Power optimization of the environmental control system for the civil more electric aircraft

Accepted Manuscript Power optimization of the environmental control system for the civil more electric aircraft Yuanchao Yang, Zichen Gao PII:

S0360-5442(19)30123-9

DOI:

https://doi.org/10.1016/j.energy.2019.01.115

Reference:

EGY 14588

To appear in:

Energy

Received Date: 14 November 2018 Revised Date:

9 January 2019

Accepted Date: 22 January 2019

Please cite this article as: Yang Y, Gao Z, Power optimization of the environmental control system for the civil more electric aircraft, Energy (2019), doi: https://doi.org/10.1016/j.energy.2019.01.115. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Power Optimization of the Environmental Control System for the Civil More Electric Aircraft

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Yuanchao Yanga, *, Zichen Gaob a. School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China

b. Flight Aviation Control Research Institute, Aviation Industry Corporation of China, Xi’an 710065, China *. Corresponding author. Tel: +86-181-9282-2395. E-mail address: [email protected]

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Abstract: As the civil aviation industry moves toward to the more-electric-aircraft (MEA) concept, environmental control system (ECS), one of the largest power consumption non-propulsive systems for civil aircraft, converts its power extraction source from bleed air to electric power and thus will introduce new challenges on ECS’s design and operation management schemes. One of

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these challenges is for the power optimization of ECS, which was an optimal sizing and planning problem for ECS air circulation pack, but under the MEA environment will be transformed into a complex power scheduling problem integrated with the constraints on the controllable electrical power generation and varying thermal power demands determined by the aircraft’s flight profiles. In this paper, we propose a complete mathematical optimization formulation for the power optimization of ECS for MEA and cast it as a non-convex nonlinear program. We provide a solution approach to tackle this problem based on a

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combination of Benders decomposition method and the successively linear programming technique and illustrate the capabilities of the proposed solution approach on an aircraft’s ECS under the civil MEA environment. The numerical results demonstrate the effectiveness of the proposed solution approach in terms of the associated computational outcomes.

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Keywords: civil more electric aircraft, environmental control system, power optimization, Benders decomposition method,

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successively linear programming.

1. Introduction

Driven by the increased awareness of the environmental and economic issues in the civil aviation industry, it becomes a tendency to develop the technologies for the aircraft design and operational management with a friendly environmental impact and efficient energy usage [1]. As significant one of these technologies, the more electric aircraft (MEA) concept has made a tremendous progress in the civil aviation industry over the last few years [2, 3]. The MEA concept basically resorts to more electrical power so as to replace the conventional non-electric power (such as pneumatic, hydraulic and mechanical resources) for the operation of the many aircraft subsystems and can thus improve their energy extraction efficiency and reduce the pollution emission and fuel consumption [4]. 1

ACCEPTED MANUSCRIPT The integration of the more electric concept into the civil aircraft will introduce several significant changes [5] and one of them will be taken for the environmental control system (ECS) [6], the largest power consumption and non-propulsive subsystems of the aircraft. The tasks of the ECS are basically to provide a physiologically comfort condition for the passengers and crews and also ensure sufficient cooling for all of the electronic equipment of the aircraft during the flight. To realize the

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above tasks, the conventional ECS for the non-MEA is designed to be able to rely on the air bled from the engine so as to achieve a required comfort temperature and pressure for the relevant compartments of the aircraft through a series of sizing and planning techniques (these techniques are also defined as power optimization of ECS [7, 8]) on the air circulation pack which is in charge of the air conditioning and pressurization functions. However, since the air bled from the engine will reduce the

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operation efficiency of the engine, it is usually set to have a constant and very high temperature and pressure which can be able to keep the engine work at an economic mode but are thus oversized than the levels of the required needs by ECS. For that

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matter, large amounts of energy will be wasted after the power optimization process of ECS and the energy operational efficiency of the conventional ECS is thus relatively low [6, 9].

Under the MEA environment, the ECS eliminates the conventional pneumatic system and bleed manifold and converts the power extraction source from the “bleed air” of the engine to the electric power which is generated by the electrical motors with adjustable speed feature and can be utilized to realize a variable temperature and pressure control directly on the imported ram air driven from the inlet of ECS [6, 10]. This “no-bleed” system architecture thus offers a number of benefits which include the

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obvious improvements in the level of the engine fuel consumption and efficiency of the energy usage by ECS. However, the integration of the “no-bleed” ECS into the civil aircraft also raises a new and complex challenge for the power optimization technique of ECS [11, 12]: the ECS basically owns the characteristic of variable air conditioning and pressurization power

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demands which are usually correlated with the aircraft’s flight profiles represented by the altitude, velocity and attitude, thus an efficient ECS power optimization should be actually able to realize an optimal scheduling of all of the controllable power

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generation resources and variable thermal power demands determined by the associated flight profiles so as to reach a minimal power usage. However, the conventional power optimization technique with the fixed temperature and pressure for the air bled from the engine does not consider the variable characteristics of the ECS power demands and controllability of the temperature and pressure of the imported (bleed) air (in other words, the conventional power optimization technique is only designed under the “worst case” [6] which is defined to have the largest thermal power consumption by ECS under some particular phases of the flight profile), and thus will introduce unnecessary energy waste and can no longer be taken as an efficient way under the new ECS environment where the temperature and pressure of the imported ram air can be regulated with the motor driven by electrical power. For that matter, it is necessary to redesign the power optimization technique for ECS under the MEA environment. 2

ACCEPTED MANUSCRIPT Some previous research efforts have been made to deal with the power optimization of ECS for MEA. The authors in [13, 14] and [15] separately present an exergy analysis and fuel penalty method applied to the modeling of an optimal sizing of air circulation pack of the aircraft’s ECS so as to realize a minimal power utilization under the “worst case” and finally validates its performance on real test cases with different phases of a typical flight profile. The authors in [16, 17] provide an integrated

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power minimization planning framework for co-optimizing the design architecture and its associated parameters of the gas turbine engine, air management system, air cycle machine and the cabin air distribution system. The above power optimization approaches are simple and effective but all face two key challenges: 1) fixed temperature and pressure of the imported ram air are still assumed, i.e. the controllability of the electric power generation is not addressed and thus the proposed approaches

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cannot fully quantity the power saving by the “no-bleed” ECS architecture for MEA; 2) they do not provide a viable schedule scheme or method that can tell what is the minimal amount of power usage under different phases of the aircraft’s flight profile.

