3.1. SVPWM Compensation Strategy Considering NPP Fluctuation
By analysing the output level status of the ESTPI, the ESTPI output voltage considering the fluctuation of the NPP can be obtained as follows:
where,
uao,
ubo, and
uco are the ESTPI output voltages of phases A, B, and C, respectively.
is the NPP fluctuation of the DC bus. Therefore, the space voltage vector is expressed as follows:
The basic voltage vectors coordinate considering NPP fluctuation can be obtained by combining Formulas (1), (12) and (13), as shown in
Table 4.
As can be seen in
Table 4 and
Figure 4, it is clear that the NPP fluctuation will result in an uneven distribution of space voltage vectors, which will cause imbalances of ESTPI output. This will affect the grid-connected power quality and narrow the linear modulation area, and the NPP fluctuation is an inherent phenomenon. Unfortunately, there are no redundant vectors available for NPP control in ESTPI. Imbalance of ESTPI output is inevitable.
It is obvious that the NPP fluctuation of the DC bus will change the distribution of basic voltage vectors. If the method in
Table 2 is still used to synthesize the reference voltage vector, the grid-connected power quality will deteriorate. At the same time, choosing the DC-link capacitor’s capacity according to the lowest limitation is a common cost-saving measure in practical industrial applications, which will worsen the fluctuation of NPP and further deteriorate the power quality. Therefore, it is necessary to consider the impact of NPP fluctuation on the synthesis of the reference voltage vector. Based on the volt-second balancing principle, the reference voltage vector synthesis should be compensated when the NPP fluctuates.
The duration time of the fundamental vector is recalculated considering NPP fluctuation. The calculation method remains the same as presented in Formulas (5) and (6), with the exception that the vector coordinates need to incorporate variations in NPP. The reference voltage vector synthesis rules are outlined in
Table 5.
To examine the compensation effect of reference voltage vector synthesis considering the NPP fluctuation, the grid-connected currents and voltages of the DC-link capacitors are tested. The main circuit parameters are shown in
Table 6, with the grid-connected currents at 6A and the capacitances of the DC-link are 820 μF. The simulation results are illustrated in
Figure 5.
As shown in
Figure 5a, if the NPP fluctuation is not considered in the synthesis of the reference voltage vector, the NPP fluctuation can remain balanced, but the distortion of the grid-connected currents is severe. As shown in
Figure 5b, when NPP fluctuation is taken into account in the synthesis of the reference voltage vector, the grid-connected currents have good sine waves at the beginning, but the NPP gradually deviates from the equilibrium point until exceeding the linear modulation region. Then, the reference voltage vector no longer satisfies the principle of volt-second balancing and results in a serious distortion of the grid-connected currents. In summary, considering the NPP fluctuation can improve the quality of three-phase grid-connected currents at the beginning, but NPP will still deviate from the equilibrium point. Thus, it is necessary to investigate the reasons for the NPP fluctuation increasing and optimize the compensation of the space vector modulation.
3.2. Optimization of SVPWM
To ensure the inverter is still running and providing power to the grid after the single bridge arm fault, it is necessary to find the reasons for the NPP’s significant fluctuation, then design the effective control strategy. Combine
Table 5, Formulas (5) and (6), and the NP current of the DC bus with NPP fluctuation can be expressed as Formula (14):
where,
K21,
K22,
K51,
K52 are defined as Formula (15).
Comparing Formulas (9) and (14), it is evident that the addition of compensation (Δ
u/
Vdc) causes the NP current to no longer satisfy the half-wave symmetry in Formula (14). Therefore, the NPP of the DC bus deviates from the equilibrium point. To maintain Formula (14) to satisfy
, the following relationship should be maintained, as shown in Formula (16):
For Equation (16) to hold true, the voltage deviation of the DC link (Δu) should not contain the DC component. Thus, the space vector modulation of adding the NPP fluctuation should be further refined.
To eliminate the DC component of Δ
u, the DC component should be extracted first. A first-order low-pass filter is designed to obtain the DC component in this work. In the s-domain, the first-order low-pass filter can be expressed as follows:
where
ωc is its cutoff frequency. According to Formula (11), it can be seen that the main AC component in the NPP fluctuation of the DC bus is the fundamental component with an angular frequency of approximately 314 rad/s. To extract the DC component, the cutoff frequency of the low-pass filter must be lower than the fundamental angular frequency. According to low-pass filter phase-frequency characteristics, when the signal passes through the filter, a large phase shift will be generated at a lower cutoff frequency, which will affect the filtering accuracy of the DC component. In this paper, the cutoff frequency is selected as 80 rad/s.
To prevent the DC bus NPP from deviating too much from the equilibrium point, a hysteresis comparator is designed to further adjust the space vector modulation compensation. The NP current is obtained by Formula (18):
According to Formula (18), in sector I, III, IV, and VI, if the value of the grid-connection system parameters (Im and φ) is determined, can be considered as a constant, and cosφ is greater than or equal to 0 within a range of φ between −90° and 90°. According to Formulas (11) and (18), NPP fluctuation can be controlled by adjusting the compensation value of the reference voltage vector synthesis in sectors I, III, IV and VI. However, it is important to note that the modulation can be considered as a proportional link throughout the entire grid-connected control, and Vref is the output of the grid-connected controller that converts into the switch driver signal via space vector modulation. Consequently, the hysteresis comparator is designed to adjust the compensation value, only when the NPP of the DC bus contains the DC component and deviates large enough from the equilibrium point, the hysteresis comparator will compensate the voltage vector. The compensation value in Formula (18) is adjusted to prevent the NPP of the DC bus from deviating from the equilibrium point, achieving the synthesis of the reference voltage vector lying in the linear modulation region.
Given
, the output of the hysteresis comparator should be a suitable value that can compensate for the NPP fluctuation, otherwise 0,
umax is the maximum allowable voltage of NPP fluctuation in the linear modulation area. Since the DC component increases as the NPP deviates from the equilibrium point, the output of the hysteresis comparator also increases, and more robust control of the NPP of the DC bus can be realised. Therefore, the output of the hysteresis comparator can be designed according to the DC component of NPP fluctuation, which can be represented as follows:
where,
K = |
A0|+1,
A0 is the DC component of NPP fluctuation.
In conclusion, the optimised compensation value is shown in Formula (20):
Based on the above analysis, a fault-tolerant control strategy with SVPWM compensation optimisation is constructed, as shown in
Figure 6.