A Novel Hybrid Active Power Filter with Multi-Coupled Coils

This paper proposes a hybrid active power filter (HAPF) with multi-coupled coils, applied to a medium- and high-voltage power grid. The passive filter of the proposed HAPF adopts the structure of multi-coupled coils to compress the traditional multiple LC branches into one branch which presents the same harmonic impedance characteristics as the former multiple LC branches. In the active power filter of the HAPF, a coupled inductor, instead of a transformer, is used to connect with the passive filter. The coupled inductor has mutual inductances with inductors of the passive filter. Through spatial magnetic coupling, the active power filter can inject compensation current into the power grid to eliminate the residual harmonics and absorb active power from the power grid to maintain the DC capacitor voltage. When the active power filter is open-circuited or short-circuited, the filtering effect of the passive filter can still be guaranteed, which improves the reliability of the filter. The benefits of the proposed HAPF with excellent harmonic filtering performance are that the inductors occupy only 1/3 space as compared with traditional three-tuned LC filter, and very small power of the active power filter. The feasibility of the proposed HAPF is verified through simulations and experiment.


Introduction
In the power grid, nonlinear loads result in harmonic problems: increasing the power loss in the power system, interfering with the communication network, affecting the performance of high-precision devices, etc. [1,2].
The power filter is usually used to deal with harmonic problems in the power system, which can be classified as passive filter (PF), active power filter (APF), and hybrid active power filter (HAPF). PF is widely used in the power grid because of its simple structure, low cost, and mature technology [3][4][5]. However, the filtering effect of PF depends on its own element parameters and the grid parameters. Besides, PF usually has large volume, large required space. Compared with PF, the filtering effect of APF is not affected by grid parameters and the control of APF is flexible [6][7][8]. However, standalone APF has limited capacity and high cost, which is not suitable for high-voltage and a large-capacity situation.
In order to combine the advantages of both PF (large capacity, high reliability) and APF (excellent control performance), HAPFs of various topologies were proposed, which can reduce the capacity of active power filter in the medium-and high-voltage power grid [9][10][11][12][13][14][15][16][17][18][19][20][21][22]. There are two types of HAPFs: series HAPFs and shunt HAPFs. In series HAPF, the active power filter is connected to the power grid in series using a transformer [9]. The series HAPF has good filtering performance, but the fundamental current flows through the transformer, limiting the feasibility of the practical application.
The common topologies of shunt HAPF can be divided into the following three types: (1) The multi-coupled coils used in the passive filter of the HAPF compress the traditional multiple LC branches into one branch which presents the same harmonic impedance characteristics as the former multiple LC branches. The inductors occupy only 1/3 of the space as compared with the traditional three-tuned LC filter. (2) A coupled inductor, instead of a transformer, is used in the active power filter of the HAPF to connect with the passive filter. When the active power filter is open-circuited or short-circuited (even if the protection does not work and not cut off from the active power filter), the filtering effect of the passive filter can still be guaranteed. The capacity of the active power filter in the proposed HAPF is small. This paper is structured as follows: Firstly, the structure of the proposed HAPF is introduced. Secondly, the working principle of the HAPF is analyzed. Thirdly, the control strategy of hybrid compensation is proposed. Then, the effectiveness of the control strategy is verified through simulations. At last, an experimental platform is built and the feasibility of the HAPF is verified by experiments.

Passive Filter with Multi-Coupled Coils
The passive filter of the proposed HAPF is shown in Figure 1. Figure 1a represents the schematic diagram of a three-tuned passive filter with multi-coupled coils. In Figure 1a, i L is the load current, i S is the supply current, and i f is the filter branch current. C 3 , C 4 , and C 5 are the capacitors in passive filter. L 1 , L 2 , L 3 , L 4 , and L 5 are the inductors in the passive filter.

filter.
L1, L2, L3, L4, L5 have mutual inductances among one another, which is t difference between the passive filter in the proposed HAPF and the conventio tuned LC filter. Conventional multi-tuned LC filters ignore the coupling effec the inductors. To ensure this, the inductors in conventional multi-tuned LC filte be arranged separately, resulting in a large required space.
In this paper, a prototype of multi-coupled coils has been designed and tured, which is shown in Figure 1b, according to the design process in [5]. The tual inductances among the inductors. The multi-coupled coils are designed as structure of multiple coaxial round wire discs, whose required space is reduced of the n-tuned filter, the required space of the multi-coupled coils is about 1/n quired space of inductors in the traditional passive filter. At present, the capaci ally manufactured multi-coupled coils ranges from dozens of kvar to several M However, the requirements for assembly of the passive filter with multi-cou in Figure 1b are higher than that of the traditional LC filter. In this structure, th of coil turns determines the self-inductance. Moreover, the number of coil tur relative positions of the coils determine the mutual inductances together. The accurate installation of this structure is crucial for obtaining the required param The passive filter with multi-coupled coils also has the inherent drawba conventional passive filter. The filtering effect of the passive filter can be affec grid parameters. In severe cases, the passive filter may resonate at a certain According to the design principle of the passive filter, considering the frequenc the power grid and the equipment manufacturing error, the tuned frequency sive filter should be set 3-15% lower than the characteristic frequency [25]. L 1 , L 2 , L 3 , L 4 , L 5 have mutual inductances among one another, which is the biggest difference between the passive filter in the proposed HAPF and the conventional multituned LC filter. Conventional multi-tuned LC filters ignore the coupling effect between the inductors. To ensure this, the inductors in conventional multi-tuned LC filters need to be arranged separately, resulting in a large required space.
In this paper, a prototype of multi-coupled coils has been designed and manufactured, which is shown in Figure 1b, according to the design process in [5]. There are mutual inductances among the inductors. The multi-coupled coils are designed as a layered structure of multiple coaxial round wire discs, whose required space is reduced. For a set of the n-tuned filter, the required space of the multi-coupled coils is about 1/n of the required space of inductors in the traditional passive filter. At present, the capacity of actually manufactured multi-coupled coils ranges from dozens of kvar to several Mvar.
However, the requirements for assembly of the passive filter with multi-coupled coils in Figure 1b are higher than that of the traditional LC filter. In this structure, the number of coil turns determines the self-inductance. Moreover, the number of coil turns and the relative positions of the coils determine the mutual inductances together. Therefore, an accurate installation of this structure is crucial for obtaining the required parameters.
The passive filter with multi-coupled coils also has the inherent drawbacks of the conventional passive filter. The filtering effect of the passive filter can be affected by the grid parameters. In severe cases, the passive filter may resonate at a certain frequency. According to the design principle of the passive filter, considering the frequency offset in the power grid and the equipment manufacturing error, the tuned frequency of the passive filter should be set 3-15% lower than the characteristic frequency [25]. Therefore, on the basis of the passive filter with multi-coupled coils, this paper also connects an active filter to the passive filter via a coupled inductor, to eliminate the remaining harmonic current after passive compensation.

