1

Dissipation of Oscillation Energy and Distributionof Damping Power in a Multimachine Power

System: A Small-signal AnalysisKaustav Chatterjee, Student Member, IEEE and Nilanjan Ray Chaudhuri, Senior Member, IEEE

Abstract—This paper revisits the concept of damping torque ina multimachine power system and its relation to the dissipationof oscillation energy in synchronous machine windings. As amultimachine extension of an existing result on a single-machine-infinite-bus (SMIB) system, we show that the total dampingpower for a mode stemming from the interaction of electromag-netic torques and rotor speeds is equal to the sum of averagepower dissipations in the generator windings corresponding tothe modal oscillation. Further, counter-intuitive to the SMIBresult, we demonstrate that, although the equality holds onan aggregate, such is not the case for individual machinesin an interconnected system. To that end, distribution factorsare derived for expressing the average damping power of eachgenerator as a linear combination of average powers of modalenergy dissipation in the windings of all machines in the system.These factors represent the distribution of damping power ina multimachine system. The results are validated on IEEE 4-machine and 16-machine test systems.

Index Terms—Damping torque, damping power, multimachinesystem, oscillation energy dissipation, synchronous machines

I. INTRODUCTION

IN small-signal analysis, stability of a power system underelectromechanical oscillations is determined by studying

the eigenvalues of the system linearized around the quasi-steady-state operating point or by performing modal estimationon the response variables. In either case, the damping ratiosobtained indicate the margin of system stability, but do notquantify the damping contribution from individual sources.In that regard, the small-signal representation of a genera-tor’s electromagnetic torque as a phasor in its synchronouslyrotating rotor speed-angle reference frame offers geometricintuition into decomposing the torque into its damping andsynchronizing components. Conceptualized by Park in his1933 paper [1] and furthered by Concordia [2], [3], Shepherd[4], and notable others [5]–[9], the damping and synchronizingtorque coefficients contribute towards insightful understandingof the stabilizing contributions coming from the machine andits associated governor and excitation systems. Consequently,some of the early designs of power system stabilizers fordamping oscillations have evolved out of these notions. How-ever, historically, these studies on damping torque have beenpresented either considering a SMIB system or by reducingthe system to a linearized single-machine equivalent. Unlikea SMIB system, damping torque of a generator in a multima-chine system depends not only on its own speed-deviation but

K. Chatterjee and N. R. Chaudhuri are with The School of Electrical Engi-neering and Computer Science, The Pennsylvania State University, UniversityPark, PA 16802, USA (e-mail: kuc760@psu.edu; nuc88@psu.edu)

Financial support from the NSF grant under award CNS 1739206 isgratefully acknowledged.

also on that of the other machines, their excitation systems,and the overall network structure and parameters. Also, theanalytical modeling of these differ in literature, for instance,authors in [10] model damping torque as a higher degreepolynomial in speed-deviations, in contrast to linear terms in[11]. The 1999 IEEE task force report investigating modelingadequacy for representing damping in multimachine stabilitystudies [12] identified eight different sources of dampingand recommended abstracting their contributions into a singleretarding torque in the swing equation of each generator.

Damping torque, thus, over the years, has largely remaineda conceptual tool for analyzing stability in power systems.Although some of the initial works listed before highlightedan intuitive link between damping torque and dissipation ofoscillation energy, it is only in the recent works [13] and[14] that a rigorous mathematical connection between thetwo has been established for a SMIB system. However, formultimachine systems such a connection is yet to be confirmed– in this paper, we make a maiden attempt to fill this gap.To that end, we use a simplified mathematical model formultimachine systems to establish an equivalence between theaverage power dissipation due to the damping torques on therotors and the average rates of oscillation energy dissipationin the machine windings. In this context, we make a noteof [15], where claims regarding the consistency of dampingand dissipation coefficients in multimachine systems are madebased on presupposition of this equality without any formalproof. Going ahead, in the paper we also demonstrate thatthe damping power of each machine stemming from theinteraction of its own speed and torque, is derived in partsfrom the rates of energy dissipation in the windings of allmachines, over and above it’s own winding.

At this point, it is important to clarify that the focus ofthis paper is not on improving the algorithmic tools of [14] or[13] for perfecting the science of locating oscillation sourcesor to present an alternate path for doing the same, but toderive further analytical insights from the findings of thesetwo seminal papers. Our primary intention is to bridge amathematical connection between the concept of transientenergy dissipation, as introduced in [14], and the notion ofdamping torque, which is nearly a century old.

The contributions of the paper are as follows: (1) we developa phasor-based small-signal-formulation for calculating themode-wise average powers (i.e. damping powers) of elec-tromechanical oscillation due to the interaction of dampingtorque and speed in each machine; (2) using this framework fora simplified system model with lossless transmission networkand constant power loads, we extend the SMIB results in [14]

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to show that the total damping power for a mode is equal to thesum of average power dissipations in the generator windingscorresponding to the modal oscillation; and finally, (3) counter-intuitive to the SMIB result in [14], we demonstrate that theaforementioned equality does not hold for individual machinesin an interconnected system – in fact the damping power ineach machine can be expressed as a weighted linear combina-tion of power dissipation in windings of different generators.These weighing factors (called ‘distribution factors’ in thepaper) are analytically derived − which essentially describethe participation of the power dissipation in different machinesin constituting the damping power of each generator.

To that end, in the next section, we derive a linearizedrepresentation of the simplified system mentioned earlier witha third-order synchronous generator model. Building on thismodel, we present contributions (1)−(3) in Sections III-V,which are followed by case studies on IEEE 4-machine and16-machine test systems in Section VI to validate the claims– both for the simplified model used in derivation, and forsystems with detailed machine models. Finally, concludingremarks are presented in Section VII.

Notations: Superscripts T , ∗, and H are respectively thetranspose, conjugate, and Hermitian operators. <{·} and ={·}denote the real and imaginary parts of a complex entity.

II. SIMPLIFIED SYSTEM MODEL: LINEARIZEDREPRESENTATION

Consider a n−bus transmission system of which, withoutloss of generality, first ng are designated as generator buses.The network is lossless and each synchronous generator isdescribed by a third-order machine model capturing the elec-tromechanical dynamics of the rotor and the field flux. Inaddition, assume manual excitation for the generators andconstant power loads at all buses.

