Model-based real-time optimization of automotive gasoline blending operations

Journal of Process Control 10 (2000) 43±58

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Model-based real-time optimization of automotive gasoline blending operations A. Singh a,1, J.F. Forbes b,*, P.J. Vermeer c, S.S. Woo d a Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, ON, Canada Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G6 c Sunoco Inc., Sarnia, ON, Canada d Imperial Oil Ltd., Toronto, ON, Canada

b

Received 1 March 1999

Abstract Gasoline blending is a key process in the successful operation of most petroleum re®neries and real-time optimization (RTO) of gasoline blend recipes has the potential to provide a competitive bene®t for oil re®ners. The trend toward the use of ``running'' tanks for blender feedstocks and the recent advances in measurement technology have provided the opportunity for improved blending performance using RTO. This paper provides an improved formulation for the gasoline blend optimization problem that incorporates both the blend horizon and a stochastic model of disturbances into the RTO problem. The proposed approach is illustrated with a blender simulation study. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Blending; Optimization; Control

1. Introduction Gasoline is one of the most important re®nery products as it can yield 60±70% of a typical re®nery's total revenue [1]. Thus, tight control of blending operations can provide a crucial edge to the pro®tability of a re®nery. As shown in Fig. 1, gasoline blending is the process of combining a number of feedstocks, produced by other re®nery process units, together with small amounts of additives (such as antioxidants, corrosion inhibitors, metal deactivators, detergents, and dyes) to make a mixture meeting certain quality speci®cations. As can be seen in Fig. 1, the crude fractionation unit distills crude oil into several ``cuts''. Only a small portion of these distillation products, the light straight run (LSR) naphtha, is suitable for blending directly into gasoline. Other ``cuts'' must be further processed by units such as catalytic cracking, hydrocracking, alkylation, and catalytic reforming. Some of these re®nery

* Corresponding author. Tel.:+1-780-492-0873; fax:+1-780-4922881. E-mail address: [email protected] (J.F. Forbes) 1 Current address is Shell Canada Ltd., Samia, ON, Canada.

products are further separated into light and heavy fractions in order to provide more ¯exibility in blending. Thus, a large re®nery can have more than 20 blender feedstocks that are blended into several grades of gasoline [2]. Speci®cations on gasoline qualities include octane number, volatility, sulphur content, aromatics content, and viscosity to ensure acceptable engine performance, as well as for environmental reasons. The octane number of a fuel is de®ned as the percentage of iso-octane (assigned an octane number of 100) in a blend with nheptane (assigned an octane number of 0) that exhibits the same resistance to knocking as the test fuel under standard conditions in a standard engine [3]. Two standard test procedures are used to characterize the antiknock properties of fuels for spark engines: (1) the ASTM D-908 test gives the research octane number (RON) and (2) the ASTM D-357 gives the motor octane number (MON). The RON represents antiknock properties under conditions of low speed and frequent accelerations while the MON represents engine performance under more severe high speed conditions. The octane number posted at gasoline stations is the arith  RON ‡ MON metic mean of the RON and MON 2

0959-1524/00/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0959-1524(99)00037-2

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A. Singh et al. / Journal of Process Control 10 (2000) 43±58

Nomenclature symbols: A Aromatics content (vol%) Ethyl RT-70 model parameters ai b Bias vector c Process economics vector d Disturbance g Blending model equations h Demand and availability constraints m Motor octane number N Normal or Gaussian distribution n Number of blend components O Ole®ns content (vol%) q Reformate qualities vector R Feedstock qualities matrix r Research octane number s Octane number sensitivity s Blended quality speci®cation vector Volume fraction of blend component i ui t Time w Weighting vector x Feedstock or component ¯ows (blend recipe) " Reformate quality innovations subscripts: f Final o Initial p Present superscript and decorations: ‰Šb predicted ‰Š Average e ‰Š Blend index [4]. The main di€erence between automotive gasoline grades is their antiknock properties. For example, regular and premium gasolines are speci®ed to have posted octane numbers of 87 and 92, respectively. Automobile engine performance is also a€ected by a fuel's volatility and boiling range. A vapour pressure that is too high for the given ambient temperature will result in vapour locking and motor stalling, while a vapour pressure which is too low will lead to diculties in engine start-up [3]. The boiling range also a€ects the engine during start-up and driving, and is particularly important for good performance during quick acceleration and high speed operations. Like vapour pressure, the boiling range has to be tailored for a given geographical region and season. There should be a fairly reasonable distribution of light (more volatile), intermediate, and heavy (relatively non-volatile) components; however, gasolines blended for use in

cold climates need more light components (front end volatility) than those intended for use in warmer climates [3]. In addition, govermnent regulations place maximum restrictions on the vapour pressure to limit the emission of volatile organic compounds into the atmosphere [5]. Other environmental restrictions include maximum compositions of aromatics, ole®n, sulphur and oxygenates. Then, the gasoline blending challenge is to produce blends in such a way as to maximize pro®t while meeting all quality speci®cations on all blends, in addition to satisfying any product demands and feedstock availability limits. As a result, the blending problem is naturally posed as a constrained optimization problem and blender control is traditionally based on optimization technology [6]. The re®nery process streams shown in Fig. 1 are typically routed through intermediate storage tanks to aid ecient gasoline blender operation; however, these process streams can be blended directly into gasoline without ®rst being routed to intermediate tanks [7]. Storage tanks can either be ``standing tanks'' or ``running tanks''. Traditionally, blender feedstocks were stored in ``standing tanks'' before being used to produce gasoline; however, the economic pressures for reducing on-site tankage and safety concerns for reducing inventory of volatile materials, have led to the practice of inline blending or blending out of intermediate ``running tanks''. While feedstock qualities remain fairly constant over time when blending out of (well-mixed) ``standing tanks'', they can vary signi®cantly during a blend when in-line blending or when blending out of ``running tanks''. This paper provides a new approach to gasoline blend optimization that can provide signi®cantly improved pro®tability. The work discussed here speci®cally addresses blend optimization when in-line blending (or blending out of ``running tanks'') and stochastic disturbances exist in feedstock qualities (e.g. disturbances due to upstream process operation changes). The paper begins by brie¯y examining blending models and their suitability for use in blend optimization. Then current blender optimization strategies are discussed and a new time-horizon based real-time optimization approach is proposed. Finally, an automotive gasoline blending case-study is used to illustrate the superior performance of this new RTO method. 2. Gasoline blending models As shown in Fig. 1, the gasoline blending process mixes various component (or feedstock) streams to produce an automotive gasoline product stream. A number of properties are used to characterize automotive gasolines and the components that are blended

A. Singh et al. / Journal of Process Control 10 (2000) 43±58

45

Fig. 1. Simpli®ed petroleum re®nery ¯owsheet.

