Control strategy for single-phase PWM ac/dc voltage-source ...

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Several methods for dc voltage control of single-phase pulse width modulation ac/ dc voltage-source converters are known. They di er in their performance and ...
INT. J. ELECTRONICS,

2000, VOL. 87, NO. 12, 1485 ± 1498

Control strategy for single-phase PWM ac/dc voltage-source converters based on Lyapunov’s direct method È MU È RCU È GIL* and OSMAN KU È KRER y HASAN KO Several methods for dc voltage control of single-phase pulse width modulation ac/ dc voltage-source converters are known. They di€ er in their performance and the complexity of the hardware and software they use. One of the basic requirements of the control method is a nearly sinusoidal input current with unity power factor. However, global stability of the converter is not guaranteed by the methods proposed so far. This paper describes a new control method based on Lyapunov’s direct method. It is shown that the converter can be stabilized globally to handle large signal disturbances. The closed-loop system not only ensures stability, but also exhibits excellent transient response for abrupt changes in the load. More importantly, the proposed control method can keep the advantages and remove the disadvantages of the existing methods. Computer simulations and experiments are presented to show the e€ ectiveness and applicability of the proposed control method for the converter.

1.

Introduction

The advent of fast high-power semiconductor switching devices has made it possible to consider pulse width modulation (PWM) techniques for ac/ dc converters in electric power systems. Conventional diode bridge and phase-controlled thyristor converters are incapable of producing input currents with low harmonic content at unity power factor. PWM converters of the voltage-source type are capable of providing near-sinusoidal currents at unity power factor. One promising version is the single-phase PWM ac/ dc voltage-source converter (Stihi and Ooi 1988, Ohnuki et al. 1996, KuÈkrer and KoÈmuÈ rcuÈgil 1997, Nishida et al. 1997). Several control strategies have been devised for single-phase PWM ac/ dc voltagesource converters. Hysteresis current control (HCC), based on the instantaneous comparison of the input current and its reference, has been presented by Stihi and Ooi (1988). The input current can be shaped well enough without knowledge of the supply voltage and the ac-side circuit parameters. The HCC technique o€ ers simple control and fast dynamic response, but su€ ers from the following disadvantage s: (a) precision current sensing is essential; and (b) switching frequency changes signi® cantly during a fundamental period, resulting in excessive stresses on the switching devices.

Received 20 July 1999. Accepted 26 May 2000. * Corresponding author. Department of Computer Engineering, Eastern Mediterranean University, G. Magosa, Mersin 10, Turkey. e-mail: hasan@ compenet.emu.edu.tr y Department of Electrical and Electronic Engineering, Eastern Mediterranean University, G. Magosa, Mersin 10, Turkey Internationa l Journal of Electronics ISSN 0020± 7217 print/ ISSN 1362± 3060 online # 2000 Taylor & Francis Ltd http:// www.tandf.co.uk /journals

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H. KoÈmuÈrcuÈgil and O. KuÈkrer

Pulse width prediction methods based on approximate discrete-time models of the converter have been proposed by Ohnuki et al. (1996) and Nishida et al. (1997). In these methods, the predicted pulse width is a function of the instantaneous value of the input current and the ac-side inductance (L ). This makes input current sensing essential for the prediction of a pulse width. However, we have recently introduced a new control method in the continuous-time domain which eliminates the need for input current sensing (KuÈkrer and KoÈmuÈ rcuÈgil 1997). In this control method, an error voltage is produced from the comparison of the output voltage with a reference voltage. This error voltage is then utilized by a PI controller to generate a command signal for the input current amplitude. The PWM converter operated with these control methods is a nonlinear dynamical system. Such a system is stable in the vicinity of the operating point but may not be stable when the system undergoes a large perturbation. Therefore, the PWM converter with the control methods proposed so far cannot guarantee system stability against large-signal disturbances (large load variations) . Thus, it is most desirable if a control method which would make the control system globally stable could be developed for the single-phase PWM converter. In this paper, a new control method based on Lyapunov’s stability theory is proposed. In this approach, the idea is to construct a scalar energy-like function (Lyapunov function) for the system and to examine the function’s time variation. The advantage of such a control method is the stable operation of the converter in the case of large-signal disturbances. In addition to this, the control method proposed here does not depend on circuit parameters. KoÈmuÈrcuÈgil and KuÈkrer (1998) introduced a Lyapunov-based control method for three-phase PWM voltage-source converters and showed that it is very useful for designing globally stable control for three-phase ac/ dc converters. Here, we investigate the applicability of the Lyapunovbased approach to the control of single-phase PWM ac/ dc converters. It is shown that a globally stable control is possible, at the expense, however, of a time-varying reference function for the output voltage. A modi® ed control law involving a constant reference for the output voltage is then proposed. An error analysis is carried out to investigate the e€ ects of this modi® cation on the input ac current. Computer simulations are undertaken to study the operation of the system with the Lyapunovbased and the modi® ed control approaches. Experimental results are also presented for the modi® ed control to verify the theoretical considerations. These results have shown that the closed-loop system with the modi® ed Lyapunov-based control method provides good transient response, unity power factor, and reduced harmonic distortion of the input current. 2.

