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Solar Energy Materials & Solar Cells 91 (2007) 1713–1725

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Modeling of a single-phase photovoltaic inverter

T.I. Marisa, St. Kourtesib, L. Ekonomouc, G.P. Fotisd,�

aDepartment of Electrical Engineering, Technological Educational Institute of Chalkida, 334 40 Psachna Evias, GreecebHellenic Public Power Corporation S.A., 22 Chalcocondyli Str., 104 32 Athens, Greece

cHellenic American University, 12 Kaplanon Str., 106 80 Athens, GreecedNational Technical University of Athens, School of Electrical and Computer Engineering, High Voltage Laboratory, 9 Iroon Politechniou St.,

Zografou, 157 80 Athens, Greece

Received 3 March 2007; received in revised form 25 May 2007; accepted 25 May 2007

Available online 27 June 2007

Abstract

The paper presents the design of a single-phase photovoltaic inverter model and the simulation of its performance. Furthermore, the

concept of moving real and reactive power after coupling this inverter model with an a.c. source representing the main power distribution

grid was studied. Brief technical information is given on the inverter design, with emphasis on the operation of the circuit used. In the

technical information section, a description of real and reactive power components is given with special reference to the control of these

power components by controlling the power angle or the difference in voltage magnitudes between two voltage sources. This a.c.

converted voltage has practical interest, since it is useful for feeding small house appliances.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Photovoltaic inverter; Modeling; Simulation; Single phase; PSCAD/EMTDC software

1. Introduction

In recent years, the need for renewable energy hasbecome more pressing. Among them, the photovoltaic (PV)system such as solar cell is the most promising energy. ThePV energy is free, abundant and distributed through theearth. PV energy applications can be divided into twocategories: one is stand-alone system and the other is grid-connected system. Stand-alone system requires the batterybank to store the PV energy and it is suitable for low-powersystem. Grid-connected system does not require the batterybank and has become the primary PV application for high-power applications. The main purpose of the grid-connected system is to transfer maximum solar arrayenergy into grid with a unity power factor. Because of thehigh cost of PV modules, PV generation systems areattractive only for remote isolated areas and for small-scale applications [1] such as PV refrigerators and water-pumping systems.

e front matter r 2007 Elsevier B.V. All rights reserved.

lmat.2007.05.027

ing author. Tel.: +306972702218; fax: +30 2107723504.

ess: gfotis@gmail.com (G.P. Fotis).

The output power of PV cell is changed by environ-mental factors, such as illumination and temperature. Sincethe characteristic curve of a solar cell exhibits a non-linearvoltage–current characteristic, a controller named maxi-mum power point tracker (MPPT) is required to match thesolar cell power to the environmental changes. Manyalgorithms have been developed for tracking maximumpower point of a solar cell [2–5]. In literature, severalmodels have been developed for the modeling andsimulation of the different components of stand-alone PVpower systems based on simulation approaches, whichperformed in various programming environments such asPspice, Matlab Simulink and Labview [6,7].Generally, it is very difficult to develop an analytical

equation or numerical model capable of predicting theperformance of the PV system under the variable clima-tic conditions (influencing the system) for the next day.These systems can be considered non-linear, which meansthat are very complex and very difficult to find a suitablemodel by classical approaches. There have been studieswhere artificial neural networks (ANN) have been consi-dered as a suitable approach for such a complex systemmodeling [8–10].

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The aim of the work presented here is to build anEMTDC model of a single-phase photovoltaic inverter andto investigate switching strategies for harmonic minimiza-tion. This inverter device is intended for domestic use andwill allow users to exploit voltage from PV cells. Theproduced AC converted voltage will be useful for feedingsmall house appliances or by employing appropriatetechniques, real and reactive power exported from theinverter can reinforce the main power stream in the‘‘distribution grids’’.

