Control optimization of air traffic emissions in a two-variable dynamic model

Abstract

Since 1999, every report released by the International Panel on Climate Change has advocated a decrease in the greenhouse gas emissions associated with aviation in order to preserve the current climate. This study used a two variable differential equations model with a non-linear control term to address several aspects of the emissions stabilization issue. By optimizing the control term parameter, several management alternatives can be obtained based on the properties of the phase plane of the model solutions, as identified by a stability analysis. The system can be stabilised around an equilibrium point that maintains the present number of passengers, or maintains the emissions level or is nearest to its present state. Each of these options entails different issues of growth or reduction in the number of passengers and/or the emissions rate, directly obtained from the model results. The last option seems especially novel and promising, since only short-distance flight passengers are severely reduced, while long-distance and international passengers are allowed to growth, and their associated emissions are reduced to below 50 percent of their actual value. Moreover, in a scenario of slow growth in air traffic, these rates could improve, with fewer reductions in the number of short-distance passengers.

Keywords: Air traffic emissions; Emissions stabilization; Flight types by distance travelled; Nonlinear feedback control; Optimization; Stability analysis; Two variable dynamic model.

Fig. 1

The portrait diagram with the asymptotes OO’ and OO”, the two equilibrium points O and E, the separatrix cycle O”O’ (depicted in blue), the two nullclines (depicted with a yellow and a red line respectively) and some relevant trajectories (depicted also in blue) for the S0 case optimized under constraint NE in the main scenario. The arrows indicate the direction of the phase diagram vectors along the nullclines.

Fig. 2

The $\beta$ parameter values obtained under the constraint PRF (marked by triangles), ERF (by squares), NE (by stars), for the different cases studied, in the main scenario. The dashed lines approximates the parameter values obtained for each constraint.

Fig. 3

The equilibrium points obtained under the three different constraints in the main scenario. Points obtained under constraint PRF are depicted with blue triangles, those obtained under ERF with red squares, the ones under NE with green stars. The initial state for each flight type is marked by black circles. The asymptotes found for the dynamical matrix of the short distance flights are depicted with a black line, for the long distance flights with a red line, and for international flights with a blue line. Continuous lines indicate the cases 0 and the dashed lines, the cases 1.

Fig. 4

(a) Variation of the emissions variable (in %) at the final equilibrium point obtained under constraints PRF (shaded blue) and NE (shaded green) in the main scenario. (b) Variation of the number of passengers variable (in %) at the final equilibrium point obtained under the ERF (shaded orange) and NE (shaded green) constraints in that scenario. (c) as in (a) but in the ’slow growth’ scenario. (d) as in (b) but in the ’slow growth’ scenario.

Figure 5

(a) The equilibrium points in the ’slow growth’ scenario (with $a_{11}^{\prime }=a_{11}/2$). Symbols and asymptotes as in Fig. 3. (b) As in (a), but focusing in S cases.