Enhancing the small-signal stability of the island microgrids under the virtual impedances-gray wolf optimization model: two distinct case studies

Abstract

1 Introduction

The energy sector has been driven to adopt renewable energy sources due to concerns regarding greenhouse gas emissions and the future of energy resources [1, 2]. Microgrids (MG) are essential in contemporary power systems as they facilitate the incorporation of renewable energy sources into traditional energy networks [3, 4]. Nevertheless, the utilization of MG presents significant obstacles such as intricate management and stability concerns, lack of inherent inertia, a greater quantity of generators, and variety of primary movers [5, 6]. MG, like traditional power systems with numerous generators, employ droop control to replicate the power-sharing capabilities of mechanical power systems [7]. Distinct control loops utilize voltage and frequency droop formulae [8, 9]. However, the interconnection of active and reactive power distribution in complicated MG has been recognized, resulting in inaccurate power distribution [10].

The technique of employing virtual impedance to rectify the imperfect distribution of reactive power (reactive mismatch) was first introduced in [11, 12]. Furthermore, virtual impedances are chosen from specified ranges, taking into account aspects such as the stability of the MG under small-signal conditions, voltage limitations, the distribution of reactive power, and the required damping [13]. In [14], virtual admittance has been employed to mitigate harmonic currents and minimize transmission losses in parallel current-controlled distributed generations, whether they are grid-connected or part of island-MG. Adjustable virtual impedance, as described in reference [15], is designed to improve the distribution of active and reactive power in a networked island MG. In [16], a unique droop control method was introduced. This method utilizes virtual impedances to address the coupling effect, reduce power oscillations between distributed generations, and stabilize the MG. The concept of complex-virtual impedance was introduced in reference [17] as a means to stabilize voltages and assure precise power distribution in islanded MG. The virtual impedance method, which was proposed in reference [18], is a strong technique aimed at enhancing the stability of the MG, minimizing voltage distortions, and improving the performance of the converter after a fault occurs.

Nevertheless, the occurrence of MG instability caused by load fluctuations or disconnections of DGs in island MGs continues to be a widespread problem [2, 19]. The stability of the MG is assessed by conducting S-SS assessment at a particular working point. The coefficients of the controller and PLL play a critical role in determining the MG’s stability [20, 21]. The stability study of both isolated and connected MGs is facilitated by the lower order small-signal models described in references [22, 23]. A study in [24] was undertaken to compare the S-SS of model-based and conventional PI voltage and current controllers in a MG system. The paper [25] identifies the optimal parameter ranges for MGs using small-signal modeling to improve the dynamic response of converters. Analyses of stability were conducted in the latter two phases. The implementation of MG reconfiguration procedures has been proven to substantially enhance stability margins, as documented in reference [26]. In a study conducted in [27], the MG on Dongao-island was examined.

The analysis focused on the frequency control of the MG, which was divided into three distinct phases: stable, precautionary, and emergency. In addition, a small-signal model was presented in reference [28] for a two-layer control structure of a micro-grid system. This structure includes an MG layer and a cluster layer, and the parameters are specifically constructed for this purpose. The MG’ S-SS is influenced by the existence of induction motors and stable-power loads, as investigated in [29]. The objective of the distributed secondary control, as presented in reference [30], is to enhance the dynamic performance of the MG by focusing on its S-SS. In addition, the benefits of DG’s control were examined in [31] employing a theoretical structure to analyze S-SS.

Today, machine/deep learning techniques are used in various fields. Liu and Bao [32] monitored the cracks using distributed fiber optic sensor and deep learning. A modified You Only Look Once model adequately interpreted distributed fiber optic sensor data. Furthermore, the transfer learning was incorporated to improve the accuracy of the deep learning model. The robustness of the approach was evaluated in different test scenarios. The [email protected] of detecting spatially-distributed cracks could reached ~0.97. In [33], automated condition assessment of pipelines was achieved by machine learning models. Detecting, locating, and quantifying anomalies of pipelines were reviewed. Further, operation data, nondestructive testing data, and computer vision data were covered. In addition, the strengths, weaknesses, opportunities, and threats (SWOT) analysis and practical recommendations are discussed. Luo et al. [34] introduce a deep neural network model that operates on an utterance basis, employing a parallel architecture that integrates convolutional neural networks and long short-term memory networks. This model is designed to extract representative features known as Audio Sentiment Vectors, which are intended to capture sentiment information from audio data to the greatest extent possible.

