Predictive Direct Power Control for Three-Phase Grid ...download.xuebalib.com/44l632rGU0WD.pdf10.1109/TPEL.2013.2289982, IEEE Transactions on Power Electronics Predictive Direct Power

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/TPEL.2013.2289982, IEEE Transactions on Power Electronics

Predictive Direct Power Control for Three-Phase Grid-Connected Converters Without Sector Information and Voltage Vector Selection

Zhanfeng Song*, Member, IEEE, Wei Chen, Member, IEEE, and Changliang Xia, Senior Member, IEEE

(*School of Electrical Engineering and Automation, Tianjin University Tianjin, 300072, China)

Abstract—In this paper, a predictive direct power control method without sector information and voltage vector selection is proposed for three-phase grid-connected converters. Different from conventional predictive direct power controllers, the proposed strategy always adopts fixed voltage vectors, instead of selecting voltage vectors according to the angular information of the grid-voltage vector or the virtual-flux vector. The converter switching time is obtained by minimizing the squared errors of the instantaneous active and reactive power. Switch duty cycles are then calculated and equivalently reconstructed. The proposed strategy not only presents rapid dynamic response due to the use of the predictive controller, but also possesses excellent steady-state performance as a result of duty cycle reconstruction. The main advantage of the proposed control scheme, compared with the classical direct power control strategies, is that it is unnecessary to use sector information of the grid-voltage vector, and selection of active voltage vectors is also not required here. Therefore, incorrect selection of voltage vectors and the resulting performance deterioration are surely avoided, without the need of any additional compensation measures. An in-depth comparison and experimental assessment are given to validate the effectiveness of the proposed strategy.

Index Terms— Predictive direct power control, three-phase converter, voltage vector selection, cost function minimization, duty cycle reconstruction.

I. INTRODUCTION

In a broad variety of application areas, such as electrical drives and distributed power generations, three-phase converters are often adopted to connect the electrical system to the unity grid. Their control strategies have garnered increasing attention over the last decades [1], [2]. Classical controllers are usually based on voltage orientation or virtual-flux orientation. These control methods regard the converter as a transfer function, and linear controllers in the synchronous rotating reference frame are designed. Outputs of linear controllers are further transferred to modulation schemes, and switch signals are finally obtained [3], [4].

Nowadays, different kinds of control strategies have been presented in technical literature. Direct power control (DPC) emerges as a new alternative for voltage-oriented vector control [5]-[8]. It adopts power controllers instead of current control loops. During its implementation process, a look-up table [9]-[11], or predefined selecting rules [12] are previously constructed based on instantaneous power

This work was supported in part by the National Key Basic Research

Program of China (973 Program) under Grant 2013CB035602, in part by the National Natural Science Foundation of China under Grant 51107084, in part by the Doctoral Program of Higher Education of China under Grant 20100032120081, and in part by the Application Foundation and Front Technology Research Program of Tianjin under grant 13JCZDJC34700.

behaviors of the converter. The optimal switching state is selected according to the errors between the measured and the reference values of active and reactive powers, as well as the information of the grid-voltage vector [13] or the estimated virtual flux vector [14]. Similar to the direct torque control method used for electrical drives, the modulation block is not required for DPC. However, the resulting variable switching frequency leads to several problems. A relatively high sampling frequency is usually required in order to obtain acceptable control performances [15]. Improved strategies based on space-vector modulation have been proposed to overcome these drawbacks [16]-[18]. Constant switching frequency is thus achieved, and sinusoidal ac currents with low harmonic distortions can be obtained with relatively low sampling frequency. Furthermore, in order to enhance the robustness of DPC and overcome problems associated with system parameter uncertainties, adaptive techniques have been included in the design of DPC laws, and therefore, the performance of the whole system is thus less sensitive to system parameters values uncertainties and variations[19], [20].

Another approach for implementing DPC for three-phase grid-connected converters is to take advantage of model predictive control strategies [21]-[23], in which the discrete nature of converters is taken into account in order to predict the future behaviors of the system[24]-[26]. The converter behaviors are predicted only for the possible switching states, and the optimal state with minimum cost is then selected and applied. This method presents fast dynamic responses as well as generating sinusoidal currents without any modulators. However, the current spectrum is distributed over a range of frequencies, which complicates the design of the grid filter. Therefore, the authors of [27] present a predictive direct power control technique using deadbeat control principle. Besides, predictive direct power control (P-DPC) algorithms based on predictive selection of voltage-vectors’ sequences are proposed in [28] and [29]. Within each sampling period, this method firstly selects two active voltage vectors according to the angular information of the grid-voltage vector [28] or the space vector position of the virtual flux [29]. The action time sequence is then calculated by minimizing the cost function, which is constructed based on the errors between the predicted and the reference values of active and reactive powers. The calculated optimal time sequence is then transferred to the modulation block, and consequently, the switch signals are obtained. By combining predictive approach with direct power control theory, this algorithm provides both high transient dynamics and constant switching frequency.

