Intelligent Automation and Soft Computing, Vol. 12, No. X, pp. 1-13, 2006
Copyright © 2006, TSI® Press
Printed in the USA. All rights reserved
LINEAR TECHNIQUES APPLIED TO SMALL-SIGNAL
ELECTROMECHANICAL STABILITY, MODEL ORDER REDUCTION AND
HARMONIC STUDIES
NELSON MARTINS1, JULIO C. R. FERRAZ1, PAULO E. M. QUINTÃO2
ALEX DE CASTRO2, SERGIO L. VARRICCHIO1, SERGIO GOMES JR.1
1
2
CEPEL, P.O. Box 68007 - Rio de Janeiro, RJ - 20001-970, Brazil
Fundação Padre Leonel Franca - Rio de Janeiro, RJ - 22451-900, Brazil
(nelson@cepel.br, jcferraz@ieee.org, quintao@momentus.com.br
alexdecastro@hotmail.com, slv@cepel.br, sgomes@cepel.br)
ABSTRACT—This paper describes recent work on small-signal stability, model order reduction
and power system harmonic analysis carried out in CEPEL, the Brazilian Electrical Energy Research
Center. CEPEL has developed, along the last three decades, a suite of power system analysis tools
that are in current use by most of the Brazilian electrical utilities.
Key Words: Small-Signal Stability, Modal Analysis, Model Reduction, Power System Harmonics.
1. INTRODUCTION
This paper describes linear techniques developed in CEPEL for the analysis of power system dynamic
models [1]. The described developments are:
1. Fast computation and effective use of linear time responses and modal time responses of large
power system models;
2. Computation of reduced models of high-order multivariable transfer functions;
3. Application of transfer function zeros to better understand and control the adverse transients on
generator terminal voltage and reactive power induced by the presence of Power System
Stabilizers (PSS);
4. Modal analysis in power system harmonic studies.
Other power system dynamics and control developments, carried out by the authors, are listed in the
last section of this paper together with the available references.
2. LINEAR TIME RESPONSES AND MODAL TIME RESPONSES OF LARGE POWER
SYSTEMS
The large system model utilized to produce the results of this section was the North-South Brazilian
interconnection. The North-Northeast and South-Southeast subsystems were interconnected in 1999
through a 1,000 km long, series-compensated 500 kV transmission line. Thyristor Controlled Series
Compensators (TCSCs) were placed at the two ends of this line, in order to damp the North-South mode, a
low-frequency, poorly damped interarea oscillation mode associated with this interconnection.
The Brazilian system model for the year 1999 has 2,370 buses, 3,401 lines/transformers, 2,519 voltage
dependent loads, 123 synchronous machines, 122 excitation systems, 46 power system stabilizers, 99
speed-governors, 4 static Var compensators, 2 TCSCs equipped with Power Oscillation Damping (POD)
controllers, 1 HVDC link with two bipoles. Each synchronous machine and associated controls is the
aggregate model of a whole power plant. All system equipment relevant to the study were modeled in
detail, and the system Jacobian matrix has 13,062 rows and 48,521 nonzero elements [2]. The system has
1,676 state variables. The North-South mode in the presence of the two TCSCs, is associated with the
complex pair of eigenvalues s = -0.31793 ± j 1.04375, having a damping ratio of 29% and frequency of
0.17 Hz, for the operating condition considered. The POD controllers of the two TCSCs are the major
source of damping for the North-South mode.
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Intelligent Automation and Soft Computing
The use of linear time response simulation has proved valuable to the oscillation damping analysis and
control of large systems. Obtaining the time response of the linear model by the sparsity-preserving,
implicit trapezoidal rule algorithm takes 50 times less CPU time than that needed by current transient
stability programs [3].
QR routines from standard mathematical libraries [4, 5], have performed quite reliably for power
system state matrices of about 2,000 states, considering practical controller parameters and operating
conditions. The CPU time for the QR eigensolution of the 1,676 state matrix is about 3 minutes on a
Pentium personal computer.
