Dynamic Modeling of Transverse Flux Permanent Magnet Generator for Wind Turbines

The transverse flux permanent magnet machines have become an interesting possibility for offshore wind turbines. These machines have the highest relation between electrical torque and weight of active materials. The pole pair modular construction could eliminate or lower the gear ratio used in conventional wind turbines. This paper presents a dynamic model of a wind turbine equipped with a transverse flux permanent magnet generator connected to a direct-current power system using a combination of 3D finite element generator model and an aerodynamic model. The results indicate that the model can give accurate response for steady-state operation and for wind speed variations.

divided according to the direction of the magnetic flux in relation to their shafts as: radial (conventional), axial and transversal. Since the transverse flux permanent magnetic machine (TFPM) has the highest relation between electrical torque and mass (their size is even smaller than conventional PMG for the same nominal torque), this concept deserves special attention [4]- [7].
The dynamic behavior of most electrical machines can be studied by the use of analytical equation, however, it is not the case of the TFPM (shown in Fig. 1). The constructive geometry of the TFG is very complex and has no 2D symmetry. The intuitive solution to model its dynamic is the approach which connects the dynamic model of the electrical power system and the controllers step-by-step to the TFG 3D finite element model. On the other hand, the required simulation time to process this analysis is impractical. For this reason, a hybrid model was developed allowing fast dynamic analysis of wind turbines equipped with TFG. This paper is organized as follows. Section II presents a brief description of the TFG. The 3D finite element analysis (FEA) of this generator is described in Section III. The wind turbine dynamic model is presented in Section IV. The dynamic analysis of the TFG is presented in Section V. In Section VI, the main conclusions are summarized.

II. TRANSVERSE FLUX PERMANENT MAGNET MACHINE
The transversal flux permanent magnet machine was first developed to operate as a motor in the 80's by H. Weh [4], [6], [7]. More than 11 geometries of transverse flux permanent magnet machines (TFPM) have been described in the literature [8]. Among them is also highlighted that fluxconcentrating TFPM provide higher force density compared to surface-mounted TFPM [8]. More research has been done in the motor operation, however, the operation as generator is very similar.
At the Institute of Electrical Machines (IEM), RWTH Aachen University, a three-phase configuration of TFPM with flux-concentrating was developed. This machine was previously designed for traction and has a water-cooled stator, an external rotor and U-shaped soft iron parts, as shown in Fig. 2. The operation of TFPM is very similar to the conventional synchronous machine allowing similar control strategies [7]. The nominal characteristics of the TFPM prototype developed in Aachen are: 25 kW; 600 rpm; 400 Nm; 205 V; 70 A.
The eddy currents losses calculated using three dimensional finite element analysis of the Aachen TFPM prototype was presented in [9]. The complex construction of the TFPM requires expensive soft magnetic composites and is justified for special and critical applications [10].
III. DYNAMIC ANALYSIS METHOD FOR TRANSVERSE FLUX PERMANENT MAGNET MACHINE The dynamic analysis of conventional electrical machines is performed using dynamic analytical equation for each phase winding. In the case of the TFPM, the most appropriate form to analyze its dynamic behavior is with the use of the finite element modeling. Duo to the complex constructive geometry, there is no symmetry considering two-dimensional axis (2D symmetry). By consequence, the analysis must be in the three-dimensional axis, some simplifications may be done.
Due to its construction, the phase windings are exactly the same and the flux linkage between each other can be neglected. This fact allows the analysis of only one phase to determine the flux linkage (λ) and the electromagnetic torque (Te). The modular configuration of the TFPM enables the finite element method to be performed in only one pole pitch (Fig. 2). The saturation effect is taking into account considering the non-linear characteristic of the core material. The flux linkage in phase A can be determined by the finite element method applied to the mesh geometry shown in Fig. 3. In addition, the general electrical machine theory can be used to determine the induced voltage in each winding using the equation (1).

A. Determination of the Flux Linkage (λ)
The terminal voltage of the TFG can be determined differentiating the flux linkage of the stator windings, as shown in equation (1).
This equation is a general form of stator voltage for alternating current generators [7]. The flux linkage is computed as function of rotor position and armature currents by 3D static finite element analysis [7]. The form of equation (1) must be expanded to implement the look-up table data: Finally, the current differential term of the equation is isolated and implemented in Matlab/Simulink: where v(t) is the terminal voltage; R is the phase resistance; i(t) is the phase current; t is time; θ is the rotor angular position of the electric system; ω m is the rotor velocity.
With the flux linkage determined by the 3D static FEA, the two differential terms of the right side of the equation (3) can be calculated. The first term of equation (3) where p is the number of pole pairs.
The interpolation of these values is shown in Fig. 6.

IV. WIND TURBINE MODELING
The TFG requires a full converter in order to operate in variable speed. This fact enables the wind turbine to maximize the generated power. The aerodynamic model, the generator model and the converter control model are described in the following topics.