Obviously, when P-DPC is implemented, selection of



2

active voltage vectors is essential and plays an important role, which is based on the knowledge of the position of the grid-voltage vector. Besides, it is found that the selection method widely used in conventional P-DPC might result in incorrect selection of voltage vectors. Therefore, the calculated value of the voltage-vector action time may be negative. Conventionally, the action time is normally forced to zero whenever a negative value is predicted, resulting in control failures and performance deterioration. In order to solve this problem, references [30], [31] and [32] proposed effective solutions. In [30], a generalized P-DPC strategy for AC/DC converters is proposed. Based on the analysis of power characteristics, a novel method of optimal voltage vector selection is developed. A least-square cost function is evaluated with respect to different sectors. Global optimal voltage vectors are thus selected by means of cost function minimization. In [31] and [32], compensation measures are added to conventional P-DPC. Whenever negative time values come out, the active voltage vectors are reselected based on the modified look-up table. In other words, active voltage vectors should be selected twice when the calculated value of the voltage-vector action time is smaller than zero. The objective of this paper is to propose an improved predictive direct power control method without sector information of the grid voltage and without voltage vector selection process. Obviously, the proposed strategy simplifies the direct power control method. Furthermore, the appearance of negative time values is completely avoided, and there is no need to take any additional compensation measures.

The rest of this paper is organized as follows. In Section II, instantaneous power behaviors of three-phase grid-connected converters are depicted briefly. Section III describes the operation principle of conventional predictive direct power control strategies based on voltage vector selection. Detailed analysis of conventional predictive direct power control strategies and the improved approach are presented in Section IV. In Section V, the effectiveness of the proposed strategy is clearly verified by experimental tests. Finally, concluding remarks are drawn in Section VI.

II. INSTANTANEOUS POWER BEHAVIOR OF THREE-PHASE

GRID-CONNECTED CONVERTERS

In this paper, it is assumed that the grid-connected converter absorbs powers from the grid. The mathematical model in α-β reference frame can be expressed as

gαgα g g gα cα

d

d

iu L R i u

t (1)

ggβ g g gβ cβ

d

d

iu L R i u

t (2)

where uc and uc denote the output voltage components of the converter; ug and ug represent the grid voltage components; ig and ig are phase current components.

The instantaneous active and reactive power inputs are given by g gα gα g g1.5( )P u i u i (3)

g g gα g g1.5( )Q u i u i (4)

Instantaneous variation of active and reactive powers can be expressed as

g gα gα gβ gβgα gα gβ gβ

d d d d d1.5( )

d d d d d

P i u i uu i u i

t t t t t (5)

g gα gβ gβ gαgβ gα gα gβ

d d d d d1.5( )

d d d d d

Q i u i uu i u i

t t t t t (6)

Grid voltage components can be written as

gα ga gb gc

2 1 1

3 3 3u u u u (7)

g gb gc

3 3

3 3u u u (8)

Therefore, the following equation can be obtained

gα ga gb gc

ga gb gc

ga gb gc

gα

d d d d2 1 1

d 3 d 3 d 3 d2 1 1

( ) ( ) ( )3 2 3 2 3 2

2 1 1( ) ( ) ( )

3 2 3 2 3 2

( )2

u u u u

t t t t

u t u t u t

u t u t u t

u t

(9)

gβ gb gc

gb gc

gb gc

gβ

d d d3 3

d 3 d 3 d3 3

( ) ( )3 2 3 2

3 3( ) ( )

3 2 3 2

( )2

u u u

t t t

u t u t

u t u t

u t

(10)

For clarity, the voltage components in (9) and (10) are defined as

2gα gα( )

2u t u

(11)

2gβ gβ( )

2u t u

(12)

Substituting (1), (2), (9), (10), (11) and (12) into (5) and (6), the power variations can thus be obtained as

g 2 2

g gα gβ gα cα gβ cβ

2 2g gα gα gβ gβ g gα gα gβ gβ

d1.5[( ) ( )

d

( ) ( )]

PL u u u u u u

t

R u i u i L u i u i

(13)

g

g gα cβ gβ cα g gβ gα gα gβ

2 2g gα gβ gβ gα

d1.5[( ) ( )

d

( )]

QL u u u u R u i u i

t

L u i u i

(14)

When three-phase voltages are balanced, the following equation can be obtained

2gα gα gβ( )

2u u t u

(15)

2gβ gβ gα( )

2u u t u

(16)

Substituting (15) and (16) into (13) and (14), the power variations can thus be rewritten as

g 2 2g gα gβ gα cα gβ cβ g gα gα gβ gβ

g gβ gα gα gβ2 2gα gβ gα cα gβ cβ g g g g

d1.5[( ) ( ) ( )

d( )]

1.5 ( ) ( )

PL u u u u u u R u i u i

tL u i u i

u u u u u u R P L Q

(1

7)



3

gg gα cβ gβ cα g gβ gα gα gβ

g gβ gβ gα gα

gα cβ gβ cα g g g g

d1.5[( ) ( )

d( )]

1.5( )

QL u u u u R u i u i

tL u i u i

u u u u R Q L P

(18)

Active and reactive power increments eP, eQ caused by the application of voltage vector Vi, are defined as

i

gP

d

d V

Pe

t (19)

i

gQ

d

d V

Qe

t (20)

