## 1 Introduction  
The environmental challenges that the world has faced in recent years have seen the introduction of stringent emission legislation in terms of emission reduction, which has resulted in advancements in internal combustion engine technology. Despite the introduction of electrification in the transport sector, internal combustion engines are still prevalent in various applications. Therefore, it is of high importance to continue decreasing harmful greenhouse gas emissions from internal combustion engines by developing and evolving low-emission combustion modes. One of the representatives of low-emission combustion is the pre-chamber spark-ignited (PCSI) combustion mode. Here, the decrease in fuel consumption caused by lower heat losses leads to a reduction in greenhouse gas emissions while reducing harmful nitrogen oxide (NOX) emissions as well. A system with a passive pre-chamber has limited ability to extend lean limit and reduce emissions in comparison with a conventional spark-ignited (SI) engine [1]. On the other hand, the active pre-chamber system with an additional injector in the pre-chamber achieves higher combustion stability at leaner mixtures when compared to conventional SI engines and engines with a passive pre-chamber system [2] and consequently lower fuel combustion and emissions. An active pre-chamber also enables better scavenging. In passive systems, the amount of residual gases that stay trapped in the pre-chamber depends only on the compression ratio (CR) and pressure during the compression stroke [3], while in an active pre-chamber system, optimal timing of fuel injection into the pre-chamber ensures better scavenging of the pre-chamber, which ensures a higher lean limit [4]. When injection timing is delayed, there is a larger pressure difference between the pre-chamber and main chamber (MC), which results in stronger jet energy, which improves the combustion process in the main chamber. Also, combustion duration is shorter when the active pre-chamber system is used in comparison to conventional SI engines [5]. Low-temperature combustion obtained when operating with lean mixtures ensures low emissions of nitrogen oxides (NOX), which mainly depends on the temperature of combustion. It has been proven that the PCSI combustion mode can achieve very low engine-out NOX emissions (NOX \(<\) 0.2 g/kWh at \(\lambda\geq\) 2.0), requiring no additional aftertreatment to satisfy current emission standards [6]. In [7], it was concluded that by using mixtures with \(\lambda\geq\) 1.6, the NOX emissions can be decreased by more than 90% compared to stoichiometric conditions. However, in those conditions, the total hydrocarbons (THC) and carbon monoxide (CO) emissions are increased because of incomplete combustion in some parts of the cylinder. According to [8], the lowest NOX emissions are achieved at \(\lambda\) = 2.2, while indicated specific fuel consumption is lowest at \(\lambda\) = 1.5-1.6. Besides using very lean mixtures, NOX emissions can also be reduced at moderately lean conditions (\(\lambda\) = 1.6) by retarding spark timing (ST) [7]. Pre-chamber injection timing has a large impact on the air/fuel mixture and on the leakage of fuel from the pre-chamber to the main chamber before ignition, which also impacts combustion stability [9]. On the other hand, stratification in the pre-chamber causes jet-to-jet variations between pre-chamber holes [10]. This implies that the combustion performance of PCSI strongly depends on pre-chamber geometry, which is defined by the pre-chamber volume and shape together with the number, diameter, and angle of pre-chamber holes. It has been shown that a pre-chamber with six orifices has higher indicated combustion efficiencies and lower THC and CO emissions in comparison to pre-chambers with a smaller number of orifices [6]. According to [11], the lowest NOX emissions are produced using pre-chamber geometry with the smallest jet holes.