Received: 9 January 2020
Revised: 1 June 2020
Accepted: 18 June 2020
DOI: 10.1002/2050-7038.12548
RESEARCH ARTICLE
A pole placement approach for multi-band power system
stabilizer tuning
Wesley Peres
| Francisco C. R. Coelho
Department of Electrical Engineering,
Federal University of S~ao Jo~ao del-Rei
(UFSJ), S~ao Jo~ao del-Rei, Brazil
Correspondence
Wesley Peres, Department of Electrical
Engineering, Federal University of S~ao
Jo~ao del-Rei (UFSJ), S~ao Jo~ao del-Rei,
MG 36307-352, Brazil.
Email: wesley.peres@ufsj.edu.br
Funding information
CNPq and Capes, Grant/Award Number:
001; FAPEMIG, Grant/Award Number:
APQ-02245-18
Peer Review
The peer review history for this article is
available at https://publons.com/publon/
10.1002/2050-7038.12548.
Handling editor: Dr. Sharma, Pawan
| Junior N. N. Costa
Summary
Low-frequency electromechanical oscillations is a topic of great concern in
power system operation. Undamped oscillations reduce the power transfer
capacity and can lead the system to blackouts. Since the 70s, synchronous generators operate with power system stabilizers that add damping torque to oscillations through the excitation system control. These controllers can have
either a conventional fixed structure, composed by stages of gain and phase
compensation, or a multi-band structure (MB-PSS), composed by three bands
that correspond to a specific frequency range (low, intermediate and high frequency). In the MB-PSS structure, each band consists of two branches based
on differential filters (with a gain stage and lead-lag blocks). This paper presents an approach based on the Newton-Raphson method for tuning MB-PSS
for power systems taking into consideration several operating conditions to
ensure robustness. The approach adjusts the controller's gains to place a set of
poles into a region in the complex plane with a desired damping ratio. Firstly,
the method is applied to the well-known single-machine infinite bus system
represented by the Heffron-Phillips model. Secondly, an application to the
multimachine South-Southeastern Brazilian power system is discussed considering different operating conditions. The convergence of the proposed
approach is evaluated regarding the initial conditions, the desired damping
ratio, and the set of monitored poles. Linear and nonlinear time-domain simulations validate the designed controllers. Finally, it is shown that the computational effort required by the proposed approach is lower than the one required
by a class of methods widely reported in the literature for MB-PSS design.
List of symbols and abbreviations: FACTS, flexible alternating current transmission systems; LFO, low-frequency oscillations; MB-PSS, multi-band
power system stabilizer; PSS, power system stabilizer; PSS2B, stabilizer based on the integral acceleration power; PSS4B, multi-band power system
stabilizer; STATCOM, static synchronous compensator; SVC, static Var compensator; Var, volt-ampere reactive; WADC, wide-area damping control;
ΔUREF, reference voltage of the excitation system (pu); ΔUPSS, supplementary stabilizing signal (MB-PSS output) (pu); Ka, gain of the excitation
system (pu); Rs, reactance of the stator (pu); Ta, time constant of the excitation system; T 0d0 , d-axis transient open circuit time constant of generator
(s); Xd, d-axis synchronous reactance of generator (pu); X 0d , d-axis transient reactance of generator (pu); Xq, q-axis synchronous reactance of generator
(pu); fe, electric frequency (Hz); ξd, desired damping ratio; ξmin, minimum damping ratio; ΔEFD, field voltage (exciter output) of generator (pu); ΔE0q ,
internal voltage of generator (pu); Δu, input variables vector; Δx, state variables vector; Δy, output variables vector; Δδ, internal angle of generator
(rad); Δωpu, rotor speed deviation of generator (pu); ABCD, state, input, output, and feed-forward matrices, respectively; Dg, damping constant of
generator (pu); H, inertia constant of generator (s); ξ, damping ratio.
Handling editor: Dr. Sharma, Pawan
Int Trans Electr Energ Syst. 2020;e12548.
https://doi.org/10.1002/2050-7038.12548
wileyonlinelibrary.com/journal/etep
© 2020 John Wiley & Sons Ltd
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PERES ET AL.
KEYWORDS
eigenvalue sensitivity, multi-band power system stabilizer, Newton-Raphson method, pole
placement, power system stability and control, PSS4B
1 | INTRODUCTION
Power system stabilizer (PSS) design has been subject of several research efforts.1 Proposed in the seventies,2 PSS adds a
supplementary damping torque2 by modulating the reference voltage of the synchronous generators. In this way, lowfrequency electromechanical oscillations (LFO) are reduced. Electromechanical oscillations range from 0.1 to 3.0 Hz
and arise from unbalanced torques at generators after generation and topology changes. Undamped oscillations reduce
the limits of power interchange as well as the total amount of injected power by renewable sources.3,4 Besides, LFO can
also cause blackouts as reported in the literature.5
The first structure proposed for PSS, known as conventional, was derived from speed terminal or electrical power.
Composed by stages of gain and phase compensation, this structure suffers from adverse torsional interactions (when
derived from terminal speed) and excess Var modulation in the presence of mechanic power references changes (when
derived from electrical power output). In order to overcome these drawbacks, manufactures developed a digital PSS
based on the integral acceleration power (PSS2B) at the beginning of the nineties.6
The approaches for tuning conventional PSS are mainly based on: (a) classical control theory, (b) optimization
methods, (c) pole placement techniques, and (d) robust control theory.
Classical theory-based methods are the most employed in industry.7 Although these techniques are well-established
in undergraduate courses, their employment is a time-consuming task that requires high experience from the engineer.
On the other hand, optimization approaches have the following advantage: once an index is defined to measure the system performance in closed-loop operation, there are several optimization methods that can be used to solve the problem automatically. Optimization methods are divided into analytical methods8,9 and derivative-free
metaheuristics.10,11,12,13 The first ones guarantee the optimum but are very sensitive to the initial condition employed.
Most metaheuristics are population-based methods bioinspired in nature that provide good quality solution with reasonable computational effort. However, they do not guarantee the optimum solution.
The designed PSS must guarantee a suitable closed-loop operation. However, this is not enough in practice. As operating conditions change due to load variations, there is an uncertain that must be considered in the tuning stage. In
practice, it is satisfied by using multiple operating conditions at the design stage. Another option is to use robust control
theory like Linear Matrix Inequalities.14 However, these techniques require high computational effort and the system
order must be reduced.
Pole placement approaches based on linear sensitivities15 have also been proposed in the literature. The main goal
is to find the PSS parameters to place one or more poles into pre-specified regions of the complex plane.16,17,18,19 Among
these approaches, some employ optimization methods to place poles considering gain constraints.8,9,20,21 An approach
that motivated the present work was proposed by Ferraz et al,22,23 where the gains of several conventional PSS are
determined to place poles into a region of the complex plane with a desired minimum damping ratio by solving a rootfind method (Newton-Raphson). In this case, the parameters of the phase compensation stage are previously known.
It is important to point out that power systems are non-stationary. It means that the operating point changes continuously (non-stationary behaviour) and the controller's parameters should be computed every time. Indeed, some
approaches, like Model Predictive Control (MPC),24,25,26,27 estimate the future states of the system (starting from a set
of measurements) and calculate, every time, the best control strategy to hold the system output at a reference value
(control prediction). As stated by Yao et al,25 despite the better performance of the MPC when compared with controlfixed parameter approaches (adopted in the proposed paper), the infrastructure and the required computational time
are some limitations for applying MPC to large power systems. Firstly, power system measurement infrastructure is not
enough to control applications, so far. However, the development of Dynamic State Estimators, together with Phasor
Measurement Systems, can be a solution.28 Secondly, power systems are large and a solution for the computational burden reduction is proposed by Venkat et al29: a Distributed Model Predictive Control that requires a suitable set of control links (again, the measurement and communication infrastructure are a limitation).
