The Geometry of Power Systems Steady-State Equations– Part I: Power Surface


  • Boris Ayuev JSC "System Operator of the Unified Power System," Moscow, Russia.
  • Viktor Davydov East Siberia State University of Technology and Management, Ulan-Ude, Russia.
  • Petr Erokhin Ural Federal University, Yekaterinburg, Russia.
  • Vladimir Neuymin JSC "Scientific and Technical Center of Unified Power System", Saint Petersburg, Russia.
  • Andrey Pazderin Ural Federal University, Yekaterinburg, Russia.


feasibility region, Jacobian, marginal state, power flow, power surface, power system, slack bus, steady state


Steady-state equations play a fundamental role in the theory of power systems and computation practice. These equations are directly or mediately used almost in all areas of the power system state theory, constituting its basis. This two-part study deals with a geometrical interpretation of steady-state solutions in a power space. Part I considers steady states of the power system as a surface in the power space. A power flow feasibility region is shown to be widely used in power system theories. This region is a projection of this surface along the axis of a slack bus active power onto a subspace of other buses power. The findings have revealed that the obtained power flow feasibility regions, as well as marginal states of the power system, depend on a slack bus location. Part II is devoted to an analytical study of the power surface of power system steady states.