Fault Diagnosis of a Grid-Forming Hybrid Energy Storage Power Station Based on a Physics-Informed Heterogeneous Temporal Graph Neural Network Constrained by Virtual Synchronous Generator Control Equations

Abstract

Fault diagnosis in grid-connected grid-forming hybrid energy storage station (GFM-HESS) systems is challenging because fault transients are jointly affected by converter control dynamics, multi-source electrical couplings, operating-condition variations, and measurement noise. To address these characteristics, this paper proposes a virtual synchronous generator (VSG)-constrained physics-informed heterogeneous temporal graph neural network (VSG-PI-HTGNN) for multi-class fault diagnosis. Based on VSG control characteristics and electrical relationships, 18-dimensional node-level features and eight-dimensional global physical features are constructed from ten monitored signals. These signals are further represented as a heterogeneous graph with five node types and eight predefined relation types, while node-type-specific transformations, heterogeneous graph convolution, learnable relation-scaling factors, and a primary–auxiliary dual-output framework are integrated for feature learning. A MATLAB/Simulink electromagnetic transient model is established to generate 3200 samples covering normal operation and nine fault conditions. At a signal-to-noise ratio (SNR) of 18 dB, the proposed model achieves 97.25% test accuracy and a macro-F1 score of 0.9727. Ablation results show that removing all physical information reduces the accuracy to 89.83%, while, under the unified experimental setting, the proposed model obtains higher values of the reported diagnostic metrics than the seven considered benchmark methods. Further evaluations show that the accuracy remains between 95.50% and 98.75% across SNR levels of 10–30 dB and reaches 95.38% with only 20% of the training data. Validation using an independently acquired hardware-in-the-loop (HIL) dataset further yields 92.75% accuracy and a macro-F1 score of 0.9282. Overall, these results indicate that the proposed method provides favorable diagnostic accuracy, noise robustness, data efficiency under limited-sample conditions, and simulation-to-HIL transferability under the evaluated conditions.

1. Introduction

With the increasing penetration of inverter-based renewable generation and the reduction in synchronous inertia, grid-forming converters (GFMCs) have received growing attention because of their capability to establish voltage and frequency references and provide inertia-like responses [1]. Hybrid energy storage systems (HESSs) combine storage technologies with complementary power, energy, and response characteristics and can provide flexible regulation over different time scales [2]. The integration of grid-forming control with HESSs provides a possible means of coordinating multiple storage units while maintaining grid-supporting capability [3]. As the application of grid-forming hybrid energy storage stations (GFM-HESSs) develops, fault diagnosis becomes relevant for identifying abnormal operating states and fault types and for supporting subsequent fault analysis and operational assessment.

Fault diagnosis of a GFM-HESS presents characteristics associated with converter control and multi-component coupling. The fault current of a GFMC is constrained by semiconductor current capability and is influenced by grid disturbances, control parameters, current-limiting actions, and grid conditions [4,5]. Moreover, a GFM-HESS involves interactions among multiple storage units, the DC bus, the GFMC, and the external grid. Therefore, DC-bus voltage, three-phase voltages and currents, frequency, and active/reactive power may exhibit coupled transient responses rather than independent variations. Diagnostic methods based mainly on individual signal amplitudes, fixed criteria, or directly learned raw time-series features may make limited use of these physical relationships, particularly under noisy or varying operating conditions [6]. Incorporating physical information and inter-variable relationships into data-driven models may therefore provide a useful direction for GFM-HESS fault classification.

Existing studies have explored grid-forming inverter fault detection using DC-link information, observer residuals, and adaptive thresholds [7,8,9]. For multi-class fault diagnosis, Mulinka et al. [10] develop a digital-twin framework for VSG-controlled grid-connected inverters and employ Bayesian optimization to improve diagnostic performance under limited data. The framework is subsequently extended by considering data quality and decision reliability [11]. These studies indicate that converter models, control information, and multi-source measurements can contribute to fault diagnosis. However, different electrical variables are generally treated as diagnostic inputs or model outputs, while their control-dependent relationships are not always explicitly represented.

Physics-informed learning provides another approach for incorporating prior knowledge into data-driven models. Ngo et al. [12] integrate physical equations with graph learning for power-system state estimation. Jabbari Zideh and Solanki [13] incorporate physical relationships into a convolutional autoencoder for anomaly detection, while Falas et al. [14] introduce physical constraints into neural-network-based state estimation. Related studies have shown the potential of physics-informed learning for improving model consistency and robustness [15]. Nevertheless, these methods mainly introduce circuit equations, state relationships, or physical constraints into estimation and anomaly-detection tasks. The use of VSG control dynamics and GFM-HESS-specific electrical relationships to construct class-discriminative fault features remains relatively limited.

Graph neural networks (GNNs) provide a suitable framework for representing relationships among multiple measurements or system components. Nguyen et al. [16] develop a spatial-temporal recurrent GNN for fault diagnosis in distribution systems, while Ngo et al. [17] employ deep graph learning to integrate electrical measurements and network information for fault detection and identification. These studies demonstrate the potential of graph-based information propagation for incorporating spatial and structural dependencies. However, most power-system GNNs are constructed primarily according to electrical-network topology and generally represent buses, branches, or similar entities using relatively uniform node and edge definitions.

Heterogeneous graph learning extends this framework by distinguishing different node categories and relation types. Chanda and Soltani [18] develop a heterogeneous graph-based multi-task model for fault-event diagnosis in smart grids. Heterogeneous graph representations have also been investigated for adaptive machine fault diagnosis [19], multi-sensor fusion in AC motor fault diagnosis [20], and multi-modal rotating-machinery fault diagnosis [21]. Liu et al. [22] further combine heterogeneous graph modeling with cross-modal information fusion for grid fault diagnosis. These studies suggest that heterogeneous graphs can provide a flexible representation of different data sources and relationships. Nevertheless, existing applications mainly focus on network topology, equipment states, or multimodal information, whereas heterogeneous relationships derived from VSG control and DC/AC electrical coupling in a GFM-HESS have received comparatively less attention.

Representative methods are compared in Table 1.

Table 1. VSG-based node-level features.

Ref.	Main Approach	Physical/Relational Information	Graph Representation	Relation to GFM-HESS Diagnosis
[7,8,9]	DC-link/observer residual/adaptive threshold	Converter states and electrical quantities	–	Mainly fault-state detection
[10,11]	Digital twin	VSG model and multi-source measurements	–	Limited explicit modeling of inter-variable relations
[12,13,14,15]	Physics-informed learning	Physical equations and constraints	Graph/neural network	Mainly estimation or anomaly-detection tasks
[16,17]	Spatial-temporal/deep GNN	Measurements and network structure	Topology-oriented graph	Limited distinction among heterogeneous measurement relations
[18]	Heterogeneous multi-task GNN	Network and fault-event relations	Heterogeneous graph	Not specifically developed for GFM-HESS control couplings
[19,20,21]	Heterogeneous graph learning	Phase/sensor/multimodal relations	Heterogeneous graph	Mainly equipment-oriented applications
[22]	Heterogeneous graph + cross-modal fusion	Multimodal fault information	Heterogeneous graph	Emphasizes multimodal information fusion

Overall, existing studies provide useful foundations for physics-informed diagnosis and graph-based relational modeling. However, two aspects remain worthy of further investigation for GFM-HESS fault diagnosis. First, physical information in existing diagnostic or physics-informed models is commonly introduced as detection criteria, state equations, or model constraints, whereas the control-dependent transient characteristics of VSG-controlled GFMCs have not been sufficiently incorporated into multi-level fault feature representations. Second, existing graph-based methods mainly define graph structures according to network topology or general data associations, while the heterogeneous relationships among DC-bus voltage, three-phase voltages and currents, frequency, and power in a GFM-HESS remain relatively less explored. Combining physics-informed feature construction with heterogeneous relation modeling may therefore provide a suitable representation of GFM-HESS fault transients.

To address these issues, this paper develops a VSG-constrained Physics-Informed Heterogeneous Temporal Graph Neural Network (VSG-PI-HTGNN) for multi-class fault diagnosis of a grid-connected GFM-HESS. The main contributions are summarized as follows:

(1)

A dual-level physics-informed feature representation is developed based on the transient characteristics of monitored electrical variables and the control relationships of the GFMC. Unlike approaches that rely primarily on raw signals or general statistical descriptors, the proposed representation incorporates node-level transient information together with system-level physical information derived from VSG dynamics, sequence components, DC-bus behavior, and power relationships, thereby introducing control-related prior knowledge into fault feature construction.

(2)

A measurement-oriented heterogeneous physical graph is constructed to describe the coupling relationships among different electrical variables. Different measurement categories and control/electrical relations are represented using different node and edge types rather than being treated uniformly. Node-type-specific transformations and learnable relation-scaling factors further allow the contributions of different predefined relation types to be adjusted during graph information propagation.

(3)

A primary–auxiliary multi-task learning framework is combined with the physics-informed heterogeneous graph to improve the representation of different fault classes under noisy and limited-sample conditions. The proposed model is evaluated through simulation-based testing, noise-robustness and limited-sample analyses, component-level comparisons, and Sim-to-HIL transferability validation, providing complementary evaluations of diagnostic performance under different test conditions.

2. VSG-PI-HTGNN Model

2.1. Dual-Branch Physical Feature Extraction

2.1.1. VSG-Based Node-Level Feature Extraction

The core principle of the VSG control mechanism is to incorporate the rotor motion equation and the active power–frequency droop characteristic of a synchronous generator into inverter control. This enables the GFMC to emulate the inertia and damping behavior of a conventional synchronous generator, thereby providing frequency support to the grid. The fundamental swing equation can be expressed as:

$${\displaystyle \frac{P_t^{\mathrm{gfmc},\mathrm{ref}}-P_t^{\mathrm{gfmc}}}{{\omega }^{\mathrm{gfmc},\mathrm{e}}}}-D^{\mathrm{gfmc},\mathrm{p}}\left({\omega }_t^{\mathrm{gfmc}}-{\omega }^{\mathrm{gfmc},\mathrm{e}}\right)=J^{\mathrm{gfmc}}{\displaystyle \frac{\mathrm{d}{\omega }_t^{\mathrm{gfmc}}}{\mathrm{d}t}}$$

(1)

where $P_t^{\mathrm{gfmc},\mathrm{ref}}$ and $P_t^{\mathrm{gfmc}}$ denote the active power reference and the actual active power of the GFMC at time t, respectively. $J^{\mathrm{gfmc}}$ is the moment of inertia of the GFMC at time t. $J^{\mathrm{gfmc}}$ is the angular frequency deviation. ${\omega }_t^{\mathrm{gfmc}}$ ${\omega }_t^{\mathrm{gfmc}}$ and ${\omega }^{\mathrm{gfmc},\mathrm{e}}$ represent the rotational angular speed and the rated angular speed of the GFMC at time t, respectively. $D^{\mathrm{gfmc},\mathrm{p}}$ is the damping coefficient of the GFMC.

