# An Effective Harmonic Suppression Method Based on Linear Active Disturbance Rejection Control for Copper Electrowinning Rectifier

Authors: Yang Cui; Cheng Liu; Yanming Cheng; Mahmoud Al Shurafa; Ilkyoo Lee; Jing Niu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Because the harmonics in the production process of copper electrowinning have an important impact on the electrical energy consumption, it is necessary to suppress the harmonics effectively. In this paper, a copper electrowinning rectifier with double inverse star circuit is selected as a study object in which a large number of harmonics mainly including the 5th, 7th, 11th, and 13th harmonics are generated and injected back into the power grid. The total harmonic distortion rate of the power grid is up to 29.19% before filtering. Therefore, a method combining the induction filtering method and the active filtering method is proposed to carry out comprehensive filtering. Simulation results demonstrate that the total harmonic distortion rate of the system decreases to 4.20%, which indicates that the proposed method can track the corresponding changes of harmonics when the load changes in real time and filter them out. In order to ensure and improve the effect of active filter, a current harmonic tracking control method based on linear active disturbance rejection control is proposed. Simulation results show that the total harmonic distortion rate decreases to 3.34%, which is also lower than that of hysteresis control. Compared with the conventional single filtering method, the new filtering method combining induction filtering with active filtering based on linear active disturbance rejection control in the copper electrowinning rectifier has obvious advantages.


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## Body

## 1. Introduction

In recent years, copper and its alloy materials are widely used in various fields, and the demand of copper is increasing in many enterprises. As one of the basic technologies of copper production, copper electrolysis and electrowinning rectification technology has attracted more and more attention. With the continuous increase of the number and capacity of copper electrowinning rectifier units, the impact of the system on the power grid in industrial application is becoming more and more serious. A large number of harmonics are injected back into the power grid, which will directly affect the quality of copper production. It is very important to ensure the safe and stable operation and power quality of power grid system [1, 2]. Many researchers and scholars at home and abroad have studied power quality problems, especially the research on current quality in power grid [3–6]. The use of a large number of power electronic converter equipment will directly produce harmonic and reactive power problems and affect the power quality of the grid [7–9]. The copper electrowinning rectifier is such a typical device.The rectifying device of the copper electrowinning device consists of voltage regulating transformer and double inverse star rectifier, which is a typical low-voltage high-current nonlinear load. In the process of operation, turning on and off semicontrolled thyristor can cause a distortion of the grid current and voltage. From the perspective of frequency domain, these waveforms contain not only the power frequency sinusoidal quantity but also some voltage or current components [2] which are positive integral multiples of power frequency. These components are called harmonics whose existence will not only affect the production quality of the system but also affect the normal operation of other equipment in the same power supply system and even lead to safety accidents in the power system. Therefore, it is an urgent problem to control the harmonics to ensure the power quality and build a clean electrical environment.Yuan et al. [10] conducted the harmonic suppression characteristic analysis of a phase-shifting reactor in a rectifier system. Takeshita et al. [11] presented the input current waveform control of the rectifier circuit which realizes simultaneously the high input power factor and the harmonic suppression of the receiving-end voltage and the source current under the distorted receiving-end voltage. Méndez et al. [12] designed an active high-power filter for the harmonic suppression in coils powered separately by 12-pulsed thyristor converters. Liu et al. [13] proposed a novel controllable inductive power filtering (CIPF) method to effectively eliminate the harmonics flowing through the transformer in the industrial DC power supply system. Liu et al. [14] proposed a novel power factor correction (PFC) controller based on proportional resonant for selective harmonic suppression of a UPS system. Ramesh and Habeebullah Sait [15] introduced harmonic removal in a switched capacitor multilevel inverter utilizing the artificial bee colony (ABC) method. In [16], an impedance reconstruction control method for the source PWM inverter was proposed, which improves the phase of the output sequence impedance of the source PWM inverter at high-frequency areas to effectively suppress the high-frequency oscillation of the island power system.Wang et al. [17] proposed a frequency-domain harmonic model for 12-pulse series-connected thyristor-controlled rectifier under unbalanced supply voltage. Wiechmann et al. [18] proposed an optimized sequential control technique to improve converter's reactive power consumption presentation for copper electrowinning high-current rectifiers. In [19], a modified DPWM (MDPWM) scheme was proposed to regulate the neutral point voltage by using redundant clamping modes in high-frequency Vienna-type rectifiers. An et al. [20] simulated diode rectifier + FC filter scheme-based large electrolytic aluminum filter device. Mao et al. [9] used the filter with single tuning in a high-power thyristor rectifier as a medium-frequency furnace and high-frequency furnace for metal smelting. Dai et al. [21] proposed a comprehensive treatment scheme of "automatic switch single tuning passive filter + high voltage side series filter inductor" in medium frequency smelting furnace. Ma et al. and Li et al. [22, 23] applied the first-order linear active disturbance rejection controller (LADRC) to the current tracking control of a 10 kV line parallel hybrid active power filter, which can well deal with the contradiction between overshoot and rapidity, and the performance is better than the traditional PI controller, which verifies the effectiveness of the linear active disturbance rejection control strategy. In [24], aiming at the disadvantages of the grid side inverter of doubly fed induction generator, such as the complexity and poor stability, and the problems of the conventional PI closed-loop control, such as poor stability and slow response speed, an ADRC control strategy was proposed, in which a resonance link is added to the linear state error feedback. Instead of PI control of current loop, it can suppress the low-order harmonics of grid connected current, improve the response speed of the system, and ensure smooth operation when the inverter is connected to the grid.Analyzing above relevant researches on harmonic suppression methods for rectifier indicates that there are few special effective researches on harmonic suppression of high-power copper electrowinning rectifier. The effect of the conventional harmonic suppression method is not ideal. It is very important to study an effective harmonic suppression technology. Therefore, in this paper, an “induction filter + active filter” method is proposed to effectively suppress the harmonics. Moreover, the LADRC technology is added to the active filter current tracking control in active filter, which further improves the harmonic suppression effect.


