# Simulation-Based Sizing of a Secondary Loop Cooling System for a Refrigerated Vehicle

^{*}

## Abstract

**:**

## 1. Introduction

_{2}[25], paraffin emulsion [26], and salt hydrates [27]) are used as secondary refrigerants. Advantages of this architecture are reduced primary refrigerant charge and leakage and the ability to use the secondary loop as a TES [23]. Storage management of the TES allows a more flexible operation of the refrigeration unit without limiting the cooling of the cooling chamber, which means that the refrigeration unit can be operated generally more energy efficiently.

## 2. Model Description

## 3. Parameter Variation

## 4. Temperature Control

#### 4.1. Model Predictive Controller

#### 4.1.1. Constraints

#### 4.1.2. Objective Function

#### 4.1.3. Implementation

#### 4.2. Rule-Based PI Controller

## 5. Simulation Setup

^{max}$\in {\mathbb{R}}_{\ge 0}$ after the door is closed. Further, it is assumed that the full state measurement or state reconstruction is available without uncertainty or measurement noise for this simulation study.

## 6. Results

#### 6.1. Controller Evaluation

#### 6.2. Simulation Study

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MPC | Model predictive controller |

PCM | Phase change material |

PI | Proportional–integral |

TES | Thermal energy storage |

## Appendix A. State-Space Representation of Hybrid Model

**Table A1.**Model parameter values (adopted from [28]).

Parameter | Value | Unit |
---|---|---|

${\alpha}_{1}$ | $9.24\xb7{10}^{-5}$ | W rpm^{−1} |

${\alpha}_{2}$ | $6.78\xb7{10}^{-4}$ | W (°C)^{−1} |

${\alpha}_{3}$ | $4.25\xb7{10}^{-3}$ | W (°C)^{−1} |

${\alpha}_{4}$ | $2.85\xb7{10}^{-2}$ | W |

${\beta}_{1}$ | $1.40\xb7{10}^{2}$ | W K^{−1} |

${\beta}_{2}$ | 4.46 | W K^{−1} |

${\gamma}_{1}$ | 34.3 | W (°C)^{−1} |

${\gamma}_{2}$ | $2.93\xb7{10}^{2}$ | W |

${\gamma}_{3}$ | 10.2 | W (°C)^{−1} |

${\gamma}_{4}$ | 0 | W |

${\kappa}_{1}$ | 68.5 | W |

${\kappa}_{2}$ | 65.0 | W |

${\kappa}_{3}$ | 43.2 | W |

${\kappa}_{4}$ | 0.178 | W rpm^{−1} |

${\kappa}_{5}$ | 10.1 | W (°C)^{−1} |

${\kappa}_{6}$ | 0.510 | W (°C)^{−1} |

${\kappa}_{7}$ | $2.28\xb7{10}^{2}$ | W |

${\chi}_{1}$ | $2\xb7{10}^{-2}$ | K W^{−1} |

${\chi}_{2}$ | $17.5$ | ${\mathrm{s}}^{-1}$ |

${\chi}_{3}$ | $7.42\xb7{10}^{-5}$ | K (Ws)^{−1} |

${\xi}_{1}$ | $6.09\xb7{10}^{-5}$ | K (Ws)^{−1} |

${\xi}_{2}$ | $2.96\xb7{10}^{-2}$ | ${\mathrm{s}}^{-1}$ |

${\xi}_{3}$ | $1.39\xb7{10}^{-2}$ | ${\mathrm{s}}^{-1}$ |

${\zeta}_{1}$ | $1.42\xb7{10}^{-2}$ | ${\mathrm{s}}^{-1}$ |

${\zeta}_{2}$ | $2.67\xb7{10}^{-3}$ | ${\mathrm{s}}^{-1}$ |

${\zeta}_{3}$ | $1.18\xb7{10}^{-3}$ | ${\mathrm{s}}^{-1}$ |

${\zeta}_{4}$ | $1.10\xb7{10}^{-4}$ | ${\mathrm{s}}^{-1}$ |

## Appendix B. Blocking Matrix

**Table A2.**Non-zero entries of the move blocking matrix ${T}_{\mathrm{mb}}$, given by its row number R, column number C, and value V.

R | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

C | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

V | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

R | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

C | 11 | 11 | 11 | 11 | 11 | 12 | 12 | 12 | 12 | 12 |

V | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

R | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

C | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 |

V | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

## Appendix C. Door-Opening Shifts

Nr. | ${\mathit{\omega}}_{1}$ | ${\mathit{\omega}}_{2}$ | ${\mathit{\omega}}_{3}$ | ${\mathit{\omega}}_{4}$ |
---|---|---|---|---|

1 | 55.1 s | −25.8 s | 42.6 s | −46.6 s |

2 | −20.7 s | −38.6 s | −118 s | 117 s |

3 | −125 s | 9.70 s | −20.7 s | 48.1 s |

4 | −49.4 s | −20.2 s | 14.7 s | 94.1 s |

5 | −88.3 s | 46.3 s | −89.9 s | 98.7 s |

6 | −102 s | −73.9 s | −75.5 s | −104 s |

7 | −78.4 s | 94.5 s | 75.2 s | −115 s |

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**Figure 1.**Schematic illustration of the refrigerated vehicle, consisting of a cooling chamber (

**a**) and a secondary loop cooling unit (

**b**). Important temperatures $\vartheta $, the air chiller heat flow ${\dot{Q}}_{\mathrm{ac}}$, switching variables s, and the compressor speed ${n}_{\mathrm{cpr}}$ are highlighted. Door openings are indicated by the warm inflowing air colored in red and the outflowing cold air colored in blue. The secondary loop refrigeration unit can be divided into a cooling loop with propane as refrigerant and a storage loop filled with a glycol mixture. Figure adopted from [28].

**Figure 2.**Illustration of the MPCs optimization problem with a future door opening, highlighted by an orange background shading. Decision variables and constraints are depicted schematically. A blue background shading highlights the temperature window, and deviations from that window are marked red. Figure adapted from [29].

**Figure 3.**Flow diagram for the calculation of the binary control variables ${s}_{\mu}$ with $\mu \in \left\{\mathrm{cu},\mathrm{acf}\right\}$.

**Figure 4.**Closed-loop simulation results with MPC and rule-based PI controller with ${K}_{\mathrm{sl}}^{\mathrm{scl}}=1$ and ${\omega}_{1}=55.1$ s, ${\omega}_{2}=$-$25.8$ s, ${\omega}_{3}=42.6$ s, and ${\omega}_{4}=$−$46.6$ s. The 4 door openings are indicated by an orange background shading, and the MPC temperature window by a blue background shading. The two upper diagrams depict the air temperature inside the cooling chamber and the glycol temperature with the MPC and rule-based PI controller. The lower two diagrams show the power consumption of the components, from which the control variables for the compressor speed, cooling unit status, and air chiller fan status can be derived.

**Figure 5.**Simulation results of the relative energy consumption ${\lambda}_{\mathrm{con}}$ and relative time for reaching the target temperature after door openings ${\lambda}_{\mathrm{t}2\mathrm{tw}}$ for different relative secondary loop thermal capacities ${\lambda}_{\mathrm{cap}}$ with the MPC and the rule-based PI controller. The mean and standard deviation of the simulations for each thermal storage capacity and controller are highlighted by a line and background shading, respectively.

**Figure 6.**Amplitude statistic of glycol temperature ${\vartheta}_{\mathrm{gly}}^{\mathrm{out}}$ plotted for the different relative thermal storage capacities with the MPC and rule-based PI controller. A circle indicates the mean glycol temperatures for each secondary loop storage size, and a dashed line highlights the minimum permissible glycol temperature ${\vartheta}_{\mathrm{gly}}^{\mathrm{out},\mathrm{min}}$.

**Figure 7.**Controller performance ${\lambda}_{\mathrm{per}}$ for three different weighting factors $\psi $ depending on the relative storage capacity of the secondary loop ${\lambda}_{\mathrm{cap}}$. Control performance is defined with $\psi =1$ solely by the relative energy consumption ${\lambda}_{\mathrm{con}}$, with $\psi =0$ by the relative time to reach the temperature window after door openings ${\lambda}_{\mathrm{t}2\mathrm{tw}}$, and with $\psi =0.5$ by equally weighting ${\lambda}_{\mathrm{con}}$ and ${\lambda}_{\mathrm{t}2\mathrm{tw}}$. A region of an optimal relative secondary loop storage capacity can be observed for the MPC, where both the minimum energy consumption and fast cooling after door openings are realized.

**Table 1.**Classification of the mode m depending on the status of the cooling unit ${s}_{\mathrm{cu}}$, the air chiller fan ${s}_{\mathrm{acf}}$, and the door ${s}_{\mathrm{door}}$.

