## 2 Methodology
### Computational Optimization Details  
Optimization Methodology  
The optimization process was carried out by using a novel approach based on the particle swarm optimization algorithm (PSO) [36]. One of the reasons for selecting this approach is because it offers a fast rate of the convergence to find the best solution. In addition, its implementation is simple and it is capable of solving problems with large search domain of candidates solutions. Nonetheless, a few drawbacks of the PSO are that the algorithm can be stuck in a local minimum not finding the global optimum, and that it has a strong sensitivity to meta-parameteric values. Although the PSO can execute, the algorithm requires some information about the particle that is being evaluated, including the position \(x_{i}\) in the search space and the velocity \(v_{i}\), according to these expressions:  
\[x_{i}(t+1)=x_{i}(t)+v_{i}(t+1), \tag{1}\] \[v_{i}(t+1)= w\cdot\beta\cdot v_{i}(t)+c_{1}\cdot\tau\cdot(p_{i}-x_{i}(t))\] (2) \[+ c_{2}\ \gamma\cdot(g-x_{i}(t)),\]  
where \(w\) means the inertia weight, \(c_{1}\) and \(c_{2}\) represent the individual and social weight respectively, and \(t\) means the iteration. Typical values for the inertia weight \(w\) are in the range of [0.5, 1.5] and the values of \(c_{1}\) and \(c_{2}\) vary in the range of [1, 3]. In addition, in the previous Equation (2) \(p_{i}\) means the current best position of \(x_{i}\) and \(g\) refers to the global best position obtained from all evaluated particles. The variables \(\beta\), \(\tau\), and \(\gamma\) are vectors where each element has a random value of a uniform distribution in the [0, 1] range. In this study, the PSO basic algorithm has been combined with a mathematical tool that is known as the novelty search concept, which improves the convergence issues of the PSO. This integration has been used in previous works with successful results [37]. The use of novelty search concepts is proposed to improve the exploration of the search space dividing the set of particles in two groups: a first family formed by standard particles,  
Figure 3: Comparison between the simulations of the reference diesel case against the baseline oxy-fuel combustion case, results of in-cylinder pressure (**top**), RoHR (**middle**) and normalized cumulative HR (**bottom**). which ruled the regular PSO equations (Equations (1) and (2)). The second family is formed by particles that \"explore\" all the search space, even the regions that provide bad results, which are not ruled by the best position information. This technique avoids that the PSO gets stuck in a local minimum. Then, all the particles generated by the algorithm are stored in a repository. It maintains all the information of the optimization process, so the new particles that explore the domain avoid the regions close to the particles already investigated. The mathematical definition of the repository is given by Equation (3):  
\[\text{MC }(t)=\frac{\sum\limits_{x\in\mathcal{R}(t)}x}{\text{card}(\mathcal{R}( t))}, \tag{3}\]  
where \(\mathcal{R}(t)\) is the repository in the iteration \(t\), \(\text{card}(\mathcal{R}(t))\) is the number of elements of \(\mathcal{R}(t)\), and \(\text{MC }(t)\) is the point that summarizes the behavior of the system in the iteration \(t\). In this expression, the \(\text{MC }(t)\) is defined to be analogous to a centre of mass. Finally, to adapt the PSO algorithm with the novelty search concept it is necessary to define a new velocity equation in order to guide the explorer particles, taking into account the information in the repository. Therefore, Equation (4) was defined to modify the particle dependency from the global best position to the new centre of mass data,  
\[ \begin{split} v_{i}(t+1)&=w\cdot\delta\cdot v_{i} (t)+c_{1}\cdot\phi\cdot(p_{i}-x_{i}(t))\ +\\ &+c_{3}\cdot\rho\cdot\exp\biggl{(}-\alpha\cdot\left|\frac{x_{i}(t )-\text{MC }(t)}{x_{max}-x_{min}}\right|\biggr{)}\cdot(x_{i}(t)-\text{MC }(t)),\end{split}