In a pioneering study [35], deep learning techniques for classifying imbalanced data were utilized for the first time in the context of cellular macromolecular complexes visualized through Cryo-electron tomography (Cryo-ET). The researchers implemented various strategies to address the challenges posed by imbalanced datasets, such as data sampling, bagging, boosting, and methods based on genetic programming. Sati et al. [36] embedded the incremental cost function into the virtual synchronous machines model. The combination enabled online economic dispatch (ED) and virtual inertia provision. Frequency restoration, online ED, and inertia emulation were achieved simultaneously. Furthermore, a dynamic outer voltage controller to improve MG stability limit was proposed. Verified via real-time experiment, small signal stability, and sensitivity analysis were developed. In [37], limiting only fundamental component of the reference current was proposed. Adaptive virtual impedance scheme was also designed to reduce control effort. DG’s voltage distortion provenience was achieved by the proposed fault ride through (FRT) approach. Furthermore, angle-droop control method was utilized for single-phase and three-phase DGs. Moreover, state space analysis was performed to confirm stability of proposed FRT approach.

Deng et al. [38] developed an adaptive virtual impedance regulation strategy to eliminate the wire impedance mismatch. Both the resistive and inductive parts of inverter impedance were regulated distributedly, which ensures accurate current distribution. The current was shared both in steady state and transient accurately. Pompodakis et al. [39] developed a method for calculating short-circuits currents in islanded networks. Inverter-based distributed generators with virtual impedance current limiters were accurately modeled. That approach combined high computation accuracy with extremely low computation time. Choudhury et al. [40] considered solar PV and fuel cell-based hybrid microsource to ensure high-energy supply reliability. The gains of the PI controller are computed through MFLC and tuned by SOA to ensure the highest free available energy delivered.

The utilization of Popov’s absolute stability criterion to ascertain the durability situations of MGs in the existence of stable power loads was elaborated in reference [41]. In order to improve the MG’s S-SS, a technique utilizing cascading lead compensators was introduced in reference [42]. The S-SS’s characteristic equation of an MG could be utilized to evaluate the stability at low frequencies. This approach, as described in reference [43], employs the dynamic-phasor and Padé approximation model. The study conducted in [44] examined the impact of power distribution on a hybrid MG’s stability that includes low-inertia distributed generations and diesel generators. Research has demonstrated that the common DG connection structure has a detrimental impact on an MG’s S-SS [45]. In [46], a bifurcation theory assessment was performed to assess the durability parameters of an MG, taking into account various types of loads. The distributed consensus method proposed in literature was developed to dynamically disperse harmonic-loads across distributed generations. The stability boundaries of an MG were determined using the differential algebraic-equation model in [47]. In [48], the researchers investigated the stability of MG under harmonic conditions.

The current paper presents a comprehensive small-signal dynamic technique for island-MG, which includes virtual impedances and phase-locked loop. Subsequently, an S-SS assessment is performed using the dynamic MG model. The suggested algorithm introduces a novel approach to optimize the configuration of virtual impedance in MG converters. Its main objective is to enhance the stability index of the MG system while reducing imbalances in reactive power. This optimization technique assesses the stability of the MG at all possible operational points. The goal function, expressed as a fraction, allows for the simultaneous achievement of various objectives. This study employs a sophisticated dynamical model of a MG that incorporates phase-locked loop dynamics and virtual impedances. This model builds upon previous research conducted in references [24, 49]. Both the virtual impedance and the phase-locked loop models are crucial for analyzing the MG’s stability. In summary, the research contributions are as follows:

• i) The paper presents a new fractional-objective problem for the modelling of virtual impedances. This objective function aims to reduce the differences in reactive power between converters and increase the vital eigenvalue of the MG.

• ii) The fractional-objective problem addresses the issue of determining weighting coefficients under a multiobjective problem commonly found in the publications.

• iii) The paper performs a comprehensive assessment of the S-SS of MG at all operating states, guaranteeing stability through the incorporation of virtual impedances.

2 Methodology

The methodology part is based on MG conceptual modelling, stability analysis, and virtual impedance design.

