## 1 Introduction  
Energy demand has gradually increased from the last decade because of the economic development and population growth around the world [1]. Liquefied natural gas (LNG) is becoming a primary energy resource in the global energy market owing to its cleanness, ease of transportability, and lower greenhouse gas emissions compared to other fossil fuels [2]. These positive advantages of LNG have attracted considerable attention in view of the current energy crisis. Liquefying natural gas and pipeline transportation are two major methods for transporting natural gas. However, the pipeline method is not economical for long-distance transportation because of unstable gas flow rates [3]. Therefore, liquefying natural gas can be the best method to transport natural gas across the ocean. The international gas union reported that various countries are constructing LNG liquefaction plants because of their rapidly growing reliance on natural gas from Asia to America [4]. Thus far, centralized large-capacity LNG liquefaction plants have been widely used; however, there has been an increasing interest toward distributed LNG liquefaction systems because of isolated small natural gas reservoirs [3], energy supply to isolated areas, and LNG bunkering systems [5]. Distributed LNG liquefaction plants can be classified into small-scale liquefaction plants with capacities of less than 40 million standard cubic feet per day or about 700 tonne per day (tpd) [6]. To satisfy the growing demand of LNG, movable and economically efficient small-scale LNG processes such as single mixed refrigerant (SMR) process and expander cycle have been studied [7]. Energy and exergy efficiencies are key factors for evaluating the performance of small-scale LNG processes [8]. [PERSON] and [PERSON] employed particle swarm optimization (PSO) on the SMR process to enhance exergy efficiency, and the result of enhanced exergy efficiency was 50.77% after optimization [9]. [PERSON] et al. [10] proposed a dual effect SMR (DSMR) LNG liquefaction cycle that has relatively lower cost and energy efficiencies. The DSMR process uses a SMR loop that separates into dual cooling and subcooling loops, and it can replace the classical dual mixed refrigerant process. Further, they reported that the exergy efficiency of the DSMR process is 36.62% and the total annualized cost is 1565 MM USD/year. [PERSON] et al. [11] attempted to replace a Joule-Thomson (JT) valve with a hydraulic turbine (HT) in the SMR process to improve energy and exergy efficiencies. The energy consumption reduced up to 16.5% and the exergy efficiency can be improved up to 10.95% compared to the conventional SMR process. The effect of the number of mixed refrigerant (MR) components in the SMR process was investigated by [PERSON] et al. ; they concluded that energy and exergy efficiencies increased with an increase in the number of MR components [12]. [PERSON] et al. [13] proposed a hybrid modified coordinate descent (HMCD) algorithm to optimize the SMR process. They compared the optimization performance of the HMCD to other optimization algorithms such as PSO, knowledge-based optimization, and genetic algorithm. The HMCD showed better performance in terms of energy savings and coefficient of performance compared to other optimization algorithms. Although the SMR process has higher energy and exergy efficiency compared to the nitrogen expansion cycle, the nitrogen expansion cycle is suited for small-scale LNG liquefaction because it is more safe and easy to operate [14]. The nitrogen expansion cycle has been studied by several researchers. [PERSON] and [PERSON] [15] proposed a novel conceptual design of parallel nitrogen expansion liquefaction for small-scale plants. They installed expanders in parallel to reduce the temperature difference, which can help achieve high exergy efficiencies. Thus, their proposed process could reduce the unit energy consumption by 4.69% compared to the conventional nitrogen expansion liquefaction.