A
Robust Damping Controller for Power Systems using Linear Matrix Inequalities
Aaron
F.
Snyder’, Student Member Mohammed
A.E.
A1Ali2 Nouredine HadjSaYd’, Member Didier Georges’, Member Thibault Margotin3 Lamine Mili4, Senior Member
‘Laboratoire
d’Electrotechnique
de
INPG
I
ENSlEG
Saint
Martin
d’H&res,
France
Laboratoire
d’
Automatique
de
INPG
I
ESISAR
Saint
Martin
d‘Hkres, France
Grenoble
Grenoble
Abstract:
This paper presents the application of
a
linear matrix inequalitiy tuning approach for the design
of
remote feedback controllers
for
power systems. In particular, the determined controllers
are
both
robust and optimal
for
a wide range of operating conditions, This duality is assured through the application of the proposed tuning process. The controllers presented
in
this paper were conceived
for and
tested on
a
4machine test power system with regard to their ability
to
damp
the
interarea oscillation present in the test system
and
to
allow
an
increase
in
tieline power flow without a loss
of
stability.
Keywords:
InterArea Oscillations, Power System Stabilizers, Synchronized Phasor Measurements, Robust Control, Linear Matrix Inequalities. I. INTRODUCTION Lowfrequency oscillations, related to the smallsignal stability of a power system, are detrimental to the goals of maximum power transfer and power system security. Automatic Voltage Regulators (AVRs) help to improve the steadystate stability of power systems, but are not as useful for maintaining stability during transient conditions. The addition of Power System Stabilizers (PSSs) in the AVR control loop provides the means to damp these oscillations [1,2]. Before its inclusion into the control scheme, the
PSS
should be tuned in a manner that allows it to be robust over the expected range of operating points of the power system 131. If possible, the PSS should also be optimized for these operating points. These two conflicting design goals of robustness coinciding with optimality are often difficult to achieve simultaneuously [4]. The added AVRs and PSSs are designed to act upon local measurements such as bus voltage, generator shaft speed, or the rotor angle of the associated machine. This type of feedback control is useful for local and control mode oscillations, but may be limited for interarea oscillations. Synchronized phasor measurements (SPMs) have been proven useful for many control applications in power systems, including state 0780344030/98/$10.00 998
IEEE
519
3ElectricitC
de
France
Direction
des
Etudes
et
Recherches ’?he Bradley Department
of
Electrical
and
Computer
Engineering
Clamart,
France
Blacksburg,
Virginia,
U.S.A.
Departement
Etudes
desRCseaux Virginia
Tech
estimation, transient stability, and FACTS device control [5,6]. A Remote Feedback Controller (RFC) that includes SPMs (both local and remote) as inputs should be able to realize a more efficient alperation based on the information received in real time from geographically separated points in the power system. However, one remaining difficulty is in determining a tuning methodology that is capable of treating these types of measurements. A new tuning methodology, called Linear Matrix Inequalities (LMIs), h(as been recently determined to provide a solution to the multiobjective controller design problem. By formulating the desired goals in the form of matrix inequalities, the numerical solution of the multiobjective problem may be achieved, even where the analytical solution is not possible. This type of solution could be considered ideal for large dynamic :systems, such as power systems, where analytical solutions are entirely out of the question. The LMI approach is also welladapted to the power system controller problem since
a
posteriori
determination of the optimality or robustness requires many long, timeconsuming simulations. By definition, the LMI solution, if it exists, is robust and optimal. A caveat of the LMI approach is that a solution may not be possible given the operatordefined constraints. In this case, no information about the solution is available
[7,8].
