basin. Optimum values for these dimensionless parameters are derived and comparison made ..... indicated by theory while Minas Basin designs use margin-.
Simple theory for designing tidal power schemes D. P R A N D L E Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead, Merseyside L43 7RA, UK
Basic parameters governing the design of tidal power schemes are identified and converted to dimensionless form by reference to (i) the mean tidal range and (ii) the surface area of the enclosed basin. Optimum values for these dimensionless parameters are derived and comparison made with actual engineering designs. A theoretical framework is thus established which can be used (i) to make a rudimentary design at any specific location or (ii) to compare and evaluate designs for various locations. Both one-way (flood or ebb) and two-way (flood and ebb) schemes are examined and, theoretically, the two-way scheme is shown to be more efficient. However, in practice, two-way schemes suffer disadvantages arising from (i) two-way flow through both turbines and sluices and (ii) lower average turbine heads. An important dimensional aspect of tidal power schemes is that, while energy extracted is proportional to the tidal amplitude squared, the requisite sluicing area is proportional to the square root of the tidal amplitude. In consequence, sites with large tidal amplitudes are best suited to tidal power development whereas for sites with low tidal amplitudes sluicing costs may be prohibitive. Key Words: tidal power, tidal energy, turbines, sluices, Bay of Fundy.
INTRODUCTION The possibilities of harnessing tidal power have attracted the attention of scientists and engineers for many years, as evidenced in the many references cited in the bibliographies (1973 and 1978) assembled by the Electricity Council of Britain. Perhaps part of the fascination derives from the multi-faceted nature of tidal power schemes. Thus conferences on this subject embrace a wide spectrum of related topics including economics, operation of electrical gridnetworks, environmental impact and oceanography in addition to the more obvious aspects of power plant design and constructional details. 1-4 A comprehensive review of the history and development of tidal power has been presented by Bernshtein. s Present interest is confined to the design and operation of tidal power plant. Dimensionless parameters are derived which aim to encapsulate the fundamental characteristics of any particular tidal power scheme. Thus for example, by using a simple theoretical derivation of the maximum available energy per tidal cycle EMAX the energy extracted in a specific design can be expressed as a fraction Of EMAx. Likewise we may compare the installed capacity (i.e. number of turbines x rated power) with the time-averaged value EMAx/P where P is the tidal period. Gibrat's (1966) 6 theoretical approach tO the design of tidal power schemes has been employed in the engineering designs for both La Ronce and Fundy. ~ Here we use a simplified version of Gibrat's approach with the aim of deriving optimum theoretical values for the basic dimensionless design parameters. Thus a framework is constructed against which designs from differing locations can be evaluated. This framework is shown to be valid for one-way power generation but in two-way schemes the Paper accepted September 1983. Discussion closes May 1984. 0309-1708/84/010021-07 $2.00 © 1984 CML Publications
engineering complexities reduce the validity of the simple theoretical approach. It should be emphasised that the present theoretical approach is not regarded as an alternative to the traditional engineering design procedure. Rather, the aim is to clarify the broad design principles involved and thereby enable the underlying mechanics of tidal power schemes to be more widely appreciated. In addition, this simple theoretical framework should also be of value in (1) making a first rough estimate for a tidal power plant and (2) in interpreting the sometimes confusing results obtained in the course of the usual step-by-step optimisation procedures. Since similar optimisation procedures are involved in many problems in the field of water resources, the simplified semi-analytic approach described here might recommend itself for other applications. THEORY
In one-way ebb generation the rising tide enters an enclosed basin through sluice gates and idling turbines, these gates are subsequently closed to maintain a maximum level in the basin and power is generated when a sufficient head develops on the falling tide. In two-way power generation, energy is extracted on both the rising and falling tide with sluicing occurring around the times of high and low water. A schematic representation of these two modes of operation is shown in Figs. l(a) and l(b). We introduce the following simplifying assumptions: (a) (b) (c) (d)
the external tide is sinusoidal and of amplitude A, the water level within the basin is horizontal, the surface area of the basin, S, is constant, during power generation, flow through the turbines is at a constant rate, Q, (e) power generation starts and stops at the same prescribed minimum head HMXN.
