design and analysis of active power control strategies for distributed generation inverters under...

8/8/2019 Design and Analysis of Active Power Control Strategies for Distributed Generation Inverters Under Unbalanced Grid

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Published in IET Generation, Transmission & DistributionReceived on 27th October 2009Revised on 19th April 2010doi: 10.1049/iet-gtd.2009.0607

ISSN 1751-8687

Design and analysis of active power controlstrategies for distributed generation invertersunder unbalanced grid faults

F. Wang J.L. Duarte M.A.M. Hendrix Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven 5600MB, The NetherlandsE-mail: [email protected]

Abstract: Distributed power generation systems are expected to deliver active power into the grid and support itwithout interruption during unbalanced grid faults. Aiming to provide grid-interfacing inverters the exibility toadapt to the coming change of grid requirements, an optimised active power control strategy is proposed tooperate under grid faults. Specically, through an adjustable parameter it is possible to change the relativeamplitudes of oscillating active and reactive power smoothly, while simultaneously eliminating the second-order active or reactive power ripple at the two extremes of the parameter range. The steering possibility of the proposed strategy enables distributed generation inverters to be optimally designed from the perspectivesof both the power-electronic converters and the power system. The proposed strategy is proved throughsimulation and further validated by experimental results.

1 IntroductionVoltage dips are short-duration decreases in rms voltage, usually caused by remote grid faults in the power system[1, 2]. Becausemost voltage dips are due to unbalanced faults, balanced voltagedips are relatively rare in practice. As shown inFig. 1, a distributed generation (DG) system (three wire or four wire)is connected to the grid. A fault occurring in a parallel feeder will cause a voltage dip on the bus until a protection action

trips to clear the fault. Conventionally, a transformer-isolatedinverter would be required to disconnect from the grid when voltage dips and reconnect to the grid when faults are cleared.However, this requirement is changing. With the increasing application of renewable energy sources, more and more DGsystems deliver electricity into the grid. Wind power generation, in particular, becomes an important electricity source in many countries. Consequently, grid codes now require wind energy systems to ride through voltage dips without interruption in order to maintain power delivery andto support the grid [3, 4]. Therefore it is necessary toinvestigate the ride-through control of wind turbine systems

and other DG systems as well. Disregarding various upstreamdistributed sources and their controls, this paper will focus onthe control of DG inverters.

Concerning the control of DG inverters under grid faults,two aspects should be noticed. First, fast system dynamicsand good reference tracking are necessary. Note that inverters use the same converter topologies as pulse-widthmodulation (PWM) rectiers. Hence, techniques that havebeen developed to control PWM rectiers[59] can also beadapted to DG inverters. It has been shown that, becauseconventional controllers cannot effectively handle thenegative-sequence voltages, systems under unbalanced

voltage dips can have unbalanced and distorted grid currentsas well as voltage variations in the dc bus[5]. This problemis discussed in[6], where dual PI current controllers in dualsynchronous rotating frames (SRF) are proposed to controlpositive-sequence and negative-sequence componentsseparately. Unfortunately, the symmetric sequences of feedback quantities have to be extracted in dc form by ltering out the components oscillating at twice the gridfrequency. The system dynamics are considerably limitedbecause of the notch lter used for this purpose. As analternative, a delayed signal cancellation method with 1/4fundamental period delay can be used to improve the

sequence detection, but because of the time delay thismethod still inuences the system dynamics[7, 8]. In order to leave out the sequence detection of feedback currents and

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further improve system dynamics, two extra resonant controllers working at twice the grid frequency are added todual SRFs, eliminating the oscillation parts by means of compensation [9]. Alternatively, stationary frame resonant controllers are found to be suitable to control all thesequences (including the zero sequence that has beendisregarded in many studies). This can simplify dual SRFs toa single stationary frame and achieve zero steady-sate error [1012]. Also, symmetric sequence detection in feedback currents becomes unnecessary when using the resonant controllers, for example the scalar resonant control structurepresented in [13]. In summary, in order to improve systemdynamics and reference tracking, controllers must be able todeal with all the symmetric-sequence components and havedirect feedback signals for closed-loop control. This isachieved by the controller employed in this paper.

