To verify the effectiveness of the proposed method, the IEEE 14 and 118 bus test systems are selected to simulate the performance generalized M-estimator with different parameters. All simulations were conducted in MATLAB, using an Intel Core i7-9750CPU (@2.6 Hz) 16 GB memory computer.
Section 5.1 describes the impact of bad data and non-Gaussian measurement noise on generalized M-estimators with different values of
β by conducting exhausted Monte Carlo simulations in the IEEE 14 bus test system.
Section 5.2 demonstrates the effectiveness of the proposed generalized M-state estimator of the optimized parameter in a large system, the IEEE 118 bus test system.
5.1. Effect of Bad Data and Measurement Noise on the Performance of the Generalized M-Estimator
To ensure the generality of the results, all of the results shown in this section are obtained by averaging the results over the 500 Monte Carlo simulations. To ensure the validity of the generalized M-state estimation method, a set of redundant conventional measurements and PMU measurements were selected in IEEE 14 bus test system as shown in
Figure 5.
For Gaussian errors, the standard deviation is set to a percentage of the measurement according to the type of measurement as follows:
SCADA: P, Q measured value (0.02), voltage (0.002)
PMU: Voltage amplitude (0.002), Voltage phase angle (0.01)
The performance of the generalized M-estimator can be adjusted by varying values of β. In this simulation test, the least value of β is 10−2. With this configuration, the generalized M-estimator can be considered to be an LAV estimator. The largest value of β is set to be 104. The corresponding estimator can be considered to be a WLS estimator. In the range of 10−2 to 104, β increases at the fixed ratio of 101/30, all estimators with different values of β are tested in the IEEE 14 bus test system. The root mean square error (RMSE) is used as the performance index of the estimators.
Case 1: Existence of bad data with fixed percentage errors. We compared the robustness of the state estimators by introducing fixed percentage errors to the voltage magnitude measurement at bus 1 and obtaining the RMSE of the estimators when the noise probability density function follows the standard Gaussian distribution. The voltage magnitude measurement at bus 1 is changed to 0, 0.2, 0.4, 0.6, 0.8 of its true value (100%, 80%, 60%, 40%, 20% errors, respectively), and tested separately. The simulation results are shown in
Figure 6.
The following conclusions can be obtained:
(1) After introducing fixed errors of different magnitudes, the trend of RMSE of each curve is roughly the same: the general trend is that RMSE increases as a larger β is selected. It will not change after a certain value;
(2) There is an optimal value of
β for each fixed percentage error, and the value decreases with the percentage of the error parameter, as summarized in
Table 2.
Case 2: Existence of bad data with random percentage of errors. The robustness of generalized M state estimators with different values of
β is compared by five individual simulations with three random bad data points in the measurements and observing their performances. The test results are shown in
Figure 7.
The following conclusions can be obtained:
(1) When there are no bad data in the measurements, the trend of RMSE with the change of β is decreasing in general; the value of β which achieves the lowest RMSE is 1.5, which is the exact value of β for normal generalized M-estimator;
(2) After the introduction of three random errors, the trend of the RMSE of each simulation with the change of β is roughly the same: The general trend is increasing. It will not change after a certain value;
(3) There is an optimal β whose corresponding estimator achieves the lowest RMSE for each simulation, which varies between 10−1 and 101 with different bad data.
Case 3: Existence of measurements with bimodal Gaussian distribution noise. When the bimodal interval size k (as introduced in
Section 3.1) is between 0~5, the trend of the RMSE with estimators with different values of β is shown in
Figure 8.
The following conclusions can be obtained:
(1) When k ≤ 1, RMSE decreases with the increase of β, and when k > 1, RMSE decreases with the increase of β;
(2) The optimal β whose corresponding estimator achieves the lowest RMSE decreases with the increase of k;
(3) In the presence of non-Gaussian measurement noise, the value range of the optimal β is between 10−1 and 101, which can roughly cover the value range of the optimal β in the presence of bad data;
(4) Since the occurrence of bad data is low and the value range of the optimal β in the bad data case can be roughly covered by that in the non-Gaussian measurement noise case, the optimal β can be determined by only considering the influence of non-Gaussian measurement noise.
