Designing a Hybrid Electric Powertrain for an Unmanned Aircraft with a Commercial Optimization Software.
Hybrid Electric Propulsion Systems (HEPS) are of increasing interest in the development of efficient aircraft powertrains , particularly for Unmanned Aerial Vehicles used for surveillance tasks. In fact, HEPSs allow the UAV to fly in electric mode thus reducing its thermal, acoustic and smoke signature.
HEPS are characterized by the usage of two energy storage systems, usually a fuel tank and a battery. The usage of the batteries also allows the optimization of the operating point of the engine that can also be downsized. In fact, engines are usually designed to meet the high requirements of thrust at takeoff but, for the most part of the mission (cruise), they work at part load. HEPS can be classified into parallel and series powertrains. In the case of series configurations , the mechanical drive path that moves the propeller is connected to an electric motor while the internal combustion engine is used to drive a generator. The electric power to the motor is the algebraic sum of the battery power and the engine/generator power. The series configuration is suitable for low-speed, high-torque applications but it is less efficient than the parallel one, where both the engine and the motor are mechanically connected to the drive train . Moreover, it requires larger batteries and electric machine .
HEPS for UAVs were studied numerically in [2, 3., 4, 5., 6] and tested experimentally in . A test flight was performed with a hybrid electric aircraft  and a reduction of fuel consumption of 30% with respect to the conventional gasoline engine was estimated but very limited information was given about these powertrains. HEPS unfortunately require an increasing of complexity, mass and volume that are critical issues in the aircraft field where reliability and power density are the main goals.
In HEPS, the power management of multi-source operation is also critical. The supervisory control strategies for hybrid electric powertrains are usually classified into four categories  : numerical optimization, analytic optimal control theory, instantaneous optimization and heuristic control techniques. Heuristic Control Techniques are rule based and require a very low computational cost . They can be based on Fuzzy Logic controllers . Other approaches use the Bellman's Principle or, in general, Dynamic Programming to perform a global Numerical Optimization. The main drawback is the heavy computational burden and the necessity to have a full knowledge of the power request versus time. Finally, the analytic optimal control searches for the optimal solution using a simple mathematical model for the powertrain . Harmon et al.  proposed a Neural Network Controller which assigns relative importance between the use of gasoline, electricity and recharging. In the present investigation, a rule-based strategy was used to keep low the computational time. More complex strategies will be taken into account as further investigation.
In the case of road vehicles, the simulation of hybrid electric powertrains is usually performed with backward simulation codes. Donateo et al [12-13] developed, implemented and validated a tool named PLA.NE.S. to evaluate the benefits and the disadvantages of advanced powertrain for aircraft by adapting the unified model approach proposed in . This approach is widely used in the automotive field [15,16] but quite new for aircraft applications. In the present investigation, the PLA.N.E.S. code was combined with a many-objective optimization environment (Modefrontier 4.4 ) to perform the optimization of powertrain specification and battery management strategy with respect to four key performance indexes. Genetic Algorithms (GAs) were chosen among the tools implemented in Modefrontier because of their robustness and their capability to deal with many-objective optimizations . Moreover, they are simple to use and to combine with existing simulation codes without significant modifications and their efficiency is independent of the nature of the problem (i.e. linearity, type of design variables, type of constrains, discontinuities of the search space, etc.). Genetic algorithms can be easily applied to multi-objective problems due to their intrinsic parallelism. Examples of application of multi-objective genetic algorithms in the design of an hybrid electric road vehicle can be found in Hu et al.  and Donateo et al. . In the previous work of the author in the automotive field , the usage of Multi-Criteria Decision Making was also proposed for the choice of the final configuration from the multi-dimensional Pareto front. In the present investigation, the choice of the final solution was performed with an elimination-by-aspect procedure thanks to the user friendly displaying tools available in Modefrontier.
