with a capacity of 225 kVA. If three generators from four main generators are interrupted, the emergency power unit will be activated. In this paper, it is assumed that the PEMFCS is mounted as an emergency power unit and its capacity or size will be determined according to the proposed optimization model.

2.2. PEMFCS Model

Figure 3 shows an overview of the fuel cell system. The system consists of a fuel cell stack and four other sub-systems, including hydrogen, air, water and a cooling section [46]. High air pressure in the air circuit is necessary to improve the PEMFCS power density, so the compressor is used to achieve the desired air pressure level. The air cooler is used to reduce the temperature of compressed air. A humidifier is used to add vapor into air flow in order to prevent dehydration of the membrane. To minimize the wastage of the hydrogen, a recirculation system is used to prevent the flooding of the PEMFCS and to limit the reactant losses. The hydrogen consumption of this PEMFCS is as Equation (1) [47].

$$P_t{}^{h,ba}=a_0{P}_t^{f,b{a}^2}+a_1{P}_t^{f,ba}+a_2$$

(1)

where $a_0$, $a_1$ and $a_2$ are the function’s coefficients. $P_t^{h,ba}$ is hydrogen power of the baseline PEMFCS. Indeed, it is a power that is produced by hydrogen as the input power of PEMFCS. Also, $P_t^{f,ba}$ is net power of the baseline PEMFCS. In order to capture the effects of PEMFCS on the system cost, the positive decision variable $d^f$ is assigned as follows:

$$P_t^f=d^f{P}_t^{f,ba};P_t^h=d^f{P}_t^{h,ba}$$

(2)

Plugging (2) in (1) yields a convex PEMFCS model [40]:

$$P_t^h=a_0\frac{P_t^{f^2}}{d_f}+a_1{P}_t^f+a_2{d}^f$$

(3)

where $P_t^h$ and $P_t^f$ are the scaled PEMFCS hydrogen power and the PEMFCS output power at time t. Also the mass of PEMFCS is scaled as follows [48]:

$$m^h=d^f(m^{f,ba}-m^h)+m^h$$

(4)

where $m^f$, $m^{f,ba}$ and $m^h$ are the total PEMFCS mass, total mass of the baseline PEMFCS and the mass of onboard hydrogen, respectively.

2.3. ESS Model

For hybridization of the system, the battery is used as an ESS. The following constraints were used to model the ESS [50].

$$0\le P_{b,q,t}^C\le b_{b,q,t}^C{P}_b^C;0\le P_{b,q,t}^D\le b_{b,q,t}^D{P}_b^D$$

(5)

$$So{C}_{b,q,NT}=So{C}_{b,q,1};So{C}_b^{\min }\le So{C}_{b,q,t}\le So{C}_b^{\max }$$

(6)

$$So{C}_{b,q,t+1}=So{C}_{b,q,t}+\Delta T(\frac{{\eta }_b^C{P}_{b,q,t}^C}{E_b}-\frac{P_{b,q,t}^D}{{\eta }_b^D{E}_b})$$

(7)

$$b_{b,q,t}^C+b_{b,q,t}^D=1;b_{b,q,t}^C,b_{b,q,t}^D\in \left\{0,1\right\}$$

(8)

In constraint (5), $P_{b,q,t}^C$ and $P_{b,q,t}^D$ are the charging and discharging power of battery b connected at bus q and at time t, respectively. They are limited by maximum chargeable and dischargeable capacity of battery b, i.e., $P_b^C$ and $P_b^D$, respectively. $b_{b,q,t}^C$ and $b_{b,q,t}^D$ are the binary variables and indicate the charging and discharging status of battery b connected at bus q and at time t. Also, $So{C}_{b,q,t}$ in (6) indicates the state of charge (SoC) battery b connected at bus q and at time t which is limited by its boundaries. The dynamic model of energy exchange in the battery is shown in (7). In this regard, ${\eta }_b^C$ and ${\eta }_b^D$ are the charging and the discharging efficiency of battery b, respectively. The constraint (8) prevents the battery from charging and discharging simultaneously.