This paper addresses the placement of a PSS and a RFC tuned using LMIs for a twoarea, fourmachine test system [9] which contains a poorly damped interarea oscillation. The test system Wiis analyzed using conventional smallsignal analysis tools through the use of Eurostag
[
101 and Matlab
[
111. The “RFCLM
I”
developed here is compared to a conventional
AVWSS
(CAPSS) to demonstrate its effectiveness. Section 2 briefly discusses the smallsignal analysis tools used in this study. Section 3 provides an introduction to the LMI theory while Section
4
describes the design choices used in its application for the controller conceived demonstrated in this paper. Section
5
describes the continuation power flow method and its results as applied to the test power system while Section 6 provides a timedomain comparison of the different controller structures. 11. POWER SYSTEM MODEL
ND
SMALLSIGNAL ANALYSIS A fourmachine test power system (Fig.
1,
following page)
was
used in this study. With a resemblance to the system in
[SI,
this system contains four generators in symmetry around a transmission network. The two loads, on Buses 3 and 13 respectively, are modeled as constant impedances. The generators were modeled following the twoaxis method from
[8,9], and the governor, turbine and constant gain excitation systems were modeled in detail [8,12].
I
I
GEN
2
GEN
12
I
Fig.
1:
Test
System
2.1
Power System Representation
Once the models were constructed, a nonreduced Jacobian formulation is finalized in order to finish the system representation. The linearized form of the differential equations is then given
in
matrix form
[
131:
AX
=
MAX NAz
0
=
PAX
+
QAz
BNRAU
(1)
Ay
=
CNRAZ
where
Ax
is the state vector representing the deviation from the operating point,
Au
the input vector,
Ay
the output vector,
z
the algebraic variable vector,
BNR
he nonreduced input matrix,
CNR
he nonreduced output matrix, and (2) the nonreduced Jacobian matrix. Once the algebraic variables have been eliminated, the classical statespace representation is obtained:
AX
=
AAx
BAu
Ay
=
CAX
+
DAu
(3)
where
A
is the state matrix,
B
the input matrix,
C
the output matrix and
D
is
the feedforward matrix.
2.2
SmallSignal Analysis
Once the system has been linearized about a certain operating point, traditional indices derived from smallsignal theory [SI, such as participation factors for machine speed and angle,
are
used in conjunction with the right eigenvectors (mode shape) to identify and classify the oscillatory modes in the test system. Table
1
provides the “oscillation profile” of the three most
weakly
damped modes for the nominal power
flow
solution
of
the test system (recall Fig.
1).
TABLE
TEST SYSTEM OSCILLATION PROFILE
From the right eigenvector entries (not presented here), it can be seen that Mode 3 is clearly an interarea mode, with generators
1
and
2
swinging against generators 11 and 12. This mode is also considered to be critical due to its instability at the nominal operating point. The system can now be defined as having two areas: the sendidg end machines (1
2)
and the receiving end machines
(11
12). In determining where
to
place a supplementary controller
PSS
or
RFC),
the controllability, observability and residue magnitudes
[S,
141
for
the critical interarea mode are shown in Table 2.
TABLE
:
CRITICAL
ODE
ANALYSIS
RESULTS
The observability factors and residues are given for controller inputs of speed and electrical power, respectively. From these indices, it should be obvious that a controller located on generator 12 (highlighted values) will provide the greatest benefit with respect to the critical oscillatory mode. 111. LINEAR MATRIX NEQUALITIES The linear matrix inequality (LMI) approach
to
control systems is a synthesis of several robust control approaches. For certain types
of
wellposed control problems, the LMI formulation allows the exact numerical solution of control problems which have no analytical solution. Another advantage of the use of LMIs is the possibility of solving socalled multi objective control problems
[7].
LMI techniques are applicable to three general types of control problems: feasibility
4),
inear objective minimization
5)
and generalized eigenvalue minimization
(6)
problems. These problems are respectively expressed
as
follows:
1.
a solution
x
to
A(x)
<
0
(4) 2. minimize
c’x
subject to
A(x)
<
0
5)
3.
minimize ubject to
B(x)
>
0
6)
A(x)
<
w 4
C X)
<
0
7)
In general, LMI constraints are of the form
A(x)
=
A0
+
x~A]
...
xJ,
<
0
where
x
=
(xI,
x2,
...