THE AIRCRAFT MODEL
The investigation was performed in collaboration with an industrial partner that furnished essential details on the specification and the performance of the UAV otherwise quite impossible to find. Because of a confidentiality agreement the performance of the aircraft will be given in a dimensionless way in the present investigation.
The UAV has an aspect ratio ([b.sup.2]/S) equal to 12 and a wing load [W.sub.0]/S=815.4 N/[m.sup.2] (where b is the wing span, S the wing area and [W.sub.0] the takeoff weight). A typical mission of the aircraft in its surveillance task was drawn from the information given by the industrial partner (available data) and is shown in Fig. 1. in arbitrary units. The mission is defined in terms of time histories of altitude, true air speed and auxiliaries power request.
The aircraft was modelled with a backward approach with a in-house simulation software named PLA.N.E.S. The simulation approach is briefly explained here. Detailed information about the code and its validation can be found in . The mission is discretized with a time-step of 60s, except for the takeoff and landing phases where a time-step of 1 s is used. The time histories of speed (V) and altitude (z) are used to calculate, at any time step, the thrust to be delivered by the powertrain. The thrust is obtained by considering the balances of the forces acting on the aircraft in the lift (L) and drag (D) directions :
L = W cos [gamma] - [W/g] V d[gamma]/dt = 1/2[c.sub.L][rho]S[V.sup.2] (1)
D = T - W sin [gamma] - [W/g] [d[gamma]/dt] - [mu]W = 1/2[c.sub.D][rho]S[V.sup.2] (2)
Where W is the instantaneous weight of the aircraft, T the required thrust, [c.sub.L] the lift coefficient, [rho] the density at the current flight level, g is the acceleration of gravity, S the wing area and [gamma] is the angle of climb (sin [gamma] = V/[dz/dt]).
The rolling force [mu]W depends on the runaway through the friction coefficient [mu] and is zero in flight while the gravitational force W sin [gamma] is zero at land and in steady flight.
The following drag polar was assumed for the UAV 21:
[c.sub.D] = 0.025 + 0.0279[c.sub.L.sup.2] (3)
The thrust obtained from (2) is used to calculate the thrust power THP at any time step as:
THP = T x V (4)
The thrust power is generated by the propeller that is connected to a planetary gear box. The gear box is used to perform the power-split between the engine and the electric machine. The flows of mechanical, electric and chemical energy in the proposed powertrain are shown in Fig. 2. The engine receives chemical energy from a fuel tank whose size is not changed in the optimization. The motor/generator is linked to a battery whose correct specification is critical for the performance of the aircraft.
At any time step, a supervisory controller has to decide among the following operational modes:
1. Thermal (the engine produces all the power required by the propeller);
2. Electric (the propeller shaft power is generated by the motor using the battery as only energy source);
3. Charging (the engine generates the power to move the propeller and to charge the battery while the electric motor works as a generator);
4. Power-split (both the engine and the motor generate mechanical power that is delivered to the propeller).
In the present investigation the following rule-base strategy was used. Mode 1 is used when the batteries' State of the Charge (SOC) is below a certain limit (SOCmin) and the engine is not able to charge the batteries. Mode 2 is used when the SOC is above the SOCmin and the electric drive is able to sustain the flight without the usage of the engine. In the case of modes 3 and 4, there is a further degree of freedom in the amount of power to be generated by the engine. This power depends on the discharge and recharge currents of the battery along the mission that are the only two parameters of the supervisory controller included in the optimization. If the engine and/or the batteries are not able to sustain the power request with the available SOC, the software returns an error flag. In the present version of the code the errors are summed up in the mission to quantify the ability of the proposed powertrain to follow the pre-loaded mission. A correction will be considered in the future to take into account the lack of power as in the automotive simulation code Advisor . The electric power is assumed to be negative when the battery is in charge. The battery state of charge is allowed to vary between 20% and 90%. The SOC limits were implemented because, in operating conditions, charging up to 100% or discharging up to 0% can involve to a plating of electrodes with consequent inhibition of active materials and irreversible loss of the internal capacity [22,23].
