A model predictive current control (MPCC) method that does not employ a cost function is proposed. The MPCC method can decrease commonmode voltages in loads fed by threephase voltagesource inverters. Only nonzerovoltage vectors are considered as finite control elements to regulate load currents and decrease commonmode voltages. Furthermore, the threephase future reference voltage vector is calculated on the basis of an inverse dynamics model, and the location of the onestep future voltage vector is determined at every sampling period. Given this location, a nonzero optimal future voltage vector is directly determined without repeatedly calculating the cost values obtained by each voltage vector through a cost function. Without utilizing the zerovoltage vectors, the proposed MPCC method can restrict the commonmode voltage within ±
V_{dc}
/6, whereas the commonmode voltages of the conventional MPCC method vary within ±
V_{dc}
/2. The performance of the proposed method with the reduced commonmode voltage and no cost function is evaluated in terms of the total harmonic distortions and current errors of the load currents. Simulation and experimental results are presented to verify the effectiveness of the proposed method operated without a cost function, which can reduce the commonmode voltage.
I. INTRODUCTION
Load currents of voltagesource inverters (VSIs) are commonly controlled by a cascaded structure with proportionalintegral (PI) controllers and distinct pulsewidth modulation (PWM) blocks
[1]
,
[2]
. Commonmode voltages generated by fast switching operation in threephase VSIs are also known to cause overvoltage stress on the winding insulation of drives and radiate electromagnetic interference noise, which can affect the functionality of other electronic systems in the vicinity
[3]
. Given that zerovoltage vectors lead to the highest commonmode voltage, studies on alleviating the commonmode voltage by avoiding the zero vectors in PWM blocks have been conducted
[3]

[8]
. Thus, PWM algorithms without zero vectors to reduce the commonmode voltage can be incorporated with PI current controllers to reduce the commonmode voltage and control the load currents. Model predictive current control (MPCC) method is a simple and effective current control technique for VSIs recently developed and used because of its simplicity without the need for any individual PWM blocks and its control flexibility
[9]

[14]
. MPCC predicts the seven possible future loadcurrent patterns based on the loadcurrent dynamics of the VSI by using the fundamental concept of the method, where only seven different voltage vectors can be applied to the loads by the VSI. All the predicted future load currents are calculated on the basis of the predefined cost function with error terms between the future reference current and the future load current to select an optimal voltage vector with the smallest cost function value. The switching algorithm with the model predictive current controller then applies an optimal switching state during the entire sampling period of the controller.
In this study, an MPCC method is proposed to reduce commonmode voltage in loads and regulate load currents without utilizing a cost function for threephase VSIs. The conventional MPCC method generates the commonmode voltage that corresponds to ±
V_{dc}
/6 or ±
V_{dc}
/2 depending on the nonzero or zerovoltage vectors selected as an optimal voltage vector. Only the nonzerovoltage vectors are employed in the proposed MPCC method to reduce the commonmode voltage and control the load currents. In addition, the proposed method develops a process for selecting a future optimal nonzerovoltage vector without employing a cost function. A future optimal nonzerovoltage vector is directly determined based on the location of the future reference voltage vector and the threephase future reference voltage vector calculated by an inverse dynamics model. The proposed method can restrict the commonmode voltage within ±
V_{dc}
/6 without utilizing the zerovoltage vectors that result in the highest commonmode voltage. This paper presents the performance of the total harmonic distortions and current errors of the load currents of the proposed method and compares them with those of the conventional method. Simulation and experimental results are included to verify the effectiveness of the proposed MPCC method operated without a cost function to reduce the commonmode voltage.