Adv. Water Resources, 1984, Volume 7, March 21
Simple theory for designing tidal power schemes: D. Prandle .A
~,X
t
~
\
Y
/ E X]'ERNAL
'l
POWER GENERATION
J ~ "~, ~
~
TIDE (Y)
time. Using (1), (2) and (3):
from (2)
T2
z.
or
Z
ZI
~l/ /~ /e /¢ ~
I /
LEVEL(ZI
T z
pgsrZ22[2
mean)eo level
Z2
i Z 2 -- Z I, A
2
\ ( ~ / ~
1//
•
(sin cor2-sin (6)
1'1
SLUICIN I
where ZI and Z2 are the basin levels at times TI and T2 respectively.
-A
0
*A
tI
TIME
t2
•
t5
[
~
I
]
eOWE.
I
//p.,.
....
-"
I
tI
~r'.
,5//
~- fromtol
MIN
I"-"
t2 t 3 TIME "-~ 14
t5
Q = eG(2ga) u2
'
....
" .....
dZ S--= dT
u,
I,
~6
17
~"/T
(7)
where e is a contraction coefficient associated with flow through the gate. Replacing eG by an effective gate area G~ and introducing (1), (2) and (3) we obtain:
.......
'//Z,,
-~
Sluicing The flow through a submerged sluice gate of area G is given by:
I
I:GENEeATION"I_ SLOIClNO 1
* ~
7 0
"
the lower case denotes the equivalent dimensionless quantity, e.g. HMAX and hMAX:
~ iTWO-WAY -
(i) Energy extracted e (= E/EMAx).
F
6
06-
2
(ii) Installed capacity c;
040200-I 0 -0 B -06 -04 -02
04 06 08 I0 > Figure 3 . Contours show sluicing capacity, ge, ((12)and (13)) required tO re-fill (or empty ) a basin between starting leve ! z3 and finishing level z4for the ease Of hMi N = 0.0 STARTING
0"0 02
LEVEL
this reflects the maximum rate of power generation possible for the number and type of turbine installed. The maximum theoretical power output is pgQHMA X where HMAX is defined below. To convert to dimensionless form we can compare this with the maximum energy output, EMAX, averaged over a tidal period, i.e.:
c = pgQHMAx x (P/4pgSA 2)
(19)
(iii) Utilisation factor u; This is the ratio of the energy extracted to the maximum plant potential given by:
u = e/c assumed and then, splitting the time axis into discrete intervals, the values o f z at each point in time were adjusted to more closely satisfy the differential equation. After 12 successive approximations the values for z showed reasonable convergence and the corresponding value for ge was adopted. Figure 3 shows the results for ge for the case OfhMi N = 0 and for
(20)
For a conventional hydro-electric plant operating continuously at the maximum design level, u = c = e = 1. (iv) Maximum head hMAX; the maximum head which occurs during power generation HMAX may be compared with the rated head of a turbine. In dimensionless form we write
rr < t 3 < 27r and
(18)
2n < t4 < 5n/2 Again, due to symmetry, the above ranges cover almost all cases of practical interest. The figure shows that the sluicing requirement increases rapidly as the finishing level approaches the maximum tidal level. Thus some compromise must be achieved between the conflicting requirements of power generation and a reasonable sluicing capacity. Figure 3 suggests a finishing level of about 0.85 would produce an acceptable sluicing capacity with gE = 7. In this range the sluicing capacity required in the one-way scheme is fairly insensitive to the starting level. By comparison, a smaller sluicing capacity is required in the two-way scheme with the requisite capacity reducing further as the starting level increases. In the case of the two-way scheme, further simulations were carried out with hMi N = 0.2 and hMl N - - 0 . 4 . The results showed a significant increase in the sluicing requirements, particularly when re-filling to near the maxinmm level. In re-filling to the suggested level of 0.85, sluicing requirements increase by up to 20% for hulN = 0.2 and by up to 60% for hulN = 0 . 4 . (In practice, in a two-way scheme sluicing may start up to 90 min before the end of generating in order to increase the available head during the next half cycle.) For these latter cases where hM1N ~ 0 , the contours in the equivalent diagrams to Fig. 3 would be discontinuous at the boundary of the one- and two-way schemes.
