Algorithm for the Optimal Design of a Fault-Tolerant Aircraft Power Transmission Network

Aircraft manufacturers aim to decrease the fuel consumption based on reducing weight and increasing the subsystem efficiency. Hence, the electric power system (EPS) acquires great relevance because it must be efficient and lightweight. Any change in the EPS must not affect the aircraft’s electrical safety, which under a traditional decentralized EPS strategy is ensured by redundancy. Recently, several decentralized EPS strategies based on the introduction of multiport power converters have arisen. Such strategies meet the established safety goals since the aforementioned devices make it possible to recalculate the path to continue powering the loads in case of failure. However, the literature does not address how to connect such multiport power converters. The main contribution of this article is to present a low-complexity algorithm that minimizing the redundancy of wiring, provides a fault-tolerant power transmission network. This is done under a decentralized EPS strategy where multiport power converters are used. The proposed strategy is evaluated on Boeing 787 aircraft, where we compare the length of the cables both under a traditional decentralized network configuration (where the redundancy option is used to ensure the safety of operation) and in the network provided by our algorithm. A saving of 66.6% is obtained.


I. INTRODUCTION
I N TRADITIONAL aircraft models, some of the basic devices installed in the fuselage required the combination of hydraulic, mechanical, and pneumatic systems to operate. Those systems were heavy, oversized, and inefficient. Furthermore, they usually produced leaks that were difficult to locate and they required ongoing maintenance [1], [2]. Thus, in the late 1950s, aircraft manufacturers started analyzing as a feasible choice the replacement of nonelectrical systems by electric ones based on their higher efficiency and lower maintenance, and along the 1970s decade, the more electric aircraft (MEA) concept was coined [3], [4].
Several approaches to the MEA concept have been adopted to increase aircraft's whole efficiency based on a reduction of aircraft's weight [5]- [8]. These approaches affect the aircraft's electric power system (EPS), and some new challenges need to be addressed [9], [10], i.e., power generation, power transmission network (EPS architecture), power electronics, or systems' safety of operation (fault-tolerant EPS). In this article, we focus on the power transmission network and the systems' safety of operation. The power transmission network transfers the energy from the engines to the workloads and the safety of operation is traditionally based on the redundancy strategy.
On the one hand, the power transmission network must be efficient and lightweight so that fuel consumption can be reduced. The reduction of weight is critical and one of the most important factors that affects the total weight of the aircraft's EPS is precisely the wiring system [11]. A relevant milestone was achieved with a multivoltage hybrid dc and ac EPS architecture that employs four different voltage levels: 230 V and 115 V in the ac power system and (±)270 V and (±)28 V in the dc power system [12], [13]. Through this proposed EPS architecture, lower currents were necessary, and consequently, the power losses through transmission were reduced and aircraft becomes lighter because smaller conductors were used [8], [10]. In conventional centralized EPS architecture, all the electrical wiring is distributed from the main bus to different loads. This centralized EPS architecture has been replaced with a semidistributed (decentralized) architecture, where a large number of power distribution units (PDUs) are located throughout the aircraft to supply the loads locally. This architecture reduces wiring and saves weight (see, e.g., [2], [5], [14]). For instance, for a large civil aircraft with high demand on the electrical network, about 46% of the electrical wiring serves to distribute electrical power from the power centers to the loads and adds up to about 2.6 tons [15], which offers potential for optimization.
On the other hand, to guarantee the safety of operation, high redundancy of wiring and critical devices is needed. It is obviousl that this redundancy increases the aircraft's weight. Hence, weight reduction and safety seem to disagree with each other. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ During the last years, milestones established by organisms such as Advisory Council for Aeronautics Research in Europe (ACARE) and the Flightpath European Commission (FPC) show that even further improvements in terms of aircraft's efficiency and safety are required [16]- [20]. According to this, current MEA's studies are cataloged into two main groups. The first group focuses on developing novel software for increasing aircraft's efficiency [21], [22]. The second group focuses on new feasible architectures to increase aircraft's efficiency and safety of operation [23], [24].
Concerning to the group that examines novel architectures, Garriga et al. [23] proposed a novel algorithm, which selects between several feasible EPS structures the optimal one for a defined aircraft model. To make this choice, the proposed algorithm iteratively balances a certain number of parameters such as new aircraft model's global weight or fuel consumption until the optimal solution is fixed. Moreover, Jia and Rajashekara [24] analyzed the advantages of designing novel aircraft's distributed EPS architectures, such as the introduction of induction generators to improve the EPS's availability and reliability.