## 2. Copper Electrowinning

### 2.1. Copper Electrowinning Process

In copper electrowinning production, cathode copper is produced through the leaching-extraction-electrowinning process; in the electrolyte, the insoluble lead-alloy plate (Pb-Ca-Sn) is used as the anode, and the thin copper starting sheet is used as the cathode. In essence, copper electrowinning is the process where copper ion is reduced by DC and deposited on the cathode [25, 26]. Main reactions are as follows.The copper sulfate solutionCuSO4 is decomposed into a copper ion Cu2+ and sulfate ion SO42− in the applied electric field, and the process is expressed as(1)CuSO4⟶Cu2++SO42−The copper ion obtains electrons on the cathode and is reduced to copper which is deposited on the cathode, and the process can be expressed as(2)Cu2++2e⟶Cu


### 2.2. Current Efficiency

According to Faraday’s law, the theory is that copper of1.1864×10−3kg should be deposited on the cathode by DC of 1 ampere-hour. In fact, compared with the theoretical value, the actual amount of produced copper is decreased by impurities, oxidation and dissolution of cathode sediment, and electrode short circuit and leakage loss. In production practice, current efficiency is proposed to evaluate effective utilization of current, and it is the percentage ratio between the actual production of copper and theoretical production of the copper and is expressed by [27](3)η=Gq⋅I⋅t⋅n×100%,where η represents current efficiency in %, G represents actual production of copper in kg during t, q represents electrochemical equivalent of copper in 1.1864×10−3kg/Ah, I represents current intensity in A, t represents the time of electrowinning in hour, and n represents the number of electrowinning electrolytic tanks.Current efficiency is an important technical and economic index of electrowinning copper production. In production practice, due to different specific conditions, the current efficiency of copper electroproduction is different, and effective measures should be taken to improve current efficiency.


### 2.3. DC Power Consumption

In copper electrowinning production, DC power consumption inkWh/kg is defined as the electricity used to produce copper of 1 kg in 1 hour, which is an important economic and technical index. DC power consumption is calculated by(4)W=vη⋅q,where W represents DC power consumption in kWh/kg, v represents electrolytic tank voltage in V, and η represents current efficiency in %.Formula (4) indicates that electrolytic tank voltage and current efficiency are the direct factors affecting energy consumption. In the industrial process, the voltage of conventional electrolytic tank voltage is in an approximate range of 1.8V∼2V, and the electrolytic tank voltage will increase with the increase of current density; consequently, the power consumption will also increase. However, if low-current-density electrowinning is used, the output will be reduced, although the power consumption can be reduced in the electrowinning process. Therefore, the actual production situation of the factory should be taken into account when a current density is determined. At present, the current density of copper electrowinning used in the world is different, fluctuating in a range; the low one is approximately 100A/m2, and the high one is approximately 270A/m2.


## 2.1. Copper Electrowinning Process

In copper electrowinning production, cathode copper is produced through the leaching-extraction-electrowinning process; in the electrolyte, the insoluble lead-alloy plate (Pb-Ca-Sn) is used as the anode, and the thin copper starting sheet is used as the cathode. In essence, copper electrowinning is the process where copper ion is reduced by DC and deposited on the cathode [25, 26]. Main reactions are as follows.The copper sulfate solutionCuSO4 is decomposed into a copper ion Cu2+ and sulfate ion SO42− in the applied electric field, and the process is expressed as(1)CuSO4⟶Cu2++SO42−The copper ion obtains electrons on the cathode and is reduced to copper which is deposited on the cathode, and the process can be expressed as(2)Cu2++2e⟶Cu


## 2.2. Current Efficiency

According to Faraday’s law, the theory is that copper of1.1864×10−3kg should be deposited on the cathode by DC of 1 ampere-hour. In fact, compared with the theoretical value, the actual amount of produced copper is decreased by impurities, oxidation and dissolution of cathode sediment, and electrode short circuit and leakage loss. In production practice, current efficiency is proposed to evaluate effective utilization of current, and it is the percentage ratio between the actual production of copper and theoretical production of the copper and is expressed by [27](3)η=Gq⋅I⋅t⋅n×100%,where η represents current efficiency in %, G represents actual production of copper in kg during t, q represents electrochemical equivalent of copper in 1.1864×10−3kg/Ah, I represents current intensity in A, t represents the time of electrowinning in hour, and n represents the number of electrowinning electrolytic tanks.Current efficiency is an important technical and economic index of electrowinning copper production. In production practice, due to different specific conditions, the current efficiency of copper electroproduction is different, and effective measures should be taken to improve current efficiency.


## 2.3. DC Power Consumption

In copper electrowinning production, DC power consumption inkWh/kg is defined as the electricity used to produce copper of 1 kg in 1 hour, which is an important economic and technical index. DC power consumption is calculated by(4)W=vη⋅q,where W represents DC power consumption in kWh/kg, v represents electrolytic tank voltage in V, and η represents current efficiency in %.Formula (4) indicates that electrolytic tank voltage and current efficiency are the direct factors affecting energy consumption. In the industrial process, the voltage of conventional electrolytic tank voltage is in an approximate range of 1.8V∼2V, and the electrolytic tank voltage will increase with the increase of current density; consequently, the power consumption will also increase. However, if low-current-density electrowinning is used, the output will be reduced, although the power consumption can be reduced in the electrowinning process. Therefore, the actual production situation of the factory should be taken into account when a current density is determined. At present, the current density of copper electrowinning used in the world is different, fluctuating in a range; the low one is approximately 100A/m2, and the high one is approximately 270A/m2.