to produce them, including octane number (ON), Reid vapour pressure (RVP), ASTM distillation points, viscosity, ¯ash point, and aniline point. Since the process control objectives for the gasoline blender are often framed in terms of meeting speci®cations on these properties for the gasoline product, process models in the form of component property mixing rules are required for e€ective process control. Barrow [8] de®nes an ideal mixing process as one in which a product quality is the volumetric average of the component properties. Further, it has been widely recognized that most gasoline properties blend in a non-ideal and nonlinear fashion, necessitating the use of more complex blending models to predict these properties [9]. This section will discuss blending models (i.e. mixing rules) that accurately predict these product properties given the blend component properties and are suitable for use in on-line blend optimization and control strategies. For the purposes of an RTO-based blend optimizer, the most important characteristics of a good blending model are: (1) accuracy for optimization purposes, (2) parsimony, and (3) ease with which model parameters can be accurately updated on-line. Modelling for RTO purposes is discussed in detail in Forbes et al. [10], and Forbes and Marlin [12]. The importance of model parsimony is discussed in Box and Jenkins [12]. Krishnan et al. [13] and Forbes and Marlin [11] discuss some RTO model updating issues. The main purpose of this paper is to present and illustrate a new method for real-time optimization of automotive gasoline blends. Thus, for clarity of presentation, the gasoline properties considered in this paper will be

limited to the research octane number (RON), the motor octane number (MON), and the Reid vapour pressure (RVP). Blending models for RON, MON and RVP are discussed brie¯y here and a more detailed discussion can be found in Singh [14]. Blending models for other qualities can be found in such literature as Gary and Handwerk [4] or McLellan [15]. It should be noted that the generality of the results presented here is in no way limited by considering only these three key gasoline properties. 2.1. Octane number The two most common octane numbers used for spark engines are the research octane number (RON) and the motor octane number (MON). These octane numbers are used to indicate the ``antiknock'' characteristics of gasoline (i.e. the ability of the gasoline to resist premature detonation in the combustion chamber of an automobile engine). The RON, de®ned by the American Society for Testing and Materials (ASTM) under the designation ASTM D-908, is intended to represent engine performance under city driving conditions; whereas the MON, as de®ned under the designation ASTM D-357, is intended to represent engine performance on the highway [4]. As stated by Rusin [9], it is well-recognized that both the RON and MON of a gasoline, blend nonlinearly and a number of empirical blending models are available in the literature for predicting blended octane numbers given component properties. The methods discussed in this section include: (1) the blending octane number method [4]; (2) the transformation method [16];

46

A. Singh et al. / Journal of Process Control 10 (2000) 43±58

(3) the Ethyl RT-70 models [17]; (4) an ole®ns content based method due to Stewart [18]; (5) the interaction method [19]; and the excess method [20]. It should be noted that this work is not intended as a comprehensive review of available octane blending models. The intent here is to present the most widely used methods for predicting the blended octane number of a gasoline mixture, given the component properties. Both the blending octane number approach [4] and the transformation method [16] attempt to convert the nonlinear blending problem to a linear problem by transforming RON and MON into quantities that blend ideally on a volumetric basis. The results of the blend calculations are then transformed back to the appropriate octane number. Although the blending octane number method requires fewer and simpler calculations than the far more complex transformation method, the blending octane numbers must be computed from experimental data. The interaction method [19,21,22] and the excess method [20] are essentially regression analysis based approaches. In the interaction method, blending nonlinearity is accounted for via two-factor interaction (or bilinear) terms; whereas, in the excess method the deviation is accounted for using an excess (or bias) term. Both of these approaches require a considerable amount of data from laboratory blending studies to estimate the large number of parameters in the models. The ethyl RT-70 models [17], and a similar method proposed by Stewart [18], represent blending nonlinearities as functions of compositions of compounds (e.g. ole®ns, aromatics and so forth). Despite their age, the ethyl RT-70 blending models are widely used and have become a standard against which many of the newer models are compared. In the ethyl RT-70 approach, the blending nonlinearity is expressed in terms of the component sensitivity (RON±MON), ole®ns content, and the aromatics content of the blend components. As previously stated, the various blending models will be compared for use in RTO based on three key factors: (1) predictive accuracy; (2) parsimony; and (3) ease of implementation. Table 1 summarizes the number of parameters contained in each of the discussed models. In Table 2 the predictive accuracy of each model is presented with respect to both interpolation and extrapolation. Interpolative accuracy is the standard error of the estimated blended quality for the data set used to estimate the model parameters and extrapolative accuracy is de®ned as the standard prediction error for some other data set selected by the given author. It is surprising that for the transformation method, Rusin et al. [16] have reported a higher prediction accuracy for extrapolation than for interpolation. The excess method, being linear, is valid only within the vicinity of the nominal blend. Also, there is little information available with regards to the predictive accuracy of the

Table 1 Parsimony of octane blending models Model

No. of parameters

Ethyl Stewart

6 2 n…n ÿ 1† !RON 2 n…n ÿ 1† !MON 2 10 n!RON n!MON

Interaction

Transformation Excess

excess or Stewart methods. The interaction method has been shown to be no more accurate than the ethyl method in predicting blended octanes on general, untested data sets even though it has many more parameters than the ethyl models. Although the transformation models seem to provide slightly better predictive accuracy than the ethyl models, they require more feedstock quality data and contain more parameters (less parsimonious). Moreover, the ethyl models are much more simple and easy to use than the transformation models. 2.2. Reid vapour pressure The Reid vapour pressure (RVP) is de®ned by the American Society for Testing and Materials under the designation ASTM D-323-56 and gives an indication of the volatility of a gasoline blend. RVP may be roughly considered as the vapour pressure of the gasoline (or blend component) at 100 F (38 C). The RVP of a gasoline blend a€ects the gasoline performance in terms of ease of starting, engine warm-up, and rate of acceleration [4]. Speci®cations on RVP are an important consideration during gasoline blending as they limit the amount of n-butane, a relatively cheap source of octane, that can be added to a blend. Two fundamental methods for predicting blended RVP are given in Stewart [23] and Vazques-Esparragoza et al. [24]. Stewart [23] presented one of the ®rst theoretical approaches for predicting blended RVP's. The method uses component data (such as feedstock composition and component volatility), thermodynamic relationships, and a set simplifying assumptions (i.e. presence of air and water vapour are ignored, absolute pressure is taken as the RVP, volatile components are assumed to have the density of butanes, and the nonvolatile components are assumed to have the thermal expansion characteristics of n-octane) to predict the blended RVP of a mixture. Vazques Esparragoza et al. [24] presented an iterative procedure that extended Stewart's method. In this approach, the additivity of liquid and gas volumes is assumed and a di€erent equation of state used. Further, the Vazques-Esparragoza et al. approach requires that the molar composition of the

A. Singh et al. / Journal of Process Control 10 (2000) 43±58

47

Table 2 Predictive accuracy of octane blending modelsa Model

Ethyl Stewart Interaction Transformation Excess a b c d e f

RON

MON

Interpolation

Extrapolation

Interpolation

Extrapolation

0.82b 0.77c 0.28d 0.51e N/Af

0.92b N/A 0.8d 0.43e N/A

0.79b 0.64b 0.33d 0.60e N/A

0.61b N/A 0.64d 0.54e N/A

Reported are standard deviations of prediction error in octane number. From Healy et al. [17]. From Stewart [18]. From Morris [19]. From Rusin et al. [16]. N/A-not available.

feedstocks be known. The computations required in both of these methods are complex in comparison to those required in other approaches. Two empirical approaches for predicting blended RVP are the interaction method [19] and the blending index approach [4]. As for predicting blended octane number, the interaction approach has been applied for predicting blended RVP [19]. The interaction method for RVP is exactly the same as for predicting blended octane numbers. Perhaps, one of the easiest to use empirical methods was developed by the Chevron Research Company and is referred to as the blending index method [4]. In this approach, blended RVP's are predicted using the Reid vapour pressure blending indices (RVPBI) which blend linearly. Other methods include that proposed by Haskell and Beavon [25], which involves the volumetric averaging of the component RVP's except those of butanes. In this approach the butanes are assigned variable ``blending pressure values'', which were calculated based on the RVP of the butanes and that of the de-butanized blend (blend with all components except the butanes). Another simple approach uses molar averaging (not volumetric) of the component RVP's based on the blend composition [23]. However, molar composition of feedstocks may not be readily available. Comparisons of predictive accuracy of some of the methods can be found in Stewart [23] who looked at the standard deviation of prediction error for 67 blends using di€erent blended RVP prediction approaches. The reported standard deviations are presented in Table 3. The methods based on fundamental principles provide more accurate predictions; however, these theoretical methods [23,24] are not suitable for use in on-line control systems due to their computational requirements. The interaction method is also not suitable for on-line implementation, since it requires numerous parameters to be updated on-line (as was discussed in the octane blending case). Although not as accurate as the theoretical meth-

