[WeD1-2] 8th International Conference on Power Electronics - ECCE Asia

May 30-June 3, 2011, The Shilla Jeju, Korea

Full Digital Control of Three-phase DCIAC

Inverter J. Leel, J. Choi2, K. H. Parkl and K. S. Hanl

1 Automation R&D Center, LSIS Co., Ltd., Cheonan, 330-741, South KOREA

2 Chungbuk National University, School of Electrical Engineering, Cheongju, 361-763, South KOREA

Abstract-- This paper presents a design of full digital

controller for three-phase DC/AC inverter to meet the

dynamic specification. The three-phase inverter system

including a LC output filter becomes a MIMO system due to

the coupling terms. These coupling terms make the

controller design difficult and complicate. In this paper,

through RGA and simulation process, it is shown that the

coupling terms can be ignored to represent the system as a

SISO one firstly. And then the full digital voltage controller

is designed directly in z-domain. It is a very simple structure

of single loop and has a good performance caused by

including no zero. Gains of designed controller are assigned

by MATLAB according to the time response such as

overshoot and settling time automatically. The simulation

results using DLL block of PSiM simulator is almost same

as the design specification. Finally, the full digital controller

is verified by the experiment using TMS320VC33 DSP.

Index Terms--Digital controller, Three-phase inverter

I. INTRODUCTION

The three-phase inverter system for renewable energy sources, such as wind power- or photovoltaic generator, is operated as stand-alone- or grid-connected mode. For the stand-alone mode operation, it keeps the voltage and frequency constant, but it is difficult to regulate the voltage and frequency constantly due to the continuous change of load. Therefore, it has been strongly recommended to design the inverter controller with fast and stable response. So far, there have been many previous works to design the robust controller for three­phase inverter. This paper also makes issues of controller design.

Firstly, it is important to make the low order of control system. Generally, the inverter controller has a double loop controller with an outer voltage controller and an inner current controller. Those are designed basically by using PI controller, and it can be implemented using the P-resonant controller[I-3l to eliminate the phase delay or the feed-forward compensator[4-6l for decoupling of coupling terms or compensating the cyclic fluctuating loads. But the more controllers are added the higher the system order. And it makes the controller design difficult and complex. Additionally, a high order system is very weak to the noise. Therefore, it is necessary to choose a controller to be low order as well as fast time response. In this paper, a PID single loop voltage controller is selected to meet the system order to be low and the control response to be fast. And two parameter construct is designed to make the robust controller.

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Second issue is how to determine the controller gains. Three-phase inverter system is usually converted into the synchronously rotating reference frame to describe the AC variables as DC ones. If DC variables are used, there is no phase delay problem for using a linear controller such as PI or PID as well. But if the three-phase inverter system with output LC filter is described on the synchronously rotating reference frame, it becomes a MIMO (multiple input multiple output) system due to the coupling of d-q components. To design the controller for a MIMO system is difficult. So we need to change the system from MIMO to SISO (Single Input Single Output). In this paper, RGA (Relative Gain Array) is derived to recognize how hard each axis is bonded firstly, and then three-phase inverter system is described as a SISO system by ignoring the coupling components. And then, gains of two parameter controller of RST are designed by using a pole-placement method to meet the time response specifications such as overshoot and settling time.

Third issue is about the digital controller which has been used generally for most of the power electronic converters recently as the spread out of high quality microprocessor. There are many benefits for using a digital controller, and it becomes much easer to implement by applying the digital simulation code provided by the common simulation tools to the microprocessor without any transformation. But the meaning of digital control has been confused in the previous works. In most studies, it was called as a digital control when it was just implemented by using a digital signal processor[7-17l. Thus the linear PID controller was designed in s-domain, and then it was adapted to a DSP system without any exchanges of domain[7-Sl. Only switching technique[9l and switching time[IOl have been considered. The system was digitalized to control but there is no exact design of a digital controller[lil. The full digital control means particular region more than the digital control because the full digital control contains to model the digital plant and design the digital controller using digital theory and then apply to the digital signal processor. Sometimes, it was called a full digital control when they change the domain from a continuous time controller to a digital controller in some studies without mentioning about the problems of added zero and pole[12-13l. Some studies designed digital controller from digitalize the plant using ZOH (zero-order hold) and z­transform[14-17l. But those techniques couldn't meet the controller design exactly to satisfY the time specifications.

