In this study, the microgrid includes static loads that are modeled using an exponential voltage-dependent load model. This model captures the relationship between load demand and system voltage, as described by Equations 1, 2. Alghamdi and Cañizares (2021): P L l = K L P l V L α p l ∀ l (1) Q L l = K L Q l V L α q l ∀ l (2)

In these equations, P L l and Q L l represent the active and reactive power demand of load l , respectively, while V L denotes the load voltage magnitude. The parameters K L P l and K L Q l are scaling factors specific to load l , and α p l and α q l are voltage exponents that define the load type characteristics.

The exponents α p l and α q l determine the sensitivity of the load to voltage changes. Common values represent different types of loads (e.g., constant power, constant current, constant impedance).

The voltage-dependent behavior of these static loads affects the power balance in the microgrid. As the system voltage decreases, the power consumption of these loads also decreases, and vice versa. This characteristic has implications for frequency regulation, as changes in voltage can influence the system’s frequency through the load demand.

Induction motor loads constitute a significant portion of the load in microgrids and are a primary cause of fault-induced delayed voltage recovery (FIDVR) events due to their stalling behavior during voltage sags (Stefopoulos and Meliopoulos, 2006). In order to accurately study the impact of induction motor loads on FIDVR in microgrids, this paper utilizes a full-order wound rotor induction machine model provided by the PSCAD simulation software PSC (Wound rotor induction machine, 2023).

Diesel generators play a crucial role in isolated microgrids, serving as reliable energy sources capable of providing continuous, uninterruptible power to meet the demand and facilitating frequency and voltage regulation. Diesel generators offer multiple advantages, such as their ability to start quickly, their dispatchability, and their capability of operating over a wide range of loading conditions. These characteristics render diesel generators particularly valuable in isolated microgrids, where other energy sources like renewable energy sources (RESs) can be intermittent and may exhibit unpredictable fluctuations (Vandoorn et al., 2011).

In isolated microgrids, diesel generators form an essential part of the primary control, responsible for maintaining the grid frequency and voltage profiles within acceptable limits. Since the diesel generators are coupled to the microgrid through their synchronous generators, their rotational speed is directly related to the system frequency. Consequently, diesel generators can adjust their mechanical input power to counteract variations in frequency due to mismatches between supply and demand, thus assisting in frequency control.

In isolated microgrids, Equation 13 describes the diesel generator’s swing equation, relating rotor inertia ( M D E R ) and damping ( D D E R ) to the balance between mechanical input power ( P m ) and electrical output power ( P D E R ) . The swing equation is given by Bevrani (2017): M DER d ω d t + D DER Δ ω = P m − P DER (13) where ω represents the rotor’s angular velocity (speed), M DER and D DER are the diesel generator’s inertia and damping coefficients respectively, P m denotes the mechanical power input to the generator, and P DER is the electrical output power.

The researchers referenced in Yeager and Willis (1993) have meticulously developed a dynamic model of a diesel generator that is tailored to perform optimally in time-domain simulations. This model is validated by its alignment with an actual field model. In the mentioned study, a comprehensive model is utilized, incorporating both the governor and exciter models. The gains in this model are aptly calibrated to the test system under examination.

Of particular significance is the choice of control mode for the governor model, which is set to Droop Mode as opposed to Isochronous Mode. Droop control is a widely used method for frequency regulation in power systems. It allows load sharing in multiple generator setups and maintains system stability under varying load conditions. The operation of the droop control governor can be quantitively represented by the droop control equation.

Consider, for instance, the following droop control equation as stated in Bevrani (2017): K DER f o − f M G = P DER − P DERo (14) Here, P DERo signifies the nominal active power of the distributed energy resource (DER), f o denotes the nominal system frequency, K DER represents the droop coefficient, and f M G is the observed operational frequency of the microgrid itself.

In microgrids, battery energy storage systems (BESSs) play a crucial role in managing the rapid fluctuations in the output power of renewable energy sources, such as solar and wind generators (Olivares et al., 2014). BESSs are integrated into the microgrid through voltage source converters (VSCs), which enable them to respond quickly to changes in the system’s frequency by delivering active power. This fast response is essential for maintaining the stability of the microgrid, as the dynamics of BESSs are significantly faster than those of conventional generators like diesel engines.

