PSSs in Multi-Machine Applications
9.2 Mode Shape Analysis
9.2.1 Example 1: Two-mass spring system
A two-mass spring system which is constrained to move freely in the positive x-direction from a reference position is shown in Figure 9.1(a). The instantaneous position and speed of the centre of mass j is (m) and (m/s), respectively, are highlighted in Figure 9.1(b). is the mass (kg), is the viscous damping coefficient (N/m/s) between the mass and the ground plane, is the spring stiffness coefficient (N/m), and is an externally applied force (N).
Figure 9.1 (a) A two-mass system free to move in the x-direction on a flat surface, (b) the general form of the parameters and variables for the jth mass.
Based on Figure 9.1(b), a general form of the equation of motion for mass can be ex- pressed as [6], [7], [8]:
. (9.3)
This equation can be rewritten in state equation form as follows:
vi
i
xi t = vi eit
xj t vj t
Mj Bj
Kjk fj t
K12 Kij
K20 Kjk
Bj
M2 M1
B2 (a)
B1
x2 x1 xj
Mj vj v1
v2
(b)
k i
f1(t) fj(t)
K01
Reference plane, node 0
Mj
fj –Kijxi+Mj dt dvj
Bjvj Kij+Kjkxj–Kjkxk
+ +
=
Sec. 9.2 Mode Shape Analysis 451
, and
. (9.4)
Applying the above relationships to the two masses in turn, a fourth-order set of state equa- tions is formed in the state variables ; the derivation of the set of equations is left as an exercise to the reader.
Consider the following parameters for the four-mass spring system:
, and
For these values of the system parameters the eigenvalues of the system are given in Table 9.1.
Table 9.1 Eigenvalues of the two-mass spring system
We note that there are two stable oscillatory modes having damping ratios of 0.067 for mode A (which is associated with the complex conjugate eigenvalue pair 1,2) and 0.055 for mode B (eigenvalue pair 3,4). However, there is no information that reveals the nature of the sys- tem performance; for example, what is the relative characteristic behaviour of the masses for mode A?
The right speed-eigenvectors for the two oscillatory modes are shown in Table 9.2. It is not- ed for mode A, when it alone is excited, that the speed states and of masses 1 and 2 are essentially in anti-phase. The mass is said to ‘swing against’ mass . The displace- ment states and , which are almost in anti-phase, lag their respective speed states by nearly . When the right eigenvectors are normalised to for the state with the largest magnitude (the speed state for mode A, for mode B), the modal behaviour of the states is interpreted more easily using the polar plots for the relevant modes as shown in Figure 9.2.
Often in mode-shape analysis only the speed elements in the right eigenvector are plotted.
In this event the plot is the same as that in Figure 9.2 except all other states are omitted.
Eigenvalue number and value
1 2 3 4
-0.214+j3.21 -0.214-j3.21 -0.098+j1.77 -0.098-j1.77 v·j Bj
Mj ---vj
– Kij
Mj
---xi Kij+Kjk Mj ---xj
– Kjk
Mj ---xk 1
Mj ---fj
+ + +
=
x·
j = vj v1 v2 x1 x2
M1 = 2 M 2=4 B 1 = 1 B 2 = 0.5 K01 = 10 K 12 = 8 K 20 = 10.
v1 v2
M1 M2
x1 x2
90 1 0
v1 v2
Table 9.2 Right eigenvectors for the oscillatory modes
Figure 9.2 Normalised right eigenvectors of speed (v) and displacement (x) for the oscillatory modes
Let us now consider the time-domain responses of the states when the mass-spring system is excited by the right eigenvector consisting of the real parts of its elements for each of the modes in Table 9.2, e.g. by the initial condition for mode A. The transient response to this initial condition is shown in Figure 9.3.