Power circuit and its operation

Figure 1 shows a schematic diagram of a single-phase PWM ac/ dc voltage-source converter. The converter consists of four sets of semiconductor switching devices that are capable of conducting current in both directions. For correct operation, it is necessary that these switching devices are turned on and o€ such that the output voltage is never shorted. The inductor L on the ac-side and the capacitor C on the dc-side act as a ® lter for the input ac current and output dc voltage, respectively. The load is a resistor R in parallel with the output capacitor. Applying Kirchho€ ’s voltage law on the ac-side and current law on the dc-side gives

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Control strategy for single-phase PW M converters

Figure 1.

Single-phase PWM ac /dc voltage-source converter.

dis ˆ v s ¡ pv o dt dv 1 C o ˆ pis ¡ v o dt R

… 1†

L

… 2†

where p is the switching function of the PWM converter and v s ˆ V m sin … !t† is the input ac source voltage with an amplitude of V m and angular frequency of !. Resistance on the ac-side is neglected. The main control objectives for such PWM converters are to produce an input ac current is with low harmonic content at a high power factor and to control the average output voltage v o . It is clear that unity power factor can be achieved if the ac input current is tracks the following reference current i *s ˆ Im sin … !t†

… 3†

where Im is the amplitude of the input current. With this reference current, the steady-state expressions for the switching function can be solved from (1) and (2) as Po… t† ˆ

1 V os

…v ¡ L didt*† s

s

and

Po… t† ˆ

1 V os dV os ‡C dt i *s R





… 4†

where V os ˆ V o ‡ v oh… t† is the steady-state output voltage with a dc reference voltage V o and a harmonic ripple content v oh… t†. Now, let x1 ˆ is ¡ i *s ;

x2 ˆ v o ¡ V os ;

p ˆ Po ‡ D p

… 5†

where x1 and x2 are the state variables and D p is the perturbation of the switching function p. Substituting (3) and (5) into (1) and then into (2) and making use of (4) in the resulting equations gives the following expressions for L x_ 1 and C x_ 2 , respectively L x_ 1 ˆ ¡Po x2 ¡ D p… V os ‡ x2† x C x_ 2 ˆ Po x1 ‡ D p… i *s ‡ x1 † ¡ 2 R

… 6† … 7†

Note that, with the switching function as control variable, (6) and (7) are nonlinear and time-varying.

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H. KoÈmuÈrcuÈgil and O. KuÈkrer

Control strategy L yapunov-base d control strategy

On the basis of the Lyapunov stability theory, a positive-de® nite scalar function as a candidate for the Lyapunov function is to be found such that the total energy of the system is continuously dissipated. In such a case, any linear or nonlinear system must eventually settle down to an equilibrium point. According to Lyapunov’s stability theorem, any linear or nonlinear system is stable if there exists a positivede® nite Lyapunov function V … x† whose time derivative V_ … x† is negative-de® nite. Then, the equilibrium point at the origin (x1 ˆ 0, x2 ˆ 0) is globally asymptoticall y stable. Asymptotic stability implies convergence of the system’ s energy to zero. Now, consider the following positive-de® nite Lyapunov function for the converter V … x† ˆ 12 L x21 ‡ 12 Cx22

… 8†

V_ … x† ˆ x1 L x_ 1 ‡ x2 C x_ 2

… 9†

Taking a derivative of (8) with respect to time gives

If we replace the expressions obtained for L x_ 1 and C x_ 2 in equations (6) and (7) into (9), we obtain 2

x V_ … x† ˆ … x2 i *s ¡ x1 V os†D p ¡ 2 R

… 10†

It is clear that V_ … x† along any system trajectory becomes negative-de® nite if D p is chosen to be

