# Applications of Kepler Algorithm-Based Controller for DC Chopper: Towards Stabilizing Wind Driven PMSGs under Nonstandard Voltages

Basiony Shehata Atia, Mohamed Metwally Mahmoud, I. Elzein, Abdel-Moamen Mohamed Abdel-Rahim, Abdulaziz Alkuhayli, Usama Khaled, Abderrahmane Beroual, Salma Abdelaal Shaaban

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Basiony Shehata Atia, Mohamed Metwally Mahmoud, I. Elzein, Abdel-Moamen Mohamed Abdel-Rahim, Abdulaziz Alkuhayli, et al.. Applications of Kepler Algorithm-Based Controller for DC Chopper: Towards Stabilizing Wind Driven PMSGs under Nonstandard Voltages. Sustainability, 2024, 16 (7), pp.2952. 10.3390/su16072952. [https://doi.org/10.3390/su16072952]

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# Applications of Kepler Algorithm-Based Controller for DC Chopper: Towards Stabilizing Wind Driven PMSGs under Nonstandard Voltages

Basiony Shehata Atia ¹D, Mohamed Metwally Mahmoud <fnref n="1" />,*D, I. M. Elzein <fnref n="2" />, Abdel-Moamen Mohamed Abdel-Rahim ¹D, Abdulaziz Alkuhayli ³D, Usama Khaled ¹D, Abderrahmane Beroual ⁴D and Salma Abdelaal Shaaban <fnref n="1" />

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**Citation:** Atia, B.S.; Mahmoud, M.M.; Elzein, I.M.; Mohamed Abdel-Rahim, A.-M.; Alkuhayli, A.; Khaled, U.; Beroual, A.; Shaaban, S.A.
Applications of Kepler Algorithm-Based Controller for DC Chopper: Towards Stabilizing Wind Driven PMSGs under Nonstandard Voltages.
*Sustainability* **2024**, *16*, 2952. https://doi.org/10.3390/su16072952

Academic Editor: Javier Serrano
González

Received: 25 February 2024

Revised: 19 March 2024

Accepted: 28 March 2024

Published: 2 April 2024

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**Copyright:** © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

**Abstract:** An optimization technique, the Kepler optimizer (KO), is presented to enable permanent magnet synchronous wind generators (PMSWG) to run safely under faults and to accomplish the goal of low-carbon efficient power delivery and sustainable development. Utility companies are struggling, which is preventing the increase in wind penetration, in spite of the grid incorporation of PMSWG. One of these undisputed concerns is the grid-side voltage dip (VD) and swell (VS) at the PCC. Converters and DCL capacitors are particularly vulnerable to PCC nonstandard voltages because of an imbalance in the DCL input–output powers. Because of this, it is essential to provide WF-GCs to support grid operations, and developing techniques to realize FRTCs has become a crucial GC need. Installing an industrial braking chopper (BC) across the DCL is the suggested technique, due to its effectiveness and low price. In addition, a new KO-based control system for BC is used to enhance its effectiveness. Four situations were examined to assess and analyze the proposed control system regarding the transient response of the system. These situations exposed the investigated system to an irregular grid condition: without BC, with BC controlled by a hysteresis controller, and with BC controlled by KO-based PI (proposed) at (a) 100% VD, (b) 70% VD, (c) 30% VD, and (d) 20% VS. To verify the advantages and efficacy of the suggested control systems in the examined circumstances, MATLAB/SIMULINK was utilized. The simulation findings confirmed the feasibility of the suggested system as a whole and the control structures in suppression of all parameter transient changes, while also achieving FRTC. Furthermore, maintaining a steady DCL voltage serves as an advantage that would lengthen the electrical converters' lifetime and shorten the time that the unit would be turned off if it happens to fail.

**Keywords:** braking chopper; DC bus voltage control; Kepler optimizer; sustainable development; unbalanced voltages; wind energy

## 1. Introduction

The demand for energy in the world today exceeds the capacity of the accessible supply. Forecast energy needs therefore need to be efficiently and affordably met. When implementing energy solutions, the usage of sustainable sources (SS) ought to come first. The current worldwide share of SSs in principal energy is not high enough to meet the world's main energy and power needs [1,2]. It seems like a lot of emerging countries are trying to improve their energy industries. It can, however, be challenging to make these discoveries a reality [3]. Sources must provide adequate energy to meet human

<footnote n="1">¹ Electrical Engineering Department, Faculty of Energy Engineering, Aswan University, Aswan 81528, Egypt</footnote>

<footnote n="2">² Department of Electrical Engineering, College of Engineering and Technology, University of Doha for Science and Technology, Doha P.O. Box 24449, Qatar; 60101973@udst.edu.qa</footnote>

<footnote n="3">³ Department of Electrical Engineering, College of Engineering, King Saud University, Riyadh 11421, Saudi Arabia</footnote>

<footnote n="4">⁴ AMPERE Lab UMR CNRS 5005, Ecole Centrale de Lyon, University of Lyon, 36 Avenue Guy de Collongue, 69130 Ecully, France</footnote>

<footnote>* Correspondence: metwally_m@aswu.edu.eg</footnote>
needs, including those for running cars, heating houses, and powering cities. Nonetheless, assessing the long-term usage of these assets is just as important. There is no end to SSs. SSs lower pollution and greenhouse gas emissions—two more factors that contribute to climate change [4–6]. SSs are natural resources including biomass, geothermal resources, sun, hydropower, wind, and the ocean that can be converted into clean, useable energy. In [7], the merits and drawbacks of various SSs were summarized in a table. To establish a better tomorrow for everybody, the United Nations created the sustainable development goals (SDGs). These objectives were set in 2015, and full adoption is expected to be completed by 2030 [8]. The SDGs’ architecture makes it possible to monitor the agenda 2030’s execution and advancement [9].

Regarding the SDGs, the growth of SSs makes energy safety possible in a number of areas [10]. In addition to helping with global preservation and building communities, SSs support energy supply demands. SSs have been promoted as a workable way to solve the energy problem in the twenty-first century, while preventing detrimental effects on the atmosphere and climate [11,12]. The production of wind energy has benefits over other SSs due to its cutting-edge technology, stable structures, and low total cost [13]. In the coming years, wind energy is likely to contribute more to the global energy supply [14].

