adiscrete-timedecoupledcurrentcontrolschemeappliedto vsc

Hindawi Publishing CorporationAdvances in Power ElectronicsVolume 2011, Article ID 605043, 9 pagesdoi:10.1155/2011/605043

Research Article

A Discrete-Time Decoupled Current Control Scheme Applied toVSC-Based Three-Phase Three-Line Systems

Hui Wang, Qingmin Li, Guangzhu Wang, and Xingguo Tan

School of Electrical Engineering, Shandong University, 73 Jing Shi Road, Shandong, Jinan 250061, China

Correspondence should be addressed to Qingmin Li, [email protected]

Received 14 February 2011; Revised 2 June 2011; Accepted 5 June 2011

Academic Editor: C. M. Liaw

Copyright © 2011 Hui Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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-timedecoupled current control strategy is proposed in the paper. With integration of the α-β transform and two independent currentcontrollers, the proposed methodology can effectively implement decoupled control of the three-phase currents, which can therebyeliminate the impact from the neutral-point voltage especially under asymmetrical loading conditions. Control performance underdigital realization was characterized with extensive tests on a shunt active power filter (SAPF) platform. Both the simulative andexperimental results have demonstrated that the SAPF could function well and thereby verified the feasibility and effectiveness ofthe proposed current control methodology.

1. Introduction

Based on fully controlled power semiconductors, voltagesource converters (VSCs), such as DC-DC, DC-AC, andAC-DC converters, are being more widely used than cur-rent source converters (CSCs), as the DC side capacitorused for energy storage in a VSC is much superior toan inductor in a CSC in terms of weight, cost, andefficiency, especially suitable for the nowadays switchingsemiconductors. For concrete applications with VSCs, suchas active power filters (APF), PWM rectifiers, power factorcorrection (PFC), HVDC light and grid-connected inverters,an accurate current control methodology with preferabledynamic performance is indispensible and critical [1–7].Current control is normally used for the inner closed loopin a VSC-based system, in which PWM signals are directlygenerated to track the current reference accurately as wellas ameliorate controllability and dynamic response of thesystem.

For a three-phase four-line circuitry of a VSC, the currentcontrol strategy makes no difference at all from that for asingle-phase system since the phase currents are independentof each other in the circuitry. However, in a three-phase

three-line system, the three-phase currents are interactionaland electromagnetically coupled. As a result, only two circuitequations are independent accordingly. In addition, theneutral point voltage of the three-line circuitry will alsoaffect the behavior of current tracking control in the VSC,especially when the three-phase voltages or loads are notsymmetrically presented. Hence, it is a pending issue to beproperly tackled of the current control strategy in a VSC-based three-phase three-line circuitry.

Most of the nowadays current control strategies are basedon the d-q synchronous rotating frame as to decouple theactive currents and the reactive currents in a three-phase VSCsystem [7–12], in which the physical concepts of the adoptedvariables are unique and clear; however, the instant value ofthe angular power frequency ω and the output inductanceof the VSC should be known in advance. Also, the sensibleimpact of the neutral point voltage was not incorporated [9,10, 12]. References [13, 14] advised using an α-β stationaryframe to transform the mathematical model of an APF, andthe VSC topology was designed in the α-β stationary frameso as to achieve current decoupling, which did facilitatethe control strategy but the corresponding hardware designbecame much more complicated than before.

2 Advances in Power Electronics

To cope with the above-mentioned issues, based on ageneral mathematical model of the three-phase three-lineVSC system, the impact of the neutral point voltage oncurrent control scheme was fully elucidated in this paper.Further, a discrete-time mathematical model in the α-βcoordinate system was established through orthogonal trans-form, with a view to developing a decoupled current controlstrategy as to eliminate the impact of the neutral pointvoltage and finally achieve decoupling of the three-phasecurrents. Simulation and experimental results on an APFprototype under nonlinear loading conditions were alsopresented for verification purpose.

