Nghiên cứu phương pháp điều khiển dự báo cho các bộ nghịch lưu đa mức

BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC BÁCH KHOA HÀ NỘI

..

--- - - ---

HOÀNG THÀNH NAM

NGHIÊN CỨU PHƯƠNG PHÁP ĐIỀU KHIỂN DỰ BÁO
CHO CÁC BỘ NGHỊCH LƯU ĐA MỨC

RESEARCH ON MODEL PREDICTIVE CONTROL FOR
MULTILEVEL CONVERTERS

LUẬN VĂN THẠC SĨ KHOA HỌC
ĐIỂU KHIỂN VÀ TỰ ĐỘNG HĨA
LỜI CAM ĐOAN
Tơi xin cam đoan đây là cơng trình của tơi. Tất cả các ấn phẩm được
công bố chung với các cán bộ hướng dẫn khoa học và các đồng nghiệp đã
được sự đồng ý của các tác giả trước khi đưa vào luận án. Cáckết quả trong
luận án là trung thực, chưa từng được công bố và sử dụng để bảo vệ trong bất
cứ một luận án nào khác.
Tác giả luận án
Trần Duy Trinh

HÀ NỘI-2018


BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC BÁCH KHOA HÀ NỘI

--- - - ---

HOÀNG THÀNH NAM

NGHIÊN CỨU PHƯƠNG PHÁP ĐIỀU KHIỂN DỰ BÁO
CHO CÁC BỘ NGHỊCH LƯU ĐA MỨC
RESEARCH ON MODEL PREDICTIVE CONTROL FOR
MULTILEVEL CONVERTERS

LUẬN VĂN THẠC SĨ KHOA HỌC
ĐIỂU KHIỂN VÀ TỰ ĐỘNG HĨA
LỜI CAM ĐOAN
Tơi xin cam đoan đây là cơng trình của tơi. Tất cả các ấn phẩm được

NGƯỜI
HƯỚNG
DẪNnghiệp
KHOA
công bố chung với các cán bộ hướng dẫn
khoa học
và các đồng
đã HỌC
được sự đồng ý của các tác giả trước khi đưa vào luận án. Cáckết quả trong
luận án là trung thực, chưa từng được công bố và sử dụng để bảo vệ trong bất
cứ một luận án nào khác.
Tác giả luận án
Trần Duy
Trinh
PGS.
TS. TRẦN TRỌNG MINH

HÀ NỘI-2018


LỜI CAM ĐOAN
Tôi xin cam đoan bản luận này là cơng trình của riêng tơi, do tơi tự thiết kế dưới
sự hướng dẫn của thầy giáo PGS. TS. Trần Trọng Minh. Các số liệu và kết quả là
hoàn toàn trung thực.
Để hồn thành luận văn này tơi chỉ sử dụng những tài liệu được ghi trong
danh mục tài liệu tham khảo và không sao chép hay sử dụng bất kỳ tài liệu nào
khác. Nếu phát hiện có sự sao chép tơi xin chịu hồn tồn trách nhiệm.
Hà Nội, ngày 10 tháng 10 năm 2018
Tác giả luận văn


TỔNG QUAN VỀ ĐỀ TÀI
1

Lý do chọn đề tài

Điều khiển các bộ biến đổi đa mức như cầu H nối tầng đặt ra nhiều vấn đề do số
lượng các module tăng nên nhiều theo số mức. Bằng các cấu trúc điều khiển thơng
thường thì các mạch vịng điều khiển sẽ rất phức tạp. Phương pháp điều khiển dự
báo FCS-MPC dựa trên tính tốn tối ưu hàm mục tiêu (cost funcion) trong khơng
gian hữu hạn các trạng thái làm việc có thể cho phép xây dựng nên một hệ thống
điều khiển có cấu trúc đơn giản hơn, lược bỏ khâu điều chế PWM, có thể đưa đến
những ứng dụng thực tế.

2

Đối tượng nghiên cứu

Nghiên cứu phương pháp điều khiển dự báo dựa trên không gian hữu hạn các trạng
thái làm việc của sơ đồ nghịch lưu đa mức cấu trúc cầu H nối tầng. Sau đó áp dụng
thuật tốn điều khiển này cho ứng dụng nghịch lưu nối lưới và điều khiển động cơ
không đồng bộ. Trong khuân khổ cuốn luận văn này, tính đúng đắn của thuật tốn
điều khiển dự báo FCS-MPC sẽ được kiểm chứng thơng qua mơ hình mơ phỏng
trên phần mềm Matlab-Simulink.

