sustainability

Article
Grid-Following Inverter-Based Resource: Numerical
State–Space Modeling
Abdullah Alassaf * , Ibrahim Alsaleh , Ayoob Alateeq and Hamoud Alafnan

Department of Electrical Engineering, College of Engineering, University of Hail, Hail 55211, Saudi Arabia;
i.alsaleh@uoh.edu.sa (I.A.); a.alateeq@uoh.edu.sa (A.A.); h.alafnan@uoh.edu.sa (H.A.)
* Correspondence: a.alassaf@uoh.edu.sa

Abstract: In the pursuit of a sustainable electric power system, the integration of renewable energy
sources and distributed energy resources is gradually replacing traditional power generation. These
new resources are integrated into the grid via inverters, which, despite their efficient performance,
present dynamic challenges to the power grid when implemented on a large scale. To maintain grid
stability and ensure effective regulation during abnormal operations, various modeling techniques
are necessary; while the dynamics of inverter-based resources (IBRs) are traditionally modeled by
transfer functions, this paper sheds light on differential-algebraic equations (DAEs) modeling and
numerical integration methods. The inherent limitations of transfer function modeling stem from its
restricted applicability, as it is exclusively suitable for linear and time-invariant systems. In contrast,
the nonlinear DAEs of the IBR system can be converted into a state–space form, which offers a
versatile framework for modeling, evaluating, and designing a diverse array of systems. In addition
to being compatible with time-varying systems and multiple-input multiple-output systems, the
state–space technique may incorporate saturation and dead zone characteristics into the dynamic
model. Our research focuses on IBR modeling in a grid-following scheme, which is current-controlled
and synchronized to the grid by a phase-locked loop (PLL). The presented state–space model consists
of the inverter, grid, control, and designed PLL. Beyond the discussion of its application to IBRs,
the presented method holds the potential to solve a wide range of DAEs. The proposed model is
compared with a benchmarked system.

Citation: Alassaf, A.; Alsaleh, I.;

Keywords: differential-algebraic equations; state–space; distributed energy resources; renewables;
Alateeq, A.; Alafnan, H.
inverter-based resource; phase-locked loop; grid-following
Grid-Following Inverter-Based
Resource: Numerical State–Space
Modeling. Sustainability 2023, 15,
8400. https://doi.org/10.3390/
su15108400 1. Introduction
Growing concerns about environmental pollution, the depletion of fossil fuels, and global
Academic Editor: Xichen Jiang
warming all serve as powerful motivators for a rapid transition away from conventional
Received: 11 April 2023 energy supplies toward renewable energy resources, which play a crucial role in promoting
Revised: 15 May 2023 sustainability [1]. As the passive distribution system is being activated, the migration
Accepted: 19 May 2023 process necessitates reconstructing the existing grid infrastructure. In the traditional grid,
Published: 22 May 2023 the power flow is central and unidirectional. Scatteredly installed distributed generators
(DG) in the grid change these two grid characteristics. In recent years, the idea of a
microgrid has become increasingly prevalent in power systems. The U.S. Department of
Energy defines the microgrid as a collection of interconnected loads and distributed energy
Copyright: © 2023 by the authors.
resources operating within properly specified electrical limits as a single controllable entity
Licensee MDPI, Basel, Switzerland.
with respect to the grid and capable to operate in both grid-connected and island modes [2].
This article is an open access article
distributed under the terms and
Dispersed generation throughout the grid reduces losses and improves reliability.
conditions of the Creative Commons
While active distribution systems provide numerous solutions, they also bring some
Attribution (CC BY) license (https://
difficulties that researchers and the industry must overcome. The significant challenge
creativecommons.org/licenses/by/ is in comprehending the system dynamics through system modeling. The focus of this
4.0/). research is on inverter-based resources (IBRs), which are extensively employed in active

Sustainability 2023, 15, 8400. https://doi.org/10.3390/su15108400 https://www.mdpi.com/journal/sustainability

Sustainability 2023, 15, 8400 2 of 18

distribution systems. An AC microgrid’s power inverter connections can be categorized as

