report 2

THE ENGINEERING ECONOMIST , VOL. , NO. , – https://doi.org/./X..

A new approach to refinery complexity factors

Mark J. Kaiser

Center for Energy Studies, Louisiana State University, Baton Rouge, Louisiana

ABSTRACT Wilbur Nelson introduced the concept of complexity factor in the 1960s to relate the performance level of refineries with different capabilities and processing technologies. Refinery complexity has been used for manyyearstodescribethesophisticationandcapital intensityofrefiner- ies, but foundational issues and the limitations associated with the met- ric have not been examined. A more precise and rigorous approach to complexity factors is presented based on the application of cost func- tions. Two alternative formulations are introduced and compared to the traditional approach and lead to new descriptive statistics and insight.

Introduction

A refinery is an industrial plant where crude oil and other feedstocks are processed into petroleum products. The main principle of refining is to separate and improve the hydro- carbon compounds that constitute crude oil to produce saleable products that satisfy regula- tory requirements. A refinery consists of three primary sections—separation, conversion, and finishing—and each section contains one or more process units that apply different technolo- gies and combinations of temperature, pressure, and catalyst to perform their function. There are a dozen or so main process operations and for each process, one or more technologies have been developed over the past 150 years (Table 1).

Separation, conversion, finishing

Before processing crude oil, refiners physically separate it into molecular weight ranges by boiling point using a distillation tower. The longer the carbon chain, the higher the temper- ature the hydrocarbon compounds will boil, and cuts of similar boiling point compounds are processed together in downstream units to allow the conversion steps to operate effi- ciently (Meyers 2004; Speight 1998). Processes downstream of distillation take these cuts and, using various chemical and physical operations, improve and change the physical properties of molecules that are blended for fuels and other products.

The primary conversion units in a modern refinery include fluid catalytic cracking (FCC), hydrocracking, and coking (Figure 1). These units are expensive to build and operate and

CONTACT Mark J. Kaiser [email protected] Center for Energy Studies, Louisiana State University, Energy, Coast and Environment Building, Nicholson Drive Extension, Baton Rouge, LA . Color versions of one or more figures in the article can be found online at http://www.tandfonline.com/utee. ©  Institute of Industrial & Systems Engineers

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Table . Refining production process operations and technologiesa.

Process operation Technology

Coking Fluid coking, delayed coking Thermal process Thermal cracking, visbreaking Catalytic cracking Fluidized bed Catalytic reforming Semiregenerative, cycle, continuous regeneration Catalytic hydrocracking Distillate upgrading, residual upgrading, lube oil Catalytic hydrotreating Pretreatment of cat reformer feeds, other naphtha desulfurization, naphtha

aromatics saturation, kerosene/jet desulfurization, diesel desulfurization, distillate aromatics saturation, other distillates, pretreatment of cat cracker feeds, other heavy gas oil hydrotreating, resid hydrotreating, lube oil polishing, posthydrotreating of FCC naphtha

Alkylation Sulfuric acid, hydrofluoric acid Polymerization/dimerization Polymerization, dimerization Aromatics BTX, hydrodealkylation, cyclohexane, cumene Isomerization C feed, C feed, C and C feed Oxygenates MTBE, ETBE, TAME Hydrogen production Steam methane reforming, steam naphtha reforming, partial oxidation Hydrogen recovery Pressure swing adsorption, cryogenic, membrane

aThe technologies listed are not exhaustive. BTX: Benzene/toluene/xylene; MTBE: Methyl tertiary butyl ether; ETBE: Ethyl tertiary butyl ether; TAME: Tertiary amyl methyl ether.

Source: Farrar ().

represent a major investment decision but, once installed, significantly enhance refinery flex- ibility and capability. Refineries with cracking and coking capacity are generally referred to as “cracking” and ”coking” refineries, or as “complex” or “very complex” refineries, to distinguish them from “simple” refineries that do not have such units. Not all refineries are configured with conversion units. In Europe and elsewhere, it is common to refer to complex refineries as “conversion” or “deep conversion” refineries.

Products from the conversion steps require treatment to make them ready for sale, which primarily involves improving product specification (e.g., octane number, vapor pressure) and reducing sulfur levels to satisfy regulatory requirements. Hydrotreatment is the most

Figure . Typical refinery configuration and simplified flow diagram for a complex refinery. Main conversion units are highlighted.

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common finishing operation worldwide. Most of the treatment and conversion processes employ catalyst to increase the speed of chemical reactions and enhance conversion rates.

Capital investment

Refineries are described by the number, size, and type of process units; technologies applied; ownership; location; level of integration; as well as the cost associated with the receipt and dispatch of feedstock and refined product. No two refineries have the same configuration or apply the same technologies because of their feedstock and product requirements, processing characteristics, and development history. As a refinery adds or expands units or installs new technologies to adapt to changing markets and feedstocks, product yields typically improve, and the refiner’s ability to process a wider slate of crude oils and flexibility in production increase, along with the value it can achieve from processing.

Refinery configurations change over time based on capital improvement projects. New construction is referred to as grassroots construction or unit addition, whereas expansion projects add incremental barrels to existing units. Both investments add capacity, of course, but the nature of the construction and capital requirements of the project types differs and, consequently, the risk and reward profile of the investment. Expansion projects are usually cheaper than new construction on a per barrel basis because existing units can be used or revamped, construction activity is less intensive and of shorter duration, permitting and regu- latory requirements are simplified, etc. Typically, the objective of replacement units and mod- ernization is to reduce energy consumption, lower operating costs, and satisfy stricter envi- ronmental regulations. New construction and capacity expansions are performed to increase operational flexibility and refining margins.

Refinery complexity

Wilbur Nelson introduced the concept of complexity factor in the 1960s to quantify the grass- roots construction cost of process units (Nelson, 1960). Nelson expressed the complexity of a process unit as the grassroots construction cost of the unit relative to the cost of the atmo- spheric distillation unit (ADU) normalized on a capacity basis:

CF(Unit) = Cost(Unit)/Capacity(Unit) Cost(ADU)/Capacity(ADU)

. (1)

Nelson applied the complexity factor of process units to describe the complexity of a refinery (Nelson 1976b). Each process unit of a refinery is assigned a complexity, and the sum of the complexity factors of all the process units, weighted by the unit capacities relative to distilla- tion capacity, defines the complexity index of the refinery:

CI ( Refinery

) = ∑ Capacity(Unit)

Capacity(ADU) ·CF(Unit). (2)

The complexity index of a refinery is a weighted average of the complexity factors of its units and quantifies the capital intensity of the refinery.

