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Nine-to-Three Phase Direct Matrix Converter with Model Predictive Control for Wind Generation System
Available online at www.sciencedirect.com
ScienceDirect Energy Procedia 42 (2013) 173 – 182
MGEF 2013
Nine-to-Three Phase Direct Matrix Converter with Model Predictive Control for Wind Generation System Omar Abdel-Rahima,b, Haitham Abu-Ruba, Abdellah Kouzouc a Texas A&M University at Qatar, Doha, Qatar, 28354. Electrical Engineering Department, Aswan Faculty of Engineering, Aswan University, Aswan, Egypt, 81542. c Djelfa University Algeria [email protected]
b
Abstract The power conversion from variable AC voltage into a desired AC voltage with fixed magnitude and fixed frequency is attracting more attention especially in case of grid-connected wind generation system. Indeed, the Direct Matrix Converter (DMC) can be used as a suitable solution for such AC/AC conversion to fulfill the requirement of the output voltage with desired magnitude and frequency. Nine to three phase matrix converter developed in this paper is proposed to ensure the conversion of nine-phase input voltage into threephase voltage, the output current magnitude and frequency control and the input current power factor control. Among the existing control techniques, The Model Predictive Control (MPC) is considered to be one of the most effective control techniques. In this paper MPC is used to control the twenty-seven switches used in the topology of the nine-to-three phase matrix converters, where the main aim is to achieve principal control functions such as, the output current magnitude and frequency control and input current control to ensure a reduced shift phase, hence a nearly unity power factor in the input side of the DMC. © The Authors. Published by Elsevier Open access under CC BY-NC-ND license. ©2013 2010 Published by Elsevier Ltd. Ltd. Selection and/or peer-review under responsibility of Selection and peer-review under responsibility of KES International Keywords: Direct Matrix Converter, Bidirectional, Multiphase, Model Predictive Control, Nine-phase.
1876-6102 © 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license.
Selection and peer-review under responsibility of KES International
doi:10.1016/j.egypro.2013.11.017
[MGEF 2013]
174
Omar Abdel-Rahim et al. / Energy Procedia 42 (2013) 173 – 182
1. INTRODUCTION The inherent advantages of multiphase generators such as higher output power, reduced phases losses due to the produced power segmentation, reduced dimensions compared to three phase generator for the same amount of output power, make it attractive, suitable and flexible solution for Wind Energy Conversion System (WECS) to go up for a higher number of the generators output phases [1-3]. It is obvious that the grid is a three-phase system and hence the deployment and the integration of multiphase generator in grid-connected applications need some sorts of AC/AC conversion system to meet the requirement of the number of phases in both sides. Indeed, this conversion can be achieved by employing back-to-back converter, with multiphase input AC/DC converter and three-phase output DC/AC converter, moreover, the insertion of a bulky DC link capacitor between the two converters is primordial. To avoid the use of two stage conversion system, to reduce the number of components and to give a higher freedom degree for the number of the input phases while keeping the output three-phase system unchanged, the matrix converter can be used to achieve these advantages all together[4]-[8] Matrix converter can be broadly classified into direct and indirect types. Direct Matrix converter is able to convert AC voltage into AC voltage with variable magnitude, frequency and number of phases. It is considered as a powerful topology for AC to AC power conversions, therefore, this kind of converters receive considerable attentions in recent years. They offer inherent advantages such as a bi-directional power flow which is very important in regeneration process, nearly sinusoidal input and output waveform, controlled input power factor, controlled output current magnitude and frequency, compact design and lack of DC-link capacitors for energy storage. Under these advantages several configurations of the matrix converter have been developed and analyzed in the literature [9]-[11]. Nine phases to three phases matrix converter is able to convert an input nine phase voltages system into controlled output three phase voltages system. This configuration has twenty-seven switches, whereas each output phase is connected to all the input phases through a bidirectional semiconductor switches, and hence, there are in total ʹଶ expected switching states. IT is important to clarify that the following constraints have to be taken into account: x Input Phases should not be short circuited together (to protect the input phases). x Output phases should not interrupt under any condition, due to the presence of inductive loads. The switching states are limited to 225. To control those twenty-seven switches and select the appropriate switching states Model Predictive Control (MPC) is used in this paper. The MPC is considered to be one of the most interesting controller [12]-[14] due to their simple implementation. The basic idea of the MPC is to perform a model of the system to be controlled to enable the prediction of the system current in the future next step based on the selection of the optimum operation according to a specified cost function. This cost function determines the required control criteria of the presented model. It is found that the presented structure has several important advantages [15]: ¾ Simple and easy to implement ¾ It can be applied to a great variety of systems. ¾ Simple treatment of constraints. ¾ However, some disadvantages have to be mentioned, like: ¾ ¾
The larger number of calculations, compared to classic controllers, hence more powerful processor is needed for real time implementation. The quality of the model has a direct influence on the quality of the resulting controller.
