Predictive Current Control with Fixed Switching Frequency for an NPC ...

Predictive Current Control with Fixed Switching Frequency for an NPC Converter M. Rivera, M. P´erez, C. Baier, J. Mu˜noz Universidad de Talca, Curic´o, CHILE Email: [email protected] http://www.utalca.cl

V. Yaramasu, B. Wu

Abstract—In this paper a Predictive Current Control strategy is proposed for constant switching frequency operation and verified with a neutral-point-clamped (NPC) converter for load current regulation and DC-link capacitor voltages balancing. The aim of this control technique is to obtain a modulated waveform at the output of the converter, whilst maintaining all the desired characteristics of FCS-MPC. The feasibility of this strategy is evaluated using simulation and compared to the classical FCSMPC strategy.

Variable 𝑣𝑑𝑐 𝑖𝑑𝑐 v i i∗ 𝑅 𝐿

N OMENCLATURE Description DC-voltage DC-current Load voltage [𝑣𝑎𝑁 𝑣𝑏𝑁 𝑣𝑐𝑁 ]𝑇 Load current [𝑖𝑎 𝑖𝑏 𝑖𝑐 ]𝑇 Load current reference [𝑖∗𝑎 𝑖∗𝑏 𝑖∗𝑐 ]𝑇 Load resistor Load inductor I. I NTRODUCTION

Multilevel diode-clamped converters are usually considered for medium-voltage, high-power applications. In fact, they can be used at high power levels with reduced common mode voltages, total harmonic distortion and electromagnetic interference [1]. These converters employ clamping diodes and cascaded DC capacitors to produce AC voltage waveforms with multiple levels [2]. The three-level diode-clamped converters is often known as the neutral-point-clamped (NPC) converters and is used in many high-power, medium-voltage industrial applications [3]. The Model Predictive Control (MPC) represents a class of receding horizon control algorithms [4]. In particular Finite Control Set Model Predictive Control (FCS-MPC) is particularly suitable for power electronics converters control. In fact, it is a simple, intuitive, powerful tool to control power converters with diverse and complex control challenges and restrictions [5], [6]. Since FCS-MPC is a model based optimization control strategy, it involves large number of calculations [7]–[9]. However, the present digital implementation platforms such as Microprocessors, DSPs and FPGAs can handle these computations as shown in recent works, where FCS-MPC is applied to a great variety of power converters [10]–[16]. However, since FCS-MPC apply one converter state for the whole sampling interval without taking advantage of 978-1-4673-7554-2/15/$31.00 ©2015 IEEE

L. Tarisciotti, P. Zanchetta, P. Wheeler

Ryerson University, CANADA University of Nottingham, Nottingham, U.K. Email: [email protected] Email: [email protected] http://www.ee.ryerson.ca/ http://www.nottingham.ac.uk

any PWM technique, the control generates waveforms with variable switching frequency. This effect produces a wide harmonic spectrum of the converter AC waveforms, decreasing the performance of the system in terms of power quality [17]– [21]. In [22] the discrete space vector modulation (SVM) is extended to be used with predictive control. Virtual vectors are considered in the control algorithm which are synthesized through an external modulator, obtaining constant switching frequency and improved performance. Similar results are obtained in [23], where traditional PI controllers were substituted by a predictive controller, but with a conventional modulation technique. In [18], [24] a deadbeat predictive controller is proposed in order to determine the duty cycle for the PWM pulses for a given reference current over the entire speed range of operation of a switched reluctance machine. In [25] a predictive controller is proposed with a cost function that includes the current error and additionally a penalization term which is used to control the switching frequency. In [17], [20], [26] the switching behaviour of PWM with triangular carriers is used to propose a predictive method with a fixed switching period which is divided into smaller evaluation steps to obtain improved performance for different power converter topologies. Several other solutions have been proposed in the literature [19], [27], [28] which allow operation at fixed switching frequency. However they result in complicated expressions for the switching time calculations and, since the switching instant calculation is related with the control cost function, it is very complicated to introduce other objectives into the cost function. In order to solve these problems, this paper proposes a Predictive Current Control strategy, to regulate to regulate the load currents and to minimize the imbalance between the DC-link capacitor voltages in an NPC converter. The proposed method has the aim to emulate a given switching pattern into the cost function minimization algorithm, in order to fix the converter switching frequency. However in this case the duty cycles are calculated from empirical relations between the cost function of the selected vectors, resulting in a straightforward control implementation. II. T OPOLOGY AND M ATHEMATICAL M ODEL OF THE T HREE L EVEL NPC C ONVERTER A three-phase NPC converter is shown in Fig. 1. The converter is composed of 12 active switches and 6 clamping