In this study, the BESS operates in a grid-feeding control mode, where it injects specified amounts of active and reactive power into the microgrid. The control strategy involves calculating the reference currents needed to achieve the desired power injection.

The model for the Type 4 wind generators (WGs) with a voltage source converter (VSC) interface operates similarly to the BESS converter described in Section 2.4. The control strategy for the WG also follows the grid-feeding control mode, where it provides active and reactive power according to reference values. However, unlike the BESS, the wind generator is set to operate at unity power factor, meaning that it provides only active power and does not contribute to reactive power support.

The primary difference from the BESS model is in how the active power reference is determined. In the wind generator model, the active power reference P r e f ( t ) is defined by Equation 19, where P r e f ( t ) = P m e a s u r e d ( t ) based on actual wind turbine measurements. P ref t = P measured t (19) where P ref ( t ) is the active power reference at time t , and P measured ( t ) represents the actual measured active power output from the wind turbine at time t , obtained from reference Farrokhabadi et al. (2018), Farrokhabadi et al. (2017).

Since the wind turbine operates at unity power factor, the reactive power reference is set to zero, i.e., Q ref = 0 . All other control aspects such as the current reference calculations and internal control loops are identical to those described in detail in Section 2.4.

The proposed distributed consensus-based VFC scheme relies on a communication graph that enables information exchange among the DERs within the microgrid. The communication graph is represented by G = ( V , E ) , where V denotes the set of vertices or nodes corresponding to the DERs, and E represents the set of edges or communication links between the DERs (Nasirian et al., 2016).

The adjacency matrix A G = [ a i j ] is used to describe the connectivity of the communication graph, where a i j = 1 if there is a communication link between DER i and DER j , and a i j = 0 otherwise. The communication links are assumed to be bidirectional, allowing for the exchange of information between neighboring DERs.

To ensure robustness against communication link failures, the proposed communication architecture employs spanning trees, which maintain graph connectivity even during a single link outage. This approach enhances the reliability and resilience of the distributed VFC scheme, ensuring uninterrupted coordination among the DERs.

The communication graph plays a crucial role in the distributed consensus-based VFC scheme, enabling the exchange of voltage, frequency, and power information among the DERs. This information exchange facilitates the coordination of the DERs’ output powers and the implementation of the proposed control algorithms, ultimately leading to improved voltage and frequency regulation, reactive power sharing, and FIDVR mitigation in the microgrid.

The proposed controller coordinates DERs to regulate microgrid frequencies through active power control and VFC, while addressing FIDVR events. It incorporates loops for DER power output regulation, considering voltage-dependent load coupling. A power-sharing mechanism ensures coordination between distributed active power and voltage controllers.

Using an exponential model for microgrid loads, a distributed consensus controller governs frequency by adjusting DER voltage setpoints. To address unequal voltage distribution and improve FIDVR response, the proposed scheme tackles reactive power sharing issues caused by varying DER-to-common point impedances. The distributed control approach is governed by Equations 20, 21. Equation 20 defines V V F C via lead–lag compensation, while Equation 21 combines frequency deviation Δ f and voltage feedback V n o m − V i ̄ through the gains α f a n d α v . V V FC = β V FC 1 + τ 1 VFC s 1 + τ 2 VFC s ⋅ (20) α f ⋅ K P VFC + α f ⋅ K I VFC s ⋅ α v ⋅ Δ f + 1 − α v ⋅ V nom − V i ̄ (21) where Δ f represents ( f 0 − f M G ) , locally measured as defined in Equation 14. β V FC , K P VFC , and K I VFC are VFC control parameters, while τ 1 VFC and τ 2 VFC denote lead-lag time constants. The nominal microgrid voltage is V nom , and V i ̄ is node i ’s estimated global average voltage. Gain adjustment factors based on frequency deviation and voltage level are α f and α v , respectively.

