# A Non-Probabilistic Solution for Uncertainty and Sensitivity Analysis on Techno-Economic Assessments of Biodiesel Production with Interval Uncertainties

^{*}

## Abstract

**:**

## 1. Introduction

_{x}, SO

_{x}, CO

_{2}, hydrocarbons, carbon monoxide, and particulate matter, resulting in various environmental pollution problems and danger on human health [1,2,3,4,5,6,7,8,9]. A great number of scientists are investigating other harmless, economic, and clean energy sources for the sake of the reduction of these adverse and negative effects. Being a valuable renewable energy resource, biodiesel is friendly to the natural environment and human health, compared to the traditional fossil fuels [10,11,12,13,14,15]. Various feedstocks-derived biodiesel production have been reported, for example, palm oil [16], waste cooking oil [17,18,19], vegetable oils [20,21], soybean oil [22,23,24,25], Jatropha curcas L. [26], algae [27,28], microalgae [28,29,30], Oleaginous yeast [31,32], lignocellulosic biomass [33], used frying oil [34], waste cottonseed oil with heterogeneous catalyst [35,36], Annona squamosa L. seed oil with heterogeneous catalyst [36,37], Butanol and pentanol [38], etc., and recent advances in biofeedstocks and biofuels have also been reviewed in [39].

## 2. Materials and Methods

#### 2.1. Several Important Concepts in the TEA for a Biodiesel Production

_{i}, FC

_{i}, OC

_{i}, BPC

_{i}, TAX

_{i}and TBS

_{i}are maintenance cost, feedstock cost, operating cost, byproduct credit, total taxation, and total biodiesel sale for the ith year, respectively.

#### 2.2. NPRI for Measuring Economically Feasible Extent of Biodiesel Production

#### 2.2.1. NPRI for Problems with Interval Parameters

#### 2.2.2. NPRI for Economically Feasible Degree in the TEA of Biodiesel Production

#### 2.3. Evaluation Procedure of the NPRI

#### 2.4. SA of NPRI for Economical Feasibility of Biodiesel Production with Regards to Uncertain Interval Parameter

## 3. Results and Discussion

#### 3.1. Evaluation of NPRI for Biodiesel Production

^{−1}by using Equation (26). A value of 1.2104 × 10

^{−1}for ${\eta}_{s}$ implies that the project will not be profitable to a great extent, in other words, a considerable part of the outcomes may be economically infeasible under the uncertain interval parameters shown in Table 1.

^{−1}, also indicating that the project is partially economically feasible, according to the discussion in Section 2.2. Thus, the two methods have the same decisions. It is noted that the introduced method in this work is more rational than that in the previous work [59], which is subjected to the assumption on probabilistic distribution and different assumptions can lead to different results for EIP.

#### 3.2. Evaluation of Sensitivity Analysis for Biodiesel Production with Respect to Interval Parameter

## 4. Conclusions

^{−1}with the interval parameters summarized in Table 1. Price of biodiesel, price of feedstock, and operating cost can cause distinct influence on the economical feasibility of biodiesel production. Compared with our previous study [59], this work has the same decision on TEA and the same importance ranking for uncertain parameters. This method is free of the assumption on distribution, but the previous method is subjected to this assumption in which different assumptions on distribution can result in different results for EIP.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

$\mathrm{BP}$ | biodiesel price |

$\mathrm{BPC}$ | byproduct credit |

${\mathrm{BPC}}_{i}$ | byproduct credit of the ith year |

$\mathrm{CC}$ | capital cost |

$\mathrm{CE}$ | conversion efficiency from feedstock to biodiesel |

$d$ | depreciation rate |

$\mathrm{FC}$ | feedstock cost |

${\mathrm{FC}}_{i}$ | feedstock cost of the ith year |

$\mathrm{FP}$ | feedstock price |

$\mathrm{FU}$ | annual total feedstock consumption |

$\mathrm{GCF}$ | glycerol conversion factor |

$\mathrm{GP}$ | glycerol price |

$\mathrm{LCC}$ | life cycle cost |

$\mathrm{MC}$ | maintenance cost |

${\mathrm{MC}}_{i}$ | maintenance cost of the ith year |

$\mathrm{MR}$ | maintenance rate |

NPRI | non-probabilistic reliability index |

$\mathrm{OC}$ | operating cost |

${\mathrm{OC}}_{i}$ | operating cost of the ith year |

$\mathrm{OR}$ | operating rate or operating cost of per-ton crude-palm-oil-derived biodiesel production |

$\mathrm{PC}$ | production capacity |

$\mathrm{PP}$ | payback period of the biodiesel production |

${\mathrm{PP}}^{u}$ | allowable upper limit of payback period |

${\mathrm{PWF}}_{n}$ | worth factor in the year n |

$\mathrm{RC}$ | replacement cost |

$r$ | interest rate |

SA | sensitivity analysis |

$\mathrm{SV}$ | salvage value |

$\mathrm{TAX}$ | annual total taxation |

$\mathrm{TBS}$ | annual total biodiesel sales |

$\mathrm{TEA}$ | techno-economic assessments |

$\mathrm{TotalProfit}$ | total profit |

$\mathrm{TR}$ | tax rate |

UA | uncertainty analysis |

$\rho $ | density of the biodiesel |

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**Figure 1.**Diagrammatic presentation for non-probabilistic reliability index (NPRI) of a system with two intervals, (

**a**) normalized limit state function (LSF) intersects with enlarged square box at one side; (

**b**) normalized limit state curve (LSC) intersects with enlarged square box at another side; and (

**c**) normalized limit state curve intersects with enlarged square box at cater-corner point.

**Figure 2.**Variation ranges of total profit (TotalProfit: USD), net present value (NPV: USD), and ${y}_{j}={g}_{j}\left(\mathit{x}\right)$ due to economic and technical uncertainties.

