## 2 Methodology
### Optimization Parameters and Objective Function  
As mentioned in previous sections, this work aims to find a suitable combustion system configuration that improves the performance of the oxy-fuel concept. Therefore, eleven parameters associated with the combustion system were selected as variables, of which six factors are geometrical parameters related to the piston bowl definition (see Figure 4), four parameters are used to characterize the injection system (nozzle holes number, spray angle, injection pressure, and start of injection), and the final parameter is the swirl number to characterize the in-cylinder gas motion. The range of all parameters used as inputs for the optimization process are presented in Table 6. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Range** \\ \hline Geometrical parameter 1 [-] & [\(-\)0.5, 1.0] \\ Geometrical parameter 2 [-] & [\(-\)1.0, 1.25] \\ Geometrical parameter 3 [-] & [\(-\)1.0, 2.0] \\ Geometrical parameter 4 [-] & [0.0, 1.0] \\ Geometrical parameter 5 [-] & [\(-\)1.4, 0.1] \\ Geometrical parameter 6 [-] & [\(-\)0.5, 1.0] \\ Number of injector holes [-] & [7, 12] \\ Spray angle [\({}^{\circ}\)] & [155, 175] \\ Swirl number at IVC [-] & [1.0, 3.0] \\ Injection pressure [bar] & [1000, 2200] \\ SOI [cad aTDC] & [\(-\)25, 0] \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Inputs parameters and their ranges considered in the optimization process. To define the first particles, the Latin hypercube methodology was used to create 100 distinct particles that were equally distributed in the range of each input. Later, the results of these parameters were used as a database for the initialization of the optimization algorithm. In the next step, PSO starts the optimization process generating the 30 standard particles and 10 particles that explore the search space. The optimization algorithm will combine all the input variables in order to maximize the engine efficiency and, at the same time, minimize the soot emissions. Additionally, the algorithm should take into account two constraints: the maximum in-cylinder pressure (in order to ensure the mechanic integrity of the cylinder, established by the manufacturer), and the CO emissions, which should be lower than the value of the baseline case. To meet these requirements, the objective function \(OF\) is defined based on the relative importance of soot, efficiency, maximum pressure (\(p_{max}\)) and CO emissions against the baseline values. The mathematical definition of the \(OF\) is presented in Equation (6),  
\[ \begin{array}{l}OF=f_{1}(soot)\cdot coef_{\text{soot}}+f_{2}(\text{eff}) \cdot coef_{\text{eff}}\\ +f_{3}(p_{max})\cdot coef_{p_{max}}+f_{4}(\text{CO})\cdot coef_{\text{CO}}, \end{array}  \tag{6}\]  
where the \(coef\) indexes are coefficients used to adjust the equation in order of importance of each term. In this work, the main objective of the optimization process is to obtain a combustion system that minimizes soot emissions and the second objective is to obtain the better efficiency value possible. For these reasons, the coefficients accustomed to these values are higher and play a major role in the objective function. The choice of lower values for the CO and maximum pressure coefficients is due to these parameters being constraints of the optimization and they should have minor influence on the objective function. In addition, the objective of soot oxidation is aligned with the objective of CO and C\({}_{x}\)H\({}_{y}\) oxidation. The absence of N\({}_{2}\) fully breaks the trade-off between oxidation and reduction reactions, in order to eliminate pollutants within the combustion chamber. In spite of this, C\({}_{x}\)H\({}_{y}\) is not within the optimization function, and it will be checked that it is not needed, because the CO optimization will also cause an C\({}_{x}\)H\({}_{y}\) oxidation below the standard air combustion levels. The values of those coefficients were \(coef_{\text{soot}}=1\), \(coef_{\text{eff}}=0.75\), \(coef_{p_{max}}=0.1\) and \(coef_{\text{CO}}=0.1\).