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𝑖𝑑𝑐

𝑣𝑑𝑐1

𝑆1𝑎

𝑆1𝑏

𝑆1𝑐

𝑆2𝑎

𝑆2𝑏

𝑆2𝑐

𝑣𝑑𝑐

𝑣𝐴𝑁 𝑂

𝑣𝐵𝑁

𝑣𝐶𝑁

𝑆 1𝑎

𝑆 1𝑏

𝑆 1𝑐

𝑆 2𝑎

𝑆 2𝑏

𝑆 2𝑐

TABLE I S WITCHING S TATES AND NPC C ONVERTER T ERMINAL VOLTAGES (𝑥 = 𝑎, 𝑏, 𝑐)

𝑖𝑎 𝑖𝑏 𝑖𝑐

𝑆𝑥

𝑆1𝑥

𝑆2𝑥

𝑆 1𝑥

𝑆 2𝑥

𝑣𝑥𝑁

1 0 −1

1 0 0

1 1 0

0 1 1

0 0 1

𝑣𝑑𝑐1 + 𝑣𝑑𝑐2 𝑣𝑑𝑐2 0

SECTOR II

𝑣𝑑𝑐2

v8

v15

Fig. 1.

SECTOR III

Power circuit of three-level NPC converter.

diodes. The switching states and the corresponding converter terminal voltages are shown in Table I, where, it can be noted that: (a) only two switches conduct at any time, and (b) switch pairs (𝑆1𝑥 , 𝑆 1𝑥 ) and (𝑆2𝑥 , 𝑆 2𝑥 ) operate in a complementary manner [2]. A total of 27 (33 ) switching combinations are available for this converter which are represented in Fig. 2. From Table. I, the voltage in any phase-𝑥 of the converter, measured from the negative point of the DC-link (𝑁 ), can be expressed in terms of switching states and DC-link capacitor voltages as follows:

v9

OPO NON

𝑣𝑥𝑁 = 𝑣𝑑𝑐1 𝑆1𝑥 + 𝑣𝑑𝑐2 𝑆2𝑥 , 𝑥 = 𝑎, 𝑏, 𝑐

Finally, assuming a passive 𝑅𝐿-load, the dynamic model can be defined as: 𝑑i 𝐿 = v − 𝑅i (3) 𝑑𝑡 In order to be implemented in a digital controller, the mathematical model of the converter must be defined in a discrete time model. From eq. (1)-(3) it is possible to obtain a discrete time model for the load current and capacitor voltages, assuming that the variables keep constant during a sampling time 𝑇𝑠 : ( ) 𝑅𝑇𝑠 𝑘 𝑇𝑠 𝑘 (4) i𝑘+1 = 1 − i + v 𝐿 𝐿 𝑇𝑠 𝑘+1 𝑖 , 𝑥 = 1, 2 𝐶𝑥 𝑐𝑥

OPP NOO

3

v3

v2

4

v7

PPO OON

2

PPP NNN

1

3

1 v0

OOO

v1

SECTOR I

PON

v13

POO ONN

PNN

2 OOP NNO

NOP

SECTOR IV v10

4

v5

v6

POP ONO

PNO

v12 SECTOR VI

NNP

v17

PNP

v11

ONP

v18

SECTOR V Fig. 2.

