Source-linked AI summary
Dynamic State Estimation for Multi-Machine Power System by Unscented Kalman Filter with Enhanced Numerical Stability
Junjian Qi, Kai Sun, Jianhui Wang, Hui Liu
TL;DR
Power-system dynamic state estimation requires filters that remain numerically stable as system size and nonlinearity increase. The paper proposes UKF-GPS and compares it with five UKF variants and EKF on WSCC and NPCC systems. All methods work well on WSCC, whereas UKF-modified and SR-UKF work very well across NPCC estimations, with UKF-GPS working well in almost all cases.
Problem
Classic UKF can lose covariance positive semidefiniteness during propagation, preventing sigma-point generation and limiting performance on large power systems.
Method
The paper proposes UKF-GPS and compares it with UKF-schol, UKF-κ, UKF-modified, UKF-ΔQ, SR-UKF, and EKF on WSCC and NPCC dynamic-state-estimation benchmarks.
Results
All methods work well on WSCC; on NPCC, UKF-GPS works well for almost all estimations, while UKF-modified and SR-UKF work very well for all estimations.
Takeaways & Limitations
UKF-modified and SR-UKF show better scalability mainly because of enhanced numerical stability, while UKF-schol, UKF-κ, and UKF-ΔQ can produce large errors.
Abstract
from arXiv · showhide
In this paper, in order to enhance the numerical stability of the unscented Kalman filter (UKF) used for power system dynamic state estimation, a new UKF with guaranteed positive semidifinite estimation error covariance (UKF-GPS) is proposed and compared with five existing approaches, including UKF-schol, UKF-$κ$, UKF-modified, UKF-$ΔQ$, and the square-root unscented Kalman filter (SR-UKF). These methods and the extended Kalman filter (EKF) are tested by performing dynamic state estimation on WSCC 3-machine 9-bus system and NPCC 48-machine 140-bus system. For WSCC system, all methods obtain good estimates. However, for NPCC system, both EKF and the classic UKF fail. It is found that UKF-schol, UKF-$κ$, and UKF-$ΔQ$ do not work well in some estimations while UKF-GPS works well in most cases. UKF-modified and SR-UKF can always work well, indicating their better scalability mainly due to the enhanced numerical stability.
NOMENCLATURE
This section defines notation for Kalman filtering, power-system dynamic state estimation, and the numerical-stability problem motivating enhanced UKF methods.
- Context: The paper studies dynamic state estimation using Kalman-filter methods for nonlinear power-system models and synchronized PMU measurements.Dynamic estimation becomes important as renewable generation increases system uncertainty and variation.
- Existing filters: EKF linearizes nonlinear transformations with Jacobian matrices, but its first-order approximation is reliable mainly in mildly nonlinear environments.The paper notes that EKF can be suboptimal, diverge, and require difficult Jacobian calculations.
- Existing filters: UKF avoids linearization and Jacobian calculations by transforming sigma points through nonlinear state-transition and measurement functions.The paper nevertheless identifies dimensionality and numerical-stability concerns for larger power systems.
- Numerical stability: The estimation-error covariance must remain positive semidefinite because its matrix square root is required to generate UKF sigma points.Loss of positive semidefiniteness can prevent the square-root calculation and limit UKF scalability.
- Paper scope: The paper compares six techniques for enhancing UKF numerical stability, including UKF-GPS and SR-UKF, on WSCC and NPCC power-system benchmarks.The methods are evaluated alongside EKF for dynamic state estimation.
II. UNSCENTED KALMAN FILTER
The UKF represents uncertainty with sigma points, propagates them through nonlinear system and measurement functions, and updates the state recursively. Its covariance can lose positive semidefiniteness in high-dimensional settings, motivating stability enhancements.
- System model: A discrete-time nonlinear system is represented by state variables, inputs, measurements, nonlinear transition and measurement functions, and Gaussian process and measurement noise.The estimated state mean and error covariance are updated recursively.
- Unscented transformation: The unscented transformation selects sigma points matching the mean and covariance, propagates them through a nonlinear function, and recomputes transformed statistics.This avoids directly approximating the nonlinear function with a linearization.
- Unscented transformation: The UKF uses 2n+1 sigma points whose spread and weights depend on scaling parameters α, β, and κ.The matrix square root of the covariance supplies the sigma-point directions.
