9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice ROBUST SLIDING MODE CONTROL OF MATRIX CONVERTERS WITH UNITY POWER FACTOR S. Ferreira Pinto, J. Fernando Silva Instituto Superior Técnico (IST), Máquinas Eléctricas e Electrónica de Potência, Av. Rovisco Pais 1096, Lisboa Codex, Portugal fax: +351-21 841 71 67; e-mail: pcsoniafp@alfa.ist.utl.pt fernandos@alfa.ist.utl.pt Košice Slovak Republic 2000 Abstract. This work presents the design of robust sliding mode controllers for three-phase ac-ac matrix converters, in order to guarantee output sinusoidal waveforms and near unity input power factor. First, the sliding mode controller, based on the state-space vectors modulation technique, is designed for the matrix converter and the obtained results are compared to those obtained using the classical Alesina / Venturini method. Then, a new sliding mode controller is designed, considering the matrix converter with a high frequency ‘LC’ filter. A new integrated approach is necessary to design the controller, using the matrix converter model in ‘αβ’ coordinates and the state-space vectors modulation technique. The obtained results show that sliding mode controllers guarantee a robust on-line control of the matrix converter output voltages, with near unity input power factor. Keywords: Matrix converters, Sliding mode control. connection of each one of the three output phases to one of the three input phases, as shown in Fig. 1. 1. INTRODUCTION With the recent progress of power devices and the development of large power integrated circuits, matrix converters have become extremely attractive as they are capable of providing simultaneously sinusoidal output voltages with varying amplitude and frequency as well as sinusoidal input currents with near unity input power factor. Besides, they allow bi-directional power flow, have noise above the audio range and have minimum energy storage requirements, that results in a higher efficiency, when compared with the previously used rectifier-inverter structures (which have an intermediate DC link) [4,6,8]. Matrix converters are useful in many applications, especially ac drives and, after the introduction of their high frequency control in 1980 by Alesina and Venturini [1,2] several control techniques have been studied [3,4,5,9,11,12,13]. One of the most recent is the Space Vector Modulation technique [6,7,8,10,14,15,16], with several advantages over the previous control methods such as immediate comprehension of the required commutation processes, simplified control algorithm and maximum voltage transfer ratio without adding third harmonic components. In this paper, the association of sliding mode controllers, known for their robustness and system order reduction, together with the State-Space vector modulation technique, will allow the design of a robust controller for the matrix converter. 2. MATRIX CONVERTER MODEL The matrix converter is made by an association of nine bidirectional switches, with turn-off capability, that allow the i va a vb vc ib S11 S21 S31 S12 S22 S32 S13 S23 S33 ic iB iA vAB iC vBC vCA Fig. 1. Matrix converter topology However, not all the switches combinations are possible: there are constraints to guarantee that the input phases are never short-circuited and that the output currents are never interrupted (due to the presence of inductive loads) [6]. Considering these constraints, only 27 switching combinations are possible [6] and are schematically represented according to a nine-element matrix (1). Each one of this matrix elements represents one switch with two possible states: ‘1’ if it is closed and ‘0’ if it is open and the referred constraints are verified if the sum of each one of this matrix rows elements is always equal to '1' [6]. S11 S12 S13 S = S 21 S 22 S 23 (1) S 31 S 32 S 33 The matrix which relates the input phase to