The VFC controller adjusts the voltage setpoints of the DERs based on the frequency deviation and voltage level, subject to the following limits:

• Voltage limits: V M G ̲ = 0.9 pu, V M G ̄ = 1.1 pu.

As shown in Equation 22, V V F C must remain between V M G and V ̄ M G , enforcing the hard voltage limits of the microgrid. V M G ̲ < V V FC < V M G ̄ (22) where the microgrid’s minimum and maximum tolerable voltages are denoted by V M G ̲ and V M G ̄ , respectively.

The controller operates based on the frequency deviation and voltage level, as summarized in Table 2. During normal conditions (i.e., when the voltage level is within limits), α v = 1 , and the VFC regulates the frequency by adjusting the voltage setpoints based on the frequency deviation Δ f . During FIDVR (i.e., when the voltage level is low), α v = 0 , and the VFC aims to bring the voltage to the nominal value by adjusting the voltage setpoints based on the voltage error ( V nom − V i ̄ ) .

The distributed consensus VFC incorporates V V FC as follows: V i * = K I V s V V FC − V i ̄ + K I Q s ∑ i = 1 n a i j Q D E R j β Q j ⋅ Q D E R j ′ − Q D E R i β Q i ⋅ Q D E R i ′ ∀ i (23) where V i ̄ represents node i ’s estimated global average voltage magnitude, obtained from the distributed voltage estimator. K I V and K I Q are distributed voltage controller parameters. Local DER i ’s reference input voltage and output reactive power are denoted by V i * and Q D E R i , respectively. Adjacent DER j ’s output reactive power is Q D E R j . Maximum reactive power capacities for local and adjacent DERs are Q D E R i ′ and Q D E R j ′ . The communication system linking DERs determines a i j , as described in Section 3.1. Reactive power support factors based on proximity to the fault location are β Q i for DER i and β Q j for DER j .

In Equation 23, the VFC incorporates two terms: one comparing V i ̄ (from the distributed voltage estimator) with the VFC reference voltage, and another for normalized reactive power error to ensure proportional reactive power sharing among DERs. The reactive power regulator compares each DER’s local normalized reactive power with its neighbors’, considering reactive power support factors ( β Q i and β Q j ) based on proximity to fault location during FIDVR events. This approach allows for adjusting DER reactive power to equalize normalized reactive power across all DERs, while prioritizing support from DERs closer to the fault. Consequently, each DER is loaded according to its rating and fault proximity, enhancing the distributed consensus VFC’s effectiveness in mitigating FIDVR. The reactive power support factors ( β Q i and β Q j ) are crucial for prioritizing reactive power contribution based on fault proximity, as detailed in the next subsection.

The proposed distributed active power coordination operates in conjunction with the distributed VFC described in Section 3.2. While both controllers use the frequency deviation term ( f o − f M G ) in their control actions, the coordination between them is achieved through the proper tuning of the control parameters and the introduction of an adaptive gain adjustment mechanism for the active power control loop.

The distributed consensus method for active power coordination is defined by Equations 29, 30, 31, and 32. Equation 29 sets the DER active power reference, while Equations 30 and 31 adapt the control gains under low-voltage conditions.

Finally, Equation 32 coordinates local power flow references among DERs in a distributed manner. P D E R r e f i = f o − f M G K P adp + K I adp s ∀ i (29) where  K P adp = K P FIDV R , if  V i ̄ < V M G ̲ K P , otherwise (30) K I adp = K I FIDV R , if  V i ̄ < V M G ̲ K I , otherwise (31) P D E R i * = K I P s ∑ j ∈ N i a i j P D E R j P D E R j ′ − P D E R i P D E R i ′ + b i P D E R r e f i − P D E R i P D E R i ′ ∀ i (32) where V i ̄ represents the estimated global average voltage at node i . The distributed active power controller’s parameters are denoted by K P , K I , K P FIDV R , K I FIDV R , and K I P .

Proper tuning of control parameters ensures coordination between the active power controllers and VFC. The gains of the VFC ( K P VFC and K I VFC ) are set to provide a fast response to frequency deviations, while the gains of the active power controller ( K P and K I ) are set to provide a slower, more sustained response. This allows the VFC to quickly adjust the voltage setpoints to regulate the frequency, while the active power controller gradually adjusts the active power setpoints to maintain frequency stability over a longer period.