A schematic of the proposed system is represented in Fig.1, as it is clearly shown, the proposed system consists of nine to three phase matrix converter used to convert nine AC phase input voltage into three
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Omar Abdel-Rahim et al. / Energy Procedia 42 (2013) 173 – 182
phase AC output, where the magnitude and the frequency of the output current are controlled. The Model predictive control is used to control matrix converter and adjust the switching sequence of the matrix converter switches. An input LC filter is used in the input side to reduce the input current harmonics. The proposed system has some inherent advantages such as; it enables the use of multiphase machine in Wind Energy Generation Systems (WEGS), in this paper a nine phase generator is used to provide the input voltages system of the matrix converter, on the other side the use of the MPC control will give a fast dynamic response.
9
Vs Is Filter Vu
Model Predictive Control
Io
Matrix
3
LLoad o ad
Figure 1: Schematic of the proposed system. 2. NINE-TO-THREE PHASE MATRIX CONVERTER Nine-to-three phase direct matrix converter is able to convert nine-phase input into three-phase output with required voltage magnitude and frequency. General topology of the Nine-to-three matrix converter is represented in Fig. 2. The converter consists of nine-phase inputs and three-phase outputs, so that there are twenty-seven bidirectional power switches; nine switches for every output phase. The Switching pattern is defined as follow: ୧୨ୀ ͳ Ͳ ,with i={a, b, c, d, e, f, g, h, i} and j={A, B, C}, and there is a switching constraint which is defined as follow: ୟ୨ ୠ୨ ୡ୨ ୢ୨ ୣ୨ ୨ ୨ ୦୨ ୧୨ ൌ ͳ
(1)
176
Omar Abdel-Rahim et al. / Energy Procedia 42 (2013) 173 – 182 SaA L
SbA
ia
ScA
a
SdA
C b
ib
L
iA
SeA
C
A SfA f
ic
L c C
id
L
SgA ShA
d SiA
C ie
e C
if
f C ig g C h
ih C
ii
i C SaB SbB ScB SdB SeB
iB
SfB f
B
SgB ShB SiB
SaC SbC ScC SdC SeC SfC f Sg gC ShC SiC
Figure 2: Nine-to-Three Phase Matrix Converter Configurations.
iC
C
Omar Abdel-Rahim et al. / Energy Procedia 42 (2013) 173 – 182
Measure Vs, Ve, Is, Io
Prediction of source and load current Eqs. (9) and (5)
Gopt=inf.
For i=1...225
Cost Function Eq. (10)
i>=225
Applying optimum switchingg states
Figure 3: MPC flowchart
3. MODEL PREDICTIVE CONTROL Due to the inherent advantages of MPC, it is used to control the operation of the matrix converter. As shown in the flowchart, Fig. 3. Input current, source voltage, and voltage after the input filter are sensed to control the input current. Output actual current are measured and the reference for the output current is also commanded to the controller, for controlling the output current. A prediction for the input and output current is performed and their values are calculated, after that the cost function is calculated and an optimization process is performed to enable the choice of the best switching state that give the value for the predicted currents. For the predicted current calculation, a model for input filter, matrix converter and load have to be performed. In the following sections those models are developed for the proposed system.
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3.1 Matrix Model Matrix Converter depicted in Fig. 2 uses a set of bidirectional switches to connect Nine-phase input supply with three-phase load and enables bidirectional power flow. Equation (1) must be satisfied to avoid open circuiting load terminal as this leads to overvoltage that can destroy the components, In addition, to avoid short circuiting supply side. Then, the relation between input and output voltage of the nine-to-three phase matrix converter is as follows: ୟ୳ ۍୠ୳ ې ێୡ୳ ۑ ሺሻ ܵ ܵ ܵ ܵௗ ܵ ܵ ܵ ܵ ܵ ێୢ୳ ۑ ۑ ێ (2) ሺሻ ൌ ܵ ܵ ܵ ܵௗ ܵ ܵ ܵ ܵ ܵ Ǥ ێୣ୳ ۑ ܵ ܵ ܵ ܵௗ ܵ ܵ ܵ ܵ ܵ ێ୳ ۑ େ ሺሻ ێ୳ ۑ ێ୦୳ ۑ ۏ୧୳ ے 3.2 Load Model
IO(t)
R
VO(t)
L
Figure 4: Inductive load modeling. Load model is very important in MPC as it is used to enable the prediction of the load current in the next sample interval. Figure. 4 shows the model of the inductive load. Load current can be obtained from Fig. 2 by applying Kirchhoff law as follows;
ୢ୧ ሺ୲ሻ ୢ୲
ൌ ୭ ሺሻ െ כ୭ ሺሻ
ሺ͵ሻ
