# Thermoeconomic Coherence: A Methodology for the Analysis and Optimisation of Thermal Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background and Conventional Approaches

#### 2.1. General Case: Optimization without Constraints

_{1}, …, x

_{m}).

_{i}may be divided into two categories: the first n variables (i from 1 to n) represent the design parameters of the thermal system, while the last m − n (i from n+1 to m) represent the economic frame. The design parameters could be any collection of variables that define univocally the system. As mentioned above, they also determine the investment required in each component:

_{j}→ f (x

_{1}, …, x

_{n}).

_{i}in Equations (4) and (5) may be substituted by the acquisition costs. Finally, if the number of components is lower than the number of design parameters, k variables may represent the investment and the other n − k are maintained as design parameters.

_{2}or other pollutant emissions) given a specific economic frame. If the thermal system is not affected by any constraint, there are n independent variables. Figure 1 depicts the topology of the problem in an example with two independent variables, assuming that the optimum exists.

#### 2.1.1. Minimization of the Generation Cost

^{*}) exists, at that point the partial derivatives regarding any variable are null:

#### 2.1.2. Maximization of the Yearly Cash Flow

#### 2.2. Optimization Subjected to Constraints

_{1}, …, x

_{r}, …, x

_{n}) = R

_{0}.

#### 2.2.1. Conventional Approach

_{r}can be solved from Equation (13), the differential form of the objective function subjected to the restriction R yields:

_{R}M

_{i}) are equal to the minimum cost in that domain. It is important to note that x

_{r}can be a design parameter but also a combination of them, so it represents any direction for varying the design. It can also be noted that, since all the restricted marginal costs are equal, the result of the externally constrained optimization is congruent with the internally constrained one, in which the constraint is intrinsically defined and, as result, all the marginal costs, internally constrained, are equal.

_{R}P

_{i}becomes zero. Therefore, this approach is not suitable with such a constraint, which may be a usual restriction if the power rate is fixed. Another singularity arises when the restriction makes the exploitation costs be constant. In such a case, constrained marginal costs are always zero.

#### 2.2.2. Lagrange Multipliers

_{i}yields:

_{i}) at the optimum are not equal but all the corrected marginal costs (

_{C}M

_{i}) equal the generation cost. Additionally, the formulation is suitable even at constrained yearly production, and only is undetermined if P

_{i}is null, which only might take place locally (in such case, equal marginal costs are ensured in the nearby of that singular point by Equations (19) and (20)) because otherwise this design parameter is not useful. It is important to note that the corrected marginal cost is not a marginal cost calculated through a specific and allowed direction, but a modification of the marginal cost following the path of the i-th design parameter. This allows

_{C}M

_{i}to be defined even at constant production. Moreover, taking into account that R

_{i}= P

_{i}, Equation (19) gives additional information and ensures that all the unconstrained marginal costs are equal in all directions, like in the case of the unconstrained optimum, although they are different from the generation cost.

_{i}= E

_{i}, then Equation (19) ensures that unconstrained marginal costs are again equal in the optimum. In both cases, the design is quite coherent since it ensures minimum exploitation costs given a yearly production or maximum production given a specific exploitation cost.

## 3. Proposed Methodology: Corrected Standardised Marginal Costs and Divergence from the Coherent Design

#### 3.1. Equivalent Standardised Marginal Costs

#### 3.2. Corrected Standardised Marginal Costs

_{r}):

_{R}∇K) in the allowed span of directions (because there is room for improvement within that constraint) while the other component is a linear combination of the constraint gradients:

_{R}∇K and each ∇R

_{r}should be zero because they are perpendicular, so the components of

_{R}∇K and the value of the λ

_{r}may be calculated with the following system:

_{R}K

_{i}is zero and this equation is equal to (21). In that case all the equivalent marginal costs are equal to the generation cost. Out of the optimum, equivalent marginal costs are defined thanks to Equation (23) and are not equal due to the existence of

_{R}∇K.

#### 3.3. Divergence from the Coherent Design

- It should be congruent for all the possible sets of design parameters of a system (independence of the coordinate system);
- It should allow the comparison of the coherence of different facilities with independence of the objective function, its value in the optimum and its number of degrees of freedom;
- The divergence should be zero at the optimum.