However, during FIDVR events, an adaptive gain adjustment mechanism is triggered in the active power control loop to maintain the frequency of the system and counteract the behavior of constant power loads, especially stalled induction motor loads. When the estimated global average voltage ( V i ̄ ) falls below the minimum tolerable microgrid voltage ( V M G ̲ ) , the gains of the active power controller are adjusted to K P FIDV R and K I FIDV R , which are designed to provide a faster and more aggressive response to frequency deviations caused by the constant power load behavior. This adaptive gain adjustment ensures that the active power control loop prioritizes frequency stability during FIDVR events.

When V i ̄ is above V M G ̲ , indicating that the voltage is within the acceptable range, the gains of the active power controller remain at their default values ( K P and K I ), providing a slower and more sustained response to maintain frequency stability under normal operating conditions.

By implementing this adaptive gain adjustment mechanism, the proposed active power control loop effectively coordinates with the VFC loop to maintain frequency stability and counteract the impact of constant power loads during FIDVR events, while also providing a stable and sustained response under normal operating conditions.

In Equation 32, neighboring DERs communicate output active powers to local DER i for inner power control loop use. P D E R i tracks P D E R r e f i for DER i , while P D E R j represents adjacent DER j ’s output active power. Maximum active power capacities for local and adjacent DERs are P D E R i ′ and P D E R j ′ . The DER communication system determines a i j (0 or 1), as detailed in Section 3.1. b i is set to 1 for local P D E R r e f i signal inclusion in the DER inner control loop, ensuring P D E R i converges to P D E R r e f i .

The proposed scheme coordinates distributed VFC and active power control through parameter tuning. This maintains frequency stability during severe disruptions while providing fast regulation. The voltage recovery coordination term enhances FIDVR event support, ensuring a coordinated response within microgrid voltage constraints.

A summarized block diagram of the entire distributed consensus-based voltage and frequency control (VFC) strategy is illustrated in Figure 2, showing how the various loops, estimators, and communications interconnect.

The modified CIGRE benchmark microgrid, which has been widely used in various studies (Farrokhabadi et al., 2018; Farrokhabadi et al., 2017; Solanki et al., 2017; Solanki et al., 2019; Farrokhabadi et al., 2016), was employed to test, validate, and compare the performance of the proposed distributed consensus-based VFC scheme described in Section 3. The microgrid topology, as illustrated in Figure 3, was implemented in PSCAD for time-domain simulations, while MATLAB was used to conduct small-perturbation analysis. Using a 12-core CPU (average 4.8 GHz per core) and 32 GB RAM, the simulations were conducted.

The microgrid system had a total load of approximately 9 MVA, which included a 2 MVA induction motor load and a mix of constant impedance (60%), constant current (30%), and constant power (10%) loads. The static loads were modeled as unbalanced, consistent with the original CIGRE microgrid system (Farrokhabadi et al., 2017; Farrokhabadi et al., 2016). As detailed in Section 2, the system’s dynamic models for static loads, induction motor load, and DERs were identical to those in the original CIGRE microgrid. Table 3 summarizes the system characteristics.

The microgrid’s Type 4 WGs, modeled according to Farrokhabadi et al. (2018), operated at unity power factor. Feeders were represented as coupled π -sections (Farrokhabadi et al., 2017). Implementing the proposed distributed VFC scheme coordinated dispatchable DERs (DUs and BESS), with a 1 μ s communication time delay Kansal and Bose (2012).

The use of the modified CIGRE benchmark microgrid allows for a comprehensive evaluation of the proposed VFC scheme under realistic operating conditions and enables a fair comparison with other control strategies reported in the literature. The diverse mix of DERs, unbalanced loads, and detailed dynamic models provide a suitable platform to assess the performance of the proposed controller in terms of frequency regulation, voltage stability, power sharing, and fault-induced delayed voltage recovery (FIDVR) mitigation.