Where L and R are the inductance and resistance of the load. The discrete form of the current equation can be expressed as follows: ୢ୧ ୢ୲
ൌ
୧ ሺ୩ାଵሻି୧ ሺ୩ሻ ౩
ሺͶሻ
Where ୱ is the sampling period. The equation for predicting the load current is obtained from substituting (4) into (3), and the resulting equation can be written as follows: ୭ ሺ ͳሻ ൌ ቀͳ െ
ୖכ౩
ቁ כ୭ ሺሻ
౩
כ୭ ሺሻ
ሺͷሻ
Where ୭ ሺ ͳሻ is the predicted value of the current for sampling interval (k+1). 3.3 Filter Model The input filter model, based on the circuit shown in Fig. 5, can be described by the following continuous-time equations:
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Omar Abdel-Rahim et al. / Energy Procedia 42 (2013) 173 – 182
ୱ ሺሻ ൌ כୱ ሺሻ ୱ ሺሻ ൌ ୳ ሺሻ
ୢ୧౩ ሺ୲ሻ ୢ୲
ୢ୴౫ ሺ୲ሻ ୢ୲
୳ ሺሻ
ሺሻ
ሺሻ
Where , and are the inductor, the resistor and the capacitor of the input filter respectively. By considering the approximation of the input current derivative: ୢ୧౩ ୢ୲
ൌ
୧౩ ሺ୩ାଵሻି୧౩ ሺ୩ሻ
౩
ሺͺሻ
The equation for predicting the source current is obtained from substituting (8) into (7), the following equation is obtained; ୱ ሺ ͳሻ ൌ ቀͳ െ
ୖ כ౩
ቁ כୱ ሺሻ
౩
כሺୱ ሺሻ െ ୳ ሺሻሻ
IS(t)
Rf
Lf
VS (t)
Cf
ሺͻሻ
Iu (t) Vu(t)
Figure 5: L-C input filter modeling. 3.4 Cost Function The key parameter of the MPC is the cost function as it determines the required control functions. In the proposed system the required control function is to control output current magnitude and frequency and to control input current to be in-phase with input voltage. For the required control criteria of the proposed system, the cost function is expressed as follows;
כ כ כ כ െ ݅ఈ ห ቚ݅ఉ െ ݅ఉ ቚ ห݅௦ఈ െ ݅௦ఈ ห ቚ݅௦ఉ െ ݅௦ఉ ቚ ݃ ൌ ห݅ఈ
(10)
Where כ୭ currents and כ୭ஒ are the real and imaginary parts of the reference output current vector. ୮ ୮ Currents ୭ and ୭ஒ are the real and imaginary parts of the predicted output current vector calculated by equation (5). Currentsכୱ and כୱஒ are the real and imaginary parts of the reference source current vector. ୮ ୮ Currents ୱ and ୱஒ are the real and imaginary parts of the predicted source current vector calculated by equation (9). The reference source current can be calculated as cited in [16]: ݅௦ כൌ ܸ כ ܩ௦ (11) ೞ (12) where, ܩis the instantaneous conductance calculated as follow ܩൌ ԡ ԡమ ೞ
Where ୱ is the average source power and || || is the 2-norm of vectorୱ . This cost function doesn’t need for weighting factor. Elimination the weighting factor from cost function make it simpler and more reliable, as the process of choosing the weighting factor not a straight way and it requires to try many times before reaching to the best value.
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Omar Abdel-Rahim et al. / Energy Procedia 42 (2013) 173 – 182
4. Simulation Results. Simulation of the proposed system has been performed under MATLAB/Simulink environment. The parameter of the proposed system is given in Table I. The main functions for the proposed system which will be investigated are: the control of the output current magnitude and frequency and the control of the input current to ensure a reduced shift phase, hence a nearly unity power factor in the input side of the DMC. The output current is fully controlled with the proposed controller; the obtained output three-phase currents are sinusoidal and follow the reference current amplitudes as it is shown in Fig. 6. It is obvious that a nearly a sine waveform is obtained with a low Total Harmonic Distortion (THD). A step change has been presented at 0.1 sec, where the output current magnitude has been changed from 3 to 6 Amp. From Fig. 7, it is shown that the MPC controller has a very fast dynamic response with high ability to control the output current, the output current follows correctly the reference current. The input current control is satisfied, where nearly a nil shift phase is presented between the input current and the input voltage as depicted in Fig. 8. Nine phase input voltages are shown in Fig.9. Table I Simulation parameters Variables Description Simulation Values Input Voltage 200 V ܸ௦ Input Frequency 50 Hz ܨ௦ Input Filter Inductance 15 mH ܮ Input Filter Resistance 0.5 Ω ܴ Input Filter Capacitance 30 μf ܥ Load Resistance 10 Ω ܴ Load Inductance 10 mH ܮ Sample Time 10 μs ܶ௦
Figure 6: Three-phase output current.
