## 3 Results and Discussion
### Computational Comparison: Real-Time Factor  
To compare the computational effort of the proposed models, the Real-Time (RT) factor is introduced. This parameter is defined as the ratio between the run-time of the simulation and the time span of the physical event. In this work, the physical event is represented by the NRTC, with a duration of 1238 s. Table 5 summarizes the evaluated RT factors for the engine models proposed. As can be seen, the detailed model performed the NRTC in 285,522 s (\(\approx\)79 h), with an RT factor of about 230. The simplified 1D model performed the same cycle in 59,312 s (\(\approx\)17 h), with an RT about of 50, whereas the map-based model was in 1161 s, with an RT of 0.9. It is worth underlining that all of the simulations were performed on a workstation with an Intel(r) i7 processor @ 3.8 GHz and dedicated GPU to improve run-time simulation.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model Type**} & **Cumulative NO\({}_{\rm x}\)** & \(\Delta\)**NO\({}_{\rm x}\)** & **Cumulative Fuel** & \(\Delta\)**Fuel** \\  & **Emissions (g)** & **(\%)** & **Consumption (g)** & **(\%)** \\ \hline Detailed 1D & 238 & / & 14,500 & / \\ Simplified 1D & 260 & +9.2 & 14,600 & +0.7 \\ Map-based & 267 & +12 & 14,036 & \(-\)3.2 \\ \hline \hline \end{tabular}
\end{table}
Table 4: NO\({}_{\rm x}\) emissions and fuel consumption results.  
Figure 10: Brake specific NO\({}_{\rm x}\) (bsNO\({}_{\rm x}\)) Comparison.  
As shown in Table 5, the RT factor of the more detailed modeling approach is the highest. As expected, the simulation run-time rises. This is due to the computational demand necessary to solve the one-dimensional equations in the ducts per each timestep. Although the simplified 1D model allows for a larger timestep compared to the previous case by adopting a predictive combustion model, it is not adequate for HiL applications due to its overly high RT factor. The map-based model is characterized by an RT factor near the unit, so the simulation duration is similar to the physical event. In terms of the simulation run-time, the map-based model is roughly 250 times faster than the detailed model and about 50 times faster than the fast-running model performing NRTC. In this modeling approach, applying an integration timestep of 10\({}^{-2}\) s, the evaluated RT factor is 0.09. Thus, the map-based model can be considered inexpensive in terms of computational requirements.  
These results can be considered coherent with the literature findings in engine simulation in terms of simulation run-time. In particular, Figure 11 displays an RT comparison among different engine models, classified according to their level of detail and speed simulation.  
As expected, 3D approaches can be considered time expensive due to their well-known reputation in terms of accuracy [29]. The detailed 1D models can be characterized by different levels of details, and thus, the computational run time depends on it [30]. The RT factor and the accuracy of the simplified 1D models can be considered acceptable. As briefly discussed above, with the proper calibration, these models can perform a simulation more quickly, and thus, can also be suitable for HiL application [31]. When simulation speed is the priority and high details are not required, map-based models can help to get an overview of the engine performance with a fast simulation time and an RT near the unit, or even faster than real-time, as reported in Table 5. However, a detailed physical description of the individual engine component cannot be incorporated in these kinds of models, and their accuracy depends on the map-building process [32; 33].