Optimal Weight Power System Design and Synthesis for More Electric Aircraft

The synthesis of a power distribution architecture for More Electric Aircraft requires weight optimization in order to reduce energy consumption. The weight of an aircraft power distribution system depends on various factors such as the functional and safety requirements, as well as component selection and location. Functional and safety requirements can be translated into a set of connectivity and reliability constraints to produce an architecture that represents an abstract topology of the power system. However, component selection and location aims to produce a solution that is closer to a final implementation. Then, this paper presents an optimization based design formulation that synthesizes a power system architecture considering component selection and location in order to shorten the gap between the topology and the physical implementation. Given the complexity in producing and solving such formulation, linear transformations are performed to enable the use powerful commercial solvers and reach a minimum weight electrical distribution system. Therefore, the design is presented as a Mixed Integer Linear Programming problem and a case study is used to exemplify the synthesis of a power distribution architecture that is optimal.


II. Introduction
The More Electric Aircraft (MEA) power distribution system consists of a set of interconnected components that performs power transmission and conversion to supply loads at the required voltage level and format (AC or DC, high, medium, and low voltage, etc.) [1], [2]. Recently, DC distribution systems have captured a lot of interest for MEA applications due to the reduced number of conductors [3], inexistence of reactive power [4], and higher dynamic stability [5] if compared to the AC counterpart. In addition, among several wiring configurations studied for MEA power system architectures, DC systems provide maximum power transfer and maximum weight to power ratio for the same voltage range [3]. Several DC distribution topologies have been proposed in MEA to provide adequate number of power conversion levels, high reliability, and high-power density [1], [5]- [7]. Hence, the main objective in determining a power system architecture for MEA is minimizing the amount of weight, i.e. reduce the system's payload, while guaranteeing safety and power transfer availability.
Safety specifications can be translated into a set of reliability constraints that ensures the power system comply with a certain reliability level (reliability metric) by assembling sufficient number of components and distribution paths [8]- [10]. Several approaches based on reliability-based network optimization problems [11] have included reliability constraints in a Mixed Integer Linear Programming (MILP) problem that synthesizes an optimum power distribution topology [10], [12]- [14]. These formulations can be solved either with iterative techniques, or sequential algorithms. In the former, the number of reliability constraints increase depending on the reliability requirements [12], so that the connectivity complexity is increased only if required by specification, otherwise, simpler topologies could be feasible candidates. In the latter, starting at an abstract representation (platform) of the system considering functional requirements, the design walks through a certain number of refinement steps until the platform is very close to a physical implementation that complies to the initial specifications [15]. Although these design frameworks have been used to synthesize an optimal power system architecture that complies with a set of requirements, further steps must be performed in order to produce a final implementation. This paper pretends to close this gap by exploring weight minimization via component selection and location while complying with a set of reliability constraints. The design exercise is formulated as a MILP to experience the advantage of reaching optimality in polynomial time (depending on the number of variables) and using powerful MILP solvers available commercially. The rest of the paper is outlined as follows. Section III develops the design formulation to synthesize a power distribution architecture considering connectivity, reliability, component selection, and location constraints. Then, Section IV presents a linear transformation of the design problem. Section V exemplifies the use of the proposed formulation in the case study. Finally, Section VI comes to the conclusion of the paper.

III. Design Formulation
The power system design must consider the functional and safety specifications from the initial design stage. In the case of a MEA power system, these specifications are based primarily on the electrical loads' power and energy requirements, the airworthiness standards, and the aircraft state-of-the-art design practices. Besides, the design outcome is inherently related to the type of aircraft, its engine capabilities, and the flight purpose or length (military, cargo, commercial short-/ long-haul, etc.). Nevertheless, the specifications and requirements can be translated into a set of connectivity and reliability constraints which are satisfied by the optimum obtained through the design framework. A set of constraints on the components' selection and location are included in an attempt to get closer to a physical power system implementation whose weight is minimum.
The MEA power distribution system comprises a set of interconnected power sources (generators, batteries, etc.) and distribution devices that performs generation, transmission, conversion, and distribution to supply power to critical and non-critical loads. For the rest of the paper, the power sources are assumed to be generators driven by the aircraft' jet engines. Given that the power conversion weight is driven by a power density value (kW/kg) [16], [17], its total weight is proportional to the amount of power to convert. Consequently, the weight of the power distribution system is independent from the generator-load pairing arrangement. Following a PBD methodology [18], the MEA power system can be synthesized in two sequential steps [15], [19]: generator selection and generator-load pairing (GS&GLP), and power distribution design (PDD). The former selects a number of power sources (generator selection) and determines which loads are supplied by those power sources at any time (generator-load pairs or distribution paths). Then, the latter synthesizes a power distribution topology whose number of components and their corresponding sizes depend on the previous generator-load pair arrangement or selected distribution paths [15], [20].