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Actually, rather than just attempting to resolve the “optimal” air circulation pack sizing at the planning level for ECS, an efficient power optimization for the ECS operation which considers the integrated scheduling of the electrical power generation and controllable ECS thermal power demands under every flight time period with its associated flight phase is more significant to finally realize a minimal power utilization. To the best of our knowledge, few research efforts have been made to deal with the above scheduling problem for the power optimization of ECS under the MEA environment. To effectively address the above challenges, we propose a new formulation and solution method for solving power

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optimization of ECS for the civil MEA. The main contributions of this paper are that: 1) we present a detailed analysis of the thermal power demands of ECS with the change of flight phase through an explicit mathematical representation and formulate the power optimization of ECS under the MEA environment as a non-convex and nonlinear constrained optimization model

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which encapsulates a co-scheduling between the electric power generation and all controllable thermal power demands of ECS with an objective for a minimal power usage; 2) we provide a computationally efficient method to solve the above problem for

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the real application on ECS of a civil MEA. We firstly divide the above formulated model into a two-level (outer and inner sub-problems) optimization problem and then obtain the efficient schedule results through a two-level Benders Decomposition method; 3) finally the numerical testing results based on the real civil aircraft case demonstrate the effectiveness of the proposed method in terms of its operation efficiency. The rest of the paper is organized as follows. In Section 2, we present the complete mathematical formulation of the power optimization of ECS for the MEA. We discuss the proposed solution approach in Section 3. In Section 4, we illustrate the application of the methodology to the test case under a real civil MEA environment. We conclude the paper in Section 5.

2. The Problem Formulation The main functions of ECS are to realize the air conditioning and cabin pressurization control [6, 18]. Since there are 3

ACCEPTED MANUSCRIPT multiple kinds of thermal power loads which will be introduced during a flight of the aircraft (such as the conductive heat loads due to heat transfer from outside of the aircraft to the inside by conduction/convection, solar heat loads by the sun radiation, internal electrical loads by the thermal emission of the electrical devices and the occupant heat loads by the aircraft crews and passengers [18]), the air conditioning function basically relies on the ram air imported from the outside

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of the aircraft and conditions the ram air within a reasonable temperature interval and transports it into the cabin which includes cockpit compartment, passenger compartment, electric device compartment and cargo compartment [4]. (Here to note that the air conditioning function in this paper only refers to the regulation of temperature and we do not consider the humidification [15] by the air conditioning function since the cabin humidification is only implemented when the aircraft

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is on ground [19].) Meanwhile, with considering the ambient pressures during the most of flight time for a civil aircraft are severely lower than the minimal survivable pressure condition for the unprotected human beings, the cabin is required to

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have pressurization control function [6, 18] which will regulate the cabin pressure at a requested physiologically comfort value through: 1) the pressure control of the imported ram air and 2) the flow amount control of the exported air from the

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cabin when the aircraft has reached above some pre-set flying altitude.

Fig. 1. Typical “no-bleed” ECS system architecture for MEA

As for the “no-bleed” ECS for MEA, a typical system architecture which will realize the above functions for ECS [10] can be depicted in Fig. 1. In Fig. 1, the system architecture of the “no-bleed” ECS is divided as four modules: electrical control module, air cycle machine module, air mixing module and air distribution module. At first, in the electrical control module, the compressor C0 is driven by a direct-current electrical motor to increase the pressure and temperature of the 4

ACCEPTED MANUSCRIPT ram air imported from the inlet of ECS. With considering the initial total temperature of the imported ram air is relatively low (especially at the high flight altitude), the imported ram air is usually pre-heated through a feedback loop where part of the compressed ram air is extracted at the valve O1 and is re-circulated back to the valve O2 and integrated with the original ram air. After that, in the air cycle machine module, through transferring part of the energy while keeping the

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pressure unchanged by the primary heat exchanger (PHX), the compressed ram air is then conditioned by a typical two-wheels (compressor C1, secondary heat exchanger (SHX) and turbine T1) and inverse-Brayton thermodynamic cycle [20] so as to reach an appropriate pressure and temperature which can be injected into the mixing unit. In the air mixing module, the fresh ram air from the upstream air cycle machine module is blended with the dried and sterilized recirculation

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air from the passenger compartment and the mixing unit allocates the mixed air separately into the cockpit and passenger compartments. Then, the mixed air is circulated sequentially through the cockpit/passenger compartment, electrical device

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compartment and the cargo compartment via the control of the air flow valves A, B and C and is finally exhausted to the outside of the aircraft by the air outflow control valve D, so as to manage the temperatures and pressures of the compartments at their associated comfortable levels (in this paper, we assume all of the interior floors and walls between the compartments are insulated and thus the heat transfer between the compartments can only rely on the air passing

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through the related airflow control valves).

Fig. 2. Flight profile represented by flight phases for a typical civil aircraft

From the above, we can find the thermal power demands of ECS are actually not fixed within the whole flight profile of the aircraft and the change of the thermal power demands of ECS are closely correlated to the change of the aircraft’s flight phases which are combined to construct the flight profile and are usually determined before the flight and up-loaded into the aircraft’s flight management computer [4, 21]. The possible flight phase belongs to one of the following five flight stages [22]: initial climb, climb, cruise, descent and finally approach and landing which are quantified by the related average altitude, velocity and pitch attitude. A typical flight profile of the civil aircraft including its associated flight phases is demonstrated in Fig. 2. In this paper, we define the flight duration from the lift-up at the initial climb stage to touchdown 5

ACCEPTED MANUSCRIPT at the approach and landing stage for a civil aircraft is Tˆ minutes and the smallest scheduling snapshot for the power optimization of ECS for MEA is one minute, then the formulated scheduling problem will have Tˆ periods. The average altitude, velocity and pitch attitude at time period t = 1, 2,K, Tˆ are defined as (h (t ), v (t ),ϑ (t )) . If the flight phases for all

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time periods 1, 2,K,Tˆ are previously given, the parameters (h (t ), v (t ),ϑ (t )) for t = 1, 2,K , Tˆ can completely address the aircraft’s flight phases of the whole flight profile; moreover, based on the associated mathematical transformation relationship [22], the parameters (h (t ), v (t ),ϑ (t )) for t = 1, 2,K , Tˆ can also reveal the values of the air static pressure and static temperature which are significant for finally calculating the associated ECS thermal power demands.

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The problem formulation of the power optimization of ECS for MEA is firstly needed to explicitly account for the objective function with a minimal power usage. Since all of the power resources of ECS basically come from the electricity,

Tˆ

min

∑ [m& t =1

i

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an effective quantification of the objective function is given in (1):

(t ) ⋅ c p ⋅ (T 2 (t ) − T 1 (t ))].

m& i (t ) , the mass flow rate of the air imported through the inlet of ECS at period γ +1

(1)

t , is equal to

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γ ig c ( γ − 1) − 2( γi i −1) P 1 (t ) ⋅ Ma (t ) ⋅ (1 + ( Ma (t )) 2 i ⋅ ⋅ A i where the associated parameters can be determined as follows (all of ) R 2 T 1 (t ) the definitions with the associated parameters and decision variables in this paper are presented in Nomenclature) [23]:

Ma (t ) =

v (t ) γ i ⋅ g c ⋅ R ⋅ T 0 (t )

, P 1 (t ) = P 0 (t ) ⋅ (1 + ( Ma (t )) 2

( γ i − 1) γ i /( γ i −1) ( γ − 1) ) ) . The decision and T 1 (t ) = T 0 (t ) ⋅ (1 + ( Ma (t )) 2 i 2 2

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making of the power optimization of ECS for MEA is to implement a co-scheduling between the controllable electrical power generations and varying thermal power loads/demands determined by the aircraft’s flight profiles in an optimal fashion. We next

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state each of the constraints we include in the problem formulation.

m& i (t ) ⋅ (T x (t ) − T 1 (t )) = m& x (t ) ⋅ (T 2 (t ) − T x (t )), ∀t = 1, 2,K , Tˆ

(2)

Eq. (2) is the constraints that the pre-heated air re-circulated from the valve O1 and the ram air imported from the inlet of

ECS will reach a common total temperature after the heat transfer at the valve O2. γ

c0

P 2 (t ) T (t ) = (1 + η c 0 ( 2 − 1)) γ c 0 −1 , t = 1, 2,K , Tˆ P 1 (t ) T x (t )

(3)

Eq. (3) is the constraints which address the isentropic change characteristic of the air state variables (temperatures and

pressures) within the compressor C0.