As WFs have become more widespread in interconnected power systems (PSs), various nations have created their own forms of interconnectedness guidelines and requirements for PSs combination of WFs, referred to as GCs [15,16]. GCs require WFs to maintain PS connection during breakdowns and to add an appropriate level of reactive power (Q) into the PCC at VDs [16,17]. After clearing the problem, the WFs must supply the PS with active power (P), following the GC. The Danish GC shown in Figure 1, states that WFs with a nominal capacity over 11 kW must maintain grid connectivity for 0.5 s in the event of PCC VD up to 80% (Area B in Figure 1a) [18]. To keep the PCC running for this brief period, the WF must inject reactive power (Figure 1b) [19]. Accordingly, reliant on the machine type and PS incorporation method, the WF may have various problems as a result of the execution of the GC. Among the various types of WFs available today, PMSWG is acknowledged as a common choice. Full-scale BTB converters (BTBCs) are used to connect the PMSWG to the grid [14]. This gives the generator some advantages, including control of Q tit, better tracking of the MPP, improve GS power quality, and more.

<figure><img src="image_4.png" /><figcaption>**Figure 1.** GC necessities that consist of (a) ride-through curve, and (b) support curve of Iq.</figcaption></figure>

While the WF generator continues to produce electricity during GS faults, the P supplied to the PS decreases as a result of both the GS-VD at PCC and the GSC’s current limiter [20]. As a result, during grid VDs, the converter’s income and outcome P is out of
balance, which leads to DC-link (DCL) over-voltage [21,22]. To achieve and improve the
FRTC of WFs, numerous strategies have been devised.

These techniques can be divided into two primary categories: (a) using external apparatus, and (b) improving pitch angle controllers and BTBCs. Each of these approaches has advantages and disadvantages, and they were discussed in [18]. A recent way of transferring the (GSC\MSC) control roles was suggested in [20]. Although this method does not need any external equipment to execute FRTC and appears to operate well for all forms of VDs, the authors of [23] demonstrated that adopting such tactics may lead to subpar performance. Even though technique (b) has received favor owing to its ease of use and affordability, it necessitates changes to the converters' and the turbines' pitch angle controls. As a result, technique (a) is considered a solid GC-compliant strategy for current WFs.

The setting up of a BC across the DCL of the BTBC is among the simplest and most
affordable actions [24], in spite of the widely used FRTC structures in (a) being complex
and expensive; for example, the positioning of FACTS and ESS [25]. One advantage of this
method is that BC exists in all kinds of WFs and can be employed as a transient and instant
store load whenever needed. Additionally, system operation was examined in [26–28] with
BC inclusion. In addition to being used in WFs, this component has additional uses, such as
in FRT implementation for solar systems [29] and AFD [30]. A list of BC engineering used
in (a) WFs and (b) AFD was presented in [31]. The BC can be managed through a variety
of control strategies. In Section 3 of this study, several control mechanisms are discussed,
together with their benefits and drawbacks.

The literature discusses a variety of techniques for managing PMSWG. For instance, reference [32] controlled BTB converters (BTBCs) during GS disturbances using a Lyapunov-based model prediction technique. This model had a reasonable transient response in DCL voltage (DCLV) regulation when its behavior was contrasted to that of a PI regulator. However, it was impossible to assess how the network performed in deep VDs because converter current limiters were absent. The MSC and GSC converters were controlled by an adaptive fuzzy RST approach in [33], and the usage of a high-selectivity filter under asymmetrical faults was analyzed. The SMC is one of the control strategies that can substantially guarantee resilience [34]. The GSC can fulfill GC requirements for feeding Q into the PS with the help of this reliable control structure, which also keeps the DCLV at its standard value and provides the appropriate normal and abnormal actions for following the desired DCLV. The SMC was utilized to manage WFs under faults. Two PMSGWs were coupled to a single DCL in [35]. The common DCL in this formation was linked to the PS by a DC-AC converter. All of these 'converter controllers were created and put into use using the SMC technique to carry out FRT. The existence of chattering in the converters' (V&I) indications is one of this method's drawbacks, and this was solved by a super-twisting SMC approach [36] and the ERL methodology [37].

The shortcomings of current BC control methods are overcome using the suggested controller (see Section 4). It is important to note that the suggested BC regulator is appropriate for any BCs deployed via a BTBC's DCL. Consequently, it works with BTBC control schemes. Moreover, this construction is not restricted to only one GC and can be matched with the GC of various nations. Of course, the Danish GC was employed as a reference GC in this investigation.

This work is organized as follows: After the introduction section, the Materials and Methods are given in Section 2, which includes a brief explanation of the PMSWG construction and model; a summary of current BC control procedures; and the proposal and design of a reliable BC controller including KOA and its application. Section 3 gives the simulation findings and analysis. Finally, Section 4 offers this study's closing observations.
## 2. Materials and Methods

### 2.1. PMSWG Description & Modeling

The PMSWG model is divided into four key components (Figure 2) [38]: (1) mechanical section; (2) equivalent electric circuit in SRF<sub>MSC</sub>; (3) BTBC, DCL, BC, and the machine Eqs., in the SRF<sub>MSC</sub> are shown in (1)–(6) [38]. The following equation symbols are fully described in [38].

$$P_m = 0.5\rho A_{tur}C_p(\lambda, \beta)v_w^3 \quad (1)$$

$$\frac{d\omega_m}{dt} = \frac{1}{J_{eq}}(T_{Tur} - T_e - B_{eq}\omega_m) \quad (2)$$

$$T_{Tur} = 0.5\rho A_{tur} \frac{C_p(\lambda, \beta)}{\lambda} v_w^2 \quad (3)$$

$$\left\{ \begin{array}{l} V_{ds} = R_s i_{ds} + L_s \frac{di_{ds}}{dt} - \omega_e L_s i_{qs} \\ V_{qs} = R_s i_{qs} + L_s \frac{di_{qs}}{dt} + \omega_e L_s i_{ds} + \omega_e \emptyset \end{array} \right. \quad (4)$$

$$P_{gen} = \frac{3}{2} \frac{p}{2} \emptyset |i_{qs}| \omega_m \quad (5)$$

$$\tau_e = \frac{3}{2} \frac{p}{2} \emptyset i_{qs} \quad (6)$$

<figure><img src="image_5.png" /><figcaption>Figure 2. Addressed system.</figcaption></figure>