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This paper presents a full digital controller which includes digital plant modeling, digital controller design, digital simulation and experiment using DSP. The digital plant of three-phase inverter is modeled by ZOH and z­transform. The digital controller is two parameter controller of RST structure without any additional zerosl181. To find the controller gain values, the pole­placement method is used, which makes the change of the settling time and overshoot demanded by user easy. Also, the designed RST controller is simulated by PSiM DLL which can be applied to DSP experiment without any conversion. Finally, this paper presents the experiment results implemented by using TMS320VC33 DSP. The simulation results and the experiment results matched well with a good performance of tracking and regulating.

IT. TECHNICAL WORK PREPARATION

A. Inverter Modeling in s-domain

The structure of three-phase DCI AC inverter with LC filter is shown in Fig. 1. The voltage and the current equations of three-phase inverter are derived as shown in Eqs. 1 and 2. The three voltage equation of Eq. 1 can be converted to Eq. 3 in synchronously rotating reference frame by using the conversion matrix, B. Then Eq. 3 can be described as Eq. 4 in s-domain. Same as above, Eq. 2 can be expressed as Eq. 5 in s-domain

Both Eqs. 4 and 5 are coupled each other. There is a current coupling term in Eq. 4 and a voltage coupling term in Eq. 5. From Eqs. 4 and 5, a dq model of three­phase inverter system is described as in Fig. 2. As it is shown in Fig. 2, the dq components are coupled each other. There are two coupling terms between d- and q­axis and those terms make the system be a MIMO system. It is difficult to design the controller for a MIMO system. But if the bonding strength of coupling is enough weak to neglect the coupling effect, then it is possible to design the controller for each d- and q-axis separately described by STSO system.

dAS �s �s

Cl-vc = IA -10 dt

Ld

B� BA BA l- 'zA = 'vA - ·vc: . dt [COS {))t sin {))t ] where B =

sin {))t cos {))t

[SIAq ] = [0 - OJ][IAq. ]+

Lj sl Ad OJ 0 I Ad 0

(1)

(2)

(3)

(4)

o [IAq -IOq ] (5) _1_ I Ad -IOd Cl

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Fig. I. Three-phase VSI inverter

Fig. 2. D-q model of three-phase VSI inverter

B. Effect o/ Coupling Term

Two kinds of analyses are proceeded to recognize the coupling effect exactly in this paper. Firstly, RGA is used to recognize how closely two axes are bonded. To solve RGA of system, the system should be described by a state-space equation as in Eq. 6.

x=Ax+BJu+B2V

y= Cx where,

r

IAq

x = Vc.oq fAd VCd

1 -ljLI I/Cf 0

A= OJ 0

0 OJ

ljLj 0

0 0 BI =

0 If Lt 0 0

,

c=[� 0 �l 0 0

-OJ 0

0 -OJ

0 - If Lf ljCI 0

B �r-l;C! 2 0

0

(6)

,

0

0

-ljet 0

And then output equation can be induced as in Eq. 7.

(7)

hi 2

( 1

)2 a/ 2m

w ereNll =--s + -- --- , N12 = --- s , LICf LfCI LICI LfCr 2m 1 2 1 2 m 2

N21 =--s , N22 = -- s +(--) - --LrCr LfCr LrCf LrC) Dll = D12 = D21 = D22

4 1 2 2 1 2 4 2m2 = s +2(-- +m )s +(--) +m - --LfCr LrCr L rCr

RGA of three-phase inverter is presented as shown in Eq. 8. The parameter AU provides an indication of how

sensible it is to pair the ith input with the /h output.

From this equation, it is given that the bonding strength is very weak to be ignored. Bonding effect of d-axis by q­axis and inverse effect is almost zero.

