Four two-level PWM rectifiers controlled by Lyapunov function for stabilisation of DC sources of five-level NPC-VSI

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1 ISSN , England, UK World Journal of Modelling and Simulation Vol. 6 (2010) No. 1, pp Four two-level PWM rectifiers controlled by Lyapunov function for stabilisation of DC sources of five-level NPC-VSI Rédha Chibani, ElMadjid Berkouk, Mohamed Seghir Boucherit Process Control Laboratory, National Polytechnic School of Algiers, Alger 16200, Algeria (Received December , Accepted October ) Abstract. The aim of this work is to propose a solution to the problem of unbalance in the DC link voltage in a five-level Neutral Point Clamped Voltage Source Inverter (NPC-VSI) by using isolated DC sources based on four two-level PWM rectifiers controlled by Lyapunov function. Keywords: five-level neutral point converter voltage source inverter, PWM rectifier, permanent magnet synchronous machine, lyapunov function control, sliding mode control 1 Introduction Multilevel converters have been receiving increased attention because they can withstand high voltage and generate a nearly sinusoidal waveform. At present, the most commonly used multilevel topologies are Neutral Point-Clamped (NPC), flying capacitor, diode clamped and cascade inverter topologies. Nowadays the NPC inverter is most popular [14, 16, 17, 20, 23]. However, Neutral-point (NP) voltage balancing problem, which is an inherent problem of three-level NPC topology, restricts the development and application of this kind of converter [17, 20]. Several publications have discussed ways to solve this balancing problem in three-level NPC- VSI [5, 6, 8, 11 13, 17 19, 24, 25]. The Space Vector Modulation (SVM) [2 6, 15, 18] involves of required voltage vector from a number of voltages vectors corresponding to switching states. Nearest three vectors (NTV) algorithm synthesizes voltage vector from nearest three vectors and is found to be the optimal solution in synthesis of required voltage vector and excellent quality. Large redundancy in three-level converters is utilized to maintain neutral point voltage balance and is an inherent part of SVM of multi-level converters. These goals are realized at the expense of a loss of performance in other areas. Nevertheless, the use of SVM beyond three-level converters is computationally intensive and becomes impractical as the number of levels increase [7]. In the second category using carrier-based pulse width modulation (PWM) scheme, a zero sequence voltage signal is added to the modulation signals [22]. In some schemes using zero sequence voltage to balance DC capacitor voltages, knowledge of load power factor (or direction of instantaneous power flow) is required which is difficult to implement under transient conditions, and in others, measurements of both capacitor voltages and load currents (magnitudes or polarities) are required [17]. Alternatively, auxiliary electronic circuitry can be added for capacitor voltage balancing [9, 10]. In this paper, a solution based on isolated DC sources is proposed to suppress this limitation for a fivelevel NPC-VSI. For this, four two-level PWM current rectifiers are used. The controller by Lyapunov function for rectifier is partitioned into two closed-loop regulators-one regulating the total dc-bus voltage and another for the net currents. Corresponding author. fax: address: redha29@yahoo.fr. Published by World Academic Press, World Academic Union

2 48 R. Chibani & E. Berkouk & et al.: Four two-level PWM rectifiers controlled by Lyapunov function This paper is organised as follows. Section 2 provides the modelling of a Permanent Magnet Synchronous Machine (PMSM) and its speed control based on sliding mode in section 3. Section 4 develops the mathematical modelling of the five-level NPC-VSI and its Pulse Width Modulation control strategy (PWM) using four bipolar carriers is given in section 5. The section 6 present the cascade used to balance DC voltages of the five-level NPC-VSI using four two-level PWM current rectifiers where their control by Lyapunov function is detailed. Simulation results are given in the section 7. 2 Permanent magnet synchronous machine modelling The model of a PM synchronous machine can be described in the rotor rotating reference frame as follows [8, 11, 12, 19] : U d U q = R s + L d s ωl q i d ωl d R s + L q s i q + 0 ωk t. (1) The electric torque is stated as: C em = K t i q + (L d + L q )i d i q, (2) J(sω) = C em C r Kω. (3) In these equations: L d, L q : self inductance of the d and q armatures equivalent winding. R s : Resistance of an armature winding. K t : Electromotive force constant. ω: Angular speed. s: Laplace operator. J: Inertia of the set machine-load. C r : Load torque. K: Viscous friction coefficient. 3 Control strategy of the actuator and speed control The model of the actuator (PMSM) presented above is non linear and strong coupling between d and q axes exists. To eliminate this coupling, we use the field oriented control. For the PM synchronous machine used, we develop the algorithm i d = 0. When the d axis current is equal to zero, the block diagram of the q axis becomes similar to that of a DC machine and the speed can be controlled by using a sliding mode controller which generates the q axis voltage (Fig. 1). We use a current regulator for the d and q axis [5, 6, 13, 15, 17, 18, 24, 25]. Fig. 1. Permanent Magnet Synchronous Machine speed control scheme based on sliding mode 4 Five level NPC-VSI modelling The general structure of the three-phase five-level NPC voltage source inverter is shown on the Fig. 2. WJMS for contribution: submit@wjms.org.uk

