**2. Method**

**Figure 2.** Parcel of owners before and after land consolidation.

owners and land users.

36 Research on Soil Erosion Soil Erosion

agement problems.

Recently, the process of complex land consolidation in the Czech Republic has provided a unique opportunity for improving the quality of the environment and sustainability of crop production through better soil and water conservation. The current process of the land con‐ solidation consists of the rearrangement of plots within a given territory, aimed at establish‐ ing the integrated land-use economic units, consistent with the needs of individual land

Integrated territory protection can be reached by controlling runoff by means of design of terraces as a soil erosion control measures. A number of mathematical models, mostly simu‐ lation ones, to solve water-management problems have been compiled, some of which in‐ clude the option of exact mathematical optimization. A certain summary of these models, including their characteristics and application possibilities, were elaborated by Kos (1992). An interesting combination of the application of a simulation and optimization model tech‐ nique in the elaboration of design of a particular water-management system was described by Major, Lenton et al. (1979), a three-model approach to solve water-management systems was used by Onta, Gupta and Harboe (1991). Benedini (1988) dealt more generally with the design and possible applications of these models. Most likely, an optimization model has not been designed, which would enable to attach territory protection and the measures to eliminate the amount and accumulation of runoff in catchments areas to solving water-man‐ The optimization process of designing the system of *integrated territory protection* (the *IOU* system) begins with the processing of the system of organisational, agrotechnical, biotechni‐ cal and technical measures at individual sites of the case study territory. It is necessary to derive hydrograms of direct runoff from extreme rainfall events for each of these variants. Then it is necessary to elaborate the variants of terraces and other conservation measures on all sections of watercourses and variant of designs of retention protection reservoirs. Not on‐ ly rivers, streams and brooks are included into the watercourse category within this proce‐ dure but also sometimes passed watercourses such as terraces, grass infiltration belts or the lines of stabilisation of concentrated runoff waterways in valley lines.

A selection of the most suitable combination of all prepared variants is listed. With respect to the fact that it is necessary to find optimal dimensions for some of the system elements, there is usually a great number, in case of a continual solving even an infinite number, of pos‐ sible combinations. It is therefore necessary to use an optimalized mathematical model to find the most suitable combination. This model was created on the basis of a mixed discrete pro‐ gramming (Korsuň et al., 2002, Dumbrovský et al., 2006). Its basic building stones are three generally formulated partial models: *A.* partial model of protective measures at individual sites of the case study area. *B.* partial model of a watercourse. *C.* partial model of a reservoir.

It is possible to shape an *optimization model of integrated territory protection* (*OMIOU*) from these partial models for any particular territory. The partial models are repeatedly inserted into the *OMIOU* as needed so as to exactly copy the modelled system structure. It is necessa‐ ry to determine in advance one criterion or more simultaneously operating optimization cri‐ teria for each optimization function. A whole range of criteria can be determined for a given purpose. These can be taken from the sphere of economy but also from those of ecology, wa‐ ter-management, social etc. However it is necessary to define the most suitable criteria as far as quality is concerned but also to have a chance to quantify the values of each defined crite‐ rion. On top of that, it is necessary, in case of several simultaneously operating optimization criteria, to assign each criterion its adequate weight with which it will enter the solving process and which will support its effect on the result, so called a compromise solution in competition with the other criteria.

In creating the procedure of the *IOU* system proposal optimization in connection with the process of territory organisation a requirement of a maximal protection of inhabited and other areas with the exertion of minimal means was formulated for the solving process on the level of land consolidation as one of the suitable optimization criteria. It is a criterion consisting of three simultaneously operating partial economic, but at the same time watermanagement and socially aimed at their impacts. Criteria include:

In case of the application of the above mentioned optimization criterion, the following indi‐ cators must be quantified for each pre-optimization processed variant of the protective

**•** its estimated effect *U* expressed financially as an average annual level of damage on land, growth, buildings, roads etc. which will occur after the variant has been realised (*residual*

**•** estimated average annual economical loss *E* in farming production related to the realisa‐

These data represent input information for the partial model *A*. In the course of the optimi‐ zation process, only one – optimal – variant with the most suitable indicators will be chos‐ en from thus prepared variants of systems of protective measures for each partial catchments area element. Residual runoffs concentrating in a watercourse runway from the water‐ course adjacent partial areas protected by optimal systems of measures will cause a gradual accretion of a flood wave passing through the watercourse. The protection from damage which could be caused by this flood wave will be provided by the protective measures on the wa‐ tercourse and retention protective reservoirs as mentioned later (the partial models *B* and *C*).

