*2.2.3. Software realisation of the parameter calculation within the DSS*

The DSS works on the base of so-called *scenarios* whereby a scenario can be understood as set of input data that represents either a real historical situation or an imaginable situation in future with respect to precipitation, exploitation, water use etc. This complete set of input data is applied to the hydrological model to obtain an answer to the question 'What are the effects to the water allocation system'. The historical scenarios usually are used for the vali‐ dation of the models whereas future scenarios should deliver guidelines for the best way to act in future under certain circumstances. In order to obtain reasonable results from the sim‐ ulation the scenario has to be complete and consistent. Otherwise the models will gain un‐ reasonable results. A scenario is complete if all data are available the hydrological model expects. In order to keep the system stable often default values are inserted if no data are assigned. But in this case the question is if the data are consistent.

The consistency of data can only be assured by the user. So if we use precipitation rates for groundwater model and for the surface water model for instance in general the information is provided in different data set and formats. In this case the user is responsible, that the da‐ ta for the groundwater model match the data for the surface water model, i.e. that at same time periods in same regions the same precipitation rates are considered.

Nevertheless, the user can be supported by a parameterization software, as it was realized within the Scenario Wizard of the Beijing DSS to ensure consistency as good as possible. With respect to the groundwater model the scenario is complete and consistent only if the before mentioned data are available for the complete simulation horizon:

**•** Initial conditions

**Figure 6.** Landuse map of the region of Beijing

34 Water Supply System Analysis - Selected Topics

*2.2.3. Software realisation of the parameter calculation within the DSS*

assigned. But in this case the question is if the data are consistent.

The DSS works on the base of so-called *scenarios* whereby a scenario can be understood as set of input data that represents either a real historical situation or an imaginable situation in future with respect to precipitation, exploitation, water use etc. This complete set of input data is applied to the hydrological model to obtain an answer to the question 'What are the effects to the water allocation system'. The historical scenarios usually are used for the vali‐ dation of the models whereas future scenarios should deliver guidelines for the best way to act in future under certain circumstances. In order to obtain reasonable results from the sim‐ ulation the scenario has to be complete and consistent. Otherwise the models will gain un‐ reasonable results. A scenario is complete if all data are available the hydrological model expects. In order to keep the system stable often default values are inserted if no data are

The consistency of data can only be assured by the user. So if we use precipitation rates for groundwater model and for the surface water model for instance in general the information is provided in different data set and formats. In this case the user is responsible, that the da‐


In order to obtain a complete and consistent set of input data for the groundwater model the user has to pass through the groundwater panel of the DSS Scenario Wizard and a so-called groundwater project is created. A groundwater project is a part of the scenario containing all groundwater relevant data.

Within the GW panel the user has to execute five subpanels such that in the end at least a complete set of input data for the groundwater model is generated.


ploitation rates from the different well fields and the upper and deeper inflow into the model area.

zation task a prediction of the hydraulic head (groundwater level) at a set of representative and fixed points is sufficient, an input-output model (e.g. a linear state space model) with considerably smaller order n (e.g. n < 50) than the original FEM model has to be derived.

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37

Methods for model reduction of those large scale systems have gained increasing impor‐ tance in the last few years [13]. Two main classes of methods for model reduction can be identified, namely methods based on singular value decomposition (SVD) and Krylov based methods. SVD based methods are suited for linear systems and nonlinear systems of an or‐ der n < 500 (e. g. balanced truncation for linear systems, proper orthogonal decomposition (POD) for nonlinear systems). Most of these methods have favourable properties like global error bounds and preservation of stability [13]. Krylov based methods are numerically very efficient as only matrix multiplications and no matrix factorization or inversion are needed. Hence they are also suited for large-scale systems. Unfortunately, global error bounds and preservation of stability cannot be guaranteed. Hence actual research is focused on the de‐ velopment of concepts which combine elements of SVD and Krylov based methods [13].

All of these approaches have in common that they aim to approximate the state vector *x* with respect to a performance criterion, e. g. minimize the deviation between original sys‐ tem and reduced system for a given test input. As a black box input-output model would be sufficient for our purposes there is no need to approximate the whole state vector *x*. Further‐ more, the dimension n of a reduced model which approximates the whole state space vector *x* would be in most cases n > 100. With this dimension, for the given optimization problem the solution time would be unacceptably high (~ hours). Last but not least the use of the commercial software FEFLOW also prevents the application of e. g. a Krylov based method

