ThermalGen Formulations

Thermal generation formulations define the optimization models that describe thermal units mathematical model in different operational settings, such as economic dispatch and unit commitment.

Table of Contents

1. ThermalBasicDispatch
2. ThermalDispatchNoMin
3. ThermalCompactDispatch
4. ThermalStandardDispatch
5. ThermalBasicUnitCommitment
6. ThermalBasicCompactUnitCommitment
7. ThermalStandardUnitCommitment
8. ThermalMultiStartUnitCommitment
9. Valid configurations

ThermalBasicDispatch

Variables:

• ActivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $p^\text{th}$

• ReactivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $q^\text{th}$

Static Parameters:

• $P^\text{th,min}$ = PowerSystems.get_active_power_limits(device).min
• $P^\text{th,max}$ = PowerSystems.get_active_power_limits(device).max
• $Q^\text{th,min}$ = PowerSystems.get_reactive_power_limits(device).min
• $Q^\text{th,max}$ = PowerSystems.get_reactive_power_limits(device).max

Objective:

Add a cost to the objective function depending on the defined cost structure of the thermal unit by adding it to its ProductionCostExpression.

Expressions:

Adds $p^\text{th}$ to the ActivePowerBalance expression and $q^\text{th}$ to the ReactivePowerBalance, to be used in the supply-balance constraint depending on the network model used.

Constraints:

For each thermal unit creates the range constraints for its active and reactive power depending on its static parameters.

\[\begin{align*} & P^\text{th,min} \le p^\text{th}_t \le P^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \\ & Q^\text{th,min} \le q^\text{th}_t \le Q^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \end{align*}\]

ThermalDispatchNoMin

Variables:

• ActivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $p^\text{th}$

• ReactivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $q^\text{th}$

Static Parameters:

• $P^\text{th,max}$ = PowerSystems.get_active_power_limits(device).max
• $Q^\text{th,min}$ = PowerSystems.get_reactive_power_limits(device).min
• $Q^\text{th,max}$ = PowerSystems.get_reactive_power_limits(device).max

Objective:

Add a cost to the objective function depending on the defined cost structure of the thermal unit by adding it to its ProductionCostExpression.

Expressions:

Adds $p^\text{th}$ to the ActivePowerBalance expression and $q^\text{th}$ to the ReactivePowerBalance, to be used in the supply-balance constraint depending on the network model used.

Constraints:

For each thermal unit creates the range constraints for its active and reactive power depending on its static parameters.

\[\begin{align} & 0 \le p^\text{th}_t \le P^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \\ & Q^\text{th,min} \le q^\text{th}_t \le Q^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \end{align}\]

ThermalCompactDispatch

Variables:

• PowerAboveMinimumVariable:

  • Bounds: [0.0, ]
  • Symbol: $\Delta p^\text{th}$

• ReactivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $q^\text{th}$

Auxiliary Variables:

• PowerOutput:
  • Symbol: $P^\text{th}$
  • Definition: $P^\text{th} = \text{on}^\text{th}P^\text{min} + \Delta p^\text{th}$

Static Parameters:

• $P^\text{th,min}$ = PowerSystems.get_active_power_limits(device).min
• $P^\text{th,max}$ = PowerSystems.get_active_power_limits(device).max
• $Q^\text{th,min}$ = PowerSystems.get_reactive_power_limits(device).min
• $Q^\text{th,max}$ = PowerSystems.get_reactive_power_limits(device).max
• $R^\text{th,up}$ = PowerSystems.get_ramp_limits(device).up
• $R^\text{th,dn}$ = PowerSystems.get_ramp_limits(device).down

Variable Value Parameters:

• $\text{on}^\text{th}$: Used in feedforwards to define if the unit is on/off at each time-step from another problem. If no feedforward is used, the parameter takes a {0,1} value if the unit is available or not.

Objective:

Add a cost to the objective function depending on the defined cost structure of the thermal unit by adding it to its ProductionCostExpression.