II. PROPOSED MODEL PREDICTIVE CURRENT CONTROL METHOD BASED ON THE REFERENCE VOLTAGE TO REDUCE COMMONMODE VOLTAGES
 A. CommonMode Voltage in the Conventional MPCC Method
The MPCC method for the VSI considers the VSI capable of applying only a finite number of voltage vectors to loads, which leads to a finite number of possible changes in the loadcurrent dynamics. According to the threephase VSI with a diode rectifier shown in
Fig. 1
(a), the output voltage vectors applied to the loads by the VSI can be expressed in the
αβ
frame as follows:
where the vector a =
e
^{j}
^{(2}
^{π}
^{/3)}
. The voltage vectors applied to the loads by the VSI include six nonzerovoltage vectors and two zerovoltage vectors, as shown in
Fig. 1
(b)
[9]
,
[10]
. The switching states of all the switches take on the binary values “1” and “0” in the open and closed states respectively. The lower switches always have the complementary values of their upper switches. Given the redundancy of the two zerovoltage vectors that generate an equal output voltage vector, only seven voltage vectors are available in the finite control set of the threephase VSI. The load current and backelectromotive force (backemf) vectors can also be expressed as follows:
(a) Threephase VSI with a dioderectifier front end. (b) Voltage vectors generated by the VSI.
The loadcurrent dynamics in the space vector form are expressed in the
αβ
frame as follows using Eqs. (1)(3):
where
R, L
, and
e
are the load resistance, inductance, and backemf vector, respectively. The derivative of the load current in the continuoustime domain model in Eq. (4) can be approximated on the basis of the forward Euler approximation with a sampling period
T_{sp}
as follows:
The loadcurrent dynamics can then be expressed in the discretetime domain model as follows:
The MPCC method is based on the concept that the seven possible future load currents can be predicted by the seven different voltage vectors of the VSI via the loadcurrent dynamics in Eq. (6). Thus, the voltage vector
v_{αβ}
(
k
), which takes on one of the seven voltage vectors generated by the VSI in
Fig. 1
(b), determines the onestep future load current
i_{αβ}
(
k
+ 1). A onestep delay compensation algorithm is included to compensate for the unavoidable control delay presented in practical controllers, where the twostep future loadcurrent dynamics are used by shifting Eq. (6) one step forward as in
[9]
:
The onestep future backemf vector in Eq. (7) can be estimated by assuming that the onestep future backemf vector is equal to the present backemf vector. This condition occurs because the backemf vector varies at a much lower frequency compared with the fast sampling frequency as follows:
where
is the estimated backemf vector at the kth time step. The twostep future reference current can be obtained by Lagrange extrapolation as in
[13]
:
Evaluating each predicted future load current using a predefined cost function results in the selection of one optimal voltage vector, which enables the future load current to track closest to the reference current, among seven available voltage vectors to minimize the errors between reference and actual currents. The cost function to measure errors between references and predicted load currents in orthogonal coordinates can be defined in terms of the current errors as follows:
An optimal voltage vector among seven possible voltage vectors, including the zerovoltage vectors, is selected every sampling period. The switching state that corresponds to the selected optimal voltage vector at the next sampling period is generated by the VSI.
In the standard threephase VSI with a dioderectifier front end in
Fig. 1
(a), the commonmode voltage is defined as the potential between the load neutral point and midpoint of the DClink voltage of the VSI. The commonmode voltage can be expressed with the pole voltages with respect to the midpoint of the DClink voltage as in
[8]
:
The VSI only takes the pole voltages at the discrete voltage levels with ±
V_{dc}
/2; thus, the commonmode voltage generated by the VSI becomes ±
V_{dc}
/6 and ±
V_{dc}
/2, which depends on the voltage vectors applied to the VSI, as summarized in
Table I
.
VOLTAGE VECTORS AND COMMONMODE VOLTAGES
VOLTAGE VECTORS AND COMMONMODE VOLTAGES
The two zero vectors
V
_{0}
and
V
_{7}
in the MPCC method result in identical output voltages, which lead to the same loadcurrent dynamics. Thus, one of the two zero vectors can be used or both zero vectors can be alternatively employed when the zero vector is selected as an optimal vector by the cost function. Only one of the zero vectors between
V
_{0}
and
V
_{7}
can be utilized in the cost function to reduce the total number of control sets and alleviate the corresponding calculation complexity. Both zero vectors can also be alternatively employed to result in a balanced loss distribution of the switching devices in the VSI
[13]
. Thus, alternatively using
V
_{0}
and
V
_{7}
when the previous vector has two “0” values (i.e.,
V
_{1}
) and two “1” values (such as
V
_{2}
) respectively in terms of the loss distribution and efficiency can be beneficial. The use of only
V
_{0}
or
V
_{7}
produces a commonmode voltage that varies between 
V_{dc}
/2 and
V_{dc}
/6 or between 
V_{dc}
/6 and
V_{dc}
/2 respectively. Therefore, the peaktopeak magnitude of the commonmode voltage that employs only one zero vector is 2
V_{dc}
/3. The commonmode voltage swings between 
V_{dc}
/2 and
V_{dc}
/2, and the peaktopeak magnitude of the commonmode voltage becomes
V_{dc}
when both
V
_{0}
and
V
_{7}
are alternatively utilized. Therefore, the conventional MPCC method that utilizes both zero vectors increases the peaktopeak magnitude of the commonmode voltage unlike the method that uses only one zero vector.