When implemented in micro-processors, the delay existed in real systems may deteriorate the performance of the proposed controller, resulting in cross-coupling phenomena[33]. This problem can be solved by introducing the following delay compensation measures. The actual active and reactive power at the kth sampling point, Pg(k) and Qg(k), can be calculated based on the actual voltage and current values at the kth sampling point, i(k), ug(k), ig(k), ug(k). However, for a small sampling period (200s), with respect to the grid fundamental frequency, it can be assumed that the actual values of grid voltage components, ug(k), ug(k) approximately equal to the sampled values ug(k), ug(k) [22]. Therefore, the actual active and reactive power can be given by

g gα gα g g( ) 1.5 ( ) ( ) ( ) ( )P k u k i k u k i k

(21)

g gβ gα gα gβ( ) 1.5 ( ) ( ) ( ) ( )Q k u k i k u k i k

(22)

where, i(k) and i(k) can be expressed as

sgα gα g gα cα gα

g

( ) ( ) ( ) ( ) ( )T

i k u k R i k u k i kL

(23)

sgβ gβ g gβ cβ gβ

g

( ) ( ) ( ) ( ) ( )T

i k u k R i k u k i kL

(24)

III. CONVENTIONAL PREDICTIVE DIRECT POWER CONTROL

BASED ON VOLTAGE VECTOR SELECTION

With respect to conventional predictive direct power control strategies, three voltage vectors including two active vectors and one zero vector are applied within each control period Ts. Active voltage vectors are often selected based on sector information of the grid-voltage vector, as shown in Table I.

TABLE I

SELECTED VOLTAGE VECTORS WITHIN DIFFERENT SECTORS

Sector Number Selected voltage vectors

I V0, V1, V2, V7or V0, V1, V6, V7

II V0, V2, V3, V7or V0, V1, V2, V7

III V0, V3, V4, V7or V0, V2, V3, V7

IV V0, V4, V5, V7or V0, V3, V4, V7

V V0, V5, V6, V7or V0, V4, V5, V7

VI V0, V6, V1, V7or V0, V5, V6, V7

The errors of predicted active and reactive power at the end of kth control period can be computed as

P gref g P0 0 Pm m Pn n2 2 2E P P e t e t e t (25)

Q gref g Q0 0 Qm m Qn n2 2 2E Q Q e t e t e t (26)

where ePm and ePn denote active power variation rates caused by the active voltage vectors Vm and Vn within one sample period Ts. eQm and eQn are reactive power variation rates caused by Vm and Vn within Ts. eP0 and eQ0 denote active and reactive power variation rates caused by zero voltage vectors, respectively. tm and tn are the application time of Vm and Vn within Ts/2, respectively. t0 represents the application time of zero vectors within Ts/2. tm, tn and t0 satisfy the boundary condition t0+tm+tn=Ts/2 [34].

For conventional predictive direct power control strategies, least-square optimization method is often adopted. The cost function is constructed as 2 2

P QW E E (27)

In order to minimize W within one sampling period, the application time t0, tm, tn satisfies the following conditions

P0 Pm g Pm m Pn n P0 0m

Q0 Qm g Qm m Qn n Q0 0

4( )( 2 2 2 )

4( )( 2 2 2 )

0

We e P e t e t e t

t

e e Q e t e t e t

(28)

p0 pn g pm m pn n p0 0n

q0 qn g qm m qn n q0 0

4( )( 2 2 2 )

4( )( 2 2 2 )

0

We e P e t e t e t

t

e e Q e t e t e t

(29)

where

g gref gP P P (30)

g gref gQ Q Q (31)

Based on (27) and (28), tm and tn can be obtained as

Qn Q0 g P0 Pn g Q0 Pn Qn P0m

Qn Q0 Pm Qm Q0 Pn Qn Qm P0

( ) ( ) ( )1

2 ( ) ( ) ( )se e P e e Q e e e e T

te e e e e e e e e

(32)

Qm Qn g Pn Pm g Qn Pm Qm Pnn

Qn Q0 Pm Qm Q0 Pn Qn Qm P0

( ) ( ) ( )1

2 ( ) ( ) ( )se e P e e Q e e e e T

te e e e e e e e e

(33)

Once tm and tn are predicted by minimizing W, switch duty cycles can thus be obtained and switch signals are generated, as demonstrated in Table II, where da, db, and dc denote three-phase duty cycles, respectively.

TABLE II

CALCULATION OF SWITCH DUTY CYCLES

Active vector Duty cycle calculation

Vm(tm) Vn(tn) da db dc

V1 V2 0.5+(tm+tn)/Ts 0.5+(-tm+tn)/Ts 0.5+(-tm-tn)/Ts

V2 V3 0.5+(tm-tn)/Ts 0.5+(tm+tn)/Ts 0.5+(-tm-tn)/Ts

V3 V4 0.5+(-tm-tn)/Ts 0.5+(tm+tn)/Ts 0.5+(-tm+tn)/Ts

V4 V5 0.5+(-tm-tn)/Ts 0.5+(tm-tn)/Ts 0.5+(tm+tn)/Ts

V5 V6 0.5+(-tm+tn)/Ts 0.5+(-tm-tn)/Ts 0.5+(tm+tn)/Ts

V6 V1 0.5+(tm+tn)/Ts 0.5+(-tm-tn)/Ts 0.5+(tm-tn)/Ts

Fig. 1 shows the block diagram of the conventional

predictive direct power control strategy. The corresponding flowchart of the control strategy is given in Fig. 2. Figs. 1 and 2 indicate that the sector information and active voltage vector selection are both essential for conventional predictive direct power controllers. Selection of two active voltage vectors is based on the sector information. Besides, when duty cycles are to be computed, sector information is also required, as demonstrated in Fig. 2.