The outputs of the supervisory control strategy are:
1. The speed and the torque of the engine;
2. The current to/from the electrochemical storage system.
In the present investigation the gearbox is modelled as a simple mechanical power split device with a mass of 20kg and a efficiency of 0.9. The sizing of the gear boxes has not been yet completed for the difficulty to retrieve literature data on mass and volume of the gearbox as a function of the nominal power and speed of engine and motor. Also the propeller is modeled as a black box with the following values of efficiency: 0.65, 0.7 and 0.8 at takeoff, climb/descent and cruise, respectively. The mass of the propeller is also assumed constant for all the designs. Scaling methods are considered for the most critical components of the hybrid powertrain, namely the engine, the motor and the battery.
Scaling the engine
The scaling parameter of the engine is assumed to be its nominal power. According to the nominal power, the model calculates the displacement, the nominal speed, the mass, the volume and the efficiency of the engine. The efficiency is obtained by using sea-level and altitude performance maps of a two-stroke diesel engine with a nominal power or 128kW . The maps contain the torque and the specific fuel consumption (bsfc) of the engine for different values of speed and Air-Fuel Ratio (AFR) with AFR=18 corresponding to the full load conditions. Fig. 3 shows the performance maps of the reference engine at 0 and 3000m. More details on the numerical simulation of the diesel engine used to obtain the maps can be found in .
To adapt the maps to a different engine size, the torque was scaled with the displacement and the engine speed with the stroke thus obtaining maps of Brake Mean Effective Pressure (BMEP) versus mean piston speed. This scaling method is widely used in literature for the design of hybrid electric powertrains . Recently Sorrentino et al.[ 25] validated the model by scaling a gasoline engine from 38.5 to 25kW and a diesel engine from 102.6 to 67.2 kW. When increasing the engine power, the nominal speed of the engine is decreased to preserve the piston mean speed, while the displacement is increased to achieve the same mean effective pressure as shown in Fig. 4.
The mass of the scaled engine can be calculated with linear correlations as proposed in literature . For four stroke diesel engines, the mass [M.sub.ICE] and the specific fuel consumption at takeoff (bsf[c.sub.t,o] are a function of the engine brake power:
[mathematical expression not reproducible] (5)
[mathematical expression not reproducible] (6)
where H[P.sub.ICE] is the brake engine power in Horse Power. These correlations  despite being based on a report of the NASA published in 1981, correctly predict the fuel consumption of the 128kW engine at 2000rpm and AFR=18 (presumed takeoff conditions) while the mass is overestimated. For the two-stroke engines used in the present investigation, the power to mass ratio of the engine will be assumed equal to 1.0 kW/kg as for the 128kw engine. The volume of the engine was assumed independent of its nominal power.
To take into account the effect of engine size on its maximum bsfc, the efficiency maps of Fig. 3 could be corrected by applying eq. (6) to every engine operating point. In the case of the 80kW engine the corrected bsfc would be about 3% larger than the bsfc of the 128kW engine. For this reason, the effect of engine size on the nominal bsfc was neglected in the present investigation but will be taken into account in the after retrieving data of modern diesel engines.
Scaling the Electric Path
The design of the battery is performed according to three parameters:
* The type of battery (1 or 2);
* The nominal capacity (Cnom);
* The number of elements in series (Nser)
Two technologies were considered for the batteries with a trade-off between higher energy density (battery 2) and higher power density (battery 1) . Their specifications are reported in of Table 1.
The following model proposed in literature for automotive applications  is used for the battery.
