review of a high performance control of ac … · 336 high performance control of ac signals based...

Suranaree J. Sci. Technol. 15(4):333-344

Department of Control System and Instrumentation Engineering King Mongkut’s University ofTechnology, Thonburi, Bangkok, THAILAND 10140; E-mail: [email protected]; Tel.: 02-470-9096;Fax.: 02-470-9092.

REVIEW OF A HIGH PERFORMANCE CONTROL OF ACSIGNALS BASED ON A PROPORTIONAL PLUS RESONANTCOMPENSATOR

Wanchak LenwariReceived: May 7, 2008; Revised: Jun 26, 2008; Accepted: Aug 25, 2008

Abstract

This article reviews a high performance control of alternative current (AC) signals based on a controllercalled the Proportional plus Resonant (P+Resonant) compensator. Recently, this type of compensatorhas been considered as a high performance controller and it can be applied to the AC control system,particularly in the field of power electronics. Applications of P+Resonant Compensator have beenproposed including the design consideration. When compared with the traditional controllers such asProportional + Integral (PI) controller, an accurate control of a high frequency AC signal can be obtainedwith this type of compensator. The characteristics and merits of the P+Resonant controller are analyzedand illustrated including the controller design in the discrete domain. Another major advantage is theability to eliminate external disturbance having the same frequency as the tuned resonance frequency.

Keywords: AC system, current control system, proportional + integral (PI) controller, proportionalplus resonant compensator

Introduction

Recently, in the field of power electronics, atten-tion has been paid to the performance of thecontrol system used in a circuit. In order to obtainhigh efficiency of power electronic devices, avery accurate control system is required. However,one of the main control problems in the field ofpower electronics, particularly when the systemis connected to the main, is that the referencesignal appears as AC signals to the controller.For example, the controller of a pulse widthmodulated (PWM) controlled rectifier (Wu et al.,1991; Rim et al., 1994) must track the AC

reference current at the fundamental frequency.Also, for a shunt active filter which is used toimprove the power quality of the supply network,the control must be very fast and accurate toinject currents which cancel the harmonic currentsdrawn by non-linear loads (Akagi, 1996). Theseharmonic currents have a frequency at least 5times that of the fundamental frequency. Thiscauses difficulty in the control design to trackthese AC reference signals with the lowestpossible error.

334 High Performance Control of AC Signals Based on a Proportional Plus Resonant Compensator

From knowledge of the control systemdesign, a very high bandwidth controller is requiredto reduce the tracking error for higher frequencyAC signals. However, due to the bandwidthlimitation of conventional Proportional + Integral(PI) or Proportional + Integral + Derivative (PID)compensators, these controllers are normallysuitable for DC input. But they cannot completelyeliminate the steady-state error when the inputis AC signals with very high frequency as shownin Figure 1, i.e. a phase difference between thereference and response signals exists. A controlsystem design to achieve such a high bandwidthrequires a very small sampling time (very highsampling frequency) (Franklin et al., 1998;Franklin et al., 2002). For a power electronicsystem, the sampling and switching frequencyof the system depends on the kVA level required,and this therefore limits the current controlsampling frequency. In practice, a high switchingfrequency causes a number of problems, suchas the device heating due to switching losses ata high power level. Additionally, the components,such as a very fast processor, a high speedanalogue to digital converter, and also a highspeed switching device, all of which are requiredfor the implementation, will increase the systemcosts.

The P+Resonant compensator has beenproposed in literature for the current control ofa power electronic system. A zero steady-stateerror can be achieved at resonant frequenciesusing this type of compensator with a properdesign. When it is applied to signals at itsresonant frequency, it can be considered to beclosely related to the PI compensator for a DCsignal. As a result, it is very effective in almost

eliminating the steady-state error when trackingan AC reference signal. The characteristics ofthe P+Resonant controller and the controlapplication based on this controller are discussedin this article. Simulation results are presentedand confirm the effectiveness of this alternativecontroller.