Table 3 Predictive accuracy of RVP blending modelsa Method

Standard deviation (psi)

Stewart Ideal blendingb Haskel and Beavon [25] Molar average

0.76 1.30 1.01 1.17

a b

All data from Stewart [25]. Ideal blending is volumetric average RVP.

ods, the simplicity of the blending index method makes it attractive for use in gasoline blender control. 3. Blender control and optimization Gasoline blending can be considered a batch process where production (i.e. blend volume) is ®xed either contractually or by the re®nery production schedule. Thus, the gasoline blend optimization and control problem is a ®xed end-point problem either in terms of blended product volume or blending time. Any gasoline blend must meet quality speci®cations, which are gasoline grade dependent. Further, the gasoline blending process is subject to a number of operating restrictions, including limits on the availability of blender feedstocks (blend components) and product storage facilities. Thus, the objectives of any gasoline blend control and optimization system are to maximize the pro®tability of a blend while satisfying all of the product quality and blender operating constraints. The performance of a gasoline blender, or blender automation system, is often discussed in terms of product ``quality giveaway'' and number of reblends required per month. ``Quality giveaway'' refers to making gasoline with qualities that either exceed the minimum required (or fall below the maximum allowed). Such

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A. Singh et al. / Journal of Process Control 10 (2000) 43±58

``quality giveaway'' may indicate that a more expensive blender feedstock was used when a lower cost feedstock might have suced, possibly (but not necessarily) resulting in a reduced blend pro®tability. It is estimated that consistent octane giveaway of 0.1 octane numbers can cost a re®nery several millions of dollars a year [2]. Re-blending is required when a blend, or portion thereof, does not meet speci®cation. Re-blends can also lead to a signi®cant reduction in re®nery revenue by taking up valuable tank space and blending time, thereby reducing overall capacity. Determining blender automation system performance solely in terms of product ``quality giveaway'' and re-blend frequency does not explicitly consider an essential third performance criterion, the pro®tability of the blend. Finally, in any blending control and optimization problem there may be more than one feasible solution with no ``quality giveaway''. In such situations, an e€ective blend automation system must consistently identify and enforce the blend recipe that is economically optimal. 3.1. Current blender automation practices There is a wide range of sophistication in gasoline blending automation technology presently employed by re®ners. A reasonably complete presentation of currently available commercial gasoline blending control and optimization systems can be found in [26]. Typically, as is shown in Fig. 2, any gasoline blending automation system is built on three levels: o€-line optimizer or scheduler, online optimizer, and regulatory control [7,27]. At the top of the hierarchy lies the planner or scheduler which uses o€-line optimization to plan re®nery

operations for the long-range (months), intermediaterange (weeks), and short-range (days). Long-range planning can be performed at either the re®nery or corporate level. Long-range forecasts of product demands, crude oil prices, and process unit performances are used to plan re®nery operations several months to a few years into the future [7,27]. Medium-range planning provides scheduling of re®nery operations on a shorter time frame (a few weeks) and is used to improve on the long-range plan. In addition to economic forecasts, forecasts of re®nery feedstock qualities and quantities, and tank availability are used to check feasibility of the long-range plan and to revise production targets set by the long-range planner. Here, the re®nery schedule is also adjusted for major unit upsets and unscheduled shutdowns. Signi®cant changes in expected crude quality and changes in operating ¯exibility (such as additional constraints) are used at this stage to revise re®nery plans as well [27]. Short-range planning works at the unit level, rather than at the re®nery level, and plans blender operations over a shorter time frame (1 to 2 days) [7]. At this stage, assumed feedstock qualities (or measured qualities, if available) are used to produce blend recipes which are then downloaded to the on-line optimizer. Usually, all levels of planning employing an o€-line optimizer based on linear programming [1,27,28]. The on-line optimizer uses on-line blended quality information to modify the initial recipe during the blend and provides ®nal blend recipes to be executed by the controller. On-line optimizers are designed to either minimize deviation from the initial blend recipe or optimize some economic performance (such as blend pro®tability), while satisfying all constraints [29].

Fig. 2. Blender control hierarchy.

A. Singh et al. / Journal of Process Control 10 (2000) 43±58

Recent developments have tended toward optimizing steady-state process economics (i.e. conventional RTO). Vermeer et al. [30,31] have expanded on the conventional steady-state RTO to include blender dynamics and provide the appropriate feedback to the LP based on-line optimizer, which can then allow the RTO layer to operate at a higher frequency. The blend recipes generated by the on-line optimizer provide component ¯ow setpoints, which are implemented at the regulatory control level. Final blend control is achieved through conventional distributed control systems (DCS). The DCS-resident regulatory controller (usually PID loops) maintains ¯ow ratio to match the recipe, maintains a target product ¯ow rate, and stops when the required amount of product has been blended [7]. The di€erent blender control and optimization subsystems in Fig. 2 have to be properly integrated to provide the required information ¯ow (e.g. tank and feedstock inventory) necessary to ensure the success of the whole system. Databases are used to keep track of feedstock inventory, tank availability, feedstock qualities, and blended qualities. The subsystems write to and/or retrieve information from these databases, thus linking the subsystems into an integrated network [7,28,29,32]. 3.1.1. Real-time optimization layer As previously discussed, at the heart of the blend automation system is a real-time optimizer. This RTO system determines the ®nal blend recipe and so the performance of the RTO system is crucial to the performance of the entire integrated gasoline blender control e€ort. The design and performance of the gasoline blender RTO system is the focus of this paper. This online optimizer is usually based on linear programming [29,33-37]. This approach neglects the widespread recognition that most important gasoline properties have been shown to blend in a nonlinear fashion [9]. Some re®ners have tried addressing this problem by using nonlinear models and employing nonlinear programming (NLP) techniques in the on-line optimizer [23,34]; however, linear programming remains as the predominant technology, owing its popularity to reliability and ease of use [38]. In an attempt to maintain accuracy of the models in the optimizer, the feedback employed is usually ``bias updating'' [34,39] Bias updating involves comparing measured blended qualities with those predicted by the models in the on-line optimizer; the di€erence between these two is then added as a constant error term to the appropriate blending models. Thus, the RTO system is most often formulated as a linear program with bias updating. 3.1.2. Limitations of existing controllers The linear programming with bias updating approach for RTO in gasoline blending has proven quite success-

