Multi-Objective Supervisory Control in More-Electric Aircraft Using Model Predictive Control: An ORCHESTRA Application

Abstract

The crucial issue of supervising and managing electrical energy in the context of aircraft electrification, known as More-Electric Aircraft (MEA), is addressed in this paper. In the pursuit of developing energy-efficient solutions with reduced environmental impact, this research contributes valuable insights into innovative control strategies crucial for advancing aircraft electrification technologies. Through optimization techniques, the management of energy aims to maximize the proposed objectives. With a focus on controlling battery power for charging, discharging, and load shedding, this study employs Model Predictive Control (MPC) alongside an optimizer solving a mixed-integer linear programming (MILP) problem. Constraints encompass various aspects, including battery charging, maximum generator power, battery absorption, discharge limits, and converter power limitations. Theoretical results and detailed simulations demonstrate the effectiveness of the proposed approach in finding a good compromise among the objectives subjected to the system constraints. Practical validation of the proposed approach is conducted through the European project ORCHESTRA, utilizing comprehensive system simulations in Matlab/Simulink (2022b).

1. Introduction

Power converters are vital components in managing power flow within electrical systems. They regulate the output to meet set values, addressing various objectives based on network conditions, overloads, and efficiency needs [1]. For instance, in aircraft systems, a DC/DC converter can be utilized to charge batteries, handle overloads, sustain voltage stability during load changes, or supply power to the grid [2]. This complexity necessitates designing multiple control algorithms and a supervisory [3] controller to select the active algorithm based on power demands.

Supervisory control plays a critical role in enhancing the performance and adaptability of these systems. By integrating predictive analytics and real-time data processing, supervisory controllers can dynamically select the most appropriate control algorithm to maintain system stability and efficiency. Studies have shown that supervisory control can significantly improve the reliability and efficiency of power management in various applications, including renewable energy systems and microgrids. According to research [4], supervisory control frameworks are essential for the effective integration of renewable energy sources, ensuring that power converters operate efficiently under varying conditions. This adaptability is crucial for modern electrical systems, where the integration of intermittent renewable energy sources and the increasing complexity of power grids demand more sophisticated control strategies. The analytic design of this control algorithm for power converters requires knowledge of the mathematical models of both the converter and the surrounding grid. This information allows for the mathematical definition of the control law and the evaluation of its performance in terms of robustness with respect to measurement noise and model uncertainties, the stability of the closed-loop system, and the capability of adapting the control action to unexpected events, such as sudden load connections or disconnections. In many cases, the control law is required to be discontinuous and provide values in the discrete set $\left\{0,1\right\}$. In other cases, instead, the control variable must be continuous in the compact interval $[0,1]$.

This knowledge enables precise control law formulation and performance evaluation concerning robustness, stability, and adaptability to unforeseen events like sudden load changes. Control strategies such as Sliding-Mode Control (SMC) [5,6], switched controllers based on Lyapunov functions [7], or Model Predictive Control [8] are commonly employed. SMC, for instance, offers accuracy, robustness, and ease of implementation, making it suitable for nonlinear systems like most power converters.

An example of applying advanced control techniques can be found in [9], where an SMC algorithm regulates a bidirectional DC/DC converter in an aircraft system, ensuring stability and efficient power management. The peculiarity of that work resides in the particular design of the SMC algorithm that, starting from the nonlinear model of the power grid, makes the closed-loop system linear and thus easier to analyze in terms of stability.

However, challenges exist in accurately estimating switching frequencies, particularly in direct switch control scenarios. Sophisticated control laws are designed based on recovered power grid models to address these challenges [10], and advanced studies have been conducted on nonlinear phenomena [11].

The mechanism concerned with changing the target is entrusted to a higher-level body called a supervisor. Its purpose could be to select one of the possible controllers of the underlying level, change its references, or choose possible gains according to the operating conditions.

One of the possible techniques for its development is a finite-state automaton, i.e., a mathematical model made up of states/transitions, where each one identifies a possible realization of an objective depending on the previous state and input. In [9], the electrical system consists of a bidirectional converter connected to one high-voltage (HV) DC bus, where a generator resides in parallel with a variable load, and a low-voltage (LV) DC bus, where a battery is linked. Here, the BBCU (buck-boost converter unit) has two main functional modalities: charging the battery, where the generator can achieve the objective of taking care of the network load, and supporting the generator in overload conditions by injecting energy taken from the battery into the grid. Hence the supervisor has a dual purpose, i.e., in normal operating conditions, it allows for control of the battery current, and in overload conditions, it allows for the control of the generator current.

Another possibility is a fuzzy network, i.e., based on logic much closer to human language than to machine language, and thus not on a Boolean degree of truth (completely true and completely false), but rather on a gradient of truth degrees. Formally, this truth degree is determined by an appropriate membership function [12]. For example, in [13], a fuzzy supervisor is used, which enables choosing a high-gain controller or a second-order sliding controller. The choice is made on the basis of fuzzy rules defined according to the state of the system (transient or steady state) and the sign of the difference of the control signals.

Another possibility is the Petri net, which consists of “places”, “transitions”, and “arcs”, which makes it very similar to a finite-state automaton. In general, it has an advantage over a finite-state automation in that it can represent an infinite number of states with a finite number of nodes and can be easily scaled. In [14], a Petri net is used for energy management in a power grid by defining a table of rules to solve overload problems or minimize control actions. The abbreviations and definitions used in this paper are listed in

Table 1. Abbreviations and definitions used in this paper.

Parameter	Value
MPC	Model Predictive Control
MEA	More-Electric Aircraft
MILP	Mixed-Integer Linear Programming
DC	Direct Current
SMC	Sliding-Mode Controller
HV	High Voltage
LV	Low Voltage
KVDC	Kilo-Voltage Direct Current
HVDC	High-Voltage Direct Current
LVDC	Low-Voltage Direct Current
DAB	Dual Active Bridge
SOC	State of Charge
EMS	Energy Management System
EPS	Energy Power System
EPC	Energy Power Controller
DERs	Distributed Energy Resources
APU	Auxiliary Power Unit

.

1.1. Electric Aircraft

In the context of aircraft electrification, an intermediate step toward the design of All-Electric Aircraft (AEA) [15] is the so-called More-Electric Aircraft (MEA) [16,17]. A pivotal challenge within the field of MEA lies in the substitution of hydraulic and pneumatic actuators with their electric counterparts [18]. This substitution is aimed at enhancing system reliability, diminishing overall weight, and introducing novel functionalities in load control. Evidently, the expanded utilization of electric energy necessitates the implementation of new automated Energy Management Systems (EMSs) on board.

The development and implementation of advanced technologies that allow for energy-efficient solutions with reduced environmental impact are of extreme importance [19]. The aviation industry is responsible for 13.9% of transportation emissions, ranking second in transport greenhouse gas emissions behind road transport.

