PowerSystems.ElectricLoad Formulations

Electric load formulations define the optimization models that describe load units (demand) mathematical model in different operational settings, such as economic dispatch and unit commitment.

Table of contents

1. StaticPowerLoad
2. PowerLoadInterruption
3. PowerLoadDispatch
4. PowerLoadShift
5. Valid configurations

StaticPowerLoad

Variables:

No variables are created

Time Series Parameters:

Uses the max_active_power timeseries parameter to determine the demand value at each time-step

Expressions:

Subtracts the parameters listed above from the respective active and reactive power balance expressions created by the selected Network Formulations.

Constraints:

No constraints are created

PowerLoadInterruption

Variables:

• ActivePowerVariable:

  • Bounds: [0.0, ]
  • Default initial value: 0.0
  • Symbol: $p^\text{ld}$

• ReactivePowerVariable:

  • Bounds: [0.0, ]
  • Default initial value: 0.0
  • Symbol: $q^\text{ld}$

• OnVariable:

  • Bounds: $\{0,1\}$
  • Default initial value: 1
  • Symbol: $u^\text{ld}$

Static Parameters:

• $P^\text{ld,max}$ = PowerSystems.get_max_active_power(device)
• $Q^\text{ld,max}$ = PowerSystems.get_max_reactive_power(device)

Time Series Parameters:

Objective:

Creates an objective function term based on the FunctionData Options where the quantity term is defined as $p^\text{ld}$.

Expressions:

• Subtract$p^\text{ld}$ and $q^\text{ld}$ terms and to the respective active and reactive power balance expressions created by the selected Network Formulations

Constraints:

\[\begin{aligned} & p_t^\text{ld} \le u_t^\text{ld} \cdot \text{ActivePowerTimeSeriesParameter}_t, \quad \forall t \in \{1,\dots, T\} \\ & q_t^\text{re} = \text{pf} \cdot p_t^\text{re}, \quad \forall t \in \{1,\dots, T\} \end{aligned}\]

on which $\text{pf} = \sin(\arctan(Q^\text{ld,max}/P^\text{ld,max}))$.

PowerLoadDispatch

Variables:

• ActivePowerVariable:

  • Bounds: [0.0, ]
  • Default initial value: PowerSystems.get_active_power(device)
  • Symbol: $p^\text{ld}$

• ReactivePowerVariable:

  • Bounds: [0.0, ]
  • Default initial value: PowerSystems.get_reactive_power(device)
  • Symbol: $q^\text{ld}$

Static Parameters:

• $P^\text{ld,max}$ = PowerSystems.get_max_active_power(device)
• $Q^\text{ld,max}$ = PowerSystems.get_max_reactive_power(device)

Time Series Parameters:

Objective:

Creates an objective function term based on the FunctionData Options where the quantity term is defined as $p^\text{ld}$.

Expressions:

• Subtract$p^\text{ld}$ and $q^\text{ld}$ terms and to the respective active and reactive power balance expressions created by the selected Network Formulations

Constraints:

\[\begin{aligned} & p_t^\text{ld} \le \text{ActivePowerTimeSeriesParameter}_t, \quad \forall t \in \{1,\dots, T\}\\ & q_t^\text{ld} = \text{pf} \cdot p_t^\text{ld}, \quad \forall t \in \{1,\dots, T\}\\ \end{aligned}\]

on which $\text{pf} = \sin(\arctan(Q^\text{ld,max}/P^\text{ld,max}))$.

PowerLoadShift

Variables:

• ShiftUpActivePowerVariable:

  • Bounds: [0.0, ShiftUpActivePowerTimeSeriesParameter]
  • Default initial value: 0.0
  • Symbol: $p^\text{shift,up}$

• ShiftDownActivePowerVariable:

  • Bounds: [0.0, ShiftDownActivePowerTimeSeriesParameter]
  • Default initial value: 0.0
  • Symbol: $p^\text{shift,dn}$

• ReactivePowerVariable (AC network models only):

  • Bounds: [0.0, ]
  • Default initial value: PowerSystems.get_reactive_power(device)
  • Symbol: $q^\text{ld}$

Static Parameters:

• $P^\text{ld,max}$ = PowerSystems.get_max_active_power(device)
• $Q^\text{ld,max}$ = PowerSystems.get_max_reactive_power(device)

Time Series Parameters:

Expressions:

• Defines the RealizedShiftedLoad expression per device per time step:

  \[p_t^\text{realized} = \text{ActivePowerTimeSeriesParameter}_t + p_t^\text{shift,up} - p_t^\text{shift,dn}, \quad \forall t \in \{1,\dots,T\}\]

• Subtracts $p_t^\text{realized}$ from the active power balance expression of the selected Network Formulations.

Objective:

Creates objective function terms based on the FunctionData Options for both shift variables:

• A cost term on $p^\text{shift,up}$ (typically zero or negative, rewarding shifting up)
• A cost term on $p^\text{shift,dn}$ (typically positive, penalizing shifting down)

Constraints:

\[\begin{aligned} & \sum_{t=1}^{T} \left( p_t^\text{shift,up} - p_t^\text{shift,dn} \right) = 0 \\ & \sum_{t=1}^{T_\text{sub}} \left( p_t^\text{shift,up} - p_t^\text{shift,dn} \right) = 0 \quad \text{(if \texttt{additional\_balance\_interval} is set)} \end{aligned}\]

\[p_t^\text{realized} \ge 0, \quad \forall t \in \{1,\dots,T\}\]

\[p_t^\text{shift,up} \le \text{ShiftUpActivePowerTimeSeriesParameter}_t, \quad \forall t \in \{1,\dots,T\}\]

\[p_t^\text{shift,dn} \le \text{ShiftDownActivePowerTimeSeriesParameter}_t, \quad \forall t \in \{1,\dots,T\}\]

\[\sum_{\tau=1}^{t} \left( p_\tau^\text{shift,dn} - p_\tau^\text{shift,up} \right) \ge 0, \quad \forall t \in \{1,\dots,T\}\]

\[q_t^\text{ld} = \text{pf} \cdot p_t^\text{realized}, \quad \forall t \in \{1,\dots,T\}\]

Valid configurations

Valid DeviceModels for subtypes of ElectricLoad include the following: