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EE404_37-45[1] | Norm (Mathematics) | Ac Power

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The Relation between the Generalized Apparent Power and the Voltage Reference
  L’Energia Elettrica - Volume 81 (2004) - “Ricerche” The Relation between the GeneralizedApparent Power and the Voltage Reference JACQUES L. WILLEMS  , Fellow IEEE  , and JOZEF A. GHIJSELEN, Member IEEE     Faculty of Engineering, Ghent University, Gent, Belgium This paper deals with the concept of apparent power as the quantity that characterizes the maximal possible utilization of a line for given circumstances, characterized by the voltage waveform and the current magnitude. It is shown that in thegeneral case of nonsinusoidal polyphase systems the apparent power depends on the definition of the magnitude of thecurrent vector and on the reference of the voltage. It is shown that these quantities cannot be chosen independently. Therecent IEEE Standard 1459-2000 is critically discussed with respect to the definition of apparent power, and aninconsistency is pointed out. 1. Introduction  The   concepts of active, reactive, and apparent power and power factor are well defined and well understood in theclassical situation of sinusoidal single-phase voltages andcurrents, and also in the case of balanced three-phasesinusoidal voltages and currents. These definitions servedthe industry well as long as the voltage and currentwaveforms remained nearly sinusoidal and balanced inthe three phases. However important changes haveoccurred in the last decades. Power electronic equipmentand clusters of personal computers represent major nonlinear loads and cause distortion. This causessignificant errors in traditional instrumentation designedfor the sinusoidal balanced situation. Moreover microprocessors enable today’s manufacturers toconstruct metering equipment capable of meteringelectrical quantities even when defined by advancedmathematical models. Therefore there is need for a newset of definitions of power quantities which are relevantfor the new situations, but which remain valid in theclassical situations. Such definitions are needed as acommon base for e.g.  electrical energy billing;  characterization of electrical energy quality;  detection of sources of deterioration of quality.A large number of research publications [1-10] discussthe generalization of the classical concepts to the moregeneral conditions. A recent IEEE Standard [5, 9] proposes a set of definitions for the measurement of electric power quantities under sinusoidal, nonsinusoidal, balanced, or unbalanced conditions.This contribution deals with the concept of apparent power. The classical and generally accepted definition of apparent power is the product of the rms values of thevoltage and current vectors. This quantity is meant tocharacterize the maximal (useful) power  1 that a line cantransmit under given constraints. Associated withapparent power is the concept of power factor as thedegree of utilization of the line. An essential point of discussion in this definition for multiterminal networks,in particular for the calculation of the effective value of the voltage, is the choice of the reference for the voltage[3, 6]. Although the choice of the reference for thevoltage does not change the power transmitted (onlyvoltage differences are essential and have physicalmeaning), the choice of the reference does indeed affectthe effective value of the voltage and hence also theapparent power according to the above definition. It is the purpose of the present paper to look deeper into this moreor less paradoxical observation. In a number of papers [3,6] it is proposed to use the virtual star point as thereference for the voltage for defining the effective valueof the voltage in the definition of the apparent power. It isshown in the present paper that the choice of the voltagereference is not a matter of convenience, but that under suitable conditions the choice should be the virtual star  point to get a physically justifiable definition of apparent power. In other situations however the reference should be the voltage of one of the terminals, in particular theneutral terminal; it is also shown that each of thesevoltage references corresponds to a particular and justifiable definition of apparent power.An important feature of electrical power transmission aswell for the instantaneous power as for the average(active) power in sinusoidal or nonsinusoidal, single- phase or polyphase, balanced or unbalanced conditions, isthat the same useful power can be transmitted for a givenvoltage waveform by various currents; this has animportant impact on the power transmission efficiency.The degree of utilization of the transmission equipmentcan be characterized by a number of concepts which have been used in power system analysis for a long time, butwhich are being carefully reconsidered in recent years because of the changing working circumstances of power systems. In particular, assuming that the active power characterizes the useful power transfer, the apparent This paper presents results of research partially supported by theBelgian Programme of Interuniversity Attraction Poles, initiated bythe Belgian State, Prime Minister’s Office for Science, Technologyand Culture, and by the Fund for Scientific Research of Flanders(FWO-Vlaanderen). The scientific responsibility rests with theauthors. 