The number of (equilibrium) steady-state solutions of models of power systems

We use powerful analytical tools and modern techniques from algebraic geometry to infer the number of (equilibrium) steady state solutions of models of power systems. The theorems developed also infer the upper bound on the number of solutions of the full-fledged (equilibrium) steady-state equations for various levels of detailed models of power systems. Sufficient conditions are provided which determine the precise number of complex solutions to the load flow. The sufficient conditions are cast in terms of properties of the admittance matrix of the power grid. Consequently, these sufficient conditions are placed on the topology (configuration) of the given power network. When the sufficient conditions are not satisfied, the determined precise number becomes an upper bound on the number of solutions.<
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