Application of the PSS Tuning Concepts to a Multi-Machine Power System
10.9 Correlation between small-signal dynamic performance and that following a major disturbance
10.9.2 The analysis of modal interactions [10], [11], [12]
Sec. 10.9 Small- and large-signal dynamic performance 513 The above example reveals the important features of small-signal analysis, that is, it furnish- es not only an understanding of the underlying modal structure of the power system and but also provides insights into a system's dynamic characteristics that cannot easily be derived from time-domain simulations for large magnitude disturbances. It is the case in Figure 10.33 that only a few of the thirteen modes appear to be excited; the nature and lo- cation of the fault does not significantly excite the local-area modes outside the faulted area at all. Understanding the nature of the small-signal modal behaviour therefore yields a syn- optic view of the system characteristics which would require many large-signal studies of faults in different locations to gain similar, but not exact, information [9].
Knowledge of the behaviour of certain local and inter-area modes has revealed the nature of the responses of the speed states following a major disturbance on the system. However, as stated earlier, the behaviour of the system is highly non-linear during the initial phase of the response. During the first 0.6 s certain exciters reach their ceiling voltages and some PSSs, together with most SVCs, hit limits on their outputs. In the context of the magnitude of rotor speed oscillations, the question is asked in Section 1.10, “how small is small?”. The peak amplitudes of the speed perturbations in Figure 10.32(a) are 1.5 to 2% which are not small. The functional non-linearities come into play and therefore the small-signal analysis is based is not strictly accurate. In the following section the applicability and validity of the small-signal analysis that has been conducted in this section is reviewed.
where , and are ‘conventional’ eigenvalues of the state matrix ; is an element of the right eigenvector corresponding to the eigenvalue or mode ; is a function of
the initial conditions; is a function of , .
Equation (10.8) reveals the relation between the state variables , the first-order system modes , and the second-order modes,
.
Note the following:
• The terms associated with the mode pairs represent “modal interactions” that arise due to the inclusion of the second-order terms.
• The second-order terms supplement information provided from the first-order linear approximation of the power system equations.
• If the system is stable, the second-order mode lies to the left of either of its constituent modes, or , in the complex s-plane; it therefore decays more rapidly than either of the individual modes.
• The “interaction coefficients”, , of the exponential terms in (10.8) provide a measure of the participation of any of the mode pairs in the state variable.
Firstly, for the first- and second-order modes discussed in the following, let us assume the linear coefficient term, , and the interaction coefficients in (10.8) are not negligible.
Secondly, we will assume and are the complex conjugate
pair of the dominant first-order mode, normally an inter-area mode. When
in (10.8), the mode pair ; likewise when , the
mode pair . Thus, due to modal interactions, a second-order mode of double the frequency and double the damping constant of the first order mode is introduced into the response; significantly, however, it decays in half the settling time of the linear mode.
Thirdly, let us assume there is some other, more heavily damped first-order mode present, . When in (10.8), the second-order mode pair is introduced. Thus the resulting complex second-order mode will be of higher frequency than the dominant first-order mode and, because , it will decay with a settling time of less than half the settling time of the linear mode.
j k l A vij
j zp0 h2klj 1k+l–j k+l–j0
xi t
1n
1+1
, 1+2, , n–1+n, n+n
k+l
k+l
k l
h2klj zk0zl0 k+l
vijzj0
j = –j+jj j+1 = –j–jj
k= and lj = j
k+l = –2j+j2j k= j+1 and l = j+1
k+l = –2j– 2j j
r r +1 = –r j r k= and lj = r
k+l = –j+r+jj+r
rj
Sec. 10.9 Small- and large-signal dynamic performance 515
Figure 10.34 Case 1. Responses of rotor angles of selected generators to a three-phase fault at the hv bus at the terminals of the transformers at BPS_2 cleared in 0.250 s. Angles
are relative to that of LPS_3. All PSS gains set to 20 pu on machine MVA rating.
The transient response of the rotor angles of generators is shown in Figure 10.34 for a three- phase fault at bus #206, the high-voltage bus at the terminals of the generator transformers at BPS_2. The system conditions are the same as those in Figure 10.32 except the fault clear- ing time has been increased from 120 ms to 250 ms. The system, which is marginally stable with rotor angle differences across the system reaching at about 1 s, is heavily stressed in the immediate post-fault period. The essential point of the analysis in this section is that in this period, during or after which no limiting by controllers occurs, one should be aware that second-order modes of some significance may arise. However, because such modes are bet- ter damped than their constituent first-order modes, they tend to decay more rapidly.
The analysis of modal interactions based on Normal Forms for a large system is compute- intensive and complex, mainly because of the size of the system and the number of combi- nations of both the second-order modes and the associated interaction coefficients.
Moreover, given the identical system conditions and type of disturbance, the latter coeffi- cients will vary depending on the instant in the transient response at which initial conditions are selected.
In the particular cases of Figures 10.32 and 10.34 it is clear from (10.8) that the first-order modes exist in the responses. However, without a detailed analysis based on Normal Forms it is unclear what modal interactions are present, and their magnitude at any instant - at least
0 1 2 3 4 5 6 7 8 9 10
−150
−100
−50 0 50 100 150 200
Rotor angles (deg.)
Time (s)
BPS−2 EPS−2 MPS−2
SPS−4 HPS−1 PPS−5
260
k+l
up to one half of the settling time of the dominant mode when the responses of second- order modes have effectively decayed away. From studies in the literature it appears that for less stressed systems the effects of modal interactions dissipate well within the latter time. For the Study Case 1, shown in Figure 10.32, this may well be the situation. In [14], [15] interesting comparisons are made between the transient response of a stressed system to major disturbances and the first- and second-order responses based on the results of Nor- mal Form analysis. For the scenarios considered the second-order responses agree closely with those derived from the transient responses based on the step-by-step simulation.
The above summary of modal interactions and their significance is necessarily very brief.
More extensive details are provided in other papers referenced in [10], [14], [15], [16].