Model Predictive Control of Cascaded Multilevel
Battery Assisted Quasi Z-Source PV Inverter
with Reduced Computational Effort
Abderezak Lashab, Dezso Sera, Josep M. Guerrero
Department of Energy Technology, Aalborg University, Aalborg DK-9220, Denmark
abl@et.aau.dk, des@et.aau.dk, joz@et.aau.dk
Abstract—In cascaded multilevel quasi Z-source inverters
(CM-qZSI), the intermittent and stochastic fluctuation of the
solar power injected to the grid can be smoothened by
connecting a battery in parallel with one of the qZ network
capacitors. However, since CM-qZSI is a special case of the
single phase qZSI, it suffers from some of its demerits, such as
double line-frequency ripple. This affects the battery current,
the second qZ network's inductor current, and the qZ network's
capacitor voltages. This issue can be alleviated either by
increasing the size of the passive elements or by developing
advanced control schemes, such as MPC. However, MPC is
computationally intensive, especially in multilevel converters. In
this paper, a low computational control method for PV-fed
battery assisted CM-qZSI is proposed, where proportionalresonant (PR) controller is introduced in the control algorithm.
The prediction is performed only for the dc side of the
converter; hence, the algorithm iterates for the shoot through
and zero states only. As shown in the obtained results, by using
the proposed method, the double line-frequency ripple can be
significantly reduced, while a less computational effort
compared to the conventional MPC is needed.
Keywords—Battery,
Computational
burden,
Grid
connected, Modular, MPC, MPPT, Photovoltaic, P&O, Znetwork.
I. INTRODUCTION
Nowadays, modular multilevel converters are the most
adopted category for high- and medium-voltage applications,
since a higher output voltage can be achieved by a series
connection of identical power modules, in which low voltage
semiconductor devices are used [1]-[2]. Modular multilevel
converter topologies have several advantages such as low
switching frequency, high output voltage quality allowing the
use of a much less sizable output filter, reduced dv/dt leading
to less electromagnetic interference (EMI), and fault
tolerance [3].
CM-qZSI presents many advantages over the conventional
multilevel converters when applied in photovoltaic (PV)
power systems. For instance, the CM-qZSI provides a
balanced dc-link voltages through its voltage boost ability, as
well as saving one-third modules [4]. The qZS-CMI allows
the integration of energy storage by adding a battery in each
module to limit the ramp rate of the solar injected power to
the grid, and to store the energy for other times.
A detailed controller parameter design for the three-phase
CM-qZSI is demonstrated in [4] by using Bode plots. The
authors in [5], proposed a control strategy for the three-phase
CM-qZSI based on a modified space vector modulation. In
XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE
[6], CM-qZSI has been used for improving the power quality
in a single phase system. A maximum power point tracking
(MPPT) algorithm based on automatic output power ratio
constraint is developed in [7], to deal with the problem of the
output current distortion in single phase CM-qZSI in case
power mismatch among the submodules. In [8], the cascaded
qZS inverter has been used in a different configuration―only
one leg is used in each power submodule, and the total
resulted dc-link voltage is inverted by using a low frequency
inverter.
On the CM-qZSI with integrated energy storage, only a
limited number of research papers have been introduced so
far [9], [10], and [12]. The first control scheme of the battery
assisted CM-qZSI has been introduced first in [9], where a
Proportional Integral (PI) controller is used for regulating the
PV voltage at its reference in each submodule, and one PR
controller is used for the regulation of the injected current to
the grid. The power delivered by each submodule is
determined by using an energy management system by taking
into account the extracted PV power and the state of charge
(SoC) of the battery. Fuzzy Logic based control strategy has
been proposed for the battery assisted CM-qZSI in [10]. In
[12], a SoC balancing algorithm is presented, the algorithm is
applied to CM-qZSI; however, it can be applied to any
cascaded multilevel converter topology.
CM-qZSI suffers from a double line-frequency ripple,
which is induced by the periodic instantaneous power
delivery to the grid. In case of battery integrated CM-qZSI,
this ripple appears in the battery current and voltage, which
increases its internal resistance and fades its capacity. A way
to mitigate this ripple is by using model predictive control
(MPC) [11]. MPC is a powerful approach, but since it
predicts the behavior of the controlled variable for all the
possible switching states at each sampling time, it is
computationally demanding.
In this paper, an MPC-based control strategy for the battery
assisted CM-qZSI is proposed. The double line-frequency
ripple in the battery current can be considerably mitigated,
where less computational effort compared to the conventional
MPC is applied.