Figure 1 and Figure 3 show the plots of the QR eigensolutions of the power system state matrix for the
North-South Brazilian system without PSSs (1,359 states) and with PSSs (1,676 states) respectively. There
are 46 major power plants with their PSSs represented, plus the POD controllers at the two TCSCs. Figure
2 and Figure 4 show the associated linear time responses for step disturbances of different polarities,
simultaneously applied to the exciter voltage references or to the mechanical power references of several
system generators. The variables being plotted are the rotor speed deviations in selected generators. The
system is seen to have several unstable modes in the absence of PSS and POD controllers (Figure 1 and
Figure 2) and stable in their presence (Figure 3 and Figure 4), showing they are essential to keep the
system operating safely at the current levels of power transfer.
0.0025
20.
10%
15%
5%
0.0015
10.
0.0005
0.
-0.0005
-10.
-20.
-5.0
-0.0015
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
-0.0025
0
2
4
6
8
10
Time (s)
Figure 1. QR eigensolution – system model
without power system stabilizers has several
unstable modes (1,359 states).
Figure 2. Time response for Brazilian system
without power system stabilizers (1,359 states).
0.0002
20.
10%
15%
5%
0.0001
10.
0
-0.0001
0.
-0.0002
-10.
-0.0003
-20.
-5.0
-0.0004
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
Figure 3. QR eigensolution – system with 46
power system stabilizers and 2 POD controllers is
stable (1,676 states).
0
5
10
Time (s)
15
20
Figure 4. Time response for Brazilian system with
46 power system stabilizers and 2 POD
controllers (1,676 states).
The North-South mode (0.17 Hz) is highly observable in Figure 4. It is quite well damped mostly due
to the presence of the TCSC POD controllers as may be inferred from the comparison of Figure 4 and
Figure 5. Figure 5 was obtained for a slightly different disturbance applied to basically the same system
model, the only change being that the two POD controllers were disabled.
Linear Techniques Applied to Small-Signal Electromechanical Stability, Model Order Reduction and Harmonic Studies
3
0.0003
0.0002
0.0001
0
-0.0001
-0.0002
-0.0003
0
4
8
12
16
20
Time (s)
Figure 5. Time response for Brazilian system with 46 power system stabilizers but no POD
controllers (1,664 states).
The next two figures illustrate the effectiveness of the model-order reduction techniques available,
which are based on dominant poles and associated residues of scalar transfer functions of the full-size
system.
Reduced Order
Complete
Reduced Order
Complete
0.0002
0.0004
0.0003
0.0001
0.0002
0.0000
0.0001
-0.0001
0.0000
-0.0002
-0.0001
-0.0002
0.
5.
10.
15.
20.
-0.0003
0.
Time (s)
5.
10.
15.
20.
Time (s)
Figure 6. Responses of the full model and the 6th
order modal equivalent.
Figure 7. Responses of the full model and the
th
10 order modal equivalent.
Figure 6 compares the rotor speed deviations for one of the generators monitored in Figure 5
(generator #5022) obtained by step-by-step numerical integration on the 1,664-state system with that
obtained by Inverse Laplace Transform of a step disturbance applied to the 6th order transfer function
modal equivalent. The 6th order model was directly obtained by using the Dominant Pole Spectrum
Eigensolver algorithm [6]. The step response of the modal equivalent was produced by computing the 6th
order approximation to y (t ) every 20 milliseconds:
6
y (t ) ≅ ∑
i =1
Ri
λi
(e λi ⋅t − 1) , where:
λ1,2 = -0.034 ± 1.079j
λ3,4 = -2.437 ± 0.054j
λ5,6 = -0.521 ± 2.881j
R1,2 = 0.015 ∠ ± 1.3°
R3,4 = 0.010 ∠ ± 77°
R5,6 = 0.001 ∠ ± 111°
The rotor speed deviations for generator #5022, obtained from the 1,676-state system model, are
compared in Figure 7 with the response of a 10th-order modal equivalent. The step response of the modal
equivalent was produced by computing the 10th order approximation to y (t ) every 20 milliseconds:
10
y (t ) ≅ ∑
i =1
Ri
λi
(e λi ⋅t − 1) , where:
λ1,2 = -7.016 ± 2.918j
λ3,4 = -2.996 ± 9.390j
λ5,6 = -0.318 ± 1.044j
λ7,8 = -0.346 ± 0.580j
λ9,10 = -0.116 ± 0.245j
R1,2 = 0.064 ∠ ± 93°
R3,4 = 0.030 ∠ ± 44°
R5,6 = 0.022 ∠ ± 7°
R7,8 = 0.013 ∠ ± 175°
R9,10 = 0.002 ∠ ± 148°
The above results are for a scalar, also referred as Single-Input-Single-Output (SISO), transfer
function. Modal equivalents for multivariable, also referred as Multi-Input-Multi-Output (MIMO), transfer
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Intelligent Automation and Soft Computing
functions are necessarily of a higher order, but can also be obtained at reasonable cost, as described in the
next section of this paper.
