Calculation Formula / Implementation of Operating Rules

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To operationalize principles for simulation, it is required to devise mathematical relations.

The previously given order of principles already indicates a useful structure for the mathematical formulation. The storage volume poses the most essential dependency. In system hydrology, this type of dependency is known as a linear single reservoir and can be thoroughly solved. Its principle is based on the assumption that the discharge is always proportional to the amount of water present in the reservoir (storage volume). The proportionality factor k is referred to as the reservoir constant. Taking into account the continuity equation and the reservoir constant gives the differential equation of the linear single reservoir. However, the reservoir equation is inapplicable to storage systems influenced by controls and operation rules. In this case, the discharges are not proportional to the storage volume, and therefore, an extension of the equation to any number of discharges is necessary.

Figure 24: Linearization of a function - section by section

Operation rules and controls need to be implemented to the set of equations by considering the relation of the storage volume to the discharge. As exemplarily shown in this chapter, this relation is usually only available in the form of supporting points. Linearly connecting supporting points leads to a characteristic curve. Here, the connection of supporting points results in a discharge function. A general demonstration of the linearization process, section by section, is given in Figure 24.

One section can be described by:

(2-1)

[math]\displaystyle{ y_{(t)}=y_{i-1}+k_i \cdot (S_{(t)} - S_{i-1}) }[/math]

with

[math]\displaystyle{ S_i \lt S_{(t)} \le S_{i+1} }[/math]

For any number of discharge functions, the equation of a linear single reservoir for a section is:

(2-2)

[math]\displaystyle{ \frac{dS}{dt} = \sum_{z=1}^n Q_z - \sum_{p=1}^m (y_{p,i-1} + k_{p,i} \cdot (S_{(t)} - S_{p,i-1})) }[/math]

with

S : Storage volume

Q_z : Inflow (independent of storage volume).

y_i-1 : Discharge value at supporting point i-1

k : Slope between the supporting points i-1 and i

n : Number of inflows

m : Number of discharges dependent on the storage volume

t : Time

After dividing the equation into a constant part and a part depending on the storage volume S, the known and solvable equation of the linear single reservoir is obtained.

(2-3)

[math]\displaystyle{ \frac{dS}{dt} = \begin{matrix} \underbrace {\sum_{z=1}^n Q_z - \sum_{p-1}^m ( y_{p,i-1} - k_{p,i} \cdot S_{p,i-1} )} \\ C_1=\mbox{constant part} \end{matrix} - \begin{matrix} \underbrace {\sum_{p=1}^m (k_{p,i})} \\ C_2=\mbox{depending on S} \end{matrix} \cdot S_{(t)} }[/math]

[math]\displaystyle{ \frac{dS}{dt} = C_1 - C_2 \cdot S_{(t)} }[/math]

As long as the reservoir volumes are within one section Si-1 to Si, the solution to the differential equation is:

(2-4)

[math]\displaystyle{ S(t) = \frac{C_1}{C_2} \cdot \left [ 1 - e^{-C_2} \cdot (t-t_0) \right ] }[/math]

If at least one discharge function exceeds the viewed section, respective changes to the reservoir volumes and to the discharges must be identified up to that point and C1 and C2 recalculated. With this method, the used time interval - the outer time step - is processed by an arbitrary number of inner time steps, depending on the closeness of the supporting points. The time it takes to undertake a section change can be determined by solving the following equation for t:

(2-5)

[math]\displaystyle{ t_1 = -\frac{1}{C_2} \cdot \ln \left ( \frac{S(t)-\frac{C_1}{C_2}}{S_0-\frac{C_1}{C_2}} \right ) + t_0 }[/math]

Whether a storage volume increases or decreases within the viewed section is to be determined by substituting S(t) by the upper section boundary, whereupon the nearest supporting point of all functions is crucial for the determination of the section boundary. The resulting value t1 determines the following three cases:

1. [math]\displaystyle{ t_1 \gt \Delta t \, }[/math] (outer time step)

   In the considered time interval, no section change takes place .

2. [math]\displaystyle{ 0 \lt t_1 \lt \Delta t \, }[/math]

   After the time t1, there is a section change . The span between t0 and t1 represents the length of the inner time step.

