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p-qTheory Power Components CalculationsJ O ~ O . Afonso, Member, IEEE, M. J. Sepulveda Freitas, and Julio s. Martins, Member. IEEE
DEI, University of Minho, Campus de Azurem, 4800-058 Guim arles, Portugale-mail: j [email protected], [email protected], jm artins~ dei.um inho .ptAbsfracf- he Generalized Theor? of the InstantaneousReactive Power in Three-phase Circuits, proposed by Akugi
er al., and also known as the p-q theory, is an interesting toolto apply to the control of active power filters, or even to ana-lyze three-phase power systems in order to detect problemsrelated to harmonics, reactive power and u nbalance.In this paper it mill be shown that in three phase electricalsystems the instantaneous power waveforni presents symme-tries of 1/6, 1/3, 112 or 1 cycle of the power system fundamen-tal frequent), depending on the system being balanced o r not,and having o r not eken harmonics (interharmonics and sub-harmon ics are n ot considered in this analysis). These symme-tries can be exploited to accelerate th e calculations fo r activefilters controllers based on the p-q theory. In the case of theconventional reactive pow er o r zero-sequence compensation, itis shown th at the theo retical control system dyn amic responsedelay is zero.Index Terms- -q Theory, Active Power Filters, DigitalController, Sliding Window, Power Quality.
I. INTRODUCTIONIn 1983 Akuyi et al . [1, 21 proposed a new theory for thecontrol of active filters in three-phase pow er system s calledGeneralized Theory of the Instantaneous Reactive Powerin Three-phase C ircuits, also known as Theory of Instan-taneous Real Power and Imaginary Power, or Theory ofInstantaneous Active Power and Reactive Power, or The-ory of Instantaneous Power, or simply as p-q Theory.The theory was initially developed for three-phase three-wire systems, with a brief mention to systems with neutralwire. Later, Wutanabe ef al . [ 3 ] an d Aredes et al . [4] ex -tended it to three-phase four-wire systems (systems withphases (I, 0, and neutral wire).Since the p-q theory is based on the time domain, it is
valid both for steady-state and transient operation, as wellas for generic voltage and current waveforms. allow ing thecontrol of the active filters i n real-time.Another advantage of this theory is the simplicity of itscalculations, since only algebraic operations are required.The only exception is in the separation of some powercomponents in their mean and alternating values. However,as it will be show n in this paper, it is possible to exploit thesymmetries of the instantaneous power waveform for eachspecific power system, achieving a calculation delay thatcan be as small as 116 and never greater than 1 cycle of thepower system frequency. It is also shown that calculationsfor reactive power and zero-sequence compensation do notintroduce any delay.Furthermore, it is possible to associate physical meaning
to the p-q theory power components, which eases the un-derstanding of the operation of any three-phase power sys-tem, balanced or unbalanced, with or without harmonics.11. p-q THEORYo \\.E R C O ~ I P O N E N T S
The p-q theory implenlents a transformation from a sta-tionary reference system i i ~ I - C oordinates, to a systemwith coordinates a-,GfI. t corresponds to an algebraictransformation, known as C / i r r .X- c~ ransformation [ 5 ] , whichalso produces a stationary reference system, where coordi-nates cz-p are orthogonal to each other, and coordinate 0corresponds to the zero-sequence component. The zero-sequence component calculated here differs from the oneobtained by the symmetrical components transfonnation, orFurtescue transformation [6 ] . y a
The voltages and current5 i n n-pO coordinates are cal-culated as follows:factor.
The p-q theory po\\.eI components are then calculatedfrom voltages and curt-enth in the a-pO coordinates. Eachcomponent can be s e p a i t c d in i t s mean and alternating val-ues (se e Fig. I ) , which prchcnt physical meanings:
Po -Mean value of th e instantaneous zero-sequencepower. I t corresponds to th e energy per time u nity thatis transferred fioni the po\ver source to the loadthrough the zero-sequence components of voltage andcurrent.y, -Alternating value o f the instantaneous zero-sequencepower. It means thc energy per time unity that is ex-changed between the power source and the loadthrough the zero-sequence components of voltage andcu [Ten .
The zero-sequence pow er exists only in three-phase sys-tems with neutral wire. Moreover. the systems must haveboth unbalanced voltages and currents, or the same third
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order harmonics, in both voltage and current, for at leastone phase. It is important to notice that Po cannot exist in apower system without the presence of [3]. Since j,, isclearly an undesired power component (it only exchangesenergy with the load, and does not transfer any energy tothe load), both P,, an d Po must be compensated.B. Instantaneous Real Power ( p )
Yp = 3, .i o + v f l if l = p + y (3 )j7 - Mean value of the instantaneous real power. It corre-
sponds to the energy per time unity that is transferredfrom the power source to the load, in a balanced way,through the a-h-c coordinates (it is, indeed, the onlydesired power component to be supplied by the powersource).5 - Alternating value of the instantaneous real pow er. It isthe energy per time unity that is exchanged betweenthe power source and the load, through the a-0-c co -ordinates. Since p does not involve any energ y trans-ference from the power source to load, it must becompensated.