_{C}M

_{i}are the corrected marginal costs described in Section 3.2. Thus, the sum of d

_{i}is the unit:

_{i}) and the optimum one (all normalized marginal costs equal to 1/n):

_{i}equal 1/n, and increases as one or several d

_{i}increase or decrease getting different from the others. Additionally, it ensures the homogeneity of the values for different systems with different amount of degrees of freedom n (a Cartesian distance does not achieve this homogeneity). Some examples of this homogeneity are shown in the appendix.

_{i}.

## 4. Application Example: Coherence in a Solar Gas Turbine

- The pressure ratio (intensive parameter) is replaced by the ratio of the compressor power to the thermal power supplied to the system (${\dot{W}}_{C}/\dot{Q}$).
- The effectiveness (intensive parameter) of the heater and cooler are replaced by the ratio of their irreversibility to the thermal power supplied to the system ($\dot{I}/\dot{Q}$).
- The maximum temperature of the solar field (intensive parameter) is replaced by the ratio of the exergy content of the thermal power to the supplied thermal power (${\dot{E}}_{Q}/\dot{Q}$).
- The mass flow (the extensive parameter) is replaced by the thermal power supplied to the system ($\dot{Q}$).

_{j,standardized}) instead the original ones (f

_{i,original}), the Jacobian matrix that contains the derivatives of these standardized parameters (y

_{j}) regarding the original ones (x

_{i}) should be assessed. The inverse of this Jacobian matrix gives the derivatives of the original parameters regarding the standardized ones, and any other derivative may be calculated as below:

_{i}). The divergence is not zero in any case because none of the design is the optimum in the unconstrained domain. However, it is close to zero in the cases of constant investment (constant exploitation cost) and constant production because all marginal costs are equal, although different from the generation cost. In those cases, the divergence measures the strength of the restriction. The value of the divergences gets higher in the base case and in the cases of constant production and constant turbine exhaust temperature or turbine inlet temperature. In the base case, it happens because the design is not an optimum. In the other cases, the marginal costs used are the unconstrained ones, so the value of the divergence measures the strength of the restriction.

## 5. Extension of the Methodology to Robustness and Uncertainty Analysis and Combined Heat and Power

#### 5.1. Robustness and Uncertainty: Optimization in Economy of Scales

_{i}that either are not deterministic or their influence on the functions have some uncertainty. With such premises, results like the generation cost (K), the cash flow (CF), the exploitation cost (E) or the yearly production (P) are of probabilistic nature and they follow certain density function. Whatever the density function, it is possible to define a variability of the results through the variance, σ, and a selected confidence level for the result. For example, given a set of variables x

_{i}it is possible to obtain a result, for example of the generation cost, which ensures with a confidence level of 95% that the actual cost is below that value. Therefore, it is possible to represent the curve of the generation cost with a confidence level of 95%.

_{95}= K +A·σ,

_{95}:

_{95}= FC − A·σ, Equation (33) becomes:

_{i}and taking into account the definition of marginal cost:

_{P,i}, the marginal costs are equal although lower than those obtained in a deterministic optimization. Thus, the coherent design is reached at the conservative side of sizing (and the contrary if variances decrease). The level of information or the strength of the restriction given by the probabilistic approach may be again calculated by comparing the thermal coherence using the corrected and unconstrained marginal costs, taking the uncertainty as a constraint-like condition.

#### 5.2. Combined Heat and Power

_{1}, and heat, P

_{2}), the most simple case is that in which both selling prices (V

_{1}, V

_{2}) are known. In this case, the thermoeconomic objective is to maximise the yearly profit or cash flow of the power plant:

_{2}) is known and the cost of the other (K

_{1}) is calculated. In such case, the minimum price at which the product should be sold is:

_{1}) in the new system (Equation (42) replacing V

_{2}by the cost of the main product, K

_{2}), and this incremental cost can be minimised. In this case, the expression is equal to Equation (44) changing V

_{2}by the cost of the main product, K

_{2}.