Binary variables can be used for modelling of individual variants of protective measure sys‐ tems in each of the partial catchments area elements in a discrete way. The total number of catchments area elements will be *m*. If, for example, *n* variants of protective measure sys‐ tems of a *d* th catchments area element are modelled by relations to binary variables *x B1dp є {0, 1}, d = 1, 2,…, m, p = 1, 2,…, n*, the effects of these measure systems for this catchments area

> *xUd* <sup>=</sup> ∑ *Udp* <sup>⋅</sup> *xB*1*dp p*

*xEd* <sup>=</sup> ∑ *Edp* <sup>⋅</sup> *xB*1*dp p*

*xNd* <sup>=</sup> ∑ *Ndp* <sup>⋅</sup> *xB*1*dp p*

into a watercourse in the individual *TI*s.

Optimization of Soil Erosion and Flood Control Systems in the Process of Land Consolidation

http://dx.doi.org/10.5772/50327

39

(1)

(2)

(3)

**•** realisation costs of a particular variant and its average annual own costs *N*,

element can be write into the model using the following equations:

the equation of protective effects (residual damage)

the equation of economic damage

the equation of own costs

measure set on a partial catchments area element:

tion of the proposed measures on arable land.

**•** the amount of residual runoff *Oi*

*damage*),


Seeing that in most cases they are average annual values, quantified for example in thou‐ sands of CZK per year, these criteria can be assigned the same weights 1:1:1 in reflection.

The optimization mathematical model is a system of equations, which model a given system behaviour, the variables in the equation describe a system structure and the dimensions of its individual elements. Non-equations found in each model are transformed into equations by means of additional variables in the course of the model solving process, therefore the term *equation* is used only. The above mentioned partial models were created in the model‐ ling and calculation system *GAMS* (*General Algebraic Modelling System*) in its general form (Charamza, 1993) so it can be used to model any integrated territory protection system. The nature of the solved problems implies that the defining process of all the variables used in the model as positive variables. They can be either continuous ones which are marked *x* herein after or binary ones (they can take on only 0 or 1 values) marked with the symbol *x <sup>B</sup>*. Other symbols are used to mark variables and coefficients. Activities proceeding in time must be modelled in the whole system according uniform timekeeping.

*The partial model A* is aimed at terraces and other biotechnical, agrotechnical, and organisa‐ tion conservation measures in the catchments area of a certain watercourse. These measures are usually designed within land consolidation to decrease overland flow of rainfall events and thus to limit the effects of soil erosion and damage in inhabited territories. The various proposals of protective measures must be elaborated in each individual case before an opti‐ mization model is designed (pre-optimization) as pragmatically created systems of various, mutually complementary interventions with the individual catchments area elements. Such a partial catchments area element could be, for example, valley and slope area above one bank of a certain watercourse section in the range from the bank line to the interstream divide line.

The part of runoff from the design rainfall events which will not be caught by the system of catchments area protective measures (*residual runoff*) will concentrate in a particular water‐ course and will create a design Q runoff or flood wave. The time *T* of passage of the design flood wave through a watercourse will be divided into *r* of equally long *time intervals* (*TI*); time *t* of the durance of one *TI* will thus be given by the relation *t = T / r*. For the individual *TI*s, partial volumes *w <sup>1</sup>* of the design flood wave are then quantified, *i = 1, 2,…, r*.

In case of the application of the above mentioned optimization criterion, the following indi‐ cators must be quantified for each pre-optimization processed variant of the protective measure set on a partial catchments area element:


These data represent input information for the partial model *A*. In the course of the optimi‐ zation process, only one – optimal – variant with the most suitable indicators will be chos‐ en from thus prepared variants of systems of protective measures for each partial catchments area element. Residual runoffs concentrating in a watercourse runway from the water‐ course adjacent partial areas protected by optimal systems of measures will cause a gradual accretion of a flood wave passing through the watercourse. The protection from damage which could be caused by this flood wave will be provided by the protective measures on the wa‐ tercourse and retention protective reservoirs as mentioned later (the partial models *B* and *C*).