Second, the generation of reference currents in case of unbalanced voltage dips is important. Assuming controllersare fast enough when voltage dips, then the only differencebefore and after a grid fault is a change in demandedcurrent. Depending on practical applications, current reference generation is objective oriented. Considering thepower-electronics converter constraints, a constant dc-link voltage is usually desirable, as in the objective set of [9, 14].In other words, the instantaneous power at the output terminal of the inverter bridge is expected to be a constant.For simplication, only the instantaneous power at the gridconnection point is considered by neglecting the part of oscillating power on account of the output lters[6].However, a constant dc bus is achieved at the cost of unbalanced grid currents, and this results in a decrease of maximum deliverable power. In[15], a power reducing scheme is used to conne the current during a grid fault.

On the other hand, the affects of the grid currents on thepower system side should also be taken into account whendesigning reference currents for DG inverters. As presentedin [16], several individual strategies are possible to obtaindifferent power quality at the grid connection point interms of instantaneous power oscillation and current distortion. These strategies show exible controlpossibilities of DG systems under grid faults. However,they only cope with specic cases. Therefore starting fromthe ideas in [16], a generalised strategy on reference current generation is designed. Further discussion and analysisbased on this strategy are also carried out in this paper.

This paper presents a generalised active power control strategy under unbalanced grid faults based on

symmetric-sequence components and shows explicitly thecontributions of symmetrical sequences to instantaneouspower. The steering possibility of the proposed strategy enables DG inverters to be optimally designed from theperspectives of both the power-electronics converters andthe grid. Thus, a further discussion on objective-orientedoptimisations of the proposed strategy is presented. Astationary frame resonant controller with direct feedback variables is employed to construct a dual-loop controlstructure. In addition, feed-forward of the grid voltage isadded to help improving system response to griddisturbances. Based on this control structure, simulationsand experiments are carried out to validate the proposedstrategy.

2 Instantaneous powercalculationIn this section, the instantaneous power theory [17, 18] isrevisited. Then instantaneous power calculation based onsymmetric sequences is developed, and the notation for thereference current design in the next sections is dened.

2.1 Instantaneous power theory For a three-phase DG system shown inFig. 1, instantaneousactive power and reactive power at the grid connection point are given by, respectively

p = v i =v a i a +v b i b +v c i c (1)

q = v i =1

3 [(v a v b )i c +(v b v c )i a +(v c v a )i b ]

with v=

1

3 0 1 11 0 11 1 0

v

(2)

where v = v a v b v c T, i = i a i b i c T, bold-italicsymbols represent vectors, and the operator . denotes thedot product of vectors. Note that the subscript is used torepresent a vector derived from the matrix transformation in(2), although vectorsvand v are orthogonal only when thethree-phase components in vector v are balanced.

To delivera given constantinstantaneous activepower ( p P )and zero reactive power, the corresponding currents can becalculated based on the instantaneous power theory. Thederived currents, denoted by i p , are expressed by

i p =P v 2

v (3)

where v 2 = v v =v 2a +v

2b +v

2c , operator . means the

norm of a vector, and subscript p represents active power-related quantities. Sincei p is in phase with v , the resulting instantaneous reactive power q in (2) equals zero. Because

Figure 1 DG inverter operates under grid faults

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calculated as

i + p =P

v + 2 k p v 2v + (19)

i p = k p P v + 2 k p v 2v (20)

Finally, the total current reference is the sum of i + p and i p ,that is

i p =P

v + 2 k p v 2(v +k p v ), 0 k p 1 (21)

3.2 Controllable oscillating reactive power Instead of compensating the oscillating active power in (13), we can similarly shape the oscillating reactive power in (14).For this purpose negative-sequence currents are imposed tomeet

v + i p = k p v i + p , 0 k p 1 (22)

