## 1 Introduction  
The hydraulic closed-circuit system (HCS) is widely used in walking machinery [1]. The oil in the closed circuit circulates within the main hydraulic circuit, unlike the hydraulic open system where it flows back to the tank during operation [2]. This implies that the inlet tubing of the hydraulic pump is directly connected to the return tubing of the actuator (hydraulic motor or hydraulic cylinder). An HCS offers several advantages. (1) Closed-circuit pumps can simultaneously regulate flow and direction, enabling bidirectional driving without the need for a directional value [3, 4]. (2) HCS is a symmetrical reversible system that automatically adapts to varying rotation and torque directions [5, 6]. (3) The system stiffness of HCS is excellent, with a fast response to regulation [7]. However, due to the fact that less oil is involved in the HCS operation and the limited space for the walking machinery, the oil temperature rise in the HCS is more noticeable [8]. The dynamic viscosity of hydraulic oil exhibits a significant reduction with an increasing temperature. The elevated oil temperatures can lead to a range of issues, such as reduced system efficiency, increased wear on hydraulic components, and compromised transmission accuracy [9]. Typically, the oil temperature within the HCS ought not surpass 80 \({}^{\circ}\)C [10]. The oil temperature represents one of the critical parameters emphasized by HCSs. The temperature of hydraulic oil is influenced by numerous factors, making it challenging to obtain precise oil temperature data through theoretical analysis [11, 12]. The process of modeling with simulation software is highly intricate, requiring the setting of many parameters based on experimental data. Moreover, the exorbitant expense of the software servesto further escalate the overall cost of simulation calculations. Hence, there is an immediate requirement for a precise, efficient, and cost-effective method to predict the oil temperature in HCS. The oil refilling flow rate (ORFR) is basically equal to the cooling flow rate [13]. Due to the challenges associated with accurately estimating oil temperature, the ORFR is typically set at 25% to 30% of the main system flow rate [14], which significantly exceeds the actual ORFR requirement. The excessively high ORFR can lead to a significant reduction in the efficiency of the HCS. Conversely, an excessively low ORFR may result in oil temperatures within the HCS exceeding specified limits. This scenario not only diminishes the performance of the hydraulic system but also poses a risk of damage to its components. So, there is a necessity to fine-tune the replenishment parameters and propose a methodology for determining the optimal ORFR under varying operational circumstances. The research on oil temperature control and parameter optimization in HCS has been a prominent focus in recent years. [PERSON] et al. investigated the impact of key operational parameters in HCS on oil temperature and developed a thermal equilibrium model based on heat generation and dissipation [15]. However, the accuracy of this model was compromised by an insufficient consideration of the environmental factors affecting the heat-transfer coefficients and oil viscosity moduli under varying working conditions. [PERSON] et al. constructed a thermodynamic model for hydraulic hybrid vehicles, enabling thermal management, stability analysis, and efficiency calculations [16]. [PERSON] et al., through exploration of the system's required ORFR and input power using thermal equilibrium equations, established matching relationships between ORFR and system flow rate [17]. However, these findings are only applicable under specific operating conditions. [PERSON] et al., through physical modeling of coupled HCS systems, examined the influence of oil viscosity and effective volumetric modulus on closed pump-motor hydraulic system stability across different oil gas content and temperature conditions [18].