This paper develops a finite approximation approach to find a non-smooth solution of an integral equation of the second kind. The equation solutions with non-smooth kernel having a non-smooth solution have never been studied before. Such equations arise frequently when modeling stochastic systems. We construct a Banach space of (right-continuous) distribution functions and reformulate the problem into an operator equation. We provide general necessary and sufficient conditions that allow us to show convergence of the approximation approach developed in this paper. We then provide two specific choices of approximation sequences and show that the properties of these sequences are sufficient to generate approximate equation solutions that converge to the true solution assuming solution uniqueness and some additional mild regularity conditions. Our analysis is performed under the supremum norm, allowing wider applicability of our results. Worst-case error bounds are also available from solving a linear program. We demonstrate the viability and computational performance of our approach by constructing three examples. The solution of the first example can be constructed manually but demonstrates the correctness and convergence of our approach. The second application example involves stationary distribution equations of a stochastic model and demonstrates the dramatic improvement our approach provides over the use of computer simulation. The third example solves a problem involving an everywhere nondifferentiable function for which no closed-form solution is available.
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Uniqueness of Power Flow Solutions Using Monotonicity and Network Topology
This article establishes sufficient conditions for the uniqueness of AC power flow solutions via the monotonic relationship between real power flow and the phase angle difference. More specifically, we prove that the P-Θ power flow problem has at most one solution for any acyclic or GSP graph. In addition, for arbitrary cyclic power networks, we show that multiple distinct solutions cannot exist under the assumption that angle differences across the lines are bounded by some limit related to the maximal girth of the network. In these cases, a vector of voltage phase angles can be uniquely determined (up to an absolute phase shift) given a vector of real power injections within the realizable range. The implication of this result for the classical power flow analysis is that under the conditions specified above, the problem has a unique physically realizable solution if the phasor voltage magnitudes are fixed. We also introduce a series-parallel operator and show that this operator obtains a reduced and easier-to-analyze model for the power system without changing the uniqueness of power flow solutions.
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- Award ID(s):
- 1808859
- NSF-PAR ID:
- 10282185
- Date Published:
- Journal Name:
- IEEE transactions on control of network systems
- ISSN:
- 2325-5870
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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