Newton Power Flow Methods for Unbalanced Three-Phase Distribution Networks

[...]central grid operators or transmission system operators (TSOs) of the power system must have different approaches for maintaining and operating the electrical grid because in this case, the main purpose of the operator has been adjusted to interconnect the various active distribution networks. In the current mismatch formulation, it is possible to choose the reactive power Qjp at the bus j for a given phase p as an unknown variable as a voltage magnitude |V| or an angle δ . Since Qjabc is an unknown variable, all the first-order partial derivatives ∂Δ(Iir)abc∂Qjabc and ∂(ΔIim)abc∂Qjabc must be computed as: ∂Δ(Iir)abc∂Qjabc=∂Δ(Iir)a∂Qja∂Δ(Iir)a∂Qjb∂Δ(Iir)a∂Qjc∂Δ(Iir)b∂Qja∂Δ(Iir)b∂Qjb∂Δ(Iir)b∂Qjc∂Δ(Iir)c∂Qja∂Δ(Iir)c∂Qjb∂Δ(Iir)c∂Qjc,∂Δ(Iim)abc∂Qjabc=∂Δ(Iim)a∂Qja∂Δ(Iim)a∂Qjb∂Δ(Iim)a∂Qjc∂Δ(Iim)b∂Qja∂Δ(Iim)b∂Qjb∂Δ(Iim)b∂Qjc∂Δ(Iim)c∂Qja∂Δ(Iim)c∂Qjb∂Δ(Iim)c∂Qjc where if i≠j ∂Δ(Iir)abc(x→)∂Qjabc=000000000,∂Δ(Iim)abc(x→)∂Qjabc=000000000 if i=j ∂Δ(Ijr)abc(x→)∂Qjabc=sinδja|Vja |sp000sinδjb|Vjb |sp000sinδjc|Vjc |sp,∂Δ(Ijr)abc(x→)∂Qjabc=−cosδja|Vja |sp000cosδjb|Vjb |sp000cosδjc|Vjc |sp When the derivatives ∂Δ(Iir)abc∂Qjabc and ∂Δ(Iim)abc∂Qjabc are added into the Jacobian matrix J(x→) , the Jacobian matrix becomes a rectangular matrix. [...]NR-c-pol and NR-c-car versions were more stable for the changes and therefore they can be applied to any unbalanced distribution networks with high R/X ratios and loading conditions. [...]of the numerical experiment, the polar current mismatch (NR-c-pol) and the Cartesian current mismatch (NR-c-car) versions developed in [65] and extended in this paper perform the best for both balanced and unbalanced distribution networks.