A Novel Current Controller for Three-Phase Voltage-Source Inverters

Mansour Mohseni Syed Islam

Department of Electrical and Computer Engineering, Curtin University of Technology, WA, Australia. mansour.mohseni@postgrad.curtin.edu.au s.islam@curtin.edu.au

Abstract- A novel vector-based hysteresis current controller for three-phase PWM voltage-source inverters is proposed in this paper. This current control scheme implements two sets of hysteresis comparators in three-phase abc and stationary α-β frames integrated with a switching table. 2-level three-phase comparators are employed as region detector to confine the optimal voltage space vectors to be applied. Then, the appropriate voltage space vector at each instant is determined using two 3-level comparators with narrow hysteresis band working in α-β coordinated frame. The proposed control scheme utilizes the advantages of both space vector modulation and hysteresis current controllers. Simulation studies are carried out to demonstrate the improvements achieved by the proposed method. Index Terms- Hysteresis current controller, three-phase PWM voltage-source inverter, vector-based methods.

1. INTRODUCTION In most applications of three-phase PWM voltage-source

inverters (PWM-VSI), such as variable speed ac drives and active power filters, the current control scheme is an essential part of the control system. Over the last a few decades, three major classes of current controller are used for three-phase PWM-VSI: hysteresis controllers, linear PI controllers, and predictive controllers [1]-[3]. Among these techniques, hysteresis current controller is a popular method because of easy implementation, very quick transient response, inherent limited maximum current, and insensitivity to the load parameters variations. Nevertheless, due to lack of coordination between three hysteresis controllers of phases, very high switching frequency at lower modulation index may happen. Hysteresis band violation up to twice of permitted bandwidth is another disadvantage of this method.

To eliminate these drawbacks, several methods were proposed in the literature. The substitution of fixed tolerance bands with variable bands is suggested to reduce the switching frequency [4]-[5]. This solution, although very simple, produces significant low-frequency current harmonics and very high switching frequency around the zero crossing points of the command current. Adaptive hysteresis band current controllers are also proposed to achieve fixed switching frequency and remove interphases dependency [6]-[10]. This method presents satisfactory frequency spectra and low current ripple. However, fixed modulation frequency is usually achieved at the expense of extra signal-processing and control complexity requirements, which jeopardises the simplicity of hysteresis current controllers. Also, these techniques might suffer from stability problems and limited transient performance. Finally, vector-based hysteresis current controllers are suggested as a successful approach to reduce the switching frequency of three-phase VSI through very simple control schemes [11]-[16]. In this method, the

current error is computed in the space vector form, and the developed control scheme is designed to keep the error within the assigned tolerance region. This current control scheme systematically employs zero and nonzero voltage space vectors to follow the current command vector with a reduced switching frequency. Very simple control configuration of vector-based method achieves very satisfactory transient and steady-state performance.

This paper presents a new vector-based hysteresis current controller to reduce the VSI switching frequency. Two sets of hysteresis comparators are employed integrated with a switching table. Three 2-level hysteresis comparators are employed as region detector to confine VSI optimal voltage space vectors. Then, the appropriate voltage space vector at each instant is determined using two 3-level hysteresis comparators working in coordinated α-β frame. The proposed control scheme keeps the current error vector inside a hexagonal tolerance region using a systematic switching pattern between zero and non-zero voltage space vectors. The proposed vector-based method presents the advantages of both hysteresis and space vector modulation (SVM) techniques. Besides considerably reduced switching frequency, it ensures minimized load current oscillation around the reference current, which results in minimized torque oscillation in the induction machine.

2. THREE-PHASE PWM-VSI The circuit diagram of a three-phase PWM-VSI

connected to a load is shown in Fig. 1.a. The load comprises of a three-phase sinusoidal counter-EMF voltage in series with L and R. This emulates an induction motor load.

Conduction states of inverter legs at each instant determine the VSI output voltage space vector, Vn. If the upper switch in phase a (S1) conducts the current then Sa

* =1; otherwise Sa

* =0 indicating S4 is conducting the current. Substituting alternative conduction states for Sa

*, Sb*, and Sc

* produces eight possible output voltage space vectors, as graphically presented in Fig. 1.b. It is observed that the available discrete space voltage vectors includes six nonzero space vectors (V1-V6) and two zero space vectors (V0

0, V01).

