Citation: Tashtoush, B.; Luo, J.;
Morosuk, T. Exergy-Based
Optimization of a CO
2
Polygeneration
System: A Multi-Case Study. Energies
2024,17, 291. https://doi.org/
10.3390/en17020291
Academic Editors: Salman Ajib
and Mohammad Ahmad Hamdan
Received: 10 December 2023
Revised: 31 December 2023
Accepted: 4 January 2024
Published: 6 January 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
energies
Article
Exergy-Based Optimization of a CO2Polygeneration System: A
Multi-Case Study
Bourhan Tashtoush 1,* , Jing Luo 2and Tatiana Morosuk 2,*
1Mechanical Engineering Department, Jordan University of Science and Technology,
P.O. Box 3030, Irbid 22110, Jordan
2Institute for Energy Engineering, Technische Universität Berlin, Marchstr. 18, 10587 Berlin, Germany;
*Correspondence: [email protected] (B.T.); [email protected] (T.M.)
Abstract: A polygeneration system for power, heat, and refrigeration has been evaluated and
optimized using exergy-based methods. CO
2
is the working fluid. The study considered two
environmental conditions for the potential implementation of the polygeneration system: cold
(Case
cold
) and hot (Case
hot
). Aspen HYSYS
®
was used to perform steady-state simulations, Python
was used for the automation of the process, and the connection of Aspen HYSYS
®
with Python was
successfully applied for single-objective and multi-objective optimizations. A wide range of decision
variables was implemented. The minimization of the average cost of a product per unit of exergy
was the goal of single-objective optimization and was included in the multi-objective optimization in
addition to the maximization of the overall exergy efficiency. Single-objective and multi-objective
optimization were applied. Both optimization algorithms result in the necessity to increase the pinch
temperature in the heat exchanger (
∆
T
pinch,HE
), maintain the pinch temperature in the gas cooler
(
∆
T
pinch,GC
), and augment this value for the evaporator (
∆
T
pinch,EVAP
). Notably, higher isentropic
efficiency for turbomachinery correlates with improved optimization outcomes. These findings
contribute to the applicability and performance of the polygeneration system, offering potential
advancements in sustainable energy solutions.
Keywords: heat recovery; solar energy; CO2; polygeneration; exergy-based method; optimization
1. Introduction
The World Energy Outlook 2023 [
1
] (International Energy Agency, IEA) reports an
average annual growth rate of 0.7% in total energy demand (Stated Policies Scenario) and
flattening total energy demand (Pledges Scenario) because of improvements in the energy
efficiency of existing technologies and advantages in technologies powered by electricity
(mainly, electricity from renewable energy sources [
2
]) over fossil-fuel-based alternatives.
In the net-zero emissions scenario, the primary energy demand has a negative growth rate
of 1.2% per year until 2030.
Regarding CO
2
emissions, the technologies still unavailable on the market, i.e., those
at the prototype or demonstration phase, delivered nearly 50% of the emissions reductions
(net zero emissions scenario in 2050). The actual status is around 35%. Particular attention
should be given to the off-grid solutions [1].
The configuration and the application of polygeneration systems depends on the
scale [2,3]:
•
Large-scale industrial systems find a wide application in complex manufacturing pro-
cesses such as petrochemicals, textiles, and food processing, where three energetic
effects (electricity, heat (in the form of steam), and cooling) are required simultaneously.
The production of chemical substances is also possible. Polygeneration is a sustainable
solution for urban areas as district energy systems. These systems can be designed as a
Energies 2024,17, 291. https://doi.org/10.3390/en17020291 https://www.mdpi.com/journal/energies
Energies 2024,17, 291 2 of 17
synergy between the energy and non-energy sectors by centralizing energy production
and distribution.
•
Middle-scale systems are well-known for their application in hospitals, university cam-
puses, and small manufacturing plants. The number of residential, business, and
commercial buildings that use air conditioning intensively has increased dramatically
in recent years. A large amount of cold is required for servers (equipment for IT-related
technologies, etc.). This is a global problem that needs attention.
•
Small-scale systems often belong to off-grid solutions [
4
]. Such systems have high
design flexibility and the highest potential for integration of renewable sources (solar
and wind) up to 100%. This approach addresses the challenges of renewable energy
variability by providing a consistent energy supply through complementary sources.
Small-scale off-grid systems contribute positively to the net zero emissions scenario. A
considerable boost in system efficiency has been observed in polygeneration systems.
The modern term “polygeneration” can be treated as a synonym for “cogeneration“,
“tri-generation”, and “multigeneration”. Not only can energetic effects be included in the
generated products list, but freshwater due to water desalination, hydrogen from water
electrolysis, syngas from biomass gasification, etc., can also be included in the generated
products (for example, [5,6]).
There are a large number of research papers dedicated to polygeneration systems. The
Scopus database (November 2023) was used to identify these publications and cluster the
keywords. Such an approach can help readers obtain an overview of the already-published
results and highlight the directions for research. Thousands of publications were identified
using the keyword “polygeneration”; later, they were filtered using the field of research
“energy” with attention to “carbon dioxide” (as emissions and as the working fluid). Only
publications in English were considered. Finally, a list of around 500 research publications
was generated. The dynamics of the publications are shown in Figure 1a. Only papers
published during the last 10 years were further used to cluster the keywords (Figure 1b).