The pertinent equations for the GS electrical components and the BTBC-DCL are provided in (7) and (8). The BC-P loss flexibility only occurs in (7) when there is a fault.
Notably, the GS (+\- sequence) electrical parts were modeled using the well-known SRF<sub>GSC</sub> approach [20].

$$ \left\{ \begin{aligned} \frac{d}{dt} \left( \frac{CV_{dc}^2}{2} \right) &= P_{gen} - P_{grid} && \text{(under normal conditions)} \\ \frac{d}{dt} \left( \frac{CV_{dc}^2}{2} \right) &= P_{gen} - P_{grid} - P_{BC} && \text{(under grid faults)} \end{aligned} \right. \qquad (7) $$

$$ \left\{ \begin{aligned} E_{dq}^{+} &= R_f I_{dq}^{+} + L_f \frac{dI_{dq}^{+}}{dt} + j\omega_f L_f I_{dq}^{+} + V_{dq}^{+} \\ E_{dq}^{-} &= R_f I_{dq}^{-} + L_f \frac{dI_{dq}^{-}}{dt} - j\omega_f L_f I_{dq}^{-} + V_{dq}^{-} \end{aligned} \right. \qquad (8) $$

### 2.1.1. Control Scheme of MSC

Two dq-axes make up the MSC controller in the SRF<sub>MSC</sub>, as shown in Figure 2b. MPPT and generator power factor management are the MSC’s two primary control goals. To cut down on power losses and stop the PMSWG from demagnetizing, the I<sub>d</sub> set-point is adjusted to zero [18]. Additionally, MPPT is performed using the OTC approach, from which the desired q-axis current is generated [25].

### 2.1.2. Control Scheme of GSC

The SRF<sub>GSC</sub> has a double current regulator installed to control the (+\-sequence) currents (Figure 2c). The method in [18] is used in this paper to separate the sequences, because the behavior of SRF-PLL and the separation of (I/V) (+\-sequence) is extremely successful in increasing the recital of the GSC control system. The second-order harmonics in the DCLV and the P supplied to the PS during failures are eliminated using this kind of controller. This is in addition to the grid’s Q supply and the management of the DCLV. Both of P<sub>0</sub><sup>ref</sup> and Q<sub>0</sub><sup>ref</sup> were calculated as described in [20]. The desired I<sub>dq</sub> (+\-sequence) was derived using (9) [39]. To improve control the DCLV, P<sub>0</sub><sup>ref</sup> was determined at each instance.

$$ \begin{bmatrix} I_{d-ref}^{+} \\ I_{q-ref}^{+} \\ I_{d-ref}^{-} \\ I_{q-ref}^{-} \end{bmatrix} = \frac{2}{3} \begin{bmatrix} V_{df}^{+} & V_{qf}^{+} & V_{df}^{-} & V_{qf}^{-} \\ V_{qf}^{+} & -V_{df}^{+} & V_{qf}^{-} & -V_{df}^{-} \\ V_{qf}^{-} & -V_{df}^{-} & -V_{qf}^{+} & V_{df}^{+} \\ V_{df}^{-} & V_{qf}^{-} & V_{df}^{+} & V_{qf}^{+} \end{bmatrix}^{-1} \begin{bmatrix} P_{0}^{ref} \\ Q_{0}^{ref} \\ 0 \\ 0 \end{bmatrix} \qquad (9) $$

### 2.2. Review of BC Control Methods

A DC chopper and resistor make up the BC’s two primary parts (Figure 2a). The resistance enters the circuit and consumes extra DCL energy to keep its voltage within acceptable bounds when the voltage rises above its threshold value. Various techniques have been suggested for regulating a BC and determining its duty cycle (D). To calculate the modulation index (m) and ultimately the D, [33,40], the modification in the DCLV and its threshold can be fed into a PI regulator (Figure 3a) [41]. The DCLV was compared to its threshold value by the authors of [42], and the output signal was fed into the gate pulse (Figure 3b) [41]. The goal of each of these controllers was to keep the DCLV at a fixed level, thus the GSC ability is not available for the Q fed. However, neither of these controllers is appropriate for usage during VDs, to mitigate the unbalance caused by VDs.

Ref. [43] discussed turning on the BC via a GSC current phasor (CP) signal. They asserted that the GSC functions to keep the P sent into the PS at a steady level during a VD. As a result, the amplitude of the CP supplied to the PS dramatically increases due to the VD. The experts claimed that another shortcoming of this approach was the possibility of fluctuations in the regulator’s behavior. Additional possible disadvantages are as follows: (1) ineffective operation when wind speed changes and modifies the value of the PS-fed CP; and (2) failure to consider a current limiter block for the converter controllers. In light of (1)
and (2), more studies were therefore thought to be required in order to accurately assess the efficacy of their control technique.

<figure><img src="image_6.png" /><figcaption>Figure 3. BC control structure.</figcaption></figure>

By dividing the variations in the P of the PMSWG, grid, and ESS by the insignificant BC power, the authors in [44,45] were able to calculate the m and, therefore, the D (Figure 3c) [41]. The ESS was implemented across the DCL in all of the aforementioned articles. The main problem with this strategy is either time postponements\delays in the BC turning on or postponements in the regulator's operational. Due to the lack of control, the DCLV is maintained at greater amplitudes and is therefore unable to maintain the desired value. The regulator's only function is to maintain the DCL's income-outcome P in balance.

The output signal (m) is expressed in Equation (10) as follows:

$$m = \frac{P_{gen} - P_{grid} - P_{ESS}}{\frac{V_{dc}^2}{R_{BC}}} \quad (10)$$

The BC approach has been integrated with other ways to improve the performance of PMSGWs during FRT execution in many studies, including in [46]. Such techniques that combine the BC system for the PMSWG pitch angle, ESS, STATCOM, and improved BTBC are shown in Table 1. The two parameters examined to make comparisons between these combinations are the ability to inject Q into the PS and the regulation of the DCLV. It is clear that very few articles examined the grid's Q injection. The combination in many of these situations, including the ones stated above, results in the imposition of extra costs on WF stakeholders. Therefore, enhancing the BC control system or combining the BC technique with improved BTBC controllers is an appealing and economical FRTC augmentation option in a sector that strives to reduce costs.