Space vector diagram for a three-level NPC converter.

(1)

The DC-link capacitor voltages can be expressed in terms of DC-link capacitor currents as follows: ⎫ 𝑖𝑑𝑐1  𝑑 𝑣𝑑𝑐1  = ⎬ 𝑑𝑡 𝐶1 . (2) 𝑖𝑑𝑐2  𝑑 𝑣𝑑𝑐2  ⎭ = 𝑑𝑡 𝐶2

𝑘+1 𝑘 = 𝑣𝑑𝑐𝑥 + 𝑣𝑑𝑐𝑥

v4

NPP

PPN

v𝑟𝑒𝑓

NPO

v16

v14

OPN

NPN

(5)

model of the system. With this information it is possible to predict the behaviour of the variables for each switching state. Each prediction is evaluated with respective to its reference in a cost function and the switching state that generates the minimum value of this evaluation is selected to be applied in the next sampling time. This control method has been extendedly implemented in different converter topologies and applications such as: DC/AC converters, AC/DC converters and AC/AC converters [5], [10], [29]. The block diagram of this control strategy is shown in Fig. 3. The control objectives are regulation of load currents and balancing of DC-link capacitor voltages. The FCS-MPC method uses discrete-model of the system, current state load currents and DC-link capacitor voltages, and all the 27 possible switching states to predict the future behavior of control variables. These predictions are evaluated using a cost function. The switching state which leads to the minimal value of the cost function is chosen and applied to the converter directly. As it is well known with this FCS-MPC, there is no need to have linear current regulators (PI controllers) and modulation. B. Cost Function Design The defined cost function has two objectives: (a) minimize the error between the predicted load currents i𝑘+1 and their references i∗ 𝑘+1 , and (b) balance the DC-link capacitor voltages. These control objectives are represented as follows:

III. C LASSIC M ODEL P REDICTIVE C ONTROL A. Control Strategy Predictive current control is a well known method which uses the finite number of possible switching states that can be generated by the power converter and the mathematical

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𝑔𝑡𝑟𝑎𝑐𝑘 𝑘+1 = [𝑖∗𝑎 − 𝑖𝑎 𝑘+1 ]2 + [𝑖∗𝑏 − 𝑖𝑏 𝑘+1 ]2 + [𝑖∗𝑐 − 𝑖𝑐 𝑘+1 ]2 𝑔𝑏𝑎𝑙 𝑘+1 = 𝜆𝑑𝑐 ∗ [𝑣𝑑𝑐1 𝑘+1 − 𝑣𝑑𝑐2 𝑘+1 ]2

(6)

𝑣𝑑𝑐1

𝑂

𝐶1 Cost Function Minimization

i∗ 3

i𝑘+1 𝑣𝑑𝑐1 𝑣𝑑𝑐2

27

𝑆1𝑎 ...𝑆 2𝑐

𝛽

𝑣𝑑𝑐2 𝐶2

v15

v9

𝑘+1 𝑘+1 𝑣𝑑𝑐1 𝑣𝑑𝑐2

Predictive Model

v8

v14

NPC

v3

v7

v2 v0

3

i𝑘

v16

v13

3

v4

𝑅

Load

𝛼

v1

𝐿 n

Fig. 3.

v5

v10

v6

v12

Block diagram of classic FCS-MPC strategy. v17

where 𝜆𝑑𝑐 is weighting factor, which can be adjusted according to the desired performance. The final cost function can be defined as follows: 𝑔 𝑘+1 = 𝑔𝑡𝑟𝑎𝑐𝑘 𝑘+1 + 𝑔𝑏𝑎𝑙 𝑘+1

v18

Fig. 4. Available vectors for the 3-level diode-clamped NPC in 𝛼 − 𝛽 plane. 𝑣𝑑𝑐1

(7) i∗

The switching state that minimizes the cost function is chosen and then applied at the next sampling instant. Additional constraints such as switching frequency reduction, current limitation and spectrum shaping can be included just by adding those conditions in the cost function [5].