the output line currents and voltages [6] (2) is obtained from (1): 1 - 157 9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice S11 − S 21 Sc = S 21 − S 31 S 31 − S11 S12 − S 22 S 22 − S 32 S 32 − S12 S13 − S 23 S 23 − S 33 S 33 − S13 (2) In order to guarantee that the output voltages follow the predefined reference, it is necessary that, in a certain switching period ‘T’, the output voltages average values (5,6) are equal to their reference average values voα and ref The matrix which relates the output line voltages to the input line voltages (3) can be obtained from (2) [16]. S11 − S21 − S12 + S22 S12 − S22 − S13 + S23 S13 − S23 − S11 + S21 3 3 3 S − S − S + S 21 31 22 32 S 22 − S 32 − S 23 + S33 S 23 − S33 − S 21 + S 31 S' = 3 3 3 S31 − S11 − S32 + S12 S32 − S12 − S33 + S13 S33 − S13 − S31 + S11 3 3 3 (3) Considering the previously defined matrices (1,2,3), it is possible to establish relations for the output line voltages [16] and for the input phase currents (4). v ab v AB v BC = S ' vbc vCA v ca ia i = S T b ic i A i B iC ref . This is valid if the sliding mode controller commutation frequency is much higher than the controlled voltage frequency. Making the derivatives of (5,6) we can write the system equations in the phase canonical form (7,8). d voα 1 = voα (7) dt T d voβ dt = 1 vo T β ±2, ±5, ±8 ±1, ±2, ±3 The sliding surfaces are expressed as linear combinations of all the phase canonical state variables that do not depend directly on the command variables (9,10). − voα = 0 S ( evo , t ) = K α v oα (9) α ref − voβ = 0 S ( evo , t ) = K β voβ β ref β ±3, ±6, ±9 ±1, ±4, ±7 (10) The two necessary sliding surfaces (11,12), expressed as functions of the output voltages and their reference values are easily obtained from (9,10). K T − voα ) dt = 0 S ( evo , t ) = α ∫ ( v oα (11) ref α T 0 S ( evo , t ) = ±7, ±8, ±9 Output line voltage vectors (8) (4) In order to use the State-Space Vector modulation technique, it is necessary to express the 27 previously referred switching states in αβ co-ordinates, using the Concordia transformation [6]. However, as six of these states have time varying phase and will not be used [6] and three other combinations correspond to the null state, this leaves only 18 possible non-null switching combinations that can be represented in the ‘αβ’ plane according to Fig. 2. [6]. ±4, ±5, ±6 voβ Kβ T − voβ ) dt = 0 ∫ ( vo T 0 β ref (12) According to the number of state-space vectors available at each time instant, two three level comparators are necessary. The results of S(evo ) and S(evo ) are applied to Input phase current vectors α Fig. 2. Representation in the ‘αβ’ plane of the output line voltage and the input current state-space vectors 3. DESIGN OF THE SLIDING MODE CONTROLLER FOR THE MATRIX CONVERTER Output voltage control The Concordia transformed output voltages (voα, voβ) are discontinuous functions of the control variables and, in the specific case of the matrix converter, with no associated dynamics. The average values of these voltages ( voα , voβ ) may be calculated, in ‘αβ’ co-ordinates, according to (5,6). voα = 1T ∫ v o dt T 0 α (5) voβ = 1T ∫ v o dt T 0 β (6) β the comparators, and according to each one of the nine possible error voltages, one of the two nearest and highest amplitude state-space vectors available at that time instant is applied to the matrix converter (Fig. 2., Tab. 1.). The existence of these two vectors, capable of guaranteeing the desired output voltage, introduces an extra degree that will be further used to control the input power factor. Input power factor control Using the Blondel-Park transformation, when the rotating frame is synchronized with the mains, the ‘q’ component is zero. In order to guarantee the near unity input power factor, it is necessary that the three input currents have exactly the same phase as the corresponding input voltages (in ‘abc’ coordinates). Using the Blondel-Park transformation, this condition is verified when the input current component ‘ i q ’ is equal to zero. The ‘ id ’ current component is automatically adjusted in order to guarantee the input/output power constraint. 