## 1 Introduction
The effect of the addition of hydrogen to a biogas was also investigated in the literature, particularly in spark ignition engines [22]. [PERSON] et al. [23] studied the combustion performance of H\({}_{2}\)/biogas in a spark ignition engine by numerical simulation; they showed that the rate of heat release increases while increasing H\({}_{2}\) content. [PERSON] et al. [24] used a quasi-dimensional multi-zone model to study the biogas/hydrogen combustion in a single cylinder engine; they concluded that the addition of increasing amounts of H\({}_{2}\) in biogas results in a decrease of the flame development period. [PERSON] et al. studied the performance and emission characteristics of biogas mixed with hydrogen in SI engine [25; 26]; they found that the presence of inert gases in the biogas can improve thermal efficiency and reduce NOx emissions; however, HC emissions and cycle variation increase. [PERSON] et al. investigated the effect of hydrogen enrichment on combustion and heat release of a biogas; they concluded that the overall reaction rate increases with H\({}_{2}\) concentration [27]. Internal combustion engines are operated very frequently at part load, especially in passenger car applications. In these operating conditions, engine efficiency is negatively affected by the combined effect of the increased relative incidence of rubbing friction, and in spark ignition engines, by intake air throttling losses [19]. An alternative to the conventional spark ignited combustion is the Controlled Auto-Ignition (CAI) combustion, allowing operation of the engine at part loads with considerable air excess, thus eliminating the need for throttling. CAI combustion is triggered by auto ignition instead of by spark ignition and subsequent flame front propagation [28]. Thanks to its characteristics, CAI combustion attracted some attention in the literature [29; 30; 31; 32]. It is characterized by the absence of a flame front propagation and by an extremely rapid evolution. This allows an efficiency improvement, limited however by the constraint that cylinder pressure rise rate cannot exceed certain values in order to prevent unacceptable mechanical and thermal loads [29]. Biogases were also used to fuel CAI engines and the influence of the amount of combustion products recycled on combustion timing and angular duration was investigated [30]; [PERSON] et al. [33] investigated the auto-ignition reactivity of natural gas with regard to the effects of intake air temperature and equivalence ratio; [PERSON] et al. [34] studied the effects of EGR ratio and engine speed on HCCI combustion; and [PERSON] et al. [35] improved the performance and extended the load range of hydrogen fuelled homogeneous charge compression ignition (HCCI) engine through charge temperature regulation and the addition of carbon dioxide in order to control the combustion phasing. Biogas, when used in spark ignition internal combustion engines, induces an unavoidable power reduction, due to the presence of inert gases. The authors here numerically investigate biogas suitability for fuelling Controlled Auto Ignition (CAI) engines that are operated in lean conditions. If CAI engines are fuelled with a biogas, a possible change of the equivalence ratio may allow for the avoiding of engine power reduction, which is always experienced in spark ignition engines. Similarly, the effect of the presence of hydrogen in the biogas fuelled to a CAI engine is here numerically investigated for the first time in the literature. The authors compare engine performance and exhaust emissions varying fuel composition.