# Extending the Dissipating Energy Flow Method to Flexible AC Transmission
Systems

Kaustav Chatterjee, Sayan Samanta, and Nilanjan Ray Chaudhuri
School of of Electrical Engineering and Computer Science
The Pennsylvania State University
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In recent years, the dissipating energy flow (DEF) method has emerged as a
promising tool for online localization of oscillation sources. In literature,
the mathematical foundations of this method is well-studied for networks with
synchronous generators. In this paper, we extend the analysis to felxible AC
transmission systems (FACTS). To that end, we derive the DEF-expressions for a
thyristor controlled series capacitor (TCSC) and a static synchronous
compensator (STATCOM) operating with conventional control strategies.
Analyzing their respective DEFs, we obtain the conditions for which these
FACTS devices could behave as the sources of oscillation energy. Our findings
are structured into propositions and are supported through numerical case
studies on IEEE test systems.

###### Index Terms:

Damping, dissipating energy flow, FACTS, oscillation energy, oscillation
source.

##  I Introduction

Sustained low-frequency oscillations may appear in a power system due to the
poorly-damped natural modes or as a result of an external periodic
disturbance. Such oscillations induce mechanical stress on the turbines, limit
utilization of existing transmission assets causing congestion, and in the
worst case, if not adequately damped, may lead to large-scale outages. An
important step in mitigating these is to identify the sources either
responsible for forcing these oscillations or contributing towards the
deterioration of the system damping. DEF-based oscillation source
localization, as proposed in [1], is one such method which has shown
significant promise when tested on real-world oscillation problems [2].
Derived from the theory of transient energy functions, the method
characterizes a power system component as a source or a sink of oscillation
energy depending on the slope of its DEF integral [1]. Despite its reasonable
success, it is important to note that, the DEF method may perform poorly when
the underlying modeling assumptions (see, [1]) are violated [3].

Moreover, while for forced oscillations, it is relatively easier to singularly
identify an element as a source from the high-value positive slope of its
DEF,111usually the amount of energy injected by the forcing function is much
higher than any other element it is more challenging for a poorly-damped
natural oscillation. In most cases, there can be more than one synchronous
generator which, depending on their damping torque characteristics and
observability of the mode, may appear as potential sources of the oscillation.
This can be further complicated by positive-sloped DEFs that may originate
from other controllable network elements like $-$ FACTS, high voltage direct
current (HVDC) systems, and converter-interfaced renewable energy sources.
While the theoretical connections between the DEF and damping contribution
from synchronous generators is well-studied in literature [4, 5, 6], no
insights have been developed relating the role of the control schemes in power
electronics-interfaced devices in shaping their respective DEFs.

To that end, in this paper, we extend the mathematical analysis of the DEF to
study the oscillation-sink/source behavior of the FACTS devices and their
controls. We derive the transient energy functions for 1) a TCSC, and 2) a
STATCOM. For each device, we then compare among the possible control schemes
to check for the presence (or absence) of path-dependent integrals in their
respective energy functions. Since the existence of a path-dependent terms
implies transient energy dissipation (or production), we can then build on
this to mathematically explain the nature of their DEFs. Establishing this
link between the control law, the path-dependent transient energy integral,
and the slope of the DEF serves two purposes: first, it helps us appreciate
how a control law influences the source/sink behavior of a given device, and
second, it gives us an intuitive understanding (if possible) of how to modify
the control law and tune the controller parameters to change the nature of the
device from source to sink of oscillation energy.

The paper is organized as follows: in Section II, we derive and analyze the
DEF-expressions for a TCSC operating under 1) a fixed compensation, and 2)
variable compensation using a damping controller in closed-loop feedback. In
Section III, we study the DEF-expression for a STATCOM operating under 1)
constant current control, and 2) reactive power $-$ voltage droop control. For
each of these, the numerical studies on a modified IEEE 2-area 4-machine
system are presented in their respective sections. Finally, the conclusions
are discussed in Section IV.

##  II DEF Analysis of a TCSC

Assume a power network, wherein between nodes $i$ and $k$, the branch $L_{ik}$
is a TCSC. The admittance of the branch be,

$\small\vspace{-0.1cm}y_{ik}=jb=j(1-k_{c})b_{0}$ (1)

where, $b$ is the variable reactance of the TCSC and $k_{c}$ is the state
variable modulating this reactance over its nominal value $b_{0}$. Further,
$k_{c}=k_{c_{0}}+\Delta k_{c}$, where, $k_{c_{0}}$ is the value in the steady-
state and $\Delta k_{c}$ is the perturbation under transients.

