applied

sciences
Article
A Novel Adaptive Model Predictive Control Based
Three-Phase Inverter Current Control Method
Mingyu Lei 1,2 , Ying Zhang 1,2 , Lexuan Meng 3 , Yibo Wang 1,2, *, Zilong Yang 1,2 and
Dufeng Cao 1,2
1 Solar Thermal Technology and Photovoltaic System Laboratory, China Academy of Sciences, Beijing 100190,
China; leimingyu@mail.iee.ac.cn (M.L.); zhangying2011@mail.iee.ac.cn (Y.Z.); yangzl@mail.iee.ac.cn (Z.Y.);
caodufeng888@126.com (D.C.)
2 Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China
3 Independent Researcher, 72482 Västerås, Sweden; lexuan.meng@gmail.com
* Correspondence: wyb@mail.iee.ac.cn; Tel.: +86-136-7135-5662




Received: 17 September 2019; Accepted: 6 November 2019; Published: 11 December 2019


Abstract: This paper proposes a novel current control method based on Model Predictive Control
(MPC) for three-phase inverters. The proposed method is based on an Adaptive MPC (A-MPC) with
a PWM modulation. An innovative model parameter estimation and modification method is also
proposed, leading to enhanced control accuracy. Comparing with traditional current control methods,
such as PI and PR control, the proposed method has better dynamic performance. The transient
dynamics, i.e., recovery time and overshoot, have been considerably improved. Simulation and
experimental results are presented to validate the effectiveness of the proposal.

Keywords: Inverters; model predictive control; current control; dynamic performance

1. Introduction
Model Predictive Control (MPC) is recently regarded as one of the most promising control methods
in controlling power converters, due to its predictive effect and robustness [1]. The fundamental idea
of MPC is to predict the optimal controls of next step(s) to minimize the cost function based on the
model of the object under control as well as a set of constraints. During normal operation, the accuracy
of predicted model parameters can be improved online, resulting in enhanced control accuracy and
better adaptability [2]. This feature is highly attractive when the operating conditions varies frequently.
In recent years, MPC has been successfully used in grid-tie converters for renewable energy systems,
where it delivers improved dynamic performances, robustness and system stability [3–8]. In most of
existing applications of MPC for power converters, the control command of next period is directly
selected from a finite set of switching states according to the cost function, namely finite-control-set
MPC (FCS-MPC) [9–12]. While in some other approaches, the output variables, such as the current
or voltage references, are continuous values, and these methods are known as continuous control set
MPC (CCS-MPC) [13,14].
Applications of FCS-MPC have attracted much attention because of several advantages, such as fast
transient response, simple implementation, and so on. Up to now, most MPC based control methods for
power converters are of this type. In [15], a FCS-MPC on three-phase voltage source inverter is proposed
to select an optimal switching state for next step to minimize the error and guarantee control accuracy.
Experimental results show that the proposed method controls the load currents effectively with a better
dynamic response comparing to conventional linear control methods. In addition, this method is simple and
easy for DSP implementation. Paper [16] proposes a FCS-MPC on controlling the flying capacitor voltage
and the grid current of the Packed-U-Cells inverter. The eight available switching states are evaluated every

Appl. Sci. 2019, 9, 5413; doi:10.3390/app9245413 www.mdpi.com/journal/applsci

Appl. Sci. 2019, 9, 5413 2 of 15

period and the optimal one is selected according to a cost function. Experimental results validate the good
dynamic performance in controlling the grid-connected system. Paper [17] proposes a FCS-MPC based
control method for four-leg indirect matrix converters to minimize the instantaneous reactive input power.
A significant advantage of this method is the reduction of control strategy complexity.
However, FCS-MPC is not suitable in the cases when the optimal switching state cannot be
directly selected or there are considerably number of switching states. The three major drawbacks of
FCS-MPC are: (i) High computational burden due to evaluation of all the switching states; (ii) low
control accuracy due to limited switching states; (iii) model dependency [13].
Paper [13] has validated that CCS-MPC based methods have better performance and reduced
computational burden in power electronic system applications. In this paper, a CCS-MPC based control
method for grid-connected photovoltaic (PV) systems. The proposed method is applied to control the power
exchange between PV and grid, while achieving the unit power factor. It is verified to have good dynamic
performance. In [18], a CCS-MPC method is applied in controlling of permanent magnet synchronous
machines. Experimental results prove the effectiveness of the proposed method in such applications.
However, for above mentioned approaches, the performance of the control largely depends on the
accuracy of the model parameters. An inherent steady-state error cannot be eliminated during the control
process since there is no adaptation on the parameters. Paper [19] analyzes the influence of model error
on the accuracy of MPC for current control in a three-phase inverter. It has been confirmed that the
mismatch of prediction inductance can cause steady-state error in inverter current. Limited research
works have been conducted in improving the model accuracy. In [20], an MPC based current control of a
switched reluctance motor is proposed, in which an adaptive controller is used to dynamically estimate
and modify the motor inductance based on Kalman Filter theory. Paper [21] proposes a FCS-MPC based
control method for voltage source inverter. This method is able to correct the correct parameters based on
ADALINE estimator. However, the adaptive methods proposed in [20,21] require additional algorithm
installation and processing in controller, which inevitably increases the computational burden.
In this paper, an Adaptive Model Predictive Control (A-MPC) based three-phase inverter current
control method is proposed. The most important contribution of this method is the novel parameter
estimation and adapting algorithm, which is simple and efficient. The parameter estimation and
modification method is decoupled with the current control loop. No extra control algorithm is used
in this estimation method. The parameter estimation can improve the control accuracy. The optimal
switching state components in two-phase rotating coordinate system is generated and used as the input
for parameter estimation and modulation. Since the variables (e.g., voltages, currents, switching states
and their references) are all DC components, the optimal switching state can be calculated directly,
avoiding the evaluation of all the available switching states. Therefore, the computational burden is
effectively reduced. Simulations and experiments validate the effectiveness of the proposed method
and the enhancement in system performance.