2. Impact of Current Coupling ina Three-Line VSC System

For a typical VSC configuration in the three-line system asshown in Figure 1, several assumptions are made as followsin establishing a mathematical model: (1) the sources andloads are in Y connection (Δ connection can be transferredinto Y connection); (2) the three-line inductance is ignoreddue to its very small value and little effect; (3) the three-lineresistances Rs and the output Inductor L f of the VSC unit aresymmetrical; (4) the mathematical model of the VSC unit isdescribed in the continuous-time system for convenience.

Given asymmetrical sources or loads in the three-phasesystem, the phase voltages uSa, uSb, and uSc can be expressedas follows:

uSa + uSb + uSc = Δu /= 0. (1)

If the sources and loads are symmetrical in the three-phase system, then uSa + uSb + uSc = Δu = 0 will be fulfilled.For the three-line voltages uab, ubc, and uca in the AC side,one gets

uab + ubc + uca ≡ 0. (2)

Based on Kirchhoff ’s Voltage Law, the circuit equation ofphase A is governed by

L fdiSadt

= uSa − RSiSa − uaN − uN0. (3)

If the switching states of S1, S2, and S3 are expressed by pa,pb, and pc, then (pa, pb,pc) ∈ (0, 1), where logic 1 representsthat the upper-leg switch is turned on while the lower-legswitch is turned off. Similarly, logic 0 represents the lower-leg switch is turned on while the upper-leg switch is turnedoff. If S1 is turned on and S2 is turned off, then pa = 1 anduaN = uDC. If S2 is turned on and S1 is turned off, then pa = 0and uaN = 0. Hence, we get uaN = uDCpa and (3) can bechanged to

L fdiSadt

= uSa − RSiSa − uDCpa − uN0. (4)

In the same way, the governing equations of phase B andphase C are obtained as follows:

L fdiSbdt

= uSb − RSiSb − uDCpb − uN0, (5)

L fdiScdt

= uSc − RSiSc − uDCpc − uN0, (6)

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 voltagedifference between point N and the neutral point.

Obviously, the three-line currents will meet

iSa + iSb + iSc = 0. (7)

Based on (4) to (7), the following relationship can bederived:(uSa − RSiSa − uDCpa − uN0

)+(uSb − RSiSb − uDCpb − uN0

)

+(uSc − RSiSc − uDCpc − uN0

) = 0.(8)

Further, we can get

uN0 = 13Δu− uDC

3

(pa + pb + pc

). (9)

If (9) is substituted into (4) to (6), the mathematicalmodel of the VSC unit can be obtained when the sources orloads are in asymmetrical conditions:

L fdiSadt

= uSa − RSiSa − uCa − 13Δu,

L fdiSbdt

= uSb − RSiSb − uCb − 13Δu,

L fdiScdt

= uSc − RSiSc − uCc − 13Δu,

(10)

where

uCa = uDC(2pa − pb − pc

)

3,

uCb = uDC(−pa + 2pb − pc

)

3,

uCc = uDC(−pa − pb + 2pc

)

3.

(11)

For the symmetrical conditions of the sources or loads inthe three-phase system, uSa + uSb + uSc = Δu = 0 will be met,which directly results in the corresponding mathematicalmodel of the VSC shown below:

L fdiSadt

= uSa − RSiSa − uCa,

L fdiSbdt

= uSb − RSiSb − uCb,

L fdiScdt

= uSc − RSiSc − uCc.

(12)

Only two governing equations are outstandingly inde-pendent since the summation of the three-phase currentsalways is equal to zero as shown in (7). With regardto the asymmetrical system, (10) shows that the neutralpoint voltage will impose direct impact on the trackingcontrol performance of output currents in the VSC system.

Advances in Power Electronics 3

a

bc

S1 S3

S2 S4

S5

S6

CDC

DCsource

orloads

VSCAC poweror loads

uSa

uSb

uSc

iSa

iSb

iSc

L f

L f

L f

0

RS

RS

RS

uDC

N

Figure 1: Typical configuration of a three-phase VSC.