From Figure 1b, we can see that a large cluster corresponds to the publications ad-
dressing the combination of heat and power generation. The topic of waste heat utilization
is also well-identified, with several large clusters. However, tri-generation systems (bottom
of Figure 1a) and “cooling” (“refrigeration”) present relatively small clusters with weak
connections to others. The research methods include theoretical research (and optimiza-
tion), laboratory experiments, and pilot plants. The papers related to the optimization and
application of exergy-based methods (thermoeconomics = exergoeconomics) form clusters
in green and yellow colors; they were published during the last 3 to 4 years.
Solar-powered, as well as solar-biomass-powered or solar-geothermal-powered, poly-
generation systems have been the subject of extensive study, analysis, and optimization
because of their potential to contribute positively to the goals of sustainability, namely,
energy efficiency, economic feasibility, and reducing emissions. For example, a compre-
hensive review of perspectives on solar-driven polygeneration systems of different scales
is reported in [
7
]. The so-called 4E analysis has been applied to evaluate and optimize
a medium-scale system based on the combination of solar and geothermal resources [
8
].
The multicriteria analysis and optimization of the medium-scale polygeneration hybrid
solar-biomass system are reported in [
9
]. A novel stochastic planning model for renewable
distributed generation (DG) in distribution networks was proposed, considering investment
in conventional assets and smart grid assets like demand-side response and coordinated
voltage control. The authors also considered active power generation curtailment. The
model used a node-variable formulation to alleviate computational burden. The study
demonstrated smart technologies’ strategic value due to flexibility in system evolution [
10
].
Energies 2024,17, 291 3 of 17
Energies 2024, 17, x FOR PEER REVIEW 3 of 17
can cover all scales, from very large (for example, [12,13], where the natural-gas-based
polygeneration systems are evaluated) to medium- and small-sale. For the generation of
the refrigeration and/or heat capacity, ejector systems [14] or their combinations [6] can
be used.
(a)
(b)
Figure 1. Bibliometric study: (a) dynamics of publications (Scopus); (b) clustering the keywords
(VOSViewer software, 1.6.16).
Vapor-compression refrigeration systems dominate in stand-alone applications
driven by electricity (including small-scale systems driven by locally generated electricity
from renewables). However, compression refrigeration machines can also be driven di-
rectly by internal combustion engines or small- or medium-scale power systems based on
Figure 1. Bibliometric study: (a) dynamics of publications (Scopus); (b) clustering the keywords
(VOSViewer software, 1.6.16).
Waste heat utilization represents a large cluster in the research on polygeneration. A
review of existing technologies and their perspectives is reported in [
11
]. Such systems
can cover all scales, from very large (for example, [
12
,
13
], where the natural-gas-based
polygeneration systems are evaluated) to medium- and small-sale. For the generation of
the refrigeration and/or heat capacity, ejector systems [
14
] or their combinations [
6
] can
be used.
Vapor-compression refrigeration systems dominate in stand-alone applications driven
by electricity (including small-scale systems driven by locally generated electricity from
Energies 2024,17, 291 4 of 17
renewables). However, compression refrigeration machines can also be driven directly by
internal combustion engines or small- or medium-scale power systems based on the organic
Rankine cycle. A combination has a classification name: “thermally-driven compression
refrigeration systems”.
This paper addresses a small- and medium-scale polygeneration system with CO
2
as
the working fluid. A conceptual design is shown in Figure 2. The authors already discussed
and evaluated the application of such a system in [15,16].
Energies 2024, 17, x FOR PEER REVIEW 4 of 17
the organic Rankine cycle. A combination has a classification name: “thermally-driven
compression refrigeration systems”.
This paper addresses a small- and medium-scale polygeneration system with CO2 as
the working fluid. A conceptual design is shown in Figure 2. The authors already dis-
cussed and evaluated the application of such a system in [15,16].
Figure 2. A conceptual design of an off-grid polygeneration system.
During the last decade, CO2 as a working fluid for refrigeration, heat pump systems,
or power generation has been extensively studied. For example, the authors contributed
to this research in [17–19].
A CO2 cogeneration system, as the combination of the vapor-compression refrigera-
tion system and the recompression Brayton cycle, was evaluated in [20] using energy, ex-
ergy, and economic analyses. The parametric sensitivity analysis was conducted to dis-
cover a set of decision variables. One-criteria optimization (maximum energy efficiency,
maximum exergy efficiency, or minimum total product cost) has been conducted. For co-
generation, the optimal (minimal) cost of the products is 3.5% lower than the cost obtained
for optimal (maximum) energy and efficiency. Under the assumption that refrigeration
capacity is the sole effect, the optimal (minimum) total product cost can be decreased by
19% compared to a base case, 4% lower than the cost obtained for optimal (maximum)
energy and efficiency.
In [21], the system for utilizing the heat from exhaust gases from the shipboard gas-
turbine system is evaluated. A cogeneration system consists of the supercritical and tran-
scritical CO2 cycles. The coefficient of performance (COP) of 2.75 was calculated for the
base case and can be improved by more than 10% because of improvements in the Brayton
(supercritical) cycle.
