2013 IEEE INTERNATIONAL CONFERENCE ON CIRCUITS AND SYSTEMS
Model Predictive Control of an Active Front End Rectifier with Unity Displacement Factor S. M. Muslem Uddin, Parvez Akter, S. Mekhilef and M. Mubin Dept. of Electrical Engineering University of Malaya 50603 Kuala Lumpur, Malaysia e-mail:
[email protected] [email protected] [email protected] [email protected]
Department of Industrial Technologies Universidad de Talca, Curico, Chile e-mail:
[email protected]
Keywords—predictive control; unity displacement factor; active front end rectifier; ac-dc power converter
INTRODUCTION
Active front end rectifier (AFE) has great advantages in the sector of industrial applications comparative to conventional diode based rectifiers. Due to the discrete capability of IGBT’s, it can be controlled precisely which offers controllable power factor, low harmonic input current and bidirectional power flow [1]. AFE rectifier can be used in active power filter and renewable energies where interconnection stages are mostly essential [2]. Different control scheme have been classified as; voltage oriented control (VOC) and direct power control (DPC) which is investigated in [3]. To control the DC-link voltage, a PI controller is used to generate the direct current references and input current control can be achieved with another linear PI controller in the rotating frame [4]. Also input output linearization and state feedback control can be implemented to control the input current [5]. The successive development and rising capabilities of digital signal processing technologies and semiconductor modern microprocessors has been made possible to implement more complex and sophisticated control algorithm precisely to satisfy the high performances in the industries. Model predictive control (MPC) is the method which attained a great contemplation for power converters
II.
MATHEMATICAL MODELLING OF THE TOPOLOGY
The AFE rectifier topology considered in this research work is shown in Fig. 1 which corresponds to a fully controlled three phase bridge with semiconductor switches connected to the three-phase voltage supply (vs) using the input filter inductances (Lf) and resistances (Rf).
The authors wish to thank the financial support from the University of Malaya through HIR-MOHE project UM.C/HIR/MOHE/ENG/17.
81 978-1-4799-1337-4/13/$31.00 ©2013 IEEE
Departamento de Electrónica, Universidad Técnica Federico Santa María (USM) Avenida España 1680, Casilla 110-V Valparaíso, Chile e-mail:
[email protected]
and drive systems. This control technique have been successfully applied to different power converters such as; neutral point clamped inverters [6], DC-DC converters [7], three phase voltage source inverter [8], cascaded H-bridge inverters [9] and in the indirect matrix converter [10] etc. In this research work, a model predictive control (MPC) technique has been proposed to control an AFE rectifier with unity power factor accurately. MPC control uses a discrete-time model of the system to make prediction of the future nature of the system variables concern to determine the cost functions for all possible switching states and selects the optimal firing actuation to converter for the next sampling period based on minimum cost function [11]. Furthermore, predictive control has been successfully investigated in different applications with the voltage source inverter and matrix converter in recent few years [8, 12-16]. Also predictive control technique have been applied in the IMC for different works such as; current control [10], current control extended by reactive power minimization [17], implementation of virtual damping resistance to mitigate the resonance effect due to low damped input filter [18] and imposed sinusoidal load and source currents [19] etc. Furthermore a comprehensive review on most exoteric control strategies and momentous impact on the resonance of the MC input filter have been discussed in [20]. This paper is organized as follows: Section II is related to mathematical background of the AFE topology. Section III presents proposed predictive control algorithm. Section IV states the simulation results and analysis of the investigations. Finally a significant conclusion is drawn in section V.
Abstract—A finite control set model predictive control of an active front end (AFE) rectifier with unity displacement of input voltage and current has been investigated and presented in this paper. In this proposed control algorithm the discrete nature of the power converter are utilized and predicts the future behavior of the input system variables. Also the method selects the switching state that minimizes a cost function for firing the converter in the next sampling period. The proposed predictive control algorithm is validated through simulation and shows potential control and tracking of input unity power factor.