- UKF procedure: UKF prediction propagates sigma points through the dynamic model, estimates the predicted state and covariance, then propagates predicted points through the measurement function.The update estimates innovation and cross-covariances, calculates the Kalman gain, and updates the state and covariance.
- Numerical stability: For high-dimensional systems, negative covariance weights can make the calculated covariance non-positive semidefinite.With α = 1 and β = 0, the center covariance weight is 1−n/3, which is negative when n > 3.
A. EKF/UKF Toolbox Approach
The UKF-schol approach lets the UKF continue when the covariance is not positive semidefinite by using an output from a modified Cholesky-factor routine, but that output is inaccurate.
- UKF-schol: When the covariance is not positive semidefinite, the schol routine can still return a lower-triangular matrix for sigma-point generation.This allows UKF estimation to continue despite the covariance-factorization problem.
- UKF-schol: UKF-schol does not solve the covariance problem because its returned factor does not satisfy P = SS⊤ when P is not positive semidefinite.The method therefore continues computation with an inaccurate Cholesky factor.
B. Selection of κ
The paper compares κ-based and modified UKF strategies for preserving covariance positive semidefiniteness, while noting trade-offs involving sigma-point locality and added process covariance.
- Selection of κ: With κ = 0, sigma-point radius grows with √n, potentially sampling nonlocal effects in strongly nonlinear systems.Thus, numerical stability is improved at the cost of a possible nonlocal-sampling problem.
- Modified UKF: The modified UKF evaluates predicted and innovation covariances about the projected mean while retaining the heuristic n + κ = 3.The modified form is reported to ensure positive semidefiniteness.
- UKF-ΔQ: UKF-ΔQ adds a positive definite ΔQ_k to the predicted covariance to improve stability, but estimation precision can decrease.The cited approaches do not provide a general design method for ΔQ_k, or use a heuristic one.
E. UKF-GPS
UKF-GPS restores a usable positive semidefinite estimation error covariance when covariance propagation loses that property, using a nearest positive semidefinite matrix procedure. The section also describes the square-root propagation and update operations used by SR-UKF.
- UKF-GPS: UKF-GPS applies the nearPD algorithm when the estimation error covariance is nonpositive semidefinite.The algorithm finds a symmetric positive semidefinite matrix nearest to the input covariance in Frobenius norm.
- UKF-GPS: The nearPD procedure converts P_k−1 or P_k− into an output covariance that guarantees positive semidefiniteness.It uses modified alternating projections and procedures that force positive definiteness and symmetry.
- nearPD implementation: The nearPD algorithm iteratively projects matrices onto the symmetric positive semidefinite matrix set using a Dykstra correction.The correction is incorporated into each projection in the modified alternating projections method.
- SR-UKF: SR-UKF directly propagates the covariance square root S, avoiding covariance refactorization at every step.Its prediction and update procedures use QR decomposition, Cholesky factor updates, and sequential Cholesky downdates.
G. Summary and Discussion
The compared filters differ in how they preserve covariance positive semidefiniteness, implementation complexity, and computational efficiency. UKF-modified and SR-UKF provide the strongest reported combination of numerical robustness and practical performance, while UKF-GPS offers covariance correction with possible precision loss.
- Covariance-stability comparison: UKF-schol does not solve non-positive-semidefinite covariance and only produces an inaccurate Cholesky factor in that case.Its numerical workaround allows computation to continue but does not restore the covariance property.
- Covariance-stability comparison: UKF-κ guarantees covariance positive semidefiniteness but discards the heuristic n + κ = 3 for n > 3 and introduces nonlocal sampling.UKF-modified also guarantees positive semidefiniteness and coincides with a modified truncated second-order filter under some conditions.
- Covariance-stability comparison: UKF-∆Q depends on selecting an appropriate extra positive definite matrix, and excessive or insufficient enlargement can respectively reduce precision or leave covariance non-positive semidefinite.The cited heuristic does not work when the estimation error covariance is non-positive semidefinite.
- Covariance-stability comparison: UKF-GPS replaces a covariance that loses positive semidefiniteness with the nearest positive semidefinite matrix, but the conversion can reduce precision when the replacement is not sufficiently close.SR-UKF instead guarantees positive semidefiniteness intrinsically by propagating the covariance square root.
- Implementation and efficiency: SR-UKF requires more extensive changes than the other UKF variants but can be more efficient through QR decomposition and Cholesky factor updating.UKF-κ and UKF-∆Q are easier to implement, whereas UKF-GPS adds the nearPD algorithm.