2.4. Loads Priority Model

The main electrical loads in the Boeing 787 are:

• Environmental control system (ECS) and pressurization. Compressed air produced by gas turbines or bleed air is used for cabin pressure, engine cooling, anti-icing system, and so on. The Boeing 787, uses electric compressors instead of bleed air.
• Wing anti-icing. The absence of bleed air makes it possible to use an electric heater in the anti-icing system which is embedded in the wing leading edge.
• Electrical motor pumps. Some of the hydraulic pumps in the aircraft have been replaced by electric pumps.

The electrical load management of an aircraft is important during emergency conditions because it must prevent the collapse of the system during operation. In this part of the paper, a priority model is presented. Various factors can affect the priority of having one load on another in an MEA electrical system. These factors include load criticality, expected energy not supplied (EENS) if a load is interrupted and penalty of interruption [51]. An efficient model must be developed to include all of these factors in the optimization formulation. The proposed model calculates a weight for each load in the system. These weights express the priority of loads and can be used in the optimization problem. Also, this weight is dynamic and time-dependent. It means this model can update its weight during different situations. The modeling process is described as follows:

2.4.1. Load Criticality

The connection of some loads during an emergency event are vital regardless of their energy consumption or cost of interruption. Therefore, these type of loads must be prioritized over all others in the system. The higher the priority of a load the later it will be shed during an event. Some Boeing 787 main electrical loads are shown in

Table 1. The Boeing 787 loads with their priorities.

Load Type	Demand (kW)	Connected Bus	Priority	${\mathit{W}}_{\mathit{j},\mathit{t}}^{\mathit{c}\mathit{r}\mathit{i}\mathit{t}\mathit{i}}$
ECS/Pressurization	320	270 VDC	High	2
Hydraulics	40	270 VDC	High	2
Equip. Cooling	40	270 VDC	High	2
ECS Fans	32	270 VDC	High	2
ICS	40	115 VAC	High	2
Flight Control	14	28 VDC	High	2
Fuel Pumps	32	230 VAC	High	2
Various AC Loads	140	115 VAC	Medium	1
Ice Protection	60	230 VAC	Medium	1
Forward Cargo AC	60	230 VAC	Low	0
Galleys	120	230 VAC	Low	0
Various DC Loads	20	28 VDC	Low	0

[52]. $W_{j,t}^{criti}$ in this table shows the importance weight of load j at time t. As shown in Table 1, the value of 2 for this weight means that load is more important than others. The loads with medium and low importance are classified in 1 and 0, respectively. This weight is a function of time, so it can be changed during the different situations.

2.4.2. EENS

The effect of an interrupted load j on the EENS is calculated as follows:

$$W_{j,t}^{ENS}={\displaystyle \sum _{k=t}^{t+MTTR}{p}_{j,q,t}}$$

(9)

where $p_{j,q,t}$ is the forecast demand of load j at time t. MTTR is the mean time to repair for the faulted component. This index shows the expected failure duration. Also, $W_{j,t}^{ENS}$ is the expected effect of load j if interrupted at time t.

2.4.3. Penalty of Interruption

The penalty of interruption is determined based on the value of lost load (VOLL). That is, the load with a higher value, in the event of an interruption, entails a higher penalty value in the objective function. The penalty associated with interruption is as follows:

$$W_{j,t}^{pen}=pe{n}_{j,t}(t,loadtype)$$

(10)

where $W_{j,t}^{pen}$ is the penalty of interruption for load j. Also, $pe{n}_{j,t}(t,loadtype)$ is the considered penalty of interruption for load type j at time t.

