**2. Water allocation model**

The water allocation model can be divided into the surface water model and the groundwa‐ ter model. The parts of the water allocation model are described in the sequel.

### **2.1. Surface water model**

largest account for about 90% of the total storage capacity of roughly 9x109 m³

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

timal water management approach are presented in section 5.

**2. Water allocation model**

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‐

The water allocation model can be divided into the surface water model and the groundwa‐

ter model. The parts of the water allocation model are described in the sequel.

of a total length of about 400 km.

24 Water Supply System Analysis - Selected Topics

is distributed to the customers using rivers and artificial transport ways (channels, pipes)

. The water

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‐ cates a lag element of the first order.

$$T\_1 \cdot \frac{dy}{dt} + y = u \tag{1}$$

where T1 is the only parameter and indicates the time lag; t indicates the time. The following relationship represents the dead-time element:

$$y = \mu(t - T\_t) \tag{2}$$

In this case the parameter is the dead-time *Tt* , which is thus the measure of the time taken for water to flow in a conduit over a known distance.

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

$$V\_j^{k+1} = V\_j^k + \Delta t^k \left(\sum\_{i \in \mathcal{E}(j)} \mathcal{Q}\_i^k + A\_{O,j}^k q\_{\text{evpt},j}^k - q\_{\text{seep},j}^k + A\_{O \text{max}} q\_{\text{pre},j}^k\right) \tag{3}$$

where *AO*, *<sup>j</sup> <sup>k</sup>* <sup>=</sup> *<sup>f</sup>* (*Vj <sup>k</sup>* ), the storage volume is denoted by *V*, the discharge into and from the 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.

Channels with a very low slope are modeled as water storage. The level dependent upper bound for the channel outflow is derived from a steady state level-flow relation like e.g. Chezy-Manning friction formula and is directly added as constraint to the optimization problem.

The surface water simulation model has been implemented in Matlab/Simulink using the toolbox "WaterLib" [5] and contains the most important elements of the drinking water sup‐ ply system of Beijing. The simulation model was developed to reach a sufficient accuracy as well as a high simulation speed. A one-year simulation is carried out in a simulation time of several seconds. This is a very important condition for using the model in the decision sup‐

The most important water resource in the considered area is groundwater that is modeled by a dynamic spatially distributed finite element groundwater model. The governing equa‐ tion for groundwater flow is Darcy's law [6] describing slow streams through unconfined aquifers. Combining Darcy's law with mass conservation yields the partial differential equa‐

In (4) denotes *h* the hydraulic head (which corresponds to the groundwater level) and *kf* the hydraulic conductivity that governs the hydrogeological properties of the soil. S0 denotes the specific storage coefficient. The terms on the right hand side of (4) summarize all sources and sinks that coincide with the time dependent groundwater exploitation due to industry, households and agriculture (Qexpl) and recharge e. g. due to precipitation and irrigation

The partial differential equation (4) is an initial-boundary value problem which has to be solved numerically for *h* in the 3 dimensional model domain Ω. The groundwater model has been implemented using FEFLOW, which is a Finite Element (FEM) software specialized on subsurface flow [7]. The initial condition is h (Ω,t0) (groundwater surface) at the initial time t0. The inflow/outflow is described by Dirichlet boundary conditions, i.e. h (∂Ω) at the boun‐ dary ∂Ω and by well boundary conditions, that define a particular volume rate into or out of Ω. The advantage of the latter one is that they are scalable. The 3D FEM model consists of more than 150,000 nodes, distributed on 25 layers (cf. Fig. 3). Huge computational costs re‐ sult from this high resolution. The simulation of 5 years needs ~15 Minutes on an Intel Core 2 Duo CPU (2.5 GHz). Hence it is very time consuming to calculate optimal water allocation strategies with the 3D FEM groundwater model. This is the motivation for model reduction

The main task with respect to the groundwater model is the parameterization of the largescaled model covering an area of 6,300 km². On the one hand the time independent soil pa‐

of measured values. On the other hand the source / sink terms Qexpl, Qrech have to be calculat‐ ed time dependent. For these calculations time dependent maps of precipitation and water demand are needed. The water demand is splitted into the three user groups households,

, S0 have to be estimated and generalized for the whole domain Ω by a (small) set

<sup>0</sup> ( *f rech l* ) exp *Sh k h Q Q* -Ñ× Ñ = - & (4)

Model Based Sustainable Management of Regional Water Supply Systems

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

27

port system.

(Qrech) in Ω.

(see subsection 2.3).

rameters kf

**2.2. Groundwater water model**

tion (4) which is a diffusion equation.

The Structure of the Beijing water supply system is shown in Fig. 2. Firstly, there are the four main reservoirs Miyun, Huairou, Baihebao and Guanting. Further sources are groundwater storages and water transfers. Secondly, there are the water transportation systems such as channels and rivers. Miyun reservoir and Huairou reservoir are connect‐ ed to Beijing by the Miyun-Beijing water diversion. In the simulation model the arrows describe hydraulic behavior of water flow. Baihebao and Guanting reservoir are connect‐ ed by tunnel and river Guishui. From Guanting water runs into the Yongding river wa‐ ter diversion system to Beijing. Existing retention areas for flood control are also considered in the simulation model.

The surface water from channels and rivers is delivered to the customers in different ways, directly or through the surface waterworks. Groundwater is distributed to the customers through ground waterworks as well as motor-pumped wells. Therefore, the waterworks build the third part of the Beijing water supply system. The last category is made up of all the customer groups (agriculture, industry, households and environment). To complete up the cycle, catchments area models are integrated in the system to take into account precipita‐ tion and evapotranspiration.

**Figure 2.** Structure of the Beijing water supply system

The surface water simulation model has been implemented in Matlab/Simulink using the toolbox "WaterLib" [5] and contains the most important elements of the drinking water sup‐ ply system of Beijing. The simulation model was developed to reach a sufficient accuracy as well as a high simulation speed. A one-year simulation is carried out in a simulation time of several seconds. This is a very important condition for using the model in the decision sup‐ port system.
