Energy management of an eVTOL aircraft with optimization based on the Equivalent Consumption Minimization Strategy (ECMS)

This article introduces an optimization-based energy management strategy developed and validated for electric vertical take-off and landing (eVTOL) aircraft. The primary objective of this strategy is to minimize hydrogen consumption while accounting for resource constraints, such as the battery and fuel cell, to enhance energy efficiency and sustainability while adhering to performance and safety requirements. The eVTOL employed in this research is an essential component of our “Viable” research project, featuring a hybrid propulsion system that combines batteries and fuel cells. To accomplish this, mathematical and computational techniques were employed using the equivalent consumption minimization strategy method to determine the optimal operational parameters for the eVTOL aircraft, considering the available energy resources. These techniques were implemented for the first time in MATLAB, enabling simulations of the aircraft’s performance under various conditions. The results demonstrate the efficacy of the energy management strategy in significantly reducing hydrogen consumption while maintaining optimal performance and safety. Graphs and comparative analyses are presented to highlight the evolution of hydrogen consumption compared to different parameters and to compare the approach with alternative methods. Furthermore, the article explores an alternative solution that offers related results and performance as MATLAB, utilizing the open-source software OpenModelica (OM). Energy management experiments using ECMS were conducted on OM, yielding highly satisfactory outcomes in terms of simulation accuracy and cost-effectiveness. The method was also evaluated in simulations of propulsive system resources, developed on OM, validating the results concerning energy distribution, compliance with constraints, and real-time feedback. In summary, this article presents a comprehensive strategy for effectively managing energy consumption in eVTOL aircraft. By leveraging simulations conducted in MATLAB and OM, the strategy’s effectiveness was assessed, with potential implications for advancing the sustainability of electric eVTOL aircraft.


Introduction
The transition to electric aviation holds immense promise for more environmentally friendly aerial operations.However, optimizing energy consumption for aircraft propulsion poses a significant challenge.This optimization demands high power density for ensuring dynamic performance and sufficient energy density to achieve optimal flight autonomy.Hybridizing energy sources, such as fuel cells and batteries, emerges as an essential solution to balance power and storage requirements.
Our research focuses on a comprehensive analysis of energy management in an electric eVTOL aircraft equipped with 20 motors, including 16 lifters and 4 pushers.Designed for rescue operations during road accidents, this aircraft utilizes a parallel topology to effectively merge its diverse energy sources.The study strongly emphasizes the management strategy and its significant role in optimizing flight performance.
These management strategies can be categorized into two main groups: rule-based methods and optimization methods.Extensive research has been undertaken in both categories.In paper [1], Zhang and al supply an overview of existing control strategies.For this purpose, they conducted a bibliometric study based on the publications referenced in the Science Citation Index Expanded and Conference Proceedings Citation Index-Science databases.In the study [1], it was proven that there are more optimization methods compared to rule-based methods, with 33 methods versus 9, respectively.

Rule-based management strategies
In rule-based management strategies, engineers rely on their ability and the analysis of performance data from various components.These strategies are relatively straightforward to implement and can operate in real time.Many of the rules described in the literature follow a cause-and-effect approach, aiming to optimize the system's efficiency.Typically, these rules are formulated without prior knowledge of the air traffic cycle.These rules are classified into two types: deterministic rules and rules based on fuzzy logic: The deterministic rules are therefore of the "if...then" type.The strategy can be summarized as a case study.Deterministic rules are both robust and easily parameterized, but they suffer from a lack of formalism and generalization, which makes them difficult to use for all kinds of situations [2].Seven distinct strategies are discerned based on deterministic rules.
Fuzzy Logic Control (FLC) is an extension of classical logic, i.e., a polyvalent logic where the truth values of variables are no longer true or false.It consists in considering various numerical factors to reach a decision that we expect to be correct.The main advantage of fuzzy logic is its robustness to measurement inaccuracies and perturbations.Furthermore, it is easily parameterized, which makes it adaptable to all kinds of scenarios [3].

Optimization-based energy management strategy
Optimization-based control methods are aimed at minimizing a cost function to ensure that the required power is always available along a mission profile.Such methods can be global or real time [1].
Global optimization strategies based on obtaining a global optimum by minimizing a cost function representing fuel economy and/or emissions over a given cycle, they require a priori knowledge of the profile, which is incompatible with a real-time system that is dependent on external factors such as weather conditions.For this reason, and because their computational cost is high, these methods are not used for real-time control.Nevertheless, they are often used in the dimensioning phase of the components of a hybrid system.
Real-time optimization methods: In contrast to global optimization methods, real-time optimization methods aim at minimizing an instantaneous cost function.This makes them less costly in terms of computing resources, making it possible to implement them in a variety of applications.Among them, the most famous are the ECMS and the MPC which are respectively the real time application of the PMP (Pontryagin's Minimum Principle) and the DP (Dynamic Programming) [7].We opted to apply the ECMS method in our case study thanks to its various advantages and the fact that, according to the state of the art, it is a better choice for our problematic.For this reason, we explain the principle of the method in the following section.