,
x,)
is an unknown scalar vector (decision or optimization variables) and
(A,,,
AI,
...
,
A,)
are the given symmetrical matrices. The inequality in
(7)
signifies that the largest real part of the eigenvalues of
A(x)
is negative. To formulate the
LMI
problem for a linear timeinvariant system, the following statespace representation is used:
dx
dt
E
=
Ax t)
Bu t)
y(t)
=
Cx t)
+
Du t)
where
A,
B,
C,
D
and
E
are real matrices and
E
is invertible. This “descriptor” formulation is useful for specifying parameterdependent systems. Recalling the traditional state space system
(3)
used for power systems, it is convenient that the LMI method is developed
in
the same general form. The
520
complete details of the LMI approach may be found in
[6]
and
U].
IV.
LMI
TUNING
ETHODOLOGY This section deals with the applied design and analysis of
a
remote feedback controller using the LMI theory presented in Section
3.
4.1 Problem
Formulation
The general multiobjective
H2/H,
statefeedback control problem is depicted in Fig. 2. This multiobjective approach includes the benefits of robustness and tracking from
H_
theory, noise insensitivity from linear quadratic Gaussian (LQG) theory and, through the use of LMIs, controllers with reasonable feedback gains and good transient response through robust pole placement
[6].
I
U
erz2
Fig.
2:
General StateFeedback Controller Problem [6]
In power systems, a statefeedback approach would never be possible since access to every state in the system is rarely, if ever, assured. Therefore, the
H_
ynthesis relies on the
so
called standard form:
X
=
Ax+
B,w+ B2u
z
=
C,x
+
D,,w
+
D u
(9)
y
=
C,,x+
DYlw
with where
P s)
and
K s)
are statespace realizations of the open loop system and controller respectively, and
z
and
z2
denote the output associated with the
H_
performance (perturbation attenuation) and the output associated with the
H2
erformance (energy minimization) respectively. The multiobjective algorithm can be used to create a
H_
controller, a
H2
controller, a mixedobjective
H2/H,
controller or a combination
of
these with regional pole placement
[6]
For our study, the
H,
approach with regional pole placement was selected
[
15,161.
In determining the actual controller via the LMIH_ synthesis approach, the use of nonstructured uncertainties was also included
in
the control loop. These uncertainties represent in some small manner the range
of
operating points over which the controller is expected to maintain functionality (closedloop stability and performance). In this paper, the use of additive uncertainties
A
was considered, such that
P s)=P, s)+A s),
where
P, s)
denotes the nominal system. The inclusion of the uncertainty transfer function
A
is achieved via a perturbed model description similar to the one used in the linear fractional transformation
[
17,181 in the case
of
structured uncertainties, seen here in Fig.
3.
I ....
.._
i
j
j
IU
Yl
I
t
.....................................
. ... .i
Fig.
3: System Representation with Uncertainties
[18]
Under this formulation, the socalled interconnection matrix
M
s determined by where
M
=
F'(P,K)
=
PII P12K(I

P22H)'P21=KSy
1
1)
and
S,
is the ouputsensitivity ransfer matrix.
Z
~~
Fig.
4:
Actual System Model
with
Uncertainties
[19]
Fig.
4
indicates the actual system model with uncertainties used
for
this paper [19]. The system
P s),
the controller
K s),
the measured output
y,
the performance output
z,
the perturbations
w
he uncertainty model
Wd,
the control
U
and the control error
e
are all represented. Then, the problem is to minimize
IIKS,wdll_
o obtain the optimal perturbation attenuation factor under a pole placement constraint. According to the wellknown smallgain theorem, stability rolbustness is guaranteed when this norm is less than 1.
