# Statistical Analysis of Design Variables in a Chiller Plant and Their Influence on Energy Consumption and Life Cycle Cost

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Research Study: Procedure to Obtain the Optimal Distribution Cooling Capacity of an Air-Condensed Chiller Plant for a Hotel Facility Conceptual Design

- Stage 1: The thermal demand values of the building were analyzed through the transfer method, using the technical description of the real estate project, the meteorological conditions of the region, and the statistical information of occupancy and operation patterns of hotels with similar characteristics;
- Stage 2: The generation of chiller plant alternatives was carried out through a statistical–mathematical procedure using the calculated thermal demand values and black box models of the water chillers. This procedure allowed different plant architectures to be established by modifying the design parameters, i.e., installed cooling capacity, number of units, and distribution of cooling capacity between chillers, regarding constraints according to design standards;
- Stage 3: As an initial state, the chiller plant was a decoupled system comprising n air-cooled chillers arranged in parallel, and only the primary circuit was involved in the analysis. The energy verification of the chiller plants generated was carried out by solving a non-linear, multivariable combinatorial optimization problem of optimal chiller loading (OCL) and optimal chiller sequence (OCS) versus building demand profiles. In order to establish the OCS, a baseline decision was made. A genetic algorithm was used for the optimization procedure. For the LCC analysis, economic and financial parameters and criteria of the region where the case study was analyzed were used.

#### 2.2. Methodology

_{i}represents the best linear unbiased estimator for σ, and P

_{(i)}are given by Equation (5).

_{XY}

_{,}as shown in Equation (11).

_{i}) of the regression under the ordinary least squares (OLS) method and regressing (û

_{i}) on the explanatory variable (X) that is supposed to be related to the heteroscedastic variance (${\sigma}_{i}^{2}$). After it is found not to be asymptotically valid under asymmetric disturbances, the function used for sample sizes larger than 50 is described in Equations (12)–(15):

_{1}and β

_{2}are the regression coefficients, (|û

_{i}|) represents the residuals of the new regression and (υ

_{i}) indicates the error term. Therefore, the Equations from (12)–(15) with the highest value of R

^{2}and the lowest value of standard error was selected to represent heteroscedasticity. Finally, a t-test needed to be performed on the selected equation, as ${\beta}_{0}=0$ and ${\beta}_{1}\ne 0$ suggested pure heteroscedasticity, whereas ${\beta}_{0}\ne 0$ and ${\beta}_{1}\ne 0$ was an indication of mixed heteroscedasticity. If the sample value of the statistic was high enough that the probability of rejecting the null hypothesis being true was less than 1%, the null hypothesis of homoscedasticity was rejected.

_{s}) is given in Equation (16), and its calculation is exactly the same as Pearson’s but on ranks rather than absolute values. Its power can be similar or only slightly lower. These are the correlation analysis, and variables are measured on a scale that is at least ordinal.

_{i}is the difference between ranks of Xi and Yi. Furthermore, n is the number of observations. Another rank coefficient, Kendall’s tau (τ) [80], is used as there are multiple independent variables. As a partial correlation coefficient, it is used as there are data on a third variable that may influence the association between two other variables of interest. It can be considered as the estimated correlation between these two variables with the same value as the third variable. Its conclusions are identical to Spearman’s correlation coefficient, as only two variables are involved. The mathematical expression of τ is presented in Equation (17):

## 3. Results and Discussion

- -
- if p-value > α (0.05) Accept H0 = accept that the data were from a normal distribution;
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- if p-value < α (0.05) Accept H1 = reject that the data were from a normal distribution.

_{i}) and the independent variable (Ŷi) was performed to assess if they exhibited any systematic pattern. The results of this statistical pre-test are shown in Table 5.

_{i}) vs. (Ŷi) using the regression line for each bivariate relationship shown in Table 5 to determine whether the estimated mean value of (Ŷi) is systematically related to the (û

_{i}).

_{i}) vs. (Ŷi).

- -
- Heteroskedasticity Test
- -
- Hypothesis:
- -
- If p-value > α (0.05) Accept H0 = Homoscedastic
- -
- If p-value < α (0.05) Accept H1 = Heteroscedastic

_{i}) in absolute value. Using the E-view statistical package [82], it was determined that the function with the highest R

^{2}value was the function described in Equation (12). Furthermore, the heteroscedasticity of the models described in Equations (39)–(44) was checked. The results are showed in Table 6.

_{ch}). In the present case study, some chiller plant configurations with four units achieved better performance than a five-unit chiller plant. Regarding the influence of (N

_{ch}) on the (LCC), Figure 2d reveals that these quadratic relationships were more evident, and the trend of increase in the number of installed chillers caused an increase in LCC. Therefore, it could be seen that an optimal configuration of three to four chillers could be found.

_{ch}and LCC vs. Cd

_{ch}tends to be exponential. This phenomenon was also influenced by the increase in the number of chillers, leading to the asymmetrical arrangement being better. This conclusion was not categorical either, as the chiller plant relies on two units and the symmetrical arrangement offered better performance than that with the asymmetrical arrangement. Therefore, the engineers need to analyze the arrangement according to the number of chillers to be installed.

