8Repetitive Current Control ofan Interleaved Grid-ConnectedInverter

8.1 Introduction

Different controllers and topologies [1–4] have been used for grid-connected invert-ers to obtain high quality output current. However, classical PID controllers and theirderivatives suffer from relatively low loop gain at the fundamental frequency andits harmonics and hence can have poor grid harmonic disturbance rejection, whichresults in poor output current THD (total harmonic distortion) if the grid voltage THDis relatively high. Proportional-resonant (PR) controllers have been widely used forgrid-connected inverters due to their ability to reject individual harmonics [5–9]. The-oretically, this introduces an infinite gain at a selected frequency. By having multiplePR controllers, higher order harmonics can be eliminated but a bank of resonant con-trollers increases the complexity of the system and the calculation burden of the digitalsignal processor (DSP).

Repetitive feedback control (RC), which is based on the concept of iterative learningcontrol, has been widely used for many practical industrial systems, such as manu-facturing [10] and robotics [11]. In these controllers error between the reference andthe output over one fundamental cycle is used to generate a new reference for the nextfundamental cycle. RC is mathematically equivalent to a parallel combination of anintegral controller, an infinite number of resonant controllers, and a proportional con-troller [12]. It has the advantage of being simpler to implement than PR controllers.However, it creates resonance gain peaks at high frequencies, which can lead to insta-bility. A low-pass filter can be used to attenuate the high frequency resonance gainpeaks but if the bandwidth of the plant is low (i.e., the gain crossover frequency is notvery high with respect to the fundamental and low harmonic frequencies) such a filterwill reduce the low frequency resonance gains and, consequently, will deteriorate theperformance of the RC [13].

Power Electronic Converters for Microgrids, First Edition. Suleiman M. Sharkh, Mohammad A. Abusara,Georgios I. Orfanoudakis and Babar Hussain.© 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd.Companion Website: www.wiley.com/go/sharkh

172 Power Electronic Converters for Microgrids

The application of RC to the control of grid-connected and stand-alone inverters isan active area of research [14–16]. A number of papers describe the effect of param-eter variations on overall system stability and transient response. However, limits ofperformance of RC with reference to system bandwidth need to be investigated, espe-cially for two-level grid-connected inverters.

The high bandwidth requirement of the inverter and its output filter dictates a highPWM (pulse width modulation) switching frequency, which becomes challenging inhigh power systems as the maximum achievable switching frequency of power elec-tronic devices reduces as their power rating increases. This limitation can be overcomeby using an interleaved inverter topology, such as that reported in [17] and Chapter 7.Interleaving is a form of paralleling technique where a single converter channel, forexample, half a bridge with an output inductor, is replaced by N smaller channels con-nected in parallel, as shown in Figure 8.1, with their switching instants phase shiftedequally over the switching period.

By introducing a phase shift between the switching instants of the parallel channelsof each phase, the amplitude of the total ripple current is N times less and its frequencyis N times greater than that of a conventional inverter. The reduction in total currentripple amplitude and the increase in its frequency reduce the size of the required fil-ter capacitance considerably, as mentioned earlier. Furthermore, sharing the currentamong a number of channels enables the use of smaller lower current power switcheswhich can switch at high frequency, thus allowing a reduction in inductor size. Thenet result is that the size of the filter and the overall size of the system are smaller thanan equivalent classical two-level bridge inverter with LCL output filter. The reductionin total current ripple and the high frequency also eliminate the need for a second out-put filter inductor (i.e., using an LCL filter instead of an LC filter) that is sometimesused in two-level and multi-level grid-connected inverters to block the switching rip-ple in cases where the grid impedance is too low. Consequently, the bandwidth of theinterleaved system is high due to the much smaller output filter capacitors and equiv-alent inductor. This makes RC, which requires a high system bandwidth, an attractiveoption for the control of interleaved inverters.

In this chapter the design and practical implementation of a repetitive controller for asix-channel interleaved inverter are discussed. Simulation and experimental results arealso presented to demonstrate the effectiveness of the proposed controller in improv-ing the THD of the output current of the inverter.

