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Master of Electric Engineering
Thesis presented to
UNIVERSIDAD DE LOS ANDES
FACULTY OF ENGINEERING
DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING
By
Daniel Sebastián Restrepo Lara
Adaptive POD for Power System with High Wind Power Penetration
Level
ASESOR
Mario Alberto Ríos Mesías, PhD, Titular Profesor,
Universidad de los Andes
Bogotá, Colombia.
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Adaptive POD for Power System 2
Content
1 Introduction .................................................................................................................... 5
2 Reactive Power Compensation ....................................................................................... 6
3 Model Reference Adaptive Control (MRAC) ................................................................ 8
3.1 MIT Rule .................................................................................................................. 9
3.2 Adjustment Mechanism ........................................................................................... 9
A. Fuzzyfication: ..................................................................................................... 10
B. Fuzzy Rule-base: ................................................................................................ 11
C. Inference Mechanism: ........................................................................................ 11
D. Defuzzyfication: ................................................................................................. 11
4 Test System .................................................................................................................. 12
4.1 HVDC line ............................................................................................................. 12
4.2 Modal Analysis ...................................................................................................... 15
4.3 POD Controller Design .......................................................................................... 15
4.4 MRAC Design ....................................................................................................... 17
5 Simulation Results ........................................................................................................ 21
5.1 Controller performance case 1 ............................................................................... 21
5.2 Controller performance case 2 ............................................................................... 23
6 Conclusions .................................................................................................................. 25
7 References .................................................................................................................... 25
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Figure Index
Figure 1. Integration STATCOM and Wind Farm [13] ......................................................... 7
Figure 2. Diagram of MRAC (Adapted from [15] ) ............................................................... 8
Figure 3. Scheme for Fuzzy controller adapted from [21] ................................................... 10
Figure 4. Fuzzyfication example .......................................................................................... 10
Figure 5. Test System ........................................................................................................... 12
Figure 6. Control strategy for substation HVDC [27] .......................................................... 13
Figure 7. Single line diagram ............................................................................................... 13
Figure 8. Supplementary controller POD ............................................................................. 16
Figure 9. MRAC with POD regulator. ................................................................................. 18
Figure 10. ESS reference model ........................................................................................... 19
Figure 11. Fuzzy Logic Controller ....................................................................................... 20
Figure 12. Error .................................................................................................................... 20
Figure 13. Delta-error ........................................................................................................... 21
Figure 14. Surface of rules-based ......................................................................................... 21
Figure 15. Power Flow line fault inter-area (MRAC) .......................................................... 22
Figure 16. Power Flow fault line inter-area (POD) .............................................................. 22
Figure 17. Adaptive parameter case 1 .................................................................................. 23
Figure 18. Power Flow line fault line SLACK (MRAC) ..................................................... 23
Figure 19. Power Flow line fault line SLACK (POD) ......................................................... 24
Figure 20. Adaptive parameter case 2 .................................................................................. 24
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Table Index
TABLE I. Requirements for the integration of non-synchronous energies. ........................... 5
TABLE II. Comparison SVC and STATCOM ...................................................................... 7
TABLE III. Electromechanical modes comparison ............................................................. 15
TABLE IV. Parameters of classic supplementary controller POD ...................................... 17
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1 Introduction
In the last decade, the use of renewable energy has presented a significant increase in the total
installed capacity throughout the world, in response to reducing dependence on fossil fuels
and their global warming problems. Wind energy is one of the most promising renewable
energy. For example, the Colombian electricity system has 16.6 GW of installed capacity to
the SIN, of which 11 GW are hydraulic, 4.6 GW thermal and approximately 1 GW of smaller
plants. However, it is expected that by the year 2023 the Colombian electricity system will
present important changes in its energy matrix, integrating at least 1.6 GW of non-
synchronous generation1 that currently have a connection concept, of which 0.3 GW will
correspond to solar generation and 1.3 GW to wind generation. XM has identified three stages
for the implementation of the proposals, which are defined based on the percentage of variable
generation integration in the system (MW) [2].
The total installed capacity of this generation represents less than 15% of the
maximum daily demand for electrical power in the system. This level of integration in
MW of the variable generation does not generate a considerable impact on the system.