_[ [ 1 0] A = Au = [G(O)]ij[G (O)]ji = 0 1

(8)

Another way to verity it, a simulation is used by comparing between the simulation results of a coupled controller and those of a decoupled controller. For decoupling, two decouplers composed of two voltage terms and two current terms are added inversely as shown in Fig. 3. The general PI controller was designed for simulation in s-domain using the PSiM 7.0. Two PI controllers for d- and q-axis have same gain values designed by pole-placement method when filter values are given by L=3mH and C=2uF and sampling time is given by 20kHz in the system.

Simulation waveforms of coupled controller are shown in Fig. 4. Reference voltages and output capacitor voltages are shown in Fig. 4(a), d-q reference voltages and output capacitor d-q voltages are shown in Fig. 4 (b), and coupling voltages and currents of waveform are shown in Fig. 4( c). Simulation waveforms of decoupled controller are shown in Fig. 5 same as those of Fig. 4. Two simulation results are very similar. The tracking pattern around the starting point of system and the regulating trend with the load change from zero to full at 20ms are almost same. Output capacitor voltage in Fig. 4(a) is bigger about two-hundred thousand times than sum of coupling voltage and coupling current in Fig. 4( c). Furthermore, magnitude of coupling terms in Fig. 4( c) and Fig. 5( c) is alike. These means that there is no decoupling effect and the coupling terms can be just ignored by disconnection.

Fig. 3. D-q model of three-phase inverter with decoupler

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Fig. 4. Simulation waveforms for coupled system: (a) Reference voltages and capacitor voltages, (b) d-q reference voltage and d-q

capacitor voltage, ( c) coupling voltage and current

Fig. 5. Simulation waveforms for decoupled system: (a) Reference voltages and capacitor voltages, (b) d-q reference voltage and d-q

capacitor voltage, ( c) coupling voltage and current

Fig. 6. Simplified three-phase inverter system

For those reason, coupling term of three-phase DC/AC inverter can be ignored. After neglecting the coupled terms, the transfer function of simplified three-phase inverter can be shown as in Fig. 6. There are no coupling terms, so d-q voltage and current in Eqs. 4 and 5 are represented simply as in Eq. 9.

(9)

m. FULL DIGITAL CONTROLLER DESIGN

To make a full digital controller, the digital form of three-phase inverter is derived by z-transform from Eq. 9, and then RST controller of two-parameter construction is designed by pole-placement method. That controller should have the fast response and minimum overshoot.

A. Digital Model of Inverter

Tn order to design the digital controller without any design process of analog controller, we must find an equivalent digital plant first from the given analog plant of Fig. 6.

The analog plant and ZOH can be modeled as a digital system as shown in Eq. 10 with a discrete transfer

Fig. 7. (a) Equivalent analog plant model. (b) Equivalent digital plant

model.

Fig. 8. (a) General control form. (b) RST control structure.

function using of z-transform ( z = e" ).

(10)

B. RST Controller Design in z-domain

Using the digital transfer function of Eq. 12, the full digital controller for three-phase inverter can be designed.

There is a general form of system and controller as shown in Fig. 8 (a). That structure can become a single feedback loop or two-parameter structure. If C, is

replaced by TIS and C2 is replaced by RI S, the

general control form can be represented by RST control structure as shown in Fig. 8. (b).

H -I _ B(q-I)R(q-l) cr (q ) -

A(q-l)S(q-l) + B(q-l)R(q-l)

B(q-l)R(q-l) (II)

p(q-I)

where, G(q-I) = B(q-I)

A(q-I ) RST control structure can be matched as a cascade

continuous PID controller when T is same as R. But the cascade PID controller has a zero in the transfer function. That zero affects badly to the performance and stability of the system. The closed loop transfer function of system is represented by Eq. 11.

The polynomial p(q-l) defines the desired closed

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loop poles linked directly to the desired regulation performance and the product B(q-l)R(q-l) defines the

closed loop zeros. The digital PID controller, in general, does not simplify the plant zeros and thus can be used for the regulation of plants having a discrete-time model with unstable zeros. Furthermore, the digital PID controller introduces additional zeros defined by R( q -') that will

depend on A( q -I ), B( q -I) and P( q -') and thus which

cannot be specified a priori. In certain situations, these zeros may produce the undesirable overshoots during the transient.