2.1 MG conceptual modelling and stability analysis

In island-MG setups, DG is commonly linked by power converters, which can be either voltage source converters or current source converters. Former converters are frequently employed in grid-forming modes to build the MG by precisely controlling the voltage and frequency parameters. This allows for influencing the operational dynamics of the island-MG. Stability in an MG refers to the capacity of the system to recover and sustain stable values within operational boundaries after experiencing any form of disruption, as reported in [50]. To effectively manage many DGs in an MG, a uniform reference frame is necessary. This reference frame can either be synchronized to a specific DG or based on a global time reference [51]. The latter approach eliminates the necessity for communication between DGs. The angle of synchronization for each distributed generation is determined with respect to a shared reference point. When one converter establishes the reference, its synchronization angle is zero, whereas other converters show relative deviation angles [52]. A phase-locked loop is crucial in this configuration to either determine the frequency or align a converter with the microgrid. Figure 1 depicts the offered schematic and control block structure for the island-MG.

Figure 1

The offered schematic and control block structure for the island-MG.

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The micro-grid operates using a cohesive reference system, which requires converting between local and global reference frames. In this study, the term ‘|${dq}^{{\prime}}$| indices refer to values that are specific to a local context, whereas ‘|${DQ}^{{\prime}}$| indices indicate values in a global context. Crucial global variables encompass bus voltages (⁠|${v}_{bD},{v}_{bQ}$|⁠), which are fundamental in creating the dynamic equations for line and load interactions. The converter-specific dynamics, as described in Equation (1), require the local bus voltages (⁠|${v}_{bd},{v}_{bq}$|⁠). Similarly, the output currents generated by converters (⁠|${i}_{od},{i}_{oq}$|⁠) must be converted to the global frame (⁠|${i}_{oD},{i}_{oQ}$|⁠) in order to accurately represent the currents supplied to the MG. Furthermore, the angle (⁠|${\delta}_{cm}$|⁠) denotes the phase disparity between the global D-axis and the local d-axis of a converter.

The subsequent sections delineate the dynamic equations that form the foundation of the control schemes illustrated in Fig. 1.

• Calculating power

The computation of instantaneous active and reactive powers (⁠|$p,q$|⁠) is obtained by analyzing the output |$dq$| components of converter voltages and currents. Subsequently, these values undergo processing via a low-pass filter (LPF) with a designated cut-off frequency (⁠|${\omega}_c$|⁠). This procedure yields two dynamic equations that are crucial for analyzing the S-SS of the MG. Figure 1 outlines the arrangement and movement of inputs and outputs through the low-pass filter and power calculator.

• Equations for virtual impedance

Equation (3) describes how virtual impedance is included into the voltage control loops to generate voltage dips. The variables (⁠|${V}_{vd}$|⁠), (⁠|${X}_{vi}$|⁠), and (⁠|${R}_{vi}$|⁠) represent the virtual-voltage drop, reactance, and resistance, respectively. The control approach guarantees that the voltage component along the d-axis remains at zero. The subsequent part will outline the optimization procedure that will ascertain the suitable values for these virtual impedances. Equation (3) also demonstrates the impact of minor changes at any given operating point on the voltage drops across various impedances.

• Droop control

The droop control is depicted in Fig. 1. In this control method, the management of frequency (⁠|${\omega}^{\ast }$|⁠) and q-axis voltage (⁠|${V}_{oq}^{\ast }$|⁠) is done using standard equations, while the voltage of d-axis = 0. The coefficients (⁠|$m$|⁠) and (⁠|$n$|⁠) reflect the droop coefficients for active and reactive power, respectively. The calculation of the virtual voltage drop (⁠|${V}_{vd}$|⁠), caused by the virtual impedance, is determined by Equation (3). When the active/reactive power settings are set to zero, there are deviations from the nominal voltage and frequency values (⁠|${\omega}_n,{V}_{oq_n}$|⁠). Nevertheless, it is imperative to incorporate the designated power levels (⁠|${P}_0,{Q}_0$|⁠) into the droop equations in order to achieve precise power regulation.