Another important use for the linear time response function is the validation of the small-signal
analysis results for large systems. This involves verifying whether the results obtained with a benchmark
transient stability program, for a sufficiently small disturbance, match those produced by the small-signal
program. Figure 8 and Figure 9 show that, for a small step disturbance simultaneously applied to several
generators, the responses from linear and non-linear simulations are in agreement. The monitored variable
is the deviations in active power flow in the North-South tie-line.
North-South Tie-Line Power
North-South Tie-Line Power
Active Power Flow Deviation - pu
Active Power Flow Deviation - pu
2
1.5
1
0.5
Time (s)
0
-0.5
0
2
4
6
8
10
12
14
16
18
20
-1
-1.5
-2
-2.5
-3
-3.5
Transient Stability
Software
Small Signal Stability
Software
Figure 8. Comparison of Linear and Non-Linear
System Response for a Small Disturbance.
System without PSSs in Paulo Afonso IV and
Xingó Power Plants and no POD controllers (100
MVA Base).
0.1
Time (s)
0
-0.1
0
2
4
6
8
10
12
14
16
18
20
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
Transient Stability Software
-0.8
Small Signal Stability Software
-0.9
Figure 9. Comparison of Linear and Non-Linear
System Response for a Small Disturbance.
System Model with Proposed PSSs in Paulo
Afonso IV and Xingó Power Plants (100 MVA
Base).
3. REDUCED MODELS OF HIGH-ORDER MULTIVARIABLE TRANSFER
FUNCTIONS
Model reduction is an important step to many control systems applications. Highly reduced models for
power system transfer functions, either scalar (SISO) or multivariable (MIMO), proved effective for
controller design [4, 7, 8, 9, 10]. The model reduction algorithms applied to power systems have been both
model-based [6, 9] and measurement-based [7, 8]. An m x m transfer function G(s) of a n-th order system,
having a complete set of eigenvectors, can be expressed as (1).
n
G ( s) =
Ri
∑s−λ
i =1
(1)
i
The above expression for G(s), involves the sum of m x m residue matrices over a first-order pole [11].
The symbol Ri denotes the residue matrix for G(s) at the pole λi ; it may be computed once the eigentriplet
(λi , xi , vi) has been obtained [12]. An efficient eigensolution algorithm [13] was recently developed to
only find the most dominant poles of high-order multivariable transfer functions. Adequate use of this
algorithm allows effective model-based system reduction.
The Sigma plot [13] of the reduced-order model (2) should be evaluated over a sufficiently large
number of ‘ω’ values within the dynamic range of G(s). Note that the order of the reduced model is
p = 2nc + nr, where nc is the number of complex-conjugate eigenvalue pairs and nr the number of real
eigenvalues retained in the model.
p
G (s) ≈
Ri
∑ jω − λ
i =1
(2)
i
The North-South Brazilian system, described in the previous section, was again utilized here. An 8 x 8
transfer function matrix G(s), relating variables from 8 power plants located in the Northeastern region of
the grid, was used in the described tests.