The basic idea of the proposed model reduction method is sketched in Fig. 8. We assume the existence of a reference scenario which means that the time dependent input parame‐ ters uref(t) of the FEM groundwater model (especially groundwater exploitation Qexpl and recharge Qrech) are determined for the whole optimization horizon. In practical cas‐ es these reference scenarios are mostly available or can be generated by plausible as‐ sumptions. Hence the task consists in the derivation of a model which approximates the behavior of the full FEM model in the case that the input parameters u differ from uref(t). This model is gained by identification techniques: Test signals (e.g. steps) are add‐ ed to the reference input uref(t) (dimension p) and the corresponding deviations from the reference output yref(t) (dimension q) are identified. Doing this separately for every com‐ ponent of the input-/output vectors u and y, we finally merge the (p x q) single input-sin‐ gle output (SISO) models to a multi input-multi output (MIMO) model. For the groundwater model, the input parameters are e. g. cumulated (e.g. spatially integrated) exploitation of certain regions or cumulated exploitations of large well fields. The output parameters of the groundwater model are the hydraulic head at representative points ("observation wells"). In our application 13 input and 13 output parameters have been defined by the users: The inputs consist by 9 counties and 4 wellfields, the 13 output pa‐ rameters are 12 observation wells and the mean hydraulic head of the whole area of the water supply system. As the slow stream groundwater flow can be interpreted as diffu‐

as no model representation (e. g. state space model) is provided by the software.

**5.** In the last subpanel the generation of the spatial and temporal distribution of the groundwater recharge data is performed. Here the user again has to provide a weight‐ ing map that determines how much of the supplied water becomes groundwater re‐ charge and how much becomes evapotranspiration. The rest of the required data was already determined in the before described subpanels. This step finishes the creation of the groundwater project and all the generated data are documented in a specific groundwater scenario report which is saved within the corresponding project directory. After the simulation of a scenario with a groundwater simulation run a report about simulation results is saved in the project directory as well.

**Figure 7.** Graphical user interface for the generation of input data like groundwater recharge maps

All computed data are saved as ASCII grid maps which can be read by any GIS system. Figure 7 shows the graphical user interface generating groundwater maps.

In order to gain the project data accessible to the FEFLOW simulation model the spatially distributed data have to be projected on the finite element mesh. But in this step no new in‐ formation is gained therefore it is not a procedure which is important for the scenario gener‐ ation but it is only a technical requirement.

### **2.3. Derivation of reduced groundwater model**

For the optimization of the water allocation system the full 3D FEM model (with > 150,000 nodes) is not very suited due to the mentioned large computational time. As for the optimi‐ zation task a prediction of the hydraulic head (groundwater level) at a set of representative and fixed points is sufficient, an input-output model (e.g. a linear state space model) with considerably smaller order n (e.g. n < 50) than the original FEM model has to be derived.

ploitation rates from the different well fields and the upper and deeper inflow into

**5.** In the last subpanel the generation of the spatial and temporal distribution of the groundwater recharge data is performed. Here the user again has to provide a weight‐ ing map that determines how much of the supplied water becomes groundwater re‐ charge and how much becomes evapotranspiration. The rest of the required data was already determined in the before described subpanels. This step finishes the creation of the groundwater project and all the generated data are documented in a specific groundwater scenario report which is saved within the corresponding project directory. After the simulation of a scenario with a groundwater simulation run a report about

simulation results is saved in the project directory as well.

**Figure 7.** Graphical user interface for the generation of input data like groundwater recharge maps

Figure 7 shows the graphical user interface generating groundwater maps.

ation but it is only a technical requirement.

**2.3. Derivation of reduced groundwater model**

All computed data are saved as ASCII grid maps which can be read by any GIS system.

In order to gain the project data accessible to the FEFLOW simulation model the spatially distributed data have to be projected on the finite element mesh. But in this step no new in‐ formation is gained therefore it is not a procedure which is important for the scenario gener‐

For the optimization of the water allocation system the full 3D FEM model (with > 150,000 nodes) is not very suited due to the mentioned large computational time. As for the optimi‐

the model area.

36 Water Supply System Analysis - Selected Topics

Methods for model reduction of those large scale systems have gained increasing impor‐ tance in the last few years [13]. Two main classes of methods for model reduction can be identified, namely methods based on singular value decomposition (SVD) and Krylov based methods. SVD based methods are suited for linear systems and nonlinear systems of an or‐ der n < 500 (e. g. balanced truncation for linear systems, proper orthogonal decomposition (POD) for nonlinear systems). Most of these methods have favourable properties like global error bounds and preservation of stability [13]. Krylov based methods are numerically very efficient as only matrix multiplications and no matrix factorization or inversion are needed. Hence they are also suited for large-scale systems. Unfortunately, global error bounds and preservation of stability cannot be guaranteed. Hence actual research is focused on the de‐ velopment of concepts which combine elements of SVD and Krylov based methods [13].