Applications

Complexity indices quantify refinery complexity and have been used in many different ways and for many different purposes (Figure 2). Complexity indices quantify refinery complexity

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Figure . Refinery complexity is applied in many different ways in refining applications.

and sophistication and, to a lesser extent, represent a proxy of refinery value. Refinery com- plexity is designed to differentiate refinery type and is frequently used to correlate with other market indicators, such as sales price and replacement cost. The more complex the refinery, the more capital was invested to achieve its configuration and, therefore, the greater the cost to insure and/or replace the process units. Larger refineries (refineries with greater distilla- tion capacity) are not necessarily more complex than smaller refineries, but a refiner’s conver- sion capacity—that is, its cracking and coking capacity—tends to be correlated to complexity. Sophisticated refineries are more expensive to build relative to simple refineries and should transact at a premium, all things being equal, because they consist of more valuable assets, provide higher yields of more valuable products, and generate greater margins. In benchmark- ing financial performance, refining operating margins tend to improve as refinery complexity increases. Complexity indices are frequently used in sales price and replacement cost models in appraisal studies, and derived measures such as complexity barrels are used in valuation and by credit rating agencies.

Purpose

The traditional way to compute complexity factors is to apply Nelson’s definition described by Equation (1) directly to adjusted and normalized cost data (Figure 3). A more precise way to formulate complexity factors is to first process the data using cost curves and to apply the cost functions at specified capacities or construct a complexity factor functional. Cost functions are well known to cost engineers and are common throughout industry (Ostwald and McLaren 2003). The cost data are the same in all three approaches and subject to the same limitations and constraints, but how the data are processed and evaluated are different and, consequently, the resultant complexity factor values will also be different.

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Figure . Traditional statistical approach to compute complexity factors and two alternative formulations based on cost functions.

The purpose of this article is to expand the formulation of complexity factor in terms of cost functions and to illustrate the procedures involved in its assessment. The complex- ity factor functional approach leads to a new probabilistic interpretation of complexity fac- tor and provides insight into the traditional methods. Computing reliable complexity fac- tors is a more difficult task than commonly acknowledged and subject to greater uncertainty than generally appreciated, and neither of these issues has been previously examined by the research community.

Outline

The outline of this article is as follows. We begin by defining an ideal refinery and the vari- ables and notation employed. Complexity factors and cross-factors are illustrated and the methodologies used in their measurement are discussed. By using cost functions and speci- fying capacities in evaluation, the complexity factor at reference capacity is defined. Extend- ing this approach, the functional that defines the complexity factor is considered via specifi- cation of capacity intervals. For cost functions described by a one-dimensional power rela- tion, closed-form solutions for the first and second moments are developed, and numerical techniques are used to construct distribution and sensitivity functions. The article concludes by comparing reference capacity and complexity factor functional averages to the traditional complexity factors based on statistical techniques and applies the methodology to illustrate how uncertainty calculations can be improved. In Appendix A, the complexity functional moments are derived, and in Appendix B, summary results for three process units (delayed coking, catalytic reforming, catalytic hydrocracking) are provided.

Ideal refinery

A refinery R is composed of n + 1 process units (U0, U1, …, Un) with charge or production capacities (Q0, Q1, …, Qn) depending on the unit type (Figure 4). Process unit U0 is designated as atmospheric distillation and Q0 denotes atmospheric distillation capacity. Charge capacity is the liquid volume of the crude that is fed to the process unit, and production capacity refers

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Figure . A refinery is described by its process units and their charge or production capacity. Complexity factors are time and region dependent and enter the complexity index via a weighted average calculation.

to the barrels of product produced, and both are expressed in barrels per stream day (bpsd) or barrels per calendar day (bpd)1 for liquid inputs/outputs. Unit capacities depend upon the size of the distillation unit and the crude slate (light or heavy crude mix) around which the refinery is designed. In gas processing and hydrogen plants, million cubic feet per day (MMcfd) of gas is the base unit; for coke and sulfur processing, long tons are applied. Outside the United States, metric tonnes and cubic meters are commonly employed for liquid and gas volumes.

The grassroots construction cost of unit Ui of capacity Qi is denoted Ci(Qi) = C(Ui, Qi) and is always referenced to a specific project, build year, and geographic location because construc- tion costs are always site, time, and location specific. Site characteristics describe attributes of the asset, including size, ownership, and contract type; location describes a specific site attribute. Project, time, and location differences translate to differences in the capital cost of construction and requires adjustment and normalization before cost data are processed. Each process unit is associated with a derived complexity factor CFi that is time and location depen- dent. Complexity factors for process units with gas or solid outputs are converted based on thermal energy or cost equivalency.

Nelson complexity

Complexity factor

The complexity factor of a process unit is defined by the cost of the unit relative to the cost of atmospheric distillation normalized on a capacity basis:

CF (Ui) = CF (Ui,U0) = C(Ui, Qi)/Qi C(U0, Q0)/Q0

. (3)

 A barrel per stream day is the nameplate (design) capacity of a unit. A barrel per calendar day represents throughput capacity taking into account downtime and related factors. Calendar day barrels vary annually, often between  and % stream day barrels, depending on the unit and operational issues at the plant.

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Table . Nelson’s complexity factors for USGC refining operations.

–  

Atmospheric distillation    Vacuum distillation    Thermal cracking, visbreaking   . Delayed coking  .  Catalytic cracking . .  Catalytic reforming    Catalytic hydrocracking    Catalytic hydrorefining    Catalytic hydrotreating  .  Alkylation    Polymerization/dimerization    Aromatics –   Isomerization    Lubes  Oxygenates  Hydrogen (MMcfd)  Asphalt .

Source: Nelson (; a), Farrar (), and Johnston ().

For example, if a 20,000 bpsd atmospheric distillation unit in Houston, Texas, cost $5 million to construct in 1960 and a 2,500 bpsd delayed coking unit cost $3 million to construct in Lake Charles, Louisiana, in 1961, then the complexity factor for delayed coking in 1960–1961 in the U.S. Gulf Coast (USGC) would be approximately five:

CF(DelayedCoking) = $3million/2, 500bpsd $5million/20, 000bpsd

= 4.8. (4)

The cost to construct a 2,500 bpsd delayed coker in the USGC in 1960–1961 was about five times more expensive per barrel than the cost to build a 20,000 bpsd atmospheric distillation unit.

If the Houston and Lake Charles units were the only units built in the region at the time, the cost data would be representative of regional cost. If more units were constructed, then potential bias in the sample selection should be considered. To increase confidence in the values of complexity factors derived, it is desirable to expand the data collection to ensure that a variety of sizes and locations and contractors are considered. Because process units are not built frequently and contractors do not publicly report the terms and conditions of their tenders, constraints always exist on data collection, which constrains the ability to provide meaningful statistics on complexity factors.

Measurement

Nelson published a list of complexity factors in the Oil & Gas Journal (OGJ) for the major pro- cess units in the 1960s, which was later updated by Gary Farrar and continued to the present day by Daniel Johnston (Table 2). The last year USGC complexity factors were reported was in 1998, before the introduction of a commercial database of refinery complexity, so reference in this article to OGJ complexity factors is with respect to 1998 values.

