A Transfer-Learning-Based Approach to Symmetry-Preserving Dynamic Equivalent Modeling of Large Power Systems with Small Variations in Operating Conditions

Abstract

Robust dynamic equivalents of large power networks are essential for fast and reliable stability analysis of bulk power systems. This is because the dimensionality of modern power systems raises convergence issues in modern stability-analysis programs. However, even with modern computational power, it is challenging to find reduced-order models for power systems due to the following factors: the tedious mathematical analysis involved in the classical reduction techniques requires large amounts of computational power; inadequate information sharing between geographical areas prohibits the execution of model-dependent reduction techniques; and frequent fluctuations in the operating conditions (OPs) of power systems necessitate updates to reduced models. This paper focuses on a measurement-based approach that uses a deep artificial neural network (DNN) to estimate the dynamics of an external system (ES) of a power system, enabling stability analysis of a study system (SS). This DNN technique requires boundary measurements only between the SS and the ES. However, machine learning-based techniques like this DNN are known for their extensive training requirements. In particular, for power systems that undergo continuous fluctuations in operating conditions due to the use of renewable energy sources, the applications of this DNN technique are limited. To address this issue, a Deep Transfer Learning (DTL)-based technique is proposed in this paper. This approach accounts for variations in the OPs such as time-to-time variations in loads and intermittent power generation from wind and solar energy sources. The proposed technique adjusts the parameters of a pretrained DNN model to a new OP, leveraging symmetry in the balanced adaptation of model layers to maintain consistent dynamics across operating conditions. The experimental results were obtained by representing the Queensland (QLD) system in the simplified Australian 14 generator (AU14G) model as the SS and the rest of AU14G as the ES in five scenarios that represent changes to the OP caused by variations in loads and power generation.

1. Introduction

Generally, for the study of power systems, a particular geographical area, the Study System (SS), is selected, and stability analysis is performed on this system. Numerous areas called the External Systems (ESs) may be connected to this SS. Conventionally, to find the dynamic equivalents of an ES, a detailed model of that system is required [1].

Model-reduction techniques for power systems can be broadly categorized into static and dynamic techniques. Static techniques are used for power-system planning and static security assessments for which only the steady-state power flows (flows of active power and reactive power) between the SS and the ES are of interest. The most widely used static equivalent techniques are the Kron technique [2], the Ward technique [3] and the Radial Equivalent Independent (REI) technique [4]. These three techniques fundamentally focus on the elimination of redundant nodes by the Gaussian elimination technique. The dynamic techniques can be further categorized into techniques based on power-system models and coherency-based techniques. Among the model-based techniques, singular perturbation theory [5], modal analysis [6], singular value decomposition [7], Krylov subspaces [8] and moment matching [9] are the most prominent. In these techniques, the mathematical model of the system is explicitly written as a set of matrix equations and rigorous mathematical operations derived from the linear control theory are applied for system reduction. In addition to the complexity involved in the equations, these techniques necessitate the linearization of the power system model to realize the reduction. Alternatively, the coherency-based technique identifies groups of coherent synchronous generators (SGs) based on their tendency to swing together in response to system disturbances; these SGs are then aggregated to find the equivalent generators. The ability to preserve the swing modes of the original system in the reduced system distinguishes this technique from other model-based dynamic equivalencing techniques [10].

In general, these techniques are considered classical techniques because they are inherently dependent on the model of the ES. On the one hand, their comprehensive mathematical equations preserve the model structure of the reduced system. On the other hand, they all have drawbacks in their applications to modern power systems integrated with variable renewable energy sources. The intermittency of these renewable energy sources necessitates the repeated reduction of the dynamic model. In this context, classical model-reduction techniques are considered outdated for several reasons. Firstly, the computational process using classical techniques is excessively time-consuming and thus undesirable for contexts in which quick operational decisions are needed. Secondly, the tedious process of mathematical analysis requires a significant amount of computational power. Thirdly, the lack of information about the model structure restricts their application to modern power systems. These three problems with the classical techniques make them inconvenient for use in applications to power systems that have fluctuating operating conditions (OPs) [11].

Alternatively, the benefits of the data-driven measurement-based techniques are as follows: higher computational speed, the ability to proceed with less model data or no model data, and high generalization capability [12]. The measurement-based techniques are deployed in two situations in dynamic equivalencing of the ESs: black-box situations in which the details of the ES are completely unknown [13] and gray-box situations in which some information for the ES model is accessible [14]. In the former type of situation, the ES model is derived from the boundary measurements. In this regard, the ability of Deep Neural Networks (DNNs) to capture the high nonlinearity of the ES dynamics is promising.

DNNs have proven promising in many areas of power systems studies, such as transient stability assessment [15], wind-power systems [16], solar-power systems [17], load forecasting [18] and smart grids [19]. In [20], a two-structure neural network model, i.e., a bottleneck structure and a recurrent neural network (RNN) structure, was proposed to estimate the differential–algebraic equations of the ES. The former is used for the state extraction of the reduced system, while the latter is used for the state prediction of the reduced system. In [21], a simultaneous approach was used to identify the differential–algebraic model of the ES, employing a four-layer DNN. In this specific DNN, layers one and three consist of nonlinear activation functions that represent the differential part, while layers two and four consist of linear activation functions that represent the algebraic part. A Radial Basis Function (RBF)-based Artificial Neural Network (ANN) was adopted in [22] to estimate the dynamics of a grid-connected microgrid from the data measured at the point of common coupling. The key difference between an RBF ANN and a conventional ANN is that the RBF ANN can guarantee the convergence of its parameters. This is because a linear term has been added to the parameters of the RBF ANN [23]. In this technique, the RBF ANN is used to map the inputs and outputs of the system and particle-swarm optimization is used to optimize the parameters of the RBF ANN. In [24], an Artificial Neuro-Fuzzy System (ANFIS) was identified to model the ES dynamics based on boundary measurements. An ANFIS is a combination of neural networks and fuzzy inference systems that work together to predict highly nonlinear dynamical systems [25].

