State equations, eigen-analysis and applications
3.9 Mode shapes
Let us assume that we able both to excite a particular mode and to evaluate the time respons- es of the states of the system. In (3.22) it was noted that the response can be separated into its natural and forced components. Assuming (i) non-zero initial conditions on the states, and (ii) no forcing signals applied at the inputs to the system, we can write the equation for the natural response of the states in the form .
In the previous section a decoupled form of the state equations is derived (3.35), ,
assuming the pseudo-states and the original states, respectively, are related by . With no external excitation at the inputs and initial conditions , the natural response is
, where is the Laplace transform of ;
i.e. . (3.42)
On expressing the latter equation in terms of the original state variables, but retaining the decoupled modes, the response becomes
,
the right and left modal matrices ( ) being defined in (3.31). An alternative form of (3.42) is
. (3.43)
If we account for the fact that the inner product of two vectors is a scalar we can rewrite (3.43) as
.
Let us assume that initial conditions on the states are set equal to the right eigenvector of the ith eigenvalue, i.e. . The latter equation becomes
X s = sI A– –1x 0
z· = z+Bˆ U z and x
x = Vz z 0
z t L–1 sI––1
z 0
L–1 diag 1 s–1 --- 1
s–2
--- 1 s–n ---
z 0
= =
Lf t f t
z t
e1t 0 0 0 e2t 0
0 0 ent
z 0
diag e1te2tent
z 0
=
=
x t V diag e1te2tent
W x 0
=
V and W
x t vh eht wh x 0
h=1 n
=
x t vhwh x 0 eht
h=1 n
=
x 0 = vi
Sec. 3.9 Mode shapes 81
. (3.44)
From (3.32), . Hence the above equation reduc-
es to
. (3.45)
Thus onlymode i is excited. Moreover,
. (3.46)
Note that each of the modal responses, , has an identical form but their shapes are determined by the initial amplitude in each response. Thus for a given mode, the relative amplitudes or shapes of the responses are determined by the associated right eigen- vector. Consequently, we can plot the mode shapes for selected modes, these shapes revealing not only the relative amplitude of the states in the mode, but also the relative phase between the responses of the states. From (3.46) the relative amplitudes of states at time t are
: , : , ... ,1, ... : , or : , : , ... ,1, ... : , (3.47) where is the element with the largest magnitude among the selected states whose mode shapes are to be displayed; this result will be employed later in Chapters 9 and 10.
A note of caution. Prior to the development of participation factors (see Section 3.10), the element of the right eigenvector was employed to determine the ‘involvement’ of the state variable in mode i. A large relative value of was assessed as representing a sig- nificant involvement of in the ith mode. However, this is misleading as the numerical val- ues of the elements depend on the units selected (e.g. speed in pu, angle in rad.) for the associated state variables, i.e. they are not dimensionless, they are scaling dependent. It should be noted, therefore, that the relative amplitude of the component re- vealed in (3.47) should not be interpreted as implying the relative participations of states j and m in mode h. The concept of ‘participation’ is considered in Section 3.10.
In a practical application, the elements in the right eigenvector corresponding to the speed states of all generators are selected to reveal the speed mode-shape, say, for an inter-area mode. For such a mode the relative phase between the speed states reveal, for example, that machines in areas A and B swing against generators in area C. The plots of mode shapes and their significance in the analysis of dynamic behaviour of power systems will be discussed in more detail in Chapters 9 and 10.
x t vh wh vieht
h=1 n
=
wh vi = 1 when h=i and is zero when h i
x t = vi eit
x1 t =v1i eit, xk t =vki eit, , xn t =vni eit x1 to xt n t
vki
x1 xk x2 xk xn xk v1i vki v2i vki vni vki vki
vki
xk vki
xk vki
vjh vmh
82 State equations; eigen-analysis Ch. 3 The significance of modal response and the mode shapes are most simply illustrated by means of a numerical example.
3.9.1 Example 3.5: Mode shapes and modal responses The state matrix of a system is given by . Determine its modal responses and mode shapes.
The eigenvalues of are evaluated from the characteristic equation, ; there
are two eigenvalues, a slower at .
Let the right modal matrix be . The right eigenvector associated with the eigen- value is found from (3.29),
, i.e. .
Likewise, for , .
Let , then .
The time response, as given by (3.43), is
, or
. The form of this response illustrates a number of significant points.
As stated earlier, if the initial value of the states is equal to either of the right eigenvectors, the associated mode is the only mode present in the response, e.g. for eigenvalue 1,
,
.
A 0 2
4 – –6
=
A det hI A– = 0
1 = –2 and a faster at 2=–4 V 1 1
2 2
=
1 = –2
Avh = hvh 0 2 4
– –6
1
2
–2 1
a2 or 2= –1
=
2 = –4 0 2
4 – –6
1
2
–4 1
2 or 2=–21
=
1 = 1 = 1 V 1 1
1
– –2 v1 v2 and W V–1 2 1
1
– –1
w1 w2
= = =
= =
x t 1 1 1 – –2
e–2t 0 0 e–4t
2 1 1
– –1
x1 0 x2 0
=
x1 t x2 t
1 1
– 2x1 0 +x2 0 e–2t 1 2
– –x1 0 –x2 0 e–4t +
=
1 = –2 x1 0 x2 0
1
2 1
1
– , x1 t x2 t
1 1
– 2x1 0 +x2 0 e–2t 1 1 – e–2t
=
=
= =
Sec. 3.10 Participation Factors 83
Correspondingly, for eigenvalue 2, :
.
Note that in each of the two modal responses, the responses are related by a constant factor for all , i.e. for 1, and for 2. We ob- serve that
(i) for both modes, the responses of the states and vary in anti-phase;
(ii) for the slow mode the relative amplitudes of the two states are same, but for the fast mode they differ by a factor of two.
Thus, for a given mode, the mode shape reveals not only relative phase between the time re- sponses of the states but also the relative amplitudes of the states in the modal responses; fur- thermore, the mode shape is determined by the right eigenvector of the associated eigenvalue.
Further insight into the physical significance of mode shapes is provided in Section 9.2.
---