# Empirical Modeling of Direct Expansion (DX) Cooling System for Multiple Research Use Cases

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Literature Review

#### 1.3. Objectives

## 2. Modeling and Evaluation

#### 2.1. Test-Bed Description

#### 2.2. RTU Modeling

#### 2.2.1. General RTU Model Structure

#### 2.2.2. Modeling Results

_{out}and T

_{coil}for each stage. The sizes of the pivot tables are 1653 and 344 for stages 1 and 2, respectively. Finally, the data of the full air flow rate were separated.

_{cap,i}(i = 1–6)) were estimated with full air flow rate data. Constrained linear regression was carried out in Matlab (lsqlin). The inequality constraint was set to normalize the multiplier so that it is close to one in rated conditions (67 °F and 95 °F for T

_{out}and T

_{WB,coil}). Then the coefficients of the air flow rate modifier (C

_{cap,i}(i = 7–9)) were estimated in the same fashion with varying air flow data.

^{2}) is 0.98. The coefficients for EIR (C

_{EIR,i}(i = 1–9)) were estimated in the same fashion. Figure 3 (center) shows the result of the regression for the power consumption. The RMSE and correlation coefficient are 0.14 kWh and 0.99, respectively.

#### 2.2.3. Model Analysis

_{WB,coil}) is fixed at 67 °F (19.4 °C). The left and right figures show stage 1 and stage 2, respectively. As one can expect, the COP becomes high when the outdoor air temperature is low. Likewise, the higher COP is obtained with higher airflow. However, this is not as sensitive as the impact from the outdoor air temperature. The experimental data for the stage 1 are obtained when the outdoor air temperature is not that high and vice versa for the stage 2 which is activated only for hot weather condition. Ideally, those two could be united to one curve by dividing the weather/operation condition between two stages. Figure 6 presents the simplified relation between the COP and outdoor air temperature. It shows the maximum, average, and minimum value from the airflow rate variation for stages 1 and 2 with T

_{WB,coil}fixed at 67 °F (19.4 °C). Typically, stage 2 is required when the outdoor air temperature is high, so stage 2 data might not be applicable with lower outdoor air temperature.

## 3. Case Study

#### 3.1. Case 1: Building Energy Modeling Study

#### 3.2. Case 2: Model-Based Predictive Control (MPC)

_{out}), one control input (Q

_{AHU}), and three disturbance inputs (Q

_{sol}, Q

_{LGT}, and Q

_{int}). The cooling rate from the supply duct to each zone and the internal heat gain are summed and input to each state (envelope and room node). The heat gain from the lighting (Q

_{LGT}) and internal equipment (a heater in this study, Q

_{int}) are distributed to the envelope and room nodes; α

_{LGT,room}+ α

_{LGT,env}= 1 and α

_{int,room}+ α

_{int,env}= 1. The corresponding state-space formulations are shown in Equations (2) and (3).

_{oe}, R

_{er}, C

_{env}, C

_{room}, α

_{sol,env}, and α

_{sol,room}) with the average (volume-weight) room air temperature. The internal Matlab function fmincon was used. The initial values of the estimate parameters, such as the convective heat transfer coefficient, were adopted from the building energy simulation program. Figure 11 presents the estimation and validation results along with the control and disturbance inputs. The RMSE of the estimated building model was 0.76 °C, which is acceptable for implementation in the simulation and actual test-bed building.

_{d,w}and B

_{d,Q}are input matrices incorporating the disturbance and control input. This was iterated with the time step, and a series of matrices were generated, including a column matrix (

**Ω**) and lower triangular matrices (

_{x}**Ω**and

_{w}**Ω**

**), as shown in Equation (7). The matrices are expressed with bold notation.**

_{u}**Q**) by the predicted COP (

_{AHU}**COP**

^{T}, where T is the transpose operator). The room air temperatures (

**T**and

_{lower}**T**) are set to 20 °C and 23 °C. They are hard constraints incorporated in an inequality constraint so that no comfort violation occurs in the room during the optimization. C

_{upper}_{T}is a pre-defined matrix for extracting the target temperature from the state (e.g., air temperature).

**Q**) is 96. The initial state temperatures for the envelope and room were set to 21.5 °C, which is the mid-point of the comfort bound. Ventilation load is not considered (i.e., no outdoor air is included in the cooling load calculation).