49

ful in practice and is the basis of many commercially available blend control systems. The success of this approach has largely been due to the practice of blending from well-mixed storage tanks; however, the current trend toward in-line blending and blending out of ``running'' tanks presents a problem that this control structure cannot adequately address. This problem results from upstream process variations such as catalyst deactivation, heat exchanger fouling, and so forth causing variations in the blender feedstock qualities. When blending from ``standing'' feedstock tanks, such quality variations are eliminated and the blend component properties can be known with a reasonable degree of accuracy. When in-line blending or blending from ``running'' tanks, the blender feedstock qualities can vary in time as a result of upstream process changes. The LP (or NLP) plus bias updating formulation may not be able to adequately handle such time-varying feedstock qualities [10]. In Forbes and Marlin [10] it was noted that the bias updating approach may still produce gasoline blends that meet quality speci®cations, but these blends may not be economically optimal. Further, most blender automation systems contain lower level feedback controllers that enforce blend speci®cations. Thus, although products can be made to meet speci®cations in this manner, the resulting blend will not necessarily be the economic optimal combination of feedstocks. Some re®ners have tried addressing this problem by feedback of laboratory analyses of feedstock qualities and then using multi-period optimization [36,40]. The interval between optimizations is usually a day and feedstock qualities are assumed to remain unchanged during these periods. Thus, this approach cannot e€ectively deal with higher frequency disturbances arising from upstream process operation changes, which have some stochastic structure to them. Vermeer [30,31] have used blended quality measurements to predict disturbances in these qualities, entering the blender. These disturbance predictions are assumed to be step disturbances that remain constant during the optimization interval and as a result, the approach does not exploit disturbances with other forms of stochastic structure. The problem addressed in this work is the development of a blend RTO system, which can e€ectively deal with stochastic disturbances in feedstock qualities. 3.2. Ideal blend optimizer The ideal blend optimizer would provide the maximum possible pro®t from the blending process, while meeting all blended quality speci®cations, as well as satisfying demand and availability limits. In order to achieve such perfect optimization, the RTO system would require perfect knowledge of the blending process. That is, it should employ blending models that capture the true blending behaviour. In addition, the

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A. Singh et al. / Journal of Process Control 10 (2000) 43±58

RTO system must have perfect knowledge of all feedstock qualities at all times throughout the entire blending time horizon (past, present and future). For batch blending of gasoline (blending to a storage tank for a given length of time), the problem can be formulated as an optimization problem where the objective is to maximize pro®t over the entire blending time horizon, subject to meeting quality constraints and demand/ availability limits at the end of the blend. Mathematically, this problem has the form: … tf

max

x

to

cT xdt

…1†

subject to: … tf to

… tf to

g…R…t†; x…t††dt4

h…x†dt  0

 … tf to



  wT …t†x…t† dt s

…1a†

…1b†

where: c is a constant vector containing process economics, x are the feedstock or component ¯ows, R is a matrix containing the feedstock qualities, g is a vector of blending equations that represent the true blending behaviour, w is a weighting vector (usually containing only ones and zeros to indicate the presence of a speci®c feedstock stream in a blend), s are the blended quality speci®cations, h contains product demand and feedstock availability constraints, and to , and tf are the initial and ®nal blend times, respectively. The controller works to maximize pro®t over the whole blend by integrating blending economics from the start of the blend (to ) to the end of the blend (tf ), rather than at a speci®c point in time as is the convention in most RTO applications. The ®rst set of constraints, given by Inequality (1a), ensure that quality speci®cations are met at the end of the blend. These constraints use perfect knowledge of the feedstock qualities (R) over the entire blend horizon (from to to tf ) as well as perfect blending models in vector g. The second set of constraints, given by Inequality (1b), integrate mass balance information over the blend horizon in order to satisfy all product demand and feedstock availability limits. Problem (1) contains integrals in the objective function as well as in the constraints and so could be solved using dynamic programming techniques [41]. Alternatively, the problem can be discretized and solved using standard NLP methods [42]. The solution to Problem (1) will provide optimal feedstock ¯ows or blend recipes (x(t)) over the whole blending horizon, which represent the optimal blend component ¯ow-rate trajectories. The value of the objective function at the

optimum represents the theoretical maximum pro®t that can be earned from the blending process. Solution of the ideal optimization Problem (1) requires future knowledge of feedstock qualities and so cannot be realized in practice; however, it can serve as a benchmark against which the performance of other blend control and optimization strategies can be evaluated, as well as provide insight into better blender control and optimization system designs. 3.3. Time-horizon based RTO As previously discussed, generally Problem (1) cannot be solved since it requires future knowledge of all feedstock qualities. Although perfect knowledge of feedstock qualities at all futures times during the blend is not possible, past measured feedstock qualities can be used to predict future values. Using predictions of future feedstock quality values for the remaining blending time-horizon will allow blend optimization results to be based on knowledge of feedstock behaviour over the entire blend horizon. This approach adaptively generates blend recipes for the remaining blend horizon and at each RTO interval, the computed recipe for the next RTO interval is implemented. This iterative process of feedstock quality prediction (model updating) and optimization would be repeated at each RTO interval in a receding horizon fashion until the end of the blend is reached. The time-horizon based RTO (THRTO) strategy discussed above and the ideal optimizer of Section 3.2 work toward meeting quality speci®cations only at the end of the blend. O€-speci®cation blends could be encountered if a blend were to be prematurely terminated. In order to guard against such situations, an additional constraint may be added to the optimization problem that would ensure that product speci®cations are met at all times in the future. This control strategy can be mathematically formulated as: tf maxX T

x

c x
t

…2†

tp

subject to: tf  tp  X X ^ g…R…t†; x…t††
t ‡ g R…t†; x…t† to


t  

(

tp

tf X



to

) t f X  wT …t†x…t†
t s h…x†dt  0





 ^ g R…t†; x…t†  wT …t†x…t† s

to

where: R is a matrix containing past and present measured feedstock qualities, RÃ is a matrix containing the predicted feedstock qualities, g is a vector of blending

A. Singh et al. / Journal of Process Control 10 (2000) 43±58

equations which may be linear or nonlinear, and to ; tp ; tf are the initial, present, and ®nal blend times, respectively,
t is the RTO interval, and h contains product demand and feedstock availability constraints. Note that most automation systems will require that the RTO interval be a ®xed time period. Thus discretization of the integral functions of Problem (1) will be based on a ®xed sampling period for most industrial applications. Note that Problem (2) is very similar to the perfect blend optimization Problem (1). The proposed blend optimization strategy di€ers from the ideal optimizer in that the blend horizon is discretized and divided into two parts (past and future). Then, optimization occurs only over the remaining blending time interval between the current time (tp ) and the blend completion time (tf ). Also, Problem (2) contains an additional constraint in order to ensure that product speci®cations are met for the product blended during the subsequent RTO interval. Since the economic value of previously blended material is ®xed and constant, the objectives of Problems (2) and (1) can be considered equivalent. The ®rst set of constraints in Problem (2) are the discrete analog of Inequality (1a) and ensure that the blend is expected to meet speci®cations upon completion of blending. This requires that feedstock qualities be predicted to the end of the blend. Such feedstock quality predictions can be obtained using conventional prediction ®lter techniques [12,43]. The second set of constraints ensure that all demand and availability limits are met for the whole blend. Finally, the third set of constraints in Problem (2) use an estimate of the feedstock qualities for next RTO interval to ensure that the product produced during this next period meets speci®cations. When computational power is available, rather than discretizing the ideal optimizer as was done in Problem (2), the ideal optimization Problem (1) can be solved using predictions of future feedstock qualities and the resulting time-horizon based optimization problem solved employing dynamic programming techniques [41]. 3.3.1. MPC analogies There are some strong analogies between the proposed gasoline blender RTO system and modern model predictive control (MPC) methods [44,45]. Although it is not the intent of this paper to fully investigate the similarities and di€erences between the proposed blender RTO strategy and MPC, some discussion is warranted for completeness. The blender optimization strategy presented in this paper adapts the blend recipe at each RTO interval in a two step procedure: (1) a model updating phase where future values of feedstock qualities are forecasted; and (2) a blend recipe trajectory is calculated for the remaining blend horizon using the updated blending model. Only the blend recipe for the next RTO interval is implemented, before the entire procedure is repeated.