A supercritical CO2 cycle with preheating was proposed for the waste heat recovery
from an engine [22]. The maximum net power output can be increased by around 7%. The
results demonstrate that improving the preheating process within the S-CO2 cycle can
achieve deeper utilization of the waste heat, leading to an increase in the entire system
performance by another 7%.
A novel two-stage transcritical CO2 refrigeration cycle was proposed in [23]. Two
ejectors are implemented in the vapor-compression cycle. The reported results show that
the improvement potential is between 20% and 80% (value of COP) and heavily depends
on the operation conditions. The experimental results of the transcritical CO2 systems are
reported in [24,25]. The first paper reports on an ejector refrigeration system, with partic-
ular attention to the design of the ejector. The second paper reviews experimental investi-
gations of the heat-to-upgraded-heat and heat-to-power systems (terminology used in
Figure 2. A conceptual design of an off-grid polygeneration system.
During the last decade, CO
2
as a working fluid for refrigeration, heat pump systems,
or power generation has been extensively studied. For example, the authors contributed to
this research in [17–19].
A CO
2
cogeneration system, as the combination of the vapor-compression refrigeration
system and the recompression Brayton cycle, was evaluated in [
20
] using energy, exergy,
and economic analyses. The parametric sensitivity analysis was conducted to discover a
set of decision variables. One-criteria optimization (maximum energy efficiency, maximum
exergy efficiency, or minimum total product cost) has been conducted. For cogeneration,
the optimal (minimal) cost of the products is 3.5% lower than the cost obtained for optimal
(maximum) energy and efficiency. Under the assumption that refrigeration capacity is the
sole effect, the optimal (minimum) total product cost can be decreased by 19% compared to
a base case, 4% lower than the cost obtained for optimal (maximum) energy and efficiency.
In [
21
], the system for utilizing the heat from exhaust gases from the shipboard
gas-turbine system is evaluated. A cogeneration system consists of the supercritical and
transcritical CO
2
cycles. The coefficient of performance (COP) of 2.75 was calculated for the
base case and can be improved by more than 10% because of improvements in the Brayton
(supercritical) cycle.
A supercritical CO
2
cycle with preheating was proposed for the waste heat recovery
from an engine [
22
]. The maximum net power output can be increased by around 7%.
The results demonstrate that improving the preheating process within the S-CO
2
cycle can
achieve deeper utilization of the waste heat, leading to an increase in the entire system
performance by another 7%.
A novel two-stage transcritical CO
2
refrigeration cycle was proposed in [
23
]. Two
ejectors are implemented in the vapor-compression cycle. The reported results show that
Energies 2024,17, 291 5 of 17
the improvement potential is between 20% and 80% (value of COP) and heavily depends
on the operation conditions. The experimental results of the transcritical CO
2
systems
are reported in [
24
,
25
]. The first paper reports on an ejector refrigeration system, with
particular attention to the design of the ejector. The second paper reviews experimental
investigations of the heat-to-upgraded-heat and heat-to-power systems (terminology used
in [
25
]). Particular attention is given to the heat transfer characteristics and the design of
the compressors and expanders.
A multi-objective optimization for a large-scale integrated energy system based on
energy hubs was conducted in [
26
]. The method considers economic analysis, energy con-
sumption, and environmental benefits and employs the
ε
-constraint-fruit fly optimization
algorithm. Results show that the proposed method reduces annual total cost, primary
energy consumption, and CO2emissions by approximately 2%, 5%, and 7%, respectively.
The bilayer model which utilizes historical data and conservativeness to mitigate
uncertainties for multi-energy building microgrids was presented in [
27
]. It determines the
energy storage dispatch, demand–response, and on–off hourly operation of a power gener-
ation system. Numerical case studies show the effectiveness of the evaluation approach
in achieving economically effective operations with high computational performance and
immunity against uncertainties.
The potential of heat-driven vapor-compression refrigeration systems is still not fully
understood. This study makes a substantial contribution to the continuing investigation
of heat-driven vapor-compression refrigeration systems, with a particular focus on the
off-grid operation. A gap in knowledge and application is addressed by describing the
optimization processes and associated algorithms. It explores the complexities of simulation
and automation processes. In addition, the study stands out because it explores various
optimization goals by conducting single-objective and multi-objective optimization.
This paper aims to do the following:
•
To describe the simulation and automation processes as the prerequisites for
optimization
;
•
To evaluate comprehensively the polygeneration system with CO
2
as the working
fluid to identify the decision variables;
•To describe the optimization procedures and associated algorithms;
•To conduct single and multi-objective optimization;
•
To conduct a comparative study between evaluated and reported solar-driven and
waste heat-driven polygeneration systems.
2. System Description
Figure 3a shows the basic design of the polygeneration system being evaluated, and
Figure 3b illustrates its thermodynamic cycle. The polygeneration system consists of nine
components within two sub-cycles:
•
A power sub-cycle is composed of a compressor for the power cycle (CM–P), a heat
exchanger (HE), and an expander (EX). The “driving energy” (for example, solar
energy, heat from biomass combustion, waste heat, etc.) for the entire system is
supplied to the HE.
