Optimization formulation

Optimal mitigation policies are formulated as the general nonlinear programming problem. Two formulations of optimal mitigation scenarios are proposed: one relying on minimizing cumulative emissions by supplying for trend demand, while changing energy carriers and aircraft within the global fleet, here called trend formulation; and other that partially avoids trend traffic and minimizes the relative ticket price increase in order to align to a target cumulative emissions, here called low-demand formulation.

The optimization variables consist of aircraft market penetration parameters (per aircraft and per market), energy mix parameters (shares of energy production per pathway per period), and, for the low-demand formulation, the supply cap parameters (share of avoided demand per market per period). Entry-Into-Service is bounded between 2035 and 2060 for all aircraft types, except for the first generation of conventional aircraft (which can be launched and deployed as soon as 2030), the energy mix variables between 0 and 1, and \(SR\) between 0 and 0.9 (at least 10 % of the trend demand is satisfied at each market).

The optimization constraints applied to all scenarios consist in limiting the sum of the control inputs among aircraft types for each market to 1, limiting the share of pathway production to positive values among each produced energy, and limiting the consumption of biomass and electricity to a fixed fair share of the global production. Also, in the case where scenarios add electric aircraft to the fleet, an extra constraint on aircraft energy consumption is added to ensure designs are feasible by EIS.

In the trend formulation the objective function is the cumulative CO2 emissions. In the low-demand formulation, cumulative emissions are constrained, and the objective is the time-discounted mean burden of demand aversion (Eq. 10).

Numerical optimization is performed with the SLSQP Algorithm [91], using forward-mode AD to obtain the gradients of objective and constraints. Forward-mode is chosen over reverse-mode, or backpropagation, due to the larger memory footprint when multiple constrains are involved (which is the case here) [92].

Robustness to background scenario

The robustness to the background scenario was also performed as a policy optimization problem. First a formulation must be chosen (results demonstrate the case for the trend formulation) and is performed in a multi-scenario simulation. The objective of the optimization problem is then considered as the mean of policy objectives under the scenario ensemble.

The optimization variables are then divided into two sets: one that is kept fixed among all scenarios (the aircraft mix variables), and one that may be specific to each scenario (the energy mix variables). This allows to ensure the robustness of the choice of aircraft mix while allowing for each scenario to individually optimize the energy production system in order to respect their respective availability constraints.