Received May 14, 2021, accepted May 23, 2021, date of publication June 4, 2021, date of current version June 14, 2021.
Digital Object Identifier 10.1109/ACCESS.2021.3086420
Review of Dynamic and Transient Modeling of
Power Electronic Converters for Converter
Dominated Power Systems
CHINMAY SHAH 1 , (Graduate Student Member, IEEE),
JESUS D. VASQUEZ-PLAZA 2 , (Graduate Student Member, IEEE),
DANIEL D. CAMPO-OSSA 2 , (Graduate Student Member, IEEE),
JUAN F. PATARROYO-MONTENEGRO 2 , (Member, IEEE),
NISCHAL GURUWACHARYA3 , (Student Member, IEEE),
NIRANJAN BHUJEL3 , (Student Member, IEEE), RODRIGO D. TREVIZAN 4 , (Member, IEEE),
FABIO ANDRADE RENGIFO 2 , (Member, IEEE), MARIKO SHIRAZI1 , (Member, IEEE),
REINALDO TONKOSKI 3 , (Senior Member, IEEE), RICHARD WIES 1 , (Senior Member, IEEE),
TIMOTHY M. HANSEN 3 , (Senior Member, IEEE), AND PHYLICIA CICILIO 1 , (Member, IEEE)
1 Department
of Electrical and Computer Engineering, University of Alaska Fairbanks, Fairbanks, AK 99775, USA
and Computer Engineering Department, University of Puerto Rico at Mayagüez, Mayagüez, PR 00682, USA
3 Department of Electrical Engineering and Computer Science, South Dakota State University, Brookings, SD 57007, USA
4 Energy Storage Technology & Systems, Sandia National Laboratories, Albuquerque, NM 87123, USA
2 Electrical
Corresponding author: Chinmay Shah (cshah@alaska.edu)
This work was supported in part by the U.S. Department of Energy Office of Science, Office of Basic Energy Sciences, Established
Program to Stimulate Competitive Research (EPSCoR) Program, in part by the Office of Electricity, Microgrid Research and Development
Program, and in part by the Office of Energy Efficiency and Renewable Energy, Solar Energy Technology Office through EPSCoR under
Grant DE-SC0020281, in part by the U.S. Department of Energy National Nuclear Security Administration under Contract
DE-NA-0003525, and in part by the U.S. Department of Energy or the United States Government under Grant SAND2021-6576 J.
ABSTRACT In response to national and international carbon reduction goals, renewable energy resources
like photovoltaics (PV) and wind, and energy storage technologies like fuel-cells are being extensively
integrated in electric grids. All these energy resources require power electronic converters (PECs) to
interconnect to the electric grid. These PECs have different response characteristics to dynamic stability
issues compared to conventional synchronous generators. As a result, the demand for validated models to
study and control these stability issues of PECs has increased drastically. This paper provides a review of the
existing PEC model types and their applicable uses. The paper provides a description of the suitable model
types based on the relevant dynamic stability issues. Challenges and benefits of using the appropriate PEC
model type for studying each type of stability issue are also presented.
INDEX TERMS Average models, data-driven models, dynamic phasor models, inverter-based resources,
large-signal models, positive-sequence models, power electronic converters, power system modeling, power
system simulation, power system stability, small-signal models, switching models.
NOMENCLATURE
ANN
CDPS
DER
DPM
EMT
Artificial neural network
Converter-dominated power system
Distributed energy resources
Dynamic phasor model
Electromagnetic transient
The associate editor coordinating the review of this manuscript and
approving it for publication was Zhilei Yao
82094
.
HVDC
IBR
LTI
LTP
NERC
PLL
PEC
PV
High-voltage direct current
Inverter-based resource
Linear time-invariant
Linear time-periodic
North American Electric Reliability Corporation
Phase-locked loop
Power electronic converter
Photovoltaic
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
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PSM
PWM
SSA
WECC
Positive-sequence model
Pulse-width modulation
State-space averaging
Western Electricity Coordinating Council
I. INTRODUCTION
The generation profile of power systems is undergoing a
fundamental shift towards converter-based resources and
reductions in large synchronous generation. Since it has
become clear that intermittent renewable energy sources
will be responsible for a large amount of power generation,
the existing methods for power systems reliability assessments will need to be modernized to account for the dynamics of wind, solar, storage, and other grid edge devices [1].
This shift introduces modeling challenges for traditional transient planning and operation and reliability practices [2], [3].
These modeling challenges include the need for electromagnetic transient (EMT) simulations and accurate power
electronic converter (PEC) models appropriate for the application of interest. Traditionally, positive-sequence simulators and phasor-based models of devices were adequate for
assessing transient stability issues due to the dominance of
synchronous generation. In systems with increasing and/or
dominating amounts of converter-based generation, numerous stability issues arise that can only be accurately captured with EMT models and simulation [4], [5]. Additionally,
numerous types of EMT models exist, each appropriate for
specific stability issues. Examples of specific stability issues
which become present in converter-dominated power systems
(CDPS), where converter-based generation exists at both distribution and transmission levels, includes but is not limited
to: a high rate of change of frequency due to low inertia
[2], [6], [7], limited fault current contribution impacting
protection coordination [2], [8]–[10], bi-directional power
flow impacting damping of inter-area modes and transient
stability margins [11]–[13], harmonic instability due to converter inner control loops [13], [14], interactions between
multiple grid-connected converters [13], [15], [16], and
lower frequency oscillations introduced by the phase-locked
loop (PLL), particularly in weak grids with short circuit ratios less than 2 [13], [17]. An excellent review of
these converter-based generation stability issues is provided
in [2], [13], [14], [18].
The recent rapid deployment and advances of converterbased generation have also lead to the development of new
models to represent these devices in various contexts. Model
development has been driven from two perspectives of opposite scales: the device-centric power electronic perspective
and the bulk power centric power systems perspective. From
a device-centric perspective, highly detailed EMT converter
models have been developed, called switching models, that
simulate the power electronic components down to the level
of detail of the pulse width modulation (PWM) signal in
the order of hundreds of kHz. Switching models accurately
represent the physical behavior of semiconductor switches
used to build PECs and are extensively used to analyze
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switching times, switching transients, switching losses,
switching faults, and instantaneous voltage and current
dynamics. These models can be further refined with manufacturer specifications to create ‘‘real-code’’ models [5]. Due
to the small (micro-second) time-step needed for simulations
with switching models, there is significant computational
burden to scale these models to larger systems. Switching and
real-code models are typically simulated in systems with only
one or a few converters due to this reason.
Levels of simplifications and linearizations are made to
create more computationally efficient models, such as average models. Both small- and large-signal average models
have been developed to assess small and large disturbances
respectively [19], [20]. Applicability of the small-signal models are limited to specific scenarios where large variations
or disturbances are not present. Further, positive-sequence
models (PSM) used in positive-sequence simulators make
additional simplifications based on the assumptions of a balanced system, and that the system operates around the fundamental frequency. PSMs have been the modeling approach
from the bulk power perspective, due to their application in
commonly used positive-sequence simulators for transient
analysis [21]–[24]. However, the increase of in front and
behind the meter converter-based generation in bulk power
systems has been shown to invalidate those assumptions [25].
Numerous studies have indicated the need to incorporate
EMT models of converters in large-scale power system
studies either through co-simulation with positive-sequence
simulators or directly in EMT simulations [14], [21], [22].
To bridge the gap between high-fidelity EMT converter
models and simplified phasor-based models, recent developments have been made with phasor models for converters, such as dynamic phasor models (DPM) and data-driven
models. Dynamic phasor models (DPMs) make it possible
to model both faster dynamics than the traditional phasor
models and to represent a sinusoidal signal of the voltage and
current at a fundamental frequency with harmonics, such as
phasors with different frequencies [26]. Data-driven models
replace the structure of previous models with mathematical
equations derived from input and output signals. For the use
cases reviewed in this paper, these data-driven models result
in expedited simulation times due to the reduced number of
components and equations to solve [27].
It is important to understand the limitations and applicability of the different types of PEC models. Other review papers
have covered topics on PEC controls and specific PEC model
types that relate to the limitation and applicability of PEC
models types [28]–[36]. The authors of [28]–[30] provided
a review on converter control methodologies to address the
challenge of stability issues in CDPS. Along with converter
control methodologies, the authors of [31] also provided
a brief review on energy management strategies in CDPS.
The review work in [32] investigated the control of PECs,
protection, and stability issues in CDPS. The authors of [33]
provided insight on the selection of existing DC-DC converter
configurations/topologies for different types of renewable
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C. Shah et al.: Review of Dynamic and Transient Modeling of PECs for Converter Dominated Power Systems
energy generation units. The advantages and disadvantages
of different PEC configurations and their applications for
wind energy systems were discussed in [34]. Along with PEC
configurations, the authors of [35] also provided a review
on PEC control and modulation strategies. A review on PEC
modeling was presented in [36], but the authors of this work
only discussed the small-signal average and state-space average PEC models. This paper does not focus on selecting PEC
configurations and PEC control methods for energy management and protection, but rather the PEC models necessary
to assess CDPS stability challenges. The contribution of this
paper is to provide a comprehensive overview of PEC model
types, including switched, small-signal average, large-signal
average, dynamic phasor, positive sequence, and data-driven,
and provide a comparison between those modeling strategies.
This work aims to highlight the relevant applications and
limitations of each of the model types. The applicability of
each model type for studying each of the dynamic power
system stability issues relevant to PECs are described in this
paper.
The paper is organized as follows. Section II outlines
the PEC relevant dynamic power system stability issues.
Section III discusses the types of converter models, their
fundamental building blocks, and examples of their application to simulate or capture dynamic power system stability
issues. Then, Section IV highlights and discusses trends in
PEC modeling. Section V finishes with concluding remarks.
FIGURE 2. Dynamic power system stability classification of
inverter-based resources organized by timescale.
and encompass the more specific stability issues listed in
Section I. The dynamic power system stability issues focused
on in this paper are also shown in Fig. 2 to highlight the
time frame in which each stability issue is present. The x-axis
in Fig. 2 is the typical simulation time-step that is needed to
simulate the identified dynamic power system stability issue.