By considering equationv + i = v + i (becausev +lagsv + by 900 and i

leads i by 900 ), the left side of (22) can be

rewritten as

v + i p = v + i p = k p v i + p (23)

where i p denotes the orthogonal vector of i p according to

(2). Then, it follows that

i p =k p v + i + p

v + 2v

(24)

Hence the negative-sequence current i p follows directly from(24) as

i p =k p v + i + p

v + 2v (25)

Solving (15) and (25), the positive-sequence currents andnegative-sequence currents are derived as

i + p =P

v + 2 +k p v 2v + (26)

i p =k p P

v + 2

+k p v 2

v (27)

Again, the total current reference is the sum of i + p and i p , that

is

i p =P

v + 2 +k p v 2(v ++k p v ), 0 k p 1 (28)

3.3 Combining strategies a and bSimple analysis reveals that (21) and (22) can be combinedand represented by

i p =P

v + 2 +k p v 2(v ++k p v ), 1k p 1 (29)

In order to analyse the variation of the power, we substitute(29) into (13) and (14), and the results are

p =P +P (1 +k p )(v v +)v + 2 +k p v 2

(30)

q =P (1 k p )(v v +)

v + 2 +k p v 2(31)

It can be seen that the variant terms of (30) and (31), that is oscillating active power and reactive power, are controlledby the coefcient k p . These two parts of oscillating power are orthogonal and equal in maximum amplitude.

Simulation results are obtained inFig. 3 by sweeping theparameter k p . Related circuit structure and controller parameters are given in the experimental part. It can beseen that either oscillating active power or oscillating reactive power can be controlled and even can beeliminated at the two extremes of thek p curve. In other words, a constant dc-link voltage can be achieved becauseof the elimination of oscillating active power whenk p 2 1; symmetric currents can be derived whenk p 0; andunity power factor can be achieved whenk p 1 wherethe three-phase currents follows the grid voltages. Therefore innite points of k p between 2 1 and 1 willallow a optimum control of the currents in order toachieve different objectives. For instance, within a certainrange of dc-voltage variation, the currents can beoptimised to get maximum power output by changing thek p with respect to the grid voltages. More discussions onthe choices of k p are presented in the next section.Furthermore, a graphic representation of the relationshipbetween these two instantaneous oscillating power isplotted in Fig. 4a . It illustrates clearly that these two part of oscillating power shift from one to another with theincrease or decrease of k p under the proposed strategy. This controllable characteristic allows to enhance systemcontrol exibility and facilitates system optimisation.

Further discussion on objective-oriented optimisations of this generalised strategy will be presented in the next section.


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4 Analysis on strategyapplications As can be observed, the proposed current referencegeneration strategy provides DG inverters control exibility,that is the coefcient k p can be optimally determined basedon optimisation objectives and be adaptively changed incase of voltage dips. Therefore the applications of theproposed strategy are discussed from different aspects.

4.1 Constraints of DG invertersConsidering the power-electronic converter constraints, a serious problem for the inverters is the second-order rippleon the dc bus, which reects to the ac side and createsdistorted grid currents during unbalanced voltage dips.Either for facilitating dc-bus voltage control or for

minimising the dc-bus capacitors, it is preferred to makek p in (29) close to2 1 so that the dc-voltage variations can bekept very small. However, the maximum deliverable power

Figure 3 Simulation results of the proposed control strategy a Phase voltages, where voltages of phase a and b dip to 80% at t 0.255 sb Injected currentsc Instantaneous p , qd Controllable coefcient k p sweeping from 2 1 to 1

Figure 4 Graphic representationa Relationship between oscillating active power p 2 v and oscillating reactive power q 2 v b Grid voltage and current trajectories before and after unbalanced voltage dips in the stationary frame, with k p as a changing parameter


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has to decrease because unbalanced phase currents reduce theoperating margin of inverters. Therefore for a maximumpower-tracking system it is preferable to shift k p towardszero to get balanced currents as long as the variation on dclink is acceptable. Note that the inverter losses andoscillating power on account of output lters are not considered, whereas the effects of output lters will becommented on later.

