65
Scientia et Technica Año XXVIII, Vol. 29, No. 02, abril-junio de 2024. Universidad Tecnológica de Pereira. ISSN 0122-1701 y ISSN-e: 2344-7214
W
Assessment of Transient Stability Indicators in
Wind-Integrated Power Systems: An Open-
Source Simultaneous Approach
Evaluación de Indicadores de Estabilidad Transitoria en Sistemas de Potencia con
Integración de Energía Eólica: un Enfoque Simultáneo de Código Abierto
J. Sosapanta-Salas ; B. J. Ruiz-Mendoza
DOI: 10.22517/23447214.25259
Artículo de investigación científica y tecnológica
Abstract—The energy transition relies on the integration of
non-conventional renewable energy sources. These disruptive
technological developments alter the functioning and operation of
the electric power system. This paper examines the impacts of
wind power on transient stability indicators of the power system,
using an implicit formulation and the nine-bus test system. The
research findings indicate that transient stability indicators are
sensitive to the location and duration of faults. Furthermore,
there is an observed trend of increasing maximum rotor speed
deviation and oscillation duration, indicating reduced stability
margins.
Index Terms—Differential-algebraic equations, power system
dynamics, power system stability, renewable energy sources,
wind power grid integration.
Resumen— La transición energética se basa en la integración
de fuentes no convencionales de energía renovable. Estos avances
tecnológicos disruptivos modifican el funcionamiento y operación
del sistema de potencia. Este documento describe los impactos de
la energía eólica en los indicadores de estabilidad transitoria del
sistema eléctrico, utilizando una formulación implícita y el
sistema de prueba de nueve barras. Los hallazgos de esta
investigación muestran que los indicadores de estabilidad
transitoria son susceptibles a la ubicación y duración de la falla.
Además, hay una tendencia creciente en los resultados para el
rotor máximo. desviación de velocidad y la duración de la
oscilación, lo que significa que los márgenes de estabilidad se
reducen.
Palabras claves—Dinámica de sistemas de potencia, ecuaciones
algebraicas diferenciales, estabilidad del sistema de potencia,
fuentes de energía renovables, integración de la red de energía
eólica.
This manuscript was sent on month day, year and accepted on month day,
year. This work was supported by the Institución Universitaria Pascual Bravo.
J. Sosapanta-Salas is a researcher of the GIIAM group, of the Institución
Universitaria Pascual Bravo, in street 73 # 73a-226 Pilarica, Medellín (email:
j.sosapantasa@pascualbravo.edu.co).
B. J. Ruiz-Mendoza is a researcher of the GIPEM group, of the National
University of Colombia, in street Av. Paralela #62236, Manizales, Caldas
(email: bjruizm@unal.edu.co).
I.
INTRODUCTION
IND power arises as an alternative to the growing demand
for electrical energy in a safe, reliable, and economical
way, that also provides benefits such as a balanced,
sustainable, environmentally friendly, and diversified energy
economy.
A.
Motivation
In the last decade, wind power generation systems have
increased their participation with an average annual growth of
15.1%. This is due to the technological improvements, that
allow energy production at a more competitive cost as
compared with other energy sources. In this order of ideas, the
need arises to investigate the impacts caused by wind power
generation on electrical power systems. In particular, transient
stability studies examine the dynamic behavior when events
that modify the topology of electrical networks occur.
B.
Literature Review
The authors [1], [2], [3], [4], [5] have discussed the
integration of wind power generation into power systems. The
impacts of the constant-speed wind turbine model on transient
stability are evaluated in [6], [7]. On the other hand, in [8] the
maximum rotor speed deviation and the oscillation duration
were proposed for the first time as transient stability
indicators. In [9] an analysis is presented for different
participation percentages of wind energy and the results of the
simulations in the time domain for the frequency of the system
and the active power of the synchronous generators are
exposed. To obtain the results in the above studies, the authors
employ commercially available software, which has a large
number of components and models with the disadvantage of
using a closed code, which makes it impossible to interact
with the source code for academic purposes.