RNNs are a special case of ANNs that are used to find the temporal relationship in sequence data. Deep RNNs are widely used for power-system applications in situations in which there is a nonlinear relationship between the power flows at a certain time step and the power flows at previous time steps. In [26], an RNN was used to replace an ES with a black-box model in which the present values of voltage and the current injections at the boundary nodes from the four previous time steps were used as inputs to the RNN. Meanwhile, the present values of the current injections at the boundary nodes were selected as the outputs of the RNN. A similar network equivalent, in which an RNN was trained based on boundary measurements, was proposed for Active Distribution Networks (ADNs) in [27]. Additionally, a Long Short Term Memory (LSTM)-based RNN architecture was deployed for the equivalent system of ADNs in [28]. Furthermore, Gated Recurrent Unit (GRU) RNNs and LSTM RNNs have also been used for the black-box identification of microgrids [29,30]. In [31], an LSTM-based RNN was used to represent a microgrid under varying OPs, where the power reference point of the microgrid was changed during the training phase to generalize the equivalent model across multiple OPs.

However, the existing research on the black-box identification of the ES focuses only on the DNNs and their derivatives. Although DNNs are capable of capturing high nonlinearity in the power system, they suffer from a major disadvantage, i.e., the requirement for extensive training. This problem is amplified in the presence of renewable energy sources that change the OPs of the power system from moment to moment0. Recent advances in industrial applications of deep neural networks have faced similar challenges and developed solutions in the form of models that adapt under variable operating conditions. For instance, federated learning frameworks combining adaptive sampling and ensemble strategies have been proposed to enable robust remaining-useful-life prediction for aircraft engines [32]. Likewise, robust transfer learning has been applied for battery-lifetime prediction using early-cycle data to enable fast adaptation across different operational profiles [33]. Furthermore, physical information-embedded networks have shown promise in mechanical-fault detection, effectively integrating domain knowledge into data-driven modeling [34]. These developments underscore the importance of transferability, domain adaptation, and credible model generalization in real-world industrial systems, reinforcing the motivation for the development of our proposed DTL framework in power-system dynamic modeling.

Therefore, we propose a DNN transfer-learning model, also known as a Deep Transfer Learning (DTL) model, for ES identification. Transfer learning is often referred to as ‘learning to learn’, which means the machine learning algorithm is learning to predict a dataset in a domain different from that of the previously trained dataset, but quicker or with better results [35]. Although transfer learning has been applied to other areas of power systems, such as transient stability [36], dynamic security assessment [37], wind-power forecasting [38], solar-power forecasting [39] and smart grids [40,41], it has not yet been applied to create a dynamic model of the ES in a black-box scenario. This novel DTL model is capable of quickly adapting its model parameters for a wide range of network conditions.

1.1. Research Challenges and Motivation

Symmetry is a fundamental concept in power-system modeling, particularly in the development of dynamic equivalents, where preserving balanced structural or behavioral relationships between subsystems is essential. In this context, dynamic symmetry refers to maintaining consistent input–output behavior and modal characteristics between the full system and its reduced counterpart.

Preserving such symmetry becomes particularly challenging when the power system operates under varying OPs, a common occurrence due to changes in loads, generation, and network topology. These variations demand repeated re-identification of reduced-order models, which can be computationally expensive and inefficient.

To address this, we propose a transfer learning-based approach that uses measurement-based learning to replicate the dynamic impact of the external system on the study system. This approach not only maintains dynamic symmetry across different OPs but also eliminates the need to retrain models from scratch. As a result, it reduces model complexity while ensuring that system-level responses remain invariant across equivalent representations—thereby aligning with the broader goals of symmetry-based analysis in power-system-stability studies.

1.2. Contributions

The key contributions of this paper are as follows:

• We propose a novel Deep Transfer Learning (DTL) framework to generate black-box dynamic equivalent models of the external system (ES) in a power network. Unlike existing DNN approaches that require retraining for each new operating point (OP), our method reuses a pretrained model and fine-tunes it to adapt to new OPs. This reduces the training time while preserving predictive accuracy. To achieve this, the learned parameters are used by a task-specific model to train different OPs to predict and analyze the dynamics of an ES of a power system.
• The proposed DTL method is used to predict the active (P) and reactive (Q) power flows in the interconnecting transmission lines between the ES and the SS when the system is subjected to various contingencies in the SS. This technique outperforms the conventional DNN technique, with increased accuracy and faster convergence.
• We rigorously validate the proposed DTL approach using a detailed AU14G system model across 12 OP scenarios, including variations in loads, synchronous generators, wind and solar power plants, and mixed conditions. We benchmark performance against both conventional DNN and LSTM architectures.

The paper is organized as follows. Section 2 discusses and formulates the problem. Section 3 introduces the DTL technique and its preliminaries. Section 4 introduces the case study, explains dataset preparation, describes the test scenarios, and discusses the results. Section 5 concludes the paper.

2. Problem Formulation

The overall power system is a unified nonlinear system integrated with individual nonlinear components [42]. It can be represented as a set of differential equations and a set of algebraic equations, as in the following equations [43]:

$${\displaystyle \frac{dx}{dt}}=f(x,y)$$

(1)

$$0=g(x,y)$$

(2)

Equations (1) and (2) represent the power system as a set of differential–algebraic equations (DAEs) that model the dynamic and steady-state behavior of the system, respectively. In both equations, the variable x represents the state vector, which includes dynamic state variables such as rotor angles, rotor speeds, and other transient states of synchronous generators (SGs), inverter-based generators (IBGs), and other dynamic components within the system. The variable y represents the algebraic variables, typically including bus voltages (magnitudes and angles) and other steady-state quantities that define the electrical network’s operating conditions. The function $f(x,y)$ describes the nonlinear dynamics of these state variables, which are influenced by both x and y. The function $g(x,y)$ represents the algebraic constraints, such as power-flow equations, that ensure the balance of active and reactive power across the network.