<table><caption>Table 1. A summary of the various controls that PMSWG uses in conjunction with the BC technique.</caption><thead><tr><th rowspan="2">The Ability of a Combination of Control Systems</th><th colspan="4">Control System Mutual with BC</th></tr><tr><th>Pitch Regulator</th><th>ESS Regulator</th><th>STATCOM Regulator</th><th>BTBC Regulator</th></tr></thead><tbody><tr><td>DCLV control</td><td>[47,48]</td><td>[48,49]</td><td>[50,51]</td><td>[45,48]</td></tr><tr><td>Q injection</td><td>[47]</td><td>[49]</td><td>[50,51]</td><td>[45,48]</td></tr></tbody></table>

In fact, the implementation of an altered GSC control system (GSCCS) [20,52] improved the operation of the PMSWG in this research. This was in addition to the investigated proposal of an improved BC regulator and its evaluation in comparison to traditional BC regulators, which is the main topic of this article. In place of generator speed measurements, the GSCCS update required determining the GSC desired P using DCLV measurements. The implemented GSCCS can perform a Q injection into the PS that complies with WF GC
in fault scenarios. In reality, the PCC voltage asymmetry is reduced by using this control approach through supplying asymmetrical current to the grid (further description in [20]).

In addition, an industrial BC was further examined in this paper to obtain insights into the BC regulator and its operation. The majority of the BCs made for WFs nowadays have six switches utilized to set the BC (Figure 4) [31]. When it comes to the quantity of the immediate load that needs to be removed and the DC resistor (DCR) thermal time constant (TTC), the first three switches (S1, S2, and S3) were set in accordance with the pertinent tables. While S5 was set in relation to the grid voltage, the (S4) was adjusted in line with the switching speed and function of the DC chopper. In actuality, the immediate standards of the load and the TTC of the DCR were required for setting the BC. The last switch (S6) was used for communication and was programmed with a specific number for each BC. It is possible to install eight BC circuits inside of a converter (for more details, see [31]). It is interesting that the BCs used in the industry nowadays rely on measurements of the DCLV. Consequently, they share the same shortcomings as BCs controlled by DCL voltage. The BC thermal capacity was fully described in [31].

<figure><img src="image_7.png" /><figcaption>Figure 4. Archetypal BC situation switches cast-off via the DCL.</figcaption></figure>

## 2.3. Proposed BC Control System

### 2.3.1. Applied Kepler Optimization Algorithm (KOA)

The KOA is based on Kepler’s rules of planetary gesticulation. During the optimization process, each planet serves as a candidate solution that is restructured at random in comparison to the sun. The KOA uses a set of preliminary objects including a stochastic ring road to start the search process. Initially, each object is located in a random orbit. The KOA evaluates the fitness of the innovative set and iterates till the expiry state is met. In this study, “time” was utilized instead of the term “iteration”. The KOA is described in six steps later in this section [53]. The flowchart of the KOA is detailed in full in Figure 5.

#### Step 1: Initialization Process

For modeling step 1, the following equation was used [54]:

$$ \vec{X}_{i,j}(0) = r_1 \times \vec{X}_{j,up} + \vec{X}_{j,low}(1 - r_1), i = 1 : N_p; j = 1 : \text{Dim} \quad (11) $$

where the studied $\vec{X}_{i,j}$, $N_p$, $\vec{X}_{j,up}$, $\vec{X}_{j,low}$, r, and Dim parameters are defined in [53].

#### Step 2: Calculating an object’s velocity

For modeling step 2, the subsequent six equations were used [54]:

$$ V_i(t) = \begin{cases} (\vec{X}_a - \vec{X}_i) \times r_4 \times H + F \times \vec{u}_2 \times (1 - R_{i-norm}(t)) \times (r_3 \vec{X}_{i,up} - \vec{X}_{i,low}) \times \vec{r}_5 & \text{if } R_{i-norm}(t) > 0.5 \\ (2 \times r_4 \times \vec{X}_i - \vec{X}_b) \rho + (\vec{X}_a - \vec{X}_b) \rho^* + F \times \vec{u}_1 \times (1 - R_{i-norm}(t)) \times (\vec{X}_{i,up} - \vec{X}_{i,low}) \times \vec{r}_5 & \text{Else} \end{cases} \quad (12) $$

where,

$$ H = \sqrt{(Ms + m_i) \times \mu(t) \left| -\frac{1}{a_i(t) + \varepsilon} + \frac{2}{R_i(t) + \varepsilon} \right|} \quad (13) $$

$$ \rho = (r_3 \times (1 - r_4) + r_4) \times \vec{u} \times H \quad (14) $$

$$ \rho^* = (r_3 \times (1 - r_5) + r_5) \times (1 - \vec{u}) \times H \quad (15) $$
$$a_i = \left[ \mu(t) \times (Ms + m_i) \times \frac{T_i^2}{4\pi^2} \right]^{\frac{1}{3}} \times r_3 \quad (16)$$

$$R_{i-norm}(t) = (R_i(t) - R_{min}(t)) / (R_{max}(t) - R_{min}(t)) \quad (17)$$

where the studied $V_i(t)$, $\vec{X}_i$, $\vec{X}_a$, $\vec{U}$, F, H, ($r_1$, $r_2$, $r_3$, $r_4$, and $r_5$), $R_{i-norm}(t)$, $\vec{X}_{i,up}$, $\vec{X}_{i,low}$, $\vec{X}_b$, $\rho$, $\rho^*$, $\mu(t)$, Ms, and $m_i$ parameters are defined in [53].

<figure><img src="image_8.png" /><figcaption>Figure 5. KOA flowchart.</figcaption></figure>

### Step 3: Escaping from the Local Optimum

The majority of solar system items spin anticlockwise around the sun, whereas some move clockwise. This characteristic is leveraged by the proposed strategy to escape resident best zones. By employing a flag F to alter the lookup path, the suggested KOA replicates this action and guarantees that agents can precisely explore the search area.