The ACARE Flightpath 2050 long-term roadmap  [20] sets specific targets for emissions per passenger per kilometer: a 75% reduction of CO₂ emissions and a 90% reduction of NOX emissions [21]. Meeting these goals necessitates rethinking future aircraft concepts, including new configurations and disruptive technologies, to achieve a major breakthrough in this crucial industry, which has remained relatively unchanged since the jet engine revolution in the 1950s.

To tackle these challenges, the More-Electric Aircraft (MEA) concept shifts the balance toward greener and more efficient electrical energy for onboard systems [22]. Many new aircraft systems are now electrically powered, replacing traditional hydraulic and pneumatic systems, and shifting the electrical power generating capacity to new and higher levels, as shown in Figure 1. Compared to traditional aircraft, MEA represent a significant increase in onboard electric power capacity. For instance, the Boeing 787 “Dreamliner”, considered the first generation of MEA, has main electrical generators with a capacity exceeding 1 MW. Additionally, the Airbus A380 provides about 600 kVA for electrified or partially electrified loads [23].

The basic architecture of the aircraft’s electric network is known as a two-busbar system.

This system comprises a high-voltage (HV) DC bus at 540/270 V, a low-voltage (LV) DC bus at 28 V, and a bidirectional converter connecting the two. The HV bus receives power from a starter generator and rectifier and supports heavy loads such as anti-icing and electro-mechanical actuators. Conversely, the LV side houses batteries and other Energy Storage System (ESS) devices like supercapacitors [24]. Traditionally, aircraft systems utilize one or two batteries, primarily serving as starters for engines or the Auxiliary Power Unit (APU) and providing power to essential loads in emergencies. Consequently, batteries often remain inactive for a significant portion of their operational lifespan. The integration of automated ESSs enables better battery utilization, especially during overloads or when priority loads require additional power. Recent advancements [25] in structural batteries offer a unique solution by combining energy storage with structural support, potentially reducing the weight-to-power density ratio. This innovation holds promise, particularly in aerospace applications, and has already found practical use in unmanned aerial vehicles. Structural batteries introduce a paradigm shift by decentralizing energy storage, requiring precise control strategies, especially in scenarios involving multiple concurrent ESS deployments within a microgrid framework, as discussed further in the subsequent section.

In MEA, where there are many electrical components and various scenarios, it is necessary to have a high-level supervisor to accompany one or more controllers managing the entire electrical system. This setup can address problems related to the distribution, conversion, limitation, or management of electrical power in increasingly electrified aircraft.

Further investigation into supervisory strategies for the development of the ’Integrated Energy Management (I-EM)’ concept is illustrated in PROSIB. Specifically, through a rigorous mathematical approach, an optimization function has been developed that processes data from some equipment in real time and returns reference signals to the various controllers to supervise their behavior in terms of electrical energy requirements.

All electrical components that are integrated into the aeronautical context represent a fairly complex scenario consisting of loads, batteries, electric motors, generators, and propellers. These electrical components are linked to two buses through DC/DC and AC/DC converters, which must be managed by controllers. The supervisor is responsible for choosing references for the controllers in order to minimize the deviation of the generator currents from a reference value. This minimization is achieved by adjusting the electrical loads of the system, whether active or passive, through the use of converters and contactors capable of managing power flows. More specifically, the supervisor acts as an optimizer that finds a bounded minimum of the cost function in real time.

1.2. MPC Control Strategy

MPC can be considered a sophisticated control strategy that finds extensive use in both electrical systems and aircraft applications due to its predictive nature and optimization capabilities [26].

Energy management has wide use and is often addressed through the use of MPC because it is perfect for solving predictive optimization problems. Energy management systems have been applied to a hybrid power system comprising a wind energy conversion system and a battery energy storage system [27], as well as electric vehicles [28], where MPC was used for optimization. MPC-based supervisory control may also be a good choice for energy management in many fields, such as railway energy storage systems [29]. In electrical systems, MPC plays a pivotal role in various domains such as power electronics, grid control, and renewable energy integration. It enables precise control of power converters like inverters and rectifiers, ensuring efficient energy conversion and stability [30].

MPC is particularly beneficial in smart grid applications, where it helps manage power flow, optimize energy usage, and maintain grid stability by considering predictive models and system constraints.

Microgrids, which are localized energy systems with distributed energy resources, benefit from the capability of MPC to balance supply and demand, incorporate renewable energy sources, and ensure grid stability.

Moreover, MPC facilitates the effective coordination and control of distributed energy resources (DERs) such as solar PV systems and wind turbines, enhancing overall grid performance and reliability.

Within electrical systems, MPC continues to evolve and make significant contributions in areas such as electric vehicle (EV) power management and energy storage systems. In EVs, MPC optimizes battery charging/discharging rates, motor control, and regenerative braking, extending battery life and improving vehicle performance. Additionally, MPC is instrumental in managing energy storage systems such as batteries and supercapacitors, maximizing their efficiency and lifespan while meeting dynamic power demands.

In the aerospace and aircraft industry, MPC is instrumental in flight control systems, autopilot functionalities, and performance optimization. It enables precise regulation of aircraft dynamics, control surface movements, and flight trajectories to ensure stability, safety, and fuel efficiency. MPC algorithms are integrated into autopilot systems to assist pilots during various flight phases, from takeoff to landing, by adjusting control inputs and flight parameters based on real-time conditions and mission requirements. Additionally, MPC enhances fault tolerance and redundancy management in aircraft systems, contributing to operational continuity and safety in critical scenarios [31].

Overall, MPC’s capability to anticipate future system behavior and optimize control actions makes it a widely used tool in modern engineering applications because it offers improved performance, efficiency, and reliability across a wide range of dynamic systems and complex environments.

This expansion of MPC’s application scope in aerospace and electrical systems underscores its versatility, adaptability, and ongoing relevance in advancing technological capabilities and addressing complex engineering challenges in these domains.

1.3. Main Contribution

The article focuses on onboard energy management in aircraft, employing Model Predictive Control (MPC) as a supervisory strategy. It defines a scenario encompassing constraints and objective functions within a complex environment. MPC tackles a mixed-integer linear programming (MILP) problem, incorporating both continuous and discrete variables. The continuous variables regulate battery power, while the discrete variables determine whether to charge or discharge batteries and decide on load-shedding operations, i.e., disconnecting or connecting loads. This study is grounded in a scenario derived from the European project ORCHESTRA which aims to further electrify aircraft by enhancing intelligent energy management operations and it is part of the Horizon 2020, an initiative that finances research projects or actions aimed at scientific and technological innovation that have a significant impact on the lives of European citizens.

1.4. Outline

The rest of this paper is organized as follows. In Section 2, preliminaries, local control, and the problem formulation are presented. In Section 3, the MPC algorithm of the ORCHESTRA application, designed to manage aeronautic energies, is presented and detailed. Simulation results showing the effectiveness of the proposed algorithm are presented in Section 3.4, with the conclusions given in Section 4.