The total installed capacity of this generation is greater than or equal to 15% and is
less than 25% of the maximum daily demand for electric power. This level of
integration in MW of variable generation begins to be relevant for the system.
The total installed capacity of the variable generation represents 25% or more of the
maximum daily demand for electrical power. This integration level in MW of the
generation compromises the flexibility and stability of the system.
TABLE I shows the technical and operational requirements for each of the integration stages
previously stated. This project is focused on the technical requirement for stage 3. Because,
there are concerns about the impacts of the wind energy on overall power system stability.
The stability of a power system it is done through small signal analysis where is defined the
capacity of the power system to keep synchronism under perturbations [1]. The high wind
power penetration level decreases the damping of weak electromechanical modes.
Nevertheless, the dynamical respond depends on the technology of the Wind Farm, precisely
the technology of the turbine.
TABLE I. Requirements for the integration of non-synchronous energies.
Stage 1 Stage 2 Stage 3
Technical
requirements
Over-frequency and sub-
frequency control
capability, Voltage
control and PQ diagram.
frequency control
through inertial
emulation for centrally
dispatched wind plants.
POD function for non-
synchronous plants with a
capacity greater than or
equal to 100 MW.
Operative
requirements
3% reserve on all plants
dispatched centrally or
greater than 20 MW for
frequency control.
Secondary reserves
3% reserve on all plants
dispatched centrally and
percentage of primary
reserve required
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Based in [3], the majority of wind farms have been based on constant-speed technology, but
with the increment of the Wind energy, the planning installations are adopting the variable-
speed technology due to its better energy capture, smoother operation, lower flicker, and
superior controllability. In [3]-[5] it is mentioned that one of the most usual types of variable
speed turbines corresponds to a double-fed induction generator (DFIG). This technology has
presented technical and economic advantages, but also presents problems with dynamic
interactions for the directly connection with the grid and through an electronic power
converter. For this reason, the modal analysis of power systems with Wind Farms based on
DFIG turbines seeks to identify the controller and networks parameters with a major impact
in the system with the participation factors. This analysis allows comparing the change of
weak electromechanical modes with high Wind Power Penetration Level. On the other hand,
for the intermittent nature of wind energy, the power output has randomness. For this reason,
the point of common coupling (PCC) presents more frequent change in voltage. Moreover,
the random behavior of wind speed can make the wind farm consume reactive power affecting
the quality, safe and stable operation of the grid [6], [7]. Because of this behavior, reactive
compensation is an inherent condition for the Wind Farm connection. Although, DFIG
technology could realize automatic power regulation. However, the Wind Farms have many
generation units which increase of complexity of reactive power output coordination, as a
result, it is necessary the use of reactive power compensation schemes [6].
The principal problem of a grid that presents weak electromechanical modes is that in the
event of contingencies or changes in the point of operation, the system can generate
oscillations and depended on the magnitude the system could lose synchronism. Thus, the
implementation of a supplementary controller is necessary. For example, in [1] was added a
classic POD in the control loop for reactive power. Nevertheless, the classic supplementary
controller could have presented inconvenient whit faults that cause a big change in the point
of operation of the system or long-distance power transmissions., different optimization
algorithms have been used for the tuning of parameters of the supplementary controls. For
example, a genetic algorithm is used as a technique of optimization for the tuning parameters
of supplementary controllers PSS and POD in FACTS devices. The time simulation result is
used as data input for the algorithm, so it is considered an offline optimization technique [8]-
[10]. Another technique of optimization corresponds to PSO (particle swarm optimization)
for the coordinated designing of the thyristor-controlled series capacitor (TCSC) damping
controller and power system stabilizer (PSS) in a multi-machine power system [11]. Whit the
use of these optimization algorithms the global optimum for the parameters change respect to
the fault condition or network conditions. Therefore, the system requires a control that selects
the parameters according to the contingency. Nevertheless, this paper proposes an adaptive
supplementary controller for increasing the damping of weak electromechanical modes by
local control directly on the selected reactive compensation device, due to coordination
problems.
2 Reactive Power Compensation
Oscillatory damping behavior is improved for DFIG based wind generators by FACTS
devices. However, it is necessary the selection between SVC and STATCOM devices
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thinking about the improvement of the dynamic response as observed in [6]-[12]. The
differences are briefly introduced in TABLE II.