## 3. Double Inverse Star Rectifying System for Copper Electrowinning Rectifier

### 3.1. Double Inverse Star Rectifying Circuit

With the continuous development of electrolysis, electroplating, electrolytic degreasing, and other industries, the demand for low-voltage and high-current DC power supply is increasing. Double inverse star rectifying circuit stands out among all kinds of circuits for its superior performance. It is a rectifying circuit composed of two three-phase half-wave circuits and a balance reactor, which is depicted in Figure1.Figure 1
Structure diagram of double inverse star rectifying circuit.Compared with the common six-phase half-wave rectifier circuit and three-phase six-pulse bridge rectifier circuit, the current effective value of the rectifier arm is smaller, and the capacity utilization rate is high. Adding the balance reactor can improve the reverse voltage drop, and the rectifier can parallel two output different instantaneous values of groups of three-phase half wave to supply power to the load. In the working process of double inverse star rectifier circuit, the rectifier is a nonlinear load, which will inevitably produce harmonics.The internal structure diagram of double inverse star transformer is illustrated in Figure2. On the primary side of the transformer, the windings are connected according to the triangle structure, and the secondary side is composed of two-star structure windings with opposite directions. On the premise of meeting the safety requirements of industrial application, Figure 3 shows that the input three-phase grid voltages ua, ub, uc are changed into six voltages ua1, ub1, uc1 and ua2 , ub2, uc2 and there is a difference of 60° between them.Figure 2
Internal structure diagram of double inverse star transformer.Figure 3
Vector relation of input and output voltages of transformer.According to the internal structure and voltage vector relationship of double inverse star rectifier transformer, the corresponding mathematical model can be established. Firstly, the input three-phase grid voltage is set as(5)ua=Umsinωt,ub=Umsinωt−2π/3,uc=Umsinωt+2π/3,where Um is defined as the magnitude of the grid voltage. According to the voltage vector relationship shown in Figure 3, the other two voltage equations can be obtained as follows:(6)ua1=Unsinωt,ub1=Unsinωt−2π/3,uc1=Unsinωt+2π/3,ua2=Unsinωt−π,ub2=Unsinωt+π/3,uac2=Unsinωt−π/3,where Un is the amplitude of secondary voltage of the transformer. Assuming that the ratio of variation is k, the relationship between Um and Un satisfies(7)Un=3Umk.


### 3.2. Analysis of Characteristic Subharmonic Current

In order to analyze the types and contents of harmonics generated by the double inverse star rectifying transformer in the working process and effectively filter out the harmonics, it is assumed that both the grid voltage and the transformer are ideal. It can be seen from Figures1 and 2 that the primary side current of the transformer can be represented by the secondary side current as(8)i1=1kia2−ia1,i2=1kib2−ib1,i3=1kic2−ic1.The system input currents are obtained as(9)ia=i1−i3,ib=i3−i2,ic=i2−i1.By substituting (9) into (10), the expressions of system input current and secondary side current are obtained as(10)ia=1kic1−ic2+ia2−ia1,ib=1kib1−ib2+ic2−ic1,ic=1kia1−ia2+ib2−ib1.According to (11), it is difficult to analyze the input current. In this study, the load will be simplified to a constant current. It is difficult to determine the instantaneous voltage difference of three-phase half-wave output Id, but its waveform is a triangular wave with three times the fundamental frequency. Since the balance reactor can make the rectifier system work independently with symmetry, the current can be supplied by two transformers equally, namely,(11)id1=id2=Id2.In this paper, the switching function is introduced to study the current relationship between AC side and DC side directly. Taking the phasea1 as an example, its switching function is Sa1=ia1/id1 whose Fourier series expansion is(12)Sa1=13+∑n=1∞sin2nπ/3nπcosnωt+1−cos2nπ/3nπsinnωt.According toSa1, Fourier series of phases b1 and c1 can be obtained, and then the expression of three-phase half-wave group I is expressed as(13)Sa1=Sa1∠0,Sb1=Sa1∠−120°,Sc1=Sa1∠+120°.On the basis of (14), the expression of three-phase half-wave group II can be obtained as(14)Sa2=Sa1∠+180°,Sb2=Sb1∠+180°,Sc2=Sc1∠+180°.The relationships between the switching function of each phase and its corresponding current input and output are as follows:(15)ia1ib1ic1=Sa1Sb1Sc1id1,(16)ia2ib2ic2=Sa2SbSc2id2.According to formulas (13)–(17), the Fourier series decomposition of each secondary side current is as follows:(17)ia1=Id213+∑n=1∞sin2nπ/3nπcosnωt+1−cos2nπ/3nπsinnωt,ib1=Id213+∑n=1∞sin2nπ/3nπcosnωt−2π3+1−cos2nπ/3nπsinnωt−2π3,ic1=Id213+∑n=1∞sin2nπ/3nπcosnωt+2π3+1−cos2nπ/3nπsinnωt+2π3,(18)ia2=Id213+∑n=1∞−1nsin2nπ/3nπcosnωt+1−cos2nπ/3nπsinnωt,ib2=Id213+∑n=1∞sin2nπ/3nπcosnωt+π3+1−cos2nπ/3nπsinnωt+π3,ic2=Id213+∑n=1∞sin2nπ/3nπcosnωt−π3+1−cos2nπ/3nπsinnωt−π3.It can be seen from (12), (18), and (19) that taking phase a as an example, its current expression is(19)ia=∑n=1,2,3,…∞Rnsinnωt,Rn=4Idnπ1+sin16nπsin12nπ.It is obvious that the harmonic content in the power grid mainly contains (6k ± 1) times (k = 1, 2, 3, ...) harmonics.The electrowinning copper rectifier used in this paper is a double inverse star rectifying system, and its system parameters are summarized in Table1.Table 1
Parameters of double inverse star rectifying system.
ParametersValuesInput voltage10kVOutput voltage60∼80VOutput current10000ATransformer capacity10kVAFrequency50HzAccording to Table1, the parameters of the double inverse star rectifying simulation circuit are set. Figure 4 shows waveforms of output voltage and output current. Figure 5 shows waveforms of voltage and current measured on the power grid. Figure 6 shows the spectrum analysis diagram of harmonic measurement on the power grid.Figure 4
Waveforms of output voltage and output current.
(a)(b)Figure 5
Waveforms of voltage and current measured on the power grid.
(a)(b)Figure 6
Spectrum of harmonic measurement on the power grid.
(a)(b)Spectrum of harmonics in Figure6 demonstrates that a large number of harmonics are generated and injected into the power grid in the rectifying process of double inverse star rectifier, and these harmonics are mainly the 5th, 7th, 11th, and 13th harmonics. Before filtering, the total measured distortion rate of the power grid is up to 29.19%.