Figs. 2
(a)(c) show the experimental waveforms of the load currents and commonmode voltage for the MPCC method that utilizes only
V
_{0}
as a zero vector, only
V
_{7}
as a zero vector, and both
V
_{0}
and
V
_{7}
, respectively. Given that the two zero vectors lead to identical loadcurrent dynamics,
Fig. 2
generates the same loadcurrent waveforms, whereas the different commonmode voltages are obtained depending on zerovector utilization.
Experimental waveforms of the load currents and commonmode voltage obtained by the conventional MPCC method using (a) V_{0}, (b) V_{7}, and (c) V_{0} and V_{7}.
 B. Proposed MPCC Method based on the Reference Voltage for Reduced CommonMode Voltage
Given that the zerovoltage vectors lead to the highest commonmode voltage that corresponds to ±
V_{dc}
/2, the proposed MPCC method realizes the current control algorithm with only six nonzerovoltage vectors to avoid utilizing the two zerovoltage vectors. Thus, the evaluation of only the six nonzerovoltage vectors based on the cost function with the absolute current error vector in Eq. (10) and the selection of an optimal nonzero vector can be utilized to achieve both reduced commonmode voltage and loadcurrent controllability. This method is called the “activevectorbased MPCC method” in this paper. Although the activevectorbased MPCC method can decrease the commonmode voltage by considering only the nonzerovoltage vectors as a possible optimal candidate, it develops a simple algorithm to directly determine the selection of a future optimal nonzerovoltage vector at every sampling period without employing a cost function to evaluate all six nonzero vectors. Assuming that the onestep future loadcurrent vector becomes equal to the onestep future reference current vector by applying the reference voltage vector, the loadcurrent dynamics in Eq. (6) can then be expressed by an inverse dynamics model as in
[15]
:
By shifting Eq. (12) one step into the future to apply the delay compensation technique, the onestep future reference voltage is obtained with the future reference current and future loadcurrent vectors as follows:
The cost function expressed with the current error terms in Eq. (10) can be replaced with the future reference voltage vector and available voltage vectors usable by the VSI on the basis of Eqs. (7), (10), and (13) as follows:
where
i
= 1, 2, …, 6. Eq. (14) shows that a future optimal voltage vector is a nonzerovoltage vector, which is closest to the future reference voltage vector at every sampling period. Once the future reference voltage vector is calculated using Eq. (13), its location and a nonzerovoltage vector closest to the future reference vector can be immediately determined. As summarized in
Table I
, the proposed MPCC method utilizes a nonzerovoltage vector placed nearest the future reference voltage vector at every sampling period to restrict the commonmode voltage within ±
V_{dc}
/6.
Fig. 3
shows six sectors divided in the complex space to directly select an optimal nonzerovoltage vector on the basis of the location of the future reference voltage vector; an optimal nonzero vector is
V_{i}
for a future reference voltage vector located in sector
S_{i}
(
i
= 16).
Table II
lists the sectors and corresponding voltage vector used in the sector.
Six sectors of the proposed MPCC method.
SECTORS AND SELECTED FUTURE VOLTAGE VECTORS
SECTORS AND SELECTED FUTURE VOLTAGE VECTORS
Thus, the proposed MPCC method can select a future optimal nonzerovoltage vector once the sector where the future reference voltage vector is located in is determined. The proposed method can be implemented with simplicity without repeatedly calculating the current errors generated by all six nonzerovoltage vectors.