## 4. Conclusions

_{ch}, CD, and GV has been carried out using non-parametric methods, as the dependent variables do not have a normal distribution. The results obtained have suggested that the total chiller design and the cooling capacity distribution among chillers have a significant influence on the energy consumption of the chiller plant with a Spearman’s Rho association index and Kendall Tau association index of −0.625 and 0.559, respectively. However, as the LCC has been considered, only the cooling capacity distribution among chillers has a substantial influence on the Kendall Tau association index (value of 0.289). Furthermore, the total installed cooling capacity has been found not to affect the performance of the chiller plant substantially. Likewise, it is proposed to define through non-parametric methods such as kernel and spline smoothing techniques, among others, the estimators that adjust the curves of the significant bivariate relationships

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Main characteristics and performance data of the chiller plants relying on 2 and 3 units (obtained from Diaz Torres [33], Diaz et al [34] Diaz et al [35]. Reprinted/adapted with permission from Ref. [34]. 2021, Elsevier and Copyright Clearance Center. Reprinted/adapted with permission from Ref. [35]. 2022, Elsevier and Copyright Clearance Center.

Chiller Plant No. | Chiller Cooling Capacity at STD (kW) | Total Units | Total Cooling Capacity (kW) | Cooling Distribution among Chillers (%) | Energy Consumption (kWh/year) | LCC MMcup | ||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | ||||||

