The necessary and sufficient condition for static stability of AC power grid is the static stability of each branch. The traditional analytical method for stable boundaries is algebraic extremum method

but it is difficult to analyze due to the high-order nonlinearity of the equation. In order to simplify analysis and enhance physical perception

a phasor geometric analysis method for the static stability boundary of power lines was proposed based on the head-end reference frame impedance model. The static stability boundary and stability domain of apparent power

voltage

power angle

and impedance were obtained. Geometric analysis reveals that the phasor triangle formed by the starting voltage ${\dot{E}}_{i}$

ending voltage ${\dot{U}}_{j}$

and ${\dot{E}}_{i}-{\dot{U}}_{j}$ of the line is geometrically related to the complex power at the end; When the complex power changes along the direction of angle φ

the ${\dot{U}}_{j}$vertex of the triangle moves on the circumcircle arc corresponding to φ

and the triangle area is proportional to the apparent power; When the triangle is isosceles (i.e. ${U}_{j}=|{\dot{E}}_{i}-{\dot{U}}_{j}|$)

the area is maximized

and the apparent power at the end is also the maximum value

which is the power boundary; Triangular isosceles is also the only geometric condition for voltage

power angle

and impedance boundaries. Subsequently

the independent power angle equation of the line and the characteristics and reasons for the variation of the terminal voltage amplitude were obtained using geometric methods. The example shows that the obtained static stable boundary and power angle characteristics are consistent with the convergence results of the PSASP program. Research shows that the geometric method can intuitively represent the process and conditions of static imbalance between complex variables

providing a more intuitive analysis basis for the safe and stable operation of the grid. The research not only has practical value

but also expands the phasor method and demonstrates the theoretical value of the phasor geometric extremum method.