In addition to the aforementioned active power–frequency control loop, the VSG control strategy also includes the reactive power–voltage control loop and the inner voltage and current control loops. These control loops jointly affect the voltage, current, frequency, and power outputs of the GFMC. Consequently, different fault conditions induce distinct variations in the multi-source measurement signals, while the VSG control loops further influence the transient evolution of these signals during the fault process.

Based on the above analysis, the ten multi-source measurement signals in each data sample are not simply treated as homogeneous time-series inputs. Instead, according to their physical attributes, they are categorized into five node types: DC, current, voltage, frequency, and power nodes. For each node signal, 18-dimensional statistical features $f_1^{\mathrm{r}}-f_{18}^{\mathrm{r}}$ are extracted from the steady-state and transient windows to characterize its local dynamic variations during fault transients. The first six dimensions are basic statistical features shared by all node types, whereas the remaining twelve dimensions are type-specific features designed according to the VSG-related physical characteristics of each node type. The detailed feature definitions are provided in Table 2.

Table 2. VSG-based node-level features.

No.	Feature Category	Feature Name	Applicable Node Types
1–6	Basic statistical features	Steady-state mean, transient mean, steady-state standard deviation, transient standard deviation, peak-to-peak value, and root mean square (RMS) value, as defined in Equations (2)–(5)	All node types
7–8	Dynamic variation features	Normalized fluctuation and normalized range, as defined in Equations (6) and (7)	All node types
9	Dynamic variation feature	Abnormal-value ratio, as defined in Equations (8)–(11)	Current nodes; voltage nodes
10	Dynamic variation feature	Limit-exceedance ratio, as defined in Equation (12)	DC nodes; current nodes; voltage nodes; frequency nodes
11–13	Distribution-shape features	Skewness, kurtosis, and mean absolute difference, as defined in Equations (13)–(15)	All node types
14–15	Pre-/post-fault difference features	Difference between the first and second halves, and steady-state-to-transient difference, as defined in Equations (16) and (17)	All node types
16–18	Power-related features	Active-power deviation index, reactive-power deviation index, and active–reactive power coupling strength, as defined in Equations (18)–(20)	Power nodes

Let the signal sequence of node v within the feature-extraction window be $x_v=\left\{x_{v,1},x_{v,2},\cdots ,x_{v,N^{\mathrm{w}}}\right\}$. The local features $f_1^{\mathrm{r}}-f_6^{\mathrm{r}}$ are defined as follows:

$$f_1^{\mathrm{r}},f_2^{\mathrm{r}}={\mu }_v={\displaystyle \frac{1}{N^{\mathrm{w}}}}{\displaystyle \sum _{k=1}^{N^{\mathrm{w}}}}{x}_{v,k}$$

(2)

$$f_3^{\mathrm{r}},f_4^{\mathrm{r}}={\sigma }_v=\sqrt{{\displaystyle \frac{1}{N^{\mathrm{w}}}}{\displaystyle \sum _{k=1}^{N^{\mathrm{w}}}}{\left(x_{v,k}-{\mu }_v\right)}^2}$$

(3)

$$f_5^{\mathrm{r}}=x_v^{\mathrm{pk}}=\max \left|x_{v,k}\right|$$

(4)

$$f_6^{\mathrm{r}}=x_v^{\mathrm{rms}}=\sqrt{{\displaystyle \frac{1}{N^{\mathrm{w}}}}{\displaystyle \sum _{k=1}^{N^{\mathrm{w}}}}{x}_{v,k}^2}$$

(5)

where N^w denotes the number of sampling points within the window. $x_{v,k}$ denotes the signal value of node v at the k-th sampling point. ${\mu }_v$, ${\sigma }_v$, $x_v^{\mathrm{pk}}$ and $x_v^{\mathrm{rms}}$ denote the mean, standard deviation, peak value, and root-mean-square (RMS) value, respectively, of the signal $x_{v,k}$ within the feature-extraction window. By applying the steady-state and transient windows separately, the steady-state mean, transient mean, steady-state standard deviation, and transient standard deviation are obtained.

The local feature $f_7^{\mathrm{r}}$ is defined as:

$$F_v^{\mathrm{fluc}}={\displaystyle \frac{{\sigma }_v}{x_v^{\mathrm{N}}+\mathsf{\epsilon }}}$$

(6)

where $F_v^{\mathrm{fluc}}$ denotes the normalized fluctuation feature of the signal $x_{v,k}$ at node v. $x_v^{\mathrm{N}}$ denotes the rated value corresponding to $x_{v,k}$. ε is a small positive constant introduced to avoid division by zero.

The local feature $f_8^{\mathrm{r}}$ is defined as:

$$R_v={\displaystyle \frac{\max \left(x_{v,k}\right)-\min \left(x_{v,k}\right)}{x_v^{\mathrm{N}}+\mathsf{\epsilon }}}$$

(7)

where $R_v$ denotes the normalized range feature of the signal $x_{v,k}$ at node v; $\max \left(x_{v,k}\right)$ and $\min \left(x_{v,k}\right)$ denote the maximum and minimum values, respectively, of $x_{v,k}$ within the feature-extraction window.

The local feature $f_9^{\mathrm{r}}$ is defined as:

$$f_9^{\mathrm{r}}=F_v^{\mathrm{un}}={\displaystyle \frac{1}{N^{\mathrm{w}}}}{\displaystyle \sum _{k=1}^{N^{\mathrm{w}}}}\mathbb{I}\left({\displaystyle \frac{{\mathrm{z}}_{2,k}}{{\mathrm{z}}_{1,k}+\epsilon }}>{\gamma }_v\right)$$

(8)

$${\mathrm{z}}_{1,k}={\displaystyle \frac{1}{3}}\left(z_{a,k}+a{z}_{b,k}+a^2{z}_{c,k}\right)$$

(9)

$${\mathrm{z}}_{2,k}={\displaystyle \frac{1}{3}}\left(z_{a,k}+a^2{z}_{b,k}+a{z}_{c,k}\right)$$

(10)

$$a={\mathrm{e}}^{j2\pi /3}$$

(11)

where $F_v^{\mathrm{un}}$ denotes the abnormal-value ratio of the signal $x_{v,k}$ at node v within the feature-extraction window; for current and voltage nodes, it corresponds to the negative-sequence ratio. ${\mathrm{z}}_{1,k}$ and ${\mathrm{z}}_{2,k}$ denote the positive- and negative-sequence components, respectively, of the three-phase currents or voltages at the corresponding current or voltage node. $z_{a,k},z_{b,k},z_{c,k}$ denote the phase-A, phase-B, and phase-C signal values, respectively. $\mathbb{I}\left(\cdot \right)$ is the indicator function, which equals 1 when the specified condition is satisfied and 0 otherwise. ${\gamma }_v$ denotes the threshold for abnormality detection.

The local feature $f_{10}^{\mathrm{r}}$ is defined as:

$$f_{10}^{\mathrm{r}}=F_v^{\mathrm{ov}}={\displaystyle \frac{1}{N^{\mathrm{w}}}}{\displaystyle \sum _{k=1}^{N^{\mathrm{w}}}}\mathbb{I}\left(x_{v,k}<x_{v,\min } \mathrm{or} x_{v,k}>x_{v,\max }\right)$$

(12)

where $F_v^{\mathrm{ov}}$ denotes the limit-exceedance ratio of the signal $x_{v,k}$ at node v within the feature-extraction window. $x_{v,\max }$ and $x_{v,\min }$ denote the upper and lower safety limits, respectively, of $x_{v,k}$.

The local features $f_{11}^{\mathrm{r}}-f_{13}^{\mathrm{r}}$ are defined as follows:

$$f_{11}^{\mathrm{r}}={\mathrm{Skew}}_v={\displaystyle \frac{1}{N^{\mathrm{w}}}}{\displaystyle \sum _{k=1}^{N^{\mathrm{w}}}}{\left({\displaystyle \frac{x_{v,k}-{\mu }_v}{{\sigma }_v+\mathsf{\epsilon }}}\right)}^3$$

(13)

$$f_{12}^{\mathrm{r}}={\mathrm{Kurt}}_v={\displaystyle \frac{1}{N^{\mathrm{w}}}}{\displaystyle \sum _{k=1}^{N^{\mathrm{w}}}}{\left({\displaystyle \frac{x_{v,k}-{\mu }_v}{{\sigma }_v+\mathsf{\epsilon }}}\right)}^4$$

(14)

$$f_{13}^{\mathrm{r}}=D_v={\displaystyle \frac{1}{N^{\mathrm{w}}-1}}{\displaystyle \sum _{k=1}^{N^{\mathrm{w}}}}\left|x_{v,k+1}-x_{v,k}\right|$$

(15)

where ${\mathrm{Skew}}_v$, ${\mathrm{Kurt}}_v$, and $D_v$ denote the skewness, kurtosis, and mean absolute difference, respectively, of the signal $x_{v,k}$ at node v within the feature-extraction window.

The local features $f_{14}^{\mathrm{r}}-f_{15}^{\mathrm{r}}$ are defined as follows:

$$f_{14}^{\mathrm{r}}=\Delta {\mathrm{half}}_v={\displaystyle \frac{\left|{\mu }_v^2-{\mu }_v^1\right|}{x_v^{\mathrm{N}}+\mathsf{\epsilon }}}$$

(16)

$$f_{15}^{\mathrm{r}}=\Delta {\mathrm{st}}_v={\displaystyle \frac{\left|{\mu }_v^{\mathrm{tr}}-{\mu }_v^{\mathrm{st}}\right|}{x_v^{\mathrm{N}}+\mathsf{\epsilon }}}$$

(17)

where $\Delta {\mathrm{half}}_v$ denotes the relative difference between the mean values of the first and second halves of the same feature-extraction window for the signal $x_{v,k}$ at node v, thereby characterizing the trend shift during the fault process. ${\mu }_v^1$ and ${\mu }_v^2$ denote the corresponding mean values of the first and second halves for the signal $x_{v,k}$ at node v, respectively. $\Delta {\mathrm{st}}_v$ denotes the relative difference between the mean values of $x_{v,k}$ in the steady-state and transient feature-extraction windows, thereby characterizing the magnitude shift induced by the fault. ${\mu }_v^{\mathrm{st}}$ and ${\mu }_v^{\mathrm{tr}}$ denote the mean values of $x_{v,k}$ in the steady-state and transient windows, respectively.