3 World Journal of Modelling and Simulation, Vol. 6 (2010) No. 1, pp Fig. 2. General structure of the three-phase five-level NPC-VSI In order to formulate a knowledge model of this inverter, we represent every pair transistor-diode by one bi-directional switch T D ks. Several complementary control laws are possible for the five-level NPC-VSI. The optimal one is given below [9] : B k1 = B k5 B k2 = B k4 B k3 = B k6 (4) B k7 = B k1 B k2 B k3 B k8 = B k4 B k5 B k6 B ks is the control signal of T D ks. k is the number of the arm and s the number of the switch. In order to deduce the knowledge model of the inverter and using the proposed complementary law, we introduce the connection function F ks of the switch T D ks which describes the state of the switch (1 = turned on and 0 = turned off). Consider now, the connection function FkM b which describes the state of a half arm with k the number of the arm and m the number of the half arm ( 1 = upper half arm and 0 = lower half arm ). The expression of the half arm connection function using the switch connection functions has the following form: Fk1 b = F k1 F k2 F k3 Fk0 b = F k4 F k5 F k6 Fk1 b = Fk1 F k2 F (5) k3 Fk0 b = F k4 F k5 F k6 The voltage of the three phases A, B, C relatively to the middle point M are given by V XM with x = point A, B or C [9, 10]. ] ] V km = [Fk1 b (U c1 + U c2 ) + F k7 U c1 [Fk0 b (U c3 + U c4 ) + F k8 U c3 (6) The input currents of the three-phase five-level inverter using the load currents are given by the following relations: i d1 = b i 1 + b i 2 + F31 b i 3 i d2 = F11 b i1 + F21 b i2 + F31 b i3 i d3 = F b 10 i1 + F b 20 i2 + F b 30 i3 i d4 = F b 10 i 1 + F b 20 i 2 + F b 30 i 3 i d0 = i 1 + i 2 + i 3 i d1 i d2 i d3 i d4 (7) WJMS for subscription: info@wjms.org.uk

4 50 R. Chibani & E. Berkouk & et al.: Four two-level PWM rectifiers controlled by Lyapunov function 5 Control strategy of the inverter In this part, we suggest a PWM strategy of the five-level NPC voltage inverter called sine triangle using four bipolar carriers. We define the three reference voltages V ref1, V ref2, V ref3 [10]. V ref1 = V m sin(ωt ϕ) V ref2 = V m sin(ωt ϕ 2π 3 ) V ref3 = V m sin(ωt ϕ 4π 3 ) (8) This strategy uses four bipolar carriers (U p1, U p2, U p3, U p4 ). It is characterised by two parameters m the index modulation and r the modulation rate. The algorithm of this strategy can be summarised as follows: Step 1. Determination of the intermediate voltages If V refk > V p1 then V km1 = +U c If V refk < V p1 then V km1 = 0 If V refk > V p2 then V km2 = +2U c If V refk < V p2 then V km2 = +U c (9) If V refk > V p3 then V km3 = 0 If V refk < V p3 then V km3 = U c If V refk > V p4 then V km4 = U c If V refk < V p4 then V km4 = 2U c Step 2. Determination of the output voltage V km = V km1 + V km2 + V km3 + V km4. (10) 6 Four two-level PWM rectifier-five-level NPC VSI-PMSM cascade We propose a cascade presented in the Fig. 3. This cascade is constituted by four two-level PWM rectifier- Five-level NPC VSI-PMSM. To avoid the unbalance problem of the medium value of the DC voltage of the intermediate bridge capacitor of this two-level current rectifier, we have to control the PWM rectifier by using a closed loop for the network currents and another one for the output voltage of the rectifier [1, 21]. 6.1 Voltage feedback control Each phase k (k = 1, 2 or 3) of the three phases network feeding the rectifier considered can be represented by an R, L circuit. V resk is the voltage of one phase k of the three phases network and V k is the voltage of the leg k of the rectifier (Fig. 4). The voltage loop imposes the effective value of the reference current of the network corresponding to the power exchanged between the network and the continue load. For the voltage tracking objective, define the tracking error as: ε U = U c U cref. In order to ensure the convergence of the tracking error to zero the Lyapunov function is chosen as: V (ε U ) = 1 2 ε2 U. The derivative of the Lyapunov function is computed as: V (ε U ) = (U c U cref ) 1 C (I red I ch ) (11) To guarantee the global asymptotic stability in the voltage loop, we have: I e = U C 3V e [I ch K U C(U C U cref )] with K U > 0. (12) WJMS for contribution: submit@wjms.org.uk