1   The concept of apparent power can also be associated with the sizeof the oscillations in the instantaneous power. This aspect is explicitlydiscussed by Ghassemi [8] and Willems [12]. This line of thought isnot pursued in the present contribution.Correspondence: J. L. Willems, Department EESA, Ghent University,Technologiepark-Zwijnaarde, 914, B-9052 Gent, Belgium, tel +32-9-2645648 fax +32-9-2645840, e-mail Jacques.Willems@ugent.be.  37  L’Energia Elettrica - Volume 81 (2004) - “Ricerche” power is generalized as follows: “The apparent power at the terminals of a power system is defined as the maximal active power that can be transferred for the given set of voltages and the given magnitude of the set of currents(or line losses)”. The degree of utilization of the power transmission or distribution line is expressed by the power factor which is the ratio between the actual active power (the useful power) transfer and the maximum possible active power which can be transferred under thegiven conditions 2 , in particular for the given line losses. Itis clear that the thus defined apparent power depends onthe way the magnitude of the current (or the line loss) ismeasured and on the choice of the voltage reference. Thisis an essential point discussed in the present paper. It isclearly shown that the choice of the definition of themagnitude of the current and the choice of the referenceof the voltage are closely related problems and cannot bedealt with separately.According to the concept introduced in the previoussection, the apparent power is defined as the maximalactive power that can be transmitted for the given voltageand the given magnitude of the current. As already statedin the previous section, the concept of apparent power requires a definition of the magnitude of the current (or the line losses). Here several possibilities can beconsidered. The magnitude of the current shouldcharacterize the power loss in the conductors or the costof the conductors. It is therefore not obvious to includethe current to terminal m+ 1 in the same way in themagnitude as the currents to the other terminals. It isinteresting to consider the two extreme casescharacteristic for the different possibilities; these casesare very often used in practical applications.  Definition A of current magnitude All currents, including the current to the   terminal m+ 1,are included in the same way in the magnitude of thecurrent, which is thus defined by: 2. Sinusoidal multiconductor power quantities ∑ += = 112 mk k  A  I  I  (3) 2.1 Definitions of apparent power    Definition B of current magnitude In this section the case of a multiterminal line (with m +1terminals) is considered in sinusoidal steady state. Inmost cases (such as polyphase power transmission) theterminals 1,…, m can be considered to play equivalentroles and the conductors connected to them can beconsidered to be identical. On the other hand, in many practical situations terminal m+ 1 plays a different role,since it is connected to the return or earth conductor,which in most practical situations is different from the phase conductors.All currents except the current to the terminal m+ 1 areincluded in the same way in the magnitude of the current,whereas the current to the terminal m+ 1 is not included.The magnitude (or norm) of the current is thus defined by: ∑ = = mk k  B  I  12 I  (4) Note that both definitions satisfy the requirements of themathematical concept of a norm 3 , because of the physicalrequirement that the sum of the currents vanishes. Hencefor expressing the magnitude of the current bothdefinitions can be equivalently used. As to the physicalmeaning the square of the current is proportional to the power loss caused in the feeding line, and alsocharacteristic for the cost of the conducting material.Definition A hence expresses that terminals 1 to m+ 1 areconnected to identical conductors, with the sameresistance, whereas Definition B expresses the fact thatterminals 1 to m are connected to identical conductors,with the same resistance, which contribute in the sameway to power loss and cost, whereas terminal m+ 1 isconnected to a conductor in which the loss and the cost of the conducting material is assumed to be negligible.In most applications the useful power is the average power or what is conventionally called the active power  ∑ +=∗∗ == 11 )Re()Re( mk k k T   I V  P  IV  (1)where and denote the phasors of the voltages and of the currents at the terminals 1 to m +1,and and denote the voltage and the current vectors. 