4

Fig. 1. Block diagram of the conventional predictive direct power control strategy.

Fig. 2. Flowchart of the conventional predictive direct power control

strategy.

IV. ANALYSIS OF CONVENTIONAL PREDICTIVE DIRECT

POWER CONTROL STRATEGY AND THE IMPROVED APPROACH

Taking Sector V for example, when the grid-voltage vector is located in Sector V, V5 and V6 are normally selected as active vectors. Their application time t5 and t6 will be calculated by minimizing W, the sum of squared power errors. Obviously, the length of active vectors can be denoted as V5·t5 and V6·t6, respectively. For clear illustrations, decomposition and reconstruction of voltage vectors in the vector space are depicted in Fig. 3.

Fig. 3. Decomposition and reconstruction of V5·t5+V6·t6 in the vector space.

Vector V6·t6 can be synthesized using a combination of the two adjacent vectors 6 6 1 6 5 6V t V t V t (34)

Therefore, the following equations can be obtained

5 5 6 6 5 5 1 6 5 6

5 5 6 1 6( )

V t V t V t V t V t

V t t V t

(35)

According to the equivalence principle, the following rules are obeyed 5 5 6 2 5 6 2 5 6( ) ( ) ( )V t t V t t V t t (36)

Substituting (36) into (35), yields 5 5 6 6 1 6 2 5 6( )V t V t V t V t t (37)

It is clearly visible in (37) that the effects of V5·t5 along with V6·t6 equals to the synthesis of V1·t6 and V2·(-t5-t6). When the conventional predictive direct power control strategy is adopted, V5 and V6 are normally selected as active vectors when the grid-voltage vector is located in Sector V. The optimal set of application time t5 and t6 can thus be obtained based on (32) and (33). However, it is implied by (37) that, if V1 and V2 are selected when the grid-voltage vector is located in Sector V, their optimal application time can also be calculated out similarly. There is no doubt that the calculated application time satisfies the minimum value conditions (28) and (29). This will surely result in minimization of W. This indicates that, when the grid-voltage vector is located in Section V, V1 and V2 can also be selected, instead of V5 and V6, which will not affect the minimization of power errors. Similarly, when the grid-voltage vector is located in other sections, V1 and V2 can always be selected, and their application time can be calculated and implemented. This is a remarkable conclusion, which builds a solid base for the proposed predictive control strategy without sector information and voltage vector selection.

For the purpose of convenience, t1 and t2 are denoted as application time for V1 and V2, respectively. Different combinations of voltage vectors can thus be expressed by V1, V2, t1, and t2as follows

1 1 2 2 2 1 2 3 1

3 2 4 1 2

4 1 5 2

5 1 2 6 1

6 2 1 1 2

( ) ( )

( )

( ) ( )

( )

( ) ( )

V t V t V t t V t

V t V t t

V t V t

V t t V t

V t V t t

(38)



5

It is implied that, within all six sectors, different combinations of voltage vectors can always be equivalently replaced by V1 and V2, with completely different application time. This, as a result, indicates that, no matter where the grid-voltage vector is located, V1 and V2 can always be selected. Their optimal application time, t1 and t2, can thus be calculated out based on cost function minimization.

According to the first line of Table II, switch duty cycles can be directly obtained based on t1 and t2.

1 2a

s

0.5t t

dT

(39)

1 2b

s

0.5t t

dT

(40)

1 2c

s

0.5t t

dT

(41)

However, it should be noted that, even though V1 and V2 are always selected and duty cycles are calculated based on the first line of Table II, V1 and V2 are not the actual active voltage vectors in most cases. Taking Sector V for example, when the grid-voltage vector is located in most part of this sector, V5 and V6 should be the actual active vectors. Their application time should be larger than zero. According to (38), if V1 and V2 are selected under this situation, the calculated application time t1 and t2 satisfy t1>0 and -t1-t2>0. For clear illustration, switch signals in this case are depicted in Fig. 4.

Fig. 4. Switch signals in case of -t1-t2>0 and t1>0.

It is demonstrated in Fig. 4 that, even though V1 and V2 are always selected and duty cycles are calculated based on the first line of Table II, the actual voltage vector implemented remains V5 (001) and V6 (101), as long as t1 and t2 satisfy t1>0 and -t1-t2>0. Similarly, based on (38), the actual active vectors and the corresponding numerical relationship among the calculated application time can be obtained, as shown in Table III.

TABLE III

THE ACTUAL ACTIVE VECTORS AND THE CORRESPONDING RELATIONSHIP

Time relationship Actual active vectors Duty cycle relationship

t1>0, t2>0 V1, V2 da>db>dc

t1+t2>0 & -t1>0 V2, V3 db>da>dc

t2>0 & -t1-t2>0 V3, V4 db>dc>da

-t1>0 & -t2>0 V4, V5 dc>db>da

-t1-t2>0 & t1>0 V5, V6 dc>da>db

-t2>0 & t1+t2>0 V6, V1 da>dc>db

Obviously, switch duty cycles should be larger than zero, and be smaller than one. However, when switch duty cycles are calculated based on (39), (41), and (41), they might exceed this scope. This phenomenon should be taken into account in real control systems. As shown in Fig. 4, the

application time -2(t1+ t2) for V5 is directly proportional to dc-da. Similarly, the application time 2t1 for V6 is proportional to da-db. Detailed relationship between duty cycles and duration time of actual active vectors are shown in Table IV.