AC power

ab

c

A

B

C

S1

S2 S4

S5

S6

CDC

Loads

APF

0

S3

uSa

uSb

uSc

iSa

iSb

iSc

iLa

iLb

iLc

iFa iFb iFc

L f L f L f

RS

RS

RS

uDC

N

Figure 2: Typical configuration of a shunt APF topology.

For an example shown in Figure 2, where a VSC unit isutilised in a shunt APF topology with nonlinear loads,simulations are carried out with the PSIM software. Thesource frequency of the APF system is 50 Hz and the sourcevoltage is 220 V (RMS). The nonlinear load is representedby a three-phase rectifier in a symmetrical system, andthe three-phase rectifier connected with an asymmetricalRLC circuitry denotes the loads within the asymmetricalsystem. Referential command currents are calculated by theinstantaneous reactive power theory, while the APF outputcurrents are controlled by a classical PID controller. Withoutconsidering the affection of current coupling under thesymmetrical or asymmetrical condition, Figure 3 shows theload and source currents in phase A after compensationof the shunt APF, while Figure 4 typically shows thatunder asymmetrical conditions including source or loadsasymmetry. By comparison of the simulated results it can beseen that the current tracking error in the asymmetric systemis 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 wellas the neutral point voltage.

3. Discrete-Time Mathematical Model of theThree-Phase Three-Line VSC in α-β Domain

The discrete-time expression of the mathematical model ofa three-phase three-line VSC can be derived from (10) asfollows:

L fiSa(k + 1)−iSa(k)

T= uSa(k)−RSiSa(k)−uCa(k)− 1

3Δu(k),

L fiSb(k + 1)−iSb(k)

T= uSb(k)−RSiSb(k)−uCb(k)− 1

3Δu(k),

L fiSc(k + 1)−iSc(k)

T= uSc(k)−RSiSc(k)−uCc(k)− 1

3Δu(k),

(13)

where T is the sampling interval, k = 0, 1, . . . ,n.


300200100

0−100−200−300

150100

500

−50−100−150

Time (s)

0.02 0.05 0.08 0.11 0.14 0.17 0.2

uSa (V) uSb (V) uSc (V)

iLa (A) iSa (A)

Figure 3: Current waveforms of a shunt APF in the symmetricalsystem.

Based on α-β transform of the instantaneous reactivepower theory, (13) can be transformed into the α-β domainthrough the orthogonal transform matrix C, which trans-forms the three-phase system to a two-phase system, asshown in (17):

C =√

23

⎡

⎢⎢⎢⎣

1 −12−1

2

0

√3

2−√

32

⎤

⎥⎥⎥⎦

, (14)

⎡

⎣uSα(k)

uSβ(k)

⎤

⎦ = C

⎡

⎢⎢⎢⎣

uSa(k)

uSb(k)

uSc(k)

⎤

⎥⎥⎥⎦

, (15)

⎡

⎣iSα(k)

iSβ(k)

⎤

⎦ = C

⎡

⎢⎢⎢⎣

iSa(k)

iSb(k)

iSc(k)

⎤

⎥⎥⎥⎦

, (16)

⎡

⎣uCα(k)

uCβ(k)

⎤

⎦ = C

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

uCa(k)− 13Δu(k)

uCb(k)− 13Δu(k)

uCc(k)− 13Δu(k)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

= C

⎡

⎢⎢⎢⎣

uCa(k)

uCb(k)

uCc(k)

⎤

⎥⎥⎥⎦. (17)

The expressions in (17) clearly indicate that the itemsassociated with the neutral point voltage have disappearedin the α-β domain, which means the same current trackingcontrol behavior of an asymmetrical system as that of asymmetrical system. Therefore, if the current control schemeof the three-line VSC is implemented directly in the α-βdomain, it would be unnecessary to consider the impactany more from the neutral point voltage, and only twocontrollers are thereby required. As a result, there comes the

300200100

0−100−200−300

150100

500

−50−100−150

Time (s)0.02 0.05 0.08 0.11 0.14 0.17 0.2

uSa (V) uSb (V) uSc (V)

iLa (A) iSa (A)

(a) Asymmetrical source system

150100

500

−50−100−150

150100

500

−50−100−150

Time (s)

