Continuous inertia optimization for PV-storage virtual synchronous machines via lightweight RBF networks: eliminating parameter discontinuity at the electromechanical edge

1 Introduction

With the rapid depletion of fossil fuels and growing environmental concerns, distributed renewable energy sources, particularly solar photovoltaic (PV) generation (Gholami et al., 2020; Zou et al., 2024), have witnessed widespread integration into modern power systems. Microgrids (Debouza et al., 2022; Micallef et al., 2023; Parol, 2024), which consolidate PV arrays, battery energy storage systems (BESS), and localized loads, offer an effective paradigm for utilizing decentralized clean energy. However, unlike traditional power grids dominated by massive synchronous generators, power electronic converter-interfaced (Nalina and Babu, 2022; Xu et al., 2020) microgrids exhibit low inertia and low damping characteristics. Consequently, highly volatile environmental variations, such as rapid changes in solar irradiance and temperature, along with sudden load fluctuations, can severely trigger frequency and voltage instabilities within the microgrid, hindering its reliable operation and stable grid integration.

To mitigate the adverse effects of low inertia, Virtual Synchronous Generator (VSG) (Gurski et al., 2026; Shi et al., 2020; Wu et al., 2026) control technology has emerged as a compelling solution for grid-forming and grid-following/grid-connected inverters. By embedding the rotor motion equations and electromagnetic transient characteristics of traditional synchronous machines into the converter control algorithms, a VSG enables power electronic interfaces to mimic the inertial support and damping capabilities of physical generators. This biomimetic control mechanism allows the microgrid to autonomously regulate active and reactive power while smoothing frequency response trajectories during disturbances (Li et al., 2023; Xie et al., 2024). Thus, the implementation of VSG control significantly enhances the frequency resilience and self-healing capability of decentralized multi-converter networks.

Despite its merits, conventional VSG control architectures typically use fixed virtual inertia (J) and virtual damping coefficients (D). Such static parameterizations create an inherent trade-off between transient overshoot and settling time; a larger virtual inertia suppresses initial frequency variations but accentuates power oscillations and prolongs regulation intervals, whereas over-damping compromises the swiftness of dynamic support (Ke et al., 2020; Ma et al., 2023). To break this limitation, various adaptive VSG control (Gurski et al., 2024; Pei et al., 2026; Wang et al., 2024) schemes have been developed. Most conventional adaptive methods rely on threshold-triggered piecewise functions that adjust parameters based on discrete boundaries of frequency deviations. However, these abrupt parameter adjustments inevitably introduce non-differentiable discontinuities into the control law, which trigger high-frequency PWM switching ripples, cause terminal current spikes, and can potentially jeopardize the small-signal stability of the system.

To overcome these structural boundaries and achieve seamless multi-objective parameter tuning, this paper proposes a smooth, continuous adaptive VSG control strategy optimized via a three-layer Radial Basis Function (RBF) neural network. The developed approach dynamically maps instantaneous frequency tracking errors and their derivatives to adjust the virtual inertia in real time using the Levenberg-Marquardt (LM) weight updating algorithm, ensuring high computational efficiency and rapid convergence. Furthermore, to maintain the system in a globally optimized damping state, the virtual damping coefficient is synchronously coupled with the optimized inertia through a continuous holonomic constraint. This intelligent synergy effectively eliminates parameter jumps, minimizes transient power and frequency overshoots, and suppresses output current harmonics under complex disturbances.

A critical prerequisite for adopting AI-driven control within energy webs is guaranteed real-time performance under strict timing constraints. Unlike cloud-centric machine learning applications, edge-resident controllers must execute inference within deterministic control periods to avoid destabilizing power-electronic dynamics. Moreover, latency-performance trade-offs must be quantitatively characterized to ensure compliance with switching frequencies and harmonic standards. This requirement positions RBF-NNs as particularly attractive: their shallow architecture enables microsecond-scale inference while preserving sufficient nonlinearity to capture complex VSG dynamics. When properly optimized, such networks can deliver continuous parameter adaptation while maintaining favorable computational efficiency for embedded control—a necessity for web-integrated microgrid deployments. Building upon these motivations, this paper proposes an edge-intelligent RBF-NN adaptive VSG control framework explicitly designed for PV–storage energy webs.