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ACCEPTED MANUSCRIPT T 2 (t ) − T x (t ) = K c 0⋅ (ω 0 (t )) 2 , t = 1, 2,K , Tˆ

(4)

Eq. (4) depicts the relationships between the rotation speed and the change of total temperature within compressor C0. P 2 (t ) = P 3 (t ) , t = 1, 2,K , Tˆ

(5)

m& i (t ) ⋅ (T 2 (t ) − T 3 (t )) = m& HX (t ) ⋅ (T 3 (t ) − T 5 (t )) , t = 1, 2,K , Tˆ

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(6)

Eq. (5) describes an isobaric characteristic of the air thermodynamic process in PHX where the pressure stays constant;

(6) states the total energy absorbed by the cold side air (usually the ram air driven by the heat exchangers) is equal to the total energy released by the hot side air (usually the ram air from the inlet of ECS) and the cold and hot side airs will reach

γ +1

γ ig c ( γ − 1) − 2( γi i −1) P 1 (t ) ⋅ Ma (t ) ⋅ (1 + ( Ma (t )) 2 i ) ⋅ ⋅ A HX where R 2 T 1 (t )

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(primary or secondary) heat exchanger at period t, is equal to

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a common total temperature after the heat transfer in PHX. m& HX (t ) which is mass flow rate of the air imported through the

A HX is the cross section area of the (primary or secondary) heat exchanger. γ

c1 P 4 (t ) T (t ) = (1 + η c1( 4 − 1)) γ c1−1 , t = 1, 2,K , Tˆ P 3 (t ) T 3 (t )

(7)

Eq. (7) is the constraints which depict the isentropic change characteristic of the air state variables within the compressor

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C1.

T 4 (t ) − T 3 (t ) = K c1⋅ (ω 1 (t )) 2 , t = 1, 2,K , Tˆ

(8)

Eq. (8) describes the relationships between the rotation speed and the change of total temperature within the compressor

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C1.

T 4 (t ) − T 3 (t ) = η m ⋅ (T 5 (t ) − T 6 (t )), t = 1, 2,K , Tˆ

(9)

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The compressor C1 and the turbine T1 are mechanically co-axial; thus (9) depicts the reduced amount of the total enthalpy driven by the turbine T1 is equal to the amount of the total enthalpy produced by the compressor C1 multiply with an efficiency parameter η m .

P 4 (t ) = P 5 (t ) , t = 1, 2,K , Tˆ

m& i (t ) ⋅ (T 4 (t ) − T 5 (t )) = m& HX (t ) ⋅ (T 5 (t ) − T 1 (t )) , t = 1, 2,K , Tˆ

(10) (11)

Eq. (10) describes an isobaric characteristic of the air thermodynamic process in SHX where the pressure stays constant;

(11) describes the total energy absorbed by the cold side air (the ram air driven by the heat exchangers) is equal to the total energy released by the hot side air (the ram air from the inlet of ECS) and the cold and hot side airs will reach a common

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ACCEPTED MANUSCRIPT total temperature after the heat transfer in SHX. T 6 (t ) P (t ) = 1 − η t1[1 − ( 6 ) T 5 (t ) P 5 (t )

γ t 1−1 γ t1

], t = 1, 2,K , Tˆ

(12)

Eq. (12) states the isentropic change characteristic of the air state variables in the turbine T1. m& i (t ) + m& rc (t ) = m& ic (t ) + m& ip (t ) , t = 1, 2,K , Tˆ pa

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m& i (t ) ⋅ T 6 (t ) + m& rc (t ) ⋅ T

(13)

(t ) = (m& ic (t ) + m& ip (t )) ⋅ T 7 (t ) , t = 1, 2,K , Tˆ

(14)

Eq. (13) and (14) separately state the mass and the energy conservation relationships within the air mixing unit.

co

(t + 1) = m co (t ) ⋅ T

(t ) + (m& ic (t ) ⋅ T 7 (t ) − m& A (t ) ⋅ T

co

co

P 6 (t ) = m co (t ) ⋅ R ⋅ T

co

(t ) / V

co

(15)

4

(t )) ⋅ (60sec) + (∑ Q coj (t )) / c

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m co (t + 1) ⋅ T

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m co (t + 1) = m co (t ) + (m& ic (t ) − m& A (t )) ⋅ (60sec) , t = 1, 2,K , Tˆ

p

j =1

, t = 1, 2,K , Tˆ

(16)

, t = 1, 2,K , Tˆ

(17)

Eq. (15) states the air mass conservation relationship in the cockpit compartment. (16) describes the change of enthalpy of

the air in cockpit compartment is equal to the sum of the imported enthalpy by the up-stream air, the exported enthalpy by

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the exhausted air from valve A and the enthalpy introduced by the multiple interior heat loads. (17) represents the pressure balance constraints in cockpit compartment.

m el (t + 1) = m el (t ) + ( m& A (t ) − m& C (t )) ⋅ (60 sec) , t = 1, 2,K , Tˆ

el

(t + 1) = m el (t ) ⋅ T

el

(t ) + ( m& A (t ) ⋅ T

co

(t ) − m& C (t ) ⋅ T

4

el

(t )) ⋅ (60 sec) + (∑ Q elj (t )) / c j =1

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m el (t + 1) ⋅ T

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P 6 (t ) = m el (t ) ⋅ R ⋅ T

(18)

p

, t = 1, 2,K , Tˆ el

(t ) / V

el

(19)

, t = 1, 2,K , Tˆ

(20)

Eq. (18) states the air mass conservation relationship in the electric device compartment. (19) describes the change of

enthalpy of the air in electric device compartment is equal to the sum of the imported enthalpy by the up-stream air from valve A, the exported enthalpy by the exhausted air from valve C and the enthalpy introduced by its multiple interior heat loads. (20) represents the pressure balance constraints in electric device compartment. m

m

pa

(t + 1) ⋅ T

pa

pa

(t + 1) = m

(t + 1) = m

pa

pa

(t ) ⋅ T

(t ) + (m& ip (t ) − m& B (t ) − m& rc (t )) ⋅ (60 sec) , t = 1, 2,K , Tˆ

pa

(t ) + (m& ip (t ) ⋅ T 7 (t ) − m& B (t ) ⋅ T

pa

(t ) − m& rc (t ) ⋅ T

(21) 4

pa

(t )) ⋅ (60 sec) + (∑ Q j =1

j pa

(t )) / c

p

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ACCEPTED MANUSCRIPT , t = 1, 2,K , Tˆ P 6 (t ) = m

pa

(t ) ⋅ R ⋅ T

pa

(t ) / V

pa

(22)