5 World Journal of Modelling and Simulation, Vol. 6 (2010) No. 1, pp Fig. 3. Cascade four two-level PWM current rectifiers-five-level NPC VSI-PMSM Fig. 4. Control algorithm of the output DC voltage of the two-level PWM current rectifiers Fig. 5. Control algorithm of the network current i resk of the two-level PWM rectifier 6.2 Current feedback control We control the network current of the phase 1 and 2 by a Lyapunov function regulator. The algorithm of this current loop is given on the Fig. 5. In this scheme, the transfer function H(p) is expressed as follows: H(p) = I resk V = 1 R + Lp. (13) For the network currents tracking objective, define the tracking error as: ε i = i resk i refk. In order to ensure the convergence of the tracking error to zero, the Lyapunov function is chosen as: V (ε i ) = 1 2 ε2 i. The derivative of the Lyapunov function is computed as: V (ε i ) = (i resk i refk )( i resk i refk ). (14) To guarantee the global asymptotic stability in the current loop, we have: N gk = 1 [ V resk RL resk I e ωl 2 cos(ωt (k 1)) 2π ] U C 3 + K il(i resk i refk ) with K i > 0. (15) 7 Simulation results 8 Results interpretation We show perfectly the problem of the unbalance of the four DC voltages of the intermediate capacitors bridge. The voltages U c1 and U c3 (Fig. 6) are increasing and the voltages U c2 and U c4 (Fig. 7) are decreasing. WJMS for subscription: info@wjms.org.uk

6 52 R. Chibani & E. Berkouk & et al.: Four two-level PWM rectifiers controlled by Lyapunov function Fig. 6. Voltages U c1 and U c3 Fig. 7. Voltages U c2 and U c4 Fig. 8. Speed and its reference Fig. 9. Electromagnetic torque Fig. 10. d axis current (i d ) Fig. 11. q axis current (i q ) The characteristics of the drive of the PM synchronous machine (Speed (Fig. 8), torque(fig. 9) and different currents (Fig. 10 and Fig. 11)) fed by a five-level PWM current rectifier-five-level NPC-VSI cascade show that the undulations of the currents i d, i q and the electromagnetic torque are very important. In order to demonstrate the feasibility of the proposed control method, the closed-loop has been tested by simulations. The Figs show the response of the DC output voltage U c1, U c2, U c3 and U c4 for a step change from 200V to 250V at t = 1s and the network currents and voltages of the two-level PWM rectifiers 1, 2, 3 and 4 respectively, controlled by Lyapunov function. The response of the output DC voltage of every rectifier exhibits a fast transient to the DC reference voltage. This means that the closed-loop system has a very good dynamic response to step changes. The network currents are nearly a sine wave with unity power factor and are in phase with the network voltages. Fig. 16 shows the performances of the speed control of a PMSM by sliding mode fed by four two-level PWM current rectifier-five-level NPC-VSI cascade from a speed reference of 400rd/s. The characteristics of the drive of the PM synchronous machine (speed, torque and different currents i d, i q and line current i 1 ) fed by a five-level NPC-VSI controlled by sine triangle strategy using four bipolar carriers for m = 12. We can see that the proposed controllers can quickly and accurately tracks the desired reference. WJMS for contribution: submit@wjms.org.uk

