Case Study

In this example, multivariate cluster analysis techniques were applied to simulation data from the Australian test system in Figure 3.10. The goal is to characterize critical associations or patterns between bus voltage magnitudes and reactive power outputs from generators and SVCs. Attention is focused on two main aspects, namely, the identification of phase relationships between reactive power and voltage signals and the computation of voltage to reactive power sensitivities to system perturbations.

For purposes of illustration, 18 reactive power output signals, including reactive power at 14 generators and 4 SVCs, and 59 bus voltage magnitudes were selected for analysis. Figure 5.7 shows some selected measurements, while Table 5.1 summarizes the main characteristics of the sets. Insight into the nature of phase relationships is given in Figure 5.8 that plots the behavior of coherent signals.

The observational data is defined as

where V, (t),j = 1,..., 59 denotes bus voltage magnitudes and Q, (f), j = 1,..., 18 are the output reactive power signals from synchronous generators and SVCs. The description of the data sets is given in Table 5.3.

FIGURE 5.7

Selected time series following a short circuit at bus 217. (a) Generators reactive power output and (b) SVC reactive power output.

FIGURE 5.8

Detail of system response.

TABLE 5.3

Selected System Measurements

Data Subtype

Features

Bus voltage magnitudes (Xv)

59 signals

Generator reactive power output signals (Хй)

15 generator reactive power output signals (Gens: 101,201,202,203,204,302,302,303, 401,402,403,404)

SVC reactive power output signals

5 SVC reactive power output signals (SVCs: ASVC_2, RSVC_3, BSVC_4, PSVC_5, SSVC_5 in Figure 3.10)

Inspection of Figures 5.7 and 5.8 reveals that:

  • • Generators 8 through 11 and SVC # 1 swing in phase.
  • • Further, generator # 2 and SVC # 3 swing in phase.

Partial least squares analysis provides a natural way to explore associations between individual data blocks or variables. As a first step to evaluate the applicability of these models each data set was analyzed independently. For reference, Figure 5.9 shows the spatial patterns extracted from Xv and XQ in equations (5.19) and (5.20) analyzed separately and obtained using PCA.

FIGURE 5.9

Real part of PCA-based reactive power (a) and bus voltage magnitude (b) modes.

Partial Least Squares Correlation

The approach in Section 5.6.2 was used to determine relationships between voltage and output reactive power from SVCs and generators. The primary goal is to determine patterns of behavior of interest for voltage control.

As a first step, voltage and reactive power data are column centered as

The correlation matrix is then computed as

Each row of matrix R, gives the correlation between a given reactive power output, Qj, and a given bus voltage magnitude, V;. Direct calculation of the covariance matrix, however, may result in numerical errors, especially for large systems and is sensitive to scaling. Figure 5.10 shows the right eigenvector plot Uxl and Uyj. Correlation analysis that SVCs # 3 and 4 swing in opposition to SVCs # 1,2, and 5.

Also of interest, Table 5.4 shows correlation measures from (5.22). This is consistent with the physical understanding of the problem.

FIGURE 5.10

Partial least squares correlation analysis of reactive power time series.

TABLE 5.4

Spatial Correlation between SVC Reactive Power Output and Bus Voltage Magnitudes

SVC Bus

Dominant Bus Voltage Magnitudes

205

209,212,208,414,202

313

314,304,307,306

412

403,411,414

507

509,505,305

509

509,505,315,207,203

Multiblock PCA Analysis of Measured Data

Further insight into the nature of dynamic patterns can be obtained from multiblock (joint) analysis of the observational matrices Xv, XQ and XSvc in Figures 5.7 and 5.8. In this case, the concatenated data is expressed in the form

Because of the different physical units, separate pre-processing methods were required for the three different types of data. Figure 5.11 shows the shape extracted from multiblock PCA analysis of the multivariate set in (5.19) and (5.20). Results are found to be in good agreement with Figures 5.9b and 5.10.

Figure 5.12 compares the frequency-based shape extracted from 19 frequency measurements extracted using DMs and Laplacian eigenmaps. Results are found to be consistent although some differences are noted.

FIGURE 5.11

Schematic illustration of dimensionality reduction.

FIGURE 5.12

(a) Diffusion maps and (b) Laplacian eigenmaps.

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