Deign of the Passive Filter with Multi-Coupled Coils
In the passive filter with multi-coupled coils, there are mutual inductances among the inductors. Therefore, in the design process, the self-inductances and mutual inductances need to be calculated and designed [5]. The basic design flow chart is shown in Figure 2.
Electronics 2021, 10, x FOR PEER REVIEW 4 of 2 Therefore, on the basis of the passive filter with multi-coupled coils, this paper als connects an active filter to the passive filter via a coupled inductor, to eliminate the re maining harmonic current after passive compensation.

Deign of the Passive Filter with Multi-Coupled Coils
In the passive filter with multi-coupled coils, there are mutual inductances amon the inductors. Therefore, in the design process, the self-inductances and mutual induct ances need to be calculated and designed [5]. The basic design flow chart is shown i Figure 2. According to Figure 2, the first step of the design process is to determine the parameters of th uncoupled passive filter considering the rated voltage, capacity. One type of uncoupled passiv filter is shown in Figure 3a. In Figure 3a, the inductors La, Lb, Lc and the capacitors C3, C4, C5 ar determined in this step.

La
Lb Lc The second step is to establish the design equations for the multi-coupled coils struc ture. Figure 3b shows the passive filter with multi-coupled coils. In Figure 3a,b, the value of capacitor C3, C4, C5 are the same. To make the network shown in Figure 3b equivalen to the network in Figure 3a, it is necessary to ensure that when the two networks have th same excitation, the voltages at point a, b, c, d and the current i1, i2, i3, i4, i5 in the two figure are the same. Moreover, according to this relationship, formulas are as follows: According to Figure 2, the first step of the design process is to determine the parameters of the uncoupled passive filter considering the rated voltage, capacity. One type of uncoupled passive filter is shown in Figure 3a. In Figure 3a, the inductors L a , L b , L c and the capacitors C 3 , C 4 , C 5 are determined in this step.
Electronics 2021, 10, x FOR PEER REVIEW 4 Therefore, on the basis of the passive filter with multi-coupled coils, this paper connects an active filter to the passive filter via a coupled inductor, to eliminate the maining harmonic current after passive compensation.

Deign of the Passive Filter with Multi-Coupled Coils
In the passive filter with multi-coupled coils, there are mutual inductances am the inductors. Therefore, in the design process, the self-inductances and mutual ind ances need to be calculated and designed [5]. The basic design flow chart is show According to Figure 2, the first step of the design process is to determine the parameters o uncoupled passive filter considering the rated voltage, capacity. One type of uncoupled pas filter is shown in Figure 3a. In Figure 3a, the inductors La, Lb, Lc and the capacitors C3, C4, C determined in this step.

La
Lb Lc The second step is to establish the design equations for the multi-coupled coils st ture. Figure 3b shows the passive filter with multi-coupled coils. In Figure 3a,b, the va of capacitor C3, C4, C5 are the same. To make the network shown in Figure 3b equiva to the network in Figure 3a, it is necessary to ensure that when the two networks have same excitation, the voltages at point a, b, c, d and the current i1, i2, i3, i4, i5 in the two fig are the same. Moreover, according to this relationship, formulas are as follows: The second step is to establish the design equations for the multi-coupled coils structure. Figure 3b shows the passive filter with multi-coupled coils. In Figure 3a,b, the values of capacitor C 3 , C 4 , C 5 are the same. To make the network shown in Figure 3b equivalent to the network in Figure 3a, it is necessary to ensure that when the two networks have The third step is to solve the nonlinear equations. The unknowns in the aforementioned equations can be worked out using the inductance calculation formula in [23,24]. If the nonlinear equations cannot be solved, the initial conditions shall be changed. In this way, the number of coil turns and the relative positions of the coils can be obtained. The passive filter with multi-coupled coils can be designed.

Structure of the HAPF with Multi-Coupled Coils
The topology of the HAPF with multi-coupled coils is shown in Figure 4, in which an active power filter is added to the passive filter. The active power filter consists of an H-bridge converter and a LCL filter. The inductor L g of the LCL filter is the coupled inductor of the active power filter. Compared with the conventional shunt HAPF topology, the proposed HAPF similar current compensation principle. The passive filter of the proposed HAPF b fundamental voltage and eliminates most of the characteristic harmonic currents  There are mutual inductances between the coupled inductor L g and the inductors in the passive filter. The coupled inductor L g is closest to inductor L 1 and most closely coupled with L 1 . When the active power filter works, the converter generates a controllable AC current i g to the coupled inductor L g . Since the spatial magnetic coupling exists among L g and the inductors in the passive filter, a current can be injected into the passive filter branch, thereby changing the current of the passive filter branch i f . Thus, the remaining characteristic harmonic currents after passive compensation can be eliminated.
Compared with the conventional shunt HAPF topology, the proposed HAPF has the similar current compensation principle. The passive filter of the proposed HAPF bears the fundamental voltage and eliminates most of the characteristic harmonic currents. The active power filter bears a very small fundamental voltage and eliminates the remaining harmonic currents.
In the proposed HAPF, the connection mode of the active power filter and the passive filter is different from the traditional HAPF. For the traditional HAPF topology in which the active power filter is in series with the passive filter through a transformer, the series connection of the transformer changes the filter loop of the passive filter. In Figure 4, the active power filter of the proposed HAPF connects with the passive filter through spatial magnetic coupling instead of a transformer. The coupled inductor L g is designed as a round wire disc, which is placed very close to the multi-coupled coils in the passive filter. The coupled inductor L g is not directly connected in series to the filter loop of the passive filter. So, the coupled inductor L g does not change the filter loop of the passive filter. Compared to a transformer, the cost of the coupled inductor L g in Figure 4 is relatively small in the entire HAPF. This connection mode ensures that whether the active converter is open-circuited or short-circuited, the filtering effect of the passive filter always exists and the safety of the power grid will not be at stake. Besides, the coaxial round wire discs inductor structure can enhance the coupling effect and increase the overall efficiency of the system.