The novelty of this study does not lie in the isolated use of adaptive virtual inertia, adaptive damping, or neural-network-based control, each of which has been investigated in previous studies. Instead, the contribution lies in their integrated implementation within a continuous parameter-scheduling framework. Specifically, the proposed approach combines (i) an RBF-NN mapping from frequency-error dynamics to virtual inertia, (ii) continuous coupling of damping and inertia to regulate the instantaneous equivalent damping characteristic, (iii) RMS(dJ/dt) as a quantitative measure of parameter smoothness, and (iv) direct DSP-level latency characterization. This integrated framework is intended to address the practical trade-off among transient performance, parameter continuity, harmonic distortion, and embedded computational cost.

2 Topology and mathematical modeling of the PV-Storage microgrid system

2.1 System configuration and topology

The architectural design of the proposed PV-storage microgrid system (Li et al., 2019; Mao et al., 2025) is illustrated in Figure 1. The system adopts a DC-coupled configuration to integrate distributed renewable energy resources and energy storage units seamlessly. The core components comprise a photovoltaic (PV) array, a battery energy storage system (BESS), dedicated power electronic converters, local AC/DC loads, and a main utility grid interface managed via a Static Transfer Switch (STS) at the Point of Common Coupling (PCC). The system operates in both grid-connected and islanded modes, with the STS enabling seamless mode transitions.

Within this grid topology, the PV array serves as the primary generation unit, coupled to the common DC bus through a unidirectional Boost converter to implement maximum power point tracking (MPPT). The BESS is linked to the same DC bus via a bi-directional DC/DC converter, acting as a bidirectional energy buffer to regulate power fluctuations and stabilize the DC link voltage. The aggregated DC power is subsequently converted to high-quality AC power via a three-phase voltage source inverter (VSI) equipped with an LC filter, suppressing high-frequency switching harmonics before supplying local loads or interacting with the utility grid.

2.2 Mathematical modeling of the photovoltaic array

The PV array (Gaurav et al., 2015) is the primary energy source of the microgrid, and its nonlinear output characteristics under varying environmental conditions must be accurately modeled to ensure stable system operation. To precisely evaluate the dynamic output behavior of the PV generation unit under volatile environmental conditions, a mathematical model based on the standard single-diode equivalent circuit is formulated. The output current-voltage (I-V) characteristic relationship of the PV cell is defined by the following implicit non-linear transcendental equation, as shown in Equation 1:

I P V = I p h − I o s e q U P V + I P V R s A K T − 1 − U P V + I P V R s R s h (1)

Where, I p h represents the photo-generated current directly proportional to solar irradiance; I o s denotes the diode reverse saturation current; q is the electron charge (1.602 × 10⁻¹⁹ C); K is the Boltzmann constant (1.381 × 10⁻²³ J/K); T signifies the absolute cell temperature in Kelvin; A is the diode ideality factor; R s and R s h represent the equivalent intrinsic series and parallel shunt resistances of the PV cell, respectively.

Given that solar irradiance S and ambient temperature T exhibit continuous variations in practical operations, environmental compensation coefficients are introduced to modify the actual output current I P V (Equation 2) and output voltage U P V (Equation 3) based on their reference values, the effect of the series resistance R s is neglected,

I P V = I p h − I o s e q U P V A K T − 1 (2)

U P V = A K T q ln I p h − I P V I o s + 1 (3)

By implementing these parameter correction equations, the PV array model accurately tracks the shift of the Maximum Power Point (MPP) under arbitrary meteorological trajectories, providing a reliable mathematical foundation for the design of the MPPT and power control loops.

2.3 Mathematical modeling of the battery energy storage system (BESS)

To address the intermittency of PV generation, the BESS (Lin and Bae, 2024) is integrated to smooth power fluctuations and maintain DC-link voltage stability, requiring an accurate electrical model for its terminal behavior and state of charge (SOC). The BESS plays a vital role in balancing power deviations, shaving peak loads, and mitigating the intermittent nature of PV generation. To characterize its thermodynamic and electrical performance without excessive computational overhead, an equivalent circuit model is established. The terminal output voltage E b of the battery cell during operation is expressed as Equation 4:

E b = E o − K Q m Q m − ∫ I b d t + A e − B ∫ I b d t (4)

Where, E o is internal electromotive force of the storage battery; Q m is rated capacity of the storage battery; K, A, B are fitting coefficients of the equivalent model; I b is battery charge/discharge current.

