## 1 Introduction  
Rechargeable lithium-ion batteries are widely used as a power source in many industrial sectors ranging from portable electronic devices to electric vehicles and power grid systems [1; 2; 3]. In the context of energy management and distribution, the rechargeable lithium-ion battery has increased the flexibility of power grid systems, because of their ability to provide optimal use of stable operation of intermittent renewable energy sources such as solar and wind energy [4; 5; 6]. In addition to storing renewable energy for later use, lithium-ion batteries are the major component dictating the performance of electric vehicle (EV) performance. Furthermore, their integration into EVs offers a promising solution to reducing CO\({}_{2}\) emissions. Rechargeable lithium-ion batteries offer many opportunities, such as high power density, long life cycle [7; 8], higher cell voltages [9; 10], lower maintenance requirements [11; 12], higher charging speeds with lower self-discharging rates [13; 14], and no memory effect compared with the previous generations of rechargeable batteries [15]. However, even if the rechargeable lithium-ion battery seems to provide a suitable solution to the limitationsof renewable energy production and utilization, there are also some existing issues related to its technology. More explicitly, one of the fundamental problems with this device is battery management systems (BMSs) [16], and among all the parameters of BMSs, state of charge was found to be the most significant parameter [17; 18]. The state of charge is one of the crucial evaluation parameters of BMSs that confirms the optimal charging or discharging profiles of batteries [2]. Moreover, the state of charge describes the remaining capacity in a battery energy storage system and is indicated in the form of a percentage, [%]. Meanwhile, one can identify the charging or discharging status of a battery by checking its SOC value [19; 20]. High precision in state-of-charge estimation is a must for power battery control strategies [21; 22]. Indeed, to ensure the optimal charging/discharging profiles of the battery, the SOC should satisfy the following condition:  
\[SOC_{min}\leq SOC(t)\leq SOC_{max}, \tag{1}\]  
where \(SOC_{min}\) is the minimum state of charge and \(SOC_{max}\) is the maximum state of charge. These conditions play an important role when designing a controller to decide when to charge and when to discharge a battery [2]. Furthermore, these box constraints on the state of charge also help the controller to cut off the power to prevent the battery from damage due to over-discharging and overcharging [23]. In ref. [24], for instance, it is reported that the overcharging of lithium-ion batteries leads to the generation of dendrites due to the deposition of lithium ions, which induces the generation of a large amount of heat due to internal short circuit, and the vaporization of the electrolyte solution. Another research work published has shown experimentally that when the lithium-ion battery is discharged to the cut-off voltage and then forced to discharge more than 20% of the state of charge, it will cause irreversible internal short-circuit damage to the battery [25; 26]. In this respect, the authors concluded that the possibility of short circuit has a non-linear relationship with the depth of over-discharging. In general, determining the SOC of rechargeable lithium-ion batteries is a complex task, due to the non-linear reaction dynamics caused by the electrochemical characteristics of lithium-ion batteries [27; 28]. This means that the SOC can only indirectly be calculated by measuring some physical parameters, such as charge/discharge current, terminal voltage, and internal resistance [29], or by using a mathematical model and algorithm [30; 31]. However, the physical parameters of lithium-ion batteries are usually affected by some uncertain factors, such as the open-circuit voltage (\(U_{oc}\)) of the battery, temperature variation, battery aging, and overall efficiency of the battery systems [32].