Working Principle
The equivalent schematic of the hybrid compensation of the proposed HAPF is shown in Figure 5, where u 1 to u 5 are the induced voltages controlled by the output current i g of the converter through spatial magnetic coupling and u g is the voltage source controlled by all the currents of the passive filter. Compared with the conventional shunt HAPF topology, the proposed HAPF has the similar current compensation principle. The passive filter of the proposed HAPF bears the fundamental voltage and eliminates most of the characteristic harmonic currents. The active power filter bears a very small fundamental voltage and eliminates the remaining harmonic currents.
In the proposed HAPF, the connection mode of the active power filter and the passive filter is different from the traditional HAPF. For the traditional HAPF topology in which the active power filter is in series with the passive filter through a transformer, the series connection of the transformer changes the filter loop of the passive filter. In Figure 4, the active power filter of the proposed HAPF connects with the passive filter through spatial magnetic coupling instead of a transformer. The coupled inductor Lg is designed as a round wire disc, which is placed very close to the multi-coupled coils in the passive filter. The coupled inductor Lg is not directly connected in series to the filter loop of the passive filter. So, the coupled inductor Lg does not change the filter loop of the passive filter. Compared to a transformer, the cost of the coupled inductor Lg in Figure 4 is relatively small in the entire HAPF. This connection mode ensures that whether the active converter is open-circuited or short-circuited, the filtering effect of the passive filter always exists and the safety of the power grid will not be at stake. Besides, the coaxial round wire discs inductor structure can enhance the coupling effect and increase the overall efficiency of the system.

Working Principle
The equivalent schematic of the hybrid compensation of the proposed HAPF is shown in Figure 5, where u1 to u5 are the induced voltages controlled by the output current ig of the converter through spatial magnetic coupling and ug is the voltage source controlled by all the currents of the passive filter. Figure 5. Equivalent schematic of the hybrid compensation. Because the coupled inductor L g is placed above the multi-coupled coils, it is closest to inductor L 1 , which is at the top of the multi-coupled coils. So, the mutual inductance M g1 between L g and L 1 is relatively large, and the effect of the induced voltage source u 1 is relatively large. For an inductor that is relatively far from L g in space, the mutual inductance between it and L g will be much smaller than M g1 .
The self-inductance and mutual inductance of each coil in Figure 1b were measured. The measurement results show that the mutual inductance M g1 (between L 1 and L g ) and M g3 (between L 3 and L g ) are much larger than M g2 (between L 2 and L g ), M g4 (between L 4 and L g ), and M g5 (between L 5 and L g ). Therefore, to simplify the theoretical analysis, the mutual inductance M g2 , M g3 , M g4 , M g5 are combined to an equivalent inductance M g' in Figure 6, where u' represents the equivalent induced voltage source generated in the filter branch by M g' . For the characteristic harmonic, the impedance of the passive filter branch is very small. L th and R th represent the equivalent inductance and resistance of the passive filter branch at harmonic frequencies, respectively. L s and R s represent the supply-system inductance and resistance, respectively. relatively large. For an inductor that is relatively far from Lg in space, the mutua ance between it and Lg will be much smaller than Mg1.
The self-inductance and mutual inductance of each coil in Figure 1b were m The measurement results show that the mutual inductance Mg1 (between L1 and Mg3 (between L3 and Lg) are much larger than Mg2 (between L2 and Lg), Mg4 (betwee Lg), and Mg5 (between L5 and Lg). Therefore, to simplify the theoretical analysis, the inductance Mg2, Mg3, Mg4, Mg5 are combined to an equivalent inductance Mg' in F where u' represents the equivalent induced voltage source generated in the filte by Mg'. For the characteristic harmonic, the impedance of the passive filter branc small. Lth and Rth represent the equivalent inductance and resistance of the pass branch at harmonic frequencies, respectively. Ls and Rs represent the supply-sy ductance and resistance, respectively. The current ifh represents the harmonic component of the filter branch curre current ifhw represents the passive current component of ifh, which is the harmonic nent when only the passive filter works; the current ifhy represents the injected component of ifh by the active power filter. iLh is the harmonic component of the l rent iL. Figure 6 is only used to simplify the theoretical analysis.
According to Figure 6, the induced voltage u1 and u' caused by ig can gene injected current ifhy on the passive filter branch. In the harmonic current loop, L and Rs will affect the value of ifhy.
At tuned frequencies, series resonance occurs in the passive filter branch and ues of Lth and Rth are very small. Therefore, at tuned frequencies, the current ig can a relatively large ifhy. By controlling the output current ig of the active power f injected current component ifhy of ifh can be adjusted, then so can the harmonic ifh.
As long as the active power filter generates a current ig of a certain freque injected current ifhy of corresponding frequency will be generated on the pass branch. Besides the tuned frequencies, the active power filter in the proposed H also filter harmonic currents at other frequencies. However, at these frequencies ues of Lth and Rth are relatively large. Therefore, the current ifhy generated on the filter branch with the same magnitude of current ig will be relatively smaller t generated at the tuned frequency.
When the amplitude of each characteristic harmonics of ifh and iLh are the sa the phases are opposite, the harmonics in the supply current iS can be complete nated.