Linear Techniques Applied to Small-Signal Electromechanical Stability, Model Order Reduction and Harmonic Studies
5
The quality of the model approximation may be assessed by comparing in Figure 10 the Sigma plots
obtained for the full-system model (1,676th-order) and the reduced-order model (41st-order modal
equivalent) for the 8 x 8 G(s) matrix.
The step response for the modal equivalent of a scalar transfer function model may be analytically
computed, as described in the previous section of this paper. The associated step response for the full
model may be computed by the implicit trapezoidal algorithm. The step responses for 12 of the 64 scalar
transfer functions that exist in the G(s) matrix are depicted in Figure 11. Note the 41st-order model of the
scalar transfer function g11(s) matches rather well the response of the 1,676th-order model. The same could
not be said about the g33(s) and a few other scalar transfer functions, which present noticeable
discrepancies. The major system oscillations, except for one faster mode in g12(s) and g21(s), were however
all captured by the reduced-models of the twelve transfer functions.
0
2
σ max - Reduced Model
-20
-60
0
-2
5
-40
σ min - Full Model
0
5
5
0
g 11 (s)
10
0
σ max - Full Model
-1
2
0
0
5
σ min - Reduced Model
0
5
10
0
-2
1
-5
1
5
10
0
5
5
5
g 13 (s)
0
-5
5
0
-1
2
10
5
10
0
g 33 (s)
g 32 (s)
10
5
g 23 (s)
10
0
0
-5
0
g 22 (s)
g 31 (s)
100
0
g 12 (s)
g 21 (s)
-5
-80
1
0
5
10
-5
5
0
5
10
120
0
140
0
0
0
g 41 (s)
5
Frequency (rad/s)
10
Figure 10. Sigma-plot for 8x8 G(s), ξ = 15%
(reduced model has order 41).
15
-1
0
5
g 42 (s)
10
-2
0
5
g 43 (s)
10
-5
0
5
10
Figure 11. Step responses for gij(s) scalar
st
transfer functions for the full model and the 41 order model. (Note: The vertical axes in the 12
plots are given in radians/s and the horizontal
axes in seconds.)
4. MODEL REDUCTION BY BALANCED TRUNCATION
This section briefly describes the model reduction of a linear multivariable dynamical system using the
balanced truncation method [14, 15, 16]. This method is then compared with the modal equivalents
alternative, considering the power system context.
Given a full-order (n-th order) model G(s) of a system, the method's objective is to compute a lowerorder (r-th order, r < n) model Gr(s), such that the H∞ norm of the model error e(s)=G(s)− Gr(s) is minimal.
Once r is chosen, which involves some degree of heuristics, the method ensures there is an upper bound for
this error given by 2
∑
n
i = r +1
σ i , where σi are the n−r lowest Hankel singular values of the full system
model.
The method provides a mathematically rigorous and systematic algorithm for model order reduction.
This method is widely used by the control system community for mid-sized systems and leads to very good
results for both SISO and MIMO transfer functions. However, the heavy computational load involved in
the calculation of the observability and controllability grammiams plus the Hankel singular values of the
full-order model makes the practical use of this method prohibitive for interconnected power system
applications.
Other possible drawbacks of the optimal order reduction algorithms include:
• the original system poles are, in principle not preserved in the reduced model, unless previously
specified;
• these algorithms require some pre-conditioning to deal with unstable poles;
• They produce results in a single shot, once the desired order of the reduced model is specified,
and do not much benefit from the knowledge of the set of poles obtained in a previous run. The
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Intelligent Automation and Soft Computing
modal equivalents, on the other hand, are preferably used in an incremental manner, when new
poles are added to the dominant pole spectrum (partial set) already found in previous runs of
specific eigensolution methods [6, 13]
The technique used for model order reduction in section III is simple and easy to apply for large
systems using efficient eigensolution algorithms. However, such a modal truncation approach suffers from
relatively large error (difference between original and reduced model) compared to the ones based on the
use of balancing transformation and Hankel singular values [17]. For large systems (thousands of states),
the balanced truncation techniques are not practical since the explicit solution of Lyapunov equations is
required. Probably a combination of the susbspace based techniques [18, 19] coupled with Schur’s
balanced truncation [20] might be better in those cases. Further investigations are needed in this area.