A genetic algorithm (GA) optimization technique was employed to tune the proposed distributed consensus-based VFC scheme’s control parameters. These include K P VFC , K I VFC , τ 1 VFC , τ 2 VFC , β V FC , K I V , K I Q , K P , K I , K P FIDV R , K I FIDV R , K I P , β est , K I est , τ 1 Q , and τ 2 Q . This approach, recognized for efficiently tuning microgrid controller settings (Bevrani, 2017), aimed to enhance overall system performance and stability under various conditions. The optimization considered multiple criteria, including minimizing voltage recovery time, ensuring system stability, achieving accurate power sharing, and maintaining frequency stability.

The optimization problem was formulated using a multi-objective function that combines the integral of the time-multiplied absolute value of the error (ITAE) for frequency deviation, voltage deviation, and reactive power sharing error, given by: J = w 1 ∫ 0 ∞ t | Δ f | d t + w 2 ∫ 0 ∞ t | Δ V | d t + w 3 ∫ 0 ∞ t | Δ Q | d t (33) where Δ f is the frequency deviation, Δ V is the voltage deviation, Δ Q is the reactive power sharing error, and w 1 , w 2 , and w 3 are weighting factors that determine the relative importance of frequency stability, voltage stability, and reactive power sharing accuracy in the optimization process. The reactive power sharing error Δ Q is included to ensure that the optimization process considers the effectiveness of the proximity-based reactive power support prioritization mechanism, which is crucial for FIDVR mitigation and overall voltage stability in the proposed VFC strategy.

The genetic algorithm optimization was implemented using PSCAD’s built-in GA optimization tool. The process involved the following steps:

• Random generation of an initial population of potential solutions.

Multiple optimization runs with different random seeds ensured solution robustness. The final control parameters were selected from the best performing solution across all runs.

To evaluate the controller’s performance and determine the optimal parameter values, the microgrid system was subjected to various disturbances, such as load changes, DER outages, fault-induced delayed voltage recovery (FIDVR) events, and communication delays. The GA minimized the objective function Equation 33 until the optimization converged, yielding the tuned controller parameters.

The tuned values of the key control parameters are presented in Table 4. These values were obtained by considering the specific characteristics of the microgrid system, the desired performance criteria, and the results from the GA optimization.

The tuned values of the control parameters ensure that the proposed VFC scheme provides effective frequency regulation, voltage stability, FIDVR mitigation, and accurate reactive power sharing among the DERs. The adaptive gain adjustment mechanism for the active power control loop, with K P FIDV R and K I FIDV R , ensures a fast and robust response to frequency deviations caused by the constant power load behavior of stalled induction motors during FIDVR events. The proximity-based reactive power sharing, governed by I Q and β Q max , prioritizes the reactive power support from DERs closer to the fault location, enhancing the effectiveness of the distributed consensus VFC in mitigating FIDVR.

The GA optimization approach allows for a systematic and efficient tuning process, considering the complex interactions between the various control loops, the specific requirements of the microgrid system, and the additional challenges posed by communication delays and FIDVR events. The resulting tuned parameters provide a balanced trade-off between the different performance objectives, ensuring the stable and reliable operation of the microgrid under various operating conditions.

To comprehensively evaluate the performance of the proposed distributed consensus-based VFC strategy, four simulation scenarios were designed to test the microgrid under various operating conditions and disturbances. These scenarios were chosen to demonstrate the capabilities of the proposed controller in regulating frequency, mitigating fault-induced delayed voltage recovery (FIDVR), maintaining stable operation in the presence of communication delays, and handling different fault durations.

Scenario 1: Sudden load increase during low wind power output.

In this introductory scenario, a sudden load increase is applied to the microgrid at t = 105 s, coinciding with the lowest output power from the wind generators. This scenario aims to demonstrate the effectiveness of the proposed VFC strategy in regulating the microgrid frequency and maintaining stable operation under challenging conditions. The performance of the proposed controller is compared to the base case with conventional droop control to highlight its superior frequency regulation capabilities.

Scenario 2: Three-phase fault at Bus 8 causing FIDVR.