Control Strategy
When the active power filter of the proposed HAPF works, the accurate h current tracking control is needed, as well as the capacitor voltage control on the  Figure 6 is only used to simplify the theoretical analysis.
According to Figure 6, the induced voltage u 1 and u' caused by i g can generate the injected current i fhy on the passive filter branch. In the harmonic current loop, L th , R th , L s , and R s will affect the value of i fhy .
At tuned frequencies, series resonance occurs in the passive filter branch and the values of L th and R th are very small. Therefore, at tuned frequencies, the current i g can produce a relatively large i fhy . By controlling the output current i g of the active power filter, the injected current component i fhy of i fh can be adjusted, then so can the harmonic i fh .
As long as the active power filter generates a current i g of a certain frequency, the injected current i fhy of corresponding frequency will be generated on the passive filter branch. Besides the tuned frequencies, the active power filter in the proposed HAPF can also filter harmonic currents at other frequencies. However, at these frequencies, the values of L th and R th are relatively large. Therefore, the current i fhy generated on the passive filter branch with the same magnitude of current i g will be relatively smaller than that generated at the tuned frequency.
When the amplitude of each characteristic harmonics of i fh and i Lh are the same, and the phases are opposite, the harmonics in the supply current i S can be completely eliminated.

Control Strategy
When the active power filter of the proposed HAPF works, the accurate harmonic current tracking control is needed, as well as the capacitor voltage control on the DC side of the active converter. Therefore, the control block diagram of hybrid compensation is shown in Figure 7, which consists of three parts: DC voltage control module, current tracking control module, and current hysteresis control module. The current tracking control module is used to make the filter branch current accurately track the load harmonic current. The output of the current tracking control module is iL0h,ref.
The sum of idc,ref and iL0h,ref is iL0,ref, which is the input of the current hysteresis control module. The output of the current hysteresis control module is the PWM control signal of the active converter. The current hysteresis control can directly track the current at high speed, so the converter can be considered as a current source.  Figure 7. The control block diagram of hybrid compensation.

DC Capacitor Voltage Control
The stability of DC capacitor voltage is the prerequisite for the proposed HAPF to work stably. In order to maintain a stable DC capacitor voltage, it is necessary for the active converter to exchange active power with the induced voltage source ug.
Current idc is the active current, which is the active component of iL0. If the phase of idc is the same as the phase of ug, the converter absorbs active power from ug, to support the DC capacitor voltage. If the phase of idc is opposite to the phase of ug, the converter exports active power to ug. So, in Figure 8, it is critical to properly set the amplitude and phase of idc,ref, which is the reference value of idc.
To obtain the amplitude Idc,ref of current idc,ref, in Figure 8, the DC capacitor voltage udc is compared with the reference value udc,ref, and the comparison result is sent to the PI controller. The output of the PI controller is Idc,ref.
The induced voltage ug cannot be measured directly, and it is controlled by all the currents in the passive filter. Since the coupled inductor Lg is closest to inductor L1 and the most closely coupled with L1, only the filter branch current if (the current on inductor L1) is considered here. Then, ug can be expressed as: where, Mg1 is the mutual inductance between the coupled inductor Lg and inductor L1. So, according to (7), a unit signal idc0,ref, which has the same waveform with ug, can be obtained by current derivation, LPF, and waveform normalization in Figure 8.  The DC voltage control module is used to stabilize the DC capacitor voltage. The output of the DC voltage control module is i dc,ref , which represents the reference value of the active current exchanged between the converter and the grid.
The current tracking control module is used to make the filter branch current accurately track the load harmonic current. The output of the current tracking control module is i L0h,ref .
The sum of i dc,ref and i L0h,ref is i L0,ref , which is the input of the current hysteresis control module. The output of the current hysteresis control module is the PWM control signal of the active converter. The current hysteresis control can directly track the current at high speed, so the converter can be considered as a current source.

DC Capacitor Voltage Control
The stability of DC capacitor voltage is the prerequisite for the proposed HAPF to work stably. In order to maintain a stable DC capacitor voltage, it is necessary for the active converter to exchange active power with the induced voltage source u g .
Current i dc is the active current, which is the active component of i L0 . If the phase of i dc is the same as the phase of u g , the converter absorbs active power from u g , to support the DC capacitor voltage. If the phase of i dc is opposite to the phase of u g , the converter exports active power to u g . So, in Figure 8, it is critical to properly set the amplitude and phase of i dc,ref , which is the reference value of i dc .  Considering the 3rd, 5th, 7th, 11th harmonic components of idc, the active current idc can be expressed as: where, an is the amplitude of each harmonic component of the unit signal idc0,ref.
When the phase of idc is the same with ug, ug can be expressed as: sin( ) g g n n n u U a n t The instantaneous power that ug exports is: To obtain the amplitude I dc,ref of current i dc,ref , in Figure 8, the DC capacitor voltage u dc is compared with the reference value u dc,ref , and the comparison result is sent to the PI controller. The output of the PI controller is I dc,ref .
The induced voltage u g cannot be measured directly, and it is controlled by all the currents in the passive filter. Since the coupled inductor L g is closest to inductor L 1 and the most closely coupled with L 1 , only the filter branch current i f (the current on inductor L 1 ) is considered here. Then, u g can be expressed as: where, M g1 is the mutual inductance between the coupled inductor L g and inductor L 1 . So, according to (7), a unit signal i dc0,ref , which has the same waveform with u g , can be obtained by current derivation, LPF, and waveform normalization in Figure 8. By multiplying i dc0,ref to I dc,ref , the reference current i dc,ref of the active current i dc can be obtained.
Considering the 3rd, 5th, 7th, 11th harmonic components of i dc , the active current i dc can be expressed as: i dc = I dc,re f ∑ n=1,3,5,7,11 a n sin(nωt + θ n ) where, a n is the amplitude of each harmonic component of the unit signal i dc0,ref .
When the phase of i dc is the same with u g , u g can be expressed as: 3,5,7,11 a n sin(nωt + θ n ) The instantaneous power that u g exports is: where, p ac is the AC component, which represents the reactive power. Additionally, the DC component in (10) represents the active power, which is influenced by I dc,ref . Therefore, in DC capacitor voltage control, by controlling I dc,ref , the active power exchanged between the converter and u g can be adjusted to support the DC capacitor voltage. In addition, from (10), it can be seen that the active power is provided by the fundamental and harmonic components in u g together.