At any time step, the supervisory control decides the power of the battery ([P.hatt]). The corresponding battery current is calculated as:
[mathematical expression not reproducible] (7)
The open circuit voltage [V.sub.OC] is calculated according to the State Of Charge of the battery at the previous time step (SOC(t-[DELTA]t)) as:
[mathematical expression not reproducible] (8)
R, E0, A, Kb and B are parameters of the model  that depend on the battery technology 27. Finally, the SOC at the current time step is:
[mathematical expression not reproducible] (9)
[C.sub.nom] is the nominal capacity of the battery, i.e. the capacity when the battery is discharged at 1[C.sub.nom] ampere. The actual capacity depends on the instantaneous discharge/recharge current. This affects the calculation of SOC. A correction will be considered as further investigation using the Peukert model  to account for the effect of current on the actual battery capacity.
The nominal power of the motor is calculated from the maximum discharge current of the battery and the selected bus voltage. The motor speed was set equal to 10000. Together with the selected cooling method ID (1 : natural convection, 2: forced air convection, 3: forced liquid convection), the nominal torque is used to predict the efficiency, mass and volume of the electric drive (including the motor and the inverter) using correlations derived from literature for permanent magnet motors. The sizing procedure is described in .
Outputs of the Model
According to the specification of engine, motor and batteries, the software calculates the takeoff weight [W.sub.0] and the overall volume of the powertrain. The takeoff weight can be used to calculate some design parameters as the required takeoff distance, the maximum climb rate at a certain altitude and speed, service ceiling, etc. The takeoff distance is an important performance parameter that was included in the optimization. It is calculated in PLA.N.E.S. using the well-known relations of flight mechanics and the specification of the aircraft .
The electric endurance of the powertrain is another important design parameters for the UAV analyzed in the present investigation. The electric flight time at an altitude where air density is p, can be calculated as proposed in literature [31-32] by:
[mathematical expression not reproducible] (10)
Where Rt is the battery hour rating (in hours), [n.sub.b] is the Peukert coefficient of the battery (see ). [[eta].sub.tot] is the overall efficiency of the electric path, V is the bus voltage. K is the lift induced parameter of the drag coefficient. It is equal to 0.0279 in the present investigation (see eq.(3)).The bus voltage is the product of the rated voltage of the battery (which depends on the battery type as in Table 1) and the number of battery elements in series (a design parameter). [U.sub.E] is the maximum endurance speed  calculated as:
[mathematical expression not reproducible] (11)
At any time step, the engine working point is chosen by the supervisory controller. This corresponds to an instantaneous fuel mass flow rate [m.sub.f].that is calculated from the engine maps.
The weight of the aircraft W is upgrades at any time step by integrating the fuel consumption as follows:
[mathematical expression not reproducible] (12)
The initial state of the carge of the battery (SOCin) was assumed equal to 100% (battery fully charged at the beginning of the mission). The final SOC, SOCfin, is an output of the simulation depending on both the size of the component and the supervisory control strategy.
Since SOCfin is necessary lower than 100%, a certain amount of fuel needs to be burned at land to recharge the battery. In the present investigation, the fuel required to recharge the batteries was estimated by assuming an overall efficiency of 0.32. This value was summed to the on-board fuel consumption to obtain the total fuel consumption M[f.sub.tot].
M[f.sub.tot]=Mf +[EE]/0.32 x LHV (13)
Where EE is the electricity consumed in the mission:
EE = 3600 xx [[SOCin-SOCfin]/100] CnomVbus[J] (14)
With [C.sub.nom] in Ah, Vbus in V. LHV is the lower heating value of the diesel fuel measured in J/kg.
The optimization was performed by changing the values of seven the design parameters reported in Table 2 together with their ranges of variation and steps.
Note that a full factorial of the design parameters with the proposed step would require about a billion of evaluations with the PLA.N.E.S. code. Using a time step of 60s, each evaluation with PLA.N.E.S. requires only 20s on a i7-4770 CPU working at 3.4GHz, but the computational load for a full factorial would be unfeasible.
The optimization was aimed at:
* Minimize the overall fuel consumption Mf_tot;
* Maximize the electric endurance E;
* Minimize additional volume Add_volume;
* Minimize the runaway length Runaway_lenght.