C. Imtuntuneoiis Z/izayiriar:irP m w 9)q = jf l . o - a . B = 4 (I (4
4 - Mean value of instantaneous imaginary power.y - Alternating value of instantaneous imaginary power.
The instantaneous imaginary power, q. has to do withpower (and corresponding undesirable currents) that is ex-changed between the system phases, and which does notimply any transference or exchange of energy between thepower source and the load.Rewriting equation (4) in a-b-c coordinates the follow-ing expression is obtained:
(5 )-vh).ir.+(vb - v ~ ) . ; ~ ,( I . , --v~,). ih]J5=This is a well known expression used in conventionalreactive power meters, in power systems without harmonicsand with balanced sinusoidal voltages. These instruments,of the electrodynamic type. display the mean value of equa-tion ( 5 ) . The instantaneous imaginary power differs fromthe conventional reactive power. because in the first case
al l the harmonics in voltage and current are considered.In the special case of a balanced sinusoidal voltage
supply and a balanced load, with or without harmonics, 4is equal to the conventional reactive power( 7 = 3.C. I , . s i n 4 ).
It is also important to note that the three-phase instanta-neous power ( y 3 ) an be written in both coordinates sys-tems, a-0-c and a-y-PO, ssuming the same value:
(6 )(7 )
Thus, to make the three-phase instantaneous power con-stant, it is necessary to compensate the alternating powercomponents p an d j o Since, as seen before, it is not pos-sible to compensate only po , all zero-sequence instantane-ous power must be compensated.
Moreover, to minimize the power system currents, theinstantaneous imaginary power, q , must also be compen-sated.
The compensation of the p-q theory undesired powercomponents ( p . p o an d q ) can bc accomplished with theuse of an active power filter. The dynamic response of thisactive filter will depend on the tim e interval required by itscontrol system to calculate these values.
p 3= vo . ,,+ vh . , + v, . , = p,,+ p, , +y,p 3 = >, . , + vf l .i p+ v,,. o = p +p o
111. EXPLOITINGOWER YMMETRI ESOnce the p-q theory power components are calculated itis important to decompose the instantaneous real power
(p ) in its mean value ( j 7 ) and alternating value ( p ). sinceonly the second quantity must be com pensated [7 , 81.The dynamic response of an active power filter control-ler depend s mainly on the time required to separate andp . With a digital controller it is possible to exp loit the in-stantaneous real power w we form symmetries, so that onlysamples of a fraction of the period of the measured voltageand current waveforms are required to separate the referredcomp onents. In fact, only 116, 113, 112 or 1 cycle is enough,depending on the system being balanced or not. and havingor not even harmonics, as shown next. These symmetriescan be exploited by a digital control system that makes useof a sliding window with a number of sam ples correspond-ing to the symmetry period.A . $VFlZ/ l?ef /~ f 1 /6 C1>c/rIn power systems with balanced voltages and cuirents,without even harmonics, the real instantaneous power pre-sents symmetry equal to 116 of the hndamental period(3.33 ms to a fundamental frequency of 50 Hz) . This meansthat the fi-equency of the instantaneous real power wave-
fonn is 6 times the fundamental frequency.B. $vimiefr;i.of 1/3 Cvclein power systems with balanced voltages and currents,
but with even harmonics in voltages an d o r currents. thereal instantaneous power pre sents symm etry equal to 113 offundamental frequcncy (6.66 ms to a fundamental fre-quency of 50 Hz). This means that the frequency of instan-taneous real power waveform is 3 times the fundamentalfrequency.Fig. 1 p-q theory power components
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C. Synzmety Ofl/2CycleIn pow er systems with unbalanced voltages and or cur-rents, without even harmon ics, the real instantaneous powerpresents symmetry equal to 112 of fundamental frequency
(10 ms to a fundamental frequency of 50 Hz). This meansthat the frequency of instantaneous real power waveform is2 times the fundamental frequency.
D. Symmet iy of I CvcleIn power systems with unbalanced voltages andlor cur-rents, and with even harm onics in voltages and/or currents,the real instantaneous power presents symmetry equal tothe fundan ~ental requency (20 ms to a fundamental fre-quency of 50 Hz).Fig. 2 shows the results of the instantaneous real powermean value calculations exploiting the four different typesof symmetry previously described, and the using of a slid-ing window approach.. Assuming that the power systemdoes not presents interharmonics or subharm onics (as resultof flicker problems, for instance), one cycle is enough tocalculate the value of p for all cases.For some active filters controllers it is important to ob-
(a) Balanced system - Without even harmonics0.-5 2 ......... .............................................--, io=. , ..,L......I.............................................p 229 c c
5 , 0~ g 2 2
s U 001 00 2 00 3 0 U4 005 0- ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .$F ,... 4
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I (a) Balanced system -Without even harmonics I
An active power filter with a control system that pre-sents a fast response, like the one based on the p-q theory,has basically two advantages:- It presents a good dynamic respons e, producing the cor-
rect values of compensating voltage or current in a shorttime after variations of the power system operatingconditions. If this response is good enough, the activefilter can be used in harsh electrical ewironinents,where the loads or the voltage system suffer intense andnumerous variations. An active filter with such a con-troller has also the capacity of performing well in powersystems with some types of subharmonics, like .flicker.It may happen becausejZickei. usually has a period (no tnecessarily constant) of some or even many powersystem cycles. and the proposed controller can respondin a time interval always inferior or equal to one cycle.- If the active filter has a fast control system, its energystorage element will suffer less to compensate thepower system parameters variation.Fig. 4 show s the voltage at the capacitor used in the DCside of the inverter of a shunt active power filter, like theone presented in Fig. 5 , when a load change occurs.
