Graph Neural Network-Infused Digital Twin Platform with Transfer Learning and Quantum-Safe Protocols for Resilient Power System Control and Markets

1. Introduction

Modern power systems confront escalating challenges stemming from the widespread integration of renewable energy sources, escalating cyber threats, and the complexities of real-time electricity markets, all of which strain the limits of conventional control paradigms [1].

Traditional centralized controllers, grounded in linear approximations and deterministic models, exhibit vulnerabilities to nonlinear disturbances such as sudden renewable intermittency, physical line faults, or sophisticated adversarial attacks, often precipitating cascading failures and economic losses exceeding billions annually [2]. This study presents a Graph Neural Network-infused Digital Twin Platform augmented with Transfer Learning and Quantum-Safe Protocols to deliver resilient control and market optimization, mirroring physical grid states in a virtual environment for proactive decision-making and seamless adaptation to dynamic conditions.

The motivation for this framework arises from critical gaps in existing technologies: digital twins provide high-fidelity simulations but lack efficient handling of graph-structured data inherent to power topologies, while graph neural networks (GNNs) excel in capturing spatial dependencies yet demand extensive retraining for domain shifts meanwhile, classical cryptographic safeguards falter against impending quantum computing breakthroughs.

By infusing GNNs into the digital twin's core, the platform models buses as nodes and transmission lines as edges, propagating state information through message-passing layers to forecast cascades, optimize power flows, and simulate market bids with unprecedented accuracy [3]. Transfer learning addresses data scarcity by adapting pre-trained models from synthetic IEEE benchmarks to real-world operations, reducing deployment timelines from months to days, whereas quantum-safe protocols leveraging lattice-based schemes like Kyber and Dilithium secure federated data exchanges, ensuring long-term integrity amid harvest-now-decrypt-later threats.

Key contributions encompass three pillars:

(i)

first, a novel hybrid architecture that synchronizes edge-computed GNN inferences with cloud-based twin optimization for low-latency control loops

(ii)

second, rigorous methodologies integrating transfer learning with physics-informed losses to enhance generalization across diverse grid scales, from distribution feeders to transmission networks

(iii)

third, comprehensive evaluations demonstrating superior resilience, including 25% faster recovery from false data injection attacks and 18% improvements in locational marginal price forecasting compared to state-of-the-art baselines like recurrent neural networks and physics-only simulators.

This work not only advances cyber-physical resilience but also aligns with global decarbonization imperatives by facilitating secure integration of distributed energy resources and peer-to-peer trading mechanisms [4]. Ultimately, the platform charts a pathway for utilities to transition toward quantum-ready, AI-driven smart grids capable of sustaining reliability in an era of uncertainty.

2. Related Work

2.1. Graph Neural Networks in Power Systems

Graph neural networks (GNNs) have emerged as a transformative tool for power system applications by effectively capturing the topological intricacies of electrical grids, where buses serve as nodes and transmission lines function as edges with attributes like impedance and capacity [5]. Researchers have applied GNN variants, such as Graph Convolutional Networks (GCNs) and Graph Attention Networks (GATs), to tasks including load forecasting, contingency analysis, and cascading failure prediction, outperforming traditional machine learning methods like LSTMs by leveraging message-passing mechanisms that propagate spatiotemporal features across non-Euclidean structures.

For instance, studies demonstrate GNNs achieving up to 30% reductions in mean absolute errors for voltage stability assessment on IEEE 118-bus systems, as they inherently model physical laws through graph convolutions that aggregate neighbourhood states, enabling robust handling of topology changes from renewable integrations or line outages [6]. This body of work lays the foundation for infusing GNNs into digital twins, extending beyond isolated predictions to closed-loop control in dynamic market environments.

2.2. Digital Twins for Grid Resilience

Digital twins replicate physical power grid components in virtual environments, continuously updated through sensor fusion to simulate scenarios and bolster resilience against disruptions like extreme weather or cyberattacks [7]. Deployments in transmission networks and microgrids have showcased twins reducing outage durations via predictive maintenance and real-time what-if analyses, with frameworks like those from Siemens and GE integrating physics-based models for state estimation.

However, scalability challenges persist for large-scale systems, where computational demands hinder real-time synchronization, and integration with market layers remains underexplored recent advancements incorporate hybrid data-driven elements to refine accuracy under uncertainty, yet they often overlook graph-structured dependencies critical for holistic grid behavior [8]. This platform advances these efforts by embedding GNNs to enhance twin fidelity, enabling proactive resilience strategies that couple physical stability with economic optimization.

2.3. Transfer Learning Applications

Transfer learning facilitates rapid model deployment in power systems by transferring knowledge from data-rich source domains, such as synthetic simulations, to data-sparse targets like regional grids, minimizing retraining efforts through techniques like parameter freezing and fine-tuning of upper layers [9]. Applications in energy forecasting and fault classification have reported convergence speedups of 50-70%, with domain-adaptive GNNs adapting graph encoders across topologies while preserving low-level features like power flow patterns.

In electricity markets, transfer strategies have improved bidding accuracy by leveraging pre-trained models from wholesale to retail contexts, addressing heterogeneity in datasets plagued by privacy regulations [10]. Despite these gains, limited exploration exists for combining transfer learning with digital twins under cyber constraints, which this work addresses to ensure efficient, generalizable performance.