3

Đóng góp khoa học trong luận văn

Đưa ra thuật toán điều khiển dự báo FCS-MPC cho bộ biến đổi 7 mức cấu trúc cầu
H nối tầng với số bước tính là hai bước, giúp bù thời gian trễ trong q trình tính
tốn, đo lường, deadtime, v.v… khi triển khai thực nghiệm. Thuật toán lựa chọn
tập hợp các vector liền kề giúp giảm đáng kể khối lượng tính tốn khi tối ưu hóa
hàm mục tiêu.


THESIS OVERVIEW
1

Problem statement

Control multilevel converters such as cascaded H-Bridge multilevel converters
pose many problems as the number of module increases. By the conventional
control strategies, the control loops will be very complex. The finite control set
model predictive control (FCS-MPC) control strategies is based on cost function
optimization in the finite number of switch states. This could allow the control

system to be simpler structure, the system does not need a modulator, can be led
to practical applications.

2

Object of the study

The FCS-MPC control strategy for three-phase CHB multilevel converter is
studied in this thesis. It is applied in grid-connected CHB as DC-AC converter for
isolated DC sources such as PV panels generating power to gird and an IM driver
application. Within the framework of the thesis, the correctness of the MPC
algorithm will be verified through Matlab-Simulink software.

3

My contributions

Proposal FCS-MPC control strategy for three-phase CHB seven level, predictive
horizon at two-steps compensate delay time. The subset of adjacent vector state
(SAVS) method is proposed to reduce computational when optimizing cost
function.


Acknowledgments

Acknowledgments

First of all, I would like to express sincere thanks to my supervisor: Assos. Prof.
Tran Trong Minh for his constant encouragement and guidance. He has walked me
through all the stages of the work of my Master of Science project. The work in

this thesis is based on research carried out at the Institute for Control Engineering
and Automation (ICEA), Hanoi University of Science and Technology (HUST).
I would like gratitude ICEA as well as the financial support provided by the
National project number: KC.05.03/16-20, “Nghiên cứu, thiết kế và chế tạo hệ
thống khắc phục nhanh sự cố tăng/giảm điện áp ngắn hạn cho phụ tải” and:
ĐTĐLCN.44/16, “Nghiên cứu thiết kế và chế tạo hệ truyền động servo xoay chiều
ba pha”.

1


Contents

Contents

Acknowledgments......................................................................................................... 1
Contents .......................................................................................................................... 2
List of figures ................................................................................................................. 4
List of tables ................................................................................................................... 5
List of abbreviations .................................................................................................... 6
1 Overview FCS-MPC for CHB multilevel converter ....................................... 7
1.1

Three-phase CHB multilevel converter ........................................................ 7

1.1.1

Structure of a three-phase CHB multilevel converter .......................... 7

1.1.2 Modulation techniques ............................................................................. 9

1.2 Modeling of three-phase CHB multilevel converter .................................11
1.3

FCS-MPC control strategy ...........................................................................14

2 FCS-MPC for gird-connected three-phase CHB ........................................... 17
2.1

FCS-MPC for grid-connected formulation .................................................17

2.1.1

Discrete-time model predictive current control ..................................18

2.1.2

Cost funcion optimization and vector state selection.........................19

2.1.3

Subset of adjacent vector state ..............................................................20

2.2

Current reference generation ........................................................................21

2.3

Simulation results ..........................................................................................22

2.4

Conclusion ......................................................................................................24

3 FCS-MPC based current control of an IM ..................................................... 25
3.1 Mathematical model of an IM ......................................................................25
3.2

FCS-MPC for IM formulation .....................................................................25

3.2.1

The required signal estimation ..............................................................27

3.2.2

Discrete-time model predictive current ...............................................27

3.2.3

Cost funcion optimization and vector state selection.........................28

3.3

Simulation results ..........................................................................................28

3.4

Conclusion ......................................................................................................31
2

Contents

4 Summary and future works ................................................................................ 32
References .................................................................................................................... 33
Appendix A Simulation FCS-MPC for a gird-connected details .................. 35
A.1

Simulation model .......................................................................................35

A.2

MPC algorithm function............................................................................36

