*4.1. LIM Performance Control; Adaptive Control*

The flux vector-oriented control is one of the most advanced and widely accepted methods used for the rotary machine torque control. It was first conceptualized by Blaschke [76] in 1972 and has been a subject of interest of many researchers ever since, e.g., [77–85]. With progress in microprocessor techniques and power electronics, the flux vector-oriented control has become a method of choice for most industrial applications, especially in the development of electric traction propulsion systems, historically based mostly on DC motors, in an effort to replace them with the much less expensive and more robust induction machines. Vector control signifies the independent, or decoupled, control of flux and torque of the motor through coordinated change in the supply voltage and frequency [83]. Flux level control is essential to avoid saturation and minimize core losses under various steady-state operating conditions. As the flux variation tends to be slow, especially with the current control, maintaining constant flux may provide precise torque response and, consequently, a desired speed response.

It is possible to distinguish three flux linkages in the induction machine complex form equations. These flux linkages are the stator flux linkage, the main, or air gap, flux linkage, and the rotor flux linkage. The current decoupling network simplifies only for the rotor flux orientation, whereas the voltage-decoupling network simplifies for the stator flux orientation. Only for a constant rotor flux orientation, the mechanical characteristic does not have a peak value and is a straight line. This linear characteristic is ideal for control application. For a given stator flux in the flux-weakening region and under steady-state operation, however, the stator flux is superior in terms of torque per unit stator current.

Direct Torque Control (DTC) is yet another vector control technique. It was introduced by Depenbrock [84,85] and Takahashi [86] and has been developed by others [87,88]. The fundamental premise of DTC is that a specific DC-link voltage and a specific stator flux establish a unique frequency of inverter operation. This is because the time required by the

time integral of the DC-link voltage to integrate up to the reference flux level is unique and represents the half-period time of the frequency of operation. Despite its simplicity, DTC can produce a fast torque response and is robust with respect to transient perturbations and motor parameter detuning [14]. It must also be noted that beside the already mentioned advantages, DTC does not use a modulator and does not employ current control loops, inherent to the vector-oriented flux control. However, during steady-state operation, a pulsation of torque, flux, and current may occur, affecting speed estimation and increasing the acoustic noise. This method is not established so well as the flux-oriented control and has not been applied in LIM controls; however, based on the up-to-date progress in its development, it shows exceedingly high application potential, particularly in the area where parameter sensitivity can be an issue.

Industrial applications of LIM motors require a relatively simple control algorithm because the parameters of industrial process LIMs are well known or can be measured in an off-line experiment. This is not so in urban transit applications since the motors are usually required to operate at peak thrust and the main parameters responsible for the precise peak tracking—the rotor resistance, *R*2, and the mutual inductance, *M*—vary in a very wide range. Thus, the controller of a transit traction LIM should be capable of tracking the maximum available thrust, independent of the air-gap length or the reaction rail construction properties and temperature. Several LIM control methods have been reported thus far, most of them based on the concept of vector control [35]; however, none of them attempt to resolve the parameter adaptation issue. To solve this problem a modified flux vector control technique has been applied [29,43].

The thrust calculated in the rotor flux reference frame compares to measurement only if the rotor parameters, *R*<sup>2</sup> and *L*2, are correctly estimated and their values do not change due to physical or environmental conditions. When these conditions are met, the secondary flux aligns with a *d*-axis and the back Electro-Motoric Force (EMF) naturally aligns with a *q*-axis. Should the rotor resistance, *R*2, change its value from the set point, the secondary flux would become misaligned and so a non-zero, *q*-axis component would develop; this means that more voltage is demanded from the supply inverter. This increased voltage generates a negative EMF *d*-component by advancing the rotor flux. Although the magnitude of the primary current vector remains constant and the secondary flux has increased, the angle between the two vectors has changed and is no longer optimal. A change of the machine secondary resistance from the reference value detunes the controls and a non-zero *d*-component of the EMF is generated. The optimal operation can be achieved again with the adapted rotor resistance reference value but at a different synchronous frequency. For the magnetizing inductance change, regardless of the cause, e.g., change in air gap, change in the reaction rail geometry, or change in the lamination coarseness or magnetization characteristics of the RR, a *q*-component of the secondary flux is generated. The induced voltage develops a negative *d*-component, such as in the case of the secondary resistance detuning. If the value of the reference magnetizing inductance is corrected to equal that of the motor, the controls would become tuned in again and a developed thrust would be optimal, although it would be lower. Since the secondary resistance compensation loop that corrects a *d*-component of the induced voltage is active, the secondary flux will be regulated to align with the *d*-axis.