Binary variables can be used for modelling of individual variants of protective measure sys‐ tems in each of the partial catchments area elements in a discrete way. The total number of catchments area elements will be *m*. If, for example, *n* variants of protective measure sys‐ tems of a *d* th catchments area element are modelled by relations to binary variables *x B1dp є {0, 1}, d = 1, 2,…, m, p = 1, 2,…, n*, the effects of these measure systems for this catchments area element can be write into the model using the following equations:

the equation of protective effects (residual damage)

$$\boldsymbol{\chi}\_{1ld} = \sum\_{p} \boldsymbol{\chi} \boldsymbol{I}\_{dp} \cdot \boldsymbol{\chi}\_{B1dp} \tag{1}$$

the equation of economic damage

$$\mathbf{x}\_{Ed} = \sum\_{p} \mathbf{E}\_{dp} \cdot \mathbf{x}\_{B1dp} \tag{2}$$

the equation of own costs

consisting of three simultaneously operating partial economic, but at the same time water-

**•** minimization of the average annual damage (material damage: it is estimated that input requirements and conditions will not allow solutions which could lead to losses of human lives) originated by overland runoffs from rainfall events and then by their concentration

**•** minimization of the average annual economic losses in farming production related to the

**•** Minimization of the average annual expenses (the sum of expenses for running and main‐ tenance plus the amortization of the capital goods) of the proposed conservation meas‐

Seeing that in most cases they are average annual values, quantified for example in thou‐ sands of CZK per year, these criteria can be assigned the same weights 1:1:1 in reflection.

The optimization mathematical model is a system of equations, which model a given system behaviour, the variables in the equation describe a system structure and the dimensions of its individual elements. Non-equations found in each model are transformed into equations by means of additional variables in the course of the model solving process, therefore the term *equation* is used only. The above mentioned partial models were created in the model‐ ling and calculation system *GAMS* (*General Algebraic Modelling System*) in its general form (Charamza, 1993) so it can be used to model any integrated territory protection system. The nature of the solved problems implies that the defining process of all the variables used in the model as positive variables. They can be either continuous ones which are marked *x* herein after or binary ones (they can take on only 0 or 1 values) marked with the symbol *x <sup>B</sup>*. Other symbols are used to mark variables and coefficients. Activities proceeding in time

*The partial model A* is aimed at terraces and other biotechnical, agrotechnical, and organisa‐ tion conservation measures in the catchments area of a certain watercourse. These measures are usually designed within land consolidation to decrease overland flow of rainfall events and thus to limit the effects of soil erosion and damage in inhabited territories. The various proposals of protective measures must be elaborated in each individual case before an opti‐ mization model is designed (pre-optimization) as pragmatically created systems of various, mutually complementary interventions with the individual catchments area elements. Such a partial catchments area element could be, for example, valley and slope area above one bank of a certain watercourse section in the range from the bank line to the interstream divide line.

The part of runoff from the design rainfall events which will not be caught by the system of catchments area protective measures (*residual runoff*) will concentrate in a particular water‐ course and will create a design Q runoff or flood wave. The time *T* of passage of the design flood wave through a watercourse will be divided into *r* of equally long *time intervals* (*TI*); time *t* of the durance of one *TI* will thus be given by the relation *t = T / r*. For the individual

*TI*s, partial volumes *w <sup>1</sup>* of the design flood wave are then quantified, *i = 1, 2,…, r*.

management and socially aimed at their impacts. Criteria include:

realisation of proposed protective measures on arable land.

must be modelled in the whole system according uniform timekeeping.

in watercourses.

38 Research on Soil Erosion Soil Erosion

ures.

$$\mathbf{x}\_{Nd} = \sum\_{p} \mathbf{N}\_{dp} \cdot \mathbf{x}\_{B1dp} \tag{3}$$

the equation of residual runoff, i.e. contribution of a *d* th catchments area element to the flood wave volume on a particular watercourse section in *i* th *TI*

$$\begin{array}{rcl}\text{x}\_{\text{Oid}} &=& \sum \text{O}\_{\text{id}p} \cdot \text{x}\_{\text{B1dp}} \\\\ &p \end{array} \tag{4}$$

*The partial model C* is outlined for a designed multipurpose water reservoir with unknown capacities of spaces protective controllable *x OO*, protective non-controllable *x ON* and total *x <sup>V</sup>*. The necessary volumes of the spaces of dead storage *S ≥ 0* and active storage capacity *Z ≥ 0* are constant – these values result from other than protective requirements. The objective of analysis is to find its dimension which, respecting the requirement to create the spaces *S* and *Z,* with its protective spaces will ensure the reduction of culminated runoff from the reser‐ voir to its optimal level during the passage of the design flood wave. In cases when the de‐ signed water reservoir has only a protective function, the value of the *Z* variable is zero; the