_{AHU}#### 3.3. Case3: Fault Diagnostics and Detection (FDD)

## 4. Conclusions and Discussion

- The estimated DX cooling model for RTU system matched well with the measurement for the two stages. Their RMSE and correlation coefficient were 0.96kW and 0.98 in the cooling capacity and 0.14kW and 0.99 in power consumption.
- A BES program validation with EnergyPlus was conducted with a 2-story unoccupied commercial building. The power consumption of the model matched well with the experiment compared to the naive adoption of the nominal curve. The NMBE and cv(RMSE) improved from −21.7~−37.1% and 25.5~41.4% to −0.2% and 6.1%, respectively.
- An MPC simulation study was carried out with an estimated grey-box building and RTU models. Simplified linear COP prediction was incorporated in the MPC formulation, and 14.3% power savings was achieved compared to the feedback control.
- Three fault tests (duct leakage, limited refrigerant, and condenser fouling) were performed with the regressed RTU model. In all cases, the delivered cooling decreased distinctively. The cv(RMSE) of faulty experimental data against the model was 7~52% while the normal experimental data against the model were 9% (baseline). However, the power consumption of the faulty condition increased slightly compared to the prediction from the model.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Front view of building (

**top left**), roof top unit (RTU) (

**top right**), and heating ventilation and air-conditioning (HVAC) diagram (

**bottom**).

**Figure 3.**Modeling results of the cooling capacity (

**left**) and power consumption without part load ratio (PLR) (

**center**) and with PLR (

**right**).

**Figure 5.**Coefficient of performance (COP) variation with outdoor air temperature and airflow rate (

**left**: stage 1,

**right**: stage 2).

**Figure 14.**Fabric installation for condenser fouling test (

**left**: 28% airflow reduction,

**right**: 58% airflow reduction).

Stage 1 | Stage 2 | Stage 1 | Stage 2 | ||
---|---|---|---|---|---|

C_{cap,1} | 1.57742940 × 10^{−1} | 1.67520969 × 10^{0} | C_{EIR,1} | 1.54543818 × 10^{0} | −6.39859076 × 10^{−1} |

C_{cap,2} | 4.37237407 × 10^{−2} | −6.92156475 × 10^{−3} | C_{EIR,2} | −2.70157221 × 10^{−2} | 1.34507138e0^{−2} |

C_{cap,3} | 7.29207705 × 10^{−4} | 5.48380617 × 10^{−5} | C_{EIR,3} | −3.46088947 × 10^{−4} | 2.34046419 × 10^{−3} |

C_{cap,4} | 3.18835887 × 10^{−2} | −5.86719003 × 10^{−2} | C_{EIR,4} | −4.03434888 × 10^{−2} | 1.01593333 × 10^{−1} |

e | −5.41107627 × 10^{−4} | 2.08504063 × 10^{−4} | C_{EIR,5} | 1.31449523 × 10^{−3} | 2.82545174 × 10^{−4} |

C_{cap,6} | −1.18166008 × 10^{−3} | 1.93809110 × 10^{−3} | C_{EIR,6} | 1.69638404 × 10^{−4} | −4.99337303 × 10^{−3} |

C_{cap,7} | 8.09989450 × 10^{−1} | 7.93045348 × 10^{−1} | C_{EIR,7} | 1.27320835 × 10^{0} | 1.20706248 × 10^{0} |

C_{cap,8} | 2.43276315 × 10^{−1} | 2.34209810 × 10^{−1} | C_{EIR,8} | −2.20756876 × 10^{−1} | −1.24473476 × 10^{−1} |

C_{cap,9} | −5.40767115 × 10^{−1} | −5.20610958 × 10^{−5} | C_{EIR,9} | 4.90756905 × 10^{−5} | 2.76826460 × 10^{−5} |

Feedback Control | MPC | Savings [%] | |
---|---|---|---|

Cooling rate [kWh] | 129.7 | 136.5 | −5.3 |

Power consumption [kWh] | 51.9 | 44.4 | 14.3 |

Cost [$] | 5.6 | 4.8 | 13.2 |

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**MDPI and ACS Style**

Joe, J.; Im, P.; Dong, J.
Empirical Modeling of Direct Expansion (DX) Cooling System for Multiple Research Use Cases. *Sustainability* **2020**, *12*, 8738.
https://doi.org/10.3390/su12208738

**AMA Style**

Joe J, Im P, Dong J.
Empirical Modeling of Direct Expansion (DX) Cooling System for Multiple Research Use Cases. *Sustainability*. 2020; 12(20):8738.
https://doi.org/10.3390/su12208738

**Chicago/Turabian Style**

Joe, Jaewan, Piljae Im, and Jin Dong.
2020. "Empirical Modeling of Direct Expansion (DX) Cooling System for Multiple Research Use Cases" *Sustainability* 12, no. 20: 8738.
https://doi.org/10.3390/su12208738