The differential equations describing the dynamics of eachgenerator are same as those given in eqns (6.132) − (6.134) of[11]. Further, the stator and network algebraic equations canbe obtained from eqns (6.142) − (6.144). Since we assume alossless network, in eqns (6.143) and (6.144) of [11], ∝ik=π/2 for i 6= k, and ∝ii= −π/2. Unless specified otherwise,all symbols have their usual meanings as in [11].

We eliminate the stator algebraic equations by substitutingIdi and Iqi obtained from eqn (6.142) into eqns (6.143) and(6.144) of [11] − with stator resistances neglected. This leavesus with 3ng differential equations and 2n algebraic equationsas functions of state variables δi, ωi, and E′qi , for i = 1 . . . ng ,and algebraic variables θi and Vi, for i = 1 . . . n as describedbelow

δi = ωi − ωs (1)

ωiωs

=Tmi2Hi−E′qiVi sin(δi − θi)

2Hi x′di+V 2i sin 2(δi − θi)

4Hi

(xqi − x′dixqix

′di

)(2)

E′qi =EfdiT ′doi

−E′qiT ′doi−E′qi − Vi cos(δi − θi)

x′di

(xdi − x′diT ′doi

)

for i = 1, . . . ng

(3)

0 = fi =

E′qiVi sin(δi − θi)x′di

− V 2i sin 2(δi − θi)

2

(xqi − x′dixqix

′di

)+ PLi −

n∑k=1,k 6=i

ViVkYik sin(θi − θk)

for i = 1, . . . ng

PLi −n∑

k=1,k 6=i

ViVkYik sin(θi − θk)

for i = ng + 1, . . . n

(4)

0 = gi =

E′qiVi cos(δi − θi)x′di

− V 2i cos2(δi − θi)

x′di+QLi − V

2i Yii

−V2i sin2(δi − θi)

xqi+

n∑k=1,k 6=i

ViVkYik cos(θi − θk)

for i = 1, . . . ng

QLi +

n∑k=1,k 6=i

ViVkYik cos(θi − θk)− V 2i Yii

for i = ng + 1, . . . n(5)

Linearizing (1) − (5) around an operating point, with Vi0as the voltage magnitude of bus i at that point and defining anew variable νi = Vi/Vi0 , we obtain ∆δ

∆ω

∆E′q

= M

∆δ∆ω

∆E′q

+ N

[∆θ∆ν

]+ B

[∆Tm

∆Efd

][

00

]= C

∆δ∆ω

∆E′q

+ D

[∆θ∆ν

] (6)

where, δ, ω, E′q , θ, and ν are the vectorized state and

algebraic variables of respective type, for instance, δ =[δi . . . δng

]Tand ν =

[νi . . . νn

]T. Finally, elim-

inating the algebraic variables, we get ∆δ∆ω

∆E′q

= A

∆δ∆ω

∆E′q

+ B

[∆Tm

∆Efd

](7)

where, A = M−ND−1

C. It follows from the equationsabove that A is of the form

A =

0 I 0A21 0 A23

A31 0 A33

(8)

Note, in (8), every Aij block is a submatrix of A whoseelements are derived later in the paper (see, Appendix B).Apart from the state variables ∆δi, ∆ωi, and ∆E′qi , theoutput variable ∆Tei , which is the electromagnetic torqueof generator i, is of specific interest to us. From the swingequation in (2) this is expressed as ∆Tei = − 2Hi

ωs∆ωi.

Following any disturbance in the system or perturbation inthe inputs, the time-evolution of these state and output vari-ables can be expressed as sum of damped sinusoids with modalfrequencies ωdr -s with differing amplitudes and phases. As aresult, for each mode r, ∆Tei,r (t), ∆δi,r(t) and ∆ωi,r(t) canbe expressed as rotating phasors − for details, see AppendixA. The notions of damping torque and damping power for agiven mode originate from the phasor representation of ∆~Tei,rin the ∆~δi,r −∆~ωi,r plane and the power resulting from theinteraction of the torque and the speed. This is explained next.

3

III. DAMPING POWER IN A MULTIMACHINE SYSTEM

In any ith machine, for a mode r, let the average powerof the electromagnetic torque ∆Tei,r (t) over a cycle startingfrom t = t0 be denoted by Wdi,r (t0), as shown below

Wdi,r (t0) =

∫ t0+ 2πωdr

t0∆Tei,r (t) ∆ωi,r(t) dt∫ t0+ 2π

ωdrt0

dt

(9)

Using the phasor notation described in Appendix A, let∆~Tei,r (t) = β1 eσrt 6 γ1 and ∆~ωi,r(t) = β2 eσrt 6 γ2.Therefore, Wdi,r (t0) =

=ωdr2π

∫ t0+ 2πωdr

t0

e2σrt β1 cos(ωdr t+ γ1) β2 cos(ωdr t+ γ2) dt

=β1 β2 ωdr

4π

∫ t0+ 2πωdr

t0

e2σrt{

cos(γ1 − γ2)

+ cos(2ωdr t + γ1 + γ2)}dt

=β1 β2 ωdr

4πcos(γ1 − γ2)

e2σrt0

2 σr

{e

4πσrωdr − 1

}+β1 β2 ωdr

4π

∫ t0+ 2πωdr

t0

e2σrt cos(2ωdr t+ γ1 + γ2) dt

(10)Now, considering that our mode of interest is poorly-damped

(as is the premise of our paper), implying |σr| << ωdr , we

may expand the exponential e4πσrωdr and neglect the second and

higher order terms. On doing so, the first term in (10) reducesto

β1 β2 ωdr4π

e2σrt0 cos(γ1 − γ2)1 + 4πσr

ωdr− 1

2 σr

=1

2β1 β2 e

2σrt0 cos(γ1 − γ2)

With the same assumption that σr is small, the second term in(10) becomes negligible, and can be ignored for mathematicaltractability. This is because, with |σr| << ωdr , for a completecycle of cos(2ωdr t), the e2σrt term remains almost constant,and therefore, the positive and negative half cycles almost addto zero. Therefore,

Wdi,r (t0) ≈ 1

2β1 β2 e

2σrt0 cos(γ1 − γ2)

=1

2<{β1 e

σrt0 6 γ1 β2 eσrt0 6 −γ2

}=

1

2<{

∆~Tei,r (t0) ∆~ω∗i,r(t0)}.

(11)

Hereafter, in the paper, assuming all phasors are computed att = t0 and powers are averaged over a cycle starting at t0, weshall drop the argument t0 from our expressions.