## 3.1. Double Inverse Star Rectifying Circuit

With the continuous development of electrolysis, electroplating, electrolytic degreasing, and other industries, the demand for low-voltage and high-current DC power supply is increasing. Double inverse star rectifying circuit stands out among all kinds of circuits for its superior performance. It is a rectifying circuit composed of two three-phase half-wave circuits and a balance reactor, which is depicted in Figure1.Figure 1
Structure diagram of double inverse star rectifying circuit.Compared with the common six-phase half-wave rectifier circuit and three-phase six-pulse bridge rectifier circuit, the current effective value of the rectifier arm is smaller, and the capacity utilization rate is high. Adding the balance reactor can improve the reverse voltage drop, and the rectifier can parallel two output different instantaneous values of groups of three-phase half wave to supply power to the load. In the working process of double inverse star rectifier circuit, the rectifier is a nonlinear load, which will inevitably produce harmonics.The internal structure diagram of double inverse star transformer is illustrated in Figure2. On the primary side of the transformer, the windings are connected according to the triangle structure, and the secondary side is composed of two-star structure windings with opposite directions. On the premise of meeting the safety requirements of industrial application, Figure 3 shows that the input three-phase grid voltages ua, ub, uc are changed into six voltages ua1, ub1, uc1 and ua2 , ub2, uc2 and there is a difference of 60° between them.Figure 2
Internal structure diagram of double inverse star transformer.Figure 3
Vector relation of input and output voltages of transformer.According to the internal structure and voltage vector relationship of double inverse star rectifier transformer, the corresponding mathematical model can be established. Firstly, the input three-phase grid voltage is set as(5)ua=Umsinωt,ub=Umsinωt−2π/3,uc=Umsinωt+2π/3,where Um is defined as the magnitude of the grid voltage. According to the voltage vector relationship shown in Figure 3, the other two voltage equations can be obtained as follows:(6)ua1=Unsinωt,ub1=Unsinωt−2π/3,uc1=Unsinωt+2π/3,ua2=Unsinωt−π,ub2=Unsinωt+π/3,uac2=Unsinωt−π/3,where Un is the amplitude of secondary voltage of the transformer. Assuming that the ratio of variation is k, the relationship between Um and Un satisfies(7)Un=3Umk.


## 3.2. Analysis of Characteristic Subharmonic Current

In order to analyze the types and contents of harmonics generated by the double inverse star rectifying transformer in the working process and effectively filter out the harmonics, it is assumed that both the grid voltage and the transformer are ideal. It can be seen from Figures1 and 2 that the primary side current of the transformer can be represented by the secondary side current as(8)i1=1kia2−ia1,i2=1kib2−ib1,i3=1kic2−ic1.The system input currents are obtained as(9)ia=i1−i3,ib=i3−i2,ic=i2−i1.By substituting (9) into (10), the expressions of system input current and secondary side current are obtained as(10)ia=1kic1−ic2+ia2−ia1,ib=1kib1−ib2+ic2−ic1,ic=1kia1−ia2+ib2−ib1.According to (11), it is difficult to analyze the input current. In this study, the load will be simplified to a constant current. It is difficult to determine the instantaneous voltage difference of three-phase half-wave output Id, but its waveform is a triangular wave with three times the fundamental frequency. Since the balance reactor can make the rectifier system work independently with symmetry, the current can be supplied by two transformers equally, namely,(11)id1=id2=Id2.In this paper, the switching function is introduced to study the current relationship between AC side and DC side directly. Taking the phasea1 as an example, its switching function is Sa1=ia1/id1 whose Fourier series expansion is(12)Sa1=13+∑n=1∞sin2nπ/3nπcosnωt+1−cos2nπ/3nπsinnωt.According toSa1, Fourier series of phases b1 and c1 can be obtained, and then the expression of three-phase half-wave group I is expressed as(13)Sa1=Sa1∠0,Sb1=Sa1∠−120°,Sc1=Sa1∠+120°.On the basis of (14), the expression of three-phase half-wave group II can be obtained as(14)Sa2=Sa1∠+180°,Sb2=Sb1∠+180°,Sc2=Sc1∠+180°.The relationships between the switching function of each phase and its corresponding current input and output are as follows:(15)ia1ib1ic1=Sa1Sb1Sc1id1,(16)ia2ib2ic2=Sa2SbSc2id2.According to formulas (13)–(17), the Fourier series decomposition of each secondary side current is as follows:(17)ia1=Id213+∑n=1∞sin2nπ/3nπcosnωt+1−cos2nπ/3nπsinnωt,ib1=Id213+∑n=1∞sin2nπ/3nπcosnωt−2π3+1−cos2nπ/3nπsinnωt−2π3,ic1=Id213+∑n=1∞sin2nπ/3nπcosnωt+2π3+1−cos2nπ/3nπsinnωt+2π3,(18)ia2=Id213+∑n=1∞−1nsin2nπ/3nπcosnωt+1−cos2nπ/3nπsinnωt,ib2=Id213+∑n=1∞sin2nπ/3nπcosnωt+π3+1−cos2nπ/3nπsinnωt+π3,ic2=Id213+∑n=1∞sin2nπ/3nπcosnωt−π3+1−cos2nπ/3nπsinnωt−π3.It can be seen from (12), (18), and (19) that taking phase a as an example, its current expression is(19)ia=∑n=1,2,3,…∞Rnsinnωt,Rn=4Idnπ1+sin16nπsin12nπ.It is obvious that the harmonic content in the power grid mainly contains (6k ± 1) times (k = 1, 2, 3, ...) harmonics.The electrowinning copper rectifier used in this paper is a double inverse star rectifying system, and its system parameters are summarized in Table1.Table 1
Parameters of double inverse star rectifying system.
ParametersValuesInput voltage10kVOutput voltage60∼80VOutput current10000ATransformer capacity10kVAFrequency50HzAccording to Table1, the parameters of the double inverse star rectifying simulation circuit are set. Figure 4 shows waveforms of output voltage and output current. Figure 5 shows waveforms of voltage and current measured on the power grid. Figure 6 shows the spectrum analysis diagram of harmonic measurement on the power grid.Figure 4
Waveforms of output voltage and output current.
(a)(b)Figure 5
Waveforms of voltage and current measured on the power grid.
(a)(b)Figure 6
Spectrum of harmonic measurement on the power grid.
(a)(b)Spectrum of harmonics in Figure6 demonstrates that a large number of harmonics are generated and injected into the power grid in the rectifying process of double inverse star rectifier, and these harmonics are mainly the 5th, 7th, 11th, and 13th harmonics. Before filtering, the total measured distortion rate of the power grid is up to 29.19%.