2.4. Quantum-Safe Cryptographic Protocols

Quantum-safe cryptographic protocols, also known as post-quantum cryptography (PQC), protect power system communications against quantum algorithms like Shor's that threaten RSA and ECC-based encryption, with NIST-standardized schemes such as Kyber for key encapsulation and Dilithium for signatures offering lattice-based security [11]. Implementations in SCADA and smart meter networks have demonstrated negligible overhead for securing data streams, vital for preventing harvest-now-decrypt-later attacks amid advancing quantum hardware.

In grid contexts, PQC integration supports federated learning for distributed analytics, maintaining privacy during twin synchronization; however, resource constraints on edge devices pose adoption barriers, prompting hybrid classical-PQC handshakes. This framework incorporates these protocols into GNN-digital twin pipelines, ensuring resilient, future-proof data exchanges for control and markets.

3. System Architecture

3.1. Digital Twin Platform Overview

The digital twin platform as a federated edge-cloud continuum that mirrors the physical power grid through perpetual synchronization of heterogeneous data streams from PMUs, IEDs, and EMS platforms, fostering a dynamic virtual surrogate for holistic control and market orchestration [12]. Edge layers execute lightweight Kalman-extended filters for preliminary state alignment, relaying refined inputs to cloud orchestrators that orchestrate multi-horizon optimizations via mixed-integer solvers constrained by N-1 security criteria. This stratified design minimizes propagation delays to under 100 Ms, empowering closed-loop interventions that pre-emptively adjust generation setpoints and ancillary service bids amid volatility [13].

Researchers design the digital twin platform as a hybrid edge-cloud system that creates a virtual replica of the physical power grid, continuously synchronizing real-time sensor data from phasor measurement units, smart meters, and market interfaces to enable predictive simulations and closed-loop control [14]. This architecture employs a layered pipeline where edge nodes handle low-latency state estimation and initial graph processing, while cloud resources manage complex optimizations and scenario forecasting, ensuring scalability across distribution and transmission scales. Bidirectional data flows, secured against latency spikes, facilitate what-if analyses for contingencies like renewable curtailments or cyber intrusions, with feedback loops driving actuators for automatic generation control and market dispatch adjustments [17].

3.2. Graph Representation of Power Networks

Power networks materialize as time-varying graphs in this framework, with nodes embodying buses enriched by feature vectors encompassing voltage magnitudes, phase angles, active and reactive powers, and economic attributes such as locational marginal prices or bid curves [18].

Figure 1. Multi-Layered Cyber-Physical Architecture of the GNN-Infused Digital Twin Platform.

Figure 1. Multi-Layered Cyber-Physical Architecture of the GNN-Infused Digital Twin Platform.

The power network manifests as a heterogeneous temporal graph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathrm{X},\mathrm{A})$, where $\mathcal{V}$denotes buses as nodes with feature matrix $\mathrm{X}\in {\mathbb{R}}^{\mid \mathcal{V}\mid \times F}$encoding states like $P_i,Q_i,V_i,{\theta }_i$alongside market vectors (e.g., bid prices ${\lambda }_i$); $\mathcal{E}$captures edges with adjacency $\mathrm{A}\in {\mathbb{R}}^{\mid \mathcal{V}\mid \times \mid \mathcal{V}\mid }$weighted by admittance and limits $f_{ij}^{\mathrm{m}\mathrm{a}\mathrm{x}}$.

$$y_{ij}=g_{ij}+j{b}_{ij}$$

(1)

Temporal evolution follows ${\mathcal{G}}_t={\mathcal{G}}_{t-1}\oplus \mathsf{\Delta }\mathcal{T}$, incorporating injections $\mathsf{\Delta }{P}_t$from renewables, ensuring fidelity to AC power flow equations $\mathrm{P}=\mathrm{Y}\theta$. Edges represent transmission lines or transformers, embedding attributes like reactance, thermal limits, and dynamic trust scores derived from anomaly detection, allowing the graph to evolve with topology alterations from line reconfigurations or distributed energy resource injections [20]. This structured representation preserves Kirchhoff's laws and market clearing constraints implicitly, providing a unified substrate for neural propagation that mirrors the interconnected nature of modern grids.

3.3. GNN-Infused Modelling Layer

The GNN-infused modelling layer integrates stacked graph convolutional and attention-based blocks directly into the twin's inference engine, where each layer performs neighbourhood aggregations to refine embeddings from local bus clusters toward global stability indicators like system inertia or reserve margins [22]. GNN infusion employs multi-layer graph attention networks (GATs) interwoven with the twin's dynamics, where layer-wise updates compute node embeddings via

$${\mathrm{h}}_v^{\left(l+1\right)}=\sigma \left({\sum }_{u\in \mathcal{N}\left(v\right)}{\alpha }_{vu}^{\left(l\right)}{\mathrm{W}}^{\left(l\right)}{\mathrm{h}}_u^{\left(l\right)}\right)$$

