## 1 Introduction  
Since throttle can restrict the air from flowing into the cylinders, pumping mean effective pressure (PMEP) is high at part load conditions for throttled internal combustion engines. Cylinder deactivation (CDA) has been demonstrated to be a reliable technology for improving fuel economy in gasoline engines under low load conditions at low and middle speed. When some of the cylinders are deactivated, the remaining firing cylinders require more air and fuel to maintain the same BMEP level. Therefore, the throttle angle is increased and the pumping losses are reduced due to higher intake manifold pressure [1; 2], and combustion is generally more complete with larger air-fuel charge motion. Moreover, when the intake and exhaust valves are deactivated, the deactivated cylinder acts as an air spring, repeatedly compressing and expanding the same air. The air spring losses are very low [2; 3], approximately 0.02 bar indicated mean effective pressure (IMEP). [PERSON] et al. found that CDA had the greatest potential to improve fuel economy for large engine sizes relative to vehicle weight, with fuel economy being improved by 6 to 11% for V8 and V10 engines [3]. The Active Fuel Management (AFM) system was included in the 3.9 L V6 engine and improved fuel consumption by 5.5-7.5% without compromising engine performance [4]. In recent years, with the restriction of environmental regulations and the wide application of small displacement turbocharged engines [5], more and more CDA technologies have been applied to small displacement engines [2; 6]. [PERSON] et al. [2] found that CDA improved fuel consumption by 6 to 25% for a 1.6 L in-line 4-cylinder gasoline engine when cylinders 2 and 3 were shut. VW has introduced CDA technology on a 1.4 L 4-cylinder TSI engine by switching the cam lobes between normal profiles and zero lift when the CDA mode was implemented [7]. In addition to the fixed two CDA modes above, three fixed-mode CDA technologies also have been developed. HONDA developed the new Variable Cylinder Management (VCM) system on a 3.5 L V6 engine. The newly developed VCM system introduces a four-cylinder mode to the conventional three-cylinder and six-cylinder modes, and makes it possible to switch between the three modes to increase practical fuel economy [8]. Moreover, the three-mode (L4-L3-L2) CDA technology was also introduced in a 1.1 L in-line four-cylinder motor engine [9]. [PERSON] et al. [10] studied the three-mode (L4-L3-L2) CDA technology on a 4-cylinder engine based on an unthrottled spark ignition (SI) engine with electromagnetic valve train, and found that the improvement in fuel consumption was about 7 to 21%. Compared with traditional fixed two-mode CDA technology, three-mode CDA can improve fuel economy further with the appropriate number of cylinders according to engine conditions. In addition to CDA, EIVC and late intake valve closing (LIVC) are also effective methods to reduce PMEP [11; 12; 13]. By combining CDA with EIVC or LIVC, PMEP can be minimized further [PERSON] et al. [6] investigated the benefits of fixed two-mode (L4-L2) CDA on a four-cylinder unthrottled engine equipped with a MultiAir system. The authors developed an innovative CDA valve strategy that combines internal Exhaust Gas Recirculation (iEGR) with EIVC. The results demonstrated a 30% reduction in pumping losses, and an 8% improvement in fuel efficiency compared with the condition of only EIVC load control strategy. [PERSON] et al. [5] proposed a novel CDA technology through double intake manifolds for four-cylinder SI engines. The results showed that after combining the CDA with LIVC, the pumping loss decreased by 58.9-65.6% and 24.5-35.3% at 2000 rpm and 4000 rpm, respectively. In addition to the combination of fixed CDA technology with EIVC or LIVC, skip fire technology is also a more attractive CDA technology that can be classified into two categories: fixed skip fire (FSF), and dynamic skip fire (DSF). [PERSON] et al. [14] designed 23 FSF modes with firing density varying from 22 to 100%. [PERSON] et al.