3.2
Modal Analysis for Control
The modal analysis of the state matrix is used for small-signal stability analysis of power systems. However, for control design, the open-loop transfer function between specific variables is of interest. For oscillation control design the use of linear control theory is possible because the oscillations of power systems about an operating point can be linearized, although they should be tested using non-linear
simulations under many different of operating conditions [30].
3.2.1 Transfer functions
The state-space representation describes the complete internal behavior of the sys- tem and the input and output properties. The transfer function representation, in contrast, is related only the input output behavior. Thus, any selection of the state variables can be done if the system is only defined by a transfer function. In contrast, the transfer function is defined if a state-space representation of a system is known [3].
In section 3.1.2the state equations in terms of the transformed variables z where
derived. Here they are repeated for convenience.
˙z = Λz + ΨB∆u (3.23)
∆y = CΦz + D∆u (3.24)
The transfer function between an input and an output, can be obtained from these equations. Substituting d/dt by s and eliminating z, the transfer function, or
transfer function matrix when y or u are vectors is [30]:
∆y
∆u = G(s) = [ CΦ(sI − Λ)−1ΨB + D ] (3.34)
The poles of G(s) are given by the eigenvalues of the state matrix A previously defined in section3.1.
3.2.2 Controllability and Observability
In order to modify a mode of oscillation by feedback, the mode must be excited by the input and it must be visible in the chosen output.
The ithmode is controllable by the jthinput if the product Ψ
iBj is not zero. The ith
mode is observable in the jthoutput if the product C
iΦj is not zero. The magnitude
of the products are defined as controllability and observability respectively.
These products can be identified in equations 3.23 and 3.24. In addition, if the
product ΨiBj is zero, the ith mode is uncontrollable if the product CiΦj is zero the
3.2. Modal Analysis for Control
3.2.3 Residues
The transfer function matrix can be expanded in partial fractions in terms of the individual modes as [30]: G(s) = n X i=1 CΦiΨiB s − λi + D = n X i=1 ri s − λi + D (3.35) the terms ri= CΦiΨiB (3.36)
are called the residues of the eigenvalues [30], [3]. If there is more than one input
or output, residues will be matrices. It can be seen from equation 3.36 that each
residue is the product of the corresponding observability and controllability. If a mode is either uncontrollable or unobservable, the corresponding residue will be zero. Residues are useful to have an idea of which modes will be affected most by feedback, since they give the sensitivity of an eigenvalue to scalar feedback between
output and input. Modes with poor damping i.e. λi with small absolute real part,
will significantly influence the magnitude of the transfer function Gr if it is scaled
up by the residue ri at around the frequency corresponding to the imaginary part
of λi. Therefore, the controllability of the input signal and the observability of
the feedback signal are very important. The chosen feedback signal must have a high degree of sensitivity around the mode to be damped out. Moreover, the output signal must have little or no sensitivity to other swing modes, expecting to minimize the interaction among other modes through the controller.
The residue is a complex variable. The angle of the ithresidue indicates the direction
in which the root locus leaves the ith pole [30]. The residue angle is related to the
phase compensation required at each of the modal frequencies needed to produce the desired damping [30],[25],[28]. Hence, selecting the most appropriate signal for damping control design could be done based on the residues, because the sensitivity of the eigenvalue to feedback can be modified with a feedback transfer function. This will modify the sensitivity of the original system by the value of the feedback transfer function evaluated at the original eigenvalue.
It is important to consider the phase angle of the residues in addition to their magnitude. The higher magnitude of the residue will, in principle, indicate the combination of input output which is most effective to damp a particular oscil- latory mode. However, the magnitude of the residue could change with different operating conditions which at the same time will modify the effectiveness of a par- ticular input/output combination. Choosing the magnitude of the residue as a
criteria is only effective for damping a single mode [25], whereas an approach based
on the phase angle of the residues, will allow a damping of multiple modes. If a
residue angle for a particular mode λi varies widely over different operating condi-
tions it is very difficult to synthesize a controller that provides an adequate phase compensation for all of them. However, the magnitude has to be also considered. It must be taken into account that the residue approach only can be considered as a sensitivity and not as the final direction in which the mode will be displaced in the complex plane. That is because of the nonlinear behavior of the mode for the
3.2. Modal Analysis for Control closed loop condition [30].
3.2.4 Selection of input/output signals
The effectiveness of the damping control may depend on the appropriate selection of feedback signals. Hence, different signals should be considered in a study for determining the most effective one and the most robust for different operating conditions.
Figure3.9 depicts the phase compensation required to damp an electromechanical
mode. θ is representing the residue angle, while ϕ represents the compensation angle to damp the mode. In this case the phase compensation will displace the eigenvalue horizontally. Damping can be introduced in the way that the eigenvalue moves in the complex plane to the left side due to the effect of the feedback controller, or in
other words, the direction of the movement should have an angle between 90◦ and
270◦ with respect to the horizontal axis, represented in figure3.9 by the shadowed
area. The ideal compensation would mode the eigenvalues in a direction parallel to the real axis.
θ φ
Im
Re
0
Figure 3.9: Representation of the eigenvalue sensitivity by means of the residue
angle and the required compensation to damp a mode
However, the effect of this compensation on a single mode to other modes should be observed. The compensation applied must not modify another eigenvalue to a poor damped or unstable situation, unless there is a beneficial result from it such as a much larger effect on the damped mode than on the other. This is a challenge especially for effectively compensate those modes close in frequencies, such as be- tween interarea modes, which would be very difficult to separate to apply different phase compensation, and therefore they will be compensated with the same angle. It is also important to evaluate the magnitude, which plays an important role in such cases. The higher the magnitude of the residue, in principle, the bigger the effect that the combination of input output will will have on the mode. Therefore, if the compensation of a mode with higher residue magnitude presents a detrimen- tal compensation for another mode with much smaller residue magnitude, control may be feasible. Then, the ideal case, would be that all the modes had the same residue angle and they were effectively damped. However, the magnitude and the angle of the residue could change with different operating conditions which at the
3.3. Power System Oscillation Damping Devices