Omar Abdel-Rahim et al. / Energy Procedia 42 (2013) 173 – 182
Figure 7: reference current and output current.
Figure 8: Input voltage divided by 10 and input current.
Figure 9: Nine-Phase Input voltage. 5. Conclusion This paper presents nine-to-three phase direct matrix converter as a solution for multiphase generator grid connected. Principle of operation of a nine-to-three phase matrix converter is discussed and analysed. The MPC is used as a powerful control technique to achieve the following goals; control input current at unity power factor, control of the output current magnitude and frequency under the nine- phase input to
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three-phase output conversion. A cost function is used to achieve the required objectives; it presents the advantage that it does not include a weighting factor. The proposed system was simulated using MATLAB software. References [1] S. Ayman, A. Khalik, and K. H. Ahmed, “Performance Evaluation of Grid Connected Wind Energy Conversion Systems with Five phase Modular Permanent Magnet Synchronous Generators having Different Slot and Pole Number Combinations” Proc. IEEE International Electric Machines and Drives conference, IEMDC-2011, 1518 May, Niagara Falls, Canada, pp. 1135-1140, 2011. [2] O. Ojo and IE. Davidson, “PWM-VSI inverter-assisted stand-alone dual stator winding induction generator”, IEEE Trans Energy Conversion, vol. 36, 2000, pp. 1604–1611. [3] O. Abdel-Rahim, M. Orabi and M.E. Ahmed," High gain single-stage inverter for photovoltaic AC modules " Applied Power Electronics Conference and Exposition (APEC), 6-11 March 2011, pp. 1961- 1967. [4] O. Abdel-Rahim, M. Orabi and M. Ahmed," Development an efficient photovoltaic (PV) configuration for low power applications” IEEE International Conference on Power and Energy (PECon), 2010, pp 622- 627. [5] O. Abdel-Rahim, H. Abu-Rub and Sk. Moin “Space vector PWM or five to three phase matrix converter” Applied Power Electronics Conference and Exposition (APEC), 2013, in press. [6] Ahmed, SK.M., Iqbal, A., Abu-Rub, H., Rodriguez, J., Rojas, C., (2010), “Simple carrier-based PWM technique for a three to nine phase matrix converter”, IEEE Trans. On Ind. Elect., vol. 58, no. 11, pp. 5014-5023, Nov. 2011. [7] Ahmed, SK., M., Iqbal, A., Abu-Rub, H., Cortes, P., (2011), “Model predictive control of a three to five-phase matrix converter”, Proc. Workshop on Predictive Control of Power electronics and Drives, 14-15 Oct., 2011, Munich, Germany, pp. 71-76. [8] Ahmed, SK. M., Iqbal. A., Abu-Rub, H, (2010), “Carrier-based PWM technique of a novel three-to-seven-phase matrix converter” Int. Conf. On Electrical Machine ICEM’10, 3-6 Sept. Rome, Italy, CD-ROM paper no RF004944. [9] P. Wheeler, J. Rodríguez, J. Clare, L. Empringham and A. Weinstein “Matrix converters: a technology review,” ”, IEEE Trans. On Ind. Elect., vol. 49, no. 2, April 2002. [10] Iqbal, A., SK, Moin. A., Abu-Rub, H., “Space Vector PWM Technique for a Three-to-Five Phase Matrix Converter”, , IEEE Trans. On Applications, Vol. 48, No. 2, March/April 201 [11] W. Kolar, F. Schafmeister, S. Round and H. Ertl” Novel three-phase ac–ac sparse matrix converters”, IEEE Trans. On Power Electronics, vol. 22, no. 5, SEPTEMBER 2007. [12] J. Rodriguez, H. Young, C. Rojas, S. Kouro, P. Cortes, and H. Abu-Rub, State of the Art of Model Predictive Control in Power Electronics and Drives, accepted, IEEE Trans. on Industrial Electronics, 2012. [13] Abu-Rub, H, Guzinski, J., Krzeminski, K. and Toliyat, H.: Predictive Current Control of Voltage Source Inverter. IEEE Trans. on Industrial Electronics, USA, Vol. 51, No. 3, June 2004, pp. 585-593. [14] Cortes, P., Wilson, A., Rodriguez, J., Kouro, S. and Abu-Rub, H. “Model Predictive Control of Multilevel Cascaded H-Bridge Inverters, accepted at IEEE Transactions on Industrial Electronics, Vol. 57, No. 8, August 2010, 2691 - 2699. [15] J. Rodriguez, P. Cortes “Predictive Control of Power Converters and Electrical Drives” Wiley, IEEE. [16] J. Rodriguez, J. Espinoza, M. Rivera, F. Villarroel, C. Rojas” Predictive control of source and load currents in a direct matrix converter” IEEE International Conference on Industrial Technology (ICIT), 2010.