3

i

IV. P ROPOSED P REDICTIVE C URRENT C ONTROL FOR THE NPC In space vector modulation (SVM), it is possible to define each available vector for the NPC converter in the 𝛼 − 𝛽 plane as shown in Fig. 4. It is possible to define different sectors which are given by two adjacent vectors, being the first sector the one between vector v1 and vector v2 , which correspond to the voltage generated by switching state number 1 and switching state number 2, respectively, based on eq. (1) and Table I. The proposed method is shown in Fig. 5. Similar to the FCS-MPC strategy, it uses the prediction of load current and capacitor voltages indicated in eq.(4) and eq. (5), respectively. Moreover the proposed technique evaluates the prediction of the two active vectors that confirm each sector at every sampling time and evaluates the cost function separately for each prediction. The cost function 𝑔 is evaluated for each case and is the same as the one considered for the classical predictive method. For example, for sector I, the first prediction and cost function 𝑔1 is evaluated for vector v1 and the second prediction and cost function 𝑔2 is evaluated for vector v2 . Each prediction is evaluated based on eq. (4) and eq. (5) and the only change is in respect to the calculation of the load voltage v. The duty cycles for the two active vectors are calculated by solving: 𝑑0 = 𝐾/𝑔0 𝑑1 = 𝐾/𝑔1 (8) 𝑑2 = 𝐾/𝑔2 𝑑0 + 𝑑1 + 𝑑2 = 𝑇𝑠

v11

Cost Function Minimization

𝑘+1

27

𝑘+1 𝑣𝑑𝑐1

𝑇0 𝑇1 𝑇2 v1 v2

N

𝑣𝑑𝑐2

𝐶1 Switching Pattern

𝑆1𝑎 ...𝑆 2𝑐

𝑘+1 𝑣𝑑𝑐2

Predictive Model 𝑣𝑑𝑐1 𝑣𝑑𝑐2

𝐶2 NPC

3

i𝑘 3

𝑅 Load

𝐿 n

Fig. 5.

Modulated model predictive current control scheme.

where 𝑑0 correspond to the duty cycle of a zero vector which is evaluated only one time. By solving the system of eq. (8), is possible to obtain the expression for 𝐾 and the expressions for the duty cycles for each vector are given as: 𝑑0 = 𝑇𝑠 𝑔1 𝑔2 /(𝑔0 𝑔1 + 𝑔1 𝑔2 + 𝑔0 𝑔2 ) 𝑑1 = 𝑇𝑠 𝑔0 𝑔2 /(𝑔0 𝑔1 + 𝑔1 𝑔2 + 𝑔0 𝑔2 ) 𝑑2 = 𝑇𝑠 𝑔0 𝑔1 /(𝑔0 𝑔1 + 𝑔1 𝑔2 + 𝑔0 𝑔2 )

(9)

With these expressions, the new cost function, which is evaluated at every sampling time, is defined as 𝑔 𝑘+1 = 𝑑1 𝑔1 + 𝑑2 𝑔2

(10)

The two vectors that minimize this cost function are selected and applied to the converter at the next sampling time. After obtaining the duty cycles and selecting the optimal two vectors to be applied, is defined the time that each vector will be applied, such as: 𝑇 0 = 𝑇𝑠 𝑑0 𝑇 1 = 𝑇𝑠 𝑑1 (11) 𝑇 2 = 𝑇𝑠 𝑑2 A switching pattern procedure, such as the one shown in Fig. 6, is adopted with the goal of applying the two active

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v1 v2 v0

1

a)

0 1

1 0 -1

0

0.01

1 0

0.02

0.03

0.04

0.05

0.02

0.03

0.04

0.05

0.02

0.03 D

0.04

0.05

1.05 𝑇1 2

𝑇0 4

𝑇2 2

𝑇0 2

𝑇2 2

𝑇1 2

𝑇0 4

b)

𝑇𝑠 Fig. 6.