To verify the above parameter compensation control concept in the simulation software, a *d–q* model of the LIM is first derived, see Figure 9. This LIM signal network clearly shows the impact of parameter detuning on the rotor flux and slip frequency estimation.

**Figure 9.** Rotor flux-oriented control LIM model [29,30].

To correlate the model and control variables of the *d–q* system with the real-time threephase values, standard Clarke and Park transformations are applied [17]. Figure 10a,b shows the simulated response of the system to a step change of *R*<sup>2</sup> and *M* both with and without the adaptive compensation control loops.

**Figure 10.** Response to a step of the secondary resistance *R*<sup>2</sup> (**a**); response to a step of the mutual inductance *M* (**b**) [29].

The adaptive algorithm improves the performance of the system by a significant margin by improving the mechanical output. The proposed algorithm successfully addresses the problem of LIM parameter detuning while preserving all the positive features of the rotor flux referenced vector control. The verification of the simulation results was performed by comparing the calculated output with measurements from a transit test vehicle fitted with the subject test LIM. This method exhibits the robustness necessary in severe transient conditions associated with the application of the LIM in transportation systems.

### *4.2. LIM Driven from the Voltage Inverter*

In all typical LIM simulation models, the symmetrical three-phase current is fed into the three-phase winding to simulate a constant current mode; however, this does not reflect reality when the LIM is driven from the voltage inverter. The differences in slip versus thrust characteristics between the simplified approach and the approach where an asymmetry of phase currents arises naturally from the real supply conditions are presented in [70].

Typically, the LIM is powered from a PWM voltage inverter, converting thrust command into current at a desired frequency. However, as the phase impedances are unequal and the three-phase currents differ in their phase and magnitude, the negative sequence currents are produced leading to a decreased motor performance. In theory, if the LIM phase impedances were known, the phase currents could be equalized, although not entirely, by a proper phase voltage control, but at a price of increased voltage harmonics. The electromagnetic fields shown below (Figure 11) are calculated considering the natural asymmetry of phase currents under symmetrical voltage excitation.

**Figure 11.** Magnetic field distribution within the LIM [70].

As can be seen from Figure 11, the magnetic field shows significant asymmetry on both ends of the machine. This results in asymmetric coupling and an asymmetric back electromotive force that leads to unequally coupled impedances and the asymmetry of phase currents. Because the phase currents are magnetically coupled with one another and additionally coupled with the induced currents of the reaction rail, these impedances are frequency and speed dependent; thus, their determination can be very involving.

The exemplary performance characteristics of the subject LIM in current and voltage modes for different speeds are shown in Figure 12.

**Figure 12.** LIM characteristics obtained for the current supply (*I* = 550 A) (**a**) and the voltage supply (*V* = 460 V) (**b**) [70].

The characteristic increase of peak-thrust slip frequency that can be seen in the figure above results from the end-effect induced magnetizing impedance change. As can be seen from above figure, it is important to account for the asymmetry of phase currents when determining the LIM performance. To do so, the electromagnetic transient FEA simulation with the symmetrical three-phase voltage source would have to be used. However, that would become prohibitively time consuming due to a need of remeshing a large solution space at every time step. Alternatively, the quasi-steady-state transient solution can be achieved by simulating in the frequency domain but only if the software allows for the modification of Ampere's law.
