Intelligence-driven Grid-forming Converter Control for Islanding Microgrids

Issarachai Ngamroo; Tossaporn Surinkaew; Yasunori Mitani

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Intelligence-driven Grid-forming Converter Control for Islanding Microgrids PDF

- ORCID：
Issarachai Ngamroo ¹
✉
- ORCID：
Tossaporn Surinkaew ²
✉
- ORCID：
Yasunori Mitani ³
✉

1. Department of Electrical Engineering, School of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand； 2. School of International and Interdisciplinary Engineering Programs, School of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand； 3. Kyushu Institute of Technology, Fukuoka 804-8550, Japan

Updated：2025-07-23

DOI：10.35833/MPCE.2024.001157

OUTLINE

Abstract

In modern microgrids (MGs) with high penetration of distributed energy resources (DERs), system reconfiguration occurs more frequently and becomes a significant issue. Fixed-parameter controllers may not handle these tasks effectively, as they lack the ability to adapt to the dynamic conditions in such environments. This paper proposes an intelligence-driven grid-forming (GFM) converter control method for islanding MGs using a robustness-guided neural network (RNN). To enhance the adaptability of the proposed method, traditional proportional-integral controllers in the GFM primary control loops are entirely replaced by the RNN. The RNN is trained by a robustness-guided strategy to replicate their robust behaviors. All the training stages are purely data-driven methods, which means that no system parameters are required for the controller design. Consequently, the proposed method is an intelligence-driven model-less GFM converter control. Compared with traditional methods, the simulation results in all testing scenarios show the clear benefits of the proposed method. The proposed method reduces overshoots by more than 71.24%, which keeps all damping ratios within the stable region and provides faster stabilization. In comparison to traditional methods, at the highest probability, the proposed method improves damping by over 14.7% and reduces the rates of change of frequency and voltage by over 59.97%. Additionally, the proposed method effectively suppresses the interactions between state variables caused by inverter-based resources, with frequencies ranging from 1.0 Hz to 1.422 Hz. Consequently, these frequencies contribute less than 19.79% to the observed transient responses.

Keywords

Distributed energy resource; grid-forming converter; microgrid; neural network; intelligence-driven control; stability

I. Introduction

GRID-FORMING (GFM) converters are crucial for grid voltage and frequency regulation, especially in 100% power electronics-based systems [

1], [2]. The GFM converter control strategy addresses various challenges in low-inertia/inertia-less microgrids (MGs) such as supporting grid frequency and voltage, providing pseudo-inertia, operating in standalone/islanding mode, and enabling black-start capability [1], [2]. However, traditional GFM controllers, specifically proportional-integral (PI) controllers embedded in tertiary, secondary, and primary control loops, encounter unsolvable and challenging problems when applied in weak MGs [3]. Additionally, these controllers face significant challenges in systems with high information uncertainties [4]. Research works including [1] and [5] have noted that fixed-parameter model-based controllers often struggle with adaptability and performance due to the reliance on accurate and up-to-date system models [1], [2]. Inadequate gain selection can negatively impact MG stability [1], [2], and the complexity of traditional GFM controllers can hinder their efficiency and practicality, which requires laborious tuning and optimization procedures [5]. These challenges become more pronounced in MGs with high penetration of distributed energy resources (DERs). Alternative methods such as intelligence-driven methods present effective solutions to these limitations. Intelligence-driven methods show promising capabilities for enhancing adaptability and optimizing performance in DER-based MG [6].

In [

7], a bi-objective control technique for MG load-frequency control is used to address uncertainties from renewable energy sources and optimize battery storage. This strategy can improve the characteristics of stability and frequency. However, the simplified load-frequency control model in [7] might neglect the impact of reactive power. A multi-layer interactive control strategy for MGs with DERs in [8] ensures the stability using optimal control parameters, but does not explicitly address DER intermittency. In [9], a novel virtual damping stabilizer in a battery enhances frequency stability and reduces capacity requirements, though potential DER intermittency drawbacks are not discussed. In [10], a control strategy is presented for managing transient power among inverter-based distributed generators, which ensures quick integration, and does not explicitly address the intermittency of DERs. Reference [11] develops a dynamic model for photovoltaic (PV)-based MGs to enhance stability, yet thoroughly explores PV generation intermittency. In [12], a settling-angle-based stability criterion is proposed, which uses the phase information of converter impedance and combines grid-following (GFL) and GFM modes. This criterion might overlook the impacts of reactive power. Reference [13] compares the control loops of GFL and GFM converters. A time-domain simulation of a 1.5 kW converter is used to analyze stability under various conditions. The results show that GFL control suits stiff grids, while GFM converter control is better for weak grids. In [14], a dual-port GFM converter control for power systems with AC/DC transmission is presented, which reduces the complexity by integrating both GFM and GFL functions and includes modeling and stability analysis for both emerging and legacy technologies. In [15], an MG control strategy that eliminates the need for a centralized GFM converter is proposed, which uses improved centralized control and droop-based power loop in DERs. The strategy supports both grid-connected and islanded modes, which ensures power sharing and smooth transitions without the need for islanding detection. In [16], an autonomous control architecture for grid-interactive inverters in MGs during utility outages is proposed, which highlights key tasks such as load sharing and voltage/frequency restoration when the MG isolates from the grid. The control architecture allows inverters to detect reconnection to the utility grid without communication, which enables a seamless transition from GFM mode to GFL mode. In [17], a GFM converter control method for solid oxide fuel cells to address voltage and frequency regulation is presented, which incorporates an incremental impedance technique to track the maximum power point and adjust output frequency during load transients.

