issn 1755-4535 repetitive voltage control of grid .../file/ietpel10.pdf · in this section, a...

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Published in IET Power ElectronicsReceived on 14th December 2009Revised on 17th March 2010doi: 10.1049/iet-pel.2009.0345

ISSN 1755-4535

H1 repetitive voltage control of grid-connected inverters with a frequencyadaptive mechanismT. Hornik Q.-C. ZhongDepartment of Electrical Engineering & Electronics, The University of Liverpool, Brownlow Hill, Liverpool L69 3GJ, UKE-mail: [email protected]

Abstract: A voltage controller is proposed and implemented for grid-connected inverters based on H 1 andrepetitive control techniques. A frequency adaptive mechanism is introduced to improve system performanceand to cope with grid frequency variations. The repetitive control, based on the internal model principle,offers excellent performance for voltage tracking, as it can deal with a very large number of harmonicssimultaneously. This leads to a very low total harmonic distortion and improved tracking performance. It turnsout that the controller can be reduced to a proportional gain cascaded with the internal model (in a re-arranged form), which can be easily implemented in real applications. The proposed controller isexperimentally tested to validate its performance, focusing on reducing tracking error and total harmonicdistortion, under different scenarios (e.g. in the stand-alone mode or in the grid-connected mode, with orwithout the frequency adaptive mechanism, with linear or non-linear loads etc).

1 IntroductionNowadays, control and stabilisation in electricity systems aretaken care of only by large centralised facilities, such as coal,nuclear or gas-fired power plants. Owing to continuouslyincreasing penetration of distributed power generationsystems (DPGS), which are mainly based on renewableenergy sources, transmission system operators have issuedmore strict interconnection requirements, called grid codecompliance. In general, the requirements are intended toensure that DPGS have the control and dynamic propertiesneeded for the operation of power systems with respect toboth short-term and long-term security of supply, voltagequality and power system stability. The requirements differsignificantly from countries to countries and depend on theproperties of each power system as well as the level of theDPGS penetration.

A common requirement in all standards is the quality ofthe distributed power. The power quality assessment ismainly based on total harmonic distortion (THD). Forboth wind turbine and photovoltaic arrays connected to the

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utility grid, the maximum THD of output voltage allowedis 5% (120 V–69 kV). Table 1 shows the maximum THDallowed in the currents [1, 2]. Since an important propertyof harmonics is that they tend to be cumulative on powersystems [3], the controllers used should have very goodcapability in harmonic rejection, in order to meet operator’srequirements.

Several feedback control schemes, for example deadbeat orhysteresis controllers [4, 5], have been proposed for invertersto minimise THD. However, these controller alone cannoteliminate the periodic distortion caused for example bynon-linear loads. Repetitive control theory [6], which isregarded as a simple learning control method, provides analternative to eliminate periodic errors in dynamic systems,using the internal model principle [7]. The internal modelis infinite-dimensional and can be obtained by connecting adelay line into a feedback loop. Such a closed-loop systemcan deal with a very large number of harmonicssimultaneously, as it has a high gain at the fundamental andall harmonic frequencies of interest. It has been successfullyapplied to constant-voltage constant-frequency (CVCF)

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PWM inverters [8–14], grid-connected inverters [15, 16]and active filters [17, 18] to obtain a very low harmonicdistortion.

For grid-connected applications, the actual grid frequencyf varies and the performance of the repetitive control can bedegraded. If the grid frequency changes slightly, then thetracking error no longer converges to zero, but it would staysmall if the frequency variations are small [19]. Thesensitivity of the control system to such frequencyvariations depends on the delay time in the internal model.The correction of the amount of delay used in the internalmodel could lead to significant improvement in theperformance; see the internal model with adjustableresonant frequencies proposed in [14]. However, followingthe discrete-time implementation with a low samplingfrequency, the adaptive delay time is difficult to implement.In [10], a fractional delay-based repetitive control schemewith fixed sampling rate is proposed with two differentdesign methods: the Lagrange interpolation method andthe least-square method. The Lagrange interpolationmethod is easy to implement but with limited performance.The least-square method provides better performance butrequires intensive computation.