1

0.95 0.01

Switching pattern for the optimal vectors.

1 c)

vectors and one zero vector. A similar idea has been first proposed for a three-phase active rectifier and a seven-level converter in [30], [31] respectively and now it is extended to a 3-level NPC converter.

-1 0.01

Fig. 7. Simulation results classical predictive control method: a) load current references and measured [pu]; b) capacitor voltages [pu]; c) load voltage [pu].

V. S IMULATION R ESULTS In order to validate the effectiveness of the proposal method, simulation results are presented during both steady-state and transient conditions. These results are compared with the results obtained with the classical FCS-MPC strategy. The simulation parameters are shown in Table II. In order to have a reasonable comparison between the proposed and classical predictive methods, a higher sampling time 𝑇𝑠 is considered for the classical predictive controller, 𝑇𝑠 = 1/60000 [s], while a 𝑇𝑠 = 1/25000 [s] is defined for the proposed predictive method.

a)

TABLE II S IMULATION PARAMETERS

𝜆𝑑𝑐 ∗

0 0.01

Value 300 [𝑘VAR] 3330 [V] 52.5 [A] 36.3 [Ω] 0.1155 [H] 4700 [𝜇F] 10.89 [Ω] 0.0126 [H] 40 [𝜇s] 25 [kHz] 1 [𝜇s] 0.1

for the proposal predictive control scheme

p.u. 1 1 1 1 1 0.3𝑍𝑏 0.11𝐿𝑏

0.02

0.03

0.04

0.05

0.02

0.03

0.04

0.05

0.02

0.03 D

0.04

0.05

1.05 b)

1

0.95 0.01

The steady-state waveforms for three-phase load currents, dc-link capacitor voltages and converter output voltage with the classical and proposed predictive control techniques are shown in Figs. 7 and 8 respectively. In both the cases it is observed that the load currents i track to their respective references i∗ very well. The reference currents are maintained at the nominal value (1 [p.u.]) with fundamental frequency of 50 [Hz]. At the same time, the capacitor voltages are very well balanced by the control approaches and the difference between 𝑣𝑑𝑐1 and 𝑣𝑑𝑐2 is negligible. It is observed that the load current ripple for the proposed predictive scheme is slightly lower than

Description Nominal power Line-line base voltage Base current Base impedance Base inductance DC-link capacitor Load resistance Load inductor Sampling time Switching frequency∗ Simulation step Weighting factor

1 -1

A. Results During Steady State

Variables 𝑆 𝑉𝑏 𝐼𝑏 𝑍𝑏 𝐿𝑏 𝐶𝑑𝑐 𝑅 𝐿 𝑇𝑠 𝑓𝑠

0

1 c)

0 -1 0.01

Fig. 8. Simulation results proposed predictive control scheme: a) load current references and measured [pu]; b) capacitor voltages [pu]; c) load voltage [pu].

the classical predictive method. This can also be seen in Fig. 8c which has a more sinusoidal load voltage 𝑣𝑎𝑛 with respect to the load voltage than in Fig. 7c. Additionally, a higher sampling frequency is required for the classical predictive control method in relation to the proposed method. As shown in Fig. 9, the classical predictive control strategy presents a spread spectrum over the frequency range where it is difficult to identify the switching frequency. On the other hand, for the proposed predictive control method, the load voltage presents a fixed switching frequency which is located around harmonic number 500 (500×50 [Hz] = 25 [kHz]). It is important to highlight that in order to have a similar THD value, the classical predictive method requires higher sampling time with respect to the proposed method. 1037

1 a) a)

1 0 -1

0.5

0.01

0.03

0.04

0.05

0.02

0.03

0.04

0.05

0.02

0.03 D

0.04

0.05

1.05

0 0

100

200

300

400

500

600

700

800

900

b)

1000

1

0.95 0.01

1 b)

0.02

1

0.5

c)

0 -1

0 0

100

200

300

400 500 600 Harmonic

700

800

0.01

900 1000

Fig. 9. FFT analysis for the load voltage 𝑣𝑎𝑛 : a) classical predictive control; b) proposed predictive control scheme.