Appendix B Simulation FCS-MPC for an IM details ...................................... 37
B.1 Simulation model ...........................................................................................37
B.2 MPC algorithm function ...............................................................................38
Appendix C List of publications ........................................................................... 40

3


List of figures

List of figures

Figure 1.1

H-Bridge switch state ........................................................................... 7

Figure 1.2

Three-phase CHB seven level converter............................................ 8

Figure 1.3

SPWM multicarrier strategy ................................................................ 9

Figure 1.4

Space vector for three-phase CHB three level ................................10

Figure 1.5

H-Bridge converter .............................................................................11

Figure 1.6

Vector state in CHB seven level converter......................................13

Figure 1.7

Classification of MPC strategies applied to power converter .......14

Figure 1.8

FCS-MPC block diagram...................................................................15

Figure 2.1

Block diagram of FCS-MPC gird-connected ..................................17

Figure 2.2

Vector state for CHB seven level three-phase ................................20

Figure 2.3

Simulation results of the proposed FCS-MPC ................................23

Figure 2.4

FFT analysis output current (phase A) .............................................24

Figure 3.1

Block diagram of FCS-MPC for IM.................................................26

Figure 3.2

Simulation results of output current and voltage ............................29

Figure 3.3

Simulation results of the proposed FCS-MPC ................................30

Figure 3.4

FFT analysis output current (phase A) .............................................31

Figure A.1

Simulation overview of FCS-MPC for a grid-connected ..............35

Figure A.2

FCS-MPC controller in subsystem ...................................................36

Figure B.1

Simulation overview of FCS-MPC for an IM .................................37

Figure B.2

FCS-MPC in subsystem .....................................................................38

4


List of tables

List of tables

Table 1.1

Switch state H-Bridge converter...........................................................11

Table 1.2

Level state CHB seven level converter ................................................12

Table 2.1

Simulation FCS-MPC for grid connected parameters .......................22

Table 3.1

Simulation FCS-MPC for IM parameters............................................28

5


List of abbreviations

List of abbreviations

NPC

Neutral diode clamped multilevel converters

FC

Flying capacitor multilevel converters

MMC

Modular multilevel converters

CHB

Cascaded H-Bridge multilevel conveters

IGBT

Insulated Gate Bipolar Transistors

DC

Direct Current

PS

Phase-shift

PD

Phase disposition

APOD

Alternative phase opposite disposition

POD

Phase opposite disposition

SVM

Space vector modulation

MPC

Model predictive control

FCS-MPC

Finite control set model predictive control

CCS-MPC

Continuous control set model predictive control

OSV-MPC

Optimal switching vector model predictive control

OSS-MPC

Optimal switching sequence model predictive control

IM

Induction motor

SAVS

Subset of adjacent vector state

RMS

Root mean square

FFT

Fast Fourier transform

THD

Total harmonic distortion

FOC

Field oriented control

6


Chapter 1. Overview FCS-MPC for CHB multilevel converter

Chapter

1

Overview FCS-MPC for CHB multilevel converter

Multilevel converters include: Neutral diode clamped (NPC), flying capacitor
(FC), modular multilevel converters (MMC) and cascaded H-Bridge (CHB).
However, technology of CHB is one of the well known, most advantageous and
basic method.

Control CHB multilevel converters will be complex when number of cells
increase. The FCS-MPC control strategy can be considered as a solution simply
handles this problem.

1.1

Three-phase CHB multilevel converter

1.1.1 Structure of a three-phase CHB multilevel converter
The Figure 1.1 shows three switch state of H-Bridge (as named is cell), each cell
make three level voltage: -1; 0 and 1.

vdc
vdc

vac
vdc

vac

STATE = -1

STATE = 1

STATE = 0

Figure 1.1

H-Bridge switch state
7

vac


Chapter 1. Overview FCS-MPC for CHB multilevel converter

In CHB multilevel converter, number of cells are connected in series. Each
cell has separate DC source which is obtained from fuel cells, batteries, capacitors,
transformers,…
Activity of m cells in each phase will make 2m+1 voltage level. Figure 1.2
is example of CHB three-phase seven level. Three-phase CHB multilevel
converter is simply like three single-phase converter connected in wye
configuration.
Z
ZA

S1

Vdc1

S3

A

B

C

vac1

C1

S2

ZC

ZB

S4

Vdc2

C2

vac2

Vdc3

C3

vac3

N

Figure 1.2

Three-phase CHB seven level converter

Advantages:
• It doesn’t need capacitors or diodes for clamping.