Concerning to the methodology for designing aircraft, Sziroczak et al. [25] claimed that the performance in future MEAs efficiency only will be achieved by redesigning both the propulsion system and the EPS. They developed a conceptual methodology where the propulsion system of a small aircraft is driven by a hybrid-electric propulsion system. The main contributions obtained with this study rely on the introduction of the recalibration of mass fractions in novel configurations as well as the hybrid-electric propulsion system.
In their study, Alexander et al. [26] made an analysis of different vehicles' EPSs and their evolution in the last years, to make a prediction about future MEAs architectures characteristics such as feasible voltage levels or equipment ratings. The research concludes that few key factors, such as energy density, weigh, propulsion systems, and EPSs' architectures, will be the main bottlenecks to achieve the goals stated to the MEA. In addition, the authors claimed that recent results obtained with lithium-air batteries and novel small-scale fullelectric aircraft makes possible the research on novel further electrified large-scale aircraft with novel architectures.
Gu et al. [27] made research on the traditional redundancy strategy, which involves increasing aircraft's weight. In order to meet FPC goals, Gu et al. [27] examined several EPS strategies based on the introduction of multiport power converters to meet established safety goals without increasing aircraft's weight. Thus, Gu et al. [27] concluded that multiport devices combined with a ring EPS strategy achieve the best results. Following the strategy proposed in [27], Rodríguez et al. [28] and Martínez et al. [29] presented a novel multiport hardware device that makes possible to recalculate the optimal path to continue powering the loads in the case of failure occurring in the aircraft's EPS network.
As well as it is proposed in [27] and [28], Buticchi et al. [30] presented a multiport quadruple active bridge (QAB) dc/dc converter to show how the architecture of the MEA can be modified. The topology proposed in [30] is traditionally applied in those EPSs where the devices connected it need to ensure flexibility and galvanic isolation.
Nevertheless, current partial decentralized EPS or aircraft's EPS architectures proposed by the Clean Sky 2 framework [31] as well as strategies proposed by Gu et al. [27] and Rodríguez et al. [28] did not address how to make an optimal connection between the developed multiport devices.
In this article, we aim to improve the aircraft's efficiency by reducing its weight while guaranteeing the safety of operation. Specifically, under a decentralized EPS strategy, we propose to minimize the redundancy of wiring. To that end, we will design a power transmission network where enough different routes are guaranteed to power the loads and with the lowest number of internal connections (cables). Once the network is established, software and hardware modifications proposed in [28] and [29] will guarantee that all connections work properly since it will recalculate the optimal path to continue powering the loads if there is a failure in the aircraft's EPS network.
The aircraft's EPS network under a decentralized strategy can be viewed as an undirected connected multigraph. There exist a variety of techniques for connecting the nodes of a graph under certain conditions. The problem addressed in this article is similar to the well-known Steiner problem in networks, which is NP-complete. This problem looks for an optimal interconnection of a given set of nodes under a certain cost function. Hakimi [32] and Levin [33] formulated this problem for the first time, but many other variants exist (see [34]). It should be mentioned that the problem considered in this article cannot be addressed neither as a Steiner problem in graphs nor as any of its variants mainly because of two reasons: in our problem, multiple connections (cables) between devices (nodes) are allowed, and not all the possible interconnections between devices are permitted (e.g., a load cannot feed other devices).
Thus, the main contribution of this article is to provide an algorithm for the optimal design of a fault-tolerant aircraft's power transmission network under a decentralized EPS strategy. Furthermore, as it was mentioned above, when designing an EPS network, goals as weight reduction and safety seem to disagree with each other. The novelty of this article is that we show that using our algorithms, it is possible to save wiring while guaranteeing the safety of operation.
The remainder of this article is organized as follows. In Section II, we state preliminary considerations regarding undirected connected multigraphs and present a technical proposition that will be used in Section III. In Section III, we present the optimal interconnection problem to be solved in order to guarantee a sufficient number of different routes to power the loads with the lowest number of cables. In Section IV, we propose two algorithms that solve, with low complexity, the minimization problem in Section III. Finally, in Sections V and VI, we give an illustrative example and some conclusions, respectively.

II. PRELIMINARIES
Aircraft's EPS network can be viewed as an undirected where V(G) is the set of vertices (devices) and E(G) is the set of edges (cables). We recall that a multigraph with no loops is a graph where multiple edges joining the same two vertices are allowed (see, e.g., [35, p. 7]).