A. Generator Selection and Generator-Load Pairing step
The design exercise requires to supply a pre-specified set of critical loads ⊆ with reliability requirements. In the GS&GLP, the functional requirement produces an abstract representation of the power system where source points deliver power to load points. Let a set of ℊ generators and a set of ℰ distribution paths (generator-load connections) be represented by a template (graph) = (ℊ, ℰ). Each generator has a set of parameters, i.e. weight , reliability , and power rating . The GS&GLP step attempts to select subsets ⊆ ℊ and ⊆ ℰ of generators and distribution paths respectively that minimizes generation weight , supply each of the critical loads (from the set ), and satisfies the connectivity and reliability constraints. Let a Boolean determine the selection ( = 1) or rejection ( = 0) of a generator of the set ℊ. The minimization of the generation's total weight can be written as: In (1), is the weight of generator and the product equals if = 1 (otherwise, generator is not selected and its weight contribution is 0). The generator weight is assumed to be a function of the power rating [16]. The weight of the distribution system is not included in (1) because in the abstraction of the generation platform, the function of the distribution system is to allocate load power to generator power according to the generator's capacity and the loads' reliability requirements. Later in the PDD step, a set of functional requirements are defined and the distribution paths are constructed to allow power transmission. Consistently, a set of connectivity and reliability constraints always ensure sufficient generation and distribution capacity to supply critical loads.
In GS&GLP, the connectivity constraints allow load to be connected to a generator . Each of the selected generators must have a power rating greater than the minimum power rating MIN but less than the maximum power rating available MAX . A set of ℎ commercial power rating values ℎ can be used to select a specific for generator . Let a Boolean ℎ select a power rating value ℎ ∈ for generator . Each generator has a unique power rating if selected, then, Let the Boolean determine the selection ( = 1) or rejection ( = 0) of the distribution path connecting generator to the load . A load is connected to a generator if that generator has been selected, then, Each load is connected at least to one generator, then, The power rating ℎ of the generator is greater than or equal to the total load connected to it, then, For each load , the reliability is considered as the availability of power supply on the load's terminals, i.e. there is a reliability target TARGET, that needs to be satisfied. Let be the reliability of the generator , and the reliability of each distribution path (generator-load connection). When a load is connected to a generator , the reliability on the load terminals is , which is the reliability of a series-system. When a load is connected to more than one generator, the reliability of a parallel series-system can be applied. Then, the reliability of a load connected to multiple generators must be greater than or equal to TARGET, , then The product in (6) allows to set to 0 if the distribution path is not selected ( = 0). As mentioned before, the outcome of the GS&GLP step is a group of generators ⊆ ℊ with their corresponding ratings ℎ and a group of distribution paths (generator-load connections) ⊆ ℰ with their corresponding reliabilities . The reliability constraint in (6) contains the distribution path's reliability variable which is used as the reliability target of the distribution path in the PDD step.