The tracking of the internal energy capacity status is executed via the classic Coulomb counting (Ampere-hour integration) method. This method is simple and widely adopted, but it requires periodic calibration to mitigate accumulated errors caused by current measurement noise. The dynamic equation governing the continuous state transition of the SOC is defined as Equation 5:

S O C = S O C o − ∫ I b Q m d t (5)

Where, S O C o represents the initial state of charge; I b is positive during discharge and negative during charge. This mathematical formulation ensures the energy management system can strictly enforce operational boundary constraints.

2.4 State-space averaged modeling of power converters

The power electronic converters (Selvi et al., 2024) act as the interface between the energy sources and the DC bus, and their dynamic behavior is critical to the overall system stability and control performance. To implement advanced linear and non-linear multi-loop control designs, the high-frequency switching characteristics of power electronic devices must be smoothed into continuous state trajectories. State-space averaged modeling is utilized for both the Boost and bi-directional DC/DC converters. This method effectively models the converter dynamics while filtering out high-frequency switching ripples, yielding a continuous-time state-space model suitable for control design.

By selecting the PV-side inductor current I P V and the DC-link filtering capacitor voltage U d c as the core state vector, the structural matrix definitions under the switch-on state and switch-off state within a single switching period are analyzed. Applying the duty cycle of the PV-side converter d as the control variable, the synthesized small-signal state-space averaged model is expressed in the following matrix differential form (Equation 6):

I ˙ P V U ˙ d c = U P V L − U d c L I P V C d c − U d c R C d c + U d c L − I P V C d c d (6)

Where, L is inductance of the PV-side inductor; C d c is capacitance of the DC-link capacitor; R is equivalent load resistance.

This matrix equation directly reveals the non-linear coupling behavior between the input PV voltage and the target DC bus voltage, serving as the plant model for designing the inner current loop.

The bi-directional DC/DC converter interfaces the battery storage with the high-voltage DC bus, operating in either Buck mode (charging) or Boost mode (discharging). By assigning the battery-side inductor current I b and the common bus capacitor voltage U d c as the state variables, and applying the duty cycle of the battery-side converter d 1 of the switching tubes, the unified state-space averaged model covering both operational modes is derived as follows (Equation 7):

I ˙ b U ˙ d c = 0 − 1 − d 1 L 1 1 − d 1 C d c 0 I b a t U d c + 1 L 1 0 0 − 1 C d c U b I l o a d (7)

Where, U b is battery terminal voltage; L 1 is inductance of the battery-side inductor; I l o a d is DC-side load current.

This multi-variable coupled state equation forms the mathematical framework for the voltage-current double closed-loop controller, enabling rapid bidirectional power tracking and robust DC bus voltage regulation.

2.5 Mathematical model and design of VSG

The core idea of the VSG control strategy is to simulate the damping, moment of inertia and power output characteristics of synchronous generators. Based on the second-order mathematical model of synchronous generators, combined with the relationship between torque and power as well as Kirchhoff’s voltage law, the rotor motion equation and stator voltage equation of the VSG are expressed as follows (Equations 8, 9):

J d ω − ω 0 d t = P m ω − P e ω − D ω − ω 0 d θ d t = ω (8)

E = U + I R v + j X v (9)

The overall control structure of the VSG is shown in Figure 2. Compared with traditional inverter control strategies, the VSG enables the inverter unit to possess damping and inertia characteristics, which improves the system robustness to a certain extent. However, with the integration of various types of new energy sources into the microgrid, the conventional VSG algorithm with fixed control parameters cannot guarantee satisfactory system performance under various disturbances. Therefore, it is necessary to further optimize the control strategy so that the VSG can adaptively adjust its parameters according to the variations in disturbances.

3 Adaptive VSG control strategy based on RBF neural network

3.1 Small-signal modeling and parameter analysis

To evaluate the dynamic performance of the Virtual Synchronous Generator (VSG) under grid-connected conditions, a small-signal model of the active power control loop is established. The closed-loop transfer function relating the output active power to the reference command is derived as:

G p s = P e P r e f = E U g J ω 0 Z s 2 + D J + k m J ω 0 s + E U g J ω 0 Z (10)

Where, P e is actual active power output of the VSG; P r e f is active power reference command; E is electromotive force amplitude of the VSG; U g is amplitude of the grid voltage at the point of common coupling (PCC); ω 0 is rated angular frequency of the power grid; Z is equivalent grid impedance; k m is frequency droop coefficient; s is laplace operator.

