Summary:375 MW of electricity is the amount currently provided by fossil fuel burning power plants for a population of about 320,000. To make this power with renewable sources (namely, solar PV) 2 GW must be generated: 375 MW for daytime use and 1.625 GW generated and stored for nighttime use. One option for storage is pumped hydroelectric power. This amount of power can be produced hydroelectrically by draining a reservoir (such as Horsetooth Reservoir near Fort Collins, Colorado) of 120 million m^{3}of water at night and pumping it back during the daytime. This would involve creating a lake downstream from the reservoir 6 m (20 ft) deep and 6 km (3-4 mi) on each side. Alternatively, 377 pairs of large tanks (500 ft in diameter 100 ft tall) slightly larger than the oil storage tanks located in Cushing, Oklahoma, one above ground and one below ground, could be filled and drained each day. Each tank takes up about 5 acres or just a bit more than 2 city blocks.

In the first *Back of the Envelope* we found that we need a 2 GW solar farm to replace the fossil fuel produced electricity currently used by a city such as Fort Collins, Loveland, Longmont, Estes Park, Colorado with a population of 320,000. 375 MW of that 2 GW will be used during the day and the rest, 1.625 GW must be stored to be used at night. The only form of energy storage reported by the Energy Information Administration is pumped hydroelectric power. “Pumped hydro” is the only large-scale energy storage in use today–99% of all storage is pumped hydro. Water is pumped into a reservoir using excess energy production and then released later to generate electricity as in a hydroelectric power plant. Some pumped hydro facilities were developed in concert with nuclear power facilities. When power production exceeded demand, the excess was used to pump water into a reservoir rather than throttle the amount of electricity produced by the nuclear power plant. This allows the nuclear power plant to operate an optimal level. This *Back of the Envelope* post will explore what would be required to provide overnight storage of daytime generated electricity using pumped hydro.

The important physics/engineering equation is that which calculates the power available in a reservoir of water. The equation is

Power (Watts) =

Efficiency x density of water (kg/m^{3}) x flow rate (m^{3}/s) x force of gravity (m/s^{2}) x height (m)

Efficiency is the efficiency of conversion of hydropower to electrical power. We’ll assume 80%. The density of water is 1000 kg/m^{3}. The force of gravity is 9.81 m/s^{2}; we’ll call it 10 m/s^{2}. The height will depend on our exact system. We’ll explore a couple of options.

**Option 1: Horsetooth Reservoir**

Horsetooh Reservoir is less than a mile away from the western edge of Fort Collins. There are four earthen dams that make up the reservoir. The reservoir holds 0.2 km^{3} (200 million m^{3}, 157,000 acre-ft) of water. At the north end is a dam that is 155 feet high. How much electricity could we generate if we let the water out? In order to pump it back we have to store the water downstream somewhere. We’ll talk about that later. Let’s estimate the water level behind the dam to be 50 meters higher than in front of the dam. However, as we’ll see we will be changing that height significantly as the water is released even in a large reservoir like Horsetooth. (Technically, we should integrate as the water is released, but as a first approximation we’ll do the calculation using a static height of 1/2 of the initial height.) Let’s say that our actual head height is 25 meters. Now we can plug numbers into our equation. We need 375 MW of power.

375 x 10^{6} W = 0.8 x 1000 kg/m^{3} x flow rate (m^{3}/s) x 10 m/s^{2} x 25 m

Solving for flow rate gives us 1875 m^{3}/s. We need to do this for 18 hours. 18 hours is 64,800 seconds.

Total volume needed is 1875 m^{3}/s x 64,800 s = 121 million m^{3}.

This is 60% of Horsetooth Reservoir drained out every night and pumped back in during the day. A flow rate of 1875 m^{3}/s is comparable to the flow rate out of other large scale hydroelectric plants. Three 6-7 m (20 feet) diameter pipes could do the trick. Pumping the released water back during the daytime needs to occur 3x faster. The giant pumps that are being installed in New Orleans since Katrina can pump at near 625 m^{3}/s. We would need nine of these monsters each costing $0.5 billion to pump all the water back each day.

Another problem is where to put the water. Without worrying for now about land use issues, let’s make a shallow lake downstream from the main dam. If it is 6 meters (20 ft) deep, we need a 33 million m^{2} area (33 km^{2}, 82,000 acres). That’s a shallow lake 6 km or 3-4 miles on a side. Interestingly, that’s the size of the solar farm needed. Perhaps we can float the solar panels on the lake.

**Option 2: Storage Tank Farm**

The previous solution requires a specific geographical setting that may only be appropriate for certain locations, and since Horsetooth Reservoir is also a recreational facility, we may not want to drain it every night. Is there a way to make this work anywhere with artificial storage tanks? The oil storage facility in Cushing, Oklahoma comes to mind. What if we built large water storage tanks, one above ground and one below ground? What would it take to store enough power to get through the night? The largest of these tanks run about 400 feet in diameter and 70 feet tall. Let’s say we can stretch the limits a bit and get to 500 feet in diameter and 100 feet tall. That would be 150 meters in diameter and 30 meters tall. Each tank holds 530,000 m^{3} of water. The average height between the levels of water is now only 15 meters (rather than the 25 meters we had for Horsetooth Reservoir). Since we have only 60% of the head, we need 1.67x as much water. Using the number from the previous calculation (121 million m^{3}) this gives us 200 million m^{3} of water.

At 530,000 m^{3} per pair of tanks, that means 377 pairs of tanks. Each tank takes up a 150 m x 150 m square (about 5 acres or 4 football fields or just over 2 city blocks). This ends up being a 10 km x 10 km (6 mi x 6 mi) water storage tank farm. That area is comparable to the area of Horsetooth Reservoir and the downstream lake that was created.

**Feasible?**

Either project is grandiose. And keep in mind that we have to do this times 1000 for the US alone. However, such projects are not necessarily engineering impossibilities. Geography might help in many instances. Coastal regions or those near the Great Lakes might be able to use the ocean or lake as the lower level storage tank. The bottom line is that this would be a massive enterprise. On the surface it looks doable, but the scale of the project makes it seem like other options might be better. These will be discussed in future BOTE blog posts.

Check out *Energy: What the World Needs Now* by Terry M. Gray and Anthony K. Rappé.

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