D p ˆ ¬… x2 i *s ¡ x1 V os † ; ¬ < 0

… 11†

D p ˆ ¬… x2 i *s ¡ V o x1† ; ¬ < 0

… 12†

where ¬ is an arbitrary real constant. Now, it is clear that the PWM converter with this proposed control method is globally stabilized against large-signal disturbances. Note that the steady-state and the perturbed expressions for the switching function given in (4) and (11) include V os . Therefore, in order to be able to generate the switching function p (or compute in a digital implementation) , it is necessary to predict the time-varying steady-state output voltage V os ˆ V o ‡ v oh… t† for the present operating point of the converter. This requires estimation of the ripple component v oh… t†. There are a number of techniques proposed in the literature aimed at achieving this estimation. However, these techniques are quite complex and limited in accuracy (Spiazzi et al. 1997, Wall and Jackson 1997). In order to avoid this problem, we modify the control strategy by ignoring the ripple component v oh… t† in (11) and obtain the following approximate expression for D p Note that in (12) x2 is to be rede® ned as x2 ˆ v o ¡ V o . Note also that V os in (4) has also to be replaced by V o . It is obvious that the closed-loop system with this modi® ed control will result in steady-state errors in the state variables (see } 4). Stability of the system is also a€ ected by this modi® cation. It is quite di cult to determine the stability properties of the system with the modi® ed control. The arbitrary real constant ¬ must be chosen in such a way that it gives a stability region as large as possible, as well as ensuring that satisfactory dynamic response is obtained over the

Control strategy for single-phase PW M converters

Figure 2.

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Response of x1 , x2 and V_ … x† for start-up. The scale of the state variable x2 is 10x2 .

operating range of the converter. Another important criterion in choosing ¬ is the ripple content of the switching function. If j¬j is chosen to be very large, this ripple increases too much and the PWM comparison process is adversely a€ ected. A typical range of ¬ for the converter in this study is found to be ¡0:002 µ ¬ µ ¡0:000 15. 3.2. Stability study A sample simulation has been carried out to demonstrate the behaviour of the converter with the modi® ed control at start-up. Note that the initial point at t ˆ 0 is x1 ˆ ¡i *s … is ˆ 0† and x2 ˆ ¡V o… v o ˆ 0†. Figure 2 shows the simulation results for the state variables (x1 and x 2 ), and the time response of V_ … x†. It can be seen that both x1 and x2 oscillate about zero as the closed-loop system converges to the stable equilibrium point. Eventually, the amplitude of x2 is equal to the amplitude of the second harmonic in v o at steady state. It can be observed that V_ … x† stays negative for all t > 0, which means that the converter under the modi® ed control is asymptotically stable for the particular operating point chosen. However, it should be mentioned that the modi® ed control may not be globally asymptotically stable, as pointed out before. Extensive simulations have been carried out for various operating p  points in a predetermined operating region of the converter (100 2 V µ V o µ 500 V; 0 µ Im µ 60 A) shown in ® gure 3. In all cases studied the operation was seen to be stable. In ® gure 3, in regions B and C the switching function becomes saturated and in region D the distortion in the input current increases beyond an acceptable level. 4.

Estimation of the steady-stat e errors resulting from modified control

In this section, we obtain expressions for the state variables that give an idea about the steady-state errors in is and v o resulting from the approximate control. In

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Figure 3.

H. KoÈmuÈrcuÈgil and O. KuÈkrer

p   Various operating points of the converter when V m ˆ 100 2 V, L ˆ 5 mH, C ˆ 200 mF and ¬ ˆ ¡0:000 25.

the steady-state, let V os ˆ V o and let x1 and x2 be the steady-state errors in is and v o , respectively. Substituting (4) and (5) in (6) and then in (7) gives the steady-state expressions L x_ 1 ˆ ¡

vs x L di *s x2 ‡ 2 ¡ ¬… x2 i *s ¡ x1 V o †… V o ‡ x2 † Vo V o dt

C x_ 2 ˆ … Po ‡ D p†… i *s ‡ x1 † ¡

… V o ‡ x2 † R

… 13† … 14†

In order to be able to solve these nonlinear equations, we need to make some approximations . Therefore, assuming that D p  0, … i *s ‡ x1† º i *s and … V o ‡ x2† º V o in the steady state, (14) can be written as C x_ 2 ˆ Po i *s ¡