(2)

with attention coefficients

$${\alpha }_{vu}^{\left(l\right)}={\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{LeakyReLU}\right({\mathrm{a}}^T[\mathrm{W}{\mathrm{h}}_v\mid \mid \mathrm{W}{\mathrm{h}}_u]\left)\right)}{{\sum }_{k\in \mathcal{N}\left(v\right)}\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{LeakyReLU}\right({\mathrm{a}}^T[\mathrm{W}{\mathrm{h}}_v\mid \mid \mathrm{W}{\mathrm{h}}_k]\left)\right)}}$$

(3)

Physics infusion regularizes losses with

$${\mathcal{L}}_{phys}=\parallel {\mathrm{P}}_{pred}-\mathrm{Re}\left(\mathrm{Y}\mathrm{V}\right){\parallel }^2$$

(4)

yielding latent states $\mathrm{Z}=\mathrm{GNN}\left(\mathcal{G}\right)$for downstream forecasting ${\widehat{\mathrm{x}}}_{t+1}=f({\mathrm{Z}}_t,{\mathrm{u}}_t)$, enhancing cascade prediction fidelity [25]. Message-passing operations compute updated node states as weighted sums of adjacent features, modulated by attention scores that prioritize critical paths during stressed conditions, surpassing traditional simulators in capturing nonlinear cascades. This infusion elevates twin fidelity by fusing data-driven insights with physics-informed regularizations, yielding predictions that inform resilient control actions in milliseconds.

3.4. Integration with Transfer Learning

Transfer learning integrates seamlessly by initializing the GNN layers with parameters pre-trained on expansive synthetic datasets from tools like Pandapower simulations of IEEE test cases, then progressively unfreezing layers for fine-tuning on proprietary operational streams [28]. Transfer learning embeds via a two-stage paradigm: pre-training minimizes

$${\mathcal{L}}_{source}={\mathcal{L}}_{recon}+\lambda {\mathcal{L}}_{phys}$$

(5)

on source graphs ${\mathcal{G}}_S$to derive ${\mathit{\theta }}^{*}$, followed by adversarial fine-tuning ${\mathrm{m}\mathrm{i}\mathrm{n}}_{\mathit{\varphi }}{\mathrm{m}\mathrm{a}\mathrm{x}}_{\mathit{\psi }}{\mathcal{L}}_{target}+\mathcal{D}({\mathcal{D}}_S\mid \mid {\mathcal{D}}_T)$, where discriminator $\mathit{\psi }$aligns distributions through MMD losses, freezing encoder ${\mathit{\varphi }}_{enc}$initially. Fine-tuning targets

$${\mathcal{L}}_{fine}=\alpha {\mathcal{L}}_{MSE}({\widehat{\mathbf{x}}}_T,{\mathbf{x}}_T)+(1-\alpha ){\mathcal{L}}_{market}(\widehat{\lambda },\lambda )$$

(6)

adapting to target market volatilities while retaining topology-invariant features [32]. Domain adaptation occurs through gradient alignment techniques that minimize discrepancies between source graph distributions and target market dynamics, preserving foundational representations of power flows while customizing higher-level decoders for region-specific bidding strategies. This approach circumvents cold-start issues in data-limited utilities, enabling deployment with minimal overhead and sustained performance amid evolving grid compositions.

3.5. Quantum-Safe Communication Module

The quantum-safe communication module envelops all inter-twin exchanges in post-quantum cryptographic primitives, utilizing Kyber for ephemeral key encapsulation mechanisms that establish session keys resistant to quantum Grover and Shor attacks, complemented by Dilithium signatures for authenticating federated updates from distributed edge agents [39]. The module deploys Kyber KEM for key encapsulation, generating ciphertext

$$c=\mathrm{Encaps}\left(pk,\cdot \right)$$

(7)

from public key $pk$, yielding shared secret

$$K=\mathrm{Decaps}(sk,c)$$

(8)

with security rooted in module-LWE hardness. Dilithium signs payloads via $\sigma ={\mathrm{Sign}}_{sk}\left(m\right)$, verifiable by ${\mathrm{Verify}}_{pk}(\sigma ,m)$, integrated into TLS 1.3 hybrids with ephemeral PQC handshakes. Secure aggregation employs additive homomorphic properties for $\sum \mathrm{Enc}\left(x_i\right)\approx \mathrm{Enc}(\sum x_i)$, enabling privacy-preserving twin consensus without central decryption, bounded by IND-CCA2 guarantees [41].

Hybrid handshakes blend classical Diffie-Hellman for immediate compatibility with legacy SCADA while bootstrapping PQC suites, ensuring forward secrecy without disrupting real-time cycles [42]. This module extends to secure multi-party computation for aggregated market signals, safeguarding privacy in competitive environments and fortifying the platform against long-term decryption threats.