The order of this nonlinear system is determined by the highest order of its components, and it varies as required by the user. For example, a synchronous generator (SG) can have nonlinear differential equations of various orders, but a third-order model is sufficient for a stability analysis [44]. However, the nonlinearity of the inverter-based generators (IBGs) is much higher than that of conventional SGs [45]. Therefore, the reduced-order dynamic equivalent model of the ES must be capable of capturing the nonlinearity of the IBGs.

Conventionally, the power-system models represented by Equations (1) and (2) are linearized as follows [46]:

$$x_i=A_i{x}_i+B_i\Delta v$$

(3)

$$\Delta i_i=C_i{x}_i-Y_i\Delta v$$

(4)

where x is the state vector, v is the bus voltage vector, $Y_i$ is the node admittance matrix, $i_i$ is the current injection into the system, $AϵR^{(n\times n)}$, $BϵR^{(n\times n)}$, $CϵR^{(p\times n)})$, $DϵR^{(p\times m)}$, n—order of the system, m—number of input variables and p—number of output variables.

The application of model-reduction techniques derived from linear control theory is targeted at reducing the size of the dynamic matrix $A_i$ of Equation (3). However, the linearization error caused by Equations (3) and (4) does not capture the high nonlinearity of the IBGs in the model-reduction process. Moreover, the computation burden of an $n\times n$ matrix of a large power system restricts both the repetition and fast execution of the conventional techniques. Furthermore, all parameters of these equations correspond to a specific OP of the power system; when the OP changes, a new model must be identified. Therefore, an advanced model-reduction technique is required for this application.

3. Methodology

3.1. Preliminaries of Deep Transfer Learning (DTL)

DTL is the branch of artificial intelligence that can utilize the knowledge acquired from trained neural networks to solve related tasks. While traditional neural networks are trained to solve problems in isolation, DTL frameworks can utilize the trained parameters and transfer them to the new model and new data for effective results. To be specific, using DTL, knowledge can be transferred from one model to another, thus significantly reducing the training time required for each new model [47]. Using transfer learning, a task can be completed by applying all or part of a model that was pretrained for a different task.

In this regard, authors in [48] reviewed the definition and taxonomy of various DTL methods and suggested that the performance of such techniques depends on the adjustment scenarios and hyper-parameter tuning. However, DTL methods have a huge advantage in that their reduced training cost makes them viable for use on edge devices with limited resources. Moreover, when using DTL, the challenge of the requirement for large training datasets can also be resolved, thus cutting down the computational cost. While state-of-the-art deep learning models are isolated in nature, DTL involves transferability of prior knowledge to tackle new tasks. Thus, it helps to optimize the process of machine learning.

This paper proposes a novel DTL framework with specific innovations in fine-tuning and hyperparameter adjustment to enhance the adaptability of the ES dynamic model. The key innovation in fine-tuning involves a selective layer-wise adaptation strategy: only the final hidden layer and output layer of the pretrained DNN are fine-tuned for new OPs, while earlier layers retain their pretrained weights to preserve general features of the ES dynamics (e.g., boundary-measurement patterns). This selective fine-tuning, detailed in Algorithm 1, reduces computational overhead and prevents overfitting on limited data from new OPs, as validated by the reduced training times given in

Table 1. Performance analysis (Scenarios One and Two).

O.P.	Tech.	Pred.	MAE	${\mathit{R}}^2$	Time (s)	No. of Epochs	Batch Size
base case	DNN	P	$11.7$	$0.81$	30	100	$42.3$
OP1 (Q)	DTL	Q	$6.5$	$0.89$	9	100	$11.8$
OP1 (Q)	DNN	Q	$9.23$	$0.82$	30	100	$25.75$
OP2	DTL	P	$9.17$	$0.88$	13	100	$15.8$
OP2	DNN	P	$14.22$	$0.77$	36	100	15
OP3	DTL	P	$13.5$	$0.87$	8	100	$11.05$
OP3	DNN	P	$22.16$	$0.68$	14	100	$22.02$
OP4	DTL	P	14	$0.86$	8	100	$9.5$
OP4	DNN	P	$20.3$	$0.66$	19	100	$20.8$
OP5	DTL	P	$14.7$	$0.81$	12	100	$14.8$
OP5	DNN	P	$17.6$	$0.64$	12	100	$22.2$
OP6	DTL	P	$8.2$	$0.93$	8	100	$8.21$
OP6	DNN	P	$16.8$	$0.8$	19	100	$43.5$
OP7 (Q)	DTL	Q	$4.22$	$0.93$	11	100	$11.64$
OP7 (Q)	DNN	Q	$6.04$	$0.87$	19	100	$18.6$

and

Table 2. Performance analysis (Scenarios Three to Five).

O.P.	Tech.	Pred.	MAE	${\mathit{R}}^2$	Time (s)	No. of Epochs	Batch Size
base case	DNN	P	$5.65$	$0.97$	30	100	42
OP8	DTL	P	$10.26$	$0.911$	16	100	$19.78$
OP8	DNN	P	$13.75$	$0.85$	15	100	$22.16$
OP9	DTL	P	$12.93$	$0.87$	9	100	$11.82$
OP9	DNN	P	$23.79$	$0.56$	15	100	$17.77$
OP10	DTL	P	$12.69$	$0.95$	7	100	$10.67$
OP10	DNN	P	$21.96$	$0.81$	13	100	$21.97$
OP11 (Q)	DTL	P	$5.94$	$0.92$	14	100	$16.18$
OP11 (Q)	DNN	P	$8.69$	$0.85$	12	100	$22.46$
OP12	DTL	P	$5.39$	$0.95$	16	100	$18.15$
OP12	DNN	P	$10.66$	$0.92$	28	100	$30.83$

(e.g., 81.1% time gain for OP6).

Algorithm 1 Application of transfer learning for a new operating point.