As can be observed, traditional methods are mainly applied in previous research on stability and control. There is little focus on intelligence-driven methods like machine learning or neural networks, which could enhance system stability, decision-making, and real-time control and highlight a gap in the application of advanced techniques to power systems. As a result, the limitations of traditional methods emphasize the need for intelligence-driven methods, which could provide an advanced level of control for modern power systems [

18], [19]. From this perspective, recent research works show promising intelligence-driven methods as follows. In [20], a decentralized control strategy for PV-based DC MGs is presented, which enables the cooperation among multiple PV sources without communication. A predictive control strategy is introduced for GFM PV units, eliminating the need for external PI controllers, enhancing MG stability, reliability, and economy, while considering various operating conditions. Reference [21] uses an intelligence-driven controller for MG frequency and power sharing, outperforming traditional methods. Reference [22] introduces an adaptive control strategy for damping oscillations in DC MGs with hybrid power sources, incorporating a multi-loop voltage controller and virtual impedance loop. Reference [23] addresses renewable energy integration challenges with an adaptive droop controller and particle swarm optimization, enhancing stability and minimizing fluctuations. Reference [24] proposes virtual inertia and master-slave control using a recurrent probabilistic wavelet fuzzy neural network. Reference [25] offers a distributed intelligent secondary control method for AC MGs with a brain emotional learning-based controller. Reference [26] develops an adaptive dynamic programming-based strategy, optimizing the power outputs of micro-turbine and battery energy storage station (BESS) with PV integration. It is shown that this strategy outperforms PI-derivative, linear quadratic regulator, and fuzzy logic controllers. Despite valuable insights, these intelligence-driven methods exhibit limitations such as insufficient exploration of replacing physical controllers in GFM or GFL control loops and inadequate adaptive learning mechanisms for dynamic adaptation. Addressing these gaps paves the way for advanced intelligence-driven methods in MG control.

A robustness-guided neural network ( $R N N$ ) framework is proposed in this paper, and the advantages of the proposed RNN framework compared with recent GFM converter control methods for DERs are summarized in Table I.

TABLE I Comparision of Proposed RNN Framework with Recent GFM Converter Control Methods for DERs

Ref.	Method	Control architecture	Adaptability to uncertainty	Supportiveness for islanding mode	Supportiveness for inertia-less MG	Historical data learning	Simplification of control loop	Computational burden
[8]	Multi-layer interactive control	PI-based GFM	Limited	$\sqrt$	$\sqrt$	$\times$	$\sqrt$	Low
[11]	Dynamic modeling for PV-based MGs	PI-based GFM	Limited	$\sqrt$	$\times$	$\times$	$\sqrt$	Moderate
[12]	Settling-angle-based stability criterion	PI-based GFL and GFM	Moderate	$\times$	$\times$	$\times$	$\sqrt$	Low
[20]	Decentralized control for PV-based DC MGs	Predictive control for GFM PV unit	High	$\sqrt$	$\times$	$\sqrt$	$\times$	Moderate
This paper	Proposed RNN	Intelligence-driven GFM	Very high	$\sqrt$	$\sqrt$	$\sqrt$	$\sqrt$	High

Note that the intelligence-driven method may require significant computational resources and deep expertise for designing and training the proposed RNN framework. However, since this process is performed offline, the computational burden is limited to the training phase. Once trained, the control or decision-making response of the proposed RNN framework is faster and more effective compared with traditional methods in the literature. To address the identified gaps in the literature, this paper presents the following contributions.

1) A new RNN framework for GFM converter control is introduced, enhancing the robustness in MG uncertainties and improving the damping performance during rapid changes of voltage and frequency in islanding MGs.

2) Physical PI controllers in the GFM converter control topology are replaced with two RNN schemes integrated into pulse width modulation (PWM), which simplifies the control architecture, eliminates additional control loops, and enhances adaptability and responsiveness to uncertainties and disturbances.

3) The intelligence-driven GFM converter control method is proposed, which involves injecting the input into the PWM, and reduces the reliance on intermediate control layers. This method improves the responsiveness and adaptiveness of DERs. Although this method shares structural similarities with traditional PI-based control loops, the novelty lies in the specific way it is implemented and the resulting performance, adaptability, and stability enhancements.

4) The proposed RNN framework continuously updates the RNN model, allowing it to learn from new data, adjust parameters, and effectively mitigate adverse effects during recurrent events.

II. Overview of GFM Converter Control in Islanding MGs

An overview of multi-MGs with DERs is shown in Fig. 1, where N_mg is the number of MG modules; $N_{L, i}$ and $N_{D E R, i}$ are the total numbers of loads and DERs of the i^th sub-MG, respectively; PCC denotes point of common coupling; and $N_{s m g}$ is the total number of sub-MGs within an MG.

Fig. 1 Overview of multi-MGs with DERs.

Overall, the MG consists of two layers, i.e., physical layer and cyber layer. The physical layer incorporates DERs equipped with control systems and other infrastructures. Specifically, the cyber layer is constructed hierarchically, consisting of centralized, local, and distributed layers. Each of the cyber layer receives information from the lower layer such as voltage, frequency, active/reactive power, and device/component statuses.

The hierarchical GFM converter control method chooses voltage and frequency references, i.e., $v^{r e f} (t)$ and $f^{r e f} (t)$ , from the GFM converter to form its MG/sub-MG. These selected references are then transmitted to other GFL converters to synchronize with the corresponding GFM converters.

In the grid-connected scenario, multi-MGs, each with sub-MGs, operate in grid-tied modes. $v^{r e f} (t)$ and $f^{r e f} (t)$ are derived from PCC using a phase-locked-loop strategy, which can ensure a synchronization with the main grid through centralized control and account for variable delays when $v^{r e f} (t) = v_{P C C} (t)$ and $f^{r e f} (t) = f_{P C C} (t)$ , where $v_{P C C} (t)$ and $f_{P C C} (t)$ are the voltage and frequency of the PCC at time $t$ , respectively. In the islanding scenario, when the MG disconnects from the grid, the voltage and frequency references change. These references are set to be the values of the DER with the highest power output. Consequently, the voltage reference is $v^{r e f} (t) = v_{D E R}^{⋆} (t)$ , and the frequency reference is $f^{r e f} = f_{D E R}^{⋆} (t)$ , where $v_{D E R}^{⋆} (t)$ and $f_{D E R}^{⋆} (t)$ are the voltage and frequency of the DER at the highest power output at time $t$ , respectively. Accordingly, the DER with the highest capacity operates in GFM mode, while others operate in GFL mode. The decentralized control is used with limited observable output signals and specific local communications.

