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%\section*
\title{Enhancing Electrical Supply by
Pumped Storage in Tidal Lagoons}
\author{ {\large David J.C. MacKay} \\
Cavendish Laboratory,
% Department of Physics\\
University of Cambridge\\
{\tt{mackay@mrao.cam.ac.uk}}}
\date{\today\ -- Draft 1.8 -- first published 5/3/07}
\maketitle
\section{Summary}
%\begin{abstract}
The principle that the net energy delivered by
a tidal pool can be increased by
pumping extra water into the pool
at high tide or by pumping extra water out of the
pool at low tide is well known in the
industry. On paper, pumping can potentially
enhance the net power delivered by a factor of
about four.
However, pumping seems generally to be viewed as
a minor optional extra, delivering only a modest
power enhancement.
Two possible reasons why pumping is not emphasized in
tidal designs are that increasing the vertical water range
introduces additional costs (for example, higher walls),
and that alternating between pumping and generating
worsens the intermittency-of-supply problem from which simple
tide pools suffer.
The intermittency-of-supply problem also causes problems
for wind. How can we switch to wind power
if the wind might stop blowing for two days at a time?
% Energy storage systems are expensive.
Chemical or kinetic-energy
storage systems are an economical way to
smooth out the fluctuations of wind power on a time-scale
of minutes, but what about hours and days?
Perhaps a shift of perspective on tidal lagoons is helpful.
I sketch designs for a large pumped-storage system
located at sea-level with a dual purpose: first, it
can turn
% intermittent power, or
power that is poorly matched
to demand into high-value demand-following power;
and second, it can simultaneously serve as a tidal
power station.
Large designs with a capacity
of several gigawatts are the most economical.
%\end{abstract}
\section{Storage and wind}
Offshore wind farms deliver, on average, about 3\,W per m$^2$ of
sea-floor area (or 3\,MW/km$^2$, if you prefer).
\begin{figure}[tbp]
\figuremargin{
\begin{center}
\mbox{\epsfxsize=\textwidth\epsfbox{../data/cambridge/Cam2006.eps}} \\[0.1in]
\mbox{\epsfxsize=\textwidth\epsfbox{../data/cambridge/Cam2006h.eps}}
\end{center}
}{
\caption[a]{Cambridge mean wind speed in metres per second, daily (heavy line),
and half-hourly (light line) during 2006. The lower figure shows detail from the upper.
Thanks to Digital Technology Group, Computer laboratory, Cambridge
% \protect\tinyurl{vxhhj}{http://www.cl.cam.ac.uk/research/dtg/weather/}.
This weather station is on the
roof of the Gates building, roughly 10\,m high.
Wind speeds at a height of 50\,m are usually about 25\% bigger.
}
\label{fig.camb.wind}
}
\end{figure}
\begin{figure}[tbp]
\figuremargin{
\begin{center}
\mbox{\epsfxsize=\textwidth\epsfbox{../data/cairngorm/CairngormWind2006.eps}}
\end{center}
}{
\caption[a]{Cairngorm mean wind speed in metres per second, daily (heavy line),
and half-hourly (light line), during six months of 2006.
Thanks to Heriot--Watt University Physics Department.
% \tinyurl{tdvml}{http://www.phy.hw.ac.uk/resrev/aws/awsarc.htm}.
}
\label{fig.cairngorm}
}
\end{figure}
Imagine that Britain had `30\,GW' of wind farms -- fifteen times as
much as today. I put quotes round `30\,GW' to emphasize that the
nominal capacity of wind farms is much bigger than the average
power delivered. The standard `capacity factor' in the UK wind industry
seems to be $1/3$, so 30\,GW of wind farms would be expected
to deliver, on average, 10\,GW.
Winds fluctuate (figures \ref{fig.camb.wind}, \ref{fig.cairngorm}).
So this average of 10\,GW would be delivered burstily: 30\,GW one hour,
and 0\,GW the next, on one day; and perhaps 0\,GW all day on the following day.
How can such bursty power be made useful to society?
The default approach is to build back-up stations
using some other sort of power -- most likely fossil fuel --
which sit idle when the wind blows, and are switched on
when it does not, or when demand peaks.
Another approach would be to manage demand --
using smart electric-car chargers, for example, which
use electricity when it is cheap; or running the Aluminium plant
and the water-purification factory only when the wind blows.
A third approach is storage.
The storage required to deliver 10\,GW for 24 hours is
240\,GWh -- twenty-six times as big as the 9\,GWh of Dinorwig.
\begin{figure}
\figuremargin{
\begin{center}
\begin{tabular}{cc}
\mbox{\epsfxsize=55mm\epsfbox{../../images/OS/Dinorwig10km.eps}} &
\mbox{\epsfxsize=55mm\epsfbox{../../images/OS/DinorwigMap.eps}} \\
\end{tabular}
\end{center}
}{
\caption[a]{Dinorwig, in the Snowdonia National Park.
