Proportional-integral adaptive relaxed-set robust model predictive control for range-extender energy management in extended-range electric vehicles

Extended-range electric vehicles pair a battery-electric drivetrain with a range-extending engine-generator whose dispatch strategy governs fuel economy and engine-cycling behavior. Fixed-parameter and nominal predictive strategies perform well near their tuning point but degrade under sensor noise or parameter variation, a gap existing approaches leave unaddressed. This work proposes a Proportional-Integral Adaptive Relaxed-Set robust model predictive controller, PI-ARS-MPC, coupling a reduced-order prediction model with a bounded receding-horizon search, an online proportional-integral-adapted equivalence factor, and relaxed-set constraint tightening absorbing forecast uncertainty without provoking infeasibility. Benchmarked against a classical predictive controller, a thermostat baseline, and an independently calibrated adaptive equivalent-consumption-minimization controller sharing the identical plant, the strategy is tested nominally and under thirty paired Monte Carlo trials at each of three disturbance intensities. Under disturbance, the proposed controller records fewer engine starts than the classical baseline in every one of the ninety paired trials, 5.9-to-6.1-fold lower at the means, though the raw, unpaired cycling ranges overlap at the highest intensity tested. A controlled ablation, evaluated at medium disturbance intensity over thirty paired trials, in which the tightening margin is forced to zero while every other mechanism is retained, reproduces the proposed controller’s behavior to within simulation precision on every trial examined, confirming that the adaptive equivalence factor, not the tightening margin, is responsible for this cycling advantage at that intensity; the tightening margin shows no measurable effect within the medium-intensity condition tested, a result that does not establish whether it would remain inactive at the low- or high-intensity conditions, and is better understood as protection against more severe or untested conditions than as a demonstrated source of the reported robustness. The thermostat baseline retains a fuel-economy advantage even after an independently calibrated, three-stage protocol reported in full in Section V-F replaces its default configuration, improving equivalent economy by approximately 5.9–10.2%, with one caveat disclosed rather than concealed: the improvement reverses at low disturbance intensity, traced to a reduced range-extender power margin. PI-ARS-MPC also finishes with a higher final state of charge than the thermostat baseline in every condition tested, 3.8-to-8.7 percentage points closer to the charge-sustaining target; because this reserved charge is purchased at the raw-fuel cost the thermostat comparison already reports, the two findings are presented together as a fuel-economy-versus-charge-retention trade-off rather than as an unconditional advantage. Proportional-integral equivalence-factor adaptation is accordingly identified as the specific, ablation-confirmed source of a measurable, statistically supported reduction in engine cycling at negligible nominal cost, compatible with embedded automotive controllers, with the relaxed-set tightening mechanism, fuel economy, and tail-event robustness identified as open trade-offs or design features awaiting further evidence rather than resolved advantages.