We also consider the time delay in all input and output signals across different sub-MGs, which accounts for the impact of communication delays and reflects how these delays influence both local and centralized control processes. As a result, the time $t$ can be represented as $t - τ_{D E R, l} (t)$ , where $τ_{D E R, l} (t)$ is the local communication delay specific to the corresponding $l^{t h}$ DER within the $s m g^{t h}$ sub-MG, and with conditions $τ_{D E R, 1} (t) \neq τ_{D E R, 2} (t) \neq τ_{D E R, 3} (t) \neq \dots \neq τ_{D E R, N_{L}} (t)$ . This adjustment accounts for the effect of time delay. For example, it accounts for the delay of the signal by shifting the original signal by $τ_{D E R, l} (t)$ . This delay arises from the exchange of information between local controllers and their associated DERs. The value of $τ_{D E R, l} (t)$ is always less than $τ_{c} (t)$ , i.e., $\forall τ_{D E R, l} (t) \leq τ_{c} (t)$ , where $τ_{c} (t)$ is the communication delay between local controllers and the centralized controller. Both $τ_{D E R, l} (t)$ and $τ_{c} (t)$ depend on time and vary over time. Accordingly, $τ_{D E R, l} (t)$ and $τ_{c} (t)$ are represented by variable time delays. The concept of variable time delays can be found in [

27], [28]. The distinction between these two delays lies in the fact that

τ_{D E R, l} (t)

is associated with localized interactions within a sub-MG, whereas

τ_{c} (t)

is the larger-scale communication delay across the entire system. Moreover, if sub-MGs are further isolated from the others, voltage and frequency references can be obtained using

v^{r e f} (t) = v_{s m g, D E R}^{⋆} (t)

and

f^{r e f} = f_{s m g, D E R}^{⋆} (t)

, where

v_{s m g, D E R}^{⋆} (t)

and

f_{s m g, D E R}^{⋆} (t)

are the voltage and frequency of the DER in the corresponding

s m g^{t h}

sub-MG with the highest power outputs at time

t

, respectively. These references are based on the DER with the highest power outputs within each sub-MG at time

t

. This scenario uses distributed control to eliminate communication issues.

III. Proposed Intelligence-driven GFM Control Method for Replacing GFM Control Loops

A. Proposed RNN Schemes

An overview of a GFM converter is shown in Fig. 2. $S_{V S C}^{1}$ - $S_{V S C}^{6}$ are the modulation signals of insulated gate bipolar transistor switches of a GFM converter. The dynamics of low-inertia MGs can be expressed using (1), which establish a connection between system outputs $y (t)$ and underlying dynamics:

y (t) = f_{y} (x (t), u (t), d (t))

(1)

Fig. 2 Overview of GFM converter.

where $x (t)$ is the state variable; $u (t) = - K y (t)$ is the input vector from the intelligence-driven method and $K$ denotes the RNN; $d (t)$ is the disturbance vector originating from wind and PV generators; and $f_{y} (\cdot)$ is the nonlinear function incorporating these inputs to determine $y (t)$ .

In the proposed RNN framework, $u (t)$ takes on a more direct role by being seamlessly integrated into the PWM process. By directly injecting $u (t)$ into the PWM, the physical PI controllers are replaced by the RNN within the GFM converter control loops. Note that the proposed RNN framework shares similarities with traditional PI-based control methods, which simplifies the control architecture and overcomes challenges of physical controllers, such as limited flexibility and reliance on accurate mathematical models. In RNN scheme 1, as shown in Fig. 3, the traditional PI controllers in the voltage and current control loops are replaced with intelligence-driven modules. These modules include $R N N_{v c, d}$ and $R N N_{v c, q}$ for voltage control, $R N N_{i c, d}$ and $R N N_{i c, q}$ for current control, and $R N N_{θ}$ for phase angle control. Here, the modules use RNNs, which provide enhanced performance and offer greater adaptability compared with traditional PI controllers.

Fig. 3 Proposed RNN scheme 1.

Accordingly, $u (t)$ of the RNN scheme 1 can be written as:

u (t) = {[\begin{matrix} v_{d q}^{*} (t) & i_{d q}^{*} (t) & θ^{*} (t) \end{matrix}]}^{T}

(2)

where $T$ is the transpose operator; $v_{d q}^{*}$ and $i_{d q}^{*}$ are the $d q$ -axis commanded signal vectors from voltage and current control loops, respectively; and $θ^{*}$ is the commanded signal from phase angle control loop.

In (2), $v_{d}^{*}$ and $v_{q}^{*}$ are transformed to $v_{a}^{*}$ , $v_{b}^{*}$ , and $v_{c}^{*}$ using the inverse Park transformation, which are consequently used to command the PWM of the GFM converters. From Fig. 3, expressing (2) will yield:

v_{d q}^{*} (t) = R N N_{i c, d q} (i_{d q}^{*} (t)) - \frac{3}{2} ω (t) L_{f} i_{q d} (t)

(3)

i_{d q}^{*} (t) = R N N_{v c, d q} (v_{d q}^{r e f} (t) - v_{d q} (t)) + i_{d q} (t) - \frac{3}{2} ω C_{f} v_{q d} (t)

(4)

θ^{*} (t) = 2 π \cdot R N N_{θ} (\int_{t}^{t + Δ t} (P (t) + k_{f} f^{r e f} (t)) d t)

(5)

where $Δ$ denotes the small deviation; $i_{q d}$ and $v_{q d}$ are the $q d$ -axis current and voltage, respectively; $v_{d q}^{r e f}$ is the reference signal of $v_{d q}$ ; $P$ is the vector representing active power output; $k_{f}$ is the frequency control gain; $ω$ is the angular velocity; $f^{r e f}$ is the vector representing frequency reference; and $C_{f}$ and $L_{f}$ are the capacitance and inductance, respectively.

Figure 4 shows RNN scheme 2, which employs a unified $R N N_{v i c}$ module integrating voltage and current control loops.

Fig. 4 Proposed $R N N$ scheme 2.

