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.
Thermal units can include multiple terms added to the objective function, such as no-load cost, turn-on/off cost, fixed cost and variable cost. In addition, variable costs can be linear, quadratic or piecewise-linear formulations. These methods are properly described in the cost function page.
The use of reactive power variables and constraints will depend on the network model used, i.e., whether it uses (or does not use) reactive power. If the network model is purely active power-based, reactive power variables and related constraints are not created.
Reserve variables for services are not included in the formulation, albeit their inclusion change the variables, expressions, constraints and objective functions created. A detailed description of the implications in the optimization models is described in the Service formulation section.
Table of Contents
ThermalBasicDispatch
ThermalDispatchNoMin
ThermalCompactDispatch
ThermalStandardDispatch
ThermalBasicUnitCommitment
ThermalBasicCompactUnitCommitment
ThermalStandardUnitCommitment
ThermalMultiStartUnitCommitment
- Valid configurations
ThermalBasicDispatch
PowerSimulations.ThermalBasicDispatch
— TypeFormulation type to enable basic dispatch without any intertemporal (ramp) constraints
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
PowerSimulations.ThermalDispatchNoMin
— TypeFormulation type to enable basic dispatch without any intertemporal constraints and relaxed minimum generation. May not work with non-convex PWL cost definitions
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
PowerSimulations.ThermalCompactDispatch
— TypeFormulation type to enable thermal compact dispatch
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
PowerSimulations.ThermalStandardDispatch
— TypeFormulation type to enable standard dispatch with a range and enforce intertemporal ramp constraints
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
PowerSimulations.ThermalBasicUnitCommitment
— TypeFormulation type to enable basic unit commitment representation without any intertemporal (ramp, min on/off time) constraints
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
PowerSimulations.ThermalBasicCompactUnitCommitment
— TypeFormulation type to enable thermal compact commitment without intertemporal (ramp, min on/off time) constraints
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
PowerSimulations.ThermalCompactUnitCommitment
— TypeFormulation type to enable thermal compact commitment
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
PowerSimulations.ThermalStandardUnitCommitment
— TypeFormulation type to enable standard unit commitment with intertemporal constraints and simplified startup profiles
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
PowerSimulations.ThermalMultiStartUnitCommitment
— TypeFormulation type to enable pg-lib commitment formulation with startup/shutdown profiles
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 DeviceModel
s for subtypes of ThermalGen
include the following:
Valid DeviceModel | Device Type | Formulation |
---|---|---|
DeviceModel(ThermalMultiStart, FixedOutput) | ThermalMultiStart | FixedOutput |
DeviceModel(ThermalStandard, FixedOutput) | ThermalStandard | FixedOutput |
DeviceModel(ThermalMultiStart, ThermalBasicDispatch) | ThermalMultiStart | ThermalBasicDispatch |
DeviceModel(ThermalStandard, ThermalBasicDispatch) | ThermalStandard | ThermalBasicDispatch |
DeviceModel(ThermalMultiStart, ThermalCompactDispatch) | ThermalMultiStart | ThermalCompactDispatch |
DeviceModel(ThermalStandard, ThermalCompactDispatch) | ThermalStandard | ThermalCompactDispatch |
DeviceModel(ThermalMultiStart, ThermalDispatchNoMin) | ThermalMultiStart | ThermalDispatchNoMin |
DeviceModel(ThermalStandard, ThermalDispatchNoMin) | ThermalStandard | ThermalDispatchNoMin |
DeviceModel(ThermalMultiStart, ThermalStandardDispatch) | ThermalMultiStart | ThermalStandardDispatch |
DeviceModel(ThermalStandard, ThermalStandardDispatch) | ThermalStandard | ThermalStandardDispatch |
DeviceModel(ThermalMultiStart, ThermalBasicCompactUnitCommitment) | ThermalMultiStart | ThermalBasicCompactUnitCommitment |
DeviceModel(ThermalStandard, ThermalBasicCompactUnitCommitment) | ThermalStandard | ThermalBasicCompactUnitCommitment |
DeviceModel(ThermalMultiStart, ThermalCompactUnitCommitment) | ThermalMultiStart | ThermalCompactUnitCommitment |
DeviceModel(ThermalStandard, ThermalCompactUnitCommitment) | ThermalStandard | ThermalCompactUnitCommitment |
DeviceModel(ThermalMultiStart, ThermalMultiStartUnitCommitment) | ThermalMultiStart | ThermalMultiStartUnitCommitment |
DeviceModel(ThermalMultiStart, ThermalBasicUnitCommitment) | ThermalMultiStart | ThermalBasicUnitCommitment |
DeviceModel(ThermalStandard, ThermalBasicUnitCommitment) | ThermalStandard | ThermalBasicUnitCommitment |
DeviceModel(ThermalMultiStart, ThermalStandardUnitCommitment) | ThermalMultiStart | ThermalStandardUnitCommitment |
DeviceModel(ThermalStandard, ThermalStandardUnitCommitment) | ThermalStandard | ThermalStandardUnitCommitment |