1 - 158 9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice evα φu >-π/6 φu <π/6 evβ -1 -1 -1 0 0 0 1 1 1 -1 0 1 -1 0 1 -1 0 1 φu >π/6 φu <3π/6 φu >3π/6 φu <5π/6 φu >5π/6 φu <7π/6 φu >7π/6 φu <9π/6 φu >9π/6 φu <11π/6 eiq=1 eiq= -1 eiq=1 eiq= -1 eiq=1 eiq= -1 eiq=1 eiq= -1 eiq=1 eiq= -1 eiq=1 eiq= -1 +3 +3; -6 -6 -9 0 +9 +6 -3; +6 -3 -1 -1; +4 +4 +7 0 -7 -4 +1; -4 +1 -2 -2; +5 +5 +8 0 -8 -5 +2; -5 +2 +3 +3; -6 -6 -9 0 +9 +6 - 3; +6 -3 +1 +1; -4 -4 -7 0 +7 +4 -1; +4 -1 -2 - 2; +5 +5 +8 0 -8 -5 +2;- 5 +2 -3 -3; +6 +6 +9 0 -9 -6 +3;-6 +3 +1 +1; -4 -4 -7 0 +7 +4 -1; +4 -1 +2 +2; -5 -5 -8 0 +8 +5 -2; +5 -2 -3 -3; +6 +6 +9 0 -9 -6 +3; -6 +3 -1 -1; +4 +4 +7 0 -7 -4 +1; -4 +1 +2 +2; -5 -5 -8 0 +8 +5 -2; +5 -2 Tab. 1. Output voltages and input currents Space Vector control (φu represents the input voltage instantaneous phase in ‘αβ’ coordinates) As for the output voltages, the input currents are also discontinuous variables with no associated dynamics. For this reason, the current controller design laws are similar to those obtained for the output voltage: assuming that it is possible to obtain an input current whose average value in a certain switching period ‘T’ is equal to the reference current, the sliding surface to the ‘ i q ’ current is designed according to (13) [15]. T S ( ei q ) = Kq ∫ eiq dt = 0 (13) 0 At each time instant, the measured ‘ i q ’ current is compared to a zero reference, then the sliding surface is applied to this current error ( ei q ) and the final result is applied to a twolevel comparator. According to the comparator output, it is chosen the best of the two state-space vectors available. Simulation results Two control methods were simulated and compared: the proposed sliding mode controller and the classical Alesina and Venturini [1,2]. In order to test the robustness of the sliding mode controller, the results were obtained to a 20% increase in the mains voltages at t=0.05s. The reference voltage was chosen to have an amplitude of 2 × 150 V and a frequency of fo=60Hz. An inductive L=10mH and resistive load R=10Ω were considered. The MATLAB/ SIMULINK simulations show that both methods present good results in steady-state operation (t<0.05s) (the output line voltages and input phase current follow the references, guaranteeing the near unity input power factor). The proposed controller (Fig. 3. a, b, c) presents a good response to the increase in the mains voltages. The input phase current amplitude decreases at t=0.05s (Fig. 3.a) as a consequence of the input-output power constraint. Also, the input power factor as well as the output line average voltages and the output phase currents (Fig. 3.b, c) keep unchanged, as expected. The results obtained using Venturini controller (Fig. 3.d, e, f) were not satisfactory. This controller does not have a good line rejection factor as a consequence of the predefined controller parameters. At the instant t=0.05s the amplitude of the output voltages and currents, increase (Fig. 3.e, f), when it should keep 40 40 600 30 30 400 20 20 vAB [V] iia [A] 0 -10 iA [A], iB [A], iC [A] 200 10 0 -200 -20 10 0 -10 -20 -400 -30 -30 -600 -40 0.02 0.03 0.04 0.05 t [s] 0.06 0.07 0.08 0.02 0.03 0.04 a) 0.05 t [s] 0.06 0.07 -40 0.02 0.08 0.03 0.04 b) 40 0.05 t [s] 0.06 0.05 t [s] 0.06 0.07 0.08 c) 40 600 30 30 400 20 20 vAB [V] iia [A] 0 -10 iA [A], iB [A], iC [A] 200 10 0 -200 -20 10 0 -10 -20 -400 -30 -30 -600 -40 0.02 0.03 0.04 0.05 t [s] 0.06 0.07 0.08 0.02 d) 0.03 0.04 0.05 t [s] 0.06 0.07 -40 0.02 0.08 e) 0.03 0.04 0.07 0.08 f) Fig. 3. AC-AC matrix converter