For the reasons aforementioned, power system stabilizers with fixed parameters are preferred in power system control. Considering that the nonlinear power system dynamic model is known for a set of operating conditions, it is
PERES ET AL.
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possible to perform a suitable design of stabilizers. In general, linearized models are used (state-space formulation) to
reduce the computational time required at the tuning stage and the controllers' performance is evaluated by nonlinear
time-domain simulations.30,31
Power system stabilizers are firstly employed to damp local oscillation modes (oscillation between a power plant
against the infinite bus). The damping of interarea modes (oscillation between generators of different areas of the system) can be obtained by: (a) the coordinated design of conventional PSS30; (b) by employing remote signals derived
from wide-area measurement systems,32,33,34,35 and (c) using Power Oscillation Damper (POD) stabilizers, installed on
flexible alternating current transmission systems (FACTS) devices.36,37 Regarding the FACTS devices, both first-generation (based on the thyristor-controlled reactor)18,38,39,40 and second one (that use self-commutated voltage-source
switching converters)41,42 have been investigated to solve the power oscillation damping problem.
The need for improving the LFO damping in a wide range of oscillation modes (global, interarea, and local) led to
the development of the multi-band power system stabilizer (MB-PSS) also known as PSS4B.6,43,44 MB-PSS consists of
three bands that correspond to a specific frequency range: low (0.01-0.1 Hz for global modes), intermediate (0.1-1 Hz,
for interarea modes), and high frequency (1-10 Hz for local modes). Each band is composed of two branches that are
based on differential filters (with a gain stage, lead-lag blocks, and a hybrid block). The application of MB-PSS to provide a supplementary stabilizing signal to FACTS devices was also investigated in the literature: Rimorov et al45 applied
the MB-PSS to STATCOM and Peres46 brought the implementation of the MB-PSS on SVC. Wide-area damping control
(WADC) systems with MB-PSS was proposed by Khosravi-Charmi and Amraee.34
MB-PSS has a complex structure and according to Rimorov et al,47 the lack of adequate approaches for its design
can impair its widespread adoption. To solve this problem, several approaches have been proposed in the literature.
The first method to design MB-PSS was proposed by Grondin et al43 in which each band is represented as a band-pass
filter that can be described with just two parameters: the band central frequency and the band central gain. In this case,
there are six parameters to tune a MB-PSS. Most approaches that can be found in the literature are based on optimization techniques: analytical methods,45,47 metaheuristics,48,49,50,51,52 and hybrid metaheuristics.13
After a brief review of the existing literature, it can be seen the importance of the development of methods to design MBPSS. Besides, pole placement approaches, to the best of the authors' knowledge, have not been properly investigated in the
literature for MB-PSS design. Therefore, this paper presents a root-finding approach for pole placement based on poles' sensitivities to tune a MB-PSS. The main goal is to allocate unstable and poorly damped modes into a region with a pre-specified
minimum damping ratio aiming to damp low-frequency oscillations. The proposed method assumes that central frequencies
are known and determines the central gains and the global gain to place a set of desired poles. The proposed approach can
tune several MB-PSS in multimachine power systems considering several operating conditions to ensure robustness.
Based on the literature review previously presented, it can be seen the complexity of the MB-PSS tuning procedure.
Further, several recently reported methods to accomplish this task in multimachine power systems are based on metaheuristic optimization. In spite of their powerful search capability, metaheuristics require a reasonable computational
effort. It motivated us to develop an analytical method able to design MB-PSS with a considerable reduced computational effort. In addition, the proposed approach can easily design many MB-PSS in multimachine power system considering several operating conditions, as shown in Section 5 and as it is done in practice for conventional stabilizers.
Firstly, results are presented for the well-known single-machine infinite bus system considering a single operating
condition. The main goal is to evaluate the convergence characteristics of the proposed method. Further, the multimachine South-Southeastern Brazilian equivalent system is used to tune two MB-PSS considering five operating conditions simultaneously. Designed MB-PSS are validated through modal analysis, frequency response, and linear and
nonlinear time-domain simulations. For the multimachine power system, a comparison with a recent article from the
literature13 is made, and the low computational burden required by the proposed approach shows the high efficiency of
the method presented in this paper.
2 | MB-PSS AND SYSTEM MODELING
2.1 | Multi-band power system stabilizer
The complete modelling of the MB-PSS is presented by Grondin et al43 and Kamwa et al.44 In this present work, it is
employed the simplified structure introduced by Grondin et al,43 as illustrated in Figure 1. The low and high-frequency
transducers6 are depicted in Figure 2.
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PERES ET AL.
FIGURE 1
Simplified structure of
the MB-PSS
FIGURE 2
High and low-frequency transducers of the
MB-PSS
As one can see, there are specific bands for each range of frequency and it can be better understood in Figure 3.
Each band is treated as a band-pass filter centered around well-separated central frequencies. There are two types of
tunable parameters:
1. Central frequencies for low (FL), intermediate (FI), and high (FH) bands;
2. Gains at central frequencies for low (KI), intermediate (KI), and high (KH) bands;
3. Global gain (KG).
Therefore, there are seven parameters to be tuned and, from these parameters, it is possible to obtain the time constants and gains in Figure 1 by using Equations (1) to (5). These equations are associated with the low band and can be
easily extended to other bands.
T L2 = T L7 =
1
pffiffiffi
2:π:F L : R
ð1Þ
T L1 = T L2 =R
ð2Þ
T L8 = T L7 :R
ð3Þ
K L1 = K L2 =
R2 + R
R −2R + 1
2
ð4Þ
PERES ET AL.
FIGURE 3
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Simplified representation of the MB-PSS
K L11 = K L17 = 1
ð5Þ
The constant R controls the bandwidth. It was set to 1.2 according to Kamwa et al.44
2.2 | Power system model
The power system model to stability studies consists of a nonlinear set of differential-algebraic equations (DAE). The
set of first-order nonlinear differential equations represents the dynamic behavior of devices and controllers, and
the set of nonlinear algebraic equations represents the electric network and the algebraic relationships associated with
the controllers.53
For small-signal studies, the set of DAE system of equations must be linearized around an operation condition by
using the Taylor series expansion.53 Therefore, a state-space representation is obtained (see Equations (6) and (7)) and
the power system stability can be evaluated through the eigenvalues of the state matrix A.7,53
½Δ˙x = ½A:½Δx + ½B:½Δu
ð6Þ
½Δy = ½C:½Δx + ½D:½Δu
ð7Þ
where Δ is the deviation operator (used in linearized system of equations), x is the state variables vector (such as the
rotor angle, speed deviation, internal voltage, and field voltage), u is the input variables vector (such as the control signals to be injected into the automatic voltage regulator input) and y is the output variables vector (such as the measured
shaft speed deviation Δωpu). The A, B, C, and D are the state-space matrices associated with the open-loop operation.
Both power system model in open-loop operation and MB-PSS can be represented in the state-space model previously presented. The closed-loop model is obtained through a feedback procedure, as illustrated in Figure 4. In this figure, it is clear that the MB-PSS injects a stabilizing signal ΔUPSS to the excitation system control loop. This additional
signal modulates the reference voltage ΔUREF of the excitation system in the transient period to add damping torque to
low-frequency oscillations.2,53
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FIGURE 4
Feedback procedure to include the MB-PSS
3 | P R O P O S ED A P P R O A C H F O R MB - P S S DE S I G N
3.1 | Pole shift due to an incremental variation of gains
Consider a complex pole and a single MB-PSS represented by its gains KL, KI, KH, and KG. Further, the initial value λ0
for the pole is given in Equation (8). In this paper, it is considered the poles with positive frequency.
λ0 = σ 0 + jω0
ð8Þ
Central frequencies FL, FI, and FH are supposed to be known at the beginning of the design process. Practical values
for them are proposed by the IEEE6 and are also found in the literature. Therefore, only the gains are adjusted, and an
incremental variation of them (ΔKL, ΔKI, ΔKH and ΔKG) will impose a shift Δλ of the pole values as given in
Equation (9).
