On the Determination of the Region Border Prior to the Limit Steady Modes of Electric Power Systems by the Analysis Method of the Tropical Geometry of the Power Balance Equations

Abstract

The analysis of the known approach (Kirshtein, B.K. and Litvinov, G.L., Autom. Remote Control, 2014, vol. 75, no. 10, pp. 1802–1813.) in which tropical geometry over complex multifields of active power balances is used to estimate the region of existence of the electric power system mode. Its limitations are shown and a new approach is proposed, a criterion is also represented for determining the boundary that precedes the violation of the stability of the energy system due to the restructuring of the tropical set of solutions. The developed approach allows to determine the approach of the power system mode to the limit by the known parameters of the lines and the dynamics of changes of the nodes voltage modules and the nodes load.

APPENDIX

Expressions (4) and (5) are obtained from (3) as follows:

$$\begin{gathered} {{U}^{2}} + \left( {{{p}^{{\operatorname{Re} }}} - j{{p}^{{\operatorname{Im} }}}} \right)(R + jX) = EU{{e}^{{ - j{{\Psi }_{U}}}}}, \\ {{U}^{2}} + {{p}^{{\operatorname{Re} }}}R + {{p}^{{\operatorname{Im} }}}X + j\left( {{{p}^{{\operatorname{Re} }}}X + {{p}^{{\operatorname{Im} }}}R} \right) = EU{{e}^{{ - j{{\Psi }_{U}}}}}. \\ \end{gathered} $$

(A.1)

The balance of modules of expression (A.1) is reduced to a quadratic equation for the unknown $\hat {U}$ = U  ²:

$$a{{\hat {U}}^{2}} + b\hat {U} + c = 0,$$

(A.2)

where

$$a = 1,\quad b = 2\left( {{{p}^{{\operatorname{Re} }}}R + {{p}^{{\operatorname{Im} }}}X} \right) - {{E}^{2}},\quad c = \left( {{{R}^{2}} + {{X}^{2}}} \right)\left[ {{{{\left( {{{p}^{{\operatorname{Re} }}}} \right)}}^{2}} + {{{\left( {{{p}^{{\operatorname{Im} }}}} \right)}}^{2}}} \right].$$

The solution to (A.2) is expression (4). The angle Ψ_U in expression (5) is determined by substituting the found expression (4) for U into Eq. (A.1).

Expression (6) of the article is obtained from the load bus power equation:

$$\dot {p} = {{p}^{{\operatorname{Re} }}} + j{{p}^{{\operatorname{Im} }}} = \dot {U}\left( {\frac{{\dot {E} - \dot {U}}}{{R + jX}}} \right){\kern 1pt} ^*{\kern 1pt} .$$

(A.3)

From (A.3) we obtain

$$\frac{{{{p}^{{\operatorname{Im} }}}}}{{{{p}^{{\operatorname{Re} }}}}} = \tan \phi = \frac{{EX\cos {{\Psi }_{U}} - UX + ER\sin {{\Psi }_{U}}}}{{ER\cos {{\Psi }_{U}} - UR - EX\sin {{\Psi }_{U}}}}.$$

(A.4)

Let’s express the voltage modulus U of the load bus from (A.4)

$$U = E\left( {\cos {{\Psi }_{U}} + \sin {{\Psi }_{U}}\left( {\frac{{R + X\tan \phi }}{{X - R\tan \phi }}} \right)} \right)$$

and put it into the expression for active power obtained from (A.3):

$${{p}^{{\operatorname{Re} }}} = \frac{U}{{({{R}^{2}} + {{X}^{2}})}}\left[ {R(E\cos {{\Psi }_{U}} - U) - EX\sin {{\Psi }_{U}}} \right].$$

(A.5)

Taking the derivative of (A.5) with respect to the angle Ψ_U and equating it to zero, we obtain the expression

$$\frac{{d{{p}^{{\operatorname{Re} }}}}}{{d{{\Psi }_{U}}}} = \frac{{\sin (2{{\Psi }_{U}})(R + X\tan \phi ) - \cos (2{{\Psi }_{U}})(R\tan \phi - X)}}{{{{{(X - R\tan \phi )}}^{2}}}} = 0,$$

(A.6)

from which we determine

$$\tan (2{{\Psi }_{U}}) = \frac{{R\tan \phi - X}}{{R + X\tan \phi }}.$$

(A.7)

The resulting expression (A.7) is equivalent to Eq. (6).