 power output at each discrete-time step along the flight path. Although P drv,i depends on the aircraft mass m i, which is itself an optimisation variable, a prior estimate of the required power output can be obtained from (15) assuming a constant mass m i = m 0 for all i. The simulation results described in Section 5 show that this assumption has a negligible effect on solution accuracy.
We define g i (·) in terms of h i (·) as is necessarily a convex, non-decreasing, one-toone function if the lower bound on P em,i is redefined as since this bound ensures that h i (·) is a one-to-one nondecreasing convex function of P em,i.
The dynamic constraints (5), (6) and the power balance (11) can be expressed using (15), (16) and (17) as These constraints are nonconvex due to their quadratic dependence on the optimisation variables P gt,i, P em,i and m i. To convexify these constraints, we first eliminate P em,i from (19) and (20) using P b,i = g i (P em,i ) and P em,i = g −1 i (P b,i ). Then (19) becomes linear, E i+1 = E i − P b,i δ. Moreover, under the assumptions on g i (·) (convex, nondecreasing and one-to-one), the inverse mapping g −1 i (·) is a concave, increasing function (e.g. East and Cannon, 2018). Note that g −1 (·) is given explicitly as if κ 2,i > 0, and by at any time steps i such that κ 2,i = 0. Therefore, by relaxing the equality constraints in (18) and (20) to inequalities, a pair of convex constraints: is obtained since g −1 i (·) is concave and f i (·) is convex. With these modifications, and noting that the objective in (12) is equivalent to minimising m 0 −m N, the optimisation to be solved to determine the optimal power profile at the kth time step can be expressed as the convex problem: min where P b,i = g i (P em,i ), P b,i = g i (P em,i ), and the constraints are imposed for i = 0,..., N − 1. The form of the objective in (23) ensures that any feasible solution that does not satisfy the constraints in (21) and (22) with equality is suboptimal. Thus the solutions of (23) and (12) are necessarily equal if (12) is feasible.

NUMERICAL RESULTS
This section uses the optimisation problem (23) to construct an energy management case study involving a representative hybrid-electric passenger aircraft. Solutions of (23) were computed using the general purpose convex optimisation solver CVX (Grant and Boyd, 2008). Since the minimisation in (23) is convex, convergence of the solver to a global optimum is ensured.

Simulation scenario
The parameters of the model used in simulations are collected in Table 1. These are based on publicly available data for the BAe 146 aircraft. The propulsion system is assumed to consist of four gas turbines and electric motors, each with the hybrid-parallel configuration shown in Fig. 1 For the purposes of this study it is assumed that velocity and height profiles are known a priori as a result of the fixed flight plan entered prior to take-off. We consider The electric loss map coefficients κ i,2, κ i,1, κ i,0 can be estimated in two steps from these profiles. First, the shaft rotation speed, ω i (= ω gt,i = ω em,i ), is interpolated from a precomputed look-up table relating measured shaft rotation speed, altitude and drive power at a given Mach number (Fig. 5). Then, the coefficients are interpolated from a precomputed record of losses in the electric motor as a function of rotation speed. This procedure requires drive power P drv to be approximated a priori, e.g. by assuming constant aircraft mass for the duration of the flight. This assumption is supported by Figure 4, which shows that the electric map coefficients are almost identical for the estimated drive power profile and for the actual drive power profile computed retrospectively. We also find that the κ 2,i coefficients are negligible for all i.
The gas turbine fuel map used in this study is approximately linear (β 2,i ≈ 0 ∀i) and furthermore the fuel consumption does not depend significantly on shaft rotation speed. Therefore the fuel map coefficients are given in Table 1 as constants (i.e. β 1,i = β 1, β 0,i = β 0 ∀i). √ T in where T in = T 0 (h) + v 2 /2c p is the temperature at inlet of the fan, c p = 1000 JK −1 kg −1 is the specific heat of air at constant pressure and T 0 (h) is the temperature of air at altitude h.