TABLE II. Comparison SVC and STATCOM
SVC STATCOM
Control scheme based on Thyristors Control schemes based on VSC
Fast response Real time response
Big losses Little loss
Great harmonics Small Harmonics
Low cost High cost
The more efficient reactive power compensation device corresponds to STATCOM, which
present better respond when the voltage is below the range of normal operation. Thus,
STATCOM is a better compensating device than SVC for stabilizing a system [12].
According to the last, the reactive compensation device selected is a STATCOM and the
integration whit the grid and Wind Farm can be seen in Figure 1.
Figure 1. Integration STATCOM and Wind Farm [13]
A STATCOM device is a great solution to improve the dynamic response of the system or
achieve a secure connection of non-synchronous energies. For example, the integration of
the Wind Farm and STATCOM coordinated between the positive and the negative sequences
of the grid voltage and evaluated the system in fault conditions [13]. On the other hand, the
STATCOM is an appropriate option for the integration of a non-synchronous system based
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in solar and wind power source through the compensation device, avoiding that the random
behavior of the sources affects directly the network [14].
3 Model Reference Adaptive Control (MRAC)
Based on [15] the general idea of MRAC is to create a closed loop controller with parameters
that can be updated to change the response to the system. These are updated based on the
error between the system output and the model reference. The goal is for the parameters
change to optimum values that cause the plant response to follow the response of the
reference model. Figure 2 shows the scheme of MRAC.
Figure 2. Diagram of MRAC (Adapted from [15] )
In [16], [17] was demonstrated that the designed optimal MRAS-PSS is capable of
guaranteeing the robust stability and robust performance of the power system over
conventional PSS. On the other hand, the methodologies most used to ensure the convergence
of the parameters are the Lyapunov stability theory and the gradient method evaluated in [16]
and [15] respectively. However, this paper is focused on gradient method.
It is worth mentioning that there is no standard methodology to find the reference model. The
reference model is defined based on the desired characteristics of the output. However, the
searched reference model must share characteristics similar to the homologous control for
the system. Generally, the model reference is a second-order transfer function as seen in (1)
to be able to ensure the desired output characteristics, such as; settling time, rise time,
overshoot.
𝐾𝑝𝑆(𝑠 + 𝑏0)
𝑆2 + 𝑎1𝑆 + 𝑎0 (1)
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Adaptive POD for Power System 9
3.1 MIT Rule
The MIT rule seeks to minimize a cost function expressed in terms of the error between the
output of the system and the reference model [18]. In this rule, the cost function is defined
as:
𝐽(𝜃) =1
2𝑒2(𝜃) (2)
Where 𝑒 indicates the error between output system and model reference and 𝜃 is the
adjustable parameter. This parameter is adjusted to minimize the cost function from the
sensibility of 𝜃 to move in a way of the negative gradient of 𝐽 as seen in (3).
𝑑𝜃
𝑑𝑡= −𝛾
𝑑𝐽(𝜃)
𝑑𝜃= −𝛾𝑒
𝑑𝑒
𝑑𝜃 (3)
Where 𝜕𝑒
𝜕𝜃 is called as the sensitivity derivative of the system [18]. Equation (3) shows how
the error is changing with the adjust of parameter 𝜃 and 𝛾 which is the adaptation gain of the
controller.
3.2 Adjustment Mechanism
The adjustment mechanism corresponds to the methodology used for the change of the
adaptive parameter looking for improvement the dynamic response of the system. In this
paper, the method selected was Fuzzy logic control. This control methodology has already
been used in POD supplementary controller. For example, [19] the fuzzy logic has been used
as an auxiliary damping signal based in angular frequency error and error derivate. The
control loop of the POD for DFIG was embedded in RSC (Rotor side converter). In another
example, fuzzy logic has been used as an auxiliary damping signal located on STATCOM
located at the midpoint of the transmission line [20]. On the other hand, fuzzy logic can also
be used as a methodology of tuning of the parameters of the supplementary control and AVR.
The terminal voltage in the bus is a great selection as one of the inputs because the controller
can be a voltage regulator. Nevertheless, the angle deviation is selected as another input to
guarantee the synchronism in the power system [21].