**Figure 3.**Original NPRI ${\eta}_{s}$ and conditional NPRI ${\eta}_{s|{x}_{i}={x}_{ij}}\left(j=1,2,\dots ,10\right)$, in which ${x}_{i}$ is fixed to a value ${x}_{ij}\in \left[{\underset{\xaf}{x}}_{i},{\overline{x}}_{i}\right]$, i.e., ${x}_{i}={x}_{ij}\left(j=1,2,\dots ,10\right)$.

**Figure 4.**Change rate between ${\eta}_{s|{x}_{i}={x}_{ij}}$ and ${\eta}_{s}$ with respect to ${\eta}_{s}$, i.e., $\left({\eta}_{s|{x}_{i}={x}_{ij}}-{\eta}_{s}\right)/{\eta}_{s}\left(j=1,2,\dots ,10\right)$, in which ${x}_{i}$ is fixed to a value ${x}_{ij}\in \left[{\underset{\xaf}{x}}_{i},{\overline{x}}_{i}\right]$, i.e., ${x}_{i}={x}_{ij}$.

**Figure 5.**Point figure for conditional NPRI ${\eta}_{s|{x}_{i}={x}_{ij}}$ with ${x}_{i}={x}_{ij}\left(i=1,2,\dots ,8;j=1,2,\dots ,10\right)$.

**Figure 6.**Point figure for change rate between ${\eta}_{s|{x}_{i}={x}_{ij}}$ and ${\eta}_{s}$ with respect to ${\eta}_{s}$, $\left({\eta}_{s|{x}_{i}={x}_{ij}}-{\eta}_{0}\right)/{\eta}_{0}$ with ${x}_{i}={x}_{ij}\left(i=1,2,\dots ,8;j=1,2,\dots ,10\right)$.

Uncertain Parameters | Variation Intervals [${\underset{\mathbf{\xaf}}{\mathit{x}}}_{\mathit{i}}$, ${\overline{\mathit{x}}}_{\mathit{i}}$] |
---|---|

Capital cost (CC: ${x}_{1}$) [42] | [$9 million, $15 million] |

Interest rate (r: ${x}_{2}$) [66] | [4.44%, 13.53%] |

Operating rate (OR: ${x}_{3}$) [42,67,68] | [$37.5/t, $225/t] |

Feedstock price (FP: ${x}_{4}$) [42] | [$200/t, $1200/t] |

Glycerol price (GP: ${x}_{5}$) [70] | [$0.08/kg, $0.2/kg] |

Maintenance rate (MR: ${x}_{6}$) [41,42] | [1%, 2%] |

Biodiesel conversion efficiency (CE: ${x}_{7}$) [69] | [96%, 99%] |

Biodiesel price (BP: ${x}_{8}$) [71] | [$0.66/L, $1.58/L] |

**Table 2.**Results of the proposed sensitivity analysis ${\mathrm{IM}}_{i}\left(i=1,2,\dots ,8\right)$ and ${\mathrm{IMR}}_{i}\left(i=1,2,\dots ,8\right)$.

Parameters | ${\mathbf{IM}}_{\mathit{i}}\mathbf{\left(}\mathit{i}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{\dots}\mathbf{,}\mathbf{8}\mathbf{\right)}$ | ${\mathbf{IMR}}_{\mathit{i}}\mathbf{\left(}\mathit{i}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{\dots}\mathbf{,}\mathbf{8}\mathbf{\right)}$ |
---|---|---|

Capital cost (CC: ${x}_{1}$) | 4.454 × 10^{−3} | 3.680 × 10^{−2} |

Interest rate (r: ${x}_{2}$) | 5.231 × 10^{−3} | 4.322 × 10^{−2} |

Operating rate (OR: ${x}_{3}$) | 4.961 × 10^{−2} | 4.099 × 10^{−1} |

Feedstock price (FP: ${x}_{4}$) | 4.858 × 10^{−1} | 4.013 × 10^{0} |

Glycerol price (GP: ${x}_{5}$) | 2.865 × 10^{−3} | 2.367 × 10^{−2} |

Maintenance rate (MR: ${x}_{6}$) | 6.643 × 10^{−4} | 5.488 × 10^{−3} |

Biodiesel conversion efficiency (CE: ${x}_{7}$) | 9.302 × 10^{−3} | 7.685 × 10^{−2} |

Biodiesel price (BP: ${x}_{8}$) | 3.257 × 10^{−1} | 2.691 × 10^{0} |

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## Share and Cite

**MDPI and ACS Style**

Tang, Z.-C.; Xia, Y.; Xue, Q.; Liu, J.
A Non-Probabilistic Solution for Uncertainty and Sensitivity Analysis on Techno-Economic Assessments of Biodiesel Production with Interval Uncertainties. *Energies* **2018**, *11*, 588.
https://doi.org/10.3390/en11030588

**AMA Style**

Tang Z-C, Xia Y, Xue Q, Liu J.
A Non-Probabilistic Solution for Uncertainty and Sensitivity Analysis on Techno-Economic Assessments of Biodiesel Production with Interval Uncertainties. *Energies*. 2018; 11(3):588.
https://doi.org/10.3390/en11030588

**Chicago/Turabian Style**

Tang, Zhang-Chun, Yanjun Xia, Qi Xue, and Jie Liu.
2018. "A Non-Probabilistic Solution for Uncertainty and Sensitivity Analysis on Techno-Economic Assessments of Biodiesel Production with Interval Uncertainties" *Energies* 11, no. 3: 588.
https://doi.org/10.3390/en11030588