_{2}) or both (V

_{1}, V

_{2}) selling prices are function on the respective productions, V

_{1}(P

_{1}) and V

_{2}(P

_{2}), and Equations (44) and (47) become:

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Acronyms | |

CGAM | CHP problem defined in [11] |

CHP | Combined heat and power |

LCOE | Levelized cost of energy |

Mu | Monetary units |

O&M | Operation & maintenance cost |

TADEUS | Thermoeconomic Approach to the Diagnosis of Energy Utility System Malfunctions [12] |

Symbols | |

A | Amortisation cost (monetary units, mu); Constant of the probability distributions |

C | Acquisition cost (mu) |

CF | Cash flow (mu) |

d | Normalized marginal cost (-) |

D | Divergence (-) |

E | Exploitation cost (mu) |

${\dot{E}}_{Q}$ | Exergy content of the thermal power (W) |

f | generic function |

F | Fuel cost (mu) |

$\dot{I}$ | Irreversibility (W) |

I_{total} | Total investment (mu) |

K | Generation cost (mu·J^{−1}) |

m | safety coefficient, number of variables including the economic frame |

$\dot{m}$ | Mass flow rate (kg·s^{−1}) |

M | Marginal cost (mu·J^{−1}) |

n | Number of degrees of freedom |

P | Yearly production of the plant (J) |

$\dot{Q}$ | Thermal power rate at the heat source (W) |

R | Restriction |

T_{it} | Turbine inlet temperature (K) |

T_{exh} | Turbine exhaust temperature (K) |

T_{max} | Maximum temperature of the solar field (K) |

UA | Product of the overall heat transfer coefficient and the heat exchange area (W·K^{−1}) |

V | Selling price of the product (mu·J^{−1}) |

${\dot{W}}_{C}$ | Compressor power (W) |

x | Original design parameters |

y | Standardized variables |

Greek letters | |

Δ | increment |

ε | Heat exchanger effectiveness (-) |

η | thermal efficiency (-) |

Λ | Lagrange multiplier |

Π | pressure ratio (-) |

Σ | Variance |

Subscripts | |

cool | Cooler |

comp | Compressor |

GT | Gas turbine |

heat | Heater |

solar | Solar field |

turb | Turbine |

## Appendix

_{0}and the other half a value of y times d

_{0}. The divergences in both cases are:

_{i}is 1/n (Equation (26)):