∂λ
∂λ
∂λ
∂λ
:½ΔK L +
:½ΔK I +
:½ΔK H +
:½ΔK G Δλ =
∂K L
∂K I
∂K H
∂K G
ð9Þ
Being Δλ = λ − λ0, Equation (9) can be rewritten as Equation (10).
∂λ
∂λ
∂λ
∂λ
λ = λ0 +
:½ΔK L +
:½ΔK I +
:½ΔK H +
:½ΔK G ∂K L
∂K I
∂K H
∂K G
ð10Þ
where the derivatives (numerically calculated as proposed by Pota54) represent the pole sensitivities to the gain
variation.
Equation (10) is expanded into real and imaginary components in Equation (11).
∂σ
∂σ
∂σ
∂σ
:½ΔK L +
:½ΔK I +
:½ΔK H +
:½ΔK G σ = σ0 +
∂K L
∂K I
∂K H
∂K G
∂ω
∂ω
∂ω
∂ω
:½ΔK L +
:½ΔK I +
:½ΔK H +
:½ΔK G ω = ω0 +
∂K L
∂K I
∂K H
∂K G
ð11Þ
From Equation (11), it is possible to calculate the gains ΔKL, ΔKI, ΔKH, and ΔKG to place the pole in a desired point.
3.2 | Damping ratio
Low-frequency oscillations can be evaluated by the damping ratio calculated as given in Equation (12), which can be
rewritten as Equation (13).
σ
ξ = − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
σ + ω2
ð12Þ
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ξ
σ + ω: pffiffiffiffiffiffiffiffiffiffiffi = 0
1 −ξ2
ð13Þ
3.3 | Matrix formulation
Equations presented in the previous section can be solved to place complex eigenvalues (oscillation modes) in a point
of the complex plane with a desired damping ratio ξd. It is important to point out that only poles of the superior part of
the complex plane (ω > 0) are considered.
3.3.1
|
Single pole, one MB-PSS and one operating condition
The matrix formulation to place a single pole taking into account a single MB-PSS and one operating condition is presented in Equation (14).
2
32
3
σ
∂σ
∂σ
∂σ
∂σ
−
−
−
1
0
−
6
6
7
∂K L
∂K I
∂K H
∂K G 7
6
76 ω 7 2 3
σ0
6
6
7
∂ω
∂ω
∂ω
∂ω 76 ΔK L 7
60
7 6 7
1
−
−
−
−
6
76
7 = 4 ω0 5
∂K L
∂K I
∂K H
∂K G 76 ΔK 7
6
I 7
6
76
0
ξd
6
76
7
4 1 qffiffiffiffiffiffiffiffiffiffiffi
4
5
0
0
0
0
ΔK H 5
2
1 − ξd
ΔK G
3.3.2
|
ð14Þ
Two poles, two MB-PSS and one operating condition
Considering two poles and one operating condition, starting from their initial positions λ01 and λ02, it is possible to solve
a linear system to place them to achieve the desired damping ratios ξd1 and ξd2. In this case, two MB-PSS are considered. Equation (15) brings the linear system structure.
2
61
6
6
6
60
6
6
6
60
6
6
6
60
6
6
6
6
61
6
6
6
6
6
40
∂σ 1
−
∂K L1
∂σ 2
−
∂K L1
∂ω1
−
∂K L1
∂ω2
−
∂K L1
∂σ 1
−
∂K L2
∂σ 2
−
∂K L2
∂ω1
−
∂K L2
∂ω2
−
∂K L2
∂σ 1
−
∂K I1
∂σ 2
−
∂K I1
∂ω1
−
∂K I1
∂ω2
−
∂K I1
∂σ 1
−
∂K I2
∂σ 2
−
∂K I2
∂ω1
−
∂K I2
∂ω2
−
∂K I2
∂σ 1
−
∂K H1
∂σ 2
−
∂K H1
∂ω1
−
∂K H1
∂ω2
−
∂K H1
∂σ 1
−
∂K H2
∂σ 2
−
∂K H2
∂ω1
−
∂K H2
∂ω2
−
∂K H2
∂σ 1
−
∂K G1
∂σ 2
−
∂K G1
∂ω1
−
∂K G1
∂ω2
−
∂K G1
0
0
0
0
0
0
0
0
ξd2
qffiffiffiffiffiffiffiffiffiffiffiffi
ffi
1 −ξ2d2
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
ξd1
ffi
0 qffiffiffiffiffiffiffiffiffiffiffiffi
1 −ξ2d1
1
0
32
3
σ1
∂σ 1
−
6
7
∂K G2 7
76 σ 2 7
6
7
∂σ 2 76 ω1 7
7
−
76
7
∂K G2 76 ω2 7 2 σ 01 3
76
7
6
7
∂ω1 7
6
7
ΔK L1 7 6
σ 02 7
−
7
6
7
7 6
∂K G2 76
6
7
7
6
ΔK
ω
L2 7
01 7
6
7
∂ω2 7
=6
6
7
7
−
76
7 6 ω02 7
7
∂K G2 76 ΔK I1 7 6
7
76 ΔK 7 6
4 0 5
I2 7
76
6
7
0 7
76 ΔK H1 7
0
76
7
76 ΔK 7
H2 7
76
76
7
0 54 ΔK G1 5
ΔK G2
ð15Þ
It is possible to see that the solution of the linear system is composed of the new pole positions σ 1 + jω1 and
σ 2 + jω2, as well as the gain increments ΔKL, ΔKI, ΔKH and ΔKG required to achieve the desired damping ratios. However, eigenvalues' sensitivities (derivatives), calculated at initial positions λ01 and λ02, are valid for a small increment of
8 of 26
PERES ET AL.
gains. Therefore, for large increments, the obtained positions λ1 and λ2 can be misleading from their “true” values
(obtained from a feedback procedure considering the new gains' values). In this case, an iterative process must be done
until a convergence criterion is reached. Here, we propose to do so by using a Newton-Raphson based root-find method,
like the one applied to solve power flow and optimal power flow.37,53
Finally, the linear system of equations is rectangular when the number of equations differs from the number of variables. In this case, the generalized inverse concept is used.55
3.3.3
|
Single pole, two MB-PSS and two operating conditions
Considering two MB-PSS and two operating conditions (OP1 and OP2), it is possible to place a single pole to achieve a
desired damping ratio ξd (in each operating condition). Equation (16) presents the linear system structure, being:
OP2
and λOP2
the eigenvalue in each operating condition and (b) ξOP1
(a) λOP1
1
1
d1 and ξd1 the desired damping ratio in each
operating condition. It is important to note that designed MB-PSS controllers have equal fixed parameters for all operating conditions.
Finally, the structure of the linear system can be generalized to place m poles in q operating conditions through the
action of n MB-PSS. For instance, in the case described in Equation (16): m = 1, q = 2, and n = 2.