From (11) it can be interpreted that Wdi,r is the averagepower due to the component of ∆~Tei,r in the direction of∆~ωi,r. Let, the electromagnetic torque be decomposed as∆~Tei,r = kdi,r∆~ωi,r + ksi,r (j∆~ωi,r). Substituting this in(11), we get

Wdi,r =1

2<{

(kdi,r + jksi,r )∆~ωi,r∆~ω∗i,r

}=

1

2kdi,r

∣∣∆~ωi,r∣∣2(12)

where, kdi,r = <{∆~Tei,r∆~ωi,r

} is the damping torque coefficientof machine i for mode r. Therefore, we shall refer to Wdi,r

as the ‘average damping power’ (or simply ‘damping power’)of ∆Tei,r (t). For a system with ng machines, (11) can beextended as follows

Wdr =

ng∑i=1

Wdi,r =1

2

ng∑i=1

<{

∆~Tei,r ∆~ω∗i,r

}(13)

Next, we express this sum of damping powers in terms ofsystem matrices by using the definition of electromagnetictorque (from the swing equation) and the linearized systemdescription obtained in (7) and (8), as shown below.

∆Te(s) = −2 H

ωs∆ω(s) = −2 H

ωs

{A21 ∆δ(s) + A23 ∆E′

q(s)}

= −2 H

ωs

{A21 + A23 (sI−A33)

−1

A31

}∆ω(s)

s∆= K(s) ∆ω(s)

(14)Defining Kr = K(jωdr ), we may re-write (13) as

Wdr =1

2<{ ng∑i=1

ng∑j=1

Kij,r∆~ωj,r∆~ω∗i,r

}=

1

2<{

∆~ωH

r Kr∆~ωr}

(15)where, Kij,r is the (i, j)th element of Kr. We call Wdr the‘total damping power’ of the system for mode r.

Next, we simplify the expression in eqn (15) using the setof claims (1) − (4) below. Claims:

(1) P−1

AT

33P = A33,

(2) AT

31P =2 H

ωsA23,

(3) 2 AT

21H = 2 H A21

(4) ∀ x ∈ Cng , <{xHKrx} = xH<{Kr} x

where, P is a diagonal matrix of machine parameters withP(i, i) =

T ′doixdi−x

′di

. Claims (1) − (3) are derived using thedifferential and algebraic equations of the system modeledin Section II. These claims are then used to establish thesymmetry of Kr, which is in-turn used in proving claim (4).Detailed proof of these claims are outlined in Appendix B.

Using claim (4) we further reduce eqn (15) as follows

Wdr =1

2∆~ω

H

r <{Kr}∆~ωr (16)

where, <{Kr} =

= −<{

2 H

jωdr ωs

(A21 + A23 (jωdrI−A33)

−1

A31

)}= −2 H

ωsA23 <

{(−jωdrI−A33)(−jωdrI−A33)

−1

(jωdrI−A33)−1

jωdr

}A31

= −2 H

ωsA23 <

{(−jωdrI−A33)(ω2

drI + A332)−1

jωdr

}A31

=2 H

ωsA23 (ω2

drI + A332)−1

A31

This, along with claim (2) when substituted in eqn (16) gives

Wdr =1

2∆~ω

H

r AT

31 P (ω2drI + A33

2)−1

A31 ∆~ωr (17)

4

IV. CONSISTENCY OF DAMPING POWER WITH POWERDISSIPATION IN SYNCHRONOUS MACHINE WINDINGS

Considering the third-order system model in Section II, theonly source of power dissipation in the machines is in thefield windings. For any mode r, we denote the average powerdissipation in the winding of machine i by Wfi,r . Followingthe phasor notation as discussed, this is expressed as

Wfi,r =1

2<{

(∆~Ifi,rRfi) ∆~I∗fi,r

}=

1

2Rfi

∣∣∣∆~Ifi,r ∣∣∣2=

1

2

T ′doixdi − x′di

∣∣∣∆ ~E′qi,r

∣∣∣2 (18)

where, ∆~Ifi,r and ∆ ~E′qi,r are the phasors of the field currentand derivative of transient e.m.f. due to field flux linkage,respectively. Further, using notations from (36) (Appendix A)we may write

∆~E′qi,r = jωdr ∆ ~E′qi,r = jωdr 2 cr e

σrt0 ψE′qi,r. (19)

We define, cr = cr eσrt0 . Next, substituting (19) in (18) we

get

Wfi,r = 2 |cr|2T ′doi

xdi − x′diω2dr

∣∣∣∣ψE′qi,r∣∣∣∣2 . (20)

We obtain the total power dissipation for the mode by addingthe dissipation in individual machines. This is shown below.

Wfr =

ng∑i=1

Wfi,r = 2 |cr|2 ω2dr ΨH

E′qr

P ΨE′qr

(21)

where, ΨE′qr=[ψE′q1,r

. . . ψE′qi,r. . . ψE′qng,r

]T.

To show that our notion of total damping power, as definedin Section III, is consistent with the power dissipated inthe system, we need to prove that for any mode r, Wfr isequal to Wdr . To do so, we make the following algebraicmanipulations.

First, since λr is an eigenvalue of the system, we maywrite AΨr = λrΨr, where, the right eigenvector Ψr =[

Ψδr Ψωr ΨE′qr

]T. Next, using the structure of matrix

A as in (8), we can split this into the following equations

ΨE′qr

= (λrI−A33)−1 A31 Ψδr and Ψδr =1

λrΨωr (22)

Using these, along with (34) describing ∆~ωr = 2 cr Ψωr , wemay re-write (21) as follows

Wfr = 2 |cr|2 ω2dr

∆~ωHr2 c∗r λ∗r

AT

31 (λ∗rI−A33T )−1 P

(λrI−A33)−1 A31∆~ωr

2 cr λr.

(23)

Finally, considering that our mode of interest is poorly-damped|σr| << ωdr – as is the premise of [14], and for consistencythis paper’s too, we substitute λr = jωdr in (23). This, alongwith use of claim (1) reduces (23) as follows

Wfr =1

2∆~ωHr A

T

31 (−jωdrI−PA33P−1)−1

P

(jωdrI−A33)−1

A31 ∆~ωr

=1

2∆~ω

H

r AT

31 P (ω2drI + A33

2)−1

A31 ∆~ωr = Wdr .

(24)

Thus, for any given mode, the equivalence of the total dampingpower and the sum of average rate of change of energydissipations in the machine windings is established.

Remarks: (1) While the concepts of damping torque anddamping power are derived using the linearlized system mod-els, the average rate of energy dissipation in windings are morefundamental and does not limit itself to small-signal analysis.

