**1. Introduction**

The sustainable management of the water resources and a safe supply of drinking water will play a key role for the development of the human prosperity in the following decades. The fast growth of many cities puts a large pressure on the local water resources, especially in regions with arid or semi-arid climate. A research project at Fraunhofer IOSB and AST has aimed to investigate ways for economic and sustainable use of the available water resources in the region of the capital of China, Beijing [1].

A main issue of the project is to develop components for a model-based decision support system (DSS), which will assist the local water authority in management, maintenance and extension of the water supply system at hand [2]. This paper deals with the derivation of suitable management strategies for a mid (till long) term horizon based on assumptions for future environmental and socio-economic conditions, which are provided by other modules of the DSS. The general structure of the proposed optimal control DSS is shown in Fig. 1.

An overview about several DSS concepts and implementations is provided in [3, 4]. A challenge of the given problem is the large area of the water supply system. The water management has to consider the total water resources of five river basins with an area of 16,800 km² as well as large groundwater storage in the plain with an area of 6,300 km². The main portion of the annual precipitation (85%) in this semi-arid region is falling from June to September leading to a highly uneven distribution throughout the year. The for‐ merly abundant groundwater resources have been overexploited over the last decades re‐ sulting in a strong decline of the groundwater head (up to 40 m). Five reservoirs are important for the management of the surface water in the considered area, where the two

<sup>© 2013</sup> Bernard et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

largest account for about 90% of the total storage capacity of roughly 9x109 m³ . The water is distributed to the customers using rivers and artificial transport ways (channels, pipes) of a total length of about 400 km.

**2.1. Surface water model**

cates a lag element of the first order.

relationship represents the dead-time element:

In this case the parameter is the dead-time *Tt*

1

*<sup>k</sup>* <sup>=</sup> *<sup>f</sup>* (*Vj*

where *AO*, *<sup>j</sup>*

for water to flow in a conduit over a known distance.

At time step *k*, the volume of a reservoir node evolves as follows:

( )

*i j*

ÎE

The surface water model has to comprise all important elements for the allocation, storage and distribution of water within the considered region. The intended field of application of the decision support system under development embraces management and upgrading strategies for the mid and long term range. This implies a significant simplification for the process model, because the retention time along the different transport elements (river reaches, channel or pipelines) is less than the desired minimum time step for decision of one day. Therefore, a simple static approach for flow processes is sufficient and the use of so‐ phisticated models for the dynamics of wave propagation (like e. g. Saint-Venant-Equations) with respect to control decisions is avoided. In this case, the flow characteristics are repre‐ sented by simple lag elements of the first order combined with dead-time elements. Given *y* and *u* as an output and input, respectively. The following mathematical relationship indi‐

1

*dy T yu dt*

where T1 is the only parameter and indicates the time lag; t indicates the time. The following

The surface water system is described as a directed graph. The edges characterize the trans‐ port elements and introduce only a variable for the discharge. The nodes represent reser‐ voirs, lakes, points of water supply or extraction and simple junction points. Every node constitutes a balance equation involving the edges linked with and possibly the storage vol‐ ume. The sole nonlinearity results from modeling the evaporation from the water surface of the storages (volume-area-curve), which is described by a piecewise polynomial approach.

> , *k k k k kk k <sup>k</sup> j j i O j evpot j seep j O prec j*

= +D + - + ç ÷

æ ö

è ø

storage is denoted by *Q* and discrete time step is denoted by *Δt*. The total evaporation from the reservoir depends on the water surface *A0* and the potential evaporation *qevpot.* q*seep* de‐ notes the seepage from the reservoir to the groundwater and *qprec* specifies the precipitation.

*V V t Q Aq q A q* <sup>+</sup>

, , , max ,

*<sup>k</sup>* ), the storage volume is denoted by *V*, the discharge into and from the

å (3)

× += (1)

Model Based Sustainable Management of Regional Water Supply Systems

http://dx.doi.org/10.5772/51973

25

( )*<sup>t</sup> y ut T* = - (2)

, which is thus the measure of the time taken

**Figure 1.** Structure of the proposed decision support system (DSS) for the region of Beijing

A common approach for policy generation in this field of application is to use mathematical programming techniques based on a dynamic model of the essential elements of the water allocation and distribution system [3]. The great impact of the groundwater storage for the supply system at hand requires a more detailed description of the groundwater flow dy‐ namics compared to other known DSS implementations. Therefore a 3D Finite-Element model of the plain region has been developed. However, a direct integration of this 3D mod‐ el into an optimal control framework is not possible due to its computational costs. A trajec‐ tory based model reduction scheme is proposed, which guarantees a very fast response of the DSS in combination with a specially tailored non-linear programming algorithm.

The chapter is organized as follows: In section 2 the essential models (surface and ground‐ water models) and the model reduction approach are described. While the formulation of the optimal control problem is subject of section 3, the numerical solution of the large scale structured non-linear programming problem is described in section 4. First results of the op‐ timal water management approach are presented in section 5.