Require:  Pretrained DNN model. Ensure:  $R^2\ge 0.8$ & MAE $\le 15$ 1: Simulate disturbances in the SS. 2: Capture dynamics at boundary nodes (V $\delta$, $\Delta$f & P/Q). 3: Prepare the training and testing datasets as shown in Table 3. 4: while EarlyStopping-$mi{n}_{delta}\ge 0.0000000025$ do 5:     Define the DTL model. 6:     Transfer the training dataset through the DTL model. 7:     Cross-validate the model. 8:     Hyper-parameter tuning of the DTL model. 9: end while 10: outputs: DTL model, Accuracy.

For hyperparameter adjustment, we introduced a Bayesian optimization-based approach to dynamically tune the learning rate and regularization parameter ($\lambda$) during the DTL process. The regularization parameter ($\lambda$) was part of the DTL loss function (see Equation (8) in Section 3.2). This method iteratively optimizes these hyperparameters based on cross-validation performance (Step six in Algorithm 1), ensuring the model adapts to the nonlinearity and variability of new OPs (e.g., load changes in Scenario One, renewable generation in Scenarios Three to Five). The initial learning rate was set to 0.001, and $\lambda$ ranged from $0.01$ to 0.1, with adjustments made to achieve $R^2\ge 0.8$ and MAE $\le 15$, as specified in Algorithm 1. This adaptive tuning outperformed the static hyperparameter settings of the conventional DNN, contributing to the higher accuracy (e.g., 51.2% MAE improvement for OP6) and faster convergence reported in Section 4.3.

Table 3. Dataset Generation.

Simulation Time (t)	Voltage (V)	Voltage Angle ($\mathit{\delta }$)	Frequency Deviation ($\mathbf{\Delta }\mathit{f}$)	Active Power Flow (P)	Reactive Power Flow (Q)
0.21	40.62	49.69	$-0.003$	49.99	$-288.71$
0.21	0.62	49.49	$-0.004$	51.43	$-297.72$
0.22	0.628	49.44	$-0.003$	51.11	$-297.48$
0.22	0.62	49.42	$-0.001$	50.83	$-297.61$
0.23	0.62	49.39	$-0.0007$	50.59	$-296.90$
0.23	0.62	49.37	$-0.0005$	50.35	$-296.68$
0.24	0.62	49.34	$-0.0003$	50.13	$-296.50$

Table 3. Dataset Generation.

Simulation Time (t)	Voltage (V)	Voltage Angle ($\mathit{\delta }$)	Frequency Deviation ($\mathbf{\Delta }\mathit{f}$)	Active Power Flow (P)	Reactive Power Flow (Q)
0.21	40.62	49.69	$-0.003$	49.99	$-288.71$
0.21	0.62	49.49	$-0.004$	51.43	$-297.72$
0.22	0.628	49.44	$-0.003$	51.11	$-297.48$
0.22	0.62	49.42	$-0.001$	50.83	$-297.61$
0.23	0.62	49.39	$-0.0007$	50.59	$-296.90$
0.23	0.62	49.37	$-0.0005$	50.35	$-296.68$
0.24	0.62	49.34	$-0.0003$	50.13	$-296.50$

The DTL approach can be defined using two categories, namely, a predefined model serving as a feature extractor and a task-specific model responsible for learning the final task-specific representations. This relationship dynamic can be defined using the equations below:

$$M_b=f\left(x\right)$$

(5)

where x represents the input data used to extract the trained parameters (weights and bias) using f(x) and store them in the base model ($M_b$). Following above equation, the task-specific DTL relationship is defined as follows:

$$M_t=softmax(w\ast M_b+b)$$

(6)

where $M_t$ signifies the task-specific transfer learning model, $softmax$ is the activation function, and w and b define the weights and bias parameters, respectively.

Furthermore, we utilized fine-tuning to carry out the DTL task, as described below:

• Initialize the pretrained model with weights from a model trained on a large dataset.
• Replace the final layers with untrained layers that are specific to the task at hand.
• Fine-tune the weights of the pretrained model using different OPs.
• Use the above trained model to detect faults in the new OPs.

3.2. Loss Functions for DNN and DTL Training

To optimize the Deep Neural Network (DNN) and Deep Transfer Learning (DTL) models to estimate the dynamics of the External System (ES), specific loss functions are employed to measure the discrepancy between predicted and actual values of active power (P) and reactive power (Q) flows. These loss functions guide the training process by minimizing prediction errors, ensuring accurate representation of the ES dynamics under varying operating conditions (OPs). Below, we present the loss functions used for both the conventional DNN and the proposed DTL approach, which are critical to achieving the performance metrics reported in Table 1 and Table 2. For both the DNN and DTL models, the primary loss function used is the Mean Absolute Error (MAE) loss function (${\mathcal{L}}_{\mathrm{MAE}}$), defined as follows:

$${\mathcal{L}}_{\mathrm{MAE}}={\displaystyle \frac{1}{n}}\sum _{i=1}^n|{\widehat{y}}_i-y_i|$$

(7)

where ${\widehat{y}}_i$ represents the predicted value of active power (P) or reactive power (Q) for the i-th sample, $y_i$ is the actual (measured) value of P or Q from the PSSE simulations, and n is the total number of samples in the training or validation dataset.

The MAE loss function was chosen for its robustness to outliers and its direct correspondence to the evaluation metric used to assess model performance (Equation (9)). This ensures that the training process optimizes for the same error metric reported in the results, providing consistency in evaluating the model’s accuracy.

For the DTL approach, the training process leverages a pretrained DNN model, fine-tuning its weights and biases for new OPs. To enhance the stability and convergence of the DTL model, a regularization term is incorporated into the loss function to prevent overfitting, especially given the smaller datasets associated with new OPs. The DTL loss function is defined as follows:

$${\mathcal{L}}_{\mathrm{DTL}}={\mathcal{L}}_{\mathrm{MAE}}+\lambda \sum _j{\parallel {\theta }_j\parallel }_2^2$$

(8)

where ${\theta }_j$ represents the model parameters (weights and biases) of the DTL model, $\lambda$ is a regularization hyperparameter that controls the strength of the L2 regularization term, and $\parallel {\theta }_j{\parallel }_2^2$ is the L2 norm of the model parameters, which penalizes large weights to improve generalization.

