Electrical Power and Energy Systems 45 (2013) 369–375

Contents lists available at SciVerse ScienceDirect

Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Electric powers that fulfill Tellegen’s theorem and those that do not: Analysisof consequences

Roberto C. Redondo a,⇑, Norberto R. Melchor a, Félix R. Quintela a, Margarita Redondo b

a Escuela Técnica Superior de Ingenierı́a Industrial, Universidad de Salamanca, 37700 Béjar, Spainb WSP House, 70 Chancery Lane, London WC2A 1AF, United Kingdom

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 August 2011Received in revised form 13 September 2012Accepted 26 September 2012Available online 1 November 2012

Keywords:Power conceptPower measurementPower systemPower system harmonicPower quality

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.09.013

⇑ Corresponding author. Tel.: +34 923408080; fax:E-mail addresses: roberrm@usal.es (R.C. Redondo

chor), felixrq@usal.es (F.R. Quintela), maritareme@gm

Several electric powers were defined during the last century with the intention of being useful in electri-cal power systems, sometimes without considering other physical properties. For this reason, a few ofthese powers may seem limited to specific purposes and distant from the general physical theory ofpower. This paper investigates which of these electrical powers satisfy the most important property ofphysical power, which is the property derived from the energy conservation law. The paper also showsthe usefulness this property confers to the electric powers that possess it and how it is possible to givethose powers broader definitions. Particularly, the power definitions for single and three-phase systemscan be extended to any multi-terminal network providing those powers satisfy the energy conservationlaw. On the contrary, this generalization is not possible for those powers that do not meet the conserva-tion law, and, what is more, some general concepts become confusing when applied to these powers.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

A line of research on electrical power which began in the nine-teenth century [1–3] and continues today [4–7] has been mainlydirected at better quantifying the behavior of electrical power sys-tems and creating concepts to optimize their design [8–10]. Thisinvestigation has produced many definitions of electric powers[11].

The properties that provide general usefulness to the thermody-namic concept of power derive from the energy conservation law,which allows to establish precise quantitative relationships in theexchange of power and energy. For example, the power absorbedby all the elements of a system is the sum of the powers absorbedby its components, regardless of the placement of those compo-nents in the system. Also, the power absorbed by some parts ofan isolated system must be delivered by the other parts of the sys-tem, regardless of where in the system they are located.

Although there are electric powers that fulfill the conservationlaw, others do not. For the former, it is true that the power ab-sorbed by any number of branches of an electrical network is thesum of the powers absorbed by these branches, regardless of theirposition in the network and their connection to one another. Butstatements like this are generally not true for all the electrical

ll rights reserved.

+34 923408127.), norber@usal.es (N.R. Mel-ail.com (M. Redondo).

powers that do not fulfill the conservation law. That is the reasonwhy the objective of this paper is to identify electrical powers withconservation properties, and those without them, and show someconsequences derived from this classification.

2. The energy conservation law and Tellegen’s theorem

The energy conservation law can be expressed as follows: thesum of the powers absorbed by all the elements of an isolated sys-tem is zero. This property requires for some of the non-zero powersabsorbed by the elements of each isolated system to be positive,and for the other non-zero powers to be negative. Negative-ab-sorbed powers are called delivered powers.

An electrical network is a system whose elements are branches.According to Tellegen’s theorem, the sum of the instantaneouspowers absorbed by all the branches of an electrical network iszero [12–14]

Xr

d¼1

pd ¼Xr

d¼1

vdid ¼ 0 ð1Þ

vd and id are the instantaneous voltage and current of the d branch,with their corresponding senses, and r is the number of branches ofthe network. Tellegen’s theorem is considered to be the formadopted by the energy conservation law in electrical networks, be-cause the instantaneous electrical power is a power in the thermo-dynamic sense [15].

(a) (b)A B

CD

4

2

1

3 E

2

-32

-1A B

CD

-1

-1

-1

3 E

3

21

4

Fig. 1. The voltages (a) fulfill KVL, but the currents (b) do not fulfill KCL.Nevertheless, this network fulfills Tellegen’s theorem.

370 R.C. Redondo et al. / Electrical Power and Energy Systems 45 (2013) 369–375

Formula (1) shows that any linear transformation applied to theset of instantaneous powers of the branches of a network providesimages that also satisfy (1). The process of calculating the averagevalues in (1), in the same time interval, is a linear transformation,and from it follows that

Xr

d¼1

Pd ¼ 0 ð2Þ

Pd is the average value of the instantaneous power pd absorbed bythe d branch. Formula (2) states that the sum of the average valuesof the instantaneous power absorbed by the branches of an electri-cal network is zero. Therefore, the instantaneous power, and itsaverage value, satisfy Tellegen’s theorem, whatever the waveformsof the voltages and currents of the network branches may be (e.g.,sinusoidal, nonsinusoidal, periodic or non-periodic). The averagevalue of the instantaneous power of a network with periodic volt-ages and currents is called active power [11]. Therefore, the activepowers of the branches of networks with sinusoidal or nonsinusoi-dal voltages and currents satisfy Tellegen’s theorem.