From 1961 to 1972, the complexity factors of vacuum distillation, thermal crack- ing/visbreaking, and catalytic hydrotreating were reported as two, indicating that during this time, average grassroots construction costs according to OGJ statistics were about twice the cost of a barrel of atmospheric distillation. Isomerization capacity was three times as expensive

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as distillation, catalytic reforming and hydrorefining four times as expensive, catalytic crack- ing and hydrocracking about six times as expensive, and so on. Aromatics were reported as the most expensive process technology on a per barrel basis and, unlike most of the other units, in subsequent years the reported values have declined. Complexity factors for lubes, oxygenates, hydrogen, and asphalt units were added in later years.

OGJ complexity factors did not change significantly for nearly four decades, either because the cost data from which the factors were computed did not change significantly or, more likely, because the cost data were simply not updated. The big jump in value for isom units might be a result of change from once-through units to partial and then full recycle of uncon- verted n-paraffin isomer. Unit names can hide large differences in process technology and cost. Complexity factors are typically rounded to the nearest whole number and class values may partially account for the lack of variation. If new projects and cost data were included over the period of evaluation, the complexity factors would certainly vary more significantly than depicted; hence, the reliability of the metrics is suspect.

Complexity cross-factor

Complexity factors are normalized with respect to atmospheric distillation because crude dis- tillation is usually the cheapest process to build with the greatest throughput capacity. How- ever, complexity factors can be defined with respect to any two process units— for example, hydrotreating and catalytic cracking—and when reference to normalization is made to a unit besides atmospheric distillation, the complexity factor is referred to as a complexity cross- factor or, more simply, cross-factor.

The complexity cross-factor CF(Ui, Uj) is defined as the normalized construction cost per barrel for process unit Ui relative to process unit Uj:

CF ( Ui,Uj

) = C(Ui, Qi)/Qi

C(Uj, Qj )/Qj . (5)

Complexity cross-factors can be inferred directly from complexity factors, if available, because the distillation normalization cancels out in the numerator and denominator terms:

CF ( Ui,Uj

) = CF(Ui)

CF(Uj ) . (6)

Conversely, if cross-factors are available, complexity factors can be inferred via the relation

CF(Ui,Uj ) CF(Uk,Uj )

= CF(Ui) CF(Uk)

. (7)

Uncertainty

If process units were built frequently and reported their construction costs consistently, com- plexity factors could be computed with a reasonably high degree of reliability and would be subject to minimal uncertainty and measurement bias, but because units are not built fre- quently or in the same region or time or with the same technologies or capacities, the data from which complexity factors are computed are often small and heterogeneous and require careful handling. Complicating matters, cost data are rarely part of the public record, and the costs that are reported may or may not apply the same categories. Cost data therefore need to be processed in a manner consistent with the available data and adjusted and normalized prior to evaluation, with a clear understanding of the limitations of analysis. Because only

344 M. J. KAISER

small data sets are usually available, small data analysis techniques are required that limit the reliability of statistical processing, and uncertainty levels are expected to be high.

Traditional approach

To compute complexity factors, cost data are collected, adjusted, and normalized prior to aggregation and processing. Each step entails selection of appropriate methodology based on industry techniques and user-defined preference. Adjustment and normalization are required to help ensure uniformity and reduce bias and uncertainty in evaluation (Ostwald and McLaren 2003). The procedures entail elements of both art and science, and rarely is the approach straightforward because of the differences associated with the input data, cost cat- egories, project characteristics, and user preferences. It is usually not possible to adjust and normalize all data in a precise fashion because of site-specific and other unobservable condi- tions.

Delineation

Grassroots refinery construction cost are normally limited to equipment inside the battery limits (ISBL) of each process, but there are no universal reporting standards among contrac- tors and companies, and when public data are presented, it is usually difficult to distinguish what categories are included or excluded. Contractor’s ISBL cost categories normally include materials, labor, design, engineering, contractors’ fees, overheads, and expense allowance but do not include working capital, taxes, insurance, interest during construction, inventories, startup expense, royalties, chemicals, catalysts, supplies, licenses, permits, and duties, which are owner’s costs. Site preparation of land, power generation, electrical substations, buildings, spare parts, off-site tankage, docks, and marine terminals are also not included in ISBL costs.

Normalization

All costs are site, time, and location specific and need to be normalized for these differences before data are grouped together and processed. Site differences are project specific and adjust- ments depend upon the experience and preference of the analyst, while time and location differences are made using industry cost indices such as the Nelson-Farrar construction cost index or Chemical Engineering Plant Cost Index. Historically, the USGC was considered base- line cost for the United States because of the widespread availability of contractors and high levels of competition (i.e., low construction costs), access to river transportation, and compet- itive labor markets. Refinery construction in the United States is still commonly referenced with respect to the USGC.

Processing

Under the best scenario, complexity factors are computed as an average from several recent construction projects in the same region, whereas under less ideal conditions, complexity factors are estimated based on only one or two projects separated in both time and space. Usually, the more data that are available the better the cost specification. If more than two data points are available, the accuracy of estimation will improve because a wider range of possibilities are presented, and the averaging process will help minimize the impact of outlier data. Sometimes median values are preferred in evaluation. However, if additional data are

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Table . USGC catalytic reformer and atmospheric distillation cost, –a.

Cost Capacity ($ million) (Mbpd)

ADU   ADU   ADU+   CCR   CCR   CCR+  

aADU+ and CCR+ represent aggregate measures and do not describe actual units.

employed to enlarge the sample but are not properly screened and adjusted for technology, time, location, or related differences or old data are included because they are the only data available, the additional data may inadvertently reduce the reliability of the assessment.2

Let {C1(U0,Q1), …, Cn(U0,Qn)}, n � 1, denote the adjusted and normalized cost data for atmospheric distillation units in a given region and time period, and let {C1(Ui,Q1), …, Cm(Ui,Qm)}, m � 1, denote adjusted and normalized cost for process technology Ui in the same region and time period. The values of n and m are small, usually less than a dozen, often less than five, and in some cases may only be one or two for a specific region and time period. The n = 1, m = 2, and n = m = 2 cases are used to illustrate how complexity factor values vary when the data are processed in different ways.

Example Two catalytic reformer units were constructed in the USGC from 2010 to 2012, a 55,000 bpd unit for $500 million and a 42,000 bpd unit for $400 million. Cost data was available for one atmospheric distillation unit built in the USGC during the same time period, a 325,000 bpd unit for $300 million. The analyst decides to create a “composite” reformer unit by summing the costs and capacities of the individual units to expand the sample data, but recognizes that the composite unit does not represent an actual built unit.

The construction cost ratios can be computed and averaged using an arithmetic or weighted average of the individual units, the composite unit can be evaluated, or the com- posite unit can be included with the base data and the two evaluated together.

Using the raw data, the complexity factor for units CCR1 and ADU1 is computed as follows:

CCR1 ADU1

= $500million/55Mbpd $300million/325Mbpd

= 9.8 (8)

and, similarly, CCR2/ADU1 = 10.3. The arithmetic average and weighted average of these two units yields a complexity factor of 10. For the composite unit, the complexity factor is computed as

CCR1+2 ADU1

= $900million/97Mbpd $300million/325Mbpd

= 10.0. (9)

The arithmetic and weighted averages of the sample do not change noticeably with the inclu- sion of the composite unit.

