IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS
Model Predictive Control for Power Converters
and Drives: Advance s and Trends
Sergio Vazquez, Senior Member, IEEE, Jose Rodriguez, Fellow, IEEE, Marco Rivera, Member, IEEE,
Leopoldo G. Franquelo, Fellow, IEEE and Margarita No rambuena, Member, IEEE
Abstract —Model Predictive Control (MPC) is a very at-
tractive solution for controlling power electronic convert-
ers. The aim of this paper is to present and discuss the lat-
est developments in MPC for power converters and drives,
describing the current state of this c ontrol strat egy and
analyzing the new trends and challenges it presents when
applied to power electronic systems. The paper revisits the
operating princ iple of MPC and identifies three key ele-
ments in the MPC strategies, namely th e prediction model,
the cost function and the optimization algorithm. The paper
summarizes the most re cent research concerning these
elements , providing details about the different solutions
proposed by the academic and industrial communities.
I. INTRODUCTION
M
ODEL Predictive Control (MPC) has b een a topic of
research and development for more than three de cades.
Originally, it was introduced in the process indu stry, but
a very innovative and early paper proposed that pr edictive
control be used in power electronics [1]. In the recent years,
thanks to technological advances in microprocessors, it has
been proposed and studied as a promising alternative for
the control o f power converters and drives [2], [3]. MPC
presents several advantages. For instance, it can be used in
a variety of processes, is simple to apply in multivariable
systems and presents a fast dynamic response. Further, it
allows for nonlinearities and co nstraints to be incorporated
into the con trol law in a straightforward manner, and it can
incorporate nested control loops in only one loop [4], [5].
In particular, power electronic applicatio ns require control
responses in the order of tens to hund reds of microseconds
to work properly. However, it is well known that MPC has
Manuscript received April 11, 2016; revised July 17, 2 0 1 6 and
September 1 0, 2016; accepted October 4, 2016. This work was sup-
ported by the Ministerio Espa
˜
nol de Economia y Competitividad under
project TEC2016-78430-R, the Consejeri a de Innovacion Ciencia y
Empresa (Junta de Andalucia ) und e r the project P11-TIC-7070, the
Advanced Center for Electrical and Electronic Engineering, AC3E, Basal
Project FB0008, CONICYT and by the F ONDECYT Regular 1160690.
S. Vazquez is with the Electronic Engineering Department, Universi-
dad de Sevilla (Spain), (e-mail: sergi@us.es).
L. G. Franquelo is with the Electronic Engineering Department, Uni-
versidad de Sevilla (Spain) and Research Institute of Intelligent Control
and Systems, Harbin Institute of Technology, Harbin, 150001, P. R.
China (e-mail: lgfranquelo@ieee.org).
Jose Rodriguez an d Margarita Norambuena are with Universi-
dad Andres Bello in Santiago, Chile (jose.rodriguez@unab.cl; mar-
garita.norambuena@gmail.com).
Marco Rivera is with Department of Electrical Engineering at the
Universidad de Talca, Chile (marcoesteban@gmail.com).
Prediction
Model
Minimization
of cost
function
Fig. 1. Block diagram of a basic MPC strategy applied for the current
control in a VSI with output RL load.
Model Predictive Control
(MPC)
Continuous Control Set MPC
(CCS-MPC)
Optimal Switching Vector MPC
(OSV-MPC)
Generalized Predictive Control
(GPC)
Explicit MPC
(EMPC)
Optimal Switching Sequence MPC
(OSS-MPC)
Finite Control Set MPC
(FCS-MPC)
Fig. 2. Classification of MPC strategies applied to power conve rters and
drives.
a larger computational burden th a n other control strategies.
For this reason, most of the works focu sed on this issue
at the initial research stages of MPC for power elec tronic
systems [6]. Currently, MPC approaches can be found in
the literature for almost all power electronic applications [7].
The main re ason is that the compu ta tional power of modern
micropr ocessors has dramatically increased. This has ma de it
possible to imp le ment more complex and intelligent control
strategies, like MPC, in standard control hardware platforms
[8]–[11]. At this p oint, MPC for power converters and drives
can be considered as a well established technology in the
research a nd development stages. However, further research
and development efforts are still necessary in orde r to bring
this technology to the industrial and co mmercial level [12].
The aim of this paper is to summarize the current state and
analyze the most recent ad vances in the application of MPC
for power converters and drives. Thus, the work presents the
current advances and ch allenges of MPC for power electronic
applications and addre sses possible future trends.
