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to the Damped Alternator
Ahmed Toumi, Mohamed Ben Ali Kamoun, Michel Poloujadoff
To cite this version:
Ahmed Toumi, Mohamed Ben Ali Kamoun, Michel Poloujadoff. Simplified Approach for the Stability Study, Application to the Damped Alternator. Journal de Physique III, EDP Sciences, 1995, 5 (10), pp.1671-1688. �10.1051/jp3:1995217�. �jpa-00249408�
Classification Physics Abstracts
41.90-86.90
Simplified Approach for the Stability Study, Application to the
Damped Alternator
Ahmed Toumi(~), Mohamed Ben Ali Kamoun(~) and Michel Poloujadoff(2)
(~) DApartement de Gdnie #lectrique (LETAU), ENIS, BP W, 3038 Sfax, Tunisia
(~) Electrotechnique, Aile 22-12, UniversitA de Paris VI, 4 Place Jussieu, 75252 Paris 05, France
(Received 4 March 1994, revised 6 June 1995, accepted 4 July1995)
R4sum4. Ce travail traite de la stabilitd d'un altemateur muni de circuits amortisseurs.
Contrairement h l'altemateur
non amorti, nous avons constatA que la zone instable autour de la valeur nulle de I'angle inteme disparait lorsque l'inertie de la partie toumante prend des valeurs courantes. Cependant, cette zone instable demeure pour de foibles valeurs du moment
d'inertie. Ce rdsultat peut Atre conclu par les approches antdrieures simplifides proposdes par
certains auteurs, en particulier celle de J. Tamura, dans laquelle l'inertie n'intervient pas dans la formulation. La m4thode propos4e peut Atie largement simplifiAe pour les valeurs de l'inertie oh l'4tude de la stabilitA
se rAduit h la discussion des signes de deux parambtres obtenus par
un calcul d14mentaire. Le r6sultat de cette m4thode simplifi4e est identique h celui donnA par
l'approche de Tamura que nous avons Atendue h l'altemateur amorti. Cette approche Atendue
reste compliquAe h utiliser, malgrA la structure matricielle que nous lui avons donnAe.
Abstract. This paper deals with the stability of the damped alternator. Unlike the un-
damped alternator, we have found that the unstable region located around the zero value of the internal angle disappears when inertia takes normal values. However, this unstable region
remains for low values of inertia This result can not be shown by the simplified approaches given by previous authors for which inertia is not taken into account in the formulae. Our
method
can be much raore simplified in the case for large values of inertia, where the stability
study is reduced to the discussion of two parameters. This result meets the one given by Tamura
approach which we extended to the damped alternator. This extensive approach remains much
more complicated than our method which turns, in this case, to an elementary calculation.
1. Introduction
In a recent paper iii, Tamura et al., have developed a new approach to the study of machine
stability with application to undamped alternator and doubly fed synchronous machine. This
approach is based on the use of a new swing equation whose coefficients are the synchronizing
and damping torques, and the inertia constant [5,9,10]. Although this method is interesting,
its theoretical basis is not very firm; as a result, it yields stability limits which do not depend
© Les Editions de Physique 1995
on the inertia constant. This has led us to search for a very firmly established method. An obvious choice is the eigenvalues method [2,10], but this requires the use of matrix algebra such
as inversion and determination of eigenvalues. Another choice was the Routh criterion [3,4],
which seems very difficult to use when considered for the first time. However, when we tried, it appeared that it was not more difficult to apply than the criterion suggested by Tamura.
In the present paper, we present our simplified approach deduced from Routh criterion and the extension of Tamura's method for salient pole damped alternator.