waveforms obtained for a 20% increase in the mains voltages with an output reference voltage of 2 × 150 V and fo = 60Hz, using the proposed sliding mode controller a) b) c) and the Alesina and Venturini controller d) e) f); a) d) Input phase current 'ia'; b) e) Output line voltage 'vAB'; c) f) Output phase currents 'iA', 'iB', 'iC' 1 - 159 9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice 0 0 1 0 l 0 i 1 1 α + − + 0 2C 2 3C iβ 0 1 1 − − 0 2C 2 3C unchanged. As a result, the input currents amplitude also increases in order to guarantee the input-output power constraint (Fig. 3.d). 4. INTEGRATED SLIDING MODE CONTROLLER FOR A MATRIX CONVERTER WITH AN INPUT ‘LC’ FILTER The introduction of the ‘LC’ filter is responsible for a new approach to control ac-ac matrix converters (Fig. 4.). The controller is now designed based on the integration of the switches with the input filter [16]. The output voltage controller will no longer depend directly on the mains voltages – it will depend on the voltages applied to the switches (the capacitor voltages) (Fig. 4.). Also, the input current controller will have to be designed considering the effect of the filter, that introduces some differences both in phase and amplitude between the currents actually consumed from the mains iia , iib , ii c and the currents at (15) The sliding mode controllers will be designed based on this model. via iia r l ia S 11 iA S 12 vab vib Cab S 21 iib r l ib r l S 31 S 32 ic S 33 iB Load S 23 Cbc iic vCA vAB S 22 vca vbc vic Load S 13 Cca the switches inputs ia , ib , i c (Fig. 4.). Matrix converter model The state-space model (14) of the input filter can be directly obtained from Fig. 4. diia r 1 1 − 0 0 − 0 dt 3l 3l l di r 1 1 i b 0 − − 0 iia 0 dt l 3l 3l di r 1 1 iib i c − − i 0 0 0 l 3l 3l ic + dt = 1 1 dv ab − 0 0 0 0 v ab 3C 3C v bc dvdt 1 1 bc − 0 0 0 0 v ca C C 3 3 dt 1 dv ca − 1 0 0 0 0 dt 3C 3C 1 0 0 0 l 0 0 0 0 0 1 0 0 0 0 0 i 1 a l via 1 − 0 1 + 3C 3C ib + (14) vi 0 0 l b 1 1 v i 0 − c 0 0 0 i c 3C 3C 0 0 0 1 1 0 − 3C 3C 0 0 0 The state-space model can be written according to (15), using the Concordia transformation [16]. diiα r 1 1 − − 0 − l 2l 2 3l dt di r 1 1 iiα iβ − − 0 l 2l iiβ dt l 2 3 = v + dv c α 1 − 1 0 0 cα dt 2C v cβ 2 3C dv 1 1 c β 0 0 2C dt 2 3C 0 1 viα l viβ 0 0 v BC iC Load Fig. 4. AC-AC Matrix Converter with input ‘LC’ filter Output voltage sliding mode control The output voltage sliding mode controller is designed as in section 3, considering that the switching vectors are calculated based on the capacitors voltages [16]. Input power factor control Following the same line of thought of section 3, in order to control of the ii q current, it is necessary to obtain the state space model (15) in ‘dq’ co-ordinates (16), using the Blondel-Park transformation. diid r 1 1 ω − − − l 2l 2 3l dt di r 1 1 iid iq − ω − − l 2l iiq dt l 2 3 dv = v + cd 1 − 1 0 ω cd dt 2C v cq 2 3C dv 1 1 cq −ω 0 2C dt 2 3C 0 0 1 0 l 0 0 1 1 id 1 vid (16) + − + 0 2C 2 3C i q l viq 0 0 1 1 − − 0 0 2C 2 3C To design the sliding mode controller, the system equations are written in the phase canonical form (17). diiq =θ dt dθ = − r θ + − ω2 − 1 i + ωr i + 1 i + i i q dt l l d 3lC 3lC q 1 - 160 9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice 1 dv iq ω ω ω v cd + v cq − v id + (17) l l l dt 3l Knowing that the system must follow a predefined reference, with an error as close to zero as possible, new state error variables, such as the input current error eii are + q defined (18). eii = ii q q ref − ii q (18) As the input current error eii command variable is the iq q current, the sliding surface is expressed as a linear combination of all the phase canonical state variables that do not depend directly on the referred command variable. As a consequence, the sliding mode controller (16) will be one order lower than the