II.
THE CONFIGURATION OF THE CASCADED MULTILEVEL
QUASI Z-SOURCE INVERTER WITH BATTERY STORAGE
An extended model of the PV fed Quasi Z-source
cascaded multilevel inverter is shown in Fig. 1, where the
second qZS network capacitor in each submodule is
paralleled to a battery. The qZS network is then connected to
the dc-link of the conventional H-bridge. In addition to the
two active states and the zero state of the H-bridge, a shoot
through state is also possible in the qZS power module as
shown in Fig. 2(c) [13]. The shoot through state consists of
gating both switches in the same leg simultaneously. The aim
of applying the later state is to boost the voltage at the dc-link
of the H-bridge. During the shoot-though state, the qZS
inductors charge at the expenses of the energy stored in the
capacitors and from the power source, which is a PV array in
this paper. As soon as the shoot through state ends, the energy
stored in the inductors charges the capacitors, thus boosting
their voltage.
The output voltage of the CM-qZSI as function of the dclink voltages and switching states can be written as
-
iL1 L1
C2
VC2 +
L2 iL2
-
iL1 L1
+
PV
vPV
-
CPV
+
vC1
C1
dig
dt
rf ig vg
(2)
where vg, ig, Lf, and rf are the grid voltage, the current injected
to the grid, the filter inductance, and the filter stray resistance.
In what follows it is assumed that the qZS network passive
vC1
-
C1
N
(b)
(a)
ib
-
iL1 L1
C2
VC2 +
L2 iL2
P
+
PV
CPV
vPV
-
+
vC1
C1
iPN
N
(c)
i 1
vout L f
PV
+
CPV
vPV
N
(1)
such as, vPNi is the voltage at the dc-link of the ith H-bridge, S1i
and S2i are the switching states of the first and second leg of
the
ith H-bridge respectively. The output voltage as a function of
the ac side parameters of the CM-qZSI can be assessed
through Kirchhoff’s voltage law as the following,
+
iPN
-
C2
+
VC2
L2 iL2 iPN
P
P
n
vout (t +Ts ) vPNi ( S1i S2i )
ib
ib
Fig. 2. PV fed Quasi Z-source submodule with integrated energy storage
during: (a) active states, (b) zero state, and (c) shoot through state.
elements in each submodule are symmetrical (C1=C2 and
L1=L2) and the total number of power modules is three.
Considering the qZS network’s inductors currents, the first
qZ network’s capacitor voltage, and the battery voltage as the
state variables, x=[iL1, iL2, vC1, vC2]t. The state space model of
each submodule during the active states can be found as
follows
x A1 x + B1 u
(3)
such as:
ib1
iL11L11
+
vPV1
PV1
CPV1
+
vC11
P1
S11
C11
-
-
rL 1
0
L
1
r
0 L2
L2
A1 =
1
0
C
1
1
0
C2
C21
VC21 +
L21iL21 iPN1
-
S21
0
+ vH1 -
S31
N1
+
S41
iL12 L12
vPV2
PV2
CPV2
C22
ig
VC22 +
L21iL22 iPN2
S12
+
vC12
C12
-
-
+ vH2 S32
N2
rf
S22
vout
S42
Lf
vg
ibn
iL1n L1n
+
vPVn
PVn
-
CPVn
C2n
+
VC2n
L2niL2n iPNn
Pn
S1n
+
vC1n
-
0
0
0
1
0
0
L
1
1
0
0
0
L2
;
1 ;
B1 = 0
0
0
C1
1
1
0
0
C2
C 2
0
t
P2
+
1
L1
u = vPV ib iPN
ib2
-
C1n
S2n
+ vHn -
S3n
Nn
-
S4n
Fig. 1. PV fed Quasi Z-source cascaded multilevel inverter with integrated
energy storage.