2. Small Perturbation Equations
In Appendix A, the classical equations of the alternator, connected to an infinite bus, are
written with the classical notations [5,6,8]. If we consider a steady state operation against
the infinite bus, and if16, u~m, #d, 4q, Iii 4D, IQ, id, iq, if, iD, iQ, Tm and Tel undergoes a
variation (hi, Au~m, Aid, A#q, A#f, A#D, A#Q, Aid, Aiq, Aif, AiD, AiQ, ATm and ATe),
then the perturbation equations are:
-VR cosboAb = -rtAid U~A#q #qoAu~m @~
-VR sin b0 hi
= -rtAiq + UJA#d #doAuJm ~~/~
0 = rfAif + i@
o = r~Ai~+AflR Ii)
0 = rqAio + @
A~ dAb
~ 7T
J ~j~
= ATm ATe
where:
#d = ltdAid + m~iAii + m~DAiD
A#q " 'tqAlq + 'IIaQA~Q
A#I = iiAii + m~iAid + mmAiD AID j~~
= IDAiD + m~DAid + mfDAii
A#q = lqAio+m~qAiq
ATe = A#diqo + #doAiq A#aide 4qoAid
3. Stability Study of the Damped Alternator
3 .I. STABILITY CONDITIONS. If Laplace transformation is applied to the above equations, the determinant of the system cancels if:
7
£m~p~ = o j3)
~=o
Table 1.
fl~ 1ll7 1ll5 1ll3 fill
fl~ 1ll6 1ll4 1ll2 1ll0
fl~ 'l5 n3 nl
fl~ n4 n2 1ll0
fl~ ~3 ~l
fl~ ~2 1ll0
fl~ Ol
fl~ 1ll0
d
1
~ 'e #
a~§ §
s
a) b)
Fig. I. Configuration and vector diagram of a damped alternator connected to an infinite bus.
This equation is known as the characteristic equation of the system. The mathematical devel- opments and explicit values of mo,
, m7 are given in Appendix B.
To apply Routh's criterion, Table I has to be considered; the new elements of the six last lines are defined as follows:
~5 ~ 5 m~$fi ~3 ~ 3 @@~m~ ~I ~ 1 @Q~m~
n4 " m4 %p n2
= m2 %~ q3
= n3 %f
~~ ~~ ~~ ~~
~~ ~ ~~ ~~ ~
The system is stable if and only if: m7, m6, n5, n4, q3, q2, oi and mu are all positive or all
negative.
Considering the values of the above coefficients, it may be seen that:
. m7 and m6, are always positive, whatever the value of inertia J.
. Coefficients n5, n4, q3 and q2 vary as shown in Figures 2 and 3. They are functions of
angle bo and of e.m.f. E (0 < E < 2 pu) for various values of J (= 2, 5, 10 s).
x1o8 E=o.1 x107 E=o.I S
1.5
o.5
-~00 -100 0 100 200 -100 o 100 200
a) b)
xio6 E=o.1 x1o4 E=o.1
~
a
----,--"~'~"-~-,---~~~~'~~-,--
3
~~~~~ ~~=~~~~
~~,
~ ~'~"~~"' ~"~'~""'~'~~~"""'~ ~~""~"~'~
2
'~",, &~~
'~~,, »',~'~
-loo o loo 200 -~oo -loo o loo 200
c) d)
Fig. 2. Coefficients n5, n4, q3 and q2 as functions of do and J, for E
= U-I pu. Solid line: J
= lo s,
interrupted line: J
= 5 s, mixed line: J
= 2 s.
. Coefficients oi and mo vary as indicated in Figure 4. They may be positive or negative.
For various values of E, oi does not vary widely as a function of J, while ma is totally independent of J.
Thus, since J is commonly larger than 2 s, we may say that for reasonably large values of J, stability depends only on oi and mo. Therefore, it is convenient to express clearly the
following quantities as functions of J (see details in Appendix C):
n5 = 15J + 14
~ s J2+s J+s
~
5 + 4
~~J3 + p2J2 + pi J + No
Q3 =
~~ j2 +
si J + so
t5J~ + t4J~ +
t§J~ + t2J~ + tl J + t0
~~ (~5J + ~4)(@3J + jL2J~ + jLl J + ~0
~
p~ J7 + p6J6 + p5J5 + p4J4 + p3J3 + p2J2 +
pi J + pu
~ (s~J'~ + si J + so)(t~J5 + t4J4 + t3J3 + t~J~ + ti J + to
~ x1o8 E=2 xlo7 E=2
o.5
-lo0 o lo0 200 -100 o loo 200
a) b)
x106 E=2 x1o4 E=2
-loo o loo 200 -~o0 -loo 0 loo 200
c) d~
Fig. 3. Coefficients n5, n4, q3 and q2 as functions of do and J, for E
= 2 pu. Solid line: J
= 10 s,
interrupted line: J
= 5 s, mixed line: J
= 2 s.