0.20.002 0.052 0.101 0.151

iLa (A)

iLa (A)

iSa (A)

iLb (B) iLc (C)

(b) Asymmetrical loads system

Figure 4: Current waveforms of a shunt APF in the asymmetricalsystem.

discrete-time mathematical model of the three-phase three-line VSC in the α-β domain, based on which decoupledcurrent control will be achieved accordingly:

L fiSα(k + 1)− iSα(k)

T= uSα(k)− RSiSα(k)− uCα(k),

L fiSβ(k + 1)− iSβ(k)

T= uSβ(k)− RSiSβ(k)− uCβ(k).

(18)

4. Implementation of the Decoupled CurrentControl Scheme

Here a classical PID controller is combined with the decou-pled current control scheme of the VSC in the α-β domain,the diagram of which is shown in Figure 5. The input of thecontroller is defined as

ej(k) = i∗S j(k)− iS j(k), (19)

where j = α,β and i∗S j denotes the referential commandcurrents of the VSC in the α-β domain. The current controlloop is realized using a DSP hardware controller, in which


the current references are calculated and given by theinstantaneous reactive power theory. The output currents ofthe VSC are sampled by the DSP element as to be the currentfeedback.

Since the sampling interval T is small, then i∗S j(k + 1) ≈i∗S j(k). From (18) and (19), one can get

ej(k + 1)− ej(k) = −iS j(k + 1) + iS j(k)

= − T

Lf

(uSj(k)− RSiS j(k)− uC j(k)

).

(20)

The incremental form of the discrete-time PID algorithmis applied, namely,

Δej(k + 1) = ej(k + 1)− ej(k)

= kP

(ej(k + 1)− ej(k) +

T

TIej(k + 1)

+TD

T

(ej(k + 1)− 2ej(k) + ej(k − 1)

)),

(21)

where kP , TI and TD represent the PID controller parameters.Combination of (20) and (21) gives the expression of theprosed control strategy:

uC j(k) = uSj(k)− RSiS j(k) +L f

TΔej(k). (22)

Firstly, the controller output is calculated according to(22). Then, based on the relationship between the outputand the switching logic, the states of the switches S1, S2, andS3 can be determined. Finally, the controlling signals of thethree-phase PWM are generated accordingly.

With substitution of (11), (17) corresponding to thecontrol strategy can be changed into

⎡

⎣uCα(k)

uCβ(k)

⎤

⎦ = C

⎡

⎢⎢⎢⎣

uCa(k)

uCb(k)

uCc(k)

⎤

⎥⎥⎥⎦

=

⎡

⎢⎢⎣

√3

2uCa(k)

√2

2(uCa(k) + 2uCb(k))

⎤

⎥⎥⎦

=√

16

⎡

⎣2

0

−1√

3

−1

−√3

⎤

⎦

⎡

⎢⎢⎢⎣

pa(k)

pb(k)

pc(k)

⎤

⎥⎥⎥⎦uDC(k).

(23)

According to (22) and (23), the following relationshipcan be obtained.

If uCα(k) < 0, pa(k) = 0; if uCα(k) > 0, pa(k) = 1.

If uCβ(k) < 0, pb(k) = 0 and pc(k) = 1; if uCβ(k) > 0,pb(k) = 1 and pc(k) = 0.

Table 1: True value table of three-phase PWM switching signal.

Output of the controller Output of the PWM signals

uCα uCβ pa pb pc

<0 <0 0 0 1

<0 >0 0 1 0

<0 0 0 1 1

>0 0 1 0 0

>0 <0 1 0 1

>0 >0 1 1 0

If uCa(k) < 0 and uCβ(k) = 0, pb(k) = 1 andpc(k) = 1; if uCa(k) > 0 and uCβ(k) = 0, pb(k) = 0and pc(k) = 0.

The true values of the three-phase PWM switchingsignals are consequently given in Table 1 for the discrete-timedecoupled current control methodology.

The switching states in the three legs can be determinedby collating the true value table according to the output of thediscrete-time controller acquired from (23). The three-phasePWM switching signals will be generated as to subsequentlyrealize decoupled control of the three-current ia, ib, and ic.