Based on the characteristic equation of the second-order system, the equivalent damping ratio ζ and the natural oscillation angular frequency ω n are formulated as:

ζ = D ω 0 + k m 2 J ω 0 J ω 0 Z E U g (11)

ω n = E U g J ω 0 Z (12)

Eigenvalue trajectory (root locus) and dynamic sensitivity analyses indicate a critical inherent trade-off between the virtual parameters: a larger virtual inertia J effectively suppresses the initial rate of change of frequency d f / d t but severely delays the settling time and amplifies power oscillations. Conversely, while a higher damping coefficient D minimizes transient overshoot, over-damping compromises the frequency support capability. Therefore, fixed parameters J and D are inherently incapable of maintaining optimal dynamic stability under highly volatile microgrid operating conditions.

3.2 RBF neural network-optimized adaptive control

Conventional adaptive VSG control methods utilize localized threshold-based piecewise functions to adjust parameters during transient events. Typically, the transient grid disturbance is divided into four distinct oscillation phases based on the sign and magnitude of the frequency deviation Δ ω and its derivative d ω / d t.

However, these threshold-triggered piecewise frameworks suffer from severe structural drawbacks. The sharp transitions between discrete parameter boundaries introduce non-differentiable discontinuities in the control law. These sudden parameter jumps inevitably induce high-frequency PWM switching ripples, trigger transient current spikes at the output terminal, and can potentially jeopardize the small-signal stability of the multi-converter microgrid interface. This motivates the development of a smooth, continuous, and universally intelligent multi-variable optimization algorithm.

To overcome the limitations of boundary discontinuities and achieve seamless multi-objective parameter tuning, a three-layer Radial Basis Function (RBF) neural network is integrated into the primary VSG control loop. The structural layout of the optimization controller consists of an input layer, a non-linear hidden layer, and a linear output layer. The input vector x t is directly mapped from the instantaneous grid tracking errors, defined as Equation 13:

x t = d ω d t , Δ ω T (13)

The number of hidden layer neurons is set to 5, and the output of the hidden layer is expressed as Equation 14:

φ j t = e − x i t − c j t 2 2 σ j t 2 , i = 1 , 2 ; j = 1 , 2 , … , 5 c j t (14)

The hidden layer utilizes the Gaussian basis function, while the output layer incorporates a hyperbolic tangent tanh activation function to enhance non-linear mapping capabilities. The input of the neural network output layer is given by Equation 15:

h j t = ∑ j = 1 5 w j t φ j t , j = 1 , 2 , … , 5 (15)

The output of the RBF neural network is expressed as Equation 16:

J = F h j t (16)

To guarantee rapid convergence and robustness against uncertainties, the Levenberg-Marquardt (LM) algorithm is employed for dynamic weight updating (Equation 17):

Δ w j t = J T t J t + α Ι − 1 g t (17)

The RBF network outputs the optimized virtual inertia J. To maintain the system at the optimal damping state (i.e., ζ = 0.707), the virtual damping coefficient D is synchronously coupled with J using the following relationship (Equation 18):

D = 2 ζ J E U g ω 0 Z − k m ω 0 (18)

Where, c j t is center vector; σ j t is width vector of Gaussian functions; w j t is weight function between the jth hidden layer neuron and the neural network output; k m is frequency regulation coefficient of synchronous generator.

This linkage ensures that the system avoids persistent oscillations while rapidly tracking the reference commands. The complete RBF-VSG control execution flowchart is illustrated in Figure 3.

3.3 Simulation studies and experimental verification

A grid-connected VSG simulation model is established to conduct comparative verification under two typical operating conditions. Condition 1: Step change of the active power reference. The initial active power reference of the VSG is set to 10 kW. It steps up to 15 kW at t = 0.5s and returns to 10 kW at t = 0.9s. The output active power and frequency under different control strategies are shown in Figures 4, 5, respectively. Condition 2: Grid frequency disturbance. The initial grid frequency is set to 50 Hz. It drops to 49.8 Hz at t = 0.5s and recovers to 50 Hz at t = 0.9s. The output active power and frequency under different control strategies are presented in Figures 6, 7, respectively.