1 | 181.3 | 360.0 | - | 2 | 538 | 33/67 | 476.3 | 799.86 |

2 | 199.8 | 360.0 | - | 2 | 557 | 36/ 64 | 487.1 | 816.14 |

3 | 203.1 | 360.0 | - | 2 | 560 | 36/64 | 487.8 | 817.13 |

4 | 229.9 | 312.2 | - | 2 | 540 | 42/58 | 483.2 | 807.17 |

5 | 273.0 | 273.0 | - | 2 | 542 | 50/50 | 545.1 | 891.09 |

6 | 273.0 | 312.2 | - | 2 | 582 | 47/53 | 533.9 | 882.30 |

7 | 98.2 | 98.2 | 360.0 | 3 | 553 | 18/18/65 | 460.8 | 801.82 |

8 | 98.2 | 119.0 | 360.0 | 3 | 574 | 17/21/62 | 432.5 | 766.46 |

9 | 98.2 | 135.1 | 312.2 | 3 | 543 | 18/25/57 | 442.8 | 778.19 |

10 | 98.2 | 151.2 | 312.2 | 3 | 559 | 17/27/56 | 449.3 | 791.61 |

11 | 98.2 | 161.7 | 312.2 | 3 | 569 | 17/28/55 | 446.8 | 790.34 |

12 | 98.2 | 181.3 | 273.0 | 3 | 549 | 18/33/49 | 432.0 | 765.91 |

13 | 98.2 | 199.8 | 273.0 | 3 | 568 | 17/35/48 | 443.7 | 783.34 |

14 | 98.2 | 203.1 | 273.0 | 3 | 571 | 17/35/48 | 444.7 | 785.09 |

15 | 98.2 | 229.9 | 229.9 | 3 | 556 | 18/41/41 | 432.4 | 764.26 |

16 | 119.0 | 119.0 | 312.2 | 3 | 548 | 22/22/57 | 431.8 | 763.54 |

17 | 119.0 | 135.1 | 312.2 | 3 | 564 | 21/24/55 | 435.4 | 772.64 |

18 | 119.0 | 151.2 | 273.0 | 3 | 540 | 22/28/50 | 436.8 | 774.93 |

19 | 119.0 | 151.2 | 312.2 | 3 | 580 | 20/26/54 | 440.3 | 783.24 |

20 | 119.0 | 161.7 | 273.0 | 3 | 551 | 22/29/49 | 427.9 | 763.51 |

21 | 119.0 | 181.3 | 273.0 | 3 | 570 | 21/32/48 | 419.6 | 753.15 |

22 | 119.0 | 199.8 | 229.9 | 3 | 546 | 22/36/42 | 416.0 | 744.67 |

23 | 119.0 | 203.1 | 229.9 | 3 | 550 | 22/37/42 | 417.5 | 747.27 |

24 | 119.0 | 229.9 | 229.9 | 3 | 577 | 20/40/40 | 414.2 | 745.79 |

25 | 135.1 | 135.1 | 273.0 | 3 | 540 | 25/25/50 | 462.1 | 809.18 |

26 | 135.1 | 135.1 | 312.2 | 3 | 579 | 23/23/54 | 460.1 | 808.53 |

27 | 135.1 | 151.2 | 273 | 3 | 556 | 24/27/49 | 447.1 | 750.89 |

28 | 135.1 | 161.7 | 273 | 3 | 566 | 24/28/48 | 436.2 | 742.44 |

29 | 135.1 | 181.3 | 229.9 | 3 | 543 | 25/33/42 | 406.6 | 777.79 |

30 | 135.1 | 181.3 | 273 | 3 | 585 | 23/31/46 | 424.6 | 737.08 |

31 | 135.1 | 199.8 | 229.9 | 3 | 562 | 24/35/41 | 423.1 | 825.32 |

32 | 135.1 | 203.1 | 203.1 | 3 | 538 | 25/38/38 | 471.3 | 753.34 |

33 | 135.1 | 203.1 | 229.9 | 3 | 565 | 24/36/41 | 424.1 | 828.08 |

34 | 151.2 | 151.2 | 273 | 3 | 572 | 26/26/47 | 457.8 | 778.04 |

35 | 151.2 | 161.7 | 229.9 | 3 | 540 | 28/30/42 | 428.5 | 798.99 |

36 | 151.2 | 161.7 | 273 | 3 | 582 | 26/28/47 | 445.3 | 768.39 |

37 | 151.2 | 181.3 | 229.9 | 3 | 559 | 27/32/41 | 424.6 | 806.38 |

38 | 151.2 | 199.8 | 199.8 | 3 | 548 | 27/36/36 | 483.9 | 757.73 |

39 | 151.2 | 199.8 | 203.1 | 3 | 551 | 27/36/37 | 480.5 | 846.01 |

40 | 151.2 | 199.8 | 229.9 | 3 | 578 | 26/34/40 | 437.5 | 844.73 |

41 | 151.2 | 203.1 | 203.1 | 3 | 554 | 27/36/36 | 482 | 775.57 |

42 | 151.2 | 203.1 | 229.9 | 3 | 581 | 26/35/39 | 438.4 | 846.2 |

43 | 161.7 | 161.7 | 229.9 | 3 | 550 | 29/29/42 | 430.7 | 803.73 |

44 | 161.7 | 181.3 | 199.8 | 3 | 539 | 30/33/37 | 458.2 | 811.80 |

45 | 161.7 | 181.3 | 203.1 | 3 | 543 | 30/33/37 | 457.4 | 805.69 |

46 | 161.7 | 181.3 | 229.9 | 3 | 570 | 28/32/40 | 424.3 | 807.89 |

47 | 161.7 | 199.8 | 199.8 | 3 | 558 | 29/36/36 | 483.5 | 758.89 |

48 | 161.7 | 199.8 | 203.1 | 3 | 561 | 29/35/36 | 479.0 | 846.8 |

49 | 161.7 | 203.1 | 203.1 | 3 | 564 | 28/36/36 | 480.1 | 841.14 |

50 | 181.3 | 181.3 | 181.3 | 3 | 540 | 33/33/33 | 475.7 | 838.53 |

51 | 181.3 | 181.3 | 199.8 | 3 | 559 | 32/32/36 | 466.4 | 831.09 |

52 | 181.3 | 181.3 | 203.1 | 3 | 562 | 32/32/36 | 465.4 | 819.71 |

53 | 181.3 | 199.8 | 199.8 | 3 | 578 | 31/34/34 | 491.2 | 820.22 |

54 | 181.3 | 199.8 | 203.1 | 3 | 581 | 31/34/35 | 487.6 | 859.85 |

55 | 181.3 | 203.1 | 203.1 | 3 | 584 | 31/35/35 | 488.4 | 855.43 |

**Table A2.**Main characteristics and performance data of the chiller plants relying on 4 units (obtained from Diaz Torres [33], Diaz et al [34] Diaz et al [35]. Reprinted/adapted with permission from Ref. [34]. 2021, Elsevier and Copyright Clearance Center. Reprinted/adapted with permission from Ref. [35]. 2022, Elsevier and Copyright Clearance Center.

Chiller Plant No. | Chiller Cooling Capacity at STD (kW) | Total Units | Total Cooling Capacity (kW) | Cooling Distribution among Chillers (%) | Energy Consumption (kWh/year) | LCC MMcup | |||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | ||||||