2. Supervisory Control Using MPC

2.1. Preliminaries

Control theory allows us to govern a system at different levels of abstraction. In the context of complex systems, we have many interacting parts that must be controlled individually with a local control algorithm.

In complex systems theory, however, putting all the subparts together gives rise to emergent behaviors that can change the overall dynamics. Therefore, in order for us to control the system appropriately, we have to operate at a higher level of abstraction to implement a global supervisor. This supervisor can, in some cases, change the references of local controllers and, in other cases, change the topology of the complex network.

In parallel, optimization theory is widely used in the development of control and supervisory systems.

In fact, in many research works of the scientific literature, e.g., [32], a control algorithm can be derived from or traced back to an optimization algorithm.

An example of a control algorithm that uses optimization theory is MPC, Model Predictive Control, which is widely used in control system development. MPC is based on a mathematical model that allows us to make predictions [33].

Specifically, it consists of the optimized prediction of the evolution of the system over a number of steps equal to the prediction horizon ($H_p$), which is a parameter that needs to be tuned.

Thus, the use of optimization algorithms can be useful in bringing predictive control algorithms to life based on models.

The supervisory algorithm provides references to converters, which are controlled by local controllers using more classical approaches, such as PID, or nonlinear approaches, such as SMC (Sliding-Mode Control).

2.2. Model Predictive Control

One of the most widely used supervision algorithms is MPC, which combines the theories of control and optimization. It is based on model predictions with the purpose of determining the optimal control action, as shown in Figure 2.

MPC is not a specific control strategy but rather a family of methodologies aimed at obtaining optimal control signals to minimize a cost function that reflects a system’s performance.

To delve deeper into this methodology, here are the main components of an MPC-based algorithm:

• Time window: An interval starting from an arbitrary time $t_0$ to $t_0+T$, with T also being arbitrary.
• Prediction horizon $H_p$: Describes the ability of the system to forecast the future, corresponding to the size T of the time window.
• Control horizon $H_c$: The number of steps within the time window over which predicted control actions are applied. Its choice affects the number of parameters used to characterize the future control sequence.
• Receding Horizon Control (RHC): Calculates the optimal trajectory for the time window at each time step but implements only the first sample of the sequence, discarding the rest.
• Information system: During prediction, relevant information about the system until time $t_0$ is needed. This information can be directly measured or estimated using an observer from the state if not accessible.
• Model: A good model that satisfactorily describes the dynamics of the system.
• Optimizer J: Defines an optimization criterion to make the best decision. The objective function variable is defined by the error between the desired response and the actual response of the system. The optimal control action is obtained by minimizing this cost function within the optimization window.

Starting from the knowledge of $x\left(k\right)$, state prediction is performed for $H_p$ samples. The predicted states at future time instants are denoted as:

$$x(k+1|k),x(k+2|k),\dots ,x(k+H_p|k)$$

(1)

A control sequence $u\left(k\right)$ for time instants $k=1,2,\dots ,H_c$ with $H_c\le H_p$ is then defined, representing the control actions used in prediction to obtain the predicted states.

Thus, based on the theoretically set control trajectory (where only the first sample of the sequence is implemented), information about the system’s subsequent states can be derived.

Given a reference signal $y_r\left(k\right)$, the objective of predictive control is to steer the predicted system output as close as possible to the reference signal within the prediction horizon $H_p$. This is achieved by finding the optimal values for the parameters characterizing the control vector U that minimize the error defined by the cost function J:

$$J=\sum _{i=0}^{H_p}\left[{(y_r(k+i|k)-y(k+i|k))}^{T}Q(y_r(k+i|k)-y(k+i|k))-u{(k+i|k)}^{T}Ru(k+i|k)\right]$$

(2)

where Q and R are weight matrices that ensure the stability and performance of the control system.

In conclusion, setting Q and R based on system requirements defines the optimization problem for MPC, ensuring stable closed-loop control. The other two fundamental parameters are $H_p$ and $H_c$. The first is the prediction horizon $H_p$, which represents the number of prediction steps of the system dynamics, and the second is the control horizon $H_c$, which represents the number of prediction steps for the control action.

In general, $H_c\le H_p$, with $H_c=20/30\%H_p$. A larger prediction horizon increases computational complexity, but it does not necessarily mean that the control action works better  [34].

So, fine-tuning the selection of these parameters is essential. Assuming a topology like the one shown in the previous section, with only one battery, one converter, one generator, and one load, local control of the converter is carried out with a dual loop of PI and Sliding-Mode Controller, while MPC is used as a global supervisor to determine the switching of the contactor and the power reference of the converter, which affect the charging and discharging of the battery.

Thus, the task implemented by MPC involves the design of the EPS, which is formulated as a mixed-integer linear problem (MILP).

Before implementing MPC, it is necessary to have a dynamic formulation of the system, which is provided in the next section.

2.3. Dynamical Model of the System

In the simplified configuration considered, that is—where there is a battery, a converter, a starter-booster generator, and only one load (Figure 3)—assuming that the power load profile $P_L\left(i\right)$ is known instant by instant and that the switch is ideal, the only dynamic equation of the system is that of the state of charge ($SOC$) of the battery, which is a real value between 0 and 1 that quantifies the charge of the battery, from 0 ($0\%$) to 1 ($100\%$).

Note that, generally speaking, the generator can be seen as a source. In Figure 3, the generator regulates the voltage on the bus, while the battery regulates the power.

So, the only state variable is the $SOC$, and its dynamic equation is as follows:

$$SOC(i+1)=SOC\left(i\right)+{\alpha }_c{P}_c\left(i\right)-{\alpha }_d{P}_d\left(i\right)$$

(3)

where $P_c\left(i\right)$ and $P_d\left(i\right)$ are, respectively, the charging and discharging power of the battery (measured in Watts $\left[W\right]$), which are control inputs given by the MPC controller; and ${\alpha }_c$ and ${\alpha }_d$ are the charging and discharging rates of the battery (measured in $\left[{\displaystyle \frac{1}{W}}\right]$).

We can also define the power of the battery as follows:

$$P_{bat}\left(i\right)=P_c\left(i\right)-P_d\left(i\right)$$

(4)

where $P_c\left(i\right)\ge 0$ and $P_d\left(i\right)\ge 0$ $\forall i$.

Therefore, having formalized the dynamics of the system, it is possible to formalize the associated optimization problem, including the constraints.

2.4. Formalization of the Generic Optimization Problem

First, MPC must determine the battery charging power ($P_c\left(i\right)$), the discharging power ($P_d\left(i\right)$) (continuous variables), and the contactor switching variable $s\left(i\right)$ (binary variable). Since both continuous and binary variables are present, we are dealing with an MILP (mixed-integer linear programming) problem.