%% 321E 098N 2 miles between lakes
The left map is a 10\,km by 10\,km area.
In the right map the blue grid is made of 1\,km\
squares.
{Dinorwig} is the home of a 9\,GWh storage system, using
Marchlyn Mawr (615E,\,620N) and Llyn Peris (590E,\,598N)
as its upper and lower reservoirs.
Images produced from Ordnance Survey's Get-a-map service
\url{www.ordnancesurvey.co.uk/getamap}.
Images reproduced with permission of Ordnance Survey.
\copyright\ Crown Copyright 2006
}
\label{fig.dinorwig}
}
\end{figure}
The Scottish island of Fair Isle (population 70, area 5.6\,km$^2$)
has pioneered several of these technologies.
To solve the demand-management problem,
Fairisle
has
for over 25 years
%% (since 1982),
had {\em{two}\/} electricity networks that distribute
power from two wind turbines
%% nominal 60kW and 100kW
and, if necessary,
a diesel electric generator.
% http://www.fairisle.org.uk/FIECo/renewed/fig_1_2.htm
Standard electricity service is provided on
one network, and electric heating is delivered by a
second set of cables. The electric heating is mainly
served by excess electricity from the turbines that would otherwise
have had to be dumped.
Remote frequency-sensitive programmable relays
control individual water heaters and
storage heaters in the individual buildings of the community.
In fact there's up to six frequency channels per household,
so the system behaves like seven networks.
Fair Isle also successfully trialled a kinetic energy
storage system (a flywheel) to store energy during
oscillations of wind strength (with a period of 12 to 20 seconds).
More recently, the combination of wind with pumped storage
has been proposed for the Canary Islands \citep{BuenoaCartab}.
\section{Designs for multi-purpose storage/tidal systems}
Here are the key ideas for an energy-enhancing pumped-storage system:
\begin{enumerate}
\item
It is said that connecting large numbers of
wind turbines to the national electricity grid could lead to
instabilities.
We thus propose decoupling wind turbines from the
grid, {\em{plugging them directly into pumped storage systems instead}}.
The wind-to-pump connection could be a flexible grid with much wider tolerances
than the national network.
\item
The pumped storage system is located in a region with large tides.
Water is pumped to and from the sea in such a way that (a)
the power delivered can respond to the grid's demand, eliminating
problems of intermittency; and (b) we get more power out than we put in.
(Yes, I mean that the energy delivered when generating {\em{exceeds}\/}
the energy received -- in contrast to Dinorwig, which has a round-trip
efficiency of about 75\%.)
\item
When the demand for pumped storage is low (during a few calm days,
say), the facility can also function as a stand-alone tidal power station.
By using multiple lagoons, it's possible to
turn the intrinsically intermittent
tidal power into always-on, demand-following capacity.
\item
The facility could also buy electricity from the national grid
for pumped storage, just like Dinorwig.
\end{enumerate}
In sum, it's a storage system that
is more than 100\% efficient.
It's a storage system that
can also produce its own power
when it's not needed for storage.
Or, it's a tidal facility that still provides a
valuable function even when the tides are small.
\section{Rough models}
Let's assume a tidal range of $2h = 4$\,m throughout.
I'll also assume that hydroelectric generators
have an efficiency of 90\% and that pumps have an efficiency
of 85\%. (These
figures are based on the pumped storage system at Dinorwig, whose round-trip
efficiency is about 75\%. I think that the best
figures for low-head tidal turbines might in fact be 90\% for
generators and 80\% for pumps.
In their paper based on La Rance, \citet{ShawWatsonRance}
assume pumping efficiencies up to 66\%, with best efficiency at large head,
and generating efficiency 80\%.)
Let's start by finding some benchmarks for energy production.
\begin{figure}
\figuremargin{
\begin{center}
\begin{tabular}{c}
{\mbox{\epsfysize=2in\epsfbox{../tide/tidepoolR.eps}}} \\
\end{tabular}
\end{center}
}{
\caption[a]{An artificial tide pool.
The pool was filled at high tide, and now it's low tide.
We let the water out through the electricity generator
to turn the water's potential energy into electricity.
% the change in potential energy of the water
% is $mgh$, where $h$ is the change in height of the centre of
% mass of the water, which is half the range.
}
\label{tidepool}
}
\end{figure}
\subsection{Production on ebb and flow (no pumping, no demand-following)}
{\sc The power of an artificial tide pool.}
To estimate the power of an artificial tide pool, imagine that it's filled
rapidly at high tide, and emptied rapidly at low tide.
Power is generated in both directions.