system order written in the phase canonical form (19). S ( eii ) = eii + k θ eθ (19) q q Doing all the necessary algebra, it is possible to express the sliding surface (19) as a combination of the input filter currents and voltages (20). r S ( eii ) = iiq + k θ θ ref − 1 − k θ iiq + q ref l + k θ ω iid − kθ nominal values but does not present a robust response concerning line supply variations. As a conclusion, the designed sliding mode controllers are robust and can be implemented on-line without any special circuits. Associated with the state space vectors they allow a quick and efficient choice of the correct switches combinations and can operate at near unity input power factor. v cd + kθ k v cq − θ viq l 2l (20) 2 3l As before (section 3), the result of S ( eii ) is applied to a q two-level comparator. According to this comparator output, it is chosen the best of the two state space-vectors available [14,15,16]. Simulation results In order to test the robustness of the proposed sliding mode controller, the results were obtained to a 20% increase in the mains voltages at t=0.05s (Fig. 5.) with a reference voltage of 2 × 150 V and fo=60Hz. An inductive L=10mH and resistive load R=10Ω, and an input filter of l=1mH, r=3Ω and C=10µF were considered. The MATLAB/ SIMULINK simulations show that the controller presents good results to the increase in the mains voltages. The amplitude of the input phase current decreases at t=0.05s (Fig. 5.a) in order to guarantee the input-output power constraint and the capacitors voltages increase, as expected (Fig. 5.b). Also, the input power factor keept unchanged, as well as the output line average voltages and the output phase currents (Fig. 5.d, e). 5. CONCLUSIONS This paper presents the design of sliding mode controllers for ac-ac matrix converters: first, for the matrix converter alone and then for the matrix converter with an input ‘LC’ filter. The introduction of this filter is responsible for a new integrated approach to control these converters. The sliding mode on-line controllers were tested and the results were the expected as they presented good and robust responses. On the contrary, the classical Alesina/Venturini method, based on predefined values, works well for the 6. REFERENCES [1] Alesina, A.; Venturini, M.; "Solid-State Power Conversion: A Fourier Analysis Approach to Generalized Transformer Synthesis"; IEEE Transactions on Circuits and Systems, Vol. CAS-28, no 4, April 1981, pp. 319-330. [2] Alesina, A.; Venturini, M.; "Analysis and Design of Optimum-Amplitude Nine-Switch Direct AC-AC Converters"; IEEE Transactions on Power Electronics, Vol. 4, no 1, January 1989, pp. 101-112. [3] Watthanasarn, C.; Zhang, L.; Liang, D. T. W.; "Analysis and DSP-based Implementation of Modulation Algorithms for AC-AC Matrix Converters"; Proc. PESC'96 Conference, pp. 1053-1058, June 1996, Baveno, Italy. [4] Kwon, W. H.; Cho, G. H.; "Static and Dynamic Characteristics of Nn-Ideal Step Up Nine Switch Matrix Converter"; Proc. EPE'91 Conference, Vol. 4, pp. 418-423, Firenze, Italy. [5] Holmes, D. G.; Lipo, T. A.; "Implementation of a Controlled Rectifier Using AC-AC Matrix Converter Theory"; IEEE Transactions on Power Electronics, Vol. 7, no 1, January 1992, pp. 240-250. [6] Huber, L.; Borojevic, D.; Burany, N.; "Analysis, Design and Implementation of the Space-Vector Modulator for Forced-Commutated Cycloconverters"; IEE Proceedings-B, Vol.139, no 2, March 1992, pp. 103-113. [7] Wiechmann, E. P.; Espinoza, J. R.; Salazar, L. D.; Rodriguez, J. R.; "A Direct Frequency Converter Controlled by Space Vectors"; Proc. PESC'93 Conference, pp. 314-320, Seattle, USA. [8] Casadei, D.; Grandi, G.; Serra, G.; Tani, A.; "Space Vector Control of Matrix Converters with Unity Input Power Factor and Sinusoidal Input/Output Waveforms"; Proc. EPE'93 Conference, Vol. 7, pp. 