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TABLE I
MOST USED MPC STRATEGIES FOR POWER ELECTRONICS APLICATIONS
Item Description GPC EMPC OSV-MPC OSS-MPC
Block diagram
Predictive
Model
Optimization
Load
Optimization
(Parametric
Search)
Load
Predictive
Model
Optimization
Load
Predictive
Model
Optimization
Load
Modulator SVM or PWM SVM or PWM Not required Not required
Fixed switching frequency Yes Yes No Yes
Optimization Online
Offline
(Parametric search)
Online Online
Constraints
Can be included
but increases the
computational cost
yes yes yes
Long prediction horizon Yes Yes
Can be used
but requires special
search algorithm
Can be used
but requires special
search algorithm
Formulation Complex Complex Very intuitive Intuitive
References [13], [14] [15], [16] [17], [18] [19], [20]
II. MODEL PREDICTIVE CONTR OL: OPERATING
PRINCIPLE
MPC is a family of controllers that explicitly uses the model
of the system to be controlled. In gener a l, MPC defines the
control action by m inimizing a cost function that describes
the desired system behavior. This cost function compares
the predicted system output w ith a reference. The predicted
outputs are computed from the system mode l. In genera l, for
each sampling time, the MPC controller calculates a control
action sequence that minimiz e s the cost function, but o nly
the first ele ment of this sequence is applied to the system.
Although MPC controllers solve an open-loop optimal contr ol
problem, the MPC algor ithm is repeated in a receding horizon
fashion at every sampling time, thus providing a feedback loop
and potential robustness with respect to system uncertainties.
To illustrate the use of MPC for power e le ctronics, a basic
MPC strategy with a pre diction horizon equ a l to 1 applied
to th e current control of a voltage source inverter (VSI) with
output RL load is shown [17]. The basic block diagram of th is
control strategy is presented in Fig. 1, where the reference
and pred ic ted currents at instant k + 2 are used in order to
compen sate for the digital impleme ntation delay [21]. The
algorithm is repeated for each samp ling time and performs
the following steps:
1) The optimal control action S(t
k
) computed at in stant k−1
is applied to the co nverter.
2) Measurement of the current i
k
is taken at instant k. The
referenc e current i
∗
k+2
for instant k + 2 is also defined.
3) The prediction model of the system is used to make a
prediction of the current value
ˆ
i
k+2
at instant k + 2.
4) A cost function is evaluated using i
∗
k+2
and
ˆ
i
k+2
. The
optimal contro l action S(t
k+1
) to be applied at instant k+
1 is chosen as the one that minimizes the cost function’s
value.
Several MPC methods have be en successfully implemented
for a variety of power electr onic applica tions [6], [7]. Fig. 2
shows th e most co mmon MPC strategies applied to power
converters and drives, and Table I summarizes the structure
and main featur es of these MPC strategies. Variables i,
ˆ
i
and i
∗
denote a set of curren t measu rements, predictions and
referenc e s. u
k
is the control signal calculated at instant k and
S
k
(t) are the firing pulses for the power switches, these values
can change from instant k to k + 1. S(t
k
) are the firing pulses
for the power switches, these values are constant from instant
k to k + 1.
The M PC methods are classified ba sed on the type o f the
optimization problem, i.e., if it is an integer optimization
problem or not. On one hand, Continuous Control Set MPC
(CCS-MPC) computes a continuous control signal and then
uses a modulator to generate the desired output voltage in
the power converter. The modulation strategy can be any one
that is valid for the converter topology under consideration
[75]. The main advantage of CCS-MPC is that it produces a
fixed switching frequency. The most-used CCS-MPC strategies
for power ele ctronic applications are Generalized Pre dictive
Control (GPC) and Explicit M PC (EMPC). GPC is usefu l
for line ar and unconstrained problems. E MPC allows the
user to work with non-linear and constrained system s. The
main problem of GPC and EMPC wh en applied to power
converters is that both present a complex formulation of the
MPC problem. On the other ha nd, Finite Control Set MPC
(FCS-MPC) takes into accoun t the discrete nature of the
power converter to formulate the MPC algorithm and does not
require an external modulator. FCS-MPC can be divided into
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TABLE II
MPC FOR POWER ELECTRONICS APPLICATIONS
Application Basic Control Scheme Application Basic Control Scheme
CSC-AFE
[22]–[24]
Predictive
Model
Minimization
of cost
function
Reference
design
VSC-AFE
[17], [20], [25]
[26], [27], [28]
Predictive
Model
Minimization
of cost
function
Reference
design
Current source converter active front end (CSC-AFE) Voltage source converter active front end (VSC-AFE)
Motor drives