• Voltage controller

The voltage control technique employs Equations (5) from the referenced literature [49], which are provided below to enhance comprehension of the system dynamics. Each converter autonomously modifies its frequency set-point (⁠|${\omega}^{\ast }$|⁠) over time. The voltage controller has a proportional-integral (PI) controller, which is defined by its proportional (⁠|${k}_{pv}$|⁠) and integral (⁠|${k}_{iv}$|⁠) coefficients. To facilitate the dynamic modeling of the system, auxiliary variables (⁠|${\varphi}_d$|⁠) and (⁠|${\varphi}_q$|⁠) are introduced. The frequency of the MG, denoted as (⁠|${\omega}_{PLL}$|⁠), is determined using a phase-locked loop system. The desired frequency set points, (⁠|${\omega}^{\ast }$|⁠) and (⁠|${V}_{oq}^{\ast }$|⁠), are obtained from the droop control equations. The first-order differentials (⁠|${\dot{\varphi}}_d$|⁠) and (⁠|${\dot{\varphi}}_q$|⁠) are crucial in the investigation of S-SS in the MG. Furthermore, this block’s outputs consist of the dq current commands (⁠|${I}_{ldq}^{\ast }$|⁠), as depicted in Fig. 1.

• Current controller

The controller uses Equations (6) to create voltage commands for the converters, employing the |$dq$| reference frame. The coefficients (⁠|${k}_{pc}$|⁠) and (⁠|${k}_{ic}$|⁠) of the PI-controller determine the dynamic response of the current controller. Furthermore, (⁠|${\omega}_n$|⁠) is the symbol used to represent the nominal frequency of the MG, while (⁠|${L}_f$|⁠) is the symbol used to represent the inductance of the filter. The dynamic modeling of the MG is improved by including auxiliary variables (⁠|${\gamma}_d$|⁠) and (⁠|${\gamma}_q$|⁠). The output of the present controller comprises the voltage commands (⁠|${v}_{id}^{\ast }$|⁠) and (⁠|${v}_{iq}^{\ast }$|⁠), assuming a straight correspondence to the voltages produced by the converters.

• The model of phase-locked loop

The PLL model, following the guidelines in [49], function within the |$dq$| frame to guarantee that the d-axis voltage becomes zero in a stable condition, thereby synchronizing the q-axis with the Q-axis. The phase-locked loop configuration consists of a low-pass filter under a cutoff frequency (⁠|${\omega}_{c, PLL}$|⁠) and a PI-controller under (⁠|${k}_{p, PLL}$|⁠) and (⁠|${k}_{i, PLL}$|⁠). The stability analysis employs the nominal frequency (⁠|${f}_n$|⁠) as specified in Equation (7). This model plays a vital role in ensuring the voltage alignment and phase synchronization within the MG.

• Loading the model

The work employs Kirchhoff’s laws within the |$DQ$| frame to generate dynamic load equations for the impedance load model. The RL load components are represented by the resistance (⁠|${R}_{\mathrm{load}}$|⁠) and inductance (⁠|${L}_{\mathrm{load}}$|⁠). The measured frequency of the MG is denoted by (⁠|${\omega}_{PLL}$|⁠). The model represents bus voltages as mathematical functions of different currents and system state variables. It employs a virtual resistor notion to facilitate the process of extracting state-space matrices. This model affects the dynamic interactions between the bus voltages and the load, which in turn affects the overall stability of the MG.

• Line model

The line model examines a standard arrangement of a resistive-inductive (RL) line between two neighboring buses. Due to the widespread employment of the shared reference frame in all MG buses, voltages that correspond to each other are utilized in the formulation of dynamic line equations. The line characteristics are determined by the resistance (⁠|${r}_{ln}$|⁠) and inductance (⁠|${L}_{ln}$|⁠), and these calculations depend on the integral frequency (⁠|${\omega}_{PLL}$|⁠) of the MG. The specified direction of line currents is crucial for accurately simulating voltage interactions between interconnected buses.

• Stability of small signal

The state variables for a typical two-node MG are defined by integrating the comprehensive dynamic formulas from the PLL, current/voltage controllers, load model, low-pass filter, and line interfaces. The variables mentioned are utilized as the foundation for S-SS equations. These equations are linearized around a predetermined operating point in order to simplify analysis. The resulting |$A$| matrix for a two-bus MG is a 36 × 36 matrix. The process for deriving the A matrix and conducting S-SS analysis stays similar for larger MG setups. Linearizing around various operating points enables a comprehensive assessment of stability across several potential MG configurations, guaranteeing a rigorous study and design of operations [17, 49].