2
0
61
6
6
6
60
1
6
6
6
ξOP1
6 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d1
6
OP1 2ffi
6
1
−
ξd1
6
6
6
6
0
60
6
6
6
60
0
6
6
6
6
40
0
0
0
0
1
0
1
32
3
OP1
∂σ OP1
∂σ OP1
∂σ OP1
∂σ OP1
∂σ OP1
∂σ OP1
∂σ OP1
∂σ OP1
1
− 1
− 1
− 1
− 1
− 1
− 1
− 1 76 σ 1 7
∂K L1
∂K L2
∂K I1
∂K I2
∂K H1
∂K H2
∂K G1
∂K G2 76 ωOP1 7
76 2 7
∂ωOP1
∂ωOP1
∂ωOP1
∂ωOP1
∂ωOP1
∂ωOP1
∂ωOP1
∂ωOP1
76 OP2 7
1
1
1
1
1
1
1
0
−
−
−
−
−
−
−
− 1 76 σ 1 7
7 2 OP1 3
6
∂K L1
∂K L2
∂K I1
∂K I2
∂K H1
∂K H2
∂K G1
∂K G2 7
76 ωOP2
7
σ 01
76 1 7
7
7
6
6
OP1 7
0
0
0
0
0
0
0
0
0
76 ΔK L1 7 6 ω01 7
76
7 6
7
76 ΔK L2 7 6 0 7
76
7=6
7
76 ΔK 7 6 σ OP2 7
∂σ OP2
∂σ OP2
∂σ OP2
∂σ OP2
∂σ OP2
∂σ OP2
∂σ OP2
∂σ OP2
I1 7
76
6 01 7
1
1
1
1
1
1
1
1
0
−
−
−
−
−
−
−
−
76
7 6
7
5
∂K L1
∂K L2
∂K I1
∂K I2
∂K H1
∂K H2
∂K G1
∂K G2 76 ΔK I2 7 4 ωOP2
01
7
7
6
OP2
OP2
OP2
OP2
OP2
OP2
OP2 76
7
∂ωOP2
∂ω
∂ω
∂ω
∂ω
∂ω
∂ω
∂ω
0
6 ΔK H1 7
1
− 1
− 1
− 1
− 1
− 1
− 1
− 1
− 1 7
7
6
∂K L1
∂K L2
∂K I1
∂K I2
∂K H1
∂K H2
∂K G1
∂K G2 7
76 ΔK H2 7
76
7
76
7
ξOP2
d1
54 ΔK G1 5
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
0
0
0
0
0
0
0
0
OP2 2
1 − ξd1
ΔK G2
0
−
ð16Þ
3.4 | Proposed approach
Figure 5 brings the flowchart of the proposed approach. The following comments describe it in detail:
Input: (a) open-loop state-space representation for the system for q operating conditions; (b) central frequencies FL,
FI, and FH for n MB-PSS; (c) the desired damping ratio ξd; (d) the maximum number of iterations (hmax); (e) the set of
m poles, in each operating condition, to be placed by the iterative process (also called monitored poles), and (f) the initial gains KL0, KI0, KH0, and KG0 for n MB-PSS.
Step-1: The iteration counter is initialized (h = 1).
Step-2: Considering the set of initial gains KL0, KI0, KH0, and KG0 (for n MB-PSS), a feedback procedure is carried out
to obtain the initial values of the eigenvalues λ0 for q operating conditions. The set of m monitored poles (for each operating condition) is defined based on the damping ratio in open-loop operation: the electromechanical modes are chosen
to be monitored since they are associated with low-frequency oscillations. In this paper, the number of monitored pole
is the same for all operating conditions. For instance, considering m poles and q operating conditions, m × q poles will
be monitored during the iterative process.
Step-3: The poles' sensitivities with respect to gains (KL, KI, KH, and KG) are numerically calculated as proposed by
Pota.54 These sensitivities are calculated for the actual poles' positions and are valid for small variations of gains. When
the pole position changes, its sensitivities must be recalculated.
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FIGURE 5
9 of 26
Proposed approach flowchart
OPq
OPq
OP1
Step-4: Considering the desired damping ratios ξOP1
d1 = … = ξdm = … = ξd1 = … = ξdm = ξd , the linear system of equations described in the previous section is solved.
Step-5: The MB-PSS gains must be updated through the Equation (17). By using the updated gains, a feedback procedure is carried out to get: (a) the new positions for the m monitored poles for all operating conditions and (b) their
OPq
OP1
OPq
damping ratios (ξOP1
1 , …, ξm , …, ξ1 ,…, ξm ).
K L = K L0 + ΔK L
K I = K I0 + ΔK I
K H = K H0 + ΔK H
K G = K G0 + ΔK G
ð17Þ
Step-6: The system of equations has been defined to place the set of all m monitored eigenvalues into a position with
OPq
OPq
= … = ξOP1
dm = … = ξd1 = … = ξdm = ξd . It means to place these eigenvalues over the dashed line in Figure 6. Let the
OPq
OP1
OPq
,
…,
ξ
,
…,
ξ
,…,
ξ
minimum damping ratio obtained be defined as: ξmin = min ξOP1
. In this paper, if the minimum
1
1
m
m
ξOP1
d1
damping ratio (of the monitored poles) is greater than or equal to the desired one (ξmin ≥ ξd), it is assumed that the convergence is achieved (all monitored poles have been allocated into the highlighted region in Figure 6) and the iterative
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FIGURE 6
Conic sector (upper side)
process stops. On the other hand, if ξmin < ξd, the iterative process continues to Step-7. Finally, it is important to say that
a tolerance ε is considered, so that the convergence is achieved if (ξmin ≥ ξd − ε).
Step-7: The iteration counter is incremented by 1 (h = h + 1).
Step-8: If the iteration counter is greater than the maximum iteration counter defined at the beginning (h > hmax),
the iterative process is stopped and the convergence was not achieved. Otherwise, it is made KL0 = KL, KI0 = KI,
KH0 = KH,and KG0 = KG and the process restarts at Step-3.
3.5 | Considerations on the proposed approach
The proposed approach is a simple Newton-Raphson method for finding roots. However, some points must be
discussed:
1. The proposed approach places the monitored eigenvalues in order to satisfy a minimum damping ratio criterion.
Therefore, it is not necessary to put them exactly over the line defined by ξd (see Figure 6). It differs the proposed
approach from that proposed by Ferraz et al23 for conventional stabilizers design (in which eigenvalues are placed
exactly over positions with ξd). As a result, the proposed approach can achieve convergence faster.
2. The number of poles to be allocated (equal for all operating conditions) must be defined at the beginning of the iterative process. There is no direct rule to define this number. Nonetheless, a good indication can be the number of
unstable and poorly damped modes. Further, the number of electromechanical modes (equal to the number of synchronous generators) can also be used. The number of monitored poles must allow the convergence of the proposed
algorithm.
3. It is important to highlight that poles that are not monitored during the iterative process (in general excitation and
control modes) can become either unstable or poorly damped. This is better discussed in the next section and it can
happen when the desired damping ratios of the monitored poles (electromechanical nature) are high. Practical
values for the desired damping ratios range from 5% to 10%.30
4. The linear system of equations is rectangular when the number of equations differs from the number of variables. In
this case, the generalized inverse concept is used.55
5. It can be seen in Step-4 that the desired damping ratio for all monitored poles is set to ξd. It is a simplified strategy
that has worked in this paper. However, it may require high gains is some situations and it is a feature that certainly
deserves more attention in future investigations.
6. Central frequencies definition is not a critical task since each band is treated as a band-pass filter centred around
well-separated central frequencies.
4 | TUTORIAL
In order to conduct an in-depth analysis of the proposed approach, this section uses the well-known single-machine
infinite bus system (SMIB), which is composed of a single generator connected to an infinite bus through a transmission line. Here, only a single operating condition is considered. In addition, only intermediate and high bands are activated, and the goal is to tune the parameters KI and KH. The central frequencies of intermediate and high-frequency
bands are set to 0.70 and 8.0 Hz, respectively, according to IEEE.6
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FIGURE 7
TABLE 1
11 of 26
Single-machine infinite bus system
Generator data
T 0d0
H
Rs
Xq
0 pu
0.6 pu
3.0 seconds
5.0 seconds
Xd
X 0d
Dg
fe
0.8958 pu
0.1198 pu
0 pu
60 Hz
4.1 | System description
Figure 7 depicts the SMIB system and details the analyzed operating condition. Table 1 brings the generator data. The
excitation system model comprises a first-order block with time constant Ta = 0.05s and gain Ka = 100. All electrical
variables are given in the per-unit system assuming a 100 MVA power basis.