The discussed dynamic power system stability issues are
outlined here:
•
II. DYNAMIC POWER SYSTEM STABILITY ISSUES
RELATING TO POWER ELECTRONIC CONVERTERS
Power system stability issues have previously been classified
by several bodies of research. Classifications most related
to this review include power systems with high penetration
of power electronic interfaced technologies [13], and for
microgrids [14]. For this paper, ‘‘dynamic’’ power system
stability is classified as that whose dynamic response occurs
in no more than several seconds. These dynamic stability
issues that relate to inverter based resources (IBRs) are identified here to provide background for the following review
of PEC model types. Building from the work in [13], [14],
the dynamic power system stability issues that are relevant to
PECs are shown in Fig. 1.
FIGURE 1. Classification of dynamic power system stability issues that
are relevant to power electronic converters.
The stability classifications defined in this review focus
on those commonly found in the literature in CDPS,
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•
Converter-driven Stability: PECs rely on control loops
and algorithms with wide ranging response times,
such as from the PLL and inner control loops. This
wide timescale can impact both electromechanical and
EMTs [13]. Slow-interaction converter-driven stability
typically occurs at less than 10 Hz and encompasses
controls for power sharing dynamics. Fast-interaction
converter-driven stability occurs roughly between tens to
thousands of Hz and encompasses switching modulation
control and voltage and current waveform dynamics.
Voltage and Frequency Stability: Frequency and voltage
stability can be impacted by large and small disturbances. Small disturbances, such as small or gradual
changes in load or generation, allow the system to
be modeled by linearizing around the operating point
of the system. Large disturbances, such as faults and
large step changes in load or generation, can result in
non-linear responses that cause linearized representations of the system to be inaccurate. Secondary voltage
and frequency controllers are used to help stabilize the
system in response to such large disturbances. Further
definitions on voltage and frequency stability are found
in [13].
A critical factor of IBR dynamics is whether the inverter
is operating in grid-forming or grid-following mode. Converter models are highly dependent on the mode of operation. In grid-following models, most approaches do not
consider variations in the main grid, which is considered
a stiff source without variations in voltage or frequency.
Additionally, in grid-following mode, the voltage waveforms
are generated by the grid. Therefore, in grid-following mode
if there are distortions in these waveforms the grid is the
source of that disturbance and that can influence the inverter’s
current dynamics. In the case of grid-forming mode, there are
dedicated controls to generate and maintain the voltage and
current waveforms, and therefore the source of voltage and
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waveform stability issues in this mode will be the inverter.
Therefore, in grid-forming mode the inverter has significant
influence on grid dynamics. These modes of operation need
to be accounted for when developing or choosing appropriate
models as they have stability implications.
III. MODEL TYPES FOR POWER
ELECTRONIC CONVERTERS
When modeling IBRs, it is essential to choose the right PEC
model type, select component values, and to estimate the
converter’s response under different scenarios. This subsection provides a review of PEC model types used to analyze
CDPS dynamic stability issues. Table 1 summarizes definitions of each of the reviewed model types, relevant simulation
time-steps, and advantages and disadvantages of each. Additionally, the ability of each model type to simulate and capture
the various dynamic stability issues is outlined in Table 2.
These tables summarize key takeaways from the following
discussions on model types and can be used as a reference.
A. SWITCHING MODELS
The switching model of a PEC includes power electronic
switching devices, such as insulated-gate bipolar transistors.
PWM techniques are used to control the gates of these
switching devices to create sinusoidal current and voltage
waveforms. The control structures to create this modulation
and generate these waveforms are also included in switching
models. The higher levels of control structures that maintain
TABLE 1. Summary of power electronic converter model type definitions, advantages, and disadvantages.
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C. Shah et al.: Review of Dynamic and Transient Modeling of PECs for Converter Dominated Power Systems
TABLE 2. Summary of power electronic converter model types and their applicability to study dynamic power system stability.
power sharing balance and voltage and frequency restoration
can also be included in these models, as illustrated in the
generic diagram of a switching model in Fig. 3. However,
due to the computational demand associated with this highly
detailed model, such higher level controls are rarely simulated
with these types of models.
average models adequately simulate and capture common
dynamic stability issues, including fault response, except in
the case of prediction of harmonic distortion and electromagnetic interference. Contrarily, the research presented in [55]
indicated it was necessary that high frequency switching
and DC link dynamics are included in inverter models to
accurately capture grid-forming and supporting modes [55],
fault conditions [56], significantly unbalanced load conditions [55], and high frequency switching ripples [53], [57].
As seen by the recent discussion, there is not complete
consensus of when a switching level model is necessary,
as often average models are adequately comparable and much
more computationally efficient. Regardless of the adequacy
of switching models in comparison to average models for
capturing specific dynamics, the high computational burden
involved in the simulation of switching models makes them
impractical for use in system-level studies or even studies
with more than one or two IBRs. However, the higher level
of detail and accuracy are critical for examining modulation
strategies, switching losses, and potentially other fast time
scale dynamics.
B. AVERAGE MODELS
FIGURE 3. Generic switching power electronic converter model
representing major control groups [177]–[179].
Examples and Applications: Switching models are commonly employed to estimate switching losses [37], determine
methods to reduce switching losses [38]–[40], or develop
modulation strategies that reduce output harmonics and/or
switching losses [41]–[45]. Similarly, in [46], a switching
model of a wind turbine was used for testing modulation control strategies for fast tracking of large changes in wind speed.
There is an immense computational burden associated with
the simulation of these detailed models, which is a reason for
the limited research that focuses solely on switching models. Real-time digital simulators using field-programmable
gate arrays have been used to increase the computational
efficiency of switching models [47].
Switching models are also commonly presented in the
literature as a benchmark to compare the accuracy of average
models [19], [23], [24], [48]–[53], many of which are discussed in greater detail in the next section. In [54], a comparison of average models and switching models determined that
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Average PEC models focus on capturing the low-frequency
behavior of the PEC without accounting for high-frequency
variations due to circuit switching. This modeling method
transforms the original discontinuous model into a continuous model that provides the best representation of
the system’s macroscopic behavior by averaging the converter state variables (state-space averaging) or averaging
the switch network terminal waveforms (average switch
modeling). A generic PEC averaged switch model is shown
in Fig. 4, which captures the general components that are
included in average models. Average models are also divided
into small-signal and large-signal models used to study
small-signal and large-signal dynamics respectively. The
following subsections discuss small-signal and large-signal
average models.
1) SMALL-SIGNAL MODELS
Small-signal analysis is the study of deviations from the
operating point for a system subject to small disturbances.
Formally the small signal response of a multi-variable system
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models is simpler than Linear Time-Variant models. When
the parameters of the system vary periodically with time,
the system is said to be Linear Time-Periodic (LTP). The
LTI and the LTP models are two of the most cited modeling
methods for predicting small-signal CDPS behavior, and they
are reviewed in this section.
a: LINEAR TIME-INVARIANT MODELS
FIGURE 4. Generalized power electronic model with average equivalent
circuit of inverter [177]–[180].
yn = fn (x1 , x2 , · · · , xm ) is expressed as in (1)


∂f1
∂f1
∂f1
···

  ∂x1


∂x2
∂xm 

 1x1
1y1
 ∂f2

∂f
∂f
2
2
  1x2 
 1y2  
···

  ∂x


∂x2
∂xm 
e 1
  .. 
 ..  =


.
.
.
 . 
..
..
..   . 
 ..
.

 1xm
1ym
 ∂fn
∂fn
∂fn 
···
∂x1
∂x2
∂xm
(1)
One of the main advantages of small-signal models is that
they can be used to perform classical stability and performance analysis methods such as Bode/Singular-value plots,
Nyquist plots, eigenvalue analysis, superposition-based analysis, and transient response analysis. However, these methods
lose accuracy when the system is highly non-linear or in
the presence of large disturbances [59]. The modeling of
CDPS is comprised of multiple dynamics and interactions
that may or may not be linear. For example, voltage and current dynamics inside the converter are typically considered
to be linear under nominal reference values. However, some
variables such as active/reactive power, frequency, or DC-bus
voltage can generate products between two or more state
variables, discontinuities, or exponential/trigonometric calculations that can affect the accuracy of the small-signal
model. When the state matrix of (1) is constant, it means that
the system’s parameters are constant and the system is said to
be Linear Time-Invariant (LTI). LTI modeling is often used
due to its ease of implementation and the low computational
cost that it involves. Also, the design of controllers for LTI
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LTI modeling is the most common modeling method for
small-signal analysis. This modeling method uses Taylor’s
theorem and assumes that the parameters are constant and
the dynamics can be approximated by its first derivative
evaluated at a certain operating point. LTI models are useful
for converter modeling if parameters such as component temperature and saturation values are considered to be constant.
Examples and Applications: There are different
approaches for obtaining small-signal models for the different
dynamics of CDPS. Each CDPS model is unique depending
on the number of nodes, the operating mode, the components, and the types of controllers used [66], [72]. These
factors could generate interactions that may or may not be
linear. Thus, most small-signal models are specific to each
CDPS configuration. Some examples with implementations
of small-signal models for CDPS are presented below.
One of the first small-signal average models of a
single-phase PEC was developed in [58] to analyze the eigenvalues of a converter in grid-following mode using droop
control as shown in Fig. 5. This work focused on obtaining
the characteristic equation that defines the eigenvalues and
the transient response of the active power, reactive power,
frequency, and voltage amplitude. The characteristic equation
was only accurate to describe the location of the eigenvalues
around a specific operating point. Also, the characteristic
equation by itself was unable to show accurate transient
FIGURE 5. Single-phase inverter in grid-following mode. Adapted
from [58].
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C. Shah et al.: Review of Dynamic and Transient Modeling of PECs for Converter Dominated Power Systems
response estimations because the system zeros were not
obtained.