, t = 1, 2,K , Tˆ

(23)

Eq. (21) states the air mass conservation relationship in the passenger compartment. (22) describes the change of enthalpy

of the air in passenger compartment is equal to the sum of the imported enthalpy by the up-stream air, the exported

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enthalpy by the exhausted air from valve B and to the air mixing unit and the enthalpy introduced by its multiple interior heat loads. (23) represents the pressure balance constraints in passenger compartment.

m ca (t + 1) = m ca (t ) + ( m& B (t ) + m& C (t ) − m& D (t )) ⋅ (60 sec) , t = 1, 2,K , Tˆ

(t + 1) = m ca (t ) ⋅ T

ca

(t ) + (m& B (t ) ⋅ T

pa

(t ) + m& C (t ) ⋅ T

el

4

(t ) − m& D (t ) ⋅ T

ca

(t )) ⋅ (60sec) + (∑ Q caj (t )) / c

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ca

j =1

, t = 1, 2,K , Tˆ

P 6 (t ) = m ca (t ) ⋅ R ⋅ T

ca

(t ) / V

ca

p

(25)

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m ca (t + 1) ⋅ T

(24)

, t = 1, 2,K , Tˆ

(26)

Eq. (24) states the air mass conservation relationship in the cargo compartment. (25) describes the change of enthalpy of

the air in cargo compartment is equal to the sum of the imported enthalpy by the up-stream air from the valve B and C, the exported enthalpy by the exhausted air from valve D and the enthalpy introduced by its multiple interior heat loads. (26)

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represents the pressure balance constraints in cargo compartment.

0 ≤ m& x (t ) ≤ m& x , 0 ≤ T x (t ) ≤ T x , 0 ≤ P 2 (t ) ≤ P 2 , 0 ≤ T 2 (t ) ≤ T 2 , 0 ≤ P 3 (t ) ≤ P 3, 0 ≤ T 3 (t ) ≤ T 3, 0 ≤ P 4 (t ) ≤ P 4 , 0 ≤ T (t ) ≤ T , 0 ≤ P (t ) ≤ P , 0 ≤ T (t ) ≤ T , max( Pˆ , P (t )) ≤ P (t ) ≤ P , 0 ≤ T (t ) ≤ T , 0 ≤ T (t ) ≤ T , 4

4

5

5

5

5

1

6

6

6

6

7

0 ≤ m& ic (t ) ≤ m& ic , 0 ≤ m& ip (t ) ≤ m& ip , 0 ≤ m& rc (t ) ≤ m& rc , 0 ≤ m& A (t ) ≤ m& A , 0 ≤ m& B (t ) ≤ m& B , 0 ≤ m& C (t ) ≤ m& C , co

≤T

0 ≤ m co (t ) ≤ m co , 0 ≤ m

co

(t ) ≤ T co , T

pa

≤T

pa

pa

(t ) ≤ T

pa

,T

el

≤T

el

(t ) ≤ T el , T

ca

≤T

ca

(t ) ≤ T ca ,

(27)

(t ) ≤ m pa , 0 ≤ m el (t ) ≤ m el , 0 ≤ m ca (t ) ≤ m ca , 0 ≤ ω 0 (t ) ≤ ω 0 , 0 ≤ ω 1 (t ) ≤ ω 1

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0 ≤ m& D (t ) ≤ m& D , T

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, t = 1, 2,K , Tˆ

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Eq. (27) generalizes that all of the decision variables are non-negative values and have their associated physically upper

bounds. Here to note, the pressures and temperatures within the four compartments are separately required to be larger than the value: max( Pˆ , P 1 (t )) ( Pˆ is the pressure value at some particular pre-set flight altitude) and the associated minimum comfortable temperatures.

We may view the above mathematical formulation in (1)-(27) is a large-scale and non-convex nonlinear constrained

programming problem. Finding an efficient method for the above problem is usually very challenging. In the next section, we will propose an efficient method based on two-level Benders Decomposition method with the successively linear programming technique. 9

ACCEPTED MANUSCRIPT 3. The Proposed Solution Approach The nonlinear characteristics of the mathematical formulation stated above by (1)-(27) come from the following three kinds of forms (here we note x 1 and x 2 are the scalar variables): 1) (3), (7) and (12) have x 1 / x 2 form which is a non-convex function; 2) (4) and (8) have ( x 1 ) 2 form which is a convex function; 3) (2), (14), (16)-(17), (19)-(20), (22)-(23) and (25)-(26) have 2

bilinear form which is a non-convex function. A general and efficient way to tackle the above non-convex and nonlinear

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x1⋅x

optimization problem can resort to Benders method [24] which can exploit the above nonlinear formulations by selecting the complicating variables which divide the original problem into the outer-level and inner-level problems and, when fixed, render

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the inner-level problem considerably more tractable [25]. As for the formulated formulation in (1)-(27), the outer-level problem will deal with only part of the continuous variables and constraints and then the variables in the outer-level problem are taken as the complicating variables and fixed as the parameters in the inner-level problem which thus usually has a linear or nonlinear

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convex optimization form where the solutions of the inner-level problem can provide the information so as to help the outer-level problem formulate the associated Benders cuts. The Benders cuts reduce the search region of the outer-level problem and usually have a linear or linearly-approximated formulation about the solution point found in the inner-level problem. Therefore, the outer-level problem is solved successively, and the inner-level problem as well, until they both reach the required convergence criterion.

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To do so, in this paper, we select the following variables as the complicating variables: m& x (t ) , T 2 (t ) , P 2 (t ) , P 3 (t ) , T 4 (t ) , T 5 (t ) , P 6 (t ) , m& ic (t ) , m& ip (t ) , m& rc (t ) , m& A (t ) , m& B (t ) , m& C (t ) , m& D (t ) , m co (t ) , m pa (t ) , m el (t ) and m ca (t ) . Then, all of the nonlinear constraints will have a convex form with given values of the complicating variables.

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Meanwhile, the outer-level problem can be represented by a simplified nonlinear programming formulation and be easily linear-approximately restated by a mixed integer and linear programming (MILP) problem which is finally solved by the

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mature commercialized software package. The above two-level decomposition scheme permits us to find an optimal electrical power utilization result with only considering the complicating variables at the outer level and to find the feasible schedule results for all of the nonlinear and convex constraints at the inner level. If the inner-level problem cannot converge or any violations of the bounds on the variables exist, the corresponding Benders cuts will be active and fed back into the next round of calculation for the outer-level problem. 3.1 Outer-level: Benders decomposition method Here we at first present a compact matrix formulation for the proposed mathematical formulation in (1)-(27) as follows:

min c T x x, y

(28)

10

ACCEPTED MANUSCRIPT s.t . Ax ≤ b ,

(29)

G ( x, y ) = g ,

(30)

x ∈ X,

(31)

y ∈ Y.

