# A Discrete-Time Decoupled Current Control Scheme Applied to VSC-Based Three-Phase Three-Line Systems

Authors: Hui Wang; Qingmin Li; Guangzhu Wang; Xingguo Tan
Journal: Advances in Power Electronics
 (2011)

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## Abstract

Accurate current control of the voltage source converters (VSCs) is one of the key research subjects in modern power electronics. To achieve a preferable solution to current coupling effect in the VSC-based three-phase three-line system, a discrete-time decoupled current control strategy is proposed in the paper. With integration of theα-β transform and two independent current controllers, the proposed methodology can effectively implement decoupled control of the three-phase currents, which can thereby eliminate the impact from the neutral-point voltage especially under asymmetrical loading conditions. Control performance under digital realization was characterized with extensive tests on a shunt active power filter (SAPF) platform. Both the simulative and experimental results have demonstrated that the SAPF could function well and thereby verified the feasibility and effectiveness of the proposed current control methodology.


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## Body

## 1. Introduction

Based on fully controlled power semiconductors, voltage source converters (VSCs), such as DC-DC, DC-AC, and AC-DC converters, are being more widely used than current source converters (CSCs), as the DC side capacitor used for energy storage in a VSC is much superior to an inductor in a CSC in terms of weight, cost, and efficiency, especially suitable for the nowadays switching semiconductors. For concrete applications with VSCs, such as active power filters (APF), PWM rectifiers, power factor correction (PFC), HVDC light and grid-connected inverters, an accurate current control methodology with preferable dynamic performance is indispensible and critical [1–7]. Current control is normally used for the inner closed loop in a VSC-based system, in which PWM signals are directly generated to track the current reference accurately as well as ameliorate controllability and dynamic response of the system.For a three-phase four-line circuitry of a VSC, the current control strategy makes no difference at all from that for a single-phase system since the phase currents are independent of each other in the circuitry. However, in a three-phase three-line system, the three-phase currents are interactional and electromagnetically coupled. As a result, only two circuit equations are independent accordingly. In addition, the neutral point voltage of the three-line circuitry will also affect the behavior of current tracking control in the VSC, especially when the three-phase voltages or loads are not symmetrically presented. Hence, it is a pending issue to be properly tackled of the current control strategy in a VSC-based three-phase three-line circuitry.Most of the nowadays current control strategies are based on the d-q synchronous rotating frame as to decouple the active currents and the reactive currents in a three-phase VSC system [7–12], in which the physical concepts of the adopted variables are unique and clear; however, the instant value of the angular power frequency ω and the output inductance of the VSC should be known in advance. Also, the sensible impact of the neutral point voltage was not incorporated [9, 10, 12]. References [13, 14] advised using an α-β stationary frame to transform the mathematical model of an APF, and the VSC topology was designed in the α-β stationary frame so as to achieve current decoupling, which did facilitate the control strategy but the corresponding hardware design became much more complicated than before.To cope with the above-mentioned issues, based on a general mathematical model of the three-phase three-line VSC system, the impact of the neutral point voltage on current control scheme was fully elucidated in this paper. Further, a discrete-time mathematical model in theα-β coordinate system was established through orthogonal transform, with a view to developing a decoupled current control strategy as to eliminate the impact of the neutral point voltage and finally achieve decoupling of the three-phase currents. Simulation and experimental results on an APF prototype under nonlinear loading conditions were also presented for verification purpose.


