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This paper is concerned with the topological space of normalized quaternion-valued positive definite functions on an arbitrary abelian group G, especially its convex characteristics. There are two main results. Firstly, we prove that the extreme elements in the family of such functions are exactly the homomorphisms from G to the sphere group S, i.e., the unit 3-sphere in the quaternion algebra. Secondly, we real a phenomenon which does not exist in the complex setting: The compact convex set of such functions is not a Bauer simplex except when G is of exponent less than or equal to 2. In contrast, its complex counterpart is always a Bauer simplex, as is well known. We also present an integral representation for such functions as an application and some other minor interesting results.

Continuation power flow (CPF) calculation is very important for analyzing voltage stability of power system. CPF calculation needs to deal with non-smooth constraints such as the generator buses reactive power limits. It is still a technical challenge to determine the step size while dealing with above non-smooth constraints in CPF calculation. In this paper, an asymptotic numerical method (ANM) based on Fischer‐Burmeister (FB) function, is proposed to calculate CPF. We first used complementarity constraints to cope with non-smooth issues and introduced the FB function to formulate the complementarity constraints. Meanwhile, we introduced new variables for substitution to meet the quadratic function requirements of ANM. Compared with the conventional predictor-corrector method combining with heuristic PV-PQ (PV and PQ are used to describe bus types. PV means that the active power and voltage of the bus are known. PQ means that the active and reactive power of bus are known.) bus type switching, ANM can effectively solve the PV-PQ bus type switching problem in CPF calculation. Furthermore, to assure high efficiency, ANM can rapidly approach the voltage collapse point by self-adaptive step size adjustment and constant Jacobian matrix used for power series expansion. However, conventional CPF needs proper step set in advance and calculates Jacobian matrix for each iteration. Numerical tests on a nine-bus network and a 182-bus network validate that the proposed method is more robust than existing methods.