B. Results During Transient Condition In order to demonstrate the performance of the proposed strategy in terms of dynamic response, transient analysis is carried out with both the control methods. Fig. 10 shows results for the classical predictive method and Fig. 11 shows the results for the proposed method under similar operating condition. A step change in the load current reference i∗ from 1 [p.u.] to 0.3 [p.u.] is applied at instant 𝑡 = 0.03 [s]. In both cases a very good dynamic response is observed. The load current ripple is observed to be lower for the proposed predictive control scheme during 1 [p.u.] to 0.3 [p.u.] load conditions. The variable switching frequency nature can be clearly seen with the classical predictive control method. The switching frequency changes in response to the step-change in load current. The switching frequency is maintained constant by the proposed predictive scheme despite the step-change in load current amplitude. C. Discussion As mentioned in [17] and [21], if the FCS-MPC controller is evaluated and the control is applied at every sampling time (which means that only one output voltage vector is applied during the whole sampling period), the resulting switching frequency becomes variable where the maximum switching frequency occurs for a reference that is equivalent to a duty cycle value of 0.5. The switching frequency decreases as the reference magnitude moves away from 0.5 value. Thus, in order to have a high switching frequency for every reference the control algorithm has to be evaluated at a higher rate than the switching frequency. This issue has been addressed in Fig. 10 where it is shown that the switching frequency varies when the reference varies. Additionally, it is necessary to consider a high sampling time in order to obtain similar average switching frequency with respect to the switching frequency. With the proposed technique it is possible to eliminate one of the main disadvantages of predictive control: variable

Fig. 10. Simulation results classical predictive control method: a) load current references and measured [pu]; b) capacitor voltages [pu]; c) load voltage [pu].

a)

1 0 -1 0.01

0.02

0.03

0.04

0.05

0.02

0.03

0.04

0.05

0.02

0.03 D

0.04

0.05

1.05 b)

1

0.95 0.01 1 c)

0 -1 0.01

Fig. 11. Simulation results proposed predictive control scheme: a) load current references and measured [pu]; b) capacitor voltages [pu]; c) load voltage [pu].

switching frequency. By considering a modulated model predictive control scheme it is possible to consider the use of two active vectors and the one zero vector during each sampling period such as is done for the space vector modulation (SVM) with the advantage that this method does not requires any linear PI controller for the current regulation as needed for the SVM. The operation at fixed switching frequency produces less ripple and a more concentrated spectrum, which is reflected in an improvement in the performance of the system. VI. C ONCLUSION In this paper, a new predictive control scheme has been proposed which allows the operation of an NPC converter with a fixed switching frequency while maintaining the advantages of the classical finite control-set model predictive control such as fast dynamic response and easy inclusion of nonlinearities.