• Entire IGBT switching in basic fundamental frequency (or near this
frequency), so that reduce power lose switch.
• The harmonics reduce because IGBT switching small frequency.

8


Chapter 1. Overview FCS-MPC for CHB multilevel converter

• The wave is quite sinusoidal in nature.
Disadvantages:
• CHB needs separate DC sources for each leg.
• Controller will be complex if number of cells increase.
Additional detail can be found in Appendix C [2], [3] and [4].

1.1.2 Modulation techniques
a. Sin-PWM (SPWM) multicarrier strategy
In the SPWM, each phase uses single sinusoidal reference. For m cells need 2m
triangular carriers. The carriers have the same frequency, the same peak to peak
amplitude. Sinusoidal reference is compared with each carrier to determine the
switching output voltages for the power converter.
1

1
Uc1

Uc2

-Uc1

-Uc2

Uc1

Uc2

0

0

-Uc1

-Uc2

-1

-1

b. PD carrier

a. PS carrier
1

1
Uc1

Uc1

Uc2

Uc2

0

0
-Uc1

-Uc1

-Uc2

-Uc2

-1

-1

c. APOD carrier
Figure 1.3

d. POD carrier
SPWM multicarrier strategy

There are four strategies of multicarrier PWM. Figure 1.3 is showed
multicarrier PWM strategy for single-phase CHB five level. It requests four
triangle carriers and only one sinusoidal reference.
9


Chapter 1. Overview FCS-MPC for CHB multilevel converter

• Phase-shift (PS) carrier PWM strategy. Each carrier is phase-shift by
360°/4=90° from it’s adjacent carrier.
• Phase disposition (PD), all carriers are in phase 0°.
• Alternative phase opposite disposition (APOD), all carriers are alternatively
in phase opposition.
• Phase opposite disposition (POD), all the carriers above the zero reference
are in phase among them.
For single-phase converter, SPWM is still a good choice, but for three-phase
converter different techniques have been developed to take the advantage of threephase systems in reducing harmonics. The most popular technique is space vector
modulation (SVM).

b. Space vector modulation (SVM)
SVM technique reduces the influence of common-mode voltages and this avoids
the use of any triangular carriers. SVM conveniently provides more flexibility such
as redundant switching sequences, adjustable duty cycles; and it is more suited to
digital implementations.
(0,1,-1)
V10

(-1,1,-1)
V11

(-1,1,0)
V12

(-1,1,1)
V13

(0,1,1)

(-1,0,0)
V4

(-1,0,1)
V14

(-1,-1,1)
V15

Figure 1.4

(1,1,-1)
V9

4

(-1,0,-1)
(0,1,0)
V3

(1,1,0)
(0,0,-1)
V2

(1,0,-1)
V8

3

1

(0,0,0)
(1,1,1)
(-1,-1,-1)
V0

(0,-1,-1)2
(1,0,0)
1 V1
(1,0,1)
(0,-1,0)
V6

(-1,-1,0)
(0,0,1)
V5

(0,-1,1)
V16

(1,-1,-1)
V7
2

(1,-1,0)
V18

(1,-1,1)
V17

Space vector for three-phase CHB three level

These advantages of SVM can lead to a significantly improved performance
of multilevel converters, especially when the level number of the converter is large.
10


Chapter 1. Overview FCS-MPC for CHB multilevel converter

The space vector of a three-phase CHB three level shows in Figure 1.4.
However, SVM for higher level converter is difficult. There generally are 6(n-1)2
triangles in the space vector diagram of a three-phase n level converter, reference
vector can be located within any triangle. SVM selects suitable switch states of the
located triangle and apply them for corresponding need duty cycles in an switching
sequence.

1.2

Modeling of three-phase CHB multilevel converter

Each cell of converter is described in Figure 1.5.