•
A refrigeration sub-cycle includes a throttling valve (TV), an evaporator (EVAP), and a
compressor for the refrigeration cycle (CM–R). The refrigeration capacity is generated
within the EVAP.
A gas cooler (GC), a mixer (MIX), and a splitter (SPLIT) link these sub-cycles. Carbon
dioxide is the working fluid for the entire system. The heat capacity is generated within the
gas cooler. The system can generate net power as the difference between the power from
EX and the power required by both compressors (CM–P and CM–R).
Energies 2024,17, 291 6 of 17
Energies 2024, 17, x FOR PEER REVIEW 6 of 17
(a)
(b)
Figure 3. CO
2
polygeneration system: (a) schematic; (b) thermodynamic cycle.
3. System Simulation and Automation
Aspen HYSYS
®
(ver. 10, AspenTech, Bedford, MA, USA) has been used as a simula-
tion tool. To apply mass and energy balances, the Span–Wagner equations of state were
selected. These equations are more accurate in predicting CO
2
thermodynamic properties
in a wide range of temperatures and pressures, including the vicinity of the critical point.
The simulation results obtained from Aspen HYSYS
®
were exported for computing
the analysis of system and component levels, evaluation, and optimization. With the aid
of programming software, the parameters and values being exported from the simulation
software and imported into the simulation software can be automated. Python was con-
nected to Aspen HYSYS
®
to automate the entire calculation process. The communication
was managed through a binary interface component object model. Automation is advan-
tageous, particularly for optimization.
The initially developed design needs to be evaluated. For this purpose, energetic and
exergy-based analyses were conducted using a large set of simulations. The economic
analysis has been conducted separately. The authors report this process in detail in [15].
Thermodynamic performance, cost of the system product(s), environmental aspects,
safety, and reliability were considered. The obtained results are used for this research.
Figure 3. CO2polygeneration system: (a) schematic; (b) thermodynamic cycle.
3. System Simulation and Automation
Aspen HYSYS
®
(ver. 10, AspenTech, Bedford, MA, USA) has been used as a simulation
tool. To apply mass and energy balances, the Span–Wagner equations of state were selected.
These equations are more accurate in predicting CO
2
thermodynamic properties in a wide
range of temperatures and pressures, including the vicinity of the critical point.
The simulation results obtained from Aspen HYSYS
®
were exported for computing
the analysis of system and component levels, evaluation, and optimization. With the aid of
programming software, the parameters and values being exported from the simulation soft-
ware and imported into the simulation software can be automated. Python was connected
to Aspen HYSYS
®
to automate the entire calculation process. The communication was
managed through a binary interface component object model. Automation is advantageous,
particularly for optimization.
The initially developed design needs to be evaluated. For this purpose, energetic and
exergy-based analyses were conducted using a large set of simulations. The economic
analysis has been conducted separately. The authors report this process in detail in [
15
].
Thermodynamic performance, cost of the system product(s), environmental aspects, safety,
and reliability were considered. The obtained results are used for this research.
4. System Evaluation
In this study, evaluation and optimization are conducted at the system level. A
component-level evaluation was already reported by the authors in [
15
,
16
]. Only physical
exergy is considered for the exergy analysis. Chemical, kinetic, and potential exergy are
Energies 2024,17, 291 7 of 17
neglected. The approach “exergy of fuel/exergy of the product” is applied to conduct the
exergy and exergoeconomic analysis.
The exergy balance for the overall system is written as
.
EF,tot =.
EP,tot +.
ED,tot +.
EL,tot, (1)
and the overall exergy efficiency is
εtot =
.
EP,tot
.
EF,tot
. (2)
where for the system being evaluated (Figure 3), the exergy of fuel is
.
EF,tot =1−T0
Tav
HS
; the ex-
ergy of product is
.
EP,tot =.
Wnet +.
EHeating +.
ECooling
with
.
Wnet =.
WEX −.
WCM−P+.
WCM−R
;
and
.
EHeating =.
E12 −
.
E11
and
.
ECooling =.
E14 −
.
E13
. Here,
Tav
HS
is the average temperature of the
heat source supplied to the system in the HE.
The cost balance for the overall system is
.
QHE1−T0
Tav
HS cHS +.
Ztot =.
Wnet +.
EHeating +.
ECoolingcav
P. (3)
For calculating the capital investment rate
.
Ztot =∑
k=9
.
Zk
the total revenue requirement
method (the application can be found in many papers, for example, in [
17
]) of the economic
analysis was applied with the following assumptions: (a) the plant’s economic lifetime
is
20 years
; (b) the effective interest rate is 10%; and (c) the average general inflation rate
is 2.5%.
For estimating the purchased equipment cost (PEC) for each component, the following
was considered (detailed explanations are reported by authors in detail [15]):
•
Heater (HE) and gas cooler (GC) are compact printed circuit heat exchangers made
of stainless steel. Therefore, the cost of such a heat exchanger is estimated by its
weight [28]:
PECHE
GC
=Costperunitmass ∗V∗ρ, (4)
with
V= .
QHE or .