Vo R

… 15†

Using the power balance relation between the input and the output of the PWM converter at unity power factor and neglecting the power loss in the converter, we write V m Im  2

V o2 R

… 16†

In equation (16), the contribution of harmonics of v o to the output power is neglected. Substituting (4) in (15) and making use of (16) in the resulting equation, we obtain

Control strategy for single-phase PW M converters

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x2  V h sin … 2!t ‡ ³†

… 17†

where V h is the amplitude of the second harmonic in v o and is given by s                                             2 2 2 Vo L Im Vh ˆ ‡ 2!RC 4CV o

… † … †

… 18†

and ³ is the phase shift of this voltage ripple, given by



³ ˆ tan ¡1 ¡

2 L Im !R 2V o2



… 19†

In order to be able to solve for x1 , we use the assumption … V o ‡ x2† º V o in (13). With this assumption, substituting v s ˆ V m sin … !t† and (3) into (13), one obtains L x_ 1 ¡ ¬V o2 x1 ˆ

…¡ VV

m



¡ ¬Im V o sin … !t† ‡

o

Substitution of (17) into (20) yields L x_ 1 ¡ ¬V o2 x1 ˆ

!L Im cos … !t† x2 Vo

…¡ VV ¡ ¬I V †V sin … !t† sin … 2!t ‡ ³† !L I ‡… V cos … !t† sin … 2!t ‡ ³† V † m

m

o

m

o

o

… 20†

h

h

Solving this ® rst-order linear ordinary di€ erential equation gives s                            s                              K12 ‡ K22 K12 ‡ K22 x1… t† ˆ sin … !t ‡ ³1 † ‡ sin … 3!t ‡ ³3† 4… !2 L 2 ‡ K32 † 4… 9!2 L 2 ‡ K32†

… 21†

… 22†

where

… ³† ; …AA sinsin …… ³³†† ¡‡ AA cos cos … ³††

³1 ˆ tan ¡1

2

1

1

2

A1 ˆ ¡K1 K3 ¡ K2 !L ;

B1 ˆ K1 K3 ¡ 3K2 !L ; K1 ˆ

…¡ VV

m o



¡ ¬Im V o V h ;

… ³† …BB sinsin …… ³³†† ¡‡ BB cos cos … ³††

³3 ˆ tan ¡1

2

1

1

2

… 23†

A2 ˆ K2 K3 ¡ K1 !L

… 24†

B2 ˆ K2 K3 ‡ 3K1 !L

… 25†

!L Im V h ; Vo

… 26†

K2 ˆ

K3 ˆ ¬V o2

The steady-state errors in v o and is can be estimated by using equations (17) and (22), respectively. Equation (17) implies that the steady-state error in the average output voltage is zero. Equation (22) means that there is a steady-state error in the fundamental component of is , with an associated error in the phase angle ³1 . Furthermore, a third harmonic component appears in is . It should be pointed out that the input current is is purely sinusoidal (with the exception of switching harmonics) with the Lyapunov-based control given by (11) (see ® gure 8). From equation (22), it can be seen that the fundamental and the third harmonic in is are functions of V h and ¬. Figure 4 shows the computed percentage of total harmonic distortion (THD) values together with the computed fundamental and third harmonic amplitude values of the input current with respect to ¬. These results have been obtained from simulation of

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Figure 4.

H. KoÈmuÈrcuÈgil and O. KuÈkrer

Computed THD (%) values together with the computed fundamental and the third harmonic amplitude values of the input current with respect to ¬.

the closed-loop system with the modi® ed control. The parameters corresponding to ® gure 4 are the same as those given in the next section. 5.