To help understanding, a vector diagram that representscurrent trajectories changing withk p under an unbalanced voltage dip is drawn inFig. 4b . For a desired power P ,unbalanced currents are generated to achieve zero p 2v whenk p equals 2 1. When the asymmetry of voltage dips becomemore pronounced, the generated currents will nally crossovercurrent levels if the system tries to maintain the samepower delivery as during the pre-fault status and keeps zero p 2v simultaneously. Thereforek p has to be adapted towards

zero, as shown inFig. 4b , that is the current trajectory starts from the attest ellipse and reshapes to a circle.Hence, k p will be somewhere in between 0 and 1 as long as p 2v is smaller than the allowed maximum value p 2v max,

which can be calculated by

p 2v max =2p f V dcC dcDV dc (32) where f is the grid frequency,V dc is the average dc voltage,C dc is the dc-link capacitance andDV dc the specied dc- voltage variation. Note that whenk p reaches zero and theamplitudes of balanced currents are still out of the

maximum current range, the delivered power has to bedecreased. As a consequence, the upstream distributedsources should also take actions to guarantee the power balance on the dc link.

4.2 Effects on the grid On the other hand, the effects of the proposed strategy on thepower system side could also impose constraints on thecontrol design, especially for large-scale DG systems.Because the oscillating reactive power causes extra power losses in the grid, it is preferred to makek p close to 1 inorder to eliminate q 2v . Again, the dc variation becomes a constraint because the amplitude of p 2v is increasing whenk p shifts to 1. Therefore similar comparison between thepredicted oscillating power given by (30) and p 2v maxgiven by 32) should be carried out. As long as the activepower oscillation is acceptable,k p can move towards 1 asclose as possible. Otherwise, delivered power downgrades inquantity.

However, the objective becomes different from voltagequality point of view. As shown in the results of Fig. 3, when k p is getting close to 1, less current is delivered intothe phases in which grid voltage is relatively low. Also, it

can be seen in Fig. 4b that the current vector tends tosynchronise with the grid voltage vector, that is thereference currents tend to follow the asymmetry of the grid

voltages. Consequently, the unbalanced voltage will be worsened because unbalanced voltage rise across gridimpedances, which is introduced by the injected gridcurrents (if the grid impedance is resistive and the injectedcurrents are large enough). On the contrary, whenk p isclose to 2 1, higher current is delivered into the phases in which grid voltage is relatively low, thereby compensating the voltage imbalance at the grid connection point. As isrevealed in the analysis above, the objectives of improving the effects on the grid side could conict with each other.Fortunately, the proposed strategy can be easily adaptedaccording to practical objectives.

4.3 Effects of output inductors The effects of output inductors are well known and thereforeare shortly presented here. The instantaneous power controldiscussed so far has been treated at the grid connection

point. When the objective is aimed at optimising dc- voltage variations, and the oscillating power on account of the output inductors cannot be neglected (in case of largeand unbalanced currents or small dc capacitance), thefundamental components of the inverter-bridge output voltages should be used in the proposed strategy for calculation instead of the grid voltages. Indirect derivationbased on grid voltages and currents or direct measurement on the output of inverter bridge can be used to get the voltages as presented in[9, 14].

5 Experimental verication To verify the proposed strategy, experiments are carried out on a laboratory experimental system constructed from a four-leg inverter that is connected to the grid through LCLlters, as shown inFig. 5. The system parameters are listedin Table 1. By using a four-leg inverter, zero-sequencecurrents can be eliminated when the grid has zero-sequence voltages. For the cases where the zero-sequence voltage of unbalanced grid dips is isolated by transformers, a three-leg inverter can be applied. A 15 kV A three-phaseprogrammable ac-power source is used to emulate theunbalanced utility grid and the distributed source isimplemented by a dc-power supply.