QGC
U∗TLMTD∗typicalareaperunitvolume !*fm
, where
Costperunitmass =
50 USD/kg;
ρ=7800 kg/m3;U= 500 W/(m2K) [TM]; and material factor fm= 0.564 m3/m3[29]
•Evaporator (EVAP) [30]:
PECE=CB*(X/XB)MfP, (5)
where CB= 32,800 USD; XB= 80 m2;M= 0.68, and fP=1.3.
•Turbomachinery of the power sub-cycle (CM-P and EX) [29]:
PECCM −P
EX
=.
W
285
TIP1.7 +0.6+TIP−0.3 +3.35 +TIT
10007.8
. (6)
•Turbomachinery of the refrigeration sub-cycle (CM_R) [30]:
PECE=CB*(X/XB)MfP, (7)
where CB= 32,800 USD; XB= 80 m2;M= 0.68, and fp=1.3.
Energies 2024,17, 291 8 of 17
•
The cost of the throttling valve (TV) equals
PECTV =
100 EUR [
17
], and the costs of
the mixer and the splitter are neglected.
5. System Optimization
Parameter optimization and structure optimization are two methods for improving
the performance of the overall system. Product (single or cumulative for a multigeneration
system) cost minimization is one of the most commonly applied objective functions for com-
mercial systems, while energy (exergy) efficiency maximization is associated with thermo-
dynamic optimizations, i.e., minimization of the primary energy for the energy-conversion
system (and associated emissions). A set of decision variables and their constraints were
discussed by the authors in [
15
]. In this study, two types of parameter optimization have
been applied: single-objective and multi-objective.
To solve the design optimization problems, stochastic algorithms—individual-based
and population-based—are selected. The population-based algorithms widely applied in
engineering are evolutionary, physical, and swarm-based algorithms. Genetic algorithms
are the earliest and most well-known evolutionary algorithms.
Two relatively new algorithms (Figure 4) were selected and applied to solve the
single-optimization problem:
•
The differential evolution (DE) algorithm was introduced in 1997 [
31
]. The DE al-
gorithm is similar to the genetic algorithm but modified for more straightforward
implementation, less computation time, reliability, and robustness.
•
The particle swarm optimization (PSO) algorithm was first suggested in 1995 [
32
].
The potential candidates in this algorithm are called particles. A group of particles
work together to continuously improve their individual and collective performance
on a given optimization task [
33
]. In addition, less computational effort is required for
solving moderate-dimensional problems. It is also robust and straightforward. PSO is
an excellent option for solving high-dimensional optimization problems.
Energies 2024, 17, x FOR PEER REVIEW 8 of 17
5. System Optimization
Parameter optimization and structure optimization are two methods for improving
the performance of the overall system. Product (single or cumulative for a multigeneration
system) cost minimization is one of the most commonly applied objective functions for
commercial systems, while energy (exergy) efficiency maximization is associated with
thermodynamic optimizations, i.e., minimization of the primary energy for the energy-
conversion system (and associated emissions). A set of decision variables and their con-
straints were discussed by the authors in [15]. In this study, two types of parameter opti-
mization have been applied: single-objective and multi-objective.
To solve the design optimization problems, stochastic algorithms—individual-based
and population-based—are selected. The population-based algorithms widely applied in
engineering are evolutionary, physical, and swarm-based algorithms. Genetic algorithms
are the earliest and most well-known evolutionary algorithms.
Two relatively new algorithms (Figure 4) were selected and applied to solve the sin-
gle-optimization problem:
• The differential evolution (DE) algorithm was introduced in 1997 [31]. The DE algo-
rithm is similar to the genetic algorithm but modified for more straightforward im-
plementation, less computation time, reliability, and robustness.
• The particle swarm optimization (PSO) algorithm was first suggested in 1995 [32].
The potential candidates in this algorithm are called particles. A group of particles
work together to continuously improve their individual and collective performance
on a given optimization task [33]. In addition, less computational effort is required
for solving moderate-dimensional problems. It is also robust and straightforward.
PSO is an excellent option for solving high-dimensional optimization problems.
(a) (b)
Figure 4. Implementation of optimization algorithms: (a) differential evolution-DE; (b) particle
swarm optimization–PSO.
For multi-objective optimization problems, the non-dominated sorting genetic algo-
rithm-II (NSGA-II) is chosen [31] as one of the most popular and successful optimization
tools. The NSGA-II can produce a Pareto frontier, or non-dominated set of solutions,
Figure 4. Implementation of optimization algorithms: (a) differential evolution-DE; (b) particle
swarm optimization–PSO.
Energies 2024,17, 291 9 of 17
For multi-objective optimization problems, the non-dominated sorting genetic algorithm-
II (NSGA-II) is chosen [
31
] as one of the most popular and successful optimization tools.
The NSGA-II can produce a Pareto frontier, or non-dominated set of solutions, which shows
the trade-off between the two objective functions without any solution being superior to
the others.
Optimization problems consist of three essential components:
•Objective functions
#Single-objective parametric optimization:
min cav
p,tot.
This involves the global optimization of decision parameters.
#Multi-objective parametric optimization:
min cav
p,totandmax εtot.