## 4. Filtering Method Combining Active Filter with Inductive Filter

Because the induction filtering method can meet the requirements of the total harmonic distortion rate after filtering and the active power filtering method can meet the requirements of tracking and filtering the harmonic changes in real time after the load changes, the two filtering methods are combined to carry out comprehensive filtering for the copper electrodeposited rectifying system.A system diagram of combining the two filtering methods is shown in Figure7, an induction filter is added to the transformer side, and the active power filter is added to the power grid side, which will make the electrowinning copper rectifier achieve a good filtering effect.Figure 7
Filter system diagram of combination of induction filter and active filter.In the simulation process, the active filter is added at 0.02 s, the inductive filter is added at 0.06 s, and the SCR trigger angle is changed from10° to 20° at 0.12 s, that is, when the load changes, the corresponding harmonic also changes. The simulation results are shown in Figures 8 and 9.Figure 8
Harmonic current waveform on power grid side after combining induction filter and active filter.
(a)(b)Figure 9
Harmonic spectrum after combining induction filter and active filter. (a) Total harmonic distortion rate at 0.08 s. (b) Total harmonic distortion rate at 0.12 s. (c) Total harmonic distortion rate at 0.16 s.
(a)(b)(c)The corresponding harmonic spectrum is shown in Figure9.From the above simulation results, it can be seen that when the active filter is added at 0.02 s, the waveform has obvious changes and the filtering effect is good. When the induction filter is added at 0.06 s, the waveform is stable after one cycle, that is, the total harmonic distortion rate is 4.83% at 0.08 s, which basically meets the requirements of the power grid. When the load changes at 0.12 s, the total distortion rate rises to 8.07%, which indicates that the harmonic has changed. After a period of 0.16 s, the total distortion rate of the system decreases to 3.76%, which indicates that the filtering system can track the corresponding changes of harmonics when the load changes in real time and filter them out.


## 5. Active Power Filter Based on Linear Active Disturbance Rejection Control

### 5.1. Construction of LADRC Applied in APF

In this paper, the current tracking control method of instantaneous value comparison is adopted. Although it has an obvious filtering effect, its tracking effect is limited by the choice of hysteresis loop width. If it is too large, the tracking effect is not good. If it is too small, the switching frequency is too high which results in higher power loss. Therefore, a new current tracking control method—linear active disturbance rejection control (LADRC)—is introduced. The LADRC technology is an improvement of linearization based on active disturbance rejection control (ADRC). The key part is the linearization of the extended state observer (ESO). Compared with ADRC, the parameter tuning of ACRC is reduced, and the control process is relatively simple, which improves its practical value in the control process.In this paper, the LADRC technology is applied in current tracking control, which means that ESO and control are linear, and the differential tracker is not needed in practical application. The control block diagram of LADRC application in APF is shown in Figure10, taking single-phase a as an example.Figure 10
Control block diagram of LADRC applied in APF.Figure11 shows the internal structure block diagram of LADRC.Figure 11
Structure block diagram of LADRC.


### 5.2. Design of Linear Expansion State Observer (LESO)

In the entire LADRC system, the LESO is the most critical part. It can calculate each state quantity and disturbance value of the unknown model system according to the input data. For any givenn-order unknown system, there is(20)ynt=fyn−1t,…,yt,wt+b0ut,where yt represents the input signal of the system, the nonlinear dynamic characteristic of the system is f•, wt represents the external disturbance of the system, ut represents the control input signal of the system, and b0 represents the gain of the controller. In LACRC, f• is the expression of total disturbance. All state spaces of the system satisfy(21)x1•=x2,⋮,x˙n−1=xn,x˙n=xn+1+b0u,x˙n+1=hX,w,y=x1,where, the state variable xn+1, which is the total disturbance passing through the LESO, can be expressed as(22)xn+1=fyn−1t,…,yt,wt.Let(23)xn+1•=hX,w.In this way, the originaln-order system is expanded into a new n+1-order linear system. The first-order linear ADRC is designed in this paper, and the state space of the system can be written as(24)x˙1•=x2+b0u,x˙2•=hX,w,y=x1.X is added to the system as an expanded state. For the convenience of design, the matrix form of (26) is(25)x˙=Ax+Bu+Eh,y=Cx,where A=0100, B=b00, E=01, and C=10. h=f••is the first derivative of the disturbance. The LESO of the system is written as(26)z˙=Az+Bu+Ly−y^,y^=Cz,where L=β1β2 is the gain of LESO.By subtracting (27) from (28), the equation of state of observation error is obtained as(27)e˙•=Aee+Eh,where Ae=−β11−β20.If there is no steady-state error, the eigenvalues of the observation error characteristic matrix should be all in the left half of the plane, and the characteristic equation can be obtained as(28)λs=s2+β1s+β2.When the bandwidth of the observer is fixed, if the following conditions are satisfied:(29)λs=s+ω02,the effect of the controller is best. According to formulas (30)and (31),(30)β1=2ω0,β2=ω02.The specific form of LESO is(31)z˙•=Aez+BLuy=−β11−β20z+b02ω00ω02uy.Inputs of LESO areut and yt. The two system states z1 and z2 track y and expand the state x2, respectively.