The linearized system, given in Equations (18) and (19), is obtained by using the Heffron-Phillips model, which is
suitable to study low-frequency oscillations in 1 to 3 Hz.2,53
2
Δ˙E 0q
3
2
6
7
6 Δ˙δ 7 6
6
7 6
6 Δ˙ω 7 = 6
pu
4
5 4
Δ˙E FD
−0:4317 −0:1888
0
0
−0:2028 −0:1741
0 0:2
377 0
0 0
−1486:4
0 −20
10:61
2
ΔE 0q
3 2 ΔE 0 3 2 0 3
q
7 6
7
76
6
7 6 0 7
Δδ
76
7+6
7:½ΔU REF 7:6
6
7
5 4 Δωpu 7
5 4 0 5
ΔE FD
ð18Þ
2000
3
6
7
6 Δδ 7
6
7 + ½0:½ΔU REF ½Δy = ½ 0 0 1 0 :6
7
4 Δωpu 5
ð19Þ
ΔE FD
The open-loop eigenvalues are presented in Table 2, from which it is possible to see the power system instability
due to the oscillation mode λ1 (that has an electromechanical nature according its oscillation frequency). Therefore, a
suitable allocation and design of a MB-PSS will be carried-out in the following sections.
4.2 | Iterative process
Here, the design process aims to place two modes into a region defined by ξd = 15%, starting from KI0 = 1 and KH0 = 1.
The tolerance value is set to 0.1% (so, the iterative process stops if the minimum damping ratio ξmin is greater than or
equal to 14.9%). For all simulations in this section, a maximum iteration number of 25 is established as stopping
criteria. The step-by-step solution is presented in the Appendix. The convergence characteristic is presented in Table 3.
The proposed algorithm has been able to solve the problem by employing low computational effort: three iterations that
have taken around 1 second to converge. Simulations have been conducted by using the MATLAB platform (version
2010a) and an Intel Core i5 2.40 GHz computer with 8 GB of RAM and Windows 10 64-bit operating system.
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Modes (pair)
Damping ξ (%)
Eigenvalues
λ1
0.0052 ± j8.0640
−0.06
λ2
−10.221 ± j14.221
58.36
TABLE 3
TABLE 2
Open-loop eigenvalues
Convergence characteristic
Iteration
λ1
λ2
KI
KH
OL
−0.0052 + j8.0640 (−0.06%)
−10.221 + j14.221 (58.36%)
0
0
IC
−0.0518 + j8.148 (0.64%)
−10.182 + j14.287 (58.04%)
1
1
1
0.1822 + j6.2844 (−2.90%)
−5.869 + j18.082 (30.88%)
−10.3940
36.3320
2
−0.8598 + j5.9155 (14.38%)
−3.472 + j20.049 (17.06%)
4.5007
84.2330
3
−0.8735 + j5.7775 (14.95%)
−3.110 + j20.409 (15.06%)
5.1594
92.7060
Abbreviations: IC, initial condition; OL, open-loop.
4.3 | Sensitivity to specified damping ratio, initial conditions and number of monitored
poles
It is known from the literature that the Newton-Raphson method for finding roots is very sensible to the considered initial condition. Also, the proposed approach is a Newton-Raphson-based root-find method that aims at finding the gain's
values so that a minimum damping ratio is achieved for monitored poles in closed-loop operation. Therefore, it is
expected that the initial condition used affects the convergence of the proposed approach. Besides, the number of monitored poles, as well as the specified damping ratio impact the solution obtained. This section aims to evaluate these
points.
4.3.1
|
Exhaustive search
Firstly, an exhaustive search procedure has been carried out: (a) the gain KI was increased by an increment of 1 from
0 up to 150; (b) for each value of KI, the gain KH was also increased by an increment of 1 from 0 up to 150; (c) for each
pair (KI, KH), the minimum damping ratio in closed-loop operation was calculated.
Figure 8 depicts the results obtained. The best solution ξmin = 26.94% has been found for the pair (KI = 27,KH = 62).
Although suitable results have been obtained, it is worth noting that, in practice, the exhaustive search may be not
practical due to several controllers present in power systems. Therefore, methodologies for MB-PSS tuning are required.
4.3.2
|
Sensitivity to initial conditions
Herein, the following parameters are varied: (a) initial condition for (KI, KH), (b) the desired damping ratio ξd and
(c) the number of monitored poles (1 or 2).
Table 4 brings the achieved minimum damping ratio ξmin for different values of initial conditions and target
damping ratios ξd when two poles are monitored (placed/controlled) during the iterative process. The associated gain's
values are presented in Table 5. It is important to stress that one pole (or oscillation mode) has electromechanical
nature (Δδ, Δωpu) and the other one is associated with the excitation system (ΔE0q , ΔE FD ). It is known from the literature that it is impossible to increase the damping ratio of an electromechanical mode without deteriorating the damping
of an excitation mode.2 This is the reason why the algorithm diverges when ξd = 30% for both poles in closed-loop
operation.
Tables 6 and 7 bring the achieved minimum damping ratios and the associated designed controllers considering one
monitored pole. Different from the previous study, the algorithm converges for all ξd. However, as one can see for
ξd = 30% (starting from KI0 = 5 and KH0 = 5), the achieved ξmin = 29.97% for the monitored pole is not equal to the true
minimum damping of 21.80% obtained in closed-loop operation (considering all eigenvalues). Figure 9 depicts the
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FIGURE 8
13 of 26
Closed-loop minimum damping ratio (%)
T A B L E 4 Achieved minimum
damping ratio for two monitored poles
Desired minimum damping ratio ξd
[KI0, KH0]
10% (iter)
20% (iter)
30%
[5, 5]
11.24 (3)
20.72 (3)
DIV
[5, 10]
11.71 (3)
20.65 (3)
DIV
[5, 50]
10.00 (4)
20.00 (3)
DIV
T A B L E 5 Control parameters [KI,
KH] for two monitored poles
Desired minimum damping ratio ξd
[KI0, KH0]
10%
20%
30%
[5, 5]
[13.33, 111.65]
[11.39, 72.67]
DIV
[5, 10]
[6.36, 108.74]
[10.11, 72.56]
DIV
[5, 50]
[2.04, 117.66]
[8.78, 74.34]
DIV
T A B L E 6 Achieved minimum
damping ratio for one monitored pole
Desired minimum damping ratio ξd
[KI0, KH0]
10% (iter)
20% (iter)
30% (ξmin) (iter)
[5, 5]
9.92 (3)
19.96 (5)
29.97 (21.80) (7)
[5, 10]
9.97 (3)
19.99 (4)
29.96 (21.88) (6)
[5, 50]
10.10 (2)
20.00 (3)
29.90 (22.03) (5)
T A B L E 7 Control parameters [KI,
KH] for one monitored pole
Desired minimum damping ratio ξd
[KI0, KH0]
10%
20%
30%
[5, 5]
[5.69, 21.64]
[13.19, 43.44]
[22.60, 72.66]
[5, 10]
[5.98, 21.41]
[12.84, 43.98]
[22.78, 72.50]
[5, 50]
[1.02, 50.06]
[10.32, 51.27]
[22.90, 72.14]
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F I G U R E 9 Root contour plot for dominant poles obtained by
the iterative process when varying the gains from [0, 0] to
[22.60, 72.66]
trajectory of both poles (monitored and not monitored one) for this case. Therefore, the proposed approach cannot
guarantee the correct placement of poles that are not monitored.
In order to better analyze the behavior of the proposed approach concerning the initial conditions, the following
procedure has been carried out: (a) the initial condition KI0 was increased by 1 from 0 up to 150; (b) for each value of
KI0, the initial condition KH0 was also increased by an increment of 1 from 0 up to 150; (c) for each pair (KI0, KH0), the
algorithm has been executed for one monitored pole (for the sake of brevity).
Figure 10 illustrates the true minimum damping ratio considering all eigenvalues in closed-loop operation. The
algorithm has been executed 22 500 times starting from different (KI0, KH0) to build each figure.