5. UNDERSTANDING THE ADVERSE IMPACTS ON VOLTAGE AND REACTIVE
POWER PERFORMANCES DUE TO PSSS
This section describes a study of the adverse impact on generator terminal voltage and reactive power
induced by the presence of a PSS. The system’s dynamic performance is evaluated for PSSs derived from
different input signals.
5.1 A Tutorial Example Using a Second Order System
Let G(s) be a second order transfer function, as described in (3), where ωn is the undamped natural
frequency and ζ is the damping ratio [21, 22].
G (s ) =
ω n2 (as + 1)
(s 2 + 2ζω n s + ω n2 )
(3)
Figure 12 presents results, produced by Matlab [5], on the performance of the second order system
when its single zero (sz) assumes different locations. This is carried out by altering the parameter a in the
numerator of G(s). Three possibilities are investigated (sz = -0.1, sz = -.05 and sz = -2.0), all having the
same pair of complex poles. The undamped natural frequency ωn was set to 7 rad/s and the damping ratio ζ
to 30%.
Figure 12 shows that in all three cases the settling values are identical (equal to 1 for a step
disturbance) and the settling times are practically the same (1.8s). Despite having practically no impact on
these two performance indices, the zero location significantly impacted the magnitude of the transient
response peak, which varied from 3.05 to 47.50. These results clearly show that the closer the zero is to the
origin of the complex plane, the greater is the peak value of the system step response.
StepaoResponse
Resposta
Degrau Unitario
50
40
Imag
Amplitude
5
b
10
0
0.11
0.05
7
6
c
5
0.46
4
4
3
3 0.64
2
1
-10
2
0.86
0
0.5
1
1.5
Tempo
Time
(s)(s)
2
2.5
3
-2.5
RespostaResponse
ao Impulso
Impulse
Magnitude (dB)
300
200
-100
b
Fase (deg)
0
c
20
-200
0
-1
-0.5
0
b
0
45
Real
a
40
-20
90
100
-1.5
1
Diagrama
de Bode
Bode Diagram
60
a
400
-2
a
b
c
0
500
Phase
(deg)
Amplitude
0.17
6
20
-300
0.25
7 0.34
30
-20
M Pole-Zero
apa de PolosMap
e Zeros
8
a
c
a
c b
-45
0
0.5
1
1.5
Tempo
Time
(s)(s)
2
2.5
3
-90
20
40
60
Frequencia(rad/sec)
(rad/sec)
Frequency
Figure 12. Time and frequency domain results for second order system.
80
100
Linear Techniques Applied to Small-Signal Electromechanical Stability, Model Order Reduction and Harmonic Studies
7
5.2 PSS Derived from Rotor Speed or Terminal Power
The test system is a Single Machine – Infinite Bus (SMIB) system having generator and AVR models
as the Xingo power plant (owned by CHESF, a major Brazilian utility). The performances of fictitious
PSSs, derived from either rotor speed or terminal power, are compared. Their tuning are such that the pole
associated with the electromechanical oscillation mode is placed at exactly the same location in the
complex plane (Table I). This ensures that PSSs derived from different input variables may be compared
on a common basis.
Table I. PSS parameters
Pole-Zero Map for (∆PT/∆PMEC)
PSSPT
PSSω
Input
Signal
PSS(s)
Rotor
Speed
(8.17) ⋅  1 + 0.174s 
Terminal
Power
Input
Signal
Rotor
Speed
Terminal
Power
2
 3s 
⋅

 1 + 0.010s   1 + 3s 
8.5
8.5
5.5
5.5
2.5
2.5
-0.5
-2.5
-2
-1.5
3s 

 1 + 0.129s   1 + 3s 
λ = -1.065 ± j7.02 ζ = 15%
λ = -1.065 ± j7.02 ζ = 15%
-0.5
0
0.5
-0.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
-0.5
0
0.5
Pole-Zero Map for (∆QT/∆PMEC)
(− 0.26) ⋅  1 + 0.153s  ⋅ 
Electromechanical Mode
-1
PSSPT
PSSω
8.5
8.5
5.5
5.5
2.5
2.5
-0.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
-0.5
-2.5
-2
-1.5
-1
Figure 13. Pole-zero maps. (Note: The poles and zeros in
Figure 13 are shown as red asterisks and blue circles
respectively.)