As derived in [1], the transient energy flowing into the branch $L_{ik}$ from
bus $i$ is $\int\Im(\vec{I}^{*}_{ik}d\vec{V_{i}})$, where $\vec{I}_{ik}$ and
$\vec{V_{i}}$ are respectively the complex-valued line current and bus voltage
phasors in the network reference frame, and $\Im(\cdot)$ denotes the imaginary
part of a complex number. Similarly, the energy flowing into branch $L_{ik}$
from bus $k$ is given by $\int\Im(\vec{I}^{*}_{ki}d\vec{V_{k}})$. Combining
these two, the total energy flowing into the TCSC, denoted by
$W_{\text{TCSC}}$, can be written as,

$\small
W_{\text{TCSC}}=\int\Im(\vec{I}^{*}_{ik}d\vec{V_{i}}+\vec{I}^{*}_{ki}d\vec{V_{k}})=\int\Im\Big{(}\vec{I}^{*}_{ik}(d\vec{V_{i}}-d\vec{V_{k}})\Big{)}.$
(2)

Denoting $\vec{V}_{ik}=\vec{V}_{i}-\vec{V}_{k}$, and thereafter, substituting
$\vec{I}_{ik}=y_{ik}\vec{V}_{ik}$ into (2), we get

$\small
W_{\text{TCSC}}=\int\Im\Big{(}y^{*}_{ik}~{}\vec{V}^{*}_{ik}~{}d\vec{V}_{ik}\Big{)}=b_{0}\int\Im\Big{(}-j(1-k_{c})~{}\vec{V}^{*}_{ik}~{}d\vec{V}_{ik}\Big{)}.$
(3)

Next, let us define, $\vec{V}_{ik}=U\text{e}^{j\phi}=U_{x}+jU_{y}$ wherein,
$U=U_{0}+\Delta U$. Following which, (3) reduces to

$\displaystyle W_{\text{TCSC}}$
$\displaystyle=b_{0}\int\Im\Big{(}-j(1-k_{c})~{}U\text{e}^{-j\phi}~{}(dU+jUd\phi)\text{e}^{j\phi}\Big{)}$
(4) $\displaystyle=-~{}b_{0}\int(1-k_{c})~{}U~{}dU$
$\displaystyle=\bigg{(}-b_{0}(1-k_{c_{0}})~{}U_{0}\int d\Delta U\bigg{)}$
$\displaystyle+\bigg{(}-b_{0}~{}(1-k_{c_{0}})\int\Delta U~{}d\Delta
U+b_{0}\int\Delta k_{c}~{}U~{}d\Delta U\bigg{)}$
$\displaystyle\overset{\Delta}{=}W_{\text{TCSC}}^{0}+W_{\text{TCSC}}^{D}.$

If the evaluation of $W_{\text{TCSC}}$ depends on the path traversed by the
variable $\Delta U$, instead of the initial and final values of $\Delta U$
alone, we call the integral path-dependent. Else, it is termed as path-
independent. Path-dependence implies, the expression of $W_{\text{TCSC}}$
contains the monotonically changing component related to energy dissipation or
production in the device. Observe that,

$\displaystyle
W_{\text{TCSC}}^{0}(t_{1})=-b_{0}(1-k_{c_{0}})U_{0}\int_{0}^{{}^{\Delta
U(t_{1})}}d\Delta U=-b_{0}(1-k_{c_{0}})U_{0}~{}\Delta U(t_{1})$

is path-independent. Also, in

$\displaystyle
W_{\text{TCSC}}^{D}(t_{1})=-b_{0}~{}(1-k_{c_{0}})\int_{0}^{{}^{\Delta
U(t_{1})}}\Delta U~{}d\Delta U+b_{0}\int_{0}^{{}^{\Delta U(t_{1})}}\Delta
k_{c}~{}U~{}d\Delta U$
$\displaystyle=-\frac{1}{2}~{}b_{0}~{}(1-k_{c_{0}})\\{\Delta
U(t_{1})\\}^{2}~{}+~{}b_{0}\int_{0}^{{}^{\Delta U(t_{1})}}\Delta
k_{c}~{}U~{}d\Delta U,$