56 | 98.2 | 98.2 | 98.2 | 273.0 | 4 | 564 | 17/17/17/48 | 445.0 | 809.77 |

57 | 98.2 | 98.2 | 119.0 | 229.9 | 4 | 543 | 18/18/22/42 | 412.6 | 763.93 |

58 | 98.2 | 98.2 | 119.0 | 273.0 | 4 | 585 | 17/17/20/46 | 416.5 | 775.88 |

59 | 98.2 | 98.2 | 135.1 | 229.9 | 4 | 559 | 17/17/24/41 | 410.6 | 763.35 |

60 | 98.2 | 98.2 | 151.2 | 199.8 | 4 | 545 | 18/18/28/36 | 434.5 | 797.67 |

61 | 98.2 | 98.2 | 151.2 | 203.1 | 4 | 548 | 18/18/27/37 | 434.2 | 797.41 |

62 | 98.2 | 98.2 | 151.2 | 229.9 | 4 | 575 | 17/17/26/40 | 411.7 | 767.57 |

63 | 98.2 | 98.2 | 161.7 | 181.3 | 4 | 536 | 18/18/30/34 | 418.6 | 773.63 |

64 | 98.2 | 98.2 | 161.7 | 199.8 | 4 | 555 | 18/18/29/36 | 432.1 | 796.08 |

65 | 98.2 | 98.2 | 161.7 | 203.1 | 4 | 558 | 18/18/28/36 | 430.5 | 794.19 |

66 | 98.2 | 98.2 | 161.7 | 229.9 | 4 | 585 | 17/17/27/39 | 407.7 | 764.42 |

67 | 98.2 | 98.2 | 181.3 | 181.3 | 4 | 556 | 18/18/32/32 | 423.9 | 783.34 |

68 | 98.2 | 98.2 | 181.3 | 199.8 | 4 | 574 | 17/17/31/35 | 433.7 | 799.89 |

69 | 98.2 | 98.2 | 181.3 | 203.1 | 4 | 577 | 17/17/31/35 | 432.5 | 798.58 |

70 | 98.2 | 119.0 | 119.0 | 203.1 | 4 | 537 | 18/22/22/38 | 399.4 | 752.36 |

71 | 98.2 | 119.0 | 119.0 | 229.9 | 4 | 564 | 17/21/21/41 | 400.6 | 750.58 |

72 | 98.2 | 119.0 | 135.1 | 199.8 | 4 | 549 | 18/22/24/36 | 411.4 | 768.97 |

73 | 98.2 | 119.0 | 135.1 | 203.1 | 4 | 553 | 18/21/24/37 | 413.1 | 771.65 |

74 | 98.2 | 119.0 | 135.1 | 229.9 | 4 | 580 | 17/20/23/39 | 406.8 | 757.75 |

75 | 98.2 | 119.0 | 151.2 | 181.3 | 4 | 547 | 18/22/28/33 | 399.3 | 750.35 |

76 | 98.2 | 119.0 | 151.2 | 199.8 | 4 | 566 | 17/21/27/35 | 407.8 | 765.13 |

77 | 98.2 | 119.0 | 151.2 | 203.1 | 4 | 569 | 17/21/26/35 | 409.3 | 767.41 |

78 | 98.2 | 119.0 | 161.7 | 161.7 | 4 | 538 | 18/22/30/30 | 402.1 | 756.76 |

79 | 98.2 | 119.0 | 161.7 | 181.3 | 4 | 557 | 18/21/29/32 | 407.1 | 759.09 |

80 | 98.2 | 119.0 | 161.7 | 199.8 | 4 | 576 | 17/21/28/35 | 421.1 | 781.27 |

81 | 98.2 | 119.0 | 161.7 | 203.1 | 4 | 579 | 17/21/28/35 | 423.0 | 784.39 |

82 | 98.2 | 119.0 | 181.3 | 181.3 | 4 | 577 | 17/21/31/31 | 418.9 | 776.52 |

83 | 98.2 | 135.1 | 135.1 | 181.3 | 4 | 546 | 18/25/25/33 | 410.0 | 765.87 |

84 | 98.2 | 135.1 | 135.1 | 199.8 | 4 | 565 | 17/24/24/35 | 418.0 | 767.00 |

85 | 98.2 | 135.1 | 135.1 | 203.1 | 4 | 568 | 17/24/24/36 | 418.8 | 781.43 |

86 | 98.2 | 135.1 | 151.2 | 161.7 | 4 | 543 | 18/25/28/30 | 419.1 | 777.68 |

87 | 98.2 | 135.1 | 151.2 | 181.3 | 4 | 562 | 17/24/27/32 | 418.6 | 773.63 |

88 | 98.2 | 135.1 | 151.2 | 199.8 | 4 | 581 | 17/23/26/34 | 432.1 | 796.08 |

89 | 98.2 | 135.1 | 151.2 | 203.1 | 4 | 584 | 17/23/26/35 | 430.5 | 794.19 |

90 | 98.2 | 135.1 | 161.7 | 161.7 | 4 | 553 | 18/24/29/29 | 407.7 | 764.42 |

91 | 98.2 | 135.1 | 161.7 | 181.3 | 4 | 573 | 17/23/28/31 | 405.7 | 761.68 |

92 | 98.2 | 151.2 | 151.2 | 151.2 | 4 | 549 | 18/27/27/27 | 453.7 | 825.76 |

93 | 98.2 | 151.2 | 151.2 | 161.7 | 4 | 559 | 17/27/27/29 | 442.2 | 812.07 |

94 | 98.2 | 151.2 | 151.2 | 181.3 | 4 | 579 | 17/26/26/31 | 429.3 | 794.90 |

95 | 98.2 | 151.2 | 161.7 | 161.7 | 4 | 569 | 17/26/28/28 | 440.7 | 810.60 |

96 | 98.2 | 161.7 | 161.7 | 161.7 | 4 | 579 | 17/28/28/28 | 442.5 | 814.10 |

97 | 119.0 | 119.0 | 119.0 | 181.3 | 4 | 537 | 22/22/22/34 | 392.2 | 736.27 |

98 | 119.0 | 119.0 | 119.0 | 199.8 | 4 | 555 | 21/21/21/36 | 413.3 | 767.54 |

99 | 119.0 | 119.0 | 119.0 | 203.1 | 4 | 558 | 21/21/21/36 | 414.8 | 769.64 |

100 | 119.0 | 119.0 | 119.0 | 229.9 | 4 | 585 | 20/20/20/39 | 411.8 | 768.95 |

101 | 119.0 | 119.0 | 135.1 | 181.3 | 4 | 552 | 22/22/24/33 | 387.7 | 718.53 |

102 | 119.0 | 119.0 | 135.1 | 199.8 | 4 | 571 | 21/21/24/35 | 405.5 | 760.66 |

103 | 119.0 | 119.0 | 135.1 | 203.1 | 4 | 574 | 21/21/23/35 | 407.9 | 764.44 |

104 | 119.0 | 119.0 | 151.2 | 151.2 | 4 | 539 | 22/22/28/28 | 411.6 | 765.70 |

105 | 119.0 | 119.0 | 151.2 | 161.7 | 4 | 549 | 22/22/27/29 | 402.3 | 754.34 |

106 | 119.0 | 119.0 | 151.2 | 181.3 | 4 | 568 | 21/21/26/32 | 394.8 | 745.03 |

107 | 119.0 | 119.0 | 161.7 | 161.7 | 4 | 559 | 21/21/29/29 | 402.5 | 755.48 |

108 | 119.0 | 119.0 | 161.7 | 181.3 | 4 | 578 | 21/21/28/31 | 392.9 | 743.90 |