Harmonic Current Control
For harmonic current elimination, at the characteristic frequencies, the amplitude and phase of the output current i g are controlled to track the harmonic current.
Taking n th load harmonic as an example, current i fhn , i fhwn , i fhyn , i Lhn , and i gn are the n th component of current i fh , i fhw , i fhy , i Lh , and i g , respectively. According to Figure 6, current i fhn , i fhwn , i fhyn can be expressed with phasor method as: I f hn ∠θ f hn = I f hwn ∠θ f hwn + I f hyn ∠θ f hyn (11) I f hwn ∠θ f hwn = −(R s + jnωL s ) R th + R s + jnω(L th + L s ) ·I Lhn ∠θ Lhn (12) where, I fhn , I fhwn , I fhyn , I Lhn , I gn are the amplitude of i fhn , i fhwn , i fhyn , i Lhn , and i gn , respectively. θ fhn , θ fhwn , θ fhyn , θ Lhn , θ gn are the phase angles of i fhn , i fhwn , i fhyn , i Lhn , and i gn , respectively. When the n th load harmonic current i Lhn is completely eliminated, we have the following relationship: I f hn ∠θ f hn + I Lhn ∠θ Lhn = 0 Substituting (11)-(13) into (14) results in: where, Z thn = R 2 th + (nωL th ) 2 , θ thn = arctan nωL th R th , M = M g1 + M g . In (15), the amplitude I gn and phase angle θ gn of current i gn are adjustable. When the n th load harmonic current i Lhn is definite, there is a unique I gn and a unique θ gn that suit Equation (15).
From (15), I gn and θ gn should be: According to (16) and (17), I gn and θ gn are only related to n th load harmonic current i Lhn . Therefore, in theory, the n th harmonic current i Lhn can be completely eliminated by controlling the amplitude and phase angle of the output current i gn of the active power filter.
Consequently, the active power filter is controlled as an adjustable current source. Each characteristic harmonic i gn of output current i g is controlled individually. Through joint control of the amplitude and phase of i gn , each characteristic harmonic i fhn of the filter branch current i f can be adjusted, thereby eliminating the load harmonic current i Lh .
As shown in Figure 9, the amplitude and phase of each characteristic harmonic component of the load current i L and filter branch current i f can be obtained by FFT. The amplitudes and phases obtained from i L are reference values. The amplitudes and phases obtained from i f are feedback signals. th In (15), the amplitude Ign and phase angle θgn of current ign are adjustable. When the n th load harmonic current iLhn is definite, there is a unique Ign and a unique θgn that suit Equation (15).
From (15), Ign and θgn should be:  (17) According to (16) and (17), Ign and θgn are only related to n th load harmonic current iLhn. Therefore, in theory, the n th harmonic current iLhn can be completely eliminated by controlling the amplitude and phase angle of the output current ign of the active power filter.
Consequently, the active power filter is controlled as an adjustable current source. Each characteristic harmonic ign of output current ig is controlled individually. Through joint control of the amplitude and phase of ign, each characteristic harmonic ifhn of the filter branch current if can be adjusted, thereby eliminating the load harmonic current iLh.
As shown in Figure 9, the amplitude and phase of each characteristic harmonic component of the load current iL and filter branch current if can be obtained by FFT. The amplitudes and phases obtained from iL are reference values.  These reference values and feedback signals are sent to their respective PI controllers, and then synthesized to obtain the reference signal for hysteresis control. Finally, the PWM control signal of the active converter is obtained. The hysteresis control can track the current iL0 well and the active power filter can generate the desired current ig. In this paper, the width of the hysteresis band is 0.2 A.
In order to improve response speed, the phase feedforward control is adopted, which is shown in Figure 9. In the phase feedforward control, pdif3 is the phase response with the These reference values and feedback signals are sent to their respective PI controllers, and then synthesized to obtain the reference signal for hysteresis control. Finally, the PWM control signal of the active converter is obtained. The hysteresis control can track the current i L0 well and the active power filter can generate the desired current i g . In this paper, the width of the hysteresis band is 0.2 A.
In order to improve response speed, the phase feedforward control is adopted, which is shown in Figure 9. In the phase feedforward control, p dif3 is the phase response with the 3rd harmonic of i L0 as the excitation and the 3rd harmonic of current i f as the output. k is the coefficient of the phase feedforward control, whose value is slightly less than 1. The phase response can be obtained through simulation, and its value does not need to be precise.

Simulation Results
In order to verify the effectiveness of the proposed scheme, simulations of a singlephase 10 kV hybrid active power filter with multi-coupled coils, which is shown in Figure 4, are carried out in MATLAB/Simulink. The load harmonic source includes 5th, 7th, and 11th harmonic current. The inductance parameters and capacitance parameters of the multi-coupled coils are calculated according to the method in [5]. The main simulation parameters except for the inductance parameters of the multi-coupled coils are shown in Table 1. The inductance matrix of the multi-coupled coils is shown in Table 2. According to the inductance and capacitance parameters in Tables 1 and 2, the impedancefrequency characteristic of the passive filter is simulated as shown in Figure 10. To improve the filtering effect of the passive filter, the active power f current ig to compensate the remaining harmonic currents. In the hybrid simulations, the control strategy shown in Figures 7-9 is used. Figure 11 shows the waveforms of supply current iS under different strategies. The waveform of iS without any compensation is shown in F Figure 10. Impedance-frequency characteristic of the passive filter. Figure 10 shows the tuned frequencies of the passive filter deviate from 250 Hz, 350 Hz, and 550 Hz (line frequency is 50 Hz in China). So, the passive filter is unable to completely eliminate the 5th, 7th, and 11th characteristic harmonics.