## References

- Bejan, A.; Tsatsaronis, G.; Moran, M. Thermal design and optimization, 1st ed.; John Wiley & sons: Hoboken, NJ, USA, 1996. [Google Scholar]
- Attala, L.; Facchini, B.; Ferrara, G. Thermoeconomic Optimization Method as Design Tool in Gas-Steam Combined Plant Realization. Energy Convers. Manag.
**2001**, 42, 2163–2172. [Google Scholar] [CrossRef] - Valdés, M.; Durán, M.D.; Rovira, A. Thermoeconomic optimization of combined cycle gas turbine using genetic algorithms. Appl. Therm. Eng.
**2003**, 23, 2169–2182. [Google Scholar] [CrossRef] - International Renewable Energy Agency. Available online: http://www.irena.org/documentdownloads/publications/re_technologies_cost_analysis-csp.pdf (assessed on 1 July 2016).
- Energy Information Administration. Available online: https://www.eia.gov/forecasts/aeo/pdf/electricity_generation.pdf (assessed on 1 July 2016).
- Evans, R.B. A Contribution to the Theory of Thermoeconomics. Master’s Thesis, University of California, Los Angeles, CA, USA, 1961. [Google Scholar]
- El-Sayed, Y.M.; Evans, R.B. Thermoeconomics and the Design of Heat Systems. Trans. ASME J. Eng. Power
**1970**, 92, 27–34. [Google Scholar] [CrossRef] - Tsatsaronis, G.; Winhold, M. Exergoeconomic Analysis and Evaluation of Energy Conversion Plants—I. A New General Methodology. Energy
**1985**, 10, 69–80. [Google Scholar] [CrossRef] - Frangopoulos, C.A. Thermo-Economic Functional Analysis and Optimization. Energy
**1987**, 12, 563–571. [Google Scholar] [CrossRef] - Lozano, M.A.; Valero, A. Theory of the Exergetic Cost. Energy
**1993**, 18, 39–60. [Google Scholar] [CrossRef] - Valero, A.; Lozano, M.A.; Serra, L.; Tsatsaronis, G.; Pisa, J.; Frangopoulos, C.A.; von Spakovsky, M. CGAM Problem: Definition and Conventional Solution. Energy
**1994**, 19, 279–286. [Google Scholar] [CrossRef] - Valero, A.; Correas, L.; Zaleta, A.; Lazzaretto, A.; Verda, V.; Reini, M.; Rangel, V. On the thermoeconomic approach to the diagnosis of energy system malfunctions—Part 1: the TADEUS problem. Energy
**2004**, 29, 1875–1887. [Google Scholar] [CrossRef] - Petrakopoulou, F.; Tsatsaronis, G.; Morosuk, T.; Carassai, A. Advanced Exergoeconomic Analysis Applied to a Complex Energy Conversion System. J. Eng. Gas Turbines Power
**2012**, 134. [Google Scholar] [CrossRef] - Li, H.; Chen, J.; Sheng, D.; Li, W. The improved distribution method of negentropy and performance evaluation of CCPPs based on the structure theory of thermoeconomics. Appl. Therm. Eng.
**2016**, 96, 64–75. [Google Scholar] [CrossRef] - Modi, A.; Kærn, M.R.; Andreasen, J.G.; Haglind, F. Thermoeconomic optimization of a Kalina cycle for a central receiver concentrating solar power plant. Energy Convers. Manage.
**2016**, 115, 276–287. [Google Scholar] [CrossRef] [Green Version] - Baral, S.; Kim, D.; Yun, E.; Kim, K.C. Experimental and Thermoeconomic Analysis of Small-Scale Solar Organic Rankine Cycle (SORC) System. Entropy
**2015**, 17, 2039–2061. [Google Scholar] [CrossRef] - Ozcan, H.; Dincer, I. Exergoeconomic optimization of a new four-step magnesiumechlorine cycle. Int. J. Hydrog. Energy. in press. [CrossRef]
- Keshavarzian, S.; Gardumi, F.; Rocco, M.V.; Colombo, E. Off-Design Modeling of Natural Gas Combined Cycle Power Plants: An Order Reduction by Means of Thermoeconomic Input–Output Analysis. Entropy
**2016**, 18. [Google Scholar] [CrossRef] [Green Version] - Piacentino, A. Application of advanced thermodynamics, thermoeconomics and exergy costing to a Multiple Effect Distillation plant: In-depth analysis of cost formation process. Desalination
**2015**, 371, 88–103. [Google Scholar] [CrossRef] - Dechamps, P.J. Incremental cost optimization of Heat Recovery Steam Generators; 95-CTP-101; The American Society of Mechanical Engineers: New York, NY, USA, 1995. [Google Scholar]
- Dechamps, P.J. The optimization of combined cycle HRSGs as a function of the plant load duty. In Proceedings of the ASME 1996 International Gas Turbine and Aeroengine Congress and Exhibition, Birmingham, UK, 10–13 June 1996.
- Kirschen, D.; Strbac, G. Fundamentals of Power System Economics; John Wiley & sons: Hoboken, NJ, USA, 2004. [Google Scholar]
- Valdés, M.; Rovira, A.; Durán, M.D. Influence of the heat recovery steam generator design parameters on the thermoeconomic performances of combined cycle gas turbine power plants. Int. J. Energy Res.
**2004**, 28, 1243–1254. [Google Scholar] [CrossRef] - Stoecker, W.F. Design of Thermal Systems, 3rd ed.; MacGraw-Hill Book Company: New York, NY, USA, 1989. [Google Scholar]
- Jaluria, Y. Design and Optimization of Thermal Systems, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2008. [Google Scholar]
- Shore, J.E.; Johnson, R.W. Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy. IEEE Trans. Inform.
**1980**, 26, 26–37. [Google Scholar] [CrossRef] - Zare, V.; Mahmoudi, S.M.S.; Yari, M. An exergoeconomic investigation of waste heat recovery from the Gas Turbine-Modular Helium Reactor (GT-MHR) employing an ammonia—water power/cooling cycle. Energy
**2013**, 61, 397–409. [Google Scholar] [CrossRef] - Ghaebi, H.; Amidpour, M.; Karimkashi, S.; Rezayan, O. Energy, exergy and thermoeconomic analysis of a combined cooling, heating and power (CCHP) system with gas turbine prime mover. Int. J. Energy Res.
**2011**, 35, 697–709. [Google Scholar] [CrossRef] - Rovira, A.; Barbero, R.; Montes, M.J.; Abbas, R.; Varela, F. Analysis and comparison of Integrated Solar Combined Cycles using parabolic troughs and linear Fresnel reflectors as concentrating systems. Appl. Energy
**2016**, 162, 990–1000. [Google Scholar] [CrossRef]