The optimal
βs obtained by simulation in the presence of measurement noises in bimodal Gaussian distribution with different values of
k are summarized in
Table 3 and plotted in
Figure 9. The optimized
βs calculated according to the proposed optimized parameter selection method (20) are also plotted in
Figure 9. It can be seen that with the increase of
k, the calculated optimized
β decreases, and are equal to the optimal
β at
k = 0 and
k ≈ 5. Although the optimized
β is larger than the optimal
β for 0 <
k < 5, the decreasing trend guarantees significantly better estimation accuracy compared to the estimator with a fixed
c value while
k increases.
5.2. Simulation Examples of the IEEE118 Bus Test System
To verify the proposed generalized M-estimation method of optimized parameters based on sampling, simulations are conducted in the IEEE 118 bus test system with the network diagram and measurement configuration shown in
Figure 10.
Among the total 186 measurements in the system, 28 of them have bimodal Gaussian distributed noises with the bimodal interval size (k) randomly set between 0~5. According to the proposed optimized parameter selection method, the optimized β, β opd, is determined as follows:
Step 1: Randomly select 80 measurements;
Step 2: For each measurement selected, test the measurement noises and record 10,000 groups of data; 12 non-Gaussian measurement noises are counted according to the statistics, and the proportion of non-Gaussian measurement, h, is 15%;
Step 3: Calculate the sagging coefficient A:
where
kmax is the maximum bimodal Gaussian error coefficient possible, and
hmax is the maximum proportion of the bimodal Gaussian distribution measurement noise possible.
Step 4: Find the average k according to the statistic bar chart of all bimodal Gaussian distributed measurements noises. To test the proposed method in different scenarios, 5 different simulations with average
k ≈ 1, 2, 3, 4, 5 are conducted with the statistical bar charts shown in
Figure 11.
To verify the effectiveness of the proposed generalized M-estimator of optimized parameters, the RMSEs of the estimators obtained from Monte Carlo simulations for different ks and against
β are shown in
Figure 12. The optimized
β,
βopd, can be calculated according to (20) with the obtained
A and
h. The optimal
β,
βopl, and the RMSEs at
βopl,
βopd,
β = 1.5 for different ks can be obtained from
Figure 12. The results are summarized in
Table 4.
The analysis performed in
Figure 12 and
Table 4 leads to the following conclusions:
(1) βopd is larger than βopl, but the RMSE is not much different from the actual RMSE;
(2) The RMSE at βopd is significantly lower than the RMSE at β = 1.5.
The conclusions above demonstrate the effectiveness of the proposed method. The underlying reason for these results is that the proposed method reduces the value of β to increase the estimator’s robustness when the level of non-Gaussian measurement noise increases. The linear drooping characteristic of β gives the proposed estimator significantly better estimator accuracy than the traditional generalized M-estimator, but slightly lower estimator accuracy than the estimator with βopl, which is acceptable considering the difficulty to obtain βopl.
5.3. Simulation Examples of the Polish 2736 Bus System
To verify that which is proposed, simulations are conducted in the Polish 2736 bus test system [
27].
Among the total 3269 measurements in the system, 280 of them have bimodal Gaussian distributed noises with the bimodal interval size (k) randomly set between 0~5. According to the proposed optimized parameter selection method, the optimized β, β opd, is determined as follows:
Step 1: Randomly select 800 measurements;
Step 2: For each measurement selected, test the measurement noises and record 10,000 groups of data; 120 non-Gaussian measurement noises are counted according to the statistics, and the proportion of non-Gaussian measurement, h is 15%;
Step 3: Calculate the sagging coefficient A;
Step 4: Find the average
k according to the statistic bar chart of all bimodal Gaussian distributed measurements noises. To verify the effectiveness of the proposed generalized M-estimator of optimized parameters, the RMSEs of the estimators obtained from Monte Carlo simulations for different ks and against
β are shown in
Table 5.
Analysis performed in
Table 5 leads to the following conclusions:
(1) The β value calculated by sagging coefficient is slightly different from the actual β value, but its RMSE is not much different from the actual RMSE;
(2) The RMSE corresponding to the calculated β value is significantly higher than that when β = 1.5, and the RMSE is significantly higher when the k value is larger.
These conclusions are almost the same with those in
Section 5.2, demonstrating the adaptability of the proposed even in very large networks.