B. Power Distribution Design step
The PDD step aims to synthesize a power distribution architecture that implements the distribution paths arrangement (generator-load connections) found in the GS&GLP step. The functional abstraction of the distribution platform considers appropriate power conversion/transmission between devices and topology reconfiguration. Consider a set of power distribution components and a set of feasible connections. Let these two sets be represented by a template (graph) = { , }. Each component has a group of design parameters, i.e. weight , reliability , and power capacity . Similarly, for each connection the group of parameters include weight , reliability , and power capacity . The PDD attempts to select a subset of components ⊆ and a subset of connections ⊆ that minimizes power distribution weight, while satisfying a group of connectivity and reliability constraints. The power distribution weight consists of a fixed weight (installation payload) and a variable weight that depends on the amount of power flow converted/transferred.
Let a Boolean select a component from the set (all selected components will form the subset ). Also, let component have a fixed weight , a variable weight KW that represents the ratio between a unit of weight and a unit of power converted/transferred, and a power flow . Similarly, let a Boolean select a connection (from the set ) between component and (all selected connections will form the subset ). Let each connection have a fixed weight , a variable weight KW that represents the ratio between a unit of weight and a unit of power transferred, and a power flow . The parameters KW and KW (in kg/kW) are the inverse of the power density (in kW/kg) for components and connections respectively. Then, the total power distribution's weight (fixed weight plus variable weight) can be written in terms of the selection variables , , and the power flows , , The first term in (7) is the summation of all fixed weights, and the second term is the summation of all variables weights (which depends on the amount of power flow converted/transferred). Given that the PDD step implements the solution of the GS&GLP (constructing the selected distribution paths ), there is a number of distribution paths that are built with selected components and connections . Each distribution path ( ∈ , exists if and only if = 1) connects the generator and the load of its corresponding path . It is possible that two or more distribution paths could share the same component and connection (depending on the reliability constraints also).
Similarly to (7), the first term in (8) is the summation of all fixed weights, and the second term is the summation of all variables weights (which depends on the amount of power flow converted/transferred). In (8), all power flows are expressed in terms of the connections' power flows , . The functional requirements are enforced with a set of connectivity constraints and the reliability requirements enforce sufficient number of components and connections to maintain the critical load of the distribution path (for which = 1) connected to the generator . The first connectivity constraint is the power flow balance on every component ∈ , i.e. nodal equation. Within each distribution path , generator supplies a power of to the load that consumes a power of . In general, each component in path is a node that can host a generator, a load, a conversion device, or a bus (step node). Thus, for the generator node, ≠ 0 and = 0, and for the load node, = 0 and ≠ 0. For the rest of cases, = = 0. Then, for every component ∈ of the distribution path ∈ , The first term in (9) represents the incoming flow, while the second term is the outgoing power flow. In the case of converters, (9) could include a loss function in terms of total power converted. By convention, incoming flow and load demand is positive, while outgoing flow and generation power is negative. For every component that is selected, some of its incoming (and outgoing) connections can be selected, then, Power flow is allowed only on selected connections and components (otherwise, it is 0). Given that distribution path delivers power demand from a generator supplying , the flow of the distribution path is limited to . For simplicity notation, let be equal to , then, , ≤ ∀( , ) ∈ , ∀ ∈ (11) ∑ , ∈ ≤ ∀ ∈ , ∀ ∈ (12) Constraints (9)-(12) allow connectivity between power distribution components and connections. Now, the reliability constraints enforce that a sufficient number of components and connections are selected in order to ensure that the distribution path supplies its critical load from the corresponding generator. Recall that the template = { , } contains all permissible interconnections between components (from generators to loads) so that the critical load receives power from an adequate power conversion level or bus. A distribution path is, on its simplest structure, a series system that connects a generator with a critical load. Then, the reliability constraint of a distribution path (series system) can be written as, where is the reliability of component , is the reliability of connection , and is the reliability target (equal to the reliability of the distribution path of the GS&GLP step). In this case, (13) implements the distribution path as a series system, which fails if any of its components fail. In order to avoid such behavior, network design approaches that can be used to build a failure resistant system which is amenable for reconfiguration. These approaches are known as resilient designs [21] and they synthesize a number of alternate paths in the case that any component or connection of the distribution path (series system) fails. Such reliability constraint can be written as, The left-hand side of (14) is the reliability of a parallel-series system. Depending on the reliability target of the path , (13) could not synthesize a distribution path but (14) can surely combine several alternate paths such that the reliability target is achieved. Up to this point, a minimum-weight topology for a power distribution system can be synthesized using (1)- (14). In order to get closer to a final implementation, additional constraints on the components' selection and location are included. These constraints allow to determine the impact of location and sizing on the overall system's construction.