A
LMI problem of type
(5)
is then computed: Minimize
f
subject to:
I
02
<O
X>O
13)
clX
+
cl
xc:
B ;
CdIX
Dcll
Y21
i
nd the
LMI
in
X,
arising from the pole placement constraint, with cl denoting the matrices of the closedloop system. It can be shown that the matrix inequality (13) may be transformed
to
a true LMI with an appropriate change of variables
[6].
521
4.2
Controller Determination
In practice, computation of LMI problems appears to be limited to systems of about order
50
when using the classical interiorpoint algorithm. With a complete representation of each generator in the system as well as their respective regulation structures, the test system can include
40
to 52 states. To accelerate the LMI algorithm calculations, the test system model was reduced to a 10th order system using the optimal Hankel approximation without balancing, described in [20]. As applied to power systems, this reduction method requires the computation of the observability and controllability Grammians and uses the left and right eigenvectors in order to perform the model reduction. By working with candidate signals that make the system observable and controllable, the reducedorder model is assured to have the same global characteristics as the srcinal system [21]. A comparison of the critical oscillatory modes
of
the srcinal system and the modes retained by the reducedorder model are shown in Table
3.
TABLE
:
SELECTED
OSCILLATORY
MODES FOR
FULL
AND
REDUCED
MODELS
5 5
4.5
Mode Full Model
Reduced
Model
0.9076 .9854 0.9308
2
6.9483



RFCLMI
2
0.5803
k
7.4152 0.5874
+
7.4080
3
0.0073
+
4.0772 0.0073
r
4.0772
The Hankel reduction thus retains the three principal oscillatory modes as targeted by the input and output signals selected before the reduction. The outputs were selected to be the input signal of a supplementary controller such as
a
PSS or RFC, and the inputs were selected to be the injection of the controller signal into the AVRs of each machine. Through this approach, the limitations of developing a controller based on the onemachinehfinitebus methodology are avoided since more dynamical information about the real system is retained in this reducedorder model of the full test system. Once the order reduction has been performed, the system matrices are expressed in the LMI formulation and the controller goals are determined. For our study, the H_ approach with regional pole placement was selected. The use of different poleplacement regions heavily influenced the
a
priori
robustness of the determined controller. Indeed,
a
controller achieving this goal provided very weak damping for the target oscillation
5
<
O.l), while the controller reported in this paper determined with a more relaxed robustness objective achieves
a
damping factor of
<
0.5.
It should be stated that this still may not be the “absolute best” controller achievable due to these constraints; howeGer, the research results have shown that the developed controller performs quite well over the investigated operating range of this small test system. An investigation of this tuning approach on a larger test system and a study of the tradeoff between robustness and performance using the LMI approach is forthcoming. Finally, being based on robust control techniques, the
LMI
approach yields controllers
of
the same order as the system investigated.
For
implementation issues, the 10th order controller was reduced using the same method as applied to the test system model to yield a 4th order controller. The lower order controller was verified under closedloop conditions using frequency domain and smallsignal analysis
to
assure that the performance goal of closedloop stability was maintained.
V.
CONTINUATION ETHOD ANALYSIS Once the modes of the system have been identified and classified, and a controller sited and tuned, the control robustness and performance may be analyzed using the continuation power flow method [22]. This method consists of increasing the power flow in the system until
a
conjugate pair of eigenvalues transversely crosses the imaginary axis, signifying the instability of the power system. This is called a Hopf Bifurcation Point [23].
’‘
CAPS>
?t
igration
Direction
i
Line Impedance
100
j
I
O‘
2.5
2
1
5
1
0.5
Real
Axis
I
Fig. 5: Eigenvalues
for
the Studied
Control Configurations
Fig.
5
shows the trace of the interarea mode for tieline power flows from
0
to 558MW for three control structures: “AVR Only”, “CAPSS” and “RFCLMI”. The
PSS
and RFC controller configurations
are
given in Appendix A. The arrow labeled “migration direction” shows how the critical mode moves in response to the increase in tieline power flow. The test system loses stability with only AVRs installed at a tieline power flow of 158MW and is clearly unstable at the nominal power flow of 381MW. The addition of the PSS
or
the RFC results in the ability of the system to attain maximum power transfer (tieline power
=
558MW)
without instability.