For the simulation of this model, the PSCAD/EMTDCsoftware [11,12] package was used and the waveforms ofinterest were taken. A low rating, mains connected devicewas designed and used to demonstrate that real andreactive power can flow in the desired direction just bychanging the phase shift or the voltage magnitude. Firstof all an inverter model that would convert the d.c. vol-tage supplied from a battery into an a.c. voltage wasdesigned, offering the capability of feeding this into thegrid through an inductance. Having obtained the coupl-ing of the inverter’s output with the grid voltage andby changing the relative phases, or voltage magnitudesof one with respect to the other, effort has been paidin order to demonstrate that phase shift between thetwo voltages leads to a real power flow and differencein magnitude of the two voltages leads to reactive powerflow.

2. Technical background information

An inverter is a d.c. to a.c. converter, i.e. it can convertd.c. voltage into a.c. for feeding into an a.c. utility network.It is possible to obtain a single-phase or a three-phaseoutput from such a device, but in this work only thebehavior of a single-phase inverter was studied. An invertersystem consists of the d.c. input, the power circuit and thecontrol circuit. The inverter finds very useful applicationsin standby power supplies or uninterruptible powersupplies (UPS) and also in a.c. motor control. Inverterscan be broadly classified either as voltage-source orcurrent-source inverters.

The d.c. input voltage into an inverter can be obtained invarious ways. In UPS systems, it is almost invariablyobtained from a storage battery. This is because some formof energy storage is required to provide for mains powerfailure. In a.c. motor control, the d.c. link voltage isobtained from rectified mains. For the case described inthis work, the voltage-fed or voltage-source inverter (VSI)was powered from a stiff, low-impedance d.c. voltagesource provided in the form of a battery supplying voltageat 38V d.c.

The inverter power circuit consists of the main switchingdevices, which carry the load current and are subjected tothe d.c. link voltage. The power circuit also includesprotection circuits, such as reverse conduction diodes,when inductive loading is expected. Such a reactivefeedback diode is characteristic of a voltage-source

inverter, providing a reverse current path for load currentthat permits an energy flow from a reactive load, throughthe inverter, to the d.c. supply. These components aresubjected to the same order of voltage magnitudes andcurrents as the main switching devices.The choice of the main devices depends on factors such

as the d.c. link voltage, the load current, the maximumoperating frequency, etc. The devices need to be force-commutated devices with high switching frequencies, forexample, insulated gate bipolar junction transistors(IGBTs), power MOSFETS or gate-turn-off thyristors(GTOs) that can provide natural turn-off facilities. Toproduce the required output waveform, the devices areswitched in a particular sequence determined by the controlcircuit. The control circuit determines the waveform of theoutput voltage and its frequency. In inverters having highpower ratings the control circuit may incorporate protec-tion circuits such as current limiting.

3. PSCAD/EMTDC simulation package description

EMTDC and PSCAD [11,12] are a group of relatedsoftware packages, which provide the user with a veryflexible power systems electromagnetic transients tool.PSCAD enables the user to design the circuit that is goingto be studied. EMTDC is the software, which actuallyperforms the electromagnetic transients analysis on theuser-defined power system. EMTDC enables the user tosimulate the circuit performance under any conditions ordisturbances of a complicated or non-linear model orprocess. The operation of such a model can be tested bysubjecting it to disturbances and parameter variations andthe stability of its response can be observed.The EMTDC provides the facility that already available

models can be interfaced with an electric circuit or controlsystem. This gives the opportunity to the user to set up andstudy DC transmission systems without too much diffi-culty, as well as making digital computer models as good asany real-time hardware based DC simulator. EMTDC canbe used to model, circuit elements (resistors, inductors andcapacitors), mutually coupled windings, cables and dis-tributed, multiphased, untransposed transmission lines.Also it can model both Thevenin and Norton sources,switches, thyristors, diodes and gate-turn-off devices.The EMTDC alone cannot provide the user with a

complete analysis of the power system under study, so theanalysis is assisted by some auxiliary programs. Graphicsplotting of output of any desired quantity can be providedin the package. Fourier analysis of any desired output ispossible, using an auxiliary program known as EMTFS.Another capability of the EMTFS program is the synthes-izing of an EMTDC output representing the response tosome complicated model, up to a fourth order linearfunction using an optimization technique. It is possible toobtain a frequency response of a model built on EMTDCfrom a single time-domain run. A perturbating input signalconsisting of either a train or pseudo-random pulses, or

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summation of sine waves can be applied. If both inputsignal and output responses are Fourier analyzed, thefrequency response can be derived.