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Figure 3. Scheme for Fuzzy controller adapted from [21]
Figure 3 shows the step-by-step scheme for a Fuzzy logic controller. A general description
is present as follows, [22]:
A. Fuzzyfication:
This step converts a crisp input into a fuzzy variable. For this conversion is necessary the
definition of the membership functions, whose standard shapes are Gaussian, trapezoidal and
triangular. This is an independent process for each input and the idea is to transform an input
value into a vector of weights associated with the different membership functions. Figure 4
shows an example of fuzzification of service (input) and 3 membership functions classified
in poor, good and excellent. For input of 7 in service, (4) shows the vector of weights
associated.
[0 0.4111 0.1353] (4)
Figure 4. Fuzzyfication example
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B. Fuzzy Rule-base:
The base of fuzzy rules is called the linguistic model of the control process, which includes
information about:
Regular and possible values of state variables.
The desired progress of the process.
the method applied to bring and keep the process at the required level.
The Fuzzy Rule-base is based on a set of if-else rules that related the inputs with the outputs.
C. Inference Mechanism:
Mamdani and Sugeno are the most frequently used model based on the fuzzy inference.
Below is shown the comparison between inference mechanism Mamdani and Sugeno [23].
The mechanism inference selected correspond to Takagi-Sugeno for the advantages in the
adaptive techniques.
MAMDANI
Are intuitive.
Have widespread acceptance.
Are well-suited to human input.
Easy formalization and interpretability.
Can be used for both MISO and MIMO systems.
Output can either be fuzzy or a crisp output.
TAKAGI-SUGENO
Are computationally efficient.
Work well with optimization and adaptive techniques.
Guarantee continuity of the output surface.
More robust when in presence of noisy input data such as sensor data.
Allowed more degrees of freedom and more flexibility in the design.
Continuous structure of output function
D. Defuzzyfication:
The main difference among inference mechanisms exposed above is the defuzzification
process. Mamdani model converts the output Fuzzy variable into a unique number using
different methods. The most popular method corresponds to Centroid that is a weighted
average. On the other hand, Takagi-Sugeno model is the functional and continuous
relationship between inputs and outputs considering the activation of individual conclusions.
The defuzzyfication expression for Sugeno model is observed in (5).
𝑍 =∑ 𝑤𝑖 ∙ 𝑓𝑖(𝑥, 𝑦)𝑛
𝑖=1
∑ 𝑤𝑖𝑛𝑖=1
(5)
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Where 𝑤𝑖 is the min{𝜇𝐴𝑖(𝑥), 𝜇𝐵𝑖
(𝑦)} , being A and B the fuzzy variable corresponding to
inputs. 𝑓𝑖(𝑥, 𝑦) is the function of each rule for example, 𝑖𝑓 𝑥 𝑖𝑠 𝐴1 𝑎𝑛𝑑 𝑦 𝑖𝑠 𝐵1 → 𝑓1(𝑥, 𝑦).
𝑍 is the output of Sugeno model.
4 Test System
The test system is an adaptation of the Kundur system of two areas taken from [24].
Additionally, the system has a VSC-HVDC line based in [25] for control strategies. This
system has 3 synchronous machines of 900 MVA and one Wind Farm of 900 MVA that
correspond to 25 percent of total installed capacity without power transmission by HVDC.
The Wind Farm is compound for 180 DFIG turbines each 5MW. There is a new generator of
900 MW, which transmitted 600 MW by the HVDC connection that is connected to bus 9.
To achieve this transmission, the load in bus 9 has an increase of the same magnitude.
Besides, the generator 2 has a PSS as a supplementary controller to get an individual control
of the two areas. The lines between the buses 7 and 9 transport 400 MW to area 2. Figure 5
shows the full test system. Dynamic simulation and test of the controllers are performed in
Power Factory DigSilent.
Figure 5. Test System
4.1 HVDC line
One of the most used controllers for the control of a substation based on VSC-HVDC
technology corresponds to the control by the vectorial method, which is based on simplifying
the representation the three-phase system through a DQ transformation. Figure 6 shows the
vector control scheme used for the control of substations.