2. Model of an Inverter
The main circuit of a full bridge AC/DC inverter studied in this work is shown in Figure 1. In
this system, it is assumed that the system is balanced, and the DC voltage of the inverter is stable.
The general mathematical model of this inverter can be obtained as [22]
 di S +S +S
 L a + Ria + ea = UDC (Sa − a 3b c )
 didtb


S +S +S


 L dt + Rib + eb = UDC (Sb − a 3b c ) (1)
 L dic + Ri + e = U (S − Sa +Sb +Sc )


dt c c DC c 3

where L and R are the filter inductance and resistance respectively. The Sa , Sb and Sc are the switching state
variables of the switching devices. The ea , eb , ec are the phase voltages of the grid. This three-phase model can
be also presented in a two-phase stationary coordinate system (αβcoordinate system). The transformation
expression of the variables between these two coordinate systems is shown as following:
 Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 16
Appl. Sci. 2019, 9, 5413 3 of 15
where L and R are the filter inductance and resistance respectively. The Sa, Sb and Sc are the switching
state variables of the switching devices. The ea, eb, ec are the phase voltages of the grid. This three-
phase model can be also"presented in
 a two-phase stationary coordinate
 a  system (αβ coordinate
 
system). The transformationαexpression 1 −0.5 −0.5
#
2  of the variables
 √ betweenthesetwo coordinate systems is
√
  
=   b  (2)
shown as following: β 3 0 2
3
− 2 3
 
c
1 −0.5 −0.5  a 
  2    
=
By transforming all the variables tothe   two-phase
3 0
3  b  coordinate system, the mathematical
3 stationary
−
(2)
 2 2  c 
model can be expressed as 
By transforming all the L didtα + Ri
 variables to the 2
two-phase stationary coordinate system, the
α + eα = 3 UDC Sα
mathematical model can be expressed
 diβ as (3)
 L + Riβ + eβ = 2 UDC Sβ

dt 3
 di 2
  L
 + Ri + e = U
DC  S
where Sα and Sβ are the switching functions  dt in two-phase 3 stationary coordinate. In this modeling
 (3)
expression, the switching variables, current L di and 2

+ Ri voltage are
+ e = U DC S  still of AC system. The model can be

 dt 3
transformed to the two-phase rotating coordinate system. The transformation expression between
where Sα and
these two systems Sβ are the
is defined asswitching functions in two-phase stationary coordinate. In this modeling
expression, the switching variables, current and voltage are still of AC system. The model can be
transformed to the two-phase" rotating
# " coordinate system. The#"transformation expression between
cos θ sin θ α
#
d
these two systems is defined as = (4)
q − sin θ cos θ β
 d  cos  sin    
 q  =  − sin  cos     
(4)
By taking (4) into (3), the   model
mathematical  of this inverter in two-phase rotating coordinate
system is shown as followings:
By taking (4) into (3), the mathematical model of this inverter in two-phase rotating coordinate
system is shown as followings:

did
 L dt +
 diRi = ωLi − e + 2 2 U Sd
L d +d Rid =  Liqq − ed d+ U3DC SDC

diq 

d (5)
 L + Riq = −ωLid − eq 3+ 2 UDC Sq
dt


dt 3 (5)
 L diq + Ri = − Li − e + 2 U S

 dt
q d q
3
DC q

where Sd and Sq are the switching functions in two-phase stationary coordinate. In this expression, all
where Sd and Sq are the switching functions in two-phase stationary coordinate. In this expression, all
of the variables (e.g., current, voltage and switching states) are of DC system. The orientation of these
of the variables (e.g., current, voltage and switching states) are of DC system. The orientation of these
systems are shown in Figure 2.
systems are shown in Figure 2.

Sa1 Sb1 Sc1

ia R L ea _
+
UDC
O

Sa2 Sb2 Sc2

Appl. Sci. 2019, 9, x FOR PEER REVIEW 1. Structure 4 of 16

Figure
Figure 1. Structureof
of aa three-phase
three-phase inverter.
inverter.

d
q

θ a

Figure 2. The orientation of abc and dq coordinate systems.

Figure 2. The orientation of abc and dq coordinate systems.

3. Adaptive Model Predictive Current Control for Inverters

The main idea of MPC is to predict the future states of the system under control and select the
control variables minimizing the given cost function according to the predicted mathematical model.
In this proposed A-MPC method, the model parameters can be updated online according to the error
between the predicted and the actual output value. The main process of this control method is given
in Figure 3.
 c

Appl. Sci. 2019, 9, 5413 4 of 15

Figure 2. The orientation of abc and dq coordinate systems.

3. Adaptive Model Predictive Current Control for Inverters

3. Adaptive Model Predictive Current Control for Inverters
The main idea of MPC is to predict the future states of the system under control and select the
The main idea of MPC is to predict the future states of the system under control and select the
control variables minimizing the given cost function according to the predicted mathematical model.
control variables minimizing the given cost function according to the predicted mathematical model.
In this proposed A-MPC method, the model parameters can be updated online according to the error
In this proposed A-MPC method, the model parameters can be updated online according to the error
between the predicted and the actual output value. The main process of this control method is given
between the predicted and the actual output value. The main process of this control method is given in
in Figure 3.
Figure 3.

J cost function
u(k) control command
y*(k) predicted output Model correction
y(k) measured output
∆y(k) error of output predicting
y*(k)
Predicted model

+
J Calculation of control command u(k)
u(k), - ∆y(k)
minimizing cost function J
y(k)
Control object

Figure 3. Main process of A-MPC current

Figure 3. current control
control method.
method.

