Compensation of unbalanced and nonlinear loads by active filter on the DC side of inverter

Abstract: An inverter DC side active filter system is introduced. This system is effective in eliminating pulsating currents caused by unbalanced and nonlinear loads at the inverter input terminals. For three-phase four-wire inverters, zero-sequence currents in the neutral line are also eliminated. This system has the characteristics of fast response speed and high efficiency. In the case of high unbalanced or nonlinear loads, excessive capacitive kVA is not required. Analysis, simulation and experiments show that this DC side active filter system is effective.

Keywords: inverter; active filter; compensation

1 Introduction

A survey shows that many three-phase variable frequency power supplies supply power to unbalanced loads or nonlinear loads to varying degrees during use. When the inverter carries an unbalanced or nonlinear load, the DC input current will have a pulsating component that is twice the inverter operating frequency. There are two ways to filter out this secondary harmonic component: one is to use L, C passive filters, and the other is to use active filters. The disadvantages of using passive filters are large size and weight, high price, high operating costs, poor dynamic characteristics and transition process tracking accuracy, and sometimes may cause system oscillation. The advantages of using active filters are:

——It can attenuate all harmonics and it is easier to suppress low-order harmonics;

——It can compensate for frequency changes and distortion effects, and improve the stability and dynamic response speed of the system;

——Good filtering effect and low cost;

——It only works on harmonics (ripples) and has a relatively small capacity;

——Small size and weight, high efficiency.

There are two types of active filters, one is series type and the other is parallel type. The former suppresses ripple voltage, while the latter suppresses ripple current. The parallel active filter suppresses DC ripple by separating the harmonic current generated by the harmonic source and offsetting it with the current generated by the active filter. Therefore, the parallel active filter is more suitable for eliminating the second harmonic current component caused by unbalanced load or nonlinear load. The simulation and experimental results show that the filtering effect is very significant.

In order to enhance the filtering effect, P. Enjeti and S. Kim proposed in the literature [2] to adopt a comprehensive filtering technology, that is, for the three-phase three-wire inverter, L, C passive filtering and single-phase full-bridge DC parallel active filtering are used; for the three-phase four-wire inverter, in addition to the above two-stage filtering, a single-phase half-bridge DC parallel active filter is added to form a three-stage filtering.

Figure 2 DC side active filter circuit of three-phase three-wire inverter

2 For three-phase three-wire inverter

The three-phase three-wire inverter using the output filter is shown in Figure 1. When the PWM control method is adopted, the Fourier series expansion of the three symmetrical switching functions (interface signals) is

SW==（1）

Using PWM control, when n=3, 5, 7, 11, 13..., we can set An=0, and the inverter output line voltage is

Un==Udc·(2)

Where: Udc is the DC voltage. The three-phase output line voltage of the inverter can be obtained by equation (2), and the input current of the inverter is

Iin=I1＋I2＋I3（3）

In the formula: I1=SW1·Ia; I2=SW2·Ib; I3=SW3·Ic.

Therefore, Iin = SW1·Ia + SW2·Ib + SW3·Ic (4)

Where: Ia, Ib and Ic are the output currents of each phase of the inverter. For a three-phase three-wire inverter

Ia＋Ib＋Ic=0（5）

For a balanced linear load, the three-phase currents Ia, Ib, and Ic are approximately sinusoidal waves with the same amplitude and a phase difference of 120°. Therefore, the input current of the inverter can be deduced from equation (4). This current will mainly consist of a DC component and harmonic components with the same switching frequency. These high-order harmonics are effectively filtered out in the DC link.

2.1 Unbalanced Load

Unbalanced load is not an ideal working state. For the unbalanced load state of the inverter "C phase disconnected" (see Figure 2). Obviously (6)

Assume that the load current is approximately sinusoidal

Ia I·sin(ωt＋θ)（7）

Where: θ is the load displacement factor angle.

For this situation, the input current Iin of the inverter can be calculated. From equations (6), (7) and (4), when n=1, we get Iin 〔cos(θ－)－cos(2ωt＋θ＋)〕（8）Equation (8) shows that for unbalanced loads (such as phase C disconnection), the input current of the inverter is composed of a DC component·cos(θ－) and a pulsating component·cos(2ωt＋θ＋) twice the inverter operating frequency. The pulsating component is caused by the unbalanced load, which generates a circulating current on the DC link capacitor and constitutes reactive volt-ampere. If the capacitance value in the DC link is large enough, there will be no distortion (ripple) of the DC voltage, resulting in a decrease in the overall performance of the inverter. The reactive volt-ampere in the DC link is proportional to the imbalance of the inverter output load. The greater the imbalance, the greater the pulsating component of the input current, the larger the value of the DC capacitor required, and the greater the impact on the overall performance of the inverter.