I Voltage at Shunt Active Power Filter DC side Capacitor I
IIII 4-J[Shunt Actixe 1'onc.r. b' i l twFig. 5 Block diagram o fa sliiint active po\ver filter
When there is a load change, the shunt active filter canact incorrectly, delivering (a s sho\vn in this case) or accu -mulating energy. If i t delivers energy, the capacitor voltagefalls. This behaviour must be considered, and is one of thefactors that influence the sizing of the capacitor. Fig. 4(a)refers to the behaviour of the capacitor voltage when a cer-tain new load is added to the po\ver system, and the controlsystem uses the alreadk described Buttenvorth filter to ob -tain the value of j ? . Fi g 1 b ) efers to the same situation,but with a digital control sys t em that calculates ,E with a1/6 cycle symmetry sliding \vindo\v. The second solutionpresents advantages: the act ive filter can compensategreater load variations. or. undcr the same operating condi-tions, the capacitor ca n hc do\\.nsized.
V. SIMULATED\ \ v i w OF COMPENSATIOSFig. 6shows an examp lc 01' compensation performed by
a shunt active power filter \\ i t l i control system based on thep-q theory. The simulation \\;is made with MotkddSinzzdinkThe voltages of t l i i h p o \ \ e ~ -ystcm are balanced and si-nusoidal. The loads a re unlial~inced nd present 2'ld orderharmonic currents. so :I slidiiig \vindow with symmetry of1 cycle was chosen to d! naniically calculate the value ofj3. Initially there are on ly tu o single-phase loads: two half-wave rectifiers with ;I rcsisti\e load. Then a three-phasefull-wave rectifier wi t h R L load i s turned-on (Fig. 6(a)).
Fig. 6(b ) shows t ha t . Ihe action of the shunt activefilter the sou rce currents hccoine hinusoidal and balanced.
The active filter control Ier coinpensates instantaneouslythe neutral wire currciit. ; incl this current is kept alwaysequal to zero in the s o t ~ r c c .15 we11 n Fig. 6(c).
The controller a lso compcnsates the instantaneousimag inary pow er iminediatcl!.. niaintain ing a null value forq a t the source (Fig. 6( e ) ) .
However. the instantaneous real power 01)must be sepa-rated in its mean an d alternating \,slue, and the sliding win-dow takes 1 cycle (2 0 ms ) to pcrfonn this task with thispower system operating conditions. It can be seen inFig. 6 (0 hat. after the load change, while the calculationof the new correct valuc of is not finished, the sourcedelivers less energy to the loads than it should do. So. dur-ing this time interval. the capacitor of the active filter m ustsupply energy to the loa d, as already explained.
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cause the instantaneous imaginary power variation can becompensated w ithout any delay. It is also possible to noticethat after a short time interval, necessary to calculate thenew value for j ? , the compensation current referencewaveform reaches steady state again.
The waveforms in Fig. 7 were obtained with the activefilter inverter tumed-off.
VII. CONCLUSIONSThe paper briefly described the calculations and physicalmeaning of the p-q theory power compo nents.In three-phase electrical systems the instantaneous realpower presents symmetries of 1/6, 113, 112 or 1 cycle of thepower system fundamental frequency, depending on thesystem being balanced or not, and having or not even har-monics. Since the instantaneous zero-sequence power pre-
sents an equal or smaller symmetry period, the instantane-ous three-phase power, which is the sum of p an d po(eq. 7) , presents the some kind of symmetries.This paper suggests the utilization of a sliding window,in a digital control system, to calculate the mean value ofthe instantaneous real power, exploiting the symmetries de-scribed above.The importance of a fast response for the active filtercontrol system is commented. Besides the improvement ofthe dynam ic response of the filter, it can also contribute to areduction in its cost, since the active filter capacitor can besmaller.Experimental results prove the good dynamic responseof the active filter controller based 011 the p-q theory, andusing a sliding window to calculate p .
This paper did not intend to discuss practical problemsrelated with the implementation of digital control systems,like the delays imposed by analog-to-digital conversiontimes and processing times. However, it is certain that thesekinds of problems are becoming each time less important,since nowadays there are already available fast ADCs (withconversion times of 80 ns, for instance) and microcontrol-lers/DSPs operating at frequencies as high a s 150 MHz.
Fig. 7 Experimental results with load changingVIII. REFERENCES
H . .4kagi. Y. Kanazawa. A. Xabae. Generalized Theory of th eInstantaneous Reactive Power in Three-phase Circuits, I fEC83 -In r . fowcr E1ectror1ic.sCor