ScienceDirect Energy Procedia 42 (2013) 173 – 182
MGEF 2013
Nine-to-Three Phase Direct Matrix Converter with Model Predictive Control for Wind Generation System Omar Abdel-Rahima,b, Haitham Abu-Ruba, Abdellah Kouzouc a Texas A&M University at Qatar, Doha, Qatar, 28354. Electrical Engineering Department, Aswan Faculty of Engineering, Aswan University, Aswan, Egypt, 81542. c Djelfa University Algeria [email protected]
b
Abstract The power conversion from variable AC voltage into a desired AC voltage with fixed magnitude and fixed frequency is attracting more attention especially in case of grid-connected wind generation system. Indeed, the Direct Matrix Converter (DMC) can be used as a suitable solution for such AC/AC conversion to fulfill the requirement of the output voltage with desired magnitude and frequency. Nine to three phase matrix converter developed in this paper is proposed to ensure the conversion of nine-phase input voltage into threephase voltage, the output current magnitude and frequency control and the input current power factor control. Among the existing control techniques, The Model Predictive Control (MPC) is considered to be one of the most effective control techniques. In this paper MPC is used to control the twenty-seven switches used in the topology of the nine-to-three phase matrix converters, where the main aim is to achieve principal control functions such as, the output current magnitude and frequency control and input current control to ensure a reduced shift phase, hence a nearly unity power factor in the input side of the DMC. © The Authors. Published by Elsevier Open access under CC BY-NC-ND license. ©2013 2010 Published by Elsevier Ltd. Ltd. Selection and/or peer-review under responsibility of Selection and peer-review under responsibility of KES International Keywords: Direct Matrix Converter, Bidirectional, Multiphase, Model Predictive Control, Nine-phase.
1876-6102 © 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license.
Selection and peer-review under responsibility of KES International
doi:10.1016/j.egypro.2013.11.017
[MGEF 2013]
174
Omar Abdel-Rahim et al. / Energy Procedia 42 (2013) 173 – 182
1. INTRODUCTION The inherent advantages of multiphase generators such as higher output power, reduced phases losses due to the produced power segmentation, reduced dimensions compared to three phase generator for the same amount of output power, make it attractive, suitable and flexible solution for Wind Energy Conversion System (WECS) to go up for a higher number of the generators output phases [1-3]. It is obvious that the grid is a three-phase system and hence the deployment and the integration of multiphase generator in grid-connected applications need some sorts of AC/AC conversion system to meet the requirement of the number of phases in both sides. Indeed, this conversion can be achieved by employing back-to-back converter, with multiphase input AC/DC converter and three-phase output DC/AC converter, moreover, the insertion of a bulky DC link capacitor between the two converters is primordial. To avoid the use of two stage conversion system, to reduce the number of components and to give a higher freedom degree for the number of the input phases while keeping the output three-phase system unchanged, the matrix converter can be used to achieve these advantages all together[4]-[8] Matrix converter can be broadly classified into direct and indirect types. Direct Matrix converter is able to convert AC voltage into AC voltage with variable magnitude, frequency and number of phases. It is considered as a powerful topology for AC to AC power conversions, therefore, this kind of converters receive considerable attentions in recent years. They offer inherent advantages such as a bi-directional power flow which is very important in regeneration process, nearly sinusoidal input and output waveform, controlled input power factor, controlled output current magnitude and frequency, compact design and lack of DC-link capacitors for energy storage. Under these advantages several configurations of the matrix converter have been developed and analyzed in the literature [9]-[11]. Nine phases to three phases matrix converter is able to convert an input nine phase voltages system into controlled output three phase voltages system. This configuration has twenty-seven switches, whereas each output phase is connected to all the input phases through a bidirectional semiconductor switches, and hence, there are in total ʹଶ expected switching states. IT is important to clarify that the following constraints have to be taken into account: x Input Phases should not be short circuited together (to protect the input phases). x Output phases should not interrupt under any condition, due to the presence of inductive loads. The switching states are limited to 225. To control those twenty-seven switches and select the appropriate switching states Model Predictive Control (MPC) is used in this paper. The MPC is considered to be one of the most interesting controller [12]-[14] due to their simple implementation. The basic idea of the MPC is to perform a model of the system to be controlled to enable the prediction of the system current in the future next step based on the selection of the optimum operation according to a specified cost function. This cost function determines the required control criteria of the presented model. It is found that the presented structure has several important advantages [15]: ¾ Simple and easy to implement ¾ It can be applied to a great variety of systems. ¾ Simple treatment of constraints. ¾ However, some disadvantages have to be mentioned, like: ¾ ¾
The larger number of calculations, compared to classic controllers, hence more powerful processor is needed for real time implementation. The quality of the model has a direct influence on the quality of the resulting controller.