Expressions:

Adds $\text{on}^\text{th}P^\text{th,min} + \Delta p^\text{th}$ to the ActivePowerBalance expression and $q^\text{th}$ to the ReactivePowerBalance, to be used in the supply-balance constraint depending on the network model used.

Constraints:

For each thermal unit creates the range constraints for its active and reactive power depending on its static parameters. It also implements ramp constraints for the active power variable.

\[\begin{align*} & 0 \le \Delta p^\text{th}_t \le \text{on}^\text{th}_t\left(P^\text{th,max} - P^\text{th,min}\right), \quad \forall t\in \{1, \dots, T\} \\ & \text{on}^\text{th}_t Q^\text{th,min} \le q^\text{th}_t \le \text{on}^\text{th}_t Q^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \\ & -R^\text{th,dn} \le \Delta p_1^\text{th} - \Delta p^\text{th, init} \le R^\text{th,up} \\ & -R^\text{th,dn} \le \Delta p_t^\text{th} - \Delta p_{t-1}^\text{th} \le R^\text{th,up}, \quad \forall t\in \{2, \dots, T\} \end{align*}\]

ThermalStandardDispatch

Variables:

• ActivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $p^\text{th}$

• ReactivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $q^\text{th}$

Static Parameters:

• $P^\text{th,min}$ = PowerSystems.get_active_power_limits(device).min
• $P^\text{th,max}$ = PowerSystems.get_active_power_limits(device).max
• $Q^\text{th,min}$ = PowerSystems.get_reactive_power_limits(device).min
• $Q^\text{th,max}$ = PowerSystems.get_reactive_power_limits(device).max
• $R^\text{th,up}$ = PowerSystems.get_ramp_limits(device).up
• $R^\text{th,dn}$ = PowerSystems.get_ramp_limits(device).down

Objective:

Add a cost to the objective function depending on the defined cost structure of the thermal unit by adding it to its ProductionCostExpression.

Expressions:

Adds $p^\text{th}$ to the ActivePowerBalance expression and $q^\text{th}$ to the ReactivePowerBalance, to be used in the supply-balance constraint depending on the network model used.

Constraints:

For each thermal unit creates the range constraints for its active and reactive power depending on its static parameters.

\[\begin{align*} & P^\text{th,min} \le p^\text{th}_t \le P^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \\ & Q^\text{th,min} \le q^\text{th}_t \le Q^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \\ & -R^\text{th,dn} \le p_1^\text{th} - p^\text{th, init} \le R^\text{th,up} \\ & -R^\text{th,dn} \le p_t^\text{th} - p_{t-1}^\text{th} \le R^\text{th,up}, \quad \forall t\in \{2, \dots, T\} \end{align*}\]

ThermalBasicUnitCommitment

Variables:

• ActivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $p^\text{th}$

• ReactivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $q^\text{th}$

• OnVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $u_t^\text{th}$

• StartVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $v_t^\text{th}$

• StopVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $w_t^\text{th}$

Static Parameters:

• $P^\text{th,min}$ = PowerSystems.get_active_power_limits(device).min
• $P^\text{th,max}$ = PowerSystems.get_active_power_limits(device).max
• $Q^\text{th,min}$ = PowerSystems.get_reactive_power_limits(device).min
• $Q^\text{th,max}$ = PowerSystems.get_reactive_power_limits(device).max

Objective:

Add a cost to the objective function depending on the defined cost structure of the thermal unit by adding it to its ProductionCostExpression.

Expressions:

Adds $p^\text{th}$ to the ActivePowerBalance expression and $q^\text{th}$ to the ReactivePowerBalance, to be used in the supply-balance constraint depending on the network model used.