The change in potential energy of the water, each six hours, is
$mgh$, where $h$ is the change in height of the centre of
mass of the water, which is half the range
(\figref{tidepool}).\label{pagetidepool}
The mass per unit land-area covered by tide-pool
is $\rho \times (2 h)$, where $\rho$ is the density
of water ($1000\,\kg/\m^3$).
So the power per unit area delivered by a tide pool is
\[
\frac{ 2 \rho h g h}{ \mbox{6\,hours} }
\]
Plugging in $h = 2\,\m$, we find%
\[
\mbox{Power per unit area of tide-pool}
% \frac{ 2 \rho h g h}{ \mbox{6\,hours} }
% = 2000 kg/m^3 * 4*9.81 m^2 m/s/s / (6*3600)s
% = 3 W/m^2
= 3.6\,\Wmm.
\]
Allowing for an efficiency of 90\% for conversion of this power to
electricity, we get
\[
\mbox{Power per unit area of tide-pool}
% \frac{ 2 \rho h g h}{ \mbox{6\,hours} }
% = 0.9* 2000 kg/m^3 * 4*9.81 m^2 m/s/s / (6*3600)s
% = 3 W/m^2
= 3.3\,\Wemm.
\]
(Or 3.3\,MW/km$^2$.)
\subsection{Tidal pools with pumping}
The pumping trick artificially increases the amplitude of the tides
in the tidal pool so as the amplify the power obtained.
The energy cost of pumping in extra water at high tide is
repaid with interest when the same water is let out at low tide;
similarly, extra water can be pumped out at low tide, then let back in at high
tide.
Let's work out the theoretical limit for this technology.
% see also storage.tex where same calcns are done
I'll assume that generation has an efficiency of $\epsilon_{\rm{g}} = 0.9$
and that pumping has an efficiency of $\epsilon_{\rm{p}} = 0.85$.
% (these
% figures are based on the pumped storage system at Dinorwig, whose round-trip
% efficiency is about 75\%).
Let the tidal range be $2h$.
I'll assume that the prices of buying and selling electricity
are the same at high tide and low tide,
so that the optimal height boost $b$ to which the pool is pumped above
high water is
given by (marginal cost of more pumping = marginal return of water):
\[
b / \epsilon_{\rm{p}} = \epsilon_{\rm{g}} ( b + 2h )
\]
Defining the round-trip efficiency
$\epsilon = \epsilon_{\rm{g}}\epsilon_{\rm{p}}$, we have
\[
b = 2h \frac{\epsilon }{1- \epsilon }
\]
For example, with a tidal range of $2h=4\,\m$, and a round-trip efficiency
of $\epsilon=76\%$, the optimal boost is $b=13\,\m$.
Let's assume the complementary trick is used at low tide. (This requires that
the basin have a vertical range of 30\,m!)
The delivered power per unit area is then
\[
\left. \left( \frac{1}{2} \rho g \epsilon_{\rm{g}} ( b + 2h )^2
- \frac{1}{2} \rho g \frac{1}{\epsilon_{\rm{p}}} b^2 \right) \right/ T ,
\]
where $T$ is the time from high tide to low tide.
We can express this as the power without pumping,
scaled up by a boost factor
\[
% \left( \frac{1}{1- \epsilon} \right)^2
%- \frac{1}{\epsilon} \left( \frac{\epsilon}{1- \epsilon} \right)^2
\left( \frac{1}{1- \epsilon} \right),
\]
which is a factor of about 4.\medskip
%% see figs/tide.gnu
\begin{center}
\begin{tabular}{cccc}\toprule
Tidal amplitude & Optimal boost height & Power & Power \\
$h$ & $b$ & with pumping & without pumping \\
(m) & (m) & (\Wmm) & (\Wmm) \\
\midrule
0.5 & 3.3 & 0.9 & 0.2\\
1.0 & 6.5 & 3.5 & 0.8\\
{\bf{2.0}} & {\bf{13}} & {\bf{14}} & {\bf{3.3}}\\
3.0 & 20 & 31 & 7.4\\
4.0 & 26 & 56 & 13 \\
\bottomrule
\end{tabular}
\medskip
\end{center}
Unfortunately, this pumping trick will rarely be exploited
to the full because of the economics of basin construction:
full exploitation of pumping requires the total height of the pool
to be roughly 4 times the tidal range, and increases the delivered
power by a factor of 4. But the material in a sea-wall of height $H$ scales
as $H^2$, so presumably the cost of constructing
a wall four times as high will be
more than four times as great.
Extra cash would probably be better spent on
enlarging a tidal pool horizontally rather than vertically.
% The economically optimal amplitude of pumping will be thus be less than a
% factor of 4.
The pumping trick can nevertheless be used for free whenever the natural
tides are smaller than the maximum tidal range.