A schematic of the proposed system is represented in Fig.1, as it is clearly shown, the proposed system consists of nine to three phase matrix converter used to convert nine AC phase input voltage into three
175
Omar Abdel-Rahim et al. / Energy Procedia 42 (2013) 173 – 182
phase AC output, where the magnitude and the frequency of the output current are controlled. The Model predictive control is used to control matrix converter and adjust the switching sequence of the matrix converter switches. An input LC filter is used in the input side to reduce the input current harmonics. The proposed system has some inherent advantages such as; it enables the use of multiphase machine in Wind Energy Generation Systems (WEGS), in this paper a nine phase generator is used to provide the input voltages system of the matrix converter, on the other side the use of the MPC control will give a fast dynamic response.
9
Vs Is Filter Vu
Model Predictive Control
Io
Matrix
3
LLoad o ad
Figure 1: Schematic of the proposed system. 2. NINE-TO-THREE PHASE MATRIX CONVERTER Nine-to-three phase direct matrix converter is able to convert nine-phase input into three-phase output with required voltage magnitude and frequency. General topology of the Nine-to-three matrix converter is represented in Fig. 2. The converter consists of nine-phase inputs and three-phase outputs, so that there are twenty-seven bidirectional power switches; nine switches for every output phase. The Switching pattern is defined as follow: ୧୨ୀ ͳ Ͳ ,with i={a, b, c, d, e, f, g, h, i} and j={A, B, C}, and there is a switching constraint which is defined as follow: ୟ୨ ୠ୨ ୡ୨ ୢ୨ ୣ୨ ୨ ୨ ୦୨ ୧୨ ൌ ͳ
(1)
176
Omar Abdel-Rahim et al. / Energy Procedia 42 (2013) 173 – 182 SaA L
SbA
ia
ScA
a
SdA
C b
ib
L
iA
SeA
C
A SfA f
ic
L c C
id
L
SgA ShA
d SiA
C ie
e C
if
f C ig g C h
ih C
ii
i C SaB SbB ScB SdB SeB
iB
SfB f
B
SgB ShB SiB
SaC SbC ScC SdC SeC SfC f Sg gC ShC SiC
Figure 2: Nine-to-Three Phase Matrix Converter Configurations.
iC
C
Omar Abdel-Rahim et al. / Energy Procedia 42 (2013) 173 – 182
Measure Vs, Ve, Is, Io
Prediction of source and load current Eqs. (9) and (5)
Gopt=inf.
For i=1...225
Cost Function Eq. (10)
i>=225
Applying optimum switchingg states
Figure 3: MPC flowchart
3. MODEL PREDICTIVE CONTROL Due to the inherent advantages of MPC, it is used to control the operation of the matrix converter. As shown in the flowchart, Fig. 3. Input current, source voltage, and voltage after the input filter are sensed to control the input current. Output actual current are measured and the reference for the output current is also commanded to the controller, for controlling the output current. A prediction for the input and output current is performed and their values are calculated, after that the cost function is calculated and an optimization process is performed to enable the choice of the best switching state that give the value for the predicted currents. For the predicted current calculation, a model for input filter, matrix converter and load have to be performed. In the following sections those models are developed for the proposed system.
177
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Omar Abdel-Rahim et al. / Energy Procedia 42 (2013) 173 – 182
3.1 Matrix Model Matrix Converter depicted in Fig. 2 uses a set of bidirectional switches to connect Nine-phase input supply with three-phase load and enables bidirectional power flow. Equation (1) must be satisfied to avoid open circuiting load terminal as this leads to overvoltage that can destroy the components, In addition, to avoid short circuiting supply side. Then, the relation between input and output voltage of the nine-to-three phase matrix converter is as follows: ୟ୳ ۍୠ୳ ې ێୡ୳ ۑ ሺሻ ܵ ܵ ܵ ܵௗ ܵ ܵ ܵ ܵ ܵ ێୢ୳ ۑ ۑ ێ (2) ሺሻ ൌ ܵ ܵ ܵ ܵௗ ܵ ܵ ܵ ܵ ܵ Ǥ ێୣ୳ ۑ ܵ ܵ ܵ ܵௗ ܵ ܵ ܵ ܵ ܵ ێ୳ ۑ େ ሺሻ ێ୳ ۑ ێ୦୳ ۑ ۏ୧୳ ے 3.2 Load Model
IO(t)
R
VO(t)
L
Figure 4: Inductive load modeling. Load model is very important in MPC as it is used to enable the prediction of the load current in the next sample interval. Figure. 4 shows the model of the inductive load. Load current can be obtained from Fig. 2 by applying Kirchhoff law as follows;
