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In order to make the principles found accessible for simulation, a suitable mathematical formulation is required.
To operationalize principles for simulation, it is required to devise mathematical relations.


The previously given order of the principles already gives a structure usable for the mathematical formulation. The main dependency is given by the storage volume. In system hydrology, this kind of dependency is known as a linear single reservoir and can thus be solved to completion. 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 called the reservoir constant. Taking into account the continuity equation and the reservoir constant, the differential equation of the linear single reservoir is given. However, the reservoir equation is unsuitable for the actual application to storage systems influenced by controls and operation rules. In this case, the discharges are not proportional to the storage volume, and therefore the equation would have to be extended to any number of discharges.
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 ]]
[[Bild:Theorie_Abb24.png|thumb|Figure 24: Linearization of a function - section by section ]]


As can be seen from the examples in this chapter, the functional relations between the storage volume and the discharge, used for operating rules, are usually only available as supporting points. The connection of the supporting points gives out the curve progression of a discharge function. A discharge function given in the form of several nodes, e.g. the characteristic curve of a spillway, can be linearly connected between its nodes (or supporting points). The general representation of functions with a linearization, section by section, is given in [[:Bild:Theorie_Abb24.png|Figure 24]].
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:
One section can be described by:
Zeile 24: Zeile 24:
:<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>  
:<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
:with
:S : storage volume
:S : Storage volume
:Q<sub>z</sub> : inflow (independent of reservoir content).
:Q<sub>z</sub> : Inflow (independent of storage volume).
:y<sub>i-1</sub> : discharge value at grid point i-1  
:y<sub>i-1</sub> : Discharge value at supporting point i-1  
:k : slope between interpolation points i-1 and i  
:k : Slope between the supporting points i-1 and i  
:n : number of inflows  
:n : Number of inflows  
:m : number of discharges dependent on the storage content
:m : Number of discharges dependent on the storage volume
:t : time
:t : Time


After division into a constant part and a part depending on the storage content S, the well-known and closed solvable equation of the linear single storage is obtained.
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>
<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{konstanter Anteil} \end{matrix} - \begin{matrix} \underbrace {\sum_{p=1}^m (k_{p,i})} \\ C_2=\mbox{von S abhaengig} \end{matrix} \cdot S_{(t)}</math>
:<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>
:<math>\frac{dS}{dt} = C_1 - C_2 \cdot S_{(t)} </math>


As long as the reservoir contents are within a section <code>S<sub>i-1</sub></code> to <code>S<sub>i</sub></code>, the solution to the differential equation is:
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>
<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>
:<math>S(t) = \frac{C_1}{C_2} \cdot \left [ 1 - e^{-C_2} \cdot (t-t_0) \right ]</math>


If the section is exceeded for at least one delivery function, the changes that have occurred up to that point in both the reservoir contents and the deliveries must be registered 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 density of the interpolation points. The time until a range change can be determined by transforming to <code>t</code> from the following equation:
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>
<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>
:<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 reservoir content increase or decrease occurs in the interval under consideration is to be determined by substituting for <code>S(t)</code> the value of the upper section boundary, where the nearest interpolation point of all functions is decisive for the determination of the section boundary. The resulting value <code>t<sub>1</sub></code> determines the following three cases:
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> (äußerer Zeitschritt)  
# <math>t_1 > \Delta t \,</math> (outer time step)  
#: No range change takes place in the considered time interval.  
#: In the considered time interval, no section change takes place .  
# <math>0 < t_1 < \Delta t \,</math>
# <math>0 < t_1 < \Delta t \,</math>
#: There is a range 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 inner time step length.
#: After the time <code>t<sub>1</sub></code>, there is a section change . 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>
# <math>t_1 < 0 \,</math>
#: There is no reservoir content increase but a decrease. Instead of the upper section limit, the lower section limit must be used and the calculation repeated.
#: 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 the range crossings are observed, the reservoir content is known at every time <code>t</code>. Consequently, the processes dependent on the reservoir contents are also known in their course over time. In general, however, it is not a time course that is required, but the mean value within a 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.
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>
<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>
:<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 summing up the values of all inner time steps passed through, the average process rate over the entire outer time interval is obtained.
After summing up the inner time step values, the average process rate over the entire outer time interval is obtained.


[[Bild:Theorie_Abb25.png|thumb|Figure 25: Range-wise linearization of a scaled discharge function]]
[[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).
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:
After introducing the scaling factors, the result for a scaled section of a function is:


<div style="float:right">(2-7)</div>
<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} \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>y_{(t)}^s = y_{i-1}^s + k_i^s \cdot \left ( S_{(t)}^s - S_{i-1}^s \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.
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 volume 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 outer time interval. The sum of the integrations over the internal time loop divided by the outer time step provides the final output value.


<div style="float:right">(2-8)</div>
<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>
:<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> .
The computation of a section-by-section linear memory with arbitrary many inputs and outputs has been described by Ostrowski (1992)<ref name="Ostrowski_1992">'''Ostrowski, M.''' (1992): Ein universeller Baustein zur Simulation hydrologischer Prozesse, Wasser und Boden, Heft 11</ref>. This solution was extended to include scaling of both the x-axis and y-axis by Ostrowski (1999)<ref name="Ostrowski_1999">'''Ostrowski, M. et al.''' (1999): Ein universeller, nicht linearer Speicherbaustein zur Simulation hydrologischer Systeme. Institutseigene Modell- und Anwendungsbeschreibung, Institut für Wasserbau und Wasserwirtschaft, TU Darmstadt, unveröffentlicht</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.
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.





Version vom 16. März 2021, 11:29 Uhr

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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
Qz : Inflow (independent of storage volume).
yi-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 summing up 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)[1]. This solution was extended to include scaling of both the x-axis and y-axis by Ostrowski (1999)[2].

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.


Literature references

  1. Ostrowski, M. (1992): Ein universeller Baustein zur Simulation hydrologischer Prozesse, Wasser und Boden, Heft 11
  2. Ostrowski, M. et al. (1999): Ein universeller, nicht linearer Speicherbaustein zur Simulation hydrologischer Systeme. Institutseigene Modell- und Anwendungsbeschreibung, Institut für Wasserbau und Wasserwirtschaft, TU Darmstadt, unveröffentlicht