The total output number of $R N N$ scheme 2 differs from that in RNN scheme 1. RNN scheme 2 forms a unified model. Additional variables are introduced as $Δ v_{d q} (t) = v_{d q}^{r e f} (t) - v_{d q} (t)$ and $Δ i_{d q} (t) = i_{d q}^{r e f} (t) - i_{d q} (t)$ . Accordingly, $i_{d q}^{r e f} (t)$ is the reference signal of $i_{d q} (t)$ . With these considerations, $u (t)$ of RNN scheme 2 can be expressed as:

u (t) = {[\begin{matrix} v_{d q}^{*} (t) & θ^{*} (t) \end{matrix}]}^{T}

(6)

v_{d q}^{*} (t) = R N N_{v i c} (Δ v_{d q} (t), Δ i_{q d} (t))

(7)

θ^{*} (t) = R N N_{v i c} (P (t), f^{r e f} (t))

(8)

B. Robust Intelligent-driven Controller Design

To replace traditional PI controllers with the proposed method, specific expressions using (3)-(5), (7), and (8) are initially defined. $K$ is introduced. Consequently, $u (t)$ can be modified as:

u (t) = f_{o} (K (W_{h, n}, b_{h, n}, b_{o}, t), y (t)) + 𝒫 (t)

(9)

u (t) = f_{o} [\sum_{h = 1}^{N_{h}} \sum_{n = 1}^{N_{n}} (W_{h, n} {\tilde{y}}_{h, n} (t) + b_{h, n}) + b_{o}] + 𝒫 (t)

(10)

where subscripts $h = 1,2, \dots, N_{h}$ and $n = 1,2, \dots, N_{n}$ are the layer and node indices, respectively; $N_{h}$ and $N_{n}$ are the total numbers of hidden layers and nodes, respectively; $𝒫$ represents any known physically-guided equation, if applicable; $f_{o} (\cdot)$ is the nonlinear activation function for the output layer; ${\tilde{y}}_{h, n}$ is the transformed output vector $y$ that passes through a node of a hidden layer (at an input layer $h = 1$ , $\tilde{y} = y$ ); $W_{h, n}$ and $b_{h, n}$ are the weight and bias vectors, respectively; and $b_{o}$ is the bias vector of the output layer.

For the RNN scheme 1 shown in Fig. 3 with (3)-(5), the output signals of all individual RNNs are explicitly expressed in the form of (10) as:

i_{d q}^{*} (t) = f_{o} \{\sum_{h = 1}^{N_{h}} \sum_{n = 1}^{N_{n}} [W_{h, n}^{1} ({\tilde{v}}_{d q, h, n} (t) - η v_{d q}^{r e f} (t)) + b_{h, n}^{1}] + b_{o}^{1}\} + i_{d q} (t) - \frac{3}{2} ω (t) C_{f} v_{q d} (t)

(11)

v_{d q}^{*} (t) = f_{o} [\sum_{h = 1}^{N_{h}} \sum_{n = 1}^{N_{n}} (W_{h, n}^{2} {\tilde{i}}_{d q, h, n}^{*} (t) + b_{h, n}^{2}) + b_{o}^{2}] - \frac{3}{2} ω (t) L_{f} i_{q d} (t)

(12)

θ^{*} (t) = f_{o} [2 π \sum_{h = 1}^{N_{h}} \sum_{n = 1}^{N_{n}} (W_{h, n}^{3} w_{θ} ({\tilde{P}}_{h, n} (t), {\tilde{f}}_{h, n}^{r e f} (t)) + b_{h, n}^{3}) + b_{o}^{3}]

(13)

w_{θ} ({\tilde{P}}_{h, n} (t), {\tilde{f}}_{h, n}^{r e f} (t)) = \int_{t}^{t + Δ t} ({\tilde{P}}_{h, n} (t) + k_{f} {\tilde{f}}_{h, n}^{r e f} (t)) d t

(14)

where ${\tilde{v}}_{d q, h, n}$ , ${\tilde{i}}_{d q, h, n}^{*}$ , ${\tilde{P}}_{h, n}$ , and ${\tilde{f}}_{h, n}^{r e f}$ are the transformed output vectors of $v_{d q}$ , $i_{d q}^{*}$ , $P$ , and $f^{r e f}$ , respectively. In (11), if $h = 1$ , $η = 1$ ; otherwise, $η = 0$ .

For the RNN scheme 2 illustrated in Fig. 4, the output signals from the $R N N_{v i c}$ are expressed in the form of (10) as:

v_{d q}^{*} (t) = f_{o} [\sum_{h = 1}^{N_{h}} \sum_{n = 1}^{N_{n}} (W_{h, n}^{1} w_{v i c} ({\tilde{v}}_{d q, h, n} (t), {\tilde{i}}_{d q, h, n} (t)) + b_{h, n}^{1}) + b_{o}]

(15)

θ^{*} (t) = f_{o} [\sum_{h = 1}^{N_{h}} \sum_{n = 1}^{N_{n}} (W_{h, n}^{2} w_{θ} ({\tilde{P}}_{h, n} (t), {\tilde{f}}_{h, n}^{r e f} (t)) + b_{h, n}^{2}) + b_{o}^{2}]

(16)

w_{v i c} ({\tilde{v}}_{d q, h, n} (t), {\tilde{i}}_{d q, h, n} (t)) = α_{d q} ({\tilde{v}}_{d q, h, n} (t) - η v_{d q}^{r e f} (t)) + β_{d q} ({\tilde{i}}_{d q, h, n} (t) - η i_{d q}^{r e f} (t))

(17)

w_{θ} ({\tilde{P}}_{h, n} (t), {\tilde{f}}_{h, n}^{r e f} (t)) = α_{θ} {\tilde{P}}_{h, n} (t) + β_{θ} {\tilde{f}}_{h, n}^{r e f} (t)

(18)

where $α_{d q}$ and $β_{d q}$ are the weighting vectors of the first layer ( $h = 1$ ) of ${\tilde{v}}_{d q, h, n}$ and ${\tilde{i}}_{d q, h, n}$ , and $η = 1$ , respectively. For other layers ( $h > 1$ ), $α_{d q} + β_{d q} = 1$ and $η = 0$ , and consequently $w_{v i c} (\cdot) = α_{d} {\tilde{v}}_{d, h, n} (t) + β_{d} {\tilde{i}}_{d, h, n} (t) + α_{q} {\tilde{v}}_{q, h, n} (t) + β_{q} {\tilde{i}}_{q, h, n} (t)$ . Here, the vectors $W_{h, n}^{1}$ , $b_{h, n}^{1}$ , $b_{o}^{1}$ , $W_{h, n}^{2}$ , $b_{h, n}^{2}$ , $b_{o}^{2}$ , $W_{h, n}^{3}$ , $b_{h, n}^{3}$ , and $b_{o}^{3}$ are trained and optimized for the RNN scheme 1. The vectors $W_{h, n}^{1}$ , $b_{h, n}^{1}$ , $b_{o}^{1}$ , $α_{d q}$ , $β_{d q}$ , $W_{h, n}^{2}$ , $b_{h, n}^{2}$ , $b_{o}^{2}$ , $α_{θ}$ , and $β_{θ}$ are trained and optimized for the RNN scheme 2. In the training stage, these parameters are automatically optimized and updated by a stochastic gradient descent method.