Further work has been developed in the literature to
account for additional dynamics found in grid-following
PECs using small-signal models. In [64], small-signal
state-space models were developed to describe the PLL
dynamics with accurate results under large disturbances.
Also, using this model, the performance of the CDPS with
the different controllers, filters, and power distribution coefficients were examined under different disturbances with
eigenvalue analysis. In [65], a phase shift control action
was added to a grid-following PEC to improve the system
dynamic response and to maintain suitable damping. This
work developed a complete state-space model that showed
accurate results estimating the transient response of the
shared power. Furthermore, in this work a new model of an
equation of state was proposed which presented a suitable
state vector to describe the behavior of the system from a
given operating point to the equilibrium point. It allowed the
initial conditions to be more practically and easily defined.
In [78], the grid voltage was modeled in the dq0 frame
to assume it was constant for small-signal analysis. The
state-space model of the PEC was used to perform open-loop
stability margin analysis because the converter model was
separated from the state-feedback controller.
One of the first small-signal PEC models in grid-forming
mode was developed in [61], and it included a power network
model and a load model. The complete CDPS state-space
model was developed in the dq0 frame assuming proportional
integral and droop controllers to describe voltage-current and
power sharing dynamics of the PEC. The authors developed
a linearized state-space model for each PEC and then merged
them using a state-space model for the network and loads.
The modeling approaches of converters in grid-forming mode
in [62]–[64], [67], [69], [70], [73], [74] used the same
modeling methodology proposed in [61]. In [63], a model
considering additional loops for the droop control was developed. The entire model developed in [64], [67] integrated
the PLL dynamics to obtain a more accurate representation
of the CDPS frequency stability. The small-signal model
developed in [62] considered a modified network model with
loads distributed across the CDPS. In [70], one of the PECs
was assumed to be working in grid-forming mode and the
rest were in grid-following mode to represent grid-following
dynamics. In [69], a small-signal model capable of switching
between CDPS modes of operation was presented. In [73],
a CDPS model considered controllers that used the internal
model principle was developed. Each of the aforementioned
approaches resulted in a single state-space model capable of
describing the interaction of all PECs and also was able to
describe the frequency, voltage stability, and other phenomena in the CDPS. However, as the number of PECs, points of
interconnection, and loads increase the CDPS becomes more
susceptible to deviations from the operating point, non-linear
dynamics, and large disturbances. Also, transient response
analysis of these models requires the initial state of all the
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variables of the model to be known, which is not always
available.
A methodology to develop a small-signal model of a CDPS
in grid-forming mode for any number of PECs in the dq0
frame was developed in [80]. The resulting model was able to
describe the interaction between all the PECs with regards to
the voltage, current, and shared power. Although the proposed
methodology was accurate for describing CDPS dynamics,
it did not consider frequency dynamics because all PECs
were assumed to be synchronized with the point of common
coupling at the nominal frequency.
Frequency and harmonic stability analysis has been performed in the literature with small-signal models. The review
performed by [77] investigated the concept and history of harmonic stability analysis and the use of linearized models identified in [25], [81]–[84] to capture harmonics. A harmonic
state-space small-signal average model was used in [71]
to examine the harmonic interaction between inverters and
the grid voltage, and it was validated for the time and frequency domain. Examination of harmonic instability due to
PLL interaction was studied in [75]. This approach used a
small-signal model of an inverter to determine a balance
between the PLL’s stability margin and the bandwidth to
prevent harmonic instability. Further work on small-signal
modeling for examining PLL induced harmonic instability
was performed in [79] and addressed the differences in these
modeling methods and stability analysis for grid-forming
and grid-following inverters. The authors of [76] stated
that the small-signal average model failed to explain the
sub-harmonic oscillations. In [68], the impact of the CDPS
frequency dynamics and the dynamics of static and dynamic
loads (induction motors) were considered. To mitigate the
oscillations caused by induction motors this approach integrated a modified droop control to the model to improve
power sharing dynamics.
Voltage and frequency stability issues have also been modeled with small-signal models. A small-signal model was
used to evaluate the impact of stability issues such as time
delay τd on the communication data link used for frequency
restoration through frequency stability analysis [85], [87],
[88], [95]. In [85], [94], a CDPS small-signal model was
analyzed to find the communication delay margins where the
CDPS maintained stability. It was found that the effect of
communication delays on the microgrid with synchronous
machines might not be significant because the reaction speed
of synchronous machines were not as fast as the IBRs.
In [86], a consensus-based method was included in the
modeling that led to the restoration of frequency to its reference value. In the same manner, [87] proposed a small-signal
model for the entire CDPS. The model used frequency
restoration at the secondary control level and executed a
consensus algorithm that worked under the assumption of
a constant time delay. In contrast, in [94] a small-signal
dynamic model was developed that considered time delays
in CDPS using the differential delay equation. Additionally,
this work presented tuning consensus algorithm parameters
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C. Shah et al.: Review of Dynamic and Transient Modeling of PECs for Converter Dominated Power Systems
for the analysis of communication delay margins. The authors
of [90] developed a distributed secondary control method
using a discrete-time model for frequency regulation in an
islanded CDPS, where the secondary control scheme was
implemented locally using a discrete-time model. This was
formulated to facilitate the iterative characteristics of the
model predictive control-based algorithm. Another way to
model and analyze the frequency dynamics of CDPS was
presented in [91], where a complete microgrid was modeled
using a small-signal state-space model. The authors analyzed
the effect of changing the nominal frequency in a voltage
source inverter, and active and reactive power was used to
investigate the stability of a CDPS. Also, a similar approach
was developed in [92], where a small-signal state-space
model was used to assess transient and steady-state frequency
stability when operating in grid-following mode.
Small-signal models are used in CDPS for analyzing voltage stability usually through the voltage margins associated with reactive power sharing [86], [89], [95]. In [88],
a small-signal model for voltage amplitude stability analysis and droop-based secondary control was developed. This
model assessed the stability of the system through eigenvalue
analysis assuming that the active power was constant and
the reactive power was variable for use in voltage amplitude
restoration. Another approach was developed in [93], where a
small-signal model for the design of a generalized droop control was presented. The model developed allowed comparison
between traditional droop control and virtual synchronous
generator control. The generalized droop control improved
transient power performance and therefore improved the
voltage dynamics.
Small-signal modeling methods applied to CDPS are useful for determining stability margins, frequency response,
transient response, and other important characteristics of
PECs and CDPS. Also, small-signal models for PECs are useful to design controllers for regulating voltage-current, power
sharing, and other phenomena. However, CDPS under large
disturbances or CDPS that have highly non-linear terms for
power sharing, frequency stability, and harmonic distortion
are not appropriate to model with small-signal models. Thus,
it is important to determine the size of the disturbances and
the linearity of the phenomena expected for the CDPS under
study to determine if large-signal models or small-signal
models are recommended for CDPS analysis.
of power system voltage and frequency stability. Thus, models developed in the LTP framework are suitable to study
phenomena such as harmonic stability, grid-synchronization
dynamics, frequency stability, voltage stability, and others.
LTP systems can be described using state-space notation:
1ẋ(t) = A(t)1x(t) + B(t)1u(t)
1y(t) = C(t)1x(t) + D(t)1u(t)
where A(t), B(t), C(t), and D(t) are time-periodic matrices
over a time period TA , and 1x(t), 1u(t), and 1y(t) are the
state, input, and output vectors, respectively. As the LTP state
equation changes over time, the conventional eigenvalue analysis cannot be performed. Thus, the State Transition Matrix
(STM) 8(t, 0) is used to assess stability for LTP systems.
The STM can be calculated by numerically solving 1ẋ(t) =
A(t)1x(t) over a time period TA with n independent initial
condition vectors x0 , where n is the size of A(t) [182]. The
union of the n solution represents the Flocker STM (FSTM)
denoted by 8(TA , 0), which is commonly used to assess
stability by analyzing its eigenvalues [183].
The LTP framework is also used to assess harmonic stability, which could be affected by the presence of non-linear
loads, grid-synchronization dynamics, and other non-linear
phenomena. Since these phenomena are mostly non-linear, it
is common to perform harmonic linearization. This method is
similar to the Taylor’s approximation, but instead of linearizing around DC values, the non-linear equations are approximated to their steady state value around a certain period TA
[184]. Then, matrix Fourier series expansions (3) are applied
to the state and input matrices, and to the state, input, and
output vectors, e.g. [77]:
A(t) =
x(t) =
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∞
X
k=−∞
∞
X
Ak ejkωA t
(3)
Xk (s)e(s+jkωA )t
(4)
k=−∞
where ωA = 2π/TA . Thus, replacing (3) and (4) in (2),
the Harmonic State-Space (HSS) model is proposed based
on the product-convolution properties of the Fourier series,
which must hold for all k:
(s + jnωA )Xk (s) =
b: LINEAR TIME-PERIODIC MODELS
Small-signal models are typically developed for LTI models. In this case, the system’s parameters are constant over
time. This allows one to perform well-known stability and
performance assessment methods such as root locus, Bode
plot, Nyquist plot, gain-phase margin estimations, etc. When
the system is linear, and its parameters change periodically
over time, it is considered LTP [124], [181]. As it is wellknown, periodicity is an important characteristic of CDPS
since it is present in the computation of instantaneous voltage,
current, and power. Also, periodicity is present in the analysis
(2)
Yk (s) =
∞
X
n=−∞
∞
X
n=−∞
Ak−n Xn +
Ck−n Xn +
∞
X
n=−∞
∞
X
Bk−n Un
Dk−n Un
(5)
n=−∞
Finally, the Harmonic Transfer Function (HTF) is obtained
using the Toeplitz transformation T [•] [77], [184], [185]:
T [Y (s)] = H(s)T [U (s)]
(6)
H = T [C][sI − (T [A] − N )]−1 T [B] + T [D]
(7)
where
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Stability analysis can be performed by computing the
eigenvalues of H(s). As the Toeplitz transformation is performed over the whole spectrum, it’s values must be truncated
to perform numerical computations on the respective vectors
and matrices.
Examples and Applications: LTP models can be more
accurate than LTI models to describe CDPS dynamics.