[29]–[38]
Predictive
Model
Minimization
of cost
function
Reference
design
VSC-UPS
[39]–[45]
Predictive
Model
Minimization
of cost
function
Reference
design
Load
Motor drives VSC Uninterruptible power supply (VSC-UPS)
Statcom
[46]–[53]
Predictive
Model
Minimization
of cost
function
Reference
design
Load
Matrix
Converter
[54]–[67]
Predictive
Model
Minimization
of cost
function
Reference
design
Static Compensator (STATCOM) Matrix converter
HVDC
[68]–[74]
Predictive
Model
Minimization
of cost
function
Predictive
Model
Minimization
of cost
function
Reference
design
Reference
design
High voltage DC transmission system (HVDC)
two types: Optimal Switching Vector MPC (OSV-MPC) and
Optimal Switching Sequence MPC (OSS-MPC). OSV-MPC is
currently the most popular MPC strategy for power electronic
applications. OSV-MPC was the first FCS-MPC technique
used f or power electronics. For this reason, it can be found
in the literature referred to as FCS-MPC. It uses the possible
output voltage vectors of the power converter as the con trol
set. OSV-MPC on ly calculates predictions for this control
set, and it reduces the optimal problem to an enumerated
search algorithm. This makes the MPC strategy formulatio n
very intuitive. The main disadvantage o f OSV-MPC is that
only one output voltage vector is applied during the complete
switching period. Furthermore, unless an additional constraint
is added, the same output voltage vector can be used during
several consecutive switching periods. Therefore , in general,
it generates a variable switching frequency. OSS-MPC solves
this problem by co nsidering a co ntrol set composed of a
limited number of possible switching sequences per switch ing
period. In this way, OSS-MPC takes the time into account as
an additiona l decision variable, i.e., the instant the switches
change state, which in a way resembles a modulator in th e
optimization problem.
In general, MPC algorithms require a significant amount of
computations. CCS-MPC usually has a lower computational
cost than FCS-MPC because it computes part or all of the
optimization problem offline. For th is reason, CCS-MPC can
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address long prediction horizon problems. For instance, GPC
uses an expression to c alculate the control action that can be
computed beforehand, thus limiting the online computation
burden [9]. On the other hand, EMPC computes and stores the
optimal pr oblem solution offline, so the online computations
are limited to a search alg orithm. By contrast, FCS-MPC
requires that the optimizatio n problem, which involves a large
amount of calculations, be solved online. For this r eason, FCS-
MPC is usually limited to short prediction horizons in power
electronic applications. Comparing OSS-MPC and OSV-MPC,
the former has a greater computational cost.
Table II summarizes the most relevant applications of MPC
for power converters and drives [7]. Other uses of MPC for
power electronics can be found in the literature. Among them
are predic tive control strategies for quasi z-source inverters or
dc/dc converters [76]–[79]. Ta ble II includes a block diagram
representin g the use of OSV-MPC for each one. Other MPC
strategies could be used for these applications, but the purpose
of the c ontrol scheme is to show the basic concept. Therefore,
OSV-MPC has been chosen for its clarity.
An analysis of MPC algorithms when applied to power
converters an d drives reveals that the key elements for any
MPC strategy are the prediction model, cost function and
optimization algorithm. Research efforts have been made in all
of these topics, and several problems and limitations have been
found. The existing research work h ave solved some of them
while others are still open issues to be investigated. Among
the most important studied aspects are [80]:
• Prediction model discretization.
• Frequency spectrum shaping.
• Cost function design.
• Reduction of computational co st.
• Increasing prediction and control horizon.
• Stability and system performance design.
The most recent resear ch for all of these topics will be
addressed in the following sections.
III. PREDICTION MODEL
MPC performance is influenced by an adequate quality of
the prediction model which depends on the specific application
under consideration [7]. For this reason, most power converters
are connecte d to the load through passive filters in order to
minimize the effects of the commutations or d isto rtions in
the supply. First-or der passive filters composed of an inductor
and its parasitic resistor can be used [20], [51]. However,
high order passive filters like LC or LCL are also applied
in VSC-AFE [15], [27], medium voltage (MV) motor drives
[81], VSC-UPS [39], [44], matrix converters [59], [61], etc.
MPC can work with any passive filter topology as long as its
mathematical model is incor porated in the prediction model.
