# Improved Large-Scale Process Cooling Operation through Energy Optimization

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- ●
- Simple linear regression model
- ●
- Bi-quadratic regression model
- ●
- Multivariate polynomial regression model
- ●
- Simpler multivariate polynomial regression model
- ●
- DOE-2 model
- ●
- Modified DOE-2 model
- ●
- Gordon-Ng universal model (based on the evaporator inlet water temperature)
- ●
- Gordon-Ng universal model (based on the evaporator outlet water temperature) *
- ●
- Modified Gordon-Ng universal model
- ●
- Gordon-Ng simplified model
- ●
- Lee simplified model
- ●
- * This model is used in the current work.

Symbol | Description | Units |
---|---|---|

D_{i} | Total cooling demand for i^{th} hour | kW |

E_{i} | Amount of stored thermal energy at i^{th} hour | kWh |

E_{max} | Maximum capacity of the TES tank | kWh |

L_{j} | Lower bound on the cooling load on j^{th} chiller | kW |

P_{AUX, ik} | Power consumed by the auxiliary equipment at k^{th} station at i^{th} hour | kW |

P_{Station, ik} | Total power consumed by k^{th} chilling station at i^{th} hour | kW |

P_{ij} | Electric power consumption of the j^{th} chiller at i^{th} hour | kW |

R_{max} | Maximum charging/discharging rate of TES tank | kW |

Condenser water inlet temperature at i^{th} hour | K | |

Chilled water outlet temperature | K | |

U_{j} | Upper bound on the cooling load on j^{th} chiller | kW |

X_{ij} | Cooling load on j^{th} chiller at i^{th} hour | kW |

M_{c,j} | Condenser heat exchanger coefficient of the j^{th} chiller | W K^{−1} |

M_{e,j} | Evaporator heat exchanger coefficient of the j^{th} chiller | W K^{−1} |

q_{c,j} | Internal condenser heat loss rate in the j^{th} centrifugal chiller | kW |

q_{e,j} | Internal evaporator heat loss rate in the j^{th} centrifugal chiller | kW |

γ_{i} | Real-time market rate of electric energy at i^{th} hour | $/kWh |

δ_{ij} | Binary variable representing on or off status of j^{th} chiller at i^{th} hour by having the value of 0 or 1 respectively | Dimensionless |

COP_{ij} | Coefficient of performance of the j^{th} chiller at i^{th} hour | Dimensionless |

DBT_{i} | Dry bulb temperature at i^{th} hour | K |

r | Number of cooling stations | Dimensionless |

m_{k} | Total number of chillers upto the k^{th} station; m_{0} = 0, m_{r} = M | Dimensionless |

M | Total number of chillers | Dimensionless |

n | Number of hours in the optimization horizon | Dimensionless |

RH_{i} | Relative humidity at i^{th} hour | Dimensionless |

WBT_{i} | Wet bulb temperature at i^{th} hour | K |

SL_{ik} | Total cooling load at k^{th} station at i^{th} hour | kW |

α | Penalty coefficient | $/kW |

Pdata | Actual power consumed by the cooling system operation in a day | MWh |

Popt | Estimated power consumption by the cooling system operation in a day for the cooling load profile resulted from solving optimization | MWh |

## 2. System Overview

**Figure 1.**Simplified schematic of the Hal C. Weaver power plant complex at the University of Texas at Austin.

^{3}) chilled water thermal energy storage tank. This tank has a storage capacity of 39,000 ton-hr (494 GJ). The tank can be filled with chilled water during the night and then discharged during the day when demand for cooling is highest. This cooling system serves over 160 buildings with approximately 17 million square feet (1.6 million m

^{2}) of space. The three active cooling stations are numbered as Station 3, Station 5, and Station 6 (Stations 1, 2, and 4 have either been decommissioned or are not currently in use). Each station includes three centrifugal chillers, a set of cooling towers, condenser water pumps, and chilled water pumps. Station 6 has variable frequency drives installed on all equipment. The chillers in any Station X are named as X.1, X.2, and X.3.

## 3. Model Development of the Cooling System

#### 3.1. Chillers

_{ij}) by each chiller in the cooling network is modeled independently as a function of its cooling load(X

_{ij}) condenser water return temperature ( ) and chilled water temperature setpoint ( ). Minimization of least squares is used to fit the plant data to the Gordon-Ng model (Equation (1)) [5] and estimate model parameters for each chiller. The parameters represented by symbols, q

_{e,j}, q

_{c,j}, M

_{e,j}and M

_{c,j}, in Equation 1 are the four model parameters that are assumed to have different values for each chiller.

_{ij}) of a chiller is defined as the ratio of its cooling load to its power consumption.

Chiller Number | Range of absolute error (%) | Mean % absolute error |
---|---|---|

3.1 | 0–6.18 | 1.41 |

3.2 | 0–9.70 | 1.36 |

3.3 | 0–30.08 | 2.25 |

5.1 | 0–7.11 | 1.61 |

5.2 | 0–6.5 | 0.99 |

5.3 | 0–13.71 | 1.19 |

6.1 | 0–23.02 | 1.34 |

6.2 | 0–31.22 | 0.93 |

6.3 | 0–3.17 | 0.64 |

#### 3.2. Auxiliaries

_{1}to β

_{10}) is obtained by fitting the year round power consumption data collected at hourly time steps from the power plant historian.