The next table gives the power delivered if the boost height
is set to $h$, that is, the range in the pool is just double the external range.
\begin{center}
\begin{tabular}{cccc}\toprule
Tidal amplitude & Boost height & Power & Power \\
$h$ & $b$ & with pumping & without pumping \\
(m) & (m) & (\Wmm) & (\Wmm) \\ \midrule
0.5 &0.5 & 0.4 &0.2\\
1.0 &1.0 & 1.6 &0.8\\
{\bf{2.0}} &{\bf{2.0}} & {\bf{6.3}} &{\bf{3.3}}\\
3.0 &3.0 & 14 &7.4\\
4.0 &4.0 & 25 &13\\
\bottomrule
\end{tabular}
\end{center}
A doubling of vertical range
is plausible at neap tides, since neap tides are typically
about half as high as spring tides.
Pumping the pool at neaps so that the full springs range
is used thus allows neap tides to deliver roughly twice as much
power as they would offer without pumping. So a system with pumping
would show two-weekly variations in power of just a factor
of 2 instead of 4.
These benchmarks -- {\bf{3.3}}\,\Wmm\ without pumping
and {\bf{6.3}}\,\Wmm\ with pumping -- assume that power is delivered
and demanded at exactly the optimal times, and that there is no limit to the
flow rate of water in the system.
Such a system is highly intermittent and spikey.
We now examine a more reasonable, smooth,
but still intermittent, benchmark.
\subsection{An intermittent solution that alternates between
steady pumping and steady generating}
Figure \ref{fig.bursty1} shows a pumping and generating schedule
where the system is always active; it spends exactly half the time pumping
(at constant power) and half generating (at constant power).
The system alternately sucks 7\,W/m$^2$
from the electricity grid (for three hours) and
delivers 20\,W/m$^2$ (for three hours).
The net energy contribution is thus {\bf{6.5}}\,W/m$^2$.
The range required is about
10\,m -- slightly more than
double the tidal range of 4\,m.
\begin{figure}
\figuremargin{
\begin{center}
\begin{tabular}{c}
{\mbox{\epsfxsize=3.5in\epsfbox{../tide/pump/figs/one2.7.20.eps}}} \\
\end{tabular}
\end{center}
}{
\caption[a]{
A bursty tidal power option using one lagoon at sea-level.
The tidal range is $2h = 4\,$m.
The system alternately sucks 7\,W/m$^2$
from the electricity grid (for three hours) and
delivers 20\,W/m$^2$ (for three hours).
The net energy contribution is thus 6.5\,W/m$^2$.
The vertical range inside the lagoon is about 10\,m.
}
\label{fig.bursty1}
}
\end{figure}
\begin{figure}
% \figuremargin{
\begin{center}
\begin{tabular}{c}
{\mbox{\epsfxsize=3.5in\epsfbox{figs/TideStore2.eps}}} \\
\end{tabular}
\end{center}
% }{
\caption[a]{
Design assumed:
one, two, or three lagoons are located at sea-level.
While one lagoon is being pumped full or pumped
empty, the other lagoon may be delivering steady, demand-following
power to the grid.
Pumping may be powered by bursty sources such as wind, by spare power
from the grid (say, in the future, from nuclear
power stations), or by the facility itself,
using one lagoon's power to pump the other lagoon to a greater height.
}
% }
\end{figure}
\subsection{Multiple-lagoon solutions}
Using multiple pools -- for example, a high pool and a low pool --
doesn't increase the deliverable power, but does increase the
flexibility of when power can be delivered, thus enhancing the
value of a facility. A two-pool facility is `always on', and
would be able to provide the same sort of valuable
service as the Dinorwig station.
\subsubsection{A two-pool facility can do its own pumping.}
Thus a tidal station can turn the intermittent tidal
power into demand-following power.
As an extreme simple case, let's assume that demand is
absolutely steady. Not realistic, but a challenging
target for a tidal source to deliver!
Figure \ref{selfpump} shows a possible schedule for a two-lagoon
system. In contrast to the single-lagoon schedule,
where pumping periods and generating
periods alternate, each lasting 3 hours (with switches from pumping
to generating at high tide and low tide),
here generating happens all the time
and pumping lasts for three hours around each high
tide and three hours around each low tide.
One lagoon's water level is always above sea-level; the other's
is always below.
The switches from pumping to generating take place at mid-tide
In this figure,
the pumping into or out of one lagoon is entirely funded by the
energy in the other lagoon. No energy is required from the
grid.
After an initial set-up period of a couple of periods, the
system delivers a steady 4.5\,W/m$^2$.
The range is about 25\,m (about six times the
tidal range).
% This is larger than the optimal range;
% I haven't carefully optimized this .
\begin{figure}
\figuremargin{
\begin{center}
\begin{tabular}{c}
{\mbox{\epsfxsize=3.5in\epsfbox{../tide/pump/figs/two2.32.0.5.eps}}} \\
\end{tabular}
\end{center}
}{
\caption[a]{
A two-lagoon system with no power input from the
grid or wind.
