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. HydroDispatchRunOfRiverBudget
3. HydroCommitmentRunOfRiver
4. HydroTurbineEnergyDispatch
5. HydroTurbineBilinearDispatch
6. HydroTurbineWaterLinearDispatch
7. HydroEnergyModelReservoir
8. HydroWaterModelReservoir
9. HydroWaterFactorModel

HydroDispatchRunOfRiver

Formulation type to add injection variables constrained by a maximum injection time series for PowerSystems.HydroGen

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.

Adds $p^\text{hy}$ to HydroServedReserveUpExpression/HydroServedReserveDownExpression expressions to keep track of served reserve up/down for energy calculations.

Constraints:

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

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

HydroDispatchRunOfRiverBudget

Formulation type to add injection variables constrained by a maximum injection time series and a budget time series for PowerSystems.HydroGen. This formulation is equivalent than HydroDispatchRunOfRiver, but with the option to include a time series with a budget.

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 $P^\text{th,max}$.

Uses the hydro_budget timeseries parameter to limit the sum of available active power throughout the horizon. An attribute hydro_budget_interval can be used to also limit the sum during a specified interval (smaller than the horizon) during a simulation.

Attributes:

During the model setup, the attribute hydro_budget_interval can be used to set-up a budget constraint with smaller interval than the full horizon of the simulation. For example if a problem has an horizon of Hour(48), with an hourly resolution, the attribute hydro_budget_interval = Hour(24) can be used to also impose a budget constraint in the first 24 hours (in addition of the full 48 hours constraint).

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.

Adds $p^\text{hy}$ to HydroServedReserveUpExpression/HydroServedReserveDownExpression expressions to keep track of served reserve up/down for energy calculations.

Constraints:

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

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

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.

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

HydroTurbineEnergyDispatch

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 turbine, computed simply as $E^\text{hy,out} = p^\text{hy} \cdot \Delta T$, 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

Objective:

Add a cost to the objective function depending on the defined cost structure of the hydro turbine 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 turbine creates the range constraints for its active and reactive power depending on its static parameters, and energy balance constraints for the reservoir.

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

HydroTurbineBilinearDispatch

Variables:

• PowerSimulations.ActivePowerVariable:

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

• PowerSimulations.ReactivePowerVariable:

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

• HydroTurbineFlowRateVariable:

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

Expressions added:

• TotalHydroFlowRateTurbineOutgoing:
  • Symbol: $f^\text{hy,out}$

The TotalHydroFlowRateTurbineOutgoing is computed as the total water flow outgoing of a turbine. This is helpful if the turbine is feed by multiple upstream reservoirs.

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
• $F^\text{hy, max}$ = PowerSystems.get_outflow_limits(device).max
• $F^\text{hy, min}$ = PowerSystems.get_outflow_limits(device).min
• $\text{powH}$ = PowerSystems.get_powerhouse_elevation(device)

Objective:

Add a cost to the objective function depending on the defined cost structure of the hydro turbine 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 turbine creates the range constraints for its active, reactive power and water flow depending on its static parameters. By defining the set $\mathcal{R}^{up}$ as the set of upstream reservoirs connected to the turbine, the turbine power output relationship can be added.

\[\begin{align*} & P^\text{hy,min} \le p^\text{hy}_t \le P^\text{hy,max}, \quad \forall t\in \{1, \dots, T\} \\ & Q^\text{hy,min} \le q^\text{hy}_t \le Q^\text{hy,max}, \quad \forall t\in \{1, \dots, T\} \\ & F^\text{hy,min} \le f^\text{hy}_t \le F^\text{hy,max}, \quad \forall t\in \{1, \dots, T\} \\ & p^\text{hy} = 10^{-6} \cdot \text{SysBasePower}^{-1} \cdot \rho g \sum_{r \in \mathcal{R}^{up}} f^\text{hy}_{r,t} \left(h_{r,t} + \text{inH} - \text{powH} \right) \end{align*}\]

where $h$ is the effective hydraulic head (above the intake), the $\text{inH}$ is the intake elevation (in meters above the sea level), $\text{powH}$ is the powerhouse elevation (in meters above the sea level), $g = 9.81~ \text{m/s}^2$ is the gravitational constant, and $\rho = 1000~ \text{kg/m}^3$ is the water density. Finally, the term $10^{-6} \cdot \text{SysBasePower}^{-1}$ is used to transform the power in Watts into per-unit.

HydroTurbineWaterLinearDispatch

Variables:

• PowerSimulations.ActivePowerVariable:

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

• PowerSimulations.ReactivePowerVariable:

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

• HydroTurbineFlowRateVariable:

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

Expressions added:

• TotalHydroFlowRateTurbineOutgoing:
  • Symbol: $f^\text{hy,out}$

The TotalHydroFlowRateTurbineOutgoing is computed as the total water flow outgoing of a turbine. This is helpful if the turbine is feed by multiple upstream reservoirs.