TABLE IV RELATIONSHIP BETWEEN DUTY CYCLES AND DURATION TIME OF ACTUAL

ACTIVE VECTORS

Actual active vectors Application time Relationship

V1: t1 da-db V1,V2 V2: t2 db-dc

V2: t1+t2 da-dc V2,V3

V3: -t1 db-da V3: t2 db-dc

V3,V4 V4: -t1-t2 dc-da V4: -t1 db-da

V4,V5 V5: -t2 dc-db V5: -t1-t2 dc-da

V5,V6 V6: t1 da-db V6: -t2 dc-db

V6,V1 V1: t1+t2 da-dc

As clearly visible in Table IV, as long as da-db and db-dc remain unchanged, the application time of actual active vectors will be the same, which can also be clearly seen from Fig. 5. This indicates that, if the calculated duty cycle exceeds the conventional scope, i.e. d<0 or d>1, they can be equivalently reconstructed, as long as da-db and db-dc remain unchanged. Besides, for the purpose of suppressing power fluctuations, application time of zero vectors is symmetrically distributed within one sampling period. The effectiveness of symmetrical distribution will be further confirmed by experimental results in the following sections.

t1 t2 t2 t1

000Sa

Sb

Sc

Ts

V0100V1

110V2

111V7

110V2

100V1

000V0

t1 t2 t2 t1

000Sa

Sb

Sc

Ts

V0100V1

110V2

111V7

110V2

100V1

000V0

da-db=2t1 Ts , db-dc=2t2 Ts

t0 t1 t2 2t0 t2 t1 t0

000Sa

Sb

Sc

Ts

V0100V1

110V2

111V7

110V2

100V1

000V0

Fig. 5. Equivalent reconstruction of switch signals in case of t1>0 and t2>0.

Based on the analysis previously mentioned, reconstruction of duty cycles can thus be outlined as follows.



6

da, db, and dc are firstly arranged in order from largest to smallest in value, dmax, d, and dmin. For example, max b min a c, ,d d d d d d (42)

The differences dmax-d and d-dmin can thus be obtained, which are proportional to the application time of active voltage vectors. Consequently, d0=1-(dmax-d)-(d-dmin) =1-(dmax-dmin) is proportional to the application time for zero vectors. For the purpose of symmetrically distributed and taking (32) into consideration, the new duty cycles da, db, and dc can be further represented as a c c a 0 b a( ) 1d d d d d d d (43)

b 01d d (44)

c b b c 0 b c( ) 1d d d d d d d (45)

For better understanding, reconstruction of duty cycles when the value db calculated out exceeds the scope (db>1) is shown in Fig. 6. Besides, power variations when voltage vectors are symmetrically and unsymmetrically distributed are shown in Figs. 7 and 8, respectively. It is clearly visible in these two figures that, when the voltage vectors are symmetrically distributed, power variations are much smaller than those obtained with unsymmetrical distribution of voltage vectors. This aspect will be further validated by experimental results in the following sections.

Sa

Sb

Sc

Ts

dmin=da

dmax=db

d=dc

Ts

Sa

Sc

Ts

Sa

Sb

Sc

Sb

Unsymmetrical distribution

Symmetrical distribution

b c

2

d d c a

2

d d b c

2

d dc a

2

d d

b c

2

d d c a

2

d d b c

2

d dc a

2

d d

b c

2

d d c a

2

d d b c

2

d dc a

2

d d0

2

d 0

4

d0

4

d

Fig. 6. Reconstruction of duty cycles when the value calculated out exceeds

the scope (db>1, d0=1-(dmax-dmin)).

Fig. 7. Active and reactive power variations when voltage vectors are

symmetrically distributed.

Fig. 8. Active and reactive power variations when voltage vectors are

unsymmetrically distributed.

Furthermore, the control region of voltage vectors is an

important aspect for power converter controllers. If the control vector is outside the control region of power converters, i.e. dmax-dmin>1, the new duty cycles da, db, and dc should be calculated by

cb max a min c

b

1, 0, a

a

d dd d d d d d

d d

(46)

For better understanding, reconstruction of duty cycles when the control vector lies outside the control region (dmax-dmin>1) is shown in Fig. 9.

b c

b a

1

2

d d

d d

c a

b a

d d

d d

b c

b a

1

2

d d

d d

Fig. 9. Reconstruction of duty cycles when the control vector lies outside the

control region.

Overall, the execution process and the block diagram of

the proposed predictive control strategy can be outlined as Fig. 10, and its flow chart is shown in Fig. 11.

Lguga

ugb

iga

igb

igc

PowerPredictive Model

PWM

ugc

d(a,b,c)

Pref Qref

udc

PI udcref

uca

ucb

ucc

Rg

Rg

Rg

abc

αβ

Lg

Lg

Grid

ig βucαβ(0,1,2)

Instant PowerCalculation

P&Q Duty cycleCalculation

ugαβ

abc

αβ

ePQ

Fig. 10. Block diagram of the proposed predictive direct power control strategy.