4. Presentation of the simulation results

4.1. Inverter design procedure

The whole design of the inverter circuit was implementedusing gate-turn-off thyristor (GTO) models. These GTOmodels are normally used as controlling switches in high-voltage devices with large power ratings, whereas in thisdesign, they are just used to provide the switching pulsesand finally produce the output. The inverter device wasintended for use alongside a photovoltaic (PV) controllerthat would act as the supply to the circuit. The PVcontroller was a battery of 38V. The rating of this invertermains connected device is about 500VA or 1 kVAmaximum.

A ‘‘triggering’’ block was used to provide the appro-priate gate triggering pulses, which when applied to thegate terminals of the thyristors would result in a squarewave output. A triggering block was created simply bycreating the new component and programming it toperform the operation required. This triggering block hasone input and five outputs out of which only four are used.The input to this triggering block was a square-pulse ofmagnitude varying between 0 and 1 and the outputs are thefour triggering pulses for the four thyristors. Triggeringpulses were obtained from the block, which were appliedon the respective gate terminals of each of the fourthyristors. This enabled the thyristors to switch ON andOFF in the pre-defined intervals and finally to produce theanticipated square waveform.

The output of the thyristor was a square wave having apeak of 36V, measured using the measuring facilities of thesimulation program. Ideally, this waveform should have apeak value of 38V, equal to the supply voltage. This is thecase because in order to get 38V as the peak value of thevoltage, the ON state voltage drops from diodes andthyristors conducting were not considered. In the simula-tion, these voltage drops were set to 1V for both thyristorsand diodes, and the final result was an output waveformwith a peak at 36V.

4.2. Representation—generation of the grid voltage

Once the correct output from the inverter was obtained,a sinusoidal wave for representing the grid voltage had tobe generated. It was simple and easy to represent this gridvoltage using the output of a low-impedance a.c. source.This a.c. source was to be used as a supply for obtaining a50Hz, 230V rms sinusoid that would represent the gridvoltage. The initial parameters of this source, i.e. magni-tude and frequency were respectively set to 230V rms and50Hz. Once this output was generated it was coupled in thecircuit as the grid voltage.

4.3. Coupling of the two circuits

After designing and implementing the inverter deviceand the a.c. source equivalent circuit, the two circuits werecoupled together through an inductance. The value of thisinductance had to be calculated taking into account therating of the device and the voltage rating of the circuit. Aninductance of value 67.35mH was used to couple the twocircuits together.Another adjustment that needed to be considered before

the two circuits could successfully be coupled together wasto ‘‘lock’’ the phase of the inverter output voltage onto thatof the grid voltage. This means that the phase of theinverter voltage had to be made equal to the phase of thegrid voltage. It is possible to achieve this task in variousways such as using a phase-locked-loop (PLL), but in thiswork a much simpler implementation technique wasemployed. This technique used a duplicate of the gridvoltage source, where its output used after being passedthrough a zero-crossing detector (ZCD) to trigger thethyristors in the inverter device. The output of the source isa sinusoidal voltage, but after being passed through theZCD this output is converted into a square pulse offrequency 50Hz and magnitude 230V rms. The ZCD, as itsname implies detects zero crossings on the input waveformand triggers at each zero crossing. As soon as the detectordetects a zero crossing, it triggers to a pre-set value and assoon as it detects a second zero crossing it triggers back tozero. In this way a sinusoidal input is easily converted intoa square wave.The ZCD output was used as the input to the triggering

block. Applying a square-pulse generated from the gridvoltage sinusoid at the input of the triggering block, thetriggering pulses obtained will eventually produce a square-wave output that will be in phase with the grid voltage. Thisphase compatibility is shown in Fig. 1, but in order to havethe two voltages in phase the triggering pulses had to beswapped around. The completed network is given in Fig. 2.The work described above was to bring the two voltages

in phase so as to prevent the large current flows in theinverter circuit. If the two voltages were out of phase whenconnected together, this would cause large power flowsleading to large current flows. The rating of the inverterlimits the current that could flow in the circuit up to amaximum value calculated equal to 2.174A rms. All thepower flow measurements, given later in this paper, weretaken with current between 1 and 2A.