If cost data from a second atmospheric unit built in the USGC during the same time period are included in the evaluation (Table 3), then the number of computational pathways increases

 In small data sets, one “outlier” data point may significantly impact the statistics obtained, but before a project is excluded from evaluation, significant work needs to be performed to document the reasons for the exclusion.

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Table . Complexity factor calculation pathwaysa.

Arithmetic average Weighted average

ADU ADU ADU+ Actual All b Actual Allb

CCR . . . . . . . CCR . . . . . . . CCR+ . . . . . . . Actual . . . . — . — Allb . . . — . — .

aBased on the data in Table . bIncludes the composite units in the calculation.

(Table 4). It is clear that the size of the data set and how it is processed will have an impact on the complexity factor calculations depending on the nature of the data.

Limitations

Unless detailed cost data are available from the contractor or owner, it is difficult to ensure that cost categories are uniform and reported consistently. If construction cost can be decom- posed into categories, then cost indices can be applied to individual categories and summed; otherwise, an aggregate cost index will be adopted for time and location differences. It is not clear what the “best” way to process a given set of cost data is or, conversely, how reported complexity factor data are computed, unless stated explicitly.

Cost functional approach

Alternative formulations

A more precise way to formulate complexity factors is to process the cost data using func- tions and then to apply the cost functions at specified capacities, or to specify the capacity interval of each unit and construct the complexity factor functional that, when integrated over its domain, yields an average value for the complexity factor. The cost data are the same in all three approaches, of course, but how the data are processed and evaluated is different and, consequently, the resultant complexity factors will also be different. The characteristics of the formulations and how the complexity factors compare provide insight into the individual metrics.

In the traditional (statistical) approach to complexity factors, cost data are normalized by capacity and statistical techniques are applied in the evaluation. Pairwise computations are easy to perform and the results are immediate, but for larger data sets ambiguity arises unless processing techniques are described clearly and precisely. It is not necessary, or necessarily desirable, however, to use data directly because this imposes constraints on processing and can lead to ambiguities. For example, one of the limitations of the traditional method is the inability to account for the impact of capacity on cost. There is no “control” in the data sam- ple, and because we are not controlling for capacity directly (only indirectly via division), its impact is unknown. Capacity is not a control variable in the statistical approach but, rather, part of the input data used to normalize the complexity factor.

In the reference capacity approach, the use of cost functions improves the reliability and transparency of the calculations and accounts for capacity variation, and because capacity

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is required in the assessment, it improves specificity. Complexity factors at reference capac- ity specify “representative” capacities in the cost functions, but the data used in constructing cost functions span a wide spectrum of costs and capacities. By selecting two specific capac- ities for the individual units, the range of capacities and their cost differences are effectively excluded. Because cost curves are constructed to describe cost as a function of capacity, it is natural to employ the cost functions directly to reveal the variation in complexity factor from its functional formulation. This also allows the statistical limitations associated with small data analysis to be managed. In the complexity factor functional approach, the properties of the functional are evaluated over the capacity intervals that define its domain.

Specification

The costs of grassroots refining units are commonly described by power expressions of the following form:

C ( Unit, Capacity

) = K · Capacitya (10)

or, in shorthand notation, C(Ui, Qi) = KQia, where the coefficients (K, a) characterize the process unit and technology, process severity,3 and feedstock requirements. For each process unit, (K, a) is determined empirically from sample data using similar technology and severity specification and is referenced with respect to a particular region and time period.

For individual equipment, the power coefficient frequently ranges between 0.5 and 0.7, reflecting scale economies and the physical relationship between surface area and volume (Peters et al. 2003; Seider et al. 2004). For process units that combine multiple components and equipment, the power coefficients range more broadly and are subject to greater uncertainty reflecting the expanded system boundaries (Kaiser and Gary 2009).

Parameter estimation

To estimate the unknown parameters (K, a), ordinary least squares using standard regression techniques is applied to the logarithm of the cost model. Most spreadsheet packages have built-in curve fitting routines for power relations. After the cost curve is computed it may be extrapolated outside the domain of the data used in its development, but this should be performed cautiously. As different time periods, geographic regions, and technologies enter the sample data, greater bias and uncertainty in parameter estimation are expected to arise.

Example Fluid catalytic cracking cost curves for 2009 USGC, C(FCC) = 24.67Q0.461, were adjusted to 2014 using Nelson-Farrar refinery construction cost indices and represented in Cartesian coordinates and log–log plots (Figure 5). The capacity range depicted is defined by active aver- age FCC units in the United States circa 2014 (59 Mbpd) plus/minus one standard deviation (43 Mbpd), yielding the interval (16, 102) Mbpd.

 Process severity typically refers to the pressure and temperature requirements of the process and/or the degree to which the input or output streams are transformed to meet various specifications.

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Figure . Equivalent representations of USGC fluid catalytic cracking cost curves. Time adjustment based on Nelson-Farrar construction indices.

Capacity range

Cost functions are estimated based on project data that specify the cost and capacity of newly built units. The capacity range reflects the interval where the cost functions are computed and can usually be extrapolated outside the range. Capacity ranges may be determined by the cost data, the installed capacity of recent projects, the active capacity of units at the time of evaluation, or an alternative selection. It is convenient to apply active average U.S. capac- ity plus/minus one standard deviation to define intervals but other selections are possible (Table 5).

Complexity factor at reference capacity

Specification

The complexity factor for unit A at reference capacity Q∗ baselined with respect to unit B at reference capacity q∗ is evaluated using the cost functions of the two units as follows:

CF(A, B) = CF(A,Q∗; B,q∗ ) = KQ∗a/Q∗ Lq∗b/q∗

= K L Q∗a−1q∗1−b, (11)

where the cost functions are described by C(A, Q) = KQa and C(B, q) = Lqb with parameters (K, a) and (L, b), and the reference capacities Q∗ and q∗ are specified. Capacities are denoted

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Table . U.S. refining unit capacity statistics circa .

Average Median SD AVG − SD AVG + SD Atmospheric distillation , , , , , Vacuum distillation , , , , , Delayed coking , , , , , Fluid catalytic cracking , , , , , Catalytic hydrocracking , , , , , Catalytic reforming , , , , , Gasoline hydrotreating , , , , , Distillate hydrotreating , , ,  , Alkylation , , , , , Polymerization , , ,  , Aromatics , , , a , Isomerization , , , , , Lubes , , , , , Oxygenates , , , , , Hydrogen (Mcfd) , , , a ,

aCapacity intervals are truncated at zero capacity when negative.

by Q and q to reinforce the notion that the capacities are specific to each unit and range over different intervals (Figure 6). Reference capacities are stated explicitly for each unit, either in shorthand notation or in words, to specify the measure.