The tidal range is $2h = 4\,$m.
Self-pumping takes place with a power of
32\,W/m$^2$.
After an initial set-up period of a couple of periods, the
system delivers a steady 4.5\,W/m$^2$.
Top graph: solid line -- power delivered to grid.
Second graph: self-pumping power.
}
\label{selfpump}
}
\end{figure}
\subsubsection{The same facility can simultaneously
be used for pumped storage.}
For simplicity and clarity, I again assume that the
demand is steady. I also assume in the computations
that the power being stored is steady, but
the system would work equally well if the incoming
power fluctuated around its average value on
a timescale of minutes or one or two hours.
\Figref{UsingWind2a} shows the result of using the
same schedule, doing self-pumping for three hours
around each high and low tide, plus pumping
5.5\,W/m$^2$ of `bursty' wind power into the
appropriate lagoon all the time.
The system is generating a steady 8.5\,W/m$^2$.
The range is roughly 20\,m (five times the tidal range).
\begin{figure}
\figuremargin{
\begin{center}
\begin{tabular}{c}
{\mbox{\epsfxsize=3.5in\epsfbox{../tide/pump/figs/two2.10.6.9.eps}}} \\
\end{tabular}
\end{center}
}{
\caption[a]{
A two-lagoon system receiving
5.5\,W/m$^2$ of bursty wind power and delivering
8.5\,W/m$^2$ of steady power.
The tidal range is $2h = 4\,$m.
Self-pumping takes place with a power of
10\,W/m$^2$.
Top graph: solid line -- power delivered to grid;
dashed line -- average power received from intermittent
source, e.g.\ wind.
Second graph: self-pumping power.
}
\label{UsingWind2a}
}
\end{figure}
\subsubsection{The facility could also be used for
pumped storage alone.}
Perhaps pumps and generators are a valuable resource
and none are available for the self-pumping
trick.
Figure \ref{nopumping} shows results for
two incoming power conditions.
In (a),
5.5\,W/m$^2$ of `bursty' wind power is
turned into
7.5\,W/m$^2$ of steady power; the
range between the high pool's maximum and the low pool's
minimum is 16\,m.
In (b) 18\,W/m$^2$ of `bursty' wind power
is turned into
19\,W/m$^2$ of steady power;
the range required is about 26\,m.
\begin{figure}
\figuremargin{
\begin{center}
\begin{tabular}{cc}
(a)&{\mbox{\epsfxsize=3.5in\epsfbox{../tide/pump/figs/two2.0.6.8.eps}}} \\
(b)&{\mbox{\epsfxsize=3.5in\epsfbox{../tide/pump/figs/two2.0.18.19.eps}}} \\
\end{tabular}
\end{center}
}{
\caption[a]{
In these simulations, no self-pumping takes place.
(a) A two-lagoon system receiving
5.5\,W/m$^2$ of bursty wind power and delivering
7.5\,W/m$^2$ of steady power.
(b) A two-lagoon system receiving
18\,W/m$^2$ of bursty wind power and delivering
19\,W/m$^2$ of steady power.
The tidal range is $2h = 4\,$m.
Top graph: solid line -- power delivered to grid;
dashed line -- average power received from intermittent
source, e.g.\ wind.
}
\label{nopumping}
}
\end{figure}
\subsection{Other designs, future work}
The next design I would like to explore uses three lagoons.
The two-lagoon solution (self-pumping) doesn't deliver
as much power per unit area, and required larger vertical
amplitudes, than the one-lagoon solution with externally-funded
pumping. I expect that there are various three- or four-lagoon
solutions in which one or two of the lagoons follow trajectories
like that of the one-lagoon solution, with most or all of the
required pumping funded internally.
\begin{figure}
\figuredangle{
\begin{center}
\begin{tabular}{c}
{\mbox{\epsfysize=2.182in\epsfbox{../../data/NationalGrid/JanJunGW.eps}}} \\
\end{tabular}
\end{center}
}{
\caption[a]{ Electricity demand
in Great Britain
% (in kWh/d per person)
(in GW)
during three winter
weeks and three summer weeks of 2006.
}
\label{fig.demand}
}
\end{figure}
An obvious piece of further work is to explore
the economics of realistic daily supply and demand inputs.
It's possible that the economically optimal pumping and generating
strategy might sometimes be to exploit just one high tide and
one low tide for pumping each night, and generate at appropriately
selected times in the day.
\begin{figure}
\figuremargin{
\begin{center}
\begin{tabular}{rc}
\raisebox{1cm}{12 January 2006} & {\mbox{\epsfysize=0.82in\epsfbox{../../data/NationalGrid/0601.eps}}} \\
\raisebox{1cm}{13 June 2006} & {\mbox{\epsfysize=0.82in\epsfbox{../../data/NationalGrid/0606.eps}}} \\
\raisebox{1cm}{24 February 2007} & {\mbox{\epsfysize=0.82in\epsfbox{../../data/NationalGrid/0702.eps}}} \\[0.05in]
& Time in hours \\
\end{tabular}
\end{center}
}{
\caption[a]{ Electricity prices
in Great Britain
(in \pounds\ per MWh)
on three days in 2006 and 2007.