•Set of decision variables:
∆Tpinch,HE
ηEX
∆Tpinch,GC
ηCM−P
∆Tpinch,EVAP
ηCM−P
pMerging
TIPmax
PRmin
Tin,CM−p,min
,
where
∆
T
pinch
represents the temperature differential (pinch point) in a heat exchanger;
and ηrepresents the isentropic efficiency of turbomachinery.
•Constraints of decision variables:
5◦C≤∆Tpinch,HE ≤40 ◦C
70% ≤ηEX ≤98%
1◦C≤∆Tpinch,GC ≤10 ◦C
70% ≤ηCM−P≤95%
1◦C≤∆Tpinch,EVAP ≤10 ◦C
70% ≤ηCM−P≤95%
75 bar ≤pMerging ≤90 bar
TIPmax =250 bar
PRmin =1.5
Tin,CM−p,min =32 ◦C
.
Two operation conditions were considered to represent the performance of the pro-
posed polygeneration system in hot and cold climates, as follows:
•
Case
hot
for the hot climate operation with an average environmental temperature of
35 ◦C;
•
Case
cold
for the cold climate operation with an average environmental temperature of
5◦C.
A medium-temperature heat source suitable for the evaluated polygeneration systems
is in the range of 320–590 ◦C [15].
Energies 2024,17, 291 10 of 17
6. Results and Discussion
6.1. Single-Objective Parametric Optimization
The results of the single-objective optimization are shown in Table 1(Case
hot
) and
Table 2
(Case
cold
). The comparison of the values obtained using both DE and PSO algo-
rithms is given as well.
Table 1. A single-objective parametric optimization utilizing DE and PSO algorithms for the polygen-
eration system at Casehot operation conditions.
Parameter Description Unit Initial
Value
Optimal Values Error
(%)
DE PSO
∆Tpinch,HE The pinch point
temperature in the HE K 20 33 26 21.2
∆Tpinch,GC The pinch point
temperature in the GC K 5 5 6 20.0
∆
T
pinch,EVAP The pinch point
temperature in the EVAP. K 5 8 9 12.5
η,EX Isentropic efficiency of EX % 90 98 98 0.0
η,CM–P Isentropic efficiency of
the CM-P % 85 94 95 1.1
η,CM–R Isentropic efficiency of
the CM-R % 85 95 95 0.0
pMerging Merging pressure, p3bar 77 80 82 2.5
PREX Pressure ration in EX - 2.6 3.1 3.0 2.7
opt cav
p,tot
Optimum average product
cost per unit of exergy USD/GJex 53.26 39.60 39.56 0.1
Execution
time time s - 1820 1633 10.3
After the optimization for Case
hot
, the average cost of a product decreased by 25% from
its original value, and the optimum solutions between DE and PSO are within the limit of
1.0%. The differences between DE and PSO optimum solutions are in the range of 2.5% for
isentropic efficiency of turbomachinery and p
Merging
. The optimal minimum temperature in
all heat exchangers,
∆
T
pinch
, has relatively large (in the limit of 25%) differences between
the initial and optimal values obtained by the DE and PSO algorithms. However, relative
values should not be the only ones to be considered. The change in the absolute values of
∆Tpinch by 1 K shows that the obtained results are within the limits of technical possibility
for improving a heat exchanger while keeping the same design.
The optimization results for Case
cold
show a decrease in the average cost of a product
by 15% from its original value, and optimum solutions between DE and PSO are at the
limit of 1.0%.
When comparing the optimization results using DE and PSO algorithms, the largest
relative difference of 60% is observed in the pinch point temperature difference for the EVAP.
Starting with the initial value of
∆
T
pinch,EVAP
= 5 K, the DE algorithm suggests increasing to
10 K, while the PSO algorithm suggests decreasing down to 4 K. Similar dynamics have
also been suggested for the value of
∆
T
pinch,GC
: a smaller value (6 K) suggested by the PSO
algorithm compared to 7 K by the DE algorithm. All values obtained by optimization are
within the limits of technical possibility for heat exchangers and turbomachinery.
The execution time for Case
hot
varied between 1820 s (DE) and 1633 s (PSO); while for
Case
cold
, it varied between 1944 s (DE) and 1699 s (PSO). The PSO algorithm completes the
optimization process roughly 200 s faster.
Energies 2024,17, 291 11 of 17
Table 2. A single-objective parametric optimization utilizing DE and PSO algorithms for the polygen-
eration system at Casecold operation conditions.
Parameter Description Unit Initial
Value
Optimal Values Error
(%)
DE PSO
∆Tpinch,HE The pinch point
temperature in the HE K 20 29 29 0.0
∆Tpinch,GC The pinch point
temperature in the GC K 5 7 6 14.3
∆
T
pinch,EVAP The pinch point
temperature in the EVAP. K 5 10 4 60.0
η,EX Isentropic efficiency of EX % 90 97 98 1.0
η,CM–P Isentropic efficiency of
the CM-P % 85 91 91 0.0
η,CM–R Isentropic efficiency of
the CM-R % 85 91 92 1.1
pMerging Merging pressure, p3 bar 77 87 88 1.1
PREX Pressure ration in EX - 2.6 2.9 2.8 1.9
opt cav
p,tot
Optimum average product
cost per unit of exergy USD/GJex 31.39 26.76 27.03 1.0
Execution
time Time s - 1944 1699 12.6
For example, the best values of decision variables obtained from both optimization
algorithms for Case
cold
are shown in Figure 5. The PSO algorithm already had a better
starting position in the first iteration and continued to lower the objective function until the
fifth iteration. A local minimum was identified within the sixth to ninth iterations, which
had further minor corrections until the tenth iteration. No better value is obtained after
the 10th iteration. Another dynamic is observed for the application of the DE algorithm.