## 2. Impact of Current Coupling in a Three-Line VSC System

For a typical VSC configuration in the three-line system as shown in Figure1, several assumptions are made as follows in establishing a mathematical model: (1) the sources and loads are in Y connection (Δ connection can be transferred into Y connection); (2) the three-line inductance is ignored due to its very small value and little effect; (3) the three-line resistances Rs and the output Inductor Lf of the VSC unit are symmetrical; (4) the mathematical model of the VSC unit is described in the continuous-time system for convenience.Figure 1
Typical configuration of a three-phase VSC.Given asymmetrical sources or loads in the three-phase system, the phase voltagesuSa, uSb, and uSc can be expressed as follows:
(1)uSa+uSb+uSc=Δu≠0.If the sources and loads are symmetrical in the three-phase system, thenuSa+uSb+uSc=Δu=0 will be fulfilled. For the three-line voltages uab, ubc, and uca in the AC side, one gets(2)uab+ubc+uca≡0.Based on Kirchhoff’s Voltage Law, the circuit equation of phase A is governed by(3)LfdiSadt=uSa-RSiSa-uaN-uN0.If the switching states ofS1, S2, and S3 are expressed by pa, pb, and pc, then (pa,pb,pc)∈(0,1), where logic 1 represents that the upper-leg switch is turned on while the lower-leg switch is turned off. Similarly, logic 0 represents the lower-leg switch is turned on while the upper-leg switch is turned off. If S1 is turned on and S2 is turned off, then pa=1 and uaN=uDC. If S2 is turned on and S1 is turned off, then pa=0 and uaN=0. Hence, we get uaN=uDCpa and (3) can be changed to(4)LfdiSadt=uSa-RSiSa-uDCpa-uN0.In the same way, the governing equations of phase B and phase C are obtained as follows:(5)LfdiSbdt=uSb-RSiSb-uDCpb-uN0,(6)LfdiScdt=uSc-RSiSc-uDCpc-uN0,
where iSa, iSb, and iSc denote the three-line currents, uaN, ubN, and ucN represent the phase voltages relative to point N, uDC expresses the DC side voltage and uN0 denotes the voltage difference between point N and the neutral point.Obviously, the three-line currents will meet(7)iSa+iSb+iSc=0.Based on (4) to (7), the following relationship can be derived:
(8)(uSa-RSiSa-uDCpa-uN0)+(uSb-RSiSb-uDCpb-uN0)+(uSc-RSiSc-uDCpc-uN0)=0.Further, we can get(9)uN0=13Δu-uDC3(pa+pb+pc).If (9) is substituted into (4) to (6), the mathematical model of the VSC unit can be obtained when the sources or loads are in asymmetrical conditions: (10)LfdiSadt=uSa-RSiSa-uCa-13Δu,LfdiSbdt=uSb-RSiSb-uCb-13Δu,LfdiScdt=uSc-RSiSc-uCc-13Δu,
where (11)uCa=uDC(2pa-pb-pc)3,uCb=uDC(-pa+2pb-pc)3,uCc=uDC(-pa-pb+2pc)3.For the symmetrical conditions of the sources or loads in the three-phase system,uSa+uSb+uSc=Δu=0 will be met, which directly results in the corresponding mathematical model of the VSC shown below:(12)LfdiSadt=uSa-RSiSa-uCa,LfdiSbdt=uSb-RSiSb-uCb,LfdiScdt=uSc-RSiSc-uCc.Only two governing equations are outstandingly independent since the summation of the three-phase currents always is equal to zero as shown in (7). With regard to the asymmetrical system, (10) shows that the neutral point voltage will impose direct impact on the tracking control performance of output currents in the VSC system. For an example shown in Figure 2, where a VSC unit is utilised in a shunt APF topology with nonlinear loads, simulations are carried out with the PSIM software. The source frequency of the APF system is 50 Hz and the source voltage is 220 V (RMS). The nonlinear load is represented by a three-phase rectifier in a symmetrical system, and the three-phase rectifier connected with an asymmetrical RLC circuitry denotes the loads within the asymmetrical system. Referential command currents are calculated by the instantaneous reactive power theory, while the APF output currents are controlled by a classical PID controller. Without considering the affection of current coupling under the symmetrical or asymmetrical condition, Figure 3 shows the load and source currents in phase A after compensation of the shunt APF, while Figure 4 typically shows that under asymmetrical conditions including source or loads asymmetry. By comparison of the simulated results it can be seen that the current tracking error in the asymmetric system is much larger than that in the symmetric system. Hence, proper measures must be taken to counteract the current tracking errors that resulted from current coupling as well as the neutral point voltage.Figure 2
Typical configuration of a shunt APF topology.Figure 3
Current waveforms of a shunt APF in the symmetrical system.Current waveforms of a shunt APF in the asymmetrical system.
(a)
Asymmetrical source system(b)
Asymmetrical loads system


## 3. Discrete-Time Mathematical Model of the Three-Phase Three-Line VSC inα-β Domain

The discrete-time expression of the mathematical model of a three-phase three-line VSC can be derived from (10) as follows:(13)LfiSa(k+1)-iSa(k)T=uSa(k)-RSiSa(k)-uCa(k)-13Δu(k),LfiSb(k+1)-iSb(k)T=uSb(k)-RSiSb(k)-uCb(k)-13Δu(k),LfiSc(k+1)-iSc(k)T=uSc(k)-RSiSc(k)-uCc(k)-13Δu(k),
where T is the sampling interval, k=0,1,…,n.Based onα-β transform of the instantaneous reactive power theory, (13) can be transformed into the α-β domain through the orthogonal transform matrix C, which transforms the three-phase system to a two-phase system, as shown in (17):(14)C=23[1-12-12032-32],(15)[uSα(k)uSβ(k)]=C[uSa(k)uSb(k)uSc(k)],(16)[iSα(k)iSβ(k)]=C[iSa(k)iSb(k)iSc(k)],(17)[uCα(k)uCβ(k)]=C[uCa(k)-13Δu(k)uCb(k)-13Δu(k)uCc(k)-13Δu(k)]=C[uCa(k)uCb(k)uCc(k)].The expressions in (17) clearly indicate that the items associated with the neutral point voltage have disappeared in the α-β domain, which means the same current tracking control behavior of an asymmetrical system as that of a symmetrical system. Therefore, if the current control scheme of the three-line VSC is implemented directly in the α-β domain, it would be unnecessary to consider the impact any more from the neutral point voltage, and only two controllers are thereby required. As a result, there comes the discrete-time mathematical model of the three-phase three-line VSC in the α-β domain, based on which decoupled current control will be achieved accordingly:(18)LfiSα(k+1)-iSα(k)T=uSα(k)-RSiSα(k)-uCα(k),LfiSβ(k+1)-iSβ(k)T=uSβ(k)-RSiSβ(k)-uCβ(k).


## 4. Implementation of the Decoupled Current Control Scheme

Here a classical PID controller is combined with the decoupled current control scheme of the VSC in theα-β domain, the diagram of which is shown in Figure 5. The input of the controller is defined as
(19)ej(k)=iSj*(k)-iSj(k),
where j=α,β and iSj* denotes the referential command currents of the VSC in the α-β domain. The current control loop is realized using a DSP hardware controller, in which the current references are calculated and given by the instantaneous reactive power theory. The output currents of the VSC are sampled by the DSP element as to be the current feedback.Figure 5
Schematic diagram of the proposed current control scheme.Since the sampling intervalT is small, then iSj*(k+1)≈iSj*(k). From (18) and (19), one can get(20)ej(k+1)-ej(k)=-iSj(k+1)+iSj(k)=-TLf(uSj(k)-RSiSj(k)-uCj(k)).The incremental form of the discrete-time PID algorithm is applied, namely,(21)Δej(k+1)=ej(k+1)-ej(k)=kP(ej(k+1)-ej(k)+TTIej(k+1)+TDT(ej(k+1)-2ej(k)+ej(k-1))),
where kP, TI and TD represent the PID controller parameters. Combination of (20) and (21) gives the expression of the prosed control strategy:(22)uCj(k)=uSj(k)-RSiSj(k)+LfTΔej(k).Firstly, the controller output is calculated according to (22). Then, based on the relationship between the output and the switching logic, the states of the switches S1, S2, and S3 can be determined. Finally, the controlling signals of the three-phase PWM are generated accordingly.With substitution of (11), (17) corresponding to the control strategy can be changed into(23)[uCα(k)uCβ(k)]=C[uCa(k)uCb(k)uCc(k)]=[32uCa(k)22(uCa(k)+2uCb(k))]=16[20-13-1-3][pa(k)pb(k)pc(k)]uDC(k).According to (22) and (23), the following relationship can be obtained.IfuCα(k)<0, pa(k)=0; if uCα(k)>0, pa(k)=1.IfuCβ(k)<0, pb(k)=0 and pc(k)=1; if uCβ(k)>0, pb(k)=1 and pc(k)=0.IfuCa(k)<0 and uCβ(k)=0, pb(k)=1 and pc(k)=1; if uCa(k)>0 and uCβ(k)=0, pb(k)=0 and pc(k)=0.The true values of the three-phase PWM switching signals are consequently given in Table1 for the discrete-time decoupled current control methodology.Table 1
True value table of three-phase PWM switching signal.
Output of the controllerOutput of the PWM signalsuCαuCβpapbpc<0<0001<0>0010<00011>00100>0<0101>0>0110The switching states in the three legs can be determined by collating the true value table according to the output of the discrete-time controller acquired from (23). The three-phase PWM switching signals will be generated as to subsequently realize decoupled control of the three-current ia, ib, and ic.


## 5. Simulation and Experiment

Performance verification of the proposed decoupled current control methodology is carried out on a developed VSC-based shunt APF system, as shown in Figure2.


### 5.1. Simulation Results

The available program PSIM is adopted to establish simulation models as to verify the effectiveness and validity of the new algorithm proposed above. This simulation topology is the same as the shunt APF system shown in Figure2, and the nonlinear load is also represented by a three-phase rectifier connected with an asymmetrical RLC circuitry. The whole system is simulated in the discrete-time domain while the sampling frequency is set to 12 kHz. The current reference signals to be tracked are acquired by the instantaneous reactive power theory, in which low-pass digital filters (LPFs) of the second-order Butterworth IIR type are incorporated [15]. The circuits for two-phase current controller are established, as well as the logic circuits to implement the decoupled control scheme. Here, the maximum switching frequency is set to 20 kHz in the simulations.Figure6 shows the three-phase load currents. The compensated currents of the shunt APF can be acquired by adjusting the parameters of the two PID controllers. As an example, both the supply voltage and current of phase A are given in Figure 7, which indicates that the supply current can be effectively compensated to be in phase with the supply voltage. A transition process as well as a little steady error may exist due to the dynamic characteristics of the low-pass digital filters being adopted; however, they can be easily avoided if the physical compensating operation starts from some instant within the steady status of the digital filters. The three-phase PWM signals shown in Figure 8 for decoupled current control can be acquired according to Table 1. It can be seen that the spectrum of phase A is deferent from that of the other two phases, which is mainly due to the decoupled process being carried out between phase B and phase C.Figure 6
The three-phase load currents.Figure 7
The supply voltage and current of phase A.Figure 8
The generated three-phase PWM signals.