A baseline case consisting of a conventional piston-prop configuration with a nominal engine power of 100 arbitrary units (a.u.) was used as reference. The baseline values for the fitness functions are Mftot=100 a.u., E=0, Add_volume=100 a.u.; Runaway_lenght=100 a.u. The additional volume of the baseline case is assumed as the volume of the secondary fuel tanks that, according to the industrial partner, could be replaced by the batteries with a minimum impact on the aircraft architecture.
The capability of the selected powertrain in completing the proposed mission is not guaranteed in the backward approach used in PLA.N.E.S. Since an error message is generated at each time step when the available power is lower than the required power, the number of errors in the mission can be used as a constrain to eliminate unfeasible designs.
Moreover, the proposed charging and discharging currents of each design are compared with the limits of Table 1. If they are beyond the limits of that corresponding battery type, the design is treated as unfeasible.
The Optimization Algorithm
The optimization was performed with the MOGAII genetic algorithm  implemented in the Modefrontier optimization environment  that was also used to display the results. The optimization was deliberately performed with the values of the parameters proposed by the software. They are reported in Table 3.
The total number of non-duplicated designs was 8607. The standard design table of Modefrontier contains all the designs analyzed by the software during the optimization, including the initial design of experiment that will be denoted later as "generation 0". The 8607 designs are shown in Fig. 5. a) as a bubble chart where x axis shows the electric endurance E in hours while the y axis is the total fuel consumption Mf.sub.tot].
The feasible designs (31% of the total) are represented as circles to distinguish them from unfeasible (asterisks). The diameter of the bubbles is proportional to the additional volume of the powertrain. The color scale represents the runaway length.
The minimum fuel consumption (74 a.u.) corresponds to an endurance of only 10 minutes while the maximum endurance (1.3 h) is achieved with a fuel consumption of 95 a.u. To achieve such an endurance, it is necessary to increase the additional volume up to 600 a.u., that is 6 times the available space.
The Pareto front of all the feasible designs found at the end of the optimization is reported in Fig. 5. b) with a similar chart. It contains 960 designs (about 10% of the total) and was used for the final choice of the powertrain as explained in the next paragraph.
Choice of the Optimal Hybrid Electric Powertrain
When the number of input parameters and goals to be achieved is very high, the choice of the final solutions among the multi-dimension Pareto front can be so difficult to require the use of Multi Criteria Decision Making (MCDM) tools . In the present investigation, the choice has been made using the "elimination by aspect" approach performed together with an industrial decision maker. First, the designs with endurance lower than 30minutes were eliminated. Then, those with fuel consumption higher than 100 a.u. (baseline value). Four configurations were chosen corresponding to the edges of the remaining Pareto front. They will be called:
* Best fuel consumption;
* Best endurance;
* Best volume;
* Best performance;
The "best fuel consumption" design allows the largest reduction of fuel consumption and guarantees an endurance of 40 minutes. The required additional volume is 1.9 times the available space. The fuel consumed on board is 79% of the baseline configuration. Considering the fuel required to charge the battery on land (4.2 a.u.), the improvement in fuel consumption is 16%. This powertrain is characterized by an additional volume 3.34 times the initial diesel engine. The runaway length required for takeoff is about 16% longer than the baseline case. The "best endurance " design allows the longest endurance (1.2 hours) with an additional volume 5.5 times the available space and the same total consumption of the baseline case. The on-board fuel consumption is 9% lower than the baseline case while the total fuel consumption is the same.
The "best volume" uses only the available secondary fuels tanks volume for the batteries. The improvement in fuel consumption is 11.8% on board and 11.2% in total.
The "best performance" is the best configuration in terms of runaway length, allows an electric endurance of 31 minutes and reduces fuel consumption by 15.9%. The additional volume is 1.6 the available space.