Reference capacity

What reference capacities should be employed in evaluation? In the traditional approach, no distinction is made because cost and capacity are normalized together via division, and what- ever data are available serve as the de facto default capacities. A more transparent approach specifies the capacities where the cost functions are evaluated. One logical choice for reference capacity is the active average (or median) capacity for all units in the region or the United

Figure . Complexity factor at reference capacity calculation for unit A relative to unit B requires specifying the capacity of evaluation for each unit.

350 M. J. KAISER

Table . Complexity factors at average and median U.S. reference capacities.

CFRCa,b

CF Median Average CFCFRC c

Crude distillation    . Vacuum distillation  . . . Delayed coking  . . . Catalytic cracking  . . . Catalytic reforming  . . . Catalytic hydrocracking  . . . Gasoline hydrotreating  . . . Distillate hydrotreating . . — Alkylation  . . . Polymerization/dimerization  . . . Aromatics  . . . Isomerization  . . . Lubes  . . . Oxygenates  . . . Hydrogen (MMcfd)  . . .

aCost functions based on Kaiser and Gary (). bComplexity factor reference capacity with reference capacities defined by active U.S. refining units circa . cComplexity factor reference capacity computed at average U.S. refining capacity circa .

States or for recently built units in the region or the United States as described previously (Table 5). Users may also prefer to select a baseline capacity to simplify the treatment.4

Example The complexity factor at reference capacity for naphtha (gasoline) hydrotreating (NHT) based on 2014 U.S. average capacities and 2009 USGC cost functions, C(NHT) = 4.96Q0.58 and C(ADU) = 8.20q0.51, yields

CF (NHT) = CF (NHT, 21; ADU, 149) = 4.96(21) 0.58

/21 8.20(149)0.51/149

= 29/21 103/149

= 1.9. (12)

Using 2014 U.S. median capacities yields

CF (NHT) = CF (NHT, 19; ADU, 117) = 4.96(19) 0.58

/19 8.20(117)0.51/117

= 27/19 93/117

= 1.8. (13)

Comparison

Complexity factors at reference capacity were computed using 2009 USGC cost functions described in Kaiser and Gary (2009) and average and median active U.S. capacities circa 2014 (Table 6). The complexity factors at average and median reference capacities are similar and indicate that small differences in the baseline capacities lead to small differences in complexity factor.

Ideally, the reference year of the cost curves and inventory evaluation period should match but, unfortunately, cost curves are not readily available except on a periodic basis,5 and simply adjusting the cost functions through indexing will not change the complexity factor values because the same adjustment occurs in both the numerator and denominator and cancel out. To match cost functions with the evaluation period, cost data should be in relatively close temporal proximity to the evaluation period.

 For example, a  Mbpd baseline capacity could be selected for alkylation, aromatics, isomerization, and lubes.  Refining cost curves are only available periodically because of the difficulty in collecting reliable cost data.

THE ENGINEERING ECONOMIST 351

For delayed coking, gasoline hydrotreating, and lubes, the complexity factors at reference capacity are similar to OGJ values and differ by less than 20%, but for other units, differences are greater and, in some cases, far greater. For vacuum distillation, catalytic cracking, reform- ing, and alkylation, for example, the OGJ values are about one and a half times larger than the complexity factor at average capacity, whereas for aromatics, isomerization, and oxygenates, the OGJ values are more than two and a half times larger. For polymerization and hydrogen plants, the OGJ complexity factors are notably smaller than the complexity factor at average reference capacity.

Cost curves are the primary element in the reference capacity approach, whereas in the traditional approach, the complexity factor is computed directly from the raw data. In the cost function approach, cost curves allow capacity impacts to be incorporated in assessment either directly through baseline capacity selection or through a more integrated treatment. Costs are determined at reference capacities that are a priori specified for both units. In the traditional approach, capacities are not controlled or specified and enter the assessment indi- rectly through the statistical processing.

Complexity factors at reference capacity computed via cost functions are expected to be a more reliable and robust assessment of complexity than the traditional statistical approach because capacities are a direct input in the evaluation and a control variable. Ref- erence capacities require specification that improves the transparency and specificity of the measures.

Complexity factor functional

Specification

The complexity factor functional CF(Q, q) of unit A baselined with respect to unit B is a two- dimensional nonlinear function of the capacities of the two units defined as follows:

CF(Q,q) = CF(A,Q; B,q) = KQ a/Q

Lqb/q = K

L Qa−1q1−b, (14)

where the cost functions of the units are again described by C(A, Q) = KQa and C(B, q) = Lqb for given parameters (K, a) and (L, b), but instead of specifying the reference capacities of evaluation (i.e., Q∗ and q∗), the range of the capacities for the cost curves are specified by IA = (c, d) and IB = (e, f). Specification of the capacity intervals for each unit is required for function definition (Figure 7).

Capacity range

Functions are defined for a specific range of variables, and for a two-dimensional function capacities require specification over two independent intervals. As before, the range may be determined in various ways, but here the specification plays a more important role because how the intervals are selected will impact the functional properties derived. As the capac- ity intervals vary, the complexity functional is defined over different domains, which will impact the shape, size, and (derived) properties of the functional such as the average and variance.

352 M. J. KAISER

Figure . The complexity factor functional is defined over the domain of the capacity intervals for the two units.

Estimation

Standard numerical techniques apply in evaluation. After the capacity intervals are specified, a grid within each interval is set up and the function is evaluated at the grid points. For illus- tration, three capacities within each interval are selected—low, average, and high values—and the complexity factor for each entry is estimated as follows:

Complexity factor function tableau.

Unit B (Mbpd)

Unit A (Mbpd) qlow qavg qhigh

Qlow Qavg Qhigh

The granularity of the intervals and the size of the tableau will determine the accuracy of the estimation, and with today’s computer processing capacity, there are no computational con- straints associated with the calculation. Capacity bounds are a user preference. The average complexity factor and standard deviation is estimated by sampling and describes the aggregate characteristics of the complexity factor functional.

Example The estimation tableau for the catalytic cracking complexity factor functional is computed using 2009 USGC cost curves, C(FCC) = 24.7Q0.46 and C(ADU) = 8.2q0.51, and capacity inter- vals defined by average active U.S. capacities circa 2014 plus/minus one standard deviation, IFCC = (16, 102) and IADU = (22, 276).

The complexity functional entries vary from 1.1 to 10.6 across the matrix. At fixed FCC capacity, the complexity factor increases with increasing ADU capacity, and for fixed ADU

THE ENGINEERING ECONOMIST 353

Catalytic cracking complexity functional.

ADU (Mbpd)

FCC (Mbpd)   

 . . .  . . .  . . .

capacity, the complexity factors decrease with increasing FCC capacity. The complexity factor functional is greatest for high ADU and low FCC capacity and smallest for low ADU and high FCC capacity. The average complexity factor computed from the matrix entries is 4.5 with a standard deviation of 2.9.

