5.1. System, Data, and Study Scenarios
The performance of the proposed PLF method was tested using a modified IEEE 118-bus system with a VSC-MTDC. As shown in
Figure 2, the wind farms WF
1, WF
2, and WF
3 were connected to the AC grid at bus 24 and bus 35 using a five-terminal DC grid incorporating VSCs. Meanwhile, the other three wind farms WF
4, WF
5, and WF
6 were directly connected to the AC grid at bus 54, bus 115, and bus 45, respectively. The parameters of the IEEE 118-bus system can be found in Matpower 6.0 as in reference [
25]. The parameters of the VSCs and DC lines are given in reference [
3]. In this paper, the control modes of the VSCs are presented in
Table 2. Note that the base capacities of the AC and DC grids were set to be 100 MVA.
The WF output model presented in reference [
24] was used for all WFs in this case study. The rated capacities of the WFs were set to be 80 MW, and the power factors of the WFs were set to be 0.95. It was assumed that the cut-in, rated, and cut-out wind speeds of the WFs were 3 m/s, 11 m/s, and 24 m/s, respectively. The correlation coefficients between the wind speeds of different WFs are given in
Table 3. It was assumed that the correlation coefficients among the bus loads were 0.1 and that the correlation coefficients between the wind speeds and loads were −0.1.
As is well known, wind speeds are affected by various factors and may follow Weibull, Burr, or Lognormal distributions. Meanwhile, some resident loads in a practical power system may follow a Weibull distribution or Gaussian distribution [
20]. To demonstrate the effectiveness of the proposed PLF method, four operation scenarios of the hybrid AC/VSC-MTDC grids were defined, as shown in
Table 4. All the uncertain sources followed normal distributions in operation scenario 1, while the wind speeds at the wind farms WF
1, WF
2, and WF
3 in operation scenario 2 were assumed to follow a Weibull distribution. In operation scenario 3, wind speeds at the different wind farms followed various distributions. Furthermore, in operation scenario 4, different types of loads were considered, and loads at buses 11, 12, 13, 14, 15, 39, 40, 41, 42, 43, 44, 74, 75, 76, 77, 78, 110, and 115 were assumed to follow a Weibull distribution. Note that the historical records of wind speeds in the four different operation scenarios can be accessed in reference [
26].
To verify the accuracy and efficiency of the proposed PLF method, the MCS method-based NATAF transformation was used to provide the reference results (note that this method was denoted as MCS-NATAF in this paper). The MCS generated 20,000 sample points from various correlated distributions, which was sufficient to yield reliable PLF results. To further demonstrate the superiority of the proposed PLF method in handling correlated random variables following different distributions, the results were compared with those from the following methods:
In reference [
14], combining the CM and Cholesky decomposition, an improved CM based PLF method was developed to consider the correlated random variables; this was denoted as CM-CD in this paper.
In reference [
15], an extended PEM based on the NATAF transformation was introduced to deal with various random variables with correlations; this was denoted as PEM-NATAF in this paper.
SRSM was combined with a CMM to deal with correlated uncertainty sources in reference [
19]; this was denoted as SRSM-CMM in this paper.
Note that the SRSM and NATAF transformation were combined to form the proposed PLF method in this paper; we denoted this as SRSM-NATAF.
5.2. Performance Evaluation
The average errors of the mean values of the DC bus voltages in the four different operational scenarios are given in
Table 5. The errors of the standard deviation (STD) values of the DC bus voltages in the four different operational scenarios are shown in
Figure 3. Meanwhile,
Table 6 presents the errors of the mean and STD values of the active power losses from the AC grid in the four operational scenarios. Note that the errors in the figures and tables refer to the percentage of relative error against the results obtained by the MCS-NATAF method.
As shown in
Table 5 and
Table 6 and
Figure 3, the responses of the PLF analysis for the AC/VSC-MTDC hybrid grids obtained by the CM-CD method present huge errors in all operation scenarios. In the four operation scenarios, the errors of the mean and STD values of the active power losses from the AC grid obtained by CM-CD, given respectively, were 3.68% and 14.13% (for operation scenario 1), 4.56% and 16.37% (for operation scenario 2), 4.89% and 17.34% (for operation scenario 3), and 4.92% and 17.89% (for operation scenario 4). The average errors of the mean and STD values of the DC bus voltages obtained by the CM-CD method, respectively, were 1.26% and 5.53% (for operational scenario 1), 1.41% and 6.28% (for operational scenario 2), 1.67% and 7.45% (for operational scenario 3), and 1.69% and 8.35% (for operational scenario 4). The improved cumulant method (CM-CD) will become less accurate if the system (such as a complex AC/VSC-MTDC hybrid grid) moves far from the linear region of behavior. The calculation errors of the CM-CD could further increase, if the system operates in stressed conditions, such as random variables following diverse distributions even with correlations.
In operation scenario 1, both the mean and STD error values obtained using SRSM-NATAF, SRSM-CMM, and PEM-NATAF were almost the same. The average errors of the mean and STD values of the DC bus voltages obtained by SRSM-NATAF, SRSM-CMM, and PEM-NATAF were 0.31%, 0.30%, and 0.32% (for the means), and 2.78%, 2.8%, and 2.73% (for the STDs). The errors of the mean and STD values of the active power losses from the AC grid, given respectively, were 1.23% and 3.22% (for SRSM-NATAF), 1.26% and 3.29% (for SRSM-CMM), and 1.28% and 3.18% (for PEM-NATAF). The reason for this was that all the input random variables in operational scenario 1 were assumed to follow normal distributions, and the SRSM-NATAF, SRSM-CMM, and PEM-NATAF methods all had a good ability to deal with correlated random variables following normal distributions in a hybrid AC/VSC-MTDC grid.