C. Data-driven Training Strategy

Let $t - τ_{c} (t)$ and $t - τ_{l} (t)$ be the time with variable delays affected centralized and local control platforms, respectively; and $y_{m} (t) \in y (t)$ be the measured time-varying output vector in a form of time-series data containing crucial MG dynamics behavior. Accordingly, the vectors $y_{m} (t)$ and $d_{m} (t)$ can be expressed as:

y_{m} (t) = Λ (t) e^{- ξ (t) t} c o s (ω_{d} (t) t + ϕ (t))

(19)

d_{m} (t) = Δ {\bar{P}}_{D E R} (t) + Δ {\bar{P}}_{D L} (t)

(20)

where $d_{m} \in d$ and $y_{m} \in y$ ; $Λ$ is the signal amplitude; $ξ$ is the damping coefficient; $ω_{d}$ is the angular frequency representing the frequency at which the poorly-damped oscillation would occur; $ϕ$ is the vector of phase shifts; $Δ {\bar{P}}_{D E R}$ is the vector of average active power output deviations of DERs; and $Δ {\bar{P}}_{D L}$ is the vector of average active power output deviations of distributed loads.

Here, $y_{m} (t) = {[Δ \bar{F} (t), Δ \bar{V} (t)]}^{T}$ , where $Δ \bar{F}$ and $Δ \bar{V}$ are the vectors of average values of bus frequency and voltage deviations of DERs, respectively. To get $y_{m} (t)$ and $d_{m} (t)$ from measurements, the 100- $m s$ stamped time (denoted by $T_{s}$ ) is regarded. In any $r^{t h}$ signal pattern within a moving window of size $k$ , the vectors $y_{m} (t)$ in (19) and $d_{m} (t)$ in (20) can be represented by their discrete-time counterparts, $y_{m}^{r}$ and $d_{m}^{r}$ , as:

y_{m}^{r} = [y_{m} [r T_{s}] y_{m} [(r + 1) T_{s}] \dots y_{m} [(r + k) T_{s}]]

(21)

d_{m}^{r} = [d_{m} [r T_{s}] d_{m} [(r + 1) T_{s}] \dots d_{m} [(r + k) T_{s}]]

(22)

Both $R N N$ scheme 1 and RNN scheme 2 are trained by prioritizing robustness and high damping as well as reducing the rates of changes of frequencies and voltages of all buses in the MG, minimizing the loss function $L$ , as:

L = \frac{1}{N_{j}} \sum_{j = 1}^{N_{j}} \sum_{r = 1}^{N_{r}} \frac{1}{M}

(23)

M = {({(γ_{r o b, j}^{r})}^{- 1} + γ_{d a m p, j}^{r} + {(γ_{R o C, j}^{r})}^{- 1})}^{2}

(24)

where $γ_{r o b, j}^{r}$ , $γ_{d a m p, j}^{r}$ , and $γ_{R o C, j}^{r}$ are the indices representing the MG robustness, damping performance, and ability to reduce the rates of changes of frequency and voltage, respectively; $N_{j}$ is the total number of training data sets; and $N_{r}$ is the total number of signal patters in each data.

The term ${(γ_{r o b, j}^{r})}^{- 1}$ serves the purpose of enhancing robustness in both frequency and voltage control loops, which is considered by:

\begin{matrix} γ_{r o b, j}^{r} = {‖{[\begin{matrix} {\tilde{G}}_{F, j}^{r} & {\tilde{G}}_{V, j}^{r} \end{matrix}]}^{T}‖}_{\infty} \end{matrix}

(25)

where operator ${‖\cdot‖}_{\infty}$ returns the $\infty$ -norm of its argument; and ${\tilde{G}}_{F, j}^{r}$ and ${\tilde{G}}_{V, j}^{r}$ denote the closed-loop systems including RNN, which are identified by $Δ \bar{F}$ , $d_{m}^{r}$ and $Δ \bar{V}$ , $d_{m}^{r}$ , using the sub-space state-space identification, respectively.

The overall damping performance is determined by:

γ_{d a m p, j}^{r} = \sum_{o s = 1}^{N_{o s}} \sum_{}^{} {\tilde{ξ}}_{j, o s}^{r} = {\tilde{ξ}}_{j, 1}^{r} + {\tilde{ξ}}_{j, 2}^{r} + \dots + {\tilde{ξ}}_{j, N_{o s}}^{r}

(26)

where $N_{o s}$ is the total number of oscillation modes; and ${\tilde{ξ}}_{j, o s}^{r}$ is the estimated damping calculated from the identified matrices ${\tilde{G}}_{F, j}^{r}$ and ${\tilde{G}}_{V, j}^{r}$ .

The rates of changes of frequency and voltage can be evaluated by:

γ_{R o C, j}^{r} = R o C o F_{j}^{r} + R o C o V_{j}^{r}

(27)

where $R o C o F_{j}^{r}$ and $R o C o V_{j}^{r}$ are the rates of changes of average frequency and voltage, respectively.

Let $\{{(γ_{r o b, j}^{r})}^{- 1}, γ_{d a m p, j}^{r}, {(γ_{R o C, j}^{r})}^{- 1}\} \in χ_{j}^{r}$ , where $χ_{j}^{r}$ can be either ${(γ_{r o b, j}^{r})}^{- 1}$ , $γ_{d a m p, j}^{r}$ , or ${(γ_{R o C, j}^{r})}^{- 1}$ , the values obtained from (25)-(27) are normalized (denoted by operator $n o r m$ ) into a range of $0$ and $1$ by:

n o r m (χ_{j}^{r}) = \frac{χ_{j}^{r} - m i n (χ_{j})}{m a x (χ_{j}) - m i n (χ_{j})} = {\bar{χ}}_{j}^{r}

(28)

where over-line “ $\bar{}$ ” means the normalized variable.

