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Power flow formulated with a distributed slack bus involves modeling the active-power output of each generator with three elements: a nominal injection modulated by a fraction of the net-load imbalance allocated via a participation factor. This setup acknowledges generator dynamics and system operations, but it has long been plagued by ambiguous and inconsistent interpretations of its constituent elemental quantities. In this paper, we establish that, with the: i) nominal active-power injections set to be the economic dispatch setpoints, ii) participation factors fixed to be the ones used in automatic generation control, and iii) net-load imbalance considered to be the total load and loss unaccounted in economic dispatch, the power flow solution best matches results from a simulation unto steady state of the system differential algebraic equation (DAE) model. Numerical case studies tailored to the New England test system validate the analysis by demonstrating that solutions obtained from a distributed slack formulation offer lower errors (with respect to DAE-model simulations) compared to the exhaustive set of all single slack bus choices. [ABSTRACT FROM AUTHOR]

Frequency dynamics during primary frequency control is closely related to pre-disturbance operating conditions, the disturbance, and equipment characteristics. How to rigorously express such a relation in a frequency stability constrained optimal dispatch problem is still an open question. This paper presents an optimal power flow (OPF) model for generation re-dispatch that considers dynamic frequency response constraints to ensure frequency stability during primary frequency regulation. The distinct feature of the model is that the dynamic frequency response is formulated as a set of differential algebraic equations (DAEs), which allows considering the transient behavior of frequency and mechanical/electromagnetic power for each generator during primary frequency control period. Based on the solution of the optimization problem, a definition of the primary reserve for each unit is proposed according to the maximum of incremental mechanical power. This definition helps meet the adequacy requirement of primary frequency response and achieve partial frequency restoration. Simulation results on WSCC 3-machine 9-bus system, New England 10-machine 39-bus system and the modified IEEE 54-machine 118-bus system validate the effectiveness of the proposed model and reveal the strong coupling between frequency dynamics and the pre-disturbance generation. [ABSTRACT FROM AUTHOR]