In contrast, different authors have developed Matlab-based
open-source software for transient stability simulations, and
some of them are described below. The Power System
Toolbox (PST) performs control studies and dynamic
simulation [10]. The Power System Analysis Toolbox (PSAT)
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Scientia et Technica Año XXVIII, Vol. 29, No. 02, abril-junio de 2024. Universidad Tecnológica de Pereira
has a Simulink interface and the ability to calculate optimal
power flows, small-signal stability, and time domain analysis
[11]. MatDyn focuses on transient stability analysis but does
not include models for wind power generation [12]. In this
previous Matlab-based software, each author implements his
own numerical integration routines. More recently, in [13] the
authors publish a Toolbox for modal analysis in the time
domain, and the differential-algebraic equations of the power
system are sorted out directly using Matlab solvers.
In [14] an explicit formulation of the differential-algebraic
equations is presented, while in [15] a semi-explicit
formulation is proposed, distinguishing its numerical
advantages. The authors in [16] show a technique that adopts a
combination of explicit and implicit methods, seeking to
exploit the advantages of each one, considering its efficiency
and numerical stability.
C.
Contributions
The contributions in this paper encompass:
▪
Implementation of an open-source in Matlab for transient
stability studies that integrates the main components of
the electrical system and the type I wind turbine model.
▪
The solution in the time domain of the differential-
algebraic equations using the Matlab solvers, through a
completely implicit formulation and a simultaneous
approach.
▪
Review of the transient stability indicators based on the
location of the failures and the level of participation of the
wind turbines.
D.
Paper Organization
In section II the wind power background and modeling are
presented with the mathematical formulation for the model
type I of the wind turbine. After that, section III describes the
maximum rotor speed deviation and the oscillation duration,
which are the transient stability indicators selected for this
study. Section IV presents the time domain simulation
approach and the solution technique in Matlab. Section V
exposes the study case and simulation scenarios. Section VI
explains the main results and their respective analysis. Finally,
the main research conclusions are indicated.
II.
WIND POWER
Non-conventional sources of renewable energy have the
following characteristics: the ability to regenerate by natural
means, reduced dependence on external supplies, few wastes,
and low influence on the environment. Particularly, the wind
turbine’s operation is based on the airflow acting over the
rotor blades and transferring the kinetic energy of the air to the
rotor shaft where it is converted into mechanical energy.
Afterward, the mechanical energy is converted into electricity
by the action of the generator.
A.
Wind Power Background
Since the beginning of the commercial use of wind turbines
in the 1980s, the global wind power installed capacity has
increased exponentially as shown in Fig. 1, with an average
annual growth of 15.1% in the last decade [17].
Fig. 1. Wind power global installed capacity [GW] 1980 – 2020 [17].
Under this scenario, the conventional operation of power
systems undergoes changes in their dynamic behavior due to
the uncertainty in the wind power generation, as a
consequence of the fluctuating nature of the wind and the
inability of wind turbines to balance the power between
generation and demand.
B.
Wind Turbine Modeling
The type I wind turbine model is defined in the IEC 61400-
27-1 standard as an asynchronous generator connected directly
to the network as indicated in Fig. 2. These generators include
capacitive compensation, in order to counteract the reactive
power that is extracted from the electrical network [18].
Fig. 2. Type I wind turbine model [19].
This document focuses on the type I wind turbine model,
which employs a squirrel cage induction generator illustrated
in Fig. 3 [2], [6]. It has fixed rotor resistance and handles
simple controls, so that the characteristics of the generator,
and the gearbox govern the speed of the rotor. Thus, the rotor
speed deviation does not exceed 2% and therefore this
generator belongs to the group of constant-speed wind
turbines [3], [7].
Scientia et Technica Año XXIII, Vol. 29, No. 02, abril-junio de año 2024. Universidad Tecnológica de Pereira.
67
𝑟
(
9
)
Fig. 3. Squirrel cage induction generator [20].
where 𝑅
𝑠
is the stator resistance, 𝑋
𝑠
is the stator reactance,
𝑋
𝑚
is the magnetizing reactance, 𝑍
𝑟𝑜𝑡
is the impedance of the
rotor. The dynamics of the rotor circuits are established by
means of slip, which, as well as the speed of the rotor, presents
changes according to the power generated. Slip is essential for
generating electromagnetic torque in a squirrel cage induction
generator, as it induces a current in the rotor when it rotates
slower than the stator's magnetic field, creating an opposing
magnetic field that produces torque. This slip allows the
turbine to adapt to varying wind speeds, adjusting the rotor
speed and thereby changing the torque and power generated.