Constraints:

For each thermal unit creates the range constraints for its active and reactive power depending on its static parameters. In addition, it creates the commitment constraint to turn on/off the device.

\[\begin{align*} & u_t^\text{th} P^\text{th,min} \le p^\text{th}_t \le u_t^\text{th} P^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \\ & u_t^\text{th} Q^\text{th,min} \le q^\text{th}_t \le u_t^\text{th} Q^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \\ & u_1^\text{th} = u^\text{th,init} + v_1^\text{th} - w_1^\text{th} \\ & u_t^\text{th} = u_{t-1}^\text{th} + v_t^\text{th} - w_t^\text{th}, \quad \forall t \in \{2,\dots,T\} \\ & v_t^\text{th} + w_t^\text{th} \le 1, \quad \forall t \in \{1,\dots,T\} \end{align*}\]

ThermalBasicCompactUnitCommitment

Variables:

• PowerAboveMinimumVariable:

  • Bounds: [0.0, ]
  • Symbol: $\Delta p^\text{th}$

• ReactivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $q^\text{th}$

• OnVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $u_t^\text{th}$

• StartVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $v_t^\text{th}$

• StopVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $w_t^\text{th}$

Auxiliary Variables:

• PowerOutput:
  • Symbol: $P^\text{th}$
  • Definition: $P^\text{th} = u^\text{th}P^\text{min} + \Delta p^\text{th}$

Static Parameters:

• $P^\text{th,min}$ = PowerSystems.get_active_power_limits(device).min
• $P^\text{th,max}$ = PowerSystems.get_active_power_limits(device).max
• $Q^\text{th,min}$ = PowerSystems.get_reactive_power_limits(device).min
• $Q^\text{th,max}$ = PowerSystems.get_reactive_power_limits(device).max

Objective:

Add a cost to the objective function depending on the defined cost structure of the thermal unit by adding it to its ProductionCostExpression.

Expressions:

Adds $u^\text{th}P^\text{th,min} + \Delta p^\text{th}$ to the ActivePowerBalance expression and $q^\text{th}$ to the ReactivePowerBalance, to be used in the supply-balance constraint depending on the network model used.

Constraints:

For each thermal unit creates the range constraints for its active and reactive power depending on its static parameters. In addition, it creates the commitment constraint to turn on/off the device.

\[\begin{align*} & 0 \le \Delta p^\text{th}_t \le u^\text{th}_t\left(P^\text{th,max} - P^\text{th,min}\right), \quad \forall t\in \{1, \dots, T\} \\ & u_t^\text{th} Q^\text{th,min} \le q^\text{th}_t \le u_t^\text{th} Q^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \\ & u_1^\text{th} = u^\text{th,init} + v_1^\text{th} - w_1^\text{th} \\ & u_t^\text{th} = u_{t-1}^\text{th} + v_t^\text{th} - w_t^\text{th}, \quad \forall t \in \{2,\dots,T\} \\ & v_t^\text{th} + w_t^\text{th} \le 1, \quad \forall t \in \{1,\dots,T\} \end{align*}\]

ThermalCompactUnitCommitment

Variables:

• PowerAboveMinimumVariable:

  • Bounds: [0.0, ]
  • Symbol: $\Delta p^\text{th}$

• ReactivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $q^\text{th}$

• OnVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $u_t^\text{th}$

• StartVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $v_t^\text{th}$

• StopVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $w_t^\text{th}$

Auxiliary Variables:

• PowerOutput:

  • Symbol: $P^\text{th}$
  • Definition: $P^\text{th} = u^\text{th}P^\text{min} + \Delta p^\text{th}$

• TimeDurationOn:

  • Symbol: $V_t^\text{th}$
  • Definition: Computed post optimization by adding consecutive turned on variable $u_t^\text{th}$

• TimeDurationOff:

  • Symbol: $W_t^\text{th}$
  • Definition: Computed post optimization by adding consecutive turned off variable $1 - u_t^\text{th}$

Static Parameters:

• $P^\text{th,min}$ = PowerSystems.get_active_power_limits(device).min
• $P^\text{th,max}$ = PowerSystems.get_active_power_limits(device).max
• $Q^\text{th,min}$ = PowerSystems.get_reactive_power_limits(device).min
• $Q^\text{th,max}$ = PowerSystems.get_reactive_power_limits(device).max
• $R^\text{th,up}$ = PowerSystems.get_ramp_limits(device).up
• $R^\text{th,dn}$ = PowerSystems.get_ramp_limits(device).down
• $D^\text{min,up}$ = PowerSystems.get_time_limits(device).up
• $D^\text{min,dn}$ = PowerSystems.get_time_limits(device).down

Objective:

Add a cost to the objective function depending on the defined cost structure of the thermal unit by adding it to its ProductionCostExpression.