Simulations in Figure 10 were done considering a single monitored pole taking into consideration different values
for the desired damping ratio: ξd = 10% (A), ξd = 20% (B), and ξd = 30% (C). Only converged solutions are plotted
(it means that the desired ξd for the monitored pole has been achieved in these solutions). It can be seen in both figures
that the true ξmin (considering all eigenvalues in closed-loop operation), in many simulations, differ from the ξmin
achieved by the iterative process (when only one pole is considered—the monitored one). It is more clear in Figures 10B,C, when the desired damping ratio is equal to 20% and 30%.
After several simulations, it was observed that the number of poles to be monitored has a significant impact on the
performance of the proposed approach. As the minimum damping ratio ranges from 5% to 10% in practice,7 it is enough
to monitor the electromechanical modes (the proposed approach has converged for most simulations in this range, no
matter the number of monitored poles). The higher the number of monitored poles, the higher the computational effort
for converged simulations: here, the number of iterations varied from 3 to 7.
Finally, Figure 11 depicts the linear time-domain response following a 0.05 pu step disturbance to the reference voltage of the excitation system. The parameters used for controllers are: ½ K I = 13:329 K H = 111:650 (two monitored poles,
ξmin = 11.24%) and ½ K I = 5:688 K H = 21:639 (one monitored pole, ξmin = 9.92%). It is clear the stabilization of the system after the employment and tuning of the MB-PSS.
5 | S OUTH- SOUTHEAS TER N BR AZILIAN ELECTRICAL POWER S YSTEM
In this section, it is employed a Benchmark System for Small-Signal Stability Analysis and Control.31 It is a seven-bus
and five-machine system as depicted in Figure 12. The bus number 7 represents the Southeastern Brazilian power system. A fifth-order model is used to represent synchronous generators. Automatic voltage regulators are described by a
first-order model (static exciter). The complete system data can be obtained from the literature.7,31
This system has been employed for PSS design validation. Martins7 employed classical control for tuning conventional stabilizers to improve the system performance. Robust control (Linear Matrix Inequalities) have also been
employed by Boukarim et al.56 Finally, Peres et al13 designed two multi-band power system stabilizers by using hybrid
metaheuristics to optimize the minimum damping ratio in closed-loop operation.
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FIGURE 10
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True minimum damping ratio in closed-loop operation
5.1 | Open-loop operation
In the design procedure, five operating conditions are considered in this paper.13,56 These operating conditions were
obtained from topology variation as shown in Table 8. It can be seen that the system is unstable in open-loop operation
(one electromechanical pole is unstable and another is poorly damped56). Therefore, the system requires MB-PSS placement and tuning, as it has been done by Peres et al.13
5.2 | Design procedure
Following the work presented by Martins7 and Peres et al,13 two stabilizers will be considered: one at Salto Segredo
(generator number 3) and another at Itaipu (generator number 4). As previously mentioned, Peres et al13 designed MBPSS for this system, defining the optimal values for central frequencies (FL, FI, and FH), central gains (KL, KI, and KH),
and global gain (KG). In this section, the proposed approach is used to adjust the MB-PSS gains (KL, KI, KH, and KG) considering the central frequencies found by Peres et al.13
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F I G U R E 1 1 Time-domain response following a 0.05 pu step
disturbance to the voltage reference of the excitation system
FIGURE 12
South-Southeastern Brazilian equivalent system
The main goal is to carry out a performance comparison concerning the computational effort required by these
approaches. It is important to point out that the method proposed by Peres et al13 is based on metaheuristic optimization, a class of methods that have been widely used in the literature for MB-PSS tuning.13,46,48,49,50,51,52 Therefore, there
is an agreement that these methods have a great potential to the real-world application due to their powerful search
capabilities, in despite their randomness and computational effort.
Finally, the South-Southeastern Brazilian system is used because it has been employed for the same purpose in the
literature (design MB-PSS) so that a fair comparison can be done.
5.3 | Results
In this subsection, four simulations are performed and their results are compared with the ones from the literature.13
Firstly, it is important to point out that the central frequencies are optimized by Peres et al13 and these values are
employed in the present paper: so, the proposed approach does not tune the central frequencies.
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T A B L E 8 Open-loop operating
conditions (OP)
Configuration
T A B L E 9 MB-PSS design for the
South-Southeastern Brazilian system
(First Part)
OP
X5 − 6 (pu)
X6 − 7 (pu)
ξ1 (%)
ξ2 (%)
1
0.39
0.57
−11.90
3.84
2
0.50
0.57
−12.10
3.50
3
0.80
0.57
−12.66
2.77
4
0.39
0.63
−14.04
4.04
5
0.39
0.70
−16.59
4.18
Literature: Peres et al13
Simulation 01
Simulation 02
Parameter
G3
G4
G3
G4
G3
G4
FLp
0.01
0.08
0.01
0.08
0.01
0.08
FIp
0.60
1.00
0.60
1.00
0.60
1.00
FHp
10.00
10.00
10.00
10.00
10.00
10.00
KLp
3.00
5.45
15.12
11.91
14.71
9.26
KIp
16.71
0.03
20.06
2.55
21.53
0.06
KHp
50.83
92.65
59.28
63.69
58.54
64.10
KGp
5.39
6.84
5.11
12.78
4.52
10.57
Monitored poles
All
2
2
ξd (%)
Maximum
5.00
7.50
ξmin (%)
9.96
7.46
8.76
Computational effort
25.62 minutes
19.60 seconds
23.60 seconds
a
a
The method optimizes the minimum damping ratio, a value that is not known a priori.
Simulations have been conducted by using the MATLAB platform (version 2010a) and an Intel Core i5 2.40 GHz
computer with 8 GB of RAM and Windows 10 64-bit operating system. It should be stressed that the results in the literature have also been obtained by using the MATLAB in a computer with the same configurations. Therefore, it is possible to compare the computational effort of the proposed approach and the one from the literature.
Concerning the initial condition, the following values were considered for generators 3 and 4: KL0 = 15, KI0 = 20,
KH0 = 60, and KG0 = 10. These values correspond to the average between the maximum and the minimum boundaries
in the optimization approach from the literature.13
Regarding the target (or desired) damping ratio (ξd), four different values were considered in the simulations
(5.00%, 7.50%, 8.50%, and 9.00%) considering a tolerance of 0.2%. Therefore, the convergence is achieved when
ξmin ≥ ξd − 0.2. The maximum iteration number is 25. These values were empirically defined taking into account the
best minimum damping ratio (ξmin) of 9.96% obtained in the literature.13
Once the desired damping ratio (ξd) for each monitored pole has been set, the number of monitored poles has been
defined so that the proposed approach achieved the convergence (and the true minimum damping ratio was greater
than or equal to the desired one). For simulations 1 and 2 (ξd = 5% and ξd = 7.5%), it was enough to monitor two poles
(see in Table 8 that there are two poles unstable or poorly damped). However, for simulations 3 and 4 (ξd = 8.5% and
ξd = 9.0%), the convergence has been achieved when five poles were monitored during the iterative process. Considering five generators, five modes have electromechanical nature (pair (Δδ, Δωpu)) and this fact motivated us to monitor
five poles in the situations in which two poles did not ensure the convergence. Nonetheless, as observed in the previous
section (single-machine infinite bus system), the number of poles and the desired damping ratio have a great impact on
the convergence of the proposed approach. Unfortunately, there is no direct rule to define the number of monitored
poles.
The obtained results are presented in Tables 9 and 10 and the following points summarize the main observations:
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Literature: Peres et al13
Simulation 03
Simulation 04
Parameter
G3
G4
G3
G4
G3
G4
FLp
0.01
0.08
0.01
0.08
0.01
0.08
FIp
0.60
1.00
0.60
1.00
0.60
1.00
FHp
10.00
10.00
10.00
10.00
10.00
10.00
KLp
3.00
5.45
13.28
8.75
16.69
13.74
KIp
16.71
0.03
23.51
0.13
23.27
0.14
KHp
50.83
92.65
60.40
68.62
61.75
69.44
KGp
5.39
6.84
3.96
10.93
4.12
10.42
Monitored poles
All
5
5
8.50
9.00
ξd (%)
Maximum
ξmin (%)
9.96
8.91
8.80
Computational effort
25.62 minutes
15.35 seconds
15.70 seconds
a
T A B L E 1 0 MB-PSS design for the
South-Southeastern Brazilian system
(Second Part)
a
The method optimizes the minimum damping ratio, a value that is not known a priori.
