Optimization Tailwater

Only a subset of available Simulation methods are supported in Optimization. These methods are: Linked or Input, Base Value Only, Base Value Plus Lookup Table, Stage Flow Lookup Table, and Coefficients Table. Selection of a method other than these will result in an error. The method selected in the Optimization Tailwater category should correspond to the one selected in the simulation Tailwater category.

Tailwater Elevation may be part of the Optimization problem and it is handled differently for each method. Below is a description of the behavior of Tailwater Elevation.

See Tailwater in Objects and Methods for details on the Simulation methods.

Opt Linked or Input

If Tailwater Elevation is not input, it is replaced as follows:

(5.29) TailwaterBaseValue: Tailwater Elevation = Tailwater Base Value

Slots Specific to This Method

Tailwater Base Value

Opt Base Value Only

If Tailwater Elevation is not input, it is replaced as follows:

(5.30) (Tailwater Base Value(t) + Tailwater Base Value(t-1) ) / 2

Slots Specific to This Method

Tailwater Base Value

Type: Series

Units: Length

Tailwater outflow relationship. Not drawn to scale.

Opt Base Value Plus Lookup Table

If Tailwater Elevation is not input, this slot is replaced by a mathematical expression based on Tailwater Base Value and a numerical approximation of Tailwater Elevation as a function of Outflow (Numerical 2-D Approximation), as defined by the user in the Tailwater Table.

Tailwater Elevation is constrained as follows:

(5.31) (tailwaterBaseValue(t) + tailwaterBaseValue(t-1))/2 + tempTWLookupValue

where tempTWLookupValue is a Numerical 2-D Approximation of increase in Tailwater Elevation, due to flow as described in the Tailwater Table. The Tailwater Table Lookup LP Param table values for outflow are used as approximation points, indexing the Tailwater Table to determine tempTWLookupValue from the Tailwater Elevation column.

Slots Specific to This Method

Tailwater Base Value

Type: Series

Units: Length

Tailwater Table

Type: Series Slot

Units: Flow vs Length

Description: Reservoir outflow vs either the tailwater elevation or the tailwater elevation increment

Information: If the Tailwater Base Value is non-zero, the Tailwater Table gives values of incremental increase in Tailwater Elevation over the Base value. If the Tailwater Base Value is zero, the table simply gives the Tailwater Elevation values. The Tailwater Table should have increasing values of outflow. Tailwater Elevation is required to be a convex function of outflow. The preferred order of approximation is substitution, piece-wise, two-point line, tangent. See Object / Simulation documentation for details.

I/O: User input

Tailwater Table Lookup LP Param

Type: Table Slot

Units: Flow, Flow, Flow

Description: LP Parameter approximation points for Outflow.

Information: The Tailwater Table Lookup LP Param table values for outflow are used as approximation points indexing the Tailwater Table to determine tempTWLookupValue from the Tailwater Elevation column.

I/O: User input only

Links: Not linkable

temp Tailwater Lookup Value

Type: Series

Units: Length

Description: Numerical 2-D Approximation of increase in Tailwater Elevation due to flow as described in the Tailwater Table.

Opt Stage Flow Lookup Table

Tailwater Elevation is numerically approximated as a function of Stage and Flow, as defined by the user in the Stage Flow Tailwater Table.

The tables are queried depending on method selection in the Tailwater Linearization Automation category, as follows.

• If the selected method in the Tailwater Linearization Automation category is None, the user-input Tailwater LP Param table values are used.

• For the Range Input Automation method, the automatically calculated points are used.

Tailwater Elevation is approximated numerically (3-D approximation), as a function of Outflow and Tailwater Base Value based on the Stage Flow Tailwater Table. The Tailwater Elevation LP Param table Tailwater Base Value and flow values are used as approximation points, indexing the Downstream Stage and Outflow columns of the Stage Flow Tailwater Table, respectively.

See Stage Flow Lookup Table in Objects and Methods for details on the Simulation method.

Slots Specific to This Method

Tailwater Base Value

Type: Series

Units: Length

Stage Flow Tailwater Table

Type: Table Slot

Units: Flow vs Length vs Length

Description: Reservoir Outflow vs Downstream Elevation (Tailwater Base Value) vs Tailwater Elevation

Information: Data must be entered into the table in increasing blocks of the same Outflow value for the three-dimensional table interpolation to work correctly. For every block of same Outflows in column 1, Downstream Stages should be listed in increasing order in column 2, and the corresponding Tailwater Elevations in column 3. Tailwater elevation is required to be a concave function of Outflow. The preferred order of approximation is substitution, piece-wise, two-point line, tangent. Internally, the Stage Flow Tailwater Table is rearranged to use the Stage as the primary index (column) with outflow as the secondary index (column). This rearranged table may be labeled as “Convolved Stage Flow Tailwater Table” in output messaging. Also, because of this rearrangement, it is desirable to repeat stage values for each outflow.