I.
J. Rodriguez
M. Rivera
Fig. 1. Active front end rectifier topology.
2 (7) (va 0 + ωvb 0 + ω 2vc 0 ) 3 The voltage vector ( v R ) is determined by the DC-link vR =
A. Rectifier Model The model of the ac side of the rectifier for each phase can be written as;
di v sa = L f sa + R f isa + v a 0 − v n 0 dt di sb v sb = L f + R f isb + vb 0 − v n 0 dt di v sc = L f sc + R f isc + v c 0 − v n 0 dt
voltage (Vdc ) and the switching state vector ( S R ) of the rectifier expressed as;
(1) (2)
(8) vR = S RVdc If S a , Sb and S c are the switching states of each leg in Fig.1
(3)
then the switching state vector ( S R ) becomes
SR =
vsa , vsb , vsc are the input phase voltages and isa , isb , isc are the input currents.
Where
2 (vsa + ωvsb + ω 2vsc ) 3 2 is = (isa + ωisb + ω 2isc ) 3
d is 1 = (vs − vR − R f is ) dt Lf
(4)
III.
j 2π / 3
−
d is 2 + R f is + (v a 0 + ω vb 0 + ω 2 v c 0 ) dt 3
(10)
PROPOSED PREDICTIVE CONTROL ALGORITHM
Model predictive control (MPC) algorithm uses the discrete nature and finite number of valid switching states of the power converter. The proposed scheme maintain the predictive values close to their respective references at the end of the sampling instants. The proposed MPC control scheme and algorithm are presented in Fig. 2 and Fig. 3, respectively. Predictive controller satisfies all the aforementioned constraints by using the following four steps;
(5)
and by substituting the values of Where ω = e equation (1), (2), (3) and (5), the supply voltage vector of equation (4) becomes
vs = L f
(9)
Hence the equation (6) becomes
The input phase voltages and input currents can define with space vector definition as;
vs =
2 ( S a + ω Sb + ω 2 S c ) 3
(6)
2 (v n 0 + ωv n 0 + ω 2 v n 0 ) 3
• •
Note that the last term of the equation (6) is zero as
(1 + ω + ω ) = 0 and the voltage vector generated by AFE 2
Rectifier can be defined as;
82
For the k th sampling instant, input current imeas, supply voltage vs, DC voltage Vdc and reference currents Iref are measured. PI controller is used to set reference currents Iref from the error signal between the measured and reference DC voltage.
Fig. 2. Predictive control scheme.
•
•
A.
for both real and imaginary parts can be combined in a single term to express the cost function.
For each possible actuating states of AFE rectifier, values of input currents I i k + 1 are predicted in the
g = is*α − ispα + is*β − ispβ
next sampling period (k+1). All the predictive values are compared with their respective reference currents and determined the cost functions of all possible switching states. The switching state corresponds to the minimum cost function is selected in the next sampling time period switching.
(13)
Discrete Time Model of Predictive Controller It is important to derive a discrete time model for the power converter system due to model predictive controller is formulated in discrete time domain. In this work, inductive input filter model is considered for the prediction of the input current. To estimate the next sampling value of the input current considering current and voltage measurements at the kth sample time, a discrete model of the input side can be employed. For kTs≤ t ≤ (k+1)Ts, with Ts being the sampling time, the state space discrete time model for input current is determined as
d is is (k + 1) − is ( k ) ≈ dt Ts
(11)
[
⎛ RT ⎞ T is ( k + 1) = ⎜1 − f s ⎟is ( k ) + s vs ( k ) − v R ( k ) ⎜ ⎟ L L f f ⎠ ⎝
B.
]
(12)
Cost Function Determination For computational simplicity in the cost function an absolute error is considered for the reference and predicted values. The predicted input current with its reference values
Fig. 3. Predictive control algorithm flow chart for the proposed control.
83
IV.