The specification of the selected designs is reported in Table 4. As for the size of the engine, all the design, except for the "best volume", use a largely downsized engine with respect to the baseline case. In particular, in the "best fuel consumption" the nominal power of the engine is about 50a.u. i.e. half the nominal power of the baseline non-hybrid case.All the selected designs use battery type 1 that has lower energy density than type 2 but higher power density. The "best fuel consumption" design uses free cooling that corresponds to the better efficiency. The air forced cooling is chosen in the other cases because it is a compromise between efficiency and volume. The nominal capacity of the battery is a key issue to optimized the endurance. The "best endurance" design is characterized by the maximum allowed value of [C.sub.nom]. The "best volume" configuration was selected as final configuration because it has the lowest impact on the aircraft architecture.
ANALYSIS OF THE OPTIMAL HYBRID ELECTRIC POWERTRAIN
The plots of Fig. 6 show the time history of power along the mission in arbitrary unit with 100 a.u. corresponding to the takeoff power of the baseline configuration. In particular, the black line shows the power requested at the propeller axis to generate the thrust demand along the mission. The magenta line represents the maximum power that the diesel engine can generate according to its nominal size and to the actual altitude. Note that it is constant as long as the aircraft fly at less than 3000m, then decreases.
The red line represents the power generated at any time by the engine. This power is zero when the powertrain work in electric mode. The blue line is the power delivered by the motor/generator while the green line is the battery power. In electric mode, the motor power is equal to the requested power while the battery power is slightly higher because of the efficiency of the electric path. Likewise, the battery power is, in module, lower than the motor power in recharge. Note that, in all operating mode, the batteries also deliver the electricity to the auxiliaries.
The "powersplit" mode is used at takeoff to fully exploit the available power of both the motor and the engine. The climb is started in "thermal mode" and continues in charge mode to recharge the batteries. In the main part of the mission (inbound/loiter/outbound) the supervisory control strategy shifts to "charging" when the batteries reach the lowest limit of SOC (20%) and then again to "electric" when the batteries achieve the maximum allowed SOC (90%). During descent, the "electric mode" is used because of the low power request. The peak of power request at landing is again satisfied using the "powersplit" mode.
To explain the improvement in the fuel consumption obtained with the optimization, the working points of the engine are shown in Fig. 7 for three powertrains:
* The baseline non-hybrid powertrain (a);
* A non-optimized powertrain (b);
* The "best volume" powertrain (c).
In the baseline case the engine works at its best efficiency (0.33) only during takeoff. At loiter, the efficiency is around 0.26 and in descent is very low (0.18) because the power request is low during this phase of the flight. The non-optimized powertrain has the same engine of the baseline case but, thanks to the addition of the electric path and the on-off strategy, efficiency is improved. The first segment of climb is in thermal mode. Then, the supervisory strategy selects the electric mode but the battery undergoes a fast discharge generating a lack of power during the second climb that causes an error in the simulation. In fact, this design belongs to the unfeasible results of Fig. 5. During the surveillance task (loiter), the engine works with an efficiency of 0.32, higher than in the baseline case. In the "best volume" case the engine works always at its best efficiency.
Performance of the Algorithm
As already explained, the optimization was performed with the optimization algorithm already implemented in the Modefrontier environment and without changing the parameters of the algorithm.
The commercial software is very user-friendly and allowed a clear representation of the overall results of the optimization and an easy choice of the final configurations. Moreover, the optimization was successful from an engineering point of view because it was able to find a design that satisfies the targets of the industrial partner. On the other, the commercial software does not allow the measurement of the performance of the selected optimization algorithm. For this reason, the results of the optimization were extracted any 10 generations and post-processed outside the optimization software. The term "all" in the following analysis will be used to refer to the overall results of the optimization. Generation 0 was the initial random design of experiment.Four metrics proposed by Ishibuchi et al.  for a many-objective maximization problem were used:
* M1 : The percentage of non-dominated individuals;
* M2: The maximum sum of the objective values (maxSum);
* M3: The sum of the maximum objective values (sumMax);
* M4: The sum of the ranges of the objective values (sumRange).
