Active research activity in power systems areas has focused on developing computational methods to solve load flow equations where a key question is the maximum number of solutions. Although several upper bounds exist, recent studies have hinted that much sharper upper bounds that depend on the topology of underlying power networks may exist. This paper provides a significant refinement of these observations. We also develop a geometric construction called adjacency polytope that accurately captures the topology of a power network and is immensely useful in the computation of the solution bound. Finally, we highlight the significant implications of the development of such solution bounds in numerically solving load flow equations. Index Terms-Power systems, power system analysis computing, power system control, load flow. E NGINEERS are regularly required to perform power flow computations for designing, operating, and controlling power systems [1] . In this, a key mathematical problem is to solve a system of multivariate nonlinear equations known as the load flow equations. In general, load flow equations may have more than one solutions [2] . There are quite a few existing methods for finding one or many solutions [3]-[29] (see [30] for a recent review). Out of the few methods that guarantee to find all load flow solutions, i.e., the interval based approach [19] , Gröbner bases technique [20]-[23] and the numerical polynomial homotopy continuation (NPHC) method [24]-[29], the NPHC method appears most promising in scalability with increasing system sizes in that it has already found all load flow solutions of up to IEEE 14-bus systems [26] (and 18 oscillators case for the Kuramoto model [29] ) and is inherently Manuscript
Meanwhile, the work in REF establishes topology dependent upper bounds on the number of power flow solutions.
On the Network Topology Dependent Solution Count of the Algebraic Load Flow Equations
IEEE Transactions on Power Systems
IEEE Transactions on Power Systems
Engineering Mathematics Computer Science