Optimization of Soil Erosion and Flood Control Systems in the Process of Land Consolidation

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41

The unknown volume of the total reservoir space is a variable, whose value which is limited from above by the maximal value *V max* corresponding with the biggest realisable variant of the reservoir design during the pre-optimization solutions. From below it is limited by the

We cannot forget a situation when building a reservoir will not be acceptable due to the used optimization criteria. It is therefore necessary to introduce a binary variable *x B2 є {0, 1}* into the set of variable values. If this variable has a zero value, the reservoir will not enter the solving process, if *x B2 = 1*, the entry of the reservoir into solving is cleared. Then the vol‐ ume of the total reservoir space (without evaporation and percolation) must correspond

The equations modelling the passage of the design flood wave through a dam profile, the calculations of the volumes of individual reservoir spaces and of necessary financial means are described in Chapter 5.1 of Patera, Korsuň et al. (2002). The partial model C can be also

The model compilation from the fore mentioned partial elements in the presented form re‐ quires the introduction of a set of concrete coefficients and variables into the model for the model equation system to copy completely a particular system of *IOU*. These coefficients and variables should be derived from the pre-optimization processed background materials. In the case of non-standard requirements of an *IOU* system structure, it is necessary to intro‐ duce other equations to the model. Such new equations would capture these requirements. The model solving process in carried out on a computer by means of some of the *GAMS*

used for already an existing reservoir with a constant volume of the total space.

*xV* = (*S* + *Z*) ⋅ *xB*<sup>2</sup> + *xOO* + *xON* (6)

*Vmin* ⋅ *xB*<sup>2</sup> ≤ *xV* ≤ *Vm* (7)

values of both variables are zero *S = Z = 0* for a dry protective reservoir.

minimal variant, still acceptable for practice, with the total volume *V min*.

with the following conditions

system tools.

for *i = 1, 2,…, r,*

*d = 1, 2,…, m,*

$$p = 1, 2, \dots, n\_r$$

where *x Ud* is the total residual damage in a *d* th catchments area element,

*xE,Ud* is the total economic loss in a *d* th catchments area element,

*xOid* is the total residual runoff from a *d* th catchments area element in *i* th *TI.*

Because only one of the protective measure system variants can enter the solving process, the following condition must be valid for the sum of all the binary variables of a *d* th catch‐ ments area element:

$$\sum\_{p} \ge\_{B1dp} \ge \mathbf{1} \tag{5}$$

*The partial model B* captures the passage of the design flood wave through the watercourse sections. The sections are either left in their present state, the optimization of a river bed or a contour furrow systems design (including the building of protective dams), or the recon‐ struction an earlier carried out adjustment or protective dams may be required. A water‐ course section can also be a water or dry protective reservoir which will be modelled in a way mentioned in the partial model C description.

Flood damage that can occur is quantified for each watercourse section during its model‐ ling. Further, runoffs from the section are calculated in the individual *TI*s of a flood wave pas‐ sage. With respect to the overland flow from the initial section profile to the last one, it is necessary to determine a time shift which will affect collisions of flood waves on the main wa‐ tercourse and at the mouths of its tributaries. The mean value of the runoff volume which can be found in a section (in a river bed or also in an inundation territory) in the course of *i* th *TI* is at the position of the basic section variable. The values of the other variables are related to this variable: the variables of the water flowing through the section, time of concentration, the level of flood damage in the section, and the level of runoff from the section. The courses of these non-linear functions are derived from the watercourse pre-optimization variant de‐ signs. They are replaced with linear function part by part in the optimization model. The for‐ mulation of particular equations is mentioned in Chapter 5.1 of Patera, Korsuň et al. (2002).