The voltages and currents are expressed in terms of their
real (𝑟) and imaginary (𝑚) components. The network bus
voltage is related to the machine stator voltages through the
set of equations
𝑣
𝑟
= 𝑉 sin
(
−𝜃
)
(
1
)
𝑣
𝑚
= 𝑉 cos
(
𝜃
)
(
2
)
where 𝑉 and 𝜃 are the magnitude and angle of the bus
voltages, respectively. The active and reactive powers are
calculated based on the stator current and voltage components,
as follows
𝑃 = 𝑣
𝑟
𝑖
𝑟
+ 𝑣
𝑚
𝑖
𝑚
(
3
)
𝑄 = 𝑣
𝑚
𝑖
𝑟
− 𝑣
𝑟
𝑖
𝑚
+ 𝑏
𝑐
(
𝑣
2
+ 𝑣
2
)
𝑟 𝑚
(
4
)
where 𝑖
𝑟
and 𝑖
𝑚
are the real and imaginary components of
the stator current, respectively, and 𝑏
𝑐
is the conductance of
the compensation capacitor. The algebraic equations
(
1
)
and
(
2
)
will be substituted in
(
3
)
and
(
4
)
, which correspond to the
network interface in a synchronously rotating reference frame.
In addition, the machine electro-magnetic differential
equations for the real and imaginary components of the
voltage behind the stator resistor are
(
𝑒
′
−
(
𝑥 − 𝑥
′
)
𝑖
)
𝑒̇
′
= 2𝜋𝑓
(
1 − 𝜔
)
𝑒
′
−
𝑟 0 𝑚
𝑟
𝑚
𝑚
𝑇
′
0
(
5
)
(
𝑒
′
−
(
𝑥 − 𝑥
′
)
𝑖
)
𝑒̇
′
= −2𝜋𝑓
(
1 − 𝜔
)
𝑒
′
−
𝑚 0 𝑟
𝑚
𝑚
𝑟
𝑇
′
0
(
6
)
𝑥
0
= 𝑥
𝑠
+ 𝑥
𝑚
(
7
)
𝑥
𝑅
𝑥
𝑚
𝑥
′
= 𝑥
𝑠
+
𝑥
𝑅
+ 𝑥
𝑚
(
8
)
where 𝑟
𝑅
is the rotor resistance, 𝑥
𝑅
is the rotor reactance, 𝑥
𝑠
is the stator reactance, and 𝑥
𝑚
is the magnetizing reactance.
The variables in
(
1
)
to
(
6
)
, and the stator current components
are related by
𝑒
′
− 𝑣 = 𝑟 𝑖 − 𝑥
′
𝑖
𝑟 𝑟 𝑠 𝑟 𝑚
(
10
)
𝑒
′
− 𝑣 = 𝑟 𝑖 − 𝑥
′
𝑖
𝑚 𝑚 𝑠 𝑚 𝑟
(
11
)
where 𝑟
𝑠
is the stator resistance. The set of differential-
algebraic equations
(
1
)
-
(
11
)
represent the behavior of a wind
turbine induction generator and will be solved according with
the methodology described in section IV.
III.
TRANSIENT STABILITY INDICATORS
Transient stability refers to the ability of the electrical
power system to remain in synchronism when it is subjected to
events or disturbances that imply large changes in the
topology of the network, introducing transition stages that lead
to changes in the state of the system [21]. In order to study the
dynamic performance, the maximum rotor speed deviation and
the oscillation duration are proposed as transient stability
indicators.
A.
Maximum rotor speed deviation
The maximum rotor speed deviation, indicated in Fig. 4, is
the maximum value that the rotor speed reaches through
transient disturbances. This indicator is calculated as [8]
|𝜔
𝑟,𝑚𝑎𝑥
− 𝜔
𝑟,𝑛𝑜𝑚
|
𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑟𝑜𝑡𝑜𝑟 𝑠𝑝𝑒𝑒𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 =
𝜔
𝑟,𝑛𝑜𝑚
(
12
)
where 𝜔
𝑟,𝑚𝑎𝑥
and 𝜔
𝑟,𝑛𝑜𝑚
are the maximum and nominal
rotor speed, respectively. When the maximum rotor speed
deviation is increased as a consequence of longer clearance
times, the transient stability margin is reduced and thus the
system becomes more susceptible to instability.