The local features $f_{16}^{\mathrm{r}}-f_{18}^{\mathrm{r}}$ are defined as follows:

$$f_{16}^{\mathrm{r}}=e_P={\displaystyle \frac{\left|P_k^{\mathrm{gfmc},\mathrm{ref}}-{\overline{P}}_k^{\mathrm{gfmc}}\right|}{\left|P_k^{\mathrm{gfmc},\mathrm{ref}}\right|+\mathsf{\epsilon }}}$$

(18)

$$f_{17}^{\mathrm{r}}=e_Q={\displaystyle \frac{\left|Q_k^{\mathrm{gfmc},\mathrm{ref}}-{\overline{Q}}_k^{\mathrm{gfmc}}\right|}{\left|Q_k^{\mathrm{gfmc},\mathrm{ref}}\right|+\mathsf{\epsilon }}}$$

(19)

$$f_{18}^{\mathrm{r}}={\rho }_{PQ}=\left|{\displaystyle \frac{{\displaystyle \sum _{k=1}^{N^{\mathrm{w}}}}(P_k^{\mathrm{gfmc}}-{\overline{P}}_k^{\mathrm{gfmc}})(Q_k^{\mathrm{gfmc}}-{\overline{Q}}_k^{\mathrm{gfmc}})}{\sqrt{{\displaystyle \sum _{k=1}^{N^{\mathrm{w}}}}{(P_k^{\mathrm{gfmc}}-{\overline{P}}_k^{\mathrm{gfmc}})}^2}\sqrt{{\displaystyle \sum _{k=1}^{N^{\mathrm{w}}}}{(Q_k^{\mathrm{gfmc}}-{\overline{Q}}_k^{\mathrm{gfmc}})}^2}+\mathsf{\epsilon }}}\right|$$

(20)

where $e_P$ and $e_Q$ denote the active-power and reactive-power deviation indices, respectively, of the GFMC output. $P_k^{\mathrm{gfmc},\mathrm{ref}}$ and $Q_k^{\mathrm{gfmc},\mathrm{ref}}$ denote the reference values of the GFMC active and reactive power, respectively, within the feature-extraction window. ${\overline{P}}_k^{\mathrm{gfmc}}$ and ${\overline{P}}_k^{\mathrm{gfmc}}$ denote their corresponding mean values within the window, calculated in the same manner as in Equation (2). ${\rho }_{PQ}$ denotes the degree of correlation between the variation trends of active and reactive power within the feature-extraction window and characterizes the coupling between the active- and reactive-power responses. $P_k^{\mathrm{gfmc}}$ and $Q_k^{\mathrm{gfmc}}$ denote the GFMC active and reactive power outputs, respectively, within the feature-extraction window.

2.1.2. Global Physical Feature Extraction

In addition to the node-level features, an 8-dimensional global physical feature vector $f_1^{\mathrm{w}}-f_8^{\mathrm{w}}$ is further extracted to characterize the overall operating state of the GFM-HESS. These features are constructed based on Kirchhoff’s laws, symmetrical component analysis, and the VSG swing equation, enabling the characterization of circuit constraint relationships and control dynamic responses at the system level.

Unlike node-level features, which focus on describing the variation characteristics of individual node signals, global physical features emphasize the overall physical consistency of the system. They can be incorporated into the model as physics-informed prior knowledge, providing supplementary constraints for subsequent feature learning. The detailed feature configuration is presented in Table 3.

Table 3. Global physical features.

Feature Name	Formula
Frequency deviation	$f_1^{\mathrm{w}}=f_k^{\mathrm{gfmc}}-f_N^{\mathrm{gfmc}}$
Swing equation residual	$f_2^{\mathrm{w}}=\sqrt{{\displaystyle \frac{1}{N^{\mathrm{w}}}}{\displaystyle \sum _{k=1}^{N^{\mathrm{w}}}{\left(\frac{P_k^{\mathrm{gfmc},\mathrm{ref}}-P_k^{\mathrm{gfmc}}}{{\omega }^{\mathrm{gfmc},\mathrm{e}}}-D^{\mathrm{gfmc},\mathrm{p}}\Delta {\omega }_k^{\mathrm{gfmc}}-J^{\mathrm{gfmc}}\frac{{\omega }_{k+1}^{\mathrm{gfmc}}-{\omega }_k^{\mathrm{gfmc}}}{\Delta t_k}\right)}^2}}$
Current unbalance factor	$f_3^{\mathrm{w}}={\displaystyle \frac{\left[\max \left(i_{k,a}^{\mathrm{gfmc}},i_{k,b}^{\mathrm{gfmc}},i_{k,c}^{\mathrm{gfmc}}\right)-\min \left(i_{k,a}^{\mathrm{gfmc}},i_{k,b}^{\mathrm{gfmc}},i_{k,c}^{\mathrm{gfmc}}\right)\right]}{{\overline{i}}_{k,abc}^{\mathrm{gfmc}}+\mathsf{\epsilon }}}$
Voltage unbalance factor	$f_4^{\mathrm{w}}={\displaystyle \frac{\left[\max \left(u_{k,a}^{\mathrm{gfmc}},u_{k,b}^{\mathrm{gfmc}},u_{k,c}^{\mathrm{gfmc}}\right)-\min \left(u_{k,a}^{\mathrm{gfmc}},u_{k,b}^{\mathrm{gfmc}},u_{k,c}^{\mathrm{gfmc}}\right)\right]}{{\overline{u}}_{k,abc}^{\mathrm{gfmc}}+\mathsf{\epsilon }}}$
Negative-sequence current ratio	$f_5^{\mathrm{w}}={\displaystyle \frac{I_{2,k}}{I_{1,k}+\epsilon }}$
DC voltage deviation	$f_6^{\mathrm{w}}=U_k^{\mathrm{dc}}-U_N^{\mathrm{dc}}$
P–Q correlation coefficient	$f_7^{\mathrm{w}}={\displaystyle \frac{\mathrm{Cov}(P_k^{\mathrm{gfmc}},Q_k^{\mathrm{gfmc}})}{{\sigma }_{P_k^{\mathrm{gfmc}}}{\sigma }_{Q_k^{\mathrm{gfmc}}}+\mathsf{\epsilon }}}$
Rate of change in frequency	$f_8^{\mathrm{w}}={\displaystyle \frac{f_{k+1}^{\mathrm{gfmc}}-f_k^{\mathrm{gfmc}}}{\Delta t_k}}$

• where $f_k^{\mathrm{gfmc}}$ and $f_N^{\mathrm{gfmc}}$ denote the GFMC frequency and its rated value, respectively, within the feature-extraction window. $\Delta {\omega }_k^{\mathrm{gfmc}}$ denotes the deviation between the VSG rotor angular velocity ${\omega }_k^{\mathrm{gfmc}}$ and the rated angular velocity ${\omega }_k^{\mathrm{gfmc},\mathrm{e}}$ within the feature-extraction window. $\Delta t_k$ is the sampling interval. $i_{k,a}^{\mathrm{gfmc}},i_{k,b}^{\mathrm{gfmc}},i_{k,c}^{\mathrm{gfmc}}$ denote the three-phase currents at the PCC within the feature-extraction window, and ${\overline{i}}_{k,abc}^{\mathrm{gfmc}}$ denotes their mean value. $u_{k,a}^{\mathrm{gfmc}},u_{k,b}^{\mathrm{gfmc}},u_{k,c}^{\mathrm{gfmc}}$ denote the three-phase voltages at the PCC, and ${\overline{u}}_{k,abc}^{\mathrm{gfmc}}$ denotes their mean value. $I_{2,k}$ and $I_{1,k}$ denote the negative- and positive-sequence components of the three-phase PCC currents, respectively. $U_k^{\mathrm{dc}}$ and $U_N^{\mathrm{dc}}$ denote the actual and rated DC-bus voltages, respectively. $\mathrm{Cov}(P_k^{\mathrm{gfmc}},Q_k^{\mathrm{gfmc}})$ denotes the covariance between the active and reactive power outputs of the GFM-HESS within the feature-extraction window, calculated according to Equation (20). ${\sigma }_{P_k^{\mathrm{gfmc}}}$ and ${\sigma }_{Q_k^{\mathrm{gfmc}}}$ denote the standard deviations of the active and reactive power outputs of the GFM-HESS, respectively, within the feature-extraction window.

2.2. Heterogeneous Temporal Graph Neural Network Model

2.2.1. Heterogeneous Graph Construction

To characterize the physical coupling relationships among the multi-source measurements of the grid-connected GFM-HESS and to enhance the physical interpretability of information propagation, a heterogeneous graph model is constructed, as illustrated in Figure 1. The ten measured signals: DC-bus voltage, phase-A current, phase-B current, phase-C current, phase-A voltage, phase-B voltage, phase-C voltage, GFMC frequency, GFMC active power, and GFMC reactive power—are represented as ten graph nodes. According to their physical attributes, these nodes are categorized into five node types: DC, current, voltage, frequency, and power nodes. Based on the physical coupling relationships among these node types, eight relation types are defined. The eight relation types are established according to the following physical principles:

Figure 1. Heterogeneous graph model.

(1)

Power-balance relationship: DC–power coupling based on the DC-side power-balance relationship;

(2)

Control-coupling relationship: frequency–power coupling associated with the active power–frequency control loop of the VSG;

(3)

Control-coupling relationship: voltage–power coupling associated with the reactive power–voltage control loop of the VSG;

(4)

Power-calculation relationship: voltage–power and current–power couplings derived from the power-calculation equations;

(5)

Same-phase electrical relationship: voltage–current coupling based on the electrical relationship between voltage and current of the corresponding phase;

(6)

AC/DC voltage-conversion relationship: DC–voltage coupling derived from the AC/DC conversion relationship of the GFMC;

(7)

Power-capacity relationship: intra-type coupling among power nodes based on the converter power-capacity constraint;

(8)

Same-type electrical relationship: intra-type coupling among electrical quantities of the same category based on the electromagnetic coupling among phases.

All edge weights are normalized, and the propagation strengths of different edge types are further adjusted by learnable scaling factors. To adaptively regulate the information propagation intensity of different relation types, a learnable parameter vector is introduced. For any target node n, the normalized propagation strength of its k-th edge is defined as the ratio of the product of the edge weight and the corresponding type-specific scaling factor to the sum over all incoming edges:

$${\alpha }_{e_k}={\displaystyle \frac{w_k{\gamma }_{r_k}}{{\displaystyle \sum _{j\in N_{n,\mathrm{in}}}}{w}_j{\gamma }_{r_j}}}$$

(21)

where ${\alpha }_{e_k}$ is the normalized propagation strength of the k-th edge. $N_{n,\mathrm{in}}$ denotes the neighbor set of node $n$. $w_j$ and $r_j$ are the raw weight and edge type of the j-th edge, respectively. ${\gamma }_{r_j}$ is the learnable scaling factor for the $r_j$-th edge type. The scaling factor is shared by all edges belonging to the same relation type and is jointly optimized with the remaining network parameters through backpropagation. It does not modify the predefined physical connectivity or introduce additional nonphysical edges. Instead, it adjusts the relative contribution of different physically defined relation types during neighborhood aggregation. For a given target node, a larger scaling factor increases the contribution of the corresponding relation type to the aggregated message, while the normalization operation maintains the relative balance among all incoming information paths. Therefore, this module functions as a relation-level gating mechanism that combines the fixed physical graph prior with data-driven feature learning.

This parameterization preserves the predefined physical coupling structure while allowing the model to identify the relative importance of different control and electrical coupling paths from the fault data, thereby facilitating the extraction of robust fault representations.

2.2.2. Graph Neural Network Layer Design

The network architecture is illustrated in Figure 2 and comprises a node-type-specific transformation module, three heterogeneous graph convolution layers with residual connections, a fully connected classification block, and a dual-output design.

Figure 2. Heterogeneous temporal graph neural network layer.

