HydroPowerSimulations Formulations

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

Table of Contents

1. HydroDispatchRunOfRiver
2. HydroDispatchReservoirBudget
3. HydroDispatchReservoirStorage
4. HydroDispatchPumpedStorage
5. HydroCommitmentRunOfRiver
6. HydroCommitmentReservoirBudget
7. HydroCommitmentReservoirStorage

HydroDispatchRunOfRiver

Variables:

• PowerSimulations.ActivePowerVariable:

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

• PowerSimulations.ReactivePowerVariable:

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

Auxiliary Variables:

• HydroEnergyOutput:
  • Symbol: $E^\text{hy,out}$

The HydroEnergyOutput is computed as the energy used at each time step from the hydro, computed simply as $E^\text{hy,out} \cdot \Delta T$, where $\Delta T$ is the duration (in hours) of each time step.

Static Parameters:

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

Time Series Parameters:

Uses the max_active_power timeseries parameter to limit the available active power at each time-step. If the timeseries parameter is not included, the power is limited by $P^\text{th,max}$.

Objective:

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

Expressions:

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

Constraints:

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

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

HydroDispatchReservoirBudget

Variables:

• PowerSimulations.ActivePowerVariable:

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

• PowerSimulations.ReactivePowerVariable:

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

Auxiliary Variables:

• HydroEnergyOutput:
  • Symbol: $E^\text{hy,out}$

The HydroEnergyOutput is computed as the energy used at each time step from the hydro, computed simply as $E^\text{hy,out} \cdot \Delta T$, where $\Delta T$ is the duration (in hours) of each time step.

Static Parameters:

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

Time Series Parameters:

Uses the hydro_budget timeseries parameter to limit the active power usage throughout all the time steps.

Objective:

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

Expressions:

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

Constraints:

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

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

HydroDispatchReservoirStorage

Variables:

• PowerSimulations.ActivePowerVariable:

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

• PowerSimulations.ReactivePowerVariable:

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

• PowerSimulations.EnergyVariable:

  • Bounds: $[0.0, E^\text{max}]$
  • Symbol: $e$

• WaterSpillageVariable:

  • Bounds: [0.0, ]
  • Symbol: $s$

• HydroEnergySurplusVariable:

  • Bounds: $[-E^\text{max}, 0.0]$
  • Symbol: $e^\text{surplus}$

• HydroEnergyShortageVariable:

  • Bounds: $[0.0, E^\text{max}]$
  • Symbol: $e^\text{shortage}$

Auxiliary Variables:

• HydroEnergyOutput:
  • Symbol: $E^\text{hy,out}$

The HydroEnergyOutput is computed as the energy used at each time step from the hydro, computed simply as $E^\text{hy,out} \cdot \Delta T$, where $\Delta T$ is the duration (in hours) of each time step.

Static Parameters:

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

Initial Conditions:

The PowerSimulations.InitialEnergyLevel: $e_0$ is used as the initial condition for the energy level of the reservoir.

Time Series Parameters:

Uses the storage_target timeseries parameter to set a storage target of the reservoir and the inflow timeseries parameter to obtain the inflow to the reservoir.

Objective:

Add a cost to the objective function depending on the defined cost structure of the hydro unit by adding it to its ProductionCostExpression. This model has shortage costs for the energy target that are also added to the objective function. In case of no shortage cost are specified, the model will turn-off the shortage variable to avoid infeasible/unbounded problems.

Expressions:

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

Constraints:

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

\[\begin{align*} & P^\text{hy,min} \le p^\text{hy}_t \le P^\text{hy,max}, \quad \forall t\in \{1, \dots, T\} \\ & Q^\text{hy,min} \le q^\text{hy}_t \le Q^\text{hy,max}, \quad \forall t\in \{1, \dots, T\} \\ & e_{t} = e_{t-1} - \Delta T (p^\text{hy}_t - s_t) + \text{InflowTimeSeriesParameter}_t, \quad \forall t\in \{1, \dots, T\} \\ & e_t + e^\text{shortage} + e^\text{surplus} = \text{EnergyTargetTimeSeriesParameter}_t, \quad \forall t\in \{1, \dots, T\} \end{align*}\]

HydroDispatchPumpedStorage

This formulation is not available with reactive power. This formulation must be used with PowerSystems.HydroPumpedStorage component.