either grid-following or grid-forming, depending on which role the inverters are designed
to perform [3,4].
Grid-following power inverters are mostly constructed to supply power to a grid
that is already energized. They can be modeled as a high-impedance, parallel current
source connected to the grid. In this application, it is important to note that this current
source should be precisely synchronized with the ac voltage at the connection point so
that the active and reactive power exchanged with the grid can be controlled accurately.
Due to this, the grid-following technique is incapable of operating autonomously in Island
Mode. The control configuration of this connection is applied to the active and reactive
power. The grid-forming technique is analogous to an ideal ac voltage source with a low
output impedance. Each of the integrated inverters of such a connection is considered
an independent source of power. Furthermore, the connected inverter is accountable for
determining the voltage and frequency of the local supplied area. Therefore, the grid-
forming scheme has the ability to operate independently in an Island Mode. The system
voltage and frequency are used to govern the configuration of this connection.
Inverters must be synchronized with the grid for all types of DERs. Grid synchro-
nization, in fact, consists in the real-time monitoring of system variables. Due to the fact
that system states may change owing to failures, disturbances, resonances, or even load
changes, instantaneous system monitoring is needed [5–8]. Furthermore, both inverter
connection and disconnection must be performed while monitoring the grid states at the
point of common coupling to ensure that the system is operating in accordance with the
grid requirement codes [9].
The phase-locked loop, or PLL, is the predominant mechanism for synchronizing
converters. Two methods are used for integrating the PLL with the grid: open-loop and
closed-loop [10]. While the open-loop connection has many applications, the closed-loop
connection has gained popularity due to its greater precision. For applications requiring a
three-phase grid-connected converter, the synchronous reference frame PLL (SRF-PLL) is
the most prevalent type of PLL [11]. The PLL device can estimate from the connection point
the voltage magnitude, frequency, and phase angle. The SRF-PLL’s mechanism proceeds as
follows [12]: the PLL transforms the three-phase grid voltage Vabc into the dq-frame voltage
Vdq , which is based on the rotating frame and appears as DC. The PI controller suppresses
the Vq to zero and aligns the grid voltage to the d-axis. The output of the PI controller and
the integrator eventually provides the grid voltage phase angle, which is likewise sent back
to the Park transformation utilized in the process.
The SRF-PLL operates effectively while operating in a balanced state, but fails to
perform well when operating in an abnormal or imbalanced state. Reference [13] proposes
a decoupled double synchronous reference frame PLL for power converters control. In their
work, the authors enable the PLL to extract the grid voltage’s fundamental component
during the inclusion of the distortion components. Other researchers in the field have
proposed improved versions of, or whole new replacements for, the standard PLL [14–20].
This paper implements the SRF-PLL and develops its controller to accommodate unbal-
anced components.
Similar to synchronous machines, inverters can be modeled via ordinary differential
equations (ODEs). The majority of the controllers can be modeled with ODEs if time-
dependent delays are approximated with lags or simply disregarded [21]. Power system
dynamics are generally driven by two types of state variables: slow and fast. The slow-state
variables involve big-time constants, whereas the fast-state variables involve small-time
constants. The dynamical variation of the slow-state variables is small and significant,
and can therefore be overlooked. The dynamics of the fast-state variables, on the other
hand, are serious and must be considered in system modeling. Due to the mixture of
such variables, the studied model is composed of a set of nonlinear differential-algebraic
equations (DAEs).
Sustainability 2023, 15, 8400 3 of 18

Modeling the dynamics of a wide variety of power system applications is a function

that can be performed with the DAE system. The numerical solution of the DAE system
can be approached using either explicit or implicit integration schemes, depending on
the element of the system being analyzed. The explicit integration scheme is renowned
for its ease of use and applicability to a wide range of applications. The implicit scheme
can achieve the same result as the explicit scheme, but it necessitates the inclusion of two
matrices that pertain to the system under consideration.
The state–space modeling of a grid-connected inverter is introduced in [22]. Expand-
ing on this existing work, our research substantially contributes to the field by detailing
the derivation process of the state–space representation. Furthermore, we have incorpo-
rated a comprehensive methodology for numerically solving the state–space equations.
The advanced understanding and tools provided through this research are expected to aid
in the design, analysis, and optimization of grid-connected inverter systems. The following
summarizes this paper’s contribution:
• Model an inverter-based resource coupled to a grid-following scheme using differential-
algebraic equations.
• Introduce a versatile synchronous reference frame phase-locked loop (SRF-PLL) that
is designed to be able to operate in the existence of the imbalanced component.
• Demonstrate how to convert the transfer functions of the system into state–space
form. The procedure, albeit implemented with the IBR model, is generic and can be
employed with other dynamical applications.
• Integrate and simulate the proposed model using the Forward Euler Method (FEM),
the Backward Euler Method (BEM), and the Implicit Trapezoidal Method (ITM).

2. Grid-Following Frequency Inverter System

Inverters facilitate the transformation of electrical energy from DC to AC. On the
DC side, the inverter is typically linked to a DC source, or alternatively, another power
electronic system. These can be effectively modeled as a combination of a DC source,
a capacitor, and a current source. The latter is a representation of the power loss attributed
to the converter. The losses incurred when the inverter is in an operational state manifest
as resistors on the AC side of the system.
In the context of this research, we delve into the dynamics of a three-phase inverter
that is engaged in the supply of a stiff, balanced grid operating at a constant frequency.
On the AC side, each phase of the inverter is interfaced with an ideal voltage source via a
series RL branch, a configuration that creates an inductive–resistive pathway. The point at
which these phases are collectively coupled with the grid is known as the Point of Common
Coupling (PCC), and the system is illustrated in Figure 1.
The role of the grid-following inverter is to "follow" the grid’s behavior by adjusting
its output voltage and frequency to align with those of the AC microgrid. This is achieved
through a mechanism known as the Phase-Locked Loop (PLL), which enables synchroniza-
tion between the inverter’s voltage and frequency and those of the grid. The PLL plays a
crucial role in maintaining grid stability and ensuring the efficient operation of the inverter
with the grid.
Sustainability 2023, 15, 8400 4 of 18

is
+

Vsabc
Vtabc R + ron L
VDC + VDC
-
iabc
Vabc
abc 𝜌
- dq abc
dq 𝜌
id iq
Vsd Vsq
mabc
abc 𝜌
dq
VDC
md mq Reference
idref Pref
Control in dq-frame iqref Signal
Generator
Qref

Figure 1. Schematic diagram of an inverter connected to the grid.