(1)

Node-Type-Specific Transformation

To address the heterogeneity in dimensional units and numerical distributions across different node types, a type-specific linear projection is first applied:

$${\mathit{H}}_{n,0}=\mathrm{ReLU}\left({\mathit{F}}_n{\mathit{W}}_{nt}+{\mathit{b}}_{nt}\right),\hspace{1em}n=1,2,\dots ,10$$

(22)

where ${\mathit{F}}_n\in {\mathbb{R}}^{N\times 26}$ is the raw feature matrix of the nodes, with N denoting the number of nodes and 26 representing the feature dimension of each node (i.e., 18-dimensional node-level VSG features plus 8-dimensional global physical feature). ${\mathit{W}}_{nt}\in {\mathbb{R}}^{26\times 26}$ is the dedicated weight matrix for the $nt$-th node type. ${\mathit{b}}_{nt}\in {\mathbb{R}}^{1\times 26}$ is the corresponding bias vector. $\mathrm{ReLU}\left(x\right)=\max \left(0,x\right)$ is the rectified linear unit activation function. 5 independent parameter sets are assigned to the 5 node types, enabling each set to learn an appropriate representation scale for the corresponding type of physical quantity. After transformation, the features of all nodes are mapped into a unified 26-dimensional space, producing the output ${\mathit{H}}_{n,0}\in {\mathbb{R}}^{N\times 10\times 26}$.

(2)

Heterogeneous Graph Convolution Layer

The node feature matrix ${\mathit{H}}_{n,0}$ after the type-specific transformation is passed through a three-layer heterogeneous graph convolutional network. The three convolution layers share the same structure, while each maintains its own independent parameters. The computation in each convolution layer consists of four steps: weighted neighbor aggregation, message transformation, node update, and residual connection.

$$\mathit{a}{\mathit{g}}_{n,l}=\sum _{m\subset N_{n,\mathrm{in}}}{\alpha }_{e_{m.n}}\cdot {\mathit{H}}_{m,l-1}$$

(23)

$${\mathit{msg}}_{n,l}=\mathrm{ReLU}\left(\mathit{a}{\mathit{g}}_{n,l}{\mathit{W}}_{gl}+{\mathit{b}}_{gl}\right)$$

(24)

$${\tilde{\mathit{H}}}_{n,l}={\mathit{H}}_{n,l-1}+\beta \cdot {\mathit{msg}}_{n,l}$$

(25)

$${\mathit{H}}_l=\mathrm{ReLU}\left({\tilde{\mathit{H}}}_l\right)+{\alpha }^{res}\cdot {\mathit{H}}_{l-1}$$

(26)

where $\mathit{a}{\mathit{g}}_{n,l}\in {\mathbb{R}}^{N\times 26}$ is the feature representation of node n after aggregating neighbor information. $N_{n,\mathrm{in}}$ denotes the set of neighboring source nodes of node $n$ (i.e., all source nodes m for which an edge m→n exists). ${\alpha }_{e_{m.n}}$ is the normalized propagation strength. ${\mathit{H}}_{m,l-1}\in {\mathbb{R}}^{N\times 26}$ is the feature vector of node m output from the l-1-th convolution layer. ${\mathit{msg}}_{n,l}\in {\mathbb{R}}^{N\times 26}$ is the transformed message feature aggregated from neighboring nodes. ${\mathit{W}}_{gl}\in {\mathbb{R}}^{26\times 26}$ is the shared weight matrix of the l-th convolution layer. ${\mathit{b}}_{gl}\in {\mathbb{R}}^{26\times 26}$ is the shared bias vector of the l-th convolution layer. ${\tilde{\mathit{H}}}_{n,l}\in {\mathbb{R}}^{N\times 10\times 26}$ is the intermediate feature of node $n$ of the l-th convolution layer. $\beta$ is the step coefficient controlling the strength of neighbor information injection. ${\mathit{H}}_l\in {\mathbb{R}}^{N\times 10\times 26}$ is the final output of the l-th convolution layer. ${\alpha }^{res}$ is the residual coefficient.

(3)

Fully Connected Classification Layer

After graph convolution, the features are flattened into ${\mathit{H}}_f=\mathrm{Flatten}\left({\mathit{H}}_l\right),{\mathit{H}}_f\in {\mathbb{R}}^{N\times 260}$ and subsequently fed into a three-layer fully connected network for feature compression and classification. Each layer sequentially performs a linear transformation, batch normalization (BN), ReLU activation, and dropout regularization:

$${\mathit{z}}_1=\mathrm{BN}\left({\mathit{H}}_f\right){\mathit{W}}_1+{\mathit{b}}_1,{\mathit{a}}_1=\mathrm{Dropout}\left(\mathrm{ReLU}\left({\mathit{z}}_1\right),p\right)$$

(27)

$${\mathit{z}}_2=\mathrm{BN}\left({\mathit{H}}_f\right){\mathit{W}}_2+{\mathit{b}}_2,{\mathit{a}}_2=\mathrm{Dropout}\left(\mathrm{ReLU}\left({\mathit{z}}_2\right),p\right)$$

(28)

$${\mathit{z}}_3=\mathrm{BN}\left({\mathit{a}}_2\right){\mathit{W}}_3+{\mathit{b}}_3,{\mathit{a}}_3=\mathrm{Dropout}\left(\mathrm{ReLU}\left({\mathit{z}}_3\right),p\right)$$

(29)

where ${\mathit{z}}_1,{\mathit{z}}_2,{\mathit{z}}_3$ denote the intermediate features obtained after the linear transformation and batch normalization in the fully connected classification layer, respectively. ${\mathit{a}}_1,{\mathit{a}}_2,{\mathit{a}}_3$ denote the activated features after ReLU activation and Dropout regularization, respectively. p is the dropout probability $\mathrm{Dropout}\left(x\right)$ used for regularization. Batch normalization $\mathrm{BN}\left(x\right)$ standardizes each feature dimension to stabilize the activation distribution during training. ${\mathit{W}}_1,{\mathit{W}}_2,{\mathit{W}}_3,{\mathit{b}}_1,{\mathit{b}}_2,{\mathit{b}}_3$ denote the weight matrix and bias vector of the l-th layer, respectively, with dimensions given by:

$${\mathit{W}}_1\in {\mathbb{R}}^{260\times 96},{\mathit{b}}_1\in {\mathbb{R}}^{1\times 96}$$

(30)

$${\mathit{W}}_2\in {\mathbb{R}}^{96\times 72},{\mathit{b}}_2\in {\mathbb{R}}^{1\times 72}$$

(31)

$${\mathit{W}}_3\in {\mathbb{R}}^{72\times 48},{\mathit{b}}_3\in {\mathbb{R}}^{1\times 48}$$

(32)

(4)

Dual-output Design

To strengthen the supervisory constraint on the shared feature representation during training and reduce the risk of overfitting associated with a single classification branch, a dual-output architecture is adopted in this study. On top of the shared features $a_3$ from the final layer, dual outputs, namely a primary classifier and an auxiliary classifier, are connected in parallel:

$${\mathit{L}}^{\mathrm{main}}={\mathit{a}}_3{\mathit{W}}_4+{\mathit{b}}_4$$

(33)

$${\mathit{L}}^{\mathrm{aux}}={\mathit{a}}_3{\mathit{W}}_a+{\mathit{b}}_a$$

(34)

where ${\mathit{L}}^{\mathrm{main}}\in {\mathbb{R}}^{N\times 10}$ and ${\mathit{L}}^{\mathrm{aux}}\in {\mathbb{R}}^{N\times 10}$ are the outputs of the primary classifier and the auxiliary classifier, respectively. The auxiliary classifier provides gradient signals only during training, and only the output of the primary classifier is used during inference. ${\mathit{W}}_4\in {\mathbb{R}}^{48\times 10}$ and ${\mathit{W}}_a\in {\mathbb{R}}^{48\times 10}$ are the weight matrices of the primary and auxiliary classifiers, respectively. ${\mathit{b}}_4\in {\mathbb{R}}^{1\times 10}$ and ${\mathit{b}}_a\in {\mathbb{R}}^{1\times 10}$ are the corresponding bias vectors.

2.2.3. Multi-Task Learning and Training Strategy

An auxiliary classifier is introduced to establish a multi-task learning framework. This design regularizes the shared feature representation and improves the generalization capability of the model. The overall loss function contains three terms: the cross-entropy loss of the primary classifier, the cross-entropy loss of the auxiliary classifier, and an L2 weight decay regularization term:

$$\mathcal{L}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{\mathcal{L}}_{CE}\left({\mathit{L}}_i^{\mathrm{main}},y_i\right)+{\lambda }^{\mathrm{aux}}{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{\mathcal{L}}_{CE}\left({\mathit{L}}_i^{\mathrm{aux}},y_i\right)+{\lambda }^{\mathrm{wd}}{‖\Theta ‖}_2^2$$

(35)

$${\mathcal{L}}_{CE}(\mathit{L},y)=-\log \left[{\displaystyle \frac{\exp \left(L_y\right)}{{\displaystyle \sum _{k=1}^{10}}\exp \left(L_k\right)}}\right]$$

(36)

where $\mathcal{L}$ denotes the total loss. ${\mathcal{L}}_{CE}\left({\mathit{L}}_i^{\mathrm{main}},y_i\right)$ and ${\mathcal{L}}_{CE}\left({\mathit{L}}_i^{\mathrm{aux}},y_i\right)$ denote the cross-entropy losses of the primary and auxiliary classifiers for the i-th sample, respectively. For a single sample, the cross-entropy loss is defined as ${\mathcal{L}}_{CE}(\mathit{L},y)$, where $y\in \left\{0,1,\dots \dots ,9\right\}$ is the ground-truth class label. $L_k$ is the raw Logits value for the k-th class. The batch loss is obtained by averaging over the N samples in the mini-batch. ${\lambda }^{\mathrm{aux}}$ is the weighting coefficient of the auxiliary task and controls the gradient contribution of the auxiliary branch. ${\lambda }^{\mathrm{wd}}$ is the weight decay coefficient. $\Theta$ denotes the set of all trainable parameters. ${‖\Theta ‖}_2^2$ denotes the squared L2 norm of all trainable parameters.

Model parameters are updated using the Adam optimizer [23]. The learning rate is scheduled with a cosine annealing warm-restart strategy:

$${\mathit{m}}_t={\beta }_1{\mathit{m}}_{t-1}+\left(1-{\beta }_1\right){\mathit{g}}_t$$

(37)

$${\mathit{v}}_t={\beta }_2{\mathit{v}}_{t-1}+\left(1-{\beta }_2\right){\mathit{g}}_t^2$$

(38)

$${\hat{\mathit{m}}}_t={\displaystyle \frac{{\mathit{m}}_t}{1-{\beta }_1}}$$

(39)

$${\hat{\mathit{v}}}_t={\displaystyle \frac{{\mathit{v}}_t}{1-{\beta }_2}}$$

(40)

$${\mathit{\theta }}_t={\mathit{\theta }}_{t-1}-lr{\displaystyle \frac{{\hat{\mathit{m}}}_t}{\sqrt{{\hat{\mathit{v}}}_t}+\mathsf{\epsilon }}}$$

(41)

$$lr(e)={\displaystyle \frac{l{r}_0}{2}}\left[1+\cos \left({\displaystyle \frac{\pi \left[(e-1)\mathrm{mod}{T}_c\right]}{T_c}}\right)\right]+l{r}_0$$

(42)

where ${\mathit{m}}_t$ and ${\mathit{v}}_t$ are the first and second-moment estimates, respectively, and ${\hat{\mathit{m}}}_t$ and ${\hat{\mathit{v}}}_t$ are their bias-corrected forms. ${\beta }_1$ and ${\beta }_2$ are the exponential decay coefficients for the first and second moments, respectively. ${\mathit{\theta }}_t$ denotes the model parameters at the t-th step. ${\mathit{g}}_t$ is the gradient of the loss function with respect to ${\mathit{\theta }}_t$. $\mathsf{\epsilon }$ is a small positive constant introduced for numerical stability. $lr$ is the learning rate at the t-th step. $lr(e)$ is the learning rate at e-th epoch. $l{r}_0$ is the minimum learning rate and $\mathrm{mod}(x)$ denotes the modulo operation.