the first term is always path-independent and represents the change in the
stored energy of the TCSC. Further, if $\Delta k_{c}=0$ (implying, fixed
compensation, $k_{c}=k_{c_{0}}$) or $\Delta k_{c}$ is an explicit algebraic
function of $\Delta U$ (e.g., $\Delta k_{c}=K_{p}\Delta U$ output of a
proportional controller with $\Delta U$ as feedback), the second term in
$W_{\text{TCSC}}^{D}$ is also path-independent. Absence of path-dependent
terms imply there is no generation or dissipation of oscillation energy. This
is summarized in the propositions below.

Proposition 1. _A TCSC-compensated lossless 222 for a resistive element, the
slope of its DEF is indefinite (see, [7]) transmission line with fixed
compensation is neither a source nor a sink of oscillation energy. _

Proposition 2. _A TCSC modulating the reactance of a transmission line as an
explicit algebraic function of the difference in its (i.e., TCSC’s) terminal
bus voltage magnitudes is neither a source nor a sink of oscillation energy._

However, when $k_{c}$ is an implicit function involving the time-variable $t$
alongside $U(t)$, $W_{\text{TCSC}}^{D}(t_{1})$ may become path dependent. To
demonstrate this, consider the closed-loop system in Fig 1.

Figure 1: Closed-loop control for modulating $k_{c}$ of a TCSC.

Unlike before, where $\Delta k_{c}$ was an algebraic function of $\Delta U$,
we now have a controller, with frequency-dependent gain and phase, modulating
$\Delta k_{c}$ as shown below.

$\small\Delta\dot{k}_{c}=-\frac{1}{T_{c}}\Delta
k_{c}+\frac{K_{p}}{T_{c}}\Delta U.\vspace{-0.1cm}$ (5)

The solution to the state equation in (5) is expressed as

$\small\Delta k_{c}(t)=\text{e}^{-\frac{t}{T_{c}}}\Delta
k_{c}(0)+\frac{K_{p}}{T_{c}}\int_{0}^{t}\text{e}^{-\frac{t-\tau}{T_{c}}}\Delta
U(\tau)~{}d\tau.\vspace{-0.2cm}$ (6)

Since, $\Delta k_{c}(0)=0$, $\Delta k_{c}(t)$ only comprises of the forced
response. Substituting this in the expression of $W_{\text{TCSC}}^{D}(t_{1})$,

$\displaystyle
W_{\text{TCSC}}^{D}(t_{1})=-\frac{1}{2}~{}b_{0}~{}(1-k_{c_{0}})\\{\Delta
U(t_{1})\\}^{2}$ (7)
$\displaystyle+b_{0}~{}\frac{K_{p}}{T_{c}}\int_{0}^{t_{1}}\int_{0}^{t}\text{e}^{-\frac{t-\tau}{T_{c}}}\Delta
U(\tau)\Big{(}U_{0}+\Delta U(t)\Big{)}~{}\Delta\dot{U}(t)~{}d\tau~{}dt.$

Our claim is that, between any two time-instants, with same initial and final
system states, the evaluation of the second term in $W_{\text{TCSC}}^{D}$
(see, (7)) depends on the trajectory traced by $\Delta U(t)$ (or $U(t)$). In
other words, for different paths of $U(t)$ between the same $U_{0}$ and
$U(t_{1})$, the value of $W_{\text{TCSC}}^{D}(t_{1})$ depends on the non-
conservative contributions in the respective paths. This is summarized in the
proposition below.

Proposition 3. _A controller with frequency-dependent gain and phase
modulating the control input_ $k_{c}$, _renders_ $W_{\text{TCSC}}^{D}$ _path-
dependent._

We verify this proposition using a simple numerical study. In (7), assume,
$U(0)=U_{0}=1$ pu with $U_{x}(0)=0.8$ pu and $U_{y}(0)=0.6$ pu. Next, consider
two different paths between time instants $t=0$ and $t=t_{1}$ s as shown
below.