So, our problem can be written in the following form:

$$\underset{x,y}{min}J(x,y),x\in {\mathbb{R}}^n,y\in {\mathbb{N}}^m$$

(5)

such that

$$\left\{\begin{array}{cc}A{\left[xy\right]}^T\hfill & \le b\hfill \\ A_{eq}{\left[xy\right]}^T\hfill & =b_{eq}\hfill \end{array}\right.$$

(6)

In the general formulation of this problem, the matrices have the following dimensions:

$$A\in {\mathbb{R}}^{p\times q},A_{eq}\in {\mathbb{R}}^{e\times q},b\in {\mathbb{R}}^p,b_{eq}\in {\mathbb{R}}^e$$

In more detail, the objective function is the sum of different components, expressed as:

$$J={\omega }_L{J}_L+{\omega }_{\delta }{J}_{\delta }+{\omega }_c{J}_c+{\omega }_d{J}_d+{\omega }_{SOC}{J}_{SOC}+{\omega }_{bat}{J}_{bat}$$

(7)

Given the prediction horizon $H_p$ and the control horizon $H_c$, we have:

$$J_L=\sum _i^{Hc}{\displaystyle \frac{(1-s(i\left)\right)}{H_c}}$$

(8)

This term minimizes the number of load-shedding instances, normalized to be between 0 and 1.

Then, we have:

$$J_{\delta }=\sum _i^{Hc}{\displaystyle \frac{\left|s(i+1)-s\left(i\right)\right|}{H_c}}$$

(9)

This term minimizes the switching frequency of the contactor connected to the load.

Then, the objective function includes components related to the battery, consisting of four parts:

$$J_c=\sum _i^{Hp}{\displaystyle \frac{|P_c^{max}-P_c\left(i\right)|}{H_p{P}_c^{max}}}$$

(10)

where $P_c^{max}$ is the maximum charging power of the battery. This term aims to maximize the charging power when the battery is being charged.

Similarly:

$$J_d=\sum _i^{Hp}{\displaystyle \frac{|P_d\left(i\right)|}{H_p{P}_d^{max}}}$$

(11)

where $P_d^{max}$ is the maximum discharging power of the battery. This term aims to minimize the discharging power when the battery is being discharged.

Then, regarding the SOC of the battery, we have:

$$J_{SOC}=\sum _i^{Hp}{\displaystyle \frac{|SO{C}^{max}-SOC(i+1)|}{H_{p}SO{C}^{max}}}$$

(12)

where $SO{C}^{max}$ is the upper bound of the battery’s SOC. It is not efficient to charge the battery to $100\%$, so to optimize battery performance, we establish an SOC upper bound $SO{C}^{max}$ and an SOC lower bound $SO{C}^{min}$. This part of the objective function aims to maximize the SOC of the battery up to $SO{C}^{max}$.

Then, the last part of the objective function is:

$$J_{bat}=\sum _i^{Hp}{\displaystyle \frac{|P_{bat}(i+1)-P_{bat}\left(i\right)|}{H_p(P_c^{max}+P_d^{max})}}$$

(13)

which maximizes the prediction of the battery power, whose equation is shown in (4).

So, $J(x,y)$ is a nonlinear objective function, and the weights of the objective functions are:

$${\omega }_L,{\omega }_{\delta },{\omega }_c,{\omega }_d,{\omega }_{SOC},{\omega }_{bat}\in {\mathbb{R}}_{+}$$

which should be appropriately tuned.

The objective function described is subject to constraints due to the topology of the electrical scheme and the dynamics of the system.

First, the battery obviously cannot be charged and discharged at the same time, so we introduce two binary variables ${\zeta }_c\left(i\right)$ and ${\zeta }_d\left(i\right)$, where ${\zeta }_c\left(i\right)=1$ if the battery is charging at time instant i and 0 if it is discharging, and ${\zeta }_d\left(i\right)$ is defined similarly. Then, we have the following constraints:

$$\left\{\begin{array}{c}0\le P_c\left(i\right)\le {\zeta }_c\left(i\right)P_c^{max}\hfill \\ 0\le P_d\left(i\right)\le {\zeta }_d\left(i\right)P_d^{max}\hfill \\ {\zeta }_c+{\zeta }_d=1\hfill \end{array}\right.$$

(14)

In addition, there are also constraints on SOC dynamics to ensure the maximum efficiency of the battery, as explained above. Therefore,

$$SO{C}^{min}\le SOC(i+1)\le SO{C}^{max}$$

(15)

where the dynamics of $SOC(i+1)$ are given by (3).

Another practical constraint of the system is the input power provided by the generator, which is always positive and never exceeds a maximum power $P_{in}^{max}$.

So, expressing the input power as the sum of the power supplied to the battery and the power supplied to the loads, we obtain:

$$P_{in}\left(i\right)=P_c\left(i\right)-P_d\left(i\right)+s{P}_L\left(i\right)$$

(16)

which leads to the constraint:

$$0\le P_c\left(i\right)-P_d\left(i\right)+s{P}_L\left(i\right)\le P_{in}^{max}$$

(17)

Summing up, the overall set of constraints is as follows:

$$\left\{\begin{array}{c}0\le P_c\left(i\right)\le {\zeta }_c\left(i\right)P_c^{max}\hfill \\ 0\le P_d\left(i\right)\le {\zeta }_d\left(i\right)P_d^{max}\hfill \\ {\zeta }_c+{\zeta }_d=1\hfill \\ SO{C}^{min}\le SOC(i+1)\le SO{C}^{max}\hfill \\ 0\le P_c\left(i\right)-P_d\left(i\right)+s\left(i\right)P_L\left(i\right)\le P_{in}^{max}\hfill \end{array}\right.$$

(18)

and assuming that $P_L\left(i\right)$ is known $\forall i$, the constraint set is linear.

In the end, the set of continuous optimization variables is:

$$x=[P_c\left(i\right)\dots P_c(i+H_c)P_d\left(i\right)\dots P_d(i+H_c)SOC\left(i\right)\dots SOC(i+H_p)]$$

while the set of binary optimization variables is:

$$y=[s\left(i\right)\dots s(i+H_c){\zeta }_c\left(i\right)\dots {\zeta }_c(i+H_p){\zeta }_d\left(i\right)\dots {\zeta }_d(i+H_p)]$$

So, the formalized optimization problem has to be solved at every time step. As $H_p$ and $H_c$ increase, it is evident that the size of the problem grows exponentially even in a simplified scenario such as the one just presented.

In the next section, the problem is formalized for a more complicated scenario, that is, the one of the ORCHESTRA project.