5.1 Control realisation The controller is realised with a double-loop current controller, which consists of an outer control loop withproportional-resonant (PR) controllers for compensating the steady-state error of the fundamental-frequency currents and an inner inductor current control loop withsimple proportional gain to improve stability. In addition, a feed-forward loop from the grid voltages is used to improvesystem response to voltage disturbances.

Furthermore, the direct feedback of currents in thestationary frame is achieved, which is desirable as presentedin Section 1. Also, the control for dual sequence


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components would be simple without transformationsbetween reference frames. A quasi-PR controller with highgain at the fundamental frequency is used

G i (s ) =K p +2K r v br s

s 2 +2v br s +v 21(33)

whereK p is the proportional gain,K r is the resonant gain,v 1denotes the fundamental radian frequency andv br theequivalent bandwidth of the resonant controller. A detaileddesign for the PR controller and its application have beenpresented in [11, 12]. Through optimising, the parametersused in the experiment are K p 2, K r 100 and

v br

10 rad/s.

The symmetric-sequence components are detected in thetime domain with a stationary frame lter in thea b frame [20]. The basic lter cell was demonstrated to beequivalent to a band-pass lter in the stationary frame, andcan be easily implemented using a multi-state-variablestructure shown in Fig. 6. Because of its robustness for small frequency variations, a phase locked loop (PLL) is

avoided in this experiment since the power supply only emulates the magnitude drop of the grid voltage.Otherwise, a PLL should be added to the lter for adapting to large frequency changes[20].

5.2 Experimental resultsBy shifting the controllable parameter k p , the system is testedunder unbalanced voltage dips with the proposed strategy. Inorder to capture the transient reaction of the system, threesituations are intentionally tested for comparison at thestart moment and nish moment of voltage dips.

5.2.1 Under voltage dips: As shown in Fig. 7a , grid voltages are emulated to be faulty at t 0.03 s wherephases A and B dip to 80%. In case of k p 2 1, theinjected grid currents and instantaneous power are shownin Fig. 8a . It can be seen that the currents on phases withlower voltage are higher than the phase with higher voltagein order to obtain a constant active power.

Figure 5 Circuit diagram and control structure of experimental four-leg inverter system

Table 1 System parameters

Description Symbol Value

output ltering inductor Lsa,b,c 2 mH

output ltering capacitor C a,b,c 5 mF

output ltering inductor La,b,c 2 mH

neutral ltering inductor Ln 0.67 mH

dC-link voltage V dc 750 V

dC-link capacitors C dc 4400 mF/900 V dc

switching frequency f sw 16 kHz

system rated power S 15 kVAtested power P 2500 W

Figure 6 Implementation diagram of sequence detection,where v 1 314 rad /s, v b 100 rad /s


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As analysed in Section 3, a set of negative-sequencecurrent opposite to the negative-sequence voltage isinjected to compensate the part of oscillating activepower introduced by negative-sequence voltage and

positive-sequence currents. Also, it can be seen that theinstantaneous reactive power has a large power rippleoscillating at twice the grid frequency. The worst case is when k p 2 1 because of an extra part of oscillating reactive power generated by the negative-sequence currentsand positive-sequence voltages.