2.1. dq-Frame
To control the grid-following converter-interfaced system, it is more practical to reduce
the system’s structure and convert the actual three-phase frame to either αβ-frame or dq-
frame. For large-scale systems, the latter is more convenient as its control is DC and it does
not require a large bandwidth. The following equation converts the three-phase frame to
dq-frame
 h i  
 2π
 4π f (t)
2  cos[ρ(t)] cos ρ(t) − 3 cosh ρ(t) − 3 i  a
 
f d (t)
= f b ( t ) , (1)
f q (t) 3 sin[ρ(t)] sin ρ(t) − 2π sin ρ(t) − 4π
 
3 3 f c (t)

where f () can represent the voltage and the current. ρ is the PCC voltage angle that
allows for the frame conversion and is achieved from PLL. The abc-frame can be retrieved
as follows:
  
 [ρ(t)]2π 
cos sin[ρ(t)]2π   f (t) 

f a (t)
 coshρ(t) − 3 i sinhρ(t) − 3 i 
 f b (t)  =  d
. (2)

4π 4π f q (t)
f c (t) cos ρ(t) − 3 sin ρ(t) − 3

2.2. Grid Modeling

This research considers an inverter that is connected to a stiff grid; the voltage is
assumed to be balanced with constant frequency. The three-phase voltages are expressed
as follows:
Vsa (t) = V
bs cos(ω0 t + θ0 ),
bs cos ω0 t + θ0 − 2π ,


Vsb (t) = V  3  (3)
4π
Vsc (t) = Vs cos ω0 t + θ0 −
b , 3

where Vbs is the peak value of the line-to-neutral voltage, ω0 is the grid angular frequency,
and θ0 is the initial phase angle. The grid is modeled in abc-frame as follows:
       
i a (t) i a (t) Vta (t) Vsa (t)
d
L  ib (t)  = −( R + ron ) ib (t)  +  Vtb (t)  −  Vsb (t) . (4)
dt
ic (t) ic (t) Vtc (t) Vsc (t)
Sustainability 2023, 15, 8400 5 of 18

Using (1), the grid can be expressed in dq-frame as follows:

L didtd = Lω (t)iq − ( R + ron )id + Vtd − V

bs cos(ω0 t + θ0 − ρ),
diq
L dt = − Lω (t)id − ( R + ron )iq + Vtq − V
bs sin(ω0 t + θ0 − ρ), (5)
dρ
dt = ω ( t ).

For this system, the control inputs Vtd , Vtq , and ω determine the dynamics of the state
variables id , iq , and ρ. It can be noticed that the system includes nonlinear components that
are ωid , ωiq , cos(ω0 t + θ0 − ρ), sin(ω0 t + θ0 − ρ). With the help of the Phase-Locked Loop
(PLL), the system phase angle can be accurately detected. ρ represents the detected angle
from PLL. With ρ = ω0 t + θ0 , the complexity of the nonlinearity disappeared

L didtd = Lω0 iq − ( R + ron )id + Vtd − Vsd ,

di (6)
L dtq = − Lω0 id − ( R + ron )iq + Vtq − Vsq ,

where Vsd = Vbs and Vsq = 0. In the current form, the system is linear and it is exited by Vs.
With Vtd and Vtq as DC, the state variables id and iq are also DC. The conversion angle ρ
is ensured to have the value of ω0 t + θ0 by the PLL mechanism, which is covered by the
following subsection. The active and reactive powers supplied to the PCC node are

Ps (t) = 32 Vsd (t)id (t) + Vsq (t)iq (t) , 

 
(7)
Qs (t) = 32 −Vsd (t)iq (t) + Vsq (t)id (t) .

As stated earlier, PLL guarantees that, at steady state, Vsq = 0. Thus, the active and reactive
power will be
Ps (t) = 23 Vsd (t)id (t),
(8)
Qs (t) = − 32 Vsd (t)iq (t).
Accordingly, Ps (t) and Qs (t) can simply be controlled by id and iq , respectively. The reference-
controlled signals of the active and reactive power can be set as follows:

idre f (t) = 3V2 Psre f (t),

sd (9)
iqre f (t) = − 3V2 Qsre f (t).
sd

With efficient control, the system variables track the reference settings.

2.3. Current-Mode Control

Two modes of control are applied to the inverter: voltage and current. The latter has
the advantage of protecting the inverter from overcurrent issues that could deteriorate
the inverter. Equation (6) describes the dynamics of the whole system and it includes the
inverter voltage terminals. Since dq-frame is implemented, the terminal voltage Vtabc is
converted to Vtd and Vtq with the help of the PLL. The applied control on the inverter
utilizes the relationship between the terminal voltage and the control modulation signals,
as follows:
Vtd (t) = VDC
2 m d ( t ), (10)
Vtq (t) = VDC
2 m q ( t ).
The inverter current-control mode that is expressed by Equation (6) has state variables id
and iq , control variables Vtd and Vtq , and disturbance inputs Vsd and Vsq . Its control block
diagram is shown in Figure 2.
Sustainability 2023, 15, 8400 6 of 18

Gff (s) Vsd

- Vtd - 1
idref kd (s) md id
/ ×
Ls + (R + ron )
-
L𝜔 0
id
L𝜔0 VDC
L𝜔 0 2
iq L𝜔0
mq Vtq - 1
iqref kq (s) / × iq
- - Ls + (R + ron )

Gff (s) Vsq

Figure 2. Control diagram.