51

Similarly, at each control interval MPC algorithms: (1) predict the e€ects of disturbances and past control actions over the time horizon of interest; (2) calculate a manipulated variable trajectory over the required time horizon; and (3) only implement the control action for the next control interval. Most MPC algorithms are considered to be receding horizon controllers, since both the prediction and control horizons are ®xed. In the proposed blender RTO strategy the endpoint of the blending operation is ®xed (i.e. the blend completion time is ®xed by the re®nery scheduling function). Thus, the proposed RTO approach is similar to the shrinking-horizon model predictive controller (SHMPC) proposed by [45] for batch processes. The key di€erence between the SHMPC algorithm and the proposed RTO strategy is that SHMPC attempts to meet the setpoints supplied to the algorithm, whereas the THRTO approach generates these setpoints (blend recipe) based on process economics. Such setpoints are then be enforced by an SHMPC algorithm or any other process control strategy. The tuning parameters in a conventional MPC algorithm include penalties for input and output variables, the length of the prediction horizon, and the length of the control horizon. In the proposed RTO strategy only the characteristics of the disturbance prediction ®lter are user speci®ed, since the optimization horizon is ®xed at each RTO interval as the remaining blending time; however, when gasoline is being blended directly to pipeline (instead of to storage tanks as in batch blending), the optimization horizon could also become a tuning parameter. The proposed blender RTO approach is unique in comparison to other blend optimization strategies as it adapts the blend recipe at each RTO interval, allowing the RTO system to look backward in time to what has been blended and forward in time to anticipate future trends in feedstock qualities. The advantages of the time-horizon formulation are that during each RTO interval the controller has an opportunity to: (1) compensate for past o€-speci®cation blended product; (2) pre-compensate for anticipated trends in feedstock qualities; and (3) re-capture any past quality ``giveaway''. Therefore, this RTO strategy should be able to e€ectively deal with stochastic disturbances in feedstock qualities and provide better performance than other available blender optimization methods. 4. Case study The case study uses an automotive gasoline blending problem to compare the performance of three blend optimization strategies: (1) the conventional linear programming with bias update approach (LP+bias); (2) a nonlinear programming with bias update approach (NLP+bias); and (3) the time horizon based RTO

52

A. Singh et al. / Journal of Process Control 10 (2000) 43±58

approach (THRTO). The blending problem studied here is based on a case study by Forbes and Marlin [10] and involves simultaneously blending two grades of automotive gasoline using ®ve feedstocks as shown in Fig. 3. For the illustrative purposes of this case study, the product quality speci®cations for each grade of gasoline will be limited to: minimum limits on RON and MON, and a maximum limit on the RVP. The blending problem is subject to both maximum and minimum limits on product demand, as well as maximum limits on feedstock availability. All problem data are provided in Tables 4±6. In this case study, a gasoline blend period of 24 h has been chosen. The RTO interval selected for the case study is two hours, providing twelve complete RTO cycles during the blend. Fig. 3. Flowsheet for gasoline blending case study.

4.1. Process model For simulation purposes, a set of equations is required to represent the blending process for octane numbers (RON and MON) and RVP. The ethyl RT-70 [17] models have been adopted to represent the mixing rules for octane numbers. As discussed in Section 2, the ethyl models were found to exhibit the best combination of predictive accuracy and parsimony for octane numbers. The ethyl RT-70 models are: RONblend ˆ r ‡ a1 …rs ÿ rs† ‡ a2

…O2

ÿ O 2 † ‡ a3 …A2

ÿ A 2 †

…3†

 s† ‡ a5 …O2 ÿ O 2 † MONblend ˆ m ‡ a4 …ms ÿ m " ‡ a6

#2 …A2 ÿ A 2 † 100

…4†

where: r is RON, m is MON, s is sensitivity (RONMON), O is ole®n context (% by volume), A is aromatic content (% by volume), and a1 ; a2 ; a3 ; a4 ; a5 ; a6 are model parameters. Quantities accented with an over-bar represent volumatric averages. Healy et al. [17] used data from 135 blends to determine the model parameter values: a1 =0.03224, a2 =0.00101, a3 =0, a4 =0.04450, a5 =0.00081, and a6 =ÿ0.0645. The Reid vapour pressure (RVP) model adopted to represent the blending process in this case study was the blending index approach [4] which has the form: "

…RVP†blend

n X ˆ ui …RVPi †1:25 iˆ1

#0:8 …5†

where: ui is the volume fraction of component i; n is the number of components in the blend. Eqs. (3±5), together with the given parameters, will be considered to adequately represent the actual blending process for the purposes of the studies contained in this paper. 4.1.1. Stochastic disturbances For the illustrative purposes of this case-study, disturbances are to be added to RON, MON, and RVP of one of the feedstocks. This gasoline blending scenario represents the case where all but one of the blender feedstocks are being stored in ``standing'' tanks and one of the feedstocks comes from a ``running'' tank. The feedstock containing the stochastic disturbances was chosen to be that which a€ected the optimal blender operation most signi®cantly, using a local parametric sensitivity analysis [46] around the nominal blend. The nominal optimal blend was obtained using MINOS in GAMS [47], the data are given in Tables 4±6 Eqs. (3±5). After scaling all the decision variables and parameters to the same order of magnitude, the linear system of Table 4 Production requirementsa

Value ($/bbl) Max. demand limit (bbl/day) Min. demand limit (bbl/day) Min. RON Min. MON Max. RVP (psi) a b

From Forbes and Marlin [10]. Chosen for this study.

Regular

Premium

33.00 8000 7000 88.5 77.0b 10.8

37.00 10000 10000 91.5 80.0b 10.8

A. Singh et al. / Journal of Process Control 10 (2000) 43±58

equations given in Ganesh and Biegler [48] was solved analytically using MAPLE [49] for the parametric sensitivity matrix (
x ). Five submatrices of
x , each corresponding to a feedstock, were obtained and their norms (largest singular values) computed. The results are summarized in Table 7. As the submatrix corresponding to the reformate qualities yielded the largest singular value, stochastic disturbances were added to the reformate stream. The correlation structure for the disturbances was represented as: 2 3 1 …6† qref;t ˆ q ref ‡ 4 0:85 5dt ÿ0:1

53

resulting RON, MON, and RVP of the reformate stream for the blend period is shown in Fig. (4). These reformate qualities were considered to be the actual values to which the blending process was subjected. The other feedstock qualities were considered to be accurately known and ®xed throughout the blend. 4.2. RTO performance

where: zÿ1 is the backward shift operator and "t are normally distributed innovations with zero mean and unit variance (i.e. "t 2 N(0, 1)). For simulation purposes, a reformate quality noise vector was generated for each RTO interval over the blend horizon and was assumed to remain unchanged during each interval. The

For the purposes of this illustrative case study, performance of an RTO strategy was measured by the pro®t earned using a given RTO approach. When a strategy produces blends that meet all quality speci®cations, the pro®t level can easily be computed using the economic data provided in Tables 4 and 5; however, if the products at the end of the blend are found to be o€speci®cation, they may not be suitable for sale as produced and will require further processing. This scenario may represent a reduction in the realized production rate of the gasoline blender and/or available product storage capacity. In such situations, evaluating the performance of the RTO system is not as simple. One option would be to include the cost of re-blending or blend correction in determining pro®t level. Such a calculation would involve the operating costs associated with the required further processing. For comparison purposes in this case study, a very simple approach to performance evaluation is adopted. For this work, the o€-speci®cation portions of blends (products made during speci®c RTO intervals that are causing the entire blend to be o€-speci®cation) can be ``removed'' from the blend (i.e. portions of the blend are

Table 5 Feedstock economic dataa

Table 7 Feedstock sensitivity

where: qref,t=[RON MON RVP]Tref,t is a vector of reformate qualities at time t; qref is a vector of nominal reformate qualities, and dt is a scalar disturbance. The scalar disturbance, dt , was modeled as steps of random height entering a well mixed tank and had the form: dt ˆ

…1 ÿ 0:56†zÿ1 1 "t …1 ÿ 0:56zÿ1 † 1 ÿ zÿ1

Reformate LSR naphtha n-Butane Catalytic gasoline Alkylate a

…7†

Available (bbl/day)

Cost ($/bbl)

Feedstock

Submatrix norm

12000 6500 3000 4500 7000

34.00 26.00 10.30 31.30 37.00

Reformate LSR naphtha n-Butane Catalytic gasoline Alkylate

3.66 1.20 0.005 1.63 0

From Forbes and Marlin [10].

