Advanced Method to Explore Voltage Stability Boundary
Voltage stability is an important problem in power system planning, operation and control. Voltage instability occurs when a power system is unable to maintain stable voltage levels. Voltage instability was responsible for several major power system collapses worldwide. Knowing precise voltage stability limits helps power system planners and operators keep their system within its secure limits, and, at the same time, fully utilize the available power transfer capability. These important tasks are influenced by the configuration of voltage stability boundary, which is actually a multidimensional hyper surface in coordinates of power system parameters. Different parts of the boundary correspond to various stress directions that can be observed in the system as a result of the changing system load, generation patterns, contingencies, market forces, variable generation, etc.
For many applications related to the voltage stability problem, the computational time becomes a critically important factor. For instance, the realtime analyses, synchrophasorbased applications, and methods based on the multidimensional security region (multidimensional nomogram) concept require really fast algorithms preferably implemented within the high performance computer environment. Static voltage and angle stability conditions (socalled saddlenode bifurcations) are often associated with singularities of the power flow Jacobian matrix. Traditional methods (e.g., continuation power flow and direction method) to calculate voltage stability boundary are computationally intensive and/or sensitive to initial guesses. Other faster methods developed for this purpose are based on simplifications of the voltage stability problem.
At Pacific Northwest National Laboratory, we are developing new methods to explore static voltage stability conditions in Cartesian coordinates instead of polar coordinates, where the singularity problem is formulated and then easily reduced to a scalar equation with respect to the stress parameter. Two methods are proposed to detect the singularity: i) direct eigenvaluebased detection of singularity points and ii) “distance function” approach to detection of singularity. The first method requires solving a generalized eigenvalue problem for power flow Jacobian matrices determined at two distinct points of the state space for each stress direction. The second one utilizes an advanced searching algorithm, where a special new scalar “distance function” of the stress parameter is defined and only a linear system needs to be solved at each step.
The proposed methods have been validated and evaluated using the Western Electricity Coordinating Council (WECC) 2020 system, which contains 21 areas, 17939 buses, 3525 generators, and 8596 loads. The boundary points are found using i) a stepbystep stressing approach using PowerWorld Simulator software, ii) direct eigenvaluebased detection of singularity, and iii) the distance function approach. The simulation time from different methods is shown in Table 1, which shows that the proposed methods are very promising in reducing calculation time. The proposed methods allow quick exploration of static stability conditions in the state space, which is essential for many applications such as developing realtime path rating and baselining power system security conditions using synchrophasors.
Table 1. Comparison of simulation time (in second) of different methods
Method 
StepbyStep Iterative 
Eigenvalue Detection 
Matrix Conditioning Detection 
Time to solve a linear system 

Single processor machine  single point  144  13.28  2.66  0.10.12 
200 point  28800  2656  532  
200processor machine  Extrapolated time    2656  532   
Actual time for 200 point    126.06  6   
Notable Publications
Y. V. Makarov, J. Ma, and Z. Y. Dong. 2007. “Determining static stability boundaries using a noniterative method,” in Proceeding IEEE Power Energy Society General Meeting, Jun. 2007.
Y. V. Makarov, J. Ma, and Z. Y. Dong, “Noniterative method to determine static stability boundaries,” in Proceeding IEEE Power Tech, Lausanne, Switzerland, Jul. 2007, pp. 349–354.
Y. V. Makarov, B. Vyakaranam, D. Wu, B. Lee, S. E. Elbert, Z. Hou, and Z. Huang, “On the Configuration of the US Western Interconnection Voltage Stability Boundary,” to be submitted.