At this point, all weights associated with each load in the system have been defined: these are $W_{j,t}^{criti}$, $W_{j,t}^{ENS}$ and $W_{j,t}^{pen}$. The final weights associated for each load can be calculated as follows:

$$W_{j,t}^L=W_{j,t}^{criti}+w_{ENS}\times W_{j,t}^{ENS}+w_{pen}\times W_{j,t}^{pen}$$

(11)

$$w_{ENS}+w_{pen}=1$$

(12)

where $W_{j,t}^L$ is the total priority weight of load j at time t. $w_{ENS}$ and $w_{pen}$ are the EENS and penalty importance weights, respectively. These weights can be assigned by the load management unit when a fault occurs. A schematic overview summarizing the proposed prioritization model is shown in Figure 4.

3. Optimization Framework

3.1. Hydrogen Consumption Cost

The first goal is the hydrogen consumption minimization. The hydrogen consumption cost is as follows:

$$C^h=\frac{c^h}{L^h}{\displaystyle \sum _{t=1}^{NT}{P}_t^h}\Delta t$$

(13)

where $c^h$ and $L^h$ are the hydrogen price in ($/g) and the lower heating value of hydrogen in (J/g). the simulation time horizon is denoted by NT. Also, $\Delta t$ is the time interval in (s).

3.2. Load Shedding Penalty

The second goal is the load shedding minimization in the emergency or abnormal conditions. Therefore, the disconnection of loads causes the penalty as following in the main objective function:

$$\begin{array}{l}{C}_{Ls}={\displaystyle \sum _{t=1}^{NT}\left({\displaystyle \sum _{q=1}^{Nq}{\displaystyle \sum _{j=1}^{Nj}{W}_{j,t}^L(1-y_{j,q,t})+{\displaystyle \sum _{g=1}^{Ng}{\displaystyle \sum _{q=1}^{Nq}{w}_{gq}^{\prime }{\alpha }_{g,q,t}}}}}\right)}\end{array}$$

(14)

Here, $y_{j,q,t}$ and ${\alpha }_{g,q,t}$ are the binary variables that describe the connection of load j at bus q at time t and the connection of generator g at bus q at time t, respectively. Also, $w_{gq}^{\prime }$ is the disconnection penalties of generator g from bus q, respectively.

3.3. Main Objective Function

In this paper, the multi-objective optimization problem is broken into two parts: hydrogen fuel consumption and load shedding. Therefore, the convex main objective function F is formed in order to minimize a weighted summation of the hydrogen consumption and the load shedding costs as follows:

$$F=\frac{\beta }{M}{C}^h+(1-\beta )C^{Ls}$$

(15)

where $0\le \beta \le 1$ is a weighting factor and expresses the importance of the two objectives. Also, M is a scaling constant and used for preventing the numerical hazards [53]. In addition to the PEMFCS and ESS constraints that were described before, the other constraints of the above objective function are as follows:

3.3.1. Power Balance

$P_{q,t}^{req}$ in (16) is the required power at bust q and at time t which is equal to the total connected loads at each time interval. $P_{q,t}^{bus}$ in (17) is the total generation of bus q at time t which is summation generation of PEMFCS and the main generators. Therefore, the power balance equation is formed as (18). $v_{f,t}$ is the commitment status of PEMFCS f at time t.

$$P_{q,t}^{req}={\displaystyle \sum _{j=1}^{Nj}{y}_{j,q,t}{p}_{j,q,t}}$$

(16)

$$P_{q,t}^{bus}={\displaystyle \sum _{g=1}^{Ng}{\alpha }_{g,q,t}{p}_{g,q,t}}+P_{f,t}{v}_{f,t}$$

(17)

$$\begin{array}{l}{\displaystyle \sum (P_{b,q,t}^C-P_{b,q,t}^D)}=P_{q,t}^{bus}-P_{q,t}^{req}\end{array}$$

(18)

3.3.2. Generators Connection

A maximum of four main generators should feed the main bus. This constraint is modeled as follows:

$${\displaystyle \sum _{g=1}^{Ng}{\alpha }_{g,q,t}}\le 4$$

(19)

3.3.3. Emergency Power Unit

In the case of failure of main generators, the fuel cell-based emergency power unit has the task of supplying the lost power. In the event of a lack of power generation by the main generators, the produced power by PEMFCS must be injected into the main bus. With regard to limitations (20) and (21), The total connection status of four main generators and the PEMFCS at any moment must reach a maximum of five and at least one. Also, the operational constraints of PEMFCS are modeled in (22)–(25).