Notwithstanding the above considerations, Tellegen’s theoremis not just a consequence of the energy conservation law, becauseother products which are not even powers in a thermodynamicsense also satisfy it. Indeed, Tellegen’s theorem is a consequenceof the currents satisfying Kirchhoff’s Current Law (KCL), and thevoltages satisfying Kirchhoff’s Voltage Law (KVL) [12–15]. It turnsout that if the instantaneous currents of the branches of a networkare replaced by their images obtained from any linear transforma-tion, then these images satisfy KCL. Likewise, if the instantaneousvoltages of a network are replaced by their images obtained froma linear transformation, then the images satisfy KVL. So, Tellegen’stheorem continues to be true in the form ’’the products of the lin-early transformed instant values of the voltages and currents of anetwork add up to zero’’, even if such products are not powers inthe thermodynamic sense [15]. For example, the Laplace trans-forms of the voltages and currents of an electrical network satisfyTellegen’s theorem: the sum of the products of these transforms iszero, even though they do not have power dimension – their unit isjoule second. Also the sum of the complex powers absorbed by thebranches of a network with sinusoidal voltages and currents of thesame frequency is zero, since their phasors and the conjugates oftheir phasors are obtained from linear transformations of each vec-tor space of sinusoidal functions of the same frequency in the set ofcomplex numbers. So,

Xr

d¼1

VdI�d ¼Xr

d¼1

Sd ¼ 0 ð3Þ

where Vd and Id are the phasors of vd and id, respectively, providingthat vd and id are sinusoidal functions of the same frequency; I�d isthe conjugate of Id, and Sd is the complex power absorbed by the dbranch. That is, the complex powers absorbed by the branches ofa sinusoidal network also satisfy Tellegen’s theorem. And, therefore,so do their real and imaginary parts

Xr

d¼1

Pd þ jXr

d¼1

Q d ¼ 0 ð4Þ

which are the sums of the active and reactive powers absorbed bythe branches, respectively. The property that the active power addsup to zero was already included in (2), since active powers are theaverage values of instantaneous powers.

Therefore, the instantaneous power and its average value – irre-spectively of the waveform of the voltage and current – the tradi-tional complex, active and reactive powers – providing that thevoltage and current waveforms are sinusoidal of the same fre-quency – and the active power – for periodic nonsinusoidal wave-

forms – they all meet Tellegen’s theorem: the sum of the powersabsorbed by the branches of a network is zero.

The practical importance of other powers meeting Tellegen’stheorem, beside the instantaneous and active powers, is well pro-ven in electrical systems. For example, it is possible to prevent agenerator from delivering the reactive power absorbed by theloads by making other parts of the system deliver it. This is calledreactive power compensation [16–22]. But also, if a power does notsatisfy Tellegen’s theorem, then the power delivered by the gener-ator will not necessarily be nullified by making a section of the net-work deliver the power absorbed by other part of the network.

As mentioned above, Tellegen’s theorem is a consequence ofboth the currents and the voltages fulfilling the two Kirchhoff laws[12–15], but this is not a necessary condition for the sum of theirproducts to be zero. For example, Fig. 1a shows voltages that fulfillKVL, and Fig. 1b indicates values as currents that do not fulfill KCL– the values of nodes A or B do not add up to zero. But the sum oftheir products is

3�3þð�1Þ4þð�1Þ2þð�1Þ1þ4ð�1Þþ3�2þ2ð�3Þþ1�2¼0

That is, the network in this example fulfills Tellegen’s theorem,even though its currents do not satisfy KCL. For this reason, show-ing that a current associated with an electrical power does not ful-fill KCL is not enough to ensure that the power does not satisfyTellegen’s theorem.

3. Instantaneous active and reactive powers

In a network with sinusoidal voltages and currents the instanta-neous power absorbed by the d branch is [11]

pd ¼ vdid ¼ pda þ pdq ð5Þ

where vd ¼ffiffiffi2p

Vd sinðxt � adÞ and id ¼ffiffiffi2p

Id sinðxt � ad � hdÞ arethe voltage and current of the d branch. The initial phase ad allowsvoltages of different branches to have different phases.

pda ¼ Pd½1� cosð2xt � 2adÞ� ð6Þ

is the instantaneous active power, pdq = �Qdsin(2xt � 2ad) is theinstantaneous reactive power, Pd = VdIdcoshd is the active power, andQd = VdIdsinhd is the reactive power absorbed by the d branch [11].