(32)

The variable x ∈ ℜ 18⋅T is a vector of the complicating variables. The variable y ∈ ℜ 12⋅T is the vector of the rest of decision ˆ

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ˆ

variables. The linear constraints in (29), involving only the variable x , contain the constraints of (5), (11), (13), (15), (18), (21) and (24). The nonlinear constraints in (30), which couple the variables x with y , include (2), (3), (4), (6), (7), (8), (9), (10), (12),

and lower bound constraints stated by (27).

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The Benders decomposition method is described as follows:

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(14), (16)-(17), (19)-(20), (22)-(23) and (25)-(26). The compact sets X and Y in (31)-(32) separately represents the upper

1) Initialization Let x *0 be the optimal solution for the following optimization problem:

min c T x x

s.t . Ax ≤ b , x ∈ X.

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Solve the inner-level optimization problem based on x *0 (the solution method will be presented in section 3.2) and let O −level = −∞ , upper y *0 be the optimal solution of the inner-level problem given x *0 . Set the outer-level lower bound L O − level

= +∞ , the iteration number k = 1 and the outer-level convergence tolerance level ζ = 0.0001 .

2) Iteration k ≥ 1 :

EP

bound U

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Step 1) Solve the outer-level problem.

The outer-level problem is the following nonlinear program: min c T x k + M Tα k

x k,α k

s.t .

α k ≥ G (x k , y *k -1 ) − g ,

Ax k ≤ b , x k ∈ X.

(33)

α k is the introduced positive slack variables and M is a column vector where all of its elements are very large positive real

value; y *k -1 is obtained at the (k-1)-th round of inner-level problem and is thus taken as parameter in (33). From (30), we 11

ACCEPTED MANUSCRIPT can find parts of the formulations in G (x k , y *k -1 ) are nonlinear corresponding to x k with given y *k -1 (the constraints (2), (3), (7) and (12)). Thus, to solve the outer-level problem in (33), we at first recast the problem in (33) as a linear-approximated MILP formulation (in this paper we use the linearization method in [26] to represent the strictly decreasing/increasing convex formulations in (3), (7) and (12) and use the linearization method in [27] to linearly

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approximate the bilinear function in (2)). The method in [27] reveals a general technique for approximating an arbitrary nonlinear function of two variables and on the basis of this method we can establish a piece-wise linear fit by means of an associated MILP representation with any selective approximation accuracy.) and then resort to the commercial optimization

L O − level = c T x *k + M T α *k .

Step 2) Check the outer level convergence.

Step 3) Solve inner-level problem.

O − level

= c T x *k + M T α *k and go to the step 3.

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If U O −level − L O −level < ζ , stop and return ( x *k , y *k ) ; Otherwise, set U

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software package to finally solve it. Let ( x *k , α *k ) be the optimal solution for the k-th round of outer-level problem and set

We will discuss the method to solve the inner-level problem in section 3.2. Let y *k be the optimal solution of the inner-level problem given x *k and input y *k as parameters into the (k+1)-th round of the outer-level problem in (33).

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Set k = k + 1 and go to the step 1. 3.2 Inner-level: successively linear programming

All of the nonlinear terms in the inner-level problem are successively linearized [24, 28] around intermediate solution points and the Benders cuts generated by solving the inner-level problem are finally added into the outer-level problem. Since the

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inner-level problem is a convex optimization problem, an optimal solution of the inner-level problem can be obtained by the successively linear programming technique and this solution can thus generate a valid cut for the outer-level problem.

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1) Initialization: Fix x *k ( k ≥ 0 ), set the inner-level lower bound L I −level = −∞ , upper bound U

I −level

= +∞ , iteration

number j = 1 and choose the inner-level convergence tolerance level δ = 0.0001 . 2) Iteration j ≥ 1 :

Step 1) Formulate the inner-level problem. The inner-level problem is equivalent to find a feasible solution of y kj within G ( x *k , y kj ) = g , y kj ∈ Y.

We can equivalently restate the above problem as the following form: 12

ACCEPTED MANUSCRIPT R j ( x *k ) =

min

j y kj ∈ Y , s 1,j k , s 2, k

1T s 1,j k + 1T s 2,j k

s.t . G (x *k , y kj ) − s 1,j k + s 2,j k = g , s 1,j k ≥ 0, s 2,j k ≥ 0,

(34)

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where s 1,j k and s 2,j k are the introduced slack continuous variables. Step 2) The j-th round of iteration for solving the inner-level problem.

From (34), we can find, if y kj is given, the optimization problem in (34) can be reformulated as a linear programming

∀µ ∈ [1, 2, K ,18 ⋅ Tˆ ] , the optimization problem in (34) is recast as 18 ⋅Tˆ

min

s

∑ (s

j , k ), µ =1 j, k )

j 1, k

[ µ ] + s 2,j k [ µ ])

(35)

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s

j 1, k [ µ ]∈Ω ( j 2, k [ µ ]∈Ω (

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D j (x *k , y kj ) , can be determined analytically as follows:

problem and its optimal value, defined as

where the constraint set Ω ( j , k ) is stated as follows:

j s 1,j k [ µ ] − s 2,k [ µ ] = g [ µ ] − G ( x *k , y% kj )[ µ ],

s 1,j k [ µ ] ≥ 0, s 2,j k [ µ ] ≥ 0, µ = 1, 2,K ,18 ⋅ Tˆ .

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The optimization problem in (35) can be divided as 18 ⋅ Tˆ of independent “two-variables” linear programming problems and the optimal value D j (x *k , y kj ) is equal to 1 T ( G (x *k , y kj ) − g ) . We define y% kj to be the linear approximation point at the j-th round of inner-level optimization problem. Let y

* k -1

be equal

EP

to the initial solution of y% kj : y% 1k and we let y% kj+1 = y% kj + dˆ kj where dˆ kj is determined by solving the following convex quadratic programming problem in (36). dˆ kj is thus a strictly decreasing direction for the value of D j (x *k , y% kj )

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as j increases.

max j dˆ k

dˆ kj T dˆ kj

s.t . ((G (x *k , y% kj )[ µ ] − g [ µ ])) ⋅ (∇ G (x *k , y% kj )dˆ kj [ µ ]) < 0 , ∀µ = 1, 2,K ,18 ⋅ Tˆ , ∇ G (x *k , y% kj )dˆ kj < (G (x *k , y% kj ) − g ) , − ∆ ≤ dˆ kj ≤ ∆, y% kj + dˆ kj ∈ Y.

(36)

The components of the vector ∆ are all very small positive real values. Step 3) Check the inner-level convergence.