109 | 119.0 | 135.1 | 135.1 | 151.2 | 4 | 538 | 22/25/25/28 | 421.5 | 780.67 |

110 | 119.0 | 135.1 | 135.1 | 161.7 | 4 | 548 | 22/24/24/29 | 411.8 | 767.60 |

111 | 119.0 | 135.1 | 135.1 | 181.3 | 4 | 567 | 21/24/24/32 | 401.6 | 755.48 |

112 | 119.0 | 135.1 | 135.1 | 199.8 | 4 | 586 | 20/23/23/34 | 415.2 | 777.81 |

113 | 119.0 | 135.1 | 151.2 | 151.2 | 4 | 554 | 21/24/27/27 | 422.5 | 783.74 |

114 | 119.0 | 135.1 | 151.2 | 161.7 | 4 | 564 | 21/24/27/28 | 411.7 | 770.18 |

115 | 119.0 | 135.1 | 151.2 | 181.3 | 4 | 583 | 20/23/26/31 | 403.4 | 760.75 |

116 | 119.0 | 135.1 | 161.7 | 161.7 | 4 | 574 | 21/23/28/28 | 412.5 | 772.18 |

117 | 119.0 | 151.2 | 151.2 | 151.2 | 4 | 570 | 21/26/26/26 | 444.5 | 816.84 |

118 | 119.0 | 151.2 | 151.2 | 161.7 | 4 | 580 | 20/26/26/28 | 432.1 | 801.51 |

119 | 135.1 | 135.1 | 135.1 | 135.1 | 4 | 537 | 25/25/25/25 | 464.4 | 841.26 |

120 | 135.1 | 135.1 | 135.1 | 151.2 | 4 | 553 | 24/24/24/27 | 447.8 | 819.40 |

121 | 135.1 | 135.1 | 135.1 | 161.7 | 4 | 563 | 24/24/24/29 | 437.1 | 804.99 |

122 | 135.1 | 135.1 | 135.1 | 181.3 | 4 | 583 | 23/23/23/31 | 424.8 | 790.61 |

123 | 135.1 | 135.1 | 151.2 | 151.2 | 4 | 569 | 24/24/26/26 | 447.3 | 820.04 |

124 | 135.1 | 135.1 | 151.2 | 161.7 | 4 | 579 | 23/23/26/28 | 435.6 | 805.29 |

125 | 135.1 | 151.2 | 151.2 | 151.2 | 4 | 585 | 23/26/26/26 | 468.0 | 851.37 |

**Table A3.**Main characteristics and performance data of the chiller plants relying on 5 units (obtained from Diaz Torres [33], Diaz et al [34] Diaz et al [35]. Reprinted/adapted with permission from Ref. [34]. 2021, Elsevier and Copyright Clearance Center. Reprinted/adapted with permission from Ref. [35]. 2022, Elsevier and Copyright Clearance Center.

Chiller Plant No. | Chiller Cooling Capacity at STD (kW) | Total Units | Total Cooling Capacity (kW) | Cooling Distribution among Chillers (%) | Energy Consumption (kWh/year) | LCC MM Cup | ||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | ||||||

126 | 98.2 | 98.2 | 98.2 | 98.2 | 151.2 | 5 | 541 | 18/18/18/18/28 | 436.8 | 825.18 |

127 | 98.2 | 98.2 | 98.2 | 98.2 | 161.7 | 5 | 551 | 18/18/18/18/29 | 435.7 | 825.45 |

128 | 98.2 | 98.2 | 98.2 | 98.2 | 181.3 | 5 | 571 | 17/17/17/17/32 | 437.5 | 830.07 |

129 | 98.2 | 98.2 | 98.2 | 119.0 | 135.1 | 5 | 546 | 18/18/18/22/25 | 410.2 | 788.54 |

130 | 98.2 | 98.2 | 98.2 | 119.0 | 151.2 | 5 | 562 | 17/17/17/21/27 | 411.3 | 793.40 |

131 | 98.2 | 98.2 | 98.2 | 119.0 | 161.7 | 5 | 573 | 17/17/17/21/28 | 408.6 | 791.19 |

132 | 98.2 | 98.2 | 98.2 | 135.1 | 135.1 | 5 | 561 | 17/17/17/24/24 | 421.6 | 806.30 |

133 | 98.2 | 98.2 | 98.2 | 135.1 | 151.2 | 5 | 578 | 17/17/17/23/26 | 418.9 | 805.36 |

134 | 98.2 | 98.2 | 119.0 | 119.0 | 119.0 | 5 | 552 | 18/18/22/22/22 | 410.5 | 787.51 |

135 | 98.2 | 98.2 | 119.0 | 119.0 | 135.1 | 5 | 567 | 17/17/21/21/24 | 402.6 | 781.18 |

136 | 98.2 | 98.2 | 119.0 | 119.0 | 151.2 | 5 | 583 | 17/17/20/20/26 | 400.6 | 781.26 |

137 | 98.2 | 98.2 | 119.0 | 135.1 | 135.1 | 5 | 583 | 17/17/20/23/23 | 405.8 | 788.54 |

138 | 98.2 | 119.0 | 119.0 | 119.0 | 119.0 | 5 | 573 | 17/21/21/21/21 | 410.5 | 791.43 |

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**Figure 1.**Scatter plot of the estimated residuals vs. (Ŷi): (

**a**) Equation (31), (

**b**) Equation (32), (

**c**) Equation (33), (

**d**) Equation (34), (

**e**) Equation (35) and (

**f**) Equation (36).

**Figure 2.**Statistical relation between (

**a**) energy consumption and total number of chillers, (

**b**) energy consumption and total installed cooling capacity, (

**c**) energy consumption and cooling distribution among chillers, (

**d**) LCC and total number of chillers, (

**e**) LCC and total installed cooling capacity, (

**f**) LCC and cooling distribution among chillers.

Variable | Symbol | Classification | Characteristic | Total of Values | Unit |
---|---|---|---|---|---|

Total of chiller | N_{ch} | Independent | Numerical | 138 | - |

Total cooling capacity installed | Q_{ch} | Independent | Numerical | 138 | kW |

Cooling distribution among chillers | Cd_{ch} | Independent | Ordinal | 138 | - |