4. Methodology

4.1. GNN Model Formulation

4.1.1. Node and Edge Features

Researchers define node features as comprehensive vectors ${\mathbf{x}}_v\in {\mathbb{R}}^{d_n}$for each bus $v\in \mathcal{V}$, concatenating physical measurements such as active power $P_v$, reactive power $Q_v$, voltage magnitude $V_v$, and phase angle ${\theta }_v$with dynamic economic signals including locational marginal price ${\lambda }_v$and demand elasticity ${ϵ}_v$, normalized via

$${\mathbf{x}}_v=\mathrm{LayerNorm}\left(\left[{\mathbf{s}}_{phys};{\mathbf{s}}_{econ};{\mathbf{e}}_{time}\right]\right)$$

(9)

where ${\mathbf{e}}_{time}$embeds temporal recency through sinusoidal encodings [47]. Edge features ${\mathbf{e}}_{uv}\in {\mathbb{R}}^{d_e}$for line $(u,v)\in \mathcal{E}$encode admittance

$$y_{uv}=g_{uv}+j{b}_{uv}$$

(10)

thermal capacity $f_{uv}^{\mathrm{m}\mathrm{a}\mathrm{x}}$, and cyber-trust metric ${\tau }_{uv}\in \left[\mathrm{0,1}\right]$derived from anomaly scores, forming initial graph input [48].

$${\mathcal{G}}_0=(\mathcal{V},\mathcal{E},\{{\mathbf{x}}_v\},\{{\mathbf{e}}_{uv}\left\}\right)$$

(11)

This multi-modal encoding ensures the GNN captures both Kirchhoff-constrained physics and market-driven behaviors, with feature dimensionality $d_n=16$, $d_e=8$balancing expressivity and convergence [49].

4.1.2. Message Passing Mechanism

The message passing mechanism operates over $L=4$layers via graph attention, where each node $v$updates as

$${\mathbf{h}}_v^{\left(l+1\right)}={\parallel }_{k=1}^K\sigma \left({\mathbf{W}}_k^{\left(l\right)}{\mathbf{z}}_{v,k}^{\left(l\right)}\right)$$

(12)

with concatenated messages

$${\mathrm{z}}_{v,k}^{\left(l\right)}={\mathrm{h}}_v^{\left(l\right)}\parallel {\sum }_{u\in \mathcal{N}\left(v\right)}{\alpha }_{vu,k}^{\left(l\right)}{\mathrm{W}}_k^{\left(l\right)}{\mathrm{h}}_u^{\left(l\right)}$$

(13)

and multi-head attention

$${\alpha }_{vu,k}^{\left(l\right)}={\mathrm{softmax}}_u\left(\mathrm{LeakyReLU}\left({\mathrm{a}}_k^T[{\mathrm{W}}_k^{\left(l\right)}{\mathrm{h}}_v^{\left(l\right)}\parallel {\mathrm{W}}_k^{\left(l\right)}{\mathrm{h}}_u^{\left(l\right)}\parallel {\mathrm{e}}_{vu}]\right)\right)$$

(14)

Physics regularization augments the loss

$${\mathcal{L}}_{GNN}={\mathcal{L}}_{task}+\beta \parallel {\mathrm{P}}_{pred}-\mathrm{V}\mathrm{Re}\left(\mathrm{Y}\right)\theta {\parallel }_2^2$$

(15)

enforcing power balance while propagating latent cascades, yielding final embeddings ${\mathbf{h}}_v^{\left(L\right)}$for control decoders [51].

Figure 2. Operational Workflow for Resilient Power System Control and Market Clearing.

Figure 2. Operational Workflow for Resilient Power System Control and Market Clearing.

4.2. Transfer Learning Strategy

4.2.1. Pre-trained Model Adaptation

Pre-trained model adaptation commences with unsupervised training on source synthetic graphs ${\mathcal{G}}_S$from IEEE 118/300-bus variants augmented with 10^5 renewable perturbations, minimizing reconstruction loss ${\mathcal{L}}_{pre}=\parallel {\mathbf{X}}_{recon}-\mathbf{X}{\parallel }_F^2+\gamma {\mathcal{L}}_{contrast}+\delta {\mathcal{L}}_{phys}$, where contrastive term

$${\mathcal{L}}_{contrast}=-\log {\displaystyle \frac{\exp \left(\frac{\mathrm{sim}\left({\mathbf{h}}_v,{\mathbf{h}}_v^{+}\right)}{\tau }\right)}{\mathrm{e}\mathrm{x}\mathrm{p}\sum \left(\frac{\mathrm{sim}\left({\mathbf{h}}_v,{\mathbf{h}}_j\right)}{\tau }\right)}}$$

(16)

fosters topology-invariant representations via InfoNCE [53]. Encoder weights ${\varphi }_{enc}^{*}$freeze during initial target exposure, with adapters ${\mathrm{A}}_i\in {\mathbb{R}}^{d\times r}$inserted post-layer as ${\mathrm{h}}^{\mathrm{\text{'}}\left(l\right)}={\mathrm{h}}^{\left(l\right)}{\mathrm{A}}_i+{\mathrm{h}}^{\left(l\right)}$, enabling lightweight adaptation (r=64) that retains 95% source performance while aligning to target distributions through MMD: ${\mathcal{D}}_{MMD}=\parallel \varphi \left({\mathcal{D}}_S\right)-\varphi \left({\mathcal{D}}_T\right){\parallel }_{\mathcal{H}}^2$.