4.4. Power measurements

With appropriate phase manipulation between the twovoltages and voltage magnitude manipulation the respec-tive transfer of real and reactive powers is feasible. In orderto measure real and reactive power, the complex power hadto be measured first. The complex power at any point in thesystem can be found by simply multiplying the correspond-ing voltage and current at that point.

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Fig. 1. Inverter output and grid voltage waveforms.

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The current at point C in the circuit was measured byusing an ammeter connected in series in the circuit. Thevoltage at that point was also measured. Multiplyinggraphically the waveforms of these two quantities, thewaveform corresponding to the complex power wasderived and from that an rms value for the complex powercan be deduced. Fig. 3 displays the waveforms of current(Ia) and voltage (V2).

Setting the d.c. supply to 250V rms, the current waslimited between the acceptable limits and it actually had anrms value of 1.3A. The current waveform was seen to bevery distorted, containing all orders of harmonics. The

inverter output waveform was also changed, since the loadbecame inductive and a ‘‘step’’ was observed in thewaveform. The voltage at point C, the grid voltage V2 isa sinusoid of peak value equal to 325.27V, but as it will bediscussed later an increase in the a.c. source’s seriesinductance will inject harmonics into this voltage wave-form.The complex power was measured using the current and

voltage values at point C. A two input–one outputmultiplier was used in order to obtain the complex powerwaveform. The complex power waveform was seen to bedistorted due to the contribution from the current wave-

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Fig. 2. The complete network.

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form. Also the rms value of the complex power wasmeasured by plotting the graph of rms complex powerversus time. The real power was measured by passing thecomplex power waveform through a first-order controltransfer function of the form G/(1+St), where G is the gainintroduced between the input and the output and t is thetime constant of the system. This transfer function has nozeroes and has only one pole that being at S ¼ �1/t.

The gain was set to 1 and the time constant t was also setto 1 s. The value of the time constant needed to be as largeas possible. A large time constant enables a more accuratemeasurement of the real power. The instantaneous valueswould not be taken into account and the output waveformindicates that real power had reached a steady-state value.For these measurements, the magnitude of the fundamen-tal of the inverter output voltage was set to 250V rmsresulting in a current flow of 1.3A through the circuit.

Real power is known to be a function of the power angled (the phase shift between the fundamental of the inverteroutput and the grid voltage). To have a real power flowbetween the two sides of the circuit, a phase shift of a fewdegrees was introduced and that effect was monitored. Thecurrent remained fixed at 1.3A rms and a phase shift of+21 was introduced resulting in the inverter output voltageleading the grid voltage by 21. The phase shift on theinverter voltage was introduced by shifting the input to thetriggering block; so shifted output triggering pulses wereobtained. These shifted pulses when applied to the thyristormodels’ gate terminals produced a square wave of the same

shape as before but this time shifted by the initial angleintroduced on the input pulse to the triggering block. Thismethod established the desired phase shift between the twovoltages and hence, in the leading mode of operation, realpower was observed to flow from the inverter side to thegrid side. The real power flow was monitored and relativegraphs showing the voltage waveform V2, the current Ia,the complex power waveform and the real power waveformwere plotted. Measurements were taken with supplyvoltage equal to 250V rms and phase shifts of +21 and�21 and the above waveforms were recorded each time.Figs. 4 and 5 give the waveforms obtained for the leadingand lagging mode of operation, respectively. In the leadingmode of operation, the inverter voltage leads the gridvoltage by 21 and real power flows from the d.c. side, theinverter side, towards the a.c. side, the a.c. source side.Similarly, in the lagging mode of operation, the invertervoltage lags the grid voltage by 21 hence real power flowsfrom the a.c. side towards the d.c. side.The waveforms given in Fig. 4, for the leading mode of

operation, are the current Ia, voltage V2 at point C, thecomplex power in MVA, and on the same set of axes therms complex power and the real power. Graph 1 shows thecurrent waveform seen to be heavily distorted and also itsrms waveform indicating that the rms value of the currentwas 1.3A. In graph 2, the voltage waveform for V2 is asinusoidal voltage as expected having a peak value of325.9V. Graph 3 shows the resulting waveform for thecomplex power. This was obtained after multiplication of

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Fig. 3. Current and grid voltage waveforms.