Expressions:

Adds $u^\text{th}P^\text{th,min} + \Delta p^\text{th}$ to the ActivePowerBalance expression and $q^\text{th}$ to the ReactivePowerBalance, to be used in the supply-balance constraint depending on the network model used.

Constraints:

For each thermal unit creates the range constraints for its active and reactive power depending on its static parameters. It also creates the commitment constraint to turn on/off the device.

\[\begin{align*} & 0 \le \Delta p^\text{th}_t \le u^\text{th}_t\left(P^\text{th,max} - P^\text{th,min}\right), \quad \forall t\in \{1, \dots, T\} \\ & u_t^\text{th} Q^\text{th,min} \le q^\text{th}_t \le u_t^\text{th} Q^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \\ & -R^\text{th,dn} \le \Delta p_1^\text{th} - \Delta p^\text{th, init} \le R^\text{th,up} \\ & -R^\text{th,dn} \le \Delta p_t^\text{th} - \Delta p_{t-1}^\text{th} \le R^\text{th,up}, \quad \forall t\in \{2, \dots, T\} \\ & u_1^\text{th} = u^\text{th,init} + v_1^\text{th} - w_1^\text{th} \\ & u_t^\text{th} = u_{t-1}^\text{th} + v_t^\text{th} - w_t^\text{th}, \quad \forall t \in \{2,\dots,T\} \\ & v_t^\text{th} + w_t^\text{th} \le 1, \quad \forall t \in \{1,\dots,T\} \end{align*}\]

In addition, this formulation adds duration constraints, i.e. minimum-up time and minimum-down time constraints. The duration constraints are added over the start times looking backwards.

The duration times $D^\text{min,up}$ and $D^\text{min,dn}$ are processed to be used in multiple of the time-steps, given the resolution of the specific problem. In addition, parameters $D^\text{init,up}$ and $D^\text{init,dn}$ are used to identify how long the unit was on or off, respectively, before the simulation started.

Minimum up-time constraint for $t \in \{1,\dots T\}$:

\[\begin{align*} & \text{If } t \leq D^\text{min,up} - D^\text{init,up} \text{ and } D^\text{init,up} > 0: \\ & 1 + \sum_{i=t-D^\text{min,up} + 1}^t v_i^\text{th} \leq u_t^\text{th} \quad \text{(for } i \text{ in the set of time steps).} \\ & \text{Otherwise:} \\ & \sum_{i=t-D^\text{min,up} + 1}^t v_i^\text{th} \leq u_t^\text{th} \end{align*}\]

Minimum down-time constraint for $t \in \{1,\dots T\}$:

\[\begin{align*} & \text{If } t \leq D^\text{min,dn} - D^\text{init,dn} \text{ and } D^\text{init,up} > 0: \\ & 1 + \sum_{i=t-D^\text{min,dn} + 1}^t w_i^\text{th} \leq 1 - u_t^\text{th} \quad \text{(for } i \text{ in the set of time steps).} \\ & \text{Otherwise:} \\ & \sum_{i=t-D^\text{min,dn} + 1}^t w_i^\text{th} \leq 1 - u_t^\text{th} \end{align*}\]

ThermalStandardUnitCommitment

Variables:

• ActivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $p^\text{th}$

• ReactivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $q^\text{th}$

• OnVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $u_t^\text{th}$

• StartVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $v_t^\text{th}$

• StopVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $w_t^\text{th}$

Auxiliary Variables:

• TimeDurationOn:

  • Symbol: $V_t^\text{th}$
  • Definition: Computed post optimization by adding consecutive turned on variable $u_t^\text{th}$

• TimeDurationOff:

  • Symbol: $W_t^\text{th}$
  • Definition: Computed post optimization by adding consecutive turned off variable $1 - u_t^\text{th}$

Static Parameters:

• $P^\text{th,min}$ = PowerSystems.get_active_power_limits(device).min
• $P^\text{th,max}$ = PowerSystems.get_active_power_limits(device).max
• $Q^\text{th,min}$ = PowerSystems.get_reactive_power_limits(device).min
• $Q^\text{th,max}$ = PowerSystems.get_reactive_power_limits(device).max
• $R^\text{th,up}$ = PowerSystems.get_ramp_limits(device).up
• $R^\text{th,dn}$ = PowerSystems.get_ramp_limits(device).down
• $D^\text{min,up}$ = PowerSystems.get_time_limits(device).up
• $D^\text{min,dn}$ = PowerSystems.get_time_limits(device).down

Objective:

Add a cost to the objective function depending on the defined cost structure of the thermal unit by adding it to its ProductionCostExpression.

Expressions:

Adds $p^\text{th}$ to the ActivePowerBalance expression and $q^\text{th}$ to the ReactivePowerBalance, to be used in the supply-balance constraint depending on the network model used.

Constraints:

For each thermal unit creates the range constraints for its active and reactive power depending on its static parameters. It also creates the commitment constraint to turn on/off the device.

\[\begin{align*} & u^\text{th}_t P^\text{th,min} \le p^\text{th}_t \le u^\text{th}_t P^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \\ & u_t^\text{th} Q^\text{th,min} \le q^\text{th}_t \le u_t^\text{th} Q^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \\ & -R^\text{th,dn} \le p_1^\text{th} - p^\text{th, init} \le R^\text{th,up} \\ & -R^\text{th,dn} \le p_t^\text{th} - p_{t-1}^\text{th} \le R^\text{th,up}, \quad \forall t\in \{2, \dots, T\} \\ & u_1^\text{th} = u^\text{th,init} + v_1^\text{th} - w_1^\text{th} \\ & u_t^\text{th} = u_{t-1}^\text{th} + v_t^\text{th} - w_t^\text{th}, \quad \forall t \in \{2,\dots,T\} \\ & v_t^\text{th} + w_t^\text{th} \le 1, \quad \forall t \in \{1,\dots,T\} \end{align*}\]

In addition, this formulation adds duration constraints, i.e. minimum-up time and minimum-down time constraints. The duration constraints are added over the start times looking backwards.

The duration times $D^\text{min,up}$ and $D^\text{min,dn}$ are processed to be used in multiple of the time-steps, given the resolution of the specific problem. In addition, parameters $D^\text{init,up}$ and $D^\text{init,dn}$ are used to identify how long the unit was on or off, respectively, before the simulation started.

Minimum up-time constraint for $t \in \{1,\dots T\}$:

\[\begin{align*} & \text{If } t \leq D^\text{min,up} - D^\text{init,up} \text{ and } D^\text{init,up} > 0: \\ & 1 + \sum_{i=t-D^\text{min,up} + 1}^t v_i^\text{th} \leq u_t^\text{th} \quad \text{(for } i \text{ in the set of time steps).} \\ & \text{Otherwise:} \\ & \sum_{i=t-D^\text{min,up} + 1}^t v_i^\text{th} \leq u_t^\text{th} \end{align*}\]

Minimum down-time constraint for $t \in \{1,\dots T\}$:

\[\begin{align*} & \text{If } t \leq D^\text{min,dn} - D^\text{init,dn} \text{ and } D^\text{init,up} > 0: \\ & 1 + \sum_{i=t-D^\text{min,dn} + 1}^t w_i^\text{th} \leq 1 - u_t^\text{th} \quad \text{(for } i \text{ in the set of time steps).} \\ & \text{Otherwise:} \\ & \sum_{i=t-D^\text{min,dn} + 1}^t w_i^\text{th} \leq 1 - u_t^\text{th} \end{align*}\]