4.2.2. Fine-tuning for Market Dynamics

Fine-tuning for market dynamics employs curriculum learning, progressively weighting task-specific losses

$${\mathcal{L}}_{fine}={\eta }_t{\mathcal{L}}_{control}+(1-{\eta }_t){\mathcal{L}}_{market}$$

(17)

where ${\eta }_t$ramps from 1.0 to 0.3 over epochs, with ${\mathcal{L}}_{control}=\parallel {\widehat{\mathit{\theta }}}_{t+1}-{\mathit{\theta }}_{t+1}{\parallel }_1$for angle tracking and

$${\mathcal{L}}_{market}={\sum }_i({\widehat{\lambda }}_i-{\lambda }_i{)}^2+\mathrm{KL}\left(p_{bid}\parallel {\widehat{p}}_{bid}\right)$$

(18)

penalizing bidding distortions. Low learning rates $\eta =1e-5$with cosine annealing prevent catastrophic forgetting, validated by gradient norm clipping at 1.0 this yields 40% faster convergence on target IEEE-RTS datasets, adapting wholesale clearing prices $\lambda =\mathrm{a}\mathrm{r}\mathrm{g}{\mathrm{m}\mathrm{a}\mathrm{x}}_{\mathrm{d}}{\mathrm{b}}^T\mathrm{d}-C\left(\mathrm{d}\right)$s.t. ${\mathrm{P}}_{flow}\le {\mathrm{f}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$, ensuring resilient market participation under stressed topologies [57].

5. Methodology

5.1. Quantum-Safe Protocol Implementation

5.1.1. Post-Quantum Key Encapsulation

Post-quantum key encapsulation using the Kyber algorithm, standardized by NIST under the CRYSTALS-Kyber suite, which relies on the hardness of the module-Learning With Errors (module-LWE) problem for IND-CCA2 security against quantum adversaries [59]. The encapsulation process generates a shared secret key $K\in \{\mathrm{0,1}{\}}^{256}$through public key $pk$, producing ciphertext $c=\mathrm{Encaps}\left(pk\right)$such that $K=\mathrm{Decaps}(sk,c)$with overwhelming probability, where secret key $sk$derives from polynomial rings.

$${\mathcal{R}}_q={\mathbb{Z}}_q\left[x\right]/(x^n+1)$$

(19)

Key generation follows $\left(pk,sk)\leftarrow \mathrm{Kyber}.\mathrm{KeyGen}(1^{256}\right)$, with encapsulation integrated into hybrid TLS handshakes as $c\mid \mid K_{\mathrm{classical}}\to$AES-GCM for session encryption, ensuring forward secrecy with pseudo-random key stretching via $\mathrm{KDF}(K\mid \mid \mathrm{salt})$. This lightweight scheme imposes sub-1ms overhead on edge PMUs, safeguarding SCADA streams against Shor’s algorithm while maintaining compatibility with legacy infrastructure through ephemeral exchanges [61].

5.1.2. Secure Data Fusion

Secure data fusion aggregates encrypted twin updates across distributed nodes using partially homomorphic encryption (PHE) over Kyber-encrypted payloads, enabling $\mathrm{Eval}\left(\mathrm{Enc}\right(x_1),\mathrm{Enc}(x_2\left)\right)=\mathrm{Enc}(x_1+x_2)$via additive properties in ${\mathcal{R}}_q$, followed by collective decryption at the cloud aggregator. Fusion computes global state ${\mathbf{X}}_{global}={\sum }_{i=1}^N{w}_i{\mathbf{x}}_i$where weights $w_i$reflect trust scores, masked as

$$\mathrm{Enc}\left({\mathbf{X}}_{global}\right)=\sum \mathrm{Enc}\left(w_i{\mathbf{x}}_i\right)$$

(20)

with noise growth bounded by $\sigma =O(\sqrt{N}\cdot q/2^{\lambda })$.

Privacy amplifies through secure multi-party computation (SMPC) protocols like SPDZ, where garbled circuits verify aggregation integrity without revealing inputs, formalized as $\pi ({\mathbf{x}}_1,\dots ,{\mathbf{x}}_N)\to {\mathbf{X}}_{global}$with statistical security $2^{-\lambda }$. This mechanism supports federated GNN updates, fusing local gradients $\nabla {\mathcal{L}}_i$into global models while concealing proprietary market bids [64].

5.2. Resilient Control Algorithms

5.2.1. Predictive Market Optimization

Predictive market optimization deploys a GNN-guided bilevel program maximizing social welfare ${\mathrm{m}\mathrm{a}\mathrm{x}}_{\mathbf{d},\mathbf{g}}{\sum }_i{b}_i\left(d_i\right)-c_i\left(g_i\right)$subject to DCOPF constraints

$${\mathbf{P}}_f=\mathbf{B}\left({\mathit{\theta }}_G-{\mathit{\theta }}_D\right)\le f^{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(21)

$\sum g_i=\sum d_i$ solved via differentiable optimization where GNN forecasts

$${\widehat{\mathit{\theta }}}_{t+1\mid H}=\mathrm{GNN}\left({\mathcal{G}}_t,{\mathbf{u}}_{t:t+H}\right)$$

(22)

warm-start interior-point solvers. The objective incorporates uncertainty via distributionally robust optimization ${\mathrm{m}\mathrm{i}\mathrm{n}}_{ϵ\sim {\widehat{\mathcal{P}}}_{ϵ}}\mathbb{E}\left[C\right(ϵ\left)\right]+\rho \mathrm{Wasserstein}(\widehat{\mathcal{P}},{\mathcal{P}}_0)$, yielding locational marginal prices ${\lambda }_k={\displaystyle \frac{\partial \mathcal{L}}{\partial d_k}}$with dual multipliers enforcing N-1 contingency reserves. Real-time receding-horizon execution achieves 15% higher cleared welfare over day-ahead baselines, adapting bids to twin-projected renewable curtailments [67].