3.1. Selection of Switching States

3.1. Selection of Switching States
The overall control scheme proposed in this paper can be divided into three steps: (i) Select the
The overall control scheme proposed in this paper can be divided into three steps: (i) Select the
optimal switching state, (ii) apply the modulation, and (iii) adaptively modify the parameters. In this
optimal switching state, (ii) apply the modulation, and (iii) adaptively modify the parameters. In this
method, the mathematical model expression in dq coordinate system is employed. In order to calculate
method, the mathematical model expression in dq coordinate system is employed. In order to
the optimal switching state, the cost function of MPC is given as
calculate the optimal switching state, the cost function of MPC is given as
J i=dre
J= f −−idid(k
| idref (k++11)) |+
+ |iqre − iiqq((kk ++1)1|)
iqreff − (6)
(6)

where iidref
where dref and
and iiqref
qref are
are the
the references
references of inverter output
of inverter output currents
currents inin two-phase
two-phase rotating
rotating coordinate
coordinate
system. The output variables (y and y ) and control variables (u and u ) are defined
system. The output variables (y1 and y2 ) and control variables (u1 and u2 ) are defined as follows:
1 2 1 2 as follows:
" #id  "Sd  #
y = y =d iq  , , =  Sd
i
uu = (7)
(7)
iq    SS
q
q

The mathematical model expression shown in the Equation (5) can be rewritten as follows after
the discretization.  i (k+1)−i (k)
 L



d
Ts
d
+ Rid (k + 1) = ωLiq (k + 1)
−ed (k + 1) + 23 UDC (k + 1)Sd (k + 1)



(8)

 iq (k+1)−iq (k)
L + Riq (k + 1) = −ωLid (k + 1)




 Ts
−eq (k + 1) + 2 U (k + 1)Sq (k + 1)


3 DC

where Ts is the length of control period, and ω is the angular frequency. In this expression, all of
the variables, including the switching state variables, are of DC system and will not change greatly
between successive two steps. Therefore, the following expression can be obtained.


 UDC (k + 1) ≈ UDC (k)

ed (k + 1) ≈ ed (k) (9)




 eq (k + 1) ≈ eq (k)

Obviously, the minimum value of cost function J can be obtained when satisfying following conditions
Appl. Sci. 2019, 9, 5413 5 of 15

(
id (k + 1) = idre f
(10)
iq (k + 1) = iqre f

Taking above values into the model expression, the switching state variables can be calculated as
 idre f (k)−id (k)
ed (k)+Rp idre f (k)−ωLp iqre f (k)+Lp

 Ts
S (k + 1) =


 d 2U

3 DC (k)

 iqre f (k)−iq (k) (11)
eq (k)+Rp iqre f (k)+ωLp idre f (k)+Lp


 Ts
 Sq (k + 1) =

 2U
3 DC (k)

This is the expression to calculate the optimal switching states of the next period. In this expression,
the model parameters Rp and Lp are of predicted values and will be modified online. During every
control period, the measured variables of currents, voltage are used to calculate the switching state
variables. In this method, only above expressions are needed to obtain the optimal switching states.
As a result, the computation is reduced comparing with other FCS-MPC inverter control methods.

3.2. Modulation
In this A-MPC current control method, conventional PWM modulation technique can be applied
to generate
Appl. Sci.the commands
2019, 9, x FOR PEERfor the switches. The instantaneous values of three-phase inverter6switching
REVIEW of 16
states variables Sa , Sb and Sc are defined as 1 when the upper switch is on, and as 0 when it is off.
the range
Therefore, of carrier is set
the equivalent as [−M, M]
sinusoidal and the value
switching stateof modulation
variable wave is A, as
in three-phase shown in Figure
coordinate system4,isthewith a
duty cycle of this switch period is
DC bias of 0.5. For the modulation process, the output Da , Db and Dc must meet following requirements:
(i) The values in two-phase rotating coordinate A +system
M A of the switching variables are equal to Sd and
D= = + 0.5 (12)
Sq , and (ii) The DC bias of switching variables 2isM0.5. 2M
The Whentwo basicM = 0.5,signals
D = A of thethe
+ 0.5, modulation
DC component process
of theare modulation
modulation outputwave
is 0.5.and carrierthewave.
Therefore,
The modulation wave can be obtained by transforming the switch state
carrier wave is ranged from −0.5 to 0.5. The modulation process helps increase the numbersvariables (S d and Sqof ) from
rotating coordinate
switching statessystem to three-phase
in one period, comparing coordinate
with othersystem
FCS-MPC to abc).
(dqbased Assuming
methods. that during
In FCS-MPC a switch
method,
period thethe
switching
range ofstate keeps
carrier is constant
set as [−M,in one
M] control
and theperiod,
value and only one switching
of modulation wave isstate,
A, aswhich
shown in
Figure minimizes
4, the duty the cycle
cost function, is selected
of this switch [23].isAs a result, the error of output current is inevitable. In
period
contrast, PWM modulation process can provide more than one switching state and their duty cycle
[24]. FCS-MPC methods can only select the +M
A switching A
state with the least current error, resulting in
D=
variable switching frequency and complicated
= + 0.5 (12)
2Mharmonic
2M content [24]. In summary, the modulation
method can achieve constant switching frequency and reduce harmonics.

A t

-M

Ton
TS t

Figure 4. Carrier
Figure 4. Carrierand
andmodulation wavesofofPWM
modulation waves PWM modulation.
modulation.

When M = 0.5,
3.3. Parameters D = A + 0.5, the DC component of the modulation output is 0.5. Therefore,
Modification
the carrier wave is ranged from −0.5 to 0.5. The modulation process helps increase the numbers of
The initial values of model parameters are given according to the acquired information about
switching states in one period, comparing with other FCS-MPC based methods. In FCS-MPC method,
the inverter. However, in many cases the actual parameters are unavailable, especially in complex
the switching statekeep
systems, and keeps constant
changing inTherefore,
[25]. one controltheperiod, and only one
model parameters needswitching state, online
to be updated whichduring
minimizes
the cost
thefunction, is selected
control process [23]. Astoa ensure
continuously result, the
the accuracy
error of output current
of the control is inevitable. In contrast, PWM
result.
modulation Theprocess
action ofcan provide modification
parameters more than one switching
is conducted state
only andthe
when their dutyhas
inverter cycle
been[24]. FCS-MPC
in steady
state. In this case the following condition has to be satisfied
methods can only select the switching state with the least current error, resulting in variable switching
 did
 L =0
 dt
 (13)
 L diq = 0

 dt

The expression of this operation state in two-phase rotating coordinate system can be rewritten
as followings:
Appl. Sci. 2019, 9, 5413 6 of 15

frequency and complicated harmonic content [24]. In summary, the modulation method can achieve
constant switching frequency and reduce harmonics.

3.3. Parameters Modification

The initial values of model parameters are given according to the acquired information about
the inverter. However, in many cases the actual parameters are unavailable, especially in complex
systems, and keep changing [25]. Therefore, the model parameters need to be updated online during
the control process continuously to ensure the accuracy of the control result.
The action of parameters modification is conducted only when the inverter has been in steady
state. In this case the following condition has to be satisfied

did
 L dt = 0


(13)
 L diq = 0


dt

The expression of this operation state in two-phase rotating coordinate system can be rewritten as
followings:
Rid = ωLiq − ed + 23 UDC Sd
(
(14)
Riq = −ωLid − eq + 23 UDC Sq
The above expression is about the state of the inverter and is satisfied in any operation situation.
The parameters L and R in (12) are both actual physical parameters. When the switching states (11) are
substituted into above state equation, it can be rewritten as

ωLiq (k) + RP (k − 1)idre f




 Rid (k) =

 idre f −id (k−1)

 −ωLP (k − 1)iqre f + LP (k − 1) Ts
(15)

Riq (k) = −ωLid (k) + RP (k − 1)iqre f





 iqre f −iq (k−1)
+ωLP (k − 1)idre f + LP (k − 1)


Ts

It can be seen that both the actual parameter values L and R and the predicted values Lp and Rp
have effect on the operation state of the inverter. Since it is under steady state, following expressions
are true. (
id (k − 1) = id (k)
(16)
iq (k − 1) = iq (k)
In this case, (15) can be simplified as

ωLiq (k) + RP idre f − ωLP iqre f




 Rid (k) =

 idre f −id (k)

 +LP Ts
(17)

Riq (k) = −ωLid (k) + RP iqre f + ωLP idre f





 iqre f −iq (k)
+LP


Ts

In above expression, the unknowns are the L and R. Therefore, the actual values of parameters
can be obtained by solving this equation.
Firstly, a series of variables are defined as followings:



 ζ1 = id iqre f − iq idre f
ζ

2 = id idre f + iq iqre f (18)



ζ3 = i2d + i2q




The expression about parameter L is shown as following

Appl. Sci. 2019, 9, 5413 7 of 15

LP (k−1)
(RP (k − 1) + TS ) ζ1 + ωLP (k − 1)ζ2
L= (19)
ωζ3

According to above analysis, the actual parameter at any state can be calculated by the measured
data and the predicted model parameters. However, it is not an acceptable way to improve the accuracy
of the predicted parameters by setting Lp equal to L directly. This is because the change of Lp or Rp
will cause the change of inverter operation state, and then change the actual value of the parameters.
Furthermore, a dramatic change of parameters may have a negative impact on the stability or power
quality of inverter current. Therefore, the method of step by step approximation is adopted here. The
difference between L and Lp can be expressed as

LP
(RP + TS )ζ1 + ωLP (ζ2 − ζ3 )
L − LP = (20)
ωζ3

Similarly, the difference between R and Rp can be calculated as

LP
−ωLP ζ1 + TS (ζ2 − ζ3 )+RP (ζ2 − ζ3 )
R − RP = (21)
ζ3

Above expressions can be used to modify the predicted model parameters. When the inverter has
been in stable operation
Appl. Sci. 2019, 9, x FORstate, the accuracy of the output current is estimated. If the 8error
PEER REVIEW of 16 between

reference and measured current is larger than the maximum permissible value, the action of parameters
Above expressions can be used to modify the predicted model parameters. When the inverter
modification
has will
been in bestable
conducted.
operation The
state,parameters
the accuracy ofcan be modified
the output current isaccording
estimated. Ifto
thefollowing expressions.
error between
reference and measured current is larger than the maximum permissible value, the action of
∆L parameters > εL

LP (kwill
parameters modification

 ) =be (k − 1) +The
LPconducted. i f Lcan
− Lbe
P modified according to following
 LP (k) = LP (k − 1) − ∆L i f L − LP < −εL

expressions. 
(22)

LP L <L L − LP < εL
 LPP(k ) = LP (k − 1) + L if i f L −−ε

 L (k ) = L (k − 1)
P

 P L (k ) = L (k − 1) − L
P P L if L − L  − (22)
RP (k) =R LPP((kk) =−L1 − 1)∆R if i f − LR −L −RLPP > ε

P ()k + L R


∆R <

R ( k ) = R ( k − 1 ) − i f R − R −ε (23)

P k ) = RP (k − 1) + R if R − RP   R
P P R
 R (kR)P (=


< P < εR

R ( k − 1 ) i
P  R (k ) = RP (k − 1) − R if R − R  − R f −ε R − R(23)
P P P R
 R (k ) = R (k − 1) −  −  
where ∆L and ∆R are the unit P changes P if
of inductance R R
R and Presistance
R respectively. εL and εR are the
maximumwhere ΔL and ΔR
acceptable are the
errors of unit changes ofand
inductance inductance and resistance
resistance respectively.
respectively. εL and εR are
The complete the
scheme of the
maximum acceptable errors of inductance and resistance respectively. The complete scheme of the
proposed A-MPC current control process is given in Figure 5.
proposed A-MPC current control process is given in Figure 5.