The instantaneous power pd satisfies Tellegen’s theorem. Butneither the instantaneous active power nor the instantaneous reac-tive power do. This statement will be verified on Fig. 2, which couldbe thought of as a generator of voltage v ¼

ffiffiffi2p

V sinðxtÞ that pro-vides power to a resistive load R through a single-phase line ofinductive reactance X. The current is i ¼

ffiffiffi2p

I sinðxt � hÞ, whereh = arctg(X/R). The voltage in X is vX ¼

ffiffiffi2p

XI sinðxt � hþ p=2Þ,and the voltage in R is vR ¼

ffiffiffi2p

RI sinðxt � hÞ. The instantaneous ac-tive powers absorbed by the branches of the network in Fig. 2 arepXa = 0[1 � cos(2xt � 2h + p)] = 0 in the inductance, pRa = RI2[1 -� cos(2xt � 2h)] in the resistance and pVa = �RI2[1 � cos(2xt)] in

v

+

A

B

R

X

i

Fig. 2. The generator delivers energy to a load R through a line of reactance X.

pXa + pRa+ pVa

pRa

pXa

pVa

Fig. 3. Instantaneous active powers absorbed by the network branches of Fig. 2 forh = 30�, and their sum.

pXq

pVq

pXq+pRq+pVq

pRq

Fig. 4. Instantaneous reactive powers absorbed by the network branches of Fig. 2for h = 30�, and their sum.

R.C. Redondo et al. / Electrical Power and Energy Systems 45 (2013) 369–375 371

the voltage source. It was taken into account that the active powerPX absorbed by X is zero, the one absorbed by R is PR = RI2, and thevoltage source absorbs PV = �RI2, because the active power doessatisfy Tellegen’s theorem: PX + PR + PV = 0. Fig. 3 shows the sumpXa + pRa + pVa, which is not zero.

Therefore, in a network where voltages and currents are sinu-soidal of the same frequency, the instantaneous active power doesnot satisfy Tellegen’s theorem. That is, in general

Xr

d¼1

pda – 0

Moreover, from (1) and (5) one obtains

Xr

d¼1

pd ¼Xr

d¼1

pda þXr

d¼1

pdq ¼ 0

which also means that, generally

Xr

d¼1

pdq – 0

The instantaneous reactive power does not satisfy Tellegen’s theo-rem either.

It is possible to verify this statement in the network of Fig. 2:the instantaneous reactive powers absorbed by the branchesare pXq = �XI2sin(2xt � 2h + p) in the inductance, pRq =0sin(2xt � 2h) = 0 in the resistance and pVq = XI2sin(2xt) in thevoltage source. It was taken into account that the reactive powerabsorbed by X is QX = XI2, the one absorbed by R is QR = 0, andthat the voltage source absorbs QV = �XI2, because the reactivepower does satisfy Tellegen’s theorem: QX + QR + QV = 0. Fig. 4 showsthe sum pXq + pRq + pVq, which is not zero.

The intrinsic power absorbed by the d branch [11] is one of thesummands of (6):

�Pd cosð2xt � 2adÞ

From (6), it follows that the sum of the instantaneous activepowers absorbed by all branches of a network is

Xr

d¼1

pda ¼Xr

d¼1

Pd �Xr

d¼1

Pd cosð2xt � 2adÞ– 0

AsPr

d¼1Pd ¼ 0, it is generally true that

Xr

d¼1

pda ¼Xr

d¼1

Pd cosð2xt � 2adÞ – 0 ð7Þ

That is, the sum of the intrinsic powers absorbed by thebranches of a network is not zero: the intrinsic power does not sat-isfy Tellegen’s theorem either. In fact, the sum of the intrinsic pow-ers of all branches of a network matches the sum of theinstantaneous active powers of the same branches, which doesnot satisfy the theorem either. Note also that this sum is a sinusoi-dal function of angular frequency 2x, since it is the sum of sinusoi-dal functions of that frequency (Fig. 3). Similarly, the sum of theinstantaneous reactive powers of all branches of the network is an-other sinusoidal function of angular frequency 2x (Fig. 4). Thesetwo conclusions will be used immediately.