In this paper, the voltage controller based on H 1 andrepetitive control techniques proposed in [15] is furtherdeveloped and experimentally tested. The attention is paidto improving power quality and tracking performance, andconsiderably reducing the complexity of the controllerdesign. Moreover, a frequency adaptive mechanism tochange the cut-off frequency of the low-pass filter in theinternal model is proposed, so that the controller can copewith grid frequency variations. This mechanism allows thecontroller to maintain very good tracking performance overa wider range of grid frequencies. Another majorimprovement in this paper with respect to [15] is that,following extensive simulations and real-time experiments,the model of the plant has been reduced and themeasurement of the current iC has been removed. Thesystem design is thus reduced to single-input–single-output(SISO) repetitive control design proposed in [20, 21]. Lessmeasurements and less weighting functions are involved,which makes the design much simpler than the oneproposed in [15] and the stability evaluation easier. Finally,

Table 1 Maximum current THD

Odd harmonics Maximum current THD

,11th ,4%

11th–15th ,2%

17th–21th ,1.5%

23rd–33rd ,0.6%

.33rd ,0.3%

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the processing delay [22] has been taken into account,which leads to slight tracking improvement.

It turns out that the controller can be reduced to aproportional gain cascaded with the internal model (in are-arranged form), which can be easily implemented in realapplications. Once again, it has been shown that advancedcontrol theories can provide insightful understanding to realapplications and lead to practically implementablecontrollers. Experimental results are presented to validatethe performance of the proposed controller.

The rest of the paper is organised as follows. The overallstructure of the system is presented in Section 2, followedby the proposed voltage controller design in Section 3. InSection 4, the experimental set-up is briefly described andexperimental results are presented and discussed. Finally,conclusions are made in Section 5.

2 System structureThe general idea of the proposed control system is to use anindividual controller for each phase in the natural frame. Thisis also called abc control. The system is implemented with aneutral point controller proposed in [23]. The controlstructure of the system is shown in Fig. 1. It consists oftwo loops: an inner voltage loop using a voltage controllerto track the reference voltage uref and an outer power loopusing a power controller to regulate active power P andreactive power Q. The inner loop is responsible for powerquality issues and the outer loop for power flow control.The power controller consists of a phase-locked loop(PLL), d, q-current controllers with necessarytransformations. The PLL is used to provide theinformation of the grid voltage, which is needed togenerate the voltage reference uref . The real power andreactive power references are determined by I ∗d and I ∗q . Theinverter is assumed to be powered by a constant DC powersource, and hence no controller is needed to regulate theDC-link voltage (otherwise, a controller can be introducedto regulate the DC-link voltage and to generate I ∗d ). Themain objective of this paper is to design the voltagecontroller, hence the d, q-current controllers are simplychosen as PI controllers.

3 Voltage controller designIn this section, a voltage controller based on the H 1 andrepetitive control techniques is designed for a grid-connected inverter after establishing its model. Therepetitive control is a technique widely used to perfectlytrack periodic signals and/or to reject periodic disturbances.The H 1 optimal control theory is an effective method todesign a controller to guarantee the performance with theworst disturbance. Its basic principle is to minimise theinfluence of the disturbances to outputs. The mainobjective of the H 1 repetitive voltage controller is tomaintain a clean and balanced grid voltage in the presence

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Figure 1 Block diagram of a grid-connected inverter with a power controller and an H1 repetitive voltage controller in thenatural frame

i:

of non-linear loads and/or grid distortion. The block diagramof the H 1 repetitive control scheme is shown in Fig. 2, whereP is the transfer function of the plant, C is the transferfunction of the stabilising compensator and M is thetransfer function of the internal model. The stabilisingcompensator C and internal model M are the twocomponents of the proposed controller. The stabilisingcompensator C, designed by solving a weighted sensitivityH 1 problem [21], assures the exponential stability of theentire system, which implies that the tracking error ebetween the voltage reference and the inverter outputvoltage will converge to a small steady-state error [24]. Theinternal model M is a local positive feedback of a delay linecascaded with a low-pass filter W (s). The external signal w

Figure 2 Block diagram of the H1 repetitive control scheme

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contains both the grid voltage (ug) and the voltagereference (uref ), which are assumed to be periodic with afundamental frequency of 50 Hz. It is worth noting thatthe current feedback needed in [15] has been removed,which reduces the controller to be SISO.