ThermalMultiStartUnitCommitment

Variables:

• PowerAboveMinimumVariable:

  • Bounds: [0.0, ]
  • Symbol: $\Delta p^\text{th}$

• ReactivePowerVariable:

  • Bounds: [0.0, ]
  • Symbol: $q^\text{th}$

• OnVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $u_t^\text{th}$

• StartVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $v_t^\text{th}$

• StopVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $w_t^\text{th}$

• ColdStartVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $x_t^\text{th}$

• WarmStartVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $y_t^\text{th}$

• HotStartVariable:

  • Bounds: $\{0,1\}$
  • Symbol: $z_t^\text{th}$

Auxiliary Variables:

• PowerOutput:

  • Symbol: $P^\text{th}$
  • Definition: $P^\text{th} = u^\text{th}P^\text{min} + \Delta p^\text{th}$

• TimeDurationOn:

  • Symbol: $V_t^\text{th}$
  • Definition: Computed post optimization by adding consecutive turned on variable $u_t^\text{th}$

• TimeDurationOff:

  • Symbol: $W_t^\text{th}$
  • Definition: Computed post optimization by adding consecutive turned off variable $1 - u_t^\text{th}$

Static Parameters:

• $P^\text{th,min}$ = PowerSystems.get_active_power_limits(device).min
• $P^\text{th,max}$ = PowerSystems.get_active_power_limits(device).max
• $Q^\text{th,min}$ = PowerSystems.get_reactive_power_limits(device).min
• $Q^\text{th,max}$ = PowerSystems.get_reactive_power_limits(device).max
• $R^\text{th,up}$ = PowerSystems.get_ramp_limits(device).up
• $R^\text{th,dn}$ = PowerSystems.get_ramp_limits(device).down
• $D^\text{min,up}$ = PowerSystems.get_time_limits(device).up
• $D^\text{min,dn}$ = PowerSystems.get_time_limits(device).down
• $D^\text{cold}$ = PowerSystems.get_start_time_limits(device).cold
• $D^\text{warm}$ = PowerSystems.get_start_time_limits(device).warm
• $D^\text{hot}$ = PowerSystems.get_start_time_limits(device).hot
• $P^\text{th,startup}$ = PowerSystems.get_power_trajectory(device).startup
• $P^\text{th, shdown}$ = PowerSystems.get_power_trajectory(device).shutdown

Objective:

Add a cost to the objective function depending on the defined cost structure of the thermal unit by adding it to its ProductionCostExpression.

Expressions:

Adds $u^\text{th}P^\text{th,min} + \Delta p^\text{th}$ to the ActivePowerBalance expression and $q^\text{th}$ to the ReactivePowerBalance, to be used in the supply-balance constraint depending on the network model used.

Constraints:

For each thermal unit creates the range constraints for its active and reactive power depending on its static parameters. It also creates the commitment constraint to turn on/off the device.

\[\begin{align*} & 0 \le \Delta p^\text{th}_t \le u^\text{th}_t\left(P^\text{th,max} - P^\text{th,min}\right), \quad \forall t\in \{1, \dots, T\} \\ & u_t^\text{th} Q^\text{th,min} \le q^\text{th}_t \le u_t^\text{th} Q^\text{th,max}, \quad \forall t\in \{1, \dots, T\} \\ & -R^\text{th,dn} \le \Delta p_1^\text{th} - \Delta p^\text{th, init} \le R^\text{th,up} \\ & -R^\text{th,dn} \le \Delta p_t^\text{th} - \Delta p_{t-1}^\text{th} \le R^\text{th,up}, \quad \forall t\in \{2, \dots, T\} \\ & u_1^\text{th} = u^\text{th,init} + v_1^\text{th} - w_1^\text{th} \\ & u_t^\text{th} = u_{t-1}^\text{th} + v_t^\text{th} - w_t^\text{th}, \quad \forall t \in \{2,\dots,T\} \\ & v_t^\text{th} + w_t^\text{th} \le 1, \quad \forall t \in \{1,\dots,T\} \\ & \max\{P^\text{th,max} - P^\text{th,shdown}, 0\} \cdot w_1^\text{th} \le u^\text{th,init} (P^\text{th,max} - P^\text{th,min}) - P^\text{th,init} \end{align*}\]