5. Simulation and Experiment

Performance verification of the proposed decoupled currentcontrol methodology is carried out on a developed VSC-based shunt APF system, as shown in Figure 2.

5.1. Simulation Results. The available program PSIM isadopted to establish simulation models as to verify theeffectiveness and validity of the new algorithm proposedabove. This simulation topology is the same as the shuntAPF system shown in Figure 2, and the nonlinear load isalso represented by a three-phase rectifier connected with anasymmetrical RLC circuitry. The whole system is simulatedin the discrete-time domain while the sampling frequencyis set to 12 kHz. The current reference signals to be trackedare acquired by the instantaneous reactive power theory,in which low-pass digital filters (LPFs) of the second-orderButterworth IIR type are incorporated [15]. The circuits fortwo-phase current controller are established, as well as thelogic circuits to implement the decoupled control scheme.Here, the maximum switching frequency is set to 20 kHz inthe simulations.

Figure 6 shows the three-phase load currents. The com-pensated currents of the shunt APF can be acquired byadjusting the parameters of the two PID controllers. As anexample, both the supply voltage and current of phase A aregiven in Figure 7, which indicates that the supply current canbe effectively compensated to be in phase with the supplyvoltage. A transition process as well as a little steady errormay exist due to the dynamic characteristics of the low-pass digital filters being adopted; however, they can be easilyavoided if the physical compensating operation starts fromsome instant within the steady status of the digital filters. The


VSC

+

AC poweror loads

PID currentcontrollers

DecoupledPWM

e−

DCsource

orloads

uS

i∗S

RSiS

iS

uDC

Figure 5: Schematic diagram of the proposed current control scheme.

150

100

50

0

−50

−100

−150

Time (s)

0.05 0.075 0.1 0.125 0.15

iLa (A) iLb (A) iLc (A)

Figure 6: The three-phase load currents.

400

200

0

−200

−400

Time (s)

0.05 0.075 0.1 0.125 0.15

iSa (A) uSa (V)

Figure 7: The supply voltage and current of phase A.

three-phase PWM signals shown in Figure 8 for decoupledcurrent control can be acquired according to Table 1. It canbe seen that the spectrum of phase A is deferent from that ofthe other two phases, which is mainly due to the decoupledprocess being carried out between phase B and phase C.

5.2. Experimental Results. An experimental platform forthe VSC-based shunt APF system has been established, inwhich the proposed decoupled current control scheme isincorporated. The converter is controlled within a closedloop by a 32-bit 150 MHz DSP element. Figure 9 shows theschematic diagram of the 400 V shunt APF circuitry forexperimental setup. Here the source voltage is in 50 Hz andrated at 220 V (RMS). In the experiments, a three-phase

Times (ms)

1

0.6

0.2

1

0.6

0.2

1

0.6

0.2

50 60 70 80 90 100

Pa

Pb

Pc

Figure 8: The generated three-phase PWM signals.

rectifier being connected with an asymmetrical RLC circuitryis used to act as the loads which are similar to that in theabove simulations.

The experimental platform, consisting of a converter anda control circuit, functions well to realize data acquisition,signal driver, generation of the compensated current, and soon. Three HALL voltage sensors as well as six HALL currentsensors are used to pick up the three-phase supply voltages,supply currents, and load currents. The detected signalsare amplified and isolated by the signal modulating unitbefore being sent into the DSP module. The most importantcomponent of the shunt APF converter is an IGBT-basedintelligent power module (IPM). Also, several DC sourcesare designed with sufficient isolation to output +5 V, +3.3 V,and +1.9 V for experimental usage. An executive program iswritten into the DSP module to complete signal sampling,calculation of the harmonic and reactive currents, outputcurrent control of the shunt APF and the DC side voltagewith a PID controller, and so forth.

All the waveforms are captured with a digital oscilloscopeAgilent DOS5034A. In spite of the supply current fed bythe current clamp directly into the oscilloscope, the restwaveforms are taken from the entry of the DSP module.

Figure 10 shows the asymmetrical load currents, whileFigure 11 shows the three-phase output currents of the shuntAPF unit. In Figures 12, 13, and 14, the experimental resultsfor the voltages and currents of phase A, B, and C aftercompensation are presented, the four waveforms of which in


iLiS

iF

L f

IPM

Sensor detecting

Signal modulating

DSPdigital

controllingDC voltage detecting

PWM driver

Powervoltages

Loadscurrents

DC voltage

CDC

APF currents

uS

uDC

Nonlinear loads

Figure 9: Schematic diagram of the VSC-based shunt APF setup for experimental study.

Agilent technologies2.00V/ 2.00V/ 2.00V/ −200.0

µs 25.0m

v

1

1 1

2

2 3

4

4 10.00ms /

Figure 10: The three-phase load currents (10 A/div).

turn from upward to downward denote the supply voltage,the compensation current of the shunt APF, the supplycurrent, and the load current.

Figure 15 shows the measured voltages and currents ofphase C when the shunt APF starts running with a loadstep, the four waveforms of which in turn denote the supplyvoltage, the compensation current, the supply current, andthe load current.

It can be seen from the experimental waveforms that,with a fast control loop based on the decoupled currentcontrol methodology, the supply current will be approx-imately compensated into a power-frequency sine wave,in which the low-frequency harmonics are significantlyreduced. Also, the supply current can be compensated to beexactly in phase with the supply voltage of base-frequency.The experimental studies demonstrate that the new controlalgorithm proposed in this paper can effectively and correctlyrealize instantaneous compensation of the harmonics andthe reactive power. The decoupled current control algorithm

Agilent technologies

25.0mv

1

1 1

2

2 3

4

41.00V/ 1.00V/ 1.00V/ 0.0s 10.00ms /

Figure 11: The output currents of the shunt APF unit (5 A/div).


25.0mv

1

1 1

2

2

4

41.00V/ 1.00V/ 0.0s 10.00ms /3

3

100mv / 2.00V/

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).



25.0mv

1

1 1

2

2

4

41.00V/ 1.00V/3

3

100mv /500m

v / −9.600ms 10.00m

s /

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).


25.0mv

1

1 1

2

2

4

41.00V/ 10.00ms /3

3

100mv /500m

v / 2.00V/ 0.0s

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).

presents a significant strategy to counteract the impactfrom either the current coupling or the nonlinear andasymmetrical loading conditions, which also helps to extendthe application of VSC-based shunt APF technology in powersystems.

6. Conclusion

The coupling effect of the three-phase currents in a three-lineVSC-based system increases the difficulty of current trackingcontrol. The paper proposed an effective decoupled currentcontrol scheme which was verified by both simulative andexperimental studies.

Only two governing equations are independent in themathematical model of a three-phase three-line VSC-basedsystem since the summation of the three-phase currentsis always zero. Further, for asymmetrical conditions, theneutral point voltage will directly affect the tracking controlbehavior of the VSC output currents. Simulations verified


25.0mv

1

1 1

2

2

4

41.00V/ 3

3

100mv /500m

v / 2.00V/ 20.00m/960.0ms s

Figure 15: The voltage and currents of phase C when SAPF startsrunning ((1) 300 V/div, (2) 5 A/div, (3) 10 A/div, and (4) 10 A/div).

that the current tracking control error would be enlarged ifthe coupling effect was ignored.

A discrete-time mathematical model in the α-β domainfor the three-phase three-line VSC is derived throughorthogonal transform, in which the impact from the neutralpoint voltage on current tracking control is eliminated. Thedecoupling of the three-phase currents can be achieved in theα-β coordinate system.

Combined with the instantaneous harmonic and reactivecurrents detection and the PID controller, an integrateddecoupled current control scheme has been established.Simulations and experiments have verified the feasibility andeffectiveness of the proposed control strategy.

Acknowledgments

This research work was supported by the National Sci-ence Foundation of China (50807033) and the ShandongProvincial Natural Science Foundation (ZR2009FQ025,ZR2009FM013).

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