S1

vdc

S3

C

vac

S2

S4

Figure 1.5

H-Bridge converter

Sign IGBT switch state: “0” corresponding IGBT is off and “1”
corresponding IGBT is on. Table 1.1 shows switch state each cell. Output voltage
obtained are 0; Vdc and –Vdc corresponding switch state is 0; 1 and -1.
Table 1.1

Switch state H-Bridge converter

Gate state

vac

Switch state

0

0

0

0

1

Vdc

1

1

1

0

-Vdc

-1

1

0

1

0

0

S1

S2

S3

S4

1

0

1

1

0

0
0

Three-phase CHB seven level converter is showed in Figure 1.2, level state
shows in Table 1.2. Output voltage vAN , vBN , vCN ; load voltage: vAZ , vBZ , vCZ and

11


Chapter 1. Overview FCS-MPC for CHB multilevel converter

common-mode voltage vZN .
Table 1.2

Level state CHB seven level converter

Switch state

vac

Level state

(1,1,1)

3Vdc

3

(1,1,0) (1,0,1) (0,1,1)

2Vdc

2

(1,0,0) (0,1,0) (0,0,1)

Vdc

1

(0,0,0)

0

0

(-1,0,0) (0,-1,0) (0,0,-1)

-Vdc

-1

(-1,-1,0) (-1,0,-1) (0,-1,-1)

-2Vdc

-2

(-1,-1,-1)

-3Vdc

-3

Assume, Vdc each cell is balance, Vdc,k = Vdc (k = 1,...n). Output voltage vAN,
vBN, vCN obtains {-3Vdc, -2Vdc, -1Vdc, 0, +1Vdc, +2Vdc, +3Vdc}, corresponding {3, 2,
1, 0, -1, -2, -3}*Vdc, this is called level state {3, 2, 1, 0, -1, -2, -3}.
Level state phase A, B and C are grand total 127 reasonable different vector
state v.
Output voltage each cell:
 0

vac = +Vdc
−V
 dc

sA = 0
sA = 1
sA = −1

(1.1)

And, output voltage CHB multilevel converters express:
vAN = k A .Vdc

vBN = kB .Vdc
v = k .V
C dc
 CN

(1.2)

where k A , kB , kC −3, −2, −1,0,1,2,3
Assuming, load is balance, output voltage each phase can be showed:
vAZ = vAN − vZN

vBZ = vBN − vZN
v = v − v
CN
ZN
 CZ

12

(1.3)

Chapter 1. Overview FCS-MPC for CHB multilevel converter
β
V103

V71

V104

V72

V105

V74

V107

V48

V75

V108

V109

V110

V50

V77

V111

V112

V79

V113

V80

V81

V64

V39

V20

V94

V63

V38

V17

V33

V93

V62

V118

V59

V92

V125

V89

V123

V87

V122

V86

V120

V126

V90

V88

V57

V119

α

V61

V124

V58

V85

V37

V60

V36

V34

V84

V19

V35

V56

V83

V117

V7

V18

V55

V115

Figure 1.6

V95

V91

V16

V54

V116

V21

V1

V32

V82

V65

V40

V8

V6

V15

V53

V96

0

V5

V31

V66

V22

V2

V0

V14

V52

V114

V3

V97

V41

V9

V10

V4

V30

V67

V23

V11

V98

V42

V24

V13

V51

V78

V43

V12

V29

V99

V68

V44

V26

V28

V100

V69

V25

V27

V49

V76

V45

V47

V101

V70

V46

V73

V106

V102

V121

Vector state in CHB seven level converter

Because of vAZ + vBZ + vCZ = 0 , so common-mode vZN as express:

vZN =

1
( vAN + vBN + vCN )
3

(1.4)

The level state can be expressed by the vector as following:
v=

where a = e

j

2
3

; a =e
2

j

2
(vA + a  vB + a2  vC )
3

(1.5)

4
3

The vector state v can be expressed in terms of complex coordinate by
using the Clarke transformation:
v = v + jv

13

(1.6)


Chapter 1. Overview FCS-MPC for CHB multilevel converter

v = vA

where: 
1
v = 3 ( vB − vC )


1.3

FCS-MPC control strategy

Model predictive control (MPC) is understood as a wide class of controller, the
main characteristic is the use of the model of the system for the prediction of the
future behavior of the controlled variables over a predictive horizon, n-steps. The
information is used by the MPC control strategy to provide the control action
sequence for the system by optimizing a user-defined cost function. It should be
noted that the algorithm is executed again every sampling period and only the first
value of the optimal sequence is applied to the system at instant k.
Model predictive control

(MPC)

Finite control set MPC
(FCS-MPC)

Continuous control set MPC
(CCS-MPC)

Optimal switching vector MPC
(OSV-MPC)

Generalized predictive control
(GPC)

Optimal switching sequence MPC
(OSS-MPC)