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Simulations results demonstrate that this is a viable alternative to avoid linear controllers and performs well during steadystate and transient conditions with a good tracking to its references with a reduced ripple while balancing the capacitor voltages. ACKNOWLEDGMENTS The authors would like to thank the financial support of FONDECYT Initiation into Research 11121492 Project and Universidad de Talca. The authors also would like to thank the support of MSc. A. Olloqui in the edition of figures. R EFERENCES [1] S. Kouro, M. Malinowski, K. Gopakumar, J. Pou, L. Franquelo, B. Wu, J. Rodriguez, M. Perez, and J. Leon, “Recent advances and industrial applications of multilevel converters,” IEEE Trans. Ind. Electron., vol. 57, no. 8, pp. 2553–2580, Aug. 2010. [2] B. Wu, High-power converters and AC drives, 1st ed., ser. Wiley-IEEE Press. John Wiley & Sons, Inc., 2006. [3] J. Rodriguez, S. Bernet, P. Steimer, and I. Lizama, “A survey on neutralpoint-clamped inverters,” IEEE Trans. Ind. Electron., vol. 57, no. 7, pp. 2219–2230, Jul. 2010. [4] W. Kwon, S. Han, and S. Han, Receding Horizon Control: Model Predictive Control for State Models, ser. Advanced Textbooks in Control and Signal Processing. Berlin, Germany: Springer-Verlag, 2005. [5] S. Kouro, P. Cortes, R. Vargas, U. Ammann, and J. Rodriguez, “Model predictive control-A simple and powerful method to control power converters,” IEEE Trans. Ind. Electron., vol. 56, no. 6, pp. 1826–1838, Jun. 2009. [6] P. Cortes, M. Kazmierkowski, R. Kennel, D. Quevedo, and J. Rodriguez, “Predictive control in power electronics and drives,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4312–4324, Dec. 2008. [7] T. Vyncke, S. Thielemans, and J. Melkebeek, “Finite-set model-based predictive control for flying-capacitor converters: Cost function design and efficient FPGA implementation,” IEEE Trans. Ind. Informat., vol. 9, no. 2, pp. 1113–1121, 2013. [8] P. Martin Sanchez, O. Machado, E. Bueno, F. J. Rodriguez, and F. J. Meca, “FPGA-based implementation of a predictive current controller for power converters,” IEEE Trans. Ind. Informat., vol. PP, no. 99, pp. 1–1, 2012. [9] A. Damiano, G. Gatto, I. Marongiu, A. Perfetto, and A. Serpi, “Operating constraints management of a surface-mounted PM synchronous machine by means of an FPGA-based model predictive control algorithm,” IEEE Trans. Ind. Informat., vol. PP, no. 99, pp. 1–1, 2013. [10] J. Rodriguez, M. P. Kazmierkowski, J. R. Espinoza, P. Zanchetta, H. Abu-Rub, H. A. Young, and C. A. Rojas, “State of the art of finite control set model predictive control in power electronics,” IEEE Trans. Ind. Informat., vol. 9, no. 2, pp. 1003–1016, May. 2013. [11] R. Morales-Caporal, E. Bonilla-Huerta, C. Hernandez, M. Arjona, and M. Pacas, “Transducerless acquisition of the rotor position for predictive torque controlled PM synchronous machines based on a DSP-FPGA digital system,” IEEE Trans. Ind. Informat., vol. 9, no. 2, pp. 799–807, 2013. [12] J. Scoltock, T. Geyer, and U. Madawala, “A comparison of model predictive control schemes for MV induction motor drives,” IEEE Trans. Ind. Informat., vol. 9, no. 2, pp. 909–919, 2013. [13] R. Aguilera, P. Lezana, and D. Quevedo, “Finite-control-set model predictive control with improved steady-state performance,” IEEE Trans. Ind. Informat., vol. 9, no. 2, pp. 658–667, 2013. [14] J. Hu, “Improved dead-beat predictive DPC strategy of grid-connected dc-ac converters with switching loss minimization and delay compensations,” IEEE Trans. Ind. Informat., vol. 9, no. 2, pp. 728–738, 2013. [15] D. du Toit, H. Mouton, R. Kennel, and P. Stolze, “Predictive control of series stacked flying-capacitor active rectifiers,” IEEE Trans. Ind. Informat., vol. 9, no. 2, pp. 697–707, 2013. [16] M. Preindl and S. Bolognani, “Model predictive direct torque control with finite control set for PMSM drive systems, Part 1: Maximum torque per ampere operation,” IEEE Trans. Ind. Informat., vol. PP, no. 99, pp. 1–1, 2013.

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