The consequences for non-linear loads (such as rectifier circuits) at the inverter output are the same as for unbalanced loads.

2.2 Filtering principle of DC side active filter

The DC side active filter circuit composed of switches S7~S10 and inductor L is shown in Figure 2. It and the passive filter composed of LoCo in the DC loop form a comprehensive filtering system to compensate for the influence of unbalanced load and nonlinear load, and use active filter to reduce the burden of LoCo passive filter. The working principle of DC active filter is introduced below.

As shown in Figure 2, the DC active filter obtains the DC link

Compensation of unbalanced and nonlinear loads by active filter on the DC side of inverter

The total input current is:

Iin.f=I4＋I5（9）

Assuming that the “C phase disconnection” described above is used as the unbalanced load, the switches S7 to S10 in the filter form a single-phase full-bridge filter and adopt PWM control. The operating frequency of the filter is the same as the operating frequency of the subsequent inverter. Then the voltage between the filter points P and Q is: UPQ = Bnsinn (ωt + ) (10)

For PWM modulation, when n=3, 5, 7, 11, 13…, Bn=0 can be set. The filter current IPQ can be considered close to sinusoidal. The harmonics of the voltage UPQ are fully attenuated to zero by the inductor L, so when n=1, it can be obtained from equation (10): IPQ·sin(ωt＋－)（11）

The resistance in the inductor L is very small and can be considered to be equal to zero, so IPQ lags behind UPQ by nearly 90°. The input current of the filter is calculated by equation (4), which is obtained by the single-phase full-bridge inverter that constitutes the filter, that is, equation (9). The input current expression of the active filter can be obtained as follows: Iin.f=·cos(2ωt＋2＋)（12）Therefore, the active filter must use equation (12) to compensate for the unbalanced load (for example, phase C is disconnected) to offset the cos(2ωt＋θ＋) component in Iin at the inverter input end (8). When fully compensated, the second harmonic components in equation (12) and equation (8) are equal, which can be obtained: cos(2ωt＋2＋)=cos(2ωt＋θ＋)

From this equation, we can solve =, B1 = (13) 2  + = θ +,  = - (14)

Therefore, as long as the active filter works according to equations (13) and (14), the purpose of eliminating the second harmonic pulsation component in equation (8) can be achieved. At this time, the input current of the inverter is Ii=Iin＋Iin.fcos(θ－)（15）

Formula (15) shows that for the unbalanced load such as "C phase disconnection", the second harmonic component contained in Iin in formula (8) is effectively eliminated in the DC input current Ii, achieving the compensation purpose. Since the impact of other forms of unbalanced loads or nonlinear loads on the DC input current is the same as that of the "C phase disconnection" unbalanced load [3], the DC side active filter method can also achieve the compensation purpose.

The control circuit block diagram of the DC side active filter is shown in Figure 3. The two-state hysteresis current tracking control is adopted. The deviation between the reference current Iref≡0 and the DC input current Ii is filtered out by the high-pass filter and used as the input of the two-state hysteresis comparator. The output is used to control the on and off of S7~S10, so that Iin.f tracks and eliminates the second harmonic component in Ii.

2.3 Simulation Results

The simulation results are shown in Figures 4, 5 and 6. Figure 4 shows

(b) Waveform of line current ia

(c) Waveform of input current iin

(d) Spectrum of input current iin

Figure 4 Figure 2 Voltage and current waveforms of the inverter

(a) Waveform of output voltage uab

Figure 2 shows the voltage and current waveforms of the inverter, where Figure 4 (a) is the waveform of the output voltage uab; Figure 4 (b) is the waveform of the line current ia; Figure 4 (c) is the waveform of the input current iin; Figure 4 (d) is the spectrum of the current iin. Figure 5 shows the waveform of the active filter in the three-phase three-wire inverter, where Figure 5 (a) is the waveform of the voltage uPQ; Figure 5 (b) is the waveform of the input current iin.f; Figure 5 (c) is the spectrum of the filter input current iin.f. Figure 6 shows the waveform and spectrum of the synthetic current ii=iin＋iin.f, where Figure 6 (a) is the waveform diagram and Figure 6 (b) is the spectrum.

3For three-phase four-wire inverter

The main circuit of the three-phase four-wire inverter is shown in Figure 7. This is a four-wire output inverter that uses the midpoint of the DC input power supply voltage as the neutral point. For a balanced linear load, the neutral line current is equal to zero, but for an unbalanced nonlinear load, there is a zero-sequence current on the neutral line that flows through the center tap of the DC link filter capacitor. This zero-sequence current is a fundamental frequency current caused by an asymmetric load, and the reactive volt-ampere formed by it can cause voltage distortion on the Udc potential.

