A Non-linear Cont troller for Single-Phase AC-AC C Power

Transcription

1 IJSI International Journal of omputer Science Issues, Vol. 9, Issue 4, No, July ISSN (Online: A Non-linear ont troller for Single-Phase A-A Power onverter to meet UPS Performance Index A.Ait elmahjoub, A.Ailane, A.Abouloiafa, A.Essadki 3, M.Rachik, E.Labriji M M.I.T.I. faculty Ben M sik,university Hassan mohamedia-asablanca, Morocco LAI, Ecole Mohammadia d Ingénieurs Rabat, Morocco 3 D.GE ENSET university Mohamed 5 Rabat Morocco Abstract This article focuses on A-A power converterr that can be used for uninterruptible power supply (UPS. The converter is built on two stages: a A-D input stage and a D-A output stage. The two blocks are connected by an intermediate D bus. The aim of control is threefold: i power factor correction ii regulation of D bus iii generating a sinusoidal voltage at the output. The synthesis of controllers has been achieved through the technique of nonlinear backstepping control. A detailedd analysis of the stability control system is presented. The performances of regulators have been validated by numerical simulation in MATLAB / SIMULINK. Keywords: UPS, A-D converter, D-A converter, power factor, Backstepping small-signal linear models [5] [6]. In this article we will order our converter by the backstepping approach. The idea is to calculate a control law to ensure that the derivative of a certain function (Lyapunov defined positive is always negative. To do this, the system is decomposed into a set of sub nested systems. The calculation of Lyapunov function is, then, starting recursively from the inside of the loop. At each step, the order of the system is increased and the non stabilized part at the previous step is processed. The law of command to apply is synthesized at the last step. It must ensure, at all times, the overall stability of the system compensated while working in pursue or regulation [7] [8] [9].. Introduction Single-phase UPS systems are widely used in low power applications such as computer systems, data storage, medical facilities and telecommunications. These systems are generally made by A-A converters which absorb currents with high levels of harmonic distortion. Several investigations have been published on the different topologies of A-A converters in order to develop control strategies to improve the energy behavior of these systems [] [] [3] [4]. Among these topologies, the one which is based on the cascade converters A-D and D-A presents a great feature: the simplicity of configuration, regardless of the values of voltage and frequency desired in the output of the converter. Figure shows the block diagram of such a system. It consists of a half-bridge rectifier with a capacitive divider that converts the input A line into a D, and an inverter which is capable of generating from D to A with the desired amplitude and frequency. Using the average model of A-A converters, several techniques for the design of controllers have been proposed, such as the classical regulators (PI or PID using i in v dc i L i out v in Fig.. A-A converter. Mathematical model of the converter The converter shown in figure consists of a rectifier and inverter each of which composed of an IGBT arm. These arms operate on the principle, widely known in the literature, pulse wih modulation (PWM. Thus, arms (k,k,k 3,k 4 and (k 5,k 6,k 7,k 8 are associated respectively with the binary switching functions et such that: if k 4 are ON, k 3 are OFF if k 4 are OFF, k 3 are ON v out opyright (c International Journal of omputer Science Issues. All Rights Reserved.