3.2. THE NEW SIMPLIFIED APPROACH. When J is very large, the following relations hold:
sign(n5) " sign(15) sign(q3) " sign(p3/s2) sign(n4) = sign(s2/15) sign(q2)
" sign(t5/(151L3))
We note that these coefficients, are always positive.
Whereas, the limit of oi, if J is large, is:
)ifll °i # o~~ =
fl7
~ ~~~5
(4)
where p7, s2 and t5 are given in Appendix C.
We call ois the simplified value of oi for large values of J. Therefore, when J is large, the
problem is reduced to the study of the signs of ma and ois. After simplifications, the last coefficient, given by equation (4), can be expressed as:
ois = ml mo o IS)
with mo, ml, co and cl are given in Appendix B.
It is very important to note that the simplified approach requires only the calculus of two coefficients. This leads to a considerable reduction of the computations and the size memory, and to the less computational time. The results are shown in Figure 5.
E=0.I E=2
iooo
soo o
-loo o loo 200 -lo0 0 100 200
aj b)
-loo 0 loo 200
C)
Fig. 4. Coefficients oi and ma as functions of do oi depends slightly of J and ma is independent of J. a) and b) solid line: J
= 10 s, interrupted line: J
= 5 s, mixed line: J = 2 s. c) Curves of ma for E = 0.1 pu (solid line), E =1 pu (interrupted line), E
= 2 pu (mixed line).
J large
U s U
-200 -loo o loo 200
Fig. 5. Stability limits of the damped salient pole alternator yielded by the simplified approach (s: stable, U: Unstable).
4. Extension of TanJura's Method to a Damped Salient Pole Alternator
In his paper iii, Tamura studied only the case of an undamped salient pole alternator; the results have been used in one of our own papers [3]. In what follows, we apply Tamura's method to a damped salient pole alternator; this is an opportunity for us to redefine this
method, which we do below. However, we use a matrix notation, while Tamura uses a more
cumbersome notation.
The electrical and mechanical equations are first written as (see Appendix A):
V = ZI and Te = I~LI (6)
respectively, where: Z
= R + u~mL + pM.
de id -rt 0 0 0 0
uqe iq 0 -rt 0 0 0
V = vi
,
I = ii
,
R = 0 0 ri 0 0
0 iD 0 0 0 rD 0
0 iQ 0 0 0 0 rQ
-ltq 0 0 -m~Q -ltd 0 -m~f -m~D 0
ltd 0 m~f m~D 0 0 -ltq 0 0 -m~Q
L = 0 0 0 0 0
,
M = m~f 0 If mm 0
0 0 0 0 0 m~D 0 mm ID 0
0 0 0 0 0 0 m~Q 0 0 IQ
Then, small sinusoidal variations ofb, creating sinusoidal variations of currents, are considered;
the torque during these movements is a function of both the internal angle b and the slip s:
Te = T(b, s). The derivatives of T with respect to b and s are called synchronizing and damping torques.
Tamura's conditions for stability are then:
K~ =
~~
< 0 and K&
"
~~
> 0
It is essential to note that the above is independent of inertia.
The mathematical development consists in the superposition of a DC and an AC steady state
operations.
. DC steady state operation
This is obtained by letting p = 0, ude
" uqe = 0, vi = rfE/(u~m~f) and u~mo
" Ii s)u~.
Introducing these equations into equation (6) yields:
VD " ZD ID and TD " I[ LID
where: ZD
" R + Ii -s)U~L.
. AC steady state
This is obtained by letting p
= -jsu~, vi = 0, vde
= -VR sin bo, uqe = VRcosbo and
u~mo " Ii s)u~. The use of complex quantities:
Vd " iVReJ~° and Vq
= VReJ~°
leads to:
VAC " ZAG iAC
JOULNAL DE PHYSIQW EL T3,N°io. ocTouER iW3 67
where:
Vd Td
Vq Tq
VAC " o TAG " If
o ID
° IQ
and ZAG
" R + Ii -s)U~L jsu~m.