If G 1 and G 2 are two subgraphs of G, the union of G 1 and G 2 is the subgraph of G of the form Let v 0 and v l be two distinct vertices of G. A path between v 0 and v l is a subgraph P = (V(P), E(P)) where and e j is an edge from v j −1 to v j for all j ∈ {1, 2, . . . , l}. Assume that P 1 , P 2 , . . . , P h are different paths between v 0 and v l . These paths are said to be edge-disjoint if they have no edges in common, that is, We finish this section with a technical proposition that will be used in Section III.
Proposition 1: Let G 1 and G 2 be two subgraphs of an undirected connected multigraph G with no loops such that by λ 1 and λ 2 , respectively. We divide the proof into seven steps.
Step 1: and therefore, l is a loop (which is absurd since G has no loops).
Step 2: We prove that if l ∈ E(G) joins v 1 , then l ∈ E(G 1 ) (analogously, it can be proved that if l ∈ E(G) joins v 2 , then l ∈ E(G 2 )). We prove it by reductio ad absurdum.
Step 3: We show that every path between v i and v * is a subgraph of G i , with i ∈ {1, 2}. This is a direct consequence of Step 2.
Step 4: 2}. This is a direct consequence of Step 3.
Step 6: We prove that if P is a path between v 1 and v 2 then v * ∈ V(P). We prove it by reductio ad absurdum. Suppose that v * / ∈ V(P). By induction, we can prove, using Step 2, that Step 7:

III. PROBLEM STATEMENT
We aim to design a fault-tolerant aircraft's power transmission network under a decentralized EPS strategy with the minimum number of cables. We consider a multivoltage hybrid dc and ac EPS architecture that employs four different voltage levels (230 V and 115 V in the ac power system and (±)270 V and (±)28 V in the dc power system). The considered decentralized EPS architecture has a high-voltage primary power center (HVPPC) that powers the high-voltage (HV) power system and a low-voltage primary power center (LVPPC) that powers the low-voltage (LV) system. The HVPPC powers the ac loads of 230 V and, by using power electronics, the dc loads of 270 V. The LVPPC powers the ac loads of 115 V and, by using power electronics, the dc loads of 28 V. Under a decentralized EPS architecture, a large number of PDUs fed from the power centers are located throughout the aircraft to supply loads locally. We here consider that the PDUs can feed the four different voltage levels or even provide power to other PDUs. Since there are four different voltage levels, there are four different subnetworks to be designed where interconnections among PDUs are allowed. Fig. 1 shows the considered EPS architecture.
Aircraft's power transmission network fault tolerance is traditionally based on the redundancy of internal connections and critical devices. In this way, the feeding of the loads during the operation phase is guaranteed but at the cost of increased weight. The number of different routes needed to feed a load is chosen by the aircraft designer 1 and depend on each load's priority level. In this article, the priority of a load is defined as the number of different routes (edge-disjoint paths) from the power center to such load. Under a traditional decentralized  EPS strategy, a PDU can feed several loads, but each load is fed by a single PDU. Therefore, the number of cables from the power center to a PDU is given by the highest of the priorities of the loads fed by such PDU. The higher priority the loads have, the higher the redundancy of internal connections, and therefore, the weight of the whole aircraft is increased as well as fuel consumption.
We aim to design an aircraft power transmission network that guarantees the number of different routes needed to keep each load's priority level with the minimum number of cables avoiding the redundancy option. To that end, the considered PDUs are also provided with software and hardware modifications proposed in [28] and [29]. Such modifications allow PDUs to be fed by other PDUs, allow loads to be fed by several PDUs, and make it possible to recalculate the optimal route to continue powering the loads in the case of failure. It should be mentioned that although the fault-tolerant algorithm developed in [28] works for every voltage level, the hardware for the power distribution of each voltage level will be different.
Without loss of generality, in this section, we state the design problem for one of the four mentioned subnetworks (the problem statement for the other subnetworks is analogous). Fig. 2 shows the considered subnetwork with a primary power center (xPPC), n PDUs (PDU 1 , . . . , PDU n ), m loads (L 1 , . . . , L m ), and the internal connections among them. In the figure, a i represents the number of cables between the xPPC and PDU i , b i, j represents the number of cables between PDU i and PDU j , and c i,k represents the number of cables between PDU i and L k , where i, j ∈ {1, . . . , n} and k ∈ {1, . . . , m}. Finally, p k represents the priority of L k . Observe that the subnetwork in Fig. 2 can be viewed as an undirected connected multigraph G with no loops. We denote by G k the subgraph of G obtained by removing in G all the loads except for L k (it is obvious that all the edges connected to the removed loads are also removed).