7

Fig. 11. Flowchart of the proposed predictive direct power control strategy.

When compared with Figs. 1 and 2, it is clearly visible in Figs. 10 and 11 that, the proposed control strategy is much simplifier than the conventional method. Sector information is not required here, and selection of active voltage vectors is also eliminated. Therefore, incorrect selection of voltage vectors and the resulting performance deterioration will surely be avoided, without the need of any additional compensation measures. These will be further confirmed by experimental results shown in the next section.

V. EXPERIMENTAL VALIDATIONS

In order to perform detailed validations, an experimental prototype with power devices IGBT FGA40N120D has been developed, as depicted in Fig. 12. A three-phase variac is adopted to connect the utility grid and the experimental system. An inductor is used to attenuate switching harmonics. Its inductance and equivalent resistance are 5mH and 0.1Ω, respectively. The dc-link voltage is 160V, and the dc-link capacitor is 2200F. The control scheme is implemented in a 150-MIPS float-point 32-bit TMS320F28335 board. The sampling period of the control algorithm is set at 200s. During the experimental validations, the YOKOGAWA digital storage oscilloscope DLM2024 is adopted for recording.

ABCiABCv

LR

dcV

Fig. 12. Diagram of the experimental prototype.

A. Steady-state performance Tests

When the proposed method is adopted, V1 and V2 are always selected wherever the grid-voltage vector is located. However, the actual active voltage vectors change

automatically with the phase angle of the grid voltage, without sector information or selection of active voltage vectors. This is the main contribution of the manuscript. With the purpose of confirming this remarkable characteristic, analysis of optimal voltage vectors should firstly performed.

Normally, three voltage vectors including two active vectors and one zero vector are applied within each sampling period Ts. Two active voltage vectors should be selected in order to minimize power ripples. The smallest power ripples can be achieved by application sequence of neighboring voltage vectors. Taking Sector V for example, when the grid voltage vector is located in this sector, V5 and V6 are always selected. These two active vectors will be used together with zero vectors V0,7. For clarity and better understanding, Figs. 13 and 14 demonstrate variation rates of active and reactive powers when the system operates at rated power with unity power factor (P=300W, Q=0Var).

Fig. 13. Variation rates of active power related to the angular location of grid

voltage vector .

3

2

3

4

3

5

3

20

Fig. 14. Variation rates of reactive power related to the angular location of

grid voltage vector.

It is shown in Fig. 14 that, when the grid voltage vector is

located in Sector V, the variation rate of reactive power resulting from V5 decreases with the grid-voltage phase, from positive value to zero, and further to negative value. However, the variation rate of reactive power corresponding to V6 and V0,7 retains positive throughout Sector V. This indicates that, with the increase of grid-voltage phase, the application of V5 firstly results in the increase of reactive power and then its decrease. And the actions of V6, V0,7 always lead to the increase of reactive power. Similarly, for the active power, when the grid-voltage vector locates in Sector V, with the increase of grid-voltage phase, the application of V5 firstly results in the decrease of active power and then its decrease.



8

The action of V6 will always decrease active power, and the action of zero vectors V0,7 will increase it all the time.

It can be inferred from the above analysis that, when the grid voltage vector is located in Sector V, V5, V6, and V0,7 can realize the flexible control of active power. The reason is that, throughout the whole sector, there always exist voltage vectors which can increase the active power as well as voltage vectors which result in its decrease. However, flexible control of reactive power can not be realized in the first part of Sector V, as V5, V6, V0,7 all lead to the increase of reactive power. With the increase of the grid-voltage phase, V5, V6, and V0,7 can achieve flexible regulation of reactive power, as the application of V5 results in its decrease.

Obviouly, in the latter part of Sector V, the optimal voltage vectors are V5, V6, V0,7. However, this is not the case in the first part of Sector V. In fact, the optimal voltage vectors for the first part of Sector V are V4, V5, and V0,7. With regard to the new combination, V4 keeps the reactive power decreasing in the whole Sector V. V5 firstly increases the reactive power then decreases it, while the action of zero vectors V0,7 increases it all the time. Therefore, this new combination can realize flexible control of the reactive power. Similar conclusion can be drawn with respective to the regulation of active power. In general, when the grid-voltage vector locates in the first part of Sector V, V4, V5, and V0,7 are the optimal vectors. With the increase of the grid-voltage phase, the optimal vectors change to V5, V6, V0,7.

When the proposed method is adopted, the optimal vectors are automatically selected, without sector information or selection of active voltage vectors. With the purpose of confirming this remarkable characteristic, experimental evaluations are performed. Fig. 15 shows the experimental results of duty cycles under the conditions of active power and reactive power being 300W and 0Var, respectively.

Fig. 15. Variation of duty cycles with the proposed method under the conditions of active power and reactive power being 300W and 0Var, respectively. (Top: 90deg/div).

dc>db>da

Sa

Sb

Sc

Ts

000V0

001V5

011V4

111V7

001V5

000V0

011V4

Fig. 16. Switch signals in case of dc>db>da.

Fig. 17. Switch signals in case of dc>da>db.