By applying (28), variables ${({\bar{γ}}_{r o b, j}^{r})}^{- 1}$ , ${\bar{γ}}_{d a m p, j}^{r}$ , and ${({\bar{γ}}_{R o C, j}^{r})}^{- 1}$ can be obtained. Consequently, these values are substituted into (23) to calculate $L$ for $\forall \{r, j\}$ . Therefore, by employing the $n o r m$ function in (28), no weighting factors are required to balance the terms ${(γ_{r o b, j}^{r})}^{- 1}$ , $γ_{d a m p, j}^{r}$ , and ${(γ_{R o C, j}^{r})}^{- 1}$ in (23). Here, the dynamics of the MG are apparent in the measured signals $y_{m}^{r}$ , as expressed in (19) and (21). In the training process of the RNN and the computation of the loss function (23), only $y_{m}^{r}$ and $d_{m}^{r}$ in the forms of (21) and (22) are used. Exact MG parameters and their updates are not required. These parameters can be uncertain or change due to variations in MG topology, disconnection/synchronization from/to the main grid, and islanding conditions. The complete process of training and data preparation is presented in Fig. 5.

Fig. 5 Complete process of training and data preparation.

IV. Performance Evaluation

A. Test MG Description

The RNN schemes have undergone the validation within the MG with DERs, as illustrated in Fig. 6. In MG1, there are three sub-MGs with a maximum load of 23 $k V A$ and a maximum generation of 69 $k V A$ . Sub-MG1 includes five loads and three wind turbines (WTs). Sub-MG2 comprises three loads. Sub-MG3 is composed of three loads, one PV, and one WT. If the MG is disconnected from the main grid, LOC1 controls all DERs with assistance from DTC1 (sub-MG1), DTC2 (sub-MG2), and DTC3 (sub-MG3). In MG2, two sub-MGs have a maximum load of 27 $k V A$ and a maximum generation of 57.5 $k V A$ . Sub-MG1 includes five loads and five PVs, while sub-MG2 comprises four loads and three WTs. If the MG is disconnected, LOC2 controls all DERs with assistance from DTC1 (sub-MG1) and DTC2 (sub-MG2). In MG3, there are four sub-MGs with a maximum load of 47 $k V A$ and a maximum generation of 105 $k V A$ . Sub-MG1 includes five loads and three WTs. Sub-MG2 consists of two loads and five PVs. Sub-MG3 comprises three loads and three WTs. Sub-MG4 includes three loads and five PVs. In islanding mode, LOC3 controls all DERs with assistance from DTC1 (sub-MG1), DTC2 (sub-MG2), DTC3 (sub-MG3), and DTC4 (sub-MG4).

Fig. 6 Test MG system with DERs and proposed hierarchical GFM converter control structure consisting of three MGs and nine sub-MGs.

Under a normal MG condition, all MGs are connected to the utility grid to assist in supporting a maximum load of 200 $k V A$ . The DERs in these MGs are modeled as GFL converters. Additionally, a 150 $k V A$ synchronous generator is connected to the utility grid. This situation falls under the classification of a grid-connected scenario, as detailed in Section II. Under these circumstances, the operations of DERs and loads are as follows.

1) At the utility grid, the synchronous generator at the PCC is operated at 80%-95% of its maximum capacity.

2) In MG1, all DERs are operated at 50%-65% of their maximum capacities, and all loads fluctuate at 75%-90% of their maximum capacities.

3) In MG2, all DERs are operated at 45%-55% of their maximum capacities, and all loads fluctuate at 60%-75% of their maximum capacities.

4) In MG3, all DERs are operated at 50%-65% of their maximum capacities, and all loads fluctuate at 80%-95% of their maximum capacities.

B. Setup RNN Parameters

First, the total number of possible scenarios for the test system shown in Fig. 6 is calculated. There is a total of $6 + 6 + 18 = 30$ possible scenarios to be trained for the RNNs. Second, for each scenario, the $50$ s time-series data in (19) and (20) are collected within $1250$ random operating points of MG.

In an individual random operating point, all loads randomly fluctuate by $\pm 50 %$ from their normal operating points specified in Section IV-A. Additionally, potential random outages and out of services of CTC, LOCs, and/or DTCs are considered at an individual random operating point. By incorporating these uncertainties into the training processes, the trained RNNs can resiliently operate under various grid-tied and islanding conditions with high robustness against communication issues. Then, the measured data sets in (19) and (20) are transformed into their discrete counterparts using (21) and (22) with $T_{s} = 100 m s$ or $0.1 s$ . A moving window length of $k = 50 s$ is used, and $L$ can be calculated as specified in (23). The maximum number of iterations is set to be 60, and the hyperbolic tangent function is used as the nonlinear activation function.

With these settings, the training processes take approximately 13 hours and 17 minutes for RNN scheme 1, and 15 hours and 52 minutes for RNN scheme 2. It is found that the training time of RNN scheme 1 is shorter than that of the RNN scheme 2 since RNN scheme 1 incorporates physically-guided terms $𝒫 (t)$ in the developed model, such as $i_{d q} (t) - \frac{3}{2} ω (t) C_{f} v_{q d} (t)$ in (11) and $\frac{3}{2} ω (t) L_{f} i_{q d} (t)$ in (12), whereas $𝒫 (t) = 0$ for RNN scheme 2. With known $𝒫 (t)$ , as a result, the training process of RNN scheme 1 converges more quickly to minimize the targeted loss function in (23). However, if certain parameters such as $i_{d q} (t)$ , $ω (t)$ , $C_{f}$ , and/or $v_{q d} (t)$ are missing or cannot be measured at time $t$ , this creates an obstacle. Such issues may arise due to communication problems or other related factors, which can hinder the design process of RNN scheme 1 and may lead to a degradation in its performance. To validate the proposed method, the trained RNN scheme 1 and RNN scheme 2 are compared with traditional GFM converter control loops using model-based PI controllers, while additional signals from the RNN are incorporated into the junction points at inner-current control and phase angle control loops. Here, the compared strategy is referred to as PI with RNN.

C. Performance of Proposed RNN Schemes in Critical Scenarios

1)　Case Study 1

In case study 1, the assumed events include periodic fluctuations of active and reactive power occurring every 1-5 s in all DERs and loads in a range of $\pm 10 %$ from their current operating points. Prior to disconnection, all DERs take $v^{r e f} (t)$ and $f^{r e f} (t)$ from the PCC. In this state, all DERs operate in the GFL modes. The disturbance $d (t)$ applied to case study 1 consists of two main factors: the unplanned disconnection event, which leads to the separation of sub-MG3 from the grid, and the fluctuations in the outputs of intermittent DERs.

Initially, MG1, MG2, and MG3 are interconnected and also connected to the main grid. At $t = 4.1 s$ , an unplanned disconnection occurs, separating sub-MG3 of MG3 from the rest of the grid. This event results in the formation of an islanding sub-MG3. Case study 1 results in the transition from grid-tied mode to sub-islanding condition. $W T_{2}$ of sub-MG3 provides the highest active power support to sub-MG3 compared with the others, which operates in GFM mode to generate references for them. After $t = 20 s$ , intermittent DERs increase in sub-MG3 within a range of $\pm 40 %$ from their current operating points.

Accordingly, Fig. 7 shows the time-domain simulation of case study 1, presenting transient responses of $R o C o V$ and $R o C o F$ during the disconnection. Considering sub-MG3, the results reveal three significant periods including: ① before the disconnection of sub-MG3; ② during the reconnection; and ③ after the disconnection with high intermittency. The PI with $R N N$ exhibits responses over different frequencies, i.e., from 0.327 Hz to 1.422 Hz. Modal analysis confirms that these modes mainly involve the state variables $i_{d q}$ and $v_{d q}$ of $W T$ s. These variables interact across all areas, contributing over 72.87% of the participation factors. The PI with $R N N$ cannot suppress high-frequency oscillations. As a result, the transient responses are dominated by these high-frequency modes. In contrast, RNN scheme 1 and RNN scheme 2 can suppress these modes effectively. While some oscillations still occur, their amplitudes are much smaller compared with the PI with $R N N$ . As a result, the PI with RNN struggles to stabilize $R o C o V$ and $R o C o F$ in case study 1, while RNN scheme 1 and RNN scheme 2 show better performance. Specifically, before the disconnection, the compared strategy (PI with RNN) exhibits higher fluctuations. In this stage, the values of $R o C o V$ and $R o C o F$ oscillate severely in the case of compared strategy. After the disturbance, both RNN schemes demonstrate better abilities to damp the fluctuations.

Fig. 7 Time-domain simulation of case study 1.

2)　Case Study 2

A time-domain analysis is conducted to explore various re-connection scenarios. Case study 2 replicates the conditions in case study 1, except for the failures in receiving input signals for LOCs across all DERs during the first period between $t = 0$ and $t = 4 s$ . These failures obstruct the transmission of local input-output signals to the RNNs, resulting in all MGs being entirely controlled by the DTCs of their corresponding sub-MGs. $d (t)$ applied in case study 2 is similar to that in case study 1. However, it includes the additional effect of a complete failure to receive signals from all local controllers at the beginning of the simulation.

Initially, from $t = 0$ to $4 s$ , each MG operates independently in an islanding mode. In MG1, $W T_{3}$ in sub-MG1, $P V_{1}$ in sub-MG2, and $W T_{1}$ in sub-MG3 operate in the GFM mode, producing the highest active power within their respective sub-MGs. Other DERs in each sub-MG operate in the GFL mode, deriving $V / f$ references from the corresponding DERs in GFM modes. Similarly, in MG2, $P V_{5}$ in sub-MG1 and $W T_{2}$ in sub-MG2 operate in GFM modes. In MG3, $W T_{3}$ in sub-MG1, $P V_{5}$ in sub-MG2, $W T_{3}$ in sub-MG3, and $P V_{5}$ in sub-MG4 also operate in GFM modes. To ensure the synchronization and form the sub-MGs, these sources generate the highest active power and produce $V / f$ references for the GFL control loops of other DERs within their respective sub-MGs.

At $t = 4.1 s$ , MG3 reconnects with MG2, establishing the islanding mode between MG2 and MG3. During this period, it is assumed that the LOCs are restored, and the information from these LOCs can be transmitted to the CTC, thereby allowing the interconnected islanding MG to be governed by the CTC. Throughout this period, the CTC detects that the $W T_{1}$ in sub-MG2 of MG2 operates in GFM mode as it provides the highest power supported to the grid. As a result, other DERs in this interconnected islanding MG derive $V / f$ references from $W T_{1}$ in sub-MG2 of MG2 for their GFL active and reactive power control loops. Later, at $t = 15.6 s$ , MG1 reconnects to the others. The MG is controlled by the CTC. During this period, the CTC detects that $W T_{1}$ in sub-MG2 of MG2 is generating the highest active power to the grid. Consequently, $W T_{1}$ in sub-MG2 of MG2 operates in the GFM mode, supplying $V / f$ references to the GFL converters of other DERs for their active and reactive power control.

Following this, at $t = 26.1 s$ , the islanding MG (comprising MG1, MG2, and MG3 connected together) reconnects to the utility grid. During this transition from islanding condition to grid-connected condition, the synchronous generator is not available, and the load is approximately $87.5 \pm 5$ $k V A$ and no inertia is supported. During this period, akin to the preceding timeframe, $W T_{1}$ in sub-MG2 of MG2 continues to operate in the GFM mode. Consequently, Fig. 8 shows time-domain simulation results of case study 2, illustrating the transient responses of $R o C o V$ and $R o C o F$ under these re-synchronization conditions. After reconnecting sub-MG3, the RNN schemes restore $R o C o V$ and $R o C o F$ within 2.18 s. They reduce oscillation amplitudes by 66.32% and 47.26% compared with the PI with RNN. When sub-MG1 reconnects to the others, the amplitudes of these signals are similarly reduced by 135.15% and 127.80%, respectively. Moreover, when all MGs reconnect to the main grid, the PI with RNN shows very high amplitudes, especially at the first peak. Eventually, the system becomes unstable. In contrast, the RNN schemes maintain the fluctuations, keeping them within acceptable ranges for $R o C o V$ and $R o C o F$ . The results clearly show that the proposed RNN schemes perform better in reducing both $R o C o V$ and $R o C o F$ during re-synchronization under all conditions.

Fig. 8 Time-domain simulation results of case study 2.

D. Probabilistic Analyses Across a Wide Range of MG Operating Conditions

In this subsection, the probability analyses are carried out across $1000$ different scenarios. It is important to note that these scenarios are entirely different from those used to train the RNNs. To verify this, in each scenario, time-series data over a duration of $50 s$ are gathered with a sampling interval of $T_{s} = 0.1 s$ . At each time step, the values for $R o C o V$ , $R o C o F$ , and $ξ$ are collected, totaling $3 \times (50 / 0.1) = 1500$ points for further analysis. After obtaining these values for all scenarios, the data sets consist of a total of $3 \times 500 \times 1000 = 1.5 \times 10^{6}$ data points ( $0.5 \times 10^{6}$ for each of $R o C o V$ , $R o C o F$ , and $ξ$ ). Subsequently, a probability analysis method is utilized to estimate the densities of each of these data sets.

First, the focus is on evaluating the performance of hierarchical GFM converter control in scenarios where CTC, LOC(s), and/or DTC(s) are unexpectedly unavailable. The conditions are kept consistent where DERs and loads randomly vary in a range of $\pm$ 25% from their normal operating points. In each scenario, we introduce additional factors of random failure, which include failures of CTC, LOC(s), DTC(s), or unexpected islanding. In this scenario, the verification is conducted on the PI with RNN while employing four GFM converter controls, which are centralized GFM converter control, local GFM converter control, distributed GFM converter control, and proposed hierarchical GFM converter control.

For the centralized GFM converter control, the assumption is made that this control has the capability to monitor global signals from all MGs with a random delay $τ_{c} \in [100,700] m s$ . However, in the event of a failure at the CTC, the GFM converter loses its ability to receive all inputs for controlling MGs. For the local GFM converter control, the delay is randomly in the range $τ_{l} \in [0,120] m s$ for each LOC. Besides, if any failure occurs at one or more LOCs, DTCs remain available to transmit signals to GFM converter control loops. However, the LOCs may restrict the ability to observe all signals. For the distributed GFM converter control, unlike centralized GFM converter control, no global signals are employed for GFM converter control. This control performs effectively, particularly in islanding scenarios without inertia support from the main grid. It is clear that the proposed hierarchical GFM converter control effectively handles unexpected islanding scenarios in a multi-MG system.

In Fig. 9, probabilistic analysis results are presented after applying random scenarios, where active and reactive power outputs vary stochastically in a range of $\pm$ 25%. The analysis details $R o C o V$ , $R o C o F$ , and $ξ$ . Under centralized GFM converter control, the ranges of $R o C o V$ , $R o C o F$ , and $ξ$ (including potential negative values of $ξ$ ) are much larger. Specifically, $R o C o V$ and RoCoF range from $[- 0.396,0.377] p . u . / s$ and $[- 0.467,0.495] H z / s$ , respectively, which show that stabilizing these parameters is more challenging during CTC failures. Local, distributed, and the proposed hierarchical GFM converter controls exhibit similar values for $R o C o V$ and $R o C o F$ , with ranges of $[- 0.056,0.084] p . u . / s$ and $[- 0.185,0.152] H z / s$ , respectively, effectively mitigating fluctuations during islanding and controller failures. However, local and distributed GFM converter controls exhibit small negative values of $ξ$ , with values of $- 0.015$ and $- 0.008$ , respectively. In contrast, the proposed hierarchical GFM converter control ensures the stability for all dominant modes without negative $ξ$ , but with $ξ$ falling within the range $[0.009,0.388]$ , and the highest probability is $9.024 %$ . As can be observed, the proposed hierarchical GFM converter control outperforms other controls.

Fig. 9 Probabilistic analysis under conditions of random delays and failures of CTC, LOC, and/or DTC. (a) RoCoV. (b) RoCoF. (c) $ξ$ .

Following this, performances of the $R N N$ scheme 1 and $R N N$ scheme 2 are evaluated. Here, MG3 is assumed to be isolated from both MG1 and MG2, creating an islanding condition for sub-MG4.

Two separate probabilistic analyses are conducted within these islanding sub-MGs of MG3: analysis 1 considers sub-MG1, sub-MG2, and sub-MG3 together, while analysis 2 focuses solely on sub-MG4. Figure 10 shows the probabilistic analyses, where active and reactive power outputs vary randomly by $\pm 50 %$ from their normal operating points. The outcomes for analysis 1 are presented in Fig. 10(a), while the outcomes for analysis 2 are shown in Fig. 10(b).

Fig. 10 Probabilistic analyses. (a) Analysis 1: results of MG3 with sub-MG1, sub-MG2, and sub-MG3. (b) Analysis 2: results of MG3 with sub-MG4.

In analysis 1, the variations of $R o C o V$ and $R o C o F$ are similar across all strategies. For the proposed RNN schemes, $R o C o V$ and $R o C o F$ fall within the ranges $[- 0.227,$ $0.198] p . u . / s$ and $[- 0.211,0.195] H z / s$ , respectively.

For the PI with $R N N$ , the ranges of $R o C o V$ and $R o C o F$ are $[- 0.305,0.311] p . u . / s$ and $[- 0.353,0.357] H z / s$ , respectively. However, the difference is observed in $ξ$ . The proposed RNN schemes do not show negative damping, while the PI with $R N N$ shows values of $ξ$ in the range $[- 0.031,0.0745]$ . In analysis 2, PI with $R N N$ and $R N N$ scheme 1 exhibit poorer performances in sub-MG4 compared with the $R N N$ scheme 2 in managing $R o C o V$ variations. The results for $R o C o V$ range from $[- 0.399,0.409] p . u . / s$ in the case of the $R N N$ scheme 1. In contrast, for $R N N$ scheme 2, the range of $R o C o V$ is only $[- 0.304,0.212] p . u . / s$ . Considering $ξ$ , the $R N N$ scheme 1 shows a significant improvement in damping, with values in the range $[0.009,0.302]$ . The highest probability occurs at a damping value of $0.1927$ for $R N N$ scheme 2. Similar to analysis 1, the worst results are observed in the PI with $R N N$ , where negative damping might occur and fall within the range $[- 0.071,0.144]$ .

V. Conclusion

This paper introduces an intelligence-driven GFM converter control method for islanding MGs with DERs. The proposed method can handle both grid-tied and islanding scenarios by adaptively adjusting references in real time to mitigate communication issues, ensuring reliable operation and optimal performance under varying conditions. The proposed RNN scheme 1 employs separate RNN modules for voltage, current, and phase angle control, while the proposed RNN scheme 2 integrates them into a single RNN module to minimize control interaction. The RNN design incorporates a convolutional neural network structure focused on robustness, damping, and minimizing voltage and frequency changes during training. The proposed method includes normalizing indices and calculating a loss function to manage uncertainties and parameter changes during critical MG operations. Time-domain simulations validate the effectiveness of the proposed RNN schemes in mode transitions and disturbances, showing superior performance over a comparative strategy. Probabilistic analysis demonstrates both RNN schemes reduce voltage and frequency fluctuations, with RNN scheme 2 particularly effective in minimizing control loop interaction.

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