Figure 13 shows the pole-zero maps of the transfer functions (TF) that relate the active and reactive
terminal power with the generator mechanical power: (∆PT/∆PMEC) and (∆QT/∆PMEC). The pole associated
with the electromechanical mode is in the same location for both stabilizers, as this was a PSS design
constraint (Table I and Figure 13). The (∆PT/∆PMEC) TF zeros are seen in Figure 13 to have practically the
same locations for both PSSs. The (∆QT/∆PMEC) TF zeros, however, are seen to have considerably different
locations for the two PSSs.
It is worth noting in Figure 13, which, for the (∆QT/∆PMEC) TF, a zero is located near the origin of the
complex plane when the PSS derived from terminal power is used. As learned from the second order
tutorial example, the closer the zero is to the origin, the bigger will be the peak value for the system step
response. Therefore, the worse performance of the PSS derived from terminal power can be anticipated
from the analysis of the pole-zero maps depicted in Figure 13.
Figure 14 depicts step responses that allow comparing the performances of the two PSSs for a
mechanical power step change (0.01 pu). The active power transient performance is the same regardless of
the input signal used by the PSS. On the other hand, the reactive power transients show much bigger
amplitude when the PSS derived from terminal power is used.
Note that in Figure 13 the ∆QT/∆PMEC TF is non-minimum phase, due to the presence of the righthand-side zero. As a result, the reactive power response depicted in Figure 14, shows a transient peak of
reverse polarity.
The use of electrical power as the input signal to the stabilizer, despite its simplicity, has been mostly
abandoned due to its comparatively poor performance. The large transients in terminal voltage and reactive
power output, shown in the simulation results of Figure 14, have usually been observed in practice
following rapid changes in active power generation. Also water turbulence (pulsating torques below 0.5
Hz) in Francis turbines have yielded unacceptable sustained oscillations in terminal voltage and reactive
8
Intelligent Automation and Soft Computing
Terminal Active Power
Terminal Reactive Power
Terminal Voltage
0,004
0.001
0,020
0,002
0.000
0,015
0,000
-0.001
-0,002
0,010
-0.002
-0,004
0,005
-0.003
-0,006
0,000
0,0
5,0
10,0
15,0
20,0
25,0
-0,008
0,0
5,0
10,0
Tempo (s)
15,0
20,0
25,0
-0.004
0.0
5.0
Tempo (s)
10.0
15.0
20.0
25.0
Tempo (s)
Figure 14. System performance following a step disturbance on generator mechanical power.
power. A third practical disadvantage of the PSS derived from terminal power is the risk of PSS output
saturation when the generator is ramping-up power, with the consequent loss of oscillation damping action.
The PSS derived from rotor speed has practical problems of another nature, associated with highfrequency noise in the speed signal. Nowadays, the preferred power system stabilizer structure is the
integral of accelerating power, incorporating a fourth or higher order ramp-tracking filter in the loop. A
detailed analysis of this PSS, in terms of pole-zero maps, is described in [23].
6. MODAL ANALYSIS IN POWER SYSTEM HARMONICS
In electromechanical stability studies, the modeling emphasis is given to the system electrical
machines and associated controllers. The electrical network is modeled by algebraic equations, a valid
assumption since the electromechanical oscillations lie in the 0.2 to 2 Hz range, which is much lower than
the natural resonant frequencies of the electrical network.
Modal analysis produces valuable information on any linearized dynamic system that would otherwise
be difficult to obtain by time domain or frequency response methods alone. In the case of electrical
networks, some of this information is the natural oscillation modes (network resonances), identification of
components that participate most in these resonances, impact of a given oscillation mode in the transient
response, sensitivities to parameter changes, etc.
