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1 Introduction

In the years 2002 and 2013, during the period of summer, the nearby region of the river Danube was flooded. In Linz on June 4th, 2013 the maximum water level was at 9.3 meters, whereas the normal gauge height is about 3.5 m.

The two pictures below give an impression of this situation: Fig. 1.

Fig. 1.
figure 1

Left: right riverbank, Ars Electronica Center and Nibelungen Bridge of Linz [1]. Right: deliberately flooded municipality area of Ottensheim [2].

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The most important goal of this work is to be able to take efficient measures to minimize the damage of an inundating Danube that could go up to millions of Euros in particular when enterprises like voestalpine in Linz are affected. See also the documentation of the above mentioned floodwater: Hochwasser Juni 2013, Donau, Ereignisdokumentation, Verbund AG [3].

2 Brief Description of the Model

The most important input data are the values of the amount of the flowing water at the power stations. Some important parameters enable the computation of the amount of water of an inundated area depending on the actual water throughputs in that area. The output is a measure of the damage in a given situation.

With the help of this model I make suggestions for an intelligent control of the water power stations in such situations. Some more flood preventing actions like eliminating sediments in the river and their impact will be investigated.

Here is a map of the affected area: Fig. 2.

Fig. 2.
figure 2

The area of the Eferdinger Becken. The yellow depicted part was seriously affected, whereas the lila zone was less afflicted [5] (Colour figure online).

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3 Mathematical Notation of the Model

This notation is made up of a DEVS (Discrete EVent System) and its functions. The most important one is the state transition function that transforms a given state s of the inundation model at some elapsed discrete time e to the next state s’ that will be valid at the time e + 1. The time slot will be one hour. This system will start at time e = 1 and stop after a given number n of hours. So we consider a certain interesting time interval within a few days.

Here is the DEVS formalism of the flooding model:

$$ {\text{DEVS}} = < {\text{X}},{\text{Y}},{\text{S}},{\varvec{\updelta}},{\varvec{\uplambda}} > $$

with

  • X is the input set: a table (e, q) where e is a number between 1 and n and q is the amount of outflowing water in m 3 at time e. This amount of water depends on the throughputs of water from the above power station and the one beneath after the saturation delay d.

  • Y is the output set: it represents the caused damage in euros.

  • S is the set of all possible states including the elapsed time: formed by a table (e, s) where s is the sum of already outflowed water at time e.

  • δ: S → S is the state transition function: it simply sums up the q values:

    • (e’, s’): = (e + 1, s + q) for (e, q) of X and (e, s) of S.

    • For (e <= d) the value of q is 0. For e = 1 the value of s starts with 0.

  • λ: S → Y is the output function: it takes the maximum value s, compares it with values in the damage table and takes the corresponding cost value from that table.

More details are stated in the next paragraph. The specific input table in this model mainly depends on the actual flowing water at the small village of Brandstatt (river km 2157). And this water flow depends on the operation mode and throughput of the power stations in Aschach and Ottensheim.

The difference QAschach – QOttensheim (Q is measured in m3 per second) is the approximate amount of outflowing water to the region called “Eferdinger Becken” which is about 50 km2 large. About half of this region is severely affected. The input function table takes values on the basis of the real situation of June 2013. It reflects the observed amount of outflowed water and with the help of the damage table one can estimate the damage that adds up to about 100 million euros.

4 The Computational Model

All the data and parameter values are stored in tables. The input table is generated by a PLSQL stored procedure. Here is a description of the most important parts:

4.1 The Tables and Their Values

There is a table that holds the data of the two power stations of interest. The next figure is a graphical representation of the input table during 27 h. At both power stations the peak of water throughput was reached about 20 h after the model starting time which is Monday 3rd of June 2013 at 5:15 am Fig. 3.

Fig. 3.
figure 3

The throughput data of Aschach and Ottensheim and the outflow.

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This diagram describes the outflowing water q in m3 per hour at time e: Fig. 4.

Fig. 4.
figure 4

The amount of outflowing water during 27 h in m3 per hour.

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The next figure shows the actual amount of outflowed water at the elapsed hour e: Fig. 5.

Fig. 5.
figure 5

The accumulated amount of outflowing water in m3.

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The following diagram shows that the overall damage is about 100 million euros Fig. 6.

Fig. 6.
figure 6

The growth of the damage in euros relative to the increase of outflowing water.

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4.2 The Parameters, Variables and Their Values

The following table lists the relevant characteristics with their names and values.

Characteristic

Variable names

Value

Name of the area

region_name

“Eferdinger Becken”

Size of this area

total_size

~50 000 000 m2

Size of the most affected area (yellow)

size_yellow

~25 000 000 m2

Size of the less affected area (lila)

size_lila

~25 000 000 m2

Stowage space of power station Aschach

stowage_A

~114 000 000 m3

Sediments increase during one year

sed_incr

~2 000 000 m3

Size of future flood polder

polder_size

~12 500 000 m2

Stowage space of future flood polder

stowage_P

~30 000 000 m3

Peak water throughput KW Aschach

max_Q_A

~9 700 m3/sec

Peak water throughput KW Ottensheim

max_Q_O

~7 500 m3/sec

Maximum water outflow June 2013

max_out

~2 200 m3/sec

Monitoring starting time

start_time

3rd June 2013, 5:15 am.