11 ,..., + m V V  I 11 ,..., + m  I  I  V The physical laws of network theory require thefollowing properties for an ( m +1)–terminal line with m +1voltages of the terminals with respect to some referenceand m +1 currents to the terminals:  Kirchhoff’s current law: the sum of the currents, or the components of the current vector is zero hence ∑ += = 11 0 mk k   I   (2)These models are now used to derive the apparent power according to the definition given above. The maximumvalue of is required for given voltages with respect tothe currents in the terminals, subject to the conditions thatthe sum of the currents vanishes and the magnitude of thecurrents (as defined in Definition A or B) is a given  P   Kirchhoff’s voltage law: this law implies thet the power only depends on the voltage differences; itis independent of the choice of the voltagereference.This means that in (1) the expression of the power isindependent of a constant added to or subtracted fromeach voltage phasor. In particular terminal m +1 can bechosen as the reference; then the last term in the sum iszero and is usually dropped. 3 The norm characterizes the magnitude of a vector. The Euclideannorm (the square root of the sum of the squares of the absolute valuesof the elements) is a possible choice, but not the only one. The reader is referred to textbooks on linear algebra for a discussion of theconcept of the norm and the required properties. 2 A n invariant voltage waveform is equivalent to an infinite bus.  38  L’Energia Elettrica - Volume 81 (2004) - “Ricerche”quantity. This problem can be solved by means of theLagrange multiplier technique. 2.2 Apparent power according to Definition A For Definition A of the current magnitude, the maximumvalue should be determined of      −    −− ∑∑∑ +=+=+= 1111211* Re2)Re( mk k mk k mk k k   I  A I  I V  µλ  with respect to the currents and the Lagrange multipliersand , where is real and complex. This leads tothe equations λ µ λ µ 0)1,...,1( 11112 ==+=+= ∑∑ +=+= mk k mk k k k   I  A I mk  I V  µλ  The solution of this set of equations is )1,...,1( 11 11211211 +=−=−==+= ∗+=+=∗∗+= ∑∑∑ mk V V  I  I V V V V m k kAmk k mk k mk k  λλµ  Hence the currents yielding the maximal active power satisfy ( ) )1,...,1( 112112 +=−−= ∗+=∗+= ∑∑ mk V V  V V  I  I  k mk k mk k kA  (5)With these currents and the given voltages the apparent power, that is the maximal active power that can bedelivered with the given voltage and the given magnitudeof the current, is obtained. The expression of the active power (1) leads to the result:  A Amk k mk k  A  I V V S  IV =−= ∑∑ +=+=∗ 112112 . (6)In this expression the voltages are measured with respectto the virtual star point (chosen such that the sum of thevoltages is zero [3, 6]). This is a well known result, wherethe norm of the voltage is the so-called collective rmsvalue. 2.3 Apparent power according to Definition B The same analysis is now performed with the magnitudeof the current vector as in Definition B, where theinteresting conclusion will be that a different result for thevoltage reference is obtained. With respect to DefinitionB, the maximum value of is required for givenvoltages with respect to the currents in the terminals,subject to the conditions that the sum of the currentsvanishes and the magnitude of the currents, as measured by Definition B, is a given quantity. This problem canalso be solved by means of the Lagrange multiplier technique. Now the maximum value should bedetermined of   P        −    −− ∑∑∑ +==+= 111211* Re2)Re( mk k mk k mk k k   I  A I  I V  µλ  with respect to the currents and the Lagrange multipliers(real) and (complex). This leads to the equations λµ 0),...,1( 11121 ====+= ∑∑ +==+ mk k mk k mk k   I  A I V mk  I V  µµλ  The solution of this set of equations is ∑∑∑ =+++==++ −==−=−== mk kB Bmmk kBmk k mk mk m  I  I mk V V  I  I V V V  1,111121211 ),...,1( λλµ  Hence the currents yielding the maximal active power satisfy ( ) ),...,1( 1121112 mk V V  V V  I  I  mk mk mk mk k kB =−−= +=++= ∑∑  (7)The current is such that the sum of the currents iszero. The maximum active power (and hence theapparent power) equals in this case:  Bm  I  ,1 +  B Bmk k mk mk  B  I V V S  IV =−= ∑∑ ==+ 12121 . (8)Here the magnitude of the currents is computed onlyfrom the components 1 to m , and the voltages are referredto the terminal m+ 1.Interestingly enough, in both cases the apparent power isgiven by the same formal expression as a function of thenorms of the voltage and the current. In the first case thevoltages should be measured with respect to the virtualstar point; the norms of the current and voltage vectors(which are of dimension m +1) should be computed withrespect to the m +1 components. In the second case,however, the last terminal should be taken as thereference for the voltages and the norms of the currentand voltage vectors should not contain the last componentof the vector (or