Significant progress was achieved in the second iteration; a local minimum was identified
within the eighth to eighth iterations, with minor corrections within the fourteenth to
fifteenth iterations.
Energies 2024, 17, x FOR PEER REVIEW 12 of 17
Figure 5. The optimization process using the DE and PSO algorithms (Casecold).
6.2. Multi-Objective Parametric Optimization
Figure 6 shows the Pareto frontier for the entire polygeneration system at warm and
cold operation conditions. As expected, an absolute optimal solution does not exist, indi-
cating that the higher the system exergy efficiency, the lower the product cost. For the
Case
hot
, an extensive flat profile is observed, which indicates a “slow” growth in the aver-
age product cost while improving the system efficiency. Such a flat area in the Case
cols
is
minimal.
In the limit of decision variables, for Case
hot
operation conditions, maximum and
minimum overall exergy efficiencies are 0.480 and 0.414, respectively, with the trade-off
of the average cost of the product per unit of exergy 0.183 USD/kWhex and 0.141
USD/kWhex. A relative difference between the limiting values is an increase in the exergy
efficiency of 16%, which corresponds to a decrease in the
average product cost of
23%.
The optimum values are 𝑜𝑝𝑡 𝜀= 0.443 and opt 𝑐,
= 0.143 USD/kWhex.
For Case
cold
operation conditions,
𝑜𝑝𝑡 𝜀= 0.549 and 𝑜𝑝𝑡 𝑐,
= 0.096
USD/kWhex. Maximum and minimum overall exergy efficiencies are 0.607 and 0.528, i.e.,
a relative difference of 15%. In contrast, the minimum and maximum values of the average
cost of the product per unit of exergy are 0.095 USD/kWhex and 0.127 USD/kWhex, respec-
tively, which is 25%.
Figure 5. The optimization process using the DE and PSO algorithms (Casecold).
Energies 2024,17, 291 12 of 17
6.2. Multi-Objective Parametric Optimization
Figure 6shows the Pareto frontier for the entire polygeneration system at warm
and cold operation conditions. As expected, an absolute optimal solution does not exist,
indicating that the higher the system exergy efficiency, the lower the product cost. For
the Case
hot
, an extensive flat profile is observed, which indicates a “slow” growth in the
average product cost while improving the system efficiency. Such a flat area in the Case
cols
is minimal.
Energies 2024, 17, x FOR PEER REVIEW 13 of 17
(a)
(b)
Figure 6. The Pareto frontiers for the polygeneration system: (a) Case
hot,
; (b) Casecold.
6.3. Comparative Analysis with Solar-Driven Polygeneration Systems
A set of six papers [25,33–37] was selected to compare the results obtained by the
authors. In these papers, the exergy study, economic evaluation, and exergoeconomic var-
iables were used for the optimization. Table 3 presents the comparative insights, which
Figure 6. The Pareto frontiers for the polygeneration system: (a) Casehot,; (b) Casecold.
Energies 2024,17, 291 13 of 17
In the limit of decision variables, for Case
hot
operation conditions, maximum and
minimum overall exergy efficiencies are 0.480 and 0.414, respectively, with the trade-off of
the average cost of the product per unit of exergy 0.183 USD/kW
hex
and 0.141 USD/kW
hex
.
A relative difference between the limiting values is an increase in the exergy efficiency of
16%, which corresponds to a decrease in the average product cost of 23%. The optimum
values are opt εtot =0.443 and opt cav
p,tot = 0.143 USD/kWhex.
For Case
cold
operation conditions,
optεtot =
0.549 and
opt cav
p,tot
= 0.096 USD/kW
hex
.
Maximum and minimum overall exergy efficiencies are 0.607 and 0.528, i.e., a relative
difference of 15%. In contrast, the minimum and maximum values of the average cost of
the product per unit of exergy are 0.095 USD/kW
hex
and 0.127 USD/kW
hex
, respectively,
which is 25%.
6.3. Comparative Analysis with Solar-Driven Polygeneration Systems
A set of six papers [
25
,
33
–
37
] was selected to compare the results obtained by the
authors. In these papers, the exergy study, economic evaluation, and exergoeconomic
variables were used for the optimization. Table 3presents the comparative insights, which
include net power production (
.
Wnet
,
kW
), overall exergy efficiency (
εtot
), and the mean
total cost of the product (
cav
p,tot
, USD/GJ
ex
). The compared systems are classified as middle-
scale off-grid polygeneration systems, with capacities between 50 and 370 kW, which is
what this comparative analysis is all about. The differences in the reported values are
caused by evaluated system configurations, selected working fluids, particular operating
conditions, and the evaluation and optimization routines in each study. Nevertheless, the
authors’ results fit the entire map of the results and contribute to the state-of-the-art.