Proportional Plus Resonant Compen-sator

The transfer function of a conventional PIcontroller which achieves zero steady-state errorfor a DC reference input is given in (1) whereKp is the proportional gain and Ki is the gain ofthe integral term:

. (1)

In the 3-phase system, the popular controlscheme is based on the coordinate transformationof the 3-phase system to 2-phase (dq) system(Bose 2001). The transformed signals will beappeared as DC values and are then easilycontrolled by using PI compensator with zerosteady-state error. The P+Resonant compensatorperforms similarly to the PI compensator in thedq reference frame in that it controls signalsalternating at the resonant frequency withquasi-zero steady-state error. It can be derivedmathematically by transforming a PI compen-sator in dq frame of reference to the 3-phasesystem frame of reference. The full derivationhas been presented by Zmood et al., 2001 whichhas been cited mostly in papers relating to thistype of controller. From a survey, there are twomain different transfer functions for a

Figure 1. Control system responses of conventional low bandwidth PI or PID controllers

335Suranaree J. Sci. Technol. Vol. 15 No. 4; October - December 2008

P+Resonant compensator. One is based on asinusoidal transfer function as in (2); similarly,the other is based on a cosinusoidal transferfunction given in (3):

, (2)

and , (3)

where Kr is the gain of the resonant term andthe resonant frequency is 0. It should be noticedthat the ideal P+Resonant controller providesan infinity gain at the resonant frequency. It isobvious that Ccos (s) in (3) has the phase margin(PM) higher than that of Csin (s) in (2) (90degrees higher), thus for a stability concern,Ccos (s) is more preferable for the design. In a

control system design, a system with positivephase margin is considered stable since a phasemargin quantifies a safety margin that the systemhas to the parameter variations. PM is the mostwidely used measure of relative stability whenworking in the frequency domain. A system withpoor phase margin can lead the system to insta-bility or even have highly oscillation behavior.Figure 2 compares two frequency responses ofthe PI controller and P+Resonant controllerhaving a resonant frequency at 50 Hz.

From Figure 2(a), it can be noticed thatthe magnitude response of the PI controller tendsto decrease when the input frequency increases.On the other hand, the P+Resonant compensatorprovides a very high gain (theoretically infinite)at its resonant frequency. This means the con-troller will react rapidly to the input or error atthis frequency due to its high gain. As a result,

Figure 2. (a) Bode plot of PI controller C(s) = 10+1/s (b) Bode plot of P+Resonantcontroller C(s) = 1+2 50.s/(s2+(2 50)2)

20

30

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50

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Mag

nitu

de

(dB

)

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101

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e (d

eg)

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nitu

de

(dB

)

101 10 2 10 3 104-450

-405

-360

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se (

deg

)

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(a)

(b)


Magnitude (abs)

Frequency (rad/sec)f0fL fH

|Kr|

0.707*|Kr|

the steady-state error can be almost completelyremoved for a closed-loop control system. Forthe PI controller, the steady-state response isdetermined by the DC gain of the system. TheDC gain can be designed to be high as the steady-state error is satisfied. However, a very high gaincontroller is not preferable in some systems inpractice, because of the limitation in the ratingof the hardware used in the circuit. Additionally,the control will amplify noises.

As mentioned, a very high gain is notalways necessary in a control system because ittends to reduce the phase margin causingdifficulty in practical work. If the phase marginis insufficient, the system is potentially sensitiveto external disturbances that can lead to insta-bility. For the P+Resonant controller, the gainof the resonant term must high enough to achievea zero or quasi-zero steady-state error. The qualityfactor Q, is introduced in (4) to change the gainat the resonant frequency. This parameter alsoaffects the band-pass region. The definition ofQ is shown in Figure 3. Q represents the ratio ofthe centre or resonant frequency, f0 , to the -3dB bandwidth which is fH

_ fL. The upper andlower frequencies, fH and fL , are defined asthe frequencies where the gain has dropped to0.707 of the gain at a centre frequency. Thetransfer function of the compensator is thereforemodified to (5) and is chosen for the control ofthe AC reference in this article. Higher valuesof Q result in higher gains at the resonantfrequency and narrower band-pass regions nearthis frequency. On the other hand, lower values

Figure 3. The definition of quality factor(Q)

of Q will result in lower gains, hence somesteady-state errors will be present:

,(4)

and .(5)

Design of P+Resonant Compensator

The stability of the current control system shouldbe considered with this type of compensatoraccording to the configuration of the resonantterm, which provides a considerably high gainin the closed loop system. Moreover thecompensator has a complex conjugated pole,which may cause a resonance in the system. Clearly,to choose the appropriate parameters of theP+Resonant compensators is not straightforward.Quality factor, resonant gain, and proportionalgain all need to be carefully selected taking intoaccount many different factors, especially systemstability and robustness. Unlike the tuning of thePI compensator, there is no principle to tuneparameters of this compensator. However,design methods have been introduced by someresearchers. Fukuda and Imamura 2005 obtainedcontroller gains empirically through theirexperiment. This is based on the rule of trail anderror. Lenwari et al. 2006 proposed the controllerdesign by using the root locus technique but thedesign much depends on the designer experiences.The criteria for choosing the proportional gain,


Kp, the resonant gain, Kr , and the quality factor,Q, is discussed as following. It should be notedthat the resonant frequency ( 0) is designed tomatch with the frequency of a sinusoidal referenceinput.