Table 6 Feedstock qualitiesa Feedstock

Reformate (C14a)

LSR naphtha (C10a)

n-Butane (C25a)

Catalytic gasoline (C12a)

Alkylate (C7a)

RON MON Ole®n (%) Aromatics (%) RVP (psi)

94.1 80.5 1.0 58.0 3.8

70.7 68.7 1.8 2.7 12.0

93.8 90.0 0 0 138.0b

92.9 80.8 48.8 22.8 5.3

95.0 91.7 0 0 6.6

a b

From Healy et al. [17]. From Forbes and Marlin [10].

54

A. Singh et al. / Journal of Process Control 10 (2000) 43±58

integrals in Problem (1), using MINOS/GAMS [47]. The theoretical maximum pro®t was found to be $66,276.55 for the blend. This pro®t was taken as the performance benchmark against which the performance of each blend RTO systems would be compared. Thus, the performance metric used for this case study will be the amount of the theoretically obtainable pro®t that a given RTO system was unable to capture. 4.3. Conventional RTO

Fig. 4. Reformate qualites during 24 h blend.

routed to tanks for re-blending). Then, the blend pro®t can be calculated based on the modi®ed smaller blend that does meet quality speci®cations. This approach measures RTO system performance solely in terms of the pro®t loss associated with the reduced production rates resulting from o€-speci®cation blending. It must be noted that in most blend control/optimization applications there is a process control system that is responsible for maintaining the blend on speci®cation and minimizing quality ``giveaway''. The blend controller performs this task by deviating from the blend recipe, generated by the RTO system, in some ``optimal'' fashion determined by the controller tuning. Then, the performance of the integrated blend automation system becomes a function of the controller structure and tuning. This further complicates RTO system performance evaluation, since RTO and controller performance become confounded. Thus, for the purposes of this case study, the e€ects of the process control system were ignored and the blend recipes were assumed to be implemented as computed by the RTO system. A practical de®nition is required for determining when a blend, or portion thereof, is o€-speci®cation. Typically, small violations of quality speci®cations can often be tolerated, since product quality measurements are limited by analyzer sensitivity. In this case study, it is assumed that violations of up to ÿ0.03 octane number for RON and MON, and +0.005 psi for RVP can be tolerated. That is, blended qualities that fall outside these tolerance limits are deemed o€-speci®cation. These tolerances were set arbitrarily for this study. Finally, in order to determine the maximum pro®t that was theoretically obtainable for the given blending problem, the ideal optimization Problem (1) was solved using perfect knowledge of all feedstock qualities throughout the blending horizon and the models used to represent the blending process. The resulting blend optimization problem was solved, after discretizing the

As previously discussed, at the heart of many commercial blender automation systems is an RTO layer that is usually based on linear programming with bias updating [38]. The objective of the on-line optimizer is usually optimization of steady-state economic performance (i.e. maximization of pro®t ¯ow). Thus the blending optimization problem is most often formulated as: max

x

cT x

…8†

subject to: T ~ x†s ‡ bh…x†40 Rx4…w

where: c is a constant vector containing process economics, RÄ is a constant matrix containing the feedstock quality blending indices, s are the blended quality speci®cations, x are the feedstock or component ¯ows, w is a weighting vector (usually containing only ones and zeros to indicate the presence of a speci®c stream in a blend), b are the biases used to update the blending model to maintain its accuracy, and h is a vector of linear equations for the maximum and minimum product ~ demands and feedstock availabilities. The matrix R contains blending indices such as blending octane numbers (BON's) that blend linearly as volumetric averages. Since linear blending models are unable to capture the true blending behaviour, structural mismatch is introduced into the optirnizer and bias updating is used to compensate for plant/model mismatch. The conventional RTO system works by: (1) taking measurements of blended qualities; (2) calculating biases as the di€erence between measured blended qualities and those predicted by the linear model; (3) solving Problem (8) using the newly calculated biases to obtain new blender feedstock ¯ows or blend recipes (xk ); and (4) implementing the blend recipe and waiting for steady-state. These four steps are carried out at each RTO interval until the end of the blend is reached. In the bias update formulation of Problem (8), the only adjustable parameters are the bias terms and all of the feedstock qualities are treated as ®xed parameters. Forbes and Marlin [10] have shown that a bias update approach can lead to a substantial loss in blender

A. Singh et al. / Journal of Process Control 10 (2000) 43±58

pro®tability given even a small variation in the feedstock qualities. The conventional LP+bias updating RTO system su€ers from two limitations: inability to incorporate the inherent blending nonlinearities, and ine€ectiveness in dealing with ¯uctuating feedstock qualities. The e€ects of these two limitations on the performance of the conventional RTO approach for this case study were determined by simulation. A closedloop simulation was produced using the models described in Section 4.1 and the conventional LP+bias update RTO system. The product qualities at the end of blend are shown in Figs. 5 and 6 as deviations from the given quality speci®cation. While there is signi®cant quality giveaway in RON and MON of regular and RON of premium gasoline, RVP's of both grades of gasoline are above the maximum speci®ed and the MON of premium grade is below the minimum allowed. In addition, RVP of both grades is o€-speci®cation throughout the blend. As a result, both grades of gasoline produced are o€-speci®cation and may not be suitable for sale. Since both grades are o€-speci®cation throughout the blend horizon, portions of blends cannot be removed to obtain blends that do meet speci®cations. Hence, a pro®t level

55

for the conventional RTO system could not be computed in this case study. In this case study the conventional RTO approach fails to make products meeting quality speci®cations. The poor performance in this case study can be attributed to a combination of plant/model structural mismatch and parametric mismatch arising from ¯uctuations in reformate qualities. Of course, in an industrial application the process control system would maintain the blend on speci®cation by deviating from the blend recipes calculated by the RTO system. Such a departure from the blend recipes would ensure a feasible blend at the cost of blend pro®tability and the exact loss of blend pro®tability would depend on the controller design, as well as its tuning. 4.4. NLP+bias RTO The e€ects of plant/model structural mismatch can be eliminated using appropriate, nonlinear blending models; however, even with the elimination of structural mismatch, the potential for plant/model parametric mismatch remains. One approach to blender optimization is based on nonlinear programming (NLP) and uses bias updating [28]: max

x

cT x

…9†

subject to: g…x†4…wT x†s ‡ bh…x†40

Fig. 5. Blended octane using LP+bias RTO.

where: g…x† is a vector of nonlinear blending models. Note that NLP+bias update Problem (9) di€ers from the LP+bias update optimization Problem (8) only in the blending models used and can be solved using standard NLP techniques [50]. The RTO cycle remains unchanged. Simulation of blending RTO was conducted as described for the LP+bias update case except that: (1) the blend recipe at each RTO interval was obtained using NLP (GAMS/MINOS); and (2) the same models were used in the RTO as were used to simulate blender operation. The only di€erence between the plant and the model in this simulation was parametric mismatch in the reformate qualities. The reformate qualities were ®xed at their nominal values given in Table 5 for the RTO system, while the actual reformate qualities were subject to stochastic disturbances as shown in Fig. 4. Upon completion of the blend, four of the blended Table 8 Deviation from speci®cation at end of blend

Fig. 6. Blended RVP using LP+bias RTO.