Station Number | Range of absolute error (%) | Mean absolute error (%) |
---|---|---|

3 | 0–40.81 | 9.96 |

5 | 0–20.31 | 2.17 |

6 | 0–23.67 | 6.98 |

Total | 0–26.48 | 5.85 |

## 4. Multi-Period Cooling System Optimization

#### 4.1. Cooling System Optimization without Storage

^{th}hour can be represented with the following set of equations:

_{ij}∈ {0,1}

_{ij}and P

_{AUX,ik}are defined by Equations (1b) and (2a) respectively.

_{ij}sets at a given time (constant i) is (2

^{M}− 1). For any fixed set of δ

_{ij}, the objective function can be written as quadratic programming (QP) formulation, i.e., in the form of the following equation, due to the nature of models.

_{ij}) was a non-linear convex formulation. It was solved for each of the (2

^{9}− 1 = 511) possible sets of δ

_{ij}in MATLAB using the sequential quadratic programming (SQP) algorithm to obtain a unique global solution always. The case resulting in the least value of the objective function was accepted as the optimal solution. The total time taken by the MATLAB algorithm in solving this QP for 511 cases in order to obtain the optimal solution varied between 1 and 2 s.

#### 4.2. Cooling System Optimization with Storage

**Figure 4.**Hourly campus cooling load values (left axis) and ambient wetbulb temperature values (right axis) over 24 h period. This data is from 11 July 2012. It serves as an example for days with more than one maxima in the cooling load profile.

_{1}= E

_{0}= 0

_{i}≥ 0, for i = 2 to n

_{i}≤ E

_{max}, for i = 2 to n

_{ij}∈ {0,1}

^{th}hour is optimally distributed among M independent chillers having different model characteristics, which is equivalent to the optimization problem without storage. Hence, the optimization problem with storage consists of n number of static optimization problems without storage.

## 5. Results and Discussion

_{i}from the objective function expression. It also made the objective function equivalent to minimizing the total power consumption (kWh) in a day for the case when α = 0. Midnight was chosen to be the initial time for each problem after iterating over other possible initial times. The 24-h cooling load profiles are compared for two chosen days in the month of September, named as Day 1 and Day 2 (Figure 5 and Figure 6 respectively). Figure 5 presents the comparison among various distributions of the optimal cooling load from the stage 1 of dynamic optimization, i.e., the redistribution of cooling load among several hours. Figure 6 presents similar results for Day 2, which has less frequent cooling load variations as compared to Day 1. For each day, the optimization problem was solved for different values for the penalty coefficient, α = 0, 0.1 and 0.5 $/kW. It is clearly visible from the Figure 5 and Figure 6 that the usage of thermal energy storage provides flexibility to shift cooling load across time and hence to opt for alternate cooling load profiles for a chosen time horizon (24 h in this case). This flexibility comes with the opportunities to save energy and/or to reduce fluctuations in the cooling load profile. These figures show various cooling load profiles for different optimization parameters, each profile independently satisfying the hourly cooling demand constraints.

**Figure 7.**Comparison of power consumption values from plant data, static optimization and dynamic optimization.

_{i}is defined as the number of chillers operating during the i

^{th}hour. The difference between the values of N

_{i}for any consecutive hours represents the number of turning on/off events occurring between those two hours. It is assumed that between any two hours, either some chillers are turned on (rise in cooling load) or some chillers are turned off (drop in cooling load) and not both.

**Table 4.**Effect of optimal chiller loading (OCL) with thermal energy storage on the frequency of cold starts.

Cooling load profile | Number of chiller turning on/off events in 24 h | Total power consumption in 24 h (MWh) |
---|---|---|

Plant data | 4 | 356.45 |

OCL Without storage | 4 | 342.34 |

OCL With storage, α = 0 | 5 | 341.99 |

OCL With storage, α = 0.1 | 1 | 342.81 |

OCL With storage, α = 0.5 | 0 | 344.51 |

**Figure 8.**Comparison of the variations in the total number of operating chillers under different cooling load profiles.

#### 5.1. Time-Varying Prices

**Figure 9.**Variation in the hourly real-time prices in the ERCOT wholesale market over the year 2012, in Austin, TX, USA.

**Figure 10.**Comparison of the cooling cost in case of time varying electricity prices—With TES (α = 0) vs. without TES.

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Kapoor, K.; Powell, K.M.; Cole, W.J.; Kim, J.S.; Edgar, T.F.
Improved Large-Scale Process Cooling Operation through Energy Optimization. *Processes* **2013**, *1*, 312-329.
https://doi.org/10.3390/pr1030312

**AMA Style**

Kapoor K, Powell KM, Cole WJ, Kim JS, Edgar TF.
Improved Large-Scale Process Cooling Operation through Energy Optimization. *Processes*. 2013; 1(3):312-329.
https://doi.org/10.3390/pr1030312

**Chicago/Turabian Style**

Kapoor, Kriti, Kody M. Powell, Wesley J. Cole, Jong Suk Kim, and Thomas F. Edgar.
2013. "Improved Large-Scale Process Cooling Operation through Energy Optimization" *Processes* 1, no. 3: 312-329.
https://doi.org/10.3390/pr1030312