A. Aerodynamic Model
For all kinds of wind turbine technologies the energy conversion principle is the same. The mechanical power (P m ) capture from the wind by a wind turbine and its related mechanical torque (T m ) can be calculated using the well-known aerodynamic equations [2]- [5], [11]: where A is the turbine rotor area, ρ is the air density, V wt is the wind speed, C p is the performance coefficient, β is the blade pitch angle, λ wt = ω wt R wt /V wt is the tip speed ratio, R wt is the radius of the turbine rotor and ω wt is turbine angular speed.
The performance coefficient depends on the blade pitch angle and the tip speed ratio (λ wt ). Usually, a group of curves are experimentally obtained by the manufacture of the wind turbine. However, for these studies we have used the analytical equation proposed in [2], [11]. The pitch angle control diminishes mechanically the C p of the wind turbine in order to limit the generated power for wind speeds higher than the nominal. The adopted pitch angle control operates as electric power regulator [11] and is activated only when the velocity of the wind turbine reaches its nominal value (21,8 rpm).
The adopted 2 MW wind turbine [11] requires a gear box (GB) ratio 1:27,5 to match the nominal generator speed.

B. Generator Model
The analytical dynamic equations of generators are sufficient for most of the studies. Therefore, there are no well-know dynamic equations for TFG and the approach using look-up tables is required, as described in section III. Additionally, the mechanical equations were implemented: where T E is the sum of the electromagnetic torque of the 3 phases, J is the combined inertia coefficient (sum of the turbine rotor and the generator rotor) and θ is the mechanical rotor angular position.
The nominal mechanical characteristics of the adopted wind turbine do not match to the nominal characteristics of the TFPM prototype, hence the concept of per unit systems (p.u. systems) must be used as suggested in [2]. The operation of the generator must follow the best rotor speed in order to maximize the mechanical power. In Fig. 8, the optimized power tracking is obtained by adjusting the turbine-generator angular speed for each wind velocity.

C. Converter Control Model
In order to maximize the generated power by the wind turbine the optimal speeds was calculated considering also the performance coefficient and the blade pitch control (Fig. 8). The more appropriate offshore transmission system to bring the electrical power to the land is not already defined, however, there is a tendency to use the HVDC transmission system in some far locations. For this reason, the converter model used in this study represents only an ideal IGBT-based machine side converter connected to an internal direct-current (dc) network of the wind farm.  The PWM (Pulse-width modulation) converter control (show in Fig. 9) employs dq0 rotating reference frame to vary the applied voltage and frequency to active optimal speed. The terms with the mark ( * ) are the reference values. In this figure, I abc are the phase currents injected by the generator into the dc network. The symbol θ is the rotor position reference to the dq0to-abc transformation. The speed controller is implemented with a proportional and integral regulator (PI), it is responsible for providing the quadrature current reference (I q * ). The direct-axis reference current (I d * ) is 0 (zero) in order to maximize the electromagnetic torque (T E ) [12], [13]. After a dq0-toabc transformation, the three-phase reference currents are compared at the current controller which gives the converter output reference voltages (V * abc ). The reference voltages are sent to the PWM signal generator and then applied to the machine by the converter.  For wind speed belong 11,8 m/s, the rotor speed assumes different values in order to achieve the optimal speed to maximize the power capture from the wind (Fig. 12). Above the nominal speed in steady-state operation, the rotor is maintained at fixed velocity. The blade pitch angle control and the rotor speed control are complementary, as can be seen in Fig. 11 and in Fig. 12.

B. Dynamic Analysis during Wind Gust Perturbation
The second part of the dynamic analyzes determines the effect of wind gust perturbations over a certain period when the wind turbine is operating at nominal speed. In this analysis, the implemented speed control maintains the rotor speed in a fixed value during the wind gust perturbation. The blade pitch angle control actuates to minimize the effect of the wind gust on the electric system. The signal adopted to control the blade pitch angle is the generated electric power. This pitch angle control strategy minimizes the fluctuations of electric power injected into the electric system. The wind gust perturbation (shown in Fig. 13) has its average value equal to 16 m/s. The rotor speed is kept constant by the rotor side converter, as shown in Fig. 14. The pitch angle control actuates to compensate the wind speed variation. When the wind speed becomes higher than the average value, the pitch angle also increases to diminish the power capture from the wind. The actuation of the blade pitch angle is less efficient when the average wind speed is closer or lower than the nominal wind speed (11,8 m/s). In Fig. 15, the variation of the pitch angle during the wind gust is shown.  One can see in Fig. 16 that the injected electric power suffers the influence of the wind gust perturbation. This fact can be explained considering the time response of the mechanical actuator used to vary the blade angle, which adds a time delay between the control order and the effective actuation.