Closed-form expressions

The average and variance of the complexity factor functional can be evaluated using a closed- form expression because the cost functions are one-dimensional power law expressions, and their ratio is similarly a power law expression. To compute the average of the complexity fac- tor function CF(Q, q) over the intervals IA = (c, d) and IB = (e, f), a double integration is performed:

E(CF) = ∫ f e

∫ d c CF(Q,q)dQdq

(d − c)( f − e) . (15)

The variance of the complexity factor functional is computed as

V (CF) = ∫ d c

∫ f e [CF(Q,q)−E(CF)]2dQdq

(d − c)( f − e) (16)

and measures the degree of variation of the complexity function around its mean over its domain. The standard deviation of the functional is SD(CF) =√V (CF) .

The integral in one of the dimensions is evaluated first, and the results of this first integra- tion are integrated in the second dimension. The order of integration is not important. The mean and variance of the complexity factor function is computed thus:

E(CF) = K(d a − ca)( f 2−b − e2−b)

La(2−b)(d − c)( f − e) (17)

V (CF) = ( K L

)2 ( f 2a−1−e2a−1)(d3−2b−c3−2b)

(3 − 2b)(2a−1)(d − c)( f − e) − 2E(CF)

× ( K L

) ( f a − ea)(d2−b−c2−b) a(2−b)(d − c)( f − e) + E(CF)

2 . (18)

See Appendix A for the derivation.

Example The complexity factor functional for catalytic cracking is constructed using MATLAB (Nat- ick, MA: MathWorks) based on 2009 USGC cost functions and capacity intervals defined by average active U.S. capacities circa 2014 plus/minus one standard deviation (Figure 8). Con- tour plots represent constant complexity factor values projected to the plane (Figure 9). The

354 M. J. KAISER

Figure . Fluid catalytic cracking complexity factor functional.

function increases rapidly for high ADU capacity and low FCC capacity and declines as ADU capacity decreases and FCC capacity increases.

The domain of the functional is defined by IA × IB and has an area of 21,844 Mbpd2. Using a sampling interval of one square 1 Mbpd, the distribution of the complexity factor is lognormal, and the mean is empirically computed to be 3.90 with a standard deviation of 1.58 (Figure 10). Formula (theoretical) values for the expected value and standard deviation yield E(CF) = 4.10 and SD(CF) = 1.55.

At point H = (200, 25) near the top of the ridge of the function, CF(H) = 7.1, and small changes in capacity will lead to large changes in the complexity value. By contrast, at point M = (150, 75) in the middle of the plateau of the function, CF(M) = 3.4, and small changes in capacity at this point will lead to much smaller changes in the functional value.

Figure . Fluid catalytic cracking complexity factor contour plot.

THE ENGINEERING ECONOMIST 355

Figure . Fluid catalytic cracking complexity factor distribution over its functional domain.

Comparison

The complexity factor functional average (CFFA) values are about 10% larger in most cases than the complexity factor at reference capacity (CFRC) values (Table 7). In the CFRC approach, reference capacities are specified, but in the CFFA approach, only the capacity inter- vals are specified. CFRC appear to be a reasonably close approximation for CFFA but, unlike the reference capacity and traditional approaches, additional useful information is available on the variation of the complexity factor. CFFA values are considered the most precise of the three formulations because they incorporate all of the functional values in an integrated fash- ion. The functional perspective provides a natural interpretation of variance for small sample sets, and the computation of the complexity factor functional, although more involved, is more rigorous and can be extended in various useful ways.

Table . Comparison of complexity factor formulations.

CF CFRCa,b CFFAa,c

Atmospheric distillation  . . () Vacuum distillation  . . (.) Delayed coking  . . (.) Catalytic cracking  . . (.) Catalytic reforming  . . (.) Catalytic hydrocracking  . . (.) Gasoline hydrotreating  . . (.) Distillate hydrotreating . . (.) Alkylation  . . (.) Polymerization/dimerization  . . (.) Aromatics  . . (.) Isomerization  . . (.) Lubes  . . (.) Oxygenates  . . (.) Hydrogen (MMcfd)  . . (.)

aCost functions based on Kaiser and Gary (). bComplexity factor at reference capacities defined by average active U.S. refining units circa . cComplexity factor functional average and standard deviation with capacity intervals defined by average U.S. refining units plus/minus one standard deviation circa .

356 M. J. KAISER

Table . Complexity factor functional average values for variable domains.

Unit AVG ± .SD AVG ± .SD AVG ± .SD AVG ± SD AVG ± SD

Vacuum distillation . (.)a,b . (.) . (.) . (.) . (.) Delayed coking . (.) . (.) . (.) . (.) . (.) Catalytic cracking . (.) . (.) . (.) . (.) . (.) Catalytic reforming . (.) . (.) . (.) . (.) . (.) Catalytic hydrocracking . (.) . (.) . (.) . (.) . (.) Gasoline hydrotreating . (.) . (.) . (.) . (.) . (.) Distillate hydrotreating . (.) . (.) . (.) . (.) . (.) Alkylation . (.) . (.) . (.) . (.) . (.) Polymerization . (.) . (.) . (.) . (.) . (.) Aromatics . (.) . (.) . (.) . (.)c . (.) Isomerization . (.) . (.) . (.) . (.) . (.) Lubes . (.) . (.) . (.) . (.) . (.) Oxygenates . (.) . (.) . (.) . (.) . (.) Hydrogen . (.) . (.) . (.) . (.)c . (.)

aCost functions based on Kaiser and Gary (). bCapacity intervals defined by active average U.S. refining units and standard deviation circa . cCapacity intervals are truncated at zero capacity when the lower bound is negative.

Sensitivity analysis

Capacity range

CFFA values based on 2009 USGC cost functions are evaluated for capacity intervals defined by U.S. average active capacities circa 2014 plus/minus standard deviation multipliers that range from 0.25 to 2 (Table 8). As the capacity intervals shrink around their mean, the aver- age complexity factors decline but in most cases do not change significantly, and uncertainty levels as measured by the standard deviation are reduced. This is intuitive and indicates that the average functional values are not highly sensitive to the size of the capacity intervals and simply become better defined as the intervals are reduced. As the capacity intervals shrink

Figure . Complexity factor functional envelope and average values. Note that CFFA values converge to CFRC as capacity intervals shrink around their average.

THE ENGINEERING ECONOMIST 357

Figure . Complexity factor functional values for variable capacity intervals.

around their mean, it is clear that the average values will converge to the reference capacity complexity factors because the intervals are converging to average capacity (Figure 11).

For FCC, for example, the average complexity factor is 4.1 with a standard deviation of 1.6 for capacity intervals defined by one standard deviation around the average, and as the intervals shrink to one-half standard deviation, the average value is reduced to 3.9

Figure . The sensitivity functional is computed as the integral of the squared difference between a point varying over the circumference of a circle relative to its center point.

358 M. J. KAISER

Figure . Fluid catalytic cracking sensitivity function for two evaluation radii. A larger radius yields larger function values over its domain because of integration effects.

with 0.7 standard deviation, and with one-quarter standard deviation intervals, the average complexity remains at 3.9 but standard deviation shrinks again to 0.3. The complexity fac- tor at average capacity is 3.7. This same pattern is observed for all of the units evaluated (Figure 12). If the capacity intervals are expanded beyond one standard deviation, the averages and standard deviations increase more substantially. For FCC, the average complexity factor increases to 4.7 with a 3.8 standard deviation for capacity intervals defined by two standard deviations.