DIMENSIONLESS POWER PLANT PARAMETERS We now use the sinmlations described in the previous section to determine some of the important parameters involved in a tidal power plant. For any particular operational sequence we consider not only the energy extracted but the following list. In the following, an upper case (or capital) letter denotes a dimensional quantity while
24 Adv. Water Resources, 1984, Volume 7, March
hMA x = HMAX/A
(21)
(v) Flow rate q The (constant) flow rate during turbining, Q, is made dimensionless by reference to the 'undisturbed' tidal flow. Assuming the tidal flow is sinusoidal with an amplitude QT sufficient to fill the basin over the tidal range 2A then QT = 2nAS/P and hence
q = Q/QT = a x P/27rAS
(22)
(vi) Duration d The duration of turbining, AT(--- T : - 7'1 in the one-way mode) is expressed as a fraction of the tidal period thus:
d = ~r/e
(23)
The following parameters were introduced in previous sections: (vii) Minimum head hM1N = HMIN/A (viii) Effective gate area ge (equation (13)).
OPTIMUM THEORETICAL VALUES All of the dimensionless quantities defined in the fourth section can be explicitly determined in the same manner as described for the energy extracted in the first part of the third section. The component parts (a) to (f) of Fig. 4 illustrate the variation of these parameters over a range of values of (i) hulN and (ii)finishing level z 2 (equation (14)). The values shown all assume a starting level of zl = 1, i.e. a full basin. To illustrate the use of these diagrams consider Fig. 4(c), the respective lines for varying values of hMi N each terminates at the value z 2 = -- 1 + hMi N. Thus, while for z2 = 0 the utilisation factor, u, is greatest for hM1N = 0.6, in fact the maximum overall value of u is obtained for hMl N = 0.4 since in the latter case generation can continue over a longer period.
Simple theory for designing tidal power schemes." D. Prandle 06
05-
04-
04
(o) energy produced [e]
06
(d)
05-
I0
maximum
head [hMAx] 02-
08-
Ol
06-
0-0
00
-0'2
-04
-06
-0'8
-I 0
=08
,
-06
,
-04
-02
j
06 0.4.
02~ 0 4
0.4-
02
oo
02oo
I0
_0.6
J
1
0.0-0'2-d,-0'6-d8-,.0-018-d6-0'4-o'2
00
0.6
04 _ _ _ ~ 0.75
J
J
(b)
installed capacity [c]
050
OO
utilisation factor [u]
(e)
o~o~
flow rate [q]
~"'~
02.
O0
0.6
04
-I
-02
.
-04
-0 6 - 08
-I 0
-0'8
] ,
-0"6 -0-4 -02
O0
000"0 -()'2 - 0 4
-0.6' -().8 -,0
- 0 8 - 0 6 -0.4 -02
oo
08-
o6
(c)
08-
02 02
~
J 025
I0
04
0204
(f) duration [d]
-
02 04
0.6-
()4
0-4-
03
O6 0;'
o.2-~
OI
O0 oo
-~2
-o'4 -o'6 -~8
-,o -o'8 -o6 -o'4 -]0'2
TWO- WAY
--oo
ONE- WAY FINJSHING LEVEL"--"-)'
(z 21
O0
],
o-o-02-o4 £ 6 - o . 8 - , o - o 8 - o 6 - o . 4 - o 2 TWO-WAY
ONE- WAY
FINISHING LEVEL. (z 2 )
oo
>
Figure 4. Component parts (a)-(f) show values obtained in discharging a tidal basin from a maximum level zl = 1 to the finishing level z2. Values for the two-way mode in {a) and (c) have been doubled to reflect the second half o f the cycle. (a) Energy produced e; (b ) installed capacity c; (c ) utilisation factor u; (d) maximum head hMAX; (e ) flow rate q; (f) duration d. Results shown for hMIN = 0.0, 0.2, 0.4 and 0.6 (as labelled). The position indicated at z2 = - 0.259 in the one-way mode corresponds approximately to the finishing level for maximum energy extraction
Figure 4(a) shows that the energy extracted, e, increases with hMl N with the increase being more pronounced for the two-way mode. However, Fig. 4(b) shows that the requisite installed capacity, c, increases in a similar fashion. Thus the ratio of e:c represented by the utilisation factor, u, in Fig, 4(c) is less sensitive to hMIN- ttowever, careful examination shows that maxinaunl efficiency, i.e. the maximum value of u, is found for a value of hMi N = 0.4. With this value OfhM[ N we found from Fig. 2 that the twoway scheme extracted up to 0.37 of the energy potential compared with 0.27 for the one-way scheme. Figure 4(b) shows that the installed capacity required to achieve these peak values of energy extraction is 0.7 in both the one-way and two-way mode. Hence the corresponding values for the utilisation factor u ( = e/c) are approximately 0.4 and 0.5 respectively as seen from Fig. 4(c). Figure 4(d) shows the maximum head developed during turbining while Fig. 4(e) shows the flow rates involved. In general, the one-way mode operates at a higher head and lower flow rate than the two-way mode. For example, with hMi N = 0.4 and at maximum energy extraction, in the one-