3. ORCHESTRA Application

This section delves into this problem and explicitly addresses it in the specific context of the Horizon 2020 project known as ORCHESTRA. As the use of electric power for non-propulsion functions in aircraft continues to rise, the next generation of aircraft is anticipated to heavily rely on electrification. This approach offers significant advantages, including reduced aircraft weight, decreased fuel consumption, overall cost savings, enhanced reliability, and simplified maintenance. In its pursuit of a substantial shift toward more electric aircraft, the EU-funded ORCHESTRA project aims to establish a comprehensive framework of innovative, adaptable, and scalable components that seamlessly integrate emerging technologies and novel design concepts. The project’s objective is to create and implement an EPS supervisory system tasked with identifying the most effective configuration for the intricate electrical system. Leveraging enhanced and manageable converters, this endeavor has the potential to decrease onboard weight and volume, thereby contributing to lower energy consumption. The next section provides detailed information about the electrical aspects of the ORCHESTRA system and how MPC is integrated into it, as shown in Figure 4.

3.1. System Description

The system depicted in Figure 5 comprises three buses: a KVDC, operating at a voltage of 1000 V; an HVDC bus with a voltage of $540/270$ V; and finally, an LVDC bus with a voltage of 28 V. The bidirectional DC/DC converters situated between these buses direct energy flow based on the supervisor’s decisions. Additionally, there are AC/DC converters that elevate the KVDC bus voltage to 1000 V from the generators, as well as DC/AC converters that supply the WTPs. The primary battery system consists of two 1300 kW batteries and bidirectional DC/DC back-to-back converters connected to two KVDC buses. These converters, controlled by current, indirectly manage power to ensure the batteries are charged or discharged according to the system requirements and the choices of MPC. The connection between the KVDC and HVDC buses is established using monodirectional Dual-Active-Bridge (DAB) converters. Specifically, there are three converters: one linked to the first KVDC bus, another connected either to the first or second KVDC bus in a ‘static’ manner, and a third connected to the second KVDC bus. The term ‘static’ implies that the connection is predetermined during the design phase and cannot be altered by the supervisor.

Downstream of the DAB converters, clusters are connected. Two clusters consist of a buck-boost bidirectional converter connected to a battery, high-voltage loads (HVDC loads), and an LVDC bus with converters for low-voltage loads (LVDC loads). Additionally, there is an ESS cluster (the middle one shown in the diagram), comprising a bidirectional buck-boost converter connected to a battery and an essential load, which must remain powered and cannot be disconnected from the network.

In the diagram, there is also a microgrid that ensures system robustness in case any of the DAB converters fail. Under normal circumstances, the first DAB converter is linked to the first cluster, the second DAB converter connects to the essential cluster, and the third DAB converter is associated with the remaining cluster. However, if one of the DAB converters malfunctions, the corresponding cluster switches to the DAB converter with the least power load. In the rare scenario where two DAB converters break down, all clusters connect to the remaining functional DAB converter.

3.2. Dynamical Model of the System

In this section, the use case for the ORCHESTRA project is addressed, expanding and defining in detail the equations and constraints for MPC. The problem is generalized with $N_{Bk}$ batteries on the KVDC bus, $N_{Bh}$ batteries on the HVDC bus, and N loads to shed. $N_C$ indicates the number of DAB converters. Furthermore, in the text, the subscript k refers to the battery on the KVDC bus, and the subscript h refers to the battery on the HVDC bus.

The dynamics of the batteries in (3) can be extended as follows:

$$\begin{array}{ccc}\hfill SO{C}_{k_j}(i+1)& =& SO{C}_{k_j}\left(i\right)+{\alpha }_c{P}_{c_{k_j}}\left(i\right)-{\alpha }_d{P}_{d_{k_j}}\left(i\right)\forall j\hfill \\ \hfill SO{C}_{h_j}(i+1)& =& SO{C}_{h_j}\left(i\right)+{\alpha }_c{P}_{c_{h_j}}\left(i\right)-{\alpha }_d{P}_{d_{h_j}}\left(i\right)\forall j\hfill \end{array}$$

(19)

where

$$\begin{array}{ccc}\hfill P_{ba{t}_{k_j}}\left(i\right)& =& P_{c_{k_j}}\left(i\right)-P_{d_{k_j}}\left(i\right)\forall j\hfill \\ \hfill P_{ba{t}_{h_j}}\left(i\right)& =& P_{c_{h_j}}\left(i\right)-P_{d_{h_j}}\left(i\right)\forall j\hfill \end{array}$$

(20)

where $P_{c_{k_j}}\left(i\right),P_{c_{h_j}}\left(i\right),P_{d_{k_j}}\left(i\right),P_{d_{h_j}}\left(i\right)\in {\mathbb{R}}_{+}$, $\forall i,j$.

3.3. Formalization of the Optimization Problem

This section reformulates the problem from Section 2.4, applying it to the case in ORCHESTRA.

So, starting from (7), the objective functions are

$$J_L,J_{\delta }\in {\mathbb{R}}_{+}$$

and

$$J_c,J_d,J_{SOC},J_{bat}\in {\mathbb{R}}_{+}^{2\times 1}$$

(21)

where

$$J_c={[J_{c_k},J_{c_h}]}^T,J_d={[J_{d_k},J_{d_h}]}^T,J_{SOC}={[J_{SO{C}_k},J_{SO{C}_h}]}^T,J_{bat}={[J_{ba{t}_k},J_{ba{t}_h}]}^T$$

(22)

The weights of the objective functions are:

$${\omega }_L,{\omega }_{\delta }\in {\mathbb{R}}_{+}$$

and

$${\omega }_c,{\omega }_d,{\omega }_{SOC},{\omega }_{bat}\in {\mathbb{R}}_{+}^{1\times 2}$$

(23)

where

$${\omega }_c=[{\omega }_{c_k},{\omega }_{c_h}],{\omega }_d=[{\omega }_{d_k},{\omega }_{d_h}],{\omega }_{SOC}=[{\omega }_{SO{C}_k},{\omega }_{SO{C}_h}],{\omega }_{bat}=[{\omega }_{ba{t}_k},{\omega }_{ba{t}_h}]$$

(24)

Compared to (7), the objective functions and variables to be optimized are separate for the batteries on the KVDC and HVDC buses. This makes it possible to assign different weights to the batteries on the different buses.

Regarding the optimization of load switching, (8) and (9) can be rewritten as follows:

$$\begin{array}{ccc}\hfill J_L& =& \sum _i^{Hc}\sum _j^N{\displaystyle \frac{{\gamma }_{L_j}(1-s_j\left(i\right))}{H_c{\sum }_j^N{\gamma }_{L_j}}}\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill J_{\delta }& =& \sum _i^{Hc}\sum _j^N{\displaystyle \frac{|s_j(i+1)-s_j\left(i\right)|}{N{H}_c}}\hfill \end{array}$$

(26)

where ${\gamma }_{L_j}$ is a vector indicating the priority of individual loads.