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Ia and V2, and it is distorted owing to the contribution ofthe current waveform. Finally, graph 4 shows, plotted onthe same pair of axes, the rms complex power and realpower variation with time. The rms waveform for thecomplex power was obtained after passing the complexpower wave through an rms conversion block with asmoothing time constant of t equal to 0.001 s. The rmsvalue of the complex power was measured to be 265VAand the real power value was 43W.

The same procedure was followed in order to obtain thegraphs for the lagging mode of operation of the model. Theinput to the triggering block was given a negative phaseshift of �21, which resulted in producing a square wavelagging the grid voltage by 21. Once again the same

quantities were measured and their respective graphs wereplotted to demonstrate the lagging mode of operation ofthe model. Graph 1 in Fig. 5 shows the current waveform Iafor the lagging mode of operation. The rms value of thiscurrent remains unchanged since the voltage magnitudes ofthe two waveforms were not altered. Graph 2 gives thesinusoidal voltage wave, which is exactly the same as forthe leading mode of operation. The shape of the waveformfor the complex power is unchanged but the rms value ofthe complex power is increased from 265 to 295MVA.The real power is reversed in sign but it is increased in

magnitude from 43W for the leading mode to �126W forthe lagging mode. The reversed sign of real power indicatesthat in the lagging mode of operation, real power flows

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Fig. 4. Leading mode waveforms for supply voltage equal to 250V.

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from the a.c. side towards the d.c. side. When the inverterwas operating in the leading mode, real power was noticedto flow exactly in the opposite direction.

4.5. Voltage magnitude manipulation—reactive power flow

The magnitude of the fundamental of the inverter outputvoltage was set to 250V rms and the magnitude of the gridvoltage to 230V rms. This had as a result a current of rmsvalue 1.3A to flow through the circuit. The current flowwas due to the voltage difference between the a.c. side andthe d.c. side and it was expected that a reactive power flowoccurred in the same direction. There was no easy way tomeasure the reactive power, so the flow of reactive power

was demonstrated by inspection of the current waveshapesfor different supply voltages that would increase ordecrease the magnitude of the fundamental of the inverteroutput.One set of measurements and graphs was obtained using

a supply voltage of 250V rms and a phase shift of +21leading. These graphs were given not only in Fig. 4 but alsoin Fig. 6 to support the reactive power flow demonstration.Another set of graphs was taken this time using a supplyvoltage of 230V rms and a phase shift of 21 leading. Theseresults are shown in Fig. 7.Comparing the two current waveforms obtained for

supply voltages equal to 250 and 230V rms, it is obviousthat in the second case, where the supply voltage was

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Fig. 5. Lagging mode waveforms for supply voltage equal to 250V.

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reduced the current spikes seem to have reduced in terms ofmagnitude. The rms value of the current was increasedfrom 1.3 to 1.6A. A shift towards the left was observed inthis waveform compared with the respective one obtainedfor supply voltage equal to 250V rms. This slight shift tothe left indicates that in the latter case reactive power wasflowing from the grid towards the inverter. This directionof reactive power flow justifies the statement that adecrease in voltage magnitude of the fundamental of theinverter output results in importing power from the a.c.side towards the d.c. side.

For a supply voltage equal to 250V, reactive power wasexported from the inverter model towards the grid. Theshape of V2 is the same as before. The complex powerwaveform did not change but there was an increase in the

complex power rms value to 362VA and real powerdecreased to a value 43W.