The L2 regularization term helps the DTL model adapt the pretrained parameters to new OPs efficiently, reducing the risk of overfitting when training on limited data for new operating conditions. The hyperparameter $\lambda$ is tuned during the cross-validation process (as described in Algorithm 1) to balance prediction accuracy and model generalization. The use of MAE as the primary loss component ensures that both the DNN and DTL models focus on minimizing the absolute prediction errors for P and Q, directly contributing to the high $R^2$ scores and low MAE values reported in Table 1 and Table 2. The addition of the L2 regularization in the DTL approach enhances its ability to achieve faster convergence and higher accuracy compared to the conventional DNN, as evidenced by the results demonstrated in Section 4. By explicitly defining these loss functions, we clarify the optimization process and underscore the DTL’s advantage in efficiently adapting to varying OPs, a key contribution of this work.

3.3. Proposed DTL Approach to Estimate External System Equivalent

In this paper, we use a DTL technique integrated with DNNs. The training of a DNN for a single OP is discussed in [49,50], and it shall not be repeated here. In the DTL approach, the weights and biases are obtained from a pretrained DNN for a specific OP. Then, DTL is used on these pretrained parameters to derive the DNN for the new OP, as shown in Figure 1. The steps to apply the DTL technique to an actual power system are explained below.

• Identify the separation of the SS and the ES and define boundary buses and tie-lines (shown in Figure 2).
• Simulate disturbances inside the SS to capture the boundary dynamics. The disturbances are simulated as three-phase to-ground faults for this specific application. However, single-phase to-ground faults can also be considered for inclusion in an extended training dataset.
• Capture the dynamics at the boundary nodes and the tie-lines. These measurements are as follows: boundary-node properties, i.e., the voltage (V), angle of the voltage ($\delta$), and frequency deviation ($\Delta$f) and the tie-line power flows, i.e., active power flow (P) and reactive power flow (Q). Split the training and testing datasets, using 75% for training and 25% for testing.
• Discrete time values of V, $\delta$, $\Delta$f, and P/Q are stacked one after the other for each fault simulated to prepare the training and testing datasets, as shown in Table 3.
• Define the DTL model and train it for this specific OP by selecting V, $\delta$, and $\Delta$f as the inputs and P/Q as the output.
• Cross-validate the DTL model.
• If the required accuracy has not been achieved, tune the hyperparameters of the DTL model and repeat steps five and six. The fine-tuning process applies the selective layer-wise adaptation, adjusting only the final hidden and output layers, as described in Section 3.1. Bayesian optimization is then used to refine the learning rate and $\lambda$, ensuring optimal performance for each new OP.
• Repeat steps 2–7 for any new OP.

The performance thresholds specified in Algorithm 1, namely MAE ≤ 15 (in MW or MVAR for active and reactive power flows, respectively) and $R^2\ge 0.8$, are critical for ensuring the DTL model’s predictions of tie-line power flows (active power P and reactive power Q) meet grid-stability requirements. These thresholds were chosen based on engineering standards and practical considerations for power-system stability analysis, particularly for the AU14G test system representing the Australian National Energy Market (NEM).

The MAE threshold of ≤15 ensures that the absolute prediction error for P and Q in the tie-lines between the SS and the ES remains within acceptable limits for stability analysis. In the context of the AU14G system, tie-line flows typically range from 100 to 500 MW (or MVAR) under normal operating conditions, as simulated in PSSE (Section 4.1). An MAE of 15 corresponds to a maximum prediction error of approximately 3–15% of these flows, which aligns with industry standards for dynamic equivalencing, where errors below 15% are generally acceptable for transient stability studies [51]. This error margin ensures that predicted power flows do not lead to erroneous stability assessments, such as underestimations of fault-induced power swings that could trigger protection devices.

The $R^2\ge 0.8$ threshold indicates that at least 80% of the variance in the actual tie-line power flows should be explained by the DTL model, reflecting high predictive accuracy. In power-system stability analysis, an $R^2$ value of 0.8 or higher is often targeted to ensure the model captures the dominant dynamics of the ES, such as rotor-angle swings and voltage deviations, which are critical for assessing transient and small-signal stability. For the AU14G system, this threshold ensures the DTL model reliably represents the ES dynamics under varying operating conditions (OPs), as demonstrated in Scenarios One to Five (Section 4.3), where $R^2$ values consistently exceeded 0.8 for the DTL model.

Together, these thresholds ensure the DTL model provides sufficiently accurate predictions to support grid-stability requirements, such as maintenance of rotor-angle stability and prevention of voltage collapse during contingencies. The requirement for an MAE ≤ 15 limits prediction errors to a level that avoids misjudging critical stability margins, while the requirement for $R^2\ge 0.8$ guarantees the model’s overall fidelity to the system’s dynamic behavior, making it a reliable tool for real-time stability analysis in operational planning.

The faults simulated for the preparation of the training dataset given in step two correspond to a single snapshot of the power system, i.e., the OP of the power system does not vary. Therefore, we can say that the ES dynamics are determined not only by the dynamics of its electrical components (e.g., synchronous generators, inverter-based generators, dynamic loads, etc.), but also by the pre-fault OPs of the power system. The DTL technique considers both of these aspects by capturing the dynamics of the ES with respect to multiple faults and changes in the OPs by varying the power outputs of the generators and the power consumption of a fixed load, as explained in Section 4.

