Parallel Coordinates and Application to Analysis and Control Application of Parallel Coordinates to Power Stability Data Visualization and Control

The overall objective of this research program is the application of the parallel coordinates data representation and visualization paradigm to the large, complex data sets associated with electric power production and distribution. Previously we have investigated parallel coordinates as a tool for data analysis and control for power plants. The current focus of the program is the application of parallel coordinates to the data visualization and management for power distribution. Lessons learned from the study of application of parallel coordinates to power plants, imply that a successful domain for parallel coordinate application will be one where the multi-dimensional data involved all have similar scaling magnitude. Data associated with power distribution fits this model better than power plant data. While there are many aspects of power distribution, we have chosen to focus on the power stability (sometimes called the power security) problem. Not only is this an important area from the point of view of practical application, but it is a problem where, from the aspect of parallel coordinates, scaling issues are minimal. Also it is a problem where high dimensionality of data sets is the norm; real power systems involve hundreds to thousands of buses.

In order to avoid getting diverted into power system modeling, per se, we have decided to make use of the work (and software) done by C. Canizares of the Univ. of Waterloo. He developed software (PFLOW) that uses power system models and computes bifurcation (stability) surfaces for these models, given appropriate input parameters (e.g. voltage loads). PFLOW uses a, so called, static power system model. In this approach, an operating point (set of loading and other parameters in the N-dimensional space describing the N bus system) is fixed, a small disturbance is imposed, and a bifurcation surface (N-1 dimensional surface) is computed that separates the stable from the unstable operating regions. As parameters are changed, one can compute different bifurcation surfaces and examine the location of the system operating point with respect to these surfaces.

Use of the above approach requires manipulation of data points in N-dimensional space. Thus data management that involves interactive visualization is out of the question for N > 3. But this is an ideal candidate for application of parallel coordinates. Because the parallel coordinate paradigm maps N-space geometry to a 2-space representation, the data points are amenable to visualization, albeit different from usual, e.g. N-space points map to polygonal lines. The application of parallel coordinates to the power stability problem involves the following:

1. Use PFLOW (or other power system simulations) to compute a bifurcation surface
2. Select an operating point for the system
3. Represent the operating point as a polygonal line in parallel coordinates
4. Represent the bifurcation surface, in a), as a set of polygonal lines in parallel coordinates
5. Compute and draw the "envelope" (least upper bound and greatest lower bound) for the setof lines that represent the bifurcation surface
6. See if the line in parallel coordinates for the operating point penetrated the envelope at any parallel axis location

Not only does the parallel coordinate method allow visualization of the operating point and bifurcation surface for N>3, but the ability to determine where along the x-axis in parallel coordinates the operating point penetrated the envelope allows one to associate specific bus(es) with the instability. (Each discrete x-axis represents one bus in the N-bus system.) For example, it would be possible for an operator, on seeing the operating point line penetrate the bifurcation envelope at some x-axis location, to immediately examine the operating parameters for the particular bus associated with that axis location. (Unfortunately, it doesn't necessarily follow that any corrective action can be restricted to that bus alone.)

Currently our emphasis is on implementing the process a) through f) above. We are currently working with a two bus system and a 32 bus system. The two bus system is for illustrative purposes only. Because the N-space (N = 2) representation is easily visualized, it can be shown together with the parallel coordinate representation. One can easily see how the bifurcation surface ( a curve in the two bus case) is represented in parallel coordinates and how the operating point is represented as a point in 2-space and a line in the parallel coordinates. The bifurcation envelope in the two bus case is easily constructed. Figure 1 shows the parameter space where the real power consumed by two loads are chosen as parameters. The curve represents the bifurcation set and it separate the two regions: operable region and inoperable region. Loading point P is in operable region where there is a stable equilibrium corresponding to the real power load distribution.

In figure 2, loading point P is in a region in which there is no stable equilibrium.

Figures 1 and 2 use traditional Cartesian coordinates for the graphical representation. This poses no problem for a simple two bus case, but in real situations where many buses are involved, this approach is not practical. However, we can represent the data using parallel coordinates. While this implies some "overkill" for the two bus example, parallel coordinates will allow us to graph the data for a many bus case using a two dimensional view. Figure 1.1 is the parallel coordinate representation of figure 1.