Regarding the optimization of the batteries, (10) and (11) can be rewritten as follows:

$$\begin{array}{c}\hfill J_{c_k}=\sum _j^{N_{Bk}}\sum _i^{Hp}{\displaystyle \frac{|P_{c_{k_j}}^{max}-P_{c_{k_j}}\left(i\right)|}{N_B{H}_p{P}_{c_{k_j}}^{max}}},J_{d_k}=\sum _j^{N_{Bk}}\sum _i^{Hp}{\displaystyle \frac{|P_{d_{k_j}}\left(i\right)|}{N_B{H}_p{P}_{d_{k_j}}^{max}}}\end{array}$$

(27)

$$\begin{array}{c}\hfill J_{c_h}=\sum _j^{N_{Bh}}\sum _i^{Hp}{\displaystyle \frac{|P_{c_{h_j}}^{max}-P_{c_{h_j}}\left(i\right)|}{N_B{H}_p{P}_{c_{h_j}}^{max}}},J_{d_h}=\sum _j^{N_{Bh}}\sum _i^{Hp}{\displaystyle \frac{|P_{d_{h_j}}\left(i\right)|}{N_B{H}_p{P}_{d_{h_j}}^{max}}}\end{array}$$

(28)

and

$$J_{SO{C}_k}=\sum _j^{N_{Bk}}\sum _i^{Hp}{\displaystyle \frac{|SO{C}_{k_j}^{max}-SO{C}_{k_j}(i+1)|}{N_B{H}_{p}SO{C}_{k_j}^{max}}},J_{SO{C}_h}=\sum _j^{N_{Bh}}\sum _i^{Hp}{\displaystyle \frac{|SO{C}_{h_j}^{max}-SO{C}_{h_j}(i+1)|}{N_B{H}_{p}SO{C}_{h_j}^{max}}}$$

(29)

and finally,

$$J_{ba{t}_k}=\sum _j^{N_{Bk}}\sum _i^{Hp}{\displaystyle \frac{|P_{ba{t}_{k_j}}(i+1)-P_{ba{t}_{k_j}}\left(i\right)|}{N_B{H}_p(P_{c_{k_j}}^{max}-P_{d_{k_j}}^{max})}},J_{ba{t}_h}=\sum _j^{N_{Bh}}\sum _i^{Hp}{\displaystyle \frac{|P_{ba{t}_{h_j}}(i+1)-P_{ba{t}_{h_j}}\left(i\right)|}{N_B{H}_p(P_{c_{h_j}}^{max}-P_{d_{h_j}}^{max})}}$$

(30)

It is assumed that the intrinsic parameters of the batteries (i.e., ${\alpha }_c$, ${\alpha }_d$, $P_{c_{k_j}}^{max}$, $P_{d_{k_j}}^{max}$, $P_{d_{h_j}}^{max}$, and $P_{d_{h_j}}^{max}$) are the same for each battery in order to simplify the mathematical formulation of the problem, although the problem can be generalized easily.

Additionally, it must be considered that the DAB converters in the system have a maximum output power, which could interfere with the results of the optimization function, and have to be incorporated into the equations. So, the output power should be less than the maximum value of $P_{conv}^{max}$.

The constraints at the i-th time instant are as follows:

$$\left\{\begin{array}{c}{0}_{N_{Bk}}\le P_{c_k}\left(i\right)-P_{d_k}\left(i\right)+{\sigma }_{in,c}\left(i\right){\sigma }_{c,bus}\left(i\right)(P_{c_h}\left(i\right)-P_{d_h}\left(i\right))+{\sigma }_{in,c}\left(i\right){\sigma }_{c,bus}\left(i\right)P_{cluster}+P_{WTP}\left(i\right)\le P_{in}^{max}\hfill \\ 0_{N_{Bh}}\le {\sigma }_{c,bus}\left(i\right)(P_{c_h}\left(i\right)-P_{d_h}\left(i\right)+P_{cluster})\le P_{conv}^{max}\hfill \\ SO{C}_{k_j}^{min}\le SO{C}_{k_j}\left(i\right)+{\alpha }_c{P}_{c_{k_j}}\left(i\right)-{\alpha }_d{P}_{d_{k_j}}\left(i\right)\le SO{C}_{k_j}^{max}\hfill \\ SO{C}_{h_j}^{min}\le SO{C}_{h_j}\left(i\right)+{\alpha }_c{P}_{c_{h_j}}\left(i\right)-{\alpha }_d{P}_{d_{h_j}}\left(i\right)\le SO{C}_{h_j}^{max}\hfill \\ 0_{N_{Bk}}\le P_{c_k}\left(i\right)\le {\zeta }_{c_k}\left(i\right)P_{c_k}^{max}\hfill \\ 0_{N_{Bk}}\le P_{d_k}\left(i\right)\le {\zeta }_{d_k}\left(i\right)P_{d_k}^{max}\hfill \\ 0_{N_{Bh}}\le P_{c_h}\left(i\right)\le {\zeta }_{c_h}\left(i\right)P_{c_h}^{max}\hfill \\ 0_{N_{Bh}}\le P_{d_h}\left(i\right)\le {\zeta }_{d_h}\left(i\right)P_{d_h}^{max}\hfill \\ {\zeta }_{c_k}+{\zeta }_{d_k}=1_{N_{Bh}}\hfill \\ {\zeta }_{c_h}+{\zeta }_{d_h}=1_{N_{Bk}}\hfill \end{array}\right.$$

(31)

with $P_{WTP},P_{in}^{max},SO{C}_k^{min},SO{C}_k^{max},\in {\mathbb{R}}^{N_{Bk}\times 1}$, $SO{C}_h^{min},SO{C}_h^{max}\in {\mathbb{R}}^{N_{Bh}\times 1}$, $P_{c_h}^{max},P_{d_h}^{max}\in {\mathbb{R}}^{N_{Bh}\times 1}$, ${\sigma }_{in,c}\in {\mathbb{R}}^{N_{Bk}\times N_C}$, ${\sigma }_{c,bus}\in {\mathbb{R}}^{N_C\times N_C}$, $P_{cluster},$ and $P_{conv}^{max}\in {\mathbb{R}}^{N_C\times 1}$, where $N_C=N_{Bh}$.

Additionally, $0_{N_{Bk}}$ is a zero vector of $N_{Bk}$ dimension, and $1_{N_{Bk}}$ and $1_{N_{Bh}}$ are column vectors of ones of $N_{Bk}$ and $N_{Bh}$ dimensions, respectively.

$$P_{cluster}={\left[\sum _t^N{s}_{1,t}\left(i\right)P_{L_{1,t}}\left(i\right)P_{ESS}\sum _t^N{s}_{2,t}\left(i\right)P_{L_{2,t}}\left(i\right)\right]}^T$$

where $P_{ESS}$ is the absorbed power by the HVDC ESS bus and $s_{i,t}$ indicates whether the contactor t in cluster i is closed.

Moreover, $P_{L_{i,t}}$ indicates the power on the t-th load of the i-th cluster.

The matrix ${\sigma }_{in,c}$ indicates which battery is connected to a given converter (1 if it is connected, 0 otherwise). Its rows represent the batteries, and its columns represent the converters.

