PSSs in Multi-Machine Applications
9.4 Determination of the PSS parameters based on the P-Vr approach with speed perturbations as the stabilizing signal
9.4.1 The P-Vr transfer function in the multi-machine environment
tem, the relative amplitudes of the states could change. On the other hand, the participation factors indicate the relative degree of involvement not only of all the states in a mode on a dimensionless basis but also of the modes in a state; the participation factors are therefore a characteristic of the system, invariant to change in units. In the two examples the ampli- tudes of the right speed (note, only speed) eigenvectors for a given mode shape appear to correlate fairly well with the participation factors for the same mode; this may lead to the misconception that the amplitudes in the mode shape represent the ‘participation’ of the speed states in the selected mode.
The application of these tools will be demonstrated in analysing the dynamic behaviour of a multi-machine power system in Chapter 10.
9.4 Determination of the PSS parameters based on the P-Vr
Sec. 9.4 PSS parameters based on the P-Vr approach 463 paths through both the power system and the other generators to the electrical torque out- put on i. A question is: Compared to the SMIB system, do these additional paths diminish the effectiveness of the P-Vr approach to PSS tuning?
Figure 9.9 (a) Model of a generator in a multi-machine power system; (b) conceptually, with shaft dynamics on all machines disabled.
The terminology used here, i.e. `P-Vr transfer function', as defined above for the multi-ma- chine context is that introduced in [9]. However, this same transfer function has been deter- mined by different techniques elsewhere. For example, for the tuning of the PSS of a generator in a multi-machine power system, phase information on the P-Vr transfer func- tion has been determined by field tests [10] or is based on SMIB models with the machine inertia constant set to a very large value on the generator of interest [11], [12]. In references [10], [11]and [12] no attempt is made to employ the P-Vr transfer function for the formal tuning of PSSs in a multi-machine system - including the concept and setting of the PSS damping gain, or as a basis for the coordination of PSSs.
The method adopted here for calculating the P-Vr transfer function is that presented in [9].
The significance of this approach is that a simple direct method is provided for determining both the magnitude and phase response of the P-Vr transfer function for each generator.
The theoretical basis for the P-Vr characteristic of generator i, , in
a multi-machine system of N generators is considered in [14], [15]. With the shaft dynamics of all generators disabled it is shown that the electrical power or torque of the N generators is given by
. (9.5)
Furthermore, it is shown that matrices Avand are essentially diagonal or block diagonal due to the diagonal dominance property of the reduced network admittance matrix into which the generator dynamic admittances are embedded as network elements. For brevity,
(b) OP
WE R YS TS ME
Machine i in a n
/(sMi)
o/s
AVRi Di
generator system
i
Pmi
i Pei
Vri
Vti
Machine i in a n
o/s
AVRi
generator system
i
i
Vri
Vti
(a)
Di
Pdi Pdi
Pei
Pei s = i Vs ri s
Pe s = Av Vs r s +B s s B
let us now consider only the first term in (9.5) associated with the P-Vr-like matrix of gen- erators in the multi-machine system, i.e.
, where from [15], (9.6)
, (9.7)
and , (9.8)
and where Y is the reduced network admittance matrix. In the first term of the summation in (9.7), - and in (9.8) - are essentially functions of the steady-state conditions and are modified by the generator operational reactances and . Moreover, in (9.7) the first term is determined mainly, and diagonally dominated, by the network admit- tance matrix Y. The Thévenin equivalent of the network as seen from the terminals of gen- erator i is not much affected by the dynamics of the other generators in the system. The second term in (9.7) depends only on the parameters of the generator i, its excitation system and a scalar multiplier vdo_i , the d-axis steady-state terminal voltage, i.e.:
, (9.9)
where Ggen_i , Gavr_iand xdi are respectively the operational transfer functions of generator i, its AVR / exciter, and its direct-axis synchronous reactance; these functions are independ- ent of the external system.
The phase characteristic of is independent of operating conditions in the external system, however, the magnitude of the low-frequency response varies only with the scalar gain vdo. The magnitude characteristic thus retains its shape over the range of operating con- ditions. Consider firstly the variation of with generator reactive power output at con- stant real power (P) and 1 pu terminal voltage as shown in Table 9.7. The low frequency gain of the P-Vr characteristics decreases with increasing lagging reactive power (Q); this obser- vation is reflected in the P-Vr characteristics of Figure 5.16 for the SMIB system.
