Optimal control of solar

energy systems

Viorel Badescu

Candida Oancea Institute

Polytechnic University of Bucharest

Contents

1. Optimal operation - systems with water

storage tanks

2. Sizing solar collectors

3. Optimal operation - maximum exergy

extraction

4. Sizing solar collection area

5. Conclusions

0. Introduction

This talk shows how the classical

methods of optimal control can be used

by the solar energy engineer.

Four applications will give a broad idea

about the usefulness of these

optimization procedures.

4. Sizing solar collection area

The optimization depends on the way the investor uses the thermal energy obtained from solar energy conversion

Two objectives:

First, to develop a sizing procedure for collection surface 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 localdesign solution;

It may be implemented by using various objective functions

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

Model

The user

may need

heat or

work fluxes

The

classical

system

may

provide

heat or

work fluxes

The optimization problem

A (primary) conventional energy transfer system

A (secondary) system based on solar energy conversion.

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.

AcFFcAc unecT 21

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

Solar collector model

“Absorbed” heat flux

Lost heat flux

Useful heat flux = “absorbed” - lost

Model

Integration of Hottel-Whillier-Bliss eq. (in J):

The time averaged form is (in W):

The time-averaged efficiency

dtdATTFUFGdtdTcmt

afiRLR

t

p

0

*****

0

**

0

*

dATUGdTcF

~~0

GTUGdAdTcF /~~/ 0

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 at temperature Ta+T and

the primary energy transfer system is a vapor compression heat pump.

(c)

The energy transferred is a heat rate extracted from a body at temperature

and the primary energy transfer system is an absorption refrigerator.

The difference consists in the factors Fnec and Fu

0 TTTT avap

Case (a) as an example

All energy

fluxes

involved are

heat fluxes

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 dA

is:

Then, the economical benefit is

dTcdF F

a

u

a

udF

dAGcdTccdFcd a

F

aa

u

aa 111$

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 solar

energy collection surface area A:

2

1

c

GcR

aa

2

1

2

02

T

T

FA

A dTG

ccdAcC

Economical indicators

the net present value (NPV)

the present value of cash inflows is

subtracted by the present value of cash

outflows.

2

1

221

2121

,,

T

T tot

a

totF

aa

red dTYG

cY

G

c

t

tc

tc

TTNPVTTNPV

Economical indicators

the internal rate of return (IRR) is the interest rate that makes NPV equal zero.

It is the return that a company would earn if they expanded or invested in themselves, rather than investing that money abroad

The

may be found by solving numerically the associated equation

cbaiTTIRR i ,,, 21

cbaiTTNPV i ,,0, 21

Examples

Results

The revenue factor R exceeds unity in case the inlet working fluid temperature exceeds a certain “threshold value”, depending on solar collector design (Fig. c).

The four threshold temperatures are lower than 50 degrees.

The temperature threshold values in case of are around 60 degrees for collectors I and II.

The other two collectors have poor economical performance as the associated NPV is negative for all operation temperatures (Fig. a).

The IRR values of Fig. b show the collector I may be used economically for T between 55 and 70 degrees while collector II is recommended for operation at more than 60 degrees.

Collectors III and IV are not recommended as the associated IRR values do not exceed the interest rate for all T values.

Results

Different economical

indicators induce

different hierarchies

over the set of solar

collectors.

Results

Let us consider a part of the collection surface consisting of a single type of collector. Integration of the efficiency definition yields the necessary surface area

The necessary collection area is slightly smaller for collector I than for collector II.

Therefore, if a single type of collector must be used, collector I should be selected.

In case both types of collectors are available, a better solution exists.

2

1

/, 21

T

TF

G

dTcTTA

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 collector

type.

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.

Optimisation

The cost

is optimised if:

One finds

2

1

~,~

~,~~

,~

0

02

0

T

T

F

A dTGU

cUcUC

0~

/~/ 0 UCC AA

0

2

20~

1~

~

~1

c

c U

c

cU~

1~

~

~1 2

2

Theorem

The following condition should be fulfilled by the optimum parameters distribution:

Theorem. The modified optical efficiency and the modified overall heat loss coefficient in an optimal collection system are distributed in such a way that the gradient of

in the bi-dimensional parametric space vanishes

0~

ln2

~,~

0

cU

2/~ln c

U~

,~0

Results

For very small values of T the unglazed solar collector is the best economical solution for both applications (Fig a).

When increases T a single transparent layer collector should be used.

The threshold temperature for which N jumps from 0 to 1 is smaller for the cold season application.

A collector without bottom thermal insulation is the best solution at very small temperatures (Fig. b).

Conclusions

The general theorem proposed here shows how the modified optical efficiency and heat loss coefficient should be distributed for cost minimization.

One finds that unglazed, single-glazed and double-glazed collectors should be used on the same collection area in order to obtain the best performance.

Also, the bottom insulation thickness should be changed accordingly.

End of part 4/4

Thank you!