Defined by: User input

Tailwater Elevation LP Param

Type: Table Slot

Units: Length, Flow, Flow, Flow

Description: Specifies the fixed tailwater base value point and the outflow points used to take the tangent, line and piecewise approximations for tailwater elevation linearization

Information: This table is used for linearization. The best Tailwater Base Value point to choose for tangent approximation would be an outflow equal to the expected value of outflow during the run; for the line approximation, the minimum and maximum values expected during the run; for piecewise approximation, the minimum and maximum values expected during the run plus additional intermediate values to more closely fit the tailwater curve.

Defined by: User input unless the selected method in the Tailwater Linearization Automation category is not None.

Tail Water Reference Elevation

Type: Table Slot

Units: Length

Description: Lowest Reservoir discharge Elevation when there are no backwater effects from a downstream pool (reservoir)

Information: Although this is part of the Stage Flow Lookup Table method (and its corresponding Opt method), this slot does not influence optimization. See Object / Simulation documentation for details.

I/O: User input

Opt Coefficients Table

This method sets up the physical Tailwater Elevation constraints using the same equation and coefficients used in the Coefficients Table tailwater method.

See Coefficients Table in Objects and Methods for details.

Slots Specific to This Method

Tailwater Base Value

Tailwater Coefficients

Type: Table

Units: multi

Defined by: If the Power Surface Approximation method is selected, then there are additional restrictions on the coefficients that can be specified. With the Power Surface Approximation, having terms for Outflow[t-1], Tailwater BaseVal[t] and Tailwater[t-1] are not allowed.

Pump Power Numerical Approximation

If the Optimization problem uses Pump Power, either directly or indirectly, this slot is added to the problem.

Before being passed to the optimization solver, this slot is numerically approximated as a function of Operating Head and Pumped Flow (Numerical 3-D Approximation). The relationship between Operating Head and Pumped Flow comes from the user-input Pump Power Lin Data table. The table is queried using the points defined in the Pump Power LP Param table. The values in the Pump Power LP Param table may be entirely user input, or may include an automatically calculated value for Operating Head. If the Operating Head is not available at the beginning of the run, this value will be assigned automatically by RiverWare.

Slots Specific to This Method

Pump Power

Type: Agg Series Slot

Units: Flow

Description: Power required to operate the pumps

Information:

Defined by: Numerical 3-D Approximation in terms of Operating Head and Pumped Flow based on the Pump Power Lin Data table. The Pump Power LP Param table contains a value for Operating head used to index the Operating Head column of the Pump Power Lin Data table. This approximated value, therefore, reduces the Pump Power to a function of Pumped Flow at the given Operating Head. The flow values in the Pump Power LP Param table are then used as approximation points indexing the Pumped Flow column of the Pump Power Lin Data table. The Pump Power Lin Data table should have increasing values of Operating Head and Pumped Flow. Pump Power should be a convex function of Operating Head, but concavity is not strictly enforced; mild non-convex regions are permissible to allow for round-off error, and so on. The preferred order of approximation is substitution, piece-wise, tangent, two-point line.

Pump Power Lin Data

Type: Table Slot

Units: Length vs Flow vs Power

Description: Table defining the relationship between Operating Head, Pumped Flow and power expended (Pump Power).

Information:

Defined by: User input

Pump Power LP Param

Type: Agg Series Slot

Units: Length, Flow, Flow, Flow

Description: Specifies the Operating Head and the flow points used to take the tangent, line and piecewise approximations for Pump Power Lin Data linearizations.

Information: This slot is the pumping equivalent of the Power LP Param table. The best Operating Head to choose should be close to the expected head during optimized period. Tangent approximation is generally not used, nor helpful as it often results in non-zero power for zero flow. The suggested points for a line approximation are 0 flow and the preferred pumping flow. The suggested points for piecewise linearization are 0 flow, 1 unit preferred pumping flow, 2 units preferred pumping flow ... n units preferred pumping flow, and maximum pumping flow.

Revised: 01/05/2024