ୢ୧ ሺ୲ሻ ୢ୲
ൌ ୭ ሺሻ െ כ୭ ሺሻ
ሺ͵ሻ
Where L and R are the inductance and resistance of the load. The discrete form of the current equation can be expressed as follows: ୢ୧ ୢ୲
ൌ
୧ ሺ୩ାଵሻି୧ ሺ୩ሻ ౩
ሺͶሻ
Where ୱ is the sampling period. The equation for predicting the load current is obtained from substituting (4) into (3), and the resulting equation can be written as follows: ୭ ሺ ͳሻ ൌ ቀͳ െ
ୖכ౩
ቁ כ୭ ሺሻ
౩
כ୭ ሺሻ
ሺͷሻ
Where ୭ ሺ ͳሻ is the predicted value of the current for sampling interval (k+1). 3.3 Filter Model The input filter model, based on the circuit shown in Fig. 5, can be described by the following continuous-time equations:
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ୱ ሺሻ ൌ כୱ ሺሻ ୱ ሺሻ ൌ ୳ ሺሻ
ୢ୧౩ ሺ୲ሻ ୢ୲
ୢ୴౫ ሺ୲ሻ ୢ୲
୳ ሺሻ
ሺሻ
ሺሻ
Where , and are the inductor, the resistor and the capacitor of the input filter respectively. By considering the approximation of the input current derivative: ୢ୧౩ ୢ୲
ൌ
୧౩ ሺ୩ାଵሻି୧౩ ሺ୩ሻ
౩
ሺͺሻ
The equation for predicting the source current is obtained from substituting (8) into (7), the following equation is obtained; ୱ ሺ ͳሻ ൌ ቀͳ െ
ୖ כ౩
ቁ כୱ ሺሻ
౩
כሺୱ ሺሻ െ ୳ ሺሻሻ
IS(t)
Rf
Lf
VS (t)
Cf
ሺͻሻ
Iu (t) Vu(t)
Figure 5: L-C input filter modeling. 3.4 Cost Function The key parameter of the MPC is the cost function as it determines the required control functions. In the proposed system the required control function is to control output current magnitude and frequency and to control input current to be in-phase with input voltage. For the required control criteria of the proposed system, the cost function is expressed as follows;
כ כ כ כ െ ݅ఈ ห ቚ݅ఉ െ ݅ఉ ቚ ห݅௦ఈ െ ݅௦ఈ ห ቚ݅௦ఉ െ ݅௦ఉ ቚ ݃ ൌ ห݅ఈ
(10)
Where כ୭ currents and כ୭ஒ are the real and imaginary parts of the reference output current vector. ୮ ୮ Currents ୭ and ୭ஒ are the real and imaginary parts of the predicted output current vector calculated by equation (5). Currentsכୱ and כୱஒ are the real and imaginary parts of the reference source current vector. ୮ ୮ Currents ୱ and ୱஒ are the real and imaginary parts of the predicted source current vector calculated by equation (9). The reference source current can be calculated as cited in [16]: ݅௦ כൌ ܸ כ ܩ௦ (11) ೞ (12) where, ܩis the instantaneous conductance calculated as follow ܩൌ ԡ ԡమ ೞ
Where ୱ is the average source power and || || is the 2-norm of vectorୱ . This cost function doesn’t need for weighting factor. Elimination the weighting factor from cost function make it simpler and more reliable, as the process of choosing the weighting factor not a straight way and it requires to try many times before reaching to the best value.
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4. Simulation Results. Simulation of the proposed system has been performed under MATLAB/Simulink environment. The parameter of the proposed system is given in Table I. The main functions for the proposed system which will be investigated are: the control of the output current magnitude and frequency and the control of the input current to ensure a reduced shift phase, hence a nearly unity power factor in the input side of the DMC. The output current is fully controlled with the proposed controller; the obtained output three-phase currents are sinusoidal and follow the reference current amplitudes as it is shown in Fig. 6. It is obvious that a nearly a sine waveform is obtained with a low Total Harmonic Distortion (THD). A step change has been presented at 0.1 sec, where the output current magnitude has been changed from 3 to 6 Amp. From Fig. 7, it is shown that the MPC controller has a very fast dynamic response with high ability to control the output current, the output current follows correctly the reference current. The input current control is satisfied, where nearly a nil shift phase is presented between the input current and the input voltage as depicted in Fig. 8. Nine phase input voltages are shown in Fig.9. Table I Simulation parameters Variables Description Simulation Values Input Voltage 200 V ܸ௦ Input Frequency 50 Hz ܨ௦ Input Filter Inductance 15 mH ܮ Input Filter Resistance 0.5 Ω ܴ Input Filter Capacitance 30 μf ܥ Load Resistance 10 Ω ܴ Load Inductance 10 mH ܮ Sample Time 10 μs ܶ௦
Figure 6: Three-phase output current.