Results
The mission is simulated with sampling interval δ = 10 s over a one-hour shrinking horizon by solving the optimisation problem (23) at each time step and implementing the first element of the optimum power split sequence as an MPC law. The closed loop energy management control strategy is shown in Figure 6, which gives the power split for a single coupled gas turbine and electric motor. Clearly the constraints on the gas turbine and electric motor power are respected. The evolution of the battery SOC and fuel consumption are shown in Figure 7. The upper plot illustrates that the constraints on SOC are respected and that the SOC reaches a minimum when the drive power becomes negative, as expected. The lower plot in Fig. 7 shows that, as expected, the rate of fuel consumption is greater during the initial climb phase when the gas turbine power output is high. The fuel consumption recorded for this simulation is F * = 1799 kg. In comparison, a fully gas turbine-powered flight with the same initial total aircraft weight has a fuel consumption of F gt = 2034 kg. We note however that this reduction is achieved at the expense of reduced available payload as a result of the weight of the electric components of the powertrain (battery storage and electric motors).
In order to evaluate the optimality of the power split solution, we compare it with the strategy of supplementing the gas turbine with the maximum electric motor power (P em ) until the battery is fully depleted, then switching to a sustaining mode in which only the gas turbine operates. In hybrid vehicles this is known as a Charge-Depleting-Charge-Sustaining (CDCS) strategy (Onori and Tribioli, 2015). Using this strategy the power split is as shown in Figure 8 and the fuel consumption is F CDCS = 1858kg.
To investigate the potential for windmilling (energy recovery when the net power demand is negative), the lower bound on electric motor power is set to P em = −2 MW, to allow transmission of power from the fan to the battery  results shown in Figures 9 and 10. The windmilling effect can be seen at the end of the flight and is characterised by negative electrical power and battery recharge.

Discussion
Intuitively, the optimal power split strategy might be expected to consume as much fuel as possible at the beginning of the flight so as to reduce the aircraft mass, and thus reduce the drive power needed during level flight and descent. However, the MPC strategy maintains an almost constant electric power over the whole flight (Fig. 6). This is explained by the relatively short flight duration and the characteristics of the aircraft model, as a result of which the change in total mass of the aircraft is relatively small (less than 5%). Despite this, the MPC strategy achieves a non-negligible reduction in fuel consumption (3.2%) over the CDCS strategy.
More radical optimal power split solutions are obtained if the change in aircraft mass during flight is more significant.
In particular, the MPC strategy allocates more electrical power at the end of the flight if the gas turbine fuel consumption is increased. For example, Figure 11 shows the power split solution for a situation in which the rate of fuel consumption is increased so that the change in aircraft total mass during flight is 15% (with all other simulation parameters unchanged). The fuel consumption for the CDCS strategy in this case is about 4% higher than that of the MPC strategy.
We next consider the case that the upper limit on the gas turbine power output is reduced to P gt = 3 MW. The MPC energy management strategy for this case is shown in Figure 12. Here the power demand is such that the gas turbine is at maximum power while the aircraft climbs. As a result, the electric motor is needed to meet the total power output requirement while the gas turbine power output is saturated. The fuel consumption for this scenario is increased slightly (by 0.1%) since the electrical power is mostly used at the beginning of the flight to compensate for the limit on the gas turbine power output.

CONCLUSIONS
This paper proposes a model predictive control law for energy management in hybrid-electric aircraft. The main contribution of the work is a convex formulation of the problem of minimising fuel consumption for a given future flight path. We provide a simulation study to illustrate the approach, and demonstrate that significant fuel savings can be achieved relative to heuristic strategies. The convexity of the formulation is crucial for computational tractability and is expected to be a basic requirement for verification by the aviation industry. Future work will consider the design of bespoke solvers. In particular, first order solution methods are expected to provide computational savings by exploiting the high degree of separability in the problem, while also being suitable