Annual energy consumption | AEC | Dependent | Numerical | 138 | MWh/year |

Life Cycle Cost | LCC | Dependent | Numerical | 138 | MMCup * |

**Table 2.**Mathematical transformation criteria for the ordinal variable cooling distribution among chillers.

Total Units | Arrangement Type | Mathematical Expression | Constrains | Equation | Classification | Scale |
---|---|---|---|---|---|---|

2 | Symmetrical | ${c}_{1}={c}_{2}$ | - | (18) | S_{1} | 48 |

Similar | ${c}_{1}\approx {c}_{2}$ | CV ≤ 7 | (19) | S_{2} | 46 | |

Asymmetrical type1 | ${c}_{1}\ne {c}_{2}$ | CV ≤ 20 | (20) | S_{3} | 44 | |

Asymmetrical type2 | ${c}_{1}\ne {c}_{2}$ | CV > 20 | (21) | S_{4} | 42 | |

3 | Symmetrical | ${c}_{1}={c}_{2}={c}_{3}$ | - | (22) | S_{1} | 38 |

Similar | ${c}_{1}={c}_{2}\approx {c}_{3}$ ${c}_{1}\approx {c}_{2}\approx {c}_{3}$ | CV ≤ 15 | (23) | S_{2} | 36 | |

Asymmetrical type1 | ${c}_{1}={c}_{2}\ne {c}_{3}$ | ${c}_{1}<{c}_{3}$ | (24) | S_{3} | 34 | |

Asymmetrical type2 | ${c}_{1}={c}_{2}\ne {c}_{3}$ | ${c}_{1}>{c}_{3}$ | (25) | S_{4} | 32 | |

Asymmetrical type3 | ${c}_{1}\ne {c}_{2}\ne {c}_{3}$ | CV ≥ 18 | (26) | S_{5} | 30 | |

4 | Symmetrical (similar) | ${c}_{1}={c}_{2}={c}_{3}={c}_{4}$ ${c}_{1}={c}_{2}={c}_{3}\approx {c}_{4}$${c}_{1}\approx {c}_{2}\approx {c}_{3}\approx {c}_{4}$ | CV ≤ 11 | (27) | S_{1} | 26 |

Asymmetrical type1 | ${c}_{1}={c}_{2}={c}_{3}\ne {c}_{4}$ | ${c}_{1}<{c}_{4}$ | (28) | S_{2} | 24 | |

Asymmetrical type2 | ${c}_{1}={c}_{2}={c}_{3}\ne {c}_{4}$ | c_{1} > c_{4} | (29) | S_{3} | 22 | |

Asymmetrical type3 | ${c}_{1}={c}_{2}\ne {c}_{3}={c}_{4}$ ${c}_{1}={c}_{2}\ne {c}_{3}\approx {c}_{4}$ | - | (30) | S_{4} | 20 | |

Asymmetrical type4 | ${c}_{1}={c}_{2}\ne {c}_{3}\ne {c}_{4}$ | - | (31) | S_{5} | 18 | |

Asymmetrical type5 | ${c}_{1}\ne {c}_{2}\ne {c}_{3}\ne {c}_{4}$ | CV > 13 | (32) | S_{6} | 16 | |

5 | Symmetrical (similar) | ${c}_{1}={c}_{2}={c}_{3}={c}_{4}={c}_{5}$ ${c}_{1}={c}_{2}={c}_{3}={c}_{4}\approx {c}_{5}$ | CV ≤ 9 | (33) | S_{1} | 12 |

Asymmetrical type1 | ${c}_{1}={c}_{2}={c}_{3}={c}_{4}\ne {c}_{5}$ | - | (34) | S_{2} | 10 | |

Asymmetrical type2 | ${c}_{1}={c}_{2}={c}_{3}\ne {c}_{4}={c}_{5}$ | - | (35) | S_{3} | 8 | |

Asymmetrical type3 | ${c}_{1}={c}_{2}={c}_{3}\ne {c}_{4}\ne {c}_{5}$ | - | (36) | S_{4} | 6 | |

Asymmetrical type4 | ${c}_{1}={c}_{2}\ne {c}_{3}={c}_{4}\ne {c}_{5}$ | - | (37) | S_{5} | 4 | |

Asymmetrical type5 | ${c}_{1}\ne {c}_{2}\ne {c}_{3}\ne {c}_{4}\ne {c}_{5}$ | - | (38) | S_{6} | 2 |

**Table 3.**Numerical transformation of the ordinal variable, cooling distribution among chiller, for the chiller plant.

Chiller Plant No. | Classification | Scale | Chiller Plant No. | Classification | Scale | Chiller Plant No. | Classification | Scale |
---|---|---|---|---|---|---|---|---|