$\displaystyle\text{Path I:}~{}~{}U_{x}(t)=U_{x}(0)+\alpha
t~{};~{}U_{y}(t)=U_{y}(0)+\alpha t$ $\displaystyle\text{Path
II:}~{}U_{x}(t)=U_{x}(0)+\alpha^{k}t^{k}~{};~{}U_{y}(t)=U_{y}(0)+\alpha^{n}t^{n}$

In the $U_{x}-U_{y}$ plane, path I traces the straight line
$U_{y}=U_{x}+\big{(}U_{y}(0)-U_{x}(0)\big{)}$, while path II traces a
nonlinear trajectory given by
$U_{y}=\big{(}U_{x}-U_{x}(0)\big{)}^{\frac{n}{k}}+U_{y}(0)$. This is shown in
Fig 2$(a)$. To ensure that the final values are same for both the paths, we
choose $t_{1}=\frac{1}{\alpha}$. Further, for this numerical study, we
arbitrarily set $k=3$ and $n=5$. The time-domain trajectories of $U(t)$ for
paths I and II are shown in Fig 2$(b)$.

Since, the area enclosed by $U(t)\big{(}=U_{0}+\Delta U(t)\big{)}$ in path I
is different from that of path II (see, Fig 2$(b)$), the evaluation of the
second term in (7) at any time $t_{1}$ is dependent on the path traced by
$U(t)$ between $t=0$ and $t_{1}$. This is shown in Fig 3 $-$ the integral in
the second term of (7) evaluated for different values of $\alpha$ (and
therefore, different endpoints $t=t_{1}$) for both paths I and II are
different. Path-dependence of the second term implies, $W_{\text{TCSC}}^{D}$
as a whole is also path-dependent.

Figure 2: Two different paths between initial and final values of $U(t)$ for
$\alpha=1$ (i.e., final value at $t_{1}=1$ s) in (a) $U_{x}-U_{y}$ plane and
(b) $t-U(t)$ plane.  Figure 3: Path-dependence of the second integral in
$W_{\text{TCSC}}^{D}$ for the two different paths in Fig 2 corresponding to
different $\alpha$ values.

The importance of proposition 3, in the context of oscillation monitoring and
source localization, is discussed next. The path dependence of the energy
function $W_{\text{TCSC}}^{D}$ implies that not all of the energy entering the
TCSC is conserved (as stored energy), and depending on how the controller
parameters are designed, there could be either dissipation or additional
production of oscillation energy. The former, improves overall damping and has
a stabilizing effect on the system, while the later, can can be a reason for
instability. The time-derivative of the second term in
$W_{\text{TCSC}}^{D}(t)$, measures the rate of energy dissipation (or
production) in the FACTS device. A negative slope333the term ‘slope’, in this
paper, refers to the average slope over a cycle (i.e, slope of the dc trend)
of $W_{\text{TCSC}}^{D}$ signifies, the TCSC is behaving as a source of
oscillation energy. Similarly, a positive slope3 of $W_{\text{TCSC}}^{D}$
signifies, the TCSC is dissipating oscillation energy (i.e., behaving as a
sink). Observe that, for the controller structure in Fig 1, the time-
derivative of $W_{\text{TCSC}}^{D}$ (see, 7) is proportional to the controller
gain $K_{p}$. Therefore, an incorrect setting of $K_{p}$ can transform a TCSC
from a sink to a source of oscillation energy. This is verified with the case
study presented next.

Case Study (A): Consider the IEEE $2-$area $4-$machine test system [8] in Fig
5$(a)$ with the following modifications $-$ (1) a TCSC is introduced in one of
the transmission lines connecting buses $8$ and $9$, (2) for the TCSC-
connected line, at steady state, $k_{c_{0}}=0.3$, and (3) a damping controller
(see, Fig 4) is introduced to modulate $k_{c}$ based on the transients in the
feedback signal $P_{10-9}$ (real power flow between buses $10$ and $9$,
measured at bus $10$, Fig 5$(a)$). The design parameters of the controller are
as follows: $T_{d_{1}}=0.4867$ s, $T_{d_{2}}=0.0543$ s, and $T_{w}=10$ s.
Three different DEF scenarios are simulated by varying $K_{p}$ values: $(i)$
$K_{p}=0$, implying, $\Delta k_{c}=0$ and the TCSC is operating under fixed
compensation, $(ii)$ $K_{p}=0.0527$, the design value to achieve the desired
pole-placement for a well-damped transient response, and $(iii)$
$K_{p}=-0.0527$, negative of the design value in $(ii)$.

