## 2 Materials and Methods
### Engine Characteristics and Modeling  
The engine used as a reference in this study is a 560 kW HD diesel engine, applied in railcars or Diesel Multiple Unit (DMU) applications. Specifically, it is a V8 direct-injection four-stroke engine, with a total displacement of 20 dm\({}^{3}\), turbocharged. The main engine characteristics are reported in Table 3.  
The engine is made up of two in-line four-cylinder banks with a bank angle of 90\({}^{\circ}\). Each cylinder bank is fed by a turbocharger system. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Stage** & **Date** & \multicolumn{6}{c}{**Emissions (g/kWh)**} \\ \hline  & & **CO** & **HC** & **NO\({}_{\mathbf{x}}\)** & **PM** & **PN** \\ \hline IIIB & 01.2011 & 3.50 & 0.19 & 2.0 & 0.025 & - \\ IV & 01.2014 & 3.50 & 0.19 & 2.0 & 0.025 & - \\ V & 2021 & 3.50 & 0.19 & 2.0 & 0.015 & \(1\times 10^{12}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Evolution of the emission standard limits for non-road diesel engines in the power range 130–560 kW. Figure 1: NRTC: Normalized Speed; Normalized Torque. Figure 2: NRTC: Actual Speed; Actual Torque. To model the engine, the 1D modelling approach was used. In particular, the reference model was derived from a previous authors' study [26], where a simulation model of a railway diesel engine was developed and validated. This model was able to perform the NRSC ISO 8178 C1 homologation cycle in pre-defined stationary speed-load points. The reference engine model is defined, in terms of \"depth\" of detail, as a \"detailed 1D model\". In this modeling approach, the whole engine layout is discretized and modelled on the base of the real sub-components characteristics. In Figure 3, a scheme of the detailed model is depicted. As shown, the inlet and outlet conditions respectively refer to the ambient conditions and to the exhaust gases state on exit from the turbines. All ducts, pipes referring to the legend in Figure 3, are joined via orifice junction connections. The orifice junction can be used to specify a flow restriction by setting the orifice diameter restriction smaller than the diameter of the mating elements. Furthermore, the solver uses the smaller diameter of the joined elements if there is no restriction to take into account. In all pipes, the flow model is based on the solution of the one-dimensional continuity (1), energy (2) and momentum (3) equations [27] that characterize the wave propagation phenomena:  
\[\frac{dm}{dt}=\sum\dot{m} \tag{1}\]  
\[\frac{d(me)}{dt}=-p\frac{dV}{dt}+\sum(\dot{m}H)-hA_{s}\Big{(}T_{fluid}-T_{wall} \Big{)} \tag{2}\]  
\[\frac{d\dot{m}}{dt}=\frac{dpA+\sum(\dot{m}u)-4C_{f}\frac{\rho u|u|}{2}\frac{dxA }{D}-K_{p}\Big{(}\frac{1}{2}\rho u|u|\Big{)}A}{dx} \tag{3}\]  
This means that all of the quantities are averages across the flow direction. The cylinders have been modelled by means of the \"filling-emptying method\". In regards to turbomachinery, both compressor and turbine have been described as simple templates based on the thermodynamics equations. In particular, for the turbine, the output power is calculated using Equation (4). \[P_{t}=\dot{m}\frac{k-1}{k}RT_{int,T}\Bigg{(}1-\frac{1}{\beta_{t}\frac{m-1}{m}} \Bigg{)} \tag{4}\]  
In Equation (4), \(\dot{m}\) is the mass flow, \(R\) represents the gas constant, \(\beta_{t}\) is the expansion ratio, \(k\) is the gas heat specific ratio, \(T_{int,T}\) the temperature upstream the turbine and \(m\) the polytropic exponent. For the compressor, the mass flow rate \(\dot{m}_{c}\) and outlet temperature are calculated from the inlet temperature (\(T_{int,C}\)), compression ratio \(\beta_{c}\) and efficiency, starting from the input power \(P_{c}\) calculated through Equation (5), considering that \(P_{t}=P_{c}\).