8.2 Proposed Controller and System Modeling

Figure 8.2 shows the block diagram of the PWM inverter, LC filter, and the controlsystem of one of the phases. This controller structure uses only one current sen-sor of the total inductors’ current as opposed to the controller structure presentedin chapter 7. However, for a protection reason, the individual inductor currents arestill measured and monitored using six small current sensors. Active damping wasnot implemented as the sampling frequency fsw (35 kHz) is quite low compared tothe filter resonance frequency. Instead, a resistor in series with the filter capacitor is

Repetitive Current Control of an Interleaved Grid-Connected Inverter 173

n

V

V

VVDC

Vin1 IL1

VcLu

IOutIL

C

R

C

R

C

R

Vua

Lu Vub

Lu Vuc

+

−

m

L

L

L

Figure 8.1 Three-phase interleaved grid-connected interleaved inverter

used to provide passive damping. Fortunately, due to the ripple cancellation featureof the interleaved topology, the capacitor current is quite small and hence the powerdissipation in R is negligible as discussed in chapter 7.

From Figure 8.2, the total inductor current IL can be shown to be given by

IL = (Vinx − B(s)Vu)A(s) (x = 1, 2, ..,N) (8.1)

where

A(s) =N(LuCs2 + RCs + 1)

LLuCs3 + RC(L + NLu)s2 + (L + NLu)s(8.2)

B(s) = RCs + 1

LuCs2 + RCs + 1(8.3)

174 Power Electronic Converters for Microgrids

+−

IL2

RCs + 1Cs

1

Lus+−

Vu

+ − IOutVcIL

PWM

∑+−

PWMIL2

+−

PWMILN1

Ls

1Ls

ZOH

ZOH

ZOH

E+−

++

KR z−n Q(z)+

*ILVin1

Vin2

VinN

K (z)

e−Tds

RC

1Ls

Figure 8.2 Block diagram of one phase and its controller

From Equation 8.2, the system natural resonance frequency is given by

fn = 1

2π

√L + NLu

LLuC(8.4)

The frequency fn depends on the grid impedance and from the system parame-ters listed in Table 8.1, fn = 19.8 kHz (for Lu = 5 μH) and 9.3 kHz (for Lu = 50 μH).This is much higher than the filter resonance frequency of a conventional two-levelgrid-connected inverter, of a similar power rating, whose fn is only 1.5 kHz [3].

Table 8.1 System parameter values used in simulation

and experiments

Description Symbol Value

Number of interleaved channels N 6

Passive damping resistor R 0.5ΩChannel inductor L 190 μH

Filter capacitor C 15 μF

Switching frequency fsw 35 kHz

Sampling frequency fs 35 kHz

Number of samples in one cycle n 700

Time delay Td 14.28 μs

Grid voltage Vu 230 V (rms)

Grid frequency fo 50 Hz

Inverter DC voltage VDC 700 V DC

Inverter rated current IOut 90 A (rms)

Repetitive Current Control of an Interleaved Grid-Connected Inverter 175

Equations 8.2 and 8.3 are represented by the block diagram in Figure 8.3. In the dis-crete time domain, the open loop transfer function, including the DSP computationaltime delay Td and zero order hold, is obtained by performing the z-transform,

G(z) = Z

[1 − e−Tss

se−TdsA (s)

](8.5)

where Ts is the sampling period. The repetitive current controller transfer function isgiven by

GRC(z) =KRQ(z)z−n

1 − Q(z)z−n(8.6)

where, Q(z) is a low pass filter, KR is the repetitive controller gain, and n is the numberof samples in one fundamental cycle. A simplified block diagram of the system isshown in Figure 8.4 where Du represents the grid disturbance B(s)Vu.

8.3 System Analysis and Controller Design

The controller design involves the determination of K(z), Q(z), and KR. The controllergain K(z) can be a phase lag to improve harmonics rejection, as was discussed in

K (z)

*IL +−

e−Tds

+− IL

B (s)

Vu

VinxA (s)ZOH

GRC (z)

++

Figure 8.3 Single channel equivalent block diagram

*IL +−

+− IL

Du

G (z)++

GRC (z)

K (z)Vinx

Figure 8.4 Simplified block diagram

176 Power Electronic Converters for Microgrids

Frequency (Hz)

101 102 103 104 105−720

0

720

1440

2160

2880

3600

4320

5040

Phas

e (d

eg)

−50

0

50

100

150

200

Mag

nit

ude

(dB

)

Q(z)=1,Lu=5e−6 H

Q(z)=1,Lu=50e−6 H

Figure 8.5 Open loop Bode diagram with Q(z)= 1

chapter 7 but, in this chapter, because the RC is supposed to take care of harmonicsrejection, K(z) is chosen to be a simple proportional gain of 1.

