No title

The nature of transport and energy use is radically changing along with the upward trend of electric vehicles. The rapid technological development of electrical vehicles opens new opportunities from the electricity distribution point of view. Efficiency can be improved by implementing energy storages to the grid and cutting the load peaks by feeding power on peak hours from the energy storages to the grid. Electric vehicles with vehicle-to-grid (V2G) properties provide an opportunity to meet this challenge. In this paper, the challenge is approached from the economic perspective of an electricity distribution company. The key target of the paper is to determine whether there is economic potential for energy storages in networks in general. To this end, a generic model is introduced to analyze the feasibility of electric vehicles as energy storages in distribution networks. The methodological framework presented in the paper provides an opportunity for distribution system planners to estimate the preliminary feasibility of energy storages. The focus is on the discharging (vehicle to grid) perspective. The paper answers, for instance, the question of how to define the feasible level of energy storages (batteries) in the distribution system. In the paper, for background information, an extensive literature review is provided on electric vehicles.


INTRODUCTION
The beach is the boundary between the rough open sea and the land. The stability of beaches is important for the protection of human settlements, for the protection of high dunes and dikes serving as storm surge walls, and for the economy in touristic regions. Stability is supported by human interference in terms of artificial sand fills and groyne construction works. To optimize their effect and to avoid negative impacts, extensive inquiries are necessary. In the past few years numerical models have been acknowledged as an instrument entitled to the same rights as physical models or natural inquiries.
Starting from the necessities of an economic design and without negative impacts, a systematic research of different geometries and properties of beach nourishment using mathematical models is presented.
This paper describes the components of numerical modelling as well as the concept of a systematic and reproducible path to find useful geometrical forms of beach nourishments.

BASIC EQUATIONS
The numerical simulation of coastal processes has a long tradition. Because the morphological change of the bathymetry near the beach is induced by the variable distribution of the surface waves as well as the nearshore currents, it is necessary to analyze the direction and height of waves, the direction and velocity of the nearshore current, and the turbulent energy. Most of the simulation systems describe the hydrodynamics near the beach based on a modular concept with an iterative indirect coupling of the various modules. In contrast, the numerical model presented here is based on the following compact system of time-dependent partial differential equations: We can say that the model consists of three submodels for calculating waves (1-4), nearshore currents (5-6), and sediment transport with bottom changes (7-9) which are fully coupled. The mathematical type of this system is a multi-dimensional advection-diffusion problem. This is of great importance rule for the numerical realization. A detail description of the components will be provided in the following subsections.
A Cartesian co-ordinate system is introduced in where the plan coordinate-axes for and can be placed arbitrarily. The co-ordinate is positive upwards. Starting from the importance of the wave distribution in the surf zone near a coast we begin with the wave components.

Wave Model
Waves approaching shallow water and the beach areas in the absence of currents are deformed due to shoaling, refraction, reflection, diffraction and breaking. The effect of reflection is negligibly small. Thus the theory of monochromatic waves may be used.
To describe the effects of interaction of waves and currents as well as the interaction with the changeable bathymetry a new hyperbolic type wave formulation has been introduced [3]. With this formulation we can use the nonlinear wave potential developed by YOO (see [4]) as well as the classical potential. The system of equations (1-4) contains the wave vector and the calculation of the wave amplitude . In these equations stands for the wave group velocity and for the energy transport velocity. The radian frequency is calculated by the dispersion relation (3), and the BATTJES relation (2) describes diffraction effects. The equations include the influence of bottom slope [4]. This enters the equations by the terms To describe the input of turbulent energy due to wave breaking in the water body, a lot of breaking criteria are realized. The wave energy loss is described by the energy dispersion coefficient . We use the D ALLY, DEAN and DALRYMPLE [1] model to assume that there is a stable wave height after breaking equal to some fraction of water depth. For wave breaking the rate of energy dissipation is proportional (with the factor ) to the difference between the actual and the stable wave energy flux ( ).
Alternatively a wave-front model (see [2]) can be used. This wave approximation can additionally describe the special effects of crossing waves and has a better approximation for diffraction. Additionally the wave-front approximation gives good intuitive pictures for wave-front distribution. An automatic refinement of computational points of the wave front with respect to the hydrodynamic requirements is calculated. The wave-front model cannot be integrated directly into the above model philosophy. The mathematical type of this model is a characteristic solution and can be coupled only iteratively with the equation of nearshore currents and sediment transport.
Next: Sediment Transport and Bottom Up: BASIC EQUATIONS Previous: Wave Model