Sensitivity function

Another measure of sensitivity is how the complexity function changes to small changes in capacity in one or both units simultaneously. Previously, we observed that the complexity factor at reference capacity only varied slightly between average and median reference capac- ities (recall Table 6), and here this is examined in a quantitative manner.

THE ENGINEERING ECONOMIST 359

Figure . Fluid catalytic cracking sensitivity function contour plots.

The sensitivity of the complexity factor is evaluated as the difference between the function evaluated at a fixed point within its domain and a variable point a small distance away. Let the point of evaluation be (Q, q), and the variable point be described by the circumference of a circle of radius r centered at (Q, q), (Q + r·cosθ, q + r·sinθ). The difference in the functional values at these two points is squared, and the average of the squared difference is computed as the point on the circumference roams around the outside of the circle (Figure 13). Division by 2π normalizes for the circular integration in a nod to mathematical formality.

The sensitivity function is written thus:

Sr ( Q, q

) = 1

2π

∫ 2π 0

[CF(Q,q) − CF(Q + rcosθ, q + rsinθ )]2dθ , (19)

where the step-out radius r is given, θ varies from 0 to 2π radians, and (Q, q) ranges over the domain defined by the intervals IA and IB. The sensitivity function is computed empirically for each point throughout the domain of IA × IB using the complexity factor function. Note that

360 M. J. KAISER

Figure . Analytic and empirical pathways for statistical measures based on cost model data. Sensitivity analysis is performed with respect to the expected complexity factor, the sensitivity function, and average sensitivity.

the computation is performed on the surface of the complexity function while the directional vector moves around a unit circle in the (Q, q) plane, and because of the circular integration, no analytic formula exists—the integral must be evaluated numerically.

The average sensitivity function E[Sr(Q, q)] provides a (global) measure of the sensitivity of the complexity factor functional over its domain:

E[Sr(Q,q)] = ∫ d c

∫ f e Sr(Q,q)dqdQ

(d − c)( f − e) . (20)

Intuitively, the sensitivity function is a local measure of how much the complexity factor changes on average at a given point in the domain for small changes in the unit capacities. As the value of r increases, the granularity of Sr(Q, q) decreases, and to be useful the value of r should be less than about 5–10% of the capacity area. The average sensitivity function is the average of the average sensitivity across its entire domain and is a global measure of the sen- sitivity of the complexity factor functional. For regions where the complexity factor function is flat, the sensitivity function will be flat and small, and in regions that are steep or changing rapidly, Sr(Q, q) will also be steep and large. E[Sr(Q, q)] represents an aggregate indicator of the shape of the complexity functional.

Example The FCC complexity factor sensitivity function is computed (Figure 14) and contour plots are derived for r = 1 Mbpd and r = 5 Mbpd (Figure 15). At point H = (200, 25) and point M = (150, 75), S5(H) = 0.30 and S5(M) = 0.001, and a change in unit capacity at point H will lead to a much larger average change in complexity factor relative to point M. E[S1(Q, q)] = 0.0032 and E[S5(Q, q)] = 0.086 indicate larger average sensitivity when the step-out is larger.

THE ENGINEERING ECONOMIST 361

Figure . Expected average complexity factor for variable a and b parameters.

Parametric sensitivity

Previously, analytic expressions were derived for E(CF) and V(CF), and numerical compu- tations were used to construct the distribution function f(CF), sensitivity function Sr(CF), and average sensitivity function E(Sr). In this final section, the results of sensitivity analy- sis on E(CF), Sr(CF), and E(Sr) are used to illustrate structural composition of the function (Figure 16).

E(CF) There are eight variables that determine the average complexity factor E(CF)—four variables from the cost function parameters, (K, a) and (L, b), and four variables describing the capacity interval endpoints, IA = (c, d) and IB = (e, f). To present the results of sensitivity analysis, it is necessary to hold a subset of these variables constant while varying the other variables. The results are illustrated on a case-by-case basis.

362 M. J. KAISER

Figure . Expected average complexity factor in terms of a and b parameter variation.

� For fixed domain IA × IB and cost model parameters a and b, E(CF) increases with increasing a (for fixed b) and decreases (for fixed a) with increasing b (Figure 17). E(CF) is a linear function K/L.

� For fixed domain IA × IB, E(CF) is a nonlinear decreasing function of b (for fixed a), a nonlinear decreasing function of a (for fixed b), and an increasing function of K/L (Figure 18).

� For fixed K/L and cost exponents a and b, the impact of reducing the size and shape of the domain is examined using homothetic transformations6 defined by parameters 0 < α, β < 1. Holding α constant, E(CF) increases as β is reduced, whereas for fixed β, E(CF) increases with increasing α (Figure 19).

Sr(Q, q) The sensitivity function is computed at three points within the capacity domain in terms of the radius: (50, 100), (75, 150), and (100, 300). K/L and the cost model exponents a and b are held fixed. The sensitivity function depends on the radius and the point selected and the shape of the functional at the point (Figure 20). The domain is defined by IA = (10, 150) and IB = (20, 300), so a 40 Mbpd radius circle within the domain will cover about 13% of the area of the domain. The rate of change of Sr(Q, q) increases with the cost model parameters and the radius selected.

E(Sr) The expected value of Sr(Q, q) is computed over the domain of the function IA × IB for given radius and cost model exponents a and b (Figure 21). For fixed K/L, the expected sensitivity function increases nonlinearly as the radius increases, and as K/L increases, the functional values also increase. The expected value function E(Sr) is different from Sr(Q, q) because it does not depend upon a specific point and is evaluated over the entire domain. E(Sr) represents an aggregate measure of the function.

 Homothetic transformations are defined by α·IA = α·(c, d) = (αc, αd), and β·IB = β·(e, f) = (βe, βf).

THE ENGINEERING ECONOMIST 363

Figure . Expected average complexity factor for variable domain via α and β parameter variation.

Application—Refinery complexity moments

Because complexity factors can be interpreted as a stochastic variable with a mean and vari- ance, refinery complexity also exhibits a mean and variance. The moments of the refinery complexity index are computed based on the complexity equation and the linearity of the operators. Process units are assumed independent. Because Qi is fixed at a point in time and complexity factors are independent random variables described by E(CFi) = CFFA(CFi) and V(CFi), i = 1, …, n, the expected value and variance of the refinery complexity is computed as follows:

E [CI (R)] = E ( 1 +

n∑ i=1

Qi Q0

CFi

) = 1 + E

( n∑

i=1

Qi Q0

CFi

)

= 1 + n∑

i=1 E ( Qi Q0

CFi )

= 1 + n∑

i=1

Qi Q0

E (CFi) (21)

364 M. J. KAISER

Figure . Sensitivity function relations for variable radius at three points {(, ), (, ), (, )} within the capacity domain.