From Fig. 2(b), the inductor voltages during the zero state
are the same as in the active state; however, the capacitors
currents change. The state space model can be expressed as:
x A0 x + B0 u
(4)
where:
rL 1
0
L
1
r
0 L2
L2
A0 =
1
0
C
1
0 1
C2
1
L1
0
0
0
1
0
L
1
1
0
L2 B = 0
; 0
0
0
0
1
0
C2
0
0
0
0
;
0
0
The state space model of each submodule during the shoot
through state can be found similarly as follows
(5)
x A st x + B st u
where:
A st
=
rL 1
L1
0
0
rL 2 1
L2 L2
1
0
0
C1
1
0
0
C2
0
1
1
0
L1
L
1
0
0
; B = 0
st
0
0
0
1
0
C2
0
IV. PROPOSED MPC FOR THE CASCADED MULTILEVEL QUASI
Z-SOURCE INVERTER WITH BATTERY STORAGE
0
0
;
0
0
The average state space model of each qZS submodule can
be obtained through the average state approach as
(6)
x A x + B u
such as
Di 1 Di
rL1
1
0
0
0
L
L
L1
L1
1
1
Di Di 1
r
L2
0
0
0 0
L2
L2
L2
A =
; B = 0 0 U i ;
Di
2Ui Di 1
0
0
C1
C1
C1
2Ui Di 1
Di
0 1 Ui
0
0
C2
C2
C2
C 2
where Ui and Di are the active state and shoot through state
duty cycles in the ith submodule, respectively.
III. CONVENTIONAL CONTROL OF CASCADED MULTILEVEL
QUASI Z-SOURCE INVERTER WITH BATTERY STORAGE
The schematic of the conventional control of the battery
assisted CM-qZSI for PV systems is shown in Fig. 3. The
injected power from each submodule is determined based on
the harvested power from the PV array and an energy
management system (EMS) through its respective switching
function (Ui). The switching function that is fed to the EMS
is the output of a PR controller, which is used to regulate the
grid current to its reference. The harvested power from the
PV array in the ith cell is maximized by using a local MPPT.
The error between the reference provided by the MPPT and
the PV voltage/current is minimized by adjusting the shoot
through duty cycle in the corresponding submodule Di using
a PI controller [9].
vPV1
iPV 2
vPV 2
MPPT
MPPT
ref
vPV
1
ref
vPV
2
D1
PI
S11
S12
S13
S14
vPV1
PI
D2
PI
D3
SoC1
vPV 3
P ref
vg
iˆg
2
PLL
igref
PR
vˆg
sin
ig
Pb1
SoC2
Pb2
SoC3
Pb3
S 22
S23
S 24
EMS
MPPT
S 21
U1
EMS
vPV 3
ref
vPV
3
U2
S31
EMS
vPV2
iPV3
PS-CBPWM
iPV 1
U3
S33
S32
S34
Fig. 3. Schematic of the conventional control of the battery assisted CMqZSI for PV systems.
MPC in power electronics has several advantages, such
as: easy implementation, constraints inclusion in nonlinear
systems, and high accuracy and fast dynamic response [14][16]. A thorough analysis of MPC approach and its
characteristics can be found in [16]. The most well-known
class of MPC in power electronics is finite control set (FCS)
MPC. FCS-MPC is summarized in the following tasks:
1- References estimation.
2- Prediction model.
3- Cost function optimization.
4- Gating signals application.
However, since FCS-MPC predicts the future conduct of
the controlled variable for all the possible switching states in
each sampling time, it is computationally demanding [17].
Hence, a control algorithm with low computational demand
for CM-qZSI has been proposed in this paper. In in the
proposed control strategy, the grid current is controlled
through a PR controller, while the PV and battery currents are
controlled through MPC.
A) References estimation:
Since the converter used here is fed by PV panels, local
maximum power point trackers are indispensable for making
the PV panels provide their maximum available power
continuously. The MPPTs cater the PV current/voltage
references that should be set for such an operating condition.
In this paper, the classical Perturb & Observe (P&O) is
adopted.
The power delivered by each submodule is determined
based on the energy management of the system. The latter
considers the power required by the grid, the power
withdrawn from the PV panels, and the SOC of the batteries.
Hence, the references are going to be the PV current, the
ref
power injected to the grid Pref and the battery current ibi . The
battery current in each submodule can be estimated as:
ibrefi
Pbi
vC 2
(7)
where Pbi is the battery exchanged power in the ith
submodule, which can be determined as follows
(8)
Pbi = Pi ref Ppv i
such as, Ppvi and Pi are the power harvested and power
reference in the ith submodule, respectively. The total active
power reference is considered to be divided equally among
the total number of submodules in order to avoid the
distortion of the current injected to the grid.
ref
B) Prediction model:
The flowchart of the shoot through state generation in the
proposed method is shown in Fig. 4. In order to control the
PV and battery currents through MPC, the model of the dcside of the system needs to be derived. This model can be
obtained from the continuous-time equations of the inverter
presented in the second section.