The Bode diagram of the open loop transfer function (GRC(z) + K(z))G(z) withK(z)= 1, Q(z)= 1 is shown in Figure 8.5. It can be noticed that the system is unstabledue to the resonant peaks near the crossover frequency, which means that Q(z) needsto be modified to attenuate the high frequency peaks. The selection of Q(z) reflectsthe trade-off between closed loop system performance and stability robustness. Azero-phase low pass filter is used; it has the following the structure:

Q(z) =𝛼1z + 𝛼o + 𝛼1z−1

𝛼o + 2𝛼1

, 𝛼o, 𝛼1 < 1 (8.7)

The normalized frequency response (i.e. the sampling period Ts is set to 1) ofEquation 8.7 is Q(ej𝜔) = 𝛼o + 2𝛼1 cos(𝜔) and𝜔 ∈ (0, π). By considering𝛼o + 2𝛼1 = 1,

Repetitive Current Control of an Interleaved Grid-Connected Inverter 177

for unit gain response, the magnitude of Q(j𝜔) can be written as follows:

|Q(j𝜔)| = ⎧⎪⎨⎪⎩𝛼o + 2𝛼1 𝜔 = 0

𝛼o + 2𝛼1 cos (𝜔) 𝜔(0, π)𝛼o − 2𝛼1 𝜔 = π

(8.8)

By selecting 𝛼o = 0.5 and 𝛼1 = 0.25, then

|Q(j𝜔)| = ⎧⎪⎨⎪⎩1 𝜔 = 0

0 < 0.5 (1 + cos𝜔) < 1 𝜔 ∈ (0, π)0 𝜔 = π

(8.9)

Equation 8.9 concludes that |Q(j𝜔)| is 1 at low frequencies and 0 at high frequencies.The filter is now given by

Q(z) = 0.25z + 0.5 + 0.25z−1. (8.10)

The Bode diagram of Q(z) as described in Equation 8.10 is shown in Figure 8.6.The value of the RC gain KR needs to be carefully selected as it is a key parameter for

error convergence and system stability. A high repetitive controller gain KR results in

−100

−50

0

50

Mag

nit

ud

e (d

B)

102 103 104−0.5

0

0.5

1

1.5

2

2.5

3x 10−9

Ph

ase

(deg

)

Frequency (Hz)

Figure 8.6 Bode diagram of Q(z) = 0.25z + 0.5 + 0.25z−1

178 Power Electronic Converters for Microgrids

Frequency (Hz)

101 102 103 104 105−270

−225

−180

−135

−90

−45

0

Phas

e (d

eg)

−50

0

50

100M

agnit

ude

(dB

)

Without RC

With RC when Lu = 5e−6 H

With RC when Lu = 50e−6 H

Figure 8.7 Bode diagram of system with Q(z) = 0.25z + 0.5 + 0.25z−1, KR = 0.1

fast error convergence but the feedback system becomes less stable. A suitable valueof KR = 0.1 is selected, which gives a satisfactory transient response while maintain-ing stability. After suitable selection of all the above parameters, the open loop Bodediagram of the system with and without RC is shown in Figure 8.7 with two differentvalues of grid impedance Lu. Stability and immunity to grid impedance variation isconfirmed. The closed loop frequency response of the disturbance transfer functionrelating IL(z)∕Du(z) is shown in Figure 8.8 which confirms the attenuation providedby RC at the grid harmonics.

8.4 Simulation Results

Detailed simulation has been carried out using the MATLAB® SimPowerSystems.The system parameters are listed in Table 8.1. Grid voltage harmonics were measuredin the laboratory and similar values were included in the simulation model. The totalgrid voltage THD is 1.9%.