In order to guarantee the number of different routes needed to keep each load's priority level with the minimum number of cables, we need to solve the following minimization problem: Now, we address the minimization problem (2) by reducing the search space according to the following assumptions.
1) In each zone z, we select a central PDU denoted by PDU d z . Only these central PDUs can be directly powered by the xPPC. Therefore, some a i will be forced to be zero. Specifically, a i = 0 whenever i = d z for all z ∈ {1, . . . , q}. 2) If two PDUs are located in different zones, they cannot be connected unless they are both central PDUs. Therefore, some b i, j will be forced to be zero. Specifically, 3) The loads in an specific zone can only be powered by PDUs located in the same zone. Therefore, some c i,k will be forced to be zero. Specifically, c i,k = 0 whenever i ∈ D z 1 and k ∈ L z 2 with z 1 = z 2 . 4) All the available PDUs located in zone z must be used to power the loads. Therefore, k∈L z c i,k ≥ 1 for all i ∈ D z . Consequently, the minimization problem (2) can be rewritten as Despite the assumptions made, the complexity of the minimization problem (3) is still very high making exhaustive search intractable. However, we can now use the divide-andconquer technique and divide the minimization problem (3) into q + 1 subproblems of the same type with much lower complexity. To that end, we first need to introduce some notation.
Fix z ∈ {1, . . . , q} and k ∈ L z . Let H k and H z be two subgraphs of G k . The subgraph H k represents the power transmission network in the zone z, that is, the vertices of H k represent the PDUs {PDU i } i∈D z together with the load L k , and the edges of H k represent the connections among them. The subgraph H z represents G k except for the power transmission network in the zone z but including PDU d z , i.e., Observe that in the subgraph H z , there are no loads, and therefore, it does not depend on k. Now, we divide the minimization problem (3) into the following minimization problems: and for each z ∈ {1, . . . , q} Since H k ∪ H z = G k and V(H k ) ∩ V(H z ) = {PDU d z }, from Proposition 1, we have that for each z ∈ {1, . . . , q} Consequently, by combining the solutions obtained in the q + 1 minimization problems (4) and (5), we obtain a solution of the minimization problem (3). It should be noticed that the minimization problem (4) provides a fault-tolerant aircraft's power transmission network with the minimum number of cables between the xPPC and each zone. Similarly, the solutions to the minimization problems (5) provide for each zone a fault-tolerant power transmission network with the minimum number of cables between loads and PDUs in the zone.

end for
We begin the section by introducing some notation. If A is a finite set, |A| denotes the number of elements in A. If x is a real number, floor(x) denotes the largest integer not greater than x.
In this section, we assume that q is the number of zones in the left-hand side of an aircraft. Let p(z) = max k∈L z p k for all z ∈ {1, . . . , q} be the maximum priority of the loads in zone z of the left-hand side of an aircraft. Without loss of generality, we assume that p(1) ≥ · · · ≥ p(q) and define p max = p(1). It should be noticed that due to the intrinsic symmetry of an aircraft, the number of zones and their priorities in the right-hand side are the same as in the left-hand side.
On the one hand, for solving the minimization problem (4), Algorithm 1 must be run for obtaining the connections between the xPPC and the zones of the left-hand side of the aircraft (observe that the connections between the xPPC and the zones of the right-hand side of the aircraft will be for all k ∈ L z do 5: if p k /|D z | > 1 then 6: for = 1 : |D z | do 7: i ← th element of D z 8: if ≤ p k mod (|D z |) then 9: c i,k ← floor( p k /(|D z |)) + 1 10: else 11:  25: end for 26: end for symmetric). Moreover, for each zone z in the left-hand side with p(z) > 1, the central PDU of that zone and its symmetric counterpart in the right-hand side must be connected with one cable.
On the other hand, for solving the minimization problem (5), Algorithm 2 must be run.
Finally, for the reader's convenience, Tables I and II describe  Algorithms 1   Typical electrical system loads at cruise condition in Boeing 787 partially recreated from [37]. in the network provided by the novel algorithms presented in this article. We use this numerical example to quantify how many meters of cable our proposal saves.