1 | s_{4} | 44 | 47 | s_{4} | 32 | 93 | s_{5} | 18 |

2 | s_{4} | 44 | 48 | s_{2} | 36 | 94 | s_{5} | 18 |

3 | s_{4} | 44 | 49 | s_{4} | 32 | 95 | s_{5} | 18 |

4 | s_{4} | 44 | 50 | s_{1} | 38 | 96 | s_{3} | 22 |

5 | s_{1} | 48 | 51 | s_{2} | 36 | 97 | s_{2} | 24 |

6 | s_{2} | 48 | 52 | s_{2} | 36 | 98 | s_{2} | 24 |

7 | s_{3} | 34 | 53 | s_{2} | 36 | 99 | s_{2} | 24 |

8 | s_{5} | 30 | 54 | s_{2} | 36 | 100 | s_{2} | 24 |

9 | s_{5} | 30 | 55 | s_{2} | 36 | 101 | s_{5} | 18 |

10 | s_{5} | 30 | 56 | s_{2} | 24 | 102 | s_{5} | 18 |

11 | s_{5} | 30 | 57 | s_{5} | 18 | 103 | s_{5} | 18 |

12 | s_{5} | 30 | 58 | s_{5} | 18 | 104 | s_{4} | 20 |

13 | s_{5} | 30 | 59 | s_{5} | 18 | 105 | s_{5} | 18 |

14 | s_{5} | 30 | 60 | s_{5} | 18 | 106 | s_{5} | 18 |

15 | s_{4} | 32 | 61 | s_{5} | 18 | 107 | s_{4} | 20 |

16 | s_{3} | 34 | 62 | s_{5} | 18 | 108 | s_{5} | 18 |

17 | s_{5} | 30 | 63 | s_{5} | 18 | 109 | s_{1} | 26 |

18 | s_{5} | 30 | 64 | s_{5} | 18 | 110 | s_{5} | 18 |

19 | s_{5} | 30 | 65 | s_{5} | 18 | 111 | s_{5} | 18 |

20 | s_{5} | 30 | 66 | s_{5} | 18 | 112 | s_{5} | 18 |

21 | s_{5} | 30 | 67 | s_{4} | 20 | 113 | s_{1} | 26 |

22 | s_{5} | 30 | 68 | s_{5} | 18 | 114 | s_{6} | 16 |

23 | s_{5} | 30 | 69 | s_{5} | 18 | 115 | s_{6} | 16 |

24 | s_{4} | 32 | 70 | s_{5} | 18 | 116 | s_{5} | 18 |

25 | s_{3} | 34 | 71 | s_{5} | 18 | 117 | s_{1} | 26 |

26 | s_{3} | 34 | 72 | s_{6} | 16 | 118 | s_{1} | 26 |

27 | s_{5} | 30 | 73 | s_{6} | 16 | 119 | s_{1} | 26 |

28 | s_{5} | 30 | 74 | s_{6} | 16 | 120 | s_{1} | 26 |

29 | s_{5} | 30 | 75 | s_{6} | 16 | 121 | s_{1} | 26 |

30 | s_{5} | 30 | 76 | s_{6} | 16 | 122 | s_{2} | 24 |

31 | s_{5} | 30 | 77 | s_{5} | 18 | 123 | s_{1} | 26 |

32 | s_{4} | 32 | 78 | s_{5} | 18 | 124 | s_{1} | 26 |

33 | s_{5} | 30 | 79 | s_{6} | 16 | 125 | s_{1} | 26 |

34 | s_{3} | 34 | 80 | s_{6} | 16 | 126 | s_{2} | 10 |

35 | s_{5} | 30 | 81 | s_{6} | 16 | 127 | s_{2} | 10 |

36 | s_{5} | 30 | 82 | s_{5} | 18 | 128 | s_{2} | 10 |

37 | s_{5} | 30 | 83 | s_{5} | 18 | 129 | s_{4} | 6 |

38 | s_{4} | 32 | 84 | s_{5} | 18 | 130 | s_{4} | 6 |

39 | s_{2} | 36 | 85 | s_{5} | 18 | 131 | s_{4} | 6 |

40 | s_{5} | 30 | 86 | s_{6} | 16 | 132 | s_{4} | 6 |

41 | s_{4} | 32 | 87 | s_{6} | 16 | 133 | s_{4} | 6 |

42 | s_{5} | 30 | 88 | s_{6} | 16 | 134 | s_{3} | 8 |

43 | s_{3} | 34 | 89 | s_{6} | 16 | 135 | s_{5} | 4 |

44 | s_{2} | 36 | 90 | s_{5} | 18 | 136 | s_{5} | 4 |

45 | s_{2} | 36 | 91 | s_{6} | 16 | 137 | s_{5} | 4 |

46 | s_{5} | 30 | 92 | s_{3} | 22 | 138 | s_{1} | 12 |

Statistical Test | AEc (Y) | LCC (Y) | ||
---|---|---|---|---|

Statistic | p-Value | Statistic | p-Value | |

Shapiro–Wilk (SW) | 0.904483 | 1.54798 × 10^{−12} | 0.957847 | 0.00242575 |

Anderson–Darling (A^{2}) | 3.49206 | 9.01993 × 10^{−9} | 1.52435 | 0.000603096 |

**Table 5.**Results of the statistical Pre-test: regression analysis between variable assuming homoscedasticity.

Equation | Bivariate Relationship (Yi = mXi + n) | Coefficients (m; n) | Standard Error (m; n) | t-Statistic (m; n) | Prob. (m; n) | R-Squared |
---|---|---|---|---|---|---|

(39) | (AEc) vs. (N_{ch}) | (−25.86455; 527.1655) | (2.56915; 9.5579) | (−10.06736; 55.15443) | (0.0000; 0.0000) | 0.427011 |

(40) | (AEc) vs. (Q_{ch}) | (−0.171896; 529.7965) | (0.162240; 91.67009) | (−1.059517; 5.779382) | (0.2912; 0.0000) | 0.008187 |

(41) | (AEc) vs. (Cd_{ch}) | (2.238676; 379.1052) | (0.168833; 4.343118) | (13.25968; 87.28871) | (0.0000; 0.0000) | 0.563850 |

(42) | (LCC) vs. (N_{ch}) | (−9.305803; 821.8668) | (3.678271; 13.68425) | (−2.529939; 60.05934) | (0.0125; 0.0000) | 0.044948 |