The actual time varying currents are then given by IA " lte(iAc) and the pulsating torque is:
TA " Ii L IA
. Resulting torque and stability
The torque resulting from the superposition of the two steady states is made of four terms;
each one may be written as: Tq
= I) L Ij, with ii or j
= A or D). Then:
K~ = £K]J and K&
"
~K/
i,j ~,j
with:
~~ " (
" #~l ~4~#
q
3Tq 31)
~ 3Ij
~~ 3b 3b ~~ ~ ~ ~
3b Evaluation is made easier if we note that )
= 0, out of its definition. This simple remark leads to:
3iD -13ZD
S " ~~~ W~~
We can verify that: §~
= -U~L. Also, §fi
= 0 yields:
~~ = -lte (2j( ~~~~ Acj
S S
with: @f
= -u~(L + jM). The following functions result:
KfD
= U~I[[(Z[~L)~ + LZj~]LID K)~
= u~lte (IiL2j~IL + jM )fAc + (2j~IL + jM)fAc)~ IAj Kf~
= U~I[lte ((Zj~L)~L + L2j(IL + jM)j TAG
K)D = u~lte (f~c (2j((L + jM)) + LZj~l) LID
Tamura Approach
u s U
-200 -io0 o loo 200
Fig. 6. Stability limits of the damped salient pole alternator yielded by extensive Tamura approach (s: stable, U: Unstable).
The DC steady state and TAG are independent of bo. This remark leads to:
~~)~ =
jVAc, ~j~ "ifAc and ~)
= -Um(iAc)
Then, we obtain these results:
KfD
= 0
j~jA = -lm (i~C~~A ~ ~~~~~~~
j~DA
= -i~Ldill(iAC)
j~AD -till(i~C)~~~
b
The numerical application yields the results represented in Figure 6, where stability
limits are depicted in the plane (bo,E).
5. Comparison of Results
Comparison of Figures 5 and 6 clearly shows that results are identical when J is very large.
However, the mathematical work required by our simplified approach is much lighter than the mathematical work required by the method of Tamura and his predecessors.
. Case of finite values of J, machine without dampers
This case cannot be studied by the method of synchronizing/damping torques. We have pub-
lished some results pertaining to it in reference [3], but we include here some unpublished
results: in Figure 7, we have plotted the stability limits in the plane (bo,In(J)) for various values of E. We note the existence of small stable domains around bo = 0° and bo = 180°, for small values of J.
E=0.i E=2
S
~ S S S S[ ~
U U
o s
s
-200 -100 0 100 200 -200 -100 0 100 200
Fig. 7. Stability limits of an undamped alternator in the plane (do, In(J)), for various values of E
(s: stable, U: Unstable).
E=0.i E=2
s s
3
~ U U ~~ U U
U
s
~
-100 0 100 200 -100 0 loo 200
Fig. 8. Stability limits of a damped alternator in the plane (do, In(J)), for various values of E
(s: stable, U: Unstable).
. Case of finite values of J, damped machine
The unstable domain around the axis bo = 0° in Figure 7 is considerably reduced in Figure 8.
It is maintained only for small values of J. We have also plotted, in Figure 9, the stability limits in the plane (bo,E) for various values of inertia.
6. Conclusion
An undamped alternator is unstable around the zero value of the internal angle (bo
" 0). The
damper circuits make this region stable for the general case of an important value of inertia.
In fact, the part of these dampers is reduced as far as the inertia decreases. This result can not be shown by the anterior approaches for which the inertia is not taken into account.