2 IJSI International Journal of omputer Science Issues, Vol. 9, Issue 4, No, July ISSN (Online: if k 6 7 are ON, k 5 8 are OFF if k 6 7 are OFF, k 5 8 are ON The cyclical and binary signals and are the control inputs of the A-A. They vary from one period to another, and their variations can determine the trajectories of the state variables of the converter such as the currents in the inductors and the voltages across the capacitors. The development of the model switched system is based on the application of Kirchhoff's laws. Thus, we obtain: diin Li ri i in vin vdc a dvdc iin iout (b diout Lo ro iout vout vdc (c dvout o iout il (d The above model proves to be unsuitable for the development of continuous control laws since it involves, as input variables, the binary signals and. To overcome this inconvenience, the equivalent average model is used. Thus, we obtain the model: Li x ri x vin ux (a x ux ux3 (b L x r x x u x (c o 3 o 3 4 ox 4 x 3 i L (d where x, x, x 3, x 4, u, u and i L represent, respectively, the means values, over a period of cutting, of variables i in, v dc, i out, v out,, and i L. Note that the mathematical model ( is nonlinear because of the products involving the state variables and input signals. The non-standard form of this model leads us to choose techniques taking into account the nonlinearities such as approach backsteeping. 3. ontroller design 3. The aim of controller Our aim is to design a controller for the A-A converter that ensures both: Power factor correction (PF (grid side; the D bus voltage regulation ; a sinusoidal voltage to the output of converter(load side The controller synthesis will be performed in three steps. First, an input current inner loop (regulator is designed to cope with the PF issue. In the second step, an output current inner loop (regulator is designed to have a sinusoidal voltage whose amplitude and frequency are fixed by the reference signal x 4=v out= Vmax sin( t. In the end an outer voltage loop (regulator 3 is built-up to generate the reference signal, which will be used by the controller, to regulate the D bus voltage at its desired reference value x =v dc. The proposed control system will have the structure shown in Figure. Both controllers and will be synthesized by a technique using Backstepping approach and the third will be done by a simple proportional-integral corrector. Figure. Basic structure of the controller 3. Input current inner loop design (regulator The PF objective means that the converter input current should be sinusoidal and in phase with the grid supply voltage. It amounts to ensuring current harmonics rejection. We therefore seek a regulator that enforces the current x to track a reference signal of the form x = vin when IR. The block diagram in Figure 3 shows the basic structure of the control loop of the input current. Figure 3. Structure of Input current regulator opyright (c International Journal of omputer Science Issues. All Rights Reserved.

3 IJSI International Journal of omputer Science Issues, Vol. 9, Issue 4, No, July ISSN (Online: onsider the tracking error z defined by: z Li ( x x its dynamics is given by: z L ( x x i (3 z4 o ( x4 x4 V.5z Let us use the Lyapunov candidate function: As its derivative with respect to time, is given by: V zz, the choice z k z Where k is a positive constant synthesis, leads to a Lyapunov candidate function whose dynamics is negative definite. So V kz. Therefore global asymptotic stability is achieved and z tends exponentially to. Using (a, (3 and (4, we will have z x (4 i x i in i k z r x v u x L x x will initially be equal to E because the capacitor of the D bus is automatically loaded by the A-D converter that acts as a rectifier diode (D, D, D 3 and D 4 at system startup if we delay the switch control (k, k, k 3, k 4. Solving the previous equation with respect to u led to the backstepping control law as follows: u ( kz ri x vin Li x (5 x Proposition. onsidérons le convertisseur A-A de la figure qui est décrit par le modèle moyen (. Si la première dérivé de β est disponible, alors la loi de commande (5 garantie la stabilité asymptotique globale du signal d erreur z. 3.3 output current inner loop design (regulator The block diagram of figure 4 shows the control loop of the output voltage. It calculates the control law of the second stage of the power converter D-A to regulate the output voltage x 4 to its desired reference value x 4 =V max sin t. The study will be developed for any grid connected to the output of the converter assuming that the current injected in the grid can be measured. Figure 4. Output current inner loop Taking into account the equations (c and (d the relative degree of the system, by respect to the variable x 4, is, the backstepping synthesis of the regulator will realized in two stages. We define an error variable (7 With (d, the dynamics of the error z 4 is given by: 4 3 L o 4 (8 onsider the following candidate Lyapunov function: 4.5 z4 V its derivative with respect to time is given by: V z z The choice z 4 k4z4 (9 Where k 4 is a positive constant synthesis, leads to a Lyapunov candidate function whose dynamics is negative definite. using (8 and (9 we will have k z x i x ( L o 4 if we choose x 3 as virtual control input we deduce the stabilizing function: x k z i x ( L o 4 As x 3 is not the control input, a new error variable z 3 between the virtual control x 3 and its desired value x 3 is introduced: z x x ( The dynamics of the error z 4 is given by: z x x ( Using (8, (9, ( and (, we will have : z z k z V k z z z Now consider the dynamics of the error z4 taking into account (c. so : o Lo Lo Lo (4a (4b r z x x u x x (5 We see appear for the first time, the true signal of the control noted u The objective now is to stabilize the system (z 4, z 3, for this we take as a candidate Lyapunov function the following function: V3 z4 z3 Its dynamics is given by V 3 z 4z4 z 3z3. Using (4a V k z z ( z z The choice z4 z 3 k3z3 (6 opyright (c International Journal of omputer Science Issues. All Rights Reserved.