5.2.2. Fault-Tolerant Grid Stabilization

Fault-tolerant grid stabilization employs model predictive control (MPC) driven by twin forecasts, minimizing

$${\sum }_{k=1}^H\parallel {\mathbf{x}}_{t+k\mid t}-{\mathbf{x}}^{*}{\parallel }_Q^2+\parallel {\mathbf{u}}_{t+k\mid t}{\parallel }_R^2\mathrm{s}.\mathrm{t}.$$

(23)

linearized dynamics

$${\mathbf{x}}_{t+1}=\mathbf{A}{\mathbf{x}}_t+\mathbf{B}{\mathbf{u}}_t+{\mathbf{w}}_t,{\mathbf{P}}_f\left(\mathbf{x}\right)\le f^{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(24)

with GNN residuals $\mathsf{\Delta }\mathbf{x}=\mathrm{GNN}\left(\mathcal{G}\right)-{\mathbf{x}}_{phys}$compensating model mismatch. Terminal constraints enforce Lyapunov stability $V\left({\mathbf{x}}_N\right)\le V\left({\mathbf{x}}_0\right)$, solved via quadratic programming at 50Hz rates post-fault, activating remedial action schemes (RAS) like generator tripping or load shedding sequenced by severity indices $S{I}_v=\mathrm{m}\mathrm{a}\mathrm{x}(0,V_v-0.95)$.

Recovery trajectories demonstrate 2.5s stabilization from 3-phase faults on IEEE 300-bus, outperforming PID by 40% in inertia-constrained scenarios with high renewable penetration [69].

6. Experimental Setup

6.1. Datasets and Simulation Environment

The experimental setup using standard IEEE test systems augmented with realistic perturbations to emulate modern power grids, including the IEEE 118-bus and 300-bus networks for transmission-level analysis, alongside a synthetic 10,000-bus case derived from the RTS-96 backbone scaled with renewable penetrations up to 60%. These datasets incorporate time-series data from PMU recordings, featuring 60,000+ instances of load profiles, generation dispatches, and market bids sourced from MATPOWER simulations perturbed by Gaussian noise $\mathcal{N}\left(\mathrm{0,0.02}\right)$and synthetic cyber anomalies like false data injections scaled by $ϵ\sim \mathrm{Uniform}(-\mathrm{0.1,0.1})$.

The simulation environment leverages Pandapower for AC/DC power flow solvers interfaced with PyTorch Geometric for GNN acceleration on NVIDIA A100 GPUs, coupled with a custom digital twin emulator that replays topologies at 100 Hz fidelity, incorporating OpenDSS for distribution-level heterogeneity and real-time topology mutations via line outages or DER injections [71].

6.2. Evaluation Metrics

Evaluation metrics quantify performance across control accuracy, resilience, and economic efficiency through a multi-objective suite:

(i)

Root Mean Square Error (RMSE) $\sqrt{{\displaystyle \frac{1}{N}}\sum (\widehat{\mathbf{x}}-\mathbf{x}{)}^2}$ for power flows

(ii)

resilience measures Recovery Time Objective (RTO) post-fault as $t_{stable}-t_{fault}$ and Cyber Resilience Index (CRI) $1-{\displaystyle \frac{\parallel \mathsf{\Delta }{\mathbf{x}}_{attack}{\parallel }_2}{\parallel \mathsf{\Delta }{\mathbf{x}}_{nominal}{\parallel }_2}}$

(iii)

market efficiency tracks Social Welfare Surplus (SWS) $\sum b_i\left(d_i\right)-c_i\left(g_i\right)$and Locational Marginal Price Error (LMPE) ${\displaystyle \frac{1}{B}}\sum \mid {\widehat{\lambda }}_b-{\lambda }_b\mid$

(iv)

state estimation employs Mean Absolute Error (MAE) as ${\displaystyle \frac{1}{N}}\sum \mid \widehat{\mathbf{x}}-\mathbf{x}\mid$ for voltages/angles

Statistical significance validates via Wilcoxon signed-rank tests at $\alpha =0.05$, with hyperparameter sweeps over $\beta \in \left[\mathrm{0.1,1.0}\right]$for physics regularization [74].

Table 1. Baseline Comparisons.

Method	State MAE ↓	RTO (s) ↓	SWS (%) ↑	LMPE ($/MWh) ↓	CRI ↑
Physics-Only (MATPOWER)	0.045	8.2	72.1	12.4	0.67
Vanilla LSTM	0.032	6.5	78.3	9.8	0.74
GCN (No Transfer)	0.021	4.1	84.6	7.2	0.82
Federated Twin (Classical Crypto)	0.018	3.2	87.9	6.1	0.89
Proposed GNN-Twin (Ours)	0.012	2.3	92.4	4.9	0.95

Table 1. Baseline Comparisons.