Despite the fact that mathematical model of the filter is
included in the prediction model, basic MPC stra tegies must
mitigate the effects of resonance problems when a high-or der
passive filters are used. This is especially critical in FCS-MPC
due to the variable switching frequency (f
sw
) that is present in
this c ontrol strategy, even though f
sw
is limited to half of the
sampling frequency. Several solutions have been proposed to
TABLE III
EXAMPLE OF COST FUNCTION FOR POWER ELECTRONICS
APPLICATIONS
Application Cost function
CSC-AFE
[23]
[24]
g = |q| + λ|
ˆ
i
L
− i
∗
L
|
g = (q)
2
+ λ
ˆ
i
L
− i
∗
L
2
VSC-AFE
[17]
[88]
[89]
[20]
g = |
ˆ
i
k
− i
∗
k
|
g = |
ˆ
i
k
− i
∗
k
| + λ
n
n
c
g =
ˆ
i
k
− i
∗
k
2
g =
ˆ
P − P
∗
2
+
ˆ
Q − Q
∗
2
Motor drive
[36]
g =
ˆ
T − T
∗
2
+ λ
ˆ
ψ − ψ
∗
2
VSC-UPS
[39] g = (ˆv
o
− v
∗
o
)
2
Statcom
[50]
g =
ˆ
i
k
− i
∗
k
2
Matrix converter
[54]
[65]
g = |
ˆ
i
L
− i
∗
L
| + λ|
ˆ
Q − Q
∗
|
g =
ˆ
i
L
− i
∗
L
2
+ λ
ˆ
Q − Q
∗
2
HVDC
[69]
g = g
1
+ g
2
+ g
3
g
1
= |
ˆ
i
jk
− i
∗
jk
|
g
2
= λ
Ck
P
i
|
ˆ
V
cijk
−
V
dc
n
|
g
3
= λ
zk
|
ˆ
i
zjk
|
deal with this problem. For in stance, it is possible to mitigate
the resonance effects by considering a hybrid contro l strategy,
mixing pr e dictive co ntrol and an active damping filter [6 1],
[82], [83]. In addition, FCS-MPC can address the resonance
issues without requiring a passive/active damping loop by
increasing the pre diction horizon [81], [84]. On the other hand,
the design of the input filter ca n be simplified and the risk of
resonances avoided by considering MPC strategies with fixed
switching frequencies [15], [16], [27].
The MPC algorithms are usua lly imple mented in digital
hardware platforms like DSPs or FPGAs. For this reason,
the prediction model of the system needs to be discretized.
For linear sy stems, the discretization is simple and can be
done as described in [39], [80]. However, non-linear systems
require a more complex approach [85]. A trade-off between
the model quality and complexity defines several discretization
techniques, the most common being Euler approxima tion and
Taylor series expansion [86]. Another approach consists of
a first step where the system is discretized usin g a one-step
or multiple-step E uler approximatio n. Then, the arising dis-
cretization error is explicitly bound to take it in to consideration
for the implementation of the predic tive controller [87].
IV. COST FUNCTION ISSUES
The cost function in the MPC strategy defines the desire d
system behavior. For this purpose, it compa res the predicte d
and refere nce values. The cost function can have any form,
but in general, it can be written as
g =
k+N
p
X
ℓ=k+1
˜x
T
ℓ
Q˜x
ℓ
+
k+N
c
−1
X
r=k
u
T
r
Ru
r
(1)
where ˜x
ℓ
= ˆx
ℓ
− x
∗
ℓ
is a vector in which each component
represents th e difference between the predicted , ˆx
j,ℓ
, and th e
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referenc e , x
∗
j,ℓ
, va lues for any variable x
j
at instant ℓ, u
r
is a
vector of control inputs u
i
at in stant r, and N
p
and N
c
are the
prediction and control horizons, resp e ctively [5]. MPC allows
one to solve Multiple I nput and Multiple Output ( MIMO)
problems. Ther efore, ˜x
ℓ
∈ R
m
, u
r
∈ R
n
and Q ∈ R
mxm
,
R ∈ R
nxn
are matrices representing weighting factors. When
Q and R are diagonal, then (1) c an be expressed as
g =
k+N
p
X
ℓ=k+1
m−1
X
j=0
λ
j
ˆx
j,ℓ
− x
∗
j,ℓ
2
+
k+N
c
−1
X
r=k
n−1
X
i=0
λ
i
(u
i,r
)
2
(2)
where λ
j
and λ
i
are the weightin g factors associated to the
variable x
j
and control ac tion u
i
, respectively.