Table 3. Key system variables of the reported solar-driven polygeneration systems and current work.
Ref. System
Specification
Results
Specification
.
Wnet (kW) εtot (−)
cav
p,tot
(USUSD/
GJex)
[33] solar-driven evaluation 145.8 0.215 77.31
[25]solar-biomass-
driven optimization 370.1 0.200 176.73
[34]solar-driven with
thermal storage evaluation 228.9 0.282 NA
[35]solar-geothermal-
gas-driven evaluation 330.4 0.267 NA
[36] desalination evaluation NA 0.249 20.97
[37] solar-driven evaluation 48.3 0.411 61.2
this work, Casehot single-objective
optimization 205.0 0.533 31.20
this work, Casecold single-objective
optimization 205.0 0.314 46.80
7. Conclusions
The emphasis of this research was placed on the evaluation and optimization of the
off-grid polygeneration system with CO
2
as the working fluid. Exergy-based methods
were applied on the system level. The simulation of the system was conducted using
Aspen HYSYS
®
, and to ensure a comprehensive assessment and optimization process, an
automated procedure was established to connect Aspen HYSYS®with Python.
The goal of the single-objective optimization was to minimize the average cost per
unit of exergy of a product. This objective transitioned into multi-objective optimization,
where the emphasis extended beyond cost considerations to maximization of overall
exergy efficiency.
This comprehensive approach contributes to the scientific understanding of polygen-
eration systems and lays the groundwork for informed decision-making in the pursuit of
sustainable and efficient energy solutions.
Energies 2024,17, 291 14 of 17
The main obtained results can be summarized as follows:
•
The polygeneration system demonstrates sustainability benefits by simultaneously
producing multiple energy effects, i.e., power, heat, and refrigeration capacities.
•
The polygeneration system exhibits adaptability for a wide range of thermally driven
applications, from solar-thermal to waste heat utilization.
•
Parametric optimization methods, encompassing single- and multi-objective optimiza-
tion, were applied to optimize the polygeneration system. Two operation conditions,
Casehot and Casecold, were selected for thorough evaluation and optimization.
•
Single-objective optimization was completed using alternative optimization algo-
rithms: DE (differential evolution) and PSO (particle swarm optimization). The
implementation and impact of these algorithms on results are reported and discussed.
•
Both optimization algorithms performed effectively, consistently indicating the need
to increase
∆
T
pinch,HE
, maintain
∆
T
pinch,GC
, and augment
∆
T
pinch,EVAP
. Higher values
for the isentropic efficiency of turbomachines correlated with improved optimization
outcomes, with PR
EX
around 3. The optimal value of p
Merging
slightly increased,
remaining significantly distant from the critical point of CO2.
•
The PSO algorithm demonstrated a shorter optimization process duration compared
to the DE algorithm.
The following study directions can be recommended for future evaluation of the
polygeneration systems:
•
Structure optimization of the polygeneration system including a polytechnological
approach with many smart technologies within optimization [38];
•The environmental assessment of the polygeneration system;
•Implementing the polygeneration systems to the local electrical grids;
•
The potential of mixtures as working fluids (including the CO
2
-based mixtures) to
enhance system performance by adjusting their type and application scenarios.
Author Contributions: J.L.: Software; Investigation. B.T.: Methodology; Investigation; Writing the
original draft. T.M.: Reviewing and Editing; Supervision. All authors have read and agreed to the
published version of the manuscript.
Funding: This work was conducted during a summer visit supported by Jordan University of Science
and Technology (JUST) at the Technical University in Berlin (TUB) with grant number: 20230183.
Data Availability Statement: Data are contained within the article.
Conflicts of Interest: The authors declare no conflicts of interest.
Nomenclature
Aheat transfer area, m2
cspecific cost (per unit of exergy), USD/GJ
.
Z,.
Ccost rate, USD/h
COP coefficient of performance, -
.
Eexergy rate, kW
ffactor, -
Mexponent, -
Ttemperature, ◦C or K
TIT temperature at the inlet of expander, ◦C or K
PEC purchased equipment cost, USD
PR pressure ratio, -
Xcharacteristic of equipment (m2or kW)
Vvolume, m3
.
Wpower, kW
εexergy efficiency, -
ηisentropic efficiency,-
Energies 2024,17, 291 15 of 17
Subscripts and superscripts
av average
Cooling cold production
Dexergy destruction
Heating heat production
ex per unit of exergy
Ffuel
kthe serial number of components
Lexergy losses
mrelated to material
Merging merging pressure of the power and refrigeration cycles
net power as the product of the overall system
Pexergy product
prelated to pressure
0 reference state
pinch pinch point
tot overall system
Abbreviations
CM–P compressor in power sub-cycle
CM–R compressor in refrigeration sub-cycle
CO2carbon dioxide
DE differential evolution
EVAP evaporator
EX expander
GC gas cooler
HE heat exchanger
MIX mixer
ORC Organic Rankine cycle
PSO particle swarm optimization
SPLIT splitter
TV throttling valve
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