Optimal control of solar energy systems

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1 Optimal ontrol of solar energy systems Viorel Badesu Candida Oanea Institute Polytehni University of Buharest

2 Contents. Optimal operation - systems with water storage tanks 2. Sizing solar olletors 3. Optimal operation - maximum exergy extration 4. Sizing solar olletion area 5. Conlusions

3 0. Introdution This talk shows how the lassial methods of optimal ontrol an be used by the solar energy engineer. Four appliations will give a broad idea about the usefulness of these optimization proedures.

4 4. Sizing solar olletion area The optimization depends on the way the investor uses the thermal energy obtained from solar energy onversion Two objetives: First, to develop a sizing proedure for olletion surfae area, with input variables: the working fluid mass flow rate and the inlet and outlet fluid temperatures Seond, propose a proedure to find the best loal design solution; It may be implemented by using various objetive funtions

5 . Sizing solar olletion area Some eonomial indies, inluding net present value and internal return rate, are examples of objetive funtions. V Badesu, Optimum size and struture for solar energy olletion systems, Energy 3 (2006)

6 Model The user may need heat or work fluxes The lassial system may provide heat or work fluxes

7 The optimization problem A (primary) onventional energy transfer system A (seondary) system based on solar energy onversion. A F F A T ne u 2 T - total energy transfer ost per unit time, - ost of one energy unit reeived/removed by using the primary system 2 - investment and operation osts of the seondary system The optimization problem: find the surfae area A whih minimizes the osts and the optimal struture of the olletion system.

8 Model The mass flow rate is fixed The fluid exits the area A at temperature T Adding area da inreases the temperature by delta_t

9 Solar olletor model Absorbed heat flux Lost heat flux Useful heat flux = absorbed - lost

10 Model Integration of Hottel-Whillier-Bliss eq. (in J): t 0 t m dt dt G F U F T T * * * * * * * * p R 0 L R fi 0 The time averaged form is (in W): The time-averaged effiieny F dt G 0 UT da F dt / 0 / GdA UT G a dadt

11 Appliations (a) The energy transferred is a heat rate reeived by a body and the primary energy transfer system is a onventional heater. (b) The energy transferred is a heat rate reeived by a body at temperature Ta+T and the primary energy transfer system is a vapor ompression heat pump. () The energy transferred is a heat rate extrated from a body at temperature T vap Ta T T 0 and the primary energy transfer system is an absorption refrigerator. The differene onsists in the fators Fne and Fu

12 Case (a) as an example All energy fluxes involved are heat fluxes

13 Case (a) Fne and Fu are heat fluxes The inrease of the heat rate supplied by the a solar energy onversion system, assoiated to the inrease of olletion area da is: df a dt u F Then, the eonomial benefit is df u a a a u a d$ df dt F a G da

14 Eonomial indiators The so alled revenue fator R ost of saved primary energy over ost of surfae area The ost C_A per unit time of the solar energy olletion surfae area A: C A R A a 0 2 da a G 2 T 2 T 2 F G dt

15 Eonomial indiators the net present value (NPV) the present value of ash inflows is subtrated by the present value of ash outflows ,, T T tot a tot F a a red dt Y G Y G t t t T T NPV T T NPV

16 Eonomial indiators the internal rate of return (IRR) is the interest rate that makes NPV equal zero. It is the return that a ompany would earn if they expanded or invested in themselves, rather than investing that money abroad The T T i a b IRR i,,, 2 may be found by solving numerially the assoiated equation NPV i T, T 0 i a, b, 2

17 Examples

Results The revenue fator R exeeds unity in ase the inlet working fluid temperature exeeds a ertain threshold value, depending on solar olletor design (Fig. ).

18 Results The revenue fator R exeeds unity in ase the inlet working fluid temperature exeeds a ertain threshold value, depending on solar olletor design (Fig. ). The four threshold temperatures are lower than 50 degrees. The temperature threshold values in ase of are around 60 degrees for olletors I and II. The other two olletors have poor eonomial performane as the assoiated NPV is negative for all operation temperatures (Fig. a). The IRR values of Fig. b show the olletor I may be used eonomially for T between 55 and 70 degrees while olletor II is reommended for operation at more than 60 degrees. Colletors III and IV are not reommended as the assoiated IRR values do not exeed the interest rate for all T values.

19 Results Different eonomial indiators indue different hierarhies over the set of solar olletors.

20 Results Let us onsider a part of the olletion surfae onsisting of a single type of olletor. Integration of the effiieny definition yields the neessary surfae area, T 2 dt A T T2 / F TG The neessary olletion area is slightly smaller for olletor I than for olletor II. Therefore, if a single type of olletor must be used, olletor I should be seleted. In ase both types of olletors are available, a better solution exists.

21 Solar olletors with optimal nonuniformly distributed parameters It was proved that systems onsisting in ombinations of different olletor types may be a better solution than systems onsisting of a single olletor type. One ould imagine the extreme ase of a olletion system with ontinuously spae variable parameters. Suh a system may be optimized from the point of view of a given eonomial indiator.

22 Optimisation The ost is optimised if: One finds 2,,, T T F A dt U G U U C 0 / / 0 U C C A A U U 2 2

23 Theorem The following ondition should be fulfilled by the optimum parameters distribution: 0 ln 0, U 2 Theorem. The modified optial effiieny and the modified overall heat loss oeffiient in an optimal olletion system are distributed in suh a way that the gradient of ln in the bi-dimensional parametri spae vanishes / 0 2,U

Results For very small values of T the unglazed solar olletor is the best eonomial solution for both appliations (Fig a). When inreases T a single transparent layer olletor should be used.

24 Results For very small values of T the unglazed solar olletor is the best eonomial solution for both appliations (Fig a). When inreases T a single transparent layer olletor should be used. The threshold temperature for whih N jumps from 0 to is smaller for the old season appliation. A olletor without bottom thermal insulation is the best solution at very small temperatures (Fig. b).

25 Conlusions The general theorem proposed here shows how the modified optial effiieny and heat loss oeffiient should be distributed for ost minimization. One finds that unglazed, single-glazed and double-glazed olletors should be used on the same olletion area in order to obtain the best performane. Also, the bottom insulation thikness should be hanged aordingly.

26 End of part 4/4 Thank you!

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