In Fig. 8b , results withk p 0 are shown. As illustrated, thecurrents are kept balanced before and after grid faults. Hence,the oscillating part of the instantaneous active and reactive

Figure 7 Emulated grid voltagesa To be faulty at t 0.03 s, where phases a and b dip to 80%b To be recovered at t 1.055 s

Figure 8 Experimental results under unbalanced voltage dips, where the waveforms from top to bottom are injected grid currents and instantaneous power

a With k p 2 1b With k p 0c With k p 1

Figure 9 Experimental results of dc-link ripple under unbalanced voltagea With k p 2 1b With k p 0c With k p 1


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power are equal in amplitude and have a 900 phase shift,as only positive-sequence currents exist. The results whenk p 1 are given inFig. 8c , it can be seen that the three-phase currents are in phase with the corresponding

voltages. This means that unity power factor can beachieved. In addition, dc-voltage ripples during voltage dipsare also observed as shown inFig. 9, where the ripplesreected by the ac-side oscillating active power are clearly illustrated. Note that the dc-voltage variations are not large,even when k p 1, because of the big dc-link capacitanceused in the system and relatively low tested power.

5.2.2 After voltage dips: Experimental results are alsomeasured when grid voltages recover after voltage dips. Asshown in Fig. 7b , the unbalanced grid voltages becomebalanced at t 1.055 s. With respect to different values of

k p , corresponding results of injected grid currents andinstantaneous power are shown inFig. 10. It can be seenthat different current waveforms and instantaneous power waveforms, which were controlled by k p under voltage dips,are the same as originally when the grid voltages becomebalanced.

6 Conclusion This paper has proposed a generalised active power controlstrategy for DG inverters operating under unbalanced gridfaults. Based on derived formulas and graphic

representations, the contributions of symmetric-sequencecomponents to the instantaneous power and theinteractions between symmetric sequences have been

explained in detail. By intentionally introducing anadjustable parameter in between the reference currents andthe grid voltages, a strategy has been found to be able tocontrol the dc-link ripples, the symmetry of currents or the

power factors in terms of controllable oscillating active andreactive powers. Then the analysis of strategy applicationsfrom the perspective of both the power-electronicconverters and the power system were discussed. Theexible adaptivity of the proposed strategy allow it to cope with multiple constraints in practical applications. Finally,the proposed control strategy was tested based on anexperimental system, and the measured results veried theperformance as expected. The next-step work is going further with on-line optimum control for specic DGsystems based on the proposed strategies.

7 References[1] MCGRANAGHAN M.F., MUELLER D.R., SAMOTYJ M.J.: Voltage sagsin industrial systems, IEEE Trans. Ind. Appl. , 1993, 29 , (2),pp. 397403

[2] ZHANG L., BOLLEN M.H.J.: Characteristic of voltage dips(sags) in power systems, IEEE Trans. Power Del. , 2000,15 , (2), pp. 827832

[3] E.O. Netz Gmbh: Grid Code for high and extra highvoltage, 2006

[4] National Grid Electricity Transmission Plc, U.K.: TheGrid Code, 2009

Figure 10 Experimental results when the voltage recovers at t 1.055 s, where the waveforms from top to bottom areinjected grid currents and instantaneous power a With k p 2 1b With k p 0c With k p 1


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[5] SANNINO A., BOLLEN M., SVENSSON J.: Voltage tolerancetesting of three-phase voltage source converters, IEEE Trans. Power Del. , 2005, 20 , (2), pp. 16331639

[6] SONG H., NAM K.: Dual current control scheme for PWMconverter under unbalanced input voltage conditions, IEEE Trans. Ind. Electron. , 1999, 46 , (5), pp. 953959

[7] SACCOMANDO G., SVENSSON J.: Transient operation of grid-connected voltage source under unbalanced voltageconditions. Proc. IEEE IAS, 2001, pp. 24192424

[8] MAGUEED F.A., SANNINO A., SVENSSON J.: Transientperformance of voltage source converter underunbalanced voltage dips. Proc. IEEE PESC, 2004,pp. 11631168

[9] SUH Y., LIPO T.A.: A control scheme in hybrid

synchronous-stationary frame for PWM AC / DC converterunder generalized unbalanced operating conditions, IEEE Trans. Ind. Appl. , 2006, 42 , (3), pp. 825 835