Current Compensation Effect
To improve the filtering effect of the passive filter, the active power filter generates current i g to compensate the remaining harmonic currents. In the hybrid compensation simulations, the control strategy shown in Figures 7-9 is used. Figure 11 shows the waveforms of supply current i S under different compensation strategies. The waveform of i S without any compensation is shown in Figure 11a. The waveform of i S with passive compensation is shown in Figure 11b. The waveform of i S with hybrid compensation is shown in Figure 11c. To improve the filtering effect of the passive filter, the active power filter generates current ig to compensate the remaining harmonic currents. In the hybrid compensation simulations, the control strategy shown in Figures 7-9 is used. Figure 11 shows the waveforms of supply current iS under different compensation strategies. The waveform of iS without any compensation is shown in Figure 11a. The waveform of iS with passive compensation is shown in Figure 11b. The waveform of iS with hybrid compensation is shown in Figure 11c.   From Figure 11a, it can be seen that without any compensation, the total harmonic distortion (THD) of iS is 26.33%. Further, the distortion of 5th, 7th, and 11th harmonics of iS are 19.57%, 14.66%, and 9.74%, respectively. From Figure 11b, it can be seen that with passive compensation, the THD of iS is reduced to 5.79%. Further, the distortion of 5th, 7th, and 11th harmonics are 4.55%, 3.36%, and 1.22%, respectively. From Figure 11c, it can be seen that the THD of iS is reduced to 0.38%. The distortion of 5th, 7th, and 11th harmonics are 0.27%, 0.26%, and 0.09%, respectively.

Current Compensation Effect
In addition, it can be seen that in Figure 11a, the amplitude of the fundamental component of iS is 204.1 A. While in Figure 11b,c, the amplitude of the fundamental component of iS is 340 A, which is larger. The reason is that after the passive filter works, the capacitive fundamental current increases, which can compensate the inductive current in the grid. From Figure 11a, it can be seen that without any compensation, the total harmonic distortion (THD) of i S is 26.33%. Further, the distortion of 5th, 7th, and 11th harmonics of i S are 19.57%, 14.66%, and 9.74%, respectively. From Figure 11b, it can be seen that with passive compensation, the THD of i S is reduced to 5.79%. Further, the distortion of 5th, 7th, and 11th harmonics are 4.55%, 3.36%, and 1.22%, respectively. From Figure 11c, it can be seen that the THD of i S is reduced to 0.38%. The distortion of 5th, 7th, and 11th harmonics are 0.27%, 0.26%, and 0.09%, respectively.
In addition, it can be seen that in Figure 11a, the amplitude of the fundamental component of i S is 204.1 A. While in Figure 11b,c, the amplitude of the fundamental component of i S is 340 A, which is larger. The reason is that after the passive filter works, the capacitive fundamental current increases, which can compensate the inductive current in the grid.

Compensation Characteristic Simulation
In hybrid compensation, the stability of DC capacitor voltage is the prerequisite. The output current i g will directly change the filter branch current i f and affect the filtering effect.
In order to study the compensation characteristic, Figure 12a-c shows the waveform of the output current i g of the active power filter, the waveform of the filter branch current i f , and the waveform of the DC capacitor voltage u dc in hybrid compensation, respectively.
In Figure 12a, the amplitude of the 5th, 7th, and 11th harmonics of i g are 27.51 A, 51.34 A, and 4.90 A, respectively. These harmonic currents are injected into the filter branch through spatial magnetic coupling, to change the filter branch current i f .
In Figure 12b

Compensation Characteristic Simulation
In hybrid compensation, the stability of DC capacitor voltage is the prerequisite. The output current ig will directly change the filter branch current if and affect the filtering effect.
In order to study the compensation characteristic, Figure 12a-c shows the waveform of the output current ig of the active power filter, the waveform of the filter branch current if, and the waveform of the DC capacitor voltage udc in hybrid compensation, respectively.  In Figure 12a, the amplitude of the 5th, 7th, and 11th harmonics of ig are 27.51 A, 51.34 A, and 4.90 A, respectively. These harmonic currents are injected into the filter branch through spatial magnetic coupling, to change the filter branch current if.
In Figure 12b, the amplitude of the 5th, 7th, and 11th harmonics of if are 39.92 A, 29.54 A, and 19.98 A, respectively. The harmonic components ifh of if in hybrid compensation includes the passive current component ifhw (the harmonic component when only the passive filter works) and the injected current component ifhy (injected by the active power filter).
From Figure 12c, it can be seen that in hybrid compensation, the DC capacitor voltage udc can be stabilized at 800 V with only small voltage fluctuations. The cause of the fluctuation is the active power exchange between the converter and the induced voltage ug.
The apparent power of the active power filter can be calculated using the following equation: in which, SAPF represents the apparent power of the active power filter; Vd represents the DC voltage of the active power filter; Igrms represents the RMS value of the output current ig.
The apparent power of the proposed HAPF can be calculated using the following equation: From Figure 12c, it can be seen that in hybrid compensation, the DC capacitor voltage u dc can be stabilized at 800 V with only small voltage fluctuations. The cause of the fluctuation is the active power exchange between the converter and the induced voltage u g .
The apparent power of the active power filter can be calculated using the following equation: (18) in which, S APF represents the apparent power of the active power filter; V d represents the DC voltage of the active power filter; I grms represents the RMS value of the output current i g . In simulation, V d is 800 V. I grms is 41.33 A. So, the apparent power S APF = 23.38 kVA, according to (18).
The apparent power of the proposed HAPF can be calculated using the following equation: in which, S HAPF represents the apparent power of the proposed HAPF; U Frms represents the RMS value of the voltage of the proposed HAPF; I frms represents the RMS value of the current i f . In simulation, I frms is 194.35 A, and U Frms is 5845 V. So, the apparent power S HAPF = 1135.975 kVA, according to (19).
S PF represents the apparent power of the passive filter. The ratio between S APF and S PF is: It can be found that the capacity ratio of the active power filter to the passive filter in the proposed HAPF is very small.