4 IJSI International Journal of omputer Science Issues, Vol. 9, Issue 4, No, July ISSN (Online: Where k 3 is a positive constant synthesis, which guarantees the negativity of the dynamics of the Lyapunov candidate function, because: V 3 k4z4 k3z3 the equations (5 and (6 lead to: ro x3 x4 ux x 3 k3z3 z4 Lo Lo Lo We deduce then the following control law: Lo ro u ( x3 x4 x 3 k3z3 z4 (7 x L L o o Proposition. onsider the A-A converter of Figure which is described by the model means (. The control law (7 guarantees the global asymptotic stability of error signals z 3 and z 4. Moreover the dynamics of these errors is described by the following model: z 4 k4 z4 z 3 k3 z3 3.4 outer voltage loop design (regulator 3 The block diagram in Figure 5 shows the structure of the regulator of D bus voltage. Figure 5. Structure for regulating the D bus voltage The aim of the outer loop is to generate a tuning law for the ratio β in such a way that the D bus voltage x be regulated to a given reference value x. The first step in designing a loop is to establish the relation (model between the ratio β (control input and the D voltage x. Hypothesis: H : The inner loop input current (regulator and the inner loop output current (regulator are supposed to have fast dynamics compared to of the outer loop of D bus voltage (regulator 3. H : The voltage drops across the different coils and their parasitic resistances are assumed negligible compared to the input and output voltages. Based on the hypotheses H and H, the control laws (5 and (7 reduce to: u v in (8a x u x 4 x if we substitute u and u given by expressions (8 in (b the model becomes: (8b x ( v x x x in 4 3 x This equation can be rewritten as follows: x x ( vin x x4x3. (9 If we set y x, its dynamics becomes E y p( t ( E with p( t cos( t x3x4 Note that the model ( can be seen as an integrator perturbed by the signal p (t. The controller design is based on the average model as follows: y k p( t ( E T with k T T T p( t x3x4 y y T T Where T is the period of the network. The control model ( has been verified by simulation. So, the step response results in a signal y (the square of the output voltage ramp type perturbed by undulations of low amplitude. The system ( can be stabilized using a simple PI controller whose transfer function is given by ( s k p ki. s The block diagram of the controlled system is shown in Figure 6. Figure 6. Equivalent loop voltage In closed loop, the output signal y depends on the reference signal y and the disturbing signal p (t by the equation: Y( s F( s. Y ( s G( s. P( s ( With: s s F( s G( s s s s s Where k k p / k i, / ki opyright (c International Journal of omputer Science Issues. All Rights Reserved.