Digital controller to match the continuous time PID and I-PD controller, R and S are considered as second order as shown in Eqs. 12 and 13. Also, the characteristic equation of the closed loop transfer function, P(q-l), can be calculated as shown in Eq. 14.

R(q-') = (r" +�q-' +r2q-2) (12)

Seq-I) = (l+s,q-' +S,q-2) = (l-q-')(l+s,'q-') (13)

P(q-') = (I + a,q-' + a2q-2)(1_q-')(1 +s,'q-')

+ (b,q-' -b2q-')(ro +r,q-' +r2q-') (14)

The pole placement method was implemented to fmd R, S and T. So the target polynomial containing 4

poles which are two dominant poles and two auxiliary poles can be made as shown in Eq. 15. In the target polynomial, PI and P2 are values by the dominant

poles and al and al are values by the auxiliary poles.

Dominant and auxiliary poles are designed by a pole­placement method. Besides, auxiliary poles can be assigned as the similar value in the z-domain.

TABLE I Three-phase T nverter Specifications

System parameters Values Lf 2 mH

Filter Cf 3 ,uF

Reference voltage 312 Vpcak DC-link 650 Vdc

Switching frequency 10 kHz

Sampling frequency 20 kHz

Overshoot <5 % Time spec.

Settling time <3 ms

For the design of a controller, the important specifications of time response for three-phase inverter system are overshoot and settling time. Those specifications are very important to drive the three-phase inverter system with constant voltage under the changing

load conditions. Designed controller gains are assigned in order that the settling time is less than 3ms and the overshoot is less than 5% as shown as Table 1. Sampling­and switching frequency is 20kHz and 10kHz, respectively. DC link voltage is 650Vdc and reference voltage is 312Vpeak. And LC filters are assigned as L = 2mH and C = 3 pF .

Pole-assignment method can control the time response such as settling time and overshoot. Four cases are considered as the different settling time (ms), t" from

0.3ms to 3.0ms as shown in Table 2. All of the controllers are designed to generate no overshoot, Mp with the resonant frequency, 0)0 ' the damping ratio, (, and

auxiliary pole, a, of the second-order controller. And also, the gain margin, GM, and the band width, BW, of each controller are shown in Table 2. It is known from the pole-zero map in z-domain as shown in Fig. 9 that the dominant pole is close to the point of (1, 0) as the settling time is longer.

TABLE II Parameters of Digital Controller

Case ts Mp 0)0 ( a GM BW 1 0.3 0. 0 15101.4 0.91 138036.5 4.3 10777.9 2 0.9 0. 0 4314.7 0.91 39739.0 11.3 3117. 2 3 1. 8 0. 0 2083.0 0.91 19039.5 14.9 1503.9 4 3. 0 0. 0 1243.0 0.91 11361.0 16.7 897. 2

Bode plots of frequency response analysis are shown in Fig. 10. Controller of Case 1 with fast time response has large BW, and there is no high point which makes overshoot at crossover frequency as shown as Fig. 10.

Unit-step responses of four cases are shown in Fig. 11. Settling times are less than 3ms and overshoots are almost zero for every case, which meet the time response specification for the controller of three-phase inverter.

TV. SIMULATION AND EXPERIMENTAL RESULTS

Four cases with the different specification shown in Table 2 are simulated by using the DLL block of PSiM 7.0 based on C++ without any analog block. After then, that same C++ source is used for the experiment.

A. Simulation Results

Simulation results are shown from Fig. 12 to Fig. 17. Each figure shows the same waveforms of the reference and capacitor voltages of three-phase in (a), the d-q reference and capacitor voltage in (b), three-phase currents in (c), d-q coupled voltages in (d), and d-q coupled currents of each case. The simulation time is 50ms but the load is changed from no load to full linear load of 1 OkW at 25ms in Figs. from 12 to 15.