Variables:

• PowerSimulations.ActivePowerOutVariable:

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

• PowerSimulations.ActivePowerInVariable:

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

• HydroEnergyVariableUp:

  • Bounds: $[0.0, E^\text{max,up}]$
  • Symbol: $e^\text{up}$

• HydroEnergyVariableDown:

  • Bounds: $[0.0, E^\text{max,dn}]$
  • Symbol: $e^\text{dn}$

• WaterSpillageVariable:

  • Bounds: [0.0, ]
  • Symbol: $s$

If the attribute reservation is set to true, the following variable is created:

• PowerSimulations.ReservationVariable:
  • Bounds: $\{0,1\}$
  • Symbol: $ss^\text{hy}$

If reservation is set to false (default), then the hydro pumped storage is allowed to pump and discharge simultaneously at each time step.

Static Parameters:

• $P^\text{out,min}$ = PowerSystems.get_active_power_limits(device).min
• $P^\text{out,max}$ = PowerSystems.get_active_power_limits(device).max
• $P^\text{in,min}$ = PowerSystems.get_active_power_limits_pump(device).min
• $P^\text{in,max}$ = PowerSystems.get_active_power_limits_pump(device).max
• $E^\text{max,up}$ = PowerSystems.get_storage_capacity(device).up
• $E^\text{max,dn}$ = PowerSystems.get_storage_capacity(device).down
• $\eta^\text{pump}$ = PowerSystems.get_pump_efficiency(device)

Initial Conditions:

The InitialHydroEnergyLevelUp: $e_0^\text{up}$ is used as the initial condition for the energy level of the upper reservoir, while InitialHydroEnergyLevelDown: $e_0^\text{dn}$ is used as the initial condition for the energy level of the lower reservoir.

Time Series Parameters:

Uses the inflow and outflow timeseries to obtain the inflow and outflow to the reservoir. inflow corresponds to the inflow into the upper (up) reservoir, while outflow corresponds to the outflow from the lower (down) reservoir.

Objective:

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

Expressions:

Adds $(p^\text{hy,out} - p^\text{hy,in})$ to the PowerSimulations.ActivePowerBalance to be used in the supply-balance constraint depending on the network model used.

Constraints:

If reservation = true, the limits are given by:

\[\begin{align*} & ss_t^\text{hy} P^\text{out,min} \le p^\text{hy,out}_t \le ss_t^\text{hy} P^\text{out,max}, \quad \forall t\in \{1, \dots, T\} \\ & (1-ss_t^\text{hy}) P^\text{in,min} \le p^\text{hy,in}_t \le (1 - ss_t^\text{hy}) P^\text{in,max}, \quad \forall t\in \{1, \dots, T\} \end{align*}\]

If reservation = false, then:

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

The remaining constraints are:

\[\begin{align*} e_{t}^\text{up} = e_{t-1}^\text{up} + \left (p_t^\text{hy,in} - \frac{p_t^\text{hy,out} + s_t}{\eta^\text{pump}} \right) \Delta T + \text{InflowTimeSeriesParameter}_t, \quad \forall t\in \{1, \dots, T\} \\ e_{t}^\text{dn} = e_{t-1}^\text{dn} + \left (p_t^\text{hy,out} + s_t - \frac{p_t^\text{hy,in}}{\eta^\text{pump}} \right) \Delta T - \text{OutflowTimeSeriesParameter}_t, \quad \forall t\in \{1, \dots, T\} \end{align*}\]

HydroCommitmentRunOfRiver

Variables:

• PowerSimulations.ActivePowerVariable:

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

• PowerSimulations.ReactivePowerVariable:

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

• PowerSimulations.OnVariable:

  • Bounds: $\{0, 1\}$
  • Symbol: $u^\text{hy}$

Auxiliary Variables:

• HydroEnergyOutput:
  • Symbol: $E^\text{hy,out}$

The HydroEnergyOutput is computed as the energy used at each time step from the hydro, computed simply as $E^\text{hy,out} \cdot \Delta T$, where $\Delta T$ is the duration (in hours) of each time step.