(2) For machines with detailed models, the power dissipa-tion in damper windings should be added to Wfr to obtain thetotal dissipation in the system. Average power dissipation inthe damper winding is expressed as [13]

Wgi,r = 2 |cr|2T ′qoi

xqi − x′qiω2dr

∣∣∣∣ψE′di,r

∣∣∣∣2. (25)

E′

di,ris the state variable describing the dynamics of the

damper winding transient e.m.f. Further, with dynamics of theexcitation systems modeled, the expression of Wfr in (21)would get modified to include the effect of ∆Efd as

Wfi,r = 2 |cr|2{

T ′doixdi − x′di

ω2dr

∣∣∣ψE′qi,r ∣∣∣2− 1

xdi − x′di<(jωdr ψE′qi,r

ψ∗Efdi,r

)}.

(26)

However, the expression for Wdr in (15) would remain thesame (with block matrices A23, A31, and A33 larger indimensions to account for the additional state variables like∆E′

d, ∆Efd, etc. now concatenated to the vector ∆E′q) and

the equality of total damping power and total power dissipationwould still be true. This will be demonstrated in Section VI.

(3) Additionally, with a power system stabilizer (PSS) andan IEEE ST1A exciter modeled, as shown in Fig. 1, threenew states will be added − one each due to the washoutblock, the lead-lag compensator, and the time constant of thetransducer. Concatenating these to the existing state variables,the block matrices A23, A31, and A33 would be furtherexpanded. Additionally, due to the speed feedback to thewashout block, A32 term would now be non-zero, implying,

Kr = 2 Hωs

(A21

jωdr+A23 (jωdrI−A33)

−1

( A31

jωdr+ A32)

).

The equality of total damping power Wdr (as calculated withthe modified Kr) and that of the sum of Wfr and Wgr willbe demonstrated in Section VI.

Ka ∆Efd

11 + s TR

∆Vt

KP (1 + s T1)1 + s T2

s Tw1 + s Tw∆ω

Vref

+

+ −

PSS

IEEE ST1A Excitation System

1

Fig. 1: PSS and IEEE ST1A excitation system.

(4) Next, let us explore the importance of our deductionsin the context of stability monitoring. From [13], we knowthat, the relative damping contribution of individual generatorscan be inferred from the power dissipations in their windings.

5

Further, our paper establishes∑Wdi,r =

∑(Wfi,r + Wgi,r ),

which building on derivations in [13] leads to the claim thatthe stability margin of the rth mode σr ∝ −

∑Wdi,r . Note

that, estimating the variables ∆Tei and ∆ωi are relativelysimpler. Because, either they can be directly measured, aswith rotor speed, or estimated from measurable outputs, liketorque from power. And therefore, damping power Wdi,r

of individual machines can be easily calculated from theirterminal measurements upon filtering for the mode of interest.This has the potential for future monitoring applications, likedetermining stability margin of a mode by measuring totaldamping power contribution from all generators. To this end,it is important to clarify that, we do not claim efficacy overthe existing mode-metering algorithms like [16] (which wouldanyways be required to identify the poorly-damped modes inthe system), instead, we propose

∑Wdi,r as a complementary

measure of stability margin for the mode, with individualWdi,r -s as indices for identifying the prospective generatorlocations for damping enhancement.

V. DISTRIBUTION FACTORS: EXPRESSING DAMPINGPOWER OF EACH MACHINE AS WEIGHTED SUM OF

DISSIPATIONS IN ALL MACHINES

With the equality in (24) now proved, it might be temptingto draw an intuitive conclusion that such a power balanceshould hold for individual generators. However, this is not thecase. This is because, the modeshape of E

′

q of each generatoris a function of modeshapes of speed-deviation of all othermachines and vice-versa (see, (22)). Therefore, the mathemat-ical representation of damping power in each machine is inreality an abstraction of dissipative effects coming from thewindings of all machines in the system including it’s own. Tothat end, we next derive the distribution factors describing thefractional contribution of windings of different machines inconstituting the damping power of a single machine.

Considering the system model described in Section II, inthis section, we express the damping power of each generatorWdi,r as a linear combination of individual Wfi,r -s. Recall,from (12)

Wdi,r =1

2kdi,r

∣∣∣∆~ωi,r∣∣∣2 = 2 |cr|2 kdi,r ΨHωr Ii Ψωr (27)

where, Ii is a ng−dimensional square matrix with the (i, i)th

entry as 1 and remaining all entries as zeros. Next, from theeigen decomposition of A as before, we may write Ψωr =

λr (λ2r I − A21)−1 A23

∆= Q ΨE′qr

. Substituting this, (27)may be expressed as

Wdi,r = 2 |cr|2 kdi,r ΨHE′qr

QH Ii Q ΨE′qr

= 2 |cr|2 kdi,rng∑`=1

ng∑j=1

Q∗i` Qij ψ∗E′q`,r

ψE′qj,r

=

ng∑j=1

2 |cr|2 kdi,rQij

∑ng`=1 Q∗i` ψ

∗E′q`,r

ψ∗E′qj,r

∣∣∣ψE′qj,r∣∣∣2

=

ng∑j=1

kdi,rPj ω2

dr

Qij ψ∗ωi,r

ψ∗E′qj,r

Wfj,r

(28)

where, Pj is the diagonal element P(j, j) =T ′doj

xdj−x′dj

. Next,

we define βij as the component of ∆~ωi,r due to ∆ ~E′qj,r

βij∆=

(Qij ∆ ~E′qj,r

∆~ωi,r

)∗=

Q∗ij ψ∗E′qj,r

ψ∗ωi,r. (29)

Since, ψ∗ωi,r =∑ngj=1 Q∗ij ψ

∗E′qj,r

, we may say Q∗ij ψ∗E′qj,r

is

the contribution of ψ∗E′qj,r

in the modeshape ψ∗ωi,r . Therefore,

Wdi,r =

ng∑j=1

kdi,r |Qij |2

Pj ω2drβij

Wfj,r (30)

Since, Wdi,r -s and Wfi,r -s are real quantities, the imaginarypart of (30) is zero. Hence,

Wdi,r =

ng∑j=1

kdi,r |Qij |2

Pj ω2dr

<( 1

βij

)Wfj,r

∆=

ng∑j=1

αij Wfj,r (31)

We call αij-s the ‘distribution factors,’ because, for a fixed i,

the ratios αijWfj,r

Wdi,r

for j = 1 to ng , are the fractions in whichthe damping power of generator i is derived from the powerdissipation in the windings of the machines 1 to ng . Further,looking from the other side, fixing a machine j, the factorsαij-s describe the ratios in which the power dissipation inthat machine winding is distributed in the ‘abstract’ dampingpower of all other machines.