Figure 4: Damping controller modulating $k_{c}$ of a TCSC.

Figure 5: $(a)$ Paths of DEF in the network, $(b)$ DEF from generators into
the network, and $(c)$ DEF from TCSC into the network, for $K_{p}=0$.

Since, the TCSC is connected to one of the lines between buses $8$ and $9$, we
calculate the DEFs out of the TCSC 444by definition (see, (2), (4)), $W^{D}$
is flowing into the device, and $-W^{D}$ out of the device, 555measured at $8$
using the $\Delta P_{ik}$ flowing into $8$ from the TCSC, and at $9$ using the
$\Delta P_{ik}$ flowing into $9$ from the TCSC into the buses $8$ and $9$. The
total DEF out of the TCSC into the network is the sum of these two individual
DEFs. The results for $(i)$ $Kp=0$ is shown in Fig 5. Observe from Fig 5$(c)$,

$\displaystyle-\frac{d}{dt}W^{D}_{\text{TCSC}}~{}=~{}\text{slope~{}of~{}}\text{DEF}_{\text{~{}TCSC
$\rightarrow$ Network}}$
$\displaystyle=~{}\text{slope~{}of~{}}\text{DEF}_{\text{~{}TCSC $\rightarrow$
8}}~{}+~{}\text{slope~{}of~{}}\text{DEF}_{\text{~{}TCSC $\rightarrow$
9}}\approx~{}0.$

This supports our proposition $1$ that a TCSC with fixed compensation is
neither a source nor a sink of oscillation energy. The DEF from all four
generators into the network, and the path of DEF in the network are shown in
Figs 5$(b)$ and $(a)$, respectively.

Next, we study scenario $(ii)$, where a damping controller modulates $k_{c}$
to achieve a desired damping. Observe from Fig 6$(c)$ that the slopes of both
the DEFs, from TCSC into buses $8$ and $9$ respectively, are negative.
Therefore, $\frac{d}{dt}W^{D}_{\text{TCSC}}>0$, implying, the TCSC is
dissipating oscillation energy. This is intuitive, since, the controller was
designed to add damping to the modal oscillation. The DEF from the generators
into the network, and the path of DEF in the network are shown in Figs. 6$(b)$
and $(a)$, respectively. Finally, in scenario $(iii)$ we reverse the sign of
gain $K_{p}$ of the damping controller. As seen from Fig 7$(c)$, the slopes of
DEF from TCSC to buses $8$ and $9$ are both positive, implying,
$\frac{d}{dt}W^{D}_{\text{TCSC}}<0$. Therefore, reversing the controller gain
alters the nature of the TCSC from sink to source of oscillation energy.

Figure 6: $(a)$ Paths of DEF in the network, $(b)$ DEF from generators into
the network, and $(c)$ DEF from TCSC into the network, for $K_{p}=0.0527$.

Figure 7: $(a)$ Paths of DEF in the network, $(b)$ DEF from generators into
the network, and $(c)$ DEF from TCSC into the network, for $K_{p}=-0.0527$.

##  III DEF Analysis of a STATCOM

Assume a lossless static synchronous compensator (STATCOM), with an internal
node $k$, is connected to the power network at bus $i$.
$\vec{V}_{i}(=V_{d_{i}}+jV_{q_{i}})$ and
$\vec{I}_{ik}(=I_{d_{ik}}+jI_{q_{ik}})$ are respectively the phasors of the
bus voltage and the current injected into the STATCOM, in its controller’s
$d-q$ reference frame. The transient energy flowing into the STATCOM is then
given by $W_{\text{STCM}}=\int\Im(\vec{I}^{*}_{ik}d\vec{V_{i}})$. Since, for a
lossless STATCOM, $I_{d_{ik_{0}}}=0$, the expression reduces to

