Journal Archive

It was previously stated that the magnitude and phase of the motor voltage are relatively stable until the next cycle, thus they are assumed to be fixed for interpretation. Additionally, for geometric interpretation/explanation of inductor current, specific values are assumed for magnitude and phase. However, during actual system application (Simulation), values detected by current sensors are utilized.^5-8)

Predicting the next cycle current based on fixed input voltage and assumed current follows the calculation process outlined below and is represented in Fig. 6.

Fig. 6 
Predict current (Ni) in the next cycle

• ① The α and β components of the motor regenerative voltage u_L are obtained.
  uLα=ULcos⁡θ,uLβ=ULsin⁡θ
• ② The α and β components of the selected inverter voltage (u_S) are obtained in the same manner as in ① above.
• ③ α, β of di of inductor are obtained through u_Lα, u_Lβ, u_Sα, u_Sβ
  diα=uLα-uSαLdt,diβ=uLβ-uSβLdt
• ④ The α and β components of the assumed inductor current are obtained in the same way as in ①
• ⑤ The α and β components of the assumed current are added to the α and β components of di, respectively, to obtain i_α and i_β in the next cycle (next cycle current prediction).

Fig. 7 shows the trajectory of the current changes when Sector 1 is selected, while Fig. 8 illustrates the scenarios when Sectors 1 through 6 are chosen. Fig. 9 depicts the trajectory of the predicted p and q values for the next cycle when Sector 1 is selected, and Fig. 10 shows the scenarios for Sectors 1 through 6.

Fig. 7 
Changes of predicted next current according to sector change

Fig. 8 
Predict current (Ni) in the next cycle @Sector 1~6

Fig. 9 
Change of P and Q according to sector change

Fig. 10 
Change of P and Q according to sector change(Sector 1~6)

Now, we are able to determine the P and Q for the next cycle. As mentioned earlier, by predicting the values of p and q according to the change in sectors and continuously selecting the sector closest to the reference P and Q values through Distance Calculation, we can effectively control p and q. This method allows for efficient control without the need for additional controllers like PID. Distance Calculation uses a standard formula, as shown in Eq. (3), to calculate the distance between two points. From the calculated distances of p and q for each sector, we select the sector with the p and q values closest to the reference P and Q. Although the block diagram may appear more complex compared to a typical PID algorithm, this simple comparison algorithm is much more advantageous in terms of computation.

Fig. 11 depicts the block diagram of the algorithm, while Fig. 12 shows the waveform of controlling P to 30 and Q to 10 using the predictive method for the next cycle without a controller. Typically, Q is controlled to 0, but it was controlled to 10 to demonstrate the characteristics of the proposed algorithm.

Fig. 11 
Block diagram of predictive PQ control

Fig. 12 
P/Control simulation (Sector selection)

The switching frequency is not determined by the control cycle of the MCU, but rather by how quickly a sector is selected and changed. Therefore, it can be confirmed through actual operation or simulation. In this system, it was measured at 7.26 kHz.

While the previous section described the most efficient method for selecting sectors to control p and q to the desired values, this section proposes a method to directly calculate vectors that satisfy the desired p and q instead of selecting sectors. Once the vectors are obtained, they can be implemented using SPWM(Sinusoidal Pulse Width Modulation).

• ① The α and β components of the motor regenerative voltage u_L are obtained.
  uLα=ULcos⁡θ,uLβ=ULsin⁡θ
• ② Obtain the α and β components of the current required to achieve the q and p command values.
  p=uLαiα+uLβiβq=uLβiα-uLαiβiα=uαp+uβquα2+uβ2iβ=uβp-uαquα2+uβ2
• ③ Find the difference between the current obtained in (2) and the present current to obtain the necessary α and β components of di.
• ④ By using the α and β components of di, the α and β components of the inverter voltage U_S are obtained.
  uSα=uLα-LdiαdtuSβ=uLβ-Ldiβdt

As shown in Fig. 13, the simulation results are shown in Fig. 14. Simulation conditions were set at switching frequency 20 kHz, P = 30, and Q = 10.

Fig. 13 
p and q control block diagram implemented by p and q prediction method

Fig. 14 
p, q control simulation implemented by P and Q prediction method

When controlling with a switching and control frequency of 20 kHz and setting P to 30 and Q to 10, both sector selection and Sinusoidal Pulse Width Modulation (SPWM) methods exhibit the characteristics/features shown in Fig. 15 below.

Fig. 15 
Sector selection (Left), sinusoidal vector calculation/SPWM (Right)

Figs. 16~19 show comparison of 2 types of control algorithm suggested above. As you can see in Fig. 17 and Fig. 19, the PQ control algorithm using sector selection method has higher harmonic distortion than SPWM method.

Fig. 16 
Harmonic wave forms(Sector selection)

Fig. 17 
Harmonic size according to order(Sector selection)

Fig. 18 
Harmonic wave forms(SPWM)

Fig. 19 
Harmonic size according to order(SPWM)

Due to the assumption that the input voltage is almost constant in the control period (50us at 20kHz switching), a small steady-state error occurs(Table 1). It can be simply improved by adding calculated offset value because the error is not a continuously changing value as following block diagram and simulation result(Fig. 20, Fig. 21).

Fig. 20 
System block diagram(Offset algorithm)

Fig. 21 
Simulation result(Offset algorithm)

As depicted in Figs. 22~25 below, a simple PID controller can be added instead of the Offset Controller.

Fig. 22 
System block diagram with PID controller(Sector selection)

Fig. 23 
Simulation result(Sector selection, PID)

Fig. 24 
System block diagram with PID controller(SPWM)

Fig. 25 
Simulation result(SPWM, PID)