Likewise, as illustrated in Table 9.8 and manifested in the P-Vr characteristics of Figure 5.22, decreases with decrease in real power output at unity power factor. At rated power out- put vdo is relatively large, but tends to zero as the real power output is reduced.
A consideration of Table 9.7 suggests that it is prudent to include a range of reactive power outputs in the set of encompassing operating conditions.
Pe s = Av Vs r s
Av s = G3 s Y G– 1 s –1G5 s +G6 s G3 Zs Gs 5 s +G6 s
=
Z s = Y G– 1 s –1 G3 s G1 s
xd s xq s
G6i s
G6i s = vdo_iGgen_i Gs avr_i s xdi s
G6i s
vd0
vd0
Sec. 9.4 PSS parameters based on the P-Vr approach 465 Table 9.7 Variation of with increasingly lagging Q at constant real power outputs
Table 9.8 Variation of with reduction in P at unity power factor
Because is the more significant term of the two in (9.7), it determines the consistently narrow bands of the frequency responses of the P-Vr characteristics of generator i given by . For example, for the SMIB system the relatively minor varia- tions in the phase of the P-Vr characteristic with steady-state operating conditions observed
in Figure 5.16 are caused by the contribution of first term in (9.7), , over the modal frequency range.
In [15] the authors imply that the P-Vr characteristic of generator i can be calculated if the network is represented by a SMIB system connected at the generator terminals. However, in multi-machine cases, it may not be clear what value should be attributed to the impedance of the Thévenin equivalent, particularly as it will change with line outages, whether electri- cally close-by machines are on/off line, the effect of close-by loads, etc. It is then simpler and more efficient to calculate the P-Vr characteristics of generator i1 for each operating condition using the complete model of the multi-machine system. Moreover, each generator may participate in a range of local- and inter-area modes as well as intra-station modes, not in a single mode as is the case in the SMIB system.
These results in [15] provide a theoretical basis for the observation in [13] that the P-Vr transfer function is relatively robust to changes in the system operating conditions in multi- machine systems. That is, for higher values of generator real power outputs both the gain, phase and the shapes of the frequency response of the P-Vr transfer functions do not vary appreciably, for practical purposes, over a wide range of operating conditions and system
Corresponding values of Q and
P=0.9, Q pu -0.2 0 0.2 0.4
pu 0.930 0.851 0.766 0.686
P=0.7, Q pu -0.2 0 0.2 0.4
pu 0.892 0.783 0.680 0.591
Corresponding values of P and
Q=0, P pu 0.9 0.7 0.5 0.3 0.1
pu 0.851 0.783 0.669 0.475 0.177
1. Or the characteristics of generating station i if there are a number of identical units in the station.
vd0
vd0
vd0
vd0
vd0
vd0
vd0
G6i s
Av_ii s = Pei Vs ri s
G3 Zs Gs 5 s
configurations. Consequently, in multi-machine systems, individual PSS designs that are based on the synthesized P-Vr transfer function using the methodology adopted in Section 5.10 are also robust over a wide range of operating conditions. Typically, this applies for generator real pow- er outputs exceeding 0.5 pu. An examination of Figures 5.21, 5.22 and Tables 5.5 and 5.6 reveals that the mode shifts are essentially real, an observation which supports the above statement.
The robustness and application of the P-Vr characteristic has been demonstrated and veri- fied for generators on very large systems [20].
For the multi-machine system the P-Vr characteristics of each generator, , are calculated in a similar fashion to that for the single- machine infinite-bus system, except that the characteristics are calculated for the entire net- work with the shaft dynamics of all machines disabled. The calculation is similar to that de- scribed in Section 5.10.3 in which rows and columns of the A, B and C matrices associated with the speed states in the states equations (3.9) are eliminated; the D matrix is usually a null matrix. The relationship between perturbations in electric power (or torque) as the output quantity and voltage reference as the input quantity can then be formed, and the frequency response evaluated for the set of encompassing operating conditions and over the range of modal frequencies.
The derivation of the synthesized transfer function,
, (9.10) which is selected from the family of P-Vr frequency response characteristics as the most suitable basis for the tuning of the PSS, has been covered in Section 5.10.6.
9.4.2 Transfer function of the PSS of generator i in a multi-machine system