In Figure 4, it can be observed that the fixed-parameter VSG control exhibits the most severe transient behavior, with the highest active power overshoot of 5.3% and the longest regulation time of 0.22 s. The conventional VSG adaptive control improves the dynamic performance, reducing the overshoot to 2.4% and shortening the regulation time to 0.15 s. In contrast, the proposed RBF-VSG adaptive control achieves the best performance, effectively suppressing the power overshoot to 0.8% and minimizing the regulation time to 0.11 s. When the reference power is restored at t = 0.9s, a similar trend is observed: the RBF-VSG adaptive control still shows the smallest fluctuation and fastest convergence. These results verify that the proposed control strategy significantly enhances the transient response of the VSG, achieving smaller overshoot and faster settling under step changes in the active power reference.

Figure 5 presents the VSG frequency response under different control strategies during the active power reference step change. As observed from the results, the fixed-parameter VSG control exhibits the poorest frequency regulation performance, with the largest frequency overshoot of 0.26% and the longest settling time of 0.21 s. By contrast, the conventional VSG adaptive control reduces the frequency overshoot to 0.21% and the settling time to 0.13 s, showing a moderate improvement in dynamic response. Notably, the proposed RBF-VSG adaptive control delivers superior performance, limiting the frequency overshoot to only 0.19% and shortening the settling time to 0.1 s. Throughout both the step-up and step-down processes of the active power reference, the RBF-VSG adaptive control consistently maintains the smallest frequency deviation and fastest convergence to the nominal value. This confirms that the proposed control scheme effectively enhances the frequency stability of the VSG under power command variations, offering better dynamic performance compared to the other two strategies.

The active power responses of the VSG under different control strategies during grid frequency disturbance is shown in Figure 6. Under the fixed-parameter VSG control, the active power exhibits the largest overshoot of 8.3% and the longest regulation time of 0.23 s. The conventional adaptive VSG control reduces the overshoot to 4.1% and shortens the regulation time to 0.14 s. In contrast, the proposed RBF-VSG adaptive control yields the best performance, with an overshoot of only 1.1% and a regulation time of 0.1 s. These results confirm that the RBF-VSG adaptive control effectively suppresses power fluctuations and accelerates convergence under grid frequency disturbances.

In Figure 7, the fixed-parameter VSG control yields the largest frequency overshoot of 0.08% and the longest settling time of 0.21 s. The conventional adaptive VSG control reduces the overshoot to 0.01% and the settling time to 0.12 s. The proposed RBF-VSG adaptive control achieves the best performance, with an overshoot of only 0.006% and a settling time of 0.11 s, demonstrating superior frequency regulation capability.

To further verify the performance of the proposed control strategy in practical applications, experiments are carried out. The experimental verification was implemented on a laboratory-scale PV-storage microgrid platform consisting of a programmable DC power supply, a photovoltaic emulator, a programmable grid simulator, a three-phase DC/AC converter, a bidirectional DC/DC converter, an electronic load, and measurement equipment including an oscilloscope. The main control units of the power electronic converters are TMS320F28335 DSP controllers, which can be programmed and configured through a host computer. The DC side of the three-phase DC/AC converter is connected to the programmable DC power supply, while its AC side is connected to the programmable grid simulator. The grid simulator can operate either in the AC source mode or in the AC load mode, corresponding to grid-connected and islanded operating conditions, respectively. The bidirectional DC/DC converter supports both boost and buck operating modes and can be connected to the photovoltaic emulator for PV-related experimental investigations. This configuration provides a flexible platform for reproducing the grid-connected and islanded operating conditions considered in the proposed VSG control framework. The main experimental parameters are as follows, PN = 3 kW, DC-side voltage Udc = 100V, Grid voltage amplitude Ug = 50V, Grid frequency = 50 Hz, the output filter includes a 400 μH filter inductor. The total harmonic distortion (THD) of the DC/AC converter output current under different control strategies is measured and summarized in Figure 8.

From the results in Figure 8, it is clear that the fixed-parameter VSG control leads to the highest current THD in both cases, while the conventional adaptive VSG control reduces harmonic distortion to a certain extent. In contrast, the proposed RBF-VSG adaptive control achieves the lowest THD values of 2.32% and 2.79%, respectively, both below 3%. These results demonstrate that the proposed strategy effectively suppresses current harmonics and maintains good power quality during dynamic transients, thus verifying its superior robustness and practical effectiveness.

The conventional adaptive controller was implemented according to Ref (Gurski et al., 2024). as the proposed method. Its adaptive parameter ranges were selected according to the operating range used in the original method and kept fixed.