### 5.3. Design of Linear State Error Feedback (LSEF)

The main function of linear error feedback control rate is not only to preserve the advantages of integral link but also to avoid the error caused by integral transformation when designing parameters. Therefore, the disturbance compensation value−z2/b0 is required to replace the integral part of PID control in the design process. For the second-order and even higher-order system, although there is no integral term, it can still reflect the function of the integral. The LADRC system used in current tracking control is a first-order system in this paper, which does not need differential tracker. Therefore, we only need to use the control rate for P.(32)u0=kpr−y,where u0 is the error feedback control quantity and kp represents the proportional constant. It can be seen from Figure 11 that the final error feedback control quantity is(33)u=u0−z2b0.For the convenience of frequency-domain analysis, Figure11 can be transformed into Figure 12.Figure 12
Block diagram of first-order LADRC.Figure12 shows the transfer function in the frequency domain. The input signal of the controller is Rs, and Ys indicates the output signal. The disturbance signal is Ws, and Ps indicates the controlled object model. The control signal is Us, and Hs and Gs are the undetermined terms of the controller, respectively.The control signal can be expressed as(34)Us=RsHs−Ys×Gs.Substituting (34) into (35) and performing Laplace transform, the following results are obtained:(35)Us=1b0kp×Rs−Ys−Z2s.The Laplace transform of (28) can be obtained as(36)Zs=1s2+β1s+β2b0sβ1s+β2−β2b0β2sUsYs.Therefore,(37)Z2s=−β2b0Us+β2sYss2+β1s+β2.By substituting equation (39) into equation (38) and combining it with equation (37), the following results can be obtained:(38)Hs=kps2+β1s+β2kps2+kpβ1+β2s+kpβ2,Gs=kps2kpβ1+β2s+kpβ2b0s2+β1s.Letkp=ωc, where ωc is the bandwidth for the controller, and from experience, ω0=3∼5ωc. By substituting equation (39) into equation (38), the following results can be obtained:(39)Hs=ωcs2+2ω0s+ω02ωcs2+ωc2ω0+ω02s+ωcω02,Gs=ωcs2+ωc2ω0+ω02s+ωcω02b0s2+2ω0s.According to equation (41), the LADRC is a single-parameter controller. In this paper, ω0=10, b0=0.1, β1=20, β2=100, and kp=2.


### 5.4. Convergence Analysis of Linear Extended State Observer

Taking the controlled object represented by equation (22) as an example, for the n-order controlled object represented by equation (23), the LESO is represented by(40)z˙1=z2−B1z1−y,⋮,z˙n=zn+1−Bnz1−y+bu,z˙n+1=−Bn+1z1−y.The parameter of the observer isL=B1,B2,…,Bn+1 which is adjusted by pole assignment method.Taking into account the general situation, if(41)B1,B2,…,Bn+1=a1ω0,a2ω02,…,an+1ω0n+1,ω0>0,then the characteristic polynomial s3+a1s2+a2s+a3 will satisfy the Herwitz criterion. In order to simplify the calculation process, setting sn+1+a1sn+⋯+ans+an+1=s+1n+1, then ai=n+1!/i!n+1−i! can be obtained, i=1,2,…,n+1. In this case, the characteristic polynomial of the LESO can be expressed as(42)λs=s+ω0n+1.It is obvious that the undetermined parameter of the whole state observer is onlyω0, setting x˜i=xi−zi, i=1,2,…,n+1. The estimation error of LESO can be obtained as(43)x˜˙1=x˜2−ω0a1x˜1,⋮x˜˙n−1=x˜n−ω0n−1an−1x˜1,x˜˙n=x˜n+1−ω0nanx˜1,x˜˙n+1=hx,w−hz,w−ω0n+1an+1x˜1,and let εi=x˜i/ω0i−1, i=1,2,…,n+1. Then, equation (45) can be converted to(44)ε˙=ω0Aε+Bhx,w−hz,wω0n,where A=−a110⋯0−a201⋮0⋮⋮⋱⋮⋮−an0⋯01−an+10⋯00, B=00⋯01.Becauseai=n+1!/i!n+1−i!, matrix A satisfies the Herwitz criterion. LESO is convergent, when hx,w satisfies Lipschitz continuity condition; that is to say, there exists ω0>0 to make limt⟶∞x˜i=0, i=0,1,2,…,n+1.Since matrix A satisfies the Herwitz criterion, there must be a positive definite matrixP such that(45)ATP+PA=−I.Matrix P selects Lyapunov function(46)Vε=εTPε.The derivation of equation (48) is obtained as(47)V˙ε=−ω0ε2+2εTPBhx,w−hz,wω02.Becausehx,w satisfies Lipschitz continuity condition for x1,x2,…xn+1, there must be a constant C, and for any x,z,w, there must exist(48)|hx,w−hz,w|≤Cx−z.Equation (50) is further deduced as(49)2εTPBhx,w−hz,wω0n≤2εTPBCx−zω0n.Whenω0>0, there is(50)x−zω0n=x˜ω0n=ε12+ε22ω02+…,εn+12ω02nω0n≤ε,where C′=1+PBC2; in combination with (49) and (51), it can be obtained as(51)V˙ε≤−ω0−C′ε2.Therefore, ifω0>C′, V˙ε≤0, that is, limt⟶∞x˜i=0, i=0,1,2,…,n+1. That is to say, the pole assignment method is used to design the convergence of the LESO.


### 5.5. Harmonic Suppression with LADRC

On the basis of the filtering method combining active filter with inductive filter, the LADRC technology is applied in current tracking control in ARF, filtering effects of which are obtained by conducting simulation. Simulation results are obtained in Figures13 and 14. The active filter is added at 0.02 s, the inductive filter is added at 0.06 s, and the SCR trigger angle changes from 10° to 20° at 0.12 s, that is, the load changes.Figure 13
Voltage and current waveforms of three-phase power system using LADRC.
(a)(b)Figure 14
Harmonic spectrum after combining active filter with inductive filter method based on LADRC. (a) Total harmonic distortion at 0.08 s under the control of LADRC. (b) Total harmonic distortion at 0.12 s under the control of LADRC. (c) Total harmonic distortion at 0.16 s under the control of LADRC.
(a)(b)(c)Comparison of total distortion rates between hysteresis control and LADRC is summarized in Table2.Table 2
Comparison of total distortion rates.
Simulation time (s)Total distortion rate (%)Hysteresis controlLADRC0.084.833.990.128.078.910.163.763.41It can be seen from Table2 that obviously the total harmonic distortion rate under the control of LADRC is 3.99% at 0.08 s, which is lower than that under hysteresis control; at 0.12 s, the trigger angle of SCR changes from 10° to 20°, that is, when the load changes, the harmonic also changes accordingly. After one cycle, the total harmonic distortion rate returns to 3.41%, which is also lower than that of hysteresis control. Therefore, the results verify that the harmonic suppression effect of LADCR is better than that of hysteresis tracking control.