### Step 4: Updating Objects' Positions

For modeling step 4, the subsequent six equation were used [54]:

$$\vec{X}_i(t + 1) = \vec{X}_i(t) + \vec{V}_i(t) \times F + (\vec{X}_s(t) - \vec{X}_i(t)) \times \vec{u} \times (Fg_i(t) + |r|) \quad (18)$$

$$Fg_i(t) = r_4 + \mu(t) \times e_i \times \frac{\vec{M}n_s \times \vec{m}n_i}{\vec{R}n_i + \varepsilon} \quad (19)$$

$$Mn_s = r_2 \times (fit_s(t) - worst(t)) / \sum_{k=1}^{N_p} (fit_k(t) - worst(t)) \quad (20)$$

$$mn_i = (fit_i(t) - worst(t)) / \sum_{k=1}^{N_p} (fit_k(t) - worst(t)) \quad (21)$$
$$Rn_i(t) = \|X_s(t) - X_i(t)\|_2 = \sqrt{\left(\sum_{j=1}^{Dim}(X_s(t) - X_i(t))^2\right)} \quad (22)$$

$$\mu(t) = \mu_o \times e^{-(t/T_{max})\gamma} \quad (23)$$

where the studied $\vec{X}_i(t+1)$, $\vec{X}_s(t)$, $F_{G_i}(t)$, $e_i$, $\vec{M}_{n_s}$, $\vec{m}_{n_i}$, $\varepsilon$, $\vec{R}_{n_i}$, $fit_i(t)$, worst(t), $\vec{X}_b$, $\rho$, $\rho^*$, $\mu(t)$, and $\mu_o$ parameters are defined in [53].

### Step 5: Updating distance with the Sun

For modeling step 5, the subsequent two equations were used [54]:

$$\vec{X}_i(t+1) = \vec{u}_i \times \vec{X}_i(t) + (1-\vec{u}_i) \times \frac{(\vec{X}_i(t)+\vec{X}_a(t)+\vec{X}_s)}{3} + \frac{1}{e^{(r\times(1+r_4\times(a_2-1)))}} \times \left( \frac{\vec{X}_i(t)+\vec{X}_a(t)+\vec{X}_s}{3} - \vec{X}_b(t) \right) \quad (24)$$

$$a_2 = -\left(1 + \frac{t}{T_{max}}\right) \quad (25)$$

where the studied $\vec{u}_i$, $\vec{X}_i(t)$, $\vec{X}_a(t)$, $\vec{X}_s$, and $\vec{X}_b(t)$ parameters are defined in [53].

### Step 6: Elitism

For modeling step 6, Equation (26) provides a summary of this process [54].

$$\vec{X}_{i,new}(t+1) = \begin{cases} X_i(t+1) & \text{if fit} \quad (\vec{X}_i(t)) \geq (\vec{X}_i(t+1)) \\ \vec{X}_i(t) & \text{Else} \end{cases} \quad (26)$$

### 2.3.2. Proposed BC Controller Design

When developing a BC controller, it is crucial to track the DCLV transient peak precisely, as well as the DCLV desired set-point [55]. Here, the KOA-PI approach was employed for this purpose. Using the typical BC structure shown in Figure 6 and applying KCL at the DCL node yielded the following output:

$$I_{dc}^{in} - I_{dc}^{out} - I_{BR} - I_{C} = 0 \quad (27)$$

where m is expressed as in (28):

$$m = \frac{I_{BR}}{I_{BR}^{max}} = \frac{R_{BR}^{min}}{R_{BR}} \quad (28)$$

where $I_{BR}^{max}$ is the maximal current of the BR, this equal splitting of the DCLV at BC function by the overall amount of BR is deployed throughout the DCL ($R_{BR}^{min}$).

<figure><img src="image_9.png" /><figcaption>Figure 6. Configuration of the DCL in the BC-BTBC.</figcaption></figure>

By extending (27) and adding (28), we can obtain the state formula associated with the DCL in (29).

$$\frac{dV_{dc}}{dt} = \frac{1}{C} \left[ \left( I_{dc}^{in} - I_{dc}^{out} \right) - \frac{V_{dc}}{R_{BR}^{min}} m \right] \quad (29)$$

In order to prevent overvoltage to the BTBC at the PS, an optimal controller is recommended to regulate the BC action. The one that is recommended is easy to use and
efficient for this purpose. The KOA yielded PI controller gains that are optimized. The obtained optimized Kp and Ki were 0.03813 and 0.002179, respectively. The goal of this optimization was to enhance the responsiveness of the structures under 'investigation during malfunctions. Equation (30) provides a formal representation of the consideration of the structure's optimization that takes control cost (CC) into mind. The F(x) utilized for CC is expressed as

$$ \text{Minimize } F(x) = \int_{0}^{T} \left( W_{1}|P - P^{*}| + W_{2}|\omega_{m} - \omega_{m}^{*}| + W_{3}|Q_{s} - Q_{s}^{*}| + W_{4}|V_{dc} - V_{dc}^{*}| + W_{5}|I_{dg} - I_{dg}^{*}| \right) \quad (30) $$

where the CS function, which in this case is $4 \times 10^5$, is estimated using the W1, W2, W3, W4, and W5 constants. T stands for average time, and the number of repetitions and agents are 100 and 6, accordingly.

The controller receives the measured and reference values as input, and its output controls the system, while taking the sawtooth signal into account. Figure 7 shows the control system of the BC.

<figure><img src="image_10.png" /><figcaption>Figure 7. Proposed strategy.</figcaption></figure>