A. Connection to the Heffron-Phillips Model

We know from the Heffron-Phillips model [17] of a SMIBsystem that, for a third-order machine model (K1 − K4model), the angle between the ∆~Tei,r and ∆~ωi,r phasorsis determined by the field circuit time-constant. Higher theresistance of the field circuit, smaller is the angle, and there-fore, higher is the damping power of the generator. This isconsistent with the results in [14] that for a SMIB system, thedamping power of the machine is derived exclusively fromthe power dissipation in its field circuit. Extending the sameto the Heffron-Phillips model for multimachine systems, wesee (from Fig. 6 in [18]) that the damping power of eachgenerator has contributions from the field circuit dissipationsin multiple other machines in the system – depending on therelative participation of those machines in the mode of interest.However, given the complexity in calculating the modewiseK1−K4 constants in a multimachine system, we, through thedistribution factors derived in this section, offer an alternativepath to express the damping powers of each generator as aweighted sum of field winding dissipations of all machines inthe system, including it’s own.

B. Potential Application in Understanding Dissipative Contri-bution from PSS and Other Controllers

The lead-lag compensators in a ∆ω-PSS are designedknowing the angular relationship between the ∆~Tei,r and∆~ωi,r phasors, and the phase-shift required to rotate ∆~Tei,rfurther in the direction of ∆~ωi,r. This phase-shift introducesadditional damping for the mode by increasing the total powerloss in the system for the mode for which the PSS is designed.However, since a PSS in one location might negatively affect

6

the damping in some other location, dissipation in someindividual generators may be reduced. While, methods like theone in [18] quantify the effect of PSS on damping powers ofindividual machines, they do not describe how the dissipationsin their windings would change. To this end, the distributionfactors connecting damping and dissipative powers becomeuseful. Also, this is not limited to PSSs in generators, if derivedfor higher-order models, the analysis can be extended to othertypes of controllers.

While distribution factors may not have direct usefulnessin terms of monitoring or controlling a mode, we believethey serve as an important tool that bridge an insightful linkbetween two apparently different frameworks.

VI. CASE STUDIES

We now verify the aforementioned claims on the fundamen-tal frequency phasor models of IEEE 4-machine [17] and 16-machine [19] test systems. In each case we consider two typesof models: (a) Simplified model with assumptions describedin Section II, and (b) Detailed model considering 4th-ordersynchronous machine dynamics (including 1 damper winding)along with exciters, where the network is still assumed to belossless and loads are of constant power nature.

A. IEEE 2−area 4−machine Kundur Test System

Consider the 4−machine system [17] shown in Fig. 2 witha total load of 2, 734 MW under nominal condition.

G3

G4

1

4

75

6

2

8

109

11 3

G1

G2

Fig. 2: Single-line diagram of 2−area 4−machine test system.

(a) Simplified model: Under nominal loading, there are threepoorly-damped modes, see Table I. Given modeling assump-tions, the only source of damping is in the field windings ofthe generators. Therefore, following our proposition, we needto show that for small perturbations in the system, for each ofthese oscillatory modes, the sum of average damping powers isnumerically equal to the sum of power dissipations in the fieldwindings, across the operating points. This is demonstratedin the Figs. 3 (a) and (b) for two of the three modes. Theoperating point is varied by progressively reducing the tie flowbetween buses 7 and 9 from 433 MW under nominal conditionto −400 MW while maintaining the total load of the systemconstant.

We now validate our claim that although the total powerdissipation is equal to the total damping power at the systemlevel, this is not necessarily true for individual machines. InFigs 4 (a) and (b), the ratios of dissipation power to dampingpower is plotted for each of the 4 machines for the 0.69 Hzand 1.04 Hz modes, respectively. Observe that the ratios showstrict monotonicity with change in operating points and underno circumstance they are equal to 1 all at once.

Fig. 3: Equality of total damping power with sum of average power dissipationin field windings of all generators across different operating points insimplified 4−machine system model for (a) 0.69 Hz and (b) 1.04 Hz modes.

Fig. 4: Ratios of average power dissipation in field windings (Wfi,r ) andthat due to damping torques (Wdi,r ) for individual machines across differentoperating points in simplified 4−machine system model for (a) 0.69 Hz and(b) 1.04 Hz modes.

Next, in Fig. 5, the relative contributions from the powerdissipations in the windings of G1 to G4 in constituting thedamping power of G1, as discussed in Section V, are plottedfor the 0.69 Hz mode . Observe that, at a given operating

point, the fractions α1jWfj,r

Wd1,r

for j = 1 to 4 add to 1. In Fig.6 the distribution factors αi1-s for i = 1 to 4 are shown.These describe fractions in which the power of oscillatoryenergy dissipation in G1 is distributed in the damping powersof G1 to G4. It can be seen that at any operating point,∑4i=1 αi1 = 1. Since the machines are nearly identical and

the network is symmetric with respect to the generators inthis system, the nature of the distribution factors are similarfor other generators, and thus are not repeated here.

(b) Detailed model: The detailed model considers DC1Aexcitation system [17] for each generator over and above the

TABLE I: POORLY-DAMPED MODES IN IEEE 4-MACHINE SYSTEM

Machine ModelEigenvalues

λr = σr ± jωdr

Modal freq.

fr (Hz)

Damp. ratio

ζr

Simplified model

−0.1183± j4.3816 0.69 0.027

−0.1444± j6.2779 1.00 0.023

−0.1544± j6.5330 1.04 0.024

Detailed model

(with DC1A exciters)−0.1231± j4.2130 0.67 0.029

Detailed model (with

ST1A exciters & PSS)−0.1477± j4.8649 0.77 0.030

7

Fig. 5: Fractions in which the damping power of G1 is distributed as powerdissipation in the windings of G1 − G4 for 0.69 Hz mode.

Fig. 6: Distribution of the power dissipation in G1 in the damping powers ofG1 − G4 for 0.69 Hz mode.

assumptions mentioned earlier – the poorly-damped mode isshown in Table I. For any mode, the power dissipation inthe system is the summation of total power dissipations infield winding Wfr and in damper winding Wgr (see, (26) and(25), respectively), where Wfr =

∑ngi=1 Wfi,r and Wgr =∑ng

i=1 Wgi,r For the detailed model, the consistency of totalpower dissipation with the total damping power is illustratedin Fig. 7.