3.1 State-space model of the plant P

The considered plant consists of the inverter bridge, an LCfilter (Lf and Cf ) and a grid interface inductor Lg. The LCfilter and the grid interface inductor form an LCL filter.The filter inductors are modelled with series windingresistance. The model of the plant is derived using thesingle-phase diagram of the system shown in Fig. 3. Thecircuit breaker SC is needed during the synchronisation andshut-down procedure. However, in this paper, the switch isconsidered to be closed (i.e. the synchronisation process isomitted). The PWM block together with the inverter aremodelled by using an average voltage approach with thelimits of the available DC-link voltage [15] so that theaverage value of uf over a sampling period is equal to u. Asa result, the PWM block and the inverter bridge can beignored when designing the controller.

The currents of the two inductors and the voltage of thecapacitor are chosen as state variables x = i1 i2 uc

[ ]T.

Figure 3 Single-phase representation of the plant P (the inverter)

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The external input w = ug uref

[ ]Tconsists of the grid

voltage ug and the reference voltage uref , and the controlinput is u. The output signal from the plant P is thetracking error e = uref − u0, where u0 = uc + Rd (i1 − i2) isthe output voltage of the inverter. The plant P can then bedescribed by the state equation

x = Ax + B1w + B2u (1)

and the output equation

y = e = C1x + D1w + D2u (2)

with

A =

−Rf + Rd

Lf

Rd

Lf

− 1

Lf

Rd

Lg

−Rg + Rd

Lg

1

Lg

1

Cf

− 1

Cf

0

⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

B1 =

0 0

− 1

Lg

0

0 0

⎡⎢⎢⎣

⎤⎥⎥⎦, B2 =

1

Lf

00

⎡⎢⎢⎣

⎤⎥⎥⎦

C1 = −Rd Rd −1[ ]

D1 = 0 1[ ]

, D2 = 0

The corresponding plant transfer function is thenP = D1 D2

[ ]+ C1(sI − A)−1 B1 B2

[ ]. In the sequel,

the following notation is used

3.2 Frequency-adaptive internal model M

The internal model M, shown in Fig. 2, is infinitedimensional and consists of a low-pass filterW (s) = vc/s + vc cascaded with a delay line e−tds . It iscapable of generating periodic signals of a givenfundamental period td so it is capable of tracking periodicreferences and rejecting periodic disturbances having thesame period. In order to improve the performance of thecontroller, the delay time td used in the internal model Mshould be slightly less than the fundamental period t [24],and is given by

td = t− 1

vc

(4)

where vc is the cut-off frequency of the low-pass filter W.

The internal model has a very high gain at frequencies pre-defined by the internal model delay line; see Fig. 4a. Whenthe actual grid frequency f varies, its performance is

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degraded. This problem could be solved by changing thedelay time td with respect to the grid frequency. However,following the discrete-time implementation and lowsampling frequency used (e.g. 5 kHz), the adaptive delaytime is impossible to be implemented without furtherdegrading the controller performance. Alternatively, inorder to maintain superior tracking performance of thecontroller, the cut-off frequency of the low-pass filter vc

can be changed with respect to the grid frequencyvariations. This adaptive mechanism is based on the formula

vc =1

td(1 − td f )

which was derived from [15]. This is to make the poles of theinternal model close to the multiples of the fundamentalfrequency on the jv-axis.