V [CI (R)] = n∑

i=1

( Qi Q0

)2 V (CFi). (22)

Example In 2014, PBF Energy’s Delaware City, Delaware, refinery reported 190 Mbpd atmospheric distillation capacity, 102 Mbpd vacuum distillation capacity, 82 Mbpd fluid catalytic crack- ing capacity, 47 Mbpd fluid coking capacity, and 18 Mbpd hydrocracking capacity (Table 9). Hydrotreaters process straight run naphtha, diesel, and middle distillates with 160 Mbpd total

THE ENGINEERING ECONOMIST 365

Figure . Expected sensitivity function for variable K/L and radii.

Table . PBF Energy’s Delaware City refinery complexity moments ().

Capacity (Mbpd) Qi/Q (Qi/Q)  CFFAi SD(CFFAi) E[CI(R)] V(CFFAi)

Atmos. distillation  . . . . . . Vacuum distillation  . . . . . . Catalytic cracking  . . . . . . Hydrotreating  . . . . . . Hydrocracking  . . . . . . Catalytic reforming  . . . . . . BTX extraction  . . . . . . C isomerization  . . . . . . Alkylation  . . . . . . Polymerization  . . . . . . Fluid coking  . . . . . . Hydrogen (MMcfd)  . . . . . . Refinery complexity . .

capacity, and there is 16 Mbpd polymerization, 11 Mbpd alkylation, and 6 Mbpd isomer- ization capacity. Using 1998 OGJ complexity factors, refinery complexity circa 2014 is com- puted to be 12.9. To compute the variance of the complexity index requires knowledge of the variance of the complexity factors, which is available through the CFFA values in Table 7. The expected value and variance of the refinery complexity is computed to be 13.0 and 3.6 (Table 9).

Conclusions

To improve the accuracy and reliability of complexity factors and to enhance their precision, it is necessary to expand data collection and processing, improve the methodology of compu- tation, and to precisely specify the computational procedures employed. The ability to expand and improve data collection is limited because cost data are not publicly reported in the detail necessary for high-quality analysis, but advances can be made by using cost functions, which facilitates more robust analysis.

The results of this article showed how improvements in methodology lead to additional insights into Nelson’s traditional complexity computations and how the sensitivity of the met- ric can be quantified in a precise fashion. These improvements translate directly to enhancing the reliability of the complexity factors used in the computation of refinery complexity. CFRA

366 M. J. KAISER

extends the traditional complexity factor metric, and the CFFA approach generalizes the met- ric so that the CFRC represents a special case.

A more precise way to formulate complexity factors is to process data using cost functions and to apply the cost functions at specified capacities or construct the complexity factor func- tional. The cost data are the same in all three approaches, but how the data are processed and evaluated is different and, consequently, the complexity factor values will also be different. Controlling for capacity enhances transparency, increases specificity, and allows for a formal quantification of uncertainty, which provides additional insight into the nature of complexity factors and their application.

Notes on contributor

Mark J. Kaiser is Professor and Director of the Research and Development Division at the Center for Energy Studies at Louisiana State University. His primary research interests are related to cost analysis, fiscal systems, and financial modeling in the oil and gas industry. Kaiser holds a Ph.D. in industrial engineering and operations research from Purdue University.

References

Farrar, G.L. (1989, October 2) Interest reviving in complexity factors. Oil & Gas Journal, p. 90. Johnston, D. (1998, May 18) Complexity index indicates refinery capability, value. Oil & Gas Journal, p.

74. Kaiser, M.J. and Gary, J.H. (2009) Refinery cost functions in the U.S. Gulf Coast. Petroleum Science and

Technology, 27(2), 168–181. Meyers, R.A. (2004) Handbook of petroleum refining processes, 3rd ed. McGraw-Hill, New York. Nelson, W.L. (1960, May 14) How to describe refining complexity. Oil & Gas Journal, p. 81. Nelson, W.L. (1976a, September 13) Complexity—1: The concept of refining complexity. Oil & Gas

Journal, p. 81. Nelson, W.L. (1976b) Guide to refinery operating cost (process costimating), 3rd ed. Petroleum Publishing,

Tulsa, OK. Ostwald, P.F. and McLaren, T.S. (2003) Cost analysis and estimation for engineering and management.

Prentice Hall, Englewood Cliffs, NJ. Peters, M.S., Timmerhaus, K.D. and West, R.E. (2003) Plant design and economics for chemical engineers,

5th ed. McGraw Hill, Boston. Seider, W.D., Seader, J.D. and Lewin, D.R. (2004) Product and process design principles, John Wiley &

Sons, New York. Speight, J.G. (1998) The chemistry and technology of petroleum, 3rd ed. Marcel Dekker, Inc., New York.

Appendix A: Complexity factor functional moments

E(CF) = ∫ d c

∫ f e CF(Q,q)dqdQ

(d − c)( f − e) = ∫ d c

∫ f e

K L Q

a−1q1−bdQdq (d − c)( f − e) =

∫ d c

K L Q

a−1 ( q1−b+1 1−b+1

) dQ

(d − c)( f − e)

= ∫ d c

K L Q

a−1 ( q2−b 2−b

) dQ

(d − c) ( f − e) = ∫ d c

K L Q

a−1 (

f 2−b−e2−b 2−b

) dQ

(d − c)( f − e) = K L

( Qa−1+1 a−1+1

)( f 2−b−e2−b

2−b

) (d − c)( f − e)

THE ENGINEERING ECONOMIST 367

= K L

[ (da−ca)( f 2−b−e2−b)

a(2−b)

] (d − c)( f − e) =

K(da − ca)( f 2−b − e2−b) La(2−b)(d − c)( f − e)

V (CF) = ∫ d c

∫ f e [CF(Q,q)−E(CF)]2dQdq

(d − c)( f − e)

= ∫ d c

∫ f e [

K L Q

a−1q1−b−E(CF)]2dQdq (d − c)( f − e)

= ∫ d c

∫ f e [(

K L )

2Q2a−2q2−2b−2( KL )Qa−1q1−b + E(CF)2]dQdq (d − c)( f − e)

= ∫ d c [(

K L )

2 ( f 2a−1−e2a−1) (2a−1) q

2−2b−2( KL ) ( f a−1−ea−1)

a q 1−b + E(CF)2( f − e)]dq

(d − c)( f − e)

= ( K L

)2 ( f 2a−1−e2a−1)(d3−2b−c3−2b)

(3 − 2b)(2a−1)(d − c)( f − e) − 2E(CF)

× ( K L

) ( f a − ea)(d2−b−c2−b) a(2−b)(d − c)( f − e) + E(CF)

2

Appendix B: Complexity factor functional examples

Figure B. Delayed coking complexity factor functional, contour map, sensitivity function, and distribution value (from top left, clockwise).

368 M. J. KAISER

Figure B. Catalytic reforming complexity factor functional, contour map, sensitivity function, and distribu- tion value (from top left, clockwise).

Figure B. Catalytic hydrocracking complexity factor functional, contour map, sensitivity function, and dis- tribution value (from top left, clockwise).

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