Component | Costing Model | Reference |
---|---|---|

Compressor | $20+\pi \xb7\mathrm{ln}(\pi )\xb7(\dot{m}/100)$ | [27,28] ^{1} |

Turbine | $20+\frac{\dot{m}}{100}\xb7\mathrm{ln}(\pi )\xb7(1+{e}^{0.1\xb7{T}_{\text{it}}-100})$ | [27,28] ^{1} |

Heat exchangers | $0.3\xb7{(\frac{\dot{m}\xb7\epsilon}{1-\epsilon})}^{0.8}$ | [2,28] ^{1,2} |

Solar field | $\frac{\dot{Q}+1000}{300}\xb7(1+{\mathrm{e}}^{0.1\xb7{T}_{\mathrm{max}}-100})$ | [29] ^{1,3} |

^{1}The model is arbitrary, based on the reference but not the one presented in it.

^{2}$\dot{m}$·ε/(1 − ε) is equivalent to UA (Product of the overall heat transfer coefficient and the heat exchange area (W·K

^{−1})) if both heat capacities are equal.

^{3}Costing models of solar fields of a specific technology (parabolic trough, linear Fresnel reflectors, central tower system, etc., each one involving characteristic operating temperatures), depend on the total reflective area of mirrors, which is proportional to the supplied thermal power. A factor depending on the maximum temperature has been included (in a similar way to the costing model of turbines) to take into account the effect of the different materials and coatings in the receiver.

Design Parameters | Marginal Cost Regarding Design Parameters | ||
---|---|---|---|

π | 8 | M_{π} (mu/GWh) | 0.076 |

ε_{heat} | 80% | M_{ε}_{heat} (mu/GWh) | 0.429 |

ε_{cool} | 95% | M_{ε}_{cool} (mu/GWh) | 2.491 |

$\dot{m}$ (kg/s) | 50 | M_{m} (mu/GWh) | 0.803 |

T_{max} (K) | 900 | M_{T}_{max} (mu/GWh) | 0.140 |

Results | |||

Compressor cost (mu) | 28.3 | ||

Turbine cost (mu) | 21.0 | ||

Heater cost (mu) | 20.8 | ||

Cooler cost (mu) | 72.3 | ||

Solar field cost (mu) | 45.5 | ||

Exploitation cost (mu) | 12.5 | ||

Yearly production (GWh) | 10.5 | ||

Generation cost (mu/GWh) | 1.198 |

Base | (1) Constant I | (2) Constant P | (3) Constant P + T_{exh} | (4) Constant P + T_{it} | |
---|---|---|---|---|---|