$\displaystyle
W_{\text{STCM}}=\int\Im\Big{(}(I_{d_{ik}}-jI_{q_{ik}})(dV_{d_{i}}+jdV_{q_{i}})\Big{)}$
(8) $\displaystyle=-\int I_{q_{ik_{0}}}d\Delta V_{d_{i}}+\Big{(}\int\Delta
I_{d_{ik}}d\Delta V_{q_{i}}-\int\Delta I_{q_{ik}}d\Delta V_{d_{i}}\Big{)}$
$\displaystyle\overset{\Delta}{=}W_{\text{STCM}}^{0}+W_{\text{STCM}}^{D}$

where, $W_{\text{STCM}}^{0}=-\int I_{q_{ik_{0}}}d\Delta V_{d_{i}}$ and
$W_{\text{STCM}}^{D}=\int\Delta I_{d_{ik}}d\Delta V_{q_{i}}-\int\Delta
I_{q_{ik}}d\Delta V_{d_{i}}$. Clearly, $W_{\text{STCM}}^{0}$ is path-
independent. Further, for constant $I_{ik}$ (implying, $\Delta I_{q_{ik}}=0$
and $\Delta I_{d_{ik}}=0$), $W_{\text{STCM}}^{D}$ is zero. Therefore, for this
mode of operation, the STATCOM cannot dissipate oscillation energy. This is
presented in proposition 4.

Proposition 4. _A STATCOM operating under constant current control is neither
a source nor a sink of oscillation energy._ This will be verified with
numerical studies later in the paper.

Next, we study the path-dependence of $W_{\text{STCM}}^{D}$ for other control
strategies. To that end, we consider the reactive power control with $\Delta
Q_{i}^{\text{ref}}-\Delta V_{i}$ droop, as shown in Fig 8.

Figure 8: Reactive power control in a STATCOM using $\Delta
Q_{i}^{\text{ref}}-\Delta V_{i}$ droop.

A proportional-integral (PI) controller is used to generate the current
reference as follows,

$\displaystyle
I_{q_{ik}}^{\text{ref}}=\Big{(}K_{p}+\frac{K_{i}}{s}\Big{)}\Big{(}Q_{i}^{\text{ref}}-Q_{i}~{}+~{}K_{\text{droop}}~{}(V_{i}^{\text{ref}}-V_{i})\Big{)}.$
(9)

In (9), we substitute, $Q_{i}=I_{d_{ik}}V_{q_{i}}-I_{q_{ik}}V_{d_{i}}$.
Further, considering that the inner current control loop in Fig 8 is much
faster than the outer voltage and reactive power control loops, while
analyzing the relatively slower electromechanical transients, we may assume
that the tracking of $I_{q_{ik}}$ is instantaneous. Implying,
$I_{q_{ik}}\approx I_{q_{ik}}^{\text{ref}}$. Therefore, we re-write (9) as

$\displaystyle
I_{q_{ik}}=\Big{(}K_{p}+\frac{K_{i}}{s}\Big{)}\Big{(}(-I_{d_{ik}}V_{q_{i}}+I_{q_{ik}}V_{d_{i}})-K_{\text{droop}}V_{i}+K_{\text{const}}\Big{)}$
(10)

where,
$K_{\text{const}}=Q_{i}^{\text{ref}}+K_{\text{droop}}V_{i}^{\text{ref}}$.
Next, without loss of generality, consider that the phase-locked-loop in the
STATCOM’s controller ensures that, in steady state, the $d$-axis aligns with
the bus voltage phasor, implying, ${V}_{i_{0}}=V_{{d_{i}}_{0}}$ and
$V_{{q_{i}}_{0}}=0$. Therefore, linearization of
$V_{i}^{2}=V_{d_{i}}^{2}+V_{q_{i}}^{2}\implies V_{i_{0}}\Delta
V_{i}=V_{{d_{i}}_{0}}\Delta V_{d_{i}}\implies\Delta V_{i}=\Delta V_{d_{i}}.$
Further, considering $I_{d_{ik_{0}}}=0$, simplifies the linearization of
(LABEL:iq) as follows,

$\displaystyle\Delta
I_{q_{ik}}=\Big{(}K_{p}+\frac{K_{i}}{s}\Big{)}\Big{(}(I_{q_{ik_{0}}}-K_{\text{droop}})\Delta
V_{d_{i}}+V_{d_{i_{0}}}\Delta I_{q_{ik}}\Big{)}.$ (11)

From (11), observe that,

1) If $K_{i}=0$ (implying, only a proportional controller in Fig 8), then,
$\Delta I_{q_{ik}}$ is an algebraic function of $\Delta V_{d_{i}}$ alone. This
implies, the integral $\int\Delta I_{q_{ik}}d\Delta V_{d_{i}}$ is path-
independent.