Figure 4 Bode plots of the discretised internal model

a Whole frequency range for f ¼ 50 Hz (log scale)b Details around 50 Hz for different cut-off frequencies (linearscale)


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Following the recommendation from [24], delay time td

should be slightly less than the fundamental period. Afterseveral experiments with values of 0.0198, 0.0196 and0.0194 s (corresponding to 99, 98 and 97 samples out of100 at 5 kHz), td has been chosen as 0.0196 s. Theproposed method makes use of the frequency estimationprovided by the PLL. In this way, the PLL estimates theactual grid frequency f and transmits to the internal modelto change vc . The frequency adaptive mechanism has beentested in a grid frequency range from 49.85 to 50.20 Hzand td might need to be changed for a wider frequencyrange. Fig. 4a shows the Bode plots of the discretisedinternal model M for different vc , with details around50 Hz shown in Fig. 4b. The change in the phase is vitalfor maintaining the performance.

3.3 Formulation of the standard H1

problem

In order to guarantee the stability of the system, an H 1

control problem, as shown in Fig. 5, is formulated tominimise the H 1 norm of the transfer functionTzw = F l (P, C) from w = [ v w ]T to z = [ z1 z2 ]T,after opening the local positive feedback loop of theinternal model and introducing weighting parameters j andm. The closed-loop system can be represented as

z

y

[ ]= P

w

u

[ ]

u = Cy

where P is the extended plant and C is the controller to bedesigned. The extended plant P consists of the originalplant P together with the low-pass filter W, the processingdelay represented by Wd and weighting parameters j andm. The additional parameters j and m are added to providemore freedom in the design.

Assume that W is realised as

Figure 5 Formulation of the H1 control problem

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and Wd is realised as

where T is the sampling period. From Fig. 5, the followingequations can be deduced

Combining equations from (5) to (7), the extended plant isthen realised as

for which the stabilising controller C can be calculated usingthe well-known results on H 1 controller design [25].

3.4 Evaluation of the system stability

According to [15, 24], the closed-loop system in Fig. 2 isexponentially stable if the closed-loop system from Fig. 5 isstable and its transfer function from a to b, denoted Tba,satisfies ‖Tba‖1 , 1.

Assume that the state-space realisation of the controller is

Note that the central controller obtained from the H1 design

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is always strictly proper. The realisation of the transferfunction from a to b, assuming that w = 0, can be found asfollows

Once the controller C is designed, the stability of the systemcan be verified by checking ‖Tba‖1.


4 Experimental validation4.1 Experimental set-up

The experimental set-up, shown in Fig. 6, consists of aninverter board, a three-phase LC filter, a three-phase gridinterface inductor, a board consisting of voltage and currentsensors, a step-up transformer (12 V/230 V/50 Hz), adSPACE DS1104 R&D controller board withControlDesk software and MATLAB Simulink/SimPowersoftware package.

The inverter board consists of two independent three-phase inverters and has the capability to generate PWMvoltages from a constant 42 V DC voltage source. Thegenerated three-phase voltage is connected to the grid via acontrolled circuit breaker and a step-up transformer. Thesampling frequency of the controller is 5 kHz and thePWM switching frequency is 10 kHz. A Yokogawa poweranalyser WT1600 is used to measure THD. Theparameters of the inverter are given in Table 2.

4.2 Controller design

The weighting functions are chosen as follows

Figure 6 Experimental set-up

a Laboratory experimental set-upb Block diagram of the experimental set-up


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and

The weighting parameters are chosen to be j = 2.5 andm = 0.8 Using the MATLAB hinfsyn algorithm, the H 1

controller C which nearly minimises the H 1 norm of thetransfer matrix from w to z is obtained as

C(s) = 735.2737(s + 1e004)(s2 + 9132s + 4.058e008)

(s + 1.109e004)(s + 2550)(s2 + 9515s + 4.232e008)

The resulting ‖Tba‖1 is 0.8426. The controller can bereduced to

C(s) = 735.27

s + 2550= KpW (s)

with Kp = 735.27/2550 without causing noticeableperformance degradation, after cancelling the poles andzeros that are close to each other. This leads to‖Tba‖1 = 0.8222, which still maintains the stability of thesystem. With this H 1 controller, the voltage controllershown in Fig. 2 is equivalent to a proportional gaincascaded with the internal model (but in a rearrangedform) shown in Fig. 7. It is worth noting that the delayline appears in the feedback path but not in the feed-forward path as normally seen in the literature. This leadsto faster responses.