For this system, a controller is applied to make the inverter’s currents, both d and q
components, follow the change of the reference signals. The difference between the current
state and the reference signal is the error e. The controller k(s) compensates for the error
and generates the signal u, which is scaled by the division by the value of the DC source
and sets the inverter modulation signal m. Next, the inverter processes the modulation
signal m to produce the terminal voltage Vt , as in Equation (10). As the model structure is
DC, the PI controller is sufficient to track the signal reference. The designed controller suits
both d and q loops.
k p s + ki
k(s) = , (11)
s
where proportional and integral gains, respectively, are denoted by k p and k i . As a result,
the system open-loop gain

kp s + k i /k p
 
`(s) = . (12)
Ls s + ( R + ron )/L

Due to the fact that the system plant pole is inherently close to the origin, the magni-
tude and phase gains of the decline are found at low frequencies. Since the plant pole is
on the left side, stability concerns are not posed. The plant pole can be canceled by setting
 
kp
the controller zero s = −k i /k p equal to the plant pole. The loop gain will be `(s) = Ls .
Consequently, the closed-loop transfer function of the system evolves to be
Id (s) 1
Idre f (s)
= Gi (s) = τi s+1 ,
k p = L/τi , (13)
k i = ( R + ron )/τi ,
where τ represents the time constant of the closed-loop system. The value of τ should be
both large enough to have a wide bandwidth and small enough to have a fast response. G f f
represents the feed-forward filter, a crucial component employed to predict variations in the
load or system conditions, thereby facilitating the necessary adjustments to the inverter’s
output. The integration of feed-forward filters into inverter control systems offers several
advantages, such as accelerated response times, reduced distortion, and improved efficiency.
The inverter is equipped with the capability to modify its output proactively in anticipation
of fluctuations in the load or system parameters.
Sustainability 2023, 15, 8400 7 of 18

2.4. Phase-Locked Loop (PLL)

The PLL synchronizes the inverter with the grid. It provides the grid angle, denoted
as ρ, that gives the ability to convert the three-phase components into dq-components.
The grid dq-components are

bs cos(ω0 t + θ0 − ρ),
Vsd (t) = V
bs sin(ω0 t + θ0 − ρ). (14)
Vsq (t) = V

As stated before, ρ = ω0 t + θ0 results in Vsd = V

bs and Vsq = 0. This is achieved by the
following steps. By incorporating feedback on the state variable ρ in Equation (5), the PLL
can be expressed as

dρ
= ω ( t ),
dt
= H ( p)Vsq (t), (15)
bs sin(ω0 t + θ0 − ρ),
= H ( p )V

where H ( p) is the transfer function of the PLL’s compensator and p is a differentiation

operator. The PLL equation is capable to set ρ to ω0 t + θ0 . However, the frequency has
to be properly initialized and constrained. The nonlinearity caused by the trigonometric
function can be eliminated if the approximation sin( x ) = x is considered as x approaches
zero. Accordingly, the PLL equation is

dρ bs (ω0 t + θ0 − ρ).
= H ( p )V (16)
dt
The difference between the input, ω0 t + θ0 , and the output, ρ, is transmitted through
the transfer function, H ( p)V
bs , to perform the management of this feedback system. The de-
tailed scheme of PLL operation is shown in Figure 3. At the PLL’s termination, a voltage-
controlled oscillator (VCO) is set up to reset the measured angle once it hits 2π.

Va
abc Vd
Vb
Vq
Vc dq
𝜌
𝜌 VCO Limiter Compensator
∫ H(s)

Figure 3. PLL operation scheme.

The dynamical performance of the PLL is mostly determined by the design of the
compensator H (s) that is defined as follows:

H (s) = H1 (s) H2 (s), (17)

where H1 (s) handles the system requirements and H2 (s) considers the stability require-
ments. The PLL has a single integrator at the origin, which is from the VCO. For that,
the constant component of the input signal, θ0 , can simply be tracked. However, since the
input signal has a ramp component that is ω0 t, another integrator is required to guarantee
that the steady-state error approaches zero. The compensator part H1 (s) may include this
pole among the other system’s design requirements.
Sustainability 2023, 15, 8400 8 of 18

To prevent malfunctioning in the control process, the PLL should also take care of the
distortion, e.g., harmonics, that are embedded in the grid voltage. Otherwise, the distortion
in the grid will be reflected in the converted frame, dq. The double harmonic component is
a significant cause for concern since it has an inherent large magnitude and a frequency
that is the closest to the fundamental frequency. The imbalanced component can also lead
to stability problems in the grid. We must consider the imbalanced component in designing
the PLL mechanism to avoid the PLL malfunctioning. The effect of the double component
distortion can be removed from the PLL when one pair of complex-conjugate zeros located
at ± j2ω0 is added to the compensator. As a result, the loop gain increases. In order to
reduce the loop gain magnitude beyond the frequency of 2ω0 , a double real pole is added
at s = −2ω0 . To date, the compensator taking into account the nominal value of the grid
voltage V bsn and the gain h becomes

s2 + (2ω0 )2
 
h
H (s) = H2 (s) (18)
Vsn
b s(s + 2ω0 )2

The additional specifications pertaining to operational and stability requirements can

be seamlessly integrated into the system’s transfer function, represented by H2 (s). In the
context of this specific case study, we opt to conceptualize H2 (s) as a lead compensator,
a vital control component that possesses the ability to inject additional phase margin into
the plant system. The choice of design is strategic, as the lead compensator modifies the
phase-frequency characteristics of the system to ensure a greater phase margin, which in
turn guarantees a more stable and reliable system performance. The integration of a lead
compensator into H2 (s) thus contributes to fortifying the overall stability of the system.
By implementing this configuration, we ensure that the system maintains an ample stability
margin in relation to the gain crossover.
This is of paramount importance as it aids in averting system instability that could
stem from an increase in gain at the crossover frequency. Thus, this design ensures that
the system is capable of operating under varying conditions while preserving its stability,
thereby resulting in a more robust and dependable control system. With δm representing
the added phase margin, the lead compensator H2 (s) is constructed as follows:

s + ( p/α)
 
H2 (s) = , (19)
s+p

where √
p = ωc α,
1 + sin δm (20)
α= ,
1 − sin δm
By considering the grid and stability requirements, the compensator becomes

s2 + (2ω0 )2 s + ( p/α)
   
h
H (s) = . (21)
V
bsn s(s + 2ω0 )2 s+p

3. Differential-Algebraic Equations Modeling

Transfer functions are employed to express the model that was provided. We convert
the given model from the s-domain to the time-domain in this section. The transformation
methodology is generalizable and can be applied to various dynamical systems. We will
analyze each component of the system, and in the end, these components will be assembled
to run the system. The grid is already expressed in differential equations as in Equation (6).