## 5.1. Construction of LADRC Applied in APF

In this paper, the current tracking control method of instantaneous value comparison is adopted. Although it has an obvious filtering effect, its tracking effect is limited by the choice of hysteresis loop width. If it is too large, the tracking effect is not good. If it is too small, the switching frequency is too high which results in higher power loss. Therefore, a new current tracking control method—linear active disturbance rejection control (LADRC)—is introduced. The LADRC technology is an improvement of linearization based on active disturbance rejection control (ADRC). The key part is the linearization of the extended state observer (ESO). Compared with ADRC, the parameter tuning of ACRC is reduced, and the control process is relatively simple, which improves its practical value in the control process.In this paper, the LADRC technology is applied in current tracking control, which means that ESO and control are linear, and the differential tracker is not needed in practical application. The control block diagram of LADRC application in APF is shown in Figure10, taking single-phase a as an example.Figure 10
Control block diagram of LADRC applied in APF.Figure11 shows the internal structure block diagram of LADRC.Figure 11
Structure block diagram of LADRC.


## 5.2. Design of Linear Expansion State Observer (LESO)

In the entire LADRC system, the LESO is the most critical part. It can calculate each state quantity and disturbance value of the unknown model system according to the input data. For any givenn-order unknown system, there is(20)ynt=fyn−1t,…,yt,wt+b0ut,where yt represents the input signal of the system, the nonlinear dynamic characteristic of the system is f•, wt represents the external disturbance of the system, ut represents the control input signal of the system, and b0 represents the gain of the controller. In LACRC, f• is the expression of total disturbance. All state spaces of the system satisfy(21)x1•=x2,⋮,x˙n−1=xn,x˙n=xn+1+b0u,x˙n+1=hX,w,y=x1,where, the state variable xn+1, which is the total disturbance passing through the LESO, can be expressed as(22)xn+1=fyn−1t,…,yt,wt.Let(23)xn+1•=hX,w.In this way, the originaln-order system is expanded into a new n+1-order linear system. The first-order linear ADRC is designed in this paper, and the state space of the system can be written as(24)x˙1•=x2+b0u,x˙2•=hX,w,y=x1.X is added to the system as an expanded state. For the convenience of design, the matrix form of (26) is(25)x˙=Ax+Bu+Eh,y=Cx,where A=0100, B=b00, E=01, and C=10. h=f••is the first derivative of the disturbance. The LESO of the system is written as(26)z˙=Az+Bu+Ly−y^,y^=Cz,where L=β1β2 is the gain of LESO.By subtracting (27) from (28), the equation of state of observation error is obtained as(27)e˙•=Aee+Eh,where Ae=−β11−β20.If there is no steady-state error, the eigenvalues of the observation error characteristic matrix should be all in the left half of the plane, and the characteristic equation can be obtained as(28)λs=s2+β1s+β2.When the bandwidth of the observer is fixed, if the following conditions are satisfied:(29)λs=s+ω02,the effect of the controller is best. According to formulas (30)and (31),(30)β1=2ω0,β2=ω02.The specific form of LESO is(31)z˙•=Aez+BLuy=−β11−β20z+b02ω00ω02uy.Inputs of LESO areut and yt. The two system states z1 and z2 track y and expand the state x2, respectively.


## 5.3. Design of Linear State Error Feedback (LSEF)

The main function of linear error feedback control rate is not only to preserve the advantages of integral link but also to avoid the error caused by integral transformation when designing parameters. Therefore, the disturbance compensation value−z2/b0 is required to replace the integral part of PID control in the design process. For the second-order and even higher-order system, although there is no integral term, it can still reflect the function of the integral. The LADRC system used in current tracking control is a first-order system in this paper, which does not need differential tracker. Therefore, we only need to use the control rate for P.(32)u0=kpr−y,where u0 is the error feedback control quantity and kp represents the proportional constant. It can be seen from Figure 11 that the final error feedback control quantity is(33)u=u0−z2b0.For the convenience of frequency-domain analysis, Figure11 can be transformed into Figure 12.Figure 12
Block diagram of first-order LADRC.Figure12 shows the transfer function in the frequency domain. The input signal of the controller is Rs, and Ys indicates the output signal. The disturbance signal is Ws, and Ps indicates the controlled object model. The control signal is Us, and Hs and Gs are the undetermined terms of the controller, respectively.The control signal can be expressed as(34)Us=RsHs−Ys×Gs.Substituting (34) into (35) and performing Laplace transform, the following results are obtained:(35)Us=1b0kp×Rs−Ys−Z2s.The Laplace transform of (28) can be obtained as(36)Zs=1s2+β1s+β2b0sβ1s+β2−β2b0β2sUsYs.Therefore,(37)Z2s=−β2b0Us+β2sYss2+β1s+β2.By substituting equation (39) into equation (38) and combining it with equation (37), the following results can be obtained:(38)Hs=kps2+β1s+β2kps2+kpβ1+β2s+kpβ2,Gs=kps2kpβ1+β2s+kpβ2b0s2+β1s.Letkp=ωc, where ωc is the bandwidth for the controller, and from experience, ω0=3∼5ωc. By substituting equation (39) into equation (38), the following results can be obtained:(39)Hs=ωcs2+2ω0s+ω02ωcs2+ωc2ω0+ω02s+ωcω02,Gs=ωcs2+ωc2ω0+ω02s+ωcω02b0s2+2ω0s.According to equation (41), the LADRC is a single-parameter controller. In this paper, ω0=10, b0=0.1, β1=20, β2=100, and kp=2.


