# Category Dynamic Modeling, Simulation and Control of Energy Generation

## Fuel Cells (PEMFCs)

In what follows we shall briefly and succinctly describe an alternate modelling approach from a different perspective. The modelling approach described in this section closely follows the flow modelling approach, to some extent, of Xue et al. (2006) and of del Real et al. (2007) within the electrode channels and for gaseous diffusion in the GDLs. The membrane is parameterized in terms of the electro­osmotic drag coefficient, the water content parameter, and the mass diffusivity of water vapor in the membrane. Although we adopt the same general approach as del Real et al. (2007) in defining the cell voltage, our model of the electrochemistry is representative of the voltage over potential due to the cell electrode charge double layer.

## Equations of Motion in Multiblade Coordinates

To express the flapping and torsion blade equations of motion in terms of multi­blade coordinates the collective and cyclic summation operators are applied to them. Thus, the flapping and torsion blade equations in multiblade coordinates are

 J 1 0 0 J 0 0 0 ‘j 1 0 0 ‘hi’ X П 0 1 0 ec + X ^ 0 0 2 jc +л 0 1 0 ec j=1 0 0 1 es j=1 0 -2 0 hs 0 0 1 .c k = 1,2,3„J,

(4.7.22)

## Combustion Chambers

In a combustion chamber, the propellants are injected through a number of coaxial injectors, with oxidant injected usually through the center units and the fuel through the others to facilitate rapid premixing. The injectors are usually of the
shear coaxial type, the swirl-coaxial type, or the impinging type. The shear coaxial injectors are supposedly effective in mixing two propellants with substantial density differences, while impinging and swirl-coaxial injectors are used for two liquid propellants with similar densities. In swirl-coaxial injectors, one of the two propellants is given an initial angular velocity. There are several variations in the geometry of injectors which all have a significant influence on combustion per­formance characteristics and stability...

## Peukert’s Equation

The actual capacity of a battery can vary significantly from the rated capacity due to the non-linear effects resulting from past charges and discharges, aging of the battery, charging or discharging history of the battery, ambient temperature vari­ations, etc. Thus, it is customary to define a nominal capacity. The nominal capacity C of the cell is related to the discharge current and the total time of discharge by Peukert’s equation (7.3.1). Peukert’s equation is given as,

So, given the nominal capacity at some discharge rate and the time of discharge in hours H,

the corresponding capacity is,

The discharge time at a given discharge current Id is then given by,

(73Л1)

It follows that the capacity for the given discharge current Id is given by,

Qid = hTs = qQQH)7 . (7.3.12)

The expo...

## Modelling a Typical Exciter

The exciter is usually a synchronous machine without damper windings and that makes its modelling relatively easier. The exciter input-amplifier’s output voltage, va, is modelled in terms of the amplifier gain ka and amplifier time constant sa as

sadva/dt + Va = ka(vt-ref – vi – V3), (3.5.9)

where the regulator output feedback v1 and the generator stabilizing feedback v3, satisfy,

sr dv1/dt + v1 = krvt, (3.5.10)

Sf dv3/dt + v3 = kfdvfd/dt. (3.5.11)

The exciter output voltage vfd satisfies

Sedvfd/dt + kevfd = va. (3.5.12)

Since the exciter feedback is via the terminal voltage vt and the reference the terminal voltage vt-ref, the terminal voltage dynamics is also essential. In partic­ular, one needs to include the voltage drop across the terminal...

## Wind Field Velocity Distributions and Spectrum

The general wind field distribution over a land mass is discussed by Burton et al. (2001). Winds are large-scale movements of air masses in the atmosphere. These movements of air are created on a global scale primarily by differential solar heating of the Earth’s atmosphere. The wind speed at a given location is contin­uously varying. Turbulence refers to wind speed fluctuations over short time scales. Wind speed can be described as the sum of a mean wind speed, U0, and a fluctuating component, Duf. The standard deviation of the turbulent component ru depends on the turbulence intensity IU and the average wind speed and is given by the product, ru = IU x U0. The turbulence intensity model is due to Rosas (2003) where IU is modelled as,

Iu = I *((a + 15/U>)/(a + 1)), (4.12.1)

where I* = ...

## Modelling and Simulation of Fuel Cells

6.1 Fuel Cell Systems

As already discussed, the fuel cell (FC) is an electrochemical energy conversion system in Chap. 1, where chemical energy is converted directly into electrical energy and heat. There are a number of advantages of fuel cell technologies which include higher efficiencies at partial loads, lower or no emissions, noiselessness operation due to the nonexistence of mechanical parts, controllability of electric and heat generation, and overall energy savings. The energy savings result from the higher conversion efficiency, typically 40 % or higher, depending on the type of fuel cell. When utilized in a cogeneration applications for recovering all the available thermal energy output, overall energy utilization efficiencies can be in the order of 85 % or more.

Physically, a FC...