In addition, this formulation adds duration constraints, i.e. minimum-up time and minimum-down time constraints. The duration constraints are added over the start times looking backwards.

The duration times $D^\text{min,up}$ and $D^\text{min,dn}$ are processed to be used in multiple of the time-steps, given the resolution of the specific problem. In addition, parameters $D^\text{init,up}$ and $D^\text{init,dn}$ are used to identify how long the unit was on or off, respectively, before the simulation started.

Minimum up-time constraint for $t \in \{1,\dots T\}$:

\[\begin{align*} & \text{If } t \leq D^\text{min,up} - D^\text{init,up} \text{ and } D^\text{init,up} > 0: \\ & 1 + \sum_{i=t-D^\text{min,up} + 1}^t v_i^\text{th} \leq u_t^\text{th} \quad \text{(for } i \text{ in the set of time steps).} \\ & \text{Otherwise:} \\ & \sum_{i=t-D^\text{min,up} + 1}^t v_i^\text{th} \leq u_t^\text{th} \end{align*}\]

Minimum down-time constraint for $t \in \{1,\dots T\}$:

\[\begin{align*} & \text{If } t \leq D^\text{min,dn} - D^\text{init,dn} \text{ and } D^\text{init,up} > 0: \\ & 1 + \sum_{i=t-D^\text{min,dn} + 1}^t w_i^\text{th} \leq 1 - u_t^\text{th} \quad \text{(for } i \text{ in the set of time steps).} \\ & \text{Otherwise:} \\ & \sum_{i=t-D^\text{min,dn} + 1}^t w_i^\text{th} \leq 1 - u_t^\text{th} \end{align*}\]

Finally, multi temperature start/stop constraints are implemented using the following constraints:

\[\begin{align*} & v_t^\text{th} = x_t^\text{th} + y_t^\text{th} + z_t^\text{th}, \quad \forall t \in \{1, \dots, T\} \\ & z_t^\text{th} \le \sum_{i \in [D^\text{hot}, D^\text{warm})}w_{t-i}^\text{th}, \quad \forall t \in \{D^\text{warm}, \dots, T\} \\ & y_t^\text{th} \le \sum_{i \in [D^\text{warm}, D^\text{cold})}w_{t-i}^\text{th}, \quad \forall t \in \{D^\text{cold}, \dots, T\} \\ & (D^\text{warm} - 1) z_t^\text{th} + (1 - z_t^\text{th}) M^\text{big} \ge \sum_{i=1}^t (1 - u_i^\text{th}) + D^\text{init,hot}, \quad \forall t \in \{1, \dots, T\} \\ & D^\text{hot} z_t^\text{th} \le \sum_{i=1}^t (1 - u_i^\text{th}) + D^\text{init,hot}, \quad \forall t \in \{1, \dots, T\} \\ & (D^\text{cold} - 1) y_t^\text{th} + (1 - y_t^\text{th}) M^\text{big} \ge \sum_{i=1}^t (1 - u_i^\text{th}) + D^\text{init,warm}, \quad \forall t \in \{1, \dots, T\} \\ & D^\text{warm} y_t^\text{th} \le \sum_{i=1}^t (1 - u_i^\text{th}) + D^\text{init,warm}, \quad \forall t \in \{1, \dots, T\} \\ \end{align*}\]

Valid configurations

Valid DeviceModels for subtypes of ThermalGen include the following: