Network Formulations

Network formulations are used to describe how the network and buses are handled when constructing constraints. The most common constraint decided by the network formulation is the supply-demand balance constraint. Available Network Models are:

PowerModels.jl available formulations:

• Exact non-convex models: ACPPowerModel, ACRPowerModel, ACTPowerModel.
• Linear approximations: DCPPowerModel, NFAPowerModel.
• Quadratic approximations: DCPLLPowerModel, LPACCPowerModel
• Quadratic relaxations: SOCWRPowerModel, SOCWRConicPowerModel, SOCBFPowerModel, SOCBFConicPowerModel, QCRMPowerModel, QCLSPowerModel.
• SDP relaxations: SDPWRMPowerModel, SparseSDPWRMPowerModel.

All of these formulations are described in the PowerModels.jl documentation and will not be described here.

CopperPlatePowerModel

Variables:

If Slack variables are enabled:

• SystemBalanceSlackUp:
  • Bounds: [0.0, ]
  • Default initial value: 0.0
  • Default proportional cost: 1e6
  • Symbol: $p^\text{sl,up}$
• SystemBalanceSlackDown:
  • Bounds: [0.0, ]
  • Default initial value: 0.0
  • Default proportional cost: 1e6
  • Symbol: $p^\text{sl,dn}$

Objective:

Add a large proportional cost to the objective function if slack variables are used $+ (p^\text{sl,up} + p^\text{sl,dn}) \cdot 10^6$

Expressions:

Adds $p^\text{sl,up}$ and $p^\text{sl,dn}$ terms to the respective active power balance expressions ActivePowerBalance created by this CopperPlatePowerModel network formulation.

Constraints:

Adds the CopperPlateBalanceConstraint to balance the active power of all components available in the system

\[\begin{align} & \sum_{c \in \text{components}} p_t^c = 0, \quad \forall t \in \{1, \dots, T\} \end{align}\]

AreaBalancePowerModel

Variables: If Slack variables are enabled:

• SystemBalanceSlackUp by area:
  • Bounds: [0.0, ]
  • Default initial value: 0.0
  • Default proportional cost: 1e6
  • Symbol: $p^\text{sl,up}$
• SystemBalanceSlackDown by area:
  • Bounds: [0.0, ]
  • Default initial value: 0.0
  • Default proportional cost: 1e6
  • Symbol: $p^\text{sl,dn}$

Objective:

Adds $p^\text{sl,up}$ and $p^\text{sl,dn}$ terms to the respective active power balance expressions ActivePowerBalance per area.

Expressions:

Creates ActivePowerBalance expressions for each area that then are used to balance active power for all buses within a single area.

Constraints:

Adds the CopperPlateBalanceConstraint to balance the active power of all components available in an area.

\[\begin{align} & \sum_{c \in \text{components}_a} p_t^c = 0, \quad \forall a\in \{1,\dots, A\}, t \in \{1, \dots, T\} \end{align}\]

PTDFPowerModel

Variables:

If Slack variables are enabled:

• SystemBalanceSlackUp:
  • Bounds: [0.0, ]
  • Default initial value: 0.0
  • Default proportional cost: 1e6
  • Symbol: $p^\text{sl,up}$
• SystemBalanceSlackDown:
  • Bounds: [0.0, ]
  • Default initial value: 0.0
  • Default proportional cost: 1e6
  • Symbol: $p^\text{sl,dn}$

Objective:

Add a large proportional cost to the objective function if slack variables are used $+ (p^\text{sl,up} + p^\text{sl,dn}) \cdot 10^6$

Expressions:

Adds $p^\text{sl,up}$ and $p^\text{sl,dn}$ terms to the respective system-wide active power balance expressions ActivePowerBalance created by this CopperPlatePowerModel network formulation. In addition, it creates ActivePowerBalance expressions for each bus to be used in the calculation of branch flows.

Constraints:

Adds the CopperPlateBalanceConstraint to balance the active power of all components available in the system

\[\begin{align} & \sum_{c \in \text{components}} p_t^c = 0, \quad \forall t \in \{1, \dots, T\} \end{align}\]

In addition creates NodalBalanceActiveConstraint for HVDC buses balance, if DC components are connected to an HVDC network.