1. 1 © A. Kwasinski, 2016 Power vs. energy delivery profile technologies Ragone chart: More information and charts can be found in Holm et. al., “A Comparison of Energy Storage Technologies as Energy Buffer in Renewable Energy Sources with respect to Power Capability.”

2. 2 © A. Kwasinski, 2016 Lead-acid batteries calculations Most calculations are based on some specific rate of discharge and then a linear discharge is assumed. The linear assumption is usually not true. The nonlinearity is more evident for faster discharge rates. For example, in the battery below it takes about 2 hours to discharge the battery at 44 A but it takes 4 hours to discharge the battery at 26 A. Of course, 26x2 is not 44. A better solution is to consider the manufacturer discharge curves and only use a linear approximation to interpolate the appropriate discharge curve. In the example below, the battery can deliver 10 A continuously for about 12 hours. Since during the discharge the voltage is around 12 V, the power is 120 W and the energy is about 14.5 kWh Discharge limit Nominal curve 10 A continuous discharge curve approximation

3. 3 © A. Kwasinski, 2016 Lead-acid batteries

4. 4 © A. Kwasinski, 2016 Lead-acid batteries efficiency Consider that during the charge you apply a constant current I C, a voltage V C during a time ΔT C. In this way the battery goes from a known state of charge to be fully charged. Then the energy transferred to the battery during this process is: E in = I C V C ΔT C Now the battery is discharged with a constant current I D, a voltage V D during a time ΔT D. The final state of charge coincides with the original state of charge. Then the energy delivered by the battery during this process is: E out = I D V D ΔT D So the energy efficiency is Hence, the energy efficiency equals the product of the voltage efficiency and the Coulomb efficiency. Since lead acid batteries are usually charged at the float voltage of about 2.25 V/cell and the discharge voltage is about 2 V/cell, the voltage efficiency is about 0.88. In average the coulomb efficiency is about 0.92. Hence, the energy efficiency is around 0.80

5. 5 © A. Kwasinski, 2016 Wind power Conversion efficiency Why is it that output power of real wind turbines do not follow a cubic relationship? Because not all the wind power is transmitted through the blades into the generator. Consider the next figure: vbvb vuvu vdvd Downwind Upwind Rotor area A

6. 6 © A. Kwasinski, 2016 Wind power conversion efficiency The wind energy “absorbed” by the wind turbine rotor equals the kinetic energy lost by the wind as it pass through the blades. Hence, the energy transmitted by the wind to the rotor blades is the difference between the upwind and the downwind kinetic energies: In the last equation it is assumed that there is no turbulence and the air passes through the rotor as a steady rate. If it is assumed that v b is the average between v u and v d, then the mass flow rate is If we define the ratio

7. 7 © A. Kwasinski, 2016 Wind power conversion efficiency Then The rotor efficiency is maximum when λ is 1/3. For this value, C p is 0.593. Still, we still need to know how much of the “absorbed” power by the blades is transmitted to the generator. This conversion stage is characterized based on the tip-speed ration (TSR): Power in the wind Fraction extracted Rotor efficiency C p

8. 8 © A. Kwasinski, 2016 Wind power conversion efficiency

9. 9 © A. Kwasinski, 2016 Concentrated solar

10. 10 © A. Kwasinski, 2016 Concentrated solar