It is clearly visible in Fig. 15 that, when the grid voltage

vector is located in the initial part of Sector V, the numerical relationship among da, db, and dc is dc>db>da. Under this situation, the voltage vectors taking actions are V4(011), V5(001), V0,7(000,111), as shown in Fig. 16. With the increase of the phase angle, this relationship changes to dc>da>db. The corresponding voltage vectors are V5(001), V6(101),

V0,7(000,111), as demonstrated by Fig. 17. This phenomena, therefore, even though sector information or selection of active voltage vectors is not required and V1 and V2 are always used to calculate duty cycles wherever the grid-voltage vector is located, the actual active voltage vectors taking actions change automatically with the phase angle of the grid voltage. In other words, based on cost function minimization and duty cycle reconstruction, the optimal voltage vectors are automatically selected.

In order to further verify the feasibility of the improved strategy, Fig. 18 depicts the phase angle of the grid voltage and the variations of t1, t2, and t1+t2 under the conditions of active power and reactive power being 300W and 0Var, respectively. Taking Sector V for example, when the grid-voltage vector is located in this sector, the value of t1 increases with the grid-voltage phase, from negative value to zero, and further to positive value, while t2 and t1+t2 both retain negative throughout the whole sector. This means that when the grid-voltage vector is located in the initial part of Sector V, -t1 and -t2 are both positive. Subsequently, t1 gradually increases to positive value. According to Table III and (28), it can be concluded that, in the initial part of Sector V, the actual active voltage vectors are V4 and V5, while with the increase of grid-voltage phase, they will be automatically replaced by V5 and V6. This, as a result, indicates that even though V1 and V2 are always selected, the actual active voltage vectors implemented change automatically with the phase angle of the grid voltage. This phenomena confirms the same remarkable conclusion implied by Fig. 15. Therefore, incorrect selection of voltage vectors and the resulting performance deterioration will automatically be avoided, as pointed out in previous sections.



9

Fig. 18. Variations of t1, t2, and t1+t2 with the proposed method adopted under the conditions of active and reactive powers being 300W and 0Var, respectively (Top: 90deg/div, Bottom: 30s/Div).

In order to illustrate and validate the effectiveness of the

proposed control strategy, comparative studies on system steady-state behavior are also conducted. The measured active and reactive power behaviors are demonstrated in Figs. 19 and 20 with Pg=450W and Qg=0Var, respectively. By comparing the results of Fig. 19(a) and the waveforms in Fig. 20(a), it is evident that the proposed strategy can achieve proper regulation of active and reactive powers even though sector information and voltage vector selection is not used here. Besides, it should be noted that, in order to improve the reliability of the THD analysis, fifty cycles of current waveforms are selected to obtain current harmonic spectra and the THD value, as demonstrated by Figs. 19(c) and 20(c). Comparison of these two figures shows that both strategies present similar current spectrum waveforms and almost the same THD, confirming that the currents are almost the same. It should be noted that, due to the non-ideal experimental conditions, even order harmonics with small amplitudes are detected in the output line-to-line voltages of the three-phase variac. This results in even order harmonics in three-phase currents, as shown in Figs.19c and 20c.

Besides, in order to evaluate system performances under unbalanced grid voltages, relative experiments were also conducted. As shown in Fig. 21, active and reactive powers can be effectively controlled. However, in this case, three-phase currents contain serious harmonic distortions. This is due to the fact that the control targets of the proposed method are active and reactive powers. Distorted currents are thus unavoidable under the condition of unbalanced grid voltages. In order to solve this problem, compensation methods developed in [35] can be adopted. Obviously, under unbalance grid voltages, improved current waveforms will surely affect the steady-state performance of active and reactive powers, as pointed out in [35].

(a)

(b)

(c)

Fig. 19. Experimental results: steady-state performance of the proposed controller. (a) Active and reactive powers (Active power: 95.8 W/div, Reactive power: 95.8Var/div). (b) Three-phase currents (3.9A/div). (c) Measured ac current harmonic spectra.

(a)

(b)



10

(c)

Fig. 20. Experimental results: steady-state performance of the conventional P-DPC. (a) Active and reactive powers (Active power: 95.8W/div, Reactive power: 95.8Var/div). (b) Three-phase currents (3.9A/div). (c) Measured ac current harmonic spectra.

(a)

(b)

(c)

Fig. 21. Experimental results: steady-state performance of the proposed controller under unbalanced grid voltages. (a) Three-phase line-to-line voltages (72V/div). (b) Active and reactive powers (Active power: 95.8W/div, Reactive power: 95.8Var/div). (c) Three-phase currents (3.9A/div).

B. Transient Response Tests

Transient response to reference changes is an important aspect for predictive control strategies. With the purpose of evaluate converter transient responses with different control strategies, an in-depth assessment and comparison are further carried out. Experimental results of the dynamic behavior of the three-phase converter are obtained for both methods under the same operating conditions. In Figs. 22 and 23, a step change is performed only on the active power from 300W to 450W. Both methods behave comparably under this situation. The active power reference is reached quickly

within 200μs, as shown in Figs. 22(a) and 23(a). Besides, no cross-coupling effects are observed. With respect to the current behaviors, the step change in active power reference clearly affects the ac currents, which rapidly change in order to follow the new power reference, as shown in Figs. 22(b) and 23(b).