170-175, Brighton, England. [9] Kaserani, M., Ooi, B. T.; "Feasibility of Both Vector Control and displacement Factor Correction by Voltage Source Type AC-AC Matrix Converter", IEEE Transactions on Industrial Electronics, Vol. 42, no 5, October 1995, pp. 524-530. [10] Huber, L.; Borojevic, D.; "Space Vector Modulated Three-Phase to Three-Phase Matrix Converter with Input Power Factor Correction"; IEEE Transactions on Industry Applications, Vol. 31, no 6, November/December 1995, pp. 1234-1246. [11] Oyama, J.; Xia X.; Higuchi, T.; Yamada, E.; "Effect of PWM Pulse Number on Matrix Converter Characteristics"; Proc. PESC'96 Conference, pp. 1306-1311, June 1996, Baveno, Italy. 1 - 161 9th International Conference on Power Electronics and Motion Control - EPE-PEMC 2000 Košice 40 40 600 30 30 400 20 10 0 -10 -20 200 10 ia [A] Capacitors voltages [V] iia [A], via/10 [V] 20 0 0 -10 -200 -20 -400 -30 -30 -600 -40 0.02 0.03 0.04 0.05 t [s] 0.06 0.07 0.08 0.02 0.03 0.04 a) 0.05 t [s] 0.06 0.07 -40 0.02 0.08 0.03 0.04 b) 0.05 t [s] 0.06 0.07 0.08 c) 40 600 30 400 20 10 iA, iB, iC [A] vAB [V] 200 0 0 -10 -200 -20 -400 -30 -600 0.02 0.03 0.04 0.05 t [s] 0.06 0.07 -40 0.02 0.08 d) 0.03 0.04 0.05 t [s] 0.06 0.07 0.08 e) Fig. 5. AC-AC matrix converter waveforms for a 20% increase in the mains voltages with an output reference voltage amplitude of 2 × 150 V and frequency fo = 60Hz; a) Input phase current iia and input phase voltage va / 10; b) Input filter capacitors voltages vab, vbc, vca; c) Input filter intermediate current ia; d) Output line voltage vAB; e) Output phase currents iA, iB, iC [12] Matsuo, T.; Bernet, S.; Colby, R. S.; Lipo, T.; "Modelling and Simulation of Matrix Converter/Induction Motor Drive"; Proc. ELECTRIMACS'96 Conference, pp. 1-10, September 1996, Saint-Nazaire, France. [13] Oyama, Jun; Xia X.; Higuchi, T.; Yamada, E.; "Displacement Angle Control of Matrix Converter"; Proc. PESC'97 Conference, pp. 1033-1039, May 1997, Saint Louis, USA. [14] Casadei, D.; Serra, G.; Tani, A.; "The use of Matrix Converters in Direct Torque Control of Induction Machines"; Proc. IECON'98 Conference, Vol. 2, pp.744-749, September 1998, Aachen, Germany. [15] Pinto, S.; Silva, J.; ‘Modeling, Simulation and Sliding Mode Control of Matrix Converters with Sinusoidal Input/Output Waveforms and Near Unity Input Power Factor’; Proc. ELECTRIMACS'99 Conference, Vol. 1, pp. 1.139-1.144, September 1999, Lisboa, Portugal. [16] Pinto, S.; Silva, J.; ‘Sliding Mode Control of Space Vector Modulated Matrix Converter with Sinusoidal Input/Output Waveforms and near Unity Input Power Factor’, Proc. EPE’99 Conference, September 1999, Lausanne, Switzerland. THE AUTHORS Sónia Ferreira Pinto received the Dipl. Ing. degree and the M.Sc. degree in electrical and computer engineering in 1992 and 1995, respectively, from the Instituto Superior Técnico, Universidade Técnica de Lisboa, Lisbon, Portugal, where she is currently working towards the Ph. D. degree. She is currently an Assistant at Electrical Engeneering Department and researcher at Centro de Automática, Instituto Superior Técnico. Her main interests are modeling, simulation and control of power converters. J. Fernando A. Silva, born in Monção Portugal in 1956, received the Dipl. Ing. in Electrical Engineering (1980) and the Doctor Degree in Electrical and Computer Engineering (Power Electronics and Control) in 1990, from Instituto Superior Técnico (IST), Universidade Técnica de Lisboa (UTL), Lisbon, Portugal. He is currently Associate Professor of Power Electronics at IST, teaching Power Electronics and Control of Power Converters, and researcher at Centro de Automatica of UTL. His main research interests include power semiconductor devices, modelling and simulation, new converter topologies and sliding mode control of power converters. He has written more than a hundred papers. 1 - 162