5 IJSI International Journal of omputer Science Issues, Vol. 9, Issue 4, No, July ISSN (Online: , k i k k k The transfer relation ( shows that the proposed controller guarantees a perfect pursuit lim ( y y and p i t a rejection disturbance since G (s contains a derivator effect. The Bode diagram of Figure 7 shows the frequency response of the transfer function F (s. r i 3mΩ 47µF D-A converter L o mh r o 3mΩ o µf Load R Ω L mh Switching frequency f PWM Khz Table. simulation parameters Bode Diagram Magnitude (db Phase (deg Frequency (rad/sec Figure 7. Bode diagram of F (s 4. SIMULATION RESULTS The performance of the proposed controller has been validated by simulation in the environment MATLAB / SIMULINK. Table shows all the parameters of the controlled system. Figures 8 to 4 shows the simulation results when reference of the D bus voltage (v dc is set at 7V and the output voltage (v out is a sinusoidal signal with a RMS amplitude equal to 3V and 5Hz. Figure 8 shows that the input current of the converter is sinusoidal. In the Figure 9, we note that current and input voltage are sinusoidal and in phase. This shows that the power factor correction is perfectly realized. Figures and show the evolution of the output voltage delivered by the converter. In particular we note that it is the sinusoidal waveform with the desired characteristics (amplitude and frequency. Figure shows that the D bus voltage follows perfectly (on average its reference. Finally in Figures 3 and 4, are presented the control signals and, it is clear that they are bounded. Item value grid E 3 V rad/s A-D converter L i 3.3mH 5. ONLUSION In this paper, we presented a new control technique using the backstepping approach to control A-A converters used in the Uninterruptible Power Systems (UPS. The control objective is threefold i correcting the power factor of the network side ii generating a perfectly sinusoidal voltage to the inverter output iii regulating the D bus voltage between the rectifier stage and inverter stage A-A converter phase used. The studied system is described by a representation of nonlinear state average of order 5. The controller has been synthesized using advanced tools of nonlinear control such as stability in the sense of LYAPUNOV. It was shown by simulation that the proposed controller guarantees the desired tracking performance and stability Figure 8. i in current Figure 9. i in current and v in (v in/6 voltage v in i in opyright (c International Journal of omputer Science Issues. All Rights Reserved.

6 IJSI International Journal of omputer Science Issues, Vol. 9, Issue 4, No, July ISSN (Online: Figure. Output voltage v out v out v outref REFERENES Figure 4. The control law Figure. Output voltage of D-A converter and its reference Figure. D bus voltage v dc and the reference v dc v dcref [] Gui-Jia Su, and Tetuhiko Ohno, A New Topology for Single Phase UPS Systems, IEEE Power onversion onference,aug. 997, Vol., pp [] Joan Salaet, Al., SVM Based ontrol of a Single-Phase Half- Bridge Rectifier Under Power Factor orrection and Balanced Operation, IEEE ISIE, Vol., pp [3] S. Begag, N. Belhaouchet, et L. Rahmani, Fonctionnement à fréquence de commutation constante d un onduleur de tension triphasé par une nouvelle technique de commande en courant par hystérésis 4th International onference on IP 7 [4] Takeshi U, Ai., A Study of the High Performance Singlephase UPS, IEEE PES, 998, Vol., pp [5] R.Ghosh G. Narayanan A Simple Analog ontroller for Single- Phase Half-Bridge Rectifier and its Application to Transformerless UPS IEEE IIS 5. [6] T.j Liang, J.L. Shyu, J.F. hen high real output power on line UPS system with built-in reactive power compensation PES. IEEE [7] Der-Fa hen; Tian-Hua Liu; he-kai Hung, "Adaptive backstepping controller design for a matrix converter based PMSM drive system", IEEE International onference on Industrial Technology, Vol, pp58 63, -4 Dec.. [8] Jin Young hoi; J.Farrell, "Observer-based backstepping control using online approximation," American ontrol onference, Vol5, 8-3 pp , June. [9] W. Bolek, J.Sasiadek, " Singularity of backstepping control for nonlinear systems," American ontrol onference, Vol 4, pp , 8- May Figure 3. The control law N.B. this work is supported by faculty Ben M sik of University Hassan, L.T. Elkhaouarizmy, Moroccan school science engineer (EMSI, the network of systems theory of Morocco. opyright (c International Journal of omputer Science Issues. All Rights Reserved.