$${\displaystyle \sum _{g=1}^{Ng}{\alpha }_{g,q,t}}+v_t^f\le 5$$

(20)

$${\displaystyle \sum _{g=1}^{Ng}{\alpha }_{g,q,t}}+v_t^f\ge 1$$

(21)

$$P_t^f{v}_t^f\le P^{f,\max }{d}^f$$

(22)

$$P_t^f{v}_t^f\ge P^{f,\min }{d}^f$$

(23)

$$d^f\le d^{f,\max }$$

(24)

$$d^f\ge d^{f,\min }$$

(25)

3.4. Transformations

The constraint (17) contains a mixed-integer quadratic constraint. Here this constraint is transformed into mixed-integer linear form, by considering $a{e}_{g,q,t}\le {\alpha }_{g,q,t}{p}_{g,q,t}$ we have the following constraints by employing the proposed techniques in [54]:

$$a{e}_{g,q,t}\le {\overline{p}}_{g,q,t}{\alpha }_{g,q,t}$$

(26)

$$a{e}_{g,q,t}\le p_{g,q,t}$$

(27)

$$a{e}_{g,q,t}\ge p_{g,q,t}-{\overline{p}}_{g,q,t}(1-{\alpha }_{g,q,t})$$

(28)

where ${\overline{p}}_{g,q,t}=\max (p_{g,q,t})$. The same transformation is applied for $P_{f,t}{v}_{f,t}$. So we replace the constraint (17) with the above constraints.

Another transformation is related to the operation of ESS. The ESS operation modes can be described by the following constraints:

$$if{P}_{q,t}^{bus}\ge P_{q,t}^{req}\to b_{b,q,t}^C=1$$

(29)

$$if{P}_{q,t}^{bus}\le P_{q,t}^{req}\to b_{b,q,t}^D=1$$

(30)

The above constraints contain logic constraints. In this paper, we replace these constraints with their K.K.T optimality condition [55]. The above constraints can be rewritten as follows:

$$1=\{\begin{array}{c}{b}_{b,q,t}^C\\ b_{b,q,t}^D\end{array}\begin{array}{l}{P}_{q,t}^{bus}-P_{q,t}^{req}\ge 0\\ P_{q,t}^{bus}-P_{q,t}^{req}<0\end{array}$$

(31)

By considering ${\mu }_{1,t}$ and ${\mu }_{2,t}$ as the Lagrangian multipliers of the constraint (31), the K.K.T optimality condition can be obtained:

$$(P_{q,t}^{bus}-P_{q,t}^{req})-{\mu }_{1,t}+{\mu }_{2,t}=0$$

(32)

$${\mu }_{1,t}(1-b_{b,t}^D)=0$$

(33)

$${\mu }_{2,t}(1-b_{b,t}^C)=0$$

(34)

$${\mu }_{1,t},{\mu }_{2,t}\ge 0$$

(35)

if $P_{q,t}^{bus}-P_{q,t}^{req}\ge 0$, according to (31), $b_{b,q,t}^C$ is equal to 1. So, the second term of (33) is not zero and its first term is zero. i.e., ${\mu }_{1,t}=0$ and ${\mu }_{2,t}$ is obtained as follows:

$${\mu }_{2,t}=-P_{q,t}^{bus}+P_{q,t}^{req}$$

(36)

Plugging (36) in (34), the equivalent of the first term of (31) can be rewritten as follows:

$$(P_{q,t}^{bus}-P_{q,t}^{req})(1-b_{b,t}^C)\le 0$$

(37)

In the same manner, the equivalent of the second term of (31) can be rewritten as follows:

$$(P_{q,t}^{bus}-P_{q,t}^{req})(1-b_{b,t}^C)\ge 0$$

(38)

It is worth mentioning that the non-equality constraints are preferable in the commercial solver [56]. So, the constraints (37) and (38) have been written in the form of non-equality. The scaled PEMFCS hydrogen power in Equation (3) is approximated by a first-order equation to prevent the problem from being concave.