Fault Simulation of the Active Power Filter
The open-circuit and short-circuit fault diagram of the active power filter is shown in Figure 13. As is introduced before, the proposed HAPF has no additional coils in series connected to the passive filter branch, and the filtering effect of the passive filter will be guaranteed whether the active power filter is open or short-circuited. To verify this, simulations in two cases (open-circuit and short-circuit) are conducted. Figure 13. As is introduced before, the proposed HAPF has no additional coils in connected to the passive filter branch, and the filtering effect of the passive filter guaranteed whether the active power filter is open or short-circuited. To verify th ulations in two cases (open-circuit and short-circuit) are conducted.
According to Figure 13, under normal working condition, point a and poin connected to each other, while point b and point c are not connected. When the con between point a and point b is broken, the active power filter is open-circuited. time, the current ig is zero and the influence of the active power filter on the passiv no longer exists. The open-circuit fault simulation result is shown in Figure 14a pared with the waveform of iS when only the passive filter works in Figure 11b, the form of iS in Figure 14a is almost the same. When point b and point c are connected, the active power filter is short-circu this case, current ig is completely generated by the induced voltage ug. At this ti influence of the current ig on the passive filter is small. The short-circuit fault sim result is shown in Figure 14b. Compared with the waveform in Figure 11b, the wa of iS in Figure 14b is very similar.
In summary, whether the active power filter is open-circuited or short-circui filtering effect of the passive filter can be guaranteed.  Figure 14a. Compared with the waveform of i S when only the passive filter works in Figure 11b, the waveform of i S in Figure 14a is almost the same.

Simulation of Three-Phase Hybrid Active Power Filter
To prove that the proposed HAPF topology is available for the three-phase system, a three-phase 10 kV hybrid active power filter simulation was carried out. In the three-phase system, three single-phase hybrid active power filters are used. Each single-phase hybrid active power filter has the same topology and the same control strategy as described above. The parameters are the same as the parameters in Tables 1 and 2. Figure 15 shows the waveforms of the supply current iS under different compensation strategies in the three-phase simulation. The waveform of iS without any compensation is shown in Figure 15a. The waveform of iS with passive compensation is shown in Figure 15b. The waveform of iS with hybrid compensation is shown in Figure 15c. When point b and point c are connected, the active power filter is short-circuited. In this case, current i g is completely generated by the induced voltage u g . At this time, the influence of the current i g on the passive filter is small. The short-circuit fault simulation result is shown in Figure 14b. Compared with the waveform in Figure 11b, the waveform of i S in Figure 14b is very similar.
In summary, whether the active power filter is open-circuited or short-circuited, the filtering effect of the passive filter can be guaranteed.

Simulation of Three-Phase Hybrid Active Power Filter
To prove that the proposed HAPF topology is available for the three-phase system, a three-phase 10 kV hybrid active power filter simulation was carried out. In the three-phase system, three single-phase hybrid active power filters are used. Each single-phase hybrid active power filter has the same topology and the same control strategy as described above. The parameters are the same as the parameters in Tables 1 and 2. Figure 15 shows the waveforms of the supply current i S under different compensation strategies in the three-phase simulation. The waveform of i S without any compensation is shown in Figure 15a. The waveform of i S with passive compensation is shown in Figure 15b. The waveform of i S with hybrid compensation is shown in Figure 15c.

Simulation of Three-Phase Hybrid Active Power Filter
To prove that the proposed HAPF topology is available for the three-phase system, a three-phase 10 kV hybrid active power filter simulation was carried out. In the three-phase system, three single-phase hybrid active power filters are used. Each single-phase hybrid active power filter has the same topology and the same control strategy as described above. The parameters are the same as the parameters in Tables 1 and 2. Figure 15 shows the waveforms of the supply current iS under different compensation strategies in the three-phase simulation. The waveform of iS without any compensation is shown in Figure 15a. The waveform of iS with passive compensation is shown in Figure 15b. The waveform of iS with hybrid compensation is shown in Figure 15c. In Figure 15a, taking phase A as an example, without any compensation, the THD of the supply current iSa is 26.29%. The distortion of 5th, 7th, and 11th harmonics of iSa is 19.56%, 14.64%, and 9.71%, respectively. In Figure 15b, with passive compensation, the THD of iSa is reduced to 5.81%. Further, the distortion of 5th, 7th, and 11th harmonics are 4.55%, 3.37%, and 1.29%, respectively. In Figure 15c, with hybrid compensation, the THD of iSa is reduced to 0.29%. In Figure 15a, taking phase A as an example, without any compensation, the THD of the supply current i Sa is 26.29%. The distortion of 5th, 7th, and 11th harmonics of i Sa is 19.56%, 14.64%, and 9.71%, respectively. In Figure 15b, with passive compensation, the THD of i Sa is reduced to 5.81%. Further, the distortion of 5th, 7th, and 11th harmonics are 4.55%, 3.37%, and 1.29%, respectively. In Figure 15c, with hybrid compensation, the THD of i Sa is reduced to 0.29%.

Experiments
In order to verify the feasibility of the proposed HAPF, a small prototype of multicoupled coils is designed and manufactured as shown in Figure 1b. Then, an experimental platform of the hybrid active power filter with multi-coupled coils is built according to the topology in Figure 4.
The structure schematic diagram of the HAPF experimental platform is shown in Figure 16. The platform is composed of the system power supply, harmonic source, and the proposed HAPF. The RMS value of the system voltage is transformed from 220 V to 36 V through a step-down transformer. The harmonic source is an adjustable harmonic current source. The HAPF with multi-coupled coils is composed of an active power filter and passive filter. The active power filter consists of the main circuit, detection circuit, and control circuit. Figure 16. The platform is composed of the system power supply, harmonic sour the proposed HAPF. The RMS value of the system voltage is transformed from 2 36 V through a step-down transformer. The harmonic source is an adjustable ha current source. The HAPF with multi-coupled coils is composed of an active pow and passive filter. The active power filter consists of the main circuit, detection circ control circuit. The experimental platform of HAPF with multi-coupled coils is shown in Fi The main experimental parameters other than the inductance parameters of the coupled coils are shown in Table 3.