3. [math]\displaystyle{ t_1 \lt 0 \, }[/math]

   There is no reservoir content increase but a decrease. Instead of the upper section boundary, the lower section boundary must be used and the calculation repeated.

If section changes are observed, the storage volume is known at every time t. Consequently, processes depending on the storage volumes are also known. In general, the development of the curve over time is not required but rather the mean value within one time interval. If equation 2-4 is inserted into equation 2-1 and integrated over the inner time step length, the average process rate in the relevant time interval is obtained.

(2-6)

[math]\displaystyle{ \bar{y} = y_{p,i-1} - k_{p,i} \cdot S_{p,i-1} + k_{p,i} \cdot \left [ \frac{C_1}{C_2} + \left ( 1-e^{-C_2} \cdot (t_1-t_0) \right ) \cdot \left ( \frac{S_0}{(t_1-t_0) \cdot C_2} - \frac{C_1}{(t_1-t_0) \cdot C_2^2} \right ) \right ] }[/math]

After building the sum of the inner time step values, the average process rate over the entire outer time interval is obtained.

Figure 25: Linearization of a scaled discharge function - section by section

The principles in the description of operating rules illustrate that discharges depend on storage volumes as well as on other system states. Thus, a one-dimensional dependency no longer exists but rather a two- or multi-dimensional relation. Considering also a time dependency the problem is extended by one more dimension. Therefore, the solution from above, and respectively a simple depiction in a diagram, are no longer possible. For clarity and adequate mathematical formulation, a conversion from multiple dependencies to a one-dimensional relation needs to be undertaken without a loss of information. Scaling the discharge functions relation achieves said conversion. It is possible for the discharge (y-axis) as well as for the reservoir volumes (x-axis).

After introducing the scaling factors, the result for a scaled section of a function is:

(2-7)

[math]\displaystyle{ y_{(t)}^s = y_{i-1} \cdot y_{\mbox{factor}} + k_i \cdot \frac{y_{\mbox{factor}}}{x_{\mbox{factor}}} \cdot \left ( S_{(t)} \cdot x_{\mbox{factor}} -S_{i-1} \cdot x_{\mbox{factor}} \right ) }[/math]

[math]\displaystyle{ y_{(t)}^s = y_{i-1}^s + k_i^s \cdot \left ( S_{(t)}^s - S_{i-1}^s \right ) }[/math]

To calculate the delivery function scaled with external system states, proceed analogously to the above method. Here, xfactor corresponds to the maximum reservoir volume and yfactor corresponds to the scaling factor from the external system state or state group. It is assumed that the factors remain constant during the outer time interval. The sum of the integrations over the internal time loop divided by the outer time step provides the final output value.

(2-8)

[math]\displaystyle{ \bar{y} = y_{\mbox{faktor}} \cdot \left [ y_{p,i-1} - k_{p,i} \cdot S_{p,i-1} + \frac{1}{x_{\mbox{faktor}}} k_{p,i} \cdot \left [ \frac{\mbox{C1}^s}{\mbox{C2}^s} + \left ( 1 - \mbox{e}^{-\mbox{C2}^s \cdot \left ( t_1 - t_0 \right )} \right ) \cdot \left ( \frac{S_0 \cdot x_{\mbox{faktor}}}{(t_1 - t_0) \cdot \mbox{C2}^s} - \frac{\mbox{C1}^s}{(t_1 - t_0) \cdot {\mbox{C2}^s}^2} \right ) \right ] \right ] }[/math]

The computation of a section-by-section linear memory with arbitrary many inputs and outputs has been described by Ostrowski (1992). This solution was extended to include scaling of both the x-axis and y-axis by Ostrowski (1999).

In summary, a reservoir can have multiple uses. For each use, there is a function depending on the reservoir volume, which must remain constant within an outer time step, but is changeable from time step to time step (time dependency). Besides, scaling the functions differently for each time step accounts for external dependencies via factors. Prerequisites for the scaling are constant factors during the time step. The calculation process is independent of the time step, as it is split according to section changes into an arbitrary number of inner time steps. As a result, the method is suitable for a wide variety of time intervals and produces results that are true to the volume. The method facilitates its use for diverse time intervals, e.g. it is suitable for a flood event with a time step of a few minutes but as well for a long-term simulation with daily values or even larger time intervals. For this purpose, it is critical, to define discharge functions with a sufficient number of supporting points.