B.
Oscillation duration
The oscillation duration refers to the time interval from the
start of the disturbance to the moment where the rotor speed
remains in a band of 10
−4
pu, for a duration greater than 2.5 s
[8]. This indicator is shown in Fig. 4 and can be calculated as
𝑂𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑡
𝑜𝑠𝑐
− 𝑡
𝑓
(
13
)
where 𝑡
𝑓
is the time when the fault is applied, and
𝑡
𝑜𝑠𝑐
= 𝑚𝑖𝑛
{
𝑡: |𝜔
𝑟
(
𝑡 + 𝑛∆𝑡
)
− 𝜔
𝑟
(
𝑡
)
| ≤ 10
−4
; 𝑛 = 1, … ,2.5/∆𝑡
}
(
14
)
where 𝜔
(
𝑡
)
is the rotor speed at time 𝑡 and ∆𝑡 is the
simulation step.
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Scientia et Technica Año XXVIII, Vol. 29, No. 02, abril-junio de 2024. Universidad Tecnológica de Pereira
0
=
𝑖
(
̅
𝑦,
𝑣̅
)
−
𝑌
̅
𝑣̅
(
18
)
Fig. 4. Transient stability indicators [8].
IV.
METHODOLOGY
The assessment of wind power’s influence on transient
stability is done through the solution of the differential-
algebraic equations in the time domain, where the models of
the electrical components and the type I wind turbine model
are incorporated (set of differential-algebraic equations
(
1
)
-
(
11
)
). The solution is obtained through an implicit method of
numerical integration using a simultaneous approach.
A.
Time Domain Power System Analysis
Transient stability studies are usually carried out through
time domain simulations, which have the advantage of
manipulating complex mathematical models and providing a
complete description of the electrical and mechanical
variables. This methodology is applied to solve the
differential-algebraic equations of the power systems, as
shown in [22]
where
𝑌
̅
is
the
bus
admittance
matrix.
The
numerical
integration of the equations
(
17
)
and
(
18
)
can be carried out
using the partitioned or simultaneous approaches. In the
partitioned approach, 𝑦 and 𝑣̅ are resolved and updated
sequentially; this means that in each integration step,
(
17
)
is
solved separately for the dynamic variables and
(
18
)
for the
algebraic variables. In the simultaneous approach, the
differential equations 𝑓 are discretized and transformed into a
set of algebraic equations, and they are solved together with
the algebraic equations in
(
18
)
, thereby eliminating the
interface errors of the partitioned approach [23]. The
simultaneous approach is numerically more stable, has better
convergence, and can only be solved using implicit integration
methods [23].
B.
Solution technique in Matlab
The time domain solution of
(
17
)
and
(
18
)
is carried out
using Matlab numerical integration solvers and the
simultaneous approach. These Matlab solvers are designed
mainly to deal with ordinary differential equations, but
specifically, the ode15s and ode23t can also solve systems of
differential-algebraic equations. This requires the definition of
a mass matrix
(
𝑀
)
, which classifies between differential and
algebraic equations, and can be calculated as
𝑀 = 𝑠𝑝𝑎𝑟𝑠𝑒
(
1: 𝑛𝑑𝑒, 1: 𝑛𝑑𝑒, 𝑜𝑛𝑒𝑠
(
1: 𝑛𝑑𝑒
)
, 𝑛𝑑𝑒 + 𝑛𝑎𝑒, 𝑛𝑑𝑒 + 𝑛𝑎𝑒
)
(
19
)
where 𝑛𝑑𝑒 indicates the number of differential equations,
nae the number of algebraic equations, 𝑠𝑝𝑎𝑟𝑠𝑒 Matlab
command to transform a sparse matrix, removing elements
equal to zero, and ones is the Matlab command to create a
series of 𝑜𝑛𝑒𝑠. For example, when 𝑛𝑑𝑒 = 2 and 𝑛𝑎𝑒 = 3 the
entire matrix 𝑀 is given by
𝑦̇ = 𝑓
(
𝑦, 𝑥
)
(
15
)
0 = 𝑔
(
𝑦, 𝑥
)
(
16
)
where 𝑦 indicates the vector of dynamic variables, 𝑥 are the
algebraic variables, 𝑓 are the differential equations, and 𝑔 are
the algebraic equations. Differential equations describe the
dynamic behavior of dynamic variables such as the angular
position and speed of the rotor [3]. Algebraic equations
describe the static components of the power system, such as
the electrical network, the loads, and the stator of the
generators. Variables that appear in both differential and
algebraic equations are known as interface variables [22].