In [186], the accuracy of the LTI and the LTP frameworks
were compared using a grid-following converter considering different grid-synchronization methods. It was demonstrated that the LTP framework is not only able to more
accurately describe the instantaneous dynamics, but also is
capable of describing the double-frequency oscillations in
the transient response of the converter. In addition, stability
assessment using Nyquist plots demonstrated that the LTP
model provides more accurate results compared to the LTI
model regarding stability margins. In [184] harmonic stability
of a grid-following converter was modeled using the LTP
framework and the harmonic linearization method. By using
Fourier transformations with an input with harmonic components, the authors found the stability boundaries considering
the grid-synchronization dynamics using the HTF with a
truncation of 2.
The LTP framework is also useful to describe the dynamics
of CDPS under non-nominal conditions or in the presence
of disturbances. In [187], [188], the non-linear LTP model
was developed using complex mathematics and linearized
using Wirtinger calculus [189] to consider unbalanced conditions in CDPS. Unbalanced conditions included positive
and negative-sequence components with oscillatory behavior.
The model, analyzed in the LTP framework, showed that the
grid-synchronization module induced harmonic resonances
for positive and negative sequences, which was not described
using conventional LTI models. In [190], the dynamics of
modular multilevel converters under unbalanced conditions
were analyzed using the LTP framework. For this purpose
the FSTM was used with the Lyapunov theory. This obtained
more accurate predictions of an unbalanced multilevel
converter compared to conventional LTI models.
The influence of the DC component on the grid voltage was
another variable that was difficult to model and analyze with
LTI modeling. In [191], this component was analyzed using
the LTP framework for three-phase and single-phase converters using the Nyquist stability criterion on the open-loop HTF.
Furthermore, the authors compared four different types of
grid-synchronization methods with DC rejection capabilities.
In [183], the stability of vector-controlled modular multilevel converters was assessed. This was done using the LTP
framework since it considered the circulating current control
loop which was more difficult to describe in the LTI framework. To assess stability, the eigenvalues of the FSTM were
used over a nominal period, TA , equal to the grid frequency.
Results of this work helped to better understand the effects of
converter parameters in the presence of circulating currents.
Also, as the LTP framework was used, unbalanced conditions
could be analyzed.
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Although LTP modeling is a well-known method, its application in CDPS is still in development due to its mathematical
complexity. Further research on this method will improve
the accuracy of small-signal models and stability analysis
methods. However, for a more broad representation of the
CDPS dynamics, large-signal modeling and stability analysis
methods are recommended.
2) LARGE-SIGNAL MODELS
Large-signal models employ non-linear mathematical
functions to describe non-linear components without
linearization [116]. The large-signal model is important for
analysis when PEC non-linearity is significant and when the
response to large perturbations causes deviations that are
substantially different from the response predicted by the
small-signal model. Fig. 6 shows two versions of large-signal
models: 1) discrete and 2) continuous-time. The discrete-time
large-signal model is befitting for simulations, whereas the
continuous-time large-signal model is suitable for analytical
calculations.
FIGURE 6. PEC average model bifurcation. Adapted from [77] and [112].
Average models typically use two averaging techniques,
either state-space averaging (SSA) or circuit averaging. The
generalized state-space representation of a continuous-time
non-linear PEC system is given by [99], [112],
d
x = f (x(t), u(t))
dt
y = h(x(t), u(t)),
(8)
where x, u, and y are the state, input, and output vectors
respectively. An example circuit averaged large-signal model
of PECs is shown in Fig. 7. The mathematical expression for
the circuit averaged large-signal model is given by,
d 0 (t)
hv2 (t)iTs
d(t)
d 0 (t)
=
hi1 (t)iTs ,
d(t)
hv1 (t)iTs =
hi2 (t)iTs
(9)
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FIGURE 7. PEC large-signal circuit average model. Adapted from [192].
where hv1 (t)iTs and hi1 (t)iTs are average input voltage and
current, hv2 (t)iTs and hi2 (t)iTs are average output voltage and
current, and Ts is the switching period.
Examples and Applications: First, large-signal models
developed with SSA are discussed with their relevance for
capturing stability issues, and then a review of large-signal
models developed using a circuit averaging technique is
provided.
a: STATE-SPACE AVERAGING
The early research on discrete and continuous-time
large-signal models for PECs using the SSA technique was
presented in [48], [97], [98], [100], which used the SSA
technique to develop DC-DC converter large-signal models.
These methods provided a basis for much of the large-signal
development using the SSA technique for inverters and
rectifiers.
The piecewise affine large-signal average model of the
power factor correction (PFC) rectifier was developed in [51]
using the SSA technique. The developed model was useful
for large-signal analysis and the design of PFC rectifier
controllers. The input current and output voltage stability analysis of the developed model was compared with a
non-linear analog model and showed good agreement. The
model was tested with a fixed AC source and resistive load
but did not consider distributed energy resources (DERs)
or grid parameters for analysis. Large-signal models have
been developed for grid-connected PV system inverters [104],
quasi-Z-source inverters of battery energy storage systems
[105], [107], semi-quasi-Z-source inverters [108], PLLsynchronized grid-connected inverters [117], three-phase
current source inverters based on Ćuk converters [111],
and single-phase PWM inverters [118]. The ability of these
large-signal models to capture the dynamic stability issues
varied.
In [104], the proposed model was used to analyze the
dynamic response of the output load voltage under three
different scenarios when the PV system was islanded from the
grid. The simulation and experimental results were comparable. However, minor differences were observed in the simulation and experimental results due to the proposed model’s
parasitic element effects. The work was only tested for three
different resistive loads and did not analyze the impact of
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line and grid parameters on the converter’s performance. The
authors of [105] and [107] designed a sliding mode controller
for a large-signal average model of a quasi-Z-source inverter,
which was stable and robust to large parameter, line, and load
variations. The authors also analyzed the proposed model’s
output voltage dynamics with input voltage and load perturbations and compared it with experimental results. The simulation and experimental results were in close agreement, and
thus the developed model was accurate enough for large perturbations. In [108] the large-signal model of a semi-quasiZ-source inverter was used to perform large-signal stability
analysis in continuous conduction mode and the results were
compared with large-signal stability analysis of the switched
model. By comparing the stability results, the authors concluded that the stability results of the proposed model held for
every possible value of the circuit inductors, capacitors, and
linear resistive load. This showed that all trajectories corresponded to the same duty cycle evolution, but different initial
conditions converged to the same steady-state trajectory. This
conclusion gave theoretical justification to the PV inverter
operation strategy proposed in [106]. Large-signal stability
analysis was also performed in [118], but for a large-signal
average model of a single-phase PWM inverter. The authors
adopted various non-linear stability analysis methods to analyze the proposed model’s fast-scale and slow-scale stability
under variations of the control parameters. The theoretical
results of the stability analysis were verified by the experimental results under resistive, inductive-resistive, and diode
rectifier load conditions.
In [111], a model of a three-phase current source inverter
was proposed. The buck-boost inherent characteristic of the
Ćuk converter, depending on the time-varying duty ratio,
provided flexibility for standalone and grid-connected applications when the required output AC voltage was lower or
greater than the DC side voltage. This property was not
found in the conventional current source inverter. The performance of the proposed model was validated using experimental results from an inverter system. Detailed control
analysis, output voltage stability during grid side imbalance,
and low-order harmonic analysis were not presented in this
paper. In [117], the large-signal model was developed for a
three-phase grid-connected voltage source inverter where the
grid voltage and angle were transformed into dq0 components
using the direct-quadrature-zero transformation technique.
The authors investigated and validated the proposed model
in the time-domain through simulation and experimental verification and compared it with the small-signal average model
results. They presented results on active and reactive power
dynamics under two scenarios. In the first scenario, less
power was produced as compared to the demand, and indirect
reserve was utilized. In the second scenario, the reactive
power compensation was used by the grid inverter as an ancillary service. Under both scenarios, the proposed large-signal
model showed higher accuracy than the small-signal models
and it was concluded that the large signal model could better
represent the system transient behavior.
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b: CIRCUIT AVERAGING
The circuit averaging technique was first developed in [49],
[50], [99] to create large-signal models of DC-DC converters. The authors of these works developed the large-signal
average model by 1) substituting the active switch with a
controlled current source iQ = αiQp , where Q was the active
switch, iQp was the current flowing through the switch when
it was ON, and α was the duty ratio, and 2) substituting
the diode with a controlled voltage source vd = αvdp ,
where vdp denoted the voltage that appeared across the diode
when it was OFF. The work of [49], [99] determined that
the experimental measurements and the theoretical predictions were very close. Additionally, [49] concluded that the
simulation time for transient analysis using the large-signal
average model was greatly reduced. The circuit averaging
technique developed by these works have been used to create
large-signal models of rectifiers and inverters for IBRs.
The literature on large-signal models for rectifiers and
inverters, developed using the circuit averaging technique,
includes three-phase buck rectifiers [101], three-phase
three-level Vienna rectifiers [103], five-level unidirectional
T-rectifiers [113], rectifiers for high voltage direct current (HVDC) systems [109], inverters in an islanded microgrid [120], zero current switching current source inverters
[102], three-phase voltage source inverters [119], [121], and
grid-forming inverters [122]. The large-signal models developed in these papers analyzed different PEC dynamic stability
issues. In [101], a large-signal average model of a three-phase
buck rectifier presented in Fig. 8 was developed. The controlled sources d̂i ILf /N , dˆij Vij /N , VCx , and h.ILf represent
input phase currents, contributions of line voltages and capacitor voltage to rectified secondary voltage, and diode current
respectively. Subscripts i and j denote phases a, b, and c. The
primary and secondary effective duty-cycles are represented
by d̂i and dˆij respectively. This large-signal average model
of the three-phase rectifier provided an accurate prediction of
input current distortion, minimized the distortion in input currents and output voltage waveforms, and was experimentally
verified using a 6 kW prototype.
FIGURE 8. Large-signal average model of three-phase rectifier. Adapted
from [101].
