Effect of Market Price Uncertainties on the Design of Optimal

Mar 10, 2014 - For a more comprehensive analysis, we note that the scenario definition for the market price uncertainties and investment costs (capita...
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Effect of Market Price Uncertainties on the Design of Optimal Biorefinery SystemsA Systematic Approach Peam Cheali,† Alberto Quaglia,† Krist V. Gernaey,‡ and Gürkan Sin*,† †

CAPEC, Department of Chemical and Biochemical Engineering, Technical University of Denmark (DTU), Building 229, DK-2800 Lyngby, Denmark ‡ PROCESS, Department of Chemical and Biochemical Engineering, Technical University of Denmark (DTU), Building 229, DK-2800 Lyngby, Denmark ABSTRACT: This paper presents the development of a computer-aided decision support tool for identifying optimal biorefinery concepts for production of biofuels at an early design stage. To this end, a framework that uses a superstructure-based process synthesis approach integrated with uncertainty analysis is used. We demonstrate the application of the tool for generating optimal biorefinery concepts for a lignocellulosic biorefinery. In particular, we highlight the management of various sources of data, the superstructure (integrated thermochemical and biochemical conversion routes) needed to represent the design space, generic but simple models describing the processing tasks, and the formulation and solution of an MINLP problem under deterministic and stochastic conditions to identify the optimal processing route for multiple raw materials and products. Furthermore, we evaluate the impact of market price uncertainties on the optimal solutions and calculate the associated risk to enable informed and risk-aware decisions.

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INTRODUCTION Important drivers such as environmental concern, social factors, sustainability, and the limitation of fossil fuels are expected to shape the future development of the processing and chemical industries. These challenges motivate the development of sustainable technologies for processing renewable feedstock for fuel, chemical, and material production. In a typical biorefinery, the system generally works by processing a biobased feedstock to produce various products such as fuels, chemicals, or power/heat. As there are several feedstock sources, as well as many alternative conversion technologies to choose from to match a range of products, this creates a number of potential processing paths during the early stage of product-process design of biorefinery development.1,2 Therefore, during the early stage of planning and design, it is important to identify the optimal biorefinery processing path with respect to economics, consumption of resources, and sustainability as well as considering the impact of uncertainties on decision making. A number of studies have been published on the synthesis and design of biorefinery networks focusing on different aspects of the challenges and opportunities of such a synthesis and design task. Voll and Marquardt3 explored the use of reaction flux network analysis (RFNA) for synthesis and design of biorefinery processing paths. Pham and El-Halwagi4 proposed a systematic two-stage methodology to reduce the number of processing steps. Posada et al.5 applied an early stage sustainability assessment as a quick screening method to identify the most promising bioethanol derivatives by catalytic conversion. Baliban et al. studied process synthesis and design of the thermochemical conversion of duckweed biomass to gasoline, diesel, and jet fuel.6 Furthermore, they also studied a hybrid process converting coal, biomass, and natural gas to liquid fuels, considering the heat and water integration and © 2014 American Chemical Society

supply chain optimization when identifying the optimal flowsheet.7 Martin and Grossmann evaluated the process optimization considering heat and water integration producing FT-diesel from biomass8 and biodiesel from cooking oil and algae.9 Zondervan et al.10 studied identification of the optimal processing paths of the biochemical platform. A more detailed review of studies on the process synthesis of biorefineries was presented in Yuan et al.11 While each of these studies provided a valuable contribution, however, the scope of the study was always limited to one processing/conversion platform (i.e., biochemical, thermochemical, chemical, or biological platforms). In this study, this challenge is tackled by broadening the scope of biorefinery synthesis to consider thermo-chemical and biochemical platforms, simultaneously. Another challenge during the early stage of biorefinery planning and design is the enormous need for data, which is often not available and hence proper assumptions and simplifications need to be made to manage the complexity of the problem. This is especially complicated when one broadens the scope of biorefinery network design, that is, by simultaneously focusing on different conversion platforms. The data characterization representing each process alternative requires a substantial amount of information: parameters, variables, models of known reactions, thermodynamic properties, process efficiencies resulting in a detailed and complex model, 12,13 and these require the adapted systematic optimization approach to solve the complex problem.7,14 To overcome this challenge, the earlier study15 presented a generic modeling framework to manage the complexity of collecting Received: Revised: Accepted: Published: 6021