Omar Abdel-Rahim et al. / Energy Procedia 42 (2013) 173 – 182
Figure 7: reference current and output current.
Figure 8: Input voltage divided by 10 and input current.
Figure 9: Nine-Phase Input voltage. 5. Conclusion This paper presents nine-to-three phase direct matrix converter as a solution for multiphase generator grid connected. Principle of operation of a nine-to-three phase matrix converter is discussed and analysed. The MPC is used as a powerful control technique to achieve the following goals; control input current at unity power factor, control of the output current magnitude and frequency under the nine- phase input to
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three-phase output conversion. A cost function is used to achieve the required objectives; it presents the advantage that it does not include a weighting factor. The proposed system was simulated using MATLAB software. References [1] S. Ayman, A. Khalik, and K. H. Ahmed, “Performance Evaluation of Grid Connected Wind Energy Conversion Systems with Five phase Modular Permanent Magnet Synchronous Generators having Different Slot and Pole Number Combinations” Proc. IEEE International Electric Machines and Drives conference, IEMDC-2011, 1518 May, Niagara Falls, Canada, pp. 1135-1140, 2011. [2] O. Ojo and IE. Davidson, “PWM-VSI inverter-assisted stand-alone dual stator winding induction generator”, IEEE Trans Energy Conversion, vol. 36, 2000, pp. 1604–1611. [3] O. Abdel-Rahim, M. Orabi and M.E. Ahmed," High gain single-stage inverter for photovoltaic AC modules " Applied Power Electronics Conference and Exposition (APEC), 6-11 March 2011, pp. 1961- 1967. [4] O. Abdel-Rahim, M. Orabi and M. Ahmed," Development an efficient photovoltaic (PV) configuration for low power applications” IEEE International Conference on Power and Energy (PECon), 2010, pp 622- 627. [5] O. Abdel-Rahim, H. Abu-Rub and Sk. Moin “Space vector PWM or five to three phase matrix converter” Applied Power Electronics Conference and Exposition (APEC), 2013, in press. [6] Ahmed, SK.M., Iqbal, A., Abu-Rub, H., Rodriguez, J., Rojas, C., (2010), “Simple carrier-based PWM technique for a three to nine phase matrix converter”, IEEE Trans. On Ind. Elect., vol. 58, no. 11, pp. 5014-5023, Nov. 2011. [7] Ahmed, SK., M., Iqbal, A., Abu-Rub, H., Cortes, P., (2011), “Model predictive control of a three to five-phase matrix converter”, Proc. Workshop on Predictive Control of Power electronics and Drives, 14-15 Oct., 2011, Munich, Germany, pp. 71-76. [8] Ahmed, SK. M., Iqbal. A., Abu-Rub, H, (2010), “Carrier-based PWM technique of a novel three-to-seven-phase matrix converter” Int. Conf. On Electrical Machine ICEM’10, 3-6 Sept. Rome, Italy, CD-ROM paper no RF004944. [9] P. Wheeler, J. Rodríguez, J. Clare, L. Empringham and A. Weinstein “Matrix converters: a technology review,” ”, IEEE Trans. On Ind. Elect., vol. 49, no. 2, April 2002. [10] Iqbal, A., SK, Moin. A., Abu-Rub, H., “Space Vector PWM Technique for a Three-to-Five Phase Matrix Converter”, , IEEE Trans. On Applications, Vol. 48, No. 2, March/April 201 [11] W. Kolar, F. Schafmeister, S. Round and H. Ertl” Novel three-phase ac–ac sparse matrix converters”, IEEE Trans. On Power Electronics, vol. 22, no. 5, SEPTEMBER 2007. [12] J. Rodriguez, H. Young, C. Rojas, S. Kouro, P. Cortes, and H. Abu-Rub, State of the Art of Model Predictive Control in Power Electronics and Drives, accepted, IEEE Trans. on Industrial Electronics, 2012. [13] Abu-Rub, H, Guzinski, J., Krzeminski, K. and Toliyat, H.: Predictive Current Control of Voltage Source Inverter. IEEE Trans. on Industrial Electronics, USA, Vol. 51, No. 3, June 2004, pp. 585-593. [14] Cortes, P., Wilson, A., Rodriguez, J., Kouro, S. and Abu-Rub, H. “Model Predictive Control of Multilevel Cascaded H-Bridge Inverters, accepted at IEEE Transactions on Industrial Electronics, Vol. 57, No. 8, August 2010, 2691 - 2699. [15] J. Rodriguez, P. Cortes “Predictive Control of Power Converters and Electrical Drives” Wiley, IEEE. [16] J. Rodriguez, J. Espinoza, M. Rivera, F. Villarroel, C. Rojas” Predictive control of source and load currents in a direct matrix converter” IEEE International Conference on Industrial Technology (ICIT), 2010.