## 1 Introduction  
According to the European legislation for heavy-duty vehicles (Regulation (EU) 2019/1242), the CO\({}_{2}\) emission level compared with the EU average in the reference period should be reduced by 15% after 2025 and by 30% after 2030 [1]. Due to increasingly stringent emission regulations, the manufacturers of internal combustion engines (ICEs) strive to reduce fuel consumption and greenhouse gas emissions by improving engine performance. Different technologies have been developed to improve engine thermal efficiency to achieve this goal, including start-stop [2; 3], rotation speed reduction [4; 5], proper selection of technological operations [6], and cylinder deactivation [7; 8; 9]. Cylinder deactivation (CDA) has received significant interest and in-depth investigations for gasoline and light-duty diesel engines, considering its convenience of implementation on existing machines. The principle of CDA is to change effective engine displacement to impel engine running with higher effective efficiency at light-load conditions without compromising output power [10]. The principle of CDA application for reduced fuel consumption in spark ignition (SI) engines is that after some cylinders are deactivated, the rest of the active cylindersrequire more fresh charge to maintain the brake mean effective pressure (BMEP); as a result, the throttle valve opens more widely, intake manifold pressure increases, and pumping losses decrease [11]. The application of CDA with variable valve actuation (VVA) in SI engines contributes to reducing the brake-specific fuel consumption (BSFC) up to 17-25% depending on engine configuration, operating points, and engine control [12; 13; 14; 15]. CDA operation combined with dynamic skip fire strategies can reduce engine vibration and bring an additional reduction in BSFC [16; 17]. The maximum BSFC improvement achieved for those engines with large displacement and multi-cylinders is attributed to pumping loss reduction under CDA operation. Currently, SI engines employing CDA technology are produced by many manufacturers, e.g., Mercedes-Benz, Honda, General Motors, Volkswagen, Ford, Mazda, and Fiat Chrysler Automobiles. In accordance with data in [18], in the 2020 model year, about 14% of light-duty vehicles sold in the United States used CDA. Even though the production of compression ignition (CI) engines with CDA has not yet been realized, much research has been dedicated to CI engines' thermal and emission performance under CDA operation. For CI engines, the advantages of CDA to improve fuel economy by reducing pumping losses are not as evident as for throttled SI engines because CI engines have no throttle but an extensive use of turbocharging. However, combustion processes in CI engines at idle or small loads are worse than those at medium and high loads due to slower evaporation and mixing rate [19]. CDA technology improves combustion quality and reduces emissions by transferring active cylinders from a low-efficiency to a high-efficiency operation range. The more significant the difference in efficiency and power from low load to medium load, the greater the possibility of CDA application on such engines [20; 21]. Therefore, CDA has application potential for high-power engines, but runs at a light load and remains idle for a long time. In CI engines, the most common way to realize CDA is to shut off the fuel injection to deactivate cylinders. This method was investigated in [22] on a 4-cylinder 2.66 L high-speed diesel engine for off-road excavators. Besides cutting off the fuel supply to deactivated cylinders, it is necessary to cooperate with different valve operation strategies to minimize engine losses, such as pumping losses and heat transfer losses. [PERSON] et al. [23] conducted an experiment to deactivate two out of four cylinders on a 2.2 L turbocharged diesel engine with two strategies for deactivating cylinders: deactivating fuel supply and deactivating both fuel supply and valve motions.