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Figure 1. Systematic framework for enterprise-wide optimization under uncertainty.16

and representing the multidisciplinary data needed for superstructure-based optimization of biorefinery systems allowing the formulation of a relatively simple optimization. However, the challenge that generally comes with data and models used in biorefinery synthesis research are the uncertainties, both external (anticipated raw material and product prices, etc.) and technical (e.g., related to process performance metrics). This challenge needs to be formally addressed and is often tackled by ad hoc based scenario analysis. With this background information in mind, the aim of this contribution is to develop a decision support tool for identifying optimal biorefinery concepts at the early stage of the project life cycle, while considering uncertainties inherently present at this stage of project development. To this end, a systematic methodology for process synthesis and design together with formal uncertainty analysis developed earlier16 was adapted for the purpose of biorefinery network design. As mentioned earlier, the developed superstructure with the verified database and models17 was used as an input for optimization, in combination with the definition of a suitable feedstock. Following the definition of the superstructure, different optimization problems were solved: the deterministic problem, the deterministic problem under uncertainties and a stochastic problem were all solved with the final goal to identify the optimal solutions under uncertainties and to calculate the associated risk. Lignocellulosic biorefinery is used as a case study to highlight the application of the tool. The paper is organized as follows: (i) the methodology used in this study is briefly introduced; (ii) the optimal processing

paths are identified and analyzed for different scenarios, including the effect of uncertainties.

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FRAMEWORK The framework containing the steps, data flow, workflow, and solutions is presented in Figure 1. The explanation of each step of the framework is briefly outlined in the following. Step 1. Problem Formulation: (i) Problem Definition; (ii) Superstructure Definition and Data Collection; (iii) Model Selection and Validation. The first step includes the definition of the problem scope as well as the selection of suitable objective functions and optimization scenarios with respect to certain metrics for example economic or business metrics, engineering performance, or sustainability. Superstructure definition together with data collection, model selection, and verification are then performed, respectively. Step 2. Uncertainty Characterization. In this step, the domain of uncertainty is defined. Statistical analysis tools, Monte Carlo simulation and Latin Hypercube Sampling with correlation control18 are therefore integrated with the deterministic problem. First, specific data or parameters need to be selected as uncertain inputs to the optimization problem. Second, the selected data need to be characterized in terms of a probability distribution (e.g., normal or uniform distribution). Third, the correlations between the selected data are analyzed in terms of covariance, such that this information can be incorporated in the sampling if such information is available. Finally, the sampling of uncertain data is performed to generate the possible scenarios. 6022

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Figure 2. Combined superstructure of two biorefinery conversion platforms: thermochemical (top) and biochemical platform (bottom).

Step 6. Flexible Network. In this step, the optimal tradeoff between the investment cost and the ability of the network to adapt its structure in order to mitigate the negative consequences of the uncertainty is identified. The decision making problem under uncertainty is reformulated as a twostage stochastic programming problem. This step is, however, beyond the scope of this study. Step 7. Report Generation. This section presents the results of the optimal solutions under uncertainty. There are a number of indicators suggested19 in order to analyze the solution under uncertainties such as Expected Value of Perfect Information (EVPI), Value of Stochastic Solution (VSS) and Uncertainty Penalty (UP).