way mode h i h x = 1.2 and q = 0.4 while in the two-way mode hMA x = 0.6 and q = 0.8. Since turbine efficiency is generally low at small heads, the approximate doubling of the operating head in the one;way scheme compared with the two.way scheme is a clear advantate. A turbine is designed to operate optimally at a rated head HRA T and a rated flow QRAT. The turbo generator performance curve, reproduced by Clark, 7 for a turbine with variable wicket gates and runner blades, shows that maximum power output can be maintained with an operating head, hop , in the range h,RAT< hop < I.ShRA T and a corresponding operating flow, qoP, in the range qRAT > qoP > 0.SqRAT. Thus operating at the maximum head conditions, i.e. 1.5hRA T and 0.5qRAT , the energy conversion is 33% more efficient than at the minimum head condition hRA T and qRAT (since from (5) E ~ HQ and 1.5hRA T X 0.5qRAT = ~hRA T X qRAT). However, one advantage in the two-way scheme is the smaller variability of the operating head over the turbining period and hence the more constant level of power generation. The flow rates in the two-way mode approach the undisturbed value (q = 1) and hence this
Adv. Water Resources, 1984, Volume 7, March 25
Simple theoryfor designing tidal power schemes: D. Prandle mode produces less change in the existing tidal flushing regime. Figure 4(f) shows the duration of the power generation phase as a fraction of the tidal period. The duration decreases as hMi N increases, thus where peak power is required a high value of hMIN may be used, whereas for more continuous power output hMl N must be reduced. When 3perating at maximum efficiency the duration d = 0.55 for both modes. Table 1 summarises the above results showing optimum theoretical values for the various dimensionless parameters when operating at maximum efficiency. Table 1 also shows the value of the effective gate area ge derived in the third section. In arriving at this value of ge it was indicated that the basin would only be filled to a level z = 0.85. Hence some slight reduction in the optimum values of e and u must follow since the values shown were derived for an initially full basin.
a factor of 4 to 1 between high and low water. Using the value of S at mean water level for converting to dimensionless parameters, the energy extracted in the one-way ebb mode increased by around 10%, the installed capacity increased by a similar amount leaving the utilisation factor more or less unchanged. Similar changes were found for the two-way mode but, assuming symmetric operation, changes on the ebb generation are effectively counterbalanced by changes on the flood. This small effect of a variable surface area found for the one-way mode may be attributed to the centering of the generation period around the time of mean water level. However, sluicing may occur over a wider range of tidal conditions and thus might be expected to be more sensitive to change in S. Tests were carried out with the same fourfold linear variation in S, in the one-way scheme, sluicing capacity increased by about 50% for the ebb generation and decreased by about 40% for flood generation. In the twoway scheme sluicing capacity at high water increased by more than 100% while at low water the decrease exceeded 50%. Hence the operational mode in a two-way scheme with variable S should be asymmetric to reflect the variation in sluicing capacity at high and low water.