*The partial model C* is outlined for a designed multipurpose water reservoir with unknown capacities of spaces protective controllable *x OO*, protective non-controllable *x ON* and total *x <sup>V</sup>*. The necessary volumes of the spaces of dead storage *S ≥ 0* and active storage capacity *Z ≥ 0* are constant – these values result from other than protective requirements. The objective of analysis is to find its dimension which, respecting the requirement to create the spaces *S* and *Z,* with its protective spaces will ensure the reduction of culminated runoff from the reser‐ voir to its optimal level during the passage of the design flood wave. In cases when the de‐ signed water reservoir has only a protective function, the value of the *Z* variable is zero; the values of both variables are zero *S = Z = 0* for a dry protective reservoir.

the equation of residual runoff, i.e. contribution of a *d* th catchments area element to the flood

Because only one of the protective measure system variants can enter the solving process, the following condition must be valid for the sum of all the binary variables of a *d* th catch‐

*The partial model B* captures the passage of the design flood wave through the watercourse sections. The sections are either left in their present state, the optimization of a river bed or a contour furrow systems design (including the building of protective dams), or the recon‐ struction an earlier carried out adjustment or protective dams may be required. A water‐ course section can also be a water or dry protective reservoir which will be modelled in a

Flood damage that can occur is quantified for each watercourse section during its model‐ ling. Further, runoffs from the section are calculated in the individual *TI*s of a flood wave pas‐ sage. With respect to the overland flow from the initial section profile to the last one, it is necessary to determine a time shift which will affect collisions of flood waves on the main wa‐ tercourse and at the mouths of its tributaries. The mean value of the runoff volume which can be found in a section (in a river bed or also in an inundation territory) in the course of *i* th *TI* is at the position of the basic section variable. The values of the other variables are related to this variable: the variables of the water flowing through the section, time of concentration, the level of flood damage in the section, and the level of runoff from the section. The courses of these non-linear functions are derived from the watercourse pre-optimization variant de‐ signs. They are replaced with linear function part by part in the optimization model. The for‐ mulation of particular equations is mentioned in Chapter 5.1 of Patera, Korsuň et al. (2002).

∑ *xB*1*dp* <sup>=</sup> <sup>1</sup>

(4)

(5)

*xOid* <sup>=</sup> ∑ *Oidp* <sup>⋅</sup> *xB*1*dp p*

where *x Ud* is the total residual damage in a *d* th catchments area element,

*xOid* is the total residual runoff from a *d* th catchments area element in *i* th *TI.*

*p*

*xE,Ud* is the total economic loss in a *d* th catchments area element,

way mentioned in the partial model C description.

wave volume on a particular watercourse section in *i* th *TI*

for *i = 1, 2,…, r,*

40 Research on Soil Erosion Soil Erosion

*d = 1, 2,…, m,*

*p = 1, 2,…, n,*

ments area element:

The unknown volume of the total reservoir space is a variable, whose value which is limited from above by the maximal value *V max* corresponding with the biggest realisable variant of the reservoir design during the pre-optimization solutions. From below it is limited by the minimal variant, still acceptable for practice, with the total volume *V min*.

We cannot forget a situation when building a reservoir will not be acceptable due to the used optimization criteria. It is therefore necessary to introduce a binary variable *x B2 є {0, 1}* into the set of variable values. If this variable has a zero value, the reservoir will not enter the solving process, if *x B2 = 1*, the entry of the reservoir into solving is cleared. Then the vol‐ ume of the total reservoir space (without evaporation and percolation) must correspond with the following conditions

$$\mathbf{x}\_V = \left< \mathbf{S} + \mathbf{Z} \right> \cdot \mathbf{x}\_{B2} + \mathbf{x}\_{\text{OO}} + \mathbf{x}\_{\text{ON}} \tag{6}$$

$$\begin{array}{ccccccccc} V\_{m\bar{m}} \cdot \mathbf{x}\_{B2} & \leq & \mathbf{x}\_{V} & \leq & V\_{m} & & & & & & & \mathbf{(7)} \end{array}$$

The equations modelling the passage of the design flood wave through a dam profile, the calculations of the volumes of individual reservoir spaces and of necessary financial means are described in Chapter 5.1 of Patera, Korsuň et al. (2002). The partial model C can be also used for already an existing reservoir with a constant volume of the total space.

The model compilation from the fore mentioned partial elements in the presented form re‐ quires the introduction of a set of concrete coefficients and variables into the model for the model equation system to copy completely a particular system of *IOU*. These coefficients and variables should be derived from the pre-optimization processed background materials. In the case of non-standard requirements of an *IOU* system structure, it is necessary to intro‐ duce other equations to the model. Such new equations would capture these requirements. The model solving process in carried out on a computer by means of some of the *GAMS* system tools.
