For an Active Power Filter (APF), it is important to control the stability of the DC-link capacitor voltage stability. As mentioned in [1A. Tarkiainen, R. Pollanen, M. Niemela, and J. Pyrhonen, "DC-link voltage effects on properties of a shunt active filter", In: IEEE Power Electronics Specialists Conference, Aachen, Germany, 2004, pp. 3169-3175.
[http://dx.doi.org/10.1109/PESC.2004.1355342] -4A. Chaoui, J.P. Gaubert, F. Krim, and L. Rambault, "On the design of shunt active filter for improving power quality", In: IEEE International Symposium Industrial Electronics, Cambridge, UK, 2008, pp. 31-37.
[http://dx.doi.org/10.1109/ISIE.2008.4677277] ], the premise condition for APF working normally is that the DC-link voltage should be greater than the line voltage peak, and the minimum DC-link voltage in the linear modulation range is given. In [1A. Tarkiainen, R. Pollanen, M. Niemela, and J. Pyrhonen, "DC-link voltage effects on properties of a shunt active filter", In: IEEE Power Electronics Specialists Conference, Aachen, Germany, 2004, pp. 3169-3175.
[http://dx.doi.org/10.1109/PESC.2004.1355342] , 2G. Zhao, J. Liu, and Y. Xin, "Analysis and specification of DC side voltage in parallel active power filter regarding compensation characteristics of generators", In: IEEE Power Electronics Specialists Conference, Rhodes, Greece, 2008, pp. 3495-3499.
[http://dx.doi.org/10.1109/PESC.2008.4592496] ] the influence of DC-link voltage reference value (the grid phase voltage RMS) on the compensation performance of APF is analysed, and the general control scheme selected for DC voltage is further proposed in [5W. Shi, Q. Wang, and Q. Xing, "Research on current tracking control method for four-leg APF", J. Electr. Instrument, vol. 12, pp. 1162-1169, 2013., 6Wenqiang Li, and Yang Lu, "Research and implementation of direct measurement technique of current speed", J. Electron. Meas. Instrum., vol. 26, no. 1, pp. 43-48, 2012.
[http://dx.doi.org/10.3724/SP.J.1187.2012.00043] ]. Two important results related to DC voltage control of APF have been gotten [7G. Jie, D. Jiang, and Y. Zhou, "AC and DC current hybrid control strategy for modular multilevel converter", Dianli Xitong Zidonghua, vol. 35, no. 7, pp. 42-47, 2011.-9M. Ban, S. Ke, and J. Shen, "A novel circulating current suppressor for modular multilevel converters based on quasi-proportional-resonant control", Dianli Xitong Zidonghua, vol. 38, no. 11, pp. 85-89, 2014.]. First, it is indicated that the control of DC voltage directly affects the compensation performance of APF. Second, the value of DC voltage influences the power loss of APF.

Taken the three-phase three-wire APF as an example, this paper firstly introduces the design of the DC voltage controller [10J. Li, X. Wu, and X. He, "Research on the active power damping control of LCL-APF based on concentrated compensation", Power Syst. Prot. Control, vol. 43, no. 5, pp. 101-106, 2015.-13Z. Qu, and Y. Wang, "Four-terminal Impedance Bridge based on current-comparator", Yiqi Yibiao Xuebao, vol. 32, no. 9, pp. 1987-1992, 2011.]. Then the relationship between DC voltage and the power loss as well as the compensation performance of APF is analysed [14X. Zhang, Y. Liu, and W. Rui, "A novel active damping control strategy for PWM converter with LCL filter", Trans. China Electro Tech. Soc., vol. 26, no. 10, pp. 188-192, 2011., 15"KAURA V. A novel control to actively damp resonance in input LC filter of a three-phase voltage source converter", IEEE Trans. Ind. Appl., vol. 33, no. 2, pp. 542-550, 1997.
[http://dx.doi.org/10.1109/28.568021] ]. Finally, a new control scheme with a droop controller is developed to regulate DC voltage [16Z. Lin, D. Jiang, and Y. Zhou, "Control and modulation for APF-STATCOM based on cascaded H-bridge converter", Power Syst. Prot. Control, vol. 42, no. 7, pp. 91-96, 2014.].

Fig. (1) shows the control scheme of three-phase three-wire APF. Here, U_{dc_ref} is the reference value of DC voltage, U_dc is the sampling value, I_dcd is a regulating signal that is the difference between U_{dc_ref} and U_dc through voltage regulator. And i_dcd is added to the d-axis component of the output current reference. In fact, the d-axis DC component i_dcd, which represents the fundamental frequency components of the active currents, causes the real power exchange between AC and DC side.