Similar experimental results are further derived under the condition of step change in reactive power, as demonstrated in Figs. 24 and 25. The reference of reactive power steps from 0Var to 150Var, while the active power remains 300W. As expected, when the proposed predictive control strategy is adopted, the reactive power can also be regulated properly, with the transient response finished within around 200μs. Moreover, the active power is unaffected by the step change of the reactive power. By comparing the transient responses demonstrated in Fig. 24 and the dynamic behaviors shown in Fig. 25, it is clearly visible that both control strategies behaves similarly. These experimental results, therefore, indicate that the sector information and selection of voltage vectors can be successfully eliminated from the predictive direct power control strategy, without influencing system behaviors.

(a)

(b)

Fig. 22 Experimental results: transient responses of the proposed controller with step-up change in active power reference (with delay compensation). (a) Active and reactive powers (Active power: 95.8W/div, Reactive power: 95.8Var/div). (b) Three-phase currents (3.9A/div).

(a)



11

(b)

Fig. 23 Experimental results: transient responses of the conventional P-DPC with step-up change in active power reference (with delay compensation). (a) Active and reactive powers (Active power: 95.8W/div, Reactive power: 95.8Var/div). (b) Three-phase currents (3.9A/div).

(a)

(b)

Fig. 24 Experimental results: transient responses of the proposed controller with step-up change in reactive power reference (with delay compensation). (a) Active and reactive powers (Active power: 95.8W/div, Reactive power: 95.8Var/div). (b) Three-phase currents (3.9A/div).

(a)

(b)

Fig. 25 Experimental results: transient responses of the conventional P-DPC with step-up change in reactive power reference (with delay compensation). (a) Active and reactive powers (Active power: 95.8W/div, Reactive power: 95.8Var/div). (b) Three-phase currents (3.9A/div).

C. Investigation of vector unsymmetrical distribution As mentioned in Section IV and demonstrated in Fig. 6,

during the reconstruction process of duty cycles, distribution of voltage vectors can be either symmetrical or unsymmetrical. Experimental results mentioned above are all obtained based on vector symmetrical distribution. In order to investigate the effect of vector unsymmetrical distribution on system performances, experiments were further conducted. Transient behaviors of the system are shown in Fig. 26. As demonstrated by Fig. 26, asymmetrical distribution of voltage vectors does not significantly influence the transient responses. The reference step change can be quickly reached, which is quite similar to the waveforms shown in Fig. 22. However, by comparing Fig.27 with Figs. 19 and 28, Fig. 27, it is evident that asymmetrical distribution of voltage vectors leads to steady-state power fluctuations (50Hz) and higher harmonics in ac currents (5kHz).

In order to analyze the reason for the steady-state power fluctuations, Fig. 28 depicts the waveforms of duty cycles when voltage vectors are asymmetrically distributed. Obviously, when vector unsymmetrical distribution is adopted, step changes can be observed for duty cycle variations, which is different from the continuous variations shown in Fig. 15 with vector symmetrical distribution. These 50Hz step changes lead to low-frequency fluctuations of active and active powers. With respect to higher harmonics in ac currents (5kHz), this phenomena can be easily explained based on Figs. 7 and 8. As pointed out in Section IV, vector unsymmetrical distribution leads to larger fluctuations of active and reactive powers in each sampling period. Therefore, higher harmonics in ac currents are surely observed in Fig. 27.

(a)

(b)

Fig. 26 Experimental results: transient responses of the proposed controller with voltage vectors asymmetrically distributed (with delay compensation). (a) Active and reactive powers (Active power: 95.8W/div, Reactive power: 95.8Var/div). (b) Three-phase currents(3.9A/div).



12

(a)

(b)

(c)

Fig. 27 Experimental results: steady-state performance of the proposed controller with voltage vectors asymmetrically distributed. (a) Active and reactive powers (Active power: 95.8W/div, Reactive power: 95.8Var/div). (b) Three-phase currents (3.9A/div). (c) Measured ac current harmonic spectra.

Fig. 28 Experimental results: Measured ac current harmonic spectra of the proposed controller with voltage vectors symmetrically distributed..

Fig. 29 Variation of duty cycles with the phase angle of grid voltage when the voltage vectors are asymmetrically distributed. (Top: 90deg/div)

VI. CONCLUSION

Normally, when predictive direct power control strategies are implemented, two active voltage vectors are firstly selected according to the angular information of the grid-voltage vector. Different from conventional predictive power control strategies, an improved predictive direct power control method is proposed in this paper. Fixed voltage vectors are always adopted, instead of selecting voltage vectors according to the angular information of the grid-voltage vector or the virtual-flux vector. The converter switching time is obtained by minimizing the squared errors of the instantaneous active and reactive power. Switch duty cycles are then calculated and equivalently reconstructed for symmetrical distribution of voltage vectors. Experimental results show that the proposed strategy not only presents rapid dynamic response due to the use of the predictive controller, but also possesses excellent steady-state performance as a result of duty cycle reconstruction. Besides, sector information is not required here, and selection of active voltage vectors is also eliminated. Therefore, incorrect selection of voltage vectors and the resulting performance deterioration are surely avoided, without the need of any additional compensation measures.

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Predictive Direct Power Control for Three-Phase Grid ...download.xuebalib.com/44l632rGU0WD.pdf10.1109/TPEL.2013.2289982, IEEE Transactions on Power Electronics Predictive Direct Power