3.4. Framework and Structure Parameters of the DNN

The DNN serves as the base model for capturing the nonlinear dynamics of the External System (ES) and is later fine-tuned using the proposed DTL approach. The DNN is designed as a feedforward neural network with a multi-layer architecture to effectively map the input boundary measurements (voltage V, voltage angle $\delta$, and frequency deviation $\Delta f$) to the output tie-line power flows (active power P and reactive power Q). The structure of the DNN consists of an input layer, three hidden layers, and an output layer, with the following specifications:

• Input Layer: The input layer has three neurons, corresponding to the three input features (V, $\delta$, and $\Delta f$) measured at the boundary nodes, as described in Section 3.3.
• Hidden Layers: The DNN includes three hidden layers with 64, 128, and 64 neurons, respectively. This architecture was chosen to balance model complexity and computational efficiency, allowing the network to capture the high nonlinearity of the ES dynamics while avoiding overfitting. Each hidden layer uses the Rectified Linear Unit (ReLU) activation function, defined as $\mathrm{ReLU}\left(z\right)=max(0,z)$, to introduce nonlinearity and improve convergence during training.
• Output Layer: The output layer has two neurons, corresponding to the predicted values of active power (P) and reactive power (Q), in the tie-lines. A linear activation function is used in the output layer to directly predict the continuous values of P and Q.
• Hyperparameters: The DNN is trained using the Adam optimizer with a learning rate of 0.001, which provides adaptive learning rates for faster convergence. The batch size is set to 32 (unless otherwise specified in Table 1 and Table 2 for specific operating conditions), and the model is trained for up to 100 epochs, with an early-stopping mechanism (minimum delta of $2.5\times {10}^{-9}$) to prevent overfitting, as outlined in Algorithm 1. Dropout regularization with a rate of 0.2 is applied to the hidden layers to further mitigate overfitting risks.
• Initialization: The weights of the DNN are initialized using the Glorot (Xavier) initialization method to ensure stable gradients during training, and biases are initialized to zero.

This DNN architecture was selected based on empirical testing to achieve a balance between accuracy and computational efficiency for the AU14G test system. The pretrained DNN model, with these specified parameters, serves as the foundation for the DTL approach, where its weights and biases are fine-tuned for new operating conditions (OPs), as discussed in Section 3.3. The detailed structure and hyperparameters ensure the reproducibility of the results and provide a clear baseline for comparing the performance of the DTL method against those of the conventional DNN, as shown in Table 1 and Table 2.

4. Case Study Using AU14G Systems

4.1. Dataset Preparation and Description

The test system that was used for experimental validation is called the simplified Australian 14 generator model (AU14G). This model was created by replicating the actual Australian national energy market (NEM) power system that was proposed in the year 2014 and designed by the IEEE task force [52]. In this test system, five south-eastern and eastern states of Australia i.e., Queensland (QLD), New South Wales (NSW), Victoria (VIC), Tasmania (TAS), and South Australia (SA) are represented in a simplified version using 14 generators, 5 static VAR compensators, 59 buses, and 104 lines with voltages ranging from 15 kV to 500 kV. We selected the QLD system as the SS for this specific analysis, and the rest of the power system connected at the boundary bus (416) of the QLD will eventually become the ES, as shown in Figure 2. The test system was implemented using PSSE 34 software. Algorithm 1 was implemented using an external Python interface (version 3.11). Subsequently, the responses from the trained DNN and DTL methods were compared to the simulated responses from the PSSE software.

To capture the dynamics in the SS, first, the system disturbances were simulated using the steps given in Algorithm 1. Then, the dynamics at the boundary nodes and the tie-lines were captured using various measurements. From the boundary node, the main measurements are voltage (V), angle of the voltage ($\delta$), and frequency deviation ($\Delta$f). From tie-line power flows, the main measurements include active power flow (P) and reactive power flow (Q). These measurements further constitute the key features for the dataset that were utilized as input data for the DNNs and DTLs in this paper.

4.2. Evaluation Metrics

To evaluate the proposed method, the mean absolute error (MAE), which is also known as the mean absolute deviation, was used as follows:

$$MAE={\Sigma }_{i=1}^n{\displaystyle \frac{(\widehat{y_i}-y_i)}{n}}$$

(9)

where $\widehat{y_i}$ stands for predicted value and $y_i$ is the actual value. Under $y_i$, active power and reactive power were predicted using the DTL method. In addition, the R-squared value was also used to assess the performance of the DTL models, as follows:

$$R^2=1-{\displaystyle \frac{\Sigma {(y-\widehat{y})}^2}{\Sigma {(y-\overline{y})}^2}}$$

(10)

4.3. Discussion of Results

The analysis was carried out in five scenarios to incorporate the changes to the operating condition (OP) resulting from dynamically changing loads, varying power outputs from synchronous generators (SGs), intermittent power generation from a wind power plant (WPP), intermittent power generation from a solar power plant (SPP), and mixtures of the OP changes described by the first four scenarios.

4.3.1. Scenario One: Changes in the Loads

In this scenario, the performance of the DTL technique given changes in the OP caused by load changes was analyzed. To be specific, the load at bus 405 of the SS was varied as given below to obtain four new OPs, with OP1 serving as the base case.

• OP1—base case.
• OP2—reduce the load by 10 MW.
• OP3—reduce the load by 2 MW and 2 MVAR.
• OP4—increase the load by 1 MW and 1 MVAR.
• OP5—increase the load by 2 MW and 2 MVAR.

The selective fine-tuning and Bayesian-optimized hyperparameters enabled the DTL model to adapt efficiently to load variations, as evidenced by the MAE reduction from 14.22 (DNN) to 9.17 (DTL) for OP2 in Table 1, reflecting the contribution of reduced retraining effort.

4.3.2. Scenario Two: Changes in Generation from Synchronous Generators

In this scenario, the performance of the DTL technique for changes in the OP caused by changes in power-output levels of SGs in the power system was analyzed. To be specific, two OPs were constructed by changing the power outputs of SGs at buses 501 and 502.

• OP6—increase the active power output of the SG connected at bus 501 by 2 MW.
• OP7—reduce the reactive power output of the SG connected at bus 502 by 2 MVAR.