2.2 Virtual impedance design

The process of creating virtual impedances is described in a complete algorithm, outlined in Fig. 2. This approach is implemented in an offline manner, enabling the incorporation of carefully designed virtual impedances into converter control systems after the initial design phase. The procedure commences by assigning voltage set-points, then establishing a positive range for virtual resistance and inductance values. Referring to [15], the common practice is to fix the virtual resistance at 0.2 times the reactance. This method ensures that the intervals are equalized in order to optimize the function efficiently. The maximum thresholds for these values are determined based on the most significant differences in impedance that have been discovered.

Figure 2

The process of creating virtual impedances based on GWOA.

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GWOA is utilized in power systems because to its robustness and efficiency in solving complicated optimization problems [53, 54]. GWOA is chosen for its favorable convergence features, ease of parameter change, and precision [55]. The sequential steps involved in the virtual impedance design process are as follows [56, 57]:

• (i) Initialization: begin by setting the optimization variables for virtual resistances (⁠|${R}_{v1},\dots, {R}_{vn}$|⁠) and inductances (⁠|${L}_{v1},\dots, {L}_{vn}$|⁠) to starting values that are within a stable and acceptable range.

• (ii) GWOA configuration: set up the GWOA by defining the population size, number of iterations, and other important parameters. After that, start the optimization process.

• (iii) Load-flow analysis: employ MATLAB to conduct load-flow analysis over a specified time period, taking into account load fluctuations at various bus locations. The quantity of unique operational points evaluated depends on the time intervals of the solution employed.

• (iv) Perform stability analysis by examining the S-SS at all operating points, taking into account the influence of virtual impedances. Propose a stability indicator that is determined by the magnitude of the real component of the most crucial eigenvalue. An MG is considered asymptotically stable if all dominant eigenvalues have negative real components, with at most one eigenvalue being zero. If this requirement is not satisfied, the process returns to stage (ii) for additional optimization.

• (v) Save solutions: after meeting the stability conditions, store the stability index and optimization solutions, and continue to the next stage.

• (vi) Simulation continuation: keep executing the simulation until the designated end time (⁠|${t}_f$|⁠). If the desired outcome is not achieved, the simulation continues to assess further operational points.

• (vii) Convergence criterion: the criteria for determining when to halt may be based on either the iterations numbers or the optimization problem’s performance. This study uses the iteration count. The objective problem’s numerator is the all converters’ reactive power-mismatches, with each mismatch multiplied by its corresponding droop coefficient. The denominator was the stability index that has the lowest value among all assessed points and eigenvalues.

• (viii) Voltage limits check: verify that all bus voltages stay under acceptable thresholds after implementing optimal virtual impedances. If there are violations in voltage restrictions, it is necessary to review stage (i) in order to make adjustments to the upper limits of virtual impedances or modify the converter voltage set-points. This is done to ensure that the voltage is delivered correctly at the load buses.

• (ix) Generate optimal values: if the voltage parameters meet the required standards, proceed to determine and produce the most efficient configurations for virtual resistances and inductances. Configure the parameters of the GWOA according to the specifications provided in Table 1. By adhering to this optimization method as depicted in Fig. 2. Implement the computed values to optimize the efficiency of the converter control system.

This systematic approach guarantees that the virtual impedances are not only optimized for performance but also adhere to operational stability and voltage standards, rendering them suitable for actual use in MG.

3 Results and discussion

The suggested optimization algorithm’s effectiveness is evaluated using the two-Bus/MG from [49] and the three-Bus/MG from [52], both developed in MATLAB, as separate cases. Comprehensive examination and deliberation of the results for each instance are presented.

• Case (I): Two-Bus/MG

Figure 3

The two-bus microgrid test.

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The two-Bus/test MG test is depicted in Fig. 3. The initial case study is comprised of four subjects and was analyzed in reference [49]. Table 2 provides comprehensive information about the controllers and the two-Bus/test MG. A virtual impedance design algorithm was used to calculate the most effective virtual inductances/resistances for this MG. Table 2 documents the calculated virtual impedances. At |$t=2\ \mathrm{s}$|⁠, a complex impedance (⁠|${R}_{pe,1}+j{X}_{pe,1}$|⁠) is added to the power system and connected in parallel under the complex impedance (⁠|${R}_{load,1}+j{X}_{load,1}$|⁠) at bus-1. In this scenario, bus-2 consistently maintains its local load as (⁠|${R}_{load,2}+j{X}_{load,2}$|⁠). The MG’s response to this change in load is then carefully examined.