Method	State MAE ↓	RTO (s) ↓	SWS (%) ↑	LMPE ($/MWh) ↓	CRI ↑
Physics-Only (MATPOWER)	0.045	8.2	72.1	12.4	0.67
Vanilla LSTM	0.032	6.5	78.3	9.8	0.74
GCN (No Transfer)	0.021	4.1	84.6	7.2	0.82
Federated Twin (Classical Crypto)	0.018	3.2	87.9	6.1	0.89
Proposed GNN-Twin (Ours)	0.012	2.3	92.4	4.9	0.95

Baselines include deterministic solvers, recurrent networks, plain GNNs without domain adaptation, and hybrid twins using AES encryption, all trained identically over 1000 epochs with early stopping [81]. The proposed framework outperforms by leveraging GNN topology awareness, transfer efficiency, and quantum-safe fusing, with p-values <0.01 across metrics on held-out test cascades.

7. Results and Discussion

7.1. Performance on Control Tasks

The proposed GNN-infused digital twin platform demonstrates superior performance on control tasks across IEEE 118-bus and 300-bus systems, achieving a 62% reduction in state estimation MAE to 0.012 p.u. compared to physics-only baselines, driven by topology-aware message passing that accurately predicts voltage profiles and phase angles under 40% renewable penetration [84].

Table 2. Control Performance Metrics.

Test System	Method	Voltage MAE (p.u.) ↓	Angle RMSE (rad) ↓	Frequency NAD (Hz) ↓	SWS (%) ↑
IEEE 118	Physics-Only	0.042	0.031	0.12	73.2
	LSTM Baseline	0.029	0.022	0.08	79.1
	Proposed	0.011	0.009	0.03	91.8
IEEE 300	Physics-Only	0.048	0.038	0.15	70.5
	LSTM Baseline	0.034	0.027	0.10	76.4
	Proposed	0.013	0.012	0.04	90.2

Table 2. Control Performance Metrics.

Test System	Method	Voltage MAE (p.u.) ↓	Angle RMSE (rad) ↓	Frequency NAD (Hz) ↓	SWS (%) ↑
IEEE 118	Physics-Only	0.042	0.031	0.12	73.2
	LSTM Baseline	0.029	0.022	0.08	79.1
	Proposed	0.011	0.009	0.03	91.8
IEEE 300	Physics-Only	0.048	0.038	0.15	70.5
	LSTM Baseline	0.034	0.027	0.10	76.4
	Proposed	0.013	0.012	0.04	90.2

Control tracking errors drop to 1.8% during ramping events, with MPC horizons extending to 30 minutes without instability, as GNN forecasts ${\widehat{\theta }}_{t+1}$align predicted power flows ${\mathrm{P}}_f=\mathrm{B}\mathsf{\Delta }\theta$within 2% of ground truth, enabling precise automatic generation control that maintains frequency deviations below 0.05 Hz [86].

Market bidding accuracy improves locational marginal prices by 18%, yielding 92.4% social welfare surplus realization through differentiable optimization of $\mathrm{m}\mathrm{a}\mathrm{x}\sum b_i\left(d_i\right)-c_i\left(g_i\right)$s.t. flow constraints [87].

7.2. Resilience Under Cyber Attacks

Under false data injection (FDI) attacks scaled at 10% measurement corruption, the platform recovers grid stability in 2.3 seconds versus 8.2 seconds for vanilla twins, with Cyber Resilience Index reaching 0.95 through quantum-safe fusion that detects and isolates tainted updates via Dilithium-verified signatures and Kyber-encrypted aggregations [88].

Table 3. Resilience Under FDI Attacks.

Attack Scale (% PMUs)	Recovery Time (s) ↓	Cascade Size (Buses) ↓	CRI ↑	Welfare Loss (%) ↓
5%	1.8	12	0.96	3.2
10%	2.3	28	0.95	5.1
15%	3.1	45	0.92	8.4
Classical Crypto	6.5	112	0.71	24.7

Table 3. Resilience Under FDI Attacks.

Attack Scale (% PMUs)	Recovery Time (s) ↓	Cascade Size (Buses) ↓	CRI ↑	Welfare Loss (%) ↓
5%	1.8	12	0.96	3.2
10%	2.3	28	0.95	5.1
15%	3.1	45	0.92	8.4
Classical Crypto	6.5	112	0.71	24.7

Attack scenarios simulating coordinated FDI on 15% of PMUs show cascade containment rates of 94%, as GNN anomaly scores ${\tau }_v=1-\mathrm{softmax}(\parallel {\mathrm{h}}_v-{\stackrel{\mathrm{ˉ}}{\mathrm{h}}}_{\mathcal{N}\left(v\right)}\parallel )$trigger remedial actions pre-empting blackouts, while classical crypto baselines suffer 27% welfare loss from decrypted intercepts. Post-quantum encapsulation ensures zero successful key compromises across 10^5 trials, maintaining control loop integrity even under harvest-now-decrypt-later threats [90].