All of these approaches have in common that they aim to approximate the state vector *x* with respect to a performance criterion, e. g. minimize the deviation between original sys‐ tem and reduced system for a given test input. As a black box input-output model would be sufficient for our purposes there is no need to approximate the whole state vector *x*. Further‐ more, the dimension n of a reduced model which approximates the whole state space vector *x* would be in most cases n > 100. With this dimension, for the given optimization problem the solution time would be unacceptably high (~ hours). Last but not least the use of the commercial software FEFLOW also prevents the application of e. g. a Krylov based method as no model representation (e. g. state space model) is provided by the software.

The basic idea of the proposed model reduction method is sketched in Fig. 8. We assume the existence of a reference scenario which means that the time dependent input parame‐ ters uref(t) of the FEM groundwater model (especially groundwater exploitation Qexpl and recharge Qrech) are determined for the whole optimization horizon. In practical cas‐ es these reference scenarios are mostly available or can be generated by plausible as‐ sumptions. Hence the task consists in the derivation of a model which approximates the behavior of the full FEM model in the case that the input parameters u differ from uref(t). This model is gained by identification techniques: Test signals (e.g. steps) are add‐ ed to the reference input uref(t) (dimension p) and the corresponding deviations from the reference output yref(t) (dimension q) are identified. Doing this separately for every com‐ ponent of the input-/output vectors u and y, we finally merge the (p x q) single input-sin‐ gle output (SISO) models to a multi input-multi output (MIMO) model. For the groundwater model, the input parameters are e. g. cumulated (e.g. spatially integrated) exploitation of certain regions or cumulated exploitations of large well fields. The output parameters of the groundwater model are the hydraulic head at representative points ("observation wells"). In our application 13 input and 13 output parameters have been defined by the users: The inputs consist by 9 counties and 4 wellfields, the 13 output pa‐ rameters are 12 observation wells and the mean hydraulic head of the whole area of the water supply system. As the slow stream groundwater flow can be interpreted as diffu‐ sion process (cf. equation (4)) only nearby located input and output parameters (e.g. re‐ gions/wellfields and the corresponding observation wells) have some correlation and a SISO model with these input-/output combinations can be gained. Due to this physical reason the number of relevant SISO models is relatively small and hence the resulting MIMO model of relatively low dimension (n < 50) which is appropriate for the optimiza‐ tion problem. This proposed approach can be called trajectory and identification based model reduction. A similar approach (for a nonlinear large scale system) is found in [14].

( ) <sup>1</sup> , , *k k k kk* <sup>+</sup> **x fxuz** = (8)

Model Based Sustainable Management of Regional Water Supply Systems

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39

( , , ) *k k kk* **hxuz 0** = (9)

( , , ) *k k kk* **gxuz 0** £ (10)

The state variables *x* are the volume content of the reservoirs and channels with small slope and the states of the reduced groundwater model. Control variables u are the discharge of the transport elements as well as the water demand of the customers. The uncontrollable in‐ puts z are the direct precipitation and the potential evaporation for the reservoirs and the flow of rivers entering the considered region, which is derived by means of rainfall-runoff-

The process equations (8) consist of the balance equations of the storage nodes and the re‐ duced groundwater model. The balance equations of the non-storage nodes are formulated as general equality constraints (9). The objective function (6) contains the goals of the water management, which are primarily the fulfillment of the customer demand, the compliance with targets for the reservoir and groundwater storage volume and the delivery of water with respect to environmental purposes. Therefore quadratic terms are formulated, which penalize the deviations from desired values, like e.g. for the demand deficit of the demand

models. The number of time steps within the optimization horizon is denoted by *K.*

( ) ( )

*j dem ref j kk kk j j k*

*k k*

*u q*

*q* -

term applies for every time step within the optimization horizon, other terms are formulated only for the final point of the horizon, like e.g. for the desired volume content of the reser‐

The inequality constraints (10) follow from the technical capabilities of the water distribu‐ tion system and rules to guarantee safe operation, which are simple bounds for the control

min max

0 2

*fu t*

where *qdem,ref,j* is the demand of and *uj*

as well as constraints for the reservoir volume *xv*:

2 , ,

=r D (11)

is the discharge delivered to the customer. While this

*k kk* **u uu** £ £ (12)

, ,

*dem ref j*

node j:

voirs.

variables:

**Figure 8.** Reduced groundwater model as a linear state space model in combination with a pre-simulated reference scenario.