Note in Figure 9.3 the instantaneous phase relationship between the states is consistent with Figure 9.2 and/or Table 9.2. From the figure it is seen that
• the time constant and the period of the response are consistent with the single mode,
; State
Right eigenvectors
Mode A: Mode B:
Magnitude Angle Magnitude Angle
0.904 180 0.492 -5.3
0.309 -9.6 0.719 0
0.281 86.2 0.277 -98.4
0.096 -103.4 0.405 -93.2
0.214
– j3.21 –0.098j1.77
v1 v2 x1 x2
-0.5 0 0.5 1
v1 x1
Mode A: -0.214+/-j3.21 Mode B: -0.098+/-j1.77
1 1
1
0.5
0.5
0
0 0
-0.5
-0.5
-1 -0.5 -1 -1-1
x2 v2
v1 v2
x2 x1
0.5
0.904
– 0.305 0.0188 0.0223–
T
0.214 – j3.21
Sec. 9.2 Mode Shape Analysis 453
Figure 9.3 First ten seconds of the transient response speed (v) and displacement states (x) for the two-mass spring system to an initial condition which excites only
mode A, .
• the speed states and as well as the displacement states and are, respec- tively, nearly in anti-phase;
• and , respectively, lag and by nearly ;
• as might be expected for mode A, , in which the masses swing against each other as shown in Figure 9.3, the amplitude of the oscillation of the smaller mass is larger.
Likewise, as seen in Figure 9.4 if the system is excited by an initial condition on the four states in Table 9.2, only mode B is excited. In this case the speed states and as well as the displacement states and are, respec- tively, nearly in-phase, i.e. the two masses ‘swing together’ with respect to the reference frame. Again, the form of the responses is consistent with the results in Figure 9.2 and/or Table 9.2.
0 1 2 3 4 5 6 7 8 9 10
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Time (s)
Amplitude
v1 v2 x1 x2
0.214 – j3.21
v1 v2 x1 x2
x1 x2 v1 v2 90
0.214 – j3.21
0.490 0.719 0.0407– 0.0224–
T
v1 v2 x1 x2
Figure 9.4 First ten seconds of the transient response of the two-mass spring system to an initial condition which excites only mode B, .
It is noted above for mode A, in which the masses swing in anti-phase, the amplitude of the oscillation of the smaller mass is larger. This suggests that the nature of the oscillations ob- served in the responses of Figure 9.3 and Figure 9.4 are associated with the interchange of energy between the energy storage elements. Let us calculate the instantaneous stored ener- gies in the masses and the spring. The instantaneous stored energy in a mass is and that in a spring is . For mode A the time responses of the stored energy in each of the five elements for the relevant initial conditions are plotted in Figure 9.5.
As is to be expected, the envelope of the decay of the stored energies decays with a time constant of one-half of that of mode A1. Further we note:
• The stored energy in each of the two masses peak more-or-less simultaneously; at that time the stored energy in each of the three springs is zero;
1. Assume that the response of a speed state of a mass is . The stored
energy will decay as .
Time (s)
Amplitudes
v1 v2 x1 x2
0 1 2 3 4 5 6 7 8 9 10
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
0.098 – j1.77
Mjvj22 Kjkxj–xk22
v t = V0e–t v2 t = V02e–2t
Sec. 9.2 Mode Shape Analysis 455
• A quarter cycle later of the modal frequency (3.21 rad/s, period approximately 2 s), the latter condition is reversed, i.e. the stored energies in the springs peak more-or-less simultaneously; at that time the stored energy in each of the masses is zero.
• If the losses during the interchange were zero (i.e. no viscous damping, ), the system would oscillate indefinitely with constant amplitude and the peaks and troughs in the responses would coincide exactly.
Figure 9.5 Stored energy response in each of the masses and springs for an initial condi- tion which excites only mode A, .
A plot of the stored energy responses, similar to Figure 9.5, for mode B ( ) can be predicted from the mode shape shown in Figure 9.2 or the amplitude responses of Figure 9.4. This is left as an exercise to the reader.
The interchange of energy between energy storage elements every quarter of a cycle of the oscillatory behaviour is explained in any text book on the fundamentals in physics or engi- neering. The significance of mode shapes in the analysis of dynamic performance is that it reveals the nature of the behaviour of the masses (or inertias) in selected modes - normally the electro-mechanical modes in power system dynamic performance.
B = 0
0 2 4 6 8 10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Time (s)
Stored energy in each of 2 masses and 3 springs (joule)
Spring 0-1 Spring 1-2 Spring 2-0 Mass 1 Mass 2
0.214 – j3.21
0.098 – j1.77
This example illustrates that, for the two-mass-spring system, there is one oscillatory mode representing the relative dynamic behaviour between the two masses. The second mode portrays the behaviour of the masses with respect to the reference frame.