Since the average current going through the PV capacitor
is zero, the PV current is estimated as the qZS network first
inductor’s current. Both models of the system during zero
state and shoot through state need to be taken into account.
2. Shoot through state, by substituting (9) into the qZS
network first inductor’s current during the shoot through state
(5), the predicted PV current can be found similarly as
follows,
Start
Initialisations
g
; :: t k _ 0
iPVi (t +Ts ) iPVi (t ) +
Measurements
No
Apply
SST1 0
Ts
vpvi (t ) iPVi (t )rL1 vC 2i (t )
L1
S11=0 & S12=0
ibi (t Ts ) iL 2 i (t )
Yes
for i=0 to 1
if i=1
No
Predictive equations Shoot through
state
Predictive equations Zero state
(10) ; (12);
(13) ; (14);
No
References
calculations
if i=1
Ts
vC 1i iPVi (t Ts ) (14)
L2
C) Cost function optimization:
Among the advantages of MPC is its multivariable
inclusion. All the controlled variables can be included in a
single cost function, which in this paper are, the PV current,
and the battery current. Hence, the cost function can be
defined as,
g pv g pv b gb
(15)
where λpv and λb, are the weighting factors for PV current
term, and the battery current. Their corresponding terms are
written as follows,
Yes
Cost function evaluation
ref
g1 i = pv i pv1
(t +Ts )- iL11i (t +Ts ) + b ib1ref (t +Ts )- ib1i (t +Ts ) ;
ref
g pv ipv
(t Ts ) ipv (t Ts ) ; gb ibref (t Ts ) ib (t Ts ) ;
;
First submodule
(13)
Then, the battery’s predicted current in each submodule can
be assessed as,
if
Yes
Apply
SST1 = Smin ( g1i )i=0,..,1 ;
D) Gating signals application:
The aforementioned cost function is calculated for both
zero state and shoot through state and the switching state that
minimizes the cost function should lead to the desired control
objective. The prediction algorithm runs only if the current
state is zero (S1i=0 & S2i=0), and if the minimized cost
function corresponds to zero state then the system will
continue in a zero state, otherwise, a shoot through state will
be applied. The shoot through state is decided as a high logic
value using the variable SST.
Second submodule
Apply
SST2 = Smin ( g 2j ) j=0,..,1 ;
Third submodule
Apply
SST3 = Smin ( g 3k )k=0,..,1 ;
The grid current is controlled through a PR controller
whose transfer function is expressed as:
where x(t+Ts) and Ts are the estimated variable for the next
sampling instant and the sampling time, respectively. By
substituting (9) into the qZS network first inductor’s current
during the zero state (3), the predicted PV current can be
found as follows:
T
iPVi (t +Ts ) iPVi (t ) + s v pvi (t ) iPVi (t )rL1 vC1i (t )
L1
(10)
The battery current is estimated as the difference between the
qZS network’s currents,
ibi iL 2i iL1i
(11)
Hence, the predicted battery current can be estimated as
follows
ibi (t Ts ) iL 2 i (t )
Ts
vC 2 i iPVi (t Ts )
L2
(12)
(16)
where is the grid frequency, and k is the resonant gain. This
transfer function can be discretized by using the Trapezoidal
approach as the following [20]:
x xz 2
(17)
H ( z)
x y 4 2 4 y z 1 x y 4 z 2
ref
ig
PR
ig
SoC 1
Pb1
SoC 2
Pb2
SoC 3
Pb3
U1
U2
U3
S ST1
S ST2
S ST3
S11
S 21
S 31
S 41
PS-PWM
(9)
k s
s k s 2
2
EMS
d x ( t ) x ( t + Ts ) - x ( t )
dt
Ts
H (s)
EMS
1. Zero state, for deriving the predicted variables, usually,
the approach of forward-difference Euler is used,
EMS
Fig. 4. Shoot through state generation in the proposed controller.
S 12
S 22
S 32
S 42
S13
S23
S 33
S 43
Fig. 6. Grid current regulation and switching state generation in the proposed
controller.
TABLE I.
A COMPARAISON OF THE PROPOSED CONTROLLER WITH FCS-MPC IN TERMS
OF THE NUMBER OF MATHEMATICAL OPERATIONS
Method
FCS-MPC
Proposed
controller
Per iteration
27 iterations
Per cell
Prediction
Per
model
iteration
2 iterations
PR
+&
18
486
11
33
×
11
297
9
27
÷
3
81
2
6
66
9
75
54
14
68
12
1
13
Fig. 7. The harvested PV power from the three cells by using the
conventional PI linear controllers.
where x =2kTs and y=(Ts)2. The grid current regulation
and switching state generation in the proposed controller are
sketched in Fig. 6.
