## 1 Introduction  
Induction machine (IMs) are the oldest electrical machines, but they are widely used in industrial and domestic operations. This machine's fundamental problem is its starting period, as starting this type represents a set of processes that occur during acceleration of the machine from zero speed to a specific speed (rated speed, no-load speed, or others) [1]. During IM direct startup, the rotor currents' frequency depends on the rotor's speed, and the machine's stator currents are very high, i.e., higher than the rated current. Hence, analyzing the starting period of IMs is necessary for the protection and stability issues. In the literature, many IMs' starting methods can be found. They can be classified into two broad categories--conventional motor starters and modern power electronic-based devices to start the machine. The first category involves methods that rely on the classic regulation of the supply voltage or practices using additional equipment [2, 3]. The direct-on-line (DOL) method belongs to this category, in which it can be realized with or without the reduced voltage technique [4], the variable rotor resistance method [3], the shunting of the stator and rotor windings [4], the star-delta method [5, 6] and the usage of auto-transformers [6, 7]. The second category involves methods that rely on power electronics devices or soft starters [2]. The advantages and disadvantages of these methodsare comprehensively discussed in [7]. The most traditional method for IM starting is the DOL method [8; 9]. However, the DOL method is much more beneficial in parameters estimation of IMs [10; 11; 12]. In this regard, speed-time characteristics calculation during IM startup is essential for the secure operation power distribution network. Observing the research works investigating the IM starting process analysis, it can be noted that this research field is vital from two perspectives. The first perspective is that the IM starting process can benefit parameter determination either with a direct startup [8; 9] or acceleration [12; 13] or acceleration-deceleration tests [14]. Many research works were dedicated to estimating motor parameters under different conditions. For instance, [PERSON] et al. in [15] have recognized IM's parameters using limited and non-intrusive observations of input voltages, stator currents, and the rotor speed and convex optimization. [PERSON] et al. in [14] proposed a complete parameter identification procedure of large IMs from no-load acceleration-deceleration tests. [PERSON] et al. in [12] have used the instantaneous power of a free acceleration test for squirrel-cage IM parameters estimation, and [PERSON] et al. in [16] proposed a hybrid simulated annealing evaporation-rate water cycle algorithm [17] for parameter estimation of single-cage and double-cage IMs models. The second perspective is that the IM starting process duration is significant for proper machine protection-relay settings, avoiding overheating and poor power quality issues [1; 18]. For instance, starting a large IM can cause substantial voltage sags in industrial areas, adversely impacting the system's sensitive and local loads (especially lightning and personal computers (PC) and power quality [19]. The no-load starting time defines the value of the maximum starting time, which in turn determines the proper current relay action (or delay time determination) for IM protection and stable and safe operation. For that reason, several manufacturers provide equations to determine torque and power and their relation with starting time [20]. The technical report of low-voltage IMs is presented in [21], and the equations, which define three-phase IMs and suggestions for the coordination of protective devices, are given in [22]. As the machine accelerates, the starting currents decrease. The critical period of time in such a startup period is the critical voltage-sag removal time (CVSRT).