4.6. Harmonic injection into the grid voltage

The waveforms in Fig. 8 were obtained to demonstratethe effect, which had on the grid voltage V2, an increase ofthe series inductance of the a.c. voltage source. Thisinductance was increased from a value of 0.001H to avalue of 0.01H, i.e. by a factor of 10 and harmonic injec-tion was evident on the grid voltage waveform V2. Fig. 8shows waveform V2 containing harmonics, alongside thecurrent waveforms, complex and real power waveforms forsupply voltage equal to 250V rms. Similarly, Fig. 9 showsresults obtained with supply voltage set to 230V rms.

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Fig. 6. Leading mode waveforms for supply voltage equal to 250V.

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The reason for this harmonic injection is that the a.c.source is active for a frequency of 50Hz (the pre-definedfrequency of the pure sinusoid generated by this source). Inthe case of higher frequency, i.e. 100Hz twice thefundamental frequency, and trying to simulate the circuitat the second harmonic the only ‘‘source’’ present would bethe inverter, which has an output containing this secondharmonic. At this frequency, the a.c. source becomes shortcircuited and the remaining circuit acts as a voltage divider,dividing the square inverter output between the seriesinductance and the coupling inductance. The larger theseries inductance the more voltage containing harmonicswill appear across it as the voltage drop. A series ofsimplified networks can be drawn in order to demonstrate

the voltage divider action at frequencies multiple of thefundamental 50Hz.

5. Conclusions

In this paper, a model of a single PV voltage inverter wasdesigned and simulated. The simulation was performedusing the PSCAD/EMTDC simulation package. Thisinverter model was used in conjunction with an a.c. voltagesource to show real and reactive power flow. The firstconcern was to establish the satisfactory operation of theinverter circuit and this was ensured obtaining a squarewave as the output. The network used to demonstratepower flow comprised of the inverter model and the a.c.

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Fig. 7. Leading mode waveforms for supply voltage equal to 230V.

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source (representing the main distribution grid). With theintroduction of a relative phase shift between the twooutput voltages, it was noticed that real power startedflowing from the most advanced (in phase terms) side ofthe network towards the least advanced. This agrees withgeneral power transmission theory stating that real poweris a function of the power angle d (phase shift between twovoltages). The leading and lagging modes of operation ofthe model were examined.

It has been proved that the inverter’s performance can beimproved using faster switching devices. Furthermore,

reliability can be also increased eliminating unwantedharmonics from the output using harmonic-eliminationtechniques. Harmonics can be injected into the networkby increasing the series inductance to the source. Thepresence of these harmonics leads to harmonic distortionthat may have a number of undesirable consequences. Forthis reason, the harmonic content of voltages in powerdistribution networks needs to be kept within acceptableproportions.Finally, the operation of the inverter device in this work

showed the model’s ability to both absorb and generate

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Fig. 8. Leading mode waveforms, harmonics injection in grid voltage with supply voltage equal to 250V.

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reactive power. It was shown that increasing the supplyvoltage at the input of the inverter, resulted in exportingreactive power from the inverter, and decreasing it, resultedin importing reactive power to the model. When the d.c.supply was increased, the magnitude of the fundamentalof the inverter output was increased with respect to thegrid voltage magnitude. The difference in voltage magni-tudes leads to VAR flows. Decreasing supply voltage leads

to exactly the opposite effects, i.e. absorption of reactivepower by the inverter. This ability may prove to be usefulif such models were installed in houses, being fed fromthe main distribution grid. With the appropriate phaseor voltage magnitude adjustments between the two‘‘sources’’, the inverter can compensate for some of thepower needed to supply such loads. If inverter deviceswere installed in many houses, for example, then the power

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Fig. 9. Leading mode waveforms, harmonics injection in grid voltage with supply voltage equal to 230V.

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demand for these loads would be reduced by a totalamount equal to the power that these devices can generate.In a case like this, the demand would drop significantly.In practical applications, the inverter will be poweredfrom PV cells, so the supply will cost less. The cells willprovide d.c. voltage that will be converted into a.c. bythe inverter. This allows less operational costs and it isanother way of exploiting the free nature resources forpower generation.

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