The switching function analysis method mentioned above is also applicable to three-phase four-wire inverters. This method is used to analyze three-phase four-wire inverters.

3.1 Unbalanced load and active filter compensation

On the DC side, the main circuit of the three-phase four-wire inverter using active filter compensation is shown in Figure 8 (a). Assuming that the inverter uses "C phase disconnection", the a-phase and b-phase loads are the same unbalanced load, and assuming that the load current is approximately sinusoidal, the input current of the inverter is

Iin=SW1·Ia＋SW2·Ib（16）

In the formula: IaI·sin(ωt+θ) (17) IbI·sin(ωt+θ-) (18)

From equations (1), (17) and (18), when n=1, equation (16) is

Figure 7 A type of three-phase four-wire inverter

(a) Waveform of voltage uPQ

(b) Waveform of input current iin.f

(c) Spectrum of input current iin.f

Figure 5 Waveform of active filter in three-phase three-wire inverter

(b)ii Spectrum

Figure 6 Waveform and spectrum of the synthetic current ii=iin＋iin.f

Compensation of unbalanced and nonlinear loads by active filter on the DC side of inverter

(a)ii waveform

(a) Main circuit diagram

(b) Vector diagram

Figure 8 Three-phase four-wire inverter using DC active filter for compensation

IinA1·I·cosθ＋sin(ωt＋θ－)－cos(2ωt＋θ＋)（19）
The value of the neutral line current IN is

IN=Ia＋Ib（20）

Substituting equations (17) and (18) into equation (20), we obtain IN = I sin (ωt + θ -) (21)

Active filter 1 uses switches S7, S8 and inductor L1 to control the generated current INf to eliminate IN. Switches S7 and S8 use PWM control, so URN = Udc·Bn·sinn(ωt+1) (22)

The current INf caused is INfsin(ωt＋1－)－sin(ωt＋1＋)(23)

From equations (21) and (23), the condition for INf to offset IN is determined as I·sin(ωt＋θ－)=sin(ωt＋1＋)I=；B1=（24）θ－=1＋；1=θ－（25）

Therefore, as long as the above conditions are met, the fundamental frequency zero-sequence current in the neutral line caused by the unbalanced load of "C phase disconnection" will be eliminated. This compensation result will also be used in the following study. The current Iin.f1 is derived using the switching function method mentioned above: Iin.f1 sin(ωt＋1－)－cos(2ωt＋21－)（26）

Substituting equations (24) and (25) into equation (26), we obtain Iin.f1 · sin(ωt＋θ－)－ · cos(2ωt＋2θ－) (27)

Due to the action of active filter 1, the input current Iin.r is

Iin.r=Iin.f1＋Iin（28）

Substituting equations (19) and (27) into equation (28), we obtain Iin.r = A1·I·cosθ－cos(2ωt＋θ＋)－cos(2ωt＋2θ－)（29）

Formula (29) shows that the fundamental component in Iin is eliminated due to the action of active filter 1. Therefore, the synthetic current Iin.r consists of a DC component and a second harmonic component. To eliminate the second harmonic component in Iin.r, a full-bridge active filter 2 composed of switches S9-S12 and inductor L2 should be used to eliminate it. The elimination method is to use active filter 2 to generate an Iin.f2 that is equal to the second harmonic component and opposite in phase, so that Iin.f2 and the second harmonic component are offset.

The last two terms in equation (29) are the sum of the second harmonics, namely -cos(2ωt＋θ＋)-cos(2ωt＋2θ－)

The value of is obtained from the vector diagram in Figure 8 (b) and the cosine theorem.

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Control Technology

(a) Control circuit of half-bridge active filter

(b) Full-bridge active filter control circuit

Figure 9 Control circuit of two active filters on the DC side

a =
(30)

From the sine theorem, we get = So sinα =

α=arcsin()

(31)

Therefore: -cos(2ωt+θ+)-cos(2ωt+2θ-)=-a·cos(2ωt+α+θ+) (32)

Substituting equation (32) into equation (29), we get: Iin.r = A1·Icosθ－a·cos（2ωt＋α＋θ＋）（33）

From formula (12), we get: Iin.f2=cos(2ωt+2+) (34)

Therefore: Ii=Iin.r+Iin.f2=A1·I·cosθ-a·cos (2ωt+α+θ+)+cos (2ωt+2+)(35)

From equation (35), the condition for eliminating the second harmonic component in Iin.r is a=; B1= (36) α + θ + = 2 +; 2 = α + θ - (37)

When the conditions of equations (36) and (37) are satisfied, the second harmonic component in equation (29) can be eliminated, making Ii=A1·I·cosθ.