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An analogous modeling technique was used in [103]
and [113], except the authors modified the three-phase
non-linear equations into the dq0 frame using the directquadrature-zero transformation. Modifying the model into
the dq0 frame reduced the simulation time. Both models were
experimentally verified using the output voltage responses
under different input perturbations. The dq0 transformation
based large-signal average model of the rectifier was also
developed in [109] for HVDC systems to analyze the system response to an abrupt change in the direct current and
the DC capacitor voltage. In [102], the large-signal model
was experimentally verified for the switching conditions and
output voltage and current dynamics of the current source
inverter. The output voltage and current of the current source
inverter had high harmonic content compared to the voltage
source inverter, but the authors of this work did not present
harmonic analysis results. The authors of [122] developed
the large-signal model using the Clarke transformation to
assess the transient stability of a dispatchable virtual oscillator controller of a grid-forming voltage source inverter.
The authors also theoretically analyzed the impact of the
dispatchable virtual oscillator controllers’ voltage amplitude
dynamics on the system, and the results were validated with a
controller hardware-in-the-loop test-bed using industry-grade
hardware. Based on the analysis, the article concluded that
the dispatchable virtual oscillator controller was superior to
droop control in terms of transient stability when subjected to
large grid disturbances. In [119], [121], the large-signal models of voltage source inverters were used to predict the resonance induced distortions in a power system. This work was
important because it identified that rooftop PV, utility-scale
PV, wind farms, and HVDC links can generate resonance
problems in the grid. Also, the large-signal model does not
linearize the hard non-linearities such as PWM saturation
that can dominate the resonance based distortions, hence the
response can be utilized to model the converter control system
to limit those distortions.
The circuit averaging technique was also used to develop
a large-signal model of a complete CDPS. In [110],
a large-signal reduced model of a CDPS in grid-forming
mode was developed. The modeling methodology was similar
to [61], except this model was not linearized. The major
disadvantage of this work was the assumption that the power
grid operated in a quasi-stationary mode. As a result, this
model cannot be used to study PEC’s performance during
grid transients. A full-order large-signal dynamic model of an
inverter-based microgrid was developed in [114] and [115],
where all the variables were transformed into the dq0 frame.
The simulation results verified the proposed model’s performance in terms of reliability, flexibility, and efficiency. It also
indicated that the proposed model could accurately reflect
the system’s dynamic characteristics under large disturbances
such as startup and sudden load changes. The major advantage of the large-signal model for CDPS is that it significantly reduces the simulation time as compared to switched
models. The circuit averaging based large-signal model of
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an inverter-dominated islanded microgrid was developed
in [120] to study reactive power sharing dynamics and voltage
stability. Additionally, the impact of load perturbations, filter
parameters, line impedance, and droop variations were also
analyzed in this work. The analysis concluded that the higher
droops improved the reactive power sharing capability, but
the steady-state operating point deviated from the nominal
set-point of the system and this was a major drawback.
As seen by the literature, large-signal models have been
used to perform numerous types of stability studies, including capturing shorter timescale output voltage dynamics and
active power sharing dynamics. These models are beneficial
for these short time scale stability analyses in comparison
to switching models due to their increased computational
efficiency gained through intelligent and validated averaging
methodologies.
C. POSITIVE-SEQUENCE MODELS
Efforts have been made to obtain models for IBRs so
that power system simulations for transmission planning
and operation could be implemented. Bulk power system dynamic analysis has historically focused on electromechanical dynamics with dynamic stability issues that
typically range on the order of milliseconds to seconds.
Examples of such dynamic stability issues include inter-area
oscillations, transient voltage, frequency stability, and protection relay settings. PSMs are representative of the dynamics
of bulk power system devices in the range of 0.1 to 3 Hz,
and up to 15 Hz for control systems [131]. These models
assume the bulk power grid is operated under three-phase balanced conditions and that system frequency deviations from
nominal are very small. On this time scale, PSMs are widely
utilized in time-domain simulations applied for assessment of
many power systems stability problems, including transient
and small-signal stability [123] due to their accuracy at those
time steps and computation efficiency for large transmission
systems.
Sequence component analysis of power systems allows
the representation of one three-phase unbalanced power
system as three balanced systems. Under balanced conditions, the negative and zero-sequence phasors are negligible.
Therefore, those two components are usually not of interest in
transmission stability studies, which can often be represented
by their positive-sequence network alone [124]. As a result,
a simpler single-phase positive-sequence network can be used
to represent the three-phase circuit, which is very useful for
the simulation of large-scale three-phase systems.
In time-domain simulations for transmission stability
assessment, it is considered that transients within the transmission network decay very fast and their dynamics can
be ignored. Therefore, the transmission network and static
loads can be represented by a positive-sequence network.
In the same context, it is considered that the dynamics of
devices such as generators and their governor and excitation
systems dominate transient (rotor angle) and small-signal
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stability problems. Therefore, those devices are represented
by differential equations [124].
Modern positive-sequence analysis tools include generic
open-source models of IBRs, such as those developed by
the Western Electricity Coordinating Council (WECC) [125],
and wind turbine models developed by the International Electrotechnical Commission (IEC) [126]. Models of converter
interfaces were developed for Type 1 through 4 wind turbine
generators with building blocks generic enough to also be
applied to model inverters for solar PV power and even battery systems [127]–[129]. In North America, the development
of these models was driven by the North American Electric
Reliability Corporation (NERC), who recognized the need
and called for standardized, non-confidential, and generic
IBR models for positive-sequence based power flow and stability analysis to assist power system planning studies [130].
These models should be generic enough so that with adequate
parameterization they could be capable of representing any
converter-based resource in commercial software for power
flow and stability analysis of bulk power systems [131].
For example, the renewable energy generator/converter
model, known as regc_a, was used with WECC’s battery
[132] and solar PV [133] dynamic models. The generic structure of the PSM regc_a and the accompanying exciter model
reec_c is shown in Fig. 9. The model’s dynamics are defined
by first-order low-pass filters to represent a time constant
of real and reactive current injection control loops, another
similar time constant for the voltage filter and the current
limitation control logic. These models have been shown to
agree with real wind turbine generators subject to tests as well
as simulations using vendor proprietary models and other
accepted models published in the research [127].
FIGURE 9. Generalized positive-sequence model of inverter-based
resource. Adapted from [193].
Examples and Applications: Previously mentioned
efforts by NERC and WECC have created standards and
driven development efforts for PSMs of IBRs. Academic
research on PSMs of renewable energy resources has indicated that these generic models might overlook some important aspects of their dynamics. A simplified IBR model
was proposed in [134] to be used with a commercial power
systems analysis software, PSLF (Positive-Sequence Load
Flow). In this work, the PSM of the power converter itself
functions as a voltage source in series with the output filter of
the inverter, which is modeled as a series resistor and inductor.
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Most converters use droop strategy in this work for active and
reactive power control and an all-IBR WECC system was
used as a test system. The results showed that such a system
was feasible and could operate after a number of different
events, such as loss of generation, faults followed by line
outages, and line connections. The voltage drops on DC buses
have shown to degrade the system’s performance compared
with cases when the DC bus voltage was constant.
In [23], the authors proposed a model using a positivesequence voltage source converter connected to a network
through a coupling inductance. The response of inner control
loops of the output currents was represented by a low-pass
filter with a time constant of 10 ms. This work used a
very simple boundary current converter representation and
a much more detailed average model for calibration of the
voltage source converter. The results showed a good agreement between both models in the time scale of power system
transient stability. Similar to the voltage source converter
model, the boundary current model contained the model for
inner current dynamics. However, its interface with the grid
was modeled as a current injection, which used the terminal
voltage of the inverter and the complex power calculated
from the terminal voltage as inputs and the output came
from the low-pass filters that emulated the dynamics of the
inner current control of dq0 current control loops. The three
converter models (proposed voltage source converter, average model, and boundary current model) were compared by
performing point-on-wave simulations and positive-sequence
simulations in a three-generator, 9-bus small test system [23].
The results show that, despite having significantly larger
simulation time steps, there was a relatively small difference
in the positive-sequence simulation results using a voltage
source converter model and the point-on-wave simulation.
However, the boundary current model tended to exhibit a
behavior during transients that was very different from the
other two.
In [135], the effect of finite DC-bus capacitance was
modeled in converters interfacing the grid and synchronous
machines. The converter included an inverter, a controlled
rectifier, and a DC-bus connecting both. This PSM was
validated by comparing it to an even more detailed, pointon-wave model. The simulations of both models suggest that
their dynamics were very similar, except during transients.
It was shown that for an all-IBR power system, these
additional dynamics produced results that were significantly different from the simplified model, thus demonstrating the importance of including such dynamics in the
simulation.
In [136], stability analysis of an all-IBR IEEE-39 bus test
system was performed. The model of each inverter followed
the voltage source inverter control proposed in [137], with
P-ω and Q-V droop controllers and a first-order, low-pass
model for its output filters. Further, in [136], an algorithm for
adjusting the droop gains based on grid sensitivity parameters
was used to place the eigenvalues to guarantee small-signal
stability in the all-IBR power system.
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Positive-sequence representations, however, prove overly
simplistic in many cases. It is found that the assumptions
regarding PSMs are violated in simulations that either contain
power electronic devices such as flexible alternating current
transmission systems and HVDC links or model fast transients, responses to faults, harmonics, or phase imbalance
[138]–[140]. Good representation of the response to faults
of unbalanced loads such as single-phase induction motors
usually require more detailed models than PSMs [141], [143].
D. DYNAMIC PHASOR MODELS
Variables of interest, such as voltages and currents, in power
systems and PECs are typically periodic in steady-state.