Current Model
The tidal motion is simulated under the assumption of vertically integrated flow. The corresponding equations of momentum are (6) and the mass conservation is described by equation (7). In these equations stands for the vertically integrated velocities, for the water depth at rest, and for the free surface. The earth acceleration is , is the water density, represents turbulent effects, and the bottom friction. Commonly the Taylor-Newton formulation is used, to which the influence of orbital motion of the waves is added.
The radiation stress term represents the momentum flow induced by the deformation of short period waves running at the surface of the water body. These effects are very small for deep water conditions, however, they are important in regions of deformation and breaking of short waves. These terms couple long and short wave propagation.
For simplicity, Coriolis, atmospheric pressure and wind terms are omitted.

Sediment Transport and Bottom Change Model
Under breaking waves and due to high flow velocities, material is being picked up from the bed and transported in the flow regime. This transport can be distributed in bed and suspended load. To determing the concentration of the suspended sediment we use a vertically integrated advection-diffusion equation (9). In this equation is the suspended sediment concentration, is the transport rate in the -direction as a sum of suspended and bed load . The diffusion coefficient is called with and is a source or sunk term, respectively. The model assumes that there is a maximum possible concentration and an actual concentration . The term is proportional to the difference between these two concentrations: with erosion and sedimentation parameters ( and ) and the sediment fall velocity .
Calculating how much material enters and leaves a computational cell leads to the bed continuity equation (7). From this the evolution of bathymetry may be calculated. The term stands for the variation of the water depth.

NUMERICAL APPROXIMATION
The given system of time-dependent partial differential equations is solved by the finite element method. Triangular grids are used with linear interpolation functions in space for the state variables.
Higher order terms are removed by partial integration. The integration follows the stream-line upwinding technique. The upwinding parameters are obtained proportional to the characteristic velocities of the different components by introducing corresponding element-PECLET-numbers.
The time integration is performed in the common step by step manner. Due to the simple structure of the software different 1st to 5th order time-integration schemes with time-step control are implemented.

OPTIMIZATION OF THE GEOMETRICAL SHAPE OF BEACH NOURISHMENT
The subject of the presented inquiry were different measures in a system of sandbars and grooves of the island Sylt. For example tombolos, double tombolos and beach parallel beach nourishments in different locations and geometries were investigated. Because of the complexity of parameters necessary to optimize such beach nourishment we selected an iterative procedure.
In a first step the tidal flow and the propagation of short waves were analyzed for the global area around the island. This inquiry generated boundary conditions for small area studies. Figure 1 shows a wave field for waves under westerly wind. At the tips of the island complicated processes of shoaling, refraction, diffraction and breaking can be seen due to extended shallow banks.

Figure 1: Large area wave simulation
Starting from the natural situation of a system of sandbars and grooves of the island Sylt an idealized bathymetry was used. Such a bathymetry allows a clear relation between cause and result of the physical processes. Figure 2 shows the calculated wave fields and velocities at one particular time of a tide for different locations of tombolos.

Figure 2: Different Tombolo locations
Applying the combined wave-current model the geometry was investigated in the first place to provide an energy dissipation as homogeneous as possibly at and in the vicinity of the beach nourishment corps, and to minimize the occurring streaming. We may now select a lot of parameters and locations for the beach nourishment, meeting the above criteria.
In a next step, with a smaller set of such measurments we can use the coast development model. Figure 3 shows a complete set of simulation results at one particular time of a tide. This figure provides the bathymetry for a sandfill reaching from the beach towards the underwater bar at a distance of about 400 meters from the beach. The flow field shows two typical eddies forming at the shoulders of the sandfill due to waves running nearly perpendicular to the coast. At last the areas of erosion are given for a simulation time of just one tide.

Figure 3: Tombolo
Using results of the idealized bathymetry and to verify the coast development model, wave-current-bottom-change simulations for a natural bathymetry were investigated. Supplementing these investigations measures at sandbars -such as elevation and broadening -were taken into account.
In addition an experimental beach nourishment near Kampen (Sylt) was performed verifying the results of the mathematical models. With the studies in the idealized bathymetry a lot of natural hydrodynamic effects are now plausible.