外文翻译--基于优化的牛顿—拉夫逊法和牛顿法的潮流计算

外文翻译--基于优化的牛顿—拉夫逊法和牛顿法的潮

流计算

英文文献

Power Flow Calculation by Combination of Newton-Raphson Method and Newton’s Method in Optimization.

Andrey Pazderin, Sergey Yuferev URAL STATE TECHNICAL UNIVERSITY ? UPI E-mail: pav@//0>., usv@//.

Abstract--In this paper, the application of the Newton’s method in optimization for power flow calculation is considered. Convergence conditions of the suggested method using an example of a three-machine system are investigated. It is shown, that the method allows to calculate non-existent state points and automatically pulls them onto the boundary of power flow existence domain. A combined method which is composed of Newton-Raphson method and Newton’s method in optimization is presented in the paper.

Index Terms?Newton method, Hessian matrix, convergence of numerical methods, steady state stability

Ⅰ.INTRODUCTION

The solution of the power flow problem is the basis on which other

problems of managing the operation and development of electrical power systems EPS are solved. The complexity of the problem of power flow calculation is attributed to nonlinearity of steady-state equations system and its high dimensionality, which involves iterative methods. The basic problem of the power flow calculation is that of the solution feasibility and iterative process convergence [1].

The desire to find a solution which would be on the boundary of the existence domain when the given nodal capacities are outside the existence domain of the solution, and it is required to pull the state point back onto the feasibility boundary, motivates to develop methods and algorithms for power flow calculation, providing reliable convergence to the solution.

The algorithm for the power flow calculation based on the Newton's method in optimization allows to find a solution for the situation when initial data are outside the existence domain and to pull the operation point onto the feasibility boundary by an optimal path. Also it is possible to estimate a static stability margin by utilizing Newton's method in optimization.

As the algorithm based on the Newton’s method in optimization has considerable computational cost and power control cannot be realized in all nodes, the algorithm based on the combination of the Newton-Raphson methods and the Newton’s method in optimization is offered to be utilized

for calculating speed, enhancing the power flow calculation.

II. THEORETICAL BACKGROUND A.Steady-state equations

The system of steady-state equations, in general, can be expressed as follows: where is the vector of parameters given for power flow calculation. In power flow calculation, real and reactive powers are set in each bus except for the slack bus. Ingeneration buses, the modulus of voltage can be fixed. WX,Y is the nonlinear vector function of steady-state equations. Variables Y define the quasi-constant parameters associated with an equivalent circuit of an electrical network. X is a required state vector, it defines steady state of EPS. The dimension of the state vector coincides with the number of nonlinear equations of the system 1. There are various known forms of notation of the steady-state equations. Normally, they are nodal-voltage equations in the form of power balance or in the form of current balance. Complex quantities in these equations can be presented in polar or rectangular coordinates, which leads to a sufficiently large variety forms of the steady-state equations notation. There are variable methods of a nonlinear system of steady-state equations solution. They are united by the incremental vector of independent variables ΔX being searched and the condition of convergence being assessed at each iteration.

B. The Newton's method in optimization

Another way of solving the problem of power flow calculation is related to defining a zero minimum of objective function of squares sum of discrepancies of steady-state

equations:2?

The function minimum 2 is reached at the point where derivatives on all required variables are equal to zero: 3

It is necessary to solve a nonlinear set of equations 3 to find the solution for the problem. Calculating the power flow, which is made by the system of the linear equations with a Hessian matrix at each iteration, is referred to as the Newton's

method in optimization [4]: 4

The Hessian matrix contains two items: 5

During the power flow calculation, the determinant of Hessian matrix is positive round zero and negative value of a determinant of Jacobian .This allows to find the state point during the power flow calculation, when initial point has been outside of the existence domain.

The convergence domain of the solution of the Newton's optimization method is limited by a positive value of the Hessian matrix determinant. The iterative process even for a solvable operating point can converge to an incorrect

solution if initial approximation has been outside convergence