}
\label{fig.moneyelec}%% fig.price
}
\end{figure}
To reduce costs associated with
high walls we could look for lop-sided schedules
where lagoons are pumped down to lower extremes and not pumped
up so high.
{\bf{Criticisms:}}
I've ignored the true dependence of generating and pumping efficiency on
head.
I've assumed the sea is an inexhaustible source or sink of water at
the current sea-level. Once the system reaches a sufficiently
large size, its sucking and blowing will have a significant
effect on local sea-level.
I've not taken account of the cost of turbines,
assuming that we can install whatever pumping and
generating capacity these schedules call for.
\begin{figure}
\figuremargin{
\begin{center}
%% To get high resolution version, switch to the second line
{\mbox{\epsfxsize=4.995in\epsfbox{../../refs/DTIAtlas/Z25.50e.pdf.jpg.eps}}}
%%{\mbox{\epsfxsize=5.3in\epsfbox{../../refs/DTIAtlas/Z25.50e.pdf.eps}}}
\end{center}
}{
\caption[a]{UK territorial waters with depth less than 25\,m (yellow)
and depth between 25\,m and 50\,m (magenta).
Data from DTI Atlas of Renewable Marine Resources.
\copyright\ Crown copyright.
}
\label{fig.Z25}
}
\end{figure}
\section{Discussion}
Some of these ranges are enormous.
Where could such a system be put?
What would it cost, and what would it be worth?
One simple observation is that the value
delivered scales as the area of the lagoons,
but the dominant part of the cost -- the walls --
scales as the circumference.
Very large systems are thus favoured
by simple economics.
Let's pick a benchmark size.
How about 10\,km $\times$ 10\,km?
\begin{itemize}
\item
A plain old intermittent tide-pool of this size
(3.3\,MW/km$^2$)
would deliver 330\,MW
on average (assuming, as usual, a 4\,m tidal range).
\item
With pumping to 5\,m above and below mean sea-level,
a single lagoon would deliver a net power of 650\,MW.
\item
The two-lagoon solution that does its own pumping
would deliver a {{\bf{steady}}} 450\,MW.
\item
It could also be used as a pumped
storage system for intermittent or unwanted
electricity.
For example, it could turn 550\,MW (average) of
bursty wind power into 750\,MW of steady power.
%% Conclusions}
(A round-trip efficiency of 135\%!)
Or it could turn 1.8\,GW of bursty wind power into
1.9\,GW of steady power.
A round-trip efficiency of 105\% compares favourably with
Dinorwig's 75\%.
\item
For comparison of pumped storage capacity with Dinorwig --
a profitable power station with a capacity to store 9\,GWh --
these tide pools would have a capacity of about 20\,GWh
%% 20.7
(assuming a height change of 13\,m).
% 0.5 * epsilon rho g h**2 A
% where h is the height above sea-level or above low water
% pr 0.5*0.9 * 1000 * 9.81 * 13**2 * 100000000 / ( 1e9 * 3600 )
So this facility would be worth two Dinorwigs.
Indeed, it would be worth more, since it would be better than 100\%
efficient, in contrast to Dinorwig's 75\%.
\end{itemize}
We need at least half of this water to have a depth of about 13\,m
below mean sea-level.
There's lots of shallow water around Britain.
We need the tidal range to be as large as possible, too.
An ideal location might be an area of shallow sea surrounding a small island,
where the pumping facilities could be built.
Alternatively the high pool could be
built on land.
Offshore lagoons have many advantages, as advocated by Tidal
Electric limited, and Friends of the Earth \cite{FOELagoonBarrage}.
I think the best two locations in the
British Isles are, on the East Coast, The Wash (where the
mean spring tidal range
exceeds 7\,m)
and, on the West Coast, anywhere in the Irish Sea
from the Mersey to the mouth of Morecambe Bay (where there
have
been proposals to build a 12-mile bridge
with built-in tidal and wind power).
The mean spring tidal range here is 7--8\,m.
Morecambe Bay already has a gas field, so there is
a precedent for energy exploitation.
From wikipedia:
`A lease has been granted for the development of two wind turbine sites in Morecambe Bay, one at Walney Island and the other at Cleveleys. Together these will have around 50 turbines.'