C. Location constraints for power distribution design
Let the 2-dimensional space availability on a MEA airplane be represented with two straight lines, one (horizontal) spanning from the left-to the right-wing tips (wing line), and the other (vertical) crossing from the cockpit to the APU at the back of the plane (fuselage line). The components can be located in any position along each one of these two lines. If selected, a component is located in a unique position ( , ), then, In (17), the length is measured as the sum of the absolute difference of the components' locations (1-Norm). There is no need to use Euclidean distance (2-Norm) because the components are only located along the two lines (the wing line, and the fuselage line). To ensure that each component has a unique location (no two components are placed on the same position), the constraint for unique location is, where is a tolerance gap (distance gap) to force either | − | ≥ or | − | ≥ when connection is selected (located either along the wing or the fuselage lines only), hence ≠ or ≠ . The selection of components is completed by including the location constraints (15)- (18). In addition, these constraints require additional conditions on cable sizing (connection) due to the reason that location affects length, voltage drop, and ampacity.

D. Constraints for cabling sizing
It is assumed that a component's location does not influence its size, i.e. the component is selected only if it satisfies the connectivity and reliability constraints. However, the connection's size is certainly influenced by the location of the components that it connects. There are two main characteristics that change in a connection depending on its length: 1) ampacity (gauge type), and 2) voltage drop. The selection of the connection's ampacity depends on the power flow transferred between any components , while the voltage drop depends on the maximum permissible voltage drop and connection's length. Consider a set of standard wire gauges that are available (commercial cables). Let a Boolean , AWG select a standard wire gauge ( ∈ ), such that , AWG = 1 for the selected wire gauge . Then, each connection that has been selected must have a unique standard wire gauge, so, Each standard wire gauge has an ampacity AWG , a resistance per-unit-of-length AWG , a weight per-unit-oflength AWG . Note that AWG is different from kW (inverse of the power density or power-to-weight ratio). Let the operating voltage of the connection be OP (this voltage has been pre-defined according to the power conversion levels considered in the template = { , }). The ampacity constraint ensures that the current flowing through connection is less than or equal to the ampacity AWG corresponding to the wire gauge , then, where / OP represent the current flowing through connection . Now, let the maximum permissible voltage drop be . . DROP (in p.u.). The voltage drop constraint ensures that the voltage drop across the connection is less than or equal to the maximum permissible voltage drop, then, OP AWG ≤ . . DROP OP + OP (1 − , AWG ) ∀( , ) ∈ , ∀ ∈ (21) where (21) is written using the Big-M method [22]. When , AWG = 1 (i.e. a wire gauge is chosen for a selected connection), constraint (21) enforces the actual voltage drop to be less than or equal to . . DROP OP ; if AWG = 0, the actual voltage drop is arbitrarily set. In summary, constraints (15)- (21) are introduced as component location and cabling selection constraints and these constraints aim to determine the cabling size of the power distribution system. If these constraints were not considered in the design, the topology's weight would still need to be adjusted after power system architecture synthesis, possibly at later refinement steps after PDD. In the following section, the linearization of constraints (6), (13), (14), and (21) will be discussed.

IV. Linear Transformations for Non-linear constraints
The reliability constraints (6), (13), and (14) are non-linear due to the products between Boolean variables. In the GS&GLP step, the reliability contribution of any generator or distribution path is 0 when it is not selected. Similarly, the reliability contribution of components and connections is 0 when they are not selected in the PDD step. In the case of the multiplications in (6), (13), and (14), logarithm functions can linearize the product series, then, where the Boolean has been taken outside the logarithm because ln(1 − ) = 0 when = 0, then if is multiplied by ln(1 − ) the result is the same for = 0 (same results are obtained for = 1). For (13), where , and , are Boolean variables that determine if a connection and component are part of the distribution path (series system). Linearization of (14) (parallel-series system) requires the selection of a specific number of alternate paths. Let a Boolean be set ( = 1) if an alternate path is constructed. Also, let the series system of the inner product have a reliability and let this series system be linearized in the same way as presented in the left-hand side of (23). Then, the linearization of (14) can be written as, In the case of the linearization of absolute values in constraints, the reader can refer to [23] to transform (17). Finally, the left hand-side of (21) presents a non-linear product of two continuous variables, (power flow) and (connection's length that depends on the location of components). One alternative is to use two-dimensional piecewise linear functions [24], [25], or McCormick envelopes [26], [27]. The main idea is to produce tight constraints in a form of grid such that a value could be located according to the selection of specific bounds (lower and upper). In any case, the product is replaced by a third variable which is constrained accordingly, then,