(c) Validation under large disturbances: Next, we considerthe 4th-order machine model with IEEE ST1A excitationsystem for validation under large disturbances. Additionally,G1 is equipped with a PSS. We simulate a 5-cycle three-phaseself-clearing fault at t = 1 s near bus 8. The detrended post-fault time-domain plots of the state and output variables ofthe two generators G1 and G3 are shown in Fig. 8. We obtainthe relative modeshapes of these signals using the approachdescribed in [20].

We use ∆ω1 as the reference signal and compute the mode-shapes for all ∆ωi, ∆Tei , ∆E′qi , ∆E′di , and ∆Efdi -s for thecritical mode in Table I. The damping and dissipative powersof all 4 generators as computed using these modeshapes areshown in Table II. It can be seen that, although for individual

Fig. 7: Equality of total damping power with sum of average power dissipationin windings of all generators across different operating points in detailed4−machine system model for the 0.67 Hz mode.

Fig. 8: Detrended post-fault time-domain plots of the state and output variablesof G1 and G3.

TABLE II: DAMPING AND DISSIPATIVE POWERS IN DETAILED4−MACHINE SYSTEM MODEL FOR 0.77 Hz MODE.

Using the eigenvectorsobtained from thesmall-signal model

Using the modeshapesestimated from the

time-domain responsesWdi,r2 |cr|2

Wfi,r2 |cr|2

Wdi,r2 |cr|2

Wfi,r2 |cr|2

G1 0.0382 0.2294 0.0361 0.2274

G2 0.0321 −0.1830 0.0340 −0.1877

G3 0.0435 0.0794 0.0405 0.0754

G4 0.0521 0.0401 0.0472 0.0390

Sum 0.1659 0.1659 0.1578 0.1541

machines the values of Wdi,r -s are different from that ofWfi,r -s, their sum totals are almost equal. Also, these valuesnearly match those calculated from the small-signal model.

TABLE III: POORLY-DAMPED MODES IN 16-MACHINE SYSTEM

Machine ModelEigenvalue

λr = σr + jωdr

Modal freq.

fr (Hz)

Damping ratio

ζr

Simplified model

−0.0336± j2.1031 0.34 0.016

−0.0338± j3.1288 0.50 0.011

−0.1062± j3.7500 0.60 0.028

−0.0264± j4.1031 0.65 0.006

Detailed model

−0.0656± j3.1137 0.49 0.021

−0.0981± j3.5169 0.56 0.028

−0.1827± j4.9627 0.79 0.037

B. IEEE 5−area 16−machine NY-NE Test System

Next, consider the 16−machine New York-New Englandtest system shown in Fig. 9. In the detailed model, G1 − G8have DC1A exciters, G9 is equipped with a ST1A exciter anda power system stabilizer (PSS), and the remaining generatorshave manual excitation. The machine and the network datacan be obtained from [19]. The poorly damped modes of thesystem under nominal loading, for both simplified and detailedmodels, are shown in Table. III. Different operating pointsare obtained by uniformly changing the total system load. Asbefore, in Figs 10 and 11, the consistency of total dampingpower and sum of power dissipation in machine windings is

8

shown respectively for the simplified and the detailed model –each corresponding to a particular mode. The negative valuesof Wfr in Fig.11 indicate that the excitation systems arecontributing towards negative damping for higher loadings.

03

G9

G3

67

22

24

21

66

37

68

27

64

65

62

26

28

09

29

G2

58

56

55

52

02

57

59

60

G8G1

01

54

25

08

63

36

1131

53

51

49

38

30

39

35

33

32

34

43

13

17

45

12

46

61

50

484047

G12

G13

G10

G11

18

14

15

16

41

42

G14

G15

G16

10

TCSC

G6

06

G7

23

07

G4

G5

19

05

20

04

NETS

NYPSAre

a #

3A

rea

#4

Are

a #

5

PSS

03

G9

G3

67

22

24

21

66

37

68

27

64

65

62

26

28

09

29

G2

58

56

55

52

02

57

59

60

G8G1

01

54

25

08

63

36

1131

53

51

49

38

30

39

35

33

32

34

43

13

17

45

12

46

61

50

484047

G12

G13

G10

G11

18

14

15

16

41

42

G14

G15

G16

10

TCSC

G6

06

G7

23

07

G4

G5

19

05

20

04

NETS

NYPSAre

a #

3A

rea

#4

Are

a #

5

PSS

Fig. 9: Single-line diagram of IEEE 5−area 16−machine NY-NE test system.

Fig. 10: Equality of total damping power with sum of average powerdissipation in field windings of all generators across different operating pointsin simplified 16−machine system model for 0.50 Hz mode.

Fig. 11: Equality of total damping power with sum of average powerdissipation in windings of all generators across different operating points indetailed 16−machine system model for 0.56 Hz mode.

Next, we present the following case study for the simplifiedmodel to validate our claims regarding the distribution ofdamping power. In this study, for G9, we compare the values

ofα9j Wfj,r

Wd9,r

(fractions in which the damping power of G9 isderived from the power dissipations in the windings of anyjth generator) as calculated from small-signal model and asestimated from time-domain responses. Under nominal loadingcondition, a 0.2 s pulse disturbance is applied to the excitationsystem of all generators and the time-domain responses of thestate and output variables are obtained. Detrended plots of thevariables of G9 are shown in Fig. 12. Using the approachin [20] as before, we next estimate the relative modeshapes

Fig. 12: Detrended time-domain responses of (a) ω, (b) Te, and (c) E′q ofG9 following a pulse disturbance in the excitation system of all generators inthe 16−machine system.

of ω9, Te9 , and E′q9 , along with E′qi for selected generators:

G3, G5, and G6 from NETS, and G11 from NYPS from theirtime-domain responses. We use ∆ω9 as the reference and themodeshapes are estimated for the 0.6 Hz mode. For this mode,the generators of NETS oscillate against those in NYPS. Next,using the estimated modeshapes, we compute the dampingand dissipative powers of the selected generators as shownin Table IV. Finally, using the αij-s from the expression in(30) and the values estimated in Table IV, we compute thefractional contributions from these selected generators towardsthe damping power of G9. These are shown in Table V. Asseen, the estimated values match those calculated from small-signal model. The variation in these fractions for change insystem loading are shown in Fig. 13.