3.1. Derivation of PLL DAEs

The methodology for deriving the gain parameter, h, involves assessing it within the
context of the loop gain, specifically at the crossover frequency of 200 rad/s. The equation
Sustainability 2023, 15, 8400 9 of 18

that characterizes the loop gain is set to equal 1, which corresponds to the magnitude at
the crossover frequency. By considering V bsn = 391V, the gain of H (s) is h = 2.68 × 105 .
Without considering the compensator gain and the stability requirements, i.e., hH2 (s) = 1,
the phase angle of the system loop gain at the gain crossover (ωc = 200 rad/s) is −210◦ .
To guarantee stability, 60◦ phase margin is added. This criterion can be met using a lead
compensator. As the system needs 90◦ , two identical lead compensators of an angle of 45◦
are used. Following (19), the applied lead compensator
2
s + 83

H2 (s) = . (22)
s + 482

Thus, the overall compensator is

685.42 s2 + 568516 s2 + 166s + 6889

H (s) = . (23)
s(s2 + 1508s + 568516)(s2 + 964s + 232324)
The system’s frequency response, which is illustrated in Figure 4, initially presents a
decreasing magnitude with a slope of −40 dB/dec. However, upon reaching the frequency
of approximately 200 rad/s, a transition occurs: the slope decelerates to −20 dB/dec. Si-
multaneously, the phase alters its course, transitioning from −180 degrees to −120 degrees.
This 60-degree shift can be attributed to the phase margin of the system, which plays a
crucial role in maintaining system stability and performance.
Magnitude [dB]

100

100 101 102 103 104

50
Phase [degrees]

100
150
200
100 101 102 103 104
Frequency [Hz]
Figure 4. The frequency response of the PLL.

Beyond the crossover frequency, the system gain continues its downward trajectory
with an unchanging slope of −40 dB/dec. This consistent decline in gain beyond the
crossover frequency highlights the system’s inherent ability to suppress higher-frequency
components and maintain stability under a range of operating conditions. The imbalanced
component corresponding to the frequency of 754 rad/s is effectively attenuated by the
designed PLL, thereby aligning Vq to zero. This effective attenuation underpins the efficacy
of the designed PLL, demonstrating its capability to maintain system balance and stability
despite the presence of an imbalanced component.
The single s in the denominator represents ω and as it passes the integrator in the VCO
it becomes ρ, the abc − dq transformation angle. For normal PLL operation, ω has to be
Sustainability 2023, 15, 8400 10 of 18

initialized with the grid frequency. To ease the state–space transformation, this integrator is
separated from the rest of H (s), as follows:

s2 + 568516 s2 + 166s + 6889

685.42
H (s) = × ,
(s2 + 1508s + 568516)(s2 + 964s + 232324) s
s2 + 568516 s2 + 166s + 6889



685.42
= 2 2
× ,
(s + 1508s + 568516)(s + 964s + 232324) s
(24)
s4 + 166s3 + 575.4 × 103 s2 + 943.73 × 104 s + 3.92 × 109 685.42
= 4 × ,
s + 2472s3 + 225.45 × 104 s2 + 898.39 × 105 s + 1.32 × 1011 s
n s4 + n2 s3 + n3 s2 + n4 s + n5 h
= 1 4 3 2
× .
d1 s + d2 s + d3 s + d4 s + d5 s

For simplicity, the numerator components are denoted n1 . . . n5 , and the denominator
components are denoted d1 . . . d5 . The process of the state–space transformation first
decomposes the transfer function into several integrators. Then, each integrator represents
a state of the system. Figure 5 clarifies the procedure.

n2 - d2 +

Va n3 - d3 +
abc Vd
Vb Vq n4 - d4 +
+
Vc dq - 1 1 1 1
𝜌 - n5 - d5 +
- x␒ 1 s x␒ 2 s x␒ 3 s x␒ 4 s
-

d2

d3

d4

d5
VCO Limiter
1 𝜔 1
h
s x␒ 6 s x␒ 5

Figure 5. PLL transformation description.

The system states are revealed by breaking down the PLL compensator’s transfer
function, H (s). Initializing the angular frequency, ω, is required for PLL operation, as was
previously explained. For this model, ẋ5 represents ω. Given that the system’s frequency is
60Hz, the initialization of ẋ5 is performed by 60 × 2π rad/s. Furthermore, to guarantee the
normal performance of the PLL, ω must be constrained so that the transformation angle
does not highly deviate from the grid angle. The transformation angle, θ, is represented by
the state ẋ6 . For that, the integrator, which is indicated as VCO, resets itself as it reaches 2π.
The purpose of the VCO is to achieve physical meaning, which is the circular rotation of
the angle. The state–space form of the PLL

x1 −d2 −d3 −d4 −d5 0 0 x1 Vq

      

 x2  
  1 0 0 0 0 0 
 x2  
  0 

d  x3  
 = 0 1 0 0 0 0 
 x3  
+ 0 
,
dt  x4   0 0 1 0 0 0  x4   0 
(25)
      
 x5   (n2 − d2) (n3 − d3) (n4 − d4) (n5 − d5) 0 0  x5   0 
x6 0 0 0 0 h 0 x6 0
ẋ5 (0) = ω0 ,
ωmin ≤ h × ẋ5 ≤ ω max .
Sustainability 2023, 15, 8400 11 of 18

While the model can be run to operate by itself as an individual system, it can be extended
to different grid configurations. For both cases, the input is abc-frame and the output is the
regulated dq-frame.