Static Parameters:

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

Time Series Parameters:

Uses the max_active_power timeseries parameter to limit the available active power at each time-step. If the timeseries parameter is not included, the power is limited by $P^\text{th,max}$.

Objective:

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

Expressions:

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

Constraints:

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

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

HydroCommitmentReservoirBudget

Variables:

• PowerSimulations.ActivePowerVariable:

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

• PowerSimulations.ReactivePowerVariable:

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

• PowerSimulations.OnVariable:

  • Bounds: $\{0, 1\}$
  • Symbol: $u^\text{hy}$

Auxiliary Variables:

• HydroEnergyOutput:
  • Symbol: $E^\text{hy,out}$

The HydroEnergyOutput is computed as the energy used at each time step from the hydro, computed simply as $E^\text{hy,out} \cdot \Delta T$, where $\Delta T$ is the duration (in hours) of each time step.

Static Parameters:

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

Time Series Parameters:

Uses the hydro_budget timeseries parameter to limit the active power usage throughout all the time steps.

Objective:

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

Expressions:

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

Constraints:

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

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

HydroDispatchReservoirStorage

Variables:

• PowerSimulations.ActivePowerVariable:

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

• PowerSimulations.ReactivePowerVariable:

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

• PowerSimulations.OnVariable:

  • Bounds: $\{0, 1\}$
  • Symbol: $u^\text{hy}$

• PowerSimulations.EnergyVariable:

  • Bounds: $[0.0, E^\text{max}]$
  • Symbol: $e$

• WaterSpillageVariable:

  • Bounds: [0.0, ]
  • Symbol: $s$

• HydroEnergySurplusVariable:

  • Bounds: $[-E^\text{max}, 0.0]$
  • Symbol: $e^\text{surplus}$

• HydroEnergyShortageVariable:

  • Bounds: $[0.0, E^\text{max}]$
  • Symbol: $e^\text{shortage}$

Auxiliary Variables:

• HydroEnergyOutput:
  • Symbol: $E^\text{hy,out}$

The HydroEnergyOutput is computed as the energy used at each time step from the hydro, computed simply as $E^\text{hy,out} \cdot \Delta T$, where $\Delta T$ is the duration (in hours) of each time step.

Static Parameters:

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

Initial Conditions:

The PowerSimulations.InitialEnergyLevel: $e_0$ is used as the initial condition for the energy level of the reservoir.

Time Series Parameters:

Uses the storage_target timeseries parameter to set a storage target of the reservoir and the inflow timeseries parameter to obtain the inflow to the reservoir.

Objective:

Add a cost to the objective function depending on the defined cost structure of the hydro unit by adding it to its ProductionCostExpression. This model has shortage costs for the energy target that are also added to the objective function. In case of no shortage cost are specified, the model will turn-off the shortage variable to avoid infeasible/unbounded problems.

Expressions:

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

Constraints:

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

\[\begin{align*} & u_t^\text{hy} P^\text{hy,min} \le p^\text{hy}_t \le u_t^\text{hy} P^\text{hy,max}, \quad \forall t\in \{1, \dots, T\} \\ & u_t^\text{hy} Q^\text{hy,min} \le q^\text{hy}_t \le u_t^\text{hy} Q^\text{hy,max}, \quad \forall t\in \{1, \dots, T\} \\ & e_{t} = e_{t-1} - \Delta T (p^\text{hy}_t - s_t) + \text{InflowTimeSeriesParameter}_t, \quad \forall t\in \{1, \dots, T\} \\ & e_t + e^\text{shortage} + e^\text{surplus} = \text{EnergyTargetTimeSeriesParameter}_t, \quad \forall t\in \{1, \dots, T\} \end{align*}\]