Quality Factor

A higher Q will result in a higher peakand narrower band-pass region around theresonant frequency. However, if the peakmagnitude is too high, the system is potentiallysensitive to external disturbances and can leadto instability. On the other hand if the peakmagnitude is too low, the steady-state errorcannot be completely removed at the chosenharmonic frequency. In addition, the qualityfactor alters the complex-conjugated polelocation which needs to be positioned inside thestable region. Too high Q will bring this poletoward the stability boundary.

Resonant Gain

Kr determines the magnitude response ofthe controller at the resonant frequency. Thechange of resonant gain Kr will change thenumerator of the resonant term which theposition of the zero is determined. In general,some distance between the zero and its poleshould be adequate to provide sufficient gainsat the resonant frequency. For example, in caseof Kr is 1, the zero is located near to the poleand this provides insufficient gain.

Proportional Gain

It should be noted that the resonant termprovides only a very small gain outside itsband-pass region. Therefore, in order to obtaina decent transient response an additionalproportional gain Kp is required. Kp plays avery important part in overall dynamic responsesof the system including a transient response. Thecontrol bandwidth is mainly determined bythis gain. Clearly a high proportional gain willresult in a faster transient response, but a valuemust be a compromise between system stabilityand steady-state error.

Nowadays, the optimization techniques

have been employed to automatically design thecontroller. These approaches range from the useof the Nelder-Mead method (Dell’Aquila et al.,2006) to the Naslin polynomial technique(Dumitrescu et al., 2007) and the Genetic Algo-rithm (Lenwari et al., 2007). These techniquescan reduce the time commissioning spent in thedesign process. The optimized controller wasobtained with the excellent performance.

Applications of P+Resonant Compen-sator on AC Current Control System

As mentioned, the popular control scheme ofthe 3-phase system is based on the coordinatetransformation of the 3-phase system to 2-phasesystem. However, the main drawback of thecoordinate transformation based methods is thatit requires a burden of complex computationsleading to difficulty in real implementation. Asa result, it can not be implemented with lesscomputation resources and this is not favorable,particularly for commercial system. In the single-phase system where the transformation theoryis not applicable, a popular control is based onPI compensator due to its simplicity. The PIbased control is relatively easy to implementbut a known disadvantage is the presence oftracking error. Although, a high bandwidth PIor PID controller can be designed by increasinga gain, mentioned earlier it is not preferable insome systems in practice. In addition, the systemstability is another concern due to the bandwidthlimitation of these types of compensators. Withgiven merits of the P+Resonant controller, thistype of controller can mitigate these problems.Therefore, it can be applied to the controlschemes of either single-phase or 3-phase powerelectronics system, i.e. control of power converterof various topologies such as current control ofsingle-phase and 3-phase inverters). It is a simpleaddition and can replace a traditional PI con-troller to obtain fast control of the AC referenceinput. Examples of application of the P+Resonantcontrol in the field of power electronics arepresented as following:

Active rectifiers: Active rectifier is widelyused as the interface circuit for power electronicdevices such as ASD (Adjustable Speed Drive)


because a unity input power factor can beobtained. The active rectifier typically consistsof a 3-phase inverter for the PWM and aninductor-capacitor-inductor (LCL) filter forenergy coupling and switching frequencyharmonic reduction as shown in Figure 4.Control of the active rectifier must be designedto obtain a near sinusoidal line current (funda-mental frequency) which results in low harmoniccontents. The famous control topology for a3-phase rectifier employs a vector control theory(Bose 2001) but coordinate transformationadds a complexity to the control. Therefore, Satoet al. 1998 proposed a new control based on (2).Zmood and Holmes 2003 and Fukuda andImamura 2005 have introduced a control basedon (3). They obtained satisfactory results withzero steady-state error. Li et al., 2008 comparedtwo current control techniques, P+Resonant andPI control, for single-phase voltage-sourcePWM rectifiers. A comparison shows the

LCL filter

uc

CDC VDC

Load-side Grid

ib

ic

ia

ub

ua

Figure 4. 3-phase active rectifier

superiority of the resonant based approach. Inaddition, it is able to eliminate the disturbanceof the fundamental supply voltage completely.Recently, a resonant based approached is appliedfor the control of multilevel active rectifiers aspresented by Dell’Aquila et al. 2008.