(43) | (LCC) vs. (Q_{ch}) | (0.147560; 704.5334) | (0.180214; 101.8261) | (0.818803; 6.918990) | (0.4143; 0.0000) | 0.0.04906 |

(44) | (LCC) vs. (Cd_{ch}) | (1.177679; 759.6843) | (0.264905; 6.814493) | (4.445666; 111.4807) | (0.0000; 0.0000) | 0.126884 |

Bivariate Relationship | Equations | p-Value | Heteroscedasticity | Mathematical Model | R-Squared |
---|---|---|---|---|---|

(AEC) vs. (N_{ch}) | (39) | 0.0240 | Yes | $\left|{\hat{u}}_{i}\right|=29.4254018319-3.278402({N}_{ch})$ | 0.036910 |

(AEC) vs. (Q_{ch}) | (40) | 0.4138 | No | $\left|{\hat{u}}_{i}\right|=68.87957-0.083524({Q}_{ch})$ | 0.004916 |

(AEC) vs. (Cd_{ch}) | (41) | 0.1418 | No | $\left|{\hat{u}}_{i}\right|=11.29616-0.148112(C{d}_{ch})$ | 0.015805 |

(LCC) vs. (N_{ch}) | (42) | 0.0189 | Yes | $\left|{\hat{u}}_{i}\right|=42.76556-4.850004({N}_{ch})$ | 0.039849 |

(LCC) vs. (Q_{ch}) | (43) | 0.6400 | No | $\left|{\hat{u}}_{i}\right|=-2.408414+0.04860({Q}_{ch})$ | 0.001585 |

(LCC) vs. (Cd_{ch}) | (44) | 0.0518 | No | $\left|{\hat{u}}_{i}\right|=16.86090+0.292584(C{d}_{ch})$ | 0.027534 |

Equation | Bivariate Relationship (Yi = mXi + n) | Coefficients (m; n) | Standard Error (m; n) | t-Statistic (m; n) | Prob. (m; n) | R-Squared |
---|---|---|---|---|---|---|

(39) * | (AEc) vs. (N_{ch}) | (−25.86455; 527.1655) | (3.210059; 12.35129) | (−8.057346; 42.68102) | (0.0000; 0.0000) | 0.427011 |

(42) * | (LCC) vs. (N_{ch}) | (−9.305803; 821.8668) | (4.344333; 16.79198) | (−2.142055; 48.94401) | (0.0340; 0.0000) | 0.044948 |

(44) * | (LCC) vs. (Cd_{ch}) | (1.177679; 759.6843) | (0.313664; 7.307557) | (3.754583; 103.9587) | (0.0030; 0.0000) | 0.126884 |

**Table 8.**Analysis of the normality condition of the estimate value (Ŷi) of the dependent variable (p-value).

Statistic | p-Value | Statistic | p-Value | Statistic | p-Value | |
---|---|---|---|---|---|---|

AEc (Ŷi) Equation (39) * | AEc (Ŷi) Equation (40) | AEc (Ŷi) Equation (41) | ||||

(SW) | 0.831547 | 2.78972 × 10^{−11} | 0.954328 | 0.000153493 | 0.960179 | 0.000483601 |

(A^{2}) | 11.5387 | <0.01 | 1.49567 | ≥0.10 | 2.44905 | ≥0.10 |

LCC (Ŷi) Equation (42) * | LCC (Ŷi) Equation (43) | LCC (Ŷi) Equation (44) * | ||||

(SW) | 0.831547 | 2.78972 × 10^{−11} | 0.954328 | 0.0001534 | 0.960179 | 0.000483 |

(A^{2}) | 11.5387 | <0.01 | 1.49567 | ≥0.10 | 2.44905 | <0.10 |

Operational Variables (Dependant) | Design Variables (Independent) | |||
---|---|---|---|---|

Cooling Distribution among Chillers | Total Chillers | Total Installed Cooling Capacity | ||

Energy consumption | r_{s} | - | −0.625 ** | −0.086 |

Sig. (bilateral) | - | 0.000 | 0.314 | |

τ | 0.559 ** | - | - | |

Sig. (bilateral) | 0.000 | - | - | |

LCC | r_{s} | - | −0.135 | 0.063 |

Sig. (bilateral) | - | 0.113 | 0.463 | |

τ | 0.289 ** | - | - | |

Sig. (bilateral) | 0.001 | - | - |

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## Share and Cite

**MDPI and ACS Style**

Torres, Y.D.; Gullo, P.; Herrera, H.H.; Torres del Toro, M.; Guerra, M.A.Á.; Ortega, J.I.S.; Speerforck, A.
Statistical Analysis of Design Variables in a Chiller Plant and Their Influence on Energy Consumption and Life Cycle Cost. *Sustainability* **2022**, *14*, 10175.
https://doi.org/10.3390/su141610175

**AMA Style**

Torres YD, Gullo P, Herrera HH, Torres del Toro M, Guerra MAÁ, Ortega JIS, Speerforck A.
Statistical Analysis of Design Variables in a Chiller Plant and Their Influence on Energy Consumption and Life Cycle Cost. *Sustainability*. 2022; 14(16):10175.
https://doi.org/10.3390/su141610175

**Chicago/Turabian Style**

Torres, Yamile Díaz, Paride Gullo, Hernán Hernández Herrera, Migdalia Torres del Toro, Mario A. Álvarez Guerra, Jorge Iván Silva Ortega, and Arne Speerforck.
2022. "Statistical Analysis of Design Variables in a Chiller Plant and Their Influence on Energy Consumption and Life Cycle Cost" *Sustainability* 14, no. 16: 10175.
https://doi.org/10.3390/su141610175