Explicit MPC
(EMPC)

Figure 1.7

Classification of MPC strategies applied to power converter

Classification of MPC strategy applied to power converter is showed in
Figure 1.7, [2]. MPC strategy can be divided into two types: continuous control
set MPC (CCS-MPC) and discrete of the power converters finite control set MPC
(FCS-MPC).
The CCS-MPC computes a continuous control signal and then uses a
modulator to generate output voltage in the power converter. The main advantage

of CCS-MPC when applied to power converter is that it produces a fixed switching
frequency. The main disadvantage of CCS-MPC is present a complex formulation
of the MPC problem.
14


Chapter 1. Overview FCS-MPC for CHB multilevel converter

The FCS-MPC based on finite number of switching state to formulate the
MPC algorithm and does not need a modulator. FCS-MPC can be divided into two
types: optimal switching vector MPC (OSV-MPC) and optimal switching
sequence MPC (OSS-MPC). OSV-MPC is the most popular MPC control strategy
for power converter. It uses the output vector state of the power converter as the
control set. The main advantage of OSV-MPC: it only calculates prediction for this
control set, therefore it reduces the optimal problem to an enumerated search
algorithm. This makes the MPC strategy formulation very intuitive. The
disadvantage of OSV-MPC is that only one output optimal vector state is applied
during the complete sampling time period, lead to uncontrolled switching
frequency.
In FCS-MPC, the prediction model of the system needs to be discretized.
Therefore, the MPC algorithms are usually implemented in digital hardware like
as DSP or FPGA. The common of FCS-MPC regularly uses Euler approximation
to discretize a one-step or multiple-step.
Conv.
x*
x

Predictive
model

Optimizaton

p

x

Sopt

J

Load
Conv.

FCS-MPC

Measurement
Estimation

Figure 1.8

FCS-MPC block diagram

Figure 1.8 shows FCS-MPC block diagram. Assume, control variable x
follow the reference variable x*, procedure design FCS-MPC following basic
steps:
• Measurement, estimation the control variable in the sampling time instant.
• For every switch states of the converters, predictive (using the mathematical
model) the behavior of variable x in the n-steps time.
• Evaluate the cost function for each prediction.

15


Chapter 1. Overview FCS-MPC for CHB multilevel converter

• Select the switch states that minimize the cost function, Sopt applied to the
converters.
In the experiment, driver, measurements and IGBT exist delay time. The
computational time is needed in the predictive control algorithm to predict the
variables, and processor delay deteriorates the performance of the predictive
control at the experimental investigation. To solve this problem, it can be
considered the predictive horizon at (k+n)th sampling time to predict the variables
which are compared with the references, and determine the cost functions. The
optimum Sopt is selected corresponding to the minimum cost function, and applied
it in the power converter.

16


Chapter 2. FCS-MPC for gird-connected three-phase CHB

Chapter

2

FCS-MPC for gird-connected three-phase CHB

2.1

FCS-MPC for grid-connected formulation

The FCS-MPC control strategy predicts behavior of the load current for each
possible vector state v generated by the power converter. The prediction of the
current is based on discretized model of system.
A, B, C
N

O

HB 1

HB 2

HB 3

r

L

Filter

CHB

Sopt
Cost function
optimization

i abc (k )

i abc (k + 2) v g ,abc (k )

Prediction
(k+2)th

i*abc (k )

Vdc (k )

Current
reference
generation

P*

Q*

FCS-MPC

Figure 2.1

Block diagram of FCS-MPC gird-connected

In abc coordinate, a block diagram of predictive current control is described
in Figure 2.1. The procedure designs FCS-MPC for grid-connected included
mainly three steps [5]:
17


Chapter 2. FCS-MPC for gird-connected three-phase CHB

• Computational current references i*abc (k ) in the sampling time instant (k)
from references of active power P* and reactive power Q*.
• Prediction horizon at (k+2)th sampling time to predictive current i abc (k + 2)
the variables which are compared with the current references i*abc (k ) .
• The optimum vector state is selected corresponding to the minimum cost
function and applied it to actuation.