### 3. Simulations and Comparative Performance Assessment of BC Controllers

During an unbalanced grid voltage, the excess value in the DCLV is the primary problem with the system. Keeping the DCLV within permissible bounds showed the success of the proposed control methods. Employing the MATLAB\Simulink tool, four simulated settings were run to provide insight on some of the FRTC augmentation strategies' effects on the DCLV. Due to the brief duration of these transient circumstances, it was assumed that the wind speed remained fixed at 12 m/s throughout the fault period. Four scenarios ((a) 100% VD, (b) 70% VD, (c) 30% VD, and (d) 20% VS) were assumed for the investigation carried out in this article. To show the impact of the KOA, the transient response of the PMSWG's parameters was simulated in three cases: without protection (C1), with hysteresis controller-based BC (C2), and with KOA-PI controller-based BC (C3) under the aforementioned scenarios. The investigated parameters were $V_{abc}$, $P_S$, $Q_S$, $I_{d-GSC}$, $I_{q-GSC}$, $T_{em}$, $V_{dc}$, and $\omega_r$. The investigated PMSWG system was assumed to be in a steady state (SS), with a $Q_S$ of $\approx 0$, a system voltage $V_{rms} = 1$ pu, and a $P_S = 0.96$ pu. The studied PMSWG and BC parameters are seen in Tables 2 and 3, respectively [28].
<table><caption>**Table 2.** Parameters for simulated PMSWG.</caption><thead><tr><th>Components</th><th>Parameter</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td rowspan="5">WT</td><td>Blade radius</td><td>R</td><td>33.05 m</td></tr><tr><td>Rated wind speed</td><td>v<sub>W</sub></td><td>12 m/s</td></tr><tr><td>Optimal power coefficient</td><td>C<sub>popt</sub></td><td>0.44</td></tr><tr><td>Optimal tip speed ratio</td><td>λ<sub>opt</sub></td><td>10.5</td></tr><tr><td>Wind density</td><td>ρ</td><td>1.225 Kg/m<sup>3</sup></td></tr><tr><td rowspan="7">PMSWG</td><td>Rated power</td><td>P</td><td>1.5 MW</td></tr><tr><td>Rated stator voltage</td><td>V</td><td>575 V</td></tr><tr><td>Pole pairs</td><td>p</td><td>40</td></tr><tr><td>Stator resistance</td><td>R<sub>s</sub></td><td>0.01 pu</td></tr><tr><td>Inductance in the d frame</td><td>L<sub>d</sub></td><td>0.7 pu</td></tr><tr><td>Inductance in the q frame</td><td>L<sub>q</sub></td><td>0.7 pu</td></tr><tr><td>A flux of the permanent magnets</td><td>ψ<sub>pm</sub></td><td>0.9 pu</td></tr><tr><td rowspan="5">BTBC</td><td>MSC frequency switching</td><td>F<sub>sw-MSC</sub></td><td>1650 Hz</td></tr><tr><td>GSC frequency switching</td><td>F<sub>sw-GSC</sub></td><td>1650 Hz</td></tr><tr><td>DCLV</td><td>V</td><td>1150 V</td></tr><tr><td>DCL capacitor</td><td>C<sub>dc</sub></td><td>10000 μF</td></tr><tr><td>Grid frequency</td><td>F</td><td>60 Hz</td></tr><tr><td rowspan="4">LC filter</td><td>GS inductance</td><td>L<sub>i</sub></td><td>0.3 pu</td></tr><tr><td>GS resistance</td><td>R<sub>i</sub></td><td>0.003 pu</td></tr><tr><td>Filter capacitor</td><td>C<sub>f</sub></td><td>0.0267 pu</td></tr><tr><td>Damping resistance</td><td>R<sub>d</sub></td><td>0.03 pu</td></tr></tbody></table>

<table><caption>**Table 3.** BC resistance parameters.</caption><thead><tr><th>Resistance Value</th><th>1.5 Ω</th></tr></thead><tbody><tr><td>Rated power</td><td>12 kW</td></tr><tr><td>Maximum temperature</td><td>150 °C</td></tr><tr><td>Thermal time constant</td><td>4 min</td></tr><tr><td>Weight</td><td>30 kg</td></tr><tr><td>Dimensions</td><td>(750 × 330 ×150) mm</td></tr></tbody></table>

### 3.1. Scenario 1: 100% VD

To assess the functioning of the system under study in such challenging circumstances, the Danish GC recommends subjecting the system to 100% VD for 150 ms for this kind of problem. The VD is reported to have started at time t = 5 s and to have lasted for 0.15 s. Figure 8's simulated parameters all demonstrate how effective C3 was in comparison to C2 and C1, with C1 being the worst-case scenario. The dynamic response of these parameters is summarized in Table 4.

### 3.2. Scenario 2: 70% VD

It was presumed that the VD originated at time t = 5 s, lasted for 0.6 s, and had a 70% VD during this time. At t = 5.6 s, the VD terminated and the system returned to SS. In comparison to C1 and C2, the characteristics in Figure 9 demonstrate the function of C3, which was to maintain the DCLV below permissible limits and attain a SS quickly. Table 4 lists the response values of the investigated parameters under this fault to show the role of the suggested approach.
<figure><img src="image_11.png" /><figcaption>(a) $V_{abc}$.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(b) $P_S$.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(c) $Q_S$.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(d) $I_{d-GSC}$.</figcaption></figure>