In the case of large value of inertia, our approach yields to a considerable simplification of the stability study, which is limited to the discussion of only two parameters. This result is
an agreement with the anterior approaches, especially the one developed by Tamura which we have generalized for damped alternator. This has been achieved using a new formulation based
J=0.i s J=0.2 s
u
' u s s u '
, u u
s
0 ' 0 '
-200 -100 0 100 200 -200 -100 0 100 200
a) b)
J=0,5 s J=I s
u s u u s u
-200 -100 0 io0 200 -200 -loo 0 100 200
c) d)
Fig. 9. Stability limits of a damped alternator in the plane (do,E), for various values of J (s: stable, U: Unstable).
on a matrix development. This formulation leads to the synchronizing and damping torques expressions. Nevertheless, this formulation remains much more complicated than our method which turns to an elementary calculation for large inertia.
Appendix A
After to apply the Park transformation for the damped alternator equations, we obtain the follows equations, written in p-u-, projected on the both directed "d" and quadrature "q" axis
(see Figure 11.
A. I. PARK TRANSFORMATION. The "dq" functions given by Park transformation are writ- ten as:
~ cos(9m cos(9m ~f cos(9m + §') ga
9d
~~ ~ ~~
sin(9m) sin(9m ~f) sin(9m + ~f g~
A.2. ALTERNATOR EquATioNs. The damped alternator is described by:
~~ ~~~~ i~~~ ~i~
Uq = -rata + iilbd #
vi = rfif +
o = rDiD + %
° " ~Q~Q + @
where:
fid " ldid + m~fif + m~DiD
~fiq " 'qiq + maoio
If = lfif + mafid + mfDiD
ID = lDiD + m~Did + mfDii
#q " lqiq+lJlaqiq
A.3. LINE EquATioNs. The alternator is connected to an infinite bus. The electrical line equations are:
l~
~ ~~~~ ~ ~ ~ j j j ~j
de R d e d m e q e
Uqe " VR COS I
# Uq Telq + uJmie1d ie~
where:
fit 9m-uJt-4~0~) and uJm
" UJ+fl
If we take about the both total resistance and inductances par phase, we have:
Ttltd "" IdTa ++ Tele
'tq " lq + le
Id " ltdid + lJlafif + lJlaDiD
#q " ltqiq + lJla~iq
Then we obtain:
-vR sin b = -rtid Ldm4q ~if
vR cos b
= -rti~ + ~am id #
A.4. MECHANICAL EquATioN. It is given by the dynamic fundamental relation:
~ duJm
W " ~~' ~
where Tm is the external torque and Te is the electromagnetic torque given by:
Te " Id lq jq id
A-S- STEADY STATE OPERATION. The steady state is obtained by letting the operator
(
= 0 in equations and it will be denoted by index "0", so that:
- VR sin bo " -rtido Ld4qo
vR cos bo = -rti~o + Ld4do
jdo = ltdido + m~fifo
4qo " 'tqiqo
with: ifo
" uf/rf = E/(u~m~f) and iDo " iqo " 0.
A.6. PARAMETERS VALUES. The parameters are expressed in per units:
rt = 0.ii lid
" 1.3 ltq = 1.0
rf = 0.0007 If = 1.16 m~f = 1.077
rD " 0.007 ID = 1.196 m~D
" 1.159
rq = 0.007 lq = 0.63 m~q = 0.6
VR " I-o u~ = I-o mm
= 1.159
Appendix B
The resolution of the equation system given by equations (I) and (2) can calculate the output currents of the alternator:
~~ ~ t ~ ~~
where:
AI = Iii AU =
tl~ and A
=
)[[ [[
and
DIP) = in + »Ld I»)Iin + »LqI»)I + Ld~LdI»)Lq1»)
with:
la
ii = VR((rt + pLq(p) cos bo u~Lq(p) sin bo)
&12 " ~~q0(Tt + flLq(fl)j ~d0UJ~q(fl)
a~i = VR((rt + pLd (P) sin(bo + u~Ld(P) cos bo)
a22 " ~d0(Tt + fl~d(fl)j ~q0UJ~d(fl)
Ld(P) and Lq(p) are the operational inductances which are expressed by:
~d(fl)
=
~~~ ~ ~lfl + b2p2
~ ~~~ ~ ~~~~ ~~~ ~~~fl~ "
~~~~~°fl
~~~~ ~~~~ ~~~ ~~ ~~~~~~~ ~~~~
(l +
QfI)(~~~ifl
+ &2fl~)