PRACTICAL APPLICATIONS In any detailed design of a tidal power plant there will be numerous constraints not considered here, see for example Swales and Wilson. 8 It is clearly impractical to discuss all such extraneous factors here, however, the initial assumption of a constant tidal amplitude A, and a constant surface area, S, are so restrictive that some comment is necessary.
COMPARISON WITH DETAILED ENGINEERING DESIGNS Table 1 summarises detailed designs for both one-way and two-way tidal power schemes published for Cumberland Basin and Minas Basin in the Bay of Fundy. These data were reduced to dimensionless quantities using the basic parameters S, surface area; A, tidal amplitude; and P ( = 1 2 . 4 2 h) tidal period. Comparisons are then easily made both between the four schemes and against the theoretical values derived here. Reasonable agreement between the theoretical values and all four design values is found for (a) sluice gate areage, (b) minimum head hMi N and (c) duration d. Designs for Cumberland Basin use slightly higher values of ge than. indicated by theory while Minas Basin designs use marginally lower values. The installed capacities are lower than the theoretical values by an average of 18% for Cumberland Basin and by 38% for Minas Basin. In the one-way mode, power outputs are reduced in similar proportions. In the two-way mode,
Variation in tidal amplitude A At almost all places the time-series of tidal elevations can be closely approximated by a sinusoidal curve between successive peaks and troughs. Since the results shown are in dimensionless form it should be possible to apply the present results to discrete portions of the time-series. While this approach may provide a useful first study, detailed design must take account of the ordering of successive operational modes to optimise energy output. Variation in surface area, S In two of the most attractive tidal power sites in the Bay of Fundy, Cumberland Basin and Minas Basin, on spring tides the surface area at high water is about three times greater than at low water. Some additional tests were made for a basin in which the surface area varied linearly by
Table 1.
Fund)'tidal power designs converted to dimensionlessform Dimensional*
Non-dimensional
Cumberland Basin
Minas Basin
86.2 5.0 23.4
261.1 6.1 105.7
Surface area, S (km 2) Tidal amplitude, A (m) EMAx = 4pgA :S (GWh)
One-way mode
One-way Two-way One-way Two-way Theory (i) power output, E (GWh) (ii) installed capacity, C (MW) (iii) utilisation factor, u = e/c (iv) rated head, HRAT (m) (v) flow rate, QRAT (m 3s- ~) (vi) duration, d (h) (vii) rain head, HMIN (m) (viii) gate area, Ge (m2)
4.8 1085
5.1 1292
17.9 3800
17.4 3782
6.5 694 6.8 2.3 8387
7.0 853
7.5 738 7.1 2.6 21 939
8.0 977
2.4 7893
* Data taken from Clark (1977)~
26 Adv. Water Resources, 1984, Volume 7, March
2.8 21619
0.27 0.7 0.4 1.2 0.4 0.55 0.4 7.0
Cumberland 0.19 0.52 0.36 1.30 0.40 0.55 0.46 8.6
Two-waymode
Minas
Theory
Cumberland
0.16 0.43 0.38 1.23 0.33
0.37 0.7 0.5 0.6 0.8 0.55 0.4 7.0
0.20 0.62 0.32 1.40 0.57 0.57 0.48 8.1
0.43 6.7
Minas 0.16 0.43 0.37 1.31 0.42 0.46 6.6
Simple theory for designing tidal power schemes: D. Prandle power outputs for the engineering designs are approximately half the theoretical values. Part of this reduction in actual output can be attributed to factors such as (a) insufficient sluicing capacity to re-fill the basin, (b) variability in tidal range and (c) power losses during turbining. However, the greater reduction in the two-way mode may perhaps be due to the use of much larger rated heads in the engineering designs than indicated by the theoretical values. These rated heads are more than double the theoretical value and in consequence the rated flows in the two-way engineering designs are almost half the theoretical values. By contrast the values for both the rated heads and rated flows in the one-way mode engineering designs are all in reasonable agreement with the theoretical values. Overall the theoretical Values are in good agreement with actual designs for the one-way mode. For the two-way mode there is similarly good agreement for several of the design parameters but a disparity exists in the choice of the rated heads (and in consequence the rated flows). The reasons for this disparity are not obvious. However, it is recognised that the turbine characteristics for two-way power generation are more complex than in the one.way mode and hence the operating conditions might be further removed from the assumptions made in the second section.