### 5.2. Experimental Results

An experimental platform for the VSC-based shunt APF system has been established, in which the proposed decoupled current control scheme is incorporated. The converter is controlled within a closed loop by a 32-bit 150 MHz DSP element. Figure9 shows the schematic diagram of the 400 V shunt APF circuitry for experimental setup. Here the source voltage is in 50 Hz and rated at 220 V (RMS). In the experiments, a three-phase rectifier being connected with an asymmetrical RLC circuitry is used to act as the loads which are similar to that in the above simulations.Figure 9
Schematic diagram of the VSC-based shunt APF setup for experimental study.The experimental platform, consisting of a converter and a control circuit, functions well to realize data acquisition, signal driver, generation of the compensated current, and so on.  Three HALL voltage sensors as well as six HALL current sensors are used to pick up the three-phase supply voltages, supply currents, and load currents. The detected signals are amplified and isolated by the signal modulating unit before being sent into the DSP module. The most important component of the shunt APF converter is an IGBT-based intelligent power module (IPM). Also, several DC sources are designed with sufficient isolation to output +5 V, +3.3 V, and +1.9 V for experimental usage. An executive program is written into the DSP module to complete signal sampling, calculation of the harmonic and reactive currents, output current control of the shunt APF and the DC side voltage with a PID controller, and so forth.All the waveforms are captured with a digital oscilloscope Agilent DOS5034A. In spite of the supply current fed by the current clamp directly into the oscilloscope, the rest waveforms are taken from the entry of the DSP module.Figure10 shows the asymmetrical load currents, while Figure 11 shows the three-phase output currents of the shunt APF unit. In Figures 12, 13, and 14, the experimental results for the voltages and currents of phase A, B, and C after compensation are presented, the four waveforms of which in turn from upward to downward denote the supply voltage, the compensation current of the shunt APF, the supply current, and the load current.Figure 10
The three-phase load currents (10 A/div).Figure 11
The output currents of the shunt APF unit (5 A/div).Figure 12
The voltage and currents of phase A ((1) 600 V/div, (2) 5 A/div, (3) 10 A/div, and (4) 10 A/div).Figure 13
The voltage and currents of phase B ((1) 300 V/div, (2) 5 A/div, (3) 10 A/div, and (4) 5 A/div).Figure 14
The voltage and currents of phase C ((1) 300 V/div, (2) 5 A/div, (3) 10 A/div, and (4) 10 A/div).Figure15 shows the measured voltages and currents of phase C when the shunt APF starts running with a load step, the four waveforms of which in turn denote the supply voltage, the compensation current, the supply current, and the load current.Figure 15
The voltage and currents of phase C when SAPF starts running ((1) 300 V/div, (2) 5 A/div, (3) 10 A/div, and (4) 10 A/div).It can be seen from the experimental waveforms that, with a fast control loop based on the decoupled current control methodology, the supply current will be approximately compensated into a power-frequency sine wave, in which the low-frequency harmonics are significantly reduced. Also, the supply current can be compensated to be exactly in phase with the supply voltage of base-frequency. The experimental studies demonstrate that the new control algorithm proposed in this paper can effectively and correctly realize instantaneous compensation of the harmonics and the reactive power. The decoupled current control algorithm presents a significant strategy to counteract the impact from either the current coupling or the nonlinear and asymmetrical loading conditions, which also helps to extend the application of VSC-based shunt APF technology in power systems.


## 5.1. Simulation Results

The available program PSIM is adopted to establish simulation models as to verify the effectiveness and validity of the new algorithm proposed above. This simulation topology is the same as the shunt APF system shown in Figure2, and the nonlinear load is also represented by a three-phase rectifier connected with an asymmetrical RLC circuitry. The whole system is simulated in the discrete-time domain while the sampling frequency is set to 12 kHz. The current reference signals to be tracked are acquired by the instantaneous reactive power theory, in which low-pass digital filters (LPFs) of the second-order Butterworth IIR type are incorporated [15]. The circuits for two-phase current controller are established, as well as the logic circuits to implement the decoupled control scheme. Here, the maximum switching frequency is set to 20 kHz in the simulations.Figure6 shows the three-phase load currents. The compensated currents of the shunt APF can be acquired by adjusting the parameters of the two PID controllers. As an example, both the supply voltage and current of phase A are given in Figure 7, which indicates that the supply current can be effectively compensated to be in phase with the supply voltage. A transition process as well as a little steady error may exist due to the dynamic characteristics of the low-pass digital filters being adopted; however, they can be easily avoided if the physical compensating operation starts from some instant within the steady status of the digital filters. The three-phase PWM signals shown in Figure 8 for decoupled current control can be acquired according to Table 1. It can be seen that the spectrum of phase A is deferent from that of the other two phases, which is mainly due to the decoupled process being carried out between phase B and phase C.Figure 6
The three-phase load currents.Figure 7
The supply voltage and current of phase A.Figure 8
The generated three-phase PWM signals.