4. Powers of networks with nonsinusoidal periodic voltages andcurrents

In a network with nonsinusoidal periodic voltages and cur-rents, the instantaneous voltage and current of each d branch arevd = vd1 + vdH, and id = id1 + idH, where vd1 ¼

ffiffiffi2p

Vd1 sinðxt � ad1Þ,and id1 ¼

ffiffiffi2p

Id1 sinðxt � ad1 � hd1Þ are the power system frequencycomponents, and vdH ¼ Vd0 þ

ffiffiffi2p P

h>1Vdh sinðhxt � adhÞ, andidH ¼ Id0 þ

ffiffiffi2p P

h>1Idh sinðhxt � adh � hdhÞ, are the remaining terms[11].

The instantaneous voltages vd of the branches of the networkfulfill KVL: their addition in each closed path is zero. But the sumof two sinusoidal functions of different frequencies with nonzeromaximum values could never be zero, nor could be zero a constantplus a sinusoidal function with nonzero maximum value. There-fore, for the nonsinusoidal instant voltages of each closed path toadd up to zero, each of the following sums must be zero:X

d

Vd0 ¼ 0;X

d

vd1 ¼ 0; andX

d

vdh ¼ 0

Each summation is obtained from each closed path of the net-work (KVL). vdh ¼

ffiffiffi2p

Vdh sinðhxt � adhÞ is every harmonic h > 1from vd. Therefore, in a nonsinusoidal periodic network the directvoltage terms, the power system frequency components of thevoltage, and the voltage harmonics of the same order h, each fulfillseparately KVL. And, as a result, the remaining terms of the voltagealso satisfy KVL:X

d

vdH ¼ 0

A similar reasoning leads to the conclusion that the direct cur-rent terms, the power system frequency current components, thecurrent harmonics of the same order h, and the remaining termsof the current satisfy KCL, since the addition of the instantaneouscurrents entering each node is zero.

(a) (b)

u2

ut

u1 j1

j2

jtid

d-branch

vd 2

t

1

••

idd-branch

vd•

•

••

O

372 R.C. Redondo et al. / Electrical Power and Energy Systems 45 (2013) 369–375

The consequence of these voltages and currents satisfying KVLand KCL respectively, is that the products of these voltages andtheir corresponding currents fulfill Tellegen’s theorem. Inparticular,

Xr

d¼1

Vd0Id0 ¼ 0 ð8Þ

Xr

d¼1

vd1id1 ¼Xr

d¼1

pd1 ¼ 0 ð9Þ

Xr

d¼1

vdhidh ¼Xr

d¼1

pdh ¼ 0 ð10Þ

Xr

d¼1

vdHidH ¼ 0 ð11Þ

But other combinations also fulfill Tellegen’s theorem [13]:

Xr

d¼1

Vd0id1 ¼ 0; andXr

d¼1

vd1idH ¼ 0

Eq. (9) is the sum of the instantaneous powers absorbed by thebranches due to the power system frequency components of thevoltage and current, and (10) is the sum of the instantaneous pow-ers due to the h harmonic. If the average values are calculated in(9), then the result is the sum of the fundamental active powers ab-sorbed by the network branches [11], which is also zero:

Xr

d¼1

Pd1 ¼Xr

d¼1

Vd1Id1 cos hd1 ¼ 0 ð12Þ

Eq. (12) shows that the fundamental active powers satisfy Telle-gen’s theorem.

If the average values of (10) are calculated, then

Xr

d¼1

Pdh ¼Xr

d¼1

VdhIdhcoshdh ¼ 0 ð13Þ

Each Pdh is the average value of the power absorbed by the dbranch due to the harmonic h, and also fulfills Tellegen’s theorem.

The active power Pd absorbed by the d branch is Pd = Pd1 + PdH,where Pd1 is the fundamental active power and PdH is the harmonicactive power (or nonfundamental active power) [10] that the dbranch absorbs. (2) shows that their sum for the r branches ofthe network is zero,

Xr

d¼1

Pd ¼ 0 ¼Xr

d¼1

Pd1 þXr

d¼1

PdH

And, taking (12) into account, it follows that

Xr

d¼1

PdH ¼ 0 ð14Þ

That is to say, the harmonic active powers absorbed by the networkbranches satisfy Tellegen’s theorem.

The instantaneous power that the d branch absorbs is

pd ¼ pda þ pdq ð15Þ

where pda ¼ Vd0Id0 þP

hPdh½1� cosð2hxt � 2adhÞ� is the instanta-neous active power absorbed by the d branch. The summation forall the r branches of a network is

Xr

d¼1

pda ¼Xr

d¼1

Vd0Id0 þXr

d¼1

Xh

Pdh �Xr

d¼1

Xh

Pdh cosð2hxt � 2adhÞ ð16Þ

From (8) and (13) it follows thatPr

d¼1Vd0Id0 ¼ 0 andPrd¼1

PhPdh ¼ 0, respectively. If the order of the summations is

changed in the last term of (16), then:

Xr

d¼1

Xh

Pdhcosð2hxt � 2adhÞ ¼X

h

Xr

d¼1

Pdhcosð2hxt � 2adhÞ

From (7) it results that for every harmonich;Pr

d¼1Pdh cosð2hxt � 2adhÞ is a sinusoidal function of angular fre-quency 2hx. Since the sum of sinusoidal functions of different fre-quencies and non-zero maximum value is never zero, in (16) itresults that generally

Xr

d¼1

pda ¼X

h

Xr

d¼1

Pdh cosð2hxt � 2adhÞ !