4.3 Experimental results

The evaluation of the proposed control strategy was made inboth stand-alone and grid-connected modes. In the stand-alone mode, two experiments with resistive and non-linearloads were carried out. In the grid-connected mode, the

Table 2 Parameters of the inverter

Parameter Value Parameter Value

Lf 150 mH Rf 0.045 V

Lg 450 mH Rg 0.135 V

Cf 22 mF Rd 1 V

Figure 7 Designed control system with the simplifiedcontroller


control strategy was evaluated in three situations: steady-state responses with local resistive and non-linear loads,transient responses with local resistive load and responses togrid frequency variations. The proposed controller wascompared with a deadbeat controller (DB) that isimplemented according to [26] as

Ui((k + 1)Ts) =1

b[4iref ((k − 1)Ts) − 3iref ((k − 2)Ts)]

− a

b

2i(kTs) − aUi(kTs) + (2 + a)

× Ug(kTs) − Ug((k − 1)Ts) (9)

where

a = e−((Rf +Rg)/(Lf +Lg))Ts

and

b = − 1

Rf + Rg

(e−((Rf +Rg)/(Lf +Lg))Ts − 1)

1. Steady-state responses at the stand-alone mode: In the stand-alone mode, the voltage reference was set to grid voltage

Figure 8 Stand-alone mode with a resistive load

a H 1 controller: voltage uA and its reference uref, trackingerror e and current iAb DB controller: voltage uA and current iA

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(inverter is synchronised and ready to be connected to thegrid). The evaluation of the proposed controller was madefor a resistive load (R ¼ 12 V) and a non-linear load (athree-phase uncontrolled rectifier loaded with an LC filterL ¼ 150 mH, C ¼ 1000 mF and a resistor R ¼ 20 V ). Theoutput voltage, current and the voltage tracking error areshown in Figs. 8 and 9, respectively. For the resistive load,the recorded local load voltage THD was 1.1% in the caseof the H 1 repetitive controller and 1.52% in the case ofthe DB controller; for the non-linear load, it was 5.05 and6.7%, respectively. The experimental results indicatesatisfactory performance with non-linear loads.

2. Steady-state responses at the grid-connected mode: Thecurrent reference of the grid output current I ∗d was set at 2 A(1.41 A RMS), after connecting the inverter to the grid.The reactive power was set at 0VAR (I ∗q = 0). Thiscorresponds to the unity power factor. The same resistiveand non-linear loads used in the previous experiment wereused again. The voltage and grid output currents are shownin Figs. 10 and 11. When the load is resistive, the recordedlocal load voltage THD was 0.99% for the proposed H 1

controller and 1.62% for the DB controller, whereas the gridvoltage THD was 1.42% and 1.25% respectively. It is worthmentioning that the quality of the local voltage in the case

Figure 9 Stand-alone mode with a non-linear load

a H 1 controller: voltage uA and its reference uref, tracking errore and current iAb DB controller: voltage uA and current iA


of the proposed controller is better than that of the gridvoltage. The grid output current THD was 6.4% for theproposed H 1 controller and 5.1% for the DB controller.The DB controller controls the output current better. Whenthe load is non-linear, the recorded local load voltage THDwas 3.08% for the proposed H 1 controller and 4.56% forthe DB controller, whereas the grid voltage THD was1.49% and 1.19%, respectively. The grid output currentTHD was 10.28% and 16.78%, respectively. The proposedH 1 repetitive controller performs better in the case of alocal non-linear load.