V. COMPUTATIONAL EFFORT IN THE PROPOSED CONTROLLER
Table I shows a comparison between the proposed MPC
controller and FCS-MPC in terms of the number of
mathematical operations. The numbers in this table have been
estimated as follows:
Proposed method: The equations used in this method are:
-(10), (12), (13), and (14), which are for the PV and battery
currents control;
-(17) is employed for the grid current control;
-A two terms cost function (15) for the optimization process.
FCS-MPC: This method uses:
-(10), (12), (13), and (14) for the PV and battery current
regulation;
-(1) and (2) in its discretized form for the grid current
control;
-A three terms cost function.
As it can be seen from this table, the number of
mathematical operation in the proposed method is 75
subtractions and addictions, 68 multiplications, and 13
divisions. Whereas, FCS-MPC algorithm performs 486
subtractions and addictions, 297 multiplications, and 81
divisions considering 27 iterations.
As it can be seen from Table I, the proposed method
performs around 85% fewer mathematical operations with
respect to the classical FCS-MPC in each sampling period.
VI. SIMULATION RESULTS AND DISCUSSION
A detailed simulation model of CM-qZS inverter
conformable to the schematic shown in Fig. 1 has been
developed for validating the theoretical analysis. The CMqZS is fed by PV panels whose specifications were taken
from the back of a real PV panel in the Lab (Universal Solar
WXS230P-US). Under the standard test conditions (STC),
these PV panels have: PMPP=230W, IMPP=7.52A, Isc=8.56A,
and voc=36.9V. To reach the desired voltage level, three PV
panels were connected in series in each power module. The
batteries were a lithium-ion type with a capacity of 6Ah and
24V nominal voltage. The qZ network passive elements were
selected to be: L1=L2=0.6mH and C1=C2=2000µF. The filter
inductor and stray resistance were 10mH and 0.1Ω,
respectively. The switching frequency was fsw=5kHz. The
MPPTs were the conventional P&O [18], with a frequency
Fig. 8. Simulation results of the CM qZSI when the conventional linear PI
controller is used, (a) the first qZ networks’ inductor currents, (b) the second
qZ networks’ inductor currents, (c) the batteries current, (d) the output
voltage, and (e) the current injected to the grid.
and step size of 10Hz and Δv=0.5V (for the current
ΔI=0.075A), respectively.
For the sake of comparison, the classical control of the
CM-qZSI has been also tested and with the same parameters.
The system has been tested under the same and different
weather conditions among the power modules. The CM-qZSI
starts with an average solar irradiance in all submodules and
after 1.5s from the starting of the test, the irradiance increases
in the first submodule and decreases in the third one. The
irradiance in the second power module remains the same
during the whole test. The results shown in Fig. 7 and Fig. 8
correspond to the conventional control of the CM-qZSI with
linear PI controllers, whereas the ones shown in Fig. 9. and
Fig 10 were obtained by the proposed control method.
As it can be seen from Fig. 8(c), the battery current in the
conventional controller suffers from a double line-frequency
ripple. Its value was measured as Δib=3.39A. In contrast, in
the proposed method this ripple has been reduced by almost
three times (Δib=1.15A) since the battery current is directly
controlled. One should note, that the double line-frequency
ripple has been reduced at the expenses of the second qZ
[2]
[3]
Fig. 9. The harvested PV power from the three cells by using the proposed
MPC controller.
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
Fig 10. Simulation results of the CM qZSI when the proposed MPC
controller is used, (a) the first qZ networks’ inductor currents, (b) the second
qZ networks’ inductor currents, (c) the batteries current, (d) the output
voltage, and (e) the current injected to the grid.
network’s inductor current iL2 as well as the first qZ network’s
capacitor voltage.
VII. CONCLUSION
A model predictive-based control method for PV fed CMqZSI with integrated energy storage has been proposed in this
paper. The double line-frequency ripple in the battery current
of CM-qZS inverter is a critical issue since it affects the
battery life time. The battery current can be smoothened by
using MPC; however, MPC is computationally demanding.
In this paper, an MPC-based control strategy has been
presented, where decreasing the battery current ripple as well
as the computational burden were the main targets. As it has
been shown in the obtained results, the battery current has
been considerably reduced, while around 85% less processing
power was applied compared to the conventional MPC.
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