– 0:

In other words, the instantaneous active power of nonsinusoi-dal periodic networks does not satisfy Tellegen’s theorem either.

If (15) is added for all branches of the network, then it resultsthat

Xr

d¼1

pd ¼Xr

d¼1

pda þXr

d¼1

pdq

AsPr

d¼1pd ¼ 0 and, usually,Pr

d¼1pda – 0, it follows that,generally

Xr

d¼1

pdq – 0 ð17Þ

The instantaneous reactive power of periodic nonsinusoidalnetworks does not fulfill Tellegen’s theorem either. Additionally

Xr

d¼1

pda ¼ �Xr

d¼1

pdq

5. Tellegen’s theorem and the power of multi-terminalnetworks

The instantaneous electric power absorbed by a t-terminal net-work is the addition of the instantaneous electric powers absorbedby all the r branches that form it, which hereafter shall be calledinternal branches (Fig. 5a):

p ¼Xr

d¼1

pd ¼Xr

d¼1

vdid ð18Þ

That power coincides with the sum of the products

p ¼Xt

k¼1

ukjk ð19Þ

where uk is the voltage between the k terminal and any potentialreference, and jk is the current entering the t-terminal networkthrough the k terminal (Fig. 5a).

The equality

p ¼Xr

d¼1

vdid ¼Xt

k¼1

ukjk ð20Þ

Fig. 5. (a) t-terminal network and (b) p is measured with t meters.

2

t

1

idd-branch

vd•

•

••

Fig. 6. Measuring p with t � 1 meters.

v

+

A

B

R

X

i

Fig. 7. The instantaneous active power absorbed by the two-terminal network isnot the addition of the instantaneous active powers absorbed by its internalbranches.

R.C. Redondo et al. / Electrical Power and Energy Systems 45 (2013) 369–375 373

shall be called hereafter multi-terminal network power theorem[15,23].

Eq. (20) allows to measure the instantaneous power absorbedby a t-terminal network with t meters located at its terminals(Fig. 5b). Each product pk = ukjk from (19) or (20) depends on thepotential reference point O chosen to determine every uk, but theiraddition p ¼

Ptk¼1pk ¼

Ptk¼1ukjk is independent of that potential

reference [14,15]. Therefore, in order to measure the power ab-sorbed by a t-terminal network, any point can be used as potentialreference, particularly the t terminal. Then (19) and (20) become

p ¼Xt�1

k¼1

ukjk ð21Þ

p ¼Xr

d¼1

vdid ¼Xt�1

k¼1

ukjk ð22Þ

Now uk is the voltage between each k – t terminal and the t ter-minal. (21) and (22) show that p can also be measured with t � 1meters (Fig. 6).

If a power fulfills Tellegen’s theorem it also fulfills the multi-terminal network power theorem [12,15,23]. Thus, the average va-lue of the power absorbed by a multi-terminal network is

P ¼Xr

d¼1

Pd ¼Xt

k¼1

Pk ð23Þ

whatever the waveforms of the voltages and currents may be. EachPd is the average value of the instantaneous power absorbed by thed internal branch of the multi-terminal network, and Pk is the aver-age value of each ukjk product, which are the voltage and current ofthe k terminal of the multi-terminal network. If the voltages andcurrents are periodic, then Pd is the active power absorbed by thed internal branch of the multi-terminal network and P is the activepower absorbed by the multi-terminal network. If the voltagesand currents are sinusoidal of the same frequency, each Pk is Pk =UkJkcoshk. Where Uk and Jk are the root-mean-square (rms) valuesof uk and jk, respectively. hk is the phase difference between uk

and jk. (20) also holds for complex powers:

S ¼Xr

d¼1

Sd ¼Xt

k¼1

UkJ�k ¼Xt

k¼1

UkJk cos hk þ jXt

k¼1

UkJk sin hk

¼Xt

k¼1

Pk þ jXt

k¼1

Qk ¼ P þ jQ ð24Þ

and, therefore, for the reactive powers:

Q ¼Xr

d¼1

Q d ¼Xt

k¼1

Q k ð25Þ

Qd is the reactive power absorbed by the d internal branch.If the voltage and current are nonsinusoidal periodic wave-

forms, then (23)–(25) hold true for the fundamental term and foreach harmonic h. For example, the fundamental active power andthe fundamental reactive power [11] absorbed by a t-terminal net-work are

P1 ¼Xr

d¼1

Pd1 ¼Xr

d¼1

Vd1Id1 cos hd1 ¼Xt

k¼1

Uk1Jk1 cos hk1 ¼Xt

k¼1

Pk1 ð26Þ

Q1 ¼Xr

d¼1

Q d1 ¼Xr

d¼1

Vd1Id1 sin hd1 ¼Xt

k¼1

Uk1Jk1 sin hk1 ¼Xt

k¼1

Q k1 ð27Þ

Similarly for the active and reactive powers of each harmonic,and for the harmonic active power (or nonfundamental activepower).

If any terminal, such as the t terminal, is taken as common ref-erence of potentials, then one may replace t for t � 1 in all formulasfrom (23)–(27), which means that each power absorbed by a multi-terminal network can be measured with t � 1 meters (Fig. 6).

So, the powers that satisfy the multi-terminal network powertheorem have an important property: that, by using the corre-sponding second members of (20) or (22), the instantaneous powerabsorbed by a multi-terminal network can be defined as the sum ofthe instantaneous powers absorbed by the internal branches of thenetwork. But also, by using the third members of (20) or (22), it canbe defined as the sum of the products of each terminal potential,which is referred to any common reference of potentials, and thecurrent of said terminal. This property gives unambiguous identityto the concept of power absorbed by a multiterminal network.

However, since the instantaneous active and reactive powers ofnetworks with periodic voltages and currents do not satisfy themulti-terminal network power theorem expressed by (20) or(22), both definitions are not equivalent in this case. For example,the two-terminal network of Fig. 7 is obtained from the right sideof AB in Fig. 2. If vAB ¼

ffiffiffi2p

V sinðxtÞ, then the sum of the instanta-neous active powers absorbed by the internal branches of the two-terminal network is pa = pXa + pRa = 0 + RI2[1 � cos(2xt � 2h)], andthe instantaneous active power the network absorbs through itsterminals is pABa = RI2[1 � cos(2xt)], where h = arctg(X/R). It resultsthat pa – pABa. So, which one of the two powers is actually theinstantaneous active power absorbed by the branch to the rightof AB?

Similarly, the sum of the instantaneous reactive power ab-sorbed by the internal branches of the two-terminal network ispq = �XI2sin(2xt � 2h + p), and the instantaneous reactive powerabsorbed by the two-terminal network through its terminals ispABq = �XI2sin(2xt).

It results that pq – pABq.So, for those powers that do not fulfill Tellegen’s theorem, these

inequalities force to choose between two nonequivalent defini-tions of power absorbed by a multi-terminal network: either as thesum of the power absorbed by its internal branches, or as thepower measured in its terminals. But, in the latter case, the mea-sure also depends on the point used as potential reference, so thisdefinition must also include said point.

6. Application to three-phase systems

As the instantaneous power satisfies Tellegen’s theorem, theinstantaneous power absorbed by any three-or-four-terminal net-

r

a

b

c

(a)

r

a

b

c

(b)

n

••

••

••

••

••

••

••

Fig. 9. Measurement of the powers that fulfill Tellegen’s theorem using an arbitrarypotential reference r.

374 R.C. Redondo et al. / Electrical Power and Energy Systems 45 (2013) 369–375

work, three-phase loads included, is the sum of the instantaneouspowers absorbed by its internal branches. If the receiver has threewires, a, b, and c, then that power coincides with

p ¼ vabia þ vcbic ¼ vbaib þ vcaic ¼ vacia þ vbcib ð28Þ

The second member of the equality results from using the b ter-minal as potential reference (Fig. 8a), the third member from usinga, and the last member from using c. (28) is valid whatever thevoltage and current waveforms may be, including sinusoidal,nonsinusoidal or constant waveforms, and balanced and unbal-anced three-phase systems.

Likewise, for a receiver with four-wires, a, b, c, and n,

p ¼ vabia þ vcbic þ vnbin ¼ vbaib þ vcaic þ vnain

¼ vacia þ vbcib þ vncin ¼ vania þ vbnib þ vcnic ð29Þ

The second member of the equality results from using the b wire aspotential reference (Fig. 8b), and the last member from using the nwire. (29) is valid whatever the voltage and current waveforms maybe.

The positive sense for all currents of (28) and (29), in included, istowards the multi-terminal network.

