State Estimation in Partially Observable Power Systems via Graph Signal Processing Tools
Abstract
:1. Introduction
- It is demonstrated that the voltages in the power system can be represented as smooth graph signals, where the graph Laplacian is the admittance matrix. This result can serve as a foundation for developing new GSP tools for other power systems applications in future research.
- New state estimation methods for PSSE in both DC-PF and AC-PF models are developed. These methods use the graph smoothness of the states, and do not require the full observability of the network. While regularization using the Laplacian quadratic form has been applied in various applications, such as image processing [44,45], principal component analysis (PCA) [46], data classification [47,48], and semisupervised learning on graphs [49,50], it has not been conducted before in the context of unobservable power systems. Additionally, the nonlinear measurement equations in the AC-PF model present a new challenge from a GSP perspective, requiring the incorporation of graphical information in the form of Laplacian regularization into the iterative method. As such, these state estimation methods contribute to the expansion of the GSP toolbox for a wide range of applications.
- A new approach for sensor placement is introduced to optimize the estimation performance. As the mean squared error (MSE) of the estimator depends on the unknown state vector, the minimization of the Cramr–Rao bound (CRB) is utilized instead. This results in a novel approach that can potentially be applied to other applications in the future.
- Numerical simulations on the IEEE 118-bus system are used to validate the merit of the new estimators and the new sensor placement method under different setups, compared to existing pseudo-measurement and matrix completion techniques.
2. Background and Model
2.1. Background: GSP Framework
2.2. DC-PF Model: State Estimation and Observability
- is the active power vector.
- is the system state vector, where is the voltage angle at bus n. In low-observability systems, it is more convenient to delay the assignment of the reference angle (p. 165 in [2]). Thus, includes the angle of the reference bus.
- is zero-mean Gaussian noise with covariance .
- is the measurements matrix, which is determined by the topology of the network, the susceptance of the transmission lines, and the meter locations [6]. In particular, the N rows of associated with the meters on the buses that measure the total power flow of the transmission lines connected to these buses together create the nodal admittance matrix (e.g., see p. 97 in [2]) with the following -th element:
3. GSP Properties of the States
3.1. Theoretical Validation—Output of a Low-Pass Graph Filter
3.2. Experimental Validation in IEEE Systems
4. GSP-WLS Estimator in DC-PF Model
4.1. GSP-WLS Estimator for the Partial Measurement Model
4.2. Remarks
- (1)
- (2)
- Large : At the other extreme, for , the coefficient matrix from (27) satisfies . Thus, in this case, the GSP-WLS estimator from (26) satisfies . This zero estimator can be interpreted as the a priori state estimator, which does not use the observations. Thus, taking too large a value of is unhelpful.
- (3)
- Relation with the pseudo-measurement WLS (pm-WLS) estimator: The pm-WLS estimator for systems that are not fully observable is based on generating pseudo-measurements of typical power injection/consumption values from historical data [1,24]. In this case, the received measurements are processed together with a priori estimated (predicted) states (without the reference bus), , which are assumed to have the error covariance matrix, . The pm-WLS estimator is the maximum a posteriori state estimator [11]:It can be seen that if and , then and the pm-WLS estimator in (29) coincides with the GSP-WLS estimator in (26). Therefore, the proposed GSP-WLS estimator can be interpreted as a special case of the pm-WLS estimator, where the GSP theory provides a mathematical strategy to determine the pseudo-data information. Moreover, in general, the GSP-WLS estimator only requires setting a single scalar parameter, , compared with the pm-WLS estimator, which requires setting both and .
4.3. Estimation of Missing Power Measurements
4.4. Detection of Bad Data in Unobservable Systems
- Largest normalized residual test (LNR) [1]:
- test with [1]:
- The GFT-based detector from [7] that was developed for the detection of false data injection (FDI) attacks. The GFT-based detection scheme calculates the GFT of an estimated grid state, , and filters the graph’s high-frequency components. By comparing the maximum norm of this outcome with a threshold, it can detect the presence of FDI attacks.