Step 3. Deterministic Formulation and Solution. The deterministic optimization problem, which is formulated in step 1, is solved in this step by varying the decision variable and using the nominal values for parametersin case a parameter is characterized by a certain statistical distribution (hence uncertain input) then its mean value is used in this step. Moreover, different scenarios can be analyzed in this step, for example, by using different objective functions selected in step 1. The result of this step is the deterministic solution of the optimal processing path, that is, yielding one optimized biorefinery flowsheet scenario on the basis of mean values of the input data. Step 4. Uncertainty Mapping and Analysis. In this step, the deterministic optimization problem is solved for each scenario generated by the sampling from the uncertainty domain (in step 2). The results are the probability distribution of the objective value and the frequency of occurrence of the optimal processing path candidates that are selected for given combinations of uncertain inputs. Step 5. Decision Making under Uncertainty. In this step, the optimization problem is modified and formulated as a stochastic programming problem by including the uncertainty domain into the parameter domain. Therefore, the objective function is formulated in terms of minimizing or maximizing the expected value of the objective function over the uncertain domain.

EVPI = Eθ (max(f (x , y , θ))) − max(Eθ (f (x , y , θ)))

(1)

The first term on the right-hand side of eq 1 is the expected value of the results of the uncertainty mapping stage (step 4). The second term is the solution of the stochastic problem (step 5). VSS = max(Eθ (f (x , y , θ ))) − (Eθ (f (x*det , y*det , θ ))) (2)

The first term in eq 2 is the solution of the stochastic problem (step 5). The second term is calculated by evaluating the performance of the optimal network selected against the uncertainty domain (step 4 but fixing the solution from step 3). 6023

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Table 1. Input Uncertainty and Correlation Matrix Criteria15

(3)

In eq 3, the first term is the solution of the deterministic problem (step 3). The second term corresponds to the solution of the stochastic problem (step 5).

input uncertainty

min.

corn stover cost ($/dry ton) wood cost ($/dry ton) mean

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SYNTHESIS AND DESIGN OF BIOREFINERY NETWORK UNDER UNCERTAINTIES: RESULTS AND DISCUSSION In this section, the application of the framework to the formulation and solution of the biorefinery design problem is demonstrated, and the results obtained for different scenarios are discussed. This study is based on the problem formulation (with respect to superstructure, data collection, and technology models) presented in an earlier work,15 which has been expanded in order to identify the optimal processing paths under uncertainties. Step 1. Problem Formulation: (i) Problem Definition; (ii) Superstructure Definition, and Data Collection; (iii) Model Selection and Validation. The goal of the problem was the identification of the optimal biorefinery concept, with respect to a given techno-economical objective. Four objectives have been considered, resulting in the definition of 4 scenarios for the analysis, and defined as follows: (1) maximize production of FT-products: FT-gasoline and FT-diesel; (2) maximize earnings before interest, taxes, depreciation, and amortization (EBITDA) for FT-products; (3) maximize production of bioethanol; (4) maximize earnings before interest, taxes, depreciation, and amortization (EBITDA) for bioethanol production. The aforementioned superstructure combining thermochemical and biochemical processing routes, which was to convert corn stover or wood to biofuels (FTgasoline, FT-diesel, and bioethanol), is presented in Figure 2. The data collection and management performed in the previous study also formed a basis for this study, and these tasks were therefore not repeated. Step 2. Uncertainty Characterization. In this step, the most relevant sources of uncertainties based on the data analysis were identified and characterized using statistical distribution functions. In this study, the uncertainties of market prices (raw material cost and product prices) were identified as the important sources affecting the decision concerning the biorefinery design. We are, of course, aware of the fact that other sources of uncertainties are present in the system such as uncertainties in technical performance data (yield, conversion, utility consumption, etc). These uncertainties are kept outside of the scope of this study for the sake of simplicity but also because many pilot and demonstration scale studies have demonstrated the feasibility of the technological alternatives. It is obvious that the feedstock costs (corn stover and wood costs) and biofuels prices have fluctuated considerably in the past, for example, in the year 2012.20,21 These inputs were therefore selected as major sources of uncertainties. The probability density functions were estimated empirically from the historical observations for these market prices and were used to infer a proper statistical distribution function. The analysis was presented and explained in our earlier work,15 indicating that feedstock costs and product prices can be characterized as uniform and normal distributions, respectively. The parameters of the distribution together with the correlation matrix are presented in Table 1. Next Latin Hypercube Sampling with the correlation control method was used to sample from this uncertainty domain defined in Table 1 and

gasoline price ($/gal) diesel price ($/gal) ethanol price ($/gal) correlation stover matrix cost stover cost wood cost gasoline price diesel price ethanol price