Converter

iL vgrid

abc θgrid
Da ,Db ,Dc Modulation
dq Modulation
Sd , S q
 i ( k ) − id ( k )
 ed + Ridref ( k ) −  Liqref (k ) + L dref id , iq dq
UDC  Sd ( k + 1) =
Ts
abc θgrid


2
U DC Main
Id-ref 
3
 i ( k ) − iq ( k )
eq + Riqref ( k ) +  Lidref ( k ) + L qref ed , eq dq Control
Iq-ref 
 Sq ( k + 1) =
Ts
abc
 2
U DC θgrid PLL
 3
id iq
L p , Rp
Id-ref
∆L , ∆R Model prediction
Model Model
Prediction Correction Iq-ref and correction

FigureFigure 5. Complete
5. Complete schemeof
scheme of the
theproposed
proposedcontrol method.
control method.

4. Simulation and Experimental Results

The proposed A-MPC based inverter control method has been validated through simulation and
experiments. Firstly, a series of simulations through MATLAB/Simulink are conducted to compare
the performances of common FCS-MPC based inverter control method and the proposed method. In
order to take the simulation and experiments to a close agreement, the simulation model of the three-
phase inverter is established according to the practical experiment parameters, which is listed as
Appl. Sci. 2019, 9, 5413 8 of 15

4. Simulation and Experimental Results

The proposed A-MPC based inverter control method has been validated through simulation and
experiments. Firstly, a series of simulations through MATLAB/Simulink are conducted to compare the
performances of common FCS-MPC based inverter control method and the proposed method. In order
to take the simulation and experiments to a close agreement, the simulation model of the three-phase
inverter is established according to the practical experiment parameters, which is listed as Table 1.

Table 1. Parameters of Simulation.

Parameters Symbol Value Unit

Voltage of grid VAC 380 V
Voltage of DC bus VDC 600 V
Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 16
Fundamental frequency f 50 Hz
Rated power of inverter Prated 10 kW
Filter Filter inductor
inductor Lf L f 1.5
1.5 mHmH
Switching
Switching frequency
frequency fs fs 66 kHzkHz

AsAsmentioned,
mentioned,ininexisting
existingFCS-MPC
FCS-MPCbased basedinverter
invertercontrol
controlmethods,
methods,the theswitching
switchingvariables
variablesare
are
directly selected according to the cost function by comparing all the possible
directly selected according to the cost function by comparing all the possible variables, known as variables, known as
exhaustive method. There are three obvious drawbacks mentioned above. To
exhaustive method. There are three obvious drawbacks mentioned above. To address these three issues,address these three
issues,
this paper this paper proposes
proposes A-MPC inverterA-MPCcurrent
invertercontrol
current controlThe
method. method. The method
proposed proposedcanmethod
overcome can
overcome these problems
these problems effectively. effectively.
InInorder
ordertotovalidate
validateit,it,two
twogroups
groupsofofsimulations
simulationsare areconducted
conductedand andthetheresults
resultsare
arecompared.
compared.
Theoutput
The outputcurrent
currentwaveforms
waveformsofofthe theFCS-MPC
FCS-MPC method
method andand proposed
proposed method
method areare shown
shown in in Figures
Figures 6
6 and 7 respectively. As seen in the figures, the total harmonic distortion (THD)
and 7 respectively. As seen in the figures, the total harmonic distortion (THD) of the proposal (4.71%) of the proposal
is(4.71%) is obviously
obviously lower
lower than thatthan that of common
of common FCS-MPC FCS-MPC
(12.19%), (12.19%),
owing toowing to the modulation
the modulation process.process.

Figure6.6.Simulation
Figure Simulationresults
resultsfor
forthe
theoutput
outputcurrent
currentwaveforms
waveformsofofFCS-MPC
FCS-MPC(a)
(a)current
currentwaveform
waveformfor
for
0.04 s, (b) current THD.
0.04 s, (b) current THD.
Appl. Sci. 2019,
Appl. Sci. 9, 5413
2019, 9, x FOR PEER REVIEW 109 of
of 15
16
Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 16

Figure
Figure 7.7.Simulation
Figure7. Simulationresults
Simulation resultsfor
results forthe
for theoutput
the outputcurrent
output current waveforms
waveformsof
currentwaveforms of the
ofthe proposed
theproposed A-MPC
proposedA-MPC (a)
A-MPC(a) current
(a)current
current
waveform
waveform
waveformforfor 0.04
for0.04 s, (b)
0.04s,s,(b) current
(b)current THD.
currentTHD.
THD.