Despite all these advantages, modal analysis has been only moderately used in electrical network
dynamic studies. This fact may be associated with the difficulties faced when using conventional state
space techniques for modeling the dynamics of large RLC networks of generic topology. Recent papers
have proposed the use of the descriptor system approach [24, 29] and the system nodal admittance matrix
in the s-domain, Y(s) [25, 26, 27, 28], to model the electrical network. These two approaches allow for an
easy modeling of the electrical network in harmonic and electromagnetic transient studies. A harmonic
analysis example extracted from [29] is reproduced below.
Consider the 3-bus test system shown in Figure 1, whose nominal frequency is 50 Hz and values for its
lumped RLC elements are given in [29]. The symbols Ih1, Ih2 and Ih3 denote harmonic current sources
connected to buses 1, 2 and 3, respectively.
Ih1
Lcc
C1
bus 1
L12
L13
R12
R13
bus 2
C2
Figure 15. System modeling.
R2
L2
bus 3
Ih2
Ih3
L3
R3
C3
Linear Techniques Applied to Small-Signal Electromechanical Stability, Model Order Reduction and Harmonic Studies
9
The frequencies (imaginary parts divided by 2 π) of the complex conjugate poles (parallel resonance)
and zeros (series resonance) of the network bus self-impedances, as well as their sensitivities [29] with
respect to the system inductances and capacitances are presented in Table II. The sensitivities are
normalized, being given in Hz/per unit change of the nominal parameter value.
Table II. Resonance frequencies and sensitivities
Zeros seen from
Bus 1
Bus 2
Bus 3
1
2
3
1
2
1
2
1
2
f(Hz) 252 489 722 425 565 332 633 382 704
LCC -101 -11 -50 0
0 -48 -80 -88 -47
L2
-3 -4 -2 0 -7 0
0 -5 -2
L3
-4 -2 0 -5 0 -5 -1 0
0
L12
-2 -78 -247 0 -290 -40 -66 -37 -289
L13 -19 -151 -78 -211 0 -75 -172 -59 -32
C1
-45 -12 -206 0
0 -38 -242 -85 -189
C2
-25 -116 -111 0 -269 0
0 -109 -145
C3
-54 -114 -28 -210 0 -127 -72 0
0
System poles
Assume that current sources Ih1 and Ih3, shown in Figure 15, have negligible moduli while source Ih2
contains significant 5th and 11th harmonic components. The impedance values seen from bus 2 at
frequencies of 250 Hz (5th harmonic) and 550 Hz (11th harmonic) are 44.48 Ω and 36.24 Ω, respectively.
Thus, the 5th and 11th harmonic voltage distortions at bus 2 will be given by (4) where I5 and I11 represent
the amplitudes of the 5th and 11th harmonic components, respectively.
V5 = 44.48 I 5 and V11 = 36.24 I 11
(4)
Assume that these distortions have exceeded their individual limits. A possible solution consists in
shifting the zeros associated with the self-impedance of bus 2, located at 332 Hz and 633 Hz to the
frequencies of 250 Hz and 550 Hz, respectively. The information on sensitivities of self-impedance zeros to
parameter changes, contained in Table II, indicates that the desired shifts would be best accomplished by
increasing the values of C1 and C3. Note that changing capacitor sizes is one of the recommended remedial
actions against harmonic distortions [30].
Applying a Newton-Raphson algorithm based on eigenvalue sensitivity coefficients required only four
iterations to obtain an accurate solution to the problem. The frequency response diagram of the bus 2 selfimpedance is shown in Figure 16 for the original (C1 = 23.9µF and C3 = 11.9µF) and new values of C1 and
C3 (C1 = 29.26µF and C3 = 22.77µF). The new frequency values of the poles and zeros are presented in
Table III.
