Dynamic State Estimation (DSE)

• Dynamic State Estimation (DSE) is a technique that fuses dynamic models with high-rate measurements to track internal states such as rotor angles and generator fluxes in power systems.
• It utilizes a range of filtering and observer-based methods—including variants of the Kalman filter, LMI-based observers, and deep learning models—to enhance estimation accuracy and robustness.
• DSE underpins critical grid functions by improving real-time monitoring, fault detection, stability control, and parameter identification in increasingly complex and cyber-resilient power networks.

Dynamic State Estimation (DSE) is a foundational methodology in the real-time monitoring, control, and protection of modern power systems. DSE extends conventional static state estimation by incorporating the dynamic behavior of generators, loads, and inverter-based resources, enabling the accurate reconstruction and tracking of internal system states—such as rotor angles, generator fluxes, and converter states—at temporal resolutions commensurate with system dynamics. In contrast to snapshot-based static estimation, DSE fuses mathematical models with high-rate synchrophasor or sampled-value measurements, providing the temporal visibility required for advanced applications in stability control, fault detection, cyber-resilience, and parameter identification.

1. Formal Models and Problem Formulation

Power system DSE is commonly formulated in either discrete or continuous-time state-space form, often arising from the time discretization of underlying differential-algebraic equations (DAEs). The generic stochastic nonlinear model is:

$$x_{k+1} = f(x_k, u_k, p) + w_k, \quad w_k \sim \mathcal{N}(0, Q_k)$$

$$z_k = h(x_k, u_k, p) + v_k, \quad v_k \sim \mathcal{N}(0, R_k)$$

where $x_k$ denotes the dynamic state vector (e.g., generator rotor angles $\delta$, rotor speeds $\omega$, excitation states), $u_k$ the known system inputs (mechanical torque, excitation voltage, reference signals), $p$ the parameter set (inertia, reactances, controller time constants), and $w_k, v_k$ are process and measurement noise, typically Gaussian white noise. For systems with algebraic constraints, the continuous-time DAE system is:

$\frac{dx(t)}{dt} = f(x(t), y(t), u(t), p)$

$0 = g(x(t), y(t), u(t), p)$

Here $y$ denotes algebraic variables (e.g., bus voltages, currents). The measurement map $h(x_k, u_k, p)$ includes both phasor and instantaneous sampled values—integral to advanced DSE implementations in protective relaying (Liu et al., 2020, Zhao et al., 2020).

Noise models may be extended to non-Gaussian, multimodal, or heavy-tailed distributions to represent practical data imperfection, notably in cyber-contingency analysis (Sarfi et al., 2020, Li et al., 2019).

2. Algorithms and Computational Frameworks

DSE is solved via a variety of recursive filtering and observer-based techniques, each offering distinct strengths in estimation accuracy, robustness, and computational efficiency.

A. Kalman Filter Family

• Extended Kalman Filter (EKF): Linearizes $f$ and $h$ about the current estimate; widely used due to moderate complexity but sensitive to strong nonlinearities (Akhlaghi et al., 2017, Liu et al., 2020).
• Unscented Kalman Filter (UKF): Propagates sigma points to capture mean and covariance exactly up to second order; exhibits lower mean-squared-error than EKF in high-noise scenarios (Jamalinia et al., 2024, Liu et al., 2020).
• Ensemble Kalman Filter (EnKF): Particle-based approach suited for non-Gaussian noise or model uncertainties; robust to outliers and multimodal measurement errors (Sarfi et al., 2020).
• Cubature Kalman Filter (CKF) / Robust CKF (RCKF): Third-order filters with robustification via M-estimation for cyber-attack resilience (Li et al., 2019).

B. Observer-Based Approaches

• $\mathcal{L}_\infty$ and $\mathcal{H}_\infty$ Observers: Synthesize dynamic estimators using Lyapunov or dissipativity arguments, providing hard error bounds and performance guarantees under unknown inputs, model errors, and measurement tampering (Nugroho et al., 2019, Nugroho et al., 2022).
• DAE Observers: Jointly recover dynamic and algebraic states for multi-machine systems with real-time PMU measurements, ensuring detectability and impulse observability (Nugroho et al., 2022).
• Deterministic Observers via Linear Matrix Inequalities (LMI): Tolerant to unknown inputs and adversarial attacks, offering formal convergence proofs (Taha et al., 2015).

C. Data-Driven and Hybrid Learning Methods

• Sparse Regression + UKF: Identifies nonlinear models via LASSO or thresholded least squares from empirical data, followed by UKF-based estimation for systems lacking full physics-based models (notably for converter-dominated PV systems) (Jamalinia et al., 2024).
• Deep Generative Model-Aided DSE: Utilizes variational autoencoders, GANs, robust encoders, and latent diffusion models to estimate states and unknown control inputs, robust to PMU data anomalies and communication loss (Pei et al., 6 Jan 2025).
• Neuro-DSE and Neural ODE-KalmanNet: Incorporates neural-ODE elements into the state transition of Kalman-type filters to address unmodeled or uncertain subsystems (Feng et al., 2022).

D. Optimization-Based Methods

• Moving Horizon Estimation (MHE): Solves an offline or sliding-window optimization to account for multi-step data, constraints, and mixed noise (Zhao et al., 2020).
• First-Order Prediction-Correction (FOPC): Efficient time-varying least-squares tracking for dynamic distribution systems, scaling to large networks with thousands of states (Song et al., 2019).