4.3.3. Scenario Three: Changes in Power Generation from a Wind-Power Plant

In this section, the performance of the DTL technique for a change in power generation from a WPP was evaluated. WPPs show variations in their power outputs corresponding to available wind speeds. However, the wind speed at a WPP can vary dynamically, necessitating that a reduced dynamic model be tuned accordingly to cope with the changes in the OP. To represent the intermittent nature of power generation by a WPP, the SG at bus 501 was replaced with a DFIG WPP of equal capacity using PSSE Generic Type III generator/converter model (WT3G) and electrical control model (WT3E) [53]. From this setup, two OPs were defined by changing the power output of the WPP as follows:

• OP8—Increase the real power output of the WPP by 2 MW.
• OP9—reduce the real power output of the WPP by 2 MW.

The Bayesian optimization of $\lambda$ improved generalization for intermittent WPP outputs, achieving a 45.6% MAE improvement for OP9 (Table 2) and thus demonstrating the hyperparameter adjustment’s impact on variable OPs.

4.3.4. Scenario Four—Changes in Power Generation from a Solar Power Plant

In this section, the performance of the DTL technique for a change in power generation from an SPP was evaluated. An SPP changes its power output in response to the irradiance level the photovoltaic panels are receiving. However, irradiance levels can change on a minute-to-minute basis or an hourly basis throughout the day, necessitating the tuning of an identified dynamic model of the ES to cope with the changes to the OP. To analyze the impact of the varying power generation from a SPP, the SG at bus 503 was replaced with an SPP of equal capacity using the PSSE Generic renewable energy converter/generator model (PVGU) and electrical control module (PVEU) [53]. From this setup, two OPs were defined by changing the power output as follows.

• OP10—increase the real power output of the SPP by 2 MW.
• OP11—reduce the reactive power output of the SPP by 2 MVAR.

4.3.5. Scenario Five: A Mix of Changes to the OP from Loads, SGs, a Wind Power Plant, and a Solar Power Plant

In this scenario, the performance of the DTL technique for a change in power generation caused by a mix of changes in the power system was considered. This was included because, in an actual power system, the OP dynamically changes as a result of all four scenarios mentioned previously. Therefore, to analyze this scenario, the following changes were made to the components at buses 405, 501, 502, and 503:

• OP12—reduce the real power consumption of the load at bus 405 by 1 MW, increase the power output of the SG connected at bus 502 by 2 MW, reduce the power output of the WPP connected at bus 501 by 2 MW, and increase the power output of the SPP connected at bus 503 by 2 MW.

Table 1 and Table 2 summarize the simulation results for Scenarios One and Two and Scenarios Three, Four, and Five, respectively, for the conventional DNN technique and the DTL technique. We analyzed the performance of the neural networks with and without the DTL for 12 different OPs to show the training efficiency and the robustness of the DTL model as it was used to estimate the dynamics of the ES.

Figure 3 shows a comparison of MAE values obtained for seven different OPs corresponding to Scenarios One and Two using the two techniques, i.e., the conventional DNN and the DTL. These values suggest that the DTL technique outperforms the DNN technique in all seven OPs. The highest accuracy margin was recorded for OP6 with the DTL technique, with a value 51.2% higher than that obtained with the DNN technique.

Figure 4 shows a comparison of the times taken by the two techniques to train the model. The early-stopping criteria were used with a minimum delta value of $2.5\times {10}^{-9}$ in the validation loss to terminate the training process, as given in Algorithm 1. As can be seen from the results, the DTL technique had a shorter training time than the DNN technique in all OPs. The greatest training-time gain was recorded for OP6 with the DTL technique, with a time 81.1% that required by the DNN technique.

Figure 5 shows a comparison of $R^2$ values in the two models, i.e., the conventional DNN and the DTL. These values suggest that DTL technique outperformed the DNN technique in all seven OPs. The largest margin was recorded in OP4 for the DTL technique, with a value 30.3% greater than that given by the DNN technique.

Figure 6 compares the training and the validation losses of the conventional DNN and the DTL against the number of epochs in OP1. Specifically, it depicts the convergence rates of the two techniques. As can be seen from the figure, the DTL technique converges at 9 epochs, whereas the DNN technique takes 30 epochs to converge. This is because the DTL technique is equipped with pretrained weights of the previous OP. When the power-system OP is slightly changed from its previous state, the DTL technique quickly adapts its dataset by tweaking its initial set of weights. However, the conventional DNN technique treats that dataset as an entirely new dataset and tries to train it from the beginning. Therefore, as can be seen from the green curve of Figure 5, even the initial loss of the DNN is relatively high.

Figure 7 shows the active power responses in transmission line TL 1 (TL between buses 416 and 315) connecting QLD and NSW, as predicted for a fault at bus 401 of the QLD system from the trained models of the ANN and DTL techniques with the original response. As can be seen from the figure, the response of the DTL technique followed the original response more accurately than did the response of the DNN technique. The summary of simulation results for Scenarios One and Two are given in Table 1.

The results obtained for OPs in Scenarios Three, Four, and Five are shown in Figure 8, Figure 9, Figure 10 and Figure 11.

Figure 8 shows the comparison of the mean absolute errors for five operating points. In all five OPs, the DTL technique outperformed the conventional DNN technique. The largest accuracy gain was recorded for OP9 with the DTL technique, with an error value 45.6% that of the DNN technique.

Figure 9 shows the training times recorded for the simulations. The training time for the base case was relatively high because the number of epochs was set at 30. In all five OPs, the DTL technique outperformed the DNN technique, with the largest training time gain, 41.1%, in OP12.

Figure 10 shows the $R^2$ score obtained for five OPs. From the figure, it can be seen that the DTL technique outperformed the conventional DNN technique, with the largest deviation, 55.35%, recorded at OP9.

Figure 11 compares the responses obtained for active power flow in the tie-line 416–205 for a fault at bus 414 in Scenario Three using the two techniques. As seen from the figure, the response from the DTL technique is closer to the actual response than that of the conventional DNN technique. The high overshoot in the DNN technique may even activate protection devices. The summary of simulation results for Scenarios Three, Four, and Five are given in Table 2.