The PLL block measures the frequency of the system and calculates a synchronization angle
𝜃 used for the DQ transform. In steady state sin 𝜃 is in phase with the fundamental component
(sequence positive) of phase A voltage, this means that if the axes dq are correctly aligned,
the component 𝑣𝑞 is equal to 0. The instantaneous information of the angle is necessary for
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Adaptive POD for Power System 13
the control independent of the active power and the reactive power, to facilitate the
approximation of the angle of synchronization on the Clark transform [27].
𝜃 = tan−1 (𝑣𝛽
𝑣𝛼) (6)
Figure 6. Control strategy for substation HVDC [27]
Figure 7. Single line diagram
Figure 7 shows the single line diagram, from the figure it is known that:
𝑣𝑠 = 𝑅𝑖𝑠 + 𝐿𝜕𝑖𝑠
𝜕𝑡+ 𝑣𝑐 (7)
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Applying the dq transform you have to
[𝑣𝑠,𝑑
𝑣𝑠,𝑞] = 𝑅 ∗ [
𝑖𝑑
𝑖𝑞] + 𝐿
𝜕
𝜕𝑡[𝑖𝑑
𝑖𝑞] + 𝐿 [
0 −𝜔𝜔 0
] [𝑖𝑑
𝑖𝑞] + [
𝑣𝑠,𝑑
𝑣𝑠,𝑞] (8)
The dynamics of the system are
𝐿𝜕𝑖𝑑
𝜕𝑡= −𝑅𝑖𝑑 + 𝜔𝐿𝑖𝑞 − 𝑣𝑐,𝑑 + 𝑣𝑠,𝑑
𝐿𝜕𝑖𝑞
𝜕𝑡= −𝑅𝑖𝑞 + 𝜔𝐿𝑖𝑑 − 𝑣𝑐,𝑞 + 𝑣𝑠,𝑞
(9)
Rewriting the previous expressions, you have to
𝐿𝜕𝑖𝑑
𝜕𝑡+ 𝑅𝑖𝑑 − 𝐿𝑖𝑞 = −𝑣𝑐,𝑑 + 𝑣𝑠,𝑑
𝐿𝜕𝑖𝑞
𝜕𝑡+ 𝑅𝑖𝑞 − 𝜔𝐿𝑖𝑑= − 𝑣𝑐,𝑞 + 𝑣𝑠,𝑞
(10)
From the previous equations, it can be observed that the model of the VSC converters in the
coordinate axis dq is a non-linear MIMO system (multiple inputs and multiple outputs), In
addition, a derivative term corresponding to a cross coupling between the two axes is
presented. This cross coupling can be considered as a disturbance from the control point of
view. In order to achieve better performance, a PI current regulator is added to improve the
response in steady state of the system. To uncouple the controller from axes d and q, the
output of each controller corresponds to the reference voltage of each axis.
𝐹(𝑠) = 𝑘𝑝 +𝑘𝑖
𝑠 (11)
𝑣𝑐,𝑑𝑟𝑒𝑓
= (𝑖𝑑,𝑟𝑒𝑓 − 𝑖𝑑)𝐹(𝑠) + 𝜔𝐿𝑖𝑞 + 𝑣𝑠,𝑑
𝑣𝑐,𝑞𝑟𝑒𝑓
= (𝑖𝑞,𝑟𝑒𝑓 − 𝑖𝑞)𝐹(𝑠) + 𝜔𝐿𝑖𝑑 + 𝑣𝑠,𝑞
(12)
Combining equations (10) and (12) respectively in the domain of Laplace you get
𝑣𝑐,𝑑𝑟𝑒𝑓
= (𝑖𝑑,𝑟𝑒𝑓 − 𝑖𝑑)𝐹(𝑠) + 𝜔𝐿𝑖𝑞 + 𝑣𝑠,𝑑 + 𝐿𝑠𝑖𝑑 + 𝑅𝑖𝑑 − 𝜔𝐿𝑖𝑞 + 𝑣𝑐,𝑑
𝑣𝑐,𝑞𝑟𝑒𝑓
= (𝑖𝑞,𝑟𝑒𝑓 − 𝑖𝑞)𝐹(𝑠) + 𝜔𝐿𝑖𝑑 + 𝑣𝑠,𝑞 + 𝐿𝑠𝑖𝑞 + 𝑅𝑖𝑞 − 𝜔𝐿𝑖𝑑 + 𝑣𝑐,𝑞
(13)
From the above it is obtained that
𝐺(𝑠) =1
𝐿𝑠 + 𝑅 (14)
𝑖𝑑 = (𝑖𝑑,𝑟𝑒𝑓 − 𝑖𝑑)𝐹(𝑠)𝐺(𝑠)
𝑖𝑞 = (𝑖𝑞,𝑟𝑒𝑓 − 𝑖𝑞)𝐹(𝑠)𝐺(𝑠)
(15)
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Adaptive POD for Power System 15
Therefore, it is possible to conclude that the current controllers are completely independent
for each axis.
