Quelltext der Seite Berechnungsschema/ Implementierung der Betriebsregeln/en

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{{Navigation|vorher=Abstraktion der Betriebsregeln|hoch=Betriebsregelkonzept|nachher=Anwendungsbeispiel: Umsetzung eines Betriebsplans}}

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.

[[Bild:Theorie_Abb24.png|thumb|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 [[:Bild:Theorie_Abb24.png|Figure 24]].

One section can be described by:

<div style="float:right">(2-1)</div>
:<math>y_{(t)}=y_{i-1}+k_i \cdot (S_{(t)} - S_{i-1})</math> 

:with 
:<math>S_i < S_{(t)} \le S_{i+1}</math>

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

<div style="float:right">(2-2)</div>
:<math>\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<sub>z</sub> : Inflow (independent of storage volume).
:y<sub>i-1</sub> : 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.

<div style="float:right">(2-3)</div>
:<math>\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>\frac{dS}{dt} = C_1 - C_2 \cdot S_{(t)} </math>

As long as the reservoir volumes are within one section <code>S<sub>i-1</sub></code> to <code>S<sub>i</sub></code>, the solution to the differential equation is:

<div style="float:right">(2-4)</div>
:<math>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 <code>C<sub>1</sub></code> and <code>C<sub>2</sub></code> 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 <code>t</code>:

<div style="float:right">(2-5)</div>
:<math>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 <code>S(t)</code> 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 <code>t<sub>1</sub></code> determines the following three cases:

# <math>t_1 > \Delta t \,</math> (outer time step) 
#: No section change takes place in the considered time interval. 
# <math>0 < t_1 < \Delta t \,</math>
#: There is a section change after the time <code>t<sub>1</sub></code>. The span between <code>t<sub>0</sub></code> and <code>t<sub>1</sub></code> represents the length of the inner time step.
# <math>t_1 < 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 <code>t</code>. 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.

<div style="float:right">(2-6)</div>
:<math>\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 values calculated from all inner time steps, the average process rate over the entire outer time interval is obtained.

[[Bild:Theorie_Abb25.png|thumb|Figure 25: Linearization of a scaled discharge function - section by section]]

From the principles for the description of operating rules it is evident that a discharge can be dependent on the storage content as well as on other system states. Thus, a one-dimensional dependency - only on the storage content - is no longer given. In such a case, a two- or multi-dimensional relationship exists for the unambiguous determination of a dischrage. If a time dependency is added, the problem is extended by one more dimension. A graphically simple representation is no longer feasible. Likewise the solution described above is not sufficient, since further ones are added to the dependence on the reservoir contents. Both for reasons of clarity and a suitable mathematical formulation, it is desirable to convert all dependencies back into a one-dimensional relationship without loss of information. This is achieved by scaling the ''discharge functions'' relationship. A scaling is possible for the discharge (y-axis) as well as for the reservoir content (x-axis).

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

<div style="float:right">(2-7)</div>
:<math>y_{(t)}^s = y_{i-1} \cdot y_{\mbox{faktor}} + k_i \cdot \frac{y_{\mbox{faktor}}}{x_{\mbox{faktor}}} \cdot \left ( S_{(t)} \cdot x_{\mbox{faktor}} -S_{i-1} \cdot x_{\mbox{faktor}} \right )</math>

:<math>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 <code>x<sub>factor</sub></code> corresponds to the maximum reservoir content and <code>y<sub>factor</sub></code> corresponds to the scaling factor from the external system state or state group. It is assumed that the factors remain constant during the external time interval. The sum of the integrations over the internal time loop divided by the external time step provides the final output value.

<div style="float:right">(2-8)</div>
:<math>\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 rangewise linear memory with arbitrarily many inputs and outputs has been described by Ostrowski (1992)<ref name="Ostrowski_1992">'''Ostrowski, M.''' (1992): A universal building block for the simulation of hydrological processes, Water and Soil, Issue 11</ref>. This solution was extended to include scaling of both the X and Y axes<ref name="Ostrowski_1999">'''Ostrowski, M. et al.''' (1999): A universal non-linear memory building block for the simulation of hydrological systems. Institute's own model and application description, Institute of Hydraulic Engineering and Water Resources Management, TU Darmstadt, unpublished</ref> .

In summary, a reservoir can have any number of uses. For each usage, there is a function dependent on the reservoir content, which must remain constant within an external time step, but can be changed from time step to time step (time dependency). In addition, these functions can be scaled differently for each time step by external dependencies via factors. Prerequisite for the scaling are constant factors during the time step. The calculation process is independent of the time step, since it is decomposed according to the section crossings into arbitrarily many internal time steps. This means that the method is suitable for a wide variety of time intervals and produces results that are true to the volume. Thus, both a flood event with a time step of a few minutes and a long-term simulation with daily values or even larger time intervals can be used. The only decisive factor is that all ''discharge functions'' are defined over a sufficient number of grid points.



==Literature references==

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