Computation of the controllable swing mode spectrum of FACTS and HVDC in large power systems

Abstract

This paper presents computation of swing modes of a large power system that could be significantly affected by power swing damping controllers in FACTS or HVDC devices at a given location. Modal controllability is a suitable measure to isolate these modes for analysis. Computation of the controllable swing mode spectrum is useful, especially in situations where the controller structure and feedback signals are not frozen (e.g., at the planning stage). This paper proposes two important steps that allow us to map the problem of finding highly controllable swing modes to the problem of finding the swing modes that have high transfer function residues (for which efficient algorithms are available). The steps are: (a) normalization of the eigenvectors corresponding to different modes and (b) identification of specific feedback signals for each type of FACTS/HVDC device such that the modal observability and modal controllability are tightly coupled. Once the mapping is done, a computationally efficient method like the Subspace Accelerated Dominant Pole Algorithm [16] (SADPA) can be adapted to find the highly controllable swing modes. The effectiveness of this approach is demonstrated by case studies of FACTS and HVDC devices in a 16-machine system and the Indian power grid.

Appendices

Appendix I. Derivation of constraint in Eq. (12)

Following a perturbation, the energy in the disturbance can be expressed as sum of potential and kinetic energy as follows:

$$\begin{aligned} E= & {} \frac{1}{2}(\Delta \delta ^TA_r\Delta \delta \;+\; \Delta \omega ^TM\Delta \omega ) \nonumber \\= & {} \frac{1}{2}\begin{bmatrix} \Delta \delta \\ \Delta \omega \end{bmatrix}^T\begin{bmatrix} A_r&[0]\\ [0]&M \end{bmatrix}\begin{bmatrix} \Delta \delta \\ \Delta \omega \end{bmatrix}. \end{aligned}$$

(A1)

It is constant for the unforced system (i.e., $u = 0$). If the $i{\hbox {th}}$ mode only is excited, then from Eq. (5) we have

$$\begin{aligned} \begin{bmatrix} \Delta \delta \\ \Delta \omega \end{bmatrix}= \begin{bmatrix} x_{\delta _i}\\x_{\omega _i} \end{bmatrix}z_{m_i} + \begin{bmatrix} x_{\delta _i}^*\\x_{\omega _i}^* \end{bmatrix}z_{m_i}^*. \end{aligned}$$

(A2)

Using Eq. (A2) in Eq. (A1), it can be shown that

$$\begin{aligned} E= & {} \frac{1}{2}[z_{m_i}^2(x_{\delta _i}^TA_rx_{\delta _i}+x_{\omega _i}^TMx_{\omega _i}) + (z_{m_i}^*)^2(x_{\delta _i}^{H}A_rx_{\delta _i}^*+x_{\omega _i}^{H}Mx_{\omega _i}^*) \nonumber \\&\quad +\, 2|z_{m_i}|^2(x_{\delta _i}^{H}A_rx_{\delta _i}+x_{\omega _i}^{H}Mx_{\omega _i})]. \end{aligned}$$

(A3)

From Eqs. (3) and (4), it can be shown that

$$\begin{aligned}&x_{\delta _i}^TA_rx_{\delta _i}+x_{\omega _i}^TMx_{\omega _i} = 0 {\text {,}} \end{aligned}$$

(A4)

$$\begin{aligned}&x_{\delta _i}^{H}tA_rx_{\delta _i}^*+x_{\omega _i}^{H}Mx_{\omega _i}^* = 0, \end{aligned}$$

(A5)

and therefore

$$\begin{aligned} E = |z_{m_i}|^2\;(x_{\delta _i}^{H}A_rx_{\delta _i}+x_{\omega _i}^{H}Mx_{\omega _i}) = E_i \end{aligned}$$

(A6)

where $E_i$ denotes the energy when only the $i{\hbox {th}}$ mode is excited. It can be shown that when several modes are excited, the disturbance energy is given by the sum of all modal energies, i.e, $E = \sum _{i = 1}^{(n_g-1)} E_i$. Note that $E_i$ is also a constant and does not depend on scaling of the eigenvectors.

The strength of the modal signal $y = v_i^Tz = z_{m_i}$ is given by its amplitude, which is equal to $|z_{m_i}|$. When only one mode is excited at a time, with the same disturbance energy $E_i$, strength of the modal signal being equal across modes implies $(x_{\delta _i}^{H}A_rx_{\delta _i}+x_{\omega _i}^{H}Mx_{\omega _i})$ is the same across all modes (as seen from Eq. (A6)). Without loss of generality, we assign its value to be 1 and obtain the following condition:

$$\begin{aligned} x_{\delta _i}^{H}A_rx_{\delta _i}+x_{\omega _i}^{H}Mx_{\omega _i} = 1. \end{aligned}$$

(A7)

Using the initial assumption of $x_{\delta _i}$ being real, we obtain the result of Eq.  (12).

Appendix II. Example of dominant pole at infinity

Let us consider the state-space matrices $E = \begin{bmatrix} I&0\\0&0 \end{bmatrix},\; A = \begin{bmatrix} \alpha&0\\ 0&\alpha \end{bmatrix} {\text {where}}\; \alpha = \begin{bmatrix} 0.1&0.2\\ -\,0.2&0.3 \end{bmatrix}$ and $b = c = [0.1\;0.1\;1.0\;1.0]^T$. Generalized eigenvalue analysis of this system yields a mode $0.2 \pm 0.17321i$ with residue of 0.01 and two poles at infinity with residue of $\infty$.

Trajectories of the eigenvalue estimate over iterations, when SADPA is applied on this system with $|\rho _i|$ (or $|\rho _i||\lambda _i|$) (case (a)), and $|\rho _i|/|\lambda _i|$ (case (b)) as selection criteria separately are shown in the following table. It is clearly seen that the selection criterion in case (b) avoids convergence to the dominant pole at $\infty$.

Iteration	Eigen-estimate
	Case (a)	Case (b)
0	0 +j 0.1700	0 +j 0.1700
1	25.2768 –j 5.1847	25.2768 –j 5.1847
2	(2.72 +j 3.50)$\times 10^5$	0.2944 +j 0.1997
3	$\infty$	0.2000 –j 0.1732

A, b, c, d:

state-space matrices

z :

state variables

$\delta , \omega$ :

rotor angle (rad), speed (rad/s)

$n_g$ :

number of generators

$\Omega _i$ :

frequency of the $i{\hbox {th}}$ swing mode (rad/s)

$x_{\delta _i}, x_{\omega _i}$ :

components of right eigenvector corresponding to rotor angle and speed, respectively

$v_{\delta _i}, v_{\omega _i}$ :

components of left eigenvector corresponding to rotor angle and speed, respectively

Re(.):

real part of a complex number

$(.)^*$ :

element-wise conjugate of a complex vector

$(.)^T$ :

transpose

$(.)^H$ :

conjugate-transpose of a vector

G(s):

SISO transfer function

SSPF:

sum of slip participation factors