2.1.1 Discrete-time model predictive current control
Grid-connected three-phase CHB converter, the following continuous time
dynamic equation for each phase current can be expressed:
 di (t )

vabc (t ) =  L abc + r.i abc (t ) + v g ,abc (t )  + vNO (t )
dt



(2.1)

where r and L is the resistance and inductance of the output filter; vabc is phase
output voltage; v g ,abc is grid voltage. Therefore, from (2.1) can be inferred:
di abc (t ) −r
1
= i abc (t ) +  vabc (t ) − vNO (t ) − v g ,abc (t ) 
dt
L
L

(2.2)

For a three-phase n cell CHB converter, the phase output voltage in become:
vabc = Vdc .vl ,abc

(2.3)

where level state vl ,abc = −n, −n + 1,...,0,..., n −1, n .
Additionally, common-mode voltage is given by:
vNO (t ) =

1
va (t) + vb (t) + vc (t)
3

(2.4)

The first order forward Euler’s approximations:
dx(t ) x(k + 1) − x(k )
=
dt
Ts

(2.5)

By applying (2.5) to (2.2) with sampling time Ts , the discrete-time of current
as bellow:
T
 rT 
i abc (k + 1) = 1 − s  i abc (k ) + s  v abc (k ) − vNO (k ) − v g ,abc (k ) 
L 
L



18

(2.6)


Chapter 2. FCS-MPC for gird-connected three-phase CHB

The discrete-time dynamic model can be expressed [10]:
x(k +1) = Ax(k ) + Bvl ,abc (k ) + Ev g ,abc (k )

(2.7)

where:

 rTs
1 − L
A=
 0


vga (k )
v g ,abc (k ) = 

vgb (k ) 

vl ,a (k )



vl ,abc (k ) = vl ,b (k ) 
v (k ) 
 l ,c 

i (k ) 
x(k ) =  a 
ib (k ) 




rTs 
1−
L 

B=

0

VdcTs
3L

 2 −1 −1
−1 2 −1



E=−

Ts

L

1 0 
0 1 



As applying the Forward Euler’s approximations, similarly (2.7), the
predictive horizon at two-steps sampling time k+2 as following [5][6]:
x(k + 2) = Ax(k +1) + Bvl ,abc (k ) + Ev g ,abc (k )

(2.8)

2.1.2 Cost function optimization and vector state selection
The last step is developing a cost function for the optimization. The cost function
is very flexible. It should be designed according to the specific control goals. The
cost function in the predictive control of grid-connected with delay compensation
can express [8][9]:
2

J = x (k + 2) − x(k + 2) 2 + x (k + 1) − x(k + 1)
*

where: a − a 2 = ( a1 − a1 ) + ... + ( ap − ap )
*

2

*

2

*

For sufficiently small

2

*

2
2

(2.9)

ia* (k )
*
x
(
k
)
=
and
 *  is the reference current.
ib (k )

sampling

time, it can be

assumed that

x* (k + 2)  x* (k +1)  x* (k ) . Therefore, cost function (2.9) can be rewritten as:
2

J = x (k ) − x(k + 2) 2 + x (k ) − x(k + 1)
*

*

2
2

(2.10)

The cost function (2.10) is evaluated for per possible three-phase level state,
and the one that minimizes it. And then, optimal level state is selected and applie d
to the load. This mean that (2.7), (2.8) and (2.10) are calculated 7 3 =343 times for

19


Chapter 2. FCS-MPC for gird-connected three-phase CHB

a seven level in order to obtain the optimal solution. However, level state can be
defined from vector state. By that way, the calculation can be still reduced 127
times. The selection criterion is to select the voltage level states that minimize the
common voltage vector.

2.1.3 Subset of adjacent vector state

In the previous section, each sampling time cost function needs to be calculated
127 times. The vector state will fastly increase follow 12m2 + 6m + 1 when the
number of cells m growth, it is very high. So that, subset of adjacent vector state
(SAVS) is proposed to reduce computational [7]. In this way, it is possible to
reduce the set of vector state to be evaluated to the vector state that are nearest to
the last applied vector, as shown in Figure 2.2.
β

Number of
redundancies for
1
each vector
2
3
4
5
6
7

α

Adjacent
vectors in time
[k]

Figure 2.2

Vector state for CHB seven level three-phase

For the calculation of the adjacent vectors to the last applied vector, the

distance between vectors can be calculated with the following function:
d ( vx , vy ) =

(v

x

− vy ) + ( vx − vy )
2

2

(2.11)

If v x near v y , the distance should be equal or less than 2Vdc/3. The
calculation of distance is made offline, and it is stored in database. In this way, the
20