3. Architectures: Centralized, Distributed, Decentralized

With the increasing deployment of PMUs and significant grid partitioning (e.g., microgrids, distribution networks), DSE architectures have diversified:

Architecture	Core Strategy	Papers
Centralized	Global data fusion and joint estimation using all measurements	(Abukhousa et al., 4 Aug 2025, Liu et al., 2020)
Distributed	Local areas estimate local states, share overlapping boundary estimates	(Nguyen et al., 2021, Kandivalasa et al., 30 Oct 2025)
Decentralized	Each device/area runs independent estimation with local measurements, fusion optional	(Sarfi et al., 2020)

Centralized DSE enables fine-grained tracking and coordinated protection but entails heavy computation and communication (Abukhousa et al., 4 Aug 2025). Distributed and decentralized schemes exploit partitioned state-space models and local Kalman filtering, sharing only minimal information at boundaries, thus enhancing scalability and resilience (Nguyen et al., 2021). Linear-time graph-based observability analysis supports compositional testing and partitioning in large systems (Kandivalasa et al., 30 Oct 2025).

4. Robustness: Uncertainties, Noise, and Cyber-Physical Threats

DSE is inherently exposed to modeling errors, noisy or corrupted measurement streams, and deliberate cyber-attacks. Recent developments include:

• Heavy-Tailed and Multimodal Noise: Ensemble and robust filters are preferred in settings with gross outlier contamination or bimodal noise due to data corruption or communication faults (Sarfi et al., 2020).
• Cyber-Attacks: False Data Injection and Denial-of-Service attacks are explicitly modeled in DSE frameworks, and robust or observer-based estimators are shown to maintain bounded estimation error under adversarial conditions (Li et al., 2019, Taha et al., 2015).
• Structured Hypothesis Testing: Centralized DSE algorithms now incorporate hypothesis-testing to discriminate between cyber-induced and true physical faults, enhancing selectivity and preventing misoperation of protective devices (Abukhousa et al., 4 Aug 2025).
• Joint Unknown Input and State Estimation: DSE algorithms are increasingly required to estimate both state variables and unknown input vectors (e.g., mechanical torque and field voltage), with methods based on joint Kalman filtering, adaptive observers, or generative models (Pei et al., 6 Jan 2025, Nguyen et al., 2021, Lorenz-Meyer et al., 2022).

5. Practical Applications in Control, Protection, and Operation

DSE underpins a spectrum of advanced grid functions:

• Stability Control: Wide-area rotor-angle estimation for centralized or distributed damping control, frequency regulation, and model predictive control (includes Koopman-MPC and reinforcement-learning-based strategies) (Liu et al., 2020).
• Protective Relaying: DSE enables setting-less protection for both lines and load buses in microgrids with inverter-dominated sources, detecting faults via model-mismatch residuals even in the absence of high fault current (Barnes et al., 2022, Barnes et al., 2022, Barnes et al., 2021). DSE-based protection is extended to dynamic load models (e.g., induction motors), outperforming traditional overcurrent relays under weak-grid conditions (Barnes et al., 2022).
• State and Parameter Identification: Joint estimation of system states and model parameters (e.g., inertia, reactances) using DREM-based or recursive EM strategies, validated on real-world PMU data (Lorenz-Meyer et al., 2022).
• Integration of Multi-Energy Systems: DSE frameworks are coupled across gas and electric networks, enabling joint estimation of pressures, mass flow, and voltages, accommodating boundary constraints and slow-varying states via exponential smoothing (Chen et al., 2021).
• Dynamic Distribution System State Estimation (DDSE): Fast gradient-based estimation on time-varying AC-linear surrogate models, robust to measurement outliers, is proposed for real-time operation and situational awareness in distribution grids (Song et al., 2019).
• Observability Analysis: Fast graph-based algorithms enable real-time, scalable certification of centralized and partitioned DSE architectures—critical for practical deployment given the size and topological variability of modern grids (Kandivalasa et al., 30 Oct 2025).

6. Implementation Considerations and Future Directions

DSE deployment necessitates consideration of sampling rates, data quality, observability, and computational cost:

• Measurement Infrastructure: High-rate PMUs and sampled-value devices are critical for capturing the fast dynamics of both conventional and converter-based resources (Zhao et al., 2020).
• Sampling and Real-Time Constraints: Typical DSE filters operate at 10–1,000 Hz; specialized Gauss–Newton solvers in protection applications achieve convergence within 1–10 ms per window (Barnes et al., 2022).
• Observability: Graph-theoretic observability criteria enable system-level and distributed partition analysis for efficient estimator synthesis (Kandivalasa et al., 30 Oct 2025).
• Integration with Cyber-Secure and AI-Driven Tools: Recommendations include incorporating encryption, authentication, attack-detection modules, and leveraging reinforcement learning or deep generative models for DSE enhancement (Liu et al., 2020, Pei et al., 6 Jan 2025).
• Transition from SSE to DSE: Best practices involve phased-in deployment, operator dashboard development, and standards evolution (IEC 61850, IEEE C37.x) to accommodate DSE in EMS/SCADA environments (Zhao et al., 2020).

DSE will continue to evolve as the backbone of real-time monitoring, robust protection, and adaptive control in increasingly complex, converter-dominated, and cyber-vulnerable power systems.