2.3. Overall Fault Diagnosis Procedure

The overall fault diagnosis procedure for the grid-connected GFM-HESS system, based on the proposed VSG-PI-HTGNN model, is illustrated in Figure 3.

Figure 3. Fault diagnosis flowchart of the grid-connected GFM-HESS system based on the VSG-PI-HTGNN model.

The specific procedure is as follows:

(1)

The operational data of the grid-connected GFM-HESS are first collected and divided into training, validation, and test sets according to the noise-seed isolation strategy described in Section 3.2, thereby reducing the influence of noise-pattern leakage on the evaluation of model generalization;

(2)

Gaussian white noise with an SNR of 18 dB is independently added to the training, validation, and test sets to emulate measurement noise in practical environments and evaluate the diagnostic performance of the model under noisy conditions;

(3)

Node-level feature extraction, global physical feature concatenation, and heterogeneous graph sample construction are performed separately for each data subset to generate the model inputs;

(4)

The VSG-PI-HTGNN model is initialized, in which the heterogeneous graph structure is used to represent different physical nodes and their coupling relationships within the GFM-HESS;

(5)

The model is trained using the graph samples in the training set, while the validation accuracy and validation loss are calculated using the validation set to determine whether the early-stopping criterion is satisfied;

(6)

If the early-stopping criterion is not satisfied, the model parameters are continuously updated based on the training loss. Otherwise, the model parameters corresponding to the best validation performance are retained;

(7)

Finally, the test-set samples are fed into the trained VSG-PI-HTGNN model with the best validation performance to perform fault diagnosis and fault-type classification, thereby obtaining the final fault classification results.

3. Scenario Settings and Data Processing

3.1. Modeling of the Grid-Connected Grid-Forming Hybrid Energy Storage Station

3.1.1. System Topology

Field fault data from GFM-HESS are difficult to obtain because actual fault events are infrequent, operating conditions vary considerably, and intentional fault injection is restricted by equipment safety requirements. Therefore, a MATLAB/Simulink electromagnetic transient model is developed to simulate the dynamic responses of the GFM-HESS under normal and various fault conditions and to generate sufficient fault data under controllable and repeatable settings. The system topology, rated parameters, and operating boundaries of the model are determined with reference to the engineering specifications of the New Energy Storage Research Demonstration Base in Guangdong Province, China. No field fault records from the demonstration project are used for model training. The 3200 samples used for training, validation, and simulation-based testing are generated using the electromagnetic transient model described in Section 3.2.

In the grid-connected GFM-HESS system, the DC side integrates four typical energy storage technologies in addition to photovoltaics (PV): lithium-ion batteries (BAT), supercapacitors (SCP), flywheel energy storage systems (FES), and an integrated hydrogen energy storage (IHES) system. The IHES consists of a hydrogen fuel cell (HFC), a hydrogen electrolyzer (HEL), and a hydrogen storage tank (HST). The system is further equipped with corresponding grid-forming converters (GFMCs) employing the virtual synchronous generator (VSG) control strategy, as well as an energy management system (EMS). The overall system topology is shown in Figure 4.

Figure 4. Topological structure of the grid-connected GFM-HESS system.

3.1.2. System Parameter Configuration

The PV unit, SCP, FES, BAT, and IHES are connected to the 1500 V DC bus through their respective power converters. The aggregated DC power is then converted into 690 V/50 Hz AC power by the GFMC. Subsequently, the AC power is delivered to the utility grid through a 2 km transmission line and a 690 V/10 kV transformer. This configuration reproduces the topology of the demonstration project and the detailed system parameters are listed in Table 4.

Table 4. Parameters of the grid-connected GFM-HESS system.

System Module	Parameter Name	Parameter Setting
DC bus	Voltage	1500 V
Photovoltaic system	Rated power	1000 kW
	Solar irradiance	1200 W/m2
Supercapacitor	Rated power	300 kW
	Initial SOC	50%
Flywheel energy storage	Rated power	400 kW
	Initial SOC	50%
Lithium-ion battery	Rated power	400 kW
	Initial SOC	50%
Integrated hydrogen energy storage system	Rated power	100 kW
	Initial SOC	50%
AC bus	Voltage	690 V
	Frequency	50 Hz
AC load	Power	300 kW
VSG control	Filter impedance parameters	0.005 Ω, 0.005 H, 1 × 10−6 F
	Line impedance parameters	0.005 Ω, 0.0012 H
	Active power reference	300 kW
	Reactive power reference	0 Var
	Moment of inertia of the VSG active-power control loop	10 kg/m2
	Damping coefficient of the VSG active-power control loop	200
	Integral gain of the VSG reactive-power control loop	86
	Droop coefficient of the VSG reactive-power control loop	58
	Proportional and integral gains of the VSG voltage control loop	40, 5 × 104
	Proportional and integral gains of the VSG current control loop	1, 100

3.2. Experimental Environment and Data Generation

3.2.1. Experimental Environment

All experiments were conducted on a Windows 11 operating system. The hardware platform was equipped with a 13-th Gen Intel(R) Core(TM) i7-13700 processor operating at 2.10 GHz and 32 GB RAM. The deep learning model was implemented in MATLAB R2022b. The dataset was generated in MATLAB R2022b/Simulink. Specifically, the grid-connected GFM-HESS system shown in Figure 4 was established, and batch simulations were performed using stratified random sampling.

3.2.2. Data Acquisition

The duration of each simulation was set to 2.00 s, and the sampling frequency was 10 kHz. All faults were imposed at 1.00 s. For each simulation case, time-series data from ten monitoring channels were extracted over the interval from 0.90 s to 1.10 s to construct a multivariate time-series sample. These channels include the DC-side bus voltage, the AC-side three-phase currents, the AC-side three-phase voltages, the AC-side frequency, the AC-side active power, and the AC-side reactive power of the GFMC.

The dataset includes ten operating conditions, namely the normal condition F0 and fault conditions F1–F9. For the normal condition F0, 80 base scenarios were generated using stratified randomization. For each fault condition from F1 to F9, the fault parameter combinations were coupled with 20 base scenarios, resulting in 720 fault scenarios. Therefore, a total of 800 base scenarios were obtained. Four independent noise realizations were then superimposed on each scenario, yielding a final dataset of 3200 samples. The randomization settings for the base scenarios under the normal condition F0 are summarized in Table 5. The variable parameter settings for fault conditions F1–F9 are given in Table 6.

Table 5. Generation settings of base scenarios.

Variable Category	Variable Name	Randomization Factor
AC load power	$P_t^{\mathrm{load}}$	80–100% (Latin hypercube sampling)
Initial energy storage SOC	$SO{C}_0^x$	30–90% (uniform sampling)
Solar irradiance	$S_t^{\mathrm{pv}}$	50–100% (uniform sampling)
Initial DC bus voltage	$U_t^{\mathrm{dc}}$	±5% (Gaussian perturbation)

Table 6. Scenario generation for different operating conditions.

No	Type	Variable Parameter	Simulation Runs
F0	Normal condition	None	80
F1	DC bus short circuit	Fault resistance (0.001, 0.01, 0.1, 1 Ω)	80
F2	Grid frequency dip	Frequency deviation (−0.3, −0.5, −1, −1.5 Hz)	80
F3	Grid phase jump	Jump angle (+10°, +30°, +60°, +90°)	80
F4	Grid voltage sag	Sag depth (−0.1, −0.3, −0.5, −0.7 p.u.)	80
F5	Grid voltage imbalance	Phase-A voltage sag depth (−0.1, −0.3, −0.5, −0.7 p.u.)	80
F6	Single-phase-to-ground fault	Fault resistance (0.001, 0.01, 0.1, 1 Ω)	80
F7	Double-phase-to-ground fault	Fault resistance (0.001, 0.01, 0.1, 1 Ω)	80
F8	Phase-to-phase fault	Fault resistance (0.001, 0.01, 0.1, 1 Ω)	80
F9	Three-phase-to-ground fault	Fault resistance (0.001, 0.01, 0.1, 1 Ω)	80

To eliminate dimensional inconsistency while preserving the discriminative information of the transient process, features were extracted from two critical time windows. The steady-state window corresponds to 40.0–47.5% of the signal duration and captures the stable operating state before fault occurrence. The transient window corresponds to 50.0–53.0% of the signal duration and captures the abrupt dynamic variation at fault inception.

3.2.3. Data Preprocessing

In the main experiments, Gaussian white noise with a signal-to-noise ratio (SNR) of 18 dB was added to all time-series data to simulate a typical industrial noise environment. Subsequently, 18-dimensional VSG node features and 8-dimensional global physical features were extracted for each channel. As a result, each sample was represented as a feature graph with 10 nodes and 26-dimensional features per node.

In addition, a dataset partition strategy based on noise-seed isolation was adopted. Each sample was tagged with the noise seed used during data generation. According to this identifier, 1600 samples generated with seeds 1 and 2 were assigned to the training set, 800 samples generated with seed 3 were used as the validation set, and 800 samples generated with seed 4 were reserved for the test set. This strategy fundamentally prevents overlap of perturbed samples generated under the same noise pattern across different subsets. It therefore eliminates noise-pattern leakage and ensures that the test set is composed entirely of noise conditions unseen during training.

On this basis, the samples within each subset were randomly shuffled, and the random seed was fixed to guarantee strict reproducibility of the experimental results. This partition strategy makes the model evaluation more consistent with practical diagnostic scenarios involving unknown noise interference. It also provides a more reliable assessment of model generalization capability and robustness.

3.3. Configuration of the VSG-PI-HTGNN Model and Evaluation Framework

3.3.1. Network Architecture and Hyperparameter Settings

The proposed VSG-PI-HTGNN model, illustrated in Figure 1, adopts a heterogeneous graph structure with 10 physical nodes and 8 edge types. The graph network consists of three GCN convolution layers with hidden dimensions of 96, 72, and 48, respectively. Neighborhood information is then aggregated through residual connections. The resulting representations are fed into three fully connected layers for classification.