3. Transient responses at the grid-connected mode: A stepchange in the grid output current Id

∗ reference from 1 A(0.707 A RMS) to 2 A (1.41 A RMS) was applied (whilekeeping Iq

∗ ¼ 0). The current iA and the voltage uA areshown in Fig. 12. Although the DB controller shows fastdynamics, the response to the change of the H 1 repetitivecontroller takes about 5 cycles, which reflects the inheritedproperty of the proposed repetitive control and the trade-off between the low current THD and the system stability.

4. Responses to grid frequency variations: The aboveexperiments were carried out with the frequency adaptivemechanism activated. In order to see the improvement of

Figure 10 Grid-connected mode with a local resistive load

a H 1 controller: voltage uA and its reference uref, tracking errore and grid output current iAb DB controller: voltage uA and grid output current iA


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the frequency adaptive mechanism, two more experimentswere carried out. In the first case, the grid frequency was49.90 Hz and in the second case the grid frequency was50.10 Hz. As can be seen from Figs. 13a and 14a, the H 1

repetitive controller without the frequency adaptivemechanism encountered problems in the voltage control. Aphase shift between voltage output uA (light grey) and thereference voltage uref (dark grey) can be clearly seen andconsequently an increased steady-state error is recorded.Since the controller was tuned for the 50.00 Hz gridfrequency, the generated voltage uA is leading the referencevoltage when the grid frequency is 49.90 Hz and thegenerated voltage uA is lagging the reference voltage whenthe grid frequency is 50.10 Hz. When the frequencyadaptive mechanism was activated, the obtained responsesare shown in Figs. 13b and 14b. The tracking error wasreduced by about 50%. As this experiment fully depends onthe change of the public grid frequency in reality, manyexperiments were carried out for different grid frequenciesbut, due to the page limit, other experimental results arenot included here. The tracking error was almost keptconstant in those experiments while the grid frequency wasvarying.

Figure 11 Grid-connected mode with a local non-linearload

a H 1 controller: voltage uA and its reference uref, tracking errore and grid output current iAb DB controller: voltage uA and grid output current iA


Figure 12 Transient responses: 1 A step change in Id∗

a H 1 controller: voltage uA and its reference uref, tracking errore and grid output current iA and its reference iref

b DB controller: voltage uA and grid output current iA and itsreference iref

Figure 13 Responses to grid frequency variations(f ¼ 49.90 Hz): voltage uA and voltage tracking error e

a Without the frequency adaptive mechanismb With the frequency adaptive mechanism

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5 ConclusionsThe H 1 repetitive control strategy has been re-consideredfor the voltage control of grid-connected inverters,focusing on reducing the complexity of the voltagecontroller and improving the control performance.The current feedback needed in previous works has beenremoved and the voltage controller only adopts the outputvoltage as feedback. Moreover, a frequency adaptivemechanism has been introduced to cope with grid-frequency variations. The designed controller has been putinto action. The experimental comparison with a DBcontroller has shown that the H 1 repetitive controlleroffers excellent performance in terms of waveform qualityand current THD, even in the case of non-linearloads connected to the system. The somewhatcomplicated design process has led to a very simplecontroller, which can be easily implemented in realapplications. Once again, it has been shown that advancedcontrol theory does offer more insightful understanding topractical applications and the controller designed usingadvanced control theory does not have to be verycomplicated.

6 AcknowledgmentsT. Hornik would like to acknowledge the financial supportfrom the EPSRC, UK under the DTA scheme and Q.-C.Zhong would like to thank the Royal Academy of

Figure 14 Responses to grid frequency variations(f ¼ 50.10 Hz): voltage uA and voltage tracking error e

a Without the frequency adaptive mechanismb With the frequency adaptive mechanism


Engineering and the Leverhulme Trust for the award of aSenior Research Fellowship. A preliminary version of thispaper was presented at the 8th World Congress onIntelligent Control and Automation (WCICA 2010) heldin Jinan, China, in July 2010.

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