13

ACCEPTED MANUSCRIPT Set L I −level = D j (x *k , y% kj ) . If U I −level = 0 or U I −level − L I −level < δ , then stop and output the current solutions y% kj as y *k and into the outer-level problem in (33); G (x k +1 , y *k ) − g is used to construct the associated Benders cuts for the outer-level problem. Otherwise, set U I −level = D j (x *k , y% kj ) , j = j + 1 and go to step 2.

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4. Numerical Tests In this section, we evaluate the performance of the proposed solution approach in Section 3 on its economic efficiency. 4.1 Test data

The test data are all collected based on a real two-engine and single-aisle civil aircraft which is the currently main type of the

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civil aircraft. The definition and description of the flight phases with the allocation of the associated scheduling time period are presented in Table 1. In the test cases, the flight duration is two hours and thus the scheduling horizon for the power optimization

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of ECS: Tˆ will be 120. Thus, based on Table 1, we can obtain the values of (h (t ), v (t ),ϑ (t )) for t = 1, 2,K , Tˆ and these values will help us to determine P 0 (t ) and T 0 (t ) for t = 1, 2,K, Tˆ (for period t, P 0 (t ) is obtained based on the transformation relationship between P 0 (t ) and h (t ) and T 0 (t ) is determined by the ideal gas equation of state [22]). From the presented equations in Section 2, we can finally collect the data of Ma (t ) , P 1 (t ) and T 1 (t ) for t = 1, 2,K, Tˆ . Flight phase allocation Phase description

Allocated time periods

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Table 1. Phase definition Initial climb

From aircraft lift-up to 450m with speed 375km/h; pitch attitude 4 o From 450m to 3000m with speed 465km/h ; pitch attitude 9.5

Climb

1

o

2-3

Constant flight altitude 3000m with speed increased from 465 to 600km/h; pitch attitude 0 o

From 3000m to cruise altitude with speed 600km/h; pitch attitude 7 ~ 10

o

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Cruise

Cruise altitude with cruise speed; pitch attitude 0

o

o

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Cruise altitude to 3000m with speed 600km/h; pitch attitude − 10 ~ − 7

Approach and landing

o

o

Flight altitude 3000m with speed decreased from 600 to 465km/h; pitch attitude 0 3000m to 1800m with speed 465km/h; pitch attitude − 9

19-36 37-72

Cruise altitude with speed decreased from cruise speed to 600km/h; pitch attitude 0

Descent

4-13 14-18

Cruise altitude with speed increased from 600km/h to cruise speed; pitch attitude 0

Time period allocation for flight phases

o

o

73-90 91-95

o

96-105 106

1800m with speed decreased from 465 to 300km/h; pitch attitude 0 1800m to 450m with speed 300km/h; pitch attitude − 4 o 450m to touchdown with speed 250km/h; pitch attitude − 1.5

o

o

o

107-112 113-116 117-120

The ECS system data includes all technical and performance parameters of the aircraft’s ECS and are given in Table 2. Based on the presented equations in Section 2, we can obtain m& i (t ) and m& HX (t ) for t = 1,2,K, Tˆ from the data in Table 2. Meanwhile, in this paper, we assume the values of the heat loads (conductive, solar, internal electric and occupants’ heat loads)

14

ACCEPTED MANUSCRIPT produced within the cockpit, passenger, electric device and cargo compartments have the following relationships with the values of (h (t ), v (t ),ϑ (t )) for t = 1, 2,K , Tˆ which are demonstrated in Table 3 based on the heat payload analysis (In Table 3, we define the total temperature of the aircraft skin T s (t ) is equal to T 1 (t ) ⋅ (1 + 0.18 ⋅ ( Ma (t )) 2 ) ) [19]. Table 2. ECS data

gc

0.286

K

1

K

c0

γ

(m2)

(m2)

0.0004

0.00001

i

1.4

m&

c1

A HX

T

x

γ

1.4

P2

x

γ

c0

T

γ

c1

1.4

P3

2

T

t1

η c0

1.3

0.9

η

0.9

P4

3

T

4

( K/(rpm)2)

(kg/min)

(K)

(kPa)

(K)

(kPa)

(K)

(kPa)

(K)

3 × 10−7

8 × 10−7

100

400

150

380

200

500

300

500

Pˆ

m& ic

m& ip

m& rc

(kg/min)

(kg/min)

(kg/min)

(kg/min)

80.8

100

100

100

100

m co

m

pa

m el

m ca

T

(kg)

(kg)

(kg)

(kg)

(K)

(K)

500

500

500

500

303

290

co

T

mB

mC

(kg/min)

(kg/min)

A

mD

(kg/min)

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m

100

co

100

c

η

t1

0.9

m

p

(kJ/ (kg·K))

P5

0.8

T

1.004

P6

5

T

6

(kPa)

(K)

(K)

300

500

150

350

310

V

V

V

co

V

pa

el

ca

(m3)

(m3)

(m3)

(m3)

23

233

20

30

100

ω0

ω1

(K)

(K)

(K)

(K)

(K)

(rpm)

(rpm)

303

290

313

290

303

283

40000

20000

T

pa

pa

T

el

T

el

7

(K)

(K)

T

T

(kPa)

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( K/(rpm)2)

(kPa)

η

c1

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R (kJ/ (kg·K))

Ai

T

ca

T

ca

Electric decive compartment Cargo compartment

Q 1co (t ) = 20 ⋅ 7.5% ⋅ (T s (t ) − T

co

(t ))

(t ) = 20 ⋅ 76% ⋅ (T s (t ) − T

pa

(t ))

Q 1el (t ) = 20 ⋅ 6.5% ⋅ (T s (t ) − T

el

(t ))

Q Q

1 pa

1 ca

(t ) = 20 ⋅ 7.5% ⋅ (T s (t ) − T

EP

Passenger compartment

Solar heat load (kJ)

Internal electrical heat load (kJ)

Occupants load (kJ)

Q 2co (t ) = 20 ⋅ h (t ) /15000

Q 3co (t ) = 180

Q 4co (t ) = 42

Conductive heat load (kJ)

Compartment Cockpit compartment

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Table 3. Heat loads produced in the associated compartments for t = 1,2,K, Tˆ

ca

Q

(t ))

2 pa

(t ) = 1900 ⋅ h (t ) /15000

Q 2el (t ) = 0 Q

2 ca

(t ) = 0

Q

3 pa

(t ) = 0

Q 3el (t ) = 1320 Q

3 ca

(t ) = 0

Q 4pa (t ) = 1122

Q 4el (t ) = 0 4 Q ca (t ) = 0

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The proposed two-level Benders decomposition algorithm for the power optimization of ECS for MEA is programmed at the Visio Studio 2016 environment [29] and is implemented under Gurobi 7.5 [30] where the approximated MILP program for the outer-level problem will be solved based on the commercialized branch and bound with cutting-plane method on a personal computer laptop with a 2.50 GHz CPU and 4 GB RAM. The related MILP gap for the outer-level problem is set to 0.0001. 4.2 Performance result analysis of the proposed solution approach We at first obtain the information of the total power utilization with the data under the different combinations of the cruise altitudes (7000m, 8000m, 9000m, 10000m, 11000m and 12000m) and cruise velocities (750km/h, 800km/h, 850km/h, 900km/h, 950km/h and 1000km/h) which are demonstrated in Fig. 3. From this figure, we can find the following two characteristics: 1) with given aircraft’s cruise altitude, a larger cruise velocity will introduce more power utilization quantity. This coincides with 15

ACCEPTED MANUSCRIPT the fact that a larger cruise velocity will create more heat generated from the friction between the air and aircraft’s skin and that will cause more cooling power to reach balanced temperatures within the compartments; 2) with fixed aircraft’s cruise velocity, a higher cruise altitude will require more power utilization quantity. This is because a higher cruise altitude will have a lower air

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pressure which needs more air pressurization effort.

