Load Flow in Multiphase Distribution Networks: Existence, Uniqueness, Non-Singularity and Linear Models

Abstract

This paper considers unbalanced multiphase distribution systems with generic topology and different load models, and extends the Z-bus iterative load-flow algorithm based on a fixed-point interpretation of the AC load-flow equations. Explicit conditions for existence and uniqueness of load-flow solutions are presented. These conditions also guarantee convergence of the load-flow algorithm to the unique solution. The proposed methodology is applicable to generic systems featuring (i) wye connections; (ii) ungrounded delta connections; (iii) a combination of wye-connected and delta-connected sources/loads; and, (iv) a combination of line-to-line and line-to-grounded-neutral devices at the secondary of distribution transformers. Further, a sufficient condition for the non-singularity of the load-flow Jacobian is proposed. Finally, linear load-flow models are derived, and their approximation accuracy is analyzed. Theoretical results are corroborated through experiments on IEEE test feeders. 1 We note that models (iii) and (iv) are the same in terms of the mathematical formulation. However, from the practical point of view, model (iv) reflects an actual mode of connection on the secondary side of the distribution transformer, whereas model (iii) pertains to the case where different distribution transformers are lumped in the same bus for network reduction purposes.

A tractable formulation for multi-period linearized optimal power flow in presence of thermostatically controlled loads

“…The opinion of the authors is that this assumption is not reasonable (for instance, it renders the model ill-defined in the nominal case with σ = 0). By contrast, condition (18) makes no assumption on the noise, it can be still satisfied in the nominal case and it is only related to the Euler approximation. It requires the discretization to be coarse enough in the temperature or fine enough in time, so that the model does not transfer more agents from a state than the ones actually present.…”

“…It is well known that the non-linear AC power flow equations lead to a nonconvex formulation of the AC OPF problem. Here, we apply a linear approximations such as the ones proposed in, e.g., [18]- [20] (and pertinent references therein), to obtain an approximate linear relationships between voltage magnitudes and net injected powers of the form:…”

Towards robustness guarantees for feedback-based optimization

“…at the PCC. In order to obtain a bound of the form (19), we sampled 10,000 operating points by randomly choosing both u and w from the power injections data and constructed the Jacobian from the controllable injections to the voltage magnitude using the method from [22]. The error ∂π(u n , w n ) − Π nom of each sample is shown in Figure 4.…”

Stochastic Hosting Capacity in LV Distribution Networks

“…To reduce the computational complexity, a linear model was developed for the load flow function f v . We utilise a linearization from [13], which we briefly summarise here. This model has the advantage that it can be linearised around a known load flow solution, increasing the model accuracy, and is valid for three phase unbalanced networks.…”

“…where v 0 ∈ C 3 is the slack bus voltage, v L ∈ C 3(N bus −1) the voltages at load buses, s 0 ∈ C 3 power injections at the slack bus, and s L ∈ C 3(N bus −1) the load injections. From [13], this implies a model of the form…”