Tn Fig. 12 which is for the case 1 with the fastest response, the output voltages of capacitor are well tracking the reference voltage waveforms without any overshoots but there are some noises under no load condition due to the sensitivity of controller.

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Fig. 9. Poles and zero in z-domain.

Fig. 10. Bode plots of controller

Fig.l1. Unit-step responses of designed controller

As the increase of the time response of controller, the settling time is delayed more and more as shown in the simulation results of Figs. 13, 14 and 15. When the load is changed, the tracking speed is faster as the settling time is designed to be shorter.

So case 1 is the fastest among them. Figures 16 and 17 show the same waveforms of cases 3 and 4 with the

Fig. 12. Wavefonns of case 1: (a) Reference and capacitor voltages of three-phase, (b) d-q reference and capacitor voltage, (c) three-phase

currents, (d) d-q coupled voltages, (e) d-q coupled currents

Fig. 13. Wavefonns of case 2: (a) Reference and capacitor voltages of three-phase, (b) d-q reference and capacitor voltage, ( c) three-phase

currents, (d) d-q coupled voltages, (e) d-q coupled currents

nonlinear load. It is verified that all controllers are well designed to have a good perfonnance and less than 5% THD though the load is nonlinear condition.

B. Experimental Results

General specification for experiment is same as in Table I. The designed controllers are experimented by TT

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Fig. 14. Waveforms of case 3: (a) Reference and capacitor voltages of three-phase, (b) d-q reference and capacitor voltage, (c) three-phase

currents, (d) d-q coupled voltages, (e) d-q coupled currents

Fig. 15. Waveforms of case 4: (a) Reference and capacitor voltages of three-phase, (b) d-q reference and capacitor voltage, (c) three-phase

currents, (d) d-q coupled voltages, (e) d-q coupled currents

DSP of TMS320VC33 and AD789 1 AID converter of 12bit, 8ch.

Capacitor voltage tracks well in cases 3 and 4 as shown in Figs. 18 and 19. There is no overshoot. But settling time is a little bit delayed comparing the design specification of controller with the simulation results due to the unknown experimental limit. Figures 20 and 21

Fig. 16. Waveforms of case 3 with nonlinear load: (a) Reference and capacitor voltages of three-phase (b) d-q reference and capacitor voltage

(c) Three-phase currents (d) d-q coupled voltages (e) d-q coupled currents

Fig. 18. Experimental results for three-phase capacitor voltage and d-q voltage of case 3

Fig. 19. Experimental results for three-phase capacitor voltage and d-q voltage of case 4

show the waveforms with the changing condition of linear load. In those cases, the output voltage is stable even though the load is changed from zero to full. Capacitor voltage and inductor current waveforms with the nonlinear load condition are presented in Figs. 22 and 23. Voltages are well regulated and there are no

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Fig. 17. Waveforms of case 4 with nonlinear load: (a) Reference and capacitor voltages of three-phase (b) d-q reference and capacitor voltage

(c) Three-phase currents (d) d-q coupled voltages (e) d-q coupled currents

Fig. 20. Experimental results for capacitor voltage, inductor current and d-q voltage of case 3 under load changing condition

Fig. 21. Experimental results for capacitor voltage, inductor current and d-q voltage of case 4 under load changing condition

overshoot as well.

V. CONCLUSIONS

This paper presents the design technology of full digital controller for three-phase DCI AC inverter to satisfy the time specification given by the demand of user.

Fig. 22. Experimental results for capacitor voltage, inductor current and d-q voltage of case 3 under nonlinear load condition

Fig. 23. Experimental results for capacitor voltage, inductor current and d-q voltage of case 4 under nonlinear load condition

The three-phase inverter system with LC filter is given by MIMO system not SISO due to two coupling terms. But from RGA and simulation results, it can be ignored for the design of controller to make the system simple. Eventually, three-phase DCIAC inverter is represented as SISO system. And then the full digital voltage controller of RST is designed directly in z-domain. It is a very simple structure of single loop and has a good performance caused by including no zero. With the digital simulation using PSiM simulator and experiments using TMS320VC33 DSP, it is verified that the proposed design algorithm for digital controller meets the design criterion very well.

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