Quality

RON (regular)

RVP (regular)

MON (premium)

RVP (premium)

Deviation

ÿ0.1

0.015 psi

ÿ0.05

0.01 psi

56

A. Singh et al. / Journal of Process Control 10 (2000) 43±58

qualities in the product tanks were found to be o€-speci®cation as summarized in Table 8. The assumed speci®cation tolerance levels were satis®ed for both grades when gasoline produced during the 1st, 9th, 10th, and 12th intervals were excluded from the ®nal products and the excluded gasoline was routed for re-blending. Eliminating products from 4 of the 12 RTO intervals resulted in a smaller production rate and the pro®t earned from the blend was $44,105.96, signi®cantly less than the theoretical maximum of $66,276.55. However, performance of the NLP+bias update RTO system proved better than that of the conventional LP+bias RTO, primarily due to the elimination of plant/model structural mismatch. In this case, the bias updating strategy is not able to adequately handle parametric mismatch arising from stochastic feedstock disturbances, resulting in the production of o€-speci®cation products. Again, an industrial blender automation system would ensure on-speci®cation blended product by deviating from the blend recipe produced by the RTO system in some user speci®ed fashion, which would e€ect blender pro®tability in some arbitrary fashion. 4.5. Time-horizon based RTO The e€ects of plant/model parametric mismatch cannot be completely eliminated in this case study due to the stochastic nature of the reformate quality disturbances; however, it is possible to decrease the amount of parametric mismatch using a prediction of future values for the reformate qualities. In Section 3.3 the proposed THRTO approach was introduced, in which the optimization Problem (2) was solved using predictions for future values of feedstock qualities. For simulation purposes all plant/model structural mismatch was eliminated from the THRTO problem (i.e. the same models were used to represent the blending process and for RTO); however, some plant/model parametric mismatch remained as predictions of future feedstock qualities cannot be predicted with perfect accuracy. In this case study, disturbances are known to be present only in reformate RON, MON, and RVP. Therefore, the feedstock qualities to be forecasted (i.e. the adjustable model parameters that will be updated online) did not have to be selected. Also, it was assumed for this case study that the reformate qualities (RON, MON, and RVP) are measured on-line. Generally, feedstock qualities may not measured and have to be estimated from other measurements. In such situations, it should be ensured that the required qualities can indeed be estimated from the available measurements. The three reformate qualities were predicted in the THRTO systems using a prediction ®lter. In this

simulation study, the actual noise model was used in designing the prediction ®lter. In general, the noise model would not be known and can be developed using identi®cation techniques ([12,51]. The ``optimal'' prediction ®lter was designed and tuned to yield minimum mean squared error forecasts [12] of the reformate qualities using the noise model given in Eq. (7). In this case study, three THRTO systems were designed, which differed in the number of steps into the future for which predictions were made (e.g. 1-step, 2-steps, and 3-steps ahead). The qualities were forecasted at each RTO interval from to , to tfÿ1 and the estimated qualities were held constant for all intervals beyond the number of steps into the future the ®lter predicted. The three RTO systems were used on the blending problem and the NLP problem was solved using GAMS/MINOS. The pro®t earned by each THRTO system for the blend period are given in Table 9. It should be noted that, while RTO systems incorporating 2- and 3-step ahead predictions yielded products meeting quality tolerance levels, the RON and RVP in regular gasoline produced using 1-step ahead prediction ®lter failed to meet quality speci®cations within the speci®ed tolerances. Excluding regular gasoline produced during the last interval brought the RON up and the RVP down to within tolerances at the cost of reducing the size of the blend, which decreased the pro®t for this blend. As can be seen in Table 9, all three controllers provide pro®t levels that are quite close to the theoretical maximum of $66,276.55, but performance of THRTO systems increase slightly as the length of the forecasting period is increased. This results from improvements in the ability of the THRTO system to e€ectively exploit more of the process noise dynamics. The performance of three THRTO systems and the NLP+bias update RTO system are compared in Fig. 7. Note that RTO performance is based on the amount of theoretical maximum pro®t captured by each system. The pro®t levels for the NLP+bias update RTO and the 1step ahead THRTO systems were calculated using modi®ed blends that meet quality speci®cations (i.e. some portion of the blend was signi®cantly o€-speci®cation). As is evident from Fig. 7, the proposed time-horizon based optimizers more eciently handle the stochastic disturbances in the reformate qualities, than does the RTO system based on the conventional bias updating Table 9 Comparison of time horizon controllers Controller

Pro®t

1-Step 2-Steps 3-Steps Ideal

$64,621.99 $65,491.29 $65,914.78 $66,276.55

A. Singh et al. / Journal of Process Control 10 (2000) 43±58

57

tion problems. The critical issues that should be addressed for other process applications of the THRTO approach are whether the product qualities of interest are both observable and predictable from available process measurements, whether there is sucient time during the blend to perform the required RTO calculations, and whether the process controllers are capable of performing at the required level. For those batch processes that meet these three requirements, the THRTO approach may o€er substantial performance improvements. Acknowledgements Fig. 7. Blender RTO performance comparison.

approach. This is primarily due to the time-horizon based RTO systems ability to update feedstock qualities directly and work to maximize pro®t over the whole blend horizon rather than at a point in time.

The authors gratefully acknowledge the ®nancial support of the Natural Sciences and Engineering Research Council of Canada, Sunoco Inc., and the Imperial Oil Charitable Foundation in the completion of this work. References

5. Summary and conclusions A novel approach to on-line blend optimization has been introduced in this work and illustrated using a gasoline blending case study. There are two key di€erences between the proposed time-horizon based RTO (THRTO) method and those currently available in the literature or in commercial use. The ®rst is that the blend recipe is optimized over the remaining blend, similar to the SHMPC approach proposed by Thomas et al. [45]. The second key di€erence is that model updating takes the form of predicting future values of the feedstock properties. The case study compared the performance of three di€erent blend RTO strategies: (1) the conventional linear programming with bias update approach (LP+bias); (2) a nonlinear programming with bias update approach (NLP+bias); and (3) the time-horizon based RTO approach (THRTO). It was shown that the LP+bias update RTO system had poor performance due to both structural and parametric plant/model mismatch. Eliminating the plant/model structural mismatch, using an NLP+bias update RTO system, improved performance signi®cantly; however, some economic opportunity remained due to plant/model parametric mismatch. Further performance improvements were obtained using a THRTO system that considered the entire remaining blend horizon and incorporated a prediction of future stochastic disturbances. Finally, discussions have been limited to gasoline blending and have shown promise for signi®cant process performance improvements by adaptively generating blend recipes; however, the proposed THRTO approach is applicable to a wide range of batch process optimiza-