Fig. 4
shows the overall block diagram of the proposed MPCC method that employs no cost function and no zero vectors.
Block diagram of the proposed MPCC method.
III. SIMULATION AND EXPERIMENTAL RESULTS
Figs. 5
and
6
show the simulation waveforms of the load currents and commonmode voltage of the conventional and proposed MPCC methods respectively with
T_{sp}
= 50 ms,
V_{dc}
= 100 V, an RLe load with
R
= 1.5 W,
L
= 15 mH, and
e
= 20 V. The load currents controlled by the proposed method accurately track the current reference, where the commonmode voltage
v_{no}
limited to ±
V_{dc}
/6 is significantly smaller than that of the conventional method because of the nonutilization of zero vectors.
Fig. 5
(a) shows that the conventional method produces a commonmode voltage that oscillates from
V_{dc}
/6 to 
V_{dc}
/2 because only
V
_{0}
is used in the control algorithm. The ripple components of the load currents in the proposed method are slightly higher than those observed in the conventional method, which is a penalty of the decreased commonmode voltage without employing the zero vectors. The frequency spectra of the load currents obtained from the proposed and conventional methods are shown in
Figs. 5
and
6
, which show that the proposed method yields a slightly higher total harmonic distortion (THD). The simulated waveforms of the activevectorbased MPCC method, which selects an optimal nonzerovoltage vector based on the cost function in Eq. (10) by evaluating six possible future current vectors generated with the six nonzero vectors, are also included in
Fig. 6
(a) under the same operating conditions for comparison. The waveforms of the load currents and commonmode voltage are exactly the same as those of the proposed method because the two methods select the same nonzerovoltage vectors at every sampling period.
Simulation waveforms for the conventional method (I = 5 A). (a) Load currents (i_{a}, i_{b}, i_{c}), aphase reference current (i_{a}^{*}), and commonmode voltage (v_{no}). (b) Frequency spectrum of the load current (i_{a}).
Simulation waveforms of the load currents (i_{a}, i_{b}, i_{c}), aphase reference current (i_{a}^{*}), commonmode voltage (v_{no}), and frequency spectrum of the load current (i_{a}) for the (a) activevectorbased MPCC method and (b) proposed method.
The dynamic responses of the proposed and conventional methods are shown in
Fig. 7
. The threephase load currents operated by the proposed method follow the change in the reference with fast dynamics, as observed in the conventional method for step changes in both frequency and magnitude.
Simulation waveforms of the transient response with a magnitude stepchange (from 3 A to 6 A) and frequency stepchange (from 60 Hz to 80 Hz) obtained for the (a) conventional method and (b) proposed method.
The proposed method was tested using the prototype setup of a threephase VSI with an Insulated Gate Bipolar Transistor module, where the entire switching algorithms with the current control methods were implemented on a Digital Signal Processor (DSP) board (TMS320F28335) to generate sinusoidal load currents with a 60 Hz fundamental output frequency. A practical consideration to compensate for the unavoidable calculation delay present in the DSP was also considered.
Fig. 8
shows the experimental waveforms for the load currents, commonmode voltage, and frequency spectrum of the load current obtained for the proposed and conventional methods with
T_{sp}
= 50 ms and
V_{dc}
= 100 V. Similar to the simulated results, the load currents of the proposed method accurately track their reference currents. The commonmode voltage of the proposed method that utilizes the nonzerovoltage vector is also limited to ±
V_{dc}
/6, whereas the conventional method generates a commonmode voltage that varies between
V_{dc}
/6 and ˗
V_{dc}
/2 because of the utilization of the zerovoltage vector
V
_{0}
.
Experimental waveforms of the load currents (i_{a} and i_{b}), aphase reference current (i_{a}^{*}), commonmode voltage (v_{no}), and frequency spectrum of the load current (i_{a}) for the (a) conventional method and (b) proposed method.
The dynamic responses of the proposed and conventional methods with the same experimental conditions are shown in
Fig. 9
. The threephase load currents for the proposed method follow the change in the reference currents with fast dynamics, as observed in the conventional MPC method for step changes in both magnitude and frequency.