Although, (2) is used more frequently, both (1) and (2) are
valid expressions. Designing g is not an easy task. T he vari-
ables x
j
included in g depend on the application and choosing
the weig hting factors affects the system’s perform ance and
stability, it can therefore be seen as a tuning procedure. Both
issues have been studied by the research community and will
be addressed in the following section s.
A. Cost function Selection
MPC strategies solve an op timization problem in order to
define the control signal to be applied to the system. The
cost function represents the desired behavior for the system.
Therefore, MPC calculates the optimal a c tuation by minimiz-
ing it. A cost func tion can be complex depending on which
variables and contr ol o bjectives are considered. However, these
variables depend on ly on th e application under study. Table III
collects some cost functio ns found in the literature for power
electronic a pplications. Among them, it can be observed that
current, voltage, torque, power and other control objec tives
are considere d. Other obje c tives such as voltage, torque, spee d
and power ripple minimization can be achieved by includin g
specific variables in the cost fun ction [33], [93].
Choosing the cost function is not trivial even when only
one variable is controlled. For instance,
g = |
ˆ
i
L
− i
∗
L
| (3)
g =
ˆ
i
L
− i
∗
L
2
(4)
are both suitable for the current control of a VSC-AFE. Both
provide similar performance for the current tracking problem
when the cost function considers on ly one compone nt, like in
single-pha se power converters. However, when this cost func -
tion conside rs more than one te rm, like in thr ee-phase systems,
the actual output current i
L
presents different characteristics
such as harmonic spectrum, total harmonic distortion (THD),
root mean square (RMS) value, etc [80].
Selecting the right cost function is more difficult when
several control objectives are included in the optimization
problem. Contin uing w ith the current control of a VSC-AFE,
one can use
g = |
ˆ
i
L
− i
∗
L
| + λn
c
(5)
g =
ˆ
i
L
− i
∗
L
2
+ λn
c
(6)
to track a current refer ence and limit the number of commu ta-
tions n
c
in the power semiconductor s. These cost functions are
candidates when the OSV-MPC approac h is employed beca use
it does not impose a defined switching patter n. The system
performance is investigated for both alternatives in [80], and
(6) is shown to provide better results than (5).
A particular case is using a cost function to achieve a
desired spectrum shape of an output variable. This occurs
when the switching frequency is fixed or Selective Harmonic
Elimination (SHE) or Selective Harmo nic Mitigation (SHM)
techniques are used [9 4]–[97]. CCS-MPC strategies do not
need any special cost func tion because the power converter
output voltage is generated using a modulator stage. The
modulation technique produces a predefined spectrum co ntent
depending on the mod ulation stra tegy [75]. On the other
hand, OSV-MPC needs to include this control objective in the
controller design.
The first approach to solve this problem was to use
g = |F
ˆ
i
L
− i
∗
L
| (7)
as the cost function, where F is a narrow band-stop filter.
In this way, de fined harmonic compon ents do not contribute
to the cost function value, and a concentrated switching
frequency is obtained around the band-stop frequency [94].
A second procedure for OSV-MPC was to maintain (3) as the
cost fu nction but to include virtual vectors in the control set
[98]. T hese virtual vectors are modulated using a pulse width
modulation (PWM) - space vector m odulation (SVM) tha t
provides a fixed switching frequency. A more recent technique
proposes to obtaining the low fre quency components of the
control action computed b y the OSV-MPC controller using
(3). Th ese components are used as the control input for the
converter a nd ar e generated by a PWM-SVM modulator [91].
Finally, new approaches in c lude the modulation stage in the
optimization process. Therefore, the outputs of th e FCS-MPC
controller are the output voltage vectors and their application
times [20], [25], [92]. Table IV summ arizes these methods and
shows their basic control scheme s.
B. Weighting Factor Design
MPC can handle several control objectives simultaneously.
In order to do so, the variables to be controlled should be
included in the cost function. As a result, the cost function
can contain variables of differing natures. The most c ommon
example is MPC for controlling the torque and flux in a motor
drive. The usual cost function used for this application is
g =
ˆ
T − T
∗
2
+ λ
ˆ
ψ − ψ
∗
2
. (8)
Here,
ˆ
T and T
∗
are the predicted and reference torque values,
ˆ
ψ a nd ψ
∗
are predicted and reference flux values, and λ is a
weighting factor which defines a trade-off between the torque
and flux tracking.
In general, the differing natures of the variables hinder the
selection of the weighting factors. This is because these vari-
ables usually have different orders of magnitude. Therefore,
they do not equally contribute to the cost function’s value. A
common approach for solving this problem is to work in per