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
• $F^\text{hy, max}$ = PowerSystems.get_outflow_limits(device).max
• $F^\text{hy, min}$ = PowerSystems.get_outflow_limits(device).min
• $\text{powH}$ = PowerSystems.get_powerhouse_elevation(device)

Objective:

Add a cost to the objective function depending on the defined cost structure of the hydro turbine 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 turbine creates the range constraints for its active, reactive power and water flow depending on its static parameters. By defining the set $\mathcal{R}^{up}$ as the set of upstream reservoirs connected to the turbine, the turbine power output relationship can be added.

\[\begin{align*} & P^\text{hy,min} \le p^\text{hy}_t \le P^\text{hy,max}, \quad \forall t\in \{1, \dots, T\} \\ & Q^\text{hy,min} \le q^\text{hy}_t \le Q^\text{hy,max}, \quad \forall t\in \{1, \dots, T\} \\ & F^\text{hy,min} \le f^\text{hy}_t \le F^\text{hy,max}, \quad \forall t\in \{1, \dots, T\} \\ & p^\text{hy} = 10^{-6} \cdot \text{SysBasePower}^{-1} \cdot \rho g \sum_{r \in \mathcal{R}^{up}} f^\text{hy}_{r,t} \left(\text{inH} - \text{powH} \right) \end{align*}\]

where $\text{inH}$ is the intake elevation (in meters above the sea level), $\text{powH}$ is the powerhouse elevation (in meters above the sea level), $g = 9.81~ \text{m/s}^2$ is the gravitational constant, and $\rho = 1000~ \text{kg/m}^3$ is the water density. Finally, the term $10^{-6} \cdot \text{SysBasePower}^{-1}$ is used to transform the power in Watts into per-unit. The method assumes that the hydraulic head is always the intake elevation for the power conversion.

HydroEnergyModelReservoir

Variables:

• PowerSimulations.EnergyVariable:

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

• WaterSpillageVariable:

  • Bounds: $[S^\text{min}, S^\text{max}]$
  • Symbol: $s$

• HydroEnergySurplusVariable:

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

• HydroEnergyShortageVariable:

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

Expressions added:

• TotalHydroPowerReservoirIncoming:

  • Symbol: $p^\text{hy,in}$

• TotalHydroPowerReservoirOutgoing:

  • Symbol: $p^\text{hy,out}$

• TotalSpillagePowerReservoirIncoming:

  • Symbol: $s^\text{in}$

Static Parameters:

• $E^\text{max}$ = PowerSystems.get_storage_level_limits(device).max
• $E^\text{min}$ = PowerSystems.get_storage_level_limits(device).min
• $E^\text{init}$ = PowerSystems.get_initial_level(device)
• $S^\text{max}$ = PowerSystems.get_spillage_limits(device).max
• $S^\text{min}$ = PowerSystems.get_spillage_limits(device).min

Initial Conditions:

The PowerSimulations.InitialEnergyLevel: $e_0 = E^\text{init}$ 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, hydro_budget timeseries parameter to set a hydro budget 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 reservoir 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:

In addition the incoming turbine power expression $p^\text{hy,in}$ handles the total power coming from upstream turbines connected to this reservoir, the outgoing turbine power expression $p^\text{hy,out}$ handles the total power going out to the downstream turbines from this reservoir. Finally, the incoming spillage expression $s^\text{in}$ handles the power coming from upstream reservoirs that was spilled into this reservoir.

Attributes:

During the model setup, the attribute hydro_budget = true can be used to set-up a budget constraint with the full horizon of the simulation. The attribute energy_target = true can be used to set-up a storage target constraint at the end of the simulation horizon. It is not recommended to set both attributes to true simultaneously to avoid infeasibility issues.

Constraints:

For each hydro reservoir creates the constraints to track the energy storage.

\[\begin{align*} & e_{t}^\text{hy} = e_{t-1}^\text{hy} + \Delta T \left(p^\text{hy,in}_t + s^\text{in}_t + \text{InflowTimeSeriesParameter}_t - p^\text{hy,out} - s^\text{hy}\right), \quad \forall t\in \{1, \dots, T\} \\ & e_T^\text{hy} + e^\text{shortage} + e^\text{surplus} = \text{EnergyTargetTimeSeriesParameter}_T, \quad \text{ if storage\_target true} \\ & \sum_{t=1}^T p^\text{hy}_t \le \sum_{t=1}^T \text{EnergyBudgetTimeSeriesParameter}_t, \quad \text{ if hydro\_budget true} \end{align*}\]

HydroWaterModelReservoir

Variables:

• HydroReservoirHeadVariable:

  • Bounds: $[L^\text{head, min}, L^\text{head, max}]$
  • Symbol: $h^\text{hy}$

• HydroReservoirVolumeVariable:

  • Bounds: $[L^\text{vol, min}, L^\text{vol, max}]$
  • Symbol: $v^\text{hy}$

• WaterSpillageVariable:

  • Bounds: $[S^\text{min}, S^\text{max}]$
  • Symbol: $s$

Expressions added:

• TotalHydroFlowRateReservoirIncoming:

  • Symbol: $f^\text{hy,in}$

• TotalHydroFlowRateReservoirOutgoing:

  • Symbol: $f^\text{hy,out}$

• TotalSpillageFlowRateReservoirIncoming:

  • Symbol: $s^\text{in}$

Static Parameters:

• $L^\text{max}$ = PowerSystems.get_storage_level_limits(device).max
• $L^\text{min}$ = PowerSystems.get_storage_level_limits(device).min
• $L^\text{init}$ = PowerSystems.get_initial_level(device)
• $S^\text{max}$ = PowerSystems.get_spillage_limits(device).max
• $S^\text{min}$ = PowerSystems.get_spillage_limits(device).min
• $\text{h2v}$ = PowerSystems.get_head_to_volume_factor(device)

The parameter $L$ for level will be dependent on the ReservoirDataType provided in each reservoir, and can bound the hydraulic head or volume variable.

Initial Conditions:

The InitialReservoirVolume: $l_0 = L^\text{init}$ is used as the initial condition for the volume level of the reservoir.

Time Series Parameters:

The inflow and outflow timeseries parameter are used to obtain the external inflow and outflow to the reservoir.

Objective:

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

Expressions:

The incoming turbine flow expression $f^\text{hy,in}$ handles the total water flow coming from upstream turbines connected to this reservoir, the outgoing turbine flow expression $f^\text{hy,out}$ handles the total water flow going out to the downstream turbines from this reservoir. Finally, the incoming spillage expression $s^\text{in}$ handles the water flow coming from upstream reservoirs that was spilled into this reservoir.

Attributes:

During the model setup, the attribute hydro_budget = true can be used to set-up a budget constraint with the full horizon of the simulation. The attribute hydro_target = true can be used to set-up a storage target constraint at the end of the simulation horizon. The storage_target is only available for HEAD type reservoir models. It is not recommended to set both attributes to true simultaneously to avoid infeasibility issues.

Constraints:

For each hydro reservoir creates the constraints to track the energy storage.

\[\begin{align*} & v_{t}^\text{hy} = v_{t-1}^\text{hy} + 3600\Delta T \left(f^\text{hy,in}_t + s^\text{in}_t + \text{InflowTimeSeriesParameter}_t - p^\text{hy,out} - s^\text{hy}- \text{OutflowTimeSeriesParameter}_t\right), \quad \forall t\in \{1, \dots, T\} \\ & v_t^\text{hy} = \text{h2v} \cdot h_t^\text{hy}, \quad \forall t\in \{1, \dots, T\} \end{align*}\]

HydroWaterFactorModel

Formulation type to constrain hydropower production with an energy block optimization representation of the energy storage capacity and water inflow time series of a reservoir for PowerSystems.HydroReservoir and PowerSystems.HydroTurbine. This model operates with water levels (volumes) and uses a bilinear power production relationship that accounts for the hydraulic head variation with reservoir level.

The formulation models the relationship between turbine flow rate, reservoir water level, and power production, allowing for more accurate representation of hydro operations when water level significantly affects power output.

Variables:

• HydroReservoirVolumeVariable:

  • Bounds: $[L^\text{min}, L^\text{max}]$
  • Symbol: $v^\text{hy}$
  • Description: Volume stored in the hydro reservoir (in $m^3$)

• WaterSpillageVariable:

  • Bounds: $[S^\text{min}, S^\text{max}]$
  • Symbol: $s$
  • Description: Water spillage from the reservoir (in $m^3/s$)

• PowerSimulations.ActivePowerVariable:

  • Bounds: $[P^\text{min}, P^\text{max}]$
  • Symbol: $p^\text{hy}$
  • Description: Active power output from the hydro turbine

• PowerSimulations.ReactivePowerVariable:

  • Bounds: $[Q^\text{min}, Q^\text{max}]$
  • Symbol: $q^\text{hy}$
  • Description: Reactive power output from the hydro turbine (only if network model includes reactive power)

• HydroTurbineFlowRateVariable:

  • Bounds: $[F^\text{min}, F^\text{max}]$
  • Symbol: $f^\text{hy}$
  • Description: Water flow rate through the turbine (in $m^3/s$)

Auxiliary Variables:

• HydroEnergyOutput:
  • Symbol: $E^\text{hy,out}$
  • Description: Energy output from the turbine computed as $E^\text{hy,out} = p^\text{hy} \cdot \Delta T$