The Wash would be big enough to fit one 10\,km by 10\,km tidal facility,
but perhaps not more. The Irish sea is bigger. Both locations have a tidal range
bigger than I assumed, so the potential power is bigger -- perhaps about twice
as big, on average.
\begin{figure}
\figuremargin{
\begin{center}
%% To get high resolution version, switch to the second line
{\mbox{\epsfxsize=2.995in\epsfbox{../../refs/DTIAtlas/BathymetryDetail.eps}}}
\end{center}
}{
\caption[a]{Two locations with plentiful shallow water and big tides:
The Wash, and the Irish Sea (marked by two circles each).
UK territorial waters with depth less than 25\,m (yellow)
and depth between 25\,m and 50\,m (magenta).
Data from DTI Atlas of Renewable Marine Resources.
\copyright\ Crown copyright.
}
\label{fig.Z25b}
}
\end{figure}
What about the cost?
Two circular lagoons enclosing 100\,km$^2$ would require
50\,km of walls.
The low pool's walls would be in water of depth about 13\,m.
And (for the most ambitious schedules described here)
we need the high pool to have a wall of height 13\,m above sea-level.
Let's look at some costs from Tidal Electric limited.
Their plan for a small tidal lagoon in Swansea Bay (where
the mean tidal range varies between 4.1\,m and 8.5\,m)
involved 9\,km of walls
% (requiring million tonnes of aggregates).
and would cost either \pounds82\,million (according to
Tidal Electric supported by AEA Technology, and W.S.\ Atkins Engineering)
or \pounds234 million (according to critics of
Tidal Electric's scheme).\footnote{
Support from AEA Technology plc Report (\#ED03922/R1),
Author: T W Thorpe, is quoted here:
\url{http://www.tidalelectric.com/News\%20AEA.htm}.
The main criticism comes from
another AEA contractor, Clive Baker \citep{BakerSwansea},
in a report authored with Peter Leach of ESS consulting.
}
%% AEA http://www.tidalelectric.com/News%20AEA.htm
The cost of the wall was estimated to be \pounds49\,million or
\pounds114\,million respectively.
Taking the larger of these two figures,
the cost per km of wall is \pounds13\,million.
This wall was of height 16\,m from sea bed to crest.
The walls I was imagining above would be slightly higher or perhaps
twice as high (if the high pool is built in water of the same
depth as the low pool). The wall, using this technology,
would thus cost at least \pounds 0.65\,billion.
Perhaps costs could be reduced by alternative
wall construction methods. And I think the wall heights could be trimmed
quite a lot without spoiling the results sketched here.
Doubling the wall's cost to allow for all the other stuff,
I'll propose \pounds 1.3\,billion as the cost for a 10\,km $\times$ 10\,km
two-lagoon system.
Dinorwig cost \pounds 0.4 billion in 1980 money, so
\pounds 1.3\,billion for a facility superior to two Dinorwigs
sounds a reasonable deal to me.
Another way of expressing the
value of the facility is to take what people
currently spend on wind turbines --
for example, \pounds 500\,million on the `650\,MW' Lewis wind farm,
plus \pounds375 million on the Lewis--Mainland electricity connection:
an expenditure of about \pounds0.9\,billion on roughly 220\,MW
(average)
of intermittent power.
Scaling this up,
550\,MW of bursty wind power seems to be
valued at \pounds 2.25\,billion.
The pumped storage solution presented
in \figref{nopumping}(a), requiring walls of height 9\,m
above mean sealevel, would turn this
550\,MW of bursty wind power into 750\,MW of
steady demand-following power. It seems plausible to me
that this service would be worth the
estimated cost of \pounds 1.3\,billion.
If the cost needs to be reduced, we simply make the
system bigger. For example, we multiply the area by four
(to 20\,km $\times$ 20\,km)
and double the length of all the walls. The estimated cost roughly doubles
(to \pounds2.6\,billion, say),
but the storage quadruples to 40\,GWh (more than four Dinorwigs).
As a source of tidal power, this quadrupled station could
deliver a steady 1.8\,GW all day and all night,
and could serve peak demand.
\subsection{Cost comparison with vanadium flow batteries}
% More recently,
For comparison,
VRB power systems have provided a 12\,MWh energy storage system
for the Sorne Hill windfarm in Ireland (currently `32\,MW', increasing
to `39\,MW').
I think VRB stands for vanadium redox battery.
This storage system is a big `flow battery',
a vanadium-based redox regenerative fuel cell, with a couple of
tanks full of vanadium in different chemical states.
This storage system can smooth the output of its windfarm on a time-scale of
minutes, but the longest time for which it could deliver one third of the
`capacity' (during a lull in the wind) is one hour.
The same company installed a 1.1\,MWh system on Tasmania.
It can deliver 200\,kW for four hours, 300\,kW for 5 minutes and 400\,kW for
10 seconds.
A 1.5\,MWh vanadium system costing \$480\,000
occupies 70\,m$^2$ with a mass of 107\,tonnes.