To evaluate the proposed VFC strategy’s ability to mitigate FIDVR, a three-phase fault is applied to Bus 8 at t = 20 s. This scenario is designed to assess the impact of FIDVR on the microgrid voltage and compare the voltage recovery performance of the proposed controller with the base case and conventional VFC strategies. By demonstrating improved voltage recovery, this scenario showcases the effectiveness of the proposed VFC strategy in enhancing the microgrid’s resilience to FIDVR events.

Scenario 3: Three-phase fault at Bus 8 with communication delays.

Building upon Scenario 2, this scenario introduces communication delays to the microgrid system while a three-phase fault is applied at Bus 8. The purpose of this scenario is to investigate the robustness of the proposed VFC strategy in the presence of communication delays, which can potentially impact the performance of distributed control schemes. By evaluating the controller’s performance under these conditions, this scenario demonstrates the resilience and practicality of the proposed VFC strategy in real-world microgrid applications.

Scenario 4: Impact of fault duration on voltage recovery.

In this scenario, the impact of fault duration on the voltage recovery time is investigated for both the proposed VFC strategy and the conventional VFC. A three-phase fault is applied at Bus 8 with varying durations, specifically 0.045s and 0.05s. The purpose of this scenario is to assess the effectiveness of the proposed VFC strategy in handling different fault durations and to compare its performance with the conventional VFC in terms of voltage recovery time and stability. By demonstrating faster voltage recovery times and improved stability under different fault durations, this scenario highlights the robustness and superiority of the proposed VFC strategy in mitigating FIDVR events and maintaining microgrid stability.

These four simulation scenarios provide a comprehensive evaluation of the proposed VFC strategy’s performance under various operating conditions, disturbances, and challenges. By assessing the controller’s effectiveness in regulating frequency, mitigating FIDVR, maintaining stable operation in the presence of communication delays, and handling different fault durations, these scenarios demonstrate the versatility, resilience, and practicality of the proposed VFC strategy in real-world microgrid applications.

The performance of the proposed distributed consensus-based VFC strategy was evaluated under a challenging scenario where the microgrid experiences a sudden load increase at t = 105 s, coinciding with the lowest output power from the wind generators. Figure 4 shows the frequency response of the microgrid system under the proposed VFC strategy (blue line) compared to the base case with conventional droop control (red line).

In the base case, the microgrid frequency drops significantly, reaching a minimum of approximately 59.55 Hz, and undergoes several oscillations before slowly recovering and eventually collapses again. In contrast, the proposed VFC strategy demonstrates superior performance, with a less severe frequency drop (minimum of around 59.85 Hz), faster and smoother recovery, and fewer oscillations.

The improved frequency response under the proposed VFC strategy is attributed to the distributed consensus-based control architecture, the coordination between the VFC and active power control loops, and the optimally tuned control parameters. These results highlight the effectiveness of the proposed VFC strategy in maintaining microgrid frequency stability and ensuring smooth operation under challenging scenarios.

In the following subsections, we will further investigate the performance of the proposed VFC strategy under different simulation scenarios, including fault-induced delayed voltage recovery (FIDVR) events, and compare its performance with existing control methods reported in the literature.

Scenario 2: Three-phase fault at Bus 8 causing FIDVR.

To evaluate the performance of the proposed VFC strategy in mitigating fault-induced delayed voltage recovery (FIDVR), a three-phase fault was applied at Bus 8 at t = 20 s, lasting for 0.045 s. The voltage response of the microgrid was observed, focusing on the average microgrid voltage (Figure 5) and the voltage at the terminal of the induction motor load (Figure 6).

In the base case with conventional VFC (red line), the average microgrid voltage (Figure 5) experiences a severe drop to approximately 0.1 pu during the fault. After the fault is cleared, the voltage recovers slowly, taking more than 2 s to reach the pre-fault level. This delayed voltage recovery is primarily caused by the stalling of induction motor loads, which draw a large amount of reactive power during the post-fault period.

Similarly, the voltage at the induction motor terminal (Figure 6) drops to nearly zero during the fault and exhibits a delayed recovery in the base case. The motor voltage remains below 0.8 pu for more than 2 s after the fault is cleared, indicating a prolonged stalling condition.

