Coupled electrical-thermal-optical simulations of a high-power AlGaInP-based red light-emitting diode (LED) are reported and compared with detailed characterization data of the device available in literature. The simulations enabled identification of the most critical factors limiting the LED performance. Among them, the following ones are found to be of primary importance: (i) absorption of emitted photons in a p+-GaAs contact layer, limiting the light extraction efficiency; (ii) device self-heating producing thermally stimulated electron leakage into the p-side of the LED structure; and (iii) current crowding around small circular p-electrodes enhancing additionally the electron leakage. Possible room for efficiency improvement is estimated by modeling. Optimization of some structural units of the LED design is discussed as well as further directions of the simulation model improvements.