*2.3. Algorithm*

Figure 5 shows a block diagram of the proposed control algorithm. Looking at the left side of Figure 5, given thrust command *FT* <sup>∗</sup> and train speed *vm*, the recommendation calculator determines the slip-frequency command that can be used within the range not exceeding the normal force limit through the previously proposed method.

**Figure 5.** Control algorithm of proposed method.

At this time, the normal force-limit value was derived from finite-element analysis and the initial train test. If derived slip-frequency command *fsl* <sup>∗</sup> and command Im<sup>∗</sup> of the total current required to drive the system are given as "reference current generator", *id* and *iq* for vector control are derived. At this time, total current command Im<sup>∗</sup> is determined through Equation (5), and the method of deriving *id* and *iq* using "reference current generator" is as follows.

$$
\omega\_{sl} = \frac{R\_r}{L\_r} \frac{i\_q}{i\_d} \tag{9}
$$

$$\|I\_{rms}\|^2 = \dot{\imath}\_d^2 + \dot{\imath}\_q^2 \tag{10}$$

Equation (9) shows the relationship between slip frequency and *d*-*q* axis reference currents *id* and *iq* in the induction motor, and Equation (10) shows the relationship between *Im* and *id*, *iq* derived through the propulsion force. These two equations are used to determine the *id*, *iq* current command in the "reference current generator".

$$i\_d = \frac{R\_r}{L\_r} \frac{i\_q}{\alpha\_{sl}} \tag{11}$$

$$I\_{rms}^2 = (\frac{R\_r}{L\_r})^2 (\frac{i\_q}{\alpha\_{sl}})^2 + i\_q^2 \tag{12}$$

If Equation (9) is summarized for *id* as in Equation (11) and substituted into Equation (10), Equation (12) representing the relationship between *iq* and *Im* can be obtained. Reorganizing this for *iq* can be expressed as Equation (13):

$$i\_q = \frac{L\_r}{R\_r} \omega\_{sl} \sqrt{\frac{1}{(1 + (\frac{L\_r}{R\_r} \omega\_{sl}))^2}} I\_{rms} \tag{13}$$

$$i\_d = \sqrt{\frac{1}{(1 + (\frac{l\_r}{R\_r}\omega\_{sl}))^2}} I\_{rms} \tag{14}$$

By arranging *id* in the same way, we can obtain Equation (14). The current command of the *d*-*q* axis can be derived from constants *Rr* and *Lr* of the train, total current *Im* required by the driving force, and slip angular velocity ω*sl* that can be obtained from the slip frequency. Therefore, when the thrust

and normal force-limit values, which are the driving conditions of the train, are proposed, it is possible to directly control the *iq* current, involved in the propulsive force of the train, and *id*, involved in the lift of the train.