## 5.2. Experimental Results

An experimental platform for the VSC-based shunt APF system has been established, in which the proposed decoupled current control scheme is incorporated. The converter is controlled within a closed loop by a 32-bit 150 MHz DSP element. Figure9 shows the schematic diagram of the 400 V shunt APF circuitry for experimental setup. Here the source voltage is in 50 Hz and rated at 220 V (RMS). In the experiments, a three-phase rectifier being connected with an asymmetrical RLC circuitry is used to act as the loads which are similar to that in the above simulations.Figure 9
Schematic diagram of the VSC-based shunt APF setup for experimental study.The experimental platform, consisting of a converter and a control circuit, functions well to realize data acquisition, signal driver, generation of the compensated current, and so on.  Three HALL voltage sensors as well as six HALL current sensors are used to pick up the three-phase supply voltages, supply currents, and load currents. The detected signals are amplified and isolated by the signal modulating unit before being sent into the DSP module. The most important component of the shunt APF converter is an IGBT-based intelligent power module (IPM). Also, several DC sources are designed with sufficient isolation to output +5 V, +3.3 V, and +1.9 V for experimental usage. An executive program is written into the DSP module to complete signal sampling, calculation of the harmonic and reactive currents, output current control of the shunt APF and the DC side voltage with a PID controller, and so forth.All the waveforms are captured with a digital oscilloscope Agilent DOS5034A. In spite of the supply current fed by the current clamp directly into the oscilloscope, the rest waveforms are taken from the entry of the DSP module.Figure10 shows the asymmetrical load currents, while Figure 11 shows the three-phase output currents of the shunt APF unit. In Figures 12, 13, and 14, the experimental results for the voltages and currents of phase A, B, and C after compensation are presented, the four waveforms of which in turn from upward to downward denote the supply voltage, the compensation current of the shunt APF, the supply current, and the load current.Figure 10
The three-phase load currents (10 A/div).Figure 11
The output currents of the shunt APF unit (5 A/div).Figure 12
The voltage and currents of phase A ((1) 600 V/div, (2) 5 A/div, (3) 10 A/div, and (4) 10 A/div).Figure 13
The voltage and currents of phase B ((1) 300 V/div, (2) 5 A/div, (3) 10 A/div, and (4) 5 A/div).Figure 14
The voltage and currents of phase C ((1) 300 V/div, (2) 5 A/div, (3) 10 A/div, and (4) 10 A/div).Figure15 shows the measured voltages and currents of phase C when the shunt APF starts running with a load step, the four waveforms of which in turn denote the supply voltage, the compensation current, the supply current, and the load current.Figure 15
The voltage and currents of phase C when SAPF starts running ((1) 300 V/div, (2) 5 A/div, (3) 10 A/div, and (4) 10 A/div).It can be seen from the experimental waveforms that, with a fast control loop based on the decoupled current control methodology, the supply current will be approximately compensated into a power-frequency sine wave, in which the low-frequency harmonics are significantly reduced. Also, the supply current can be compensated to be exactly in phase with the supply voltage of base-frequency. The experimental studies demonstrate that the new control algorithm proposed in this paper can effectively and correctly realize instantaneous compensation of the harmonics and the reactive power. The decoupled current control algorithm presents a significant strategy to counteract the impact from either the current coupling or the nonlinear and asymmetrical loading conditions, which also helps to extend the application of VSC-based shunt APF technology in power systems.


## 6. Conclusion

The coupling effect of the three-phase currents in a three-line VSC-based system increases the difficulty of current tracking control. The paper proposed an effective decoupled current control scheme which was verified by both simulative and experimental studies.Only two governing equations are independent in the mathematical model of a three-phase three-line VSC-based system since the summation of the three-phase currents is always zero. Further, for asymmetrical conditions, the neutral point voltage will directly affect the tracking control behavior of the VSC output currents. Simulations verified that the current tracking control error would be enlarged if the coupling effect was ignored.A discrete-time mathematical model in theα-β domain for the three-phase three-line VSC is derived through orthogonal transform, in which the impact from the neutral point voltage on current tracking control is eliminated. The decoupling of the three-phase currents can be achieved in the α-β coordinate system.Combined with the instantaneous harmonic and reactive currents detection and the PID controller, an integrated decoupled current control scheme has been established. Simulations and experiments have verified the feasibility and effectiveness of the proposed control strategy.



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*Source: 605043-2011-08-21.xml*