Since
SincethetheA-MPC
A-MPCmethodmethodremoves
removesthe theprocess
processof ofevaluating
evaluatingall theoptional
allthe optionalswitching
switchingstates,
states,the
the
computation During above simulations, the processing time
computation is reduced. During above simulations, the processing time of each method is shownin
is reduced. of each method is shown in
Figure
Figure
Figure8.8.8.The
The total
Thetotal recorded
totalrecorded
recorded time
time
timeofof simulation
simulation
of simulation under
under
under control
control of
ofthe
of the
control proposal
proposal
the (265.62
(265.62
proposal s) iss)
(265.62 s)isisthan
less less
lessthan
that
than
that
of
thatof
ofFCS-MPC
FCS-MPCFCS-MPCbasedbased
method
based method
(338.16
method (338.16
s). The
(338.16 s). The
Thelisted
s).listed total total
totalrecorded
listedrecorded time
timeincludes
time includes
recorded both the
includes both the
time
both time
cost
the bycost
time the
cost
by
bythe
thecontrol
control controlmethod
method and by and
method andby
otherby other
parts ofparts
other ofofthe
thesimulation
the simulation
parts model.
model. Since
simulation model. Since
there is there
Since no isisnonodifference
difference
there in
inthe
in the inverter
difference the
inverter
inverter models of each simulation, all of the reduced time (72.54 s) is caused by the simplificationof
models ofmodels
each of each
simulation,simulation,
all of the all of the
reduced reduced
time time
(72.54 s)(72.54
is s)
caused is caused
by the by the simplification
simplification of control
of
control
method. method. Therefore,
Therefore, the the computation
computation reduction reduction
in control in control
method
control method. Therefore, the computation reduction in control method is significant.method
is is significant.
significant.

(a)
(a)

(b)
(b)
Figure 8. Comparison of simulation time between the proposed method and FCS-MPC based method
Figure
Figure 8.8.Comparison
Comparison of simulation time between the
theproposed method and andFCS-MPC based method
(a) recorded time of theof simulation
A-MPC basedtime between
method, proposed
(b) recorded timemethod FCS-MPC
of the FCS-MPC basedbased method
method.
(a) recorded time of the A-MPC based method, (b) recorded time of the FCS-MPC based
(a) recorded time of the A-MPC based method, (b) recorded time of the FCS-MPC based method. method.
Appl. Sci. 2019, 9, 5413 10 of 15
Appl.
Appl.Sci.
Sci.2019,
2019,9,9,x xFOR
FORPEER
PEERREVIEW
REVIEW 1111ofof1616

InInorder
In order
orderto totoanalyze
analyze
analyzethe the performance
theperformance
performanceof ofofthis
this MPC
thisMPC based
MPCbased control
basedcontrol method
controlmethod under
methodunder grid
undergrid transients,
transients,aaa
gridtransients,
group of simulations
groupofofsimulations
group simulationsare are conducted
areconducted
conductedon on the
onthe condition
thecondition of single-phase
conditionofofsingle-phase ground
single-phaseground fault.
groundfault. The
fault.The current
Thecurrent ofof
currentof
inverter
inverter are
inverterare shown
areshown
shown inininFigure
Figure
Figure 9.9.It
9.Itcan
Itcanbebe
can beseen
seen thethe
seen the amplitude
amplitude
amplitude ofofcurrent
current
of current changes
changes
changes during
during this this
during thisperiod
periodperiod
and
and
andits
itsTHD
THD increases.
increases. Although
Although the
the fault
fault has
has negative
negative effect
effectononthe
the
its THD increases. Although the fault has negative effect on the power quality, the current is underpower
power quality,
quality, the
the current
currentisis
under
control control
under and
control and
return return
andtoreturn to
stableto stable
stable
state afterstate
stateafter
the after the
fault the fault is eliminated.
fault is eliminated.
is eliminated. The performance
The performance
The performance of the proposed
of the proposed
of the proposed method
method
method isissimilar
similar toto that
that of
ofother
other control
control method,
is similar to that of other control method, such as PI control. method, such
such asasPI
PIcontrol.
control.

Figure Current
Figure9.9.Current performance
Currentperformance under
performanceunder single-phase
undersingle-phase ground
single-phaseground fault
groundfault
fault(a)
(a)current
currentwaveform (b)THD
waveform(b) THDofof
current within the fault period.
current within the fault period.

In
InInorder
orderto
order totoverify
verifythe
verify theeffectiveness
the effectivenessof
effectiveness ofofthe
theproposal
the proposalin
proposal ininpractical
practicalapplications
practical applicationsand
applications andcompare
and comparethe
compare the
the
performances
performances with other traditional control methods, a series of experiments are conducted inaaa
performances with
with other
other traditional
traditional control
control methods,
methods, aa series
series of
of experiments
experiments are
are conducted
conducted in
in
microgrid
microgridlaboratory,
microgrid laboratory,
laboratory, as shown
asasshown
shown ininin
Figure
Figure10.
Figure 10.Parameters
10.Parameters
Parameters ofofthe
ofthecontrolled
the inverter
controlled
controlled inverterand
inverter andother
and devices
other
otherdevicesof
devices
the experimental
ofofthe
theexperimental system
experimentalsystem are
systemare shown
areshown as Table
shownasasTable 1. The
Table1.1.Theexperimental
Theexperimental result
experimentalresult is collected
resultisiscollected by
collectedby ScopeCorder
byScopeCorder
ScopeCorder
DL850E,
DL850E, which combines a mixed oscilloscope and data acquisition recorder.
DL850E, which combines a mixed oscilloscope and data acquisition recorder. In thisway,
which combines a mixed oscilloscope and data acquisition In this
recorder. way,
In the
this collected
way,the
the
data can
collected be
collecteddata inputted
datacancanbe into a
beinputtedcomputer
inputtedinto and processed.
intoaacomputer
computerandandprocessed.
processed.