4.2 Modal Analysis
TABLE III shows a comparison of electromechanical modes between the system whit 25
percent of wind energy and the system with all generators as a synchronous machine. It is
important to clarify that the electromechanical modes are those that have a frequency
oscillation between 0.5 and 3 Hz. On the other hand, a damping percentage of less than 20%
is assumed for a mode to be considerable. TABLE III shows that the high penetration level
of wind energy presents a significant change in the inter-area electromechanical mode. This
mode presents positive real part which makes it unstable.
TABLE III. Electromechanical modes comparison
All generators synchronous machine
Mode Eigen Value Frequency (Hz) Damping 𝜻(%)
Area 1 -0.5395±6.7135 1.0685 8.01
Area 2 -0.6843±6.9959 1.1134 9.74
Inter-area -0.7476±3.5194 0.5601 20.78
Penetration level wind energy 25 percent
Mode Eigen Value Frequency (Hz) Damping 𝜻(%)
Area 1 -0.6318±6.7348 1.0719 9.34
Area 2 -1.2176±17.6447 2.8082 6.88
Inter-area 0.1023± 3.5527 0.5654 -2.88
4.3 POD Controller Design
The structure of the supplementary controller POD is shown in Figure 8, this structure has a
proportional gain, a washout filter and a phase lead-lags filter [26].
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Adaptive POD for Power System 16
Figure 8. Supplementary controller POD
Based on [26], the washout filter acts like a low-pass filter that prevents changes in the
frequency, speed or power flow that can affect the voltages. A washout time constant of 10
seconds is desirable for inter-area oscillations. For the case of the lead-lags filters, it is
recommendable to select their constants in function of a compensation angle. Where is the
exponent for the washout filter that takes a value of 1 if the input signal of the POD is the
rotor speed. On the other hand, n is the exponent for the lead-lags filter for a compensation
angle less than 45°. The parameters of the controller are α, the pole (𝑇2) and the zero (𝑇1),
were calculated from the following equations:
𝜙𝑚 =180° − Θ𝑑𝑒𝑝
𝑛 (16)
𝛼 =1 + sin 𝜙𝑚
1 − sin 𝜙𝑚 (17)
𝑇2 = √𝛼𝜔𝐶 (18)
𝑇1 =𝑇2
𝛼 (19)
Where ∅𝑚, 𝜃𝑑𝑒𝑝 and 𝜔𝑐 represents the required compensation phase, the output angle of the
inter-area mode and the mode frequency in rad / s, respectively. TABLE IV shows the
parameters of the supplementary controller POD for the inter-area mode with a 25 percent
penetration level of wind energy presented in TABLE III.
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TABLE IV. Parameters of classic supplementary controller POD
Parameter Value
m 1
n 2
𝚯𝒅𝒆𝒑 88.3509
𝝓𝒎 45.8246
𝜶 6.0724
𝑻𝟐 8.7545
𝑻𝟏 1.4417
Tw 10
K 200
4.4 MRAC Design
The error between the model reference and the plant can be observed in
𝑒(𝑡) = 𝐺𝑝(𝑡)𝑈𝑐(𝑡) − 𝐺𝑚(𝑡)𝑈(𝑡) (20)
Where, 𝐺𝑝, 𝐺𝑚 correspond to the function that describes the behavior plant and the behavior
model reference respectively. Besides, 𝑈𝑐(𝑡) = 𝜃(𝑡) ∗ 𝑈(𝑡) where 𝜃(𝑡) is the adaptive
parameter. The error can be rewritten in Laplace form as in (21).