Uninterruptible power supplies (UPS):The P+Resonant control has been applied tocurrent control algorithms for UPS systemswhich are applicable to both single-phase andthree-phase systems. For example, Loh et al.,2003 developed a high performance UPScontrol algorithm by adding a P+Resonantcompensator into the outer voltage regulationloop to achieve zero steady-state error.

Active power filters: The harmonicproblems in electrical power systems havesparked research in the field of active filteringand led to the development of the shunt activefilter shown in Figure 5. The active filter is apower electronic device and is controlled to

ua

ub

uc

CDC VDC

Grid

ih b

ih c

ih a

Figure 5. 3-phase shunt active filter


inject current (ih) into the grid, which cancelsthe harmonic currents caused by disturbingnon-linear loads. The 5th, 7th, 11th, and 13th arethe frequencies which are of the most concernsince they are commonly the main harmonicfrequencies generated by the non-linear load.The difficulty is that the frequencies of theseharmonics are considerably high. If the controlis not fast enough, a shunt active filter mayactually exacerbate the grid harmonics problem.The standard PI controllers can be applied withvector control topology but they cannot controlharmonic current injection accurately, particu-larly for high order harmonics. The P+Resonantcontroller was applied and obtained an excellentcontrol of harmonic currents (Mattavelli 2001;Yuan et al., 2002; Fukuda and Imamura 2005;Liserre et al., 2006; Lenwari et al., 2006; Lascuet al., 2007). Some controllers were designedto have more than 1 resonant frequency.

Harmonic reference generation: Since theresonant terms provide high gains at resonantfrequencies, they can be modified to use as har-monic voltage or current extraction method forgenerating harmonic reference for the activepower filter. These applications have been reported,for example, by Newman et al., 2002.

Photovoltaic (PV) system: In a PV system,the control of grid-connected inverter can bedesigned with P+Resonant controllers, for ex-ample, reported by Ciobotaru et al., 2005 andXiaoqiang et al., 2006. The control was improvedin the performance and achieved zero steady-stateerror. It is not limited only to PV system applica-tions but also in different power generationsystems in the future such as hydropower, windpower, etc.

Distributed generation system: Anotherapplication is on the power quality improvementalong the distributed line. For example, to improvethe power quality of a grid-connected system,the control based on the P+Resonant wereproposed by Liserre et al., 2006 and Li et al.,2006. In the latter, the P+Resonant controllerswere used to maintain the grid-connected inverterand compensate both positive and negativesequences components in a micro-grid system.The results show the improvement of voltagecontrol and power quality on a micro-grid system.

Future application: Nowadays, researchtrend in renewable energy has become morepopular particularly with the focus on energyefficiency improvement. Renewable energytechnologies have been growing rapidly formany forms of renewable energy. A powerconverter plays a very important part on thesystem efficiency including the utility interfac-ing. The high performance control of powerconverter circuit will therefore enhance thesystem performance and if resonant control isapplicable to parts of the control system, it willbe possible to boost a system performance witha precise control.

Digital Implementation

Nowadays, the control strategy is digitallyimplemented using microcontrollers or a digitalsignal processor. To design the P+Resonantcontroller, the simple current control loop indiscrete domain is shown in Figure 6. A compu-tation delay to represent the processor delay isincluded and set to be a 1 interrupt period. Thedigital controller is shown in the dotted area ofFigure 6. The continuous-time controller (5) isdiscretized by the forward rectangular ruleapproximation (Franklin et al., 1998) given in(6) where Ts is the sampling time. The digitalcontroller parameters are then derived as in (7):

, (6)

(7)

and

. (8)

In order to evaluate the control performancebased on the P+Resonant controller, it is comparedwith a simply designed PI controller. A controlloop in Figure 6 is modeled by MATLAB/Simulink. The plant is an inductor with littleresistance, a common type in the field of powerelectronics. In this paper, the plant inductanceand resistance used in the design were 7.5mH


and 0.4 respectively. The sampling frequencyis 5kHz. A P+Resonant compensator wasdesigned to have a resonant frequency at 250 Hzand a PI controller was designed with a zerocancelling the plant pole. The control responsesare compared as shown in Figure 7. The reference(green color) is an AC signal of 250 Hz with amagnitude of 1A and the feedback signal is blue.Figure 7 clearly shows that the P+Resonantcontroller (bottom) even with a lower gain (1)has a better performance when compared with

the PI controller (top) with a higher gain (5) asthe phase error is almost perfectly eliminated.Further investigation was under taken when thegain of the PI controller was increased to 37(just before the loop is unstable at the gain of38). A small phase error can be seen in theresult. This clearly shows the superiority of theP+Resonant controller and strengthens what theauthor mentioned earlier about the merit of theresonant compensator in practice.