Mode $\mathit{m}\left(\mathit{t}\right)$ | ${\mathit{s}}_{\mathbf{cu}}\left(\mathit{t}\right)$ | ${\mathit{s}}_{\mathbf{acf}}\left(\mathit{t}\right)$ | ${\mathit{s}}_{\mathbf{door}}\left(\mathit{t}\right)$ |
---|---|---|---|

1 | 0 | 0 | 0 |

2 | 0 | 0 | 1 |

3 | 0 | 1 | 0 |

4 | 0 | 1 | 1 |

5 | 1 | 0 | 0 |

6 | 1 | 0 | 1 |

7 | 1 | 1 | 0 |

8 | 1 | 1 | 1 |

Parameter | Value |
---|---|

${T}_{\mathrm{s}}$ | 20 s |

${N}_{\mathrm{pr}}$ | 60 |

${N}_{\mathrm{ctr}}$ | 31 |

${n}_{\mathrm{cpr}}^{\mathrm{min}}$ | 700 rpm |

${n}_{\mathrm{cpr}}^{\mathrm{max}}$ | 5000 rpm |

${\vartheta}_{\mathrm{gly}}^{\mathrm{out},\mathrm{min}}$ | −35 °C |

${N}_{\mu}^{\mathrm{up}},{N}_{\mu}^{\mathrm{down}}$ | 5 |

${T}_{\mathrm{mb}}$ | see Appendix B |

M | 13 |

${\vartheta}_{\mathrm{cc}}^{\mathrm{ref}}$ | 5 °C |

${\vartheta}_{\mathrm{tw}}^{\mathrm{min}}$ | 4.5 °C |

${\vartheta}_{\mathrm{tw}}^{\mathrm{max}}$ | 5.5 °C |

Q | $8\xb7{10}^{6}$ |

${R}_{1}$ | 0.1 |

${R}_{2}$ | $1\xb7{10}^{-5}$ |

${T}_{1}$ | $5\xb7{10}^{3}$ |

${T}_{2}$ | $1\xb7{10}^{3}$ |

Parameter | Value |
---|---|

${T}_{\mathrm{s}}$ | 20 s |

${n}_{\mathrm{cpr}}^{\mathrm{min}}$ | 700 rpm |

${n}_{\mathrm{cpr}}^{\mathrm{max}}$ | 5000 rpm |

${N}^{\mathrm{up}},{N}^{\mathrm{down}}$ | 5 |

${\vartheta}_{\mathrm{cc}}^{\mathrm{ref}}$ | 5 °C |

${H}_{\vartheta}$ | 0.4 °C |

${P}_{\mathrm{PI}}$ | $-764$ |

${I}_{\mathrm{PI}}$ | $-16.2$ |

Condition | Value |
---|---|

Simulation study: | |

${K}_{\mathrm{sl}}^{\mathrm{scl}}$ | {0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1, 1.125, … 1.25, 1.375, 1.5, 1.75, 2, 2.25, 2.5, 3, 3.5, 4} |

# of scaling factors | 19 |

# of door opening realizations | 7 |

# of controllers | 2 |

# of simulations | $19\xb77\xb72=266$ |

Individual simulation: | |

${T}_{\mathrm{sim}}$ | 120 min |

${\vartheta}_{\mathrm{amb}}$ | 22 °C |

Door openings: | |

Start | $[20\phantom{\rule{3.33333pt}{0ex}}\mathrm{min}+{\omega}_{1},45\phantom{\rule{3.33333pt}{0ex}}\mathrm{min}+{\omega}_{2},70\phantom{\rule{3.33333pt}{0ex}}\mathrm{min}+{\omega}_{3},90\phantom{\rule{3.33333pt}{0ex}}\mathrm{min}+{\omega}_{4}]$ |

Duration | $\left[3\phantom{\rule{3.33333pt}{0ex}}\mathrm{min},1\phantom{\rule{3.33333pt}{0ex}}\mathrm{min},4\phantom{\rule{3.33333pt}{0ex}}\mathrm{min},2\phantom{\rule{3.33333pt}{0ex}}\mathrm{min}\right]$ |

T2TW^{max} | [100 s, 60 s, 120 s, 80 s] |

Initial conditions: | |

${x}_{\mathrm{c}}$ | [2.70 °C, 5 °C, 5.27 °C, 6.69 °C]^{T} |

${u}_{\mathrm{c}}$ | 1080 rpm |

m | 7 |

Computation: | |

Processor | Intel Core i9-10850K [42] |

RAM | 32 GB |

Solver | Matlab ode15s [43] |

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## Share and Cite

**MDPI and ACS Style**

Lösch, M.; Fallmann, M.; Poks, A.; Kozek, M.
Simulation-Based Sizing of a Secondary Loop Cooling System for a Refrigerated Vehicle. *Energies* **2023**, *16*, 6459.
https://doi.org/10.3390/en16186459

**AMA Style**

Lösch M, Fallmann M, Poks A, Kozek M.
Simulation-Based Sizing of a Secondary Loop Cooling System for a Refrigerated Vehicle. *Energies*. 2023; 16(18):6459.
https://doi.org/10.3390/en16186459

**Chicago/Turabian Style**

Lösch, Maximilian, Markus Fallmann, Agnes Poks, and Martin Kozek.
2023. "Simulation-Based Sizing of a Secondary Loop Cooling System for a Refrigerated Vehicle" *Energies* 16, no. 18: 6459.
https://doi.org/10.3390/en16186459