On the other hand, three-phase VSI output voltage (vabc) can be transformed into a space vector in the 2-D stationary frame of α-β. The space vector V(t)=vα(t)+jvβ(t) is defined by

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−−=⎥

⎦

⎤⎢⎣

⎡

c

b

a

vvv

vv

23

230

21

211

32

β

α (1)

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Fig. 1. (a) Circuit diagram of a PWM-VSI feeding an induction motor (b) Space vector representation of VSI output voltage

3. PRINCIPLE OF VECTOR-BASED METHODS Using Kirchhoff’s voltage law for Fig. 1.a. produces

( ) ( ) ( )[ ]

( ) ( ) ( )[ ]

( ) ( ) ( )[ ]⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−−−−−−=

−−−−−−=

−−−−−−=

cbbaaccc

bccaabbb

accbbaaa

iLRevevev

Ldtdi

iLRevevev

Ldtdi

iLRevevev

Ldtdi

231

231

231

(2)

Using (1), vector representation of (2) is given by

( ) ono

LR

Ldtd ieVi

−−= 01 (3)

where io is the output current space vector and e0 is the counter-EMF voltage space vector. If iref is the current command space vector and ie is the current error space vector,

orefe iii −= (4) From (3) and (4), the differential equation of current error vector is derived as

( )0eVii

ii

−−+=+ nrefref

ee R

dtd

LRdt

dL (5)

Therefore, the current error vector ie changes with L/R time constant. Moreover, it is influenced by the current command vector and its derivative (iref and diref /dt), VSI output voltage vector (Vn), and counter-EMF voltage vector (e0). Neglecting R value, the desired output voltage Vn

* to achieve zero current error is calculated by

dtd

L refn

ieV += 0

* (6)

Substituting (6) in (5) produces

nne

dtd

L VVi

−= * (7)

It is concluded form (6) and (7) that the information about the counter-EMF voltage space vector and command current vector is decisive to achieve zero current error. However, the calculation of Vn

* is not practical since it includes the counter-EMF voltage vector e0.

Vector-based approaches are presented to solve this problem. Suppose that the desired output voltage vector Vn

* is located in Region I in Fig. 1.b. Based on (7), current error derivative vectors respect to space voltage vectors V0-V6 can be represented by Fig. 2 [16]. Small die/dt is required to follow the current command with minimum switching

Fig. 2. Derivative vectors of current error when Vn

* is located in Region I

frequency. Therefore, only discrete voltage space vectors adjacent to the desired voltage vector Vn

* are to be applied in order to achieve an optimal switching pattern, e.g. V1, V2, V0

0

and V01 in this case. Note that the zero space vectors are the

only vector except non-zero adjacent voltage vectors which are applied to reduce the magnitude of current error; exactly similar to the well-known Space Vector Modulation (SVM) technique. One the other hand, for quick current response during the transient conditions, non-optimal voltage vectors with high current error derivatives must be applied to force the current error vector into the tolerance region as fast as possible. Therefore, to reduce the VSI switching frequency, the set of optimal voltage vectors at each instant must be determined based on the position of the desired voltage vector Vn

*. Furthermore, since only eight discrete VSI output voltage space vectors are available, the region of counter-emf voltage vector is decisive for optimal hysteresis-based switching pattern not the exact vector value.

4. CONVENTIONAL HYSTERESIS METHOD In this current control method, three hysteresis bands of

width δhys are defined around each reference value of the three-phase currents. Accordingly, each phase current is controlled with a 2-level hysteresis comparator, as shown in Fig. 3.a. When the line current becomes greater (or less) than the current reference by the hysteresis band δhys, the respective inverter leg is switched in the negative (or positive) direction. This results in a control scheme targeted to keep the current error within the hexagonal band shown in Fig. 3.b [3]. Note that in this figure the tolerance region is drawn extremely large for the sake of clarity. The hexagon

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centre is pointed by the command current vector at each instant, and it circulates around the origin of the coordinated α-β system with the fundamental frequency. Once the output current vector touches the hexagon surface, the current error exceeds δhys and the VSI output voltage space vector must be switched in the direction of the current error vector to force the current back into the hexagon area.

Nevertheless, there is a mutual interaction between three phase currents in the case of isolated load neutral. Under this interphases dependency, when the conduction state of an inverter leg changes, the resulting voltage space vector is dependent on the states of the other two inverter legs. This causes unnecessary high switching frequency at low counter-emf voltage and the actual current error reaching twice of the permitted hysteresis band (2δhys), shown by the dashed area in Fig. 3.b [3].