4. Optimization Results

The PEMFCS/battery hybrid emergency power unit was considered as a case study to demonstrate the proposed multi-objective optimization results. The system parameters are given in

Table 2. Proton exchange membrane fuel cell (PEMFC) and energy storage systems (ESS) characteristics [47].

Parameter	Value	Parameter	Value
Hydrogen price $c^h$ ($/g)	0.00341	battery capacity E (kWh)	50
Hydrogen lower heating value $L^h$ (J/g)	120,000	Maximum charging power $P^C$ (kW)	25
Baseline PEMFC mass $m^{f,ba}$ (kg)	223	Maximum discharging power $P^D$ (kW)	25
On-board hydrogen mass $m^h$ (kg)	73	Maximum battery state of charge (SoC) $So{C}^{\min }$ (%)	30
Maximum PEMFC scaling $d^{f,\max }$	3	Minimum battery SoC $So{C}^{\min }$ (%)	90
Minimum PEMFC scaling $d^{f,\min }$	0.1	Charging/discharging efficiency ${\eta }^C$/${\eta }^D$ (%)	90/85

. The hydrogen price was 0.00341 ($/g) [57]. The hydrogen lower heating value was 120,000 (J/g). The baseline PEMFCS mass and the onboard hydrogen mass were considered 223 and 73 kg, respectively [58]. Also, the maximum and minimum PEMFCS scale parameter (i.e., $d^{f,\max }$ and $d^{f,\min }$) were 3 and 0.1, respectively. The capacity of each main generator was 250 kVA [58]. The CPLEX solver was used to solve the proposed MIQP in the GAMS software [59]. Also, the model was simulated during the cruise condition of Boeing 787. The aircraft distribution system was considered as a single-bus network. The execution time for solving the proposed model was 0.114 s.

4.1. Case A: Normal Mode

In this mode, all of the four main generators were in their ideal condition. Figure 5 shows the output power of PEMFCS with $\beta =0$. The zero value for $\beta$ means that there was no importance on the PEMFCS capacity and hence on the amount of hydrogen consumption. In other words, no cost was considered for it. Therefore, the amount of dispatched PEMFCS power was equal to its maximum capacity, i.e., 300 kW.

For the rest values of the weighting parameter $\beta$, the dispatched power was zero. In other words, as the hydrogen consumption cost increased, the emergency power unit did not inject any power into the main bus during normal condition. Figure 6 shows the changes in PEMFCS size by increasing the weighting parameter $\beta$. As can be seen, with the increasing importance on the hydrogen consumption cost, the size decreased from 300 kW to 30 kW. Although this change was happening very fast between $\beta =0$ and $\beta =0.1$, it remained constant for values greater than 0.1. It can be said that during normal condition, the optimum size of the PEMFCS was very sensitive to its cost rises.

Figure 7 shows the SoC of batteries during normal mode. As it shows, the ESS did not operate in this mode. All loads were powered from main bus and no loads were interrupted. This result was valid for all values of the weighting parameter $\beta$.

4.2. Case B: Abnormal Mode with Fixed Priorities

In this case, the optimal solutions of the optimization problem were investigated during emergency conditions, i.e., main generator outages. Also, the priorities of loads were considered only based on their interruption penalty, i.e., $W_{j,t}^{pen}$. It means the higher interruption penalty of a load caused it to be shed later during an outage. The value of $W_{j,t}^{pen}$ for the high priority loads, medium priority loads and low priority loads were considered 10³, 10² and 10, respectively.

Table 3. The load status connection in the moment of three outages for $\beta =0$.