V. Case Study
A case study is presented to exemplify the use of the design formulation detailed in Section III for a short-haul small MEA application. Two different MEA power distribution systems are synthesized to satisfy the requirements of the demands listed in Table 1. In both cases, total load is 100kW. L2 and L3 are critical in case 1, while all loads are critical in case 2. The permissible voltage drop across any connection is 2.5% of the nominal HV or LV DC value. The reliability target for all the loads is 1.0 × 10 −3 , which is the probability of not being supplied with electrical power, i.e. 1 − TARGET, . Loads 2 , 3 are LV DC, and 1 , 4 are HV DC loads. The design templates = {ℊ, ℰ} for the GS&GLP and = { , } for the PDD steps are shown in Fig. 1. Due to the reason that there are LV DC and HV DC loads, the system is assumed to have two voltage levels (high-and low-voltage DC), which require power conversion devices between HV and LV DC levels. Hence, there are three types of power distribution components: source matrix contactor (HV box) on HV level, HV/LV DC converter, and LV bus (LV DC level). In the LV DC side, LV DC buses distribute the power to the LV DC loads. In the HV DC side, the HV box distributes the generation power which is assumed to be rectified (e.g. AC generator with rectifier unit). Power conversion is performed by HV/LV DC converters (e.g. dual active bridge or DAB) which can be fed from several generators via the HV box and their LV DC output can be paralleled to other converters to supply the same load bus.

Fig. 1 Templates for the GS&GLP step (left), and PDD step (right)
The HV box is a fast switching device that receives power from multiple generators, and routes it either to the HV/LV DC converters, or to the HV DC loads directly. The HV box switches generation power in such a way that there is no risk of cross-connecting generators. The GS&GLP and PDD optimization problems are in Table 2. The components' parameters of the abstraction platforms for GS&GLP and PDD are listed in Table 3. The components' power capacities are expressed as a range [•] between minimum and maximum ( for the generators). For the generators' weight , a function in terms of the power rating ( ) is used, while for the distribution platform, there are fixed weight values ( , ) and values for the inverse of the power density ( kW , kW ) [(kW/kg) −1 ]. Finally, the components' reliabilities are presented as the probability of not supplying power, i.e. 1 − in the case of GS&GLP, and 1 − or 1 − for the PDD. In the GS&GLP, the abstraction of the distribution path requires a model in which the only parameter is the reliability , which is related to the allocation of the load's power to a specific generator (i.e. the probability that the distribution path is able to deliver power from generator to load ). On the other hand, the location of the generators and loads is fixed. The two LV DC loads are located in the wings (one on the left, and the other on the right wing), one LV DC load is located in the airplane's cockpit, and the HV DC load is located at the back of the aircraft. The generators are located next to the engines (left and right), and there is also one generator in the tail of the aircraft (e.g. driven by the auxiliary power unit). The rest of the power distribution components are free to be located along the wing or the fuselage lines only.  The selected components and connections from the GS&GLP and PDD assessments produce an optimal power system that has minimum weight. The case study is solved using a Windows High Spec PC Intel Xeon 64-bit 3.60GHz running CPLEX Studio IDE 12.9.0 [28]. The optimal topologies and location for cases 1 and 2 are shown in Fig. 2. 5.0 × 10 −6 1.0 × 10 −4 5.0 × 10 −6 2.0 × 10 −6 * Fixed weight, ** variable weight in terms of the inverse of the power density It can be seen from Fig. 2 that topologies for cases 1 and 2 have different number of components and these components are located differently in the aircraft (except the generators and loads). The results reflects that the initial requirements, i.e. load demand, load criticality, and type of supply (HV or LV DC), produces different optimal architectures. In this case, both power systems were synthesized to reach minimum total weight.

VI. Conclusion
An optimization-based formulation for the design and synthesis of a MEA power system architecture that has minimum weight has been proposed. Functional and safety requirements have been translated to a set of connectivity constraints and reliability constraints. In an attempt to close the gap between architecture and physical implementation, a set of location and cabling sizing constraints were introduced because system's weight depends directly on these parameters. Given the non-linearity of some of the reliability, location, and cabling sizing constraints, linearization is introduced as an enabler to obtain global optimum via commercial Mixed Integer Linear Programming solver. The results have shown that the initial load requirements influences heavily on the topology synthesized.