Model training is performed using a cosine annealing learning-rate schedule, with the initial learning rate set to 4 × 10⁻⁴. The batch size and dropout rate are set to 32 and 0.30, respectively. The auxiliary-classifier loss weight is set to 0.25. In addition, the weight decay coefficient and gradient clipping threshold are set to 2 × 10⁻⁴ and 1.5, respectively. The early-stopping patience is fixed at 80 epochs. To improve the reliability of the reported results, three independent training runs are conducted using different random seeds, namely seed42, seed123, and seed456.

3.3.2. Evaluation Metrics

The following metrics were used to comprehensively evaluate model performance: Accuracy (Acc), Macro-Precision (P), Macro-Recall (R), Macro-F1 Score (F1), Cohen’s Kappa coefficient (κ), and Matthews Correlation Coefficient (MCC). Class-wise metrics were aggregated using macro averaging. The κ coefficient measures the degree of agreement beyond random classification. MCC, extended from binary to multiclass classification, provides a balanced evaluation by jointly considering all entries of the confusion matrix.

4. Simulation Experiments and Results Analysis

4.1. Analysis of Model Training and Test Performance

4.1.1. Training Process and Convergence Behavior

Figure 5 shows that the VSG-PI-HTGNN model converges stably under different random seeds. The training loss decreases rapidly within the first 60 epochs and then gradually stabilizes. The validation accuracy reaches saturation after approximately 60 epochs and ultimately exceeds 95% in all runs. Under the early-stopping mechanism, training terminates at around epoch 150, and the highest validation accuracy reaches 97.62%.

Figure 5. Training results of the VSG-PI-HTGNN model under different random seeds.

As reported in Table 7, the test accuracies obtained with random seeds seed42, seed123, and seed456 are 97.25%, 97.00%, and 96.88%, respectively. The mean test accuracy is 97.00% ± 0.25%, and the difference between the maximum and minimum values is only 0.37%. These results demonstrate good stability and reproducibility of the proposed model.

Table 7. Test accuracy of the VSG-PI-HTGNN model under different random seeds.

Random Seed	Test Acc (%)
seed42	97.25
seed123	97.00
seed456	96.88

Furthermore, Table 8 presents the test results under different training-set proportions. When the training-set proportion is 20%, the model already achieves a test accuracy of 95.38% and an F1-score of 0.9540. As the proportion increases to 60%, the test accuracy and F1-score improve to 97.00% and 0.9700, respectively, which are already close to the performance obtained with the full dataset. With 100% of the training data, the highest accuracy and F1-score reach 97.25% and 0.9727, respectively. These results indicate that the proposed model has high data efficiency and can maintain strong diagnostic performance even under limited-sample conditions.

Table 8. Test accuracy and F1-score of the VSG-PI-HTGNN model under different training-set proportions.

Training-Set Proportion	Test Acc (%)	F1-Score
20%	95.38	0.9540
40%	96.25	0.9628
60%	97.00	0.9700
80%	97.12	0.9715
100%	97.25	0.9727

4.1.2. Overall Test Performance

Using the model parameters corresponding to the optimal validation seed, seed42, under a noise level of SNR = 18 dB, the overall fault diagnosis performance of the VSG-PI-HTGNN model on the test set is summarized in Table 9. All evaluation metrics are approximately 0.97, which is substantially higher than the expected performance of a random classifier. This result indicates a high degree of agreement between the model predictions and the true labels, demonstrating strong classification accuracy and reliability.

Table 9. Overall fault diagnosis performance of the VSG-PI-HTGNN model.

Metric	Value
Test Acc (%)	97.25
P	0.9740
R	0.9725
F1-score	0.9727
κ	0.9694
MCC	0.9700

4.1.3. Test Performance Under Different Operating Conditions

Figure 6 shows the normalized confusion matrix for all operating conditions. The results indicate high recall across most classes. Specifically, the recall values for conditions F0, F1, and F2 are 0.93, 0.84, and 0.96, respectively, whereas all conditions from F3 to F9 achieve a recall of 1.00.

Figure 6. Normalized confusion matrix for all classes.

The main confusion occurs between F0 and F1. Approximately 6% of the F0 samples are misclassified as F1, while 16% of the F1 samples are misclassified as F0. This is mainly because, at the initial stage of the fault and under relatively large short-circuit impedance, F1 exhibits only slight reductions in DC bus voltage and small current increments. As a result, part of its feature distribution overlaps with that of F0. In addition, about 4% of the F2 samples are misclassified as F0, whereas no misclassification occurs between F2 and conditions F3–F9. Overall, the model achieves completely correct identification for conditions F3–F9, demonstrating strong classification capability.

In addition, Table 10 summarizes the precision (P), recall (R), and F1-score for the 10 operating conditions. The F1-scores of all conditions exceed 0.87, indicating consistently strong class-wise diagnostic performance. For conditions F3–F9, all three metrics reach 1.00, which confirms perfect identification for these fault categories. For condition F0, the recall is higher than the precision, suggesting that samples of this class are more susceptible to false-positive assignment than to false-negative omission. In contrast, conditions F1 and F2 exhibit higher precision than recall, indicating a certain tendency toward missed detection.

Table 10. Class-wise performance metrics under different operating conditions.

Condition	P	R	F1-Score
F0	0.8222	0.9250	0.8706
F1	0.9306	0.8375	0.8816
F2	0.9872	0.9625	0.9747
F3	1.0000	1.0000	1.0000
F4	1.0000	1.0000	1.0000
F5	1.0000	1.0000	1.0000
F6	1.0000	1.0000	1.0000
F7	1.0000	1.0000	1.0000
F8	1.0000	1.0000	1.0000
F9	1.0000	1.0000	1.0000

4.1.4. Feature Space and Discriminability

Figure 7 compares the two-dimensional t-SNE projections of the raw signal features and the features produced by the VSG-PI-HTGNN model. In the raw feature space, the class distributions are relatively scattered. A clear overlap is observed between F0 and F1, and the cluster boundaries among F6, F7, and F8 are closely spaced. These observations indicate limited separability of the raw features. After processing by VSG-PI-HTGNN, the class clusters become more compact, and the inter-cluster distances increase markedly. Overall separability is therefore significantly improved. This result demonstrates that the proposed model can effectively enhance fault discriminability through physical feature extraction and heterogeneous graph-based information fusion. Nevertheless, a certain degree of overlap still remains among F0, F1, and F2, which is consistent with the test accuracy of 97.25%.

Figure 7. Two-dimensional t-SNE projections of the raw signal features and the output features from the VSG-PI-HTGNN model.

The above results indicate that the VSG-PI-HTGNN model achieves relatively balanced recognition performance across different operating conditions. The integration of dual-branch physical features and heterogeneous graph convolution effectively improves both the discriminative capability of the model and the separability of the learned feature space.

4.1.5. Analysis of the Learnable Relation-Scaling Mechanism

To further analyze the contribution of the learnable relation-scaling mechanism, a controlled comparison is conducted between the proposed model and a fixed-scaling variant. In the fixed-scaling variant, the scaling factors of all relation types are fixed at one, whereas the physical graph topology, input features, network architecture, data partition, hyperparameter settings, and random seed remain unchanged. Therefore, the only difference between the two models is whether the relative propagation strengths of different relation types are adaptively learned from the training data.

The comparison is conducted using seed42 under the default noise condition of SNR = 18 dB. As shown in Table 11, the test accuracy with learnable relation-scaling factors is 97.25%, compared with 96.72% for the fixed-scaling variant, corresponding to a difference of 0.53%. Because the predefined physical connectivity remains unchanged in both settings, the observed difference is associated with the adaptive adjustment of the relative contributions of different physically defined relation types during neighborhood aggregation rather than with changes in graph connectivity.

Table 11. Effect of the learnable relation-scaling mechanism on diagnostic performance.

Relation-Scaling Setting	Test Acc (%)	P	R	F1-Score	κ	MCC
Learnable scaling factors	97.25	0.9740	0.9725	0.9727	0.9694	0.9700
$\mathrm{Fixed} \mathrm{scaling} \mathrm{factors} ({\gamma }_{r_j}$ = 1)	96.72	0.9690	0.9672	0.9678	0.9639	0.9644

These results suggest that assigning identical propagation strengths to all relation types may not fully reflect their different contributions to fault-feature extraction. The learnable relation-scaling mechanism allows relation-dependent information propagation to be adjusted according to the training data while preserving the predefined physical graph structure.

4.2. Comparative Analysis of Noise Robustness

4.2.1. Experimental Design

To evaluate the robustness of the VSG-PI-HTGNN model under different noise intensities, additive white Gaussian noise was injected into the original noise-free signals according to predefined signal-to-noise ratio (SNR) levels. Six noise levels were considered, namely 10, 15, 18, 20, 25, and 30 dB, among which 18 dB was used as the default noise setting in the main experiments. For each SNR level, the model was trained independently using the same feature extraction scheme, network architecture, and hyperparameter configuration as those adopted in the main experiment. An early stopping strategy was applied, with a patience of 80 epochs. The optimal validation seed, seed42, was used throughout the experiments. For each noise condition, the best validation accuracy, test accuracy, F1-score, and early stopping epoch were recorded to assess the convergence behavior and noise robustness of the model.

4.2.2. Experimental Results

Figure 8 presents the experimental results under different SNR conditions. The model maintains high performance across all noise levels. At SNR = 10 dB, the test accuracy and F1-score still reach 95.50% and 0.9555, respectively. Under the default noise level of 18 dB, the test accuracy and F1-score are 97.25% and 0.9727, respectively. As the SNR further increases, the model performance improves gradually. At 25 dB and 30 dB, the test accuracy reaches 98.12% and 98.75%, with corresponding F1-scores of 0.9813 and 0.9876, respectively.

Figure 8. Diagnostic performance under different noise levels.

Overall, both test accuracy and F1-score increase monotonically with increasing SNR and become relatively stable after 18–20 dB. Across the wide noise range of 10–30 dB, the fluctuation in test accuracy is only 3.25%, which demonstrates strong noise robustness of the proposed model. This robustness can be mainly attributed to the noise-suppression capability of the VSG-based physical features and the weighted multi-channel information fusion enabled by heterogeneous graph convolution.

4.3. Ablation Experiments

4.3.1. Experimental Design

To verify the effectiveness of the core components in the proposed VSG-PI-HTGNN model, five study cases were designed for the ablation analysis. The input features and graph structure configurations of these cases are listed in Table 12. All cases were conducted under the same data partition, network architecture, hyperparameter setting, noise level of SNR = 18 dB, and the optimal validation seed, seed42, in order to ensure a fair comparison.

Table 12. Input feature and graph structure configurations of the ablation cases.

Case	Description	Input Features	Graph Structure	Auxiliary Classifier
Case1	Full VSG-PI-HTGNN model	18-dimensional VSG physical node features + 8-dimensional global physical constraints = 26-dimensional features	Heterogeneous graph	Enabled
Case2	Without global physical information	18-dimensional VSG physical node features	Heterogeneous graph	Disabled
Case3	Without VSG node features	6-dimensional minimal statistical features + 8-dimensional global physical constraints = 14-dimensional features	Heterogeneous graph	Enabled
Case4	Without heterogeneous graph structure	18-dimensional VSG physical node features + 8-dimensional global physical constraints = 26-dimensional features	Homogeneous graph	Enabled
Case5	Without all physical information	6-dimensional minimal statistical features	Homogeneous graph	Disabled

4.3.2. Experimental Results

Under the model setting corresponding to the optimal validation seed, seed42, and the noise level of SNR = 18 dB, the ablation results for the core components of the VSG-PI-HTGNN model are presented in Table 13, and the training loss curves are shown in Figure 9. The full model, Case1, achieves the best performance, with a test accuracy, precision, recall, and F1-score of 97.25%, 0.9740, 0.9725, and 0.9727, respectively.