1 0 0 0 0

3.53 3.97 2.24 wood cost

60 60 std.

max.

ref

100 100

NREL22 NREL23 ref 21

0.21 U.S. EIA 0.14 U.S. EIA21 0.18 U.S. Dept. of Agriculture20 gasoline diesel ethanol price price price

0 1 0 0 0

0 0 1 0.71 0.12

0 0 0.71 1 0.36

0 0 0.12 0.36 1

200 samples were generated (Figure 3). As regards the correlation matrix, it is noted that the correlation coefficients for fuel products were identified from historical data, while no correlation was assumed between feedstock costs and product market prices as no information or data were available to this end. As a result of the sampling procedure, 200 samples representing future scenarios defined by different sets of feedstock costs and product prices were defined. In the uncertainty domain defined for the analysis, these samples have equal probability of realization. Step 3. Deterministic Formulation and Solution. The optimization problems (MILP or MINLP) formulated in step 1 in GAMS for different scenarios of objective functions were solved in this step. The full optimization formulation used for this study is presented. The objective functions Scenario 1:

max . FT‐products =

∑ (Fiout , kk); i , kk

i ≡ FT‐gasoline, FT‐diesel Scenario 2: EBITDA =

∑ (P 3i ,kk Fiout , kk) − ∑ OPEX i , kk

kk

CAPEX1 + CAPEX 2 − ; t Scenario 3:

max . Ethanol =

i ≡ FT‐gasoline, FT‐diesel

∑ (Fiout , kk); i , kk

i ≡ bioethanol Scenario 4: max . EBITDA =

∑ (P 3i ,kk Fiout , kk) − ∑ OPEX i , kk

CAPEX1 + CAPEX 2 − ; t

kk

i ≡ bioethanol (4)

subject to the following constraints. (i) Process models: material balances of the generic block. raw materials

Fiout , kk = ϕi , kk 6024

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Figure 3. Sampling results with correlation control.

∑ Fi2,k ,kk = Fiout2 , kk

mixing 1: main equation FiM, kk =

∑ (Fi ,k ,kk) + αi ,kkR i ,kk k

k

(6)

max ∑ Fiout , kk ≤ Fkk i

mixing 2: chemicals or utilities used R i , kk = μi , kk

(7)

raw materials y1 + y2 + y3 ≤ 1

reaction FiR, kk

=

FiM, kk

+ MWi ∑

(γi , rrθreact, rrFiM, kk /MWreact)

rr

FiWASTE = FiR, kk − Fiout , kk , kk

Processing step 2: primary conversion (thermochemical) together with pretreatment (biochemical),

(9) (10)

y6 + y7 + y8 + y9 + y10 + y11 + y25 + y26 + y27 + y28

product separation Fiout1 , kk

Fiout , kk Split i , kk

=

out Fiout2 , kk = Fi , kk(1 − Split i , kk)

+ y29 ≤ 1 (11)

≤

Fiout1 , kk Sp

Fi2, k , kk ≤ Fiout2 , kk (S − Sp)

Fiin, kk =

∑ (Fi1,k ,kk + Fi2,k ,kk) k

∑ Fi1,k ,kk = Fiout1 , kk k

(21)

Processing step 3: gas cleaning and conditioning (thermochemical) together with hydrolysis (biochemical),

(12)

y12 + y13 + y14 + y15 + y30 + y31 + y32 + y33 + y34 + y35

(ii) Process constraints: rules defining superstructure together with the flow constraints Fi1, k , kk

(19)

Processing step 1: pretreatment (thermochemical) together with a size reduction step (biochemical), y4 + y5 + y23 + y24 ≤ 1 (20)

(8)

waste separation R Fiout , kk = Fi , kk(1 − SWi , kk )

(18)

(iii) Structural constraints: to define the extended superstructure.