2) If $K_{i}\neq 0$, the solution $\Delta I_{q_{ik}}(t)$ is an implicit
function of both $t$ and $\Delta V_{d_{i}}(t)$. Therefore, following the
discussions preceding proposition 3, the term $\int\Delta I_{q_{ik}}d\Delta
V_{d_{i}}$ is path-dependent. This implies, $W_{\text{STCM}}^{D}$ is also
path-dependent (see, (8)). This is summarized in the proposition presented
next.

Proposition 5. _In a STATCOM with $\Delta Q_{i}^{\text{ref}}-\Delta V_{i}$
droop, a proportional-integral controller generating the current reference_
$I_{q}^{\text{ref}}$, _renders $W_{\text{STCM}}^{D}$ path-dependent._

Path-dependence of $W_{\text{STCM}}^{D}$ signifies that, the FACTS device,
depending on the design of the controller parameters, can either dissipate
oscillation energy (aiding in damping) or can behave as a source of
oscillation energy. The value of the droop coefficient $K_{\text{droop}}$ can
influence this source-sink behavior of the STATCOM. This is presented in the
proposition $6$ and in the case study following that.

Proposition 6: _Tuning the droop coefficient_ $K_{\text{droop}}$ _transforms
the STATCOM from source to sink of oscillation energy._

We validate propositions $4$ and $6$ through the case study, presented next.

Case Study (B): We consider the modified $2$-area $4$-machine system in Fig 9.
A STATCOM is connected at bus $7$, at the midpoint of the double-circuit tie-
line connecting the two areas, as shown. The generator, load, and the line
data are same as in [8].

Figure 9: Modified $2$-area $4$-machine system with a STATCOM.

First, we study the DEF for the STATCOM considering the reactive power control
strategy in Fig 8. The DEF from the generators and the STATCOM into the
network, for different values of $K_{\text{droop}}$, are shown in Figs
10$(a)-(c)$.

Figure 10:  DEF from the STATCOM and the generators into the network for $(a)$
$K_{\text{droop}}=-1$ VAr/kV, $(b)$ $K_{\text{droop}}=1$ VAr/kV, $(c)$
$K_{\text{droop}}=3$ VAr/kV, $(d)$ constant $I_{q}^{\text{ref}}$.

As seen from Fig 10, for (a) $K_{\text{droop}}=-1$ VAr/kV,
$\frac{d}{dt}W^{D}_{\text{STCM}}>0$, implying, the STATCOM is dissipating
oscillation energy and contributing towards stabilizing the system mode; for
(b) $K_{\text{droop}}=1$ VAr/kV, $\frac{d}{dt}W^{D}_{\text{STCM}}=0,$
implying, the STATCOM is neither a source nor a sink of oscillation energy and
therefore, has no damping contribution; and for (c) $K_{\text{droop}}=3$
VAr/kV, $\frac{d}{dt}W^{D}_{\text{STCM}}<0$, implying, the STATCOM is
injecting oscillation energy into the network destabilizing the system. These
studies support propositions $5$ and $6$, that a controller with frequency
dependent phase and gain introduces path-dependent terms in the energy
function (observed from nonzero slopes of $W^{D}_{\text{STCM}}$) and that by
tuning $K_{\text{droop}}$ the damping contribution from a STATCOM can be
improved. Finally, in Fig 10$(d)$, the DEFs for constant $I_{q}$ control are
shown. Observe, $\frac{d}{dt}W^{D}_{\text{STCM}}=0$. This verifies proposition
$4$.

##  IV Conclusions

The DEF analysis for a TCSC and a STATCOM was presented in the paper. It was
observed that a TCSC operating with a fixed compensation can not be a source
(or sink) of oscillation energy. However, a frequency-dependent feedback
controller modulating its compensation can introduce dissipative effects.
Further, an incorrect gain of the controller can transform the TCSC from sink
to source of oscillation energy. For the STATCOM, it was shown that under
constant current control, the device is neither a sink nor a source. Under
droop control, however, the device could behave either as a source or sink,
depending on the droop coefficient.

## Acknowledgements

Financial support from NSF grants under awards CNS 1739206 and ECCS 1656983 is
gratefully acknowledged.

## References

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