7 World Journal of Modelling and Simulation, Vol. 6 (2010) No. 1, pp Fig. 12. Output voltage U c2, Voltage V res1, network current i res1 and its reference i ref1 Fig. 13. Output voltage U c1, Voltage U res1, network current j res1 and its reference j ref1 Fig. 14. Output voltage U c3, Voltage W res1, network current k res1 and its reference k ref1 Fig. 15. Output voltage U c4, Voltage X res1, network current l res1 and its reference l ref1 The Fig. 17 give the output voltage obtained by controlling the five-level NPC-VSI by a sine triangle strategy using four bipolar carriers for m = 12. Fig. 18 show the different voltages U c1, U c2, U c3 and U c4 obtained for a voltage reference of 200V. The parameter perturbations introduced in the control were set to the following values: 300% in the network resistance, 500% in the inductor at t = 1s. We can see clearly that stable operation and good performances are preserved against the inaccuracy in the modelling of the parameter and their variations. The proposed control scheme is unaffected by these variations. WJMS for subscription: info@wjms.org.uk

8 54 R. Chibani & E. Berkouk & et al.: Four two-level PWM rectifiers controlled by Lyapunov function Fig. 16. Performances of the PMSM for a speed reference of 400rd/s 9 Conclusion The paper proposed a simple method for controlling the four input DC capacitor voltages of a five-level NPC-VSI. It intends to demonstrate that permanent magnet synchronous machine control based on sliding mode control when applied with four two-level PWM current rectifier-five-level PWM NPC-VSI may contribute both for functional performances improvement and attenuation of some technological limitations of multilevel NPC-VSI. We have developed a knowledge model of a five level NPC voltage source inverter and have presented a sine triangle strategy using four bipolar carriers. We have studied the performances of the speed control by Sliding Mode Control of the Permanent Magnet Synchronous Machine fed by a five-level NPC-VSI. The input DC voltages are generated by a four two-level PWM current rectifiers controlled by Lyapunov function. References [1] H. Kömürcgil, O. Kükrer. Lyapunov-based Control for Three-phase PWM AC/DC Voltage-source converters. IEEE Transactions on Power Electronics, 1998, 13(5): [2] M. Prats, J. Solís, L. García Franquelo. New space vector modulation algorithms applied to multilevel converters with balanced DC-link voltage. HAIT Journal of Science and Engineering B, 2005, 2: WJMS for contribution: submit@wjms.org.uk

9 World Journal of Modelling and Simulation, Vol. 6 (2010) No. 1, pp Fig. 17. Output voltage V A of the five-level NPC-VSI controlled by a sine triangle strategy using four bipolar carriers for m = 12 Fig. 18. Voltages U c1, U c2, U c3 and U c4 for a reference of 200V (at t = 1s, we multiply the value of the network resistor by 3 and the inductor value by 5) [3] M. Baiju, K. Gopakumar. A New Multilevel Inverter Topology with a Hybrid Approach. EPE Journal, 2003, 3(2): [4] A. Beig, G. Narayanan, V. Ranganathan. Modified svpwm algorithm for three level VSI with synchronized and symmetrical waveforms. IEEE Transactions on Industrial Electronics, 2007, 54(1): [5] A. Bendre, S. Krsti. Comparative evaluation of modulation algorithms for Neutral-Point-Clamped converters. IEEE Transactions on Industry Applications, 2005, 41(2): [6] A. Bendre, G. Venkataramanan, et al. Modelling and design of neutral-point voltage regulator for a three-level diode-clamped inverter using multiple-carrier modulation. IEEE Transactions on Industrial Electronics, 2006, 53(3): [7] O. Bouhali, E. Berkouk. New direct space vector modelling and control of five-level three-phase inverters. Archives of electrical engineering, 2005, 212(2): [8] N. Celanovic, D. Borojevic. A comprehensive study of neutral point voltage balancing problem in three-level Neutral-Point-Clamped voltage source PWM inverters. IEEE Transactions on Power Electronics, 2000, 15: [9] R. Chibani, E. Berkouk. Five-level PWM current rectifier-five-level NPC VSI-permanent magnet synchronous machine cascade. European Physical Journal-Applied Physics, 2005, (30): WJMS for subscription: info@wjms.org.uk

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