3.2. Derivation Grid-Following Inverter DAEs

To reveal the state of feed-forward and feedback compensators, their transfer functions
are expanded, as shown in Figure 6. The states x7 and x8 belong to feed-forward compen-
sators. The states x9 and x10 refer to the states of the feedback compensators. The states x11
and x12 represent the inductor of the grid. The state–space model follows

−1
 Vsd 
0 0 0 0 0
 
gf gf
x7 x7
   
−1
 Vsq 
  0 0 0 0 0
 

 x8   gf 
 x8  
  gf


d x9   0
 = 0 0 0 − Ki 0  x9  
+ Ki idre f

. (26)
  
dt 
 x10   0
  0 1 0 0 − Ki 
 x10  
  Ki iqre f
 
x11 −(K p + Rt ) x11   −(Vsd −K p idre f )
  1 
1 
0 0 0
 
 L L L

L
x12 x12
 
1 1 −(K p + Rt ) −(Vsq −K p iqre f )
0 L 0 L 0 L L

1 1
s gf Vsd
Kp x␒ 7 -
ron + R

- Vtd - -
idref 1 md 1 1
id
Ki
s
/ × L s
x␒ 9 x␒ 11
-
L𝜔0
id L𝜔0 VDC
L𝜔0 2
iq L𝜔0
mq Vtq -
iqref Ki
1
/ ×
1
L
1
iq
x␒ 10 s
- x␒ 12 s
- -

1 1
Kp ron + R
s
x␒ 8
gf Vsq
-

Figure 6. Control model transformation description.

4. Numerical Integration Methods

Dynamical systems that are characterized by differential-algebraic equations can be
resolved and simulated through three prominent numerical integration methods: the
Forward Euler Method (FEM), the Backward Euler Method (BEM), and the Implicit Trape-
zoidal Method (ITM). Each of these methods possesses its unique set of advantages and
limitations; hence, the choice of an appropriate method should be tailored to align with the
specific attributes and requirements of the application at hand. The standard form of the
explicit ordinary differential equation is

ẋ = ( x, u), (27)

where x and u denote the state and the input of the system, respectively. FEM is a first-order
technique, and the process of its integration is straightforward. The state step is mainly
based on the preceding state step. FEM is generally formulated as follows:
 
x (i+1) = x (i) + ∆t f x (i) , u(i) , (28)
Sustainability 2023, 15, 8400 12 of 18

where x (i) is available from the prior step and ∆t is the time step. Despite its reputation for
simplicity and stability, FEM is the least precise method.
BEM is an implicit method and is known for its high stability. BEM integrates the
previous state with the current state, which is attained by numerical approaches, such
as the Newton method if the system contains nonlinear components. The BEM standard
form is  
x (i+1) = x (i) + ∆t f x (i+1) , u(i+1) . (29)

As previously mentioned, BEM is particularly useful when the system is nonlin-

ear, which is a fundamental property of models of power systems. The following is the
implementation of the Newton technique on the BEM. The standard form is set as
 
0n,1 = x (i) + ∆t f x (i+1) , u(i+1) − x (i+1) , (30)

which is equivalent to
 
0n,1 = h x (i) , x (i+1) , u(i+1) , ∆t = h(i+1) . (31)

The k-th iteration of the Newton solving process is

( i +1) ( i +1) ( i +1)

x ( k +1) = x ( k ) − H ( k ) h ( i +1) , for k = 1, 2, . . . , (32)

where
( i +1) ∂h(i+1) ∂ f ( i +1)
H(k) = ( i +1)
= ∆t ( i +1)
− In . (33)
∂x(k) ∂x(k)
( i +1)
The initial value of the state iteration is x(0) = x (i) .
The implicit Trapezoidal Method (ITM) combines the BEM and the FEM to provide
excellent stability and precision in the calculation. The ITM is attained by combining the
FEM and BEM as follows:
1   1  
h(i+1) = x (i) + ∆t f x (i) , u(i) + ∆t f x (i+1) , u(i+1) − x (i+1) , (34)
2 2
and
( i +1) ∂h(i+1) 1 ∂ f ( i +1)
H(k) = ( i +1)
= ∆t − In . (35)
∂x(k) 2 ∂x (i+1)
(k)

The ITM is one of the most trustworthy and efficient techniques. In fact, it is the solver of
preference for commercial and noncommercial power system software products. The ITM
standard form is
∆t  (i+1) (i+1)  ∆t  (i) (i) 
x ( i +1) = x ( i ) + f x ,u + f x ,u . (36)
2 2

