Reservoirs
Reservoir balance
The reservoir balance is expressed as
where the reservoir volume, \(Volume(i,k)\), is the sum of up to three variables, depending on the volume boundary configuration.
Physical Reservoir boundaries
A reservoir is bound by its physical max volume boundary. If no other boundary is specified, then the volume is represented by the variable \(res(i,k)\):
with the bounds
Minimum Reservoir Boundary
Hard boundary
A reservoir can be configured with a hard or absolute minimum volume restriction. With this configuration we introduce a variable \(res_{min}(i,k)\) that represents the reservoir volume up to the minimum boundary:
The bounds and cost for the variables are as follows:
where \(res_{min}(i,k)\) is given an incentive to be used in the form of a negative cost: $\( Cr_{min}(i,k) < 0 \quad \forall i, k.\)$
Soft boundary
A reservoir can be configured with a soft volume restriction instead of a hard one. There are no additional variables added and no incentive or cost for violating the restriction. Instead downstream discharge is shut off for the following decision problem if the initial reservoir volume is below the minimum limit. This is done by setting all discharge variable bounds to zero.
See the following pseudocode:
if ResInit(i) > ResMin(i,k):
No effect
elif ResInit(i) <= ResMin(i,k):
dis(i,k) = 0
Maximum Reservoir Boundary
A reservoir can be configured with maximum volume restriction. With this configuration we introduce a variable \(res_{max}(i,k)\) that represents the reservoir volume above the maximum boundary. This variable is given a penalty in the form of a positive cost.
The following equations describe the reservoir volume with a maximum volume restriction:
where
If a hard minimum volume restriction is also configured, the equations used are instead
where
Hard and soft boundary
A reservoir can be configured to have a hard or soft maximum volume boundary. The difference is which timeseries is to be used for the penalty values. By default, the hard boundary has a higher penalty.