Fig. 3. Conventional three-phase hysteresis method (a) Schematic diagram

(b) Designated error surface

5. PROPOSED VECTOR-BASED METHOD This paper presents a novel vector-based hysteresis

controller using a region detector. Two sets of hysteresis comparators are employed in three-phase abc and coordinated α-β frames, as shown in Fig. 4.a. Three 2-level hysteresis comparators with wide band of δ+h work as the region detector and determine the region of desired voltage vector Vn

*. The relationship between the digital outputs of three-phase comparators (Da, Db, Dc) and Vn

* region is given by Table. I. Therefore, the optimal set of three non-zero and one zero voltage space vectors is determined at each instant using three-phase comparators. Then, the appropriate voltage space vectors among the optimal set is determined using two 3-level comparators used in α-β frame. The practical implementation of 3-level hysteresis comparator is shown in Fig. 5.

To describe the operation of the proposed current controller scheme, suppose that digital outputs of three-phase comparators are Da=1, Db=0, and Dc =0. This indicates Vn

*

is located in Region I; thus adjacent voltage vectors of V6, V1, V2, and V0

1 are selected to be applied. Under this condition, if eα= iα-ref - iα-o and eβ= iβ-ref - iβ-o touches the highest and lowest hysteresis bands respectively, then Dα=2 and Dβ= 0. This implies that the voltage space vector V6 must be applied to increase the α-component and simultaneously decrease the β-component of the current vector. Similarly, V1 and V2 must be respectively applied if (Dα=2, Dβ= 1) or (Dα=2 and Dβ=2). Zero voltage space

vector V01 is selected in other cases. Since the Sa

* is always equal to one for non-zero voltage space vectors in Region I, the selection of V0

1 causes further reduction in VSI switching

frequency. Using similar control strategy for other regions, the switching table given in Table I is extracted. The proposed control scheme keeps the current error vector inside the hexagonal tolerance region shown in Fig. 4.b.

As long as the region is correctly determined by the three-phase comparators, the α-β comparators are able to keep the current error inside the square tolerance region. However, once the Vn

* changes, the incorrect implementation of optimal voltage set causes the current error moves from the outer square band towards the hexagonal band. The region change is finally detected when the error vector touches the hexagonal band. Afterwards, the current error vector is forced back into the square tolerance region by the implementation of correct optimal voltage set. This procedure repeats for the every region change.

The proposed control scheme produces the switching pattern shown in Fig. 6. It is observed that the switching pattern constituted of six consecutive 4-state optimal voltage vectors sets, which roughly matches up with six 60°-regions of one period. The VSI switching frequency is significantly reduced by systematic implementation of zero and non-zero voltage space vectors, similar to SVM technique.

Fig. 4. Proposed vector-based method (a) Schematic block diagram

(b) Assigned tolerance region

Fig. 5. Practical implementation of 3-level α-β hysteresis comparator

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Fig. 7. Iα-β and Vn for (a) Proposed vector-based method (b) Conventional Hysteresis method

Table I Region definition and switching table for the proposed method

DαDβ

Da Db Dc Reg. 00 01 02 10 11 12 20 21 22

100 I V01 V0

1 V01 V0

1 V01 V0

1 V6 V1 V2 110 II V0

0 V00 V3 V0

0 V00 V0

0 V00 V1 V2

010 III V01 V4 V3 V0

1 V01 V0

1 V01 V0

1 V2 011 IV V5 V4 V3 V0

0 V00 V0

0 V00 V0

0 V00

001 V V5 V4 V01 V0

1 V01 V0

1 V6 V01 V0

1 101 VI V5 V0

0 V00 V0

0 V00 V0

0 V6 V1 V00

Fig. 6. VSI switching pattern of for the proposed vector-based method

6. SIMULATION STUDIES Simulation studies are conducted to verify the validity of

the proposed control scheme and to compare its performance with conventional hysteresis method. The circuit diagram shown in Fig. 1.a. is simulated with the parameters of the induction motor given in Appendix I. The dc-link voltage of the PWM-VSI is 200 V. Note that since the proposed method

is an improved hysteresis-based current controller, some general features, such as limited maximum current and insensitivity to the load parameters variations, are structural properties of both methods. Therefore, only load current trajectory, VSI switching pattern, and current error trajectory will be investigated in simulation studies.