Algebraic equations correspond to the expressions for the
current injections 𝑖̅ in the network buses, and the algebraic
variables correspond to the vector of complex bus voltages
𝑣̅ as follows [23]
Then the mass matrix is multiplied by the implicit form of
(
17
)
, as follows
𝑀𝑦̇ = 𝑓
(
𝑦, 𝑣̅
)
(
21
)
V.
STUDY CASE AND SIMULATION SCENARIOS
The nine-bus test system is widely used for dynamic studies
in power systems. This system is shown in Fig. 5 and its main
characteristics are indicated in Table I. The study is performed
by applying a three-phase fault on load buses 5, 6, and 8, with
a clearance time of 100 ms.
𝑦̇ = 𝑓
(
𝑦, 𝑣̅
)
(
17
)
1
0
0
0 0
𝖥
0
1
0
0 0
1
𝑀 = 0
0
0
0 0
I0
0
0
0 0I
[
0
0
0
0 0
]
(
20
)
Scientia et Technica Año XXIII, Vol. 29, No. 02, abril-junio de año 2024. Universidad Tecnológica de Pereira.
69
TABLE I
Characteristics of nine-bus test system.
System characteristics
Values
Buses quantity
9
Generators quantity
3
Loads quantity
3
Transmission lines quantity
9
Total generation
319.64 MW
33.7 MVAR
Total load
315 MW
125.86 MVAR
Fig. 5. Nine-bus test system one-line diagram.
The simulation scenarios seek to equate the increase in the
active power of the load with an equivalent amount of wind
power. The wind power penetration level, given in equation
22, is increased in steps of 5% up to 25%; in this way, six sub-
scenarios are obtained with wind power penetration levels of
0, 5, 10, 15, 20, and 25%. For the base case, the increase in
load is supplied with conventional synchronous generators,
which serve as a reference point [8].
when wind power generation is used. The differences in this
indicator are more pronounced for the generator connected in
bus 2 compared to the one located in bus 1. Additionally, as
the wind power penetration level increases, these absolute
differences are attenuated as a consequence of the dynamic
response of induction generators.
In the first two simulation scenarios, the oscillation duration
is longer when wind turbines are used and for the remaining
scenarios, the indicator is lower than the base case employed
as a reference. The results obtained for the transient stability
indicators are similar to those found in [8], where it is
proposed that these indicators can be improved with the
implementation of network voltage and frequency control.
𝑃
𝑊𝑃
𝑃𝑒𝑛𝑒𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑙𝑒𝑣𝑒𝑙 = (
) 100%
𝑃
𝑊𝑃
+ 𝑃
𝐶𝐺
(
22
)
where 𝑃
𝑊𝑃
and𝑃
𝐶𝐺
are the total active power generated by
wind turbines and conventional synchronous generators,
respectively.
VI.
RESULTS ANALYSIS
Figs. 6 to 8 present the results of the transient stability
simulations for the maximum rotor speed deviation and the
oscillation duration when the wind power penetration level
increases according to the simulation scenarios.
A.
Fault at bus 5
The results for the generators connected to buses 1 and 2 are
presented in Fig. 6 when the fault is applied to bus 5. The
maximum rotor speed deviation is greater in all scenarios
Fig. 6. (a) Maximum rotor speed deviation and (b) oscillation
duration when the fault is applied at bus 5 [24].
B.
