# Solar Field Output Temperature Optimization Using a MILP Algorithm and a 0D Model in the Case of a Hybrid Concentrated Solar Thermal Power Plant for SHIP Applications

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## Abstract

**:**

## 1. Introduction

## 2. Envisaged System and Algorithm Structure

#### 2.1. Concentrated Solar Thermal System under Consideration

#### 2.1.1. Schematic and System Description

#### 2.1.2. Main Hypothesis

- Some thermal and parasitic losses are neglected: heat losses from night’s recirculation to avoid freezing, line pumping power...
- Perfectly mixed hot tank: uniform temperature in the tank.
- Heat losses’ approximation in the SF: they are classically taken as the module’s losses, with the average temperature of the SF.
- Constant fluid properties with temperature.
- Constant boiler efficiency. No limitations on the power (maximal or minimal) are considered.
- No heat losses in the cold tank: no calculations were made on the cold tank, although heat losses do happen in it as well.
- Perfect solar forecast: this hypothesis is chosen in order to exclude the influence of the forecasting model.

#### 2.2. Algorithm Structure

## 3. Physical Model

#### 3.1. 0D Solar Field Model

- Sun position

- Optical efficiency

- Heat losses

^{2}of aperture area, ${T}_{SF,out}$ the outlet temperature of the solar field in °C, ${T}_{SF,in}$ the inlet temperature of the solar field in °C and ${T}_{amb}$ the ambient temperature in °C, and ${a}_{1}$ and ${a}_{4}$ the heat losses correlation coefficients. For the LF-11 Fresnel collector, ${a}_{1}=0.032913\frac{W}{{m}^{2}.K}$ and ${a}_{4}=1.48\ast {10}^{-9}\frac{W}{{m}^{2}.{K}^{4}}$. Those coefficients are taken from a single module and extrapolated to the entire SF, assuming that the temperature evolves linearly in the SF.

- Heat from the solar field

^{2}, ${\eta}_{opt}$ the optical efficiency, and ${\dot{q}}_{losses}$ the heat losses in W/m

^{2}.

- Mass flow rate of the SF from outlet temperature, considering inertia.

- Validation

#### 3.2. Plant Model

#### 3.2.1. Mass and Mass Flow Rates Balance (Storage and Demand)

- Case 1: ${M}_{h+1}>{M}_{max}$. Too much mass flow rate at the inlet, need to defocus. Implementation:$${\dot{m}}_{SF,h}\leftarrow {\dot{m}}_{SF,h}-\frac{{M}_{h+1}-{M}_{max}}{\Delta t}\mathrm{and}{M}_{h+1}\leftarrow {M}_{max}$$
- Case 2: ${M}_{h+1}<0$. Demand was not answered correctly, need to increase the auxiliary mass flow rate. Implementation:$${\dot{m}}_{aux,h}\leftarrow {\dot{m}}_{aux,h}-\frac{{M}_{h+1}}{\Delta t}\mathrm{and}{M}_{h+1}\leftarrow 0$$

#### 3.2.2. Temperature of the Storage

#### 3.2.3. Boiler Model

#### 3.2.4. Plant Metrics

## 4. Control Models

#### 4.1. MILP Algorithm

#### 4.1.1. General Description of MILP Algorithm and Solver

#### 4.1.2. Methodology: How Is the Mathematical Problem Posed

#### 4.1.3. From Algorithm Parameters to MILP Algorithm’s Parameters

**bold**.

**t**is the time set; hourly resolution is chosen, with a time horizon of 48 h.

- Temperature range discretization:

- Absorbed heat from the sun, from Equation (3):

- Mass flow rate calculation:

#### 4.1.4. From MILP Parameters to MILP Formulation

#### 4.2. Control Models for Comparison

#### 4.2.1. Control Approach 1 (CA1)

#### 4.2.2. Control Approach 2 (CA2)

## 5. Case study and Results

#### 5.1. Case Study

#### 5.2. Hourly Results

#### 5.3. Yearly Results

## 6. Sensitivity Analysis

#### 6.1. Methodology

- -
- the number of modules in one loop is determined with the nominal mass flow rate of the module, the inlet and outlet temperature at design, under a condition of 900 W/m
^{2}:

- -
- the number of loops in parallel is determined with the solar field area necessary to answer the need with the available DNI on a user-chosen clear-sky day, the design efficiency, and the solar multiple:

- -
- the size of the storage is calculated from the number of storage hours and the average mass flow rate demand in kg/h:

#### 6.2. Process Temperature

#### 6.3. Heat Losses

#### 6.4. Other Sensibility Analyses

## 7. Discussion and Perspectives

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Nomenclature | |

$A$ | Area (m^{2}) |

${a}_{1},{a}_{4}$ | $\mathrm{Heat}\mathrm{losses}\mathrm{coefficient}(\mathrm{W}/{\mathrm{m}}^{2}\xb7\mathrm{K},\mathrm{W}/{\mathrm{m}}^{2}\xb7{\mathrm{K}}^{4}$) |

${C}_{p}$ | Specific heat capacity at constant pressure (J/kg·°C) |

$DNI$ | Direct Normal Irradiation (W/m^{2}) |

$h$ | Specific enthalpy (J/kg) |

IAM | Incidence Angle Modifier (-) |

${I}_{t}$ | Inertia of the fluid (J/K) |

$M$ | Mass (kg) |

$\dot{m}$ | Mass flow rate (kg/s) |

$N$ | Integer number (-) |

$Q$ | Heat (J or Wh) |

$\dot{q}$ | Heat flux (W) |

$S{F}_{a}$ | Solar Fraction (-) |

$SM$ | Solar Multiple (-) |

$\overrightarrow{s}$ | Solar vector |

$T$ | Temperature (°C) |

UA | Heat losses of the storage (W/K) |

Greek Symbols | |

$\alpha $ | Azimuth of the sun (0° North, 90° East) |

$\eta $ | Efficiency |

$\gamma $ | Elevation of the sun (0° Horizon, 90° Zenith) |

$\theta $ | Incidence angle on collector |

$\varphi $ | Latitude |

Subscripts | |

abs | Absorbed |

amb | Ambient |

aux | Auxiliary |

boiler | Boiler |

h | Hour h |

in | Inlet of the solar field |

L | Longitudinal |

opt | Optical |

out | Outlet of the solar field |

SF | Solar Field |

St | Storage |

T | Transversal |

$x\to y$ | Flow from component x to component y |

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**Figure 4.**Heat losses in W/m

^{2}of collector aperture area as a function of the outlet temperature of the LF-11 Fresnel collector, with an inlet temperature of 50 °C and ambient temperature of 15 °C.

**Figure 5.**Heat balance over a node. Taken from Wagner et al. [26].

**Figure 11.**Evolution of Solar Fraction as a function of process temperature for the 4 demand curves and 3 control strategies.

**Figure 12.**Annual solar fraction as a function of heat losses’ coefficient

**a**

_{1}(Equation (2)) for the 4 demand curves and 3 control strategies.

Variable | Value (Unit) | Description |
---|---|---|

$\varphi $ | 41.117° | Latitude |

${Q}_{daily}$ | 451 MWh/day | Daily heat demand |

${T}_{process}$ | 350 °C | Process temperature |

${T}_{return}$ | 50 °C | Return temperature |

${\eta}_{design}$ | 0.6 | Design efficiency (defined in Section 4.1) |

SM | 1.5 | Solar Multiple (defined in Section 4.1) |

${A}_{SF}$ | 113,367 m^{2} | Solar field area |

${N}_{Module}$ | 31 | Number of modules in one loop |

${N}_{loop}$ | 159 | Number of loops |

$\Delta {t}_{storage}$ | 8 h | Storage hours |

${M}_{max}$ | 864 t | Tank capacity |

$UA$ | 570 W/K | Heat losses of the storage |

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**MDPI and ACS Style**

Kamerling, S.; Vuillerme, V.; Rodat, S.
Solar Field Output Temperature Optimization Using a MILP Algorithm and a 0D Model in the Case of a Hybrid Concentrated Solar Thermal Power Plant for SHIP Applications. *Energies* **2021**, *14*, 3731.
https://doi.org/10.3390/en14133731

**AMA Style**

Kamerling S, Vuillerme V, Rodat S.
Solar Field Output Temperature Optimization Using a MILP Algorithm and a 0D Model in the Case of a Hybrid Concentrated Solar Thermal Power Plant for SHIP Applications. *Energies*. 2021; 14(13):3731.
https://doi.org/10.3390/en14133731

**Chicago/Turabian Style**

Kamerling, Simon, Valéry Vuillerme, and Sylvain Rodat.
2021. "Solar Field Output Temperature Optimization Using a MILP Algorithm and a 0D Model in the Case of a Hybrid Concentrated Solar Thermal Power Plant for SHIP Applications" *Energies* 14, no. 13: 3731.
https://doi.org/10.3390/en14133731