Equivalent Consumption Minimisation Strategy
The ECMS is based on the PMP since the purpose is also to minimize a Hamiltonian function but this time at each time step.The principle consists in reducing the global criterion to an instantaneous optimization problem, by introducing a cost function, which depends only on the variables of the system at the present time.The output variable is the power distribution.Mathematically, the aim of the strategy is to reduce the equivalent consumption.This is defined as the sum of the fuel cell (FC) power   and the battery power   .

Optimization problem definition
The power management problem is therefore to find the best distribution of power between the system's energy sources.However, this distribution must satisfy the power profile needed for each flight as well as the operating constraints.As previously mentioned, the eVTOL considered in our project has two power sources.The energy management problem can be formulated as a dynamic optimization problem in which the system, represented by a dynamic equation (1), is controlled to minimize a cost criterion (2) while respecting equality (3) and inequality (4) constraints.
The overall goal of our study is to optimize the energy distribution between the energy sources to minimize the hydrogen consumption of the fuel cell, while respecting the constraints of the power limits of the FC and the State of Charge (SOC) of the battery.The criterion most used in the literature is fuel consumption for internal combustion engines and hydrogen consumption for fuel cells, as in present application.Then the optimization problem for figuring out the equivalent hydrogen consumption is formulated as follows: Find the optimal solution x (5) which minimizes (6): (5) () = [  () + ()  ()] (6) with   and   are, respectively, the power of the battery and the FC. is the sampling time and  is the penalty coefficient empirically confirmed [5].
+  (7) This optimization problem formulation of the different hybrid system sources imposes equal, maximum, and minimum constraints on the power output and energy levels that can be achieved as shown below: is the power required for a flight mission and µ is the SOC balance coefficient; in this case, µ = 0.6; obtained from experiments to achieve a minimum SOC of 60% at the end of the mission profile [4]. _ and  _ are the minimum and maximum power of the fuel cell, respectively. _ and  _ are respectively the minimum and maximum power of the battery.For the battery SOC we opted for a value between 20% and 90%.Lastly,  is the dynamics of the fuel cell, i.e., the incrementing or decrementing of the power.

Modeling and programming in OpenModelica
OpenModelica is a versatile open-source platform designed for modeling and simulating complex systems, rooted in the Modelica language.Its applicability extends across a wide range of domains, encompassing disciplines from mechanics to biology.With a trio of optimization strategies at its disposal, OpenModelica caters to a diverse set of optimization needs.These strategies include integrated dynamic optimization using annotations, dynamic optimization eased by CasADi, and traditional optimization using OMOptim for static scenarios.OpenModelica supplies flexible solutions to tackle optimization challenges of varying complexities.
In our research, we tackled the optimization challenge by employing specific annotations to define various constraints.Additionally, we harnessed OpenModelica's tools to formalize the optimization problem, as showed in Figure 1 that follows.

Figure 1. Formalization of ECMS with OM tools
Thanks to the code developed within OpenModelica, we have effectively integrated the ECMS method.This integration has led to improved results, and we supply a comprehensive breakdown of these outcomes in the next section.In the final segment of this open-source implementation, we assessed the robustness and reliability of our method by integrating it into the propulsion system designed by our team using OM.This system consists of two batteries, each equipped with its specific Buck-Boost converter, and two fuel cells with their respective Boost converters.

Experimental results and validation
To assess the effectiveness of the developed strategy, we conducted a simulation using MATLAB and OM software to model a half-eVTOL.We focus on the results of this simulation during a typical flight mission.In Figure 2 and 4, the load profile displays the red curve.It's important to note that our study concentrates on half of the propulsion system of a full-sized eVTOL, referred to as the "half-eVTOL".Specifically, we examine 10 motors, including 8 lifters and 2 pushers, powered by two distinct sources: a fuel cell and a battery.The initial parameters for our experiments were as follows: • SOC of the battery = 90%; •  _ =  and _ = 0.

Experimental results on MATLAB
At first, a great result is illustrated in Figure 2 where the sum of the distributed power is equal to the value of the power demand   throughout the flight, proving that the developed strategy is correct and perfectly satisfies the power demand.We notice that the results obtained use only the batteries for the take-off of the aircraft considering that the power of these last is sufficient for this first phase of flight.Once the SOC is lower than the average of the SOC max and SOC min in this case the value 55%, the optimization program starts charging the batteries, and we can see that it starts charging progressively respecting the dynamics of the fuel cells.We also note that the fuel cells are solicited at the end of the flight to assist landing, but also to charge the batteries before landing, to always minimize hydrogen consumption.
Figure 3 provides evidence of adherence to all limits, which remain well within their permissible ranges.It illustrates the real-time SOC variation for the two batteries, with an initial SOC of 90% and a concluding SOC of 39%, in compliance with the α constraint, ensuring that the SOC remains below 60% at the flight's end to enable an immediate subsequent flight.The fluctuation range of the penalty coefficient is also maintained within the specified 0 to 2 boundaries, as per the limits established.
Moreover, the performance characteristics of the FC have been rigorously examined, supported by the calculation of the power derivative of the fuel cell, as depicted in Figure 3, showing a power derivative ranging from 0 to 800 W/s.To further bolster the validation of our strategy, we computed the battery current, designated as i(t), based on real-time measurement of the SOC.This computation considered the maximum charge capacity of the cells within the battery, utilizing the provided equation.It was ensured that the current consistently stayed within operational limits, and that our power values maintained their physical significance throughout the validation process.
As illustrated in Figure 3, the current also falls well within its designated operational range, and its power characteristics closely align with our expectations, proving an exact response.These collective outcomes serve as a robust validation of the strategy we have employed.The next step involves the assessment of hydrogen consumption and conducting the associated analysis, which will supply further insights into the system's performance.