SIMULATION RESULTS AND ANALYSIS
The proposed control scheme is simulated in MATLAB Simulink to validate the performance of the active front end rectifier. The simulation parameters are depicted in Table I and the simulation has been carried out with Ts =50 µS. DC-link reference is taken as 346 V which compared with output DC-link voltage. The error calcualed from the reference and output DC voltage is used as an input to PI controller. The reference current is calculated with this PI controller. This reference current is compared with predicted input current for all possible switching states for the active front end rectifier. Hence cost functions are found for 8 valid switching states of the converter. The switching state corresponds to minimum cost function is selected for the next sampling time firing actuation. (a)
Fig. 4. Supply voltage [V] and input current [A] with unity power factor.
(b) Fig. 6. (a) DC voltage [V] and DC voltage reference [V]; (b) DC current [A].
Fig. 4 shows the simulation output of both supply voltage and input current which reveals the unity power factor because the angle beween input voltage and current is zero degree (00). Also, Fig. 5 shows the balanced and low harmonic three phase input current which is expected. Fig. 6(a) shows a very good tracking of DC-link reference voltage. It maintains smooth magnitude around 346 V DClink compare to the reference DC-link which is taken as 346 V. Fig. 6(b) is the presentation of DC current. In Fig. 6(b), the average DC current is around 21.5 A. So, the output power becomes 7.439 kW. The input power is calculated as 7.5 kW for three phase supply. Therefore, the efficiency of the power transfer becomes 99.18% which is very encouraging and this is due to the unity power factor control in power transfer.
Fig. 5. Three phase input current [A].
84
TABLE I. SL.No
SIMULATION PARAMETERS
[8]
Parameters values and unit Variables and Parameters
Values
Unit
1.
Supply Voltage (vs)
200
V
2.
Supply Frequency (fs)
50
Hz
3.
Reference DC-link (Vdc)
346
V
4.
Input Filter Inductance (Lf)
10
mH
5.
Input Filter resistance (Rf)
0.1
Ω
6.
Sampling Time (Ts)
50
µS
7.
Load resistance (RLoad)
15.5
Ω
8.
Capacitor value ( C)
7
µF
[9]
[10]
[11]
[12] [13]
V.
CONCLUSION
A powerful and simple MPC control algorithm has been presented in this paper in order to control input unity power factor and output DC-link voltage with active front end rectifier. The control scheme has turned to advantages the discrete nature of power converter to predict the future behavior of the input current of the system. It has also employed to get the cost function for all valid switching states of the AFE rectifier and the switching state corresponding to minimum cost function is selected for the next operating period. Simulation study has been proved with good power transfer efficiency and encouraging results.
[14]
REFERENCES
[18]
[1] [2]
[3]
[4] [5]
[6]
[7]
[15]
[16]
[17]
J. R. Rodríguez, J. W. Dixon, J. R. Espinoza, J. Pontt, and P. Lezana, "PWM regenerative rectifiers: state of the art," IEEE Transactions on Industrial Electronics vol. 52, pp. 5-22, 2005. P. Salmeron and S. P. Litrán, "A control strategy for hybrid power filter to compensate four-wires three-phase systems," IEEE Transactions on Power Electronics, vol. 25, pp. 19231931, 2010. M. Malinowski, M. P. Kazmierkowski, and A. M. Trzynadlowski, "A comparative study of control techniques for PWM rectifiers in AC adjustable speed drives," IEEE Transactions on Power Electronics, vol. 18, pp. 1390-1396, 2003. M. Perez, R. Ortega, and J. R. Espinoza, "Passivity-based PI control of switched power converters," IEEE Transactions on Control Systems Technology, vol. 12, pp. 881-890, 2004. L. Yacoubi, K. Al-Haddad, L.-A. Dessaint, and F. Fnaiech, "Linear and nonlinear control techniques for a three-phase three-level NPC boost rectifier," IEEE Transactions on Industrial Electronics, vol. 53, pp. 1908-1918, 2006. J. D. Barros and J. F. Silva, "Optimal predictive control of three-phase NPC multilevel converter for power quality applications," EEE Transactions on Industrial Electronics, vol. 55, pp. 3670-3681, 2008. A. G. Beccuti, S. Mariéthoz, S. Cliquennois, S. Wang, and M. Morari, "Explicit model predictive control of dc–dc switchedmode power supplies with extended Kalman filtering," IEEE Transactions on Industrial Electronics, vol. 56, pp. 1864-1874, 2009.