Finally, the matrix ${\sigma }_{c,bus}$ indicates which converter is connected to a given bus (1 if it is connected, 0 otherwise). Its rows represent the converters, and its columns represent the buses. The matrix is also manipulated based on DAB converter failures, whereby this functionality is taken into account in the equations. The constraints (31) define the matrices A, b, $A_{eq}$, and $b_{eq}$ in (6).

Finally, the x optimization variables in (5) are as follows:

$$\begin{array}{c}\hfill x=[P_{c_{k}1}\left(i\right)P_{c_{k}1}(i+1)\dots P_{c_{k}1}(i+H_p)\dots P_{c_k{N}_{Bk}}\left(i\right)P_{c_k{N}_{Bk}}(i+1)\dots P_{c_k{N}_{Bk}}(i+H_p)\\ \hfill P_{d_{k}1}\left(i\right)P_{d_{k}1}(i+1)\dots P_{d_{k}1}(i+H_p)\dots P_{d_k{N}_{Bk}}\left(i\right)P_{d_k{N}_{Bk}}(i+1)\dots P_{d_k{N}_{Bk}}(i+H_p)\\ \hfill P_{c_{h}1}\left(i\right)\dots P_{d_{h}1}\left(i\right)\dots P_{d_h{N}_{Bh}}(i+H_p)\\ \hfill SO{C}_{k1}\left(i\right)SO{C}_{k1}(i+1)\dots SO{C}_{k1}(i+H_p)\dots SO{C}_{k{N}_{Bk}}\left(i\right)SO{C}_{k{N}_{Bk}}(i+1)\dots SO{C}_{k{N}_{Bk}}(i+H_p)\\ \hfill SO{C}_{h1}\left(i\right)\dots SO{C}_{h{N}_{Bh}}(i+H_p){]}^T\in {\mathbb{R}}^n\end{array}$$

and, considering the vectors

$$\begin{array}{ccc}\hfill s_t& =& [s_{t,1}\left(i\right)s_{t,1}(i+1)\dots s_{t,1}(i+H_c)s_{t,2}\left(i\right)s_{t,2}(i+1)\dots s_{t,2}(i+H_c)\dots s_{t,N}\left(i\right)\hfill \\ & & s_{t,N}(i+1)\dots s_{t,N}(i+H_c){]}^T\hfill \\ \hfill s& =& [s_1{s}_2\dots s_{N_C}]\hfill \\ \hfill {\zeta }_{k,t}& =& {[{\zeta }_{c_k}\left(i\right){\zeta }_{c_k}(i+1)\dots {\zeta }_{c_k}(i+H_p){\zeta }_{d_k}\left(i\right){\zeta }_{d_k}(i+1)\dots {\zeta }_{d_k}(i+H_p)]}^T\hfill \\ \hfill {\zeta }_{h,t}& =& {[{\zeta }_{c_h}\left(i\right){\zeta }_{c_h}(i+1)\dots {\zeta }_{c_h}(i+H_p){\zeta }_{d_h}\left(i\right){\zeta }_{d_h}(i+1)\dots {\zeta }_{d_h}(i+H_p)]}^T\hfill \\ \hfill \zeta & =& [{\zeta }_{k,1}{\zeta }_{k,2}\dots {\zeta }_{k,N_{Bk}}{\zeta }_{h,1}\dots {\zeta }_{h,N_{Bh}}]\hfill \end{array}$$

where $s_{t,j}\left(i\right),{\zeta }_{c_k}\left(i\right),{\zeta }_{d_k}\left(i\right),{\zeta }_{c_h}\left(i\right),{\zeta }_{d_h}\left(i\right)\in \{0,1\}$, the y optimization variable in (5) is set as follows:

$$y={\left[s\zeta \right]}^T\in {\mathbb{N}}^m.$$

Therefore, the calculation process can be summarized using Algorithm 1 and the flowchart in Figure 6.

Algorithm 1 Calculation process

1: [Start] The process begins with the initialization of the system. 2: [Input Electrical Variables] Electrical variables such as the SoC, load powers, and status of the converters are read and processed. 3: [Input Optimization Variables] The optimization variables required for the system are collected. These include the constraints, cost functions, and performance metrics. 4: [Calculate Matrices] Using the electrical and optimization variables, the matrices $A,b,A_{eq}$, and $b_{eq}$ are calculated as in (5) using the constraints (31) and objectives (25)–(30). These matrices are essential for the optimization process. 5: [Optimization Solver] The matrices $A,b,A_{eq}$, and $b_{eq}$ are input for the optimizer. The optimizer uses these matrices to find the optimal solution based on the given constraints and objectives. 6: [Determine MPC Steps] The optimizer outputs the next steps for MPC. These steps include the actions that need to be taken to optimize the system’s performance. 7: [Implement MPC Steps] The determined MPC steps are implemented in the system. The system follows the optimal path determined by the optimizer. 8: [Monitor and Feedback] The system is monitored and feedback is collected. This feedback is used to update the electrical and optimization variables for the next cycle. 9: [Loop Back] The process loops back to the input of the electrical and optimization variables, creating an optimization cycle.

3.4. Simulation Results

The simulated electrical system considers five batteries: two primary batteries on the KVDC bus (denoted as $P_{bat{t}_{k_1}}$ and $P_{bat{t}_{k_2}}$), and three batteries on the HVDC bus, one on each of the three HVDC buses (denoted as $P_{bat{t}_{h_1}}$, $P_{bat{t}_{h_2}}$, and $P_{bat{t}_{h_3}}$). There are three loads on $HVDCbus1$ and three loads on $HVDCbus2$, each with corresponding load-shedding switches (denoted as $s_1$, $s_2$, $s_3$, $s_4$, $s_5$, and $s_6$) and power values ($P_{L_{1,1}}$, $P_{L_{1,2}}$, $P_{L_{1,3}}$, $P_{L_{2,1}}$, $P_{L_{2,2}}$, and $P_{L_{2,3}}$). Additionally, there is an Energy Storage System (ESS) load on the $HVDCESSbus$ (denoted as $P_{ESS}$). The $HVDCESSbus$ is statically connected for the whole simulation time to $HVDCbus2$. $P_{i{n}_1}$ and $P_{i{n}_2}$ are the power generators of $KVDCbus1$ and $KVDCbus2$, respectively.

Simulink/MATLAB 2022b was used for system simulation, using the Simscape and Optimization toolboxes.

Table 2. Electrical and optimization parameters.

Parameter	Value
$H_p$	10
$H_c$	3
$P_{in}^{max}$	10,000	[W]
$P_{WTP}$	4000	[W]
$P_{ESS}$	400	[W]
$P_{conv}^{max}$	6000	[W]
$P_{c_{k_j}}^{max}$	5000	[W]
$P_{d_{k_j}}^{max}$	5000	[W]
$SO{C}_{k_j}^{max}$	0.9
$SO{C}_{k_j}^{min}$	0.3
$SO{C}_{h_j}^{max}$	0.9
$SO{C}_{h_j}^{min}$	0.3
${\sigma }_{in,c}$	[1 0 0;0 1 1]

presents the data for the MPC optimization problem.