% Energy density in Wh/litre:
% Lead acid 12--18;
% (70 in theory)
% VRB 16--33.
% Efficiency: lead acid 45\%, VRB
Its efficiency is 70--75\%, round-trip.
Scaling this up, and translating into British,
a 10\,GWh system using vanadium would cost
% \$3.2\,billion
% (
\pounds 1.64\,billion; a 20\,GWh system would cost
\pounds3.3\,billion.
%% 1.64 billion
The tidal-pumped-storage system thus looks competitive with
the storage technology currently used for
large windfarms.
[Scaling up the Vanadium technology to
10 or 20\,GWh might have a noticeable effect on the world Vanadium
market, but it is probably feasible.
Current worldwide production of
Vanadium is 40\,000 tonnes per year.
% http://www.indexmundi.com/en/commodities/minerals/vanadium/vanadium_t7.html
% atomic mass 51.
A 10\,GWh system, assuming 1-molar Vanadium solution,
would contain 36\,000 tonnes of Vanadium -- one year's worth
of current production.
% weigh 700\,000 tonnes, and thus contain 700 \times 10$^6$
% moles of Vanadium, weighing 36 thousand tonnes.]
%% 700e6 * 51g = 700
Vanadium is currently produced as a by-product of other processes,
and the total world Vanadium resource is estimated to be 63\,million tonnes.]
%% http://www.mii.org/Minerals/photovan.html
\subsection{Compressed-air storage}
Compressed air storage
is said to be significantly less expensive than other
large-scale storage options
\citep{citeulike:998687}.
``Energy is stored by compressing air in an airtight
underground storage cavern. To extract stored energy,
compressed air is drawn from the storage vessel, heated,
and then expanded through a high-pressure turbine, which
captures some of the energy in the compressed air. The air
is then mixed with fuel and combusted, with the exhaust
expanded through a low-pressure gas turbine. The turbines
are connected to an electrical generator. Turbine exhaust
heat and gas burners are used to preheat cavern air entering
the turbines.
CAES can be considered a hybrid generation/storage
system because it requires combustion in the gas turbine.
The storage benefit of precompressed air is the elimination
of the turbine input compressor stage, which uses approximately 60\% of the mechanical energy produced by a
standard combustion turbine. By utilization of precompressed air, CAES effectively "stores" the mechanical energy
that would be required to turn the input compressor and
uses nearly all of the turbine mechanical energy to drive the
electric generator.''
1\,kWh of electricity
generated by the CAES turbine requires 4649 kJ of fuel (1.3\,kWh)
plus
0.735\,kWh of compressor electricity.
This is said to be five times more efficient
than the most efficient plain fossil combustion
technology.
Further details including a life-cycle analysis
are in the paper \citep{citeulike:998687}.
I haven't found a figure for the cost of such a storage system.
\subsection{Additional opportunities}
A pair of lagoons in the sea with 13\,m-high walls
and electrical plumbing installed would be a
good place to locate wind turbines. The turbines would be offshore, which is
good,
but erection and maintenance of turbines on the walls
would be much easier and cheaper than
for regular offshore turbines.
100\,m diameter turbines (with `capacity' 3.5\,MW) could be placed
every 500\,m -- 100 turbines in total, with a `capacity' of 350\,MW.
A good combination: wind, pumped storage, and tidal energy, all
enhancing each other.
Perhaps to kill four birds with one stone,
we could sequester carbon too:
the walls could be built out of artificial limestone, or coal!
\nocite{ShawWatsonShort}
\nocite{ShawWatsonRance}
\nocite{ShawWatsonTrading}
\nocite{DTIAtlas}
\nocite{BakerSwansea}
\nocite{Hammons}
\nocite{Kowalik04}
\nocite{Salter2}
\nocite{BickleyRyrie}
\nocite{DTISevern}
\nocite{DinorwigDiscussion}
\nocite{Dinorwig8739}
\subsection*{Acknowledgements}
I thank Stephen Salter,
Denis Mollison, Adrian Wrigley, Tim Jervis,
Marcus Frean, and Trevor Whittaker for helpful discussions.
\bibliography{cftbibs}%/home/mackay/bibs}
\newpage
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\input{PowerGrid.tex}
\end{document}
Paragraphs 236-239 of the following select committee report discuss Tidal
Electric's barrage and objections:
http://www.publications.parliament.uk/pa/cm200506/cmselect/cmwelaf/876/87610.htm#a52
Their scheme has no pumping, renewable or generation matched to the grid -
it's just a plain tidal lagoon with turbines that are inactive under 4m of
head.
%% http://www.publications.parliament.uk/pa/cm200506/cmselect/cmwelaf/876/87610.htm#n435
\noindent
\rule{\textwidth}{1pt}
%\ENDfullpagewidth
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