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Multi-objective approach for a combined heat and power geothermal plant optimization

Fabien Marty ORCID logo, Sabine Sochard, Sylvain Serra ORCID logo and Jean-Michel Reneaume


This paper presents the simultaneous optimization of the design and operation in nominal conditions of a geothermal plant where the geothermal fluid is split into two streams to feed an Organic Rankine Cycle (ORC) and a District Heating Network (DHN). The topology of the DHN is also investigated. A Mixed Integer Non-Linear Programming (MINLP) optimization problem is formulated and solved using the GAMS software in order to determine the ORC sizing and the DHN topology. In this study, only R-245fa is used as ORC working fluid, an optional Internal Heat Exchanger (IHE) is considered in the ORC and consumers in DHN can be definite or optional. A multi-objective optimization is performed by maximizing the annual net profit and minimizing the total exergy losses in the plant. The weighted sum of objective functions is used to solve the problem. By varying the weight factor, a Pareto front is obtained and the distance to the ideal, but infeasible, solution enabled to choose the best compromise. Four different DHN topologies are observed depending on the weight factor. Using a suitable criterion to make a decision, the selected configuration corresponds to the most expanded DHN with the smallest value of profit. A sensitive analysis shows that, in case of lower geothermal temperature, it is possible to obtain a unique DHN topology whatever the weight factor.

Corresponding author: Sylvain Serra, Université de Pau et des Pays de l’Adour, E2S UPPA, LaTEP, Pau, France, E-mail:

Funding source: ADEME : Agence de la transition écologique


heat transfer area, m2


high value coefficient


refer to all constraints.

C p

specific heat capacity, J/kg/K


investment cost, €


annual working cost, € by year

d o

outside tube diameter, m

D direct

direct distance to the ideal solution

D normalized

normalized distance to the ideal solution


distance, m


specific mass exergy, J/kg

E ˙ x

exergy flow rate, W


existence (binary variable)


enthalpy, J/kg


tube length, m

m ˙

mass flow rate, kg/s

N t

number of tubes per pass


multi-objective function


pressure, Pa


difference temperature at pinch the point in heat exchanger, K


annual profit, € by year

Q ˙

heat transfer rate, W


entropy, J/kg/K


sale, € by year


temperature, K

tx imp

corporate tax rate

Tx div

division rate

V ˙

volumetric flow rate, m3/h


refer to all continuous variables

W ˙

mechanical work, W

Greek symbols

coefficient of decrease of temperature, K/km


pressure drop, Pa

pressure drop (in pipe)


temperature decrease in pipe, K


density, kg/m3


efficiency, %


weight factor between the two objective functions

Subscripts and superscripts



cold/hot fluid






plant components


cooling water


destruction (refer to exergy)


District Heating Network

District Heating Network








electrical generator


geothermal water




refer to DHN nodes (producer and consumers)


Internal Heat Exchanger

Internal Heat Exchanger








losses (refer to exergy)






net electricity produced


outward path/return path


Organic Rankine Cycle

Organic Rankine Cycle

path, ij

path between i and j








heat losses (in pipe)


geothermal water reinjection

return, i

outlet of consumer iafter mixing in return path






DHN user




working fluid


pressure drop, Pa

pressure drop (in pipe)


Combined Heat and Power


District Heating Network


Internal Heat Exchanger


Mixed Integer Non-Linear Programming


Non-Linear Programming


Objective Function


Organic Rankine Cycle


Specific Investment Cost


The authors thank the ADEME through the “Appel à Manifestation d’Intérêts (AMI)”. They also thank the Enertime society, for its expertise about the ORC systems, and the “FONROCHE Géothermie” society, which is the coordinator of the FONGEOSEC project.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: The study was funded by ADEME : Agence de la transition écologique.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A: Development of equations presented in Table 5

Equations (6) and (7) are used to determine exergy flow destruction for each component of the plant.

For heat exchangers:

E ˙ x d , e x c h = m ˙ h [ h h , i n h 0 T 0 ( s h , i n s 0 ) ] + m ˙ c [ h c , i n h 0 T 0 ( s c , i n s 0 ) ] m ˙ h [ h h , o u t h 0 T 0 ( s h , o u t s 0 ) ] m ˙ c [ h c , o u t h 0 T 0 ( s c , o u t s 0 ) ]

After simplification and considering m ˙ h ( h h , i n h h , o u t ) = m ˙ c ( h c , o u t h c , i n ) equation becomes

E ˙ x d , e x c h = T 0 [ m ˙ h ( s h , o u t s h , i n ) + m ˙ c ( s c , o u t s c , i n ) ]

For turbine:

E ˙ x d , t u r b i n e = m ˙ w f [ h t u r b i n e , i n h t u r b i n e , o u t T 0 ( s t u r b i n e , i n s t u r b i n e , o u t ) ] W ˙ e l e c

with W ˙ e l e c corresponds to the produced electricity equal to

W ˙ e l e c =   η g e n W ˙ t u r b i n e = η g e n m ˙ w f ( h t u r b i n e , i n h t u r b i n e , o u t )


E ˙ x d , t u r b i n e = m ˙ w f [ ( 1 η g e n ) ( h t u r b i n e , i n h t u r b i n e , o u t ) T 0 ( s t u r b i n e , i n s t u r b i n e , o u t ) ]

For ORC pump:

E ˙ x d , O R C p u m p = m ˙ w f [ h O R C p u m p , i n h O R C p u m p , o u t T 0 ( s O R C p u m p , i n s O R C p u m p , o u t ) ] + W ˙ O R C p u m p

With W ˙ O R C p u m p = m ˙ w f ( h O R C p u m p , o u t h O R C p u m p , i n ) then

E ˙ x d , O R C p u m p = m ˙ w f T 0 ( s O R C p u m p , o u t s O R C p u m p , i n )

For DHN pump:

In the same way than for ORC pump, the obtained expression is

E ˙ x d , D H N p u m p = m ˙ D H N T 0 ( s D H N p u m p , o u t s D H N p u m p , i n )

For water, entropy difference is obtained by Equation (8)

E ˙ x d , D H N p u m p = m ˙ D H N T 0 T D H N p u m p , i n T D H N p u m p , o u t C p l , w a t e r T d T

First, C p l , w a t e r is assumed constant between the inlet and the outlet of the pump, expression is then

E ˙ x d , D H N p u m p = m ˙ D H N T 0 C p l , w a t e r ln T D H N p u m p , o u t T D H N p u m p , i n

Next, the increase temperature in pump is low (0.05 K), logarithm can be approximated by a first-order limited development around 1.

E ˙ x d , D H N p u m p = m ˙ D H N T 0 C p l , w a t e r T D H N p u m p , o u t T D H N p u m p , i n T D H N p u m p , i n

Since C p l , w a t e r is assumed constant, this corresponds to

E ˙ x d , D H N p u m p = W ˙ D H N p u m p T 0 T D H N p u m p , i n

For pressure drop in pipes:

The destroyed exergy linked to pressure drops is due to the work dissipated by friction and is determined by:

E ˙ x d , Δ P = m ˙ D H N [ h D H N p u m p , o u t h D H N p u m p , i n T 0 ( s D H N p u m p , o u t s D H N p u m p , i n ) ]

Which corresponds to

E ˙ x d , Δ P =   W ˙ D H N p u m p E ˙ x d , D H N p u m p

By replacing with the previous expression

E ˙ x d , Δ P =   W ˙ D H N p u m p   ( 1 T 0 T D H N p u m p , i n )


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Received: 2020-02-07
Accepted: 2020-05-29
Published Online: 2020-08-06

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