A sinusoidal three-phase current command with a step change from 10A to 18A at t=20 msec is considered. The tolerance region is formed with h=δ=0.3A and Δδ=0.1A for the proposed control scheme. The VSI output current Iα-β and voltage space vector Vn are shown in Fig. 7.a. For the sake of clarity, V0

0 and V01 are both presented by Vn=0. Fig. 7.b.

shows similar waveforms for the conventional hysteresis current controller. Since the actual current error in conventional hysteresis method reaches twice of the permitted hysteresis band δhys, the tolerance region is formed with δhys=0.5(δ+h)=0.3A to achieve the same current control accuracy. Comparing Fig. 7.a. and Fig. 7.b., it is clear that the proposed method successfully follows the command current with much less oscillations since it controls the current in α-β frame. The torque pulsation in the induction motor load is consequently minimized. Besides, more regular switching pattern can be observed for the proposed method. Fig. 8. shows that the switching frequency is noticeably reduced using the proposed current controller. The conclusion is that the proposed vector-based method considerably improves the steady-state performance of hysteresis current controller using a switching pattern similar to the SVM technique. The superior transient behaviour of conventional hysteresis controller, on the other hand, is retained by the proposed control scheme. That is, it takes almost 1 msec for both methods to follow the current command. Non-optimal voltage space vectors are applied during the transient time.

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Fig. 8. Number of switching for phase a

Fig. 9. Error current trajectory

Fig. 10. a. Three-phase currents and respective hexagonal bands

b.-d. Digital outputs of three-phase 2-level comparators c. Region of desired voltage space vector

Fig. 9 shows the current error trajectory and its assigned tolerance region for both current control schemes. As expected, the actual current error for conventional hysteresis controller reaches to 2δhys=0.6A because of interphases dependency. The proposed method, on the other hand, keeps the current error vector inside the square tolerance region for most of the time. However, once the region of Vn

* changes, the current error vector moves from the square tolerance region to the hexagonal tolerance region. That is, one of three-phase current signals touches the δ+h band. The error vector is then forced back into the square tolerance region as the region is updated and suitable optimal voltage space vectors are applied thereafter. Note that a few observed tolerance region violations are related to negligible transient time that takes for the current controller to follow the step change of current command.

Fig. 10.a. shows three-phase current signals and their respective δ+h=0.6A band. Once each phase current touches the hysteresis band, the status of its respective 2-level hysteresis comparators changes, as shown in Fig.10.b.-10.d. This updates the region of Vn

*; thus the suitable optimal set of voltage space vectors is determined. The current error is then forced back inside the square tolerance region until the next region change. Each phase current hits the δ+h bands two times in one period, which forms six regions of one period. However, Fig. 10.e. shows that regions are not evenly distributed throughout one period since the distances of square tolerance sides to the hexagonal tolerance sides are not equal for different regions. This causes unequal heating of three-phase inverter legs due to unbalanced switching losses.

It is finally worth mentioning that the proposed method is also compared with other vector-based method presented by [11], [12] and [16]. Observably from the simulation results, the load current oscillations in α-β frame are not effectively suppressed using the current controller proposed by [16] as it works with current signals represented in three-phase system frame. The number of switching, on the other hand, is almost the same for both current controllers; although there is a minor unbalance in three-phase switching frequency where the proposed current controller is applied. This unbalanced three-phase switching can cause unequal heating in inverter switches. Comparing to current controller presented by [11] and [12], the number of switching for the proposed method is about 15% less. This is originated the fact that the presence of redundant situations in the control action of [11] and [12] yields non-optimal switching pattern in some occasions. Comparative simulation results are not shown here due to space limitation.

7. CONCLUSION An improved vector-based hysteresis current controller is

presented in this paper. Three 2-level hysteresis comparators are employed as region detector to define the optimal voltage space vectors to be applied. That is, three-phase comparators determine the region of desired voltage space vector. Afterwards, at each instant, two 3-level comparators working in α-β stationary frame determine the suitable voltage space vector among three adjacent non-zero and one zero voltage

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space vectors defined by the region detector. The proposed current controller keeps the current error vector inside a hexagon tolerance region with a significantly reduced switching frequency. Besides that, the load current vector oscillation is noticeably suppressed due to implementation of current controller in α-β coordinated frame.