If any other point r is used as potential reference, the instanta-neous power for a three-wire receiver is (Fig. 9a)

p ¼ varia þ vbrib þ vcric ð30Þ

And for a four-wire receiver (Fig. 9b)

p ¼ varia þ vbrib þ vcric þ vnrin ð31Þ

whatever the voltage and current waveforms may be – that is, sinu-soidal or nonsinusoidal – or whether the three-phase system mightbe balanced or not. Note the need for vnrin in (31).

Formulas (28)–(31) can be applied to any power that satisfiesTellegen’s theorem. For example, the fundamental active powerP1 absorbed by a three-phase receiver is the sum

Prd¼1Pd1 of the

fundamental active powers absorbed by its r internal branches.According to (26), for a receiver with three wires, a, b, and c, andusing b as a potential reference,

P1 ¼ Vab1Iab1 cos hab1 þ Vcb1Icb1 cos hcb1 ð32Þ

Similarly for the rest of the powers that fulfill the multi-termi-nal network power theorem, which are also the ones that satisfyTellegen’s theorem.

7. Apparent power and other powers

The apparent power of each branch of a network is Sd = VdId.Each Sd is a positive real number because the rms values Vd andId of the instantaneous voltage and current of the branches are po-sitive real numbers. Therefore, the apparent power does not fulfillTellegen’s theorem, since the addition of positive numbers cannotbe zero. Neither does it fulfill the multi-terminal network powertheorem, that is

(a)

idd-branch

vd

••

••

a

b

c

(b)

idd-branch

vd

••

••

a

b

c

n•

•

Fig. 8. Any point can be used as potential reference in order to measure the powersthat satisfy Tellegen’s theorem. In these two cases the reference is the b terminal.

Xr

d¼1

Sd –Xt

k¼1

VkIk ð33Þ

Vk and Ik are the rms values of the voltage and current of the k ter-minal. In addition, the second member of (33) is not independent ofthe potential reference. The same is true for all power obtained as aproduct of rms values, as are the fundamental, nonfundamental,harmonic and effective apparent powers, and current and voltagedistortion powers.

The vector apparent power can be defined for any multi-terminalnetwork as the modulus of its complex power:

SV ¼Xr

d¼1

Sd

���������� ¼

Xt

k¼1

VkI�k

���������� ¼ jP þ jQ j ð34Þ

Sd is the complex power absorbed by the internal d branch. This def-inition includes the vector apparent power for three-phase sinusoi-dal systems [11]. SV can be obtained for any multi-terminal networkby measuring the complex power at its terminals, and the resultwill be independent of the potential reference.

The definition of the arithmetic apparent power is specific forthree-phase sinusoidal unbalanced systems [11]

SA ¼ VaIa þ VbIb þ VcIc ð35Þ

SA depends on the potential reference, so its definition must includewhat that reference is, that is, what points must be used to measurethe rms values Va, Vb, and Vc.

8. Conclusions

The amount of defined electric powers is already considerable,but not all of these powers have similar physical properties. So,well-established statements considered true for many electricpowers, may not be true for others.

This paper has examined definitions of electric powers in orderto find out which of these powers satisfy the conservation propertyexpressed by Tellegen’s theorem. If a power satisfies Tellegen’stheorem, it means that the receivers that absorb this power fromthe electrical systems require the generator to deliver it, whichin some cases leads to increasing energy losses. It is possible toavoid this situation by designing compensation devices that deliverthose powers and that could be placed near the receivers, in thesame way it is already being done for reactive power compensa-tion. But this compensation technique cannot be generally appliedto powers that do not satisfy Tellegen’s theorem: introducing a de-vice in a network in order to deliver one of these powers does notmean that another part of the network will not deliver that powertoo, because the sum of these powers absorbed by the branches ofa network is not zero, so the network as a whole can absorb – ordeliver – a non-zero amount of any power that does not fulfill Telle-gen’s theorem.

R.C. Redondo et al. / Electrical Power and Energy Systems 45 (2013) 369–375 375

Moreover, the powers that fulfill Tellegen’s theorem also fulfillthe multi-terminal network power theorem. This allows to defineunambiguously the power absorbed by any multi-terminal net-work as the sum of the powers that its internal branches absorb,which coincides with the sum of the powers measured in its termi-nals, independently of the chosen potential reference, which, inturn, leads to the ability to measure this power with t or t � 1 me-ters. In general, this is not the case for powers that do not fulfillTellegen’s theorem.