Connection Status of Loads
Time	ECS	Hydraulics	Equip. Cooling	ECS Fans	ICS	Flight Control	Fuel Pumps	Various AC Loads	Ice Protection	Forward Cargo AC	Galleys	Various DC Loads
15	0	1	1	1	1	1	1	1	1	1	0	1
16	0	1	1	1	1	1	1	1	1	1	0	1
17	0	1	1	1	1	1	1	1	1	1	0	1
18	0	1	1	1	1	1	1	1	1	1	0	1
19	0	1	1	1	1	1	1	1	1	1	0	1
20	0	1	1	1	1	1	1	1	1	1	1	1

and

Table 4. The load status connection in the moment of three outages for $\beta =0.5$.

Connection Status of Loads
Time	ECS	Hydraulics	Equip. Cooling	ECS Fans	ICS	Flight Control	Fuel Pumps	Various AC Loads	Ice Protection	Forward Cargo AC	Galleys	Various DC Loads
15	0	1	1	1	1	1	1	1	1	0	0	1
16	0	1	1	1	1	1	1	1	1	0	0	1
17	0	1	1	1	1	1	1	1	1	0	0	1
18	0	1	1	1	1	1	1	1	1	0	0	1
19	0	1	1	1	1	1	1	1	1	1	0	1
20	0	1	1	1	1	1	1	1	1	0	1	1

contain the connection status of loads during three generators outages and for $\beta =0$ (i.e., without considering hydrogen cost, we named this the load importance mode) and for $\beta =0.5$, respectively. The simultaneous disconnection of three generators is an extremely critical state, so the solutions of optimization problem can be robust in this case. It was assumed that the outages time was in time slot 15. As can be seen from Table 3, the high priority load ECS and the low priority galleys are interrupted at time slot 15 because of their high power magnitude (see Table 1).

Table 5. Pareto results for the four states with fixed priorities.

	First Outage		Second Outage		Third Outage		Fourth Outage
Weighting Factor	Load Shedding at the Moment of Outage (kW)	PEMFC Power (kW)	Load Shedding at the Moment of Outage (kW)	PEMFC Power (kW)	Load Shedding at the Moment of Outage (kW)	PEMFC Power (kW)	Load Shedding at the Moment of Outage (kW)	PEMFC Power (kW)
$\beta =0$	0	300	120	300	440	300	640	300
$\beta =0.1$	0	179	120	300	440	300	640	300
$\beta =0.2$	0	179	120	300	440	300	640	300
$\beta =0.3$	0	144	180	256	440	300	640	300
$\beta =0.4$	120	45	180	256	500	295	640	300
$\beta =0.5$	120	45	180	256	500	180	640	300
$\beta =0.6$	120	45	320	101	500	145	660	279
$\beta =0.7$	120	45	320	101	500	111	720	212
$\beta =0.8$	120	25	320	66	640	23	720	212
$\beta =0.9$	120	11	340	31	640	18	760	168
$\beta =1$	918	0	918	0	918	0	918	0

contains the Pareto results for the proposed two-objective optimization problem. The optimal results of load shedding and PEMFCS power are shown at the beginning time of different generators outages. We can see for $\beta =0$ and three generators outage, the PEMFCS optimum size was 300 kW and the optimum amount of interrupted load is 440 kW which was equal to the sum of ECS and galleys. This optimum size of PEMFCS, although it prevents a greater amount of load shedding, according to Figure 8, weighs 377 kg, which is heavy.

We expect that by increasing the weighting parameter $\beta$, the size and weight of the PEMFCS will decrease, and more load will be cut off. The contents of Table 4 are similar to Table 3, with the difference that the results are reported for $\beta =0.5$ (we name this the compromise mode) in Table 4. As it turns out, in addition to the previous loads, the Forward Cargo AC was interrupted too. So, based on Table 5, the total load shedding was 500 kW and also the optimum size of PEMFCS was 180 kW. This size of PEMFCS has less weight (225 kg according to Figure 8) and looks more suitable for vehicles applications such as aircraft. So, as Table 5 turns out, by increasing $\beta$ the PEMFCS optimum size was decreased and the total load shedding was increased. For $\beta =1$ (we name this the hydrogen economy mode) all the loads were interrupted and the PEMFCS power was zero because it was the most attention to minimizing the hydrogen consumption.