Phasor-based models utilize complex values to provide a
convenient mathematical representation of those variables
and circuit parameters such as the electrical network circuits’
impedances and elements. In the literature, the phasor model
can encompass both static and dynamic phasors [26]. Static
phasor modeling assumes that the changes in fundamental
frequency can be neglected; thus, it results in a simpler model
that is well-suited for steady-state analysis and modeling
transmission lines and loads in large power systems to which
slow dynamics are attributed. From the point of view of
dynamic analysis, it is important to model how those deviate
from the steady-state [20]. DPMs are capable of modeling
harmonics, and they provide a more accurate model for representing variations of phasors over time. However, DPM
shows difficulties in performing classical small-signal stability assessment methods [154]. The use of DPMs makes it possible to represent a sinusoidal signal of voltage or current at a
fundamental frequency with harmonics, such as phasors with
different frequencies [26]. DPMs are based on the property
that a (possibly complex) time-domain waveform x (t) can
be represented within the interval τ ∈ (t − T , t) by the
following complex Fourier series [20], [148], [149]:
x(τ ) =
∞
X
Xk (t)ejkωs τ
(10)
k=−∞
where Xk is the coefficient of the k th harmonic and ωs
is the fundamental frequency of the periodic variable. The
dynamic time-varying phasor Xk can be calculated using (11)
[20], [149].
Z t
1
(11)
x(τ )e−jkωs τ dτ = hxik (t)
Xk (t) =
T t−T
where hxik (t) is the k th phasor over period T . A distinctive
feature of dynamic phasors is their time derivative (12) [149].
dXk (t)
dx(t)
=
− jkωs Xk (t)
(12)
dt
dt k
It is interesting to note how (12) relates to the differential equations in time-domain and static phasor representations. For example, if we apply it to the differential
equation that relates voltage (vL (t)) and current (iL (t)) of
an inductor, the k th component of the DPM is given by
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hVL ik = Ld hiL (t)ik /dt + jkωs hiL (t)ik [142], which includes
an average of the derivative of the differential equation
in the time-domain, vL (t) = LdiL (t)/dt, and the phasor
representation, VL = jωLIL = jXL IL .
Examples and Applications: DPMs have been used to
analyze power dynamics and faults in power systems [144],
[145]. Other examples of DPMs used to successfully perform
CDPS modeling and analysis are demonstrated in [20], [146],
[148], [149], [151]–[153]. DPMs provide good system representation when the system dynamics can be approximated
by a low-order approximation of the system that uses only a
few coefficients [20]. One of the main advantages of DPMs
are their flexibility. DPMs can also be used to model the
dynamics of power system components such as single-phase
induction motors [143], power lines, transformers, and capacitors [149]. DPMs can also be used to achieve more accurate
models of microgrids dominated by PECs, which have faster
dynamics due to low system inertia [26].
DPMs have been utilized to study whether grid-forming
droop-controlled inverters are impacted by high harmonics
or unbalance [153]. This work demonstrated that six-step
switching harmonics and unbalance did not significantly
impact a small systems’ inverter dynamics. Additional
work by [150], [155] also used DPMs to study multiple
harmonic and unbalanced conditions in microgrids with
droop-controlled inverter-based DERs. In [149], a DPM was
used for analysis of a droop-controller using the eigenvalues
to determine stability margins of a CDPS. Another approach
was developed in [151], where a DPM was developed under
the stationary ABC reference frame of an inverter-based
unbalanced CDPS. In this paper, the power dynamics were
analyzed, taking advantage of the DPM representing EMT
behavior and unbalanced CDPS. A new approach to a DPM of
a modular multilevel converter was proposed in [152], where
the modular multilevel converter dynamics were modeled
considering the harmonic spectrum of internal and external
variables caused by the operation of the converter with an
extended frequency range. Similarly, in [146] a DPM was
developed that allowed the model to analyze power dynamics
and stability of a modular multilevel converter after linearization obtained a large number of dynamic equations. Another
DPM of modular multilevel converters was used to analyze
power dynamics, as proposed in [147]. They compared both
the small-signal model and DPM in parallel-connected inverters to predict the instabilities of the system, however one
aspect to consider is that the proposed model was not based
in the dq0 frame.
E. DATA-DRIVEN MODELS
Data-driven modeling, in contrast to physics-based modeling,
uses data to derive a model or parameters of a specific system.
Physics-based modeling is performed through laws of physics
that govern the components of the system. Physics-based
modeling requires detailed knowledge of the system that
makes it difficult for complex systems. However, data-driven
modeling requires no or partial information about the system.
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The relationship between input and output is inferred from
data. The models developed from this approach are called
data-driven models. These models rely upon computational
intelligence, classical statistics (ordinary least square or maximum likelihood estimation), machine learning, etc., assuming the data contains sufficient information to describe the
modeled system’s physics [194]. Examples of data-driven
algorithms include artificial neural network (ANN), support
vector machines, random forest, etc. These models capture
the dynamics with no or incomplete prior knowledge of the
system’s physical behavior.
Data-driven modeling can determine the structure, parameters, and temporal behaviors of a system or component
of the system such as a PEC. Generally, the modeling of
a dynamic system is classified into two approaches: first
principle modeling and data-driven modeling [156]. First
principle modeling utilizes the system’s physics to derive
the mathematical representation using established equations
of the system or component. When the system is complex,
model derivation using first principle modeling can be complicated due to the many components that may need to be
modeled and parameters obtained. Additionally, preliminary
information about the system or component may be unknown.
Data-driven modeling is used to extract the model and/or
parameters from the collected data without any prior knowledge or partial knowledge of the system. There are three
types of data-driven models, which are classified in terms of
known parameters and structured as a black-box, grey-box,
or white-box model [195].
A black-box model refers to a model where no prior
information about the system is known. Different transfer
functions with different numbers of poles and zeros or neural
networks are considered and fit with the linear black-box
model’s collected data. In a grey-box model, one or more
of the system’s dynamic equations and/or parameters are
known, and the remaining part is unknown. To identify the
remaining part(s), the observed input-output data are fitted
in a model. There are different techniques for parameter
identification, such as minimizing the sum of squares of
error, maximum likelihood estimation, and subspace system
identification [158]. In a white-box model, the model and
parameter values of the system are fully represented. Fig. 10
shows the comparison of different types of data-driven models where the models are classified based on known and
unknown parameters and structure.
When dynamic systems are not easily modeled from first
principle modeling, the model can be built with system identification approaches from measured input-output data. System
identification is a process where measured input and corresponding output data are fed into it to derive an unknown
system’s parameters. With the help of system identification
tools, a PEC data-driven model representing the dynamics of interest can be designed without (full) knowledge of
the underlying control structure and/or control parameters.
Fig. 11 illustrates the basic concepts of a system identification
process. The input signal u(t) and the output signal y(t) are
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The model of the dynamical system is more accurate when
the FPE value is low, and fit of the model can be calculated using normalized root mean square error as defined
in (14) [197]:
!
y(t) − ŷ(t | θ)
fit = 100 × 1 −
(14)
ky(t) − mean y(t)k
FIGURE 10. Comparison of different types of data-driven models.
Adapted from [156].
FIGURE 11. Fundamental idea of system identification where measured
inputs and outputs are fed into system identification algorithm to identify
the unknown dynamical system. Adapted from [171].
first measured from the unknown dynamic process to be
identified. The resulting dataset is then fed into a system
identification algorithm, which typically minimizes a defined
cost-function to estimate the reduced-order system model
Ĝ(s). This reduced-order model is then used to study the
system-level dynamics.
Examples and Applications: Many approaches for developing high-fidelity data-driven models have been performed
in the literature. Models obtained from these different methods need to be accurate enough to use. For that purpose,
error metrics such as Akaike’s final prediction error (FPE)
or normalized root mean square error fitness value (fit) need
to be calculated. FPE [196] can be calculated as defined
in 13:
!
!
N
T
1+ D
1 X
N
FPE = det
e(t, θˆN ) e(t, θˆN )
N
1− D
t=1
N
(13)
where e(t) represents the prediction errors, θ represents the
set of the unknown parameters/coefficients of the dynamical system, N represents the total number of measured
input-output data in the time interval 1 ≤ t ≤ N , and D
represents the total number of estimated parameters.
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where ŷ (t) is the estimated output data.
In the literature, data-driven black-box models have been
used as a useful approach for modeling PECs of power
systems [159], [160]. Early in black-box modeling, DC-DC
converters had been widely discussed [161]–[165] and then
shifted towards modeling of DC-AC converters [160], [166],
[167]. However, black-box models alone were not always
accurate for large operating ranges [160]. Integrating several
models to represent the dynamics over a range-of-interest
could result in a better fit over a wide operating range,
for example, in a polytopic structure [160], [167] discussed
later. The trend of data-driven modeling is shifting towards
system-level studies versus specific components [168], [169].
The review of these system-level studies is out of the scope
of this paper. We will specifically be reviewing data-driven
models of PECs.
One of the first data-driven black-box models to determine the unknown information of PECs was developed
in [159]. The paper proposed the black-box modeling
approach to identify different types of PECs, including resonant converters, PWM converters, and zero voltage-switched
PWM converters. This was done by collecting data from
time-domain simulations and a hardware setup. The input
voltage, input current, and duty cycle were the input variables,
and the output inductor current and output capacitor voltage
were the output variables. Then, a system identification procedure was applied to the input/output data streams and a
model was created for each type of converter. This approach’s
main advantage was to identify the transfer function of PECs
in power systems regardless of knowing the internal structure,
and it was useful in developing the reduced-order model to
represent the complex subsystem.
For black-box modeling, tools such as those provided by
the MATLAB’s system identification toolbox [197], python’s
package–SysIdentPy [175], and an R library–sysid [176]
are used. The modeling methods available range from simple linear models based on transfer functions to non-linear
models using methods such as the Hammerstein-Wiener
model [157], [170]. In [171], a data-driven, black-box
approach was used to model the current dynamics of a PEC.
The methodology used identified reduced-order dynamics
of the inverter interfaced with the grid. A reduced-order
model (transfer function) of the inverter was developed using
MATLAB’s system identification toolbox [198], for which
the inverter output current data was recorded when the grid
voltage was perturbed. A set of linear dynamic models for the
converter were developed to improve the fit of the model. The
authors of this paper compared different transfer functions
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C. Shah et al.: Review of Dynamic and Transient Modeling of PECs for Converter Dominated Power Systems
and selected the best one based on FPE and fit. In [157]
regression analysis and curve fitting had been used to design
LTI black-box models.