Table 13. Results of the ablation study.

Case	Test Acc (%)	P	R	F1-Score	Early Stopping Epoch
Case1	97.25	0.9740	0.9725	0.9727	137
Case2	96.67	0.9672	0.9662	0.9664	194
Case3	96.58	0.9673	0.9637	0.9636	176
Case4	96.38	0.9679	0.9637	0.9644	162
Case5	89.83	0.9143	0.9037	0.9067	218

Figure 9. Training loss curves of the ablation experiments.

After removing the heterogeneous graph structure, the test accuracy of Case 4 decreases from 97.25% to 96.38%, representing the largest accuracy reduction among the partial ablation cases. This controlled comparison suggests that heterogeneous relation modeling contributes to the diagnostic representation under the present experimental setting. For Case 3, where the VSG-based node features are removed, the test accuracy decreases to 96.58% and the F1-score decreases to 0.9636. For Case 2, where the global physical information is removed, the test accuracy is 96.67%. Although the performance degradation is relatively small, the early-stopping epoch increases markedly, suggesting that the global physical constraints may contribute to more stable and efficient model training.

After removing all physical information, the test accuracy of Case5 decreases to 89.83%, and the F1-score falls to 0.9067. Compared with the full model, the reductions reach 7.42% and 0.0660, respectively. These results suggest that the heterogeneous graph structure, VSG-based node features, and global physical constraints each contribute to the diagnostic performance of the model. In particular, the larger performance reduction observed after removing all physical information indicates the importance of incorporating the overall physical prior into the diagnostic representation.

4.4. Comparative Experiments

4.4.1. Experimental Design

To compare the diagnostic performance of the proposed method with representative benchmark methods, eight comparative cases were designed. The input features and model architectures of all cases are summarized in Table 14. The comparison methods cover three categories.

Table 14. Input feature and model architecture configurations of the comparative cases.

Case	Description	Input Features	Model Architecture	Physical Information
Case1	Full VSG-PI-HTGNN model	18-dimensional VSG physical node features + 8-dimensional global physical constraints = 26-dimensional features	Heterogeneous graph convolutional network + auxiliary classifier	Enabled
Case6	1D-CNN	Segmented temporal statistical features	One-dimensional convolutional network	Disabled
Case7	BiLSTM	Segmented temporal statistical features	Bidirectional long short-term memory network	Disabled
Case8	GCN	6-dimensional minimal statistical features	Homogeneous graph convolutional network	Disabled
Case9	GraphSAGE	6-dimensional minimal statistical features	Inductive homogeneous graph neural network	Disabled
Case10	PhyGCN	6-dimensional minimal statistical features + 8-dimensional global physical constraints = 14-dimensional features	Homogeneous graph convolutional network	Enabled
Case11	Phy-LSTM	Segmented temporal statistical features + 8-dimensional global physical constraints	Bidirectional long short-term memory network	Enabled
Case12	SVM	Handcrafted features designed based on physical knowledge	Nearest-centroid classifier	Enabled

The first category consists of deep sequential models, including the one-dimensional convolutional neural network (1D-CNN) and the bidirectional long short-term memory network (BiLSTM). Both models take segmented temporal statistical features as input and learn temporal patterns automatically.

The second category includes graph neural network variants, namely the homogeneous graph convolutional network (GCN), the inductive graph model based on sampling and aggregation (GraphSAGE), the physics-guided graph convolutional network (PhyGCN), and the physics-guided long short-term memory network (Phy-LSTM). These models adopt different input features and graph structures, so as to examine the independent contributions of physical information and heterogeneous graph modeling.

The third category is a traditional machine learning method, namely the support vector machine (SVM) based on a nearest-centroid classification strategy. It uses manually designed physical features for classification and serves as a non-deep-learning baseline.

All cases were evaluated under the same data partition, network architecture, hyperparameter setting, noise level of SNR = 18 dB, and the optimal validation seed, seed42, to ensure a fair comparison.

4.4.2. Experimental Results

Under the model setting corresponding to the optimal validation seed, seed42, and the noise level of SNR = 18 dB, the comparative results are presented in Table 15. The proposed VSG-PI-HTGNN obtains a test accuracy of 97.25% and an F1-score of 0.9727.

Table 15. Results of the comparative experiments.

Case	Test Acc (%)	P	R	F1-Score	Early Stopping Epoch
Case1	97.25	0.9740	0.9725	0.9727	137
Case6	92.12	0.9287	0.9262	0.9270	-
Case7	90.88	0.9243	0.9112	0.9149	-
Case8	89.83	0.9143	0.9037	0.9067	306
Case9	90.92	0.9264	0.9175	0.9196	241
Case10	91.29	0.9355	0.9275	0.9292	281
Case11	93.08	0.9365	0.9287	0.9309	-
Case12	75.88	0.8608	0.7587	0.7885	-

The corresponding test accuracies of 1D-CNN and BiLSTM are 92.12% and 90.88%, respectively. Among the graph-based methods, GCN, GraphSAGE, and PhyGCN achieve test accuracies of 89.83%, 90.92%, and 91.29%, respectively, while Phy-LSTM achieves 93.08%. The SVM obtains a test accuracy of 75.88% and an F1-score of 0.7885.

Under this unified experimental configuration, the proposed model obtains higher values of accuracy, precision, recall, and F1-score than the considered benchmark methods. These controlled comparisons support the contribution of combining physics-informed feature construction with heterogeneous relation modeling under the present dataset and experimental settings. Because repeated multi-seed experiments are not conducted for all baseline methods, the observed between-model differences are not interpreted as evidence of statistical significance or universal superiority.

4.5. Sim-to-HIL Transferability Validation

4.5.1. HIL Platform and Real-Time Data Acquisition

To further evaluate the generalization capability of the proposed method under real-time operating conditions, a hardware-in-the-loop (HIL) platform is established, as shown in Figure 10. The platform consists of a host computer, an RT-LAB OP5600 real-time simulator (OPAL-RT Technologies, Inc., Montréal, QC, Canada), a TMS320F28335 digital signal processor (DSP) (Texas Instruments Incorporated, Dallas, TX, USA), and signal-conditioning and data-acquisition interfaces. The main circuit of the grid-connected GFM-HESS, the AC grid, fault modules, and measurement components operate in real time on the OP5600, whereas the VSG, virtual impedance, and energy management strategy are implemented on the DSP. The measured voltage and current signals are transmitted to the DSP, and the resulting modulation commands are fed back to the OP5600, thereby forming a closed-loop HIL system.

Figure 10. The HIL platform.

It should be noted that the VSG-PI-HTGNN diagnostic network is not deployed on the OP5600 or DSP and does not participate in the real-time HIL control loop. The HIL platform is used for real-time execution of the GFM-HESS plant and control system and for acquisition of the corresponding electrical signals. After each HIL test, the acquired signals are transferred to the host computer, where the diagnostic processing, including feature extraction, heterogeneous graph construction, and VSG-PI-HTGNN inference, is performed offline. Therefore, the present HIL experiment evaluates the transferability of the simulation-trained diagnostic model to HIL-acquired data rather than the real-time implementation of the diagnostic network.

Consistent with the offline simulation settings, the real-time model uses a fixed step size of 1 μs, while the powergui module operates in discrete mode with the same sampling time. The monitored variables include the DC-bus voltage, three-phase AC currents, three-phase AC voltages, frequency, active power, and reactive power. The data-acquisition frequency is set to 10 kHz, and the interval from 0.90 to 1.10 s is extracted from each test to construct a diagnostic sample.

The HIL tests cover the normal condition F0 and fault conditions F1–F9. Each test lasts 2.00 s, with the fault applied at 1.00 s and maintained until the end of the test. Representative oscilloscope waveforms for the F4 symmetric voltage sag and F5 asymmetric voltage sag are shown in Figure 11. Owing to space limitations, only these two representative conditions are presented because they respectively characterize the system responses under balanced and unbalanced voltage disturbances.

Figure 11. Representative oscilloscope waveforms under F4 and F5 fault conditions.

4.5.2. Sim-to-HIL Transferability Test Protocol

The VSG-PI-HTGNN model used in this experiment is trained exclusively on the MATLAB/Simulink dataset. No HIL samples are used for model training, hyperparameter selection, or model fine-tuning. The trained model is directly evaluated on an independently acquired HIL dataset. This protocol evaluates whether the diagnostic representations learned from the simulation data, together with the embedded physical features and heterogeneous coupling relationships, remain applicable to independently acquired data from the HIL environment.

4.5.3. Experimental Results

Table 16 compares the diagnostic performance of the proposed model on the MATLAB/Simulink test dataset and the independently acquired HIL dataset. The HIL dataset contains 800 samples covering the normal condition F0 and fault conditions F1–F9, with 80 samples collected for each operating condition. During the HIL tests, the electrical signals are generated and acquired in real time from the HIL platform. After data acquisition, the diagnostic samples are transferred to the host computer, where feature extraction, heterogeneous graph construction, and VSG-PI-HTGNN inference are performed offline. The trained model, hyperparameters, and classification settings remain unchanged during this evaluation, and no HIL samples are used for model fine-tuning.

Table 16. Diagnostic performance on the MATLAB/Simulink and HIL test datasets.

Test Dataset	Number of Samples	Test Acc (%)	P	R	F1-Score	κ	MCC
MATLAB/Simulink test dataset	800	97.25	0.9740	0.9725	0.9727	0.9694	0.9700
HIL test dataset	800	92.75	0.9300	0.9275	0.9282	0.9194	0.9200

As shown in Table 16, the proposed model achieves a test accuracy of 92.75%, a macro-F1 score of 0.9282, a Cohen’s kappa coefficient of 0.9194, and an MCC of 0.9200 on the HIL dataset. Compared with the MATLAB/Simulink test dataset, the test accuracy decreases by 4.50%. This performance reduction may be associated with differences between the offline simulation and HIL-acquired data, including real-time numerical execution of the plant and controller, hardware-interface delays, signal quantization, and measurement noise. Nevertheless, the model retains a test accuracy above 90% on the independently acquired HIL dataset.

These results indicate that the diagnostic representation learned from the simulation data retains a degree of transferability to HIL-acquired data. Because the diagnostic inference is performed offline on the host computer rather than being embedded in the real-time HIL control loop, the present experiment should be interpreted as a Sim-to-HIL transferability validation rather than as a real-time implementation of the VSG-PI-HTGNN diagnostic network.

5. Discussion

5.1. Computational Complexity and Scalability

The computational complexity of the proposed VSG-PI-HTGNN is mainly associated with node-type-specific feature transformation, heterogeneous graph convolution, and fully connected classification. Unlike topology-oriented GNNs in which graph nodes generally correspond to buses or physical devices, the graph adopted in this study is constructed from monitored electrical variables. The current graph contains 10 physical nodes belonging to five node types and eight predefined relation types. Therefore, its graph size is relatively small and does not increase directly with the number of buses in the external power network.