The weight of PEMFCS was an important factor to select the best choice. Given this factor and according to Table 5, it can be said that the power of 279 kW for $\beta =0.6$ in the event of four generators outage was a good choice. Because, according to Figure 8, this size for PEMFCS had 340 kg weight. This amount of power and weight compared to 300 kW and 377 kg was a more appropriate choice. The amount of 279 kW for the PEMFCS as an emergency power unit led to interruption of 660 kW loads in the case of four generators outage. Which is, of course large, but the probability of simultaneous outages of four generators is also very small. If this value is selected, in the case of three generators outage, 500 kW will be cut off. With the outage of two generators, this PEMFCS power (i.e., 279 kW) leads to interrupt the load below 180 kW. Also, if only one generator is disconnected, then the 279 kW PEMFCS will not interrupt loads at any time.

Figure 9 shows the changes in the SoC of ESS during emergency modes (time slots 15–20). As it shows, the ESS released more energy in the compromise mode (i.e., for $\beta =0.5$). In this case, the battery was discharged to 30% whereas it was discharged to 60% in the load importance and hydrogen economy modes (i.e., for $\beta =0,\beta =1$).

Therefore, due to the weight limitation, a little higher value of compromise between the use of hydrogen and load (i.e., for $\beta =0.6$) led to a good choice both in terms of power and weight. Also, the ESS was more important in compromise mode. By increasing the ESS size, the amount of load shedding will be reduced for the determined PEMFCS size. However, increasing the ESS size leads to its weight gain.

4.3. Case C: Abnormal Mode with Proposed Dynamic Priorities

The purpose of this section is to investigate the impact of the proposed priority model on the optimal solutions mentioned in Case B. In this regard, Equation (11) is considered to prioritize the loads in the objective function (15). The values of $w_{ENS}$ and $w_{pen}$ were taken to be 0.5 to generalize this paper by assigning equal importance weights for each factor EENS and interruption penalty. The value of $W_{j,t}^{pen}$ for the high priority loads, medium priority loads and low priority loads were considered 10³, 10² and 10, respectively. Also, the values of $W_{j,t}^{criti}$ were equal to those mentioned in Table 1. The MTTR for each generator is assumed to be five time slots.

Table 6. The load status connection in the moment of three outages for $\beta =0$ with dynamic priorities.

Connection Status of Loads
Time	ECS	Hydraulics	Equip. Cooling	ECS Fans	ICS	Flight Control	Fuel Pumps	Various AC Loads	Ice Protection	Forward Cargo AC	Galleys	Various DC Loads
15	1	1	1	0	1	1	1	0	1	0	0	1
16	0	1	1	1	1	1	1	1	1	1	0	1
17	0	1	1	1	1	1	1	1	1	1	0	1
18	1	1	1	0	1	1	1	0	1	0	0	1
19	0	1	1	1	1	1	1	1	1	1	0	1
20	0	1	1	1	1	1	1	1	1	0	1	1

is the same as Table 3 but the proposed priority model was considered for it. As can be seen, unlike Table 3, the high priority load ECS at time slot 15 and 18 was connected. But instead of that, the ECS fans which have less power were cut off during these hours. Also, various AC loads and forward AC cargo were disconnected which have medium and low priorities, respectively. Because in the proposed model, in addition to the penalty interruption of loads, the EENS and the importance of loads was also considered. Therefore, the disconnection of the load with high power and high priority was prevented and replaced by less important loads. Also,

Table 7. The load status connection in the moment of three outages for $\beta =0.5$ with dynamic priorities.

