Transient stability analysis examines the dynamic behavior of power system electrical distribution networks, electrical loads, and the electro-mechanical equations of motion of the interconnected generators for as much as several seconds following a disturbance [1,3,46]. Under normal operating conditions, an electrical power system is near equilibrium, with only minor deviations from true steady-state conditions caused by small, nearly continuous, changes in the loads. When a short circuit occurs in the power distribution network, there are significant, nearly instantaneous, changes in the loads at some generators in the system.
Mechanical controls in the generators react slower than the electromagnetic loads, so there is the possibility that instead of the power system returning to a steady-state condition after the disturbance, one or more generators may encounter sufficient variations in rotational speed that they loose synchronization with the power network and must be taken off-line to avoid catastrophic problems. If a generator must be taken off-line because it has lost synchronization, there will be a decrease in available generator capacity and another source of disruption will be injected into the power system. Cascading system failures can cause wide spread power outages, reduce the interconnected power grid to islands of power service, and even cause physical damage to generating equipment [7].
There is a simple mechanical analog to the transient stability problem that will assist in understanding the nature of the problem. Consider a number of masses, that represent the electrical generators in a power system, suspended within a network of elastic strings, that represent the electrical transmission lines. A sudden loss of a transmission line can be modeled by cutting a string in the steady state network. The forces in the remaining strings will fluctuate, and the masses will experience coupled motion that is dependent on the network and the tensions in the strings. This disturbance may cause two effects:
The reaction of a single generator to variations in its load can be modeled using ordinary differential equations (ODEs):
However, there are multiple generators supplying power to the network and the generators are coupled through the power network. The power network can be modeled by non-linear algebraic equations:
and this entire system of differential-algebraic equations (DAEs) must be solved simultaneously (or nearly so). Transient stability analysis is computationally intensive because the large systems of DAEs must be solved at small time increments to ensure that errors are minimized in the numerical integration of the potentially computationally-stiff ODEs [41]. The dynamics of each generator are individually modeled with as few as two and with as many as forty differential equations while the generators are coupled via the algebraic equations that describe the electrical network. This requires the solution of large systems of simultaneous sparse non-linear equations.
The number of equations required to solve the transient stability DAEs are extremely large and very sparse. For example, if there are twenty differential equations to describe each generator and two equations to describe the complex voltage/current at each network bus, then a state-wide or regional interconnected power system with 2000 buses and 300 generators could generate a sparse, irregular system of 10,000 non-linear algebraic equations that must be solved simultaneously. In this example, the matrix would be formed by 300 blocks of generator equations along the diagonal with additional blocks formed from the network equations. After reordering the matrix, all equations with variables outside of the blocks along the diagonal will be relocated into narrow borders along the bottom and right side of the matrix. Each generator equation may have as many as 20 variables, and each network equation would have only between two and ten variables due to the limited number of transmission lines at any bus.
An example of the block-bordered diagonal matrix for a smaller power system with only ten generators is depicted in figure 1. Generator equations occur in the blocks along the diagonal starting at the upper-left-hand side of the diagram, and are labeled G1 to G10. The network equations occur in the lower-right-hand side of the matrix, and in this representation, the network equations have been reordered into block-bordered-diagonal form according to the natural hierarchy encountered in electrical power distribution networks. The blocks along the diagonals, in this portion of the matrix, are independent portions of the network, while the inner borders contain network equations that depict the interconnections between independent portions of the electrical network. The outer borders contain equations that relate generator current to voltages at the buses connected to generators.
Figure 1: Block-Bordered Diagonal Matrix Form Derived from the Power System
Transient Stability Differential-Algebraic Equations
There are several levels of inherent hierarchy in a state or regional power network due to the clustering of loads within power systems, interconnections between load clusters for system reliability within an electrical utility, and the interconnections between utilities for added reliability and the inter-utility sale of electricity [22,31]. A simple two-level hierarchical network and block-bordered diagonal matrix is illustrated in figure 2.
Figure 2: Hierarchical Power System Network Mapping to a Block-Bordered Diagonal Matrix Form
Solutions of simultaneous nonlinear equations are performed using iterative techniques such as Newton's method that require the solution of one or more linear systems of equations
The Jacobian matrix used in the Newton method has the same sparsity pattern as the original matrix. Examples in the literature discuss using time-step sizes that would require between 12 and 100 time-steps per second of simulation time [10,25,33,34,35,47]. Consequently, a real-time or faster-than-real time transient stability analysis program could require the solution of systems of 10,000 non-linear algebraic equations a hundred times per second, and each solution of the systems of nonlinear equations could require multiple solutions of a similar number of linear equations that require forward reductions and backward substitutions for a number of vectors equal to the number of linear equations [20].
At this point, it would be desirable to develop estimates of the number of floating-point operations required to solve these linear equations, in order to calculate a detailed analysis of the potential performance on a target parallel architecture, however, such an estimate would be highly dependent on the number of equations in the actual sparse matrix, and the sparsity structure in the matrix [14]. The sparsity structure can directly affect the number of calculations; although there are many techniques available to reorder a matrix in order to reduce the number of calculations and/or increase the amount of calculations that can be performed concurrently [23]. Estimates of the number of floating point operations in the calculations of the transient stability DAEs will be used in subsequent research as portions of objective functions to determine optimal parallel processing load balancing, however, without more details on a specific, large interconnected power system in this example, it is not possible to give more detailed estimates of the number of floating point operations. Nevertheless, it is possible to develop estimates of potential algorithmic and concurrent performance improvements using other approximate techniques based on the amount of overhead in a concurrent algorithm and the fraction of sequential operations in a concurrent implementation, using Amdahl's law [32,37,38]. Such performance estimates will be discussed later in this paper.