Now, two scenarios are shown to validate the approaches in this paper.

3.4.1. Scenario 1: Priority and Failure of DAB Converters

Figure 7 shows the loads priorities, ${\gamma }_{L_j}$, in (25) (denoted by ${\gamma }_{L_1},{\gamma }_{L_1},{\gamma }_{L_{2}2},{\gamma }_{L_3},{\gamma }_{L_4},$ and ${\gamma }_{L_5}$, corresponding to the weights for $P_{L_{1,1}},P_{L_{1,2}},P_{L_{1,3}},P_{L_{2,1}},P_{L_{2,2}},$ and $P_{L_{2,3}}$, respectively). Figure 8 and Figure 9 show the MPC outputs, i.e., the load switches and the battery power chosen through optimization. In both figures, $P_{L_{i,j}}$ and the status of the DAB converters are also plotted to provide greater clarity on the optimizer’s choices. Specifically, the first figure shows the load powers, which vary equally $\forall i,j$ from 1000 W to 2000 W, and vice versa, while the second figure indicates with 1 that the converter is functioning correctly, and with 0 that the converter has failed.

In particular, in Figure 9, the red curves represent the references for $P_{bat{t}_{k_i}}$ and $P_{bat{t}_{h_i}}$ provided by the MPC to the controllers, and the sky-blue curves represent the power feedback controlled by the system. By convention, we refer to positive powers as the battery’s charging powers and negative powers as its discharging powers.

In these figures, the background indicates a higher priority for batteries (shown in blue) and a higher priority for loads (shown in red). Moreover, the background edges are violet if the priority of the KVDC batteries is higher than that of HVDC Batteries; otherwise, they are green.

Finally, in Figure 10, the power of the generators is plotted. It is important to note that in the steady state, $P_{i{n}_1}$ and $P_{i{n}_2}$ remain below their maximum threshold, $P_{in}^{max}$.

The simulation is divided into three parts: the first part demonstrates a shift in priority, initially focusing on loads and then on batteries; the second part involves changes in load powers; and finally, the third part showcases the effectiveness of the method proposed in this article in addressing $DAB$ (KVDC-HVDC converters) failures. Following this, the simulation graphs are described.

Initially, within the time frame from 0 to $0.26$ s, batteries take precedence over loads. Specifically, batteries connected to the KVDC bus are favored over those linked to the HVDC bus. Consequently, the charging power of KVDC batteries is increased to meet system requirements.

As time progresses into the interval between $0.26$ and $0.36$ s, a shift in priority occurs, with loads gaining precedence over batteries. Notably, loads associated with the $s_1$ and $s_2$ switches become the focus, while KVDC batteries maintain their priority status. In response, MPC adjusts by reducing battery power and supplying the prioritized loads.

Subsequently, from $0.36$ to $0.46$ s, loads continue to take precedence over batteries, with a preference for HVDC batteries. MPC adapts by allocating more charging to HVDC batteries compared to their KVDC counterparts.

At 0.46 s, a simultaneous increase in load gamma (${\gamma }_{L_j}$) prompts all loads to draw power concurrently, testing the capacity of the system.

The dynamics intensify further at $0.56$ s, when all load powers surge from 1000 W to 2000 W, resulting in a corresponding decrease in battery power.

By $0.67$ s, the priorities revert to those in the initial phase (0–0.26 s), reinstating battery superiority in the system’s operation. Moreover, at $0.76$ s, the load power returns to 1000 W.

However, the system encounters disruptions at 0.87 s, when $DA{B}_3$ malfunctions, followed by the failure of $DA{B}_2$ at $0.96$ s. Notably, MPC promptly reacts to these breakdowns, showcasing its adaptability in managing unexpected events.

After $0.96$ s, the system configuration adjusts accordingly, with the generator on $KVDCbus2$ exclusively powering the WTP and its corresponding battery on the same bus, while the generator on $KVDCbus1$ assumes the responsibility of supplying all HVDC buses.

Overall, this simulation highlights MPC’s capability to navigate dynamic power system conditions, ensuring efficient operation and resilience in the face of challenges.

3.4.2. Scenario 2: SoC of Batteries

In Scenario 2, the priorities of the objective functions are fixed, as well as the weights of the loads and batteries, similar to Scenario 1. The load power consumption values are set to a constant 1000 W, and the power values for the water treatment plant (WTP) and ESS load are fixed according to the values in Table 2. Also, in this case, the priority remains on managing the batteries. The focus of this scenario is to evaluate how the algorithm interacts with the constraint related to the state of charge (SoC) of the batteries, specifically if the maximum SoC is reached. Figure 11 illustrates the temporal behavior of the load switches, while Figure 12 shows the battery power levels along with the reference values, depicted by the blue and red curves, respectively. Finally, Figure 13 shows the state of charge of all batteries. As a result, the simulation starts in the same initial state as Scenario 1, but at $0.32$ s, the first HVDC battery reaches its maximum SoC of $0.9$, causing its charging power to drop to zero, meaning it ceases to charge. This excess power is then redirected to $P_{bat{t}_{h_1}}$, which increases its power from 1000 W to 5000 W. The additional available power also allows for the activation of $s_1$, corresponding to the first load. This simulation aims to demonstrate the effectiveness of the algorithm in managing the battery’s SoC constraint.

In modern electrical systems, maintaining a balance between supply and demand is essential for stability and efficiency. Traditional control systems, which rely solely on real-time data without predictive capabilities, often struggle with optimal performance. Supervisory control systems like ORCHESTRA, on the other hand, leverage advanced algorithms and historical data to predict future load requirements, enabling proactive adjustments that prevent overloads or shortages. This predictive foresight not only enhances system reliability but also optimizes generator operation, reducing fuel consumption and operational costs. Consequently, supervisory control systems are more effective in managing load connections and disconnections, ensuring a more efficient and stable electrical system compared to traditional, non-predictive methods.

4. Conclusions

The article addresses the issue of supervising and managing electrical energy, specifically controlling the power of batteries, which can be used for both charging and discharging, as well as load shedding. The problem is tackled using Model Predictive Control (MPC) and an optimizer that solves a mixed-integer linear programming (MILP) problem. The constraints relate to battery charging, maximum generator power, maximum battery absorption, and discharge, and also take into account the maximum power of converters. The solution is able to satisfy all the constraints of the flight mission and to find a good trade-off among the objectives according to the chosen priorities. The approach is practically demonstrated through the application of the proposed method to the test rig considered by the EU project ORCHESTRA, where the validity of the method is verified using complex system simulations in Matlab/Simulink 2022b. Future developments include verifying the method on a physical system and generating the corresponding supervision code in C to be loaded onto an electronic board.