Experimental results in OM
This section highlights the results obtained using OM.To verify the integration of the optimization program within the entire propulsion system, the system includes two batteries and two fuel cells.It is equipped with a Buck-boost converter for the batteries and a second boost converter for the fuel cells.Figure 4 displays four curves.The red curve is the power demand   , while the green and yellow curves depict the power generated by the fuel cells and batteries, respectively, because of the optimization.The summation of these two curves perfectly aligns with the demand, thus confirming the successful validation of the solution.The second part of the validation involves checking if the constraints have been adhered to.This is presented in Figure 5.The state of charge of the batteries aligns with the required ranges, as do the penalty factors, alpha.As for the dynamics, it also follows the constraints and is upheld.In addition to integration, we assessed the robustness by applying it to various load profiles, recognizing that, in certain applications, ECMS is influenced by the load profile.The eVTOL developed within the Viable project is specifically designed to accommodate such a load profile.In our study, we introduced a power peak to challenge the optimization behavior in this scenario, and the expected results proved to be highly effective.The power distribution, adherence to imposed constraints, and the principle of minimizing hydrogen consumption were all successfully met.
For the comparison, the evaluation between free and paid software aims to enhance our simulation capabilities without incurring more expenses.The primary disparity lies in the initial constraint of fuel cell dynamics, initially confined to ± ∆P.This refinement targets the reduction of computation time.In the OM version, this constraint has been fine-tuned to allow for a wider range of values while remaining within the ∆P boundaries.Furthermore, different numerical solving methods were employed in both scenarios.These modifications primarily improve computational efficiency and broaden the range of viable values within the predefined limits, without directly altering the solution itself.

Hydrogen consumption
To present the energy management strategies based on optimization, it is necessary to figure out the equation that defines the hydrogen consumption.The mass of fuel consumed from the fuel tank can be expressed as a function of the net power output of the cell   and the total efficiency of the fuel in our case is hydrogen, this is expressed according to the following equation.
  and ΔH are respectively the hydrogen efficiency coefficient in the FC and the hydrogen heating value equal to 33.33 kWh/kg.For the efficiency factor we have found that for a new FC the efficiency is 51-55% and for a slightly used FC it is 47-49%.In this work we have therefore opted for the value 52%.

EASN-2023
Journal of Physics: Conference Series 2716 (2024) 012013 Accordingly, at the end of the flight, the hydrogen consumption value is "6.92 Kg", a very suitable value in terms of kg for such a flight profile, using a rule-based energy management strategy (work already conducted by the Viable project also [6]).In these studies, the hydrogen mass value around 8 Kg.This result is therefore important as it supplies an important solution to the level of fuel consumption in the urban area as by minimising the consumption of hydrogen, we consequently reduce our impact upon the environment.

Conclusion
This study introduces the implementation of the Equivalent Consumption Minimization Strategy for a hybrid propulsion system in a half-eVTOL aircraft.ECMS, a real-time optimization method, aims to reduce hydrogen consumption by minimizing FC usage through an instantaneous cost function.
The first implementation of ECMS was conducted using MATLAB.The hybrid system forms lithium-ion batteries with specific charge and discharge constraints and fuel cells with dynamic characteristics.The study effectively validated ECMS through simulations of flight mission load profiles, proving promising results in terms of energy distribution, constraint compliance, and simulation duration.Tests on various profiles confirmed the method's stability.The study calculated hydrogen consumption during the flight mission, showing that ECMS resulted in the lowest consumption.
In the second part, ECMS was implemented and assessed in OpenModelica, an open-source software, with a comparison of the two approaches.ECMS was successfully applied to the hybrid system, delivering highly satisfactory results in terms of energy distribution, constraint adherence, and simulation duration.In conclusion, ECMS has once again proven its robustness, especially in a free and open-source environment.

Figure 2 .
Figure 2. Optimal power distribution achieved by the ECMS strategy.

Figure 4 .
Figure 4. Distribution of power with ECMS in OM

Figure 3 .
Figure 3. Real-time variation of the SOC, penalty factor, Current value during the flight mission and FC power derivation

Figure 5 .
Figure 5. Verification of constraints observed with ECMS in OM.