## 5.4. Convergence Analysis of Linear Extended State Observer

Taking the controlled object represented by equation (22) as an example, for the n-order controlled object represented by equation (23), the LESO is represented by(40)z˙1=z2−B1z1−y,⋮,z˙n=zn+1−Bnz1−y+bu,z˙n+1=−Bn+1z1−y.The parameter of the observer isL=B1,B2,…,Bn+1 which is adjusted by pole assignment method.Taking into account the general situation, if(41)B1,B2,…,Bn+1=a1ω0,a2ω02,…,an+1ω0n+1,ω0>0,then the characteristic polynomial s3+a1s2+a2s+a3 will satisfy the Herwitz criterion. In order to simplify the calculation process, setting sn+1+a1sn+⋯+ans+an+1=s+1n+1, then ai=n+1!/i!n+1−i! can be obtained, i=1,2,…,n+1. In this case, the characteristic polynomial of the LESO can be expressed as(42)λs=s+ω0n+1.It is obvious that the undetermined parameter of the whole state observer is onlyω0, setting x˜i=xi−zi, i=1,2,…,n+1. The estimation error of LESO can be obtained as(43)x˜˙1=x˜2−ω0a1x˜1,⋮x˜˙n−1=x˜n−ω0n−1an−1x˜1,x˜˙n=x˜n+1−ω0nanx˜1,x˜˙n+1=hx,w−hz,w−ω0n+1an+1x˜1,and let εi=x˜i/ω0i−1, i=1,2,…,n+1. Then, equation (45) can be converted to(44)ε˙=ω0Aε+Bhx,w−hz,wω0n,where A=−a110⋯0−a201⋮0⋮⋮⋱⋮⋮−an0⋯01−an+10⋯00, B=00⋯01.Becauseai=n+1!/i!n+1−i!, matrix A satisfies the Herwitz criterion. LESO is convergent, when hx,w satisfies Lipschitz continuity condition; that is to say, there exists ω0>0 to make limt⟶∞x˜i=0, i=0,1,2,…,n+1.Since matrix A satisfies the Herwitz criterion, there must be a positive definite matrixP such that(45)ATP+PA=−I.Matrix P selects Lyapunov function(46)Vε=εTPε.The derivation of equation (48) is obtained as(47)V˙ε=−ω0ε2+2εTPBhx,w−hz,wω02.Becausehx,w satisfies Lipschitz continuity condition for x1,x2,…xn+1, there must be a constant C, and for any x,z,w, there must exist(48)|hx,w−hz,w|≤Cx−z.Equation (50) is further deduced as(49)2εTPBhx,w−hz,wω0n≤2εTPBCx−zω0n.Whenω0>0, there is(50)x−zω0n=x˜ω0n=ε12+ε22ω02+…,εn+12ω02nω0n≤ε,where C′=1+PBC2; in combination with (49) and (51), it can be obtained as(51)V˙ε≤−ω0−C′ε2.Therefore, ifω0>C′, V˙ε≤0, that is, limt⟶∞x˜i=0, i=0,1,2,…,n+1. That is to say, the pole assignment method is used to design the convergence of the LESO.


## 5.5. Harmonic Suppression with LADRC

On the basis of the filtering method combining active filter with inductive filter, the LADRC technology is applied in current tracking control in ARF, filtering effects of which are obtained by conducting simulation. Simulation results are obtained in Figures13 and 14. The active filter is added at 0.02 s, the inductive filter is added at 0.06 s, and the SCR trigger angle changes from 10° to 20° at 0.12 s, that is, the load changes.Figure 13
Voltage and current waveforms of three-phase power system using LADRC.
(a)(b)Figure 14
Harmonic spectrum after combining active filter with inductive filter method based on LADRC. (a) Total harmonic distortion at 0.08 s under the control of LADRC. (b) Total harmonic distortion at 0.12 s under the control of LADRC. (c) Total harmonic distortion at 0.16 s under the control of LADRC.
(a)(b)(c)Comparison of total distortion rates between hysteresis control and LADRC is summarized in Table2.Table 2
Comparison of total distortion rates.
Simulation time (s)Total distortion rate (%)Hysteresis controlLADRC0.084.833.990.128.078.910.163.763.41It can be seen from Table2 that obviously the total harmonic distortion rate under the control of LADRC is 3.99% at 0.08 s, which is lower than that under hysteresis control; at 0.12 s, the trigger angle of SCR changes from 10° to 20°, that is, when the load changes, the harmonic also changes accordingly. After one cycle, the total harmonic distortion rate returns to 3.41%, which is also lower than that of hysteresis control. Therefore, the results verify that the harmonic suppression effect of LADCR is better than that of hysteresis tracking control.


## 6. Conclusions

Since the induction filtering method can meet the requirements of the total harmonic distortion rate after filtering and the active power filtering method can meet the requirements of tracking and filtering the harmonic changes in real time when the load changes, in order to effectively suppress a large number of harmonics mainly including the 5th, 7th, 11th, and 13th harmonics generated during operation of a copper electrowinning rectifier with double inverse star circuit, firstly, Firstly, this paper proposes a method combining the induction filtering method and the active filtering method to carry out comprehensive filtering. Simulation results indicate that the total distortion rate of the system decreases to 4.83% from 29.19% before filtering, which indicates that the proposed method can track the corresponding changes of harmonics when the load changes in real time and filter them out. Secondly, a current harmonic tracking control method based on LADRC is proposed to ensure and improve the effect of the active filter. Simulation results show that the total harmonic distortion rate can be decreased to 3.41%, which is also lower than that of hysteresis control. Compared with the conventional single filtering method, it is obvious that the proposed filtering method combining induction filtering with active filtering based on LADRC in the copper electrowinning rectifier has obvious advantages in setting parameter and debugging system.



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*Source: 9933558-2021-12-23.xml*