[19]
[20]
85
J. Rodriguez, J. Pontt, C. A. Silva, P. Correa, P. Lezana, P. Cortés, and U. Ammann, "Predictive current control of a voltage source inverter," IEEE Transactions on Industrial Electronics, vol. 54, pp. 495-503, 2007. M. A. Pérez, P. Cortés, and J. Rodríguez, "Predictive control algorithm technique for multilevel asymmetric cascaded Hbridge inverters," IEEE Transactions on Industrial Electronics, vol. 55, pp. 4354-4361, 2008. P. Correa, J. Rodríguez, M. Rivera, J. R. Espinoza, and J. W. Kolar, "Predictive control of an indirect matrix converter," IEEE Transactions on Industrial Electronics, vol. 56, pp. 1847-1853, 2009. P. Cortes, M. P. Kazmierkowski, R. M. Kennel, D. E. Quevedo, and J. Rodríguez, "Predictive control in power electronics and drives," IEEE Transactions on Industrial Electronics, vol. 55, pp. 4312-4324, 2008. P. Correa, M. Pacas, and J. Rodriguez, "Predictive torque control for inverter-fed induction machines," IEEE Transactions on Industrial Electronics, vol. 54, pp. 1073-1079, 2007. P. Cortés, J. Rodríguez, D. E. Quevedo, and C. Silva, "Predictive current control strategy with imposed load current spectrum," IEEE Transactions on Power Electronics, vol. 23, pp. 612-618, 2008. M. Rivera, R. Vargas, J. Espinoza, J. Rodriguez, P. Wheeler, and C. Silva, "Current control in matrix converters connected to polluted AC voltage supplies," in Power Electronics Specialists Conference, 2008. PESC 2008. IEEE, 2008, pp. 412-417. M. E. Rivera, R. E. Vargas, J. R. Espinoza, and J. R. Rodriguez, "Behavior of the predictive DTC based matrix converter under unbalanced AC supply," in 42nd IAS Annual Meeting Industry Applications Conference, 2007. Conference Record of the 2007 IEEE, 2007, pp. 202-207. M. Rivera, J. Rodriguez, P. W. Wheeler, C. A. Rojas, A. Wilson, and J. R. Espinoza, "Control of a matrix converter with imposed sinusoidal source currents," IEEE Transactions on Industrial Electronics, vol. 59, pp. 1939-1949, 2012. J. Rodriguez, J. Kolar, J. Espinoza, M. Rivera, and C. Rojas, "Predictive current control with reactive power minimization in an indirect matrix converter," in IEEE International Conference on Industrial Technology (ICIT), 2010, 2010, pp. 1839-1844. M. Rivera, J. Rodriguez, B. Wu, J. R. Espinoza, and C. A. Rojas, "Current control for an indirect matrix converter with filter resonance mitigation," IEEE Transactions on Industrial Electronics, vol. 59, pp. 71-79, 2012. M. Rivera, J. Rodriguez, J. R. Espinoza, T. Friedli, J. W. Kolar, A. Wilson, and C. A. Rojas, "Imposed sinusoidal source and load currents for an indirect matrix converter," IEEE Transactions on Industrial Electronics, vol. 59, pp. 3427-3435, 2012. J. Rodriguez, M. Rivera, J. W. Kolar, and P. W. Wheeler, "A review of control and modulation methods for matrix converters," IEEE Transactions on Industrial Electronics, vol. 59, pp. 58-70, 2012.