TABLE IV: DAMPING AND DISSIPATIVE POWERS FOR THE 0.6 HzMODE ESTIMATED FROM TIME-DOMAIN RESPONSES

Wd9,r2|cr|2

Wf3,r2|cr|2

Wf5,r2|cr|2

Wf6,r2|cr|2

Wf9,r2|cr|2

Wf11,r2|cr|2

0.0806 0.0905 0.0663 0.0791 0.1340 0.0334

TABLE V: FRACTIONAL CONTRIBUTION FROM SELECTED GENER-ATORS TOWARDS DAMPING POWER OF G9 FOR THE 0.6 Hz MODE

frac. contr. from G9 G3 G5 G6 G11

small-signal model 0.2064 0.1382 0.1063 0.1102 0.0520

time-domain estimation 0.2046 0.1369 0.1003 0.1198 0.0501

Now, we consider the other poory-damped mode at 0.5 Hz,for which generators outside NETS − G14 and G16 oscillateagainst each other with marginal participation from generatorsin other areas. The damping and dissipative powers of thesetwo generators, for the mode, are shown in Table VI. Notethat, since other generators do not participate in the mode, thesummation of damping powers of G14 and G16 is approx-imately equal to the summation of their dissipative powers.Finally, in Fig. 14, the relative dissipative contributions fromthe generators in different areas in constituting the damping

9

Fig. 13: Fractions in which the damping power of G9 is derived from thepower dissipations in the windings of selected generators in NETS: G3, G5,G6, and G9 for the 0.60 Hz mode.

power of G16 are shown. For G1 − G9 and G10 − G13, theiraggregates

9∑j=1

α16,j

Wfj,r

Wd16,r

and13∑j=10

α16,j

Wfj,r

Wd16,r

are shown as contributions from NETS and NYPS, respec-tively. We note that G14 and G16 have the highest participationin the mode, which is aligned with the observation that thesegenerators have the highest dissipative contribution.

TABLE VI: DAMPING AND DISSIPATIVE POWERS OF G14 and G16FOR THE 0.5 Hz MODE

Calculated fromsmall-signal model

Estimated from thetime-domain responses

Wdi,r2 |cr|2

Wfi,r2 |cr|2

Wdi,r2 |cr|2

Wfi,r2 |cr|2

G14 0.2150 0.1642 0.2070 0.1503

G16 0.2665 0.3212 0.2904 0.3418

Sum 0.4815 0.4854 0.4974 0.4921

Fig. 14: Fractions in which the damping power of G16 is derived from thepower dissipations in the windings of generators in NETS (G1 − G9), NYPS(G10 − G13), and G14 − G16 for the 0.50 Hz mode.

VII. CONCLUSIONS

A mathematical proof for the equality of total dampingpower of the system and the total power dissipation ingenerators was presented for multimachine systems. It wasdemonstrated that the equality while true when added over allgenerators, does not hold for individual machines. Thereafter,distribution factors were derived representing the dissipativecontributions from different generators in constituting thedamping power of each machine.

APPENDIX A: PHASOR NOTATION

Let the small-signal dynamics of an autonomous system berepresented by the state space model x(t) = Ax(t). Next,assuming there are m oscillatory modes in the response, eachdue to a complex-conjugate eigenvalue pair λr (= σr + jωdr )and λ∗r of A, the time evolution of any ith state variable xi(t)can be expressed as the sum of m modal constituents, as shownin eqn (32)

xi(t) =

m∑r=1

xi,r(t) =

m∑r=1

{eλrtcrψi,r + eλ

∗rtc∗rψ

∗i,r

}(32)

where, cr = φTr x(0), and φTr and Ψr are respectively the leftand right eigenvectors of A corresponding to the eigenvalueλr with ψi,r as the ith entry of Ψr.

Denoting 2crψi,r∆= βi,re

jγi,r , xi,r(t) reduces to

xi,r(t) = 2 <{eλrtcrψi,r

}= βi,r e

σrt cos(ωdr t+ γi,r). (33)

This sinusoidal variation is represented in the dynamicphasor (mentioned as ‘phasor’ going forward) notation usingthe magnitude and phase of the signal, as shown in eqn (34).

~xi,r(t)∆= βi,r e

σrt 6 γi,r = 2 cr ψi,r eσrt (34)

~xi,r(t) is a phasor rotating at the modal frequency ωdr with itsamplitude having an exponential decay. It represents the timeevolution of xi,r(t). We denote ~xr as the vector of ~xi,r(t)-s.

Next, from (33),

xi,r(t) = βi,r eσrt{σr cos(ωdr t+ γi,r) −

ωdr sin(ωdr t+ γi,r)}.

(35)

Considering the mode to be poorly-damped, |σr| << ωdr ,this reduces to xi,r(t) ≈ −βi,r eσrt ωdr sin(ωdr t + γi,r).Therefore,

~xi,r(t) ≈ j ωdr ~xi,r(t). (36)

APPENDIX B: PROOF OF CLAIMS

Observe that, from (6), M, N, C, and D can be structured as

M =

[M11 M12 M13

M21 M22 M23

M31 M32 M33

]N =

[N11 N12

N21 N22

N31 N32

]

C =

[C11 C12 C13

C21 C22 C23

]D =

[D11 D12

D21 D22

].

(37)

Following the notation that, Dkl(i, j) is the (i, j)th elementof the (k, l)th submatrix Dkl of D, we may write ∀ j 6= i

D12(i, j) = Vj0∂fi∂Vj

∣∣∣∣0

= −Vi0Vj0Yij sin(θi0 − θj0) (38a)

D21(i, j) =∂gi∂θj

∣∣∣∣0

= Vi0Vj0Yij sin(θi0 − θj0) = D12(j, i) (38b)

D11(i, j) =∂fi∂θj

∣∣∣∣0

= Vi0Vj0Yij cos(θi0 − θj0) = D11(j, i)

(38c)

D22(i, j) = Vj0∂gi∂Vj

∣∣∣∣0

= Vi0Vj0Yij cos(θi0 − θj0) = D22(j, i).