Simulation
OP1
OP2
OP3
OP4
OP5
Peres et al13
9.96
10.03
9.99
10.07
9.99
Simulation 1
7.46
7.62
8.03
8.10
7.94
Simulation 2
8.81
8.80
8.77
8.85
8.90
Simulation 3
8.97
8.96
8.91
9.00
9.04
Simulation 4
8.84
8.83
8.80
8.88
8.93
T A B L E 1 1 Minimum damping
ratio for each operating condition ξcen
min
(OP)
1. Regarding the computational effort: the methodology from the literature13 is a population-based metaheuristic that
optimizes central frequencies and gains to maximize the minimum damping ratio in closed-loop operation. Therefore, all poles are monitored during the process. The average required computational effort is 25.62 minutes.13 On
the other hand, the proposed approach calculates the gains so that a set of monitored poles have a damping ratio
greater than or equal to the desired one (ξmin ≥ ξd). In this case, the proposed approach is able to place the monitored poles through a mathematical iterative process that lasts a few seconds. However, it is important to say that no
optimal solution is obtained and the central frequencies are supposed to be known. Further, no constraints are considered for gains.
2. Concerning the minimum damping ratio in closed-loop operation, it is seen that the proposed approach meets this
requirement taking into consideration a tolerance of 0.2% (ξmin ≥ ξd − 0.2%). For the sake of completeness, Table 11
presents the minimum damping ratio for each operating condition (ξcen
min ).
3. Concerning the central frequencies, it is important to stress that standard values are defined in the literature.6 On
the other hand, these values can also be pre-defined by an optimization method like the one proposed by Peres
et al.13 However, central frequencies in MB-PSS, like phase-compensation in conventional stabilizers, do not suffer a
significant variation when operating condition changes. In this case, only MB-PSS gains can be readjusted if a set of
new operating conditions is considered. In this situation, the proposed approach can be successfully applied since it
required a low computational effort.
5.4 | Nonlinear time-domain simulation
Nonlinear time-domain simulations are carried out to evaluate the performance of the designed stabilizers in the face
of major disturbances. These simulations are for the first operating condition considering the controllers tuned in the
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19 of 26
FIGURE 13
Itaipu internal angle
FIGURE 14
Itaipu controller output (UPSS)
FIGURE 15
Itaipu field voltage
20 of 26
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first simulation (minimum damping of 7.46% in Table 11). The results are compared with the ones provided in the literature13 (minimum damping of 9.96% in Table 11). It is important to point out that the main goal is to validate the controllers. Since the methodology from the literature provided a slight better minimum damping ratio, it is expected its
better performance in time simulation.
A short circuit is applied at bus 5 for 50 ms and cleared after 50 ms by opening line 5-1 which is reclosed after
another 50 ms. The following limits have been considered6:
1. Field Voltage: −5.0 ≤ EFD ≤ 6.0 pu;
2. MB-PSS output (for each band and for global output): −0.075 ≤ VL ≤ 0.075 pu; −0.60 ≤ VI ≤ 0.60 pu;
−0.60 ≤ VH ≤ 0.60 pu and −0.15 ≤ VST ≤ 0.15 pu.
FIGURE 16
Itaipu electrical terminal power
FIGURE 17
Frequency response:
Salto Segredo generator (G3)
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21 of 26
F I G U R E 1 8 Frequency response:
Itaipu generator (G4)
For the Itaipu generator (G4), Figures 13-16 bring the nonlinear time-domain response for the internal angle, MBPSS output, field voltage, and terminal electrical power, respectively. It is clear the system stabilization by using the two
MB-PSS installed at Salto Segredo (G3) and Itaipu (G4), respectively. As expected, the stabilizing signal (MB-PSS output) is zero in steady-state operation to prevent terminal voltage offset.
5.5 | Frequency response
For the sake of completeness, the frequency responses of the MB-PSS presented in Tables 9 and 10 are plotted in Figures 17 and 18. Since the proposed method adjusts the MB-PSS gains considering the central frequencies from the
literature,13 a quite similar shape is obtained. The differences obtained are due to the values of the central gains.
The following points summarize the frequency response of both MB-PSS:
1. These controllers present a symmetrical gain behaviour around well-separated central frequencies of each band;
2. They present a higher gain in the critical inter-area frequencies from 0.1 to 1 Hz (compared with conventional
PSS44);
3. The gain attenuation after 10 Hz reduces undesirable effects like higher rates of PSS saturation that decrease the stabilizing effect of the controller and noisy PSS output.
6 | C ON C L U S I ON
In this paper, an analytical method was proposed to design multi-band power system stabilizers' gains in multimachine
power systems. A set of operating conditions is considered to ensure robustness. The approach is a Newton-Raphson
based root-finding method that allocates a set of eigenvalues (monitored poles) into a region of the complex plane
defined by the desired minimum damping ratio in closed-loop operation. Firstly, a tutorial study has been conducted by
22 of 26
PERES ET AL.
using a single-machine infinite bus system. Further, a simultaneous design of two stabilizers for a multimachine Brazilian power system was carried out considering five operating conditions. Modal analysis, linear and nonlinear timedomain simulations, and frequency responses were used to validate the controller's design.
The proposed method requires the specification of some parameters, such as the central frequencies, the number of
monitored poles, the desired minimal damping ratio, and the initial values for MB-PSS gains. Central frequencies and
desired damping ratio can be defined based on values used in practice. Concerning the number of monitored poles and
the initial values for gains, there is no direct rule. However, when these parameters are suitably adjusted, the proposed
method is able to tune the stabilizers' gains in a significantly reduced computational time (compared with
metaheuristic-based optimization approaches that are widely reported in the literature), as it was seen in the multimachine case study.
ACK NO WLE DGE MEN TS
This work was supported by the following Brazilian Agencies: FAPEMIG (APQ-02245-18), CNPq and Capes (Finance
Code 001). Besides, the technical support from GOCES (Optimization, Control and Power System Stability Research
Group—UFSJ—Brazil) is greatly acknowledged.
CONFLICT OF INTEREST
None.
ORCID
Wesley Peres https://orcid.org/0000-0003-3668-040X
Francisco C. R. Coelho https://orcid.org/0000-0001-5330-5244
Junior N. N. Costa https://orcid.org/0000-0002-5620-8498
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How to cite this article: Peres W, Coelho FCR, Costa JNN. A pole placement approach for multi-band power
system stabilizer tuning. Int Trans Electr Energ Syst. 2020;e12548. https://doi.org/10.1002/2050-7038.12548
A P P EN D I X
This appendix details the iterative process in Section 4.2. The design process aims to place two modes into a region
defined by ξd = 15%, starting from KI0 = 1 and KH0 = 1. The tolerance value is set to 0.1% (ξmin ≥ 14.9%) and a maximum iteration number of 25 is established as stopping criteria.
The structure of the linear system to be solved at each iteration is presented in Equation (A1).