For a graph convolution layer l, let N and E denote the numbers of nodes and edges, respectively, and let d_l−1 and d_l denote the input and output feature dimensions. The main computational operations consist of neighbor aggregation and feature transformation, which can be approximately expressed as O(Ed_l−1 + Nd_l−1d_l). The additional learnable relation-scaling operation assigns only one trainable coefficient to each predefined relation type and therefore introduces limited additional computation relative to the graph convolution itself.

To further quantify the computational requirements of the proposed model, the number of trainable parameters and the average inference time are evaluated. The VSG-PI-HTGNN contains 42,580 trainable parameters. The inference-time evaluation is conducted on the same host computer used for the diagnostic experiments, equipped with a 13th-Gen Intel(R) Core(TM) i7-13700 processor operating at 2.10 GHz and 32 GB RAM, with the model implemented in MATLAB R2022b. The average inference time is 4.83 ms per sample. These results indicate that the computational burden of the current model is relatively limited for the post-event fault diagnosis task considered in this study. However, since the diagnostic inference in the present HIL validation is performed offline on the host computer rather than being embedded in the real-time HIL control loop, the reported inference time should be interpreted as a quantitative characterization of the current host-computer implementation and as a reference for potential future online deployment, rather than as evidence of real-time implementation of the diagnostic network.

The measurement-oriented graph structure also provides a certain degree of scalability with respect to the external network configuration. If the external grid topology changes while the monitored variables and the corresponding physical relationships remain unchanged, the number of graph nodes does not necessarily increase. In contrast, if additional storage units, measurement channels, or control variables are introduced, the heterogeneous graph can be extended by adding the corresponding node types and physically meaningful relations. However, substantial changes in the measurement configuration or converter control structure may alter the physical relationships represented by the graph and may therefore require graph reconstruction or model adaptation. Thus, the proposed architecture is scalable in terms of measurement-oriented graph extension, whereas its applicability to substantially different GFM-HESS configurations still requires further validation.

5.2. Model Interpretability

The interpretability of the proposed model mainly originates from two aspects: physics-informed feature construction and physically predefined heterogeneous relations. At the feature level, the input representation is not constructed solely from generic statistical descriptors. The node-level features incorporate transient characteristics associated with VSG dynamics, while the global physical features describe system-level relationships involving sequence components, DC-bus behavior, frequency variation, and active/reactive power. Therefore, part of the diagnostic representation retains an explicit correspondence with measurable electrical quantities and control-related physical characteristics.

At the graph level, the connections among different measurement nodes are predefined according to control and electrical relationships rather than determined using unconstrained fully connected associations. The heterogeneous graph distinguishes different types of physical quantities and relation paths, including the coupling between active power and frequency, reactive power and voltage, DC-bus voltage and power balance, and same-phase voltage and current. The graph structure therefore provides a physically interpretable basis for information propagation among different measurements.

The ablation results further support the contribution of these physics-informed components. Removing all physical information reduces the test accuracy from 97.25% to 89.83%, corresponding to a decrease of 7.42%. Removing the heterogeneous graph structure while retaining the physical features results in an accuracy of 96.38%. These results suggest that both physics-informed feature construction and heterogeneous relation modeling contribute to the diagnostic representation, although their contributions are not identical.

In addition, the learnable relation-scaling mechanism provides further insight into relation-dependent information propagation. When all relation-type scaling factors are fixed at one while the remaining model settings are unchanged, the test accuracy decreases from 97.25% to 96.72%, and the F1-score decreases from 0.9727 to 0.9678. Since the predefined graph connectivity remains unchanged in this comparison, the difference is associated with the adaptive adjustment of the relative propagation strengths of different physical relation types rather than with changes in graph topology. The learned scaling factors should not be interpreted as strict causal importance scores; instead, they provide an indication of how the heterogeneous graph adjusts the contributions of different predefined physical relations during feature aggregation.

5.3. Generalization and Statistical Reliability

The generalization capability of the proposed method is evaluated from several complementary perspectives rather than from a single test condition. First, noise-seed isolation is adopted during dataset partitioning, so that samples generated using the same noise realization are not distributed across the training, validation, and test sets. This setting reduces the possibility that the reported performance is associated with repeated noise patterns across different subsets. Second, the noise-robustness experiments cover SNR levels from 10 to 30 dB. The test accuracy varies from 95.50% at 10 dB to 98.75% at 30 dB, with an overall variation of 3.25%. Third, the limited-sample experiments show that the model achieves a test accuracy of 95.38% when only 20% of the training data are used, and the accuracy gradually increases as more training samples are introduced. These results indicate that the proposed representation maintains relatively stable diagnostic performance under variations in noise intensity and training-data availability.

The influence of training randomness is further examined using three independent random seeds. The test accuracies obtained with seed42, seed123, and seed456 are 97.25%, 97.00%, and 96.88%, respectively, and the variation among the three runs is relatively small. These repeated experiments provide additional evidence that the reported performance is not dependent on a single initialization. Nevertheless, the comparative and ablation experiments are conducted using the same optimal validation seed to maintain consistent experimental settings. Therefore, the performance differences among different model architectures should primarily be interpreted as controlled comparative results under the present dataset and experimental configuration, rather than as formal claims of statistical significance across repeated independent trials.

The Sim-to-HIL evaluation further examines the applicability of the learned diagnostic representation outside the offline MATLAB/Simulink data environment. In the HIL tests, the GFM-HESS plant and control system operate in real time and the corresponding electrical signals are acquired from the HIL platform, whereas the diagnostic inference is subsequently performed offline on the host computer. Therefore, this experiment evaluates the transferability of the simulation-trained model to independently acquired HIL data rather than the real-time implementation of the VSG-PI-HTGNN. The observed performance differences may reflect the effects of real-time plant and controller execution, hardware interfaces, signal quantization, and measurement noise in the HIL data-acquisition process.

However, robustness to noise, training-data variation, and simulation-to-HIL differences does not necessarily imply generalization to arbitrary GFM-HESS configurations. The current simulation and HIL datasets are based on the same basic system structure and control principle. Variations in storage-device combinations, converter parameters, measurement locations, control strategies, or external network structures may modify the fault-feature distributions and physical relationships represented by the heterogeneous graph. Therefore, cross-configuration generalization remains an aspect requiring further investigation.

5.4. Limitations and Future Work

Although the results indicate the applicability of the proposed VSG-PI-HTGNN under the considered operating conditions, several limitations remain. First, the model is trained primarily using fault samples generated from an electromagnetic transient simulation model. The HIL tests introduce real-time plant and controller execution as well as signal-acquisition effects, but they do not fully represent the variability encountered in field operation. Moreover, the diagnostic inference in the present HIL validation is performed offline on the host computer rather than being embedded in the real-time HIL control loop. Therefore, the current HIL experiment evaluates Sim-to-HIL transferability rather than real-time implementation of the diagnostic network. Future work will further investigate online deployment and execution performance on embedded or real-time diagnostic platforms. Further validation using experimental-platform data and field fault records, when available, would also provide a broader assessment of the proposed method.

Second, the heterogeneous graph is constructed according to the monitored variables and physical relationships of the VSG-controlled GFM-HESS considered in this study. Although the graph can be extended by introducing additional node and relation types, substantial changes in storage configuration, control strategy, or measurement arrangement may require corresponding modification of the graph structure and model parameters. Future work can investigate graph adaptation, transfer learning, and domain adaptation methods to improve diagnostic adaptability across different system configurations.

Third, the present work focuses on fault-type classification and does not further estimate fault location or fault severity. Extending the multi-task framework to incorporate fault localization and severity assessment may provide additional information for post-event analysis. In addition, incremental learning could be considered to update the diagnostic model when new operating conditions or previously unseen fault samples become available.

Finally, the present comparison mainly evaluates diagnostic performance under a unified experimental configuration. Although repeated training with three independent random seeds is conducted for the proposed model, repeated multi-seed experiments are not performed for all baseline methods. Therefore, the reported between-model differences should be interpreted as controlled comparative results under the present dataset and experimental settings rather than as formal evidence of statistical significance. Repeated multi-seed comparisons across all baseline methods, together with more extensive cross-configuration tests, would allow a more comprehensive statistical assessment of the observed performance differences in future work.

6. Conclusions

This paper develops a VSG-PI-HTGNN for multi-class fault diagnosis in grid-connected GFM-HESS systems by integrating control-related physical information with heterogeneous relational learning. Rather than treating the ten monitored electrical signals as homogeneous time-series inputs, the proposed method constructs node-level features according to their physical attributes and supplements them with global physical information associated with VSG dynamics and system-level electrical relationships. A measurement-oriented heterogeneous graph is further established to represent the couplings among DC-bus voltage, three-phase voltages and currents, frequency, and active/reactive power. This representation enables fault characteristics to be learned from both individual measurement responses and their physically defined interactions.

Under the default SNR of 18 dB, the proposed model achieves a test accuracy of 97.25% and a macro-F1 score of 0.9727. The ablation results provide further insight into the contributions of the proposed components. Removing all physical information reduces the test accuracy to 89.83%, whereas replacing the heterogeneous graph with a homogeneous graph results in an accuracy of 96.38%. Moreover, fixing all relation-scaling factors to one reduces the accuracy from 97.25% to 96.72%, suggesting that adaptive adjustment of relation-dependent propagation contributes to the heterogeneous feature aggregation while preserving the predefined physical graph structure. The model also exhibits relatively stable performance across SNR levels of 10–30 dB and under reduced training-data conditions. In the controlled comparative experiments conducted under the same data partition and experimental settings, the proposed model obtains higher values of the reported diagnostic metrics than the considered 1D-CNN, BiLSTM, GCN, GraphSAGE, PhyGCN, Phy-LSTM, and SVM baselines. These between-model differences are interpreted as controlled comparative results under the present experimental configuration rather than as formal evidence of statistical significance.

The Sim-to-HIL evaluation provides an additional assessment beyond offline simulation. The proposed model achieves an accuracy of 92.75% and a macro-F1 score of 0.9282 on the independently acquired HIL dataset, indicating that the diagnostic representation learned from simulation retains a degree of transferability to HIL-acquired data. In the present validation, the GFM-HESS plant and controller operate in real time on the HIL platform, whereas diagnostic inference is subsequently performed offline on the host computer. Therefore, the HIL results should be interpreted as evidence of Sim-to-HIL transferability rather than as a real-time implementation of the VSG-PI-HTGNN diagnostic network.

The present results are obtained for a specific GFM-HESS configuration and control structure, and therefore do not establish generalization to arbitrary storage configurations, converter controls, measurement arrangements, or external grid topologies. In addition, the current task focuses on fault-type classification rather than fault localization or severity assessment. Future work will investigate graph adaptation, transfer learning, and domain adaptation for different system configurations, together with fault localization, severity assessment, incremental learning, online deployment and execution-performance evaluation on embedded or real-time diagnostic platforms, and validation using broader experimental or field fault datasets.