4.5. Optimization of the Sampling Policy
Algorithm 1 Greedy selection of the measured buses |
Input: (1) Laplacian matrix, , and noise covariance matrix, (2) Number of buses with sensors, q (3) Regularization parameter, Output: Subset of q buses,
|
5. Extension to the AC-PF Model
5.1. Model, State Estimation, and Observability
- is the measurement vector that includes the active and reactive branch power flows and power injections.
- is the measurement function, which is determined by the sensor types and their locations in the network.
- , is the state vector here, where bus 1 is the reference bus, and thus, and is known (e.g., see Chapter 4 in [1]).
5.2. GSP-Based Gauss–Newton Algorithm
Algorithm 2: Regularized Gauss–Newton (GSP-WLS) |
Input: (1) Laplacian matrix, , and noise covariance matrix, (2) Tuning parameters: , , and number of iterations, l (3) Measurement vector, , and the function, Output: State estimator, |
5.3. Remarks
6. Results
6.1. Simulations Platform and Parameters
- (i)
- Random bus selection policy (rand.)—the measured buses are randomly chosen independently from , where for more than 72 buses only observable systems are taken.
- (ii)
- Experimentally designed sampling (E-design) [38]—the buses are chosen to maximize the smallest singular value of the matrix , where R is set to 48. The basic assumption behind this method (which was suggested in [66] for power systems) is that the measured graph signal (here, the power signal) is an R-bandlimited signal in the graph frequency domain. That is, the GFT of satisfies , , where R is the cutoff frequency. As can be seen in Figure 3, in practice, the R-bandlimitness assumption does not hold for the power signal.
- (iii)
- Minimum CRB (Algorithm 1)—the proposed bus selection policy from Algorithm 1.
6.2. State Estimation and Sampling under the DC-PF Model
- 1.
- The pm-WLS estimator from [11], generated with , and where is randomly chosen from a zero-mean Gaussian distribution with covariance .
- 2.
- 3.
6.3. State Estimation and Sampling under the AC-PF Model
- (1)
- (2)
- The mc method from [5], implemented using Equations (6), (8), (12a), and (12c) from [5], where the low-rank matrix used in this method, composed of the real and the imaginary parts of . In addition, we added to this method the current measurements as inputs (instead of the power flow measurements) for a fair comparison. The implementation was conducted by the SDP solver of CVX [60].
- (3)
- The proposed regularized Gauss–Newton method for implementing the GSP-WLS estimator from Algorithm 2, with the regularization parameters and .
6.4. Detection of Bad Data
7. Discussion
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
GSP | Graph signal processing |
WLS | Weighted least squares |
PF | Power flow |
DC | Direct current |
AC | Alternating current |
PSSE | Power systems state estimation |
EMS | Energy management system |
GFT | Graph Fourier transform |
DSP | Digital signal processing |
KKT | Karush–Kuhn–Tucker |
pm | Pseudo-measurements |
MSE | Mean squared error |
CRB | Cramr–Rao bound |
LNR | Largest normalized residual |
FDI | False data injection |
ROC | Receiver operating characteristic |
GNN | Graph neural network |
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Measure | IEEE Test Case System | ||||
---|---|---|---|---|---|
14-Bus | 30-Bus | 57-Bus | 118-Bus | 300-Bus | |
0.6617 | 0.3015 | 0.3714 | 1.1740 | 1.2371 | |
0.0036 | 0.0022 | 0.008 | 0.0082 | 0.0199 | |
16.4079 | 18.3307 | 50.8035 | 56.1047 | 138.8024 |
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Dabush, L.; Kroizer, A.; Routtenberg, T. State Estimation in Partially Observable Power Systems via Graph Signal Processing Tools. Sensors 2023, 23, 1387. https://doi.org/10.3390/s23031387
Dabush L, Kroizer A, Routtenberg T. State Estimation in Partially Observable Power Systems via Graph Signal Processing Tools. Sensors. 2023; 23(3):1387. https://doi.org/10.3390/s23031387
Chicago/Turabian StyleDabush, Lital, Ariel Kroizer, and Tirza Routtenberg. 2023. "State Estimation in Partially Observable Power Systems via Graph Signal Processing Tools" Sensors 23, no. 3: 1387. https://doi.org/10.3390/s23031387