The MG stability index (MG-SI) is a metric used to measure the stability of a MG. The MG stability index is of utmost importance as it is determined by the absolute value of the most susceptible eigenvalue to instability. The installation of virtual impedance boosts the absolute real part of the stability index, hence improving this mode. Figure 4 displays the MG’s principal eigenvalues are displayed in two different scenarios, with a total of five showcases. The stability index, first introduced in [49] and denoted as |$\lambda$|⁠, and the improved index employing the present control method, denoted as (⁠|${\lambda}_v$|⁠), as illustrated in Figure 4. The index increases from 1.41 to 3.94 following the introduction of the specifically designed virtual impedance. The incorporation of appropriate virtual impedance into the new control architecture improves the stability of the MG. Table 3 presents the MG eigenvalues, with the eigenvalues and damping ratios displayed in the second and third columns, respectively. All modes demonstrate damping and stability, with certain modes exhibiting significant damping (e.g. modes 1 and 2 with 100% damping). The damping ratio associated with the MG-SI is 16.316%. Since one eigenvalue is 0 and the rest have negative real-components, the system stays asymptotically stable, as reported in [58].

The frequency within the MG is monitored using phase-locked loops installed on the buses. These loops include proportional-integral controllers in their control loops. After a load change is posted at |$t=2\ \mathrm{s}$|⁠, the frequency drops in accordance with the active power/frequency droop specifications. Figure 5 depicts the MG-frequency following the load adjustment is measured using both the established and the new control methodologies. The Nadir (minimum point) and the rate of change of frequency display approximately similar patterns in both strategies; however, the Nadir is slightly lower in the new strategy compared to the previous way.

Figure 6 illustrates the data presented in this study show the active and reactive power outputs from the converters in two different situations: one using the newly implemented control strategy and the other using the way described in reference [49]. The active power output of converters under the new framework exhibits faster resolution and less overshoot compared to the prior method. The equilibrium active powers are the same for both strategies.

Prior to the load shift at |$t=2\ \mathrm{s}$|⁠, the reactive powers (⁠|${Q}_o$|⁠) from the converters are essentially identical. After the load is applied, both converters evenly divide the reactive power, with (⁠|${Q}_1={Q}_2=100$|⁠) Var. Due to the converters being similar, the distribution is completely smooth and uninterrupted. In contrast, the method described in the previous study [49] demonstrated inefficient allocation of reactive power after a load change, resulting in a significant disparity in power distribution, with one component receiving 150 Var and the other receiving 50 Var.

The (⁠|${i}_{ld}\&{i}_{lq}$|⁠) parameters (terminal currents) correspond to the dq components of (⁠|${i}_{l, abc}$|⁠) shown in Fig. 1, quantified prior to filtration. The currents generated by converters utilizing both the innovative and the previous control approaches, as explained in reference [49], are depicted in Fig. 7. An examination of the d-axis currents demonstrates that the innovative control mechanism ensures consistency with a change in load, in contrast to the varied currents in the previous method. The implementation of a uniform distribution ensures that one converter does not become overloaded while another remains underloaded. Furthermore, an examination of the q-axis currents reveals decreased variability and faster stabilization times with the new approach, although the stable q-axis currents remain unchanged in both situations. As depicted in Fig. 8, the current-sharing efficiency of the new control technique is emphasized. The d-axis currents from both converters exhibit a high degree of similarity, measuring approximately 0.81 A, both prior to and during the load modification. In contrast, the previous work [49] documented different d-axis currents of 1.2 and 0.4 A from two comparable converters.

Figure 4

The MG’s principal eigenvalues are displayed in two different scenarios.

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Figure 5

The MG-frequency following the load adjustment is measured using both the established and the new control methodologies.

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Figure 6

The data presented in this study show the active and reactive power outputs from the converters in two different situations.

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Figure 7

The currents generated by converters utilizing both the innovative and the previous control approaches.