π | 8 | 4.92 | 4.85 | 7.25 | 4.29 |

ε_{heat} | 80% | 80.9% | 80.3% | 62.2% | 71.4% |

ε_{cool} | 95% | 86.4% | 86.0% | 89.6% | 85.7% |

$\dot{m}$ (kg/s) | 50 | 64.7 | 38.9 | 64.9 | 46.6 |

T_{max} (K) | 900 | 966.7 | 966.9 | 957.0 | 959.4 |

T_{exh} (K) | 499.4 | 595.7 | 596.1 | 499.4 | 580.5 |

T_{it} (K) | 836.7 | 887.7 | 885.6 | 817.3 | 836.7 |

C_{comp} (mu) | 28.3 | 25.1 | 23.0 | 29.3 | 22.9 |

C_{turb} (mu) | 21.0 | 21.0 | 20.6 | 21.3 | 20.7 |

C_{heat} (mu) | 20.8 | 26.7 | 17.3 | 12.6 | 13.5 |

C_{cool} (mu) | 72.3 | 37.1 | 23.9 | 47.3 | 27.1 |

C_{solar} (mu) | 45.5 | 78.1 | 48.1 | 53.7 | 51.9 |

I_{total} (um) | 188.0 | 188.0 | 132.9 | 164.2 | 136.1 |

E (um/year) | 12.5 | 12.5 | 8.86 | 10.9 | 9.08 |

P (GWh) | 10.5 | 17.7 | 10.5 | 10.5 | 10.5 |

K (mu/GWh) | 1.198 | 0.707 | 0.847 | 1.046 | 0.868 |

η_{GT} | 20.7% | 20.5% | 20.2% | 17.5% | 18.3% |

(mu/GWh) | Base | (1) Constant I | (2) Constant P | (3) Constant P + T_{exh} | (4) Constant P + T_{it} | |
---|---|---|---|---|---|---|

Design parameters | M_{π} | 0.076 | 0.496 | 0.518 | 0.066 | 0.997 |

M_{εheat} | 0.429 | 0.496 | 0.518 | 0.235 | 0.364 | |

M_{εcool} | 2.491 | 0.496 | 0.518 | 0.871 | 0.582 | |

M_{m} | 0.803 | 0.496 | 0.518 | 0.694 | 0.539 | |

M_{T}_{max} | 0.140 | 0.496 | 0.518 | 0.235 | 0.364 | |

Standardised | M_{W}_{c} | 1.045 | 0.496 | 0.518 | 0.093 | 0.334 |

M_{I}_{heater} | 0.891 | 0.496 | 0.518 | 0.196 | 0.232 | |

M_{I}_{cool} | 4.786 | 0.496 | 0.518 | 1.299 | 0.483 | |

M_{EQ} | 0.063 | 0.496 | 0.518 | 0.196 | 0.232 | |

M_{Q} | 0.803 | 0.496 | 0.518 | 0.694 | 0.539 |

(mu/GWh) | (1) Constant I | (2) Constant P | (3) Constant P + T_{exh} | (4) Constant P + T_{it} | |
---|---|---|---|---|---|

Optimised | _{c}M_{comp} | 0.707 | 0.847 | 1.046 | 0.868 |

_{c}M_{turb} | 0.707 | 0.847 | 1.046 | 0.868 | |

_{c}M_{heat} | 0.707 | 0.847 | 1.046 | 0.868 | |

_{c}M_{cool} | 0.707 | 0.847 | 1.046 | 0.868 | |

_{c}M_{solar} | 0.707 | 0.847 | 1.046 | 0.868 | |

λ_{r} | −1.6 × 10^{−3} | −3.2 × 10^{−2} | −3.4 × 10^{−2} (P)−6.2 × 10 ^{−3} (T_{exh}) | −3.1 × 10^{−2} (P)−7.7 × 10 ^{−4} (T_{it}) | |

Base case | _{c}M_{W}_{c} | 0.345 | 0.919 | 2.353 | –1.343 |

_{c}M_{I}_{heater} | 0.294 | 0.765 | 1.603 | 1.617 | |

_{c}M_{I}_{cool} | 1.578 | 4.660 | 1.203 | 1.251 | |

_{c}M_{EQ} | 0.021 | –0.063 | 0.775 | 0.789 | |

_{c}M_{Q} | 0.265 | 0.677 | –0.394 | –3.305 | |

λ_{r} | 4.3 × 10^{−3} | 1.2 × 10^{−2} | 0.11 (P) −2.0 × 10 ^{−2} (T_{exh}) | 0.39 (P) −1.2 × 10 ^{−2} (T_{it}) |

(mu/GWh) | Base | (1) Constant I | (2) Constant P | (3) Constant P + T_{exh} | (4) Constant P + T_{it} |
---|---|---|---|---|---|

D_{W}_{c} | –0.001 | –0.067 | –0.088 | –0.054 | –0.120 |

D_{I}_{heater} | 0.002 | –0.067 | –0.088 | –0.064 | –0.111 |

D_{I}_{cool} | 1.405 | –0.067 | –0.088 | 0.121 | –0.120 |

D_{EQ} | 0.035 | –0.067 | –0.088 | –0.064 | –0.111 |

D_{Q} | 0.004 | –0.067 | –0.088 | –0.079 | –0.114 |

D | 1.084 | 0.059 | 0.108 | 0.353 | 0.281 |

(mu/GWh) | (1) Constant I | (2) Constant P | (3) Constant P + T_{exh} | (4) Constant P + T_{it} | |
---|---|---|---|---|---|