Before designing the voltage controller, the conversion factors from AC input current i_d`i_q to DC output current i_dcd must be known. According to the mathematical model of APF, it has i_dc=1.5(S_di_d+S_qi_q). Here, the relationship between AC and DC current is time-varying because of two variables S_d and S_q. To simplify the process and design it with thelinear method, it is believed that i_q has completed the transient process reaching to 0 before a large change on the DC voltage, in which the value of i_q equals 0 in steady state, and i_q changes slightly in a dynamic process due to the current closed-loop. Based on this, the d-axis block diagram of the voltage outer-loop control can be seen in Fig. (2)

Fig. (1) The control scheme of three-phase three-wire APF.

Fig. (2) Axis block diagram of the voltage.

Where G_c(s) is the closed-loop transfer function of inner current, T_vf means the delay time constant of voltage feedback path (switching frequency is 9600Hz, or delay time is 104 us). K_vf means the amplification factor of the voltage feedback path (taking 1). K_if means the sampling coefficient of the output current (taking 16.384), and the capacitor C takes 20 mF.

Due to the second-order of the transfer function of current closed-loop G_c(s), it is better to lower the order of G_c(s) so as to make the synthesis of the voltage loop easier. The process is shown as follows:

The transfer function of the inner current control loop is:

(1)

The open-loop transfer function without the controller is:

(2)

Because T_vf and T_sf are time constant, so equation (2) can be simplified as bellow:

(3)

The transfer function of PI controller is:

(4)

Thus, the open-loop transfer function with PI controller is:

(5)

From the analysis above, it can be seen that equation (5) is a typical type II system, the controller parameters can be designed according to Oscillation index method. The key point of the Oscillation index method is to select parameters according to the practical engineering requirements for the closed-loop relative resonance peak. Based on this, it can be ensured the system performance index does not exceed a specified value, i.e. the maximum overshoot is not greater than a certain limit value, the corresponding phase margin is the biggest, and the amplitude margin is the best, also the adjusting time is the shortest. Fig. (3) shows the Bode diagram of the open-loop voltage with PI controller.

Fig. (3) Bode diagram of the open-loop voltage with PI controller.

From the Bode diagram in Fig. (3), a best fit between the two parameters can be found based on the minimum peak norm of amplitude-frequency characteristics in Oscillation index method. When the closed-loop resonant peak M_r reaches to the minimum, the relationship of each variable is obtained as:

(6)

The key to design the voltage loop is determining the bandwidth h. Practical experience shows that the width of medium-frequency h varies generally between 3 to 10. In addition, a larger h does no significant effects on decreasing M_r. When h=constant, it can be derived as:

(7)

So the parameters of the voltage controller can be gotten.

The PI controller of the voltage outer-loop is:

(8)

The power loss of APF mainly includes the following two aspects: the switching loss of PWM inverter and the output inductor hysteresis loss, copper loss, etc.

The output current of APF is higher harmonics, and including some switching ripple current. In general, the switching frequency is much higher than that of the output current, so the output inductor hysteresis loss is mainly proportional to the size of the switching ripple current as well as the area surrounded by the magnetization curve of the magnetic material. When magnetic material with small magnetization curve area and high permeability, such as amorphous material, is selected, the output inductor hysteresis loss will be reduced to a relatively low percentage.

However, switching loss of PWM inverter actually contains IGBT switching loss, the diode reverse recovery loss, and the conduction loss of IGBT as well as the diode.

Furthermore, the IGBT conduction loss is proportional to the drop voltage and the flowing current, and the drop voltage increases with the flowing current. So it can be concluded that the conduction loss of IGBT increases as the APF output current rises. And this is also suitable to the diode conduction loss.

The IGBT switching loss includes turn-on and turn-off loss, and both the two components increases with DC voltage in case of the constant flowing current and switching frequency. It does also apply to the diode reverse recovery loss.

The main components of these losses mentioned above are IGBT switching loss and diode reverse recovery loss. With the constant switching frequency and output current, any increase/decrease in DC voltage causes a relative increase/decrease in the power loss of APF.

In addition, due to the complexity of the switching loss simulation, the relationship between DC voltage and the power loss of APF will be verified depending on experiments.

Without considering the influence of current regulator and the error of harmonic extraction algorithm, the capability of compensating harmonic currents of nonlinear load is important to be considered in order to analyse the compensation performance of APF. Based upon the topology structure of APF and neglecting the inverter loss, according to equation (9), the output current of APF is relevant to the inductor drop voltage which can be easily obtained by a simple subtraction of the midpoint voltage of PWM inverter arms and the grid voltage.