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Figure 8

Comparison of the current-sharing efficiency of the new control technique.

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Regarding q-axis currents, the new control technique significantly reduces the current overshoot and settling durations, as shown in the bottom graph of Fig. 8. The voltage components produced by converters are compared in Fig. 9 under the scenarios that used the new control strategy and the earlier approach [49]. Both solutions have the objective of keeping the d-axis voltage components at zero, resulting in minimal variation at |$t=2\ s$|⁠, which quickly stabilizes afterwards. Implementing virtual impedances leads to detectable voltage decreases on these components; a decrease of around 1.48 V is evident on the exit voltages of the q-axis with the new approach. In order to maintain a stable exit voltage, the set-point of voltage was modified to ~90 V, as specified in one of the optimization procedures illustrated in Fig. 2.

• Case (II): Three-Bus/MG

Figure 9

Comparison of the voltage components produced by converters.

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An esteemed low-voltage (220 V RMS) three-Bus/MG is demonstrated in Fig. 10. The second case study focuses on a complex system incorporating resistive-inductive transmission lines among distributed generation components with changing characteristics. This system was chosen based on its exploration in [52] and its complexity. This machine-learning model has undergone validation by several researchers since 2007. Table 4 lists the characteristics and virtual impedances obtained by the suggested optimization algorithm for the three-Bus/MG. These impedances are designed to provide stability and efficiently suppress all kinds of oscillation while reducing differences in reactive power across the converters. Significantly, at |$t=2\ \mathrm{s}$|⁠, a 30-kilowatt load (⁠|${R}_{pe,2}=5\ \Omega \&{X}_{pe,2}=1\ k\Omega$|⁠) was added to bus-1, and the reaction of the MG to this alteration is then analyzed.

Figure 10

Three-bus/MG.

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The MG-SI is calculated based on the non-zero eigenvalues with the lowest real portion. The optimization algorithm is specifically intended to maximize the value in question. Figure 11 displays the vital eigenvalues of the three-Bus/MG, where, they are compared in terms of contrasts between the cases with and without the application of virtual impedances from Table 4. The specific eigenvalues of the MG, obtained using the optimized virtual impedances, are presented in Table 5. The minimal damping ratio, corresponding to modes 25 and 26, is seen to be 25.272%, whereas the bulk of other modes exhibit a damping ratio of 100%. Prior to the installation of these impedances, the stability index of the MG was (⁠|$\lambda =0.314$|⁠). However, after their application, the stability index improved to (⁠|${\lambda}_v=2.0301$|⁠), greatly enhancing the stability of the MG.

Figure 11

The vital eigenvalues of the three-bus/MG.

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The three-Bus MG monitors the frequency response to load changes, specifically in situations when appropriate virtual impedances are used compared to the control approach described in [49]. As illustrated in Fig. 12, at |$t=2\ \mathrm{s}$|⁠, when a 30 kW load is added to bus-1, the frequency reaches its lowest point of 49.91 Hz using ideal virtual impedances. This is a better level of stability compared to the previous control strategy, which obtained a frequency of 49.71 Hz. This highlights the effectiveness of the suggested control and impedance adjustment in improving frequency stability. The frequency’s rate of change is more advantageous when using the ideal configuration. The frequency dip is more noticeable at bus-1 because of its direct load impact. However, the suggested approach and the method from [49] both result in steady-state frequencies that are very near to each other, specifically 49.93 and 49.92 Hz, respectively.

Figure 12

The frequency response to load changes of the three-bus MG.

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Figure 13

The power outputs from the converters under two distinct control regimes.

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The power outputs from the converters under two distinct control regimes are displayed in Fig. 13. When using optimal virtual impedances, converter-1 (⁠|${P}_{1, new}$|⁠) has a smoother transition and lower power overshoot compared to the previous technique. At a condition of equilibrium, each converter generates an identical power output of 9.83 kW, regardless of the control approach used. The proposed method results in a notable enhancement in the reactive power values, with a combined contribution of roughly 288 VAr from all three converters. This reactive power is mostly absorbed by the transmission lines. By comparison, the previous approach in [49] results in inconsistent reactive power outputs among the converters, varying from 5.08 to −3.103 kVAr. The optimization efficiently eliminates internal reactive power transfers and guarantees fair distribution across the converters.