In contrast, the proposed VFC strategy (blue line) demonstrates a significantly improved voltage recovery performance. The average microgrid voltage (Figure 5) drops to approximately 0.1 pu during the fault. However, after the fault is cleared, the voltage recovers much faster, reaching the pre-fault level within 1 s. This rapid voltage recovery is achieved by the proposed VFC’s ability to detect the FIDVR condition and adjust the voltage setpoints of the DERs to provide additional reactive power support.

The voltage at the induction motor terminal (Figure 6) also exhibits a faster recovery under the proposed VFC strategy. The motor voltage reaches 0.8 pu within 1 s after the fault is cleared, indicating a shorter stalling duration and improved motor performance.

Figure 7 compares the voltage recovery time of the proposed VFC strategy and the conventional VFC. The proposed VFC demonstrates a significantly faster voltage recovery, taking approximately 0.5 s, while the conventional VFC requires around 2 s to restore the voltage to its nominal value. This highlights the superior performance of the proposed VFC strategy in mitigating FIDVR and ensuring rapid voltage recovery.

The superior voltage recovery performance of the proposed VFC strategy can be attributed to several factors:

1. The modified VFC equation Equation 21, which includes a voltage error term and a switching mechanism based on the estimated global average voltage, enables the controller to prioritize voltage recovery during FIDVR events.

These results demonstrate the effectiveness of the proposed VFC strategy in mitigating FIDVR and ensuring a faster and more stable voltage recovery compared to conventional VFC methods. By maintaining higher voltage levels during faults and reducing the duration of induction motor stalling, the proposed VFC strategy enhances the microgrid’s resilience to disturbances and improves overall system stability.

In Scenario 2, the reactive power sharing performance of the proposed VFC strategy was evaluated during a three-phase fault at Bus 8, which caused a fault-induced delayed voltage recovery (FIDVR) event. Figure 8 shows the reactive power output of BESS 1 and BESS 3 during the post-fault period.

The reactive power sharing mechanism in the proposed VFC strategy is based on the proximity of each DER to the fault location, as described in the power sharing Equation 23. After the fault is cleared, BESS 3, which is closer to the fault location at Bus 8, provides a higher reactive power output compared to BESS 1. This is in accordance with the proximity-based reactive power sharing mechanism, where DERs closer to the fault location are assigned higher reactive power support factors ( β Q i ) and, consequently, contribute more to the voltage recovery process.

The gradual reduction in reactive power output of both BESS units after the fault is cleared indicates the effectiveness of the proposed VFC strategy in mitigating the FIDVR event and restoring the microgrid voltage to its nominal value. The proximity-based reactive power sharing ensures that the DERs closer to the fault location bear a larger share of the reactive power burden, thereby minimizing the overall voltage deviations and enhancing the speed of voltage recovery.

These results demonstrate the successful implementation of the proximity-based reactive power sharing mechanism in the proposed VFC strategy, which prioritizes the contribution of DERs closer to the fault location during FIDVR events. By effectively coordinating the reactive power support provided by the DERs, the proposed VFC strategy ensures a faster and more stable voltage recovery, enhancing the microgrid’s resilience to disturbances.

Figure 9 shows the speed of the induction motor load connected to the microgrid during Scenario 2. The motor speed experiences a sudden drop due to the voltage sag caused by the three-phase fault at Bus 8. In the case of conventional VFC (red line), the motor speed drops to approximately 0.92 pu and takes a long time to recover after the fault is cleared, indicating the impact of fault-induced delayed voltage recovery (FIDVR) on the motor’s performance.

In contrast, the proposed VFC strategy (blue line) demonstrates a significantly improved motor speed response. Although the motor speed drops during the fault, it recovers much faster after the fault is cleared compared to the conventional VFC case. The faster recovery of the motor speed under the proposed VFC strategy can be attributed to the improved voltage recovery performance, as discussed in the previous sections.

The proposed VFC strategy demonstrates superior performance in restoring the system frequency and voltage to nominal values compared to the conventional VFC approach, as evident from Figures 1, 2.