Figure
Figure10.
10.Microgrid
Microgridlaboratory.
laboratory.
Appl. Sci. 2019, 9, 5413 11 of 15
Appl.Sci.
Appl. Sci. 2019,
2019,9,
9, xx FOR
FOR PEER
PEER REVIEW
REVIEW 12 of
12 of 16
16

Firstly, a group
Firstly, of ofexperiments
experimentsare areconducted
conductedaiming to show validityof ofthe
themodel
modelparameters
parameters
Firstly, aa group
group of experiments are conducted aiming to
aiming to show
show validity
validity of the model parameters
modification process
modification process in
process in the
in the proposal.
the proposal.
proposal. AtAt the
At the beginning
the beginning
beginning the the predicted
the predicted parameters
predicted parameters
parameters are are given
are given
given as asas L= 0.80.8
modification LL == 0.8
mHmH andandR =R3= Ω, and reference
referenceRMS RMSof ofoutput
outputcurrent is 10 A. The
The modification
modificationactionactionisisperformed
performed
mH and R = 33 Ω,Ω, and
and reference RMS of output current is
current is 10
10 A.A. The modification action is performed
once every
once every 0.1
every 0.1 s. Recorded
0.1 s.s. Recorded output
Recorded output current
output current during
current during
during the the modification
the modification process
modification process
process is is shown
is shown
shown in in Figure
in Figure
Figure 11a, 11a,
11a,
once
while its RMS
while its
while its RMS waveform
RMS waveform
waveform is is shown
is shown
shown inin Figure
in Figure 11b,
Figure 11b, which
11b, which
which is is calculated
is calculated
calculated byby computer.
by computer. It
computer. ItIt cancan
can be be
be seenseen
seen the the
the
accuracy
accuracy
accuracy of of
RMS
of RMS
RMS ofof the
of output
the
the outputcurrent
output currentisis
current iscontinuously
continuously improved
continuously improved with
improved with themodification.
with the
the modification.When
modification. When
When the
the
the
error between
error between reference
reference andand collected
collected current
current is
is less
less than
than the threshold,
threshold, the
the
error between reference and collected current is less than the threshold, the modification is stopped, modification
modification is is stopped,
stopped,
and
andand
willwill not
notnot
will bebebe started
started
started until
until
until thetheerror
the errorisis
error islarger
largerthan
larger thanthe
than thethreshold.
the threshold.
threshold.

Figure
Figure
Figure 11.
11.11. Experimental
Experimental
Experimental results
results for
forfor
results model
model parameters
parameters
model modificationprocess
modification
parameters modification processofof
process ofthe
theproposed
the proposedmethod
proposed method(a)
method
(a)
Current Current waveform,
waveform,
(a) Current (b) current
(b) current
waveform, RMSRMS
(b) current RMS waveform.
waveform.
waveform.

In In
this system,
In this
this system,
system,
the control
the
the controlfrequency
control frequency666kHz
frequency kHzis
kHz isrelatively
is
low because
relatively low
relatively
becauseof
because ofthe
of thelimit
the limitby
limit byby the
the
the
switching
switching
switching
frequency of
frequency of
frequency inverter.
of inverter.In
inverter. Inorder
In order to
order to verify
to verify the
verify the effectiveness
the effectiveness of the proposed
effectiveness of the proposed
proposed A-MPCA-MPC
A-MPC based based method
based method
method ininin
such
sucha practical
such application,
aa practical
practical application,
application, thethe
steady
the state
steady
steady performance
state
state performance
performance isisanalyzed.
is analyzed.When
analyzed. Whenthe
When theinverter
the inverteris
inverter isoperated
is operatedin
operated
in
a stablea stable
state, state,
the the
currentcurrent
waveformwaveform
and and
its THDits THD
are are
shown shown
in in
FigureFigure
in a stable state, the current waveform and its THD are shown in Figure 12. The THD 4.70%,12. The 12. The
THD THD
4.70%, 4.70%, can
can satisfy
can
satisfy
thesatisfy
current the current
theharmonics harmonics
requirement
current harmonics requirement
of IEEE of
requirement of IEEE
standard standard
IEEE1547
IEEE standard IEEE1547 [26].
[26]. [26].
IEEE1547

Output Current
Current [200ms/div]
[200ms/div]
Output

[5ms/div]
[5ms/div]

Current THD=4.70%
Current THD=4.70%

Figure
Figure
Figure 12.
12.12. Experimentalresults
Experimental
Experimental resultsfor
results forsteady-state
for steady-statecurrent
steady-state current of
of the
of the proposed
the proposed A-MPC
proposedA-MPC based
A-MPCbased method.
basedmethod.
method.
Appl. Sci. 2019, 9, 5413 12 of 15

Appl. Sci.
Appl. Sci. 2019,
2019, 9,
9, xx FOR
FOR PEER
PEER REVIEW
REVIEW 13 of
13 of 16
16

ToTo analyze dynamic

analyze dynamic
performance
dynamic performance
performance of
of the
of the
proposal,
the proposal,
anothergroup
proposal, another
groupofofexperiments
another group
experiments is carried
is carried
carried out.
out.
out.
To analyze of experiments is
During the
During the experiment,
the experiment, the
experiment, the reference
the reference current
reference current RMS
current RMS changed
RMS changed from
changed from
from 5A5A
5A toto
to 10A10A
10A atat some
at some moment.
some moment.
moment. This This
This
During
test is carried
test is
is carriedout
carried outin
out inorder
in orderto
order to compare
to compare the
compare the response
the response speed
response speed
speed andand output
and output current
output current overshoot
current overshoot
overshoot under under
under stepstep
step
test
change. When
change. When the
When the inverter
the inverteris
inverter isoperated
is operated under
operated under
under PIPI control,
PI control, the
control, the current
the current waveform
current waveform
waveform and and
and itsits
its RMSRMS waveform
RMS waveform
waveform
change.
areare
presented
presented inin
Figures
Figure 13a
13a and
and 13b respectively
Figure 13b respectively
are presented in Figure 13a and Figure 13b respectively

Figure
Figure 13. Experimental
13.
Figure 13. Experimental results
Experimentalresults forfor
results
for stepstep
step current of PI
PI control
current
current of control (a) real-time
real-time
of PI control
(a) current waveform,
(a) real-time
current waveform, (b) RMS
RMS
current waveform,
(b)
current.
(b)current.
RMS current.

