A simple, effective electric load leveling technique is to use excess produced electricity to pump water into a hilltop reservoir during off peak hours and then discharge it during peak electrical...

A simple, effective electric load leveling technique is to use excess produced electricity to pump water into a hilltop reservoir during off peak hours and then discharge it during peak electrical demand periods. This simple, old style energy storage concept is increasingly being used to have more effective power generation system by addressing the peak power demand. The drawback to this technique is one has to operate it in a hilly or mountainous region with a water supply. Someone has suggested an alternative that can be applied at any location. In this alternative, a reclaimed concrete mass is to be raised using excess generated electrical power. You can assume density of the reclaimed concrete is the same as concrete. This mass will be raised using a pulley-motor combination when there is excess electrical power production. During the peak power demands the mass will be lowered using a pulley-electric generator to produce power. The effect of the friction in the pulley is a tangential force that operates at the outer pulley circumference that interfaces with the cable. This force, Ffric, is proportional to the mass being lifted by the following relationship: Ffric= 0.005 (lifted weight) Note: the force is given as a function of weight (a force in N) not the mass (in kg). There is a similar loss in the cable-winding drum, but the relationship is Fpulley= 0.004 (lifted weight). These frictional losses are ultimately converted to heat into the environment. The mass will be raised or lowered into a pit dug into the earth’s surface. The depth of this pit is 100 m and the mass cannot extend above the earth’s surface. Assume the device isothermal during these processes.

1. Determine the footprint (the land area needed) of the pit required to store an excess uniform power production of 800 kW over a 12-hour period if the electric motor has an efficiency of 90%. The stored energy is used to meet a peak power demand over a 4-hour period. The power delivered is not uniform over this 4-hour period and follows a function such that power generated = 400 (kW) +bsin (t/a), wheret= time measured in hours, a=(4 hour)/
   
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   is constant, andbis another constant measured in kW that you should figure out according to given information. The generator efficiency is 90%. NOTE: It is up to you to determine the geometric shape of the concrete mass. The choice of this variable will affect the lift height and the footprint. You must decide this shape. But do not try to optimize it (optimization will be a different part).
   


2. Determine the maximum power output from the storage system within the 4 hour usage period.


3. Determine the energy storage efficiency of this concept. The energy storage efficiency is defined as the ratio of the energy output of the system (energy delivered to be used outside of this system) to the energy input (energy used to charge to the system). Please note that both energy input (used to charge) and energy output (when discharging) of this storage system will be different that total energy storage capacity of the system due to inefficiencies in charging and discharging. Discuss the feasibility of this concept. Your discussion should include technically and qualitative economic points. For example, how does it compare to the efficiency of a battery storage system or the hydro power storage systems.


4. Is there a best geometric shape that maximizes efficiency or reduce the real estate costs? Hint: Solve for the storage efficiency in terms of algebraic parameters; not numbers. You can use numerical or iterative solutions, as well as using an analytical solution. All techniques are acceptable. For example, If the geometric shape is a rectangular shaped solid in which the height is L and the sides are multiples of the height, such N*L. You can determine if there is a value of N which would maximize the energy storage efficiency. Discuss your result.


5. Is this device competitive with battery storage? Consider the battery packs you used in homework 3 to determine how many battery packs would be required to store this energy. How does it compare to the size and weight of the proposed system?


6. Which factors, the friction coefficients, motor efficiency or generator efficiency should be modified to make this system competitive with battery energy storage systems from an efficiency viewpoint. Do not include costs in this question.


7. Is the energy output from this device dependent on the geometry of the concrete slab? To answer this question, it may be helpful to plot the energy output from the device as a function of mass of the concrete. You can also use the variable “N” described in part D in place of mass, if you used that approach in part D.


8. Estimate the maximum allowable initial investment to purchase land, machinery, and construction costs if the purchase cost of the electricity into this system is $0.08/kWh during the 12-hour period and if the sell back price over the 4-hour period is $0.20/kWh. Assume the lifetime of the system is 15 years and it would be used 360 days per year.