Optimised | D_{W}_{c} | −2.1 × 10^{−6} | 1.6 × 10^{−6} | 1.3 × 10^{−6} | 2.6 × 10^{−8} |

D_{I}_{heater} | −2.1 × 10^{−6} | 1.2 × 10^{−7} | 9.5 × 10^{−7} | 1.1 × 10^{−6} | |

D_{I}_{cool} | 2.1 × 10^{−8} | 2.6 × 10^{−6} | −4.3 × 10^{−8} | −2.7 × 10^{−8} | |

D_{EQ} | 2.1 × 10^{−6} | −1.9 × 10^{−6} | −1.0 × 10^{−6} | −1.2 × 10^{−6} | |

D_{Q} | 1.4 × 10^{−6} | 3.7 × 10^{−6} | 4.0 × 10^{−6} | 1.5 × 10^{−6} | |

D | 7.6 × 10^{−11} | 1.1 × 10^{−10} | 8.3 × 10^{−11} | 2.4 × 10^{−11} | |

Base case | D_{W}_{c} | –0.029 | –0.001 | 0.466 | 0.023 |

D_{I}_{heater} | –0.025 | 0.002 | 0.105 | 0.062 | |

D_{I}_{cool} | 0.740 | 1.694 | 0.017 | 0.014 | |

D_{EQ} | 0.004 | 0.034 | –0.011 | 0.001 | |

D_{Q} | –0.022 | 0.005 | 0.003 | 0.876 | |

D | 0.620 | 1.383 | 0.325 | 0.608 |

Homogeneous Variance Variation | Variance Variation with T_{max} | Homogeneous Variance variation | Variance Variation with T_{max} | ||
---|---|---|---|---|---|

π | 4.82 | 4.78 | K (mu/GWh) | 0.972 | 0.849 |

ε_{heat} | 80.0% | 80.3% | K_{95} | 1.192 | 1.247 |

ε_{cool} | 85.7% | 85.9% | η_{GT} | 20.1% | 20.0% |

$\dot{m}$ (kg/s) | 28.3 | 39.8 | M_{π} | 0.532 | 0.519 |

T_{max} (K) | 967.0 | 960.0 | M_{ε}_{heat} | 0.532 | 0.519 |

T_{exh} (K) | 596.3 | 594.1 | M_{ε}_{ref} | 0.532 | 0.519 |

T_{it} (K) | 884.4 | 879.3 | M_{m} | 0.532 | 0.519 |

C_{comp} (mu) | 22.1 | 23.0 | M_{T}_{max} | 0.532 | 0.348 |

C_{turb} (mu) | 20.4 | 20.6 | D_{π} | –0.036 | –0.032 |

C_{heat} (mu) | 13.2 | 17.6 | D_{εheat} | –0.036 | –0.032 |

C_{cool} (mu) | 18.3 | 24.3 | D_{εref} | –0.036 | –0.032 |

C_{solar} (mu) | 35.9 | 47.8 | D_{m} | –0.036 | –0.032 |

I_{total} (mu) | 109.9 | 133.3 | D_{T}_{max} | –0.036 | –0.035 |

E (mu/year) | 7.33 | 8.88 | D | 0.153 | 0.140 |

P (GWh) | 7.54 | 10.5 | m | 0.827 | 0.253 |

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**MDPI and ACS Style**

Rovira, A.; Martínez-Val, J.M.; Valdés, M.
Thermoeconomic Coherence: A Methodology for the Analysis and Optimisation of Thermal Systems. *Entropy* **2016**, *18*, 250.
https://doi.org/10.3390/e18070250

**AMA Style**

Rovira A, Martínez-Val JM, Valdés M.
Thermoeconomic Coherence: A Methodology for the Analysis and Optimisation of Thermal Systems. *Entropy*. 2016; 18(7):250.
https://doi.org/10.3390/e18070250

**Chicago/Turabian Style**

Rovira, Antonio, José María Martínez-Val, and Manuel Valdés.
2016. "Thermoeconomic Coherence: A Methodology for the Analysis and Optimisation of Thermal Systems" *Entropy* 18, no. 7: 250.
https://doi.org/10.3390/e18070250