For three-phase four-wire inverters, two active filters must be used, namely active filters 1 and 2. The combination of these two filters can eliminate the low-frequency pulsating current and reactive volt-ampere caused by unbalanced loads in the DC link. At the same time, the current INr in the neutral line is also eliminated, so there is no fundamental frequency zero-sequence current flowing through the DC link capacitor. As mentioned earlier, nonlinear loads and unbalanced loads have the same impact on the DC link input current, so the above method also compensates for the impact of nonlinear loads. For detailed descriptions, please refer to relevant literature.

The control circuits of the half-bridge active filter 1 and the full-bridge active filter 2 in the three-phase four-wire inverter are shown in FIG9 . FIG9 (a) is the control circuit of the half-bridge filter 1, and FIG9 (b) is the control circuit of the full-bridge filter 2. Similar to FIG3 , the control circuit shown in FIG9 also adopts two-state hysteresis current tracking control. FIG9 (a) makes INr track Iref.1≡0 to eliminate the neutral line current; FIG9 (b) makes the second harmonic component in Ii track Iref.2≡0 to eliminate the second harmonic component in Ii.

3.2 Simulation Results

The three-phase four-wire inverter using half-bridge active filter 1, full-bridge active filter 2 and passive filter LoCo for comprehensive filtering compensation is simulated, and the results of Figures 10 to 18 are obtained. Figure 10 shows the voltage and current waveforms of the three-phase four-wire inverter, where Figure 10 (a) is the waveform of the output voltage uaN; Figure 10 (b) is the waveform of the line current ia; Figure 10 (c) is the waveform of the line current ib. Figure 11 shows the current waveform of the neutral line current iN=ia+ib of "C phase disconnection" and the spectrum of iN. Figure 12 shows the waveform of the input current iin and its spectrum. Figure 13 shows the waveform of the active filter 1, where Figure 13 (a) is the waveform of the voltage uRN, and Figure 13 (b) is the waveform of the current iNf. Figure 14 shows the waveform of the synthetic neutral line current iNr=iNf+iN and its spectrum. Figure 15 shows the waveform of the current iin.f1 provided by the DC link to the active filter 1 and its spectrum. Fig. 16 shows the waveform and spectrum of the DC link current iin.r=iin+iin.f1. Fig. 17 shows the waveform of the current iin.f2 provided by the DC link to the active filter 2. Fig. 18 shows the waveform and spectrum of the synthetic input current ii.

4 Conclusion

When the three-phase inverter carries an unbalanced or nonlinear load, the DC output current will have a pulsating component that is twice the inverter operating frequency.

Compensation of unbalanced and nonlinear loads by active filter on the DC side of inverter

(a) Waveform of output voltage uaN

(b) Waveform of line current ia

Figure 10 Voltage and current waveforms of three-phase four-wire inverter

(c) Waveform of line current ib

(a) iN waveform

(b) iN spectrum

(a) iin waveform

(b) Spectrum of iin

Figure 12 Waveform and spectrum of input current iin

(a) Voltage uRN waveform

(b) Waveform of current iNf

Figure 13 Waveform of active filter 1

Figure 11 Waveform and spectrum of neutral line current (iN=ia＋ib) when “phase C is disconnected”

The use of a DC side active filter can effectively eliminate the pulsation component. The reason why a half-bridge active filter 1 is used in the three-phase four-wire inverter shown in Figure 8 (a) is that it can eliminate the neutral line current.

The method proposed by P. Enjeti and S. Kim in 1991 [2] to use comprehensive filtering technology to eliminate the influence of unbalanced and nonlinear loads is a very practical and effective method. In fact, many three-phase variable frequency power supplies or UPS have unbalanced or nonlinear loads to varying degrees during use. In order to improve the overall performance of the inverter, it must be compensated. Therefore, the comprehensive filtering compensation technology of Enjeti and Kim introduced in this article is of promotion significance.

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Control Technology

Figure 15 Waveform and spectrum of current iin.f1

(a) Waveform of iNr

(b) Spectrum of iNr

Figure 14 Waveform and spectrum of the synthetic neutral line current iNr=iNf+iN

(a) Waveform of iin.f1

(b) Spectrum of iin.f1

Figure 18 Waveform and spectrum of synthetic input current

(a) Waveform of iin.r

(b) Spectrum of current iin.r

Figure 16 Waveform and spectrum of DC link current iin.r=iin＋iin.f1

Figure 17 Waveform of current iin.f2

(a) Waveform of ii

(b) Spectrum of ii

Compensation of unbalanced and nonlinear loads by active filter on the DC side of inverter