𝑒(𝑠) = 𝐺𝑝(𝑠) ∗ 𝑈(𝑠) − 𝐺𝑚(𝑠) ∗ 𝑈(𝑠) ∗ 𝜃(𝑠) (21)
Based in the supplementary controller POD, Equation (22) shows the adaptive controller
𝜃(𝑠).
𝜃(𝑠) = (𝐾𝑠𝑇𝑤
1 + 𝑠𝑇𝑤 (𝛼
1 + 𝑠𝑇1
1 + 𝑠𝑇2)
2
) (22)
Where 𝐾 is the adaptive parameter and the data to change. Replacing (22) in (21).
𝑒(𝑠) = 𝐺𝑝(𝑠)𝑈(𝑠) (𝐾𝑠𝑇𝑤
1 + 𝑠𝑇𝑤 (
1 + 𝑠𝑇1
1 + 𝑠𝑇2)
2
) − 𝐺𝑚(𝑠)𝑈(𝑠) (23)
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Adaptive POD for Power System 18
The derivative of sensitivity is obtained by deriving the controller with respect to the
adaptive parameter.
𝛿𝑒(𝑠)
𝛿𝐾(𝑠)= (
𝑠𝑇𝑤
1 + 𝑠𝑇𝑤) (
1 + 𝑠𝑇1
1 + 𝑠𝑇2)
2
(24)
Next, the MIT rule is applied.
𝑠 ∗ 𝐾(𝑠) = −𝛾 ∗ 𝑒(𝑠) ∗ (𝑠𝑇𝑤
1 + 𝑠𝑇𝑤) (
1 + 𝑠𝑇1
1 + 𝑠𝑇2)
2
(25)
Equation (26) is obtained by clearing the adaptive gain from (25).
𝐾(𝑠) = −𝛾 ∗ 𝑒(𝑠) ∗ (𝑇𝑤
1 + 𝑠𝑇𝑤) (
1 + 𝑠𝑇1
1 + 𝑠𝑇2)
2
(26)
Figure 9 shows the model reference adaptive controller based in a POD regulator. Where 𝛾
is the adaptive gain.
Figure 9. MRAC with POD regulator.
On the other hand, the reference model is based on the second-order transfer function shows
in (27). How the damping of the most unstable mode is negative, it is necessary that the
reference model does not present a damping of more than 9% because it requires a greater
effort in the turbine and a greater inertial mass.
𝑦𝑚 =𝜔𝑛
2
𝑠2 + 2𝜁𝜔𝑛𝑠 + 𝜔𝑛2 (27)
The frequency of the unstable mode was 0.5654 Hz. whit this is possible calculated the peak
time as in (18)
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Adaptive POD for Power System 19
𝑡𝑝 =1
𝑓𝑜𝑠𝑐≈ 1.7686 [𝑠] (28)
The peak time and the damping selected were used for the calculation of oscillation frequency
and the establishment time of the reference model as it is presented in the following.
𝜔𝑛 =𝜋
𝑡𝑝 ∙ √1 − 𝜁2= 1.7846 [
𝑟𝑎𝑑
𝑠] (29)
𝑡𝑠 =4
𝜁 ∙ 𝜔𝑛= 24.9189 [𝑠] (30)
Replacing 𝜔𝑛 and 𝜁 in (27), the transfer function is shown in (31).
𝑦𝑚 =3.1811
𝑠2 + 0.3210𝑠 + 3.1811 (31)
Equation (32) shows the eigenvalues of reference model present a positive real part, which
implies that the damping is positive.
𝜆1,2 = −0.1605 ± 𝑖1.7763 (32)
Besides, is necessary add a gain of 𝑘𝑝 = 1 ∗ 10−3 so that the output resembles the
response of a synchronous machine.
𝑦𝑚 =3.1811 ∗ 10−3
𝑠2 + 0.3210𝑠 + 3.1811 (33)
Figure 10 shows the error stationary state of the reference model. It is important that the
establishment time is congruent on the value found previously.