The
CAPSS and the RFCLMI
are
also robust over the same range of operating points previously examined with only AVRs in the system; therefore, the Hopf Bifurcation Point conditions could not be met due to the voltage collapse limit. Finally, the RFC LMI input signals
are
assumed to be synchronized and delay
free,
which is not too boastful
if
a
system employing phasor
measurement
units
[
241
and
a
highspeed communications network [25] is considered. In an attempt to avoid the voltage collapse limit (singular Jacobian matrix, implying loss of a load flow solution), the impedance of the lines connecting the two halves of the test system was varied between
50
and
150
of its nominal value. However, while this attempt was unsuccessful under this range of impedance values, this test provided another view of the robustness
of
the controller for a variety of operating points. The plots of the controllers remain roughly the same for whatever impedance value and are therfore not presented in this
52 2
paper.
A
plot of the damping factors for the controllers
is
shown
in
Fig.
6.
Here, the clear advantage
of
the RFCLMI controller over the
CAPSS
in terms
of
damping is demonstrated.
2
0.4
c”
0.3


.
 
_ .
0
. U _
0
100
200
300
400
500
600
TieLine
Power
Plow
(MW)
Fig.
6:
Damping Factor
for
the Studied Control Configurations
VI.
TIMEDOMAIN
NALYSIS
To validate the controllers under extreme operating conditions, timedomain simulations using Eurostag were performed. In order to excite the interarea oscillation, a short circuit
of
lOOms is initiated and cleared on the line
3102
(recall Fig.
I).
Fig.
7
shows, through the plot of the electrical power of generator
12,
that the system is clearly oscillatory and unstable with only the AVRs installed. Fig.
8
provides a comparison between the CAPSS and the RFCLMI controllers under the same conditions for the same generator. It is visible that the RFC tuned using the LMI approach provide
a
remarkable improvement over the srcinal system,
as
well as the CAPSS. The effectiveness of the proposed RFCLMI is also clearly evident through a plot of the speed of generator
12
(Fig.
9). A
view of the tieline power oscillation (Fig.
10)
shows that the RFCLMI achieves a higher level of damping while requiring a much smaller increase in the amount of energy transferred between the two zones of the test power system. VII.
CONCLUSIONS
In this paper, a Remote Feedback Controller
RFC)
employing synchronized phasor measurements has been designed and placed in a fourmachine test power system. The exploitation of model reduction and Linear Matrix Inequality (LMI) techniques resulted in a robust, optimal control structure. The proposed controller has been shown to improve the damping of the lowfrequency interarea oscillation present in the test system in comparison with a conventional
AVRJPSS
controller for a wide range of operating points. VIII.
ACKNOWLEDGEMENTS
The authors would like to thank
Electricite‘ de France
and the
Fede‘ration Elesa
for their continued financial support of this research under CERD/R45/1 K8495BR350. The authors would also like to thank Dr. Mario Rios for his contributions towards the power system model reduction and controller parametrization methodologies used during this project.
.Oo0
]Power
I
Time setjd
4 1~~
320
3iO
9
do
980
Fig.
7:
Generator
12
Electrical Power,
AVR
Only Case
Time sec)
400
800.0
302.6
305.0
507.5
310.0
312.5
Slb.0
~~
Fig.
8:
Generator 12 Electrical Power, All Controllers
1.00e
II
CAPSS
I
Time
sec)
920
530
do
Fig. 9: Generator
12
Speed,
All
Controllers
Time
sec)
308
ai2
310
3dO
Fig.
10:
TieLine Power, All Controllers
IX. REFERENCES
[l]
IEEE
Power Engineering Society System Oscillations Work Group. “InterArea Oscillations
in
Power Systems,”
IEEE
#95TP101, October 1994.
523