Direct load angle control

4.4 4.5 4.6 4.7

0 3

6 δt,max=81.8°deg

if,tmax=4.68

0°

40°

δ 80°

P

Qcap

P,Q [pu] δ [°deg]

if [pu]

Figure 4.17 Capasitive reactive power Qcap, active power P and load angle δ as a function of field current if. Flux linkage |ψs| = 0.33, maximal stator current modulus |is|max=4.0, speed n=3.0, reference motor parameters (see Appendix A). The maximal torque working point (unity power factor) lies on the load angle δ_t,max = 81.8°deg.

4.4.2 Direct load angle control

DTC requires high speed control algorithms for the stator flux linkage estimation and for the modulation control. Tiitinen et al (1995) have reported the control cycle of 25 µs. Such a speed in the controller gives possibilities for the direct control of the stator flux linkage angle as well.

The basic functional principle of DTC should be considered again. The modulation is based on three variables, which are the torque error, the flux linkage modulus error and the stator flux linkage vector location in the stator reference frame. As a result of the torque comparison the modulator selects a voltage vector, which increases or decreases the stator flux linkage vector rotational speed in the stator reference frame. When the torque is increased, the load angle changes to the same direction, but with a lower speed. This observation gives the idea of the direct load angle control, where load angle is directly controlled by means of the torque error variable.

The direct stator flux linkage angle control is very unsymmetrical in different directions. Fig. (4.18) shows a situation, where the stator flux linkage angle has reached its maximal value. If the modulator tries to increase the torque, the stator voltage is used mainly for the compensation of the electro motive force and the change in the load angle is slow

∂δ

∂τ ^≈

− u_{s av} e

ψs ^{. (4.36)}

When a too high load angle value is observed, such a voltage vector can be selected, which reduces the load angle as fast as possible. The change in the load angle is much faster than in Eq. (4.36), because the electro motive force and the voltage vector have approximately the same direction

∂δ

∂τ ^≈

− u_{s av} − e

ψs ^{. (4.37)}

If the maximal speed in the load angle reduction is used according to Eq. (4.37), the load angle controller must work with a very short control cycle, and a high switching frequency is required.

Another possibility is to select a voltage vector parallel to the stator flux linkage vector. In that case the DTC adjusts the stator flux linkage modulus and only the electro motive force e reduces the load angle. This type of load angle reduction avoids large changes in the stator flux linkage angle and in the torque, and the load angle controller can work with approximately half a speed compared to the previous case. The different cases are shown in Fig. (4.18).

u_s(τ₀)

Figure 4.18 Change of the stator flux linkage load angle in the rotor reference frame from the initial value ψ0 when the positive torque producing voltage vector is used (∆ψ1), the load angle is reduced by the electro motive force e alone (∆ψ2) or the load angle is reduced by the maximal speed (∆ψ3).

DTC works in the stator reference frame, and thus the fastest control loops work in the stator reference frame as well. The presented load angle control method works in the rotor reference frame, where fast estimates are not available for the load angle. Co-ordination transformations require a lot of computation power due to the trigonometric functions included. This method requires, however, fast estimates of the stator flux linkage components in the rotor reference frame.

A shortcut for the co-ordination transformation equations can be found with the Taylor series. The stator flux linkage components in the rotor reference frame are

( ) ( )

If the values of a function f and its k. derivatives f ^(k)(x) are known in a certain point a, the Taylor series gives an approximation for the function value in the neighbourhood of the known function value f(a),

In the case of the co-ordination transformation according to Eq. (4.39) Taylor series is very useful, because the transformation itself contains the values of the function and its derivative. Thus time consuming trigonometrical calculations can be avoided in the fast control cycles. The estimates for the trigonometric functions are

sin( ( )) sin( ( )) cos( ( )) cos( ( )) cos( ( )) sin( ( ))

b m

b m

ϑ ϑ ϑ ω ω

ϑ ϑ ϑ ω ω

t t t t t

t t t t t

+ = + ⋅ ⋅

+ = − ⋅ ⋅

∆ ∆

∆ ∆ (4.40)

Mechanical time constants are large compared to those of the electrical system, and the angular speed ωm can be assumed to be constant during the approximation in Eq. (4.40). A typical value for the maximal value for ∆t in Eq. (4.40) is 1 ms. With Eqs. (4.38) and (4.40) fast estimates for the stator flux linkage components can be calculated in the rotor reference frame

( ) ( ) ( ( ) ) ( ) ( ( ) )

( ) ( ) ( ( ) ) ( ) ( ( ) )

ψ ψ ϑ ψ ϑ

ψ ψ ϑ ψ ϑ

sd sx sy

sq sy sx

cos sin

cos sin

t t t t t t t t t t

t t t t t t t t t t

+ = + + + + +

+ = + + − + +

∆ ∆ ∆ ∆ ∆

∆ ∆ ∆ ∆ ∆ (4.41)

Eq. (4.41) can be calculated with high speed in a real time control, e.g. within a few hundred microseconds cycle, since it includes only a few DSP processor operations.

In the field weakening range another possibility for the stability control is to increase the stator flux linkage modulus reference. If the voltage reserve is reduced to the theoretical minimum, the stator flux linkage vector can no more rotate with an over synchronous speed, and thus the load angle can no more increase. Further comparison between these two methods is presented in the simulations.

The presented methods assume, that the synchronous machine is working in the motor mode, where the electrical power is converted to the mechanical power. If the machine is working in the generator mode, the previous assumption is not valid, and the methods require some modifications.

The generator mode is however not analysed in this work.