<caption>Figure 8. *Cont.*</caption>
<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /><figcaption>Figure 8. Transient system parameters (SP) as a result of 100% VD.</figcaption></figure>
<table><caption>**Table 4.** Simulated parameters under studied cases.</caption><thead><tr><th rowspan="2">Parameters</th><th colspan="3">Studied Cases at 100% VD</th></tr><tr><th>C<sub>1</sub></th><th>C<sub>2</sub></th><th>C<sub>3</sub> (Proposed)</th></tr></thead><tbody><tr><td>Overvoltage at V<sub>dc</sub></td><td>5.8 pu</td><td>1.3 pu</td><td>1.05 pu</td></tr><tr><td>P<sub>S</sub> change</td><td>0.59 pu → 1.109 pu</td><td>0.59 pu → 0.954 pu</td><td>0.59 pu → 0.957 pu</td></tr><tr><td>I<sub>d-GSC</sub> ripple</td><td>0.318 pu</td><td>0.2653 pu</td><td>0.2652 pu</td></tr><tr><td>I<sub>q-GSC</sub> ripple</td><td>±0.06 pu</td><td>±0.037 pu</td><td>±0.03 pu</td></tr><tr><td>ω<sub>r</sub></td><td>1.091 pu</td><td>1.091 pu</td><td>1.091 pu</td></tr><tr><td>T<sub>em</sub></td><td>0.8975 pu</td><td>0.8962 pu</td><td>0.8956 pu</td></tr><tr><td colspan="4">Studied Cases at 70% VD</td></tr><tr><td>Parameters</td><td>C<sub>1</sub></td><td>C<sub>2</sub></td><td>C<sub>3</sub> (proposed)</td></tr><tr><td>Overvoltage at V<sub>dc</sub></td><td>9.15 pu</td><td>1.3 pu</td><td>1.035 pu</td></tr><tr><td>P<sub>S</sub> change</td><td>0.45 pu → 1.19 pu</td><td>0.45 pu → 0.9571 pu</td><td>0.44 pu → 0.957 pu</td></tr><tr><td>I<sub>d-GSC</sub> ripple</td><td>0.3302 pu</td><td>0.2731 pu</td><td>0.269 pu</td></tr><tr><td>I<sub>q-GSC</sub> ripple</td><td>±0.06 pu</td><td>±0.022 pu</td><td>±0.02 pu</td></tr><tr><td>ω<sub>r</sub></td><td>1.091 pu</td><td>1.091 pu</td><td>1.091 pu</td></tr><tr><td>T<sub>em</sub></td><td>0.8976 pu</td><td>0.8975 pu</td><td>0.8955 pu</td></tr><tr><td colspan="4">Studied Cases at 30% VD</td></tr><tr><td>Parameters</td><td>C<sub>1</sub></td><td>C<sub>2</sub></td><td>C<sub>3</sub> (proposed)</td></tr><tr><td>Overvoltage at V<sub>dc</sub></td><td>5.9 pu</td><td>1.3 pu</td><td>1.043</td></tr><tr><td>P<sub>S</sub> change</td><td>0.82 pu → 1.149 pu</td><td>0.85 pu → 0.9516 pu</td><td>0.84 pu → 0.957 pu</td></tr><tr><td>I<sub>d-GSC</sub> ripple</td><td>0.2824 pu</td><td>0.2677 pu</td><td>0.2608 pu</td></tr><tr><td>I<sub>q-GSC</sub> ripple</td><td>±0.04067 pu</td><td>±0.01 pu</td><td>±0.015 pu</td></tr><tr><td>ω<sub>r</sub></td><td>1.091 pu</td><td>1.091 pu</td><td>1.091 pu</td></tr><tr><td>T<sub>em</sub></td><td>0.8974 pu</td><td>0.8976 pu</td><td>0.8963 pu</td></tr><tr><td colspan="4">Studied Cases at 20% VS</td></tr><tr><td>Parameters</td><td>C<sub>1</sub></td><td>C<sub>2</sub></td><td>C<sub>3</sub> (proposed)</td></tr><tr><td>Overvoltage at V<sub>dc</sub></td><td>1.022 pu</td><td>1.022 pu</td><td>1.022 pu</td></tr><tr><td>P<sub>S</sub> change</td><td>0.9566 pu → 0.9581 pu</td><td>0.9565 pu → 0.9581 pu</td><td>0.9566 pu → 0.9583 pu</td></tr><tr><td>I<sub>d-GSC</sub> ripple</td><td>-0.1532 pu</td><td>-0.1532 pu</td><td>-0.1731 pu</td></tr><tr><td>I<sub>q-GSC</sub> ripple</td><td>0 pu</td><td>0 pu</td><td>0 pu</td></tr><tr><td>ω<sub>r</sub></td><td>1.091 pu</td><td>1.091 pu</td><td>1.091 pu</td></tr><tr><td>T<sub>em</sub></td><td>0.8941 pu</td><td>0.8942 pu</td><td>0.8942 pu</td></tr></tbody></table>

<figure><img src="image_19.png" /><figcaption>**Figure 9.** *Cont.*</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Figure 9. *Cont.*</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>(f) $T_{em}$.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>(g) $V_{dc}$.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>(h) $\omega_r$.</figcaption></figure>

<caption>Figure 9. Transient SP as a result of 70% VD.</caption>

### 3.3. Scenario 3: 30% VD

It was presumed that the VD occurred at time $t = 5$ s, lasted for 1.25 s, and had a 30% VD during this time. At $t = 6.25$ s, when the VD terminated, the system returned to a SS. The parameters in Figure 10 demonstrate the function of C3 in lowering the DCLV below the permissible bounds and quickly achieving SS. Table 4 lists the values of the investigated parameters under this fault to show the role of the suggested approach.
<figure><img src="image_24.png" /><figcaption>Figure 10. *Cont.*</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Figure 10. Transient SP as a result of 30% VD.</figcaption></figure>
### 3.4. Scenario 4: 20% VS

It was presumed that the vs. occurred at time t = 5 s, lasted for 1.25 s, and had a 20% increase in voltage during this time. At t = 6.25 s, in Figure 11, when the fault terminated, the system returned to SS. Table 4 lists the values of the investigated parameters under this fault to show the role of the suggested approach.

<figure><img src="image_26.png" /><figcaption>Figure 11. *Cont.*</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>(d) $I_{d-GSC}$.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>(e) $I_{q-GSC}$.</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>(f) $T_{em}$.</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>(g) $V_{dc}$.</figcaption></figure>

<caption>Figure 11. *Cont.*</caption>
<figure><img src="image_31.png" /><figcaption>Figure 11. Transient SP as a result of 20% VS.</figcaption></figure>

In order to evaluate the suggested technique, Figure 12 shows the DCLV values for each of the situations that were examined. Additionally, Table 5 presents a comparison with prior published methods based on FLC, MPC, SMC, or optimizers. The more sophisticated and modern varieties exhibited acceptable transient responsiveness, but because of their low switching frequency, FCS-MPC were prone to complexities and fluctuations in currents and DCLV after fault removal. In every case of the VD scenario, the DCLV values failed to meet the criteria (greater than 1.1 pu). In addition to improving the abrupt responsiveness of the PMSWG settings, the suggested control systems maintained the DCLV value in all investigated scenarios.