[1] C.W. DeWitt, L.S. Lasdon, A.D. Waren, D.A. Brenner, S.A. Melhem, Omega: an improved gasoline blending system for Texaco, Interfaces 19 (1989) 85±101. [2] L. Givens, Modern petroleum re®ning: an overview, Automotive Engineering 93 (1985) 69±77. [3] F.H. Palmer, A.M. Smith, The performance and speci®cation of gasolines In: E.G. Hancock (Ed.) Technology of Gasoline, Blackwell Scienti®c London, 1985. [4] J.H. Gary, G.E. Handwerk, Petroleum Re®ning Technology and Economics, third ed., Marcel Dekker, New York, 1994. [5] P.T. Lo, On-line RVP analysis improves gas blending, INTECH September (1994) 31±33. [6] W.L. Leer, Petroleum Re®ning for the Non-Technical Person. PennWell, Oklahoma, 1985. [7] S.S. Agrawal, Integrated blending control, optimization and planning. Hydrocarbon Processing August (1995) 129-139. [8] G.M. Barrow, Physical Chemistry, MeGraw-Hill, New York, 1961. [9] M.H. Rusin, The structure of nonlinear models, Chem. Eng. Sci. 30 (1975) 937±988. [10] J.F. Forbes, T.E. Marlin, Model accuracy for economic optimizing controllers: the bias update case, Ind. Eng. Chem. Res. 18 (1994) 497±510. [11] J.F. Forbes, T.E. Marlin, Design cost: a systematic approach to technology selection for model-based real-time optimization systems, Computers chem. Engng. 20 (1996) 717±734. [12] G.E.P. Box, G.M. Jenkins, Time Series Analysis: Forecasting and Control. Holden-Day, Oakland, 1976. [13] S. Krishnan, G.W. Barton, J.D. Perkins, Robust parameter estimation in on-line optimization-part I: methodology and simulated case studies, Computers chem. Engng. 16 (1992) 545±562. [14] A. Singh, Modeling and Model Updating in the Real-Time Optimization of Gasoline Blending, M.Sc. thesis, University of Toronto, Canada, 1997. [15] P.J. McLellan, A Nonlinear Re®nery Model for Scheduling and Economic Evaluation, M.A.Sc. thesis, University of Waterloo, Canada, 1983. [16] M.H. Rusin, H.S. Chung, J.F. Marshall, A transformation method for calculating the research and motor octane numbers of gasoline blends, Ind. Eng. Chem. Fund. 20 (1981) 195±204.

58

A. Singh et al. / Journal of Process Control 10 (2000) 43±58

[17] W.C. Healey Jr., C.W. Maasen, R.T. Peterson, A new approach to blending octanes, Proc. 24th Meeting API Re®ning Division, vol. 39, New York, 1959. [18] W.E. Stewart, Predict octanes for gasoline blends, Petroleum Re®ner 38 (1959) 135±139. [19] W.E. Morris, The interaction approach to gasoline blending, NPRA 73rd Annual Meeting, Paper no. AM-75-30, Mar., 1975. [20] A. Muller, New method produces accurate octane blending values, Oil and Gas Journal March 23, (1992) 80-90. [21] W.E. Morris, Optimum blending gives best pool octane, Oil and Gas Journal, January 20 (1986). 63±66. [22] W.E. Morris, W.E. Smith, R.D. Snee, Interaction blending equations enhance reformulated gasoline pro®tability, Oil and Gas Journal January 17(1994) 54-58. [23] W.E. Stewart, Predict RVP of blends accurately, Petroleum Re®ner 38 (1959) 231±234. [24] J.J. Vazques-Esparragoza, G.A. Iglesias-Silva, M.W. Hlavinka, J. Bullin, How to estimate RVP of blends, Hydrocarbon Processing, August (1992) 135-138. [25] N.B. Haskell, D.K. Beavon, Front end volatility of gasoline blends, Ind. Eng. Chem. 2 (1942) 167±170. [26] Anonymous, Advanced control and information systems '97, Hydrocarbon Processing 76, 9, 1997. 98-104. [27] T.L. Sullivan, Re®nery-wide blend control and optimization, Hydrocarbon Processing, May (1990) 93-96. [28] J.R. Ramsey Jr., P.B. Truesdale, Blend optimization integrated into re®nery-wide strategy, Oil and Gas Journal, March, 19 (1990) 40-44. [29] T.F. Michalek, K. Nordeen, R. Rys, Using relational databases for blend optimization, Hydrocarbon Processing, September (1994) 47-49. [30] P.J. Vermeer, C.C. Pedersen, W.M. Canney, J.S. Ayala, Design and integration issues for dynamic blend optimization, NPRA Computer Con¯. Paper no. CC_96-130, November 1996. [31] P.J. Vermeer, C.C. Pedersen, W.M. Canney, J.S. Ayala, Blendcontrol system all but eliminates reblends for Canadian re®ner, Oil Gas Journal, July 28, 1997. [32] R.E. Palmer, J. Dar-Dov, and S. Whitacker, Use an integrated approach to product in-line blending, Hydrocarbon Processing, February (1995) 62±66. [33] C.S. Bene®eld, R.R. Broadway, Re®nery blends gasoline for the unleaded era, Oil and Gas Journal, 18 March (1985) 92-98. [34] A. Diaz, J.A. Barsamian, Meeting changing fuel requirements with on-line optimization, Hydrocarbon Processing, February (1996). 71±76.

[35] J. White, F. Hall, Gasoline blending optimization cuts use of expensive components, Oil and Gas Journal, November 9 (1992) 8184. [36] J.M. McDonald, L.M. Grupa Jr., M.P. McKippen, D.E. Roe, Gasoline blending using NIR spectroscopy and LP optimization, NPRA Computer Conf., Paper no. CC-92-37, Nov., 1992. [37] Y. Serpemen, F.W. Wenzel, A. Hubel, Blending technology key to making new gasolines, Oil and Gas Journal, March 18 (1991) 62-74. [38] K. Magoulas, D. Marinos-Kouris, A. Lygeros, Instructions are given for building a gasoline blending LP, Oil and Gas Journal, July 4 (1988) 32-37. [39] M.L. Bain, K.W. Mansfeild, J.G. Maphet, R.W. Szoke, W.H. Bosler, J.P. Kennedy, Gasoline blending and integrated on-line optimization, scheduling, and control system, NPRA Comp. Con¯. Paper no. CC-93-191, Nov., 1993. [40] A. Rigby, L.S. Lasdon, A.D. Warren, The evolution of Texaco's blending systems: from OMEGA to StarBlend, Interfaces 25 (1995) 64±83. [41] R.E. Bellman, E.S. Lee, History and development of dynamic programming, Control Systems, Nov, 24, 1984. [42] P. Tanartkit, L.T. Biegler, Stable decomposition for dynamic optimization, Ind. Eng. Chem. Res. 34 (1995) 1253. [43] C.K. Chiu, Kalman Filtering with Real-Time Applications, Springer-Verlag, New York, 1987. [44] S.J. Qin, T.A. Badgwell, An overview of industrial model predictive control technology, Proceedings of CPC-V, Tahoe City, CA, January 7±12, (1996) 232-256. [45] M.M. Thomas, B. Joseph, J.L. Kardos, Batch chemical process quality control applied to curing of composite materials, AIChE J. 43 (10) (1997) 2535±4255. [46] P. Seferlis, Collocation Models for Distillation Units and Sensitivity Analysis Studies in Process Optimization, Ph.D. thesis, MeMaster University, Canada, 1995. [47] A. Brooke, D. Kendrick, A. Meeraus, GAMS: A Users Guide. Boyd and Raser, MA, 1992. [48] N. Ganesh, L.T. Biegler, A reduced hessian strategy for sensitivity analysis of optimal ¯owsheets, AIChE J. 33 (1987) 282± 296. [49] B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monogan, W.M. Watt, Maple Language Reference Manual, Waterloo Maple Publishing. Waterloo, Canada, 1991. [50] D.P. Bertsekas, Nonlinear Programming, Athena Scienti®c. Belmont, MA, 1995. [51] L. Ljung, System Identi®cation: Theory for the User. PrenticeHall, New Jersey, 1991.