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Fig. 3. Total power under different combinations of aircraft’s cruise altitude and velocity

Fig. 4. Power usage analysis with time periods under different combinations of aircraft’s cruise altitude and velocity

Besides of the above results, the performance with respect to the economic efficiency and computational time are also compared under different time periods associated with different flight phases. We present the associated results of the power usage analysis with time periods under different combinations of cruise altitude and velocity in Fig. 4 where “HA”, “LA”, “LV” and “SV” separately represent high altitude (11000m), low altitude (8000m), large velocity (950km/h) and small velocity (800km/h). From Fig.4, we can find: the total power usage quantity is nearly positively linear-correlated with the cruise velocity when the cruise altitude is fixed and with the cruise altitude when the cruise velocity is fixed. The curves in Fig. 4 demonstrate a 16

ACCEPTED MANUSCRIPT unified curve shape which reveals a rising tendency of the power usage as the cruise altitude or velocity increases. Finally, the average computation time of the above sets of test samples is 58.6 sec.

5. Conclusions Under the MEA environment, the conventional ECS converts the power extraction source from the bleed air of engine to the

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electric power and thus introduces new challenge on the power optimization for ECS. Rather than an optimal sizing and planning problem for the air circulation pack, the power optimization of ECS under the MEA environment will become a complex power scheduling problem integrated with all of the constraints on the controllable electrical power generation and varying thermal power demands determined by the aircraft’s flight profiles. In this paper, in order to effectively address the above challenge,

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we propose a new formulation and solution method for solving power optimization of ECS for the civil MEA and cast it as a non-convex nonlinear program. After that, we provide a computationally efficient method based on a combination of

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Benders decomposition method and the successively linear programming technique so as to solve the above problem for the

real application on ECS of a civil MEA. Finally, we demonstrate the capabilities of the proposed solution approach on a real aircraft’s ECS and the numerical results reveal the effectiveness of the proposed problem formulation and solution approach. In the future, the work on the power optimization of ECS can be extended and integrated as one part of a more general and complex energy optimization problem -- power management problem [31-33] for the multiple sub-systems of a MEA which, beside of

None.

Acknowledgements

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Declarations of interest

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ECS, includes the engine control system, primary flight control system, wings/elevators anti-icing system and etc..

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Parameters:

t: R:

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Nomenclature

time period index; variation range t = 1, 2,K , Tˆ gas constant of the ram air (J/ (kg·K))

h (t ) :

average altitude value at period t (m)

v (t ) :

average speed value at period t (m/sec)

ϑ (t ) :

pitch angle value at period t ( o )

17

ACCEPTED MANUSCRIPT air static pressure before the ECS inlet at period t (Pa)

T 0 (t ) :

air static temperature before the ECS inlet at period t (K)

P 1 (t ) :

air total pressure before the ECS inlet at period t (Pa)

T 1 (t ) :

air total temperature before the ECS inlet at period t (K)

gc:

gravitational conversion factor

Ma (t ) :

Mach number at period t

γ i:

specific heats ratio in ECS inlet/ (primary or secondary) heat exchanger

m& i (t ) :

mass flow rate of the air imported through the ECS inlet at period t (kg/sec)

m& HX (t ) :

mass flow rate of the air imported through the (primary or secondary) heat

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P 0 (t ) :

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exchanger at period t (kg/sec) cross section area of the ECS inlet (m2)

A HX :

cross section area of the (primary or secondary) heat exchanger (m2)

γ

:

specific heats ratio of compressor C0

γ c1 :

specific heats ratio of compressor C1

γ t1 :

specific heats ratio of turbine T1

η c0 :

operation efficiency of compressor C0

η c1 :

operation efficiency of compressor C1

η t1 :

operation efficiency of turbine T1

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c0

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Ai :

η m:

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mechanical efficiency between compressor C1 and turbine T1

c p:

specific heat at constant pressure of the compressor C0 and C1 (J/(kg K))

K c0 :

transfer coefficient between the rotation speed and the change of total

temperature for the compressor C0 (K/(rpm)2)

K c1 :

transfer coefficient between the rotation speed and the change of total temperature for the compressor C1 (K/(rpm)2)

V

co

V

pa

:

volume of the cockpit compartment

:

volume of the passenger compartment

(m3) (m3)

18

ACCEPTED MANUSCRIPT V el :

volume of the electric device compartment

V

volume of the cargo compartment

ca

:

(m3)

(m3)

conductive heat load imported into the cockpit compartment at period t (J)

2 Q co (t ) :

solar heat load imported into the cockpit compartment at period t (J)

Q 3co (t ) :

internal electrical heat load imported into the cockpit compartment at period t (J)

4 Q co (t ) :

occupants load imported into the cockpit compartment at period t (J)

Q 1pa (t ) :

conductive heat load imported into the passenger compartment at period t (J)

Q 2pa (t ) :

solar heat load imported into the passenger compartment at period t (J)

Q 3pa (t ) :

internal electrical heat load imported into the passenger compartment at period t (J)

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Q 1co (t ) :

occupants load imported into the passenger compartment at period t (J)

Q 1el (t ) :

conductive heat load imported into the electric device compartment at period t (J)

Q el2 (t ) :

solar heat load imported into the electric device compartment at period t (J)

Q 3el (t ) :

internal electrical heat load imported into the electric device compartment at

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Q 4pa (t ) :

period t (J)

occupants load imported into the electric device compartment at period t (J)

Q 1ca (t ) :

conductive heat load imported into the cargo compartment at period t (J)

2 Q ca (t ) :

solar heat load imported into the cargo compartment at period t (J) internal electrical heat load imported into the cargo compartment at period t (J)

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Q 3ca (t ) :

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Q el4 (t ) :

4 Q ca (t ) :

occupants load imported into the cargo compartment at period t (J)

Decision Variables: m& x (t ) :

recirculation air flow amount for pre-heating at period t (kg/sec)

T x (t ) :

pre-heated total temperature before compressor C0 at period t (K)

P 2 (t ) :

total pressure after compressor C0 at period t (Pa)

T 2 (t ) :

total temperature after compressor C0 at period t (K)

19

ACCEPTED MANUSCRIPT total pressure after primary heat exchanger (PHX) at period t (Pa)

T 3 (t ) :

total temperature after primary heat exchanger (PHX) at period t (K)

P 4 (t ) :

total pressure after compressor C1 at period t (Pa)