To apply the metrics to the present case, the optimization problem was reformulated as follows:
[[Maximize [F.sub.1]] = [min[M.sub.f tot] / [M.sub.f tot]]] (15)
[Maximize [F.sub.2] = [E / Max(E)]] (16)
[Maximize [F.sub.3] = [min](add_volume) / [add_volume] (17)
[Maximize [F.sub.4] = [min](runaway_lenght) / [runaway_lenght] (18)
The minimum and maximum in the above equations were calculated over the whole set of feasible designs In this way, all the fitness functions have a maximum equal to 1. The results of the application of the first metric to the generations are shown in Fig. 8. Considering all the designs analyzed in the 100 generations (all), the unfeasible, feasible and Pareto design were 59%, 31% and 10%, respectively. Along the mission, the number of unfeasible designs was almost equal to the population size (100%) at the beginning, then rapidly decreased to about 60% the population size and remained constant up to generation 60. Finally, it started to decrease again. The percentage of dominated feasible individuals was always very low. The analysis of the other metrics was performed only for the feasible solutions and the results are shown in Fig. 9. To better analyze the results, the maximum and minimum values of the four fitness functions are plotted in Fig. 10, where the markers represent the maximum values and the error band the range of the same variable. Considering the whole set of solutions, each function reaches 1, i.e. the maximum value. Along the generations, each function reached 1 in a different generation.
In particular [F.sub.3] and [F.sub.4] reach 1 at generation 1, [F.sub.1] at generation 14 and [F.sub..sub.2] at generation 55. The sumMax metric is equal to 4 when considering all the design. This metric measures the convergence toward the Pareto front  around its edges. The values of this metric is expected to increase with generations. Instead it showed an oscillating behavior in the first 50 generations and was almost constant for the rest of the optimization performed with Moga-II.
This metric is obtaining by summing the maximum values of the four fitness functions shown in Fig. 10. Along the generations, the maximum values of [F.sub.1] and [F.sub.4] were almost constant. The maximum value of [F.sub.2] tend to increase up to generation 80, and then slowly decreased. On the contrary, the maximum value of fitness function [F.sub.3] decreased with generation even if the algorithm should maximize it. This can have seriously affected the "engineering" optimization since [F.sub.3] is related to the additional volume, the ultimate target for the decision maker.
The metric M2 (maxSum) measures the convergence toward the Pareto front  around its central region and should also increase with generation while it is almost constant in Fig. 9. As for the range (Fig. 10), it is higher for [F.sub.2] (endurance) and [F.sub.3] (additional volume). There is less variability for [F.sub.1] and [F.sub.4]. For all the four fitness functions, the range tend to decrease with generations. This results in a lower diversity of the last generation. In fact the "sumRange" metric decreased with generation.
Summing up, along the generations the MOGA-II algorithm evolved towards a better definition of the Pareto front (the number of non-dominated individuals increased) and a lower diversity of the population (reduction of sumRange). However, the convergence versus either the edges or the center of the Pareto front is not clearly visible with the proposed metrics. In particular, the convergence of the algorithm was unsatisfactory with respect to the fitness function [F.sub.3] whose maximum value decreased with generation. The values of the metrics calculated over the whole set of solutions are very different from the results found with the last generation (100). Fortunately the decision maker was allowed to pick the solution from the overall Pareto front and not from the last generation only.
Similarly to the approach used for the fitness values, the input variables were scaled so to have a maximum value equal to 1.