To further examine the theoretical role of the damping ratio independently of the specific nonlinear simulation cases discussed above, a separate analytical sensitivity analysis is conducted using the second-order small-signal model. The purpose of this analysis is not to reproduce the transient responses in Figures 4–7, but to illustrate the general theoretical relationship between the damping ratio, overshoot, and settling time. Accordingly, a separate illustrative parameter set is used in Figure 9. Therefore, the numerical overshoot values obtained from Figure 9 should not be directly compared with the nonlinear simulation values reported in Figures 4, 6, because the two analyses use different model assumptions and parameter configurations.

The parameter set used in Figure 9 is an illustrative analytical configuration for sensitivity analysis and is not identical to the laboratory hardware operating point. In particular, the analytical values of E and Ug = 311 V and Z = 1.5 Ω are used solely to examine the theoretical second-order damping-ratio sensitivity and should not be interpreted as the nominal voltage and impedance values of the experimental platform. Sensitivity of dynamic performance to damping ratio ζ in the RBF-VSG second-order control loop (as in Figure 9). (a) Overshoot σ% and settling time t_s as functions of ζ, computed from the closed-form step-response formulas of a second-order system. The orange vertical line marks the optimal operating point ζ = 0.707, which provides the selected compromise between overshoot suppression and settling-time performance. The gray triangle indicates the fixed-parameter case (J = 0.1, ζ = 0.45), showing significantly higher overshoot. (b) Superimposed view showing the trade-off between overshoot suppression and settling time across the full ζ range.

To validate the selection of the optimal damping ratio, a sensitivity analysis was conducted on the second-order small-signal model, the closed-form expressions for overshoot and settling time are Equations 19, 20: Where ω_n = √(EU_g/(J·ω₀·Z)) is the natural frequency. Figure 9 plots σ% and t_s as functions of ζ ∈ [0.10, 1.80]. Three key observations support the selection of ζ = 0.707:

σ % = exp − π ζ / ( 1 − ζ 2 × 100 % (19)

t _ s = 3.5 / ζ · ω _ n (20)

• At ζ = 0.707, the overshoot reaches the well-known theoretical minimum of 4.3%, and the settling time is 0.158 s.
• Within the analytical sensitivity study, the illustrative case with J = 0.1 and ζ = 0.45 gives a theoretical overshoot of 20.7%. This value belongs to the analytical benchmark and should not be interpreted as the simulated overshoot of the fixed-parameter VSG in Figure 4.
• For ζ > 0.707, t_s decreases monotonically but σ% is already negligible; for ζ < 0.707, σ% grows rapidly (e.g., σ = 16.3% at ζ = 0.40), while t_s increases due to the lower effective bandwidth.

Therefore, ζ = 0.707 represents the selected compromise between overshoot suppression and settling-time performance.

To establish the causal link between RBF-NN smoothing and harmonic suppression, steady-state output current spectra were analyzed via FFT for all three control strategies. The spectra in Figure 10 reveal a clear progression in harmonic mitigation:.

• Fixed-parameter VSG (gray): The 3rd harmonic component is only ∼8 dB below the fundamental, with significant energy in the 5th and 7th harmonics. The total harmonic distortion (THD) extracted from the spectrum is 8.25%, consistent with the experimental measurement
• Conventional adaptive VSG (blue): The 3rd harmonic is suppressed to ∼14 dB below fundamental, reducing THD to 3.76%. However, visible spectral lobes remain at the 5th and 7th harmonics.
• RBF-NN VSG (red): All harmonic components are suppressed below −20 dB. The 3rd harmonic is −22 dB relative to the fundamental, and the THD extracted from the spectrum is 2.32%, which is consistent with the measured mean THD reported in Table 1.

TABLE 1. Quantitative performance comparison between RBF-NN and conventional control.

The spectral analysis directly supports the causal chain: Continuous J(t) → eliminated parameter discontinuities → reduced PWM spectral spreading → lower harmonic injection → leading to a lower measured mean THD of 2.31%. This explains why the RBF-NN achieves the lowest measured THD. The RBF-NN controller achieves the lowest measured mean THD among the three compared strategies.

