# Abstract

It is not feasible to store meaningful amounts of energy using underwater vacuum containers.

# Introduction

On March 3, 2017, a team of German researchers from the Fraunhofer Institute successfully concluded a 4-week test of an underwater pumped-storage hydroelectric plant at Lake Constance.

The news were reported on at ArsTechnica and discussed on Hacker News.

The basic idea is to have a hollow sphere at great depth in a lake or ocean. At the inlet of the sphere there is a turbine and a motor/generator. If energy needs to be stored, the turbine acts as a pump, emptying the sphere of water and creating a vacuum. If energy needs to be recovered, the flow is reversed and the turbine drives the generator until the sphere is completely full of water.

In the future, the research group plans to build a full-sized plant in an ocean, 700m below the surface. The full-sized sphere is projected to have an inner diameter of 30m, a capacity of 20MWh and a power rating of 5MW.

Curiously absent from both the official press release and the ArsTechnica article is the storage capacity of the prototype, so let’s investigate.

# Pumped Storage Hydroelectric Plants

A pumped storage hydroelectric plant stores energy in the form of gravitational potential energy. The amount of energy stored is E = mgh where E is the energy, m the mass, g the acceleration of gravity, and h the height.

To get an intuition of the energy density of gravity batteries, let’s consider a 1m x 1m x 1m cube of water at a height of 10m. We know that g = 9.81ms⁻² and water has a density of 1000kg/m³. The cube thus has a mass of m = 1000kg. Now we can calculate the potential energy:

E = 1e3kg x 9.81ms⁻² x 10m ~ 1e5J = 0.1MJ

This is approximately 0.028kWh, which is the amount of energy stored in 0.002l (a tenth of a shot glass) of diesel or two 18650 Li-Ion cells.

To store meaningful amounts of potential energy, m or h must be large. Usually this is done by using a lake as storage (large m) and long pressure lines leading down to the turbines (large h).

It is not particularly easy to find the energy capacity of the largest pumped storage plants in the world. On this wikipedia article, only one of the plants has got an estimate of the amount of energy that can be stored: the Pumpspeicherwerk Goldisthal at roughly 8.5GWh.

# The Hollow Sphere Underwater Storage is a Potential Energy Storage

At first it may be unclear why the vacuum sphere used by the researchers at Fraunhofer is a potential energy storage.

The following schematic should clarify the situation. Imagine a cube with a side length s at a depth of 10s. The water column pushing down on the cube has a volume 9s³. If the cube is full of water, the upper level is at 10s. If water is pumped out of the cube, the water level in the column rises to 11s. There are two ways to interpret this:

1. the cube at the very bottom was lifted up 10s.
2. the 10 cubes above the ground were lifted up 1s.

The two ways are equivalent in terms of energy. (10m x g x h = m x g x 10h)

Of course, the surface area of Lake Constance (563km²) is large compared to the footprint of the sphere, so the level of the lake will not rise measurably when water is pumped out of the sphere.

This leaves us with the following model, where the upper water level is not influenced by the state of the cube: # So, How Much Energy Can be Stored in the Prototype?

The prototype has a radius of r = 1.5m and is located at a depth of h=100m.

The volume of the sphere is V = 4/3πr³ = 14.14m³ and can contain water with a mass of 14.14e3kg.

Thus, the energy is E = m x g x h = 14.14e3kg x 9.81m/s² x 100m = 13.87MJ = 3.8kWh

Remember, the project took years to build and uses a 20 tons of concrete. As an alternative, I can buy 4.8kWh worth of lead acid deep-cycle batteries on Amazon for 340€.

# But the Capacity Scales With the Cube of the Radius!

Yes it does! Also, it scales linearly with the depth of the sphere.

The researchers propose a 10x larger diameter and a 7x greater depth. This leaves us with 26.6MWh (7000 times the energy of the prototype), which is in line with what the researchers propose one full size model can store.

If we can trust wikipedia articles on the subject of top secret depth ratings of US nuclear attack submarines, it is possible to build a steel vessel with a collapse depth of 730m.

I’m no structural expert and have little insight in static analysis of underwater concrete structures, but given that it took from the 70s to 2014 to scale wind turbines from 22kW to 8MW I find it hard to believe that we will be able to store meaningful amounts of electrical energy using underwater energy storage spheres in the foreseeable future.

On a side-note, it would cost 1.4 million € to store 20MWh with the above-mentioned deep cycle batteries (batteries only), so 70k€ to store 1MWh.

Note that the Goldisthal Pumpspeicherwerk cost 600 million € to build, that is 75 million € per GWh (75 k€ per MWh, basically on par with lead acid batteries).

# Further Concerns

Apart from the technical difficulties of building concrete structures that can withstand the extreme pressures, we haven’t touched the issues that the turbines need to operate unattended in the corrosive sea water, or that the efficiency of pumped energy storage is around 70%-80%.

# Final Words

I don’t mean to belittle the engineering effort that went into this project by any means.

However, I remain critical of how this means of storage, which has been hyped by the media and positively received by the Hacker News crowd, can be even marginally important in the future energy storage landscape.