<table><caption>Table 5. Comparative analysis with previously published techniques.</caption><thead><tr><th rowspan="2">Ref.</th><th rowspan="2">Publisher</th><th rowspan="2">Year</th><th rowspan="2">Technique</th><th rowspan="2">Smooth Operation</th><th rowspan="2">Transient Response</th><th rowspan="2">Complexity</th><th colspan="4">DCLV during Faults</th></tr><tr><th>DCLV (pu)</th><th>VD (pu)</th><th>DCLV (pu)</th><th>VS (pu)</th></tr></thead><tbody><tr><td>[56]</td><td>Elsevier</td><td>2017</td><td>ZN-PI</td><td>X</td><td>Low (L)</td><td>L</td><td>1.347</td><td>0.9</td><td>X</td><td>X</td></tr><tr><td>[57]</td><td>Springer</td><td>2020</td><td>WOA-PI</td><td>✓</td><td>High (H)</td><td>H</td><td>1.304</td><td>0.85</td><td>X</td><td>X</td></tr><tr><td>[56]</td><td>Elsevier</td><td>2017</td><td>ISMC</td><td>✓</td><td>H</td><td>H</td><td>1.347</td><td>0.9</td><td>X</td><td>X</td></tr><tr><td>[58]</td><td>MDPI</td><td>2021</td><td>Modified MPC</td><td>X</td><td>Very High (VH)</td><td>VH</td><td>1.091</td><td>0.9</td><td>X</td><td>X</td></tr><tr><td>[59]</td><td>Springer</td><td>2021</td><td>ALO-PI</td><td>X</td><td>H</td><td>L</td><td>1.372</td><td>0.9</td><td>X</td><td>X</td></tr><tr><td>[56]</td><td>Elsevier</td><td>2017</td><td>FCS-MPC</td><td>X</td><td>VH</td><td>VH</td><td>1.347</td><td>0.9</td><td>X</td><td>X</td></tr><tr><td>[60]</td><td>MDPI</td><td>2023</td><td>NLESO and GFTSMC</td><td>✓</td><td>VH</td><td>H</td><td>1.154</td><td>1</td><td>X</td><td>X</td></tr><tr><td>[28]</td><td>Taylor&Francis</td><td>2023</td><td>WHO-PI</td><td>✓</td><td>VH</td><td>L</td><td>1.043</td><td>0.9</td><td>X</td><td>X</td></tr><tr><td>[61]</td><td>MDPI</td><td>2022</td><td>Mode-shift control</td><td>✓</td><td>H</td><td></td><td>1.2</td><td>0.7</td><td>X</td><td>X</td></tr><tr><td>[62]</td><td>MDPI</td><td>2022</td><td>ESS and BC</td><td>✓</td><td>VH</td><td>VH</td><td>1.01</td><td>0.4</td><td>X</td><td>X</td></tr><tr><td>[63]</td><td>MDPI</td><td>2023</td><td>ILP-PLL</td><td>✓</td><td>VH</td><td>VH</td><td>1.004</td><td>0.35</td><td>X</td><td>X</td></tr><tr><td>[64]</td><td>MDPI</td><td>2022</td><td>MRFO-PI</td><td>✓</td><td>VH</td><td>L</td><td>1.059</td><td>0.85</td><td>X</td><td>X</td></tr><tr><td>[52]</td><td>SAGE</td><td>2021</td><td>FLC</td><td>X</td><td>H</td><td>H</td><td>1.047</td><td>0.9</td><td>X</td><td>X</td></tr><tr><td rowspan="3">[65]</td><td rowspan="3">Elsevier</td><td rowspan="3">2024</td><td rowspan="3">Modified MSC</td><td rowspan="3">✓</td><td rowspan="3">H</td><td rowspan="3">L</td><td>1.4</td><td rowspan="3">asymmetrical grid faults</td><td rowspan="3">X</td><td rowspan="3">X</td></tr><tr><td>1.05, 1</td></tr><tr><td>1.043, 0.7, 0.3</td></tr><tr><td></td><td></td><td></td><td>KO-PI (proposed)</td><td>✓</td><td>VH</td><td>L</td><td>1.043, 0.7, 0.3</td><td>1.022</td><td>0.2</td><td></td></tr></tbody></table>
<figure><img src="image_32.png" /><figcaption>Figure 12. DCLV values under investigated scenarios.</figcaption></figure>

**4. Conclusions**

This article addressed the dynamic response of PMSWG using a developed BC with the KOA under VD and VS situations. Mathematical modelling of the investigated system and application of the KOA was presented. Four cases were examined when operating the system with C1, C2, and C3 (proposed). The obtained results are summarized in Table 4. The system was operated with different control strategies to show the efficacy and effectiveness of C3. The results showed that C3 not only enhanced the dynamic response of the parameters, it also enhanced FRTC by keeping the DC under the permissible values. In the worst scenario (100% VD), with C3 the over voltage at DCL was reduced by 93.5% and 19.231% compared to C1 and C2, respectively. C3 was effective compared to C2 and C1 in smoothing the power delivered to the utility grid and damping the oscillations in the rotor angular speed, electromagnetic torque of the generator, and direct and quadrature currents. In addition, a comparison study (Table 5) involving over a dozen published articles from reputable publishers was carried out to demonstrate the contribution of the suggested approach to sustainable development. Finally, it can be concluded that the proposed strategy will help in achieving high penetration scenarios for wind energy.

**Author Contributions:** Conceptualization, M.M.M. and B.S.A.; formal analysis, A.-M.M.A.-R. and S.A.S.; investigation, M.M.M., I.M.E. and A.A.; resources, A.B. and U.K.; writing—original draft preparation, M.M.M. and B.S.A.; writing—review and editing, M.M.M., B.S.A., A.A. and I.M.E.; visualization, A.A.; supervision, A.-M.M.A.-R., S.A.S. and M.M.M. Funding, A.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Researchers Supporting Project number (RSP2024R258), King Saud University, Riyadh, Saudi Arabia.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data are available on request from the authors.

**Acknowledgments:** The authors are grateful for the support by the Researchers Supporting Project (number RSP2024R258), King Saud University, Riyadh, Saudi Arabia.

**Conflicts of Interest:** The authors declare no conflicts of interest.
Abbreviations

<table><tr><td>AFD</td><td>adjustable frequency drives</td><td>SDGs</td><td>sustainable development goals</td></tr><tr><td>BC</td><td>braking chopper</td><td>SS</td><td>sustainable sources</td></tr><tr><td>BR</td><td>braking resistance</td><td>WF</td><td>wind farm</td></tr><tr><td>BTBC</td><td>back-to-back converter</td><td>PMSWG</td><td>permanent magnet synchronous wind generator</td></tr><tr><td>PI</td><td>proportional-integral</td><td>SMC</td><td>sliding mode control</td></tr><tr><td>ERL</td><td>exponential reaching low</td><td>SPWM</td><td>sinusoidal pulse width modulation</td></tr><tr><td>ESS</td><td>energy storage system</td><td>SRF</td><td>synchronous reference frame</td></tr><tr><td>GC</td><td>grid-code</td><td>PLL</td><td>phase-locked loop</td></tr><tr><td>GSC</td><td>grid-side converter</td><td>VD</td><td>voltage dip</td></tr><tr><td>KCL</td><td>Kirchhoff's current law</td><td>MSC</td><td>machine-side converter</td></tr><tr><td>FRTC</td><td>fault ride-through capability</td><td>OTC</td><td>optimal torque control</td></tr><tr><td>MPPT</td><td>maximum power point tracking</td><td>PCC</td><td>point of common coupling</td></tr></table>

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