Edit on GitHub

Network Formulations

This section gives an overview of the three-phase power flow formulations that are available to perform state estimation in PowerModelsDistributionStateEstimation. All formulations except the Reduced ones are imported from PowerModels or PowerModelsDistribution. These are only a subset of the formulations available in these two packages. For further information please refer to their official documentation.

Type Hierarchy

Formulations (or "PowerModels") follow the type hierarchy of PowerModels and PowerModelsDistribution, reported here for convenience for the relevant cases. At the top of the type hierarchy are abstract types. Three exact nonlinear (non-convex) models are available at the top level:

abstract type PowerModelsDistribution.AbstractUnbalancedACPModel <: PowerModelsDistribution.AbstractUnbalancedPowerModel end
abstract type PowerModelsDistribution.AbstractUnbalancedACRModel <: PowerModelsDistribution.AbstractUnbalancedPowerModel end
abstract type PowerModelsDistribution.AbstractUnbalancedIVRModel <: PowerModelsDistribution.AbstractUnbalancedACRModel end

Abstract Models types are then used as the type parameter for PowerModels:

mutable struct PowerModelsDistribution.ACPUPowerModel <: PowerModelsDistribution.AbstractUnbalancedACPModel PowerModelsDistribution.@pmd_fields end
mutable struct PowerModelsDistribution.ACRUPowerModel <: PowerModelsDistribution.AbstractUnbalancedACRModel PowerModelsDistribution.@pmd_fields end
mutable struct PowerModelsDistribution.IVRUPowerModel <: PowerModelsDistribution.AbstractUnbalancedIVRModel PowerModelsDistribution.@pmd_fields end

A "reduced" version of each of the three formulations above is derived in PowerModelsDistributionStateEstimation:

mutable struct PowerModelsDistributionStateEstimation.ReducedACPUPowerModel <: PowerModelsDistribution.AbstractUnbalancedACPModel PowerModelsDistribution.@pmd_fields end
mutable struct PowerModelsDistributionStateEstimation.ReducedACRUPowerModel <: PowerModelsDistribution.AbstractUnbalancedACRModel PowerModelsDistribution.@pmd_fields end
mutable struct PowerModelsDistributionStateEstimation.ReducedIVRUPowerModel <: PowerModelsDistribution.AbstractUnbalancedIVRModel PowerModelsDistribution.@pmd_fields end

AbstractReducedModel = Union{ReducedACRUPowerModel, ReducedACPUPowerModel}

The reduced models are still exact for networks like those made available in the ENWL database, where there are no cable ground admittance, storage elements and active switches. A positive semi-definite (SDP) relaxation is also made available for state estimation in PowerModelsDistributionStateEstimation. The SDP model belongs to the following categories: conic models and branch flow (BF) models, and there relevant type structure is the following:

abstract type PowerModelsDistribution.AbstractUBFModel <: PowerModelsDistribution.AbstractUnbalancedPowerModel end
abstract type PowerModelsDistribution.AbstractUBFConicModel <: PowerModelsDistribution.AbstractUBFModel end
abstract type PowerModelsDistribution.AbstractConicUBFModel <: PowerModels.AbstractBFConicModel end
abstract type PowerModelsDistribution.SDPUBFModel <: PowerModelsDistribution.AbstractConicUBFModel end

mutable struct PowerModelsDistribution.SDPUBFPowerModel <: PowerModelsDistribution.SDPUBFModel PowerModelsDistribution.@pmd_fields end

where UBF stands for unbalanced branch flow. Finally, a linear unbalanced branch flow model is available for state estimation: the LPUBFDiagModel, better known as LinDist3FlowModel.

abstract type PowerModelsDistribution.AbstractLPUBFModel <: PowerModelsDistribution.AbstractNLPUBFModel end
abstract type PowerModelsDistribution.LPUBFDiagModel <: PowerModelsDistribution.AbstractLPUBFModel end
const PowerModelsDistribution.LinDist3FlowModel = PowerModelsDistribution.LPUBFDiagModel

mutable struct PowerModelsDistribution.LPUBFDiagPowerModel <: PowerModelsDistribution.LPUBFDiagModel PowerModelsDistribution.@pmd_fields end
const PowerModelsDistribution.LinDist3FlowPowerModel = PowerModelsDistribution.LPUBFDiagPowerModel

Details on the Formulations

This sub-section reports for convenience the relevant literature for the formulations used in PowerModelsDistributionStateEstimation and is again a reduced version of the official PowerModelsDistribution documentation, available here.

AbstractACPModel

  • Formulation without shunts: Mahdad, B., Bouktir, T., & Srairi, K. (2006). A three-phase power flow modelization: a tool for optimal location and control of FACTS devices in unbalanced power systems. In IEEE Industrial Electronics IECON (pp. 2238–2243).

See also:

  • Carpentier, J. (1962) Contribution to the economic dispatch problem. In Bulletin de la Societe Francoise des Electriciens, vol. 3 no. 8, pp. 431-447.
  • Cain, M. B., O' Neill, R. P. & Castillo, A. (2012). History of optimal power flow and Models. Available online

AbstractACRModel

See:

  • Cain, M. B., O' Neill, R. P. & Castillo, A. (2012). History of optimal power flow and Models. Available online

AbstractIVRModel

  • O' Neill, R. P., Castillo, A. & Cain, M. B. (2012). The IV formulation and linear approximations of the ac optimal power flow problem. Available online

SDPUBFModel

  • Gan, L., & Low, S. H. (2014). Convex relaxations and linear approximation for optimal power flow in multiphase radial networks. In PSSC (pp. 1–9). Wroclaw, Poland. doi:10.1109/PSCC.2014.7038399

LPUBFDiagModel or LinDist3FlowModel

  • Sankur, M. D., Dobbe, R., Stewart, E., Callaway, D. S., & Arnold, D. B. (2016). A linearized power flow model for optimization in unbalanced distribution systems. arXiv:1606.04492v2