Fuel Cell–Battery Cargo Drone: ECMS Power Split and Hydrogen Use

The controller follows the instantaneous ECMS principle of Paganelli et al. (2002) and Serrao, Onori and Rizzoni (2011), Eqs. 26–27. For each feasible candidate it minimizes $J=\dot m_{H2}+s(z)V_{oc}(z)I_b/LHV$, where discharge current is positive and $s(z)=\mathrm{clip}[s_0+k_z(z_{goal}-z),0.05,12]$. Chemical battery power, rather than terminal power, includes the effect of the electrical losses on charge withdrawal. Negative chemical power receives a charging credit. A bounded search uses up to fourteen trials, with bisection and safeguarded secant interpolation between feasible brackets. Only a feasible full mission may become the best trial or stop the search. The early-stop SOC error is 0.0002; a feasible fit within 0.002 is accepted. Failure to find a feasible fit is reported as a tuning failure, rather than proof of insufficient hardware.

The comparison policy requests $P_{load}+P_{bias}+K_z(z_{goal}-z)$. Its requested endpoint power is tested over a complete ramp; if necessary, a feasible grid candidate nearest that request is selected. Its bias is tuned independently to the same SOC target. This is a simple charge-sustaining load-following rule.

Rotor demand uses the profile, induced and parasite terms of Zeng et al. (2019), Appendix A. Total disk area is the sum of rotor areas. Axial climb uses actuator-disk inflow; descending forward flight solves $v_i\sqrt{V^2+(v_z+v_i)^2}=T/(2\rho A)$ by 24 bisections. Shaft demand is $P_0[1+3V^2/U_{tip}^2]+k_iTv_i+Tv_z+\rho A_DV^3/2$. Take-off adds $m v_{climb}/t_{take}$ to thrust. Demand is calculated at interval boundaries and interpolated linearly within each interval. Negative shaft power is rejected because windmilling and regenerative conversion are outside this model.

The PEM fit is $v_c=E_0-a\ln(1+I/I_0)-r_cI+b\ln(1-I/I_{lim})$. Net stack power is $\eta_{dc}(N I v_c-P_{aux})$. Hydrogen supplied is $N I M_{H2}/(2F U_{H2})$; utilization includes mean purge loss. A 101-point map converts net power into current and flow on its increasing branch.

The battery obeys $V_b=V_{oc}-v_p-R_0I_b$, $\dot z=-I_b/Q$ and $\dot v_p=I_b/C_p-v_p/(R_pC_p)$ (MathWorks equivalent-circuit equations). Classical fourth-order Runge–Kutta integrates the two states. At every RK stage and at the updated endpoint, current is recomputed from the low-current root of $P_b=(V_{oc}-v_p-R_0I_b)I_b$. Current is not held constant as voltage changes.

Every candidate stack trajectory starts at the previous actual output and ramps linearly to its proposed endpoint, with $|dP_{fc}/dt|\leq R_{fc}$. Battery demand is the difference between bus demand and that same trajectory. The initial instant, all RK stages and the updated endpoint are checked against current, voltage, SOC and real-root limits. Infeasible candidate trajectories are excluded from selection. Hydrogen is integrated along the ramp by Simpson quadrature; charge withdrawal uses the RK4 current quadrature. Stored endpoint states and stage values have separate DC-balance checks. A rejected policy run stops at its first detected violation.

Defaults define a generic 25 kg cargo multirotor with a warm 72-cell stack and a 12 Ah battery. The polarization coefficients and battery fit are engineering assumptions, not a manufacturer's measured data; replace them with a fit at the intended temperature, pressure and aging state. The fitted no-load voltage absorbs operating-condition effects.

Rotor disks do not overlap; density, tip speed, mass, motor efficiency and component parameters stay constant. Quasi-steady thrust equals weight outside take-off; fuselage tilt is small. Descent retains cruise horizontal speed. Windmilling cases with negative shaft demand are rejected. Rotor regeneration and battery-converter loss are excluded. Initial battery polarization is zero. The stack stays on above minimum current; startup and shutdown fuel are excluded.

ECMS minimizes equivalent hydrogen over the tested ramp candidates. SOC feedback, a finite power grid and ramp constraints mean this is not a proof of a global mission minimum. The base equivalence factor and rule-based charging bias are fitted to the known mission. Feasible trials have priority; an SOC match from an infeasible run is never accepted. A failed finite search may reflect the grid, feedback policy or target as well as the source limits.

Default resolution uses at least six intervals per nonzero phase, nine uniformly spaced ECMS endpoint powers, and additional load-following and neutral-mean-power candidates. Intervals are shortened to at most 10 s and one third of the battery RC time constant; the default mission has 73 integration intervals. Reducing the phase and power grids keeps browser evaluation below a second in the measured default run. This costs switching resolution and can change small hydrogen differences. Increase the grids and reduce the maximum integration step to test convergence, especially near a battery limit.

The continuous stack ramp is resolved in every interval. SOC and polarization follow the coupled ODE with power-dependent current. Limits are checked at all Runge–Kutta stages and at stored endpoints, including both sides of a phase-demand change. These are numerical trajectory checks, not an analytic certification of every substep extremum. Flight demand is piecewise linear between aerodynamic samples; faster real demand changes require measured profiles and finer time resolution. The 8 s constant-power battery regression agrees with an independent SciPy DOP853 integration within 0.0001 V polarization and 0.000001 SOC.

Forward descent requires horizontal speed at least twice hover induced velocity, following the wake-convection limitation discussed by Johnson (2004). Negative shaft power is outside the nonregenerative domain, even when avionics would make net bus power positive. The model does not assess vortex-ring avoidance, handling qualities, approach, flare or landing. Forward-flight rotor theory is limited to advance ratio at most 0.3 and modest fuselage tilt.

An initial electrical violation is a direct source-limit failure before ramp support arrives. Later search failures are identified as policy/tuning failures. A feasible landing-SOC mismatch above 0.002 is flagged; compare hydrogen only with matched reserves. Published checks verify Faraday flow and hover inflow; the added regressions check numerical dynamics and previously reported failures. They do not validate a particular aircraft or determine dispatch fuel reserves. Playback only interpolates stored trajectories and does not rerun the ODE.