(9)

In order to quantitatively analyse further, assuming the system is symmetrical, U_sm is the peak of the grid phase voltage, and ω is the power system angular velocity. And the three-phase source voltage can be defined as:

(10)

Suppose the load is the three-phase uncontrolled rectifier with LR connection. According to the characteristic of the nonlinear load, it only exists 6n±1 harmonic orders in the load current, where 6n+1 is the positive sequence component, and 6n-1 is the negative one. Without considering the influence of current regulator and the error of harmonic extraction algorithm, the APF output current and the PWM inverter output voltage can be established as bellow:

(11)

(12)

Here, I_f1m is the peak of APF output fundamental current, U_f1m is the peak of PWM inverter output fundamental voltage, I_fkm and I_flm mean the peak of APF output the k-th and l-th harmonic currents respectively, U_fkm and U_flm represent the peak of PWM inverter output the k-th and l-th harmonic voltages respectively, θ and φ signify the initial phase angle of each order component.

According to the definition of Park transform, a space vector synthesized by any three-phase variables x_a`x_b and x_c can be illustrated mathematically as:

(13)

Then equations (4-1) can be transfer to equation (14):

(14)

Substitute equations (10) ` (11) ` (12) into equations (13) ` (14), we can get:

(15)

Obviously, the real and the imaginary part are equal in both sides respectively, and it is true for any ωt. So equation (15) can be simplified as:

(16)

(17)

(18)

According to equations (16 - 18), the harmonics of APF output current depends on each harmonic of PWM inverter output voltage and the voltage of the power grid.

Moreover, PWM inverter amplitude modulation ratio M is the ratio of the length of output voltage resultant vector U_fr and half of the average DC voltage U_dc.

(19)

(20)

Considering compensating the load in the worst case, the maximum value of U_fr can be expressed by equation (21).

(21)

The maximal U_fr is the biggest PWM inverter output voltage necessary to compensate harmonic currents of nonlinear load under the worst condition. As for PWM inverter, an APF must output the voltage vector exceeding the maximum, so as to completely compensate harmonic currents of nonlinear load. Based on the above analysis and combining equations (19) and (21), we can get:

(22)

(23)

Equation (23) indicates the linearity range of space vector modulation. Suppose that the amplitude modulation ratio M is constant and equals to or more than 1.1547during the operation of APF, then formula (22) can be transformed to:

(24)

(25)

From the analysis of the above two equations, it can be concluded that:

(1) U_Δ determines the capacity of APF output harmonic current;

(2) With the same harmonic current RMS, the higher the order of harmonics is, the bigger PWM inverter output harmonic voltage is required;

(3) The influence on PWM inverter output harmonic voltage caused by the size of output inductor L is a factor to be considered when designing L.

Here to analyse the influence of U_Δ on the capability of APF output harmonic current. Assume that the designed output inductor L=0.3mH, the rated grid phase voltage RSM U_sn=220V, and the nominal DC voltage U_dcn=700V.Based on the above design conditions, the compensation performance of APF can be achieved to meet the requirements. If the grid voltage fluctuates in a range of 90%-110%, the capability of APF output harmonic current varies with the changed U_Δ.

As shown in Fig. (4), the vertical axis shows the peak of the maximum k-th harmonic current that APF can output with the corresponding grid voltage. When U_s =220V, the peak of maximum 5-th harmonic current is 197.5A, and when U_s=1.1U_sn=242V, the peak one is 131.4A, and it is 263.6Awhen U_s=0.9U_sn=198V. Comparing the grid voltage 1.1U_sn with U_sn, the capacity of APF output the maximum 5-th harmonic current declines by almost 33.5%. There is a similar relationship for other harmonics.

Fig. (4) The capability of APF output harmonics when the grid voltage varies.

In conclusion, without considering the impact of current regulator and the error of harmonic extraction process, U_Δ will influence on the compensation performance of APF. In the linear modulation range and with the constant modulation ratio, increasing DC voltage can enhance the compensation performance of APF, while decreasing DC voltage can reduce the performance. Similarly, any increase/decrease in the grid voltage leads to a relative decrease/increase on the APF compensating performance.

This paper firstly describes the design of DC voltage regulator in detail, and then the influence on the power loss and compensation performance of APF caused by DC voltage is analysed. According to the analysis`simulations and experiments, some important results have been gotten. Any increase/decrease in DC voltage can cause increase/decrease in the power loss of APF. Besides, without considering the influence of current regulator and the error of harmonic extraction process, the compensation performance of APF increases with the rising of DC voltage, as well as it decreases as the grid voltage rises. A higher DC voltage can improve compensation performance with the increased grid voltage. Also, on the basis of guaranteeing the compensation performance, a lower DC voltage can reduce power loss when the grid voltage decreases. Therefore, a new control strategy for adjusting DC voltage with the droop regulator is proposed. Adjusting DC voltage with the use of droop regulator can keep U_Δ constant as the grid voltage fluctuates, and what can be achieved the integrative optimization of APF’s power loss and compensation performance.