Experimental waveforms of the transient response with a magnitude stepchange (from 3 A to 6 A) and frequency stepchange (from 60 Hz to 80 Hz) obtained for the (a) conventional method and (b) proposed method.
The performance of the VSI operated with the proposed method is compared with that of the VSI operated with the conventional method in terms of the THDs and current errors of the load currents as a function of the sampling period. This approach was used because the performance of the model predictive control algorithm is largely dependent on the sampling frequency. The comparative results of the VSIs with the proposed and conventional methods are shown in
Fig. 10
for a loadcurrent magnitude of 5 A. The percent THD is defined as follows:
where
i
_{x}
_{1}
and
i_{xn}
are the fundamental and
n
thharmonic components in the load current of phase
x
respectively.
n
was set to 8333 in this comparison. The percentage of the loadcurrent error is defined as the absolute difference between the reference current and load current normalized to the respective root mean square value of the reference current in phase x as follows:
where
N
was set to 10,000 per fundamental period. The percent THD and current errors observed in the proposed method to reduce the commonmode voltage are slightly higher than those in the conventional method because of the reduced voltage vector utilization.
Comparative results of the proposed and conventional methods: (a) THD and (b) current error as a function of the sampling periods.
The effects of the RL model errors on the current errors and THD values of the load currents obtained from the proposed method are investigated and shown in
Figs. 11
and
12
. The current errors and THD values are more affected by the load inductance than by the load resistance in the proposed method. In addition, the experimental results with the reduced inductance to 5 mH are shown in
Fig. 13
. The commonmode voltage is still limited to ±
V_{dc}
/6 by the proposed method, although the current ripples and THD values are increased because of the reduced load inductance.
Effect of model inductance errors on the (a) current errors and (b) THD values of load currents obtained from the proposed method.
Effect of model resistance errors on the (a) current errors and (b) THD values of load currents obtained from the proposed method.
Experimental waveforms of the proposed method with 5 mH inductance (a) load currents (i_{a}, i_{b}, i_{c}), aphase reference current (i_{a}^{*}), and commonmode voltage (v_{no}), and (b) the frequency spectrum of the load current (i_{a}).
IV. CONCLUSION
An MPCC method that operates without a cost function was proposed. This method can reduce the commonmode voltage for threephase VSIs. The commonmode voltage generated by the VSI in the proposed MPCC method, restricted within ±
V_{dc}
/6, was significantly smaller than that of the conventional method because the proposed method utilized only nonzero voltage vectors. Furthermore, the proposed method developed a process of selecting a future optimal nonzero voltage vector without employing a cost function. On the basis of the threephase future reference voltage vector calculated by an inverse dynamics model, an optimal nonzero voltage vector was directly determined on the basis of the location of the future reference voltage vector. As a result, the proposed method can find a future optimal nonzero voltage vector without evaluating all the nonzero vectors in terms of the future current errors based on the cost function at every sampling period, which can lead to simple implementation.
Acknowledgements
This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (2014R1A2A2A01006684) and the Chungang University Research Scholarship Grants in 2014.
BIO
Sungki Mun received the B. S. degree in electrical and electronics engineering from Chungang University, Seoul, Korea in 2014. Currently, he is working toward M.S degree in electrical and electronics engineering from Chungang University, Seoul, Korea. His research interests are control and analysis for multilevel inverters and voltage source inverters.
Sangshin Kwak (S’02M’05) received the Ph.D. degree in electrical Engineering from Texas A&M University, College Station, Texas in 2005. He worked as a research engineer at LG Electronics, Changwon, Korea from 1999 to 2000. He was also with Whirlpool R&D Center, Benton Harbor, MI, in 2004. He worked as a senior engineer in Samsung SDI R&D Center, Yongin, Korea from 2005 to 2007. He worked as an assistant professor at Daegu University, Gyeongsan, Korea from 2007 to 2010. He has been with Chungang University, Seoul, Korea since 2010, and is currently an associate professor. His research interests are topology design, modeling, control, and analysis of AC/DC, DC/AC, AC/AC power converters, including resonant converters for adjustable speed drives and digital display drivers, as well as modern control theory applied to DSPbased power electronics.
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