Figure 10. ESS reference model
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Adaptive POD for Power System 20
Figure 11 shows the Fuzzy logic controller selected for this paper. The controller is
compound for two inputs that corresponding to error and derivate of error of the frequency
in the bus 9. The rule-based has 81 rules according to 9x9 Fuzzyfication inputs. For a better
solution, the tuning of Sugeno model is possible through the tool “Anfis” of matlab.
Figure 11. Fuzzy Logic Controller
Figure 12 and Figure 13 show the Fuzzyfication schemes for the inputs in the Fuzzy
system. Additionally, Figure 14 shows the surface that related the inputs with the output
based on the rule-based.
Figure 12. Error
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Adaptive POD for Power System 21
Figure 13. Delta-error
Figure 14. Surface of rules-based
5 Simulation Results
In order to compare the supplementary control of the MRAC based in a regulatory POD and
the classic POD, there are two cases to measure their performances.
5.1 Controller performance case 1
In the first case, the system is subjected to a three phases fault at a time of 0.5 s on the line
8-9A and the failure clearance 5 cycles after. Figure 15 and Figure 16 show the power flow
in the line 7-8A and 7-8B whit the MRAC and POD supplementary controller respectively.
It is possible to observe that a POD controller presents less error in the stationary state than
the MRAC controller. Nevertheless, the overshoot of the MRAC controller is lower. On the
other hand, the establishing time is approximately the same.
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Adaptive POD for Power System 22
Figure 15. Power Flow line fault inter-area (MRAC)
Figure 16. Power Flow fault line inter-area (POD)
Figure 17 shows the change of the adaptive parameter γ through time. The time interval is of
6 to 12 seconds because the initial values present great magnitude and the parameter after 4
seconds appears to be zero. It is possible to observe that the data tends to stabilize close to
20.
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Adaptive POD for Power System 23
Figure 17. Adaptive parameter case 1
5.2 Controller performance case 2
In the second case, the system is subjected to a three phases fault at a time of 0.5 s on the line
10-11A and the failure clearance 5 cycles after. It is important to clarify that this line connects
the slack node with the rest of the system. Figure 18 and Figure 19 shows the power flow in
the line 7-8A and 7-8B whit the MRAC and POD supplementary controller respectively.
Figure 18. Power Flow line fault line SLACK (MRAC)
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Adaptive POD for Power System 24
Figure 19. Power Flow line fault line SLACK (POD)
It is possible to observe that a POD controller is unable to maintain the stability of the
system for the proposed failure. On the other hand, the MRAC controller presents large
oscillations but maintains the stability of the system after the second 6. Figure 20 shows the
change of the adaptive parameter γ through time. Figure 20 shows that the value presents
more oscillations in comparison whit case 1, but tends to a set point.
Figure 20. Adaptive parameter case 2
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Adaptive POD for Power System 25
6 Conclusions
It is important to highlight that the Small-Signal Stability was the main analysis to
demonstrate that the high penetration of wind energy considerably affects the damping of the
most unstable mode by the low inertia of the turbine and the random behavior of the wind.
Besides, it was important to clarify the need to STATCOM as a reactive power compensation
for the complexity of coordinated the reactive output of all the turbines.
This paper had proposed a methodology for design and implementing a supplementary
controller on a STATCOM connected in the point of common coupling whit the Wind Farm
and the network. It was demonstrated that an adaptive supplementary control by a reference
model based on a POD regulator presents better results than a classic POD regulator based
on a SISO system. It is important to clarify that a comparison was made between the two
most commonly used inference mechanisms for Fuzzy controller to determine the
mechanism that best suits the needs of an adaptive control. Besides, the supplementary
control was probed in three-phase fault inter-area and three-phase fault line that connect the
Slack generator. This last fault is the most critical for the system according to the results.
Future extension to this work could be the implementation of a methodology for the tuning
the Sugeno model because the data training was selected for multiple tests made to the system
and taking the data of the best results. Which does not ensure a global optimum in the
adaptive parameter. Furthermore, the Wind Farm could replace the generator 5 in order to
evaluate the effect of an HVDC connection that presents a random behavior. On the other
hand, future work may also be to apply the methodology to an equivalent Colombian
transmission system. In order to verify the feasibility of each of the stages proposed by XM.
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