Table III. Frequencies of poles and zeros for the new values of capacitances at bus 1 and 3
Zeros seen from
Bus 1
Bus 2
Bus 3
1
2
3
A
B
A
B
A
B
f(Hz) 205 422 675 308 565 250 550 363 671
System poles
For these new values of C1 and C3 the impedance values seen from bus 2 at the frequencies of 250 Hz
and 550 Hz (5th and 11th harmonics, respectively) have changed from 44.48 Ω to 12.73 Ω and from
36.24 Ω to 2.81 Ω. This means a reduction in voltage distortion at bus 2 of 70 % at the 5th and 90 % at the
11th harmonics. It must be pointed out that the zeros can be shifted without changing the system operating
point at fundamental frequency (maintaining the original power flow condition) as described in [29].
10
Intelligent Automation and Soft Computing
80
Impedance Modulus (Ω)
70
Original
New
60
50
40
30
20
10
0
0
100
200
300
400
500
600
700
800
900 1000
Frequency (Hz)
Figure 16. Self-impedance seen from bus 2 for both the original and the new values of capacitances
at buses 1 and 3.
7. FINAL REMARKS
This paper described applications of modal analysis and other linear techniques to electromechanical
oscillation damping and harmonic problems of large electrical networks. The dynamic information
provided by modal analysis is highly useful and complementary to those obtained by other methodologies.
A well-designed graphical user interface and visualization of results allows a more effective utilization
of the small-signal stability package [31, 32].
Additional work carried out by the authors, in modal analysis and related topics, are listed below:
• Partial eigensolution algorithms [6, 13, 33];
• Frequency response [34, 35];
• Modal sensitivities and mode shapes (transfer function residues, controllability and observability
factors and participation factors) [12, 31];
• Root-locus and root-contour plots [2];
• Coordinated tuning of damping controllers [36];
• Hopf bifurcations [37];
• Assessing PSS robustness through a synthetic system with variable impedance [38];
• Subsynchronous resonance studies [39];
• Modeling power systems directly in the s-domain [26, 27, 28];
• Small-signal voltage stability [40];
• Electromagnetic transients [28].
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ABOUT THE AUTHORS
N. Martins B.Sc. (1972) in Elec. Eng. from UnB, Brazil, M.Sc. (1974) and Ph.D.
(1978) degrees from UMIST, England. Dr. Martins works at CEPEL since 1978 in
the development of computer tools for power system dynamics and control. He is
convenor of the CIGRE TF 38.02.23 on Coordinated Voltage Control in
Transmission Networks and current Chair of the Power System Stability Controls
Subcommittee, PSDP Committee, IEEE/PES.
J. C. R. Ferraz B.Sc. (1997) in Elec. Eng. from UFJF, Brazil, M.Sc. (1998) and
Ph.D (2002) degrees from COPPE/UFRJ, Brazil He works in CEPEL since 1997,
and his general research interest is in the area of computer tools for power system
dynamics and control.
Linear Techniques Applied to Small-Signal Electromechanical Stability, Model Order Reduction and Harmonic Studies
13
P. E. M. Quintão B.Sc. (1993) in Elec. Eng. from UFRJ, Brazil, M.Sc. (1999) from
COPPE/UFRJ, Brazil. He works for Fundação Padre Leonel Franca since 1994, in
the development of power system analysis software including new algorithms and
graphical user interfaces.
A. de Castro B.Sc. (1994) in Elec. Eng. from UFRJ. Mr. Castro works for
Fundação Padre Leonel Franca since 1994, in the development of power system
analysis software including new algorithms and graphical user interfaces.
S. L. Varricchio B.Sc. (1987) and M.Sc. (1994) both in Elec. Eng. from Catholic
University of Petrópolis, Petrópolis, Brazil, and Federal University of Rio de
Janeiro (UFRJ), Rio de Janeiro, Brazil, respectively. He has been with CEPEL since
1989, and his research interests include power system operation, power system
quality and electromagnetic transients.
S. Gomes Jr. B.Sc. (1992) in Elec. Eng. from UFF, Brazil, M.Sc. (1995) and
Ph.D., in COPPE/UFRJ, Brazil. He works in CEPEL since 1994, and his general
research interest is in the area of computer tools for power system dynamics and
control.