# Volume dependent discharge

For some modules there is a significant physical relationship between head and maximum discharge capacity.

The following is assumed, and input data will be concaveified.

- The outlet is not submerged (or that the tailwater level is constant). 
- Head can be expressed as a concave function of volume.

Under these assumptions maximum discharge capacity can be expressed as a concave function of reservoir volume.

Using the input curve of n points of the discharge(volume) function, n-1 cuts are calculated like this:

d: discharge for point n
v: volume for point n
RHS: b value of the ax + b function
GRAD: a value for the ax + b function

$$\\{c \in curve\ points-1\\}$$

$$\textrm{GRAD}(c) = \frac{\textrm{d}(c + 1) - \textrm{d}(c)}{\textrm{v}(c + 1) - \textrm{v}(c)}$$

$$\textrm{RHS}(c) = \textrm{d}(c) - \textrm{GRAD}(c) * \textrm{v}(c)$$

These point pairs are calculated once at the start of the simulation and used for the rest of the run.

$$ \\{i \in hydro\ plants,\ k \in \\{0, 1, ..., timesteps - 1\\},\ c \in \\Bigr(\textrm{RHS},\textrm{GRAD}\Bigr)\\}$$

$$dis(i,k) \leq \textrm{RHS}(i,c) + \textrm{GRAD}(i,c) * res(i,k)  \qquad \forall i,k,c \qquad (1)$$