Graph Neural Ordinary Differential Equations for Power System Identification with Transfer Learning Applications

The gist The proposed MPG-NODE framework develops a novel control-oriented message passing graph neural ordinary differential equation for power system identification that allows for heterogeneous node dynamics and edge couplings through learned embeddings.

Motivation and Problem Context

The increasing complexity of power systems due to decentralized energy generation makes physics-based modeling challenging, while the growing availability of measurement data enables data-driven system identification <ref:2604.24210#pg2>. Traditional methods require synchronized voltage phase angle measurements, which can be hard to obtain in practice <ref:2604.24210#pg2>. Graph neural networks (GNNs) leverage power systems’ inherent network structure, but many GNN approaches are discrete time forcasting architectures that lack control inputs <ref:2604.24210#pg2>. This work combines the continuous-time formulation of NODEs and the structural inductive bias from GNNs to propose MPG-NODEs for power system identification <ref:2604.24210#pg2>.

MPG-NODE Architecture

The MPG-NODE is formulated as a graph where node dynamics and edge couplings are represented by neural networks shared across the graph topology <ref:2604.24210#pg4>. Specifically, the MPG-NODE reads x˙(i)(t) = fθf(x(i)(t), cθc(ξ(i)), m(i)(t), θ(i)node), where mθm is a message passing function <ref:2604.24210#pg5>. This formulation extends vanilla NODEs by incorporating augmentation, data control, and an output function <ref:2604.24210#pg6>. The node dynamics fθf and the message passing function mθm use fully connected MLPs <ref:2604.24210#pg6>.

Key Model Components

The MPG-NODE incorporates several novel concepts to handle system complexities <ref:2604.24210#pg6>:

1. Augmentation involves inferring a (partial) latent initial state representation with hθh from past measurements and control inputs contained in the input features ξ ∈ R nξ <ref:2604.24210#pg6>.

2. Data control conditions the dynamics on input features through an embedding cθc(ξ), which is assumed to be a non-learnable selector of the current control input, cθc(ξ) = πu(ξ) = u(i)(ts), i.e., a non-learnable selector of the current control input at node i <ref:2604.24210#pg6>.

3. Heterogeneity is accounted for by using learnable node embeddings θ(i)node to condition the dynamics function fθf, which allows for different dynamical properties of nodes <ref:2604.24210#pg6>.

4. Edge couplings are handled by edge embeddings θ(ij)edge that condition the message passing function mθm, allowing for different line properties such as conductances and susceptances <ref:2604.24210#pg6>.

Model Training and Application

The model is trained on a dataset D = 1 VN samples with the loss function L = 1/VNK Σ i Σ n Σ k y(i)(tsn+k) − yˆ(i)(tsn+k)2 squared <ref:2604.24210#pg6>. The training employs minibatch gradient descent with the Adam optimizer <ref:2604.24210#pg6>. The case study applies the MPG-NODE to identify voltage and frequency dynamics of power systems, specifically using the IEEE 9-bus system <ref:2604.24210#pg6>.

Results and Transfer Learning

The comparison against a monolith NODE baseline on the evaluation dataset D4 shows that while the MPG-NODE does not reach the same level of accuracy as a monolith NODE, it captures system dynamics reasonably well <ref:2604.24210#pg8>. A key advantage is its flexibility in topology changes, where nodes and edges can be added or removed without retraining an entire model <ref:2604.24210#pg8>. The transfer learning experiment demonstrated that the MPG-NODE could be successfully adapted to a system with an added node and edge with only a fraction of the original training data and effort <ref:2604.24210#pg8>. Furthermore, retrained models were obtained after less than 100 epochs compared to over 1000 epochs for the original model <ref:2604.24210#pg8>.

Conclusion

In this study, the MPG-NODE framework develops a novel graph NODE for heterogeneous node dynamics and edge couplings to identify the coupled dynamics of power systems. The model was shown to offer a much more flexible structure that allows for easy adaptation to changes in the system topology. In future work, the authors plan to investigate potential improvements in prediction accuracy by employing semilocal encoders and physics-inspired regularization in edge couplings. The MPG-NODE could be successfully adapted to a system with an added node and edge with only a fraction of the original training data and effort <ref:2604.24210#pg8>.

How it works

The MPG-NODE is formulated as x˙(i)(t) = fθf(x(i)(t), cθc(ξ(i)), m(i)(t), θ(i)node), where mθm is a message passing function <ref:2604.24210#pg6>. The message passing function is defined as m˙m = Pj∈Ni mθm(x˙(i)(t), x(j)(t), θijedge) <ref:2604.24210#pg6>. Initial state estimation is achieved via the initial state encoder hθh, which uses TCNs to encode a sequence through multiple residual blocks <ref:2604.24210#pg6>. The initial state is obtained by applying a projection head, i.e., a linear layer, to obtain the latent initial state <ref:2604.24210#pg6>.

Results and Transfer Learning

The comparison against a monolith NODE baseline on the evaluation dataset D4 shows that while the MPG-NODE does not reach the same level of accuracy as a monolith NODE, it captures system dynamics reasonably well <ref:2604.24210#pg8>. A key advantage is its flexibility in topology changes, where nodes and edges can be added or removed without retraining an entire model <ref:2604.24210#pg8>.

1. A novel control-oriented message passing graph neural ordinary differential equation (MPG-NODE) for coupled systems with heterogeneous node dynamics and edge couplings can be developed by using learned node and edge embeddings to account for heterogeneity as well as an autoregressive scheme to allow for piecewise constant control inputs. This improved system can identify the voltage and frequency dynamics of power systems by inferring latent phase angle representations from local historic observations, thereby avoiding the need for synchronized phasor measurements.

2. The MPG-NODE can be enhanced with a more robust initial state estimation mechanism by using an observer-inspired approach where the augmented state is defined as x(i)(ts) = [x˜(i)(ts), h˜θh(ξ(i)), u(i)(ts−H) · · · u(i)(ts−1) y(i)(ts−H+1) · · · y(i)(ts)] to "infer latent representations of their true initial values with a function of the form hθh(x).

3. The system can be adapted for topology changes efficiently, as the MPG-NODE allows for flexibility by stating that the topology of the model can be easily modified and requiring only updating the adjacency information, which is less demanding than retraining an entirely new model.