When
When
When the
thethe inverter
inverter is operated
operated
is operated
inverter is under
under PR control,
control,
PR control,
under PR thecurrent
the step
the step current
step current waveform
waveform and its
and its and
waveform RMS its RMS
waveform
RMS
waveform
arewaveform are
presentedare presented
inpresented in
Figure 14a,b Figure 14a,b respectively.
respectively.
in Figure 14a,b respectively.

Figure
Figure
Figure 14. Experimental
14.
14. Experimental results
Experimentalresults forfor
for
results stepstep
step current
current of PR
of
currentPR of
control
control (a) real-time
(a) real-time
PR control current waveform,
current
(a) real-timewaveform, (b) RMS
(b) RMS
current waveform,
current.
(b)current.
RMS current.
Appl. Sci. 2019, 9, 5413 13 of 15
Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 16

The response of the proposed A-MPC based control method to the same change is shown in
Figure 15a,b.
15a,b. Since
Sincethe
thetesting
testingtime
time
is is long,
long, above
above waveform,
waveform, which
which spans
spans 2 s, is2 just
s, isajust
partaofpart of the
the whole
whole
currentcurrent
waves.waves.
That isThat
whyisthe
why the
step step change
change time intime
theseintwo
these two figures
figures are different.
are different.

Figure 15. Experimental

Experimental results
results for step current of the proposed A-MPC control method (a) real-time
current waveform, (b) RMS current.

As shown
As shown inin the
the figures,
figures, the
the response
response speed
speed of
of the
the proposal
proposal to
to step
step change
change isis much
much higher
higher than
than
that of PI and PR control. At the same time, the current overshoot is also reduced substantially
that of PI and PR control. At the same time, the current overshoot is also reduced substantially in this in this
A-MPC method. Generally,
A-MPC method. Generally, in
in many
many of of traditional
traditional control
control methods,
methods, suchsuch as
as PI
PI and
and PR
PR control,
control, the
the
response speed and overshoot are two conflicting goals when designing control parameters.
response speed and overshoot are two conflicting goals when designing control parameters. By using By using
the proposal
the proposalinincontrolling an an
controlling inverter, bothboth
inverter, thesethese
performances are improved.
performances Therefore,
are improved. this method
Therefore, this
has significant
method advantages
has significant in applications
advantages with frequent
in applications withchange
frequentof current,
change such as photovoltaic
of current, such as
inverters andinverters
photovoltaic energy storage converters.
and energy storage converters.

5. Conclusions
5. Conclusions
In this
In this paper,
paper,ananA-MPC
A-MPCbased based current
current control
control for for three-phase
three-phase converters
converters is proposed.
is proposed. This
This approach takes use of inverter mathematical model in two-phase rotating
approach takes use of inverter mathematical model in two-phase rotating coordinate system coordinate system to
to
calculate the
calculate the optimal
optimal switching
switching state
state in
in each
each cycle. Comparing to
cycle. Comparing to existing
existing FCS-MPC
FCS-MPC based
based inverter
inverter
control methods, the proposed method reduces the computational burden and
control methods, the proposed method reduces the computational burden and control complexity, control complexity,
and improves
and improves power
power quality.
quality. A
A parameter
parameter estimation
estimation and
and adaptive
adaptive control
control method
method isis also
also integrated
integrated
into the
into the proposed
proposed method.
method. TheThe overall
overall control
control scheme
scheme isis straightforward
straightforward and and simple
simple toto implement.
implement.
Experimental results are presented to validate the improvement of dynamic
Experimental results are presented to validate the improvement of dynamic performances. performances. Comparing
to conventional PI and PR control, the response speed of the A-MPC
Comparing to conventional PI and PR control, the response speed of the A-MPC is fasteris faster and the overshoot
and the
is smaller. isFuture
overshoot research
smaller. Futureaims to apply
research aimsthe method
to apply thetomethod
other converter topologies,
to other converter e.g., modular
topologies, e.g.,
multi-level
modular converters,
multi-level etc.
converters, etc.
Author Contributions: Conceptualization, M.L.; methodology, M.L. and L.M.; software, Z.Y.; validation, M.L., Y.Z.
Author Contributions:
and D.C.; formal analysis,Conceptualization,
Z.Y.; investigation,M.L.;
L.M.;methodology,
resources, M.L.M.L.
andand
Z.Y.;L.M.; software,M.L.;
data curation, Z.Y.; writing—original
validation, M.L.,
Y.Z. and D.C.; formal analysis, Z.Y.; investigation, L.M.; resources, M.L. and Z.Y.; data curation,
draft preparation, M.L. and L.M.; writing—review and editing, M.L.; visualization, Y.Z.; supervision, M.L.; writing—
Y.W.; project
administration, Y.W.; funding acquisition, Y.W.
original draft preparation, M.L. and L.M.; writing—review and editing, M.L.; visualization, Y.Z.; supervision,
Y.W.; project
Funding: Thisadministration, Y.W..; funding
research was funded acquisition,
by Qinghai Y.W.
Science and Technology Project “Key technologies for renewable
energy and energy storage integration applications”: 2019-GX-A9
Funding: This research was funded by Qinghai Science and Technology Project “Key technologies for renewable
energy and energy storage integration applications”: 2019-GX-A9
Appl. Sci. 2019, 9, 5413 14 of 15

Conflicts of Interest: The authors declare no conflict of interest.

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