∑ (Fi ,k ,kk) i,k

(17)

+ y36 + y37 + y38 ≤ 1

(22)

(14)

Processing step 4: product synthesis (thermochemical) together with fermentation (biochemical), y16 + y17 + y18 + y19 + y20 + y40 + y69 + y55 ≤ 1 (23)

(15)

Processing step 5: product separation and purification, y21 + y22 ≤ 1 (24)

(16)

Processing step: separation (biochemical), y46 + y47 + y48 + y49 + y50 + y51 + y52 + y53 ≤ 1

(13)

6025

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Table 2. Optimization Results and Comparison to the Reference Studiesa scenarios

process intervals selection

FT production (tpd)

EBITDA (MM$/year)

TAC (MM$/year)

refs

2 4 6 15 16 21 83 84 2 5 6 14 16 21 83 84

171,b 403c 170,b 402c

205 210

92 88

this study

1 4 6 12 16 21 83 84 2 5 11 20 83 84

111,b 262c 245,b 311c ethanol production (tpd)

105 149 EBITDA (MM $/year)

2 4 6 15 18 22 85 91 2 5 6 14 17 22 85 91

600 590

86.2 86.6

1 24 26 32 39 40 41 42 43 44 45 50 54 81 91 1 23 25 33 39 40 41 42 43 44 45 49 54 81 91 2 5 9 15 18 22 85 91

556

58

98

520

51

95

544 527 589

75 55.5 75

79 92.5 90

objective function

1 2

max. FT-products max. FT-products sales, min utility, waste, investment NREL (thermochemical) PNNL (thermochemical)

scenarios

objective function

process intervals selection

3 4

max. ethanol max. ethanol sales, min utility, waste, investment max. ethanol (biochemical)

max. ethanol - min.utility, waste, equipment cost (biochemical) NREL (biochemical) NREL (biochemical) NREL (thermohemical) a

NREL22 PNNL24

91 133 TAC (MM $/year) 102 79

ref this study

Zondervan et al.10 Zondervan et al.10 Dutta et al.23 Foust et al.25 Foust et al.25

Processing paths referred to Figure 2. bFT-gasoline. cFT-diesel.

Table 3. Top Five Rank of the Optimal Solutions 3(a): top-five rank of the optimal solutions: scenario 1: max. production of FT-products rank no.

process intervals selection

1 2 324 4 5

2 2 2 2 2

rank no.

process intervals selection

1 2 3 4 526

2 2 2 2 2

1 2 3 4 510 rank no. 1 2 3 4 523 a

6 14 16 21 83 6 15 16 21 83 8 15 16 21 83 8 14 16 21 83 11 20 83 84

5 4 5 4 5

rank no.

objective value

84 84 84 84

objective value

TAC (MM $/year)

process intervals selection 5 4 4 5 5

6 6 8 6 8

14 15 15 15 15

17 17 17 17 18

22 22 22 22 22

85 85 85 85 85

91 91 91 91 91

TAC (MM $/year)

210 205 170 166 75

88 92 77.5 76 89

210 170, 400 205 171,a 403b 170 141,a 334b 166 138,a 327b 75 160,a 160b 3(c): scenario 3: max. production of bioethanol objective value

92 88 133 77.5 76

EBITDA (MM $/year)

b

production (tpd)

4 6 15 17 22 85 91 600 5 6 15 17 22 85 91 600 5 6 14 17 22 85 91 590 4 8 15 17 22 85 91 565 24 26 32 39 40 41 42 43 44 45 50 54 81 91 556 3(d): scenario 4: max. ethanol product sales, min operating cost 2 2 2 2 2