Time to fill stowage space KW Ottensheim

saturation_delay

10 h

Monitoring period from start time

duration

27 h

Tailback water amount

tailback_water

~5 000 000 m3

Sum of water outflow + tailback water

sum_out

~92 500 000 m3

Percentage of ground water

percent_gw

~50 %

Sum of water above ground

sum_over

~46 250 000 m3

Average water hight in yellow area

avg_yellow

1,6 m

Average water hight in lila area

avg_lila

0,25 m

4.3 Dependencies Between the Variables

The following equations must be satisfied:

total_size = size_yellow + size_lila

max_out = max_Q_A - max_Q_O

sum_out = sum_over + sum_out * percent_gw/100

sum_over = size_yellow * avg_yellow + size_lila * avg_lila

4.4 The Definition of a Typical View Written in SQL

Here is one typical view, used to get the results above:

4.5 The Input Data Generator

For this purpose I wrote a PLSQL procedure to generate a plausible curve of throughputs at the power stations Aschach and Ottensheim. If available, this code can be replaced with statistical input data to store into the table “ flow_rates_m3_sec ”.

5 Case Studies

Now it is possible to consider some typical flood water situations. The first one describes an inundation of type “June 2013”, the second one assumes the elimination of sediments before this event and the third one includes a yet not existing flood polder of a plausible size.

5.1 The June 2013 Inundation

Taking the results of the model described above, this inundation caused an amount of outflowed water of approximately 92.5 million m 3 in this region and a damage that rounds up to 100 million euros. It is a fact that the sediments above the power station of Aschach were constantly increasing and no flood preventing constructions were installed.

5.2 Same Inundation After Eliminating Sediments

Let us consider the constant elimination of the sediments. According to the above parameter of sediments increase, after 30 years about half of the stowage space of the power station Aschach was filled with stones, gravel and sludge. So it is not surprising that the company Via Donau, that manages the power stations, wanted to increase the water level upstream of (not only) this hydropower station. Because of the increase of the sediments the water amount in the stowage space decreases and therefore the performance of this power station also reduces to a certain extent.

After the elimination of the sediments we would have had 60 million m 3 more stowage space upstream this power station, and then the amount of outflowed water could obviously have been 60 million m3 less which would have reduced the amount of the inundation to approximately 32.5 million m 3 in this region. So the damage could have been reduced from about 100 million euros to only about 28 million euros according to the damage table.

5.3 Same Inundation Having Flood Preventing Constructions

Additionally after the construction of a flood polder almost all the remaining 32.5 million m 3 of water can be hold in it and the flow off can be controlled so that the downriver areas are protected to that dimension. As a result, there is no need to deliberately flood the Eferdinger Becken and there will be no damage at all.

All the assumed parameter values are realistic and based on official documents and a discussion on this subject in a conference of the Landtag Oö held by Landesrat Rudolf Anschober [3, 4].

6 Recommendations

So far I have not mentioned the possibility of reducing the water amount in all stowage spaces before an obvious flooding event. This will also help a lot to minimize the devastating consequences of a natural catastrophe of this kind. The below stated recommendations are a summary of what I have learned by studying this subject.

6.1 Define Appropriate Rules of Operation

The reduction of the amount of water in all stowage spaces should begin more than 2 days before a critical rainy weather situation in the affecting areas. In such a situation rain probably will continue and therefore the flowing speed of the river is to accelerate to make sure that a maximum amount of water flows downstream, still without causing serious damage. A weather forecast with high quality prediction will facilitate decisions of this kind. These rules should operate in two modes:

(A) pre-emptying all stowage spaces upriver at least 54 h before and

(B) holding back as much water as possible when the wave arrives.

Last time the prediction was based on a 24 h time window, but during 30 h it is possible to transport approximately 1 000 000 000 m 3 (one cubic kilometer) of water downriver, so it is recommendable to extend this period to about 54 h.

6.2 Eliminate Sediments

The elimination of sediments along the whole river Danube in Bavaria and Austria and its tributary streams will contribute a lot to prevent floodwaters. There will be more space to temporarily store water and the throughput of water will be significantly higher as in the actual situation. There is one disadvantage, namely, during the period of pre-emptying the stowage spaces the power stations are almost out of service. In his speech on the 12th of June 2014, Landesrat Rudolf Anschober [4] made clear that flood prevention had a much higher priority than electricity generation.

6.3 Build Flood Polders

The existence of appropriate flood polders allows that a high amount of water can be stored temporarily and after a certain time it can be released without causing any damage. These constructions will make it possible to cope with extreme floodwaters caused by heavy precipitations.

6.4 Develop an App for Smart Phones

This “Flood Warner” application should alert the users and display important data such as water throughputs and water levels at all power stations and important places of residences. Moreover, it should also give a prognosis of the expected maximum water level and when it will be reached at a given user location.

7 Conclusion

Applying the above written recommendations will result in an intelligent treatment of floodwaters. I hope that in the near future we always will be able to say: “We have been lucky, not even that heavy rainfall has caused any serious damage!”