Fault at bus 6
The results for the generators connected to bars 1 and 2 are
presented in Fig. 7 when the fault is applied to bus 6. The
maximum rotor speed deviation has a greater discrepancy with
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Scientia et Technica Año XXVIII, Vol. 29, No. 02, abril-junio de 2024. Universidad Tecnológica de Pereira
the increase of the wind power penetration level and the
results of the oscillation duration are contrasted with those of
Fig. 6 since in general terms the inclusion of wind power
implies higher oscillation duration values.
Fig. 7. (a) Maximum rotor speed deviation and (b) oscillation
duration when the fault is applied at bus 6 [24].
C.
C. Fault at bus 8
The results for the generators connected to buses 1 and 2 are
presented in Fig. 8 when the fault is applied to bus 8. Figs. 6 to
8 show that the stability indicators in some cases increase and
in others decrease, depending on the fault location and the
wind power penetration level. Likewise, the results exhibit an
increasing trend in terms of the maximum rotor speed
deviation and the oscillation duration, which means that the
stability margins are reduced.
In order to improve the transient stability indicators,
constant-speed wind turbines can be equipped with a pitch
control system in such a way that the temporary unbalance
Fig. 8. (a) Maximum rotor speed deviation and (b) oscillation
duration when the fault is applied at bus 8 [24].
between the input mechanical power and the output
electrical power can be minimized. Another option involves
modifying some critical parameters of the wind turbine in the
manufacturing phase, with the disadvantage of increased
complexity and construction costs. Moreover, when wind
turbines are considered as distributed generation sources, this
increases the strengths of this type of technology, since power
consumption at generation points decreases power flows along
the lines. The reduction of the power flows in the lines has the
consequence of an increase in the damping of the oscillations
and makes it possible to improve the stability margins.
VII.
CONCLUSIONS
An open-source in Matlab for dynamic studies integrating
constant speed wind turbines is developed and a completely
implicit formulation with the inclusion of the simultaneous
approach has proven to solve the transient stability problem,
Scientia et Technica Año XXIII, Vol. 29, No. 02, abril-junio de año 2024. Universidad Tecnológica de Pereira.
71
with the advantage to eliminate interface errors. Transient
stability indicators are computed, which showed to be strongly
influenced by the location and duration of the faults. There is
an increasing trend in the results for the maximum rotor speed
deviation and the oscillation duration, which means that the
stability margins are reduced with wind power integration.
From the results, fault at bus 6 exhibit lower values for the
oscillation duration and the maximum rotor speed deviation.
On the other hand, the results for faults at buses 5 and 8
manifest similarities which means more susceptibility to lose
rotor angle stability. In order to improve the transient stability
indicators, constant-speed wind turbines can be equipped with
a pitch control system in such a way that the temporary
imbalance between the input mechanical power and the output
electrical power can be minimized.
VIII.
REFERENCES
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systems with large amount of wind power,” Doctoral Thesis,
Universidad Técnica de Dinamarca, Lyngby, 2003.
https://findit.dtu.dk/en/catalog/537f0c897401dbcc120014da
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72
Scientia et Technica Año XXVIII, Vol. 29, No. 02, abril-junio de 2024. Universidad Tecnológica de Pereira
Joseph Sosapanta Salas, Institución
Universitaria Pascual Bravo
Joseph Sosapanta Salas is a researcher and
professor in the GIIAM group at
Institución Universitaria Pascual Bravo.
He received the degree in electrical
engineering from Universidad Nacional
de Colombia, in 2014 and the degree of master in electrical
engineering from the same university in 2023. He also
received the MBA degree in 2021. His areas of interest are
power systems, economics and energy policy.
ORCID: https://orcid.org/0000-0002-2035-9323
Belizza Janet Ruiz Mendoza,
Universidad Nacional de Colombia
Belizza Janet Ruiz Mendoza is vice-rector
and director of GIPEM group at
Universidad Nacional de Colombia – Sede
Manizales. She received the degree in
electrical engineering from Universidad
Nacional de Colombia, in 2002, and the master and PhD
degree from Universidad Nacional Autónoma de México. She
also participated as vice-minister of energy in 2022 and 2023.
Her areas of interest are energy policy and renewable energy.
ORCID: https://orcid.org/0000-0003-3016-7787