In [172], a behavioral black-box model of a DC-AC
three-phase voltage source inverter was analysed based on
its transient response. The paper presented a parametric
method to identify the transfer function of a three-phase
voltage source inverter. A step voltage input was generated
at the input to record the d-axis output current (i0d ). The
least-square method was applied to fit the transfer function from the transient response [157]. The approach’s main
advantage was that the model development was simple and
the time required for the simulation was short.
Conventional linear data-driven models are appropriate for
small operating ranges, and thus these models cannot capture non-linear behaviors. The black-box polytopic model is
used for PECs to capture significant dynamic non-linearities.
A polytopic model explains the behavior of non-linear systems with small-signal models at different operating points
and integrates them into a non-linear structure using weighting functions. In [173], [174], a polytopic model scheme was
used to capture the dynamic behavior of the converter where
the input variables were the input voltage, the dq0 component
of three-phase output voltage, the input current, and the frequency of the grid voltage, and the output variables were the
dq0 components of the output current. In [160], a data-driven
large-signal model of a grid-connected inverter was proposed
using the polytopic approach. First, the operating range was
divided into multiple small operating ranges. These obtained
small-signal models were averaged such that the weight was
taken based on their respective distance from the operating
point. These different operating points had different distances
from the actual operating point. So, the model’s operating point that was closer to the actual operating point was
given higher weight and vice-versa. The main advantages
of this approach were that it used a linear model that could
be easily identified and that it could incorporate non-linear
dynamics.
To solve the small range applicability of conventional
linear data-driven model, ANN based models have been
proposed to capture the non-linear behavior. In [27],
a data-driven ANN based black-box model for a PV microinverter was proposed. The model incorporated a large operating range within a single model using an ANN. The model
captured inverter dynamics of the large operating range
including burst mode. Burst mode refers to the operating
mode where the power supply control circuit is intermittently
disabled at light load conditions and non-burst mode refers
to all other operating modes. The ANN model was created
by extracting root mean square data from the PV side current
and grid voltage as the input, and the output was the frequency
components (magnitude, phase) of the grid current. The ANN
provided the predicted dynamics in the frequency domain,
which later were converted to the time domain. The main
advantage of this method was that the large operating range of
the inverter could be incorporated into a single model without
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prior knowledge about the inverter components or control.
The error (which is defined as the difference between the
measurement and model values in terms of rated value) on
the model was found to be 0.6% for magnitude and 0.66◦
for phase when calculated on training data. With testing data,
the error was found to be less than 1.2% for magnitude and
0.9◦ for phase. For PV rated power, the error for magnitude
and phase was found to be 2.2% and 3.5◦ , respectively. This
higher error was because of lower power.
Other data-driven models include the impedance model
where the output impedance is obtained by measurements
through intrusive or non-intrusive methods [199], [200]. The
impedance model is used for modeling and stability analysis
of power systems, including CDPS. This technique considers the system as a cascaded system consisting of a source
subsystem and a load subsystem. The impedance model of
each source and load is obtained individually before cascading to perform the impedance-based stability analysis of the
CDPS [201], [202].
Data-driven models are advantageous when the system is
very complex. Depending upon the accuracy of the model
desired, different techniques can be used. System identification techniques can give simplified models, whereas ANN
models can be used when higher accuracy is desired.
IV. DISCUSSION AND TRENDS
Determining the best model type is dependent on defining the
application and phenomenon of interest. Switching models
include every component of the PEC down to the switching
IGBT and diodes. With these models, the modulation signal
needs to be developed and tuned. Inadequate tuning of the
control of the modulation signal can result in high frequency
harmonics. Additionally, switching level models are the only
model that can accurately mimic exact hardware implementations. This is typical because actual controllers sample current
and voltage waveforms for feedback control at a specific
instance during the switching period, often optimized to minimize the delay. When using an average model, there will
effectively be a one switching period delay in the feedback
system. Switching models are the most accurate; however,
they have numerous parameters to define and tune and are the
most computationally burdensome. Switching models will be
continued to be used for examination or design of switching
modulation strategies and determination of switching losses.
With improvements in real time digital simulation and other
simulation acceleration techniques, switching models may be
more prominently used when modeling small systems with
a handful of devices. However, there is ongoing research
investigating when switching models are necessary to use
instead of average models.
Typically, small-signal models in the Laplace domain or
in state-space are used to perform classic small-signal performance analysis such as Bode/singular-value diagrams,
eigenvalue analysis, Nyquist diagrams, stability margin estimation, etc. These analyses are useful because they not
only provide information about transient response but also
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C. Shah et al.: Review of Dynamic and Transient Modeling of PECs for Converter Dominated Power Systems
help to identify stability margins and frequency response.
Small-signal models are commonly used to design linear controllers such as proportional-integral, proportional-resonant,
linear quadratic regulator, etc. Due to these advantages,
small-signal average models are a common model type found
in the literature and could be applied to every timescale
and phenomenon excluding switching modulation. However,
because some converter dynamics are highly non-linear, such
as large disturbances like faults, small-signal models can
result in inaccuracies when the states are not close to the operating point. Due to the mathematical complexity involved
in the development of mathematical models for CDPS with
numerous converters, research is tending to develop models
reduced to small-signal [203]–[205]. Most of these reduced
models are based on the premise that converter instantaneous dynamics (voltage/current and filter) can be neglected
or transformed because they are on the order of microseconds/milliseconds. For, this reason, these models consider
only the power dynamics that are the slowest and most
important in CDPS [110], [206]. Small-signal models with
predictive mechanisms are a future trend on CDPS modeling
for improving the analysis of frequency and voltage stability.
[85], [87], [88], [90], [94], [96].
PEC large-signal models are used where PEC non-linearity
is significant and when the response to large perturbations deviates substantially from the response predicted by
the small-signal model. Large-signal models generate more
accurate results than the small-signal models under large
disturbances and load changes. As a result, the scientific
community has been adopting the circuit-average based
large-signal model of PECs to study dynamic phenomena in
CDPS. In the future, with high penetrations of PEC-based
DERs, large-signal models will be used to model PECs in the
grid-connected mode since PEC non-linearities dominate the
resonance-based distortions in power grids. This will help
develop better control strategies for PECs. Also, with the
advancement in simulation tools and an increase in computational power, the simulation time to simulate large-signal
models has been drastically reduced which will aid increased
adoption of large signal models in the future.
PSMs are a type of large-signal model that are based solely
on the positive-sequence component, with a single phase
representation. PSMs are most commonly used to study the
bulk power system dynamics in positive-sequence simulators.
These models implemented in positive-sequence simulators
are best used to simulate large disturbance events and capture
non-linear components of PECs, such as discontinuous protection settings. Due to the simplified representation of PECs
in PSMs, these models are highly computationally efficient
which is why they are used in simulating large transmission
systems. However, PSMs are unable to capture inner control
loops, harmonics, and switching dynamics associated with
PECs. Additionally, PSMs assume that the system is balanced
and operates near the fundamental frequency. PSMs will
continue to be used, especially for large system modeling, and
future advancements of these models such as incorporating
82110
multiple protection settings for these aggregated models will
improve accuracy in system studies [21], [22].
DPMs simulate the dynamics of an electrical network’s
circuit elements for positive and negative sequence signals.
Modeling with dynamic phasors allows one to analyze both
harmonics and stability in CDPS. However, this method has
historically been used for modeling conventional power system dynamics. More research about using this model to simulate PECs and perform stability and performance analysis is
expected in future research works as this is an active field of
research.
A relevant trend in CDPS modeling is data-driven models.
These models are becoming popular because they obtain simplified mathematical models of complex systems. A CDPS
may be very computationally burdensome to simulate using
parameterized models, such as the other model types discussed in this paper. Data-driven models, such as black-box
or grey-box models, are implemented using partial or no
information of each component’s parameters. These models
require time-series data from the device and/or system to
create a model that simulates the dynamics captured in the
time-series data. However, a data-driven model’s accuracy
is limited to the amount, diversity, and time-step of the
data collected. Also, because the model depends on data,
a data-driven model can only represent implemented devices
where operational field data can be collected. In the future,
data-driven models will be used to investigate the dynamics of
various inverters under various operating conditions or modes
of operation [168], [169], [171]. The various linear transfer
function models derived from the data-driven model will be
combined using a statistical approach to derive a generalized
non-linear model that captures the inverter’s most important
dynamics. These generalized non-linear models can be used
to develop and analyze CDPS.
Developing new forms of average and simplified PEC
models, such as average, DPMs, PSMs, and data-driven models, is an active area of research. This research is driven by the
need to create highly accurate yet computationally efficient
methods to simulate large power systems that are installing
increasing amounts of IBRs and DERs that are producing
new and more pronounced fast dynamics in the system. New
developments of these models will greatly benefit from careful considerations of what power system dynamic stability
issues are of greatest concern for the specific systems of
interest.
V. CONCLUSION
There is an increasing amount of IBRs in electrical grids.
The appropriate selection of the PEC model type is essential to simulate and study the impact of IBRs in power
systems, particularly CDPS. This paper provides a detailed
review of PEC model types and their applications to study
different power system dynamic stability issues. This review
paper provides examples of applications of the PECs model
types for simulating various power system dynamic stability
issues that are either created by or responded to by IBRs.
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C. Shah et al.: Review of Dynamic and Transient Modeling of PECs for Converter Dominated Power Systems
A discussion is provided on the key takeaways of the use
of PEC model types in relevance to those dynamic stability
issues. Additionally, this review identifies advantages and
limitations of each PECs model type. We conclude that identifying the specific dynamic power system stability issues of
interest, its relevant timescale, and the mode of operation of
CDPS is the key to choosing the right model type.