Simulation results confirm the considerable improvement achieved by the proposed vector-based hysteresis control scheme. Besides minimized oscillations in load current trajectory, the proposed control scheme reduces the switching frequency by 35% in comparison with the conventional hysteresis method. On the other hand, very good transient performance of conventional hysteresis current controller is retained. The only disadvantage of the proposed method is unequal switching frequency in three legs of VSI which causes unequal heating of VSI switches. Appendix I Squirrel-cage induction motor parameters: 10 hp, 220 V, 1470 r/min, 50 Hz, three-phase, four poles. Stator Parameter - resistance: 0.68 Ω, leakage inductance: 0.005 H; Rotor Parameter - resistance: 0.45 Ω, leakage inductance: 0.0025 H; Magnetizing inductance: 0.1486 H.

REFERNCES [1] M. P. Kazmierkowski and L. Malesani, “Current control techniques for three-phase voltage-source PWM converters: a survey,” IEEE Trans. Ind.Electron., vol. 45, no. 5, pp. 691–703, Oct. 1998. [2] J. Holtz, “Pulsewidth modulation—A survey,” IEEE Trans. Ind. Electron., vol. 39, pp. 410–420, Oct. 1992. [3] D. M. Brod and D. W. Novotny, “Current control of VSI-PWM inverters,” IEEE Trans Ind. Applicat., vol. IA-21, no. 4, pp. 562–570, May/Jun. 1985. [4] A. Tripathi and P. C. Sen, “Comparative analysis of fixed and sinusoidal band hysteresis current controllers for voltage source inverters,” IEEE Trans. Ind. Electron., vol. 39, no. 1, pp. 63–73, 1992. [5] K. M. Rahman, M. R. Khan, M. A. Choudhury, and M. A. Rahman, “Variable band hysteresis current controllers for PWM voltage source inverters,” IEEE Trans. Power Electron., vol. 12, pp. 964–970, Nov. 1997. [6] B. K. Bose, “An adaptive hysteresis-band current control technique of a voltage-fed PWM inverted for machine drive system,” IEEE Trans. Ind. Electron., vol. 37, pp. 402–408, Oct. 1990. [7] L. Malesani and P. Tenti, “A novel hysteresis control method for current controlled VSI PWM inverters with constant modulation frequency,” IEEE Trans. Ind. Applicat., vol. 26, pp. 88–92, Jan./Feb. 1990. [8] L. Malesani, L. Rossetto, L. Sonaglioni, P. Tomasin, and A. Zuccato, “Digital, adaptive hysteresis current control with clocked commutations and wide operating range,” IEEE Trans. Ind. Applicat., vol. 32, pp. 1115–1121, Mar./Apr. 1996. [9] L. Malesani, P. Mattavelli, and P. Tomasin, “Improved constant-frequency hysteresis current control for VSI inverters with simple feedforward bandwidth prediction,” IEEE Trans. Ind. Appl., vol. 33, no. 5, pp. 1194–1202, Sep./Oct. 1997. [10] P. N. Tekwani, R. S. Kanchan, and K. Gopakumar, “Novel current error space phasor based hysteresis controller using parabolic bands for control of switching frequency variations,” IEEE Trans. Ind. Electron., vol. 54, pp. 2648–2656, Oct. 2007. [11] M. P. Kazmierkowski, M. A. Dzieniakowski, and W. Sulkowski, “Novel space vector based current controllers for PWM-inverters,” IEEE Trans. Power Electron., vol. 6, pp. 158–166, Jan. 1991. [12] C. T. Pan and T. Y. Chang, “An improved hysteresis current controller for reducing switching frequency,” IEEE Trans. Power Electron., vol. 9, pp. 97–104, Jan. 1994.

[13] A. Tilli and A. Tonielli, “Sequential design of hysteresis current controller for three-phase inverter,” IEEE Trans. Ind. Electron., vol. 45, pp. 771–781, Oct. 1998. [14] J. F. A. Martins, A. J. Pires, and J. F. Silva, “A Novel and Simple Current Controller for Three-Phase PWM Power Inverters,” IEEE Trans. Ind. Electron., vol. 45, pp. 802-805, Oct. 1998. [15] E. Aldabas, L. Romeral, A. Arias, and M.G. Jayne, “Software-based digital hysteresis-band current controller,” in Proc. Electric Power Applications, March 2006, pp. 184 – 190. [16] B.-H. Kwon, T.-W. Kim, and J.-H. Youn, “A novel SVM-based hysteresis current controller,” IEEE Trans. Power Electron., vol. 13, pp. 297–307, Mar. 1998.

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