The Formulation of a Power Flow Using $d\text{--}q$ Reference Frame Components—Part II: Unbalanced $3\phi$ Systems

Abstract

This paper presents the formulation and testing of the extended $d\text{--}q$ -axis power flow (DQPF) method to analyze power systems that have buses with unbalanced $3\phi$ voltages. The extended DQPF method is based on converting a $3\phi$ system into three networks, which are defined using the $d$ -axis, $q$ -axis, and 0-axis voltage and current components. Each of the three networks is modeled by nodal equations, where the nodal voltages and admittance matrix determine the currents flowing in that network. Moreover, the apparent power mismatches are used (instead of active and reactive power mismatches) in order to reduce computational requirements. This approach offers an accurate representation of buses with unbalanced $3\phi$ voltages resulting from load unbalances or asymmetrical impedances of $3\phi$ transmission lines. The extended DQPF method is implemented for performance testing on different power systems that have buses with unbalanced $3\phi$ voltages. Performance results show good accuracy, fast convergence, and minor sensitivity to the source of voltage unbalance. In addition, performance results reveal that the extended DQPF requires less iterations and lower memory requirements to obtain power flow solutions than other methods.