IV. POWER SYSTEM DYNAMIC STATE ESTIMATION
The dynamic state-estimation model represents multi-machine generators in state-space form and uses PMU voltage and current phasors as outputs. The study assumes mechanical torque and excitation voltage are constant and known to focus on UKF numerical stability.
- System model: The multi-machine measurement model supports fourth-order transient and second-order classical generator models, with terminal voltage and current phasors from PMUs as outputs.Generators with fourth-order models include four states, while second-order models use the first two equations.
- System model: The number of system states is n = 4g4 + 2g2, where g4 and g2 count fourth-order and second-order generators.The state count follows directly from the model order assigned to each generator set.
- Measurements and inputs: For generators with PMUs, terminal voltage and current phasors are measured, while mechanical torque and excitation voltage serve as inputs.The measured phasors are represented through their real and imaginary components.
- State-space formulation: The generator dynamics are rewritten in general state-space form with state, input, and output vectors, then discretized for dynamic state estimation.The discrete state transition uses a modified Euler method.
- Model assumption: The analysis assumes Tm and Efd are constant and known because its main objective is enhancing UKF numerical stability.Dynamic state estimation with unknown inputs is deferred to prior and future work.
V. SIMULATION RESULTS
The experiments test UKF-GPS and SR-UKF on WSCC and NPCC benchmark systems using implementations based on the EKF/UKF toolbox. The tests run on a 3.2-GHz Intel Core i7-4790S desktop.
- Test systems: UKF-GPS and SR-UKF are tested on the WSCC 3-machine 9-bus and NPCC 48-machine 140-bus systems.The benchmark systems are extracted from the Power System Toolbox.
- Implementation: The EKF and classic UKF use the EKF/UKF toolbox, while UKF-GPS and SR-UKF are implemented based on that toolbox.All tests are performed on the same desktop platform.
A. Settings
The study evaluates dynamic state estimation under simulated postcontingency conditions using specified noise, PMU-placement, initialization, and covariance settings. Estimation performance is quantitatively compared with a defined state-estimation error index.
- Measurement configuration: PMU placement uses one terminal-bus PMU for WSCC and 24 terminal-bus PMUs for NPCC, selected by maximizing the empirical observability Gramian determinant.
- Filter settings: Dynamic state estimation is performed over [0, 10 s] after fault clearing, initialized with the pre-contingency state.
- Filter settings: All methods use α = 1 and β = 0; UKF-κ uses κ = 0, while other methods use κ = 3 −n.
- Noise and covariance settings: Process and measurement noise covariances are diagonal, with measurement-noise diagonal entries 0.012; UKF-∆Q uses ∆Q = 0.0052In.
- Evaluation metric: Estimation results are quantitatively compared using a defined system state estimation error index based on estimated and corresponding true states over time.
B. WSCC 3-Machine System
On the six-state WSCC 3-machine system, all tested Kalman-filter methods produce good estimates without obvious numerical stability problems. Across twelve fault scenarios, the methods have small average estimation errors and standard deviations.
- WSCC 3-Machine System: All tested methods work well on the six-state WSCC 3-machine system.The system has no obvious numerical stability problem during estimation.
- WSCC 3-Machine System: The estimation error covariance can maintain positive semidefiniteness during propagation in this case.This observation is reported for the WSCC setting with κ = −3.
- WSCC 3-Machine System: Twelve fault scenarios are evaluated, with average state-estimation errors and standard deviations reported for each method.The scenarios apply three-phase faults to either of two buses on six eligible branches.
- WSCC 3-Machine System: All methods achieve small average estimation errors and standard deviations on the WSCC system.
VI. CONCLUSION
The paper compares six approaches for improving UKF numerical stability and scalability in dynamic state estimation. Results show that UKF-GPS generally performs well, while UKF-modified and SR-UKF work very well across estimations.
- Six approaches, including the proposed UKF-GPS, are introduced and compared to enhance UKF numerical stability and scalability.
- The approaches are tested on WSCC 3-machine and NPCC 48-machine power systems alongside the EKF.
- For the WSCC system, classic UKF has no numerical stability problem and all methods work well.
- For the NPCC system, EKF cannot converge and classic UKF encounters numerical stability problems.
- UKF-schol, UKF-κ, and UKF-∆Q can produce large errors, whereas UKF-GPS works well for almost all estimations.
- UKF-modified and SR-UKF work very well for all estimations, reflecting better numerical stability and scalability.