The exit current elements from the converters are described in detail in Fig. 14, where the novel optimum virtual-impedances (⁠|${i}_{\mathrm{od},\mathrm{new}}$|⁠) are compared to the old technique (⁠|${i}_{od}$|⁠) [49]. The q-axis currents using the new method demonstrate reduced peaks at 38.51 A, in contrast to a maximum of 50.88 A achieved with the old approach, hence emphasizing the enhanced efficiency of the new method in current management. The diagram demonstrates that by using ideal virtual impedances, the q-axis currents of all converters stabilize at a consistent value of 0.755 A. This is in contrast to the previous technique, where the q-axis currents varied significantly from 3.349 to −8.12 A. This harmonization efficiently avoids the necessity of q-component current balance among the converters.

Figure 14

The exit current elements from the converters.

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Both control solutions ensure that the d-axis voltage (⁠|${v}_{od}$|⁠) remains at zero throughout steady state. After the load is applied, the q-axis voltages of converters-1 to 3 adjust to 371.6, 382.8, and 391.3 V, respectively, from an initial value of 400 V. This small decrease in voltage is within acceptable ranges, demonstrating adequate voltage regulation. The q-axis voltages were initially set at 1.04 per unit to compensate for the decrease caused by virtual impedances. With the older method [49], the q-axis voltages show slightly less variation after the load change. They stabilize at values ranging from 374.3 to 385.1 V. Both methods effectively maintain the voltage within acceptable limits, proving that strategic adjustments to voltage set points can successfully reduce costs associated with eliminating reactive power exchanges and enhancing MG stability.

4 Conclusions

This study presented a concise model for a self-contained MG, which includes phase-locked loops and virtual impedances, to analyze its tiny signal behavior. This study presented a system that utilizes GWOA to customize the converters’ virtual impedances in an MG. The main objectives of this methodology were to improve the S-SS and optimize the distribution of reactive power. The study assessed the durability of tiny signals under different operational settings, guaranteeing the stability of the MG by strategically implementing virtual impedances. The MG-SI was measured at all working points derived from load-flow calculations, demonstrating significant enhancements with the incorporation of the planned virtual impedances. When the proposed control methodology was used to implement a load change case in the MG, it was noticed that the voltage and frequency of the MG stay within acceptable boundaries. The key outcomes are:

• i) The MG stability index was much increased, reactive power imbalances were eliminated, and transient responses in active powers and current components were dramatically improved compared to earlier studies. The work highlighted that the use of optimum virtual impedance can greatly enhance the stability of MG tiny signals, especially in systems that involve PLL.

• ii) The outcomes from two distinct cases confirm the effectiveness of the suggested approach. In case (I), the MG’s S-SS increases from 1.41 to 3.94, indicating a significant improvement in stability with the new method compared to a previous control scheme.

• iii) The second case study provided additional evidence of the efficacy of the strategy, as indicated by the increase in the stability index from 0.314 to 2.0301. In addition, the methodology greatly improved the distribution of reactive power. In case (I), converters 1 and 2 inject reactive power in a more balanced manner, with each injecting 91.18 Var. This was in contrast to the previous way where the injections were 148.2 and 53.4 Var, respectively.

• iv) In case (II), each converter evenly distributes a reactive power load of 288 Var, which is a significant improvement compared to the uneven values of 5.08 to −3.103 kVAr recorded in the previous method. One potential area for future investigation could be the incorporation of the existing technique into a live, internet-based tuning system for optimizing virtual impedance in MG. This would improve flexibility and operational effectiveness.

Author contributions

Hamidreza Shahravi (Conceptualization [equal], Data curation [equal], Formal Analysis [equal], Resources [equal], Software [equal], Supervision [equal]), Javad Olamaei (Conceptualization [equal], Data curation [equal], Formal Analysis [equal], Resources [equal], Software [equal], Supervision [equal]), Morteza Kheradmandi (Conceptualization [equal], Data curation [equal], Formal Analysis [equal], Resources [equal], Software [equal], Supervision [equal]), and Ali Sadr (Conceptualization [equal], Data curation [equal], Formal Analysis [equal], Resources [equal], Software [equal], Supervision [equal]).

Conflict of interest

None declared.

Funding

None declared.

References