The underlying impetuous for our research is to improve the performance of electrical power system applications to provide real-time power system control and real-time support for proactive decision making. Our research has focused on load-flow and transient stability applications [2,29]. Sparse linear solvers are employed in both applications and linear solvers account for the majority of floating point operations encountered. Scalability, or the ability to apply more processors to larger problems, is desired when developing multi-processor implementations because load-flow and transient stability applications have the potential to be utilized across different sized geographical areas, from single electrical power utilities to regional power authorities.
Load-flow analysis examines steady-state equations based on the positive definite network admittance matrix that represents the power system distribution network. Load-flow analysis is used for identifying potential network problems in contingency analyses, for examining steady-state operations in network planning and optimization, and also for determining initial system state in transient stability calculations [29]. Load flow analysis entails the solution of non-linear systems of simultaneous equations, which are performed by repeatedly solving sparse linear equations. Load flow is calculated using the network admittance matrices, which are symmetric positive definite and have sparsity defined by the power system distribution network. The size of these matrices is limited because individual power systems generally use networks with less than 2,000 sparse complex equations in their operations centers, while regional power authority operations centers would also be limited to sparse load-flow matrices with less than 10,000 sparse complex equations. Power systems planning studies often incorporate larger networks as lower voltage distribution lines are included in these studies. Sparse matrices employed in planning studies can have from 10,000 to 50,000 sparse equations. This paper presents data for power system networks of 1,723, 5,300, and 9,430 nodes. The last network is from a power system planning study.
Transient stability analysis is a detailed simulation of the power system, that models the dynamic behavior of the electrical distribution networks, electrical loads, and the electro-mechanical equations of motion of the interconnected generators [2]. Transient stability analysis can be used to perform selective detailed analyses of generator commitment stability, and to support crisis decision-making during network recovery. The transient stability problem is modeled by differential algebraic equations (DAEs) with differential equations representing the generators and non-linear algebraic equations representing the power system network that interconnects the generators. The DAEs are in natural non-symmetric block-diagonal-bordered form, with diagonal blocks of generator equations coupled by the power system distribution network. In this representation, there are as many coupling equations as the entire sparse admittance matrix. However, it it possible to order the admittance matrix to block-diagonal-bordered form to order to increase available parallelism. This is illustrated in figure 1. The size of the sparse matrices representing the DAEs have as many as 10,000 complex equations for an individual power system, while regional power authorities could have as many as 50,000 sparse complex equations in the matrix formed from the DAEs.
Figure 1: Ordering the Admittance Sub-Matrix in the Transient Stability Differential-Algebraic Equations
The parallel block-diagonal-bordered Choleski algorithm, presented in this paper, addresses the most difficult of these application to implement on multi-processors. Load-flow has the smallest matrices and the fewest calculations due to symmetry and lack of requirements for pivoting to ensure numerical stability. Load-flow calculations are included in decoupled solutions to transient stability differential-algebraic equations. Instead of the common practice of decoupling the generator and network calculations in a transient stability simulation, we will examine using more powerful differential-algebraic equation solvers for transient stability analysis that do not decouple the generator and network equations. The differential-algebraic equations will offer more potential for good load balancing and offer substantially more calculations because