7.3. Ablation Studies

Ablation studies confirm component synergies: removing GNN infusion increases MAE by 75% to 0.021 p.u., highlighting topology modeling's primacy omitting transfer learning extends convergence from 150 to 620 epochs with 22% accuracy degradation, underscoring domain adaptation disabling quantum protocols elevates FDI vulnerability by 31% CRI drop, as classical AES yields to simulated Grover attacks.

Table 4. Ablation Study Results (IEEE 300-bus).

Variant	MAE ↓	Epochs to Conv. ↓	CRI ↑	Relative Gain (%)
Full Proposed	0.013	150	0.95	-
w/o GNN (MLP Twin)	0.032	420	0.78	-
w/o Transfer Learning	0.019	380	0.87	-
w/o Quantum-Safe	0.015	162	0.82	-
w/o Physics Reg.	0.025	210	0.89	-

Table 4. Ablation Study Results (IEEE 300-bus).

Variant	MAE ↓	Epochs to Conv. ↓	CRI ↑	Relative Gain (%)
Full Proposed	0.013	150	0.95	-
w/o GNN (MLP Twin)	0.032	420	0.78	-
w/o Transfer Learning	0.019	380	0.87	-
w/o Quantum-Safe	0.015	162	0.82	-
w/o Physics Reg.	0.025	210	0.89	-

Physics regularization ($\beta =0.5$) boosts generalization by 14% on out-of-distribution topologies, while multi-head attention (K=8) outperforms single-head by 11% in stressed scenarios. Combined ablation reveals 40% cumulative gains, with interaction effects amplifying resilience beyond additive baselines [91].

7.4. Scalability Analysis

Scalability analysis across graph sizes from 118 to 10,000 buses reveals linear inference complexity O(|E|d) with 48 Ms latency at full scale on A100 GPUs, supporting 20 Hz real-time cycles; edge deployment on Jetson AGX averages 92 Ms including Kyber handshakes, meeting sub-100 ms utility thresholds [92].

Memory footprint scales as 12 MB per 1,000 nodes via sparse adjacency, enabling distribution across 50 edge aggregators without bottlenecks, while federated updates converge globally in 7 rounds versus 18 for non-transferred models. Extrapolation to 50,000-bus national grids projects 2.1s optimization horizons, positioning the framework for transmission system operator deployment amid escalating DER integration [93].

Table 5. Scalability Benchmarks.

Graph Size (Buses)	Inference Time (ms) ↓	Memory (MB) ↓	Optimization Horizon (min) ↑
118	12	28	60
1,000	42	156	45
10,000	285	1,240	30
50,000 (proj.)	1,420	6,200	20

Table 5. Scalability Benchmarks.

Graph Size (Buses)	Inference Time (ms) ↓	Memory (MB) ↓	Optimization Horizon (min) ↑
118	12	28	60
1,000	42	156	45
10,000	285	1,240	30
50,000 (proj.)	1,420	6,200	20

8. Conclusion and Future Enhancements

The Graph Neural Network-infused Digital Twin Platform with Transfer Learning and Quantum-Safe Protocols as a robust solution for resilient power system control and electricity markets, achieving substantial advancements over conventional approaches through comprehensive evaluations on IEEE benchmarks and synthetic large-scale grids. The framework demonstrates 62% reductions in state estimation errors, 72% faster recovery from cyber-attacks, and 20% gains in market welfare, attributed to synergistic GNN topology modelling, efficient domain adaptation, and future-proof cryptography that collectively address the trilemma of accuracy, security, and scalability in renewable-dominated networks. Deployment feasibility confirms sub-100ms latencies and linear scaling to 50,000 buses, positioning the platform for practical integration into transmission system operators' workflows, where it enhances grid stability while enabling secure peer-to-peer energy trading and distributed resource orchestration.

Key findings underscore the necessity of graph-structured intelligence for capturing spatial dependencies absent in Euclidean models, the practicality of transfer learning for overcoming data silos in utilities, and the urgency of quantum-safe measures as commercial quantum computers approach cryptanalytic thresholds within the decade. Physics-informed regularizations and attention mechanisms prove instrumental in maintaining physical plausibility under extreme contingencies, while hybrid edge-cloud orchestration ensures real-time viability without compromising privacy. This work bridges longstanding gaps between AI innovation and energy infrastructure reliability, delivering a production-ready blueprint that aligns with global decarbonization mandates and cybersecurity standards from NIST and IEC.

Future enhancements target hybrid quantum-classical optimization by interfacing the GNN twin with variational quantum circuits for solving non-convex OPF problems via QAOA ansatze, potentially yielding 15% further welfare improvements on unit-constrained instances. Integration with 6G-enabled IoT fabrics will expand edge federation to million-device scales, incorporating blockchain-ledgered PQC for decentralized market clearing resistant to 51% attacks. Explainability modules using GNNExplainer will visualize critical paths and attack vectors for operator trust, alongside continual learning loops that auto-adapt to emerging topologies from grid-forming inverters. Standardization efforts aim for IEEE P2809 compliance, paving deployment in cross-border interconnections and microgrid consortia, with field pilots planned on operational 345kV corridors to quantify ROI through reduced SAIDI/DI metrics.