5. Case Study
5.1. Robust PLL Performance under Voltage Imbalance Conditions
In the design of modern power systems, an essential consideration is the ability to
maintain a stable operation in the presence of voltage imbalances. This case study presents a
thorough evaluation of the proposed PLL model (26), demonstrating its robust performance
under various conditions, including the presence of voltage imbalances. The PLL model,
grounded in the natural behavior of the operating system, is specifically designed to address
the challenges posed by imbalanced components, a common occurrence in distribution
systems. A comprehensive simulation is conducted, assessing the model’s response to
three distinct scenarios, as illustrated in Figure 7.
Sustainability 2023, 15, 8400 13 of 18

Initially, the PLL operates under ideal conditions, with pure voltage composed solely
of the fundamental component. This baseline evaluation serves to validate the model’s
performance when no imbalances are present. Following this, at t = 0.02 s, an imbalance
is introduced to the system to assess the PLL’s ability to adapt and maintain accurate and
stable operation. The simulation results show that the PLL continues to function effec-
tively, with the distortion caused by the imbalance manifesting primarily in the direct- and
quadrature-axis voltage components, Vd and Vq . This observation confirms the model’s
ability to handle voltage imbalances while producing minimal distortion in the output
voltages. Finally, at t = 0.08 s, the imbalance is removed from the system, allowing for
the evaluation of the PLL’s response to the restoration of balanced conditions. The sim-
ulation demonstrates that the PLL promptly returns to smooth operation, reaffirming its
adaptability and resilience in the face of varying operating conditions.

500
Vabc

−500

391
Vdq

377
ω

2π
ρ

0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
Time (s)

Figure 7. Illustrating the performance of the designed PLL model under various conditions, including
the presence and removal of voltage imbalances. The first subplot illustrates the three-phase voltage
signals, with the imbalance component evident in the blue phase. The second subplot represents the
Vd and Vq variables, depicted in blue and orange, respectively. The third subplot reveals the grid’s
angular frequency, while the fourth subplot discloses the detected grid’s angle.

The case study under consideration was also subjected to tests involving the or-
dinary Phase-Locked Loop (PLL) to evaluate its performance and effectiveness under
circumstances of imbalanced operational conditions. This meticulous examination was
orchestrated to assess the resilience of the PLL when it encounters conditions that veer
from the balanced state, thus testing the robustness of its design. The outcomes of the
simulation are visually represented in Figure 8. The graphic illustration provides a clear
representation of the system’s behavior during and after the intrusion of the imbalanced
component. A careful inspection of the figure reveals that the performance of the PLL
begins to exhibit signs of deterioration immediately following the onset of the imbalanced
component. This indicates that the system’s stability and efficiency are adversely affected
by such disruptions.
In the specific context of operations under the influence of the imbalanced component,
it is noteworthy that the PLL encounters difficulty in accurately detecting the fundamental
Sustainability 2023, 15, 8400 14 of 18

frequency of the grid. This is a significant observation as it highlights the limitations of the
PLL under non-ideal conditions. Furthermore, it is observed that, once the imbalanced
component is eliminated from the system, the PLL is unable to revert to its normal operation.
This suggests a lack of adaptive recovery mechanisms in the PLL to restore balance and
resume normal functioning after the system experiences instability. This inability to self-
correct and re-establish normal operating parameters underscores the need for further
improvements in the design and control strategy of the PLL to enhance its resilience to
imbalances and other operational anomalies.

500
Vabc

−500

391
Vdq

377
ω

360

2π
ρ

0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
Time (s)

Figure 8. The performance of the ordinary PLL model under various conditions, including the
presence and removal of voltage imbalances. The first subplot displays the three-phase voltage
signals, wherein the blue phase contains the imbalance component. The second subplot illustrates
the Vd and Vq signals, represented in blue and orange, respectively. The third subplot exhibits the
grid’s angular frequency, while the fourth subplot presents the detected angle of the grid.

5.2. Grid-Following Inverter

In this study, a grid-following Inverter-Based Resource (IBR) is employed to evaluate
the effectiveness of the proposed approach. To establish a reliable comparison, parameters
from [23] are adopted, as presented in Table 1. The system simulation is implemented using
the Implicit Trapezoidal Method, a numerical integration technique discussed in a previous
section. The Python-coded simulation results are illustrated in Figure 9.
Figure 9 comprises several subplots, each representing various aspects of the inverter’s
performance. The first subplot highlights the voltage at the AC side of the inverter, specif-
ically at the Point of Common Coupling (PCC), which connects the inverter to the grid.
As expected, during normal operation, the three-phase grid voltage remains balanced.
The second subplot depicts the dq-frame voltage corresponding to the three-phase grid
voltage. The Phase-Locked Loop (PLL) accurately transforms the abc-frame to the dq-frame,
as evidenced by the plot. The simulation of this part is modeled using Equation (25).
Sustainability 2023, 15, 8400 15 of 18

Table 1. Model parameters.

Parameter Value
ω0 377 rad/s
Vs d 391 V
L 100 µL
R 0.75 Ω
ron 0.88 Ω
VDC 1250 V
fs 3420 Hz
gf 8 × 10−6
τi 2 ms
kp 0.05
ki 0.815

391
Vabc

−391
391
Vdq

377
ω

2π
ρ

1000
id ,idref

2000
iq ,iqref

0
0.1 0.2 0.3 0.4 0.5
Time (s)

Figure 9. The simulation of an inverter connected to the grid with changes in the references. The
first subplot illustrates the three-phase voltage signals, with the imbalance component present in the
blue phase. The second subplot represents the Vd and Vq variables, indicated in blue and orange,
respectively. The third subplot displays the grid’s angular frequency, while the fourth subplot
discloses the detected grid’s angle. The fifth subplot depicts id and its control signal, and, the sixth
subplot exhibits iq and its associated control signal.