The vertical axis scale is the power load value, and the left and right hand vertical axes are Bus 1 and Bus 2 of figure 1, respectively. (The numbers along the horizontal axis in figure 1.1 are arbitrary. They are artifacts of the way in which we set up parallel coordinates. ) Because points in Euclidean space map to lines in parallel coordinates, the stable operating point in figure 1 is represented as a straight line in figure 1.1. The curved line is the envelope formed by the tangents at each point of the bifurcation set (curve in figure 1). The region between the horizontal axis and envelope is the stable operating region for this example. The loading point is stable if the straight line is within this region, is unstable if it crosses the envelope, and is itself a bifurcation point if it is tangent to envelope. Figure 2.2 is the parallel coordinate representation corresponding to figure 2.

In the case shown in figure 2.2, the operating line (corresponding to the operating point in figure 2) crosses the bifurcation (stability) envelope) in two places. (This is just a consequence of the simplicity of the example being used. There are two tangents to the bifurcation surface in figure 2 that pass through the operating point.) Because the operating line in parallel coordinates passes above the stability envelope, this set of operating parameters is unstable.

For a more realistic case, we are using a 32 bus model that is supplied as a working example in the PFLOW package. (There is also a 300+ bus model included in the package.) The model has 32 buses, 5 areas, 9 generators, 25 lines, 15 transformers (7 LTCs), and was originally created to study the voltage collapse of the northern part of Belgium in 1982. In order to get the bifurcation set of this system in loading space, we use PFLOW to calculate the bifurcation point corresponding to input of an operating point. PFLOW has been designed to calculate local bifurcations characterized by a singularity in the system Jacobian. The program also generates a series of output files that allow further analyses, such as left and right eigenvectors at the bifurcation point, Jacobians, power flow solutions at different loading levels, some voltage stability factors, etc.

Figure 3 shows the parallel coordinate representation of one operating point for the 32 bus example. The horizontal axis is the bus number (in order) and the vertical axis is the power load. The green area represents the stable operating region and is bounded by a polygonal line which is the parallel coordinate representation of the bifurcation curve in the N=32 dimensional geometric space. The red line inside the green area is the parallel coordinate representation of a stable operating condition, i.e. stable state for the 32 buses.

In this example, the operation is stable but there is a possible marginal situation indicated by the fact that the operating point is quite close to the bifurcation envelope in the vicinity of bus 16.

Figure 4 shows a (hypothetical) case where the operating point has clearly exceeded the stability region.

The dashed red line represents a portion of the (hypothetical) power perturbation which results in a power load at Bus 5 which is greater than the bifurcation envelope value at Bus 5. Using software we have developed (SCENE),which couples visualization and data management systems, it is straight forward for a user (dispatcher) to click a mouse on the Bus 5 axis and retrieve (in graphical or textual format) all the relevant information to examine possible causes of the Bus 5 overload. (Of course, because of the nonlinearity of the power system, there is no assurance that the overload at Bus 5 is a result of only those factors local to Bus 5.

The "bottom line" is that the parallel coordinate representation allows visualization of the power loads and potential stability situations that would be impossible in traditional Cartesian coordinates; we just can't obtain a visualization of the whole 32 dimensional space at once. Real power systems have more than 32 buses; thousands of buses are not uncommon. But the same principals of parallel coordinate visualization will apply. Because the spacing between the axis shown along the horizontal is arbitrary, one can reduce this space to allow visualization of all buses at once. One can use pan and zoom techniques to expand the view to get more detail at any local.

Work in Progress

Using a Unix version of PFLOW, we have implemented software procedures to map the operating point and points along the bifurcation surface into parallel coordinates. The difficult part of the task is establishing the bifurcation surface envelope. While we can do this by "brute force" (translate a great many bifurcation surface points to polygonal lines), this can be inefficient in cases where the bifurcation surface changes as a result of power grid operational changes. We are investigating interpolation and computational geometry techniques that would allow us to get a good approximation to the envelopes with less computational effort. Once we obtain some bifurcation points, we hope to be able to interpolate the whole bifurcation set. This process can be done in parallel coordinates; that is, the interpolation can actually be done in two dimensional space as opposed to N-dimensional (N=32 for figure 3) space. Once the bifurcation envelopes are in place for a given N bus system, any operating point can be quickly checked for stability by examination for any region where it touches or passes through the stability envelope.

At this stage of the study, we are using commercial software packages for visualization, i.e. production of parallel coordinate graphs. This together with use of PFLOW allows us to concentrate on the major issues, namely methods to establish the bifurcation surface envelopes. At a later stage we plan to return to the use of our own interactive parallel coordinate tools, and to develop models that do not bind us to the PFLOW example models, and are more adaptable to real power systems. We are also looking into parallel coordinate visualization as a tool for power flow optimization. Among the questions of interest is the relation between optimal load distribution and stability. That is, there a way to use parallel coordinate visualization to check that an optimal load distribution is also stable?