[10] YUAN X., MERK W., STEMMLER H., ALLMELING J.: Stationaryframe generalized integrators for current control of activepower lters with zero steady-state error for currentharmonics of concern under unbalanced and distortedoperating conditions, IEEE Trans. Ind. Appl. , 2002, 38 , (2),pp. 523532

[11] ZMOOD D., HOLMES D.: Stationary frame currentregulation of PWM inverters with zero steady-state error,IEEE Trans. Power Electron. , 2003, 18 , (3), pp. 814822

[12] TEODORESCU R., BLAABJERG F., LISERRE M., LOH P.C.:Proportional-resonant controllers and lters for grid-connected voltage-source converters, IEE Proc. Electron.Power Appl. , 2006, 153 , (5), pp. 750762

[13] ETXEBERRIA-OTADUI I., VISCARRET U., CABALLERO M., RUFER A.,BACHA S.: New optimized PWM VSC controlstuctures and strategies under unbalanced voltagetransients, IEEE Trans. Ind. Electron. , 2007, 54 , (5),pp. 29022914

[14] YIN B., ORUGANTI R., PANDA S., BHAT A.: An output-power-control strategy for a three-phase PWM rectier underunbalanced supply conditions, IEEE Trans. Ind. Electron. ,2008, 55 , (5), pp. 21402151

[15] CHONG H., LI R., BUMBY J.: Unbalanced-grid-fault ride-through control for a wind turbine inverter, IEEE Trans.Ind. Appl. , 2008, 44 , (3), pp. 845856

[16] RODRIGUEZ P., TIMBUS A.V., TEODORESCU R., LISERRE M., BLAABJERGF.: Flexible active power control of distributed power

generation systems during grid faults, IEEE Trans. Ind.Electron. , 2007, 54 , (5), pp. 25832592

[17] AKAGI H., WATANABE E.H., AREDES M.: Instantaneous powertheory and applications to power conditioning (IEEEPress, 2007)

[18] PENG F.Z., LAI J.S.: Generalized instantaneous reactivepower theory for three-phase power systems, IEEE Trans.Instrum. Meas. , 1996, 45 , (1), pp. 293297

[19] ANDERSSON P.M.: Analysis of faulted power systems (IEEEPress, 1995)

[20] WANG F., DUARTE J., HENDRIX M.: High performancestationary frame lters for symmetrical sequences orharmonics separation under a variety of grid conditions.Proc. IEEE APEC, 2009, pp. 15701576

8 Appendix 1 An alternative expression in the stationary a b g frame for the instantaneous power calculation can be derived, that is

p = v i = (v +ab +v ab ) (i +ab + i ab ) +v g i g (34)q = v i = (v +ab +v ab ) (i +ab + i ab ) (35)

with v g = 3 v 0a , i g = 3 i 0a , v ab +, =T ab v +, andi ab +, =T ab i +,, where

T ab = 2

3 1

12

12

0 3 2 3 2

The orthogonal matrix transformation is

v +,ab =

0 1

1 0 v +,ab (36)

9 Appendix 2 The dot product of a positive-sequence voltage vector v + anda negative-sequence vector i 2 can be calculated in thea b c reference frame as

v + i =v +a i a +v +b i b +v +c i c =V +cos(v t +w 1) I cos(v t +w 2)

+V +cos v t +w 1 23

p I cos v t +w 2 +23 p +V +cos v t +w 1 +

23

p I cos v t +w 2 23 p =

3

2V + I cos(2v t

+w

1 +w

2) (38)

where symbolsw 1 and w 2 are the phase angles of v +a and i a ,


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respectively. It can be seen that the dot product of v + and i 2

leads to an instantaneous power oscillations at twice thefundamental frequency.

As dened in Section 3,i + p and i p are in phase withv +

and v , respectively. Thusi + p =k1v +, i p =k2v , wherek1

and k 2 are scalars. Hence, we can obtain

v i + p =1k2

i p k1v + =k1k2

(v + i p ) (39)

Likewise, we can also nd that the dot products of v

i + p and v + i p have a similar relationship.


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