Figure 10 shows that the active power output of BESS 1 and 3 using the proposed VFC strategy exhibits a more rapid and coordinated response to the fault-induced disturbance compared to BESS 1 using conventional VFC. The proposed VFC enables the BESS units to quickly inject active power into the system to support frequency recovery.

Figure 11 illustrates the system frequency response under both the proposed and conventional VFC strategies. The proposed VFC achieves a faster frequency recovery, with the frequency returning to its nominal value within approximately 2 s after the fault. In contrast, the conventional VFC exhibits a slower and more oscillatory frequency response, taking longer to stabilize.

The effectiveness of the proposed VFC strategy can be attributed to the adaptive gain adjustment mechanism implemented in the active power control loop. During the FIDVR event, the gains of the active power controller are dynamically adjusted to provide a faster and more aggressive response to frequency deviations caused by the constant power load behavior of stalled induction motor loads. This adaptive approach ensures that the active power control loop prioritizes frequency stability, enabling a more rapid and robust frequency recovery.

The coordinated action of the BESS units, facilitated by the distributed consensus control architecture, further enhances the overall system performance. The BESS units collectively contribute to frequency regulation and voltage support, leading to a more stable and resilient microgrid operation.

Scenario 3: Three-phase fault at Bus 8 with communication delays.

The impact of communication delays on the performance of the proposed VFC strategy was evaluated by applying a three-phase fault at Bus 8 and observing the average microgrid voltage recovery under different time delay conditions (150 µs, 50 m, 100 m, and 500 m), as shown in Figure 12.

The results demonstrate that the proposed VFC strategy can restore the average microgrid voltage to its nominal value, even in the presence of communication delays. However, the voltage recovery time increases with larger time delays. In the worst-case scenario (500 m delay), the voltage recovery is significantly slower, taking approximately 1.5 s to reach the nominal value after the fault is cleared. Despite the slower recovery, no instability is observed, highlighting the robustness of the proposed VFC strategy.

The slower voltage recovery with larger time delays is attributed to the delayed communication and coordination among the DERs. Nevertheless, the proposed VFC strategy’s adaptive gain adjustment mechanism and the coordination between the VFC and active power control loops help maintain stability and eventually restore the voltage to its nominal value.

These results demonstrate the resilience and effectiveness of the proposed VFC strategy in handling communication delays, ensuring reliable operation of the microgrid under realistic conditions where communication infrastructure may introduce latencies.

Scenario 4: Impact of fault duration on voltage recovery.

Figure 13 shows the voltage at the terminal of the induction motor (IM) load for fault durations of 0.045s and 0.05s, comparing the proposed VFC and the conventional VFC.

The proposed VFC achieves faster voltage recovery compared to the conventional VFC for both fault durations. With a fault duration of 0.05s, the conventional VFC fails to recover the voltage, and the IM load stalls, while the proposed VFC restores the voltage within approximately 1 s after the fault is cleared. For a fault duration of 0.045s, the proposed VFC exhibits a faster voltage recovery than the conventional VFC. Longer fault durations lead to slower voltage recovery for both control strategies.

Figure 14 compares the voltage recovery time of the proposed VFC strategy and the conventional VFC for different fault durations. The proposed VFC consistently achieves faster voltage recovery times compared to the conventional VFC, with recovery times of approximately 1.1 s and 1.3 s for fault durations of 0.045s and 0.05s, respectively. In contrast, the conventional VFC exhibits significantly slower recovery times of around 2.5 s for a fault duration of 0.045s and fails to recover the voltage for a fault duration of 0.05s, indicating potential voltage instability.

To provide a consolidated overview of the improvements achieved by the proposed VFC strategy, Table 5 presents a comparison of key performance metrics between the proposed VFC and conventional methods.

As observed from Table 5, the proposed VFC strategy reduced the voltage recovery time after a 0.045s fault from 2.5 s to 1.1 s, representing a 56% improvement. Similarly, under sudden load increases, the frequency deviation was limited to ± 0.15 Hz with the proposed VFC, compared to ± 0.45 Hz with conventional droop control, resulting in a 66% reduction. The table also highlights the controller’s robustness to communication delays of up to 500 m and improved reactive power sharing among DERs.