In order to evaluate the proposed algorithms, we select Boeing 787 aircraft during cruise conditions since it has many novel MEA features and is an example of a decentralized EPS architecture (see, e.g. [38]). In this example, we consider a few electrical system loads in Boeing 787 (see Fig. 3), and we use the algorithms to design the subnetwork for powering some of the (±)270 V loads in the dc power system.
In the literature, it can be found some descriptions of the loads in an aircraft (see, e.g., [5], [37], [39]). The properties of the airplane's electrical system loads that we consider in this numerical example are given in Tables III and IV. According to [5, p. 268], these loads are fed by 21 PDUs located throughout the aircraft. This can be visualized in Fig. 4. In this figure, the airplane is divided into eight different zones (four in the right-hand side and four in the left-hand side). Fig. 5 shows the connections between the HVPPC and the PDUs under a traditional decentralized network configuration where the redundancy option is employed in order to guarantee the fault-tolerance of the network. Fig. 6 shows the resulting interconnections between the HVPPC and the PDUs according to Section IV. Moreover, after running Algorithm 2 for each   zone, we would obtain the connections between the devices in each zone. As an example, Fig. 7 shows the connections provided by Algorithm 2 between the devices in the right-hand side zones.  Regarding the connections between the PDUs and the loads, the number of cables that feed a particular load depends exclusively on its priority, and it remains the same whether we use the traditional decentralized network (where each load is fed by a single PDU) or the network obtained after running Algorithm 2 (where each load can be fed by several PDUs). Hence, in order to quantify how many meters of cable our proposal saves, we only need to compare the connections between the HVPPC and the PDUs, that is, the connections shown in Figs. 5 and 6. In order to quantify the length of the cables, we have combined the information shown in the figure in [5, p.268] (where the location of the 21 PDUs is sketched) with the dimensions of Boeing 787 aircraft. Specifically, we have considered the aircraft as a 2-D object and we have assumed that the length of the cables that connect two devices is the same as the straight line distance between such devices. Fig. 8 shows the distances among the central PDUs of each zone and the HVPPC. Moreover, we assume that the average distance among the PDUs located in the same zone is 5 m.
For the considered example, under a traditional decentralized network configuration where the redundancy option is  It should be noticed that for the numerical example, we have considered that there are 21 ECS/pressurization loads, four hydraulics loads, four equipment cooling loads, and eight ECS fans. Observe that considering more loads of these types (with the same priorities and in the same zones) does not affect the network between the HVPPC and the PDUs, that is, the networks shown in Figs. 5 and 6 would remain the same. If there were more loads of these types, there would be more connections between the PDUs and the loads, but the network between the HVPPC and the PDUs would be the same. Consequently, the savings obtained using our algorithms would remain the same unless we change the considered distances among DPUs and HVPPC.
Finally, Fig. 9 bears evidence of the reliability of the aircraft's EPS network. Specifically, if, for example, Load 10 is considered (priority three), the figure shows that there exist three different edge-disjoint paths from the HVPPC to the load.

VI. CONCLUSION
In recent years, based on the MEA concept, researchers have proposed several approaches related to the aircraft's EPS architecture to increase the whole aircraft's efficiency and their fault-tolerant ability. In this article, we have addressed the problem of achieving the target reliability with the minimum wiring under a decentralized EPS strategy, considering that the PDUs are provided with software and hardware modifications proposed in [28] and [29].
In order to guarantee the reliability with the minimum number of cables, we need to solve a minimization problem. However, the complexity of such minimization problem grows exponentially with the number of devices, making exhaustive search intractable. To handle the former minimization problem, we have made some assumptions regarding the physical location of devices, we have used the divide-and-conquer technique and designed two algorithms to solve the resulting minimization problems. Specifically, we have presented two very low-complexity algorithms to connect the devices with the minimum number of cables and guaranteeing the number of different routes needed to keep each load's priority level. One algorithm establishes the connections between the xPPC and each zone of the aircraft, while the second algorithm establishes the connections between the PDUs and the loads in each zone.
In order to evaluate our algorithms, we have compared the length of the cables under a traditional decentralized network configuration and the length of the cables in the network provided by our algorithms. To that end, a few loads of the (±)270 V (dc) power transmission subnetwork of the Boeing 787 aircraft have been examined. In the considered example, we saved 66.6% of wiring between the xPPC and the PDUs. The main conclusion of our work is that, by using the proposed algorithms, the total weight of the aircraft's EPS can be significantly reduced.