The maximum and minimum values of each variable were calculated any 10 generations and displayed in Fig. 11. Again, the markers represent the maximum values while the error bars are the ranges of variation. The nominal power of the engine converged toward the lowest feasible value. Both the discharge and recharge currents tend to increase with generation but with a consistent variability (their error bars are quite large). The motor cooling ID and the number of battery elements in series also contributed to the diversity of the final generations. Note that the optimization converges towards a nominal capacity of 100 Ah and a battery type equal to 1, i.e. towards the "best endurance" design (Table 4). All in all, the MOGA-II algorithm seems to give the highest weight to the optimization of the electric endurance and the lowest weight to the minimization of the additional volume. This is probably due to the different order of magnitude of these goals and could be solved by re-defining the fitness functions used in the Modefrontier optimization. The usage of a commercial software limited the possibility to further understand (and improve) the behavior of the genetic algorithm. As a further investigation, the same problem will be solved with other (open-source) multi-objective algorithms .
The design of a hybrid electric powertrain for an unmanned aerial vehicle was performed with a many-objective optimization of the components of the powertrain and the energy management strategy. The design variables included in the present investigation were the size of the piston engine, the cooling method of the motor and the specification of the batteries. The mass and the volume of the main components of the powertrain were calculated according to its specification.
The optimization was aimed at maximizing electric endurance, minimizing fuel consumption, minimizing the powertrain size and maximizing performance (measured through the takeoff run ground). Accordingly, the outcome of the optimization was a four-dimension Pareto front. Four powertrains were selected from the Pareto front, analyzed in details and compared with a baseline configuration consisting of a simple piston-prop configuration. The optimization was successful from an engineering point of view because it was able to find a design that satisfies the targets of the industrial partner. The optimal design was analyzed in detail.
The optimization was performed with a multi-objective genetic algorithm available in a commercial optimization software. The results were post-processed to analyze the evolutionary performances of the algorithm using four metrics proposed in literature. Along the generations, the optimization algorithm evolved towards a better definition of the Pareto front and a lower diversity of the population. However, the convergence versus either the edges or the center of the Pareto front was unsatisfactory, particularly with respect to the additional volume. The usage of commercial software limited the possibility to understand and improve the behavior of the genetic algorithm. As a further investigation, the same problem will be solved with other multi-objective algorithms.
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Investigation funded by the Italian Ministry for Education, University and Research (project code PON03_00067_8) and part of a research project of the Aerospace Technological District (DTA-Scarl).
Teresa Donateo and Antonio Ficarella University of Salento
Table 1. Specification of the batteries Parameter Battery 1 Battery 2 Rated Voltage [V] 3.7 3.6 Max Voltage [V] 4.2 4 Cut-off Voltage [V] 2.7 2.8 Max continuous current in 5[C.sub.nom] 2.2[C.sub.nom] discharge [A] Peak current [A] 10[C.sub.nom] 5[C.sub.nom] Max continuous current in 3[C.sub.nom] 1[C.sub.nom] charge [A] Energy density [Wh/kg] 134.5 149 Energy volumetric density 140 172 [k\Vh/m3] Power density [kW/kg] 0.67 0.33 Table 2. Range of variation and steps of the design parameters Variable Unit Min Max Step Nominal Engine kW 50 150 2 power Motor cooling ID - 1 3 1 Battery ID - 1 2 1 Battery nominal Ah 10 100 5 capacity [C.sub.nom] Battery elements - 10 100 1 in series Discharge current A 10 210 5 Recharge current A 10 210 5 Table 3. Parameters of the Moga-II algorithm  Variable Value Size of the population 100 Max number of generation 100 Probability of crossover 0.5 Probability of mutation 0.1 Table 4. Specification of the selected designs Design best best fuel best best endurance cons. performance volume Nominal Engine [a.u.] 59.9 50.5 69.4 113.6 power Motor cooling ID 2 1 2 2 Battery ID 1 1 1 1 Battery nominal [Ah] 100 50 45 40 capacity Battery elements in 91 88 78 92 series Battery discharge [A] 125 100 115 50 current Battery charge [A] 170 90 70 120 current
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|Author:||Donateo, Teresa; Ficarella, Antonio|
|Publication:||SAE International Journal of Aerospace|
|Date:||Sep 1, 2017|
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