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Optimal control of solar
energy systems
Viorel Badescu
Candida Oancea Institute
Polytechnic University of Bucharest
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Contents
1. Optimal operation - systems with water
storage tanks
2. Sizing solar collectors
3. Optimal operation - maximum exergyextraction
4. Sizing solar collection area
5. Conclusions
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4. Sizing solar collection area
The optimization depends on the way the investoruses the thermal energy obtained from solar energyconversion
Two objectives:
First, to develop a sizing procedure for collectionsurface area, with input variables:
the working fluid mass flow rate and
the inlet and outlet fluid temperatures
Second, propose a procedure to find the best local design solution;
It may be implemented by using various objectivefunctions
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1. Sizing solar collection area
Some economical indices, including
net present value and
internal return rate,
are examples of objective functions. V Badescu, Optimum size and structure for solar energy
collection systems, Energy 31 (2006) 1483-1499
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Model
The user
may need
heat or
work fluxes
The
classical
system
may
provideheat or
work fluxes
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The optimization problem
A (primary) conventional energy transfer system
A (secondary) system based on solar energyconversion.
cT - total energy transfer cost per unit time,
c1 - cost of one energy unit received/removed by using the primary system
c2 - investment and operation costs of the secondary system The optimization problem:
find the surface area A which minimizes the costs
and the optimal structure of the collection system.
Ac F F c Ac unecT 21
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Model
The mass flow rate is
fixed
The fluid exits the area
A at temperature T
Adding area dA
increases the
temperature by delta_T
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Solar collector model
“Absorbed” heat flux
Lost heat flux
Useful heat flux = “absorbed” - lost
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Model
Integration of Hottel-Whillier-Bliss eq. (in J):
The time averaged form is (in W):
The time-averaged efficiency
dt dAT T F U F Gdt dT cmt
a fi R L R
t
p
0
*****
0
**
0
*
dAT U GdT c F ~~
0
GT U GdAdT c F /~~/ 0
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Applications
(a)
The energy transferred is a heat rate received by a body and
the primary energy transfer system is a conventional heater.
(b)
The energy transferred is a heat rate received by a body attemperature Ta+T and
the primary energy transfer system is a vapor compressionheat pump.
(c)
The energy transferred is a heat rate extracted from a bodyat temperature
and the primary energy transfer system is an absorptionrefrigerator.
The difference consists in the factors Fnec and Fu
0 T T T T avap
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Case (a) as an example
All energy
fluxes
involved are
heat fluxes
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Case (a)
Fnec and Fu are heat fluxes
The increase of the heat rate supplied by the
solar energy conversion system,
associated to the increase of collection area dAis:
Then, the economical benefit is
dT cdF F
a
u
a
udF
dAGcdT ccdF cd a
F
aa
u
aa 111$
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Economical indicators
The so called “revenue” factor R
cost of saved primary energy over cost of
surface area
The cost C_A per unit time of the solarenergy collection surface area A:
2
1
c
Gc R
a
a
2
1
2
02
T
T
F A
A dT G
ccdAcC
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Economical indicators
the net present value (NPV)
the present value of cash inflows is
subtracted by the present value of cash
outflows.
2
1
22
1
21
21
,,
T
T tot
a
tot F
aa
red dT Y G
cY
G
c
t
t c
t c
T T NPV T T NPV
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Economical indicators
the internal rate of return (IRR) is the interest rate that makesNPV equal zero.
It is the return that a company would earn if they expanded orinvested in themselves, rather than investing that money abroad
The
may be found by solving numerically the associated equation
cbaiT T IRR i,,, 21
cbaiT T NPV i
,,0, 21
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Examples
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Results
The revenue factor R exceeds unity in casethe inlet working fluid temperatureexceeds a certain “threshold value”,depending on solar collector design (Fig.c).
The four threshold temperatures arelower than 50 degrees.
The temperature threshold values in
case of are around 60 degrees forcollectors I and II.
The other two collectors have pooreconomical performance as theassociated NPV is negative for alloperation temperatures (Fig. a).
The IRR values of Fig. b show thecollector I may be used economically forT between 55 and 70 degrees while
collector II is recommended for operationat more than 60 degrees.
Collectors III and IV are notrecommended as the associated IRRvalues do not exceed the interest rate forall T values.
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Results
Different economical
indicators induce
different hierarchies
over the set of solar
collectors.
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Results
Let us consider a part of thecollection surface consistingof a single type of collector.Integration of the efficiencydefinition yields thenecessary surface area
The necessary collectionarea is slightly smaller forcollector I than for collectorII.
Therefore, if a single type ofcollector must be used,collector I should beselected.
In case both types ofcollectors are available, a
better solution exists.
2
1
/, 21
T
T F
G
dT cT T A
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Solar collectors with optimal non-
uniformly distributed parameters
It was proved that systems consisting in
combinations of different collector types may
be a better solution
than systems consisting of a single collectortype.
One could imagine the extreme case of a
collection system with continuously space
variable parameters.
Such a system may be optimized from the
point of view of a given economical indicator.
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Optimisation
The cost
is optimised if:
One finds
2
1
~,~
~,~
~,~
0
02
0
T
T
F
A dT GU
cU cU C
0~
/~/ 0 U C C A A
0
2
20
~1
~
~
~1
c
c U
c
cU ~1
~
~
~1 2
2
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Theorem
The following condition should be fulfilled bythe optimum parameters distribution:
Theorem. The modified optical efficiencyand the modified overall heat loss coefficient
in an optimal collection system are distributedin such a way that the gradient of
in the bi-dimensional parametric spacevanishes
0
~
ln 2
~,~0
cU
2/~ln c
U ~
,~0
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Results
For very small values of Tthe unglazed solarcollector is the besteconomical solution forboth applications (Fig a).
When increases T asingle transparent layercollector should be used.
The thresholdtemperature for which N jumps from 0 to 1 is
smaller for the coldseason application.
A collector without bottomthermal insulation is thebest solution at very smalltemperatures (Fig. b).
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Conclusions
The general theorem proposed here showshow the modified optical efficiency and heatloss coefficient should be distributed for costminimization.
One finds that unglazed, single-glazed anddouble-glazed collectors should be used onthe same collection area in order to obtain thebest performance.
Also, the bottom insulation thickness shouldbe changed accordingly.
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End of part 4/4
Thank you!
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