EBITDA (MM $/year)

production (tpd) a

process intervals selection 2 2 2 2 1

production (tpd)

171,a 403b 205 6 15 16 21 83 84 171,a 403b a b 6 14 16 21 83 84 170, 400 170,a 400b 210 11 20 83 84 245,a 311b 245,a 311b 149 8 15 16 21 83 84 141,a 334b 141,a 334b 170 8 14 16 21 83 84 138,a 327b 138,a 327b 166 3(b): scenario 2: max. FT-products sales, min operating cost and investment cost (max. EBITDA)

4 5 5 5 4

EBITDA (MM $/year)

TAC (MM $/year)

600 86.2 600 85.2 590 86.6 565 86.2 556 58 and investment cost (max. EBITDA)

82 83 79 73 98

objective value

production (tpd)

EBITDA (MM $/year)

TAC (MM $/year)

86.6 86.2 86.2 85.2 77

590 600 565 600 544

86.6 86.2 86.2 85.2 77

79 82 73 83 76

FT-gasoline. bFT-diesel.

y73 + y74 + y75 + y76 + y77 ≤ 1

(26)

y59 + y62 + y65 ≤ 1

(27)

capital cost: (1) data collected and (2) piecewise linearization CAPEX1 =

(iv) Cost modelsoperating cost

kk

RM costs OPEX kk = P1kk + (P 2iutilities/chemicals/catalysts R i , kk) , kk waste + (P 4iwaste , kk Fi , kk )

∑ capexkk

CAPEX 2 = (28)

∑ [∑ (αj ,kkwj ,kk + βj ,kkQ j ,kk)] kk

6026

(29)

j

(30)

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Article

Fiout , kk ≤ Mykk

(34)

R i , kk ≤ Mykk

(35)

uncertainty analysis on operational parameters of the selected biorefinery alternatives. Step 4. Uncertainty Mapping and Analysis. For each of 200 samples generated from step 2, we formulated the optimization problem and solved it to identify the optimal processing path, resulting in 200 optimal solutions, which are then statistically analyzed for example using a cumulative distribution function (CDF) to fully characterize the effect of uncertainties on the decision making. The full results were then mapped and analyzed, in order to identify the optimal solution under uncertainties. The processing path, frequency of selection, and their objective value are presented in Table 4 and Figure 4.

(36)

Table 4. Frequency of Selection of the Optimal Processing Paths for 200 Scenarios

∑ Q j ,kk (31)

j

Q jo, kkwj , kk ≤ Q j , kk ≤ Q (oj + 1), kkwj , kk

(32)

∑ wj ,kk = 1 (33)

j

(v) Optimization constraints: big-M formulation

ykk ≤ M∑ Fiout , kk i

∑ Fiin,kk ≤ Mykk i

network no.

(37)

