From the mathematical modeling point of view, a power flow solution consists of solving the set of non-linear algebraic equations that describe the electrical power network under steady-state conditions. In this paper, a general discussion of ill-conditioning of a system based on the condition number of the Jacobian matrix is presented. Load flow studies are carried out on ill-conditioned test systems with Flexible AC Transmission System (FACTS) device (Unified Power Flow Controller (UPFC)) incorporated. The variation of the condition number and the rate of convergence are presented for both well-conditioned and ill-conditioned test systems.

Load flow calculations are essential in steady-state analysis and system planning studies. For several decades, the Newton-Raphson method with its quadratic convergence characteristics has proved invaluable in solving the power flow problem. The additional burden imposed on the numerical solution due to various constraints does not affect the Newton-Raphson method on its convergence. Newton-Raphson method is used for load flow calculations of ill-conditioned systems with MATLAB environment. A system is said to be ill-conditioned when a small change in input parameter produces a large change in the solution.

The UPFC is one such device, which can provide real time control and dynamic compensation of AC transmission systems providing multi-functional flexibility required to solve many of the problems facing the power delivery industry. The UPFC is able to control simultaneously or selectively all the parameters affecting the power flow in the transmission line (voltage, impedance and phase angle). Alternatively, it can provide the unique functional capability of independently controlling both real and reactive power flow in the line. These basic capabilities make UPFC the most powerful device presently available for transmission system control. As mentioned in Hingaroni and Gyugyi (2000), the FACTS devices influence the power system network parameters extensively. This paper presents the effect of UPFC on load flow studies of ill-conditioned systems.

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