Computer simulation and experimental verification

In order to demonstrate the feasibility of the proposed control method, the closed-loop system has been tested by both simulations and experiments. In all the simulationp and  experimental results given, unless otherwise stated, V o ˆ 200 V, V m ˆ 100 2 V, ! ˆ 100p rad s¡1 , R ˆ 50 «, L ˆ 5 mH, C ˆ 200 mF, f s ˆ 3:6 kHz and ¬ ˆ ¡0:000 25. Simulations were performed using the Simulink toolbox of Matlab. The solution method chosen is Runge± Kutta with order 5 with a ® xed step size (4 ms). 5.1. Computer simulations Figure 5 shows the switching function (p ˆ Po ‡ D p) of the converter obtained with the modi® ed Lyapunov-base d control method. This function is used to generate the PWM control signals for the switching devices of the converter. Figure 6 shows the steady-state input current is with the converter source voltage. Note that unity power factor operation is achieved. The computed third harmonic amplitude is 0.176 A (the computed fundamental amplitude is 11.34 A), whereas the theoretical third harmonic amplitude from equation (22) is 0.242 A. The computed total harmonic distortion of is is THD ˆ 8:54%. The main contribution to this comes from harmonics caused by switching. The ripple in the output voltage causes additional lower-order harmonics in is which are negligible. The input voltage of the converter, v i , is displayed in ® gure 7. Simulation results have also been obtained for the Lyapunov-based control (equation (11)). Figure 8 shows the input current is with

Control strategy for single-phase PW M converters

Figure 5.

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Simulated switching function waveform obtained with modi® ed control.

this control. It has been observed that is does not contain low-order harmonics (3rd, 5th, . . .) in this case. The computed total harmonic distortion of is is THD ˆ 8:42%.

Figure 6.

Simulated response of the steady-state is with v s obtained with modi® ed control. Source voltage scale is V m =25.

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Figure 7.

Figure 8.

Simulated response of v i with the modi® ed control.

Simulated response of steady-state is with v s obtained with Lyapunov-based control. Source voltage scale is V m =25.

Control strategy for single-phase PW M converters

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Figure 9 shows the transient responses of output voltage and input current for a step change in R from to 50 « to 25 «. The output voltage exhibits a fast transient response. It can be seen that there is a ripple amplitude (for R ˆ 50 «) of approximately 31.3 V, in good agreement with the theoretical value of V h ˆ 32:08 V. After the step change in R, the value of V h is almost double, in agreement with equation (18). The input current is nearly sinusoidal with unity power factor, before and after the step change.

5.2. Experimental results Operation of the proposed control strategy was experimentally veri® ed on a hardware simulation circuit. This circuit is an analogue circuit involving op-amps, analogue multiplexers and multipliers. Analogue multiplexers are digitally controlled switches and are used to implement the electronic switches of the single-phase PWM converter. A block diagram of the simulation circuit is shown in ® gure 10. Note that the L and C components in this diagram are emulated using op-amp integrators and adders. In the simulation circuit voltages and currents are scaled by a factor of 25. Experimental results were obtained that correspond to the simulation cases given in ® gures 5, 6, 7 and 9. Figures 11, 12 and 13 show the experimental switching function, steady-state input current with source voltage, and converter input voltage waveforms respectively. Figure 14 shows the experimental result that corresponds to the simulation case given in ® gure 9. It can be observed that all the experimental results are in good agreement with the simulation results.

Figure 9.

Simulated responses for a step change in R from 50 « to 25 « with modi® ed control.

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Figure 10.

Figure 11.

Block diagram of the closed-loop system.

Experimental switching function waveform.

Control strategy for single-phase PW M converters

Figure 12.

Experimental response of steady-state is with v s .

Figure 13.

Experimental response of v i .

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Control strategy for single-phase PW M converters

Figure 14.

6.

Experimental responses of v o (top scale) and is (bottom scale) for a step change in R from 50 « to 25 «.

Conclusions

A Lyapunov-based control strategy has been proposed for single-phase PWM ac/ dc voltage-source converters. It is stressed that global stability of the closed-loop converter can be achieved with this strategy. Owing to some di culties in implementing the Lyapunov-base d control, a modi® cation has been made to the control law. Analysis of the modi® ed control law has shown that negligible degradation occurs in the performance of the system. Furthermore, the proposed control strategy is robust, meaning that stability is una€ ected by changes in the parameters of the system. It is apparent from the results that the modi® ed Lyapunov-base d strategy exhibits excellent dynamic response, comparable with those of the methods proposed in the literature. Theoretical and simulation results have been veri® ed through experiments.

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KUï KRER, O., and K Oï MUï RCUï GI_L, H ., 1997, Control strategy for single-phase PWM recti® ers. Electronics L etters, 33, 1745± 1746.

N ISHIDA, Y., M IYASHITA, O., H ANEYOSHI, T., TOMITA, H ., and M AEDA, A., 1997, A predictive

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