Nevertheless, wind speeds in a practical power system may not follow normal distributions but may follow diverse distributions, even with correlations. In operation scenarios 2 and 3 (wherein the wind speeds were assumed to follow various distributions), it could be observed from
Table 5 and
Table 6, and
Figure 3, that the relative errors obtained using SRSM-NATAF and PEM-NATAF were much smaller than those obtained using SRSM-CMM. For example, in operation scenario 2, the STD errors in the DC bus voltage obtained using SRSM-NATAF, PEM-NATAF, and SRSM-CMM were 3.02%, 3.3%, and 6.08%, respectively. Meanwhile, in operation scenario 3, the STD errors in the DC bus voltage were 3.41%, 3.88%, and 7.38%. SRSM-CMM cannot accurately consider different correlated distributions, resulting in a decrease in the computational accuracy of the PLF calculation. However, the proposed SRSM-NATAF method could solve this problem. It is of value to note that the calculation errors obtained using SRSM-NATAF were smaller than those obtained using PEM-NATAF.
In operation scenario 4, different types of loads were considered. In AC/VSC-MTDC hybrid grids, the number of correlated random variables following diverse distributions further increases compared with that in operation scenario 3. Hence, the computational errors from the SRSM-CMM method increase as well. In operation scenarios 3 and 4, the SRSM-CMM method’s errors in the mean and STD values of the active power losses from the AC grid, given respectively, were 5.88% and 10.19% (for operation scenario 3), and 6.89% and 16.56% (for operation scenario 4). However, the proposed SRSM-NATAF method can accurately handle different distributions with correlations in the PLF analysis. For example, in operation scenario 4, the average errors in the mean and STD values of the DC bus voltage obtained by SRSM-NATAF were 0.42% and 3.8%, respectively, and the errors in the mean and STD values of the active power losses from the AC grid obtained using SRSM-NATAF were 2.89% and 4.59%, respectively. The key reason for this lay in the fact that CMM, based on the assumption of normal distribution, will result in inaccurate conclusions in the PLF analysis. However, the results in
Table 5 and
Table 6, and
Figure 3, show that the proposed SRSM-NATAF method was quite effective in all examined operation scenarios.
The cumulative probability curves of the voltage magnitude at bus 24 in the four different operation scenarios are shown in
Figure 4. Although the cumulants and moments of the PLF output could be respectively obtained using PEM-NATAF and CM-CD, these two methods had no ability to directly estimate the PDFs and CDFs of the PLF results. Therefore, PEM-NATAF and CM-CD were combined with the popular Gram–Charlier series expansion to estimate the CDFs of the PLF responses.
It can be seen from
Figure 4 that the results obtained using SRSM-CMM, CM-CD, and PEM-NATAF were obviously biased compared with those obtained by using MCS-NATAF. The reasons for this were as follows: (1) SRSM-CMM could not accurately consider the correlated different distributions, leading to inaccurate results of the PLF analysis; (2) PEM-NATAF could not accurately estimate the high-order moments of the PLF responses, resulting in poor accuracy of the cumulative probability curves; and (3) CM-CD assumed that the original DLF model was linear, which would impair its accuracy. Furthermore, as shown in
Figure 4, the cumulative probability curves obtained by CM-CD and PEM-NATAF in operation scenarios 3 and 4 were evidently biased when the probability was near 0 or 1. The key reason lay in that SEMs, like the Gram–Charlier series, may become unreliable when the probability was near 0 or 1 [
16]. Compared with those from SRSM-CMM, CM-CD, and PEM-NATAF, the cumulative probability curves obtained using the proposed SRSM-NATAF method were much closer to those obtained using the MCS-NATAF. Hence, by using the proposed SRSM-NATAF method, the means, STDs, and the cumulative probability curves of the PLF results could be accurately obtained.
5.3. Computational Efficiency
The CPU usage times for the PLF analysis of the modified IEEE 118-bus system under the four different operation scenarios using the MCS-NATAF, SRSM-CMM, PEM-NATAF, CM-CD, and the proposed SRSM-NATAF methods are shown in
Table 7. The average CPU usage times for the MCS-NATAF, CM-CD, PEM-NATAF, SRSM-CMM, and the proposed SRSM-NATAF method were 3757.89 s, 24.80 s, 33.81 s, 33.65 s, and 33.82 s, respectively. Although CM-CD could accelerate the PLF analysis, the required linearization process of this method seriously impaired its accuracy, especially for the complex AC/VSC-MTDC hybrid grids.
The CPU usage times of the PEM-NATAF, SRSM-CMM, and the proposed SRSM-NATAF method were almost the same. Compared with the MSC-NATAF, the proposed SRSM-NATAF method allowed an extreme reduction in the calculation time required for the PLF analysis of AC/DC hybrid grids. Hence, the proposed SRSM-NATAF method not only had good computational accuracy but it also had high computational efficiency. Note that all operation scenarios were implemented in MATLAB on an Intel i5 3.30 GHz PC with 8 GB RAM. Meanwhile, the AC power flow was solved by Matpower as in Reference [
25].