# Numerical methods for integrated modeling of electrical drives

The research focuses on numerical methods for the time-domain simulation of a combined electrical network/finite element model for electrical machines connected to a power-electronic supply. Conventional time-stepping schemes such as the beta-scheme are widely used in the case of devices which are connected to a sinusoidal voltage supply. However, special demands are put on the numerical integration of the model when advanced control strategies are to be taken into account. In the case of high-dynamic electrical drives, a pulse-width modulated (PWM) supply is used to control the machine phase currents. Due to the current feedback loop, the switching instants of the converter are not a priori known. A conventional fixed-step time-stepping method, such as the beta-scheme, is therefore not suitable for the simulation of the model.

An embedded Runge-Kutta (RK) scheme forms a good alternative for the numerical integration of the combined machine-converter model. At every time step the scheme yields two solutions with a different order of accuracy. The difference between the solutions can be used to control the step size. Another advantage of the Runge-Kutta scheme is the use of interpolator polynomials. These can be used to calculate phase currents at prescribed sampling rates, regardless of the step size used. The polynomials are also extremely valuable to detect a priori unknown switching instants, e.g. in the case of a diode in which the current falls to zero. By making use of analytical functions, the computations of current samples and switching instants can be done in a computationally efficient way. Figure 1: Geometry of 6x4 switched reluctance motor and finite element mesh

As an example, an embedded Runge Kutta scheme has been applied to a switched reluctance motor drive. Figure 1 shows the geometry of the motor. Figure 2 shows the part of the converter supplying one motor phase, and the current control feedback loop. In figure 3, the current control of one motor phase is depicted. Figures 4 and 5 show details, in which the calculated Runge-Kutta steps are indicated, as well as the current samples and switching instants, calculated by interpolation. Figure 2: Converter and current control loop for SRM (single leg) Figure 4: Detail of PWM control: calculated RK steps, periodic current samples and interpolators Figure 5: Detail of phase switch-off: calculated RK steps, samples and interpolation points

## Relevant Publications

Kristof R. Geldhof, Thomas J. Vyncke, Frederik M.L.L. De Belie, Lieven Vandevelde, Jan A.A. Melkebeek and René K. Boel, "Embedded Runge-Kutta Methods for the Integration of a Current Control Loop in a SRM Dynamic Finite Element Model," IET Sci. Meas. Technol., 2007, 1, (1), pp. 17-20.