Figure 7. Comparison of control response between PI controller and Resonant controller:(Top) 5(z-0.9894)/(z-1); (Bottom) (z2-1.545z+0.6396)/(z2-1.844z+0.9391)

1.492 1.493 1.494 1.495 1.496 1.497 1.498 1.499 1.5-1.5

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p(A

)

Reference signal

Feedback signal

Reference signal

Reference signal

Feedback signal

Figure 6. A simple current control loop in z-domain


Disturbance Rejection Property ofP+Resonant Control System

One of the most important concerns in a feedbackcontrol system is the effect of the disturbancesignals. A disturbance signal is an unwanted inputsignal that affects the system output. It is intro-duced by any distorted voltages or currents inthe system such as the harmonics in the mainsupply. Some control systems are subject todisturbance signals that cause the system toprovide an inaccurate output or even make thesystem unstable (Franklin et al., 2002). Thedisturbance rejection property of the currentcontrol system based on a P+Resonant compen-sator was investigated in this paper with boththe simulation and experimental methods. Sincethe P+Resonant controller provides a very highgain at the resonant frequency, it is able to rapidlyrespond to any disturbances at this frequency

and eliminate them. Theoretically, the overallclosed loop gain is high enough to eliminatethese main disturbances. This excellent propertycan also been applied to the drive system toimprove the system performance (Mihalache,2004).

To begin, the current loop with respect tothe presence of a disturbance (Figure 8) ismodeled by MATLAB/Simulink. The voltagedisturbance is an AC signal 50 Hz with theamplitude of 10 V. It is applied to the currentcontrol loop at the time 0.5 second. Simulationresults presented in Figure 9 shows the feedbacksignal (blue), the output of the delay block inFigure 8, and the input voltage disturbance(green). It is clearly seen from Figure 9 that thecurrent control quickly reacts to the voltagedisturbance. At the steady-state, the voltagedisturbance is controlled.

Figure 8. Current control loop with respect to disturbance

+

_

i

VoltageDisturbances

P+ResonantCompensatordelay

Plant

*

_

+

Figure 9. Simulation of the current control loop with respect to the voltage disturbanceshowing the disturbance rejection property

0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62-15

-10

-5

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plit

ud

e(V

)

Voltage Disturbance

Response to the disturbance


In practice, the current control system canbe seriously impaired by external disturbances,particularly the background harmonics on thesupply voltage, and the effect of non-linearitiesin the power electronic converter such asdeadtime and even the PWM algorithm itself.The deadtime has a considerable effect on theinverter output voltage. The distortion in theoutput voltage at the current zero crossing resultsin low order harmonics such as 3rd, 5th, 7th andso on, of the fundamental frequency in the inverteroutput. Similar distortions occur in the line-to-line output voltages of the 3-phase PWM inverterwhere low-order harmonics are of the order6m+1 (m = 1, 2, . . .) of fundamental frequency(Mohan et al.,1995). The main voltage distur-bances corresponding to the supply harmonics,and deadtime effects inevitably exist in thesystem.

To experimentally verify the disturbancerejection property, the comparison had beenmade with the fundamental current control usingthe simple PI controller and the P+Resonantcontroller having resonant frequencies at 5th,7th, 11th, and 13th of the fundamental frequency

as shown in Figure 10. In Figure 10(a), the PIcontroller is unable to reject the disturbancesintroduced by the supply voltage harmonics, andthe inverter non-linearities, and significant 5th

and 7th harmonic distortion can be seen. Bycontrast, the operation of the P+Resonant con-troller is illustrated in Figure 10(b) and the harmonicreduction in the presence of the same externaldisturbances is obvious. This confirms the excep-tional property of the resonant based controlsystem.

Conclusions

This article has reviewed a high performancecontrol of AC signals by using the P+Resonantcontroller. The characteristic of the suggestedcontroller has been described in detail. Recently,research trends favor fully digital control. Thus,a modified transfer function of the P+Resonantcontroller is proposed and discretized, whichallows digital implementation. In order to showthe advantage of using this type of controller,applications to an AC current control system arepresented showing the precise control of the AC

Figure 10. Fundamental Current Control (a) PI controller (b) P+Resonant controller

(a)

(b)


reference having a frequency at the resonantfrequency. As the zero steady-state error can beachieved at resonant frequencies, this controlleris very fascinating and the author believes thatsuch an advantage may be of interest to a controlsystem designer, particularly in field of powerelectronics where the AC control system iswidely used.

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review of a high performance control of ac … · 336 high performance control of ac signals based...

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