(38d)

10

Similarly, it can be shown that,

D12(i, i) = Vi0∂fi∂Vi

∣∣∣∣∣0

=∂gi∂θi

∣∣∣∣∣0

= D21(i, i). (39)

Therefore, from eqns (38a), (38b), and (39), it can beinferred that D

T

12 = D21. Additionally, from eqns (38c) −(38d), D

T

11 = D11 and DT

22 = D22. Thus,

DT

=

D11 D12

D21 D22

T =

DT

11 DT

21

DT

12 DT

22

= D. (40)

Further, D being real and symmetric implies D−1

is also realand symmetric

=⇒ D−T

= D−1

. (41)

Proof of Claim (1) : Recall, A = M−ND−1

C. Therefore,

A33 = M33 −[N31 N32

]D−1[

C13

C23

]

=⇒ AT

33 = MT

33 −[CT

13 CT

23

]D−T

NT

31

NT

32

(42)

From eqns (3) and (4) observe that, ∀ j 6= i,

N31(i, j) =∂E′qi∂θj

∣∣∣∣0

= 0 ; C13(i, j) =∂fi∂E′qj

∣∣∣∣0

= 0 ; (43)

and for elements on the principal diagonal,

N31(i, i) =∂E′qi∂θi

∣∣∣∣0

=Vi0 sin(δi0 − θi0)

x′di

(xdi − x

′di

T ′doi

)(44a)

C13(i, i) =∂fi∂Eqi

∣∣∣∣0

=Vi0 sin(δi0 − θi0)

x′di(44b)

Further note, N31 and C13 are rectangular matrices of dimen-sions Rng×n and Rn×ng respectively. Therefore, combiningeqns (43) and (44) we get

P−1

C13

T

= N31 (45)

where, P is a diagonal matrix with P(i, i) =T ′doi

xdi−x′di

.Similarly, from eqns (3) and (5), ∀ j 6= i,

N32(i, j) = Vj0∂E′qi∂Vj

∣∣∣∣0

= 0 ; C23(i, j) =∂gi∂E′qj

∣∣∣∣0

= 0 ; and

N32(i, i) = Vi0∂E′qi∂Vj

∣∣∣∣0

=Vi0 cos(δi0 − θi0)

x′di

(xdi − x

′di

T ′doi

)C23(i, i) =

∂gi∂Eqi

∣∣∣∣0

=Vi0 cos(δi0 − θi0)

x′di

Therefore, following arguments as before,

P−1

C23

T

= N32 (46)

Finally, observe that M33 ∈ Rng×ng with M33(i, j) =∂E′qi∂δj

∣∣∣∣0

= 0 ∀ j 6= i. This implies M33 is diagonal.

Using the results (41), (45) and (46), eqn (42) can berewritten as

P−1

AT

33P = P−1

MT

33P−P−1[CT

13 CT

23

]D−T

NT

31

NT

32

P

= M33 −[N31 N32

]D−1[

C13

C23

]= A33.

This concludes the proof.

Proof of Claim (2) : It can be seen from eqns (2) and (3)that blocks M31 and M23 are diagonal. Also,

M31(i, i) =∂E′qi∂δi

∣∣∣∣0

= −Vi0 sin(δi − θi)x′di

(xdi − x

′di

T ′doi

)(47a)

M23(i, i) =∂ωi∂E′qi

∣∣∣∣0

= −Vi0 sin(δi − θi)2Hi x′di

ωs (47b)

Therefore, we my write

MT

31 P =2 H

ωsM23 (48)

where H is diagonal with H(i, i) = Hi.Now, as before, for N and C matrices,

N21(i, j) =∂ωi∂θj

∣∣∣∣0

= 0 ; C11(i, j) =∂fi∂δj

∣∣∣∣0

= 0. Also,

N21(i, i) =∂ωi∂θi

∣∣∣∣0

=ωs E

′qi0Vi0 cos(δi0 − θi0)

2Hi x′di

−ωsV

2i0 sin 2(δi0 − θi0)

2Hi

(xqi − x′dixqix

′di

)C11(i, i) =

∂fi∂δi

∣∣∣∣0

=E′qi0Vi0 cos(δi0 − θi0)

x′di

−V 2i0 sin 2(δi0 − θi0)

(xqi − x′dixqix

′di

)=

2 Hiωs

N21(i, i)

Combining these with the fact that, N21 ∈ Rng×n and C11 ∈Rn×ng we may write,

CT

11 =2 H

ωsN21. (49)

Similarly, it can be shown that

CT

21 =2 H

ωsN22. (50)

Now, recall A = M−ND−1

C. Therefore,

A31 = M31 −[N31 N32

]D−1[

C11

C21

]

=⇒ AT

31P = MT

33P−[CT

11 CT

21

]D−T

NT

31

NT

32

P

(51)

Next, substituting eqns (41) and (45) − (50) in (51)

AT

31P =2 H

ωsM23 −

2 H

ωs

[N21 N22

]D−1

[C13

C23

]=

2 H

ωsA23

(52)

This concludes the proof.

Proof of Claim (3) : As before, observe from eqn (2) thatM21 is diagonal. Therefore, we may write

H M21 = MT21 H. (53)

Also, A21 = M21 −[N21 N22

]D−1[C11

C21

]

=⇒ 2 AT

21H = MT

21H −[CT

11 CT

21

]D−T

2 NT

21 H

2 NT

22 H

(54)

11

Finally, substituting eqns (41), (49), (50), and (53) in (54)

2 AT

21H = 2 H M21 − 2 H[N21 N22

]D−1[C11

C21

]= 2 H A21.

(55)

This concludes the proof.

Proof of Claim (4) : From the definition of Kr and (14),

KTr = −

{2

ωsAT

21H + AT

31 (jωdrI−AT

33)−1

(2

ωsAT

23 H)

}1

jωdr

= −{

2

ωsAT

21H + AT

31 P (jωdrI − P−1AT

33P)−1

P−1(2 H

ωsA23)T

}1

jωdr

Next, using claims (1) − (3) we can re-write KTr as

KTr = −2 H

ωs

{A21 + A23 (jωdrI−A33)

−1

A31

}1

jωdr= Kr

(56)Now, ∀ x ∈ Cng , let us decompose it into its real andimaginary parts as shown: x = x1 + jx2. Therefore,

xHKrx = (xT1 − jxT2 )(<(Kr) + j=(Kr)

)(x1 + jx2) (57)

Further, using eqn (56), we can infer on the symmetry of boththe real and imaginary parts of Kr. Hence,

xT1 =(Kr)x2 = xT2 =(Kr)x1 and xT1 <(Kr)x2 = xT2 <(Kr)x1.(58)

This reduces the real part of eqn (57) as follows

<{xHKrx} = xT1 <(Kr)x1 + xT2 <(Kr)x2. (59)

Finally, using the real part of eqn (58)

<{xHKrx} = xT1 <(Kr)x1 + xT2 <(Kr)x2

+ jxT1 <(Kr)x2 − jxT2 <(Kr)x1

= xH<{Kr} x.(60)

This concludes the proof.

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