Figure 8.9 shows the output current without RC for a 10 A (rms) demand. The cur-rent THD is 22%. It should be noted that the current THD is very high because thecurrent demand is only 11% of the rated current of the inverter and the controller pro-portional gain K(z) is set to 1. The current THD could also be improved if K(z) wasset to a higher value or modified to be a phase lag. Figure 8.10 shows the steady-stateoutput current (the time scale has been adjusted and the starting time shown in thefigure is after the controller has reached steady state). The current THD is reduced

Repetitive Current Control of an Interleaved Grid-Connected Inverter 179

Frequency (Hz)

101 102 103 104 105−45

0

45

90

135

180

225

270

Ph

ase

(deg

)

−80

−60

−40

−20

0M

agn

itu

de

(dB

)Without RC

With RC

Figure 8.8 Frequency response of the disturbance transfer functionIL(z)Du(z)

0 0.01 0.02 0.03 0.04 0.05 0.06−20

−10

0

10

20

Time (s)

Cu

rren

t (A

)

Figure 8.9 Simulated output current without RC

to only 2.2%. The effectiveness of the RC in improving the current THD is clearlydemonstrated. Figure 8.11 shows the current error and it can be noticed that the errorconverges within about 0.25 s.

8.5 Experimental Results

The proposed controller was tested experimentally with an interleaved inverter. Thesystem parameters are listed in Table 8.1. The controller was implemented usingthe Texas Instrument TMS320F2808 32-bit DSP. This processor has the capabilityof generating six interleaved PWM outputs. The internal counter of the first PWMcarrier is set to give the required switching frequency. The second PWM counter is

180 Power Electronic Converters for Microgrids

0 0.01 0.02 0.03 0.04 0.05 0.06

−10

−5

0

5

10

Time (s)

Curr

ent

(A)

Figure 8.10 Simulated output current with RC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (s)

Cu

rren

t (A

)

Figure 8.11 Error convergence with RC

synchronized with the first counter and delayed by Ts∕6. The third PWM counter issynchronized with the second counter and delayed by Ts∕6, and so on. This producessix interleaved triangular carriers to generate carrier-based PWM outputs. One DSPper phase was used and the low speed communications between the controllers, suchas start/stop and total current commands, were implemented using the controller areanetwork (CAN) protocol. Synchronization with the grid was implemented by havingeach phase controller measuring its corresponding grid phase voltage to detect the zerocrossing. The reference sine waves were generated internally by the individual con-trollers using look-up tables of 700 samples (35 kHz/50 Hz). The sine wave amplitudeis set externally (by a setting in the user interface) and sent via CAN-bus to the threephase controllers. The input DC is regulated by an external boost circuit to 700 V DC.

The RC was implemented indirectly by creating an array of 700 entries representinga state variable x which is the difference between RC input and output such as x(i) =KRe(i) + y(i), as illustrated in Figure 8.12.

Repetitive Current Control of an Interleaved Grid-Connected Inverter 181

z−699 z−1 z−1 0.25

0.5

0.25 +

+

+

+

+

e(i) RCoutput

y(i)

RCinput

x(i)KR

Figure 8.12 RC implementation

Figure 8.13 shows the output current when RC is de-activated. The demand currentis set to 15 A (rms). The current THD is measured to be 16.0%. Figure 8.14 shows theoutput current but when the RC is activated. The current THD is measured to be only2.0%. In the system reported in [17] (which is similar to this one but without RC and

Figure 8.13 Output current and its spectrum without RC

182 Power Electronic Converters for Microgrids

Figure 8.14 Output current and its spectrum with RC

K(z) is a phase lag), such low THD was only achievable when the output current wasequal to the rated current, that is, 90 A (rms).

8.6 Conclusions

The interleaved topology offers higher bandwidth than a classical two-level inverterdue to smaller filter components thanks to the ripple cancellation feature of this topol-ogy, and higher switching frequency of its lower power devices. The high bandwidthmakes the repetitive controller an attractive solution for rejecting grid harmonics.The design and practical implementation of a digital repetitive controller scheme foran interleaved grid-connected inverter was discussed. Simulation and experimentalresults reveal that the repetitive controller improves the current THD level consider-ably.

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