The third subplot illustrates the grid angular frequency measured by the PLL. In the
state–space model, the angular frequency is denoted by x5 . The grid angle, represented by
x6 , is computed by integrating the limited angular frequency. While regular integrators
can perform this integration, resettable integrators are necessary to restrict the angle range
between 0 and 2π.
The fifth subplot demonstrates the inverter current aligned to the d-axis and its
reference. The active power, as explained in Equation (8), is related to this current and
can be primarily regulated using it. To assess the effectiveness of the control applied to
id , the reference current, idre f , starts with a value of 0 and is changed to 1000 at t = 0.1 s.
As observed in the plot, the response of id adequately follows its reference.
Sustainability 2023, 15, 8400 16 of 18

Similarly, the sixth subplot shows the inverter current that is aligned to the q-axis and
its reference. The reactive power is associated with it, iq , as in Equation (8). Furthermore, its
response is tested to validate the system control. The reference current, iqre f , starts with a
value of 0, and is then changed at t = 0.2 s to 2000. As displayed, the response appropriately
changes and matches its reference.

5.3. Grid-Following Inverter under Grid Voltage and Frequency Variations

To evaluate the robustness of the system under investigation, we subjected it to a
series of stress tests designed to mimic abnormal conditions which may realistically occur
in an actual power system. This approach was undertaken with the intent of assessing how
well the system could maintain its operational integrity under unexpected circumstances,
a critical aspect of real-world system resilience. Among the various adverse conditions
considered, we focused on two key events that frequently transpire in power grids: drops
in voltage and frequency. These scenarios were carefully incorporated into our case study,
the results of which are visually represented in Figure 10.

391
Vabc

−391

391
Vdq

377
ω

200
id ,idref

50
iq ,iqref

0
0.1 0.2 0.3 0.4 0.5
Time (s)

Figure 10. Simulation with voltage and frequency change events that occur in the grid. The first
subplot shows the three-phase voltage signals, with the imbalance component appearing in the
blue phase. The second subplot displays the variables Vd and Vq , represented in blue and orange,
respectively. The third subplot illustrates the grid’s angular frequency. The fourth subplot presents
the detected angle of the grid. In the fifth subplot, we can see id and its control signal and the sixth
subplot exhibits iq and its corresponding control signal.

In the conducted simulation, the inverter initially operates under normal conditions.
However, at t = 0.08 s, we introduced a disruption by reducing the grid voltage to 85% of
its nominal value. This sudden drop persisted until 0.2 s, at which point we allowed the
grid to recover to its nominal voltage. The response of the designed PLL to this sudden
change was prompt and effective. The PLL swiftly tracked the alteration in grid voltage
and mirrored it in its DC values, thereby demonstrating its ability to respond to voltage
variations. As depicted in Figure 10, the PLL was also capable of restoring normal operation
in sync with the grid recovery, further illustrating its adaptability and resilience.
Sustainability 2023, 15, 8400 17 of 18

Next, at t = 0.3 s, we implemented another disruption by altering the grid frequency

from its standard 60 Hz to 59.95 Hz. This frequency variation lasted for a brief duration of
0.01 s. The impact of this frequency change was observable in the PLL output, primarily af-
fecting the Vq component. Despite these disturbances, the PLL demonstrated its robustness
by swiftly regaining its normal state after a recovery period of 0.2 s. This rapid recovery
underscores the PLL’s ability to maintain system stability and performance even under
challenging conditions, thus confirming its efficacy and robustness in the face of real-world
grid disturbances.

6. Conclusions
The penetration of IBRs that interface renewables and DERs changes the power system
infrastructure. The importance of modeling can never be overstated. Transfer functions are
frequently used to model the grid connection of IBRs. On the other hand, this paper models
it with differential-algebraic equations which can be converted into state–space form.
The state–space modeling is superior to the transfer function modeling in addition to being
in the time domain. Multiple-input, multiple-output systems, and time-variant systems can
both be handled by the state–space approach. State–space models can also be employed to
implement Kalman filters. We also develop the state–space model of a PLL customized to
address imbalanced components. Reaching improved methods of analysis and simulation
is the purpose of achieving a different sort of modeling. Furthermore, preparing the system
for the deployment of state–space optimal control strategies is another objective of the grid-
following IBR state–space model development. By employing primary integration methods
capable of addressing nonlinear systems, our proposed model has been numerically solved,
demonstrating a precise correspondence between the transfer function model and the
state–space model. Through the case studies presented, the validity and robustness of the
state–space approach have been further substantiated, providing a strong foundation for
future research and development in the field of power electronics and grid integration.

Author Contributions: Conceptualization, A.A. (Abdullah Alassaf) and I.A.; methodology, A.A.
(Abdullah Alassaf) and I.A.; software, A.A. (Abdullah Alassaf) and I.A.; validation, A.A. (Ayoob
Alateeq) and H.A.; writing—original draft preparation, A.A. (Abdullah Alassaf); formal analysis,
A.A. (Ayoob Alateeq) and I.A.; investigation, A.A. (Ayoob Alateeq) and H.A.; writing—review and
editing, A.A. (Ayoob Alateeq) and H.A.; project administration, I.A.; funding acquisition, I.A. All
authors have read and agreed to the published version of the manuscript.
Funding: This research has been funded by Scientific Research Deanship at University of Ha’il, Saudi
Arabia through project number BA-2110.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data used to perform the case studies are available upon request from
the corresponding author.
Conflicts of Interest: The authors declare no conflict of interest.

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