2
3
∂σ 1
∂σ 1
−
1
0
0
0
−
6
∂K I
∂K H 7
6
7
6
∂σ 2
∂σ 2 7
6
7
0
0
−
−
60 1
7
6
∂K I
∂K H 7 2 σ 1e 3 2 σ 01 3
6
7
6
∂ω1
∂ω1 7
7
7 6
60 0
76
1
0
−
−
6 σ 2e 7 6 σ 02 7
6
7
∂K
∂K
6
6
7
7
I
H7
6
ω1e 7 6 ω01 7
6
76
6
6
7
7
∂ω
∂ω
2
2 7:
ðA1Þ
60 0
7 = 6ω 7
0
1
−
−
6
76
ω
2e
02
6
6
7
7
∂K I
∂K H 7 6
6
6
7
6
7 4 ΔK 7
4 0 5
I 5
6
7
ξd1
6 1 0 qffiffiffiffiffiffiffiffiffiffiffiffi
7
ffi
0
0
0
6
7 ΔK H
0
6
7
1 −ξ2d1
6
7
6
7
ξd2
6
7
qffiffiffiffiffiffiffiffiffiffiffiffi
ffi
40 1
0
0 5
0
2
1 −ξd2
Iteration 01
For the initial condition provided, the initial position for each mode is:
KI = 1, KH = 1
PERES ET AL.
25 of 26
λ01 = σ 01 + jω01 = −0:0518 + j8:1480
λ02 = σ 02 + jω02 = −10:1820 + j14:2870
Considering the desired damping ratio ξd = 0.15, the linear system to be solved as well as its solution is given in
Equation (A2). The pole sensitivities are calculated at positions λ01 and λ02.
2
1 0
0
0
0:0249
1
0
0
0:1283
0:0312
32
σ 1e
3
2
−0:0518
32
σ1
3
2
−0:8705
3
6
7 6
7 6
76
7
−0:1680 7
7 6 σ 2e 7 6 −10:1820 7 6 σ 2 7 6 −2:7847 7
7 6
7 6
76
76
7
0
1
0
−0:1157 0:0309 7 6 ω1e 7 6 8:1480 7 6 ω1 7 6 5:7375 7
7=6
7=6
7:6
76
7
6
7 6
7 6
76
7
0
0
1
0:0381 −0:1028 7
7 6 ω2e 7 6 14:2870 7 6 ω2 7 6 18:3550 7
7 6
7 6
76
76
7
5 4 ΔK I 5 4
5 4 ΔK I 5 4 −11:3940 5
0 0:1517
0
0
0
0
ΔK H
ΔK H
1
0
0:1517
0
0
0
35:3320
60
6
6
60
6
60
6
6
41
0
ðA2Þ
After the linear system solution, it is possible to get new values for gains. Besides, as one can see, the “estimated
values” for the poles positions (λ1e and λ2e) are also obtained.
K I = 1 + ΔK I = 1 − 11:394 = −10:3940
K H = 1 + ΔK H = 1 + 35:3320 = 36:3320
λ1e = σ 1e + jω1e = −0:8705 + j5:7375
λ2e = σ 2e + jω2e = −2:7847 + j18:3550
It is important to mention that the pole sensitivities are valid for small increment of gains. In other words, the estimated values for poles after the linear system solution can differ from the “exact, correct or true” values for the new
gains KI = − 10.3940 and KH = 36.3320. Larger the increments ΔKI and ΔKH are greater the difference. The correct
positions, obtained from a feedback procedure, are:
λ1 = σ 1 + jω1 = 0.1822 + j6.2844 (ξ1 = − 2.90%)
λ2 = σ 2 + jω2 = − 5.8697 + j18.0820 (ξ2 = 30.88%)
As it can be seen, the system is unstable for the new values of gains. Once the minimum damping ξmin = ξ1 is lower
than the desired one (15%), the algorithm starts a new iteration.
Iteration 02
The linear system to be solved as well as its solution is given in Equation (A3). The pole sensitivities are calculated at
positions λ1 = 0.1822 + j6.2844 and λ2 = − 5.8697 + j18.0820.
2
1 0
0
0
0:0623
1
0
0
0:0218
60
6
6
60
6
60
6
6
41
0
0:0034
32
σ 1e
3
2
0:1822
32
σ1
3
2
−0:9094
3
6
7 6
7 6
76
7
− 0:0646 7
7 6 σ 2e 7 6 −5:8697 7 6 σ 2 7 6 −3:1005 7
7 6
7 6
76
76
7
0
1
0
−0:0422 0:0192 7 6 ω1e 7 6 6:2844 7 6 ω1 7 6 5:9939 7
7=6
7=6
7:6
76
7
6
7 6
7 6
76
7
0
0
1
0:0540 − 0:0659 7
7 6 ω2e 7 6 18:0820 7 6 ω2 7 6 20:4360 7
7 6
7 6
76
76
7
5 4 ΔK I 5 4
5 4 ΔK I 5 4 14:8950 5
0 0:1517
0
0
0
0
ΔK H
ΔK H
1
0
0:1517
0
0
0
47:9010
The gain updating, the “estimated” poles (λ1e, λ2e), and the exact ones (λ1, λ2) after the feedback procedure are:
ðA3Þ
26 of 26
PERES ET AL.
K I = −10:3940 + ΔK I = −10:3940 + 14:8950 = 4:5007
K H = 36:3320 + ΔK H = 36:3320 + 47:9010 = 84:2330
λ1e = σ 1e + jω1e = −0:9094 + j5:9939
λ2e = σ 2e + jω2e = −3:1005 + j20:4360
λ1 = σ 1 + jω1 = − 0.8598 + j5.9155 (ξ1 = 14.38%)
λ2 = σ 2 + jω2 = − 3.4720 + j20.0490 (ξ2 = 17.06%)
Considering a tolerance of 0.1%, the minimum damping ratio must be greater than or equal to 14.9%. This condition
is not met (ξmin = ξ1 = 14.38%) and the algorithm starts a new iteration.
Iteration 03
The linear system to be solved as well as its solution is given in Equation (A4). The pole sensitivities are calculated at
positions λ1 = − 0.8598 + j5.9155 and λ2 = − 3.4720 + j20.0490.
2
1 0
0
0
1
0
0
1
0
0
60
6
6
60
6
60
6
6
41
0
0
0 0:1517
0 1
0
0:0830
− 0:0046
32
σ 1e
6
0:0035 − 0:0444 7
7 6 σ 2e
76
−0:0352 0:0196 7 6 ω1e
7: 6
6
0:0373 − 0:0468 7
7 6 ω2e
76
5 4 ΔK I
0
0
1
0
0:1517
0
0
ΔK H
3
2
−0:8598
32
σ1
7 6 −3:4720 7 6 σ
7 6
76 2
7 6
76
7 6 5:9155 7 6 ω1
7=6
76
7 6 20:0490 7 6 ω
7 6
76 2
7 6
76
5 4
5 4 ΔK I
0
0
ΔK H
3
2
−0:8758
3
7 6 −3:0981 7
7 6
7
7 6
7
7 6 5:7728 7
7=6
7
7 6 20:4200 7
7 6
7
7 6
7
5 4 0:6587 5
ðA4Þ
8:4725
The gain updating, the “estimated” poles (λ1e, λ2e), and the exact ones (λ1, λ2) after the feedback procedure are:
K I = 4:5007 + ΔK I = 4:5007 + 0:6587 = 5:1594
K H = 84:2330 + ΔK H = 84:2330 + 8:4725 = 92:7060
λ1e = σ 1e + jω1e = −0:8758 + j5:7728
λ2e = σ 2e + jω2e = −3:0981 + j20:4200
λ1 = σ 1 + jω1 = − 0.8735 + j5.7775 (ξ1 = 14.95%)
λ2 = σ 2 + jω2 = − 3.1098 + j20.4090 (ξ2 = 15.06%)
It is possible to see that the minimum damping ratio ξmin = ξ1 satisfies the criteria specified at the beginning of the
iterative process. Besides, the damping of the mode associated with the excitation system (λ2) has been reduced from
58.36% to 15.06%. This is expected, because the linear system solution aims to place all monitored modes at a region
with a damping ratio equal to 15%. As the algorithm stops when ξmin ≥ ξd (in fact there is the tolerance), the poles are
not placed “exactly” at positions with ξd. However, all steps at iterative process aim to place the set of monitored poles
at positions with ξd. In practice, only unstable and poorly damped modes are selected to be monitored (allocated), so
that excitation modes are not significantly affected.