The stability and convergence rate of the LM-based weight update are critical for demonstrating the online learning feasibility of the RBF-NN controller. Figure 11 shows the MSE loss as a function of LM iteration number. The convergence proceeds in three distinct phases:

Phase 1 (iterations 1–30): Steep exponential decay. The MSE drops from ∼0.12 to ∼0.015, representing a 87.5% reduction in 30 iterations. The damping parameter α decreases from 0.5 to ∼0.08, indicating the algorithm transitions from gradient-descent-dominated to Gauss-Newton-dominated optimization.

Phase 2 (iterations 30–100): Moderate refinement. MSE decreases from ∼0.015 to ∼0.0015. The rate of decay slows as the weights approach their optimal values. α continues to decrease, reaching ∼0.02.

Phase 3 (iterations 100–170): Fine-tuning. MSE reaches 10^–4 at approximately iteration 128, satisfying the convergence criterion. Beyond this point, the loss plateaus with only minimal fluctuation (standard deviation <5 × 10⁻⁶).

The total computational cost per LM update is O (m²) where m = 5 is the number of hidden neurons, requiring only 25 matrix operations per iteration. On the F28335 DSP platform (150 MHz, single-cycle FPU), each iteration completes in approximately 8 μs. This supports the computational feasibility of the online RBF-NN implementation.

Figure 12 presents the core innovation of the RBF-NN adaptive VSG control strategy—the continuous adjustment of virtual inertia J and the corresponding regulation of the instantaneous equivalent damping ratio around the target value ζ = 0.707.

(a) Upper subplot—Virtual Inertia J Dynamic Trajectory: RBF-NN Adaptive J (red solid line): The RBF neural network takes Δω and dω/dt as inputs and produces a smooth, continuous J(t) trajectory via tanh-activated Gaussian basis functions. During the power reference step-up (10 kW→15 kW at t = 0.5s) and step-down (15 kW→10 kW at t = 0.9s), J increases smoothly from the base value J₀ = 0.05 to ∼0.33, avoiding any discontinuous jumps. - Conventional Segmented Adaptive J (blue dashed line): Based on threshold-triggered Bang-Bang logic, J exhibits discrete step changes between 0.03 and 0.40, creating four distinct jump events per oscillation cycle. - Fixed Parameter J = 0.1 (green dot-dashed line): Constant value with no adaptive capability.

(b) Lower subplot—Real-Time Damping Ratio ζ(t) Evolution: RBF-NN (red solid line): the instantaneous equivalent damping ratio is regulated around 0.707. - Conventional (blue dashed line): ζ oscillates between ∼0.45 and ∼1.8, spending significant time outside the optimal band, causing either underdamped ringing (ζ < 0.5) or overdamped sluggishness (ζ > 1.2). - Fixed (green dot-dashed line): ζ = 0.45, permanently suboptimal. - The orange highlighted band marks the optimal damping region ζ ∈ [0.607, 0.807] (±0.1 around 0.707). Key quantitative results are as follow in Table 2.

TABLE 2. Comparative quantitative metrics: RBF-NN against conventional approach.

The second-order small-signal model in Equations 10–12 is used as an instantaneous local approximation of the VSG dynamics. Since the proposed controller continuously updates J, the damping ratio calculated from Equation 11 represents an instantaneous equivalent damping characteristic under the quasi-static assumption rather than a rigorous invariant damping ratio of the complete time-varying nonlinear system. The continuously varying J trajectory and the corresponding bounded variation of the virtual parameters are therefore important for maintaining the validity of this local interpretation.

J-Smoothness Analysis: The RMS value of dJ/dt for RBF-NN is 14.57, which is 76.1% lower than the conventional segmented approach (60.87). This quantitative result demonstrates that the RBF neural network substantially reduces the parameter variation associated with threshold-based adaptive control. The smooth J trajectory directly translates to eliminated high-frequency excitation of the power electronics switching dynamics.

Root Locus Interpretation: From the second-order small-signal model, the closed-loop poles are located at: s = -ζω_n ± jω_n√(1-ζ²) With ζ = 0.707 (RBF-NN), the poles are at -ω_n/√2 ± jω_n/√2, providing: • 4.3% overshoot (theoretically optimal for step response) • Minimal settling time without ringing • Equal real and imaginary parts → maximally flat response. The 4.3% value represents the theoretical overshoot of the canonical second-order benchmark at ζ = 0.707. In contrast, the 0.8% value is obtained from the complete nonlinear VSG simulation under the specific power-reference trajectory in Figure 4. The analytical benchmark is therefore used to motivate the damping-ratio selection, rather than to predict or impose a lower bound on the nonlinear simulation overshoot.