All of the powers examined in this paper meet Tellegen’s theo-rem, except the instantaneous active and reactive powers and theintrinsic power, as well as those which are products of rms volt-ages and rms currents – such as the apparent powers. For all theseexceptions, the power absorbed by a multi-terminal network hasto be defined either as the sum of the powers absorbed by its inter-nal branches, or the sum of the powers absorbed by its terminals,since these two sums are not the same. Furthermore, in the lattercase the power depends on the potential reference, so that refer-ence must be included in the definition.

Since three-phase systems are a particular case of multi-termi-nal networks, all the above conclusions also hold true for three-phase systems. Particularly, instantaneous active and reactivepowers, and apparent powers, require the voltage reference to beclearly established for the phases and neutral voltage measure-ments involved in the formulas of those powers, especially inunbalanced three-wire three-phase systems.

References

[1] Shallenberger OB. The energy of alternating currents. Electr World1888:114–5.

[2] Stanley Jr WM. Phenomena of retardation on the retardation in the inductioncoil. Am Inst Electr Eng 1888;5(4).

[3] Steinmetz CP, with the assistance of Berg EJ. Theory and calculation ofalternating current phenomena. New York: The W.J. Johnston Co.; 1897.

[4] Preliminary Report of Special Joint Committee. Power factor in polyphasecircuits. J AIEE 1920:1449–50.

[5] Proceeding of the international workshop in power definitions undernonsinusoidal conditions. Como, Italy: Polytechnic of Milan and IEEE NorthItaly; 1991 [chapter].

[6] IEEE Working Group on Nonsinusoidal Situations. Practical definitions forpower systems with nonsinusoidal waveforms and unbalanced loads: adiscussion. IEEE Trans Power Del 1996;11:79–87.

[7] Morsi WG, El-Hawary ME. Defining power components in nonsinusoidalunbalanced polyphase systems: the issues. IEEE Trans Power Del2007;22:2428–38.

[8] Akagi H, Watanabe EH, Aredes M. Instantaneous power theory andapplications to power conditioning (IEEE press series on powerengineering). New York: Wiley-IEEE Press; 2007.

[9] Kawadya TA, Taalaba AI, Ahmedb ES. Dynamic performance of the powerdifferential relay for transmission line protection. Int J Electr Power EnergySyst 2010;32:390–7.

[10] Singha B, Jayaprakashb P, Kotharic DP. New control approach for capacitorsupported DSTATCOM in three-phase four-wire distribution system undernon-ideal supply voltage conditions based on synchronous reference frametheory. Int J Electr Power Energy Syst 2011;33:1109–17.

[11] IEEE standard definitions for the measurement of electric power quantitiesunder sinusoidal, nonsinusoidal, balanced, or unbalanced conditions, vol.1459. IEEE Std; 2010.

[12] Tellegen BDH. A general network theorem, with applications. Philips Res Rep1952;7:259–69.

[13] Penfield Jr P, Spence R, Duinker S. A generalized form of Tellegen’s theorem.IEEE Trans Circ Theory 1970;17:302–5.

[14] Penfield Jr P, Spencer R, Duinker S. Tellegen’s theorem and electricalnetworks. Cambridge: MIT Press; 1970.

[15] Quintela FR, Redondo RC, Melchor NR, Redondo M. A general approach toKirchhoff’s laws. IEEE Trans Educ 2009;52:273–8.

[16] Lee S, Wu CJ, Chang WN. A compact control algorithm for reactive powercompensation and load balancing with static VAr compensator. Elect PowerSyst Res 2001;58:63–70.

[17] Quintela FR, Arévalo JMG, Redondo RC. Power analysis of static VArcompensators. Int J Electr Power Energy Syst 2008;30:376–82.

[18] Quintela FR, Arévalo JMG, Redondo RC, Melchor NR. Four-wire three-phaseload balancing with Static VAr Compensators. Int J Electr Power Energy Syst2011;33:562–8.

[19] Quintela FR, Redondo RC, Arévalo JMG, Melchor NR. Single-phase powersupply to balanced three-phase loads through SVAr compensators. Int J ElectrPower Energy Syst 2011;33:715–20.

[20] Pires DF, Antunes CH, Martins AG. NSGA-II with local search for a multi-objective reactive power compensation problem. Int J Electr Power Energy Syst2012;43:313–24.

[21] Sirjani R, Mohamed A, Shareef H. Optimal allocation of shunt VArcompensators in power systems using a novel global harmony searchalgorithm. Int J Electr Power Energy Syst 2012;43:562–72.

[22] Roy PK, Ghoshal SP, Thakur SS. Optimal VAr control for improvements involtage profiles and for real power loss minimization using biogeographybased optimization. Int J Electr Power Energy Syst 2012;43:830–8.

[23] Quintela FR, Melchor NR. Multi-terminal network power measurement. Int JElectr Eng Educ 2002;39:148–61.