Categories
Variables Value  The experimental platform of HAPF with multi-coupled coils is shown in Figure 17. The main experimental parameters other than the inductance parameters of the multicoupled coils are shown in Table 3. Figure 16. The platform is composed of the system power supply, harmonic source, an the proposed HAPF. The RMS value of the system voltage is transformed from 220 V t 36 V through a step-down transformer. The harmonic source is an adjustable harmoni current source. The HAPF with multi-coupled coils is composed of an active power filte and passive filter. The active power filter consists of the main circuit, detection circuit, an control circuit. The experimental platform of HAPF with multi-coupled coils is shown in Figure 17 The main experimental parameters other than the inductance parameters of the multi coupled coils are shown in Table 3.    The inductance parameters of the multi-coupled coils in the platform are shown in Table 4. The impedance-frequency characteristic of the passive filter in the experiment is measured by voltammetry as shown in Figure 18. From Figure 18, it can be seen that the measured tuned frequencies of the passive filter are 145 Hz, 247 Hz, and 347 Hz, which slightly deviate from the frequencies of the 3rd, 5th, and 7th harmonics.

L1
155.0 80. The impedance-frequency characteristic of the passive filter in the experiment is measured by voltammetry as shown in Figure 18. From Figure 18, it can be seen that the measured tuned frequencies of the passive filter are 145 Hz, 247 Hz, and 347 Hz, which slightly deviate from the frequencies of the 3rd, 5th, and 7th harmonics. In the experiment, the load harmonic current source injects the 3 rd harmonic current into the power grid, whose amplitude is 14.1 A. Figure 19 shows the experimental waveform of the supply current iS under different compensation strategies. The experimental waveform of iS without compensation is shown in Figure 19a. The experimental waveform of iS with passive compensation is shown in Figure 19b. The experimental waveform of iS with hybrid compensation is shown in Figure 19c. In the experiment, the load harmonic current source injects the 3rd harmonic current into the power grid, whose amplitude is 14.1 A. Figure 19 shows the experimental waveform of the supply current i S under different compensation strategies. The experimental waveform of i S without compensation is shown in Figure 19a. The experimental waveform of i S with passive compensation is shown in Figure 19b. The experimental waveform of i S with hybrid compensation is shown in Figure 19c.

L1
155.0 80. The impedance-frequency characteristic of the passive filter in the experiment is measured by voltammetry as shown in Figure 18. From Figure 18, it can be seen that the measured tuned frequencies of the passive filter are 145 Hz, 247 Hz, and 347 Hz, which slightly deviate from the frequencies of the 3rd, 5th, and 7th harmonics. In the experiment, the load harmonic current source injects the 3 rd harmonic current into the power grid, whose amplitude is 14.1 A. Figure 19 shows the experimental waveform of the supply current iS under different compensation strategies. The experimental waveform of iS without compensation is shown in Figure 19a. The experimental waveform of iS with passive compensation is shown in Figure 19b. The experimental waveform of iS with hybrid compensation is shown in Figure 19c. From Figure 19a, it can be seen that iS contains the obvious 3rd harmonic. The amplitude of the fundamental component of the supply current iS is 23.65 A and the THD is 60.95%. The distortion of the 3rd harmonic of supply current iS is 60.68%. From Figure 19a, it can be seen that i S contains the obvious 3rd harmonic. The amplitude of the fundamental component of the supply current i S is 23.65 A and the THD is 60.95%. The distortion of the 3rd harmonic of supply current i S is 60.68%.
From Figure 19b, it can be seen that amplitude of the fundamental component of i S is 43.9 A, which is larger than that before compensation because of the increase of capacitive fundamental current. The THD of i S is reduced to 15.93%. The distortion of the 3rd harmonic is reduced to 13.22%. At the same time, the distortion of the 5th harmonic is 5.92%, and the distortion of the 7th harmonic is 6.16%.
In passive compensation, the 5th and 7th harmonic currents appear in the supply current i S , which is caused by the very small impedance of the system transformer in the experiment. For the 5th and 7th harmonics, the impedance of the passive filter branch is very small, so the 5th and 7th harmonics are introduced from the power grid.
From Figure 19c, it can be seen that the amplitude of the fundamental component of the supply current i S is 43.26 A and the THD is reduced to 3.08%. The 3rd, 5th, 7th harmonics of i S are greatly eliminated. Thus, it is reasonable to conclude that the proposed hybrid compensation method is feasible.

Conclusions
This paper proposes a novel hybrid active power filter with multi-coupled coils. The passive filter of the proposed HAPF adopts the structure of multi-coupled coils, which can reduce the required space of the inductors. The active power filter of the proposed HAPF adopts a coupled inductor instead of a transformer to connect with the passive filter. The active power filter can inject the compensation current into the passive filter branch through spatial magnetic coupling. At the same time, through spatial magnetic coupling, the active power filter can absorb active power from the passive filter branch to maintain the capacitor voltage.
To verify the effectiveness of the proposed scheme, a simulation model of a 10 kV hybrid active power filter with multi-coupled coils is built in MATLAB/Simulink. Furthermore, an experimental platform of hybrid active power filter with multi-coupled coils is built. Simulations and experiments in this paper verify the feasibility of the proposed HAPF.
(1) The passive filter with multi-coupled coils can save the required space of the inductors on the premise of eliminating characteristic harmonics. (2) The active power filter of the proposed HAPF can improve the filtering effect of the passive filter. The current injection through spatial magnetic coupling is effective and feasible. (3) Since the connection method of spatial magnetic coupling, whether the active power filter is open-circuited or short-circuited, the filtering effect of the passive filter can still be guaranteed, which enhances the reliability of the filter.
In the simulation and experiment, it can be found that the inductance matrix of the multi-coupled coils directly affects the performance of the filter. Therefore, the parameter optimization of the multi-coupled coils is worthy of further research.
Funding: This research received no external funding.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.