ACKNOWLEDGMENT
The contributions to this research work were achieved in part
through the U.S. Department of Energy Office of Science,
Office of Basic Energy Sciences, EPSCoR Program; Office of
Electricity, Microgrid Research and Development Program;
and Office of Energy Efficiency and Renewable Energy, Solar
Energy Technology Office. The views expressed in the article
do not necessarily represent the views of the DOE or the
U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges
that the U.S. Government retains a nonexclusive, paid-up,
irrevocable, worldwide license to publish or reproduce the
published form of this work or allow others to do so, for the
U.S. Government purposes. Sandia National Laboratories is a
multi-mission laboratory managed and operated by National
Technology and Engineering Solutions of Sandia, LLC.,
a wholly owned subsidiary of Honeywell International, Inc.
This paper describes objective technical results and analysis.
Any subjective views or opinions that might be expressed in
the paper do not necessarily represent the views of the U.S.
Department of Energy or the United States Government.
The authors would like to thank Dr. Ujjwol Tamrakar for
reviewing the paper.
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CHINMAY SHAH (Graduate Student Member,
IEEE) received the B.Tech. degree in instrumentation and control engineering from Nirma
University, India, in 2012, and the M.S. degree
in electrical engineering from the University of
Houston, Houston, TX, USA, in 2017. He is currently pursuing the Ph.D. degree in electrical engineering with the University of Alaska Fairbanks.
From 2012 to 2014, he worked as an Instrumentation and Control Engineer with Dodsal Engineering and Construction, Dubai, United Arab Emirates. He worked as a
Research Intern with the National Renewable Energy Laboratory, in summer
2019. He currently works as a Research Assistant with the Alaska Center
for Energy and Power (ACEP). His research interests include modeling and
control of power electronic converter-based DERs, distributed optimization,
distributed controls for the power grid, power system resiliency and reliability, application of blockchain, and the Internet of Things (IoT) in the power
distribution networks.
82116
JESUS D. VASQUEZ-PLAZA (Graduate Student
Member, IEEE) received the B.S. degree in electronic engineering from the Universidad Del Valle,
Cali, Colombia, in 2017, and the M.S. degree
in electrical engineering from the University of
Puerto Rico, Mayagüez Campus, Mayagüez, PR,
USA, in 2020, where he is currently pursuing the
Ph.D. degree.
His main research interests include optimal and
robust control systems, modeling of converterdominated power systems, analysis, design, and control of power electronic
converters.
DANIEL D. CAMPO-OSSA (Graduate Student
Member, IEEE) was born in Cali, Colombia.
He received the B.Sc. degree in electronic engineering and the M.Sc. degree in engineering in
automatic control from the Universidad del Valle,
Cali, Colombia, in 2008 and 2013, respectively.
He is currently pursuing the Ph.D. degree in electrical engineering with the University of Puerto
Rico at Mayagüez Campus (UPRM), Mayagüez,
PR, USA.
His main research interests include the modeling, analysis, and control of
power electronic converters, sequential logic, control systems, applications
of power electronics in renewable energy systems, and converter-dominated
microgrid applications.
JUAN F. PATARROYO-MONTENEGRO (Member, IEEE) received the B.S. degree in electronics engineering from the University of Quindio,
Armenia, Colombia in 2011, and the M.S.
and Ph.D. degrees in automatic control from
the University of Puerto Rico at Mayaguez
(UPRM), Mayaguez, PR, USA, in 2015 and 2019,
respectively.
He joined the Sustainable Energy Center (SEC),
UPRM, as a Laboratory Coordinator, in 2017.
He is currently working as a Postdoctoral Researcher with the SEC, UPRM.
His main research interests include optimal and robust control systems,
embedded systems, and modeling, analysis, control, and design of power
electronic converters, principally dc/ac power conversion.
NISCHAL GURUWACHARYA (Student Member,
IEEE) received the B.E. degree in electrical engineering and the M.Sc. degree in energy systems
planning and management from Tribhuvan University, Nepal, in 2013 and 2019, respectively.
He is currently pursuing the Ph.D. degree in electrical engineering with South Dakota State University (SDSU), Brookings, SD, USA.
His research interests include data-driven modeling, power electronics and control, and grid
integration of renewable energy systems.
NIRANJAN BHUJEL (Student Member, IEEE)
received the B.E. degree in electrical engineering
from Tribhuvan University, Nepal, in 2017. He is
currently pursuing the Ph.D. degree in electrical
engineering with South Dakota State University
(SDSU), Brookings, SD, USA.
His research interests include optimization, optimal control, optimal estimation, and
multi-time scale control in microgrids.
VOLUME 9, 2021
C. Shah et al.: Review of Dynamic and Transient Modeling of PECs for Converter Dominated Power Systems
RODRIGO D. TREVIZAN (Member, IEEE)
received the M.Sc. degree in power systems engineering from the Grenoble Institute of Technology
(ENSE3), in 2011, the B.S. and M.Sc. degrees
in electrical engineering from the Federal University of Rio Grande do Sul, Brazil, in 2012 and
2014, respectively, and the Ph.D. degree in electrical engineering from the University of Florida,
in 2018.
He is currently a Senior Member of Technical
Staff with Sandia National Laboratories. His research interests include information security, control of energy storage systems, demand response, power
system state estimation, and detection of nontechnical losses in distribution
systems.
FABIO ANDRADE RENGIFO (Member, IEEE)
received the B.Sc. degree in electronic engineering and the master’s degree in engineering with
emphasis on automatic control from the Universidad Del Valle, Cali, Colombia, in 2004 and 2007,
respectively, and the Ph.D. degree from the Universitat Politècnica de Catalunya (UPC), Barcelona,
Spain, in 2013.
He joined the Motion Control and Industrial
Centre Innovation Electronics (MCIA), in 2009.
He worked as a Postdoctoral Researcher with UPC and Aalborg University,
Denmark, in 2014. He is currently working as the Director of the Sustainable Energy Center (SEC) and an Associate Professor of power electronics
applied to renewable energy with the University of Puerto Rico, Mayaguez
campus. His main research interests include modeling, analysis, design,
and control of power electronic converters, principally for dc/ac power
conversion, grid-connection of renewable energy sources, and microgrid
application.
MARIKO SHIRAZI (Member, IEEE) received
the B.S. degree in mechanical engineering from
the University of Alaska Fairbanks (UAF),
in 1996, and the M.S. and Ph.D. degrees in electrical engineering from the University of Colorado
Boulder, in 2007 and 2009, respectively.
She was an Engineer with the National Renewable Energy Laboratory for a period of 15 years,
working on early efforts to integrate wind into village power systems, and later on power electronics
design for microgrid applications. She currently works as the President’s
Professor of Energy with the UAF, where she is interested in bridging power
electronics and power systems research to understand the performance of
converter-dominated microgrids.
REINALDO TONKOSKI (Senior Member,
IEEE) received the B.A.Sc. degree in control and automation engineering and the M.Sc.
degree in electrical engineering from the Pontifícia Universidade Católica do RS (PUC-RS),
Brazil, in 2004 and 2006, respectively, and the
Ph.D. degree from Concordia University, Canada,
in 2011.
He is currently the Harold C. Hohbach Endowed
Professor with the Electrical Engineering and
Computer Science Department, South Dakota State University, USA, and
a Visiting Professor with Sandia National Laboratories. He has authored
over one hundred technical publications in peer-reviewed journal articles
and conference papers. His research interests include grid integration of
sustainable energy technologies, energy management, power electronics, and
control systems. He is also an Editor of IEEE TRANSACTIONS ON SUSTAINABLE
ENERGY, IEEE ACCESS, and IEEE SYSTEMS JOURNAL.
VOLUME 9, 2021
RICHARD WIES (Senior Member, IEEE)
received the B.S., M.S., and Ph.D. degrees in
electrical engineering from the University of
Wyoming, Laramie, WY, USA, in 1992, 1995, and
1999, respectively.
Since 1999, he has been with the University
of Alaska Fairbanks, Fairbanks, AK, USA, where
he is currently a Professor with the Electrical
and Computer Engineering Department, with a
concentration in electric power systems. He leads
research focused on the engineering challenges of renewable energy integration in remote islanded microgrids in collaboration with the Alaska
Center for Energy and Power. His research interests include the development
of advanced distributed generation and load control schemes and optimal
power dispatch strategies for remote islanded microgrids employing high
penetrations of renewable energy, grid-forming operation with standalone
asynchronous renewable generation, impacts of renewable power on food
and water systems, and stability of converter-dominated grids.
Dr. Wies has served on the PES Power Systems Dynamic Performance
and Power Engineering Education Committees, contributed to two task force
reports, and been invited to present on a number of panels at IEEE sponsored
conferences about his research work with remote islanded microgrids. He is
a Licensed Professional Engineer in the State of Alaska.
TIMOTHY M. HANSEN (Senior Member,
IEEE) received the B.S. degree in computer engineering from the Milwaukee School of Engineering, Milwaukee, WI, USA, in 2011, and the
Ph.D. degree in electrical engineering degree from
Colorado State University, Fort Collins, CO, USA,
in 2015.
He is currently an Assistant Professor with
the Electrical Engineering and Computer Science Department, South Dakota State University, Brookings, SD, USA. His research interests include optimization,
high-performance computing, and electricity market applications to sustainable power and energy systems, low-inertia power systems, smart cities, and
cyber-physical-social systems.
Dr. Hansen is also an Active Member in ACM SIGHPC. He was a
recipient of the 2019 IEEE-HKN C. Holmes MacDonald Outstanding
Teaching Award. He was an inaugural recipient of the Milwaukee School of
Engineering Graduate of the Last Decade Award, in 2020. Within IEEE he
has been the IEEE Siouxland Section Chair, since 2019. He is active within
the IEEE PES Power Engineering Education Committee, also serving as the
Awards Subcommittee Chair.
PHYLICIA CICILIO (Member, IEEE) received
the B.S. degree in chemical engineering from
the University of New Hampshire, Durham, NH,
USA, in 2013, and the M.S. and Ph.D. degrees in
electrical and computer engineering from Oregon
State University, Corvallis, OR, USA, in 2017 and
2020, respectively.
She is currently a Research Assistant Professor
with the Alaska Center for Energy and Power, University of Alaska Fairbanks. Her research interests
include power system reliability and dynamic power system modeling particularly of loads, inverter-based resources, and distributed energy resources.
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