T 4 (t ) :

total temperature after compressor C1 at period t (K)

P 5 (t ) :

total pressure after secondary heat exchanger (SHX) at period t (Pa)

T 5 (t ) :

total temperature after secondary heat exchanger (SHX) at period t (K)

P 6 (t ) :

total pressure after turbine T1 at period t (Pa)

T 6 (t ) :

total temperature after turbine T1 at period t (K)

T 7 (t ) :

total temperature of the air imported into the cockpit and passenger compartment

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P 3 (t ) :

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at period t (K) m& ic (t ) :

flow amount of the air imported into the cockpit compartment at period t (kg/sec)

m& ip (t ) :

flow amount of the air imported into the passenger compartment at period t (kg/sec)

m& rc (t ) :

flow amount of the air re-circulated from the passenger compartment at period t

T

co

(t ) :

m& B (t ) : T

pa

(t ) :

T

el

(t ) :

m& D (t ) : T

ca

total temperature of the air in the cockpit compartment at period t (K) flow amount of the air passing through the valve B at period t (kg/sec) total temperature of the air in the passenger compartment at period t (K) flow amount of the air passing through the valve C at period t (kg/sec)

AC C

m& C (t ) :

flow amount of the air passing through the valve A at period t (kg/sec)

EP

m& A (t ) :

TE D

(kg/sec)

(t ) :

total temperature of the air in the electric device compartment at period t (K) flow amount of the air passing through the valve D at period t (kg/sec) total temperature of the air in the cargo compartment at period t (K)

m co (t ) :

mass of the air in the cockpit compartment at period t (kg)

m pa (t ) :

mass of the air in the passenger compartment at period t (kg)

m el (t ) :

mass of the air in the electric device compartment at period t (kg)

20

ACCEPTED MANUSCRIPT m ca (t ) :

mass of the air in the cargo compartment at period t (kg)

ω 0 (t ) :

rotation speed of the compressor C0 at period t (rpm)

ω 1 (t ) :

rotation speed of the compressor C1 at period t (rpm)

References

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[1] Edwards HA, Hardy DD, Wadud Z. Aircraft cost index and future of carbon emissions from air travel. Applied Energy 2016; 164: 553-562. https://doi.org/10.1016/j.apenergy.2015.11.058.

[2] Sarlioglu B, Morris CT. More electric aircraft: review, challenges, and opportunities for commercial transport aircraft. IEEE Transactions on Transportation Electrification 2015; 1(1): 54-64. https://doi.org/10.1109/TTE.2015.2426499.

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[3] Baharozu E, Soykan G, Ozerdem MB. Future aircraft concept in terms of energy efficiency and environmental factors. Energy 2017;

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[4] Moir I, Seabridge A. Aircraft systems: mechanical, electrical, and avionics subsystems integration. 3rd ed. UK: John Wiley & Sons; 2008.

[5] Roboam X, Sareni B, Andrade AD. More electricity in the air: toward optimized electrical networks embedded in more-electrical aircraft. IEEE Industrial Electronics Magazine 2012; 6(4): 6-17. https://doi.org/10.1109/MIE.2012.2221355. [6] Herzog J. Electrification of the environmental control system. 25th International Congress of the Aeronautical Sciences 2009; 1:1-4. [7] Grande IP, Leo TJ. Optimization of a commercial aircraft environmental control system. Applied Thermal Engineering 2002; 22:

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1885-1904. https://doi.org/10.1016/S1359-4311(02)00130-8.

[8] Ordonez JC, Bejan A. Minimum power requirement for environmental control of aircraft. Energy 2003; 28: 1183-1202. https://doi.org/10.1016/S0360-5442(03)00105-1.

Meeting 1995; 1:1-8.

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[9] Hunt EH, Reid DH, Space DR, Tilton FE. Commercial airliner environmental control system. Aerospace Medical Association Annual

[10] Sinnett M. 787 no-bleed systems: saving fuel and enhancing operational efficiencies. Boeing Quarterly Commercial Aero-magazine

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2007; 28(4): 9-14.

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ACCEPTED MANUSCRIPT [15] Chen L, Zhang X, Wang C, Yang C. A novel environmental control system facilitating humidification for commercial aircraft. Building and Environment 2017; 126: 34-41. https://doi.org/10.1016/j.buildenv.2017.09.013. [16] Shi M, Chakraborty I, Tai J, Mavris DN. Integrated gas turbine and environmental control system pack sizing and analysis. AIAA Aerospace Sciences Meeting 2018; 1:1-29. https://doi.org/10.2514/6.2018-1748. [17] Shi M, Chakraborty I, Cai Y, Tai J, Mavris DN. Mission-level study of integrated gas turbine and environmental control system

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architectures. AIAA Aerospace Sciences Meeting 2018; 1:1-14. https://doi.org/10.2514/6.2018-1751.

[18] FAA. AMT airframe handbook chapter 16: cabin environmental control systems. https://www.faa.gov/regulations_policies/ handbooks_manuals/aircraft/amt_airframe_handbook/media/ama_Ch16.pdf; 2014.

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[21] Jones A, Childs T, Chen R, Murray A. Thermal sensitivity analysis of avionic and environmental control subsystems to variations in flight condition. AIAA Aerospace Sciences Meeting 2016; 1:1-15. https://doi.org/10.2514/6.2016-1980.

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[26] Carrion M, Arroyo JM. A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem. IEEE Transactions on Power Systems 2006; 21(3): 1371-1378. https://doi.org/10.1109/TPWRS.2006.876672. [27] Yang Y, Wang J, Guan X, Zhai Q. Subhourly unit commitment with feasible energy delivery constraints. Applied Energy 2012; 96: 245-252. https://doi.org/10.1016/j.apenergy.2011.11.008.

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[28] Castillo A, Lipka P, Watson JP, Oren SS, O’Neill RP. A successive linear programming approach to solving the IV-ACOPF. IEEE Transactions on Power Systems 2016; 31(4): 2752-2763. https://doi.org/10.1109/TPWRS.2015.2487042. [29] Microsoft Visual Studio 2016. http://www.microsoft.com/en-us.

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[30] Gurobi 7.5. http://www.gurobi.com/downloads/download-center. [31] Schroeter T, Schulz D. Aircraft power management – algorithms and interactions. International Symposium on Power Electronics, Electrical Drives, Automation and Motion 2012; 1:432-439. https://doi.org/10.1109/SPEEDAM.2012.6264629. [32] Yang Y, Gao Z. Novel power management method for power system of civil more electric aircraft Part I: problem formulation. submitted to IEEE Transactions on Power Systems 2018. [33] Yang Y, Gao Z. Novel power management method for power system of civil more electric aircraft Part II: algorithm and numerical tests. submitted to IEEE Transactions on Power Systems 2018

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ACCEPTED MANUSCRIPT Highlights: power optimization of ECS for MEA is formulated as a nonlinear optimization problem

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an efficient method is provided to solve the above problem on real ECS of civil MEA

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test results are demonstrated based on real civil MEA revealing the effectiveness

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