otherwise said the voltage and currentvectors are of dimension m and should not include thecurrent and voltage at terminal m+ 1).As far as the measurement issue is concerned it is clear that the norm of the voltage according to the Definition Bcan be derived from measurements of the voltages between the terminals 1,…, m and terminal m+1. It is39  L’Energia Elettrica - Volume 81 (2004) - “Ricerche”interesting to note that the results corresponding toDefinition A can be rearranged to avoid the physicalrealization of the virtual star point. It is indeed readilychecked that the norm of the voltage according toDefinition A can be rewritten as ∑ += = 112 mk k k   A I  R  (10)where denotes the rsistance of condctor  k.   k   R  Using the Lagrange multiplier technique this is equivalentto the finding the maximum value of the function ∑ ∑ =++= −+= mk mk l l k  A V V m 1112 11 V  (9)     −    −− ∑∑∑ +=+=+= 1111211* Re2)Re( mk k mk k k mk k k   I  A I  R I V  µλ  (11)It is thus expressed as a function of only the voltagedifferences between the terminals without having toconsider (and realize) explicitly the virtual star point.with respect to the currents and the Lagrange multipliers(real) and (complex). This leads to the equations λ µ 2.4 The particular case of a single-phase system 0)1,...,1( 11112 ==+=+= ∑∑ +=+= mk k mk k k k k k   I  A I  Rmk  I  RV  µλ  As a particular case we consider the two-terminal line (or single-phase system). Here the usual approach consists of considering one voltage quantity, the voltage difference between the two terminals, and one current quantity, thecurrent entering the active terminal and returning throughthe other terminal. This leads to the apparent power corresponding to Definition B. It is however readily seenthat for this situation the apparent power does not dependon the definition used. Indeed the norm of the voltage is2 times smaller and the norm of the current is 2times larger if Definition A is used, compared to theresults when Definition B is used. However the product,which is the apparent power, is the same such that thetwo definitions do not lead to different results. Actuallythe reason is that in Definition A the current losses arecharacterized by the losses in both conductors. The lossmay clearly also be characterized by the loss in oneconductor, as in Definition B, by considering a resistancein this single conductor which is the sum of theresistances in both conductors of the two-terminalsystem.The solution of this set of equations is )1,...,1( /)/1( / 1121121111 +=−=−=== ∑∑∑∑ +=+=+=+= mk  RV V  I  I  R RV V V  R RV  k ref k kGmk k k mk k ref k ref mk k k mk k  λλµ  (12)Hence the currents yielding the maximal power satisfy )1,...,1( / 112112 +=    −−= ∑∑ +=+= mk  RV V  RV V  I  R I  k ref k mk k ref k mk k k kG  (13) 3. A more general analysis  The apparent power, the maximal instantaneous power that can be delivered with the given voltage and the givenmagnitude of the current, characterized by the conductor losses or a proportional quantity, is 3.1 Different conductor resistances The analysis of the maximization of the power in the casethat the losses in the return conductor are negligibly smallalready shows that the virtual star point is not an obviousreference point and not always the reference point to beused. An objection to the analysis in the previous sectionis that the two situations considered (identity of theresistances of the neutral conductor and of the other conductors on the one hand, and no resistance in theneutral conductor on the other hand) are extreme cases;the real situation is often somewhere in between.Therefore in the present section a more general analysis isdiscussed, where all conductor resistances may bedifferent. ∑∑ +=+= −= 112112 ./ mk k k mk k ref k G  I  R RV V S   (14)This expression of the apparent power contains a product of a voltage-dependent factor and a current-dependent factor. It is clear from the voltage-dependentfactor that the voltages of the terminals should bereferred to the reference value V  obtained in theabove analysis. Expression (14) contains the two casesconsidered in the previous section as special cases,namely for all conductor resistances equal (includingthe neutral conductor) or for all conductor resistancesequal except the neutral conductor of which theresistance is assumed to be zero. More interpretation of this result is discussed in the next section. In theremainder of the present section we look at this resultfrom a more fundamental point of view. ref    3.2 Apparent power in the general case In that case the maximum value of the active power,given by (1), is required with respect to the currents in theterminals for given voltages and given conductor losses,subject to the conditions that the sum of the currentsvanishes. If we do not assume the conductors to beidentical the conductor losses are given by an expressionof the following form:40
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