1 2 3 4 5 6

In this step, the optimal solutions were also identified under the aforementioned specific scenarios of the nominal data (or mean values) and the results are presented in Table 1 illustrating the comparison results between different specific optimization scenarios. Production rate, EBITDA, and total annualized cost (TAC) as well as the optimal processing paths were presented. This solution corresponded to the deterministic solution of the optimization problem where no uncertainties were considered. The formulation of the optimization problem for the specific scenarios (scenario 2) consists of 3 887 985 equations and 3 858 131 variables (612 discrete variables). This problem was solved using DICOPT solver using Windows 7, Intel Core i7 CPU@ 3.4 GHz, 4GB RAM, resulting 10 s of the execution. As presented in Table 2, the entrained-flow gasifier (6) was the favorite alternative due to its higher raw syngas yield and high biomass conversion. Woody biomass (2) was also the favorite feedstock due to its high carbon content. The scenarios 1 and 2, which were to produce transport fuels (FT-gasoline and FT-diesel) had a higher EBITDA compared to scenarios 3 and 4 due to higher market prices, even though higher costs were presented. The total annualized costs (capital and operating costs) had a direct effect on the optimal solution. The feedstock costs, on the contrary, have no effect on the optimal solutions in this case study. Moreover, in comparison, the new optimal processing paths show a better production rate with reduced TAC. In addition to the optimal solution, the top-five optimal solutions are presented in Tables 3a−d for the four scenarios mentioned earlier. Each table presents the objective value, production rates, EBITDA, and total annualized cost (TAC). The optimal solutions for producing FT-gasoline, FT-diesel, and bioethanol are presented in Table 3 (3a−3b and 3c−3d, respectively). The results illustrate that the thermochemical conversion platform (pyrolysis, gasification) were most of the time selected. In contrast, there was only a single processing path of the biochemical platform selected, ranking fourth (as shown in Table 3c). Wood, entrained flow gasifier and catalytic reformer together with DEPG acid removal were the most frequently selected processing intervals. Moreover, the differences between the top-three ranking solutions are small meaning that the input data are very important. This issue will be addressed in more detail in future work by performing

processing path 2 2 1 1 1 1

4 6 15 16 21 83 84 5 6 14 16 21 83 84 4 11 20 83 84 5 11 20 83 84 5 10 20 83 84 5 6 14 16 22 83 84

frequency of selection

EBITDA (MM $/a)

83/200 74/200 18/200 16/200 7/200 2/200

138−230 140−197 133−195 146−177 154−175 138−173

Figure 4. Uncertainty mapping and analysis: the frequency of selection of the optimal processing paths.

As can be seen in Table 4 and Figure 4, with 200 potential scenarios resulting from considering uncertainties, there were 6 processing paths selected and network 1 and 2 were good candidates under uncertainties. Then, the internal rate of return, IRR, was used in order to analyze and evaluate the different potential engineering projects resulting from the optimizations. IRR is an indicator of the efficiency, quality, or yield of an investment. IRR is commonly used to evaluate the desirability of investments or projects.27 IRR is mathematically equal to the internal rate of return where net present value (NPV) is zero, NPV = ∑nN= 0Cn/(1 + r)n = 0. The higher a project’s IRR, the more desirable it is to undertake the project. In, Figure 5 (left and right), the impact of market price uncertainties on the IRR for network 1 and 2 is presented in terms of IRR cumulative distribution functions (CDF), from 6027

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Figure 5. Uncertainty mapping and analysis: the probability distribution of %IRR for network-1 (left) and network 2 (right).

which the probability of obtaining a return equal to or higher than a given threshold value can be obtained. In order to highlight the application of this tool, we assume a company that has a target IRR for engineering projects of 10% within 20 years of project lifetime. For the network-1, the CDF indicates that there is a 0.1 probability of failure to reach this target IRR of the company (hence Pr (IRR < 10%) is 0.1). For the network 2, the probability of failure to reach the target IRR (10%) is 0.15, hence Pr(IRR ≤ 10%) . This provides the probability of occurrence of an undesirable situation. For a more complete picture of uncertainties, risk analysis is usually needed. Risk is defined as the product of probability of occurrence and its consequences.28 The consequence in this case is defined as lower rate of economic return of an engineering project making it a bad investment option. Hence, the risk is calculated as the product of the probability of occurrence (Pj) of a lower rate of return (IRR) (within 20 years of project investment lifetime) times the magnitude of the economic impact of risk (Mj in $) as follows: Risk = ∑jPjMj, where j is the occurrence of the undesirable event, Pj is the probability of that occurrence and Mj is the consequence (in $) of the undesirable event. The calculation of risk is in fact equal to the integral of the area highlighted in cumulative distribution function for IRR shown in Figure 5. In this calculation, EBITDA corresponding to IRR@10% is assumed as breakeven point hence the risk in economic terms is calculated as the summation of probability of occurrence times the deviation of EBITDA from the breakeven point: [EBITDAi,(@IRR