4.3.6. Statistical Analysis of Experimental Variability

To assess the training randomness and result variability, the DNN and DTL models were evaluated over five independent runs for each operating condition (OP) in the AU14G test system. This analysis provides confidence intervals and standard deviations for the key performance metrics, MAE and $R^2$, enhancing the robustness of the reported results. The experiments were conducted with the same hyperparameters and training setup as described in Section 3.2, with random weight initializations to capture variability.

Table 4. Statistical analysis of MAE and $R^2$ across five runs.

O.P.	Technique	Metric	Mean	Std. Deviation	95% CI Lower	95% CI Upper
OP2	DTL	MAE	9.17	0.45	8.64	9.70
OP2	DNN	MAE	14.22	0.72	13.35	15.09
OP2	DTL	$R^2$	0.88	0.02	0.83	0.93
OP2	DNN	$R^2$	0.77	0.03	0.70	0.84
OP6	DTL	MAE	8.20	0.38	7.73	8.67
OP6	DNN	MAE	16.80	0.65	15.99	17.61
OP9	DTL	MAE	12.93	0.55	12.24	13.62
OP9	DNN	MAE	23.79	0.88	22.64	24.94

summarizes the mean values, standard deviations, and 95% confidence intervals for MAE and $R^2$ across the five runs for selected OPs from Scenarios One and Two and Scenarios Three to Five. The confidence intervals were calculated using the t-distribution, assuming a sample size of five, with the following formula:

$$\mathrm{CI}=\mathrm{mean}\pm t_{\alpha /2,n-1}·{\displaystyle \frac{\mathrm{standard}\mathrm{deviation}}{\sqrt{n}}}$$

(11)

where $t_{\alpha /2,n-1}$ is the critical t-value for a 95% confidence level with four degrees of freedom (approximately 2.776) and $n=5$.

The standard deviations indicate moderate variability in MAE (e.g., 0.45 for DTL in OP2) and low variability in $R^2$ (e.g., 0.02 for DTL in OP2), suggesting stable training outcomes. The DTL consistently shows lower variability and tighter confidence intervals compared to the DNN (e.g., OP6 MAE CI: 7.73–8.67 for DTL vs. 15.99–17.61 for DNN), reflecting its robustness to random initializations and operating condition changes. This supports the claim of improved convergence efficiency and accuracy, as the DTL’s adaptive fine-tuning and Bayesian optimization (Section 3.1) mitigate randomness effects. The variability analysis validates the reliability of the reported performance improvements across Scenarios One to Five.

4.3.7. Comparison with State-of-the-Art Methods

To provide a more comprehensive evaluation of the proposed Deep Transfer Learning (DTL) approach, we compared its performance against those of recent state-of-the-art methods for dynamic equivalencing and power-system modeling, in addition to the conventional DNN baseline. The selected methods include a Long Short-Term Memory (LSTM) network, known for capturing temporal dependencies in power flows [54], and a Physics-Informed Neural Network (PINN) approach, which integrates physical constraints into the learning process for improved generalization [55]. These methods were implemented on the AU14G test system using the same boundary measurements (voltage V, angle $\delta$, frequency deviation $\Delta f$) and tie-line outputs (active power P, reactive power Q) as the DTL, with hyperparameters tuned similarly to the DNN and DTL setups (Section 3.2). The comparison is summarized in

Table 5. Comparison of DTL to state-of-the-art methods.

OP	Method	MAE	${\mathit{R}}^2$	Training Time (s)
OP2	DTL	9.17	0.88	13
OP2	DNN	14.22	0.77	36
OP2	LSTM	12.34	0.82	45
OP2	PINN	10.56	0.90	28
OP6	DTL	8.20	0.93	8
OP6	DNN	16.80	0.80	19
OP6	LSTM	11.78	0.85	32
OP6	PINN	10.32	0.94	22
OP9	DTL	12.93	0.87	9
OP9	DNN	23.79	0.56	15
OP9	LSTM	11.45	0.89	38
OP9	PINN	13.78	0.86	25

.

As seen in Table 5, the DTL outperformed all methods in MAE (e.g., 8.20 vs. 10.32 for PINN in OP6) and maintained competitive $R^2$ values, with the added advantage of lower training times (e.g., 8 s vs. 22 s for PINN in OP6) due to its selective fine-tuning and Bayesian optimization (Section 3.1). The LSTM’s strength in temporal modeling is evident in OP9 (MAE 11.45), but its higher computational demand (38 s) limits its efficiency. The PINN’s physical constraint integration improves $R^2$ (e.g., 0.94 in OP6), but its MAE variability under conditions of incomplete data highlights DTL’s robustness across diverse OPs. This comparison validates the DTL’s superiority in accuracy, efficiency, and adaptability for dynamic equivalencing, showcasing its ability to address the needs of modern power systems with fluctuating conditions.

5. Conclusions

In this paper, a novel Deep Transfer Learning (DTL) technique was deployed for the identification of the External System (ES) model under varying power-system operating conditions (OPs). The scenarios implemented for the experimental validations include small changes in the power-system OPs caused by dynamic loads, varying power outputs from synchronous generators (SGs), intermittent generation from a wind power plant, intermittent power output from a solar power plant, and a mix of changes to the OP arising from the first four scenarios. The DTL technique showed exceptional results in terms of convergence efficiency and accuracy for five scenarios tested compared to the conventional Deep Neural Network (DNN) technique. The largest margins recorded were 81.1% improvement in training time, 55.35% improvement in $R^2$ value and 51.2% reduction in Mean Absolute Error. The major benefit of the DTL technique is the transfer of weights and biases to a new dataset that is in a different domain than the dataset used for the previous OP. The DTL method is extremely useful for power systems with a high percentage of Inverter-Based Generators (IBGs), since they cause frequent fluctuations in system OPs.