Static Parameters:

• $L^\text{max}$ = PowerSystems.get_storage_level_limits(device).max
• $L^\text{min}$ = PowerSystems.get_storage_level_limits(device).min
• $L^\text{init}$ = PowerSystems.get_initial_level(device)
• $S^\text{max}$ = PowerSystems.get_spillage_limits(device).max
• $S^\text{min}$ = PowerSystems.get_spillage_limits(device).min
• $P^\text{min}$ = PowerSystems.get_active_power_limits(device).min
• $P^\text{max}$ = PowerSystems.get_active_power_limits(device).max
• $Q^\text{min}$ = PowerSystems.get_reactive_power_limits(device).min
• $Q^\text{max}$ = PowerSystems.get_reactive_power_limits(device).max
• $F^\text{min}$ = PowerSystems.get_outflow_limits(device).min
• $F^\text{max}$ = PowerSystems.get_outflow_limits(device).max
• $\eta$ = PowerSystems.get_efficiency(device): Turbine efficiency
• $H^\text{elev}$ = PowerSystems.get_intake_elevation(device) - PowerSystems.get_powerhouse_elevation(device): Elevation head (in $m$)
• $\text{h2v}$ = PowerSystems.PowerSystems.get_head_to_volume_factor(device): Head to volume conversion factor
• $\rho$ = $1000 kg/m^3$: Water density
• $g$ = $9.81 m/s^2$: Gravitational constant
• $K$ = $\eta \rho g$: Energy block constant

Initial Conditions:

The InitialReservoirVolume: $v_0^\text{hy} = L^\text{init}$ is used as the initial condition for the volume level of the reservoir.

Time Series Parameters:

Uses the InflowTimeSeriesParameter to track the water inflow to the reservoir at each time-step (in $m^3/s$).

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.

If service models are included, adds $p^\text{hy}$ to HydroServedReserveUpExpression and HydroServedReserveDownExpression expressions to track served reserves for energy calculations.

Objective:

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

Constraints:

For each hydro reservoir, the reservoir inventory constraint tracks the water volume balance at every time step, based on volume balance in the previous time step, sum of water flows into the conencted downstream turbines, and the water spillage.

\[\begin{align*} & v_{t}^\text{hy} = v_{t-1}^\text{hy} + 3600 \cdot \Delta T \left(\text{InflowTimeSeriesParameter}_t - \sum_{i \in \mathcal{R}} f_{i,t}^\text{hy} - s_t \right), \quad \forall t\in \{2, \dots, T\} \\ & v_{1}^\text{hy} = L^\text{init} + 3600 \cdot \Delta T \left(\text{InflowTimeSeriesParameter}_1 - \sum_{i \in \mathcal{R}} f_{i,1}^\text{hy} - s_1 \right)\\ & v_t^\text{hy} = \text{h2v} \cdot h_t^\text{hy}, \quad \forall t\in \{1, \dots, T\} \end{align*}\]

where $\mathcal{R}$ is the set of downstream turbines connected to the reservoir, $f_i^\text{hy}$ is the turbine flow rate for turbine $i$, and $\Delta T$ is the duration (in hours) of each time step. The factor $3600$ converts hours to seconds.

An optional reservoir level target constraint can be added to enforce a target level at the end of the simulation horizon:

\[v_T^\text{hy} \ge L^\text{target}\]

where $L^\text{target}$ = PowerSystems.get_level_targets(device) $\times$ PowerSystems.get_storage_level_limits(device).max.

For each hydro turbine, the hydro power constraint relates the power output to the water flow rate and hydraulic head. The power production is calculated using a bilinear formulation that accounts for the variation in hydraulic head as the reservoir level changes:

\[\begin{align*} & p^\text{hy}_1 = \frac{\Delta T}{P^\text{base}} \cdot K \cdot f^\text{hy}_1 \cdot \left(0.5 \cdot \text{h2v} \cdot (v^\text{hy}_1 + L^\text{init}) + H^\text{elev} \right), \\ & p^\text{hy}_t = \frac{\Delta T}{P^\text{base}} \cdot K \cdot f^\text{hy}_t \cdot \left(0.5 \cdot \text{h2v} \cdot (v^\text{hy}_t + v^\text{hy}_{t-1}) + H^\text{elev} \right), \quad \forall t\in \{2, \dots, T\} \end{align*}\]

where $P^\text{base}$ is the system base power, and the hydraulic head is approximated using the average reservoir volume between consecutive time steps. The first time step uses the average of the initial volume and the volume at time $t=1$.

For each hydro turbine creates the range constraints for its active, reactive power and water flow depending on its static parameters:

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

Note: This formulation does not support piecewise head to volume factor curves. Only linear (proportional) head to volume relationships are supported.