Figure 13 establishes the practical feasibility and performance advantage of the RBF-NN approach from two critical engineering perspectives: training convergence stability and real-time inference performance. (a) Upper subplot—LM Algorithm Weight and Center Convergence: Five hidden neuron weights w_j (solid lines, colors 1–5) evolve from initial random values toward their converged values [0.35, −0.28, 0.42, −0.15, 0.31]. - Five hidden neuron centers c_j (dashed lines, same color scheme) remain near zero with minimal adjustment, confirming the design choice c_j = [0,0,0,0,0]^T is appropriate. - All weights converge within ∼150 L M iterations (MSE <1 × 10⁻⁴). - The Levenberg-Marquardt algorithm combines the fast convergence of Gauss-Newton with the stability of gradient descent, requiring only O (5²) = 25 matrix operations per update—trivial for modern DSPs.

(b) Lower subplot—Inference Latency vs. THD Distribution: Box-whisker plots (left offset, colored boxes) show the distribution of single inference latencies measured on the DSP TMS320F28335 platform.

• Fixed VSG (green): μ = 12.0 μs, σ = 2.1 μs, max = 17.2 μs
• Conventional (blue): μ = 34.4 μs, σ = 4.6 μs, max = 49.8 μs
• RBF-NN (red): μ = 85.8 μs, σ = 12.1 μs, max = 115.7 μs

Box-whisker plots (right offset, dark red boxes) show the corresponding THD distribution.

• Fixed VSG: μ = 8.45%, σ = 1.22%, max = 12.00%
• Conventional: μ = 3.91%, σ = 0.76%, max = 6.21%
• RBF-NN: μ = 2.31%, σ = 0.39%, max = 3.32%

Scatter points (translucent) represent individual measurements; diamond/square markers indicate mean values.

THD Causal Chain: The results are consistent with the proposed mechanism linking smoother J(t) evolution with reduced harmonic distortion: NN weight smoothness → continuous J(t) trajectory → eliminated parameter discontinuities → reduced PWM spectral spreading → lower harmonic injection with a measured mean THD of 2.31%. The RBF-NN achieves the lowest measured THD. Quantitative comparison of latency and THD performance was in Table 1.

4 Conclusion

This paper addresses the critical issue of parameter discontinuity in adaptive control of virtual synchronous machines (VSMs), core electromechanical interfaces for PV-storage energy webs, by proposing a continuous inertia optimization framework based on a lightweight RBF neural network. By replacing threshold-triggered piecewise logic with a smooth RBF-NN mapping aligned with the rotor motion characteristics of synchronous machinery, the proposed strategy replaces threshold-triggered piecewise scheduling with a continuous virtual-inertia mapping, thereby avoiding the abrupt parameter jumps associated with the conventional threshold-based strategy. The integration of a holonomic damping constraint continuously couples D with J so that the instantaneous equivalent damping ratio is regulated toward the target value of 0.707, achieving an 84.9% reduction in active-power overshoot under the power-reference step condition and a 76.1% decrease in the RMS of dJ/dt, directly improving the transient stability of the electromechanical energy conversion process. From an implementation standpoint, the algorithm was profiled on a TMS320F28335 edge controller, with an average inference time of 85.8 μs. This study does not claim strict IEEE 519 compliance, which requires evaluation at the applicable PCC conditions and system-dependent current distortion limits. The measured average inference latency of the RBF-NN controller is 85.8 μs, which is shorter than the nominal 100 μs control period. The maximum measured latency is 115.7 μs and therefore exceeds one nominal control period. Accordingly, the present implementation demonstrates favorable average real-time feasibility on the TMS320F28335 platform, while a strict hard real-time guarantee for every control cycle is not claimed. Further implementation-level optimization will be considered in future work.

The framework effectively balances transient stability, power quality, and real-time computational constraints for electromechanical systems, marking a feasible pathway for deploying data-driven control in edge-resident microgrids. Future work will explore federated RBF-NN training across multi-node energy webs to enable wide-area collaborative control of distributed electromechanical energy conversion assets, and extend the approach to hybrid mechanical energy storage systems such as flywheel-PV coupled architectures.

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Footnotes

Edited by: Mohamed Arezki Mellal, University of Boumerdés, Algeria

Reviewed by: Fikri Waskito, Gadjah Mada University, Indonesia

Plamen Stanchev, Sofia University, Bulgaria