Connection Status of Loads
Time	ECS	Hydraulics	Equip. Cooling	ECS Fans	ICS	Flight Control	Fuel Pumps	Various AC Loads	Ice Protection	Forward Cargo AC	Galleys	Various DC Loads
15	1	1	1	0	1	1	1	0	1	0	0	1
16	0	1	1	1	1	1	1	1	1	1	0	1
17	0	1	1	1	1	1	1	1	1	0	0	1
18	1	1	1	0	1	1	1	0	1	0	0	1
19	0	1	1	1	1	1	1	1	1	0	0	1
20	0	1	1	1	1	1	1	1	1	0	0	1

verifies these results for $\beta =0.5$. However, the total amount of load shedding was increased for forward cargo AC load during time slots 17–20.

The effect of the proposed priority model on PEMFCS size and load shedding can be seen in

Table 8. Pareto results for the four states with dynamic priorities.

	First Outage		Second Outage		Third Outage		Fourth Outage
Weighting Factor	Load Shedding at the Moment of Outage (kW)	PEMFC Power (kW)	Load Shedding at the Moment of Outage (kW)	PEMFC Power (kW)	Load Shedding at the Moment of Outage (kW)	PEMFC Power (kW)	Load Shedding at the Moment of Outage (kW)	PEMFC Power (kW)
$\beta =0$	0	300	120	300	352	300	640	300
$\beta =0.1$	0	179	120	300	352	300	640	300
$\beta =0.2$	0	179	120	300	352	300	640	300
$\beta =0.3$	0	179	120	300	352	300	640	300
$\beta =0.4$	0	179	120	300	352	300	640	300
$\beta =0.5$	0	172	120	293	352	300	640	300
$\beta =0.6$	0	158	180	228	352	295	672	265
$\beta =0.7$	0	131	212	200	500	150	672	265
$\beta =0.8$	120	45	320	72	532	103	732	199
$\beta =0.9$	120	26	352	45	640	23	732	194
$\beta =1$	918	0	918	0	918	0	918	0

. As it turns out, the optimum power of the PEMFCS is 265 kW for $\beta =0.7$. This power in the modes of four, three and two generator outage led to interruptions of 672 kW, less than 500 kW and less than 180 kW, respectively. Also, in the mode of only one generator outage, it does not result in interruption of loads. So, the load shedding was decreased compared to Table 5. Also, according to Figure 8, the weight of the PEMFCS with the power of 265 kW was 325 kg. Therefore, prioritizing loads in critical situations has an important impact on the fuel cell weight and the consumption of hydrogen. According to the simulation results, the proposed priority model reduces the consumption of hydrogen while decreasing the load shedding.

5. Conclusions

In this paper, optimization of both load shedding and PEMFCS size in a commercial aircraft was investigated. The proposed multi-objective optimization was cast as MIQP. A model to better prioritize system’s loads was presented, then its effect on the optimum size of the fuel cell was studied and compared with the simple priority method. The proposed model also takes into account the cost of interruption, EENS and load criticality. The main results of the numerical analysis are as follows:

• With the proposed two-objective optimization model, it is possible to establish a tradeoff between choosing the size of the fuel cell (in terms of weight and consumption of hydrogen) and load shedding in the critical modes. The output of the proposed two-objective problem is to provide a series of solutions that help the user to select the appropriate fuel cell size. The optimal solution is to reduce hydrogen consumption and load shedding during emergency condition simultaneously;
• Prioritizing loads will definitely affect the size of the fuel cell as a source of emergency power unit. By choosing a suitable model to prioritize the loads in critical situations, the optimal fuel cell size can be selected. With the proposed priority model an optimum size of fuel cell which uses lower hydrogen and reduces the load shedding is achieved;
• Paying more attention to the cost of hydrogen consumption rather than the cost of load shedding leads to better solutions. Because for $\beta =0.6$ or $\beta =0.7$, the more appropriate solutions are achieved;
• The amount of battery capacity has a significant impact on the optimal solutions. Without considering the weight limits of the ESS, the load shedding during emergency situations will be reduced with the increase in its capacity;
• Using the transformations performed in Section 3.1, the optimization model is solved with CPLEX, which makes it possible to efficiently and quickly solve the problem. The execution time of the model in GAMS was 0.114 s.
• In future research, the degradation of ESS and PEMFCS will be considered in this model.