SUMMARY AND CONCLUSIONS An important dimensional aspect of tidal power schemes is that, while energy extracted is proportional to the tidal amplitude squared, the requisite sluicing area is proportional to the square root of the tidal amplitude. Thus, as the tidal amplitude increases, the unit cost of sluicing capacity decreases. Since turbine efficiency also increases for larger tidal amplitudes then, clearly, sites with large tidal range are most attractive for tidal power development. The simplified approach used here provides a description of tidal power schemes in terms of eight dimensionless parameters. Optimal theoretical values for these parameters have been deduced. While these values cannot be regarded as an alternative to detailed engineering designs they do serve as a useful basis for evaluating the latter. In addition, the dimensionless parameters provide an interesting means f o r comparing engineering designs made for differing locations. Theoretical simulations of tidal power schemes suggest (possibly incorrectly from the practical viewpoint)that two-way, ebb and flood, operation can extract up to 37% more energy than a one-way, ebb or flood, scheme using the same intalled capacity. In addition, compared with the one-way mode, the two-way mode can be operated at a more constant head and the flushing regime in the enclosed tidal basin remains closer to the undisturbed state. However, the operating heads in the one-way mode may be double those in the two-way mode and hence turbine efficiency should be greater. Turbine efficiency in the twoway mode is further reduced by the basic turbine design compromises necessary to accommodate two-way flow. Thus, the theoretical advantage of the two-way scheme shown here may well be more than counter-balanced by practical engineering realities.
ACKNOWLEDGEMENT This study was partially completed while the author was on leave at The Hydraulics Laboratory, National Research Council, Ottawa, Canada.
NOTATION A mean tidal amplitude C installed capacity E energy extracted g gravitational acceleration G area of sluice gate G e = eG effective gate area H turbine head P tidal period POW power produced Q flow through turbines Qo flow through turbines when used for sluicing Qs flow through sluices for 1 m head QT sinusoidal flow amplitude to fdl tidal basin QTOT total sluicing flow S surface area of tidal basin T time Y tidal level outside basin Yp annual power output Z water level in basin e contraction coefficient for sluices p water density co 2n/P Subscripts 1 at start of power production 2 at end of power production 3 at start of sluicing 4 at end of sluicing MIN minimum value for power production MAX maximum value during power production RAT turbine rated value Non-dimensional parameters c installed capacity (equation (19)) d (7"2-- T)/P duration of power production e E/Emax fraction of maximum energy extracted ge area of sluice gate (equation (13)) h H/A turbine head m number of sluices n number of turbines q Q/QT flow through turbines u e/c utilisation factor
REFERENCES 1 Gray, T. J. and Gashus, O. K. Tidal Power, Plenum Press, New York, 1972. 2 Darborn, G. R. Fund)' Tidal Power and the Environment, Workshop Proceedings, Acadia University, Wolfville, Nova Scotia, 1977. 3 Waveand Tidal Energy Symposium. B.H.R.A. Fluid Engineering, Cranfield, Bedford, 1978. 4 Severn, R. T., Dineley, D. and Hawker, L. E. Tidal Power and Estuary Management, Colston Papers. No. 30. Scientechnica, Bristol, 1979. 5 Bernshtein, L. B. l~dal Energy for Electric Power Plants, Israel Program for Scientific Translations, Jerusalem, 1965. 6 Gibrat, R. L'6nergiedes mar6es, Presses Universitaires de France, Paris, 1966. 7 Clark, R. H. Reassessment of Fundy Tidal Power, Reports of the Bay of Fundy Tidal Power Review Board and Management Committee, Ministry of Supply and Services, Canada, 1977. 8 Swales, M. C. and Wilson, E. M. Optirnisation of tidal power generation, WaterPower 1968, 20, 109.
BIBLIOGRAPHY B.74 TidalPower. 1973. B74a Wave and Tidal Power, The Electricity Council, Millbank, London, 1978.
Adv. Water Resources, 1984, Volume 7, March 27
