# Synchronous Homopolar Generator without Permanent Magnets for Railway Passenger Cars
## 3 Operating Points of the Generator, Variable Geometric Parameters, and Objective Function  
In the traction drive, motors with a wide speed range at a constant mechanical power (CPSR) are common [25]. An unavoidable feature of CPSR is an increased load on the mechanical part (torque) at low speeds. As a natural generalization for a generator, there could be a requirement for a given range of speeds with a constant output power (active power minus the power of the excitation winding). However, at low speeds and high torques, machine saturation and losses increase, and efficiency also decreases. Therefore, even more mechanical power and even more torque are required. Therefore, in this paper, the generator with a given CPSR is considered. The rotational speed of the undercar generator increases as the speed of the train increases. The undercar generator described in [31] provides a constant power of 35 kW in the speed range from 750 rpm to 3450 rpm. This paper describes the generator with a CPSR from \(n_{2}\) = 750 rpm to \(n_{1}\) = 3450 rpm and the rated mechanical power of \(P_{max}\) = 40 kW (Figure 3). It is expected that the averaged value of the output power estimates as the average losses in the modes with the speeds of 750 rpm and 3450 rpm is close to 35 kW. Mechanical losses, namely bearing and windage losses, were neglected. Figure 3: The dependences of the mechanical power \(P_{mech}\) (blue line) and torque \(T\) (red line) on the generator shaft on the rotation speed \(n\). Figure 2: Schematics of the three-phase-controlled rectifier with a DC chopper to supply the excitation winding. The uppercase letters indicate the phases A, B, C of the generator. During the optimization, it is necessary to minimize the following characteristics of the generator (the most important characteristics are listed first):  
1. Average losses <\(P_{loss}\)>, which were estimated as the average losses in the modes with the speeds of 750 rpm and 3450 rpm;
2. Maximum value of the stator current \(I_{arm}\) (reached at 750 rpm mode);
3. Maximum symmetrized torque ripple max(_TRsym_) in both modes;
4. Maximum nonsymmetrized torque max(_TR_) in both modes. When optimizing, both the torque ripple _TR_ of a separate SRSC and the torque ripple _TRsym_ resulting from the addition of the torque waveforms of all SRSCs of the SHG were considered [9]. The material for the housing and the sleeve is solid steel 1010. Because the magnetic properties of structural steel are not reported and are not guaranteed by the manufacturer, in order to guarantee that the drop of the magnetomotive force on the axial magnetic cores will be small compared to the drop on the SRSC, in the course of optimization, the flux density in the stator housing and in the rotor sleeve was limited to 1.6 T.  
In this study, the single-criteria unconditional Nelder-Mead method was used to optimize the SHG design. Therefore, the objective function was given as a product of individual characteristics raised to a certain power, reflecting the importance of the characteristic. Optimization constraints cannot be set by assigning an infinite value to the optimization function if these conditions are not met, as this would lead to a rapid decrease in the volume of the simplex and convergence to an undesirable local minimum. Therefore, the maximum flux density constraint was set as a soft constraint; that is, the corresponding multiplier begins to increase rapidly if the constraint is not complied. In view of the above, the objective function is defined as follows:  
\[ \begin{split} F=<P_{loss}>\max(I_{arm})^{0.7}\max(TRsym)^{0.025} \max(TR)^{0.01}f\bigg{(}\bigg{(}\frac{B_{h}}{16}\bigg{)}^{5}\bigg{)},\\ f(x)=\left\{\begin{array}{l}x,x>1,\\ 1,\ otherwise,\end{array}\right.\end{split}